Decomposition of the axial-vector current in a finite box

We consider the matrix element of the axial-vector current between two nucleon states in a finite box. Starting from the chiral Lagrangian density with nucleon and $\Delta$-isobar degrees of freedom, we study the finite-volume effects at the one-loop level. We show that the standard decomposition into the axial-vector and pseudoscalar form factor is incomplete in a finite box. We derive expressions for the complete set of form factors at one loop. We verify that the axial Ward identity holds in the chiral limit. Selected numerical results are shown for two flavor-SU(2) lattice ensembles. Sizable finite-volume effects are observed, with an important role for the $\Delta$-isobar. We discuss the implications of our results for lattice studies of the axial-vector current. We conclude that full finite-box results are crucial for a precise determination of the form factors.