Complete analysis on QED corrections to $B_{q}\, \to\, \tau^+\, \tau^-$

TL;DR

This paper calculates complete next-to-leading order QED corrections to $B_{d,s} o au^+ au^-$ decays using SCET, revealing small electromagnetic effects due to the large tau mass.

Contribution

It extends previous QED correction analyses from muonic to tauonic B decays, accounting for the large tau mass and different power counting in SCET.

Findings

QED corrections to $B_{d,s} o au^+ au^-$ are approximately 0.04%.

Hard-collinear effects are less significant in tau decays than in muon decays.

Electromagnetic effects slightly reduce the branching ratios of $B_{d,s} o au^+ au^-$.

Abstract

Motivated by a dynamical enhancement of the electromagnetic corrections by a power of $\Lambda_{\mathrm{QCD}}/m_b$ in $B_{d,s}\, \to\, \mu^+\, \mu^-$ at next-to-leading order (NLO), we extend the QED factorization effects on the leptonic $B$ meson decays with light muon leptons to tauonic final states, $B_{d,s} \, \to\, \tau^+\, \tau^-$, using soft-collinear effective theory (SCET). This extension is necessary owing to the appearance of the large $\tau$ mass, which will lead to different power counting in SCET and also different results. We provide a complete NLO electromagnetic corrections to $B_{d,s} \, \to\, \tau^+\, \tau^-$, which include hard functions and hard-collinear functions below the bottom quark mass scale $\mu_b$. The power enhanced electromagnetic effects from hard-collinear contributions on $B_{d,s}\, \to\, \mu^+\, \mu^-$ discussed before also exist in $B_{d,s} \, \to\, \tau^+\, \tau^-$. However the logarithm term arising from contributions of hard-collinear photon and lepton virtualities for $B_{d,s}\, \to\, \tau^+\, \tau^-$ is not large as it is in muon case due to the hard-collinear scale of $\tau$ mass, which lead to only approximately $0.04\%$ QED corrections to the branching fraction of $B_{d,s} \, \to\, \tau^+\, \tau^-$ compared with overall reduction about $0.5\%$ in $B_{d,s}\, \to\, \mu^+\, \mu^-$.