Chargino contribution to the rare decay b ->s s bar{d}

The rare decay b ->s s bar{d} is studied in the supersymmetric standard model by considering the contribution from the chargino box diagrams. We find that this contribution amounts to 10^{-9} in the branching ratio.

A very important aspect in the study of B physics is to expose possible virtual effects due to physics beyond the Standard Model(SM). These virtual effects are likely to be hidden in the processes induced by flavor changing neutral current (FCNC) interactions, since the SM contributions come from loop diagrams so that they do not always dominate over the new physics contributions. Furthermore, since the SM contributions serve as the background, the rare decays with unobservable SM contributions are more suitable to probe the new physics signals. As was pointed out in [1] and [2], the rare decay b → ssd is a clean channel with less SM background. The quark-level transition materializes as decays of B mesons into S = 2 final states, of which a fraction 1/4 are states with two charged kaons which can be identified at future experiments. Several exclusive channels of b → ssd transition have been searched by different groups [3].
More data will improve the bounds on these transitions and on the possible new physics.
The rare decay b → ssd in the SM proceeds with a box diagram which is strongly suppresses by the second order weak interactions and by the Glashow-Maiani-Illiopoulous (GIM) cancellations. In the SM this branching ratio is too small to be observable. This may help to expose the possible signal of new physics clearly from the background of the SM. Motivated by this observation, several new physics models has been studied for this process [1,2,4].
In this Letter we will consider the chargino contribution to b → ssd within the Minimal Supersymmetric Standard Model (MSSM). In the MSSM, the charged Higgs contribution is negligible [2]. The gluino-squark contribution has been studied in [1] which can be as large as 10 −8 in the branching ratio, depending on the parameter space. The chargino contribution, which is usually another important source of the FCNC interactions, is to be studied below.
In the MSSM, the chargino contribution to b → ssd is from the chargino and the up-type squark diagrams. These up-type squarks are not degenerate and do not align with the up-type quarks in the super-multiplets. We use the mass insertion approach to estimate the chargino contribution. Following the notations of [5], the couplings of the left-handed down-type quarks, the left-handed up-type squarks and the charginos are flavor diagonal. In this basis, the mass-squared matrix for the uptype squarks is not diagonal. The off-diagonal elements, which will be denoted as ∆ LL ij (i, j = 1, 2, 3), are taken as the two-point vertices. Consequently, the insertions of these two-point vertices on the up-type squarks propagators inside the loops induce the transitions between different external down-type quarks. Similarly, since the righthanded stop couples with the higgsino components of the charginos with a strength which is proportional to large Yukawa coupling of the top quark, there are also the diagrams with the insertions of the left-handed up-type squarks and the right-handed stop ∆ LR i3 (i = 1, 2, 3) inside the loops. Note that the possible large splitting in the masses of the left-handed and the right-handed stops induces another source of FCNC.
The relevant effects are parameterized as [5] R where K is the CKM matrix, andm is the averaged up-type squark mass. The repeating indices i, j in the same equation indicate sum over from 1 to 3. The quantities R's are constrained directly by several experimental observables, e.g. R LL sb , R LR st and R RL tb are constrained by the mass difference ∆M B S and by the branching ratio of b → sγ, and R LL sd , R LR st , R RL td are constrained by the mass difference ∆M K . The Feynman diagrams relevant to the process b → ssd are shown in Fig. 1. We calculate the decay width from the chargino contribution to be , V is the matrix to diagonalize the chargino mass matrix X chargino with UX chargino V −1 = diag(mχ± 1 , mχ± 2 ), and the Yukawa couplings are defined as The loop functions are [5] f We would like to mention that the chargino contribution of the box diagrams is pro- If both charginos are heavy, their contribution to Br(b → ssd) is small, so the main contribution comes from the lighter charginoχ ± 1 with a small mass (say < 200GeV).
Numerically we take into account the constraints of ∆M B S ≥ 14.4ps −1 [6], 2 × 10 −4 ≤ Br(B → X s γ) ≤ 4.5 × 10 −4 , and the lower bounds on the superparticles [7]. As for the constraint of ∆M K , we demand the chargino contribution to ∆M K does not exceed the experimental value of ∆M exp K = (3.489 ± 0.008) × 10 −15 GeV. The calculation of this process contains 14 parameters, 6 of them are ∆ LL ij with ij = 11, 12, 13, 22, 23, 33 which contribute to R LL sb,sd , 3 of them are ∆ LR ij with ij = 13, 23, 33 which contribute to R RL tb,td and R LR st . The other parameters are tan β, M 2 , µ in chargino mass matrix, a common up-type squark massm and the mass of the (light) right-handed stop mt R . We scan the parameter space in the region 4 ≤ tan β ≤ 50, 250GeV ≤m ≤ 500GeV, 90GeV ≤ mt R ≤ 200GeV, 100GeV ≤ M 2 ≤ 500GeV, −500GeV ≤ µ ≤ 500GeV. All the 9 ∆'s are normalized bym 2 varying in the range between −1 and 1. We find in the calculation that a common feature is that Br(b → ssd) does not depend sensitively on tan β, but rather strongly on the mass parameters m, mt R and mχ± 1 , the lighter chargno mass .
We find that the contribution to Br(b → ssd) from ∆ LL ij 's (with ij = 11, 12, 13, 22, 23, 33) and thus from R LL sb,sd is always below 10 −11 if we use the experimental constraints mentioned above. The dominant contribution to Br(b → ssd) comes from ∆ LR 13,23 , while ∆ LR 33 is less important. We give in Tab. 1 two representative points with non-vanished ∆ LR,LL 23 to show the main contribution of ∆ LR ij . We show the ∆ LR 13,23,33 contribution in Fig. 2. From Fig. 2, Br(b → ssd) can be as large as 10 −9 . In some region, with the increasing of Br(b → ssd), ∆M Bs can be as large as 60ps −1 . In conclusion, we have studied the chargino contribution to the rare decay b → ssd.
We find that the dominant contribution is from ∆ LR 13,23 and the branching ratio can amount to 10 −9 . This may be an important source of b → ssd in the MSSM.