Modified St$\ddot{u}$ckelberg Formalism: Massive
4D Abelian 2-Form Theory

A. K. Rao(a), R. P. Malik(a,b)
(a) Physics Department, Institute of Science,
Banaras Hindu University, Varanasi - 221 005, (U.P.), India

(b) DST Centre for Interdisciplinary Mathematical Sciences,
Institute of Science, Banaras Hindu University, Varanasi - 221 005, India
e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract: We demonstrate that the celebrated St$\ddot{u}$ckelberg formalism
gets 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 St$\ddot{u}$ckelberg-modified massive Abelian
2-form gauge theory. The modification in the usual St$\ddot{u}$ckelberg
technique is backed by the precise mathematical arguments from differential
geometry where the (co-)exterior derivatives play decisive roles. The modified
version of the St$\ddot{u}$ckelberg technique remains invariant under the
discrete duality symmetry transformations which also establish a precise and
deep connection between the nilpotent (anti-)BRST and (anti-)co-BRST symmetry
transformations of our 4D theory.

PACS numbers: 03.70.+k; 11.30.-j; 02.40.-k.

Keywords: Modified St$\ddot{u}$ckelberg technique; (anti-)BRST symmetries;
(anti-)co-BRST symmetries; discrete duality symmetries; conserved and
nilpotent charges; fields with negative kinetic terms; possible candidates of
dark matter/dark energy

##  1 Introduction

The concepts 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 in the realm of modern-day science. In particular,
the recent developments in 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 domain of theoretical research activities related to the
gauge theories, gravitational theories, (super)string theories, topological
field theories, etc. In this context, it is pertinent to point out that the
celebrated St$\ddot{u}$ckelberg-technique of compensating field(s) [1],
responsible for the massive field theories (e.g. Proca theory) to acquire the
beautiful gauge symmetry invariance, is also based on the ideas of the
differential geometry (see, e.g. [2-5] for details). In particular, the
exterior derivative ($d=d\,x^{\mu}\,\partial_{\mu},\,d^{2}=0$) plays a key
role [see, e.g. Eq. (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 Proca theory) which converts the second-class
constraints of the massive theory to the first-class constraints (see, e.g.
[6,7] for details). The latter appear in the expression of generator for the
gauge symmetry transformations (of the St$\ddot{u}$ckelberg-modified theory).

One of central purposes of our present endeavor is to demonstrate that the
standard St$\ddot{u}$ckelberg-technique gets modified in the context of
massive Abelian $p$-form ($p=1,2,3,...$) 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
appear in 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. the massive Abelian 1-form) theory. In fact,
we have been able to demonstrate that, due to the modified version of the
St$\ddot{u}$ckelberg formalism (SF), we obtain the nilpotent (anti-)BRST and
(anti-)co-BRST symmetries for the gauge-fixed Lagrangian density of the
St$\ddot{u}$ckelberg-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 the spacetime) have been proven to be field-theoretic
models of Hodge theory (see, e.g. [9] and reference therein). Further, we have
been able to show that the St$\ddot{u}$ckelberg-modified massive 2D Abelian
1-form (i.e. Proca theory) and 4D Abelian 2-form theory are, once again, very
interesting examples of Hodge theory within the framework of BRST formalism
(see, e.g. [10-12]).

The central theme of our present investigation is to show that the
St$\ddot{u}$ckelberg-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 $(i)$ the modification in the
standard St$\ddot{u}$ckelberg technique [cf. Eq. (2)] where an axial-vector
field $(\tilde{\phi}_{\mu})$ also appears explicitly [cf. Eq. (7)] backed by
the precise mathematical arguments, and $(ii)$ the existence of a set of
discrete duality symmetry transformations under which the modified
St$\ddot{u}$ckelberg-technique [cf. Eq. (9)] as well as the 4D Lagrangian
density ${\cal L}$ [cf. Eq. (29)] both remain invariant. The generalization of
these discrete duality symmetry transformations, within the realm of BRST
formalism [cf. Eqs. (48), (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-)BRST and
(anti-)co-BRST transformations have their own identities as they provide the
physical realizations [12] of the (co-)exterior derivatives of the
differential geometry [2-5] 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. Sec. 6) that is the generalization
of ${\cal L}_{(b_{1})}$ [cf. Eq. (43)] and establish a connection between the
BRST and co-BRST symmetry transformations due to the existence of a couple of
discrete duality symmetry transformations (48) plus (54). In exactly similar
fashion, the generalization of the Lagrangian density ${\cal L}_{(b_{2})}$
[cf. Eq. (45)] can be obtained at the quantum level (within the framework of
BRST formalism) as ${\cal L}_{\bar{\cal B}}$ which will be anti-BRST as well
as anti-co-BRST invariant [12]. 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) and (54) at the quantum level
(see, e.g. [12, 9] for details). Besides this connection, there exists another
relationship between the co-BRST and BRST symmetry transformations [cf. Eq.
(60)] which provide the physical realization of the relationship between co-
exterior and exterior derivative of differential geometry [2-5].

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 St$\ddot{u}$ckelberg-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 out 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 of our present
theory because such kinds of exotic fields are the possible candidates for the
dark matter and dark energy and they play an important role in the context of
the cyclic, bouncing and self-accelerated cosmological models of Universe
(see, e.g. [13-15] and references therein). Third, we desire to establish a
connection between the nilpotent (anti-)BRST and (anti-)co-BRST
transformations on the basis of discrete duality symmetry transformations [cf.
Eqs. (48), (54)] in our theory. Fourth, the higher $p$-form ($p=2,3,...$)
gauge theories of massless and massive varieties are interesting from the
point of view of (super)string theories as they appear in their excitations.
Finally, we wish to find out the physical realizations of the Hodge duality
operator of differential geometry [2-5] in terms of the discrete duality
symmetry transformations.

The theoretical material of our present endeavor is organized as follows. In
Sec. 2, we discuss the bare essentials of the gauge symmetry transformations
for the standard St$\ddot{u}$ckelberg-modified Lagrangian density in any
arbitrary D-dimension of spacetime. Our Sec. 3 deals with the modification of
the St$\ddot{u}$ckelberg-formalism where the (co-)exterior derivatives of
differential geometry play decisive roles. The subject matter of Sec. 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 outside. Our Sec. 5
contains the theoretical discussion on the linearized versions of the gauge-
fixed Lagrangian densities and Curci-Ferrari (CF) type restrictions. In Sec.
6, we establish a relationship between the BRST and co-BRST symmetry
transformations due to the discrete duality symmetry transformations [cf. Eqs.
(48), (54)] in our BRST-invariant theory. Finally, in Sec. 7, we make some
concluding remarks and point out a few future theoretical directions for
further investigation(s).

In our Appendix A, we very briefly recapitulate the bare essentials of our
earlier work [8] on the St$\ddot{u}$ckelberg-modified 2D Proca theory where
modified-SF has been used. The theoretical contents of our Appendix B is
devoted to the generalization of the classical symmetry transformations (37)
and (35) to their quantum counterparts (co-)BRST symmetry transformations for
the appropriate (co-)BRST invariant Lagrangian density.

Convention and Notations: We follow the convention of the left-derivative
w.r.t. all the fermionic (i.e.
$\bar{C}_{\mu},\,C_{\mu},\,\bar{C},\,C,\,\rho,\,\lambda,$) fields of our
theory in the context of the derivation of the equations of motions,
definition of the conjugate momenta and deduction of the Noether conserved
currents and charges. The 4D Levi-Civita tensor is denoted by
$\varepsilon_{\mu\nu\lambda\xi}$ with conventions:
$\varepsilon_{0123}=+1=-\,\varepsilon^{0123}$ and
$\varepsilon_{\mu\nu\lambda\xi}\,\varepsilon^{\mu\nu\lambda\xi}=-\,4!$,
$\varepsilon_{\mu\nu\lambda\xi}\,\varepsilon^{\mu\nu\lambda\rho}=-\,3!\,\delta_{\xi}^{\rho}$,
$\varepsilon_{\mu\nu\lambda\xi}\,\varepsilon^{\mu\nu\rho\sigma}=-\,2!\,(\delta_{\lambda}^{\rho}\,\delta_{\xi}^{\sigma}-\delta_{\xi}^{\rho}\,\delta_{\lambda}^{\sigma})$,
etc., where the Greek indices $\mu,\,\nu,\,\lambda,...=0,1,2,3$ stand for the
time and space directions and the Latin indices $i,\,j,\,k...=1,2,3$
correspond to space directions only. Hence, the 3D Levi-Civita tensor is
$\epsilon_{ijk}=\varepsilon_{0ijk}$. 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: $P\cdot
Q=\eta_{\mu\nu}\,P^{\mu}\,Q^{\nu}=P_{0}\,Q_{0}-P_{i}\,Q_{i}$. We denote the
(anti-)BRST transformations by $s_{(a)b}$ and the notation $s_{(a)d}$ stands
for the (anti-)dual [i.e. (anti-)co]-BRST transformations. In the same way,
the conserved and nilpotent (anti-)BRST and (anti-)co-BRST charges are denoted
by $Q_{(a)b}$ and $Q_{(a)d}$, respectively (corresponding to the above
nilpotent symmetries).

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 $(\delta)\,d$
are called as the (co-)exterior derivatives that are connected with each-other
by the relationship: $\delta=\pm\,*\,d\,*$ 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 [2-5]. This algebra (which is not a
Lie algebra) is popularly known as the Hodge algebra and $\Delta$ behaves like
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 in our
present endeavor.

##  2 Preliminaries: St$\ddot{u}$ckelberg Formalism in Any Arbitrary Dimension
of Spacetime

We begin with the D-dimensional Lagrangian density $({\cal L}_{0})$ for the
massive Abelian 2-form $(B^{(2)}=[(d\,x^{\mu}\wedge
d\,x^{\nu})/{2!}]\,B_{\mu\nu})$ 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)

$\displaystyle{\cal
L}_{0}=\frac{1}{12}\,H^{\mu\nu\lambda}\,H_{\mu\nu\lambda}-\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu},$
(1)

where $H^{(3)}=d\,B^{(2)}\equiv[(d\,x^{\mu}\wedge d\,x^{\nu}\wedge
d\,x^{\lambda})/3!]\,H_{\mu\nu\lambda}$ defines the kinetic term (with
$H_{\mu\nu\lambda}=\partial_{\mu}\,B_{\nu\lambda}+\partial_{\nu}\,B_{\lambda\mu}+\partial_{\lambda}\,B_{\mu\nu}$
for the anti-symmetric tensor field $B_{\mu\nu}$ where the Greek indices
$\mu,\,\nu,\,\lambda...=0,1,...,D-1$). It is straightforward to check that the
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 $m^{2}\neq 0$. As a consequence, we observe that
$B_{\mu\nu}$ field obeys the Klein-Gordon equation
$(\Box+m^{2})\,B_{\mu\nu}=0$. 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 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 St$\ddot{u}$ckelberg-formalism (SF) [17] related with
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]

$\displaystyle B_{\mu\nu}\longrightarrow
B_{\mu\nu}\mp\frac{1}{m}\,(\partial_{\mu}\,\phi_{\nu}-\partial_{\nu}\,\phi_{\mu}),$
(2)

where the Abelian 1-form $\Phi^{(1)}=d\,x^{\mu}\,\phi_{\mu}$ defines the
vector field $\phi_{\mu}$. 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 $\phi_{\mu}$ fields are same in the D-dimensional spacetime when we use
the natural units: $\hbar=c=1$. Hence, the rest mass $m$ should be present in
the denominator of Eq. (2) to balance the mass dimension on the l.h.s and
r.h.s. of Eq. (2) in the above natural units. The mass term in Eq. (1) changes
as follows, due to (2), namely;

$\displaystyle-\,\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}\longrightarrow-\,\frac{m^{2}}{4}\big{[}B_{\mu\nu}\mp\frac{1}{m}\,(\partial_{\mu}\,\phi_{\nu}-\partial_{\nu}\,\phi_{\mu})\big{]}\,\big{[}B^{\mu\nu}\mp\frac{1}{m}\,(\partial^{\mu}\,\phi^{\nu}-\partial^{\nu}\,\phi^{\mu})\big{]}.$
(3)

Let us define an Abelian 2-form $F^{(2)}=d\,\Phi^{(1)}=[(d\,x^{\mu}\wedge
d\,x^{\nu})/2!]\,\Phi_{\mu\nu}$ where
$\Phi_{\mu\nu}=\partial_{\mu}\,\phi_{\nu}-\partial_{\nu}\,\phi_{\mu}$ is the
antisymmetric field strength tensor for the vector field $\phi_{\mu}$. With
all these inputs, we obtain the St$\ddot{u}$ckelberg-modified Lagrangian
density ${\cal L}_{S}$ from ${\cal L}_{0}$ as

$\displaystyle{\cal L}_{0}\longrightarrow{\cal
L}_{S}=\frac{1}{12}\,H^{\mu\nu\lambda}\,H_{\mu\nu\lambda}-\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}\pm\frac{m}{2}\,B_{\mu\nu}\,\Phi^{\mu\nu}-\frac{1}{4}\,\Phi_{\mu\nu}\,\Phi^{\mu\nu},$
(4)

which (i.e. the Lagrangian density ${\cal L}_{S}$) respects the following
continuous and infinitesimal gauge symmetry transformations $\delta_{g}$,
namely;

$\displaystyle\delta_{g}\,H_{\mu\nu\lambda}=0,\qquad\qquad\qquad\qquad\;\;\delta_{g}\,\phi_{\mu}=\pm\,(\partial_{\mu}\,\Lambda-m\,\Lambda_{\mu}),$
$\displaystyle\delta_{g}\,B_{\mu\nu}=-\,(\partial_{\mu}\,\Lambda_{\nu}-\partial_{\nu}\,\Lambda_{\mu}),\qquad\delta_{g}\,\Phi_{\mu\nu}=\mp\,m\,(\partial_{\mu}\,\Lambda_{\nu}-\partial_{\nu}\,\Lambda_{\mu}),$
(5)

where the Lorentz-scalar $\Lambda(x)$ and Lorentz-vector $\Lambda_{\mu}(x)$
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:
$\phi_{\mu}\rightarrow\phi_{\mu}\pm\partial_{\mu}\,\Lambda$ can be
accommodated in the standard St$\ddot{u}$ckelberg-technique [considered in
(2)] without changing it in any way. This is why, in the gauge transformation
of $\phi_{\mu}$ field [cf. Eq. (5)], we have the local Lorentz scalar
transformation parameter $\Lambda(x)$. It is straightforward to check that:
$\delta_{g}\,{\cal L}_{S}=0$ implying that the St$\ddot{u}$ckelberg-modified
Lagrangian density ${\cal L}_{S}$ respects the infinitesimal local gauge
symmetry transformations (5) in a perfect manner. In other words, the
St$\ddot{u}$ckelberg modified Lagrangian density $({\cal L}_{S})$ does not
transform to a total spacetime derivative that will ensure the gauge symmetry
invariance of the action integral. We mention, in passing, that the second-
class constraints of ${\cal L}_{0}$ have been converted into the first-class
constraints [due to the introduction of the St$\ddot{u}$ckelberg pure vector
field $\phi_{\mu}$ in (2)]. The ensuing first-class constraints are the
generators for the infinitesimal gauge symmetry transformations $(\delta_{g})$
in (5). These statements are true in any arbitrary D-dimension of spacetime
[17].

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

In the differential-form terminology, the standard St$\ddot{u}$ckelberg-
technique (2), defined for any arbitrary D-dimension of spacetime, can be re-
expressed as [17]:

$\displaystyle B^{(2)}\longrightarrow B^{(2)}\mp\frac{1}{m}\,F^{(2)}\equiv
B^{(2)}\mp\frac{1}{m}\,d\,\Phi^{(1)}.$ (6)

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

$\displaystyle B^{(2)}\longrightarrow
B^{(2)}\mp\frac{1}{m}\,d\,\Phi^{(1)}\mp\frac{1}{m}\,*\,d\,\tilde{\Phi}^{(1)},$
(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
$\tilde{\Phi}^{(1)}=d\,x^{\mu}\,\tilde{\phi}_{\mu}$ with the axial-vector
field $\tilde{\phi}_{\mu}$. A pseudo 2-form
$\tilde{F}^{(2)}=d\,\tilde{\Phi}^{(1)}=[(d\,x^{\mu}\wedge
d\,x^{\nu})/{2!}]\,\tilde{\Phi}_{\mu\nu}$ has been constructed from
$\tilde{\Phi}^{(1)}$ by applying an exterior derivative on it so that we
obtain
$\tilde{\Phi}_{\mu\nu}=\partial_{\mu}\,\tilde{\phi}_{\nu}-\partial_{\nu}\,\tilde{\phi}_{\mu}$.
To bring the parity of $B^{(2)},\,F^{(2)}=d\,\Phi^{(1)}$ and the pseudo 2-form
$\tilde{F}^{(2)}$ on equal footing***We assume that the parity symmetry is
respected in our discussion on the massive 4D Abelian 2-form theory (unlike
the parity violation in the realm of weak interactions)., it is essential to
obtain an ordinary 2-form from the pseudo 2-form $\tilde{F}^{(2)}$ by
operating a single Hodge duality operation $*$ on it. This mathematical
technique (on the 4D spacetime manifold) leads to

$\displaystyle*\,\tilde{F}^{(2)}=*\,\Big{(}\frac{d\,x^{\mu}\wedge
d\,x^{\nu}}{{2!}}\Big{)}\,\tilde{\Phi}_{\mu\nu}\equiv\frac{1}{2\,!}\,(d\,x^{\mu}\wedge
d\,x^{\nu})\,\tilde{f}_{\mu\nu},$ (8)

where
$\tilde{f}_{\mu\nu}=\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,\tilde{\Phi}^{\lambda\xi}\equiv\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\lambda}\,\tilde{\phi}^{\xi}$.
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 4D St$\ddot{u}$ckelberg technique (7),
namely;

$\displaystyle B_{\mu\nu}$ $\displaystyle\longrightarrow$ $\displaystyle
B_{\mu\nu}\mp\frac{1}{m}\,(\partial_{\mu}\,\phi_{\nu}-\partial_{\nu}\,\phi_{\mu})\mp\frac{1}{m}\,\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\lambda}\,\tilde{\phi}^{\xi}$
(9) $\displaystyle\equiv$ $\displaystyle
B_{\mu\nu}\mp\frac{1}{m}\,\Phi_{\mu\nu}\mp\frac{1}{m}\,\tilde{f}_{\mu\nu}\equiv
B_{\mu\nu}\mp\frac{1}{m}\,\Big{(}\Phi_{\mu\nu}+\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,\tilde{\Phi}^{\lambda\xi}\Big{)}.$

It is very interesting to state that the above modified 4D
St$\ddot{u}$ckelberg technique remains form-invariant under the following
discrete duality symmetry transformations

$\displaystyle B_{\mu\nu}\rightarrow\mp
i\,\tilde{B}_{\mu\nu}\;\equiv\;\mp\frac{i}{2!}\,\varepsilon_{\mu\nu\lambda\xi}\,B^{\lambda\xi},\qquad\phi_{\mu}\rightarrow\pm
i\,\tilde{\phi_{\mu}},\qquad\tilde{\phi_{\mu}}\rightarrow\mp i\,\phi_{\mu},$
(10)

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

$\displaystyle\tilde{B}^{(2)}=\Big{(}\frac{d\,x^{\mu}\wedge
d\,x^{\nu}}{{2!}}\Big{)}\,\Big{[}\frac{1}{2!}\,\varepsilon_{\mu\nu\lambda\xi}\,B^{\lambda\xi}\Big{]}\equiv\Big{(}\frac{d\,x^{\mu}\wedge
d\,x^{\nu}}{{2!}}\Big{)}\,\tilde{B}_{\mu\nu}.$ (11)

We shall see that the discrete duality symmetry transformations in (10) will
play very important role, later on, as its generalization (within the
framework of BRST formalism) will provide the analogue of Hodge duality $*$
operation 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
$(B^{(2)}=[(d\,x^{\mu}\wedge d\,x^{\nu})/{2!}]\,B_{\mu\nu})$ in the physical
four (3+1)-dimensions of spacetime.

It is an elementary exercise to note that the mass term of Eq. (1) transforms,
under the modified St$\ddot{u}$ckelberg-technique [cf. Eq. (9)], as

$\displaystyle-\,\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}\longrightarrow$
$\displaystyle-$
$\displaystyle\,\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}\pm\frac{m}{2}\,B_{\mu\nu}\,(\Phi^{\mu\nu}+\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}\,\tilde{\Phi}_{\rho\sigma})$
(12) $\displaystyle-$
$\displaystyle\,\frac{1}{4}\,\Phi_{\mu\nu}\,\Phi^{\mu\nu}+\frac{1}{4}\tilde{\Phi}_{\mu\nu}\,\tilde{\Phi}^{\mu\nu},$

modulo a total spacetime derivative
$\partial_{\mu}\,[-\,\varepsilon^{\mu\nu\lambda\xi}\,{\phi_{\nu}}\,\partial_{\lambda}\,\tilde{\phi}_{\xi}]$
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). 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

$\displaystyle H_{\mu\nu\lambda}\longrightarrow
H_{\mu\nu\lambda}\mp\frac{1}{m}\,(\partial_{\mu}\,\Phi_{\nu\lambda}+\partial_{\nu}\,\Phi_{\lambda\mu}+\partial_{\lambda}\,\Phi_{\mu\nu})\mp\frac{1}{m}\,(\partial_{\mu}\,\tilde{f}_{\nu\lambda}+\partial_{\nu}\,\tilde{f}_{\lambda\mu}+\partial_{\lambda}\,\tilde{f}_{\mu\nu}),$
(13)

where $\Phi_{\mu\nu}=\partial_{\mu}\,\phi_{\nu}-\partial_{\nu}\,\phi_{\mu}$
and
$\tilde{f}_{\mu\nu}=\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\lambda}\,\tilde{\phi}^{\xi}$.
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:

$\displaystyle\Sigma_{\mu\nu\lambda}$ $\displaystyle=$
$\displaystyle(\partial_{\mu}\,\tilde{f}_{\nu\lambda}+\partial_{\nu}\,\tilde{f}_{\lambda\mu}+\partial_{\lambda}\,\tilde{f}_{\mu\nu})$
(14) $\displaystyle\equiv$
$\displaystyle(\varepsilon_{\mu\nu\rho\sigma}\,\partial_{\lambda}+\varepsilon_{\nu\lambda\rho\sigma}\,\partial_{\mu}+\varepsilon_{\lambda\mu\rho\sigma}\,\partial_{\nu})\,(\partial^{\rho}\,\tilde{\phi}^{\sigma}).$

Thus, we have to find out the explicit value of the following (for the changes
in the kinetic term due to the modified version of 4D St$\ddot{u}$ckelberg
technique [cf. Eq. (9)]), namely;

$\displaystyle\frac{1}{12}\,H_{\mu\nu\lambda}\,H^{\mu\nu\lambda}\;\Longrightarrow\;\frac{1}{12}\,H_{\mu\nu\lambda}\,H^{\mu\nu\lambda}\mp\frac{1}{6\,m}\,H_{\mu\nu\lambda}\,\Sigma^{\mu\nu\lambda}+\frac{1}{12\,m^{2}}\,\Sigma_{\mu\nu\lambda}\,\Sigma^{\mu\nu\lambda},$
(15)

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

$\displaystyle\mp\frac{1}{6\,m}\,H^{\mu\nu\lambda}\,\Big{[}\varepsilon_{\mu\nu\rho\sigma}\,\partial_{\lambda}+\varepsilon_{\nu\lambda\rho\sigma}\,\partial_{\mu}+\varepsilon_{\lambda\mu\rho\sigma}\,\partial_{\nu}\Big{]}\,(\partial^{\rho}\tilde{\phi}^{\sigma}).$
(16)

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

$\displaystyle\pm\frac{1}{6\,m}\,(\partial_{\lambda}\,H^{\lambda\mu\nu})\,\varepsilon_{\mu\nu\rho\sigma}\,(\partial^{\rho}\tilde{\phi}^{\sigma}).$
(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 [12], the higher derivatives terms
have been ignored. This is why, relevant terms in the Lagrangian density have
been obtained by trial and error method. However, we note in our present
endeavor that one can get rid of a single derivative by using the on-shell
condition: $\partial_{\lambda}\,H^{\lambda\mu\nu}+m^{2}\,B^{\mu\nu}=0$. All
the three terms, on the r.h.s. of (16), individually contribute to the same
result which can be added together to yield the following

$\displaystyle\mp\frac{m}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,B^{\mu\nu}\,(\partial^{\lambda}\,\tilde{\phi}^{\xi})\equiv\mp\frac{m}{4}\,\varepsilon^{\mu\nu\lambda\xi}\,B_{\mu\nu}\,\tilde{\Phi}_{\lambda\xi},$
(18)

where
$\tilde{\Phi}_{\lambda\xi}=\partial_{\lambda}\,\tilde{\phi}_{\xi}-\partial_{\xi}\,\tilde{\phi}_{\lambda}$.
It is interesting to point out that the above term has been incorporated into
the BRST invariant Lagrangian density of our earlier work [12] on the basis of
the trial and error method. However, as it self-evident, we have derived this
term correctly in our present endeavor which is motivated by our earlier work
on 2D Proca theory [8] where we have exploited similar trick to get rid of the
higher derivative terms in the 2D massive Abelian 1-form (i.e. 2D Proca)
theory. The mass term in Eq. (18) is just like the topological mass term of
the $B\wedge F$ theory. In the latter theory, the 4D Abelian 2-form gauge
theory also incorporates the Maxwell Abelian 1-form
($A^{(1)}=d\,x^{\mu}\,A_{\mu}$) gauge field with curvature 2-form
$F^{(2)}=d\,A^{(1)}$. There are many ways to derive (18) from (16). However,
we have chosen one of the simplest methods to derive the equation (18) which
is not a higher derivative culprit term for our 4D massive Abelian 2-form
theory.

We now focus on the 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 also be total 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 Eq. (16). However, it
turns out that only three of them contribute to the Lagrangian density and the
rest of six terms are found to be total spacetime derivatives. Let us focus on
the first existing term that is equal to:

$\displaystyle\frac{1}{12\,m^{2}}\,\varepsilon^{\mu\nu\rho\sigma}\,\partial^{\lambda}\,(\partial_{\rho}\,\tilde{\phi}_{\sigma})\,\varepsilon_{\mu\nu\alpha\beta}\,\partial_{\lambda}\,(\partial^{\alpha}\,\tilde{\phi}^{\beta})$
$\displaystyle=$
$\displaystyle-\,\frac{1}{6\,m^{2}}\,[\partial^{\lambda}\,(\partial_{\alpha}\,\tilde{\phi}_{\beta}-\partial_{\beta}\,\tilde{\phi}_{\alpha})]\,[\partial_{\lambda}\,(\partial^{\alpha}\,\tilde{\phi}^{\beta})]$
(19) $\displaystyle\equiv$
$\displaystyle\frac{1}{6\,m^{2}}\,\partial^{\lambda}\,(\partial^{\alpha}\,\tilde{\Phi}_{\alpha\beta})\,(\partial_{\lambda}\,\tilde{\phi}^{\beta}),$

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

$\displaystyle-\,\frac{1}{6}\,(\partial^{\lambda}\,\tilde{\phi}_{\beta})\,(\partial_{\lambda}\,\tilde{\phi}^{\beta})\equiv\frac{1}{6}\,\tilde{\phi}_{\beta}\,\Box\,\tilde{\phi}^{\beta},$
(20)

where we have dropped a total spacetime derivative term. Using the Klein-
Gordon equation†††It can be seen that the equation of motion:
$\partial_{\alpha}\,\tilde{\Phi}^{\alpha\beta}+m^{2}\tilde{\phi}^{\beta}=0$
implies that $\partial\cdot\tilde{\phi}=0$ for $m^{2}\neq 0$ due to the
antisymmetric $(\tilde{\Phi}^{\alpha\beta}=-\,\tilde{\Phi}^{\beta\alpha})$
property of he $\tilde{\Phi}^{\alpha\beta}$. As a consequence, we obtain
$(\Box+m^{2})\,\tilde{\phi}^{\beta}=0$.: $(\Box+m^{2})\,\tilde{\phi}_{\mu}=0$,
we obtain the following (from the first contribution), namely;

$\displaystyle\frac{1}{12\,m^{2}}\,\varepsilon^{\mu\nu\rho\sigma}\,\partial^{\lambda}\,(\partial_{\rho}\,\tilde{\phi}_{\sigma})\,\varepsilon_{\mu\nu\alpha\beta}\,\partial_{\lambda}\,(\partial^{\alpha}\,\tilde{\phi}^{\beta})\equiv-\,\frac{m^{2}}{6}\,\tilde{\phi}_{\beta}\,\tilde{\phi}^{\beta}.$
(21)

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

$\displaystyle\frac{1}{12\,m^{2}}\,\Sigma_{\mu\nu\lambda}\,\Sigma^{\mu\nu\lambda}=-\,\frac{m^{2}}{2}\,\tilde{\phi}_{\mu}\,\tilde{\phi}^{\mu}.$
(22)

It is clear that the above term is not a culprit term and it is useful for us
for our further discussions. The total terms on the r.h.s. of Eq. (15) can be
re-expressed as follows:

$\displaystyle\frac{1}{12}\,H^{\mu\nu\lambda}\,H_{\mu\nu\lambda}\mp\frac{m}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,B^{\mu\nu}\,(\partial^{\lambda}\,\tilde{\phi}^{\xi})-\frac{m^{2}}{2}\,\tilde{\phi}_{\mu}\,\tilde{\phi}^{\mu}$
$\displaystyle\equiv-\,\frac{1}{8}\,\varepsilon^{\mu\nu\lambda\xi}\,(\partial_{\nu}\,B_{\lambda\xi})\,\varepsilon_{\mu\alpha\beta\gamma}\,(\partial^{\alpha}B^{\beta\gamma})\pm\frac{m}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,(\partial^{\mu}\,B^{\nu\lambda})\,\tilde{\phi}^{\xi}-\,\frac{m^{2}}{2}\,\tilde{\phi}_{\mu}\,\tilde{\phi}^{\mu},$
(23)

where we have dropped a total spacetime derivative term and used the
following:

$\displaystyle\frac{1}{12}\,H^{\mu\nu\lambda}\,H_{\mu\nu\lambda}=-\,\frac{1}{8}\,\varepsilon^{\mu\nu\lambda\xi}\,(\partial_{\nu}\,B_{\lambda\xi})\,\varepsilon_{\mu\alpha\beta\gamma}\,(\partial^{\alpha}\,B^{\beta\gamma}).$
(24)

The correctness of the above equality can be checked explicitly by using the
well-known property of the 4D Levi-Civita tensor. It is straightforward to
observe that the final expression for (23) can be written as

$\displaystyle-\,\frac{1}{2}\,\Big{[}\frac{1}{4}\,\varepsilon^{\mu\nu\lambda\xi}\,(\partial_{\nu}\,B_{\lambda\xi})\,\varepsilon_{\mu\alpha\beta\gamma}\,(\partial^{\alpha}\,B^{\beta\gamma})\pm
m\,\tilde{\phi}_{\mu}\,\varepsilon^{\mu\nu\lambda\xi}\,(\partial_{\nu}\,B_{\lambda\xi})+m^{2}\,\tilde{\phi}_{\mu}\,\tilde{\phi}^{\mu}\Big{]},$
(25)

where we have used the following

$\displaystyle\pm\frac{m}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,(\partial^{\mu}\,B^{\nu\lambda})\,\tilde{\phi}^{\xi}=\mp\,\frac{m}{2}\,\tilde{\phi}_{\mu}\,\varepsilon^{\mu\nu\lambda\xi}\,(\partial_{\nu}\,B_{\lambda\xi}),$
(26)

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

$\displaystyle\frac{1}{12}\,H_{\mu\nu\lambda}\,H^{\mu\nu\lambda}\mp\frac{1}{6\,m}\,H_{\mu\nu\lambda}\,\Sigma^{\mu\nu\lambda}+\frac{1}{12\,m^{2}}\,\Sigma_{\mu\nu\lambda}\,\Sigma^{\mu\nu\lambda}=-\,\frac{1}{2}\,\Big{[}\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\partial^{\nu}\,B^{\lambda\xi}\pm
m\,\tilde{\phi}_{\mu}\Big{]}^{2},$ (27)

which is nothing but the explicit expression for Eq. (25). Thus, we note that
the final version of the Lagrangian density (with the modified SF [cf. Eq.
(9)]) is‡‡‡It is worthwhile to mention here that the kinetic terms for the
$\phi_{\mu}$ and $\tilde{\phi}_{\mu}$ fields have a relative sign difference.
In other words, one of the above fields has a negative kinetic term which is
interesting.

$\displaystyle{\cal L}_{S}^{(m)}$ $\displaystyle=$
$\displaystyle-\,\frac{1}{2}\,\Big{[}\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\partial^{\nu}\,B^{\lambda\xi}\pm
m\,\tilde{\phi}_{\mu}\Big{]}^{2}-\,\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}$
(28) $\displaystyle\pm$
$\displaystyle\frac{m}{2}\,\Big{[}\Phi^{\mu\nu}+\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}\,\tilde{\Phi}_{\rho\sigma}\Big{]}\,B_{\mu\nu}\,-\frac{1}{4}\,\Phi_{\mu\nu}\,\Phi^{\mu\nu}+\frac{1}{4}\tilde{\Phi}_{\mu\nu}\,\tilde{\Phi}^{\mu\nu},$

where we have taken the inputs from Eqs (12) and (27) and superscript $(m)$ on
this Lagrangian density denotes that we have taken the help of modified SF
[cf. Eq. (9)].

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

$\displaystyle{\cal L}={\cal L}_{S}^{(m)}+{\cal L}_{gf}$ $\displaystyle\equiv$
$\displaystyle-\,\frac{1}{2}\,\Big{[}\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\partial^{\nu}\,B^{\lambda\xi}\pm
m\,\tilde{\phi}_{\mu}\Big{]}^{2}-\,\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}$
(29) $\displaystyle\pm$
$\displaystyle\frac{m}{2}\,\Big{[}\Phi^{\mu\nu}+\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}\,\tilde{\Phi}_{\rho\sigma}\Big{]}\,B_{\mu\nu}\,-\frac{1}{4}\,\Phi_{\mu\nu}\,\Phi^{\mu\nu}+\frac{1}{4}\tilde{\Phi}_{\mu\nu}\,\tilde{\Phi}^{\mu\nu}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\,\Big{[}\partial^{\nu}\,B_{\nu\mu}\pm
m\,\phi_{\mu}\Big{]}^{2}+\frac{1}{2}\,(\partial\cdot\tilde{\phi})^{2}-\frac{1}{2}\,(\partial\cdot\phi)^{2},$

where the gauge-fixing term [$\frac{1}{2}\,(\partial^{\nu}\,B_{\nu\mu}\pm
m\,\phi_{\mu})^{2}$] is like the t’Hooft gauge in the context of modified
Proca theory [17, 8]. 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 Eq. (9). The equations of motion, satisfied by the basic fields
($B_{\mu\nu},\,\phi_{\mu},\,\tilde{\phi}_{\mu}$), are the Klein-Gordon
equations:
$(\Box+m^{2})\,B_{\mu\nu}=0,\,(\Box+m^{2})\,\phi_{\mu}=0,\,(\Box+m^{2})\,\tilde{\phi}_{\mu}=0$
which emerge out from the Lagrangian density (29) which shows that all the
fields have the rest mass $m$.

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

A close look and careful observation of the modified SF in (7) demonstrates
that we have the freedom to introduce a pure-scalar ($\phi$) and pseudo-scalar
($\tilde{\phi}$) fields in our 4D massive theory. The nilpotency ($d^{2}=0$)
of the exterior derivative ($d=d\,x^{\mu}\,\partial_{\mu}$) ensures that the
following transformations are permitted in our theory, namely;

$\displaystyle\Phi^{(1)}\;\longrightarrow\;\Phi^{(1)}\pm
d\,\Phi^{(0)}\qquad\Longrightarrow\qquad\phi_{\mu}\;\longrightarrow\;\phi_{\mu}\pm\kappa\,\partial_{\mu}\,\phi,$
$\displaystyle\tilde{\Phi}^{(1)}\;\longrightarrow\;\tilde{\Phi}^{(1)}\pm
d\,\tilde{\Phi}^{(0)}\qquad\Longrightarrow\qquad\tilde{\phi}_{\mu}\;\longrightarrow\;\tilde{\phi}_{\mu}\pm\kappa\,\partial_{\mu}\,\tilde{\phi},$
(30)

where $\kappa$ is a numerical constant. The latter can be chosen to fit our
understanding of the CF-type restrictions emerging out from the superfield
approach to BRST formalism [18]. As a result of the existence of the above
pure-scalar and pseudo-scalar fields, the kinetic and gauge-fixing terms of
the gauge field ($B_{\mu\nu}$) and the gauge-fixing terms for ($\phi_{\mu}$)
and ($\tilde{\phi}_{\mu}$) fields will be modified (where the mass dimension
of these fields will be taken into consideration). It is pretty obvious that
the mass dimension of $\phi_{\mu}$ and $\tilde{\phi}_{\mu}$ fields is $[M]$.
Hence, the mass dimension of fields $\phi$ and $\tilde{\phi}$ is zero (i.e.
$[M]^{0}$). As a consequence, we have the following modifications of the
gauge-fixed Lagrangian density (29), namely;

$\displaystyle{\cal L}\longrightarrow{\cal L}_{(1)}$ $\displaystyle=$
$\displaystyle-\,\frac{1}{2}\,\Big{[}\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\partial^{\nu}\,B^{\lambda\xi}\pm
m\,\tilde{\phi}_{\mu}\mp\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\Big{]}^{2}-\,\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}$
$\displaystyle\pm$
$\displaystyle\frac{m}{2}\,\Big{[}\Phi^{\mu\nu}+\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}\,\tilde{\Phi}_{\rho\sigma}\Big{]}\,B_{\mu\nu}\,-\frac{1}{4}\,\Phi_{\mu\nu}\,\Phi^{\mu\nu}+\frac{1}{4}\tilde{\Phi}_{\mu\nu}\,\tilde{\Phi}^{\mu\nu}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\,\Big{[}\partial^{\nu}\,B_{\nu\mu}\pm
m\,\phi_{\mu}\mp\frac{1}{2}\,\partial_{\mu}\,\phi\Big{]}^{2}+\frac{1}{2}\,\Big{(}\partial\cdot\tilde{\phi}+\frac{m}{2}\,\tilde{\phi}\Big{)}^{2}$
(31) $\displaystyle-$
$\displaystyle\frac{1}{2}\,\Big{(}\partial\cdot\phi+\frac{m}{2}\,\phi\Big{)}^{2},$
$\displaystyle{\cal L}\longrightarrow{\cal L}_{(2)}$ $\displaystyle=$
$\displaystyle-\,\frac{1}{2}\,\Big{[}\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\partial^{\nu}\,B^{\lambda\xi}\pm
m\,\tilde{\phi}_{\mu}\pm\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\Big{]}^{2}-\,\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}$
(32) $\displaystyle\pm$
$\displaystyle\frac{m}{2}\,\Big{[}\Phi^{\mu\nu}+\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}\,\tilde{\Phi}_{\rho\sigma}\Big{]}\,B_{\mu\nu}\,-\frac{1}{4}\,\Phi_{\mu\nu}\,\Phi^{\mu\nu}+\frac{1}{4}\tilde{\Phi}_{\mu\nu}\,\tilde{\Phi}^{\mu\nu}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\,\Big{[}\partial^{\nu}\,B_{\nu\mu}\pm
m\,\phi_{\mu}\pm\frac{1}{2}\,\partial_{\mu}\,\phi\Big{]}^{2}+\frac{1}{2}\,\Big{(}\partial\cdot\tilde{\phi}-\frac{m}{2}\,\tilde{\phi}\Big{)}^{2}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\,\Big{(}\partial\cdot\phi-\frac{m}{2}\,\phi\Big{)}^{2},$

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 $\phi$ and $\tilde{\phi}$ 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 $\tilde{\phi}$ and $\phi$, respectively. The latter have been
incorporated into ${\cal L}_{(1)}$ and ${\cal L}_{(2)}$ 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
$\tilde{\phi}_{\mu}$ and $\phi_{\mu}$, respectively, are fixed that lead to
the EL-EoMs§§§We lay emphases on the fact that the axial-vector field
($\tilde{\phi}_{\mu}$) and pseudo-scalar field ($\tilde{\phi}$) possess
negative kinetic terms. However, they satisfy proper Klein-Gordon equation.
Hence, these fields correspond to the exotic relativistic particles with well-
defined rest mass. As a consequence, they are the possible candidates of dark
matter [19, 20] and they are like the “phantom” fields in the realm of
cosmology [13-15].:
$(\Box+m^{2})\,\phi_{\mu}=0,\,(\Box+m^{2})\,\tilde{\phi}_{\mu}=0,\
(\Box+m^{2})\,\phi=0,\,(\Box+m^{2})\,\tilde{\phi}=0$.

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

$\displaystyle B_{\mu\nu}\longrightarrow\mp
i\,\tilde{B}_{\mu\nu}\;\equiv\;\mp\,\frac{i}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,B^{\lambda\xi},\qquad\phi_{\mu}\longrightarrow\pm\,i\,\tilde{\phi}_{\mu},\qquad\tilde{\phi}_{\mu}\longrightarrow\mp\,i\,\phi_{\mu},$
$\displaystyle\phi\longrightarrow\pm\,i\,\tilde{\phi},\qquad\quad\tilde{\phi}\longrightarrow\mp\,i\,\phi,\qquad\qquad
B_{\mu\nu}\,B^{\mu\nu}\longrightarrow B_{\mu\nu}\,B^{\mu\nu},$ (33)

is respected by the completely gauge-fixed Lagrangian densities ${\cal
L}_{(1)}$ and ${\cal L}_{(2)}$ and all the fields (i.e.
$B_{\mu\nu},\,\phi_{\mu},\,\tilde{\phi}_{\mu},\,\phi,\,\tilde{\phi}$) satisfy
the following Klein-Gordon equation¶¶¶We stress on the fact that the discrete
duality symmetry transformations (33) and equations of motion (34) are true
for both the Lagrangian densities ${\cal L}_{(1)}$ and ${\cal L}_{(2)}$ [cf.
Eqs. (31),(32)].

$\displaystyle(\Box+m^{2})\,B_{\mu\nu}=0,\;\qquad(\Box+m^{2})\,\phi_{\mu}=0,\qquad\;(\Box+m^{2})\,\tilde{\phi}_{\mu}=0,$
$\displaystyle(\Box+m^{2})\,\phi=0,\qquad\quad\;(\Box+m^{2})\,\tilde{\phi}=0,$
(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
$[B_{\mu\nu}\longrightarrow\mp(i/2)\,\varepsilon_{\mu\nu\lambda\xi}\,B^{\lambda\xi}]$.
The latter has its origin in the self-duality condition [cf. Eq. (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 $(\delta_{g})$ for the basic fields of the Lagrangian density
${\cal L}_{(1)}$, namely;

$\displaystyle\delta_{g}\,B_{\mu\nu}=-(\partial_{\mu}\,\Lambda_{\nu}-\partial_{\nu}\,\Lambda_{\mu}),\qquad\quad\delta_{g}\,\phi_{\mu}=\pm(\partial_{\mu}\,\Lambda-m\,\Lambda_{\mu}),$
$\displaystyle\delta_{g}\,\Phi_{\mu\nu}=\mp
m\,(\partial_{\mu}\,\Lambda_{\nu}-\partial_{\nu}\,\Lambda_{\mu}),\qquad\delta_{g}\,\phi=\pm\,2\,[(\partial\cdot\Lambda)+m\,\Lambda],$
$\displaystyle\delta_{g}\,[H_{\mu\nu\lambda},\,\tilde{\phi}_{\mu},\,\tilde{\phi},\,\tilde{\Phi}_{\mu\nu}]=0,$
(35)

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

$\displaystyle\delta_{g}\,{\cal L}_{(1)}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,\Big{[}\mp\,m\,\varepsilon^{\mu\nu\lambda\xi}\,\Lambda_{\nu}\partial_{\lambda}\,\tilde{\phi}_{\xi}\Big{]}\mp\Big{(}\partial\cdot\phi+\frac{m}{2}\,\phi\Big{)}\,\big{[}\Box+m^{2}\big{]}\,\Lambda$
(36) $\displaystyle-$ $\displaystyle\Big{[}\partial_{\nu}\,B^{\nu\mu}\pm
m\,\phi^{\mu}\mp\frac{1}{2}\,\partial^{\mu}\,\phi\Big{]}\,\big{[}\Box+m^{2}\big{]}\,\Lambda_{\mu}.$

Thus, it is crystal clear that if we impose the restrictions on the gauge
transformation parameters as:
$(\Box+m^{2})\,\Lambda=0,\;(\Box+m^{2})\,\Lambda_{\mu}=0$ from outside, the
transformations (35) will become the symmetry transformations for the
Lagrangian density ${\cal L}_{(1)}$. We shall see that, under 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.
$\delta_{g}\,H_{\mu\nu\lambda}=0,\;\delta_{g}\,\tilde{\phi}_{\mu}=0,\;\delta_{g}\,\tilde{\phi}=0,\;\delta_{g}\,\tilde{\Phi}_{\mu\nu}=0$),
oweing their origin to the exterior derivative $d=d\,x^{\mu}\,\partial_{\mu}$
(with $d^{2}=0$) of differential geometry [2-5], remain invariant. In addition
to the gauge transformations (35), we have another set of infinitesimal and
continuous transformations $(\delta_{dg})$ in the theory, namely;

$\displaystyle\delta_{dg}\,B_{\mu\nu}=-\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\lambda}\,\Sigma^{\xi},\qquad\qquad\delta_{dg}\,\tilde{\phi}_{\mu}=\pm(\partial_{\mu}\Omega-m\,\Sigma_{\mu}),$
$\displaystyle\delta_{dg}\,\tilde{\phi}=\pm
2\,[\partial\cdot\Sigma+m\,\Omega],\qquad\qquad\delta_{dg}\,\tilde{\Phi}_{\mu\nu}=\mp\,m\,(\partial_{\mu}\,\Sigma_{\nu}-\partial_{\nu}\,\Sigma_{\mu}),$
$\displaystyle\delta_{dg}\,[\partial^{\nu}\,B_{\nu\mu},\phi_{\mu},\,\phi,\,\Phi_{\mu\nu}]=0,$
(37)

which imply that the total gauge-fixing term [i.e.
$\frac{1}{2}\,(\partial_{\nu}\,B^{\nu\mu}\pm
m\,\phi^{\mu}\mp\frac{1}{2}\,\partial^{\mu}\,\phi)$], owing its origin to the
co-exterior derivative: $\delta=\pm\,*\,d\,*$, 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 ${\cal L}_{(1)}$ transforms under the infinitesimal and
continuous transformations (37) as follows

$\displaystyle\delta_{dg}\,{\cal L}_{(1)}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,\Big{[}\mp\,m\,\varepsilon^{\mu\nu\lambda\xi}\,\Sigma_{\nu}\,\partial_{\lambda}\,\phi_{\xi}\Big{]}\pm\Big{(}\partial\cdot\tilde{\phi}+\frac{m}{2}\,\tilde{\phi}\Big{)}\,\big{[}\Box+m^{2}\big{]}\,\Omega$
(38) $\displaystyle+$
$\displaystyle\Big{[}\frac{1}{2}\,\varepsilon^{\mu\nu\lambda\xi}\,\partial_{\nu}\,B_{\lambda\xi}\pm
m\,\tilde{\phi}^{\mu}\mp\frac{1}{2}\,\partial^{\mu}\tilde{\phi}\Big{]}\,\big{[}\Box+m^{2}\big{]}\,\Sigma_{\mu},$

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

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

$\displaystyle\delta_{dg}\,B_{\mu\nu}=-\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\lambda}\,\Sigma^{\xi},\qquad\delta_{dg}\,\tilde{\phi}_{\mu}=\pm(\partial_{\mu}\Omega-m\,\Sigma_{\mu}),$
$\displaystyle\delta_{dg}\,\tilde{\phi}=\mp
2\,[\partial\cdot\Sigma+m\,\Omega],\qquad\delta_{dg}\,\tilde{\Phi}_{\mu\nu}=\mp\,m\,(\partial_{\mu}\,\Sigma_{\nu}-\partial_{\nu}\,\Sigma_{\mu}),$
$\displaystyle\delta_{dg}\,[\partial^{\nu}\,B_{\nu\mu},\phi_{\mu},\,\phi,\,\Phi_{\mu\nu}]=0,$
(39)
$\displaystyle\delta_{g}\,B_{\mu\nu}=-(\partial_{\mu}\,\Lambda_{\nu}-\partial_{\nu}\,\Lambda_{\mu}),\qquad\quad\delta_{g}\,\phi_{\mu}=\pm(\partial_{\mu}\,\Lambda-m\,\Lambda_{\mu}),$
$\displaystyle\delta_{g}\,\Phi_{\mu\nu}=\mp
m\,(\partial_{\mu}\,\Lambda_{\nu}-\partial_{\nu}\,\Lambda_{\mu}),\qquad\delta_{g}\,\phi=\mp\,2\,[(\partial\cdot\Lambda)+m\,\Lambda],$
$\displaystyle\delta_{g}\,[H_{\mu\nu\lambda},\,\tilde{\phi}_{\mu},\,\tilde{\phi},\,\tilde{\Phi}_{\mu\nu}]=0,$
(40)

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

$\displaystyle\delta_{dg}\,{\cal L}_{(2)}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,\Big{[}\mp\,m\,\varepsilon^{\mu\nu\lambda\xi}\,\Sigma_{\nu}\,\partial_{\lambda}\,\phi_{\xi}\Big{]}\pm\Big{(}\partial\cdot\tilde{\phi}-\frac{m}{2}\,\tilde{\phi}\Big{)}\,\big{[}\Box+m^{2}\big{]}\,\Omega,$
$\displaystyle+$
$\displaystyle\Big{[}\frac{1}{2}\,\varepsilon^{\mu\nu\lambda\xi}\,\partial_{\nu}\,B_{\lambda\xi}\pm
m\,\tilde{\phi}_{\mu}\pm\frac{1}{2}\,\partial_{\mu}\tilde{\phi}\Big{]}\,\big{[}\Box+m^{2}\big{]}\,\Sigma_{\mu}$
$\displaystyle\delta_{g}\,{\cal L}_{(2)}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,\Big{[}\mp\,m\,\varepsilon^{\mu\nu\lambda\xi}\,\Lambda_{\nu}\,\partial_{\lambda}\,\tilde{\phi}_{\xi}\Big{]}\mp\Big{(}\partial\cdot\phi-\frac{m}{2}\,\phi\Big{)}\,\big{[}\Box+m^{2}\big{]}\,\Lambda$
(41) $\displaystyle-$ $\displaystyle\Big{[}\partial_{\nu}\,B^{\nu\mu}\pm
m\,\phi^{\mu}\pm\frac{1}{2}\,\partial^{\mu}\,\phi\Big{]}\,\big{[}\Box+m^{2}\big{]}\,\Lambda_{\mu}.$

It is evident that if we impose the restrictions

$\displaystyle(\Box+m^{2})\,\Sigma_{\mu}=0,\,\qquad\qquad(\Box+m^{2})\,\Lambda_{\mu}=0,\,$
$\displaystyle(\Box+m^{2})\,\Omega=0,\;\ \qquad\qquad(\Box+m^{2})\,\Lambda=0,$
(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 ${\cal L}_{(2)}$. We note that the outside restrictions
(36), (38) and (42) are exactly same on the (dual-)gauge transformation
parameters of our theory. Hence, when we elevate the Lagrangian densities
${\cal L}_{(1)}$ and ${\cal L}_{(2)}$ to their counterparts at the quantum
level (within the framework of BRST formalism), we shall find out 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
${\cal L}_{(1)}$ which can be written as

$\displaystyle{\cal L}_{(1)}\longrightarrow{\cal L}_{(b_{1})}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\,{\cal B}_{\mu}\,{\cal
B}^{\mu}-{\cal
B}^{\mu}\,\Big{[}\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\nu}\,B^{\lambda\xi}\pm
m\,\tilde{\phi}_{\mu}\mp\frac{1}{2}\,\partial_{\mu}\tilde{\phi}\Big{]}-\,\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}$
(43) $\displaystyle-$
$\displaystyle\frac{1}{4}\,\Phi_{\mu\nu}\,\Phi^{\mu\nu}+\frac{1}{4}\tilde{\Phi}_{\mu\nu}\,\tilde{\Phi}^{\mu\nu}\pm\frac{m}{2}\,B_{\mu\nu}\,\Big{[}\Phi^{\mu\nu}+\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}\,\tilde{\Phi}_{\rho\sigma}\Big{]}$
$\displaystyle+$ $\displaystyle B^{\mu}\,\Big{[}\partial^{\nu}\,B_{\nu\mu}\pm
m\,\phi_{\mu}\mp\frac{1}{2}\,\partial_{\mu}\,\phi\Big{]}\,-\frac{1}{2}\,B^{\mu}\,B_{\mu}+B\,(\partial\cdot\phi+\frac{m}{2}\,\phi)$
$\displaystyle+$ $\displaystyle\frac{1}{2}\,B^{2}-{\cal
B}\,(\partial\cdot\tilde{\phi}+\frac{m}{2}\,\tilde{\phi})-\frac{1}{2}\,{\cal
B}^{2},$

which leads to the following equations of motion:

$\displaystyle{\cal
B}_{\mu}=\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\nu}\,B^{\lambda\xi}\pm
m\,\tilde{\phi}_{\mu}\mp\frac{1}{2}\,\partial_{\mu}\tilde{\phi},\qquad{\cal
B}=-\,(\partial\cdot\tilde{\phi}+\frac{m}{2}\,\tilde{\phi}),$ $\displaystyle
B_{\mu}=\partial^{\nu}\,B_{\nu\mu}\pm
m\,\phi_{\mu}\mp\frac{1}{2}\,\partial_{\mu}\,\phi,\qquad\quad\qquad
B=-\,(\partial\cdot\phi+\frac{m}{2}\,\phi).$ (44)

In the above, the auxiliary fields $({\cal B}_{\mu}\;B_{\mu},\;{\cal B},\;B)$
are the Nakanishi-Lautrup auxiliary fields which have been invoked for the
linearization purposes. For instance, the auxiliary field ${\cal B}_{\mu}$ 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
$(B_{\mu},\,B,\,{\cal B})$ have been introduced to linearize the gauge-fixing
terms for the $B_{\mu\nu},\phi_{\mu}$ and $\tilde{\phi}_{\mu}$ fields. In
exactly similar fashion, we can linearize the Lagrangian density ${\cal
L}_{(2)}$ by invoking a different set of Nakanishi-Lautrup type auxiliary
fields $(\bar{{\cal B}}_{\mu}\;\bar{B}_{\mu},\;\bar{{\cal B}},\;\bar{B})$ as
follows:

$\displaystyle{\cal L}_{(2)}\longrightarrow{\cal L}_{(b_{2})}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\,\bar{{\cal B}}_{\mu}\,\bar{{\cal
B}}^{\mu}+\bar{{\cal
B}}^{\mu}\,\Big{[}\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\nu}\,B^{\lambda\xi}\pm
m\,\tilde{\phi}_{\mu}\pm\frac{1}{2}\,\partial_{\mu}\tilde{\phi}\Big{]}-\,\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}$
(45) $\displaystyle-$
$\displaystyle\frac{1}{4}\,\Phi_{\mu\nu}\,\Phi^{\mu\nu}+\frac{1}{4}\tilde{\Phi}_{\mu\nu}\,\tilde{\Phi}^{\mu\nu}\pm\frac{m}{2}\,B_{\mu\nu}\,\Big{[}\Phi^{\mu\nu}+\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}\,\tilde{\Phi}_{\rho\sigma}\Big{]}$
$\displaystyle-$
$\displaystyle\bar{B}^{\mu}\,\Big{[}\partial^{\nu}\,B_{\nu\mu}\pm
m\,\phi_{\mu}\pm\frac{1}{2}\,\partial_{\mu}\,\phi\Big{]}\,-\frac{1}{2}\,\bar{B}^{\mu}\,\bar{B}_{\mu}-\bar{B}\,(\partial\cdot\phi-\frac{m}{2}\,\phi)$
$\displaystyle+$ $\displaystyle\frac{1}{2}\,\bar{B}^{2}+\bar{{\cal
B}}\,(\partial\cdot\tilde{\phi}-\frac{m}{2}\,\tilde{\phi})-\frac{1}{2}\,\bar{{\cal
B}}^{2}.$

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

$\displaystyle\bar{{\cal
B}_{\mu}}=-\,\Big{[}\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\nu}\,B^{\lambda\xi}\pm
m\,\tilde{\phi}_{\mu}\pm\frac{1}{2}\,\partial_{\mu}\tilde{\phi}\Big{]},\qquad\bar{{\cal
B}}=(\partial\cdot\tilde{\phi}-\frac{m}{2}\,\tilde{\phi}),$
$\displaystyle\bar{B}_{\mu}=-\,\Big{[}\partial^{\nu}\,B_{\nu\mu}\pm
m\,\phi_{\mu}\pm\frac{1}{2}\,\partial_{\mu}\,\phi\Big{]},\qquad\quad\qquad\bar{B}=(\partial\cdot\phi-\frac{m}{2}\,\phi).$
(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 equations of motion (44) and (46), namely;

$\displaystyle{\cal B}_{\mu}+{\bar{\cal
B}_{\mu}}\pm\partial_{\mu}\,\tilde{\phi}=0,\qquad B+\bar{B}+m\,\phi=0,$
$\displaystyle B_{\mu}+\bar{B_{\mu}}\pm\partial_{\mu}\,\phi=0,\qquad{\cal
B}+{\bar{\cal B}}+m\,\tilde{\phi}=0,$ (47)

which are nothing but the CF-type restrictions on our theory (see, e.g. [12,
21] for details).

We end this section with the following remarks. First of all, the Lagrangian
densities ${\cal L}_{(b_{1})}$ and ${\cal L}_{(b_{2})}$ have been derived in a
completely different manner in our present endeavor if we compare our present
derivation against the derivation in our earlier work [12] where we have
exploited the method of trial and error. Second, the CF-type restrictions:
$B+\bar{B}+m\,\phi=0$ and ${\cal B}+\bar{{\cal B}}+m\,\tilde{\phi}=0$ are same
as in our earlier work [12, 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 context of 4D
Abelian 2-form massless and massive gauge theories [18, 21], we find that the
other two restrictions of (47) are: ${\cal B}_{\mu}+{\bar{\cal
B}_{\mu}}+\partial_{\mu}\,\tilde{\phi}=0$ and
$B_{\mu}+\bar{B_{\mu}}+\partial_{\mu}\,\phi=0$. Hence, the $(\pm)$ signs
associated with thee (pseudo-)scalar fields (e.g.
$\pm\frac{1}{2}\,\partial_{\mu}\,\phi,\;\pm\frac{1}{2}\,\partial_{\mu}\;\tilde{\phi}$)
are fixed. As a consequence, we find that, in the Lagrangian density ${\cal
L}_{(b_{1})}$, we have only the minus sign for the scalar and pseudo-scalar
fields
($i.e.-\,\frac{1}{2}\,\partial_{\mu}\,\phi,\;-\,\frac{1}{2}\,\partial_{\mu}\;\tilde{\phi}$)
and the plus sign
($\frac{1}{2}\,\partial_{\mu}\,\phi,\;\frac{1}{2}\,\partial_{\mu}\;\tilde{\phi}$)
for the exact expression of Lagrangian density ${\cal L}_{(b_{2})}$. Fourth,
it is straightforward to note that the duality symmetry transformations (33)
are now generalized in the following forms:

$\displaystyle
B_{\mu\nu}\longrightarrow\mp\frac{i}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,B^{\lambda\xi},\quad\phi_{\mu}\longrightarrow\pm
i\,\tilde{\phi}_{\mu},\quad\tilde{\phi}_{\mu}\longrightarrow\mp\,i\,\phi_{\mu}\quad\phi\longrightarrow\pm
i\,\tilde{\phi},$ $\displaystyle\tilde{\phi}\longrightarrow\mp
i\,\phi,\quad{\cal B}_{\mu}\longrightarrow\mp i\,B_{\mu},\quad
B_{\mu}\longrightarrow\pm i\,{\cal B}_{\mu},\quad B\longrightarrow\pm i{\cal
B},\quad{\cal B}\longrightarrow\mp i\,B,$ $\displaystyle\bar{{\cal
B}}_{\mu}\longrightarrow\mp
i\,\bar{B}_{\mu},\quad\bar{B}_{\mu}\longrightarrow\pm i\,\bar{{\cal
B}}_{\mu},\quad\bar{B}\longrightarrow\pm i\bar{{\cal B}},\quad\bar{{\cal
B}}\longrightarrow\mp i\,\bar{B}.$ (48)

Finally, in the next section, we shall take only the simplest choices of signs
for the (pseudo-) scalar fields within the framework of BRST formalism where
the Lagrangian density ${\cal L}_{(b_{1})}$ will be generalized to incorporate
into them the Faddeev-Popov ghost terms.

##  6  Nilpotent (co-)BRST Invariant Lagrangian Density

We have generalized the Lagrangian densities ${\cal L}_{(b_{1})}$ and ${\cal
L}_{(b_{2})}$ to their counterparts nilpotent (anti-)BRST and (anti-)co-BRST
invariant Lagrangian densities ${\cal L}_{\cal B}$ and ${\cal L}_{\bar{\cal
B}}$ that incorporate the Faddeev-Popov ghost terms. Such a set of coupled
(but equivalent) Lagrangian densities have been written in our earlier works
[12, 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, too. However, we shall not dwell on them as
it will be only an academic exercise. We would like to emphasize that, in our
earlier works [12, 21], such kinds of relationships have not been performed.

Towards the above goal in mind, we begin with the following (co-)BRST
invariant Lagrangian density∥∥∥For the sake of brevity, we have taken only one
specific sign in the kinetic energy and gauge-fixing terms for the
$B_{\mu\nu}$ and associated fields. This is true for the (anti-)ghost fields,
too. In our Appendix B, we take the most general form of the (co-)BRST
invariant Lagrangian density which respects the generalized forms of (co-)BRST
symmetry transformations corresponding to classical transformations (37) and
(35). (where ${\cal L}_{(b_{1})}\longrightarrow{\cal L}_{\cal B}$) (see, e.g.
[9, 12, 21] for details)

$\displaystyle{\cal L}_{\cal B}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,{\cal B}_{\mu}\,{\cal B}^{\mu}-{\cal
B}^{\mu}\,\Big{[}\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\nu}\,B^{\lambda\xi}+m\,\tilde{\phi}_{\mu}-\frac{1}{2}\,\partial_{\mu}\tilde{\phi}\Big{]}-\,\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}$
(49) $\displaystyle-$
$\displaystyle\frac{1}{4}\,\Phi_{\mu\nu}\,\Phi^{\mu\nu}+\frac{1}{4}\tilde{\Phi}_{\mu\nu}\,\tilde{\Phi}^{\mu\nu}+\frac{m}{2}\,B_{\mu\nu}\,\Big{[}\Phi^{\mu\nu}+\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}\,\tilde{\Phi}_{\rho\sigma}\Big{]}$
$\displaystyle+$ $\displaystyle
B^{\mu}\,\Big{[}\partial^{\nu}\,B_{\nu\mu}+m\,\phi_{\mu}-\frac{1}{2}\,\partial_{\mu}\,\phi\Big{]}\,-\frac{1}{2}\,B^{\mu}\,B_{\mu}+B\,\Big{(}\partial\cdot\phi+\frac{m}{2}\,\phi\Big{)}$
$\displaystyle+$ $\displaystyle\frac{1}{2}\,B^{2}-{\cal
B}\,\Big{(}\partial\cdot\tilde{\phi}+\frac{m}{2}\,\tilde{\phi}\Big{)}-\frac{1}{2}\,{\cal
B}^{2}-\,\frac{1}{2}\,\partial_{\mu}\,\bar{\beta}\,\partial^{\mu}\,\beta+\frac{m^{2}}{2}\,\bar{\beta}\,\beta$
$\displaystyle-$
$\displaystyle(\partial_{\mu}\,\bar{C}_{\nu}-\partial_{\nu}\,\bar{C}_{\mu})\,(\partial^{\mu}\,C^{\nu})+(\partial_{\mu}\,\bar{C}-m\,\bar{C}_{\mu})\,(\partial^{\mu}\,C-m\,C^{\mu})$
$\displaystyle-$
$\displaystyle\frac{1}{2}\,\Big{(}\partial\cdot\bar{C}+m\,\bar{C}+\frac{\rho}{4}\Big{)}\,\lambda-\frac{1}{2}\,\Big{(}\partial\cdot
C+m\,C-\frac{\lambda}{4}\Big{)}\,\rho,$

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

The above Lagrangian density respects the following off-shell nilpotent
$(s_{b}^{2}=0)$ BRST symmetry transformations $(s_{b})$, namely;

$\displaystyle
s_{b}\,B_{\mu\nu}=-\,(\partial_{\mu}\,C_{\nu}-\partial_{\nu}\,C_{\mu}),\quad
s_{b}\,C_{\mu}=-\,\partial_{\mu}\,\beta,\quad s_{b}\bar{C}_{\mu}=B_{\mu},$
$\displaystyle s_{b}\,\bar{\beta}=-\,\rho,\quad
s_{b}\,\phi_{\mu}=+(\partial_{\mu}C-m\,C_{\mu}),\quad s_{b}\,\bar{C}=B,\quad
s_{b}\,\phi=+\lambda,$ $\displaystyle s_{b}\,C=-\,m\,\beta,\quad
s_{b}\,[H_{\mu\nu\lambda},\,B,\,\lambda,\,\rho,\,B_{\mu},\,{\cal
B}_{\mu},\,\beta,\,{\cal
B},\,\tilde{\phi}_{\mu},\,\tilde{\phi}_{\mu\nu},\,\tilde{\phi}]=0,$ (50)

because the Lagrangian density ${\cal L}_{(\cal B)}$ transforms as [12]

$\displaystyle s_{b}\,{\cal L}_{\cal B}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,\Big{[}-m\,\varepsilon^{\mu\nu\lambda\xi}\,\tilde{\phi}_{\nu}\,\partial_{\lambda}\,C_{\xi}-\,(\partial^{\mu}\,C^{\nu}-\partial^{\nu}\,C^{\mu})\,B_{\nu}-\frac{1}{2}\,\lambda\,B^{\mu}$
(51) $\displaystyle+$ $\displaystyle
B\,(\partial^{\mu}\,C-m\,C^{\mu})+\frac{1}{2}\,\rho\,(\partial^{\mu}\,\beta)\Big{]},$

which implies that the action integral $S=\int d^{4}\,x\,{\cal L}_{(\cal B)}$
remains invariant under the infinitesimal, continuous and nilpotent BRST
symmetry transformations (50). This happens because of Gausss’s divergence
theorem due to which all the physical fields vanish-off as
$x\longrightarrow\pm\,\infty$. In addition to $s_{b}$, the Lagrangian density
${\cal L}_{\cal B}$ also respects the infinitesimal, continuous and nilpotent
$[s_{d}^{2}=0]$ co-BRST symmetry transformations $(s_{d})$ [12]:

$\displaystyle
s_{d}\,B_{\mu\nu}=-\,\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\lambda}\,\bar{C}^{\xi},\quad
s_{d}\,\bar{C}_{\mu}=-\,\partial_{\mu}\,\bar{\beta},\quad s_{d}\,C_{\mu}={\cal
B}_{\mu},$ $\displaystyle s_{d}\,\beta=-\,\lambda,\quad
s_{d}\,\tilde{\phi}_{\mu}=+(\partial_{\mu}\,\bar{C}-m\,\bar{C}_{\mu}),\quad
s_{d}\,C={\cal B},\quad s_{d}\,\bar{C}=-m\,\bar{\beta},$ $\displaystyle
s_{d}\,\tilde{\phi}=-\rho,\quad s_{d}\,[\partial^{\nu}\,B_{\nu\mu},\,{\cal
B}_{\mu},\,B_{\mu},\,{\cal
B},\,\phi_{\mu},\,\Phi_{\mu\nu},\,\phi,\,\bar{\beta},\,\lambda,\,\rho]=0.$
(52)

It is straightforward to check that ${\cal L}_{(\cal B)}$ transforms, under
$(s_{d})$, as the total spacetime derivative in the four (3+1)-dimensional
spacetime, namely;

$\displaystyle s_{d}\,{\cal L}_{\cal B}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,\Big{[}-m\,\varepsilon^{\mu\nu\lambda\xi}\,\phi_{\nu}\,\partial_{\lambda}\,\bar{C}_{\xi}+\,(\partial^{\mu}\,\bar{C}^{\nu}-\partial^{\nu}\,\bar{C}^{\mu})\,{\cal
B}_{\nu}-\frac{1}{2}\,\rho\,{\cal B}^{\mu}$ (53) $\displaystyle-$
$\displaystyle(\partial^{\mu}\,\bar{C}-m\,\bar{C}^{\mu})\,{\cal
B}+\frac{1}{2}\,\lambda\,(\partial^{\mu}\,\bar{\beta})\Big{]}.$

As a consequence of the above observation, we find that the action integral
$S=\int d^{4}\,x\,{\cal L}_{(\cal B)}$ remains invariant under the co-BRST
transformation $s_{d}$ for all the physical fields that vanish-off as
$x\longrightarrow\pm\infty$ due to Gauss’s divergence theorem.

In addition to discrete duality symmetry transformations (48) in the bosonic
(i.e. non-ghost) sector of the Lagrangian density ${\cal L}_{(\cal B)}$, we
have the following discrete symmetry transformations in the ghost-
sector******It can be readily checked that the Faddeev-Popov ghost part of the
Lagrangian density ${\cal L}_{\cal B}$ remains invariant under a couple of
discrete symmetry transformations (54) of our theory, namely;

$\displaystyle C_{\mu}\longrightarrow\pm
i\,\bar{C}_{\mu},\quad\bar{C}_{\mu}\longrightarrow\pm i\,C_{\mu},\quad
C\longrightarrow\pm i\,\bar{C},\quad\bar{C}\longrightarrow\pm i\,C,$
$\displaystyle\rho\longrightarrow\mp
i\,\lambda,\qquad\lambda\longrightarrow\mp
i\,\rho,\qquad\beta\longrightarrow\pm
i\,\bar{\beta},\qquad\bar{\beta}\longrightarrow\mp i\,\beta.$ (54)

Under the full discrete 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 $s_{b}\longrightarrow s_{d}$, we obtain

$\displaystyle
s_{b}\,(*\,B_{\mu\nu})=-\,*\,(\partial_{\mu}\,C_{\nu}-\partial_{\nu}\,C_{\mu})\quad\Longrightarrow\quad
s_{d}\,B_{\mu\nu}=-\,\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\lambda}\,\bar{C}^{\xi},$
(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 exactly similar fashion, we note the following
(with replacement: $s_{d}\longrightarrow s_{b}$), for the transformations
$s_{d}\,B_{\mu\nu}=-\,\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\nu}\,\bar{C}^{\xi}$,
namely;

$\displaystyle
s_{d}\,(*\,B_{\mu\nu})=-\,*\,\varepsilon_{\mu\nu\lambda\xi}\,(\partial^{\lambda}\,\bar{C}^{\xi})\quad\longrightarrow\quad
s_{b}\,B_{\mu\nu}=-\,(\partial_{\mu}\,C_{\nu}-\partial_{\nu}\,C_{\mu}),$ (56)

when, 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 field. To corroborate this assertion, let
us focus on the symmetry transformation:
$s_{b}\,\phi_{\mu}=+(\partial_{\mu}\,C-m\,C_{\mu})$ on a bosonic vector field
$(\phi_{\mu})$. By exploiting the strength of the full discrete duality
symmetry transformations (48) and (54), we observe the following
transformations on the axial-vector field (with input: $s_{b}\longrightarrow
s_{d}$), namely;

$\displaystyle
s_{b}\,(*\,\phi_{\mu})=+*\,(\partial_{\mu}\,C-m\,C_{\mu})\quad\Longrightarrow\quad
s_{d}\,\tilde{\phi}_{\mu}=+(\partial_{\mu}\,\bar{C}-m\,\bar{C}_{\mu}).$ (57)

This happens because, under discrete duality symmetry transformations (48), we
have: $\phi_{\mu}\rightarrow\pm\,i\,\tilde{\phi}_{\mu}$ and
$\tilde{\phi}_{\mu}\rightarrow\mp\,i\,\phi_{\mu}$. In exactly, similar
fashion, we obtain the reverse symmetry transformations as follows (with
input: $s_{d}\longrightarrow s_{b}$), namely;

$\displaystyle
s_{d}\,(*\,\tilde{\phi}_{\mu})=+*\,(\partial_{\mu}\,\bar{C}-m\,\bar{C}_{\mu})\quad\Longrightarrow\quad
s_{b}\,\phi_{\mu}=+(\partial_{\mu}\,C-m\,C_{\mu}).$ (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
($B_{\mu\nu}\longrightarrow\mp\,\,(i/2)\,\varepsilon_{\mu\nu\lambda\xi}\,B^{\lambda\xi},\;\phi_{\mu}\longrightarrow\pm\,i\,\tilde{\phi}_{\mu},\;\tilde{\phi}_{\mu}\longrightarrow\mp\,i\,\phi_{\mu}$)
of the modified St$\ddot{u}$ckelberg formalism [cf. Eqs (9), (10)]. To
complete our present discussion, let us focus on a transformation on a
fermionic field $s_{d}\,\bar{C}=-\,m\,\bar{\beta}$. Using the strength of the
discrete duality symmetry transformations (54), we obtain the following (with
the input: $s_{d}\longrightarrow s_{b}$), namely;

$\displaystyle
s_{d}\,[*\,(\bar{C})]=-\,m\,*\,\bar{\beta}\quad\Longrightarrow\quad
s_{b}\,C=-\,m\,\beta.$ (59)

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

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 been
seen that the interplay of the discrete and continuous symmetry
transformations provides the physical realization of $\delta=\pm*\,d\,*$ that
exist [2-5] between the (co-)exterior derivatives of differential geometry.
This interesting and beautiful relationship between $s_{d}$ and $s_{b}$
is††††††The $(\pm)$ signs on the r.h.s. of (60) are dictated by the successive
operations of the discrete duality symmetry transformations (48) and (54) on
the generic field
$\Phi=B_{\mu\nu},\,\phi_{\mu},\,\tilde{\phi}_{\mu},\,C_{\mu},\,\bar{C}_{\mu},\,\phi,\,\tilde{\phi},$
etc. In other words, the signs on the r.h.s. of $*\,(*\,\Phi)=\pm\,\Phi$
dictate the signs on (60) (see, e.g., [5, 12] for details).

$\displaystyle s_{d}=\pm*\,s_{b}\,*,$ (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 $(d)$ and co-exterior $(\delta)$
derivatives of differential geometry [2-5] even though these derivatives are
connected with each-other by the relationship: $\delta=\pm*\,d\,*$. Finally,
it can be seen that the exactly similar kind of relationships exist between
the nilpotent anti-co-BRST symmetry and anti-BRST symmetry transformations
that exist for the Lagrangian density ${\cal L}_{\bar{\cal B}}$ which turns to
be the generalization of ${\cal L}_{b_{2}}$ [cf. Eq. (45)] (see, e.g. [12, 9]
for details).

##  7 Conclusions

The St$\ddot{u}$ckelberg-modified massive 4D free Abelian 2-form theory has
already been proven to be a massive model of Hodge theory [12] 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 [2-5] of the differential geometry at the algebraic
level within the framework of BRST formalism [12]. However, the full coupled
(but equivalent) Lagrangian densities of this theory have been obtained by
trial and error method. In our present investigation, we have 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. Sec. 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. [12]) within the framework of BRST
formalism.

One of the key results of our present investigation is the modification [cf.
Eqs. (7), (9)] of the St$\ddot{u}$ckelberg-formalism (SF) in 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. Eq. (10)] whose generalizations [cf. Eqs. (48), (54)], within the realm
of BRST formalism, provide the physical realizations of the Hodge duality $*$
operation of the differential geometry. As the gauge-fixed Lagrangian density
(29) remains invariant under the discrete duality symmetry transformations
(10), in exactly similar fashion, the (co-)BRST invariant Lagrangian density
(49) remains invariant under the generalization of the discrete duality
symmetry transformations (10) $(i)$ to the equation (48) in the non-ghost
sector, and $(ii)$ to the equation (54) in the ghost-sector of the Lagrangian
density (49). In addition, we have been able to establish a 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). The latter symmetry transformations also play an important role
in providing the analogue of relationship: $\delta=\pm\,*\,d\,*$ in the
terminology of symmetry transformations of our present massive 4D theory [cf.
Eq. (60)].

It is worthwhile to mention that the modified SF [cf. Eq. (9)] is invariant
under the discrete duality symmetry transformations (10) and they lead to the
combination of the polar-vector and axial-vector fields ($\phi_{\mu}$ and
$\tilde{\phi}_{\mu}$) in the form:
$\partial_{\mu}\,\phi_{\nu}-\partial_{\nu}\,\phi_{\mu}+\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\lambda}\,\tilde{\phi}^{\xi}$.
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:
$F_{\mu\nu}=\partial_{\mu}\,V_{\nu}-\partial_{\nu}\,V_{\mu}+\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\lambda}\,A^{\xi}$
where $V_{\mu}$ and $A_{\mu}$ are the polar-vector and axial-vector
potentials. We have discussed the local duality invariance [23] of the
Maxwell’s theory with these potentials and shown the existence of an axial-
photon which mediates the spin-spin universal long-range interaction (see,
e.g. [24], [23]). 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 $\tilde{\phi}_{\mu}$ and $\phi$ turn up with
negative kinetic terms in our theory which are interesting in the sense that
they belong to the class of exotic fields that are supposed to be one of
possible candidates for the dark matter/dark energy [19, 20] and “phantom”
and/or “ghost” fields in the context of cyclic, bouncing and self-accelerated
cosmological models of Universe [13-15].

In a set of very nice works [25-27], the St$\ddot{u}$ckelberg-modified (SUSY)
quantum electrodynamics and other aspects of (non-)interacting Abelian gauge
theories have been considered where an ultralight dark matter candidate has
been proposed and St$\ddot{u}$ckelberg-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 [25-27]. 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 St$\ddot{u}$ckelberg-modified
Proca (i.e. massive Abelian 1-form) theory [8] as well as our present work on
the St$\ddot{u}$ckelberg-modified massive 4D Abelian 2-form theory to study
the St$\ddot{u}$ckelberg-modified massive 6D Abelian 3-form theory within the
framework of BRST formalism. We shall report on our progress in the above
mentioned theoretical directions in our future publication(s) [28].

Acknowledgments

One of us (AKR) thankfully acknowledges the financial support from the BHU-
fellowship program of the Banaras Hindu University (BHU), Varanasi, under
which the present research work has been carried out. Fruitful discussions
with Mr. S. Kumar, Mr. B. Chauhan and Mr. A. Tripathi are gratefully
acknowledge, too. The authors dedicate their present work to late Prof. T.
Pradhan who was the Ph. D. advisor to one (RPM) of them.

Appendix A: On Modified 2D Proca Theory

For our present paper to be self-contained, we dwell a bit on the massive 2D
Abelian 1-form (i.e. Proca) theory which has been at the heart of our present
investigation on the massive 4D Abelian 2-form theory. We start off with the
Proca Lagrangian density $[{\cal L}_{(P)}]$ for a vector boson ($A_{\mu}$)
with rest mass $m$ as follows (see, e.g. [17])

${\cal
L}_{(P)}=-\,\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu}+\frac{m^{2}}{2}\,A_{\mu}\,A^{\mu},$
$None$

where the 2-form $F^{(2)}=d\,A^{(1)}=[(d\,x^{\mu}\wedge
d\,x^{\nu})/{2!}]\,F_{\mu\nu}$ defines the field strength tensor
$F_{\mu\nu}=\partial_{\mu}\,A_{\nu}-\partial_{\nu}\,A_{\mu}$ for the vector
field $A_{\mu}$ that is defined through an Abelian 1-form
($A^{(1)}=d\,x^{\mu}\,A_{\mu}$). Here the symbol
$d=d\,x^{\mu}\,\partial_{\mu}$ (with $d^{2}=0$) stands for the exterior
derivative of differential geometry [2-5]. The standard St$\ddot{u}$ckelberg
formalism (valid in any arbitrary D-dimensional spacetime) gets modified in 2D
case as (see, e.g. [8] for details)

$A_{\mu}\longrightarrow
A_{\mu}\mp\frac{1}{m}\,(\partial_{\mu}\phi+\varepsilon_{\mu\nu}\,\partial^{\nu}\,\tilde{\phi}),$
$None$

where $\phi$ is a pure-scalar field and $\tilde{\phi}$ is a pseudo-scalar
field in 2D spacetime which is endowed with the Levi-Civita tensor
$\varepsilon_{\mu\nu}$ (with
$\varepsilon_{01}=\varepsilon^{10}=+1,\,\varepsilon_{\mu\nu}\,\varepsilon^{\mu\nu}=-\,2!,\,\varepsilon_{\mu\nu}\,\varepsilon^{\mu\rho}=-\,1!\,\delta_{\nu}^{\rho},\,E=-\,\varepsilon^{\mu\nu}\,\partial_{\mu}\,A_{\nu}=F_{01},$
etc.). It can be readily checked that the modified 2D St$\ddot{u}$ckelberg
formalism is invariant under the discrete symmetry transformations:
$A_{\mu}\to\mp\,i\,\varepsilon_{\mu\nu}A^{\nu},\;\phi\to\mp\,i\,\tilde{\phi},\;\tilde{\phi}\to\mp\,i\,\phi$
which play a very important role in establishing a relationship with the Hodge
duality $*$ operation of the differential geometry (see, e.g. [8]).

We observe that, under the modified St$\ddot{u}$ckelberg formalism (A.2), the
field-strength tensor transforms as (see, e.g. [8])

$F_{\mu\nu}\longrightarrow
F_{\mu\nu}\mp\frac{1}{m}\,(\varepsilon_{\nu\rho}\,\partial_{\mu}-\varepsilon_{\mu\rho}\,\partial_{\nu})\,(\partial^{\rho}\,\tilde{\phi}).$
$None$

We can introduce a notation
$\Sigma_{\mu\nu}=(\varepsilon_{\mu\rho}\,\partial_{\nu}-\varepsilon_{\nu\rho}\,\partial_{\mu})\,(\partial^{\rho}\,\tilde{\phi})$
to re-express the above transformation for the field strength tensor as
follows

$F_{\mu\nu}\longrightarrow F_{\mu\nu}\pm\frac{1}{m}\,\Sigma_{\mu\nu},$ $None$

which leads to the following transformations for the kinetic term
[$-\,(1/4)\,F_{\mu\nu}\,F^{\mu\nu}$] of the Proca (i.e. massive Abelian
1-form) theory, namely;

$-\,\frac{1}{4}\,F^{\mu\nu}\,F_{\mu\nu}\longrightarrow-\,\frac{1}{4}\,F^{\mu\nu}\,F_{\mu\nu}\mp\frac{1}{2\,m}\,F^{\mu\nu}\,\Sigma_{\mu\nu}-\,\frac{1}{4\,m^{2}}\,\Sigma^{\mu\nu}\,\Sigma_{\mu\nu}.$
$None$

It is straightforward to note that the second and third terms, on the r.h.s.
of (A.5), are higher order derivative terms for a 2D theory of a vector boson.
This is due to the fact that there are three and four derivative terms in the
second and third terms, respectively, of the transformation for the field
strength tensor [c.f. Eq. (A.5)] under (A.2) .

We can get rid of the higher derivative terms by exploiting the on-shell
conditions: $\partial_{\mu}\,F^{\mu\nu}+m^{2}\,A^{\nu}=0$ and
$(\Box+m^{2})\,\tilde{\phi}=0$. The latter implies that the on-shell condition
$(\Box+m^{2})\,\partial^{\mu}\,\tilde{\phi}=0$ is also true. The second term,
on the r.h.s. of (A.5), can be explicitly expressed as follows:

$\mp\frac{1}{2\,m}\,F^{\mu\nu}\,(\varepsilon_{\mu\rho}\,\partial_{\nu}-\varepsilon_{\nu\rho}\,\partial_{\mu})\,(\partial^{\rho}\,\tilde{\phi}).$
$None$

Dropping the total spacetime derivative terms, we note that we have the
following explicit form of (A.6), namely;

$\pm\,\frac{1}{2\,m}\,(\partial_{\nu}\,F^{\mu\nu})\,\varepsilon_{\mu\rho}\,(\partial^{\rho}\,\tilde{\phi})\mp\,\frac{1}{2\,m}\,(\partial_{\mu}\,F^{\mu\nu})\,\varepsilon_{\nu\rho}\,(\partial^{\rho}\,\tilde{\phi}),$
$None$

where both the terms are equal and they lead to the following (due to the use
of $E=-\,\varepsilon^{\mu\nu}\,\partial_{\mu}\,A_{\nu}$ and on-shell
condition: $\partial_{\mu}\,F^{\mu\nu}=-\,m^{2}\,A^{\nu}$), namely;

$\pm\,m\,A^{\nu}\,\varepsilon_{\nu\rho}\,\partial^{\rho}\,\tilde{\phi}\;\;\equiv\;\;\pm\,\varepsilon^{\rho\nu}\,(\partial_{\rho}\,A_{\nu})\,\tilde{\phi}\;\;\equiv\;\;\mp\,m\,E\,\tilde{\phi}.$
$None$

In the above, we have dropped a total spacetime derivative term (as it is a
part of the Lagrangian density and its presence does not change the dynamics).

We concentrate now on the third term (with four derivatives) on the r.h.s. of
equation (A.5) which is explicitly expressed as:

$-\,\frac{1}{4\,m^{2}}\,\Sigma^{\mu\nu}\,\Sigma_{\mu\nu}=-\,\frac{1}{4\,m^{2}}\,[(\varepsilon^{\mu\rho}\,\partial^{\nu}-\varepsilon^{\nu\rho}\,\partial^{\mu})\,(\partial_{\rho}\,\tilde{\phi})]\,[(\varepsilon_{\mu\sigma}\,\partial_{\nu}-\varepsilon_{\nu\sigma}\,\partial_{\mu})\,(\partial^{\sigma}\,\tilde{\phi}).$
$None$

The above expression leads to the following (modulo total spacetime
derivatives)

$-\,\frac{2}{4\,m^{2}}\,[\varepsilon^{\mu\rho}\,\partial^{\nu}\,(\partial_{\rho}\,\tilde{\phi})\,\varepsilon_{\mu\sigma}\,\partial_{\nu}\,(\partial^{\sigma}\,\tilde{\phi})]\;\;\equiv\;\;\frac{1}{2\,m^{2}}\,\partial^{\nu}\,(\partial_{\sigma}\,\tilde{\phi})\partial_{\nu}\,(\partial^{\sigma}\,\tilde{\phi}),$
$None$

where we have used
$\varepsilon^{\mu\rho}\,\varepsilon_{\mu\sigma}=-\,\delta_{\sigma}^{\rho}$.
Dropping, once again, the total spacetime derivative term, we obtain the
following

$\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\,\partial^{\mu}\,\tilde{\phi}\;\;\equiv\;\;+\frac{1}{2}\,m^{2}\,\tilde{\phi}^{2},$
$None$

where we have used the on-shell conditions:
$(\Box+m^{2})\,\partial^{\sigma}\,\tilde{\phi}=0,\,(\Box+m^{2})\,\tilde{\phi}=0$.
Since the field-strength tensor $F_{\mu\nu}$ has only one non-vanishing
component in 2D (which is nothing but the pseudo-scalar electric field
$E=F_{01}=-\,\varepsilon^{\mu\nu}\,\partial_{\mu}\,A_{\nu}$), we note that the
explicit form of (A.5), with the help of (A.8) and (A.11), is as follows

$-\,\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu}\longrightarrow\frac{1}{2}\,E^{2}\mp\,m\,E\,\tilde{\phi}+\frac{1}{2}\,m^{2}\,\tilde{\phi}^{2}\equiv\frac{1}{2}\,(E\,\mp\,m\,\tilde{\phi})^{2},$
$None$

which has been derived in a different manner‡‡‡‡‡‡In our earlier work [8], we
have taken into account
$F_{01}=\partial_{0}\,A_{1}-\partial_{1}\,A_{0}\equiv-\,\varepsilon^{\mu\nu}\,\partial_{\mu}\,A_{\nu}$
and obtained $F_{01}=E\rightarrow(E\pm\frac{1}{m}\,\Box\,\tilde{\phi})$ which
has been converted into: $E\rightarrow(E\mp m\,\tilde{\phi})$ because of the
use of mass-shell condition: $\Box\,\tilde{\phi}=-\,m^{2}\,\tilde{\phi}$. The
latter emerges out as an EoM from the Lagrangian density (A.15). in our
earlier work [12]. It is straightforward to note that the mass term of (A.1)
transforms under (A.2) as

$\frac{m^{2}}{2}\,A_{\mu}\,A^{\mu}\longrightarrow\frac{m^{2}}{2}\,A_{\mu}\,A^{\mu}\mp
m\,A_{\mu}\,\partial^{\mu}\,\phi+\frac{1}{2}\,\partial_{\mu}\,\phi\,\partial^{\mu}\,\phi-\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\,\,\partial^{\mu}\,\tilde{\phi}\pm
m\,E\,\tilde{\phi},$ $None$

modulo some total spacetime derivative terms. Here we have used
$E=-\,\varepsilon^{\mu\nu}\,\partial_{\mu}\,A_{\nu}$. Thus, the total
Lagrangian density for the modified version of 2D Proca theory is

${\cal L}^{(2D)}_{(P)}=\frac{1}{2}\,{(E\mp m\,\tilde{\phi})}^{2}\pm
m\,E\,\tilde{\phi}-\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\,\,\partial^{\mu}\,\tilde{\phi}+\frac{m^{2}}{2}A_{\mu}\,A^{\mu}\mp
m\,A_{\mu}\,\partial^{\mu}\,\phi+\frac{1}{2}\,\partial_{\mu}\,\phi\,\partial^{\mu}\,\phi,$
$None$

which has been taken into account in our earlier works [8, 10]. It is
important to point out that the kinetic terms of the pure-scalar and pseudo-
scalar have positive and negative signs, respectively. The latter (i.e. the
pseudo-scalar) field is interesting from the point of view of the fact that it
provides a field for a possible candidate of dark matter/dark energy. Such
exotic fields are also useful in the context of cyclic, bouncing and self-
accelerated cosmological models of the Universe [13-15] where these (i.e.
fields with negative kinetic terms) have been called as the “phantom” and
“ghost” fields.

We end this Appendix with final comment that one can add the gauge-fixing term
$({\cal L}_{gf})$ to the above Lagrangian density (A.14) in the ’t Hooft gauge
as follows (see, e.g. [17])

${\cal L}^{(2D)}_{(S)}+{\cal L}_{(gf)}=\frac{1}{2}\,{(E\mp
m\,\tilde{\phi})}^{2}\pm
m\,E\,\tilde{\phi}-\frac{1}{2}\,\partial_{\mu}\,\tilde{\phi}\,\,\partial^{\mu}\,\tilde{\phi}+\frac{m^{2}}{2}A_{\mu}\,A^{\mu}$
$\mp\,A_{\mu}\,\partial^{\mu}\,\phi+\frac{1}{2}\,\partial_{\mu}\,\phi\,\partial^{\mu}\,\phi-\frac{1}{2}\,(\partial\,\cdot
A\pm m\,\phi)^{2},$ $None$

which respects the discrete duality symmetry transformations on the basic
fields of the theory as:
$A_{\mu}\to\mp\,i\,\varepsilon_{\mu\nu}A^{\nu},\;\phi\to\mp\,i\,\tilde{\phi},\;\tilde{\phi}\to\mp\,i\,\phi$.
The Lagrangian density (A.15) has been taken into account for the BRST
analysis in our earlier work [8, 10].

Appendix B: On the Generalized (co-)BRST Symmetries

The central purpose of our present Appendix is to generalize the classical
(dual-)gauge symmetry transformations (37) and (35), respectively, to their
counterparts quantum (co-) BRST symmetry transformations for the appropriate
generalized form of the (co-)BRST invariant Lagrangian density [that is more
general than the Lagrangian density (49)]. First of all, we generalize the
classical gauge symmetry transformations (35) to the following off-shell
nilpotent ($s_{b}^{2}=0$) quantum BRST symmetry transformations, namely;

$~{}~{}~{}~{}s_{b}\,B_{\mu\nu}=-\,(\partial_{\mu}\,C_{\nu}-\partial_{\nu}\,C_{\mu}),\qquad
s_{b}\,C_{\mu}=-\,\partial_{\mu}\,\beta,\qquad s_{b}\,\bar{C}_{\mu}=B_{\mu},$
$~{}s_{b}\,\phi_{\mu}=\pm(\partial_{\mu}C-m\,C_{\mu}),\quad\qquad
s_{b}\,C=-\,m\,\beta,\qquad s_{b}\,\bar{C}=B,$
$s_{b}\,\Phi_{\mu\nu}=\mp\,m\,(\partial_{\mu}\,C_{\nu}-\partial_{\nu}\,C_{\mu})\qquad
s_{b}\,\phi=\pm\lambda,\qquad s_{b}\,\bar{\beta}=\mp\,\rho,$
$s_{b}\,[H_{\mu\nu\lambda},\,\rho,\,\lambda,\,\beta,\,{\cal
B}_{\mu},\,B_{\mu},\,B,\,\tilde{\phi},\,\tilde{\phi}_{\mu},\,\tilde{\Phi}_{\mu\nu}]=0,$
$None$

which transform the following generalized (co-)BRST invariant Lagrangian
density $({\cal L}_{\cal B}^{(g)})$, with appropriate $(\pm)$ signs, namely;

$\qquad{\cal L}_{\cal B}^{(g)}\quad=\frac{1}{2}\,{\cal B}_{\mu}\,{\cal
B}^{\mu}-{\cal
B}^{\mu}\,\Big{[}\frac{1}{2}\,\varepsilon_{\mu\nu\lambda\xi}\,(\partial^{\nu}\,B^{\lambda\xi})+m\,\tilde{\phi}_{\mu}-\frac{1}{2}\,\partial_{\mu}\tilde{\phi}\Big{]}-\,\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu}$
$-\frac{1}{4}\,\Phi_{\mu\nu}\,\Phi^{\mu\nu}+\frac{1}{4}\tilde{\Phi}_{\mu\nu}\,\tilde{\Phi}^{\mu\nu}\pm\frac{m}{2}\,B_{\mu\nu}\,\Big{[}\Phi^{\mu\nu}+\frac{1}{2}\varepsilon^{\mu\nu\lambda\xi}\,\tilde{\Phi}_{\lambda\xi}\Big{]}$
$\qquad\qquad+B^{\mu}\,\Big{[}(\partial^{\nu}\,B_{\nu\mu})+m\,\phi_{\mu}-\frac{1}{2}\,\partial_{\mu}\,\phi\Big{]}\,-\frac{1}{2}\,B^{\mu}\,B_{\mu}+B\,\Big{(}\partial\cdot\phi+\frac{m}{2}\,\phi\Big{)}$
$\qquad+\frac{1}{2}\,B^{2}-{\cal
B}\,\Big{(}\partial\cdot\tilde{\phi}+\frac{m}{2}\,\tilde{\phi}\Big{)}-\frac{1}{2}\,{\cal
B}^{2}-\,\frac{1}{2}\,\partial_{\mu}\,\bar{\beta}\,\partial^{\mu}\,\beta+\frac{m^{2}}{2}\,\bar{\beta}\,\beta$
$\;\qquad-(\partial_{\mu}\,\bar{C}_{\nu}-\partial_{\nu}\,\bar{C}_{\mu})\,(\partial^{\mu}\,C^{\nu})\pm(\partial_{\mu}\,\bar{C}-m\,\bar{C}_{\mu})\,(\partial^{\mu}\,C-m\,C^{\mu})$
$\mp\,\frac{1}{2}\,\Big{(}\partial\cdot\bar{C}+m\,\bar{C}+\frac{\rho}{4}\Big{)}\,\lambda\mp\frac{1}{2}\,\Big{(}\partial\cdot
C+m\,C-\frac{\lambda}{4}\Big{)}\,\rho,$ $None$

to the total spacetime derivative as:

$s_{b}\,{\cal L}_{\cal B}^{(g)}=\partial_{\mu}\,\Big{[}\mp
m\,\varepsilon^{\mu\nu\lambda\xi}\,\tilde{\phi}_{\nu}\,\partial_{\lambda}\,C_{\xi}-\,(\partial^{\mu}\,C^{\nu}-\partial^{\nu}\,C^{\mu})\,B_{\nu}\mp\frac{1}{2}\,\lambda\,B^{\mu}$
$\pm\,B\,(\partial^{\mu}\,C-m\,C^{\mu})\pm\frac{1}{2}\,\rho\,(\partial^{\mu}\,\beta)\Big{]},$
$None$

As a consequence of the above observation, it is clear that the action
integral $S=\int d^{4}x\,{\cal L}_{\cal B}^{(g)}$ remains invariant
($s_{b}\,S=0$) under the infinitesimal, continuous and off-shell nilpotent
$(s_{b}^{2}=0)$ BRST symmetry transformations (B.1). A noteworthy point, at
this juncture, is the observation that $(\pm)$ signs, associated with
($\pm\,m\,\tilde{\phi}_{\mu},\,\pm\,m\,\phi_{\mu}$) in the kinetic term and
gauge-fixing term, respectively, have been changed to
($+m\,\tilde{\phi}_{\mu},\,+m\,\phi_{\mu}$) because only this choice of sign
is allowed by the nilpotent BRST symmetry transformations (B.1).

The generalized Lagrangian density $({\cal L}_{\cal B}^{(g)})$ also respects a
set of off-shell nilpotent ($s_{d}^{2}=0$) dual-BRST (i.e. co-BRST) symmetry
transformations $(s_{d})$ as quoted below

$~{}~{}~{}\qquad\qquad
s_{d}\,B_{\mu\nu}=-\,\varepsilon_{\mu\nu\lambda\xi}\,\partial^{\lambda}\,\bar{C}^{\xi},\quad\quad
s_{d}\,\bar{C}_{\mu}=-\,\partial_{\mu}\,\bar{\beta},\qquad\quad
s_{d}\,C_{\mu}={\cal B}_{\mu},$ $s_{d}\,\tilde{\phi}=\mp\rho,\quad\qquad\qquad
s_{d}\,\tilde{\phi}_{\mu}=\pm(\partial_{\mu}\,\bar{C}-m\,\bar{C}_{\mu}),\quad$
$~{}~{}~{}s_{d}\,\beta=\mp\,\lambda,\qquad\qquad s_{d}\,C={\cal
B},\qquad\qquad s_{d}\,\bar{C}=-m\,\bar{\beta},$
$s_{d}\,[(\partial^{\nu}\,B_{\nu\mu},\,{\cal B}_{\mu},\,B_{\mu},\,{\cal
B},\,\phi,\,\phi_{\mu},\,\Phi_{\mu\nu},\,\bar{\beta},\,\lambda,\,\rho]=0.$
$None$

because ${\cal L}_{\cal B}^{(g)}$ transforms to a total spacetime derivative
as follows:

$s_{d}\,{\cal L}_{\cal B}^{(g)}=\partial_{\mu}\,\Big{[}\mp
m\,\varepsilon^{\mu\nu\lambda\xi}\,\phi_{\nu}\,\partial_{\lambda}\,\bar{C}_{\xi}+\,(\partial^{\mu}\,\bar{C}^{\nu}-\partial^{\nu}\,\bar{C}^{\mu})\,{\cal
B}_{\nu}\mp\frac{1}{2}\,\rho\,{\cal B}^{\mu}$ $\mp\,{\cal
B}(\partial^{\mu}\,\bar{C}-m\,\bar{C}^{\mu})\pm\,\frac{1}{2}\,\lambda\,(\partial^{\mu}\,\bar{\beta})\Big{]}.$
$None$

As a consequence, it is crystal clear that the infinitesimal, continuous and
off-shell nilpotent ($s_{d}^{2}=0$) co-BRST transformations (B.4) are the
symmetry transformation for the action integral $S=\int d^{4}x\,{\cal L}_{\cal
B}^{(g)}$ due to the validity of the Gauss’s divergence theorem, because of
which, all the physical fields vanish-off as $x\longrightarrow\pm\,\infty$.

We end this Appendix with the final remark that the modified SF [cf. Eq. (9)]
and Lagrangian density (B.2) remain invariant under the discrete duality
symmetry transformations [cf. Eqs. (48), (54)] at the quantum  level.
Furthermore, these latter discrete symmetry transformations provide a
connection between the BRST symmetry transformations (B.1) and co-BRST
symmetry transformations (B.4) in exactly the same manner as we have discussed
such kind of relationship in the simpler case of Lagrangian density (49) in
Sec. 6. To take a simple example, let us focus on $s_{b}\,\phi=\pm\,\lambda$.
If we take the input $s_{b}\rightarrow s_{d}$ and the discrete symmetry
transformations:
$\phi\longrightarrow\pm\,i\,\tilde{\phi},\;\lambda\longrightarrow\mp\,i\,\rho$
[cf. Eqs. (48), (54)], we obtain $s_{d}\,\tilde{\phi}=\mp\,\rho$ from
$s_{b}\,\phi=\pm\,\lambda$. Reciprocal relationship, it can be readily
checked, is also true where we obtain $s_{b}\,\phi=\pm\,\lambda$ from
$s_{d}\,\tilde{\phi}=\mp\,\rho$ if we take into account: $s_{d}\rightarrow
s_{b}$ and the discrete symmetry transformations (48) and (54) $together$.

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