Modified Stückelberg Formalism: Free Massive Abelian 2-Form Theory in 4D

Abstract

We demonstrate that the celebrated Stückelberg formalism is modified in the case of a massive four (3 + 1)-dimensional (4D) Abelian 2-form theory due to the presence of a self-duality discrete symmetry in the theory. The latter symmetry entails upon the modified 4D massive Abelian 2-form gauge theory to become a massive model of Hodge theory within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism where there is the existence of a set of (anti-)co-BRST transformations corresponding to the usual nilpotent (anti-)BRST transformations. The latter exist in any arbitrary dimension of spacetime for the usual Stückelberg-modified massive Abelian 2-form gauge theory. The modification in the Stückelberg technique is backed by the precise mathematical arguments from the differential geometry where the exterior derivative and Hodge duality operator play the decisive roles. The modified version of the Stückelberg technique remains invariant under the discrete duality transformations which also establish a precise and deep connection between the off-shell nilpotent (anti-)BRST and (anti-)co-BRST transformations. We have clarified a simple trick of using the equations of motion to remove the higher derivative terms in the appropriate Lagrangian densities so that our 4D theory can become consistent.

1. Introduction

The basic concepts and ideas behind the subject of pure mathematics and their applications in the progress of theoretical physics have been intertwined together in a meaningful manner since the advent of physics as a specific branch of (the all-encompassing broad field of modern-day) science which incorporates, into its ever-widening folds, other branches, as well. In particular, the recent developments in the domain of theoretical high-energy physics owe a great deal to some of the key ideas and concepts behind pure mathematics. For instance, we know that the concepts of differential geometry have found decisive applications in the realm of theoretical research activities related to the specific topics of gauge theories, gravitational theories, (super)string theories, topological field theories, higher-spin gauge theories, etc. In this context, it is pertinent to point out that the celebrated Stückelberg technique of compensating field(s) [1], responsible for the massive field theories (e.g., the Proca theory) to acquire the beautiful gauge symmetry invariance, is also based on the ideas of the differential geometry (see, e.g., [2,3,4,5]). In particular, the exterior derivative ($d=d{x}^{\mu }{\partial }_{\mu },d^2=0$) plays a key role (see, e.g., Equation (6) below) in the replacement/modification of the gauge field due to the presence of some compensating field(s) (e.g., a pure scalar field in the context of the Proca theory) which converts the second-class constraints of the original massive field theory into the first-class constraints (see, e.g., [6,7]). The latter appear in the expression for the generator of the gauge symmetry transformations (existing in the case of the Stückelberg-modified theory where the mass and gauge symmetry co-exist together).

One of the central purposes of our present endeavor is to demonstrate that the standard Stückelberg-technique is modified in the context of massive Abelian p-form ($p=1,2,3,\dots$) gauge theories in $D=2p$ dimensions of spacetime because such kinds of massive theories respect, in addition to the gauge symmetry transformations (that are generated by the first-class constraints in the terminology of Dirac’s prescription for the classification of the constraints [6,7]), the dual-gauge symmetry transformations which exist for the gauge-fixed Lagrangian densities of the above kinds of theories. In a very recent work [8], we have been able to corroborate the above claim in the context of a 2D Proca (i.e., a massive 2D Abelian 1-form) theory. In fact, we have been able to demonstrate that, due to the modified version of the Stückelberg formalism (SF), we obtain the (anti-)BRST and (anti-)co-BRST symmetries for the gauge-fixed Lagrangian density of the Stückelberg-modified 2D Proca theory within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism (cf. Appendix A below). It is worthwhile to mention that the massless Abelian p-form ($p=1,2,3$) gauge theories, in $D=2p$ dimensions of spacetime, have already been proven to be the field-theoretic models of Hodge theory (see, e.g., [9] and reference therein). Further, we have been able to show that the Stückelberg-modified massive 2D Abelian 1-form (i.e., Proca theory) [8] (see, e.g., Appendix A) and 4D massive Abelian 2-form theory (see, e.g., [10]) are, once again, very interesting examples of the massive models of Hodge theory within the framework of BRST formalism (see, e.g., [10,11,12]).

The central theme of our present investigation is to show that the Stückelberg-modified Lagrangian density of the massive 4D Abelian 2-form theory respects the (anti-)BRST symmetry transformations in any arbitrary dimension of spacetime. However, in the physical four (3 + 1)-dimensions of spacetime, it respects the (anti-)BRST as well as the (anti-co-)BRST symmetry transformations due to $\left(i\right)$ the modification in the standard Stückelberg technique [cf. Equation (2)] where an axial-vector field $({\tilde{\varphi }}_{\mu })$ also appears explicitly (cf. Equation (7)) backed by the precise mathematical arguments, and $\left(ii\right)$ the existence of a set of discrete duality symmetry transformations under which the modified Stückelberg technique (cf. Equation (9)) as well as the 4D Lagrangian density $\mathcal{L}$ (cf. Equation (29)) both remain invariant. The generalization of these discrete duality symmetry transformations, within the realm of BRST formalism (cf. Equations (48) and (54)), also establish a precise connection between the (anti-)BRST and (anti-)co-BRST symmetry transformations [9]. We provide proper arguments, however, to demonstrate that the nilpotent (anti-)co-BRST and (anti-)BRST transformations have their own identities as they provide the physical realizations [10] of the (co-)exterior derivatives of the differential geometry [2,3,4,5], which are also independent of each-other.

In our present endeavor, for the sake of brevity, we consider only the (co-)BRST invariant Lagrangian density (cf. Section 6) that is the generalization of ${\mathcal{L}}_{\left(b_1\right)}$ (cf. Equation (43)) and establish a direct connection between the BRST and co-BRST symmetry transformations due to the existence of a couple of discrete duality symmetry transformations (48) and (54). In an exactly similar fashion, the generalization of the Lagrangian density ${\mathcal{L}}_{\left(b_2\right)}$ (cf. Equation (45)) can be obtained at the quantum level (within the framework of BRST formalism) as ${\mathcal{L}}_{\overline{\mathcal{B}}}$. The latter will be anti-BRST as well as anti-co-BRST invariant [10]. Once again, we shall be able to establish the interconnection between the anti-BRST and anti-co-BRST symmetry transformations by exploiting the theoretical potential and power of the discrete duality symmetry transformation (48) plus (54) at the quantum level (see, e.g., [9,10] for details). In addition to this direct connection (which is a novel observation), there exists another relationship between the co-BRST and BRST transformations (cf. Equation (60)) which provide the physical realization of the relationship between the co-exterior and exterior derivatives of the differential geometry [2,3,4,5]. In this context, it is worthwhile to mention that that the discrete duality symmetry transformations (48) plus (54) provide the physical realization of the Hodge duality operator of the differential geometry (cf. Equation (60)).

The following key factors have been at the heart of our present investigation. First and foremost, in a very recent work [8], we have discussed the modification of the standard Stückelberg formalism in the context of a massive Abelian 1-form (i.e., Proca) theory in two (1 + 1)-dimensions of spacetime. Hence, we have been curious to find its analogue in the context of a massive Abelian 2-form theory in the physical four (3 + 1)-dimensions of spacetime. Second, we envisage to find out the existence of fields with negative kinetic terms on the basis of symmetry properties (as has happened in the case of a modified 2D Proca theory) because such kinds of exotic fields are the possible candidates for dark matter and dark energy, and they play an important role in the context of the cyclic, bouncing, and self-accelerated cosmological models of the Universe (see, e.g., [13,14,15] and references therein). Third, we desire to establish a direct connection between the nilpotent (anti-)BRST and (anti-)co-BRST transformations on the basis of a set of discrete duality symmetry transformations (cf. Equations (48) and (54)) alone which has not been accomplished in our earlier works [9,10]. Fourth, we have developed a simple theoretical trick of using the EL-EoMs to remove the higher derivative terms so that our 4D theory can become renormalizable (cf. Section 3 for details). Fifth, the higher p-form ($p=2,3,\dots$) gauge theories of massless and massive varieties are interesting from the point of view of the (super)string theories as they appear in their quantum excitations. Finally, we wish to find the physical realizations of the Hodge duality operator of differential geometry [2,3,4,5] in terms of the discrete duality symmetry transformations within the framework of BRST formalism.

The theoretical material of our present endeavor is organized as follows. In Section 2, we discuss the bare essentials of the gauge symmetry transformations for the standard Stückelberg-modified Lagrangian density in any arbitrary D-dimension of spacetime. Section 3 deals with the modification of the Stückelberg formalism where the exterior derivative and the Hodge duality operator of differential geometry play decisive roles. The subject matter of Section 4 concerns itself with the derivation of the 4D Lagrangian densities that respect the (dual-)gauge symmetry transformations together for the gauge-fixed Lagrangian densities, provided exactly similar kinds of restrictions are imposed on the (dual-)gauge transformation parameters from the outside. Section 5 contains the theoretical discussion on the linearized versions of the gauge-fixed Lagrangian densities and Curci–Ferrari (CF)-type restrictions. In Section 6, we establish a relationship between the BRST and co-BRST symmetry transformations due to the discrete duality symmetry transformations (cf. Equations (48) and (54)) in our BRST-invariant theory. Finally, in Section 7, we make some concluding remarks and point out a few future theoretical directions for further investigation(s).

In Appendix A, we very briefly recapitulate the bare essentials of our earlier work [8] on the Stückelberg-modified 2D Proca theory (where the modified SF has been used). The theoretical content of Appendix B is devoted to the generalization of the classical symmetry transformations (37) and (35) to their quantum counterparts and (co-)BRST symmetry transformations for the appropriate (co-)BRST invariant Lagrangian density. It turns out, however, that the Lagrangian density (49) is appropriate and unique as it satisfies all the essential requirements of a properly gauge-fixed and (anti-)BRST invariant theory.

• Convention and Notations: We follow the convention of the left-derivative w.r.t. all the fermionic (i.e., ${\overline{C}}_{\mu },C_{\mu },\overline{C},C,\rho ,\lambda ,$) fields of our theory in the context of the derivation of the equations of motions, definition of the conjugate momenta, deduction of the Noether conserved currents and charges, etc. The 4D Levi–Civita tensor is denoted by ${\epsilon }_{\mu \nu \lambda \xi }$ with conventions: ${\epsilon }_{0123}=+1=-{\epsilon }^{0123}$ and ${\epsilon }_{\mu \nu \lambda \xi }{\epsilon }^{\mu \nu \lambda \xi }=-4!$, ${\epsilon }_{\mu \nu \lambda \xi }{\epsilon }^{\mu \nu \lambda \rho }=-3!{\delta }_{\xi }^{\rho }$, ${\epsilon }_{\mu \nu \lambda \xi }{\epsilon }^{\mu \nu \rho \sigma }=-2!({\delta }_{\lambda }^{\rho }{\delta }_{\xi }^{\sigma }-{\delta }_{\xi }^{\rho }{\delta }_{\lambda }^{\sigma })$, etc., where the Greek indices $\mu ,\nu ,\lambda ,\dots =0,1,2,3$ stand for the time and space directions, and the Latin indices $i,j,k\dots =1,2,3$ correspond to the space directions only. Hence, the 3D Levi–Civita tensor is ${ϵ}_{ijk}={\epsilon }_{0ijk}$. The background flat 4D Minkowskian spacetime manifold is endowed with a flat metric tensor ${\eta }_{\mu \nu }=$ diag $(+1,-1,-1,-1)$ so that the dot product between two non-null 4-vectors $P_{\mu }$ and $Q_{\mu }$ is represented by: $P·Q={\eta }_{\mu \nu }{P}^{\mu }{Q}^{\nu }=P_0{Q}_0-P_i{Q}_i$. We denote the nilpotent (anti-)BRST symmetry transformations by $s_{\left(a\right)b}$, and the notations $s_{\left(a\right)d}$ stand for the nilpotent (anti-) dual (i.e., (anti-)co)-BRST symmetry transformations.
• Standard Definition: On a compact manifold without a boundary, we have a set of three operators ($d,\delta ,\Delta$) which are known as the de Rham cohomological operators of differential geometry. The operators $\left(\delta \right)d$ are called the (co-)exterior derivatives that are connected with each other by the algebraic relationship: $\delta =\pm \ast d\ast$ where ∗ is the Hodge duality operator on the above manifold. These operators satisfy the algebra: $d^2={\delta }^2=0,\Delta ={(d+\delta )}^2=\{d,\delta \},[\Delta ,d]=[\Delta ,\delta ]=0$, where $\Delta$ is the Laplacian operator [2,3,4,5]. This algebra (which is not a Lie algebra) is popularly known as the Hodge algebra and $\Delta$ behaves similar to a Casimir operator for the whole algebra (but not in the Lie algebraic sense). We shall be frequently using the names of these cohomological operators (see, e.g., [2,3,4,5]) of differential geometry in our present endeavor in appropriate places.

2. Preliminaries: Stückelberg Formalism in Any Arbitrary Dimension of Spacetime

We begin with any arbitrary D-dimensional Lagrangian density $\left({\mathcal{L}}_0\right)$ for the massive Abelian 2-form $(B^{\left(2\right)}=[(d{x}^{\mu }\wedge d{x}^{\nu })/2!]B_{\mu \nu })$ theory with the anti-symmetric tensor $(B_{\mu \nu }=-B_{\nu \mu })$ field (carrying the rest mass m) as follows (see, e.g., [16] and references therein)

$$\begin{array}{c}\hfill {\mathcal{L}}_0=\frac{1}{12}{H}^{\mu \nu \lambda }{H}_{\mu \nu \lambda }-\frac{m^2}{4}{B}_{\mu \nu }{B}^{\mu \nu },\end{array}$$

(1)

where $H^{\left(3\right)}=d{B}^{\left(2\right)}\equiv [(d{x}^{\mu }\wedge d{x}^{\nu }\wedge d{x}^{\lambda })/3!]H_{\mu \nu \lambda }$ defines the kinetic term (with the field-strength tensor $H_{\mu \nu \lambda }={\partial }_{\mu }{B}_{\nu \lambda }+{\partial }_{\nu }{B}_{\lambda \mu }+{\partial }_{\lambda }{B}_{\mu \nu }$ for the anti-symmetric tensor field $B_{\mu \nu }$ where the Greek indices $\mu ,\nu ,\lambda \dots =0,1,\dots ,D-1$). It is straightforward to check that the Euler–Lagrange (EL) equation of motion (EoM): ${\partial }_{\mu }{H}^{\mu \nu \lambda }+m^2{B}^{\nu \lambda }=0$ implies the subsidiary conditions: ${\partial }_{\nu }{B}^{\nu \lambda }={\partial }_{\lambda }{B}^{\nu \lambda }=0$, which emerge out from it for $m^2\ne 0$. As a consequence, we observe that $B_{\mu \nu }$ field obeys the Klein–Gordon equation $(\square +m^2)B_{\mu \nu }=0$ with a definite rest mass m. We note that the massive Lagrangian density (1) does not respect the gauge transformation due to the fact that it is endowed with the second-class constraints in the terminology of Dirac’s prescription for the classification scheme of constraints (because the gauge symmetries are generated by the first-class constraints [6,7]).

The gauge symmetry transformations can be restored for the modified version of the standard Lagrangian density (1) if we exploit the basic theoretical methodology of the Stückelberg formalism (SF) related to the compensating field(s). In other words, due to SF, we replace the basic antisymmetric Abelian 2-form field $B_{\mu \nu }$ as follows [16,17]:

$$\begin{array}{c}\hfill B_{\mu \nu }⟶B_{\mu \nu }\mp \frac{1}{m}({\partial }_{\mu }{\varphi }_{\nu }-{\partial }_{\nu }{\varphi }_{\mu }),\end{array}$$

(2)

where the Abelian 1-form ${\Phi }^{\left(1\right)}=d{x}^{\mu }{\varphi }_{\mu }$ defines the vector field ${\varphi }_{\mu }$. It is straightforward to check that $H_{\mu \nu \lambda }={\partial }_{\mu }{B}_{\nu \lambda }+{\partial }_{\nu }{B}_{\lambda \mu }+{\partial }_{\lambda }{B}_{\mu \nu }$ remains invariant under (2). We note that the mass dimension of $B_{\mu \nu }$ and ${\varphi }_{\mu }$ fields are the same in the D-dimensional Minkowskian flat spacetime when we use the natural units: $\hslash =c=1$. Hence, the rest mass m should be present in the denominator of Equation (2) to balance the mass dimension on the l.h.s and r.h.s. of Equation (2). The mass term in Equation (1) changes as follows, due to (2), namely;

$$\begin{array}{c}\hfill -\frac{m^2}{4}{B}_{\mu \nu }{B}^{\mu \nu }⟶-\frac{m^2}{4}[B_{\mu \nu }\mp \frac{1}{m}({\partial }_{\mu }{\varphi }_{\nu }-{\partial }_{\nu }{\varphi }_{\mu })][B^{\mu \nu }\mp \frac{1}{m}({\partial }^{\mu }{\varphi }^{\nu }-{\partial }^{\nu }{\varphi }^{\mu })].\end{array}$$

(3)

Let us define an Abelian 2-form $F^{\left(2\right)}=d{\Phi }^{\left(1\right)}=[(d{x}^{\mu }\wedge d{x}^{\nu })/2!]{\Phi }_{\mu \nu }$ where ${\Phi }_{\mu \nu }={\partial }_{\mu }{\varphi }_{\nu }-{\partial }_{\nu }{\varphi }_{\mu }$ is the antisymmetric field strength tensor for the vector field ${\varphi }_{\mu }$. With all these inputs, we obtain the Stückelberg-modified Lagrangian density ${\mathcal{L}}_S$ from ${\mathcal{L}}_0$ as

$$\begin{array}{c}\hfill {\mathcal{L}}_0⟶{\mathcal{L}}_S=\frac{1}{12}{H}^{\mu \nu \lambda }{H}_{\mu \nu \lambda }-\frac{m^2}{4}{B}_{\mu \nu }{B}^{\mu \nu }\pm \frac{m}{2}{B}_{\mu \nu }{\Phi }^{\mu \nu }-\frac{1}{4}{\Phi }_{\mu \nu }{\Phi }^{\mu \nu },\end{array}$$

(4)

which respects the following local, continuous and infinitesimal classical gauge symmetry transformations ${\delta }_g$, namely

$$\begin{array}{ccc}& & {\delta }_g{H}_{\mu \nu \lambda }=0,{\delta }_g{\varphi }_{\mu }=\pm ({\partial }_{\mu }\Lambda -m{\Lambda }_{\mu }),\hfill \\ & & {\delta }_g{B}_{\mu \nu }=-({\partial }_{\mu }{\Lambda }_{\nu }-{\partial }_{\nu }{\Lambda }_{\mu }),{\delta }_g{\Phi }_{\mu \nu }=\mp m({\partial }_{\mu }{\Lambda }_{\nu }-{\partial }_{\nu }{\Lambda }_{\mu }),\hfill \end{array}$$

(5)

where the Lorentz scalar $\Lambda \left(x\right)$ and Lorentz vector ${\Lambda }_{\mu }\left(x\right)$ are the infinitesimal local gauge symmetry transformation parameters. It is important to point out that there is a stage-one reducibility in the theory because the transformation ${\varphi }_{\mu }\to {\varphi }_{\mu }\pm {\partial }_{\mu }\Lambda$ can be accommodated in the standard Stückelberg technique (considered in Equation (2)) without changing it in any way. This is why, in the gauge transformation of ${\varphi }_{\mu }$ field (cf. Equation (5)), we have the local Lorentz scalar transformation parameter $\Lambda \left(x\right)$. It is straightforward to check that ${\delta }_g{\mathcal{L}}_S=0$, implying that the Stückelberg-modified Lagrangian density ${\mathcal{L}}_S$ respects the infinitesimal and continuous local gauge symmetry transformations (5) in a perfect manner. We mention, in passing, that the second-class constraints of ${\mathcal{L}}_0$ have been converted into the first-class constraints (due to the introduction of the Stückelberg polar vector field ${\varphi }_{\mu }$ in (2)). The ensuing first-class constraints are the generator for the infinitesimal gauge symmetry transformations $\left({\delta }_g\right)$ in (5). These statements are true for our theory in any arbitrary D-dimension of Minkowskian flat spacetime [17].

3. Massive 4D Abelian 2-Form Theory: Modified SF

In the differential form terminology, the standard Stückelbergtechnique (2), defined for any arbitrary D-dimension of spacetime, can be re-expressed as follows [17]:

$$\begin{array}{c}\hfill B^{\left(2\right)}⟶B^{\left(2\right)}\mp \frac{1}{m}{F}^{\left(2\right)}\equiv B^{\left(2\right)}\mp \frac{1}{m}d{\Phi }^{\left(1\right)}.\end{array}$$

(6)

This also establishes the invariance of $H^{\left(3\right)}=d{B}^{\left(2\right)}$ under (2) because of the nilpotency $(d^2=0)$ of the exterior derivative $\left(d\right)$. In the physical four (3 + 1)-dimensional (4D) flat spacetime, the theoretical technique (6) of the standard Stückelberg formalism can be modified in the following manner (in the language of differential forms), namely;

$$\begin{array}{c}\hfill B^{\left(2\right)}⟶B^{\left(2\right)}\mp \frac{1}{m}d{\Phi }^{\left(1\right)}\mp \frac{1}{m}\ast d{\tilde{\Phi }}^{\left(1\right)},\end{array}$$

(7)

where the first two terms of the r.h.s. have already been explained. In the third term, on the r.h.s., we have taken the axial-vector 1-form ${\tilde{\Phi }}^{\left(1\right)}=d{x}^{\mu }{\tilde{\varphi }}_{\mu }$ with the axial-vector field ${\tilde{\varphi }}_{\mu }$. A pseudo 2-form ${\tilde{F}}^{\left(2\right)}=d{\tilde{\Phi }}^{\left(1\right)}=[(d{x}^{\mu }\wedge d{x}^{\nu })/2!]{\tilde{\Phi }}_{\mu \nu }$ has been constructed from ${\tilde{\Phi }}^{\left(1\right)}$ by applying an exterior derivative on it so that we obtain ${\tilde{\Phi }}_{\mu \nu }={\partial }_{\mu }{\tilde{\varphi }}_{\nu }-{\partial }_{\nu }{\tilde{\varphi }}_{\mu }$. To bring the parity of $B^{\left(2\right)},F^{\left(2\right)}=d{\Phi }^{\left(1\right)}$ and the pseudo 2-form ${\tilde{F}}^{\left(2\right)}$ on equal footing1, it is essential to obtain an ordinary 2-form from the pseudo 2-form ${\tilde{F}}^{\left(2\right)}$ by operating a single Hodge duality operator ∗ on it. This mathematical technique (on the 4D spacetime manifold) leads to

$$\begin{array}{c}\hfill \ast {\tilde{F}}^{\left(2\right)}=\ast \left(\frac{d{x}^{\mu }\wedge d{x}^{\nu }}{2!}\right){\tilde{\Phi }}_{\mu \nu }\equiv \frac{1}{2!}(d{x}^{\mu }\wedge d{x}^{\nu }){\tilde{f}}_{\mu \nu },\end{array}$$

(8)

where ${\tilde{f}}_{\mu \nu }=\frac{1}{2}{\epsilon }_{\mu \nu \lambda \xi }{\tilde{\Phi }}^{\lambda \xi }\equiv {\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\lambda }{\tilde{\varphi }}^{\xi }$. Thus, in the language of a set of antisymmetric tensors ($B_{\mu \nu },{\Phi }_{\mu \nu },{\tilde{f}}_{\mu \nu }$), we have obtained the following from the modified version of the 4D Stückelberg technique (7), namely;

$$\begin{array}{ccc}\hfill B_{\mu \nu }& ⟶& B_{\mu \nu }\mp \frac{1}{m}({\partial }_{\mu }{\varphi }_{\nu }-{\partial }_{\nu }{\varphi }_{\mu })\mp \frac{1}{m}{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\lambda }{\tilde{\varphi }}^{\xi }\hfill \\ & \equiv & B_{\mu \nu }\mp \frac{1}{m}{\Phi }_{\mu \nu }\mp \frac{1}{m}{\tilde{f}}_{\mu \nu }\equiv B_{\mu \nu }\mp \frac{1}{m}\left({\Phi }_{\mu \nu }+\frac{1}{2}{\epsilon }_{\mu \nu \lambda \xi }{\tilde{\Phi }}^{\lambda \xi }\right).\hfill \end{array}$$

(9)

It is very interesting to state that the above modified 4D Stückelberg technique remains form-invariant under the following discrete duality symmetry transformations:

$$\begin{array}{c}\hfill B_{\mu \nu }\to \mp i{\tilde{B}}_{\mu \nu }\equiv \mp \frac{i}{2!}{\epsilon }_{\mu \nu \lambda \xi }{B}^{\lambda \xi },{\varphi }_{\mu }\to \pm i\widetilde{{\varphi }_{\mu }},\widetilde{{\varphi }_{\mu }}\to \mp i{\varphi }_{\mu },\end{array}$$

(10)

where ${\tilde{B}}_{\mu \nu }=\frac{1}{2!}{\epsilon }_{\mu \nu \lambda \xi }{B}^{\lambda \xi }$ emerges out from the self-duality condition: $\ast B^{\left(2\right)}=[(d{x}^{\mu }\wedge d{x}^{\nu })/2!]B_{\mu \nu }$, which leads to the derivation of the dual Abelian 2-form (in 4D) as follows:

$$\begin{array}{c}\hfill {\tilde{B}}^{\left(2\right)}=\left(\frac{d{x}^{\mu }\wedge d{x}^{\nu }}{2!}\right)\left[\frac{1}{2!}{\epsilon }_{\mu \nu \lambda \xi }{B}^{\lambda \xi }\right]\equiv \left(\frac{d{x}^{\mu }\wedge d{x}^{\nu }}{2!}\right){\tilde{B}}_{\mu \nu }.\end{array}$$

(11)

We shall see that the discrete duality symmetry transformations in (10) will play a very important role, later on, as its generalization (within the framework of BRST formalism) will provide the analogue of the Hodge duality ∗ operator of differential geometry. We would like to lay emphasis on the fact that the root cause behind the existence of the discrete duality symmetry transformations in (10) is the self-duality condition on the Abelian 2-form $(B^{\left(2\right)}=[(d{x}^{\mu }\wedge d{x}^{\nu })/2!]B_{\mu \nu })$ in the physical four (3 + 1)-dimensions of spacetime.

It is an elementary exercise to note that the mass term of Equation (1) transforms, under the modified Stückelberg technique (cf. Equation (9)), as

$$\begin{array}{ccc}\hfill -\frac{m^2}{4}{B}_{\mu \nu }{B}^{\mu \nu }⟶& -& \frac{m^2}{4}{B}_{\mu \nu }{B}^{\mu \nu }\pm \frac{m}{2}{B}_{\mu \nu }({\Phi }^{\mu \nu }+\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{\tilde{\Phi }}_{\rho \sigma })\hfill \\ & -& \frac{1}{4}{\Phi }_{\mu \nu }{\Phi }^{\mu \nu }+\frac{1}{4}{\tilde{\Phi }}_{\mu \nu }{\tilde{\Phi }}^{\mu \nu },\hfill \end{array}$$

(12)

modulo a total spacetime derivative ${\partial }_{\mu }[-{\epsilon }^{\mu \nu \lambda \xi }{\varphi }_{\nu }{\partial }_{\lambda }{\tilde{\varphi }}_{\xi }]$ which emerges out from a term $(-\frac{1}{2}{\tilde{f}}_{\mu \nu }{\Phi }^{\mu \nu })$ that appears in Equation (12) due to the substitution (9) for the modified version of the antisymmetric field $B_{\mu \nu }$. We also point out that the kinetic term ($(1/12)H_{\mu \nu \lambda }{H}^{\mu \nu \lambda }$) also transforms under (9) because it is straightforward to note that we have the following

$$\begin{array}{c}\hfill H_{\mu \nu \lambda }⟶H_{\mu \nu \lambda }\mp \frac{1}{m}({\partial }_{\mu }{\Phi }_{\nu \lambda }+{\partial }_{\nu }{\Phi }_{\lambda \mu }+{\partial }_{\lambda }{\Phi }_{\mu \nu })\\ \hfill \mp \frac{1}{m}({\partial }_{\mu }{\tilde{f}}_{\nu \lambda }+{\partial }_{\nu }{\tilde{f}}_{\lambda \mu }+{\partial }_{\lambda }{\tilde{f}}_{\mu \nu }),\end{array}$$

(13)

where ${\Phi }_{\mu \nu }={\partial }_{\mu }{\varphi }_{\nu }-{\partial }_{\nu }{\varphi }_{\mu }$ and ${\tilde{f}}_{\mu \nu }={\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\lambda }{\tilde{\varphi }}^{\xi }$. We note that the second term on the r.h.s. of the above equation turns out to be zero. However, the third term exists as:

$$\begin{array}{ccc}\hfill \mp \frac{1}{m}{\Sigma }_{\mu \nu \lambda }& =& \mp \frac{1}{m}({\partial }_{\mu }{\tilde{f}}_{\nu \lambda }+{\partial }_{\nu }{\tilde{f}}_{\lambda \mu }+{\partial }_{\lambda }{\tilde{f}}_{\mu \nu })\hfill \\ & \equiv & \mp \frac{1}{m}({\epsilon }_{\mu \nu \rho \sigma }{\partial }_{\lambda }+{\epsilon }_{\nu \lambda \rho \sigma }{\partial }_{\mu }+{\epsilon }_{\lambda \mu \rho \sigma }{\partial }_{\nu })({\partial }^{\rho }{\tilde{\varphi }}^{\sigma }).\hfill \end{array}$$

(14)

Thus, we have to find the exact value of the following (for the changes in the kinetic term due to the modified version of 4D Stückelberg technique (cf. Equation (9))), namely;

$$\begin{array}{c}\hfill \frac{1}{12}{H}_{\mu \nu \lambda }{H}^{\mu \nu \lambda }⟹\frac{1}{12}{H}_{\mu \nu \lambda }{H}^{\mu \nu \lambda }\mp \frac{1}{6m}{H}_{\mu \nu \lambda }{\Sigma }^{\mu \nu \lambda }+\frac{1}{12m^2}{\Sigma }_{\mu \nu \lambda }{\Sigma }^{\mu \nu \lambda },\end{array}$$

(15)

where we have taken into account $H_{\mu \nu \lambda }⟶H_{\mu \nu \lambda }\mp \frac{1}{m}{\Sigma }_{\mu \nu \lambda }$ (cf. Equations (13) and (14)). We focus on the second term $[\mp (1/6m)H^{\mu \nu \lambda }{\Sigma }_{\mu \nu \lambda }]$, which can be explicitly written as:

$$\begin{array}{c}\hfill \mp \frac{1}{6m}{H}^{\mu \nu \lambda }[{\epsilon }_{\mu \nu \rho \sigma }{\partial }_{\lambda }+{\epsilon }_{\nu \lambda \rho \sigma }{\partial }_{\mu }+{\epsilon }_{\lambda \mu \rho \sigma }{\partial }_{\nu }]({\partial }^{\rho }{\tilde{\varphi }}^{\sigma }).\end{array}$$

(16)

The first term on the r.h.s. of the above equation contributes the following (modulo a total spacetime derivative), namely;

$$\begin{array}{c}\hfill \pm \frac{1}{6m}({\partial }_{\lambda }{H}^{\lambda \mu \nu }){\epsilon }_{\mu \nu \rho \sigma }({\partial }^{\rho }{\tilde{\varphi }}^{\sigma }).\end{array}$$

(17)

It is self-evident that there are three derivatives in the above expression because $H_{\mu \nu \lambda }$ contains one derivative. Thus, the expression in (17) belongs to a higher derivative term for our 4D theory. It is worthwhile to mention here that in our earlier work [10], the higher derivative terms have been ignored. This is why relevant terms in the Lagrangian density have been obtained by the trial and error method. However, we note, in our present endeavor, that one can remove a single derivative by using the on-shell condition: ${\partial }_{\lambda }{H}^{\lambda \mu \nu }+m^2{B}^{\mu \nu }=0$, which is equivalent to the EL-EoM: $(\square +m^2)B_{\mu \nu }=0$. It is due to the presence of the higher derivative term that the substitution of the on-shell condition does not make this term (in the Lagrangian density) equal to zero. This should be contrasted against the use of the on-shell conditions in the context of the simple cases of (i) the Dirac Lagrangian density and (ii) the pure (Klein–Gorden) scalar field Lagrangian density (where there are no higher order derivatives). All the three terms, on the r.h.s. of (16), individually contribute to the same result, which can be added together to yield

$$\begin{array}{c}\hfill \mp \frac{m}{2}{\epsilon }_{\mu \nu \lambda \xi }{B}^{\mu \nu }({\partial }^{\lambda }{\tilde{\varphi }}^{\xi })\equiv \mp \frac{m}{4}{\epsilon }^{\mu \nu \lambda \xi }{B}_{\mu \nu }{\tilde{\Phi }}_{\lambda \xi },\end{array}$$

(18)

where ${\tilde{\Phi }}_{\lambda \xi }={\partial }_{\lambda }{\tilde{\varphi }}_{\xi }-{\partial }_{\xi }{\tilde{\varphi }}_{\lambda }$. It is interesting to point out that the above term has been incorporated into the BRST invariant Lagrangian density of our earlier work [10] on the basis of the trial and error method. However, as is self-evident, we have derived this term correctly in our present endeavor, which is motivated by our earlier work on the Stückelberg-modified 2D Proca theory [8], where we have exploited a similar kind of trick to remove the higher derivative terms. The mass term in Equation (18) is similar to the topological mass term of the $B\wedge F$ theory. In the latter theory, the 4D Abelian 2-form theory also incorporates the Maxwell Abelian 1-form ($A^{\left(1\right)}=d{x}^{\mu }{A}_{\mu }$) gauge field with the curvature 2-form $F^{\left(2\right)}=d{A}^{\left(1\right)}$. There are many ways to derive (18) from (16). However, we have chosen one of the simplest methods to derive Equation (18), which is not a higher derivative culprit term for our 4D Abelian 2-form massive theory.

We now focus on the exact and explicit computation of the third term on the r.h.s. of (15). It is evident that, for a 4D Abelian 2-form theory, this third term is a higher derivative term because it contains four derivatives in it. A close look at (14) shows that there will be a total of nine terms when we take into account ($(1/12 m^2){\Sigma }_{\mu \nu \lambda }{\Sigma }^{\mu \nu \lambda }$) and write the expression for ${\Sigma }_{\mu \nu \lambda }$ from Equation (16). However, it turns out that only three of them contribute to the Lagrangian density, and the rest of the six terms are found to be total spacetime derivatives. Hence, they can be ignored as the dynamics of the theory does not depend on them. Let us focus on the first existing term, which is equal to the following:

$$\begin{array}{ccc}& & \frac{1}{12m^2}{\epsilon }^{\mu \nu \rho \sigma }{\partial }^{\lambda }({\partial }_{\rho }{\tilde{\varphi }}_{\sigma }){\epsilon }_{\mu \nu \alpha \beta }{\partial }_{\lambda }({\partial }^{\alpha }{\tilde{\varphi }}^{\beta })\hfill \\ & & =-\frac{1}{6m^2}[{\partial }^{\lambda }({\partial }_{\alpha }{\tilde{\varphi }}_{\beta }-{\partial }_{\beta }{\tilde{\varphi }}_{\alpha })][{\partial }_{\lambda }({\partial }^{\alpha }{\tilde{\varphi }}^{\beta })]\hfill \\ & & \equiv \frac{1}{6m^2}{\partial }^{\lambda }({\partial }^{\alpha }{\tilde{\Phi }}_{\alpha \beta })({\partial }_{\lambda }{\tilde{\varphi }}^{\beta }),\hfill \end{array}$$

(19)

where we have dropped a total spacetime derivative term, as it will not affect the dynamics At this stage, to remove the derivatives, we use the EoM: ${\partial }_{\alpha }{\tilde{\Phi }}^{\alpha \beta }+m^2{\tilde{\varphi }}^{\beta }=0$. Thus, the final expression on the r.h.s. of Equation (19) is

$$\begin{array}{c}\hfill -\frac{1}{6}({\partial }^{\lambda }{\tilde{\varphi }}_{\beta })({\partial }_{\lambda }{\tilde{\varphi }}^{\beta })\equiv \frac{1}{6}{\tilde{\varphi }}_{\beta }\square {\tilde{\varphi }}^{\beta },\end{array}$$

(20)

where we have dropped a total spacetime derivative term. Using the Klein–Gordon equation2: $(\square +m^2){\tilde{\varphi }}_{\mu }=0$, we obtain the following (from the first contribution), namely;

$$\begin{array}{c}\hfill \frac{1}{12m^2}{\epsilon }^{\mu \nu \rho \sigma }{\partial }^{\lambda }({\partial }_{\rho }{\tilde{\varphi }}_{\sigma }){\epsilon }_{\mu \nu \alpha \beta }{\partial }_{\lambda }({\partial }^{\alpha }{\tilde{\varphi }}^{\beta })\equiv -\frac{m^2}{6}{\tilde{\varphi }}_{\beta }{\tilde{\varphi }}^{\beta }.\end{array}$$

(21)

It is obvious that there are three such contributions in the total evaluation of the third term ($(1/12m^2){\Sigma }_{\mu \nu \lambda }{\Sigma }^{\mu \nu \lambda }$) on the r.h.s. of Equation (15). Thus, ultimately, we obtain the following exact and explicit expression from all three existing terms, namely;

$$\begin{array}{c}\hfill \frac{1}{12m^2}{\Sigma }_{\mu \nu \lambda }{\Sigma }^{\mu \nu \lambda }=-\frac{m^2}{2}{\tilde{\varphi }}_{\mu }{\tilde{\varphi }}^{\mu }.\end{array}$$

(22)

Going from Equation (20) to Equation (22) is essential and interesting for our purpose. It is clear that the above term is not a culprit term, and it is useful to us for our further discussions. The total terms on the r.h.s. of Equation (15) can be re-expressed as follows:

$$\begin{array}{ccc}& & \frac{1}{12}{H}^{\mu \nu \lambda }{H}_{\mu \nu \lambda }\mp \frac{m}{2}{\epsilon }_{\mu \nu \lambda \xi }{B}^{\mu \nu }({\partial }^{\lambda }{\tilde{\varphi }}^{\xi })-\frac{m^2}{2}{\tilde{\varphi }}_{\mu }{\tilde{\varphi }}^{\mu }\hfill \\ & & \equiv -\frac{1}{8}{\epsilon }^{\mu \nu \lambda \xi }({\partial }_{\nu }{B}_{\lambda \xi }){\epsilon }_{\mu \alpha \beta \gamma }({\partial }^{\alpha }{B}^{\beta \gamma })\hfill \\ & & \pm \frac{m}{2}{\epsilon }_{\mu \nu \lambda \xi }({\partial }^{\mu }{B}^{\nu \lambda }){\tilde{\varphi }}^{\xi }-\frac{m^2}{2}{\tilde{\varphi }}_{\mu }{\tilde{\varphi }}^{\mu },\hfill \end{array}$$

(23)

where we have dropped a total spacetime derivative term and used the following exact equality, namely;

$$\begin{array}{c}\hfill \frac{1}{12}{H}^{\mu \nu \lambda }{H}_{\mu \nu \lambda }=-\frac{1}{8}{\epsilon }^{\mu \nu \lambda \xi }({\partial }_{\nu }{B}_{\lambda \xi }){\epsilon }_{\mu \alpha \beta \gamma }({\partial }^{\alpha }{B}^{\beta \gamma }).\end{array}$$

(24)

The correctness of the above equality can be checked explicitly by using the well-known property of the 4D Levi–Civita tensor, where one index (i.e., $\mu$) is contracted. It is straightforward to observe that the final expression for (23) can be written as

$$\begin{array}{c}\hfill -\frac{1}{2}\left[\frac{1}{4}{\epsilon }^{\mu \nu \lambda \xi }\left({\partial }_{\nu }{B}_{\lambda \xi }\right){\epsilon }_{\mu \alpha \beta \gamma }\left({\partial }^{\alpha }{B}^{\beta \gamma }\right)\pm m{\tilde{\varphi }}_{\mu }{\epsilon }^{\mu \nu \lambda \xi }\left({\partial }_{\nu }{B}_{\lambda \xi }\right)+m^2{\tilde{\varphi }}_{\mu }{\tilde{\varphi }}^{\mu }\right],\end{array}$$

(25)

where we have used the following

$$\begin{array}{c}\hfill \pm \frac{m}{2}{\epsilon }_{\mu \nu \lambda \xi }\left({\partial }^{\mu }{B}^{\nu \lambda }\right){\tilde{\varphi }}^{\xi }=\mp \frac{m}{2}{\tilde{\varphi }}_{\mu }{\epsilon }^{\mu \nu \lambda \xi }\left({\partial }_{\nu }{B}_{\lambda \xi }\right),\end{array}$$

(26)

to express (15) (and/or (23) and/or (25)) as a squared term, namely;

$$\begin{array}{c}\hfill \frac{1}{12}{H}_{\mu \nu \lambda }{H}^{\mu \nu \lambda }\mp \frac{1}{6m}{H}_{\mu \nu \lambda }{\Sigma }^{\mu \nu \lambda }+\frac{1}{12m^2}{\Sigma }_{\mu \nu \lambda }{\Sigma }^{\mu \nu \lambda }\\ \hfill =-\frac{1}{2}{\left[\frac{1}{2}{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\nu }{B}^{\lambda \xi }\pm m{\tilde{\varphi }}_{\mu }\right]}^2,\end{array}$$

(27)

which is nothing but the explicit expression for Equation (25). Thus, we note that the final version of the Lagrangian density (with the modified SF (cf. Equation (9))) is as follows3

$$\begin{array}{ccc}\hfill {\mathcal{L}}_S^{\left(m\right)}& =& -\frac{1}{2}{\left[\frac{1}{2}{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\nu }{B}^{\lambda \xi }\pm m{\tilde{\varphi }}_{\mu }\right]}^2-\frac{m^2}{4}{B}_{\mu \nu }{B}^{\mu \nu }\hfill \\ & \pm & \frac{m}{2}\left[{\Phi }^{\mu \nu }+\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{\tilde{\Phi }}_{\rho \sigma }\right]B_{\mu \nu }-\frac{1}{4}{\Phi }_{\mu \nu }{\Phi }^{\mu \nu }+\frac{1}{4}{\tilde{\Phi }}_{\mu \nu }{\tilde{\Phi }}^{\mu \nu },\hfill \end{array}$$

(28)

where we have taken the inputs from Equations (12) and (27), and the superscript $\left(m\right)$ on this Lagrangian density denotes that we have taken into account the help of the modified SF (cf. Equation (9)) in the modified version of the Lagrangian density (28).

We end this section with the final remark that we can add a gauge-fixing term for the Abelian 2-form field $\left(B_{\mu \nu }\right)$, the axial-vector field $\left({\tilde{\varphi }}_{\mu }\right)$, and the polar vector field $\left({\varphi }_{\mu }\right)$ so that we can quantize the theory (described by the Lagrangian density (28)). At this stage, the role of the co-exterior derivative ($\delta =\pm \ast d\ast ,{\delta }^2=0$) becomes quite essential as we note that $\delta B^{\left(2\right)}=\left({\partial }^{\nu }{B}_{\nu \mu }\right)d{x}^{\mu },\delta {\Phi }^{\left(1\right)}=(\partial ·\varphi ),\delta {\tilde{\Phi }}^{\left(1\right)}=(\partial ·\tilde{\varphi })$ where $\delta =-\ast d\ast$ is the co-exterior derivative defined on the 4D spacetime (which is an even dimensional Minkowskian spacetime manifold). The full Lagrangian density, with the gauge-fixing terms, is

$$\begin{array}{ccc}\hfill \mathcal{L}={\mathcal{L}}_S^{\left(m\right)}+{\mathcal{L}}_{gf}& \equiv & -\frac{1}{2}{\left[\frac{1}{2}{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\nu }{B}^{\lambda \xi }\pm m{\tilde{\varphi }}_{\mu }\right]}^2-\frac{m^2}{4}{B}_{\mu \nu }{B}^{\mu \nu }\hfill \\ & \pm & \frac{m}{2}\left[{\Phi }^{\mu \nu }+\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{\tilde{\Phi }}_{\rho \sigma }\right]B_{\mu \nu }-\frac{1}{4}{\Phi }_{\mu \nu }{\Phi }^{\mu \nu }+\frac{1}{4}{\tilde{\Phi }}_{\mu \nu }{\tilde{\Phi }}^{\mu \nu }\hfill \\ & +& \frac{1}{2}{\left[{\partial }^{\nu }{B}_{\nu \mu }\pm m{\varphi }_{\mu }\right]}^2+\frac{1}{2}{(\partial ·\tilde{\varphi })}^2-\frac{1}{2}{(\partial ·\varphi )}^2,\hfill \end{array}$$

(29)

where the gauge-fixing term ($\frac{1}{2}{({\partial }^{\nu }{B}_{\nu \mu }\pm m{\varphi }_{\mu })}^2$) is similar to the t’Hooft gauge in the context of the Stückelberg-modified Proca theory [8,17]. We point out that the above gauge-fixed Lagrangian density respects the duality symmetry transformations (10). The latter, it goes without saying, are also respected by the modified SF that has been defined in Equation (9). The equations of motion, satisfied by the basic fields ($B_{\mu \nu },{\varphi }_{\mu },{\tilde{\varphi }}_{\mu }$), are the Klein–Gordon equations: $(\square +m^2)B_{\mu \nu }=0,(\square +m^2){\varphi }_{\mu }=0,(\square +m^2){\tilde{\varphi }}_{\mu }=0$, which emerge out from the Lagrangian density (29). This observation establishes that (i) all the fields have the rest mass m and our gauge-fixing procedure is correct, where the fields $({\varphi }_{\mu },{\tilde{\varphi }}_{\mu })$ have been incorporated into (29) on the basis of the consideration of the proper mass dimension in 4D, and (ii) the EL-EoMs used, in this section, to remove the higher derivative terms are not far-fetched.

4. Final Forms of the Gauge-Fixed Lagrangian Densities: Massive Free 4D Abelian 2-Form Theory

The gauge-fixing term and the kinetic term that have been obtained in (29) can be further generalized. It is a textbook4 material that one can incorporate a pure scalar field $\left(\varphi \right)$ with mass dimension one (i.e., $\left[M\right]$) in the massless case of Abelian 2-form gauge theory in the following explicit manner:

$$\begin{array}{c}\hfill B^{\mu }({\partial }^{\nu }{B}_{\nu \mu }-{\partial }_{\mu }\varphi )-\frac{1}{2}{B}^{\mu }{B}_{\mu }\equiv \frac{1}{2}{({\partial }^{\nu }{B}_{\nu \mu }-{\partial }_{\mu }\varphi )}^2,\end{array}$$

(30)

where, in the 4D Minkowskian spacetime, the mass dimensions of the Nakanishi–Lautrup type auxiliary field $B^{\mu }$ and $({\partial }^{\nu }{B}_{\nu \mu }-{\partial }_{\mu }\varphi )$ are two (i.e., ${\left[M\right]}^2$) in the natural units ($\hslash =c=1$). In the case of our massive 4D Abelian 2-form theory, we can choose the analogue of (30) by generalizing the gauge-fixing term of (29). However, since our theory is duality-invariant, we have to generalize the kinetic term, too, by incorporating a pseudo-scalar field $\left(\tilde{\varphi }\right)$ with a mass dimension of one (i.e., $\left[M\right]$). To be consistent with the Curci-Ferrari restriction that has been derived by the superfield approach to BRST formalism in the context of Abelian 2-form theory (see, e.g., [18] for details), we choose the pure scalar and pseudo-scalar fields with a factor of ($\pm 1/2$) to begin with. However, only one sign will be taken into consideration for a specific Lagrangian density of our theory (later on).

As a consequence of the above arguments, we have the following modifications of the gauge-fixed Lagrangian density (29), namely;

$$\begin{array}{ccc}\hfill \mathcal{L}⟶{\mathcal{L}}_{\left(1\right)}& =& -\frac{1}{2}{\left[\frac{1}{2}{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\nu }{B}^{\lambda \xi }\pm m{\tilde{\varphi }}_{\mu }\mp \frac{1}{2}{\partial }_{\mu }\tilde{\varphi }\right]}^2-\frac{m^2}{4}{B}_{\mu \nu }{B}^{\mu \nu }\hfill \\ & \pm & \frac{m}{2}\left[{\Phi }^{\mu \nu }+\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{\tilde{\Phi }}_{\rho \sigma }\right]B_{\mu \nu }-\frac{1}{4}{\Phi }_{\mu \nu }{\Phi }^{\mu \nu }+\frac{1}{4}{\tilde{\Phi }}_{\mu \nu }{\tilde{\Phi }}^{\mu \nu }\hfill \\ & +& \frac{1}{2}{\left[{\partial }^{\nu }{B}_{\nu \mu }\pm m{\varphi }_{\mu }\mp \frac{1}{2}{\partial }_{\mu }\varphi \right]}^2+\frac{1}{2}{\left(\partial ·\tilde{\varphi }+\frac{m}{2}\tilde{\varphi }\right)}^2\hfill \\ & -& \frac{1}{2}{\left(\partial ·\varphi +\frac{m}{2}\varphi \right)}^2,\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill \mathcal{L}⟶{\mathcal{L}}_{\left(2\right)}& =& -\frac{1}{2}{\left[\frac{1}{2}{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\nu }{B}^{\lambda \xi }\pm m{\tilde{\varphi }}_{\mu }\pm \frac{1}{2}{\partial }_{\mu }\tilde{\varphi }\right]}^2-\frac{m^2}{4}{B}_{\mu \nu }{B}^{\mu \nu }\hfill \\ & \pm & \frac{m}{2}\left[{\Phi }^{\mu \nu }+\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{\tilde{\Phi }}_{\rho \sigma }\right]B_{\mu \nu }-\frac{1}{4}{\Phi }_{\mu \nu }{\Phi }^{\mu \nu }+\frac{1}{4}{\tilde{\Phi }}_{\mu \nu }{\tilde{\Phi }}^{\mu \nu }\hfill \\ & +& \frac{1}{2}{\left[{\partial }^{\nu }{B}_{\nu \mu }\pm m{\varphi }_{\mu }\pm \frac{1}{2}{\partial }_{\mu }\varphi \right]}^2+\frac{1}{2}{\left(\partial ·\tilde{\varphi }-\frac{m}{2}\tilde{\varphi }\right)}^2\hfill \\ & -& \frac{1}{2}{\left(\partial ·\varphi -\frac{m}{2}\varphi \right)}^2,\hfill \end{array}$$

(32)

where the mass dimensions of the fields have been taken into account, and we have taken into consideration both the signs that are present in (30) and chosen the constant numerical factor to be $1/2$. We shall corroborate the logic behind the choice of the terms containing $\varphi$ and $\tilde{\varphi }$ in the modified Lagrangian densities (31) and (32). We shall also dwell a bit on our choice of the factor $(1/2)$ in the kinetic and gauge-fixing terms that contain fields $\tilde{\varphi }$ and $\varphi$, respectively. The latter have been incorporated into ${\mathcal{L}}_{\left(1\right)}$ and ${\mathcal{L}}_{\left(2\right)}$ at appropriate places (e.g., the kinetic and gauge-fixing terms) with proper mass dimensions. It is worthwhile to mention that the signs of the last two terms, corresponding to the gauge-fixing of the axial-vector and polar-vector fields ${\tilde{\varphi }}_{\mu }$ and ${\varphi }_{\mu }$, respectively, are fixed, which leads to the EL-EoMs5: $(\square +m^2){\varphi }_{\mu }=0,(\square +m^2){\tilde{\varphi }}_{\mu }=0,(\square +m^2)\varphi =0,(\square +m^2)\tilde{\varphi }=0$.

At this juncture, we would like to point out that the generalization of the discrete duality symmetry transformations (10), namely:

$$\begin{array}{ccc}& & B_{\mu \nu }⟶\mp i{\tilde{B}}_{\mu \nu }\equiv \mp \frac{i}{2}{\epsilon }_{\mu \nu \lambda \xi }{B}^{\lambda \xi },{\varphi }_{\mu }⟶\pm i{\tilde{\varphi }}_{\mu },{\tilde{\varphi }}_{\mu }⟶\mp i{\varphi }_{\mu },\hfill \\ & & \varphi ⟶\pm i\tilde{\varphi },\tilde{\varphi }⟶\mp i\varphi ,B_{\mu \nu }{B}^{\mu \nu }⟶B_{\mu \nu }{B}^{\mu \nu },\hfill \end{array}$$

(33)

is respected by the completely gauge-fixed Lagrangian densities ${\mathcal{L}}_{\left(1\right)}$ and ${\mathcal{L}}_{\left(2\right)}$, and all the fields (i.e., $B_{\mu \nu },{\varphi }_{\mu },{\tilde{\varphi }}_{\mu },\varphi ,\tilde{\varphi }$) satisfy the following Klein–Gordon equation6

$$\begin{array}{ccc}& & (\square +m^2)B_{\mu \nu }=0,(\square +m^2){\varphi }_{\mu }=0,(\square +m^2){\tilde{\varphi }}_{\mu }=0,\hfill \\ & & (\square +m^2)\varphi =0,(\square +m^2)\tilde{\varphi }=0,\hfill \end{array}$$

(34)

which is the signature of the completely (and correctly) gauge-fixed Lagrangian density. It should be noted that the mass term for the $B_{\mu \nu }$ field (i.e.,$-(m^2/4)B_{\mu \nu }{B}^{\mu \nu }$) remains invariant under the transformation $[B_{\mu \nu }⟶\mp (i/2){\epsilon }_{\mu \nu \lambda \xi }{B}^{\lambda \xi }]$. The latter has its origin in the self-duality condition (cf. Equation (11)). This observation is crucial because it forces the whole theory to have a single mass parameter m. We point out that both the signs, chosen in the kinetic and gauge-fixing terms as well as in the third term of (31) and (32), are allowed, and they do not violate the Klein–Gordon equations in (34). It is very interesting to highlight the following infinitesimal and continuous gauge transformations $\left({\delta }_g\right)$ for the basic fields of the Lagrangian density ${\mathcal{L}}_{\left(1\right)}$, namely

$$\begin{array}{ccc}& & {\delta }_g{B}_{\mu \nu }=-({\partial }_{\mu }{\Lambda }_{\nu }-{\partial }_{\nu }{\Lambda }_{\mu }),{\delta }_g{\varphi }_{\mu }=\pm ({\partial }_{\mu }\Lambda -m{\Lambda }_{\mu }),\hfill \\ & & {\delta }_g{\Phi }_{\mu \nu }=\mp m({\partial }_{\mu }{\Lambda }_{\nu }-{\partial }_{\nu }{\Lambda }_{\mu }),{\delta }_g\varphi =\pm 2[(\partial ·\Lambda )+m\Lambda ],\hfill \\ & & {\delta }_g[H_{\mu \nu \lambda },{\tilde{\varphi }}_{\mu },\tilde{\varphi },{\tilde{\Phi }}_{\mu \nu }]=0,\hfill \end{array}$$

(35)

which are nothing but the generalization of the gauge symmetry transformations (5). Under these transformations, we observe that the Lagrangian density ${\mathcal{L}}_{\left(1\right)}$ transforms as follows:

$$\begin{array}{ccc}\hfill {\delta }_g{\mathcal{L}}_{\left(1\right)}& =& {\partial }_{\mu }\left[\mp m{\epsilon }^{\mu \nu \lambda \xi }{\Lambda }_{\nu }{\partial }_{\lambda }{\tilde{\varphi }}_{\xi }\right]\mp \left(\partial ·\varphi +\frac{m}{2}\varphi \right)\left[\square +m^2\right]\Lambda \hfill \\ & -& \left[{\partial }_{\nu }{B}^{\nu \mu }\pm m{\varphi }^{\mu }\mp \frac{1}{2}{\partial }^{\mu }\varphi \right]\left[\square +m^2\right]{\Lambda }_{\mu }.\hfill \end{array}$$

(36)

Thus, it is crystal clear that if we impose the restrictions on the gauge transformation parameters as: $(\square +m^2)\Lambda =0,(\square +m^2){\Lambda }_{\mu }=0$ from outside, the transformations (35) will become the symmetry transformations for the Lagrangian density ${\mathcal{L}}_{\left(1\right)}$. We shall see that, within the framework of the BRST approach to this theory, there will be no imposition of any kind of restriction from outside on the theory. We christen the infinitesimal and continuous transformations (35) as the gauge transformations because we observe that the total kinetic terms (i.e., ${\delta }_g{H}_{\mu \nu \lambda }=0,{\delta }_g{\tilde{\varphi }}_{\mu }=0,{\delta }_g\tilde{\varphi }=0,{\delta }_g{\tilde{\Phi }}_{\mu \nu }=0$), owing their origin basically to the exterior derivative $d=d{x}^{\mu }{\partial }_{\mu }$ (with $d^2=0$) of differential geometry [2,3,4,5], remain invariant. In addition to the gauge transformations (35), we have another set of infinitesimal and continuous transformations $\left({\delta }_{dg}\right)$ in the theory, namely;

$$\begin{array}{ccc}& & {\delta }_{dg}{B}_{\mu \nu }=-{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\lambda }{\Sigma }^{\xi },{\delta }_{dg}{\tilde{\varphi }}_{\mu }=\pm ({\partial }_{\mu }\Omega -m{\Sigma }_{\mu }),\hfill \\ & & {\delta }_{dg}\tilde{\varphi }=\pm 2[\partial ·\Sigma +m\Omega ],{\delta }_{dg}{\tilde{\Phi }}_{\mu \nu }=\mp m({\partial }_{\mu }{\Sigma }_{\nu }-{\partial }_{\nu }{\Sigma }_{\mu }),\hfill \\ & & {\delta }_{dg}[{\partial }^{\nu }{B}_{\nu \mu },{\varphi }_{\mu },\varphi ,{\Phi }_{\mu \nu }]=0,\hfill \end{array}$$

(37)

which imply that the total gauge-fixing term (i.e., $\frac{1}{2}({\partial }_{\nu }{B}^{\nu \mu }\pm m{\varphi }^{\mu }\mp \frac{1}{2}{\partial }^{\mu }\varphi )$), owing its origin primarily to the co-exterior derivative: $\delta =\pm \ast d\ast$, remains invariant. Here, the infinitesimal transformation parameters ${\Sigma }_{\mu }$ and $\Omega$ are the Lorentz axial vector and pseudo-scalar, respectively. We observe that the Lagrangian density ${\mathcal{L}}_{\left(1\right)}$ transforms under the infinitesimal and continuous transformations (37) as follows:

$$\begin{array}{ccc}\hfill {\delta }_{dg}{\mathcal{L}}_{\left(1\right)}& =& {\partial }_{\mu }\left[\mp m{\epsilon }^{\mu \nu \lambda \xi }{\Sigma }_{\nu }{\partial }_{\lambda }{\varphi }_{\xi }\right]\pm \left(\partial ·\tilde{\varphi }+\frac{m}{2}\tilde{\varphi }\right)\left[\square +m^2\right]\Omega \hfill \\ & +& \left[\frac{1}{2}{\epsilon }^{\mu \nu \lambda \xi }{\partial }_{\nu }{B}_{\lambda \xi }\pm m{\tilde{\varphi }}^{\mu }\mp \frac{1}{2}{\partial }^{\mu }\tilde{\varphi }\right]\left[\square +m^2\right]{\Sigma }_{\mu },\hfill \end{array}$$

(38)

which shows that, if we impose the conditions: $(\square +m^2){\Sigma }_{\mu }=0$ and $(\square +m^2)\Omega =0$ from outside, the infinitesimal and continuous transformations (37) will become the symmetry transformations for the completely gauge-fixed Lagrangian density ${\mathcal{L}}_{\left(1\right)}$. We christen the infinitesimal transformations in (37) as the dual-gauge transformations (${\delta }_{dg}$) because the gauge-fixing term for the $B_{\mu \nu }$ (and associated fields ${\varphi }_{\mu }$ and $\varphi$) remain invariant.

Before we end this section, we very concisely highlight a few key points connected with the continuous symmetries of the Lagrangian density ${\mathcal{L}}_{\left(2\right)}$ (cf. Equation (32)). In this context, it is very illuminating to point out that the following local, infinitesimal and continuous (dual-)gauge symmetry transformations $\left[{\delta }_{\left(d\right)g}\right]$, namely;

$$\begin{array}{ccc}& & {\delta }_{dg}{B}_{\mu \nu }=-{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\lambda }{\Sigma }^{\xi },{\delta }_{dg}{\tilde{\varphi }}_{\mu }=\pm ({\partial }_{\mu }\Omega -m{\Sigma }_{\mu }),\hfill \\ & & {\delta }_{dg}\tilde{\varphi }=\mp 2[\partial ·\Sigma +m\Omega ],{\delta }_{dg}{\tilde{\Phi }}_{\mu \nu }=\mp m({\partial }_{\mu }{\Sigma }_{\nu }-{\partial }_{\nu }{\Sigma }_{\mu }),\hfill \\ & & {\delta }_{dg}[{\partial }^{\nu }{B}_{\nu \mu },{\varphi }_{\mu },\varphi ,{\Phi }_{\mu \nu }]=0,\hfill \end{array}$$

(39)

$$\begin{array}{ccc}& & {\delta }_g{B}_{\mu \nu }=-({\partial }_{\mu }{\Lambda }_{\nu }-{\partial }_{\nu }{\Lambda }_{\mu }),{\delta }_g{\varphi }_{\mu }=\pm ({\partial }_{\mu }\Lambda -m{\Lambda }_{\mu }),\hfill \\ & & {\delta }_g{\Phi }_{\mu \nu }=\mp m({\partial }_{\mu }{\Lambda }_{\nu }-{\partial }_{\nu }{\Lambda }_{\mu }),{\delta }_g\varphi =\mp 2[(\partial ·\Lambda )+m\Lambda ],\hfill \\ & & {\delta }_g[H_{\mu \nu \lambda },{\tilde{\varphi }}_{\mu },\tilde{\varphi },{\tilde{\Phi }}_{\mu \nu }]=0,\hfill \end{array}$$

(40)

transform the Lagrangian density ${\mathcal{L}}_{\left(2\right)}$ as follows:

$$\begin{array}{ccc}\hfill {\delta }_{dg}{\mathcal{L}}_{\left(2\right)}& =& {\partial }_{\mu }\left[\mp m{\epsilon }^{\mu \nu \lambda \xi }{\Sigma }_{\nu }{\partial }_{\lambda }{\varphi }_{\xi }\right]\pm \left(\partial ·\tilde{\varphi }-\frac{m}{2}\tilde{\varphi }\right)\left[\square +m^2\right]\Omega ,\hfill \\ & +& \left[\frac{1}{2}{\epsilon }^{\mu \nu \lambda \xi }{\partial }_{\nu }{B}_{\lambda \xi }\pm m{\tilde{\varphi }}_{\mu }\pm \frac{1}{2}{\partial }_{\mu }\tilde{\varphi }\right]\left[\square +m^2\right]{\Sigma }_{\mu }\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\delta }_g{\mathcal{L}}_{\left(2\right)}& =& {\partial }_{\mu }\left[\mp m{\epsilon }^{\mu \nu \lambda \xi }{\Lambda }_{\nu }{\partial }_{\lambda }{\tilde{\varphi }}_{\xi }\right]\mp \left(\partial ·\varphi -\frac{m}{2}\varphi \right)\left[\square +m^2\right]\Lambda \hfill \\ & -& \left[{\partial }_{\nu }{B}^{\nu \mu }\pm m{\varphi }^{\mu }\pm \frac{1}{2}{\partial }^{\mu }\varphi \right]\left[\square +m^2\right]{\Lambda }_{\mu }.\hfill \end{array}$$

(41)

It is evident that if we impose the restrictions

$$\begin{array}{ccc}& & (\square +m^2){\Sigma }_{\mu }=0,(\square +m^2){\Lambda }_{\mu }=0,\hfill \\ & & (\square +m^2)\Omega =0,(\square +m^2)\Lambda =0,\hfill \end{array}$$

(42)

on the dual-gauge transformation parameters (${\Sigma }_{\mu },\Omega$) and the gauge transformation parameters (${\Lambda }_{\mu },\Lambda$) from outside, we obtain the (dual-)gauge symmetry transformations (39) and (40) for the Lagrangian density ${\mathcal{L}}_{\left(2\right)}$. We note that the outside restrictions (36), (38), and (42) are exactly the same on the (dual-)gauge transformation parameters of our theory. Hence, when we elevate the Lagrangian densities ${\mathcal{L}}_{\left(1\right)}$ and ${\mathcal{L}}_{\left(2\right)}$ to their counterparts at the quantum level (within the framework of BRST formalism), we shall observe that the Faddeev–Popov ghost terms will be the same for the coupled (but equivalent) (anti-)BRST and (anti-)co-BRST invariant Lagrangian densities. The (anti-)ghost fields will not be restricted from outside for the quantum version of our theory within the ambit of BRST formalism (as the EoMs for the (anti-)ghost fields will take care of them).

5. Linearized Versions of the Lagrangian Densities: Auxiliary Fields and CF-Type Restrictions

We linearize the kinetic term for the $B_{\mu \nu }$ (and associated fields) and all the gauge-fixing terms by invoking the Nakanishi–Lautrup-type auxiliary fields. In this context, first of all, let us focus on the Lagrangian density ${\mathcal{L}}_{\left(1\right)}$, which can be written as:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\left(1\right)}& ⟶& {\mathcal{L}}_{\left(b_1\right)}=\frac{1}{2}{\mathcal{B}}_{\mu }{\mathcal{B}}^{\mu }-{\mathcal{B}}^{\mu }\left[\frac{1}{2}{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\nu }{B}^{\lambda \xi }\pm m{\tilde{\varphi }}_{\mu }\mp \frac{1}{2}{\partial }_{\mu }\tilde{\varphi }\right]\hfill \\ & -& \frac{1}{4}{\Phi }_{\mu \nu }{\Phi }^{\mu \nu }+\frac{1}{4}{\tilde{\Phi }}_{\mu \nu }{\tilde{\Phi }}^{\mu \nu }\pm \frac{m}{2}{B}_{\mu \nu }\left[{\Phi }^{\mu \nu }+\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{\tilde{\Phi }}_{\rho \sigma }\right]\hfill \\ & +& B^{\mu }\left[{\partial }^{\nu }{B}_{\nu \mu }\pm m{\varphi }_{\mu }\mp \frac{1}{2}{\partial }_{\mu }\varphi \right]-\frac{1}{2}{B}^{\mu }{B}_{\mu }-\frac{m^2}{4}{B}_{\mu \nu }{B}^{\mu \nu }\hfill \\ & +& B(\partial ·\varphi +\frac{m}{2}\varphi )+\frac{1}{2}{B}^2-\mathcal{B}(\partial ·\tilde{\varphi }+\frac{m}{2}\tilde{\varphi })-\frac{1}{2}{\mathcal{B}}^2.\hfill \end{array}$$

(43)

The above Lagrangian density leads to the following equations of motion:

$$\begin{array}{ccc}& & {\mathcal{B}}_{\mu }=\frac{1}{2}{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\nu }{B}^{\lambda \xi }\pm m{\tilde{\varphi }}_{\mu }\mp \frac{1}{2}{\partial }_{\mu }\tilde{\varphi },\mathcal{B}=-(\partial ·\tilde{\varphi }+\frac{m}{2}\tilde{\varphi }),\hfill \\ & & B_{\mu }={\partial }^{\nu }{B}_{\nu \mu }\pm m{\varphi }_{\mu }\mp \frac{1}{2}{\partial }_{\mu }\varphi ,B=-(\partial ·\varphi +\frac{m}{2}\varphi ).\hfill \end{array}$$

(44)

where the auxiliary fields $({\mathcal{B}}_{\mu },B_{\mu },\mathcal{B},B)$ are the Nakanishi–Lautrup auxiliary fields, which have been invoked for linearization purposes. For instance, the auxiliary field ${\mathcal{B}}_{\mu }$ has been invoked for the linearization of the kinetic term for the 2-form field $B_{\mu \nu }$ and associated fields. On the other hand, the auxiliary fields $(B_{\mu },B,\mathcal{B})$ have been introduced to linearize the gauge-fixing terms for the $B_{\mu \nu },{\varphi }_{\mu }$ and ${\tilde{\varphi }}_{\mu }$ fields, respectively. In an exactly similar fashion, we can linearize the Lagrangian density ${\mathcal{L}}_{\left(2\right)}$ by invoking a different set of Nakanishi–Lautrup-type auxiliary fields $({\overline{\mathcal{B}}}_{\mu },{\overline{B}}_{\mu },\overline{\mathcal{B}},\overline{B})$ as follows:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\left(2\right)}& ⟶& {\mathcal{L}}_{\left(b_2\right)}=\frac{1}{2}{\overline{\mathcal{B}}}_{\mu }{\overline{\mathcal{B}}}^{\mu }+{\overline{\mathcal{B}}}^{\mu }\left[\frac{1}{2}{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\nu }{B}^{\lambda \xi }\pm m{\tilde{\varphi }}_{\mu }\pm \frac{1}{2}{\partial }_{\mu }\tilde{\varphi }\right]\hfill \\ & -& \frac{1}{4}{\Phi }_{\mu \nu }{\Phi }^{\mu \nu }+\frac{1}{4}{\tilde{\Phi }}_{\mu \nu }{\tilde{\Phi }}^{\mu \nu }\pm \frac{m}{2}{B}_{\mu \nu }\left[{\Phi }^{\mu \nu }+\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{\tilde{\Phi }}_{\rho \sigma }\right]\hfill \\ & -& {\overline{B}}^{\mu }\left[{\partial }^{\nu }{B}_{\nu \mu }\pm m{\varphi }_{\mu }\pm \frac{1}{2}{\partial }_{\mu }\varphi \right]-\frac{1}{2}{\overline{B}}^{\mu }{\overline{B}}_{\mu }-\overline{B}(\partial ·\varphi -\frac{m}{2}\varphi )\hfill \\ & +& \frac{1}{2}{\overline{B}}^2+\overline{\mathcal{B}}(\partial ·\tilde{\varphi }-\frac{m}{2}\tilde{\varphi })-\frac{1}{2}{\overline{\mathcal{B}}}^2-\frac{m^2}{4}{B}_{\mu \nu }{B}^{\mu \nu }.\hfill \end{array}$$

(45)

The above Lagrangian density leads to the following equations of motion w.r.t. the Nakanishi–Lautrup-type auxiliary fields, namely;

$$\begin{array}{ccc}& & \overline{{\mathcal{B}}_{\mu }}=-\left[\frac{1}{2}{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\nu }{B}^{\lambda \xi }\pm m{\tilde{\varphi }}_{\mu }\pm \frac{1}{2}{\partial }_{\mu }\tilde{\varphi }\right],\overline{\mathcal{B}}=(\partial ·\tilde{\varphi }-\frac{m}{2}\tilde{\varphi }),\hfill \\ & & {\overline{B}}_{\mu }=-\left[{\partial }^{\nu }{B}_{\nu \mu }\pm m{\varphi }_{\mu }\pm \frac{1}{2}{\partial }_{\mu }\varphi \right],\overline{B}=(\partial ·\varphi -\frac{m}{2}\varphi ).\hfill \end{array}$$

(46)

It is crystal clear that we can derive the following very useful and interesting relationships amongst the Nakanishi–Lautrup-type auxiliary fields and (pseudo-)scalar fields from the above equations of motion (44) and (46), namely;

$$\begin{array}{c}\hfill {\mathcal{B}}_{\mu }+{\overline{\mathcal{B}}}_{\mu }\pm {\partial }_{\mu }\tilde{\varphi }=0,B+\overline{B}+m\varphi =0,\\ \hfill B_{\mu }+\overline{B_{\mu }}\pm {\partial }_{\mu }\varphi =0,\mathcal{B}+\overline{\mathcal{B}}+m\tilde{\varphi }=0,\end{array}$$

(47)

which are nothing but the (anti-)BRST and (anti-)co-BRST invariant CF-type restrictions on our theory (see, e.g., [10,21] for details).

We end this section with the following remarks. First of all, the Lagrangian densities ${\mathcal{L}}_{\left(b_1\right)}$ and ${\mathcal{L}}_{\left(b_2\right)}$ have been derived in a completely different manner in our present endeavor if we compare our present method of derivation against the derivation in our earlier work [10], where we have exploited the method of trial and error. Second, the CF-type restrictions $B+\overline{B}+m\varphi =0$ and $\mathcal{B}+\overline{\mathcal{B}}+m\tilde{\varphi }=0$ are the same as in our earlier work [10,21], but the other two restrictions in (47) are different. Third, if we stick with the CF-type restrictions that have been derived from the superfield approach to BRST formalism in the context of 4D Abelian 2-form massless and massive gauge theories [18,21], we find that the other two restrictions of (47) are: ${\mathcal{B}}_{\mu }+{\overline{\mathcal{B}}}_{\mu }+{\partial }_{\mu }\tilde{\varphi }=0$ and $B_{\mu }+\overline{B_{\mu }}+{\partial }_{\mu }\varphi =0$. Hence, the $(\pm )$ signs associated with the (pseudo-)scalar fields (e.g., $\pm \frac{1}{2}{\partial }_{\mu }\varphi ,\pm \frac{1}{2}{\partial }_{\mu }\tilde{\varphi }$) are fixed. As a consequence, we find that, in the Lagrangian density ${\mathcal{L}}_{\left(b_1\right)}$, we have only the minus signs for the scalar and pseudo-scalar fields (i.e., $-\frac{1}{2}{\partial }_{\mu }\varphi ,-\frac{1}{2}{\partial }_{\mu }\tilde{\varphi }$) and the plus signs ($\frac{1}{2}{\partial }_{\mu }\varphi ,\frac{1}{2}{\partial }_{\mu }\tilde{\varphi }$) for the exact expression for the Lagrangian density ${\mathcal{L}}_{\left(b_2\right)}$. Fourth, it is straightforward to note that the duality transformations (33) are now generalized in the following form:

$$\begin{array}{ccc}& & B_{\mu \nu }⟶\mp \frac{i}{2}{\epsilon }_{\mu \nu \lambda \xi }{B}^{\lambda \xi },{\varphi }_{\mu }⟶\pm i{\tilde{\varphi }}_{\mu },{\tilde{\varphi }}_{\mu }⟶\mp i{\varphi }_{\mu }\varphi ⟶\pm i\tilde{\varphi },\hfill \\ & & \tilde{\varphi }⟶\mp i\varphi ,{\mathcal{B}}_{\mu }⟶\mp i{B}_{\mu },B_{\mu }⟶\pm i{\mathcal{B}}_{\mu },B⟶\pm i\mathcal{B},\mathcal{B}⟶\mp iB,\hfill \\ & & {\overline{\mathcal{B}}}_{\mu }⟶\mp i{\overline{B}}_{\mu },{\overline{B}}_{\mu }⟶\pm i{\overline{\mathcal{B}}}_{\mu },\overline{B}⟶\pm i\overline{\mathcal{B}},\overline{\mathcal{B}}⟶\mp i\overline{B}.\hfill \end{array}$$

(48)

Under the above discrete duality symmetry transformations, the coupled Lagrangian densities ${\mathcal{L}}_{\left(b_1\right)}$ and ${\mathcal{L}}_{\left(b_2\right)}$ are found to remain invariant even with the fixed choice of signs for the (pseudo-)scalar fields $\tilde{\varphi }$ and $\varphi$. Finally, in the next section, we shall take only the simplest choices of the signs for the (pseudo-)scalar fields within the framework of BRST formalism where the Lagrangian density ${\mathcal{L}}_{\left(b_1\right)}$ will be generalized to incorporate into it the Faddeev–Popov ghost terms by following the standard technique [10,18,21].

6. Nilpotent (co-)BRST Invariant Lagrangian Density

We have generalized the Lagrangian densities ${\mathcal{L}}_{\left(b_1\right)}$ and ${\mathcal{L}}_{\left(b_2\right)}$ to their counterpart nilpotent (anti-)BRST and (anti-)co-BRST invariant Lagrangian densities ${\mathcal{L}}_{\mathcal{B}}$ and ${\mathcal{L}}_{\overline{\mathcal{B}}}$ that incorporate the Faddeev–Popov ghost terms. Such a set of coupled (but equivalent) Lagrangian densities have been written in our earlier works [10,21]. However, we shall focus on only one Lagrangian density and discuss the importance of discrete duality symmetry transformations (48) (and (54) below) which will connect the BRST transformations with the co-BRST transformations and vice versa. This kind of connection exists for the anti-BRST and anti-co-BRST symmetries, as well. However, we shall not dwell on the latter as it will be only an academic exercise. We would like to emphasize that, in our earlier works [10,21], such kinds of relationships have not been established where only the analogue of the Hodge duality operator (i.e., the set of discrete duality symmetry transformations) play a decisive role (along with the replacements: $s_b\iff s_d$). This observation is totally different from (60).

Towards the above goal in mind, we begin with the following (co-)BRST invariant Lagrangian density7 (where ${\mathcal{L}}_{\left(b_1\right)}⟶{\mathcal{L}}_{\mathcal{B}}$) (see, e.g., [9,10,21])

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\mathcal{B}}& =& \frac{1}{2}{\mathcal{B}}_{\mu }{\mathcal{B}}^{\mu }-{\mathcal{B}}^{\mu }\left[\frac{1}{2}{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\nu }{B}^{\lambda \xi }+m{\tilde{\varphi }}_{\mu }-\frac{1}{2}{\partial }_{\mu }\tilde{\varphi }\right]-\frac{m^2}{4}{B}_{\mu \nu }{B}^{\mu \nu }\hfill \\ & -& \frac{1}{4}{\Phi }_{\mu \nu }{\Phi }^{\mu \nu }+\frac{1}{4}{\tilde{\Phi }}_{\mu \nu }{\tilde{\Phi }}^{\mu \nu }+\frac{m}{2}{B}_{\mu \nu }\left[{\Phi }^{\mu \nu }+\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{\tilde{\Phi }}_{\rho \sigma }\right]\hfill \\ & +& B^{\mu }\left[{\partial }^{\nu }{B}_{\nu \mu }+m{\varphi }_{\mu }-\frac{1}{2}{\partial }_{\mu }\varphi \right]-\frac{1}{2}{B}^{\mu }{B}_{\mu }+B\left(\partial ·\varphi +\frac{m}{2}\varphi \right)\hfill \\ & +& \frac{1}{2}{B}^2-\mathcal{B}\left(\partial ·\tilde{\varphi }+\frac{m}{2}\tilde{\varphi }\right)-\frac{1}{2}{\mathcal{B}}^2-\frac{1}{2}{\partial }_{\mu }\overline{\beta }{\partial }^{\mu }\beta +\frac{m^2}{2}\overline{\beta }\beta \hfill \\ & -& ({\partial }_{\mu }{\overline{C}}_{\nu }-{\partial }_{\nu }{\overline{C}}_{\mu })\left({\partial }^{\mu }{C}^{\nu }\right)+({\partial }_{\mu }\overline{C}-m{\overline{C}}_{\mu })({\partial }^{\mu }C-m{C}^{\mu })\hfill \\ & -& \frac{1}{2}\left(\partial ·\overline{C}+m\overline{C}+\frac{\rho }{4}\right)\lambda -\frac{1}{2}\left(\partial ·C+mC-\frac{\lambda }{4}\right)\rho ,\hfill \end{array}$$

(49)

where $\left(\overline{\beta }\right)\beta$ are the bosonic (anti-)ghost fields with ghost numbers $(-2)+2$, respectively, and $\left({\overline{C}}_{\mu }\right)C_{\mu }$ are the fermionic$({\overline{C}}_{\mu }{C}_{\nu }+C_{\nu }{\overline{C}}_{\mu }=0,{\overline{C}}_{\mu }{\overline{C}}_{\nu }+{\overline{C}}_{\nu }{\overline{C}}_{\mu }=0,C_{\mu }^2={\overline{C}}_{\mu }^2=0,$ etc.) (anti-)ghost fields with ghost numbers $(-1)+1$, respectively. In addition, we have Lorentz scalar fermionic ($C\overline{C}+\overline{C}C=0,C^2={\overline{C}}^2=0$, etc.) (anti-)ghost fields with ghost numbers $(-1)+1$, respectively. Our theory also contains the auxiliary fermionic (${\rho }^2={\lambda }^2=0,\rho \lambda +\lambda \rho =0$) fields $\left(\rho \right)\lambda$ that carry the ghost numbers $(-1)+1$, respectively.

The above Lagrangian density respects the following off-shell nilpotent $(s_b^2=0)$ BRST symmetry transformations $\left(s_b\right)$, namely;

$$\begin{array}{ccc}& & s_b{B}_{\mu \nu }=-({\partial }_{\mu }{C}_{\nu }-{\partial }_{\nu }{C}_{\mu }),s_b{C}_{\mu }=-{\partial }_{\mu }\beta ,s_b{\overline{C}}_{\mu }=B_{\mu },\hfill \\ & & s_b\overline{\beta }=-\rho ,s_b{\varphi }_{\mu }=+({\partial }_{\mu }C-m{C}_{\mu }),s_b\overline{C}=B,s_b\varphi =+\lambda ,\hfill \\ & & s_{b}C=-m\beta ,s_b[H_{\mu \nu \lambda },B,\lambda ,\rho ,B_{\mu },{\mathcal{B}}_{\mu },\beta ,\mathcal{B},{\tilde{\varphi }}_{\mu },{\tilde{\varphi }}_{\mu \nu },\tilde{\varphi }]=0,\hfill \end{array}$$

(50)

because the Lagrangian density ${\mathcal{L}}_{\mathcal{B}}$ transforms as [10]

$$\begin{array}{ccc}\hfill s_b{\mathcal{L}}_{\mathcal{B}}& =& {\partial }_{\mu }[-m{\epsilon }^{\mu \nu \lambda \xi }{\tilde{\varphi }}_{\nu }{\partial }_{\lambda }{C}_{\xi }-({\partial }^{\mu }{C}^{\nu }-{\partial }^{\nu }{C}^{\mu })B_{\nu }-\frac{1}{2}\lambda B^{\mu }\hfill \\ & +& B({\partial }^{\mu }C-m{C}^{\mu })+\frac{1}{2}\rho \left({\partial }^{\mu }\beta \right)],\hfill \end{array}$$

(51)

which implies that the action integral $S=\int d^{4}x{\mathcal{L}}_{\mathcal{B}}$ remains invariant (i.e., $s_{b}S=0$) under the infinitesimal, continuous, and nilpotent BRST transformations (50). This happens because of Gauss’s divergence theorem, due to which all the physical fields vanish off as $x⟶\pm \infty$. In addition to $s_b$, the Lagrangian density ${\mathcal{L}}_{\mathcal{B}}$ also respects the infinitesimal, continuous, and nilpotent $[s_d^2=0]$ co-BRST (i.e., dual-BRST) transformations $\left(s_d\right)$ [10]:

$$\begin{array}{ccc}& & s_d{B}_{\mu \nu }=-{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\lambda }{\overline{C}}^{\xi },s_d{\overline{C}}_{\mu }=-{\partial }_{\mu }\overline{\beta },s_d{C}_{\mu }={\mathcal{B}}_{\mu },\hfill \\ & & s_d\beta =-\lambda ,s_d{\tilde{\varphi }}_{\mu }=+({\partial }_{\mu }\overline{C}-m{\overline{C}}_{\mu }),s_{d}C=\mathcal{B},s_d\overline{C}=-m\overline{\beta },\hfill \\ & & s_d\tilde{\varphi }=-\rho ,s_d[{\partial }^{\nu }{B}_{\nu \mu },{\mathcal{B}}_{\mu },B_{\mu },\mathcal{B},{\varphi }_{\mu },{\Phi }_{\mu \nu },\varphi ,\overline{\beta },\lambda ,\rho ]=0.\hfill \end{array}$$

(52)

It is straightforward to check that ${\mathcal{L}}_{\mathcal{B}}$ transforms, under $\left(s_d\right)$, as the total spacetime derivative in the four (3 + 1)-dimensional (4D) spacetime, namely;

$$\begin{array}{ccc}\hfill s_d{\mathcal{L}}_{\mathcal{B}}& =& {\partial }_{\mu }[-m{\epsilon }^{\mu \nu \lambda \xi }{\varphi }_{\nu }{\partial }_{\lambda }{\overline{C}}_{\xi }+({\partial }^{\mu }{\overline{C}}^{\nu }-{\partial }^{\nu }{\overline{C}}^{\mu }){\mathcal{B}}_{\nu }-\frac{1}{2}\rho {\mathcal{B}}^{\mu }\hfill \\ & -& ({\partial }^{\mu }\overline{C}-m{\overline{C}}^{\mu })\mathcal{B}+\frac{1}{2}\lambda \left({\partial }^{\mu }\overline{\beta }\right)].\hfill \end{array}$$

(53)

As a consequence of the above observation, we find that the action integral $S=\int d^{4}x{\mathcal{L}}_{\mathcal{B}}$ remains invariant (i.e., $s_{d}S=0$) under the co-BRST symmetry transformation $s_d$ for all the physical fields that vanish off as $x⟶\pm \infty$.

In addition to discrete duality symmetry transformations (48) in the bosonic (i.e., non-ghost) sector of the Lagrangian density ${\mathcal{L}}_{\mathcal{B}}$, we have the following discrete symmetry transformations in the ghost-sector8:

$$\begin{array}{ccc}& & C_{\mu }⟶\pm i{\overline{C}}_{\mu },{\overline{C}}_{\mu }⟶\pm i{C}_{\mu },C⟶\pm i\overline{C},\overline{C}⟶\pm iC,\hfill \\ & & \rho ⟶\mp i\lambda ,\lambda ⟶\mp i\rho ,\beta ⟶\pm i\overline{\beta },\overline{\beta }⟶\mp i\beta .\hfill \end{array}$$

(54)

Under the full discrete duality symmetry transformations (48) and (54), it can be checked that the (co-)BRST symmetry transformations (32) and (50) are interconnected. To corroborate this claim, let us begin with $s_b{B}_{\mu \nu }=-({\partial }_{\mu }{C}_{\nu }-{\partial }_{\nu }{C}_{\mu })$. If we apply the discrete symmetry transformations (48) and (54) on it and take the replacement: $s_b⟶s_d$, we obtain the following explicit relationship:

$$\begin{array}{c}\hfill s_b(\ast B_{\mu \nu })=-\ast ({\partial }_{\mu }{C}_{\nu }-{\partial }_{\nu }{C}_{\mu })⟹s_d{B}_{\mu \nu }=-{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\lambda }{\overline{C}}^{\xi },\end{array}$$

(55)

where ∗ is nothing but the full discrete duality symmetry transformations (48) plus (54). In other words, we have obtained the co-BRST symmetry transformation $s_d$ operating on $B_{\mu \nu }$ field from the operation of $s_b$ on $B_{\mu \nu }$. In an exactly similar fashion, we note the following (with the replacement: $s_d⟶s_b$), for the transformations $s_d{B}_{\mu \nu }=-{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\nu }{\overline{C}}^{\xi }$, namely;

$$\begin{array}{c}\hfill s_d(\ast B_{\mu \nu })=-\ast {\epsilon }_{\mu \nu \lambda \xi }\left({\partial }^{\lambda }{\overline{C}}^{\xi }\right)⟶s_b{B}_{\mu \nu }=-({\partial }_{\mu }{C}_{\nu }-{\partial }_{\nu }{C}_{\mu }),\end{array}$$

(56)

where, once again, the ∗ operation is nothing but the total discrete duality symmetry transformations (48) plus (54). This observation is not limited only to the bosonic antisymmetric tensor gauge field. To corroborate this assertion, let us focus on the symmetry transformation: $s_b{\varphi }_{\mu }=+({\partial }_{\mu }C-m{C}_{\mu })$ on a bosonic vector field $\left({\varphi }_{\mu }\right)$. By exploiting the strength of the full discrete duality symmetry transformations (48) plus (54), we observe the following transformations on the axial-vector field (with input: $s_b⟶s_d$), namely;

$$\begin{array}{c}\hfill s_b(\ast {\varphi }_{\mu })=+\ast ({\partial }_{\mu }C-m{C}_{\mu })⟹s_d{\tilde{\varphi }}_{\mu }=+({\partial }_{\mu }\overline{C}-m{\overline{C}}_{\mu }).\end{array}$$

(57)

This happens because, under discrete duality symmetry transformations (48), we have: ${\varphi }_{\mu }\to \pm i{\tilde{\varphi }}_{\mu }$ and ${\tilde{\varphi }}_{\mu }\to \mp i{\varphi }_{\mu }$. In an exactly, similar fashion, we obtain the reciprocal symmetry transformations as follows (with inputs: $s_d⟶s_b$ and use of the discrete duality symmetry transformations), namely;

$$\begin{array}{c}\hfill s_d(\ast {\tilde{\varphi }}_{\mu })=+\ast ({\partial }_{\mu }\overline{C}-m{\overline{C}}_{\mu })⟹s_b{\varphi }_{\mu }=+({\partial }_{\mu }C-m{C}_{\mu }).\end{array}$$

(58)

The above kind of exercise can be repeated with all the fields of our theory. We observe that the discrete duality symmetry transformations (48) and (54) are the generalization of our basic discrete duality symmetry transformations ($B_{\mu \nu }⟶\mp (i/2){\epsilon }_{\mu \nu \lambda \xi }{B}^{\lambda \xi },{\varphi }_{\mu }⟶\pm i{\tilde{\varphi }}_{\mu },{\tilde{\varphi }}_{\mu }⟶\mp i{\varphi }_{\mu }$) of the modified Stückelberg formalism (cf. Equations (9) and (10)). To complete our present discussion, let us focus on a transformation on a fermionic field $s_d\overline{C}=-m\overline{\beta }$. Using the strength of the discrete duality symmetry transformations (54), we obtain the following (with the input: $s_d⟶s_b$), namely;

$$\begin{array}{c}\hfill s_d[\ast \left(\overline{C}\right)]=-m\ast \overline{\beta }⟹s_{b}C=-m\beta .\end{array}$$

(59)

Thus, we are able to obtain the BRST symmetry transformation: $s_{b}C=-m\beta$ from the co-BRST symmetry transformation: $s_d\overline{C}=-m\overline{\beta }$ by exploiting the strength of the discrete duality symmetry transformations (54). Hence, our observation is true for fermionic field, as well. It goes without saying that, repeating the same procedure, we can obtain: $s_d\overline{C}=-m\overline{\beta }$ from the given BRST symmetry transformation: $s_{b}C=-m\beta$. Thus, the discrete duality symmetry transformations (48) plus (54) connect the BRST and co-BRST transformations for the bosonic as well as the fermionic fields of our theory.

We end this section with the following remarks. First, the discrete duality symmetry transformations (48) and (54) are able to provide a connection between the symmetry transformations $s_b$ and $s_d$. Second, it can be seen that the interplay of the discrete and continuous symmetry transformations provides [10] the physical realization of $\delta =\pm \ast d\ast$ that exist [2,3,4,5] between the (co-)exterior derivatives ($\left(\delta \right)d$) of differential geometry. This interesting and beautiful relationship between $s_d$ and $s_b$ is9

$$\begin{array}{c}\hfill s_d=\pm \ast s_b\ast ,\end{array}$$

(60)

where ∗ is nothing but the complete set of discrete duality symmetry transformations (48) and (54). Third, despite the above connections between the BRST and co-BRST symmetry transformations in the language of the symmetry properties of our theory, these symmetries are independent of each-other in the same manner as do the exterior $\left(d\right)$ and co-exterior $\left(\delta \right)$ derivatives of differential geometry [2,3,4,5] even though these derivatives are connected with each other by the relationship: $\delta =\pm \ast d\ast$. Finally, it can be seen that the exactly similar kinds of relationships exist between the nilpotent anti-co-BRST symmetry and anti-BRST symmetry transformations that exist for the Lagrangian density ${\mathcal{L}}_{\overline{\mathcal{B}}}$ (which turns out to be the generalization of the Lagrangian density ${\mathcal{L}}_{b_2}$ (see e.g., [9,10] for details)).

7. Conclusions

The Stückelberg-modified massive 4D free Abelian 2-form theory has already been proven to be a massive model of Hodge theory [10], where its discrete and continuous symmetry transformations (and corresponding conserved charges) have been shown to provide the physical realizations of the de Rham cohomological operators [2,3,4,5] of the differential geometry at the algebraic level within the framework of BRST formalism [10]. However, the full coupled (but equivalent) Lagrangian densities of this theory have been obtained by the trial and error method. In our present investigation, we have theoretically derived the correct forms of the coupled (but equivalent) Lagrangian densities. To be precise, we have concentrated only on the (co-)BRST invariant Lagrangian density (cf. Section 6) for the sake of brevity but indicated the theoretical methodology for the derivation of the coupled (but equivalent) Lagrangian densities that respect six continuous and a couple of useful discrete duality symmetry transformations (see, e.g., [10]) within the framework of BRST formalism. The above set of symmetries entail upon this model (i.e., the 4D massive Abelian 2-form theory) to become a massive field-theoretic example of Hodge theory.

One of the key results of our present investigation is the modification (cf. Equations (7) and (9)) of the Stückelberg formalism on the 4D flat Minkowskian spacetime manifold where the ideas from the differential geometry have played very important roles. It has been demonstrated that the modified SF remains form-invariant under the discrete duality symmetry transformations (cf. Equation (10)), whose generalizations (cf. Equations (48) and (54)), within the realm of BRST formalism, provide the physical realizations of the Hodge duality ∗ operator of the differential geometry. As the gauge-fixed Lagrangian density (29) remains invariant under the discrete duality symmetry transformations (10), in an exactly similar fashion, the (co-)BRST invariant Lagrangian density (49) remains invariant under the generalization of the discrete duality symmetry transformations (cf. Equation (10)): $\left(i\right)$ to Equation (48) in the non-ghost sector and $\left(ii\right)$ to Equation (54) in the ghost-sector of the Lagrangian density (49). In addition, we have been able to establish a direct connection between the BRST and co-BRST symmetry transformations (i.e., $s_b\leftrightarrow s_d$) due to the existence of the discrete duality symmetry transformations (48) and (54), which is a novel result in our present investigation. The latter symmetry transformations ($s_b$ and $s_d$) also play an important role [10] in providing the analogue of relationship: $\delta =\pm \ast d\ast$ in the terminology of nilpotent symmetry transformations of our present massive 4D theory (cf. Equation (60)).

It is worthwhile to mention that the modified SF (cf. Equation (9)) is invariant under the discrete duality symmetry transformations (10), and they lead to the combination of the polar vector and axial-vector fields (${\varphi }_{\mu }$ and ${\tilde{\varphi }}_{\mu }$) in the form: ${\partial }_{\mu }{\varphi }_{\nu }-{\partial }_{\nu }{\varphi }_{\mu }+{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\lambda }{\tilde{\varphi }}^{\xi }$. Exactly the same combination has been taken by Zwanziger [22] in the description of the (electromagnetic global duality invariant) 4D Maxwell theory of electrodynamics with double potentials with the field strength tensor as: $F_{\mu \nu }={\partial }_{\mu }{V}_{\nu }-{\partial }_{\nu }{V}_{\mu }+{\epsilon }_{\mu \nu \lambda \xi }{\partial }^{\lambda }{A}^{\xi }$, where $V_{\mu }$ and $A_{\mu }$ are the polar vector and axial-vector potentials, respectively. We have discussed the local duality invariance [23] of the Maxwell theory with these potentials and shown the existence of an axial photon which mediates the spin-spin universal long-range interaction (see, e.g., [23,24] for details). However, we have not discussed the applications of the axial-vector potential $A_{\mu }$ in the context of dark energy/dark matter. On the contrary, a close and careful look at the Lagrangian densities (31) and (32) demonstrates that the fields ${\tilde{\varphi }}_{\mu }$ and $\tilde{\varphi }$ turn up with negative kinetic terms in our theory, which are interesting in the sense that they belong to a class of exotic fields that are supposed to be one of the possible set of candidates for the dark matter/dark energy [19,20] and the “phantom” and/or “ghost” fields in the context of the modern developments in the cyclic, bouncing and self-accelerated cosmological models of the Universe [13,14,15], which take care of the modern experimental observation of the accelerated expansion of the Universe.

In a set of very nice works [25,26,27], the Stückelberg-modified (SUSY) quantum electrodynamics and other aspects of the (non-)interacting Abelian gauge theories have been considered, where an ultralight dark matter candidate has been proposed (and the Stückelberg boson has been able to cure the infrared problem in QED). It will be an interesting idea to apply our BRST approach to the examples that have been considered in [25,26,27]. Furthermore, we have already established that the 6D Abelian 3-form gauge theory is a model of Hodge theory within the ambit of BRST formalism [9]. It will be a nice future endeavor to extend our understandings of the 2D Stückelberg-modified Proca (i.e., the massive Abelian 1-form) theory [8] as well as our present work (on the Stückelberg-modified massive 4D free Abelian 2-form theory) to study the Stückelberg-modified massive 6D Abelian 3-form theory within the framework of BRST formalism. In a very recent work [28], a prototype system of first-class constraints and various kinds of BRST-type symmetries and their relationships have been established. It will be interesting to see weather the brand-new BRST-type symmetries (that have been pointed out in [28]) can be accommodated within the framework of field-theoretic models of Hodge theory. We are involved with these ideas at present, and we shall report on our progress elsewhere in our future publication(s).