Isospin-breaking in the $ππ$ scattering amplitude I: Effects due to the pion mass difference

TL;DR

This work develops a dispersive, model-independent framework to quantify isospin-breaking effects in ππ scattering arising from the charged-neutral pion mass difference by generalizing Roy equations to nondegenerate pions. It couples a dispersive representation with χPT$_ ext{γ}$ to obtain subtraction constants and matches low-energy behavior, while solving a coupled-channel, multi-wave system for the S- and P-waves below the inelastic thresholds. The analysis finds sizable but controlled corrections near thresholds (up to ~12% in some neutral-channel cases) and small shifts in resonance poles, with f0(500) and f0(980) largely unchanged and a tiny rho mass splitting (~1 MeV) due to mpi-difference. These results establish a rigorous baseline for including additional isospin-breaking effects (QED real/virtual photons) toward accurate hadronic vacuum polarization contributions to g-2 and related observables. The planned extensions will enable more precise assessments of isospin-breaking corrections in vector form factors and τ-decay data for hadronic g-2 calculations.

Abstract

This is the first of a series of papers devoted to a detailed analysis of isospin-breaking effects in the $ππ$ scattering amplitude and the vector form factor of the pion. Isospin breaking originates from the mass difference between up and down quarks and from electromagnetic effects. The latter can be further split into effects due to the pion-mass difference and the remaining virtual and real photonic effects. In this paper, we derive the modifications to the Roy equations for $ππ$ scattering due to the mass difference between the charged and the neutral pion. We solve the equations numerically after matching them to the chiral representation of the $ππ$ scattering amplitude evaluated in the same approximation, which is also provided. Numerical results are presented and discussed in detail.

Figures (8)

• Figure 1: Result for the real part of the $t_S^{+0}(s)$ partial wave, with matching conditions imposed at $s_1=(1.15\,\,\text{GeV})^2$ to the isospin-limit result (dashed gray line). Roy equations are imposed only in the $\pi\pi$ elastic region so that the LHS (blue) and RHS (red) curves start deviating above $s_\text{max}=(0.975\,\,\text{GeV})^2$.
• Figure 2: Results for the real (left panel) and imaginary (right panel) parts of the $t_S^{n}(s)$ partial wave. For the real part, we display the isospin-breaking parameterization, i.e., the LHS of Roy equations (solid blue), and the dispersive representation, i.e., the RHS (dashed red), along with the isospin-limit result (gray-dotted line). For the imaginary part, since unitarity is exactly satisfied, we plot only the isospin-breaking parameterization and the isospin-limit result. The blue band represents the uncertainty in the isospin-breaking parameterization due to variations in the asymptotic value of $\text{Im}\,t_0^2$. The inset figures in both panels highlight the low-energy region, where the effect of the neutral-pion threshold becomes visible. At the bottom of both panels, we show the difference between the isospin-breaking and isospin limit parameterizations (gray-dotted line), along with the uncertainty band of the isospin-breaking parameterization. The isospin-limit result is evaluated using the variable $\bar{s}(s)$, defined in \ref{['eq:sbarmap']}, which maps the charged-pion threshold into the neutral one, allowing for a direct comparison of both results at the same energies. For the real part, we also plot at the bottom the difference between the LHS and RHS of Roy equations for $\Delta_\pi\neq 0$.
• Figure 3: We compare the isospin-breaking and isospin-limit results for the real (left panel) and imaginary part (right) of the $t_S^{c}(s)$ partial wave. The different curves follow the conventions in Fig. \ref{['fig:tnS']}. In this case, the physical region for both the isospin-limit and isospin-breaking parameterizations starts at the charged-pion threshold. Thus, the shift observed at low energies in the inset figures originates from the $\chi$PT$_\gamma$ correction to the scattering length.
• Figure 4: Results for the real (left) and imaginary (right) part of the $t_S^{x}(s)$ partial wave. The curves follow the same conventions as in Fig. \ref{['fig:tnS']}. The imaginary part of both the isospin-breaking and isospin-limit partial waves opens at the charged-pion threshold. At this energy, the pion-mass difference correction is given by $\Delta a_x^{+-}$, whose value in $\chi$PT$_\gamma$ leads to the small shift observed in the inset figures.
• Figure 5: We show the isospin-breaking and isospin-limit results for the real (left panel) and imaginary (right panel) of the $t_P^{c}(s)$ partial wave. The bottom of each figure displays the difference between the isospin-breaking and isospin-limit results, highlighting the minimal impact of pion-mass corrections in this case. For the real part, we also depict the LHS and RHS of Roy equations. Both curves almost coincide in the entire elastic region ($s_{+-}\le s\le s_\text{in}$) with their difference---displayed in the bottom panel---remaining well below the deviation from the isospin-limit result.