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?(? = 0, 1, 2, 3) and a pair of Grassmannian variables ? and ¯? (with ?2 = ¯?2 = 0, ?¯? + ¯?? = 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) (? and/or ¯?) 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. MSC 2010: 81T80, 81T13, 58J70.
Highlights
The usual superfield approach [1,2,3,4,6,19,20,21] to BRST formalism has been very successfully applied to the case of 4DAbelian 1-form (A(1) = dxμAμ) gauge theories
We find that the gauge invariant restriction (GIR) on the matter (Dirac) superfields (defined on the above (4, 2)-dimensional supermanifold) enables us to derive the exact nilpotentBRST symmetry transformations for the matter (Dirac) fields which can never be obtained by exploiting the horizontality condition1 (HC) alone
As in our earlier work [15] on the 4D freeAbelian 1-form gauge theories, we find that the Grassmannian independence of the super Lagrangian densities is a sure guarantee that the corresponding 4D Lagrangian density would respect the nilpotentBRST symmetry invariance
Summary
This restriction, does not shed any light on the derivation of the nilpotent (anti-)BRST symmetry transformations that are associated with the matter (e.g. Dirac, complex scalar, etc.) fields of an interacting 4D (non-)Abelian 1-form gauge theory. In a set of a couple of papers [13,14], we have been able to generalize the HC by a gauge invariant restriction (GIR) on the matter superfields (defined on the above (4, 2)-dimensional supermanifold) which enables us to derive the nilpotent (anti-)BRST symmetry transformations together for the gauge, (anti-)ghost and matter fields of a 4D interacting (non-)Abelian gauge theory in one stroke.
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