The Pion Sum Rule beyond the Chiral Limit

TL;DR

The paper addresses the electromagnetic contribution to the pion mass difference and extends the classic pion sum rule beyond the chiral limit to include linear in light-quark masses. It develops a framework that combines dispersion theory, chiral perturbation theory with electromagnetism, and the operator product expansion to derive a finite-energy generalization with a dispersive left-right correlator and an effective CHPT description. Through a careful matching of the dispersive and chiral representations, the authors obtain a physical finite-energy pion sum rule, Eq. ($eq:pionsumrulephys$), that remains valid at finite energy and includes quark-mass effects. Testing against hadronic tau-decay data from ALEPH shows good agreement with the next-to-leading order prediction, supporting the approach and highlighting its potential for refining isospin-breaking analyses and electromagnetic low-energy constants.

Abstract

The difference between the charged and neutral pion masses can be predicted from a well-known dispersion relation involving an infinite-energy integral over experimental data, the pion sum rule. This relation, however, holds only in the chiral limit. Here we combine several nonperturbative techniques to determine the form of the physical finite-energy integral, thereby generalizing this sum rule to include effects linear in the light-quark masses. We test the dispersion relation using hadronic tau-decay data.

Figures (6)

• Figure 1: Integration contours in the complex $k^2$ plane used to derive the finite-energy representation of $\Pi(-Q^2)$. The jagged line denotes the physical cut along the positive real axis, split at $s_0$; the cross marks the pole at $k^2=-Q^2$ enclosed by $\mathcal{C}_1(s_0)$, while $\mathcal{C}_2(s_0)$ denotes the contour used in the treatment of duality-violating contributions.
• Figure 2: The diagrams at lowest order that lead to counter terms needed. Dashed lines indicate an instertion of the spurion terms of Eq. (\ref{['eq:lagone']}), the arrowed lines indicate quarks and the wiggly line a photon. (a) The leading contribution to the $\langle Q_L^2+Q_R^2\rangle$ operator. (b) The type of diagrams contributing to the LR case.
• Figure 3: The diagrams contributing to $\Pi ^{Q,\chi}_{\textrm{LR}}(q^2)$ through NLO in the chiral counting.
• Figure 4: $\mathrm{Im}\Pi$ as obtained from ALEPH hadronic tau decay data.
• Figure 5: $I(s_0)$ neglecting duality violations, using or not the Weinberg Sum Rules, compared to the predicted NLO value obtained in this work.