Super No-Scale F-SU(5): A Dynamic Determination of M_{1/2} and tan beta
Tianjun Li, James A. Maxin, Dimitri V. Nanopoulos, Joel W. Walker
aa r X i v : . [ h e p - ph ] S e p Super No-Scale F - SU (5): A Dynamic Determination of M / and tan β Tianjun Li a,b , James A. Maxin a , Dimitri V. Nanopoulos a,c,d , Joel W. Walker e, ∗ a George P. and Cynthia W. Mitchell Institute for Fundamental Physics,Texas A & M University, College Station, TX 77843, USA b Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, P. R. China c Astroparticle Physics Group, Houston Advanced Research Center (HARC),Mitchell Campus, Woodlands, TX 77381, USA d Academy of Athens, Division of Natural Sciences,28 Panepistimiou Avenue, Athens 10679, Greece e Department of Physics, Sam Houston State University, Huntsville, TX 77341, USA
Abstract
We study the Higgs potential in No-Scale F - SU (5), a model built on the tripodal foundations of the F -lipped SU (5) × U (1) X Grand Unified Theory, extra F -theory derived TeV scale vector-like particle multiplets,and the high scale boundary conditions of no-scale supergravity. V min , the minimum of the potential followingradiative electroweak symmetry breaking, is a function at fixed Z-Boson mass of the universal gauginoboundary mass M / and tan β , the ratio of Higgs vacuum expectation values. The so-scale nullificationof the bilinear Higgs soft term B µ at the boundary reduces V min ( M / ) to a one dimensional dependency,which may be secondarily minimized. This “Super No-Scale” condition dynamically fixes tan β and M / atthe local minimum minimorum of V min . Fantastically, the walls of this theoretically established secondarypotential coalesce in descent to a striking concurrency with the previously phenomenologically favored“Golden Point” and “Golden Strip”. Keywords:
No-Scale Supergravity, F-Theory, Vector-like Multiplets, Flipped SU(5), Grand Unification, HiggsPotential, Gauge Hierarchy
PACS:
1. Introduction and Background
We have recently demonstrated [1, 2] the uniquephenomenological consistency and profound predic-tive capacity of a model dubbed No-Scale F - SU (5),constructed from the merger of the F -lipped SU (5)Grand Unified Theory (GUT) [3–5], two pairs ofhypothetical TeV scale vector-like supersymmet-ric multiplets with origins in F -theory [6–10], andthe dynamically established boundary conditionsof no-scale supergravity [11–15]. It appears that ∗ Corresponding author:
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Email addresses: [email protected] (TianjunLi), [email protected] (James A. Maxin), [email protected] (Dimitri V. Nanopoulos), [email protected] (Joel W. Walker) the no-scale scenario, particularly vanishing of theHiggs bilinear soft term B µ , comes into its own onlywhen applied at an elevated scale, approaching thePlanck mass [16]. M F , the point of the second stage SU (5) × U (1) X unification, emerges in turn as asuitable candidate scale only when substantially de-coupled from the primary GUT scale unification of SU (3) C × SU (2) L via the modification to the renor-malization group equations (RGEs) from the extra F -theory vector multiplets [1, 2].Taking a definition of M V = 1 TeV for the newvector-like fields as an elemental model feature, weshowed [1] that the viable parameter space consis-tent with radiative electroweak symmetry breaking(EWSB), limits on the flavor changing neutral cur-rent ( b → sγ ) process and on contributions to themuon anomalous magnetic moment ( g − µ , runs Preprint submitted to Physics Letters B November 5, 2018 ufficiently perpendicular to both the B µ ( M F ) = 0and centrally observed WMAP 7 cold dark matter(CDM) relic density contours that the non-trivialmutual intersection is a narrowly confined “GoldenPoint” with a universal gaugino boundary mass M / around 455 GeV, and a ratio tan β = 15 ofHiggs vacuum expectation values (vevs). Insomuchas the collision of top-down model based constraintswith bottom-up experimental data effectively ab-sorbs the final dynamic degree of freedom, this waslabeled a No-Parameter Model.Advancing from the “Golden Point” to the“Golden Strip” [2], we relaxed the definition of thevector-like mass and studied the impact of fluctu-atating key electroweak reference data ( α s , M Z ) andthe top quark mass m t about the error margins.The most severe variation occurred for m t , via itsconnection to the large Yukawa coupling essentialto radiative EWSB. We recognized this dependenceby effectively treating m t as an additional input,selecting the appropriate value to restore a vanish-ing B µ ( M F ) at each point in the ( M / , tan β, M V )volume. The ( g − µ and ( b → sγ ) constraints,both at their lower limits, were found to exert op-posing pressures on M / due to operation of theformer in alignment with, and the latter in counter-balance of, the Standard Model (SM) leading term.Cross cutting by the WMAP CDM measurementcompleted demarcation of the strip, running diag-onally from about ( M / , M V ) = (455,1020) GeV,to (481,691) GeV, with tan β = 15 independentlyenforced for the full space. With parameterizationfreedom exhausted, the model was finally requiredto make a correlated postdiction for the top quarkmass. The result, m t = 173 . . . ± . M V is testable atthe LHC, and the partial lifetime for proton decayin the leading ( e | µ ) + π channels is 4 . × years,testable at the future Hyper-Kamiokande [18] andDUSEL [19] experiments.
2. The Super No-Scale Mechanism
In the present work we volunteer a small stepbackward to emphasize a giant leap forward. Hav-ing established practical bounds on the vector-like mass, we revert to a single conceptual uni-verse, ostensibly our own or one of sufficient phe-nomenological proximity, with M V = 1000 GeV,and m t = 173 . H u and H d imposesa pair of constraint equations which may be usedto eliminate any two free parameters of the set M / , B µ , tan β ≡ h H u i / h H d i , and the supersym-metry (SUSY) preserving bilinear Higgs mass term µ . The overall magnitude of the Higgs vev v ≡ p h H u i + h H d i ≃
174 GeV is considered to beexperimentally constrained by measurement of thegauge couplings and Z-Boson mass. Typically, onewill solve for µ ( M Z ) and B µ ( M Z ) in terms of theconstrained Higgs vevs and tan β , at fixed M / .We consider though that the no-scale boundarycondition B µ ( M F ) = 0 fixes the value of B µ atall other scales as well via action of the renormal-ization group. Restricting then to just the solutionsubset for which B µ ( M Z ) given by EWSB stitchescleanly onto that run down under the RGEs from B µ ( M F ) = 0, tan β (or alternatively µ ) becomesan implicit function of the single modulus M / .Concretely, we shall consider that the first EWSBconstraint absolutely establishes µ , and that thesecond gives a line of parameterized solutions forthe functional relationship between M / and tan β .We therefore distinguish the residual freedom in thedynamic modulus M / and parameter tan β by theability to exert direct influence on the Higgs poten-tial within a single physical parameterization.The crucial observation is that the minimizationof the Higgs potential is therefore at this stageincomplete. In no-scale supergravity, the specificstructure of the K¨ahler potential K leads to a con-tribution to the scalar potential which is is zero andflat at tree level, so that the gravitino mass M / ,or by proportional equivalence M / , is to be de-termined dynamically by radiative corrections. Inorder to finish specification of the physical vacuum,we must then secondarily minimize the Higgs po-tential with respect to the dependency on M / , adependency which is embodied in the bulk propor-tionality of the full low energy mass spectrum tothis SUSY breaking parameter [12, 15]. At this lo-cally smallest value of V min ( M / ), which we dubthe minimum minimorum , the dynamic determi-nation M / is established. Moreover, the implicitdependence of the parameter tan β on M / meansthat its value is also simultaneously provided bythe system dynamics. Henceforth, the impositionof dV min /dM / = 0 on the Higgs potential will bereferred to as the “Super No-Scale” condition.We emphasize that the justification for this pro-cedure traces back to the fact that the soft SUSYbreaking mass M / is related to the F-term of a2ynamic modulus. For example, in the weakly cou-pled heterotic E × E string theory, or in M-theoryon S /Z , M / is related to the F-term of a K¨ahlermodulus T . In string models, there exists a fun-damental question of how any such moduli are tobe stabilized. Thus, the physical motivation of theSuper No-Scale condition is the stabilization of theF-term of the modulus. Again, for each M / , wewill have an electroweak symmetry breaking vac-uum corresponding to minimization of the scalarHiggs potential. Among these minima, the mini-mum minimorum is the dynamically preferred lo-cally smallest minimum of the Higgs potential.We openly recognize that the potential affords anadditional dimensionality along the degree of free-dom which has been locked out by the fixing of v , and that minimization with respect to this ad-ditional parameter remains a question of interest.However, this is a delicate point of ongoing research,and beyond the scope of the current study. If oneaccepts, for the sake of argument, that the currentmodel fairly represents the physics of our Universe,then current experimental measurements guaran-tee that the potential along this direction is indeedbounded, not running away from the adopted con-stant value of v . It is therefore only the secondarybounding along the degree of freedom associatedwith M / which is experimentally unknown to us,and which may be predicted according to modelformulations such as the one here presented. F - SU (5) Models In the Flipped SU (5) GUTs, the gauge group is SU (5) × U (1) X , which embeds in SO (10). Gaugecoupling unification near 10 GeV strongly sug-gests the existence of a Grand Unified Theory(GUT). In minimal SUSY SU (5) models there areproblems with doublet-triplet splitting and dimen-sion five proton decay by colored Higgsino ex-change [5]. These difficulties are elegantly overcomein Flipped SU (5) GUT models via the missing part-ner mechanism [5]. The generator U (1) Y ′ is definedfor fundamental five-plets as − / / Q Y = ( Q X − Q Y ′ ) /
5. There arethree families of Standard Model (SM) fermions, apair of ten-plet Higgs for breaking the GUT sym-metry, and a pair of five-plet Higgs for EWSB.Historically, the first flipped F-theory SU (5)GUT was constructed in Ref. [20], and further as-pects of flipped SU (5) F-theory GUTs have been considered in [21–23]. We introduce in addition,vector-like particle multiplets, derived likewise inthe context of F-theory model building [6], to ad-dress the “little hierarchy” problem, altering thebeta coefficients of the renormalization group to dy-namically elevate the secondary SU (5) × U (1) X uni-fication at M F to near the Planck scale, while leav-ing the SU (3) C × SU (2) L unification at M closeto the traditional GUT scale. In other words, oneobtains true string-scale gauge coupling unificationin free fermionic string models [6, 24] or the decou-pling scenario in F-theory models [7, 8]. To avoida Landau pole for the strong coupling constant, weare restricted around the TeV scale to one of thefollowing two multiplet sets [6]. (cid:16) XF ( , ) ≡ ( XQ, XD c , XN c ) , XF ( , − ) (cid:17)(cid:0) Xl ( , − ) , Xl ( , ) ≡ XE c (cid:1) (1)Prior, XQ , XD c , XE c , XN c have the same quan-tum numbers as the quark doublet, right-handeddown-type quark, charged lepton, and neutrino, re-spectively. We have argued [2] that the eminentlyfeasible near-term detectability of these hypotheti-cal fields in collider experiments, coupled with thedistinctive flipped charge assignments of the multi-plet structure, represents a smoking gun signaturefor Flipped SU (5), and have thus coined the term flippons to collectively describe them. Immediately,our curiosity is piqued by the announcement [25] ofthe DØ collaboration that vector-like quarks havebeen excluded up to a bound of 693 GeV, corre-sponding to the lower edge of our golden strip. Wehere consider only the Z Z
4. No-Scale Supergravity
The Higgs boson, being a lorentz scalar, is notstable in the SM against quadratic quantum masscorrections which drive it toward the dominantPlanck scale, some seventeen orders of magnitudeabove the value required for consistent EWSB. Su-persymmetry naturally solves this fine tuning prob-lem by pairing the Higgs with a chiral spin-1 / M / , scalar mass M , trilinear coupling A , and thelow energy ratio tan β ), plus the sign of the Higgsbilinear mass term µ .No-Scale Supergravity was proposed [11–15] toaddress the cosmological flatness problem. It maybe verified for the simple example K¨ahler potential K = − T + T − X i Φ i Φ i ) , (2)where T is a modulus field and Φ i are matter fields,that the no-scale boundary conditions M = A = B µ = 0 are enforced automatically, while M / > i.e. running under the RGEs,to the classically flat potential. Additionally, thetree level vacuum energy vanishes automatically.The fiercely reductionist no-scale picture moreoverinherits an associative weight of motivation from itsrobustly generic and natural appearance in stringbased constructions.The simple from of Eq. (2) has been indepen-dently derived in both weakly coupled heterotic E × E string theory [26] and for strong coupling,in the leading order compactification of M-theoryon S /Z [27]. In both cases, the Yang-Mills fieldsspan a ten dimensional space-time. However, thispotential is not obtained directly out of F-theory, asrepresented for example by the strong coupling liftfrom Type IIB intersecting D-brane model build-ing with D7- and D3-branes [20, 28–30], wherethe Yang-Mills fields on the D7-branes occupy aneight dimensional space-time. Nevertheless, it iscertainly possible in principle to calculate a gaugekinetic function, Kahler potential and superpoten-tial in the context of Type IIB interecting D-branemodel building, and the F-theory could thus admita more general definition of no-scale supergravity, as realized by a K¨ahler potential like K = − ln( S + S ) − ln( T + T ) − ln( T + T ) − ln( T + T ) , (3)where only three of the moduli fields S and T i mayyield non-zero F-terms.However, the F - SU (5) type models under discus-sion have been constructed locally in F-theory [7, 8],and without a corresponding consistent global con-struction, we do not know the concrete K¨ahler po-tential of the SM fermions and Higgs fields, andcannot by this means explicitly calculate the su-persymmetry breaking scalar masses and trilinearsoft terms. Essentially then, we aim to study anF-theory inspired variety of low energy SUSY phe-nomenology, remaining agnostic as to the detailsof the K¨ahler structure. By studying the simplestno-scale supergravity, we may still however expectto encapsulate the correct leading order behavior.Should the favorable qualitative phenomenology ofthis lowest order analysis prove persistent, our fu-ture attention will be directed toward quantita-tively specific no-scale supergravity generalizations.
5. The Higgs Minimum Minimorum
We now proceed to specifically implement, withinthe context of the F - SU (5) construction, the Su-per No-Scale mechanism described in Section (2).Again, for a given Higgs vev, i.e. for a fixed Z-Boson mass, we establish tan β , by application ofthe two EWSB consistency conditions, to be an im-plicit function of the universal gaugino boundarymass M / , along a continuous string of minima ofthe broken Higgs potential V min , which are likewiselabeled by their value of M / . It is with respectto this line of solutions that we seek to establish alocal secondary minimum minimorum of the Higgspotential V min ( M / ).We employ an effective Higgs potential in the’t Hooft-Landau gauge and the DR scheme, givensumming the following neutral tree ( V ) and oneloop ( 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) (4)Prior, m H u and m H d are the soft SUSY breakingmasses of the Higgs fields H u and H d , g and g Y are4
200 400 600 800 100051015202530 8001000120014001600Electroweak Vacua (B (M F )=0) Gold Point
Minimum Minimorum tan Vmin(h)M
15 481
Minimum Minimorum Golden Strip t an M (GeV) Electroweak Vacua (B (M F )= 0) Figure 1: A ) The minimum V min of the Higgs effective potential or more precisely, the signed fourth root of the energy densitySign( V min ) × | V min | / , is plotted (green curve, GeV) as a function of M / (GeV) and tan β , emphasizing proximity of the“golden point” of Ref. [1] to the dynamic region of the V min minimorum . B ) The projection onto the ( M / ,tan β ) plane isfurther detailed in the second figure, expanding to span the boundary cases of the Ref. [2] “golden strip”. The symmetry axisof the B µ = 0 parabola is rotated slightly above the M / axis. the gauge couplings of SU (2) L and U (1) Y , n i and m i ( φ ) are the degree of freedom and mass for φ i ,and Q is the renormalization scale. In particular,the soft breaking parameters m H u and m H d are notfree parameters, but rather functions of the univer-sal gaugino boundary mass M / , run down to thepoint of electroweak symmetry breaking under therenormalization group. We include the completeMinimal Supersymmetric Standard Model (MSSM)contributions to one loop, following Ref. [31], al-though the result is phenomenologically identicalaccounting only the leading top and partner stopterms. Since the minimum of the electroweak (EW)Higgs potential V min depends implicitly on M / ,the gravitino mass is determined by the Super No-Scale condition dV min /dM / = 0. Being, however,that M / is proportional to M / , it is equivalentto employ M / directly as our modulus parameter,as previously described. All other SUSY breakingsoft terms will subsequently be derived from thissingle dynamically determined value.Factors explicit within the potential are obtainedfrom our customized extension of the SuSpect2.34 [32] codebase, including a self-consistency as-sessment [1] on B µ = 0. We apply two-loop RGErunning for the SM gauge couplings, and one-looprunning for the SM fermion Yukawa couplings, µ term and soft terms.Studying V min generically in the ( M / , tan β ) plane, no point of secondary minimization is readilyapparent in the strong, roughly linear, downwardtrend with respect to M / over the region of in-terest. However, the majority of the plane is notin physical communication with our model, disre-specting the fundamental B µ = 0 condition. Isolat-ing only the compliant B µ = 0 contour within thissurface, a parabola is traced, the nadir of whichis in excellent agreement with our original goldenpoint, as shown in Fig. (1A). Restoring parameteri-zation freedom to ( M V , m t ), we may scan across thecorresponding golden point of each nearby universevariant, reconstructing in their union the previouslyadvertised golden strip, as in Fig. (1B). Notably,the theoretical restriction on tan β remains stableagainst variation in these parameters, exactly as itsexperimental counterpart.We find it quite extraordinary that the phe-nomenologically preferred region rests precisely atthe curve’s locus of symmetric inflection. Note inparticular that it is the selection of the parabolic B µ = 0 contour out of the otherwise uninterest-ing V min ( M / , tan β ) inclined surface which allowsa clear minimum minimorum to be established. Wereiterate that consistency of the dynamically posi-tioned M / and tan β with the golden strip im-plies broad consistency with all current experimen-tal data, within the resolution of the methodologyand numerical tools employed.5 strongly linear relationship is observed be-tween the SUSY and EWSB scales with M EWSB ≃ . M / , such that a corresponding paraboliccurve may be visualized. There is a charged stauLSP for tan β from 16 to 22, and we connect pointswith correct EWSB smoothly on the plot in this re-gion. If tan β is larger than 22, the stau is moreovertachyonic, so properly we must restrict all analysisto tan β ≤
6. Additional Phenomenology
No-Scale F - SU (5) features, quite stably, the dis-tinctive mass hierarchy m ˜ t < m ˜ g < m ˜ q of a lightstop and gluino, both comfortably lighter than allother squarks. Typical ballpark mass values con-sistent with the dynamic determination of M / ∼
450 GeV are m ˜ t ∼
500 GeV, m ˜ g ∼
625 GeV, and m ˜ q ∼ ∼ .
8% Bino, may feature a mass somewhat less than100 GeV. For direct comparison, we reprint the de-tailed spectrum of the original “Golden Point” ofRef. [1] in Table (1).We suggest that the spectrum so described thusfar survives the advancing detection limits beingposted by early LHC results, which we further pointout are typically tuned to the mSUGRA/CMSSMcontext, often also with particular assumptions ap-plied to tan β . However, the margin of escape maybe narrow, even for the meager 35 pb − of in-tegrated luminosity heretofore described. Specif-ically, Figure (2) of Ref. [33] seems to imply a95% lower exclusion boundary of slightly more than1 TeV for gluino masses in our favored range.The approach taken by this example analysis doesat least take a step toward probing No-Scale F - SU (5) by the claim of model independence from themSUGRA/CMSSM orthodoxy, but there are sev-eral peculiar assumptions made with regards to thespectrum that suggest the prudence of a certain cir-cumspection in interpretation of any quoted bottomline results. In particular, the lightest neutralinois made massless, and all SUSY fields besides thegluino and the first two squark generations, i.e. allsleptons, all Higgs, all other neutralino componentsand the third generation of squarks, are decoupledby assignment of an ultra-heavy 5 TeV mass. Acloser inspection of the data files published by theATLAS collaboration along with the cited reportconfirms that all decay modes are eliminated be-sides those to the massless neutralino plus hadronicjets or leptons. We maintain some ever present anticipation thatthe discovery of supersymmetry at the LHC couldbe imminent, a sentiment which an optimistic read-ing of the early reports from ATLAS and CDFmight be taken to reinforce. We should remark,however, that relaxation of the fixed M V and m t mass values adopted here for simplicity and con-creteness will allow the migration, if necessary, to asomewhat heavier spectrum. This may be accom-plished without wholesale rejection of the underly-ing model (No-Scale F - SU (5)) or method of anal-ysis (the Super No-Scale mechanism) which havebeen our focus in the present work. We defer forfuture work a comprehensive mapping of such alter-native configurations, which in their union composethe complete viable model space.
7. The Gauge Hierarchy Problem
The “gauge hierarchy problem” represents, in ac-tuality, the clustering of multiple related difficultiesinto a single amalgamation, rather than a singleisolated problem with a correspondingly isolatedsolution. Not only must we explain stabilizationof the EW scale against quantum corrections, butwe must also explain why this scale and TeV-sizedSUSY breaking soft-terms are “initially” positionedso far below the Planck mass. These latter com-ponents of the gauge hierarchy problem are themore subtle. In their theoretical pursuit, we do notthough feign ignorance of established experimentalboundaries, taking the phenomenologist’s perspec-tive that pieces fit already to the puzzle stipulate apartial contour of those yet to be placed. Indeed,careful knowledge of precision EW scale physics, in-cluding the strong and electromagnetic couplings,the Weinberg angle and M Z are required even torun the one loop RGEs. In the second loop, one re-quires also minimally the leading top quark Yukawacoupling, as deduced from m t , and the overall mag-nitude of the Higgs vev v , established in turn frommeasurement of the effective Fermi coupling, orfrom M Z and the electroweak couplings.Reading the RGEs up from M Z , we take unifica-tion of the gauge couplings as evidence of a GUT.Reading them in reverse from a point of high en-ergy unification, we take the heaviness of the topquark, via its large Yukawa coupling, to dynam-ically drive the term M u + µ negative, trigger-ing spontaneous collapse of the tachyonic vacuum, i.e. radiative electroweak symmetry breaking. Aswe have elaborated in Section(2), the minimization6 able 1: Sparticle and Higgs spectrum (in GeV) for the M / = 455 GeV and tan β = 15 “Golden Point” of Ref. ([1]). Here,Ω χ = 0 . σ SI = 1 . × − pb, and h σv i γγ = 1 . × − cm / s. The central prediction for the p → ( e | µ ) + π protonlifetime 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 of this potential with respect to the neutral compo-nents of H u and H d at fixed Z-Boson mass allowsone to absolutely establish a numerical value for µ , in addition to a line of of continuously parame-terized solutions for the functional relationship be-tween M / and tan β .Strictly speaking though, we must recognize thathaving effectively exchanged input of the Z-massfor a constraint on µ ( M F ), we dynamically estab-lish the SUSY breaking soft term M / and tan β within the electroweak symmetry breaking vacua, i.e. with fixed v ≃
174 GeV. By employing only val-ues of µ consistent with the physically constrainedHiggs vev, the current construction does not thenintrinsically address the µ problem, i.e. the reasonfor the proximity of the SUSY preserving Higgsmass parameter µ to the electroweak scale andthe soft SUSY breaking mass term M / . Thisproblem is however ubiquitous to all supersym-metric model constructions, and there is no rea-son to prevent a parallel embedding of the usualproposals for addressing the µ problem alongsidethe Super No-Scale mechanism. Likely candidatesfor the required suppression relative to the Planckscale would include the invocation of powers of F-term vevs h F i /M Pl via the Giudice-Masiero mech-anism [34], or the introduction of a SM singletHiggs field as in the next-to-minimal supersymmet-ric standard model (NMSSM), or as a final exam-ple, the consideration of an anomalous U (1) A gaugesymmetry to be realized out of a string theoreticmodel building approach [35].Acknowledging that we have not here fundamen-tally explained the TeV-scale correlation of µ and M V to the modulus M / , we are nevertheless con-tent to justify the values employed by the successof the globally consistent picture which they facil-itate. In any event, a clear conceptual distinctionshould be maintained between the simple param-eters µ and M V and the string theoretic modu-lus M / , the latter being uniquely eligible for dy-namic stabilization under application of the Super No-Scale mechanism. The current proposal mayreach somewhat farther though, than even it firstappears. Having predicted M F as an output scalenear the reduced Planck mass, we are licensed toinvert the solution, taking M F as a high scale in-put and dynamically address the gauge hierarchythrough the standard story of radiative electroweaksymmetry breaking. This proximity to the elemen-tal high scale of (consistently decoupled) gravita-tional physics, arises because of the dual flippedunification and the perturbing effect of the TeVmultiplets, and is not motivated in standard GUTs.Operating the machinery of the RGEs in re-verse, we may transmute the low scale M Z for thehigh scale M F , emphasizing that the fundamentaldynamic correlation is that of the ratio M Z /M F ,taking either as our input yardstick according totaste. For fixed M F ≃ × GeV, in a sin-gle breath we receive the order of the electroweakscale, the Z-mass, the Higgs bilinear coupling µ andthe Higgs vevs, all while dynamically tethering thisderived scale to the soft SUSY breaking parame-ter M / via the action of the secondary minimiza-tion dV min /dM / = 0. All other dependent di-mensional quantities, including the full superparti-cle mass spectrum likewise then fall into line. It isin this sense that the Super No-Scale mechanism,as applied to the present No-Scale F - SU (5) con-struction, may contribute to an understanding ofthe issues composing the gauge hierarchy problem.
8. Conclusion
In this Letter, we have explored the Super No-Scale condition, that being the dynamic localiza-tion of the minimum minimorum of the Higgs po-tential, i.e. a locally smallest value of V min ( M / ),such that both tan β and M / are determined. Thestabilized supersymmetry breaking and electroweakscales may both be considered as dependent out-put of this construction, thus substantively address-ing the gauge hierarchy problem in the No-Scale7 - SU (5) context. We have furthermore demon-strated the striking concurrence of this theoreticalresult with the previously phenomenologically fa-vored “golden point” and “golden strip”.By comparison, the standard MSSM construc-tion seems a hoax, requiring horrendous fine tuningto stabilize if viewed as a low energy supergravitylimit, and moreover achieving TeV scale EW andSUSY physics as a simple shell game by manual se-lection of TeV scale boundaries for the soft terms M / , M , and A . It is remarkable that despite fea-turing more freely tunable parameters, these con-structions are finding it increasingly difficult to rec-oncile their phenomenology with early LHC data.
9. Acknowledgments
This research was supported in part by the DOEgrant DE-FG03-95-Er-40917 (TL and DVN), by theNatural Science Foundation of China under grantNo. 10821504 (TL), and by the Mitchell-HeepChair in High Energy Physics (TL).
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