Using the general formalism presented in [Phys. Rev. D 94, 013008 (2016); Phys. Rev. D 100, 034511 (2019)], we study the finite-volume effects for the $\mathbf{2}+\mathcal{J}\to \mathbf{2}$ 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 $\mathcal{O}(1/L^5)$ 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, $1/L^3$, in contradiction to Phys. Rev. D 91, 074509 (2015), which found a $1/L^2$ 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.

• Received 13 February 2020
• Accepted 28 April 2020

DOI:https://doi.org/10.1103/PhysRevD.101.094508

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Published by the American Physical Society