Improving the electromagnetic form factor of the pion at large $Q^2$ using the Feynman-Hellmann theorem

TL;DR

This paper enhances lattice QCD calculations of the pion electromagnetic form factor at high momentum transfer by applying noise reduction techniques, notably all-mode averaging and momentum smearing, within the Feynman-Hellmann framework.

Contribution

It introduces a novel combination of all-mode averaging and momentum smearing tailored for the Feynman-Hellmann method to improve high-Q^2 form factor computations.

Findings

All-mode averaging improves statistical precision at equal computational cost.

Momentum smearing reduces uncertainties in high-momentum states.

Combined techniques yield significant preliminary improvements in form factor calculations.

Abstract

At large momentum transfer, it becomes increasingly difficult to access the form factor of the pion $F_\pi(Q^2)$ using lattice QCD simulations. Two of the limiting factors include the increased computational cost of adding more statistics to overcome gauge noise, as well as suppressed overlap with the ground state of the boosted pion. Here we apply two noise reduction techniques, all-mode averaging (AMA) and momentum smearing, to the computation of $F_\pi(Q^2)$ at high momentum transfers using the Feynman-Hellmann (FH) theorem. First, we show that all-mode averaging by itself produces good improvement compared to previous results, at an equal computational cost. We also implement a momentum smearing technique to further reduce statistical uncertainties. In contrast to conventional smearing approaches, our Feynman-Hellmann method requires combining back-to-back momentum states, and hence we adapt a version of smearing involving a superposition of back-to-back smearing operations. This method is then implemented to compute $F_\pi(Q^2)$ at $Q^2 = 6.6 \;\mathrm{GeV^2}$, demonstrating good improvement over the regular smeared counterpart. Finally both all-mode averaging and momentum smearing are combined to determine $F_\pi(Q^2)$ at $Q^2 = 6.6 \;\mathrm{GeV^2}$ showing an excellent preliminary improvement over previous calculations.