Superfield approach to symmetries for matter fields in Abelian gauge theories

Abstract

The derivation of the nilpotent Becchi-Rouet-Stora-Tyutin (BRST)-and anti-BRST symmetries for the matter fields, present in any arbitrary interacting gauge theory, has been a long-standing problem in the framework of superfield approach to BRST formalism. These nilpotent (anti-)BRST symmetries for the Dirac fields are derived in the superfield formulation for the interacting Abelian gauge theory in four (3 + 1)-dimensions (4D) of spacetime. The same type of symmetries are deduced for the 4D complex scalar fields having a gauge invariant interaction with the U(1) gauge field. The above interacting theories are considered on a six (4 + 2)-dimensional supermanifold parametrized by four even spacetime coordinates and a couple of odd elements of the Grassmann algebra. The invariance of the conserved matter (super)currents and the horizontality condition on the (super)manifolds play very important roles in the above derivations. The geometrical origin and interpretation for all the above off-shell nilpotent symmetries are provided in the framework of superfield formalism. .in † The set (d, δ, ∆) of operators, defined on a compact manifold without a boundary, is called the set of de Rham cohomological operators where δ = ± * d * , d = dx µ ∂ µ , ∆ = (d + δ) 2 are called the (co-)exterior derivatives ((δ)d) and the Laplacian operator (∆) respectively. Here * is the Hodge duality operation on the manifold. These operators obey an algebra: d 2 = δ 2 = 0, ∆ = {d, δ}, [δ, ∆] = 0, [d, ∆] = 0 showing that the Laplacian operator ∆ is the Casimir operator for the whole algebra (see, eg, [17,18] for details).

Augmented superfield approach to exact nilpotent symmetries for matter fields in non-Abelian theory

“…In our earlier works [8][9][10][11][12][13], we have consistency extended the horizontality condition by requiring the equality of the supersymmetric versions of the conserved currents/charges with the ordinary local conserved corrents/charges. In one of our recent works [13], in addition to the horizontality condition, any conserved quantities are required to be invariant on the supermanifold.…”

“…In a recent set of papers [8][9][10][11][12][13], the above horizontality condition has been consistently extended 4 by requiring the equality of (i) the conserved currents/charges, and (ii) the gauge (i.e. BRST) invariant quantities that 3 For the 1-form non-Abelian gauge theory, the super curvature 2-formF (2) =dà (1) + ià (1) ∧ A (1) is equated with the ordinary 2-form F (2) = dA (1) + iA (1) ∧ A (1) due to the horizontality condition that leads to the derivation of the nilpotent (anti-) BRST symmetry transformations for the gauge and (anti-)ghost fields (see, e.g., [3] for details).…”

Nilpotent Symmetries for a Free Relativistic Particle in Augmented Superfield Formalism

“…All the above attempts [6][7][8][9][10][11][12][13][14][15][16][17][18][19], however, have not yet been able to shed any light on the nilpotent symmetries that exist for the matter fields of an interacting gauge theory. Thus, the results of the above approaches [6][7][8][9][10][11][12][13][14][15][16][17][18][19] are still partial as far as the derivation of all the symmetry transformations are concerned.Recently, in a set of papers [20][21][22], the restriction due to the horizontality condition has been augmented with the requirement of the invariance of matter (super)currents on the (super)manifolds † . The latter restriction produces the nilpotent (anti-)BRST symmetry transformations for the matter fields of a given interacting gauge theory.…”

“…The salient features of these requirements are (i) there is a beautiful consistency and complementarity between the nilpotent transformations generated by the horizontality restriction and the requirement of conserved matter (super)currents on the (super)manifolds. The purpose of the present paper is to derive the nilpotent (anti-)BRST transformations for all the fields present in the description of a free scalar relativistic particle (moving on a world-line) in the framework of augmented superfield formulation [20][21][22]. This study is essential primarily on three counts.…”

“…Of course, the latter theories were considered on the four (2 + 2)-dimensional-and six (4 + 2)-dimensional supermanifolds, respectively. Second, to generalize our earlier works [20][21][22] (which were connected only with the gauge symmetries) to the case where the gauge symmetries as well as the reparametrization symmetries co-exist in the theory. Finally, to tap the potential and power of the above restrictions for the case of a new physical system defined on a new three (1+2)-dimensional supermanifold which is somewhat different from our earlier considerations of the interacting (non-)Abelian gauge theories on four (2 + 2)-and six (4 + 2)-dimensional supermanifolds.…”