Leptonic Decay of Upsilon, a Possible Signature of New Physics

We calculate the inclusive decay width of $\Upsilon \to l^+ l^-$. Then we get the ratio $R_{\tau\mu}=\Gamma[\Upsilon \to \tau^+\tau^-]/ \Gamma[\Upsilon \to \mu^+\mu^-]$ to ${\cal{O}}(\alpha)$ and ${\cal{O}}(\alpha_s^2)$ within the Standard Model(SM). Comparing with the recent Babar's data $R_{\tau \mu}=1.005\pm0.013\pm 0.022$, we find that SM prediction $R_{\tau\mu}$ is not consistent with the experimental data in the error bar. The discrepancy is about $1.25\sigma$. So leptonic decay of $\Upsilon$ may be a possible signature of New Physics(NP). We present a better approach to test the Standard Model, $R_{\tau\mu}(E_{soft})=.\Gamma[\Upsilon \to \tau^+\tau^-+X]/ \Gamma[\Upsilon \to \mu^+\mu^-+X]|_{E_X<E_{soft}}$. After resumming the large logarithms, we get $R_{\tau\mu}(E_{soft})$ at the precision level of 0.1%. It can be compared with experimental data more precise. We also consider the impact of $R_{\tau \mu}(E_{soft})$ and $R_{\tau \mu}$ from light Higgs $h$ and pseudoscalar Higgs $A_0$.