Two-Pion Matrix Elements in SU(2) Chiral Perturbation Theory: Analytic Structures and Implications for Lattice QCD

Two-particle matrix elements play a central role in understanding nucleon structure, for instance in determining the form factors of unstable resonances such as the ρ meson, which are nearly inaccessible experimentally. While their general features have been analyzed in various theoretical frameworks, a complete lattice-QCD determination of the corresponding amplitudes remains ongoing. Because the process involves four independent external momenta, one key difficulty lies in identifying the functional form suitable for global fits, which motivates our work.

Using SU(2) chiral perturbation theory, we study the two-pion matrix element of the electromagnetic current, relevant to the process ππ → γ* + ππ at leading (LO) and next-to-leading order (NLO), in terms of partial-wave expansions. Based on the formal structure, we isolate and subtract the pole and branch-cut contributions and extract the analytic part, which also provides a cross-check of the theoretical formalism. Our work offers intuition into the low-energy region and multi-dimensional global fitting for a full lattice QCD calculation.