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

Abstract

We develop a strategy for computing the $B\to K\ell^+\ell^-$ and $\bar{B}_s\toγ\ell^+\ell^-$ 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.

Figures (19)

• Figure 1: Connected quark-flow diagrams for the process $B\to K\ell^+\ell^-$ which contain a charm-quark loop. The shaded circle, marked $O_{1,2}^{(c)}$, represents either of the two current-current operators $O_1^{(c)}$ or $O_2^{(c)}$ defined in Eq. (\ref{['eq:O12def']}). The charged leptons $\ell^\pm$, which couple to the virtual photon, $\gamma^\ast$, are not shown. Panels (a)-(d) correspond, respectively, to photon emission from the charm, strange, down, and bottom quarks.
• Figure 2: Schematic illustrations of the two time-orderings in Eq. (\ref{['eq:H12def0']}) contributing to $B\to K\ell^+\ell^-$ decays. The dashed lines between the operators represent all propagating states with the indicated $B$ and $S$ quantum numbers. The emission of the virtual photon is not shown. Panel (a) corresponds to the time ordering $t<0$, while panel (b) corresponds to $t>0$.
• Figure 3: Schematic illustrations of the three time-orderings in Eq. (\ref{['eq:HmunuBtogammagamma']}) which contribute an imaginary part to the $B\to \gamma\ell^+\ell^-$ decay amplitude. The dashed lines between the operators represent all propagating states with the indicated $B$ quantum number. The emission of the real and virtual photons is not shown. Panel (a) corresponds to the time ordering $t_W<0<t$, panel (b) to $t_W<t<0$, while panel (c) to $t<t_W<0$.
• Figure 4: Schematic diagram illustrating the time ordering described in the text. The dashed lines represent the particles coupling to the operators $O_{P_i},\,(i=1,\cdots,\!n)$ and carrying the momentum $k_{P_i}$.
• Figure 5: Schematic diagram of the decay $\bar{B}_s\to\gamma\mu^+\mu^-$, labelled with the notation used in the text.