The a-function for gauge theories

Abstract

Abstract:The a-function is a proposed quantity defined for quantum field theories which has a monotonic behaviour along renormalisation group flows, being related to the β-functions via a gradient flow equation involving a positive definite metric. We construct the a-function at four-loop order for a general gauge theory with fermions and scalars, using only one and two loop β-functions; we are then able to provide a stringent consistency check on the general three-loop gauge β-function. In the case of an N = 1 supersymmetric gauge theory, we present a general condition on the chiral field anomalous dimension which guarantees an exact all-orders expression for the a-function; and we verify this up to fifth order (corresponding to the three-loop anomalous dimension).

Constraints on 3- and 4-loop β-functions in a general four-dimensional Quantum Field Theory

“…Up to two loops, it is possible to show directly that such a decomposition of W I and Q I exists, and that T IJ is indeed symmetric, but it is unclear how the program proceeds at higher loop orders. This is perhaps not surprising, as it has already been shown that the first off-diagonal contributions to T IJ in a general theory occur at three loops [6]. When deriving constraints on the β(B)-functions, imposing symmetry on T IJ would correspond to identifying the coefficients of each pair of tensors related by the interchange of open indices I and J, and setting them equal-in six-dimensional φ 3 theory at sufficiently high loop order, it has been shown that this does indeed force an additional CC, which (rather amazingly) turn out to be satisfied by the MS coefficients, and renormalizationscheme independent [21].…”

“…If the kinetic terms of the theory possess a global symmetry, then β I in (1.1) is replaced by a generalized function B I , which begins to differ at three loops [5], and is related to the possible existence of theories with limit cycles [20]. While the effects of this shift have been included for the simple case of six-dimensional φ 3 theory [21] and the more complicated four-dimensional scalar-fermion theory [5], equivalent analyses for general gauge theories have not yet reached the point where considering this shift is necessary [6]. All quantities in Osborn's equation are given as functions of the couplings and group structures of the theory.…”

“…At the risk of adding an additional layer of abstraction, this will help us keep our expressions more manageable later on. Following [6], we define a new multiplet Ψ i of Majorana spinors Ψ = ψ ψ † , taking advantage of the unitary equivalence between Weyl and Majorana fermions 14 . This allows us to assemble the Yukawa couplings and fermionic generators into larger matrices,…”

“…This is known as the "3-2-1" ordering, since the coefficients of the gauge β-function are related to the Yukawa coefficients one loop lower, and the scalar coefficients two loops lower. Previous attempts at investigating these constraints have been limited in different ways: some analyses of Osborn's equation have been done using the β-functions for a general scalar-fermion theory without gauge interactions [5], or a general theory with a simple gauge group [6]; another analysis included the three gauge groups of the Standard Model (SM), but considered only the top Yukawa and Higgs self-interaction couplings, and limited its investigations to constraints that do not involve higher-order corrections [7]. The power of Osborn's equation comes with the realization that one can investigate the associated constraints algebraically, without actually knowing the β-function coefficients.…”

N=1 Euler Anomaly from RG-dependent metric-Background

“…Ever since it has served as a fruitful laboratory for testing different techniques by rederiving the result. Examples include four-loop perturbation theory [4], the local renormalisation group (RG) [5] and employing superspace techniques assuming a gradient flow equation [6] conforming an older conjecture [7].…”