Radiative $B$ decays in supersymmetry without $R$ parity

We present a systematic analysis of all the contributions at the leading log order to the branching ratio of the inclusive radiative decay $B\to X_s+\gamma$ in the framework of supersymmetry without $R$ parity. The relevant set of four-quark operators involved in QCD running are extended from six (within the standard model and the minimal supersymmetric standard model) to 24, with also many new contributions to the Wilson coefficients of (chromo)magnetic penguins for either chiral structure. We present complete analytical results here without any a priori assumptions on the form of $R$-parity violation. Mass-eigenstate expressions are given; hence the results are free from the commonly adopted mass-insertion approximation. In the numerical analysis, we focus here only on the influence of the trilinear ${\lambda }_{ijk}^{\prime }$ couplings and report on the possibility of a few orders of magnitude improvement for the bounds on a few combinations of the ${\lambda }^{\prime }$ couplings. Our study shows that the Wilson coefficients of the current-current operators due to $R$-parity violation dominate over the direct contributions to the penguins. However, the interplay of various contributions is complicated due to the QCD corrections which we elaborate here.

Figure 1

Magnetic penguin diagram with Higgs in the loop. Here the chirality flip is on the external $b$ quark. Note that the photon can be attached to any electrically charged particle inside the loop but not outside the loop. The diagram with a photon outside the loop does not contribute to the magnetic transition. There are similar diagrams with other SUSY particles in the loop. For the chromomagnetic penguin one attaches a gluon instead of a photon to the quarks in the loop.

Figure 2

A magnetic penguin diagram but with the chirality flip inside the loop.

Figure 3

Branching ratio as a function of $\tan \beta$ in MSSM with $A_t$ positive and $\mu$ negative. The “$+$” sign is for the QCD corrected rate and the “$\times$” sign for the rate without QCD corrections. The two horizontal lines in this figure and all the following figures for the branching fraction is the experimental uncertainty at $1\sigma$. See text for the values of the various parameters.

Figure 4

Various contributions to Wilson coefficients versus $\tan \beta$. SM (line with symbol $+$) and charged-Higgs (denoted as CS-L, line with $*$ sign) are of the same sign (positive) whereas the chargino contribution (denoted as CH-L, line with $\times$ sign) has a strong $\tan \beta$ dependence and is of opposite sign (negative) because $A_t\mu <0$.

Figure 5

Same as Fig. 3, but with the sign of $\mu$ positive.

Figure 6

Branching fraction as a function of ${\lambda }_{132}^{\prime }(={\lambda }_{122}^{\prime })$.

Figure 7

Wilson coefficients as a function of ${\lambda }_{132}^{\prime }(={\lambda }_{122}^{\prime })$. CH, CS, and NS stand for the chargino, charged-Higgs, and sneutrino contributions, respectively. The letter “L” indicates that these contributions go into $C_7$. Similar notation would apply to the following figures. The letter “R” (L) would indicate that the contribution goes into coefficients with (without) tilde.

Figure 8

Branching fraction as a function of ${\lambda }_{333}^{\prime }(={\lambda }_{323}^{\prime })$.

Figure 9

Relevant Wilson coefficients as a function of ${\lambda }_{333}^{\prime }(={\lambda }_{323}^{\prime })$.

Figure 10

Branching ratio as a function of ${\lambda }_{233}^{\prime }(={\lambda }_{232}^{\prime })$.

Figure 11

Branching ratio as a function of ${\lambda }_{113}^{\prime }(={\lambda }_{112}^{\prime })$.

Figure 12

Relevant Wilson coefficients as a function of ${\lambda }_{233}^{\prime }(={\lambda }_{232}^{\prime })$.

Figure 13

Relevant Wilson coefficients as a function of ${\lambda }_{113}^{\prime }(={\lambda }_{112}^{\prime })$.