Superspace Unitary Operator in Superfield Approach to Non-Abelian Gauge Theory with Dirac Fields

Abstract

Within the framework of augmented version of the superfield approach to Becchi-Rouet-Stora-Tyutin (BRST) formalism, we derive the superspace unitary operator (and its Hermitian conjugate) in the context of four (3 + 1)-dimensional (4D) interacting non-Abelian 1-form gauge theory with Dirac fields. The ordinary 4D non-Abelian theory, defined on the flat 4D Minkowski spacetime manifold, is generalized onto a (4, 2)-dimensional supermanifold which is parameterized by the spacetime bosonic coordinatesxμ(withμ=0,1,2,3) and a pair of Grassmannian variables (θ,θ-) which satisfy the standard relationships:θ2=θ-2=0and θθ-+θ-θ=0. Various consequences of the application of the above superspace (SUSP) unitary operator (and its Hermitian conjugate) are discussed. In particular, we obtain the results of the application of horizontality condition (HC) and gauge-invariant restriction (GIR) in the language of the above SUSP operators. One of the novel results of our present investigation is the derivation of explicit expressions for the SUSP unitary operator (and its Hermitian conjugate) without imposing any Hermitian conjugation condition fromoutsideon the parameters and (super)fields of the supersymmetric version of our 4D interacting non-Abelian 1-form theory with Dirac fields.

BRST and Superfield Formalism—A Review

“…The superfield formalism applies well to this enlarged symmetry, provided we introduce another anticommuting coordinate,θ:θ 2 = 0, ϑθ +θϑ = 0. Here, we do not repeat the full derivation as in the previous case but simply introduce the supergauge transformation [15,[40][41][42][43]:…”

BRST and superfield formalism. A review

“…The superfield formalism applies well to this enlarged symmetry provided we introduce another anticommuting coordinate θ: θ2 = 0, ϑ θ + θϑ = 0. Here we do not repeat the full derivation as in the previous case, but simply introduce the supergauge transformation, [10,23],…”

Superunitary operator and BRST transformations of a non-Abelian two-form

“…To include interacting systems where the gauge field couples to matter fields, the superfield approach has been consistently generalised to obtain the BRST transformations for the matter fields as well, which is called the augmented superfield approach [21][22][23][24], where, in addition to the horizontality condition, some gauge-invariant restrictions are also exploited. The mapping of ordinary fields on the Minkowski spacetime to the superfields on the superspace can also be carried out via a superspace unitary operator [13][14][15]25,26], or superunitary operator for short. The superunitary operator upgrades the fields and gauge connections to their superspace counterparts in the same fashion as the unitary gauge operator maps the fields and gauge connections to their gauge-transformed counterparts.…”

“…BRST transformations and superunitary operator approach for 1-form gauge theories. -In this section we present a brief review of the BRST transformations, the augmented superfield approach [1,2,21] and the superunitary operator approach [13][14][15]25,26], which also helps to set up notations. We outline the general procedure and state important results without going into detailed derivations.…”

“…These transformations are anti-commuting, s b s ab +s ab s b = 0, and off-shell nilpotent, s 2 b = s 2 ab = 0. One way to obtain the symmetry transformations (2) is the augmented superfield approach [1,2,[13][14][15]21,25,26]. This approach relies on the extension of the ordinary spacetime to a superspace which has a space of Grassmann-odd coordinates, θ and θ (satisfying θ 2 = θ2 = 0, θ θ + θθ = 0), attached to each and every point of ordinary spapcetime.…”