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On the inclusion of the pion form factor in $e^+e^- \to π^+π^-$ beyond leading order

Francesco P. Ucci

TL;DR

The paper addresses the precision challenges in predicting the hadronic vacuum-polarization contribution to $(g-2)_$, highlighting the dominance of the $pipi$ channel and the need for radiative corrections beyond the leading-order factorised $F_pi(q^2)$ treatment. It compares ab-initio lattice and dispersive approaches, emphasizing recent CMD-3 results that bring theory and experiment closer, while still requiring next-to-leading order (NLO) radiative corrections. The authors implement three strategies—generalised vector meson dominance (GVMD), a dispersive FsQED approach, and a factorised sQED baseline—within an updated version of the BabaYaga@NLO Monte Carlo to include the pion form factor inside loop amplitudes. Numerical results show sizable NLO effects (up to $\pm 8\%$ on integrated cross sections) and nontrivial forward-backward asymmetries, underscoring the importance of accounting for pion compositeness at loop level for precision hadronic predictions and MC event generation.

Abstract

The pion form factor plays a crucial role in the determination of the contribution of the hadronic vacuum polarisation to the muon anomalous magnetic moment. In order to measure this quantity, energy-scan experiments rely on Monte Carlo generator to simulate the $e^+e^- \to π^+π^-(γ)$ process. For the theoretical accuracy to match the experimental precision, next-to-leading order calculations and the resummation of multiple photon emissions are needed. In this context, the inclusion of the pion form factor beyond the leading order approximation is crucial to reproduce some observables, like the pion charge asymmetry. We present the impact of the inclusion of the pion form factor in loop diagrams with three approaches and the interplay with radiative corrections.

On the inclusion of the pion form factor in $e^+e^- \to π^+π^-$ beyond leading order

TL;DR

The paper addresses the precision challenges in predicting the hadronic vacuum-polarization contribution to , highlighting the dominance of the channel and the need for radiative corrections beyond the leading-order factorised treatment. It compares ab-initio lattice and dispersive approaches, emphasizing recent CMD-3 results that bring theory and experiment closer, while still requiring next-to-leading order (NLO) radiative corrections. The authors implement three strategies—generalised vector meson dominance (GVMD), a dispersive FsQED approach, and a factorised sQED baseline—within an updated version of the BabaYaga@NLO Monte Carlo to include the pion form factor inside loop amplitudes. Numerical results show sizable NLO effects (up to on integrated cross sections) and nontrivial forward-backward asymmetries, underscoring the importance of accounting for pion compositeness at loop level for precision hadronic predictions and MC event generation.

Abstract

The pion form factor plays a crucial role in the determination of the contribution of the hadronic vacuum polarisation to the muon anomalous magnetic moment. In order to measure this quantity, energy-scan experiments rely on Monte Carlo generator to simulate the process. For the theoretical accuracy to match the experimental precision, next-to-leading order calculations and the resummation of multiple photon emissions are needed. In this context, the inclusion of the pion form factor beyond the leading order approximation is crucial to reproduce some observables, like the pion charge asymmetry. We present the impact of the inclusion of the pion form factor in loop diagrams with three approaches and the interplay with radiative corrections.
Paper Structure (7 sections, 11 equations, 4 figures)

This paper contains 7 sections, 11 equations, 4 figures.

Table of Contents

  1. Introduction
  2. NLOPS accuracy in factorised SQED
  3. Inserting $F_\pi(q^2)$ in the loop
  4. Generalised vector meson dominance
  5. Dispersive approach
  6. Phenomenological results
  7. Conclusions

Figures (4)

  • Figure 1: Feynman diagrams for the $e^+e^-\to\pi^+\pi^-$ process. From left to right: tree level diagram, example of real ISR, virtual FSR vertex correction and virtual IFI.
  • Figure 2: Virtual FSR and ISR Feynman diagrams, where the blob amounts to the insertion of $F_\pi(q^2)$ inside the loop integral with the proper virtuality.
  • Figure 3: The left plot shows the absolute value squared of the pion form factor, fitted with the GS and BW parametrisations, the bottom panel displays $\Delta = |F_\pi^\text{GS}|/|F_\pi^\text{BW}|-1$. To the right, the integrated cross section as a function of $\sqrt{s}$ at three levels of accuracy.
  • Figure 4: Forward-backward asymmetry as a function of the centre of mass energy with the F$\times$sQED, GVMD and FsQED approaches at two levels of precision. Black crosses are the experimental CMD3 data.