Imprint of SUSY in radiative $B$-meson decays

We study supersymmetric (SUSY) effects on $C_7({\mu }_b)$ and $C_7^{\prime }({\mu }_b)$ which are the Wilson coefficients (WCs) for $b\to s\gamma$ at b-quark mass scale ${\mu }_b$ and are closely related to radiative $B$-meson decays. The SUSY-loop contributions to $C_7({\mu }_b)$ and $C_7^{\prime }({\mu }_b)$ are calculated at leading order (LO) in the Minimal Supersymmetric Standard Model (MSSM) with general quark-flavor violation (QFV). For the first time we perform a systematic MSSM parameter scan for the WCs $C_7({\mu }_b)$ and $C_7^{\prime }({\mu }_b)$ respecting all the relevant constraints, i.e., the theoretical constraints from vacuum stability conditions and the experimental constraints, such as those from $K$- and $B$-meson data and electroweak precision data, as well as recent limits on SUSY particle masses and the 125 GeV Higgs boson data from LHC experiments. From the parameter scan we find the following: (1) The MSSM contribution to $\mathrm{Re}(C_7({\mu }_b))$ can be as large as $\sim \pm 0.05$, which could correspond to about $3\sigma$ significance of New Physics (NP) signal in the future LHCb and Belle II experiments. (2) The MSSM contribution to $\mathrm{Re}(C_7^{\prime }({\mu }_b))$ can be as large as $\sim -0.08$, which could correspond to about $4\sigma$ significance of NP signal in the future LHCb and Belle II experiments. (3) These large MSSM contributions to the WCs are mainly due to (i) large scharm-stop mixing and large scharm/stop involved trilinear couplings $T_{U23}$, $T_{U32}$ and $T_{U33}$, (ii) large sstrange-sbottom mixing and large sstrange-sbottom involved trilinear couplings $T_{D23}$, $T_{D32}$ and $T_{D33}$ and (iii) large bottom Yukawa coupling $Y_b$ for large $\tan \beta$ and large top Yukawa coupling $Y_t$. In case such large NP contributions to the WCs are really observed in the future experiments at Belle II and the LHCb Upgrade, this could be the imprint of QFV SUSY (the MSSM with general QFV) and would encourage to perform further studies of the WCs $C_7^{\prime }({\mu }_b)$ and $C_7^{\mathrm{MSSM}}({\mu }_b)$ at higher order (NLO/NNLO) level in this model.

Figure 1

The SM and MSSM one-loop contributions to the WCs $C_{7,8}({\mu }_W)$ and $C_{7,8}^{\prime }({\mu }_W)$ at the weak scale ${\mu }_W$ for the transitions $b_R\to s_L{\gamma }_L,g_L$ and $b_L\to s_R{\gamma }_R,g_R$, respectively [see Eqs. (5)–(7)]. Here ${\gamma }_L$, $g_L$ and ${\gamma }_R$, $g_R$ are left-handed photon, gluon and right-handed photon, gluon, respectively. The photon is emitted from any electrically charged line and the gluon from any color charged line. For the SM one-loop contributions $(\mathrm{X},\mathrm{Y})=(\mathrm{t}/\mathrm{c}/\mathrm{u},W^{+})$. For the MSSM one-loop contributions $(\mathrm{X},\mathrm{Y})=(\mathrm{stop}/\mathrm{scharm}/\sup ,\mathrm{chargino})$, (sbottom/sstrange/sdown, gluino), (sbottom/sstrange/sdown, neutralino) and (t/c/u, $H^{+}$), where stop/scharm/sup denotes top-, charm-, up-squark mixtures and so on.

Figure 2

Schematic illustration of important parts of the up-type squark-chargino loop contributions to $C_{7,8}({\mu }_W)$ in terms of the mass-insertion approximation.

Figure 3

Schematic illustration of an important part of the down-type squark-gluino loop contributions to (a) $C_{7,8}({\mu }_W)$ and (b) $C_{7,8}^{\prime }({\mu }_W)$ in terms of the mass-insertion approximation.

Figure 4

Schematic illustration of the top quark—$H^{+}$ loop contribution to $C_{7,8}({\mu }_W)$.

Figure 5

The scatter plot of the scanned parameter points within the ranges given in Table 1 in the planes of (a) $\mathrm{Re}(C_7^{\prime }({\mu }_b))\text{−}\mathrm{Im}(C_7^{\prime }({\mu }_b))$, (b) $\mathrm{Re}(C_7^{\mathrm{MSSM}}({\mu }_b))\text{−}\mathrm{Im}(C_7^{\mathrm{MSSM}}({\mu }_b))$, and (c) $\mathrm{Re}(C_7^{\mathrm{MSSM}}({\mu }_b))\text{−}\mathrm{Re}(C_7^{\prime }({\mu }_b))$.

Figure 6

The scatter plots of the scanned parameter points within the ranges given in Table 1 in the planes of (a) $T_{U23}\text{−}\mathrm{Re}(C_7^{\prime }({\mu }_b))$, (b) $T_{U32}\text{−}\mathrm{Re}(C_7^{\prime }({\mu }_b))$, (c) $T_{U33}\text{−}\mathrm{Re}(C_7^{\prime }({\mu }_b))$ and (d) $\tan \beta \text{−}\mathrm{Re}(C_7^{\prime }({\mu }_b))$.

Figure 7

The scatter plots of the scanned parameter points within the ranges given in Table 1 in the planes of (a) $T_{D23}\text{−}\mathrm{Re}(C_7^{\prime }({\mu }_b))$, (b) $T_{D32}\text{−}\mathrm{Re}(C_7^{\prime }({\mu }_b))$ and (c) $T_{D33}\text{−}\mathrm{Re}(C_7^{\prime }({\mu }_b))$.

Figure 8

The scatter plot of the scanned parameter points within the ranges given in Table 1 in the planes of (a) $T_{U23}\text{−}\mathrm{Re}(C_7^{\mathrm{MSSM}}({\mu }_b))$, (b) $T_{U32}\text{−}\mathrm{Re}(C_7^{\mathrm{MSSM}}({\mu }_b))$, (c) $T_{U33}\text{−}\mathrm{Re}(C_7^{\mathrm{MSSM}}({\mu }_b))$ and (d) $\tan \beta \text{−}\mathrm{Re}(C_7^{\mathrm{MSSM}}({\mu }_b))$.

Figure 9

The scatter plot of the scanned parameter points within the ranges given in Table 1 in the $T_{D23}\text{−}\mathrm{Re}(C_7^{\mathrm{MSSM}}({\mu }_b))$ plane.

Figure 10

Contour plots of $\mathrm{Re}(C_7^{\prime }({\mu }_b))$ around the benchmark point P1 in the parameter planes of (a) $T_{U23}\text{−}{T}_{U32}$, (b) $T_{U23}\text{−}{T}_{U33}$, (c) $T_{U23}\text{−}\tan \beta$, and (d) $T_{U23}\text{−}{M}_{U23}^2$. The parameters other than the shown ones in each plane are fixed as in Table 2. The “X” marks P1 in the plots. The red hatched region satisfies all the constraints in Appendix pp1. The red solid lines, the nearly vertical blue dashed lines in (c),(d), the red dashed lines in (a),(b),(c) and the blue dash-dotted lines show the $m_{h^0}$ bound, the $\mathrm{B}(b\to s\gamma )$ bound, the $\mathrm{B}(B_s\to {\mu }^{+}{\mu }^{-})$ bound, and the $m_{{\tilde{d}}_1}$ bound, respectively.

Figure 11

Contour plots of $\mathrm{Re}(C_7^{\prime }({\mu }_b))$ around the benchmark point P1 in the parameter planes of (a) $T_{D23}\text{−}{T}_{D32}$, (b) $T_{D23}\text{−}{T}_{D33}$, (c) $T_{D23}\text{−}\tan \beta$, and (d) $T_{D23}\text{−}{M}_{D23}^2$. The parameters other than the shown ones in each plane are fixed as in Table 2. The “X” marks P1 in the plots. The red hatched region satisfies all the constraints in Appendix pp1. The definitions of the bound lines are the same as in Fig. 10. In addition to these the blue solid lines in (a) and the green solid lines in (c) show the $\mathrm{\Delta }{M}_{B_s}$ bound and the vacuum stability bound on $T_{D23}$, respectively.

Figure 12

Contour plots of $\mathrm{Re}(C_7^{\mathrm{MSSM}}({\mu }_b))$ around the benchmark point P1 in the parameter planes of (a) $T_{U23}\text{−}{T}_{U32}$, (b) $T_{U23}\text{−}{T}_{U33}$, (c) $T_{U23}\text{−}\tan \beta$, and (d) $T_{U23}\text{−}{M}_{U23}^2$. The parameters other than the shown ones in each plane are fixed as in Table 2. The “X” marks P1 in the plots. The red hatched region satisfies all the constraints in Appendix pp1. The definitions of the bound lines are the same as those in Fig. 10.

Figure 13

Contour plots of $\mathrm{Re}(C_7^{\mathrm{MSSM}}({\mu }_b))$ around the benchmark point P1 in the parameter planes of (a) $T_{D23}\text{−}{T}_{D32}$, (b) $T_{D23}\text{−}{T}_{D33}$, (c) $T_{D23}\text{−}\tan \beta$, and (d) $T_{D23}\text{−}{M}_{D23}^2$. The parameters other than the shown ones in each plane are fixed as in Table 2. The “X” marks P1 in the plots. The red hatched region satisfies all the constraints in Appendix pp1. The definitions of the bound lines are the same as those in Fig. 11.