Pion Interactions in Chiral Field Theories∗

In this paper we study the interactions of observed pions with inferred scalar σ meson [1,2] and fermion quark SU(2) fields in a chiral-invariant manner at low energies. Specifically we consider two chiral theories: a) A chiral quark model (CQM) dynamically inducing [2] the entire quark-level SU(2) linear σ model (LσM) but depending on no free parameters. b) Chiral perturbation theory (ChPT) involving ten strong interaction parameters L1 − L10 [3-5], now called low energy constants (LECs). Following the surveys of Donoghue and Holstein [6,7], we compare the predictions of the above two theories with the measured values of the i) pion charge radius, ii) πe3 form factor ratio FA/FV at zero invariant momentum transfer and the π◦ → γγ amplitude, iii) charged pion polarizabilities, iv) γγ → π◦π◦ amplitude at low energies, v) ππ s-wave I = 0 scattering length. We begin in Sec.II with the quark-level Goldberger-Treiman relation (GTR), (its meson analog) the KSRF relation [8] and the link to vector meson dominance (VMD). Then in Sec.III we examine the pion charge radius rπ in the above two chiral theories. Next in Sec IV, we first review π◦ → γγ decay and then study its isospin-rotated semileptonic weak analog π → eνγ, giving rise to the form factor ratio FA/FV ≡ γ at zero invariant momentum transfer. This naturally leads in Sec.V to the charged pion electric polarizability απ+ due to the model-independent relation [9,6] between απ+ and the above πe3 ratio γ. Finally in Sec.VI we review the Weinberg soft-pion prediction [10] for the s-wave I = 0 ππ scattering length and its chiral-breaking corrections. In all of the above cases the predictions of the CQM-LσM and ChPT chiral theories are compared with the measured values of rπ, γ, απ+ , a (0) ππ . We review these results in Sec.VII.