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

TL;DR

The paper tackles the challenge of computing the pion electromagnetic form factor $F_\pi(Q^2)$ at large $Q^2$ on the lattice, where gauge noise and excited-state contamination hinder precision. It combines the Feynman-Hellmann theorem with two noise-reduction techniques—All-Mode Averaging (AMA) and momentum smearing (including a novel superposition approach for back-to-back momenta)—to extract $F_\pi(Q^2)$ efficiently. AMA improves statistical precision at reduced cost, momentum smearing enhances ground-state overlap for boosted pions, and their combination (MSAMA) yields a substantial reduction in uncertainty at $Q^2=6.6$ GeV$^2$, enabling a robust determination of $F_\pi(Q^2)$ at high momentum transfers. The methods offer a pathway to higher-$Q^2$ lattice QCD insights and can generalize to related quantities such as the Compton amplitude of hadrons.

Abstract

At large momentum transfer, it becomes increasingly difficult to access the form factor of the pion $F_π(Q^2)$ 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 $F_π(Q^2)$ 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 $F_π(Q^2)$ 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 $F_π(Q^2)$ at $Q^2 = 6.6 \;\mathrm{GeV^2}$ showing an excellent preliminary improvement over previous calculations.

Figures (7)

• Figure 1: Insertion of an electromagnetic current into a pion
• Figure 2: Equal cost comparison of the effective mass of a non-AMA correlator utilising $4$ strict sources compared to an AMA correlator utilising $N_\text{strict} = 1$ and $N_\text{appx} = 14$
• Figure 3: A comparison of the determined energy shifts using FH for $F_\pi(Q^2)$ using various precisions and perturbation strengths. Matching $\lambda$ values are shifted for visibility and the $\lambda = 10^{-5}$ blue point is above the graph. An enlarged view is provided for the larger $\lambda$ cases for clarity in identifying the signal behavior.
• Figure 4: Comparison between $Q^2 F_\pi(Q^2)$ computed with AMA (orange points with $Q^2<7$ are shifted for clarity) for a roughly equal computation time comparison with results from Chambers et al. Ref. QCDSF:2017ssq (blue). VMD prediction shown for $m_\rho = 932\;\mathrm{MeV}$ (solid blue line) and pQCD prediction, Eq. (\ref{['eq:pQCD']}), shown for $f_\pi = 175\;\mathrm{MeV}$QCDSF:2017ssq and $N_f = 3$ (dotted blue line).
• Figure 5: Comparison between effective mass plots for baryons with momentum $\mathbf{p}_1=\frac{2\pi}{L}(-3,0,0)$ (left) and $\mathbf{p}_2=\frac{2\pi}{L}(3,0,0)$ (right). Each plot shows effective energy of the baryon when smeared with Jacobi smearing (purple crosses), momentum smearing with $\mathbf{k}_1 = \frac{2\pi}{L}(-1,0,0)$ (blue circles) and superposition of both $\mathbf{k}_1$ and $\mathbf{k}_2$ (red dots).