Nilpotent Symmetries for QED in Superfield Formalism

In the framework of superfield approach, we derive the local, covariant, continuous and nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations on the U(1) gauge field $(A_\mu)$ and the (anti-)ghost fields $((\bar C)C)$ of the Lagrangian density of the two $(1 + 1)$-dimensional QED by exploiting the (dual-)horizontality conditions defined on the four $(2 + 2)$-dimensional supermanifold. The long-standing problem of the derivation of the above symmetry transformations for the matter (Dirac) fields $(\bar\psi, \psi)$ in the framework of superfield formulation is resolved by a new set of restrictions on the $(2 + 2)$-dimensional supermanifold. These new physically interesting restrictions on the supermanifold owe their origin to the invariance of conserved currents of the theory. The geometrical interpretation for all the above transformations is provided in the framework of superfield formalism.


Introduction
One of the most attractive and intuitive geometrical approaches to gain an insight into the physics and mathematics behind the Becchi-Rouet-Stora-Tyutin (BRST) formalism is the superfield formulation [1][2][3][4][5][6]. In this scheme, a D-dimensional gauge theory (endowed with the first-class constraints in the language of Dirac [7,8]) is considered on a (D + 2)dimensional supermanifold parameterized by D-number of spacetime (even) co-ordinates x µ (µ = 0, 1, 2....D − 1) and a couple of (odd) Grassmannian variables θ andθ (with θ 2 =θ 2 = 0, θθ +θθ = 0). In general, the (p + 1)-form super curvatureF constructed from the super exterior derivatived (withd 2 = 0) and the super p-form connectionÃ of a p-form (p = 1, 2, 3....) gauge theory through the Maurer-Cartan equation (i.e.dÃ+Ã∧Ã = F ) is restricted to be flat along the Grassmannian directions of the (D + 2)-dimensional supermanifold due to the so-called horizontality condition † . Mathematically, this condition impliesF = F where F = dA + A ∧ A is the (p + 1)-form curvature defined on the ordinary D-dimensional spacetime manifold. The horizontality condition, where only one of the three de Rham cohomological operators ‡ is exploited, leads to the derivation of the nilpotent (anti-)BRST symmetry transformations on the gauge-and (anti-)ghost fields of the (anti-)BRST invariant Lagrangian density of a given D-dimensional p-form gauge theory.
In a recent set of papers [15][16][17], all the three (super) de Rham cohomological operators have been exploited, in the generalized versions of the horizontality condition, to derive the (anti-)BRST, (anti-)co-BRST and a bosonic symmetry (which is equal to the anticommutator(s) of the (anti-)BRST and (anti-)co-BRST symmetries) transformations for the free one-form Abelian gauge theory in two-dimensions (2D) of spacetime. For the derivation of the above nilpotent symmetries, the super (co-)exterior derivatives (δ)d have been exploited in the (dual-)horizontality conditions on the four (2 + 2)-dimensional supermanifold. The Lagrangian formulation of the above symmetries has also been carried out in a set of papers [18][19][20] where it has been shown that this theory presents (i) an example of a tractable field theoretical model for the Hodge theory, and (ii) an example of a new class of topological field theory where the Lagrangian density turns out to be like Witten type topological field theory but the symmetries of the theory are that of Schwarz type. Similar symmetries for the self-interacting 2D non-Abelian gauge theory have also been obtained in the framework of 2D Lagrangian formalism [21] as well as in the four (2 + 2)-dimensional superfield formulation [22]. Furthermore, the above type of symmetries have been shown to exist for the 4D 2-form free Abelian gauge theory in the Lagrangian formalism [23,24]. † Nakanishi and Ojima call it the "soul-flatness" condition which amounts to setting the Grassmannian components of a (p + 1)-form super curvature tensor (for a p-form gauge theory) equal to zero [9].
One of the most difficult and long-standing problems in the realm of superfield approach to BRST formalism has been to derive the (anti-)BRST symmetry transformations on the matter (e.g. Dirac, complex scalar etc.) fields for a given interacting p-form gauge theory. The purpose of the present paper is to demonstrate that an additional set of restrictions, besides the (dual-)horizontality conditions w.r.t super (co-)exterior derivatives (δ)d, are required on the (D + 2)-dimensional supermanifold for the derivation of the (anti-)BRST and (anti-)co-BRST transformations on the matter fields. For this purpose, as a prototype field theoretical model, we choose the two-dimensional interacting U(1) gauge theory (i.e. QED § ) and show that the (anti-)BRST and (anti-)co-BRST symmetry transformations on the matter fields, derived in our earlier works [25,26] in the framework of Lagrangian formalism, can be obtained by exploiting the invariance of the conserved (super) currents constructed by the (super) Dirac fields of the theory on a (super)manifold. In a more precise and sophisticated language, the equality of the supercurrentsJ µ (x, θ,θ) andJ (5) µ (x, θ,θ) constructed by the superfields (cf. eqns. (4.2) and (4.9) below) on the four (2 + 2)dimensional supermanifold with the conserved currents J µ (x) = (ψγ µ ψ)(x) and J (5) µ ((x) = (ψγ µ γ 5 ψ)(x) constructed by the ordinary Dirac fields on the 2D ordinary manifold leads to the derivation of the (anti-)BRST and (anti-)co-BRST symmetry transformations on the Dirac fields, respectively. The above equality emerges automatically and is not imposed by hand. We also provide, in the present paper, the geometrical interpretations for the nilpotent symmetries and the corresponding nilpotent generators.
The outline of our present paper is as follows. In Section 2, we recapitulate the salient features of our earlier works [25,26] on the existence of the off-shell nilpotent (anti-)BRSTand (anti-)co-BRST symmetries in the Lagrangian formulation for the interacting U(1) gauge theory in two-dimensions of spacetime. Section 3 is devoted to the derivation of the above symmetry transformations on the gauge field A µ and the (anti-)ghost fields (C)C by exploiting the (dual-)horizontality conditions on the four (2 + 2)-dimensional supermanifold [17,22]. This exercise is carried out for the sake of this paper to be self-contained. The central of our paper is Section 4 where we derive the above symmetry transformations for the matter (Dirac) fields by invoking the invariance of the conserved currents as the physical restriction on the supermanifold. Finally, we make some concluding remarks and pinpoint a few future directions in Section 5 for further investigations.

Preliminary: (anti-)BRST-and (anti-)co-BRST symmetries
To recapitulate the bare essentials of our earlier works [25,26] on QED in two-dimensions, let us begin with the (anti-)BRST invariant Lagrangian density L b for the interacting two (1 + 1)-dimensional (2D) U(1) gauge theory in the Feynman gauge [27][28][29] As is evident, the latter is constructed by the application of the exterior derivative d = dx µ ∂ µ (with d 2 = 0) on the 1-form A = dx µ A µ (which defines the vector potential A µ ). It will be noted that in 2D, F µν has only the electric component (i.e. F 01 = E) and there is no magnetic component associated with it. The gauge-fixing term (∂ · A) is derived through the operation of the co-exterior derivative δ (with δ = − * d * , where * is the Hodge duality operation. The fermionic Dirac fields (ψ,ψ), with the mass m and charge e, couple to the U(1) gauge field A µ (i.e. −eψγ µ A µ ψ) through the conserved current J µ =ψγ µ ψ. The anticommuting (CC +CC = 0, C 2 =C 2 = 0, Cψ + ψC = 0 etc.) (anti-)ghost fields (C)C are required to maintain the unitarity and "quantum" gauge (i.e. BRST) invariance together at any arbitrary order of perturbation theory . The kinetic energy term ( 1 2 E 2 ) of (2.1) can be linearized by invoking an auxiliary field B which is the analogue of the Nakanishi-Lautrup auxiliary field B that is required to linearize the gauge-fixing term − 1 2 (∂ · A) 2 in (2.1). The above Lagrangian density (2.2) respects the following off-shell nilpotent (s 2 The noteworthy points, at this stage, are (i) under the (anti-)BRST and (anti-)co-BRST transformations, it is the kinetic energy term (more precisely E itself) and the gauge-fixing term (more accurately (∂ · A) itself) that remain invariant, respectively. (ii) The electric ¶ We adopt here the conventions and notations such that the 2D flat Minkowski metric is: The Dirac γ matrices in two-dimensions are chosen to be: Here σ's are the usual 2 × 2 Pauli matrices and the Greek indices: µ, ν, ρ... = 0, 1 correspond to the spacetime directions on the manifold. The full strength of the (anti-)ghost fields turns up in the discussion of the unitarity and gauge invariance for the perturbative computations in the realm of non-Abelian gauge theory where the loop diagrams of the gauge (gluon) fields play a very important role (see, e.g., [30] for details). * * We adopt here the notations and conventions followed in [29]. In fact, in its full glory, a nilpotent

Nilpotent symmetries for the gauge-and (anti-)ghost fields
We begin here with a four (2+2)-dimensional supermanifold parametrized by the superspace coordinates Z M = (x µ , θ,θ) where x µ (µ = 0, 1) are a couple of even (bosonic) spacetime coordinates and θ andθ are the two odd (Grassmannian) coordinates (with θ 2 =θ 2 = 0, θθ+ θθ = 0). On this supermanifold, one can define a supervector superfieldÃ M (i.e.Ã M = (B µ (x, θ,θ), Φ(x, θ,θ),Φ(x, θ,θ)) with B µ , Φ,Φ as the component multiplet superfields [4]. The superfields B µ , Φ,Φ can be expanded in terms of the basic fields (A µ , C,C) and auxiliary fields (B, B) of (2.2) and some extra secondary fields as follows It is straightforward to note that the local fields R µ (x),R µ (x), C(x),C(x), s(x),s(x) are fermionic (anti-commuting) in nature and the bosonic (commuting) local fields in (3.1) are: . It is unequivocally clear that, in the above expansion, the bosonic-and fermionic degrees of freedom match. This requirement is essential for the validity and sanctity of any arbitrary supersymmetric theory in the superfield formulation. In fact, all the secondary fields will be expressed in terms of basic fields due to the restrictions emerging from the application of horizontality condition (i.e.F = F ), namely; where the super exterior derivatived and the connection super one-formÃ are defined as

(3.3)
In physical language, this requirement implies that the physical field E, derived from the curvature term F µν , does not get any contribution from the Grassmannian variables. In other words, the physical electric field E for 2D QED remains intact in the superfield formulation. Mathematically, the condition (3.2) implies the "flatness" of all the components of the super curvature (2-form) tensorF M N that are directed along the θ and/orθ directions of the supermanifold. To this end in mind, first we expanddÃ as Ultimately, the application of soul-flatness (horizontality) condition (dÃ = dA) yields [17] The insertion of all the above values in the expansion (3.1) leads to the derivation of the (anti-)BRST symmetries for the gauge-and (anti-)ghost fields of the Abelian gauge theory. In addition, this exercise provides the physical interpretation for the (anti-)BRST charges Q (a)b as the generators (cf. eqn. (2.5)) of translations (i.e. Limθ →0 (∂/∂θ), Lim θ→0 (∂/∂θ)) along the Grassmannian directions of the supermanifold. Both these observations can be succinctly expressed, in a combined way, by re-writing the super expansion (3.1) as Φ (x, θ,θ) = C(x) + θ (s ab C(x)) +θ (s b C(x)) + θθ (s b s ab C(x)), Φ (x, θ,θ) =C(x) + θ (s abC (x)) +θ (s bC (x)) + θθ (s b s abC (x)).
detail discussions). With the above inputs, it can be checked that the superscalar superfield δÃ = − ⋆d ⋆Ã, turns out to bẽ δÃ = (∂ · B + ∂ θΦ + ∂θΦ) + s θθ (∂ θ Φ) + sθθ (∂θΦ). (3.8) Ultimately, the dual-horizontality restrictionδÃ = δA produces the following restrictions on the component superfields (see, e.g., [31] for details) where, as is evident, the r.h.s. of the last entry in the above equation is due to δA = (∂ · A). Exploiting the super expansions of (3.1), we obtain It is clear from the above that we cannot get a unique solution for R µ ,R µ and S µ in terms of the basic fields of the Lagrangian density (2.2). This is why there are non-local and non-covariant solutions for these in the case of QED in 4D (see, e.g., [31]). It is interesting, however, to point out that for 2D QED, we have the local and covariant solutions as With the above insertions, it can be easily checked that the expansion (3.1) becomes Φ (x, θ,θ) =C(x) + θ (s adC (x)) +θ (s dC (x)) + θθ (s d s adC (x)). (3.12) Thus, the geometrical interpretation for the generators Q (a)d of the (anti-)co-BRST symmetries is identical to that of the (anti-)BRST charges Q (a)b . However, there is a clear-cut distinction between Q (a)d and Q (a)b when the transformations on the (anti-)ghost fields are considered. For instance, the BRST charge Q b generates a symmetry transformation such that the superfieldΦ(x, θ,θ) becomes anti-chiral and the superfield Φ(x, θ,θ) becomes an ordinary local field C(x). In contrast, the co-BRST charge Q d generates a symmetry transformation under which just the opposite of the above happens. Similarly, the distinction between Q ab and Q ad can be argued where one of the above superfields becomes chiral.

Nilpotent symmetries for the Dirac fields
In contrast to the (dual-)horizontality conditions that rely on the (super-)co-exterior derivatives (δ)δ, the (super-)exterior derivative (d)d and the (super-)one-form (Ã)A for the derivation of the (anti-)BRST and (anti-)co-BRST symmetry transformations on the gauge field A µ and the (anti-)ghost fields (C)C, the corresponding nilpotent symmetries for the matter (Dirac) fields (ψ,ψ) are obtained due to the invariance of the conserved currents of the theory. To corroborate this assertion, first of all, we start off with the super expansion of the superfields (Ψ,Ψ)(x, θ,θ)), corresponding to the ordinary Dirac fields (ψ,ψ)(x), as It is clear and evident that, in the limit (θ,θ) → 0, we get back the Dirac fields (ψ,ψ) of the Lagrangian density (2.1). Furthermore, the number of bosonic fields (b 1 ,b 1 , b 2 ,b 2 ) match with the fermionic fields (ψ,ψ, f,f ) so that the above expansion is consistent with the basic tenets of supersymmetry. Now one can construct the supercurrentJ µ (x, θ,θ) from the above superfields with the following general super expansioñ where the above components (i.e.K µ , K µ , L µ , J µ ), along the Grassmannian directions θ andθ as well as the bosonic directions θθ and identity1 of the supermanifold, can be expressed in terms of the components of the basic super expansions (4.1), as To be consistent with our earlier observation that the (co-)BRST transformations (s (d)b ) are equivalent to the translations (i.e. Lim θ→0 (∂/∂θ)) along theθ-direction and the anti-BRST (s ab ) and anti-co-BRST (s ad ) transformations are equivalent to the translations (i.e. Limθ →0 (∂/∂θ)) along the θ-direction of the supermanifold, it is straightforward to re-express the expansion in (4.2) as follows It can be checked explicitly that, under the (anti-)BRST transformations (2.3), the conserved current J µ (x) remains invariant (i.e. s b J µ (x) = s ab J µ (x) = 0). This statement, with the help of (4.2) and (4.3), can be mathematically expressed as One of the possible solutions of the above restrictions, in terms of the components of the basic expansions in (4.1) and the basic fields of the Lagrangian density (2.2), is (4.6) At the moment, it appears to us that the above solutions are the unique solutions to all the restrictions in (4.5) † † . Ultimately, the restriction that emerges on the (2 + 2)-dimensional supermanifold isJ µ (x, θ,θ) = J µ (x). (4.7) † † Let us focus on b 1 γ µ ψ =ψγ µ b 2 . It is evident that the pair of bosonic components b 1 and b 2 should be proportional to the pair of fermionic fieldsψ and ψ, respectively. To make the latter pair bosonic in nature, we have to include the ghost field C of the Lagrangian density (2.2) to obtain: b 1 ∼ψC, b 2 ∼ Cψ. Rest of the choices in (4.6) follow exactly similar kind of arguments.

Conclusions
In the present investigation, we set out to derive the off-shell nilpotent (anti-)BRST and (anti-)co-BRST symmetries for the matter (Dirac) fields in the framework of geometrical superfield approach to BRST formalism We chose the two-dimensional interacting U(1) gauge theory (i.e. QED) for our discussion primarily for two reasons. First and foremost, this theory provides one of the simplest gauge theory and a unique interacting field theoretical model for the Hodge theory. Second, the Lagrangian density (2.2) of this theory is endowed with a local, covariant, continuous and nilpotent (anti-)co-BRST symmetries which is not the case for the four dimensional QED where the (anti-)co-BRST transformations are nonlocal and non-covariant (see, e.g., [31] for details). We have been able to derive the off-shell nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations on the Dirac fields by invoking a couple of restrictions (i.e.J µ (x, θ,θ) = J µ (x) andJ (5) µ (x, θ,θ) = J (5) µ (x)) on the (2 + 2)-dimensional supermanifold. In contrast to the (dual-)horizontality conditions, these restrictions are not imposed by hand from the outside. Rather, they appear very naturally because of the fact that s (a)b J µ (x) = 0, s (a)d J (5) µ (x) = 0 in the super expansion of the super currentsJ µ (x, θ,θ) andJ (5) µ (x, θ,θ) (cf. eqns. (4.4) and (4.9)). Physically, these conditions imply nothing but the conservation of the electric charge for the massive Dirac fields and the conservation of the spin (i.e. helicity in 2D spacetime) for the massless Dirac fields, respectively. These conservation laws persist even in the superfield formulation of the theory. This is why, automatically, we get the conditionsJ µ (x, θ,θ) = J µ (x) andJ (5) µ (x, θ,θ) = J (5) µ (x). We would like to comment that our method of derivation of the (anti-)BRST transformations for the matter fields, in the framework of the superfield formalism, can be generalized to the physical 4D Abelian as well as non-Abelian gauge theories (see, e.g., [31,32] for transformations). It would be also interesting to obtain the on-shell nilpotent version of the above symmetries in the framework of the superfield formulation. These are some of the open problems which are under investigation and our results would be reported elsewhere [33].