aa r X i v : . [ h e p - ph ] N ov LAUR-07-7160SLAC-PUB-12967
Minimal Direct Gauge Mediation
Masahiro Ibe and Ryuichiro Kitano Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 andPhysics Department, Stanford University, Stanford, CA 94305 Theoretical Division T-8, Los Alamos National Laboratory, Los Alamos, NM 87545
We propose a simple model of gauge mediation where supersymmetry is broken by a strongdynamics at O (100) TeV. A. Introduction
One of the simplest and the most natural scenarios forsupersymmetry breaking is to assume dynamical super-symmetry breaking at an energy scale of O (100) TeV.The electroweak scale comes out as a one-loop factorlower scale than O (100) TeV via gauge mediation [1, 2, 3,4, 5, 6, 7]. The scenario contains only a single scale as op-pose to other scenarios where the supersymmetry break-ing scale and the messenger scale are generated by differ-ent mechanisms. However, it has been known that a con-crete model building of such a one-scale scenario is theo-retically challenging (see Refs. [8, 9, 10, 11, 12, 13, 14, 15]for earlier attempts). In this letter, we propose a simple(and possibly the simplest) model with such low scaledynamics where not only the supersymmetry breakingbut also the masses of the messenger particles are gen-erated by the effects of the strong dynamics of a gaugeinteraction. B. Dynamical supersymmetry breaking/messengersector
A model of the dynamics of the supersymmetry break-ing is based on a supersymmetric SU(5) H gauge theorywith five flavors ( F and ¯ F ) [16], where the subgroup ofa global SU(5) F symmetry ( F : 5 and ¯ F : ¯5) is identi-fied with the gauge groups of the standard model. As wesee later, dynamical supersymmetry breaking is thereforedirectly communicated to the standard model sector di-rectly [1, 4]. The only other ingredient of the model isa singlet superfield S which couples to F and ¯ F in thesuperpotential, W = kSF ¯ F , (1)where k is a coupling constant. F and ¯ F (or theircomposite particles) serve as the messenger fields onceboth the scalar- and the F -term of the singlet obtainnon-vanishing vacuum expectation values; h S i 6 = 0 and F S = 0 [31]. Recently, models of the sweet spot super-symmetry [17] based on this dynamical supersymmetrybreaking/messenger model have been analyzed in theelementary messenger regime [18] and in the compos-ite messenger regime [19]. In the sweet spot supersym-metry , the supersymmetry breaking local minimum at h S i 6 = 0 is realized by the gravitational stabilization mechanism [20]. In this letter, we seek another possi-bility of making h S i 6 = 0.To see how the supersymmetry breaking occurs, letus consider the region where the “messenger mass”, M mess ≡ kS , is smaller than the scale Λ around which theSU(5) H gauge interaction becomes strong. In this region,the model can be described by using mesons, M ∼ F ¯ F and baryons B ∼ F and ¯ B ∼ ¯ F . Here, we omit theindices of the flavor SU(5) F symmetry. In terms of themesons and baryons, the full effective superpotential isgiven by W eff = kS · Tr M + X (det M − B ¯ B − (Λ / ) , (2)where X is a Lagrange multiplier field which guaranteesthe quantum modified constraint between the mesonsand baryons [21]. Λ dyn denotes the dynamical scale ofSU(5) H gauge interaction and it is naively related to thescale Λ by Λ dyn ∼ √ N c Λ / (4 π ) with N c = 5 [22, 23]. Byexpanding the meson and baryon fields around a solutionto the above constraint: M = Λ dyn (Λ dyn δ ij / δ ˆ M ),(Tr δ ˆ M =0), ˆ B ∼ B/ Λ and ˆ¯ B ∼ ¯ B/ Λ , we obtain asuperpotential, W eff ∼ k Λ S + S (cid:18) k δ ˆ M + k ˆ B ˆ¯ B + · · · (cid:19) , (3)which has a linear term of the singlet S . Here, the el-lipses denote the higher dimensional operators of mesonsand baryons, and we have neglected non-calculable cor-rections to the coupling constants which are naively ex-pected to be O (1). This superpotential shows that thereis a supersymmetric minimum at S = 0 and δ ˆ M = 0 orˆ B ˆ¯ B = 0. However, if the singlet S has a local minimumat S = h S i & Λ dyn , the mesons and baryons have posi-tive masses squared and the spontaneous supersymmetrybreaking is achieved by F S ∼ k Λ .Now, the question is: is there a possibility for the sin-glet S to have a local minimum at S = h S i > Λ dyn ? Toaddress this question, notice that there are non-calculablecontributions to the effective K¨ahler potential of the sin-glet S from the strong interactions below the scale Λ, K eff = S † S + 25Λ (4 π ) δK (cid:18) kS Λ (cid:19) . (4)Here, we have used the “naive dimensional analysis” [22,23], and we expect that the non-calculable contribution δK ( x ) has no small parameter. From this K¨ahler poten-tial, we obtain a scalar potential, V ( S ) ∼ | F S | k/ π ) δK (2) ( kS/ Λ) , (5)where δK (2) denotes the second derivative of δK with re-spect to kS/ Λ. This potential shows that there is a pos-sibility that the potential has a local minimum around h S i ∼ Λ /k which is larger than Λ dyn for k . π/ √ S has a local minimumaround Λ /k by the effect of the non-calculable contribu-tions from the strong dynamics (see Fig. 1).Note that calculable radiative corrections to the K¨ahlerpotential through the diagrams in which the mesons andbaryons circulate dominate over the non-caluculable con-tribution δK for a small value of the singlet S , kS ≪ Λ.The potential curves up by this contribution [32]. Themasses of the mesons and baryons, however, become com-parable to the scale Λ around S ∼ Λ /k . Hence, theireffects can be overwhelmed by the non-calculable contri-bution in the region of S ∼ Λ /k . Therefore, the pos-sibility of the local minimum around S ∼ Λ /k can notbe excluded by these effects (see Ref. [25] for a similardiscussion).For M mess = kS ≫ Λ, the dynamics can be describedby using F and ¯ F as elementary fields. In this region, itcan be shown that the potential curves up in the S direc-tion by radiative corrections to the Kahler potential [26].Therefore, the singlet S cannot have a local minimum at S ≫ Λ /k .The above discussion does not exclude the possibilityof having a local minimum around S ∼ Λ /k where allthe calculable contributions are comparable to the non-calculable ones. Put it all together positively, we hereassume that there is a local and supersymmetry break-ing minimum around h S i ∼ Λ /k , aside from the super-symmetric minimum at S = 0 [33]. In the following, weconsider a model with gauge mediation around the localminimum at h S i ∼ Λ /k ∼ π Λ dyn / ( √ k ) & Λ dyn , (6) F S ∼ k Λ , (7) D δ ˆ M E = D ˆ B E = D ˆ¯ B E = 0 . (8) C. Spectrum of supersymmetric standard modelparticles
The mesons δ ˆ M transform as the adjoint represen-tation under SU(5) F whose subgroup is identified withthe standard model gauge group, (i.e. (8 , , (1 , ,(3 , − / and (¯3 , ¯2) / under the standard group). Thus,the mesons mediate the effects of the supersymmetrybreaking to the standard model sector via gauge interac-tions of the standard model. Through loop diagrams of L (cid:144) L L Log @ kS D V H S L Non - Calculable Perturbative H ~ Log L Perturbative H ~ Log L FIG. 1: Schematic picture of the scalar potential V ( S ). Inthe dark shaded region, there is a possibility that V ( S ) hasa local minimum due to the non-caluculable contribution tothe K¨ahler potential from the strong dynamics. the mesons, we obtain masses of the gauginos and scalarparticles in the standard model sector, m gaugino = N m g (4 π ) F S h S i (cid:16) O (cid:16) ( √ k/ π ) (cid:17)(cid:17) , (9) m = 2 N m C η (cid:18) g (4 π ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) F S h S i (cid:12)(cid:12)(cid:12)(cid:12) , (10)where N m = 5 is the sum of the Dynkin index of themesons, C is the quadratic Casmir invariant of the scalarparticles, and g denotes the gauge coupling constant ofthe standard model gauge group. As we discussed above, F S / h S i is given by F S / h S i ∼ √ k Λ dyn / π . The O (1)coefficient η for the scalar masses denotes non-caluculable O ( F S / h S i ) contributions from the heavy hadrons whichare charged under the standard model gauge groups,while gaugino masses do not get O ( F S / h S i ) contribu-tions from them. The O (( √ k/ π ) ) contribution comesfrom the diagrams with more S inserted. As a result,we achieve a model with gauge mediation with a low dy-namical scale,Λ dyn ∼ GeV × k − (cid:16) m gluino (cid:17) , (11)for k = O (1).As an interesting prediction, the gravitino mass can beas small as O (1) eV, m / ∼ × k − (cid:16) m gluino (cid:17) , (12)for Λ dyn ∼ GeV[34]. Here, we have used the defini-tion of the gravitino mass, m / = F S √ M pl ∼ k Λ √ M pl , (13)where M pl ≃ . × GeV denotes the reduced Planckscale.Now several comments are in order. The perturba-tivity of the standard model gauge interactions up tothe scale of the Grand Unification Theory (GUT) put abound on the sum of the Dynkin index of the messengerfield N m as N m . / ln( M GUT /M mess ) , (14)where M GUT ≃ × GeV denotes the scale of GUT.For M mess ∼ Λ ∼ GeV, this condition requires N m as N m ≤
6. Therefore, the Dynkin index of the presentmodel, N m = 5, satisfies the perturbative condition ofthe standard model gauge interactions up to the GUTscale.We should also mention the perturbativity of the cou-pling constant k . The coupling constant k becomes smallat the high energy scale as a result of the large renor-malization effect from the strong gauge interaction ofSU(5) H . Thus, we can expect the coupling constant k stays perturbative up to around the GUT scale, althoughit is not necessarily required [35].The tunneling rate to the supersymmetric vacuumat S = 0 per unit volume is roughly given byΓ /V ∼ h S i e −S E , where S E is estimated by S E ∼ π h S i /V ( S ) ∼ π (4 π ) / (5 k ) ∼ for k ∼
1. Onthe other hand, the vacuum stability condition withinthe observable volume and over the age of the universeonly requires S E & S . The leading effect of thesupergravity comes from the linear term of the singlet S in the superpotential which leads to a linear term in thescalar potential, V ( S ) linear = 2 m / k Λ S + h.c. (15)The linear term, however, is negligible compared with thescalar potential in Eq. (5) around S ∼ Λ /k as long as, k & (cid:18) Λ dyn M pl (cid:19) / . (16)Here we have used ∂V ( S ) /∂S ∼ √ k/ π ) | F S | / Λ dyn for S ∼ π Λ dyn / ( √ k ). The condition is easily satisfiedfor Λ dyn ∼ GeV ( k = O (1)) (Eq. (11)), and hence, thelocal minimum we chose is stable against the supergravityeffects.On the other hand, the linear term plays an importantrole to generate the mass of the R -axion which corre-sponds to the spontaneous breaking of the R -symmetry by S = 0. Since the R -symmetry is broken explicitly bythe linear term, the R -axion obtains a mass [28], m a ≃ m / (cid:18) M pl h S i (cid:19) / (17) ∼
10 MeV × k − / (cid:16) m gluino (cid:17) / . The R -axion couples to the standard model particlesthrough the loop diagrams of mesons which are relevantfor the gauge mediation. As a result, it decays mainlyinto photons at the cosmic temperature T ∼ O (10) MeV.Note that for the axion with mass m a ∼ O (10) MeV, fi-nal states with hadrons or electroweak gauge bosons arekinematically forbidden. The cosmic abundance of the R -axion before the decay (both from thermal and non-thermal production) is estimated to be small enough thatthe decay does not cause a large entropy production [36].The decay temperature is also high enough not to spoilthe success of the Big-Bang Nucleosynthesis. Besides,the above R -axion marginally satisfies an astrophysicalconstraint based on stellar cooling rate and supernovadynamics: m a & O (10) MeV [29]. Therefore, we findthat the R -axion in our model does not cause any cos-mological and astrophysical problems. D. Conclusions
We find a very simple model with gauge mediationwhere the supersymmetry breaking/mediation is real-ized by a dynamics at Λ dyn ∼ GeV. Furthermore,the model predicts a very small gravitino mass ( m / ∼ O (1) eV) for Λ dyn ∼ GeV, which can be measuredat the future collider experiments such as LHC/ILC, bymeasuring the branching ratio of the decay rate of thenext to lightest superparticle [30].Finally, it should be noted that the present model isalso applicable for a wide range of the dynamical scaleup to Λ dyn ∼ GeV, ( k ∼ − , m / ∼ O (10) MeV),where the condition in Eq. (16) is saturated. Acknowledgments