Second-order effects in chiral-invariant pion Lagrangians and the use of superpropagators

Nonlinear realization of chiral symmetries and localizability

“…The first three SU(ή) components of Ψ become proportional to (ξ 1 ,ξ 2 ,ξ 3 ) = :η as this is the only possible remaining SU(2) vector (Ψ 1 9 Ψ 2 9 Ψ 3 ) = η(f 0 +Σf a P a (ξ\...ξ" +1 )) = :ηg. 2 Glaser and Epstein [4] have shown that the localizability of the twopoint function is sufficient to render Green's functions localizable in all orders of perturbation theory.…”

“…With (x) we denote the normal 3-dimensional vector product. Evaluating the vector products and using Lehmann and Trute's technique [1] Z Z 0 Z η = η(x), ξ 4 = ξ 4 (y)...ξ" +1 = ξ" + 1 (y)…”

“…The antisymmetric structure of the vector product d μ η®η implies (cf. [1]): <Z 1 Z 2 > = 0. On account of the positivity of the spectral functions of (Lγ^L^ and (L 2 , L 2 yL 1 and L 2 must be separately localizable.…”

“…In this paper we generalize a theorem due to Lehmann and Trute [1] which states that the nonlinear realization of the chiral SU (2)xSU (2) symmetry is uniquely determined by the requirement of localizability. We extend this theorem to S U(n) x S U(ή) (n ^ 2).…”

Towards a unified nonlinear theory of massless bosons

“…Another recent example has been in chiral pion dynamics (Davies 1972), where non-polynomial Lagrangians (resulting from non linear realizations of pions) may have renormalization advantages over polynomial types (resulting from linear representations). Techniques include use of resummation methods (Delbourgo et al 1969;Keck & Taylor 1971) and th a t of the superpro pagator and summation over a minor coupling constant (Salam & Strathdee 1970;Lehmann & Trute 1973). Some more detailed aspects of the quantum field theory for non-polynomial Lagrangians have been developed (Charap 1973;Davies 1973;Keck & Taylor 1973;Taylor 1974).…”