Consistency checks for two-body finite-volume matrix elements: II. Perturbative systems

Using the general formalism presented in [Phys. Rev. D 94, 013008 (2016)PRVDAQ2470-001010.1103/PhysRevD.94.013008; Phys. Rev. D 100, 034511 (2019)PRVDAQ2470-001010.1103/PhysRevD.100.034511], we study the finite-volume effects for the 2+J→2 matrix element of an external current coupled to a two-particle state of identical scalars with perturbative interactions. Working in a finite cubic volume with periodicity L, we derive a 1/L expansion of the matrix element through O(1/L5) and find that it is governed by two universal current-dependent parameters, the scalar charge and the threshold two-particle form factor. We confirm the result through a numerical study of the general formalism and additionally through an independent perturbative calculation. We further demonstrate a consistency with the Feynman-Hellmann theorem, which can be used to relate the 1/L expansions of the ground-state energy and matrix element. The latter gives a simple insight into why the leading volume corrections to the matrix element have the same scaling as those in the energy, 1/L3, in contradiction to Phys. Rev. D 91, 074509 (2015)PRVDAQ1550-799810.1103/PhysRevD.91.074509, which found a 1/L2 contribution to the matrix element. We show here that such a term arises at intermediate stages in the perturbative calculation, but cancels in the final result.