The No-Scale Multiverse at the LHC
Tianjun Li, James A. Maxin, Dimitri V. Nanopoulos, Joel W. Walker
aa r X i v : . [ h e p - ph ] A p r ACT-05-11, MIFPA-11-11
The No-Scale Multiverse at the LHC ∗ Tianjun Li,
1, 2
James A. Maxin, Dimitri V. Nanopoulos,
1, 3, 4 and Joel W. Walker George P. and Cynthia W. Mitchell Institute for Fundamental Physics,Texas A & M University, College Station, TX 77843, USA Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, P. R. China Astroparticle Physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, Woodlands, TX 77381, USA Academy of Athens, Division of Natural Sciences,28 Panepistimiou Avenue, Athens 10679, Greece Department of Physics, Sam Houston State University, Huntsville, TX 77341, USA
We present a contemporary perspective on the String Landscape and the Multiverse of plausiblestring, M- and F-theory vacua, seeking to demonstrate a non-zero probability for the existence ofa universe matching our own observed physics within the solution ensemble. We argue for theimportance of No-Scale Supergravity as an essential common underpinning for the spontaneousemergence of a cosmologically flat universe from the quantum “nothingness”. Our context is ahighly detailed phenomenological probe of No-Scale F - SU (5), a model representing the intersectionof the F -lipped SU (5) × U (1) X Grand Unified Theory (GUT) with extra TeV-Scale vector-likemultiplets derived out of F -theory, and the dynamics of No-Scale Supergravity. The latter in turnimply a very restricted set of high energy boundary conditions.We present a highly constrained “Golden Point” located near M / = 455 GeV and tan β = 15in the tan β − M / plane, and a highly non-trivial “Golden Strip” with tan β ≃ m t = 173 . . M / = 455-481 GeV, and M V = 691-1020 GeV, which simultaneously satisfies all theknown experimental constraints, featuring also an imminently observable proton decay rate. Wesupplement this bottom-up phenomenological perspective with a top-down theoretical analysis of theone-loop effective Higgs potential. A striking consonance is achieved via the dynamic determinationof tan β and M / for fixed Z -boson mass at the local minimum minimorum of the potential, thatbeing the secondary minimization of the spontaneously broken electroweak Higgs vacuum V min .By also indirectly determining the electroweak scale, we suggest that this constitutes a completeresolution of the Standard Model gauge hierarchy problem.Finally, we present the distinctive collider level signatures of No-Scale F - SU (5) for the √ s = 7 TeVLHC, with 1 fb − of integrated luminosity. The characteristic feature is a light stop and gluino,both sparticles lighter than all other squarks, generating a surplus of ultra-high multiplicity ( ≥ F - SU (5) events, while readily suppressing the contribution of all Standard Modelprocesses, and allowing moreover a clear differentiation from competing models of new physics, mostnotably minimal supergravity. Detection by the LHC of the ultra-high jet signal would constitute asuggestive evocation of the intimately linked stringy origins of F - SU (5), and could provide a glimpseinto the underlying structure of the fundamental string moduli, possibly even opening a darkenedglass upon the hidden workings of the No-Scale Multiverse. PACS numbers: 11.10.Kk, 11.25.Mj, 11.25.-w, 12.60.Jv
I. INTRODUCTION
The number of consistent, meta-stable vacua of string,M- or (predominantly) F-theory flux compactificationswhich exhibit broadly plausible phenomenology, includ-ing moduli stabilization and broken supersymmetry [1–6], is popularly estimated [7, 8] to be of order 10 . Itis moreover currently in vogue to suggest that degen-eracy of common features across these many “universes” ∗ Submitted to the Fifth International Workshop DICE 2010:Space, Time, Matter - Current Issues in Quantum Mechanics andBeyond, September 13-17, 2010, Castello Pasquini, Italy, based onthe invited talk by D.V.N. might statistically isolate the physically realistic universefrom the vast “landscape”, much as the entropy functioncoaxes the singular order of macroscopic thermodynam-ics from the chaotic duplicity of the entangled quantummicrostate. We argue here though the counter point thatwe are not obliged a priori to live in the likeliest of alluniverses, but only in one which is possible. The exis-tence merely of a non-zero probability for our existenceis sufficient.We indulge for this effort the fanciful imagination thatthe “Multiverse” of string vacua might exhibit some lit-eral realization beyond our own physical sphere. A sin-gle electron may be said to wander all histories throughinterfering apertures, though its arrival is ultimately reg-istered at a localized point on the target. The journey tothat destination is steered by the full dynamics of thetheory, although the isolated spontaneous solution re-flects only faintly the richness of the solution ensemble.Whether the Multiverse be reverie or reality, the con-ceptual superset of our own physics which it embodiesmust certainly represent the interference of all navigableuniversal histories.Surely many times afore has mankind’s notion of theheavens expanded - the Earth dispatched from its cen-tral pedestal in our solar system and the Sun renderedone among some hundred billion stars of the Milky Way,itself reduced to one among some hundred billion galax-ies. Finally perhaps, we come to the completion of ourOdyssey, by realizing that our Universe is one of at least10 so possible, thus rendering the anthropic view ofour position in the Universe (environmental coincidencesexplained away by the availability of 10 × solarsystems) functionally equivalent to the anthropic view ofthe origin of the Universe (coincidences in the form andcontent of physical laws explained away by the availabil-ity, through dynamical phase transitions, of 10 uni-verses). Nature’s bounty has anyway invariably trumpedour wildest anticipations, and though frugal and equani-mous in law, she has spared no extravagance or whimsyin its manifestation.Our perspective should not be misconstrued, however,as complacent retreat into the tautology of the weak an-thropic principle. It is indeed unassailable truism that anobserved universe must afford and sustain the life of theobserver, including requisite constraints, for example, onthe cosmological constant [9] and gauge hierarchy. Ourpoint of view, though, is sharply different; We should beable to resolve the cosmological constant and gauge hier-archy problems through investigation of the fundamentallaws of our (or any single) Universe, its accidental andspecific properties notwithstanding, without resorting tothe existence of observers. In our view, the observer is theoutput of, not the raison d’ˆetre of, our Universe. Thus,our attention is advance from this base camp of our ownphysics, as unlikely an appointment as it may be, to thesummit goal of the master theory and symmetries whichgovern all possible universes. In so seeking, our first halt-ing forage must be that of a concrete string model whichcan describe Nature locally. II. THE ENSEMBLE MULTIVERSE
The greatest mystery of Nature is the origin of theUniverse itself. Modern cosmology is relatively clear re-garding the occurrence of a hot big bang, and subsequentPlanck, grand unification, cosmic inflation, lepto- andbaryogenesis, and electroweak epochs, followed by nucle-osynthesis, radiation decoupling, and large scale struc-ture formation. In particular, cosmic inflation can ad-dress the flatness and monopole problems, explain ho-mogeneity, and generate the fractional anisotropy of thecosmic background radiation by quantum fluctuation ofthe inflaton field [10–14]. A key question though, is from whence the energy of the Universe arose. Interestingly,the gravitational field in an inflationary scenario can sup-ply the required positive mass-kinetic energy, since its po-tential energy becomes negative without bound, allowingthat the total energy could be exactly zero.Perhaps the most striking revelation of the post-WMAP [15–17] era is the decisive determination thatour Universe is indeed globally flat, i.e. , with the netenergy contributions from baryonic matter ≃ ≃ ≃
73% finely balanced against the gravitationalpotential. Not long ago, it was possible to imagine theUniverse, with all of its physics intact, hosting any ar-bitrary mass-energy density, such that “ k = +1” wouldrepresent a super-critical cosmology of positive curvature,and “ k = −
1” the sub-critical case of negative curvature.In hindsight, this may come to seem as na¨ıve as the no-tion of an empty infinite Cartesian space. The observedenergy balance is highly suggestive of a fundamental sym-metry which protects the “ k = 0” critical solution, suchthat the physical constants of our Universe may not bedivorced from its net content.This null energy condition licenses the speculative con-nection ex nihilo of our present universe back to the pri-mordial quantum fluctuation of an external system. In-deed, there is nothing which quantum mechanics abhorsmore than nothingness. This being the case, an extrauniverse here or there might rightly be considered no ex-tra trouble at all! Specifically, it has been suggested [10–12, 18, 19] that the fluctuations of a dynamically evolvedexpanding universe might spontaneously produce tunnel-ing from a false vacuum into an adjacent (likely also false)meta-stable vacuum of lower energy, driving a local in-flationary phase, much as a crystal of ice or a bubbleof steam may nucleate and expand in a super-cooled orsuper-heated fluid during first order transition. In this“eternal inflation” scenario, such patches of space willvolumetrically dominate by virtue of their exponentialexpansion, recursively generating an infinite fractal arrayof causally disconnected “Russian doll” universes, nestingeach within another, and each featuring its own uniquephysical parameters and physical laws.From just the specific location on the solution “target”where our own Universe landed, it may be impossible todirectly reconstruct the full theory. Fundamentally, itmay be impossible even in principle to specify why ourparticular Universe is precisely as it is. However, super-string theory and its generalizations may yet present tous a loftier prize - the theory of the ensemble Multiverse. III. THE INVARIANCE OF FLATNESS
More important than any differences between variouspossible vacua are the properties which might be invari-ant, protected by basic symmetries of the underlyingmechanics. We suppose that one such basic propertymust be cosmological flatness, so that the seedling uni-verse may transition dynamically across the boundaryof its own creation, maintaining a zero balance of somesuitably defined energy function. In practice, this im-plies that gravity must be ubiquitous, its negative poten-tial energy allowing for positive mass and kinetic energy.Within such a universe, quantum fluctuations may notagain cause isolated material objects to spring into ex-istence, as their net energy must necessarily be positive.For the example of a particle with mass m on the surfaceof the Earth, the ratio of gravitational to mass energy ismore than nine orders of magnitude too small (cid:12)(cid:12)(cid:12)(cid:12) − G N M E mR E (cid:12)(cid:12)(cid:12)(cid:12) ÷ mc ≃ × − , (1)where G N is the gravitational constant, c is the speedof light, and M E and R E are the mass and radius ofthe Earth, respectively. Even in the limiting case of aSchwarzschild black hole of mass M BH , a particle of mass m at the horizon R S = 2 G N M BH /c has a gravitationalpotential which is only half of that required. (cid:12)(cid:12)(cid:12)(cid:12) − G N M BH mR S (cid:12)(cid:12)(cid:12)(cid:12) = 12 mc (2)It is important to note that while the energy density forthe gravitational field is surely negative in Newtonianmechanics, the global gravitational field energy is notwell defined in general relativity. Unique prescriptionsfor a stress-energy-momentum pseudotensor can be for-mulated though, notably that of Landau and Lifshitz.Any such stress-energy can, however, be made to vanishlocally by general coordinate transformation, and it isnot even entirely clear that the pseudotensor so applied isan appropriate general relativistic object. Given thoughthat Newtonian gravity is the classical limit of generalrelativity, it is reasonable to suspect that the properly de-fined field energy density will be likewise also negative,and that inflation is indeed consistent with a correctlygeneralized notion of constant, zero total energy.A universe would then be in this sense closed, an is-land unto itself, from the moment of its inception fromthe quantum froth; Only a universe in toto might sooriginate, emerging as a critically bound structure pos-sessing profound density and minute proportion, each asaccorded against intrinsically defined scales (the analo-gous Newton and Planck parameters and the propaga-tion speed of massless fields), and expanding or inflatinghenceforth and eternally. IV. THE INVARIANCE OF NO-SCALE SUGRA
Inflation, driven by the scalar inflaton field is itselfinherently a quantum field theoretic subject. However,there is tension between quantum mechanics and generalrelativity. Currently, superstring theory is the best candi-date for quantum gravity. The five consistent ten dimen-sional superstring theories, namely heterotic E × E , het- erotic SO (32), Type I, Type IIA, Type IIB, can be uni-fied by various duality transformations under an eleven-dimensional M-theory [20], and the twelve-dimensionalF-theory can be considered as the strongly coupled for-mulation of the Type IIB string theory with a varyingaxion-dilaton field [21]. Self consistency of the string (orM-, F-) algebra implies a ten (or eleven, twelve) dimen-sional master spacetime, some elements of which – six(or seven, eight) to match our observed four large di-mensions – may be compactified on a manifold (typicallyCalabi-Yau manifolds or G manifolds) which conservesa requisite portion of supersymmetric charges.The structure of the curvature within the extra dimen-sions dictates in no small measure the particular phe-nomenology of the unfolded dimensions, secreting awaythe “closet space” to encode the symmetries of all gaugedinteractions. The physical volume of the internal spatialmanifold is directly related to the effective Planck scaleand basic gauge coupling strengths in the external space.The compactification is in turn described by fundamen-tal moduli fields which must be stabilized, i.e. , givensuitable vacuum expectation values (VEVs).The famous example of Kaluza and Klein prototypesthe manner in which general covariance in five dimensionsis transformed to gravity plus Maxwell theory in four di-mensions when the transverse fifth dimension is cycledaround a circle. The connection of geometry to particlephysics is perhaps nowhere more intuitively clear thanin the context of model building with D − seconds in our Universe. Expan-sion and inflation appear to uniquely require propertieswhich arise naturally only in the No-Scale SUGRA for-mulation [22–26].SUSY is in this case broken while the vacuum energydensity vanishes automatically at tree level due to a suit-able choice of the K¨ahler potential, the function whichspecifies the metric on superspace. At the minimum ofthe null scalar potential, there are flat directions whichleave the compactification moduli VEVs undeterminedby the classical equations of motion. We thus receivewithout additional effort an answer to the deep ques-tion of how these moduli are stabilized; They have beentransformed into dynamical variables which are to be de-termined by minimizing corrections to the scalar poten-tial at loop order. In particular, the high energy grav-itino mass M / , and also the proportionally equivalentuniversal gaugino mass M / , will be established in thisway. Subsequently, all gauge mediated SUSY breakingsoft-terms will be dynamically evolved down from thisboundary under the renormalization group [27], estab-lishing in large measure the low energy phenomenology,and solving also the Flavour Changing Neutral Current(FCNC) problem. Since the moduli are fixed at a falselocal minimum, phase transitions by quantum tunnelingwill naturally occur between discrete vacua.The specific K¨ahler potential which we favor has beenindependently derived in both weakly coupled heteroticstring theory [28] and the leading order compactificationof M-theory on S /Z [29], and might be realized in F-theory models as well [30–33]. We conjecture, for thereasons given prior, that the No-Scale SUGRA construc-tion could pervade all universes in the String Landscapewith reasonable flux vacua. This being the case, intelli-gent creatures elsewhere in the Multiverse, though sep-arated from us by a bridge too far, might reasonably soconcur after parallel examination of their own physics.Moreover, they might leverage via this insight a deeperknowledge of the underlying Multiverse-invariant mastertheory, of which our known string, M-, and F-theoriesmay compose some coherently overlapping patch of thegarment edge. Perhaps we yet share appreciation, acrossthe cords which bind our 13.7 billion years to their corre-sponding blink of history, for the common timeless prin-ciples under which we are but two isolated condensationsupon two particular vacuum solutions among the physi-cal ensemble. V. AN ARCHETYPE MODEL UNIVERSE
Though we engage in this work lofty and speculativequestions of natural philosophy, we balance abstractionagainst the measured material underpinnings of concretephenomenological models with direct and specific con-nection to tested and testable particle physics. If thesuggestion is correct that eternal inflation and No-ScaleSUGRA models with string origins together describewhat is in fact our Multiverse, then we must as a prereq-uisite settle the issue of whether our own phenomenologycan be produced out of such a construction.In the context of Type II intersecting D-brane mod-els, we have indeed found one realistic Pati-Salam modelwhich might describe Nature as we observe it [34–36]. Ifonly the F-terms of three complex structure moduli are non-zero, we also automatically have vanishing vacuumenergy, and obtain a generalized No-Scale SUGRA. Itseems to us that the string derived Grand Unified Theo-ries (GUTs), and particularly the Flipped SU (5) × U (1) X models [37–39], are also candidate realistic string mod-els with promising predictions that can be tested at theLarge Hadron Collider (LHC), the Tevatron, and otherfuture experiments.Let us briefly review the minimal flipped SU (5) × U (1) X model [37–39]. The gauge group of the flipped SU (5) model is SU (5) × U (1) X , which can be embeddedinto SO (10). We define the generator U (1) Y ′ in SU (5)as T U(1) Y ′ = diag (cid:18) − , − , − , , (cid:19) . (3)The hypercharge is given by Q Y = 15 ( Q X − Q Y ′ ) . (4)In addition, there are three families of SM fermions whosequantum numbers under the SU (5) × U (1) X gauge groupare F i = ( , ) , ¯ f i = ( ¯5 , − ) , ¯ l i = ( , ) , (5)where i = 1 , , H = ( , ) , H = ( , − ) , (6) h = ( , − ) , h = ( ¯5 , ) . Interestingly, we can naturally solve the doublet-tripletsplitting problem via the missing partner mechanism [39],and then the dimension five proton decay from the col-ored Higgsino exchange can be highly suppressed [39].The flipped SU (5) × U (1) X models have been constructedsystematically in the free fermionic string constructionsat Kac-Moody level one previously [39–43], and in theF-theory model building recently [30–33, 44, 45].In the flipped SU (5) × U (1) X models, there are twounification scales: the SU (3) C × SU (2) L unification scale M and the SU (5) × U (1) X unification scale M F . Toseparate the M and M F scales and obtain true string-scale gauge coupling unification in free fermionic stringmodels [43, 46] or the decoupling scenario in F-theorymodels [44, 45], we introduce vector-like particles whichform complete flipped SU (5) × U (1) X multiplets. In or-der to avoid the Landau pole problem for the strong cou-pling constant, we can only introduce the following twosets of vector-like particles around the TeV scale [46] Z XF = ( , ) , XF = ( , − ) ; (7) Z XF , XF , Xl = ( , − ) , Xl = ( , ) , (8)where XF ≡ ( XQ, XD c , XN c ) , Xl ( , ) ≡ XE c . (9)In the prior, XQ , XD c , XE c , XN c have the same quan-tum numbers as the quark doublet, the right-handeddown-type quark, charged lepton, and neutrino, respec-tively. Such kind of the models have been constructedsystematically in the F-theory model building locally anddubbed F − SU (5) within that context [44, 45]. In thispaper, we only consider the flipped SU (5) × U (1) X mod-els with Z Z e | µ ) + π channels falls around 5 × years, testable at the future Hyper-Kamiokande [47] andDeep Underground Science and Engineering Laboratory(DUSEL) [48] experiments [49–51]. The lightest CP-evenHiggs boson mass can be increased [52], hybrid inflationcan be naturally realized, and the correct cosmic primor-dial density fluctuations can be generated [53]. VI. NO-SCALE SUPERGRAVITY
In the traditional framework, supersymmetry is brokenin the hidden sector, and then its breaking effects aremediated to the observable sector via gravity or gaugeinteractions. In GUTs with gravity mediated supersym-metry breaking, also known as the minimal supergrav-ity (mSUGRA) model, the supersymmetry breaking softterms can be parameterized by four universal parame-ters: the gaugino mass M / , scalar mass M , trilinearsoft term A , and the ratio of Higgs VEVs tan β at lowenergy, plus the sign of the Higgs bilinear mass term µ .The µ term and its bilinear soft term B µ are determinedby the Z -boson mass M Z and tan β after the electroweak(EW) symmetry breaking.To solve the cosmological constant problem, No-Scalesupergravity was proposed [22–26]. No-scale supergrav-ity is defined as the subset of supergravity models whichsatisfy the following three constraints [22–26]: (i) Thevacuum energy vanishes automatically due to the suitableK¨ahler potential; (ii) At the minimum of the scalar po-tential, there are flat directions which leave the gravitinomass M / undetermined; (iii) The super-trace quantityStr M is zero at the minimum. Without this, the largeone-loop corrections would force M / to be either zero orof Planck scale. A simple K¨ahler potential which satisfiesthe first two conditions is K = − T + T − X i Φ i Φ i ) , (10)where T is a modulus field and Φ i are matter fields. Thethird condition is model dependent and can always besatisfied in principle [54].The scalar fields in the above K¨ahler potential parame-terize the coset space SU ( N C +1 , / ( SU ( N C +1) × U (1)), where N C is the number of matter fields. Analogousstructures appear in the N ≥ N C = 4 for N = 5, whichcan be realized in the compactifications of string the-ory [28, 29]. The non-compact structure of the sym-metry implies that the potential is not only constantbut actually identical to zero. In fact, one can easilycheck that the scalar potential is automatically positivesemi-definite, and has a flat direction along the T field.Interestingly, for the simple K¨ahler potential in Equa-tion (10) we obtain the simplest No-Scale boundary con-dition M = A = B µ = 0, while M / may be non-zeroat the unification scale, allowing for low energy SUSYbreaking.It is important to note that there exist several meth-ods of generalizing No-Scale supergravity, for instance,the previously mentioned Type II intersecting D-branemodels [34–36], and the compactifications of M-theoryon S /Z with next-to-leading order corrections, wherewe have obtained a generalization employing modulusdominated SUSY breaking [56–60]. Similarly, miragemediation for the flux compactifications can be consid-ered as another form of generalized No-Scale supergrav-ity [61, 62]. In this paper we concentrate on the simplestNo-Scale supergravity, reserving any such generalizationsfor future study. VII. THE GOLDEN POINT
First, we would like to review the Golden Point of No-Scale and no-parameter F - SU (5) [63]. In the No-Scalecontext, we impose M = A = B µ = 0 at the unifica-tion scale M F , and allow distinct inputs for the singleparameter M / ( M F ) to translate under the RGEs todistinct low scale outputs of B µ and the Higgs mass-squares M H u and M H d . This continues until the point ofspontaneous breakdown of the electroweak symmetry at M H u + µ = 0, at which point minimization of the bro-ken potential establishes the physical low energy valuesof µ and tan β . In practice however, this procedure is atodds with the existing SuSpect 2.34 code [64] base fromwhich our primary routines have been adapted. In orderto impose the minimal possible refactoring, we have in-stead opted for an inversion wherein M / and tan β floatas two effective degrees of freedom. Thus, we do not fix B µ ( M F ). We take µ > g µ − MicrOMEGAs2.1 [65] wherein the revised
SuSpect
RGEs have alsoimplemented. We use a top quark mass of m t = 173.1GeV [66] and employ the following experimental con-straints: (1) The WMAP 7-year measurements of thecold dark matter density [15–17], 0.1088 ≤ Ω χ ≤ χ to be larger than the upper bound due toa possible O (10) dilution factor [67] and to be smallerthan the lower bound due to multicomponent dark mat-ter. (2) The experimental limits on the FCNC process, b → sγ . We use the limits 2 . × − ≤ Br ( b → sγ ) ≤ . × − [68, 69]. (3) The anomalous magnetic mo-ment of the muon, g µ −
2. We use the 2 σ level bound-aries, 11 × − < ∆ a µ < × − [70]. (4) The pro-cess B s → µ + µ − where we take the upper bound to be Br ( B s → µ + µ − ) < . × − [71]. (5) The LEP limiton the lightest CP-even Higgs boson mass, m h ≥ FIG. 1: Viable parameter space in the tan β − M / plane.The “Golden Point” is annotated. The thin, dark green linedenotes the WMAP 7-year central value Ω χ = 0 . p → ( e | µ ) + π proton lifetime pre-dictions, in units of 10 years. In the tan β − M / plane, B µ ( M F ) is then calcu-lated along with the low energy supersymmetric parti-cle spectrum and checks on various experimental con-straints. The subspace corresponding to a No-Scalemodel is clearly then a one dimensional slice of this man-ifold, as demonstrated in Figure 1. It is quite remarkablethat the B µ ( M F ) = 0 contour so established runs suffi-ciently perpendicular to the WMAP strip that the pointof intersection effectively absorbs our final degree of free-dom, creating what we have labeled as a No-ParameterModel. It is truly extraordinary however that this in-tersection occurs exactly at the centrally preferred relicdensity, that being our strongest experimental constraint.We emphasize again that there did not have to be an ex-perimentally viable B µ ( M F ) = 0 solution, and that theconsistent realization of this scenario depended cruciallyon several uniquely identifying characteristics of the un-derlying proposal. Specifically again, it appears that theNo-Scale condition comes into its own only when appliedat near the Planck mass, and that this is naturally identi-fied as the point of the final F - SU (5) unification, which is naturally extended and decoupled from the primary GUTscale only via the modification to the RGEs from theTeV scale F -theory vector-like multiplet content. Theunion of our top-down model based constraints with thebottom-up experimental data exhausts the available free-dom of parameterization in a uniquely consistent andpredictive manner, phenomenologically defining a trulyGolden Point near M / = 455 GeV and tan β = 15 GeV. TABLE I: Spectrum (in GeV) for the Golden Point of Fig-ure 1. Here, Ω χ = 0.1123, σ SI = 1 . × − pb, and h σv i γγ = 1 . × − cm /s . The central prediction for the p → ( e | µ ) + π proton lifetime is 4 . × years. e χ e χ ± e e R e t e u R m h . e χ e χ ± e e L e t e u L m A,H e χ e ν e/µ e τ e b e d R m H ± e χ e ν τ e τ e b e d L e g Our Golden point features M / = 455 . β =15 .
02, and is in full compliance with the WMAP 7-year results, with Ω χ = 0.1123. It also satisfies theCDMS II [74], Xenon 100 [75], and FERMI-LAT spacetelescope constraints [76], with σ SI = 1 . × − pband h σv i γγ = 1 . × − cm /s . The proton lifetime isabout 4 . × years, which is well within reach of theupcoming Hyper-Kamiokande [47] and DUSEL [48] ex-periments. Inspecting the supersymmetric particle andHiggs spectrum for the Golden Point of Table I revealsthat the additional contribution of the 1 TeV vector-likeparticles lowers the gluino mass quite dramatically. Thegluino mass M runs flat from the M unification scaleto 1 TeV as shown in Figure 2, though, due to supersym-metric radiative corrections, the physical gluino mass atthe EW scale is larger than M at the M scale. This istrue for the full parameter space. For our data point, theLSP neutralino is 99.8% Bino. Similarly to the mSUGRApicture, our this point is in the stau-neutralino coannihi-lation region, but the gluino is lighter than the squarksin our models.We plot gauge coupling and gaugino mass unifica-tion for the Golden Point in Figure 2. The figure ex-plicitly demonstrates the two-step unification of flipped SU (5) × U (1) X . In addition, we present the RGE runningfor the µ term, the SUSY breaking scalar masses, trilin-ear A-terms, and bilinear B µ term in Figure 3. Note inparticular that the EW symmetry breaking occurs when H u + µ goes negative. VIII. THE GOLDEN STRIP
Second, we shall review the Golden Strip of correlatedtop quark, gaugino, and vectorlike mass in No-Scale, no-parameter F - SU (5) [77]. From the above discussions,only a small portion of viable parameter space is consis-tent with the B µ ( M F ) = 0 condition, which thus consti-tutes a strong constraint. Since the boundary value of M M i Q (GeV)M M i ( G e V ) FIG. 2: RGE Running of the SM gauge couplings and gaug-ino masses from the EW scale to the unification scale M F .There is a discontinuity in the running at M due to remix-ing of the hypercharge with the residual Abelian phase fromthe breaking of SU(5) the universal gaugino mass M / , and even the unifica-tion scale M F ≃ . × GeV itself, are established bythe low energy experiments via RGE running, we are notleft with any surviving scale parameters in the presentmodel. The floor of the “valley gorge” in Figure 4 rep-resents accord with the B µ = 0 target for variations in( M / , M V ). We fix tan β = 15, as appears to be rathergenerically required in No-Scale F - SU (5) to realize ra-diative EWSB and match the observed CDM density.We have allowed for uncertainty in the most sensitiveexperimental input, the top quark mass, by effectivelyredefining m t as an independent free parameter. Lessersensitivities to uncertainty in ( α s , M Z ) are included inthe ± B µ = 0.We have established that there is a two dimensional sheet(of some marginal thickness to recognize the mentioneduncertainty) defining | B µ ( M F ) | ≤ M / , M V , m t ) volume, as shown inFigures (5,6). This sheet is inclined in the region of in-terest at the very shallow angle of 0 . ◦ to the ( M / , M V )plane, such that m t is largely decoupled from variationin the plane.The ( g − µ and b → sγ constraints vary most stronglywith M / . The two considered effects are each at theirlower limits at the boundary, but they exert pressurein opposing directions on M / due to the fact that theleading gaugino and squark contributions to Br ( b → sγ )enter with an opposing sign to the SM term and Higgscontribution. For the non-SM contribution to ∆ a µ , theeffect is additive, and establishes an upper mass limit on M / . Incidentally, the same experiment forms the cen-tral rationale for the adoption of sign( µ ) >
0, such thatappropriate interference terms between SM and SUSY -1250-1000-750-500-250025050075010001250 10 -2500-2000-1500-1000-50005001000150020002500 T r ili nea r A - t e r m s ( G e V ) m Hu + ~ m uR ~ m uL ~ m tR ~ m eL ~ m L ~ m eR ~ m R ~ A ~ A e~ A t ~ A u~ A b~ A d ~ m dR ~ m bR ~ m tL M M EWSB m Hd M a ss P a r a m e t e r s ( G e V ) Q (GeV) m Hu B M F FIG. 3: RGE Running of the µ term and SUSY breaking softterms from the EW scale to the unification scale M F . M M V |B ( M F )| . . . . . . FIG. 4: The B µ = 0 target for variations in ( M / , M V ), withtan β = 15. The specific m t which is required to minimize | B µ ( M F ) | is annotated along the solution string. contributions are realized. Conversely, the requirementthat SUSY contributions to Br ( b → sγ ) not be overlylarge, undoing the SM effect, requires a sufficiently large, i.e. lower bounded, M / . The WMAP-7 CDM measure-ment, by contrast, exhibits a fairly strong correlationwith both ( M / , M V ), cross-cutting the M / bound,and confining the vectorlike mass to 691-1020 GeV. Wenote that the mixing of the SM fermions and vector par- > WMAP 7-yr Br(b s) m t M M V |B(M F )|<1 Hyper-surface (thickness 0.3 GeV & inclined 0.2 above M -M V plane) Golden Strip { WMAP 7-yr a Br ( b s ) |B (M F )|<1 Hyper-surface } a . . FIG. 5: With tan β ≃
15 fixed by WMAP-7, the residual pa-rameter volume is three dimensional in ( M / , M V , m t ), withthe | B µ ( M F ) | ≤ . ◦ ) incline above the ( M / , M V ) plane.FIG. 6: A flattened presentation of the the ( M , M V ) planedepicted in Figure 5. The overlayed blue contours mark the p → ( e | µ ) + π proton lifetime prediction, in units of 10 years. ticles may give additional contributions to Br ( b → sγ )and ∆ a µ , but we do not consider them here.The intersection of these three key constraints with the | B µ ( M F ) | ≤ F - SU (5). All of theprior is accomplished with no reference to the experimen-tal top quark mass, redefined here as a free input. How-ever, the extremely shallow angle of inclination (0 . ◦ ) ofthe | B µ ( M F ) | ≤ M / , M V ) plane andinto the m t axis causes the Golden Strip to imply an ex-ceedingly narrow range of compatibility for m t , between173 . . m t = 173 . ± . σ SI = 1 . . × − pb. Likewise, the allowed regionsatisfies the Fermi-LAT space telescope constraints [76],with the photon-photon annihilation cross section h σv i γγ ranging from h σv i γγ = 1 . . × − cm /s . TABLE II: Spectrum (in GeV) for the benchmark point.Here, M / = 464 GeV, M V = 850 GeV, m t = 173.6 GeV,Ω χ = 0.112, σ SI = 1 . × − pb, and h σv i γγ = 1 . × − cm /s . The central prediction for the p → ( e | µ ) + π proton lifetime is 4 . × years. The lightest neutralino is99.8% Bino. e χ e χ ± e e R e t e u R m h . e χ e χ ± e e L e t e u L m A,H e χ e ν e/µ e τ e b e d R m H ± e χ e ν τ e τ e b e d L e g IX. THE SUPER NO-SCALE MECHANISM
In the following sections, we would like to review ourstudy on Super No-Scale F - SU (5) [78, 79]. The singlerelevant modulus field in the simplest string No-Scale su-pergravity is the K¨ahler modulus T , a characteristic ofthe Calabi-Yau manifold, the dilaton coupling being ir-relevant. The F-term of T generates the gravitino mass M / , which is proportionally equivalent to M / . Ex-ploiting the simplest No-Scale boundary condition at M F and running from high energy to low energy under theRGEs, there can be a secondary minimization, or min-imum minimorum , of the minimum of the Higgs poten-tial V min for the EWSB vacuum. Since V min depends on M / , the gaugino mass M / is consequently dynami-cally determined by the equation dV min /dM / = 0, aptlyreferred to as the “Super No-Scale” mechanism [78, 79].It could easily have been that in consideration of theabove technique, there were: A) too few undeterminedparameters, with the B µ = 0 condition forming an in-compatible over-constraint, and thus demonstrably false,or B) so many undetermined parameters that the dy-namic determination possessed many distinct solutions,or was so far separated from experiment that it could notpossibly be demonstrated to be true. The actual state ofaffairs is much more propitious, being specifically as fol-lows. The three parameters M , A, B µ are once againidentically zero at the boundary because of the definingK¨ahler potential, and are thus known at all other scalesas well by the RGEs. The minimization of the Higgsscalar potential with respect to the neutral elements ofboth SUSY Higgs doublets gives two conditions, the firstof which fixes the magnitude of µ . The second condition,which would traditionally be used to fix B µ , instead hereenforces a consistency relationship on the remaining pa-rameters, being that B µ is already constrained.In general, the B µ = 0 condition gives a hypersurfaceof solutions cut out from a very large parameter space.If we lock all but one parameter, it will give the finalvalue. If we take a slice of two dimensional space, ashas been described, it will give a relation between twoparameters for all others fixed. In a three-dimensionalview with B µ on the vertical axis, this curve is the “flatdirection” line along the bottom of the trench of B µ = 0solutions. In general, we must vary at least two parame-ters rather than just one in isolation, in order that theirmutual compensation may transport the solution alongthis curve. The most natural first choice is in some sensethe pair of prominent unknown inputs M / and tan β ,as was demonstrated in Ref. [78, 79].Having come to this point, it is by no means guaranteedthat the potential will form a stable minimum. It mustbe emphasized that the B µ = 0 No-Scale boundary con-dition is the central agent affording this determination, asit is the extraction of the parameterized parabolic curveof solutions in the two compensating variables which al-lows for a localized, bound nadir point to be isolated bythe Super No-Scale condition, dynamically determining both parameters. The background surface of V min for thefull parameter space outside the viable B µ = 0 subset is,in contrast, a steadily inclined and uninteresting func-tion. Although we have remarked that M / and tan β have no directly established experimental values, they areseverely indirectly constrained by phenomenology in thecontext of this model [63, 77]. It is highly non-trivialthat there should be accord between the top-down andbottom-up perspectives, but this is indeed precisely whathas been observed [78, 79]. F )=0) Gold Point
Minimum Minimorum tan Vmin(h)M
FIG. 7: The minimum V min of the Higgs effective potential(green curve, GeV) is plotted as a function of M / (GeV)and tan β , emphasizing proximity of the “Golden Point” ofRef. [63] to the dynamic region of the V min minimorum .
15 481
Minimum Minimorum Golden Strip t an M (GeV) Electroweak Vacua (B (M F )= 0) FIG. 8: The projection onto the ( M / ,tan β ) plane of Fig-ure 7 is further detailed, expanding to span the boundarycases of the Ref. [77] “Golden Strip”. The symmetry axis ofthe B µ = 0 parabola is rotated slightly above the M / axis. X. THE MINIMUM MINIMORUM OF THEHIGGS POTENTIAL: FIXING M Z We employ an effective Higgs potential in the ’t Hooft-Landau gauge and the DR scheme, given summing thefollowing neutral tree ( V ) and one loop ( V ) terms. V = ( µ + m H u )( H u ) + ( µ + m H d )( H d ) − µB µ H u H d + g + g Y (cid:2) ( H u ) − ( H d ) (cid:3) V = X i n i π m i ( φ ) (cid:18) ln m i ( φ ) Q − (cid:19) (11)Above, m H u and m H d are the SUSY breaking soft massesof the Higgs fields H u and H d , g and g Y are the gaugecouplings of SU (2) L and U (1) Y , n i and m i ( φ ) are thedegree of freedom and mass for φ i , and Q is the renor-malization scale. We include the complete Minimal Su-persymmetric Standard Model (MSSM) contributions toone loop, following Ref. [80], although the result is phe-nomenologically identical accounting only the leadingtop and partner stop terms. Since the minimum ofthe electroweak (EW) Higgs potential V min depends im-plicitly on M / , the gravitino mass is determined by dV min /dM / = 0. Being that M / is proportional to M / , it is equivalent to employ M / directly as ourmodulus parameter, from which all other SUSY break-ing soft terms here derive. In our numerical results inthe figures, we shall designate differences in the fourth-root of the effective Higgs potential as ∆ V min ( h ) ≡ V / ,measured in units of GeV relative to an arbitrary overallzero-offset.Factors explicit within the potential are obtained from0our customized extension of the SuSpect 2.34 [64] code-base, including a self-consistency assessment [63] on B µ = 0. We apply two-loop RGE running for theSM gauge couplings, and one-loop running for the SMfermion Yukawa couplings, µ term and soft terms.Studying V min generically in the ( M / , tan β ) plane,no point of secondary minimization is readily apparentin the strong, roughly linear, downward trend with re-spect to M / over the region of interest. However, themajority of the plane is not in physical communicationwith our model, disrespecting the fundamental B µ = 0condition. Isolating only the compliant contour withinthis surface, mirabile dictu , a parabola is traced withnadir alighting gentle upon our original Golden Point,as in Figure (7). Restoring parameterization freedom to( M V , m t ), we may scan across the corresponding GoldenPoint of each nearby universe variant, reconstructing intheir union the previously advertised Golden Strip, as inFigure (8). Notably, the theoretical restriction on tan β remains stable against variation in these parameters, ex-actly as its experimental counterpart. We find it quiteextraordinary that the phenomenologically preferred re-gion rests precisely at the curve’s locus of symmetric in-flection. Note in particular that it is the selection of theparabolic B µ = 0 contour out of the otherwise uninter-esting V min ( M / , tan β ) inclined surface which allows aclear minimum minimorum to be established. We reiter-ate that consistency of the dynamically positioned M / and tan β with the Golden Strip, implies consistency withall current experimental data.A strongly linear relationship is observed between theSUSY and EWSB scales with M EWSB ≃ . M / , suchthat a corresponding parabolic curve may be visualized.There is a charged stau LSP for tan β from 16 to 22, andwe connect points with correct EWSB smoothly on theplot in this region. If tan β is larger than 22, the stauis moreover tachyonic, so properly we must restrict allanalysis to tan β ≤ XI. THE GAUGE HIERARCHY PROBLEM
Not only must we explain stabilization of the elec-troweak scale against quantum corrections, but we mustalso explain why the electroweak scale and TeV-sizedSUSY breaking soft-terms are “initially” positioned sofar below the Planck mass. These latter componentsof the “gauge hierarchy” problem are the more subtle.In their theoretical pursuit, we do not though feign ig-norance of established experimental boundaries, takingthe phenomenologist’s perspective that pieces fit alreadyto the puzzle stipulate a partial contour of those yet tobe placed. Indeed, careful knowledge of precision elec-troweak scale physics, including the strong and electro-magnetic couplings, the Weinberg angle and the Z-massare required even to run the one loop RGEs. In thesecond loop one requires also minimally the leading topquark Yukawa coupling, as deduced from m t , and the Higgs VEV v ≡ p h H u i + h H d i ≃
174 GeV, estab-lished in turn from measurement of the effective Fermicoupling, or from M Z and the electroweak couplings.Reading the RGEs up from M Z , we take unification ofthe gauge couplings as evidence of a GUT. Reading themin reverse from a point of high energy unification, we takethe heaviness of the top quark, via its large Yukawa cou-pling, to dynamically drive the term M u + µ negative,triggering spontaneous collapse of the tachyonic vacuum, i.e. radiative electroweak symmetry breaking. Minimiza-tion of this potential with respect to the neutral compo-nents of H u and H d yields two conditions, which may besolved for µ ( M Z ) and B µ ( M Z ) in terms of the constrainedHiggs VEVs, which are in turn functions of M Z (consid-ered experimentally fixed) and tan β ≡ h H u i / h H d i (con-sidered a free parameter).Restricting to just the solution subset for which B µ ( M Z ) given by EWSB stitches cleanly onto that rundown under the RGEs from B µ ( M F ) = 0, tan β , orequivalently µ , becomes an implicit function of the sin-gle moduli M / ( M F ). The pinnacle of this construc-tion is the Super No-Scale condition dV min /dM / = 0,wherein M / , and thus also tan β , are dynamically es-tablished at the local minimum minimorum . By com-parison, the standard MSSM construction seems a hoax,requiring horrendous fine tuning to stabilize if viewed asa low energy supergravity limit, and moreover achievingTeV scale EW and SUSY physics as a simple shell gameby manual selection of TeV scale boundaries for the softterms M / , M , and A .Strictly speaking, having effectively exchanged inputof the Z-mass for a constraint on µ ( M F ), we dynami-cally establish the SUSY breaking soft term M / andtan β within the electroweak symmetry breaking vacua, i.e. with fixed v ≃
174 GeV. However, having predicted M F as an output scale near the reduced Planck mass,we are licensed to invert the solution, taking M F as ahigh scale input and dynamically addressing the gaugehierarchy through the standard story of radiative elec-troweak symmetry breaking. This proximity to the el-emental high scale of (consistently decoupled) gravita-tional physics, arises because of the dual flipped unifica-tion and the perturbing effect of the TeV multiplets, andis not motivated in standard GUTs. Operating the ma-chinery of the RGEs in reverse, we may transmute thelow scale M Z for the high scale M F , emphasizing thatthe fundamental dynamic correlation is that of the ratio M Z /M F , taking either as our input yardstick accordingto taste. For fixed M F ≃ × GeV, in a single breathwe receive the order of the electroweak scale, the Z-mass,the Higgs bilinear coupling µ , the Higgs VEVs, and allother dependent dimensional quantities, including pre-dictions for the full superparticle mass spectrum. It isin this sense that we claim a complete resolution of thegauge hierarchy problem, within the context of the SuperNo-Scale F - SU (5) model.1 XII. THE GUT HIGGS MODULUS
An alternate pair of parameters for which one may at-tempt to isolate a B µ = 0 curve, which we consider forthe first time in this work, is that of M / and the GUTscale M , at which the SU (3) C and SU (2) L couplingsinitially meet. Fundamentally, the latter corresponds tothe modulus which sets the total magnitude of the GUTHiggs field’s VEVs. M could of course in some sensebe considered a “known” quantity, taking the low energycouplings as input. Indeed, starting from the measuredSM gauge couplings and fermion Yukawa couplings atthe standard 91 .
187 GeV electroweak scale, we may cal-culate both M and the final unification scale M F , andsubsequently the unified gauge coupling and SM fermionYukawa couplings at M F , via running of the RGEs. How-ever, since the VEVs of the GUT Higgs fields H and H are considered here as free parameters, the GUT scale M must not be fixed either. As a consequence, the lowenergy SM gauge couplings, and in particular the SU (2) L gauge coupling g , will also run freely via this feedbackfrom M .We consider this conceptual release of a known quan-tity, in order to establish the nature of the model’s de-pendence upon it, to be a valid and valuable technique,and have employed it previously with specific regards to“postdiction” of the top quark mass value [77]. Indeed,forcing the theoretical output of such a parameter is onlypossible in a model with highly constrained physics, andit may be expected to meet success only by interventionof either grand coincidence or grand conspiracy of Na-ture.For this study, we choose a vector-like particle mass M V = 1000 GeV, and use the experimental top quarkmass input m t = 173 . M V = 1000 GeV is not an arbitrary one, sincea prior analysis [77] has shown that a 1 TeV vector-likemass is in compliance with all current experimental dataand the No-Scale B µ =0 requirement.In actual practice, the variation of M is achieved inthe reverse by programmatic variation of the Weinbergangle, holding the strong and electromagnetic couplingsat their physically measured values. Figure 9 demon-strates the scaling between sin ( θ W ), M (logarithmicaxis), and the Z -boson mass. The variation of M Z is at-tributed primarily to the motion of the electroweak cou-plings, the magnitude of the Higgs VEV being held es-sentially constant. We ensure also that the unified gaugecoupling, SM fermion Yukawa couplings, and specificallyalso the Higgs bilinear term µ ≃
460 GeV, are each heldstable at the scale M F to correctly mimic the previouslydescribed procedure.The parameter ranges for the variation depicted in Fig-ure 9 are M Z = 91 . − .
64, sin ( θ W ) = 0 . − . M = 1 . × − . × GeV, and like-wise also the same for Figures (10-16), which will fea-ture subsequently. The minimum minimorum falls atthe boundary of the prior list, dynamically fixing M ≃ FIG. 9: The interrelated variation of sin ( θ W ), the GUT scale M (logarithmic axis), and the Z -boson mass M Z is demon-strated for the parameter strips which preserve B µ = 0 and µ = 460 GeV at M F . The variation in M Z is linked dom-inantly to motion of the EW couplings via sin ( θ W ). Alsoshown is the corresponding predicted proton lifetime in theleading ( e | µ ) + π channels, in units of 10 years, with the cur-rent lower bound of 1 . × years indicated by the dashedhorizontal purple line. . × GeV and placing M / again in the vicinity of450 GeV. The low energy SM gauge couplings are simul-taneously constrained by means of the associated Wein-berg angle, with sin ( θ W ) ≃ . e | µ ) + π modes is2 . × − . × years [49, 50]. If the GUT scale M becomes excessively light, below about 7 × GeV,then proton decay would be more rapid than allowed bythe recently updated lower bound of 1 . × years fromSuper-Kamiokande [81].We are cautious against making a claim in preciselythe same vein for the dynamic determination of M Z ≃ . conjunction with the radiative electroweak symmetrybreaking [82, 83] numerically implemented within the SuSpect 2.34 code base [64], the fixing of the HiggsVEV and the determination of the electroweak scale mayalso plausibly be considered legitimate dynamic output, if one posits the M F scale input to be available a priori .The present minimization, referencing M / , M andtan β , is again dependent upon M V and m t , while thepreviously described [78] determination of tan β was, bycontrast, M V and m t invariant. Recognizing that a min-imization with all three parameters simultaneously ac-tive is required to declare all three parameters to havebeen simultaneously dynamically determined, we empha-size the mutual consistency of the results. We againstress that the new minimum minimorum is also con-sistent with the previously advertised Golden Strip, sat-isfying all presently known experimental constraints toour available resolution. It moreover also addresses theproblems of the SUSY breaking scale and gauge hierar-2chy [78], insomuch as M / is determined dynamically. tan V min (h)M Z Minimum Minimorum EWSB Vacua
FIG. 10: Three-dimensional graph of ( M Z , tan β, ∆ V min ( h ))space (green curve). The projections onto the three mutuallyperpendicular planes (red curves) are likewise shown. M Z and ∆ V min ( h ) are in units of GeV. The dynamically preferredregion, allowing for plausible variation, is circled and tippedin gold. M V min (h)M Z Minimum Minimorum
FIG. 11: Three-dimensional graph of ( M Z , M / , ∆ V min ( h ))space (green curve). The projections onto the three mutuallyperpendicular planes (red curves) are likewise shown. M Z , M / , and ∆ V min ( h ) are in units of GeV. The dynamicallypreferred region, allowing for plausible variation, is circledand tipped in gold. XIII. THE MINIMUM MINIMORUM OF THEHIGGS POTENTIAL: FIXING YUKAWACOUPLINGS AND µ AT M F We have revised the
SuSpect 2.34 code base [64] to in-corporate our specialized No-Scale F - SU (5) with vector-like mass algorithm, and accordingly employ two-loopRGE running for the SM gauge couplings, and one-loopRGE running for the SM fermion Yukawa couplings, µ term, and SUSY breaking soft terms. For our choice of M V = 1000 GeV, m t = 173 . µ ( M F ) ≃
450 452 454 456 458 460101520253035 750760770780790 tan V min (h)M
Minimum Minimorum
FIG. 12: Three-dimensional graph of ( M / , tan β, ∆ V min ( h ))space (green curve). The projections onto the three mutuallyperpendicular planes (red curves) are likewise shown. M / and ∆ V min ( h ) are in units of GeV. The dynamically preferredregion, allowing for plausible variation, is circled and tippedin gold. M V min (h)v = (v + v ) Minimum Minimorum
FIG. 13: Three-dimensional graph of ( v, M / , ∆ V min ( h ))space (green curve). The projections onto the three mutuallyperpendicular planes (red curves) are likewise shown. M / , v , and ∆ V min ( h ) are in units of GeV. The dynamically pre-ferred region, allowing for plausible variation, is circled andtipped in gold. GeV, we present the one-loop effective Higgs potential∆ V min ( h ) in terms of M Z and tan β in Figure 10, interms of M Z and M / in Figure 11, in terms of M / and tan β in Figure 12, and in terms of v and M / in Fig-ure 13, where v = p v u + v d , v u = h H u i , and v d = h H d i .These figures clearly demonstrate the localization of the minimum minimorum of the Higgs potential, corroborat-ing the dynamical determination of tan β ≃ −
20 and M / ≃
450 GeV in [78].Additionally, we exhibit the ( M Z , M / , M ) space inFigure 14, the ( M Z , M / , v ) space in Figure 15, and the( M Z , M / , g ) space in Figure 16, where g = p g + g Y .Figure 14 demonstrates that M ≃ . × GeV atthe minimum minimorum , which correlates to M Z ≃ M M M Z FIG. 14: Three-dimensional graph of ( M Z , M / , M ) space(blue curve). The projections onto the three mutually per-pendicular planes (red curves) are likewise shown. M Z , M / ,and M are in units of GeV. M v = (v + v ) M Z FIG. 15: Three-dimensional graph of ( M Z , M / , v ) space(purple curve). The projections onto the three mutually per-pendicular planes (red curves) are likewise shown. M Z , M / ,and v are in units of GeV. . ( θ W ) ≃ . M Z , M / , and M scales, as well as the electroweakgauge couplings, and the Higgs VEVs. The curves in eachof these figures represent only those points that satisfythe B µ = 0 requirement, as dictated by No-Scale super-gravity, serving as a crucial constraint on the dynamicallydetermined parameter space. Ultimately, it is the signif-icance of the B µ = 0 requirement that separates the No-Scale F - SU (5) with vector-like particles from the entirecompilation of prospective string theory derived models.By means of the B µ = 0 vehicle, No-Scale F - SU (5) hassurmounted the paramount challenge of phenomenology,that of dynamically determining the electroweak scale,the scale of fundamental prominence in particle physics.We wish to note that recent progress has been madein incorporating more precise numerical calculations into M g = (g + g ) M Z FIG. 16: Three-dimensional graph of ( M Z , M / , g ) space(royal blue curve). The projections onto the three mutuallyperpendicular planes (red curves) are likewise shown. M Z and M / are in units of GeV. our baseline algorithm for No-Scale F - SU (5) with vector-like particles. Initially, when we commenced the task offully developing the phenomenology of this model, the ex-treme complexity of properly numerically implementingNo-Scale F - SU (5) with vector-like particles compelled agradual strategy for construction and persistent enhance-ment of the algorithm. Preliminary findings of a precisionimproved algorithm indicate that compliance with the 7-year WMAP relic density constraints requires a slightupward shift to tan β ≃ −
20 from the value com-puted in Ref. [63], suggesting a potential convergence toeven finer resolution of the dynamical determination oftan β given by the Super No-Scale mechanism, and thevalue demanded by the experimental relic density mea-surements. We shall furnish a comprehensive analysis ofthe precision improved algorithm at a later date. XIV. PROBING THE BLUEPRINTS OF THENO-SCALE MULTIVERSE AT THE COLLIDERS
We offer here a brief summary of direct collider, de-tector, and telescope level tests which may probe theblueprints of the No-Scale Multiverse which we have laidout. As to the deep question of whether the ensemble beliteral in manifestation, or merely the conceptual super-set of unrealized possibilities of a single island Universe,we pretend no definitive answer. However, we have ar-gued that the emergence ex nihilo of seedling universeswhich fuel an eternal chaotic inflation scenario is particu-larly plausible, and even natural, within No-Scale Super-gravity, and our goal of probing the specific features ofour own Universe which might implicate its origins in thisconstruction are immediately realizable and practicable.The unified gaugino M / at the unification scale M F can be reconstructed from impending LHC events by de-termining the gauginos M , M , and M at the elec-troweak scale, which will in turn require knowledge of4the masses for the neutralinos, charginos, and the gluino.Likewise, tan β can be ascertained in principle from a dis-tinctive experimental observable, as was accomplished formSUGRA in [84]. We will not undertake a comprehen-sive analysis here of the reconstruction of M / and tan β ,but will offer for now a cursory examination of typicalevents expected at the LHC. We will present a detailedcompilation of the experimental observables necessary forvalidation of the No-Scale F - SU (5) at the LHC in thefollowing section.For the benchmark SUSY spectrum presented in Ta-ble III, we have adopted the specific values M / = 453,tan β = 15 and M Z = 91 . minimum minimorum a little bit, for example, withinthe encircled gold-tipped regions of the diagrams in theprior section. We have selected a ratio for tan β at thelower end of this range for consistency with our previousstudy [78], and to avoid stau dark matter. TABLE III: Spectrum (in GeV) for the benchmark point.Here, M / = 453 GeV, M V = 1000 GeV, m t = 173.1 GeV, M Z = 91.187 GeV, µ ( M F ) = 460.3 GeV, ∆ V min ( h ) = 748GeV, Ω χ = 0.113, σ SI = 2 × − pb, and h σv i γγ =1 . × − cm /s . The central prediction for the p → ( e | µ ) + π proton lifetime is around 4 . × years. The lightest neu-tralino is 99.8% Bino. e χ e χ ± e e R e t e u R m h . e χ e χ ± e e L e t e u L m A,H e χ e ν e/µ e τ e b e d R m H ± e χ e ν τ e τ e b e d L e g At the benchmark point, we calculate Ω χ = 0 .
113 forthe cold dark matter relic density. The phenomenologyis moreover consistent with the LEP limit on the lightestCP-even Higgs boson mass, m h ≥
114 GeV [72, 73], theCDMS II [74] and Xenon 100 [75] upper limits on thespin-independent cross section σ SI , and the Fermi-LATspace telescope constraints [76] on the photon-photon an-nihilation cross section h σv i γγ . The differential cross-sections and branching ratios have been calculated with PGS4 [85] executing a call to
PYTHIA 6.411 [86], usingour specialized No-Scale algorithm integrated into the
SuSpect 2.34 code for initial computation of the spar-ticle masses.The benchmark point resides in the region of theexperimentally allowed parameter space that generatesthe relic density through stau-neutralino coannihilation.Hence, the five lightest sparticles for this benchmarkpoint are e χ < e τ ± < e e R < e χ ∼ e χ ± . Here, thegluino is lighter than all the squarks with the exceptionof the lighter stop, so that all squarks will predominantlydecay to a gluino and hadronic jet, with a small per-centage of squarks producing a jet and either a e χ ± or e χ . The gluinos will decay via virtual (off-shell) squarksto neutralinos or charginos plus quarks, which will fur-ther cascade in their decay. The result is a low-energy tau through the processes e χ → e τ ∓ τ ± → τ ∓ τ ± e χ and e χ ± → e τ ± ν τ → τ ± ν τ e χ .The LHC final states of low-energy tau in the F - SU (5)stau-neutralino coannihilation region are similar to thosesame low-energy LHC final states in mSUGRA, however,in the stau-neutralino coannihilation region of mSUGRA,the gluino is typically heavier than the squarks. TheLHC final low-energy tau states in the stau-neutralinocoannihilation regions of F - SU (5) and mSUGRA willthus differ in that in F - SU (5), the low-energy tau stateswill result largely from neutralinos and charginos pro-duced by gluinos, as opposed to the low-energy tau statesin mSUGRA resulting primarily from neutralinos andcharginos produced from squarks.Also notably, the TeV-scale vector-like multiplets arewell targeted for observation by the LHC. We haveargued [77] that the eminently feasible near-term de-tectability of these hypothetical fields in collider exper-iments, coupled with the distinctive flipped charge as-signments within the multiplet structure, represents asmoking gun signature for Flipped SU (5), and have thuscoined the term flippons to collectively describe them.Immediately, our curiosity is piqued by the recent an-nouncement [87] of the DØ collaboration that vector-likequarks have been excluded up to a bound of 693 GeV,corresponding to the immediate lower edge of our antic-ipated range for their discovery [77]. XV. THE ULTRA-HIGH JET SIGNAL OFNO-SCALE F - SU (5) AT THE √ s = 7 TEV LHC
The Large Hadron Collider (LHC) at CERN has beenaccumulating data from √ s = 7 TeV proton-proton col-lisions since March 2010. It is expected to reach an in-tegrated luminosity of 1 fb − by the end of 2011, all insearch of new physics beyond the SM. SUSY, which pro-vides a natural solution to the gauge hierarchy problem,is the most promising extension of the SM. Data corre-sponding to a limited 35 pb − has already establishednew constraints on the viable parameter space [88–90]due to the unprecedented center of mass collision energy.The search strategy for SUSY signals in early LHC datahas been actively and eagerly studied by quite a fewgroups [91–95], with particular focus on the parameterspace featuring a traditional mass relationship betweensquarks and the gluino, such as a gluino heavier than allsquarks or a gluino lighter than all squarks.A question of great interest is whether there existSUSY models which are well motivated by a fundamentaltheory such as string theory, which can be tested in theinitial LHC run, permitting a probe of the UV physicsclose to the Planck scale. In this Section we presentsuch a model. It is well known that the supersymmetricflipped SU (5) × U (1) X models can solve the doublet-triplet splitting problem elegantly via the missing part-ner mechanism [40–42]. To realize the string scale gaugecoupling unification, two of us (TL and DVN) with Jiang5proposed the testable flipped SU (5) × U (1) X models withTeV-scale vector-like particles [46], where such modelscan be realized in the F -ree F -ermionic string construc-tions [43] and F -theory model building [44, 45], dubbed F - SU (5). In particular, we find the generic phenomeno-logical consequences are quite interesting [44, 45, 51].In the simplest No-Scale supergravity, all the SUSYbreaking soft terms arise from a single parameter M / .The spectra in the entire Golden Strip are therefore verysimilar up to a small rescaling on M / , with equivalentsparticle branching ratios. This leaves invariant most ofthe “internal” physical properties, whereas this rescal-ing ability on M / is not apparent in alternative SUSYmodels. For our analysis here, we use a vector-like par-ticle mass of M V ∼ F - SU (5) at the earlyLHC run.To represent our model for this phase of analysis, weselect the No-Scale F - SU (5) benchmark point of Ta-ble IV. The optimized signatures presented here offeran alluring testing vehicle for the stringy origin of F - SU (5). This point is again representative of the entirehighly constrained F - SU (5) viable parameter space. TheSUSY breaking parameters for this point slightly differfrom previous F - SU (5) studies [63, 77, 79] insomuch asmore precise numerical calculations have been incorpo-rated into our baseline algorithm. The masses shift afew GeV from the spectra given in previous work, butwhere different, we believe this to be the more accuraterepresentation. The branching ratios and decay modes ofthe spectrum in Table IV and the spectra in [63, 77, 79]are identical, so the physical properties are consistent be-fore and after code improvements. Thus, the signaturesstudied here will be common to the spectra provided pre-viously. TABLE IV: Spectrum (in GeV) for M / = 410 GeV, M V =1 TeV, m t = 174.2 GeV, tan β = 19.5. Here, Ω χ = 0.11 andthe lightest neutralino is 99.8% bino. e χ e χ ± e e R e t e u R m h . e χ e χ ± e e L e t e u L m A,H e χ e ν e/µ e τ e b e d R m H ± e χ e ν τ e τ e b e d L e g For the initial phase of generation of the low orderFeynman diagrams which may link the incoming beam tothe desired range of hard scattering intermediate states,we have used the program
MadGraph 4.4 [96]. Thesediagrams were subsequently fed into
MadEvent [96] forappropriate kinematic scaling to yield batches of MonteCarlo simulated parton level scattering events. The cas-caded fragmentation and hadronization of these eventsinto final state showers of photons, leptons, and mixed jets has been handled by
PYTHIA [86], with
PGS4 [85]simulating the physical detector environment. We im-plement MLM matching to preclude double counting offinal states, and use the CTEQ6L1 parton distributionfunctions to generate the SM background. All 2-bodySUSY processes are simulated. The b-jet tagging algo-rithm in
PGS4 is adjusted to update the b-tagging effi-ciency to ∼ p T <
100 GeV for the two leadingjets; p T <
350 GeV for all jets; pseudorapidity | η | > E/ T <
150 GeV; isolatedphoton with p T >
25 GeV; or isolated electron or muonwith p T >
10 GeV. Likewise, we discard any single jetwith | η | >
3. These cuts are quite standard, but alonethey are insufficient to reveal the ultra-high multiplicityjet event signature; We must also investigate the eventcut on the number of jets and the p T cut on a single jetto preserve ultra-high jet events.The detector simulations use the spectrum for the F - SU (5) point in Table IV. The most significant asset ofthe spectrum for our analysis is the relationship betweenthe stop, gluino, and other squarks. The distinctive masspattern of m e t < m e g < m e q is the smoking gun signatureand possibly a unique characteristic of only F - SU (5). Togain a comparison of the model studied here with morestandardized SUSY models, we examine the ten “Snow-mass Points and Slopes” (SPS) benchmark points [97]for suitable samples. We find that none of the ten SPSbenchmarks support the m e t < m e g < m e q mass pattern.This critical element is indicative of how unique the F - SU (5) signal could be. Previous minimal supersymmet-ric SM studies focused on signals from a low-multiplicityof jets, whereas the aforementioned mass pattern is ex-pected to show a very high-multiplicity of jets. For theSPS benchmarks, we only consider those spectra not lightenough to have been excluded by the initial phase of LHCdata, or those not too heavy for early LHC production.A few points satisfy these criteria, though we select onlyone since we anticipate the corollary points to exhibitanalogous characteristics. For our analysis here, we usethe SPS SP3 benchmark.Considering the large number of hadronic jets we areexamining for our signatures, there is little intrusion fromSM background processes after post-processing cuts. Weexamine the background processes studied in [91–95, 98]and our conclusion is that only the tt + jets possessesthe requisite minimum cross-section and sufficient num-ber of jets to intrude upon the F - SU (5) signatures. Pro-cesses with a higher multiplicity of top quarks can gen-erate events with a large number of jets, however, thecross-sections are sufficiently suppressed to be negligible,bearing in mind the large number of ultra-high jet eventswhich our model will generate. The same is true forthose more complicated background processes involvingcombinations of top quarks, jets, and one or more vec-tor bosons, where the production counts for 1 f b − ofluminosity are again sufficiently small. Furthermore, weneglect the QCD 2,3,4 jets, one or more vector bosons,6 -1 @7TeV Number of Jets / Event
Single Jet p T > 20 GeV F -SU(5) with vector-like particles mSUGRA Snowmass Benchmark SP3 Standard Model tt + jets Install cut @ 9 jets N u m be r o f E v en t s CMS Cuts (arXiv:1101.1628 [hep-ex]) -1 @7TeV 1fb -1 @7TeV Single Jet p T > 10 GeVInstall cut @ 11 or 12 jets FIG. 17: Distribution of events per number of jets. For clarity of the peaks, polynomials have been fitted over the histograms. and bb processes since none of these can sufficiently pro-duce events with 9 or more jets after post-processing cutshave been applied.The F - SU (5) with vector-like particles mass patternproduces events with a high multiplicity of virtual stops,which concludes in events with a very large number ofjets through the dominant chains e g → e t t → tt e χ → W + W − bb e χ and e g → e t t → bt e χ +1 → W − bb e τ +1 ν τ → W − bbτ + ν τ e χ , as well as the conjugate processes e g → e t t → tt e χ and e g → e t t → bt e χ − , where the W bosons willproduce mostly hadronic jets and some leptons. Addi-tionally, the heavy squarks will produce gluinos by meansof e q → q e g . In Figure 17 we plot the number of jets perevent versus the number of events for three distinct sce-narios. We suppress the noise on the histogram contourto admit a more lucid distinction of the peaks in thenumber of jets, and fit polynomials over the data pointsand conceal the histograms. This allows us to gauge anappropriate selection cut for the number of jets to max-imize our signal to background ratio, while assessing theimpact of the selection cuts implemented by the CMSCollaboration in [88, 99]. As depicted in Figure 17, thefirst pane displays a comparison of the number of jetswhen employing the prior CMS cuts, while the remainingtwo panes present the results for the post-processing se-lection cuts defined in this paper, discriminating betweentwo explicit cuts of the minimum p T for a single jet. Fig-ure 17 demonstrates that the CMS cuts of [88, 99] discardall the high-multiplicity jets, converting the events withat least 9 jets to events with few jets, thus, all informa-tion on these events with a large number of jets is lost.To retain the events with a high multiplicity of jets, weexplore alternative cuts by shifting the minimum p T fora single jet lower to the two cases of 10 GeV and 20GeV. A minimum jet p T of 20 GeV is secure from in-terfering with jet fragmentation, which typically occursin the realm below 10 GeV, indicating that 10 GeV is certainly fringe. We see in Figure 17 that both the 10GeV and 20 GeV jet p T cuts preserve the high numberof jets, permitting an obvious choice for location of thecut on the minimum number of jets. We thus adopt arevised cut of single jet p T >
20 GeV and total numberof jets greater than 9. To assess the discovery potential,we plot the number of events per 200 GeV versus H T ,where H T = P N jet i =1 E j i T . Figure 18 delineates the con-vincing separation between the F - SU (5) signal and theSM tt + jets and the SP3 point. The total number ofevents are summarized in Table V. We also include onemeasure of discovery threshold that compares the num-ber of signal events S to the number of background eventsB, where we require S √ B >
5. Notice that F - SU (5) com-fortably surpasses this requirement. TABLE V: Total number of events for 1 fb − and √ s = 7TeV. Minimum p T for a single jet is p T >
20 GeV. F - SU (5) SP tt + jetsEvents . . S √ B . . The spectrum of Table IV exceeds the LEP constraintson the lightest neutralino e χ and lightest stau e τ , andeven more tantalizing, the close proximity of the staumass beyond the LEP reach suggests imminent discov-ery at LHC. The stau presence can be reconstructed, forinstance, from the dominant F - SU (5) process e g → e t t → bt e χ +1 → W − bb e τ +1 ν τ → W − bbτ + ν τ e χ . The inference ofthe short-lived stau in the F - SU (5) SUSY breaking sce-nario from tau production assumes fruition of the ex-pected much improved tau detection efficiency at LHC.7
500 1000 1500 2000 2500 3000 3500051015202530 E v en t s / G e V H T (GeV) F -SU(5) with vector-like particles mSUGRA Snowmass Benchmark SP3 Standard Model tt + jets -1 @ 7 TeV p jT > 20 GeV FIG. 18: Counts for events with ≥ XVI. CONCLUSION
The advancement of human scientific knowledge andtechnology is replete with instances of science fictiontransitioning to scientific theory and eventually scientificfact. The conceptual notion of a “Multiverse” has longfascinated the human imagination, though this specula-tion has been largely devoid of a substantive underpin-ning in physical theory. The modern perspective pre-sented here offers a tangible foundation upon which le-gitimate discussion and theoretical advancement of theMultiverse may commence, including the prescription ofspecific experimental tests which could either falsify orenhance the viability of our proposal. Our perspectivediverges from the common appeals to statistics and theanthropic principle, suggesting instead that we may seekto establish the character of the master theory, of whichour Universe is an isolated vacuum condensation, basedon specific observed properties of our own physics whichmight be reasonably inferred to represent invariant com-mon characteristics of all possible universes. We havefocused on the discovery of a model universe consonantwith our observable phenomenology, presenting it as con-firmation of a non-zero probability of our own Universetranspiring within the larger String Landscape.The archetype model universe which we advance inthis work implicates No-Scale supergravity as the ubiq-uitous supporting structure which pervades the vacua ofthe Multiverse, being the crucial ingredient in the ema-nation of a cosmologically flat universe from the quan-tum “nothingness”. In particular, the model dubbedNo-Scale F - SU (5) has demonstrated remarkable consis-tency between parameters determined dynamically (thetop-down approach) and parameters determined throughthe application of current experimental constraints (thebottom-up approach). This enticing convergence of the- ory with experiment elevates No-Scale F - SU (5), in ourestimation, to a position as the current leading GUT can-didate. The longer term viability of this suggestion islikely to be greatly clarified in the next few years, basedupon the wealth of forthcoming experimental data.We have presented a highly constrained “GoldenPoint” located near M / = 455 GeV and tan β = 15in the tan β − M / plane, and a highly non-trivial“Golden Strip” with tan β ≃ m t = 173 . . M / = 455-481 GeV, and M V = 691-1020 GeV, whichsimultaneously satisfies all the known experimental con-straints, featuring moreover an imminently observableproton decay rate. In addition, we have studied the one-loop effective Higgs potential, and considered the “Super-No-Scale” condition. With a fixed Z -boson mass, we dy-namically determined tan β and M / at the local mini-mum minimorum of the Higgs potential, while simultane-ously indirectly determining the electroweak scale, thussuggesting a complete resolution of the gauge hierarchyproblem in the Standard Model (SM). Furthermore, fix-ing the SM fermion Yukawa couplings and µ term at the SU (5) × U (1) X unification scale, we dynamically deter-mine the ratio tan β ≃ −
20, the universal gauginoboundary mass M / ≃
450 GeV, and consequently alsothe total magnitude of the GUT-scale Higgs VEVs, whileconstraining the low energy SM gauge couplings. In par-ticular, these local minima minimorum lie within the pre-viously described “Golden Strip”, satisfying all currentexperimental constraints.The LHC era has long been anticipated for the ex-pected revelations of physics beyond the Standard Model,as the quest for experimental evidence and insight intothe structure of the underlying theory at high energiesis enticingly close at hand. Consequently, the field ofprospective supersymmetry models has grown as finger-prints of these models at LHC are studied. Nevertheless,our exploration of recently published signatures for su-persymmetry discovery reveals a common focus towardlow-multiplicity jet events. However, we showed herethat manipulation of LHC data skewed toward these lowjet events could mask an authentic supersymmetry sig-nal. We have offered a clear and convincing ultra-highjet multiplicity signal for events with at least nine jets,unmistakable for the Standard Model or minimal super-gravity. Notably, the optimized post-processing selectioncuts outlined here are essential for discovery of super-symmetry if F - SU (5) is indeed proximal to the physicalmodel. Our revised cuts are not drastic, with the twochief adjustments being lowering the minimum p T for asingle jet to 20 GeV, and raising the minimum numberof jets in an event to nine. Recognition of such a signalof stringy origin at the LHC could not only reveal theflipped nature of the high-energy theory, but might alsoshed light on the geometry of the hidden compactified six-dimensional manifold in the string derived models, andeven possibly on the hidden structure of the No-ScaleMultiverse.The blueprints which we have outlined here, integrat-8ing precision phenomenology with prevailing experimen-tal data and a fresh interpretation of the Multiverse andthe Landscape of String vacua, offer a logically connectedpoint of view from which additional investigation may bemounted. As we anticipate the impending stream of newexperimental data which is likely to be revealed in en-suing years, we look forward to serious discussion andinvestigation of the perspective presented in this work.Though the mind boggles to contemplate the implica-tions of this speculation, so it must also reel at eventhe undisputed realities of the Universe, these acknowl-edged facts alone being manifestly sufficient to humbleour provincial notions of longevity, extent, and largess. The stakes could not be higher or the potential revela-tions more profound. Acknowledgments
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