The asymmetry of the dimension 2 gluon condensate: the zero temperature case

TL;DR

This paper algebraically studies the renormalization of specific gluon condensate operators in Landau gauge, develops a formalism for the zero temperature effective potential, and sets the stage for finite temperature analysis related to electric-magnetic symmetry.

Contribution

It provides a renormalizable decomposition of the gluon condensate operators and develops a formalism for the effective potential applicable to finite temperature.

Findings

Confirmed <A^2> eq 0 at zero temperature

Established <A_ u A_ u - rac{ ext{trace}}{d} A^2> = 0, consistent with Lorentz symmetry

Derived three-loop renormalization group functions for the operators

Abstract

We provide an algebraic study of the local composite operators A_\mu A_\nu-\delta_{\mu\nu}/d A^2 and A^2, with d=4 the spacetime dimension. We prove that these are separately renormalizable to all orders in the Landau gauge. This corresponds to a renormalizable decomposition of the operator A_\mu A_\nu into its trace and traceless part. We present explicit results for the relevant renormalization group functions to three loop order, accompanied with various tests of these results. We then develop a formalism to determine the zero temperature effective potential for the corresponding condensates, and recover the already known result for <A^2> \neq 0, together with <A_\mu A_\nu-\delta_{\mu\nu}/d A^2>=0, a nontrivial check that the approach is consistent with Lorentz symmetry. The formalism is such that it is readily generalizable to the finite temperature case, which shall allow a future analytical study of the electric-magnetic symmetry of the <A^2> condensate, which received strong evidence from recent lattice simulations by Chernodub and Ilgenfritz, who related their results to 3 regions in the Yang-Mills phase diagram.