Constraint on Right-Handed Squark Mixings from B_s - B_sbar Mass Difference

We point out that the right-handed squark mixings can sizably enhance SUSY contributions to Delta M_s by taking into account renormalization group effects via the CKM matrix. The recent result of Delta M_s from the D0 experiment at the Tevatron thus implies a strong constraint on the right-handed mixings.

Effects of the underlying physics at high energy scale are imprinted not only in flavor structures of the matters in the standard model (SM), but also in those of their superpartners by extending the SM to include supersymmetry (SUSY). They evolve from the cutoff scale to a low energy via the renormalization running, and are recognized through signals of the flavor-changing neutral currents (FCNCs).
Theb −s mixings of the right-handed squarks have attracted a lot of interests [1] in the light of the discovery of the neutrino oscillations with large mixing angles [2], and the success of the supersymmetric grand unification. The mixings are parameterized by the flavorchanging components betweenb R −s R andb L −s R in the mass matrices, which are called the RR and RL mixings, respectively. At the weak scale, they contribute to b → s transition processes. Some golden modes have been already measured precisely in experiments, and their results are compared to the SM predictions. Particularly, the measured branching ratio of the inclusive B d → X s γ decay is known to agree well with the SM value [3,4]. Thus it provides one of the severest constraints on the down-type squark mixings, including the RR and RL mixings. Even with the constraint from Br(b → sγ), there is still left large possibility to detect sizable effects on some b → s processes, especially, from the right-handed squark mixings.
Recently, the DØ collaboration have reported the updated result of the mass difference of the B s mesons [5]: which is the first result with a direct two-side bound. Although the data includes large uncertainties, this result is in agreement with the SM predictions, which are estimated as 21.3 ± 2.6 ps −1 by the UTfit group [6] and 20.9 +4.5 −4.2 ps −1 by the CKMfitter group [7]. In the supersymmetric SM, it is known that a combination of the LL and RR mixings, 23 , can enhance the SUSY contributions to ∆M s sizably [8]. If the LL mixing is suppressed sufficiently, the current data of ∆M s remains insensitive to the right-handed mixings [9].
Among the squark mixings, the LL mixing at the weak scale generally receives a correction of at least O(0.01). This is because the SUSY breaking effects are usually mediated to the visible sector at the high energy scale. Actually in general supersymmetric SM at the weak scale, we might choose the down-type LL squark mixing to vanish. However, in realistic models, the mixing will be induced because the CKM matrix affects the left-handed mass matrix of the down-type squarks during the renormalization group evolutions. In a class of supergravity mediations, the LL mixing generally gets a correction of (δ d LL ) 23 ∼ λ 2 , where λ ∼ 0.2 is the Wolfenstein parameter. In this letter, we want to emphasize that this tiny mixing is significant for ∆M s when we discuss the mixings in the right-handed sector.
We will show that even without any imprinted LL mixing, the RR squark mixing can affect ∆M s to the level of the SM value satisfying with the bound from Br(b → sγ).
Let us first review the SUSY contributions to ∆M s . The B s −B s transition is represented by the transition matrix element, In terms of R, which corresponds to the SUSY contributions, the mass difference between B s andB s becomes ∆M s = ∆M SM s |1 + R|. Although the estimation of the SM value contains large hadronic uncertainties, the ratio ∆M s /∆M d can be predicted more cleanly. Then R is given by with including a 40 % uncertainty from the SM estimations, which is mainly due to the ratio of the CKM matrix elements V td /V ts [6,7].
The SM contribution, M SM 12 , is obtained by exchanging the W boson and top quark. On the other hand, the SUSY contribution M SUSY 12 is by exchanging the gluino and downtype squarks. In the following analysis, we evaluate the SUSY contributions by the full expressions, namely, without the mass-insertion approximation [11,12] (see Ref. [13] for the explicit forms which are relevant for the right-handed mixings). In contrast, it is known that the chargino contributions are much suppressed compared to the gluino ones [8]. Thus we neglect them in the following. to mention that the ratio R is enhanced especially by a pair of the LL and RR mixings, The LL squark mixing receives the radiative corrections via the CKM matrix during the renormalization group evolution. The running from the cutoff scale M X to the weak scale M W gives a mixing such as where Y t is the top Yukawa coupling, m 0 ∼ mq are typical values of diagonal components of the scalar and squark mass matrices, and a 0 is a trilinear coupling, A t ≡ a 0 Y t . We stress that the mixing arises even when the mass matrix is diagonal at the cutoff scale. Actually, in supergravity mediations with M X at the Planck scale or the GUT scale M G ≃ 2 × 10 16 GeV, the renormalization group evolution induces (δ d LL ) 23 ≃ 0.04. With this LL mixing, the RR squark mixing can contribute effectively to the ratio R. It should be stressed that the LL mixing of O(0.01) is rather general independent of the details of the squark mass matrices at the cutoff scale, and it is hard to suppress the LL mixing at the weak scale unless the mixing is tuned at the cutoff scale. In the following analysis, we will use (δ d LL ) 23 ≃ 0.04 as a representative value 2 .
Let us consider the bound of the squark mixings from Br(b → sγ). The SUSY contributions to this mode are induced through the gluino, chargino and charged-Higgs loops 3 .
They significantly depend on tan β and a sign of the higgsino mass parameter, µ H . In fact, the SUSY contributions are enhanced by large tan β, and sign(µ H ) determines signs of the chargino and gluino contributions compared to the SM and charged-Higgs ones. As well as the evaluations of ∆M s , we evaluate the SUSY contributions including the gluino ones by the full expressions [16,17,18] 4 . In the following analysis, we take the bound, which is rather conservative after taking into account both the experimental and theoretical uncertainties.
We calculated the SUSY contributions to ∆M s as well as Br(b → sγ). In Fig. 1, we displayed the regions which is favored by the current result of ∆M s , and that is excluded by Br(b → sγ) for a range of the real and imaginary parts of the RR mixing, (δ d RR ) 23 ≡ (m 2 d RR ) 23 /m 2 q , where m 2 q is a typical mass of the squarks. Here we assume all relevant soft parameters including µ H are m soft = 500 GeV, and tan β = 10. In order to clarify the effects of the renormalization group evolutions, we show the result of (δ d LL ) 23 = 0 in Fig. 1(a), and that of (δ d LL ) 23 = 0.04 in Fig. 1(b). Consequently, we find that the radiative corrections in of the details of the models, and the following constraint on the right-handed squark mixing is rather generic. 3 Those diagrams also contribute to the decay amplitude of b → sll, which is sensitive to the sign of C 7γ , and has been already limited by the experiment [14]. The bound is always satisfied here because the LL mixing is small enough, (δ d LL ) 23 = 0.04 [15]. 4 The single mass-insertion approximation of the gluino-mediated diagrams is not consistent with the full estimations for the dipole operators, C 7γ and C 8G , when we study the LL and/or RR squark mixings.
Rather the dominant contributions are provided by so-called the double mass-insertion diagrams even for small tan β, like tan β = 5. See Ref. [19] for the explicit forms of the double-mass insertions.
the LL mixing can enhance the SUSY contributions for ∆M s extremely to the extent of the magnitude which is implied by the current data ( Fig. 1(b)), compared to the result without the effects (Fig. 1(a)).
We also considered other two sets of the mass spectrum of the gluino and squarks. The first pattern is mg ≪ mq. This type is interesting for the FCNCs whose amplitude is dominated by the Wilson coefficient of the gluon-dipole operator, O 8G . Since the relevant contribution to Br(b → sγ) comes from the photo-dipole operator, O 7γ , the mass pattern is favored to enhance the SUSY contributions for such FCNCs with satisfying the bound from Br(b → sγ), namely, enhance C 8G compared to C 7γ [13]. We estimated numerically ∆M s and Br(b → sγ) in this case: mg ≪ mq. In Fig. 2, the parameters are set as the same as Fig. 1(b), but the gluino and squark masses are mg = 300 GeV and mq = 1 TeV, respectively.
We find that although both the contributions are suppressed by the heavy squarks, the ∆M s region remains inside that excluded by Br(b → sγ).
The second mass spectrum is mq L ≫ mq R . Such a pattern can suppress the SUSY contributions to the electric dipole moments (EDMs). It has been pointed out that the strong bounds are imposed on the CP-violatingb −s mixings from the hadronic EDMs [20,21].
Especially it is very strong for the right-handed sector even when we allow large hadronic uncertainties. In the above discussions, we implicitly assumed accidental cancellations among the additional SUSY contributions for the EDMs. Another way out of the EDMs is to suppress them by large squark masses. When the left-handed squarks decouple, the EDM bound can be relaxed to the negligible level. The numerical result is in Fig. 3, where the parameters are the same as Fig. 1(b) but mq L ≫ m soft . We find the favored regions by ∆M s in the graph, though all the constraints are satisfied because the SUSY contributions to Br(b → sγ) as well as the EDMs can be neglected by large mq L .
Even when we take other sets of parameters, the results remain similar. Let us first flip the sign of µ H parameter. Then the cancellation becomes worse among the SUSY contributions to Br(b → sγ). In contrast, ∆M s is insensitive to the sign(µ H ). Then the favored region from ∆M s starts to be restricted by Br(b → sγ). Instead, when we take tan β = 40, the remaining contributions after the cancellation become enhanced for Br(b → sγ), while ∆M s is insensitive to tan β. We checked that the favored regions by the observables become narrower for the RR mixing in both cases.
There might be a large mixing in the RL component at the cutoff scale. As already known, the flavor mixings with chirality flipping cannot induce large ∆M s at the weak scale [8]. This is because the mixings are constrained by Br(b → sγ) to the extent of the negligible level for ∆M s . On the contrary, the RL mixing at the high energy scale can contribute sizably to the RR mixing at the weak scale through the renormalization group running [22]. Thus the similar arguments may be applicable for the RL mixing at the cutoff scale as those of the RR case, though the detailed discussion depends on the parameters. Consequently, we conclude that the SUSY contributions to ∆M s is sensitive to the right-handed squark mixings at the cutoff scale in the light of the current result of ∆M s by considering the radiative correction for the LL squark mixing.
Let us comment on implications to other b → s transition processes. The mixing-induced CP asymmetry of b → qqs decays has been measured by BaBar and Belle [3]. Although the results still contain large theoretical/experimental uncertainties, we obtain an important implication from the measurements that all the values tend to shift to the same side from the SM values, which is determined by b → ccs modes. This feature is observed independently of the parity of the final states. If the displacements are due to SUSY effects, the results naturally imply additional CP-violating mixings in the left-handed squark sector [19]. On the contrary, in order to realize the same feature by the right-handed squark mixings, rather large SUSY contributions are required [23]. We checked that such mixings induce too large ∆M s to stay within the current data as long as we consider the renormalization group effects. This means that although the SUSY SU(5) GUT + ν R model is one of the best candidates which naturally induce large SUSY contributions to b → qqs decays [1], the model is disfavored in order to explain the current experimental result of ∆M s as well as those modes. Other interesting b → s observables are mixing-induced CP asymmetry of the b → sγ decay and B s → J/ψφ. Since they are sensitive to the right-handed squark mixings, we can still expect to detect signals of new physics at LHC/super B-factory. We thus stress that measurements of ∆M s have impacts on the squark-flavor mixings. ∆M s . The experimental error was reduced very well, and the result is found to be consistent with the DØ value, thus the SM estimation. Even taking into account the CDF result, the analysis in this letter does not change because the uncertainties in the analysis dominantly come from the SM estimation. Although the uncertainty may be reduced by combining the other measurements of the flavor changing processes, such an analysis is beyond the scope of this letter and should be studied elsewhere.   Fig. 1(b), but mg = 300 GeV and mq = 1 TeV. Figure 3: Same as Fig. 1(b), but mq L decoupled.