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, $R^{\mu e}_K$ are also discussed.
Highlights
Flavor changing neutral currents, being both loop and Cabibbo-Kobayashi-Maskawa (CKM) suppressed within the Standard Model (SM), are an ideal hunting ground for physics beyond the SM
There are three kind of diagrams contributing to virtual corrections: (i) photon starting and ending at the same charged particle leg [Fig. 2(a)]; (ii) photon line between two different charged particles [Figs. 2(b) and 2(c)]; (iii) photon from the contact term ending on a charged particle leg [Fig. 2(d)]
While calculating the virtual corrections due to the contact term, there are UV divergences that should cancel against similar divergences of higher dimensions, in particular two photon contact terms
Summary
Flavor changing neutral currents, being both loop and Cabibbo-Kobayashi-Maskawa (CKM) suppressed within the Standard Model (SM), are an ideal hunting ground for physics beyond the SM. If the kinematical range is chosen such that the dilepton invariant mass is way larger than for either of the leptons chosen, it is expected that the ratio of the two branching fractions is unity to a high accuracy To this end, the following quantity is often considered as a clean test of the LFU and SM itself [25]: RμKe. Within SM, this ratio is unity while experimentally it has shown deviations from this expectation [23]: RμKejexp 1⁄4 0.846−þ00..005640−þ00..001146: ð2Þ. Following the standard approach of operator product expansion and integrating out heavy degrees of freedom, an effective Hamiltonian is built out of relevant degrees of freedom that are evolved down to the scale of a b quark with the help of renormalization group equations With this set of quark level operators, the physical hadronic matrix elements are computed and it is this step that involves the introduction of the form factors. The other factors entering the amplitude above, depending on the combinations of the Wilson coefficients (Ce7ff, Ce9ff, and C10) and form factors (fþ, f−, and fT), are given as
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