The Hierarchy Problem: From the Fundamentals to the Frontiers
UUniversity of CaliforniaSanta Barbara
The Hierarchy Problem:From the Fundamentals to the Frontiers
A dissertation submitted in partial satisfactionof the requirements for the degreeDoctor of PhilosophyinPhysicsbySeth KorenCommittee in charge:Professor Nathaniel Craig, ChairProfessor Mark SrednickiProfessor Claudio CampagnariSeptember 2020 a r X i v : . [ h e p - ph ] S e p he Dissertation of Seth Koren is approved.Professor Mark SrednickiProfessor Claudio CampagnariProfessor Nathaniel Craig, Committee ChairAugust 2020 DocuSign Envelope ID: 232997C1-A8F9-4F34-82CF-CD7D1CB5F9EE he Hierarchy Problem:From the Fundamentals to the FrontiersCopyright c (cid:13)
You seem very clever at explaining words, Sir”, said Alice.“Would you kindly tell me the meaning of the poem ‘Jabberwocky’ ?”“Let’s hear it”, said Humpty Dumpty.“I can explain all the poems that ever were invented— and a good many that haven’t been invented just yet.”Lewis Carroll
Through the Looking Glass (1871) [1] iv cknowledgements My thanks go first and foremost to Nathaniel Craig for his continual support and encour-agement. From Nathaniel I learned not only an enormous amount of exciting physics,but also how to prepare engaging lessons and talks, how to be an effective mentor, how tothink intuitively about the natural world and determine the particle physics to describeit, and how to be a caring, welcoming, community-focused academic. It goes withoutsaying that Nathaniel’s influence pervades every word written below and the physicalunderstanding behind them.I would not have made it here were it not for the many senior academics who havecharitably given their own time to encourage and support me. As a young undergraduate,Tom Lubensky’s patient help during many office hours and his explicit encouragementfor me to continue studying physics were vital. Cullen Blake took a risk on me as anundergrad with little to show but excitement and enthusiasm, taught me how to problem-solve like a physicist and a researcher, and truly made a huge difference in my life. Andthere were many such professors who kindly gave me their time and support—MirjamCvetič, Larry Gladney, Justin Khoury, H. H. “Brig” Williams, Ned Wright, and others—and if I listed all the ways they have all supererogatorily supported me and my educationthis section would be too long.In graduate school I have also benefited from the kindness of a cadre of senior aca-demics. Dave Sutherland and John Mason spent hours engaging me in many elucidatingconversations. Don Marolf generously included me in gravity theory activities and an-swered my endless questions. Nima Arkani-Hamed has graciously given me his time andmade me feel welcome at every turn. And there have been many other particle theo-rists who have offered me their advice and support over the years—Tim Cohen, PatrickDraper, Matthew McCullough, and Flip Tanedo, among others.vf course I have not worked alone, and little of this science would have been accom-plished were it not for my many collaborators who have helped me learn and problem-solve and provided guidance and been patient when I was overwhelmed with being pulledin too many directions. Let me especially mention those undergrads I have spent sig-nificant time mentoring during my time in graduate school, namely Aidan Herderschee,Samuel Alipour-fard, and Umut Can Öktem. Indeed, they each took a chance on me aswell, and working with them has taught me how to be a better teacher and physicist—notto mention all of the great science we worked out together.My physics knowledge would have also been stunted were it not for the countless hoursspent discussing all manner of high energy theory with friends and peers—primarilyMatthew Brown, Brianna Grado-White, Alex Kinsella, Robert McGehee, and GabrielTreviño Verastegui. And my enjoyment of graduate school would have been stuntedwere it not for the board games, hiking, trivia, art walks, biking, and late-night philo-sophical discussions with them and with Dillon Cisco, Neelay Fruitwala, Eric Jones,Farzan Vafa, Sicheng Wang, and others. No one has enriched my life here moreso thanNicole Iannaccone—a summary statement of far too few words.Nor would this have been possible were it not for my ‘medical support team’ consistingprimarily of endocrinologists Dr. Mark Wilson and Dr. Ashley Thorsell, student healthphysician Dr. Miguel Pedroza, and my therapist Dr. Karen Dias, who have helped meimmensely in my time here. Even putting aside physiological issues, graduate school canbe and has been incredibly mentally taxing. I’m not sure I could not have made it throughwere it not for the psychological assistance I have received, both psychotherapeutic andpharmacological.Of course a special ‘shout-out’ goes to Mother Nature. The idyllic weather, scenicocean, beautiful mountains, and clear night skies of Santa Barbara may have just beentoo distracting for me to progress through my degree were it not for the Rey fire, thevihittier fire, the Cave fire, the Santa Barbara microburst, the recurring January flooding,the Ridgecrest earthquakes, the month of unbreathable air from the Thomas fire, thethreat of the Thomas fire itself, the Montecito mudslides, and of course the COVID-19pandemic, all of which have kept me indoors thinking about physics.Finally, I would like to thank all those who kindly read drafts of (parts of) this the-sis and provided invaluable feedback, including Samuel Alipour-fard, Ian Banta, ManuelBuen-Abad, Changha Choi, Nathaniel Craig, David Grabovsky, Adolfo Holguin, SamuelHomiller, Lucas Johns, Soubhik Kumar, Umut Can Öktem, Robert McGehee, AlexMeiburg, Gabriel Treviño Verastegui, Farzan Vafa, and George Wojcik.vii urriculum Vitæ
Seth Koren
Education
Publications “Supersoft Stops”T. Cohen, N. Craig, S. Koren, M. McCullough, J. Tooby-SmithAccepted to Phys. Rev. Lett., [arXiv:2002.12630 [hep-ph]] [2]“IR Dynamics from UV Divergences: UV/IR Mixing, NCFT, and the Hierarchy Prob-lem”N. Craig and S. KorenJHEP 03 (2020) 037, [arXiv:1909.01365 [hep-ph]] [3]“Freezing-in Twin Dark Matter”S. Koren and R. McGeheePhys. Rev. D101 (2020) 055024, [arXiv:1908.03559 [hep-ph]] [4]“The Weak Scale from Weak Gravity”N. Craig, I. Garcia Garcia, S. KorenJHEP 09 (2019) 081, [arXiv:1904.08426 [hep-ph]] [5]“Exploring Strong-Field Deviations From General Relativity via Gravitational Waves”S. Giddings, S. Koren, G. TreviñoPhys. Rev. D100 (2019) 044005, [arXiv:1904.04258 [gr-qc]] [6]“Neutrino - DM Scattering and Coincident Detections of UHE Neutrinos with EM Sources”S. KorenJCAP 09 (2019) 013, [arXiv:1903.05096 [hep-ph]] [7]“Constructing N=4 Coulomb Branch Superamplitudes”A. Herderschee, S. Koren, T. Trott viiiHEP 08 (2019) 107, [arXiv:1902.07205 [hep-th]] [8]“Massive On-Shell Supersymmetric Scattering Amplitudes”A. Herderschee, S. Koren, T. TrottJHEP 10 (2019) 092, [arXiv:1902.07204 [hep-th]] [9]“The second Higgs at the lifetime frontier”S. Alipour-fard, N. Craig, S. Gori, S. Koren, D. RedigoloJHEP 07 (2020) 029, [arXiv:1812.09315 [hep-ph]] [10]“Discrete Gauge Symmetries and the Weak Gravity Conjecture”N. Craig, I. Garcia Garcia, S. KorenJHEP 05 (2019) 140, [arXiv:1812.08181 [hep-th]] [11]“Long Live the Higgs Factory: Higgs Decays to Long-Lived Particles at Future LeptonColliders”S. Alipour-fard, N. Craig, M. Jiang, S. KorenChin. Phys. C43 (2019) 053101, [arXiv:1812.05588 [hep-ph]] [12]“Cosmological Signals of a Mirror Twin Higgs”N. Craig, S. Koren, T. TrottJHEP 05 (2017) 038, [arXiv:1611.07977 [hep-ph]] [13]“The Low-Mass Astrometric Binary LSR1610-0040”S. C. Koren, C. H. Blake, C. C. Dahn, H. C. HarrisThe Astronomical Journal 151 (2016) 57, [arXiv:1511.02234 [astro-ph.SR]] [14]“Characterizing Asteroids Multiply-Observed at Infrared Wavelengths”S. C. Koren, E. L. Wright, A. MainzerIcarus 258 (2015) 82-91, [arXiv:1506.04751 [astro-ph.EP]] [15]ix bstract
The Hierarchy Problem:From the Fundamentals to the FrontiersbySeth KorenWe begin this thesis with an extensive pedagogical introduction aimed at clarifyingthe foundations of the hierarchy problem. After introducing effective field theory, wediscuss renormalization at length from a variety of perspectives. We focus on conceptualunderstanding and connections between approaches, while providing a plethora of ex-amples for clarity. With that background we can then clearly understand the hierarchyproblem, which is reviewed primarily by introducing and refuting common misconcep-tions thereof. We next discuss some of the beautiful classic frameworks to approach theissue. However, we argue that the LHC data have qualitatively modified the issue into‘The Loerarchy Problem’—how to generate an IR scale without accompanying visiblestructure—and we discuss recent work on this approach. In the second half, we presentsome of our own work in these directions, beginning with explorations of how the NeutralNaturalness approach motivates novel signatures of electroweak naturalness at a varietyof physics frontiers. Finally, we propose a New Trail for Naturalness and suggest that thephysical breakdown of EFT, which gravity demands, may be responsible for the violationof our EFT expectations at the LHC. x ermissions and Attributions
1. The content of Chapter 5 is the result of collaboration with Nathaniel Craig andTimothy Trott, and separately with Robert McGehee. This work previously ap-peared in the Journal of High Energy Physics (JHEP (2017) 038) and PhysicalReview D (Phys. Rev. D101 (2020) 055024), respectively.2. The content of Chapter 6 is the result of collaboration with Samuel Alipour-fard,Nathaniel Craig, and Minyuan Jiang, and previously appeared in Chinese PhysicsC (Chin. Phys.
C43 (2019) 053101).3. The content of Chapter 7 is the result of collaboration with Nathaniel Craig andpreviously appeared in the Journal of High Energy Physics (JHEP (2020) 037).xi reface The first four chapters of this thesis are introductory material which has not previouslyappeared in any public form. My intention has been to write the guide that would havebeen most useful for me toward the beginning of my graduate school journey as a fieldtheorist interested in the hierarchy problem. My aim has been to make these chaptersaccessible to beginning graduate students in particle physics and interested parties inrelated fields—background at the level of a single semester of quantum field theory shouldbe enough for them to be understandable in broad strokes.Chapter 1 introduces fundamental tools and concepts in quantum field theory whichare essential for particle theory, spending especial effort on discussing renormalizationfrom a variety of perspectives. Chapter 2 discusses the hierarchy problem and how tothink about it—primarily through the pedagogical device of refuting a variety of commonmisconceptions and pitfalls. Chapter 3 introduces in brief a variety of classic strategiesand solutions to the hierarchy problem which also constitute important frameworks intheoretical particle physics beyond the Standard Model. Chapter 4 discusses more-recentideas about the hierarchy problem in light of the empirical pressure supplied by the lackof observed new physics at the Large Hadron Collider. Throughout I also make note ofinteresting research programs which, while they lie too far outside the main narrative forme to explain, are too fascinating not to be mentioned.The first half of this thesis is thus mostly an introduction to and review of materialI had no hand in inventing. As always, I am ‘standing on the shoulders of giants’, andI have benefited enormously from the pedagogical efforts of those who came before me.When my thinking on a topic has been especially informed by a particular exposition, orwhen I present an example which was discussed in a particular source, I will endeavor tosay so and refer to that presentation. As to the rest, it’s somewhere between difficult andxiimpossible to distinguish precisely how and whose ideas I have melded together in my ownunderstanding of the topics—to say nothing of any insight I may have had myself—butI have included copious references to reading material I enjoyed as a guide. Ultimatelythis is a synthesis of ideas in high energy theory aimed toward the particular purpose ofunderstanding the hierarchy problem, and I have attempted to include the most usefuland pedagogical explanations of these topics I could find, if not invent.I then present some work on the subject by myself and my collaborators. Chapter 5contains work constructing a viable cosmological history for mirror twin Higgs models, anexemplar of the modern Neutral Naturalness approach to the hierarchy problem. Chapter6 focuses on searching for long-lived particles produced at particle colliders as a discoverychannel for a broad class of such models. Chapter 7 is an initial exploration of a newapproach to the hierarchy problem which follows a maximalist interpretation of the lackof new observed TeV scale physics, and so relies on questioning and modifying some coreassumptions of conventional particle physics. In Chapter 8 we conclude with some briefparting thoughts.If you enjoy reading this work, or find it useful, or have questions, or comments, orrecommendations for good references, please do let me know—at whatever point in thefuture you’re reading this. As of autumn 2020, I can be reached at [email protected] ontents
Acknowledgements vCurriculum Vitae viiiAbstract xPermissions and Attributions xiPreface xii1 Effective Field Theory 1 φ . . . . . . . . . . . . . . . . 22Renormalization and Locality . . . . . . . . . . . . . . . . . . 281.2.2 To Repair Perturbation Theory . . . . . . . . . . . . . . . . . . . 31Renormalization Group Equations . . . . . . . . . . . . . . . 31Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Renormalized Perturbation Theory . . . . . . . . . . . . . . . 35Continuum Renormalization . . . . . . . . . . . . . . . . . . 37Renormalization Group Improvement . . . . . . . . . . . . . 411.2.3 To Relate Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 43Mass-Independent Schemes and Matching . . . . . . . . . . . 43Flowing in Theory Space . . . . . . . . . . . . . . . . . . . . 49Trivialities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551.2.4 To Reiterate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56xiv.3 Naturalness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581.3.1 Technical Naturalness and Fine-Tuning . . . . . . . . . . . . . . . 59Technical Naturalness and Masses . . . . . . . . . . . . . . . 641.3.2 Spurion Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 651.3.3 Dimensional Transmutation . . . . . . . . . . . . . . . . . . . . . 67 Non solutions to the Hierarchy Problem . . . . . . . . . . . . . . . . . . . 732.2.1 An End to Reductionism . . . . . . . . . . . . . . . . . . . . . . . 732.2.2 Waiter, there’s Philosophy in my Physics . . . . . . . . . . . . . . 762.2.3 The Lonely Higgs . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.2.4 Mass-Independent Regulators . . . . . . . . . . . . . . . . . . . . 862.3 The Hierarchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Γ -plet Higgs . . . . . . . . . . . . . . . . . . . . 1404.2 The Loerarchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434.3 Violations of Effective Field Theory . . . . . . . . . . . . . . . . . . . . . 1454.3.1 Gravity and EFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Tw in Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . 2215.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2215.3.2 The Mirror Twin Higgs & Cosmology . . . . . . . . . . . . . . . . 2225.3.3 Kinetic Mixing & A Massive Twin Photon . . . . . . . . . . . . . 2245.3.4 Freezing- Tw in Dark Matter . . . . . . . . . . . . . . . . . . . . . 2265.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 φ Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2757.3.1 Dimensional Regularization . . . . . . . . . . . . . . . . . . . . . 2817.4 Yukawa Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2847.4.1 Motivation: Strong UV/IR Duality . . . . . . . . . . . . . . . . . 2847.4.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2867.4.3 Scalar Two-Point Function . . . . . . . . . . . . . . . . . . . . . . 2897.4.4 Fermion Two-Point Function . . . . . . . . . . . . . . . . . . . . . 2927.4.5 Three-Point Function . . . . . . . . . . . . . . . . . . . . . . . . . 2937.5 Softly-broken Wess-Zumino Model . . . . . . . . . . . . . . . . . . . . . . 2967.6 Whence UV/IR Mixing? . . . . . . . . . . . . . . . . . . . . . . . . . . . 3017.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 xvi
How to Formulate Field Theory on a Noncommutative Space 313C Wilsonian Interpretations of NCFTs from Auxiliary Fields 317
C.1 Scalar Two-Point Function . . . . . . . . . . . . . . . . . . . . . . . . . . 318C.2 Fermion Two-Point Function . . . . . . . . . . . . . . . . . . . . . . . . . 319C.3 Three-Point Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
Bibliography 323 xvii hapter 1Effective Field Theory
The formulation and understanding of the hierarchy problem is steeped heavily in theprinciples and application of effective field theory (EFT) and renormalization, so we beginwith an introductory overview to set the stage for our main discussion. As is clear fromthe table of contents, I have prioritized clarity over brevity—especially when it comes torenormalization. The reader with a strong background in particle physics may find muchof this to be review, so may wish to skip ahead directly to Chapter 2 and circle back tosections of this chapter if and when the subtleties they discuss become relevant.We will endeavor to discuss the conceptual points which will be useful later in un-derstanding the hierarchy problem, and more generally to clarify common confusionswith ample examples. Of course we will be unable to discuss everything, and will tryto provide references to more detailed explanations when we must needs say less thanwe would like. Some generally useful introductions to effective field theory can be foundfrom Cohen [16] and Georgi [17], and useful, pedagogical perspectives on renormalizationare to be found in Srednicki [18], Peskin & Schroeder [19], Zee [20], Polchinski [21], andSchwartz [22], among others. 1 ffective Field Theory Chapter 1
Effective field theory is simply the familiar strategy to focus on the important degreesof freedom when understanding a physical system. For a simple example from an intro-ductory Newtonian mechanics course, consider studying the motion of balls on inclinedplanes in a freshman lab. It is neither necessary nor useful to model the short-distancephysics of the atomic composition of the ball, nor the high-energy physics of special rela-tivity. Inversely, it is also unnecessary to account for the long-distance physics of Hubbleexpansion or the low-energy physics of air currents in the lab. In quantum field theoriesthis intuitive course of action is formalized in decoupling theorems, showing preciselythe sense in which field theories are amenable to this sort of analysis: the effects ofshort-distance degrees of freedom may be taken into account as slight modifications tothe interactions of long-distance degrees of freedom, instead of including explicitly thosehigh-energy modes.Of course when one returns to the mechanics laboratory armed with an atomic clockand a scanning tunneling microscope, one begins to see deviations from the Newtonianpredictions. Indeed, the necessary physics for describing a situation depends not only onthe dynamics under consideration but also on the precision one is interested in attainingwith the description. So it is crucial that one is able to correct the leading-order descrip-tion by systematically adding in subdominant effects, as organized in a suitable powerseries in, for example, ( v/c ) , where v is the ball’s velocity and c is the speed of light. Ofcourse when the full description of the physics is known it’s in principle possible to justuse the full theory to compute observables—but I’d still rather not begin with the QEDLagrangian to predict the deformation of a ball rolling down a ramp.The construction of an appropriate effective description relies on three ingredients.The first is a list of the important degrees of freedom which specify the system under2 ffective Field Theory Chapter 1 Figure 1.1: A cartoon of an experimental setup to look at miniature balls rolling downramps, for which proper accounting of subleading effects may be necessary.consideration—in particle physics this is often some fields { φ i } . The second is the setof symmetries which these degrees of freedom enjoy. These constrain the allowed inter-actions between our fields and so control the dynamics of the theory. Finally we needa notion of power counting, which organizes the effects in terms of importance. Thiswill allow us to compute quantities to the desired precision systematically. Frequently ineffective field theories of use in particle physics this role is played by E/ Λ , where E isan energy and Λ is a heavy mass scale or cutoff above which we expect to require a newdescription of the physics. We will often be interested in determining the appropriate description of a system atsome scale, so it is necessary to understand which degrees of freedom and which interac-tions will be important as a function of energy. We can gain insight into when certainmodes or couplings are important by studying the behavior of our system under scale3 ffective Field Theory Chapter 1 transformations. Consider for example a theory of a real scalar field φ , with action S = (cid:90) d d x (cid:18) − ∂ µ φ∂ µ φ − m φ − λφ − τ φ + . . . (cid:19) , (1.1)where d is the dimensionality of spacetime, m, λ, τ are couplings of interactions involvingdifferent numbers of φ , the . . . denote terms with higher powers of φ , and we’ve imposeda Z : φ → − φ symmetry for simplicity. We’ve set c = (cid:126) = 1 leaving us with solelythe mass dimension to speak of. From the fact that regardless of spacetime dimensionwe have [ S ] = 0 and [ x ] = − , where [ · ] denotes the mass dimension, we first calculatefrom the kinetic term that [ φ ] = d − , and then we can read off [ m ] = 2 , [ λ ] = 4 − d , [ τ ] = 6 − d . These are known as the classical dimension of the associated operatorsand are associated to the behavior of the operators at different energies, for the followingreason.If we wish to understand how the physics of this theory varies as a function of scale, wecan perform a transformation x µ → sx (cid:48) µ and study the long-distance limit s (cid:29) with x (cid:48) µ fixed. The measure transforms as d d x → s d d d x (cid:48) and the derivatives ∂ µ → s − ∂ µ (cid:48) .Then to restore canonical normalization of the kinetic term such that the one-particlestates are properly normalized for the LSZ formula to work, we must perform a fieldredefinition φ ( x ) = s − d φ (cid:48) ( x (cid:48) ) , and the action becomes S = (cid:90) d d x (cid:48) (cid:18) − ∂ µ (cid:48) φ (cid:48) ∂ µ (cid:48) φ (cid:48) − m s φ (cid:48) − λs − d φ (cid:48) − τ s − d φ (cid:48) + . . . (cid:19) . (1.2)As a reminder, in the real world (at least at distances (cid:38) µm ) we have d = 4 . As you lookat the theory at longer distances the mass term becomes more important, so is knownas a ‘relevant’ operator. One says that the operator φ has classical dimension ∆ φ = 2 .The quartic interaction is classically constant under such a transformation, so is knownas ‘marginal’ with ∆ φ = 0 , and interactions with more powers of φ shrink at low energiesand are termed ‘irrelevant’, e.g. ∆ φ = − . We have been careful to specify that these4 ffective Field Theory Chapter 1 are the classical dimension of the operators, also called the ‘engineering dimension’ or‘canonical dimension’, which has a simple relation to the mass dimension as ∆ O = d − [ O ] for some operator O . If the theory is not scale-invariant then quantum corrections modifythis classical scaling by an ‘anomalous dimension’ δ O ( m , λ, τ, . . . ) which is a function ofthe couplings of the theory, and the full behavior is known as the ‘scaling dimension’.The terms ‘marginally (ir)relevant’ are used for operators whose classical dimension iszero but whose anomalous dimensions push them to one side.The connection to the typical EFT power counting in ( E/ Λ) is immediate. In anEFT with UV cutoff Λ , it’s natural to normalize all of our couplings with this scale andrename e.g. τ → ¯ τ Λ − d where ¯ τ is now dimensionless. It’s easy to see that the long-distance limit is equivalently a low-energy limit by considering the derivatives, which pulldown a constant p (cid:48) µ and scale as s − —or by simply invoking the uncertainty principle.Operators with negative scaling dimension contribute subleading effects at low energiesprecisely because of these extra powers of a large inverse mass scale. Then he made the tank of cast metal, 10 cubits across frombrim to brim, completely round; it was 5 cubits high, and itmeasured 30 cubits in circumference.God on the merits of working to finite precision1 Kings 7:23, Nevi’imNew Jewish Publication Society Translation (1985) [23]
The procedure of writing down the most general Lagrangian with the given degreesof freedom and respecting the given symmetries up to some degree of power countingis termed ‘bottom-up EFT’ as we’re constructing it entirely generally and will have5 ffective Field Theory Chapter 1 to fix coefficients by making measurements. A great example is the Standard ModelEffective Field Theory (SMEFT), of which the Standard Model itself is described bythe SMEFT Lagrangian at zeroth order in the power counting. It is defined by being an SU (3) × SU (2) × U (1) gauge theory with three generations of the following representationsof left-handed Weyl fermions: x Fermions SU (3) C SU (2) L U (1) Y Q ¯ u ¯3 - − ¯ d ¯3 - L - 2 − ¯ e - - 1In addition the Standard Model contains one scalar, the Higgs boson, which is responsi-ble for implementing the Anderson-Brout-Englert-Guralnik-Hagen-Higgs-Kibble-’t Hooftmechanism [24, 25, 26, 27, 28, 29] to break the electroweak symmetry SU (2) L × U (1) Y down to electromagnetism U (1) Q at low energies: H - 2 − The Standard Model Lagrangian contains all relevant and marginal gauge-invariant op-erators which can be built out of these fields, and has the following schematic form L kin = − F µν F µν − i ¯ ψ /Dψ − ( D µ H ) † ( D µ H ) (1.3) L Higgs = − yH Ψ ψ + h.c. + m H † H − λ H † H ) , (1.4)with F a gauge field strength, ψ a fermion, D the gauge covariant derivative in thekinetic term Lagrangian on the first line, and the second line containing the Higgs’Yukawa couplings and self-interactions. If a refresher on the Standard Model would beuseful, the introduction to its structure toward the end of Srednicki’s textbook [18] will6 ffective Field Theory Chapter 1 suffice for our purposes, while further discussion from a variety of perspectives can befound in Schwartz [22], Langacker [30], and Burgess & Moore [31].The SMEFT power-counting is in energies divided by an as-yet-unknown UV scale Λ ,so the dimension- n SMEFT Lagrangian consists of all gauge-invariant combinations ofthese fields with scaling dimension n − d . At dimension five there is solely one operator, L (5) = ( LH ) / Λ + h.c., which contains a Majorana mass for neutrinos. In even-more-schematic form, the dimension six Lagrangian contains operators with the field content − Λ L (6) = F + H + D H + ψ H + F H + ψ F H + ψ DH + ψ , (1.5)where for aesthetics we have multiplied through by the scale Λ and haven’t botheredwriting down couplings. After understanding the structure of the independent symmetry-preserving operators (see e.g. [32, 33, 34]), the job of the bottom-up effective fieldtheorist is to measure or constrain the coefficients of these higher-dimensional operators[35]. Useful data comes from both the energy frontier with searches at colliders for theproduction or decay of high-energy particles through these higher-dimensional operatorsand from the precision frontier measuring fundamental processes very well to look fordeviations from the Standard Model predictions (e.g. [36, 37, 38]). For more detail, seethe introduction to SMEFT by Brivio & Trott [39].Another approach is possible when we already have a theory and just want to focuson some particular degrees of freedom. Then we may construct a ‘top-down EFT’ bytaking our theory and ‘integrating out’ the degrees of freedom we don’t care about—forexample by starting with the Standard Model above and getting rid of the electroweakbosons to focus on processes occurring at lower energies (e.g. Fermi’s model of the weakinteraction [40]). We can’t necessarily just ignore those degrees of freedom though; whatwe need to do is modify the parameters of our EFT such that they reproduce the results7 ffective Field Theory Chapter 1 of the full theory (to some finite precision) using only the low-energy degrees of freedom.Such a procedure can be illustrated formally by playing with path integrals. Considerthe partition function for a theory with some light fields φ and some heavy fields Φ : Z = (cid:90) D φ D Φ e iS ( φ, Φ) . (1.6)This contains all of the physics in our theory, and so in principle we may use it to computeanything we wish. But if we’re interested in low-energy processes involving solely the φ fields, we could split up our path integral and first do the integral over the Φ fields. Thelight φ fields are the only ones left, so we can then write the partition function as Z = (cid:90) D φ e iS eff ( φ ) , (1.7)where this defines S eff . Thus far this still contains all the same physics, as long as wedon’t want to know about processes with external heavy fields . But having decided thatwe are interested in the infrared physics of the φ fields, we can say that the effects ofthe heavy Φ fields will be suppressed by factors of the energies of interest divided by themass of Φ , and we should expand the Lagrangian L eff in an appropriate series: Z = (cid:90) D φ exp (cid:34) i (cid:90) d d x (cid:32) L ( φ ) + N (cid:88) n =0 M d − n Φ (cid:88) i λ ( n ) i O ( n ) i ( φ ) (cid:33)(cid:35) , (1.8)where L ( φ ) is the part of the full Lagrangian that had no heavy fields in it, O ( n ) i ( φ ) isan operator of classical dimension n , λ ( n ) i is a dimensionless coupling, and N ≥ d definesthe precision to which one works in this effective theory. This is the procedure to find atop-down effective field theory in the abstract. Since we haven’t made any approximations and have the same object Z , one may be confused asto why we’ve lost access to the physics of the Φ fields. In fact I’ve been a bit sloppy. If we wantto compute correlation functions of our fields φ , we must couple our fields to classical sources J φ as L ⊃ φ ( x ) J φ ( x ) . Physically, those sources allow us to ‘turn on’ particular fields so that we can thencalculate their expectation values. Mathematically, we really need the partition function as a functionalof these sources Z [ J φ ] , and we take functional derivatives with respect to these sources as a step tocalculating correlation functions or scattering amplitudes. In integrating out our heavy field Φ , we nolonger have a source we can put in our Lagrangian to turn on that field, as it no longer appears in theaction. ffective Field Theory Chapter 1 Figure 1.2: Schema of how the SM fits into SU (5) . The SU (5) vector field V decomposesinto the gluons G AB , the SU (2) L vectors W αβ , hypercharge B along the diagonal, and thenew as-yet-unobserved leptoquarks X αβ carrying both color and electroweak charge. Theright-handed down-type quarks and the left-handed leptons are unified into an antifun-damental. Decomposing the to find the rest of the SM fermions is left as an exercise.A great example of a top-down EFT is in studying the Standard Model fields in thecontext of a Grand Unified Theory (GUT). Broadly, Grand Unification is the hope thatthere is some simpler, more symmetric theory behind the Standard Model which explainsits structure. A GUT is a model in which the gauge groups of the SM are (partially [41])unified in the UV. If there is a full unification to a single gauge factor, then this requires‘gauge coupling unification’ in the UV until the symmetry is broken down to the SMgauge group at a high scale [42]. While one’s first exposure to this idea today may be inthe context of a UV theory like string theory which roughly demands such unification,this was in fact first motivated by the observed infrared SM structure. It is franklyamazing that not only are the values of the SM gauge couplings consistent with this idea,and not only does SU (3) × SU (2) × U (1) fit nicely inside SU (5) , but the SM fermionrepresentations precisely fit into the ⊕ ¯5 representations of SU (5) (see Figure 1.2).It’s hard to imagine a discovery that would have felt much more like one was obviouslylearning something deep and important about Nature than when Georgi realized hownicely all of this worked out. I’m reminded of Einstein’s words on an analogous situation9 ffective Field Theory Chapter 1 in the early history of electromagnetism—the original unified theory:The precise formulation of the time-space laws of those fields was the work ofMaxwell. Imagine his feelings when the differential equations he had formu-lated proved to him that electromagnetic fields spread in the form of polarisedwaves, and with the speed of light! To few men in the world has such an ex-perience been vouchsafed. At that thrilling moment he surely never guessedthat the riddling nature of light, apparently so completely solved, would con-tinue to baffle succeeding generations.— Albert Einstein Considerations concerning the Fundaments of Theoretical Physics , 1940 [43]And just as with Maxwell, the initial deep insight into Nature was not the end of the story.As of yet, Grand Unification remains an unproven ideal, and indeed further empiricaldata has brought into question the simplest such schemes. But it’s hard to imagine all ofthis beautiful structure is simply coincidental, and I would wager that most high energytheorists still have a GUT in the back of their minds when they think about the UVstructure of the universe, so this is an important story to understand. To learn generallyabout GUTs, I recommend the classic books by Kounnas, Masiero, Nanopoulos, & Olive[44] and Ross [45] or the recent book by Raby [46] for the more formally-minded. Shorterintroductions can be found in Sher’s TASI lectures [47] or in the Particle Data Group’sReview of Particle Physics [48] from Hebecker & Hisano.The structure of the simplest SU (5) GUT is that the symmetry group breaks down tothe SM at energies M GUT ∼ GeV via the Higgs mechanism. More generally, unifica- The scale M GUT is determined from low-energy data by computing the scale-dependence of the SMgauge couplings, evolving them up to high energies, and looking for an energy scale at which they meet.Since we have three gauge couplings at low energies, it is quite non-trivial that the curves g i ( µ ) , i = 1 , , meet at a single scale µ = M GUT . The M GUT so computed is approximate not only due to experimentaluncertainties on the low-energy values of parameters in the SM, but also because additional particles with ffective Field Theory Chapter 1 tion may proceed in stages as, for example, SO (10) → SU (4) c × SU (2) L × SU (2) R → SM ,and the breaking may occur via other mechanisms, as we’ll discuss further in Section 3.Back to our simple single-breaking example, as is familiar in the SM this means that thegauge bosons corresponding to broken generators get masses of order this GUT-breakingscale. As this is a far higher scale than we are currently able to directly probe, it isneither necessary nor particularly useful to keep these degrees of freedom fully in ourdescription if we’re interesting in understanding the effects of GUT-scale fields. Ratherthan constructing the complete top-down EFT of the SM from a GUT, let’s focus on oneparticularly interesting effect.One of the best ways to indirectly probe GUTs is by looking for proton decay. TheGUT representations unify quarks and leptons, so the extra SU (5) gauge bosons havenonzero baryon and lepton number and fall under the label of ‘leptoquarks’. It’s worthconsidering in detail why proton decay is a feature of GUTs and not of the SM, as it’s asubtler story than is usually discussed. While U (1) B , the baryon number, is an accidentalglobal symmetry of the SM , it’s an anomalous symmetry and so is not a symmetry ofthe quantum world. The ‘baryon minus lepton’ number, U (1) B − L , is non-anomalous, butthis is a good symmetry both of the SM and of a GUT and clearly does not prevent e.g. p + → π e + . What’s really behind the stability of the proton is that, though U (1) B and U (1) L are not good quantum symmetries, the fact that they are good classical symmetriesmeans their only violation is nonperturbatively by instantons. Such configurations yield SM charges affect slightly how the couplings evolve toward high energies. Indeed, adding supersymmetrymakes the intersection of the three curves even more accurate than it is in the SM itself. ‘Accidental’ here means that imposing this symmetry on the SM Lagrangian does not forbid anyoperators which would otherwise be allowed. The SM is defined, as above, by the gauge symmetries SU (3) C × SU (2) L × U (1) Y and the field content. Writing down the most general dimension-4 La-grangian 1.3,1.4 invariant under these symmetries gives a Lagrangian which is automatically invariantunder U (1) B . This no longer holds at higher order in SMEFT, and indeed the dimension-6 Lagrangian(Equation 1.5) does contain baryon-number-violating operators. If one wants to study a baryon-number-conserving version of SMEFT, one needs to explicitly impose that symmetry on the dimension-6 La-grangian, so U (1) B is no longer an accidental symmetry of SMEFT. ffective Field Theory Chapter 1 Figure 1.3: A representative diagram contributing to proton decay due to exchange of aheavy GUT gauge boson X.solely baryon number violation by three units at once, corresponding to the number ofgenerations, and thus the proton with B = 1 is stable .But in GUTs, baryon number and lepton number are no longer accidental symmetries,so no such protection is available and the GUT gauge bosons mediate tree-level protondecay processes as in Figure 1.3. We can find the leading effect by integrating these out— Convincing yourself fully that ∆ B = 3 is the smallest allowed transition is not straightforward,but let me try to make it believable for anyone with some exposure to anomalies and instantons. Theexistence of a mixed U (1) B SU (2) L anomaly—equivalently a nonvanishing triangle diagram with two SU (2) L gauge legs and a baryon current insertion—means that the baryon current will no longer bedivergenceless, ∂ µ j µB ∝ g π tr ˜ W W , where W µνa is the field strength and ˜ W its Hodge dual. Instantonsare field configurations interpolating between vacuum field configurations of different topology, and thereare nontrivial instantons in 4d Minkowski space for SU (2) L but not for U (1) Y as a result of topologicalrequirements on the gauge group. So while there is also a mixed anomaly with hypercharge, we canignore this for our purposes. The SM fermions contributing to the U (1) B SU (2) L anomaly are then only Q a (the left-handed quark doublet with B = 1 / , with a a color index) and similarly for the leptonanomaly only the doublet L matters. There are three generations of each, which leads simply to a factorof three in the divergence of the global currents. Thinking about it in terms of triangle diagrams, this issimply because there are thrice as many fermions running in the loop. The extent to which an instantonsolution changes the topology is given by the integral of g π tr ˜ W W over spacetime, which as a totalderivative localizes to the boundary, and furthermore turns out to be a topological invariant of the gaugefield configuration known as the winding number, an integer (technically the change in winding numberbetween the initial and final vacua). Then the anomaly, by way of the nonvanishing divergence, relatesthe winding number of such a configuration to the change in baryon and lepton number it induces.The factor of the number of generations means that each unit of winding number ends up producing ∆ B = − ∆ L = n g = 3 . And that is why the proton is stable. Classic, detailed references on anomaliesand instantons include Coleman [49], Bertlmann [50], and Rajamaran [51]. ffective Field Theory Chapter 1 in particular we’ll here look just at a four-fermion baryon-number-violating operator. Thetree-level amplitude is simply i M ( uude ) = ( ig ) v u γ µ u u − i (cid:16) η µν + p µ p ν M GUT (cid:17) p + M GUT v e γ ν u d ≈ ig v u γ µ u u η µν M GUT v e γ ν u d + O ( 1 M GUT ) , (1.9)so integrating out the gauge boson from this diagram gives us one of the contributionsto the low-energy operators in Equation 1.8 O = g M GUT (cid:15) αβγ ( u Rα γ µ u Lβ ) ( e R γ µ d Lγ ) , (1.10)where, in the notation of Equation 1.8, O (6) i = (cid:15) αβγ ( u Rα γ µ u Lβ ) ( e R γ µ d Lγ ) , λ (6) i = g / and M Φ = M GUT . The calculation of the proton lifetime from this operator is quitecomplicated, but the dimensional analysis estimate of τ p ∼ M GUT /g m p actually workssurprisingly well.The job of the top-down effective field theorist is to calculate the effects of someparticular UV physics on IR observables and by doing so understand how to searchfor their particular effects. While the effects will, by necessity, be some subset of theoperators that the bottom-up effective field theorist has written down, the patterns andcorrelations present from a particular UV model can suggest or require particular searchstrategies. In the present context, a GUT may suggest the most promising final statesto look for when searching for proton decay. If we wanted to calculate the lifetime andbranching ratios more precisely we would have to deal with loop diagrams (among manycomplications), which of course is a generic feature. So we now turn our attention to thenew aspects and challenges of field theory that appear once one goes beyond tree-level.13 ffective Field Theory Chapter 1 Renormalization is a notoriously challenging topic for beginning quantum field theoriststo grok, and explanations often get bogged down in the details of one particular per-spective or scheme or purpose and ‘miss the forest for the trees’, so to speak. We’llattempt to overcome that issue by discussing a variety of uses for and interpretationsof renormalization, as well as how they relate. And, of course, by examining copiousexamples and pointing out a variety of conceptual pitfalls.
Loops are necessary
At the outset the only fact one needs to have in mind is that renormalization is a pro-cedure which lets quantum field theories make physical predictions given some physicalmeasurements. Such a procedure was not necessary for a classical field theory, which isroughly equivalent to a quantum field theory at tree-level. A natural question for be-ginners to ask then is why we should bother with loops at all: Why don’t we just startoff with the physical, measured values in the classical Lagrangian and be done with it?That is, if we measure, say, the mass and self-interaction of some scalar field φ , let’s justdefine our theory L exact ? = − ∂ µ φ∂ µ φ − m phys φ − λ phys φ , (1.11)for some definitions of these physical parameters, and compute everything at tree-level.However, this does not constitute a sensible field theory, as the optical theorem tells usthis is not consistent. We define S = 1 + i M as the S-matrix which encodes how states inthe theory scatter, where the is the ‘trivial’ no-interaction part. Quantum mechanics Not that I begrudge QFT textbooks or courses for it, to be clear. There is so much technology tointroduce and physics to learn in a QFT class that discussion of all of these various perspectives andissues would be prohibitive. ffective Field Theory Chapter 1 Figure 1.4: Schematic description of the optical theorem Equation 1.12 as applied toa → process. On the right hand side one must sum over all possible intermediatestates, including both various state labels X and their phase space Π X .turns the logical necessity that probabilities add up to 1 into the technical demand of‘unitarity’, S † S = 1 , which tells us the nontrivial part must satisfy i (cid:0) M † − M (cid:1) = M † M . (1.12)Sandwiching this operator equation between initial and final states, we find that theleft hand side is the imaginary part of the amplitude M ( i → f ) , which is nonzero solelydue to loops. This is depicted schematically in Figure 1.4. We can see why this is byexamining a scalar field propagator. Taking the imaginary part one findsIm p + m + i(cid:15) = − (cid:15) ( p + m ) + (cid:15) . (1.13)This vanishes manifestly as (cid:15) → except for when p = − m , and an integral to find thenormalization yields Im p + m + i(cid:15) = − πδ ( p + m ) . (1.14)So internal propagators are real except for when the particle is put on-shell. In a tree-leveldiagram this occurs solely at some measure-zero set of exceptional external momenta,but in a loop-level diagram we integrate over all momenta in the loop, so an imaginarypart is necessarily present. Now we see the necessity of loops solely from the conservationof probability and the framework of quantum mechanics . A natural question to ask is whether this structure can be perturbed at all, but in fact it really is ffective Field Theory Chapter 1 The lesson to take away from this is that classical field theories produce correlationfunctions with some particular momentum dependence, which can be essentially read offfrom the Lagrangian. But a consistent theory requires momentum dependence of a sortthat does not appear in such a Lagrangian, which demands that calculations must includeloops. In particular it is the analyticity properties of these higher-order contributionsthat are required by unitarity, and there is an interesting program to understand theset of functions satisfying those properties at each loop order as a way to bootstrap thestructure of multi-loop amplitudes (see e.g. [54, 55, 56, 57, 58, 59]).So far from being ‘merely’ a way to deal with seemingly unphysical predictions, renor-malization is very closely tied to the physics. We begin in the next section with under-standing its use for removing divergences, as this is the most basic application and isoften the first introduction students receive to renormalization. We will then move on todiscuss other, more physical interpretations of renormalization.
Physical input is required
As a first pass, let’s look again at a φ theory L = − ∂ µ φ∂ µ φ − m φ − λ φ , (1.15)and now treat it properly as a quantum field theory. As a simple example, let us consider → scattering in this theory, our discussion of which is particularly influenced by Zee[20]. At lowest-order this is extremely simple, and the tree-level amplitude is i M ( φφ → φφ ) = − iλ . But if we’re interested in a more precise answer, we go to the next order in quite rigid. After Hawking—motivated by black hole evaporation—proposed that the scattering matrixin a theory of quantum gravity should not necessarily obey unitarity [52], the notion of modifying theS-matrix to a non-unitary ‘ $ ’-matrix (pronounced ‘dollar matrix’) received heavy scrutiny. This wasfound to necessarily lead to large violations of energy conservation, among other maladies [53]. ffective Field Theory Chapter 1 Figure 1.5: One-loop diagrams contributing to → scattering in the φ theory. Sincethe external legs are all the same, these three different internal processes must be summedover, corresponding to momentum exchanged in the s, t, u channels, where these are theMandelstam variables [60].perturbation theory and we have the one-loop diagrams of Figure 1.5.Defining P µs ≡ p µ + p µ as the momentum flowing through the loop in the s -channeldiagram, that diagram is evaluated as i M ( φφ → φφ ) one-loop s-channel = ( − iλ ) ( − i ) (cid:90) d k (2 π ) P s + m P s + k ) + m . (1.16)Just from power-counting, we can already see that this diagram will be divergent. Inthe infrared, as k → , the diagram is regularized by the mass of the field, but in theultraviolet k → ∞ , the integral behaves as ∼ (cid:82) d k/k ∼ (cid:82) dk/k which is logarithmicallydivergent.Though one might be tempted now to give up, we note that this divergence is ap-pearing from an integral over very high energy modes—far larger than whatever energieswe’ve verified our φ model to, so let’s try to ignore those modes and see if we can’t geta sensible answer. The general term for removing these divergences is ‘regularization’and we will here regularize (or ‘regulate’) this diagram by imposing a hard momentumcutoff Λ in Euclidean momentum space, which is the maximum energy of modes we letpropagate in the loop. The loop amplitude may then be calculated with elementarymethods detailed in, for example, Srednicki’s textbook [18]. First we introduce Feynmanparameters to combine the denominators, using ( AB ) − = (cid:82) dx ( xA +(1 − x ) B ) − , which17 ffective Field Theory Chapter 1 here tells us P s + m P s + k ) + m = (cid:90) dx (cid:2) x (cid:0) ( P s + k ) + m (cid:1) + (1 − x )( P s + m ) (cid:3) − (1.17) = (cid:90) dx (cid:2) q + D (cid:3) − , (1.18)where we’ve skipped the algebra letting us rewrite this with q = k + xP s and D = x (1 − x ) P s + m . The change of variables k → q has trivial Jacobian, so the next step isto Wick rotate—Euclideanize the integral by defining q = i ¯ q d , such that q ≡ q µ η µν q ν =¯ q µ δ µν ¯ q ν ≡ ¯ q . The measure simply picks up a factor of i , d d q = id d ¯ q , and we can thengo to polar coordinates via d d ¯ q = ¯ q d − d ¯ qd Ω d . Lorentz invariance then means the angularintegral gives us the area of the unit sphere in d dimensions, Ω d = 2 π d/ / Γ( d/ , where Ω = 2 π , and the radial integral becomes (cid:90) Λ0 d ¯ q ¯ q (cid:2) q + D (cid:3) − = 12 (cid:20) D ¯ q + D + log (cid:0) ¯ q + D (cid:1)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) Λ0 (1.19) = − (cid:20) Λ Λ + D + log D Λ + D (cid:21) . (1.20)In fact it is possible to do the x integral analytically here, but we’ll take Λ (cid:29) | P | (cid:29) m to find a simple answer − (cid:90) dx (cid:20) Λ Λ + D + log D Λ + D (cid:21) = 12 (cid:18) − log P s Λ (cid:19) + . . . (1.21)Now putting all that together and including all the diagrams up to one-loop, we get theform M ( φφ → φφ ) = − λ + Cλ (cid:20) log (cid:18) Λ s (cid:19) + log (cid:18) Λ t (cid:19) + log (cid:18) Λ u (cid:19)(cid:21) + subleading , (1.22)where s, t, u are the Mandelstam variables and C is just a numerical coefficient. Nowwe see explicitly that the divergence has led to dependence of our amplitude on ourregulator Λ . Of course this is problematic because we introduced Λ as a non-physicalparameter, and it would not be good if our calculation of a physical low-energy observable18 ffective Field Theory Chapter 1 depended sensitively on how we dealt with modes in the far UV. But let us try to connectthis with an observable anyway. We note that the theory defined by the Lagrangian inEquation 1.15 can not yet be connected to an observable because we have not yet givena numerical value for λ . So let’s imagine an experimentalist friend of ours preparessome φ s and measures this scattering amplitude at some particular angles and energiescorresponding to values of the Mandelstam variables s , t , u . They find some value λ phys , which is a pure number. If our theory is to describe this measurement accurately,this tells us a relation between our parameters and a physical quantity − λ phys = − λ + Cλ (cid:20) log (cid:18) Λ s (cid:19) + log (cid:18) Λ t (cid:19) + log (cid:18) Λ u (cid:19)(cid:21) + O ( λ ) . (1.23)This is known as a ‘renormalization condition’ which tells us how to relate our quantumfield theories to observations at non-trivial loop order. Since the left hand side is aphysical quantity, it may worry us that the right hand side contains a non-physicalparameter Λ . But we still haven’t said what λ is, so perhaps we’ll be able to finda sensible answer if we choose λ ≡ λ (Λ) in a correlated way with our regularizationscheme. We call this ‘promoting λ to a running coupling’ by changing from the ‘barecoupling’ λ to one which depends on the cutoff. So let’s solve for λ in terms of λ phys and Λ . Rearranging we have λ = λ phys + Cλ (cid:20) log (cid:18) Λ s (cid:19) + log (cid:18) Λ t (cid:19) + log (cid:18) Λ u (cid:19)(cid:21) + O ( λ ) (1.24) λ (Λ) = λ phys + Cλ phys (cid:20) log (cid:18) Λ s (cid:19) + log (cid:18) Λ t (cid:19) + log (cid:18) Λ u (cid:19)(cid:21) + O ( λ phys ) (1.25)where in the second line the replacement λ (cid:55)→ λ phys modifies the right side only at higher-order and so that is absorbed into our O ( λ ) uncertainty. To see what this has done forus, let us plug this back into our one-loop amplitude Equation 1.22. This will impose ourrenormalization condition that our theory successfully reproduces our experimentalist19 ffective Field Theory Chapter 1 friend’s result. We find M ( φφ → φφ ) = − λ phys − Cλ phys (cid:20) log (cid:18) Λ s (cid:19) + log (cid:18) Λ t (cid:19) + log (cid:18) Λ u (cid:19)(cid:21) (1.26) + Cλ phys (cid:20) log (cid:18) Λ s (cid:19) + log (cid:18) Λ t (cid:19) + log (cid:18) Λ u (cid:19)(cid:21) + O ( λ phys ) where again we liberally shunt higher-order corrections into our uncertainty term. Takingadvantage of the nice properties of logarithms, we rearrange to get M ( φφ → φφ ) = − λ phys − Cλ phys (cid:20) log (cid:18) ss (cid:19) + log (cid:18) tt (cid:19) + log (cid:18) uu (cid:19)(cid:21) + O ( λ phys ) . (1.27)We see that our renormalization procedure of relating our theory to a physical observablehas enabled us to write the full amplitude in terms of physical quantities, and remove thedivergence entirely. This one physical input at some particular kinematic configurationhas enabled us to fully predict any → scattering in this theory.We thus see how the renormalization procedure removes the divergences in a naïveformulation of a field theory and allows us to make finite predictions for physical ob-servables. While we did need to introduce a regulator, once we make the replacement λ → λ (Λ) as defined in Equation 1.25 (and similar replacements for the coefficients ofthe other operators), the one-loop divergences are gone. We are guaranteed that anyone-loop correlation function we calculate is finite in the Λ → ∞ limit, which removesthe regulator. If we wanted to increase our precision and calculate now at two loops,we would first renormalize the theory at two loops analogously to the above, and wouldfind a more precise definition for λ (Λ) which included terms of order O ( λ phys ) . At eachloop order, replacing the bare couplings with running couplings suffices to entirely ridthe theory of divergences. 20 ffective Field Theory Chapter 1 Renormalizability
An important question is for which quantum field theories do a finite set of physicalinputs allow the theory to be fully predictive, in analogy to the example above. Such atheory is called ‘renormalizable’ and means that after some finite number of experimentalmeasurements, we can predict any other physics in terms of those values. Were thisnot the case, and no finite number of empirical measurements would fix the theory, itwould not be of much use. Within the context of perturbation theory, a theory willbe renormalizable if its Lagrangian contains solely relevant and marginal operators, andindeed for our φ theory three renormalization conditions are needed—one for each suchoperator.The simplest way to understand why we must restrict to relevant and marginal op-erators is that irrelevant operators inevitably lead to the generation of a tower of more-and-more irrelevant operators. To see this, imagine now including a φ interaction, asdepicted in Figure 1.6a. At one loop this leads to a → scattering process with thesame sort of divergence we saw in our previous loop diagram. So this loop is probingthe UV physics, but we cannot absorb the unphysical divergence into a local interactionin our Lagrangian unless we now include a φ term. But then we can draw a similarone-loop diagram with the φ interaction which will require a φ interaction, and so on.Note that in our φ theory we also have → scattering at one loop, seen in Figure1.6b, but there it comes from a box diagram which is finite, and so there is no need toinclude more local operators.However, we emphasized above that the most useful description of a system dependson the precision at which one wishes to measure properties of the system. Thus in thestudy of effective field theories a broader definition of renormalizability should be used.For a theory with cutoff Λ , one decides to work to precision O ( E/ Λ) n where E is a21 ffective Field Theory Chapter 1 (a) (b) Figure 1.6: In (a), a pictoral representation of the nonrenormalizability of theories de-scribed by Lagrangians with irrelevant operators. A tree-level six-point coupling leadsto a new divergence in → scattering, which must be absorbed into a renormalizedeight-point coupling, which would then beget a divergent → amplitude . . . . Notethat this does not mean there are not one-loop contributions to → scattering. Such adiagram is depicted in (b), but it is finite and does not require additional local operators.typical energy scale of a process and n is some integer. There are then a finite numberof operators which contribute to processes to that precision—only those up to scalingdimension n —and so there is a notion of ‘effective renormalizability’ of the theory. Westill require solely a finite number of inputs to set the behavior of the theory to whateverprecision we wish, but such a theory nevertheless fails the original criterion, which maybe termed ‘power counting renormalizability’ in comparison. Wilsonian renormalization of φ Above we characterized our cutoff as an unphysical parametrization of physics at highscales that we do not know and we found that its precise value dropped out of ourphysically observable amplitude. To some extent this is rather surprising, as it’s telling usthat the high energy modes in our theory have little effect on physics at long distances—we can compensate for their effects by a small shift in a coupling. We can gain insightinto the effects of these high energy modes by taking the cutoff seriously and looking22 ffective Field Theory Chapter 1 at what happens when the cutoff is lowered. This brilliant approach due to Wilson [61]is aimed at providing insight as to the particular effects of these high-energy modes byintegrating out ‘shells’ of high-energy Euclidean momenta and looking at the low-energyeffects. This discussion is closely inspired by that in Peskin & Schroeder’s chapter 12[19], as well as Srednicki’s chapter 29 [18].It is easiest to see how to implement this by considering the path integral formulation.We can equally well integrate over position space paths as over momentum modes: Z = (cid:90) D φ ( x ) e i (cid:82) L ( φ ) = (cid:32)(cid:89) k (cid:90) d φ ( k ) (cid:33) e i (cid:82) L ( φ ) , (1.28)and here it is clear that we may integrate over particular momentum modes separately ifwe so choose. In the condensed matter application in which Wilson originally worked, acutoff appears naturally due to the lattice spacing a giving an upper bound on momenta k max ∼ /a . In a general application we can imagine defining the theory with a funda-mental cutoff Λ by including in the path integral only modes with Euclidean momentum k ≤ Λ . The theory is defined by the values of the parameters in the theory with thatcutoff—our familiar relevant and marginal operators m (Λ) , λ (Λ) and in principle all ofthe ‘Wilson coefficients’ of irrelevant operators as well, since the theory is manifestlyfinite. The idea is to effectively lower the cutoff by explicitly integrating out modes with b Λ ≤ k ≤ Λ for some b < . This will leave us with a path integral over modes with k ≤ b Λ —which is a theory of the same fields now with a cutoff b Λ . By integratingout the high-energy modes we’ll be able to track precisely their effects in this low-energytheory.Peskin & Schroeder perform this path integral explicitly by splitting the high energymodes into a different field variable and quantizing it, but since we’ve already introduced It is necessary to define this cutoff in Euclidean momentum space for the simple fact that inLorentzian space a mode with arbitrarily high energy k may have tiny magnitude by being close to thelight cone | k | (cid:39) | (cid:126)k | . It is left as an exercise for the reader to determine what deep conclusion should betaken away from the fact that we generally perform all our QFT calculations in Euclidean space. ffective Field Theory Chapter 1 the conceptual picture of integrating fields out we’ll take the less-involved approach ofSrednicki. To repeat what we discussed above, the diagrammatic idea of integratingout a field is to remove it from the spectrum of the theory and reproduce its effects onlow-energy physics by modifying the interactions of the light fields. Performing the pathintegral over some fields does not change the partition function, so the physics of theother fields must stay the same. We want to do the same thing here, but integrate outsolely the high energy modes of a field and reproduce the physics in terms of the lightmodes.We’ll continue playing around with φ theory and define our (finite!) theory with acutoff of Λ , which in full generality looks like: L (Λ) = − Z (Λ) ∂ µ φ∂ µ φ − m (Λ) φ − λ (Λ) φ − ∞ (cid:88) n =3 c n (Λ)(2 n )! φ n . (1.29)For simplicity we will decree that at our fundamental scale Λ we have a canonicallynormalized field Z (Λ) = 1 and no irrelevant interactions c n (Λ) = 0 , but just someparticular m (Λ) and λ (Λ) .Let’s look first at the one-loop four-point amplitude, which we must ensure is thesame in both the theory with cutoff Λ and that with cutoff b Λ . In the original theory,the amplitude at zero external momentum is iV Λ4 (0 , , ,
0) = − iλ (Λ) + 32 ( − iλ (Λ)) (cid:90) | k | < Λ d k (2 π ) ( − i ) ( k + m (Λ)) + O ( λ ) (1.30)When we evaluate this in the theory with a lowered cutoff b Λ , the modification is simplyto everywhere make the replacement Λ (cid:55)→ b Λ . In order for the physics to remain thesame without the high-energy modes, the vertex function must not change. We’ll takefull advantage of the perturbativity of the result—that is, λ (Λ) − λ ( b Λ) ∼ O ( λ ) , m (Λ) − m ( b Λ) ∼ O ( λ ) —to swap out quantities evaluated at b Λ in the second-order term for24 ffective Field Theory Chapter 1 those evaluated at Λ at the cost solely of higher-order terms which we ignore. ≡ V Λ4 (0 , , , − V b Λ4 (0 , , , (1.31) = − λ (Λ) + 32 λ (Λ) (cid:90) | ¯ k | < Λ d ¯ k (2 π ) k + m (Λ)) + λ ( b Λ) − λ ( b Λ) (cid:90) | ¯ k |
We have seen that the need to remove divergences in our theory led to the introduction ofrunning couplings which change as a function of scale. In our example above we see thatrenormalization has the operational effect of transferring loop-level physics into the tree-level parameters. This is an interesting perspective which bears further exploration—ifthere is hidden loop-level physics that really has the same physical effects as the tree-level bare parameters, perhaps this is a sign that there is a better way to organize ourperturbation theory. Indeed, at some very general level renormalization can be thoughtof as a method for improving the quality of perturbation theory. For useful discussionsat this level of abstraction of how renormalization operates, see [63] for its natural ap-pearance whenever infinities are encountered in naïve perturbative calculations, and [64]for its usefulness even when infinities are not present. We’ll discuss this perspective onrenormalization further in the next section.However it’s clear that loops also give rise to physics that is starkly different fromthe lowest-order result (e.g. non-trivial analytic structure), so how do we know what28 ffective Field Theory Chapter 1
Figure 1.7: A position space Feynman diagram with vertices labeled.higher-order physics we can stuff into tree-level? In a continuum quantum field theory, aLagrangian is a local object L ( x ) —that is, it contains operators like φ ( x ) which give aninteraction between three φ modes at a single spacetime point x . Such effects are knownas ‘contact interactions’, but even at tree-level a local Lagrangian can clearly producenon-local (that is, long-range) physics effects. For example, consider the amplitude for → scattering in a φ theory at second order in the coupling.In position space the non-locality here is obvious, as in Figure 1.7: A simple tree-leveldiagram corresponds to a particle at point x and a particle at point y exchanging a φ quantum, but one may forget this important fact when working in momentum space.There the result is i M ( φφ → φφ ) = ig p + m , (1.42)and indeed, Fourier transforming the cross-section for this process yields a Yukawa scat-tering potential for our φ particles, showing that they mediate a long-range force overdistances r ∼ /m . We obviously cannot redefine the Lagrangian to put this effect intothe lowest order of perturbation theory since this is not a local effect.But if we do have a continuum quantum field theory, then because of locality itdescribes fluctuations on all scales. When we go to loop-level, we must integrate overall possible internal states, which includes integrating over arbitrarily large momenta or29 ffective Field Theory Chapter 1 equivalently fluctuations on arbitrarily small scales. Heuristically, when the loop integralis sensitive to the ultraviolet of the theory, it is computing effects that operate on allscales—that is, it gives a contribution to local physics. This tells us that the pieces of thishigher-order contribution which we can reshuffle into our Lagrangian are connected withultraviolet sensitivity, leading to a close connection of renormalization with divergences.A couple notes are warranted about the notion of locality we rely on here. Firstly, it’sclear that this criterion of local effects appears because we began with a local Lagrangian.If we postulated that our fundamental theory contained nonlocal interactions, say L ( x ) ⊃ gφ ( x ) (cid:0)(cid:82) dyφ ( y ) (cid:1) (cid:0)(cid:82) dzφ ( z ) (cid:1) , then we could clearly absorb further nonlocalities with thesame structure into this coupling as well. However, this sort of nonlocality is differentfrom the nonlocality we saw appearing out of the local theory at tree-level. In particularit would break the standard connection between locality and the analytic structure ofamplitudes—see e.g. Schwartz [22] or Weinberg [65] on ‘polology’ and locality.Secondly, our notion of locality should be modified in a low-energy theory with anenergy-momentum cutoff Λ , as can be seen in hindsight in our Wilsonian discussion above.As Λ defines a maximum energy scale we can probe, there is equivalently a minimumtime and length scale we can probe due to the uncertainty principle, heuristically ∆ x µ (cid:38) / ∆ p µ (cid:38) / Λ . As a result, any fluctuations on shorter length scales are effectively localfrom the perspective of the low-energy theory. An exchange of a massive field with M > Λ or of a light field with high frequency ω > Λ appears instantaneous to low-energy observers. This explains how it’s sensible to use renormalization techniques in,for example, condensed matter applications, where systems are fundamentally discrete.We can see this concretely by imagining the light φ s in the tree-level example aboveinstead exchanged a heavier scalar Φ with mass M (cid:29) m . While the amplitude M = g / ( p + M ) is still nonlocal in the continuum theory, if we’re only interested in physicsat energies E (cid:28) M we may Taylor expand the result M = g /M + g p / M + . . . .30 ffective Field Theory Chapter 1 We may then absorb the leading effects of this heavy scalar into an effectively localinteraction g φ /M among the light fields—as long as we work at energies below M .In the next section we’ll explore concretely how these insights enable us to transferloop-level physics to tree-level physics, and so improve our calculations. Renormalization group equations
The astute reader may notice a potential issue with our one-loop results in the φ theory,Equations 1.32,1.34,1.35. We’ve derived this behavior as the first subleading terms in aseries expansion in the number of loops. In relation to the tree-level results, the one-loop contributions are suppressed by a factor ∼ λ/ π log(Λ / Λ) , where I’ve switchednotation to now have Λ as a high scale and Λ as the lower scale we integrate down to.Higher n -loop contributions will be further suppressed by n loop factors. But what if wewanted to study physics at a scale far lower than Λ ? Eventually this factor becomeslarge enough that we will need to compute many loops to obtain high precision, and thenlarge enough that we have reason to question the convergence of the series . Keep inmind that we are in the era of precision measurements of the Standard Model, so theseone-loop expressions are very restrictive.For a concrete example, say we wanted to check the SM prediction for a measurementof a coupling λ (Λ) with λ (Λ ) = 1 whose experimental uncertainty was . Let’s define Please excuse my slang. Perturbative series in QFT are quite generally not convergent but wecan trust the answers anyway to order n ∼ exp 1 / expansion parameter because they are asymptotic series. So really when this parameter becomes large enough we worry that our series is not even asymp-totic. Thinking deeply about this leads to many interesting topics in field theory, from accounting fornonpertubative instanton effects which are (partially) behind the lack of convergence; to the programof ‘resurgence’, the idea that there are secret relations between the perturbative and nonperturbativepieces. This is all far outside my purview here, but some introductions aimed at a variety of audiencescan be found in [66, 67, 68, 69, 70, 71]. ffective Field Theory Chapter 1 a theoretical uncertainty on a perturbative calculation to n th order as (cid:15) n ≡ n th order result − estimated size of ( n + 1) th order result n th order result , (1.43)where our Wilsonian calculation in Section 1.2.1 gave at first order, as a reminder (andwith modified notation) λ (Λ) = λ (Λ ) − π λ (Λ ) log Λ Λ + O ( λ ) , (1.44)and our heuristic estimate for the size of the second order correction is ( π ) λ (Λ ) log Λ .When the result is simply a series, the uncertainty is very simple to calculate, as the nu-merator is then our estimate of the ( n + 1) th order correction, which is roughly the squareof the first order correction in this case for n = 1 . Here we have (cid:15) = π λ (Λ ) log Λ Λ .Then in order for our theoretical uncertainty to be subdominant to the experimentalprecision, (cid:15) < . , we must go past the one-loop result if we wish to look at energiesbelow Λ ∼ exp( − π × . / ∼ Λ / , the two-loop result is only sufficient until Λ ∼ Λ / , and if we can manage to calculate the three-loop corrections that only getsus down to something like Λ ∼ Λ / × . If we’re interested in taking the predictionsof a Grand Unified Theory defined at Λ ∼ GeV and comparing them to predictionsat SM energies, how in the world are we to do so?Fortunately, we can do better by applying our one-loop results more cleverly. It isclear by looking at Equations 1.32,1.34,1.35 that the results have the same form no matterthe values of Λ , Λ . So if we take Λ to be only slightly smaller than Λ (corresponding to − b (cid:28) in our previous notation) the expansion parameter becomes very small and theone-loop result becomes very trustworthy. What we would like is some sort of iterativeprocedure to gradually lower the cutoff, which we could then use to find the one-loopresult for energies far lower than the range of our perturbative series. This is in factprecisely the sort of problem that a differential equation solves, and we can derive such32 ffective Field Theory Chapter 1 an equation by differentiating both sides by ln Λ and then taking Λ infinitesimally closeto Λ . That exercise yields d Z (Λ) d log Λ = 0 (1.45)d m (Λ) d log Λ = λ (Λ)8 π (cid:0) m (Λ) − Λ (cid:1) (1.46)d λ (Λ) d log Λ = 316 π λ (Λ) (1.47)These are known as ‘renormalization group equations’ and they indeed allow us to evolvethe coupling down to low energies—one says we use them to ‘resum the logarithm’. Thento study physics at a very low scale we can bring these couplings down to a lower scaleand do our loop expansion using those couplings, which is known as ‘renormalizationgroup improved perturbation theory’, and which we will discuss in more detail soon.Explicitly solving with our boundary condition at Λ yields λ (Λ) = λ (Λ )1 + λ (Λ )16 π log Λ Λ (1.48)Turning back to our effective field theory language, we see that quantum correctionshave generated an anomalous dimension for λ , δ φ = 3 λ / (16 π ) , correcting the leadingorder scaling behavior. Since δ φ > , we’ve determined that the quartic interaction ismarginally irrelevant, which we will return to later.We can now look at the theoretical uncertainty in this one-loop resummed calculationby including an estimate of the next order correction to the running of the quartic d λ (Λ) d log Λ = π λ (Λ) + (cid:0) π (cid:1) λ (Λ) and resumming that expression. This can no longer bedone analytically, but numerical evaluation easily reveals that the theoretical uncertaintyhere stays below for many, many orders of magnitude below Λ . Resumming thelogarithmic corrections allows us to use our loop results to far greater effect.33 ffective Field Theory Chapter 1 Decoupling
The physical meaning and technical statement of the decoupling theorem commonlyconfuse even prominent practitioners of effective field theory, so it’s worth going clearlythrough an example to refine our understanding. Indeed, one may be confused just atzeroth order about how decoupling is sensible against the background of the hierarchyproblem—which is an issue of sensitivity of a scalar mass to heavy mass scales. Howcan we claim QFT obeys a decoupling theorem and then go on to worry at length aboutquantum corrections δm ∝ M ?The correct way to think about the decoupling theorem is not whether a top-downcalculation could yield a result that depends on heavy mass scales, but whether a bottom-up effective field theorist and low-energy observer could gain information about the heavymass scales through low-energy measurements. We can clarify this important differenceby looking at a one-loop mass correction to a light scalar φ of mass m from a heavy scalar Φ of mass M through the interaction λφ Φ . We again take a Wilsonian perspective andbegin at a scale Λ > M . In close analogy to what we had before, we now find ≡ Σ Λ (0) − Σ Λ (0) (1.49) = m (Λ ) + λ (Λ ) (cid:90) | k | < Λ d k (2 π ) k + M (Λ )) − m (Λ) − λ (Λ) (cid:90) | k | < Λ d k (2 π ) k + M (Λ))= m (Λ ) − m (Λ) + λ (Λ ) (cid:90) Λ | k | =Λ d k (2 π ) k + M (Λ ))= (cid:2) m (Λ ) − m (Λ) (cid:3) + λ (Λ )16 π (cid:2) Λ − Λ (cid:3) − λ (Λ ) M (Λ )16 π log (cid:18) Λ + M (Λ )Λ + M (Λ) (cid:19) ⇒ m (Λ) = m (Λ ) + λ (Λ )16 π (cid:18)(cid:2) Λ − Λ (cid:3) + M (Λ ) log Λ + M (Λ )Λ + M (Λ) (cid:19) + O ( λ ) . (1.50)And we may already exhibit the confusion. If we use this to calculate the mass at alow scale Λ (cid:28) M , we see that m (Λ) does depend on the heavy mass scale, and gets a34 ffective Field Theory Chapter 1 contribution which goes like m (Λ) ∼ M (Λ ) log (1 + Λ /M (Λ )) .However, the effect on the light scalar is an additive shift of the mass. If we go outand measure the mass at a single scale m (Λ) we can’t tell empirically which ‘parts’ ofthat came from m (Λ ) and which came from M (Λ ) or whatever else is in there, sowe have no idea of how this low-energy measurement depends on heavy scales. To getinformation about the various contributions to the light scalar mass, we can measure it atdifferent scales and look at how it changes. Of course this information is contained in therenormalization group equation for m (Λ) . At O ( λ ) , we can find this by differentiatingthe above, and we find d m (Λ) d log Λ = λ (Λ)8 π Λ (cid:20) M (Λ)Λ + M (Λ) − (cid:21) . (1.51)Now we can see the difference. If we perform low-energy observations where we can takethe cutoff below the mass of the heavy scalar Λ (cid:28) M , then the physics of the heavyscalar decouples from the running of the light scalar mass. It is only by studying thisrunning at low energies that we can gain information about the ultraviolet, and we seethat this information is contained solely in small corrections scaling as Λ /M . At lowenergies, to learn about short-distance physics we must make very precise measurementsof the low-energy physics. This is the sense in which heavy mass scales decouple fromthe theory in the infrared.
Renormalized perturbation theory
Now let us study another, slightly more complex theory and apply renormalization tech-niques to simplify our calculations. We avoid the complication of gauge symmetries andfocus instead on a Yukawa theory of a Dirac fermion interacting with a parity-odd scalar.Our first improvement to perturbation theory will be to switch from ‘bare’ to ‘renor-malized perturbation theory’. Let’s first recap our procedure in Section 1.2.1. We began35 ffective Field Theory Chapter 1 with a Lagrangian with bare parameters m , λ , . . . , introduced a regulator, computedthe physical parameters m phys , λ phys in terms of the bare ones, inverted those relation-ships, and then plugged in for the bare parameters in terms of the renormalized ones,after which we were left with an amplitude which remains finite as we remove the regu-lator. This procedure works to remove the divergences in any renormalizable theory, butis obviously rather cumbersome.Furthermore one may question the validity of performing a perturbative expansionin a bare parameter which we later discover is formally infinite in the continuum theory λ ∼ log(Λ ) → ∞ . It is both conceptually and computationally easier to instead startoff by performing perturbation theory in terms of the renormalized parameters which weknow to be finite by definition. Fortunately we can improve our accounting simply byreshuffling the Lagrangian as follows.In terms of the bare parameters and fields, the Lagrangian reads L = ¯ ψ (cid:0) i /∂ − M (cid:1) ψ + 12 φ (cid:0) (cid:3) − m (cid:1) φ (1.52) L = ig φ ¯ ψ γ ψ − λ φ (1.53)where we’ve split up the free and interaction parts. Just as in our earlier example,when we compute at one-loop these parameters will get corrections such that the bareparameters are no longer the physical parameters we measure. Anticipating that fact,let us rewrite the Lagrangian to explicitly account for those corrections from the outset.Although it was not a feature of our simple example above, in general there will be‘wavefunction renormalization’ which changes the normalization of our field operators,so we define φ = Z / φ φ, ψ = Z / ψ ψ where ψ, φ are now renormalized fields, We do thesame to define renormalized masses related to the bare masses as M = Z M Z − ψ M, m = Z − φ Z m m , and for the couplings g = Z − / φ Z − ψ Z g g, λ = Z − φ Z λ λ . Next we use thebrilliant strategy of adding zero to split these Z -factors into a piece with the same form36 ffective Field Theory Chapter 1 we started with and a ‘counterterm’ proportional to ( Z − . Since at tree-level there’sno renormalization needed, we know Z = 1 + O ( couplings ) . At nontrivial loop level, wemust choose the Z -factors to implement our chosen renormalization scheme.The Lagrangian now takes the form L = ¯ ψ (cid:0) i /∂ − M (cid:1) ψ + 12 φ (cid:0) (cid:3) − m (cid:1) φ (1.54) L = iZ g gφ ¯ ψγ ψ − Z λ λφ + L ct (1.55) L ct = i ( Z ψ −
1) ¯ ψ /∂ψ − ( Z M − M ¯ ψψ −
12 ( Z φ − ∂ µ φ∂ µ φ −
12 ( Z m − m φ (1.56)where we’ve split off the counterterms into L ct . We can now treat the terms in L ct simplyas additional lines and vertices contributing to our Feynman diagrams. We’ll see howuseful this is once we begin renormalizing the theory. This is done in full in Srednicki’schapters 51-52 [18], so we will not go through every detail. Continuum renormalization
We’ll regulate this theory using ‘dimensional regularization’ (dim reg) which analyticallycontinues the theory to general dimension d = 4 − (cid:15) . That this will regulate our theoryis not obvious, but I recommend Georgi [17] to convince yourself of this and Collins[72] for a full construction of dim reg; we’ll content ourselves with seeing it in action.Our renormalization scheme will be ‘modified minimal subtraction’ and denoted MS,where ‘minimal subtraction’ means we’ll choose our counterterms solely to cancel offthe divergent pieces (rather than to enforce some relation to physical observables, aswe did previously) and ‘modified’ means that actually it’s a bit nicer if we cancel off acouple annoying constants as well. Since we’re using MS, the mass parameters m, M will not quite be the physical masses, which are always the locations of the poles in thepropagators, and the fields will not be normalized to have unity residue on those poles.So we’ll have to relate these parameters to the physical ones later.37 ffective Field Theory Chapter 1 Figure 1.8: Diagrams giving the one-loop correction to the scalar propagator in Yukawatheory.We’ll briefly go through renormalizing the scalar two-point function at one loop toevince dim reg and MS. In our one-loop diagrams we use propagators given by L ,since we know the counterterms begin at higher order. The full details of the one-looprenormalization of this theory can be found in Srednicki’s Chapter 51.At one-loop, the scalar two-point function gets corrections due to both interactions,as seen in Figure 1.8. There is the diagram we had in the φ theory above, but we mustrecompute this in dim reg i Σ φ loop ( p ) = − i λ (cid:90) d k (2 π ) − ik + m (1.57) = − i λ ˜ µ (cid:15) (cid:90) d d ¯ k (2 π ) d k + m (1.58) = − i λ (cid:16) − (cid:15) (cid:17) m (4 π ) (cid:18) π ˜ µ m (cid:19) (cid:15)/ (1.59) (cid:39) i λ π ) m (cid:18) (cid:15) + 1 − γ E (cid:19) (cid:18) (cid:15) π ˜ µ m (cid:19) (1.60) (cid:39) i λ π ) m (cid:18) (cid:15) + 1 − γ E − log 4 π ˜ µ m (cid:19) (1.61) = i λ (4 π ) m (cid:18)
12 + 1 (cid:15) + 12 log µ m (cid:19) + O ( (cid:15) ) , (1.62)where we have analytically continued to d = 4 − (cid:15) dimensions including replacing λ (cid:55)→ λ ˜ µ (cid:15) ,with ˜ µ a mass scale, to keep λ dimensionless; performed the integral in general dimension;expanded for (cid:15) (cid:39) ; and defined µ ≡ π ˜ µ e − γ E , where γ E is the Euler-Mascheroniconstant, to simplify the expression. Details on these calculational steps are laid out in38 ffective Field Theory Chapter 1 Srednicki’s Chapter 14, but there’s an important and naïvely surprising feature of theabove that we should discuss.In previous sections our regularization scheme explicitly introduced a mass scale Λ which we could think of as having a physical interpretation as some sort of short-distancecutoff, if we wished. We then saw that by cleverly studying how the parameters in thetheory are modified as we change Λ and demand the physics stays the same, we couldmake a variety of things easier to calculate and make the physical content of the theorymore transparent. Note that in doing so, we’re stretching the meaning of the scale Λ awayslightly from that physical picture—we don’t care what the ‘real’ cutoff of our system is,or if there really is any sort of cutoff; we simply know that allowing such scale-dependencein our couplings and studying the theory at different values of Λ makes our lives easier,so we imagine varying it.Now the way this new regularization scheme works is somewhat opaque, but it stillnecessitates the introduction of a new scale. In this case, the unphysical scale µ isrequired to ensure that our couplings remain dimensionless away from d = 4 . This schemethus invites us to further broaden our notion of varying a scale to study the theory atdifferent energies—this time the scale explicitly never had a physical interpretation. Wecan view µ as labeling a one-parameter family of calculational schemes. We’ve ensuredby construction that the physics is the same no matter what µ we choose, but by cleverlyusing the scale-dependence we can make our lives far easier. The intuition should be thesame as in the previous case, and lowering µ does likewise transfer loop-level physics totree-level physics and can be used to improve the convergence of perturbative calculations.The connection is now slightly more opaque, which is why we began by discussing a cutoffin Euclidean momentum space, but the calculations become far simpler.39 ffective Field Theory Chapter 1 There’s another diagram with a ψ loop i Σ ψ loop ( p ) = ( − ig ) (cid:90) d k (2 π ) ( − i ) Tr (cid:2) ( − /k + M ) γ ( − /k − /p + M ) γ (cid:3) ( k + M )(( k + p ) + M ) (1.63) = − g π (cid:20) (cid:15) ( k + 2 M ) + 16 k + M − (cid:90) dx (3 x (1 − x ) k + M ) ln Dµ (cid:21) , (1.64)with D = x (1 − x ) k + m , whose evaluation follows similar steps but we skip for brevity.Adding these together, MS tells us the φ counterterms must take the values Z φ = 1 − g π (cid:15) (1.65) Z m = 1 + (cid:18) λ π − g π M m (cid:19) (cid:15) . (1.66)For the fermion, evaluating the one-loop diagrams gives us the counterterms Z ψ = 1 − g π (cid:15) (1.67) Z M = 1 − g π (cid:15) . (1.68)Since we didn’t choose the counterterm to keep the location of the pole in the propagatorfixed, m is no longer the physical scalar mass. But we can find the physical, ‘pole’ massprecisely from that condition: ≡ ∆ − ( k = − m phys ) (1.69) = k + m + Σ( k ) (cid:12)(cid:12) k = − m phys (1.70) = − m phys + m − Σ( − m phys ) (1.71) ⇒ m phys = m − Σ( − m ) + O ( λ , g , λg ) , (1.72)where we have used our favorite trick to replace m phys with m in the one-loop correction,since it is already higher order in couplings.40 ffective Field Theory Chapter 1 (a) (b) Figure 1.9: Some one-loop diagrams in Yukawa theory correcting the interactions. In (a),a triangle diagram correcting the Yukawa interaction. In (b), a box diagram correctingthe scalar quartic interaction.As for the interactions, we have a triangle diagram for the Yukawa coupling and anew contribution to the quartic with a fermion running in the loop, as depicted in Figures1.9a and 1.9b respectively. These lead to the counterterms Z g = 1 + g π (cid:15) (1.73) Z λ = 1 + (cid:18) λ π − g π λ (cid:19) (cid:15) , (1.74)from which we’ll be able to understand how the strength of the interactions changes asa function of the energy at which the theory is probed. Renormalization group improvement
Now the second improvement to perturbation theory is the RG-improved perturbationtheory we mentioned above. This takes on an even more useful role in our continuumrenormalization scheme here. In the Wilsonian picture, Λ was a high cutoff and weensured the physics was invariant under evolution of Λ , but this scale still needed to stayfar above the scales of interest in the problem Λ (cid:29) m, M, − k , . . . . Now the scale µ isentirely unphysical and we are free to bring it all the way down to the scales of kinematic41 ffective Field Theory Chapter 1 interest—in fact doing so will vastly simplify calculations. As a result we are able tomake even more use of the RG-improvement than we could above.We find the running couplings by again using the fact that the bare parametersare independent of the unphysical renormalization scale µ . Having utilized a mass-independent regulator, a Wilsonian interpretation of couplings running with the value ofthe regulator is nonsensical here and so renormalization group improvement is the wayto extract predictions from this theory. We already have the relations between the bareand renormalized quantities, e.g. g = Z − / φ Z − ψ Z g g ˜ µ (cid:15)/ . (1.75)And since we know that the bare parameters are independent of µ by definition, we have ln g = ln (cid:18) g π (cid:15) (cid:19) + ln (cid:18) g π (cid:15) (cid:19) + ln (cid:18) g π (cid:15) (cid:19) + ln g + 12 (cid:15) ln ˜ µ (1.76) = 5 g π (cid:15) + ln g + 12 (cid:15) ln ˜ µ + O ( g ) (1.77)d ln g d ln µ = 0 (1.78) = 10 g π (cid:15) d g d ln µ + d ln g d ln µ + 12 (cid:15) (1.79) = d g d ln µ (cid:18) g π (cid:15) (cid:19) + (cid:15)g. (1.80)If we expand d g d ln µ = a (cid:15) + a + . . . order by order in (cid:15) , then matching the O ( (cid:15) ) termsgives a = − g/ and matching the O ( (cid:15) ) terms tells us that, in the (cid:15) → limit, we haved g d ln µ = 516 π g + O ( g ) . (1.81)This (cid:15) -independent piece is known as the ‘beta function’ for the coupling, β ( g ) = π g .Of course there are higher-order terms in d g d ln µ which are needed to match the O ( (cid:15) − n ) terms, and which one can solve for. But these vanish in the (cid:15) → limit, so will notcontribute to the running of g ( µ ) . 42 ffective Field Theory Chapter 1 We can now resum this logarithm to find the evolution of this coupling with renor-malization scale g ( µ ) = ¯ g (cid:113) − g π log µ ¯ µ , (1.82)where we’ve used the boundary condition g (¯ µ ) = ¯ g . As before, the resummed version willallow us to maintain precision to far lower scale than we could with simply its leadingorder approximation.It’s useful to keep in mind the Wilsonian picture as a clearer example because ourregulator had a physical interpretation. The point is that the logarithms are really what’sencoding how couplings change as a function of scale; in the Wilsonian calculation it wasobvious that the logarithmic contribution log b is present no matter the initial cutoff.One says that couplings which receive logarithmic quantum corrections ‘get contributionsfrom all scales’. Then it’s clear why this RG-improvement is sensible—though we maystart at some particular Λ or µ , a one-loop calculation offers information on the lowest-order logarithmic running over all momenta, and we may sum up those modifications toimprove our perturbation theory. Mass-independent schemes and matching
We’ve seen already the necessity of renormalization when a theory produces naïvely diver-gent results, and its enormous use in improving the precision of perturbative calculationsin a given theory. The last facet we’ll discuss is its use in connecting theories. This isnecessary to use the computationally-simple scheme of dim reg with MS in theories withdifferent mass scales, and is very closely related to the effective field theory philosophywe discussed in Section 1.1. Cohen’s monograph [16] goes into far more depth than Iwill be able to, and is a fantastic introduction to these ideas and their application. This43 ffective Field Theory Chapter 1 perspective on renormalization has also been of enormous use in condensed matter tounderstand behaviors that appear in many distinct systems in the long-distance limit,and has applications in formal field theory to understand better the properties of QFTitself.In the previous section we derived the beta function for Yukawa theory in the MSscheme. As promised by our terming of this as a ‘mass-independent’ scheme, the betafunctions indeed have no reliance on the masses. But this should seem remarkably pecu-liar, as it suggests that there is no decoupling at all. Were that the case, by measuringthe low-energy beta functions of QED we could tell how many charged particles existedup to arbitrarily large mass scales! What has gone wrong is that MS does not meet thecriterion of a physical scheme which is necessary for the Appelquist-Carrazone theoremto operate. In MS the renormalization condition has nothing to do with physical valuesof the parameters so, while it makes calculations far simpler, MS has broken decoupling.To restore decoupling and allow us to properly use a mass-independent scheme, wemust implement the mass-dependence ourselves by ‘matching’ the Yukawa theory atenergies above the fermion mass to a theory of solely light scalars at energies below thespinor mass. To ‘match’, we consider some process which exists in both theories—forexample, φ scattering—and ensure that at the matching scale M both theories agree onthe physics.In the high-energy Yukawa theory we can run the RG scale all the way down from ahigh scale ¯ µ to µ = M phys , the physical, pole mass of the fermion. To get simple closed-form expressions, we’ll take the couplings small enough that working to lowest order givesa good approximation. We’ll denote all of the UV values with bars, e.g. M ( µ = ¯ µ ) = ¯ M .44 ffective Field Theory Chapter 1 Firstly, we use the counterterms to find the anomalous dimension of the fermion mass d log Md log µ = − dd log µ (cid:0) Z − ψ Z M (cid:1) (1.83) = − g π (1.84) ⇒ M ( µ ) (cid:39) ¯ M (cid:18) ¯ µµ (cid:19) ¯ g π (1.85) (cid:39) ¯ M (cid:18) g π log ¯ µµ (cid:19) + . . . (1.86)We then find the fermion pole mass as − M phys + M ( µ = M phys ) − Σ( − M phys ) (1.87) M phys = ¯ M (cid:18) g π log ¯ µ ¯ M (cid:19) − Σ( − ¯ M ) + subleading (1.88) = ¯ M (cid:18) g π log ¯ µ ¯ M (cid:19) − ¯ g π ¯ M (cid:90) dxx log (cid:18) x ¯ M + (1 − x ) ¯ m ¯ M (cid:19) + . . . (1.89) = ¯ M (cid:20) g π (cid:18)
12 + log ¯ µ ¯ M (cid:19)(cid:21) + . . . (1.90)Now we need the value of the other parameters at that mass threshold m dmd log µ = 14 π (cid:18) λ − g M m + 12 g (cid:19) (1.91) ⇒ m ( M phys ) = ¯ m (cid:18) − π (cid:20)
18 ¯ λ − ¯ g ¯ M ¯ m + 12 ¯ g (cid:21) log ¯ µ ¯ M (cid:19) + . . . (1.92) dgd log µ = 516 π g (1.93) ⇒ g ( M phys ) = ¯ g (cid:18) − g π log ¯ µ ¯ M (cid:19) + . . . (1.94) dλd log µ = 116 π (cid:0) λ + 8 λg − g (cid:1) (1.95) ⇒ λ ( M phys ) = ¯ λ (cid:18) − π (cid:2) λ + 8¯ λ ¯ g − g (cid:3) log ¯ µ ¯ M (cid:19) + . . . (1.96)Now we are ready to proceed to even lower energies. We enforce decoupling by matching45 ffective Field Theory Chapter 1 to the low-energy theory of just a self-interacting scalar. We have L = 12 φ (cid:0) (cid:3) − m (cid:1) φ (1.97) L = − Z λ λφ + L ct (1.98) L ct = −
12 ( Z φ − ∂ µ φ∂ µ φ −
12 ( Z m − m φ (1.99)The counterterms and beta functions in this theory can be conveniently found by trun-cating those found above. Of course, by construction, we find that the heavy fermion ψ no longer contributes to the running of parameters at low-energies. To make sure we’regetting the physics correct, we must impose the boundary condition that the predictionsmatch at µ = M phys , which here is quite simple—we just use the values at M phys in theUV theory as literal boundary conditions for our running in the IR theory. In the IRtheory, for µ ≤ M phys , we have dλd log µ = 316 π λ (1.100) ⇒ λ ( µ ) = λ ( M phys )1 + π λ ( M phys ) log M phys µ (1.101) (cid:39) ¯ λ (cid:18) − π (cid:2) λ + 8¯ λ ¯ g − g (cid:3) log ¯ µ ¯ M (cid:19) (cid:18) − π ¯ λ log ¯ Mµ (cid:19) + . . . (1.102) (cid:39) ¯ λ (cid:18) − π (cid:2) λ + 8¯ λ ¯ g − g (cid:3) log ¯ µ ¯ M − π ¯ λ log ¯ Mµ (cid:19) + . . . (1.103)The benefit is now clear. While the RGEs in the UV theory were very complicated, therunning of λ in the low energy theory is simple. Our mass-independent scheme allows usto explicitly factorize these and contain all the UV physics in the boundary condition,which lets us study the low-energy theory in a simple manner.The general procedure of renormalization group evolution in mass-independent schemesis called ‘running and matching’. The parameters in the Lagrangian run as you evolvedown in energies, but at a mass threshold M we must match the UV theory at µ = M from above to a theory without this field at µ = M from below. When we match we46 ffective Field Theory Chapter 1 ensure that the physics of the low-energy fields stays constant as we cross that thresholdand remove that particle from the spectrum of our theory. This becomes less trivialwhen we have multiple mass scales, so consider now upgrading our Yukawa theory withadditional fermions. L ⊃ N (cid:88) i =1 M i ¯ ψ i ψ i + g ( N ) φ N (cid:88) i =1 ¯ ψ i γ ψ i (1.104)Now if we are given the theory at very high energies µ (cid:29) M i and we want to understandwhat it looks like at very low energies, there is a cascade of EFTs we evolve through. Asdepicted in Figure 1.10, we run the parameters down to the largest fermion mass, matchto a theory with one less fermion, run down again until the next mass scale, and so on.Figure 1.10: A schematicdepiction of the cascadeof effective field theoriesas one transitions froma Yukawa theory of N fermions at high energiesthrough integrating theseout sequentially until onefinds an effective φ theoryat low energies.As an example which is closer to the real world, considerQED with our three generations of leptons (and ignoring thestrong sector for simplicity). At low energies we measurethe asymptotic value α ( µ (cid:39) (cid:39) / , and in colliderswe measure the value of the gauge coupling at the Z mass m Z (cid:39) GeV. To compare these, we must run the high-energy value all the way down into the infrared. Above m τ (cid:39) . GeV we have a theory where all of e, µ, τ run in loopsand giving an MS beta function β α = 2 α /π . But at m τ we should remove the τ from our theory, such that from m τ down to m µ (cid:39) MeV we have β α = 4 α / π . Below the µ we solely have the electron and recover the textbook β α =2 α / π , and finally as we cross the electron mass threshold m e (cid:39) keV we remove the electron from the spectrumand find that the gauge coupling stops running β α ≡ .Physically this corresponds to the fact that pure QED is47 ffective Field Theory Chapter 1 scale-invariant, meaning that the coupling will not evolve at all in a theory with nocharged particles. This is the regime in which classical electrodynamics holds precisely(up to the presence of additional interactions suppressed by powers of m e , that is).A possible confusion is to conflate the mass-independence of the regularization schemewith that of the renormalization scheme, and conclude that dimensional regularizationcannot be used if one wants decoupling without having to integrate out and match. Solest one confuse the roles let’s quickly look at an example of using dimensional regular-ization with a renormalization scheme which does satisfy decoupling, known as ‘off-shellmomentum subtraction’. For simplicity, we’ll look at the anomalous dimension of ourYukawa scalar φ , and we’ll perform wavefunction renormalization by subtracting thevalue of the graphs at the off-shell momentum scale k = µ E . In symbols this amount tothe prescription Z φ = 1 + Σ loop ( k = µ E ) , where Π loop ( k ) = k Σ loop ( k ) + mass renorm pieces . (1.105)Since we’re still using the same regularization scheme, we have the same result for Π loop ( k ) as above. We can then simply calculate the anomalous scaling dimension asdefined by γ φ ≡ d log Z φ d log µ E , γ φ = 12 d Σ loop ( µ E ) d log µ E (1.106) = 3 g π (cid:90) d x x (1 − x ) µ E M + (1 − x ) xµ E . (1.107)This integral can be performed analytically, but the full expression is unilluminating.However, it is useful to look at the limits γ φ = g π , µ E (cid:29) M g π µ E M , µ E (cid:28) M ffective Field Theory Chapter 1 to check if they agree with our expectations. At energies far above the fermion mass itscontribution to the scalar anomalous dimension cannot know about that scale, and atenergies far below its mass we expect inverse dependence on the mass for decoupling tooccur. This is precisely what we find, so the lesson is that even if we didn’t want to gothrough the trouble of integrating out the fermion and matching, we could still make useof the magical regularization scheme that is dim reg. Flowing in theory space
Our interpretation of the renormalization group thus far has been as a way of under-standing what a particular theory looks like at different energies. But there is anotherway of looking at it that is also useful, for which we shall follow an example of Peskin& Schroeder, though I recommend Skinner’s lecture notes [73] for clear explanation ofthese concepts which goes farther than we have time to. Let’s return to the idea of theWilsonian path integral and successively integrating out Euclidean momentum shells. Inthe previous section we began with a scalar field theory Z = (cid:32)(cid:90) Λ (cid:89) k =0 d φ ( k ) (cid:33) exp (cid:20) − (cid:90) d d x (cid:18)
12 ( ∂ µ φ ) + 12 m φ + 14! λφ (cid:19)(cid:21) . (1.108)We then integrated over momentum shells from Λ down to b Λ with < b < Λ , and foundwe could express our result as (schematically; see 1.32,1.34,1.35,1.41) Z = (cid:32)(cid:90) b Λ (cid:89) k =0 d φ ( k ) (cid:33) exp (cid:20) − (cid:90) d d x L eff (cid:21) (1.109) L eff = (cid:18)
12 (1 + ∆ Z )( ∂ µ φ ) + 12 ( m + ∆ m ) φ + 14! ( λ + ∆ λ ) φ + 16! ∆ c φ + . . . (cid:19) . (1.110)Above we interpreted this in terms of looking at the same theory at lower energies, havingcoarse-grained over the largest momentum modes, which is a useful way of comparingthe two path integrals. Another useful way to compare is to get them to a form where49 ffective Field Theory Chapter 1 they look similar, so let’s now define a change of variables k (cid:48) = k/b, x (cid:48) = xb , in terms ofwhich the path integral now looks like Z = (cid:32)(cid:90) Λ (cid:89) k (cid:48) =0 d φ ( k (cid:48) ) (cid:33) exp (cid:20) − (cid:90) d d x (cid:48) L eff (cid:21) (1.111) L eff = b − d (cid:18)
12 (1 + ∆ Z ) b ( ∂ µ (cid:48) φ ) + 12 ( m + ∆ m ) φ + 14! ( λ + ∆Λ) φ + 16! ∆ c φ + . . . (cid:19) . (1.112)We can transform the kinetic terms back to the canonical form with the field redefinition φ (cid:48) ( x (cid:48) ) = b (2 − d ) / (1 + ∆ Z ) / φ ( x (cid:48) ) , after which we can write the effective Lagrangian as L eff = 12 ( ∂ µ (cid:48) φ (cid:48) ) + 12 m (cid:48) φ (cid:48) + 14! λ (cid:48) φ (cid:48) + 16! c (cid:48) φ (cid:48) + . . . . (1.113)with m (cid:48) = ( m + ∆ m )(1 + ∆ Z ) − b , λ (cid:48) = ( λ + ∆ λ )(1 + ∆ Z ) − b d − , c (cid:48) = ∆ c (1 +∆ Z ) − b d − , . . . , and it’s clear that we could write such an effective action regardless ofwhat sort of coefficients we began with before integrating out this momentum shell.Now our series of transformations has effected the change (up to normalization) Z = (cid:32)(cid:90) Λ (cid:89) k =0 d φ ( k ) (cid:33) exp (cid:20) − (cid:90) d d x L (cid:21) → Z = (cid:32)(cid:90) Λ (cid:89) k (cid:48) =0 d φ (cid:48) ( k (cid:48) ) (cid:33) exp (cid:20) − (cid:90) d d x (cid:48) L (cid:48) (cid:21) . (1.114)Since all of our dynamical variables are integrated over in calculating the partition func-tion, we can view this as a transition in the space of Lagrangians, L → L (cid:48) . So this givesus an interpretation of the renormalization group as a flow in ‘theory space’.This interpretation invites us to conceptualize renormalization group flow as a paththrough theory space between two conformal field theories (CFTs), as depicted in Figure1.11. CFTs are quantum field theories with an enlarged spacetime symmetry group , This statement may appear confusing if you have come across the Coleman-Mandula theorem [74],which roughly says that the most amount of symmetry you can have in a QFT is the direct product ofthe Poincaré spacetime symmetry and whatever internal symmetries you have. However, that beautifulresult relies on properties of the S-matrix, and CFTs do not have S-matrices because they do not havemass gaps, meaning this enlarged spacetime symmetry group does not violate the theorem. We’ll seeanother, even more interesting loophole in this theorem exploited in Section 3.1, which comes fromenlarging our notion of what sorts of symmetries an S-matrix could possess. ffective Field Theory Chapter 1 consisting essentially of a scaling symmetry. CFTs are fixed points of RG flows—sincethey possess scaling symmetry they look the same at all energy scales, so if an RG flowis to have an endpoint it clearly must be a CFT .There’s a terminological confusion here, which is that the ‘renormalization group’isn’t actually a group at all, since the operation of integrating out a momentum shellis irreversible. This came up already above when we saw that integrating out heavyfields means we can no longer compute processes which have them as external fields(see Footnote 1). Flowing to lower energies, or toward the IR, is really a coarse-grainingoperation which does lose information about small scales, in precise analogy to decreasingthe resolution of an image. This means that RG evolution is a directed flow, so there isa difference between fixed points in the UV and in the IR.Quantum field theories can have different sorts of fixed points. As a familiar example,if the theory has a ‘mass gap’—no zero-energy excitations in the infrared, which maybe because one began solely with massive fields or through dynamical mechanisms likeHiggsing and confinement—then one finds a ‘trivial’ fixed point. In the far infrared,everything has been integrated out and there is not enough energy to excite any modes.We know phenomenologically that this happens in QCD. In the other familiar case onecan have a ‘Gaussian’ or ‘free’ fixed point if the theory contains massless fields whichdon’t interact, such as in QED. At energies far below the mass of the lightest chargedparticle this is a theory of free electromagnetism, though one can still excite photons ofarbitrarily-long wavelength.Such a Gaussian fixed point occurs in the UV for QCD—the celebrated result of‘asymptotic freedom’ [77, 78]—because the strong coupling flows toward zero, giving I’ve elided a subtlety here, which is that it is not entirely known whether scale invariance in factimplies conformality in four-dimensional QFTs, the latter of which includes also invariance roughlyunder inversion of spacetime through a point. No counterexamples have been found, despite mucheffort. Polchinski’s early paper on the topic is a classic [75], and a recent review can be found fromNakayama [76]. ffective Field Theory Chapter 1 Figure 1.11: Schematic two-dimensional projection of some RG flow trajectories from aninteracting UV fixed point which has been perturbed by one or another relevant operator,and which flow to IR fixed points that are either interacting or free. Perturbing the UVCFT by an irrelevant operator would lead to a flow directly back to the same CFT.a free theory. This famously cannot occur for U (1) gauge theories whose couplingsnecessarily grow with increasing energies, leading them to herald their own breakdownwith the prediction of a ‘Landau pole’ [79], a finite UV energy where the perturbativetheory predicts the coupling becomes infinite. From one perspective this is an inverseto the prediction of a confinement scale in QCD, where the perturbative prediction is ablowup of the coupling at low energies, as we’ll discuss further in Section 1.3.3. In eithercase the theory cannot make predictions for energies above the Landau pole or below theconfinement scale, respectively, and so can be called inconsistent.It’s clear that the divergence of a coupling either in the IR or the UV is problematicfor a complete, consistent interpretation of a QFT. This is precisely why the formalperspective on a well-defined QFT is that it describes an RG flow between two CFTs,such that in neither direction does a coupling grow uncontrollably. In fact this provides52 ffective Field Theory Chapter 1 an incredibly important perspective on renormalizability, which we’ll get to momentarily.First, let us introduce somewhat of a generalization of the ‘relevant/irrelevant’ ter-minology which we introduced in Section 1.1. We implicitly had in mind that we werestudying a theory in the vicinity of the Gaussian fixed point which we perturbed withvarious field operators—indeed, this is precisely how we normally carry out perturbativecalculations—and our terminology depended on that. Generally one wishes to imagineperturbing a CFT by a particular operator, flowing down in energy, and seeing whetherthe operator grows in importance—a relevant operator—or shrinks—an irrelevant op-erator. Perturbing a CFT by an irrelevant operator does not induce an RG flow (to adifferent CFT), so interesting dynamical RG flows come from CFTs perturbed by relevantoperators. This clearly agrees with our power-counting notion of relevance when we’renear the Gaussian fixed point, but works also if one is near a strongly-coupled, interact-ing fixed point where one may not know how to do such power-counting and anomalousdimensions of operators may grow to overpower their classical dimensions.The importance of this language was realized in particular by Polchinski in his pi-oneering article [21]. He showed that in fact the intuitive notion of ‘power-countingrenormalizability’ that the field had been building—that for a theory near the Gaussianfixed point we could see whether it was renormalizable merely by checking whether it hasany coefficients of negative mass dimensions—in fact maps on to a very general state-ment. This is enormously powerful, as prior arguments for renormalizability were madeon a case by case basis and were complicated and messy and graph-theoretic. His deriva-tion of this fact is brilliant but requires much work, so we’ll merely try to get a sense forwhy it should be true by building on our intuition from our Wilsonian renormalizationof φ theory in Section 1.2.1.So long as your theory has a finite number of fields, all of which have mass dimen-sion [ φ i ] > when canonically normalized, then there are a finite number of relevant53 ffective Field Theory Chapter 1 or marginal operators. As we flow down in energy through theory space, we saw abovethe sense in which those coefficients are UV-sensitive—the IR coefficients of those oper-ators are determined primarily by their UV values and IR corrections are subdominant.Contrariwise, the coefficients of the infinite number of possible irrelevant operators areUV-insensitive, being determined primarily by the IR physics of the relevant and marginaloperators. That is, the RG flows are attracted to a finite-dimensional submanifold of thespace of Lagrangians.Consider a Wilsonian RG flow where we start off, as above, by specifying a cutoff Λ and the values of coefficients λ i of all the n marginal and relevant operators atthat scale, as well as the coefficients c i of however many irrelevant operators we wishto turn on. We can then flow downwards in energy as normal, and at an energy scale Λ R (cid:28) Λ let’s say we measure the relevant and marginal coefficients λ iR at that scale. Thecoefficients of the irrelevant operators are then dominated by the infrared λ iR and Λ R upto precision (Λ R / Λ ) ∆ i from subleading corrections, with ∆ i > the scaling dimension ofthe irrelevant operator. So indeed, the RG flow is attracted to the n -dimensional surfacedescribed by c i = c iR ( λ iR ; Λ R ) and separate trajectories through theory space as a functionof scale Λ which reach the same λ iR will differ only by positive powers of (Λ R / Λ ) . Forless abstract discussion, Polchinski goes through a simple example which may providefurther insight, and Schwartz discusses the same example in Chapter 23 of his textbook[22].To see why this implies renormalizability, recall that the program of Wilsonian renor-malization is to define renormalized, ‘running’ couplings as a function of scale to keepinfrared physics at Λ R fixed while ‘removing the cutoff’. A bit more formally, we wanta family of Lagrangians L (Λ ; λ i , c i ) with coefficients chosen as a function of Λ suchthat each Lagrangian yields the same low energy physics λ iR , in terms of which all the IRobservables can be calculated up to subleading corrections in Λ R / Λ . When the cutoff is54 ffective Field Theory Chapter 1 removed by taking Λ → ∞ , one thus recovers precisely the correct physics, specified bythose chosen values of the λ iR , which are the renormalization conditions. If we can findsuch a family of Lagrangians, then we say this theory is power counting renormalizable.Polchinski’s argument shows that this can be done so long as one wishes solely to fix theIR values of relevant and marginal couplings. Trivialities
The attentive reader may at this point notice an inconsistency due to imprecise language.We’ve seen now that the criterion for renormalizability, which Polchinski provided arobust basis for, is power-counting of the operators near the IR fixed point. This wouldsuggest that the λφ theory we’ve studied by means of an example is renormalizable.However, recall the result of resumming its renormalization group equation: λ (Λ) = λ (Λ )1 + λ (Λ )16 π log Λ Λ , (1.115)which has a Landau pole for Λ = Λ exp( π λ (Λ ) ) , preventing us from taking the limit werequired above. A mathematical physicist would say λφ theory is ‘trivial’ or ‘quantumtrivial’, as if we demand the existence of a continuum limit, that sets λ (Λ) = λ (Λ ) = 0 .The issue is that the tree-level, classical scaling dimension captures only the scaling ofthe operators infinitesimally close to the infrared Gaussian fixed point. If we move a finitedistance upward in energy scale, we’ve seen above that the φ operator gets an anomalousdimension δ φ > and so is marginally irrelevant. So it’s clear that Polchinski’s pictureof renormalization is only getting at a perturbative sense of renormalizability, and cannottell us whether there truly exists an RG flow from a UV fixed point down to the IR theorywe want to study.So what are we to make of λφ theory—or for that matter of QED, which has thesame problem? Of course we know empirically that QED works fantastically well and55 ffective Field Theory Chapter 1 we can absolutely make finite, accurate predictions after a finite number of inputs. Tounderstand this, we must appeal the language of effective field theory, which we’ve alreadydiscussed. In fact the feature we’re really relying on is effective renormalizability, whichtells us we require solely a finite number of inputs to set the behavior of the theory to agiven precision. It’s clear that in this sense QED itself is an effective field theory whosevalidity breaks down somewhere below its Landau pole.Finally, let me mention another reason not to be too worried that our most belovedquantum field theories don’t exist in the continuum limit: Our universe is not describedby a QFT at its smallest scales! It’s indeed true that only RG flows between a UV CFTand an IR CFT can hope to define fully consistent and mathematically well-defined QFTs.But the existence of gravity—and the very strong evidence that a quantum field theory ofgravity is inconsistent—means that at some energy scale effects not present in quantumfield theory must become relevant. And since gravity couples universally to everything[80, 81], we have no strict empirical need for a UV complete, interacting quantum fieldtheory that does not include gravity. It is entirely consistent, and overwhelmingly likely,that a quantum-field-theoretic description of the world works only approximately andsome inherently quantum gravitational theory provides a sensible UV complete theory. Before moving on, let us reiterate what we’ve discussed about renormalization. As we’veseen, renormalization is so important and so useful and fulfills so many purposes thatan entirely general statement risks becoming vague. But if a single sentence summary isdemanded: Renormalization reveals for us the scale-dependence of a quantum mechanicalfield theory.The effects of this seemingly innocuous statement, however, are powerful and mani-56 ffective Field Theory Chapter 1 fold, including: • Correctly accounting for this scale-dependence is necessary to have well-definedquantum field theories, which otherwise appear nonpredictive. • Bringing the non-scale-invariance of the quantum mechanical theory into clear scale-dependence of the couplings makes it simple to read off the qualitative behavior atdifferent scales from the renormalized Lagrangian. • Including this scale-dependence in the couplings allows us to reorganize our pertur-bative series such that we can efficiently calculate the behavior of the theory overa far wider range of energies than a naïve treatment allows. • Properly accounting for the scale dependence allows us to harness the full power ofeffective field theory, as we can study a theory of low-energy fields which correctlyaccounts for corrections from the high-energy physics. • Understanding the perspective of single quantum field theories as flows throughtheory space as a function of the scale allowed us to develop a nonperturbativedefinition of a fully UV-complete quantum field theory and how it behaves.All of these various perspectives will be of use in the following chapters as we applythis technology to understanding what the hierarchy problem is and how we can solve it.57 ffective Field Theory Chapter 1
Naturalness is the notion that we have the right to ask about the origins of the dimen-sionless numbers in our theories—past solely fitting them to the data. It was Dirac whofirst introduced such a notion to particle physics in 1938 [82]. In modern language, whatis referred to as ‘Dirac naturalness’ consists of the idea that dimensionless parameters ina fundamental physical theory should take values of order 1. In the language of EFT,in a theory with a cutoff Λ and an operator O with scaling dimension ∆ , we expectits coefficient to take a value c O ∼ O (1)Λ d − ∆ . As stated this is essentially dimensionalanalysis, as Λ is the only scale we’ve introduced, but we will discuss below that quantummechanics gives additional credence to this expectation—indeed we’ve already seen thisfeature in our one-loop examples above.’t Hooft pointed out a refinement of this principle [83], which has come to be called‘technical naturalness’. If the operator O breaks a symmetry which is respected by theaction in the limit c O → , then one says it is ‘technically natural’ for c O to take on asmall value. The reasoning here is simple—as we saw above, in a quantum field theorydefined at a high scale, one finds corrections δc O to such coupling constants as they runto low energies. If there is an enhanced symmetry of the theory in the limit that c O → ,then such quantum mechanical corrections cannot generate that operator and break thesymmetry, so we know that δc O ∝ c O . The low-energy effective field theorist says of suchcouplings that one can ‘set it and forget it’: if one has c O (cid:28) at the cutoff Λ , thatcoupling will remain small as one flows to lower energies.Conversely, we can emphasize the connection to Dirac naturalness by looking at thispicture in reverse. We know of mechanisms to generate small technically natural couplingsat low energies from Dirac natural ones, as we will discuss in detail below. Imagine onemeasures a small coupling c O at long distances in the low-energy theory with cutoff Λ ffective Field Theory Chapter 1 that does not have a Dirac natural explanation. If that parameter is technically natural,it remains small up to the cutoff, and so the next generation of physicists can explainits small size at Λ in the UV theory, as emphasized nicely by Zhou [84]. If c O is nottechnically natural, then its RG evolution up to the cutoff yields a value to which thelow-energy physics is very sensitive, and we must explain why it has a very specific valuesuch that the correct physics emerges at long distances. A model is fine-tuned if a plot of the allowed parameterspace makes you wanna puke. David E. Kaplan (2007)
It’s useful to make this less abstract by looking at a simple example. Consider a d = 6 dimensional effective field theory of a real scalar field φ of mass m which is odd undera Z symmetry, which we expect is a good description of our system up to a cutoff Λ with m (cid:28) Λ . If we add a small explicit breaking σφ with σ (cid:28) at low energies, σ istechnically natural and stays small up to the cutoff, so we can easily write down a UVcompletion which generates this small value Dirac naturally.However, if we add another Z -odd real scalar Φ and give it a large Z -breakinginteraction with φ , then σ is no longer technically natural. Its low-energy value becomesextremely sensitive to the values of the parameters at the cutoff. It then becomes difficultto understand an ultraviolet reason for why those values take the precise values they needto realize small σ in the far IR. Consider the bare action S = (cid:90) d x (cid:20) −
12 ( ∂φ ) − m φ −
12 ( ∂ Φ ) − m Φ − σ φ − y φ Φ (cid:21) , (1.116)where we’ve given the two fields the same mass for simplicity. This is not stable under59 ffective Field Theory Chapter 1 radiative corrections, but that’s a higher-order effect which will not come into our one-loop calculation of the RG evolution of the cubic couplings.We will renormalize this theory at one loop using dim reg with MS. As discussedabove, we will compute the one-loop 1PI diagrams and add counterterms to cancel solelythe (cid:15) pieces of the results. With counterterms, the action is S = (cid:90) d d x (cid:20) − Z φ ( ∂φ ) − Z mφ m φ − Z Φ ( ∂ Φ) − Z m Φ m Φ + σ Z σ φ + y Z y φ Φ (cid:21) (1.117)where these parameters and fields are the renormalized parameters, and for compactnesswe have not written down the split of these terms as we did above in Equation 1.97.At tree level the relation to the bare quantities is trivial and so Z = 1 + . . . . To getan accurate picture of how the strength of the interactions vary as they’re probed atdifferent energy scales, we must fully renormalize the theory. Since our focus is on theinteractions, we simply state the results for the quadratic part of the action, where wehave Z φ = 1 − π ) (cid:0) σ + y (cid:1) (cid:15) + . . . (1.118) Z Φ = 1 − π ) (cid:0) σ + y (cid:1) (cid:15) + . . . (1.119) Z mφ = 1 − π ) (cid:0) σ + y (cid:1) (cid:15) + . . . (1.120) Z m Φ = 1 − π ) y (cid:15) + . . . (1.121)These tell us that the physical mass of the fields and the normalization of their one-particle states has changed. The relation to these can be found using the quantum-corrected propagator ∆( k ) − as ∆( − m phys ) − ≡ to define the mass and R − = ddk [∆( k ) − ] (cid:12)(cid:12) k = − m phys to define the normalization R , but solving for these relationsexplicitly will not be necessary for our purposes.The one-loop three-point functions each have two diagrams, whose evaluation only60 ffective Field Theory Chapter 1 (a) (b) Figure 1.12: Diagrams contributing to the one-loop corrections to the three-point func-tions in our d scalars with cubic interactions. Dashed lines denote φ and full lines denote Φ . In (a), the diagrams renormalizing σ , and in (b), the diagrams renormalizing y .differs in the coupling constants. For the correction to σ , we have triangle diagrams witheither φ or Φ running in the loop. We can evaluate them in d = 6 − (cid:15) dimensions as i σ V σ = 1 σ (cid:2) ( iσ ) + ( iy ) (cid:3) (cid:90) d q (2 π ) ( − i ) ( q + m ) (1.122) = iσ (cid:2) σ + y (cid:3) ˜ µ (cid:15) (cid:90) d d ¯ q (2 π ) d q + m ) (1.123) = iσ (cid:2) σ + y (cid:3) ˜ µ (cid:15) Γ (cid:0) (cid:15) (cid:1) π ) (cid:18) m π (cid:19) − (cid:15) (1.124) = i π ) σ (cid:2) σ + y (cid:3) (cid:18) (cid:15) − γ E (cid:19) (cid:18) (cid:15) π ˜ µ m (cid:19) (1.125) = i π ) σ (cid:2) σ + y (cid:3) (cid:18) (cid:15) + log µ m (cid:19) . (1.126)The counterterm vertex contributes to this as − i ( Z σ − , meaning that MS prescribeswe set Z σ = 1 + 1(4 π ) (cid:18) σ + y σ (cid:19) (cid:15) (1.127)For the other vertex correction there are diagrams with either one or two of each internal61 ffective Field Theory Chapter 1 line, which give i y V y = 1 y (cid:2) ( iy ) + ( iσ )( iy ) (cid:3) (cid:90) d q (2 π ) ( − i ) ( q + m ) (1.128) = i π ) (cid:2) y + σy (cid:3) (cid:18) (cid:15) + log µ m (cid:19) , (1.129)leading to the counterterm Z y = 1 + 1(4 π ) (cid:0) y + σy (cid:1) (cid:15) . (1.130)This gives us the beta functions β σ = 14(4 π ) (cid:0) − σ − y + y σ (cid:1) + . . . (1.131) β y = 112(4 π ) (cid:0) σ y − σy − y (cid:1) + . . . (1.132)Now we are finally in a place to mathematically evince our physics point about technicalnaturalness. Without Φ , the coupling σ is the only one which violates the Z and so thebeta function is necessarily proportional to σ . Let’s say we recruit an experimentalistfriend of ours to measure the → scattering cross-section of φ s and we find that at µ = m , the theory fits the data for σ ( m ) = σ with σ (cid:28) . Solving the beta function,we find that to lowest order σ ( µ ) (cid:39) σ (cid:18) − π ) σ log (cid:104) µm (cid:105)(cid:19) . (1.133)So σ is indeed radiatively stable. If σ ( m ) = σ is small, then it takes until the enormousscale µ (cid:39) m exp ((4 π ) /σ ) for σ to change by an order one fraction. So running σ ( µ ) up to wherever the cutoff Λ of our theory is, σ (Λ) will still be small. If by Λ we haven’tdiscovered any explanation for the size of σ ( m ) , we can ask the theory above Λ to producethis small value of σ (Λ) from Dirac natural parameters at yet higher energies. Perhaps This is trivial in our case as σ would be the only interaction, but you’ll find the feature persistsif you add other symmetry-respecting interactions, for example a Z -even scalar ψ with an interaction φ ψ . The general argument for this fact is given in the next section. ffective Field Theory Chapter 1 the high-energy Dirac natural value of σ has been relaxed toward zero by an axion-typemechanism, or perhaps σ is the vev of another Z -odd field which spontaneously brokethe Z via confinement. We don’t need to know a particular mechanism; the fact that σ is technically natural means that if we don’t find an explanation for its size there ishope yet that our academic descendants will. One says that here σ is UV-insensitive asits low-energy value does not depend strongly on the physics at high energies.On the other hand, if σ is not technically natural, we have a much more difficultissue. If we now have the theory with both φ and Φ , and our experimentalist measures σ ( m ) = σ (cid:28) and y ( m ) = y = O (1) , then to lowest order the RG evolution of σ willbe β σ (cid:39) − y / (4 π ) leading to σ ( µ ) = σ − π ) y log (cid:104) µm (cid:105) . (1.134)And we can see the issue, as we are no longer guaranteed that a small σ ( m ) is relatedto a small σ (Λ) . For concreteness, if σ = 10 − , y = − and Λ = 10 m , then (using thefull one-loop RG), we have σ (Λ) (cid:39) . and y (Λ) (cid:39) . . How are we to ensure these values at Λ ? We know how to produce small numbers, but not incredibly specific ones.To see that we do need to produce these values very precisely, let’s switch directionsand consider the RG evolution down in energy from Λ to m . In the theory with solely φ , the coupling σ runs incredibly slowly, so an O (1) change in σ (Λ) , evolved down tothe scale m , results in an O (1) change to σ ( m ) . But in the theory with two sources ofbreaking, σ ( m ) is enormously sensitive to the values of the couplings at Λ . With thesame cutoff Λ = 10 m , if we very slightly change the input value to σ (Λ) = 0 . andleave y (Λ) as above, evolving down now gives us σ ( m ) (cid:39) − —a < change in inputparameters has resulted in a change in our low energy observable! It’s even moresensitive to the input value of y ; a ∼ modification solely to y (Λ) = − . tricklesdown to give σ ( m ) (cid:39) × − , a change. In this theory σ is now a UV-sensitive63 ffective Field Theory Chapter 1 parameter, whose low-energy behavior depends strongly on the high-energy physics. Tosay the least, it seems difficult to find a natural way to achieve the precise values neededto reproduce the observed low-energy physics in this theory. We’ll return to this issue atlength in Section 2.2.2. Technical naturalness and masses
Our understanding of technical naturalness allows us to already see another warning signof the hierarchy problem. An elementary spin-1 field comes along with a gauge symmetry A µ → A µ − ∂ µ α ( x ) which is broken by a mass term m A A µ A µ . So a mass for a gauge bosonis technically natural and one necessarily finds δm A ∝ m A . Similarly, a massive Diracfermion Ψ = ( ψ, χ † ) has a U (1) global symmetry under which ψ → e iα ψ, χ → e − iα χ . Inthe m → limit, the symmetry is enhanced to U (1) as arbitrary rephasings of the twoWeyl fermions become symmetries, so again δm Ψ ∝ m Ψ . But an elementary scalar doesnot automatically come with any such protective symmetry, and we’ve already seen in allour examples above that scalar mass corrections indeed get contributions not proportionalto the mass itself.In fact for discussing the technical naturalness of masses there is an even simplerargument: A massless spin-1 boson has two degrees of freedom and a massive one hasthree. Quantum corrections cannot generate a degree of freedom ex nihilo , so a masslessgauge boson must be protected. Similarly a massless chiral fermion has two degrees offreedom, but a massive Dirac fermion has four. So for charged spinors and for vectors,it is simply the representation theory of the Lorentz group that is responsible for thestability of their masses. In either of these cases mass must arise from interactions ofthe field with a scalar (very broadly defined) as in the Higgs mechanism, which can pairup chiral spinors together and lend vectors another degree of freedom. But a masslessscalar and a massive scalar have the same number of degrees of freedom. If we want a64 ffective Field Theory Chapter 1 scalar mass to be technically natural, it must come from some symmetry past simply theLorentz group. We’ll see some examples of how to arrange this in Chapter 3. An important tool for understanding symmetries and their violation is known as ‘spurionanalysis’. The basic idea is simple: for a theory which respects a symmetry except for thecoupling c , this coupling parametrizes the breaking of the symmetry and any effects whichviolate the symmetry are proportional to c . More concretely, one assigns such couplingsspurious transformation properties under the symmetry such that the action becomesinvariant under the symmetry. Physically one can imagine that the observed values ofsuch couplings come from the vacuum expectation values of some heavy fields which areabove the cutoff of the theory. This can be viewed as imagining a UV completion wherethe explicit symmetry breaking in the low-energy effective theory comes microscopicallyfrom some spontaneous symmetry breaking, but the value of spurions is not dependentupon a particular realization of the UV completion.We can quickly see the utility of this by looking at a simple example of a complex scalarfield φ with an interaction which explicitly breaks the U (1) global symmetry φ → φe iα . S = (cid:90) d x (cid:18) − ∂ µ φ † ∂ µ φ − m φ † φ − λφ + h.c. (cid:19) , (1.135)where “ + h.c. ” denotes the addition of the Hermitian conjugate of the non-Hermitianinteraction term. A naïve effective field theorist would say that our Lagrangian has nosymmetries, and so we have no control and should expect that quantum corrections giveus any polynomials in φ, φ † at low energies.However, we may note that if we assign λ a charge of − such that λ → λe − iα , thenthe theory is invariant under that U (1) global symmetry. So quantum corrections cannotviolate our spurious symmetry, and as a result we know that we can only generate terms65 ffective Field Theory Chapter 1 like λ φ , and there are no φ or φ interactions generated at any order in perturbationtheory.We can also usefully apply this to the example of technical naturalness studied in thepreceding section. If (cid:15) is given the spurious transformation Z : (cid:15) → − (cid:15) then the (cid:15)φ term is invariant. Then it’s simple to see that no matter what other sorts of interactionswe add, so long as they respect the Z symmetry we must have δ(cid:15) ∝ (cid:15) . But having added yφ Φ we see that this term can also be made invariant with y → − y , and this allows δ(cid:15) ∝ y as well.An important real-world example where spurion analysis is useful is in understandingthe flavor structure of the SM. With all masses turned off, the SM has a large globalsymmetry group U (3) = U (3) Q × U (3) u × U (3) d × U (3) L × U (3) e = SU (3) × U (1) ,corresponding to arbitrary unitary reshufflings of the three generations of each fermionrepresentation. These symmetries are explicitly broken by the Yukawa matrices whichgenerate hierarchically different masses for the three generations. L ⊃ − Y ijd Q i Hd j − Y iju Q i H † u j − Y ije L i He j (1.136)where i, j = 1 , , are generation indices, and the Yukawa couplings are matrices in thisgeneration space.We don’t understand why these hierarchies are present, but we can carry out a spurionanalysis to see how worried we should be. We see that our theory will be invariantunder the full flavor group if we assign the Yukawa matrices the following transformationproperties under the various SU (3) symmetry groups Y d ∼ (3 , ¯3 , under SU (3) Q × SU (3) d × SU (3) u (1.137) Y u ∼ (3 , , ¯3) under SU (3) Q × SU (3) d × SU (3) u (1.138) Y e ∼ (3 , ¯3) under SU (3) L × SU (3) e (1.139)66 ffective Field Theory Chapter 1 Since these are the only flavor-violating couplings in the SM and they are all in distinctspurious flavor representations, this tells us the quantum corrections to these matricesmust be proportional to the matrices themselves e.g. δY e ∝ Y e . Thus this pattern ofYukawa couplings is stable under RG evolution to higher scales, and we are justified inthinking that the generation of this pattern may take place at large, currently-inaccessiblescales.This eases our minds about when we need to discover the origin of these flavor hierar-chies, but this holds true only as long as these remain the only flavor-violating couplings.Fantastic work in precision flavor measurements and theory has provided lower boundson the scale at which additional flavor violation can occur. Searches for flavor-violatingprocesses have constrained these violations to take place at scales enormously higher thanscales we are able to directly probe at colliders, which poses a puzzle. If there is newphysics near the TeV scale, how is it arranged to respect the flavor structure of the SM?A phenomenological approach known as Minimal Flavor Violation [85] demands that all flavor violation is proportional to these Yukawa couplings, but no fundamental explana-tion for this is known. For recent introductions to flavor in the Standard Model, see e.g.[86, 87, 88]. Perhaps the most important example of a Dirac natural field theory generating a smallnumber is that of ‘dimensional transmutation’. In particular, in quantum chromody-namics (QCD) the theory is ‘asymptotically free’—meaning that the interaction strengthvanishes in the far UV—and the gauge coupling g grows as one goes to lower energies.We skip the Nobel-worthy calculation (see e.g. Srednicki’s chapter 73 [18]) and simplyquote the results for the beta function of QCD (here parametrized via α s = g / π ), which67 ffective Field Theory Chapter 1 dictates the dependence of the gauge coupling on energy. In MS, the calculation finds β ( α s ) ≡ d α s d ln µ = − α s π (11 − n f ) + O ( α s ) (1.140)where µ is the energy scale of interest and n f is the number of quarks with massesbelow µ , which at high energies is n f = 6 . Then if we know the gauge coupling at afundamental scale like M pl , we can follow the procedure discussed in Section 1.2.3 ofrunning and matching to sequentially evolve the coupling down to low energies. We endup with a result like α ( M pl ) − α ( µ ) = b π ln M pl µ (1.141)where b is somewhere between the − × / value it takes above the top quarkmass and the − × / value it has below the charm quark mass. This tellsus that eventually the QCD coupling blows up in the infrared, and the theory becomesstrongly coupled—we expect our perturbative understanding of the theory to break down.While there is no proof of the precise effects of this, there is strong evidence that this isresponsible for the observed phenomenon of ‘color confinement’— at low energies coloredparticles form bound states which are color-neutral. The intuition being that the gluoninteraction is so strong that trying to pull quarks in a color-singlet apart from each otherrequires so much energy that it is energetically favorable for a quark-antiquark pair tobe created out of the vacuum and to end up with two color singlets. We may define anew scale Λ QCD as being the energy at which α diverges Λ QCD ≡ M pl e − πb α ( Mpl ) (1.142)So for some reasonable fundamental coupling α ( M pl ) at high energies, the theory gener-ates a new scale which is exponentially far removed from the fundamental physics. Sincethe mass of the proton is mainly from QCD binding energy, m p ∼ Λ QCD this explains thehuge hierarchy m p ≪ M pl . This is an extremely important mechanism and historically68 ffective Field Theory Chapter 1 one of the first suggestions for how to generate the electroweak scale was by copying thisstrategy, as we’ll discuss in Section 3.3. 69 hapter 2The Hierarchy Problem The physical question of the hierarchy problem is how to get an infrared scale v out ofa microscopic theory whose degrees of freedom live at the much higher scale Λ , with v/ Λ (cid:28) . The tools introduced in Section 1 have already provided a window into whythis can be difficult in a quantum field theory. Our aim in this section is to expand on thatnotion for the generation of the electroweak symmetry breaking scale in the StandardModel, where this scale is provided by the Higgs mass. In the Standard Model the Higgsmass is not technically natural, so the example discussed in Section 1.3 suggests the issuethat may appear.It is well-known that the Higgs plays a central role in the Standard Model, but thetagline ‘it provides mass’ doesn’t go far enough in underscoring its importance. TheHiggs is needed because the fermions of the Standard Model have a chiral spectrum:There are no representations with opposite charges under the full SM gauge group. Thismeans that there are no gauge-invariant fermion bilinears, so no fermions can be pairedup to form mass terms. If the Standard Model were not chiral, we could write mass terms70 he Hierarchy Problem Chapter 2 directly in the Lagrangian. In such a case there’s no reason to expect small masses forthe fermions, and indeed in the familiar case of right-handed neutrinos—which are SMgauge-neutral themselves, so can have Majorana masses—we generally expect them tobe very heavy. In some sense the natural expectation for such ‘vector-like’ (non-chiral)fermions would be to have Planck-scale masses, as in the absence of other particle physics,this is the only scale.So macroscopic structure in the universe is solely made possible by the chiral natureof the Standard Model. There may well be other vector-like sectors which indeedcontain Planck-scale masses. The fact that the particles which comprise us have a weakly-coupled description where quantum gravitational effects are suppressed—and so notionslike locality and Riemannian geometry work well—would not hold in such a vector-likesector.Thus the fact that the Standard Model is chiral and so requires the Higgs mechanismto provide masses answers a deep and important question about our place in the universe.But it doesn’t provide a full answer, as the scale at which the Standard Model sits stillneeds to be generated somehow. And in the absence of a mechanism to make it light, wemust worry once more about losing our macroscopic existence with a mass scale whichis again naturally of order the only other mass scale, the Planck scale.So, far more than the hierarchy problem being a small detail to clean up after havingempirically verified the structure of the Standard Model, the question of why m H (cid:28) M pl has serious physical importance. If we want an answer to why we live in a world with We note that it’s also true that the existence of a macroscopic universe relies on the smallness ofthe cosmological constant, which is the other pressing fine-tuning issue present in the Standard Model.This way of viewing the naturalness problems of the Standard Model has been beautifully articulatedby Nima Arkani-Hamed in [89] and in many seminars. It’s worth noting that in the complete absence of the Higgs, electroweak symmetry is broken bythe QCD chiral condensate [90, 91]. It’s interesting to ponder why Nature did not choose to let QCDconfinement solely fill the role, but we know empirically that there is EWSB at ∼ GeV scales, so weneed to understand the generation of that separate scale. he Hierarchy Problem Chapter 2 macroscopic structure, we must grapple with the hierarchy problem.This section is devoted to understanding the technical statement of this physical ques-tion in the framework of effective field theory. We pursue this by introducing, discussing,and refuting some common confusions about the hierarchy problem.72 he Hierarchy Problem Chapter 2 Non solutions to the Hierarchy Problem
In this section I will introduce a few common confusions and misconceptions about thehierarchy problem. Discussion and refutation of these arguments provides a naturalbackdrop for introducing how the hierarchy problem should be properly understood andwhy it is important.
A first point of confusion is that the Higgs mass is a free input parameter in the StandardModel, so a natural objection is that we should just set m H = 125 GeV and call it a day.Indeed, this hits on a basic and important point:
There is no hierarchy problem in theStandard Model.
The hierarchy problem exists for a more-fundamental theory which predicts the Higgs mass—that is, one in which the Higgs mass is an output parameter.We can evince this in a simple toy model of a scalar φ interacting with other generalfields ψ i where a tree-level, ‘bare’ mass term is allowed by the symmetries. This is ourtoy version of the Standard Model, in which the scalar mass is likewise an input. S = (cid:90) d d x (cid:20) −
12 ( ∂ µ φ ) − m φ − V ( φ ) − φg O ( ψ i ψ j ) (cid:21) (2.1)Of course, as is familiar, m itself is not measurable. When one calculates the two-point function of φ perturbatively in couplings one finds quantum corrections Γ (2) ( p ) = p + m + g ( m + . . . ) + O ( g ) , where these are generically large because the massis not technically natural—there’s no symmetry protecting it. When one measures themass of φ with e.g. some scattering experiment, it is m phys = m + g m + . . . whichone measures. And luckily so, because m may well be formally infinite in a continuum Perturbative calculations are an expansion in couplings, not (cid:126) [92], though this subtlety is commonlyelided. he Hierarchy Problem Chapter 2 quantum field theory, and a similarly infinite bare mass term is necessary to end upwith the correct finite physical mass. Indeed, we are justified in this theory in choosing m such that m phys matches the measured value. This is just the familiar procedure ofrenormalization, stretching back many decades and first understood in the context ofquantum electrodynamics. To define QED one needs to input some definitions of theelectron mass and the electromagnetic coupling based on experimental data, and thesetwo inputs then determine all other predictions of the theory—e.g. the differential crosssection of Coulomb scattering, the lifetime of positronium, and anything else you couldhope to measure.However, consider now a theory which has a global SU (2) symmetry which is spon-taneously broken at a high scale M . We want to understand how to get a light scalardegree of freedom out of this theory—that is, we’ve measured m phys (cid:28) M . This is a toymodel of a Grand Unified Theory, where the microscopic physics exists at a high scale M GUT , and indeed it was in this guise that the hierarchy problem was first recognized.Let’s say our light scalar degree of freedom φ originated from a doublet Φ a = ( ϕ, φ ) (cid:124) .Our microscopic theory now does not have a bare mass term for φ but rather solelyfor Φ as a whole, since it must respect the symmetry. A difference in the masses of φ and ϕ can only come from the spontaneous breaking of the symmetry—let’s say whenanother fundamental scalar Σ gets a vev ν = M . This vev is a physical, measurableparameter related to the mass of Σ and its self-interactions. Our action is controlled bythe symmetries, as ever. S = (cid:90) d d x (cid:20) −
12 ( ∂ µ Φ) † ( ∂ µ Φ) − M Φ † Φ − λ Φ † ΣΣ † Φ − V (Φ) (cid:21) (2.2)where M is the bare mass of the Φ doublet and λ is its bare interaction strength. In theabsence of any other scales we generally expect M ∼ M and λ ∼ O (1) . In this theorythere is no reason for the values of these parameters to be connected to each other in any74 he Hierarchy Problem Chapter 2 way.When Σ gets a vev, (cid:104) Σ (cid:105) = (0 , ν ) (cid:124) , it breaks the SU (2) symmetry and gives a masssplitting between the two degrees of freedom in Φ , since (cid:104) Σ (cid:105) † Φ = νφ . We then have themasses m ϕ = M , m φ = M + λ ν . In this theory our tree-level inputs are M and λ (and the interactions controlling the value of ν , which for simplicity we don’t writedown) and the scalar mass m φ is an output . In fact here the hierarchy problem occurs attree-level, simply as a result of wishing to produce a small mass via splitting a multiplet.If we wish to have, say, M (cid:39) GeV and m φ (cid:39) GeV—the values of the GUTscale and the electroweak scale in the real world—we need to fine-tune λ enormously soit takes a value like − . × M ν .Of course when we look at our theory at loop level there will again be quantumcorrections to our tree-level parameters m and λ , and again it will be their correctedvalues which are physical and measurable M phys = M + gM + . . . , λ phys = λ + gλ + . . . .But if our theory is renormalizable, we know that quantum corrections will merely changethe values of these parameters, and not the operators we have. The point is that thequantum corrections are SU (2) invariant, so the masses of both φ and ϕ will receive thesame loop contributions. We will then still predict the mass of φ as m φ = M phys + λ phys ν .Now at the level of inventing the theory we may still tune these parameters to get a small m φ , but we’re tuning physical, observable parameters.In the theory described by Equation 2.1, one might have also said that we needed tofine-tune m against m in order to get a small m φ , especially if we calculated that thequantum correction m was large. But there the fine-tuning was of a different sort, since m and m were only ever observable in the combination m phys . Here the fine-tuning hasa much sharper meaning. This tuning translates into a physical demand on our theorythat at high energies the strength of the interaction between our two scalars Φ , Σ forsome reason has a value extremely close to − M phys /ν , despite having nothing to do75 he Hierarchy Problem Chapter 2 with either of these parameters. The tuning is now a physical feature of our theory anddemands explanation. So we see explicitly that the hierarchy problem is present when the light scalar mass isan output of the theory, rather than an input. If one is so inclined, one can say the wordsthat the Higgs mass is simply an input, but this possibility spells the end of scientificreductionism. It is indeed conceivable that this is how the universe works. However,we know there is physics beyond the Standard Model at smaller length scales, and ourbest ideas for what those could be involve theories where the inputs are defined in theultraviolet and the Higgs mass comes out. Whether the Higgs ultimately originates asa component of a larger multiplet, or a bound state of fermions, or an excitation of astring, we expect that the Higgs mass is a parameter that comes out in the low-energytheory.
You can always use the history of physics to illustrateany polemic point you want to make.Nima Arkani-Hamed
Now having exhibited that getting a light scalar truly does involve some fine-tuningof physical parameters, one may still say ‘
So? ’. In the real world, we’ve observed (theanalogue of) m φ , but the physics we’ve discussed at the heavy scale M is new physics, Let me mention parenthetically a confusion one may encounter if one reads older literature on thehierarchy problem in GUTs. It was common to speak of having to ‘re-tune the parameters at everyorder in perturbation theory’, as if imagining an algorithmic process where one first computed a tree-level prediction, tuned that to be correct, and then computed the one-loop corrections, re-tuned thoseparameters to get it right again, etc. This is framed as being ‘worse’ than just requiring ‘one’ set oftunings. This is moronic, for the simple physical reason that Nature does not compute via perturbationtheory. There is a physical problem, which is how to get the electroweak scale out of other physicalparameters in the theory. Whether you compute the predictions in perturbation theory, or on a lattice,or whilst standing on your head is immaterial. he Hierarchy Problem Chapter 2 and we don’t have experimental measurements of λ telling us the value is not thatperfect value to get our light scalar. One might thus object that there’s no fundamentalinconsistency, and as long as our theory can fit the data anything else is just philosophy .But the criterion of it being not literally impossible for a theory to fit the data is anincredibly low bar, and scientists always use additional criteria to select theories. Whilethere is ultimately a degree of subjectivity in any notion of ‘naturalness’, this is really thesame subjectivity that one constantly uses in science to decide which of two explanationsfor some data to accept.A simple (approximately) historical example of this can be seen in epicyclic theoriesof the motion of the planets. The Ptolemaic, geocentric model of the universe predictedat first that the heavenly bodies orbited the Earth in circles, but eventually astronomicaldata was accurate enough to show that the motion of the planets and sun around the earthwas not circular. In the Ptolemaic model, this was dealt with by adding an epicycle , thesuggestion that the heavenly bodies moved on smaller circular orbits about their circularorbit around the Earth. As astronomical observations became more and more detailedover the ensuing centuries, multiple layers of epicyclic motion were needed to explainthe data—circles on circles on circles, as depicted in Figure 2.1. The description neverstopped working though; if you’d like to, you may describe the orbit of any solar systemobject via r ( θ ) = N (cid:80) n =1 r n sin θn , and for some large but finite N you’ll be able to fit anyorbit to within observational precision.So why did sixteenth century physicists favor Kepler’s laws and heliocentrism? Thediscriminating factor is manifestly not which model better-fitted the data. Rather, thechoice comes down to Occam’s razor, to explanatory power, to simplicity and to fine-tuning of parameters. Physicists favor theories which do more with less—theories whichexplain more about the world while requiring fewer inputs. This is a subjective biasabout how we think the universe should work, and it’s possible that this philosophy will77 he Hierarchy Problem Chapter 2 Figure 2.1: Schematic progression of epicycles needed in the Ptolemaic model of Marsorbiting the Earth as astronomical measurements increased in precision.ultimately fail—but it’s been working well thus far. While keeping the above intuition firmly in the back of our minds, it can be usefulto introduce a mathematical classification of this fine-tuning, with the understandingthat no such measure is god-given and so what to do with such a measure is up to us.We’ll discuss a couple such schemes, the first being a mathematical formalization of thedependence of an output of interest on the values of the inputs. This has the benefitthat it is intuitive and simple to compute, so it is widely used in the particle physicsliterature. However, it lacks independence under how variables are parametrized so canlead to misleading conclusions if used without care.Furthermore, it will assign a measure of fine-tuning to individual points in the pa-rameter space of a model, whereas we’d like to characterize the naturalness of a modelas a whole—if a model only produces predictions that match the real world in a smallregion of its parameter space, that’s another important element of fine-tuning [93]. Forexample, if new data removes all but a small fraction of the viable parameter space in a As a semi-autobiographical aside, I had the honor and pleasure of being in the inaugural cohortof the Integrated Studies Program for Benjamin Franklin Scholars in the School of Arts & Sciences atthe University of Pennsylvania. The program, founded and spearheaded by the classicist Prof. PeterStruck, offered a dedicated interdisciplinary experience wherein, each semester, three diverse fields gavecourses offering perspectives around a central topic, which were concurrently collectively compared andcontrasted. In my year we studied biology, anthropology, classics, political science, physics, and litera-ture, all taught by preeminent professors in their respective fields. After noticing a pattern, the groupkept track of (among other things) how many times each professor mentioned, discussed, or appealedto ‘beauty’. The winner in this regard was Prof. Vijay Balasubramanian—lecturing on the way theuniverse works—by a country mile. he Hierarchy Problem Chapter 2 given model, we want to regard that model as being less natural afterward. An approachbased on Bayesian statistics allows us to incorporate these issues and gives unambiguouscomparisons of the relative naturalness of models upon the collection of new data [94],but loses out on simplicity.A simple and often-used measure was introduced by Giudice and Barbieri [95], whosuggested the definition ∆ X ≡ (cid:12)(cid:12)(cid:12)(cid:12) d ln m d ln X (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Xm d m d X (cid:12)(cid:12)(cid:12)(cid:12) (2.3)which may be called a measure of the fine-tuning of the input parameter X necessary toget out the correct output parameter m . The logarithmic dependence naturally givesa measure of relative sensitivity and removes dependence on overall scale or choice ofunits. If ∆ X is large, this denotes a large sensitivity of m to the value of X , and impliesthat one must choose the value of X very carefully to get out the right physics. Inthe example in Equation 2.2 above we have ∆ λ = λm φ ν (cid:39) M m φ (cid:39) , indeed signalingenormous fine-tuning, and likewise ∆ ν = λ ν m φ . Contrast this with the familiar case of aseesaw mechanism where the light neutrino mass is given by a formula like m ν = m /M ,with M a heavy mass scale and m a weak scale mass. We can check whether thismechanism requires fine-tuning with ∆ M = M m ν m M = m ν m M = 1 , and we find thatseesaw mechanisms are natural. So ∆ X matches our intuition here, and can be quiteuseful.However, for a model with free parameters a notion of fine-tuning at a single point inparameter space does not capture the full picture, and we should incorporate a notion ofthe volume of viable parameter space into our naturalness criterion [96], as diagrammedin Figure 2.2. This necessity should be intuitive in the context of constraining modelsof new physics. Models which achieve their aim throughout parameter space are viewedmore favorably than models which only work in some small region of their parameter79 he Hierarchy Problem Chapter 2 Figure 2.2: A schematic representation of the LHC ruling out previously-viable parameterspace in some model. Surely afterwards we should view this model as more fine-tunedcompared to a model on which the LHC results had no impact.space. For a relevant example, there are still corners of the MSSM parameter space thatare natural under the Giudice-Barbieri measure. But the fact that the LHC has ruledout large swathes of this parameter space means that we should surely view weak-scalesupersymmetry as less natural now than we did a decade ago. A pointwise measure offine-tuning misses this.An approach based on Bayesian statistics can be used to better match what we wantfrom a measure of naturalness, as is discussed well in [84]. The definition of a model ina Bayesian framework requires priors on its free parameters { θ i } , denoted p ( θ i |M ) , anddifferent choices of { p ( θ i |M ) } should be considered different models. The probabilist’snotation p ( A | B ) may be interpreted as ‘the probability of A given that B is true’. Wealso require a prior probability p ( M ) that the model M is true as a whole. After wereceive data d , Bayes’ theorem gives us the posterior probability for the model as p ( M| d ) = p ( d |M ) p ( M ) p ( d ) (2.4)where p ( d |M ) is known as the likelihood. Both p ( M ) and p ( d ) are explicitly subjective,but if we take the ratio of the posterior probabilities for two models M and M we find80 he Hierarchy Problem Chapter 2 p ( M | d ) p ( M | d ) = p ( d |M ) p ( d |M ) p ( M ) p ( M ) (2.5)This expresses how the ratio of the likelihood of these two models changes after receivingnew data. So while different physicists may disagree on the prior and posterior probabil-ities, the ‘Bayes update factor’ B ≡ p ( d |M ) /p ( d |M ) is unambiguous and shows thatthe physicists agree on how the relative naturalness of the two models is affected by thenew data.The likelihood in a model with free parameters { θ i } is calculated as p ( d |M ) = (cid:90) (cid:32)(cid:89) i dθ i (cid:33) p ( d |M , θ i ) p ( θ i |M ) (2.6)This is an integration over parameter space of the likelihood of producing the data in thismodel, weighted by the prior probability distribution we’ve placed on our parameters.This balances the competing effects of how well parameter points fit the data with theprinciple of parsimony—models with large regions of parameter space which don’t fitthe data are penalized. The need to compare models to define naturalness in a Bayesianformalism is easily seen by the fact that the likelihood for any new physics model decreasesmonotonically as more data is collected and previously-viable regions of parameter spaceare ruled out (in the absence of a discovery, of course).An interesting playground for these ideas is the strong CP problem. In brief, this isthe smallness of the so-called ‘theta angle’ θ in QCD, which controls the amount of CPbreaking in the strong sector. While θ ∈ [0 , π ) , empirical measurements now constrain θ (cid:46) − . Although θ is not technically natural, in the Standard Model it runs veryslowly—since the other source of CP breaking is from the CKM matrix—such that ifone sets it tiny in the UV, it stays small down to the IR. The Guidice-Barbieri measurewould thus produce ∆ θ UV (cid:39) , as the measured value is insensitive to small changesin the UV value. And yet, the small theta angle is regarded as a naturalness problem,81 he Hierarchy Problem Chapter 2 which can be justified in a Bayesian approach. The necessity of a prior does help quantifyone’s surprise at the small value of θ in the context of the Standard Model, but to reallythink about naturalness we need to have a comparison. We know of simple theorieswhich produce vanishing θ starting from generic values of the parameters—for example,an axion naturally produces θ = 0 . Then if we have new data which pushes down theupper bound on θ , axion models receive large Bayesian update factors in relation to theStandard Model. When we know of a simple model which automatically explains somedata, it’s puzzling if our current model requires precise choices for free parameters inorder to explain the data.We can see these notions play out for the hierarchy problem by comparing our toymodel of a GUT to one with weak-scale supersymmetry. Our toy model in Section 2.2.1produced a scalar through a cancellation of GUT-scale ∼ GeV contributions. Atoy model with supersymmetry (to be discussed in Section 3.1) would remove sensitivityto the ultraviolet, and replace the scale of GUT-breaking with the effective scale ofsupersymmetry-breaking in the SM sector ˜ m . And while the GUT scale is (more orless) fixed by the running of SM couplings, the SUSY-breaking scale in the SM canbe far lower. We use ˜ m here as a one-parameter avatar of the scale of superpartners.The general prediction for supersymmetry before the LHC was ˜ m ∼ O (100 GeV ) , withmultiple species of superpartners appearing below the TeV scale. Such a model producesan improvement in Giudice-Barbieri tuning of δ ∆ ∼ over our model without SUSY.Let’s take p ( ˜ m | weak scale SUSY ) to be a logarithmic prior from m low to m high , wherean upper limit m high ∼ GeV − TeV is justified by the requirement that the modelgive small Higgs mass corrections and a weak-scale dark matter candidate. If we collectcollider data for a decade and find that much of the parameter space is ruled out, saywith a limit ˜ m ≥ m we should update our thoughts on the naturalness of the model. Thisdata has no effect on our non-SUSY GUT model, as it predicts no new light particles,82 he Hierarchy Problem Chapter 2 but there has been a large effect on our SUSY model as its most favored parameter spacehas been ruled out. So we have a large Bayesian update factor. B ≡ p ( LHC data | non-SUSY GUT ) p ( LHC data | weak scale SUSY GUT ) (2.7) = 1 (cid:16) log m high m low (cid:17) − (cid:82) m high m d ˜ m ˜ m = log m high m low / log m high m Of course supersymmetric extensions of the SM have a large number of parameters, andhow to translate to an upper bound m on our one-parameter version isn’t well-defined,but certainly much of the previously-favored parameter space has been ruled out. Thegeneral lesson is that models which don’t predict new visible states near the weak scalehave received large Bayesian update factors from the LHC. This doesn’t give a strictmandate for our overall relative belief, as one may argue there are good reasons to take alarge prior for supersymmetry and expect always to find superpartners right around thecorner. After all, even if ˜ m ∼ TeV, it would still have δ ∆ ∼ Giudice-Barbierituning better than the GUT without SUSY. But it does motivate further investigationof models which don’t succumb to this issue, be they neutral naturalness modules whichpush the scale of new visible states up by a loop factor or more radical ideas about theorigin of the electroweak scale.Furthermore, we can directly input physics into making a sensible choice of the priorone places on a model, and there has been much discussion of justifying simple choices.The dictum of Dirac naturalness mandates priors peaked at O (1) values. But with amodel which makes some parameter technically natural one, a prior that allows smallvalues is justified, such as a logarithmic prior. An explicit example of this modificationof priors by physics can be seen in the application of the Weak Gravity Conjecture to thehierarchy problem [97, 98, 99, 100, 101, 102, 103, 104, 11, 5, 105], which will be discussed83 he Hierarchy Problem Chapter 2 in more detail in Section 4.3.1. This mechanism explicitly modifies the prior one shouldhave on UV theories by linking the notion of the Swampland—that some string vacuadon’t admit universes like ours—to the allowed range of Higgs masses. This addressesthe hierarchy problem by constructing a model where the priors are forced by UV physicsto favor a light Higgs.While these measures of naturalness are useful to help us clarify our expectations,we emphasize again that they must be used sensibly. But it’s clear that some notionof naturalness appears solely from the axioms of probability, and is indeed baked-in tothe practice of science inquiry. That’s not to say that we should be epistemologicallycommitted to the naturalness of the universe, but we can still see it as a useful guidetoward new physics which has worked well in the past. All models are wrong, but some are useful.George E. P. Box
Robustness in the Strategy ofScientific Model Building (1979) [106]
There is another obvious suggestion that is useful to discuss: perhaps the Higgs doesnot interact with any physics at higher mass scales, such that despite in principle worriesabout the hierarchy problem there is no hierarchy problem in practice . The physics whichcan destabilize the Higgs mass must, as seen in the above example, both be heavier thanthe Higgs and interact with it in order to generate a contribution. Since the Higgs massin the SM is not technically natural (that is, it is UV sensitive), large mass correctionsfrom such particles are generic, as we saw in Section 1.3. One can explore the idea thatperhaps there are no such particles. This faces a number of challenges.84 he Hierarchy Problem Chapter 2 (a) (b)
Figure 2.3: In (a), a representative two-loop diagram giving gravitational corrections tothe Higgs mass from a new dark fermion ψ . These diagrams do not destabilize the Higgsmass. In (b), a representative three-loop diagram in which the gravitons couple to anoff-shell top quark, which is no longer proportional to the Higgs mass.The first difficulty is that we know there must be new physics. The Standard Modeldoes not explain neutrino masses nor dark matter, though it is possible that both ofthese be resolved without introducing new heavy particles. But a deeper issue is that weknow the SM cannot be a fundamental theory because the Landau pole in hyperchargemakes it inconsistent. This demands that something must happen to rid the theory ofthis pole, and it will interact with the Higgs because the Higgs is hypercharged. This isone motivation for thinking that something like Grand Unification must take place, andof course its breaking introduces a heavy fundamental scale. One can try to get aroundthis by appealing to quantum gravity coming in at scales below that of the Landau pole.But the Higgs certainly interacts gravitationally, so for this program to succeed one needsa quantum gravitational theory which does not introduce any scales. This is interestingto explore, but does not seem to be the way the universe works, though we leave detailedcriticism of this idea to the gravity theorists.Furthermore, even if somehow all heavy particles are neutral under the SM gaugegroups, there is no way for them to escape gravitational interactions with the Higgs.85 he Hierarchy Problem Chapter 2 This leads to irreducible three-loop corrections to the Higgs mass [107], as depicted inFigure 2.3. Consider a fermion ψ with mass M ψ . The obvious two-loop diagram wherethe graviton couples to the Higgs directly gives a correction proportional to m H , as thegraviton coupling to a massless on-shell particle vanishes at zero momentum. However,we can draw a diagram where a ψ loop talks gravitationally to an off-shell top loopcontribution to the Higgs two-point function, and integrating this particle out yields acorrection δm H (cid:39) y t (16 π ) M ψ M pl M ψ + subleading . (2.8)The powers of M ψ appear because the only other possible mass scale for the numerator is m t = y t v , and the top loop power-law correction to the Higgs mass does not vanish in the v → limit. The sensitivity of the Higgs mass is softened by three loop factors as well asby the Planck mass from the gravitational couplings, but insisting that δm H (cid:46) (1 TeV ) places an upper ‘naturalness’ limit on such fermions of ∼ GeV. While far betterthan the ∼ TeV limit for SM-charged particles, this is still well below the Planck scaleand amounts to an enourmous constraint on UV physics. In fact the problem is a bitworse than this estimate, as we should sum over all SM loops that couple to the Higgs,but this suffices already to see the issue.So asking for the Higgs to be lonely enough to cure the hierarchy problem is a humon-gous requirement on the ultraviolet of the universe, and this approach faces a number ofimportant hurdles. We mention that there is work on interesting theories which touchon some of these points, but as a whole Nature seems not to have taken this approach.
One may hear the statement that the hierarchy problem disappears if you use a mass-independent regularization scheme, for example dimensional regularization. Unlike the86 he Hierarchy Problem Chapter 2 prior nonsolutions we’ve considered, this one is definitively incorrect. The mass scale oneintroduces in EFT is a stand-in for genuine physical effects of any sort which appear atshorter distance scales. So the cutoff regularization is useful for seeing an avatar of thehierarchy problem even when one does not know the ultraviolet theory. With a mass-independent scheme, one must instead put in specific short-distance physics to see theproblem, but we can easily see the general issue.As a simple example, take a theory with two real scalars - our light φ and a heavier ϕ . If we impose a Z symmetry for simplicity, the action is S = (cid:90) d x (cid:20) −
12 ( ∂ µ φ ) − m φ − M ϕ − λ φ ϕ (cid:21) (2.9)Doing continuum effective field theory with dimensional regularization and the MS renor-malization scheme, we must upgrade the masses and couplings to running parameterswhich depend on the renormalization scale µ , as usual (introduced in Section 1.2.2).Since our renormalization scheme is mass-independent, if we want to study physics at anenergy scale µ ∼ m φ (cid:28) M ϕ , we should implement the decoupling theorem by hand. Weintegrate out the heavy degree of freedom at the scale µ ∼ M ϕ and match to a low-energyeffective field theory which only contains the low-energy degree of freedom φ (introducedin Section 1.2.3). So let us go ahead and integrate out the heavy scalar ϕ . The effect onthe φ mass comes from the simple diagram of Figure 2.4.We go to general dimension d = 4 − (cid:15) and replace λ → λ ˜ µ (cid:15) , where ˜ µ is an arbitraryscale which soaks up the mass dimension of λ away from d = 4 . The resources needed tocompute the integrals for general dimension and to take the limit (cid:15) → can be found in87 he Hierarchy Problem Chapter 2 Figure 2.4: The one-loop diagram for a scalar quartic interaction contributing to a scalarmass.Srednicki’s textbook [18]. − iδm = − iλ ˜ µ (cid:15) (cid:90) d d k (2 π ) d − i ( k + M ϕ )= − iλµ (cid:15) (cid:90) d d ¯ k (2 π ) d k + M ϕ )= − i λ ˜ µ (cid:15) π ) Γ( − (cid:15) π ) (cid:15) ( M ϕ ) − (cid:15) = i λ π ) M ϕ (cid:20) (cid:15) − γ E + 1 + ln 4 π − ln M ϕ + 2 ln ˜ µ + O ( (cid:15) ) (cid:21) = i λ (4 π ) M ϕ (cid:20) (cid:15) + 12 + ln µ M ϕ + O ( (cid:15) ) (cid:21) (2.10)Where γ E is the Euler-Mascheroni constant, and we have Wick rotated k → i ¯ k d , inte-grated in general dimension, expanded in the limit (cid:15) → and defined µ ≡ π ˜ µ e − γ E to soak up the annoying constants. Indeed, we see that there’s no quadratic divergence,which ultimately is due to the fact that scaleless integrals vanish in dimensional regular-ization (cid:82) d d kk n = 0 , as must be true simply by dimensional analysis.Now we follow the MS renormalization scheme by adding a counterterm which cancelsoff the divergent piece and we match at µ = M ϕ to ensure that our low-energy EFTproduces the same predictions as the UV theory, as discussed in Section 1.2.3. This gives88 he Hierarchy Problem Chapter 2 us S = (cid:90) d d x (cid:20) −
12 ( ∂ µ φ ) − (cid:18) m + λ (4 π ) M ϕ (cid:19) φ − . . . (cid:21) (2.11)We see that the high-energy degree of freedom ϕ contributes a threshold correction whenwe flow to lower energies and remove it from the spectrum. While there was never aquadratic divergence, we still found a large quadratic correction to the mass of the scalar φ which is proportional to the scale of new physics.This underscores the importance of not getting confused by unphysical features ofrenormalization. There is a physical issue, which is the sensitivity of the physical low-energy scalar mass to the physics in the ultraviolet. Indeed if we tame the UV in differentways, we find different avatars of this sensitivity. It’s true that a Wilsonian cutoff actsas a stand-in for arbitrary scaleful physics, which is why it’s more direct to see the issuein that picture, but the same physical problem appears regardless of the regularization.89 he Hierarchy Problem Chapter 2 After discussing those misconceptions, we may give a one-sentence description of thehierarchy problem:In a theory of physics beyond the Standard Model where the Higgs mass is an output,physical parameters must be finely-tuned in order to produce a mass which is far belowthe scale of new physics, in tension with the principle of parsimony.With that in hand, we are prepared to delve in to how the hierarchy problem maybe solved in the next section. Our discussion below will not take place within a UVcomplete extension of the Standard Model, so one might worry that we are attacking aproblem without knowing its source. While true, the point from Section 2.2.3 is that thesensitivity to UV physics is so general that we expect to need a mechanism which stabilizesthe Higgs mass to whatever new heavy physics is out there. Our toy calculation of therelative naturalness of SUSY already evinces this point—SUSY tamps UV sensitivity nomatter what it is, so the details of the UV completion are immaterial. As a result ofthis idea, we will mostly worry just about finding a way to produce a light scalar andassume that it can be embedded into whatever physics exists at high scales, rather thancommitting to a particular framework of grand unification or what have you. Of courseit’s possible that interesting mechanisms to produce an IR scale do rely on particularproperties of the UV, and we’ll discuss this important idea in Section 4.3. But eventhere, our initial goal is simply to produce a light scalar which is compatible with theStandard Model, rather than to write down a full theory of the universe on all scales. Ifwe can first solve the problem in a toy model which shares some features of our universe,then we can hope to abstract what we learn from that to solve the real problem.90 hapter 3The Classic Strategies
Without going out of my doorI can know all things on earthWithout looking out of my windowI could know the ways of heavenGeorge Harrison satirizing thepast decades of particle theoryin light of LHC data
The Inner Light (1968) [108] he Classic Strategies Chapter 3 Fantastic Symmetries and How To Break Them
In large part the story of particle physics over the past decades is the story of attemptsto solve the hierarchy problem. Much theoretical effort has been put into understandinginteresting symmetries and mechanisms for breaking them, and more generally waysthat small numbers can pop out of physical theories; and much experimental effort hasfocused on locating empirical hints of these ideas. However, the past few years have seenmany practitioners turn their attention toward topics like dark matter, cosmology, andastrophysics. And for good reason—on the experimental side, this is largely where thenew data is and will be for the foreseeable future, and at the purely theoretical level thehierarchy problem has become a lot more challenging, as we will argue below. But thishas lead to a new generation of particle theorists who are largely unfamiliar with thefantastic and brilliant ideas which drove the field in the prior couple decades.Despite the fact that we will argue below that these ideas largely appear to not be theway the world works at the weak scale, understanding this prior work can be enormouslyhelpful for inventing novel ideas in the future. It is with this in mind that we introducebelow the basics of a variety of interesting ideas and methods in particle physics againstthe backdrop of their relevance to the hierarchy problem. These are ideas that havenot yet made their way into standard textbooks on field theory, but are neverthelessessential topics for students of particle theory to absorb. We will endeavor to explicatethe core of these ideas in the simplest models possible, and will largely avoid discussingphenomenological considerations past producing a light scalar. The discussion will notbe at the level of depth required for research in the field, but will hopefully be a niceoverview of interesting topics for which references to serious introductions and reviewswill be provided as well.So how does one solve the hierarchy problem? The classical solutions may be concep-92 he Classic Strategies Chapter 3 tually divided into two steps. First one introduces some structure above the electroweakscale which protects the Higgs mass from large contributions due to UV physics. Thiscould be something like a new symmetry which forbids a scalar mass term, or a modifi-cation to spacetime on small length scales, or the dissolution of a non-fundamental Higgsinto component fields.However, the Higgs is not exactly massless, which is due to the fact that whateverstructure we add is not a feature of the low-energy Standard Model. There must thus besome IR dynamics that break that UV structure at the electroweak scale to ensure thatwe end up with the Standard Model at low energies. Depending on the UV structurethis may be something like spontaneous symmetry breaking or moduli stabilization ordimensional transmutation.There are two big categories of classical solutions. One is to find a field-theoreticmechanism which prevents contributions to the Higgs mass in the UV. Supersymmetry isthe prime example here. The other is to bring the fundamental cutoff of the theory downto the infrared, such that in the UV there’s no Higgs to talk about. This is exemplifiedby composite Higgs theories or theories where the cutoff of quantum gravity is loweredto the weak scale.To evince these strategies, we’ll go through a couple examples of ways to forbid scalarmasses and to break those structures. Our aim here is not to construct realistic theoriesof the Higgs but rather to understand these general principles, so we’ll study simple toymodels which allow us to appreciate the essential points.93 he Classic Strategies Chapter 3
Superpartners aren’t essentialBut would have been consequentialSuch wasted superpotentialSuper once, super twiceSuper chicken soup with riceMaurice Sendak on hisdisappointment with the LHC dataLost Stanza of
Chicken Soup withRice (1962) [109]
Supersymmetry exploits a loophole in the classic Coleman-Mandula theorem [74] byintroducing fermionic symmetry generators, which in layman’s terms turn bosons intofermions and vice-versa. By the Haag–Łopuszański–Sohnius theorem [110], this is theunique extension to the Poincaré algebra. Since we know that symmetries tend to makephysics easier, it is not surprising that supersymmetry is an indispensable tool in highenergy theory, regardless of how or whether it is realized in the real world. Some usefulgeneral introductions to supersymmetry in d = 4 and its application to the real world areTerning’s book [111], Martin’s periodically-updated lecture notes [112], and Shih’s videolectures [113], in roughly increasing order of friendliness to neophytes.In a supersymmetric theory fields come in multiplets which include particles of dif-ferent spins (so called ‘supermultiplets’) all having the same mass and quantum num-bers. We add fermionic generators Q α and Q † ˙ α , called supercharges, with the defining94 he Classic Strategies Chapter 3 (anti)commutation relations (cid:110) Q α , Q † ˙ α (cid:111) = 2 σ µα ˙ α P µ { Q α , Q β } = 0 = (cid:110) Q † ˙ β , Q † ˙ α (cid:111) (3.1) [ Q α , P µ ] = 0 = (cid:104) P µ , Q † ˙ α (cid:105) (3.2)where P µ is the generator of spacetime translations and σ µ = (1 , (cid:126)σ i ) with σ i the Paulimatrices. These may be determined simply by writing down all objects with the correctindex structure. For later use, recall that spinor indices are raised/lowered with theinvariant antisymmetric symbols (cid:15) αβ , (cid:15) ˙ α ˙ β , as used for example in defining the conjugateinvariants ( σ µ ) ˙ αα ≡ σ µβ ˙ β (cid:15) αβ (cid:15) ˙ α ˙ β .We want to find irreducible representations of the supersymmetry algebra, calledsupermultiplets. Since P µ P µ = m commutes with the generators Q α , Q † ˙ α , the differentparticles in a supermultiplet will have the same mass. As an example of how to generatesupermultiplet states, consider a massive particle. We can go to its rest frame, whereit has momentum P µ = ( m, , , with m its mass. Then the supersymmetry algebragreatly simplifies to (cid:110) Q α , Q † ˙ α (cid:111) = 2 m α ˙ α , and we see that this is just a Clifford algebraof raising and lowering operators. We define a lowest weight state, or Clifford vacuum | Ω s (cid:105) such that it is annihilated by the undotted generators | Ω s (cid:105) = Q Q | m, s, s z (cid:105) (3.3) ⇒ Q | Ω s (cid:105) = 0 = Q | Ω s (cid:105) (3.4)Now we can use the dotted generators as raising operators to generate the entire multiplet. | Ω s (cid:105) (3.5) Q † ˙1 | Ω s (cid:105) , Q † ˙2 | Ω s (cid:105) (3.6) Q † ˙1 Q † ˙2 | Ω s (cid:105) (3.7)A single fermionic supersymmetry generator must change the spin of a state by . Start-ing at the top with a spin j particle gives us states of spin j − and j + on the middle95 he Classic Strategies Chapter 3 line, and another state of spin j on the bottom line. In d = 4 the supermultiplet formedfrom a vacuum state of spin is called a ‘vector multiplet’ and contains four states withspins (0 , , , . That formed from spin 0 is called a ‘chiral multiplet’, and has stateswith spins (0 , , ) . Note that this is fewer degrees of freedom, since negative spins arenot allowed. Beginning with spins higher than leads to states with spins greater than , which will take us into supergravity and will not be necessary for our purposes.We could repeat this exercise for massless supermultiplets, labelling states by theirenergy and helicity | E, λ (cid:105) . We would find a Clifford algebra with only one set of rais-ing/lowering operators, and find supermultiplets with helicities λ and λ + for somestarting λ . Then CPT invariance would force us to add states of helicity − λ and − λ − .Merely from the definition of the symmetry group there are already a few interestingimmediate results. For a start, we show that physical states have nonnegative energyin a supersymmetric theory, and the vacuum energy is an order parameter for super-symmetry breaking. First, let’s give a simple expression for the Hamiltonian operator ofsupersymmetry. We act on our anticommutation relation with ( σ ν ) ˙ αα and recall variousidentities to note that σ µα ˙ α ( σ ν ) ˙ αα = 2 η µν , which gives us P ν = ( σ ν ) ˙ αα (cid:110) Q α , Q † ˙ α (cid:111) ⇒ P = 4 H = ˙ αα (cid:110) Q α , Q † ˙ α (cid:111) H = Q Q † ˙1 + Q † ˙1 Q + Q Q † ˙2 + Q † ˙2 Q , where we have used the fact that the zeroth component of the generator of spacetimetranslations is the generator of time translations, which is the Hamiltonian operator.Then we can write the energy of some state state S as (cid:104) S | H | S (cid:105) = 14 (cid:16) || Q | S (cid:105) || + || Q † ˙1 | S (cid:105) || + || Q | S (cid:105) || + || Q † ˙2 | S (cid:105) || (cid:17) ≥ , (3.8)so the energy of S is non-negative. Furthermore, consider a vacuum state | (cid:105) of our96 he Classic Strategies Chapter 3 theory. In a standard QFT, the vacuum energy (cid:104) | H | (cid:105) = E vac is non-physical—we canjust shift the Hamiltonian arbitrarily to remove it. But here, the supersymmetry algebragives a preferred frame. If a vacuum state | (cid:105) is supersymmetric then it is annihilatedby the supercharges Q α | (cid:105) = 0 , Q † ˙ α | (cid:105) = 0 , otherwise the vacuum would not be invariantunder supersymmetry transforms. This implies that it will have vanishing total energy (cid:104) | H | (cid:105) = 0 . Conversely, if the vacuum state is non-supersymmetric, then its energyis strictly positive. We say supersymmetry is broken in such a state. Thus the vacuumenergy acts as an order parameter for SUSY breaking.Connected to that fact is that each supermultiplet contains the same number offermionic and bosonic degrees of freedom. We can see this by defining an operator F which counts the fermion number of a state, so that bosonic states have eigenvalue under ( − F , and fermionic states have eigenvalue − . Since the SUSY generators interchangebosonic and fermionic states, they must anticommute with ( − F .Now, for a given supermultiplet consider the states | a (cid:105) with the same given four-momentum p µ , p = E (cid:54) = 0 . Since the supercharges commute with P µ , we know thatthese must form a complete set of states in this subspace (cid:80) a | a (cid:105) (cid:104) a | = 1 . Now considerthe trace of the weighted energy operator ( − F H/ . (cid:88) a (cid:104) a | ( − F H | a (cid:105) = (cid:88) a (cid:104) a | ( − F QQ † | a (cid:105) + (cid:88) a (cid:104) a | ( − F Q † Q | a (cid:105) = (cid:88) a (cid:104) a | ( − F QQ † | a (cid:105) + (cid:88) a (cid:88) b (cid:104) a | ( − F Q † | b (cid:105) (cid:104) b | Q | a (cid:105) = (cid:88) a (cid:104) a | ( − F QQ † | a (cid:105) + (cid:88) b (cid:104) b | Q ( − F Q † | b (cid:105) = (cid:88) a (cid:104) a | ( − F QQ † | a (cid:105) − (cid:88) b (cid:104) b | ( − F QQ † | b (cid:105) = 0 (3.9)where we have suppressed the contracted spinorial indices. This implies that the number97 he Classic Strategies Chapter 3 of bosonic degrees of freedom is the same as the number of fermionic degrees of freedomin our supermultiplet, which we found to be true in the example we considered above.There is a beautiful formalism of ‘superspace’ which can be used to make super-symmetric theories far more transparent, but introducing this would be too large of adigression for our purposes. We simply want to see the effects of supersymmetry on(in)sensitivity of low-energy physics to the ultraviolet, for which studying a simple the-ory of chiral superfields will do. The Wess-Zumino model is the simplest such examplewhich is not free, consisting of a single self-interacting chiral supermultiplet, and washistorically the first non-trivial four-dimensional theory proved to be supersymmetric.We may write down the Wess-Zumino Lagrangian as L = (cid:90) d x (cid:18) − ∂ µ φ (cid:63) ∂ µ φ − m φ (cid:63) φ − iψ † ¯ σ µ ∂ µ ψ − mψψ − yφψψ − ymφ φ (cid:63) − | y | φφφ (cid:63) φ (cid:63) (cid:19) , (3.10)where for compactness we’ve left off the Hermitian conjugate terms. To avoid introducingcertain technical complications we eschew the proof that this is indeed invariant undera supersymmetric transformation and instead evince its UV insensitivity. For fun we’llcompute without assuming the masses are the same m φ (cid:54) = m ψ , as can happen in thepresence of soft breaking of the symmetry.Let’s look at the vacuum energy, which we’ll calculate generally but schematically. Aquantum harmonic oscillator has ground state energy ± (cid:126) ω for bosonic and fermionicstates respectively, with the sign being familiar from the Casimir effect. If we consider a Martin’s notes [112] serve as a good introduction to traditional ‘off-shell’ superspace for N = 1 , d = 4 theories, and Thaler’s TASI lecture notes [114] are also a fantastic resource. There is a related butdistinct formalism of ‘on-shell’ superspace, which falls under the heading of the amplitudes/on-shell/S-matrix program. This was first introduced very early on by Nair [115] and was used to great effect byArkani-Hamed, Cachazo, & Kaplan [116] much later. A pedagogical introduction to on-shell techniquesincluding superspace can be found in the textbook by Elvang & Huang [117]. Until recently, the on-shellprogram was mostly restricted to massless particles. As it so happens, after Arkani-Hamed, Huang,& Huang [118] introduced a beautiful extension of the formalism to include massive particles, it wasTimothy Trott, my undergrad mentee Aidan Herderschee, and myself who formulated an extension ofthe on-shell superspace formalism for massive particles [9]. The on-shell program is another fascinatingline of work that I suggest any aspiring particle or field theorist learn about. he Classic Strategies Chapter 3 box of side length V /D , the energy of the fields inside it is E = bosons (cid:88) (cid:126)k (cid:126) ω (cid:126)k − fermions (cid:88) (cid:126)k (cid:126) ω (cid:126)k , (3.11)where (cid:126)k = ( k , k , . . . , k D )(2 π/V /D ) , k i ∈ Z . Now in QFT each mode has energy ω (cid:126)k = √ k + m , and as we make the box bigger V /D → ∞ , the sum turns into anintegral E = V (cid:90) d D k (2 π ) D (cid:18) (cid:113) k + m B − (cid:113) k + m F (cid:19) , (3.12)with m B a boson mass and m F a fermion mass, where the sum over species is implicit.We also recognize E /V as the vacuum energy density, denoted Λ , which we can writeas Λ (cid:39) π ) D (cid:90) d D k (cid:18)(cid:113) k + m B − (cid:113) k + m F (cid:19) . (3.13)Now if we specialize to D = 4 and introduce a cutoff k max up to which we’re confidentthat our description of particle physics holds, the schematic form is simply Λ ∼ k max (cid:32) bosons (cid:88) − fermions (cid:88) (cid:33) + k max (cid:32) bosons (cid:88) m B − fermions (cid:88) m F (cid:33) + . . . (3.14)Now we see quite generally and explicitly that in a supersymmetric state the vacuumenergy vanishes, since there are equal numbers of bosonic and fermionic fields with de-generate masses. Furthermore, spontaneous breaking of supersymmetry breaks the de-generacy but does not change the numbers of fields, so softly broken supersymmetryretains protection from the largest contribution.Let’s look now more sharply at the one-loop contributions to the scalar mass in theWess-Zumino model, regularized with a hard cutoff Λ . Take care that we’ve writtenthe Lagrangian in terms of two-component spinors, an exhaustive guide to which can befound in [119]. The three diagrams are shown in Figure 3.1, and their evaluation proceeds99 he Classic Strategies Chapter 3 Figure 3.1: The one-loop diagrams contributing to the scalar mass correction in theWess-Zumino model.as − iδm = ( − im ψ y ) (cid:90) d k (2 π ) ( − i ) ( k + m φ ) + ( − iy ) (cid:90) d k (2 π ) ( − i )( k + m φ )+ ( −
1) 12 ( − iy ) (cid:90) d k (2 π ) ( − i ) Tr [ σ µ k µ ¯ σ ν k ν ]( k + m ψ ) (3.15) = i | y | (cid:90) d ¯ k (2 π ) (cid:34) m ψ (¯ k + m φ ) − k + m φ ) + ¯ k (¯ k + m ψ ) (cid:35) (3.16) = i | y | π (cid:90) Λ0 d ¯ k ¯ k (cid:34) m ψ − m φ − ¯ k (¯ k + m φ ) + ¯ k (¯ k + m ψ ) (cid:35) (3.17) = i | y | π ( m φ − m ψ ) log Λ + finite . (3.18)We see that the UV sensitivity of the scalar mass in this theory has disappeared, evenif the two fields have different masses. In the limit of unbroken supersymmetry, thecontribution vanishes identically.This fact of removing the UV sensitivity of the mass of a scalar persists generally,no matter which other superfields are added, so long as supersymmetry is at most softlybroken. The connection to the hierarchy problem is clear: If, in the UV, all fields comein supermultiplets, then the Higgs mass is protected from UV contributions.Of course we do not observe mass-degenerate superpartners, so this soft supersymme-try breaking is a necessary feature of any implementation of supersymmetry to the realworld. The Minimal Supersymmetric Standard Model [120] embeds each of our fermionsin a chiral supermultiplet and each gauge boson in a vector multiplet. The Higgs sector100 he Classic Strategies Chapter 3 must be enlarged to two chiral multiplets containing the up and down Yukawas respec-tively, as is necessary for anomaly cancellation .The question of supersymmetry-breaking is a very non-trivial one. At the level of aphenomenological accounting of possible soft breaking terms in the MSSM, there are 105physical parameters [122]. However, constraints on flavor-violating couplings and on CPviolation tell us empirically that the soft terms that appear must be very non-generic. Infact, looking at the MSSM in detail it turns out there are no places for supersymmetry-breaking to enter directly, and indeed there are general arguments that such breakingmust take place in another, hidden sector and be indirectly communicated to the MSSMfields (see e.g. Martin’s Section 7.4 [112]).The origins of supersymmetry-breaking being a separate sector does force us to expandour model of particle physics, but on the upshot this sequestering means we can exploreinteresting phenomenology in sectors which are unconstrained. One can write downmodels where supersymmetry breaking is mediated by supergravity effects [123, 124,125, 126, 127, 128, 129], communicated to the SM fields by our gauge bosons from asector with new, massive SM-charged particles [130, 131, 132, 133, 134, 135], or takesplace at a physically separate location in an extra dimension [136, 137, 138, 139, 140, 141,142, 143, 144, 145], for a few examples. A full discussion of the mechanisms and strategiesfor models of supersymmetry-breaking is beyond our scope, but we highly recommendIntriligator & Seiberg’s lecture notes [146] as a general reference along with Martin’snotes [112].Of course we would like this phase transition to originate as spontaneous symme- Perhaps the more urgent reason for needing two Higgs multiplets is that the interactions in su-persymmetric theories are highly constrained by ‘holomorphy’, a full explanation of which here wouldrequire too much machinery but which leads to the conclusion that the same multiplet cannot haveYukawa interactions with both the up- and down-type quarks. However see [121] for the interestingpossibility that at high energies only the up-type Yukawa interactions exist, and the down-type andcharged lepton masses are induced by supersymmetry-breaking. he Classic Strategies Chapter 3 try breaking, rather than explicitly putting it in by hand, since we want the far UV tobe supersymmetric. Such spontaneous breaking requires the generation of a scale, andso it would be great if such a scale were generated dynamically , as in the dimensionaltransmutation we saw in QCD in Section 1.3.3. This would then be a natural mecha-nism for SUSY breaking. This phenomenological prospect lead to and benefited from afantastic body of work understanding the details of supersymmetric gauge theories e.g.[147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157], which can be found reviewed intextbooks by Terning [111] and Shifman [158] and in TASI notes by Strassler [159].We stated above that fields in a given supermultiplet share all the same quantumnumbers, but there is in fact one exception. The supersymmetry algebra in Equation 3.1is invariant under opposite rephasings of the supercharges Q α → Q α e − iα , Q † ˙ α → Q † ˙ α e iα .So there is a generator of a global internal symmetry that we may add that has nontrivialcommutation relations with the supercharges: [ R, Q ] = − Q, (cid:2) R, Q † (cid:3) = Q † . (3.19)This generator is known as an R -symmetry generator, and if the theory is invariant underan R -symmetry that means that each supermultiplet Φ can be assigned an R -charge r Φ and the theory is invariant under transformations of each multiplet Φ → Φ e ir Φ α ,schematically, where Φ is the collection of fields in that multiplet. Since R does notcommute with the supercharges, the different fields in the supermultiplet have different R -charges. For example if Φ is a chiral superfield consisting of ( φ, ψ ) then under a globalrotation by α they transform as φ → φe ir Φ α , ψ → ψe i ( r Φ − α , (3.20)which is simply because | ψ (cid:105) ∼ Q | φ (cid:105) .It’s important to emphasize that this R -symmetry is not part of the supersymmetryalgebra, so one may have supersymmetric theories which do or do not implement R -102 he Classic Strategies Chapter 3 symmetry. Nelson & Seiberg showed a fascinating connection between R -symmetry andsupersymmetry-breaking [160]. They show roughly that for a low-energy Wess-Zuminomodel (possibly after having integrated out confined strong dynamics) as long as one hasa ‘generic’ potential (in the sense that a generic set of n equations in n unknowns has asolution), then a vacuum spontaneously breaks supersymmetry if and only if it sponta-neously breaks R -symmetry. The reasoning is simply that such a symmetry imposes anadditional constraint on the potential minimization equations, leading to a solution nolonger being generically present.For later use we mention the possibility of ‘extended supersymmetry’, where addi-tional supercharges are added (cid:110) Q Aα , Q † B ˙ α (cid:111) = 2 σ µα ˙ α P µ , (cid:8) Q Aα , Q Bβ (cid:9) = 0 = (cid:110) Q † A ˙ β , Q † B ˙ α (cid:111) (3.21)where A, B = 1 .. N index the supercharges. The construction of supermultiplets proceedsas before, but there are now more nonvanishing combinations of supercharges to act onthe Clifford vacuum, so supermultiplets are enlarged. In four dimensions, the most su-persymmetry we can have without gravity is N = 4 , which contains enough superchargesto relate the helicity − vector all the way to the helicity +1 vector; any more super-charges would necessarily yield particles of spins / , . If we are willing to include thesedegrees of freedom we can only go up to N = 8 ‘supergravity’ (SUGRA), as more chargeswould lead to a theory with fundamental particles of spins s > which is pathological .The classic references for supergravity are Wess & Bagger [162] and Freedman & VanProeyen [163]. Ultimately if supersymmetry is a field-theoretic feature of the ultravioletof our universe we must have SUGRA as well, as it simply results from applying thesupercharges to the graviton field, but we won’t discuss SUGRA any further. There’s an important exception here, which is a theory which includes particles of all spins. Thisis necessary for string theory to operate, but has also led to the formulation of novel field theoriescommonly called ‘Vasiliev gravity’ [161]. he Classic Strategies Chapter 3
When we enlarge our superalgebra we also enlarge the (potential) R-symmetry group—the group of symmetries which do not commute with the supercharges—since we can nowshuffle around the supercharges in addition to rephasing them. We’ll revisit this in Sec-tion 3.2.3 in the context of utilizing non-trivial R-symmetry representations to breaksupersymmetry. 104 he Classic Strategies Chapter 3
I exist in the hope that these memoirs, in somemanner, I know not how, may find their way tothe minds of humanity in Some Dimension,and may stir up a race of rebels who shallrefuse to be confined to limited Dimensionality.Edwin Abbott Abbott,
Flatland , 1884 [164]
One of the most important ideas in theoretical physics developed in the latter partof the th century is that there may be additional spatial dimensions past the familiarthree of our everyday experiences. Theories in which additional spatial dimensions arepresent were first studied in the context of unifying gravity and electromagnetism, first byNordström [165] ( before General Relativity!) and then by Kaluza [166] and Klein [167].These ideas saw a resurgence of interest some half-century later with the advent of stringtheory, and the vision of all features of the universe being fundamentally geometrized.Against that backdrop, it is clearly prudent to consider the interplay of such theorieswith the puzzle of the hierarchy problem. As we shall see, extra dimensional theories canproduce terribly interesting physics, and the possibilities are manifold . To consider the possibility that there are additional microscopic dimensions, we developa picture of the effects of fundamentally D -dimensional fields where D = 4 + d . On our105 he Classic Strategies Chapter 3 manifold M = M × K , where K is compact, we write down an action S = (cid:90) M (cid:90) K (cid:112) − g ( + d ) L ( φ , A M , g MN ) (3.22)where our fields are in irreducible representations of the D -dimensional Poincaré alge-bra, and the Lagrangian manifestly obeys D -dimensional Lorentz invariance. We’ll useboldface for D -dimensional fields and Latin letters for D -dimensional Lorentz indices.However, since K is compact, it places constraints on the mode expansions of our fields.Then when we want to study the effective four-dimensional theory we first need to de-compose our fields into irreducible representations of the -dimensional Poincaré algebra.The D -dimensional vectors and tensors will become multiplets of -dimensional fields.Then we can explicitly integrate over the compact manifold K , which will produce atower of states.As an easy, explicit example, take the compact manifold to be the circle S of length πR , and consider a single complex scalar. We start with S = (cid:90) d x (cid:90) πR d y (cid:20) − ∂ M φ ∗ ∂ M φ − m φ ∗ φ (cid:21) (3.23)From introductory quantum mechanics, we know that the boundary conditions of beingsingle-valued around the circle constrains the mode expansion of φ . We can write φ ( x µ , y ) = 1 √ πR + ∞ (cid:88) n = −∞ φ n ( x µ ) e iny/R (3.24)where the normalization will produce canonically normalized kinetic terms in the -dimensional action, and should generally be (cid:112) Vol ( K ) . Plugging this into the actiongives S = 12 πR (cid:90) d x (cid:88) mn (cid:90) πR (cid:20) − ∂ µ φ ∗ m ∂ µ φ n − (cid:18) − imR (cid:19) (cid:18) inR (cid:19) φ ∗ m φ n − m φ ∗ m φ n (cid:21) e i ( n − m ) y/R (3.25)106 he Classic Strategies Chapter 3 Figure 3.2: A schematic Kaluza-Klein spacetime for intuition on the appearance of atower of four-dimensional ‘Kaluza-Klein modes’. The compact extra dimension demandsquantized momentum along the fifth dimension. For an observer on large scales, theextra component of momentum appears as a four-dimensional mass term.where the first and second terms come from either the M derivatives or the K derivativesacting on the mode expansion. The integral over the compact direction now gives, usingorthogonality (cid:82) πR d ye i ( n − m ) y/R = 2 πRδ nm , a -dimensional action S = (cid:90) d x (cid:88) n (cid:20) − ∂ µ φ ∗ n ∂ µ φ n − (cid:18) m + n R (cid:19) φ ∗ n φ n (cid:21) (3.26)We see that we now have a tower of -dimensional states that have arisen from the one -dimensional scalar, as shown schematically in Figure 3.2. There is a ‘zero-mode’ whichhas a mass given by the -dimensional mass term, and then there are states with largermasses. If R is small, these will be heavy, and so we would only see them at, say, a highenergy particle collider. Note that all of these higher levels are doubly-degenerate, since n ranges over all the integers.If we had included the graviton in this compactification procedure, we would see the107 he Classic Strategies Chapter 3 -dimensional graviton split up into g MN = g µν · · · g µ ... . . . ... g ν · · · g so we see the emergence of a -dimensional scalar, vector, and two-index symmetrictensor. And it was in this context that compactification was originally studied as aunification of general relativity and electromagnetism.There’s another very important effect of the compactification. Let’s consider theEinstein Hilbert action S = 116 πG (4+ d ) N (cid:90) M × K R (cid:112) − g ( + d ) d d x (3.27)Noting that the Ricci scalar R ∼ ∂ always has mass dimension [ R ] = 2 , we see thatthe (4 + d ) -dimensional Newton’s constant must have (cid:104) G (4+ d ) N (cid:105) = 2 − D = − − d , so torewrite the action in natural units we must define a (4 + d ) -dimensional Planck mass as /G (4+ d ) N = ( M (4+ d ) pl ) d . If we take the metric to be independent of the K coordinates,then the integration over K just gives a factor of the volume V K of the compact space S = ( M (4+ d ) pl ) d π V K (cid:90) M R (cid:112) − g (4) d x (3.28)Putting this into the form of the four-dimensional Einstein-Hilbert action, we find M pl =( M (4+ d ) pl ) d V K . Taking V K (cid:39) R d for some radius R , we see that if the compact dimensionsare not Planck-sized R ∼ /M pl but larger for whatever reason, then the effective Planckmass at long distances can be much larger than the fundamental Planck mass.Now we don’t see zero-mode, different spin partners of our particles which fill outrepresentations of the higher-dimensional symmetry, so this simple compactification can-not be the real way the world is. If we have compact dimensions, they must be such that108 he Classic Strategies Chapter 3 the symmetry of zero-modes is somehow broken for the SM fields. This is important—we don’t just want to have different d fields with zero modes or not, we need differentfour-dimensional components of them to have or not have zero modes. We’ll solve thisproblem in Section 3.2.3. In the previous section we noticed that extra dimensions dilute the fundamental Planckmass in the higher-dimensional theory to produce a weaker effective four-dimensionalPlanck mass. So perhaps we can fix the hierarchy between the electroweak scale andPlanck scale by lowering the fundamental scale of quantum gravity. Arkani-Hamed,Dimopoulos, and Dvali proposed that the fundamental Planck scale can be M (4+ d ) pl ∼ TeVwith the weakness of four-dimensional gravity resulting from the dilution of gravitationalflux into the extra d dimensions [168]. Using (1 TeV ) (2+ d ) / (cid:39) ( M (4+ d ) pl ) (2+ d ) / (cid:39) M (4) pl /R d/ (3.29)where R is the radius of the extra dimensions, we find that a parsimonious d = 3 sphericaldimensions of radius R (cid:39) nm suffice to remove the hierarchy problem and accord withEöt-Wash constraints on the behavior of gravity at scales down to ∼ O ( µ m ) [169, 170,171].Now a nanometer is tiny compared to human scales, but the associated energy scaleis /R (cid:39) eV, which is a scale we have quite a bit of information about. In particular,if the Standard Model fields were to propagate in all d dimensions then it wouldbe easy to excite the ‘winding modes’ with momentum in the extra dimensions, and weshould have observed many finely-space Kaluza-Klein resonances with the same quantumnumbers. This is obviously not how the universe works, so to dilute gravity with largeextra dimensions one must trap Standard Model fields to the four-dimensional manifold109 he Classic Strategies Chapter 3 we know and love.This can be done by imagining we live on a (3 + 1) -dimensional topological defectwhich is embedded in the larger (4 + d ) -dimensional space. ‘Topological defect’ soundsexotic, but these are just (semi-)familiar non-perturbative objects such as the branes ofstring theory [172, 173, 174], or the cosmic strings or domain walls that can appear inHiggsed gauge theories and which are introduced well in Shifman’s textbook [158]. Theoriginal proposal suggests a weak-scale vortex in which zero-modes of our familiar fieldsare trapped, which is super cool.Note that the new physics appearing at the TeV scale in this scenario is about asviolent as you could imagine: quantum gravity appears at a TeV! This leads to a variety offascinating signatures and constraints, even in the absence of concrete model of quantumgravity, though embeddings into string theory have also been found [175, 176, 177]. Verygenerically there are corrections to the Newtonian gravitational laws [178], all sorts ofeffects on precision observables [179], a loss of flux of high energy particles into theambient space either astrophysically [180] or at a collider [144], violations of the globalsymmetries of the SM since quantum gravity does not respect them [181, 182, 183],and production of Kaluza-Klein gravitons [184] and black holes at TeV-scale colliders[185, 186, 187, 188]. This is a fascinating field which is well worth studying in detail, butunfortunately we do not have the space to do it justice. For more detail we refer to thereviews by Rubakov [189] and Maartens & Koyama [190], and the more introductory notesfrom Csáki [191], Kribs [192], Pérez-Lorenzana [193], Cheng [194], and Csáki, Hubisz, andMeade [195].However, shrewd readers will be eager to point out that we haven’t actually solvedthe hierarchy problem; we’ve merely traded the m H (cid:28) M pl hierarchy for a /R (2+ d ) / (cid:28) M (4+ d ) pl hierarchy. And indeed, it can be difficult to stabilize the size of the extra dimen-sions in this scheme. But the conceptual leap of considering geometric solutions to the110 he Classic Strategies Chapter 3 hierarchy problem is incredibly important and leads to many further interesting direc-tions. The next ingredient we need is to control the existence of opposite-spin partners. Let us consider a spacetime manifold M = M × K G with M four-dimensional Minkowskispace and K G a d -dimensional compact ‘orbifold’. An orbifold is constructed by ‘mod-ding out’ a manifold K by a discrete symmetry group G . A manifold is a space thatlooks locally like Euclidean space; an orbifold is one which locally looks like the quotientspace of Euclidean space quotiented by a finite group. In layman’s terms, quotientingor modding out is just identifying points which are transformed into each other under G —our space becomes the space of equivalence classes of K under the action of elementsof G .As an illustrative example, consider the line R and the action of the discrete symmetry Z : x (cid:55)→ − x . When we form the quotient space and identify points under this Z action,we find the half line R ≥ . You’ll notice that this space has a boundary at x = 0 , which wasa fixed point of the symmetry group. Mathematicians would say that the Z acts freelyexcept at this point. This is a generic feature, and in fact what makes orbifolds interestingfor our purposes. We could have also imagined ‘orbifolding’ R by the translation T (2 πR ) ,but this would have produced a compact manifold without boundary, S , because thissymmetry has no fixed points. Part of the power of orbifolds comes because the quotientgroup structure gives us some information about what happens at fixed points - you couldimagine just studying the half-line, but it seems natural to consider smooth structureson the manifold and then look at the effects under the identification. This should beevinced later in our examples.Compactifying on orbifolds also gives us a way to cure our missing partner ills. Con-111 he Classic Strategies Chapter 3 Figure 3.3: Schematic of the construction of an orbifold from the real line. The real linehas a Z (0) : x (cid:55)→ − x symmetry and a translational T (2 πR ) : x (cid:55)→ x + 2 πR symmetry.Modding out by the former yields an orbifold because it has a fixed point, while moddingout by the latter returns a regular manifold.Figure 3.4: Schematic of the construction of an orbifold from a circle. Quotienting by Z corresponds to folding the circle over on itself. We can produce independent fixed pointsby uplifting first to the real line, or equivalently by folding the circle into quarters.sider quotienting the circle by a Z which folds the circle over onto itself, y (cid:39) πR − y .This produces a line segment with boundaries at both ends y = 0 , πR . We can alter-natively think of this as modding out the real line by both translation and mirroring,producing R /T (2 πR ) × Z (0) . There are then two different sorts of fixed points— y = 0 is fixed by Z (0) , and y = πR is fixed by T (2 πR ) Z (0) ∼ Z ( πR ) . You can then en-vision this as the circle of length πR with these two discrete identifications, that is S (4 πR ) / Z (0) × Z ( πR ) , as depicted in Figure 3.4.How does this affect the resulting compactification? As a first step should take our112 he Classic Strategies Chapter 3 results and rewrite them in terms of eigenfunctions of our Z symmetry. We write φ ( x µ , y ) = 1 √ πR (cid:34) √ φ (+)0 ( x µ ) + + ∞ (cid:88) n =1 φ (+) n ( x µ ) cos ny/R + + ∞ (cid:88) n =1 φ ( − ) n ( x µ ) sin ny/R (cid:35) (3.30)where all we have done is rearrange things by defining φ (+)0 ≡ φ , φ (+) n> ≡ √ φ n + φ − n ) , φ ( − ) n> ≡ i √ φ n − φ − n ) , (3.31)where the superscript denotes their eigenvalue under the Z . Now the way to changeour S compactification to an S / Z compactification is to impose in our -dimensionalaction that φ transform with definite parity, which must be the case if the Z is a goodsymmetry . You’ll notice our free action does not demand a particular choice of parityfor φ , so we are free to choose φ either even or odd. But in either case we are forcedto get rid of half of our states. If φ is even we must set φ ( − ) n = 0 , and if φ is odd wemust set φ (+) n = 0 —including, importantly, getting rid of the zero mode. If we wantedto include certain interactions in the higher-dimensional theory, that could dictate thetransformation of φ —for example, the interaction term φ would necessitate an even φ .This is just the same as we’re familiar with in four dimensions.More interestingly, let us consider the effect of the orbifold on larger Lorentz repre-sentations. Imagine a -dimensional gauge field with free action S = − (cid:90) M F MN F MN (3.32)where F MN = ∂ M A N − ∂ N A M is the field strength. The mixed terms here read F µ = ∂ µ A − ∂ A µ , and this must transform coherently under the symmetry in order for thekinetic term to be invariant. But notice that Z : ∂ (cid:55)→ − ∂ . So A and A µ are forcedto have opposite transformations under the reflection symmetry! Thus in this example Note that since we’re getting our eigenmodes through this orbifold reduction our reduced action willstill need to be produced by integrating from y = 0 to y = 2 πR because it’s this domain over which oureigenfunctions are orthonormal. he Classic Strategies Chapter 3 one and only one of the four-dimensional vector and scalar multiplet has a masslesszero mode; so orbifold compactifications generally enable us to have light fields withoutpartners.For another example of the usefulness of orbifolding, consider a five dimensionaltheory with the minimal amount of supersymmetry and we want to ensure our four-dimensional theory has only N = 1 supersymmetry. First let’s recall why we cannothave more than N = 1 in four dimensions. As emphasized above, the SM is a chiraltheory, wherein the different chiral components of its Dirac fields are in different gaugesymmetry representations. This means that N = 2 supersymmetry in four dimensions istoo much for us, since the irreducible representations of super-Poincaré in that case don’tallow for chiral matter. In particular, the N = 2 vector multiplet must transform in theadjoint, and the N = 2 hypermultiplet must transform in a real representation in orderfor it to be CPT self-conjugate, which means the two Weyl fermions must transformin conjugate representations. Thus if we want to imagine that the world came from ahigher-dimensional supersymmetric theory, orbifold compactification is necessary.The same problem appears just for five dimensional spinor fields, since there is nosuch thing as chirality in odd spacetime dimensions. So under dimensional reduction,one five-dimensional spinor breaks into two conjugate four-dimensional spinors, and youcannot get chiral matter. This is really the same problem as the above, since the possiblesupersymmety comes exactly from the possible spinor representations.Compactifying a theory on an orbifold, rather than a manifold, will allow us to solveboth of these problems—obtaining chiral matter, and reducing the amount of supersym-metry we have. First note that we can build N = 2 supermultiplets out of two N = 1 supermultiplets, which is really just thinking about a particular ordering for the construc-tion of the supermultiplet by acting with supercharges. So for each N = 1 superfieldwe want to have, we need to arrange for it to be even under the parity, and so have114 he Classic Strategies Chapter 3 a zero-mode, and its partner superfield to be odd under the parity, and so only have n > Kaluza-Klein modes. In this way we get a set of zero modes which are chiral and N = 1 supersymmetric, while the towers reflect the full N = 2 supersymmetry and arenon-chiral.As an explicit example, consider an N = 2 vector multiplet, which consists of a realscalar Σ , two fermions ψ a with a = 1 , , and a vector A M . Under the SU (2) R symmetry,the scalar and vector are singlets and the fermions form a doublet. As in the exampleabove, the N = 2 Lagrangian for this multiplet dictates that the two N = 1 multipletsliving inside it transform differently under the Z . So on the four-dimensional boundaryone gets a zero-mode for either ( A µ , ψ ) or ( ψ , A + i Σ) , corresponding to a vectormultiplet or a chiral multiplet respectively. In slightly more group-theoretic language wecan say that we’ve embedded the Z in the SU (2) R , and have been forced by the physicsto put the doublet in a two-dimensional representation Z : ψ a → σ az b ψ b , σ z = − (3.33)and to pair each of these fermions with two bosonic degrees of freedom, Z : A µ →± A µ , A → ∓ A , Σ → ∓ Σ . This gives us either an N = 1 chiral multiplet or an N = 1 vector multiplet on the boundary. More discussion and details can be found in Quirós’TASI notes [196], and in e.g. [197, 198, 199].In fact orbifolding can do even more symmetry-breaking for us. As mentionedabove, we can alternatively consider compactification on an interval as the orbifold S (4 πR ) / ( Z (0) × Z ( πR )) , which then means the two boundaries are fixed points of independent Z symmetries. In particular this means we can choose different embed-dings of the Z in the full symmetry on the two ends y = 0 , πR [200, 201]. Now let’sapply this technology to the N = 2 case we considered above. We saw we had twodifferent choices for nontrivial embeddings of the N = 2 into the R-symmetry to break115 he Classic Strategies Chapter 3 the supersymmetry down to N = 1 at the boundary. These correspond to choosingwhich half of the fields are even under the Z and so get zero-modes. Now that we haveindependent Z s on either end, we can choose these to leave different N = 1 symmetriesunbroken.In a microscopic picture, what we end up with is a theory on R , × I [0 , πR ] wherethe bulk is ( N = 2 )-supersymmetric and the boundaries respect different N = 1 super-symmetries. When we look at the effective four-dimensional theory at distances muchlarger than R , we have fully broken supersymmetry with the breaking being a nonlocaleffect—one must be sensitive to physics on both boundaries in order to see the full break-ing. As a result of this nonlocality, supersymmetry-breaking is guaranteed to be ‘soft’and the effects cannot depend on positive powers of UV scales, as we will discuss furthermomentarily. This leads to fantastically predictive models of BSM physics from aroundthe turn of the millennium which really do read like they have it all figured out (see e.g.[202, 197, 203, 204, 205, 198, 206, 199]). It’s worth understanding these in some detailsimply because of how beautiful they are, but they uniformly lead to lots of (thus far)unseen structure near the TeV scale as KK partners become excited. Consider a single extra circular dimension of radius R with gravity and some gauge field.Upon restriction to the four-dimensional Lorentz group, the five-dimensional gravitonbreaks up into a four-dimensional graviton, vector, and scalar, while the five-dimensionalvector breaks up into a four-dimensional vector and scalar. Each of these must be masslessby five-dimensional gauge-invariance above the scale /R , but below that they can pickup mass corrections up to that cutoff. Either of these possibilities then amounts to amechanism for UV protection of a scalar mass. In implementing this in the SM, the first116 he Classic Strategies Chapter 3 possibility is known as the Higgs being a radion—the scalar which controls fluctuations ofthe size of the fifth dimension, h − δh = (cid:104) h (cid:105) = 1 /R , and the latter is denoted ‘gauge-Higgsunification’ for obvious reasons. A particular motivation for gauge-Higgs unification isas an extension of the strategy of grand unification. As mentioned in Section 1, the SMgauge bosons and fermions beautifully fit into representations of larger gauge groups, butin 4d GUTs the Higgs is left out in the cold as an extra puzzle piece. But ‘grand gauge-Higgs unification’ in higher dimensions may allow even further frugality of ingredients[207, 208, 209, 210].A very interesting feature of this sort of construction is that the symmetry-breakingwhich is responsible for producing the light scalar is nonlocal —one must traverse aroundthe fifth dimension to see the effects of the breaking. This should be intuitively clear, asat distances small compared to R the theory looks like five-dimensional Minkowski space.If it isn’t obvious, I recommend musing by analogy on how we could tell whether or notthe universe is a sphere with radius far, far larger than the Hubble scale R (cid:29) /H .As a result of this nonlocal symmetry-breaking, the scalar mass must be finite andcalculable in the low-energy theory below /R . There cannot be any sensitivity to ultra-violet energy scales Λ UV , as this corresponds to a ‘local counterterm’, but in the theoryabove /R we know that the mass vanishes identically by gauge invariance, so sucha counterterm cannot occur (recall our discussion in Section 1.2.1). This is powerfulbecause symmetry-breaking generally leaves residual logarithmic dependence on largescales even when quadratic dependence has been eliminated, as we saw in the exampleof supersymmetry above.Of course the theory can generate finite corrections to the mass of such a scalar atand below the scale /R . If we look only at the low-energy effective theory then theselook divergent, but we know they must get cut off at /R . Since we know the high-energytheory we can ask how the scalar gets a mass at all, which seems to be impossible from117 he Classic Strategies Chapter 3 gauge invariance. In fact the scalar mass comes from the Wilson loop wrapping the non-trivial cycle around the fifth dimension [211, 212, 213, 214, 215, 216, 217]. In any gaugetheory there is a gauge-invariant operator called a ‘Wilson loop’, W C = P exp ig (cid:73) C A M d x M , (3.34)where g is the gauge coupling, C is some closed path through spacetime, and P denotes‘path ordering’ of the operators along C in similarity to the time ordering in the definitionof the Feynman propagator. In an Abelian theory, the gauge field transforms as A µ → A µ − ∂ µ Γ( x ) and we see that (cid:72) C A µ d x µ → (cid:72) C A µ d x µ − (cid:72) C ∂ µ Γ d x µ . Integration by partsleaves us only with gauge transformations at the endpoints of C , of which there are noneif C is a loop (equivalently if we started more generally looking at a ‘Wilson line’, wecan say that for a loop the endpoints are connected and so their gauge transformationscancel each other). This remains true in non-Abelian gauge theories, though we do notgo through the proof.In our case we can consider a path which goes solely around the fifth dimension, inwhich case the Wilson loop contains only our 4d scalar A M d x M → A d x and yet is fullygauge-invariant. Quantum corrections can thus generate the operator L ⊃ σ Tr P exp ig (cid:73) A ( x µ , x ) d x + h.c. (3.35)We note that on a circle the four- and five-dimensional gauge couplings are related as g = Rg . Proceeding naïvely and expanding A ( x µ , x ) = √ πR + ∞ (cid:80) n = −∞ φ ( n ) ( x µ ) e inx /R ,we see that this operator includes a four-dimensional mass for the zero-mode, L ⊃ σ Tr P exp ig (cid:73) φ (0) ( x µ ) d x + h.c. (cid:39) σ cos (cid:104) g √ πR φ (0) (cid:105) ⊃ σ g R (cid:0) φ (0) (cid:1) , (3.36)where we note that the natural size for the Wilson coefficient σ (cid:39) /R yields a scalarmass expectation which is m φ (cid:39) g /R , unsurprisingly as R is the only scale in the118 he Classic Strategies Chapter 3 problem, and the dependence on the gauge coupling reveals the scalar’s five-dimensionalorigins.For a successful such model of gauge-Higgs unification, we need not only to get alight scalar but also to endow that scalar with dynamics pushing it to break electroweaksymmetry [218, 202, 219, 220, 221, 222, 223, 224, 225, 226]. There is far too much richphysics involved here to mention, but discussions of the calculation of the one-loop effec-tive potential in these models can be found in [227, 228, 229]. Gauge symmetry-breakingby a higher-dimensional gauge field component getting a vev is sometimes referred to asthe ‘Hosotani mechanism’ [215, 216, 217].As opposed to the Large Extra Dimensional model of the previous section, in thesemodels the extra dimensions are ‘universal’—all of the SM fields can propagate aroundthe small dimension, not just gravitons. This means that all of our familiar fields are thezero modes of KK towers with spacing ∼ /R ∼ m H . These KK partners have the samegauge charges, and so should be produced copiously in interactions with √ s (cid:29) /R , yetnone have been observed at the LHC (see e.g. [230, 231, 232, 233] for some constraints).As a result, the lower bound on the KK scale is now far above the Higgs mass, whichmakes all of these sorts of models increasingly less attractive as solutions to the hierarchyproblem. 119 he Classic Strategies Chapter 3 We have saved for last what is, in some sense, the most obvious strategy to pursue. Asdiscussed in Section 1.3, the Standard Model already breaks a symmetry and generates amass scale in a natural manner with the chiral condensate of QCD. Perhaps Nature hasonly this one trick, and repeats the same mechanism to generate the weak scale. Afterall, we mentioned above that the QCD condensate does in fact break the electroweaksymmetry, just not at the right scale.
The idea is to introduce a new gauge sector which is asymptotically free and so becomesstrong and confines at the electroweak scale. If the condensate has electroweak quantumnumbers, then it breaks electroweak symmetry just as a Higgs field would, but now with-out any Higgs. This strategy is known as ‘technicolor’, and was proposed in its simplestform by Weinberg [234] and Susskind [235]. A modern pedagogical introduction can befound in TASI lectures from Chivukula [236] or Contino [237], which has heavily influ-enced this discussion, and a more detailed classic review is from Hill & Simmons [238].We introduce a technicolor sector which is an SU ( N T C ) gauge group with N D techni-color fundamentals which are electroweak doublets and their singlet partners, togetherenjoying a global SU (2) L × SU (2) R symmetry which is broken in the infrared to SU (2) V by confinement in the SU ( N T C ) sector at Λ T C ∼ v . This structure exactly matches thatof the SM QCD sector (save for the values of N T C , N D ), so we expect all the same phe-nomenology, for example with composite technipions appearing close to the electroweakscale. But in this case since the technicolor chiral condensate is the leading breakingof the gauged SU (2) L , the technipions will be predominantly ‘eaten’ and appear as the120 he Classic Strategies Chapter 3 longitudinal polarizations of our W, Z bosons. Technicolor can ‘Higgs’ the electroweakgauge symmetry just like a fundamental Higgs field would, but without having any scaleat which it looks like there is a scalar field breaking electroweak symmetry.However, the SM Higgs not only breaks electroweak symmetry but also gives mass tothe quarks, and thus far we haven’t introduced any coupling between the quarks and thetechnicolor sector. To get the appropriate couplings we can embed both of these gaugegroups in a larger ‘extended technicolor’ group, SU ( N ET C ) ⊃ SU (3) c × SU ( N T C ) . Afterthis extended group undergoes spontaneous symmetry breaking at Λ ET C , the brokengauge bosons generate the appropriate four-Fermi interactions
L ⊃ g ET C Λ ET C (¯ qq )( ¯ ψ T C ψ T C ) + O (cid:18) ∂ Λ ET C (cid:19) , (3.37)and then when the technicolor group confines at a scale Λ T C , we see the emergence ofquark masses m q (cid:39) g ET C Λ ET C (cid:10) ¯ ψ T C ψ T C (cid:11) ∼ Λ T C (cid:18) Λ T C Λ ET C (cid:19) . (3.38)But this clearly suffices solely to generate a single quark mass scale, since the Yukawacoupling is originating from a single gauge coupling, so it was quickly realized thataccounting for flavor physics required far more structure and significantly larger gaugegroups [239, 240, 241, 242]. To generate the variety of quark mass scales with thismechanism would require a cascade of breakings from SU ( N ET C ) down to SU (3) c × SU ( N T C ) . Furthermore the same massive gauge boson exchanges which generate thoseneeded four-Fermi operators also generate four-quark interactions (¯ qq )(¯ qq ) which can leadto large flavor violation.While there were insights on how various aspects of this could be tackled, the death-knell for technicolor came with Peskin & Takeuchi’s parametrization of oblique correctionsto the two-point functions of the electroweak vector bosons from BSM physics [243].These efficiently parametrize deviations from the tree-level form of the vector boson121 he Classic Strategies Chapter 3 propagators, and can be simply connected with experiment. Ensuing estimates for thesizes of these parameters in strongly-interacting models were very far from empiricalmeasurements [244]. The program of technicolor lives on with ‘walking technicolor’, theidea that the confining dynamics may be due to a strongly-coupled gauge theory whichbehaves very differently from QCD [245, 246, 247, 248, 249, 250, 251]. This is too largea digression for us to introduce, but we mention that there is interesting recent workrelating the existence of ‘walking’ dynamics to proximity of the theory to a fixed pointat complex value of the coupling [252, 253, 254, 255, 256]. However, there is another way that compositeness can be useful for us in securing alight electroweak-symmetry-breaking scalar: It will allow us to realize the dream of apseudo-Nambu-Goldstone Higgs. Recall that whenever a continuous global symmetryis spontaneously broken, there appear scalar Goldstone bosons π i ( x ) which parametrizeexcitations about the vacuum state in the direction of the broken generators. Since thetheory had a global symmetry, the potential along these directions is flat, and thus theGoldstones are massless. More strongly, they contain a shift symmetry: L ( π i ( x )) = L ( π i ( x ) + ξ ) (3.39)where ξ is independent of x . This means they may only be ‘derivatively coupled’; theymay appear in the Lagrangian solely as ∂ µ π i ( x ) . Thus, a non-zero mass for such a scalaris technically natural—if such an operator is present, π i is said to be a pseudo -Goldstoneand corresponds to the breaking of an approximate symmetry. This is a familiar storyin the context of the QCD condensate breaking the approximate chiral symmetry of thequarks, leading to light but not massless pions.In the case of technicolor, the phenomenon of confinement of a strongly interacting122 he Classic Strategies Chapter 3 sector is directly responsible for breaking the gauged electroweak symmetry, leading to noseparation between the two scales Λ T C and v . This means the technipions are immediatelyeaten, and there is no scale at which it looks like a scalar field is responsible for symmetry-breaking, which leads to large electroweak precision constraints. In a general compositeHiggs scenario, we’ll attempt to arrange for separation between the scale of confinementand the scale of electroweak symmetry breaking by having confinement break an enhanced global symmetry, leading to pseudo-Nambu Goldstone bosons whose masses are protected.We then want to radiatively generate a potential for these pNGBs, leading to them gettinga vev and breaking electroweak symmetry at a lower scale. A degree of separation betweenconfinement at the scale f and EWSB at v will reduce the difficulties with electroweakprecision constraints, as this scenario returns to the SM elementary Higgs sector in thelimit v/f → . More pressingly, now that we have gained experimental access to energiesclose to v it’s even more clear that scale separation is needed—confining dynamics leadgenerically to lots of resonances at the scale f , which would be seen in all sorts of ways.The general setup is a group G of (approximate) global symmetries, of which a sub-group H is gauged. We will have strong dynamics at the scale f break the globalsymmetry to H , and in full generality allow for some part of the gauge symmetries to bebroken, such that the unbroken gauge symmetry is H = H ∩ H . This leads to [ G ] − [ H ] Goldstone bosons (where [ · ] is the dimension of a group) of which [ H ] − [ H ] are eaten bythe broken gauge bosons. The uneaten, light pNGBs transform non-trivially under theremaining gauge symmetry H ⊃ SU (2) L × U (1) Y , so we can hope to arrange for themto break this symmetry. In a realistic minimal model, we can have the strong dynamicsnot break any gauge symmetry, H ⊂ H ⇒ H = H . Such a minimal model may beconstructed with G = SO (5) × U (1) → H = SO (4) × U (1) ⊃ H = SU (2) L × U (1) Y [257]. We diagram the general structure in Figure 3.5a and the minimal model in 3.5b.To evince the ideas in the simplest scenario possible, we’ll discuss an even simpler123 he Classic Strategies Chapter 3 (a) (b) (c) Figure 3.5: In (a), the general symmetry setup for a composite Higgs model. Solid linesare the UV symmetries, with the global symmetry enclosed in the smooth circle and thegauge symmetry in the loopy circle. The symmetry group after confinement is dashed.In (b), the same structure applied to the ‘minimal’ model of [257]. In (c), the compositeAbelian Higgs toy model discussed in [258], where the symmetry group after confinementcoincides with the gauge group. 124 he Classic Strategies Chapter 3 model for a composite pNGB which then breaks a U (1) gauge symmetry. This is just atoy model to understand the features, which has already been kindly worked out in theextensive review from Panico & Wulzer [258], and which we’ll call a ‘composite AbelianHiggs’ model. To find the most minimal model we’ll ask for a composite pNGB whichbreaks the smallest continuous gauge symmetry, U (1) , for which we simply need twouneaten degrees of freedom (since a charged scalar is necessarily complex). We canmake the even more minimal choice H = H = H , as we’re not worried about havingadditional unbroken global symmetries. Since [ U (1)] = 1 , we then just need to choosea group G with at least [ G ] = 3 to get the right number of pNGBs. We’ll study thebreaking SO (3) → SO (2) (cid:39) U (1) , which is especially nice because we have geometricintuition for the Lie algebras of these groups.We’ll study this using a ‘linear sigma model’, of which the ‘chiral Lagrangian’ de-scribing the QCD pions is the most familiar example. Much of the general technologywas developed by Callan, Coleman, Wess & Zumino [259, 260], and some modern intro-ductions to this technology can be found in Schwartz’ textbook [22], in a pedagogicalreview of Little Higgs models by Schmaltz & Tucker-Smith [261], and in exhaustive detailin the review by Scherer [262]. The big idea is one of bottom-up effective field theory:Given knowledge of the symmetry-breaking structure, we can cleverly parametrize ourfields to easily see the structure of the Lagrangian which is demanded both before andafter symmetry-breaking.In our case we must start with an SO (3) -invariant Lagrangian of a fundamental field (cid:126) Φ , which is a familiar d vector. −L = 12 ∂ µ (cid:126) Φ (cid:124) ∂ µ (cid:126) Φ + g (cid:63) (cid:16) (cid:126) Φ (cid:124) (cid:126) Φ − f (cid:17) , (3.40)where SO (3) rotations act as (cid:126) Φ → g · (cid:126) Φ with g = exp iα A T A , where T A , A = 1 .. arethe generators of SO (3) . The potential of (cid:126) Φ is minimized for (cid:68)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:69) = f , so (cid:126) Φ gets125 he Classic Strategies Chapter 3 a nonzero vev and the symmetry is broken down to rotations keeping (cid:68) (cid:126) Φ (cid:69) fixed, whichis simply U (1) . There are a two-sphere worth of vacua corresponding to possible anglesfor (cid:68) (cid:126) Φ (cid:69) , which parametrize the Goldstone directions. We can make the split betweenbroken and unbroken generators explicit by parametrizing our field as (cid:126) Φ = exp (cid:32) i √ f Π i ( x ) ˆ T i (cid:33) f + σ ( x ) = ( f + σ ) sin (cid:16) Π f (cid:17) (cid:126) ΠΠ cos (cid:16) Π f (cid:17) (3.41)where in the first equality ˆ T i are the two broken generators, Π i ( x ) are the masslessGoldstones corresponding to fluctuations along the vacuum manifold, and σ ( x ) is themassive ‘radial mode’ giving fluctuations about the vev. We eschew writing down thegenerators explicitly and assert that in this case one finds the compact latter expression,with Π = (cid:112) (cid:126) Π (cid:124) (cid:126) Π . We can find an explicit expression for the interaction of the Goldstonesand the radial modes by simply plugging this parametrization into the Lagrangian above.We indeed find a mass for σ of m σ = g (cid:63) f , massless pions (cid:126) Π and a tower of all possibleinteractions between these fields which are consistent with the symmetries.We now have a theory of massless scalars transforming under an unbroken symmetry;our pions form a doublet of SO (2) transforming as (cid:126) Π → exp ( iασ ) (cid:126) Π , correspondingto rotations about the unbroken SO (3) generator. We can complexify by introducing H ≡ √ (Π − i Π ) to view SO (2) (cid:39) U (1) as acting H → exp ( iα ) H . As it stands, H is an exact Goldstone boson, so cannot pick up a potential to then itself get a vevand break U (1) . However, when we gauge a U (1) subgroup of our original SO (3) globalsymmetry we’re introducing explicit breaking of the symmetry and resultingly H becomesa pNGB and can pick up a mass. The tree-level effect of this gauging is simply to upgradederivatives to gauge covariant derivatives, ∂ µ (cid:126) Π → D µ (cid:126) Π = ( ∂ µ − ieA µ σ ) (cid:126) Π .Now as a result of SO (3) rotations no longer being an exact symmetry, H is free126 he Classic Strategies Chapter 3 Figure 3.6: Representative diagrams contributing to the one-loop effective potential fora charged scalar.to pick up a potential, which in general will be radiatively generated as a result of thisgauging. It is easy to draw diagrams in which loops of our U (1) gauge bosons generate anonzero mass and quartic for H , as in Figure 3.6, and in a realistic model the correctionsfrom loops of top quarks will be especially important. These loop diagrams come alongwith an obvious cutoff: Above the compositeness scale f , the fields (cid:126) Φ no longer exist inthe spectrum. There is a beautiful general method for deriving the radiatively-generatedpotential for H by resumming the one-loop diagrams with various numbers of external H legs, known as the Coleman-Weinberg potential [263].Unfortunately, absent any of other structure, finding v/f (cid:28) still requires somedegree of cancellation between various contributions to the potential of H . However anyamount of v/f < will help alleviate the pressure from electroweak precision observableswhich thus far look empirically as expected for an elementary Higgs field, while stillretaining the benefit of forbidding corrections to the Higgs mass above f . But one stillcan’t get away from requiring lots of structure near the electroweak scale, which has not(yet) been observed.There are a variety of important aspects and interesting directions we do not have thespace to discuss. Even past understanding the best sorts of group structures to whichto apply this strategy, it is clearly important to understand the sorts of field theorieswhich can confine to break G → H , as well as the detailed structure of the potentialradiatively-generated by SM fields. Then it’s important to explore the possibility of a127 he Classic Strategies Chapter 3 natural structure which dictates v/f < —while there has been much work on this, wemention in particular the interesting strategy of ‘collective’ symmetry breaking in whicha symmetry is broken only by an interplay between different couplings. This class ofmodels is known as the ‘Little Higgs’ [261, 264, 265, 266, 267, 268], and can be seenas a purely four-dimensional application of the strategy of nonlocal symmetry breakingthrough ‘nonlocality in theory space’ [269, 270], which is a fascinating topic. The TASInotes by Csaki, Lombardo, Telem [271] provide a pedagogical introduction to these topics.Finally, let me mention that composite Higgs models may be understood as beingdual to a novel class of extra-dimensional models known as Randall-Sundrum models[272, 273]. Unlike in the simple cases we discussed in Section 3.2, in these models thespacetime does not have a product structure (as did M × K ) and the geometry is saidto be ‘non-factorizable’. In this scenario our four-dimensional universe is seen as a braneliving on one end of a five-dimensional orbifold of anti-de Sitter space. The minuteness ofthe electroweak scale compared to the Planck scale is a result of a large fifth-dimensionalAdS ‘warp factor’ between the brane we live on (the ‘IR brane’) and the brane on theother end of the space (the ‘UV brane’). As in Section 3.2.2, this trades the electroweakhierarchy into a geometric hierarchy, but now in AdS we can find novel, natural ways ofgenerating such a hierarchy of scales [274, 275]. Furthermore, embedding our universeinto an AdS spacetime means we can take advantage of the enormously powerful programof AdS/CFT holography, which enables us to study the strong-coupling phenomena ofcomposite Higgs models via their weakly-coupled gravitational duals. Pedagogical intro-ductions to holography can be found in lecture notes from Sundrum [276] and Kaplan[277], and with more background in the textbook by Ammon & Erdmenger [278]. Thatmachinery is not all necessary to appreciate the workings of Randall-Sundrum models,though, and there are a wide variety of great lectures notes aimed at particle theorists,for example those of Sundrum [279], Csaki & Tanedo [280], and Gherghetta [281].128 hapter 4The Loerarchy Problem The great tragedy of science — the slaying of abeautiful hypothesis by an ugly fact.Thomas Henry Huxley
Biogenesis and Abiogenesis (1870) [282] he Loerarchy Problem Chapter 4
We’ve seen in Chapter 3 a cadre of theories which can produce a light scalar naturally,and there’s one feature all the classic approaches have in common: they predict newstates with Standard Model charges close to the mass of the Higgs. This is seeminglyinevitable simply from the structure of effective field theory—whatever extra structureprotects the Higgs mass at UV scales must be broken close to the electroweak scale toallow the Standard Model, which does not have that extra structure. This feature meansthat smashing together protons at scales much greater than the electroweak scale wouldsurely reveal the physics of whatever mechanism solves the hierarchy problem. And sothe Large Hadron Collider was eagerly awaited to tell us which of these ideas was correct.Yet even years before the LHC turned on, those who could clearly read the tealeaves were realizing that something was amiss with our naturalness expectations (seee.g. the ‘LEP paradox’ [283], also [284]), and exploring the idea that supersymmetrywould not show up to solve the hierarchy problem (e.g. ‘split supersymmetry’ [285, 286,287]). Perhaps there was something else present which made weak-scale supersymmetryunnecessary for protecting the Higgs.Of course the LHC has been a fantastic success. It has confirmed for us the existenceof a light Higgs resonance that looks SM-like, and made many great measurements of theSM besides. But rather than revealing to us which TeV-scale new physics kept the Higgslight, we’ve instead had a march of increasingly powerful constraints on new particleswhich couple to the Standard Model.These null results for physics beyond the Standard Model from run 1 of the LHCrapidly popularized the idea that something else might be responsible for stabilizing theHiggs mass at the electroweak scale up to a higher scale where, say, supersymmetry camein. This line of thinking is termed the ‘Little Hierarchy Problem’—the idea being that130 he Loerarchy Problem Chapter 4 one of those classic solutions would appear at Λ ∼ TeV to solve the ‘Big HierarchyProblem’ and explain why the Higgs mass was not at the Planck scale, leaving a smallerhierarchy of a couple orders of magnitude between m H and Λ unexplained. Perhapsrather than minimal supersymmetry there was another module which provided this lastbit of protection. But this has to be a special module to protect the Higgs mass withoutintroducing new colored particles. The first such proposal in fact appeared before the LHC had even turned on. The mirrortwin Higgs (MTH) [288] introduces a second copy of the SM gauge group and statesrelated to ours by a Z symmetry. Since these ‘twin’ states are neutral under the SMgauge group, they are subject only to indirect bounds from precision Higgs couplingmeasurements. The two sectors are connected solely by Higgs portal-type interactionsbetween the two SU (2) doublet scalars. Subject to conditions on the quartic coupling,the Higgs sector enjoys an approximate SU (4) global symmetry, and the breaking of thissymmetry leads naturally to a pseudo-Nambu Goldstone boson. Seemingly magically, thisstructure is accidentally respected by the quadratically-divergent one-loop corrections tothe Higgs potential, and the pNGB continues to be protected through one-loop fromlarge corrections to its mass. The Twin Higgs thus allows the postponement of a solutionto the ‘Big Hierarchy Problem’ until scales a loop factor π ∼ O (100) above the Higgsmass.While the space of Neutral Naturalness models has now been explored more thor-oughly and we will discuss some generalities below, the mirror twin Higgs remains per-haps the most aesthetically pleasing of all these approaches and serves as a useful avatar We return in Section 5.3 to the prospect of kinetic mixing between the two U (1) Y factors, which isalso allowed by the symmetries. he Loerarchy Problem Chapter 4 for this general strategy. As a result, in Chapter 5 we consider cosmological signaturesof the MTH specifically, so we give here a more-detailed introduction to the twin Higgsin particular. This is necessarily slightly more technical than the rest of this chapter, sothe reader who is not planning on reading Chapters 5 or 6 in detail may skip ahead ∼ pages to Section 4.1.2 without loss of continuity.The scalar potential in this model is best organized in terms of the accidental SU (4) symmetry involving the SU (2) Higgs doublets of the SM and twin sectors, H A and H B .The general tree-level twin Higgs potential is given by (see e.g. [289]) V ( H A , H B ) = λ ( | H A | + | H B | − f / + κ ( | H A | + | H B | ) + σf | H A | . (4.1)The first term respects the accidental SU (4) global symmetry, as can be seen by writingit in terms of H = ( H A , H B ) (cid:124) , which transforms as a complex SU (4) fundamental. Thesecond term breaks SU (4) but preserves the Z and the final term softly breaks the Z .In order for the SU (4) to be a good symmetry of the potential, we require κ, σ (cid:28) λ .However, the gauging of an SU (2) × SU (2) subgroup constitutes explicit breakingof the SU (4) , so we should worry about whether quantum corrections reintroduce largemasses for the would-be Goldstones when SU (4) is broken. But writing down the one-loop corrections reveals a fortuitous accidental symmetry. The one-loop effective scalarpotential gets the following leading corrections from the gauge bosons at the quadraticlevel: V − loop ( H A , H B ) ⊃ g ,A π Λ | H A | + 9 g ,B π Λ | H B | + subleading → Z g π Λ |H| , (4.2)where we see explicitly that if the Z symmetry is good at the level of the gauge couplings,then these largest one-loop corrections continue to respect the SU (4) . It is easy to seefrom here by power counting that this holds for all the quadratically-divergent pieces solong as the Z is a good symmetry for the interactions involved.132 he Loerarchy Problem Chapter 4 There is radiative SU (4) -breaking at the level of the quartic, since the Z symmetryno longer suffices to form the Higgses into an SU (4) invariant. The coupling κ shouldnaturally be of the order of these corrections, the largest of which comes from the Yukawainteractions with the top/twin top, κ ∼ y t / (8 π ) log(Λ /m t ) ∼ . for a cut-off Λ ∼ TeV ( y t being the top quark Yukawa coupling and m t its mass). Requiring λ (cid:29) κ there-fore implies λ (cid:38) . As the SM and twin isospin gauge groups are disjoint subgroups ofthe SU (4) , the spontaneous breaking of the SU (4) coincides with the SM and twin elec-troweak symmetry breaking. This gives seven Goldstone bosons, six of which are ‘eaten’by the SU (2) gauge bosons of the two sectors, which leaves one Goldstone remaining.This will acquire mass through the breaking of the SU (4) that is naturally smaller thanthe twin scale f . For future reference, it is convenient to define the real scalar degreesof freedom in the gauge basis as h A = √ (cid:60) ( H A ) − v A and h B = √ (cid:60) ( H B ) − v B , where (cid:104) H A (cid:105) = v A and (cid:104) H B (cid:105) = v B .The surviving Goldstone boson should be dominantly composed of the h A gaugeeigenstate in order to be SM-like. The soft Z -breaking coupling σ is required to tunethe potential so that the vacuum expectation values (vevs) are asymmetric and that theGoldstone is mostly aligned with the h A field direction. The (unique) minimum of theTwin Higgs potential (4.1) occurs at v A ≈ f (cid:113) λ ( κ − σ ) − κσλκ and v B ≈ f (cid:113) σ + κκ . The requiredalignment of the vacuum in the H B direction occurs if σ ≈ κ , which has been assumed inthese expressions for the minimum. The consequences of this are that v A ≈ v/ √ and v B ≈ f / √ (cid:29) v (where v is the vev of the SM Higgs, although v A ≈ GeV is the vevthat determines the SM particle masses and electroweak properties), so that the SM-likeHiggs h is identified with the Goldstone mode and is naturally lighter than the otherremaining real scalar, a radial mode H whose mass is set by the scale f . The componentof h in the h B gauge eigenstate is δ hB ≈ v/f (to lowest order in v/f ). Measurementsof the Higgs couplings restrict f (cid:38) v [290, 289], and the Giudice-Barbieri tuning of the133 he Loerarchy Problem Chapter 4 weak scale associated with this asymmetry is of order f / v .The spectrum of states in the broken phase consists of a SM-like pseudo-GoldstoneHiggs h of mass m h ∼ κv , a radial twin Higgs mode H of mass m H ∼ λf , aconventional Standard Model sector of gauge bosons and fermions and a correspondingmirror sector. The masses of quarks, gauge bosons, and charged leptons in the twinsector are larger than their Standard Model counterparts by ∼ f /v , while the twin QCDscale is larger by a factor ∼ (1 + log( f /v )) due to the impact of the higher mass scaleof heavy twin quarks on the renormalisation group (RG) evolution of the twin strongcoupling. The relative mass of twin neutrinos depends on the origin of neutrino masses,some possibilities being ∼ f /v for Dirac masses and ∼ f /v for Majorana masses fromthe Weinberg operator. Mixing in the scalar sector implies that the SM-like Higgs couplesto twin sector matter with an O ( v/f ) mixing angle, as does the radial twin Higgs mode toStandard Model matter. These mixings provide the primary portal between the StandardModel and twin sectors.The Goldstone Higgs is protected from radiative corrections from Z -symmetric physicsabove the scale f . While the mirror Twin Higgs addresses the little hierarchy problem,it does not address the big hierarchy problem, as nothing stabilizes the scale f againstradiative corrections. However, the scale f can be stabilized by supersymmetry, compos-iteness, or perhaps additional copies of the twin mechanism [291] without requiring newstates beneath the TeV scale. Minimal supersymmetric UV completions can furthermoreremain perturbative up to the GUT scale [292, 293].As mentioned, the collider constraints on twin Higgs models are very mild and pertainmostly to a lower bound on the soft breaking of the Z . In this respect, the Twin Higgsnaturally reconciles the observation of a light Higgs with the absence of evidence for newphysics thus far at the LHC. The primary challenge to these models comes from cosmologydue to the effects of additional light particles on the cosmic microwave background. We134 he Loerarchy Problem Chapter 4 will discuss these issues in Chapter 5 and propose a natural resolution. More broadly, the twin Higgs is just the simplest example of the more general ‘NeutralNaturalness’ paradigm in which the states responsible for stabilizing the electroweak scaleare not charged under (some of) the Standard Model (SM) gauge symmetries [288, 294,295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305], thus explaining the lack of expectedsignposts of naturalness.In the symmetry-based solutions to the hierarchy problem discussed in Chapter 3,modifications of the Higgs mass were technically natural as a result of a continuoussymmetry which commuted with the SM gauge groups. This naïvely seems necessary toensure that the necessary degrees of freedom are present and couple to the Higgs with theright strength to cancel divergences. How is the top quark contribution ∝ N c y t (where N c is the number of colors) to be canceled if not by another colored particle with the samecoupling to the Higgs, which gives the opposite contribution? Well we saw above thatthe twin Higgs nevertheless works with a quark charged under a separate gauge group.Indeed, at one-loop N c is really just a ‘counting factor’ and we are free to get those threeopposite-sign contributions in a variety of ways. To some extent the space of NeutralNaturalness models is an exercise in interesting ways to find that color factor. We’ll beable to see that picture more clearly in the language of orbifold projection, which willalso give us good reason to expect that these models with naïvely strange symmetries infact do have nice UV completions.In Section 3.2.3 we saw how orbifolds could be useful dynamically—that is, in af-fording a higher-dimensional, symmetric theory which at low energies looks like a lesssymmetric, four-dimensional theory due to boundary conditions imposed by the orbifold135 he Loerarchy Problem Chapter 4 discrete symmetries. But these theories had interesting properties in their low-energybehavior below the scale of compactification; we didn’t need to make reference to theirorigins in studying them.Now we want to understand the variety of low energy theories we can get very gener-ally, but only at the level of the zero-mode spectrum. Rather than decomposing our fieldsinto modes and integrating over the compact manifold and noticing that only those fieldsinvariant under the orbifold symmetry are left with zero-modes, we’re going to skip tothe answer and look at the spectrum of fields left invariant under our discrete symmetry.We’ll find that these theories have enhanced symmetry properties at one loop.The underlying reason lies in the ‘orbifold correspondence’ in large N gauge theories[306, 307, 308]. Given a ‘mother’ field theory, you can create a ‘daughter’ theory byembedding some given discrete symmetry in the symmetries of the mother theory andprojecting out states which are not invariant under that discrete symmetry; we callthis process ‘orbifolding’. Then at leading order in large N , the correlation functionsof the daughter theory match those of the mother theory. This is nothing short ofamazing—a theory with no supersymmetry to speak of can nevertheless ‘accidentally’exhibit supersymmetric behavior at leading order.The general case of the orbifold correspondence and how to construct Neutral Nat-uralness models is beautiful and I do recommend reading [306, 295, 309], but the grouptheory required for a full discussion would be too large a detour from our main narrative.Fortunately we can get a good sense for what’s going on by considering a few explicitexamples, which will not require much mathematical machinery. Example 1: Folded Supersymmetry
Let’s first consider the example of ‘Folded Supersymmetry’ [295] which was the firstNeutral Naturalness model constructed explicitly via orbifolding. The idea is precisely136 he Loerarchy Problem Chapter 4 to consider a supersymmetric theory and orbifold project onto a theory with no explicitsupersymmetry but in which supersymmetric cancellations still occur at one loop. Asa pedagogical example, consider an N = 1 supersymmetric U (2 N ) C gauge theory with N flavors of left and right fundamental chiral superfields Q = (˜ q, q ) which enjoy a U (2 N ) F,L × U (2 N ) F,R global flavor symmetry. Let’s decree also that the theory respects R -symmetry. We will orbifold by the discrete group Z as before, but we must choose howto embed the Z in each of these symmetry groups. That is, our original ‘mother’ theoryis invariant under a large symmetry group U (2 N ) C × U (2 N ) F,L × U (2 N ) F,R × U (1) R ,which contains many Z subgroups, and we must decide precisely which Z we want toorbifold by. We choose the following embeddings: C = N − N , F = N − N , R = ( − F , (4.3)where the first matrix is in color-space, the second is in flavor-space, and the thirdtransformation is by fermion number. Each of these generates a Z subgroup of one ofthe symmetry factors of the mother theory. To see which fields are invariant under this Z , we must only act them on the fields and see how they transform. Gauge bosons A µ and their superpartner gauginos λ are in the adjoint of the gauge group. Indexing A, B = 1 ..N separately over the two halves of the gauge indices, we have A µ = A µ,AA A µ,AB A µ,BA A µ,BB → C A µ,AA A µ,AB A µ,BA A µ,BB C † R = + A µ,AA − A µ,AB − A µ,BA + A µ,BB (4.4) λ = λ AA λ AB λ BA λ BB → C λ AA λ AB λ BA λ BB C † ( − ) R = − λ AA + λ AB + λ BA − λ BB (4.5)where the difference here is because of the different R transformations. We see thatif we project down to only those states invariant under this transformation, our gaugegroup dissolves from U (2 N ) C down to disconnected pieces U ( N ) C × U ( N ) C . We see that137 he Loerarchy Problem Chapter 4 embedding the Z non-trivially in the R -symmetry group means the daughter theory willbe non-supersymmetric. The superpartners of our gauge fields are no longer present, butrather the gauginos have been twisted into bifundamentals under the two gauge factors.In the matter sector, letting a, b = 1 ..N similarly index the two halves of the flavor indicesand writing the fields as matrices in a combined color and flavor space, we have ˜ q = ˜ q Aa ˜ q Ab ˜ q Ba ˜ q Bb → C ˜ q Aa ˜ q Ab ˜ q Ba ˜ q Bb F † R = +˜ q Aa − ˜ q Ab − ˜ q Ba +˜ q Bb (4.6) q = q Aa q Ab q Ba q Bb → C q Aa q Ab q Ba q Bb F † ( − ) R = − q Aa + q Ab + q Ba − q Bb . (4.7)Again we see that we have broken supersymmetry. The flavor group has also broken downto U ( N ) F,L × U ( N ) F,L and the invariant states are squarks which are bifundamentalsunder ‘diagonal’ combinations of the gauge and flavor groups, and quarks which arebifundamentals under the ‘off-diagonal’ combinations.It is not too hard to roughly see the magic of how the orbifold-projected theorycontinues to protect scalar masses. Draw the one-loop the diagrams in the mother theorywhich would contribute to a calculation of the mass of, say, ˜ q Aa , as in Figure 4.2. Inthe mother theory we know there are no quadratic corrections by supersymmetry. In thedaughter theory, half of each sort of diagram will be eliminated by the orbifold projection,so it’s clear that there will still be no quadratic divergences. But because it is differentinternal states that have been eliminated for different classes of diagrams, the daughtertheory has no supersymmetry to speak of! Again, as we saw in the example of the twinHiggs, the magic is in that at one-loop we really only need to get the counting right, andso we can use orbifolding to construct theories which do that in clever ways.The structure of the theory can be succinctly summarized in a ‘quiver’ or ‘moose’diagram where the various symmetry groups correspond to nodes in a graph and the138 he Loerarchy Problem Chapter 4 (a) (b) Figure 4.1: In (a), a quiver diagram for the (left-chiral fields in) the mother theory. Acircle corresponds to a gauge symmetry and a square to a flavor symmetry. Lines betweendenote bifundamentals, which here represents a full chiral supermultiplet. In (b), a quiverdiagram for the orbifold daughter theory described in the text. The gauge and flavorgroups have been divided into two distinct groups each, and the degrees of freedom inthe chiral and vector supermultiplet have been shuffled around.Figure 4.2: Representative diagrams contribution to the one-loop mass correction of ascalar in the supersymmetric mother theory. The daughter theory projects out someof the internal states of each diagram in a pattern that is non-supersymmetric but stillenforces cancellations. 139 he Loerarchy Problem Chapter 4 matter is represented by links between these groups which represent their charges, as inFigure 4.1.Despite the fact that our daughter theory has no supersymmetry, the orbifold cor-respondence guarantees that in the N → ∞ limit the correlation functions have fullsupersymmetric protection. For finite N the supersymmetric relations are broken, butonly by /N corrections. Going through this explicitly is useful, and pedagogical discus-sions of it can be found in [306, 298]. Example 2: The Γ -plet Higgs We can understand the twin Higgs as the simplest example of a Γ -siblings Higgs whichis the orbifold projection SU (3Γ) × SU (2Γ) × SU (Γ) F / Z Γ , where the first two factorsare gauged and the last is a flavor symmetry. We focus on the sector which containsthe Yukawa interaction of the top quark, as the top partners have the most effect on thenaturalness of the Higgs. In the mother theory the matter content is: SU (3Γ) SU (2Γ) SU (Γ) F H - (cid:3) (cid:3) Q (cid:3) (cid:3) - u (cid:3) - (cid:3) where ‘ (cid:3) ’ denotes the fundamental representation and ‘ (cid:3) ’ denotes the antifundamental,and we note that the charges allow the operator y t HQu . The Abelian generalization of thetwin Higgs is found by embedding the Z Γ in these groups as the block-diagonal element S n = diag [ n , n exp(2 πi/ Γ) , . . . , n exp(2 πki/ Γ) , . . . , n exp(2 π (Γ − i/ Γ)] , where n =1 , , corresponds to which SU ( n Γ) factor the element belongs to and n is the n × n identity matrix.The pattern of orbifolding is extremely simple in this example. We write the Higgsfield as a matrix of Γ columns and rows in blocks of two, with the field in block i, j he Loerarchy Problem Chapter 4 being H ( i ) A j where i simply labels the column and A j = 1 , is an SU (2) index. Now wecan look at how this field transforms under the chosen Z Γ , H = H (0) A . . . H (Γ − A ... . . . ... H (0) A Γ − . . . H (Γ − A Γ − → S H S † (4.8) → A H (0) A . . . A H (Γ − A e − π (Γ − i/ Γ ... . . . ... A Γ − e π (Γ − i/ Γ H (0) A Γ − . . . A Γ − e π (Γ − i/ Γ H (Γ − A Γ − e − π (Γ − i/ Γ , (4.9)and we see immediately that the only invariant elements are those on the diagonal.It is simple to repeat this for the Q, u fields to find the same feature. Then in thisexample there are no off-diagonal fields at all, and the daughter theory consists of Γ SM-like sectors. These sectors are all identical and so have a S Γ rearrangement symmetry,leading to a one-loop quadratic potential of the form V − loop ∼ c Λ π − (cid:88) i =0 (cid:12)(cid:12)(cid:12) H ( i ) A i (cid:12)(cid:12)(cid:12) , (4.10)That is, just as in the twin Higgs, the SU (2Γ) symmetry of the scalar potential is re-spected by the one-loop quadratic corrections as a result of the discrete symmetry despitebeing explicitly broken by the gauge groups. This clearly opens up a much wider spaceof Neutral Naturalness models where the Higgs is a pseudo-Nambu Goldstone boson andso receives some protection of its mass without new colored particles at the electroweakscale.The general orbifolding approach to constructing models of Neutral Naturalness wasfully laid out in [309, 298], where they in particular explore a ‘regular representation’embedding of the discrete group into the continuous symmetries of the mother theory.This approach of orbifolding to find low-energy models with accidental symmetries isuseful also because such models come along with guides for how to UV-complete them.141 he Loerarchy Problem Chapter 4 When we wrote down the twin Higgs model above, it was perhaps not obvious that thereis a nice UV completion of this theory. But now we see that the twin Higgs is an orbifoldprojection of a SU (6) × SU (4) gauge theory by Z , so we expect we can uplift thisto a five-dimensional UV completion where the twin Higgs emerges dynamically at lowenergies from orbifold boundary conditions. So we can confidently study solely the low-energy effective theory of the zero-modes we’ve projected out without worrying explicitlyabout whether a UV completion exists. 142 he Loerarchy Problem Chapter 4 Modern particle theorists must now confront a new version of the issue of electroweaknaturalness. When originally understood, the pressing problem was understanding whatsorts of UV structure could protect a scalar from large mass corrections. Of course thisstructure needed to be broken to get the SM structure in the IR and there’s lots ofinteresting physics and subtleties on that end as well. But with the structure of the weakscale barely explored, the ways in which this could be done were abundant.Over the ensuing decades we explored electroweak physics with increasing precision,which has provided invaluable guidance for how the SM structure must appear. Graduallythe IR dynamics became more and more constrained to the point where now, as we haveemphasized, we have enormous constraints on any appearance of new physics with SMquantum numbers up to mass scales that are often many times the electroweak scale.We thus have a qualitative change in the electroweak naturalness issue over the pastdecades. We term the modern, empirical, low-energy puzzle of electroweak naturalnesswithout visible structure around the weak scale as ‘The Loerarchy Problem’, for obviousreasons. In this language, the little hierarchy problem is just one approach toward this problem,which assumes that one of the classic solutions is just out of reach and another moduleis needed to postpone the appearance of SM charged particles. While it’s more thanworthwhile to continue looking for and exploring those theories, in the face of increasinglypowerful LHC data in excellent agreement with the Standard Model it’s worth thinkingtransversely. As intriguing as the Neutral Naturalness models are, these classes of models For readers who do not share my sense of humor, the reasoning is an implied fake etymology for theword ‘hierarchy’ as ‘high + erarchy’, and a retcon of the term ‘The Hierarchy Problem’ as emphasizingthe ‘high’ energy, UV aspects of the issue, by which we are comparatively emphasizing the ‘low’ energy,IR aspects, suggesting that a natural parallel term would similarly combine ‘low + erarchy’ to form‘loerarchy’, which is not a dictionary word. he Loerarchy Problem Chapter 4 rely on one-loop accidents and so do not completely relieve the empirical pressure froma lack of new physics at the energy frontier. Such models are gradually also beingconstrained by the LHC, so what else are we to do? The maximalist interpretation ofthe LHC data is that Nature may be leading us to the conclusion that there is no newphysics at the weak scale . With every inverse femtobarn of LHC data without a signal ofnew physics, the impetus for such a paradigm shift becomes stronger.But how can we generate a scale without additional structure appearing surroundingit? Within the context of effective field theory, decoupling theorems demand that if theRG evolution is to change around a scale M it must be due to fields close to the scale M .This was perhaps clearest in dim reg with MS, where the beta functions explicitly changesolely at mass thresholds, though the Wilsonian approach is more useful for physicalintuition. So how are we to break this feature? This undertaking is the maximalistapproach to the Loerarchy Problem. 144 he Loerarchy Problem Chapter 4 There are more things in heaven and earth, Horatio,Than are dreamt of in your philosophy.William Shakespeare
Hamlet , c. 1600 [310]
The line of thought we suggest here is that perhaps the apparent violation of EFTexpectations at the weak scale is a sign of the breakdown of EFT itself. Dependingon how much background in particle physics one has this statement may seem more orless heretical, but the idea is not as radical as it may at first seem—for one reason, thecosmological constant problem has inspired sporadic reexaminations of the validity ofeffective field theory in our universe for decades.The cosmological constant problem is the fine-tuning issue with the other dimension-ful parameter in the Standard Model. Just as with the Higgs mass, there is no protectivesymmetry in the Standard Model for the vacuum energy, and so the natural expectationis Λ ∼ M pl , some orders of magnitude higher than observations suggest. Howeverthere’s an important difference in the severity of these problems—for the Higgs, as em-phasized in Section 2.2.1, the worrisome mass corrections are those from new physics.This is why the severity of the problem has only ratcheted up in recent years, as we haveseen nothing to protect the Higgs from BSM mass corrections. But for the vacuum en-ergy, there are finite, calculable, physical contributions in the Standard Model itself! Fora start, EWSB by the Higgs yields a contribution ∼ − v , and chiral symmetry-breakingyields a contribution ∼ − Λ QCD . How is it that these can be nearly perfectly canceled offin the late universe?There have been important attempts to address the cosmological constant problem145 he Loerarchy Problem Chapter 4 with a violation of effective field theory, from Coleman’s suggestion [311] that nonlocalityinduced by wormholes may allow the early universe to be sensitive to late-time require-ments to Cohen-Kaplan-Nelson’s suggestion [312] that the Bekenstein bound demandsan infrared cutoff on the validity of any EFT. From one perspective, our suggestion toextend this philosophy to the hierarchy problem appears natural in light of its apparentneed in cosmology. We can point to an even-more-general motivation with the realizationthat gravity necessarily violates EFT.
The perturbative quantum field theory of the Einstein-Hilbert Lagrangian [313] is quiteclearly an effective field theory of a symmetric two-index tensor field which obeys dif-feomorphism invariance and with power counting in /M pl [314], as we can easily see bywriting it out and expanding around flat space g µν = η µν + h µν : −L EH = 12 M pl √ g R ∼ ∂ h + 1 M pl ∂ h + 1 M pl ∂ h + . . . (4.11)where we have only given the schematic form of the operators in terms of the number ofderivatives and linearized gravitational fields they contain, as the full expressions quicklybecome complicated [315, 316]. Then we may expect that this effective field theory willbe a good approximation to infrared gravitational physics until we get to energies closeto the Planck scale, at which point the higher-dimensional operators are unsuppressedand we need a UV completion. From this bottom-up approach it’s not clear why gravityshould be particularly special. But we can get some insight at a very basic level bythinking about gravitational scattering.Let’s compare two effective field theories: the four-Fermi theory of the weak interac-tion below the weak scale G − / F and the perturbative theory of quantum gravity belowthe Planck mass. If we consider scattering two leptons at √ s (cid:28) G − / F , we can make146 he Loerarchy Problem Chapter 4 very precise predictions by computing in the four-Fermi theory—the possible final states,the differential cross-section; whatever we’d like. And similarly if we scatter two particlesat √ s (cid:28) M pl , we can compute to high precision in quantum gravitational correctionswhat will happen.Now inversely, imagine scattering two leptons at √ s (cid:29) G − / F in the four-Fermitheory. At a scale like G − / F , the EFT has obviously broken down and we can sayessentially nothing about what this process will look like—any calculation we tried wouldbe hopelessly divergent, and we have no idea what sorts of states might exist at energiesthat large.However, what if we scatter two particles at √ s (cid:29) M pl , say a ridiculously trans-Planckian scale like M pl ? In this case, in fact, we know what will happen to incredi-ble precision—a black hole will form! This will be a ‘big’ black hole of mass (cid:29) M pl witha lifetime of order days. It will be well-described by classical general relativity for somemacroscopic time, and then semiclassical GR for an O (1) fraction of the full lifetime. Infact, unless you are interested in incredibly detailed measurements which involve collect-ing essentially every emitted Hawking quanta (of which there will be n ∼ in thiscase!) and finding their entanglement structure, we know how to describe the evaporationnearly completely. So something profoundly weird is going on. The key point being that in gravity,the far UV of the theory is controlled by classical, infrared physics. This is obviously afeature that we do not see in other EFTs.This is not a new idea; it has long been known that gravity contains low-energy effectswhich cannot be understood in the context of EFT. The fact that black holes radiateat temperatures inversely proportional to their masses [318] necessitates some sort of I recommend Giddings’ Erice lectures [317] for more on the perspective of quantum gravitationalbehavior as a function of the Mandelstam variables. he Loerarchy Problem Chapter 4 ‘UV/IR mixing’ in gravity—infrared physics must somehow ‘know about’ heavy massscales in violation of a naïve application of decoupling. As a perhaps-more-fundamental raison d’être for such behavior, the demand that observables in a theory of quantumgravity must be gauge-(that is, diffeomorphism-)invariant dictates that they must benonlocal (see e.g. [319, 320, 321, 322, 323]), again a feature which standard EFT tech-niques do not encapsulate. In view of this, the conventional position is that EFT shouldremain a valid strategy up to the Planck scale, at which quantum gravitational effectsbecome important. But once locality and decoupling have been given up, how and whyare violations of EFT expectations to be sequestered to inaccessible energies? Indeed, the‘firewall’ argument [324] evinces tension with EFT expectations in semiclassical gravityaround black hole backgrounds at arbitrarily low energies and curvatures, as does recentprogress finding the Page curve from semiclassical gravity [325, 326, 327, 328, 329, 330].That quantum gravitational effects will affect infrared particle physics is likewisenot a new idea. This has been the core message of the Swampland program [331],which has been cataloging—to varying degrees of concreteness and certainty—ways inwhich otherwise allowable EFTs may conjecturally be ruled out by quantum gravitationalconsiderations. These are EFTs which would look perfectly sensible and consistent to aninfrared effective field theorist, yet the demand that they be UV-completed to theorieswhich include Einstein gravity reveals a secret inconsistency. While this is powerfulinformation, the extent to which the UV here meddles with the IR is relatively minor—just dictating where one must live in the space of infrared theories. Even so, they havebeen found to have possible applications to SM puzzles, including the hierarchy problem[97, 98, 99, 100, 101, 102, 103, 104, 11, 5, 105].Let us review briefly the approach to connect the Weak Gravity Conjecture (WGC)[332] to the hierarchy problem. The WGC is one of the earliest and most well-testedSwampland conjectures, its formulation is relatively easy to understand, and it’s em-148 he Loerarchy Problem Chapter 4 blematic of the way one might try to connect the hierarchy problem to Swamplandconjectures in general.The prime motivation for formulating the WGC was the well-known folklore thatquantum gravity does not respect global symmetries. The simple argument for this factis that ‘black holes have no hair’—the only quantum numbers a black hole has correspondto its mass, spin, and gauge charges. This means that if we make a black hole by smashingtogether a bunch of neutrons, there is no reason why it cannot decay into solely photons,which violates the global U (1) B − L symmetry of the SM but no gauge symmetries. Theauthors suggested that the fact that Abelian gauge symmetries smoothly become globalsymmetries as the gauge coupling vanishes means that something must go wrong withvery small gauge couplings as well, so as for the physics to be smooth in this limit.Another line of thinking for quantum gravity not respecting global symmetries, whichconnects more closely to the WGC formulation, is based on entropic arguments. If globalsymmetries could be exact, then we could create a big black hole with an arbitraryglobal charge, and wait a very very long time while it Hawking evaporates down to thePlanck scale. Unlike gauge charges, the global charge does not affect the metric, so theblack hole does not shed this charge as it evaporates. We then end up with a Planck-sizedblack hole with an arbitrarily large global charge, which would necessitate the existence ofarbitrarily-many black hole microstates for a fixed-mass black hole. But this is a disaster!In calculating a scattering amplitude one has to sum over all possible intermediate states,in principle including black holes. These effects are surely Boltzmann-suppressed byenormous amounts, but if there are an infinite number of possible black holes then anynonzero contribution from a single black hole will lead to a divergence. A cuter (albeitsomewhat tongue-in-cheek) ‘hand waving argument’ is provided by [333]: Were thereincredibly large numbers of super-Planckian states which could populate a thermal bath,a vigorous wave of your hand would produce them in Unruh radiation, and you could149 he Loerarchy Problem Chapter 4 feel them against your hand as they evaporated.Now if we have a gauge symmetry, there are no longer Planck-sized black holes witharbitrarily large charge because there is an extremality bound: g | Q | ≤ M/M pl , with Q the gauge charge and M the mass, in units of the gauge coupling g and the Planck mass.If this were disobeyed a naked singularity would appear, which would violate cosmiccensorship [334]. But for a very tiny gauge coupling, while we no longer have strictlyinfinitely-many black holes at each mass, there are still an enormous number of blackhole microstates for a Planck-sized black hole as g → . So a worry like the hand-wavingargument still applies.This sort of reasoning motivated Arkani-Hamed, Motl, Nicolis, and Vafa to conjecturethat just as quantum gravitational theories must not have exact global symmetries, theymust also suffer some physical malady which disallows the limit where Abelian gaugesymmetries become global symmetries. Their conjecture has two forms: The ‘magnetic’conjecture dictates that a quantum gravitational theory with a U (1) gauge symmetrymust be enlarged into a theory allowing magnetic monopoles by a cutoff Λ (cid:46) gM pl ,which connects to such a description never being valid in the limit g → . The ‘electric’form of the WGC is that such a theory must contain a particle which is ‘super-extremal’—it has charge greater than its mass gqM pl > m . The existence of such a particle woulddestabilize the extremal, charged black holes, allowing them to decay (though the extentto which this really soothes our entropic worries is unclear).Well this super-extremality bound should look very interesting to us, as it provides anupper bound on a mass scale. While we cannot apply this directly to the Higgs becauseit is not charged under any unbroken Abelian gauge symmetries, we know that one ofthe Higgs’ jobs is to provide mass to other particles. So if the weak gravity conjecturebound must apply to some state φ with mass m φ = yv , with v the Higgs vev, then thisstill amounts to a bound on the electroweak scale.150 he Loerarchy Problem Chapter 4 This was suggested in the context of gauged U (1) B − L with very tiny coupling givingan upper bound on the lightest neutrino mass [97], but the magnetic form of the WGCis difficult to deal with in this context. This can be circumvented by introducing a newdark Abelian gauge group U (1) X and charged states which get (some of) their mass fromthe Higgs [5].The way such a model solves the hierarchy problem is by changing the shape of ourprior for the electroweak scale, as mentioned in Section 2.2.2. As naïve effective fieldtheorists with no information about quantum gravity, we assumed some sort of flat priorin [ − M pl , M pl ] . But in fact much of this space is ruled out by quantum gravitationalconstraints, consisting of theories ‘in the Swampland’, meaning that our prior should bereshaped to include only values which can actually be produced by a theory of quantumgravity. This is how much of the connection from Swampland conjectures to the realworld has worked schematically: one says that Quantum Gravity demands one live in asubregion of the EFT parameter space.In theory far more flagrant violations of low-energy expectations are permissible—thatis, the extent to which quantum gravitational violation of EFT will affect the infraredof our universe is not at all certain. Of course any proposal to see new effects froma breakdown of EFT must contend with the rampant success of the SM EFT in theIR—though not in the far IR, recalling the cosmological constant problem. Certainly aviolation of EFT must both come with good reason and be deftly organized to spoil onlythose observed EFT puzzles. For the former, the need for quantum gravity is obviouslycompelling. As to the latter, it is interesting to note that the most pressing mysteriesinvolve the relevant parameters in the SM Lagrangian.Ultimately, our ability to address the hierarchy problem through quantum gravita-tional violations of EFT is limited by our incomplete understanding of quantum gravity.This motivates finding non-gravitational toy models that violate EFT expectations on151 he Loerarchy Problem Chapter 4 their own, providing a calculable playground in which to better understand the potentialconsequences of UV/IR mixing. In Chapter 7 we pursue the idea that UV/IR mixingmay have more direct effects on the SM by considering noncommutative field theory(NCFT) as such a toy model. These theories model physics on spaces where translationsdo not commute [335, 336], and have many features amenable to a quantum gravitationalinterpretation—indeed, noncommutative geometries have been found arising in variouslimits of string theory [337, 338, 339, 340].152 hapter 5Neutral Naturalness in the Sky
In the beginning the Universe was created.This has made a lot of people very angry andbeen widely regarded as a bad move.Douglas Adams
The Restaurant at the End of the Universe (1980) [341] eutral Naturalness in the Sky Chapter 5
It is an amazing and serendipitous fact that the universe started off hot. As a result of theinitially high energies and densities, the details of microscopic physics greatly affected thelarge-scale evolution of the universe. Since the speed of light is finite, by looking out inthe sky at enormous distances we can not only learn about the history of the universe butwe can use this information to learn about particle physics. While cosmology doesn’tgive us probes of arbitrarily high temperatures, there’s still a humongous amount tobe learned—in part due to further serendipity. The fact that the universe transitionsfrom radiation domination to matter domination shortly before it becomes transparentto photons means that the cosmic microwave background (CMB) encodes informationboth about the light, radiation-like degrees of freedom as well as the matter density in theearly universe. Had radiation domination ended far before recombination, it would be farmore difficult to use the CMB to constrain light degrees of freedom like extra neutrinos.Had radiation domination ended far after recombination, there would be little evidence ofdark matter in the CMB, which is the strongest evidence for particle dark matter insteadof, say, a modification of gravity at large distances. In fact such a ‘cosmic coincidence’also occurs much later, as there is a very long epoch of matter domination before theuniverse transitions to dark energy domination. Were there just slightly less dark energy,its effects would be essentially invisible thus far in the history of the universe, and itwould be very difficult to measure dark energy at all.All that is to say that there is enormous value in collaboration between particle physicsand cosmology. In this chapter we investigate this connection for the twin Higgs modelin particular, though our findings are relevant for general Neutral Naturalness theories aswell. In Section 4.1.1 we noted that the energy frontier does not effectively probe thesetheories. Since they do not introduce new particles with Standard Model charges, it is154 eutral Naturalness in the Sky Chapter 5 only precision electroweak measurements made at colliders that constrain them at all.However, such theories are in fact probed very well by cosmology, as they introduce newlight degrees of freedom. Despite the fact that these do not directly interact with normalmatter, their gravitational effects still contribute to the evolution of the universe, and sothe CMB provides a powerful constraint on new light particles.It is this cosmological effect that provided the biggest obstacle to the original twinHiggs proposal [288], which became an urgent issue after the null results of run 1 ofthe LHC and the increased interest in models where the lightest states responsible forHiggs naturalness were SM-neutral. The landmark approach taken in [289] was to parethe model down to a ‘minimal’ version where only those states necessary for Higgs nat-uralness appeared in the twin spectrum. This revived the twin Higgs as a solution tothe little hierarchy problem, and their ‘fraternal’ version brought about many interestingphenomenological possibilities.The ‘fraternal twin Higgs’ has a twin sector consisting—at energies below its cutoff—solely of the third generation of fermions, and with ungauged twin hypercharge. Thisbrilliantly removes all light particles from the spectrum, so their effects would not causetrouble in the early universe. But this approach leaves perhaps a niggling unpleasanttaste for those worried about parsimony. Yes the fraternal twin Higgs introduces fewernew particles than the mirror twin Higgs, so a naïve desire to solve problems with fewingredients might suggest that this is a windfall. However, the mirror twin Higgs reallyconsists of only two ‘ingredients’: a Z symmetry and some soft breaking to misalign theresulting Higgs vevs—whereas the fraternal twin Higgs has much more structure.While those statements seem to be of a very subjective sort, we can ground thisunease in physics by considering what’s needed to UV-complete such a model. At thecutoff of this model Λ ∼ TeV, we need the Z to still be a relatively good symmetryamong the largest couplings in the twin sector—that is, the gauge couplings g , g and the155 eutral Naturalness in the Sky Chapter 5 top Yukawa y t —such that the cancellation of contributions from the two sectors to theHiggs potential works well. Yet in other parts of the theory we have done great violenceto the structure of the theory, having broken twin hypercharge and removed parts of thespectrum. How are we to ensure firstly that the correct degrees of freedom gain largemasses, and secondly that this does not radiatively feed into the remaining light degreesof freedom?In the midst of this digression, I should mention a terminological confusion. In theliterature, the phrase ‘mirror twin Higgs’ often refers to models in which the full collectionof twin degrees of freedom are present in the low-energy theory, regardless of how much Z -breaking is present. Judicious introduction of such asymmetries has been used tocreate models which avoid cosmological issues by making all the twin fermions heavy,while still staying within the technical definition of the mirror twin Higgs. But this is anoverreliance on a definition; the fraternal twin Higgs is merely a limit of these theoriesin which the Z -breaking is severe enough to push some degrees of freedom above thecutoff. The real distinction between classes of Neutral Naturalness models should bebetween those which break the Z only minimally and those which do greater violenceto the symmetry. It is this distinction which classifies the difficulty involved in finding aUV completion.Now let me emphasize that this is not to undercut the value of such a model. After all,the Yukawa interactions in the Standard Model badly break the large global symmetries itwould otherwise have. Indeed, the fraternal twin Higgs showcased interesting phenomena,pointed to new general experimental probes, and provided a basis for many intriguinglines of research. In fact we will return to this model in Chapter 6 to study a novelcollider search strategy to which it lent credence and which turns out to be a broadlyuseful probe of many theories of BSM physics. Despite the fact that the vast, vastmajority of theory papers written in particle physics will not ultimately be the exact156 eutral Naturalness in the Sky Chapter 5 right model of the universe, they still contain value. They may guide experimentalsearches toward interesting classes of signals to look for, or teach us new things aboutthe range of particle phenomenology or quantum field theories. Regardless, we obviouslydon’t know in advance which model will be correct, so exploring all possible directions iscrucial.Yet when there is the possibility for a more parsimonious model, it’s certainly worthpursuing that option. This is the philosophy that led to my collaborators and me lookinginto the prospect of attaining a realistic twin Higgs cosmology that respected the Z symmetry. 157 eutral Naturalness in the Sky Chapter 5 The primary challenge to the mirror Twin Higgs comes not from LHC data, but fromcosmology. An exact Z exchange symmetry predicts mirror copies of light StandardModel neutrinos and photons states, which contribute to the energy density of the earlyuniverse. In particular, twin neutrinos and a twin photon provide a new source of darkradiation that is strongly constrained by CMB and BBN measurements [342, 343]. Whilethese constraints could be avoided if the two sectors were at radically different tempera-tures, the Higgs portal couplings required by naturalness keep the two sectors in thermalequilibrium down to relatively low temperatures.Constraints on dark radiation in the mirror Twin Higgs have motivated models inwhich the Z symmetry is approximate [298, 309, 344, 345, 346, 289, 302, 347, 348, 349,350], in which case the dark radiation component can be made naturally small. Thesemodels have proved to be a boon for phenomenology. Among other things, they quitegenerally motivate looking for Higgs decays to long-lived particles at colliders [351, 352,353, 354, 12, 355, 10, 356] and contain well motivated dark matter (DM) candidates [357,358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368]. However, such cosmological fixescome at the cost of minimality, as models with approximate Z symmetries require aconsiderable amount of additional structure near the TeV scale.In this work we take an alternative approach and investigate ways in which earlyuniverse cosmology can reconcile the mirror Twin Higgs with current CMB and BBNobservations. In doing so, we find compelling scenarios that transfer the signatures ofelectroweak naturalness from high-energy colliders to cosmology. We consider severalpossibilities in which the energy density of the light particles in the twin sector is diluted158 eutral Naturalness in the Sky Chapter 5 by the out-of-equilibrium decay of a new particle after the two sectors have thermallydecoupled. Crucially, the new physics in the early universe respects the exact (albeitspontaneously broken) Z exchange symmetry of the mirror Twin Higgs. This symmetrymay be used to classify representations of the particle responsible for this dilution. Weconcentrate on two minimal cases: In the first, the long-lived particle is Z -even and theasymmetry is naturally induced by kinematics. In the second, there is a pair of particleswhich are exchanged by the Z symmetry and which may be responsible for inflation. Moreover, in these cases the new physics does not merely reconcile the existence of amirror twin sector with cosmological constraints, but predicts contributions to cosmolog-ical observables that may be probed in current and future CMB experiments. This raisesthe prospect of discovering evidence of electroweak naturalness first through cosmology,rather than colliders, and provides natural targets for future cosmological constraints onminimal realizations of neutral naturalness.The next sections are organized as follows: In Section 5.2.2 we discuss the thermalhistory of the mirror Twin Higgs, with a particular attention to the interactions keepingthe Standard Model and twin sector in thermal equilibrium and the cosmological con-straints on light degrees of freedom. In Section 5.2.3 we present a simple model where theout-of-equilibrium decay of a particle with symmetric couplings to the Standard Modeland twin sector leads to a temperature difference between the two sectors after theydecouple. We turn to inflation in Section 5.2.4, constructing a model of “twinflation”in which the softly broken Z -symmetry extends to the inflationary sector and leads totwo periods of inflation. The first primarily reheats the twin sector, while the secondprimarily reheats the Standard Model sector. We conclude in Section 5.2.5. A third case exists, in which the particle is Z -odd. This may additionally be related to the spon-taneous Z -breaking in the Higgs potential, although we find that a realisation of such a scenario isdependent upon the UV completion of the model. eutral Naturalness in the Sky Chapter 5 The primary challenge to the mirror Twin Higgs comes from cosmology, rather thancollider physics. The mirror Twin contains not only states responsible for protectingthe Higgs against radiative corrections (such as the twin top), but also a plethora ofextra states due to the Z symmetry that are irrelevant to naturalness. The lightest ofthese, namely the twin photon and twin neutrinos, contribute significantly to the energydensity of the early universe around the era of matter-radiation equality, since they havea temperature comparable to that of the Standard Model plasma at all times. This isbecause the same Higgs portal coupling that makes the Higgs natural also keeps thetwo sectors in thermal equilibrium down to O (GeV) temperatures. Then the identicalparticle content in the twin and Standard Model sectors guarantees that they remain atcomparable temperatures even after they decouple - for every massive Standard Modelspecies that becomes non-relativistic and transfers its entropy to the rest of the plasma,its twin counterpart does the same within a factor of f /v in temperature.In this section we undertake a detailed study of the decoupling between the StandardModel and twin sectors as well as the constraints from precision cosmology. Twin Degrees of Freedom
In thermal equilibrium, each relativistic degree of freedom has roughly the same energydensity. In general, we express the energy density of the universe ρ during the radiation-dominated era as ρ ≡ g (cid:63) π T , where we define g (cid:63) through this relation as the effectivenumber of relativistic degrees of freedom and T the temperature of the SM photons. Thisthen determines the evolution of the scale factor through the first Friedmann equation H = 1 M pl (cid:20) π g (cid:63) T (cid:21) / (5.1)160 eutral Naturalness in the Sky Chapter 5 (assuming spatial flatness), where M pl is the reduced Planck mass. In general, the energydensity of a particular species i may be computed from ρ i = g i (cid:82) d p (2 π ) f i ( p, T i ) E ( p ) , where g i are the number of internal degrees of freedom, E ( p ) is the energy as a function ofmomentum p , while f i ( p, T i ) is the phase-space number density and is a Bose-Einstein orFermi-Dirac distribution if the species is in equilibrium at temperature T i . The number ofeffective relativistic degrees of freedom may then be defined for each sector separately as g SM (cid:63) ( T ) and g t (cid:63) ( T ) satisfying ρ SM ( T ) = π g SM (cid:63) ( T ) T and ρ t ( T ) = π g t (cid:63) ( T ) T , respectively,where ρ SM ( T ) and ρ t ( T ) are the total energy densities of SM and twin particles. Thevalues of g (cid:63) ( T ) for the SM and twin sectors are shown in Figure 5.1, where all specieswithin each sector are in thermal equilibrium. These can then be used to calculate thetotal number g (cid:63) as a function of temperature, by weighting twin sector energy densityby its temperature: g (cid:63) ( T ) = g SM (cid:63) ( T ) + g t (cid:63) ( ˆ T )( ˆ T /T ) , where ˆ T is the twin sector photontemperature when the SM photon temperature is T .Likewise, entropy densities for each sector i are defined as s i ( T ) = π g i (cid:63) ( T ) T . Weneglect the small differences between the number of relativistic degrees of freedom definedfrom energy and entropy densities, which are not significant over the range of tempera-tures of interest here. Decoupling
In the early universe, the two sectors are thermally linked by interactions mediated by theHiggs, which, through mixing with both h A and h B components, allows for SM fermionsand weak bosons to scatter off or annihilate into their twin counterparts. However,once the temperature drops sufficiently for this Higgs-mediated interaction to becomerare on the expansion time-scale, the sectors decouple and thereafter thermally evolveindependently. More precisely, thermal decoupling will occur once the rate at whichenergy can be exchanged between SM and twin particles (through the Higgs) falls below161 eutral Naturalness in the Sky Chapter 5 -2 -1 Temperature (GeV)020406080100120 g Effective Number of Relativistic Degrees of FreedomSMf/v = 3f/v = 10
Figure 5.1: The effective number of relativistic degrees of freedom for mirror Twin Higgsmodels for different values of f /v . The dash-dotted line is the for the Standard Model g SM (cid:63) ( T ) and the dashed lines are the twin sector degrees of freedom g t (cid:63) ( T ) . The evolutionof g (cid:63) during the QCD phase transition (QCDPT) is not well-understood, so we assign theSM QCDPT a central value of MeV and a width of MeV and interpolate linearlybetween the values of g (cid:63) at MeV for free partons and at
MeV for pions. Furtherdiscussion may be found in [369]. For the twin sector we use a central value and widthwhich are (1 + log( fv )) times larger than the SM values. Note that new mass thresholds,expected to appear at energies ∼ TeV in UV completions of the twin Higgs, have notbeen included. 162 eutral Naturalness in the Sky Chapter 5 the Hubble rate.Thermal decoupling is traditionally formulated from the Boltzmann equations de-scribing the evolution of single-particle phase space number densities, wherein collisionsinduce instantaneous changes to the shape of these distributions. When the collisionsoccur faster than the expansion rate, the phase space probability density functions ofthe interacting species are expected to relax to an equilibrium distribution (Boltzmann,neglecting quantum statistics, will be applicable to our case). However, once the rateof collisions falls below the expansion rate, collisions become rare on cosmological timescales and the phase space distributions depart from equilibrium. The decoupling tem-perature is determined as that at which the scattering rate of a participating particle, Γ ,drops below the Hubble rate, assuming that this occurs instantaneously across the entirephase space where the number density is significant. This formulation can be used todetermine the time at which a particular species of particle will cease to scatter off twinparticles on cosmological time scales.In the case of interest here, however, both sectors of particles remain thermalisedwithin themselves while the interactions between sectors freeze-out. This implies that thephase space number densities are still Boltzmann distributions throughout decoupling,with a different temperature for each sector. As it is the twin sector temperature thatultimately determines the impact of the light twin degrees of freedom on the cosmologicalobservables (discussed below in Section 5.2.2), we wish to describe the thermal evolutionof the two sectors by that of their entire energy or entropy content and the bulk heatflows between them. They may then be identified as thermally decoupled once the rateat which they exchange energy falls below the expansion rate.If the SM and twin sector plasmas have temperatures T and ˆ T respectively, thencalling q the net heat flow density from the SM to the twin sector, the rate at which the163 eutral Naturalness in the Sky Chapter 5 twin entropy densities s t and s SM evolve is determined by ds t dt + 3 Hs t = 1ˆ T dqdt = 1ˆ T (cid:16) dq in dt − dq out dt (cid:17) (5.2) ds SM dt + 3 Hs SM = − T dqdt = − T (cid:16) dq in dt − dq out dt (cid:17) . (5.3)Here, H is the Hubble rate. The heat flow rate has been decomposed into the sum of theenergy transferred into and out of the twin sector by collisions in the second equality ineach line, where dq in dt and dq out dt are both positive.The rate of heat flow q may be calculated by performing a phase space average ofthe rate that energy is transferred from the SM to the twin sector through particleinteractions. Since the decay rates of top quarks or weak bosons are fast compared totheir scattering rate and the Hubble rate, energy transferred to them is instantaneouslytransferred to the rest of the plasma. Similarly, the scattering rate of lighter fermionsoff other particles of the same sector (such as photons or gluons) is much faster thantheir interaction rate with twin fermions. Energy transferred to the lighter fermionstherefore quickly diffuses throughout their respective plasmas. The rate of heat flowbetween sectors may therefore be well approximated by the rate at which energy istransferred from SM particles to twin particles in Higgs mediated interactions. Thismay occur through elastic scattering of SM particles off twin particles or annihilationsof SM particle/antiparticle pairs into twin particles (or the reverse). The energy densitytransferred to twin particle i from SM particle j in scattering is given by dq ij → ij dt = g i g j (2 π ) (cid:90) (cid:90) d k E i ( k ) d h E j ( h ) f i ( k, ˆ T ) f j ( h, T )4 E i ( k ) E j ( h ) (cid:90) v rel ( E i ( p ) − E i ( k )) dσ ij → ij d Ω d Ω , (5.4)where p is the outgoing 4-momentum of particle i . In the cosmic comoving frame, thephase space number densities f i and f j are just Boltzmann factors, although evaluatedat the different temperatures of each sector. The factor g i is the number of internal164 eutral Naturalness in the Sky Chapter 5 degrees of freedom of particle i , which here includes colour (the cross section should notbe colour averaged, as each colour of quark is present in the plasma in equal abundancesand each mediates the exchange of energy, so have their contributions summed). Finally, E i ( k ) is the on-shell energy of particle i with momentum k , while dσ ij → ij d Ω is the differentialscattering cross section for species i scattering off j per solid angle Ω and v rel is the usualrelative speed of the incoming particles. As described in [370], the factor in the integrandgiving the energy transferred per reaction is simply a component of a 4-vector, X = 4 E i ( k ) E j ( h ) (cid:90) ( p − k ) v rel dσ ij → ij d Ω d Ω . (5.5)This may be calculated in the centre-of-mass frame and then boosted back into the cosmiccomoving frame where the integrals in (5.4) can be evaluated, similarly to the thermalaveraging procedure described in [371].The integral (5.4) may be decomposed into two terms giving the positive and negativeenergy changes of the twin particle, which respectively contribute to dq in dt and dq out dt . Whenevaluated in the centre-of-mass frame, these terms correspond to the cases where thescattering angle of the twin particle is respectively less than and greater than the anglebetween its initial momentum and the total momentum of the system. However, when T (cid:54) = ˆ T , we find the integrals involved in this decomposition substantially more arduousthan when they are evaluated together.Energy transferred through annihilations may be similarly calculated as dq j ¯ j → i ¯ i dt = g j (2 π ) (cid:90) (cid:90) d k E j ( k ) d h E j ( h ) f j ( k ) f j ( h )4 E j ( k ) E j ( h ) (cid:90) v rel ( E j ( h ) + E j ( k )) dσ j ¯ j → i ¯ i d Ω d Ω − g i (2 π ) (cid:90) (cid:90) d k E i ( k ) d h E i ( h ) f i ( k ) f i ( h )4 E i ( k ) E i ( h ) (cid:90) v rel ( E i ( h ) + E i ( k )) dσ i ¯ i → j ¯ j d Ω d Ω , (5.6)165 eutral Naturalness in the Sky Chapter 5 where dσ j ¯ j → i ¯ i d Ω is now the differential annihilation cross section. This rate may be evaluatedas described above and is more directly amenable to the factorisation of the integralsobserved in [371]. See also [372] for further details of similar calculations. The first termof (5.6) is the energy transferred from the SM to the twin sector and contributes to dq in dt in (5.2), while the second term is the energy transferred from the twin sector to the SMand contributes to dq out dt .In thermal equilibrium, the rate of energy transferred through collisions into onesector will be balanced by that of energy transferred out of it so that there is negligiblenet heat flow. This state will be rapidly attained (compared to the age of the universe)if dq in,out dt (cid:29) H ˆ T s t . However, as the universe expands and the plasma cools, the energytransfer rates fall faster than the Hubble rate. This is demonstrated in the Figure 5.2below. Once they drop below the Hubble rate, energy exchange ceases on cosmologicaltime scales and the sectors thermally decouple, thereafter thermodynamically evolvingindependently.To determine the decoupling temperature of the sectors, we calculate the rates of pos-itive energy exchange for the twin particles interacting with the SM particles. The crosssections are calculated using a tree-level effective fermion-twin fermion contact interac-tion that, in the full twin Higgs model, would be UV completed by a SM Higgs exchange(the heavier mass of the radial mode would make its exchange subdominant). The in-teraction strength is determined by the masses of the fermions through their Yukawacouplings, as well as the mixing angle of the SM-like mass state h with the gauge eigen-state h B , giving a 4-fermion coupling of strength m f m ˆ f m h f (here m f and m ˆ f are the massesof fermions f and ˆ f ). See [292], [347] for a more detailed discussion of the cross sections.This effective interaction is appropriate for the temperatures of interest here and helps tosimplify the integrals of (5.4). In order to further simplify the integrations of (5.4) whenit is to be decomposed into terms in which the energy exchange is positive and negative,166 eutral Naturalness in the Sky Chapter 5 we calculate dq in dt under the assumption that the sectors have the same temperature (thisensures that the rate dq out dt is identical). This is then combined with the rate of energytransferred from annihilation. A similar calculation of these rates was recently performedin [347], for cases where the Yukawa couplings do not respect the Z twin symmetry.In Figure 5.2 we compare the energy transfer rate to the Hubble rate in order todetermine when decoupling occurs. As long as the energy exchange rate exceeds the ex-pansion rate, the sectors will be thermalised and have the same temperature. Decouplingthen occurs once this rate drops below the Hubble rate. From Figure 5.2, this occursat a temperature ∼ GeV. However, even after the energy exchange rate drops belowthe Hubble rate, the sectors will remain at the same temperature unless some event thateither injects or redistributes entropy occurs within a sector (such as the temperaturedropping below a mass threshold). As the heavy quark masses roughly coincide withthe decoupling temperature, these do cause the twin sector to be mildly reheated withrespect to the SM below decoupling. However, the resulting temperature difference issmall and the energy exchange rates are expected to continue to be well-approximatedby the rates presented in Figure 5.2 beyond decoupling.The lower plot of Figure 5.2 illustrates the decomposition of the energy exchangerates into contributions from interactions involving different SM quarks. The interactioncross sections are proportional to the Yukawa couplings of the interacting fermions. Thegreatest heat exchange is therefore expected to be mediated by the most massive particles,provided that their abundances are not too Boltzmann suppressed. As expected, attemperatures ∼ GeV, the bottom quark is the best conduit of thermal equilibration,followed by the charm quark and then the τ (with colour factors enhancing the former twowith respect to the latter). The rate of heat flow that the top quarks and weak bosons canmediate at these temperatures (or below) is negligible because of Boltzmann suppression.The bend in the curves at temperatures ∼ GeV in the lower plot corresponds to167 eutral Naturalness in the Sky Chapter 5 a transition from temperatures where the dominant energy exchange rate is throughscatterings to those where it occurs through annihilations, as can be seen in the upperplot. The annihilation rate into twin bottom quarks is the dominant component at highenough energies (again because of the larger Yukawa coupling), but this becomes rapidlythreshold suppressed as the temperature drops. As can also be inferred in the upper plot,the energy exchange rate through annihilations involving the twin charmed quarks andtau leptons overtakes that of twin bottom quarks at similar temperature, but are stillsubdominant to scatterings.The decoupling temperature depends upon f /v , which sets both the mass scale ofthe twin sector and the strength of the Higgs-mediated coupling. As f /v is increased,decoupling occurs earlier because of the greater Boltzmann suppression, although this isonly a relatively small effect that, for f /v = 10 , increases the decoupling temperature byonly GeV.When the twin sector is colder than the SM (which will be important for much ofwhat follows) the heat flow is typically dominated by annihilations of SM into twinparticles. However, the energy exchange from elastic scattering can be comparable tothat from annihilations, as illustrated in Figure 5.2. Although the energy exchange in anannihilation will generally exceed that of a scattering because all of the energy involvedin the process must be transferred, the annihilation rate also becomes more Boltzmannor threshold suppressed when the temperature drops below the mass of the heavier twinparticles. It is therefore not always clear that energy transfer through annihilationsdominates.Decoupling is not exactly instantaneous and there is some range of temperatures overwhich the rate of heat flow freezes-out. The net heat flow rate dqdt is greater for larger tem-perature differences between sectors. The generation of a potentially large temperaturedifference within this brief epoch of sector decoupling, such as those discussed below in168 eutral Naturalness in the Sky Chapter 5
AnnihilationScatteringH0.5 1 5 1010 - - - - - Temperature ( GeV ) R a t e ( G e V ) bc τ H0.5 1 5 1010 - - - - - Temperature ( GeV ) R a t e ( G e V ) Figure 5.2: Rates of energy density exchange per twin entropy density ( s t ˆ T dq in dt ) de-composed into contributions from scattering and annihilation (top) and for interactionsinvolving different species of SM fermions (bottom), along with the Hubble parameter,for f /v = 4 . The decoupling temperature is that where the sum of the energy exchangerates equals the Hubble rate, which occurs at T decoup ≈ GeV.169 eutral Naturalness in the Sky Chapter 5
Section 5.2.3, may be cut off when the heat flow rate becomes comparable to the Hubblerate. For a given SM temperature T , the minimum twin-sector temperature ˆ T min duringthe decoupling period may be roughly estimated as that which satisfies H ∼ s t ˆ T dqdt (cid:12)(cid:12)(cid:12) ˆ T = ˆ T min . (5.7)Twin temperatures colder than ˆ T min will partially thermalise back to this value. Asthe participating fermions are not non-relativistic, instantaneous decoupling is not asaccurate an approximation as it is, for example, for chemical decoupling of a WIMP,although it is still reliable.In Figure 5.3, we show the minimum temperature that the twin sector may have asa function of SM temperature for heat flow to freeze out, estimated using (5.7). Onlyannihilations have been included in the determination of the minimum temperature,although we have verified that, for these temperatures, the scatterings contribute only (cid:46) to the heat flow. Note that while the energy exchange rate, such as T dq in dt in(5.2), in scattering processes may be faster, the net energy flow rate, or heat flow ( T dqdt in(5.2)), which is the difference between energy exchange rates into and out of the sector,is actually dominated by annihilations. Generally, we find that decoupling begins attemperatures ∼ GeV. The temperature difference can reach an order of magnitudewithout relaxing once the SM temperature drops to ∼ GeV.While the extent of thermal decoupling is temperature dependent, the maximumtemperature difference that will not relax grows quickly as the SM temperature drops.Then we may describe the two sectors as being decoupled if, in a given cosmology, allevents that raise the temperature of one sector relative to the other (such as the crossingof a mass threshold and the resulting entropy redistribution, the most significant of whichis the confinement of colour) induce temperature differences that are too small to partiallyrelax. 170 eutral Naturalness in the Sky Chapter 5
SMTwin1.0 1.5 2.0 2.5 3.0 3.5 4.001234 SM Temperature ( GeV ) T e m pe r a t u r e ( G e V ) Figure 5.3: Minimum temperature of the twin sector that will not be heated by inter-actions with a hotter SM plasma, as a function of SM temperature, for f /v = 4 . Alsoshown is the SM temperature, for reference.At energies (cid:46) GeV in Figure 5.2, the reliability of the calculation of the heat flowrate diminishes because of the strengthening of the strong coupling and the eventualconfinement of colour. Fortunately, for a cooler twin sector, which will be of interest insubsequent sections, annihilations from the SM dominate other processes over most ofthe parameter space. These are the least sensitive to higher order corrections and non-perturbative effects because of their higher temperature, and hence energy, comparedto the potentially cooler twin sector. The range of temperatures illustrated in Figures5.2 and 5.3 have been selected to roughly illustrate the duration of decoupling, butmay extend below the range where the perturbative calculation of the heat flow rate isvalid. For example, at temperatures below the twin sector QCDPT, which occurs at ∼ (cid:0) fv ) (cid:1) higher temperatures than in the SM, the partonic calculation of twinquark/anti-quark pair production must be replaced by a hadronic one. Furthermore, thegrowth of the twin strong coupling necessitates that the quark-Higgs Yukawa couplings171 eutral Naturalness in the Sky Chapter 5 be RG evolved to the scale of the energy exchanged, which can induce an O (1) changeto the cross section, although this has only a relatively small effect on the decouplingtemperature. It is nevertheless clear that decoupling is mostly complete by then and thatthese uncertainties are not large enough to affect this conclusion.In the standard mirror Twin Higgs cosmology, knowing the decoupling temperaturetells us how the temperatures of the two sectors will be related at subsequent times.The sectors separately evolve adiabatically after decoupling, though they redshift in thesame way and differences in temperature only arise from events that redistribute entropy.Non-minimal cosmological events that could potentially cause the temperatures of eachsector to diverge can therefore only be effective if they leave each sector colder than thisapproximate decoupling temperature. Cosmological Constraints
Given that the twin and Standard Model sectors remain in thermal equilibrium to O ( GeV ) temperatures, the simplest mirror Twin Higgs scenario is cosmologically in-viable due to the presence of light twin species (photons and neutrinos) with abundancescomparable to those of the SM. The cosmological observables through which evidence oflight species may be inferred are typically represented by N eff , the “effective number ofneutrino species” in the early universe; their individual masses, which determine theirfree-streaming distances; and the “effective mass” m eff ν , which parameterises their contri-bution to the present-day energy density of non-relativistic matter. These observablesare probed by both the CMB and large scale structure (LSS). Effective number of neutrinos
The parameter N eff describes the amount of radiation-like energy density during the evolution of the CMB anisotropies before photon decou-pling. It is defined as the effective number of massless neutrinos with temperature as172 eutral Naturalness in the Sky Chapter 5 predicted in the standard cosmology that would give equivalent energy density in radia-tion: ρ r = ρ γ + 78 (cid:18) (cid:19) / N eff ρ γ , (5.8)where ρ r is the energy density of radiation and ρ γ is the energy density of photons (thefactor of (cid:16) (cid:17) / arises from the relative reheating of the photons from electron/positronannihilation, which occurs after most of the neutrinos have decoupled, and the factor of / is from the opposite spin statistics). A deviation from the Standard Model predic-tion of . [373] is denoted by ∆ N eff = N eff − . . This definition of radiation, orequivalently, relativistic degrees of freedom, becomes less clear if the new fields have anon-negligible mass, as we discuss further below.We here review the CMB physics of dark radiation, summarising the discussion in[374]. See also [342] for further review. The angular size and scale of the first acousticpeak is well-measured and this approximately fixes the scale factor at matter-radiationequality a eq . If we imagine fixing all other Λ CDM parameters, extra radiation woulddelay the epoch of matter-radiation equality. This would have a pronounced effect onthe power spectrum in the vicinity of the first acoustic peak through the early IntegratedSachs-Wolfe (eISW) effect. The modes corresponding to this feature entered the horizonclose to matter-radiation equality and the evolution of their potentials is highly sensitiveto the radiation energy density. However, the impact of a ∆ N eff ∼ O (1) deviation on thepeak height can be simultaneously balanced by increasing the amount of non-relativisticmatter, to the extent to which other observations providing independent constraintsupon Ω c permit (for Λ CDM + N eff , a variation of ∼ in Ω c h is consistent with presentCMB+BAO measurements [342], although these variations must be consistent with otherobservables). This degeneracy is not expected to be broken by CMB-S4 [375].Given that a eq is approximately fixed, the utility of N eff arises because, in simple ex-173 eutral Naturalness in the Sky Chapter 5 tensions of the Λ CDM model, it approximately corresponds to the suppression of powerin the small scale CMB anisotropies that arises from Silk damping. The reason for this isroughly that, although the greater expansion rate induced by the extra radiation reducesthe time that CMB photons have to diffuse before decoupling, it also reduces the soundhorizon size more severely. As the angular size of the sound horizon is determined bythe location of the acoustic peaks and is also well measured, the reduction in the soundhorizon must be compensated for by a reduction in the angular diameter distance to theCMB. This effectively raises the angular distance over which photon diffusion proceedsand results in a prediction of smoother temperature anisotropies at small scales. Thiscorrespondence with the Silk damping allows N eff to be approximately factorised fromother parameters and constrained independently, providing a direct observational avenuefor detecting the presence of new, massless fields [374] (see [376] for further implicationsfor model building). This relationship arises because the fixing of a eq implies that N eff effectively determines the energy density of the universe, and hence the Hubble rate, dur-ing CMB decoupling. Note, however, that further extensions of Λ CDM may complicatethis correspondence, in particular deviations from the standard Big Bang Nucleosynthesisprediction of the primordial helium abundance.The contribution to N eff (or ∆ N eff ) in the mirror Twin Higgs arises from two sources:the twin photons, which can be treated as massless dark radiation with an appropriatetwin temperature T teq at the time of matter-radiation equality, and the twin neutrinos,whose non-zero masses may need to be accounted for. For the twin photons, the con-tribution to N eff is simple; their equation of state is always w = 1 / and their energydensity is given by g π (cid:0) T teq (cid:1) , where g = 2 . The twin temperature at matter-radiationequality is found from the SM temperature using comoving entropy conservation, T teq T SMeq = (cid:18) g t (cid:63) ( T decoup ) g SM (cid:63) ( T decoup ) (cid:19) / (cid:32) g SM (cid:63) ( T SMeq ) g t (cid:63) ( T teq ) (cid:33) / , (5.9)174 eutral Naturalness in the Sky Chapter 5 where the two sectors have the same number of thermalized degrees of freedom by thistime. Here, T SMeq is the SM photon temperature at matter-radiation equality and T decoup is the sector decoupling temperature.Since neutrinos are massive, their behavior is more complicated. Their equation ofstate parameter takes on a scale factor dependence which is controlled by their mass.In the Standard Model, this sensitivity is negligible because present CMB bounds implythat neutrinos are ultra-relativistic at a eq to good approximation [342]. However, thefactor by which the twin neutrino masses are enhanced may raise them to order T t eq orgreater (see Section 4.1.1 for discussion of the scaling of the masses with f /v ).To better describe the impact of the extra twin (semi-)relativistic degrees of freedomon the CMB, we choose to define N eff through the effects of neutrinos at matter-radiationequality, when the impact on the expansion rate of the universe for most of the periodrelevant for the evolution of the CMB is greatest. Note that, in their presentation of jointexclusion bounds on N eff and (cid:80) m ν (the sum of SM neutrino masses) or m eff ν (effectivemass contributing to the present-day non-relativistic matter density of an extra sterileneutrino), the Planck collaboration define N eff as the value in (5.8) at temperaturessufficiently high that the neutrinos are fully relativistic. Our values cannot be directlycompared with their analysis, although we consider ours to be a reasonable rough estimatethat is more representative of the CMB constraints. The ensuing correction from the finiteneutrino masses is, in the cases considered in this work, a small effect anyway.To determine this correction and provide a definition of N eff that better describes theimpact of quasi-relativistic particles on the CMB, we first define the epoch of matter-radiation equality as the time at which the average equation of state parameter of theuniverse is ¯ w = 1 / (the equation of state is defined as ρ = ¯ wP , where ρ is energy density175 eutral Naturalness in the Sky Chapter 5 and P is pressure). We can express this condition as d ln Hd ln a (cid:12)(cid:12)(cid:12)(cid:12) a eq = − , (5.10)as in [377].Call the quasi-relativistic neutrino energy density ˜ ρ ( a ) with time-evolving equation ofstate parameter w ( a ) , which is to be balanced against some extra non-relativistic energydensity ∆ ρ CDM ( a ) ∝ a − to keep a eq the same. This amount of non-relativistic energydensity ∆ ρ CDM is ∆ ρ CDM ( a eq ) = ρ r ( a eq ) − ρ m ( a eq ) − a eq d ˜ ρda (cid:12)(cid:12)(cid:12)(cid:12) a eq − ρ ( a eq ) , (5.11)where ρ r and ρ m are the energy densities of the radiation and non-relativistic matter.For a perfect fluid, d ˜ ρda = − w ( a )) ˜ ρ/a (neglecting the anisotropic stress that isexpected only to contribute to a weak phase shift in the CMB [378]), this results in aHubble parameter of H ( a eq ) = 23 M pl [ ρ r ( a eq ) + 3 w ( a eq ) ˜ ρ ( a eq )] . (5.12)This suggests a definition of the effective number of neutrinos, N eff , via H ( a eq ) = 23 M pl (cid:0) ρ γ + N eff ρ thν,m =0 (cid:1)(cid:12)(cid:12) a eq (5.13) N eff ≡ (cid:88) i w i / ρ i ρ thν,m =0 , (5.14)where ρ i is the contribution to the energy density from some species i with equation ofstate parameter w i and ρ thν,m =0 is the energy density of a massless neutrino with a thermaldistribution in the standard cosmology. Then w gives the ‘relativistic fraction’ of theenergy density. Note that this is simply a ratio of the pressure exerted by the new fieldsto that of a massless neutrino. The effectiveness of this approximation was discussed in[379] in the context of thermal axions (while effective at keeping a eq fixed, changes to176 eutral Naturalness in the Sky Chapter 5 odd peak heights subsequent to the first are imperfectly cancelled and require furtherchanges to H to compensate - see Section 5.2.2 below).Calling T iν the temperature at which the neutrinos in sector i freeze-out and a iν thecorresponding scale factor, then assuming instantaneous decoupling, the phase spacenumber density for scale factor a is given by a redshifted Fermi-Dirac distribution [380] f iα ( p ) ≈ (cid:104) e pa/ ( a iν T iν ) (cid:105) − (5.15)for the α neutrino mass eigenstate in the i sector ( m iα (cid:28) T iν , so has been dropped). Theenergy density and pressure are ρ iν α = g α π (cid:90) ∞ dp p (cid:113) p + ( m iα ) f iα ( p ) (5.16) P iν α = g α π (cid:90) ∞ dp p (cid:113) p + ( m iα ) f iα ( p ) , (5.17)where g α = 2 is the number of degrees of freedom for a neutrino species.Since the neutrino decoupling temperature depends on the strength of the weak in-teraction as T ν ∝ G − / F , while G F ∝ v , then the twin neutrino decoupling temperature T t ν is related to the SM neutrino decoupling temperature T SM ν by T t ν = ( f /v ) / T SM ν . (5.18)We can then simply use (5.16) and (5.17) at matter-radiation equality to find ∆ N eff (assuming instantaneous decoupling). We thus obtain H ( a eq ) = 23 M pl (cid:32) ρ SM γ + 3 . ρ thν,m =0 + ρ t γ + (cid:88) α w ν α ρ t ν α (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a eq (5.19)and ∆ N eff = (cid:18) (cid:19) / π ( T SM ) (cid:32) ρ t γ + (cid:88) α w tν α ρ t ν α (cid:33) , (5.20)where we now have equation of state parameters w ν α for each neutrino, while ρ SM γ and ρ t γ are the SM and twin photon energy densities, ρ thν,m =0 and ρ t ν α are the neutrino energydensities. 177 eutral Naturalness in the Sky Chapter 5 Neutrino masses
Because they are so weakly interacting, the neutrinos have a longfree-streaming scale given by the distance travelled in a Hubble time v ν /H , with v ν ∝ m − ν the speed of the neutrino once it becomes non-relativistic. This defines a free-streamingmomentum scale k fs = (cid:113) aHv ν ∝ m ν , above which neutrinos do not cluster. Belowthis scale, perturbations in the matter density consist coherently of neutrinos and othermatter, but well above it only non-neutrino matter contributes to density perturbations.This results in a suppression of the matter power spectrum on large scales which isproportional to the fraction of energy density in the free-streaming matter. Since thisoccurs at late times when neutrinos are non-relativistic, the energy density is simply ρ ν α = n ν α m ν α for each neutrino species α , where n ν α is the number density. Constraintson the sum of neutrino masses then come from the observations of power on small scales,which is suppressed relative to that expected for massless neutrinos by a factor ∝∼ − f ν ,where f ν = Ω ν / Ω m is the fraction of non-relativistic energy in neutrinos at late times[381].More generally, inferences of the matter power spectrum constrain the present-dayenergy density fraction of free-streaming species that do not cluster on small scales andhave since become non-relativisitic, Ω ν = ( (cid:80) m ν + m eff ν ) / (94 . eV ) , where (cid:80) m ν is thesum of SM neutrino masses and m eff ν is the sum of twin neutrino masses weighted by theirnumber density m eff ν = n t ν n SM ν (cid:88) α m t ν α . (5.21)Here n t ν is the number density of a relic twin neutrino flavour and n SM ν is that for a SMneutrino. It is assumed that the neutrinos have been thermally produced as hot relics.The relic abundance of a neutrino species is given by its number density when itdecoupled, diluted by the factor by which the universe has since expanded. The scalefactors at which neutrino decoupling occurs in the two sectors, a SM ν and a t ν can be deter-178 eutral Naturalness in the Sky Chapter 5 mined from (5.18), the relative temperatures in the two sectors and comoving entropyconservation, to obtain a t ν = a SM ν (cid:18) vf (cid:19) / (cid:18) g t (cid:63) ( T decoup ) g SM (cid:63) ( T decoup ) (cid:19) / (5.22)where the same mass thresholds have been assumed in each sector below their neutrinodecoupling temperatures, so that g SM (cid:63) (cid:0) T SM ν (cid:1) = g t (cid:63) ( T t ν ) . The neutrino number densitiesare then n t ν n SM ν = (cid:18) T t ν a t ν T SM ν a SM ν (cid:19) = g t (cid:63) ( T decoup ) g SM (cid:63) ( T decoup ) . (5.23)For f /v from to and using T decoup ∼ − GeV from Section 5.2.2, we find g t (cid:63) ( T decoup ) / g SM (cid:63) ( T decoup ) ∼ . and thus arrive at m eff ν ≈ . (cid:18) fv (cid:19) n (cid:88) α m SM ν α , (5.24)where n = 1 for Dirac masses and n = 2 for Majorana masses.If they are sufficiently light and hot, the twin neutrinos only affect the CMB as darkradiation and their masses may then only be inferred from tests of the matter powerspectrum. However, if heavier and colder, they are better described as a hot dark mattercomponent. Their impact on the CMB is discussed in [382], where the shape of thepower spectrum can depend upon the individual neutrino kinetic energies through theircharacteristic free-streaming lengths. The early Integrated Sachs-Wolfe effect (eISW) isalso sensitive to the masses if the neutrinos become non-relativistic during decoupling(thereby affecting the radiation energy density and the growth of inhomogeneities) [381].There is a significant degeneracy in cosmological fits to the CMB between Ω m and H (the Hubble constant) [383], where raising the non-relativistic matter fraction, suchas with nonrelativistic neutrinos, can be accommodated by a decrease in H (or equiva-lently, the dark energy density), which keeps the angular diameter distance to the CMB179 eutral Naturalness in the Sky Chapter 5 approximately fixed. This degeneracy can be broken by measurements of the baryonacoustic oscillations (BAOs), which are sensitive to the expansion rate of the late uni-verse and provide an independent measurement of Ω m and H . It is through combinationwith these results that bounds from Planck on neutrino masses are strongest [342]. Bounds
The authors are unaware of any specialised analysis of the present and pro-jected future cosmological constraints on scenarios with both massless dark radiationand additional light, semi-relativistic sterile neutrinos. In the absence of this, we usebounds from [342] as a rough indication of the present level of sensitivity to these pa-rameters, which we nevertheless expect to be a reliable indication of the (in)viability ofthis model. The 95% confidence limits on these parameters are N eff = 3 . ± . and (cid:80) m ν < . eV when each are constrained separately with the other fixed. This, ofcourse, overlooks correlations between the impacts of masses and ∆ N eff on the CMB andLSS. Bounds on an additional sterile neutrino as the only source of dark radiation arealso presented with number density, or equivalently, contribution to ∆ N eff , left to float.These are similar to the limit on (cid:80) m ν . It was found in [384] that, allowing (cid:80) m ν and m eff ν to float independently for a single extra sterile neutrino, the bound mildly relaxesto m eff ν (cid:46) eV, although the bound may be stronger depending on the combination ofdata sets chosen (the lensing power spectrum presently prefers higher neutrino massesand raises the combined bounds if included). Other bounds from LSS on (cid:80) m ν exist andare potentially stronger than those placed from the CMB, possibly as low as m eff ν (cid:46) . eV, again depending on data sets combined (see [385], [386]), although these are subjectto greater uncertainties in the inference of the power spectra of dark matter halos fromgalaxies surveys and the Ly α forest.It must also be noted that the shape of the CMB temperature anisotropies dependsupon both the mass of individual neutrino components (through their free-streaming180 eutral Naturalness in the Sky Chapter 5 distance) and their contribution to the energy density of the nonrelativistic matter thatdoes not cluster on small scales. However, it is not expected that improvements in boundson the former will be made from improved measurements of the primary CMB itself,but rather from weak lensing of the CMB, in conjunction with future measurementsfrom DESI of the BAOs to break degeneracy with Ω m . The lensing spectrum, likeinferences of the matter power spectrum made in galaxy surveys, is expected to measurethe suppression of small scale power and therefore to strengthen constraints upon m eff ν ,rather than the individual neutrino masses. One of the goals of CMB-S4 will be thedetection of neutrino masses, given the present lower bound (cid:80) m ν (cid:38) . eV fromoscillations. Projected bounds are as low as ∼ . eV [375], although this assumesno extra dark radiation or sterile neutrinos. A projection of the joint bound on N eff (from extra massless dark radiation) and m eff ν combining improved measurements CMBtemperature measurements, lensing and BAOs indicates a limit of m eff ν (cid:46) . eV at σ [375]. Any contribution from additional states to m eff ν may therefore be testable andbounded by the excess of the neutrino mass inference over the minimum neutrino mass,although laboratory measurements or measurements of ∆ N eff will be required to furtherascertain the contribution from the new particles.Constraints on ∆ N eff from improved measurements of the damping tail as part ofCMB-S4 are projected to be ∼ . − . at σ [375]. In the following sections, we usean optimistic estimate of . for its reach in order to identify as much of the potentiallytestable parameter space as possible.To estimate the impact of current and projected CMB limits on the mirror Twin Higgs,we consider two scenarios: the minimal Standard Model neutrino mass spectrum of m ν =[0 . , . eV , . eV ] and a degenerate spectrum of m ν = [0 . eV , . eV , . eV ] / from[342]. In Figure 5.4 we plot the predictions of the mirror Twin Higgs for ∆ N eff and m eff ν for both types of spectra, as well as for both Dirac and Majorana masses (which scale181 eutral Naturalness in the Sky Chapter 5 X m ν + m effν (eV)4.55.05.56.06.5 ∆ N e ff Standard Mirror Twin Higgs CosmologyMinimal, DiracMinimal, MajoranaDegenerate, DiracDegenerate, Majorana f / v Figure 5.4: Predicted values of ∆ N eff and (cid:80) m ν + m eff ν for minimal and degenerateneutrino mass spectra with both Dirac and Majorana masses for f /v from 3 to 10. ThePlanck 2015 constraint[342] is the dashed line; the corresponding N eff upper bound iswell below the bottom of the plot. All points are excluded by the combination of boundson ∆ N eff and (cid:80) m ν + m eff ν .differently with f /v ). As is plainly evident, the mirror Twin Higgs is ruled out cosmolog-ically, no matter the choices of neutrino masses one makes, if only for the presence of thetwin photon. In the standard cosmology, the twin sector will have roughly the same tem-perature as the SM, giving . (cid:46) ∆ N eff (cid:46) . for f /v < , according to the definition of(5.20). This range depends upon f /v through the twin neutrino decoupling temperature182 eutral Naturalness in the Sky Chapter 5 (5.18), which determines the extent to which the twin photons are reheated relative to thetwin neutrinos after twin electron/positron annihilations. This is sufficiently large thateven the cold dark matter fraction cannot be adjusted to keep matter-radiation equalityfixed, resulting inevitably in changes to the height and shape of the first acoustic peak.The energy density in neutrinos is predicted to be above the present observational upperbounds for most neutrino mass configurations, with the exception of the minimal valuespermitted by neutrino oscillation measurements with f /v (cid:46) . We therefore discusscosmological mechanisms in which the twin radiation is diluted to levels compatible withthese observational bounds in the subsequent sections of this paper. We now turn to simple scenarios that reconcile the mirror Twin Higgs with cosmologicalbounds, while taking care to respect the softly-broken Z symmetry. We begin with theout-of-equilibrium decay of a particle with symmetric couplings to the Standard Modeland twin sectors, in which the desired asymmetry is generated kinematically. That is tosay, the dimensionless couplings between the decaying particle and the two sectors areequal, and asymmetric energy deposition into the two sectors is a direct consequence ofthe asymmetric mass scales. In this respect, the scenario is philosophically similar to N naturalness [387], albeit with a parsimonious N = 2 sectors. See also [388], [389] and[372] for other recent related ideas of using long-lived particles for the dilution of darksectors.For simplicity, here we will focus on the case of a real scalar X coupled symmetricallyto the A and B sector Higgs doublets. Due to the difference in masses between the sectorsafter electroweak symmetry breaking, simple kinematic effects give X a larger branchingratio into the Standard Model. This occurs over a range of X masses within a few decades183 eutral Naturalness in the Sky Chapter 5 of the weak scale. If X decays out-of-equilibrium below the decoupling temperature ofthe two sectors, this injects different amounts of energy into the two sectors, effectivelysuppressing the temperature of the twin sector relative to the Standard Model. Thisrelative cooling suppresses the contribution of the light degrees of freedom of the mirrorTwin Higgs to below cosmological bounds. Insofar as the asymmetry is driven entirelyby kinematic effects arising from v (cid:28) f , the resulting temperature inequality betweenthe two sectors is proportional to powers of v/f .The requisite suppression of the twin sector temperature relative to the StandardModel temperature necessitates that the X dominate the cosmology before it decays.Our main discussion will follow the simplest case of an X which dominates absolutelybefore it decays, comprising all of the energy density of the universe and effectively actingas a ‘reheaton’. Afterwards, we will discuss the possibility of a ‘thermal history’ for X – a scenario where X is in thermal equilibrium with the two sectors, then chemicallydecouples at some high temperature and grows to dominate the cosmology before itdecays. This scheme will result in additional stringent constraints on the viable parameterspace. Asymmetric Reheating A Z -even scalar X which is a total singlet under the SM and twin gauge groups admitsthe renormalisable interactions V ⊃ λ x X ( X + x ) (cid:0) | H A | + | H B | (cid:1) + 12 m X X , (5.25)where m X is the mass of X (neglecting corrections from mixing that will be shown belowto be tiny), λ x is a dimensionless coupling and x is a dimensionful parameter, which onemay imagine identifying as a vacuum expectation value (vev) of X in an UV theory.Note that these interactions preserve the accidental SU (4) symmetry of the Twin Higgs.184 eutral Naturalness in the Sky Chapter 5 The X field may additionally possess self-interactions, which we omit here as they donot play a significant role in what follows.The interactions in (5.25) allow X to decay into light states in the Standard Model andtwin sectors. If X reheats the universe through out-of-equilibrium decays, the reheatingtemperatures of the two sectors will be determined by its partial decay widths, assumingthat the decay products do not equilibrate. In the instantaneous decay approximation, X decays when the Hubble parameter falls to its decay rate Γ X ∼ H . As we will show inSection 5.2.3, in order to evade cosmological constraints we need the X to decay mostlyinto the SM, so we may estimate Γ X ∼ Γ( X → SM ) . Then the energy that was containedin the X is transferred into radiation energy density, with the resulting temperature ofthe radiation given by (see [390]) T ∼ . (cid:115) Γ X M pl √ g (cid:63) (5.26)where g (cid:63) is the effective number of relativistic degrees of freedom, as defined in Section5.2.2, of the particles that are being reheated. Our numerical calculation of the reheatingtemperature, which will be presented in Section 5.2.3, indicates that the approxima-tion T ∼ . (cid:112) Γ X M pl reliably reproduces the reheating temperature over the range ofinterest.As shown in Section 5.2.2, the two sectors thermally decouple when the temperaturefalls below T decoup ∼ GeV, so reheating must take place to below this temperature. Ateven lower temperatures, big bang nucleosynthesis (BBN) places strong constraints onenergy injected into the SM at temperatures below O (1 − MeV [391]. Requiring thatthe SM reheating temperature is above ∼ MeV, these constraints on the SM reheatingtemperature become constraints on the decay rate of the X into the SM, which in theabove approximation becomes × − GeV (cid:46) Γ X (cid:46) × − GeV . (5.27)185 eutral Naturalness in the Sky Chapter 5 This then constrains the couplings λ x and x of the X to the Higgs sector. Importantly,it means that X must couple very weakly, in order to be long-lived enough to reheat toa low temperature, as will be shown below.The asymmetry in partial widths arises from different effects depending upon themass of X . For masses below the SM Higgs threshold, it is predominantly differences inmass mixing with the two Higgs doublets that produces the asymmetry, where the sizeof the mixing angles determines the effective coupling of X to the SM and twin particlesand therefore its branching fractions. For masses below the twin scale, the relative sizeof the mixing scales inversely with the vevs in each sector. Thus the hierarchy v (cid:28) f already present in the Higgs sector can automatically gives rise to a hierarchy in partialwidths. Note that additional threshold effects can enhance the asymmetry further, inparticular when X has mass above threshold for a significant decay channel in the SM, butbelow the corresponding mass threshold in the twin sector. Decays into on-shell Higgsescomplicate this picture further. In what follows, we first give an analytic calculation ofthe mass mixing effect, then present a more precise calculation of the decay widths intoeach sector.To lowest order, X decays via its interactions with the SM and twin Higgs, and onlyto other fermions and gauge bosons through its mass mixing with the Higgs scalars.Expanding the X potential after the SU (4) is spontaneously broken, the mixing termbetween X and h A in the scalar mass matrix is √ λ x xv A , while that between X and h B is √ λ x xv B . The h A and h B components of the X mass eigenstate, which we denoterespectively as δ XA and δ XB , can then be determined. The expressions for the mixingangles are in general complicated, but they simplify in limits m X < f and m X (cid:29) f : ( δ XA , δ XB ) ≈ λ x xv A m X − m h (cid:16) √ , v A f (cid:17) m X < f λ x xfm X (cid:16) √ v A f , (cid:17) m X (cid:29) f (5.28)186 eutral Naturalness in the Sky Chapter 5 to lowest order in ( v/f ) and κ/λ . The partial width for the decay of X into SM states(excluding the Higgs) is Γ( X → SM ) ≈ | δ XA | Γ h ( m h = m X ) , (5.29)where Γ h ( m h = m X ) denotes the decay width of a SM Higgs if it were to have mass m X . Note that the Higgs partial width must be computed using the vev v A ≈ v/ √ todetermine the masses and couplings of the SM particles. The partial width of the X intotwin states is computed the same way using δ XB and the vev v B ≈ f / √ .From the mixing angles (5.28), it is already apparent over what mass range asym-metric reheating from X decays will work. These give Γ( X → SM )Γ( X → Twin ) ∼ f /v A (cid:29) m X < fv A /f (cid:28) m X (cid:29) f. (5.30)Thus when the mass of X is less than the twin scale, the Standard Model will be reheatedto a higher temperature than the twin sector, but in the large mass limit this mechanismworks in the opposite direction and would appear to lead to preferential reheating of thetwin sector.More precise statements about the relative branching ratios and resulting temper-atures require additional care. In addition to decaying through mass mixing, X candecay into the Higgs mass eigenstates themselves if above threshold. As the energy isultimately transferred to the SM and twin sectors, we then need to consider how thesestates decay and account for the further mixing of the Higgs mass eigenstates into Higgsgauge eigenstates.For m X > m h , decay can occur into the lighter (SM-like) Higgs mass eigenstate h with partial width Γ( X → hh ) ≈ λ x x πm X (cid:115) − (cid:18) m h m X (cid:19) . (5.31)187 eutral Naturalness in the Sky Chapter 5 Similarly, for m X > m H , decays can proceed into HH with a similar partial width,but with the h mass replaced with that of the H . Above the intermediate threshold m X > m h + m H , there is also the mixed decay Γ( X → hH ) ≈ λ x πm X (cid:115) − (cid:18) m H + m h m X (cid:19) ( f δ AX + 2 v A δ BX ) . (5.32)Here, δ AX ≈ − δ hA δ XA − δ hB δ XB is the component of the h A gauge eigenstate in the X mass eigenstate and δ BX ≈ δ hB δ XA − δ hA δ XB is the corresponding component of the h B gauge eigenstate, where δ hA and δ hB are, respectively, the components of the SM Higgsin the h A and h B gauge eigenstates to zeroth order in λ x . Combining all ingredients, thisdecay width is of order λ x x . Since it is only the total decay width that is constrained tobe small by the demand that the SM reheating temperature lie in the required window,this fixes only a product of λ x and x . If x ∼ v , then the mixed decay to hH is effectivelysecond order in the small coupling λ x and can be neglected relative to the other partialwidths. Conversely if x (cid:28) v , then λ x is much larger and this decay cannot be neglected.In what follows we will work in the region of parameter space where mixed decays to hH are negligible.The rate of heat flow into each sector may be well approximated by adding the decayrates of X into each channel and weighting these by the fraction of energy transferredinto the particular sector. Of course, when X decays into Higgs particles, these in turndecay out of equilibrium into both the Standard Model and twin sectors. As the Higgsdecays are almost instantaneous, the fraction of energy transferred into each sector issimply that carried by the Higgs decay products multiplied by their branching fractionsfor each sector. The total rate at which X particles are transferred into the SM plasma188 eutral Naturalness in the Sky Chapter 5 is W ( X → SM ) ≈ Γ( X → SM ) + Γ( X → hh ) Br ( h → SM )+ Γ( X → HH )( Br ( H → SM ) + Br ( H → hh ) Br ( h → SM )) . (5.33)The corresponding rate for energy deposition into the twin sectors is simply given by thereplacement of SM (cid:55)→ Twin. The first term is the rate at which X decays directly intothe SM through mass mixing with the Higgs. The second is the fraction of X energythat is transferred into lighter Higgs states that subsequently decay into the SM. Thethird is the analogous term for decays into the heavy Higgs, where cascade decays of the H into the h and subsequently other SM particles must be included. Note that decaysof the heavy Higgs into the light Higgs make up a majority of decay width, because ofthe large quartic coupling required for the twin Higgs potential.Below the hh threshold, it is possible for X to decay via one on-shell and one off-shell Higgs boson. The partial width for off-shell Higgs production was calculated for X → hh ∗ → hb ¯ b and found to be negligible compared to two-body decays through massmixing and so we omit three-body decay widths in what follows.Ultimately, the complete partial widths for the decay of X into the Standard Modeland twin sectors includes the sum of decays into Higgs bosons h and H and direct decaysinto the fermions and gauge bosons of the two sectors. We compute the latter to anintended level of accuracy of ∼ (including, e.g., NLO QCD corrections to decaysinto light-flavor quarks), mostly following [392]. The resulting partial widths into theStandard Model and twin sectors are shown as a function of m X in Figure 5.5 with theratio of branching fractions displayed in Figure 5.6.Over much of the space below the Higgs mass, the branching ratio exhibits the ex-pected ( f /v ) scaling from the mass mixing. Below ∼ GeV, suppression of the twinpartial width arises because the twin bottom quark pair production threshold is crossed.189 eutral Naturalness in the Sky Chapter 5
SMTwin10 50 100 500 100010 - - - - m X ( GeV ) D e c a y W i d t h ( λ X x ) - ( G e V ) - Figure 5.5: The partial widths of the X into the SM (solid blue line) and twin sector(dashed orange) for f /v = 3 in units of ( λ x x ) . The light gray bands indicate regionsof QCD-related uncertainty in the SM calculation, while the darker gray bands indicatethe corresponding regions of uncertainty for the twin calculation.As m X nears m h , the SM branching fraction grows by ∼ orders of magnitude as the W W ∗ , ZZ ∗ , and then W W and ZZ decays go above threshold. Since the analogousthresholds are at much higher energies in the twin sector, the enhancement is not paral-leled by decays into the twin sector until m X is close to the twin scale. There is thereforea large range of masses m h (cid:46) m X (cid:46) m H over which the SM branching fraction dominatesby several orders of magnitude.Above the X → hh threshold, the ratio of decay widths is roughly constant in massup to the HH threshold. The twin sector decay rate is dominated by decays of on-shelllight Higgs into twin states, Γ( X → Twin ) ≈ Γ( X → hh ) Br ( h → Twin ) ∝ /m X asin (5.31). If the SM were also predominantly reheated through this channel, then theratio of branching fractions would again be approximately δ hA /δ hB ≈ ( f /v ) . However,190 eutral Naturalness in the Sky Chapter 5
10 50 100 500 100010 - - m X ( GeV ) Γ t / Γ S M Figure 5.6: The ratio of branching fractions of the X into the SM and twin sectorsat f /v = 3 . The dashed line gives the expected ( v/f ) scaling from the mass mixing;deviations are due to various mass threshold effects.the SM decay width also receives a larger contribution from decays through mass mixingbetween the X and the Higgs gauge eigenstates.For masses m X > m h , decays through mass mixing are dominated by the SM W W and ZZ channels. In this mass region, the decay rate of a Higgs into longitudinallypolarized vector bosons scales as Γ( h → W W, ZZ ) ∼ m X , but the mixing angle scalesas δ AX ∼ /m X (as in (5.28)), resulting in the same ∼ /m X scaling and thus a roughlyconstant ratio in this range of masses. Near m X ∼ TeV, decays into twin vector bosonsthrough mass mixing begin to dominate, and there is no favourable asymmetry in thebranching fractions, as discussed in this section. Even at higher masses, the effects ofheavy Higgs decays into light Higgs do not compensate sufficiently, as this partial widthscales with m X in the same way as the partial width for longitudinally polarised weakbosons.The constraint on the decay width from the required reheating temperature (5.27)191 eutral Naturalness in the Sky Chapter 5 translates into a constraint on the size of the coupling λ x x . For m X (cid:38) m h , this gives − . GeV (cid:46) λ x x (cid:46) − GeV, while for lower masses, this range increases to − GeV (cid:46) λ x x (cid:46) − . GeV at m X ∼ GeV.The gray bands in Figure 5.5 highlight regions where our analytic estimates of thepartial widths encounter enhanced uncertainties arising from the bottom and charmthresholds in both sectors. Over most of these ranges, we estimate the size of theseuncertainties to be either ∼ or confined to very small subregions. The thicknesses ofthese bands have been chosen conservatively, and ultimately the branching ratios shouldbe accurate to within a factor of ± Λ QCD of the bottom and charm mass thresholds.In particular, the prescription of [393] has been followed for approximating the bottompartial width close to the open flavour threshold. Resonant decay into gluons frombottomonia mixing has been neglected, although these resonant mass ranges are expectedto be only ∼ MeV wide at the CP-even, spin-0 bottomonia masses m X = m χ bi (see [393]and [394]). It should be noted, however, that at temperatures above that of the QCDphase transition, the quark decay products behave differently compared to that expectedin a low temperature environment. In particular, for hot enough temperatures, the b or c quarks may not hadronise and the partonic partial widths may more reliable. Theapplicability of the treatment of the flavour thresholds used here may therefore not bevalid if the decay occurs in the hot early universe. However, it is only very close to thethreshold itself (within several GeV) that this uncertainty becomes significant. Finally,quark masses have been neglected in the gluon partial width. For m X close to the flavourthresholds, this approximation breaks down, but the gluon branching fraction is only ∼ and so the error does not contribute to the uncertainty of the total width by morethan this order (it is this uncertainty that is responsible for most of the extension of thelength of the gray bands about the flavour threshold).Close to the charm threshold, the analogous uncertainties are even more poorly under-192 eutral Naturalness in the Sky Chapter 5 stood. Below the charm threshold, hadronic decays of a light scalar are highly uncertain(see [395] for discussion). We avoid these regions altogether by restricting our consider-ations to m X roughly above the twin charm threshold. Note that below the SM charmthreshold, the smaller decay rate of a Higgs-like scalar necessitates larger couplings λ X x for X to have a lifetime within the required reheating window. The larger couplings thenimply potentially stronger constraints from invisible mesonic decays. See [394, 395, 396]for further discussion and recent analysis of the pertinent experimental constraints.Taken together, the results in Figures 5.5 and 5.6 bear out the expectation that ascalar X with symmetric couplings to the Standard Model and twin sectors may nonethe-less inherit a large asymmetry in partial widths from the hierarchy between the scales v and f . Across a wide range of masses m X , the asymmetry is proportional to (or greaterthan) v /f , tying the reheating of the two sectors to the hierarchy of scales.Before proceeding to our computation of cosmological observables, we comment onan alternative variation on the reheating mechanism presented here that involves having X odd under the twin parity. This permits two renormalisable interactions with theHiggses to give a Higgs potential of the form: V ⊃ m (cid:0) | H A | + | H B | (cid:1) + λ (cid:0) | H A | + | H B | (cid:1) + (cid:15)X (cid:0) | H A | + | H B | (cid:1) + ˜ (cid:15)X (cid:0) | H A | − | H B | (cid:1) . (5.34)If X then acquires a vev at some scale, it may be possible to arrange for the resultingspontaneous breaking of the Z to give that required in the Higgs potential. However,we find that, in order for X to be long-lived and reheat the universe, its couplings to theHiggs must be highly suppressed and therefore that the resulting vev of X required toexplain the soft Z -breaking in the Higgs potential must be many orders of magnitudeabove the twin scale. If this is to be identified with the characteristic mass scale of X ,then a UV-completion of the twin Higgs is required for anything further to be said of theprospects of this possibility. However, if such a UV completion has similar structure to193 eutral Naturalness in the Sky Chapter 5 the couplings in (5.34), then asymmetric reheating may require a cancellation betweenthe odd and even couplings of X to the Higgs potential in order to suppress its twin-sectorbranching fraction (because the odd coupling appears with opposite signs in the couplingbetween X and the h A and h B states). We do not consider this possibility further. Imprints on the CMB
For appropriate values of m X , the out-of-equilibrium decay of X reheats the two sectorsto different temperatures and effectively dilutes the energy density in the twin sector.We obtain an analytic estimate of the effects of the X decay on the number of lightdegrees of freedom observed from the CMB by approximating both the decay of X andthe decoupling of species as instantaneous in Section 5.2.3. We then demonstrate thatthis estimate is reliable over most of the parameter space of interest with a numericalcalculation in Section 5.2.3. In Section 5.2.3 we consider neutrino masses and their jointconstraints with N eff . Analytic estimate of N eff If X dominates the energy density of the universe and thendecays, depositing energy ρ SM and ρ t into the SM and twin sectors respectively, then thetemperature ratio is determined by ρ t ρ SM = g t (cid:63) ( T treheat ) g SM (cid:63) ( T SMreheat ) (cid:18) T treheat T SMreheat (cid:19) ≈ Γ( X → Twin )Γ( X → SM ) , (5.35)where T SMreheat and T treheat are the reheating temperatures for each sector, while g SM (cid:63) and g t (cid:63) are the SM and twin effective number of relativistic degrees of freedom, respectively.We have assumed that the two sectors are cool enough that they have already decoupled.We point out that not only does the number of effective degrees of freedom in each sectorneed to be evaluated at the temperature of that sector, but that g t (cid:63) and g SM (cid:63) differ asfunctions of temperature due to the differences in the spectra of the sectors, as seen in194 eutral Naturalness in the Sky Chapter 5 Figure 5.1. As is well-known [390], reheating is a protracted process that occurs overa time-scale given by the lifetime of the reheaton. During this time, the temperatureof the plasma cools slowly because, while the energy is being replenished by the decayof the reheaton, it is simultaneously diluted and redshifted with the expansion of theuniverse. It is assumed in (5.35) that any primordial energy density in either sector issubdominant.The temperatures of both sectors then redshift in the same way, so the only additionaldifferences between their temperatures arise from changes to the effective number ofdegrees of freedom in each sector. By conservation of comoving entropy within eachsector, each evolves as T ieq /T i reheat = (cid:0) g i(cid:63) ( T i reheat ) /g i(cid:63) ( T ieq ) (cid:1) / a ( T reheat ) /a ( T eq ) where T ieq isthe temperature of the sector at matter-radiation equality, which the CMB probes asexplained in Section 5.2.2, and a ( T ) is the scale factor as a function of temperature. Inthe mirror Twin Higgs model, the two sectors have the same number of light degrees offreedom at recombination (three neutrinos and a photon, assuming that the neutrinosare still relativistic), so (cid:18) T t eq T SM eq (cid:19) = (cid:18) T treheat T SMreheat (cid:19) (cid:18) g t (cid:63) ( T treheat ) g SM (cid:63) ( T reheat ) (cid:19) / = Γ( X → Twin )Γ( X → SM ) (cid:18) g t (cid:63) ( T treheat ) g SM (cid:63) ( T reheat ) (cid:19) / . (5.36)As our range of reheat temperatures encompasses the QCD phase transitions of bothsectors, the factors of g (cid:63) can be important.Given the temperatures of the two sectors after X decays, we can obtain a simple es-timate of the contribution to N eff that neglects the impact of masses of the twin neutrinosdiscussed in Section (5.2.2), (∆ N eff ) m ν =0 = 47 (cid:18) (cid:19) / g SM (cid:63) ( T SM eq ) ρ t ( T t eq ) ρ SM ( T SM eq ) (5.37) ≈ . × Br ( X → Twin ) Br ( X → SM ) (cid:18) g t (cid:63) ( T treheat ) g SM (cid:63) ( T SMreheat ) (cid:19) / . (5.38)In this limit the most recent Planck data give a σ bound of ∆ N eff (cid:46) . assuming pure195 eutral Naturalness in the Sky Chapter 5 Λ CDM+ N eff [342]. This translates into the requirement ρ t ( T t eq ) ρ SM ( T SM eq ) ≈ Γ( X → Twin )Γ( X → SM ) (cid:46) . ,ignoring possible differences in g (cid:63) .Of course, as discussed in Section 5.2.2, the twin neutrino masses are relevant at thetemperature of matter-radiation equality, so we can obtain a more meaningful estimateof ∆ N eff using the results of Section 5.2.2 evaluated at the twin temperature determinedabove: ∆ N eff = (cid:18) (cid:19) / π (cid:0) T SM eq (cid:1) (cid:32) ρ t γ (cid:0) T t eq (cid:1) + (cid:88) α w tν α (cid:0) T t eq (cid:1) ρ t ν α (cid:0) T t eq (cid:1)(cid:33) (5.39) T t eq = T SM eq (cid:18) Γ( X → Twin )Γ( X → SM ) (cid:19) / (cid:18) g t (cid:63) ( T treheat ) g SM (cid:63) ( T SMreheat ) (cid:19) / (5.40)with T SM eq ≈ . eV [342] the photon temperature. While the right-hand side of thisequality has implicit dependence on T t eq through g t (cid:63) , this is only important if the reheat-ing occurs between the SM and twin QCDPTs and the neglecting of the factors of g (cid:63) is otherwise reliable. With the further inclusion of Standard Model neutrino masses oran extra sterile neutrino, the bound described above weakens to ∆ N eff (cid:46) . . As dis-cussed in Section 5.2.2, we are not aware of any analyses specific to our model involvingboth pure dark radiation and three sterile neutrinos with masses of order the photondecoupling temperature of the CMB and possibly cooler temperatures. In the absence ofsuch an analysis, we use the inequality ∆ N eff (cid:46) . to indicate where the present CMBmeasurements are likely to constrain the light degrees of freedom of this model, leavinga more detailed analysis of the CMB constraints as future work. In this case, the boundon the decay width ratio is Γ( X → Twin )Γ( X → SM ) (cid:46) . . The next generation of CMB experimentsare projected to strengthen this constraint to ∆ N eff (cid:46) . at the σ level [397]. Numerical Calculation of N eff A more precise study of the effect of X decay on thenumber of effective neutrino species at recombination may be performed by numericallysolving a system of differential equations for the entropy in X and the two sectors as a196 eutral Naturalness in the Sky Chapter 5 function of time. Following the analysis of Chapter 5.3 of [390] we have H = 1 a dadt = (cid:115) M P l ( ρ X + ρ SM + ρ t ) (5.41) dρ X dt + 3 Hρ X = − Γ X ρ X (5.42) ρ i = 34 (cid:18) π g i(cid:63) (cid:19) / S / i a − (5.43) S / i dS i dt = (cid:18) π g i(cid:63) (cid:19) / a (cid:16) ρ X Γ X → i + dq j → i dt (cid:17) , (5.44)where S i are comoving entropy densities and it has been assumed that X is cold by thetime it decays so that ρ X = m X n X with number density n X (this is reliable as we onlyconsider m X > GeV, which is above the decoupling temperature of ∼ GeV). Therate of heat flow from sector j to i per proper volume, dq j → j dt , is defined in (5.6). Toaccount for the temperature-dependence of the effective number of relativistic degrees offreedom in each sector, these equations are solved iteratively in the profiles of g i(cid:63) ( T i ) .The equations are solved in three stages: before, during and after the decoupling ofthe SM and twin sectors. The ratio f /v is fixed to for this analysis. Initial conditionswere chosen with ρ = 10 − ρ X , for combined SM and twin energy densities ρ . However,it is only the requirement that the initial energy density of X dominates over that ofthe SM and twin sectors that is important for simulating the cosmology over the timesof interest here, as the entirety of the latter is then generated by the subsequent decay.The results close to the decoupling and reheating epochs are otherwise insensitive to theinitial conditions and ultimately match onto the standard outcome [390] expected byequating the Hubble rate with the decay rate of X. The sectors are assumed to be inthermal equilibrium and sharing entropy until a temperature of GeV, below whichthey are evolved separately with the heat flows dq i → j dt switched on. Elastic scatteringswere neglected from the heat flow rate to accelerate the computation. It was verified forthe results found below that their contribution to the heat flow was always (cid:46) while197 eutral Naturalness in the Sky Chapter 5 the heat flow was itself not dominated by the Hubble rate. Heat flow was switched offagain once the twin temperature reaches . GeV, by which time thermal decoupling islong-since complete, and the sectors are subsequently evolved separately. Again, althoughthe strengthening of the colour force and the QCDPT make the perturbative tree-levelcomputation of the scattering rates unreliable at temperatures below ∼ GeV, as found inSection 5.2.2 and also in the results below, the sectors decouple above these temperatures.Notably, the impact of X on the expansion rate causes decoupling to occur at slightlyhotter temperatures than expected from the analysis of Section 5.2.2 for the decouplingin the standard cosmology. - GeV10 - GeV10 - GeV10 - GeV10 - GeV Γ t / Γ SM - - - - - - - ( s ) E ne r g y D en s i t y r a t i o Figure 5.7: Ratio of twin to SM energy densities throughout decoupling and reheating,for different decay rates Γ X . The dashed line corresponds to the prediction of from theratio of decay widths, here selected to be / .The ratio of energy densities in each sector determines N eff , from (5.39). A plot of thisratio over time is shown in Figure 5.7, with the expectation under the approximations ofthe previous section shown as well. This approximation is reliable as long as the lifetime198 eutral Naturalness in the Sky Chapter 5 of X is much longer than the temperature at which decoupling concludes, here ∼ GeV.The larger asymptotic value of the ratio of the blue line arises because the lifetime liesclose to the decoupling period, so that a significant fraction of the energy is transferredwhile the sectors are thermalised or partially thermalised and does not contribute towardasymmetric reheating. Equivalently, as will be discussed below, insufficient time elapsesbetween decoupling and reheating for the twin energy density to dilute and be repopulatedby the decays to the level predicted by (5.35). The subsequent bump represents the periodbetween the reheating of the twin sector by its QCD phase transition followed by thatof the SM. The green and orange lines correspond to reheating temperatures that liebetween SM and twin QCD phase transitions. In these cases, the reheating of the SMfrom the subsequent SM QCD phase transition raises its energy density relative to thetwin sector above that expected from the ratio of branching fractions. As this occursafter the lifetime of the reheaton, the estimate of the reheating temperatures presentedin (5.36) is still good as subsequent changes in the ratio due to the evolution of g (cid:63) areaccounted for in our analysis of the reheating scenarios.The steep drop in the energy density ratio corresponds to the brief period duringwhich the energy density of the twin sector present at decoupling dilutes and redshifts,which continues until it reaches a comparable size to the energy density that is beingreplenished by reheating. If the twin-sector branching fraction is highly suppressed, ascan occur in the “valley” region in Figure 5.6 with m h (cid:46) m X (cid:46) m h , then a longer time isrequired for this to happen, especially close to the decay epoch where the diminishing ofthe X population also contributes to a reduced reheating rate. These effects can prolongthe time required for the energy density ratio to converge to the asymptotic predictionof (5.35).Contour plots of ∆ N eff as a function of m X and f /v appear in Figure 5.8, along withcurrent and predicted bounds using the analytic results of Section 5.2.3. The minimum199 eutral Naturalness in the Sky Chapter 5 neutrino mass configuration with Dirac masses has also been assumed, although theresults are relatively insensitive to this provided that the twin neutrino masses are notwell above the eV scale. A SM reheating temperature of . GeV has been assumed. Atthis temperature, we have verified using the numerical calculation of Section 5.2.3 that thetwin sector reheating temperature is always roughly above the twin neutrino decouplingtemperature over the parameter space of the figure, ensuring that the neutrinos thermaliseonce produced in the decays and hence that the predictions of Section 5.2.3 are valid. Atreatment of the case in which the twin neutrinos are produced below their decouplingtemperature is beyond the scope of this analysis, but would involve the computation ofthe phase space spectrum of the neutrino decay products of the X .Also, as discussed in Section 5.2.2, a large temperature difference may partially relaxback if reheating occurs close to sector decoupling. However, a reliable calculation of theheat flow at the temperatures of interest here must incorporate non-perturbative effects.We do not perform such a computation, but note that, at a slightly higher SM reheatingtemperature of GeV where this computation is more reliable, ∆ N eff in Figure 5.8 canbe raised by up to an order of magnitude in the region with f /v (cid:46) and GeV (cid:46) m X (cid:46) GeV, notably where the twin sector partial width is suppressed relative to theSM by several orders of magnitude. The resulting ∆ N eff prediction is, nevertheless, stillout of observable reach. At the lower SM reheating temperature assumed in Figure 5.8,it is expected that decoupling will be further advanced and the enhancement in ∆ N eff would be weaker.We emphasize that, if the lifetime of X is sufficiently close to the time of decoupling,or equivalently, that the reheating temperature is sufficiently close to the decouplingtemperature, then the residual twin energy density left-over may be comparable to orgreater than that regenerated by reheating. Consequently, the suppression in ∆ N eff wouldbe less than that predicted in (5.36). In this respect, the projection of Figure 5.8 should200 eutral Naturalness in the Sky Chapter 5 be regarded as a lower bound on ∆ N eff . In the regions of high suppression, such as the“valley” region, the full asymmetry may not be generated before the complete decay of X when the reheating temperature is of similar order as the decoupling temperature. Inparticular, for the reheating temperature chosen here of . GeV and branching fractionBr ( X → Twin ) ∼ − , the numerical calculation of the energy density ratio saturatesat ∼ × − . We do not include this effect in Figure 5.8 as its only impact is to mildlyshift the unobservably small ∆ N eff = 10 − contour. Lower reheating temperatures wouldagree with the prediction of (5.35) were it not for the caveat that the twin neutrinosmay be produced out of equilibrium. However, this minimum value at which ∆ N eff issaturated can grow significantly with hotter reheating temperatures upon which it ishighly dependent.CMB-S4 observations will be able to probe a large portion of the most natural pa-rameter space, save the region m h (cid:46) m X (cid:46) m h where decays into the Standard Modeldominate well beyond the ratio f /v , as previously discussed. Significantly, precisionHiggs coupling measurements at the LHC are unlikely to probe the mirror Twin Higgsmodel beyond f ∼ v , so that the observation of additional dark radiation may be the first signature of a mirror Twin Higgs. Neutrino Masses
In addition to the bounds on N eff , we must also respect the boundson neutrino masses. The analysis remains nearly the same as in Section 5.2.2, butnow with the twin neutrinos at a lower temperature, as determined above. As men-tioned above, for large enough f /v and SM reheating temperature sufficiently close tothe lower bound, the reheating temperature of the twin sector may be below the twinneutrino decoupling temperature and the resulting energy density would be more diffi-cult to compute. For simplicity, we choose λ x x large enough such that the twin reheatingtemperature is always above the twin neutrino decoupling temperature.201 eutral Naturalness in the Sky Chapter 5 - - - - - - - - - - - - - - - - - - - Log m X ( GeV ) f / v Figure 5.8: Contours of log ∆ N eff as a function of m X and f /v , for T SMreheat = 0 . GeV.The dark blue region is in tension with Planck, while the light blue region will be testedby CMB-S4. Gray regions are where the X mass is below the twin charm threshold andour calculation of the twin sector partial width is unreliable.As before, we compute m eff ν as m eff ν = n t ν n SM ν (cid:88) α m t ν α . (5.45)In relating the scale factors at neutrino decoupling in each sector, we now have to use202 eutral Naturalness in the Sky Chapter 5 the above temperature ratio to find, analogously to Section 5.2.2, that m eff ν = (cid:18) Γ t Γ SM (cid:19) / (cid:18) g t (cid:63) ( T treheat ) g SM (cid:63) ( T SMreheat ) (cid:19) / (cid:18) fv (cid:19) n (cid:88) α m SM ν α , (5.46)where, again, n = 1 for Dirac masses and n = 2 for Majorana masses. Interestingly, ifthe branching ratios scale as Γ t / Γ SM = ( v/f ) , then we have m eff ν ∝ ( f /v ) − / n , so thecontribution grows with f /v for Majorana masses, but is suppressed for Dirac masses.As before, we consider the minimal mass spectrum of m ν = [0 . , . , . eV ] anda degenerate spectrum of m ν = [0 . eV , . eV , . eV ] / . In Figure 5.9 we plot thepredictions of the X reheating for ∆ N eff and m eff ν for both spectra and both Dirac andMajorana masses using the approximations of Section 5.2.2, for f /v from 3 to 10 andassuming the Γ t Γ SM ∼ ( v/f ) scaling; there are regions in the space of m X where thesuppression of m eff ν would be much higher.Dashed lines indicate the rough locations of present experimental limits from Planck2015, and projected bounds from CMB-S4. As mentioned in Section 5.2.2, we are unawareof any study of bounds on both m eff ν and ∆ N eff treated jointly. In the absence of this, weshow present and projected constraints on N eff and (cid:80) m ν from [398] and [375], ignoringcorrelations, as described in Section 5.2.2. Thermal Production
In our discussion up to this point, we have been agnostic about the origin of the cosmicabundance of X and have operated under the assumption that it absolutely dominatesthe cosmology before it decays. Here, we consider the possibility that X was thermallyproduced through freeze-out and subsequently dominates the universe as a relic beforedecaying. This thermal history is viable, but places strong constraints on the mass andcouplings of the X .The energy density of relativistic species redshifts as ρ r ∝ a − ∝ T , while the energy203 eutral Naturalness in the Sky Chapter 5 X m ν + m effν (eV)0.00.10.20.30.40.50.60.70.80.9 ∆ N e ff X Reheaton CosmologyMinimal, DiracMinimal, MajoranaDegenerate, DiracDegenerate, Majorana f / v Figure 5.9: Predicted values of ∆ N eff and (cid:80) m ν + m eff ν for minimal and degenerateneutrino mass spectra with both Dirac and Majorana masses for f /v from 3 to 10. ThePlanck 2015 [342] bounds on (cid:80) m ν and N eff , as discussed in Section 5.2.2, are representedby the dashed lines, and the projected CMB-S4 constraints are given by the dotted lines.It has been assumed that Γ t Γ SM ∼ ( v/f ) . Note however, that, from Figure 5.8, this scalingof the partial widths holds only for the mass range GeV (cid:46) m X (cid:46) GeV, outside ofwhich the twin partial width is more suppressed and the model is only testable through ∆ N eff over a smaller range in f /v .density of non-relativistic, chemically decoupled matter scales as ρ m ∝ a − . The energydensity contained in the X can therefore only grow relative to the energy density inthe thermal bath once it becomes non-relativistic. We found in Section 5.2.3 that byrecombination, ρ t /ρ SM (cid:46) . is needed to evade current bounds on ∆ N eff . Thus we204 eutral Naturalness in the Sky Chapter 5 need to have the energy density in the X dominate over the SM and twin plasmas bymore than this factor when it decays. If X becomes non-relativistic instantaneously atthe moment that its temperature reaches some fraction c ∼ O (0 . of its mass, then, as T ∝ /a and ρ X is ∼ /g (cid:63) of the total energy density, the mass is required to satisfy m X (cid:38) /c × g (cid:63) ( T = m X ) T SM X reheat . Since the SM reheating temperature is stronglyconstrained to be above BBN, this effectively puts a lower limit on the mass of the X .Importantly, X must freeze-out when relativistic or its energy density will be furtherBoltzmann suppressed. The lower limit on the mass of the X becomes an upper limiton the X ’s couplings - if it couples too strongly to the thermal bath, then it won’t freezeout early enough to be hot.In fact the situation is somewhat less favorable than the above analysis suggests,because it is relevant operators that must keep X in thermal equilibrium. For an X with the interactions introduced in Section 5.2.3, the annihilations have rates that scalewith temperature as Γ ∼ n X (cid:104) σv (cid:105) ∼ T for T (cid:38) m X , m h (where n X is the numberdensity of X and (cid:104) σv (cid:105) is its thermally averaged annihilation cross section). However, ina radiation-dominated universe, H ∼ T . Thus, at high enough temperatures, X is notin thermal equilibrium with the plasma and it is only once the universe cools enoughthat it may thermalise. Then, as the temperature drops, XX → q ¯ q annihilations becomesuppressed by the Higgs mass and subsequent Boltzmann suppression causes X to freeze-out. Note that the rates of these annihilation processes are controlled by the coupling λ x , independently of x , which is unconstrained by itself (other processes mediated by λ x x are found to be subdominant in the ensuing analysis, for the range of λ x over whichthermal production is successful). If the coupling is too weak to begin with, then the X never thermalises and thermal production cannot happen. Thermal production thereforerequires a careful balancing of parameters - small coupling λ x is preferred for X tofreeze-out hot and as early as possible, but the coupling is bounded from below by the205 eutral Naturalness in the Sky Chapter 5 requirement that X reach thermal equilibrium. This combination of constraints severelyrestricts the size of the parameter space over which thermal production is viable to casesin which the coupling is selected so that X enters and departs from thermal equilibriumat close to the same temperature.To obtain numerical predictions for this scenario, the calculation of Section 5.2.3was modified to account for the time after the freeze-out of X before it becomes non-relativistic. During this period we use (5.15) and (5.16) for the energy density of the X , approximating decays as being negligible, before switching over to (5.42) when thetemperature drops below the mass of the X . The approximation that the X does notdecay appreciably while it is relativistic must be good if there is to be sufficient timefor it to grow to dominate between becoming non-relativistic and decaying. The decaywidth of X was fixed to × − GeV, corresponding to a reheating temperature closeto the ∼ MeV lower limit, in order to maximise the amount of time over which theenergy density of X may grow relative to the SM plasma, thereby providing the greatestpossible reheating.The predictions for ∆ N eff from a thermally produced X are shown in Figure 5.10 forthe small regions of parameter space where this is viable, with f /v = 4 . We find that thedominant annihilation channels over this region are XX → t ¯ t and XX → b ¯ b , mediatedby the light Higgs, as well as their twin analogues, mediated by the heavy Higgs. Asexpected, the primordial energy density in the twin sector is too large compared to thatgenerated by the X for the asymmetric reheating to be effective when m X is too light( (cid:46) GeV in this case). Similarly, when the coupling is too strong, the X is heldin equilibrium for longer and freezes-out underabundant compared to the twin energydensity. However, when the coupling is too weak (the gray region), X never thermalisesto begin with (close to the boundary with this region, X freezes-out almost immediatelyafter thermalising). The peak in the contours occurs because of the “ H -funnel” in which206 eutral Naturalness in the Sky Chapter 5 the twin Higgs resonantly enhances annihilations into twin quarks. All of this region willbe testable by CMB-S4.
200 400 600 800 1000 1200 14006.06.26.46.66.87.07.2 mX ( GeV ) - Log λ X Figure 5.10: Parameter space where thermal production of X gives a large enough relicabundance to dilute the twin sector, for f /v = 4 . In the gray region, the coupling is tooweak for X to ever reach thermal equilibrium. The blue region is in tension with recentPlanck measurements of ∆ N eff , whereas all of the white region will be tested by CMB-S4. Predictions presented here for ∆ N eff close to the gray boundary are more uncertainbecause of the high sensitivity of the freeze-out temperatures to the coupling.207 eutral Naturalness in the Sky Chapter 5 As an alternative to the model presented above of late, out-of-equilibrium decays of a Z -symmetric scalar, one may imagine that the field driving primordial inflation reheats onlythe Standard Model to below the decoupling temperature of the two sectors. Productionof the twin particles then ceases at some time after the temperature drops below thedecoupling temperature during reheating.To make this consistent with a softly-broken Z symmetry, we extend the inflationarysector and introduce a ‘twinflaton’ that couples solely to the twin sector. The combinedinflationary and twinflationary sectors respect the Z symmetry. However, if the twosectors are entirely symmetric then one generally expects both inflationary dynamics tohappen coincidentally, which would result in identical reheating. We therefore rely on soft Z -breaking to give an asymmetry between the two sectors that causes the twinflationarysector to dominate the universe first. With the right arrangement we can end up withtwo distinct periods of inflation - a first caused by the “twinflaton” and a second thatthen reheats the Standard Model to below the decoupling temperature, having dilutedthe sources of twin-sector reheating from the first period.One simple mechanism for Z -breaking which is well-suited for introducing asymme-try to inflationary sectors is to introduce an additional Z -odd scalar field η (as was donein [399]). This admits linear and quadratic interactions to antisymmetric and symmetriccombinations of the inflationary sector fields, respectively. When η acquires a vev, thisintroduces an asymmetry in the fields to which it was coupled, dependent on the combi-nation of its vev and its couplings. If η is coupled to both the inflationary sectors and theHiggs sectors, it could be the sole source of Z -breaking in a twinflationary theory. Onemay generally imagine that, in some UV completion, the mechanism that softly breaksthe symmetry in the Higgs potential could also be the origin of the soft breaking of the208 eutral Naturalness in the Sky Chapter 5 inflationary sector.Cosmologically, this possibility may have similar observational signatures as the modeldiscussed in Section 5.2.3, where the amount of twin-sector dark radiation is determinedby the partial widths of the inflaton of the second inflationary epoch. If this dominantlycouples to the SM, then ∆ N eff will be suppressed which, while successfully resolving thecosmological problems of the Mirror Twin Higgs, may also be observationally inaccessible.However, additional, distinctly inflationary signatures may make this potentially testableby other cosmological observations.The mechanism of twinflation completes a catalog of models of asymmetric reheatingby late decays, which may be indexed by representations of the twin parity: the caseof a Z -even particle, in which a kinematic asymmetry in the partial widths providesthe reheating asymmetry, the case of a Z -odd particle, which can also provide thespontaneous Z -breaking required in the Higgs potential, and the case where two distinct,long-lived particles couple to each sector, which may also be related to inflation. Toy Model
As a toy model we here consider ‘twinning’ the simple ϕ chaotic inflation scenario.The inflationary dynamics in this case are easy to understand and we have the additionalbenefit that this inflationary model has been considered in the literature before as ‘DoubleInflation’ (see [400], [401] and [402]). We furthermore specialize to ‘double inflation witha break’, where there are two distinct periods of inflation which produces a step in thepower spectrum, and we consider the constraints that this places on our model. In thiscase, it is assumed that each inflaton field couples and therefore decays dominantly intothe sector to which it belongs. We will comment briefly on the case without a break andthe additional signals one could look for in that case.209 eutral Naturalness in the Sky Chapter 5 The potential of the inflationary sector for inflaton ϕ A and twinflaton ϕ B is V = 12 m A ϕ A + 12 m B ϕ B , (5.47)where m A (cid:54) = m B may arise from soft Z - breaking, perhaps related to the soft Z -breaking in the Higgs potential. In order for the ‘twinflation’ to occur first, we requirethat the energy of the B field initially dominates the energy density of the universe. Wetake the initial positions of the fields to be the same and m B (cid:29) m A . Call ϕ A (0) = ϕ B (0) = n √ M pl = nϕ c , where ϕ c is the critical value at which inflation stops and m B = rm A = rm with n, r > . The inflationary dynamics are then those of slowly-rolling scalar fields. At some point in the early universe we imagine that the slow-rollapproximation holds for both fields and the inflationary sector dominates the universe.The dominating field then slow-rolls down its potential for n − e -folds, while the lighterfield’s velocity is suppressed by approximately ϕ A r ϕ B . Solving the system numericallyreveals that the motion of ϕ A during this period can be neglected entirely.After ϕ B reaches the critical value √ M pl , it stops slow-rolling and begins oscillatingaround the minimum of its potential. For there to be two distinct periods of inflation,there must be a period where these oscillations dominate the universe, which requires thatthe energy densities of each inflaton ρ A and ρ B satisfy ρ B ( ϕ c ) = r m M pl > ρ A ( ϕ (0)) = n m M pl and therefore r > n . For a ϕ B potential, the energy in these oscillations redshiftsas ρ B ∼ a − . Eventually, the energy density in ϕ B drops below that of ϕ A and a newepoch of inflation, driven by ϕ A , begins. This provides a further n − e -folds of inflationto give n − in total, while the B -sector energy density is diluted away.Note that in order for our toy model to reheat below the decoupling temperature of thetwo sectors, reheating must occur well after the end of inflation. If, during the coherent Note that merely giving the twin field a much larger initial condition does not instigate twinflation.The dynamics of the subdominant field in this case are such that it will track the dominant field andboth will reach the critical value at the same time. This is easily confirmed numerically. eutral Naturalness in the Sky Chapter 5 oscillation of an inflaton, it becomes the case that the inflaton decay width Γ ∼ H , thenreheating will occur and result in temperature T reheat ∼ . (cid:112) Γ M pl . However, if Γ (cid:29) H when inflation ends, then all of the energy in the inflaton is immediately transferred andwe instead have reheating temperature T reheat ∼ . (cid:112) m α M pl for an inflaton of mass m α .But in order for T reheat (cid:46) GeV, it is required that m α (cid:46) − eV, so this possibility thatthe inflaton is short lived is not viable. The procedure of twinning inflationary potentialsmay be generalised to other, more realistic models, provided that this constraint uponthe reheating temperature can be satisfied. Observability
One could always make a twinflationary scenario consistent with observational constraintsby letting the second inflationary period of inflation last long enough. In our toy model,this would correspond to setting n high enough that the momentum modes which leftthe horizon during the first inflation have not yet re-entered the horizon - such a scenariowould look exactly like single-field chaotic inflation.Alternatively, we may also allow for n small enough that all the momentum modesthat left the horizon during the second inflation are currently sub-horizon. In this case,fluctuations at large enough wavenumbers (equivalently, small enough length scales) are‘processed’ (cross the horizon) at a different inflationary energy scale than those thatwere processed earlier, giving a step in the power spectrum. While Planck has measuredthe primordial power spectrum for modes with − Mpc − (cid:46) k (cid:46) . Mpc − (where thelower bound is set by the fact that smaller modes have not yet re-entered the horizon),proposed CMB-S4 experiments will increase this range [375] somewhat, as will be dis-cussed further below. We wish to show that the power spectrum of our toy model is notruled out and, furthermore, may be observed in the coming decades.The height of the step in the primordial power spectrum is determined by the energy211 eutral Naturalness in the Sky Chapter 5 scale of each period of inflation, so modes crossing the horizon in the second inflationaryperiod should be suppressed by a factor of r > n (cid:38) compared to those exitingin the first period. This degree of suppression is ruled out by Planck for the range ofmodes over which it has reconstructed the power spectrum [398]. A computation of theprimordial power spectrum for double inflation was given in [401]. It was found thatsignificant damping does not occur for modes which cross outside the horizon during thefirst inflationary period, re-enter during the inter-inflationary period and again cross thehorizon during the second inflationary period. It is only those scales which first cross thehorizon during the second inflationary period that are significantly damped (althoughother features in the shape, such as oscillations, may be present for modes that aresubhorizon during the intermediate period).The relation of this characteristic scale to present-day observables is easily done usingthe framework given in [403]. Let the subscripts a, b, c, d, e respectively correspond tothe beginning of the first inflationary period, the end of that period, the beginning ofthe second inflationary period, the end of that period, and the beginning of radiationdomination. During the coherent oscillation periods, the inflaton acts as matter and theenergy density falls as ρ ∝ a − . Let k i be the momentum whose mode is horizon-size atthe i epoch; k i = a i H i . The scales k i can be related using the number of e -folds in eachperiod, which are themselves determined from the first Friedmann equation. Denoting N ij = ln a j a i , we have k a = e − N ab nk b , k b = e N bc k c and similarly for the other characteristicmodes, where, in particular, slow-roll inflation predicts that N ab = N cd = n − . Theevolution of the characteristic momentum scales is shown schematically in Figure 5.11.Finally, k e can be determined using the conservation of comoving entropy: k e = πg / (cid:63) ( T ) g / (cid:63) ( T reheat ) T T reheat √ M pl , (5.48)where T and a are the temperature and scale factor today and T reheat is the reheating212 eutral Naturalness in the Sky Chapter 5 Figure 5.11: Schematic evolution of the characteristic scales in Twinflation, as seen bycomparing wavenumbers to the Hubble radius over time. Note that the time axis is nota linear scale.temperature (which is sufficiently low that only SM particles are produced). We workexplicitly with the convention a = 1 . The characteristic modes associated with the breakcan then be determined.As mentioned above, [401] shows that damping occurs for modes that exit the horizononly during the second inflationary period, so we should take the characteristic dampingscale to be the smallest such scale, which here corresponds roughly to k b This can be213 eutral Naturalness in the Sky Chapter 5 determined as k b = ne N bc − N cd + N de k e = n (cid:16) rn (cid:17) / exp (cid:18) − n − (cid:19) (cid:34) m M pl π g (cid:63) ( T reheat ) T reheat (cid:35) / πg / (cid:63) ( T ) g / (cid:63) ( T reheat ) T T reheat √ M pl (5.49)where k c only differs by the factor of ( r/n ) / (which is roughly close to unity). Onceagain, between k b and k c are oscillatory features, so k b should merely be taken as therough characteristic scale of the damping. - l og [ k b / ( M p c - ) ] Ruled Out by PlanckNot Enough InflationObservationally Single - Stage Inflation
Figure 5.12: The prediction for the characteristic suppression scale as a function of theinitial values of the fields. The mapped regions should be interpreted not as havinghard boundaries, but rather fuzzy endpoints where they break down. Here we have used T reheat = 10 MeV and r = 2 n .Now the characteristic damping scale is determined by m , n , r , and T reheat . Ourobservational bound on k b is that Planck has not seen this suppression on momentumscales at which it has been able to reconstruct the primordial power spectrum from theangular temperature anisotropy power spectrum, which is roughly k (cid:46) . Mpc − . Wehave constraints on the reheating temperature from rethermalization of the twin sectoror interrupted big bang nucleosynthesis MeV (cid:46) T reheat (cid:46) GeV, on having a period214 eutral Naturalness in the Sky Chapter 5 of intermediate matter domination between the two inflations r > n and on the totalnumber of e -folds n − (cid:38) to solve cosmological problems. Note that we requirefewer e -folds of inflation than is typically assumed in the standard cosmology. Since thelow reheating temperature gives fewer e -folds from reheating up to today, less inflationis needed to explain the large causal horizon and flatness.The normalization of the spectrum provides a further constraint, the most recentmeasurement of which come from Planck [398]. The scalar power spectrum at k (cid:63) =0 . Mpc − is measured to be P R ( k (cid:63) ) = e . ± . × − . Then for k (cid:63) < k c (i.e. k (cid:63) having left the horizon during the first period of inflation and not re-entered beforethe second, so no deviation from single-field inflation would be seen at this scale), thespectrum of [401] yields the constraint . × − = r m M pl ln (cid:18) k b k (cid:63) (cid:19) (cid:18) ln k b k (cid:63) + n (cid:19) . (5.50)The characteristic scale (5.49) depends much more strongly on n than it does on anyof the other parameters. In Figure 5.12, we give a rough idea of the scale as a function of n , having set T reheat = 10 MeV and r = 2 n , while m is chosen to satisfy the normalizationcondition. We also show the constraint on k b set by Planck. Note again that the regiondescribed as “observationally single-stage inflation" does still provide a solution to theproblem of reconciling cosmology with the mirror Twin Higgs.CMB-S4 will improve the constraint on k b through its improved measurement of polar-ization anisotropies [375]. With only precision measurements of temperature anisotropies,the un-lensed power spectrum cannot be so easily reconstructed from the lensed spec-trum. The effects of gravitational lensing of CMB place an upper limit on the sizeof primordial temperature anisotropies that can be measured [404], which Planck hassaturated. However, the polarization anisotropy power spectrum allows the removal oflensing noise from the temperature spectrum so that higher primordial modes can be215 eutral Naturalness in the Sky Chapter 5 detected. The polarization power spectrum itself also gives us another window into thehigh- (cid:96) modes of the primordial power spectrum, as the signal does not become dominatedby polarized foreground sources until higher scales near (cid:96) ∼ . CMB-S4 is projectedto make cosmic variance limited measurements of both the temperature and polarizationanisotropy power spectra up to the modes where they become foreground-contaminatedand so provide additional information on the shape of the primordial power spectrum[375]. The map from measurements of angular modes (cid:96) to contraints on spatial modes k depends on the evolution of the power spectrum between inflation and the CMB, soforecasting constraints requires careful study. However, these improvements will not testmost of the parameter space presented in Figure 5.12, where the step is predicted onextremely small distance scales.We have discussed a twinflationary model of double inflation with a break for sim-plicity, but there is a parametric regime where double inflation without a break gives therequired amount of asymmetric reheating into the Standard Model. With two periodsof inflation, the second period dilutes the energy density of the heavier field sufficientlythat there is no observable signal of it produced in reheating. However, even with onlyone period, inflation can continue for long enough after the inflaton turns the corner infield space such that, at late times, the fraction of the inflaton in the B state relativeto the A state is small enough that the expected energy densities that are transferredinto each sector satisfy ρ B /ρ A < . . This occurs as long as r (cid:38) . , assuming thatthe mixing angle of the slow-rolling field with the ϕ A and ϕ B fields entirely determinesthe fraction of its energy that reheats each sector. There is thus a much larger range of r where this toy model of inflation passes N eff bounds than our above analysis shows.The resulting imprint on the CMB could resemble that of the long-lived decay model ofSection 5.2.3, with ∆ N eff again being related to the ratio of branching fractions, althoughthis is dependent upon the UV completion of the Twin Higgs.216 eutral Naturalness in the Sky Chapter 5 When there is only one period of inflation, the step is smoothed out and less pro-nounced and it is necessary to locate the feature numerically. Furthermore, having mul-tiple degrees of freedom available allows for non-trivial evolution of momentum modesafter they become super-horizon, which does not occur in single-field inflation but maybe calculated from the full solution to the field equations [402]. While a twinned poten-tial leading to two periods of inflation generally predicts a step in the power spectrum,when there is no break the predictions, and thus constraints, this prediction become moremodel-dependent. Therefore we leave detailed predictions in that case for future studyusing realistic models and merely state that the range of r = 1 to n interpolates betweenthe single field spectrum and that with a step, as one would expect.There are also at least two other detectable effects one might expect in double inflationwithout a break and in general realistic twinflationary models. Interactions betweeninflaton fields may produce primordial non-Gaussianities, while the presence of additionaloscillating degrees of freedom may produce isocurvature perturbations. These do notappear in our toy model because the heavy field is exponentially damped during thesecond inflation. CMB-S4 is projected to improve Planck’s bounds on non-Gaussianitiesby a factor of ∼ and on isocurvature perturbations by perhaps an order of magnitude(though model-independent projections have not been made), so may be able to detector place useful constraints on realistic twinflationary models [375].We have introduced twinflation as a mirror Twin Higgs model which suppresses thecosmological effects of twin light degrees of freedom. It extends the mirror symmetry tothe inflationary sector. The soft Z symmetry-breaking of the Higgs sector may be usedin the inflationary sector to cause distinct periods of inflation. There exists a parametricregion where this is cosmologically indistinct from single-stage inflation, but also anotherin which it may be observable. As the direct product of inflation and the Mirror TwinHiggs, this is in some sense a minimal solution.217 eutral Naturalness in the Sky Chapter 5 In this section we have considered scenarios in which cosmology provides meaningfulinsight on solutions to the electroweak hierarchy problem. In particular, we have demon-strated several simple mechanisms in which the cosmological history of a mirror TwinHiggs model is reconciled with current CMB constraints and provides signatures accessi-ble in future CMB experiments. In the case of out-of-equilibrium decays, we have foundthat decays of Z -even scalars sufficiently dilute the energy density in the twin sectorwithout the addition of any new sources of Z -breaking. In much of the parameter space,the residual contribution to ∆ N eff is directly proportional to the ratio of vacuum expec-tation values v /f parameterizing the mixing between Standard Model and twin sectors(as well as the tuning of the electroweak scale), and may be within reach of CMB-S4experiments. In the case of twinflation, we have found that a (broken) Z -symmetric in-flationary sector may successfully dilute the energy density in the twin sector, as well aspotentially leave signatures in the form of a step in the primordial power spectrum or indepartures of primordial perturbations from adiabaticity and Gaussianity. In both cases,these models raise the tantalizing possibility that signatures of electroweak naturalnessmay first emerge in the CMB, rather than the LHC.There are a variety of possible directions for future work. Here we have focused onthe cosmological consequences of late-decaying scalars and twinned inflationary sectorswithout specifying their origin in a microscopic model. It would be interesting to con-struct complete models (where, e.g., supersymmetry or compositeness protect the scale f from UV contributions) in which the existence and couplings of late-decaying scalarsarise as intrinsic ingredients of the UV completion. Likewise, we have considered onlya toy model of twin chaotic inflation; it would be interesting to see if twinflation maybe realized in complete inflationary models that match the observed spectral index and218 eutral Naturalness in the Sky Chapter 5 constraints on the tensor-to-scalar ratio.While we have taken care to ensure that our scenarios respect the well-measured cos-mological history beneath T ∼ ∼ GeV. In the case oftwinflation, inflationary dilution of pre-existing baryon asymmetry requires that baryo-genesis occur in association with reheating or via another mechanism at temperaturesbelow ∼ GeV. It would be worthwhile to study models for the baryon asymmetryconsistent with these scenarios. Steps in this direction have been taken in [362], whichattempted to relate this to asymmetric dark matter in the twin sector.Likewise, any investigation of dark matter, be it related directly to the twin mech-anism or otherwise, must also address implications of the dilution. Previous work at-tempting to construct dark matter candidates in the twin sector [347, 357, 358, 359, 360,361, 362, 363]) has relied upon explicit Z -breaking that is not present in the mirrormodel. Dark matter may alternatively be unrelated to the Twin Higgs mechanism, suchas a a WIMP in some minimal extension of the electroweak sector that freezes-out as anoverabundant thermal relic and is then diluted to the observed density during reheating.Alternatively, it may be that the dark matter abundance is produced directly duringreheating. It would be interesting to study extensions of our scenarios that incorporatedark matter candidates directly related to the mechanism of dilution.Finally, we have only approximately parameterized Planck constraints and the reachof CMB-S4 on twin neutrinos and twin photons. Ultimately, more precise constraintsand forecasts may be obtained via numerical CMB codes. This strongly motivates the219 eutral Naturalness in the Sky Chapter 5 future study of CMB constraints on scenarios with three sterile neutrinos and additionaldark radiation whose temperatures differ from the Standard Model thermal bath.220 eutral Naturalness in the Sky Chapter 5 Tw in Dark Matter In this work, we build on a MTH framework where ‘hard’ breaking of the Z is absent.In [13, 405], it was realized that late-time asymmetric reheating of the two sectors couldarise naturally in these models if the spectrum were extended by a single new state. Thisasymmetric reheating would dilute the twin energy density and so attune the MTH withthe cosmological constraints. This dilution of twin energy density to negligible levelswould seem to hamper the prospect that twin states might constitute the dark matter,and generating dark matter was left as an open question. This presents a major challengetoward making such cosmologies realistic. However, we show that asymmetric reheatingperfectly sets the stage for a MTH realization of the ‘freeze-in’ mechanism for dark matterproduction [406, 407, 408, 409, 410, 411, 412, 413].Freeze-in scenarios are characterized by two assumptions: 1) DM has a negligibledensity at some early time and 2) DM interacts with the SM so feebly that it neverachieves thermal equilibrium with the SM bath. This second assumption is motivatedin part by the continued non-observation of non-gravitational DM-SM interactions. Bothassumptions stand in stark contrast to freeze-out scenarios.Freeze-twin dark matter is a particularly interesting freeze-in scenario because bothassumptions are fulfilled for reasons orthogonal to dark matter considerations: 1) thenegligible initial dark matter abundance is predicted by the asymmetric reheating alreadynecessary to resolve the MTH cosmology, and 2) the kinetic mixing necessary to achievethe correct relic abundance is of the order expected from infrared contributions in the We note that the feeble connection between the two sectors may originate as a small dimensionlesscoupling or as a small ratio of mass scales, either of which deserves some explanation. eutral Naturalness in the Sky Chapter 5
MTH. To allow the frozen-in twin electrons and positrons to be DM, we need onlybreak the Z by a relevant operator to give a Stueckelberg mass to twin hypercharge.Additionally, the twin photon masses we consider can lead to dark matter self-interactionsat the level relevant for small-scale structure problems [414].The next sections are organized as follows: In Section 5.3.2, we review the MTH andits cosmology in models with asymmetric reheating, and in Section 5.3.3 we introduce ourextension. In Section 5.3.4, we calculate the freeze-in yield for twin electrons and discussthe parameter space to generate dark matter and constraints thereon. We discuss futuredirections and conclude in Section 5.3.5. For the interested reader, we include somediscussion of the irreducible IR contributions to kinetic mixing in the MTH in AppendixA. The mirror twin Higgs framework [288] introduces a twin sector B , which is a ‘mirror’ copyof the Standard Model sector A , related by a Z symmetry. Upgrading the SU (2) A × SU (2) B gauge symmetry of the scalar potential to an SU (4) global symmetry adds aHiggs-portal interaction between the A and B sectors: V = λ (cid:0) |H| − f / (cid:1) , (5.51)where H = H A H B is a complex SU (4) fundamental consisting of the A and B sectorHiggses in the gauge basis. The SM Higgs is to be identified as a pseudo-Goldstone modearising from the breaking of SU (4) → SU (3) when H acquires a vacuum expectationvalue (vev) (cid:104)H(cid:105) = f / √ . Despite the fact that the global SU (4) is explicitly brokenby the gauging of SU (2) A × SU (2) B subgroups, the Z is enough to ensure that thequadratically divergent part of the one-loop effective action respects the full SU (4) . The222 eutral Naturalness in the Sky Chapter 5 lightness of the SM Higgs is then understood as being protected by the approximateaccidental global symmetry up to the UV cutoff scale Λ (cid:46) πf , at which point newphysics must come in to stabilize the scale f itself.We refer to twin particles by their SM counterparts primed with a superscript ’, andwe refer the reader to [288, 289] for further discussion of the twin Higgs mechanism.The thermal bath history in the conventional MTH is fully dictated by the Higgsportal in Eq. (5.51) which keeps the SM and twin sectors in thermal equilibrium down totemperatures O ( GeV ) . A detailed calculation of the decoupling process was performedin [13] by tracking the bulk heat flow between the two sectors as a function of SMtemperature. It was found that for the benchmark of f /v = 4 , decoupling begins at aSM temperature of T ∼ GeV and by ∼ GeV, the ratio of twin-to-SM temperaturesmay reach (cid:46) . without rebounding. While heat flow rates become less precise below ∼ GeV due to uncertainties in hadronic scattering rates, especially close to color-confinement, decoupling between the two sectors is complete by then for f /v (cid:38) . Forlarger f /v , the decoupling begins and ends at higher temperatures.As mentioned above, one class of solutions to this N eff problem uses hard breaking ofthe Z at the level of the spectra [289, 347, 348, 349, 350] while keeping a standard cos-mology. An alternative proposal is to modify the cosmology with asymmetric reheating todilute the energy density of twin states. For example, [405] uses late, out-of-equilibriumdecays of right-handed neutrinos, while [13] uses those of a scalar singlet. These newparticles respect the Z , but dominantly decay to SM states due to the already-presentsoft Z -breaking in the scalar sector. In [405], this is solely due to extra suppression by f /v -heavier mediators, while in [13], the scalar also preferentially mass-mixes with theheavier twin Higgs. [13] also presented a toy model of ‘Twinflation’, where a softly-broken Z -symmetric scalar sector may lead to inflationary reheating which asymmetrically re-heats the two sectors to different temperatures. In any of these scenarios, the twin sector223 eutral Naturalness in the Sky Chapter 5 may be diluted to the level where it evades Planck bounds [415] on extra radiation, yetis potentially observable with CMB Stage IV [416].We will stay agnostic about the particular mechanism at play, and merely assumethat by T ∼ GeV, the Higgs portal interactions have become inefficient and somemechanism of asymmetric reheating has occurred such that the energy density in thetwin sector has been largely depleted, ρ twin ≈ . This is consistent with the results ofthe decoupling calculation in [13] given the uncertainties in the rates at low temperatures,and will certainly be true once one gets down to few × MeV.One may be concerned that there will be vestigial model-dependence from irrelevantoperators induced by the asymmetric reheating mechanism which connect the two sectors.However, these operators will generally be suppressed by scales above the reheating scale,as in the example studied in [13]. Prior to asymmetric reheating, the two sectors are inthermal equilibrium anyway, so these have little effect. After the energy density in twinstates has been diluted relative to that in the SM states, the temperature is far below theheavy masses suppressing such irrelevant operators, and thus their effects are negligible.So we may indeed stay largely agnostic of the cosmological evolution before asymmetricreheating as well as the details of how this reheating takes place. We take the absenceof twin energy density as an initial condition, but emphasize that there are external,well-motivated reasons for this to hold in twin Higgs models, as well as concrete modelsthat predict this occurrence naturally.
In order to arrange for freeze-in, we add to the MTH kinetic mixing between the SM andtwin hypercharges and a Stueckelberg mass for twin hypercharge. At low energies, these If asymmetric reheating leaves some small ρ twin > , then mirror baryon asymmetry can lead totwin baryons as a small subcomponent of dark matter [417]. eutral Naturalness in the Sky Chapter 5 reduce to such terms for the photons instead, parametrized as L += − (cid:15) F µν F (cid:48) µν − m γ (cid:48) A (cid:48) µ A (cid:48) µ . (5.52)This gives each SM particle of electric charge Q an effective twin electric charge (cid:15)Q . The twin photon thus gives rise to a ‘kinetic mixing portal’ through which the SM bathmay freeze-in light twin fermions in the early universe.The Stueckelberg mass constitutes soft Z -breaking, but has no implications forthe fine-tuning of the Higgs mass since hypercharge corrections are already consistentwith naturalness [289]. We will require m γ (cid:48) > m e (cid:48) , to prevent frozen-in twin electron/-positron annihilations, and m γ (cid:48) > m e (cid:48) , to ensure that resonant production through thetwin photon is kinematically accessible. Resonant production will allow a much smallerkinetic mixing to generate the correct relic abundance, thus avoiding indirect boundsfrom supernova cooling. We note that while taking m γ (cid:48) (cid:28) f does bear explanation, theparameter is technically natural.On the other hand, mixing of the twin and SM U (1) s preserves the symmetries ofthe MTH EFT, so quite generally one might expect it to be larger than that needed forfreeze-in. However, it is known that in the MTH a nonzero (cid:15) is not generated throughthree loops [288]. While such a suppressed mixing is phenomenologically irrelevant formost purposes, here it plays a central role. In Appendix A, we discuss at some length thevanishing of infrared contributions to kinetic mixing through few loop order. If nonzero Note that twin charged states do not couple to the SM photon. Their coupling to the SM Z bosonhas no impact on freeze-in at the temperatures under consideration. Furthermore, the miniscule kineticmixing necessary for freeze-in has negligible effects at collider experiments. See Ref. [418] for details. While we are breaking the Z symmetry by a relevant operator, the extent to which a Stueckelbergmass is truly soft breaking is not clear. Taking solely Eq. (5.52), we would have more degrees of freedomin the twin sector than in the SM, and in a given UV completion it may be difficult to isolate this Z -asymmetry from the Higgs potential. One possible fix may be to add an extremely tiny, experimentallyallowed Stueckelberg mass for the SM photon as well [419], though we note this may be in violationof quantum gravity [420, 421] or simply be difficult to realize in UV completions without extreme fine-tuning. We will remain agnostic about this UV issue and continue to refer to this as ‘soft breaking’,following [418]. eutral Naturalness in the Sky Chapter 5 contributions appear at the first loop order where they are not known to vanish, kineticmixing of the order (cid:15) ∼ − − − is expected.The diagrams which generate kinetic mixing will likely also generate higher-dimensionaloperators. These will be suppressed by (twin) electroweak scales and so, as discussedabove for the irrelevant operators generated by the model-dependent reheating mecha-nism, freeze-in contributions from these operators are negligible. Tw in Dark Matter As we are in the regime where freeze-in proceeds while the temperature sweeps overthe mass scales in the problem, it is not precisely correct to categorize this into either“UV freeze-in” or “IR freeze-in”. Above the mass of the twin photon, freeze-in proceedsthrough the marginal kinetic mixing operator, and so a naive classification would saythis is IR dominated. However, below the mass of the twin photon, the clearest approachis to integrate out the twin photon, to find that freeze-in then proceeds through anirrelevant, dimension-six, four-Fermi operator which is suppressed by the twin photonmass. Thus, at temperatures T SM (cid:46) m γ (cid:48) , this freeze-in looks UV dominated. This leadsto the conclusion that the freeze-in rate is largest at temperatures around the mass of thetwin photon. Indeed, this is generally true of freeze-in — production occurs mainly attemperatures near the largest relevant scale in the process, whether that be the largestmass out of the bath particles, mediator, and dark matter, or the starting thermal bathtemperature itself in the case that one of the preceding masses is even higher.As just argued, freeze-in production of dark matter occurs predominantly at andsomewhat before T ∼ m γ (cid:48) . We require m γ (cid:48) (cid:28) GeV so that most of the freeze-in yieldcomes from when
T < GeV, which allows us to retain ‘UV-independence’ in that weneed not care about how asymmetric reheating has occurred past providing negligible226 eutral Naturalness in the Sky Chapter 5 density of twin states at T = 1 GeV. Specifically, we limit ourselves to m γ (cid:48) < m π , bothfor this reason and to avoid uncertainties in the behavior of thermal pions during theepoch of the QCD phase transition. However, we emphasize that freeze-in will remain aviable option for producing a twin DM abundance for even heavier dark photons. Butthe fact that the freeze-in abundance will be generated simultaneously with asymmetricreheating demands that each sort of asymmetric reheating scenario must then be treatedseparately. Despite the additional difficulty involved in predicting the abundance forlarger twin photon masses, it would be interesting to explore this part of parameterspace. In particular, it would be interesting to consider concrete scenarios with twinphotons in the range of tens of GeV [422].To calculate the relic abundance of twin electrons and positrons, we use the Boltzmannequation for the number density of e (cid:48) : ˙ n e (cid:48) + 3 Hn e (cid:48) = (cid:88) k,l − (cid:104) σv (cid:105) e (cid:48) ¯ e (cid:48) → kl ( n e (cid:48) n ¯ e (cid:48) − n eq e (cid:48) n eq ¯ e (cid:48) ) , (5.53)where (cid:104) σv (cid:105) e (cid:48) ¯ e (cid:48) → kl is the thermally averaged cross section for the process e (cid:48) ¯ e (cid:48) → kl , thesum runs over all processes with SM particles in the final states and e (cid:48) ¯ e (cid:48) in the initialstate, and n eq e (cid:48) is the equilibrium number density evaluated at temperature T . As we arein the parametric regime in which resonant production of twin electrons through the twinphoton is allowed, → annihilation processes ¯ f f → γ (cid:48) → ¯ e (cid:48) e (cid:48) , with f a charged SMfermion, entirely dominate the yield.In accordance with the freeze-in mechanism, n e (cid:48) remains negligibly smaller than itsequilibrium number density throughout the freeze-in process, and so that term is ignored.It is useful to reparametrize the abundance of e (cid:48) in terms of its yield, Y e (cid:48) = n e (cid:48) /s where s = π g ∗ s T is the entropy density in the SM bath. Integrating the Boltzmann equation227 eutral Naturalness in the Sky Chapter 5 using standard methods, we then find the yield of e (cid:48) today to be Y e (cid:48) = (cid:90) T i dT (45) / √ π √ g ∗ g ∗ s M P l T (cid:18) T + ∂ T g ∗ s g ∗ s (cid:19) × (cid:88) ¯ ff → ¯ e (cid:48) e (cid:48) (cid:104) σv (cid:105) ¯ ff → ¯ e (cid:48) e (cid:48) n eq ¯ e (cid:48) n eq e (cid:48) , (5.54)where T i = 1 GeV is the initial temperature of the SM bath at which freeze-in begins inour setup, g ∗ ( T ) is the number of degrees of freedom in the bath, and M P l is the reducedPlanck mass. We will calculate this to an intended accuracy of 50%. To this level ofaccuracy, we may assume Maxwell-Boltzmann statistics to vastly simplify the calculation[423]. As a further simplification, we observe that the ∂ T g (cid:63)s term is negligible comparedto /T except possibly during the QCDPT - where uncertainties on its temperaturedependence remain [369] - and so we ignore that term. The general expression for thethermally averaged cross section of the process → is then (cid:104) σv (cid:105) n eq n eq = T π s (cid:90) ∞ Max ( m m T , m m T ) dxx (5.55) × (cid:112) [1 , (cid:112) [3 , K ( x ) (cid:90) d (cos θ ) |M| → , where s is 1 if the final states are distinct and 2 if not, x = √ s /T , (cid:112) [ i, j ] = (cid:113) − (cid:0) m i + m j xT (cid:1) (cid:113) − (cid:0) m i − m j xT (cid:1) , and |M| → is the matrix element squared summed(not averaged) over all degrees of freedom. To very good approximation, the yield re-sults entirely from resonant production, and so we may analytically simplify the matrixelement squared for ¯ f f → ¯ e (cid:48) e (cid:48) using the narrow-width approximation (cid:90) d (cos θ ) |M| ff → ¯ e (cid:48) e (cid:48) ≈ π α (cid:15) (cid:0) m f + m γ (cid:48) (cid:1) (5.56) × (cid:0) m e (cid:48) + m γ (cid:48) (cid:1) Γ γ (cid:48) m γ (cid:48) T δ ( x − m γ (cid:48) /T ) . Γ γ (cid:48) is the total decay rate of the twin photon.228 eutral Naturalness in the Sky Chapter 5 For the range of m γ (cid:48) we consider, the twin photon can only decay to twin electronand positron pairs. Thus, its total decay rate is Γ γ (cid:48) = α (cid:0) m γ (cid:48) + 2 m e (cid:48) (cid:1) m γ (cid:48) (cid:115) − m e (cid:48) m γ (cid:48) . (5.57)Its partial widths to SM fermion pairs are suppressed by (cid:15) , and so contribute negligiblyto its total width.The final yield of twin electrons is then Y e (cid:48) ≈ m γ (cid:48) π (45) / M P l √ π (cid:88) f (cid:90) T i T f dT Γ γ (cid:48) → ¯ ff K ( m γ (cid:48) T ) √ g ∗ g ∗ s T , (5.58)where T f = Λ QCD for quarks, T f = 0 for leptons, Γ γ (cid:48) → ¯ ff is the partial decay width ofthe twin photon to f ¯ f , and the sum is over all SM fermion-antifermion pairs for which m γ (cid:48) > m f .Since we have approximated the yield as being due entirely to on-shell productionand decay of twin photons, the analytical expression for the yield in Eq. (5.58) exactlyagrees with the yield from freezing-in γ (cid:48) via ‘inverse decays’ ¯ f f → γ (cid:48) , as derived in [410].We have validated our numerical implementation of the freeze-in calculation by success-fully reproducing the yield in similar cases found in [423, 424]. We have furthermorechecked that reprocessing of the frozen-in dark matter [411, 425] through e (cid:48) ¯ e (cid:48) → e (cid:48) ¯ e (cid:48) e (cid:48) ¯ e (cid:48) is negligible here, as is the depletion from e (cid:48) ¯ e (cid:48) → ν (cid:48) ¯ ν (cid:48) .An equal number of twin positrons are produced as twin electrons from the freeze-inprocesses. Requiring that (cid:15) reproduce the observed DM abundance today, we find (cid:15) = (cid:115) Ω χ h ρ crit /h m e (cid:48) ˜ Y e (cid:48) s , (5.59)where Ω χ h ≈ . , ρ crit /h ≈ . × − GeV / cm , and s ≈ / cm [48]. ˜ Y e (cid:48) is thetotal yield with the overall factor of (cid:15) removed. This requisite kinetic mixing appears To be conservative, we calculate the rate assuming all interactions take place at the maximum √ s (cid:39) m γ (cid:48) and find that it is still far below Hubble. We perform the calculation of the cross sectionusing MadGraph [426] with a model implemented in Feynrules [427]. eutral Naturalness in the Sky Chapter 5 Figure 5.13: Contours in the plane of twin photon mass m γ (cid:48) and kinetic mixing (cid:15) whichfreeze-in the observed DM abundance for two values of f /v . The dip at high massescorresponds to additional production via muon annihilations. In the dashed segments,self-interactions occur with σ elastic /m e (cid:48) (cid:38) cm / g. Also included are the combined super-nova cooling bounds from [428, 429].in Fig. 5.13 as a function of the twin photon mass m γ (cid:48) for the two benchmark f /v values 4 and 10. In grey, we plot constraints from anomalous supernova cooling. Tobe conservative, we include both, slightly different bounds from [428, 429]. The dashedregions of the lines show approximately where self-interactions through Bhabha scatteringare relevant in the late universe, σ elastic /m e (cid:48) (cid:38) cm / g. Self-interactions much largerthan this are constrained by the Bullet Cluster [430, 431, 432] among other observations.Interestingly, self-interactions of this order have been suggested to fix small-scale issues,230 eutral Naturalness in the Sky Chapter 5 and some claimed detections have been made as well. We refer the reader to [414] for arecent review of these issues.As mentioned above and discussed further in Appendix A, the level of kinetic mixingrequired for freeze-in is roughly of the same order as is expected from infrared contri-butions in the MTH. It would be interesting to develop the technology to calculate thehigh-loop-order diagrams at which it may be generated. In the context of a completemodel of the MTH where kinetic mixing is absent in the UV, (cid:15) is fully calculable anddepends solely on the scale at which kinetic mixing is first allowed by the symmetries.Calculating (cid:15) would then predict a minimal model at some m γ (cid:48) to achieve the right darkmatter relic abundance, making this effectively a zero-parameter extension of MTH mod-els with asymmetric reheating. Importantly, even if (cid:15) is above those shown in Fig. 5.13,that would simply point to a larger value of m γ (cid:48) which would suggest that the parame-ter point depends in more detail on the mechanism of asymmetric reheating. We notethat in the case that the infrared contributions to (cid:15) are below those needed here, therequired kinetic mixing may instead be provided by UV contributions and the scenariois unaffected. The mirror twin Higgs is perhaps the simplest avatar of the Neutral Naturalness program,which aims to address the increasingly severe little hierarchy problem. Understandinga consistent cosmological history for this model is therefore crucial, and an importantstep was taken in [13, 405]. As opposed to prior work, the cosmology of the MTH wasremedied without hard breaking of the Z symmetry by utilizing asymmetric reheatingto dilute the twin energy density. Keeping the Z as a good symmetry should simplifythe task of writing high energy completions of these theories, but low-scale reheating may231 eutral Naturalness in the Sky Chapter 5 slightly complicate cosmology at early times. These works left as open questions howto set up cosmological epochs such as dark matter generation and baryogenesis in suchmodels. We have here found that at least one of these questions has a natural answer.In this work, we have shown that twin electrons and positrons may be frozen-inas dark matter following asymmetric reheating in twin Higgs models. This requiresextending the mirror twin Higgs minimally with a single free parameter: the twin photonmass. Freezing-in the observed DM abundance pins the required kinetic mixing to a levelexpected from infrared contributions in MTH models. In fact, the prospect of calculatingthe kinetic mixing generated in the MTH could make this an effectively parameter-freeextension of the MTH. Compared to generic freeze-in scenarios, it is interesting in thiscase that the “just so” stories of feeble coupling and negligible initial density were alreadypresent for reasons entirely orthogonal to dark matter.This minimalism in freeze-twin dark matter correlates disparate signals which wouldallow this model to be triangulated with relatively few indirect hints of new physics. Ifdeviations in Higgs couplings are observed at the HL-LHC or a future lepton collider,this would determine f /v [433, 434, 12, 10], which would set the dark matter mass.An observation of anomalous cooling of a future supernova through the measurement ofthe neutrino ‘light curve’ might allow us to directly probe the m γ (cid:48) , (cid:15) curve [428, 429],though this would rely on an improved understanding of the SM prediction for neutrinoproduction. Further astrophysical evidence of dark matter self-interactions would pointto a combination of f /v and m γ (cid:48) . All of this complementarity underscores the value of arobust experimental particle physics program whereby new physics is pursued via everyimaginable channel. We thank Jae Hyeok Chang for a discussion on this point. hapter 6Neutral Naturalness in the Ground
There is nothing like looking, if you want tofind something. You certainly usually findsomething, if you look, but it is not alwaysquite the something you were after.J.R.R. Tolkien
The Hobbit , 1937 [435] eutral Naturalness in the Ground Chapter 6
Long-Lived Particles
When we introduced models of Neutral Naturalness in Section 4.1.2, we were motivatedby the lack of signals of new, SM-charged particles at colliders. However, experimentalistsare very clever, and it is in fact possible to observe the effects of such particles if youknow where to look.In particular, Neutral Naturalness models usually exhibit a ‘hidden valley’ type phe-nomenology, wherein a dark sector is connected to the SM only through some heavystates. At a collider, a high energy collision can transfer energy from our sector to theseheavy dark sector states, which may then decay just as heavy SM particles do. If the darksector contains absolutely stable particles, then the energy may cascade into those states,which simply leave the detector. But without a symmetry dictating that stability, darksector states may be destabilized by the effects of that same high-energy link to the SM.That connection may induce higher-dimensional operators which allow the decay of thedark sector states back into SM states. Since this decay channel comes from interactionsof heavy states, the width may be highly suppressed, leaving to a macroscopically longlifetime, even when the scale set by the particle’s mass is microscopically small. Thisleads to the appearance of SM particles out of nowhere inside a detector a macroscopicdistance away from the interaction point in a collider.In fact there are many reasons an unstable particle may be long-lived, and we see amultitude of examples in the SM itself. In analogy with the hidden valley phenomenology,charged pions are long-lived compared to the QCD scale because their decay must takeplace through a heavy W boson. Neutrons are similarly very long-lived as a result of theirsmall mass splitting with protons. Protons themselves are very long-lived as a result ofan approximate global symmetry (see Section 1.1.2 for some little-appreciated subtletiesin this reasoning). The SM Higgs is long-lived compared to the electroweak scale because234 eutral Naturalness in the Ground Chapter 6
Figure 6.1: Schematic depiction of hidden valley phenomenology. When another sectoris connected to the SM only through heavy particles, it is possible to produce particlesfrom that sector in high energy collisions. If the other sector has phenomenology similarto QCD, this may lead to the production of many states in a dark shower. If (someof) the lowest-lying states in the dark sector may decay back to SM particles, these willgenerically be displaced decays because they must go through the higher-dimensionaloperator. In the Twin Higgs, this may be production of either Higgs leading to darkshowers with many twin glueballs, some of which mix with the Higgs and decay into SMstates. Figure adapted from [436]. 235 eutral Naturalness in the Ground Chapter 6 its leading decay mode proceeds through the small bottom quark Yukawas. Given thegenericity of long-lived particles in our sector, it’s entirely reasonable to imagine that somedark sector will have similar phenomenology. So even aside from our Neutral Naturalnessmotivation, such searches are generically useful and interesting things to look for.In this section we forecast how well an electron-positron collider will be able to probeHiggs decays to long-lived particles, using the parameters for some machines which havebeen proposed by the community and search strategies of our own design. Such forecast-ing is crucially important at the present time, as the community is still discussing whatthe next collider is that we will build . It’s clearly necessary to know what sort of physicsprogram we expect we can carry out before we build a machine to do it. And these stud-ies are used to motivate different types of detectors, their detailed design features andhow trigger bandwidth is allocated. Note that I have no idea what the ‘present time’ is for you, the reader, but I am confident thisstatement remains true regardless. eutral Naturalness in the Ground Chapter 6
Following the discovery of the Higgs boson in 2012 [437, 438], the precision study of itsproperties has rapidly become one of the centerpieces of the physics program at the LHC.The expansion of this program beyond the LHC has become one of the key motivators forproposed future accelerators, including lepton colliders such as CEPC [439, 440], FCC-ee [441], the ILC [442, 443], and CLIC [444, 445] that would operate in part as Higgsfactories.The potential gains of a precision Higgs program pursued at both the LHC and fu-ture colliders are innumerable. Confirmation of Standard Model predictions for Higgsproperties would mark a triumphant validation of the theory and illuminate phenomenanever before seen in nature. The observation of deviations from Standard Model predic-tions, on the other hand, would point the way directly to additional physics beyond theStandard Model. Such deviations could take the form of changes in Higgs couplings toitself or other Standard Model states, or they could appear as exotic decay modes notpredicted by the Standard Model. The latter possibility has been extensively exploredfor prompt exotic decay modes in the context of both the LHC (see e.g. [446, 447]) andfuture Higgs factories [448].However, an equally compelling possibility is for new physics to manifest itself inexotic decays of the Higgs boson to long-lived particles (LLPs). Such signals were firstconsidered in the context of Hidden Valleys [449, 450, 451] and subsequently found toarise in a variety of motivated scenarios for physics beyond the Standard Model, includingsolutions to the electroweak hierarchy problem [289] and models of baryogenesis [452];for an excellent recent overview, see [453]. The search for exotic Higgs decays into LLPsnecessarily involves strategies outside the scope of typical analyses. The non-standardnature of these signatures raises the compelling possibility of discovering new physics237 eutral Naturalness in the Ground Chapter 6 that has been heretofore concealed primarily by the novelty of its appearance.There is a rich and rapidly growing program of LLP searches at the LHC. A varietyof existing searches by the ATLAS, CMS, and LHCb collaborations (e.g. [454, 455, 456];for a recent review see [457]) constrain Higgs decays into LLPs at roughly the percentlevel across a range of LLP lifetimes. Significant improvements in sensitivity are possiblein future LHC runs with potential advances in timing [458], triggers [459, 460, 461], andanalysis strategies [353, 352]. Most notable among these is the possible implementationof a track trigger [459, 460], which would significantly lower the trigger threshold forHiggs decays into LLPs and potentially allow sensitivity to branching ratios on the orderof − in zero-background scenarios.While studies of prompt exotic Higgs decays at future colliders [448] have demon-strated the potential for significantly improved reach over the LHC, comparatively littlehas been said about the prospects for constraining exotic Higgs decays to long-lived par-ticles at the same facilities. In this work we take the first steps towards filling this gapby studying the sensitivity of e + e − Higgs factories to hadronically-decaying new particlesproduced in exotic Higgs decays with decay lengths ranging from microns to meters. Forthe sake of definiteness we restrict our attention to circular Higgs factories operating ator near the peak rate for the Higgsstrahlung process e + e − → hZ , namely CEPC andFCC-ee, while also sketching the corresponding sensitivity for the √ s = 250 GeV stageof the ILC. While essentially all elements of general-purpose detectors may be broughtto bear in the search for long-lived particles, the distribution of decay lengths for a givenaverage lifetime make it advantageous to exploit detector elements close to the primaryinteraction point. We thus focus on signatures that can be identified in the tracker. Inorder to provide a faithful forecast accounting for realistic acceptance and background A notable exception is CLIC, for which a study of tracker-based searches for Higgs decays to LLPshas been recently performed [462]. For preliminary studies of other non-Higgs LLP signatures at futurelepton colliders, see e.g. [463]. For studies of LLP signatures at future electron-proton colliders, see [464]. eutral Naturalness in the Ground Chapter 6 discrimination, we employ a realistic (at least at the level of theory forecasting) approachto the reconstruction and isolation of secondary vertices.A key question is the extent to which future Higgs factories can improve on the LHCsensitivity to Higgs decays to LLPs, insofar as the number of Higgs bosons producedat the LHC will outstrip that of proposed Higgs factories by more than two orders ofmagnitude. Higgs decays to LLPs are sufficiently exotic that appropriate trigger andanalysis strategies at the LHC should compensate for the higher background rate andmessier detector environment. As we will see, there are two natural avenues for improvedsensitivity at future lepton colliders: improved vertex resolution potentially increasessensitivity to LLPs with relatively short lifetimes, while lower backgrounds and a cleanerdetector environment improves sensitivity to Higgs decays into lighter LLPs whose decayproducts are collimated.This chapter is organized as follows: In Section 2 we present a simplified signal modelfor Higgs decays into pairs of long-lived particles, which in turn travel a macroscopic dis-tance before decaying to quark pairs. We further detail the components of our simulationpipeline and lay out an analysis strategy aimed at eliminating the majority of StandardModel backgrounds. In Section 3 we translate this analysis strategy into the sensitivityof future lepton colliders to long-lived particles produced in Higgs decays as a functionof the exotic Higgs branching ratio and the mass and decay length of the LLP. Whilethese forecasts are generally applicable to any model giving rise to the signal topology, weadditionally interpret the forecasts in terms of the parameter space of several motivatedmodels in Section 4. We summarize our conclusions and highlight avenues for futuredevelopment in Section 5. 239 eutral Naturalness in the Ground Chapter 6
Exotic decays of the Higgs to long-lived particles encompass a wide variety of intermediateand final states. The decay of the Higgs itself into LLPs can proceed through a varietyof different topologies. Perhaps the most commonly-studied scenario is the decay of theHiggs to a pair of LLPs, h → XX , though decays involving additional visible or invisibleparticles (such as h → X + invisible or h → XX + invisible ) are also possible. Thelong-lived particles in turn may have a variety of decay modes back to the StandardModel, including X → γγ, jj, (cid:96) ¯ (cid:96), or jj(cid:96), including various flavor combinations. Thesedecay modes may also occur in the company of additional invisible states. Moreover, agiven long-lived particle may possess a range of competing decay modes, as is the case forLLPs whose decays back to the Standard Model are induced by mixing with the Higgs.Our aim here is to be representative, rather than comprehensive, as each productionand decay mode for a long-lived particle is likely to require a dedicated search strategy.For the purposes of this study, we adopt a simplified signal model in which the Higgsdecays to a pair of long-lived scalar particles X of mass m X , which each decay in turnto pairs of quarks at an average “proper decay length” cτ . Both the mass m X andproper decay length cτ are treated as free parameters, though they may be related inmodels that give rise to this topology. For the sake of definiteness, for m X > GeVwe take a branching ratio of . to b ¯ b and equal branching ratios of . to each of u ¯ u, d ¯ d, s ¯ s, c ¯ c , though the precise flavor composition is not instrumental to our analysis.For m X ≤ GeV we take equal branching ratios into each of the lighter quarks. Wefurther restrict our attention to Higgs factories operating near the peak of the e + e − → hZ cross section, for which the dominant production process will be e + e − → hZ followed Of course, “proper decay length” is a bit of a misnomer, but we use it as a proxy for c times themean proper lifetime τ . eutral Naturalness in the Ground Chapter 6 by h → XX . The associated Z boson provides an additional invaluable handle forbackground discrimination. Here we develop the conservative approach of focusing onleptonic decays of the Z , though added sensitivity may be obtained by incorporatinghadronic decays.Given the signal, there are a variety of possible analysis strategies sensitive to Higgsdecays to long-lived particles, exploiting various parts of a general-purpose detector.Tracker-based searches are optimal for decay lengths below one meter, with sensitivity toshorter LLP decay lengths all the way down to the tracker resolution. Timing informationusing timing layers between the tracker and electromagnetic calorimeter offers optimalcoverage for slightly longer decay lengths, while searches for isolated energy depositionin the electromagnetic calorimeter, hadronic calorimeter, and muon chambers providessensitivity to decay lengths on the order of meters to tens of meters. In principle, in-strumenting the exterior of a general-purpose detector with large volumes of scintillatormay lend additional sensitivity to even longer lifetimes. In this work we will focus ontracker-based searches at future lepton colliders, as these may be simulated relativelyfaithfully and ultimately are among the searches likely to achieve zero background whileretaining high signal efficiency.We define our signal model in FeynRules [427] and generate the signal e + e − → hZ → XX + (cid:96) ¯ (cid:96) at √ s = 240 GeV using
MadGraph 5 [426]. Where appropriate, we willalso discuss prospects for Higgs factories operating at √ s = 250 GeV (potentially withpolarized beams) such as the ILC by rescaling rates with the appropriate leading-ordercross section ratios. In order to correctly simulate displaced secondary vertices, the decayof the LLP X and all unstable Standard Model particles is then performed in Pythia 8 [465].In addition to the signal, we consider some of the leading backgrounds to our signalprocess and develop selection cuts aimed at achieving a zero-background signal region.241 eutral Naturalness in the Ground Chapter 6
The most significant irreducible backgrounds from Standard Model processes include e + e − → hZ with Z → (cid:96) ¯ (cid:96) and h → b ¯ b as well as e + e − → ZZ → (cid:96) ¯ (cid:96) + b ¯ b . Unsurprisingly,there are a variety of other Standard Model backgrounds, but they are typically well-controlled by imposing basic Higgsstrahlung cuts, and we do not simulate them with highstatistics. In addition to irreducible backgrounds from hard collisions, there are possiblebackgrounds from particles originating away from the interaction point, including cosmicrays, beam halo, and cavern radiation; algorithmic backgrounds originating from effectssuch as vertex merging or track crossing; and detector noise. Such backgrounds are wellbeyond the scope of the current study, and will require dedicated investigation with fullsimulation of the proposed detectors.Correctly emulating the detector response to LLPs using publicly-available fast sim-ulation tools is notoriously challenging. In particular, we have found that the defaultclustering algorithms in the detector simulator Delphes [466] tends to cluster calorime-ter hits from different secondary vertices into the same jets, significantly complicatingthe realistic reconstruction of secondary vertices. As such, we develop an analysis strat-egy using only ingredients from the
Pythia output, although we do further run eventsthrough
Delphes and utilize
ROOT [467] for analysis.We implement two distinct tracker-based analyses with complementary signal param-eter space coverage, which we denote as the ‘large mass’ and ‘long lifetime’ pipelines. Weshall eventually see that the former will be effective for m X (cid:38) GeV down to properdecay lengths cτ (cid:38) µ m, while the latter is able to push down in m X by a factor ofa few though is only fully effective for cτ (cid:38) cm. Full cut tables for both irreduciblebackgrounds and a variety of representative signal parameter points appear in Tables 6.1and 6.2, respectively. 242 eutral Naturalness in the Ground Chapter 6 Table 6.1: Cut flow of the ‘large mass’ analysis for the CEPC with entries of accep-tance × efficiency. The top set of rows gives the cut flow on 500k Z ( b ¯ b ) Z ( (cid:96) ¯ (cid:96) ) eventsand 100k h ( b ¯ b ) Z ( (cid:96) ¯ (cid:96) ) background events, which are used to confirm our analysis is in theno-background regime. The next sets of rows give cut flows on 5k signal events at repre-sentative parameter points, where the different columns are labeled by m X / GeV , cτ / m.The full row labels are given in the top set of rows and the labels below are abbreviationsfor the same cuts or selections.Cut/Selection ZZ Background hZ BackgroundDilepton Invariant Mass 0.97 0.98Recoil Mass 0.006 0.94Displaced Cluster ( ≥ resolution) 0.004 0.94Invariant Charged Mass (6 GeV) 0 0.00005Invariant ‘Dijet’ Mass 0 0.00005Pointer Track 0 0.00001 m X , cτ − − − − M (cid:96)(cid:96) M recoil | (cid:126)d cluster | M charged M cluster eutral Naturalness in the Ground Chapter 6 m X , cτ − − − − M (cid:96)(cid:96) M recoil | (cid:126)d cluster | M charged M cluster × efficiency. The top set of rows gives the cut flow on 500k Z ( b ¯ b ) Z ( (cid:96) ¯ (cid:96) ) eventsand 100k h ( b ¯ b ) Z ( (cid:96) ¯ (cid:96) ) background events, which are used to confirm our analysis is in theno-background regime. The next sets of rows give cut flows on 5k signal events at repre-sentative parameter points, where the different columns are labeled by m X / GeV , cτ / m.The full row labels are given in the top set of rows and the labels below are abbreviationsfor the same cuts or selections.Cut/Selection ZZ Background hZ BackgroundDilepton Invariant Mass 0.97 0.98Recoil Mass 0.006 0.94Displaced Cluster ( ≥ cm) 0.004 0.62Charged Invariant Mass (2 GeV) 0 0.002‘Dijet’ Invariant Mass 0 0.002Pointer Track 0 0.001Isolation 0 0.00005244 eutral Naturalness in the Ground Chapter 6 m X , cτ − − − − M (cid:96)(cid:96) M recoil | (cid:126)d cluster | M charged M cluster m X , cτ − − − − M (cid:96)(cid:96) M recoil | (cid:126)d cluster | M charged M cluster ≤ M ee ≤ GeV ( ≤ M µµ ≤ GeV) and with recoil mass M recoil ≡ (cid:0) ( √ s , (cid:126) µ − p µ(cid:96)(cid:96) (cid:1) in therange ≤ M recoil ≤ GeV, with p µ(cid:96)(cid:96) the momentum of the lepton pair. This allows usto limit our background considerations to the irreducible backgrounds mentioned aboveand cuts down severely on the e + e − → ZZ background, as seen in Tables 6.1 and 6.2.We next identify candidate secondary vertices using a depth-first ‘clustering’ algo-rithm, which roughly emulates that performed in the CMS search [468]. We performthis clustering using all particles in the event because at later points in the analysis we245 eutral Naturalness in the Ground Chapter 6 need this truth-level assignment of neutral particles to clusters, but we expect that this(admittedly unrealistic) inclusion does not significantly modify the performance of thisalgorithm. Beginning with a single particle as the ‘seed’ particle for our algorithm, welook through all other particles in the event and create a ‘cluster’ of particles consistingof the seed particle and any others whose origins are within (cid:96) cluster = 7 µ m (the projectedtracker resolution of CEPC [440]) of the seed particle. We then add to that cluster anyparticles whose origins are within (cid:96) cluster of any origins of particles in the cluster, and dothis step iteratively until no further particles are added to the cluster. We then choose anew seed particle which has not yet been assigned to a cluster and begin this clusteringprocess again. We repeat this process until all particles in the event have been assignedto clusters. We assign to each cluster a location (cid:126)d cluster which is the average of the ori-gins of all charged particles in the cluster. To ensure that our events contain displacedvertices, we impose a minimum bound on the displacement from the interaction point | (cid:126)d cluster | > d min , and clusters satisfying this requirement constitute candidate secondaryvertices. For our ‘large mass’ analysis we set d min to be the impact parameter resolution( (cid:39) µ m for both CEPC and FCC-ee [440]), and so retain sensitivity to very short X lifetimes. For our ‘long lifetime’ analysis we set d min = 3 cm, which removes the vastmajority of clusters coming from B hadron decays in background events, as seen in Table6.2. An upper bound | (cid:126)d cluster | < r tracker is imposed by the outer radius of the tracker,where r tracker = 1 . m for CEPC and r tracker = 2 . m for FCC-ee are proposed.At this point an experimental analysis might sensibly examine dijets containing can-didate secondary vertices and impose an upper bound on the dijet invariant mass toremove backgrounds coming from Standard Model H or Z decays. As discussed abovewe are limited to Pythia objects, but to mock up the (small) penalty to signal ofsuch a selection we implement a selection on the total invariant mass of the clusters M cluster ≡ ( (cid:80) i ∈ cluster p µi ) . Since this is truth-level information, to turn it into an analog246 eutral Naturalness in the Ground Chapter 6 for the dijet invariant mass we apply a Gaussian smearing with a standard deviation of GeV to account for the dijet resolution. We then select only candidate secondary ver-tices with M cluster < m h / . This has no effect on background in our simulation pipelineas the background candidate secondary vertices are the result of hadronic decays, so theinvariant masses of these clusters are not analogs for dijet invariant masses. As empha-sized above, the imposition of this cut is strictly to account for possible selections thatmight appear in a more realistic experimental analysis.While the total invariant mass of the clusters is not an experimental observable, theinvariant mass of charged particles in the clusters M charged = ( (cid:80) i charged p µi ) is experi-mentally accessible. For our ‘large mass’ analysis we select candidate secondary verticeswith M charged > GeV, which gets rid of nearly all clusters from hadronic decays, as seenin Table 6.1. For the ‘long lifetime’ analysis, while the increased displacement require-ment removes b hadrons it still allows c, s hadrons, and so we select M charged > GeV toaddress this, which Table 6.2 shows is again very effective.Next we select the cluster closest to the beamline which passes the above selectionrequirements as our secondary vertex for the event. Choosing the closest one preferen-tially selects X decay clusters over hadronic decay clusters in the jets to which the X decays, though this can be fooled by a non-zero fraction of ‘back-flowing’ quarks in X decays (quarks with momenta pointing toward the beamline).To remove displaced vertices coming from the decays of charged b hadrons we im-plement a ‘pointer track’ cut in both analyses as follows. For the cluster selected asthe secondary vertex, we consider a sphere of radius r = 0 . mm around the position (cid:126)d cluster . We look for any charged particles whose origins are outside this sphere and whosemomenta (at the point at which they were created) point into it, and veto the event ifthere are any such particles. The main effect of this cut is to remove clusters which wereproduced from the decay of a charged hadron. The sphere size has been chosen to maxi-247 eutral Naturalness in the Ground Chapter 6 mize this effect, though this allows a small effect on signal due to geometric coincidence.Since this cut is only on charged particles, roughly ∼ of background clusters areunaffected. For this cut we ignore the effect of the magnetic field in the tracker, whichshould not highly impact the trajectories on short scales.For the ‘long lifetime’ analysis we further implement an ‘isolation’ cut to removeneutral hadronic background decays. Given the cluster selected as the secondary vertex,we consider the plane perpendicular to the sum of momenta of charged particles in thecluster which passes through (cid:126)d cluster . We project the paths of prompt (vertex within µ m of the primary vertex, the planned CEPC vertex resolution [440]) charged particlesonto this plane (again ignoring the magnetic field) and veto the event if any come within R = 10 cm of the position of the secondary vertex. This radius was chosen to maximallyreduce background, and does have a deleterious effect on short decay lengths (cid:46) mm,as can be seen in Table 6.2. This cut is not perfectly effective at rejecting background dueto the non-negligible presence of jets whose prompt components have neutral fraction .248 eutral Naturalness in the Ground Chapter 6 To confirm that our analysis pipelines put us in the zero-background regime we runboth the ‘long lifetime’ and the ‘large mass’ analyses on 500k e + e − → Z ( b ¯ b ) Z ( (cid:96) ¯ (cid:96) ) eventsand 100k e + e − → h ( b ¯ b ) Z ( (cid:96) ¯ (cid:96) ) background events. For both pipelines we find that zero e + e − → Z ( b ¯ b ) Z ( (cid:96) ¯ (cid:96) ) events remain, while for e + e − → h ( b ¯ b ) Z ( (cid:96) ¯ (cid:96) ) we find efficiencies of × − and × − respectively. We then run each analysis on 5k signal events to getacceptance × efficiencies for each ( m X , cτ ) point, for a selection of points with m X = 2 . from GeV to GeV and cτ from µ m to m. In Table 6.2 we give a cut table forboth backgrounds and some representative signal parameter points for the ‘long lifetime’analysis, and in Table 6.1 we do the same for the ‘large mass’ analysis.In the zero-background regime, Poisson statistics rules out model points which predict3 or more signal events to confidence (or better) if no signal is detected. We maythen find a projected upper limit on branching ratio asBr ( h → XX ) = N sig L × σ ( e + e − → hZ ) × Br ( Z → (cid:96)(cid:96) ) × A × ε , (6.1)with N sig = 3 and A × ε the result of our simulations. For both the CEPC and FCC-ee,the most recent integrated luminosity projections [440, 469] give L × σ ( e + e − → hZ ) =1 . × Higgses produced.In Figure 6.2 we show projected 95% upper limits on Br ( h → XX ) as a functionof X mass and proper decay length. While we plot separate lines for both CEPC andFCC-ee, we only use one set of signal events generated at √ s = 240 GeV and onlyaccount for the difference in tracker radii, so these overlap entirely at smaller lifetimes.Approximate limits for the ILC can be obtained by multiplying the above branchingratio limits by a factor of ∼ . (i.e. weakening the limit) to account for the leadingorder differences in center-of-mass energy, polarization, and integrated luminosity at the249 eutral Naturalness in the Ground Chapter 6 √ s = 250 GeV ILC run, assuming comparable acceptance and efficiency. The ILC limitsweaken slightly further for large decay lengths, as its proposed tracker radius is . m.Of course, adding the higher-energy ILC runs should significantly improve sensitivitygiven analyses suitable for the W W fusion production relevant at those energies.For small masses we are only able to use the ‘long lifetime’ analysis, which requireslarge displacement from the beamline to cut out the SM b hadron background. As aresult we only retain good sensitivity to X decay lengths comparable with the trackersize, though the fact that we only require one displaced vertex (out of two X s per signalevent) significantly broadens our sensitivity range. This fact also helps us retain efficiencyat low masses, as we are able to get down to a projected branching ratio limit of × − for m X = 2 . GeV despite our GeV cut on charged invariant mass of the decay cluster.For larger masses this cut has less effect, which allows it to push down to even lowerbranching ratios ∼ × − .The ‘large mass’ analysis begins working well for masses not far above the GeVcharged invariant mass cut and provides sensitivity to far shorter decay lengths, reachingall the way down to the impact parameter resolution and below. For m X = 10 GeV,where we are aided by the boost factor, we project a limit of × − for a proper decaylength of micron. The sensitivity to extremely small decay lengths drops for largermasses, but at m X = 50 GeV we cross below the − threshold by . µ m. For X masses high enough that the charged invariant mass cut does not remove a large amountof signal events, this analysis projects a branching ratio limit of ∼ × − across roughlythe entire range of decay lengths corresponding to the geometric volume of the detector.There is a slight dip in sensitivity for cτ ∼ mm, where the pair of dijets from the two X decays are most likely to overlap and trigger the cut on ‘pointer’ tracks.The notable region of this parameter space to which our analyses do not provide goodsensitivity is the low mass ( m X (cid:46) GeV) and short proper decay length ( cτ (cid:46) cm)250 eutral Naturalness in the Ground Chapter 6 regime. The difficulty is that, from the perspective of the tracker, the X here looks moreand more like a neutral SM hadron. An analysis making use of the impact parameterdistribution of particles in clusters may help here [468], but we leave this to futurework. Taking advantage of calorimeter data to distinguish between clusters in single jetsversus dijets is also likely to provide good sensitivity, but we again leave this to futureexploration.Broadly speaking, our results suggest a peak sensitivity of Br( h → XX ) ∼ × − ,weakening to ∼ − for lower-mass LLPs. Significant additional improvement couldbe expected with the inclusion of hadronic Z decays, but this requires further studyto ensure the control of corresponding Standard Model backgrounds. These limits arecompetitive with LHC forecasts based on conventional Higgs triggers [353, 352], notingthat these latter forecasts assume zero background. However, the lepton collider limitsare potentially superseded by an efficient CMS track trigger [459, 460] for higher-massLLPs, again assuming zero background is achievable with high signal efficiency acrossa range of lifetimes. In this respect, the primary strengths of the Higgs factories insearching for exotic Higgs decays to LLPs are the potential to push down to shorterdecay lengths and lighter LLPs. In particular, the relatively clean and low-backgroundenvironment of lepton colliders should enable efficient LLP searches even when the LLPdecay products become collimated, which remains a weakness of the corresponding LHCsearches. 251 eutral Naturalness in the Ground Chapter 6 - × - × - × - % B r ( h -> XX ) L i m i t m X = - × - × - × - m X = - × - × - × - % B r ( h -> XX ) L i m i t m X = - × - × - × - m X =
10 GeV - × - × - × - Proper Decay Length ( m ) % B r ( h -> XX ) L i m i t m X =
25 GeV - × - × - × - Proper Decay Length ( m ) m X =
50 GeV
Figure 6.2: Projected 95% h → XX branching ratio limits as a function of properdecay length for a variety of X masses. Blue lines are for CEPC and orange lines arefor FCC-ee, and where only one is visible they overlap. The larger dashes are the ‘longlifetime’ analysis and the smaller dashes are the ‘large mass’ analysis.252 eutral Naturalness in the Ground Chapter 6 While the bounds presented in the previous section apply to any scenario in which theHiggs decays into pairs of long-lived particles which in turn decay (at least in part) intopairs of quarks, it is also useful to interpret these bounds in the context of specific modelsthat relate the Higgs branching ratio to LLPs (and the LLP lifetime) to underlyingparameters. This illustrates the potential for LLP searches at future lepton collidersto constrain motivated scenarios for physics beyond the Standard Model and allows usto explore the potential complementarity between LLP searches and precision Higgscoupling measurements. To this end, we consider the implications of the LLP limitspresented here in the context of both the original Higgs portal Hidden Valley model anda variety of models of neutral naturalness.
As a general proxy model for Higgs decays into LLPs, we first consider the archetypalHiggs portal Hidden Valley [450]. This entails the extension of the Standard Model by anadditional real singlet scalar φ , which couples to the Standard Model through the Higgsportal [470, 471, 472] via L ⊃ −
12 ( ∂ µ φ ) − M φ − A | H | φ − κ | H | φ − µφ − λ φ φ − λ H | H | . (6.2)If φ respects a Z symmetry under which φ → − φ , this additionally sets µ = A = 0 , suchthat the singlet scalar only couples to the Standard Model via the quartic interaction | H | φ . After electroweak symmetry breaking, in unitary gauge H = (cid:16) , √ ( h + v ) (cid:17) , butthe CP-even scalars h and φ do not mix. Nonetheless, the quartic interaction nonethelessprovides a significant portal for the production of φ , as φ may be pair produced via the253 eutral Naturalness in the Ground Chapter 6 decay h → φφ for m φ < m h / . Of course, φ is stable if the Z symmetry is exact,rendering it a potential (albeit highly constrained) dark matter candidate [473, 474, 475].This model gives rise to long-lived particle signatures [450] if the Z is broken by asmall amount, such that A (cid:54) = 0 but e.g. A /M (cid:28) κ . The relative smallness of A istechnically natural, as the Z symmetry is restored when A → . This then leads to massmixing between the CP even scalars. As long as A is small compared to M and v , themass eigenstates consist of an SM-like Higgs h SM and a mostly-singlet scalar s , relatedto the gauge eigenstates by h SM = h cos θ + φ sin θ (6.3) s = − h sin θ + φ cos θ, (6.4)where θ (cid:28) is the mixing angle. There are now two parametrically distinct processes:pair production of the scalar s via Higgs decays, governed by the size of the Z -preservingcoupling κ , and decay of the s scalar back to the Standard Model, governed by the size ofthe Z -breaking coupling A . In the limit of small mixing, the former process is of order Γ( h → ss ) ≈ κ v πm h (cid:115) − m s m h , (6.5)where we are neglecting subleading corrections proportional to λ H sin θ . The latterprocess proceeds into whatever Standard Model states Y are kinematically available,with partial widths Γ( s → Y Y ) = sin θ × Γ( h SM [ m s ] → Y Y ) , (6.6)where h SM [ m s ] denotes a Standard Model-like Higgs of mass m s . This naturally leads toa scenario in which the s scalars may be copiously produced via Higgs decays but travelmacroscopic distances before decaying back to Standard Model particles.This scenario may be constrained not only by direct searches for Higgs decays toLLPs (with the scalar s playing the role of the LLP), but also by precision Higgs coupling254 eutral Naturalness in the Ground Chapter 6 measurements. Higgs coupling deviations in this scenario arise from two parametricallydistinct effects: tree-level deviations proportional to θ due to Higgs-singlet mixing, andone-loop deviations proportional to κ due to s loops. Both effects result in a universalmodification of Higgs couplings, which is best constrained at lepton colliders via theprecision measurement of the e + e − → hZ cross section [476, 477]. The net deviation inthe e + e − → hZ cross section due to these effects in the limit of small mixing is δσ hZ σ SM hZ ≈ − θ − Re d M hh dp (cid:12)(cid:12)(cid:12)(cid:12) p = m h , (6.7)where the radiative correction [477] d M hh dp (cid:12)(cid:12)(cid:12)(cid:12) p = m h = − π κ v m h (6.8) × (cid:32) m s m h (cid:115) m h m h − m s tanh − (cid:34)(cid:115) m h m h − m s (cid:35)(cid:33) is approximated at θ = 0 . Either effect can dominate depending on the relative sizeof A/M and κ .Constraints from a direct search for Higgs decays to LLPs and precision Higgs mea-surements as a function of the underlying parameters θ and κ are shown in Figure 6.3for the illustrative benchmarks m s = 2 . , , and 50 GeV. Unsurprisingly, in the regimewhere s is long-lived, the bounds from precision Higgs coupling measurements are modestand direct searches provide the leading sensitivity. Higgs decays to LLPs are also motivated by naturalness considerations, arising frequentlyin models of neutral naturalness that address the hierarchy problem with SM-neutraldegrees of freedom [288, 298]. In these models, partially or entirely SM-neutral partner255 eutral Naturalness in the Ground Chapter 6 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) log Θ l o g Κ CEPC (cid:144)
FCC (cid:45) ee Figure 6.3: Projected 95% limits on the Higgs portal Hidden Valley model in the κ, θ plane for three choices of m s ; green lines correspond to m s = 2 . GeV, blue to m s = 10 GeV, and red to m s = 50 GeV. The solid lines are the projected lower limits fromprecision Higgs measurements, taking the CEPC projections [440] for definiteness. Thedashed lines are projected limits from this work, which are essentially identical for CEPCand FCC-ee. Long dashes are from the ‘long lifetime’ analysis and short dashes from the‘large mass’ analysis. 256 eutral Naturalness in the Ground Chapter 6 particles that couple to the Higgs boson are charged under an additional QCD-like sector.Confinement in the additional QCD-like sector leads to a variety of bound states thatcouple to the Higgs and may be pair-produced in exotic Higgs decays with predictivebranching ratios. The bound states with the same quantum numbers as the physicalHiggs scalar typically decay back to the Standard Model by mixing with the Higgs.These decays occur on length scales ranging from microns to kilometers, making them amotivated target for LLP searches at colliders [289, 352].For simplicity, here we will restrict our focus to scenarios with the sharpest predictionsfor the Higgs branching ratio to LLPs. In these cases, the LLPs in question are typicallyglueballs of the additional QCD-like sector, of which the J P C = 0 ++ is typically thelightest. The coupling of the SM-like Higgs to these LLPs is predominantly due to toppartner loops, for which the scales and couplings are directly related to the naturalnessof the parameter space. In the Fraternal Twin Higgs [289], the entirely SM-neutralfermionic partners of the top quark induce Higgs couplings to twin gluons, which thenform glueballs; the ++ states are the lightest in the twin QCD spectrum only if theother twin quarks are sufficiently heavy. In addition, there are tree-level deviations inHiggs couplings due to the pseudo-goldstone nature of the SM-like Higgs. In FoldedSUSY [295], the scalar top partners carry electroweak quantum numbers, leading toradiative corrections to standard Higgs decays as well as the existence of exotic decaymodes. Loops of the scalar top partners again induce Higgs couplings to twin gluons, andwithout light folded quarks the ++ glueball is generically the lightest state in the foldedQCD spectrum. While there are no tree-level Higgs coupling deviations in this case, theelectroweak quantum numbers of the scalar top partners induce significant correctionsto the branching ratio h → γγ . Finally, in the Hyperbolic Higgs [303] (see also [304]),the scalar top partners are entirely SM-neutral, and induce couplings to ++ glueballsthat are generically the lightest states in the hyperbolic QCD spectrum. As with the257 eutral Naturalness in the Ground Chapter 6 Fraternal Twin Higgs, however, there are also tree-level Higgs coupling deviations due tomass mixing among CP-even neutral scalars.In each of these scenarios, the branching ratio of the SM-like Higgs can be parame-terized as follows:
Br( h → ++ ++ ) ≈ (cid:18) v α (cid:48) s ( m h ) α s ( m h ) (cid:20) y M (cid:21)(cid:19) × Br( h → gg ) SM × (cid:115) − m m h (6.9)Here α (cid:48) s denotes the coupling of the additional QCD-like sector (whether twin, folded, orhyperbolic), which is necessarily of the same order as the SM QCD coupling α s , and m is the mass of the glueball, which is determined in terms of the QCD-like confinementscale. Adopting the schematic notation of [352], the parameter (cid:104) y M (cid:105) encodes the model-dependence of the Higgs coupling to pairs of gluons in the QCD-like sector, with (cid:20) y M (cid:21) ≈ − v v f Fraternal Twin Higgs v m t m t Folded SUSY v vv H sin θ Hyperbolic Higgs (6.10)For the Fraternal Twin Higgs, f denotes the overall twin symmetry-breaking scale f = v + v (cid:48) in terms of the SM weak scale v and the fraternal weak scale v (cid:48) . For FoldedSUSY, m ˜ t denotes the mass of the scalar top partners, neglecting possible mixing effects.For the Hyperbolic Higgs, v H is the hyperbolic scale and tan θ ≈ vv H encodes tree-levelmixing effects. In each case, the scales appearing in the effective coupling are relatedto the fine-tuning of the model, drawing a direct connection between the Higgs exoticbranching ratio and the naturalness of the weak scale.In each case, the ++ glueballs of the additional QCD-like sector decay back to theStandard Model by mixing with the SM-like Higgs, with a partial width to pairs of SM258 eutral Naturalness in the Ground Chapter 6 particles Y given by Γ(0 ++ → Y Y ) = (cid:18) π (cid:20) y M (cid:21) vm h − m (cid:19) (cid:0) πα Bs F S ++ (cid:1) × Γ( h SM [ m ] → Y Y ) , (6.11)where πα Bs F S ++ ≈ . m and, as before, h SM [ m ] denotes a Standard Model-like Higgsof mass m .Constraints on each model from a direct search for Higgs decays to LLPs and precisionHiggs measurements are shown in Figure 6.4 as a function of the LLP mass m and therelevant scale ( f, v H , and m ˜ t , respectively). For precision Higgs measurements we use theCEPC projections from [440]. In the Fraternal Twin Higgs and Hyperbolic Higgs, thedominant indirect constraint is from σ Zh , while for Folded SUSY it is from Br( h → γγ ) .For both the Fraternal Twin Higgs and the Hyperbolic Higgs, tree-level Higgs couplingdeviations make precision Higgs measurements the strongest test of the model. However,the sensitivity of LLPs searches provides valuable complementarity in the event thatHiggs coupling measurements yield a discrepancy from Standard Model predictions. Inparticular, the size of an observed Higgs coupling deviation would single out the relevantoverall mass scale ( f or v H ), providing a firm target for LLP searches that would thenvalidate or falsify these models as an explanation of the deviation. Note also that inthe Fraternal Twin Higgs there may be additional contributions to the Higgs branchingratio into LLPs coming from the production of twin bottom quarks, which could leadto sensitivity in the LLP search comparable to that of Higgs couplings. In the caseof Folded SUSY, the absence of tree-level Higgs coupling deviations and the relativelyweaker constraints on Br( h → γγ ) make the LLP search the leading test of this modelat Higgs factories. 259 eutral Naturalness in the Ground Chapter 6
10 20 30 40 5050010001500200025003000 m (cid:72) GeV (cid:76) f (cid:72) G e V (cid:76) Fraternal Twin Higgs
10 20 30 40 50050010001500200025003000 m (cid:72) GeV (cid:76) v H (cid:72) G e V (cid:76) Hyperbolic Higgs
10 20 30 40 5002004006008001000 m (cid:72) GeV (cid:76) m t (cid:142) (cid:72) G e V (cid:76) Folded SUSY
Figure 6.4: Projected 95% limits on the underlying scale as a function of the LLP mass m in three models of neutral naturalness: the Fraternal Twin Higgs ( f ), the HyperbolicHiggs ( v H ), and Folded SUSY ( m ˜ t ). The blue dashed line denotes the limit coming fromprecision Higgs coupling measurements, taking for definiteness the CEPC projectionsfrom [440]. For the Fraternal Twin Higgs and Hyperbolic Higgs, the dominant indirectconstraint is from σ Zh , while for Folded SUSY it is from Br( h → γγ ) . The shaded regiondenotes the projected limits from direct LLP searches obtained in this work.260 eutral Naturalness in the Ground Chapter 6 The exploration of exotic Higgs decays is an integral part of the physics motivation forfuture lepton colliders. New states produced in these exotic Higgs decays may them-selves decay on a variety of length scales, necessitating a range of search strategies.While considerable attention has been devoted to the reach of future lepton collidersfor promptly-decaying states produced in exotic Higgs decays, the reach for long-livedparticles is relatively unexplored.In this paper we have made a first attempt to study the reach of proposed circularHiggs factories such as CEPC and FCC-ee (as well as approximate statements for the √ s = 250 GeV run of the ILC) for long-lived particles produced in exotic Higgs decays,focusing on the pair production of LLPs and their subsequent decay to pairs of quarks.We have developed a realistic tracker-based search strategy motivated by existing LHCsearches that entails the reconstruction of displaced secondary vertices. Rather thanrelying on existing public fast simulation tools, which do not necessarily give a sensibleparameterization of signal and background efficiencies for long-lived particle searches, wehave implemented a realistic approach to clustering and isolation. This allows us to char-acterize some of the leading irreducible Standard Model backgrounds to our search anddetermine reasonable analysis cuts necessary for a zero-background analysis. We obtainforecasts for the potential reach of CEPC and FCC-ee on the Higgs branching ratio tolong-lived particles with a range of lifetimes. The projected reach is competitive withLHC forecasts and potentially superior for lower LLP masses and shorter lifetimes. In ad-dition to our branching ratio limits, which may be freely interpreted in a variety of modelframeworks, we interpret our results in the parameter space of a Higgs portal Hidden Val-ley and various incarnations of neutral naturalness, demonstrating the complementaritybetween direct searches for LLPs and precision Higgs coupling measurements.261 eutral Naturalness in the Ground Chapter 6
There are a variety of directions for future work. While we have attempted to in-vestigate some of the leading irreducible backgrounds and impose realistic cuts, we havenot attempted to estimate possible backgrounds coming from cosmic rays; algorithmic,detector, or beam effects; or other contributions. Our tracker-based analysis has focusedon Higgs decays to pairs of hadronically-decaying LLPs, but a comprehensive pictureof exotic Higgs decays would also suggest the investigation of Higgs decays to variousLLP combinations as well as the consideration of additional LLP decay modes. More-over, tracker-based searches for displaced vertices are but one of many possible avenuesto discover long-lived particles. Analogous searches based on timing or on isolated en-ergy deposition in outer layers of the detector (including either the electromagnetic orhadronic calorimeter, the muon chambers, or potentially instrumented volumes outsideof the main detector) would be valuable for building a complete picture of LLP sensitivityacross a range of lifetimes.More broadly, it is an ideal time to study the potential sensitivity of future Higgsfactories to long-lived particles, as the results are likely to inform the design of detectorsfor these proposed colliders. This is a necessary step in motivating the physics case offuture Higgs factories and ensuring that they enjoy optimal coverage of possible physicsbeyond the Standard Model. 262 hapter 7New Trail for Naturalness
Is commuting dead?
The Connecticut Mirror
HeadlineJune 2, 2020 [478] ew Trail for Naturalness Chapter 7
At its heart, the electroweak hierarchy problem is a question of how an infrared (IR)scale can emerge from an ultraviolet (UV) scale without fine-tuning of UV parameters.Given the sensitivity of the Standard Model Higgs mass to UV scales, the expectation ofeffective field theory (EFT) is that the two should coincide. Conventional solutions to thehierarchy problem introduce both symmetries that control UV contributions to the Higgspotential and dynamics that generate IR contributions, leading to considerable structureat the weak scale and correspondingly sharp experimental tests. Ongoing explorationof the weak scale has given no evidence for these solutions, despite their theoreticalsoundness.In the face of increasingly powerful LHC data in excellent agreement with the Stan-dard Model, it’s worth taking seriously the possibility that Nature may be leading us tothe conclusion that there is no new physics at the weak scale . While this is often taken tosuggest the existence of considerable fine-tuning in the Higgs potential, here we pursue analternative idea. Perhaps the apparent violation of EFT expectations at the weak scaleis a sign of the breakdown of EFT itself. We’ll use the broad term ‘UV/IR mixing’ todenote any effects that the UV has on low-energy physics which goes past that expectedin EFT.In this work we pursue the idea that such UV/IR mixing may have more directeffects on the SM by considering noncommutative field theory (NCFT) as a toy model.These theories model physics on spaces where translations do not commute [335, 336],and have many features amenable to a quantum gravitational interpretation—indeed,noncommutative geometries have been found arising in various limits of string theory[337, 338, 339, 340]. Noncommutative branes arising in gauge theory matrix models have also been found to contain ew Trail for Naturalness Chapter 7
This noncommutativity bears out the general expectation that the general-relativisticnotion of spacetime should break down in a theory of quantum gravity [490]. Its realiza-tion here leads directly both to UV/IR mixing in the form of a violation of decouplingand to nonlocal effects in interactions. This gives rise to many interesting effects, butparticularly fascinating for our purposes is that UV divergences present in the S-matrixelements of QFTs on commutative spaces can be transmogrified into new infrared poles inthe corresponding field theory on noncommutative space [491]. An effective field theoristliving in a noncommutative space would have no way to understand the appearance ofthis infrared scale; its existence is intrinsically linked to the geometry of spacetime andto the far UV of the theory. Such an effective field theorist would see a surprising lackof new physics accompanying this pole to explain its presence.It is clear from the outset that the direct application of NCFT to understand thehierarchy problem is immediately hindered by the Lorentz invariance violation which isinherent to these theories. Precisely how fatal this might be is not entirely clear; resultsregarding the extent to which ‘generic’ Lorentz violation is empirically ruled out [492]are partly circumvented here by the fact that the Lorentz violation is not generic, butcomes as part of some larger structure. In this case the novel effects of UV/IR mixing infact only appear in nonplanar loop diagrams [493] and care is required when interpretingEFT constraints on Lorentz violation—a point we will emphasize in Section 7.2. Even so,it is difficult to imagine that observed properties of the weak scale and the wide range ofconstraints on Lorentz violation leave room for NCFT to be directly relevant to puzzlesof the Standard Model.Thus we make no claim about having solved the hierarchy problem. The value ofthis work is in the exploration of this toy model of UV/IR mixing, which possesses emergent gravitational effects, and so have been suggested as novel quantum theories of gravity [479,480, 481, 482, 483, 484, 485, 486, 487, 488]. We do not pursue this perspective here, but refer the readerto [489] for a review of this approach. ew Trail for Naturalness Chapter 7 the intriguing feature that ultraviolet dynamics generate a scale whose lightness wouldbe baffling to an effective field theorist. As this is the only model (of which we areaware) with this feature—and this feature, at the level of words, increasingly matchesthe experimental situation with the Higgs—it’s worth understanding its appearance inas much detail as possible.To make this work self-contained for the contemporary particle theorist, we beginwith an extensive introduction. In Section 7.2, we review quantum field theory on non-commutative spaces with an emphasis on the violation of EFT expectations. In Section7.3 we use this technology to go over the classic result of [491] which first identifiedthis emergent infrared pole in a Euclidean φ theory. We compute also the effect in di-mensional regularization to evince the regularization-independence of the UV/IR mixingeffects.In Section 7.4 we ask how general the effect of UV/IR mixing is within NCFT, whichleads us to study noncommutative Yukawa theory in detail. We find that the scalar prop-agator again develops a new infrared pole at one loop, in contrast with previous work.Intriguingly, the pole in this case is accessible in s -channel scattering in the Lorentziantheory, making Yukawa theory a promising setting for probing phenomenological conse-quences of UV/IR mixing.In Section 7.5 we upgrade our model to the softly-broken Wess-Zumino model tostudy the interplay between UV-finiteness and UV/IR mixing effects. When the fermionis kept in the spectrum of the theory below the cutoff, the lack of UV sensitivity ofthe field theory removes the light pole. As the fermion is taken above the cutoff, aneffective theorist again sees effects past those observed in Wilsonian EFT. These resultsare expected, but this model affords us a concrete demonstration that UV/IR mixingcan only have interesting low-energy effects if the field theory is UV sensitive, and putsthis naturalness strategy in stark contrast to conventional approaches. Of course, this266 ew Trail for Naturalness Chapter 7 also makes addressing the hierarchy problem with UV/IR mixing a potentially Pyrrhicvictory: to generate an IR scale, the field theory alone cannot be fully predictive.Finally, in Section 7.6 we examine the appearance of the emergent light pole in NCFTfrom more general arguments, so as to ascertain the relative importance of nonlocalityand Lorentz-violation for these effects. The conclusion is inevitably that in this casethe two are inexorably linked, and no strong conclusion about the possibility of findinga light pole in a theory with only one or the other is available. However, we providesome direction toward future explorations into both of these possibilities. We wrap upin Section 7.7. 267 ew Trail for Naturalness Chapter 7 In this section we review the salient features of the formulation of noncommutative fieldtheories and the standard formalism for studying their perturbative physics. Usefulgeneral references for this background include [494, 495]. Readers familiar with NCFTmay wish to skip to Section 7.3, but we emphasize that our interest is necessarily non-perturbative in the parameter controlling the noncommutativity, unlike much of theearlier phenomenological literature.Physics on noncommutative spaces involves the introduction of a nonzero commutatorbetween position operators [ˆ x µ , ˆ x ν ] = iθ µν , (7.1)where we will refer to θ µν = − θ νµ as the noncommutativity tensor, and we emphasizethat it is covariant under Lorentz transformations. So while it does break Lorentz in-variance, it only does so in the way that turning on a magnetic field in your lab choosesa preferred frame, and it can indeed be thought of as simply a background field. Thisbasic definition is reminiscent of the introduction of a nonzero commutator in passingfrom classical mechanics to quantum mechanics. Indeed much of the structure is pre-cisely analogous, including importantly the construction of noncommutative versions offamiliar commutative theories via a quantization map. At an even more basic level, theabove nonzero commutator induces an uncertainty relation ∆ˆ x µ ∆ˆ x ν ≥ | θ µν | , (7.2)which immediately makes apparent the presence of UV/IR mixing in this theory. If youattempt to create a wavepacket which is very small in one direction it will necessarily beelongated in another, and so we see already the non-trivial mixing of UV and IR modes.This clearly violates the separation of scales which is baked in to EFT. Thus purely from268 ew Trail for Naturalness Chapter 7 the defining relation of noncommutative geometry, we see already an indication thatnoncommutative theories should violate EFT expectations.Field theories on this space may be conveniently formulated in terms of fields thatare functions of commuting coordinates imbued with a new field product, known as aGroenewold-Moyal product (or star-product), with position-space representation f ( x ) (cid:63) g ( x ) = exp (cid:18) i θ µν ∂ µy ∂ νz (cid:19) f ( y ) g ( z ) (cid:12)(cid:12)(cid:12)(cid:12) y = z = x = f ( x ) exp (cid:18) i ←− ∂ µ θ µν −→ ∂ ν (cid:19) g ( x ) . (7.3)We derive this procedure in Appendix B. It is important to observe that this is a nonlocalproduct, since it contains an infinite series of derivative operators. So we see again thatone of the tenets of EFT has been violated by our basic definition of field theory onnoncommutative spaces.With this in hand we may now write down noncommutative versions of familiartheories in terms of commuting coordinates , which will then allow us to use normal QFTmethods to analyze them. First note that this noncommutative quantization will notaffect the quadratic part of the tree-level action due to momentum conservation and theantisymmetry of the noncommutativity tensor. For the interacting part of the actionthe effects of noncommutative quantization are not so trivial, but are easy to analyzeclassically. As an example, for a simple φ n theory we find L ( NC ) int = λn ! n copies (cid:122) (cid:125)(cid:124) (cid:123) φ ( x ) (cid:63) φ ( x ) (cid:63) · · · (cid:63) φ ( x ) . (7.4)Note, importantly, that the star-product has endowed our vertices with a notion ofordering, as it is only cyclically invariant. If we now Fourier transform the action tomomentum space, we find that we can account for the effects of quantization on thetree-level action with a simple modification of the momentum-space vertex factor: ˜ V ( k , . . . , k n ) = δ ( k + · · · + k n ) exp (cid:32) i n (cid:88) i This failure of unitarity is well-understood from the stringy perspective. Spatialnoncommutativity appears from a background magnetic field and the field theory limitto a spacelike NCFT is smooth [339]. In the case of timelike noncommutativity, however,approaching the field theory limit forces an electric field to supercritical values whencepair-production of charged strings destabilizes the vacuum [504]. Study of string theorieswith timelike noncommutativity (e.g. ‘noncommutative open string theory’ [504, 505]) isoutside our scope, but there are at least some hints of similar UV/IR mixing effects asthose in the NCFT [506]. We note in passing that there are further interesting connectionsbetween NCFTs and string theories—not only do particles on noncommutative spaces actin many ways like rods of size L ∼ pθ (see e.g. [507, 508, 509, 510, 511]), mimicking thebehavior of extended objects, but there have been many hints in the spacelike theoriesthat the curious IR effects in the NCFT are reproducing effects from closed strings,despite the fact that these have been decoupled (e.g. [491, 512, 513, 514, 515, 516, 517,506, 518, 519]).Within the realm of field theory, there have long been suggestions that this difficultyis pointing to the need for a modified definition of quantum field theories on timelikenoncommutative spaces (for some early references, see [520, 521, 522, 523, 524, 525, 526,527]). From this perspective, the issue is that such field theories are non-local in time,which renders nonsensical the normal time-ordering involved in the perturbative Dysonseries (at the least). That is, our effective definition of these theories above via thediagrammatic expansion may be too naïve. An interesting line of work is to formulate amodification of the standard quantum field theory machinery to non-local-in-time theorieswhich avoids the unitarity issue by construction. We note that the same UV/IR mixingeffects of interest in the two-point function have been seen to persist in at least some ofthese approaches (e.g. [522]). For some recent work on the formulation and propertiesof nonlocal field theories, see e.g. [528, 529, 530, 498, 531, 532].273 ew Trail for Naturalness Chapter 7 Below we will begin in Euclidean space, where k ◦ k ≥ is guaranteed for any θ µν ,but will then venture into Lorentzian signature. All of our calculations and the generalfeatures we find, including finding new infrared poles, will hold robustly in spacelike non-commutative theories. However we will comment also on how these features are modifiedwhen timelike noncommutativity is turned on, taking license from the aforementionedhints that unitary completions/reformulations of timelike NCFT may retain the UV/IRmixing exhibited in the naïve approach. 274 ew Trail for Naturalness Chapter 7 φ Theory In this section we review the perturbative physics of the noncommutative real scalar φ theory at one loop, which was first studied in detail by Minwalla, Van Raamsdonk, andSeiberg in [491]. In four Euclidean dimensions the action on noncommutative space becomes S = (cid:90) d x (cid:18) ∂ µ φ∂ µ φ + 12 m φ + g φ (cid:63) φ (cid:63) φ (cid:63) φ (cid:19) , (7.11)where we have already used the fact that the quadratic part of the noncommutativeaction is the same as the commutative theory to eliminate the star product there. Ourobject of interest will be the one-loop correction to the two-point function. In the com-mutative theory this is given by a single Feynman diagram, but the noncommutativetheory contains both a planar diagram and a nonplanar diagram. − Γ (2)1 = p k + p k The expressions for these two diagrams now differ—not only in symmetry factor but alsodue to the phase in the integrand. We find Γ (2)1 , planar = g π ) (cid:90) d kk + m Γ (2)1 , nonplanar = g π ) (cid:90) d kk + m e ik µ θ µν p ν . (7.12)We may already see that something interesting should happen, as in the nonplanar di-agram the phase mixes the internal and external momenta. One may intuit that therapidly oscillating phase in the UV of the loop integration will dampen the would-be Some early results in this model may also be found in [533, 534]. ew Trail for Naturalness Chapter 7 divergence, and indeed we will see that nonplanar diagrams are finite. However, unlikein the case where the vertex factor vanishes rapidly for large Euclidean momenta and soensures UV-finiteness [532], here the damping is in some sense ‘marginal’. This fact willbe responsible for the interesting feature we will find presently.The simplest method to evaluate noncommutative diagrams is to use Schwinger pa-rameters, recalling the identity k + m = (cid:82) ∞ d α e − α ( k + m ) . The presence of the phasein the nonplanar diagram means we must complete the square before going to sphericalcoordinates to get a Gaussian integral. This means that after the momentum integralswe end up with Γ (2)1 , planar = g π (cid:90) d αα e − αm Γ (2)1 , nonplanar = g π (cid:90) d αα e − αm − p ◦ p α (7.13)where again p ◦ q = − p µ θ µν q ν . Moving to Schwinger space trades large- k divergences forsmall- α divergences, which we now smoothly regulate by multiplying the integrands by exp ( − / (Λ α )) so that the small α region will be driven to zero. Note that a term ofthis form already exists in the expression for the nonplanar diagram. After introducingthe regulator, we can evaluate the integrals to find Γ (2)1 , planar = g π (cid:18) Λ − m log (cid:18) Λ m (cid:19) + O (1) (cid:19) Γ (2)1 , nonplanar = g π (cid:18) Λ eff − m log (cid:18) Λ eff m (cid:19) + O (1) (cid:19) , (7.14)where we’ve defined Λ eff ≡ / Λ + p ◦ p/ , (7.15)which is the effective cutoff of the nonplanar diagram.The first thing to note is that it seems the UV divergence of the nonplanar diagramhas disappeared—the graph is finite in the limit Λ → ∞ , and so appears to have beenregulated by the noncommutativity of spacetime. In fact the effect is more subtle, as276 ew Trail for Naturalness Chapter 7 alluded to earlier, and now the UV and IR limits of this amplitude do not commute. If wefirst take an infrared limit p ◦ p → we find that Λ eff → Λ and the ultraviolet divergenceof the commutative theory reappears. If we take the UV limit Λ → ∞ first we find anIR divergence p ◦ p , so the noncommutativity has transmogrified the UV divergence intoan IR one. Turning to the question of renormalizability, one may naïvely ask if we can absorball UV divergences into a finite number of counterterms. Under this criterion, it is clearthat this procedure works in the noncommutative theory at least when the commutativeversion is renormalizable. In the current case, we may absorb the UV divergences ofthis correction to the two-point function into a redefinition of the physical mass, M = m + g Λ π − g m π log Λ m , and so write down a one-particle irreducible quadratic effectiveaction which has a finite Λ → ∞ limit: S (2)1 PI = (cid:90) d p (2 π ) (cid:32) p + M + g π (cid:0) p ◦ p + (cid:1) (7.16) − g M π log 1 M (cid:0) p ◦ p + (cid:1) + · · · + O ( g ) (cid:33) φ ( p ) φ ( − p ) . However, in the Λ → ∞ limit one finds that at one loop the propagator now has two poles. The first is a standard radiative correction to the free pole, but the second hasappeared ex nihilo at one loop: p = − m + O ( g ) p ◦ p = − g π m + O ( g ) , (7.17) We note here that the failure of a ‘correspondence principle’ between commutative and noncom-mutative theories as θ µν → is clearly intrinsically linked to the appearance of UV/IR mixing. Thisfailure doesn’t violate Kontsevich’s proof of the existence of deformation quantization for any symplecticmanifold [535], as that is confined solely to ‘formal’ deformation quantization—that is, the productionof a formal power series expansion of the algebra of observables in terms of the deformation parameter.As was noted in Section 7.2 and is now on prime display, the physics of the theory with nonperturbative θ -dependence is starkly different from that of any truncation. ew Trail for Naturalness Chapter 7 where we have assumed that θ µν is full rank. The former is to be interpreted as theon-shell propagation of the particles associated to our fundamental field φ . If θ µν hasonly one eigenvalue / Λ θ —with Λ θ thought of as the scale associated with the breakdownof classical geometry—we have p ◦ p = p Λ θ . We see that the new pole appears at p ∝ g θ m , and so if our field φ lives in the deep UV of the theory, our new pole appears atparametrically low energy scales. To the extent that poles are particles, we appear tohave generated a new light particle from ultraviolet dynamics.The interpretation of the new pole can be sharpened by considering more carefullythe criteria for renormalizability in Wilsonian EFT. In a Wilsonian picture, we upgradeour Lagrangian parameters to running parameters, and define our theory at the scale Λ as S W ilson (Λ) = (cid:90) d x (cid:18) Z (Λ) ∂ µ φ∂ µ φ + 12 Z (Λ) m (Λ) φ + Z (Λ) g (Λ)4! φ (cid:63) φ (cid:63) φ (cid:63) φ (cid:19) . (7.18)It is immediately apparent from the above calculation that we cannot write the ac-tion at a lower scale Λ < Λ in this same form by choosing appropriate definitions for Z (Λ) , m (Λ) , g (Λ) —there’s nowhere to put the p ◦ p term! Stated more precisely, for Wilsonian renormalizability we require that we can definethe running couplings such that correlation functions computed from this action convergeuniformly to their Λ → ∞ limits. However, this requirement is flatly violated by thenoncommutation of the UV and IR limits of the diagrams. For any finite value of Λ , There has been much work on understanding renormalizability of NCFTs, especially with an eyetoward finding a mathematically well-defined four-dimensional quantum field theory with a non-trivialcontinuum limit. Renormalizability has been proven for modifications of NCFTs where the free actionis supplemented by an additional term which adjusts its long-distance behavior. Such an action ismanufactured either by requiring it manifest ‘Langman-Szabo’ duality [536] p µ ↔ θ − ) µν x ν [537, 538]or by adding a /p ◦ p term to the free Lagrangian [539], the latter of which directly has the interpretationof adding ‘somewhere to put the /p ◦ p counterterm’. For recent reviews of these and related effortswe refer the reader to [540, 541]. It would be interesting to understand fully the extent to which thephysics of these schemes agrees with the interpretation of the IR effects as coming from auxiliary fields[491, 515]. ew Trail for Naturalness Chapter 7 the effective action of Equation 7.16 differs significantly from its limiting value for smallmomenta p ◦ p (cid:28) . This is the precise sense in which the violation of Wilsonian EFTappears in this one-loop correction.This brings up the question of how an effective field theorist would describe theuniverse if they unknowingly lived on a noncommutative space. A consistent Wilsonianinterpretation can be regained by including a degree of freedom which can absorb thenew infrared dynamics of the quadratic effective action. Since we need this to involvethe φ momentum, this new particle must mix linearly with the φ field. We manufactureits tree-level Lagrangian such that the problematic inverse p ◦ p term in the quadraticeffective action of φ is replaced with its Λ → ∞ value for all values of Λ , to satisfy ourprecise condition for Wilsonian renormalizability. To see how this works, we add to ourtree level Wilsonian action ∆ S (Λ) = (cid:90) d x (cid:18) ∂χ ◦ ∂χ + 12 Λ ∂ ◦ ∂χ ) + i √ π gχφ (cid:19) . (7.19)Since χ appears quadratically, we may integrate it out exactly at tree level to find acontribution to the effective action ∆ S PI (Λ) = (cid:90) d p (2 π ) (cid:32) − g π (cid:0) p ◦ p + (cid:1) + g π p ◦ p (cid:33) φ ( p ) φ ( − p ) (7.20)This precisely subtracts off the problematic term in the original 1PI quadratic effectiveaction and adds back its Λ → ∞ limit, as we had wanted. Ignoring the logarithmic term, we are left with an effective action which is manifestly independent of the cutoff Λ , andso satisfies our criterion for Wilsonian renormalizability. We discuss the generalizationof this procedure in Appendix C. Discussion of the interpretation of logarithmic singularities as being due to auxiliary fields propa-gating in extra dimensions may be found in [515]. In Equation 7.19, the four-derivative quadratic action of the auxiliary field can be rewritten astwo fields with two-derivative actions, one of which is of negative norm and may be thought of as the‘Lee-Wick partner’ of the positive norm state [542], viz. L = 12 ∂χ (cid:48) ◦ ∂χ (cid:48) − ∂ ˜ χ ◦ ∂ ˜ χ − 12 4Λ ˜ χ + i √ π g ( χ (cid:48) − ˜ χ ) φ, χ (cid:48) ≡ χ + ˜ χ (7.21) ew Trail for Naturalness Chapter 7 Now while we have written down an action which identifies the new observed IR polewith a field and in doing so gives our effective action a Wilsonian interpretation, theextent to which χ can be taken seriously as a fundamental degree of freedom is unclear. The new pole is inaccessible in Euclidean space—so one does not immediately concludethere is a tachyonic instability—and relatedly, when we naïvely analytically continue thisresult to Lorentzian spacetime this new pole is inaccessible in the s -channel. However,its presence is still enough to break unitarity for this theory [499], and in fact may still beinterpreted as being due to the presence of tachyons [503]. As discussed in Section 7.2, itis possible this may be resolved if analytical continuation is adjusted for nonlocal-in-timetheories, or it may be that a UV theory cures this apparent violation.Separately, it is not obvious much has been gained by attributing the new pole to anew, independent field, past acting as a formal tool to regain a notion of renormalizability.Since the only interaction of χ above is linear mixing, its action is not renormalized—any One may then wonder if the lightness of the new IR pole may be understood through the regularizationperformed by the Lee-Wick field, as is done for the Higgs in the ‘Lee-Wick standard model’ [543].However, in that theory the Higgs is kept light because every particle comes with a Lee-Wick partner,and so all diagrams contributing to corrections to the Higgs mass are made finite. The presence of theHiggs’ Lee-Wick partner alone is not enough to keep it light. Here, the lightness of χ can be understooddiagrammatically as being simply due to the fact that its only interaction is linear mixing with φ , and soany correction to its two-point function is absorbed into that of the two-point function of φ . A furtherissue with the Lee-Wick rewriting is that the seeming perturbative unitary of the theory is normallyguaranteed by the Lee-Wick partner being heavy and unstable. But as we take the Λ → ∞ limit in ourWilsonian action, we see that the Lee-Wick partner becomes massless as well, in accordance with theresult that this theory is non-unitary [499]. We note that in matrix models containing dynamical noncommutative geometries it has been arguedthat emergent infrared singularities should be associated with the dynamics of the geometry (see e.g. [544,489]). As our field theories are formulated on fixed noncommutative backgrounds, this interpretation isunavailable to us. Note that this peculiar connection regarding (in)accessibility is due to the Lorentz violation. Whilethe normal pole which is inaccessible in Euclidean signature becomes accessible for timelike momenta inLorentzian signature, the Wick rotation affects the noncommutative momentum contraction differently.When taking x → − ix , one also rotates θ ν → − iθ ν such that Equation 7.1 continues to holdfor the same numerical θ µν . For the simplest configuration of full-rank noncommutativity with θ µν block-off-diagonal and only one eigenvalue / Λ θ , the Euclidean p ◦ p = p / Λ θ becomes a Lorentzian p ◦ p = ( p − p + p + p ) / Λ θ . So a noncommutative pole which is inaccessible in the Euclidean theorybecomes accessible in the Lorentzian theory for spacelike momenta, while a noncommutative pole whichcan be accessed in the Euclidean theory becomes accessible in the s -channel in Lorentzian signature. ew Trail for Naturalness Chapter 7 divergences are instead absorbed into the running of φ parameters—and so no interactionsare generated. Furthermore one is obstructed from integrating out the heavy field φ tocome up with an effective action of χ at low energies by the fact that the kinetic termsof χ are non-standard, which prevents diagonalization of the quadratic terms in theLagrangian. Thus it seems it is intrinsically linked with the heavy scalar which begat it.There are further obstructions to asking that this specific mechanism be responsiblefor the lightness of an observed particle such as the Higgs. Prime among these is themodified dispersion relation of the new field, p ◦ p = O ( g ) , which means that the freepropagation of this field would be Lorentz violating. We will explore these issues furtherin the next sections, as in the Yukawa theory of Section 7.4 the new pole will appearwith the opposite sign and so will offer the prospect of appearing as an s -channel pole.We emphasize that a new infrared scale whose lightness is unexplained in the contextof Wilsonian effective field theory is an exciting feature that makes further exploration ofUV/IR mixing an interesting pursuit. The fact that it here appears as the scale of a polein a propagator makes the connection to the hierarchy problem captivating, but askingthat this toy model—where Lorentz violation is at the fore—literally solve the problemfor us would be too much. We proceed without further hindrance in exploring NCFT soas to learn more about the appearance and effects of UV/IR mixing here. A good question to ask is whether, or to what extent, these effects are an artifact of ourchoice of regularization. To demonstrate their physicality, we repeat the calculation ofthe one-loop correction to the two-point function now in dimensional regularization. Weset up our integral in d = 4 − (cid:15) dimensions, having defined g = ˜ g ˜ µ (cid:15) , and we again go This dispersion relation means that χ only propagates in noncommutative directions, and so at-tempts to use hidden extra-dimensional noncommutativity to avoid four-dimensional Lorentz violationconstraints seem a phenomenological nonstarter. ew Trail for Naturalness Chapter 7 to Schwinger space: Γ (2)1 , planar = ˜ g ˜ µ (cid:15) π ) d (cid:90) d d k d α e − α ( k + m ) Γ (2)1 , nonplanar = ˜ g ˜ µ (cid:15) π ) d (cid:90) d d k d α e − α ( k + m )+ ik µ θ µν p ν . (7.22)After completing the square in the nonplanar integral, the momentum integral andthe Schwinger integral may then be performed analytically, with the results: Γ (2)1 , planar = ˜ g ˜ µ (cid:15) π ) d/ ( m ) d − Γ(1 − d (2)1 , nonplanar = ˜ g ˜ µ (cid:15) π ) d/ d ( m ) ( d − ( √ p ◦ p ) − d K d − ( m √ p ◦ p ) . (7.23)If we expand the planar graph in the limit (cid:15) → , which should be thought of as probingthe ultraviolet, we recover Γ (2)1 , planar = − ˜ g m π ) (cid:20) (cid:15) + ln µ m (cid:21) , (7.24)where in MS we would subtract off the pole and find the renormalization group evolutionof m from the logarithmic term, as usual.The question of dimensional regularization for the nonplanar diagram is a subtle one[545]. If we first take the (cid:15) → limit of Equation 7.23, we see this manifestly has nodivergences, and we are simply left with the finite, (cid:15) term Γ (2)1 , nonplanar = g m π ) (cid:20) m p ◦ p − ln 4 m p ◦ p − γ (cid:21) , (7.25)which we have expanded near p ◦ p → to manifest the IR divergence. We have againtransmogrified our UV divergence into an IR pole. We now expect to see that the IRlimit does not commute with the above UV limit. To do so, we expand Equation 7.23around p ◦ p → to find Γ (2)1 , nonplanar = ˜ g m π ) π (cid:15)/ ˜ µ (cid:15) m (cid:15) Γ (cid:16) − (cid:15) (cid:17) + ˜ g π ˜ µ (cid:15) π (cid:15)/ Γ (cid:16) − (cid:15) (cid:17) p ◦ p − (cid:15)/ + O ( p ◦ p ) . (7.26)282 ew Trail for Naturalness Chapter 7 If we were to now blindly take the (cid:15) → limit of this expression, we would again getEquation 7.25, contrary to our expectations. However, we notice that if the dimension ofspacetime over which we had performed the integral was particularly low (cid:15) > , then wehave incorrectly kept the second term in Equation 7.26, as that term would be at least O ( p ◦ p ) . If we were to work in d < , expand in p ◦ p → and so ignore that term, and then analytically continue back to d = 4 , we would instead find the (cid:15) − pole Γ (2)1 , nonplanar = − ˜ g m π ) (cid:20) (cid:15) + ln µ m (cid:21) , (7.27)and now we recover the UV divergence that was present in the commutative theory, sothat once again we find the UV and IR limits don’t commute.The key to understanding clearly this seemingly ambiguous dimensional regularizationprocedure is that while Γ (2)1 , nonplanar ( p ◦ p ) ∼ (cid:82) d d q d α e − α ( q + m ) − p ◦ p α is convergent in d > for p ◦ p > , at p ◦ p = 0 it is only convergent for d < . Since it is a property ofdimensional regularization that if an integral converges in δ dimensions, it converges tothe same value in d < δ dimensions [72], we may thus perform the integral at d < for all p ◦ p and correctly find Equation 7.23. It is only when taking the IR limit that wemust remember the integral was performed in d < dimensions, and so our expansionto get Equation 7.27 is unambiguously correct. Thus our conclusion that the UV and IRlimits of the two-point function do not commute here is robust.It is thus clear that the UV/IR mixing we have observed in this model is not an artifactof a choice of regularization, and is in fact a physical feature of this noncommutative fieldtheory. 283 ew Trail for Naturalness Chapter 7 We observed in our first example that the UV divergences of the real φ commutative the-ory are transmogrified into infrared poles in the noncommutative theory. It is natural toask whether this “strong UV/IR duality” [546] is a common feature of all noncommutativetheories.The answer is no, and the simplest counterexample is provided in the case of a complexscalar field with global U (1) symmetry and self-interaction [546]. In the quantization ofthe scalar potential we have two quartic terms which are noncommutatively-inequivalentdue to the ordering non-invariance, so the general noncommutative potential is V = m | φ | + λ φ ∗ (cid:63) φ (cid:63) φ ∗ (cid:63) φ + λ φ ∗ (cid:63) φ ∗ (cid:63) φ (cid:63) φ, (7.28)where λ and λ are now different couplings. By doodling some directed graphs, onesees simply that the one-loop correction to the scalar two-point function contains planargraphs with each of the λ , λ vertices, but the only nonplanar graph has a λ vertex.There is thus no necessary connection of the ensuing nonplanar IR singularity to the UVdivergence in the θ → limit, as the coefficients are unrelated (and in particular, we arefree to turn off the IR singularity at one loop by setting λ = 0 ).Another important counterexample is that of charged scalars, the simplest exampleof which is noncommutative scalar QED, which was first constructed in [547]. Thereis a very rich and interesting structure of gauge theories on noncommutative spaces,a full discussion of which is far beyond the scope of this paper. We refer the reader While we only presented the calculation of the one-loop correction to the two-point function, [491]goes through corrections to the two- and n -point functions for φ n with n = 3 , and finds the samefeatures in all cases. ew Trail for Naturalness Chapter 7 to [548, 549, 550, 551, 552, 497, 553] for discussions of some features relevant to SMmodel-building. We here satisfy ourselves with the simplest case, for which we have thenoncommutative Lagrangian L = 14 g F µν (cid:63) F µν + ( D µ φ ) ∗ (cid:63) ( D µ φ ) + V ( φ, φ ∗ ) , (7.29)where even though we’re quantizing U (1) we have F µν = ∂ µ A ν − ∂ ν A µ − i [ A µ ∗ , A ν ] dueto the noncommutativity, where [ · ∗ , · ] is the commutator in our noncommutative algebra.The vector fields transform as A µ (cid:55)→ U (cid:63) A µ (cid:63) U † + i∂ µ U (cid:63) U † , where U ( x ) is an elementof the noncommutative U (1) group, which consists of functions U ( x ) = (cid:0) e iθ ( x ) (cid:1) (cid:63) , whichis the exponential constructed via power series with the star-product.The potential and the covariant derivative both depend on the representation wechoose for the scalar. In contrast to commutative U (1) gauge theory, where we merelyassign φ a charge, our only choices now are to put φ in either the fundamental or theadjoint of the gauge group. Note that an adjoint field smoothly becomes uncharged inthe commutative limit. Such a field φ transforms as φ (cid:55)→ U (cid:63) φ (cid:63) U † . The covariantderivative is thus D µ φ = ∂ µ φ − ig [ A µ ∗ , φ ] . The gauge-invariant potential then includesboth quartic terms in Equation 7.28, in addition to others such as φ ∗ (cid:63) φ (cid:63) φ (cid:63) φ , since theadjoint complex scalar is uncharged at the level of the global part of the gauge symmetry.Strong UV/IR duality then should not hold here either.The situation is even worse if φ is in the fundamental, where it transforms as φ (cid:55)→ U (cid:63)φ and φ ∗ (cid:55)→ φ ∗ (cid:63) U − with covariant derivative D µ φ = ∂ µ φ − iA µ (cid:63) φ . It is easy to see in thiscase that the λ interaction term is no longer gauge invariant, and a charged scalar mayonly self-interact through V = λ φ ∗ (cid:63) φ (cid:63) φ ∗ (cid:63) φ . Purely from gauge invariance we thussee that a fundamental scalar has no nonplanar self-interaction diagrams in the one-loop It is important to note that many fundamental concepts which one normally thinks of as dependingupon Lorentz invariance still hold on noncommutative spaces, due to a ‘twisted Poincaré symmetry’[554, 555, 556, 557]. This includes the unitary irreducible representations, so it is sensible to speak of avector field. ew Trail for Naturalness Chapter 7 correction to its two-point function, and so there is no remnant of strong UV/IR dualityto speak of. The question is then whether there are other examples where this strong UV/IRduality does occur, or whether it is perhaps a peculiar feature of real φ n theories onnoncommutative spaces. To answer this, we will study in detail another case of especialphenomenological significance: Yukawa theory. Noncommutative Yukawa theory wasfirst studied in [560]. Our result on the presence of strong UV/IR mixing differs, forreasons we will explain henceforth. For reasons that will soon become clear, we will now work directly in Minkowski space,and begin with a commutative theory of a real scalar ϕ and a Dirac fermion ψ withYukawa interaction: L ( C ) = − ∂ µ ϕ∂ µ ϕ − m ϕ + iψ /∂ψ − ψM ψ + gϕψψ. (7.30)When constructing a noncommutative version of this theory, the quadratic part ofthe action does not change. However, ordering ambiguities appear for the interactionterm, and we in fact find two noncommutatively-inequivalent interaction terms whichgenerically appear: L ( NC ) int = g ϕ (cid:63) ψ (cid:63) ψ + g ψ (cid:63) ϕ (cid:63) ψ. (7.31)These terms are inequivalent because the star product is only cyclically invariant. In Noncommutative QED also has strange behavior in the gauge sector that runs counter to strongUV/IR duality—the photon self-energy correction gains an infrared singularity from nonplanar one-loopdiagrams, even though the commutative quadratic power-counting divergence is forbidden by gauge-invariance. The theory is constructed in detail in [558], while more physical interpretation is given in[559], and the possible relation to geometric dynamics in the context of matrix models is discussed in[544]. Aspects of noncommutative Yukawa theory have also been studied recently in d=3 in [561], and witha modified form of noncommuativity in [562]. ew Trail for Naturalness Chapter 7 the analysis of [560], only the g interaction was included. As a result, it was concludedthat this theory contains no nonplanar diagrams at one loop, and the first appear at twoloops as in Figure 7.1. This immediately tells us that the one-loop quadratic divergenceof the scalar self-energy will not appear with a one-loop IR singularity, and so rules outthe putative strong UV/IR duality of the theory they studied. g ₂ g ₂ g ₂ g ₂ g ₂ g ₂ g ₂ g ₂ Figure 7.1: Representative leading nonplanar corrections to the self-energies in the non-commutative Yukawa theory of [560]. Fermion lines have arrows and dashing denotesnonintersection.However, we must ask whether we actually have the freedom to choose g and g inde-pendently. To address that question, we must understand the role of discrete symmetriesin noncommutative theories. For ease of reference we here repeat our definition of thenoncommutativity parameter [ x µ , x ν ] = iθ µν (7.1)It is manifest that the noncommutativity tensor does not transform homogeneously un-der either parity or time-reversal, but only under their product: P T : x µ → − x µ ⇒ P T : θ µν → θ µν . So while any Lagrangian with full-rank noncommutativity unavoidablyviolates both P and T , it may preserve P T .Since both ϕ and the scalar fermion bilinear are invariant under all discrete sym-metries, these symmetries naïvely play no further role in this theory. However, the287 ew Trail for Naturalness Chapter 7 time-reversal operator is anti-unitary, and thus negates the phase in the star-product: ( P T ) − ( f ( x ) (cid:63) g ( x )) P T = g ( x ) (cid:63) f ( x ) . (7.32)Armed with this, we may now apply CPT to our interaction Lagrangian, to find ( CP T ) − L ( NC ) int CP T = g ψ (cid:63) ϕ (cid:63) ψ + g ϕ (cid:63) ψ (cid:63) ψ. (7.33)Comparing with Equation 7.31, we see that our interactions have been re-cycled! Re-quiring that our interactions preserve CPT amounts to imposing ( CP T ) − L ( NC ) int CP T = L ( NC ) int = ⇒ g = g (7.34)And so the theory of [560] appears to violate CPT. When we instead include bothorderings of interactions the nonplanar diagrams now occur at the first loop order. Fur-thermore, with both couplings set equal the planar and nonplanar diagrams will have thesame coefficients, which reopens the question of strong UV/IR duality for this theory.In the following we will keep g and g distinguished merely to evince how the differentvertices appear, but in drawing conclusions about the theory we will set them equal. We note that while the CPT theorem has only been proven in NCFT without space-time noncom-mutativity [563, 564, 565, 566], the difficulty in the general case is related to the issues with unitaritydiscussed in Section 7.2, and we expect it should hold in a sensible formulation of the space-time caseas well. We should note that in the construction of noncommutative QED it has been argued that it is sensibleto assign θ the anomalous charge conjugation transformation C : θ µν → − θ µν ([567] and many otherssince). The argument is that charged particles in noncommutative space act in some senses like dipoleswhose dipole moment is proportional to θ , and so charge conjugation should naturally reverse thesedipole moments. Here, however, our particles are uncharged, and thus we have no basis for arguing inthis manner. Furthermore, such an anomalous transformation makes charge conjugation relate theorieson different noncommutative spaces M θ → M − θ . The heuristic picture of the CPT theorem (that is,the reason we care about CPT being a symmetry of our physical theories) is that after Wick rotating toEuclidean space, such a transformation belongs to the connected component of the Euclidean rotationgroup [568], and so is effectively a symmetry of spacetime. So it is at the least not clear that defining aCPT transformation that takes one to a different space accords with the reason CPT should be satisfiedin the first place. ew Trail for Naturalness Chapter 7 First we consider the planar diagrams, of which there are two: − i Γ ,s,p ( p ) = g ₁ g ₁ p k + p/2k - p/2 + g ₂ g ₂ p k - p/2k + p/2 The ‘symmetrization’ of the momenta of the internal propagators is an importantcalculational simplification. This calculation is textbook save for our Schwinger-spaceregularization, so we will be brief and merely point out the salient features. The sum ofthese diagrams gives Γ (2) ,s,p ( p ) = i ( − (cid:0) ( ig ) + ( ig ) (cid:1) (cid:90) d k (2 π ) ( − i ) Tr (cid:2)(cid:0) M − /k − /p/ (cid:1) (cid:0) M − /k + /p/ (cid:1)(cid:3) (( k + p/ + M ) (( k − p/ + M ) . (7.35)To evaluate this, we must now introduce two Schwinger parameters α , α and thenswitch to ‘lightcone Schwinger coordinates’ which effects the change (cid:82) ∞ d α (cid:82) ∞ d α → (cid:82) ∞ d α + (cid:82) + α + − α + d α − . Regulating the integral by exp (cid:2) − / √ α + Λ (cid:3) , we may then evaluateand isolate the divergences as Λ → ∞ to find Γ (2) ,s,p ( p ) = − ( g + g )2 π (cid:20) Λ − M + p (cid:18) Λ M + p / (cid:19) + . . . (cid:21) (7.36)Turning now to the nonplanar diagrams, there are again two − i Γ (2) ,s,np = g ₂ p g ₁ + g ₂ p g ₁ Each now has one g vertex and one g vertex, which makes it clear why the analysisof [560] found no such diagrams. The two diagrams will come with opposite phase factors, e ip ∧ k and e ik ∧ p , so we can compute one and then find the other by taking p (cid:55)→ − p . In289 ew Trail for Naturalness Chapter 7 this case it’s obvious that after completing the square we will only be left with termswhich are quadratic in p , and so the two diagrams give the same contribution. We canthus compute both terms at the same time.The phase factor in the integrand will modify our change of variables, as it did in the φ case, to give again an effective cutoff for this diagram due to the noncommutativity.We find Γ (2) ,s,np ( p ) = g g π (cid:90) d q d α d α q (cid:18) M − q + α α ( α + α ) p + p ◦ p α + α ) (cid:19) × e − ( α + α ) ( q + M ) − α α α α p − p ◦ p α α . (7.37)We can now follow the same steps to regulate and integrate this, and again find aclosed-form expression for the pieces which contain divergences. Note that unlike the φ calculation, we can already see that the nonplanar expression will not merely be givenby Λ → Λ eff , as the change of variables has here modified the numerator of the integrandto give an extra piece to the momentum polynomial multiplying the exponential. Andso integration gives us Γ (2) ,s,np ( p ) = g g π (cid:34) (cid:0) M + p p ◦ p + 40(4 M + p ) p ◦ p Λ eff (cid:1) K (cid:32) (cid:112) M + p Λ eff (cid:33) + 20 (cid:112) M + p Λ eff (cid:0) − 96 + p p ◦ p + 12 p ◦ p Λ eff (cid:1) K (cid:32) (cid:112) M + p Λ eff (cid:33)(cid:35) . (7.38)We must now think slightly more carefully about what we want to add to the quadraticeffective action to find a Wilsonian interpretation of this theory. We may isolate the IRdivergence that appears when the cutoff is removed by first taking the limit Λ → ∞ with p ◦ p held fixed, and then expanding around p ◦ p = 0 . We may then ask that this samedivergence appears at any value of Λ . To account for this IR divergence, we must add toour effective action ∆ S PI (Λ) = − (cid:90) d p (2 π ) g g π (cid:18) Λ eff − p ◦ p (cid:19) ϕ ( p ) ϕ ( − p ) , (7.39)290 ew Trail for Naturalness Chapter 7 which can easily be done through the addition of an auxiliary scalar field as was done inSection 7.3 and is discussed in more generality in Appendix C. After having added this toour action, for small p ◦ p the scalar two-point function now behaves as Γ s ( p ) = − g g π p ◦ p + . . . for any value of Λ . The new pole in this case has the opposite sign as that in 7.19, and sowill be accessible in Euclidean signature, clearly signaling a tachyonic instability. Whilethis puts the violation of unitarity in this theory on prime display, it also means that thispole will be accessible in the s -channel in the Lorentzian theory if we allow for timelikenoncommutativity.We emphasize that any conclusions about the Lorentzian theory with timelike non-commutativity are speculative and dependent upon a solid theoretical understanding ofa unitary formulation of the field theory, and in principle such a formulation could findradically different IR effects than this naïve approach. However, it was found in [522] thata modification of time-ordering to explicitly make the theory unitary (at the expense ofmicrocausality violation) leaves the one-loop correction to the self-energy unchanged in φ theory, and the same might be expected to hold true for Yukawa theory. This makesit worthwhile to at least briefly consider the potential phenomenological consequences ofthe new pole.At low energies, the propagator is here modified to m + ( p i + p j ) − g g π p i + p j ) ◦ ( p i + p j ) .If we consider scattering of fermions through an s -channel ϕ and take the simple case of anoncommutativity tensor which in the lab frame has one eigenvalue / Λ θ with m (cid:29) Λ θ ,then the emergent pole appears at s = g g π − β β Λ θ m . Here s = − ( p i + p j ) is the invariantmomentum routed through the propagator, and β is the boost of the ( p i + p j ) system withrespect to the lab frame. The Lorentz-violation here then has the novel effect of smearingout the resonance corresponding to the light pole for a particle which is produced at avariety of boosts. This is in contrast to the pole at m , which gives a conventionalresonance at leading order. Of course, we have not constructed a fully realistic theory in291 ew Trail for Naturalness Chapter 7 any respect, and ultimately it may well be that other Lorentz-violating effects providethe leading constraint. Nonetheless, the lineshape of resonances may be an interestingobservable in this framework.A further feature of this opposite sign of the new pole compared to that in the φ theory is that the unusual momentum-dependence of the two-point function will lead toordered phases which break translational invariance [569, 491, 570, 571, 572]. While aLorentz-violating background field may possibly be very well constrained, the detailedconstraint depends on its wavelength and the ways in which it interacts with the SM.But this is another obvious line of exploration for constraining realistic NCFTs. There are again two planar diagrams: − i Γ (2) ,f,p = p g ₁ g ₁ + p g ₂ g ₂ No new features appear in the evaluation of these diagrams, so we merely quote the finalresult: Γ (2) ,f,p = − g + g π (cid:18) M − /p (cid:19) log 4 p Λ m + 2 m ( p − M ) + ( M + p ) + . . . (7.40)We also have two nonplanar diagrams, which again mix the two vertices − i Γ (2) ,f,np = p g ₁ g ₂ + p g ₁ g ₂ Here we find that the different phase factors for each diagram, which we saw wereinconsequential for the nonplanar corrections to the scalar, have an important role. Whenwe complete the square in each of the two cases, we find that one of the diagrams hasan integrand proportional to (cid:16) M − /p α α + α − p µ θ µν γ ν α + α (cid:17) and the other is proportional292 ew Trail for Naturalness Chapter 7 to (cid:16) M − /p α α + α + p µ θ µν γ ν α + α (cid:17) , so the would-be divergence in pθ will cancel manifestlybetween the two diagrams. After this everything proceeds as before, and we find Γ (2) ,f,np = − g g π (cid:18) M − /p (cid:19) log 4 p Λ eff m + 2 m ( p − M ) + ( M + p ) + . . . (7.41)We see that with g = g ≡ g , the fermion quadratic effective action also behaves asexpected from ‘strong UV/IR duality’. The logarithmic divergence of the commutativetheory has been transmogrified in the nonplanar diagrams into IR dynamics via thesimple replacement Λ → Λ eff , and so a p ◦ p → pole will emerge when we remove thecutoff. We discuss the use of an auxiliary field to restore a Wilsonian interpretation herein Appendix C.2. The correction to the vertex function constitutes further theoretical data toward theWilsonian interpretation of the noncommutative corrections. We calculate the one-loopcorrection in this section and delay the discussion of the use of auxiliary fields to accountfor them until Appendix C.3. We will find that while we can use the same fields toaccount for the modifications to both the propagators and the vertices, the physicalinterpretation of such fields is unclear.We can compute corrections for each fixed ordering of external lines separately sincethey’re coming from different operators. For simplicity we’ll compute the g ordering,which we will denote Γ ϕψψ ( r, p, (cid:96) ) . There are four diagrams in total: one planar diagramwith two insertions of the g vertex, one nonplanar diagram with two insertions of the g vertex, and two nonplanar diagrams with one insertion of each. It is easy to see bylooking at the diagrams that the same expressions with g ↔ g compute the correctionto the other ordering, Γ ψϕψ ( r, p, l ) . 293 ew Trail for Naturalness Chapter 7 The new feature of this computation is that we now need three Schwinger parameters,and this presents a problem for our previous computational approach. We won’t be ableto perform the two finite integrals before expanding in a variable which isolates thedivergences when α + α + α → , analogously to what we did in d Schwinger space.Instead we slice d Schwinger space such that we can perform the integral which isolatesthe leading divergences first, and then—as long as we’re content only to understand thisdivergence—we can discard the rest without having to worry about performing the othertwo integrals.The planar diagram is i Γ ϕψψ ,p ( p, (cid:96) ) = p g ₂ g ₂ g ₁ l , and corresponds to the expression Γ ϕψψ ,p ( p, (cid:96) ) = − i ( ig )( ig ) × (7.42) (cid:90) d k (2 π ) ( − i ) (cid:16) M − ( /k + /p + /(cid:96) ) (cid:17) (cid:16) M − ( /k − /p − /(cid:96) (cid:17)(cid:0) ( k + p + (cid:96) ) + M (cid:1) (cid:0) ( k − p − (cid:96) ) + M (cid:1) (cid:0) ( k + p − (cid:96) ) + m (cid:1) . After moving to Schwinger space, integrating over the loop momentum, and introducinga cutoff exp ( − / (Λ ( α + α + α ))) , we switch variables to α = ξ η, α = ξ η, α = (1 − ξ − ξ ) η, (7.43)under which (cid:82) ∞ d α (cid:82) ∞ d α (cid:82) ∞ d α → (cid:82) d ξ (cid:82) − ξ d ξ (cid:82) ∞ d η η . Performing the mo-mentum integral transfers the divergence for large k to a divergence in small α + α + α = η . This will allow us to find the leading divergent behavior immediately by carrying outthe η integral and then expanding in Λ → ∞ . This yields Γ ϕψψ ,p ( p, (cid:96) ) = g g π log (cid:0) Λ (cid:1) + finite , (7.44)where we are unable to determine the IR cutoff of the logarithm, but this suffices for ourpurposes. 294 ew Trail for Naturalness Chapter 7 The three nonplanar graphs now each receive a different phase corresponding to whichexternal line crosses the internal line i Γ ϕψψ ,np ( p, (cid:96) ) = p g ₁ lg ₁ g ₁ + p lg ₁ g ₂ g ₁ + p lg ₁ g ₂ g ₁ , (7.45)where the first gets exp [ − i ( k ∧ p + k ∧ (cid:96) + p ∧ (cid:96) )] , the second exp [ − i ( k ∧ p + p ∧ (cid:96)/ ,and the third exp [ − i ( k ∧ (cid:96) + p ∧ (cid:96)/ . The evaluation of these diagrams proceeds as inthe previous examples. If we take the IR limit p, (cid:96) → of the nonplanar contributionsto this ordering of the three-point function and then expand in large Λ we find lim p,(cid:96) → Γ ϕψψ ,np ( p, (cid:96) ) = g ( g + 2 g )16 π log (cid:0) Λ (cid:1) + finite . (7.46)However, if we first take the UV limit Λ → , and then expand in small momenta, wefind lim Λ →∞ Γ ϕψψ ,np ( p, (cid:96) ) = g π (cid:34) g log (cid:18) p + (cid:96) ) ◦ ( p + (cid:96) ) (cid:19) (7.47) + g log (cid:18) p ◦ p (cid:19) + g log (cid:18) (cid:96) ◦ (cid:96) (cid:19) (cid:35) + finite , where we again see UV/IR mixing, and we note that each nonplanar diagram has beeneffectively cutoff by the momenta which cross the internal line. We discuss the use ofauxiliary fields to restore a Wilsonian interpretation to this vertex correction in AppendixC.3. 295 ew Trail for Naturalness Chapter 7 We now turn our attention to the softly-broken noncommutative Wess-Zumino model asa controllable example of the interplay between UV/IR mixing and the finiteness of thefield theory. We will restrict ourselves to calculating the one-loop correction to the scalartwo-point function. Since the new poles appearing in the quadratic effective action inthe scalar and Yukawa theories are intimately related to the quadratic divergences of thecommutative theories, we will not be surprised to find that this feature will disappearwhen both the scalar and the fermion are present in the EFT below the cutoff. Bystudying the softly-broken theory we can take the fermion above or below the cutoff tosmoothly see the relation between the finiteness of the field theory and the effects ofUV/IR mixing. The exactly supersymmetric noncommutative Wess-Zumino model wasfirst discussed in detail in [573], and the absence of an infrared pole in a softly-brokentheory was first noted in [559]. The softly-broken Wess-Zumino model was first consideredin [496]. The noncommutative Wess-Zumino theory can be suitably formulated in off-shellsuperspace as L = (cid:90) d θ Z Φ † Φ + (cid:90) d θ (cid:18) M Φ + 16 y Φ (cid:63) Φ (cid:63) Φ (cid:19) + h.c. , (7.48)where Φ is a chiral superfield and we have included a wavefunction renormalization factorin the Kähler potential Z = 1 + O ( y ) . We can introduce soft supersymmetry breakingby promoting this factor to a spurion Z = 1 + ( | M | − m ) θ θ † , the only effect of whichis to modify the scalar mass spectrum. Our one-loop results agree with those of [496] save for their claim that logarithmic IR divergences areabsent in the exactly supersymmetric theory, which contradicts [573]. We will below find a logarithmicIR divergence in the wavefunction renormalization which is independent of the soft-breaking, which isconsistent with the expectations of strong UV/IR duality. ew Trail for Naturalness Chapter 7 Formulating the noncommutative theory including the auxiliary F fields makes itmanifest that we have preserved supersymmetry off-shell. This procedure is in factprecisely the same as quantizing after integrating out F , and so we end up with a star-product version of the familiar Lagrangian: −L NCWZ = Z∂ µ φ ∗ ∂ µ φ − iZψ † ¯ σ µ ∂ µ ψ + Z − m φ ∗ φ + 12 M ψψ + 12 M ∗ ψ † ψ † + 12 Z − yφ (cid:63) ψ (cid:63) ψ + 12 Z − y ∗ φ ∗ (cid:63) ψ † (cid:63) ψ † + 12 Z − yM ∗ φ (cid:63) φ (cid:63) φ ∗ + 12 Z − y ∗ M φ ∗ (cid:63) φ ∗ (cid:63) φ + 14 Z − | y | φ (cid:63) φ (cid:63) φ ∗ (cid:63) φ ∗ (7.49)where φ is a complex scalar and ψ is a Weyl fermion. Of course, now that we’ve introducedsupersymmetry breaking we expect to find that there is further renormalization beyondthat associated with Z , but keeping the manifest factors of Z will allow us to easilycompare to our expectations for the supersymmetric limit.The calculation of the one-loop correction to the two-point function goes much as thepreviously-demonstrated examples. The presence of the three-scalar interaction gives anew class of diagrams, whose evaluation is routine. The two-component fermions yieldslightly different factors than did the Dirac fermions [119]. Finally, it is important tonote that the results for the diagrams computed in Section 7.3 cannot be used here, aswe must here regulate uniformly using exp( − / (Λ ( α + α ))) like we did in Section 7.4.This may be easily accommodated by writing the integrand in the quartic diagrams as k + m k + m k + m .Adding up all these diagrams and taking the limit where Λ , Λ eff are large, we find297 ew Trail for Naturalness Chapter 7 that the one-loop scalar two-point function may be organized as Γ (2) ,s ≡ Zp + Z − ( m + δm ) (7.50) Z = 1 + y π log (cid:20) ΛΛ eff M (cid:21) + . . . (7.51) δm = y π (cid:0) M − m (cid:1) log (cid:20) ΛΛ eff M (cid:21) + . . . , (7.52)where we make manifest the presence of supersymmetric nonrenormalization in the limit m → M , which acts as a non-trivial check. As expected, the absence of the quadratic UVdivergence in the Wess-Zumino model has led to the absence of an infrared pole from thenoncommutativity, even as the fermion is made arbitrarily heavy relative to the scalar.However, logarithmic UV/IR mixing still occurs.We may repeat this calculation using dimensional regularization and taking note ofthe issues which arose in Section 7.3.1. Using the same parametrization of the one-looptwo-point function as above, the planar diagrams contribute Z planar = 1 + y π (cid:18) (cid:15) + log µ M (cid:19) + . . . (7.53) δm planar = y π ( M − m ) (cid:18) (cid:15) + log µ M (cid:19) + . . . , (7.54)as expected. The full form of the nonplanar diagrams is unenlightening, but if we takethe IR limit p ◦ p → first, they give precisely the same contribution as the planardiagrams, since the diagram degeneracies are all the same in this case. Taking the UVlimit (cid:15) → first (and staying in d < ), we instead find Z nonplanar = 1 + y π log 4 M p ◦ p + . . . (7.55) δm nonplanar = y π ( M − m ) log 4 M p ◦ p + . . . , (7.56)which has precisely the same correspondence with the Schwinger-space regularization aswe saw for the φ case. 298 ew Trail for Naturalness Chapter 7 We thus see clearly the conflict between supersymmetry and the use of UV/IR mixingto explain low-energy puzzles. UV/IR mixing transmogrified UV momentum dependenceinto IR momentum dependence, and so depended crucially on the sensitivity of our fieldtheory to UV modes. For a theory which is finite as a field theory, the dependence onthe UV physics has been removed, and so we see no interesting IR effects.Of course, in the presence of a cutoff Λ it is also possible to study the behavior of thescalar two-point function when M (cid:29) Λ (cid:29) | M − m | as the fermion is taken abovethe cutoff while keeping the scalar light. This corresponds to taking M/ Λ , M/ Λ eff > and then expanding in the limit where Λ , Λ eff are large. This gets rid of the nonplanarYukawa-type diagrams and, as one might expect, results in a return of UV sensitivity inthe scalar EFT below the cutoff, foreshadowing a return of the UV/IR mixing effects.The scalar mass-squared in this limit becomes δm = y π (cid:0) M + 16Λ + 8Λ eff (cid:1) + . . . . (7.57)and UV/IR mixing reappears at the quadratic level. So our EFT intuition isn’t totallyout the window; it’s been broken in a controlled way, and we can smoothly interpolatebetween theories with and without UV/IR mixing by taking the states responsible forfiniteness above the cutoff. This sharpens the sense in which UV/IR mixing can dosomething interesting in the IR as long as the field-theoretic description of our universeis never finite.Ultimately, this highlights a central challenge for approaching the hierarchy problemvia UV/IR mixing. The hierarchy problem is particularly sharp when the full theoryis finite and scale separation is large, in which case the sensitivity of the Higgs massto underlying scales is unambiguous. But UV/IR mixing effects potentially relevant tothe hierarchy problem are absent in this case, and emerge only when finiteness is lost.This tension is not necessarily fatal to UV/IR approaches to the hierarchy problem—299 ew Trail for Naturalness Chapter 7 ultimately the UV sensitive degrees of freedom are not the ones we would wish to identifywith the Higgs—but it bears emphasizing.Moreover, there is a possible loophole in the general argument that finiteness must besurrendered in order to generate a scale from UV/IR mixing. The presence of interestingeffects in the IR here depends solely on the UV sensitivity of the nonplanar diagrams.The ‘orbifold correspondence’ [308, 307, 306] provides non-supersymmetric field theoriesconstructed via orbifold truncation of N > theories whose planar diagrams agree withthose of the supersymmetric theory and so are finite. A noncommutative orbifold fieldtheory [574] may then provide a theory which is fully predictive, yet which still generatesan infrared scale via UV/IR mixing. Generally, it may be possible that UV/IR mixingappears in such a way that it is the sole effect sensitive to short distances.300 ew Trail for Naturalness Chapter 7 To attempt to formulate a realistic theory which uses UV/IR mixing to solve extanttheoretical puzzles, it would be useful to have an understanding of which features ofNCFT were responsible for the curious infrared effects discussed above. This wouldbe helpful whether one wishes to test out these ideas in any of the many proposedmodifications of NCFT, or to write down other toy models which share some features ofNCFT but are based upon different principles.Qualitatively, the two unusual features involved in the formulation of NCFT areLorentz invariance violation and nonlocality. However, it is obvious that one may havetheories with one or both of these features without the interesting effects we have seen.The answer then is not so simple as pointing to one axiom or another of EFT which hasbeen broken, but depends sensitively on the way in which they are broken. We brieflyexplore two ways we may better understand the interplay here between nonlocality andLorentz-violation and how they come together to cause surprising low-energy effects. Wefirst give a general argument based on the way nonlocality appears to postdict the formof the violation of EFT expectations. We then phenomenologically examine the loopintegration appearing in our NCFT calculations to diagnose what caused the appear-ance of the IR pole. This will lead us to discuss an avenue toward investigating (ormanufacturing) such effects in nonlocal, Lorentz-invariant theories.To see how EFT expectations may be violated, consider the peculiar way in which thenoncommutative effects in the one-loop action (e.g. Equation 7.16) induce nonlocality.In Wilsonian EFT, integrating out momentum modes p (cid:38) Λ produces a nonlocal theoryat those scales, or equivalently on distances x (cid:46) / Λ . However, particles on a noncom-mutative space can be thought of as rods of size L ∼ pθ [507, 508, 509, 510, 511]. Thistells us that in a NCFT we should expect nonlocality to be present for scales x (cid:46) pθ .301 ew Trail for Naturalness Chapter 7 Comparing the two scales, we see that we should find nonlocal effects past those expectedin Wilsonian EFT for < pθ . Here this momentum-dependent nonlocality occurs in aLorentz-violating way. This expectation was exactly borne out in the examples above,where we saw that the one-loop effective action in momentum space is nonlocal for p ◦ p (cid:29) / Λ [491].Purely from this analysis of the form of nonlocality, we may conclude there will be abreakdown of Wilsonian renormalization. After we remove the cutoff, the theory shouldbe nonlocal on all scales p ◦ p > . But if we compute a correlation function at a large-but-finite Λ , the theory will still be local for momenta p ◦ p < / Λ , and so will greatly differfrom the continuum result. So our surprising discovery of the non-uniform convergenceof correlation functions in the examples above is understood easily from this picture.While this sort of momentum-dependent nonlocality may seem ad hoc , it has beensuggested previously for separate purposes. It has been argued [575] that quantum gravityshould obey a ‘Generalized Uncertainty Principle’ ∆ x (cid:38) (cid:126) ∆ p + (cid:96) p ∆ p , with (cid:96) p the Plancklength, based on the use of Hawking radiation to measure the horizon area of a black hole.This gives precisely the same sort of momentum-dependent nonlocality as we saw above.We refer the reader to [576] for a review of the Generalized Uncertainty Principle, [577,578] for similar conclusions within string theory, and [579] for a more general review ofthe appearance of an effective minimal length in quantum gravity. It would be interestingto investigate other field theories which obey such uncertainty principles and determinewhether UV/IR mixing causes similar features as appear in NCFT. For theories whichviolate Lorentz invariance, care must be taken to avoid arguments that even Planck-scaleLorentz violation is empirically ruled out [492, 580].We may also attempt to phenomenologically diagnose what caused the appearanceof the IR pole from the form of the loop integration. The presence of an exponential ofmomenta was clearly crucial, and this implies a necessity of nonlocality. It’s also clear302 ew Trail for Naturalness Chapter 7 that the modification of the cutoff in the nonplanar diagrams Λ (cid:55)→ Λ eff , which renderedthe diagrams UV finite in a way that brought UV/IR mixing, was a result of the contrac-tion between the loop momentum and the external momentum. Less obviously, one maysee that any quadratic term in loop momentum in the exponential would have erasedthis feature, as after momentum integration one would find an integrand ∼ α + , andany divergence will have disappeared. Heuristically, the quadratic suppression in loopmomentum is too strong and regulates the UV divergence entirely independently of thecutoff, so no UV/IR mixing appears. NCFT disallows such terms as a result of mo-mentum contractions being performed with an antisymmetric tensor, and this particularmechanism seems to imply the necessity of Lorentz invariance violation. However, thisargument only considers small deviations from the form of the integral in NCFT. Furtherdiscussions of the form of loop integrals with generalizations of the star-product may befound in [581, 582].Likely a better approach to understand the prospect for finding features similar to thatof NCFT in a Lorentz invariant theory is to back up and study formulations of Lorentzinvariant extensions of NCFT. This is accomplished by upgrading the noncommutativitytensor θ µν from a c -number to an operator. This was proposed already by Snyder in1947 [335], and this approach has been revived a number of times more recently (e.g.[583, 584, 585, 586, 587]). Schematically, this results in an action containing an integralover θ µν S = (cid:90) d x d θ W ( θ ) L ( φ, ∂φ ) , (7.58)where W ( θ ) is a ‘weighting function’, and the Lagrangian is still defined using the star-product. The challenge in this approach for our purposes is in devising a method fornonperturbative calculations in θ , which as we saw above was necessary to preserve thefeatures of UV/IR mixing. 303 ew Trail for Naturalness Chapter 7 Searching more generally for Lorentz invariant theories which contain UV/IR mix-ing will likely allow more promising phenomenological applications. That such theoriesshould exist can be broadly motivated by quantum gravity, as any gravitational theoryis expected both to be nonlocal and to have UV/IR mixing. That Lorentz violationshould be present is less clear. A particularly interesting line of development is to thenunderstand in detail the class of nonlocal theories that would have UV/IR mixing of asort similar to that discussed here. Recent work toward placing nonlocal quantum fieldtheories on solid theoretical ground [498, 532] is clearly of sharp interest here, thoughthe larger goal is quite distinct. The nonlocality studied in these works is designed torender the field theory UV-finite, and so the nonlocal vertex kernels are chosen preciselyto avoid the introduction of new poles by ensuring these are momentum-space entirefunctions which vanish rapidly in Euclidean directions. The nonlocal vertices of NCFTmanage to introduce new poles by oscillating as p → ∞ , which presumably allows for theappearance of new ‘endpoint singularities’ [588, 589], though a full examination of theLandau equations in NCFT has not (to our knowledge) been performed. Our interest isthus in a disjoint class of nonlocal theories, where new poles can appear in interestingways. Classifying the space of such theories and developing an approach to systematicallyunderstand their unitarity properties seems well motivated.304 ew Trail for Naturalness Chapter 7 The lack of evidence for conventional solutions to the hierarchy problem has placedparticle physics at a crossroads. While it is possible that the answer ultimately liesfurther down the well-trodden path of existing paradigms, the appeal of less-travelledpaths grows greater with every inverse femtobarn of LHC data.In this work we have ventured to take seriously the apparent failure of expectationsfrom Wilsonian effective field theory regarding the hierarchy problem by investigatinga concrete framework—noncommutative field theory—in which Wilsonian EFT itselfbreaks down. Not only does noncommutative field theory violate Wilsonian expectations,it provides a sharp instance of UV/IR mixing: ultraviolet modes of noncommutativetheories can generate an infrared scale whose origin is opaque to effective field theory. Tothe extent that UV/IR mixing has any relevance to the hierarchy problem, the emergenceof an infrared scale seems to be among the most promising effects. Although the real-world applicability of these theories is likely limited by their Lorentz violation, theynonetheless provide valuable toy models for exploring the potential relevance of UV/IRmixing to problems of the Standard Model.To this end, we have surveyed existing results on noncommutative theories with an eyetowards ‘strong UV/IR duality’—the transmogrification of UV divergences into infraredpoles at the same order. This led us to a detailed analysis of noncommutative Yukawatheory, perhaps the most useful toy model for thinking about the hierarchy problem(insofar as the Yukawa sector of the Standard Model is responsible for the largest UVsensitivity of the Higgs mass, and highlights the relative UV insensitivity of the fermionmasses). In the noncommutative theory, the presence of both inequivalent Yukawa cou-plings implies the same strong UV/IR duality exhibited by real φ theory: a quadraticdivergence in the one-loop correction to the scalar mass from fermion loops gives rise to305 ew Trail for Naturalness Chapter 7 a simple IR pole, while a logarithmic UV divergence in the one-loop correction to thefermion mass from scalar loops give rise to only a logarithmic IR divergence. Intriguingly,the infrared pole in the scalar two-point function appears accessible in the s -channel inthe Lorentzian theory, a feature which gives it particular phenomenological relevance.We then introduced softly-broken supersymmetry as a way to explore the interplaybetween (in)finiteness and UV/IR mixing. Choosing soft terms in order to keep thescalar light as the fermion mass is varied concretely illustrates several expected features.Strong UV/IR duality is preserved in the sense that both UV and IR divergences areabsent at quadratic order (and persist at logarithmic order) when both the scalar and thefermion are in the spectrum. However, infrared structure reappears as the fermion massis raised above a fixed cutoff and (quadratic) finiteness is lost. This underlines the sensein which UV/IR mixing may only ever play an interesting role when the field theory isquadratically UV sensitive at all scales, a scenario in which the hierarchy problem is lessconcrete.Finally, building on the lessons from the toy models considered here, we have high-lighted a variety of interesting lines of exploration in theories featuring nonlocality withor without Lorentz violation that may be of relevance to the hierarchy problem.While the prospect that UV/IR mixing will solve outstanding theoretical problemsin the low-energy universe is possibly fanciful, now is the time for such reveries. Theparadigms of the past few decades of particle theory are under considerable empiricalpressure, and innovative approaches are needed. At the very least, by pushing the limitsof EFT we stand to learn more about the broad spectrum of phenomena possible withinquantum field theory. 306 hapter 8Conclusion Scientists are baffled: What’s up with the universe? The Washington Post HeadlineNovember 1, 2019 [590] We end the way we began: Declaring it to be an exciting time in particle physics.The picture we have painted above on the state of the field is one of uncertainty—and indeed we have barely even touched on many of the important problems of theStandard Model. Dark matter and neutrino masses, while having had canonical, obvious,beautiful solutions in the context of supersymmetric grand unified theories, are also as yetmysterious. These fields have likewise turned their focus toward alternative mechanismsin the past few years as a result of the lack of observational evidence for their standardsolutions. But these facts all make the universe a more exciting place to study. Imagineif we had found weak-scale supersymmetry at the LHC, and our job now was simplyto interpret the data in terms of which of the supersymmetric extensions proposed andwell-studied in the past decades were correct. Or even worse, if technicolor had reallybeen the answer and we had to watch Nature repeat the same trick she used at the strongscale again at the weak scale. How dreadfully boring!307 onclusion Chapter 8 Yes, yes, this attitude is selfish and a bit flippant, but what we now have is the chanceto learn more about the universe and about the spectrum of possibilities in physics, andto explore new, radical ideas.Let me end with a reminder of another, prior era in which theoretical physicistshad thought they had everything figured out, recalled by no less than Max Planck ina 1924 talk at the University of Munich, and bring to your mind the outcome of thosepredictions:As I began my university studies [in 1878] I asked my venerable teacherPhilipp von Jolly for advice regarding the conditions and prospects of mychosen field of study. He described physics to me as a highly developed, nearlyfully matured science, that through the crowning achievement of the discoveryof the principle of conservation of energy it will arguably soon take its finalstable form. It may yet keep going in one corner or another, scrutinizing orputting in order a jot here and a tittle there, but the system as a whole issecured, and theoretical physics is noticeably approaching its completion tothe same degree as geometry did centuries ago. That was the view fifty yearsago of a respected physicist at the time. — Max PlanckAs translated in Wells (2016) [591] from Planck (1933) [592]May the universe continue to surprise us. (cid:3) In fairness to von Jolly (1809-1884), he really was a respected experimental physicist in his day—enough so to have been knighted—and earlier in his life made important contributions to the under-standing of gravity and of osmosis [591]. This attitude was not rare at the time, and he wouldn’t beremembered for it were it not for a student of his having played a role in revolutionizing physics. ppendix AKinetic Mixing in the Mirror TwinHiggs Since kinetic mixing plays a central role in freeze-twin dark matter, we discuss here atsome length the order at which it is expected in the low-energy EFT. Of course, theremay always be UV contributions which set (cid:15) to the value needed for freeze-in. However,if the UV completion of the MTH disallows such terms - for example, via supersymmetry,an absence of fields charged under both sectors, and eventually grand unification in eachsector (see e.g. [593, 594, 292, 293, 595, 596])- then the natural expectation is for mixingof order these irreducible IR contributions.To be concrete, we imagine that (cid:15) = 0 at the UV cutoff of the MTH, Λ (cid:46) πf . Tofind the kinetic mixing in the regime of relevance, at momenta µ (cid:46) GeV, we must rundown to this scale. As we do not have the technology to easily calculate high-loop-orderdiagrams, our analysis is limited to whether we can prove diagrams at some loop orderare vanishing or finite, and so do not generate mixing. Thus our conclusions are strictlyalways ‘we know no argument that kinetic mixing of this order is not generated’, andthere is always the possibility that further hidden cancellations appear. With that caveat309 inetic Mixing in the Mirror Twin Higgs Chapter A divulged, we proceed and consider diagrammatic arguments in both the unbroken andbroken phases of electroweak symmetry.Starting in the unbroken phase, we compute the mixing between the hyperchargegauge bosons. Two- and three-loop diagrams with Higgs loops containing one gaugevertex and one quartic insertion vanish. By charge conjugation in scalar QED, the three-leg amplitude of a gauge boson and a complex scalar pair must be antisymmetric underexchange of the scalars. However, the quartic coupling of the external legs ensures thattheir momenta enter symmetrically. As this holds off-shell, the presence of a loop whichlooks likecauses the diagram to vanish. However, at four loops the following diagram can be drawnwhich avoids this issue:where the two hypercharges are connected by charged fermion loops in their respectivesectors and the Higgs doublets’ quartic interaction. This diagram contributes at leastfrom the MTH cutoff Λ (cid:46) πf down to f , the scale at which twin and electroweaksymmetries are broken. We have no argument that this vanishes nor that its unitarity cutsvanish. We thus expect a contribution to kinetic mixing of (cid:15) ∼ g c W / (4 π ) , with g thetwin and SM hypercharge coupling and c W = cos θ W appearing as the contribution to thephoton mixing operator. In this estimate we have omitted any logarithmic dependenceon mass scales, as it is subleading. 310 inetic Mixing in the Mirror Twin Higgs Chapter A In the broken phase, we find it easiest to perform this analysis in unitary gauge. TheHiggs radial modes now mass-mix, but the emergent charge conjugation symmetries inthe two QED sectors allow us to argue vanishing to higher-loop order. The implicationsof the formal statement of charge conjugation symmetry are subtle because we have twoQED sectors, so whether charge conjugation violation is required in both sectors seemsunclear. However, similarly to the above case, there is a symmetry argument whichholds off-shell. The result we rely on here is that in a vector-like gauge theory, diagramswith any fermion loops with an odd number of gauge bosons cancel pairwise. Thus, eachfermion loop must be sensitive to the chiral nature of the theory, so the first non-vanishingcontribution is at five loops as in:where the crosses indicate mass-mixing insertions between the two Higgs radial modeswhich each contribute ∼ v/f . Thus, both the running down to low energies and the finitecontributions are five-loop suppressed. From such diagrams, one expects a contribution (cid:15) ∼ e g A g V ( v/f ) / (4 π ) , where with g V and g A we denote the vector and axial-vectorcouplings of the Z , respectively. We note there are other five loop diagrams in whichHiggses couple to massive vectors which are of similar size or smaller.Depending on the relative sizes of these contributions, one then naturally expectskinetic mixing of order (cid:15) ∼ − − − . If (cid:15) is indeed generated at these loop-levels,then mixing on the smaller end of this range likely requires that it becomes disallowednot far above the scale f . However, we note that our ability to argue for higher-looporder vanishing in the broken versus unbroken phase is suggestive of the possibility thatthere may be further cancellations. We note also the possibility that these diagrams,311 inetic Mixing in the Mirror Twin Higgs Chapter A even if nonzero, generate only higher-dimensional operators. Further investigation of thegeneration of kinetic mixing through a scalar portal is certainly warranted.312 ppendix BHow to Formulate Field Theory on aNoncommutative Space In this appendix we provide detail on how to formulate field theories on a space which isdefined by Equation 7.1, which we repeat here for convenience: [ˆ x µ , ˆ x ν ] = iθ µν . (B.1)To construct a field theory on this space we must specify the algebra of observables. Firstwe briefly recall the familiar, commutative case. For simplicity, we consider a scalar fieldtheory on flat Euclidean space. We denote by Alg (cid:0) R d [ x ] , · (cid:1) the commutative, C ∗ -algebraof Schwartz functions of d -dimensional Euclidean space with the standard point-wiseproduct, and this constitutes our algebra of observables. A convenient basis for thevector space is that of plane waves e ip · x .The case of interest here is noncommutative flat Euclidean space, on which we defineAlg (cid:0) R dθ [ˆ x ] , · (cid:1) . This now consists of such functions of d variables ˆ x µ related by Equation7.1, and so is a noncommutative algebra, although we’ve specified again the normal‘point-wise’ product. A useful basis will again be that of plane waves, which we may313 ow to Formulate Field Theory on a Noncommutative Space Chapter B define as the eigenfunctions of appropriately-defined derivatives on the noncommutativespace, and which look familiar e ip · ˆ x . To can get a sense for this algebra it is useful tocarry out the simple exercise of multiplying two plane waves by simply applying Baker-Campbell-Hausdorff e ik · ˆ x · e ik (cid:48) · ˆ x = exp (cid:18) ik · ˆ x + ik (cid:48) · ˆ x − k µ k (cid:48) ν [ˆ x µ , ˆ x ν ] (cid:19) = e − i θ µν k µ k (cid:48) ν e i ( k + k (cid:48) ) · ˆ x . (B.2)As in quantum mechanics, we will wish to study noncommutative versions of familiarcommutative theories, and so it will be useful to view R dθ as a ‘deformation’ of R d . Wethen wish to construct a map from our commutative algebra to our noncommutative onewhich returns smoothly to the identity as θ µν → . The standard such choice is theWeyl-Wigner map ˆ W , which one may roughly think of as merely replacing x s with ˆ x s.The procedure is simply to Fourier transform from commutative space to momenta,and then inverse Fourier transform to noncommutative space. Given a commutativespace Schwartz function f , we may compose the two operations and write ˆ W [ f ] = (cid:90) d d xf ( x ) ˆ∆( x ) , ˆ∆( x ) = (cid:90) d d k (2 π ) d e ik i · ˆ x i e − ik i · x i . (B.3)Note that this is an injective map of Schwartz functions on R d to those on R dθ whichrespects the vector space structure but not the structure of the algebra. This propertyis familiar from quantum mechanics.We may now construct noncommutative versions of field variables, but we still don’tknow how to do physics on these spaces. That is, we can write down the Lagrangian fornoncommutative φ theory, and we could even determine an action after we formulate anotion of an integral over a noncommutative space. But our familiar results about howto go from the action of a field theory to a calculation for a physical observable mostcertainly depended implicitly on living on a commutative space, and so it seems we mustre-formulate physics from the bottom up. 314 ow to Formulate Field Theory on a Noncommutative Space Chapter B Fortunately, such a drastic measure may not be necessary, as one may formulateQFT on noncommutative spaces as a simple modification of our normal field theorystructure. The core idea is to find an algebra of functions on R d which is isomorphicto Alg (cid:0) R dθ [ˆ x ] , · (cid:1) by pushing the noncommutativity into a new field product, known as aGroenewold-Moyal product (or star-product). We diagram the structure we wish to lookfor in Figure B.1. Alg (cid:0) R d [ x ] , · (cid:1) Alg (cid:0) R dθ [ˆ x ] , · (cid:1) Alg (cid:0) R d [ x ] , (cid:63) θ (cid:1) ˆ W : R d [ x ] → R dθ [ˆ x ] Id R d [ x ] ˆ W an isomorphism of algebras Figure B.1: The relations between the algebra of a commutative field theory, the noncom-mutative algebra one finds by applying the Weyl-Wigner map, and the noncommutativealgebra most useful for field theory making use of the Groenewold-Moyal product.Thevertical arrows respect only the vector space structure, and one should think of construct-ing a new algebra by applying a vector space map and then endowing the vectors with amultiplication operation.In particular, we may do this by demanding that our quantization map ˆ W is upgradedto an isomorphism between Alg (cid:0) R dθ [ˆ x ] , · (cid:1) and an algebra on the vector space of functionsof commutative Euclidean space, with a multiplication operation which is chosen topreserve the algebraic structure. That is, we must satisfy ˆ W [ f (cid:63) θ g ] = ˆ W [ f ] · ˆ W [ g ] , (B.4)for any Schwartz functions f, g on commutative Euclidean space. But we can guaranteethis by ensuring it for plane waves, the calculation of which we’ve essentially already315 ow to Formulate Field Theory on a Noncommutative Space Chapter B done above in Equation B.2: e ikx (cid:63) θ e ik (cid:48) x = ˆ W − (cid:104) ˆ W (cid:2) e ikx (cid:3) · ˆ W (cid:104) e ik (cid:48) x (cid:105)(cid:105) = ˆ W − (cid:104) e − i θ ij k i k (cid:48) j e i ( k + k (cid:48) ) · ˆ x (cid:105) e ikx (cid:63) θ e ik (cid:48) x ≡ e − i θ µν k µ k (cid:48) ν e i ( k + k (cid:48) ) · x , (B.5)where the θ subscript merely tells us the star-product will depend on the noncommutativ-ity tensor, and this will henceforth be dropped. This gives a position-space representationof the star-product, f ( x ) (cid:63) g ( x ) = exp (cid:18) i θ µν ∂ µy ∂ νz (cid:19) f ( y ) g ( z ) (cid:12)(cid:12)(cid:12)(cid:12) y = z = x = f ( x ) exp (cid:18) i ←− ∂ µ θ µν −→ ∂ ν (cid:19) g ( x ) . (B.6)The general procedure to construct a noncommutative field theory from a commutativeone is then by application of the Weyl-Wigner map. As an example, for a simple φ n theory we find L ( NC ) int = λn ! ˆ W − (cid:104) ˆ W ( φ ( x )) n (cid:105) = λn ! n copies (cid:122) (cid:125)(cid:124) (cid:123) φ ( x ) (cid:63) φ ( x ) (cid:63) · · · (cid:63) φ ( x ) . (B.7)316 ppendix CWilsonian Interpretations of NCFTsfrom Auxiliary Fields In this appendix we discuss various generalizations of the procedure introduced in [491,515] to account for the new structures appearing in the noncommutative quantum effec-tive action via the introduction of additional auxiliary fields.317 ilsonian Interpretations of NCFTs from Auxiliary Fields Chapter C C.1 Scalar Two-Point Function It is simple to generalize the procedure discussed in Section 7.3 to add to the quadraticeffective action of φ any function we wish through judicious choice of the two-pointfunction for an auxiliary field σ which linearly mixes with it. In position space, if wewish to add to our effective Lagrangian ∆ L eff = 12 c φ ( x ) f ( − i∂ ) φ ( x ) , (C.1)where f ( − i∂ ) is any function of momenta, and c is a coupling we’ve taken out for con-venience, then we simply add to our tree-level Lagrangian ∆ L = 12 σ ( x ) f − ( − i∂ ) σ ( x ) + icσ ( x ) φ ( x ) , (C.2)where f − is the operator inverse of f . It should be obvious that this procedure is entirelygeneral. As applied to the Euclidean φ model, we may use this procedure to add a secondauxiliary field to account for the logarithmic term in the quadratic effective action as ∆ L = 12 σ ( x ) 1log (cid:2) − ∂ ◦ ∂ (cid:3) σ ( x ) − gM √ π σ ( x ) φ ( x ) , (C.3)where we point out that the argument of the log is just / (Λ eff p ◦ p ) in position space. Wemay then try to interpret σ also as a new particle. As discussed in [515], its logarithmicpropagator may be interpreted as propagation in an additional dimension of spacetime.Alternatively, we may simply add a single auxiliary field which accounts for boththe quadratic and logarithmic IR singularities by formally applying the above procedure.But having assigned them an exotic propagator, it then becomes all the more difficult tointerpret such particles as quanta of elementary fields.318 ilsonian Interpretations of NCFTs from Auxiliary Fields Chapter C C.2 Fermion Two-Point Function To account for the IR structure in the fermion two-point function, we must add anauxiliary fermion ξ . If we wish to find a contribution to our effective Lagrangian of ∆ L eff = c ¯ ψ O ψ, (C.4)where O is any operator on Dirac fields, then we should add to our tree-level Lagrangian ∆ L = − ¯ ξ O − ξ + c (cid:0) ¯ ξψ + ¯ ψξ (cid:1) , (C.5)with O − the operator inverse of O . In the Lorentzian Yukawa theory of Section 7.4, ifwe add to the Lagrangian ∆ L = − ξ M − i /∂/ M − ∂ / (cid:20) log (cid:18) − ∂ ◦ ∂ (cid:19)(cid:21) − ξ + g √ π (cid:0) ξψ + ψξ (cid:1) . (C.6)we again find a one-loop quadratic effective Lagrangian which is equal to the Λ → ∞ value of the original, but now for any value of Λ .319 ilsonian Interpretations of NCFTs from Auxiliary Fields Chapter C C.3 Three-Point Function We may further generalize the procedure for introducing auxiliary fields to account for IRpoles to the case of poles in the three-point effective action. It’s clear from the form of theIR divergences in Equation ?? that they ‘belong’ to each leg, and so naïvely one mightthink this means that the divergences we’ve already found in the two point functionsalready fix them. However those corrections only appear in the internal lines and werealready proportional to g , and so they will be higher order corrections. Instead we mustgenerate a correction to the vertex function itself which only corrects one of the legs.To do this we must introduce auxiliary fields connecting each possible partition ofthe interaction operator. However, while an auxiliary scalar χ coupled as χϕ + χψψ would generate a contribution to the vertex which includes the χ propagator with the ϕ momentum flowing through it, it would also generate a new ( ψψ ) contact operator,which we don’t want. To avoid this we introduce two auxiliary fields with off-diagonaltwo-point functions, a trick used for similar purposes in [515]. By abandoning minimality,we can essentially use an auxiliary sector to surgically introduce insertions of functionsof momenta wherever we want them.We can first see how this works on the scalar leg. We add to our tree-level Lagrangian ∆ L = − χ ( x ) f − ( − i∂ ) χ ( x ) + κ χ ( x ) ϕ ( x ) + κ χ ( x ) (cid:63) ψ ( x ) (cid:63) ψ ( x ) . (C.7)Now to integrate out the auxiliary fields we note that for a three point vertex, one mayuse momentum conservation to put all the noncommutativity between two of the fields.That is, χ ( x ) (cid:63) ψ ( x ) (cid:63) ψ ( x ) = χ ( x )( ψ ( x ) (cid:63) ψ ( x )) = ( ψ ( x ) (cid:63) ψ ( x )) χ ( x ) as long as thisis not being multiplied by any other functions of x . So we may use this form of the320 ilsonian Interpretations of NCFTs from Auxiliary Fields Chapter C interaction to simply integrate out the auxiliary fields. We end up with ∆ L eff = κ κ ψ (cid:63) ψ (cid:63) f ( − i∂ ) ϕ (C.8)which is exactly of the right form to account for an IR divergence in the three-pointfunction which only depends on the ϕ momentum.For the fermionic legs, we need to add fermionic auxiliary fields which split the Yukawaoperator in the other possible ways. We introduce Dirac fields ξ, ξ (cid:48) and a differentialoperator on such fields O − ( − i∂ ) . Then if we add to the Lagrangian ∆ L = − ξ O − ξ (cid:48) − ξ (cid:48) O − ξ + c ( ξ(cid:63)ψ(cid:63)ϕ + ψ(cid:63)ξ(cid:63)ϕ )+ c ( ξ(cid:63)ϕ(cid:63)ψ + ψ(cid:63)ϕ(cid:63)ξ )+ c ( ξ (cid:48) ψ + ψξ (cid:48) ) , (C.9)we now end up with a contribution to the effective Lagrangian ∆ L eff = c c (cid:0) ¯ ψ (cid:63) O ( ψ ) (cid:63) ϕ + ¯ ψ (cid:63) O ( ψ (cid:63) ϕ ) (cid:1) + c c (cid:0) ¯ ψ (cid:63) ϕ (cid:63) O ( ψ ) + ¯ ψ (cid:63) O ( ϕ (cid:63) ψ ) (cid:1) , (C.10)where we have abused notation and now the argument of O specifies which fields it actson. These terms have the right form to correct both vertex orderings.Now that we’ve introduced interactions between auxiliary fields and our original fields,the obvious question to ask is whether we can utilize the same auxiliary fields to correctboth the two-point and three-point actions. In fact, using two auxiliary fields with off-diagonal propagators per particle we may insert any corrections we wish. The new trickis to endow the auxiliary field interactions with extra momentum dependence.For a first example with a scalar, consider differential operators f , Φ , and add to theLagrangian ∆ L = − χ f − ( − i∂ ) χ + κ χ ϕ + κ χ ψ (cid:63) ψ + gϕ Φ( − i∂ ) χ . (C.11)We may now integrate out the auxiliary fields and find ∆ L eff = gκ ϕf (Φ( ϕ )) + κ κ ψ (cid:63) ψ (cid:63) f ( ϕ ) (C.12)321here we’ve assumed that f and Φ commute. If we take Φ = then we have theinterpretation of merely inserting the χ two-point function in both the two-and three-point functions. But we are also free to use some nontrivial Φ , and thus to make thecorrections to the two- and three-point functions have whatever momentum dependencewe wish. It should be obvious how to generalize this to insert momentum dependenceinto the scalar lines of arbitrary n − point functions.The case of a fermion is no more challenging in principle. For differential operators O , F , we add ∆ L = − ξ O − ξ (cid:48) − ξ (cid:48) O − ξ + c ( ξ (cid:63) ψ (cid:63) ϕ + ψ (cid:63) ξ (cid:63) ϕ ) + c ( ξ (cid:63) ϕ (cid:63) ψ + ψ (cid:63) ϕ (cid:63) ξ )+ c ( ξ (cid:48) ψ + ψξ (cid:48) ) + g (cid:0) ¯ ξ O − F ψ + ¯ ψ O − F ξ (cid:1) , (C.13)and upon integrating out the auxiliary fields we find ∆ L eff = gc ¯ ψ F ψ + c c (cid:0) ¯ ψ (cid:63) O ( ψ ) (cid:63) ϕ + ¯ ψ (cid:63) O ( ψ (cid:63) ϕ ) (cid:1) + c c (cid:0) ¯ ψ (cid:63) ϕ (cid:63) O ( ψ ) + ¯ ψ (cid:63) O ( ϕ (cid:63) ψ ) (cid:1) , (C.14)where the generalization to n -points is again clear. Note that in the fermionic case it’scrucial that we be allowed to insert different momentum dependence in the correctionsto the two- and three-point functions, as these have different Lorentz structures.Now we cannot quite implement this for the two- and three-point functions calculatedin Section 7.4, for the simple reason that we regulated these quantities differently. 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