Finite-box effects in the axial-vector form factor: a case study for the nucleon

Abstract We consider the axial-vector form factor of the nucleon in a finite box. Starting from the chiral Lagrangian with nucleon and $$\Delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math> -isobar degrees of freedom, we address, at the one-loop level, the impact of two types of finite-volume effects. On the one hand, there are the implicit effects from the in-box values of the nucleon and $$\Delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math> -isobar masses. On the other hand, there are the explicit effects caused by computing the in-box loop integrals with the values of the nucleon and $$\Delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math> -isobar masses obtained in the infinite-volume limit. Selected numerical results are shown for three lattice ensembles. We show that the implicit effects dominate the in-box form factor. Our results are presented in terms of a set of basis functions that generalize the Passarino–Veltman reduction scheme to the finite-box case, such that only scalar loop integrals have to be performed. The techniques we developed are more generally relevant for lattice studies of hadronic quantities.