On Uplifted SUSY-Breaking Vacua and Direct Mediation in Generalized SQCD
aa r X i v : . [ h e p - t h ] M a r On Uplifted SUSY-Breaking Vacua and Direct Mediationin Generalized SQCD
Roberto Auzzi, Shmuel Elitzur and Amit Giveon
Racah Institute of Physics, The Hebrew UniversityJerusalem 91904, Israel.We search for viable models of direct gauge mediation, where the SUSY-breaking sectoris (generalized) SQCD, which has cosmologically favorable uplifted vacua even when thereheating temperature is well above the messenger scale. This requires a relatively largetadpole term in the scalar potential for the spurion field X and, consequently, we arguethat pure (deformed) SQCD is not a viable model. On the other hand, in SQCD with anadjoint, which is natural e.g. in string theory, assuming an appropriate sign in the K¨ahlerpotential for X , such metastable vacua are possible.January 2010 . Introduction If supersymmetry (SUSY) provides an explanation to the hierarchy between the weakscale and the Planck scale, its discovery is just around the corner. Breaking supersymme-try entirely in the Minimal Supersymmetric extension of the Standard Model (MSSM) isexcluded by experiment. Thus, in addition to the MSSM we must have a SUSY-breakingsector, and SUSY breaking is mediated by messengers to the MSSM (for a review, see e.g.[1]). Especially appealing models are those of direct mediation, where the messengers arepart of the SUSY-breaking sector.Supersymmetry must be broken spontaneously, but to explain the hierarchy it shouldbetter be broken dynamically, e.g. in Supersymmetric QCD (SQCD) and its generaliza-tions. Moreover, to obtain non-vanishing gaugino masses, R-symmetry must be brokenand, consequently, metastable states are highly favored phenomenologically over globalminima, in the landscape of vacua of SQCD (for a review, see e.g. [2]).Interestingly, metastable vacua are generic in SQCD, as was shown recently in [3].We shall refer to metastable states of the type found in [3] as the ISS-like vacua. Thereis however a serious problem with ISS-like vacua: even when R-symmetry is broken (e.g.explicitly as in [4,5,6]), the gaugino masses remain anomalously small relative to the softscalar masses, leading to the undesired necessity of a high fine tuning of the Higgs mass.The generality of this phenomenon was understood very recently in [7], and its impor-tant consequence is the following (see also [8]). For supersymmetry to provide an expla-nation to the hierarchy problem, we are likely to live in an uplifted metastable vacuum (ofthe type studied e.g. in [9,10,11,12]). By an uplifted vacuum we mean a SUSY-breakingmetastable one, which is part of an approximate pseudo-moduli space, along which somemessengers are unstable at some other point. The ISS-like metastable vacua are not up-lifted.Yet, the existence of uplifted vacua was shown to be a generic feature of (deformed)SQCD in [8], and such vacua were used for direct gauge mediation of SUSY breaking in[8,13]. However, the parameters space of the examples in [8] is rather constrained, and moregeneric phenomenologically viable uplifted vacua in the landscape of SQCD are desired. The purpose of this work is to investigate the landscape of SQCD in order to identifyuplifted metastable vacua which can be used as the gauge mediation minima in modelswith interesting phenomenology. Concretely, we inspect models with a non-trivial K¨ahler Other types of uplifted vacua were studied more recently, e.g. in [14,15]. and its properties. Aninitial study of this issue will be presented in this work.Finally, given the rich landscape of generalized and deformed SQCD, a natural ques-tion is: which of the local minima in the landscape is favored during the cosmologicalevolution of the universe? We shall address this question as well. Concretely, we shall firstassume that the reheating temperature is well above the messenger scale, and that theuniverse cools down adiabatically. This will allow us, as in [21,22,23,24,25], to use thethermal potential as a tool to inspect the possible phase transitions as the universe coolsdown, and to evaluate the likelihood to end up in a phenomenologically interesting statein the landscape.This note is organized as follows. In section 2, we consider massive SQCD in the freemagnetic phase, with a higher-dimension (“Planck” suppressed) deformation which leadsto a tadpole term, − aX , in the scalar potential for the spurion of SUSY breaking, X .The latter pushes the system away from the origin in pseudo-moduli space. We assumethat the sign in front of the leading correction to the K¨ahler potential is such that it canbalance the tadpole “repulsion,” and find the conditions that the theory has long-lived uplifted metastable vacua away from the tachyonic regime.Moreover, we embed the MSSM in the flavor symmetry group of SQCD, and requirethat the gaugino masses, generated via direct gauge mediation, are at the weak scale. Wefind that if the Yukawa coupling of the spurion to the magnetic quarks is of order one,then the value of the tadpole is rather tuned. However, if the Yukawa coupling is small,then uplifted vacua satisfying all the requirements are generic.In section 3, we inspect the thermal history of this model, as the universe coolsdown from a high reheating temperature towards the messenger scale. We find that thecosmological evolution may favor uplifted vacua when the tadpole term is relatively large.However, we argue that one cannot find cosmologically favored vacua, which satisfy all ourconditions naturally in pure (deformed) SQCD. On the other hand, as we show in section4, cosmologically favored uplifted metastable states in viable models of gauge mediationare generic in SQCD with adjoint matter.Finally, we discuss some aspects of this work in section 5, and present some technicaldetails in an appendix. Perturbatively stable ISS-like vacua in SQCD with adjoint matter were investigated e.g.in [19,20]. In [25] it was argued e.g. that the uplifted vacua of [8] are disfavored cosmologically. . Uplifted vacua and gauge mediation in deformed SQCD Consider deformed SU ( N c ) SQCD with N f flavors in the free magnetic phase [26](namely, N c < N f < N c ; for a review, see e.g. [2]). In this subsection we are onlyinterested in the gross understanding of the physics, and thus we shall ignore indices,traces, N c,f factors, etc.; a more precise analysis will be done in the following subsection.At low energies this theory is described by the Seiberg dual “magnetic theory,” whichis IR free. Concretely, in the IR we consider a theory with a superpotential W = h (cid:0) X q ˜ q − µ X + ǫµX (cid:1) , (2 . K = X ¯ X − γ Λ ( X ¯ X ) + . . . , (2 . X is given in terms of the electric quarks Q, ˜ Q by (see [3] and the review[2]) X = 1 h Λ Q ˜ Q , (2 . N f magnetic quarks q, ˜ q are in the fundamental and anti-fundamental of themagnetic gauge group SU ( N f − N c ). The constant γ in (2.2) is an uncalculable realnumber. We assume here that γ is positive .We look for local minima in which q = ˜ q = 0 and X is proportional to the identitymatrix , and denote its common eigenvalue also by X . To leading order, the potential is1 h V = (cid:12)(cid:12) µ − ǫµX (cid:12)(cid:12) (cid:18) γ | X | Λ + h π log (cid:0) | X | (cid:1) + . . . (cid:19) + . . . , (2 . If ˜Λ = Λ, where ˜Λ is the magnetic scale, then X = √ α Λ Q ˜ Q , where α is a real number oforder one in the K¨ahler potential, K = α | Λ | T rM † M + . . . , M ≡ Q ˜ Q , and h = √ α ΛˆΛ , where ˆΛ isgiven in terms of Λ and ˜Λ by the scale matching Λ N c − N f ˜Λ N f − N c ) − N f = ˆΛ N f (see e.g. [3]; herewe ignore phases). In the notation of [6,10,12], these are the vacua with k = 0 and n = N f . These are themost uplifted vacua. Vacua with k Higgsed magnetic quarks have a larger hierarchy between thegaugino and sfermion masses [8]. In particular, in the ISS-like vacua, k = N f − N c , the gauginomasses vanish to leading order in SUSY breaking, even though R-symmetry is broken. X Λ and in the loop expansion parameter h π , and for non-perturbative corrections. From now on we shall take ǫ, h, µ and Λ to bereal and positive.A few comments are in order:(i) In the UV, the quadratic term in (2.1) is quartic in the electric quarks, and hencesuppressed by the high-energy scale M ∗ (which can be taken to be e.g. the Planckscale): hǫµX ≃ M ∗ ( Q ˜ Q ) . (2 . hǫ ≃ ( h Λ) µM ∗ . (2 . m q ± = h (cid:0) X min ± µ + O ( ǫ ) (cid:1) , (2 . X min is not sufficiently large. We arethus looking for a local minimum at X min > µ . (2 . X ≫ µ it is justified to use the log approximation for the one-loop Coleman-Weinberg potential (see appendix A), as was done in eq. (2.4). Although we onlyrequire X min to satisfy (2.8), we shall still use the log approximation since we areonly interested in a gross understanding of the physics.Since we take all the parameters to be real and positive, we can treat X as real. Wealso take ǫ < p γ (cid:16) µ Λ (cid:17) , (2 . V ∼ µ − ǫµ X + 4 γµ X Λ + 2 (cid:18) h π (cid:19) µ log X + . . . . (2 . If, instead, ǫµ > µ / Λ, such that the correction to K is negligible, the general argumentin appendix A of [8] implies that the log term from one-loop prevents the existence of a localminimum at X which is bigger than the largest mass scale in V , ǫµ . So in this case there is nolocal minimum (2.8). Strictly speaking, the potential is well approximated by this equation only when ǫ ≪√ γ (cid:0) µ Λ (cid:1) , but we are only interested in the gross understanding of the physics, for which (2.9) isfine. X > h π Λ √ γ , (2 . ǫ > p γ h π µ Λ . (2 . X min ≃ ǫ Λ γµ . (2 . X min we must have X min ≪ Λ . (2 . S susy ∼ X susy V min ≃ X susy h µ . (2 . µǫ is relatively large, then the location of the super-symmetric minima is dictated by the non-perturbative superpotential to be: X susy ≃ Λ h (cid:16) µ Λ (cid:17) Nf − Nc ) Nc , (2 . h (cid:18) Λ µ (cid:19) Nc − Nf ) Nc ≫ , (2 . N f /N c is not too close to 3 / µǫ is relatively small, then X susy = µǫ , (2 . h ǫ ≫ , (2 . S meta ∼ X min h µ , (2 . X min , and ∆ V ≃ h µ . Hence, metastability requires that either X min issufficiently bigger than µ and/or h ≪ h (cid:18) X min µ (cid:19) ≫ . (2 . X min < X susy , (2 . SU ( N f ) flavor symmetry of the model aboveand discuss the direct mediation of SUSY breaking (for a review, see e.g. [1]). Thespurion for SUSY breaking is X and the messengers are (some of) the q, ˜ q . To leadingorder in SUSY breaking, the masses of the gauginos are m λ r = α r π Λ G , (2 . G ≡ F X X min ≃ hµ X min . (2 . m f = 2 X r =1 C r ˜ f (cid:16) α r π (cid:17) Λ S , (2 . The hidden sector thus has an unbroken SU ( N f − N c ) gauge symmetry and consequentlylight particles; these are very weakly coupled to the MSSM. For N f − N c = 1, the dynamicsof these light d.o.f. should be taken into account, but below, we will usually consider the case N f − N c = 1. Taken to be e.g. in the 5 , ¯5 of SU (5) GUT. S and Λ G is given in terms of the messenger’s number N mess : N mess = Λ G Λ S . (2 . To obtain gaugino masses near 100 GeV we should thus require thatΛ G ≃ γhµ ǫ Λ ≃ GeV . (2 . µ Λ ≃ (cid:18) GeV8 γM ∗ (cid:19) . (2 . h, γ ≃
1, which are their natural values in SQCD, and M ∗ ≃ M P lanck ,we want to satisfy all the constraints and requirements. The system is highly constrained,but has solutions in parameters space, for instance: µ Λ ≃ − . , ǫ ≃ µ Λ , Λ ≃ . GeV, X min ≃ Λ25 < X susy ≃ Λ / Nf − Nc ) Nc is satisfied if N f < N c . In this case, the bounceaction (2.15) is large, (2.17), as well as the bounce action (2.20), so longevity is secured.We shall end this subsection with a few comments:(1) We have verified numerically, by using the exact tree-level and one-loop potentials,that indeed minima of this type are easily obtained; this is described in the nextsubsection.(2) A particularly interesting case is the case of a much larger tadpole term. A sufficientlylarge ǫ will allow the cosmological evolution to favor an uplifted vacuum, as we shallsee later. Consider for instance ǫ ≃ /
10. Equation (2.9) implies that in this case µ/ Λcannot be much smaller than 1 /
10 as well. The approximations after eq. (2.9) showthat one can find in this case a local minimum. We have verified numerically thatuplifted minima may exist in this range (see the next subsection). However, now the Definitions of Λ G , Λ S and an effective number of messengers, as well as its notion for theoriesthat respect just the SM gauge symmetry SU (3) C × SU (2) W × U (1) Y , in a general framework,are given in [28], but we will not need these here. This is not true in vacua where k of the quarks are Higgsed. In particular, in the latter vacua, N mess (2.26) will be smaller than the number of messengers. Actually, it will even be smallerthan the number of messengers which contribute to the gaugino masses, as explained in [8]. M ∗ . This implies that for a relativelylarge tadpole term, uplifted vacua satisfying all our constraints are impossible in pureSQCD. However, by considering SQCD with an adjoint, we will see that the landscapeof possibilities is much richer, in particular allowing viable uplifted vacua which arefavored cosmologically.(3) Finally, recall that everything done above assumes that the constant γ is positive. InSQCD we do not know how to calculate γ . However, by embedding the theory aboveon brane systems in string theory, and going to the perturbative string regime, one cancalculate the K¨ahler potential (as well as higher derivative D-terms). This was donein [12], where it was found that γ is positive, giving rise to the uplifted metastablevacua of the type analyzed above. We shall now present a more precise analysis. The superpotential is W = h (cid:18) X ji q ia ˜ q aj − µ T r ( X ) + 12 ǫµT r ( X ) (cid:19) , (2 . i, j = 1 , . . . , N f are flavor indices, T r is over flavor indices, and a = 1 , . . . , N , N ≡ N f − N c , (2 . The K¨ahler potential is K = T r ( X † X ) − c Λ T r ( X † X ) − c Λ ( T r ( X † X )) + . . . , (2 . U ( N f ) symmetry and higherorder terms in X/ Λ. We look for solutions of the form X ji = X δ ji , (2 . We may also add a double-trace deformation (
T r ( X )) to (2.29) which, in particular, givesdifferent masses to the adjoint and singlet of SU ( N f ). This is sometimes a necessity, e.g. in orderto give sufficiently large masses to the adjoint fermions from the X superfield [29] and to push theLandau-pole well above Λ (as e.g. in [8]). This term does not affect our gross understanding ofthe physics, but it will be interesting to add it when inspecting the whole features of our models. The terms omitted in (2.31) have negligible contributions. Similar corrections to K wereconsidered for N f = N c SQCD in [30]. X ∈ R + . (2 . V = V + V + . . . = h N f (cid:18) γ X Λ (cid:19) (cid:0) µ − ǫµX (cid:1) + h N f N π h ( X + µ − ǫµX ) log (cid:18) h ( X + µ − ǫµX )Λ (cid:19) + ( X − µ + ǫµX ) log (cid:18) h ( X − µ + ǫµX )Λ (cid:19) − X log (cid:18) h X Λ (cid:19) i + h O ( h , h ( X/ Λ) , ( X/ Λ) ) . (2 . q = ˜ q = 0, (2.32),(2.33), X < µ/ǫ and h, µ, ǫ, Λ ∈ R + . (2 . V = (cid:18) ∂∂X ji ∂∂ ¯ X ¯ j ¯ i K (cid:19) − ∂W∂X ji ∂ ¯ W∂ ¯ X ¯ j ¯ i is obtained from (2.29) and (2.31),and the one-loop potential is discussed in appendix A. The real parameter γ is a certaincombination of the c ’s in (2.31), γ = c + N f c + . . . , (2 . X , the one-loop potential can be approximated by: V ≈ h N N f µ π log X . (2 . X and X ) of the tree-level piece in (2.34),then in order to have a minimum one gets: ǫ > h √ N γπ µ Λ , (2 . X min ≈ ǫ Λ γµ . (2 . ǫ . When ǫ is suffi-ciently larger than the bound (2.38), the location of X min is large enough to ignore the10og X term in V . So, now we ignore the one-loop term, but we keep the whole tree-levelpotential in (2.34). In order to have a minimum we get: ǫ < r γ µ Λ , (2 . X min ≈ µ ǫ . (2 . ǫ in the window h √ N γπ µ Λ < ǫ < r γ µ Λ , (2 . h π s Nγ Λ < X min < √ γ Λ . (2 . h ∼
1, which is its natural value in SQCD, and for small N , thevalue of the tadpole ǫ must be rather tuned to have a local minimum. Consequently,the corresponding location of the minimum is highly restricted and, in particular, itslocation is very far from the origin. On the other hand, if h ≪
1, there is a large rangeof ǫ for which a minimum exists, and thus uplifted vacua are generic. Moreover, thecorresponding location of the minimum is in a wide range and, in particular, can beclose to X tachyon ≈ µ .(b) In order to trust the calculation we should impose X min Λ ≪ γ , (2 . K is valid. So we can just marginally trust the valuesof ǫ near the upper bound of the window (2.42),(2.43).Taking into account the number of messengers, N , the scale Λ G in (2.23),(2.24) isΛ G ≃ N hµ X min . (2 . We will usually restrict to the case N = 1, for which the dynamics of the unbroken SU ( N )gauge theory is trivial (see also footnote 8). Yet, we shall keep the explicit N -dependence in theequations below. ǫ is close to its smaller bound, from (2.39) we haveΛ G ≃ N γhµ ǫ Λ ≃ GeV , (2 . µ Λ ≃ (cid:18) GeV8 γN M ∗ (cid:19) . (2 . ǫ is closer to its upper bound, from (2.41) we haveΛ G ≃ N ǫhµ ≃ GeV , (2 . M ∗ < γ N ǫ GeV . (2 . ǫ is large the high-energy scale M ∗ is anoma-lously small.This will be discussed further in the next sections, where we will argue, in particular,that this problem is cured by adding adjoint matter to SQCD. However, we shall firstshow that a large tadpole term is necessary for the universe to end its early evolution in anuplifted vacuum, once the reheating temperature is well above the messenger scale. This isthe topic of the next section, where we shall inspect the thermal potential in our deformedSQCD.
3. The Thermal Potential
We now turn to study the thermal history of the model in the previous section. Wewill assume that after inflation and reheating the universe is in a thermal state with atemperature well above the messenger scale, and that it cools down adiabatically. Thisallows us to investigate the evolution of the universe from the end of reheating till a(metastable) vacuum is populated, by inspecting the thermal effective potential. Thepurpose of this section is to find the conditions allowing the cosmological evolution tofavor an uplifted metastable state, as it cools down after inflation and reheating.12e shall thus begin by inspecting the thermal potential. The expression for the one-loop effective thermal potential is obtained by adding to the zero-temperature potential(2.34) the term [31]: V th = X ± T π Z ∞ dw w log ∓ e − q w + m X,q ) T , (3 . X and the magnetic quarks q, ˜ q .Now, if the temperature is lower than the mass of the particles in the system, thecontribution is essentially zero. On the other hand, in the regime where T is much biggerthan the mass of the particles, we can expand in m /T : V th ≈ − π T (cid:18) N bos + 78 N fer (cid:19) + T X bos m ( X, q ) + X fer m ( X, q )2 ! . (3 . q = ˜ q = 0 , X ji = X δ ji , (3 . q = ˜ q † = (cid:18) q I N × N (cid:19) , X = 0 , (3 . SU ( N ) gauge symmetry is broken.On the mesonic direction, using the spectrum presented in appendix A, one finds that V th ( X, q = 0) ≈ X ≫ Th , (3 . V th ( X, q = 0) ≈ N f N (cid:18) − π T + T h X + . . . (cid:19) for X ≪ Th . (3 . Strictly speaking, this is true only for N = 1. For bigger N , we should also take into accountthe effects of the SU ( N ) gauge dynamics, e.g. the contribution of the magnetic gauge bosons andD-terms (see [23]). These give corrections which are negligible in the limit of a small magneticgauge coupling. We also ignore the MSSM interactions for similar reasons (see [24] and [25]). V ( T ), is obtained by adding the T -dependent piece V th to the zero-temperature potential (2.34): V ( T ) ≡ V + V + V th + . . . . (3 . V ( T ) has a minimum at X th ≈ ǫµ N T . (3 . ǫ term in (2.10).Evolving into a thermal minimum, which is located away from the origin as the universecools down from T reheat towards the symmetry breaking scale, thus requires a large tadpoleterm.We now turn to the thermal potential in the squarks direction (3.4). Using the spec-trum presented in appendix A, one finds: V ( T ; X = 0 , q ) ≈ h (cid:0) N ( q − µ ) + ( N f − N ) µ (cid:1) + T h (cid:18) N f ǫ µ + 12 N N f q (cid:19) , (3 . q is defined in (3.4). Hence, at a sufficiently low temperature, squarks becometachyonic and a second order phase transition, towards an ISS-like vacuum, occurs. The critical temperature for the second order phase transition is: T c ≈ µ N f , (3 . X min ( T = T c ) ≈ ǫµN f N . (3 . The new basin of attraction at X ≫ T and q ∼ µ , taken into account in [25], is irrelevanthere since the messengers masses, required for this to exist, behave like q and thus emerge onlyafter the phase transition to the ISS-like vacuum already occurred. This is slightly different from the results of [23], who get (in the limit of small magnetic gaugecoupling): T c = µ N f − N . µ , in order to avoid the phase transition to the ISS-like vacuum. Thisrequires the condition ǫ > NN f . (3 . N f ≈
10 and N = 1, for which the smallest tadpole allowed bythe requirement (3.12) is ǫ ≈ /
10. In this case, the zero-temperature uplifted minimumis located at X min ≈ µ (see eq. (2.41)), so the universe is likely to slide towards thisuplifted state as it continues to cool down.There is however a serious drawback in this scenario. From eq. (2.49) we see that sucha large tadpole leads, in particular, to an anomalously small high-energy scale, M ∗ < GeV, and one cannot satisfy all our constraints . We shall next cure this problem byconsidering uplifted vacua in (deformed) SQCD with adjoint matter.
4. Uplifted vacua in SQCD with an adjoint
We shall now show that since the landscape of uplifted vacua in SQCD with an adjointis richer, in particular, it allows the existence of viable uplifted metastable states in thepresence of a large tadpole term (large ǫ ), that are favored by the cosmological evolutioneven when the reheating temperature is well above the messenger scale. In this work, weshall focus on a rather limited regime in the landscape of possibilities. Hence, one shouldregard our investigation in this section only as the initial step towards a more thoroughstudy of uplifted metastable vacua in SQCD with an adjoint, and the mediation of SUSYbreaking to the MSSM.Before adding the adjoint though, let us recall the problems with having cosmologicallyfavored uplifted vacua in pure (deformed) SQCD. The electric mesons M ≡ Q ˜ Q are relatedto the canonically normalized IR fields X via M = √ α Λ X , where α is a positive realparameter of order one, appearing in K = α | Λ | M † M + . . . . Now, when the magneticscale is chosen to be equal to the electric scale, then h = √ α , thus h is also of orderone, and since hǫ = α Λ µM ∗ , this cannot be satisfied unless the high-energy scale M ∗ is verysmall, in which case one cannot satisfy all our constraints. And even if it turns out that If h is of order one, then the gauge mediation vacuum is not metastable, (2.21), while forsmall h , eq. (2.48) sets the symmetry-breaking scale µ above the cutoff scale M ∗ . α = h ≪
1, the relation hǫ = h Λ µM ∗ is still problematic (since the high-energy scale M ∗ isway too small).To be able to resolve these problems, we need a model where the value of the Yukawacoupling h is separated from the value of the parameter α . For instance, if h ≪ α is order one, we are able to satisfy all the constraints in sections 2 and 3.This is the situation in SQCD with an adjoint, as we shall now see. First we recallsome results of Kutasov, Schwimmer and Seiberg [32,33,34] (see also [20]). Consider an N = 1 SU ( N c ) SYM with N f flavors ( Q, ˜ Q ), an adjoint Φ and a superpotential W el = T r k +1 X n =2 g n n Φ n . (4 . SU ( N ) SYM, where N = kN f − N c , (4 . N = 1 , (4 . N f flavors( q, ˜ q ) and k meson fields X j , which are given in terms of the UV variables by X j = 1 √ α j Λ j +1 Q Φ j ˜ Q , j = 0 , , . . . , k − , (4 . SU ( N c ) gauge theory and α j are orderone positive numbers appearing in the K¨ahler potential K = P k − j =0 1 α j | Λ | T r ( M † j M j ) + . . . , M j ≡ Q Φ j ˜ Q , and there is a superpotential W mag = k − X j =0 h j X j q ˜ q . (4 . X j , q, ˜ q are canonically normalized.As before, we choose the magnetic scale to be equal to the electric one. The Yukawacouplings are functions of the dimensionless parameters of the theory h j = h j ( α i , g n Λ n − ) , (4 . The scale matching in SQCD with an adjoint is Λ N c − N f ˜Λ N − N f = ( ˆΛ /g k +1 ) N f (see [34]for the definitions of the various scales). h j are thus free parameters, which can takein particular very small values, even though the α i are order one parameters. This aspectis an important difference with respect to the situation in pure SQCD.To break supersymmetry in uplifted vacua we deform the theory to W = k − X j =0 h j (cid:18) X j q ˜ q − µ j X j + 12 ǫ j µ j X j (cid:19) , (4 . ǫ j terms are high-dimension terms in the UVfields, and thus h j ǫ j ≃ α j Λ j +2 µ j M j +1 ∗ , (4 . M ∗ being a high-energy scale. Unlike pure SQCD, now relatively large tadpole termsare easily obtained, even for large M ∗ , and consequently, cosmologically favored upliftedvacua satisfying all the other constraints are generic even when the reheating temperatureis well above the messenger scale.We shall now focus on a concrete example, the k = 2 case. In this case, W el = T r (cid:16) g − m Φ (cid:17) , (4 . g ≡ g is a dimensionless coupling and m Φ ≡ − g is the mass of the adjoint. For N = Λ / ˜Λ = α j = 1, one finds (following [34,20]) in this case h = 1 g , h = N c − g m Φ Λ . (4 . h j , and thus large ǫ j , are possible, and cosmologically favored uplifted vacua,which satisfy all the constraints, can thus exist.For instance, if g is large, then h is small, and since h ≪ h , the X sector is almostdecoupled from SUSY breaking. Consequently, the analysis of sections 2 and 3 applies.In particular, if ǫ is sufficiently large, and since α i are order one, the uplifted vacua areviable, cosmologically favored gauge mediation minima. Similarly, if SUSY breaking isalmost entirely in the X sector, and if h is small (e.g. if m Φ ≪ Λ), then for sufficientlylarge tadpole terms, cosmologically favored uplifted vacua may exist. From these examples,it seems possible that this is generic in SQCD with an adjoint. This is the minimal deformation required for our purpose; we could have added more termsat this order, but their effect will be negligible in what we shall do next. h i ∼ g SM , one should alsotake into account the effect of the MSSM interactions (see [24] and [25]). It appears thoughthat this effect will help in favoring the uplifted metastable vacua, for the following reason.The supersymmetric vacuum has more light particles than the non-supersymmetric ones(even at finite temperature) because the degeneracy between the SM particles and theirsuperpartners is bigger, so this effect reduces the relative number of light particles nearthe origin, and thus decreases the probability for the second order phase transition to theISS-like vacuum to occur (e.g., it decreases the value of the critical temperature (3.10)).This effect is bigger the larger the SM couplings are relative to our Yukawa couplings, h .So in our case, of a relatively large tadpole ǫ and thus small h , it seems that this effectincreases the probability that the cosmological dynamics will favor the uplifted metastablevacua.
5. Discussion – uplifted metastable vacua in the cooling universe
The main result of this work can be summarized as follows. Assuming that the signin front of the leading correction to the K¨ahler potential for the spurion of SUSY breaking X is appropriate, we argued that generalized SQCD – concretely, deformed SQCD withan adjoint – has generic, cosmologically favored, long-lived metastable vacua, which canbe used in viable models of direct gauge mediation.Explicitly, for theories in the free magnetic phase, when the Yukawa coupling of X to the magnetic quarks is small, a relatively large tadpole term for X is naturally gener-ated (from “Planck” suppressed high-dimension operators), giving rise to cosmologicallyfavorable uplifted metastable vacua. By embedding the MSSM in the flavor group of thisgeneralized SQCD, and setting the messenger scale appropriately, one can obtain phe-nomenologically interesting models.After inflation and reheating, as the universe cools down adiabatically towards themessenger scale, the tadpole term pushes the thermal minimum away from the origin in X . And when the tadpole term is sufficiently large, the universe is likely to populate anuplifted metastable state, of the type desired phenomenologically, before the second orderphase transition towards the undesired metastable ISS-like vacuum occurs.On the other hand, when the tadpole term ǫ is small, our study gives a bound on thereheating temperature, as in [25]. Namely, in this case T reheat must be smaller than thesymmetry breaking scale, such that the universe may be trapped in a phenomenologically18iable uplifted vacuum already during inflation. This is a rather severe constraint forcosmology if we require a low-scale gauge mediation of SUSY breaking.As discussed in the introduction, the (generalized) deformed SQCD models of thetype considered here have simple embedding in string theory, e.g. on systems of D4-branesstretched between NS5-branes, and the effective theory for the spurion X is remarkablysimilar to the one assumed here. Vacuum selection in such systems was analyzed recentlyin [35], by considering the dynamics of the D4-brane probes in the background of non-extremal NS5-branes. The analog of the cooling universe in this picture is decreasing thenon-extremal scale.The results in [35] are compatible with the results in this work. Namely, when thetadpole term is sufficiently large, the dynamics select uplifted metastable states. We shouldemphasize that while the models studied here and in [35] are connected continuously instring theory [18,10], this is not sufficient for neither the early universe dynamics, nor thezero temperature one to be identical. Nevertheless, we do find striking similarities.Actually, the theories investigated in [35] correspond only to the embedding of (de-formed) pure SQCD in the type IIA brane construction. It will be interesting to repeatthe analysis of [35] to the brane embedding of SQCD with an adjoint.Moreover, it should be interesting to investigate the dynamical vacuum selection inmodels with generic k and N in section 4, and to consider more general theories, e.g.including couplings of the adjoint Φ to the quarks Q, ˜ Q , as well as adding more matter tothe theory, both in gauge theory and in its string embeddings.Finally, it will also be interesting to use our uplifted vacua in deformed SQCD withan adjoint, to reconsider aspects of flavor physics along the lines of the recent work [20].The authors of [20] embedded the first and second SM generations inside the compositefields Q Φ ˜ Q and Q ˜ Q of section 4, respectively, to provide an explanation to the flavorhierarchy within a “single-sector” SUSY-breaking model [36,37]. However, they studiedthe theory in an ISS-like vacuum, which suffers from split supersymmetry. On the otherhand, investigating such models in our uplifted metastable vacua may lead to interestingflavor physics, in models that also provide an explanation to the hierarchy between theweak scale and the Planck scale. Acknowledgements:
We thank Andrey Katz, Zohar Komargodski, David Kutasov andTomer Volansky for discussions. This work was supported in part by the BSF – American-Israel Bi-National Science Foundation, by a center of excellence supported by the Israel19cience Foundation (grant number 1468/06), DIP grant H.52, and the Einstein Center atthe Hebrew University.
Appendix A. Potentials and spectra
The superpotential of the Seiberg dual of deformed SU ( N c ) SQCD with N f flavors is: W = hq i X ji ˜ q j + h Tr (cid:18) − µ X + 12 ǫµX (cid:19) , (A.1)where i, j = 1 , . . . , N f , the meson superfields X ji are singlets, and the magnetic quarks q i (˜ q j ) are in the (anti-)fundamental of the magnetic gauge group SU ( N ) , N ≡ N f − N c . (A.2)The tree-level potential, in the case of a canonical K¨ahler potential, is: V = | h | (cid:0) | q i ˜ q j − µ δ ij + ǫµX ij | + | X q | + | ˜ qX | (cid:1) . (A.3)Next we present the tree-level spectrum and the one-loop effective potential [29] (see also[38]). A.1. Mesonic direction: q = ˜ q = 0 , X ji = X δ ji The tree-level spectrum in this direction includes: N f N fermions , m = h X , N N f scalars , m = h ( X + µ − X ǫµ ) , N N f scalars , m = h ( X − µ + X ǫµ ) , The mass squared of the last group of scalars is negative for
X < µ . There are also2 N f scalars , m = h ǫ µ , N f fermions , m = h ǫ µ , From now on we take ǫ, h, µ to be positive real numbers. X .The one-loop contribution reads: V = N f N π (cid:18) h ( X + µ − X ǫµ ) log (cid:18) h ( X + µ − X ǫµ ) M c (cid:19) ++ h ( X − µ + X ǫµ ) log (cid:18) h ( X − µ + X ǫµ ) M c (cid:19) − h X log (cid:18) h X M c (cid:19)(cid:19) , (A.4)where M c is a cutoff scale. Changing the cutoff scale, M c → ˜ M c , amounts to a renormal-ization of the overall constant of the tree-level potential, h N f N π ( µ − ǫµX ) log ˜ M c M c ! , (A.5)which is negligible. In this work we took M c = Λ. A.2. Squark direction: X = 0We consider here the maximally Higgsed case (when SU ( N ) is completely broken) –the ISS-like vacuum. Using flavor symmetry, the VEV of the squarks can be chosen to be q = ˜ q † = (cid:18) q I N × N (cid:19) , X = 0 . (A.6)The potential along this direction is: V = h (cid:0) N ( q − µ ) + ( N f − N ) µ (cid:1) . (A.7)Next, let us present the classical spectrum.There are the following scalars:2( N f − N ) m = h ǫ µ ,N m = ± h ( µ − q ) , N ( N f − N ) m = h (cid:18) q + ǫ µ + µ ± q q ǫ µ + ǫ µ − ǫ µ + µ (cid:19) , N ( N f − N ) m = h (cid:18) q + ǫ µ − µ ± q q ǫ µ + ǫ µ + 2 ǫ µ + µ (cid:19) , m = h (cid:18) q + ǫ µ + µ ± q q + 10 q ǫ µ − q µ + ǫ µ − ǫ µ + µ (cid:19) ,N m = h (cid:18) q + ǫ µ − µ ± q q + 6 q ǫ µ − q µ + ǫ µ + 2 ǫ µ + µ (cid:19) . Finally, there are the following fermionic degrees of freedom:2 N m = 0 , N f − N ) m = h ǫ µ , N ( N f − N ) m = h (cid:18) q + ǫ µ ± ǫµ q q + ǫ µ (cid:19) , N m = h (cid:18) q + ǫ µ ± ǫµ q q + ǫ µ (cid:19) . eferences [1] G. F. Giudice and R. Rattazzi, “Theories with gauge-mediated supersymmetry break-ing,” Phys. Rept. , 419 (1999) [arXiv:hep-ph/9801271].[2] K. A. Intriligator and N. Seiberg, “Lectures on Supersymmetry Breaking,” Class.Quant. Grav. , S741 (2007) [arXiv:hep-ph/0702069].[3] K. A. Intriligator, N. Seiberg and D. Shih, “Dynamical SUSY breaking in meta-stablevacua,” JHEP , 021 (2006) [arXiv:hep-th/0602239].[4] K. A. Intriligator, N. Seiberg and D. Shih, “Supersymmetry Breaking, R-SymmetryBreaking and Metastable Vacua,” JHEP , 017 (2007) [arXiv:hep-th/0703281].[5] N. Haba and N. Maru, “A Simple Model of Direct Gauge Mediation of MetastableSupersymmetry Breaking,” Phys. Rev. D , 115019 (2007) [arXiv:0709.2945 [hep-ph]].[6] A. Giveon and D. Kutasov, “Stable and Metastable Vacua in SQCD,” Nucl. Phys. B , 25 (2008) [arXiv:0710.0894 [hep-th]].[7] Z. Komargodski and D. Shih, “Notes on SUSY and R-Symmetry Breaking in Wess-Zumino Models,” JHEP , 093 (2009) [arXiv:0902.0030 [hep-th]].[8] A. Giveon, A. Katz and Z. Komargodski, “Uplifted Metastable Vacua and GaugeMediation in SQCD,” JHEP , 099 (2009) [arXiv:0905.3387 [hep-th]].[9] R. Kitano, H. Ooguri and Y. Ookouchi, “Direct mediation of meta-stable supersym-metry breaking,” Phys. Rev. D , 045022 (2007) [arXiv:hep-ph/0612139].[10] A. Giveon and D. Kutasov, “Stable and Metastable Vacua in Brane Constructions ofSQCD,” JHEP , 038 (2008) [arXiv:0710.1833 [hep-th]].[11] B. K. Zur, L. Mazzucato and Y. Oz, “Direct Mediation and a Visible MetastableSupersymmetry Breaking Sector,” JHEP , 099 (2008) [arXiv:0807.4543 [hep-ph]].[12] A. Giveon, D. Kutasov, J. McOrist and A. B. Royston, “D-Terms and SupersymmetryBreaking from Branes,” Nucl. Phys. B , 106 (2009) [arXiv:0904.0459 [hep-th]].[13] D. Koschade, M. McGarrie and S. Thomas, “Direct Mediation and Metastable Super-symmetry Breaking for SO(10),” arXiv:0909.0233 [hep-ph].[14] S. A. Abel, J. Jaeckel and V. V. Khoze, “Gaugino versus Sfermion Masses in GaugeMediation,” arXiv:0907.0658 [hep-ph].[15] J. Barnard, “Tree Level Metastability and Gauge Mediation in Baryon DeformedSQCD,” arXiv:0910.4047 [hep-ph].[16] A. Giveon and D. Kutasov, “Brane dynamics and gauge theory,” Rev. Mod. Phys. ,983 (1999) [arXiv:hep-th/9802067].[17] A. Giveon, J. McOrist and A. B. Royston, in progress.[18] A. Giveon and D. Kutasov, “Gauge symmetry and supersymmetry breaking fromintersecting branes,” Nucl. Phys. B , 129 (2007) [arXiv:hep-th/0703135].2319] A. Amariti, L. Girardello and A. Mariotti, “Non-supersymmetric meta-stable vacua inSU(N) SQCD with adjoint matter,” JHEP , 058 (2006) [arXiv:hep-th/0608063].[20] N. Craig, R. Essig, S. Franco, S. Kachru and G. Torroba, “Dynamical SupersymmetryBreaking, with Flavor,” arXiv:0911.2467 [hep-ph].[21] S. A. Abel, C. S. Chu, J. Jaeckel and V. V. Khoze, “SUSY breaking by a metastableground state: Why the early universe preferred the non-supersymmetric vacuum,”JHEP , 089 (2007) [arXiv:hep-th/0610334].[22] N. J. Craig, P. J. Fox and J. G. Wacker, “Reheating metastable O’Raifeartaigh mod-els,” Phys. Rev. D , 085006 (2007) [arXiv:hep-th/0611006].[23] W. Fischler, V. Kaplunovsky, C. Krishnan, L. Mannelli and M. A. C. Torres, “Meta-Stable Supersymmetry Breaking in a Cooling Universe,” JHEP , 107 (2007)[arXiv:hep-th/0611018].[24] S. A. Abel, J. Jaeckel and V. V. Khoze, “Why the early universe preferred the non-supersymmetric vacuum. II,” JHEP , 015 (2007) [arXiv:hep-th/0611130].[25] A. Katz, “On the Thermal History of Calculable Gauge Mediation,” JHEP , 054(2009) [arXiv:0907.3930 [hep-th]].[26] N. Seiberg, “Electric - magnetic duality in supersymmetric nonAbelian gauge theo-ries,” Nucl. Phys. B , 129 (1995) [arXiv:hep-th/9411149].[27] M. J. Duncan and L. G. Jensen, “Exact tunneling solutions in scalar field theory,”Phys. Lett. B , 109 (1992).[28] C. Cheung, A. L. Fitzpatrick and D. Shih, “(Extra)Ordinary Gauge Mediation,” JHEP , 054 (2008) [arXiv:0710.3585 [hep-ph]].[29] R. Essig, J. F. Fortin, K. Sinha, G. Torroba and M. J. Strassler, “Metastable super-symmetry breaking and multitrace deformations of SQCD,” JHEP , 043 (2009)[arXiv:0812.3213 [hep-th]].[30] A. Katz, Y. Shadmi and T. Volansky, “Comments on the meta-stable vacuum inN(f) = N(c) SQCD and direct mediation,” JHEP , 020 (2007) [arXiv:0705.1074[hep-th]].[31] L. Dolan and R. Jackiw, “Symmetry Behavior At Finite Temperature,” Phys. Rev. D , 3320 (1974).[32] D. Kutasov, “A Comment on duality in N=1 supersymmetric nonAbelian gauge the-ories,” Phys. Lett. B , 230 (1995) [arXiv:hep-th/9503086].[33] D. Kutasov and A. Schwimmer, “On duality in supersymmetric Yang-Mills theory,”Phys. Lett. B , 315 (1995) [arXiv:hep-th/9505004].[34] D. Kutasov, A. Schwimmer and N. Seiberg, “Chiral Rings, Singularity Theory andElectric-Magnetic Duality,” Nucl. Phys. B , 455 (1996) [arXiv:hep-th/9510222].[35] D. Kutasov, O. Lunin, J. McOrist and A. B. Royston, “Dynamical Vacuum Selectionin String Theory,” arXiv:0909.3319 [hep-th].2436] N. Arkani-Hamed, M. A. Luty and J. Terning, “Composite quarks and leptons fromdynamical supersymmetry breaking without messengers,” Phys. Rev. D , 015004(1998) [arXiv:hep-ph/9712389].[37] M. A. Luty and J. Terning, “Improved single sector supersymmetry breaking,” Phys.Rev. D , 075006 (2000) [arXiv:hep-ph/9812290].[38] A. Giveon, D. Kutasov and O. Lunin, “Spontaneous SUSY Breaking in Various Di-mensions,” Nucl. Phys. B822