CP Violation in B_s Mixing in the SUSY SU(5) GUT with Right-handed Neutrinos

It is recently announced by the {\bf Ut}{\it fit} collaboration that the CP phase of the $B_s$ mixing amplitude, $\phi_{B_s}$, deviates more than $3 \sigma$ from the standard-model prediction. In this paper we discuss how large correction to $\phi_{B_s}$ is possible in the supersymmetric SU(5) ground unified model (SUSY SU(5) GUT) with right-handed neutrinos. Here, we assume the supergravity-like boundary condition for the SUSY-breaking terms. We found that the 95% probability region derived by the {\bf Ut}{\it fit} collaboration is marginal in this model.

In the supersymmetric standard model (SUSY SM), the SUSY-breaking terms are introduced, and they are sources of flavor and/or CP violation. We may get insights to physics affecting the SUSY-breaking terms beyond the SUSY SM by studies of flavor and/or CP violating phenomena in the hadronic and leptonic sectors.
In the supersymmetric ground unified models (SUSY GUTs), new flavor-violating interactions are introduced, and rich flavor-violating structure in sfermion mass matrices may be generated by the GUT interactions. When the SUSY-breaking terms in the SUSY SM are originated from dynamics above the GUT scale, the GUT interactions affect them radiatively [1]. In the SUSY GUTs, quarks and leptons are embedded in common GUT multiplets. Thus, one of tests of the SUSY GUTs is studies of correlations among flavorviolating phenomena in the leptonic and hadronic sectors.
Recently, it is announced by the Utfit collaboration that the phase of the B s mixing amplitude deviates more than 3σ from the SM prediction [2]. They parametrized the new-physics contribution to the B s mixing as and performed the model-independent analysis using the available observables of the B s system. They found that the 68% (95%) probability regions of φ Bs and C Bs are given as The phase φ Bs deviates from zero at 3.7σ. The further studies are necessary so that the deviation is confirmed, since their result comes from a combination of various observables.
They also used the SU(3) symmetry in order to evaluate unknown strong phases. However, their result encourages us to reanalyze the new-physics contribution to various flavorchanging neutral current (FCNC) processes.
In this paper we discuss how large deviation of the phase φ Bs is possible in the SUSY SU(5) GUT with right-handed neutrinos. Here, we assume the supergravity-like (or CMSSM-like) boundary condition for the SUSY-breaking terms. The large mixing angle observed in the atmospheric neutrino experiments suggests that the sizable deviations from the SM may appear in the bottom-strange quark transitions in the SUSY GUTs.
The neutrino Yukawa interaction radiatively generates the bottom-strange component in the right-handed down-squark mass matrix [3]. On the other hand, the contribution to the bottom-strange quark transition is constrained from the upperbound on Br(τ → µγ) in the model [4]. The neutrino Yukawa interaction also induces the lepton-flavor violating terms in the left-handed slepton mass matrix so that lepton-flavor violating processes are predicted [5], and Br(τ → µγ) is also enhanced by the large atmospheric neutrino-mixing angle [6]. We evaluate φ Bs in the model, and it is found that it can deviate at most ∼ (8 • − 9 • ) from zero under the experimental constraint. Thus, it is difficult to get such a large deviation in φ Bs from the SM as in the 68% probability regions derived by the Utfit collaboration in Eq. (2), while one of the 95% probability region is marginal.
The SUSY contribution to the phase in the B s mixing is also evaluated by Refs. [7,8,9,10] in a framework of the SUSY GUTs. In Ref. [7] the constraints on the mass insertion parameters for the right-handed squarks are derived from the lepton-flavor violating processes using the GUT relation among the squark and slepton mass matrices at the GUT scale. The authors in Refs [8,9] evaluate the phase in the B s mixing in the SUSY GUTs, though they do not include the Higgs boson mass bound in their analysis. The analysis in Ref. [10] is similar to ours, though they do not evaluate the maximum value of the phase in the B s mixing in the model.
First, we briefly explain the SUSY SU(5) GUT with right-handed neutrinos. In the model, quarks and leptons in the SUSY SM are 10-and 5 ⋆ -dimensional multiplets, while the right-handed neutrinos are singlets. The Yukawa couplings for quarks and leptons and the Majorana mass terms for the right-handed neutrinos in this model are given as where Ψ and Φ are the 10-and 5 ⋆ -dimensional multiplets, respectively, and N is the where The unitary matrix V is the CKM matrix in the extension of the SM to the SUSY SU(5) GUT. When the Majorana mass matrix for the right-handed neutrinos M is diagonal in the above basis, U is the the MNS matrix (with the Majorana phases), since the left-handed neutrino mass matrix is Here, H f is a doublet Higgs in H. In the following, we assume for simplicity that M is The colored-Higgs multiplets H c and H c are introduced in H and H as SU (5)  are given universally by m 0 and A 0 , respectively. In this case, the off-diagonal terms (i = j) in the sfermion mass matrices at low energy are approximated as where (m As mentioned above, the bottom-strange quark and τ -µ transitions are correlated in the SUSY SU(5) GUT with right-handed neutrinos. This is because a following simple relation is valid from Eq. (6), under an assumption, f 2 ντ ≫ f 2 νµ ≫ f 2 νe . The corrections to this relation are possible, when the the right-handed neutrino mass matrix is not diagonal. However, it is shown in Ref. [4] that this relation is approximately valid when the right-handed neutrino masses are hierarchical. The hierarchical spectrum is welcome from a phenomenological viewpoint, since the muon-neutrino Yukawa coupling is so small that Br(µ → eγ) is suppressed below the experimental bound. Here, we also assume U e3 < ∼ 0.01 in order to suppress Br(µ → eγ). In such a case the large corrections to (m 2 D ) 23 and (m 2 L ) 23 by the colored-Higgs interactions may be allowed.
We evaluate the SUSY mass spectrum and the interactions by numerically solving the renormalization-group equations for the SUSY breaking parameters in the SUSY GUT and the SUSY SM (with right-handed neutrinos).
In the following, we give constraints on the above parameter space from the experimental bounds on the Higgs boson mass m h in addition to Br(τ → µγ). The SUSY contribution to the B s mixing is dominated by the box diagrams including gluino exchanges when both left-and right-handed down-type squarks have flavor-violating mass terms [11]. Thus, the SUSY correction is less sensitive to tan β. On the other hand, Br(τ → µγ) is proportional to tan 2 β since it is generated by the effective dipole operators. The constraint on (m 2 D ) 23 is more severe for larger tan β due to the null results in BaBar and Belle, respectively, [13]. The combined upperbound of them is 1.6 × 10 −8 [14]. The SM Higgs boson mass lowerbound is 114.4 GeV [15]. In the following we impose m h > 111.4 GeV [16], taking theoretical uncertainties in the Higgs boson mass evaluation.
The constraint from Br(b → sγ) is also included in our analysis, though we found that it is not significant compared with Br(τ → µγ).  Fig. 3.
Here, we take tan β = 5 and 10. While ∆M s is precisely measured by CDF [17], the hadronic uncertainties in the theoretical prediction are large. The constraint on C Bs is given in Eq. (2). It is found that this constraint is not significant in Fig. 3. Next, the CP asymmetry in B d → φK s , which is induced by the b-s penguin diagrams, is also shown in Fig. 3. The correction to the process can be as large as ∼ 30%. The experimental and theoretical uncertainties [18,19] in the b-s penguin processes are still large, so that we could not give a rigid constraint on the parameters from it.
When the flavor-violating mass terms for the right-handed squarks are non-vanishing, the hadronic EDMs are generated at one-loop [20] and two-loop levels [21]. The non-zero (m 2 D ) 23 generates the strange-quark chromoelectric dipole moment, which contributes to the hadronic EDMs [22,23]. When evaluating the neutron EDM using the formula in Ref. [23], it can be as large as ∼ 10 −25 e cm even in the region allowed by the Br(τ → µγ) and m h . (See Fig. 3). The current experimental bound is 2.9 × 10 −26 e cm [24]. Thus, the sizable deviation of φ Bs from the SM prediction is possible if the neutron EDM suffers from the hadronic uncertainties or it is suppressed due to an accidental cancellation.
In the above discussion we took M N 3 = 6×10 14 GeV for the right-handed tau neutrino mass. When M N 3 is larger, the neutrino Yukawa coupling is larger with the left-handed tau neutrino mass fixed. However, the scalar mass squareds for the 5 ⋆ -dimensional multiplets may become negative above the GUT scale. In the case, the flavor-violating mass terms of the 5 ⋆ -dimensional multiplets rather become smaller so that the deviation of φ s is not enhanced. In fact, we found it difficult to get larger deviation of φ Bs by raising M N 3 . In which are wider than in Eq. (8).
GUT with right-handed neutrinos. Here, we assumed the supergravity-like boundary condition for the SUSY-breaking terms. We found that the 95% probability region derived by the Utfit collaboration is marginal in this model.