Phenomenological aspects of supergravity theories in de Sitter vacua
AARENBERG DOCTORAL SCHOOL
Faculty of Science
Phenomenological aspects ofsupergravity theories in deSitter vacua
Rob Knoops
Dissertation presented in partialfulfillment of the requirements for thedegree of Doctor in Science (Phd):PhysicsAugust 2016Supervisor:Prof. dr. I. AntoniadisProf. dr. A. Van Proeyen a r X i v : . [ h e p - ph ] A ug henomenological aspects of supergravity theoriesin de Sitter vacua Rob KNOOPS
Examination committee:Prof. dr. C. Maes, chairProf. dr. I. Antoniadis, supervisorProf. dr. A. Van Proeyen, supervisorProf. dr. N. BobevProf. dr. M. FannesProf. dr. T. HertogProf. dr. T. Van RietProf. dr. J.-P. Derendinger(Bern University) Dissertation presented in partialfulfillment of the requirements forthe degree of Doctor in Science(Phd): PhysicsAugust 2016 ankwoord
Een thesis schrijf je niet alleen, daarom ook een kort dankwoord. Graag wil ikhier iedereen bedanken die op de één of andere manier heeft bijgedragen aandeze scriptie.Ten eerste zou ik graag mijn ouders bedanken, die me altijd gemotiveerd hebbenom verder te studeren en om mijn dromen te volgen. Zonder hen zou ik hierniet staan, letterlijk en figuurlijk.Graag zou ik ook mijn promotors Antoine Van Proeyen en Ignatios Antoniadisbedanken. Enerzijds is er Toine, wie voor mij een grote inspiratiebron is geweestin mijn keuze om een doctoraatsopleiding te volgen, en wie altijd klaar stondom mij te begeleiden met zowel mijn administratieve als fysicavragen. I’d alsolike to thank Ignatios who, despite having a very loaded travel schedule, alwaysfound a way to advice me on this thesis. Naast mijn promotors, zou ik ook graag de andere leden van mijn examen-commissie bedanken, namelijk Nikolay Bobev, Mark Fannes, Thomas Hertog,Christian Maes, Thomas Van Riet en Jean-Pierre Derendinger, voor hetdoornemen van mijn thesis, het bijwonen van mijn interne verdediging, ende bijhorende feedback op een eerste versie van deze thesis.Next, I’d like to thank the mathematics department at Geneva University, within particular Anton Alekseev, for their hospitality during the last two years ofmy thesis.Being at CERN, I have also had the opportunity to discuss with (or rather,learn from) Fabio, Jean-Pierre, John, Luis, and Sergio. Besides the endlessdiscussions about physics, I have also had the opportunity to meet the mostamazing people in Geneva that educated me in the school of life. In particular, In one occasion, we even met on the train from Geneva to Bern when Ignatios wastraveling from Paris, on a holiday (Jeûne Genevois), just so we could discuss our project. Theresult of this train ride is section 5.3. ii DANKWOORD many of the most amazing weekends of my life were spent with Bruno, Carolyn,Claire, Dani, Garoe, Jabu, Cadi, Iikka, Javi, Jose, Helina, Luis, MCRaf, Sabri,Tincho, Tito, Tom, and many more. A special thanks goes to Kevin, Marco,Matthi, Pedro and Subodh for their support during the final stages of thisthesis.Finally, I’d like to thank my brother Gert, who is also pursuing a Ph.D. inPhysics, for not finishing his thesis before mine, although it took me nearly fiveyears.This work was supported in part by the European Commission under the ERCAdvanced Grant 226371. This work was also partly supported by the NCCRSwissMAP, funded by the Swiss National Science Foundation. bstract
Supersymmetry is the most promising theory that extends the Standard Model.Theories with local supersymmetry (supergravity) appear as the low energyeffective physics of string theory. Recently there has been great interest infinding de Sitter vacua of supergravity and string theory. In this thesis, wepresent a new class of de Sitter vacua in N = 1 supergravity, and we focus onthe phenomenological implications of these models.In the first part of the thesis we introduce supersymmetry and supergravity.Particular emphasis is put on spontaneous supersymmetry breaking and theMinimally Supersymmetric Standard Model (MSSM). We introduce a modelbased on the gauged shift symmetry of a single chiral multiplet, which can beidentified with the string dilaton or a compactification modulus. The modelallows for a tunably small and positive value of the cosmological constant.The gravitino mass parameter and the dilaton Vacuum Expectation Value areseperately tunable.For certain values of the parameters the vacuum is metastable, and it is shownthat the tunneling rate to the false vacuum exceeds the lifetime of the universe.Moreover, the scalar potential corresponds with one derived from D-branes innon-critical strings. Finally, a dual version of this model is presented in termsof a linear multiplet containing a 2-index antisymmetric tensor.In a second part we analyze the quantum consistency of these models. Recentwork by Elvang, Freedman and Körs on anomalies in supergravity theories withFayet-Iliopoulos terms was extended, such that their results can be interpretedfrom a field-theoretic point of view. Next, we show that for certain valuesof the parameters the anomaly cancellation conditions are inconsistent witha TeV gravitino mass. We elaborate on the relations between the Green-Schwarz counterterms and the gaugino mass parameters, and show how (by anappropriate Kähler transformation) the resulting contribution to the gauginomass can be identified with one-loop gaugino mass parameters that are generated iiiv ABSTRACT by anomaly mediation.In the third part the above model (in the parameter range which is consistentwith the anomaly cancellation conditions) is used as a hidden sector wherespontaneous supersymmetry breaking occurs. This supersymmetry breaking isthen communicated to the visible sector (MSSM) by gravity effects, and theresulting soft supersymmetry breaking terms are calculated. In its minimalversion, the model leads to tachyonic scalar soft masses. This problem canhowever be circumvented by the introduction of an extra Polónyi-like hiddensector field, or by allowing for non-canonical kinetic terms for the StandardModel fields, while maintaining the desirable features of the model. The resultinglow energy spectrum consists of very light neutralinos, charginos and gluinos,while the squarks remain heavy, with the exception of the stop squark whichcan be as light as 2 TeV.Finally, we discus possibility that the shift symmetry is identified with knownglobal symmetries of the MSSM. The particular cases where this global symmetryis Baryon minus Lepton number ( B − L ), or 3 B − L , which contain the knownR-parity or matter parity of the MSSM, are analyzed in great detail. Thelatter combination has also the advantage of forbidding all dimension-four anddimension-five operators violating baryon or lepton number in the MSSM. It isshown that the phenomenology is similar to the above case where the MSSMfields are inert, with the exception of the stop squark, which can be as lightas 1 . bbreviations AdS Anti-de SitterB-L Baryon minus Lepton numberCERN European Council for Nuclear ResearchCP Charge ParitydS de SitterFI Fayet-IliopoulosGCS Generalized Chern-Simons (term)GeV Giga ElectronVoltGS Green-Schwarz (mechanism)GUT Grand Unified TheoryLHC Large Hadron Colliderlhs left-hand sideLSP Lightest Supersymmetric ParticlemAMSB minimal Anomaly Mediated SupersymmetryBreakingMSSM Minimally Supersymmetric Standard ModelmSUGRA minimal (super)gravity mediated (supersymme-try breaking)QCD Quantum Chromo DynamicsRGE Renormalization Group Equationrhs right-hand side vi ABBREVIATIONS
SM Standard ModelSUGRA supergravitysuper-BEH super-Brout-Englert-HiggsSUSY supersymmetryTeV Tera ElectronVoltUV Ultra-VioletVEV Vacuum Expectation ValueWIMP Weakly Interacting Massive Particle ontents
Abstract iiiContents vii1 Introduction 1 N = 1 supersymmetric theory . . . . 18 viiiii CONTENTS p = 1 . . . . . . . . . . . . . . . . 454.2.2 Tunable dS vacua for p = 2 . . . . . . . . . . . . . . . . 464.3 A few remarks on p = 2 . . . . . . . . . . . . . . . . . . . . . . 504.3.1 Metastability of the de Sitter vacuum . . . . . . . . . . 504.3.2 Connection with string theory . . . . . . . . . . . . . . . 514.3.3 Linear-Chiral duality . . . . . . . . . . . . . . . . . . . . 52 ONTENTS ix p = 2 . . . . . . . . . . . . . . . . . . 625.3 Anomalies and gaugino masses . . . . . . . . . . . . . . . . . . 64 A.1 Calculation of the fermion mass matrices in the various models 103A.1.1 Fermion masses for p = 1 . . . . . . . . . . . . . . . . . 104A.1.2 Fermion masses for p = 2 . . . . . . . . . . . . . . . . . 105A.1.3 Fermion masses for the model including an extra Polónyi-like field . . . . . . . . . . . . . . . . . . . . . . . . . . . 106A.2 Mass of the U (1) R gauge boson . . . . . . . . . . . . . . . . . . 107A.3 Scalar masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 B Linear-Chiral duality 111
B.1 Tensor-scalar duality in global supersymmetry . . . . . . . . . . 111
CONTENTS
B.2 Tensor-scalar duality in supergravity . . . . . . . . . . . . . . . 116B.3 Proof of an important identity . . . . . . . . . . . . . . . . . . 122
C R-symmetries and anomalies 125D Comments the case with p = 1 and non-zero β D.1 On the small value of β . . . . . . . . . . . . . . . . . . . . . . 129D.2 On the inconsistency of γ = 0 . . . . . . . . . . . . . . . . . . . 132 Bibliography 135 hapter 1
Introduction
What is the smallest thing you can imagine? A grain of salt, a hair, or perhapsit is an atom? Ever since ancient history human kind has been fascinatedwith questions like ”What is matter made of?”, and ”which are nature’s mostfundamental building blocks?”. Everything around you, from the book (orthesis) in front of you to the chair you are sitting on, is made of tiny atomsand molecules. These atoms however, consist of even smaller constituents,called protons, neutrons and electrons. While it is assumed that the electronis a fundamental particle, protons and neutrons on the other hand consist ofeven smaller particles called quarks. As particle physicists it is our goal totry to understand these most fundamental building blocks of nature and theinteractions that govern them.One approach is that of the theorist who applies mathematical rigor based onphysical motivations and appropriate questions to establish a theory of natureat the most fundamental level. On the other hand there is the experimenter,who puts these theories to the test. The content of this thesis is of the formerapproach. Our goal is to establish a mathematically consistent model based onsimple symmetry principles, and to identify possible experimental signaturesthat could distinguish it from other models.
One of the most impressive theories describing the fundamental particles andtheir interactions is the Standard Model. The Standard Model unifies three ofthe fundamental forces of nature, namely the electromagnetic interaction, theweak force, which is responsible for the nuclear decay that is for example thesource of energy in a nuclear power plant, and the strong interaction, whichgets its name because it is so strong that it can keep a nucleus consisting of(neutrons and) positively charged protons together, even though they repel eachother due to the electromagnetic interaction. Perhaps one of the most bafflingaccomplishments of the Standard Model is its incredibly accurate predictionsthat have been confirmed by numerous experiments. For example, the magneticmoment (or g-value) of the electron has been confirmed up to 13 significantdigits. Moreover, the Standard Model successfully predicted the existence ofthe W and Z bosons, the gluons, which are responsible for the weak and strongforce respectively, and the top and charm quarks as well as the Higgs Boson,even before these particles were observed. Even their predicted experimentalproperties, as for example the masses of the W and Z bosons, were confirmedwith an incredible precision. Despite its incredible experimental successes, physicists are convinced that theStandard Model is not the end of the story:• First, although it neatly describes the electromagnetic, the weak andthe strong interactions, the Standard Model does not describe a fourthfundamental force of nature, namely gravity. Although gravity is quitewell understood at a long distance scale by Einstein’s theory of GeneralRelativity, physicists can only guess how gravity ’works’ when look at itvery closely, at the smallest scales. In order words, although physicists canvery well describe ’how fast the apple falls on the floor’ or ’how the speedand orbit of a planet around the sun are related’, we fail to understandhow gravity is described at the quantum level.• Second, certain elementary particles, called neutrinos, are massless inthe Standard Model. Since neutrinos interact only very weakly (andonly through the weak force) with other matter and therefore tend tojust fly through most particle detectors without leaving any trace, they
HE STANDARD MODEL 3 are extremely difficult to detect and an accurate mass measurement ofthese neutrinos has never been accomplished. However, neutrinos come inthree lepton flavors, namely electron neutrinos, muon neutrinos and tauneutrinos, and it is observed a neutrino can change flavor after travelingsome distance. These ’flavor oscillations’ can be used to calculate themass differences between these neutrinos of different lepton flavor. Variousneutrino oscillation experiments have indeed confirmed a very small massdifference between the three neutrinos, which is inconsistent with theStandard Model, where all neutrinos are assumed to be massless.• The particle content of the Standard Model, which in this context oftenreferred to as baryonic matter (even though the Standard Model technicallydoes not only contain baryons), makes up only about 5 percent of theknown matter of the universe. The rest is dark matter (about 27 percent)and dark energy (about 68 percent). Although dark matter can not bedirectly seen with telescopes since it does not (or very little) interact withordinary matter, its existence can be confirmed from its gravitationaleffects on visible matter [1]. While mass of large astronomical objects canbe calculated from visible effects, like stars, dust or gas, their gravitationalinteractions suggest that the mass of these objects is much larger. Darkenergy on the other hand, is hypothesized to accommodate for recentobservations that the universe seems to be expanding at an acceleratingrate.• Perhaps one of the most compelling arguments in favor of the existence ofa theory beyond the standard model, at least from a theoretical point ofview, is the hierarchy problem [2–6]. The hierarchy problem is the largediscrepancy between the weak force and gravity, the former being about10 times stronger than gravity. The discovery of the Higgs boson atCERN in 2012, which led to a Nobel Prize for Francois Englert and PeterHiggs , also meant the first experimental confirmation of the existenceof a scalar particle. In the Standard Model, the measured Higgs bosonmass of roughly 125 GeV is the sum of the ’bare’ (classical) mass squared m and its quantum corrections δm , such that m = m + δm .Here, the bare mass m is an input parameter for the theory which shouldbe determined by experiment, while its quantum corrections δm canbe calculated. The problem however is, that these quantum correctionsturn out to be huge. If one assumes that the Standard Model is a goodtheory of nature all the way up to an energy where gravitational effectsbecome important, say the Planck scale M P ∼ GeV, this implies that Actually, besides Francois Englert another Belgian physicist, namely Robert Brout, madesignificant contributions to the Brout-Englert-Higgs mechanism and if he were still alive wouldmost probably have participated in the Nobel Prize.
INTRODUCTION m should be chosen to be very big and very precise such that one canrecover the measured value of the Higgs mass of about 125 GeV. This anaturalness problem, in the sense that it seems unnatural that m shouldbe fine-tuned so precisely to result in a low m measured , and is regarded asa clear indication that the Standard Model is not a fundamental theoryof nature.• If the recent measured values for the top quark mass of the Higgs massesare naively extrapolated to high scales, without introducing any newphysics beyond the Standard Model, it turns out that the electroweakvacuum is metastable, or even unstable [7], depending on the result offuture greater accuracy measurements for example for the mass of thetop quark. Even if experimental measurements turn out in favor of ametastable electroweak vacuum, this could still pose several questions.For example, if there exists another vacuum and our vacuum is not thetrue one, why is the vacuum energy in our vacuum so low? Or perhapsmore importantly, why has not most of the universe already fallen intothis lower energy state due to the large Higgs fluctuations in the earlyuniverse [8]? These questions could be solved by dropping the assumptionthat the Standard Model remains valid at high energies, and point in thedirection of the existence of new physics before the Planck scale.For more information and a more complete list of problems with the StandardModel, including also for example the strong CP problem and the matter-antimatter asymmetry, see for example [9]. There are many theories that extend the Standard Model and potentially solve atleast a subset of the problems outlined before. Perhaps one of the most appealingcandidates is supersymmetry. Supersymmetry extends the symmetries of spaceand time (the Poincaré algebra) by adding extra transformations that relate thetwo kinds of fundamental particles, namely fermions which have a half-integerspin and bosons which have an integer spin. Such theories typically predict theexistence of supersymmetric particles, or sparticles in short. Each particle inthe Standard Model, would have a supersymmetric counterpart (superpartner)whose spin differs with 1/2 and whose mass is typically much larger than itsStandard Model counter part. Although supersymmetric particles have yet tobe discovered, certain theories like the Minimally Supersymmetric Standard See also section 1.6 on naturalness.
EYOND THE STANDARD MODEL 5
Model (MSSM) include promising candidates that might be discovered withinthe next run of the LHC (Large Hadron Collider, see below) or a future collider.Before we continue to introduce certain technical aspects involving supersym-metric theories, let us readdress the problems with the Standard Model outlinedin the previous section in view of supersymmetry :• Supersymmetry provides a potential solution to the hierarchy problem.For every fermion in a supersymmetric theory, there is a correspondingboson which is related by a supersymmetry transformation. Moreover, theinvariance of the Lagrangian under supersymmetry transformations impliesthat for every fermion contribution to δm , its bosonic superpartner givesexactly the same quantum correction to the mass of the Higgs boson, butwith an opposite sign, which drastically reduces the value of δm and thussolves the hierarchy problem [11–14]. This cancellation however, is onlyexact when supersymmetry is unbroken and the mass of each sparticleequals that of their Standard Model counter part. Recent experimentalbounds are pushing lower limits on the sparticle masses upward, whichreintroduces the question of naturalness (see below). Although somenew ideas are emerging [15, 16], none of them is yet as compelling assupersymmetry. For a recent review on naturalness in supersymmetrictheories, see for example [17, 18].• In order to prevent the proton in the theory from decaying, a discretesymmetry called R-parity is imposed on the MSSM. This R-parity impliesthat the Lightest Supersymmetric Particle (LSP) is stable. If additionallythis LSP also interacts very weakly with all the Standard Model particles,it is a natural dark matter candidate in the form of a WIMP (weaklyinteracting massive particle) [19, 20].• Perhaps one of the most convincing arguments in favor of supersymmetry isthe apparent gauge coupling unification at a high energy scale. The gaugecouplings of the three fundamental interactions of the Standard Modeldepend on the energy scale that they are probed, and their dependence onthe energy scale can be described by a set of differential equations calledthe RGEs (Renormalization Group Equations). When corrections to theseRGEs from the particle content of the MSSM are included, they seemto intersect quite accurately (within experimental bounds) at a certainenergy scale. This energy scale can then be interpreted as the energy scaleof some Grand Unified Theory (GUT). For a recent review and current experimental status of these points, see for example [10].
INTRODUCTION • Although other solutions to the electroweak vacuum instability exist (seefor example [21, 22] based on dimension 6 operators), supersymmetry alsoprovides a solution to the stability of the electroweak vacuum [23].• Supersymmetry appears naturally in superstring theory, which is a possiblecandidate for a theory that quantizes gravity.• According to Coleman-Mandula [24] there are no nontrivial extensionsof the Poincaré algebra, which are the symmetries of spacetime, thatare consistent with a non-vanishing scattering amplitude. However,theHaag–Lopuszanski–Sohnius theorem [25] provides a loophole in thisargument by allowing graded Lie algebras. In this case, the possiblesymmetries of a consistent 4-dimensional quantum field theory do not onlyconsist of internal symmetries and Poincaré symmetry, but they can alsoinclude supersymmetry as a nontrivial extension of the Poincaré algebra.It is important to realize that supersymmetry is a mathematical framework,rather than a theory. A quantum field theory, like the Standard Model, isoften defined by its Lagrangian. In a supersymmetric theory, this Lagrangianis not only invariant under the symmetries of spacetime and possible internalsymmetries, but also under supersymmetry transformations. Obviously,there are many different possible supersymmetric Lagrangians, and thussupersymmetric theories. The simplest such model that is consistent with theparticle content and internal symmetries of the Standard Model is the MSSM.However, even is we restrict ourselves to the MSSM, the exact properties andexperimental signatures of these supersymmetric theories are very dependenton the underlying supersymmetry breaking mechanism. Acquiring a betterunderstanding of supersymmetry breaking mechanisms and the potentialunderlying theories is therefore of great importance.If supersymmetry were realized as an exact symmetry of nature, this wouldimply that the masses (as well as the couplings) of all particles would be equal tothat of their superpartners. If this were the case, sparticles would have showedup in experiments already a long time ago. One is therefore forced to concludethat supersymmetry can only be realized in nature as a broken symmetry,i.e. there are terms present in the Lagrangian which violate the invariance undersupersymmetry transformations. From a practical point of view, it can be usefulto simply parametrize our ignorance of any underlying mechanism that breakssupersymmetry by adding extra terms to MSSM Lagrangian that explicitlybreak supersymmetry. The extra terms are called soft supersymmetry breakingterms, where ’soft’ means these terms are of a positive mass dimension andtherefore it can be argued (see for example [26]) that they maintain a hierarchybetween the electroweak and the Planck scale. This way, there are no largequantum corrections ( δm ) to the Higgs mass and the theory to still provides EYOND THE STANDARD MODEL 7 a solution to the hierarchy problem, even in the presence of these extra termsthat break supersymmetry.These soft supersymmetry breaking terms however, are not very appealingfrom a theoretical point of view. While in quantum field theories one usuallyprefers to write down the most general Lagrangian based on certain symmetryprinciples, these terms are not only ’added by hand’, they even violate thesymmetries (supersymmetry) that we started with in the first place. However,it might be possible that while supersymmetry is an exact symmetry of nature,it is broken spontaneously. In other words, the Lagrangian of the underlyingmodel is invariant under supersymmetry transformations, but the vacuumis not. If one of the scalar fields in the MSSM were to be responsible forthe spontaneous breaking of supersymmetry by acquiring a VEV (VacuumExpectation Value), this would additionally break at least one of the StandardModel gauge symmetries. It is therefore necessary to introduce additional fieldsto the model, which are responsible for spontaneous supersymmetry breaking.These fields are in a so-called ’hidden sector’, since it is assumed that these fieldsdo not interact (or interact very weakly) with the Standard Model. This leavesus with the question how supersymmetry is communicated from the hiddensector to the MSSM.One such idea is to promote supersymmetry to a local (gauge) symmetry.Theories with local supersymmetry necessarily contain gravity and are thereforecalled supergravity theories. The idea is that besides the Standard Modelparticles and their superpartners, there also exists a ’hidden sector’ of fieldsthat does not interact with the Standard Model sector. supersymmetry isthen broken spontaneously in the hidden sector when a (hidden sector) scalarfield acquires a VEV due to a nontrivial scalar potential, similar to the Brout-Englert-Higgs mechanism, which breaks the Standard Model electroweak gaugesymmetry. Next, gravitational effects are responsible for the communicationof the supersymmetry breaking from the hidden sector to the visible (MSSM)sector. This can result in the necessary soft supersymmetry breaking terms ofthe MSSM, thus providing a framework for a possible origin for these terms.On the other hand, certain supergravity theories arise as a low energy limit(where this is where the limit where the string length goes to zero) of stringtheory. String theory, although it is still poorly understood, provides a quantumtheory of gravity and is a candidate for a theory that unifies all interactionsin a consistent quantum framework. It is therefore interesting whether such a Other ideas include Gauge Mediation [27, 28], where supersymmetry is mediated from a’hidden sector’ to the MSSM by gauge interactions, or dynamical supersymmetry breaking [29](or see [30–32] for more recent works) where fermion condensates of a strongly coupled gaugegroup are responsible for the breaking of supersymmetry.
INTRODUCTION supergravity theory can correspond to a certain string theory at a very highenergy.We conclude that trying to recover the MSSM from a supergravity theory cantherefore be useful for two reasons: On the one hand, by breaking supersymmetryspontaneously one can obtain the desirable soft supersymmetry breaking terms,which in turn provides information on the low energy spectrum of this theory.On the other hand, if one manages to identify this supergravity theory as thelow energy limit of a certain string theory, this could give some useful insight inpossible UV-completions of this theory.
In cosmology, the cosmological constant, usually denoted as Λ, is the value of theenergy density of the vacuum in space. In the context of his theory of GeneralRelativity, the cosmological constant was introduced by Albert Einstein in orderto make to the theory consistent with the world view at that time. Since it wasbelieved that we live in a static universe, an additional (constant) contributionΛ was introduced to allow for static solutions of the Einstein equations. Thisidea was abandoned however, when Georges Lemaître in 1927 [33] and EdwinHubble in 1929 [34] discovered that we live in an expanding universe. Sincethe 1990s, more and more experimental evidence confirmed that the universe isnot only expanding, but it appears to be expanding at an accelerating rate .This lead to the reintroduction of the cosmological constant. It turns out thatthis cosmological constant is very small and positive (Λ ≈ (10 − eV) ), wherea universe with a positive cosmological constant is called a de Sitter (dS)universe .In a quantum field theory without gravity, like the Standard Model, the absolutevalue of the minimum of the potential has no significance. A constant termcan be added to the Lagrangian without changing the laws of physics thatit describes. However, this changes drastically when gravity is taken intoaccount. In a theory containing gravity, like supergravity or string theory, thevalue of the minimum of the potential for the scalar fields is a contributionto the cosmological constant, as it results in a similar constant contributionto the resulting Einstein equations. It is still one of the biggest challenges ofstring theory at this date to accommodate for a positive cosmological constant.Although interesting progress has been made in the last decade [37–45], the For a recent review, see for example [35]. For the latest measurements, see [36]. A vanishing cosmological constant would correspond to flat space (Minkowski), while anegative cosmological constant corresponds to an Anti-de Sitter (AdS) universe.
UANTUM ANOMALIES 9 hunt for de Sitter spaces in string theory remains a challenging topic. At istherefore interesting to study supergravity theories which allow for a positive(and small) value of the minimum of the potential.In this thesis we aim to obtain a supergravity theory with a tunably small andpositive cosmological constant based on a minimal particle content and simplesymmetry principles. We then relate such a theory to a string theory resultthat was obtained in the literature, and we continue to calculate the MSSMsoft supersymmetry breaking terms and low energy spectrum that allows us todistinguish such a model from other models. In doing so, we find that anomalycancellation conditions, which will be introduced in the next section, imposevery stringent constraints on the consistency of such a theory.
A quantum field theory, as was already mentioned above, is defined by itsLagrangian. Such a Lagrangian is usually constructed from symmetry principles:The Lagrangian, or rather its spacetime integral called the action, should beinvariant under the symmetry transformations. Sometimes however, it canoccur that although the classical action is invariant under a symmetry, thissymmetry is violated when quantum corrections are included. In this case thesymmetry is said to be anomalous.An anomalous global symmetry of the Lagrangian usually does not pose aproblem. It merely indicates that the classical selection rules related to thissymmetry are not obeyed in the quantum theory. If these symmetries are local(gauge) symmetries on the other hand, this would be disastrous since the gaugesymmetry is required in order to cancel the unphysical degrees of freedom witha negative norm. In this case, the theory is said to be anomalous and is, inprinciple, useless. It is therefore of tremendous importance to check that atheory is anomaly-free. Typically, the anomalous contributions from variousfermion triangle diagrams corresponding to all fermions in the theory can canceleach other, provided that the anomaly cancellation conditions are satisfied.Or, occasionally, counterterms can be introduced in order to cancel certainanomalies.
A good theory should provide a natural explanation for the experimental data.What is exactly meant with ’naturalness’ is something many physicists disagree about. However, in particle physics one could define naturalness as the propertythat the physical constants and parameters in the theory should not differ bytoo many orders of magnitude. One example of a so-called naturalness problemis the gauge hierarchy problem, which concerns the question why the weak scale( m weak ∼ . − m p ≈ . × TeV. Answering this question has been one of the guiding principles motivatingproposals for new particles and interactions during the last decades.The Higgs hierarchy problem, which as already introduced above, is that thequantum corrections to the Higgs mass ( δm ) are quadratically sensitive to thescale of new physics. This scale can be the mass of a new (and yet undiscovered)particle, the UV cut-off scale after integrating out any new physics, or eventhe Planck scale if gravity is included in a quantum theory. The latter givesan alternative facet of the question why gravity is so weak compared to otherforces.Another example of a naturalness problem involves the cosmological constant.It will be explained later in more detail how the cosmological constant in relatedto the vacuum energy in the theory. Although particle physicists do not knowhow to calculate the vacuum energy density ρ vac exactly, theory allows one toestimate its value. Depending on how this estimate is made , its value differsbetween 40 to 120 orders of magnitude with its measured value. This discrepancyis sometimes called ’The worst prediction in the history of physics’ [47]. Thecosmological constant problem can then be stated as follows: There are severalindependent contributions to the vacuum energy density (for example fromthe potential energy of each field in the theory, or from a bare cosmologicalconstant in the Lagrangian). Each of these contributions are estimated to bemuch larger than the observed value of the cosmological constant, and yet theycombine to result in an uncanny small (measured) value. This is widely seen asan indication that some unknown physics plays a decisive role.The strong CP problem (CP standing for charge parity) is an example of afine-tuning problem in physics. This problem has its origin in the fact that thesymmetries of the Lagrangian describing the strong interactions a priori allowfor a term that violates CP. However, since there is no experimentally knownCP violation in the strong interactions, this term should be severely supressed(or even vanish). The strong CP problem then questions why this CP-violatingterm vanishes (or is so small), while from a theoretical point of view there is noreason why this term should be absent.It was already mentioned above that supersymmetry provides a solutionto the Higgs hierarchy problem. Since bosons and fermions are related by For a more precise statement of the cosmological constant problem and on how thisestimate is made, the reader is refered to the literature. See for example [46].
ATURALNESS 11 supersymmetry transformations in the Lagrangian, their quantum correctionsto the Higgs mass cancel out, which would in principle solve the Higgs hierarchyproblem. However, since the superpartners should be much heavier than theirStandard Model counterparts to escape experimental bounds, supersymmetrymust be a broken symmetry of nature (if it is a symmetry of nature at all).This reintroduces corrections to the Higgs mass that depend on the sparticlemasses. If these masses are too large, this results in a new fine-tuning problem,which questions the theoretical motivatons behind supersymmetry as a solutionto the hierarchy problem. While the current experiments are pushing the lowerbounds of the sparticle masses upwards, this is indeed a problem that shouldbe addressed.Although numerous different definitions of naturalness exist in this context,it is commonly agreed that sparticle masses around the TeV-scale can still beconsidered as ’natural’. A particular idea is that of ’split supersymmetry’ [48].In a split supersymmetry scenario, certain superpartners (the gauginos andhiggsinos) are relatively light, while for example the squarks and gravitino areallowed to be heavier ( ∼
10 TeV). Although by most definitions of ’naturalness’,such a scenario is considered to be ’more natural’, split supersymmetry is notconsidered to be a solution to the Higgs hierarchy problem.Supersymmetry, in particular supergravity, also has an impact on thecosmological constant problem. It was already explained above, and it will bequantified in later chapters, that the value at the minimum of the potentialgoverning the scalar fields in the theory can be interpreted as the cosmologicalconstant. The cosmological constant problem can in this case be interpreted asthe question why the value of the minimum of the potential is so incredibly small.The scalar potential in supergravity has two contributions: A so-called F-termcontribution which can be negative due to the presence of a contribution comingfrom the elimination of a gravity auxiliary field, and a D-term contribution whichis positive (nonnegative). A very small (and positive) cosmological constant canthen be obtained by a nearly exact cancellation between the F-term and theD-term contributions of the scalar potential. This results in a scenario wheresupersymmetry is broken exactly in the right way, such that the F-term andthe D-term contributions cancel to a incredible accuracy. However, this comesat the cost of a new and unexplained fine-tuning.The fine-tuning problem of the cosmological constant has then been reducedto a new fine-tuning problem: What is the mechanism behind this miraculouscancellation between the F-terms and the D-terms?The approach in this thesis is indeed of the former, where the minimum ofthe potential is tuned by a cancellation between an F-term contribution and aD-term contribution to the scalar potential. The resulting sparticle spectrum will turn out to be similar to the spectrum of split supersymmetry.Another approach has its origin in the fact that there are an extraordinarylarge amount of different vacua in string theory. In each of these vacua, thelow energy physics (and in particular the cosmological constant) is different.Finding realistic vacua in the string theory landscape that correspond to thelaws of physics as we known them is a very contemporary problem. Underthe assumption that there exists at least one particular of those vacua, onecould explain the smallness of the cosmological constant through an anthropicprinciple (which is liked as much as disliked among physicists) [49].
Although this work is of a theoretical nature, most of it was carried out atCERN, the European Council for Nuclear Research near Geneva, Switzerland,which is one of the biggest particle physics laboratories in the world. CERNhosts the Large Hadron Collider, which is undeniably one of the most impressivemachines that was ever made: The LHC is constructed in a circular tunnel witha circumference of 27 km and about 100 m below the ground on the borderbetween France and Switzerland. Its goal is to accelerate protons to nearly thespeed of light in two separate tubes, in opposite directions in this tunnel. Whenthe two beams of protons reach an energy of about 7 TeV each, they are collidedinside four particle detectors. It is then possible to look for possible signaturesin the particle detectors that could indicate the existence of any new physicsbeyond the known Standard Model, say, perhaps a supersymmetric particle..
Our first goal is to obtain a supergravity model with an infinitesimally small,but positive, value of the minimum of the scalar potential. It was argued abovethat in the presence of gravity this can be interpreted as the cosmologicalconstant, which is indeed small and positive. After such a model is obtainedbased on simple symmetry principles and a minimal particle content, our nextgoal is two-fold: On the one hand we aim to find possible UV completionsfrom which this model can be obtained. In particular, we explain how sucha potential can arise in the context of string theory. On the other hand, thismodel is used as a hidden sector responsible for supersymmetry breaking, which
VERVIEW OF THE THESIS 13 is then communicated to the visible sector (MSSM) through gravity effects. Wecalculate the resulting soft supersymmetry breaking terms and the resultinglow energy phenomenology and particle spectrum. In particular, we explainhow this model distinguishes itself from other scenarios. Finally, we show howglobal symmetries of the MSSM can be identified with a gauge symmetry in ourmodel and we work out the corresponding implications on the phenomenologyof this model.Our main result includes a model in work with I. Antoniadis [50], which wasoriginally proposed in [51,52], based on a single chiral multiplet S with a gaugedshift symmetry. The Lagrangian of a supergravity theory is determined bythree functions, namely a Kähler potential, a superpotential and gauge kineticfunctions. The shift symmetry of the dilaton implies that the Kähler potentialof the theory should be a function of the real part of S , which is taken to bea logarithm, where in the context of string theory S can then be identifiedwith the heterotic string dilaton or a compactification modulus. The mostgeneral superpotential consistent with the shift symmetry is an exponential of S , and the gauge kinetic function can be at most linear in S . In the case of anexponential superpotential, the shift symmetry becomes a gauged R-symmetrywhich results in a (positive) D-term contribution to the scalar potential. Thisway, the F-term contribution to the scalar potential (which can be negativebecause of a contribution from a gravity auxiliary field) can be fine-tuned withrespect to the positive D-term contribution in order to allow for a very smalland positive value of the minimum of the potential. The resulting model allowsfor a tunable small (and positive) cosmological constant, while keeping thegravitino mass parameter and thus the supersymmetry breaking scale as anindependent parameter.The resulting model consists of one complex scalar field whose imaginary partbecomes the longitudinal component of the U (1) gauge boson, which in turnbecomes massive due to a Stückelberg mechanism, a gravitino which becomesmassive due to a super-Brout-Englert-Higgs mechanism by eating the Goldstino,which in turn is a linear combination of the chiral fermion (the superpartnerof the complex scalar s ) and the R-gaugino (the superpartner of the U (1)gauge boson), and one physical fermion which is orthogonal to the Goldstino.It is however known that for theories with a shift symmetry there exists analternative formulation in terms of a 2-index antisymmetric tensor which is dualto Im S (under Poincaré duality of its field strength). In this work we provide adual formulation of the model introduced above, which is to our knowledge thefirst dual formulation of a model with a field dependent superpotential. Theimportance of this result lies in the fact that the 2-form field appears naturallyin the string basis. Another interesting question is whether this model canbe obtained in the context of string theory. Indeed, it turns out [50] that the scalar potential can be identified with the one in [53] derived from D-branes innon-critical strings.It is important to check whether the new gauge symmetry that was introduced,namely the gauged shift symmetry (which is an R-symmetry when thesuperpotential is an exponential of S ), is not violated at the quantum level.The work by Elvang, Körs and Freedman [54, 55], on quantum anomalies insupergravity theories with gauged R-symmetries, was extended, in work withI. Antoniadis and D. Ghilencea [56], and their final result can be interpretedin from a field-theoretic point of view. We show how in the above model, thecubic anomalies, corresponding to the gauged shift symmetry, generated by thecorresponding triangle diagrams are canceled by a Green-Schwarz mechanism.It turns out that the requirement of gravitino mass to be in the TeV range,in combination with the anomaly cancellation conditions, severely restrictsthe allowed parameter range. There exist certain choices for the parametershowever, for which the model is consistent and can be successfully used as ahidden sector supersymmetry breaking sector coupled to the MSSM.The anomaly-consistent model was analyzed in full detail in work withI. Antoniadis [57]. Here, it was shown that when this model is coupled to theMSSM, these fields suffer from a negative scalar mass squared. This problemcan however be avoided by introducing a non-canonical Kähler potential for theStandard Model fields, or by including an extra Polónyi-like field. The model isthen coupled to the MSSM, leading to calculable soft supersymmetry massesand a distinct low energy phenomenology that allows to differentiate it fromother models of supersymmetry breaking.Finally, in work with I. Antoniadis [58], the gauged shift symmetry is identifiedwith a global symmetry of the Standard Model (or rather, the MSSM)and the corresponding phenomenology is worked out in full detail. Oneparticularly interesting possibility is to identify the gauged shift symmetry witha combination of baryon and lepton number that contains the known matterparity (or R-parity) of the MSSM, and guarantees the absence of dimension-fourand five operators that violate baryon and lepton number, implying protonstability and smallness of neutrino masses at phenomenologically acceptablelevels. In chapter 2 we introduce supersymmetry along with the necessary conventionsand notation which will thoroughly be used in the following chapters. Inparticular, we will focus on the necessary ingredients to build a supersymmetrictheory: The particle content in the form of supermultiplets on the one hand,
VERVIEW OF THE THESIS 15 and the Kähler potential, the superpotential and gauge kinetic functions whichdefine the supersymmetric Lagrangian on the other hand. Next, we introducethe MSSM. We focus on its particle content, its superpotential and the softsupersymmetry breaking terms. We close this chapter by introducing R-parity,which in the MSSM is sufficient to guarantee the stability of the proton. Weshow its relation with the global symmetry B − L (baryon minus lepton number)of the MSSM and show how certain (baryon and lepton number violating)dimension 5 operators are still allowed.In chapter 3 we introduce supergravity theories, which are theories with localsupersymmetry. We readdress the Kähler potential, the superpotential andgauge kinetic functions in this context, and we introduce the scalar potential.We comment on the importance of R-symmetries and Kähler transformations insupergravity theories. Next, we discuss spontaneous supersymmetry breaking,where we focus on the super-Brout-Englert-Higgs mechanism. We elaborateon hidden sector models where supersymmetry is mediated to the visiblesector (MSSM) due to gravity effects and we relate this discussion to the softsupersymmetry breaking terms of the MSSM.The presentation of our own work starts in chapter 4, where a model basedon a gauged shift symmetry is introduced. We discuss in full detail for whichvalues of the parameters a (meta)stable de Sitter vacuum exists and how theseparameters can be tuned to obtain a small but positive cosmological constant.For particular values of the parameters, the scalar potential can be obtainedin the context of string theory. Finally, we provide a dual description of thismodel in terms of a linear multiplet. This is the first time a such a duality hasbeen performed in the presence of a nontrivial superpotential. This chapter ismainly based on work with I. Antoniadis [50].In chapter 5, we briefly review the work of Elvang, Körs and Freedman [54, 55]on anomalies in supergravity theories with Fayet-Iliopoulos terms. In ourresults, based on work with I. Antoniadis and D. Ghilencea [56], their anomalycancellation conditions are rewritten in a form that is more intuitive from a“naive“ field theory point of view. These results are then applied to the abovemodel. It turns out that the anomaly cancellation conditions force the gravitinomass parameter to exceed the Planck scale for a certain value of the parametersin the model. Finally, We introduce the necessary notation concerning Green-Schwarz counterterms and how the gaugino mass parameters can be generatedat one loop, either due to the Green-Schwarz counterterms, or due to an effectcalled ’anomaly mediation’.In chapter 6, we calculate the soft supersymmetry breaking terms for theparameters where this model is anomaly-free. It is shown that in the minimalcase the model suffers from negative soft scalar masses squared. These can however be cured by introducing another (Polónyi-like) field or by allowing non-canonical kinetic terms for the MSSM superfields, while the desirable propertiesof the model such as a tunably small and positive cosmological constant and aTeV gravitino mass remain intact. We discuss the low energy particle spectrumof this model and show how it can be distinguished from other minimal scenarios.In chapter 7, we demonstrate how the shift symmetry of the model can beidentified with a known global symmetry of the MSSM. We work out theparticularly interesting case where this symmetry is a combination of baryonand lepton number, since in this case it reduces to the known matter parity (orR-parity) of the MSSM. Following, the phenomenology of this model is workedout in full detail, and it is compared with the case of the previous chapterwhere the Standard Model fields are inert under the shift symmetry. Finally,we demonstrate how a third possibility to avoid the tachyonic masses based ona D-term contribution proportional to the charge of the MSSM fields under theshift symmetry does not lead to a viable electroweak vacuum.Our conclusions are listed in chapter 8.The masses of the hidden sector scalars, fermions and gauge field are calculatedin Appendix A. In Appendix B we present the full details of the calculationinvolving the linear-chiral duality in section 4.3.3. In Appendix C we presentthe anomaly cancellation conditions for models with a gauged R-symmetry froma field-theoretic point of view. Finally, in Appendix D we comment on thecertain values of the parameters that were ignored in the rest of the work. hapter 2 Supersymmetry
In this chapter we introduce supersymmetry. Supersymmetry is, as was alreadyexplained in the introduction, perhaps one of the most promising theoriesthat extends the Standard Model. From a technical point of view it providesthe only ’loophole’ in the Coleman-Mandula theorem [24], which under veryreasonable assumptions prohibits the symmetries of spacetime (the Poincaréalgebra) and internal symmetries from being extended in a nontrivial way. TheHaag-Lopuszanski-Sohnius theorem [25] evades this problem by allowing forso-called graded Lie algebras. Indeed, they demonstrate that supersymmetryallows the symmetries of spacetime and internal symmetries to be combinedconsistently : The Poincaré algebra is extended with supersymmetry generators Q α , which obey the (anti-)commutation relations (cid:8) Q α , Q β (cid:9) = −
12 ( γ µ ) βα P µ , [ M µν , Q α ] = −
12 ( γ µν ) βα Q β , [ P µ , Q α ] = 0 . (2.1)Here (and below), Greek indices in the middle of the alphabet ( µ, ν ) indicatespacetime indices, where the metric is assumed to be ’mostly plus’, i.e. η µν =diag( − + ++), and we assume four spacetime dimensions. Greek indices inthe beginning of the alphabet ( α, β ) are two-index spinor indices which will be In fact, for theories including only massless fields, the Coleman-Mandula theorem allowsfor an extension involving the conformal group, and the Haag-Lopuszanski-Sohnius theoremallows the superconformal group.
178 SUPERSYMMETRY dropped whenever possible , P µ is the generator for spacetime translations, and M µν is the generator for Lorentz transformations. The squared brackets standfor commutation relations, while curly brackets stand for anti-commutators. The γ − matrices are defined in [59]. Although, several copies of the supersymmetrygenerators (extended supersymmetry) are allowed, i.e. Q iα , i = 1 . . . N , welimit ourselves for phenomenological reasons to so-called simple supersymmetrywhere there is only one copy of the supersymmetry generators ( N = 1).Although supersymmetry is a very interesting and technical subject, we willintroduce merely its aspects that will turn out important for our work. For amore detailed introduction to supersymmetry, the reader is referred to standardtextbooks, such as [59] whose notation will be used throughout this thesis.In section 2.1 we introduce our notation and the basic ingredients that areneeded to construct a supersymmetric theory. In particular, we introduce thevarious multiplets and discuss how a supersymmetric Lagrangian is constructedby specifying a Kähler potential, a superpotential and gauge kinetic functions.In section 2.2 we introduce the MSSM, with in particular its particle contentand the soft supersymmetry breaking terms that parametrize the breaking ofsupersymmetry. We introduce global R-symmetries in supersymmetric theories,and a discrete R-parity that is usually imposed on the MSSM to ensure protonstability. It is shown how R-parity can be reformulated in terms of anotherdiscrete symmetry called matter parity, and how it is related to certain globalsymmetries of the MSSM, such as Baryon minus Lepton number ( B − L ). N = 1 supersymmet-ric theory Before we can present the basic ingredients of a supersymmetric theory, wefirst specify its particle content. Since supersymmetry relates bosons andfermions, they appear together in so-called supermultiplets . A supermultiplet Below, Greek indices α, β will instead label the various chiral supermultiplets in a theory,see section 2.1. In the literature, a superfield method is very often used to combine bosons and fermions ina single superfield by the use of anti-commuting Grassmann variables. Since some concepts, asfor example R-symmetry can be interpreted more conveniently in this formalism, a superfieldmethod will be used in certain appendices. We will however postpone the necessary notationinvolving the superfield method to appendices and refer the reader to standard textbooks, asfor example [60].
HE BASIC INGREDIENTS OF A N = 1 SUPERSYMMETRIC THEORY 19 is a set of boson and fermion fields which transform among themselves undersupersymmetry transformations. In this thesis however, we will not introducethe supersymmetry transformations, and we merely list the various multiplets.For the SUSY variations and an introduction to supermultiplets the reader isreferred to standard textbooks, such as [59].A chiral multiplet consists of a complex scalar Z ( x ), a Majorana spinor χ ( x ) anda complex scalar auxiliary field F ( x ). The complex scalar field F is auxiliary inthe sense that the Lagrangian does not contain any kinetic terms for F suchthat it can therefore be eliminated by its equations of motion. The field F doesnot correspond to a physical field. Its elimination by its equations of motionresults in a potential for the physical scalars in the theory, called the scalarpotential, which will be very relevant in later chapters. The complex conjugateof a chiral multiplet is an anti-chiral multiplet. We will denote a (anti-)chiralmultiplet as Chiral multiplet: ( Z, χ, F ) , Anti-chiral multiplet: (cid:0) ¯ Z, ¯ χ, ¯ F (cid:1) . (2.2)A gauge (or vector) multiplet consists of a spin-1 gauge boson A µ ( x ), a spin-1 / λ ( x ) (or thecorresponding Weyl field P L λ ), and a real scalar auxiliary field D ( x ). As above,the kinetic terms of the auxiliary field D are absent in the Lagrangian and it canbe eliminated by its equations of motion, which results in another contributionto the scalar potential. A gauge multiplet is denoted asGauge multiplet: ( A µ , λ, D ) . (2.3)A third (and lesser known) multiplet is a linear multiplet. These are usuallyomitted in the literature since due to a linear-chiral duality (see section 4.3.3)a theory containing a linear multiplet can be rewritten in terms of a chiralmultiplet. A linear multiplet consists of a real scalar field l ( x ), a Majoranafermion χ ( x ) and divergenceless field strength v µ ( x ), satisfying ∂ µ v µ = 0 . (2.4) In a superspace formalism, in the conventions of [60], a linear multiplet can be obtainedby imposing the constraints D L = ¯ D L = 0 , on a general superfield, where ¯ D and D are the chiral and anti-chiral projection operators. It follows that v µ can be rewritten in terms of an antisymmetric tensor b µν bya Poincaré duality v µ = 1 √ (cid:15) µνρσ ∂ ν b ρσ , (2.5)which has a gauge symmetry b µν −→ b µν + ∂ µ b ν − ∂ ν b µ . (2.6)A linear multiplet is denoted asLinear multiplet: ( l, χ, v µ ) , with ∂ µ v µ = 0 . (2.7) For a certain set of chiral multiplets (labeled by a Greek index α, β, . . . ), andtheir respective gauge transformations with the corresponding gauge multiplets(labeled by capital Roman indices
A, B, . . . ), a supersymmetric Lagrangian canbe constructed in terms of the following functions of the scalar component fieldsof the chiral multiplets:• A Kähler potential K ( Z, ¯ Z ), which is a gauge invariant function of thescalar components of the chiral multiplets and their conjugates, gives riseto the kinetic terms for the components of the chiral multiplets.• A gauge invariant superpotential W ( Z ), which is a holomorphic functionof the scalars and therefore only depends on Z , and not on ¯ Z . Thesuperpotential contributes to the various interaction terms as well as thescalar potential.• Gauge kinetic functions f AB ( Z ), whose (among other contributions) realcomponent Re f AB multiplies the kinetic terms of the gauge bosons, thegauginos and the D -auxiliary fields.The Lagrangian is given by L = L kin,chir + L kin,gauge + L pot,chir , (2.8)where L kin,chir and L kin,gauge include the kinetic terms for the component fieldsof the chiral multiplets and the gauge multiplets respectively, and L pot,chir includes the interaction terms. For the full Lagrangian, together with therelevant conventions and notation, the reader is referred to the literature (see HE BASIC INGREDIENTS OF A N = 1 SUPERSYMMETRIC THEORY 21 for example eq. (14.57) in [59]). Below, we focus only on the resulting scalarpotential.As was already anticipated in the previous section, the auxiliary fields F α and D A in the supersymmetric Lagrangian (2.8) do not appear with kineticterms and can therefore be eliminated by their equations of motion. After theelimination of the auxiliary fields, one obtains (among other terms) a potentialfor the scalar fields in the theory. The scalar potential is given by V = g α ¯ β W α ¯ W ¯ β + 12 (Re f ) − AB P A P B , (2.9)where W α is short for ∂W∂Z α and the Kähler metric is given by g α ¯ β = ∂ α ∂ ¯ β K ( Z, ¯ Z ) . (2.10)The moment maps are P A = ik αA K α + p A , (2.11)where k αA are the Killing vectors, which appear in the gauge transformations(with gauge parameters θ A ) of the scalar fields δZ α = θ A k αA ( Z ) , (2.12)and the p A are the so-called Fayet-Iliopoulos (FI) constants [61]. These constantsturn out very useful to break supersymmetry and have their origin from thefact that the integral S FI = − R d x p A D A is invariant under supersymmetrytransformations as well as gauge transformations, provided that the gaugesymmetry A is abelian, and can therefore be freely added to the theory for anyconstant p A . This is however in contrast with theories with local symmetry inthe next chapter, where the value of the Fayet-Iliopoulos constant parameter isseverely constrained.Notice that the scalar potential (2.9) satisfies V ≥
0. An important questionarises whether the symmetries of the action are broken in the vacuum. If thebroken symmetry is a global symmetry, then according to the Goldstone theoremthe theory contains a massless scalar particle. In case the broken symmetry is agauge symmetry, the Goldstone boson will disappear from the spectrum, and,by the Higgs mechanism, a spin-1 gauge boson becomes massive.If supersymmetry would be a symmetry of nature, it can only appear as abroken symmetry (see next section). Although we will not present a proof inthis work, it can be shown that supersymmetry is broken if and only if theminimum of the potential is non-zero, V min > In the next chapter, we show that in a theory with local supersymmetry thereis an additional negative F-term contribution to the scalar potential which hasits origin from a gravity auxiliary field. This way, the scalar potential can havenegative values. Moreover, this allows one to construct theories with a vanishingminimum of the scalar potential, while supersymmetry is spontaneously brokeneven though the vacuum energy vanishes. Since supersymmetry is a localsymmetry, the massless Goldstino disappears from the spectrum by the super-Brout-Englert-Higgs mechanism, and the spin-3 / Supersymmetry provides a mathematical framework to construct theoriesthat have desirable properties, as for example solving the hierarchy problemor including a dark matter candidate in the sparticle spectrum. Thesupersymmetric theory with the minimal particle content that includes theStandard Model is the MSSM [63, 64] . In this section we review the basicstructure of the MSSM, its particle content and the superpotential. Following,we introduce R-symmetries, which are internal symmetries that do not commutewith supersymmetry. Finally, we discuss a discrete symmetry called R-parity,which is usually imposed on the MSSM to exclude terms in the Lagrangianthat may lead to proton decay. We show how one can equivalently imposematter parity to prohibit the same terms from the Lagrangian, and we discussits relation with known global symmetries of the MSSM. The MSSM is the minimal supersymmetric model that includes the StandardModel. An obvious question arises to identify the supermultiplets in whichthese Standard Model fields reside. Since all components of a supermultipletcarry the same quantum numbers, the Standard Model fermions can not residein the same multiplet with any of the Standard Model gauge bosons. Instead,the Standard Model fermions are the fermionic components of a chiral multiplet,together with their scalar superpartners, called sfermions , which have the samequantum numbers corresponding to the Standard Model gauge groups. TheStandard Model gauge bosons on the other hand are the spin-1 components in a For more recent reviews, see for example [65, 66]. For example, the superpartner of the electron is a selectron and the superpartner of a topquark is called a stop squark.
SSM 23 gauge multiplet. Their fermionic superpartners are called gauginos . However,in contrast with the Standard Model which contains a single Higgs doublet, theMSSM contains two Higgs chiral supermultiplets called H u and H d .The chiral multiplets together with their Standard Model quantum numbers aresummarized in table 2.1. Here, Q stands for the SU (2) L -doublet chiral multipletcontaining the left-handed up quark (or since family indices are suppressed in Q i , ¯ u i , ¯ d i , ¯ e i , L i , it could represent the multiplet containing the charm or thetop quark), ¯ u ( ¯ d ) stands for the SU (2) L -singlet chiral multiplet containing u † R ( d † R ). L are the lepton doublets and ¯ e are the antilepton singlets. Q ¯ u ¯ d L ¯ e H u H d U (1) Y SU (2) 2 1 1 2 1 2 2 SU (3) 3 ¯3 ¯3 1 1 1 1Table 2.1: The MSSM particle content and their representations in the StandardModel gauge groups. The bar on ¯ u, ¯ d, ¯ e is part of the name and does not representany sort of complex conjugation. Family indices i are surpressed.It is important to note that the above superpartners are not necessarily themass eigenstates of the theory. For example, there is a mixing between theelectroweak gauginos and the Higgsinos when electroweak symmetry breaking(and supersymmetry breaking) effects are included. Similarly, mass mixing canoccur between squarks, sleptons and Higgs scalars that carry the same electriccharge. Such masses and mixing of the superpartners contain very valuableinformation for the experimentalists. We will however not discuss this here, andrefer to standard textbooks [26, 67] instead.The Kähler potential is assumed to be canonical, i.e. for the MSSM chiralsuperfields ϕ α , one has K ( ϕ, ¯ ϕ ) = P α ϕ α ¯ ϕ α . The MSSM superpotential isgiven by W MSSM = y iju ¯ u i Q j · H u − y ijd ¯ d i Q j · H d − y ije ¯ e i L j · H d + µH u · H d , (2.13)where the family indices i, j are explicitly written and all gauge indices aresurpressed. The Yukawa matrices y u , y d and y e are 3 × µ -term and shouldbe read out as µ ( H u ) α ( H d ) β (cid:15) αβ , where (cid:15) αβ is a two-index antisymmetric tensorused to tie together SU (2) L isospin indices. Likewise, the first term is short for In particular, the fermionic superpartners of the Standard Model gauge bosons are calledphotino, bino, wino and gluiono. A single chiral multiplet containing the Higgs boson would lead to quantum anomalies. ¯ u ia ( y u ) ji Q jαa ( H u ) β (cid:15) αβ (and similarly for the second and third term), where a is the color index.The gauge kinetic functions are assumed to be constant and inverselyproportional to the square of their respective gauge couplings, f A = 1 /g A .With the particle content of the MSSM in table 2.1, together with the Kählerpotential, the superpotential and gauge kinetic functions above, one can inprinciple now write down the MSSM Lagrangian L SUSY . However, as we arenot concerned with the exact form of L SUSY in this thesis, the reader is referredto standard textbooks such as [26] or [67] for a more complete treatment of theMSSM Lagrangian.
In the previous section we introduced the particle content of the MSSM. It wasalready mentioned that the various components fields of a supermultiplet sharethe same representations of the (Standard Model) gauge groups. Moreover, theinvariance of the Lagrangian L SUSY under supersymmetry transformations alsoimplies that the different components of a multiplet have the same mass. Ifthis were the case, this would imply that for example the selectron, the scalarsuperpartner of the electron has a mass of about 0 .
511 MeV (of course, similarstatements are true for the squarks, gluinos, photinos, . . . ). Such a scalarparticle is very easy to detect, and should have shown up in collider experimentsif it existed. Since no superpartners were found yet, one can conclude that ifsupersymmetry is a symmetry is nature, it has to be a broken symmetry.In practice, one usually ’manually’ adds soft supersymmetry breaking terms [68]to the supersymmetric MSSM Lagrangian. The effective Lagrangian of theMSSM is given by L = L SUSY + L soft . (2.14)Here, L SUSY is the Lagrangian of the MSSM defined above, which is invariantunder supersymmetry transformations. The possible soft supersymmetrybreaking interactions are L soft = − (cid:18) M A λ A λ A + 16 a ijk φ i φ j φ k + 12 b ij φ i φ j (cid:19) + c.c. − ( m ) ij φ j † φ i , (2.15) Tadpole contributions t i φ i to the soft supersymmetry breaking Lagrangian are notincluded in eq. (2.15) since they require the existence of a gauge singlet in the MSSM. SSM 25 where M A are the mass parameters for the gauginos λ A for each gauge group.The scalar fields of the MSSM are labeled by i, j, k , a ijk are the trilinearcouplings, b ij is the ’B µ -term’ and ( m ) ij are the scalar soft masses squared.The terms in L soft are soft in the sense that they have a positive mass dimensionand their presence does not spoil supersymmetry as a potential solution to thehierarchy problem.Without further justification, the addition of the soft supersymmetry breakingterms to the Lagrangian might seem anything but elegant and a rather arbitraryrequirement. However, as was already mentioned before, supersymmetry can bespontaneously broken if one of the scalar fields acquires a VEV in an appropriatescalar potential. While supersymmetry is spontaneously broken it can be shownthat, given an appropriate scalar potential, the soft supersymmetry breakingterms (2.15) appear after the scalar fields responsible for supersymmetrybreaking are replaced with their VEVs. This scalar field can however notbe the superpartner of one of the Standard Model fermions, since this wouldadditionally break some of the Standard Model gauge symmetries. It is thereforerequired to introduce an additional ’hidden sector’ sector, whose scalar fieldsare responsible for supersymmetry breaking. These fields are usually singletsunder the Standard Model gauge groups and do not communicate with theStandard Model at low energies. Many supersymmetric theories are invariant under a (global) chiral U (1)symmetry called R-symmetry (which is consequently denoted by U (1) R ). AnR-symmetry is characterized by the fact that (at the level of the algebra) itsgenerator T R does not commute with the supersymmetry generators. Instead,it acts on the supersymmetry generators Q α as[ T R , Q α ] = − i ( γ ∗ ) βα Q β , (2.16)where γ ∗ is given by γ ∗ = iγ γ γ γ . This has the consequence that, in contrastwith other internal symmetries, like the Standard Model gauge groups underwhich all components of a chiral multiplet carry the same charge, an R-symmetryacts differently on each component of a chiral multiplet. The scalar component Z , the fermionic component P L χ and the auxiliary field F carry R-charges r, r χ = r − , r F = r − U (1) R are summarized as δ R Z = iρrZ,δ R P L χ = iρ ( r − P L χδ R F = iρ ( r − F. (2.17)In global supersymmetry there is an interesting connection between theorieswhich have a global R-symmetry and supersymmetry breaking [69]. It shouldalso be noted that by eq. (2.16) in global supersymmetry an R-symmetry can notbe gauged. This is not the case however in theories with local supersymmetry.Moreover, in local supersymmetry a gauged R-symmetries implies the presenceof a Fayet-Iliopoulos D-term contribution to the scalar potential (see section 3.1). Although in the literature R-parity is usually introduced as a discrete R-symmetry, we point out in this section that R-parity can be formulated insuch a way that it not an R-symmetry at all (in the definitions of the previoussection).The most general gauge-invariant and renormalizable superpotential for theMSSM would not only include the usual terms eq. (2.13), but also the followingbaryon- and lepton-number violating interactions W ∆ L =1 = 12 λ ijk L i L j ¯ e k + λ ijk L i Q j ¯ d k + µ i L i H u ,W ∆ B =1 = 12 λ ijk ¯ u i ¯ d j ¯ d k . (2.18)The chiral superfields carry baryon number B = 1 / Q i , B = − / u i , ¯ d i and B = 0 for all the others. Similarly, L i and ¯ e i carry lepton number +1and −
1, respectively, while all other superfields have vanishing lepton number.Since the baryon and lepton number violating processes (2.18) are not seenexperimentally, these terms should be absent (or sufficiently suppressed). Thisis usually done by imposing that a discrete R-parity is preserved. The R-parityof a field is given by P R = ( − B − L )+2 s , (2.19)with s its spin. Note that the Standard Model particles and Higgs bosons carry P R = +1, while the ’sparticles’ (squarks, sleptons, gauginos and Higgsinos) SSM 27 have P R = −
1. Also, since every interaction vertex contains an even number of P R = − P R = − P M of a superfield (as opposed to R-parity, which is defined separately on eachcomponent field) is defined as P M = ( − B − L ) . (2.20)Note that since the matter parity of all fields within a given supermultipletis the same, this symmetry does commute with supersymmetry. Since for thescalar components ( s = 0) the matter parity is the same as the R-parity, it iscompletely equivalent to impose either matter parity or R-parity as a symmetryon the theory. Moreover, R-parity and matter parity only differ by the fermionnumber, which is an exact parity symmetry by itself.We conclude that imposing matter parity or R-parity is completely equivalent.While the R-parity interpretation can be useful to easily abstract itsphenomenological consequences, from a model building point of view it isfar more natural to impose matter parity (2.20), since (in contrast with R-parity) it commutes with supersymmetry. In fact, since R-parity is equivalentto the (non-R) matter parity, this shows that there is nothing intrinsically ’R’about R-parity.In chapter 4, we will introduce a model based on a gauged U (1) (shift) symmetry.Since matter parity is nothing else but 3( B − L ). It therefore seems an obviouschoice in this context to see whether one can consistently identify it with the U (1) symmetry we need in our toy model of supersymmetry breaking. Thiscan be done by giving a charge proportional to B − L to the MSSM superfields.As a result, the terms in eqs. (2.18) are excluded from the superpotential, andthus the U (1) symmetry takes over the role of R-parity. This will be discussedin full detail in chapter 7.It should however be noted that in principle one can also have dimension 5operators that violate baryon and/or lepton number (see for example [74] and references therein) W dim 5 = κ (0) ij H u L i H u L j + κ (1) ijkl Q i Q j Q k L l + κ (2) ijkl ¯ u i ¯ u j ¯ d k ¯ e l + κ (3) ijk Q i Q j Q k H d + κ (4) ijk Q i ¯ U j ¯ e k H d + κ (5) i L i H u H u H d . (2.21)Here the various couplings κ ( n ) have inverse mass dimensions and can begenerated by a high-energy microscopic theory, such as a supersymmetric grandunified theory or string theory. While R-parity forbids the terms in the lastline of eq. (2.21), all terms in the first line are still allowed. Imposing a B-L symmetry additionally forbids the terms proportional to κ (0) ij . The termsproportional to κ (1) ijkl and κ (2) ijkl are still allowed. It should however be noted thata 3 B − L symmetry (which has the same parity ( − B − L = ( − B − L onMSSM fields) forbids all the above dimension 5 operators. However, in contrastwith a gauged B − L which can be made anomaly-free upon the inclusion ofthree right-handed neutrinos to the MSSM, a gauged 3 B − L contains a cubic U (1) B − L , and mixed U (1) B − L × SU (2) and U (1) B − L × U (1) Y anomalieswhich should be canceled by a Green-Schwarz mechanism (see chapter 7). hapter 3 Supergravity
In the previous section we introduced supersymmetry, along with the necessarynotation, and its minimal model containing the Standard Model, the MSSM.While supersymmetry is realized as a global symmetry in the MSSM, weintroduce in this chapter theories where supersymmetry is realized as local(gauge) supersymmetry. If a bosonic symmetry is promoted to a local symmetry,the gauge theory requires the presence gauge bosons to preserve gauge invarianceof the Lagrangian under this symmetry. Local supersymmetry on the otherhand is a fermionic symmetry and requires the presence spin-3/2 fermion calledthe gravitino. Since the gravitino appears together with a spin-2 graviton in agravity multiplet, theories with local supersymmetry naturally contain gravity.In this chapter we explain how a supergravity theory is constructed, where weparticularly focus on gauged R-symmetries and Fayet-Iliopoulos terms, whichplay an important role in later chapters. The scalar potential is introduced,together with the relevant conventions and notation. We discuss spontaneoussupersymmetry breaking, and in particular how the soft supersymmetry breakingterms arise in a gravity mediated supersymmetry breaking scenario. We finishthis chapter by explaining the relation between the cosmological constant andthe minimum of the potential. As before, we follow the notation of [59].
As was the case in a rigid supersymmetric theories (see section 2.1), the particlecontent of a supergravity theory consists, besides the gravity multiplet, of chiral
290 SUPERGRAVITY multiplets (see eq. (2.2)) and gauge multiplets (see eq. (2.3)). One can constructa Lagrangian invariant under local supersymmetry transformations by defininga Kähler potential K , a superpotential W and gauge kinetic functions f AB . Theresulting Lagrangian however, is rather lengthy, and the reader is referred tostandard textbooks (see for example eq. (18.6) in [59]) for its full form. Belowhowever, we will introduce several relevant terms in the Lagrangian, such as thescalar potential, fermion mass terms and possible Green-Schwarz countertermswhich can arise.Although a supergravity theory is uniquely determined (up to Chern-Simonsterms) by a Kähler potential K ( z, ¯ z ), a superpotential W ( z ), and the gaugekinetic functions f AB ( z ), there is a certain degeneracy in the definitions of theKähler potential and the superpotential. Namely, a theory is invariant under aKähler transformations K ( z, ¯ z ) −→ K ( z, ¯ z ) + J ( z ) + ¯ J (¯ z ) ,W ( z ) −→ e − κ J ( z ) W ( z ) , (3.1)where J ( z ) ( ¯ J (¯ z )) is an (anti-)holomorphic function, and κ is the inverse of thereduced Planck mass, m p = κ − = 2 . × TeV, and the z α are the scalarcomponents of the chiral multiplets of the theory.A supergravity theory can be described by a gauge invariant function (fortheories with a non-vanishing superpotential) G = κ K + log( κ W ¯ W ) . (3.2)The gauge transformations (with gauge parameters θ A ) of the chiral multipletscalars are given by holomorphic Killing vectors eq. (2.12). In contrast withglobal supersymmetry, the Kähler potential and superpotential in a supergravitytheory need not be invariant under a gauge transformation, but can leave aKähler transformation. The gauge transformation of the Kähler potentialsatisfies δ K = θ A [ r A ( z ) + ¯ r A (¯ z )] , (3.3)for some holomorphic function r A ( z ), provided that the gauge transformationof the superpotential satisfies δW = − θ A κ r A ( z ) W . One then has from δW = W α δz α that the superpotential satisfies W α k αA = − κ r A W, (3.4)where W α = ∂ α W and α labels the chiral multiplets. OW TO BUILD A SUPERGRAVITY THEORY 31
The scalar potential is given by [75] V = V F + V D ,V F = e κ K (cid:16) − κ W ¯ W + ∇ α W g α ¯ β ¯ ∇ ¯ β ¯ W (cid:17) ,V D = 12 (Re f ) − AB P A P B , (3.5)where the Kähler covariant derivative of the superpotential is ∇ α W = ∂ α W ( z ) + κ ( ∂ α K ) W ( z ) . (3.6)The moment maps P A are given by P A = i ( k αA ∂ α K − r A ) . (3.7)While the scalar potential of a globally supersymmetric theory (see eq. (2.9)) isalways positive (or rather nonnegative), the F-term contribution to the scalarpotential of a supergravity theory contains a negative contribution proportionalto − κ W ¯ W . This negative contribution, which has its origin in the eliminationof a gravity auxiliary field, makes it possible to construct spontaneously brokensupergravity with a vanishing minimum of the scalar potential.In this thesis we will be concerned with theories with a gauged R-symmetry , forwhich r A ( z ) is given by an imaginary constant r A ( z ) = iκ − ξ . In this case, κ − ξ is a Fayet-Iliopoulos (FI) [61] constant parameter (where ξ is dimensionless).Note that in contrast with a globally supersymmetric theory, the FI constantcan not be chosen at will, but it is constrained by eq. (3.4). Since a FI term givesa (positive) D-term contribution to the scalar potential, these terms can be veryinteresting to study in the context of spontaneous supersymmetry breaking.The mass terms for the fermions in the Lagrangian are given by L m = 12 m / ¯ ψ µ P R γ µν ψ ν − m αβ ¯ χ α χ β − m αA ¯ χ α λ A − m AB ¯ λ A P L λ B + h.c. , (3.8)where ψ µ is the spin-3/2 gravitino (which is the superpartner of the graviton)and its mass parameter is given by m / = κ e κ K / W. (3.9) In fact, we are interested in theories where the minimum of the potential is at a verysmall and positive value as to accommodate for the cosmological constant. For an early paper on gauged R-symmetries, see for example [76]. The generation of its physical mass will be discussed below in section 3.2
The spin-1/2 mass matrices of the chiral fermions χ α and the gauginos λ A aregiven by m αβ = e κ K / (cid:2) ∂ α + (cid:0) κ ∂ α K (cid:1)(cid:3) ∇ β W − e κ K / Γ γαβ ∇ γ W, (3.10) m αA = i √ (cid:20) ∂ α P α − f ABα (Re f ) − AB P C (cid:21) , (3.11) m AB = − e κ K / f ABα g α ¯ β ¯ ∇ ¯ β ¯ W , (3.12)where f ABα = ∂ α f AB , and the connection components Γ γαβ are given byΓ αβγ = g α ¯ δ ∂ β g γ ¯ δ , Γ ¯ α ¯ β ¯ γ = g δ ¯ α ∂ ¯ β g δ ¯ γ . (3.13)All other connection components vanish (i.e. Γ αβ ¯ γ = Γ ¯ α ¯ βγ = 0). In thesupergravity models in the following chapters it will often occur that thenon-diagonal elements of the gaugino mass matrix m AB vanish. In this case,the diagonal elements will be denoted by a single index, m A . Similarly for thegauge kinetic functions f A . It is a well-known fact that when a gauge symmetry is spontaneously broken, aBrout-Englert-Higgs (BEH) mechanism occurs; For example, in the StandardModel, the SU (2) L × U (1) Y gauge symmetry is spontaneously broken to a U (1).This occurs because the potential of a scalar field (the Higgs boson) forces itto acquire a non-zero vacuum expectation value (VEV). For each of the threebroken generators of the gauge groups, there exists a so-called Goldstone boson,which is subsequently ’eaten’ by the corresponding W + , W − and Z gaugebosons which in turn become massive. The fermions in the theory then acquiremasses when the Higgs field is replaced with its VEV in its interactions withthe fermions. The gauge bosons ’eat’ the Goldstone bosons in the sense that they absorb their degreeof freedom. They indeed need an extra degree of freedom to become massive since a masslessgauge boson has two (on-shell) degrees of freedom, while a massive one has three.
PONTANEOUS SUPERSYMMETRY BREAKING 33
A similar mechanism, called the super-BEH mechanism, occurs when localsupersymmetry is spontaneously broken by the VEV of some scalar field(s).An important difference with the above however, is that the supersymmetrygenerators (see eqs. (2.1)) are fermionic. Therefore, spontaneous supersymmetrybreaking will lead to a Goldstone fermion (instead of a boson), called theGoldstino P L ν , which in general is a linear combination of the fermionicsuperpartners of all fields that contribute to the supersymmetry breaking P L ν = χ α δ s χ α + P L λ A δ s P R λ A , (3.14)where the ’fermion shifts’ (the scalar parts of the supersymmetry transformationsof the fermions) are given by δ s χ α = − √ e κ K / ∇ α W,δ s P R λ A = − i P A . (3.15)Due to the super-BEH effect, the elimination of the Goldstino will give a mass tothe gravitino given by eq. (3.9), and the mass matrix for the fermions becomes m = m αβ + m ( ν ) αβ m αB + m ( ν ) αB m Aβ + m ( ν ) Aβ m AB + m ( ν ) AB ! , (3.16)where the corrections to the fermion mass terms due to the elimination of theGoldstino are given by m ( ν ) αβ = − κ m / ( δ s χ α )( δ s χ β ) ,m ( ν ) αA = − κ m / ( δ s χ α )( δ s P R λ A ) ,m ( ν ) AB = − κ m / ( δ s P R λ A )( δ s P R λ B ) . (3.17)Since the elimination of the Goldstino results in a reduction of the rank of m ,its determinant vanishes and the physical masses of the fermions correspond tothe non-zero eigenvalues of m . It was already explained in section 2.2.2 that the MSSM superfields can not beresponsible for supersymmetry breaking since a VEV of a scalar superpartner of a Standard Model fermion would additionally break a Standard Model gaugesymmetry. Instead, the origin of supersymmetry breaking should be in a ’hiddensector’ of particles. Although there are several mechanisms available by whichsupersymmetry breaking can be communicated to the visible sector (MSSM),we will focus here on gravity mediation [77]. For other mechanisms, such asgauge mediation [78] of anomaly mediation [79–83], the reader is referred tothe literature.In a gravity mediated supersymmetry breaking scenario [77], the chiral multipletsof a model are divided into two sectors: An observable sector, which includesthe chiral multiplets ϕ i of the MSSM (as well as its gauge multiplets), and ahidden sector where the scalar fields are labeled by z α . The Kähler potentialand the superpotential can be decoupled K = K ( z, ¯ z ) + X ϕ ¯ ϕ,W = W h ( z ) + W MSSM ( ϕ ) , (3.18)where the indices labeling chiral multiplets have been omitted. As in theMSSM, the observable sector Kähler potential is taken to be canonical andthe MSSM superpotential is given in eq. (2.13). In the limit κ → M P → ∞ ) one simply has a model with two decoupled sectors.When gravitational effects are included however, the scalar potential is given by V = e κ K ( z, ¯ z )+ P κ ϕ ¯ ϕ (cid:16) − κ | W h + W MSSM | + g z ¯ z |∇ z W | + |∇ ϕ W | (cid:17) , (3.19)where ∇ z W = ∂ z W h + κ ∂ z K ( z, ¯ z ) ( W h + W MSSM ) , ∇ ϕ W = ∂ ϕ W MSSM + κ ¯ ϕ ( W h + W MSSM ) , (3.20)by the definition in eq. (3.6).The hidden sector superpotential W h is constructed such that supersymmetryis broken in this sector and the scalar fields z α in the hidden sector acquirea VEV (typically of order ∼ m p = 1 /κ ). The couplings in the hidden sectorhowever, carry a high scale (usually assumed to be at the GUT-scale or near thePlanck mass), such that its effects are not detectable at low (collider) energies.We will show below that, when the hidden sector scalars are replaced with theirVEVs, one retrieves exactly the soft supersymmetry breaking terms (2.15). PONTANEOUS SUPERSYMMETRY BREAKING 35
The derivative with respect to a visible sector field ϕ of the scalar potential isgiven by ∂ ϕ V = e κ K ( z, ¯ z )+ P κ ϕ ¯ ϕ h κ ¯ ϕ (cid:16) − | W h + W MSSM | + g z ¯ z |∇ z W | + |∇ ϕ W | (cid:17) − κ ¯ W W ϕ + κ g z ¯ z K z W ϕ ( ¯ W ¯ z + κ K ¯ z ¯ W )+ ( W ϕϕ + κ ¯ ϕW ϕ )( ¯ W ¯ ϕ + κ ϕ ¯ W ) + ( W ϕ + κ ¯ ϕW ) ¯ W i , (3.21)where W ϕ = ∂ ϕ W , K z = ∂ z K are shorthand notations. Under the assumptionthat the visible sector fields ϕ do not contribute to the supersymmetry breaking,we have h ϕ i = h ¯ ϕ i = 0, and therefore for W MSSM , given by eq. (2.13), we have h W ϕ i = h W ¯ ϕ i = 0. The soft supersymmetry breaking masses are then given by ∂ ϕ ∂ ¯ ϕ V (cid:12)(cid:12)(cid:12) ϕ = ¯ ϕ =0 = e κ K ( z, ¯ z ) (cid:16) − κ W h ¯ W h + κ g z ¯ z |∇ z W h | (cid:17) , (3.22)where the hidden fields z are replaced with theirs VEVs. The Bµ − term can becalculated similarly from ∂ ϕ ∂ ϕ V (cid:12)(cid:12)(cid:12) ϕ = ¯ ϕ =0 = κ e κ K ( z, ¯ z ) (cid:2) g z ¯ z K z (cid:0) ¯ W ¯ z + κ K ¯ z ¯ W (cid:1) − ¯ W (cid:3) W ϕϕ , (3.23)as well as the trilinear terms ∂ ϕ ∂ ϕ ∂ ϕ V (cid:12)(cid:12)(cid:12) ϕ = ¯ ϕ =0 = κ e κ K ( z, ¯ z ) g z ¯ z K z (cid:0) ¯ W ¯ z + κ K ¯ z ¯ W (cid:1) W ϕϕϕ . (3.24)The gaugino mass terms can be calculated from eq. (3.12). Note however that aconstant gauge kinetic function implies a vanishing gaugino mass since eq. (3.12)is proportional to the derivative of its respective gauge kinetic function. Wewill show in section 5.3 that, even for a constant gauge kinetic function, thegaugino mass terms can be generated at one loop.The MSSM is defined however in rigid supersymmetry, where the scalar potentialis given by V = P W MSSM ,ϕ ¯ W MSSM , ¯ ϕ . In the discussion above, this correspondsto the last term in eq. (3.19). Note however that, after the hidden sector fieldsare replaced with their VEVs, this term is proportional to e κ K . Therefore, inorder to interpret the last term in eq. (3.19) as the usual MSSM scalar potentialat low energies, a rescaling is needed:ˆ W MSSM = e κ K ( z, ¯ z ) / W MSSM , (3.25)such that at low energies the last term in eq. (3.19) reduces to P ˆ W MSSM ,ϕ ˆ¯ W MSSM , ¯ ϕ . As a result, all trilinear terms a ijk (see eq. (2.15)) are the same and are givenby A ˆ y i , where y i are the Yukawa couplings in the MSSM superpotential (2.13)(rescaled by eq. (3.25)) and A = κ e κ K ( z, ¯ z ) / g z ¯ z K z (cid:0) ¯ W ¯ z + κ K ¯ z ¯ W (cid:1) . (3.26)Similarly, the ’ Bµ -term is given by B ˆ µ , where ˆ µ = e κ K/ µ by eq. (3.25), and B = κ e κ K ( z, ¯ z ) / (cid:2) g z ¯ z K z (cid:0) ¯ W ¯ z + κ K ¯ z ¯ W (cid:1) − ¯ W (cid:3) , (3.27)Note that one has the relation [84] A = B + m / , (3.28)where the gravitino mass term is given by eq. (3.9).Note however above that we only discussed the F-term contribution to thescalar potential. Indeed, a D-term contribution to the scalar potential can alsocontribute to the soft supersymmetry breaking terms. This will be discussedfurther in section 7.2. In this section we show the relation between the minimum of the potential andthe cosmological constant Λ. In the Einstein field equations, the cosmologicalconstant arises as R µν − g µν R + Λ g µν = 8 πG T µν , (3.29)where R µν is the Ricci tensor, R is the curvature scalar R = g µν R µν , T µν is thestress-energy tensor, G is the gravitational constant, and we put the speed oflight in the vacuum c = 1.On the other hand, one can write the action of a supergravity theory as theEinstein-Hilbert term plus a term L M involving any matter fields in the theory S = Z d x √− g (cid:20) κ R + L M − V (cid:21) . (3.30)Here, g = det g µν is the determinant of the metric tensor, and κ = 8 πG .A constant contribution V has been added that corresponds to the value at Any terms involving the gravitino are also assumed to be included in L M . ELATION BETWEEN THE MINIMUM OF THE POTENTIAL AND THE COSMOLOGICAL CONSTANT37 the minimum of the scalar potential. We will show below how the constantcontribution to the Lagrangian V gives rise to a contribution in the Einsteinfield equations (3.29) that can be interpreted as the cosmological constant Λ.The variation of the action eq. (3.30) gives δS = Z d x (cid:20) κ δ (cid:0) √− gg µν (cid:1) R µν + 12 κ √− gg µν δR µν + δ (cid:0) √− g L M (cid:1) − δ (cid:0) V √− g (cid:1) (cid:21) . (3.31)Since the variation of the Ricci tensor is given by δR µν = ∇ ρ δ Γ ρµν − ∇ µ δ Γ ρνρ , (3.32)the second term in eq. (3.31) is a total derivative and vanishes. Further, we use δg µν = − g µρ δg ρσ g σν ,δ √− g = 12 √− gg µν δg µν = − √− gg µν δg µν , (3.33)to reduce eq. (3.31) to δS = 12 κ Z d x δg µν √− g " R µν − g µν R − κ (cid:18) − δ ( √− g L M ) δg µν √− g (cid:19) + κ V g µν . (3.34)One now defines the stress-energy tensor T µν = − δ ( √− g L M ) δg µν √− g , (3.35)and solve for δS = 0 to find the Einstein field equations (3.29) withΛ = 8 πGV . (3.36)Alternatively, the constant contribution V can be absorbed in the stress-energytensor by defining T µν = − δ ( √− g L M ) δg µν √− g + T vac µν , (3.37) where T vac µν = − ρ vac g µν , with ρ vac = V = κ − Λ . (3.39)The above equivalence shows the relation between the cosmological constantand the energy of the vacuum, and is the reason the terms ’vacuum energy’ and’cosmological constant’ are used interchangeably. Moreover, the latest resultsfrom the Planck collaboration [85] showΛ obs = 3Ω Λ H c ≈ . × − m − , (3.40)where the Hubble constant is given by H = 67 . kms Mpc , the dark energydensity Ω Λ = 0 . c . The stress-energy tensor can be modeled as a perfect fluid, T µν = ( ρ + p ) U µ U ν + pg µν , (3.38)where ρ is the energy density and p is the isotropic pressure in its rest frame, U µ is the fluidfour-velocity. One obtains eq. (3.39) by putting ρ vac = − p vac . hapter 4 A model with a tunably small(and positive) cosmologicalconstant
It was already mentioned in the introduction that experimental evidence suggeststhat the cosmological constant is very small and positive [35, 36]. Therefore, ifstring theory were to be a realistic theory of nature, its vacuum energy shouldtherefore be positive as well. This is called a de Sitter (dS) vacuum. Thehunt for de Sitter vacua in string theory with a tunably small value of thecosmological constant is a very interesting and challenging subject (for recentwork, see for example [37–45]).In this chapter we start the presentation of our own work, where this problemis addressed in the context of supergravity. This is interesting since certainsupergravity theories appear as the low energy limit of string theory. The goalof this chapter is to obtain a supergravity theory, based on a chiral multipletwith a gauged shift symmetry, which has a tunably small and positive valueof the minimum of the scalar potential. In the string theory context, thischiral multiplet can be identified with the string dilaton or an appropriatecompactification modulus. We show that for suitable values of the parametersthe scalar potential can be identified with the one obtained in [53].The model [50], originally proposed in [51, 52], consists of (besides the gravitymultiplet) a chiral multiplet S and a vector multiplet associated with a gauged U (1) symmetry which acts as a shift along the imaginary part of the scalarcomponent of S . By gauge invariance of the theory, the Kähler metric can
390 A MODEL WITH A TUNABLY SMALL (AND POSITIVE) COSMOLOGICAL CONSTANT therefore only be a function of the real part of S , which is taken to be alogarithm K = − κ − p log( s + ¯ s ), where the constant p is assumed to be aninteger . The most general gauge kinetic function, consistent with the shiftsymmetry, contains a constant and a linear contribution, while the most generalsuperpotential is either a constant, or an exponential of S . In the case ofan exponential superpotential, the gauged shift symmetry becomes a gaugedR-symmetry , which is therefore labeled U (1) R below (even in the case wherethe superpotential is constant).As a result, the R-symmetry fixes the form of the Fayet-Iliopoulos term, leadingto a supergravity action with two independent parameters that can be tunedsuch that the scalar potential possesses a (metastable) de Sitter vacuum with atunably small (and positive) cosmological constant. A third parameter fixes theVEV of the string dilaton, while the mass of the gravitino (which is essentiallythe supersymmetry breaking scale) is separately tunable.An alternative (dual) description of this model is given in terms of a linearmultiplet. The string dilaton appears naturally in a linear multiplet, where it isaccompanied by a 2-index antisymmetric tensor potential which is dual to Im S under Poincaré duality of its field-strength. We show that this dual descriptionpersists even in the presence of a field dependent (exponential) superpotential.The model is introduced in section 4.1. Our own contributions start in section 4.2where we examine and quantify the tunability of the scalar potential in thismodel. It turns out that for p ≥ p = 1 and p = 2 are presented in full detail in subsections 4.2.1 and4.2.2 respectively. Following, in section 4.3 we make a few remarks on the case p = 2: Firstly, since for p = 2 the vacuum is metastable, quantum tunneling toanother vacuum solution can in principle occur. It is checked in section 4.3.1that this tunneling probability is indeed extremely low. Next, in section 4.3.2we briefly comment on a possible connection with string theory and we closethis chapter by rewriting the model in terms of a linear multiplet in section 4.3.3which is a more natural description from a string theory point of view.This chapter is based on work together with I. Antoniadis [50, 57] and on workwith I. Antoniadis and D. Ghilencea [56]. In particular, section 4.2.1 is basedon work [57], sections 4.2.2, 4.3.2 and 4.3.3 are based on [50], while section 4.3.1is based on [56]. For example, for the total volume-axion one has p = 3 (which results in no-scalesupergravity), or for the dilaton-axion p = 1. For other studies based on a gauged R-symmetry, see for example [86–89].
NTRODUCTION OF A MODEL 41
In order to obtain a model with a tunably small (and positive) value of thecosmological constant, one could try to cancel a (negative) F-term contributionto the scalar potential by a positive D-term contribution. The simplest modelone can look at then has a chiral multiplet S whose superpotential gives rise toan F-term contribution to the scalar potential, and a vector multiplet whosecorresponding gauge symmetry gives rise to a D-term contribution. We willassume that the gauge symmetry is abelian, and that the scalar component s of the chiral multiplet transforms under this U (1) R (where the subscript R willbe clear below) as a shift s −→ s − icθ, (4.1)where θ is the gauge parameter and c is a parameter of the model. The real scalarcomponent of s can be identified with the string dilaton or a compactificationmodulus , while its imaginary part is an axion. In supersymmetric theories thestring dilaton and the axion a can be described as the real and imaginary partof the scalar component s of a chiral multiplet S = ( s, ψ, F ) s = 1 g + ia, (4.2)where g is the four dimensional gauge coupling. In perturbation theory, theaxion has an invariance under a Peccei-Quinn symmetry which shifts s by animaginary constant.As outlined in section 3.1, a supergravity theory is defined by a Kähler potential,a superpotential and gauge kinetic functions. The shift symmetry eq. (4.1)forces the Kähler potential to be a function of s + ¯ s . Moreover, while keepingthe interpretation of the string dilaton (or a compactification modulus) in theback of our mind, the Kähler potential is taken to be of the form K = − κ − p log( s + ¯ s ) , (4.3)where p is a parameter of the model and the factor κ − is needed to restore theappropriate mass dimensions of the Kähler potential . Of course, this Kählerpotential results in a non-renormalizable theory. It should therefore be viewed In superfields the shift symmetry (4.1) is given by δS = − ic Λ, where Λ is the superfieldgeneralization of the gauge parameter. The gauge invariant Kähler potential below in eq. (4.15)is then given by K ( S, ¯ S ) = − κ − p log( S + ¯ S + cV R ) + κ − b ( S + ¯ S + cV R ), where V R is thegauge superfield of the shift symmetry. Indeed, we will show below in section 4.3.2 how s corresponds to the string dilaton for aparticular value of the parameters in the model. Note that the field s has mass dimension 0. either as a classical field theory, or as a quantum field theory with a certaincutoff.Since the superpotential W ( s ) is a holomorphic function, it can not be afunction of the combination s + ¯ s and thus any (non-constant) superpotentialwill transform under the shift symmetry. The only nontrivial possibility is thento allow the superpotential to transform with a phase according to eq. (3.4).The most general superpotential turns out to be an exponential W = κ − ae bs , (4.4)where a (not to be confused with the axion) and b are parameters in the model.Note that the superpotential is not invariant under the shift symmetry. Insteadit transforms as δW = − ibcθW, (4.5)in which case the shift symmetry becomes an R-symmetry and is thereforelabeled with a subscript R . This is consistent if eq. (3.4) is satisfied, which gives r A = iξκ − , (4.6)where ξ = bc (4.7)and κ − ξ is the Fayet-Iliopoulos constant [61] in the theory. It is interesting thatthis superpotential appears naturally in the context of gaugino condensation(see for example [90–93]) and in recent works on supersymmetric extensions ofStarobinsky models of inflation [94–96].Following, the most general gauge kinetic function is either a constant or linearin s f ( s ) = γ + βs. (4.8)It is however important to note that if the parameter β is non-zero, theLagrangian contains a Green-Schwarz [97–101] contribution L GS = i f ( s )) F µν ˜ F µν = 18 Im ( f ( s )) (cid:15) µνρσ F µν F ρσ , (4.9) Below, the parameter β will often carry a subscript R , such that β R can be distinguishedfrom similar parameters ( β A ) corresponding to similar Green-Schwarz contributions for theStandard Model gauge groups A . NTRODUCTION OF A MODEL 43 where the dual of an antisymmetric tensor is defined as˜ F µν = − i (cid:15) µνρσ F ρσ . (4.10)The term (4.9) however, is not gauge invariant, since its gauge transformationis given by δ L GS = − θβc (cid:15) µνρσ F µν F ρσ . (4.11)To restore gauge invariance, this term should be canceled by another contribution.We postpone this discussion until chapter 5, where the gauge invariance of thismodel at the quantum level is studied in full detail.The scalar potential is given by V = − | a | ( s + ¯ s ) p e b ( s +¯ s ) + (cid:18) b − ps + ¯ s (cid:19) (cid:18) a p e b ( s +¯ s ) ( s + ¯ s ) p − + c β ( s + ¯ s ) + 2 γ (cid:19) . (4.12)Below, we will assume a > K = − κ − p log( s + ¯ s ) ,W = κ − ae bs ,f ( s ) = γ + βs, (4.13)where the moment map P , defined in eq. (3.7), is given by κ P = c (cid:18) b − ps + ¯ s (cid:19) . (4.14)This can be rewritten by performing a Kähler transformation (see eqs. (3.1))with J = κ − bs in the form K = − κ − p log( s + ¯ s ) + κ − b ( s + ¯ s ) ,W = κ − a,f ( s ) = γ + βs. (4.15) One can therefore absorb the exponential part of the superpotential in theKähler potential. In this case the gauge symmetry is not an R-symmetryanymore, which has important consequences that will be discussed in section 5.3.Nevertheless, we will continue to refer to the shift symmetry as U (1) R , even inthe ’Kähler frame’ eq. (4.15) where the shift symmetry is technically not anR-symmetry. It is important to note that indeed both eqs. (4.13) and eqs. (4.15)result in the same supergravity Lagrangian, and in particular the same scalarpotential. In this section we investigate the parameter range of the model defined aboveby eqs. (4.13) (or equivalently by eq. (4.15)) and (4.8) for which there exists apositive and small minimum of the scalar potential eq. (4.12). We then presentthe relations between the parameters which ensure a (tunably) small value ofthe cosmological constant.For b >
0, there always exists a supersymmetric AdS (anti-de Sitter) vacuumat h s + ¯ s i = b/p , while for b = 0 supersymmetry is broken in AdS space. Sincewe are interested in supersymmetry breaking de Sitter vacua, we thereforefocus on b <
0. For p ≥ p <
3, the F-term contribution V F is unbounded frombelow when s + ¯ s →
0. On the other hand, the D-term contribution to thescalar potential V D is positive and diverges when s + ¯ s → p in front of the logarithm in the Kähler potential is assumed tobe a positive integer. Below we show that for p = 1, a tunable vacuum can befound (see subsection 4.2.1) when γ >
0, while for p = 2 tunable vacua canbe found when γ = 0 and β >
0. We show how the parameters a and c can betuned such that the scalar potential has a minimum at an infinitesimally small(and positive) value. Moreover, it will turn out that the parameter b can betuned independently to determine the VEV of the scalar component s of thechiral multiplet. A negative or vanishing p would indeed not lead to consistent kinetic terms in theLagrangian. The real part of the gauge kinetic function at the minimum of the potential should bepositive, for positivity of the kinetic energy of the gauge field.
UNABILITY OF THE MODEL 45 p = 1 For p = 1, a stable de Sitter vacuum can be found on the condition that theconstant contribution to the gauge kinetic function is non-zero. In this sectionwe present the relations between the parameters a, b, c that are necessary toensure a positive and tiny cosmological constant for the case β = 0. Although anon-zero β is in principle allowed, it is shown in the Appendix D.1 that due toconstraints from anomaly cancellations its contribution turns out to be verysmall. In Appendix D.2 we show the relations between the parameters to obtaina vanishing cosmological constant for a vanishing constant contribution ( γ = 0, β >
0) to the gauge kinetic function. However, since this case is inconsistentwith anomaly cancellation conditions, similar to the case for p = 2 in section 5.2,it will not be discussed below.In the case β = 0, the constant γ in eq. (4.8) can be absorbed in other constantsof the theory by an appropriate rescaling of the vector field, and we can put γ = 1. The model is then given by K = − κ − log( s + ¯ s ) + κ − b ( s + ¯ s ) ,W = κ − a,f ( s ) = 1 . (4.16)The scalar potential is V = V F + V D , V F = κ − a e b ( s +¯ s ) s + ¯ s σ s , σ s = − b ( s + ¯ s ) − , V D = κ − c (cid:18) b − s + ¯ s (cid:19) . (4.17)The minimization of the potential ∂ s V = 0 gives c a = h s + ¯ s i (2 − b h s + ¯ s i ) e b h s +¯ s i . (4.18)By plugging this relation into V min = Λ ≈ (10 − eV) , which ensures theminimum of the potential to be at a very small but positive value, one finds κ e − b h s +¯ s i h s + ¯ s i Λ a = − b h s + ¯ s i − (cid:20) − b h s + ¯ s i (cid:21) . (4.19) An infinitesimally small cosmological constant Λ can then be obtained by tuningthe parameters a, b, c such that b h s + ¯ s i = α ≈ − . , (4.20) bc a = A ( α ) + 2 κ Λ α a b ( α − , A ( α ) = 2 e α α − ( α − ( α − ≈ − . , where α is the negative root of − α − (2 − α /
2) = 0 close to − .
23. Theother roots are either imaginary or would not allow for a real solution of thesecond constraint. Note that lim s +¯ s →∞ V = κ − ξ > , (4.21)where κ − ξ is the FI constant defined above in eq. (4.7). It follows that for κ − ξ / > Λ, the scalar potential allows for a stable de Sitter (dS) vacuumwith an infinitesimally small (and tunable) value for the cosmological constant.The gravitino mass (3.9) is given by m / = κ − p | b | a e α/ α ≈ . p | b | a × TeV . (4.22)One can obtain a TeV gravitino mass by tuning a p | b | ≈ . × − . A plotof the scalar potential is given in figure 4.1.We conclude that for p = 1 there can exist indeed a dS vacuum with a tunablysmall cosmological constant. Moreover, the parameter b can be used to fix theVEV of h s + ¯ s i , while a relation between the parameters a and c ensures asmall cosmological constant. In chapter 6 however, we will show that when thisminimal model is coupled to the MSSM, it will lead to tachyonic soft scalarmasses for the MSSM fields. We therefore postpone a careful treatment of thismodel, its possible solutions to avoid tachyonic masses and its the resultingenergy sparticle spectrum to chapter 6. p = 2 For p = 2, metastable de Sitter vacua can be found for various values of theparameters if the gauge kinetic function is linear in s ( γ = 0). Moreover, as In fact, a very small γ is still consistent with the existence of a dS vacuum, provided β = 0. The resulting corrections however to the scalar potential and the discussion below arevery small. We therefore take γ = 0. UNABILITY OF THE MODEL 47
60 80 100 120 140 160 180 s (cid:43) s (cid:180) (cid:180) (cid:180) (cid:180) V Figure 4.1: A plot of the scalar potential (4.17) for p = 1, b = α/ a = 10 − (blue) and a = 0 . × − (red), and c is determined by eq. (4.20) with Λ = 0.The VEV of s + ¯ s is specified by choosing b = α/
50, where we have taken h s + ¯ s i = 50 in this plot. The values of a are chosen such that the gravitino massis of the order of TeV. Note that the scalar potential has a stable dS minimumat h s + ¯ s i = α/b .far as the minimization of the potential is concerned, the constant β in eq. (4.8)can be absorbed in other constants of the theory . The model is then given by K = − κ − log( s + ¯ s ) + κ − b ( s + ¯ s ) ,W = κ − a,f ( s ) = s. (4.23)The scalar potential (4.12) then reduces to V = κ − a e b ( s +¯ s ) (cid:18) b − bs + ¯ s − s + ¯ s ) (cid:19) + κ − c s + ¯ s (cid:18) s + ¯ s − b (cid:19) . (4.24) For a non-trivial β , the D-term contribution to the scalar potential is given by κ V D = c β s + ¯ s (cid:16) s + ¯ s − b (cid:17) . In this case the results of this section hold true with c replaced by c , given by c = c /β . As in the previous section, we first look for a Minkowski minimum and solvethe equations V ( s ) = 0 ,d V ds ( s ) = 0 . (4.25)This leads to the following relations between the parameters at the minimum ofthe potential : b h s + ¯ s i = α ≈ − . ,bc a = A ( α ) , (4.26)where α is the root of the polynomial − x + 7 x − x − x + 40 x + 8 closeto − .
18 and A ( α ) is given by A ( α ) = e α α α − α − − α − α ! ≈ − . . (4.27)Note that the above polynomial has five roots, four of which are unphysical:two roots are imaginary, one is positive, which is incompatible with the firstline in eqs. (4.26), since h s + ¯ s i should be positive by eq. (4.2), and a fourthroot gives rise to a positive A ( α ), which is incompatible with the last line ineqs. (4.26) since b < A Λ ( α ) = a bc − κ Λ b c α e − αα − α − ! . (4.28)It follows that by carefully tuning a and c , Λ can be made positive and arbitrarilysmall independently of the supersymmetry breaking scale. The gravitino massterm is given by m / = κ − abα e α/ . (4.29)We conclude that one can obtain a tunable de Sitter vacuum with the minimumof the potential at h s + ¯ s i = α/b . As in the previous section, the parameter b can The definition of A ( α ) is the inverse of the one used in [50,56,58] in order to be consistentwith the rest of this work. UNABILITY OF THE MODEL 49 be used to determine the value of Re( s ) at the minimum of the potential. Sincethe gauge kinetic function is linear in the dilaton s , we have h s +¯ s i = 1 /g s , where g s is the string coupling constant, which can thus be tuned by the parameter b ,independently of the minimum of the potential and the gravitino mass. A TeVgravitino mass can be obtained by tuning a | b | ∼ . × − , and the parameter c should satisfy eq. (4.26). The scalar potential is plotted in figure 4.2 for a = 3 × − and a = 6 × − , for b = α/
50 and c determined by eq. (4.26).Note that, in contrast with the p = 1 case in section 4.2.1, the de Sitter vacuumis metastable since V → s + ¯ s → ∞ . One should therefore check whetherthis vacuum is sufficiently long lived. This is confirmed below in section 4.3.1.
100 150 200 250 300 s (cid:43) s (cid:180) (cid:180) (cid:180) (cid:180) (cid:180) V Figure 4.2: A plot of the scalar potential (4.24) for p = 2, b = α/ a = 3 × − (blue) and a = 6 × − (red), and c determined by eq. (4.26). The VEV of s + ¯ s is specified by choosing b = α/
50, where we have taken h s + ¯ s i = 50 inthis plot. The values of a are chosen such that the gravitino mass is of theorder of TeV. Note that the scalar potential has a metastable dS minimum at h s + ¯ s i = 50. The metastability is discussed below.Due to a Stückelberg mechanism, the imaginary part of s (the axion) is eatenby the gauge field, which acquires a mass. On the other hand, the Goldstino P L ν , which is a linear combination of the fermion of the chiral multiplet χ and the gaugino λ , is eaten by the gravitino (see section 3.2). As a result, thephysical spectrum of the theory consists (besides the graviton) of a massivescalar, namely the dilaton, a Majorana fermion, a massive gauge field and amassive gravitino. A calculation of the masses of these fields is presented in Appendix A. The results are m s = κ − b cα r
48 + 192 α − α − α + 24 α − α + α α − α ,m A µ = κ − √ b / cα / ,m f = κ − a b e α (cid:0)
116 + 68 α − α − α + 8 α (cid:1) α . (4.30)Note that by eqs. (4.26) and eq. (4.29) all masses are proportional to the gravitinomass, which is proportional to the constant a (or c related by eq. (4.26)). Thus,these masses vanish in the supersymmetric limit where the charge of the shiftsymmetry is taken to zero c →
0, or equivalently, in the limit m / → p = 2 Classically, the above vacua are stable. For the case p = 2 however, thescalar potential tends to zero for Re( s ) → ∞ . Although the ground state ineq. (4.26) is separated by a barrier (see Fig. 4.2) from this runaway direction atRe( s ) = ∞ , quantum tunneling can in principle occur between these solutions.One thus needs to estimate the probability (Γ) for the current vacuum V min =Λ = κ − (cid:15) > V min = 0 along the Re( s ) direction,through the potential barrier (Fig. 4.2). This is to ensure that the ground stateis long-lived enough. This probability is (per unit of time and volume)Γ = A e − B (cid:126) (1 + O ( (cid:126) )) , (4.31)where (cid:126) is the reduced Planck constant (we set (cid:126) = 1), and A and B dependon the model. The value of A plays a minor role in comparison with theexponential suppression; B is fixed by the Euclidean action of the instanton(bounce) solution ( S ) which in the limit of a very small energy differencebetween the two minima is [102–104] B = 27 π S (cid:15) . (4.32) In the case where p = 1, the ground state is stable when κ − ξ / > Λ (see eq. (4.21))and such analysis is not necessary.
FEW REMARKS ON P = 2 We redefine the field s into (for standard kinetic terms; s is dimensionless) φ s = κ − log( s + ¯ s ) , (4.33) S is given by [102] S = Z dφ s p V ( φ s ) = κ − | a | S ( b ) , (4.34)where V is the potential of φ s and S ( b ) = √ Z ∞ ln ( αb ) dx h e b e x (cid:18) b − be − x − e − x (cid:19) + e − x A ( α ) b (cid:0) e − x − b (cid:1) i / , which can be computed numerically. A ( α ) is given by eq. (4.27) (where one hasset Λ = 0 to a good approximation).If we interpret Re( s ) as the (inverse of the) 4D coupling (1 /g s ) to a GUT-likevalue, Re( s ) ≈
25, this fixes b via b = α/ h s + s i ≈ − × − . For this value of b one finds S ≈ . a , see previous section) one finds B ≈ . Therefore Γ isextremely small (largely due to the small difference between the two minima) ;the ground state is long lived enough to use this model as a starting pointfor building realistic models, by adding physical fields, that have such groundstate along this field direction. However, we will show in section 5.2 that otherconstraints coming from quantum anomalies render the model with p = 2 uselessfor this purpose. As supergravity theories appear as a low energy limit (where the string lengthgoes to zero) of string theory, it is an interesting question whether one can finda UV-completion of this model in the context of string theory. Indeed, it turnsout that for b = 0, the scalar potential (4.24) reduces to V = − κ − a ( s + ¯ s ) + 4 κ − c ( s + ¯ s ) . (4.35)The potential (4.35) coincides with the one derived in [53] from D-branes innon-critical strings. Indeed the second term corresponds to a disc contributionproportional to the D-brane tension deficit δ ¯ T induced by the presence ofmagnetized branes in type I orientifold compactifications, while the first term Usually values of
B ≥
400 are regarded as metastable enough [105]. corresponds to an additional contribution at the sphere-level, induced fromgoing off criticality, proportional to the central charge deficit δc . More precisely,in the Einstein frame ( M p = 1), the scalar potential in [53] is given by V nc = e ϕ δc + e ϕ (2 π ) v / δ ¯ T , (4.36)where ϕ is the four-dimensional dilaton related to the ten-dimensional dilaton ϕ by e − ϕ = e − ϕ v , and v is the six-dimensional volume given by v = V (4 π α ) .By identifying e − ϕ = Re( s ), one sees that the scalar potential (4.36) is indeedof the form of equation (4.35). One can then identify δc and δ ¯ T as δc = − κ − a ,δ ¯ T = (2 π ) v / κ − c . (4.37)Note that δc can become infinitesimally small only if it is negative [53], as isneeded for the existence of an anti-de Sitter vacuum in equation (4.36). Thepotential has a minimum at h s + ¯ s i = c a , given by V ( h s i ) = − κ − a c < b implied by the most generalsuperpotential consistent with the shift symmetry allows one to find (metastable)de Sitter vacua for certain values of the parameters, as is shown in section 4.2.2. Although the model (4.23) is formulated in terms of a chiral multiplet and avector multiplet, we show in this section how it can be reformulated in terms ofa linear multiplet (see section 2.1) and a vector multiplet. Because of the shiftsymmetry (4.1) of the axion there exists a dual description where the axion isdescribed by an antisymmetric tensor b µν , which has a gauge symmetry givenby eq. (2.6), repeated here for convenience of the reader b µν −→ b µν + ∂ µ b ν − ∂ ν b µ . In the context of string theory, it is this antisymmetric tensor that appearsin the massless spectrum as the supersymmetric partner of the dilaton. Theantisymmetric tensor is related to a field strength v µ by eq. (2.5) from whichdivergenceless of the field strength (which is the Bianchi identity) eq. (2.4)follows. Together with a real scalar l and a Majorana fermion χ , this real vector v µ belongs to a linear multiplet L . It should be noted that the shift symmetry FEW REMARKS ON P = 2 (4.1) of the axionic partner of the dilaton is a crucial ingredient for the duality towork, since it is related to the gauge symmetry (2.6) which in turn is responsiblefor the divergenceless of the field strength eq. (2.4).Below, we present the dual version of the model (4.13) in the representationwhere the superpotential is an exponential of S below for p = 2, which is toour knowledge the first time the linear-chiral duality has been performed inthe presence of a non-trivial superpotential. The details of this calculation canbe found in Appendix B, where a superfield formalism is used in the globallysupersymmetric case, and the chiral compensator formalism is used in the locallysupersymmetric case.The result eq. (B.49), repeated here for convenience of the reader, is given by L = (cid:20) − (cid:0) S e − gV R ¯ S (cid:1) L − (cid:21) D − [ cLV R ] D + 1 b (cid:20)(cid:0) T ( L ) − α W (cid:1) − (cid:0) T ( L ) − α W (cid:1) ln (cid:18) T ( L ) − α W abS (cid:19)(cid:21) F + h.c.+ (cid:2) β W (cid:3) F + h.c. , (4.38)where the operations [ ] F and [ ] D are defined in eqs. (B.26) and (B.27)respectively. Although in eq. (4.38) the multiplet L is an a priori unconstrainedmultiplet, it is shown in the Appendix B that L indeed contains the degrees offreedom of a linear multiplet. Moreover, it is shown that the (non-gravitational)spectrum of this theory can be described, except for a gauge multiplet, interms of two real scalars l and w , where w is defined as Z = ρe iw , with Z the θ -component of L , and a Majorana fermion.In particular, the F-term contribution to the scalar potential is given in termsof l by V F = κ − a e b/l (cid:18) b − bl − l (cid:19) , (4.39)which on substitution of s + ¯ s = 1 /l (see eq. (B.30)) indeed gives the F-termcontribution to the scalar potential in eq. (4.24). hapter 5 Anomalies in supergravitytheories with FI terms
Symmetries, and in particular gauge symmetries, play an important role inquantum field theories. A symmetry of the classical Lagrangian is a fieldtransformation that leaves the Lagrangian, or rather the action, invariant.Examples of such symmetries are the SU (3) × SU (2) × U (1) Standard Modelgauge group, Poincaré transformations or supersymmetry transformations. It isan important question whether these symmetries still persist at the quantumlevel. If the symmetry is violated in the quantum theory, it is said to beanomalous. There are several ways to understand how such anomalies can arise.For example, a violation of the classical action at the quantum level can beseen in the functional integral formulation of quantum field theory. Here, thesymmetries of the (classical) action can be expressed in the Ward identities forthe correlation functions. An important assumption, however, is that not onlythe Lagrangian, but also the integral measure is invariant under the symmetry.When this is not the case, the symmetry is violated at the quantum level andthe symmetry is said to be anomalous and the Ward identities are violated .Alternatively, one can check whether a classically conserved current remainsconserved in the quantum theory. A current that is not conserved at thequantum level indicates the presence of an anomalous contribution to thequantum effective action. In practice, one can check whether fermion triangle(Feynman) diagrams spoil current conservation at the quantum level [107]. For an introduction to anomalies in quantum field theory, see for example [106].
556 ANOMALIES IN SUPERGRAVITY THEORIES WITH FI TERMS
Finally, in computing Feynman diagrams, one is forced to introduce aregularization. Often, it may happen that there exists no regularization thatpreserves all symmetries. In this case, one can not guarantee that the Wardidentities (and thus the classical symmetries) are displayed in the renormalizedGreen’s functions. If they do not, the theory is anomalous.At this point it is important to distinguish between global and local symmetries:An anomalous global symmetry of the action does not pose a problem to theconsistency of a theory; it merely implies that the classical selection rules arenot obeyed in the quantum theory. Consequently, certain classically forbiddenprocesses may occur.The presence of an anomalous local symmetry on the other hand is disastrousfor the quantum consistency of any quantum field theory . We should thereforecheck, before investigating its low energy phenomenology in chapter 6, whetherthe extra U (1) R gauge group introduced in the previous chapter in order toobtain a tunable cosmological constant, can be consistently combined with theMSSM in an anomaly-free way. Moreover, anomaly cancellation conditions canimpose strong constraints on a theory. Luckily, since our model has a scalar fieldwith a shift symmetry, we have the terms (4.9)) whose gauge transformationcan cancel certain anomalies, which is the Green-Schwarz mechanism.The starting point in this chapter will be the work of Elvang, Freedman andKörs [54, 55] on anomalies in supergravity theories with FI terms by usingthe Fujikawa method [109]. We briefly summarize their results in section 5.1.Although these results are correct, they are obscure from a field-theoretic pointof view. We present our work with I. Antoniadis and D. Ghilencea [56], wherethe anomaly cancellation conditions of [55] are reformulated in such a way thattheir interpretation can be interpreted and compared with a naive field-theoreticapproach, which is presented in Appendix C. These results are then applied insection 5.2 to our the model with p = 2 (see section 4.2.2), and it is shown thatthe anomaly cancellation condition are inconsistent with a TeV gravitino masswhich is necessary to have a viable low energy phenomenology. We concludethis chapter in section 5.3 where we elaborate on the relation between quantumanomalies and the generation of gaugino mass terms at one loop.Section 5.1 is based on [56], while section 5.2 is based on [56,58], and section 5.3is a result from [57]. Luckily for the Standard Model, where one can a priori have anomalies for each of thetriangle diagrams corresponding to SU (3) × U (1), SU (2) × U (1), U (1) gauge bosons onthe external lines, and a mixed gravitational- U (1) anomaly, some miraculous cancellationsoccur between the various anomalous contributions of its gauge symmetries within a fermiongeneration (See for example [108], where this was originally shown for a four-quark version ofthe Standard Model). NOMALIES IN SUPERGRAVITY THEORIES WITH FI TERMS 57
In the previous chapter we introduced a model that allows for a tunablecosmological constant, while leaving the gravitino mass separately tunable. Thisdiscussion however was limited to the classical level. As was elaborated inthe introduction, quantum consistency can lead to additional constraints on atheory by imposing anomaly cancellation conditions. Such conditions can havean important impact on a theory.Anomaly cancellation in supergravity with an anomalous U (1) gauge symmetrywith Fayet-Iliopoulos terms and a Green-Schwarz mechanism were discussed inthe past in a general setting in [54, 55]. This subject is certainly not new, but itis worth revisiting since supergravity theories with Fayet-Iliopoulos couplingsappear for example as the effective four-dimensional effective theory in fluxcompactifications of string theory [76].In supergravity, the metric g α ¯ β is invariant under Kähler transformations (3.1).In certain theories the Kähler potential is not a global scalar. Instead, itcan be defined locally in each coordinate chart, and the definitions of theKähler potentials in the different charts are related by a Kähler transformation.Such a Kähler transformation however is accompanied by an appropriate axialgauge transformation of the fermions in the model, which can be anomalous.This so-called Kähler anomaly does not make the theory inconsistent at thequantum level. Instead, in this section we focus on the situation where a gaugetransformation does not leave the Kähler potential invariant, but leaves a Kählertransformation. This typically leads to Fayet-Iliopoulos terms (see section 3.1).The accompanying transformations of the fermions can be anomalous and inthis section we review their anomaly cancellation conditions.Here, we review the results of [55]. The relation and agreement of such resultsto the “naive” field theory approach however, was not examined for the ratherspecial case of a gauged U (1) R . In fact, in global SUSY U (1) R can not evenbe gauged. This can easily be seen from the commutation relations of thesupersymmetry algebra eqs. (2.16). In this section, we rewrite the results of [55]in such a way that they can successfully be interpreted from a field-theoreticpoint of view (which is reviewed in the Appendix C). This section, based on workwith D. Ghilencea and I. Antoniadis [56] is also relevant on its own, independentof the rest of this thesis.Below, we assume that the field content is that of minimal supergravity, includinga gauged U (1) R , and a dilaton multiplet S whose shift symmetry gives rise to aGreen-Schwarz mechanism (as for example in chapter 4, where in the Kähler basis with an exponential superpotential eq. (4.13) the gauged shift symmetryis indeed a gauged R-symmetry). Additionally, we assume the presence ofadditional MSSM-like superfields, and that the gauge symmetries are thoseof the Standard Model gauge group times U (1) R . The anomaly cancellationconditions for the cubic anomaly U (1) R ((a) below), and the mixed anomaliesof U (1) R with the Kähler connection K µ ((b), (c)), with the SM gauge group(d) and gravity (e) are given by in eqs. (4.4) in Section 4 of [55]. These are ( a ) C ˜ C : 0 = − Tr (cid:2) ( Q + ξ/ Q (cid:3) − ξ a KCC − c b C , ( b ) C ˜ K : 0 = Tr (cid:2) ( Q + ξ/ Q (cid:3) + ξ a CKK − a KCC − c b CK , ( c ) K ˜ K : 0 = − Tr (cid:2) Q (cid:3) + 12 ξ ( n λ + 3 − n χ ) + 4 a CKK − c b K , ( d ) ( F ˜ F ) A : 0 = Tr (cid:2) Q ( τ a τ b ) A (cid:3) − ξ (cid:2) T G − T R (cid:3) A δ ab + 13 c b A δ ab , ( e ) R ˜ R : 0 = − Tr (cid:2) Q (cid:3) + ξ n λ − − n χ ) + 8 c b grav , (5.1)where ˜ labels the dual field strength defined in eq. (4.10). Here, the gaugefield (strength) of U (1) R is indicated by C µ ( C µν ) to distinguish its role fromthe Standard Model gauge fields (strengths) A µ ( F µν ) (with group generators τ a ); A is a group index that runs over U (1) Y , SU (2) L , SU (3). The Kählerconnection K µ essentially fixes the coupling of the gravitino. These threefields are involved in the conditions (a), (b), (c), (d). Condition (e) is forthe mixed, U (1) R -gravitational anomaly. There is also the usual conditionTr[ τ a { τ b , τ c } ] = 0 for the Standard Model group generators.In eqs. (5.1), n λ is the number of gauginos present (or equivalently, the numberof vector multiplets), n χ is the number of chiral multiplets which include boththe matter fermions and the dilatino . The U (1) R -charges Q of the (scalar)matter superfields are such that the Killing vector corresponding to U (1) R of a(complex) scalar field z α in a chiral multiplet is given by k αR = iQz α . (5.2)Moreover in eqs. (5.1), κ − ξ is the Fayet-Iliopoulos term corresponding with In [55, 56], the transformation of a field z with charge Q under an abelian gauge group isdefined as δz = exp( − iθQ ) z . In this thesis, we follow the conventions of [59], where one uses δz = exp( iθQ ) z . As a result, in eq. (5.1) the signs of the charges (and FI-term) are flippedcompared to [55, 56]. Refs. [54, 55], as well as [59] use anti-hermitian T a = − iτ a , so we added a minus in frontof Tr in line (d) of eqs. (5.1). The dilatino is the fermionic component of the S -multiplet. Note that the contributions to eqs. (5.1) proportional to ξ are Planck suppressed comparedto the FI-term κ − ξ . NOMALIES IN SUPERGRAVITY THEORIES WITH FI TERMS 59 U (1) R . Since U (1) R is an R-symmetry, the Standard Model gauginos, the U (1) R gaugino and the gravitino are (unlike other types of anomalous U (1)’s)all charged under U (1) R and their contribution to the anomalies should betaken into account. Next, Tr r ( τ a τ b ) = T R δ ab is the trace over the irreduciblerepresentations R , and f acd f bcd = δ ab T G , with T G = N for SU ( N ) and 0 for U (1). The terms in eqs. (5.1) proportional to ξ with coefficients +3 and − c is the Green-Schwarz coefficient and a KCC , a CKK are coefficients of local counterterms that can be present (defined below).The Kähler connection K µ = i κ (cid:0) ∂ µ z α ∂ α K − ∂ µ ¯ z ¯ α ∂ ¯ α K (cid:1) − A Aµ P A = i κ (cid:16) ˆ ∂ µ z α ∂ α K − ˆ ∂ µ ¯ z ¯ α ∂ ¯ α K (cid:17) + A Aµ ( r A − ¯ r A ) (5.3)transforms under gauge ( δz α = k αA θ A ) and Kähler transformations (seeeqs. (3.1)) as δK µ = − κ ∂ µ (cid:0) θ A Im( r A ) + Im( J ) (cid:1) . (5.4)In eq. (5.3), the covariant derivative is defined asˆ ∂ µ z α = ∂ µ − A Aµ k αA . (5.5)To understand the role of the Kähler connection it is instructive to write downthe covariant derivatives of the gravitino ψ µ , the gauginos λ and (MSSM-like)matter fermions χ D µ ψ ν = (cid:18) ∂ ν − i K µ γ ∗ (cid:19) ψ ν ,D µ λ A = (cid:18) ∂ µ − i K µ γ ∗ (cid:19) λ A − A Cµ λ B f ABC ,D µ χ α = (cid:18) ∂ µ + 12 iK µ (cid:19) χ α − A Aµ ∂k αA ∂z β χ β , (5.6)where we have omitted terms involving the ψ -torsion, the spin connection andthe Christoffel connection. As was discussed in chapter 3, the Kähler potentialand the superpotential are not invariant under an R-symmetry. Instead, a The Kähler connection differs with a factor 1 / U (1) R gauge transformation leaves a Kähler transformation, given by eqs. (3.3)and (3.4). Indeed, the contributions from the Kähler connection K µ are takeninto account in the anomaly cancellation conditions eqs. (5.1). We will howevershow below that the conditions (a), (b) and (c) can be reduced to the familiarcubic anomaly condition, while the remaining two (independent) conditions canbe satisfied by a suitable choice of the counterterms a CKK and a KCC .The local counterterms that come with coefficients a KCC , a CKK are [54, 55] L KC = 124 π (cid:15) µνρσ [ a CKK C µ K ν ∂ ρ K σ + a KCC K µ C ν ∂ ρ C σ ] , (5.7)and b C , b CK , b K , etc, of eqs. (5.1) are coefficients present in the Chern-Simonsterms [111–113] L CS = 148 π Im( s ) (cid:15) µνρσ ∂ µ Ω νρσ (5.8)= 196 π Im( s ) (cid:15) µνρσ [ b C C µν C ρσ + b CK C µν K ρσ + b K K µν K ρσ + b A ( F µν F ρσ ) A + b grav R µν R ρσ ] . In the context of string theory some of the coefficients b C , b K , b CK , b grav can berelated, as is the case for example in heterotic string theory (for b C and b grav ) andthey can therefore not be adjusted at will. However, since it is difficult to derivea U (1) R from strings, this applies only to non-R anomalous U (1)’s. We thereforerelax this constraint and consider them to be independent. Additionally, weassume that the dilaton ( S ) is the only (super)field that transforms nonlinearlyunder the gauged U (1) R and it implements the Green-Schwarz mechanism. Thecanonical normalization of the gauge kinetic term for the U (1) R gauge field ( C µ )gives b C = 12 π , while we assume that b K and b CK can in general be non-zero.The supergravity anomaly cancellation conditions (5.1) are not transparent fromthe “naive” field theory point of view for the anomalies of U (1) R . So let us clarifythe link between these conditions and the field theory result in Appendix C.First, the U (1) R -charges of the fields are shown below (see Appendix C) anddepend on the FI term(s): R χ = Q + ξ/ , R λ = R / = − ξ/ , R χ s = ξ/ , (5.9)where Q are the charges of the scalar components of the matter superfields (seeeq. (5.2)); R λ , R / and R χ s are the charges of the gauginos λ A , the gravitino ψ µ and the dilatino χ s , respectively. Using this information, the first threerelations in eqs. (5.1) can be combined, after multiplying them by 4, − ξ and ξ NOMALIES IN SUPERGRAVITY THEORIES WITH FI TERMS 61 respectively, and then adding them by using that Tr1 = n χ −
1. The result is Tr[ R χ ] + n λ R λ + 3 R / + R χ s = − c (cid:2) b C − ξb CK + ξ b K (cid:3) . (5.10)On the lhs one recognizes the usual field theory cubic U (1) R anomaly cancellationcondition in the presence of FI terms and Green-Schwarz mechanism, in whichall fermionic contributions are added and compensated by a Green-Schwarzshift on the rhs: The trace includes all contributions from matter fermions. Thenumber of gauginos is n λ = 1+(8+3+1) = 13 for the U (1) R × Standard Modelgauge group. Each gaugino has a contribution R λ . The gravitino contribution(+3) R / is three times larger than that of one gaugino, as was shown in [110].The above result has (with b C = 12 π and b CK = b K = 0) the same form as the“naive” field theory result, eqs. (C.11) in Appendix C. This is interesting sincein global SUSY U (1) R can not even be gauged. Note however the difference inthe rhs due to ξb CK + ξ b K . The terms in L CS of coefficients b K , b CK are notpresent in the naive field theory case, and give extra freedom in canceling thisanomaly.The two remaining independent conditions of constraints (a), (b), (c) in eq. (5.1),refer to Kähler and mixed U (1) R -Kähler connection. They can always berespected by a suitable choice of a KCC and a CKK of the local countertermsshown and are not further discussed. One finds (by combining these tworemaining constraints with condition (e) in eq. (5.1)), that a CKK = − ξ + c ( b K + 2 b grav ), while a KCC is found from one of equations (a), (b), (c) ineq. (5.1).The last two conditions in eq. (5.1) can be re-written as( F ˜ F ) A : Tr (cid:2) R χ ( τ a τ b ) A (cid:3) + T G δ ab R λ = − (1 / δ ab b A c, R ˜ R : Tr [ R χ ] + n λ R λ + ( − R / + R χ s = 8 b grav c. (5.11)The lhs of the first equation is exactly the naive field theory contribution fromthe (MSSM) matter fermions, of R χ = Q + ξ/
2, and the gauginos. In the secondequation, there are contributions of: the n λ gauginos of the Standard Modeland U (1) R gauge groups; the gravitino contribution (( −
21) times that of agaugino [110]), the dilatino, and the trace is over all matter fermions. Theseequations agree with the naive field theory result, eqs. (C.12) and (C.16) forthe corresponding anomalies. The rhs of the first equation also contains acontribution b A from the counterterm due to the Green-Schwarz mechanismin eq. (5.8). By supersymmetry, the gauge kinetic functions corresponding to “-1” in Tr1 = n χ − Q of thedilaton is 0), since n χ is the total amount of chiral multiplets, which includes the dilatonmultiplet S . the Standard Model gauge groups contain a linear contribution β A s , with β A ≡ b A / (12 π ). The rhs of the second condition in eq. (5.11) indeed agreeswith eqs. (C.16) for b grav = 12 π β grav . while in supergravity one is free to adjustthis coefficient (unlike in heterotic string case). This ends the relation of theanomaly cancellation conditions to the naive field theoretical results (globalSUSY) obtained using the Tr over the charged states. p = 2 The above results can, in principle, be applied the model introduced in theprevious section, since in the Kähler frame where the superpotential is anexponential of the scalar field s (see eq. (4.13)), the U (1) gauge symmetryis an R-symmetry and the above results apply. Moreover, when the gaugekinetic function has a linear contribution in s , which transforms under the shiftsymmetry (4.1), the constant c can indeed be identified with the constant c inthe above section.However, as we will show below, the anomaly cancellation conditions are muchsimpler in the Kähler frame with a constant superpotential (see eq. (4.15)), andwe will therefore continue in this frame . Here, in the case p = 2 the requirementof the existence of a vanishing (or infinitesimally small and positive) minimumof the potential forces the constant contribution to the gauge kinetic function tovanish, which leaves only a linear contribution in s . The Lagrangian thereforecontains a Green-Schwarz counterterm eq. (4.9) and is not gauge invariant: Thegauge variation of the Lagrangian leaves a non-vanishing contribution given byeq. (4.11).One can however introduce another field z , which can either be a hidden sectorfield or a MSSM field. This extra field z is assumed to have a canonical Kählerpotential and has a charge q under the U (1) and its anomalous contribution tothe Lagrangian cancels the Green-Schwarz contribution (4.11). The model isgiven by K = − κ − log( s + ¯ s ) + κ − b ( s + ¯ s ) + z ¯ z,W = κ − a,f ( s ) = β R s. (5.12) The quantity b A / π = β A plays the role of Kac-Moody levels in the heterotic string. Note however, that anomalies should cancel in both cases, which in fact has very interestingconsequences on the mass terms of the gauginos. This will be discussed in section 5.3.
ONSEQUENCES FOR THE CASE P = 2 The scalar potential is given by V = V F + V D , V F = κ − a e K (cid:0) σ s + κ z ¯ z (cid:1) , V D = 12 β R ( s + ¯ s ) (cid:18) κ − c (cid:18) b − s + ¯ s (cid:19) − qz ¯ z (cid:19) . (5.13)where σ s is defined in eq. (4.24). The mass of the field z is given by m z = m / ( σ s + 1) + κ − cqβ ( s + ¯ s ) (2 − α ) , (5.14)in the notation of section 4.2.2, where α = h s + ¯ s i b ≈ − . β R bc β R a = A ( α ) , (5.15)where A ( α ) ≈ − . U (1) R anomaly is given by δ L − loop = − θ π C R (cid:15) µνρσ F µν F ρσ , (5.16)where C R = Tr[ Q ]. The total variation of the Lagrangian vanishes, i.e. δ L − loop + δ L GS = 0 (where δ L GS is given in eq. (4.11)) if β R = − q π c . (5.17)This fixes the sign of q and results in a negative D-term contribution to thescalar mass squared of z , which by using eqs. (4.26) and (5.17) is m z = m / ( σ s + 1) − κ − / m / / R ( α ) ,R ( α ) = (cid:18) π A ( α )(2 − α ) e − α α (cid:19) / ≈ . . (5.18) Alternatively, this result can be obtained from eq. (5.10) by putting ξ = 0 (since thesymmetry is not an R-symmetry here), and by using Tr (cid:2) R χ (cid:3) = q , Tr (cid:2) R λ (cid:3) = Tr h R / i = 0,and by identifying b C = 12 π β R . In fact, this fixes the sign of cq/β R in eq. (5.14), which is indeed the relevant quantity.We however assume below that c > The constraint that the mass of z remains non-tachyonic, m / ( σ s + 1) > κ − / m / / R ( α ) , (5.19)forces the mass of the gravitino to exceed the Planck scale m / > R ( α ) σ s + 1 κ − ≈ . κ − . (5.20)One concludes that the model with p = 2 can not be consistently made gaugeinvariant by the introduction of an extra field charged under U (1) R . The aboveargument can easily be generalized to include several charged fields (unless theirnumber becomes extremely large). Since the case p = 1 is consistent with aconstant contribution to the gauge kinetic function, this problem does not occurhere, and this model can be used to calculate the low energy spectrum of themodel in chapter 6. In this section we comment on the relation between the quantum anomalies andthe gaugino mass terms that are generated at one loop as a consequence of thenecessary counterterms. Although this work, based on [57], will turn out veryuseful when calculating the gaugino masses of our model in chapter 6, it is alsoa very relevant result, independently of the rest of the thesis. We explain howat one loop there are two contributions to the gaugino mass parameters. Onthe one hand, one has a contribution necessary to maintain invariance undersupersymmetry transformations, accompanying the Green-Schwarz terms thatare present to cancel the mixed U (1) R anomalies with the Standard Model gaugegroups. On the other hand, there are contributions to the mixed anomaliesdue to a mechanism called anomaly mediation [79–83]. The total one-loopcontribution to the gaugino masses is the sum of both contributions.In a similar way as was done for the cubic U (1) R anomaly and its correspondingGreen-Schwarz counterterm in the previous section, one also has to take intoaccount the mixed U (1) R × G anomalies, where G stands for the StandardModel gauge groups. The anomalous contributions to the gauge variation ofthe Lagrangian are proportional to C A , C A δ ab = Tr (cid:2) R χ ( τ a τ b ) A (cid:3) + T G δ ab R λ , (5.21)where as above A = Y, , U (1) R , and postpone the discussion on the case where the MSSM superfields NOMALIES AND GAUGINO MASSES 65 carry a non-zero charge to chapter 7. In this case, in the Kähler frame with aconstant superpotential all charges vanish and all C A = 0. In the Kähler framewith an exponential superpotential (4.13) however, the R-charge of the matterfermions is R χ = bc/
2, while the gauginos carry a charge R λ = − bc/
2, suchthat eq. (5.21) can be rewritten as C A = bc T R − T G ) . (5.22)As a reminder, T G is the Dynkin index of the adjoint representation, normalizedto N for SU ( N ) and 0 for U (1), and T R is the Dynkin index associated withthe representation R of dimension d R , equal to 1 / SU ( N ) fundamental.For U (1) Y we have T G = 0 and T R = 11, for SU (2) we have T G = 2 and T R = 7,and for SU (3) we have T G = 3 and T R = 6.Anomaly cancellation (see section 5.1) then requires that β A = C A π c , (5.23)where β A appear in the gauge kinetic terms for the Standard Model gaugebosons as f A ( s ) = 1 g A + β A s, and g A are the gauge couplings. Since the gaugino masses are proportionalto derivatives of their respective gauge kinetic functions (see eq. (3.12)), theGreen-Schwarz counterterms lead to contributions to the gaugino mass terms M A = − g A e κ K/ ∂ α f A ( s ) g α ¯ β ¯ ∇ ¯ β ¯ W = − g A e κ K/ β A g s ¯ s ¯ ∇ ¯ s ¯ W . (5.24)By using the anomaly cancellation conditions (5.23), this can be rewritten as M A = − g A π b ( T G − T R ) e κ K / g s ¯ s ¯ ∇ ¯ s ¯ W . (5.25)where it is taken into account that the masses of the MSSM gauginos calculatedby (3.12) need a rescaling proportional to g A due to their non-canonical kineticterms: L /e = −
12 Re( f ) A ¯ λ A (cid:26)(cid:26) Dλ A = − (cid:18) g A + β A αb (cid:19) ¯ λ A (cid:26)(cid:26) Dλ A , (5.26) where β A αb << g − A if the gauge coupling is in the perturbative region. Oneconcludes that an ’anomalous’ U (1) R can give a contribution (at one loop) tothe mass of the gauginos.It is assumed that the Standard Model superfields are inert under U (1) R . Inthis case, their only contribution to the anomaly cancellation conditions comesfrom the respective R-charges of the fermions in the multiplets. In the Kählerrepresentation with a constant superpotential eqs. (4.15) the shift symmetryis not an R-symmetry. As a result, β A = 0 and the gaugino masses vanish.On the other hand, in the representation with an exponential superpotentialeqs. (4.13) the shift symmetry is an R-symmetry. In general, the R-charges ofthe Standard Model fermions contribute to the anomaly cancellation conditionswhich results in a non-zero β A , and consequently non-vanishing gaugino masses.It is curious that the gaugino masses vanish for the model (4.15), while theclassically equivalent model (4.13) which is related by a Kähler transformationhas non-zero gaugino masses. This creates a puzzle on the quantum equivalenceof these models. The answer to this puzzle is based on the fact that gauginomasses are present in both representations and are generated at one loop levelby an effect called anomaly mediation [79–83]. Indeed, it has been arguedthat gaugino masses receive a one-loop contribution due to the super-Weyl-Kähler and sigma-model anomalies . These contributions are different in bothrepresentations, and we will show below that the difference accounts exactly forthe contributions (5.26).The contribution from anomaly mediation to the gaugino masses is given by [83] g π (cid:20) (3 T G − T R ) m / + ( T G − T R ) K α F α + 2 T R d R (log det K| R ) ,α F α (cid:21) , (5.27)where an implicit sum over all matter representations R is understood. Theexpectation value of the auxiliary field F α , evaluated in the Einstein frame isgiven by F α = − e κ K / g α ¯ β ¯ ∇ ¯ β ¯ W . (5.28)In eq. (5.27), the first term proportional to the one-loop beta function coefficient3 T G − T R is always present in a gravity mediated supersymmetry breakingscenario, and is in fact the main ingredient of anomaly mediated supersymmetrybreaking scenarios [79–83]. The second term is only present when a hiddensector field acquires a Planck scale VEV. In the third term, K R is the Kähler This can also be seen in [114]: since a Kähler transformation is anomalous, there are ingeneral additional contributions to the effective action at the quantum level.
NOMALIES AND GAUGINO MASSES 67 metric restricted to the representation R , and (log det K| R ) ,α therefore vanishesfor any MSSM-like fields with a canonical Kähler potential.Since the Kähler potential in eqs. (4.13) and (4.15) differ by a linear term b ( s + ¯ s ), the contribution of the second term in eq. (5.27) differs by a factor δM A = − g A π ( T G − T R ) be κ K / g α ¯ β ¯ ∇ ¯ β ¯ W , (5.29)which exactly coincides with eq. (5.25).We conclude that even though the models (4.13) and (4.15) differ by a (classical)Kähler transformation, they generate the same gaugino masses at one loopgiven by the sum of the Green-Schwarz contribution eq. (5.25) and the anomalymediated contribution eq. (5.27). While the one-loop gaugino masses for themodel (4.15) are generated entirely by eq. (5.25), the gaugino masses for themodel (4.13) which is related by a Kähler transformation have a contributionfrom eq. (5.27) as well as from a field dependent gauge kinetic term whosepresence is necessary to cancel the mixed U (1) R × G anomalies. This is due tothe fact that the extra U (1) has become an R-symmetry giving an R-charge toall fermions in the theory. hapter 6 Soft terms andphenomenology
In chapter 4, a model was introduced based on a gauged shift symmetry of thedilaton that allows for a tunably small and positive value for the cosmologicalconstant. Moreover, in section 4.3, we made contact between this model and apossible UV-completion in the context of string theory for a particular valueof the parameters. By studying the quantum consistency of such models inchapter 5, we showed that for p = 2 the gravitino mass becomes unacceptablylarge (since it exceeds the Planck scale) due to the anomaly cancellationconditions. For p = 1 however, a value of the gravitino mass of the TeVscale is allowed, and we therefore continue with p = 1.In this chapter, we calculate the low energy phenomenology and look for possibleexperimental signatures of this model. We assume that the MSSM fields areinert under the shift symmetry, and postpone a discussion on a charged MSSMto chapter 7. The model which was introduced in chapter 4 will be used as ahidden sector for supersymmetry breaking, where the breaking of supersymmetryis communicated to the visible sector (MSSM) via gravity mediation. In thesimplest case however the resulting scalar masses are tachyonic. We show thatthis can be avoided, without modifying the main properties of the model, byintroducing either a new ‘hidden sector’ field participating in the supersymmetrybreaking, similar to the Polónyi field [115], or by introducing dilaton dependentmatter kinetic terms for the MSSM fields. In both cases an extra parameteris introduced with a narrow range of values in order to satisfy all requiredconstraints. All scalar soft masses and trilinear A-terms are generated at thetree-level and are universal under the assumption that matter kinetic terms
690 SOFT TERMS AND PHENOMENOLOGY are independent of the ‘Polónyi’ field, while gaugino masses are generated atthe quantum level, via the so-called anomaly mediation contributions, and arenaturally suppressed compared to the scalar masses.It follows that the low energy spectrum is very particular and can bedistinguished from other models of supersymmetry breaking and mediation,such as mSUGRA and mAMSB. It consists of light neutralinos, charginos andgluinos, where the experimental bounds on the (mostly bino-like) LSP and thegluino force the gravitino mass to be above 15 TeV and the squarks to be veryheavy (above 13 TeV), with the exception of the stop squark which can be aslight as 2 TeV.This chapter is organized as follows: In section 6.1 we show that the simplestmodel with p = 1 results in negative scalar soft masses squared. A first solutionto this problem, based on an extra Polónyi-like field is presented in section 6.2.The resulting low energy spectrum is then calculated in section 6.3 for variousvalues of the extra parameter γ and it is shown that the spectrum can bedistinguished from other minimal models of supersymmetry breaking. Finally,in section 6.4 another possible solution to the tachyonic masses is presented.However, since the low energy spectrum is expected to be very similar, we donot elaborate on its low energy phenomenology. This chapter is based on worktogether with I. Antoniadis [57]. The model with p = 1 was introduced in section 4.2.1. There it was shown thatthe relations between the parameters a, b, c in eqs. (4.20) ensure an infinitesimallysmall and positive cosmological constant, while the gravitino mass, given byeq. (4.22), is separately tunable.If one now adds an MSSM-like field ϕ with a canonical Kähler potential, andinvariant under U (1) R , K = − κ − log( s + ¯ s ) + κ − b ( s + ¯ s ) + X ϕ ¯ ϕ,W = κ − a + W MSSM , (6.1)where W MSSM is the MSSM superpotential defined in eq. (2.13). The softscalar mass squared at h ϕ i = h ¯ ϕ i = 0 is negative, given by ∂ ϕ ∂ ¯ ϕ V| h ϕ i =0 = | a | b e α α ( h σ s i + 1) < , (6.2) N EXTENDED MODEL WITH EXTRA POLÓNYI-LIKE FIELD 71 where σ s is defined in eq. (4.17), α ≈ − .
233 and b is assumed to be negativewhich is necessary to allow for a de Sitter minimum. Since h σ s i ≈ − .
48, anynon-zero solutions h ϕ i 6 = 0 of ∂ ϕ V = 0 would mean that the field ϕ contributesin general to the supersymmetry breaking. We conclude that the model onits own can not be consistently extended to include the MSSM with canonicalkinetic terms. To circumvent this problem, one can add an extra hidden sectorfield which contributes to (F-term) supersymmetry breaking.This will be worked out in full detail in the following sections. However, we willshow in section 6.4 that the above problem of tachyonic soft masses can also besolved if one allows for a non-canonical Kähler potential in the visible sector,which gives an additional contribution to the masses through the D-term. As described above, the model (with p = 1 and a field independent gaugekinetic function) presented there would give a tachyonic mass to any MSSM-likefields (that are invariant under the shift symmetry and have a canonical Kählerpotential). In this section we add an extra hidden sector field z (similar to theso-called Polónyi field [115]) to circumvent this problem. Note that this choiceis not unique and that the problem can also be circumvented by allowing anon-canonical Kähler potential for the MSSM fields (see section 6.4).The Kähler potential, superpotential and gauge kinetic function are given by K = − κ − log( s + ¯ s ) + κ − b ( s + ¯ s ) + z ¯ z,W = κ − a (1 + γκz ) ,f ( s ) = 1 , (6.3)with γ an additional constant parameter. The scalar potential is V = V F + V D , V F = κ − | a | e b ( s +¯ s )+ κ z ¯ z s + ¯ s ( σ s A ( z, ¯ z ) + B ( z, ¯ z )) , V D = κ − c (cid:18) b − s + ¯ s (cid:19) , (6.4) where A ( z, ¯ z ) = | γκz | ,B ( z, ¯ z ) = (cid:12)(cid:12) γ + κ ¯ z + γκ z ¯ z (cid:12)(cid:12) . (6.5)We focus on real z = ¯ z = κ − t (we will confirm below that the imaginary partof the VEV of z indeed vanishes): A ( t ) = (1 + γt ) ,B ( t ) = ( γ + t + γt ) ; (6.6) ∂ t V = 0 then gives0 = γ ( σ s + 1) + ( σ s + 1 + γ ( σ s + 2)) t + γ (2 σ s + 5) t + (1 + γ ( σ s + 4)) t + 2 γt + γ ,σ s = − α − , α = b h s + ¯ s i . (6.7)As before, ∂ s V = 0 implies c a = αb e α + t (cid:2) A ( t )(2 − α ) − B ( t ) (cid:3) . (6.8)This can be combined with V = 0 c a = − αb e α + t (cid:20) σ s A ( t ) + B ( t )( α − (cid:21) , (6.9)to give0 = A ( t ) (cid:18) σ s −
12 ( α − ( α − (cid:19) + B ( t ) (cid:18) −
12 ( α − (cid:19) . (6.10)In principle for any value of γ , a Minkowski minimum can be found by solvingeqs. (6.7) and (6.10) for α and t , and then tuning the parameters a , b and c byusing the relation (6.9).The role of the extra hidden sector field z is to give a (positive) F-termcontribution to the scalar potential, which in turn gives a positive contribution(proportional to |∇ z W | ) to the soft mass squared of any MSSM-like field ineq. (6.2). It turns out that the addition of the extra hidden sector field z indeedresults in positive soft masses squared. N EXTENDED MODEL WITH EXTRA POLÓNYI-LIKE FIELD 73
It is however necessary that z contributes to the supersymmetry breaking. Theexistence of any minimum of the potential with |∇ z W | = 0 can be troublesome and we therefore require ∇ z W = ∂ z W + κ K z W = a ( γ + ¯ z (1 + γz )) = 0 . (6.11)Since γ is real, any root of ∇ z W = 0 is also real. To ensure the condition (6.11)we must ensure that the roots Re( z ) = ( − ± √ − γ ) / γ are complex. Thisrequires | γ | > / γ the solution ( α, t ) of the set of equations (6.7) and (6.10) shouldgive a positive right-hand side of eq. (6.8) (or equivalently, eq. (6.9)). Thisconstraint leads to γ < . γ ∈ [0 . , . . (6.12)For example, for γ = 1, we have b h s + ¯ s i = α ≈ − . h t i = 0 . b can be chosen freely to fix the VEV of Re( s ). Theparameters a and c should be tuned carefully according to bc a = − αe α + t (cid:20) σ s A ( t ) + B ( t )( α − (cid:21) ≈ − . . (6.13)Note that the number on the right-hand side changes when γ is varied. Theremaining free parameter a can be used to tune the supersymmetry breakingscale and (as shown below) the soft masses for the MSSM-like fields comparedto the gravitino mass depend slightly on γ (provided c and a are also tunedaccording to eq. (6.8)). We summarize the VEVs of α and t , together with theabove constraint on the parameters for the particular choice γ = 1 below forfuture reference γ = 1 , α ≈ − . , h t i ≈ . , bc a ≈ − . . (6.14)For γ in the allowed parameter range (6.12), the scalar potential is positivedefinite for all Re( s ) > , z, ¯ z , including the imaginary part of z , which justifiesour assumption to look for a Minkowski minimum with Im( z ) = 0. In fact, forthe allowed values of γ , the solution of the set of equations (6.7) and (6.10)together with ∂ Im( z ) V = 0 gives Im( z ) = 0 as a solution. The Polónyi-like field z was originally introduced to circumvent the problem of thetachyonic soft scalar masses outlined in section 6.1. Its role is to give an extra contribution tothe scalar potential, proportional to |∇ z W | , such that the scalar soft masses squared (seeeq. (6.2)) become positive. The existence of a minimum of the potential with |∇ z W | = 0can therefore be troublesome since in this vacuum one has tachyonic soft scalar masses forany MSSM-like fields. Finally, note that this Minkowski minimum can be lifted to a dS vacuum withan infinitesimally small cosmological constant by a small increase in c . Acosmological constant Λ can be obtained by replacing the condition (6.13) with c a = − αb e α + t (cid:20) σ s A ( t ) + B ( t )( α − (cid:21) + 2 α ( α − κ Λ a b . (6.15) The gravitino mass is given by m / = κ − a r bα e α/ t / (1 + γt ) . (6.16)Note that this can be arranged to be at the TeV scale (to obtain a viable lowenergy sparticle spectrum) by suitably tuning a . For example, for γ = 1, suchthat α and t are given by eq. (6.14) and m / = 1 TeV, we have a √ b ≈ . × − . (6.17)Since the VEV of Im( z ) vanishes, it does not mix with the other hidden sectorscalars and its mass is given by m z ) = m / f Im( z ) , (6.18) f Im( z ) = 2 (cid:0) t γ + t γ + σ s + 2 tγ (2 + σ s ) + γ (3 + σ s ) + t (cid:0) γ (4 + σ s ) (cid:1)(cid:1) (1 + γt ) . However, the masses of the scalars Re( s ) and Re( z ) mix, so one shoulddiagonalize their mass matrix (with eigenvalues m ts and m ts ) while taking inaccount the non-canonical kinetic term for s . We omit the details and merelystate the result for the particular choice of parameters γ = 1 in eq. (6.14): m Im( z ) ≈ . m / ,m ts ≈ . m / ,m ts ≈ . m / . (6.19) N EXTENDED MODEL WITH EXTRA POLÓNYI-LIKE FIELD 75
The imaginary part of s is eaten by the U (1) gauge boson, which becomesmassive. Its mass is given by : m A µ = κ − √ bcα ≈ . m / , (6.20)where the last line was obtained by the relation between the parameters eq. (6.13)and by substituting the numerical values for γ = 1 eq. (6.14).The Goldstino, which is a linear combination of the gaugino, the z-fermion andthe s-fermion, is eaten by the gravitino, which in turn becomes massive. Themasses of the remaining two hidden sector fermions are calculated in AppendixA.1.3 and their values for γ = 1 are given by m χ ≈ . m / ,m χ ≈ . m / . (6.21) We are now ready to couple the model above, which allows for a TeV gravitinoand an infinitesimally small cosmological constant, to the MSSM and to calculateits soft breaking terms.As was already mentioned, for simplicity, we take the MSSM-like fields ϕ α tobe chargeless under U (1) R . These can then be coupled to the above model inthe following way: K = − κ − log( s + ¯ s ) + κ − b ( s + ¯ s ) + z ¯ z + X α ϕ ¯ ϕ,W = κ − a (1 + γκz ) + W MSSM ( ϕ ) ,f R ( s ) = 1 , f A ( s ) = 1 /g A . (6.22)The various chiral multiplets in the MSSM are labeled by an index α , which isomitted for simplicity. As before, the Standard Model gauge groups are labeledby an index A, while the extra U (1) is referred to with an index R . All gaugekinetic functions are taken to be constant. The result (6.20) is obtained by substituting h s + ¯ s i = α/b in eq. (A.16) and by putting p = 1. For more details, see the Appendix A.2. For a charged MSSM, see section 7.1
The scalar potential is now given by V = V F + V D , V F = κ − e b ( s +¯ s )+ z ¯ z + ϕ ¯ ϕ s + ¯ s σ s A ( z, ¯ z, ϕ, ¯ ϕ ) + B ( z, ¯ z, ϕ, ¯ ϕ ) + κ X α |∇ α W | ! , V D = κ − c (cid:18) b − s + ¯ s (cid:19) , (6.23)where A ( z, ¯ z, ϕ, ¯ ϕ ) = (cid:12)(cid:12) a + aγκz + κ W MSSM (cid:12)(cid:12) B ( z, ¯ z, ϕ, ¯ ϕ ) = (cid:12)(cid:12) aγ + κ ¯ z ( a + aγz + κ W MSSM ) (cid:12)(cid:12) |∇ α W | = (cid:12)(cid:12) ∂ α W MSSM + κ ¯ ϕW (cid:12)(cid:12) . (6.24)It can be easily seen that the resulting scalar potential has a minimum at h ϕ i = h W MSSM i = 0, in which case the potential of last section is reproduced andits conclusions are still valid. For example, A ( z, ¯ z, ϕ, ¯ ϕ ) | h z i = κt, h ϕ i =0 = a A ( t )and B ( z, ¯ z, ϕ, ¯ ϕ ) | h z i = κt, h ϕ i =0 = a B ( t ), where A ( t ) and B ( t ) are defined ineqs. (6.6). The second derivatives of the potential, evaluated on the groundstate are given by ∂ ϕ ∂ ¯ ϕ V = κ − a be α + t α (cid:2) ( σ s + 1) A ( t ) + B ( t ) + κ W ϕϕ ¯ W ¯ ϕ ¯ ϕ (cid:3) ,∂ ϕ ∂ ϕ V = κ − abW ϕϕ e α + t α (cid:2) ( σ s + 2)(1 + γt ) + t ( γ + t + γt ) (cid:3) . (6.25)There is no mass mixing between the different ϕ α (except of course for the B term defined below) and between the MSSM fields with z and s . The MSSMsuperpotential is defined in eq. (2.13). Note that in the scalar potential eq. (6.23)the MSSM F-terms P α |∇ α W | come with a prefactor exp( α + t ) b/α (wherethe fields have been replaced by their VEVs). To bring this into a conventionalform, one should rescale the MSSM superpotential (see also eq. (3.25))ˆ W MSSM = r bα e α/ t / W MSSM . (6.26)Then the squark and slepton soft masses are given by m Q = m u = m d = m L = m e = m I ,m = κ − ba e α + t α [ A ( t ) ( σ s + 1) + B ( t )] . (6.27) N EXTENDED MODEL WITH EXTRA POLÓNYI-LIKE FIELD 77
Here, I is the unit matrix in family space. The trilinear couplings are given by a u = A ˆ y u , a d = A ˆ y d , a e = A ˆ y e ,A = κ − a r bα e ( α + t ) / (cid:2) ( σ s + 3)(1 + γt ) + t ( γ + t + γt ) (cid:3) , (6.28)where ˆ y u , ˆ y d and ˆ y e are the Yukawa couplings of the MSSM superpotential afterthe rescaling of eq. (6.26). Also, m H u = m H d = m , (6.29)and B = κ − a r bα e ( α + t ) / (cid:2) ( σ s + 2)(1 + γt ) + t ( γ + t + γt ) (cid:3) , (6.30)where B generates a term proportional to − ˆ µB H u · H d + h.c., where ˆ µ is therescaled µ -parameter (in the sense of eq. (6.26)). Summarized, in terms of thegravitino mass (eq. (6.16)), the MSSM soft terms are given by m = m / (cid:20) ( σ s + 1) + ( γ + t + γt t ) (1 + γt ) (cid:21) ,A = m / (cid:20) ( σ s + 3) + t ( γ + t + γt )1 + γt (cid:21) ,B = m / (cid:20) ( σ s + 2) + t ( γ + t + γt )(1 + γt ) (cid:21) . (6.31)Note the relation [84] A = B + m / , as in eq. (3.28).At tree level, the gaugino mass terms are given by eq. (3.12), which isproportional to the derivative of the gauge kinetic function. Since the gaugekinetic functions are constant, they vanish at tree-level m AB | tree = 0 . (6.32)At one loop however, the ’anomaly mediated’ gaugino mass contribution, givenby eq. (5.27), becomes M A = − g π m / (cid:20) (3 T G − T R ) − ( T G − T R ) (cid:18) ( α − + t γ + t + γt γt (cid:19)(cid:21) , where the relation for the gravitino mass eq. (6.16) was used. The differentgaugino mass parameters this gives (in a self-explanatory notation) are M = 11 g Y π m / (cid:20) − ( α − − t ( γ + t + γt )1 + γt (cid:21) ,M = g π m / (cid:20) − α − − t ( γ + t + γt )1 + γt (cid:21) ,M = − g π m / (cid:20) α − + t ( γ + t + γt )1 + γt (cid:21) . (6.33)For example, if we choose γ = 1 (as in eq. (6.14)) the above equations give M ≈ . g Y m / ,M ≈ . g m / ,M ≈ . g m / . (6.34)If we now assume that the gauge couplings unify at some unification scale g Y ≡ g = g = g = 0 .
51, we get the gaugino masses at this scale M ≈ . m / ,M ≈ . m / ,M ≈ . m / . (6.35)The gaugino masses for other values of γ are listed in table 6.1 below.Note that in a similar way, the trilinear terms A also receive correctionsproportional to δA ijk = −
12 ( γ i + γ j + γ k ) m / , (6.36) The gaugino masses eqs. (6.33) can equivalently be calculated in the Kähler representationof the model where the superpotential is an exponential, as in eqs. (4.13). Since in thisrepresentation the shift symmetry becomes an R-symmetry, all fermions in the model (includingStandard Model fermions) carry an R-charge and one has to take into account the anomalycancellation conditions of chapter 5. A Green-Schwarz mechanism is necessary to cancelanomalies, which requires a linear contribution to the gauge kinetic functions. This field-dependent gauge kinetic function results in a contribution to the gaugino mass parameters,given by [57]ˆ M = 1116 π bg Y e α/ ( α − , ˆ M = 516 π bg e α/ ( α − , ˆ M = 316 π bg e α/ ( α − . The total gaugino mass parameters are then given by the sum of the above contribution andeq. (5.27), which give the same result given in eq. (6.33). It was shown in section 5.3 thatthis equivalence is true in a more general case.
OW ENERGY SPECTRUM 79 where the γ ’s are the anomalous dimensions of the corresponding cubic termin the superpotential. These contributions however are small compared to thetree-level value in eq. (6.31).Although the gaugino masses are generated at one-loop, our model is verydifferent from an mAMSB [79–83] scenario: In mAMSB, the second and thirdterm in eq. (5.27) are missing due to the absence of hidden sector fields with aPlanck scale VEV. In our model however, the second term in eq. (5.27) is presentbecause of the non-vanishing F-terms of the s and z fields, and has the effectthat it raises the gaugino masses slightly to the order M / ≈ × − m / compared to M / ≈ − − − m / for a mAMSB where only the firstterm in eq. (5.27) is non-vanishing. Another important difference is that wehave M < M which results in a mostly bino-like LSP, compared with amostly wino-like LSP in mAMSB. Note also that we do not have any danger oftachyonic scalar soft masses because of the presence of a tree-level soft mass m in eqs. (6.31). We also have tree-level trilinear couplings A , which are notpresent in the mAMSB.Our model is also different from the minimal supergravity mediated scenario(mSUGRA) [77]. Indeed, in mSUGRA gaugino masses are imposed to be equalat tree-level at the GUT unification scale M : M : M = g : g : g of the order m (plus or minus an order of magnitude), while our model hasvanishing tree-level gaugino masses. They are generated at one-loop and do notsatisfy the above relation. Since the gaugino masses are generated at one-loopthey are much smaller than the other soft terms.We conclude that although the soft terms m , A and B = A − m / aresimilar to an mSUGRA scenario, the anomaly mediated gaugino masses (whichhave on top of the usual AMSB contribution proportional to the beta functionanother contribution from the Planck scale VEVs of s and z ) are not universaland are much smaller. Therefore, the particle spectrum will resemble muchmore the spectrum of a mAMSB scenario, with the important difference thatthe lightest neutralino is bino-like instead of wino-like (See section 6.3). The results for the soft terms calculated in the previous section, evaluated fordifferent values of the parameter γ , are summarized in table 6.1. For every γ ,the corresponding t and α are calculated by imposing a vanishing cosmologicalconstant by eqs. (6.8) and (6.9). The scalar soft masses and trilinear terms arethen evaluated by eqs. (6.31) and the gaugino masses by eqs. (6.33). Note that the relation (3.28), namely A = B − m / , is valid for all γ . We therefore donot list the parameter B . γ t α m A M M M tan β tan βµ > µ < m / ) for various values of γ . If a solutionto the RGE exists, the value of tan β is shown in the last columns for µ > µ < B is substituted for tan β , the ratio betweenthe two Higgs VEVs, as an input parameter for the RGEs that determine thelow energy spectrum of the theory. Since B is not a free parameter in ourtheory, but is fixed by eq. (3.28), this corresponds to a definite value of tan β .For more details see [116,117] (and references therein). The corresponding tan β for a few particular choices for γ are listed in the last two columns of table 6.1for µ > µ < γ (cid:46) .
1, for bothsigns of µ .Some characteristic masses [118] for γ = 1 . m / (cid:38)
15 TeV. On the other hand,for µ > µ <
0) no viable solution for the RGE was found when m / (cid:38) m / (cid:38)
35 TeV). We conclude that (for γ = 1 . (cid:46) m / (cid:46)
30 TeV for µ > ,
15 TeV (cid:46) m / (cid:46)
35 TeV for µ < . (6.37)As we will see below, these upper bounds can differ for different choices of γ .In figure 6.2, the same spectrum is plotted as a function of γ for m / = 25 TeV.As one can see, the stop mass varies heavily with γ , and can become relativelylight when γ ≈ .
1. For all values of γ the LSP is given by the lightest neutralinoand since M < M (see table 6.1) the lightest neutralino is mostly bino-like,in contrast with a typical mAMSB scenario, where the lightest neutralino ismostly wino-like [82].To get a lower bound on the stop mass, the sparticle spectrum is plotted infigure 6.3 (left) as a function of the gravitino mass for γ = 1 . µ > OW ENERGY SPECTRUM 81
15 20 25 30 m32 (cid:72)
TeV (cid:76) µ >
15 20 25 30 35 m32 (cid:72)
TeV (cid:76) µ < γ = 1 . µ > µ < m / = 10 TeV and 138 GeV (149 GeV) for m / = 30 TeV for µ > µ < m / ≈
15 TeVindicates the exclusion limit (lower bound) on the gluino mass. Γ µ > Γ µ < γ for m / = 25 TeV and for µ > µ < γ < .
1. Notice thatfor γ → . µ < m / (cid:38)
15 TeV. In this limit the stop mass can be as low as 2 TeV.To obtain an upper bound on the stop mass on the other hand, the sparticlespectrum is plotted in figure 6.3 (right) for γ = 1 . µ >
0. Above a gravitinomass of (approximately) 30 TeV, no solutions to the RGE were found. In this limit the stop mass is about 15 TeV.
20 25 30 35 40 45 m32 (cid:72)
TeV (cid:76) γ = 1 . µ >
16 18 20 22 24 26 28 30 m32 (cid:72)
TeV (cid:76) γ = 1 . µ > m / for γ = 1 . γ = 1 . µ > γ = 1 . m / (cid:38)
45 TeV,while for γ = 1 . m / (cid:38)
30 TeV. Thelower bound corresponds in both cases to a gluino mass of 1 TeV.To conclude, the lower end mass spectrum consists of (very) light charginos(with a lightest chargino between 250 and 800 GeV) and neutralinos, with amostly bino-like neutralino as LSP (80 −
230 GeV), which would distinguish thismodel from the mAMSB where the LSP is mostly wino-like. These upper limitson the LSP and the lightest chargino imply that this model could in principle beexcluded in the next LHC run. In order for the gluino to escape experimentalbounds, the lower limit on the gravitino mass is about 15 TeV. The gluino massis then between 1-3 TeV. This however forces the squark masses to be very high(10 −
35 TeV), with the exception of the stop mass which can be relatively light(2 −
15 TeV).
As was shown in section 6.1, the model (4.16) has tachyonic soft scalar massesfor the MSSM fields. In section 6.2 we proposed a solution by adding an extrafield to the hidden sector. However, we will show in this section that the problemcan also be circumvented by allowing non-canonical kinetic terms for the MSSMfields.
ON-CANONICAL KINETIC TERMS 83
We consider the following model K = − κ − log( s + ¯ s ) + κ − b ( s + ¯ s ) + ( s + ¯ s ) − ν X ϕ ¯ ϕ,W = κ − a + W MSSM ,f ( s ) = 1 , f A ( s ) = 1 /g A , (6.38)where a sum over all visible sector fields ϕ is understood in the Kähler potential.Here, ν is considered to be an additional parameter in the theory, where ν = 1corresponds with the leading term in the Taylor expansion of − log( s + ¯ s − ϕ ¯ ϕ )and the model resembles a no-scale model [119–121]. The gauge kinetic functionsfor the Standard Model gauge groups f A ( s ) are taken to be constant.The scalar potential is given by V = V F + V D , V F = κ − e b ( s +¯ s )+ P κ ( s +¯ s ) − ν ϕ ¯ ϕ s + ¯ s − W ¯ W + g s ¯ s |∇ s W | + X ϕ ( s + ¯ s ) ν |∇ ϕ W | ! , V D = c (cid:18) κ − b − κ − s + ¯ s − ν ( s + ¯ s ) − ν − X ϕ ¯ ϕ (cid:19) . (6.39)Since the visible sector fields appear only in the combination ϕ ¯ ϕ , their VEVsvanish provided that the scalar soft masses squared are positive. Moreover, forvanishing visible sector VEVs, the scalar potential reduces to eq. (4.17) andthe non-canonical Kähler potential for the visible sector fields does not changethe discussion on the minimization of the potential in section 4.2.1. Therefore,the non-canonical Kähler potential does not change the fact that the F-termcontribution to the soft scalar masses squared is negative. One has as in eq. (6.2) ∂ V F ∂ϕ∂ ¯ ϕ (cid:12)(cid:12)(cid:12)(cid:12) h ϕ i =0 = κ − a e α (cid:18) bα (cid:19) ν +1 ( h σ s i + 1) < . (6.40)However, the visible fields will enter in the D-term scalar potential through thederivative of the Kähler potential with respect to s . Even though this has noeffect on the ground state of the potential, the ϕ -dependence of the D-termscalar potential does result in an extra contribution to the scalar masses squared ∂ V D ∂ϕ∂ ¯ ϕ (cid:12)(cid:12)(cid:12)(cid:12) h ϕ i =0 = νκ − c (cid:18) bα (cid:19) ν +2 (1 − α ) . (6.41) The total soft mass squared is then the sum of these two contributions m = κ − a (cid:18) bα (cid:19) (cid:18) e α ( σ s + 1) + ν A ( α ) α (1 − α ) (cid:19) , (6.42)where eq. (4.20) has been used to relate the constants a and c , and correctionsdue to a small cosmological constant have been neglected. A field redefinitiondue to a non-canonical kinetic term g ϕ ¯ ϕ = ( s + ¯ s ) − ν is taken into account. Thesoft mass squared is now positive if ν > − e α ( σ s + 1) αA ( α )(1 − α ) ≈ . . (6.43)The gravitino mass is given by eq. (4.22). In the hidden sector, the imaginarypart of s is eaten by the gauge boson corresponding to the shift symmetry,which becomes massive (similar to eq. (A.16)) m A µ = κ − bcα ≈ . m / . (6.44)The mass of the real part of s squared is given by 2( α/b ) ∂ s ∂ s V evaluated atthe ground state, where the factor 2( α/b ) comes from the non-canonical kineticterm, m s = 2 (cid:18) α − α + 4 α + e − α (3 − α ) Aα − (cid:19) m / ≈ . m / . (6.45)Finally, the Goldstino is given by a linear combination of the fermionicsuperpartner of s and the gaugino, which is eaten by the gravitino by theBEH mechanism. The mass of the remaining fermion is given by (see AppendixA.1.1) m f ≈ . m / . (6.46)Note that in the scalar potential eq. (6.39) the MSSM F-terms P ϕ |∇ ϕ W | comewith a prefactor e κ K g ϕ ¯ ϕ (where the hidden fields are replaced by their VEVs).To bring this into a more recognizable (globally supersymmetric) form where L ∼ − g ϕ ¯ ϕ ∂ µ ϕ∂ µ ¯ ϕ − g ϕ ¯ ϕ W ϕ ¯ W ¯ ϕ , one should rescale the MSSM superpotential(defined in eq. (2.13)) ˆ W MSSM = exp( α ) ( b/α ) W MSSM . (6.47) ON-CANONICAL KINETIC TERMS 85
However, another rescaling is needed to take into account the non-canonicalKähler potential for the visible sector . The trilinear couplings are given by A = m / ( s + ¯ s ) ν/ ( σ s + 3) , (6.50)and B = m / ( s + ¯ s ) ν/ ( σ s + 2) . (6.51)The main phenomenological properties of this model are not expected to bedifferent from the one we analyzed in section 6.3 with the parameter ν replacing γ . Gaugino masses are again generated at one-loop level while mSUGRA appliesto the soft scalar sector. We therefore do not repeat the phenomenologicalanalysis for this model. After the rescaling (6.47), the Lagrangian contains (very schematically) the followingterms L = − ( s + ¯ s ) − ν ∂ µ ¯ ϕ∂ µ ϕ − ( s + ¯ s ) − ν ∂ µ ¯ h∂ µ h + ˆ µ ¯ hh + ˆ y ˆ µ ¯ hϕϕ + ˆ y ¯ ϕ ¯ ϕϕϕ + . . . + 16 A ˆ yϕϕϕ + 12 B ˆ µhh, (6.48)where h stands for the Higgsinos and ϕ labels the other scalar superpartners and all indicesare suppressed for clarity. y stands for the Yukawa couplings and µ is the usual µ -parameter.The first line contains the kinetic terms and the F-terms coming from ˆ W MSSM . The last linecontains the trilinear supersymmetry breaking terms (A-terms) and the B-term. In order toobtain canonical kinetic terms, one needs a rescaling ϕ → ϕ = ( s + ¯ s ) − ν/ ϕ (and similarlyfor h ). However, to bring the MSSM superpotential back into its usual form one also needs toredefine ˆ µ → ˆ µ = ( s + ¯ s ) ν/ ˆ µ and ˆ y → ˆ y = ( s + ¯ s ) ν ˆ y . One then obtains L = − ∂ µ ¯ ϕ ∂ µ ϕ − ∂ µ ¯ h ∂ µ h + ˆ µ ¯ h h + ˆ y ˆ µ ¯ h ϕ ϕ + ˆ y ¯ ϕ ¯ ϕ ϕ ϕ + . . . + 16 ( s + ¯ s ) ν/ A ˆ y ϕ ϕ ϕ + 12 ( s + ¯ s ) ν/ B ˆ µ h h . (6.49) hapter 7 Gauging global symmetries ofthe MSSM
In the previous chapter, the soft supersymmetry breaking terms were calculatedfor the model introduced in chapter 4. It was shown that the minimal modelsuffers from tachyonic scalar soft masses, and two possible solutions to thisproblem were presented: One is based on the introduction of an extra hiddensector (Polónyi-like) field z , which introduces an extra parameter γ in the model.A second solution involves a s -dependent Kähler potential for the MSSM fieldsand an extra parameter ν . Since the low energy spectra of both cases are verysimilar, only the spectrum of the former case was calculated. While the gauginomasses vanish at tree level, they are generated at one loop due to anomalymediation, and are therefore suppressed compared to the scalar soft masses. Itwas shown that the low energy spectrum is very particular, and how it can bedistinguished from other minimal supersymmetry breaking scenarios.In the discussion in chapter 6 however, the Standard Model fields were assumedto be inert under the shift symmetry. In this chapter we study the possibilitythat the gauged shift symmetry is identified with a known global symmetry ofthe Standard Model, or more generally its supersymmetric extension. This isdone while keeping the desirable properties of the model, namely the existenceof a metastable dS vacuum with a tunable cosmological constant and a viablelow energy spectrum of the superpartners. A particular attractive possibility isto use a symmetry that contains the usual R-parity, or matter-parity (dependingon the Kähler basis) of the MSSM (see section 2.2.3). We find that this isindeed possible and analyze explicitly the anomaly free symmetry B − L . Acharge q proportional to B − L is introduced for the Standard Model fields, that
878 GAUGING GLOBAL SYMMETRIES OF THE MSSM allows to extrapolate between the current analysis and the one of chapter 6. Itturns out though that q is bounded from the requirement of existence of theelectroweak vacuum. Moreover, the spectrum is very similar to the one wherethe MSSM fields are inert (see section 6.3), with the exception that the stopsquark mass can now be as low as 1 . U (1). However, the answer turns out to be negative due toconstraints arising from the existence of the usual electroweak vacuum. Finally,we analyze the phenomenological implications of the extra U (1) and we find thatits coupling is too small to have possible experimental signatures in colliders atpresent energies.This chapter is organized as follows. In section 7.1 we analyze the possibility ofidentifying the gauged shift symmetry with the B − L . We work out the modeland its phenomenology; we also comment on the case of 3 B − L , but we do notrepeat the analysis since the results are very similar. In Section 7.2, we considerthe most general global symmetry and address the question of tachyonic scalarmasses without extra field or modification of the matter kinetic terms.This chapter is entirely based on work with I. Antoniadis [58]. As discussed above, we now explore the possibility to give the MSSM superfields(denoted by ϕ α ) a charge q α under the extra U (1) R , proportional to B − L ,extending the model (6.3). This means that Q, ¯ u and ¯ d have charges q/ , − q/ − q/
3, respectively. The Higgs superfields do not carry a charge and theleptons L and ¯ e carry a charge − q and + q respectively.First, this gives contributions to the D-term part of the scalar potential, andone should check that this does not ruin its stability. The scalar potential isnow given by V = V F + V D , V D = 12 (cid:18) − κ − cs + ¯ s + κ − bc − X q α ϕ α ¯ ϕ α (cid:19) , (7.1) HE MODEL EXTENDED WITH B-L CHARGES 89 where V F is the same as in eq. (6.23). The D-term part will give an extracontribution to the soft scalar masses of the matter fields ϕ α . The restrictionthat these remain non-tachyonic gives0 < ∂ ϕ α ∂ ¯ ϕ α V| ϕ =0 = κ − a e b ( s +¯ s )+ t s + ¯ s ( A ( t )( σ s + 1) + B ( t )) + κ − q α c (cid:18) s + ¯ s − b (cid:19) , where A ( t ) and B ( t ) are given in eqs. (6.6). Since q α can be either positive ornegative, and the first term on the r.h.s. is positive for the VEVs of t and s (see section 6.2), it follows that | q | < a c e α + t A ( t )( σ s + 1) + B ( t )1 − α , (7.2)which can be rewritten as (by use of eqs. (6.9) and (6.16)) | q | < q max = κm / A ( t )( σ s + 1) + B ( t ) p | A ( t ) σ s + B ( t ) | √ γt ) . (7.3)However, we will show below that one actually needs | q | /q max < .
013 in orderto find a viable solution to the RGE. Note that the constraint (7.3) can berewritten (by using the relation (6.9)) as | q | < q max = bc A ( t )( σ s + 1) + B ( t ) A ( t ) σ s + B ( t ) 1 − αα , (7.4)where κ − bc is the Fayet-Iliopoulos constant in the scalar potential (7.1).While the mixed U (1) R × U (1) Y , U (1) R × SU (2) and U (1) R × SU (3) anomaliesvanish, the cubic anomaly vanishes only upon the inclusion of three right-handedneutrinos which are singlets under the Standard Model gauge groups. Otherwise,the cubic anomaly is proportional toTr Q = − q , (7.5)but the mixed anomalies still vanish. In this case, the cubic anomaly should becanceled by a Green-Schwarz counterterm (4.9), provided f ( s ) = 1 + β R s,β R = − q π c . (7.6) The gaugino masses are generated at one loop from anomaly mediaton, givenby eqs. (6.33), while the other soft supersymmetry breaking terms are given by m ,i = m / (cid:20) ( σ s + 1) + ( γ + t + γt ) (1 + γt ) (cid:21) + κ − q α bc (cid:18) α − (cid:19) ,A = m / (cid:20) ( σ s + 3) + t ( γ + t + γt )1 + γt (cid:21) ,B = m / (cid:20) ( σ s + 2) + t ( γ + t + γt )(1 + γt ) (cid:21) . (7.7)Or, by using bc = m / κ p − A ( t ) σ s + B ( t ))1 + γt α − α , (7.8)the soft terms can be written as m ,i = m / (cid:20) ( σ s + 1) + ( γ + t + γt ) (1 + γt ) (cid:21) + κ − m / q α p − A ( t ) σ s + B ( t ))1 + γt ,A = m / (cid:20) ( σ s + 3) + t ( γ + t + γt )1 + γt (cid:21) ,B = m / (cid:20) ( σ s + 2) + t ( γ + t + γt )(1 + γt ) (cid:21) . (7.9)Note that the relation (3.28) still holds.In chapter 6 the special case q = 0 was analyzed in full detail; it was shown thatfor γ < . γ → . m ˜ t can become verysmall. By imposing a lower bound of about m / ≥
15 TeV on the gravitinomass, which originates from a lower bound of about 1 TeV on the gluino mass,it was shown that the mass of the lightest stop can be as low as about 2 TeV,while the masses of the other squarks remain high ( >
10 TeV).As it turns out, the only considerable effect of a non-zero charge q to thesparticle spectrum is for the lightest stop, whose dependence on the inputparameter q/q max for m / = 15 TeV and γ = 1 . q/q max > . . q/q max → . HE MODEL EXTENDED WITH B-L CHARGES 91 (cid:45) (cid:45) (cid:45) q (cid:144) q max m t (cid:72) GeV (cid:76)
Figure 7.1: The mass of the lightest stop squark as a function of the charge q/q max for γ = 1 . m / = 15 TeV. The gravitino mass is chosen such thatthe gluino mass is right above the experimental bound of 1 TeV (while otherexperimental bounds such as the neutralino and charginos are also satisfied). Apositive q corresponds to the scalar soft masses m Q and m e being heavier than m L and m d = m u (see eq. (7.9)). For q/q max > .
013 no solutions to the RGEwere found.for the MSSM fields which allow the terms in the superpotential (2.13), whileforbidding the Baryon and Lepton violating terms (2.18) and the dimension fiveoperators (2.21). As was mentioned in section 2.2.3, a gauged B − L forbidsthe terms in eq. (2.18), but it still allows certain dimension five operators. Thiscan be solved by gauging 3 B − L . A gauged 3 B − L is anomalous and its U (1) B − L × U (1) Y and U (1) B − L × SU (2) anomalies are proportional to C = − q, C = 6 q, (7.10)while the U (1) B − L × U (1) Y and U (1) B − L × SU (3) anomalies vanish. As wasexplained in section 5.3, this results in a contribution to the gaugino masseseq. (5.24) given by M = − g Y α ( α − π qbc m / ,M = g α ( α − π qbc m / . (7.11) (cid:45) (cid:45) (cid:45) q (cid:144) q max M A (cid:144) m (cid:144) Figure 7.2: The gaugino masses M /m / (blue), M /m / (red), M /m / (black) in the 3 B − L model, as a function of q/q max , where g Y = g = g = 0 . | q/q max | < . M = g Y π m / (cid:18) (cid:20) − ( α − − t ( γ + t + γt )1 + γt (cid:21) − α ( α −
1) 6 qbc (cid:19) ,M = g π m / (cid:18)(cid:20) − α − − t ( γ + t + γt )1 + γt (cid:21) + α ( α −
1) 12 qbc (cid:19) ,M = − g π m / (cid:20) α − + t ( γ + t + γt )1 + γt (cid:21) . (7.12)By using (from eqs. (7.3) and (7.8)) qbc = qq max A ( t )( σ s + 1) + B ( t ) A ( t ) σ s + B ( t ) α − α , (7.13)the corrections to the gaugino masses proportional to q/q max can be calculatedfor every γ . It turns out that these corrections are very small, as can be seenin figure 7.2, where the gaugino masses are plotted as a function of q/q max for γ = 1 .
1. The low energy spectrum is then expected to be similar to that ofthe B − L case described above and we therefore do not perform a separateanalysis for the 3 B − L case.The kinetic terms of the U (1) R gauge boson are given by L kin /e = − F µν F µν . (7.14) Alternatively, one can define the gauge kinetic function as f ( s ) = 1 /q , such that thecharge of the fermions is given by (instead of being proportional to) B − L . -TERM CONTRIBUTIONS TO THE SCALAR SOFT MASSES 93 Its mass is given by eq. (A.16), M R = κ − bcα = m / " (1 + γt ) e α t s σ s A ( t ) + B ( t )( α − . (7.15)In the allowed parameter range this corresponds to M R ∈ ]25 . , .
4[ TeV. Thecovariant derivative of the Standard Model fermions χ α (with charge q ) is D µ χ α = ( ∂ µ − iqA µ ) χ α , (7.16)where we have omitted the spin connection and the Kähler connection. Thecharge q of the MSSM fermions satisfies | q | < . q max ≈ O (10 − ). Weconclude that the U (1) R gauge boson is (unfortunately) well beyond the currentexperimental Z bounds or the corresponding compositeness limits [124]. In this section we show that another possible solution to the problem of negativesoft scalar masses squared in section 6.1, based on a D-term contribution tothe scalar soft masses, does not lead to consistent electroweak vacua. As in theother solutions proposed in chapter 6, solving the problem of tachyonic massescomes at the cost of introducing an extra parameter, b . In this case the modelis given by K = − κ − log( s + ¯ s ) + κ − b ( s + ¯ s ) + K MSSM ,W = κ − a + e b s W MSSM ,f ( s ) = 1 + βbs. (7.17)Because of the shift symmetry (4.1), the MSSM superpotential needs totransform (with gauge parameter θ ) as W MSSM −→ W MSSM e ib cθ . (7.18) For a review on Z s, see for example [122, 123]. Note that the coefficient β in the gauge kinetic function has a slightly different definitionthan in the rest of this thesis. The reason lies in the fact that this definition allows oneto eliminate the parameter b from the vacuum conditions (D.2), similar to the case with aconstant gauge kinetic function in section 4.2.1. The scalar potential is given by V = V F + V D , V F = e K (cid:2) − W ¯ W + g s ¯ s |∇ s W | + |∇ ϕ W | (cid:3) V D = 12 (cid:18) κ − bc − κ − cs + ¯ s − q α ϕ α ¯ ϕ α (cid:19) , (7.19)where ϕ α stands for the various MSSM fields and the linear part in the gaugekinetic function has been neglected. Indeed, it is shown in Appendix D.1 that β (cid:28) q α from the D-term scalar potential. This implies that all MSSM fieldsmust have a positive charge under this extra U (1), which is the motivationbehind the transformation (7.18) and the factor e b s in eq. (7.17). The softsupersymmetry breaking terms can be calculated (with respect to a rescaledsuperpotential ˆ W MSSM = e K / e b s W MSSM ) to be m = m / ( σ s + 1) + κ − q α bc − αα ,A = m / ρ s , (7.20) B = m / ( ρ s + 1) , ρ s = − α − α − b ( s + ¯ s )) , where α and σ s are defined in eqs. (6.7). The gravitino mass is given by m / = κ − ae α/ r bα . (7.21)By using the relation for the gravitino mass eq. (6.16), the relations (6.9) canbe rewritten as bc = − κm / e − α/ p αA ( α ) , (7.22)which can be used to rewrite the D-term contribution to the mass in the form m = m / ( σ s + 1) + m / κ − q α Q ( α ) ,Q ( α ) = α − α e − α/ p αA ( α ) ≈ . . (7.23)To avoid tachyonic masses, the charges of the MSSM fields should satisfy q α > q = − κm / σ s + 1 Q ( α ) ≈ . κm / , (7.24) -TERM CONTRIBUTIONS TO THE SCALAR SOFT MASSES 95 which corresponds to q (cid:38) . × − for m / ≈
20 TeV. The (non-universal)scalar soft masses and trilinear terms can be summarized as m ,i = m / (cid:20) ( σ s + 1) (cid:18) − q α q (cid:19)(cid:21) ,A = m / (cid:20) − α − (cid:18) α − qq P ( α ) (cid:19)(cid:21) ,P ( α ) = 2 αe α/ ( σ s + 1) p αA ( α ) Q ( α ) ≈ . , (7.25)where eq. (6.9) was used and q = b c/
2. The charges q α are given in terms ofthree independent parameters θ , θ Q and θ L by q H u = θq,q H d = (2 − θ ) q,q L = θ L q,q ¯ e = ( θ − θ L ) q,q Q = θ Q q,q ¯ u = (2 − θ − θ Q ) q,q ¯ d = ( θ − θ Q ) q, (7.26)such that q H u + q H d = 2 q,q ¯ e + q L + q H d = 2 q,q ¯ d + q Q + q H d = 2 q,q ¯ u + q Q + q H u = 2 q, (7.27)and eq. (7.18) is satisfied for the MSSM superpotential (2.13). Next, the cubic and mixed anomalies are proportional to C R = q f ( θ, θ L , θ Q ) , C = 3 q − θ Q − θ L ) , C = q (2 + 3 θ L + 9 θ Q ) , C = 6 q, (7.28)where f ( θ, θ L , θ Q ) = 3 (cid:0) θ Q + 3(2 − θ Q − θ L ) + 3( θ − θ Q ) + 2 θ L + ( θ − θ L ) (cid:1) + 2 (cid:0) (2 − θ ) + θ (cid:1) . (7.29)The Green-Schwarz counterterms are then proportional to β = − q f ( θ, θ L , θ Q )12 π bc ,β = − q − θ Q − θ L )8 π c ,β = − q (2 + 3 θ L + 9 θ Q )4 π c ,β = − q π c . (7.30)This results in contributions to the gaugino masses M A = − g A α ( α − b m / β A . (7.31)The Green-Schwarz contributions to the gaugino masses are given by M = − g Y π − θ Q − θ L ) α ( α − e α p αA ( α ) κ − q,M = − g π θ L + 9 θ Q ) 2 α ( α − e α p αA ( α ) κ − q,M = − g π α ( α − e α p αA ( α ) κ − q. (7.32) -TERM CONTRIBUTIONS TO THE SCALAR SOFT MASSES 97 The anomaly mediated contribution to the gaugino masses are given by eq. (5.27).As was explained in section 5.3, the total one-loop gaugino mass is the sum ofthese contributions M = − g Y π (cid:18) m / (cid:2) − α − (cid:3) −
32 (6 − θ Q − θ L ) ( α − Q ( α ) κ − q (cid:19) ,M = − g π (cid:18) m / (cid:2) − α − (cid:3) − (3 + θ L + 9 θ Q ) ( α − Q ( α ) κ − q (cid:19) ,M = − g π (cid:18) m / (cid:2) α − (cid:3) − α − Q ( α ) κ − q (cid:19) . (7.33)The soft terms can be summarized as (where ξ = q/q >
2, and eqs. (7.23) and(7.22) were used to rewrite the gaugino masses) m ,i = m / [( σ s + 1) (1 − θ α ξ )] ,A = m / [ − α −
1) ( α − ξP ( α ))] ,B = A + m / ,M = − g Y π m / (cid:18) (cid:2) − α − (cid:3) + 32 (6 − θ Q − θ L ) ( α − Q ( α ) ( σ s + 1) (cid:19) ,M = − g π m / (cid:18)(cid:2) − α − (cid:3) + (3 + θ L + 9 θ Q ) ( α − Q ( α ) ( σ s + 1) (cid:19) ,M = − g π m / (cid:18) (cid:2) α − (cid:3) + 6 ( α − Q ( α ) ( σ s + 1) (cid:19) . (7.34)The above soft terms depend on five parameters, namely m / , ξ , θ , θ L and θ Q . Following, a parameter scan has been performed [118] for m / ∈ [15 TeV ,
40 TeV], ξ ∈ [2 , θ , θ L , θ Q ∈ ]0 , β ∈ [1 , m / < ξ > A is negative and monotonicallydecreasing with ξ , such that for ξ >
10 the trilinear term becomes A < − m / .In principle, the value of tan β (which is the ratio between the two Higgs VEVs)is fixed by B [116, 117] (as in chapter 6), however we performed a scan overall possible values of tan β instead. Such a high value for | A | would contributeto the RGE for the stop mass parameter, so that it runs to a negative valuebefore the electroweak symmetry breaking scale is reached . Or the stau becomes tachyonic for a very small region of the parameter space.
In this parameter range, no viable electroweak symmetry breaking conditionswere found. We conclude that, even though the above idea is very appealingfrom a theoretical point of view, one can not (at least in this model) use aD-term contribution to the scalar soft masses proportional to the charge of aMSSM field under an extra U (1) factor to solve the problem with tachyonicmasses. hapter 8 Conclusions
Supersymmetry is one of the most promising theories that extends the StandardModel. Not only does it provide a possible solution to the hierarchy problem, itnaturally contains a dark matter candidate in certain supersymmetric theorieslike the MSSM, and the gauge couplings seem to unify at a certain energy whensupersymmetric corrections are taken into account.If a theory is invariant under local supersymmetry variations, it contains gravityand is called a supergravity theory. Moreover, since certain supergravity theoriesappear in the low energy limit of string theory, one can address certain openquestions in string theory in supergravity. One such active topic of research instring theory is the endeavor to find de Sitter vacua. In terms of a supergravitytheory one can address this problem by searching for theories where the minimumof the scalar potential has a positive but small value.In this work we introduced a model based on a single chiral multiplet S ,which can be identified with the string dilaton or a compactification modulus.The scalar component s of this chiral multiplet is invariant under a gaugedshift symmetry, with an associated vector multiplet. From gauge invariance,the Kähler potential should be a function of s + ¯ s , which is chosen to be K = − κ − p log( s + ¯ s ). The most general superpotential is an exponential of thechiral multiplet W = a exp( bs ), while the most general gauge kinetic functionhas a linear as well as a constant contribution. As a result, the scalar potentialcontains a positive D-term contribution including a Fayet-Iliopoulos term, andan F-term contribution which can be negative. For positive b there always existsa supersymmetric AdS vacuum, while for negative b an infinitesimally smalland positive value for the minimum of the potential can be found for p < in order to obtain a tunably small value for the cosmological constant for thetwo cases p = 1 and p = 2, and showed that the gravitino mass term (which isessentially the supersymmetry breaking scale) is separately tunable.Further in chapter 4, the case p = 2 is studied in great detail. While for p = 1the dS vacuum is stable, for p = 2 the scalar potential has another minimum atthe runaway direction Re s → ∞ , in principle allowing for tunneling to this otherminimum and rendering our desired vacuum metastable. It was shown that thetunneling rate is very low and exceeds the lifetime of the universe. Following,it was shown that in the limit where b →
0, the scalar potential correspondswith the one derived in [53] from D-branes in non-critical strings. While b = 0results in a supersymmetry breaking vacuum in AdS space, a non-zero b allowsfor a positive cosmological constant. It is therefore a very interesting (and open)question how a non-zero b can be realized in string theory. Finally, the presenceof a shift symmetry indicates that the theory can in principle be reformulatedin terms of a linear multiplet. In the literature however, this duality was neverperformed in the presence of a field dependent superpotential. We showedthat the above model can be reformulated in terms of a (unconstrained) vectormultiplet, which indeed has the degrees of freedom corresponding to a linearmultiplet.Quantum consistency can impose stringent constraints on quantum field theoriesin terms of anomaly cancellation conditions. In chapter 5 we extend the workby Elvang, Freedman and Körs [55] on quantum anomalies in supergravitytheories with Fayet-Iliopoulos terms and we show that their results can beinterpreted from a field-theoretic point of view. Moreover, we show that in thecase p = 2 the anomaly cancellation conditions are inconsistent with a TeVgravitino mass term, which excludes this case since it does not lead to a viablelow energy phenomenology. Finally, we demonstrated the relation betweenquantum anomalies and the gaugino mass terms that are generated at one loop.We showed that two contributions should be included in the calculation of thegaugino masses. Firstly, there is a contribution due to anomaly mediation.Secondly, there is a contribution due to the anomaly cancellation conditions,since anomaly cancellation can require the presence of a Green-Schwarz counterterm which originates from a linear contribution to the gauge kinetic function.This contribution however, also results in a contribution to the gaugino massterms. Finally, we showed that by an appropriate Kähler transformation, thelatter contribution to the gaugino mass terms can be reformulated in terms ofan anomaly mediated contribution.An interesting question arises whether this model leads to a distinguishablelow energy spectrum for the sparticles in the theory. In chapter 6 the casewith p = 1 is used as a hidden sector where supersymmetry breaking occurs.This supersymmetry breaking is communicated to the visible sector (MSSM) ONCLUSIONS 101 via gravity mediation. However, it is shown that the resulting scalar softsupersymmetry breaking terms are tachyonic. This can be cured by theintroduction of an extra Polónyi-like hidden sector field, or by allowing fornon-canonical kinetic terms of the Standard Model fields, while maintaining thedesirable features of the model. This however introduces an extra parameter γ (or ν in the second case), which turns out to be heavily constrained: γ shouldbe in the range [1 .
1; 1 . B is related to thetrilinear coupling by B = A − m / , the ratio between the two Higgs VEVs β is not an independent parameter and the model turns out to be very predictive.The low energy spectrum of the theory consists of (very) light neutralinos,charginos and gluinos, where the experimental bounds on the (mostly bino-like)LSP, the lightest chargino and the gluino mass force the gravitino mass to beabove 15 TeV. This in turn implies that the squarks are very heavy, with theexception of the stop squark which can be as light as 2 TeV when the parameterapproaches its lowest limit γ → .
1. It follows that the resulting spectrum canbe distinguished from other models of supersymmetry breaking and mediationsuch as mSUGRA and mAMSB.In chapter 7 it is shown that the shift symmetry can be identified with knownglobal symmetries of the MSSM. We analyzed the phenomenological implicationsin great detail for the particular case where the global symmetry is B − L , or3 B − L which contains the known matter parity of the MSSM as a subgroup.The latter combination has also the advantage of forbidding all dimension-fourand dimension-five operators violating baryon or lepton number in the MSSM.We showed that the phenomenology is similar to the one obtained in chapter 6,where the MSSM fields are inert under the shift symmetry, with the exceptionof the stop mass which can become lighter to about 1 . ppendix A Masses of the scalar, thefermion and the gauge field
In this appendix we calculate the masses of the fermions (section A.1) for thecases p = 1, p = 2, and for the model involving an extra Polónyi-like field insection 6.2. We demonstrate how the U (1) R gauge boson acquires a mass bythe Stueckelberg mechanism [125–129] for general p in section A.2. Finally, weshow how the scalar mass is calculated in the case p = 2 in section A.3. A.1 Calculation of the fermion mass matrices inthe various models
For convenience of the reader, we here repeat the fermion mass matrices andits corrections due to the super-BEH mechanism, which were discussed insections 3.1 and 3.2. The fermion mass matrix is given by m = m αβ + m ( ν ) αβ m αB + m ( ν ) αB m Aβ + m ( ν ) Aβ m AB + m ( ν ) AB ! , (3.16) where m αβ , m αB and m AB are given by eqs. (3.10-3.12). The corrections to thefermion mass terms due to the elimination of the Goldstino P L ν are given by m ( ν ) αβ = − κ m / ( δ s χ α )( δ s χ β ) ,m ( ν ) αA = − κ m / ( δ s χ α )( δ s P R λ A ) ,m ( ν ) AB = − κ m / ( δ s P R λ A )( δ s P R λ B ) , (3.17)and the fermion shifts are given by δ s χ α = − √ e κ K / ∇ α W,δ s P R λ A = − i P A . (3.15)The kinetic involving the fermions in the model however, are (in general) notcanonically normalized. Instead, they are given by L kin /e = − g α ¯ β χ α (cid:26)(cid:26) D (0) ¯ χ ¯ β − Re( f AB )2 ¯ λ A (cid:26)(cid:26) D (0) λ B , (A.1)where D (0) is the fermion covariant derivative. For diagonal g α ¯ α and diagonalgauge kinetic functions f A , canonically normalized kinetic terms for the fermionscan be obtained by making a field redefinition χ α = √ g α ¯ α χ α and λ A = p Re( f A ) λ A (note that there is no summation over indices here). As a result,the components in the fermion mass matrix above should be rescaled accordingly.The resulting mass matrix then has one zero eigenvalue corresponding to theGoldstino, while the other eigenvalues of Tr( m † m ) correspond to the masses ofthe physical fermions. A.1.1 Fermion masses for p = 1 The fermion mass matrix for case p = 1 in section 4.2.1 (as well as the model inthe model in section 6.4) for the fermionic superpartner of s and the gauginocorresponding to the shift symmetry (4.1) is given by m = κ − (cid:0) αb (cid:1) ae α/ ( α +4 α − ) α/b ) / − (cid:0) αb (cid:1) i √ b c ( α − α − ) α − (cid:0) αb (cid:1) i √ b c ( α − α − ) α c e − α ( α − a ( α/b ) / , (A.2) ALCULATION OF THE FERMION MASS MATRICES IN THE VARIOUS MODELS 105 where the factors (cid:0) αb (cid:1) have been taken into account due to non-canonical kineticterms for the chiral fermions. The gaugino already has canonical kinetic termssince f ( s ) = 1. The hidden sector fermions do not mix with the fermions of theMSSM. Also, the determinant of m is proportional to (2 + 8 α − α − α + α ),which indeed has a root at α ≈ − . m † m . Since one of theeigenvalues (the Goldstino)of the 2 × m f = Tr (cid:2) m † m (cid:3) = m / f χ , (A.3)where f χ = (cid:16) e α α (cid:0) α + 4 α − (cid:1) + ( α − A ( α ) + 4 e α α (cid:0) α − α − (cid:1) A ( α ) (cid:17) α e α ≈ . , (A.4)and we have used the relations between the parameters and the numerical valuesfor α and A ( α ) in eqs. (4.20). A.1.2 Fermion masses for p = 2 For p = 2, the fermion mass terms are given by m αβ = κ − ae bs ( s + ¯ s ) (cid:0) b ( s + ¯ s ) − b ( s + ¯ s ) + 2 (cid:1) ,m αA = κ − i √ c (cid:18) s + ¯ s ) − b s + ¯ s ) (cid:19) = m Aα ,m AB = κ − − a e b ¯ s ( s + ¯ s ) (cid:18) b − s + ¯ s (cid:19) , (A.5)where ( s + ¯ s ) should be evaluated in the minimum of the potential, i.e. eqs. (4.26)hold. The corrections due to the BEH-effect are given by m ( ν ) αβ = κ − (cid:18) s + ¯ s − b (cid:19) − as + ¯ s e b ( s +¯ s ) ,m ( ν ) αA = κ − (cid:18) s + ¯ s − b (cid:19) −√ ic,m ( ν ) AB = κ − (cid:18) s + ¯ s − b (cid:19) c a ( s + ¯ s ) e − b ( s +¯ s ) , (A.6)
06 MASSES OF THE SCALAR, THE FERMION AND THE GAUGE FIELD
Next, due to a field redefinition (see the discussion below eq. (A.1)) in order toobtain canonically normalized kinetic terms, the entries in the fermion massmatrix corresponding to m α ¯ β , m αA and m AB should be multiplied by a factor α / b , p α/ b and b/α respectively. As a result, the full (2 ×
2) mass matrixis κ − times α b ab α e α/ (cid:0) − α + α (cid:1) − p α b icb √ α (cid:0) − − α + 2 α (cid:1) − p α b icb √ α (cid:0) − − α + 2 α (cid:1) bα e − α/ ( α − aα (cid:0) − a αe α + 4 bc ( α − (cid:1)! By inserting equation (4.26), the determinant of m isDet( m ) = − κ − b c (cid:0) − − α − α + 94 α + 15 α − α + 2 α (cid:1) α ( − − α + α )= 0 (A.7)The determinant has a root at the earlier defined value of α = − . ... , sothat the mass matrix has indeed a zero mode, corresponding to the reductionof the rank of the full mass matrix m , because the Goldstino disappears. The(squared) mass of the fermion is thus given by the trace of the mass matrixsquared m f = Tr( m † m )= κ − a b e α (cid:0)
116 + 68 α − α − α + 8 α (cid:1) α . (A.8)By using the numerical value of α and eq. (4.29), the fermion mass can bewritten in terms of m / m f ≈ . m / . (A.9) A.1.3 Fermion masses for the model including an extraPolónyi-like field
We now calculate the fermion masses for the model with the extra hidden sectorfield z in section 6.2. This model contains one extra hidden sector fermion. Itsmass matrix is given by m = m ss + m ( ν ) ss m st + m ( ν ) st m sR + m ( ν ) sR m st + m ( ν ) st m tt + m ( ν ) tt m tR + m ( ν ) tR m sR + m ( ν ) sR m tR + m ( ν ) tR m RR + m ( ν ) RR , ASS OF THE U (1) R GAUGE BOSON 107 where m ss + m ( ν ) ss = κ − (cid:16) αb (cid:17) ae ( t + α ) (cid:0) − α + α (cid:1) (1 + tγ )3 (cid:0) αb (cid:1) / ,m st + m ( ν ) st = κ − (cid:16) αb (cid:17) ae ( t + α )( − α ) (cid:0) t + γ + t γ (cid:1) (cid:0) αb (cid:1) / ,m sR + m ( ν ) sR = − κ − (cid:16) αb (cid:17) i √ b c (cid:0) − − α + α (cid:1) α ,m tt + m ( ν ) tt = κ − ae ( t + α ) (cid:0) tγ + 2 t γ − γ + t γ + t (cid:0) γ (cid:1)(cid:1) p αb (1 + tγ ) ,m tR + m ( ν ) tR = − κ − i √ bc ( − α ) (cid:0) t + γ + t γ (cid:1) α (1 + tγ ) ,m RR + m ( ν ) RR = κ − b c p αb e ( − t − α ) (cid:0) − α (cid:1) a (1 + tγ ) . (A.10)It has been checked that the determinant of this matrix vanishes for α and t satisfying eqs. (6.8) and (6.9). The masses of the physical fermions are the twonon-zero eigenvalues of this matrix. The result however is quite tedious and weonly state the numerical values for γ = 1: m χ ≈ . m / ,m χ ≈ . m / . (A.11) A.2 Mass of the U (1) R gauge boson The kinetic term L s of the scalar field s can be written as L s /e = − g s ¯ s ˆ ∂ µ s ˆ ∂ µ ¯ s, (A.12)where the covariant derivative is defined asˆ ∂ µ s = ∂ µ s − k s A µ = ∂ µ s + icA µ , (A.13)
08 MASSES OF THE SCALAR, THE FERMION AND THE GAUGE FIELD where k s is the Killing vector associated with the shift symmetry given by k s = − ic (see eq. (4.1)). Equation (A.12) can then be written as L s /e = − pκ − ( s + ¯ s ) (cid:0) ∂ µ s∂ µ ¯ s − icA µ ∂ µ ( s − ¯ s ) + c A µ A µ (cid:1) . (A.14)The local shift symmetry allows us to gauge fix s − ¯ s = 0, which results in amass term L m /e = − m A µ A µ A µ for the gauge boson. This mass term however,needs a rescaling due to a non-canonical kinetic term L kin /e = − Re f ( s )4 F µν F µν . (A.15)The result is m A µ = s p Re f ( s ) κ − c h s + ¯ s i , (A.16)where Re f ( s ) = 1 for p = 1, and Re f ( s ) = s +¯ s for p = 2. A.3 Scalar masses
In this section we demonstrate how the mass of the scalar fields is calculatedfor p = 2. The masses of the scalar fields for the other models is calculatedanalogously.We isolate the quadratic contribution of the scalar potential eq. (4.24), which isrepeated here for convenience V = κ − a e b ( s +¯ s ) (cid:18) b − bs + ¯ s − s + ¯ s ) (cid:19) + κ − c s + ¯ s + 2 d (cid:18) s + ¯ s − b (cid:19) . The quadratic term V quad = V ( α/b ) (cid:0) s + ¯ s − αb (cid:1) in the series expansion of V around its minimum at h s + ¯ s i = αb is given by (simplified by using equation(4.26)) V quad = κ − (cid:0)
48 + 192 α − α − α + 24 α − α + α (cid:1) b c ( s + ¯ s − αb ) α (2 + 4 α − α ) . This rescaling was absent in an earlier version of this thesis. We thank A. Chatrabhutifor pointing this out.
CALAR MASSES 109
Taking into account the non-canonical form of the kinetic terms , the mass ofthe dilaton Re( s ) = s +¯ s is then given by m s = − κ − b c (cid:0)
48 + 192 α − α − α + 24 α − α + α (cid:1) α ( − − α + α ) . (A.19)For the earlier calculated numerical value of α (see eq. (4.26)) this expression isindeed positive: m s > The kinetic term of Re( s ) is given by L kin = − κ − s ) ∂ µ Re( s ) ∂ µ Re( s ) . (A.17)Expanding around the minimum φ = Re( s ) − S , with S = ακ b , gives L = − S (cid:16) − φS + . . . (cid:17) ∂ µ φ∂ µ φ. (A.18)We should thus rescale the mass with a factor S . ppendix B Linear-Chiral duality
In this appendix we present the details of the linear-chiral (or tensor-scalar)duality which lead to the result presented in section 4.3.3. This duality ispresented in global supersymmetry in section B.1. Since a superspace formalismis more convenient in this context, the conventions of Wess and Bagger [60] wereused in this section. Following, a dual version of the theory in section 4.2.2 inthe Kähler frame with an exponential superpotential is presented in section B.2.An important identity is proven in section B.3. This appendix is based on workwith I. Antoniadis [50].
B.1 Tensor-scalar duality in global supersymmetry
In this section we will use the superfield notation and conventions of [60]. It isimportant to note that in this notation, a linear multiplet L is defined from ageneral superfield by imposing the relation D L = ¯ D L = 0 , (B.1)where the differential operators ¯ D and D are given by D α = ∂∂θ α + iσ µα ˙ α ¯ θ ˙ α ∂ µ , ¯ D ˙ α = − ∂∂ ¯ θ ˙ α − iσ µα ˙ α ∂ µ . (B.2)Here, θ α (¯ θ ˙ α ) are the fermionic coordinates of the (anti-)chiral superspace. Easy warm-up: No superpotential
As a warm-up, we will show how one can describe a (globally) supersymmetrictheory, given by a single chiral multiplet S with vanishing superpotential anda Kähler potential K ( S + ¯ S ) given by eq. (4.3) in terms of a linear multiplet.The Lagrangian of this theory is given by L = Z d θd ¯ θ K ( S + ¯ S ) . (B.3)To obtain the dual version of this theory, we consider the Lagrangian L = Z d θd ¯ θ (cid:0) F ( L ) − pL ( S + ¯ S ) (cid:1) , (B.4)where L is a priori an unconstrained real vector superfield. For purposes thatbecome clear in an instant, we take F ( L ) to be given by F ( L ) = p log( L ). Theequations of motion for the (anti-)chiral superfields S ( ¯ S ) give D L = ¯ D L = 0,which is exactly the condition (B.1) for L to be a linear multiplet. The secondterm in (B.4) is thus a generalised Lagrange multiplier that ensures the linearityof L . The resulting theory in terms of the linear multiplet is given by Z d θd ¯ θ F ( L ) . (B.5)On the other hand, the equations of motion of (the now unconstrained realsuperfield) L give F ( L ) = p (cid:0) S + ¯ S (cid:1) , (B.6)such that S + ¯ S = 1 L . (B.7)Some basic algebra then shows that F ( L ) = p log (cid:18) S + ¯ S (cid:19) = − p log (cid:0) S + ¯ S (cid:1) . (B.8)The chiral theory is then given by Z d θd ¯ θ K ( S + ¯ S ) , (B.9)with K ( S + ¯ S ) given in (4.3).Since the natural partner of the graviton in the massless sector of superstringsis the scalar dilaton and an antisymmetric tensor, the above exercise shouldbe repeated in supergravity (local supersymmetry). Below, we first performthe duality in the presence of a superpotential given by eq. (4.4) in globalsupersymmetry, and consequently in supergravity. ENSOR-SCALAR DUALITY IN GLOBAL SUPERSYMMETRY 113
With a superpotential
We are now ready to perform the above duality in the presence of a non-trivial(exponential) superpotential, given by eq. (4.4). This superpotential has thefollowing property: under the shift symmetry (4.1) the superpotential transformsas W ( S ) −→ W ( S ) e − iαb , where the gauge parameter is now denoted as α todistinguish it from the superspace coordinates θ . This transformation canbe compensated by the R-transformation θ −→ θe − i αb such that R d θW ( S )is invariant under the combined shift and R-transformations. Note that onecan not gauge the shift symmetry in this model, since it is entangled with anR-symmetry, which can not be gauged in rigid supersymmetry, in contrast withsupergravity.We are now interested in the dual description of the theory L = Z d θ K ( S + ¯ S ) + Z d θW ( S ) + Z d ¯ θ ¯ W ( ¯ S ) , (B.10)where the Kähler potential is given by eq. (4.3) and the superpotential given by(4.13). To find the dual theory, we start from the Lagrangian L = Z d θ F ( L ) − p Z d θL ( S + ¯ S ) + Z d θW ( S ) + Z d ¯ θ ¯ W ( ¯ S ) , (B.11)where as before, S and ¯ S are chiral and anti-chiral superfields, respectively, and L is an a priori unconstrained real superfield. As we will see below, it turnsout that in the dual theory, L will still be a real superfield off-shell. However,it will contain the same number of propagating degrees of freedom as those ofa linear multiplet. As in the previous example, we take F ( L ) = p log( L ). Theequations of motion for L give, as before S + ¯ S = 1 L . (B.12)By substituting this relation into the Lagrangian (B.11) one retrieves (B.10).We now derive the equations of motion for the chiral superfield. This results inmodified linearity conditions p ¯ D L = − W ( S ) , (B.13) pD L = − W ( ¯ S ) , (B.14)
14 LINEAR-CHIRAL DUALITY and thus L is still an unconstrained real multiplet. One can use these equationsto write S and ¯ S in terms of LS = 1 b log (cid:18) − p ab ¯ D L (cid:19) , (B.15)¯ S = 1 b log (cid:18) − p ab D L (cid:19) . (B.16)Note that one can rewrite the Lagrangian (B.11) as L = Z d θ F ( L ) + Z d θ (cid:16) p D LS + W ( S ) (cid:17) + Z d ¯ θ (cid:16) p D L ¯ S + ¯ W ( ¯ S ) (cid:17) . (B.17)By substituting (B.15) and (B.16) into (B.17) one obtains the dual theory interms of the superfield L L L = Z d θ F ( L )+ Z d θ (cid:18)(cid:16) p b ¯ D L (cid:17) log (cid:18) − p ab ¯ D L (cid:19) − p b ¯ D L (cid:19) + Z d ¯ θ (cid:18)(cid:16) p b D L (cid:17) log (cid:18) − p ab D L (cid:19) − p b D L (cid:19) . (B.18)Note that L is a real vector superfield instead of linear superfield. However,as we show below, the vector multiplet L contains auxiliary fields that can beeliminated by their equations of motion, and the physical content of the theorycan still be described in terms of only a real scalar l , a Majorana fermion χ and a field strength vector field v µ . In fact, a more natural description of thistheory will be given by a real scalar l , a Majorana fermion χ and a pseudoscalar w , which is the phase of the θ -component of L .In the conventions of [60], the general expression of the real superfield L isdefined as follows L = l + iθχ − i ¯ θ ¯ χ + θ Z + ¯ θ ¯ Z − θσ µ ¯ θv µ + iθ ¯ θ (¯ λ − i σ µ ∂ µ χ ) − i ¯ θ θ ( λ − i σ µ ∂ µ ¯ χ ) + 12 θ ¯ θ ( D − (cid:3) l ) . (B.19) ENSOR-SCALAR DUALITY IN GLOBAL SUPERSYMMETRY 115
Rewriting Z = ρe iw and ¯ Z = ρe − iw , it follows that the Lagrangian (B.18) isgiven in components by L = F ( l )2 (cid:18) D − (cid:3) l (cid:19) + F ( l )2 (cid:18) − χ (cid:18) λ − i σ µ ∂ µ ¯ χ (cid:19) − ¯ χ (cid:18) ¯ λ − i σ µ ∂ µ χ (cid:19) + 2 ρ − v µ v µ (cid:19) + 14 F ( l ) (cid:0) χ ρe − iw + ¯ χ ρe iw + χσ µ ¯ χv µ (cid:1) + F ( l )16 χ ¯ χ + pb D log (cid:16) pab ρ (cid:17) + pb w∂ µ v µ − p b λ ρ e iw − p b ¯ λ ρ e − iw . (B.20)Note that the fields D, λ, v µ and ρ appear without kinetic terms in theLagrangian. They are therefore auxiliary fields that can be eliminated bytheir equations of motion. These are ρ = abp e − b p ( F ( l )) ,p λbρ e iw = −F ( l ) χ, F v µ = F χσ µ ¯ χ − pb ∂ µ w, − pb D = 2 F ρ + ρ F (cid:0) χ e − iw + ¯ χ e iw (cid:1) + p b ( λ ρ e iw + ¯ λ ρ e − iw ) . (B.21)Indeed, these equations of motion can be used to eliminate these fields fromthe Lagrangian by substituting them back into eq. (B.20) L = − F ( l ) (cid:3) l + i F ( l ) ( χσ µ ∂ ¯ χ + ¯ χ ¯ σ µ ∂ µ χ ) + a b p F ( l ) e bp F ( l ) + 1 F ( l ) (cid:18) −F ( l )2 χσ µ ¯ χ∂ µ w + p b ∂ µ w∂ µ w (cid:19) + (cid:18) F ( l )16 − F ( l ) F ( l ) (cid:19) χ ¯ χ + ab p e bp F ( l ) / (cid:0) χ e − iw + ¯ χ e iw (cid:1) (cid:18) F ( l ) − bp F ( l ) (cid:19) . (B.22)It follows that the Lagrangian can be written only in terms of the physicalpropagating fields l, χ and w . Note that it is also possible to write the Lagrangian
16 LINEAR-CHIRAL DUALITY in terms of l, χ and v µ . The above calculation proves that the field strength v µ contains only one degree of freedom.Note that the resulting scalar potential for the field l (last term of the first lineof the above expression) is: V = − a b p F ( l ) e bp F ( l ) = a b p l − e bl . (B.23)As a check, we can substitute (B.12) into the above equation to find V = a b p ( s + ¯ s ) e b ( s +¯ s ) , (B.24)which corresponds to the scalar potential that can be directly obtained in thechiral formulation from equation (B.10). B.2 Tensor-scalar duality in supergravity
We now calculate the same duality in supergravity by using the chiralcompensator formalism [59]. In order to simplify the notation, we put κ = 1 inthis section. We start with the Lagrangian L = (cid:20) − (cid:0) S e − gV R ¯ S (cid:1) L − (cid:21) D + (cid:2) S W ( S ) (cid:3) F + (cid:2) f W (cid:3) F + h.c. − (cid:2) L ( S + ¯ S + cV R ) (cid:3) D , (B.25)where, as before, S ( ¯ S ) is an (anti-)chiral multiplet with zero Weyl weight, L is a(unconstrained) real multiplet with Weyl weight 2 and V R is a vector multiplet ofzero Weyl weight associated with a gauged R-symmetry with coupling constant g given by g = bc . S ( ¯ S ) is the (anti-)chiral compensator superfield with Weylweight +1 ( − F is defined on a chiral multiplet X = ( s, P L ζ, F ) with weight 3[ X ] F = e Re (cid:18) F + 1 √ ψ µ γ µ P L ζ + 12 s ¯ ψ ν γ µν P R ψ ν (cid:19) , (B.26)where e is the determinant of the vierbein. From now on, we write P L,R ζ = ζ L,R .A D-term [ ] D is defined on a real multiplet C = ( C, χ, Z, v a , λ, D ) with weight ENSOR-SCALAR DUALITY IN SUPERGRAVITY 117 C ] D = e (cid:18) D −
12 ¯ ψ · γiγ ∗ λ − CR ( ω ) + 16 (cid:0) C ¯ ψ µ γ µρσ − i ¯ χγ ρσ γ ∗ (cid:1) R ρσ ( Q )+ 14 (cid:15) abcd ¯ ψ a γ b ψ c (cid:18) v d −
12 ¯ ψ d χ (cid:19)(cid:19) , (B.27)where R ( ω ) and R ( Q ) are the graviton and gravitino curvatures. Thesuperpotential is given by eq. (4.13) and W = W α W α contains the gaugekinetic terms for V R with gauge kinetic function f , which is assumed here tobe constant f = γ . A linear gauge kinetic function, as is necessary to obtain atunable cosmological constant (see section 4.2.2), will be discussed below.Note that since the R-symmetry generator T R does not commute with thesupersymmetry generators Q α by eq. (2.16), repeated here for convenience ofthe reader [ T R , Q α ] = − i ( γ ∗ ) βα Q β , the global R-symmetry in the previous section necessarily becomes gauged whenone couples this theory to gravity. The above Lagrangian thus has a gaugedR-symmetry (coupled to the gauged shift symmetry), under which the variousfields transform as S −→ S − ic Λ , ¯ S −→ ¯ S + ic ¯Λ ,S −→ e ibc Λ S , ¯ S −→ S e − ibc ¯Λ ,V R −→ V R + i (cid:0) Λ − ¯Λ (cid:1) ,L −→ L, (B.28)where the gauge parameter Λ(¯Λ) is a (anti-)chiral multiplet.The equations of motion for L are given by L = S e − gV R ¯ S ( S + ¯ S ) / . (B.29)Below, we fix the conformal gauge by choosing the lowest components s and ¯ s of S and ¯ S as s ¯ s = l (see equation (B.43)), so that the lowest componentsof equation (B.29) reads l = 1 s + ¯ s . (B.30)
18 LINEAR-CHIRAL DUALITY
If one substitutes equation (B.29) back into the Lagrangian (B.25), one retrieves L = (cid:20) − S e − gV R ¯ S ( S + ¯ S + cV R ) / (cid:21) D + (cid:2) S W ( S ) (cid:3) F + (cid:2) f W (cid:3) F + h.c. , (B.31)which corresponds to a theory with Kähler potential K = − (cid:0) S + ¯ S (cid:1) , asuperpotential W ( S ) and a gauged (shift) R-symmetry in the old minimalformalism.We are now ready to derive the analogue of the Lagrangian (B.18) in supergravityand confirm that the propagating degrees of freedom are still two real scalarsand a Majorana fermion, as was the case in rigid supersymmetry.To obtain the theory in terms of the superfield L , we use the following identitywhich is proven in section B.3 (cid:2) L ( S + ¯ S ) (cid:3) D = [ T ( L ) S ] F + h.c. , (B.32)where T is the chiral projection operator. In global supersymmetry, the chiralprojection is in superspace the operator ¯ D , where ¯ D ˙ α is defined in equation(B.2). In supergravity one can only define such an operator T on a real multipletwith Weyl weight w and chiral weight c given by ( w, c ) = (2 , L with components L = ( l, χ, Z, v a , λ, D ), where a from nowon is used to indicate a flat frame index, related to a curved Lorentz index µ by v a = e µa v µ . Since a chiral multiplet Σ can be defined by its lowest component,we define the chiral projection of L via the operator T as a chiral multiplet withlowest component − ¯ ZT ( L ) : L (2 , −→ Σ( L ) = T ( L ) , ( w, c ) = (3 , . (B.33)In global supersymmetry, this definition of the chiral projection operator T ( L )would correspond to T ( L ) = − ¯ D L .By using the relation (B.32), one can now calculate the equation of motion for S S = 1 b log (cid:18) T ( L ) abS (cid:19) , (B.34)and insert this back into the original Lagrangian (B.25) to obtain L = (cid:20) − (cid:0) S e − gV R ¯ S (cid:1) L − (cid:21) D − [ cLV R ] D + 1 b (cid:20) T ( L ) − T ( L ) log (cid:18) T ( L ) abS (cid:19)(cid:21) F + (cid:2) f W (cid:3) F + h.c. (B.35) ENSOR-SCALAR DUALITY IN SUPERGRAVITY 119
Below, it is shown that apart from the obvious extra fields compared to globalsupersymmetry, namely the graviton, the gravitino, a gauge boson and agaugino , this theory does not contain any additional degrees of freedom andthe (non-gravitational) spectrum can be described in terms of two real scalars l and w , where w is defined as Z = ρe iw , and a Majorana fermion χ .We now look for a component expression of the Lagrangian (B.35). Since weare interested in comparing the F-term scalar potential of the above theory, wewill from now on neglect the vector multiplet V R putting g = c = f = 0. Thecomponents of the real multiplet L are given by L = ( l, χ, Z, v a , λ, D ), while thecomponents of the (auxiliary) chiral multiplet are given by S = ( s , P L ζ, F ),where P L ( R ) is the left-handed (right-handed) projection operator. The Weyland chiral weights of the Z -component of a real linear multiplet are givenby ( w, c ) = ( w + 1 , − w = c for the superconformal algebra to close, it is clear thatthe operation T ( L ), given by equation (B.33), can only be defined on a realmultiplet with w = 2, so that ¯ Z has weights ( w, c ) = (3 ,
3) and can thus be usedas lowest component of a chiral multiplet. This agrees with the assumptionthat L has Weyl weight 2. The auxiliary field S is chosen to have weight 1.The chiral multiplet obtained by the projection operator T has components T ( L ) = ( − ¯ Z, −√ iP L (cid:0) λ + (cid:0) D χ (cid:1) , D + D a D a l + i D a v a ), where the covariantderivatives are given by D a l = ∂ a l − B a l − i ¯ ψ a γ ∗ χ, D a v b = ∂ a v b − B a v b + 12 ¯ ψ a ( γ b λ + D b χ ) −
32 ¯ ϕ a γ b χ, D a P L χ = (cid:18) ∂ a + 14 ω bca γ bc − B a + 32 i A a (cid:19) χ − P L (cid:0) iZ − (cid:1) v − i (cid:0) D l (cid:1) ψ a + 2 ilP L ϕ a , (B.36)where ψ µ is the gravitino, ϕ µ is the gauge field corresponding to the conformalsupercharge S , B µ is the gauge field of dilatations and A µ is the gauge field ofthe T-symmetry (which is the U (1) R-symmetry of the superconformal algebra). Actually, a linear combination of the two 2-component fermions present in this model isthe Goldstino which will be eaten by the gravitino according to the super-BEH mechanism,see section 4.2.2.
20 LINEAR-CHIRAL DUALITY
To obtain a real multiplet from a real function φ ( s, ¯ s ) of chiral superfields( s, ζ L , F ) and their anti-chiral counterparts, we use [130] (cid:16) φ ( s, ¯ s ); −√ iφ s ζ L ; − φ s F + φ ss ¯ ζ L ζ L ; iφ s ∂ a s − iφ ¯ s ∂ a ¯ s + iφ s ¯ s ¯ ζ L γ a ζ R ; − √ iφ s ¯ s F ζ R + √ iφ s ¯ s (cid:1) ∂ ¯ sζ L + i √ φ ss ¯ s ζ R · ¯ ζ L ζ L ; φ s ¯ s (cid:2) F ¯ F − ∂ a s∂ a ¯ s − ∂ a ¯ ζ L γ a ζ R + ¯ ζ L (cid:1) ∂ζ R (cid:3) − φ ss ¯ s (cid:2) ¯ ζ L ζ L ¯ F + ¯ ζ L (cid:1) ∂sζ R (cid:3) − φ s ¯ s ¯ s (cid:2) ¯ ζ R ζ R F + ¯ ζ R (cid:1) ∂ ¯ sζ L (cid:3) + φ ss ¯ s ¯ s ζ L ζ L · ¯ ζ R ζ R (cid:19) , (B.37)where φ s = ∂φ∂s , φ ¯ s = ∂φ∂ ¯ s , etc. To obtain a real superfield from a function f ( C )of real multiplets with components C i , χ Li , Z i , v ai , λ Ri , D i we use (cid:18) f ( C ); f i χ Li ; f i Z i − f ij ¯ χ Li χ Lj ; f i v ai + i f ij ¯ χ Li γ a χ Rj ; f i λ Ri + 12 P L (cid:2) Z i − i (cid:1) v i − (cid:1) ∂C i (cid:3) χ j − f ijk χ Ri · ¯ χ Lj χ Lk ; f i D i − f ij (cid:2) χ i λ j − Z i ¯ Z j + v ai v aj + ∂ a C i ∂ a C j − ( ∂ a ¯ χ Li ) γ a χ Rj + ¯ χ Li (cid:1) ∂χ Rj (cid:3) − f ijk (cid:2) ¯ χ Li ¯ Z j χ Lk + ¯ χ Ri Z j χ Rk + 2 i ¯ χ i (cid:1) v j χ k (cid:3) + 18 f ijkl ¯ χ Li χ Lj · ¯ χ Rk χ Rl (cid:19) . We are now ready to write down the (non-gauge) bosonic part of the Lagrangian(B.35) as L /e = 12 ( s ¯ s ) l − R + ( s ¯ s ) l − D + 32 ( s ¯ s ) l − v µ v µ − b log (cid:18) ρ a b ( s ¯ s ) (cid:19) D + 1 b v µ ∂ µ w + L kin − V F , (B.38)where we used that (cid:3) l = D a D a l = ∂ a ∂ a l − lR (B.39)to rewrite a term involving (cid:3) l in the third term in (B.35). The scalar field w isdefined as Z = ρe iw , (B.40) ENSOR-SCALAR DUALITY IN SUPERGRAVITY 121 and it will turn out that this field inherits the gauged shift symmetry. L kin and V F in equation (B.38) are given by L kin = 3( s ¯ s ) l − ∂ µ s ∂ µ ¯ s + 32 ( s ¯ s ) l − ∂ µ ( s ¯ s ) ∂ µ ( s ¯ s ) − s ¯ s ) l − ∂ µ l∂ µ ( s ¯ s ) + 32 ( s ¯ s ) l − ∂ µ l∂ µ l − (cid:16) s s l (cid:17) ∂ µ ∂ µ l, V F = 9( s ¯ s ) l − F ¯ F + 3( s ¯ s ) l − (cid:0) ¯ s Z ¯ F + s ¯ ZF (cid:1) + 3 Z ¯ Fb ¯ s + 3 ¯ ZFbs + 32 ( s ¯ s ) l − Z ¯ Z = − ρ ( s ¯ s ) l (cid:18) b + ( s ¯ s ) l − (cid:19) + 32 ρ ( s ¯ s ) l − , (B.41)where we solved the equations of motion of the auxiliary field F and usedequation (B.40) to go from the first to the second line of V F . On the otherhand, the equations of motion for D and v µ are given by ρ = a b ( s ¯ s ) e b ( s s l ) , ( s ¯ s ) l − v µ = − b ∂ µ w. (B.42)We now fix the conformal gauge as( s ¯ s ) = l (B.43)to obtain a correctly normalized Einstein-Hilbert term. Upon this choice L kin reduces to L kin = −
12 1 l ∂ µ l∂ µ l and we obtain the final result L = 12 R −
12 1 l ∂ µ l∂ µ l − b ∂ µ w∂ µ w − V F , (B.44)with V F = a e bl (cid:18) b − bl − l (cid:19) . (B.45)Upon substituting the relation (B.30), one indeed finds the correct F-term scalarpotential computed with the chiral formulation (see equation (4.24)), namely V F = a e b ( s +¯ s ) (cid:18) b − bs + ¯ s − s + ¯ s ) (cid:19) . (B.46)
22 LINEAR-CHIRAL DUALITY
In order to obtain a tunable dS vacuum (see section 4.2.2), one has to extend theabove formalism to include a linear gauge kinetic function f ( s ). By followingthe same method as above, but by including an extra term (cid:2) βS W (cid:3) F + h.c. , (B.47)which contains the gauge kinetic terms of V R with gauge kinetic function f ( S ) = βS + γ to the Lagrangian (B.25), one finds that the result (B.35) is stillvalid upon the substitution T ( L ) −→ T ( L ) − β W . (B.48)The theory dual to the one described in section 4.2.2 is given by L = (cid:20) − (cid:0) S e − gV R ¯ S (cid:1) L − (cid:21) D − [ cLV R ] D + 1 b (cid:20)(cid:0) T ( L ) − β W (cid:1) − (cid:0) T ( L ) − β W (cid:1) log (cid:18) T ( L ) − β W abS (cid:19)(cid:21) F + h.c.+ (cid:2) γ W (cid:3) F + h.c. . (B.49) B.3 Proof of an important identity
In this Appendix we prove equation (B.32), repeated here for convenience (cid:2) L ( S + ¯ S ) (cid:3) D = [ T ( L ) S ] F + h.c. . (B.50)To do this, we calculate both sides in components and see that they coincide. Itis sufficient to show this only for the bosonic terms. If the (bosonic) componentsof the chiral multiplet S are given by ( s, , F ) and the (bosonic) components ofthe real multiplet L are given by ( l ; 0 ; Z ; v µ ; 0 ; D ), then the components of S + ¯ S can be calculated using (B.37) S + ¯ S = (cid:0) s + ¯ s ; 0 ; − F + ¯ F ) ; i∂ µ ( s − ¯ s ) ; 0 ; 0 (cid:1) . (B.51)The chiral multiplet T ( L ) is defined as a chiral multiplet with lowest component − ¯ Z and has components T ( L ) = ( − ¯ Z ; 0 ; D + (cid:3) l + i D µ v µ ) . (B.52)The left-hand side of equation (B.50) can be written in components by usingequations (B.37) and (B.27) (neglecting all terms involving fermions) (cid:2) L ( S + ¯ S ) (cid:3) D = − l ( s + ¯ s ) R + ( s + ¯ s ) D − ¯ ZF − Z ¯ F − iv µ ∂ µ ( s − ¯ s ) − ∂ µ ∂ µ ( s + ¯ s ) . (B.53) ROOF OF AN IMPORTANT IDENTITY 123
Similarly, by using equation (B.37), [ T ( L ) S ] F can be written in components as[ T ( L ) S ] F = − ¯ ZF + sD + s (cid:3) l + is∂ µ v µ . (B.54)The right-hand side is thus given by[ T ( L ) S ] F + h.c. = − ¯ ZF − Z ¯ F + ( s + ¯ s ) D + ( s + ¯ s ) (cid:3) l + i ( s + ¯ s ) ∂ µ v µ . The equality (B.50) is then proven by using equation (B.39) for (cid:3) l and thenpartially integrating the last two terms. ppendix C R-symmetries and anomalies
In rigid supersymmetry, the Lagrangian of the model in eqs. (4.13) is given by L = (cid:2) K ( S + ¯ S ) (cid:3) D + (cid:0) [ W ( S )] F + (cid:2) f ( S ) W (cid:3) F + h.c. (cid:1) . (C.1)Here, S = ( s, χ, F ) is the dilaton multiplet and the operations [ ] F and [ ] D aredefined in eqs. (B.26) and (B.27) respectively. The Kähler potential K ( S + ¯ S ) isgiven by eq. (4.3), the superpotential is given by eq. (4.4), and the gauge kineticfunction is given by eq. (4.8), summarized here for convenience of the reader K = − κ − p log( s + ¯ s ) ,W = κ − ae bs ,f R ( s ) = γ + β R s. (C.2)The subscript R is explicitly written to distinguish the gauge kinetic functioncorresponding to U (1) R from the gauge kinetic functions of the Standard Modelgauge groups.In eq. (C.1), W is the multiplet whose scalar component is ¯ λP L λ , where λ is the R-gaugino. For example, the last term in eq. (C.1) contains the U (1) R kinetic terms, given by L ⊃ −
14 Re( f ( s )) F µν F µν + 18 Im( f ( s )) (cid:15) µνρσ F µν F µν . (C.3)Since the superpotential transforms under the shift symmetry eq. (4.1), thiscannot be a gauge symmetry in a rigid supersymmetric theory (in contrast with supergravity). However, it can be considered as a global R-symmetry. When s → s − ic , the superpotential transforms as W ( s ) → W ( s ) e − ibc . (C.4)It follows that under this global symmetry, the chiral fermions and gauginostransform as P L χ → P L χe ibc/ ,P L λ → P L λe − ibc/ , (C.6)Respectively. With ξ = bc , the R-charges of the chiral fermions, the gauginosand the gravitino in the theory are then given by R χ = Q + ξ/ ,R λ = − ξ/ ,R / = − ξ/ , (C.7)where we included a superfield charge Q for the chiral multiplets (see eq. (5.2)).This implies that the components of a matter multiplet ( z, χ, F ) carry a charge R z = Q , R χ = Q + ξ/ R F = Q + ξ respectively. The scalar component ofthe dilaton multiplet transforms as a shift under U (1) R , and has therefore avanishing charge Q .Even though an R-symmetry can not be gauged in a rigid supersymmetrictheory, since the R-symmetry generator does not commute with supersymmetry This can be more intuitively seen in the superfield formalism, where the Lagrangiancontains a term
L ⊃ Z d θW ( S ) . (C.5)Since the superpotential transforms as (C.4), dθ should transform as dθ → dθe ibc/ such thatthe Lagrangian remains invariant. The Grassmann variable then transforms as θ → θe − ibc/ .In the conventions of [60], a chiral superfield is given by Φ = C + √ θχ + θ F . To compensatefor the transformation of θ , the component fields should transform as C → C , χ → χe ibc/ and F → F e ibc . In a gauge multiplet however, the gauge boson appears as the term proportionalto θ ¯ θ and the D-component field appears proportional to θ ¯ θ . They are therefore invariantunder the global R-symmetry. Finally the gaugino, appearing in a vector superfield as the fieldproportional to ¯ θ θ , has a transformation with opposite sign to the chiral fermions λ → λ − ibc . Although the gravitino does not appear in the spectrum of a rigid supersymmetric theory,its contribution to the anomalies should be taken into account in section 5.1. It can be seenfrom eqs. (5.6) that the R-charge of the gravitino is the same as that of a gaugino. Thecontributions of the gravitino to the cubic anomaly and the mixed gravitational anomaly arediscussed in [110]. -SYMMETRIES AND ANOMALIES 127 (see eq. 2.16), we continue by analyzing the anomalous contributions to theLagrangian originating from a gauge symmetry under which the fermions carrya charge given by eqs. (C.7).We now define C A δ ab = Tr A (cid:2) R χ ( τ a τ b ) A (cid:3) + T G δ ab R λ , (C.8) C R = Tr (cid:2) R χ (cid:3) + n λ R λ + 3 R / , (C.9)where C A denotes the mixed anomaly of U (1) R with the Standard Model gaugegroups A = U (1) Y , SU (2) L , SU (3). In eq. (C.8), one has T G δ ab = f acd f bcd ,with T G = N for SU ( N ) and 0 for U (1). Tr R ( τ a τ b ) = δ ab T R is the traceover irreducible representations. The contribution of the gravitino to the cubicanomaly is three times that of a gaugino [110].The anomalous U (1) R generates a cubic anomaly. As a result, the variation ofthe Lagrangian does not vanish. Instead, it is given by δ L = − (1 / C R θ π (cid:15) µνρσ F µν F ρσ . (C.10)This anomaly can however be canceled by a Green-Schwarz mechanism (seeeqs. (4.9) and (4.11)). Cancellation of δ L GS + δ L = 0 gives β R c = − C R π . (C.11)This anomaly cancellation condition is to be compared with the result ineq. (5.10). Similarly, we have for the mixed anomalies ( U (1) R times StandardModel gauge groups) β A c = − C A π . (C.12)Regarding the mixed gravitational anomaly, one has δ L = − iθ C grav π R µν ˜ R µν , (C.13)where C grav = Tr[ R χ ] + n λ R λ + ( − R / . (C.14) The presence of Generalized Chern-Simons (GCS) terms [111–113] is assumed in eqs. (C.11)and (C.12). Note that there is an additional symmetry factor 1 /
28 R-SYMMETRIES AND ANOMALIES
Here it was used that the contribution of the gravitino is ( −
21) times thecontribution of a gaugino [110]. We assume that an (unknown) UV-completionof the theory cancels this anomaly due to a term in the Lagrangian L grav = iβ grav Im( s )4 R µν ˜ R µν . (C.15)The anomaly is canceled δ L + δ L grav = 0 if C grav = 96 π β grav c. (C.16)This indeed agrees with eqs. (5.11) for β grav = b grav / π . ppendix D Comments the case with p = 1 and non-zero β D.1 On the small value of β In this Appendix we first demonstrate how tunable dS vacua can be found forthe model (7.17) for a general β . In section 7.2 it was assumed that β (cid:28) β to bevery small.If β is not neglected, the D-term contribution to the scalar potential is given by V D = 12 + βb ( s + ¯ s ) (cid:18) κ − bc − κ − cs + ¯ s − q α ϕ α ¯ ϕ α (cid:19) . (D.1)The solution to ∂ V = V = 0 gives( α − (cid:0) α − (cid:1) + (cid:18) βα ( α − βα − (cid:19) (cid:0) − α − (cid:1) = 0 . (D.2)For β = 0, this gives indeed α ≈ − . β the result is plotted in figure D.1. The relation betweenthe parameters to obtain a vanishing cosmological constant becomes bc a = − αe α (2 + βα )( − α − )( α − = A ( α, β ) . (D.3) P = 1 AND NON-ZERO β
20 40 60 80 100 Β (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Α Figure D.1: The solutions to eq. (D.2) are plotted with α as a function of β .Although up to four (complex) solutions can exist for every β , only a singlesolution satisfies α < b ) and A ( α, β ) < β is plotted since the positivity is required to satisfythe anomaly cancellation conditions (see below).One concludes that for every (finite) β the vacuum is tunable. The gravitinomass eq. (7.21) is repeated here for convenience of the reader m / = κ − ae α/ r bα . The scalar soft mass squared and the trilinear couplings are given by m = m / ( σ s + 1) + κ − q α bc βα/ − αα ,A = m / ρ s ,ρ s = − α −
1) ( α − P ( α, β ) ξ ) ,P ( α, β ) = 2 αe α ( σ s + 1)( α − A ( α, β ) (cid:18) βα (cid:19) . (D.4)The relations (D.3) can be written as bc = − κm / e − α/ p αA ( α, β ) , (D.5)which are used to write the scalar soft mass squared as m = m / ( σ s + 1) + m / κ − q α Q ( α, β ) ,Q ( α, β ) = α − α e − α/ p αA ( α, β )1 + βα/ . (D.6) N THE SMALL VALUE OF β To avoid tachyonic masses, the charges of the MSSM fields should satisfy q α > q = − κm / σ s + 1 Q ( α, β ) . (D.7)Next, the anomaly cancellation conditions (see eqs. (7.30)) for the cubic anomalygive β = − f ( θ, θ L , θ Q ) q π bc = f ( θ, θ L , θ Q ) ξ ( κm / ) g ( α, β ) , (D.8)where eqs. (D.5) and (D.7) were used, and g ( α, β ) = − e α/ ( σ s + 1) π Q ( α, β ) p αA ( α, β ) . (D.9)From eq. (7.29) it follows that 9 . ≤ f ( θ, θ L , θ Q ) ≤
91, for all θ, θ L , θ Q ∈ ]0 , β > P ( α, β ) is plotted asa function of β , it follows that the trilinear coupling A has only a very slightdependence on β . We can therefore assume ξ < O (10) (as in section 7.2) sinceotherwise the trilinear coupling A would be too large to allow for a realisticelectroweak vacuum.In figure D.3, g ( α, β ) is plotted as a function of β . As is shown, we canapproximate g ( α, β ) by a linear function of β (In fact, g ( β ) is also plotted infigure D.3 and completely overlaps with the actual function g ( α, β )) g ( α, β ) ≈ g ( β ) = g (0) + ω β, (D.10)where g (0) = 0 . ,ω = 0 . . (D.11)For a given m / , ξ and f ( θ, θ L , θ Q ), one can then solve β = f ( θ, θ L , θ Q ) ξ ( κm / ) g ( β ) , to find β = g (0)1 / A − ω , (D.12)
32 COMMENTS THE CASE WITH P = 1 AND NON-ZERO β where A = f ( θ, θ L , θ Q ) ξ ( κm / ) . (D.13)For ξ < m / <
40 TeV and f ( θ, θ L , θ Q ) <
91, this corresponds to β (cid:46) O (10 − ). Note that for small β one can approximate eq. (D.12) by β (cid:46) g (0) A which leads to the same result.We conclude that the anomaly cancellation condition can only be satisfied forvery small β . D.2 On the inconsistency of γ = 0 In this Appendix we address the tunability of the scalar potential (4.12) forwhere p = 1, γ = 0 and non-zero β . In this case, the constraints ∂ V = V = 0give ( α − (cid:0) α − (cid:1) + ( α − (cid:0) − α − (cid:1) = 0 . (D.14)The real and negative solutions to this equation are α ≈ − .
47 and α ≈ − . h σ s i >
0, while the F-termcontribution to the scalar potential, which is at the minimum of the potentialgiven by V F = κ − | a | e α h s +¯ s i σ s , should be negative.Indeed, a vanishing cosmological constant can be found when b h s + ¯ s i = α ≈− .
39, while the other parameters are related by b c βa = − α e α ( − α − )( α − ≈ . . (D.15)However, due to the presence of a Green-Schwarz contribution (4.9) because ofthe gauge kinetic function which is linear in s , the model is not gauge invariant.One could argue that this can be solved by introducing an extra field z whichis charged under the U (1), where its charge is carefully chosen such that itsanomalous contribution to the variation of the Lagrangian cancels the Green-Schwarz contribution (4.11). However, an argument similar to the one presentedin section 5.2 for p = 2 shows that this is inconsistent. N THE INCONSISTENCY OF γ = 0 Β P (cid:72) Α , Β (cid:76) Figure D.2: A plot of P ( α, β ) given by eq. (D.4) as a function of β , where α and β are solutions of eq. (D.2). It follows that the trilinear coupling A (asa function of ξ ) has only a very small dependence on β . One should thereforefocus on ξ (cid:46)
10 (as in section 7.2) since otherwise | A | becomes too large toallow for a viable electroweak vacuum.
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