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The axial-vector form factor of the nucleon in a finite box

Felix Hermsen, Tobias Isken, Matthias F. M. Lutz, Rob G. E. Timmermans

TL;DR

This paper addresses the challenge of determining the nucleon axial-vector form factor $G_A(q^2)$ from lattice QCD in finite volume by combining a flavor-$SU(2)$ chiral Lagrangian that includes the $$-isobar with a novel, minimal set of in-box scalar loop functions. The authors extend the Passarino-Veltman reduction to finite volume and express one-loop finite-box corrections in terms of renormalized tadpole, bubble, and triangle basis functions, carefully handling the $$-isobar contributions via coefficients that encode nucleon-Delta transitions. They find that implicit finite-volume effects (mass shifts in the box) typically dominate the in-box form factor, while explicit finite-box loop corrections introduce nontrivial momentum dependence and can be substantial at smaller volumes. The framework is numerically illustrated on three lattice ensembles with $m_pprox 250$ MeV, showing how in-box masses and the Delta inside loops shape the finite-volume corrections and how these results can inform extrapolations and constrain low-energy constants. Overall, the work provides a general, efficient method for lattice studies of hadronic form factors in finite volume and establishes a pathway to more precise determinations of axial properties across hadronic observables.

Abstract

We consider the axial-vector form factor of the nucleon in a finite box. Starting from the chiral Lagrangian with nucleon and Delta-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-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-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.

The axial-vector form factor of the nucleon in a finite box

TL;DR

This paper addresses the challenge of determining the nucleon axial-vector form factor from lattice QCD in finite volume by combining a flavor- chiral Lagrangian that includes the -isobar with a novel, minimal set of in-box scalar loop functions. The authors extend the Passarino-Veltman reduction to finite volume and express one-loop finite-box corrections in terms of renormalized tadpole, bubble, and triangle basis functions, carefully handling the -isobar contributions via coefficients that encode nucleon-Delta transitions. They find that implicit finite-volume effects (mass shifts in the box) typically dominate the in-box form factor, while explicit finite-box loop corrections introduce nontrivial momentum dependence and can be substantial at smaller volumes. The framework is numerically illustrated on three lattice ensembles with MeV, showing how in-box masses and the Delta inside loops shape the finite-volume corrections and how these results can inform extrapolations and constrain low-energy constants. Overall, the work provides a general, efficient method for lattice studies of hadronic form factors in finite volume and establishes a pathway to more precise determinations of axial properties across hadronic observables.

Abstract

We consider the axial-vector form factor of the nucleon in a finite box. Starting from the chiral Lagrangian with nucleon and Delta-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-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-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.

Paper Structure

This paper contains 7 sections, 55 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The axial-vector form factor of the nucleon in a box
  3. A new set of in-box basis loop functions
  4. Numerical results
  5. Summary
  6. The basis of loop functions in the finite volume
  7. List of $\alpha$ factors

Figures (2)

  • Figure 1: The tree and one-loop diagrams contributing to the axial-vector form factor of the nucleon up to chiral order $Q^4$ as given in Eq. \ref{['res-FA']}. The wavy, dotted solid, and double lines represent the axial current, pion, nucleon, and $\Delta$-isobar, respectively. The chiral order of all vertices is indicated.
  • Figure 2: The nucleon bubble (top) and nucleon-pion-$\Delta$ loop (bottom) corrections to the axial-vector form factor as a function of $t = (\bar{p} - p)^2)$. The lattice length $L$ increases from the left to the right. The white circles correspond to the infinite-volume case, with no finite-box corrections in the masses or in the form factor. The blue triangles contain only the finite-box corrections of the hadron masses. These "implicit" finite-box effects were already considered in Ref. Lutz:2020dfi. The red diamonds contain only the "explicit" finite-box corrections to the form factor, by using the infinite-volume hadron masses. The green dots give the full results, taking both types of finite-box corrections into account.