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

Understand

Dispersive representations of the pi-pi 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 pi-pi scattering amplitudes alone, and reduces to the phase satisfying Watson's theorem in the isospin limit.