On the Impact of Soft Photons on $B\to K{\ell }^{+}{\ell }^{-}$

Abstract

We give detailed results of $\mathcal{O}(\alpha )$ QED corrections (both real emission and virtual corrections) to $B\to K{\ell }^{+}{\ell }^{-}$ modes. Requiring the real emission to be gauge invariant, the structure of the contact term(s) is fixed. The calculation is done with a fictitious photon mass as the IR regulator and results are shown to be independent of it, establishing cancellation of the soft divergences. Results are presented for a choice of photon energy, $k_{max}$, and photon angle ${\theta }_{cut}$. The QED effects are negative, thereby reducing the rate compared to that without QED effects. Electron channels are shown to receive large corrections ($\mathcal{O}(20\%)$). Impact on lepton flavour universality ratio, $R_K^{\mu e}$ are also discussed.

Comments: 27 pages, 6 figues