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

Dipartimento di Fisica and INFN, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, I-00133 Roma, Italy

Dipartimento di Fisica, Università Roma Tre and INFN, Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy

INFN, Sezione di Roma, Piazzale Aldo Moro 2, I-00185 Roma, Italy and Dipartimento di Fisica, Università di Roma La Sapienza, Piazzale Aldo Moro 2, I-00185 Roma, Italy

Department of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom

Istituto Nazionale di Fisica Nucleare, Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy

INFN, Sezione di Roma, Piazzale Aldo Moro 2, I-00185 Roma, Italy

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Phys. Rev. D 113, 034509 – Published 24 February, 2026

DOI: https://doi.org/10.1103/qk4f-1dv5

Abstract

We develop a strategy for computing the $B \to K ℓ^{+} ℓ^{-}$ and ${\bar{B}}_{s} \to γ ℓ^{+} ℓ^{-}$ decay amplitudes using lattice QCD (where $ℓ^{\pm}$ 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 ultraviolet 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 nonperturbatively. We apply the spectral density methods in an instructive exploratory computation of the charming penguin diagram in $B \to K ℓ^{+} ℓ^{-}$ 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.

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Phys. Rev. D 113, 034509– Published 24 February, 2026

Vol. 113, Iss. 3 — 1 February 2026

Received 1 September 2025
Accepted 22 January 2026

DOI: https://doi.org/10.1103/qk4f-1dv5

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

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Theoretical framework for lattice QCD computations of B→Kℓ+ℓ− and B¯s→ℓ+ℓ−γ decays rates, including contributions from charming penguin diagrams

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Physics Subject Headings (PhySH)

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References (93)

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