Leading logarithms of the two-point function in massless O(N) and SU(N) models to any order from analyticity and unitarity

Abstract

Leading (large) logarithms in non-renormalizable theories have been investigated in the recent past. Besides some general considerations, explicit results for the expansion coefficients (in terms of leading logarithms) of partial wave amplitudes and of scalar and vector form factors have been given. Analyticity and unitarity constraints haven been used to obtain the expansion coefficients of partial waves in massless theories, yielding form factors and the scalar two-point function to five-loop order in the O(4)/O(3) model. Later, the all order solutions for the partial waves in any O(N+1)/O(N) model were found. Also, results up to four-loop order exist for massive theories. Here we extend the implications of analyticity and unitarity constraints on the leading logarithms to arbitrary loop order in massless theories. We explicitly obtain the scalar and vector form factors as well as to the scalar two-point function in any O(N) and SU(N) type models. We present relations between the expansion coefficients of these quantities and those of of the relevant partial waves. Our work offers a consistency check on the published results in the O(N) models for form factors, and new results for the scalar two-point function. For the SU(N) type models, we use the known expansion coefficients for partial waves to obtain those for scalar and vector form factors as well as for the scalar two-point function. Our results for the form factor offer a check for the known and future results for massive O(N) and SU(N) type models when the massless limit is taken. Mathematica notebooks which can be used to calculate the expansion coefficients are provided as ancillary files.Large logarithms are a trademark of radiative loop calculations and often signal substantial modifications of tree level results. It is well known that in renormalizable theories, the so called leading logs (lowest power of the coupling constant) can be conveniently summed up by renormalization group techniques and the result of a simple one-loop calculation. Indeed, this often leads to large corrections and a substantial reduction of the (artificial) scale dependence. The next-to-leading logarithms can be treated based on two-loop calculations; and similarly for higher loop orders.It was pointed out by Weinberg [1] that the coefficients of the leading logarithms in effective (nonrenormalizable) field theories at two-loop order could also be obtained from a one-loop calculation. An early application was in demonstrating how to derive these leading logs coefficients from renormalization group (RG) equations for effective field theories by Colangelo [2]. Even earlier, Kazakov [3] argued that renormalization group techniques could be used to relate higher order leading logs to a one-loop calculation in a general setting for non-renormalizable Lagrangians, following notions put forward in ref.[4] in a different context. Soon afterwards, the double chiral logs of the chiral Lagranigan were evaluated along these lines by Bijnens, Colangelo and Ecker [5].Encouraged by these...

Chiral Perturbation Theory at NNNLO

“…Here the chiral logarithm is ii can be determined without knowing the higher order Lagrangians and are known up to six-loop order [29]. For massless O(N) and SU(N) models, the leading logarithms are known to any order [30]. That the leading logarithms can be calculated from one-loop diagrams only is true for any non-renormalizable theory [31].…”

Chiral Perturbation Theory at NNNLO

“…The leading logarithmic coefficients a M,F ii can be determined without knowing the higher order Lagrangians and are known up to six-loop order [29]. For massless O(N) and SU(N) models, the leading logarithms are known to any order [30]. Since the NNNLO Lagrangian only contributes at tree level, the NNNLO LECs cannot multiply a chiral logarithm.…”

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

“…refs. [3][4][5][6]. Moreover, the function Ω(z) may contain the information on the non-perturbative spectrum of masses of a non-renormalizable EFT.…”

“…We also note that using the technique of refs. [4], [6] one can compute the LL-approximation for various form factors as well as the correlation functions of two currents in the bi-quartic theory (1.6) in terms of the LL-amplitude (2.20).…”