Decay amplitudes to three hadrons from finite-volume matrix elements

Abstract

We derive relations between finite-volume matrix elements and infinite-volume decay amplitudes, for processes with three spinless, degenerate and either identical or non-identical particles in the final state. This generalizes the Lellouch-Lüscher relation for two-particle decays and provides a strategy for extracting three-hadron decay amplitudes using lattice QCD. Unlike for two particles, even in the simplest approximation, one must solve integral equations to obtain the physical decay amplitude, a consequence of the nontrivial finite-state interactions. We first derive the result in a simplified theory with three identical particles, and then present the generalizations needed to study phenomenologically relevant three-pion decays. The specific processes we discuss are the CP-violating K → 3π weak decay, the isospin-breaking η → 3π QCD transition, and the electromagnetic γ* → 3π amplitudes that enter the calculation of the hadronic vacuum polarization contribution to muonic g − 2.

Multi-particle systems on the lattice and chiral extrapolations: a brief review

“…Recently, important progress has been achieved addressing decay processes with three-particle final states. Namely, a formula that relates the decay amplitudes in a finite and in the infinite volume, has been derived in the NREFT approach [139] and, later, in the RFT setting [140]. In addition, a generalization to the case of non-rest frames, non-identical particles and partial-wave mixing was discussed within the RFT approach.…”

“…First, the momentum dependence of the short-range part of the decay vertex should be parametrized, e.g., via polynomials of a given order. This parametrization is already built in the NREFT approach [139], and can be conveniently introduced in the RFT setting [140], expanding the vertex in the vicinity of the decay threshold. Second, the long-range part can be systematically calculated (in both approaches) within effective theory in a finite volume, leading to the momentum-dependent LL factor one is looking for.…”

“…This will be discussed in Sec. 5, where we reflect on the latest developments related to the derivation of a three-body analog of the Lellouch-Lüscher formula that enables one to measure three-body decay amplitudes on the lattice [139,140].…”

Relativistic-invariant formulation of the NREFT three-particle quantization condition

“…Furthermore, in quite a few recent papers, the theoretical approaches mentioned above have been successfully used to analyze data from lattice calculations [38,53,[55][56][57][58][59][60][61][62][63][64][65]. Last but not least, a three-body analog of the Lellouch-Lüscher formula for the finite-volume matrix elements [66] has been recently derived in two different settings [67,68]. These developments are extensively covered in the latest reviews on the subject, to which the reader is referred for further details [69,70].…”

Higher partial wave contamination in finite-volume formulae for 1-to-2 transitions

“…Generalizations for transitions with three-particle final states have also been recently derived[17,18].…”