Hypercharged Conformally Sequestered Gauge Mediation
aa r X i v : . [ h e p - ph ] F e b Preprint typeset in JHEP style - HYPER VERSION
Hypercharged Conformally Sequestered GaugeMediation
Hae Young Cho
FPRD and Department of Physics and Astronomy, Seoul National University, Seoul 151-747,KoreaE-mail: [email protected]
Abstract:
The
Bµ/µ solution in GMSB via the hidden sector dynamics is simple and natural.However, it has some obstacles to be physical. To circumvent this situation, we introduce the visibleand the hidden branes, each of which has its own U (1) symmetry, in a five dimensional setup. Inthe bulk we allow Chern-Simons coupling between the visible and the hidden U (1)s which gives anenhancement of the mass of bino. If this gives a considerable contribution to the mass of bino, wecan get a proper radiative electro weak symmetry breaking with the boundary conditions, in which Bµ and squared scalar masses are suppressed at the scale, where the hidden sector is integratedout. Keywords:
MSSM, µ problem in GMSB, RG effect of the Hidden Sector. ontents
1. Introduction 12. Sequestering in Gauge Mediation 23. Introducing a Chern-Simons Term in the Bulk 54. Phenomenology 75. Conclusion 7
1. Introduction
Sequestering is a mechanism that can suppress the amplitudes of the unwanted operators. Thisis usually used when we want to suppress the tree level flavor changing neutral current (FCNC)in the mediation mechanism, where the gravity may have a significant contribution. It can beunderstood in the geometrical sense via the string theory [1]. Sequestering gives a number ofinteresting phenomenological features so it is worth studying.There are a variety of mediation mechanisms in MSSM. Among them the most famous ones,which are free from FCNC problem are gauge mediated supersymmetry breaking (GMSB) andanomaly mediated supersymmetry breaking (AMSB). In GMSB, generating an electro weak scale µ is not serious by itself however, the requirement for the low energy electro-weak symmetry break-ing (EWSB) makes a very unnatural situation [2]. Among a number of possible explanations toameliorate this [3, 4, 5], the sequestering idea suggests a very simple solution [4, 5]. By the way,once the information of the hidden sector renormalization effect is imposed, we have an additionaleffect: the gravitino has an enhanced mass. This aspect also makes the sequestering idea interestingin GMSB. The idea to solve Bµ/µ problem in GMSB via the conformal sequestering is clear andsimple. However, the simplest form appears not to have physical case, i.e. it does not seem toprovide a physical solution to a proper EWSB [6, 9]. It is because the boundary conditions given atan intermediate scale have relatively small scalar squared masses including higgs. When we followalong the MSSM RG equation, it is hard to satisfy the EWSB conditions. In other words, theparameter space where we have a proper EWSB is not compatible with the boundary conditionsat the intermediate scale. Therefore some modification to the simplest case is necessary. Here weconsider the 5 dimensional setup. We introduce two 3 branes: one is where the visible sector residesand the other is for hidden sector. We introduce U (1) h at the hidden brane and a five dimensionalChern-Simons coupling in the bulk. As an effect of five dimensional Chern-Simons term, the massof bino on the visible brane is enhanced. This is the idea in [11] to solve the tachyonic slepton * L CFT :CFT breakingHidden sector Integrated out L : SUSY Breaking M EW ConformalSequestering
Figure 1: Scale and the dynamics of Ref.[4, 5]problem in a pure AMSB setup. Just like [11], in this setup, all the scalars charged under U (1) Y get the radiative correction by the mass of bino. With a numerical study we find a physical solution,i.e. get a proper EWSB.This paper is organized as follows. In section 2, we discuss on the sequestering in GMSB. Insection 3, we take the viewpoint of the five dimensional setup and introduce a Chern-Simons termin the bulk to find a physical solution. In section 4, we discuss the phenomenological implication.Finally we make a conclusion.
2. Sequestering in Gauge Mediation
Strictly speaking, sequestering in GMSB is not a necessary condition to circumvent FCNC problembecause supersymmetry breaking occurs at a rather low energy scale so that undesirable gravitycontribution is negligible. The role of sequestering in GMSB, however, appears to be interesting andattractive because of its unique feature[10]. In GMSB there is a problem known as
Bµ/µ problem,and it is recently suggested that if we consider the sequestering effect in GMSB, then we can solve
Bµ/µ problem [4, 5]. Here we will briefly review the idea of sequestering as a solution to
Bµ/µ problem. In the supersymmetry conserving part, µ is the unique dimensionful parameter so that itis necessary to link it with supersymmetry breaking to ensure the low energy supersymmetry. InGMSB, it is possible to introduce the superpotential as W = λX · ¯ + ξ d H d · + ξ u H u ¯ · ¯ , (2.1)where , ¯ are messenger fields, X denotes the field which breaks supersymmetry and, λ and ξ u,d are O (1) appropriate dimensionless couplings. After integrating out the massive messenger fields,– 2 –e get µ ∼ ξ u ξ u π H H (cid:18) M † M F † M † (cid:19) = ξ u ξ d π Λ (cid:20) O ( F M mess ) (cid:21) ,B µ ∼ ξ u ξ d π H H (cid:18) M † M ( F F † M M † ) (cid:19) = ξ u ξ d π Λ (cid:20) O ( F M mess ) (cid:21) = Λ µ,A H , ∼ ∂ log Z H u,d ∂ log X = − | ξ u,d | π − | ξ u,d | π log XX † FM ∼ − | ξ u,d | π FM ,m H u,d ∼ ∂ log Z H u,d ∂ log X∂ log X † = − (cid:18) | ξ u,d | π (cid:19) − | ξ u,d | π log XX † ) F F † M M † ∼ − (cid:18) | ξ u,d | π (cid:19) F F † M M † , (2.2)Here we see Bµ = Λ µ , which is undesirable for the phenomenological requirement. This is the Bµ/µ problem in GMSB. Here we see that the relation between A and µ , i.e. A u A d = | µ | . Notto conflict to the perturbation, the coupling ξ u,d should not be large. For a convenience, we take ξ u = ξ d , but later we will consider a general case in range, where the perturbation of the messengercoupling is guaranteed.The basic idea to solve this problem, which we concern, is using the 1PI effect on the propagatorof X in the strongly interacting hidden sector. As a result, the operators which are proportional to XX † are suppressed relative to those which are proportional to either X or X † . The operators forsupersymmetry breaking masses are generated at the original messenger scale just in the case ofusual GMSB setup. As we go down to low energy, the theory goes through a conformal window. Atan intermediate scale Λ CF T , the conformal symmetry is broken and the supersymmetry breakingoperators get their values, which are affected by the hidden sector RG effect described above. Thisimposes that Bµ can be made of O ( µ ) or smaller, therefore Bµ/µ problem in GMSB can besolved with a simple assumption. In addition to this, there can be another effect, coming from theanomalous dimension of X , which makes the masses of ordinary superpartners suppressed relativeto the mass of the gravitino. In other words, the amount that the gravitino feels by supersymmetrybreaking is not the same as the others.Here we want to make it clear whether this mechanism spoils the nice feature of GMSB insolving FCNC problem or not. The gravitational contribution to soft breaking parameters appearsto be proportional to the mass of gravitino, which can be a source of the FCNC problem. As denotedabove, the gravitino mass is enhanced by the hidden sector RG effect and the gauge contributionsuppressed by a factor of anomalous dimensions. Here we can see that a tension between FCNC and Bµ/µ problem. As denoted in [5] if the gravity contribution is enhanced, then one of the virtuesof GMSB is lost. Since the problematic contribution to FCNC appears in soft scalar mass squares,we should be careful about this inequality, m gravity ∼ F † FM P = m / < m gauge . (2.3) This anomalous dimension should be considered independently i.e. the effect which suppresses the operatorscontaining XX † is another. To make this difference clear, see [5, 9] – 3 –y the language appearing in [5], the constraint on the mass parameters is given by(16 π ) λ M mess M p (cid:18) Λ ∗ Λ CF T (cid:19) γ X < O (1) , (2.4)where λ is the coupling given in (2.1), Λ ∗ is a scale where the hidden sector gets conformal and γ X is an anomalous dimension of X . This can be easily satisfied if (cid:16) Λ ∗ Λ CFT (cid:17) γ X is not large.Now we investigate the low energy physics with a numerical tool. m i ∼ α i π Λ (i=1,2,3) , A H u,d ∼ − µ,m φ ∼ , m H u,d + µ ∼ With these boundary conditions, we use softsusy to investigate the low energy spectrum [13].The idea is very simple and natural however, it appears to have an unnatural situation in RGimproved studies [6, 9]. In the MSSM RG equations, we see the scalar masses are determined bythe gaugino contribution, the trilinear terms and their masses. In this analysis, we just use thevalue of µ , which is given by the low energy requirement, that is we can not handle µ and A u,d at Λ CF T . This makes the analytic approach to this problem difficult. Since A u,d grow via mainlythe gluino contribution and give considerable radiative correction to higgs mass because of largeyukawa couplings especially top yukawa at small tan β region. From the RG equation of MSSM, wesee that gauginos give positive contribution and, A u,d and scalar masses give negative contributionto higgs masses as we go down to the low energy scale. Since A u,d get larger than wino and bino,the higgs masses can get negative. On the other hand, because of A l , the lightest stau gets negativesquared mass, which is not favored. If we restrict Bµ to a positive definite quantity at the lowenergy scale in order not to have a tadpole problem, Bµ/µ , which is under the control of onlygauginos and A u,d,l can grow large enough to threaten the stability of the higgs potential in someparameter space. Unfortunately Bµ is also a given quantity in this analysis, we check whetherit is from the boundary condition, which we assume at Λ CF T . And the result was the boundarycondition is not compatible with the low energy physics [6]. It also appears that it is hard to makethings better if Λ is larger than about 200TeV. Though it is not easily seen, if we have a large Λ,then all the soft terms get the radiative corrections from rather massive gauginos, of which massesare proportional to Λ at Λ
CF T . As denoted above, the gluino gives large correction to A u,d so that Bµ gets negative at electro weak scale. In [9], the authors try to evade this problem by introducingadditional messenger masses by an adjoint chiral multiplet, which is supposed to break the grandunified theory. As a result, they are free from that unnatural situation however, the scalar massesappear small so that the experimental bound might be dangerous. In the next section, we suggestanother way out. There is a confusion on this boundary conditions [7, 8]. Since µ and A are generated via supersymmetry breaking F terms, a survey on the effective Kahler potential shows that higgs mass is given as m h + | µ | . By the hidden sectorRG effects higgs masses is vanishes like the other scalar. – 4 – . Introducing a Chern-Simons Term in the Bulk Recently, it is found that a hidden U (1) h has interesting phenomena in the low energy physics.Among a number of solution to solve the tachyonic slepton problem, there is a study where hidden U (1) h takes an important role [11, 12]. There they consider two three branes, which have theirown U (1) gauge theory: U (1) v for the visible brane and U (1) h for the hidden brane. In the fivedimensional bulk we introduce a Chern Simons coupling R C p − ∧ trF , where C p − is Ramond-Ramond p − U (1) gauge fields get mixed, so that there can betwo linearly independent combinations. One of them remains massless, which will be U (1) Y , andthe other gets massive. By this mechanism, bino can get an additional contribution, and the scalarpartners of fermions as well as the higgs get radiative correction δm i = − π g Y i M log µM . (3.1)Therefore, the boundary condition at UV scale can be changed. Here we want to do the same jobin the conformally sequestered GMSB setup. m i ∼ α i π Λ , (i=2,3) A H u,d ∼ − µ,m φ ∼ , m H u,d + µ ∼ ,m ∼ α π Λ + (enhancement by CS interaction: ˜ M ) (3.2)Then we use a package softsusy again to check whether our modification works. For simplicity,we will consider Λ CF T to be close to the original messenger scale because we do not want toconsider the visible sector RG effect. This also help us to consider the more suppressed case thana naive expectation of 16 π , because it can minimize the possible visible sector contribution. Theboundary condition (3.2) is used at the effective messenger scale, and below that scale, it is agood approximation to use MSSM RG. We set the effective messenger scale as 10 GeV, varyingtan β and Λ. The important ingredient is the mass of bino correction from CS interaction. Thisdepends on our choice, and we assume that it is order of µ ˜ M ∼ O (1) . One question may ariseon the additional CP phase. Though we assume that supersymmetry breaking is up to a singlefield X , there can be a misalignment between other gauginos and bino so that there can exist theadditional CP phase. For simplicity, however, we assume there is no additional CP phase. Withthese boundary conditions, we run softsusy . In the right panel of Fig. (2) we see that there exists parameter space where the low energyEWSB requirements and the consistency of mechanism are satisfied. The region can be changed,when we allow correction to the mass of bino, i.e. this can be tuned by appropriate ˜ M andsuppression factor, which is given as boundary conditions at Λ CF T . To see the effect of ˜ M , here Unlike the original idea, the gravitino mass is not an order parameter in this case. So we make a use of µ , whichis made for a proper EWSB. There can be an error of ± – 5 – L H GeV L T a n Β Not Allowed L H GeV L T a n Β Allowed
Figure 2: We choose Λ
CF T = 10 GeV and set sign of µ to be positive. The blue region is excludedby the mass bound for lightest higgs, the red by the inconsistency and the orange by the stau massbound. The left panel is the result of ˜ M = 0 and the right one is the result of ˜ M = 5 µ . In theright, the colored with gradient is allowed region, and the brighter is the better.we do a simple numerical analysis. In addition to that, we consider a general case for ξ u,d . Wedo a numerical analysis varying ˜ Mµ and A u µ , then search for the valid region, which satisfies theboundary conditions. In Fig. (3) we collected the valid points which satisfy at least the low energyrequirements and (cid:12)(cid:12)(cid:12) Bµµ (cid:12)(cid:12)(cid:12) < .
01. The bright part is the allowed region in the previous analysis. Herewe see that the variation on ˜ M and ξ u is restricted by the boundary conditions. If Λ CF T is as low as10 GeV, the result is slightly deformed. But the conclusion that we can find the parameter region,where the low energy requirements are satisfied with appropriate ˜ Mµ and A u µ , is not changed. - - - M ŽΜ A u Μ Figure 3: The region which satisfies the boundary conditions at the electro weak scale and Λ
CF T .It is projected to the ˜ Mµ and A u µ plane. – 6 – . Phenomenology In the parameter space which passes the low energy requirement, we pick a typical point, andanalyze it. As we discussed in the previous section, here we have three dimensionful parameters:Λ, Λ
CF T and ˜ M . In addition to them, we have 2 dimensionless parameter: tan β and A u µ . Theseare under the control of the low energy constraints. Here we investigate the caseΛ = 1 . × GeV, Λ CF T = 10 GeV, ˜ M = 5 µ, tan β = 4 . , A u = − µ. (4.1) χ χ +1 ˜ t ˜ b ˜ u L ˜ c L ˜ d L ˜ s L ˜ g h H d R ˜ s R ˜ b ˜ ν τ ˜ ν e ˜ ν µ ˜ τ ˜ µ L ˜ e L χ A χ +2 χ ˜ t ˜ u R ˜ c R ˜ τ ˜ e R ˜ µ R χ H ± µ µ as large as few TeV therefore the absolute value of trilinear couplinggrows as µ at Λ CF T . Here we assume that the trilinear coupling generated by the messenger higgscoupling, so that the values of the trilinear couplings A u,d are obtained with an ambiguity A u A d . Ifwe look the RG flow of the trilinear coupling, A u and A d grow monotonically as we go down to thelow energy scale. Moreover small tan β means that top yukawa coupling is large relatively to otherstherefore, large off diagonal term in the stop mass matrix makes stop lighter than any other scalarpartners. Now we consider some physical constraints, especially the decay rate of the rare process B → X s γ and the anomalous magnetic moment of muon. Here is our result at the point given in(4.1), using microOmegas [15]. ( g − µ = 5 . × − ,Br ( B → X s γ ) = 3 . × − . (4.2)The anomalous magnetic moment of muon is reported to be △ a µ = (30 . ± . × − in [16], andthe rare process B → X s γ is reported as Br ( B → X s γ ) = (355 ± +9 − ± × − in [17]. The LSPis definitely gravitino, though the gravitino receive the correction form the anomalous dimension of X . It is because the gravity contribution is not negligible if very large correction is applied to thegravitino mass as shown in (2.4). As we accept large Λ, the masses are increased. If we take theanomalous dimension effect on gravitino into consideration, the gravitino remains to be the LSP.
5. Conclusion
The idea using the hidden sector dynamics to solve
Bµ/µ problem in GMSB is quite simple andnatural. However, it needs some modification to get a physical solution. There might be a number ofways to circumvent this situation. We propose a simple method via five dimensional setup, which– 7 –ncludes a bulk CS interaction between two U (1) gauge fields. Within a reasonable parameterregion, we get a physical solution. The typical region appears to have small tan β , which is near 4. Acknowledgments
We thank to H.D. Kim and D.Y. Kim for useful discussion. This work was supported by the KoreaResearch Foundation grants (KRF-2008-313-C00162).
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