On the three-particle analog of the Lellouch-Lüscher formula

30 Jul 2021, 06:30
15m
Oral presentation Hadron Spectroscopy and Interactions Hadron Spectroscopy and Interactions

Speaker

Fabian Müller (University of Bonn)

Description

Back in 2000, Lellouch and Lüscher derived a formula, relating the
matrix element of the weak $K \rightarrow 2\pi$ decay in a finite
volume to its infinite-volume counterpart.
In contrast, albeit latest theoretical developments enable the
extraction of three-body scattering amplitudes on the lattice,
a three-particle analog of the Lellouch-Lüscher equation has not
been available until very recently. In this talk, we report on the
first attempt to close this gap.

The interest in the study of three-body decays on the lattice is large. While
the most obvious candidates for such a study are provided by the
three-pion decays of low-mass light-flavored mesons, like the weak
process $K \rightarrow 3\pi$, also the candidates for exotica,
$X(3872)$ and $X(4260)$, decay largely into three-particle final states
as well. Moreover, the proper treatment of the three-particle decay
channel of the Roper resonance might improve the extraction of its
parameters. Last but not least, the study of the electromagnetic process
$\gamma^* \rightarrow 3\pi$, contributing to the anomalous magnetic
moment of the muon, is certainly very interesting.

In order to avoid unnecessary technical complications in the derivation
of the three-particle analog of the Lellouch-Lüscher formula,
we consider the simplest case of a decay into three identical spinless
particles, which interact only in the S-wave. The derivation
is carried out within the explicitly covariant version of the non-relativistic
effective field theory, where relativistic corrections in the internal lines
are summed up to all orders. The non-relativistic formalism provides a very
transparent and simple framework -- especially, the use of the
particle-dimer picture drastically reduces the number of relevant
diagrams needed to describe the final-state interactions in the
three-particle decay. Further developments concerning particles
with spin, partial wave mixing, moving frames and an so on are
already in progress. These modifications will not affect our result at
the leading order which, as expected, will be sufficient for the first
generation of lattice calculations.

We demonstrate that, similar to
the two-particle sector, the relation between
the finite- and infinite-volume decay matrix elements is described by an
overall multiplicative factor, depending on the size of the cubic box and
the parameters of the final-state interactions only. In contrast, at higher
orders, the factor will become a matrix with the dimension equal
to the number of independent couplings, describing the three-particle
decay at this order.

Primary authors

Fabian Müller (University of Bonn) Dr Akaki Rusetsky (Univerity of Bonn)

Presentation materials