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Dispersive representations of the pi-pi scattering amplitudes and pion formfactors, valid at two-loop accuracy in the low-energy expansion, areconstructed in the presence of isospin-breaking effects induced by thedifference between the charged and neutral pion masses. Analytical expressionsfor the corresponding phases of the scalar and vector pion form factors arecomputed. 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 isentirely determined in terms of the pi-pi scattering amplitudes alone, andreduces to the phase satisfying Watson's theorem in the isospin limit. Thesecond one can be sizeable, although it vanishes in the same limit. Thedependence of these isospin corrections with respect to the parameters of thesubthreshold expansion of the pi-pi amplitude is studied, and an equivalentrepresentation in terms of the S-wave scattering lengths is also brieflypresented and discussed. In addition, partially analytical expressions for thetwo-loop form factors and pi-pi scattering amplitudes in the presence ofisospin breaking are provided.

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