# Charms of Strongly Interacting Conformal Gauge Mediation

CCharms of Strongly Interacting Conformal Gauge Mediation

Gongjun Choi, , ∗ Tsutomu T. Yanagida, , , † Norimi Yokozaki, ‡ Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China Kavli IPMU (WPI), UTIAS, The University of Tokyo,5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan Zhejiang Institute of Modern Physics and Department of Physics,Zhejiang University, Hangzhou, Zhejiang 310027, China (Dated: February 10, 2021)By extending a previously proposed conformal gauge mediation model, we construct a gauge-mediated SUSY breaking (GMSB) model where a SUSY-breaking scale, a messenger mass, the µ -parameter and the gravitino mass in a minimal supersymmetric (SUSY) Standard Model (MSSM)are all explained by a single mass scale, a R-symmetry breaking scale. We focus on a low scaleSUSY-breaking scenario with the gravitino mass m / = O (1)eV, which is free from the cosmologicalgravitino problem and relaxes the ﬁne-tuning of the cosmological constant. Both the messenger andSUSY-breaking sectors are subject to a hidden strong dynamics with the conformality above themessenger mass threshold (and hence the name of the model “strongly interacting conformal gaugemediation”). In our model, the Higgs B-term is suppressed and a large tan β is predicted, resultingin the relatively light second CP-even Higgs and the CP-odd Higgs with a sizable production crosssection. These Higgs bosons can be tested at future LHC experiments. I. INTRODUCTION

Field theories containing a scalar ﬁeld with an interme-diate mass scale have a common pathology even if theyare renormalizable; for a quantum correction to the scalarmass squared ( m ), there appears the quadratic di-vergence proportional to Λ where Λ UV is a UV-cutoﬀ.Notoriously, the Standard Model (SM) suﬀers from sucha pathology due to the presence of the Higgs boson.However, imposing supersymmetry (SUSY) can cure thispathology in an elegant manner by introducing balancebetween bosonic and fermionic degrees of freedom intheir contributions to m . In addition, concerningthe renormalization procedure of a theory, one needsonly wave function renormalization for individual ﬁeldsin SUSY theories. Thus, supersymmetrized theories aretheoretically advantageous as compared to theories with-out SUSY as far as the issue of softening or removingdivergences is concerned.In spite of these merits that one can enjoy in a SUSYtheory, there remain phenomenological problems in theminimal supersymmetric standard model (MSSM) suchas the ﬂavor changing neutral current (FCNC) and theCP problem [1–3]. In addition, the observed vanishinglysmall cosmological constant demands severe ﬁne-tunedcancellation between a F-term SUSY-breaking contribu-tion and a R-symmetry breaking contribution to a scalarpotential. As for these diﬃculties, the low scale gaugemediated SUSY-breaking (GMSB) scenario seems to bealmost a unique solution [4–6]; a gauge mediation modelcan escape from the former two problems. And the lowera SUSY-breaking scale becomes, the milder ﬁne-tuning ∗ [email protected] † [email protected] ‡ [email protected] for the observed vanishingly small cosmological constantis required.Bearing these merits of GMSB models, in this paper,we give our special attention to a class of GMSB modelsdubbed “conformal gauge mediation (CGM)” [7, 8] (seealso the follow-up works in [9, 10]). Diﬀering from manyof GMSB models, this type of GMSB model has the inter-esting property of having less number of free parameters.To be more speciﬁc, the model has a messenger mass as asingle fundamental parameter which determines a SUSY-breaking scale and SUSY particle mass spectrum. Abovethe messenger mass scale, gauge coupling and Yukawacoupling constants are assumed to be near infrared ﬁxedpoints (IRFP) and hence the name “conformal gauge me-diation”.In this work, we construct and study an extendedstrongly interacting conformal gauge mediation model byintroducing a new R -symmetry breaking chiral superﬁeldΦ. The new ingredient of our model as compared to theprevious CGM is that an energy scale of R -symmetrybreaking serves as the fundamental single parameter ofthe model. Thereby, we attribute not only a SUSY-breaking and messenger mass scale but also R-symmetrybreaking scale to a common origin. For the purpose ofsoftening the degree of ﬁne-tuning in producing the ob-served tiny cosmological constant, we consider the sce-nario with the gravitino mass around m / = O (1)eV.The gravitino in this mass range is also favored fromthe view point of cosmology: we do not have an up-per bound on the reheating temperature [11], and hence,the thermal leptogenesis [12] can explain the observedbaryon asymmetry in a naive way. We shall see thatthe model is featured by the R-symmetry breaking scalenear 10 GeV, which in turn provides us with the mes-senger mass and the SUSY-breaking scale amounting to O (100)TeV and O (1)TeV Higgsino mass .In Sec. II, we ﬁrst make a brief review for the ex- a r X i v : . [ h e p - ph ] F e b isting primordial form of the CGM model. After that,we discuss the key essence of our extended model whichis the introduction of a chiral superﬁeld Φ breaking R-symmetry in the model in the spontaneous way and howwe infer the R-symmetry breaking scale. In Sec. III, wediscuss several interesting consequences and features ofthe model including how the model can give us a natu-ral explanation for O (100)TeV messenger mass, how themodel generates µ and B -terms, how the model helps usunderstand the electroweak scale, and why the introduc-tion of Φ is harmless in cosmology and the dark mattercandidate of the model. Finally, in Sec. IV, we concludethis paper by summarizing the structure and resultantaspects of the model. II. STRONGLY INTERACTING CONFORMALGAUGE MEDIATION MODELA. Hidden strong dynamics

In a supergravity model, the scalar potential at theleading order is determined by a diﬀerence between | F | and 3 W /M P where F is the order parameter for theSUSY-breaking, W is R symmetry-breaking constantterm in a superpotential and M P (cid:39) . × GeV is thereduced Planck mass. Now that the observed energy den-sity of cosmological constant turns out to be negligiblytiny, we expect | F | (cid:39) √ W /M P = √ m / M P . Giventhis, it is realized that the smaller gravitino mass makesthe degree of ﬁne-tuned cancellation between two contri-butions to the scalar potential less severe. The desire toavoid unnatural miraculous ﬁne-tuning in parameters inthe model prefers lower values of | F | and m / , posing aquestion: how low m / could be?In [13], the stringent upper bound of the gravitino mass m / < . On the other hand, | F | being directly related to m / , we expect that m / cannot be arbitrarily small tobe consistent with the null observation of any SUSY par-ticles in the LHC to date. Along this line of reasoning,we consider a scenario with m / = O (1)eV correspond-ing to a SUSY-breaking scale ∼ O (100)TeV.Such a low SUSY-breaking scale might imply a GMSBscenario. Yet, we need m / (cid:38) O (100)eV to explain theobserved Higgs boson mass 125GeV in a perturbativegauge mediation model [16, 17]. Therefore, it becomesnecessary to consider the strongly coupled gauge medi-ation scenario [17]. As an exemplary model satisfying See also Ref. [14] where the matter power spectrum resultingfrom the Lyman- α forest data was used to obtain the weakerconstraint m / < ω / /ω CDM (cid:46) . For instance, one may refer to Ref. [15] where an O (1)eV lowerbound on m / is found in the direct SUSY searches in the LHC. this feature, we give our special attention to the stronglyinteracting conformal gauge mediation model [7–10].Therein SUSY-breaking is induced by the presence of aconformal phase of a hidden non-Abelian gauge theory ofwhich the strong dynamics accounts for the interactionsin the messenger sector and SUSY-breaking sector. Be-low, we ﬁrst go through a review of the existing stronglyinteracting conformal gauge mediation model [7–10] andthen in the next subsection we extend it by introducinga new ﬁeld content.For the promised strong interactions in the SUSY-breaking and messenger sectors, we consider SU ( N ) hid-den gauge group with N Q pairs ( i, ˜ i = 1 − N Q ) of ( Q i , ˜ Q ˜ i )and N P pairs of ( a, ˜ a = 1 − N P ) of ( P a , ˜ P a ) [9, 10].Here ( Q i , P a ) and ( ˜ Q ˜ i , ˜ P a ) are assumed to transform asthe fundamental and anti-fundamental representationsunder SU ( N ) respectively. Aside from these matterﬁelds, we introduce N Q singlet ﬁelds S i ˜ i which coupleto ( Q i , ˜ Q ˜ i ). When a choice of ( N, N Q , N P ) satisﬁes thecondition (3 N ) / < N Q + N P < N , there exists anIRFP [18]. For our purpose, we focus on the case with( N, N Q , N P ) = (4 , ,

5) by which the theory lies at theconformal window.With these ﬁeld contents, now we consider the follow-ing superpotential W = λS i ˜ i Q i ˜ Q ˜ i + m P P a ˜ P a , (1)where m P is a mass of P a and ˜ P a , and λ is a dimensionlesscoupling. Q i and ˜ Q ˜ i shall become massive once S i ˜ i ac-quires a vacuum expectation value (VEV). The massivematter ﬁelds P a and ˜ P a are taken to serve as messen-gers in the model. P a and ˜ P a respect the gauged ﬂavor SU (5) F symmetry which we identify with SU (5) GUT inthe SM.We assume that the theory is in the close proximityof an IRFP of SU (4) at the energy scale above m P andthe scalar component of S i ˜ i acquires a non-zero VEV.For the simplicity of discussion hereafter, we take S i ˜ i to be diagonal in the ﬂavor space, i.e. S i ˜ i = Sδ i ˜ i . Oncethe massive ( Q i , ˜ Q ˜ i ) and ( P a , ˜ P a ) are integrated-out, S i ˜ i -dependence of the eﬀective superpotential induced by thegaugino condensation reads [9, 10] W eﬀ ∼ det[ S i ˜ i ] N = S NQN . (2)On the other hand, since the theory has an approxi-mate superconformal symmetry at an energy scale higherthan m P , a low energy Kahler potential of the model isexpected to have the term ∼ ( S † S ) / ∆ S together withother m P -dependent terms where ∆ S is a scaling dimen-sion of S . Along with the superpotential in Eq. (2), thisterm in a Kahler potential yields the scalar potential ofthe following form V scalar ∼ ( S † S ) ∆ S − S − NQN , (3)where we used the notation S here to denote the scalarcomponent of the chiral superﬁeld S . We can inferfrom Eq. (3) that V scalar can be stabilized provided∆ S > N/N Q holds. For S -ﬁeld, the scaling dimensionand the anomalous dimension at the IRFP ( γ S ∗ ) are re-lated via ∆ S = 1 + γ S ∗ and γ S ∗ is found to be ∼ . N, N Q , N P ) = (4 , ,

5) [9, 19]. Thus we see that∆ S > N/N Q is indeed satisﬁed in the model, guaran-teeing the spontaneous SUSY-breaking. Notice that themodel could have IRFP thanks to the presence of ( P a , ˜ P a )ﬁelds, which is the essential point allowing for the SUSY-breaking.By solving the coupled equations β g = β λ = 0 with β g and β λ beta functions of g and λ respectively, IRFPsin the coupling space ( g , λ ) can be found and they read( g ∗ , λ ∗ ) (cid:39) (3 . , .

2) at the one-loop level [9, 19]. Usingthis g ∗ , we can obtain the relation between m P and anenergy scale Λ at which g blows up via the followingdimensional transmutation8 π g ∗ (cid:39) b ln (cid:16) m P Λ (cid:17) → m P ∼ Λ , (4)where b = 3 N − N Q = 9 is the one-loop coeﬃcient of β g after ( P a , ˜ P a ) are integrated-out. The empirical relationΛ QCD ∼ πf π between the dynamical (composite) scaleΛ QCD of QCD and the pion decay constant f π can pro-vide us with the rough estimate of the dynamical scaleΛ of SU (4) in our model in terms of Λ in Eq. (4), i.e.Λ ∼ π Λ. We notice, however, that there can be am-biguity in the factor 4 π in the relation Λ ∼ π Λ sinceour strong dynamics is diﬀerent from QCD in the num-ber of colors and matter ﬁelds, and exploring this factorrigorously is beyond the scope of the current paper. Sowhatever the factor is, the greater one among m P andΛ is to be used for estimating the soft masses basedon the naive dimensional analysis (NDA) [20, 21]. Un-der this circumstance, we assume m P (cid:38) Λ from hereon. Now along this line of reasoning, Eq. (4) informs usthat the model has the non-trivial feature that a mes-senger mass equals the dynamical scale up to an O (1)factor and thereby it becomes the one parameter theory.With the SUSY-breaking induced dynamically, we inferthe SUSY-breaking scale F S ∼ Λ / (4 π ) where F S is theSUSY-breaking order parameter ( F -term of the chiral su-perﬁeld S ) [20, 21].Guided by an eﬀective operator analysis [7, 8] andNDA, one can try to make an estimate for the sfermionand gaugino masses. Using λS ∼ Λ + Λ θ obtainedfrom NDA and parametrizing the ambiguity arising fromthe strong dynamics in the hidden sector by N eﬀ and N (cid:48) eﬀ for each of sfermion and gaugino masses, the softSUSY-breaking masses can be expressed as [9, 10] m ∼ N eﬀ C , SM ( ) (cid:16) α SM π (cid:17) Λ m P , Here one loop level means that one-loop anomalous dimensionsof ﬁelds appearing in Eq. (1) and one-loop β λ were used togetherwith NSVZ beta function for g . m gaugino ∼ N (cid:48) eﬀ (cid:16) α SM π (cid:17) Λ m P , (5)where C , SM ( ) is a quadratic Casimir of a SM gaugegroup, α SM ≡ g / (4 π ) with g SM a gauge coupling of aSM gauge group. Now that the relation between Λ and m P is uncertain in our model, the ratio Λ /m P is treatedas a free parameter. For the later discussion in Sec. III B,we take a scale of the gaugino mass as a free parameter,imposing a GUT relation among the gaugino masses.Recalling that SUSY-breaking scale around O (100)TeV is considered in the model, now we en-counter a fundamental question for an origin of m P .Namely, m P = O (100)TeV needs an explanation forwhy it is much smaller than the Planck scale as a massparameter of the vector-like ﬁelds ( P a , ˜ P a ). B. R-symmetry breaking ﬁeld Φ In this subsection, we show that we can answer thequestion concerning an origin of m P and its magnitude O (100)TeV by relying on a R -symmetry breaking ﬁeld.What we discuss below is not the unique answer to thequestion. Nevertheless, it may lead on to non-trivial as-pects of the model as explained in the next section, whichmotivates our exploration for the possibility presentedbelow.In any supergravity model where a SUSY-breakingscale deviates from the Planck scale, R-symmetry mustbe assumed in order to prevent W from being M P and to reproduce the vanishingly small cosmological con-stant. With that being said, the gauged discrete Z R isa particularly interesting R-symmetry because it satis-ﬁes the anomaly-free conditions for Z NR ⊗ SU (2) L and Z NR ⊗ SU (3) c in the MSSM with three families of quarksand leptons [22], and has Z R as a subgroup to hinder theproton decay at the renormalizable level. Furthermore,once Z R is assumed, the model can beneﬁt from the au-tomatic suppression of the dangerous dimension 5 protondecay operator

10 10 10 5 ∗ [23, 24] and an intermediatescale Higgsino mass in a natural manner. Motivated bythis winning attributes, in our model, we impose the dis-crete gauged Z R as a way to control the constant termin the superpotential. In Table. I, we show R-charge as-signment assumed in our model that makes the mixedanomalies of Z R ⊗ SU (2) L and Z R ⊗ SU (3) c free andYukawa interactions in the MSSM respect Z R .Above all, for Z R to be anomaly-free with respect to SU (4), R-charges of ( P a , ˜ P a ) ﬁelds denoted by R [ P ] and From the anomaly-free conditions for the mixed anomalies of Z NR ⊗ SU (2) L and Z NR ⊗ SU (3) c ( N ∈ N ), and constraintsfrom Yukawa interactions, it can be shown that R-charge of op-erators

10 10 10 5 ∗ and H u H d are 0 modulo N and 4 modulo N respectively. ∗ H u H d P ˜ P Φ R -charge 3 -1 -2 0 2 2 2TABLE I. R-charge assignment. For denoting quarks andleptons in the SM, we borrow their representations in a GUTmodel with the gauge group SU (5) GUT . H u ( H u ) is the chi-ral superﬁeld for the up (down) type Higgs. For S i ˜ i and( Q i , ˜ Q ˜ i ) ﬁelds, we assign any arbitrary R-charges satisfying R [ S i ˜ i ] + R [ Q i ] + R [ ˜ Q ˜ i ] = 2 modulo 6 with R [ Q i ] + R [ ˜ Q ˜ i ] aneven number. For ( P a , ˜ P a ), another R-charge assignment ispossible as far as R [ P ] + R [ ˜ P ] = 4 is satisﬁed. R [ ˜ P ] should satisfy [25–27]4 + 52 ( R [ P ] + R [ ˜ P ] −

2) = 62 k (k ∈ Z ) . (6)Note that the mixed anomaly of Z R ⊗ SU (4) does notconstrain R-charges of ( Q i , ˜ Q ˜ i ) ﬁelds since N Q = 3. Sowe omitted the contribution of ( Q i , ˜ Q ˜ i ) in Eq. (6). FromEq. (6), we obtain R [ P ]+ R [ ˜ P ] − k − × (2 /

5) whichimplies that R [ P ] + R [ ˜ P ] can never be 2 modulo 6. Inregard to the mass term of ( P a , ˜ P a ) in Eq. (1), this tellsus that either Z R is explicitly broken at an energy scaleabove m P or m P needs to be understood as a spurionﬁeld with a proper R -charge. In our work, we considerthe later case with the choice of R [ P ] + R [ ˜ P ] = 4, i.e. k = 3.Specifying m P as a spurion ﬁeld, m P may stem fromcondensation of a ﬁeld coupled to ( P a , ˜ P a ). For thatpurpose, we introduce a chiral superﬁeld Φ with R-charge 2 whose condensation induces the spontaneous R-symmetry breaking. In the next section, we shall makean explicit explanation how introducing Φ can explain O (100)TeV messenger mass. Before discussing m P , nowwe discuss a R-symmetry breaking scale in what follows.We may assume that Φ is a low energy degree of free-dom of a hidden strong dynamics and thus has the fol-lowing self-interaction terms in the superpotential W ⊃ − ξ Φ + κ (4 π ) Φ M P + ... , (7)where ξ is a dimensionful parameter and the ellipsisstands for the higher order terms. The factor (4 π ) ispresent in the second term of Eq. (7) on account of theassumed strong dynamics from which Φ results [20, 21].After condensation of Φ, terms in Eq. (7) are expected to For S i ˜ i and ( Q i , ˜ Q ˜ i ) ﬁelds, we assign any arbitrary R-chargessatisfying R [ S i ˜ i ]+ R [ Q i ]+ R [ ˜ Q ˜ i ] = 2 modulo 6 with R [ Q i ]+ R [ ˜ Q ˜ i ]an even number. We anticipate the formation of the meson ﬁeld M i ˜ i ∼ Q i ˜ Q ˜ i in the low energy conﬁned phase of SU (4) and themeson ﬁeld should not break Z R . To see this, we may assume that (3 k − × (2 /

5) is an integermultiple of 6. Then we encounter an inconsistent conclusion that k cannot be an integer. produce constant terms in the superpotential. For deter-mining a constant term in W , there are two possibilitiesdepending on which is greater among the two terms inEq. (7).If the ﬁrst term is greater than the second one, we ex-pect that Z R -breaking scale is equal to m / ∼ O (1)eVsince the natural scale for (cid:112) | ξ | is the Planck scale with-out any additional symmetry to suppress the ﬁrst term.This possibility, however, predicts an inconsistent verylow SUSY-breaking scale via m P ∼ (cid:112) πF S and Eq. (8).Therefore, we consider the case where the second term isgreater than the ﬁrst term in Eq. (7), assuming a sym-metry suppressing the ﬁrst term, e.g. a discrete Z underwhich Φ and the spurion ξ have the charge 2 and -2 re-spectively. By comparing the dominating second termin Eq. (7) to W = m / M P , we ﬁnd that m / = O (1)eVrequires (cid:104) φ (cid:105) (cid:39) GeV with κ = O (1) where φ is thescalar component of Φ.We end this section by commenting on the contributionto the mixed anomaly of Z R ⊗ SU (2) L and Z R ⊗ SU (3) c made by ( P a , ˜ P a ) ﬁelds. ( P a , ˜ P a ) are the only non-MSSM ﬁelds charged under SU (2) L and SU (3) c andthus their presence spoils the cancellation of the mixedanomaly of Z R ⊗ SU (2) L and Z R ⊗ SU (3) c . For eachanomaly, the contribution from ( P a , ˜ P a ) is identically2( R [ P ] + R [ ˜ P ] −

2) = 4. For completeness of the model,it suﬃces to introduce a pair of new chiral superﬁelds( P (cid:48) , ˜ P (cid:48) ) that transform as and ∗ under SU (5) GUT andserve as singlets under SU (4). Their R-charges satisfythe condition, R ( P (cid:48) ) + R ( ˜ P (cid:48) ) = 0, to guarantee vanish-ing Z R ⊗ SU (2) L and Z R ⊗ SU (3) c anomalies. Then,( P (cid:48) , ˜ P (cid:48) ) ﬁelds obtain the mass as heavy as 10 GeV fromthe operator ∼ Φ P (cid:48) ˜ P (cid:48) whose presence does not cause anyharm. III. CHARMS OF THE MODELA. Messenger mass

Since Φ is assigned R-charge 2, the mass term in Eq. (1)can be originated from W ⊃ κ Φ M P P a ˜ P a , (8)where κ is a dimensionless coupling. Thus with κ = O (1), ( P a , ˜ P a ) ﬁelds acquire the mass m P = O (1)TeV atthe energy scale around (cid:104) φ (cid:105) (cid:39) GeV when Z R getsspontaneously broken. On the other hand, at an energy We checked the presence of a consistent charge assignment under Z . scale E lower than (cid:104) φ (cid:105) (cid:39) GeV, we have [7] m P ( E ) = m P ( M IRFP ) Z P ( E ) = (cid:18) EM IRFP (cid:19) γ P ∗ m P ( M IRFP ) , (9)where Z P ( E ) and γ P ∗ are the wavefunction renormal-ization constant and the anomalous dimension at theIRFP for ( P a , ˜ P a ) respectively. In addition, M IRFP isthe energy scale at which the UV to IR evolution of g reaches its IRFP. With γ P ∗ (cid:39) − . m P = O (100)TeV can be obtained at the messenger massscale provided we choose M IRFP (cid:39) GeV.In summary, Φ condensation imposes O (1)TeV mass tothe messengers on the spontaneous Z R -breaking. How-ever, thanks to the non-trivial large anomalous dimensionof ( P a , ˜ P a ) ﬁelds which is attributable to the strong dy-namics of SU (4), m P at the messenger threshold becomesas large as O (100)TeV. B. µ -term and B -term Referring to the R-charge assignment of the model inTable. I, we see that the µ -term can arise from W ⊃ κ Φ M P H u H d , (10)where κ is a dimensionless parameter. Given (cid:104) φ (cid:105) (cid:39) GeV, the model produces the Higgsino mass µ = O (1)TeV for κ = O (1).On the other hand, the SUSY-breaking being at-tributed to the non-vanishing F-term of S i ˜ i , we noticethat the F-term of Φ should vanish since Φ is decoupledfrom S i ˜ i . Thus the term in Eq. (10) cannot generate anon-vanishing contribution to the mixing term for H u and H d Higgs bosons ( B -term). In addition, the wave-function renormalization of H u and H d cannot generatea signiﬁcant non-zero B -term at the scale around m P since there are no S i ˜ i -dependent terms of the wavefunc-tion renormalization constants of H u and H d at one-looplevel. The same argument applies for the MSSM trilinearscalar coupling terms ( A -term) to make them suppressed.Note that the suppressed A and B terms are favorablefor resolving the CP problem appearing in the MSSM.Given the suppressed B -term at the messenger massscale, the model is expected to result in a small valueof B -parameter and a large tan β at the electroweakscale [28]. With this large tan β , the masses of the secondCP-even Higgs ( H ) and the CP-odd Higgs ( A ) can be assmall as 2-3 TeV as shown in Ref. [29]. Figure 1 showsthe predicted m A ( ≈ m H ) and tan β as functions of thephysical gluino mass for Λ S eﬀ = 1000 TeV and 1200 TeV,where (Λ S eﬀ ) ∼ N eﬀ Λ /m P . In the plots, we estimatethe scalar masses as m (cid:39) C , SM ( ) (cid:16) α SM π (cid:17) (Λ S eﬀ ) . (11)We take the messenger scale to be 200 TeV, where thenon-zero scalar masses and gaugino masses are given m gluino (GeV)1000150020002500300035004000 m A ( G e V ) m gluino (GeV)30354045505560 t a n FIG. 1. The plots of the CP odd Higgs mass ( m A ) and tan β as functions of the physical gluino mass. We take µ < S eﬀ = 1000 and1200 TeV, respectively. m gluino (GeV)10 H / A ( pb ) FIG. 2. The b-associated production cross-section of

H/A asa function of the physical gluino mass. The black lines and redlines correspond to Λ S eﬀ = 1000 and 1200 TeV, respectively. as boundary conditions of renormalization group equa-tions of MSSM. In the plots, we assume a GUT rela-tion among the gaugino masses, i.e. m bino : m wino : m gluino = g : g : g as in the minimal gauge media-tion, where g , g and g are gauge coupling constantsof U (1) Y , SU (2) L and SU (3) c , respectively. The massspectrum of SUSY particles is calculated using SOFTSUSY4.1.9 [30]. The lightest CP-even Higgs mass computedby

FeynHiggs 2.18.0 [31–38] is 124.5 GeV (125.3 GeV)for Λ S eﬀ = 1000 TeV (1200 TeV) and m gluino = 3 TeV. InFig. 2, we show the b-associated production cross-sectionof H/A . The cross section is computed using

SusHi pack-age [39, 40]. The Higgs masses, m A and m H , are smallerthan 3 TeV for m gluino < C. Understanding the electroweak scale

As was discussed in the previous subsection, the modelis featured by a large tan β at the electroweak scale. Be-cause of this, one of the electroweak symmetry breaking(EWSB) conditions yields the relation m Z (cid:39) − m H u − µ , (12)where m Z , m H u and µ are the masses of Z-boson, theup-type Higgs boson and the Higgsino at the electroweakscale. In the GMSB models with a large tan β at the elec-troweak scale, there is no reason for µ to be of the sameorder as − m H u , which makes the ﬁne-tuned cancellationbetween the two to produce m Z of the smaller order ofmagnitude incomprehensible.To put it another way, it is challenging to under-stand why the uncorrelated parameters can be of thesame order to produce a smaller scale , i.e. the elec-troweak scale. In the strongly interacting conformalgauge mediation model, however, we realize that the rela-tion O ( |− m H u | ) = O ( | µ | ) which is necessary for Eq. (12)is a natural consequence of the model and the tiny cos-mological constant rather than the conspiracy betweenparameters. Below we demonstrate why this is the caseby expressing − m H u and µ in terms of a common pa-rameter and showing the resulting expressions are verycomparable.Since m H u is dominantly determined by stop loops, itcan be related to m P via − m H u ∼ π y t m t log m P m ˜ t ∼ . m t ∼ (cid:16) α π (cid:17) m P , (13)where α = g / (4 π ), y t is the top Yukawa coupling and m ˜ t is the stop mass.Meanwhile, µ -parameter can be related to m P by thecosmological constant as shown by what follows. (cid:104) φ (cid:105) determines the constant term in the superpotential W throughout Eq. (7) and thus the vanishingly small cos-mological constant demands | F S | (cid:39) √ κ (4 π ) (cid:104) φ (cid:105) M P = √ κ κ (4 π ) µ , (14)where F S is the F-term of S -ﬁeld. For the last equality,we used µ = κ (cid:104) φ (cid:105) /M P which is indicated by the factthat the µ -term is generated from the operator shown inEq. (10). Finally, combined with Eq. (14), the relation F S (cid:39) m P / (4 π ) yields µ (cid:39) κ (4 × √ × (4 π ) × κ ) / m P . (15)Now taking α (cid:39) /

10 in Eq. (13) and κ , κ = O (1)in Eq. (15), we encounter the notable consequence of themodel: both of − m H u and µ are parametrized by thecommon parameter m P and very close to each other asa result of the model’s structure.Therefore, the closeness of independent parametersin their magnitudes can be understood natural in thestrongly interacting conformal gauge mediation modelwith the aid of the empirical condition for the vanish-ing cosmological constant. And ultimately this helps usunderstand the separation between the EWSB scale andthe soft SUSY-breaking mass scale. D. Cosmologically safe Φ The chiral superﬁeld Φ was newly introduced inSec. II B as the ﬁeld whose condensation is responsiblefor the spontaneous breaking of Z R down to Z R . Thismeans that its condensation produces three distinct de-generate vacua. Hence, the model predicts the formationof domain walls at the time when Z R breaks down andreduces to Z R . To avoid the situation where the energybudget of the universe is dominated by that of these do-main walls, it is required for the model to assume thatthe breaking of Z R takes place prior to the end of theinﬂation.Yet, the simple assumption for breaking of Z R dur-ing the inﬂation era does not fully resolve the potentialdomain wall problem. Even if the scalar component ( φ )of Φ sits at a global minimum of its potential duringthe inﬂation, it is still probable to have the symmetryrestoration at the post inﬂation era in the case where theﬁeld ﬂuctuation is too large. When the inﬂaton ﬁeld os-cillates around the origin of its ﬁeld space at the end ofthe inﬂation, φ may begin its oscillation as well and thisoscillation of φ can induce the growth of its own ﬂuc-tuation via parametric resonance depending on a formof the potential. Then given Eq. (7), one may wonderwhether the model suﬀers from such a dangerous sym-metry restoration or not.To see whether the restoration of Z R after the inﬂa-tion is likely to happen, here we examine the potential of φ in the early universe. During the inﬂation, we may havethe Hubble induced mass term for φ due to its couplingto the inﬂaton ﬁeld. We assume a suﬃciently large Hub-ble expansion rate during the inﬂation ( H inf ) so that theHubble induced mass term and φ -term in V ( φ ) domi-nate over the cubic term. The reason for this assumptionis to become clear soon. Accordingly, we have the follow-ing eﬀective potential of φ during the inﬂation V ( φ ) eﬀ , inf = − c H H φ + 16 κ (4 π ) φ M P , (16)where the negative Hubble induced mass is necessary tobreak Z R during the inﬂation and c H > c H , κ = O (1), applicationof ∂V ( φ ) eﬀ /∂φ = 0 to Eq. (16) yields (cid:104)| φ inf |(cid:105) (cid:39) √ H inf M P / (4 π ) (cid:39) × × (cid:114) H inf GeV GeV . (17)After the inﬂation ends, while the inﬂaton ﬁeld ( σ ) os-cillates around a global minimum of its potential V ( σ ) inf ,the eﬀective potential of φ will be modiﬁed to V ( φ ) eﬀ , osc = − c H V ( σ ) inf M P φ + 16 κ (4 π ) φ M P . (18)It is expected that following the oscillation of σ , φ beginsits oscillation which can possibly drive the growth of theﬁeld ﬂuctuation. Now regarding this point, of particularinterest in Eq. (18) is the presence of φ (sextet) term.We notice that the growth of the ﬂuctuation of φ during φ -oscillation due to parametric resonance [42, 43] is ex-pected to be ineﬃcient for the sextet potential, which wasboth analytically and numerically conﬁrmed in Ref. [44]. In spite of this, one may still wonder that at a cer-tain time during the oscillation of φ , the cubic term ofthe potential due to Eq. (7) can become greater than φ -term so that the growth of the ﬂuctuation of φ dueto the parametric resonance becomes eﬃcient. Even inthis case, however, we see that the restoration of Z R isavoided since the ﬁeld value ( φ (cid:63) ) at this transition time(the cubic term becomes as large as φ -term) satisﬁes [45] φ (cid:63) = (cid:18) ξM P κ (4 π ) (cid:19) / < × (cid:104) φ (cid:105) (cid:39) GeV , (19)where (cid:104) φ (cid:105) (cid:39) GeV is the required VEV of φ at thepost inﬂationary era for the breaking of Z R . Note that ξM P can be at most 10 GeV to make the ﬁrst term inEq. (7) smaller than the second one. So Eq. (19) is indeedeasily satisﬁed. Therefore, basically Eq. (18) and Eq. (19) Note that for the quartic (or cubic) potential case, there existboth of negative friction on the motion of φ and the eﬃcientparametric resonance of ﬁeld. This allows for the symmetryrestoration at the post inﬂationary era for the quartic (or cu-bic) potential case [44]. preclude our concern for the potential restoration of Z R symmetry to cause the domain wall problem at the postinﬂationary era.We end this subsection by pointing out a lower boundon H inf to make our discussion made thus far valid.Recall that Eq. (16) was obtained by assuming a largeenough H inf to make the Hubble induced mass term andthe sextex term much greater than the cubic term. Wenotice that this assumption is justiﬁed when (cid:104)| φ inf |(cid:105) inEq. (17) is greater than φ (cid:63) in Eq. (19). Thus from thisrequirement, we obtain the following lower bound on H inf (cid:104)| φ inf |(cid:105) > φ (cid:63) → H inf > GeV . (20) E. Dark Matter

As one of the essential ingredients of the model, thegravitino is assumed to have the mass as light as O (1)eV.As such, it is relativistic when produced from the decayof MSSM particles or scattering events among MSSMparticles. Therefore, it contributes to the current darkmatter (DM) population as a warm dark matter (WDM)as long as it is the lightest supersymmetry particle (LSP)in the model. Indeed, it is LSP in the strongly interactingconformal gauge mediation model and explains O (1)% ofthe current DM abundance [14]. On the other hand, as a stable and neutral low energydegree of freedom of SU (4) gauge theory, we can havethe following composite states as the candidate of thecold dark matter (CDM) [10] Φ DM ⊃ QQQ ˜ Q † , QQ ˜ Q † ˜ Q † , Q ˜ Q † ˜ Q † ˜ Q † ... , (21)which is SU (4) invariant via the omitted contractionwith anti-symmetric (cid:15) -tensor. As discussed in Sec. II A, m / = O (1)eV implies the SUSY-breaking scale within O (100)TeV. Since the mass of the above CDM candi-date is expected to be around Λ (cid:39) (cid:112) πF S , we see thatit satisﬁes the upper bound on a thermal DM obtainedfrom the partial wave unitarity [47, 48].Hence, basically the strongly interacting conformalgauge mediation is the very model which naturally em-bodies the ΛCWDM model studied in Ref. [13, 14]. Aswas pointed out in those references, the scenario (espe-cially m / ) will be tested by future probes of the matterpower spectrum at scales k = O (0 . − O (0 . h − Mpcwith a higher resolution. Since the fraction of DM abundance attributed to the light grav-itino, f / = ρ / , / ( ρ / , + ρ CDM , ), is proportional to m / ,one obtains 1 /

10 of the constraint on f / (cid:46) .

12 given inRef. [14] as the expected f / for m / = O (1)eV. For another composite DM candidate in a low-scale gauge medi-ation model, see Ref. [46].

IV. CONCLUSION

In this paper, by extending the existing CGM model,we constructed a GMSB model featured by a hiddenstrong dynamics with the conformal phase at a high en-ergy. Motivated by the phenomenological problems inSUSY models, we assumed a light gravitino mass sce-nario with m / = O (1)eV (corresponding to O (100)TeVSUSY-breaking scale) which is allowed by the current as-trophysical and the LHC constraint. For the strong dy-namics, a hidden SU (4) gauge symmetry with three pairsof matter ﬁelds ( Q i , ˜ Q ˜ i ) is assumed. The matter sector ofthe SU (4) gauge theory is extended with additional ﬁvepairs of messenger ﬁelds ( P a , ˜ P a ) whose presence ensuresthe presence of an IRFP. As a consequence, both theSUSY-breaking scale and the messenger mass ( m P ) be-come equal to one another up to an O (1) factor, makingthe invisible sector parametrized by a single parameter,i.e. m P .On top of this, guided by the question for an origin of m P , we introduced the R-symmetry ( Z R ) breaking ﬁeldΦ with R-charge 2. Remarkably, thanks to the nontriv-ial structure of the model (strong dynamics, conformalphase in a high energy regime), m P = O (100)TeV wasshown to stem from R-symmetry breaking scale given bythe cosmological constant. Therefore, the model uniﬁesorigins of the SUSY-breaking scale, messenger mass andR-symmetry breaking scale with a great self-consistencyalthough nothing enforces such a correlation in princi- ple. We emphasize that the value of the model preciselylies at this point: the model is parametrized by a singledimensionful parameter like QCD.We also presented unexpected charms of the model:the origin of the messenger mass, O (1)TeV µ -term, thesuppressed B -term at the electroweak scale, the possibil-ity of having the light second Higgs bosons, the natural(comprehensible) electroweak scale, cosmologically safeintroduction of Φ and dark matter candidates. We no-tice that these rich and appealing aspects of the modelexcept the last one is totally ascribed to the introduc-tion of R-symmetry breaking Φ ﬁeld with R-charge 2and thus new merits of our model as compared to theexisting CGM. The presence of O (1) ambiguities inher-ent in values of quantities for which we estimate basedon the naive dimensional analysis is considered the weakpoint of the theory, which we hope to be improved with anon-perturbative approach like the lattice computation. ACKNOWLEDGMENTS

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