On the $ R_{K} $ Theory Error

TL;DR

This paper assesses the theoretical uncertainties in the $R_K$ ratio, focusing on QED corrections and structure-dependent effects, concluding that $R_K$ remains a reliable observable for testing lepton flavor universality.

Contribution

The study provides a detailed analysis of QED correction uncertainties in $R_K$, demonstrating that structure-dependent effects do not introduce significant additional errors.

Findings

QED corrections do not produce large non-logarithmic terms.

Structure-dependent corrections are shown to be negligible.

$R_K$ remains a theoretically safe observable.

Abstract

To quantify the theory error on $R_K$, essentially means to quantify the uncertainty due to QED corrections since the latter breaks lepton flavour universality through the lepton masses. Since experiment uses photon shower programs, e.g. \texttt{PHOTOS}, to capture QED effects, assessing the uncertainty involves investigating effects not captured by the specific use of these tools. This includes structure-dependent corrections, potentially large non-logarithmic terms and charmonium resonances entering the lower bin by migration of radiation. We are able to close in on these loopholes. For example, using gauge invariance, we show that structure-dependent QED corrections do not lead to additional (sizeable) hard-collinear logs of the form ${\cal O}(\alpha) \ln m_\ell/m_B$. Hence, from the theory point of view $R_K$ is a safe observable.