Leading infrared logarithms from unitarity, analyticity, and crossing

Abstract

We derive nonlinear recursion equations for the leading infrared logarithms in massless nonrenormalizable effective field theories. The derivation is based solely on the requirements of the unitarity, analyticity and crossing symmetry of the amplitudes. That emphasizes the general nature of the corresponding equations. The derived equations allow one to compute leading infrared logarithms to essentially unlimited loop order without performing a loop calculation. For the implementation of the recursion equation one needs to calculate tree diagrams only. The application of the equation is demonstrated on several examples of effective field theories in four and higher space-time dimensions.

Exact summation of leading infrared logarithms in 2D effective field theories

“…In this Appendix, following the method of ref. [2], relying on the fundamental QFT requirements of unitarity, analyticity, and crossing symmetry, we derive the recurrence relation for leading IR logarithms in a general Φ 4 -type theory in two dimensions defined by the action:…”

“…The LL-coefficients ω n are related to each other by the non-linear recurrence relations derived in refs. [1,2]. The summation of the LL-series in Eq.…”

“…Moreover, the function Ω(z) may contain the information on the non-perturbative spectrum of masses of a non-renormalizable EFT. For the 4D massless EFTs the explicit expressions for the function Ω(z) was obtained only for the case of the large-N limit in the O(N + 1)/O(N ) [1,2] and SU(N )/(SU(N − M ) × SU(M )) [7] σ-models. For the O(N + 1)/O(N ) σ-model in the limit N → ∞ the exact result for the generating function Ω(z) (corresponding to the isospin zero, S-wave scattering amplitude) has the form:…”

“…In analogy with the general reasoning of ref. [2], we would like to work out the recurrence relations for the LL-coefficients of amplitudes (A.11). Obviously, one has where, we employ that, according to our conventions, Note that for even κ (once the number of derivatives in the interaction is a multiple of 4) the recursive equation has the form which is different from the usual RG-equation type due to the presence of (−1) n factor in the equation.…”

“…O(NLL terms). (A.15)The dispersion relations for the transmission and reflection amplitudes have the following form:M I (s, t = 0, u = 4m 2 − s) ≡ M I, T (s) I (s, t = 4m 2 − s, u = 0) ≡ M I, R (s)Here where the crossing matrices C IJ xy were defined for a general symmetry group 8 in ref [2]…”