No-Scale Solution to Little Hierarchy
aa r X i v : . [ h e p - ph ] M a y MIFP-07-15ACT-03-07May, 2007
No-Scale Solution to Little Hierarchy
Bhaskar Dutta , Yukihiro Mimura and Dimitri V. Nanopoulos , , Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA 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
Abstract
We show that the little hierarchy problem can be solved in the no-scale supergravity frame-work. In this model the supersymmetry breaking scale is generated when the electroweak sym-metry breaking condition is satisfied and therefore, unlike usual supersymmetric models, thecorrelation between the electroweak symmetry breaking scale and the average stop mass scalecan be justified. This correlation solves the little hierarchy puzzle. Using minimal supergravityboundary conditions, we find that the parameter space predicted by no-scale supergravity isallowed by all possible experimental constraints. The predicted values of supersymmetric par-ticle masses are low enough to be very easily accessible at the LHC. This parameter space willalso be probed in the upcoming results from the dark matter direct detection experiments.
Introduction
Supersymmetry (SUSY) is one of the key ingredients to consider physics beyond the StandardModel (SM). The large scale hierarchy between the Planck scale and the weak scale is stabilizedonce the hierarchy is generated. In the minimal SUSY extension of the standard model (MSSM),the electroweak symmetry breaking condition is satisfied by the renormalization group runningof the SUSY breaking mass for Higgs fields, and therefore, a large hierarchy can be generatedradiatively [1]: m W /M P ∼ exp( − π ). In this picture we come across three different scales.The scale Q , where one of the eigenvalues of Higgs mass squared becomes negative, is muchsmaller than the Planck scale. However, to generate the electroweak symmetry breaking vacuaradiatively a typical SUSY breaking scale Q S , where loop correction from the Higgs potentialvanishes, is needed to be smaller than the scale Q . In addition, there is another scale, Q st ,where the electroweak potential is destabilized in the D -flat direction. The SUSY breakingscale Q S needs to be within the window between Q st and Q , i.e., Q st < Q S < Q . In theSUSY breaking models, the scale Q S is just an input to obtain a phenomenological model asan anthropic selection.The recent SUSY particle search limits seem to demand an unnatural constraint on theradiative electroweak symmetry breaking vacua to generate the correct Z boson mass. Thesearch attempts have already exceeded the Z boson mass scale. This means that Q S is pushedup, and a little hierarchy between the Z boson mass and SUSY breaking masses gets created.Naively, if an unnatural fine-tuning is not allowed, the electroweak symmetry breaking conditionleads to the fact that Q S , typically the average stop mass, is not very large compared to the Z boson mass. Surely, if we allow fine-tuning, there is no problem. The fine-tuning is encodedin the fact that the two unrelated scales Q S and Q are close. The scale Q is obtained to behierarchically small from the Planck scale, and the hierarchy is determined by dimensionlessparameters. While the SUSY breaking scale Q S is a dimensionful parameter of the model.Why are two such unrelated scales destined to be close? Does there exist any relation between Q and Q S ? These are fundamental questions and require urgent attention since the recentexperimental constraints have caused a little hierarchy between the Z boson and SUSY breakingmasses.It is well known that SUSY is an attractive candidate of physics beyond the SM since it cansolve the unnatural tuning for the quadratic divergence of Higgs mass. It also provides a darkmatter candidate, the neutralino, to explain the 23% of the content of the Universe [2]. It istherefore important to understand whether there exists a physics reason behind the selectionof the electroweak symmetry breaking vacua with little hierarchy. One of the recent attemptsis to reduce the fine-tuning in the symmetry breaking condition by selecting a SUSY breaking1cenario [3, 4, 5]. Another is to consider statistically probable vacua among the electroweaksymmetry breaking vacua [6, 7]. Such landscape idea can nicely explain the little hierarchy.However, the selection of the symmetric breaking vacua is still due to the anthropic reason.No-scale supergravity (SUGRA) model [8, 9], on the other hand, can explain not only theselection of the electroweak symmetry breaking window, but also the little hierarchy betweenthe Z boson mass and SUSY breaking scale [10]. In no-scale SUGRA, the gravitino mass is notdetermined due to the flat potential and this continues until the gravitino mass or the SUSYbreaking feels the electroweak potential. The gravitino mass is determined dynamically due tothe radiative electroweak symmetry breaking. In this sense, the radiative symmetry breakingvacua are automatically selected. Therefore, the reason why Q S is in the symmetry breakingwindow is explained by its own mechanism. Besides, the closeness of Q and Q S is also realizedby the feature of no-scale electroweak potential. So, the no-scale structure is a golden solutionof the little hierarchy problem.No-scale SUGRA is well studied and has been well known for more than twenty years[8, 9]. However, the no-scale structure is often used only as a boundary condition at theunification scale. In this paper, we discuss the no-scale structure of the dynamical determinationof the SUSY breaking scale as a natural solution of the little hierarchy. The electroweaksymmetry breaking leads to two conditions corresponding to the minimization by Higgs vacuumexpectation values (VEVs). The dynamical determination of the SUSY breaking scale givesone more relation between the Z boson mass and the SUSY breaking scale. The relation iswritten in terms of the renormalization group equations (RGEs) of the Higgs boson mass. Wedescribe the model constraints to generate radiative electroweak symmetry breaking vacua, andfind the prediction of no-scale SUGRA. Importantly, we find that the SUSY breaking mass,typically the stop and the gluino masses have upper bounds which are very easy to reach atthe upcoming collider experiments. We also describe the phenomenological constraints andshow the interesting prospect of discovering this model at the upcoming dark matter detectionexperiments.This paper is organized as follows. In section 2, we discuss the Higgs potential and see whatkind of tuning is needed. In section 3, we describe symmetry breaking vacuum and no-scaleSUGRA. In section 4, we discuss no-scale supergravity model and phenomenology. Section 5contains our conclusion. 2 Higgs potential and Little Hierarchy
The tree-level neutral Higgs potential is V (0) = m v d + m v u − ( m v u v d + c.c. ) + g Z v u − v d ) , (1)where v u and v d are the VEVs of the neutral Higgs bosons, H u and H d . The quartic coupling isobtained from D -term and thus the coupling is related to the gauge couplings : g Z = g + g ′ .The quadratic terms are given by SUSY breaking Higgs masses, m H d and m H u , Higgsino mass µ , and SUSY breaking bilinear Higgs mass Bµ : m = m H d + µ , m = m H u + µ and m = Bµ .The Z boson mass is g Z √ v , where v = q v u + v d . Minimizing the tree-level Higgs potential (i.e., ∂V /∂v u = 0, ∂V /∂v d = 0), we obtain M Z m − m tan β tan β − , sin 2 β = 2 m m + m , (2)where tan β = v u /v d . The Z boson mass can be also expressed as M Z − µ + m H d − m H u tan β tan β − ≡ − µ + M H . (3)The SUSY breaking Higgs mass M H is approximately − m H u for tan β > ∼
10. The electroweaksymmetry can be broken by RGE flow of the Higgs mass [1]. Since the scale of M H is naivelygoverned by colored SUSY particles, it is not comparable to the Z boson mass using thecurrent experimental bounds on uncolored SUSY particles, if the universal boundary conditionis applied at the GUT or the Planck scale. Therefore, fine-tuning is required between µ and M H . So, naturalness demands a model which generates smaller values of µ (corresponds tosmaller M H ) to reduce the fine-tuning [11].Since the mass parameters run by RGEs, it is important to note the scale where the fine-tuning is needed. Let us rewrite the expression of the Z boson mass to see what kind of tuningis needed. The tree-level expression of Z boson mass depends on scale Q , and thus, let us definethe Q dependent m Z , m Z ( Q ) ≡ m ( Q ) − m ( Q ) tan β ( Q )tan β ( Q ) − . (4)Taking into account the 1-loop correction of the potential [12] in DR ′ scheme [13], V (1) = 164 π X i ( − J i (2 J i + 1) m i ln m i Q − ! , (5)where J i is a spin of the particle i with mass m i , we obtain M Z = m Z ( Q ) + 1 v cos 2 β v u ∂V (1) ∂v u − v d ∂V (1) ∂v d ! . (6)3his expression of M Z does not depend on Q up to the wave function renormalization for v u and v d at one-loop order. Therefore the proper Z boson mass is obtained approximately at Q = Q S where ∂V (1) ∂v u = cot β ∂V (1) ∂v d is satisfied, namely eQ S = Y i ( m i ) XiX , (7)where X i = ∂m i ∂v u − cot β ∂m i ∂v d ! ( − J i (2 J i + 1) , X = X i X i . (8)The scale Q S is naively the average of the stop masses. Let us define the scale Q where thefunction m Z ( Q ) is zero, which is equivalent to the scale m m = m . Then the Z boson massis expressed as M Z ≃ ln Q S Q dm Z d ln Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q = Q , (9)and dm Z d ln Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q = Q = − β dm d ln Q sin β + dm d ln Q cos β − dm d ln Q sin 2 β ! . (10)For large tan β > ∼ M Z ≃ ln Q Q S ! dm d ln Q . (11)From this expression, one can find that the Z boson mass is proportional to the stop mass upto a loop factor, and Q and Q S need to be close as needed by the little hierarchy between thestop mass and Z boson mass. It is important to note that the smallness of the µ parameteris not important in this expression since µ and − m H u are canceled in RGE at Q = Q .Therefore, the little hierarchy is characterized only by the spectrum of stop masses in RGE ofHiggs mass and the closeness of Q and Q S . For example, in the focus point solution [14] ofminimal supergravity (mSUGRA), it may give rise to a solution of the naturalness problem if Q S is fixed at TeV scale (just below the focus point) since the µ parameter is small. However,the little hierarchy problem is not solved since the Z boson mass is sensitive to ln Q /Q S andthe stop masses are heavy in this solution.The radiative symmetry breaking elegantly explains the smallness of Q and the focus pointscale compared to the Planck scale. However, the hierarchy is determined irrespective of theoverall scale parameter since RGEs are homogenous differential equations, and there is noreason that Q and the focus point scales are close to Q S (which is proportional to the overall Once Q S is fixed, the smallness of µ (naturalness) is important for less fine-tuning in the Z boson mass.However, there is no reason that Q S is fixed in general SUSY breaking model. If Q S is free, the tuning parameteris ln Q /Q S , and then the smallness of µ is not important for the tuning. Q and Q S is probable among the electroweak symmetrybreaking vacua in the landscape picture [7]. However, in such a picture, the vacua wherethe electroweak symmetry is not broken (namely Q < Q S ) are also enormously probable,and the electroweak symmetry breaking vacuum has a special existence among the multiverse.Obtaining the electroweak symmetry breaking vacua could be just for anthropic reason at thisstage.In this paper, we stress that electroweak symmetry breaking vacua with a little hierarchyis naturally obtained in no-scale SUGRA. In this section, we study the origin of the electroweak symmetry breaking vacuum [8] and thenatural occurrence of the closeness of Q and Q S in no-scale electroweak potential.In supergravity [15, 16], the SUSY breaking scale is obtained in the hidden sector physics,and thus the scales Q and Q S are intuitively different and there is no reason that Q S is selectedin the electroweak symmetry breaking region. In no-scale supergravity, on the other hand, theSUSY breaking scale is not determined since the potential for the moduli T and their F -termsare completely flat. The SUSY breaking scale, which is a function of T , is determined bythe radiative effect of the Higgs potential. Since the dynamical determination of the SUSYbreaking scale is due to the electroweak radiative effect, Q and Q S can be related in theno-scale SUGRA.The Q -independent electroweak potential is given as V ( v u , v d ) = V (0) ( v u , v d ; Q ) + ∆ V ( v u , v d ; Q ) , (12)where ∆ V is loop correction and the Higgs VEVs-independent pieces need to be subtracted,∆ V = V (1) ( v u , v d ; Q ) − V (1) (0 , Q ) . (13)When Q S is larger than Q , the electroweak symmetry does not break, and thus v u = v d = 0and V = 0. If Q S is smaller than Q , the Q -independent potential can be negative due to thetree-level potential term. In other words, at the minimal point of the Q -independent potential V ( v u , v d , Q S ( T )) (i.e., ∂V /∂v u = 0, ∂V /∂v d = 0 and ∂V /∂T = 0), the electroweak symmetryis broken. Therefore, if there is no other hidden sector term to determine the scale Q S , thebreaking condition Q > Q S is automatically satisfied in this framework. Besides, as we will5
100 200 300 400-1.0-0.8-0.6-0.4-0.20.00.2 V / ( G e V ) m (GeV) m =A =B =0 Figure 1: The no-scale electroweak potentialsee later, Q S is just below the scale Q , and thus the scale Q S can be larger than the stability-violating scale Q st .Now let us consider a more concrete situation. We assume that every mass parameter inthe supergravity model is proportional to one mass parameter (typically the gravitino mass).For example, in mSUGRA, the mass parameters are ( m , m / , A , µ , B ), which are SUSYbreaking scalar mass, gaugino mass, trilinear scalar coupling, Higgsino mass and SUSY breakingbilinear Higgs mass parameter, respectively. Since the electroweak potential does not dependon gravitino mass explicitly, it is useful to use the gaugino mass as an overall scale. A givenno-scale model gives dimensionless parameters ( ˆ m , ˆ A , ˆ µ , ˆ B ) and ˆ m / e.g., ˆ m = m /m / ,ˆ A = A /m / , and so on. The overall scale m / is determined by the electroweak potential.In figure 1, we show the numerically calculated potential minimized by v u and v d as a functionof m / when ˆ m = ˆ A = ˆ B = 0. The ˆ µ parameter is chosen to obtain the proper Z bosonmass at the minimum. In this choice, tan β ∼ Q S and Q are close at the minimal point. The potential isobtained using the minimizing conditions by v u and v d as V = − g Z M Z cos β + ∆ V − v u ∂ ∆ V∂v u + v d ∂ ∆ V∂v d ! . (14)Substituting Eq.(6), we obtain V = − g Z m Z ( Q ) cos β + ∆ V − sin 2 β v d ∂ ∆ V∂v u + v u ∂ ∆ V∂v d ! + · · · . (15)Since the potential is Q -independent, let us choose the scale Q to make terms beyond the secondterm to be zero. Naively, it is the scale where ∆ V = 0 when tan β is large. We call this scale6 V . The potential can be written as V ≃ − g Z dm Z d ln Q ln Q V Q ! cos β. (16)Since dm Z d ln Q is approximately proportional to the overall scale which is related to Q V , the po-tential is V ∝ − Q V ln Q Q V ! . (17)Minimizing the potential by Q V , we obtain Q V = Q /e / . Thus the scale Q V is just below thesymmetry breaking scale Q . When we write Q S = kQ V , the Z boson mass at the minimum isobtained from Eq.(11) M Z ≃ (1 − ln k ) dm d ln Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q = Q . (18)In the MSSM mass spectrum, the stop masses are important to determine Q V and Q S . Thus,these two scales are close and k ∼
1. In the numerical calculation, k depends on stop mixingsetc, but ln k is about 0 . − .
2. Note that the low energy particle spectrum ratio does notdepend on overall scale (we choose it as m / ), when ( ˆ m , ˆ A , ˆ µ , ˆ B ) are fixed as boundarycondition. Therefore, Q V is naively proportional to m / , and thus, the minimization of thepotential by an overall scale is rationalized.The parameter ˆ µ is consumed to fix Z boson mass at the minimum, and ˆ B is determinedwhen tan β is fixed. So, the model parameters in the minimal supergravity are ˆ m , ˆ A , tan β and the signature of µ .Since the RGE of m at Q = Q is almost determined by stop mass parameters with a loopfactor, dm d ln Q ≃ π ( y t ( m t L + m t R ) + A t ) (19)the little hierarchy between the Z boson and stop masses is obtained by a minimization of theno-scale electroweak potential. Numerically one finds that the gaugino mass at the GUT scaleis about 200 GeV for small ˆ m . This result does not depend on tan β very much unless tan β is small. In this section, we study the no-scale supergravity model to realize the no-scale electroweakpotential in the previous section, and find the phenomenological consequence of the model.There are several ways to realize the no-scale structure [8, 9]. Here, we consider the simplestmodel to realize what we have described in the previous section.7n the simplest no-scale model, the K¨ahler potential is given as [9] G = − T + ¯ T − φ i ¯ φ i ) + ln | W | , (20)where φ i ’s are matter and Higgs fields. In this choice, m and A are zero as boundary con-ditions. The µ term can be proportional to gravitino mass when bilinear term H u H d is in theK¨ahler potential but not in the superpotential W . More generally, one can write down theK¨ahler potential with modular weights λ i and ρ as [9, 17] G = − T + ¯ T ) + φ i ¯ φ i ( T + ¯ T ) λ i + h H u H d ( T + ¯ T ) ρ + h.c. ! + ln | W | . (21)Then, we obtain m i = (1 − λ i ) m / , (22) A ijk = (3 − λ i − λ j − λ k ) m / , (23) B = (2 − λ H u − λ H d + ρ ) m / , (24)and µ term is proportional to 1 − ρ . To make that the Higgsino mass µ is proportional to m / ,we need λ H u + λ H d = 2 ρ . The gravitino mass is m / = | W | M P T + ¯ T ) / . The modular weights λ i and ρ are determined in a concrete model [17, 18].The gauge kinetic function to determine the gaugino mass is f A = k A T ℓ A . (25)In our assumption, every weak scale mass parameter is proportional to one dimensionful mass.In order to achieve this, the gauge kinetic function should depend only on the real part of T .Then the modular weight ℓ A needs to be 1 (or 0). Therefore, all (kinetic normalized) gauginomasses are unified at the boundary, while the gauge coupling constants can be different since k A can be different. The gaugino mass is same as the gravitino mass at the cutoff scale.If there are fields which acquire heavy scale VEVs such as GUT Higgs fields, these fieldsneed to be inside the log as in Eq.(20) so that the flat potential is not destabilized.Even if the potential is flat at the tree-level, the quantum effects may destroy the flatness[17]. The dangerous term which destabilizes the electroweak scale is Λ Str M / (32 π ), whereΛ is a cutoff scale. The supertrace is proportional to m / and thus, it destroys the dynamicaldetermination of m / by electroweak potential. In a simplest case, Str M is negative, and thenthe gravitino mass goes to infinity. Therefore, Str M needs to be zero including moduli and thehidden sector fields. Here after, we assume that the supertrace is zero, which can be realized.8hough we have to forbid the Λ m / term, there can be a harmless correction such as αm / term in the potential. Such a term can arise due to Casimir effects which is related tothe SUSY breakings, or due to a correction in the K¨ahler potential [17] − T + ¯ T ) → − ln(( T + ¯ T ) + c ) . (26)When such a correction in the potential is taken into account, the result in the previous sectionis modified. The potential with the αm / term is given, naively, as V ∝ − Q V ln Q Q V ! + ¯ αQ V , (27)where ¯ α is proportional to α . Then, minimizing the potential with respect to Q V , we obtainln Q Q V = 1 + √ α , (28)and ln Q Q V > by using ∂ V∂Q V >
0. Therefore, we write M Z > ∼
12 (1 − ln k ) dm d ln Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q = Q , (29)which provides an upper bound of the overall SUSY breaking scale ( m / ) for given ˆ m , ˆ A , andtan β . The upper bound of the gaugino mass at the minimum is about √ α = 0 case Eq.(18). In figure 2, we show the numerical result of the minimization of m / with the experimental constraints. We emphasize that the no-scale bound we have obtaineddoes not depend on the detail of the no-scale model constructed from string theory. We obtainthe no-scale bound, as long as there is no a priori scale around the weak scale and the potentialis flat.In drawing the figure, we assume universal scalar mass m and universal trilinear coupling A at the GUT scale ∼ × GeV. We use 2-loop RGE between the GUT and the weakscale to determine the weak scale spectrum and 1-loop corrected potential for minimization.The 1-loop potential has a slight Q -dependence and it may change the result by a few percent.We choose the evaluation scale Q to be about 500 GeV so that the result is insensitive to Q . Ifthe SUSY breaking terms are universal, m and A are related, ˆ A = 3 ˆ m , due to Eqs.(22,23),but we do not assume such relations in drawing the figure because of the reason which we willdescribe later.As we have noted, the parameters are ˆ m , ˆ A , tan β and also a signature for µ . We choose µ > b → sγ constraint. We show the case for tan β = 10 at the minimal point of thepotential so that the region is allowed by the Br[ b → sγ ] which we take to be 2 . × − < Br[ b → m [GeV] m [ G e V ] A =1.5 m , m> b =10, m h ≤ G e V m c ˜ > m t ˜ a m ≤ × -10 d a r k m a tt er a ll o w e d a = a = - . a = - . Figure 2: We show minimization contours of the potential for different values of α (definedin the text) in the mSUGRA parameter space. The blue narrow bands are allowed by darkmatter constraints. The lightest Higgs mass m H ≤
114 GeV is in the pink shaded region. a µ ≤ × − in the light blue shaded region. sγ ] < . × − [19]. We choose ˆ A = 1 . b → sγ ] constraint. Then, by changing ˆ m , we obtain m / by numericalminimization of the potential, and the solid lines are drawn. The three lines corresponds to thevalue α = 0 , − . , − . α is the coefficient of the correction of the potential αm / .When α >
0, the m / value at the minimal point of the potential becomes smaller. As one canunderstand from the discussion above, the minimal value of m / is almost determined by theratio m / /Q V . The ratio is determined by the low energy spectrum, typically by stop mass.Therefore, the solid lines are naively obtained by the trajectories for constant average stopmass, and thus, they are elliptic curves in the m - m / plain. The solid lines are insensitive tothe top quark mass, but depends on strong gauge coupling α . We use α ( M Z ) MS = 0 . m t = 172 . a µ [21], dark matter [22]. We also show the region whereneutralino is the lightest SUSY particle.It is interesting to note that the no-scale allowed region is within the reach of the LHC, andin the mSUGRA model it is allowed by all the experimental constraints. It is also importantto note that the dark matter ˜ χ - p cross-sections are (in 10 − pb) 1.6-5, 1-2.7 and 0.3-1.8 for10 = 0, − .
01 and − .
015 respectively. (The ranges in the cross-sections are obtained for theexperimental range of σ πN , strange quark content of proton and strange quark mass [23]). Therecent upper limit on the neutralino proton cross-section is 5 . × − pb from the XENON 10experiment [24]. We see that the no-scale SUGRA allowed region will be probed very soon inthese direct detection experiments.The phenomenological constraints so far we have discussed are for mSUGRA models ofsoft SUSY breaking terms. Though we use the universal boundary conditions for m and A for simplicity to draw figure 2, the no-scale prediction does not depend on the detail of theboundary conditions Eqs.(22,23), as well as the cutoff scale very much because the predictionis determined by the low energy stop mass spectrum via Eqs.(18,29). On the other hand, theexperimental constraints depends on the location of the cutoff scale as well as the universalityconditions, especially for the dark matter allowed region and the stau LSP region.The important prediction of the no-scale structure is Eq.(18) for true flat potential α = 0,and the bound is obtained from Eq.(29). Eq.(29) gives upper bound to stop mass, and therefore,generates upper bound to the gluino mass, as well. The gluino mass is bounded as m ˜ g < ∼ m , m / , A , tan β m ˜ g m ˜ t , m h,H m ˜ τ , m ˜ χ , , , m ˜ χ ± , , and the cutoff scale is chosen to be around a few times 10 GeV to avoid stau LSP. (2) Choose the modular weights for squarks and up-type Higgs fieldto generate the trilinear coupling with suitable values to satisfy the lightest Higgs boson massbound and the b → sγ constraint especially for the case of true flat no-scale potential α = 0. Since the LEP and the Tevatron data do not show any direct evidence for SUSY, the SUSYparticle mass scale has become larger compared to the M Z scale, therefore a little hierarchy iscreated between this scale and the Z boson mass scale. In order to implement the little hierar-chy, two apparently unrelated scales Q (where the electroweak symmetry breaks radiatively)and Q S (the scale where the correct Z boson mass gets produced) need to be close satisfying There may be an exception when the Boltzmann equation is modified [25]. S < Q , and the closeness is characterized by stop mass spectrum. In this paper, we haveinvestigated the no-scale SUSY breaking models and found that the dynamical determinationof the SUSY breaking scale in these models provides a natural solution of the little hierarchy.The two scales Q and Q S get related in the no-scale model since the electroweak symmetrybreaking vacuum also fixes Q S as a minimal of the electroweak potential. Since the potential,minimized by Higgs VEVs, is naively proportional to − m Z , a larger overall scale is favored anda large Q S provides a small value for the potential. However, when Q S becomes very close to Q (which is independent of the overall scale), the Z boson mass becomes smaller by definitionand the potential becomes larger. As a result, Q S is stabilized just below the scale Q .We considered a no-scale potential where the potential is flat up to the gravitino mass andassumed that all the weak scale parameters are proportional to a single scale, which is natural inno-scale supergravity models. Then we found that the lighter stop and gluino masses can be aslarge as 480 and 730 GeV respectively. These masses can be easily accessed at the LHC. Further,the parameter space is allowed by the Higgs mass bound and the Br[ b → sγ ] using the mSUGRAboundary conditions. It is also interesting to note that the dark matter detection cross sectionis in the range from 0.3 to 5 × − pb. The future dark matter detection experiments caneasily probe these cross-sections. The model also can be fit with proper modular weight factorsfor the quark and lepton fields.We now note what happens when we do not assume the single scale proportionality factorfor the parameters. Suppose that both µ and m / are free and ˆ m and ˆ A are fixed. Then Q can be changed by varying µ , while the scale Q H , where M H becomes zero, is independent of µ and m / . As in the case where Q is fixed in the single scale proportionality, Q S can be aslarge as Q H (but Q S < Q H ). By definition, Q S < Q < Q H is satisfied, and thus, all threescales are close at the minimal value of the potential. The closeness of Q and Q H means that µ is small by definition and we find µ < M Z at the minimal point, which is already excludedby the chargino mass bound. When both µ and B are free from the other SUSY breakingparameters, one finds a non-stabilized direction to the D -flat direction. Thus the µ - B ratioshould be fixed in this case. Therefore, the single scale proportionality is a rational assumptionin no-scale models.In no-scale models, the potential for moduli T is almost flat even if we include the elec-troweak potential, and therefore, the mass of the moduli is tiny, i.e., m T ∼ m W /M P ∼ − eV. This moduli mass m T ∼ m W /M ∗ (where M ∗ is a fundamental scale) does not depend onthe detail of the model when the no-scale structure is broken by the radiative effect. Suchlight moduli overclose universe if the misaligned of the moduli from its minimal value is ofthe O ( M P ) after the inflation [26]. In order to avoid this problem, the misalignment from the12inimal value should be much less than the Planck scale [27]. In other words, the modulican be a part of dark matter. There are other interesting cosmological implications of no-scalemodel [28], which are out of scope of this paper.Another point about the no-scale model is that the electroweak potential is − O ( m W ) at theminimum. Therefore, we need to add possible contribution to cancel the vacuum energy. Suchcontribution can come from other stabilized moduli or hidden sector fields from F or D term.However, such fields may destroy the no-scale structure. In general, such contribution generates m − γX m γ / term in the potential. For example, we obtain γ = 4 / m X is around the weak scale, it can avoid the destabilization ofthe no-scale electroweak potential and the vacuum energy can be canceled. However, by m / minimization, such positive terms require the overall scale to be smaller which is disfavoredby experimental results. In order to make the model viable, we need to make γ = 0, which ispossible from a D -term contribution, so that the correction to tune vacuum energy should notdepend at all on no-scale moduli T . This work was supported in part by the DOE grant DE-FG02-95ER40917.
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