Theoretical framework for lattice QCD computations of $B\to K \ell^+ \ell^-$ and $\bar{B}_s\to \ell^+\ell^- \gamma$ decays rates, including contributions from charming penguin diagrams

TL;DR

This paper proposes a lattice QCD strategy to compute complex decay amplitudes of B mesons, including charming penguin contributions, which are crucial for accurate phenomenological predictions but challenging to calculate.

Contribution

It introduces a spectral density-based approach to evaluate complex decay contributions, especially charming penguins, in lattice QCD, addressing a significant gap in current computational methods.

Findings

Developed a spectral density method for complex amplitude calculations

Applied the method to a charm-quark loop diagram in B decays

Discussed non-perturbative subtraction of divergences

Abstract

We develop a strategy for computing the $B\to K\ell^+\ell^-$ and $\bar{B}_s\to\gamma\ell^+\ell^-$ decay amplitudes using lattice QCD (where $\ell^\pm$ are charged leptons). We focus on those terms which contain complex contributions to the amplitude, due to on-shell intermediate states propagating between the weak operator and electromagnetic current(s). Such terms, which are generally estimated using model calculations and represent significant uncertainties in the phenomenological predictions for these decays, cannot be computed using standard lattice QCD techniques. It has recently been shown that such contributions can be computed using spectral-density methods and our proposed strategy, which we discuss in detail, is built on this approach. The complex contributions include the ``charming penguins" (matrix elements of the current-current operators $O_1^{(c)}$ and $O_2^{(c)}$ defined in Eq. (6) below), in which the charm-quark loop can propagate long distances, particularly close to the region of charmonium resonances. They also include the contributions from the chromomagnetic operator ($O_8$ in standard notation, defined in Eq. (8) below). We discuss the renormalization of the ultra-violet divergences, and in particular those which arise due to ``contact" terms, and explain how those which appear as inverse powers of the lattice spacing can be subtracted non-perturbatively. We apply the spectral density methods in an instructive exploratory computation of the charming penguin diagram in $B\to K\ell^+\ell^-$ decays in which the virtual photon is emitted from the charm-quark loop (the diagram in Fig. 1(a) below) and discuss the prospects and strategies for the reliable determination of the amplitudes in future dedicated computations; computations which are however, beyond the scope of the present paper.