Abstract
In this work we develop a Lorentz-covariant version of the previously derived formalism for relating finite-volume matrix elements to $\textbf 2 + \mathcal J \to \textbf 2$ transition amplitudes. We also give various details relevant for the implementation of this formalism in a realistic numerical lattice QCD calculation. Particular focus is given to the role of single-particle form factors in disentangling finite-volume effects from the triangle diagram that arise when $\mathcal J$ couples to one of the two hadrons. This also leads to a new finite-volume function, denoted $G$, the numerical evaluation of which is described in detail. As an example we discuss the determination of the $\pi \pi + \mathcal J \to \pi \pi$ amplitude in the $\rho$ channel, for which the single-pion form factor, $F_\pi(Q^2)$, as well as the scattering phase, $\delta_{\pi\pi}$, are required to remove all power-law finite-volume effects. The formalism presented here holds for local currents with arbitrary Lorentz structure, and we give specific examples of insertions with up to two Lorentz indices.
Highlights
In recent years, interest in hadron spectroscopy has increased significantly, primarily due to various experimental discoveries of unconventional excitations.1 This has led to an abundance of theoretical proposals as to the underlying nature of the unexpected states
II we present a slightly modified version of this formalism in which all infinite-volume quantities are Lorentz covariant and the single-particle matrix elements that enter, abbreviated 1 þ J → 1, are expressed in terms of standard Lorentzinvariant form factors
In this work we have presented a modified version of the finite-volume formalism for studying 2 þ J → 2 transition amplitudes
Summary
Interest in hadron spectroscopy has increased significantly, primarily due to various experimental discoveries of unconventional excitations. This has led to an abundance of theoretical proposals as to the underlying nature of the unexpected states. The idea has been generalized to all possible two-body systems, in particular to multiple two-particle channels built form any number of particle species, including particles with any intrinsic spin [7, 17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32] These formal ideas, together with significant algorithmic developments, have resulted in a proliferation of scattering amplitudes determined directly from lattice QCD [33,34,35,36,37,38,39,40,41,42,43,44,45,46]. Appendix B includes various technical aspects regarding the evaluation of the finite-volume functions discussed in the main text
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