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

This is the first of a series of papers devoted to a detailed analysis of isospin-breaking effects in the $\pi \pi$ 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 $\pi\pi$ 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 $\pi \pi$ scattering amplitude evaluated in the same approximation, which is also provided. Numerical results are presented and discussed in detail.