The like-sign dimuon charge asymmetry in SUSY models

We study the new physics (NP) implications of the recently reported 3.2 sigma Standard Model (SM) deviation in the like-sign dimuon asymmetry at the Tevatron. Assuming that new physics only enters the B(s) mixing amplitude we explore the implications for generic new physics, general supersymmetric (SUSY) models and also SUSY SU(5). In the case of SUSY SU(5) we exploit the GUT scale relationship between slepton and squark soft masses to predict rates for lepton flavour violation (LFV). The predicted rates for tau -->mu + gamma are found to be detectable at future Super-B factories.


I. INTRODUCTION
The Standard Model (SM), with the phase of the Cabibbo-Kobayashi-Maskawa (CKM) matrix as the sole source of CP violation, predicts a very small violation of CP in B meson mixing. While observations of B d mixing conform to this paradigm, in the B s system there have begun to emerge hints of physics beyond the SM. The first measurements of B s −B s mixing and its associated CP phase have been observed by the Tevatron's D∅ [1][2][3] and CDF [4,5] collaborations. Although the mass difference ∆M s shows little deviation from what is expected in the SM, the CP phase φ s observed in B s → ψφ decays is found to deviate from the SM by almost 3σ.
More recently, the D∅ collaboration reported a 3.2σ SM deviation in the like-sign dimuon charge asymmetry [6], = −(9.57 ± 2.51 ± 1.46) × 10 −3 (1) where N ±± b is the number of semileptonic events bb → µ ± µ ± X, and the SM prediction is given as a b,SM sl = (−2.3 +0. 5 −0.6 ) × 10 −4 [7]. If we assume that there is negligible CP violation in the tree-level decay amplitude then this measurement can be interpreted as further evidence for large CP violation in B s mixing.
An earlier result by the CDF collaboration found [8], a b sl = (8.0±9.0±6.8)×10 −3 , which has larger uncertainty yet is still compatible with the D∅ result. As the Tevatron produces both B d and B s mesons this measurement is a linear combination of the individual asymmetries, Using the average of the CDF and D∅ results for a b sl and assuming that new physics doesn't enter B d mixing we find, The direct measurement by D∅ [9], a s sl = −(1.7 ± 9.1) × 10 −3 , has larger uncertainty yet still agrees with the value derived above. Taking the average of these we arrive at the combined result, which we shall use in the analysis which follows.
The dimuon asymmetry has already been investigated in the context of generic new physics [10] and specific new physics models, for example models with flavour changing neutral Higgs [11], flavour changing Z ′ [12], or Axigluons [13] (see also [14]). In this work we examine the implications of the recent measurement of a dimuon asymmetry in SUSY models. We first investigate the implications of the recent measurement of a b sl in generic new physics before studying the dominant SUSY contributions in the mass insertion approximation (MIA). The preferred mass insertion parameter region is determined for two sample SUSY input points. In the context of SUSY SU(5) we then exploit the GUT relationship of squark and slepton mass insertions to make predictions for the rates of τ → µγ in light of this latest measurement.

II. CONSTRAINTS ON NEW PHYSICS FROM
Bs MIXING AND a s sl The ∆B = 2 transition between B s andB s mesons is defined as, where M Bs is the mass of the B s meson. We can then define the B s mass eigenstate difference as, and its associated CP phase, In the Standard Model M s 12 is given by, where G F is Fermi's constant, M W the mass of the W boson,η B = 0.551 is a short-distance QCD correction. The bag parameterB Bs and decay constant f Bs are nonperturbative quantities and contain the majority of the theoretical uncertainty. V ts and V tb are elements of the CKM matrix, and S 0 ( The Standard Model predictions for the other parameters relevant to B s mixing are [7] ∆Γ s,SM = (0.096 ± 0.039)ps −1 Generic NP contributions to B s mixing may be parameterized as, denotes the relative size of the NP contribution.
From the definition of eq. (11) we have the constraint, for r s and σ s . The UTfit analysis [15] gives ρ s at the 95% C.L. to be, From eq. (12) the phase associated with NP can also be written in terms of r s and σ s , Here [15] gives the 95% C.L. constraint, with a two-fold ambiguity.
In order to consistently apply the above constraints all input parameters are chosen to match those used in the analysis of the UTfit group [15,17] with the nonperturbative parameters, Finally, the asymmetry a s sl is defined as, where w s = 2|M s 12 |/Γ s and z s = |Γ s 12 |/Γ s . We shall assume that NP enters into the mixing amplitude only and take Γ s 12 ≡ Γ s,SM

12
. In our numerical analysis we use the combined value of a s sl in eq. (4) to constrain the NP parameter space. Fig. 1 shows the r s -σ s NP parameter space allowed by the constraints of ∆M s and φ N P s . The shaded blue/black region depicts the parameter space determined by the 95% C.L. bound from the ∆M s measurement, while the two yellow/light grey regions represent the combined 95% C.L. bounds of both ∆M s and φ s . Also displayed are red contour lines for the dimuon asymmetry a s sl . The solid red curve corresponds to the central value given in eq. (4) while the 1σ and 2σ regions are shown by the inner most and outer most dashed red curves respectively. At the 2σ level we can see that although the two-fold ambiguity in the CP phase φ s isn't resolved by the measurement of a s sl , the parameter space is further restricted. If the CP violation observed in the dimuon asymmetry indeed corresponds to NP in B s mixing the tension between a s sl and φ s implies that the central value of a s sl should increase somewhat in the future. Recently D∅ and CDF released new results based on 6.1f b −1 [18] and 5.2f b −1 [19] of integrated luminosity. These results are still preliminary and have not yet been combined together. As a result we shall not use these results in the present analysis.
In the follow section we study the implications of the dimuon asymmetry for the allowed mass insertion parameter space of SUSY models and also for the predictions of large τ → µγ rates in SUSY SU(5).

III. THE DIMUON CHARGE ASYMMETRY IN SUPERSYMMETRIC MODELS
The dominant SUSY contribution to B s mixing comes from the gluino contribution which may be written as [20], where x denotes the ratio of the squared gluino and downsquark masses, x = m 2 g /m 2 d . The functions a s 1,4 can be found in [20,21]. Here we have ignored terms proportional to δ d RL,LR mass insertions as they are expected to be heavily suppressed due to constraints from b → sγ.
In fig. 2 we show the constraints on the mass insertion parameter space of (δ d RR ) 23 from the 95% C.L. bounds of the mass difference ∆M s (blue/black), the CP phase φ s (yellow/light grey) and the dimuon asymmetry a s sl (red curves). Again, the solid red curve corresponds to the central value of a s sl from eq. (4), while the 1σ and 2σ bounds are shown by the inner most and outer most dashed red curves respectively. At the 2σ level the new value of a s sl agrees well with both of the regions allowed by the combined ∆M s and φ s bounds shown by the yellow/light grey shaded areas. This is similar to what we have found already in fig. 1. At the 1σ level there is a slight tension between a s sl and the CP phase φ s which indicates that we should either expect the central value of a s sl to increase somewhat in the future, or that new physics also enters into the width difference, as discussed in [23].
In SU(5) the quarks and leptons are placed into 10 = (Q, u c , e c ) and5 = (L, d c ) representations. Due to the symmetry of this simple SUSY GUT there exists relations amongst the slepton and squark soft SUSY breaking masses, These relations hold for scales at and above the GUT scale. Interestingly this implies that left-handed slepton mixing and right-handed down squark mixing are related. This relation can still be felt very strongly at the Electroweak scale in the correlation of LFV rates and FCNCs. Due to these GUT scale relations the (δ d RR ) 23 contributions to FCNCs and (δ l LL ) 23 contributions to LFV are clearly correlated. As a result we may explicitly write the rate of τ → µγ in the form, where md is the average down squark mass and M S is the typical SUSY scale. We shall also consider the RGE effects of the CKM mixings in the left-handed down squark matrices. At the FIG. 2: Mass insertion parameter space of (δ d RR )23 constrained by the 2σ measurements of ∆Ms (blue/black) and the CP phase φs (yellow/light grey), see eq. (16,14), plotted with mg = md = 300 GeV (upper) and mg = md = 500 GeV (lower). The red curves show the central value (solid curve), 1σ (inner dashed curve) and 2σ (outer dashed curve) constraints from a s sl as given in eq. (4).
SUSY scale M SUSY these effects can be approximated as, Here m 0 is the universal scalar mass, A 0 the universal A-term and M GUT is the scale of SU(5) unification.
In fig. 3 we explore the correlation between B s mixing and the branching ratio for τ → µγ in SUSY SU(5) where we have taken tan β = 10 [26]. The predicted rates for τ → µγ can be scaled by (tan β/10) 2 for different values of tan β. In [21] it was found that the large CP phase φ s restricts the mass insertion parameter space such that large rates for τ → µγ are expected in the case of SUSY SU(5) as shown by the yellow/light grey points in fig. 3. From the yellow/light grey points we find an approximate  (5). The plotted points agree with the 2σ bounds of ∆Ms (blue/black), φs (yellow/light grey) and a s sl (red/dark grey). The bold horizontal line is the experimental bound of Br(τ → µγ) < 4.5 × 10 −8 [22], while the narrow line shows the proposed Super-B factory bound Br(τ → µγ) < 2 × 10 −9 [24,25].
lower bound of, The new measurement of a b sl further constrains the range of predictions for Br(τ → µγ). The red/dark grey dots plotted in fig. 3 are those points which agree at the 2σ confidence level with each of ∆M s , φ s and a s sl . Including the new measurement we find that the allowed points cluster towards the middle of the yellow/light grey band, moving away from the highest and lowest rates for τ → µγ. Interestingly the majority of the preferred parameter space lies in the region between the present experimental bound for Br(τ → µγ) and the reach of the proposed Super flavour factories [24,25].

IV. SUMMARY
In this work we have investigated the impact on the new physics parameter space of the recent measurement of the like-sign dimuon charge asymmetry at the Tevatron. Assuming that new physics enters only into B s −B s mixing and not into the lifetime difference, we first saw that this measurement is in reasonably good agreement with existing observations of large CP violation in the B s system. The viable NP parameter region is further reduced by the latest Tevatron results although the twofold ambiguity in the CP phase φ s remains unresolved. If new physics present in B s mixing is indeed responsible for the dimuon asymmetry we should expect the central value to increase in the future.
In the case of generic supersymmetric models and supersymmetric SU(5) we have explored the mass insertion parameter space and the correlation between FCNCs and LFV rates in light of the recent dimuon asymmetry measurement. The (δ d RR ) 23 mass insertion parameter space is restricted to two small regions which under the SU(5) GUT relations predict a τ → µγ branching ratio just below the present experimental bound and within the reach of the proposed Super flavour factories.