Zero modes, beta functions and IR/UV interplay in higher-loop QED

Abstract

We analyze the relation between the short-distance behavior of quantum field theory and the strong-field limit of the background field formalism, for QED effective Lagrangians in self-dual backgrounds, at both one and two loop. The self-duality of the background leads to zero modes in the case of spinor QED, and these zero modes must be taken into account before comparing the perturbative β function coefficients and the coefficients of the strong-field limit of the effective Lagrangian. At one-loop this is familiar from instanton physics, but we find that at two-loop the role of the zero modes, and the interplay between IR and UV effects in the renormalization, is quite different. Our analysis is motivated in part by the remarkable simplicity of the two-loop QED effective Lagrangians for a self-dual constant background, and we also present here a new independent derivation of these two-loop results.

An addendum to the Heisenberg-Euler effective action beyond one loop

“…In addition to the well-known leading-log terms [4,65,67,72], eqs. (3.6) and (3.7) also account for the strongly suppressed contribution ∼G 2 which is of relevance for the photon are of the same structure and only differ in the specific numerical coefficients.…”

Heisenberg–euler Effective Lagrangians: Basics and Extensions

“…In this section I review the general argument [8,86,88,90,[131][132][133]144] relating the strong-field asymptotic behavior of the effective Lagrangian to the perturbative β function. I present the argument for QED, but it is more general.…”

“…This has the consequence that simply doing the Dirac traces in the spinor two loop effective Lagrangian (4.1), one finds that it can be written as the sum of two terms involving matrix elements of the scalar propagator, and moreover these are the same two matrix elements of the scalar propagator that appear in the scalar QED effective Lagrangian, but with different numerical coefficients [144] : This structure explains why the two loop answers (6.2) and (6.3) for spinor and scalar QED have such a similar form, involving just two terms with different numerical coefficients.…”

“…However, for spinor QED comparing with (5.13) we see that the correspondence appears to fail. The reason is that in a self-dual background spinor QED has zero modes, and these preclude the massless limit on which is based the argument in Section 5.1 relating the β-function and the strong field limit of L. This argument must be modified to account for the zero modes [144]. At one loop the mismatch is due to the infrared divergence of the unrenormalized L spinor , as is familiar from instanton physics [108,109].…”

Running of theU(1)coupling in the dark sector

“…The 2-loop beta function of U (1) d , with n F fermions and n S scalars of unit charge, is given by (see, for example, Refs. [21][22][23][24])…”