Light Higgsino from $A_t$ Dilemma in Rare $B$-decays

In the Minimal Supersymmetric Standard Model (MSSM), large chiral symmetry breaking term $A_t$, which plays an important role in Higgs mass, may significantly contribute in flavor changing neutral current (FCNC) processes $B\to X_{s}\gamma$ and $B_{s}\to \mu^{+}\mu^{-}$. Though the above processes can both be categorized as $b\to s$ transitions, the two rare decays behave completely different in MSSM. With an on-shell photon in the final state, helicity of initial state $b$-quark and final state $s$-quark must be flipped in $B\to X_{s}\gamma$, which corresponds to the simultaneous breaking of chiral symmetry and electroweak symmetry. The common feature is shared by fermion mass generation. Same as radiative mass generation in MSSM, Peccei-Quinn and $R$ symmetry breaking contributions, for example from a Higgsino-stop loop when $\mu A_t<0$, may significantly cancel the contribution from charged Higgs and reduce the prediction of $B\to X_{s}\gamma$. For the latter process, including Babu-Kolda FCNC proportional to $\mu A_t$, $B_{s}\to \mu^{+}\mu^{-}$ is mediated by a scalar $H_d$ boson which corresponds to chiral symmetry breaking. In addition, as a result of interference among the Higgs extension sector and $Z$ contributions, in the region of $\mu A_t <0$ which is favored by $B\to X_{s}\gamma$, there may simultaneously exist large enhancement in $B_{s}\to \mu^{+}\mu^{-}$. However, we still find viable parameter region with light Higgsino of a few hundreds GeV when charged Higgs contribution is not negligible with $M_A$ of 400 GeV.