Superfield approach to symmetry invariance in quantum electrodynamics with complex scalar fields

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We show that the Grassmannian independence of the super-Lagrangian density, expressed in terms of the superfields defined on a (4,2)-dimensional supermanifold, is a clear-cut proof for the Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST invariance of the corresponding four (3+1)-dimensional (4D) Lagrangian density that describes the interaction between the U(1) gauge field and the charged complex scalar fields. The above 4D field theoretical model is considered on a (4,2)-dimensional supermanifold parametrized by the ordinary four space-time variables x µ (with µ = 0, 1, 2, 3) and a pair of Grassmannian variables θ and $$ \bar \theta $$ (with θ 2 = $$ \bar \theta $$ 2 = 0, θ $$ \bar \theta $$ + $$ \bar \theta $$ θ = 0). Geometrically, the (anti-)BRST invariance is encoded in the translation of the super-Lagrangian density along the Grassmannian directions of the above supermanifold such that the outcome of this shift operation is zero.