On How QCD Gauge Invariance Gets Realized in the Context of Effective Locality

Received Aug 23, 2017; Accepted Mar 29, 2018

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP 3 .

1. Introduction

In some recent articles [1–5], a property, which bears on the nonperturbative fermionic Green’s functions of QCD, has been put forth under the name of effective locality (EL). This property can be summarized as follows.

For any fermionic $2n$ -point Green’s functions and related amplitudes, the full gauge-fixed sum of cubic and quartic gluonic interactions, with fermionic loops included, results in a local contact-type interaction. This local interaction is mediated by a tensorial field which is antisymmetric both in Lorentz and color indices. Moreover, the resulting sum appears to be fully gauge-fixing independent, that is, gauge-invariant.

This is an unexpected result because integrations of elementary degrees of freedom ordinarily result in highly nonlocal structures. The “effective locality” denomination, which sounds like an oxymoron, accounts for this unusual circumstance. It is worth pointing out that in the pure euclidean Yang Mills case and up to the first nontrivial orders of a semiclassical expansion, effective locality was observed as a welcome property in an attempt to construct a formulation dual to the original Yang Mills theory [6–8].

Now, apart from a supersymmetric extension, QCD is not known for possessing any dual formulation and the full EL functional expressions certainly attest to this difficulty. It remains that, like in the pure Yang Mills situation of [6, 7], the EL property may allow one to learn something about the nonperturbative regime of QCD and this from first principles.

In the next section a comparison of the QED and QCD Green’s functions generating functional is given, while Section 3 displays the property of effective locality in the simpler situation of eikonal and quenching approximations where it was first noticed and then in full generality. Eventually, Section 4 presents some concluding remarks.

2. Contrasting Generating Functionals

In QED it has been known for quite a long time that manifest covariance and manifest gauge invariance are competing aspects of a generating functional construction. For short, starting from a photonic...

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