A Master Formula for Chiral Symmetry Breaking

Abstract

We derive a master formula for chiral SU(2) × SU(2) breaking, based on the VeltmanBell equations and the Peierls-Dyson relation. Our approach does not rely on the use of the soft pion limit or an expansion around the chiral limit, and yields exact results for on-shell pions. Threshold theorems for πN → πN, γN → πN, πN → ππN, γN → γπN, γN → ππN and πN → πγN are recovered, and corrections to them are given. The reactions π → eνγ, π → eνe + e − , γπ → γπ and γγ → ππ are also discussed. A general formula for ππ scattering and a new one loop effective action are obtained. The new effective action reproduces the KSFR relation, and yields specific estimates for the pion polarisabilities. A detailed comparison with baryon-free chiral perturbation theory to one loop is made. An extension of our effective action to two loops is outlined.

Dilepton and photon emission rates from a hadronic gas. II

“…Taking q 2 = 0 and σ πN → 0 reduces the loop result to that given in [19]. Conformity with the Wardidentities [11] requires an on-shell expansion [13] as taken into account here.…”

“…This behavior is expected to occur for all terms in W N πN and so we may neglect the terms of order κ N κ π . For completeness, we quote the result for the first term in (11) in Appendix C. Qualitatively, we note that the full ππ scattering amplitude as well as terms of photonpion scattering similar to W π appear with an additional suppression factor of κ. In the soft pion limit most of the correlation functions in the pionic state are amenable to correlations in the vacuum, some of which were assessed in I and found to be small.…”

Δ(1232)inπN→ππNreaction

“…is an annihilation (creation) operator of the pion with the isospin component a and the four-momentum k.Ŝ =Ŝ[φ] is the extended S matrix operator which is a functional of φ = (a, v, s, J); the axial vector, vector, scalar, and pseudoscalar c-number external fields [13]. At φ = 0,Ŝ is reduced to the ordinary S matrix operator.…”

“…[13], the form factor S(t) is equal to −σ πN (t)/f π m 2 π , where σ πN (t) is the pion-nucleon sigma term which becomes independent of t at tree level.…”

“…Recently, a general framework analyzing hadron reactions has been developed by Yamagishi and Zahed [13] on the basis of chiral symmetry. This framework introduces a new type of reduction formula, i.e.…”