Chiral perturbation theory with vector mesons and the pion electromagnetic form factor

Abstract

We formulate chiral perturbation theory with vector mesons kept as explicit intermediate degrees of freedom, in the regime where the momentum carried by a vector meson is of order $Q$ while its mass is of order $Λ_χ$. This is the regime relevant to vector-meson exchange in nuclear forces and low-energy nuclear electroweak processes. We use the corresponding power counting and show that it gives a consistent one-loop expansion, with the divergences generated by the leading-order Lagrangian absorbed by next-to-leading-order counterterms. As an application, we calculate the pion electromagnetic form factor. The vector mesons introduce a single new leading-order parameter, $a$. For the KSRF value $a=2$, the nonlocal one-loop contribution to the direct photon--pion vertex vanishes at NLO, while the remaining local term is found to be consistent with zero when fixed from the pion charge radius and $ρ$-meson decay data. Vector-meson dominance therefore emerges as a good numerical approximation at this order. The explicit $ρ$ propagator, kept in its resummed form, also reproduces the observed curvature of the form factor more efficiently than vector-meson-less one-loop ChPT. We also present a phenomenological extension of the form factor into the $ρ$ resonance region, using a resummation of the $ρ$ self-energy. The relation to other formulations with explicit vector mesons is also discussed.