Electromagnetic corrections to ππ scattering lengths: Some lessons for the construction of effective hadronic field theories

Abstract

Using the framework of effective chiral Lagrangians, we show that, in order to correctly implement electromagnetism (EM), as generated from the Standard Model, into effective hadronic theories (such as meson-exchange models) it is insufficient to consider only graphs in the low-energy effective theory containing explicit photon lines. The Standard Model requires the presence of contact interactions in the effective theory with are electromagnetic in origin, but which involve no photons in the effective theory. We illustrate the problems which can result from a “standard” EM subtraction: i.e., from assuming that removing all contributions in the effective theory generated by graphs with explicit photon lines fully removes EM effects, by considering the case of the s-wave ππ scattering lengths. In this case it is shown that such a subtraction proceedure would lead to the incorrect conclusion that the strong interaction isospin-breaking contributions to these quantities were large when, in fact, they are known to vanish at leading order in m d-m u . The leading EM contact corrections for the channels employed in the extraction of the I = 0.2 s-wave ππ scattering lengths from experiment are also evaluated.