Symmetry invariance, anticommutativity and nilpotency in BRST approach to QED: superfield formalism

TL;DR

This paper offers a geometric superfield interpretation of BRST and anti-BRST symmetries in 4D U(1) gauge theory with matter, demonstrating invariance, anticommutativity, and nilpotency through supermanifold translations.

Contribution

It provides a novel geometric superfield framework that explains BRST invariance, anticommutativity, and nilpotency in an interacting U(1) gauge theory with matter fields.

Findings

BRST invariance corresponds to superfield translations along Grassmann directions

Anticommutativity and nilpotency of BRST charges are derived from superfield formalism

Superfield approach simplifies understanding of symmetry properties in gauge theories

Abstract

We provide the geometrical interpretation for the Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry invariance of the Lagrangian density of a four (3 + 1)-dimensional (4D) interacting U(1) gauge theory within the framework of superfield approach to BRST formalism. This interacting theory, where there is an explicit coupling between the U(1) gauge field and matter (Dirac) fields, is considered on a (4, 2)-dimensional supermanifold parametrized by the four spacetime variables x^\mu (\mu = 0, 1, 2, 3) and a pair of Grassmannian variables \theta and \bar\theta (with \theta^2 = \bar \theta^2 = 0, \theta \bar\theta + \bar \theta \theta = 0$). We express the Lagrangian density and (anti-)BRST charges in the language of the superfields and show that (i) the (anti-)BRST invariance of the 4D Lagrangian density is equivalent to the translation of the super Lagrangian density along the Grassmannian direction(s) (\theta and/or \bar\theta) of the (4, 2)-dimensional supermanifold such that the outcome of the above translation(s) is zero, and (ii) the anticommutativity and nilpotency of the (anti-)BRST charges are the automatic consequences of our superfield formulation.