Virtual photons to fourth order CHPT with nucleons # 1

Baryon chiral perturbation theory offers another possibility of investigating isospin violation. As first stressed by Weinberg [1], reactions involving nucleons and neutral pions can lead to gross violations of isospin, e.g. in the scattering length difference a(π 0 p) − a(π 0 n) he predicted an effect of the order of 30%. This is because chiral symmetry and isospin breaking appear at the same order and the leading isospin symmetric terms involving neutral pions are suppressed due to chiral symmetry. It is, however, known that precise and complete one loop calculations in the baryon sector should be carried out to fourth order since it has also been shown that in many cases one loop graphs with exactly one dimension two insertion are fairly large. Most calculations in baryon CHPT are performed in heavy baryon chiral perturbation theory (HBCHPT) [5, 6]. This is based on the observation that a straightforward extension of CHPT with baryons treated fully relativistically leads to a considerable complication in the power counting since only nucleon three–momenta are small compared to typical hadronic scales, as discussed in detail in ref.[7]. However, one has to be careful with strict non–relativistic expansions since in some cases they can conflict structures from analyticity, as discussed e.g. in ref.[8]. Therefore, in ref.[9], a novel scheme was proposed which is based on the relativistic formulation but circumvents the power counting problems (to one loop) by a clever separation of the loop integrals into IR singular and regular parts. In this formulation, all analytic constraints are fulfilled by construction. The starting point of our approach is to construct the most general chiral invariant Lagrangian built from pions, nucleons and external scalar, pseudoscalar, vector, axial–vector sources and virtual photons, parametrized in terms of the vector field A µ (x). The Goldstone bosons are collected in a 2×2 matrix-valued field U(x) = u 2 (x). The nucleons are described by structureless