Isospin breaking in pion Compton scattering

Abstract

Using chiral perturbation theory we calculate for pion Compton scattering the isospin-breaking effects induced by the difference between the charged and neutral pion mass. At one-loop order this correction is directly proportional to $\ensuremath{m_{\pi^\pm}^2-m_{\pi^0}^2}$ and free of (electromagnetic) counterterm contributions. The differential cross-section for charged pion Compton scattering $\ensuremath{\pi^-\gamma \rightarrow \pi^-\gamma}$ gets affected (in backward directions) at the level of a few permille. At the same time the isospin-breaking correction leads to a small shift of the pion polarizabilities by $\ensuremath{\delta(\alpha_\pi- \beta_\pi) \simeq 1.3 \cdot 10^{-5}}$ fm^3. In case of the low-energy $\ensuremath{\gamma\gamma \rightarrow \pi^0\pi^0}$ reaction isospin breaking manifests itself through a cusp effect at the $\ensuremath{\pi^+\pi^-}$ threshold. We give an improved estimate for it based on the empirical $ \pi$ $ \pi$ -scattering length difference $\ensuremath{a_0-a_2}$ .