Chiral Logarithms in the Massless Limit Tamed

Abstract

We derive nonlinear recursion relations for the leading chiral logarithms (LLs) in massless theories. These relations not only provide a very efficient method of computation of LLs (e.g., the 33-loop contribution is calculated in a dozen of seconds on a PC) but also equip us with a powerful tool for the summation of the LLs. Our method is not limited to chiral perturbation theory only; it is pertinent to any nonrenormalizable effective field theory such as, for instance, the theory of critical phenomena, low-energy quantum gravity, etc.

Running couplings and operator mixing in the gravitational corrections to coupling constants

“…In his influential paper on effective field theory [16], Weinberg showed that the content of the renormalization group in these theories is to relate the highest powers of q 2 logðÀq 2 = 2 Þ to each other. This behavior has been explored subsequently in more detail [17][18][19][20][21]. This is due to the power counting relations of effective field theory which tell us that loop processes generate higher order operators that involve more powers of the derivatives and/or fields.…”

“…Because of the increasing powers of q 2 in the factors of q 2n log n ðÀq 2 = 2 Þ, this does not lead to a renormalization of the leading coupling constant, but rather the higher order couplings are renormalized. In addition, the logarithms enter differently in different processes or even in two form factors for the same process [21]. 1 Therefore, attempts to define a running coupling necessarily involve definitions which fall outside of the usual renormalization group.…”

Leading chiral logarithms for the nucleon mass

“…In the bosonic ChPT this set of problems has been solved in refs. [8,9,10,11,12]. The main idea of solution was to reconstruct the LLog coefficient by evaluation of the fixed chiral order solution of the system of renormalization group equations on the amplitude and on LECs.…”

“…The base of the method is the formal solution of the renormalization group equation for an amplitude, which can be written in the form [9,10]…”

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

“…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:…”