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$K π$ scattering as a step towards $B \to K^* \ell^+ \ell^-$ from Lattice QCD

Felix Erben, Matthew Black, Peter Boyle, Matteo Di Carlo, Vera Gülpers, Maxwell T. Hansen, Nelson Pitanga Lachini, Rajnandini Mukherjee, Antonin Portelli, J. Tobias Tsang

Abstract

Rare $b\to s\ell^+\ell^-$ decays provide some of the most sensitive tests of the Standard Model and require precise and systematically improvable hadronic input from lattice QCD. For the phenomenologically important channel $B\to K^*\ell^+\ell^-$ this entails a first-principles treatment of a resonant $Kπ$ final state together with controlled heavy-quark dynamics. We present the status of a new exploratory lattice calculation that combines a variational determination of finite-volume $Kπ$ states with the $1+J\to2$ finite-volume formalism to access the relevant matrix elements. The computation is carried out on an RBC/UKQCD domain-wall fermion ensemble with $a^{-1} \approx 2.7\,\mathrm{GeV}$ and employs a dual heavy-quark strategy, using both a relativistic heavy-quark action tuned to the physical $b$ mass and domain-wall heavy masses extrapolating from charm. All correlation functions are computed using (stochastic) distillation, providing a versatile setup that supports a broad range of heavy-to-light transitions into resonant final states. We show first two-point results for the $K^*\leftrightarrow Kπ$ system and discuss the accessible kinematic region, which allows for a controlled study at high $q^2$. The outlook for extending the calculation to lower $q^2$ and for incorporating effects from charmonium resonances is outlined.

$K π$ scattering as a step towards $B \to K^* \ell^+ \ell^-$ from Lattice QCD

Abstract

Rare decays provide some of the most sensitive tests of the Standard Model and require precise and systematically improvable hadronic input from lattice QCD. For the phenomenologically important channel this entails a first-principles treatment of a resonant final state together with controlled heavy-quark dynamics. We present the status of a new exploratory lattice calculation that combines a variational determination of finite-volume states with the finite-volume formalism to access the relevant matrix elements. The computation is carried out on an RBC/UKQCD domain-wall fermion ensemble with and employs a dual heavy-quark strategy, using both a relativistic heavy-quark action tuned to the physical mass and domain-wall heavy masses extrapolating from charm. All correlation functions are computed using (stochastic) distillation, providing a versatile setup that supports a broad range of heavy-to-light transitions into resonant final states. We show first two-point results for the system and discuss the accessible kinematic region, which allows for a controlled study at high . The outlook for extending the calculation to lower and for incorporating effects from charmonium resonances is outlined.
Paper Structure (6 sections, 8 equations, 2 figures)

This paper contains 6 sections, 8 equations, 2 figures.

Table of Contents

  1. Introduction and motivation
  2. From form factors to finite-volume matrix elements
  3. Exploratory computation: ensemble and heavy-quark strategy
  4. Distillation for electroweak transitions into resonances
  5. First status: $K^*\leftrightarrow K\pi$ two-point functions
  6. Kinematic considerations and outlook

Figures (2)

  • Figure 1: Expected finite-volume energy levels (free spectrum) in the $K\pi$ channel on the F1M ensemble around the $K^*$ region, shown relative to the nominal resonance mass $M_{K^*}$. The resonance region lies well above the $K\pi$ threshold: on F1M we have $M_{K^*}\simeq 960~\mathrm{MeV}$ and $M_\pi+M_K\simeq 742~\mathrm{MeV}$.
  • Figure 2: Matrix of two-point correlation functions entering the $5\times5$ GEVP in the $K^*$ channel in the rest frame on the F1M ensemble. Each panel corresponds to one element $C_{ij}(t)$ of the correlation matrix built from the operator basis $\{V,\,K_1\pi_1,\,K_2\pi_2,\,K_3\pi_3,\,K_4\pi_4\}$. All correlators are normalized to the $V$–$V$ two-point function so that a common vertical scale can be used across panels. The data shown are based on 8 gauge configurations with 24 source times per configuration; for this exploratory look three bins per configuration are treated as statistically independent, so the uncertainties are expected to be underestimated.