Beyond leading-order corrections to B¯→Xsγ at large tan β: The charged-Higgs-boson contribution

Among the $O\left({\alpha }_s\tan \beta \right)$ contributions to the Wilson coefficients $C_7$ and $C_8,$ relevant for the decay $\overline{B}\to X_s\gamma ,$ those induced by two-loop diagrams with charged-Higgs-boson exchange and squark-gluino corrections are calculated in supersymmetric models at large $\tan \beta .$ The calculation of the corresponding Feynman integrals is exact, unlike in previous studies that are valid when the typical supersymmetric scale $M_{\mathrm{SUSY}}$ is sufficiently larger than the electroweak scale $m_{\mathrm{weak}}\left(\sim m_W{,m}_t\right)$ and the mass of the charged Higgs boson $m_{H^{\pm }}.$ Therefore, the results presented here can be used for any value of the various supersymmetric masses. These results are compared with those of an approximate calculation, already existing in the literature, that is at the zeroth order in the expansion parameter ${(m}_{\mathrm{weak}}^2{,m}_H^2{)/M}_{\mathrm{SUSY}}^2,$ and with the results of two new approximate calculations in which the first and second order in the same expansion parameter are retained, respectively. This comparison allows us to assess whether the results of these three approximate calculations can be extended beyond the range of validity for which they were derived, i.e., whether they can be used for $m_H\gtrsim M_{\mathrm{SUSY}}$ and/or $M_{\mathrm{SUSY}}\sim m_{\mathrm{weak}}.$ It is found that the zeroth-order approximation works well even for $m_H\gtrsim M_{\mathrm{SUSY}},$ provided $M_{\mathrm{SUSY}}^2\gg m_{\mathrm{weak}}^2.$ The inclusion of the higher-order terms improves the zeroth-order approximation for $m_H^2\ll M_{\mathrm{SUSY}}^2,$ but it worsens it for $m_H\gtrsim M_{\mathrm{SUSY}}.$