Two-loop representations of low-energy pion form factors and ππ scattering phases in the presence of isospin breaking

Dispersive representations of the ππ scattering amplitudes and pion form factors, valid at two-loop accuracy in the low-energy expansion, are constructed in the presence of isospin-breaking effects induced by the difference between the charged and neutral pion masses. Analytical expressions for the corresponding phases of the scalar and vector pion form factors are computed. It is shown that each of these phases consists of the sum of a “universal” part and a form-factor dependent contribution. The first one is entirely determined in terms of the ππ scattering amplitudes alone, and reduces to the phase satisfying Watson’s theorem in the isospin limit. The second one can be sizeable, although it vanishes in the same limit. The dependence of these isospin corrections with respect to the parameters of the subthreshold expansion of the ππ amplitude is studied, and an equivalent representation in terms of the S-wave scattering lengths is also briefly presented and discussed. In addition, partially analytical expressions for the two-loop form factors and ππ scattering amplitudes in the presence of isospin breaking are provided.