Massive 4D Abelian 2-Form Theory: Nilpotent Symmetries from the (Anti-)Chiral
Superfield Approach

S. Kumar(a), B.Chauhan(a), A. Tripathi(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><EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract: The off-shell nilpotent and absolutely anticommuting (anti-)BRST
symmetries are obtained by using the (anti-)chiral superfield approach (ACSA)
to Becchi-Rouet-Stora-Tyutin (BRST) formalism for the four (3+1)-dimensional
(4D) St$\ddot{u}$ckelberg-modified massive Abelian 2-form gauge theory. We
perform exactly similar kind of exercise for the derivation of the off-shell
nilpotent (anti-)co-BRST symmetry transformations, too. In the above
derivations, the symmetry invariant restrictions on the superfields play very
important and decisive roles. To prove the sanctity of the above nilpotent
symmetries, we generalize our 4D ordinary theory (defined on the 4D flat
Minkowskian spacetime manifold) to its counterparts (4,1)-dimensional
(anti-)chiral super sub-manifolds of the (4, 2)-dimensional supermanifold
which is parameterized by the superspace coordinates
$Z^{M}=(x^{\mu},\theta,\bar{\theta})$ where $x^{\mu}(\mu=0,1,2,3)$ are the
bosonic coordinates and a pair of Grassmannian variables
$(\theta,\bar{\theta})$ are fermionic:
$\theta^{2}=\bar{\theta^{2}}=0,\,\,\theta\,\bar{\theta}+\bar{\theta}\,\theta=0$
in nature. One of the novel observations of our present endeavor is the
derivation of the Curci-Ferrari (CF)-type restrictions from the requirement of
the symmetry invariance of the coupled (but equivalent) Lagrangian densities
of our theory within the framework of ACSA to BRST formalism. We also exploit
the standard techniques of ACSA to capture the off-shell nilpotency and
absolute anticommutativity of the conserved (anti-)BRST as well as the
conserved (anti-)co-BRST charges. In a subtle manner, the proof of the
absolute anticommutativity of the above conserved charges also implies the
existence of the appropriate CF-type restrictions on our theory. This proof is
also a novel result.

PACS numbers: 11.15.-q; 11.30.-j; 03.70.+k; 95.35.+d.

Keywords: 4D massive Abelian 2-form gauge theory; (anti-)BRST symmetries;
(anti-)co-BRST symmetries; coupled Lagrangian densities; Curci–Ferrari type
restrictions; conserved charges; ACSA to BRST formalism; symmetry invariant
restrictions; nilpotency and absolute anticommutativity properties.

##  1 Introduction

The superfield approach to Becchi-Rouet-Stora-Tyutin (BRST) formalism [1-8]
provides the geometrical origin and interpretation for the nilpotency and
absolute anticommutativity properties that are associated with the quantum
BRST and anti-BRST symmetries corresponding to a given classical local gauge
symmetry transformation for a given $p$-form ($p$ = 1, 2, 3…) gauge theory
which is generated by the first-class constraints [9, 10] that characterize
such a theory. One of the key features of the usual superfield approach (USFA)
to BRST formalism [4-6], proposed by Bonora and Tonin (BT), is the derivation
of (i) the exact off-shell nilpotent and absolutely anticommuting (anti-)BRST
symmetry transformations, and (ii) the (anti-) BRST invariant CF-type
restriction(s). The existence of the latter is the signature [11, 12] of the
quantum version of a gauge theory that is discussed and described within the
framework of BRST formalism. The other characteristic feature of USFA
(proposed by BT) is the observation that it leads only to the derivation of
the nilpotent (anti-)BRST symmetries for the gauge and associated (anti-)ghost
fields of the given $p$-form quantum gauge theory.

The above USFA has been systemically generalized in our earlier works (see,
e.g. [13-16]) which lead to the derivation of proper (anti-)BRST symmetry
transformations for the matter fields in addition to the gauge and
(anti-)ghost fields of an interacting $p$-form (non-)Abelian gauge theory. The
generalized version of the USFA has been christened as the augmented version
of superfield approach (AVSA) to BRST formalism (see, e.g. [13-16]). In a very
recent set of works (see, e.g. [17-20]), we have incorporated the
diffeomorphism transformation in the BT-superfield formalism which has been
called as the modified BT-superfield approach to BRST formalism. One of the
common features of the above superfield approaches [1-8, 13-20] is the
observation that the fields of an ordinary D-dimensional gauge/diffeomorphism
invariant theory have been generalized onto a suitably chosen (D,
2)-dimensional supermanifold which is parameterized by the super space
coordinates $Z^{M}=(x^{\mu},\theta,\bar{\theta})$ where coordinates
$x^{\mu}(\mu=0,1,2....D-1)$ define the bosonic D-dimensional ordinary
coordinates of the D-dimensional flat Minkowskian sub-manifold of the above
(D, 2)-dimensional supermanifold and the Grassmannian variables
($\theta,\bar{\theta})$ satisfy:
$\theta^{2}=\bar{\theta}^{2}=0,\theta\,\bar{\theta}+\bar{\theta}\,\theta=0$
demonstrating that they are fermionic in nature. Furthermore, it has been
observed that the full expansions of the superfields have been taken along all
the possible Grassmannian directions of the (D, 2)-dimensional supermanifold
in all the above superfield approaches to BRST formalism [1-8, 13-20].

In our very recent set of works (see, e.g. [21-23] for details), we have
applied the simplified version of the AVSA/USFA as well as the modified BT-
superfield approaches where the ordinary fields of a given ordinary
D-dimensional gauge theory have been generalized onto a couple of (D,
1)-dimensional (anti-)chiral super sub-manifolds of the general (D,
2)-dimensional supermanifold that has been considered in [1-8, 13-20]. The
purpose of our present investigation is to apply the simplified version (see,
e.g. [21-23]) of the superfield approach to BRST formalism where only the
(anti-)chiral super expansions are taken into account for the (anti-)chiral
superfields [defined on the (D, 1)-dimensional (anti-)chiral super sub-
manifolds] and apply the symmetry invariant restrictions on them to obtain the
(anti-)BRST as well as the (anti-)co-BRST transformations for our 4D
St$\ddot{u}$ckelberg-modified massive Abelian 2-form theory***Our present 4D
massive Abelian 2-form theory is interesting because there is some physical
relevance of this theory in the context of dark matter, dark energy and
cosmological models of the universe (cf. Sec. 7 below for detail). It is
precisely because of this reason that we have concentrated seriously on the
proof of its quantum symmetries as well as the CF-type restrictions within the
framework of ACSA to BRST formalism. which has been proven by us to be a
massive model for the Hodge theory in physical (3+1)-dimensional (4D)
spacetime [24].

In our present investigation, we briefly mention the most general forms of the
Lagrangian densities [cf. Eqs. (10), (11) below] by linearizing the kinetic
and gauge-fixing terms for the fields
$(B_{\mu\nu},\phi_{\mu},\tilde{\phi}_{\mu})$ by invoking the Nakanishi-Lautrup
type auxiliary fields. These coupled (but equivalent) Lagrangian densities are
the generalizations of the ordinary Lagrangian density [cf. Eq. (6) below]
where the kinetic term for the $B_{\mu\nu}$ field and gauge-fixing terms for
$(B_{\mu\nu},\phi_{\mu},\tilde{\phi}_{\mu})$ fields are not linearized. We
focus on the (anti-)BRST and (anti-)co-BRST symmetries of these Lagrangian
densities and obtain the Euler-Lagrange equations of motion (EL-EOMs) as well
as the CF-type restrictions from them. The main results of our present
investigation are the derivations of the (anti-)BRST, (anti-)co-BRST
symmetries and CF-type restrictions by using the ACSA to BRST formalism.
Furthermore, we express the coupled Lagrangian densities and (anti-)BRST as
well as (anti-)co-BRST charges in terms of the (anti-)chiral superfields which
are obtained after the applications of the (anti-)BRST and (anti-)co-BRST
invariant restrictions on the (anti-)chiral superfields. We prove the
existence of the CF-type restrictions on our theory by demanding the (anti-)
BRST and (anti-)co-BRST symmetry invariance of the super Lagrangian densities
as well as by proving the absolute anticommutativity of the (anti-)BRST as
well as (anti-) co-BRST charges that are present in our theory.

The following motivating factors have spurred our curiosity in perusing our
present investigation. First of all, we have demonstrated the existence of the
proper (anti-)BRST, (anti-)co-BRST symmetries and CF-type restrictions in our
earlier work [24] on the 4D massive Abelian 2-form gauge theory. To prove the
sanctity of the above continuous symmetries and the CF-type restrictions, it
is essential to verify them within the framework of ACSA to BRST formalism.
Second, we have taken (anti-)ghost part of the coupled (but equivalent)
Lagrangian densities of our 4D theory to be the same [24]. We provide the
precise arguments for the above correct choice in our present endeavor
starting from the (dual-)gauge symmetry transformations (cf. Sec. 2). Finally,
we know, from our earlier works [11, 12], that the existence of the CF-type
restriction(s) is a decisive feature of a gauge theory (described within the
framework of BRST formalism). We verify their existence, using the ACSA to
BRST formalism in our present endeavor.

Our present paper is organized as follows. In Sec. 2, we discuss the bare
essentials of the continuous symmetry properties of the St$\ddot{u}$ckelberg-
modified Lagrangian density for the massive 4D Abelian 2-form theory. Our Sec.
3 is devoted to the discussion of the off-shell nilpotent (anti-)BRST and
(anti-)co-BRST symmetries for the coupled (but equivalent) Lagrangian
densities. We derive the (anti-)BRST as well as the (anti-)co-BRST symmetries
within the framework of ACSA to BRST formalism and comment on their absolute
anticommutativity property in Sec. 4. In Sec. 5 of our present endeavor, we
show the existence of the CF-type restrictions by proving the invariance of
the super Lagrangian densities within the framework of ACSA to BRST formalism.
Our Sec. 6 deals with the proof of nilpotency and absolute anticommutativity
of the (anti-)BRST and (anti-)co-BRST conserved charges within the framework
of ACSA. We also demonstrate the existence of the CF-type restrictions, in a
subtle manner, in the proof of the absolute anticommutativity property of the
conserved and nilpotent charges. Finally, in Sec. 7, we make some concluding
remarks and point out a few future directions.

In our Appendices A and B, we perform some explicit computations which
complement the theoretical contents of our sub-sections 5.2 and 6.1 in the
main body of the text of our present investigation.

Convention and Notations: We adopt the convention of the left derivative
w.r.t. all the fermionic fields of our theory. We take the 4D flat Minkowskian
metric tensor $\eta_{\mu\nu}$ as: $\eta_{\mu\nu}$ = diag $(+1,-1,-1,-1)$ so
that the dot product between two non-null 4D vectors $P_{\mu}$ and $Q_{\mu}$
is defined as: $P\cdot Q=\eta_{\mu\nu}P^{\mu}\,Q^{\nu}\equiv
P_{0}\,Q_{0}-P_{i}\,Q_{i}$ where the Greek indices
$\mu,\nu,\lambda...=0,1,2,3$ stand for the time and space directions and Latin
indices $i,j,k...=1,2,3$ correspond to the 3D space directions only. We denote
the nilpotent (anti-)BRST and (anti-)co-BRST symmetry transformations (of all
kinds) by $s_{(a)b}$ and $s_{(a)d}$, respectively. The corresponding conserved
charges are represented by $Q_{(a)b}$ and $Q_{(a)d}$. The 4D Levi-Civita
tensor $\varepsilon_{\mu\nu\eta\kappa}$ is chosen such that
$\varepsilon_{0123}=+1=-\varepsilon^{0123}$ and
$\varepsilon_{\mu\nu\eta\kappa}\varepsilon^{\mu\nu\eta\kappa}=-\,4!$,
$\varepsilon_{\mu\nu\eta\kappa}\varepsilon^{\mu\nu\eta\rho}=-\,3!\,\delta^{\rho}_{\kappa}$,
$\varepsilon_{\mu\nu\rho\sigma}\varepsilon^{\mu\nu\eta\kappa}=-\,2!\,(\delta^{\eta}_{\rho}\delta^{\kappa}_{\sigma}-\delta^{\eta}_{\sigma}\delta^{\kappa}_{\rho})$,
etc. We also adopt the convention: $(\delta B_{\mu\nu}/\delta
B_{\rho\sigma})=\frac{1}{2!}\,\big{(}\delta^{\rho}_{\mu}\,\delta^{\sigma}_{\nu}-\delta^{\rho}_{\nu}\,\delta^{\sigma}_{\mu}\big{)}$,
etc. for the tensorial dififferentiation/variation.

##  2 Preliminaries: Lagrangian Formulation

In this section, we discuss the infinitesimal and continuous (dual-)gauge
symmetry transformations of the St$\ddot{u}$ckelberg-modified Lagrangian
densities before their generalizations to the off-shell nilpotent and
continuous (anti-)BRST and (anti-)co-BRST symmetry invariant coupled (but
equivalent) Lagrangian densities for our present 4D massive Abelian 2-form
theory. Our present section is divided into two  parts as discussed and
described below:

###  2.1 Infinitesimal Gauge Symmetry Transformations

We begin with the four $(3+1)$-dimensional (4D) Kalb-Ramond Lagrangian density
${\cal L}_{(0)}$ for the free Abelian 2-form massive theory (with the rest
mass equal to $m$) as follows (see, e.g. [25] for details)

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

where the antisymmetric $(B_{\mu\nu}=-B_{\nu\mu})$ tensor field $B_{\mu\nu}$
is the 4D massive Abelian 2-form $\big{[}B^{(2)}=\frac{1}{2!}\,(dx^{\mu}\wedge
dx^{\nu})\,B_{\mu\nu}\big{]}$ field and the curvature (i.e. field strength)
tensor
$H_{\mu\nu\eta}=\partial_{\mu}B_{\nu\eta}+\partial_{\nu}B_{\eta\mu}+\partial_{\eta}B_{\mu\nu}$
is derived from the 3-form
$\big{[}H^{(3)}=dB^{(2)}\equiv\frac{1}{3!}\,(dx^{\mu}\wedge dx^{\nu}\wedge
dx^{\eta})\,H_{\mu\nu\eta}\big{]}$ where $d=dx^{\mu}\partial_{\mu}$ (with
$d^{2}=0$) is the exterior derivative of differential geometry. We note that
the mass dimension of $B_{\mu\nu}$ is $[M]$ in the natural units (where we
take $\hbar=c=1$) for our 4D theory. Due to the existence of mass term
$(-\frac{m^{2}}{4}\,B_{\mu\nu}\,B^{\mu\nu})$, the gauge invariance is lost
because the above Lagrangian density is endowed with the second-class
constraints [26] in the terminology of Dirac’s prescription for the
classification scheme of constraints [9, 10]. We can find out the Euler-
Lagrange equation of motion (EL-EOM) from ${\cal L}_{(0)}$ as:
$\partial_{\mu}H^{\mu\nu\eta}+m^{2}\,B^{\nu\eta}=0$. At this stage, it is
evident that, for $m^{2}\neq 0$, we have
$\partial_{\mu}\,B^{\mu\nu}=0=\partial_{\nu}\,B^{\mu\nu}$. The latter
conditions (i.e. $\partial_{\mu}\,B^{\mu\nu}=0,\partial_{\nu}\,B^{\mu\nu}=0$)
emerge out because of the totally antisymmetric nature of $H^{\mu\nu\eta}$
(present in the original equation:
$\partial_{\mu}H^{\mu\nu\eta}+m^{2}\,B^{\nu\eta}=0$). Ultimately, we obtain
the usual Klein-Gordan equation [i.e. $(\Box+m^{2})\,B_{\mu\nu}=0$] for the
massive Abelian 2-form field $(B_{\mu\nu})$. This observation, in a subtle
way, implies that all the numerical factors in Eq. (1) are correct with their
proper signatures.

By the application of Stückelberg’s technique, it can be checked that, we have
the following appropriate transformation [24] of the antisymmetric tensor
field $B_{\mu\nu}$:

$\displaystyle B_{\mu\nu}\longrightarrow
B_{\mu\nu}-\frac{1}{m}\,\Phi_{\mu\nu}-\frac{1}{2\,m}\,\varepsilon_{\mu\nu\eta\kappa}\,\tilde{\Phi}^{\eta\kappa}$
$\displaystyle\equiv$ $\displaystyle
B_{\mu\nu}-\frac{1}{m}\,\big{(}\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu}+\varepsilon_{\mu\nu\eta\kappa}\,\partial^{\eta}\tilde{\phi}^{\kappa}\big{)}$
(2) $\displaystyle\equiv$ $\displaystyle
B_{\mu\nu}-\frac{1}{m}\,\Phi_{\mu\nu}-\frac{1}{m}\,{\cal F}_{\mu\nu}.$

In the above, the Abelian 2-form $\Phi^{(2)}=\frac{1}{2!}\,(dx^{\mu}\wedge
dx^{\nu})\,\Phi_{\mu\nu}\equiv d\,\Phi^{(1)}$ (with vector 1-form
$\Phi^{(1)}=dx^{\mu}\,\phi_{\mu}\Rightarrow\Phi_{\mu\nu}=\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu}$)
is obtained from a vector field $\phi_{\mu}$. On the contrary, the dual
antisymmetric tensor
$\tilde{\Phi}_{\mu\nu}=\partial_{\mu}\tilde{\phi}_{\nu}-\partial_{\nu}\tilde{\phi}_{\mu}$
is constructed with the help of an axial-vector field $\tilde{\phi}_{\mu}$
which is defined through the axial-vector 1-form
$\tilde{\Phi}^{(1)}=dx^{\mu}\,\tilde{\phi}_{\mu}$. In other words, the axial
Abelian 2-form:
$\tilde{\Phi}^{(2)}=d\,\tilde{\Phi}^{(1)}\equiv\big{(}\frac{d\,x^{\mu}\wedge
d\,x^{\nu}}{2!}\big{)}\,\tilde{\Phi}_{\mu\nu}$ leads to the derivation:
$\tilde{\Phi}_{\mu\nu}=\partial_{\mu}\,\tilde{\phi}_{\nu}-\partial_{\nu}\,\tilde{\phi}_{\mu}$.
We would like to add that we have ${\cal
F}_{\mu\nu}=\frac{1}{2!}\varepsilon_{\mu\nu\eta\kappa}\,\tilde{\Phi}^{\eta\kappa}$
where
$\tilde{\Phi}_{\mu\nu}=\partial_{\mu}\tilde{\phi}_{\nu}-\partial_{\nu}\tilde{\phi}_{\mu}$
is defined, as argued earlier, from the axial Abelian 2-form
$\tilde{\Phi}^{(2)}$ to maintain the parity invariance in our Abelian 2-form
massive gauge theory. We shall comment on the specific structure of the
antisymmetric tensor:
$\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu}+\varepsilon_{\mu\nu\eta\kappa}\,\partial^{\eta}\tilde{\phi}^{\kappa}$
and its connection with the source-free Maxwell’s theory in our Conclusions
section (cf. Sec. 7 below). Thus, we observe that, under the above
transformations (2), the Lagrangian density ${\cal L}_{(0)}$ transforms (i.e.
${\cal L}_{(0)}\rightarrow{\cal L}_{(1)}$) into the following form†††It should
be noted that the kinetic term (i.e.
$\frac{1}{12}H_{\mu\nu\eta}H^{\mu\nu\eta}$) transforms, under (2), to itself
plus extra terms [25]. The latter terms, however, turn out to be derivatives
of third and fourth order. Hence, they have been ignored keeping in mind the
renormalizibility of our theory [in four $(3+1)$-dimensions of spacetime].:

$\displaystyle{\cal L}_{(0)}\to{\cal L}_{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\,H^{\mu\nu\eta}H_{\mu\nu\eta}-\frac{m^{2}}{4}\,B^{\mu\nu}B_{\mu\nu}-\frac{1}{4}\,\Phi^{\mu\nu}\Phi_{\mu\nu}+\frac{1}{4}\,\tilde{\Phi}^{\mu\nu}\tilde{\Phi}_{\mu\nu}$
(3) $\displaystyle+$
$\displaystyle\frac{m}{2}\,B^{\mu\nu}\Phi_{\mu\nu}+\frac{m}{4}\,\varepsilon^{\mu\nu\eta\kappa}\,B_{\mu\nu}\tilde{\Phi}_{\eta\kappa},$

The above St$\ddot{u}$ckelberg’s modified Lagrangian density respects the
following continuous and infinitesimal gauge symmetry transformations
$(\delta_{g})$

$\displaystyle\delta_{g}B_{\mu\nu}=-\,(\partial_{\mu}\Lambda_{\nu}-\partial_{\nu}\Lambda_{\mu}),\quad\delta_{g}\phi_{\mu}=(\partial_{\mu}\,\Lambda-\,m\,\Lambda_{\mu}),\quad\delta_{g}\tilde{\phi}_{\mu}=0,$
(4)

where $\Lambda_{\mu}$ and $\Lambda$ are the vector and scalar gauge
transformation parameters. Under the continuous gauge symmetry transformations
$(4)$, the Lagrangian density ${\cal L}_{(1)}$ transforms to the following
total spacetime derivative, namely;

$\displaystyle\delta_{g}\,{\cal
L}_{(1)}=\partial_{\mu}\Big{[}-\,m\,\varepsilon^{\mu\nu\eta\kappa}\Lambda_{\nu}\,\partial_{\eta}\,\tilde{\phi}_{\kappa}\Big{]}.$
(5)

As a consequence, the action integral $S=\int d^{4}x\,{\cal L}_{(1)}$ remains
invariant under (4) for the physically well-defined fields which vanish off at
infinity.

We would like to end this sub-section with the following remarks. First, we
observe that the kinetic term
$\big{(}\frac{1}{12}H_{\mu\nu\eta}H^{\mu\nu\eta}\big{)}$ remains invariant
under the infinitesimal gauge symmetry transformations (4). To be precise, we
note that the curvature (i.e. the field strength) tensor $H_{\mu\nu\eta}$,
owing its origin to the exterior derivative
$d=d\,x^{\mu}\partial_{\mu}\;(d^{2}=0)$ of the de Rham cohomological operators
[27-31], remains invariant (i.e. $\delta_{g}H_{\mu\nu\eta}=0$) under the
infinitesimal gauge symmetry transformations (4). Second, we do note that the
axial-vector field $\tilde{\Phi}_{\mu}$ and, hence
$\tilde{\Phi}_{\mu\nu}=\partial_{\mu}\,\tilde{\phi}_{\nu}-\partial_{\nu}\,\tilde{\phi}_{\mu}$
as well as the kinetic term for this field, remain invariant (i.e.
$\delta_{g}\tilde{\phi}_{\mu}=0,\delta_{g}\tilde{\Phi}_{\mu\nu}=0$) under the
gauge symmetry transformations (4). Third, we have not invoked any new fields
in the theory besides the St$\ddot{u}$ckelberg fields (e.g. vector field
${\cal\phi_{\mu}}$ and axial-vector field $\tilde{\phi}_{\mu}$) due to the
dimensionality of the spacetime and antisymmetric $(B_{\mu\nu}=-B_{\nu\mu})$
nature of the gauge field $(B_{\mu\nu})$. Finally, we note that, so far, our
gauge symmetry transformations (4) are classical. However, they can be
elevated to their quantum counterparts [i.e. nilpotent and anticommuting
(anti-)BRST symmetries] within the framework of BRST formalism (see, Sec. 3
below).

###  2.2 Infinitesimal Dual-Gauge Symmetry Transformations

To discuss the dual-gauge symmetry transformations, we have to add the gauge-
fixing term to the St$\ddot{u}$ckelbrg-modified Lagrangian density ${\cal
L}_{(1)}$ [cf. Eq. (3)] which respects the classical gauge symmetry
transformations quoted in Eq. (4). Furthermore, we have to modify the kinetic
term as well as the gauge-fixing term by invoking some new  additional fields.
This has already been done systematically in our earlier work [24] where
physical and mathematical arguments have been provided for the same. The total
Lagrangian density with the modified kinetic term, gauge-fixing term and a
mass term is as follows (see, e.g. [24] for details)

$\displaystyle{\cal L}_{(1)}\to{\cal L}_{(2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\bigg{(}\frac{1}{2}\varepsilon_{\mu\nu\eta\kappa}\,\partial^{\nu}B^{\eta\kappa}-\frac{1}{2}\partial_{\mu}\tilde{\varphi}+m\,\tilde{\phi}_{\mu}\bigg{)}^{2}-\frac{m^{2}}{4}\,B^{\mu\nu}B_{\mu\nu}$
(6) $\displaystyle-$
$\displaystyle\frac{1}{4}\,\Phi^{\mu\nu}\Phi_{\mu\nu}+\frac{m}{2}\,B^{\mu\nu}\Phi_{\mu\nu}+\frac{1}{4}\,\tilde{\Phi}^{\mu\nu}\tilde{\Phi}_{\mu\nu}+\frac{m}{4}\,\varepsilon^{\mu\nu\eta\kappa}B_{\mu\nu}\tilde{\Phi}_{\eta\kappa}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left(\partial^{\nu}B_{\nu\mu}-\frac{1}{2}\,\partial_{\mu}\varphi+m\,\phi_{\mu}\right)^{2}-\frac{1}{2}\left(\partial_{\mu}\phi^{\mu}+\frac{1}{2}\,{m}\,\varphi\right)^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left(\partial_{\mu}\tilde{\phi}^{\mu}+\frac{1}{2}m\,\tilde{\varphi}\right)^{2},$

where the (pseudo-)scalar fields $(\tilde{\varphi})\varphi$ are the new fields
and the gauge-fixing term $(\partial^{\nu}B_{\nu\mu})$ for the antisymmetric
tensor field owes its origin to the co-exterior derivative
$\delta=\pm\ast\,d\,\ast(\delta^{2}=0)$ of the de Rham cohomologial operators
of differential geometry [26-30] because $\delta
B^{(2)}\equiv(\partial^{\nu}B_{\nu\mu})\,dx^{\mu}$. Here $\ast$ is the Hodge
duality operator. It should be noted that the last two terms in Eq. (6) are
nothing but the gauge-fixing terms for the vector and axial-vector fields
$\phi_{\mu}$ and $\tilde{\phi}_{\mu}$, respectively. At this stage, it can be
noted that the new fields
($\varphi,\tilde{\varphi},\phi_{\mu},\tilde{\phi}_{\mu}$) have mass dimension
$[M]$ in the natural units (i.e. $\hbar=c=1$) for our present theory.

It is obvious that the above modified Lagrangian density (6) would not have
the perfect gauge symmetry transformations (4) because of the additional
terms. However, it is very interesting to note that under the following
infinitesimal and continuous transformations

$\displaystyle\delta_{dg}B_{\mu\nu}=-\,\varepsilon_{\mu\nu\eta\kappa}\partial^{\eta}\Sigma^{\kappa},\qquad\delta_{dg}\tilde{\phi}_{\mu}=(\partial_{\mu}\,\Sigma-\,m\,\Sigma_{\mu}),\qquad\delta_{dg}\phi_{\mu}=0,$
$\displaystyle\delta_{dg}\,\tilde{\varphi}=-\;\sigma,\qquad\delta_{dg}\,\varphi=0,\qquad\delta_{dg}\big{[}\partial^{\nu}B_{\nu\mu}-\frac{1}{2}\,\partial_{\mu}\varphi+m\,\phi_{\mu}\big{]}=0,$
$\displaystyle\delta_{g}B_{\mu\nu}=-\,(\partial_{\mu}\Lambda_{\nu}-\partial_{\nu}\Lambda_{\mu}),\qquad\delta_{g}\phi_{\mu}=(\partial_{\mu}\,\Lambda-\,m\,\Lambda_{\mu}),\qquad\delta_{g}\tilde{\phi}_{\mu}=0,$
$\displaystyle\delta_{g}\,\varphi=\tilde{\lambda},\quad\delta_{g}\,\tilde{\varphi}=0,\quad\delta_{g}\Big{[}\frac{1}{2}\varepsilon_{\mu\nu\eta\kappa}\,\partial^{\nu}B^{\eta\kappa}-\frac{1}{2}\partial_{\mu}\tilde{\varphi}+m\,\tilde{\phi}_{\mu}\Big{]}=0,$
(7)

we obtain the following transformations for the Lagrangian density (6):

$\displaystyle\delta_{dg}\,{\cal L}_{(2)}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Big{[}\frac{1}{2}\,\tilde{\varphi}\,\partial^{\mu}\big{(}(\partial\cdot\Sigma)+\frac{1}{2}\,\sigma+m\,\Sigma\big{)}$
$\displaystyle-$
$\displaystyle\big{(}(\partial\cdot\Sigma)+\frac{1}{2}\,\sigma
m\,\Sigma\big{)}\Big{(}\frac{1}{2}\,\varepsilon^{\mu\nu\eta\kappa}\partial_{\nu}B_{\eta\kappa}+m\,{\tilde{\phi}}^{\mu}\Big{)}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\,\tilde{\varphi}\,(\Box+m^{2})\,\Sigma^{\mu}+\frac{m}{2}\,\varepsilon^{\mu\nu\eta\kappa}\Sigma_{\nu}\Phi_{\eta\kappa}\Big{]}$
$\displaystyle+$
$\displaystyle\Big{(}\frac{1}{2}\,\varepsilon^{\mu\nu\eta\kappa}\partial_{\nu}B_{\eta\kappa}+m\,{\tilde{\phi}}^{\mu}\Big{)}(\Box+m^{2})\,\Sigma_{\mu}$
$\displaystyle+$
$\displaystyle(\partial\cdot\tilde{\phi})\;(\Box+m^{2})\,\Sigma-\frac{1}{4}\,\tilde{\varphi}\;(\Box+m^{2})\,\sigma,$
$\displaystyle\delta_{g}\,{\cal L}_{(2)}$ $\displaystyle=$
$\displaystyle-\,\partial_{\mu}\Big{[}\frac{1}{2}\,{\varphi}\,\partial^{\mu}\big{(}(\partial\cdot\Lambda)-\frac{1}{2}\,\tilde{\lambda}+m\,\Lambda\big{)}$
(8) $\displaystyle-$
$\displaystyle\big{(}(\partial\cdot\Lambda)-\frac{1}{2}\,\tilde{\lambda}m\,\Lambda\big{)}\Big{(}\partial_{\nu}B^{\nu\mu}+m\,{{\phi}}^{\mu}\Big{)}$
$\displaystyle-$
$\displaystyle\frac{1}{2}{\varphi}\,(\Box+m^{2})\,\Lambda^{\mu}+\frac{m}{2}\,\varepsilon^{\mu\nu\eta\kappa}\Lambda_{\nu}\tilde{\Phi}_{\eta\kappa}\Big{]}$
$\displaystyle-$
$\displaystyle\Big{(}\partial_{\nu}B^{\nu\mu}-m\,{{\phi}}^{\mu}\Big{)}(\Box+m^{2})\,\Lambda_{\mu}$
$\displaystyle-$
$\displaystyle(\partial\cdot{\phi})\;(\Box+m^{2})\,\Lambda-\frac{1}{4}\,{\varphi}\;(\Box+m^{2})\,\tilde{\lambda}.$

We christen the above transformations [cf. Eqs. (7), (4)] as the (dual-)gauge
symmetry transformations because of the following arguments. First of all, we
note that under the gauge symmetry transformations $(\delta_{g})$, the total
kinetic term remains invariant. On the other hand, it is the total gauge-
fixing term that remains invariant under the dual-gauge symmetry
transformations $(\delta_{dg})$. Second, as argued earlier, the kinetic term
has its origin in the exterior derivative $d=dx^{\mu}\,\partial_{\mu}$ (with
$d^{2}=0$) but the gauge-fixing term owes its origin to the dual-exterior
(i.e. co-exterior) derivative $\delta=\pm\ast\,d\,\ast$ (with $\delta^{2}=0$).
Thus, the nomenclature (dual-)gauge symmetry transformation $\delta_{(d)g}$
sounds appropriate. Finally, even though there are three individual terms in
the kinetic and gauge-fixing terms, the origin of the additional terms like
$[-\frac{1}{2}\partial_{\mu}\tilde{\varphi}+m\,\tilde{\phi}_{\mu}]$ and
$[-\frac{1}{2}\partial_{\mu}\varphi+m\,\phi_{\mu}]$ is the
$H^{(3)}=d\,B^{(2)}$ and
$\delta\,B^{(2)}=(\partial^{\nu}B_{\nu\mu})\,dx^{\mu}$ which owe their origins
to $d=d\,x^{\mu}\partial_{\mu}$ and $\delta=\pm\star d\star$, respectively. It
is evident that the transformation parameters $(\Sigma_{\mu},\Sigma,\sigma)$
are the axial-vector and pseudo-scalars (i.e. $\Sigma,\sigma)$ for the full
dual-gauge symmetry transformations $(\delta_{dg})$. On the other hand, we
have already noted that the Lorentz vector ($\Lambda_{\mu}$) and Lorentz
scalars ($\Lambda,\tilde{\lambda}$) are the transformation parameters for the
full gauge symmetry transformations ($\delta_{g}$).

At this stage, we now comment on the transformations (8) which have been
obtained after the applications of $\delta_{dg}$ and $\delta_{g}$. It is
straightforward to note that under the following restrictions on the
(dual-)gauge transformation parameters, namely;

$\displaystyle(\Box+m^{2})\,\Sigma_{\mu}=0,\quad\qquad(\Box+m^{2})\,\Sigma=0,\quad\qquad(\Box+m^{2})\,\sigma=0,$
$\displaystyle(\Box+m^{2})\,\Lambda_{\mu}=0,\quad\qquad(\Box+m^{2})\,\Lambda=0,\qquad\quad(\Box+m^{2})\,\tilde{\lambda}=0,$
(9)

we achieve the perfect (dual-)gauge symmetry invariance of the Lagrangian
density ${\cal L}_{(2)}$. It is very interesting to pinpoint that the
mathematical structure of the restrictions in Eq. (9) is exactly the same on
the (dual-)gauge symmetry transformation parameters. Thus, it is very clear
that, within the framework of BRST approach, the (anti-)ghost part of the
Lagrangian density would be exactly the same for the coupled (but equivalent)
Lagrangian densities (cf. Sec. 3 below). The bosonic nature of the
transformation parameters
$(\Sigma_{\mu},\Sigma,\sigma,\Lambda_{\mu},\Lambda,\tilde{\lambda})$ implies
that, at the quantum level, these parameters would be replaced by the
fermionic (anti-)ghost fields within the framework of BRST formalism.

We wrap up this sub-section with the following remarks. We note, as pointed
out earlier, that total gauge-fixing term remains invariant under the dual-
gauge symmetry transformations $(\delta_{dg})$. Furthermore, the vector and
scalar fields ($\phi_{\mu}$ and $\varphi$) do not transform at all under
$\delta_{dg}$. Hence, the kinetic term
($-\,\frac{1}{4}\Phi_{\mu\nu}\,\Phi^{\mu\nu}$) for the vector field
$(\phi_{\mu})$ also does not transform under the dual-gauge symmetry
transformations. Whereas the gauge symmetry transformations $(\delta_{g})$
exist at the classical level, we note that the dual-gauge symmetry
transformations $(\delta_{dg})$ exist only when the gauge-fixing term is
incorporated for the purpose of quantization (or the derivation of the
propagator) for the gauge field $(B_{\mu\nu})$ of our massive 4D theory.
Finally, we observe that new  fields $(\varphi,\tilde{\varphi})$ have been
invoked for the discussion of the dual-gauge symmetries transformations
$(\delta_{dg})$ in our theory [in contrast to the gauge symmetry
transformations $(\delta_{g})$ where no such fields have been invoked (see,
e.g., the discussions after Eq. (5))].

##  3 Coupled Lagrangian Densities: Off-Shell Nilpotent (Anti-)BRST and
(Anti-)co-BRST Symmetry Transformations

In this section, we concisely mention the off-shell nilpotent symmetries and
the CF-type restrictions for the most generalized version of the Lagrangian
density (6) where the Nakanishi-Lautrup type auxiliary fields are invoked for
the linearizations of the kinetic and gauge-fixing terms for the $B_{\mu\nu}$
field and gauge-fixing terms for the fields $\phi_{\mu}$ and
$\tilde{\phi}_{\mu}$. The central theme and purpose of this section is to
mention all the appropriate equations in the 4D ordinary spacetime which are
important in the context of superfield approach to BRST formalism (cf. Secs.
4, 5, 6 below). We begin with the following coupled (but equivalent)
(dual-)BRST invariant Lagrangian densities (see, e.g. [24] for details)

$\displaystyle{\cal L}_{(B,\cal B)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\cal B}_{\mu}\;{\cal B}^{\mu}-{\cal
B}^{\mu}\left(\frac{1}{2}\varepsilon_{\mu\nu\eta\kappa}\,\partial^{\nu}B^{\eta\kappa}-\frac{1}{2}\,\partial_{\mu}\tilde{\varphi}+m\;\tilde{\phi}_{\mu}\right)-\frac{m^{2}}{4}\,B^{\mu\nu}B_{\mu\nu}$
(10) $\displaystyle-$
$\displaystyle\frac{1}{4}\,\Phi^{\mu\nu}\Phi_{\mu\nu}+\frac{m}{2}\,B^{\mu\nu}\Phi_{\mu\nu}+\frac{1}{4}\,\tilde{\Phi}^{\mu\nu}\tilde{\Phi}_{\mu\nu}+\frac{m}{4}\,\varepsilon^{\mu\nu\eta\kappa}B_{\mu\nu}\tilde{\Phi}_{\eta\kappa}-\frac{1}{2}\,B^{\mu}B_{\mu}$
$\displaystyle+$ $\displaystyle
B^{\mu}\left(\partial^{\nu}B_{\nu\mu}-\frac{1}{2}\,\partial_{\mu}\varphi+m\;\phi_{\mu}\right)+\frac{1}{2}\,B^{2}+B\left(\partial_{\mu}\phi^{\mu}+\frac{m}{2}\,\varphi\right)$
$\displaystyle-$ $\displaystyle\frac{1}{2}\,{\cal B}^{2}-{\cal
B}\left(\partial_{\mu}\tilde{\phi}^{\mu}+\frac{m}{2}\,\tilde{\varphi}\right)+\big{(}\partial_{\mu}\bar{C}-m\;\bar{C}_{\mu}\big{)}\big{(}\partial^{\mu}C-m\;C^{\mu}\big{)}$
$\displaystyle-$
$\displaystyle\big{(}\partial_{\mu}\bar{C}_{\nu}-\partial_{\nu}\bar{C}_{\mu}\big{)}\big{(}\partial^{\mu}C^{\nu}\big{)}-\frac{1}{2}\,\partial_{\mu}\bar{\beta}\,\partial^{\mu}\beta+\frac{1}{2}\,m^{2}\,\bar{\beta}\beta$
$\displaystyle-$
$\displaystyle\frac{1}{2}\left(\partial_{\mu}\bar{C}^{\mu}+m\,\bar{C}+\frac{\rho}{4}\right)\lambda-\frac{1}{2}\left(\partial_{\mu}C^{\mu}+m\,C-\frac{\lambda}{4}\right)\rho,$
$\displaystyle{\cal L}_{(\bar{B},\bar{\cal B})}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bar{\cal B}_{\mu}\bar{\cal B}^{\mu}+\bar{\cal
B}^{\mu}\left(\frac{1}{2}\varepsilon_{\mu\nu\eta\kappa}\,\partial^{\nu}B^{\eta\kappa}+\frac{1}{2}\,\partial_{\mu}\tilde{\varphi}+m\;\tilde{\phi}_{\mu}\right)-\frac{m^{2}}{4}\,B^{\mu\nu}B_{\mu\nu}$
(11) $\displaystyle-$
$\displaystyle\frac{1}{4}\,\Phi^{\mu\nu}\Phi_{\mu\nu}+\frac{m}{2}\,B^{\mu\nu}\Phi_{\mu\nu}+\frac{1}{4}\,\tilde{\Phi}^{\mu\nu}\tilde{\Phi}_{\mu\nu}+\frac{m}{4}\,\varepsilon^{\mu\nu\eta\kappa}B_{\mu\nu}\tilde{\Phi}_{\eta\kappa}-\frac{1}{2}\,\bar{B}^{\mu}\bar{B}_{\mu}$
$\displaystyle-$
$\displaystyle\bar{B}^{\mu}\left(\partial^{\nu}B_{\nu\mu}+\frac{1}{2}\,\partial_{\mu}\varphi+m\,\phi_{\mu}\right)+\frac{1}{2}\,\bar{B}^{2}-\bar{B}\left(\partial_{\mu}\phi^{\mu}-\frac{m}{2}\,\varphi\right)$
$\displaystyle-$ $\displaystyle\frac{1}{2}\,\bar{\cal B}^{2}+\bar{\cal
B}\left(\partial_{\mu}\tilde{\phi}^{\mu}-\frac{m}{2}\,\tilde{\varphi}\right)+\big{(}\partial_{\mu}\bar{C}-m\,\bar{C}_{\mu}\big{)}\big{(}\partial^{\mu}C-m\,C^{\mu}\big{)}$
$\displaystyle-$
$\displaystyle\big{(}\partial_{\mu}\bar{C}_{\nu}-\partial_{\nu}\bar{C}_{\mu}\big{)}\big{(}\partial^{\mu}C^{\nu}\big{)}-\frac{1}{2}\,\partial_{\mu}\bar{\beta}\,\partial^{\mu}\beta+\frac{1}{2}\,m^{2}\,\bar{\beta}\beta$
$\displaystyle-$
$\displaystyle\frac{1}{2}\left(\partial_{\mu}\bar{C}^{\mu}+m\,\bar{C}+\frac{\rho}{4}\right)\lambda-\frac{1}{2}\left(\partial_{\mu}C^{\mu}+m\,C-\frac{\lambda}{4}\right)\rho,$

where the auxiliary fields $({\cal B}_{\mu},B_{\mu},B,{\cal B},\bar{\cal
B}_{\mu},\bar{B}_{\mu},\bar{B},\bar{\cal B})$ are nothing but the bosonic
Nakanishi-Lautrup type auxiliary fields. The fermionic (anti-)ghost
fields‡‡‡These fermionic (anti-)ghost fields are the generalizations of the
bosonic (dual-)gauge symmetry transformation parameters
($\Sigma_{\mu},\Sigma,\sigma,\Lambda_{\mu},\Lambda,\tilde{\lambda}$) which
have been mentioned at the fag end of Sec. 2 of our present endeavor. are:
$(\bar{C}_{\mu})C_{\mu}$, $(\bar{C})C$, $(\rho)\lambda$ and bosonic
(anti-)ghost fields are $(\bar{\beta})\beta$. Because of the stage-one
reducibility in our theory, we have the ghost-for-ghost bosonic fields
$(\bar{\beta})\beta$. It should be noted that the ghost-part of the Lagrangian
densities (10) and (11) are same. We have provided some arguments regarding it
(cf. Sec. 2) in the language of the (dual-)gauge symmetry transformations and
the restrictions on the transformation gauge parameters [cf. Eq. (9)] for
their invariance. We also note here that the fields $(\rho)\lambda$ are
auxiliary fields but they are fermionic in nature and they carry the ghost
number (-1)+1, respectively. In addition, the ghost numbers for the fermionic
(anti-)ghost fields $(\bar{C}_{\mu})C_{\mu}$ and $(\bar{C})C$ are (-1)+1 and
that of the bosonic (anti-)ghost fields $(\bar{\beta})\beta$ are (-2)+2,
respectively.

We observe that the following, nilpotent ($s_{(a)b}^{2}=0$) (anti-)BRST
transformations ($s_{(a)b}$)

$\displaystyle
s_{ab}B_{\mu\nu}=-(\partial_{\mu}\bar{C}_{\nu}-\partial_{\nu}\bar{C}_{\mu}),\quad
s_{ab}\bar{C}_{\mu}=-\partial_{\mu}\bar{\beta},\quad
s_{ab}C_{\mu}=\bar{B}_{\mu},\quad s_{ab}\beta=-\lambda,$
$\displaystyle\;\;s_{ab}\bar{C}=-m\,\bar{\beta},\;\;s_{ab}C=\bar{B},\;\;s_{ab}B=-\,m\,\rho,\;\;s_{ab}\varphi=\rho,s_{ab}B_{\mu}=-\partial_{\mu}\rho,\;\;$
$\displaystyle
s_{ab}\phi_{\mu}=\partial_{\mu}\bar{C}-m\,\bar{C}_{\mu},\;\;s_{ab}[\bar{B},\rho,\lambda,\bar{\beta},\bar{B}_{\mu},{\cal
B}_{\mu},{\bar{\cal B}}_{\mu},\tilde{\phi}_{\mu},\tilde{\varphi},{\cal
B},\bar{\cal B},H_{\mu\nu\kappa}]=0,$ (12) $\displaystyle
s_{b}B_{\mu\nu}=-(\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}),\quad
s_{b}C_{\mu}=-\partial_{\mu}\beta,\quad s_{b}\bar{C}_{\mu}=B_{\mu},\quad
s_{b}\bar{\beta}=-\rho,$ $\displaystyle
s_{b}\phi_{\mu}=\partial_{\mu}C-m\,C_{\mu},\;\;s_{b}C=-\,m\,\beta,\;\;s_{b}\bar{C}=B,\;\;s_{b}\bar{B}=-\,m\,\lambda,\;s_{b}\varphi=\lambda,$
$\displaystyle s_{b}\bar{B}_{\mu}=-\partial_{\mu}\lambda,\quad
s_{b}[B,\rho,\lambda,\beta,B_{\mu},{\cal B}_{\mu},{\bar{\cal
B}}_{\mu},\tilde{\phi}_{\mu},\tilde{\varphi},{\cal B},\bar{\cal
B},H_{\mu\nu\kappa}]=0,\quad$ (13)

leave the action integrals $S_{1}=\int d^{4}x\;{\cal L}_{(B,{\cal B})}$ and
$S_{2}=\int\,d^{4}x\;{\cal L}_{(\bar{B},\bar{\cal B})}$ invariant because the
Lagrangian densities transform to the total spacetime derivatives under the
$s_{(a)b}$ as follows [24]:

$\displaystyle s_{ab}{\cal L}_{(\bar{B},\bar{\cal B})}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\,\tilde{\phi}_{\nu}\big{(}\partial_{\eta}\bar{C}_{\kappa}\big{)}-\bar{B}_{\nu}\big{(}\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu}\big{)}+\frac{1}{2}\,\bar{B}^{\mu}\,\rho$
(14) $\displaystyle+$
$\displaystyle\bar{B}\big{(}\partial^{\mu}\bar{C}-m\,\bar{C}^{\mu}\big{)}-\frac{1}{2}\,\big{(}\partial^{\mu}\bar{\beta}\big{)}\,\lambda\bigg{]},$
$\displaystyle s_{b}{\cal L}_{(B,{\cal B})}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\,\tilde{\phi}_{\nu}\big{(}\partial_{\eta}C_{\kappa}\big{)}+B_{\nu}\,\big{(}\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu}\big{)}+\frac{1}{2}\,B^{\mu}\,\lambda$
(15) $\displaystyle-$ $\displaystyle
B\,\big{(}\partial^{\mu}C-m\,C^{\mu}\big{)}-\frac{1}{2}\,\big{(}\partial^{\mu}\beta\big{)}\,\rho\bigg{]}.$

We point out that the above action integrals remain invariant due to Gauss’s
divergence theorem (as all the physical fields vanish off at infinity). It is
very interesting to note that the above coupled (but equivalent) Lagrangian
densities ${\cal L}_{(\bar{B},\bar{\cal B})}$ and ${\cal
L}_{(\bar{B},\bar{\cal B})}$ also respect another set of off-shell nilpotent
($s_{(a)d}^{2}=0$) symmetries which are known as the (anti-)co-BRST [or
(anti-)dual BRST] symmetries [$s_{(a)d}$]. This is due to the fact that we
observe the following transformations for the coupled (but equivalent)
Lagrangian densities:

$\displaystyle s_{ad}{\cal L}_{(\bar{B},{\cal{\bar{B}}})}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\,\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\phi_{\nu}\,\big{(}\partial_{\eta}C_{\kappa}\big{)}+\bar{\cal
B}_{\nu}\big{(}\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu}\big{)}+\frac{1}{2}\,\bar{\cal
B}^{\mu}\,\lambda$ (16) $\displaystyle-$ $\displaystyle\bar{\cal
B}\,\big{(}\partial^{\mu}C-m\,C^{\mu}\big{)}+\frac{1}{2}\,\big{(}\partial^{\mu}\beta\big{)}\,\rho\bigg{]},$
$\displaystyle s_{d}{\cal L}_{(B,{\cal B})}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\,\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\phi_{\nu}\big{(}\partial_{\eta}\bar{C}_{\kappa}\big{)}-{\cal
B}_{\nu}\,\big{(}\partial^{\mu}{\bar{C}}^{\nu}-\partial^{\nu}\bar{C}^{\mu}\big{)}+\frac{1}{2}\,{\cal
B}^{\mu}\,\rho$ (17) $\displaystyle+$ $\displaystyle{\cal
B}\,\big{(}\partial^{\mu}\bar{C}-m\;\bar{C}^{\mu}\big{)}-\frac{1}{2}\,\big{(}\partial^{\mu}\bar{\beta}\big{)}\,\lambda\bigg{]}.$

Hence, the action integrals $S_{1}=\int d^{4}x\;{\cal L}_{(B,{\cal B})}$ and
$S_{2}=\int d^{4}x\;{\cal L}_{(\bar{B},\bar{\cal B})}$ remain invariant under
the following infinitesimal and continuous (anti-)co-BRST symmetry
transformations [$s_{(a)d}$]:

$\displaystyle
s_{ad}B_{\mu\nu}=-\,\varepsilon_{\mu\nu\eta\kappa}\partial^{\eta}C^{\kappa},\quad
s_{ad}\bar{C}_{\mu}={\bar{\cal
B}}_{\mu},\quad\;s_{ad}C_{\mu}=\partial_{\mu}\beta,\quad\;s_{ad}\tilde{\varphi}=-\,\lambda,$
$\displaystyle s_{ad}\tilde{\phi}_{\mu}=\partial_{\mu}C-m\,C_{\mu},\quad
s_{ad}C=m\,\beta,\qquad s_{ad}{\cal B}=m\,\lambda,\quad
s_{ad}{\cal{B}}_{\mu}=\partial_{\mu}\lambda,$ $\displaystyle
s_{ad}\bar{\beta}=\rho,\,s_{ad}{\bar{C}}=\bar{\cal
B},\,s_{ad}[\partial^{\nu}B_{\nu\mu},B_{\mu},{\bar{\cal
B}}_{\mu},\bar{B}_{\mu},\bar{\cal
B},B,\bar{B},\varphi,\phi_{\mu},\rho,\lambda,\beta]=0,$ (18) $\displaystyle
s_{d}B_{\mu\nu}=-\varepsilon_{\mu\nu\eta\kappa}\partial^{\eta}\bar{C}^{\kappa},\qquad
s_{d}C_{\mu}={\cal B}_{\mu},\qquad
s_{d}\bar{C}_{\mu}=-\partial_{\mu}\bar{\beta},\qquad s_{d}\beta=-\lambda,$
$\displaystyle
s_{d}\tilde{\phi}_{\mu}=\partial_{\mu}\bar{C}-m\,\bar{C}_{\mu},\qquad
s_{d}C={\cal B},\qquad\;\quad s_{d}\bar{C}=-\,m\,\bar{\beta},\qquad
s_{d}\bar{\cal B}=m\,\rho,$ $\displaystyle s_{d}{\bar{\cal
B}}_{\mu}=\partial_{\mu}\rho,\;s_{d}\tilde{\varphi}=-\,\rho,\;\;s_{d}[\partial^{\nu}B_{\nu\mu},B_{\mu},{\cal{B}}_{\mu},\bar{B}_{\mu},{\cal
B},B,\bar{B},\varphi,\phi_{\mu},\rho,\lambda,\bar{\beta}]=0.$ (19)

It is straightforward to note that the (anti-)BRST and (anti-)co-BRST symmetry
transformations are off-shell nilpotent $(s_{(a)b}^{2}=0,s_{(a)d}^{2}=0)$ of
order two and, hence, are fermionic in nature. We note that the total gauge-
fixing term for the massive gauge field $B_{\mu\nu}$ remains invariant under
$s_{(a)d}$. This observation should be contrasted with the (anti-)BRST
symmetry transformations where the total kinetic term for the massive gauge
field $B_{\mu\nu}$ is found to be invariant. Furthermore, it is worth pointing
out that the BRST as well as anti-co-BRST transformations increase the ghost
number by one and anti-BRST as well as the co-BRST symmetry transformations
decrease the ghost number by one when they operate on an ordinary field.

A few comments are in order as far as the absolute anticommutativity
$({\\{s_{b},s_{ab}}\\}=0,{\\{s_{d},s_{ad}}\\}=0)$ of the (anti-)BRST
$(s_{(a)b})$ and (anti-)co-BRST $(s_{(a)d})$ symmetry transformations are
concerned. It can be checked that the following anticommutators, namely;

$\displaystyle\\{s_{b},\,s_{ab}\\}\,B_{\mu\nu}=-\partial_{\mu}\big{(}B_{\nu}+\bar{B}_{\nu}\big{)}+\partial_{\nu}\big{(}B_{\mu}+\bar{B}_{\mu}\big{)},$
$\displaystyle\\{s_{b},\,s_{ab}\\}\,\phi_{\mu}=\partial_{\mu}\big{(}B+\bar{B}\big{)}-m\,\big{(}B_{\mu}+\bar{B}_{\mu}\big{)},$
$\displaystyle{\\{s_{d},s_{ad}\\}}\,B_{\mu\nu}=-\,\varepsilon_{\mu\nu\eta\kappa}\partial^{\eta}({\cal
B}^{\kappa}+\bar{\cal B}^{\kappa}),$
$\displaystyle{\\{s_{d},s_{ad}\\}}\,\tilde{\phi}_{\mu}=\partial_{\mu}({\cal
B}+\bar{\cal B})-m\,({\cal B}_{\mu}+{\bar{\cal B}}_{\mu}),$ (20)

are equal to zero only when the following physically allowed CF-type
restrictions [24] are imposed from outside, namely;

$\displaystyle B_{\mu}+\bar{B}_{\mu}+\partial_{\mu}\varphi=0,\qquad\qquad
B+\bar{B}+m\,\varphi=0,$ $\displaystyle{\cal B}_{\mu}+\bar{\cal
B}_{\mu}+\partial_{\mu}\tilde{\varphi}=0,\qquad\qquad\;{\cal B}+\bar{\cal
B}+m\,\tilde{\varphi}=0.$ (21)

We can explicitly check that the above CF-type restrictions are (anti-)BRST as
well as (anti-)co-BRST invariant. To corroborate the above statement, we point
out the following precise observations:

$\displaystyle
s_{b}\big{[}B_{\mu}+\bar{B}_{\mu}+\partial_{\mu}\varphi\big{]}=0,\qquad\;\,s_{d}\big{[}{\cal
B}_{\mu}+\bar{\cal B}_{\mu}+\partial_{\mu}\tilde{\varphi}\big{]}=0,$
$\displaystyle
s_{ab}\big{[}B_{\mu}+\bar{B}_{\mu}+\partial_{\mu}\varphi\big{]}=0,\qquad
s_{ad}\big{[}{\cal B}_{\mu}+\bar{\cal
B}_{\mu}+\partial_{\mu}\tilde{\varphi}\big{]}=0,$ $\displaystyle
s_{d}\big{[}B+\bar{B}+m\,\varphi\big{]}=0,\qquad\quad\,s_{b}\big{[}{\cal
B}+\bar{\cal B}+m\,\tilde{\varphi}\big{]}=0,$ $\displaystyle
s_{ad}\big{[}B+\bar{B}+m\,\varphi\big{]}=0,\qquad\;\;\;s_{ab}\big{[}{\cal
B}+\bar{\cal B}+m\,\tilde{\varphi}\big{]}=0.$ (22)

Thus, for a model of the Hodge theory (i.e. 4D massive Abelian 2-form gauge
theory), the CF-type restrictions (21) are physical constraints on the theory
because the restrictions (21) are (anti-)BRST as well as (anti-)co-BRST
invariant, together. It would be worthwhile to note that some of the pertinent
equations of motion from the coupled Lagrangian densities ${\cal L}_{(B,{\cal
B})}$ and ${\cal L}_{(\bar{B},\bar{\cal B})}$ are:

$\displaystyle{\cal
B}_{\mu}=\left(\frac{1}{2}\varepsilon_{\mu\nu\eta\kappa}\,\partial^{\nu}B^{\eta\kappa}-\frac{1}{2}\,\partial_{\mu}\tilde{\varphi}+m\;\tilde{\phi}_{\mu}\right),\quad
B=-\left(\partial_{\mu}\phi^{\mu}+\frac{m}{2}\,\varphi\right),$ $\displaystyle
B_{\mu}=\left(\partial^{\nu}B_{\nu\mu}-\frac{1}{2}\,\partial_{\mu}\varphi+m\;\phi_{\mu}\right),\qquad\quad{\cal
B}=-\left(\partial_{\mu}\tilde{\phi}^{\mu}+\frac{m}{2}\,\tilde{\varphi}\right),$
$\displaystyle\bar{\cal
B}_{\mu}=-\left(\frac{1}{2}\varepsilon_{\mu\nu\eta\kappa}\,\partial^{\nu}B^{\eta\kappa}+\frac{1}{2}\,\partial_{\mu}\tilde{\varphi}+m\;\tilde{\phi}_{\mu}\right),\qquad\bar{B}=\left(\partial_{\mu}\phi^{\mu}-\frac{m}{2}\,\varphi\right),$
$\displaystyle\bar{B}_{\mu}=-\left(\partial^{\nu}B_{\nu\mu}+\frac{1}{2}\,\partial_{\mu}\varphi+m\;\phi_{\mu}\right),\qquad\quad\;\;\bar{\cal
B}=\left(\partial_{\mu}\tilde{\phi}^{\mu}-\frac{m}{2}\,\tilde{\varphi}\right).$
(23)

We note that the (anti-)BRST and (anti-)co-BRST invariant restrictions (21)
can be derived from the above equations (23). Thus, in a subtle manner, we
provide the derivation of the CF-type restrictions (21) from the coupled (but
equivalent) Lagrangian densities ${\cal L}_{(B,{\cal B})}$ and ${\cal
L}_{(\bar{B},\bar{\cal B})}$ in the sense that the appropriate EL-EOM lead to
their existence on our theory.

In the context of the existence of the above CF-type restrictions (21), we
note the following transformations for the (anti-)BRST and (anti-)co-BRST
invariant Lagrangian densities ${\cal L}_{(B,{\cal B})}$ and ${\cal
L}_{(\bar{B},\bar{\cal B})}$:

$\displaystyle s_{ab}{\cal L}_{(B,{\cal B})}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\,\tilde{\phi}_{\nu}\big{(}\partial_{\eta}\bar{C}_{\kappa}\big{)}+\Big{(}\partial_{\nu}B^{\nu\mu}+\frac{1}{2}\,\bar{B}^{\mu}+m\,\phi^{\mu}\Big{)}\rho$
(24) $\displaystyle+$ $\displaystyle
B_{\nu}\big{(}\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu}\big{)}-B\big{(}\partial^{\mu}\bar{C}-m\;\bar{C}^{\mu}\big{)}-\frac{1}{2}\,\big{(}\partial^{\mu}\bar{\beta}\big{)}\,\lambda\bigg{]}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\,\big{[}B_{\mu}+\bar{B}_{\mu}+\partial_{\mu}\varphi\big{]}\big{(}\partial^{\mu}\rho\big{)}+\partial_{\mu}\big{[}B_{\nu}+\bar{B}_{\nu}+\partial_{\nu}\varphi\big{]}\big{(}\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu}\big{)}$
$\displaystyle+$ $\displaystyle
m\big{[}B_{\mu}+\bar{B}_{\mu}+\partial_{\mu}\varphi\big{]}\big{(}\partial^{\mu}\bar{C}-m\,\bar{C}^{\mu}\big{)}-\frac{m}{2}\,\big{[}B+\bar{B}+m\;\varphi\big{]}\rho$
$\displaystyle-$
$\displaystyle\partial_{\mu}\big{[}B+\bar{B}+m\;\varphi\big{]}\big{(}\partial^{\mu}\bar{C}-m\,\bar{C}^{\mu}\big{)},$
$\displaystyle s_{b}{\cal L}_{(\bar{B},\bar{\cal B})}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\,\tilde{\phi}_{\nu}\big{(}\partial_{\eta}C_{\kappa}\big{)}-\Big{(}\partial_{\nu}B^{\nu\mu}-\frac{1}{2}\,B^{\mu}+m\,\phi^{\mu}\Big{)}\lambda$
(25) $\displaystyle-$
$\displaystyle\bar{B}_{\nu}\big{(}\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu}\big{)}+\bar{B}\big{(}\partial^{\mu}C-m\,C^{\mu}\big{)}-\frac{1}{2}\,\big{(}\partial^{\mu}\beta\big{)}\,\rho\bigg{]}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\,\big{[}B_{\mu}+\bar{B}_{\mu}+\partial_{\mu}\varphi\big{]}\big{(}\partial^{\mu}\lambda\big{)}-\partial_{\mu}\big{[}B_{\nu}+\bar{B}_{\nu}+\partial_{\nu}\varphi\big{]}\big{(}\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu}\big{)}$
$\displaystyle-$ $\displaystyle
m\big{[}B_{\mu}+\bar{B}_{\mu}+\partial_{\mu}\varphi\big{]}\big{(}\partial^{\mu}C-m\,C^{\mu}\big{)}-\frac{m}{2}\,\big{[}B+\bar{B}+m\,\varphi\big{]}\lambda$
$\displaystyle+$
$\displaystyle\partial_{\mu}\big{[}B+\bar{B}+m\;\varphi\big{]}\big{(}\partial^{\mu}C-m\,C^{\mu}\big{)},$
$\displaystyle s_{d}{\cal L}_{(\bar{B},\bar{\cal B})}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\,\phi_{\nu}\big{(}\partial_{\eta}\bar{C}_{\kappa}\big{)}-\Big{(}\frac{1}{2}\,\varepsilon^{\mu\nu\eta\kappa}\,\partial_{\nu}B_{\eta\kappa}-\frac{1}{2}\,{\cal
B}^{\mu}+m\,\tilde{\phi}^{\mu}\Big{)}\rho$ (26) $\displaystyle+$
$\displaystyle\bar{\cal
B}_{\nu}\big{(}\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu}\big{)}-\bar{\cal
B}\big{(}\partial^{\mu}\bar{C}-m\,\bar{C}^{\mu}\big{)}-\frac{1}{2}\,\big{(}\partial^{\mu}\bar{\beta}\big{)}\,\lambda\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{1}{2}\,\big{[}{\cal B}_{\mu}+\bar{\cal
B}_{\mu}+\partial_{\mu}\tilde{\varphi}\big{]}\big{(}\partial^{\mu}\rho\big{)}+\partial_{\mu}\big{[}{\cal
B}_{\nu}+\bar{\cal
B}_{\nu}+\partial_{\nu}\tilde{\varphi}\big{]}\big{(}\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu}\big{)}$
$\displaystyle+$ $\displaystyle m\big{[}{\cal B}_{\mu}+\bar{\cal
B}_{\mu}+\partial_{\mu}\tilde{\varphi}\big{]}\big{(}\partial^{\mu}\bar{C}-m\,\bar{C}^{\mu}\big{)}-\frac{m}{2}\,\big{[}{\cal
B}+\bar{\cal B}+m\,\tilde{\varphi}\big{]}\rho$ $\displaystyle-$
$\displaystyle\partial_{\mu}\big{[}{\cal B}+\bar{\cal
B}+m\;\tilde{\varphi}\big{]}\big{(}\partial^{\mu}\bar{C}-m\,\bar{C}^{\mu}\big{)},$
$\displaystyle s_{ad}{\cal L}_{(B,{\cal B})}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\,\phi_{\nu}\big{(}\partial_{\eta}C_{\kappa}\big{)}+\Big{(}\frac{1}{2}\,\varepsilon^{\mu\nu\eta\kappa}\,\partial_{\nu}B_{\eta\kappa}+\frac{1}{2}\,\bar{\cal
B}^{\mu}+m\,\tilde{\phi}^{\mu}\Big{)}\lambda$ (27) $\displaystyle-$
$\displaystyle{\cal
B}_{\nu}\big{(}\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu}\big{)}+{\cal
B}\big{(}\partial^{\mu}C-m\,C^{\mu}\big{)}+\frac{1}{2}\,\big{(}\partial^{\mu}\beta\big{)}\,\rho\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{1}{2}\,\big{[}{\cal B}_{\mu}+\bar{\cal
B}_{\mu}+\partial_{\mu}\tilde{\varphi}\big{]}\big{(}\partial^{\mu}\lambda\big{)}-\partial_{\mu}\big{[}{\cal
B}_{\nu}+\bar{\cal
B}_{\nu}+\partial_{\nu}\tilde{\varphi}\big{]}\big{(}\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu}\big{)}$
$\displaystyle-$ $\displaystyle m\big{[}{\cal B}_{\mu}+\bar{\cal
B}_{\mu}+\partial_{\mu}\tilde{\varphi}\big{]}\big{(}\partial^{\mu}C-m\,C^{\mu}\big{)}-\frac{m}{2}\,\big{[}{\cal
B}+\bar{\cal B}+m\,\tilde{\varphi}\big{]}\lambda$ $\displaystyle+$
$\displaystyle\partial_{\mu}\big{[}{\cal B}+\bar{\cal
B}+m\;\tilde{\varphi}\big{]}\big{(}\partial^{\mu}C-m\,C^{\mu}\big{)}.$

It is crystal clear, at this stage, that both the Lagrangian densities ${\cal
L}_{(B,\cal B)}$ and ${\cal L}_{(\bar{B},\bar{\cal B})}$ are equivalent in the
sense that both of them respect the (anti-)BRST as well as (anti-)co-BRST
symmetry transformations together provided our whole theory is considered on
the submanifold of space of fields which is defined by the field equations
(21). The latter are nothing but the (anti-) BRST and (anti-)co-BRST invariant
[cf. Eq. (22)] CF-type restrictions on our theory. We observe that, besides
the perfect symmetry invariance(s) [cf. Eqs. (14), (15), (16), (17)], we have
the following, namely;

$\displaystyle s_{ab}{\cal L}_{(B,{\cal B})}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\,\tilde{\phi}_{\nu}\big{(}\partial_{\eta}\bar{C}_{\kappa}\big{)}+\Big{(}\partial_{\nu}B^{\nu\mu}+\frac{1}{2}\,\bar{B}^{\mu}+m\;\phi^{\mu}\Big{)}\rho$
$\displaystyle+$
$\displaystyle\big{(}\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu}\big{)}\;B_{\nu}-B\;\big{(}\partial^{\mu}\bar{C}-m\;\bar{C}^{\mu}\big{)}-\frac{1}{2}\,\big{(}\partial^{\mu}\bar{\beta}\big{)}\,\lambda\bigg{]},$
$\displaystyle s_{b}{\cal L}_{(\bar{B},\bar{\cal B})}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\,\tilde{\phi}_{\nu}\big{(}\partial_{\eta}C_{\kappa}\big{)}-\Big{(}\partial_{\nu}B^{\nu\mu}-\frac{1}{2}\,B^{\mu}+m\;\phi^{\mu}\Big{)}\lambda$
$\displaystyle-$
$\displaystyle\big{(}\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu}\big{)}\;\bar{B}_{\nu}+\bar{B}\;\big{(}\partial^{\mu}C-m\;C^{\mu}\big{)}-\frac{1}{2}\,\big{(}\partial^{\mu}\beta\big{)}\,\rho\bigg{]},$
$\displaystyle s_{d}{\cal L}_{(\bar{B},\bar{\cal B})}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\,\phi_{\nu}\big{(}\partial_{\eta}\bar{C}_{\kappa}\big{)}-\Big{(}\frac{1}{2}\,\varepsilon^{\mu\nu\eta\kappa}\,\partial_{\nu}B_{\eta\kappa}-\frac{1}{2}\,{\cal
B}^{\mu}+m\;\tilde{\phi}^{\mu}\Big{)}\rho$ $\displaystyle+$
$\displaystyle\big{(}\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu}\big{)}\;\bar{\cal
B}_{\nu}-\bar{\cal
B}\;\big{(}\partial^{\mu}\bar{C}-m\;\bar{C}^{\mu}\big{)}-\frac{1}{2}\,\big{(}\partial^{\mu}\bar{\beta}\big{)}\,\lambda\bigg{]},$
$\displaystyle s_{ad}{\cal L}_{(B,{\cal B})}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\,\phi_{\nu}\big{(}\partial_{\eta}C_{\kappa}\big{)}+\Big{(}\frac{1}{2}\,\varepsilon^{\mu\nu\eta\kappa}\,\partial_{\nu}B_{\eta\kappa}+\frac{1}{2}\,\bar{\cal
B}^{\mu}+m\tilde{\phi}^{\mu}\Big{)}\lambda$ (28) $\displaystyle-$
$\displaystyle\big{(}\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu}\big{)}\;{\cal
B}_{\nu}+{\cal
B}\;\big{(}\partial^{\mu}C-m\;C^{\mu}\big{)}+\frac{1}{2}\,\big{(}\partial^{\mu}\beta\big{)}\,\rho\bigg{]},$

which are also perfect symmetry transformations on a submanifold of the space
of fields where the (anti-)BRST and (anti-)co-BRST invariant CF-type
restrictions are satisfied. Hence, the latter are physical constraints on our
theory.

We end this section with the remark that the most appropriate generalized
versions of the Lagrangian density (6) (that are nothing but ${\cal
L}_{(\bar{B},\bar{\cal B})}$ and ${\cal L}_{(B,{\cal B})}$) respect both types
of off-shell nilpotent symmetries [i.e. (anti-)BRST and (anti-)co-BRST
symmetries] provided the whole theory is restricted to be defined on the
submanifold of space of fields where the CF-type restrictions (21) are
respected. In fact, on this submanifold, the (anti-)BRST and (anti-)co-BRST
symmetries are found to be absolutely anticommutating, too. Hence, the
submanifold of the space of fields, defined by the (anti-)BRST and (anti-)co-
BRST invariant [cf. Eq. (22)] field equations (21), are physical subspace of
the quantum fields where the proper off-shell nilpotent symmetries and
corresponding proper (i.e. coupled and equivalent) Lagrangian densities ${\cal
L}_{(\bar{B},\bar{\cal B})}$ and ${\cal L}_{(B,{\cal B})}$) are defined.

##  4  Off-Shell Nilpotent (Anti-)BRST and (Anti-)co-BRST Symmetries: ACSA to
BRST Formalism

In this section, we exploit the basic tenets of ACSA to BRST formalism [21-23]
to derive the (anti-)BRST and (anti-)co-BRST symmetries for the coupled (but
equivalent) Lagrangian densities ${\cal L}_{(\bar{B},\bar{\cal B})}$ and
${\cal L}_{(B,{\cal B})}$) [(cf. Eqs. (10),(11))] of our theory by invoking
the symmetry invariant restrictions on the (anti-)chiral superfields. Our
present section is divided into two sub-sections as described in 4.1 and 4.2.

###  4.1 (Anti-)BRST Symmetries: ACSA

First of all, we derive the BRST symmetries [cf. Eq. (13)]. For this purpose,
we generalize the ordinary fields of the Lagrangian density ${\cal
L}_{(B,{\cal B})}$) onto their counterpart anti-chiral superfields on the (4,
1)-dimensional (anti-) chiral super sub-manifold as:

$\displaystyle
B_{\mu\nu}(x)\longrightarrow\;\;{\tilde{B}_{\mu\nu}}(x,\bar{\theta})\;=B_{\mu\nu}(x)+\bar{\theta}\,R_{\mu\nu}(x),$
$\displaystyle C_{\mu}(x)\;\longrightarrow\;\;{\tilde{\cal
F}}_{\mu}(x,\bar{\theta})=C_{\mu}(x)+\bar{\theta}\,B^{(1)}_{\mu}(x),$
$\displaystyle\bar{C}_{\mu}\;(x)\longrightarrow\;\;{\tilde{\bar{\cal
F}}}_{\mu}(x,\bar{\theta})=\bar{C}_{\mu}(x)+\bar{\theta}\,B^{(2)}_{\mu}(x),$
$\displaystyle\beta(x)\;\;\longrightarrow\;\;{\tilde{\beta}}(x,\bar{\theta})\;=\beta(x)+\bar{\theta}\,f_{1}(x),$
$\displaystyle\bar{\beta}(x)\;\;\longrightarrow\;\;{\tilde{\bar{\beta}}}(x,\bar{\theta})=\bar{\beta}(x)+\bar{\theta}\,f_{2}(x),$
$\displaystyle\varphi(x)\;\;\longrightarrow\;\;{\tilde{\Phi}}(x,\bar{\theta})=\varphi(x)+\bar{\theta}\,f_{3}(x),$
$\displaystyle\phi_{\mu}(x)\;\longrightarrow\;\;{\tilde{\Phi}}_{\mu}(x,\bar{\theta})=\phi_{\mu}(x)+\bar{\theta}\,R_{\mu}(x),$
$\displaystyle C(x)\;\;\longrightarrow\;\;{\tilde{\cal
F}}(x,\bar{\theta})=C(x)+\bar{\theta}\,B_{1}(x),$
$\displaystyle\bar{C}(x)\;\;\longrightarrow\;\;{\tilde{\bar{\cal
F}}}(x,\bar{\theta})=\bar{C}(x)+\bar{\theta}\,B_{2}(x),$ $\displaystyle
B_{\mu}(x)\;\longrightarrow\;\;\tilde{B}_{\mu}(x,\bar{\theta})=B_{\mu}(x)+\bar{\theta}\,f^{(1)}_{\mu}(x),$
$\displaystyle\bar{B}_{\mu}(x)\;\longrightarrow\;\;\tilde{\bar{B}}_{\mu}(x,\bar{\theta})=\bar{B}_{\mu}(x)+\bar{\theta}\,f^{(2)}_{\mu}(x),$
$\displaystyle
B(x)\;\;\longrightarrow\;\;\tilde{B}(x,\bar{\theta})=B(x)+\bar{\theta}\,f_{4}(x),$
$\displaystyle\bar{B}(x)\;\;\longrightarrow\;\;\tilde{\bar{B}}(x,\bar{\theta})=\bar{B}(x)+\bar{\theta}\,f_{5}(x),$
$\displaystyle{\cal B}_{\mu}(x)\;\longrightarrow\;\;\tilde{{\cal
B}}_{\mu}(x,\bar{\theta})={\cal B}_{\mu}(x)+\bar{\theta}\,f^{(3)}_{\mu}(x),$
$\displaystyle\bar{\cal B}_{\mu}(x)\;\longrightarrow\;\;{\tilde{\bar{\cal
B}}}_{\mu}(x,\bar{\theta})=\bar{\cal
B}_{\mu}(x)+\bar{\theta}\,f^{(4)}_{\mu}(x),$
$\displaystyle\tilde{\varphi}(x)\;\;\longrightarrow\;\;{\it{\tilde{\Phi}}}(x,\bar{\theta})=\tilde{\varphi}(x)+\bar{\theta}\,f_{6}(x),$
$\displaystyle\tilde{\phi}_{\mu}(x)\;\longrightarrow\;\;{\it{\tilde{\Phi}}}_{\mu}(x,\bar{\theta})=\tilde{\phi}_{\mu}(x)+\bar{\theta}\,R_{\mu}^{(1)}(x),$
$\displaystyle\lambda(x)\;\;\longrightarrow\;\;\tilde{\lambda}(x,\bar{\theta})=\lambda(x)+\bar{\theta}\,B_{3}(x),$
$\displaystyle\rho(x)\;\;\longrightarrow\;\;\tilde{\rho}(x,\bar{\theta})=\rho(x)+\bar{\theta}\,B_{4}(x),$
$\displaystyle{\cal B}(x)\;\;\longrightarrow\;\;\tilde{\cal
B}(x,\bar{\theta})={\cal B}(x)+\bar{\theta}f_{7}(x),$ $\displaystyle{\bar{\cal
B}}(x)\;\;\longrightarrow\;\;{\tilde{\bar{\cal B}}}(x,\bar{\theta})={\bar{\cal
B}}(x)+\bar{\theta}f_{8}(x).$ (29)

In the above anti-chiral super expansions, it is worthwhile as well as
pertinent to point out that the secondary fields
$(R_{\mu\nu},f_{1},f_{2},f_{3},R_{\mu},f^{(1)}_{\mu},f^{(2)}_{\mu},f_{4},$
$f_{5},f^{(3)}_{\mu},f^{(4)}_{\mu},f_{6},R_{\mu}^{(1)},f_{7},f_{8})$ are
fermionic and $({B_{\mu}^{(1)}},B_{\mu}^{(2)},B_{1},B_{2},B_{3},B_{4})$ are
bosonic in nature due to the fermionic $(\bar{\theta}^{2}=0)$ nature of the
Grassmannian variable $\bar{\theta}$ that characterizes the anti-chiral super
sub-manifold [along with the bosonic coordinates $x^{\mu}(\mu=0,1,2,3)$].

We note that the following non-trivial quantities are BRST invariant in view
of the symmetry transformations (13), namely;

$\displaystyle s_{b}(\lambda\,\varphi)=0,\quad\qquad
s_{b}(\bar{\beta}\,\lambda-\varphi\,\rho)=0,\quad\qquad
s_{b}(C_{\mu}\partial^{\mu}\rho-\partial_{\mu}\bar{\beta}\,\partial^{\mu}\beta)=0,$
$\displaystyle
s_{b}(C_{\mu}\partial^{\mu}{C}+\phi_{\mu}\partial^{\mu}\beta)=0,\qquad
s_{b}(\bar{B}_{\mu}+\partial_{\mu}\varphi)=0,\qquad
s_{b}(\bar{B}+m\,\varphi)=0,$ $\displaystyle
s_{b}\big{[}\bar{C}_{\mu}(m\,C^{\mu}-\partial^{\mu}{C})+B^{\mu}\,\phi_{\mu}\big{]}=0,\qquad
s_{b}\big{[}\bar{C}(m\,C_{\mu}-\partial_{\mu}{C})+B\,\phi_{\mu}\big{]}=0,$
$\displaystyle
s_{b}\big{[}m\,B_{\mu\nu}-(\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu})\big{]}=0,\quad\qquad
s_{b}\big{[}m\,\bar{\beta}\,\beta+\rho\,C\big{]}=0.$ (30)

In addition, we have some trivial BRST invariant quantities as:
$s_{b}[B,B_{\mu},\rho,\lambda,\beta,{{\cal B}_{\mu}},{\bar{\cal B}_{\mu}},\\\
\tilde{\varphi},\tilde{\phi}_{\mu},$ ${\cal B},{\bar{\cal
B}},H_{\mu\nu\lambda}]=0$ [cf. Eq. (13)]. It is because of the latter
observation that we have the following trivial super-expansions and equalities
for the appropriate anti-chiral superfields

$\displaystyle\tilde{B}^{(b)}(x,\bar{\theta})=B(x),\qquad\tilde{\rho}^{(b)}(x,\bar{\theta})=\rho(x),\qquad\tilde{\lambda}^{(b)}(x,\bar{\theta})=\lambda(x),$
$\displaystyle{{\tilde{\bar{\cal B}}}}_{\mu}^{(b)}(x,\bar{\theta})=\bar{\cal
B}_{\mu}(x),\quad{\it{\tilde{\Phi}}}^{(b)}(x,\bar{\theta})=\tilde{\varphi}(x),\quad{{\it{\tilde{\Phi}}}}_{\mu}^{(b)}(x,\bar{\theta})=\tilde{\phi}_{\mu}(x),$
$\displaystyle{\tilde{\cal B}}_{\mu}^{(b)}(x,\bar{\theta})={\cal
B}_{\mu}(x),\quad{\tilde{\bar{\cal B}}}^{(b)}(x,\bar{\theta})={\bar{\cal
B}}(x),\quad\tilde{B}_{\mu}^{(b)}(x,\bar{\theta})=B_{\mu}(x),$
$\displaystyle\tilde{\beta}^{(b)}(x,\bar{\theta})=\beta(x),\qquad{\tilde{\cal
B}}^{(b)}(x,\bar{\theta})={\cal
B}(x),\quad\;\tilde{H}_{\mu\nu\lambda}^{(b)}(x,\bar{\theta})=H_{\mu\nu\lambda}(x),$
(31)

where the superscript $(b)$ denotes the superfields that have been obtained
after the applications of the BRST invariant trivial restrictions. The above
equation (31) also implies that we have some of the secondary fields trivially
equal to zero:
$B_{3}=B_{4}=0,f_{1}=f_{4}=f_{6}=f_{7}=f_{8}=f_{\mu}^{(1)}=f_{\mu}^{(3)}=f_{\mu}^{(4)}=R_{\mu}^{(1)}=0.$
The above expansions (31) can now be utilized in the non-trivial BRST-
invariant equalities as follows:

$\displaystyle\tilde{\lambda}^{(b)}(x,\bar{\theta})\;\tilde{\Phi}(x,\bar{\theta})=\lambda(x)\,\varphi(x),\quad{\tilde{\bar{B}}}_{\mu}(x,\bar{\theta})+\partial_{\mu}\tilde{\Phi}(x,\bar{\theta})=\bar{B}_{\mu}(x)+\partial_{\mu}\varphi(x),$
$\displaystyle\tilde{\bar{\beta}}(x,\bar{\theta})\;\tilde{\lambda}^{(b)}(x,\bar{\theta})-\tilde{\Phi}(x,\bar{\theta})\;\tilde{\rho}^{(b)}(x,\bar{\theta})=\bar{\beta}(x)\,\lambda(x)-\varphi(x)\,\rho(x),$
$\displaystyle
m\,\tilde{\bar{\beta}}(x,\bar{\theta})\;\tilde{\beta}^{(b)}(x,\bar{\theta})+\tilde{\rho}^{(b)}(x,\bar{\theta})\;{\tilde{\cal
F}}(x,\bar{\theta})=m\,\bar{\beta}(x)\,\beta(x)+\rho(x)\,C(x),$
$\displaystyle{\tilde{\cal
F}}_{\mu}(x,\bar{\theta})\;\partial^{\mu}{\tilde{\cal
F}}(x,\bar{\theta})+{{\tilde{\Phi}}}_{\mu}(x,\bar{\theta})\;\partial^{\mu}\tilde{\beta}^{(b)}(x,\bar{\theta})=C_{\mu}(x)\partial^{\mu}{C}(x)$
$\displaystyle+\phi_{\mu}(x)\;\partial^{\mu}\beta(x),\quad{\tilde{\bar{B}}}(x,\bar{\theta})+m\,\tilde{\Phi}(x,\bar{\theta})=\bar{B}(x)+m\,\varphi(x),$
$\displaystyle{\tilde{\bar{\cal
F}}}_{\mu}(x,\bar{\theta})\;\big{[}m\,{\tilde{\cal
F}}^{\mu}(x,\bar{\theta})-\partial^{\mu}{\tilde{\cal
F}}(x,\bar{\theta})\big{]}+{\tilde{{B}}}^{\mu(b)}(x,\bar{\theta})\;{\tilde{\Phi}}_{\mu}(x,\bar{\theta})$
$\displaystyle=\bar{C}_{\mu}(x)\,\big{[}(m\,C^{\mu}(x)-\partial^{\mu}{C}(x)\big{]}+B_{\mu}(x)\;\phi^{\mu}(x),$
$\displaystyle{\tilde{\bar{\cal F}}}(x,\bar{\theta})\big{[}m\,{\tilde{\cal
F}}_{\mu}(x,\bar{\theta})-\partial_{\mu}{\tilde{\cal
F}}(x,\bar{\theta})\big{]}+{\tilde{{B}}}^{(b)}(x,\bar{\theta})\;{\tilde{\Phi}}_{\mu}(x,\bar{\theta})=\bar{C}(x)\big{[}m\,C_{\mu}(x)$
$\displaystyle-\partial_{\mu}{C}(x)\big{]}+B(x)\,\phi_{\mu}(x),\,\qquad
m\,{\tilde{B}}_{\mu\nu}(x,\bar{\theta})-\big{[}\partial_{\mu}\tilde{\Phi}_{\nu}(x,\bar{\theta})-\partial_{\nu}\tilde{\Phi}_{\mu}(x,\bar{\theta})\big{]}$
$\displaystyle=m\,B_{\mu\nu}(x)-\big{[}\partial_{\mu}\phi_{\nu}(x)-\partial_{\nu}\phi_{\mu}(x)\big{]},\quad\tilde{\cal
F}_{\mu}(x,\bar{\theta})\;\partial^{\mu}\tilde{\rho}^{(b)}(x,\bar{\theta})$
$\displaystyle-\partial_{\mu}\tilde{\bar{\beta}}(x,\bar{\theta})\;\partial^{\mu}\tilde{\beta}^{(b)}(x,\bar{\theta})=C_{\mu}(x)\;\partial^{\mu}\rho(x)-\partial_{\mu}\bar{\beta}(x)\,\partial^{\mu}\beta(x).$
(32)

Here we have utilized the basic tenets of ACSA to BRST formalism and imposed
the condition that the BRST invariant (physical) quantities should be
independent of the fermionic ($\bar{\theta}^{2}=0$) Grassmannian variable
$\bar{\theta}$. The above restrictions on the superfields lead to the
derivation of all the rest of the secondary fields of the super expansion (29)
in terms of the basic and auxiliary fields of the Lagrangian density ${\cal
L}_{(B,{\cal B})}$ as:

$\displaystyle
R_{\mu\nu}(x)=-(\partial_{\mu}C_{\nu}(x)-\partial_{\nu}C_{\mu}(x)),\;B_{\mu}^{(1)}(x)=-\,\,\partial_{\mu}\,\beta(x),\;B_{\mu}^{(2)}(x)=B_{\mu}(x),$
$\displaystyle
f_{2}(x)=-\,\rho(x),\;\;\;R_{\mu}(x)=\partial_{\mu}C(x)-m\,C_{\mu}(x),\;\;\;B_{1}(x)=-\,m\,\beta(x),$
$\displaystyle
f_{7}(x)=-\,m\;\lambda(x),\;f_{\mu}^{(2)}(x)=-\,\partial_{\mu}\lambda(x),\;\,f_{3}(x)=\lambda(x),\;B_{2}(x)=B(x).$
(33)

Plugging in these inputs into the anti-chiral super expansion (29), we obtain
the coefficients of $\bar{\theta}$ as the non-trivial BRST symmetry
transformations (13) as illustrated below:

$\displaystyle
B_{\mu\nu}(x)\longrightarrow{\tilde{B}_{\mu\nu}}^{(b)}(x,\bar{\theta})$
$\displaystyle=$ $\displaystyle
B_{\mu\nu}(x)+\bar{\theta}\,[-(\partial_{\mu}C_{\nu}(x)-\partial_{\nu}C_{\mu}(x))]$
$\displaystyle\equiv$ $\displaystyle
B_{\mu\nu}(x)+\bar{\theta}\,[s_{b}\,B_{\mu\nu}(x)],$ $\displaystyle
C_{\mu}(x)\;\,\longrightarrow\;\;{\tilde{\cal F}}_{\mu}^{(b)}(x,\bar{\theta})$
$\displaystyle=$ $\displaystyle
C_{\mu}(x)+\bar{\theta}\,(-\,\,\partial_{\mu}\,\beta(x))\equiv
C_{\mu}(x)+\bar{\theta}\,[s_{b}\,C_{\mu}(x)],$
$\displaystyle\bar{C}_{\mu}(x)\longrightarrow\;{\tilde{\bar{\cal
F}}}_{\mu}^{(b)}(x,\bar{\theta})$ $\displaystyle=$
$\displaystyle\bar{C}_{\mu}(x)+\bar{\theta}\,(B_{\mu}(x))\equiv\bar{C}_{\mu}(x)+\bar{\theta}\,[s_{b}\,\bar{C}_{\mu}(x)],$
$\displaystyle\beta(x)\longrightarrow\;\;{\tilde{\beta}}^{(b)}(x,\bar{\theta})\;$
$\displaystyle=$
$\displaystyle\beta(x)+\bar{\theta}\,(0)\equiv\beta(x)+\bar{\theta}\,[s_{b}\,\beta(x)],$
$\displaystyle\bar{\beta}(x)\longrightarrow\;\;{\tilde{\bar{\beta}}}^{(b)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\bar{\beta}(x)+\bar{\theta}\,\,(-\,\rho(x))\equiv\bar{\beta}(x)+\bar{\theta}\,\,[s_{b}\,\bar{\beta}(x)],$
$\displaystyle\varphi(x)\;\;\longrightarrow\;\;{\tilde{\Phi}}^{(b)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\varphi(x)+\bar{\theta}\,(\lambda(x))\equiv\varphi(x)+\bar{\theta}\,[s_{b}\,\varphi(x)],$
$\displaystyle\phi_{\mu}(x)\quad\longrightarrow{\tilde{\Phi}}_{\mu}^{(b)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\phi_{\mu}(x)+\bar{\theta}\,[\partial_{\mu}C(x)-m\,C_{\mu}(x)]$
$\displaystyle\equiv$
$\displaystyle\phi_{\mu}(x)+\bar{\theta}\,[s_{b}\phi_{\mu}(x)],$
$\displaystyle C(x)\;\;\longrightarrow\;\;{\tilde{\cal
F}}^{(b)}(x,\bar{\theta})$ $\displaystyle=$ $\displaystyle
C(x)+\bar{\theta}\,(-\,m\,\beta(x))\equiv C(x)+\bar{\theta}\,[s_{b}\,C(x)],$
$\displaystyle\bar{C}(x)\;\;\longrightarrow\;\;{\tilde{\bar{\cal
F}}}^{(b)}(x,\bar{\theta})$ $\displaystyle=$
$\displaystyle\bar{C}(x)+\bar{\theta}\,(B(x))\equiv\bar{C}(x)+\bar{\theta}\,[s_{b}\bar{C}(x)],$
$\displaystyle
B_{\mu}(x)\;\;\;\longrightarrow\tilde{B}_{\mu}^{(b)}(x,\bar{\theta})$
$\displaystyle=$ $\displaystyle B_{\mu}(x)+\bar{\theta}\,(0)\equiv
B_{\mu}(x)+\bar{\theta}\,[s_{b}B_{\mu}(x)],$
$\displaystyle\bar{B}_{\mu}(x)\;\longrightarrow\tilde{\bar{B}}_{\mu}^{(b)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\bar{B}_{\mu}(x)+\bar{\theta}\,(-\partial_{\mu}\lambda(x))\equiv\bar{B}_{\mu}(x)+\bar{\theta}\,[s_{b}\,\bar{B}_{\mu}(x)],$
$\displaystyle B(x)\;\;\longrightarrow\;\;\tilde{B}^{(b)}(x,\bar{\theta})$
$\displaystyle=$ $\displaystyle B(x)+\bar{\theta}\,(0)\equiv
B(x)+\bar{\theta}\,[s_{b}\,B(x)],$
$\displaystyle\bar{B}(x)\;\;\longrightarrow\;\;\tilde{\bar{B}}^{(b)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\bar{B}(x)+\bar{\theta}\,(-\,m\lambda(x))\equiv\bar{B}(x)+\bar{\theta}\,[s_{b}\,\bar{B}(x)],$
$\displaystyle{\cal B}_{\mu}(x)\;\longrightarrow\tilde{{\cal
B}}_{\mu}^{(b)}(x,\bar{\theta})$ $\displaystyle=$ $\displaystyle{\cal
B}_{\mu}(x)+\bar{\theta}\,(0)\equiv{\cal
B}_{\mu}(x)+\bar{\theta}\,[s_{b}\,{\cal B}_{\mu}(x)],$ $\displaystyle\bar{\cal
B}_{\mu}(x)\;\longrightarrow{\tilde{\bar{\cal
B}}}_{\mu}^{(b)}(x,\bar{\theta})$ $\displaystyle=$ $\displaystyle\bar{\cal
B}_{\mu}(x)+\bar{\theta}\,(0)\equiv\bar{\cal
B}_{\mu}(x)+\bar{\theta}\,[s_{b}\,\bar{\cal B}_{\mu}(x)],$
$\displaystyle\tilde{\varphi}(x)\;\;\;\longrightarrow{\it{\tilde{\Phi}}}^{(b)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\tilde{\varphi}(x)+\bar{\theta}\,(0)\equiv\tilde{\varphi}(x)+\bar{\theta}\;[s_{b}\,\tilde{\varphi}(x)],$
$\displaystyle\tilde{\phi}_{\mu}(x)\;\;\;\longrightarrow{\it{\tilde{\Phi}}}_{\mu}^{(b)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\tilde{\phi}_{\mu}(x)+\bar{\theta}\,(0)\equiv\tilde{\phi}_{\mu}(x)+\bar{\theta}\;[s_{b}\,\tilde{\phi}_{\mu}(x)],$
$\displaystyle\lambda(x)\;\;\longrightarrow\;\;\tilde{\lambda}^{(b)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\lambda(x)+\bar{\theta}\,(0)\equiv\lambda(x)+\bar{\theta}\,[s_{b}\,\lambda(x)],$
$\displaystyle\rho(x)\;\;\longrightarrow\;\;\tilde{\rho}^{(b)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\rho(x)+\bar{\theta}\,(0)\equiv\rho(x)+\bar{\theta}\,[s_{b}\,\rho(x)],$
$\displaystyle{\cal B}(x)\;\;\longrightarrow\;\;{\tilde{\cal
B}}^{(b)}(x,\bar{\theta})$ $\displaystyle=$ $\displaystyle{\cal
B}(x)+\bar{\theta}\,(0)\equiv{\cal B}(x)+\bar{\theta}\,[s_{b}\,{\cal B}(x)],$
$\displaystyle{\bar{\cal B}}(x)\;\;\longrightarrow\;\;{\tilde{\bar{\cal
B}}}^{(b)}(x,\bar{\theta})$ $\displaystyle=$ $\displaystyle{\bar{\cal
B}}(x)+\bar{\theta}\,(0)\equiv{\bar{\cal
B}}(x)+\bar{\theta}\,[s_{b}\,{\bar{\cal B}}(x)].$ (34)

In the above, the superscript $(b)$ denotes the fact that the super expansions
have been obtained after exploiting the basic tenets of ACSA and coefficients
of $\bar{\theta}$ lead to the determination of the BRST symmetry
transformations (13). It is elementary to note that
$\partial_{\bar{\theta}}\,\Omega^{(b)}(x,\bar{\theta})=s_{b}\,\omega(x)$ where
$\Omega^{(b)}(x,\bar{\theta})$ is the generic superfield obtained after the
application of the BRST invariant restrictions (31) and (32) and $\omega(x)$
is the ordinary (4D) generic field of the Lagrangian density ${\cal
L}_{(B,{\cal B})}$. Thus, we note that the nilpotency ($s_{b}^{2}=0$) property
of $s_{b}$ is deeply connected with the nilpotency
$(\partial_{\bar{\theta}}^{2}=0)$ of the translational generators
($\partial_{\bar{\theta}}$) along the $\bar{\theta}$-direction of the (4,
1)-dimensional anti-chiral super sub-manifold of the (4, 2)-dimensional
general supermanifold (on which our 4D ordinary massive Abelian 2-form theory
is generalized).

We now concentrate on the derivation of the off-shell nilpotent anti-BRST
symmetry transformations (12) for ${\cal L}_{(\bar{B},{\bar{\cal B})}}$ within
the framework of ACSA to BRST formalism. It may be mentioned here that the
anti-BRST transformations (12) are perfect symmetry transformations for ${\cal
L}_{(\bar{B},\bar{\cal B})}$ [cf. Eq. (14)]. Keeping this objective in our
mind, first of all, we note that the following interesting and useful
quantities are anti-BRST invariant in view of the quantum nilpotent,
infinitesimal and continuous anti-BRST symmetry transformations (12), namely;

$\displaystyle s_{ab}[\rho\,\varphi]=0,\quad
s_{ab}[{\beta}\,\rho-\varphi\,\lambda]=0,\quad
s_{ab}[\bar{C}_{\mu}\partial^{\mu}\lambda-\partial_{\mu}{\beta}\,\partial^{\mu}\bar{\beta}]=0,$
$\displaystyle
s_{ab}[\bar{C}_{\mu}\partial^{\mu}{\bar{C}}+\phi_{\mu}\partial^{\mu}\bar{\beta}]=0,\quad
s_{ab}[{B}_{\mu}+\partial_{\mu}\varphi]=0,\quad s_{ab}[{B}+m\,\varphi]=0,$
$\displaystyle
s_{ab}[{C}_{\mu}(m\,\bar{C}^{\mu}-\partial^{\mu}\bar{C})+\bar{B}^{\mu}\,\phi_{\mu}]=0,\quad
s_{ab}[{C}(m\,\bar{C}_{\mu}-\partial_{\mu}{\bar{C}})+\bar{B}\,\phi_{\mu}]=0,$
$\displaystyle
s_{ab}[m\,B_{\mu\nu}-(\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu})]=0,\quad
s_{ab}[m\,\bar{\beta}\,\beta+\lambda\,\bar{C}]=0.$ (35)

According to the basic tenets of ACSA to BRST formalism, the above quantities
should be independent of the Grassmannian variable $\theta$ of the (4,
1)-dimensional chiral super sub-manifold on which the ordinary fields of
${\cal L}_{(\bar{B},{\bar{\cal B})}}$ are generalized in the following manner

$\displaystyle
B_{\mu\nu}(x)\longrightarrow\;\;{\tilde{B}_{\mu\nu}}(x,\theta)\;=B_{\mu\nu}(x)+\theta\,\bar{R}_{\mu\nu}(x),$
$\displaystyle C_{\mu}(x)\;\longrightarrow\;\;{\tilde{\cal
F}}_{\mu}(x,\theta)=C_{\mu}(x)+\theta\,{\bar{B}}^{(1)}_{\mu}(x),$
$\displaystyle\bar{C}_{\mu}\;(x)\longrightarrow\;\;{\tilde{\bar{\cal
F}}}_{\mu}(x,\theta)=\bar{C}_{\mu}(x)+\theta\,{\bar{B}}^{(2)}_{\mu}(x),$
$\displaystyle\beta(x)\;\;\longrightarrow\;\;{\tilde{\beta}}(x,\theta)\;=\beta(x)+\theta\,\bar{f}_{1}(x),$
$\displaystyle\bar{\beta}(x)\;\;\longrightarrow\;\;{\tilde{\bar{\beta}}}(x,\theta)=\bar{\beta}(x)+\theta\,\bar{f}_{2}(x),$
$\displaystyle\varphi(x)\;\;\longrightarrow\;\;{\tilde{\Phi}}(x,\theta)=\varphi(x)+\theta\,\bar{f}_{3}(x),$
$\displaystyle\phi_{\mu}(x)\;\longrightarrow\;\;{\tilde{\bf\Phi}}_{\mu}(x,\theta)=\phi_{\mu}(x)+\theta\bar{R}_{\mu}(x),$
$\displaystyle C(x)\;\;\longrightarrow\;\;{\tilde{\cal
F}}(x,\theta)=C(x)+\theta\,\bar{B}_{1}(x),$
$\displaystyle\bar{C}(x)\;\;\longrightarrow\;\;{\tilde{\bar{\cal
F}}}(x,\theta)=\bar{C}(x)+\theta\,\bar{B}_{2}(x),$ $\displaystyle
B_{\mu}(x)\;\longrightarrow\;\;\tilde{B}_{\mu}(x,\theta)=B_{\mu}(x)+\theta\,\bar{f}_{\mu}^{(1)}(x),$
$\displaystyle\bar{B}_{\mu}(x)\;\longrightarrow\;\;\tilde{\bar{B}}_{\mu}(x,\theta)=\bar{B}_{\mu}(x)+\theta\,\bar{f}_{\mu}^{(2)}(x),$
$\displaystyle
B(x)\;\;\longrightarrow\;\;\tilde{B}(x,\theta)=B(x)+\theta\,\bar{f}_{4}(x),$
$\displaystyle\bar{B}(x)\;\;\longrightarrow\;\;\tilde{\bar{B}}(x,\theta)=\bar{B}(x)+\theta\,\bar{f}_{5}(x),$
$\displaystyle{\cal B}_{\mu}(x)\;\longrightarrow\;\;\tilde{{\cal
B}}_{\mu}(x,\theta)={\cal B}_{\mu}(x)+\theta\,\bar{f}_{\mu}^{(3)}(x),$
$\displaystyle\bar{\cal B}_{\mu}(x)\;\longrightarrow\;\;{\tilde{\bar{\cal
B}}}_{\mu}(x,\theta)=\bar{\cal B}_{\mu}(x)+\theta\,\bar{f}_{\mu}^{(4)}(x),$
$\displaystyle\tilde{\varphi}(x)\;\;\longrightarrow\;\;{\it{\tilde{\Phi}}}(x,\theta)=\tilde{\varphi}(x)+\theta\,\bar{f}_{6}(x),$
$\displaystyle\tilde{\phi}_{\mu}(x)\;\longrightarrow\;\;{\it{\tilde{\Phi}}}_{\mu}(x,\theta)=\tilde{\phi}_{\mu}(x)+\theta\,\bar{R}_{\mu}^{(1)}(x),$
$\displaystyle\lambda(x)\;\;\longrightarrow\;\;\tilde{\lambda}(x,\theta)=\lambda(x)+\theta\,\bar{B}_{3}(x),$
$\displaystyle\rho(x)\;\;\longrightarrow\;\;\tilde{\rho}(x,\theta)=\rho(x)+\theta\,\bar{B}_{4}(x),$
$\displaystyle{\cal B}(x)\;\;\longrightarrow\;\;\tilde{\cal B}(x,\theta)={\cal
B}(x)+\theta\bar{f}_{7}(x),$ $\displaystyle{\bar{\cal
B}}(x)\;\;\longrightarrow\;\;{\tilde{\bar{\cal B}}}(x,\theta)={\bar{\cal
B}}(x)+\theta\,\bar{f}_{8}(x),$ (36)

where the fermionic $(\theta^{2}=0)$ Grassmannian variable $\theta$ [along
with the bosonic coordinates $x^{\mu}(\mu=0,1,2,3)$] characterize the chiral
super sub-manifold. We note that the secondary fields
$(\bar{R}_{\mu\nu},\bar{f}_{1},\bar{f}_{2},\bar{f}_{3},\bar{R}_{\mu},\bar{f}_{\mu}^{(1)},\bar{f}_{\mu}^{(2)},\bar{f}_{4},\bar{f}_{5},\bar{f}_{\mu}^{(3)},\bar{f}_{\mu}^{(4)},\bar{f}_{6},\bar{R}_{\mu}^{(1)},\bar{f}_{7},\bar{f}_{8})$
are fermionic in nature and the other secondary fields
$(\bar{B_{\mu}^{(1)}},\bar{B}_{\mu}^{(2)},\bar{B}_{1},\bar{B}_{2},\bar{B}_{3},\bar{B}_{4})$
are bosonic. These secondary fields are to be determined in terms of the basic
and auxiliary fields of the Lagrangian density ${\cal L}_{(\bar{B},{\bar{\cal
B})}}$. At this stage, as argued earlier, the following restrictions on the
specific combination of the chiral superfields, namely;

$\displaystyle\tilde{\rho}^{(ab)}(x,\theta)\;\tilde{\Phi}(x,\theta)=\rho(x)\,\varphi(x),\;\;{\tilde{{B}}}_{\mu}(x,\theta)+\partial_{\mu}\tilde{\Phi}(x,\theta)={B}_{\mu}(x)+\partial_{\mu}\varphi(x),$
$\displaystyle\tilde{{\beta}}(x,\theta)\;\tilde{\rho}^{(ab)}(x,\theta)-\tilde{\Phi}(x,\theta)\;\tilde{\lambda}^{(ab)}(x,\theta)={\beta}(x)\,\rho(x)-\varphi(x)\,\lambda(x),$
$\displaystyle
m\,\tilde{\beta}(x,\theta)\;\tilde{\bar{\beta}}^{(ab)}(x,\theta)+\tilde{\lambda}^{(ab)}(x,\theta)\;{\tilde{\bar{\cal
F}}}(x,\theta)=m\,\bar{\beta}(x)\,\beta(x)+\lambda(x)\,\bar{C}(x),$
$\displaystyle{\tilde{\bar{\cal
F}}_{\mu}}(x,\theta)\;\partial^{\mu}{\tilde{\bar{\cal
F}}}(x,\theta)+{{\tilde{\Phi}}}_{\mu}(x,\theta)\;\partial^{\mu}{\tilde{\bar{\beta}}}^{(ab)}(x,\theta)=\bar{C}_{\mu}(x)\partial^{\mu}{\bar{C}}(x),$
$\displaystyle+\phi_{\mu}(x)\partial^{\mu}\bar{\beta}(x),\;{\tilde{\cal
F}}_{\mu}(x,\theta)\;\big{[}m\,{\tilde{\bar{\cal
F}}}^{\mu}(x,\theta)-\partial^{\mu}{\tilde{\bar{\cal
F}}}(x,\theta)\big{]}+{\tilde{{\bar{B}}}}^{\mu(ab)}(x,\theta)\;{\tilde{\Phi}}_{\mu}(x,\theta),$
$\displaystyle={C}_{\mu}(x)\,[m\,\bar{C}^{\mu}(x)-\partial^{\mu}{\bar{C}}(x)]+\bar{B}^{\mu}(x)\phi_{\mu}(x),\;\;{\tilde{{\cal
F}}}(x,\theta)\big{[}m\,{\tilde{\bar{\cal F}}}_{\mu}(x,\theta)$
$\displaystyle-\partial_{\mu}{\tilde{\bar{\cal
F}}}(x,\theta)\big{]}+{\tilde{{\bar{B}}}}(x,\theta)\;{\tilde{\Phi}}_{\mu}(x,\theta)={C}(x)\big{[}m\,\bar{C}_{\mu}(x)-\partial_{\mu}{\bar{C}}(x)\big{]}+\bar{B}(x)\,\phi_{\mu}(x),$
$\displaystyle
m\,{\tilde{B}}_{\mu\nu}(x,\theta)-\big{[}\partial_{\mu}\tilde{\Phi}_{\nu}(x,\theta)-\partial_{\nu}\tilde{\Phi}_{\mu}(x,\theta)\big{]}=m\,B_{\mu\nu}(x)-[\partial_{\mu}\phi_{\nu}(x)$
$\displaystyle-\partial_{\nu}\phi_{\mu}(x)],\quad{\tilde{\bar{\cal
F}}}_{\mu}(x,\theta)\;\partial^{\mu}\tilde{\lambda}^{(ab)}(x,\theta)-\partial_{\mu}\tilde{\beta}(x,\theta)\;\partial^{\mu}{\tilde{\bar{\beta}}}^{(ab)}(x,\theta)=$
$\displaystyle\bar{C}_{\mu}(x)\partial^{\mu}\lambda(x)-\partial_{\mu}{\beta}(x)\,\partial^{\mu}\bar{\beta}(x),\;{\tilde{{B}}}(x,\theta)+m\,\tilde{\Phi}(x,\theta)={B}(x)+m\,\varphi(x),$
(37)

are to be satisfied due to the basic tenets of ACSA to BRST formalism. The
above equalities [cf. Eq. (37)] are nothing but the generalizations of the
useful and interesting anti-BRST invariant quantities (35) onto the chiral
super sub-manifold [of the general (4, 2)-dimensional supermanifold] with the
following inputs [due to the trivial anti-BRST invariant quantities:
$s_{ab}[\bar{B},\rho,\lambda,\bar{\beta},\bar{B}_{\mu},{\cal
B}_{\mu},{\bar{\cal B}}_{\mu},\tilde{\phi}_{\mu},\tilde{\varphi},{\cal
B},\bar{\cal B},H_{\mu\nu\kappa}]=0$ that are useful and important for our
purpose], namely;

$\displaystyle\tilde{\bar{B}}^{(ab)}(x,\theta)=\bar{B}(x),\qquad\tilde{\rho}^{(ab)}(x,\theta)=\rho(x),\qquad\tilde{\lambda}^{(ab)}(x,\theta)=\lambda(x),$
$\displaystyle{\tilde{\bar{\cal B}}}_{\mu}^{(ab)}(x,\theta)=\bar{\cal
B}_{\mu}(x),\qquad{\it{\tilde{\Phi}}}^{(ab)}(x,\theta)=\tilde{\varphi}(x),\qquad{{\it{\tilde{\Phi}}}_{\mu}}^{(ab)}(x,\theta)=\tilde{\phi}_{\mu}(x),$
$\displaystyle{\tilde{\cal B}}_{\mu}^{(ab)}(x,\theta)={\cal
B}_{\mu}(x),\qquad{\tilde{\bar{\cal B}}}^{(ab)}(x,\theta)={\bar{\cal
B}}(x),\qquad{\tilde{\cal B}}^{(ab)}(x,\theta)={\cal B}(x),$
$\displaystyle\,\tilde{H}_{\mu\nu\eta}^{(ab)}(x,\theta)=H_{\mu\nu\eta}(x),\quad\bar{B}_{\mu}^{(ab)}(x,\theta)=\bar{B}_{\mu}(x),\quad\tilde{\bar{\beta}}^{(ab)}(x,\theta)=\bar{\beta}(x).$
(38)

The above equalities/restrictions [i.e. Eq. (38)] also imply that the
secondary fields:
$\bar{f}_{2}=\bar{f}_{3}=\bar{f}_{5}=\bar{f}_{6}=\bar{f}_{7}=\bar{f}_{8}=\bar{B}_{3}=\bar{B}_{4}=\bar{f}_{\mu}^{(2)}=\bar{f}_{\mu}^{(3)}=\bar{f}_{\mu}^{(4)}=\bar{R}_{\mu}^{(1)}=0$
in the chiral super eaxpansions of the chiral superfilelds in (36). In other
words, the coefficients of $\theta$ in the chiral super expansions (36) (that
correspond to the anti-BRST symmetry transformations ($s_{ab}$) are trivially
zero for all the ordinary fields that are present on the r.h.s. of (38).

The anti-BRST invariant restrictions (37) lead to the following precise
expressions for the secondary fields in terms of the basic and auxiliary
fields of Lagrangian density ${\cal L}_{(\bar{B},{\bar{\cal B})}}$:

$\displaystyle\bar{R}_{\mu\nu}(x)=-[\partial_{\mu}\bar{C}_{\nu}(x)-\partial_{\nu}\bar{C}_{\mu}(x)],\;\;\bar{B}_{\mu}^{(2)}(x)=-\,\partial_{\mu}\,\bar{\beta}(x),\;\;\bar{B}_{\mu}^{1}(x)=\bar{B}_{\mu}(x),$
$\displaystyle\bar{f}_{1}=-\,\lambda(x),\;\;\;\bar{R}_{\mu}(x)=\partial_{\mu}\bar{C}(x)-m\,\bar{C}_{\mu}(x),\;\;\;\bar{B}_{2}(x)=-\,m\;\bar{\beta}(x),$
$\displaystyle\bar{B}_{1}(x)=\bar{B}(x),\;\,\bar{f}_{4}(x)=-\,m\;\rho(x),\;\,\bar{f}^{(1)}_{\mu}(x)=-\partial_{\mu}\rho(x),\;\,\bar{f}_{3}(x)=\rho(x).$
(39)

The above derivations are straightforward as there are no complicated tricks
involved in their deductions. Substitutions of the above precise values of the
secondary fields into the chiral super expansions (36) lead to the
determination of anti-BRST symmetries $(s_{ab})$ [cf. Eq. (12)] as the
coefficients of $\theta$ as illustrated in the following super expansions:

$\displaystyle
B_{\mu\nu}(x)\longrightarrow{\tilde{B}_{\mu\nu}}^{(ab)}(x,\theta)$
$\displaystyle=$ $\displaystyle
B_{\mu\nu}(x)+\theta\,[-(\partial_{\mu}\bar{C}_{\nu}(x)-\partial_{\nu}\bar{C}_{\mu}(x))]$
$\displaystyle\equiv$
$\displaystyle\;B_{\mu\nu}(x)+\theta\,[s_{ab}\,B_{\mu\nu}(x)],$ $\displaystyle
C_{\mu}(x)\;\,\longrightarrow\;\;{\tilde{\cal F}}_{\mu}^{(ab)}(x,\theta)$
$\displaystyle=$ $\displaystyle C_{\mu}(x)+\theta\,(\,\bar{B}_{\mu}(x))\equiv
C_{\mu}(x)+\theta\,[s_{ab}\,C_{\mu}(x)],$
$\displaystyle\bar{C}_{\mu}(x)\longrightarrow\;{\tilde{\bar{\cal
F}}}_{\mu}^{(ab)}(x,\theta)$ $\displaystyle=$
$\displaystyle\bar{C}_{\mu}(x)+\theta\,(-\,\partial_{\mu}{\bar{\beta}}(x))\equiv\bar{C}_{\mu}(x)+\theta\,[s_{ab}\,\bar{C}_{\mu}(x)],$
$\displaystyle\beta(x)\longrightarrow\;\;{\tilde{\beta}}^{(ab)}(x,\theta)\;$
$\displaystyle=$
$\displaystyle\beta(x)+\theta\,(-\,\lambda(x))\equiv\beta(x)+\theta\,[s_{ab}\,\beta(x)],$
$\displaystyle\bar{\beta}(x)\longrightarrow\;\;{\tilde{\bar{\beta}}}^{(ab)}(x,\theta)$
$\displaystyle=$
$\displaystyle\bar{\beta}(x)+\theta\,\,(0)\equiv\beta(x)+\theta\,\,[s_{ab}\,\bar{\beta}(x)],$
$\displaystyle\varphi(x)\;\;\longrightarrow\;\;{\tilde{\Phi}}^{(ab)}(x,\theta)$
$\displaystyle=$
$\displaystyle\varphi(x)+\theta\,(\rho(x))\equiv\varphi(x)+\theta\,[s_{ab}\,\varphi(x)],$
$\displaystyle\phi_{\mu}(x)\;\;\longrightarrow{\tilde{\Phi}}_{\mu}^{(ab)}(x,\theta)$
$\displaystyle=$
$\displaystyle\phi_{\mu}(x)+\theta\,[\partial_{\mu}\bar{C}(x)-m\,\bar{C}_{\mu}(x)]$
$\displaystyle\equiv$
$\displaystyle\phi_{\mu}(x)+\theta\,[s_{ab}\phi_{\mu}(x)],$ $\displaystyle
C(x)\;\;\longrightarrow\;\;{\tilde{\cal F}}^{(ab)}(x,\theta)$ $\displaystyle=$
$\displaystyle C(x)+\theta\,(\bar{B}(x))\equiv C(x)+\theta\;[s_{ab}\,C(x)],$
$\displaystyle\bar{C}(x)\;\;\longrightarrow\;\;{\tilde{\bar{\cal
F}}}^{(ab)}(x,\theta)$ $\displaystyle=$
$\displaystyle\bar{C}(x)+\theta\,(-\,m\,\bar{\beta}(x))\equiv\bar{C}(x)+\theta\,[s_{ab}\bar{C}(x)],$
$\displaystyle
B_{\mu}(x)\;\;\;\longrightarrow\tilde{B}_{\mu}^{(ab)}(x,\theta)$
$\displaystyle=$ $\displaystyle
B_{\mu}(x)+\theta\,(-\,\partial_{\mu}\,\rho)\equiv
B_{\mu}(x)+\theta\,[s_{ab}B_{\mu}(x)],$
$\displaystyle\bar{B}_{\mu}(x)\;\longrightarrow\tilde{\bar{B}}_{\mu}^{(ab)}(x,\theta)$
$\displaystyle=$
$\displaystyle\bar{B}_{\mu}(x)+\theta\,(0)\equiv\bar{B}_{\mu}(x)+\theta\,[s_{ab}\,\bar{B}_{\mu}(x)],$
$\displaystyle B(x)\;\;\longrightarrow\;\;\tilde{B}^{(ab)}(x,\theta)$
$\displaystyle=$ $\displaystyle B(x)+\theta\,(-\,m\rho(x))\equiv
B(x)+\theta\,[s_{ab}\,B(x)],$
$\displaystyle\bar{B}(x)\;\;\longrightarrow\;\;\tilde{\bar{B}}^{(ab)}(x,\theta)$
$\displaystyle=$
$\displaystyle\bar{B}(x)+\theta\,(0)\equiv\bar{B}(x)+\theta\,[s_{ab}\,\bar{B}(x)],$
$\displaystyle{\cal B}_{\mu}(x)\;\longrightarrow\tilde{{\cal
B}}_{\mu}^{(ab)}(x,\theta)$ $\displaystyle=$ $\displaystyle{\cal
B}_{\mu}(x)+\theta\,(0)\equiv{\cal B}_{\mu}(x)+\theta\,[s_{ab}\,{\cal
B}_{\mu}(x)],$ $\displaystyle\bar{\cal
B}_{\mu}(x)\;\longrightarrow{\tilde{\bar{\cal B}}}_{\mu}^{(ab)}(x,\theta)$
$\displaystyle=$ $\displaystyle\bar{\cal
B}_{\mu}(x)+\theta\,(0)\equiv\bar{\cal B}_{\mu}(x)+\theta\,[s_{ab}\,\bar{\cal
B}_{\mu}(x)],$
$\displaystyle\tilde{\varphi}(x)\;\;\;\longrightarrow{\it{\tilde{\Phi}}}^{(ab)}(x,\theta)$
$\displaystyle=$
$\displaystyle\tilde{\varphi}(x)+\theta\,(0)\equiv\tilde{\varphi}(x)+\theta\,[s_{ab}\,\tilde{\varphi}(x)],$
$\displaystyle\tilde{\phi}_{\mu}(x)\;\;\longrightarrow{\it{\tilde{\Phi}}}_{\mu}^{(ab)}(x,\theta)$
$\displaystyle=$
$\displaystyle\tilde{\phi}_{\mu}(x)+\theta\,(0)\equiv\tilde{\phi}_{\mu}(x)+\theta\,[s_{ab}\,\tilde{\phi}_{\mu}(x)],$
$\displaystyle\lambda(x)\;\;\longrightarrow\;\;\tilde{\lambda}^{(ab)}(x,\theta)$
$\displaystyle=$
$\displaystyle\lambda(x)+\theta\,(0)\equiv\lambda(x)+\theta\,[s_{ab}\,\lambda(x)],$
$\displaystyle\rho(x)\;\;\longrightarrow\;\;\tilde{\rho}^{(ab)}(x,\theta)$
$\displaystyle=$
$\displaystyle\rho(x)+\theta\,(0)\equiv\rho(x)+\theta\,[s_{ab}\,\rho(x)],$
$\displaystyle{\cal B}(x)\;\;\longrightarrow\;\;\tilde{\cal
B}^{(ab)}(x,\theta)$ $\displaystyle=$ $\displaystyle{\cal
B}(x)+\theta\,(0)\equiv{\cal B}(x)+\theta\,[s_{ab}\,{\cal B}(x)],$
$\displaystyle{\bar{\cal B}}(x)\;\;\longrightarrow\;\;{\tilde{\bar{\cal
B}}}^{(ab)}(x,\theta)$ $\displaystyle=$ $\displaystyle{\bar{\cal
B}}(x)+\theta\,(0)\equiv{\bar{\cal B}}(x)+\theta\,[s_{ab}\,{{\bar{\cal
B}}}(x)].$ (40)

It is self-evident that we have an interesting relationship:
$\partial_{\theta}\,\Omega^{(ab)}(x,\theta)=s_{ab}\,\omega(x)$ where the
generic superfield $\Omega^{(ab)}(x,\theta)$ represents nothing but the chiral
superfields present on the l.h.s. of Eqs. (38) as well as (40) and $\omega(x)$
denotes nothing but the generic 4D field which stands for the ordinary basic
and auxiliary fields of the 4D ordinary Lagrangian density ${\cal
L}_{(\bar{B},{\bar{\cal B})}}$ (that respects anti-BRST symmetry
transformations (12) in a perfect manner [cf. Eq. (14)]). In other words, the
translation of the superfields (obtained after the application of anti-BRST
invariant restrictions) along the chiral $\theta$-direction of the chiral (4,
1)-dimensional super sub-manifold generates the anti-BRST symmetry
transformations $(s_{ab})$ in the ordinary 4D space [cf. Eq. (12)]. We also
observe that the nilpotency $(s_{ab}^{2}=0)$ of $s_{ab}$ and the nilpotency
$(\partial_{\theta}^{2}=0)$ of the translational generator
$(\partial_{\theta})$ (along the $\theta$-direction of the chiral super sub-
manifold) are deeply related to each-other.

###  4.2 (Anti-)co-BRST Symmetries: ACSA

We focus now on the derivation of the (anti-)co-BRST symmetry transformations
$s_{(a)d}$ by applying the ACSA to BRST formalism. For this purpose, we use
the (anti-)chiral super expansions (29) and (36) for the sake of brevity§§§We
can choose a (4, 4)-dimensional supermanifold for the generalization of our
ordinary 4D theory where the Grassmannian variables can be chosen to be
$(\theta_{1},\bar{\theta}_{1},\theta_{2},\bar{\theta}_{2})$ with the fermionic
properties:
$\theta_{1}^{2}=\theta_{2}^{2}=0,\,\bar{\theta}_{1}^{2}=\bar{\theta}_{2}^{2}=0,\,\theta_{1}\,\bar{\theta}_{1}+\bar{\theta}_{1}\,\theta_{1}=0,\,\theta_{2}\,\bar{\theta}_{2}+\bar{\theta}_{2}\,\theta_{2}=0$,
etc. The pair $(\theta_{1},\bar{\theta}_{1})$ and corresponding derivatives
$(\partial_{\theta 1},\partial_{\bar{\theta}1})$ can be associated with the
BRST and anti-BRST symmetries [keeping the pair
$(\theta_{2},\bar{\theta}_{2})$ intact]. On the other hand, the pair
$(\theta_{2},\bar{\theta}_{2})$ and corresponding derivatives
$(\partial_{\theta 2},\partial_{\bar{\theta}2})$ could be associated with the
co-BRST and anti-co-BRST symmetries [keeping the pair
$(\theta_{1},\bar{\theta}_{1})$ intact]. This is required because the
(anti-)BRST and (anti-)co-BRST symmetries are independent of each-other as are
the exterior and co-exterior derivatives of differential geometry. However,
for the sake of brevity, we have considered only a single pair [i.e.
$(\theta,\bar{\theta})$] of Grassmannian variables $\theta$ and $\bar{\theta}$
for the discussions of the off-shell nilpotent (anti-)BRST and (anti-)co-BRST
symmetries together in our present endeavor.. First of all, we derive the co-
BRST symmetry transformations $(s_{d})$ by taking into account the chiral
super expansions (36). The basic tenets of ACSA to BRST formalism states that
the following useful and interesting co-BRST invariant quantities, namely;

$\displaystyle s_{d}\big{[}\rho\,\tilde{\varphi}\big{]}=0,\qquad
s_{d}\big{[}{\beta}\,\rho+\tilde{\varphi}\,\lambda\big{]}=0,\qquad
s_{d}\big{[}\bar{C}_{\mu}\partial^{\mu}\lambda-\partial_{\mu}{\beta}\,\partial^{\mu}\bar{\beta}\big{]}=0,$
$\displaystyle
s_{d}\big{[}\bar{C}_{\mu}\partial^{\mu}\bar{C}+\tilde{\phi}_{\mu}\partial^{\mu}\bar{\beta}\big{]}=0,\quad
s_{d}\big{[}\bar{\cal B}_{\mu}+\partial_{\mu}\tilde{\varphi}\big{]}=0,\quad
s_{d}\big{[}\bar{\cal B}+m\,\tilde{\varphi}\big{]}=0,$ $\displaystyle
s_{d}\big{[}{C}_{\mu}\;(m\,\bar{C}^{\mu}-\partial^{\mu}\bar{C})+{\cal
B}^{\mu}\tilde{\phi}_{\mu}\big{]}=0,\quad
s_{d}\big{[}{C}\;(m\,\bar{C}_{\mu}-\partial_{\mu}\bar{C})+{\cal
B}\;\tilde{\phi}_{\mu}\big{]}=0,$ $\displaystyle
s_{d}\big{[}m\,B_{\mu\nu}-\varepsilon_{\mu\nu\eta\kappa}\partial^{\eta}\tilde{\phi}^{\kappa}\big{]}=0,\quad
s_{d}\big{[}m\,\bar{\beta}\,\beta+\lambda\,\bar{C}\big{]}=0,$ (41)

should be independent of the Grassmannian variables $\theta$ when they are
generalized onto the (4, 1)-dimensional chiral super sub-manifold [of the
general (4, 2)-dimensional supermanifold]. In other words, we demand the
following conditions on the chiral superfields

$\displaystyle\tilde{\rho}^{(d)}(x,\theta)\;{\it{\tilde{\Phi}}}(x,\theta)=\rho(x)\,\tilde{\varphi}(x),\;\;{\tilde{{\bar{\cal
B}}}}_{\mu}(x,\theta)+\partial_{\mu}{\it{\tilde{\Phi}}}(x,\theta)={\bar{\cal
B}}_{\mu}(x)+\partial_{\mu}\tilde{\varphi}(x),$
$\displaystyle\tilde{\beta}(x,\theta)\;\tilde{\rho}^{(d)}(x,\theta)+{\it{\tilde{\Phi}}}(x,\theta)\;\tilde{\lambda}^{(d)}(x,\theta)={\beta}(x)\,\rho(x)+\tilde{\varphi}(x)\,\lambda(x),$
$\displaystyle+{{\cal B}}^{\mu}(x)\;\tilde{\phi}_{\mu}(x),\;\;{\tilde{{\cal
F}}}(x,\theta)\big{[}m\,{\tilde{\bar{\cal
F}}}_{\mu}(x,\theta)-\partial_{\mu}{\tilde{\bar{\cal
F}}}(x,\theta)\big{]}+{\tilde{{\cal
B}}}^{(d)}(x,\theta)\;{\it{\tilde{\Phi}}}_{\mu}(x,\theta)$
$\displaystyle={C}\big{[}m\,\bar{C}_{\mu}(x)-\partial_{\mu}{\bar{C}}(x)\big{]}+{\cal
B}(x)\,\tilde{\phi}_{\mu}(x),\;\;{\tilde{{\bar{\cal
B}}}}(x,\theta)+m\,{\it{\tilde{\Phi}}}(x,\theta)$ $\displaystyle={\bar{\cal
B}}(x)+m\,\tilde{\varphi}(x),\;\;{\bar{\cal
F}}_{\mu}(x,\theta)\;\partial^{\mu}\tilde{\lambda}^{(d)}(x,\theta)-\partial_{\mu}\tilde{\beta}(x,\theta)\;\partial^{\mu}\;\tilde{\bar{\beta}}^{(d)}(x,\theta)$
$\displaystyle=\bar{C}_{\mu}(x)\partial^{\mu}\lambda(x)-\partial_{\mu}{\beta}(x)\,\partial^{\mu}\bar{\beta}(x),\;m\,\tilde{\beta}(x,\theta)\;\tilde{\bar{\beta}}^{(d)}(x,\theta)+\tilde{\lambda}^{(d)}(x,\theta)\;{\tilde{\bar{\cal
F}}}(x,\theta)$
$\displaystyle=m\,\bar{\beta}(x)\,\beta(x)+\lambda(x)\,\bar{C}(x),\;\;{\tilde{\bar{\cal
F}}_{\mu}}(x,\theta)\;\partial^{\mu}{\tilde{\bar{\cal
F}}}(x,\theta)+{\it{\tilde{\Phi}}}_{\mu}^{(d)}(x,\theta)\;\partial^{\mu}\;{\tilde{\bar{\beta}}}^{(d)}(x,\theta)$
$\displaystyle=\bar{C}_{\mu}(x)\;\partial^{\mu}{\bar{C}}(x)+\tilde{\phi}_{\mu}(x)\;\partial^{\mu}\bar{\beta}(x),\;m\,{\tilde{B}}_{\mu\nu}(x,\theta)-\varepsilon_{\mu\nu\eta\kappa}\partial^{\eta}{\it{\tilde{\Phi}}}^{\kappa}(x,\theta)=$
$\displaystyle
m\,{B}_{\mu\nu}(x)-\varepsilon_{\mu\nu\eta\kappa}\partial^{\eta}\tilde{\phi}^{\kappa}(x),\;\;{\tilde{\cal
F}}_{\mu}(x,\theta)\;\big{[}m\,{\tilde{\bar{\cal
F}}}^{\mu}(x,\theta)-\partial^{\mu}{\tilde{\bar{\cal F}}}(x,\theta)\big{]}$
$\displaystyle+{\tilde{{\cal
B}}}^{\mu(d)}(x,\theta)\;{\it{\tilde{\Phi}}}_{\mu}^{(d)}(x,\theta)={C}_{\mu}(x)\,[m\,\bar{C}^{\mu}(x)-\partial^{\mu}{\bar{C}}(x)],$
(42)

where the superfields with the superscript $(d)$ have been derived from our
earlier observation:
$s_{d}[\partial^{\nu}B_{\nu\mu},B_{\mu},{\bar{B}}_{\mu},B,\bar{B},{\cal
B}_{\mu},{\cal B},\phi_{\mu},\bar{\beta},\varphi,\rho,\lambda]=0$ [cf. Eq.
(19)] which imply the following trivial restrictions on the chiral
superfields:

$\displaystyle\partial^{\nu}\tilde{B}_{\nu\mu}^{(d)}(x,\theta)=\partial^{\nu}B_{\nu\mu}(x),\quad\tilde{\rho}^{(d)}(x,\theta)=\rho(x),\quad\tilde{\lambda}^{(d)}(x,\theta)=\lambda(x),$
$\displaystyle\tilde{\cal B}_{\mu}^{(d)}(x,\theta)={\cal
B}_{\mu}(x),\quad\Phi_{\mu}^{(d)}(x,\theta)=\phi_{\mu}(x),\quad\tilde{\cal
B}^{(d)}(x,\theta)={\cal B}(x),$
$\displaystyle\tilde{B}^{(d)}(x,\theta)=B(x),\quad\tilde{\bar{B}}^{(d)}(x,\theta)=\bar{B}(x),\quad\tilde{B}_{\mu}^{(d)}(x,\theta)=B_{\mu}(x),$
$\displaystyle\tilde{\bar{\beta}}^{(d)}(x,\theta)=\bar{\beta}(x),\quad\quad\tilde{\Phi}^{(d)}(x,\theta)=\varphi(x),\quad{\tilde{\bar{B}}}_{\mu}^{(d)}(x,\theta)=\bar{B}_{\mu}(x).$
(43)

A close look at the chiral super expansions (36) demonstrates that the
secondary fields
$(\bar{R}_{\mu},\bar{B}_{3},\bar{B}_{4},\bar{f}_{2},\bar{f}_{3},\bar{f}_{4},\bar{f}_{5},\bar{f}_{7},\bar{f}_{\mu}^{(1)},\bar{f}_{\mu}^{(2)},\bar{f}_{\mu}^{(3)})$
are all trivially equal to zero due to the straightforward co-BRST invariant
restrictions in (43).

The restrictions (42) on the chiral superfields, along with the inputs from
(43), lead to the determination of the secondary fields [in terms of the basic
and auxiliary fields of ${\cal L}_{(B,{\cal B})}$] of the chiral expansions in
(36) as:

$\displaystyle\bar{R}_{\mu\nu}=-\varepsilon_{\mu\nu\eta\kappa}\partial^{\eta}\bar{C}^{\kappa},\;\bar{B}_{\mu}^{(1)}={\cal
B}_{\mu},\;\bar{B}_{\mu}^{(2)}=-\partial_{\mu}\bar{\beta},\;\bar{f}_{1}=-\lambda,\;\bar{f}^{(4)}_{\mu}=\partial_{\mu}\rho,$
$\displaystyle\bar{R}_{\mu}^{(1)}=\partial_{\mu}\bar{C}-m\,\bar{C}_{\mu},\;\,\bar{B}_{1}={\cal
B},\quad\bar{B}_{2}=-\,m\,\bar{\beta},\;\,\bar{f}_{8}=m\,\rho,\;\bar{f}_{6}=-\,\rho.$
(44)

Substitutions of these precise values into the chiral expansions (36) lead to
the following in terms of the super expansions along $\theta$, namely;

$\displaystyle
B_{\mu\nu}(x)\longrightarrow{\tilde{B}_{\mu\nu}}^{(d)}(x,\theta)$
$\displaystyle=$ $\displaystyle
B_{\mu\nu}(x)+\theta\,[-\varepsilon_{\mu\nu\eta\kappa}\partial^{\eta}\bar{C}^{\kappa}(x)]$
$\displaystyle\equiv$ $\displaystyle
B_{\mu\nu}(x)+\theta\,[s_{d}\,B_{\mu\nu}(x)],$ $\displaystyle
C_{\mu}(x)\;\,\longrightarrow\;\;{\tilde{\cal F}}_{\mu}^{(d)}(x,\theta)$
$\displaystyle=$ $\displaystyle C_{\mu}(x)+\theta\,(\,{\cal B}_{\mu}(x))\equiv
C_{\mu}(x)+\theta\;[s_{d}\;C_{\mu}(x)],$
$\displaystyle\bar{C}_{\mu}(x)\longrightarrow\;{\tilde{\bar{\cal
F}}}_{\mu}^{(d)}(x,\theta)$ $\displaystyle=$
$\displaystyle\bar{C}_{\mu}(x)+\theta\,(-\,\partial_{\mu}{\bar{\beta}}(x))\equiv\bar{C}_{\mu}(x)+\theta\,[s_{d}\,\bar{C}_{\mu}(x)],$
$\displaystyle\beta(x)\longrightarrow\;\;{\tilde{\beta}}^{(d)}(x,\theta)\;$
$\displaystyle=$
$\displaystyle\beta(x)+\theta\,(-\,\lambda(x))\equiv\beta(x)+\theta\,[s_{d}\,\beta(x)],$
$\displaystyle\bar{\beta}(x)\longrightarrow\;\;{\tilde{\bar{\beta}}}^{(d)}(x,\theta)$
$\displaystyle=$
$\displaystyle\bar{\beta}(x)+\theta\,\,(0)\equiv\beta(x)+\theta\,\,[s_{d}\,\bar{\beta}(x)],$
$\displaystyle\varphi(x)\;\;\longrightarrow\;\;{\tilde{\Phi}}^{(d)}(x,\theta)$
$\displaystyle=$
$\displaystyle\varphi(x)+\theta\,((0))\equiv\varphi(x)+\theta\,[s_{d}\,\varphi(x)],$
$\displaystyle\phi_{\mu}(x)\;\;\longrightarrow{\tilde{\Phi}}_{\mu}^{(d)}(x,\theta)$
$\displaystyle=$
$\displaystyle\phi_{\mu}(x)+\theta\,(0)\equiv\phi_{\mu}(x)+\theta\,[s_{d}\phi_{\mu}(x)],$
$\displaystyle C(x)\;\;\longrightarrow\;\;{\tilde{\cal F}}^{(d)}(x,\theta)$
$\displaystyle=$ $\displaystyle C(x)+\theta\,({\cal B}(x))\equiv
C(x)+\theta\;[s_{d}\,C(x)],$
$\displaystyle\bar{C}(x)\;\;\longrightarrow\;\;{\tilde{\bar{\cal
F}}}^{(d)}(x,\theta)$ $\displaystyle=$
$\displaystyle\bar{C}(x)+\theta\,(-\,m\,\bar{\beta}(x))\equiv\bar{C}(x)+\theta\,[s_{d}\bar{C}(x)],$
$\displaystyle B_{\mu}(x)\;\;\;\longrightarrow\tilde{B}_{\mu}^{(d)}(x,\theta)$
$\displaystyle=$ $\displaystyle B_{\mu}(x)+\theta\,(0)\equiv
B_{\mu}(x)+\theta\,[s_{d}B_{\mu}(x)],$
$\displaystyle\bar{B}_{\mu}(x)\;\longrightarrow\tilde{\bar{B}}_{\mu}^{(d)}(x,\theta)$
$\displaystyle=$
$\displaystyle\bar{B}_{\mu}(x)+\theta\,(0)\equiv\bar{B}_{\mu}(x)+\theta\;[s_{d}\,\bar{B}_{\mu}(x)],$
$\displaystyle B(x)\;\;\longrightarrow\;\;\tilde{B}^{(d)}(x,\theta)$
$\displaystyle=$ $\displaystyle B(x)+\theta\,(0)\equiv
B(x)+\theta\,[s_{d}\,B(x)],$
$\displaystyle\bar{B}(x)\;\;\longrightarrow\;\;\tilde{\bar{B}}^{(d)}(x,\theta)$
$\displaystyle=$
$\displaystyle\bar{B}(x)+\theta\,(0)\equiv\bar{B}(x)+\theta\;[s_{d}\,\bar{B}(x)],$
$\displaystyle{\cal B}_{\mu}(x)\;\longrightarrow\tilde{{\cal
B}}_{\mu}^{(d)}(x,\theta)$ $\displaystyle=$ $\displaystyle{\cal
B}_{\mu}(x)+\theta\,(0)\equiv{\cal B}_{\mu}(x)+\theta\;[s_{d}\,{\cal
B}_{\mu}(x)],$ $\displaystyle\bar{\cal
B}_{\mu}(x)\;\longrightarrow{\tilde{\bar{\cal B}}}_{\mu}^{(d)}(x,\theta)$
$\displaystyle=$ $\displaystyle\bar{\cal
B}_{\mu}(x)+\theta\;(\partial_{\mu}\,\rho(x))\equiv\bar{\cal
B}_{\mu}(x)+\theta\;[s_{d}\,\bar{\cal B}_{\mu}(x)],$
$\displaystyle\tilde{\varphi}(x)\;\;\;\longrightarrow{\it{\tilde{\Phi}}}^{(d)}(x,\theta)$
$\displaystyle=$
$\displaystyle\tilde{\varphi}(x)+\theta\;(-\,\rho(x))\equiv\tilde{\varphi}(x)+\theta\;[s_{d}\,\tilde{\varphi}(x)]$
$\displaystyle\tilde{\phi}_{\mu}(x)\;\;\longrightarrow{\it{\tilde{\Phi}}}_{\mu}^{(d)}(x,\theta)$
$\displaystyle=$
$\displaystyle\tilde{\phi}_{\mu}(x)+\theta\,[\partial_{\mu}\bar{C}(x)-m\,\bar{C}_{\mu}(x)]$
$\displaystyle\equiv$
$\displaystyle\tilde{\phi}_{\mu}(x)+\theta\;[s_{d}\,\tilde{\phi}_{\mu}(x)],$
$\displaystyle\lambda(x)\;\;\longrightarrow\;\;\tilde{\lambda}^{(d)}(x,\theta)$
$\displaystyle=$
$\displaystyle\lambda(x)+\theta\,(0)\equiv\lambda(x)+\theta\,[s_{d}\,\lambda(x)],$
$\displaystyle\rho(x)\;\;\longrightarrow\;\;\tilde{\rho}^{(d)}(x,\theta)$
$\displaystyle=$
$\displaystyle\rho(x)+\theta\,(0)\equiv\rho(x)+\theta\,[s_{d}\,\rho(x)],$
$\displaystyle{\cal B}(x)\;\;\longrightarrow\;\;\tilde{\cal
B}^{(d)}(x,\theta)$ $\displaystyle=$ $\displaystyle{\cal
B}(x)+\theta\,(0)\equiv{\cal B}(x)+\theta\,[s_{d}\,{\cal B}(x)],$
$\displaystyle{\bar{\cal B}}(x)\;\;\longrightarrow\;\;{\tilde{\bar{\cal
B}}}^{(d)}(x,\theta)$ $\displaystyle=$ $\displaystyle{\bar{\cal
B}}(x)+\theta\,(m\,\rho(x))\equiv{\bar{\cal B}}(x)+\theta\,[s_{d}\,{{\bar{\cal
B}}}(x)],$ (45)

where the superscript $(d)$ denotes the chiral super expansions of the
superfields that lead to the derivation of the co-BRST transformations (19) as
the coefficients of $\theta$. It is crystal clear that
$\partial_{\theta}\,\Omega^{(d)}(x,\theta)=s_{d}\,\omega(x)$ where the generic
superfield $\Omega^{(d)}(x,\theta)$ stands for all the chiral superfields that
are present on the l.h.s. of Eqs. (43) as well as (45) and the ordinary
generic field $\omega(x)$ corresponds to all the basic and auxiliary fields of
the Lagrangian density ${\cal L}_{(B,{\cal B})}$. We also note that the off-
shell nilpotency $(s_{d}^{2}=0)$ of $s_{d}$ is deeply connected with the
nilpotency $(\partial_{\theta}^{2}=0)$ of the translational generator
$({\partial_{\theta}})$ along the $\theta$-direction of the (4, 1)-dimensional
chiral super sub-manifold [of the general (4, 2)-dimensional supermanifold].

We devote time on the derivation of the anti-co-BRST symmetry transformations
$s_{ad}$ by applying the basic tenets of ACSA to BRST formalism. In this
connection, first of all, we observe that the following anti-co-BRST invariant
quantities of interest, namely;

$\displaystyle s_{ad}\big{[}\rho\,\bar{\beta}\big{]}=0,\qquad
s_{ad}\big{[}\bar{\beta}\,\lambda+{\tilde{\varphi}}\,\rho\big{]}=0,\qquad
s_{ad}\big{[}C_{\mu}\partial^{\mu}\rho-\partial_{\mu}\bar{\beta}\,\partial^{\mu}\beta\big{]}=0,$
$\displaystyle
s_{ad}\big{[}C_{\mu}\partial^{\mu}{C}+\tilde{\phi}_{\mu}\partial^{\mu}\beta\big{]}=0,\quad
s_{ad}\big{[}{\cal B}_{\mu}+\partial_{\mu}\tilde{\varphi}\big{]}=0,\quad
s_{ad}\big{[}{\cal B}+m\,\tilde{\varphi}\big{]}=0,$ $\displaystyle
s_{ad}\big{[}\bar{C}_{\mu}\;(m\,C^{\mu}-\partial^{\mu}{C})+\bar{{\cal
B}}^{\mu}\tilde{\phi}_{\mu}\big{]}=0,\quad
s_{ad}\big{[}\bar{C}\;(m\,C_{\mu}-\partial_{\mu}{C})+\bar{{\cal
B}}\tilde{\phi}_{\mu}\big{]}=0,$ $\displaystyle
s_{ad}\big{[}m\,B_{\mu\nu}-\varepsilon_{\mu\nu\eta\kappa}\partial^{\eta}{\tilde{\phi}}^{\kappa}\big{]}=0,\quad
s_{ad}\big{[}m\,\bar{\beta}\,\beta+\rho\,C\big{]}=0,$ (46)

can be generalized onto the suitably chosen (4, 1)-dimensional anti-chiral
super sub-manifold where we have to take into account the anti-chiral
superfield expansions (29). However, before we perform that, we note that (due
to the trivial anti-co-BRST symmetry invariance
$s_{ad}[\partial^{\nu}B_{\nu\mu},B_{\mu},{\bar{B}}_{\mu},B,\bar{B},\bar{\cal
B}_{\mu},\bar{\cal B},\phi_{\mu},\beta,\varphi,\rho,\lambda]=0$), we have the
following

$\displaystyle\partial^{\nu}\tilde{B}_{\nu\mu}^{(ad)}(x,\bar{\theta})=\partial^{\nu}B_{\nu\mu}(x),\quad\tilde{\rho}^{(ad)}(x,\bar{\theta})=\rho(x),\quad\tilde{\lambda}^{(ad)}(x,\bar{\theta})=\lambda(x),$
$\displaystyle{\tilde{\bar{\cal B}}}_{\mu}^{(ad)}(x,\bar{\theta})={\bar{\cal
B}}_{\mu}(x),\quad\tilde{\Phi}_{\mu}^{(ad)}(x,\bar{\theta})=\phi_{\mu}(x),\quad{\tilde{\bar{\cal
B}}}^{(ad)}(x,\bar{\theta})={\bar{\cal B}}(x),$
$\displaystyle\tilde{B}^{(ad)}(x,\bar{\theta})=B(x),\quad\tilde{\bar{B}}^{(ad)}(x,\bar{\theta})=\bar{B}(x),\quad\tilde{B}_{\mu}^{(ad)}(x,\bar{\theta})=B_{\mu}(x),$
$\displaystyle\tilde{\beta}^{(ad)}(x,\bar{\theta})=\beta(x),\quad\tilde{\Phi}^{(ad)}(x,\bar{\theta})=\varphi(x),\quad{\tilde{\bar{B}}}_{\mu}^{(ad)}(x,\bar{\theta})=\bar{B}_{\mu}(x),$
(47)

where the superscript $(ad)$ denotes the anti-chiral superfields which have
been obtained due to the trivial anti-co-BRST invariance [cf. Eq. (18)].
Another way of saying the fact that the secondary fields
$(R_{\mu},B_{3},B_{4},f_{1},f_{3},f_{4},f_{5},f_{8},f_{\mu}^{(1)},f_{\mu}^{(2)},f_{\mu}^{(3)})$
are trivially equal to zero so that we can have
$s_{ad}[\partial^{\nu}B_{\nu\mu},B_{\mu},{\bar{B}}_{\mu},B,\bar{B},\bar{\cal
B}_{\mu},\bar{\cal B},\phi_{\mu},\beta,\\\ \varphi,\rho,\lambda]=0$ as the
coefficients of $\bar{\theta}$ in the super expansions that have been listed
in Eq. (29) and represented in the super expansions (47).

Using the trivial equalities (47), we have the following generalizations of
the anti-co-BRST invariant quantities (46) [located inside the square
brackets] in terms of the anti-chiral superfields, namely;

$\displaystyle\tilde{\rho}^{(ad)}(x,\bar{\theta})\;{\tilde{\bar{\beta}}}(x,\bar{\theta})=\rho(x)\,\bar{\beta}(x),\quad{\tilde{\cal
B}}_{\mu}(x,\bar{\theta})+\partial_{\mu}{\it{\tilde{\Phi}}}(x,\bar{\theta})={\cal
B}_{\mu}(x)+\partial_{\mu}\tilde{\varphi}(x),$
$\displaystyle\tilde{\bar{\beta}}(x,\bar{\theta})\;\tilde{\lambda}^{(ad)}(x,\bar{\theta})+{\it{\tilde{\Phi}}}(x,\bar{\theta})\;\tilde{\rho}^{(ad)}(x,\bar{\theta})=\bar{\beta}(x)\,\lambda(x)+\tilde{\varphi}(x)\,\rho(x),$
$\displaystyle
m\,\tilde{\bar{\beta}}(x,\bar{\theta})\;\tilde{\beta}^{(ad)}(x,\bar{\theta})+\tilde{\rho}^{(ad)}(x,\bar{\theta})\;{\tilde{\cal
F}}(x,\bar{\theta})=m\,\bar{\beta}(x)\,\beta(x)+\rho(x)\,C(x),$
$\displaystyle{\tilde{\cal
F}}_{\mu}(x,\bar{\theta})\;\partial^{\mu}{\tilde{\cal
F}}(x,\bar{\theta})+{\it{\tilde{\Phi}}}_{\mu}(x,\bar{\theta})\;\partial^{\mu}\tilde{\beta}^{(ad)}(x,\bar{\theta})=C_{\mu}(x)\partial^{\mu}{C}(x)+\tilde{\phi}_{\mu}(x)\partial^{\mu}\beta(x),$
$\displaystyle{\tilde{\cal
B}}(x,\bar{\theta})+m\,{\it\tilde{\Phi}}(x,\bar{\theta})={\cal
B}(x)+m\,\tilde{\varphi}(x),\quad{\tilde{\bar{\cal
F}}}_{\mu}(x,\bar{\theta})\;\big{[}m\,{\tilde{\cal
F}}^{\mu}(x,\bar{\theta})-\partial^{\mu}{\tilde{\cal
F}}(x,\bar{\theta})\big{]}$ $\displaystyle+\;{\tilde{\bar{\cal
B}}}^{\mu}(x,\bar{\theta})\;{\it{\tilde{\Phi}}}_{\mu}(x,\bar{\theta})=\bar{C}_{\mu}(x)\big{[}m\,C^{\mu}(x)-\partial^{\mu}{C}(x)\big{]}+{\bar{\cal
B}}_{\mu}(x)\;\tilde{\phi}^{\mu}(x),$ $\displaystyle{\tilde{\bar{\cal
F}}}(x,\bar{\theta})\big{[}m\,{\tilde{\cal
F}}_{\mu}(x,\bar{\theta})-\partial_{\mu}{\tilde{\cal
F}}(x,\bar{\theta})\big{]}+\;{\tilde{\bar{\cal
B}}}^{(ad)}(x,\bar{\theta})\;{\it{\tilde{\Phi}}}_{\mu}(x,\bar{\theta})=\bar{C}(x)\big{[}m\,C_{\mu}(x)$
$\displaystyle-\partial_{\mu}{C}(x)\big{]}+\;{\bar{\cal
B}}(x)\,\tilde{\phi}_{\mu}(x),\,\qquad
m\,{\tilde{B}}_{\mu\nu}(x,\bar{\theta})-\varepsilon_{\mu\nu\eta\kappa}\;\partial^{\eta}{\it\tilde{\Phi}}^{\kappa}(x,\bar{\theta})=m\,B_{\mu\nu}(x)$
$\displaystyle-\varepsilon_{\mu\nu\eta\kappa}\;\partial^{\eta}{\it\tilde{\Phi}}^{\kappa}(x),\;\;\tilde{\cal
F}_{\mu}(x,\bar{\theta})\;\partial^{\mu}\tilde{\rho}^{(ad)}(x,\bar{\theta})-\partial_{\mu}\tilde{\bar{\beta}}(x,\bar{\theta})\;\partial^{\mu}\tilde{\beta}^{(ad)}(x,\theta)$
$\displaystyle=C_{\mu}(x)\;\partial^{\mu}\rho(x)-\partial_{\mu}\bar{\beta}(x)\,\partial^{\mu}\beta(x).$
(48)

At this stage, we substitute the anti-chiral super expansions (29) into the
above equalities which lead to the determination of secondary fields in terms
of the basic and auxiliary fields of the Lagrangian density ${\cal
L}_{(\bar{B},{\bar{\cal B}})}$ as:

$\displaystyle
R_{\mu\nu}=-\varepsilon_{\mu\nu\eta\kappa}\partial^{\eta}C^{\kappa},\quad
B_{\mu}^{(2)}=\bar{\cal B}_{\mu},\quad B_{\mu}^{(1)}=\partial_{\mu}\beta,\quad
f_{6}=-\lambda,\quad\bar{f}^{(3)}_{\mu}=\partial_{\mu}\rho,$ $\displaystyle
R_{\mu}^{(1)}=\partial_{\mu}C-m\,C_{\mu},\quad B_{2}={\bar{\cal B}},\quad\quad
B_{1}=\,m\,\beta,\quad f_{7}=m\,\lambda,\quad f_{2}=\rho.$ (49)

The substitutions of (49) and observations in (47) enable us to write the
anti-chiral super expansions (29), in terms of the anti-co-BRST symmetry
transformations (18), as:

$\displaystyle
B_{\mu\nu}(x)\longrightarrow{\tilde{B}_{\mu\nu}}^{(ad)}(x,\bar{\theta})$
$\displaystyle=$ $\displaystyle
B_{\mu\nu}(x)+\bar{\theta}\,[-\varepsilon_{\mu\nu\eta\kappa}\partial^{\eta}\bar{C}^{\kappa}(x)]$
$\displaystyle\equiv$ $\displaystyle
B_{\mu\nu}(x)+\bar{\theta}\,[s_{ad}\,B_{\mu\nu}(x)],$ $\displaystyle
C_{\mu}(x)\;\,\longrightarrow\;\;{\tilde{\cal
F}}_{\mu}^{(ad)}(x,\bar{\theta})$ $\displaystyle=$ $\displaystyle
C_{\mu}(x)+\bar{\theta}\,(\partial_{\mu}\beta(x))\equiv
C_{\mu}(x)+\bar{\theta}\,[s_{ad}\,C_{\mu}(x)],$
$\displaystyle\bar{C}_{\mu}(x)\longrightarrow\;{\tilde{\bar{\cal
F}}}_{\mu}^{(ad)}(x,\bar{\theta})$ $\displaystyle=$
$\displaystyle\bar{C}_{\mu}(x)+\bar{\theta}\,(\bar{\cal
B}_{\mu}(x))\equiv\bar{C}_{\mu}(x)+\bar{\theta}\,[s_{ad}\,\bar{C}_{\mu}(x)],$
$\displaystyle\beta(x)\longrightarrow\;\;{\tilde{\beta}}^{(ad)}(x,\bar{\theta})\;$
$\displaystyle=$
$\displaystyle\beta(x)+\bar{\theta}\,(0)\equiv\beta(x)+\bar{\theta}\,[s_{ad}\,\beta(x)],$
$\displaystyle\bar{\beta}(x)\longrightarrow\;\;{\tilde{\bar{\beta}}}^{(ad)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\bar{\beta}(x)+\bar{\theta}\,\,(\rho(x))\equiv\bar{\beta}(x)+\bar{\theta}\,\,[s_{ad}\,\bar{\beta}(x)],$
$\displaystyle\varphi(x)\;\;\longrightarrow\;\;{\tilde{\Phi}}^{(ad)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\varphi(x)+\bar{\theta}\,((0))\equiv\varphi(x)+\bar{\theta}\,[s_{ad}\,\varphi(x)],$
$\displaystyle\phi_{\mu}(x)\;\;\longrightarrow{\tilde{\Phi}}_{\mu}^{(ad)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\phi_{\mu}(x)+\bar{\theta}\,(0)\equiv\phi_{\mu}(x)+\bar{\theta}\,[s_{ad}\phi_{\mu}(x)],$
$\displaystyle C(x)\;\;\longrightarrow\;\;{\tilde{\cal
F}}^{(ad)}(x,\bar{\theta})$ $\displaystyle=$ $\displaystyle
C(x)+\bar{\theta}\,(m\,\beta(x))\equiv C(x)+\bar{\theta}\;[s_{ad}\,C(x)],$
$\displaystyle\bar{C}(x)\;\;\longrightarrow\;\;{\tilde{\bar{\cal
F}}}^{(ad)}(x,\bar{\theta})$ $\displaystyle=$
$\displaystyle\bar{C}(x)+\bar{\theta}\,(\bar{\cal
B}(x))\equiv\bar{C}(x)+\bar{\theta}\,[s_{ad}\bar{C}(x)],$ $\displaystyle
B_{\mu}(x)\;\;\;\longrightarrow\tilde{B}_{\mu}^{(ad)}(x,\bar{\theta})$
$\displaystyle=$ $\displaystyle B_{\mu}(x)+\bar{\theta}\,(0)\equiv
B_{\mu}(x)+\bar{\theta}\;[s_{ad}B_{\mu}(x)],$
$\displaystyle\bar{B}_{\mu}(x)\;\longrightarrow\tilde{\bar{B}}_{\mu}^{(ad)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\bar{B}_{\mu}(x)+\bar{\theta}\,(0)\equiv\bar{B}_{\mu}(x)+\bar{\theta}\;[s_{ad}\,\bar{B}_{\mu}(x)],$
$\displaystyle B(x)\;\;\longrightarrow\;\;\tilde{B}^{(ad)}(x,\bar{\theta})$
$\displaystyle=$ $\displaystyle B(x)+\bar{\theta}\,(0)\equiv
B(x)+\bar{\theta}\;[s_{ad}\,B(x)],$
$\displaystyle\bar{B}(x)\;\;\longrightarrow\;\;\tilde{\bar{B}}^{(ad)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\bar{B}(x)+\bar{\theta}\,(0)\equiv\bar{B}(x)+\bar{\theta}\,[s_{ad}\,\bar{B}(x)],$
$\displaystyle{\cal B}_{\mu}(x)\;\longrightarrow\tilde{{\cal
B}}_{\mu}^{(ad)}(x,\bar{\theta})$ $\displaystyle=$ $\displaystyle{\cal
B}_{\mu}(x)+\bar{\theta}\;(\partial_{\mu}\lambda(x))\equiv{\cal
B}_{\mu}(x)+\bar{\theta}\;[s_{ad}\,{\cal B}_{\mu}(x)],$
$\displaystyle\bar{\cal B}_{\mu}(x)\;\longrightarrow{\tilde{\bar{\cal
B}}}_{\mu}^{(ad)}(x,\bar{\theta})$ $\displaystyle=$ $\displaystyle\bar{\cal
B}_{\mu}(x)+\bar{\theta}\;(0)\equiv\bar{\cal
B}_{\mu}(x)+\bar{\theta}\;[s_{ad}\,\bar{\cal B}_{\mu}(x)],$
$\displaystyle\tilde{\varphi}(x)\;\;\;\longrightarrow{\it{\tilde{\Phi}}}^{(ad)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\tilde{\varphi}(x)+\bar{\theta}\;(-\,\lambda)\equiv\tilde{\varphi}(x)+\bar{\theta}\;[s_{ad}\,\tilde{\varphi}(x)],$
$\displaystyle\tilde{\phi}_{\mu}(x)\;\;\longrightarrow{\it{\tilde{\Phi}}}_{\mu}^{(ad)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\tilde{\phi}_{\mu}(x)+\bar{\theta}\,[\partial_{\mu}C(x)-m\,C_{\mu}(x)]$
$\displaystyle\equiv$
$\displaystyle\tilde{\phi}_{\mu}(x)+\bar{\theta}\;[s_{ad}\,\tilde{\phi}_{\mu}(x)],$
$\displaystyle\lambda(x)\;\;\longrightarrow\;\;\tilde{\lambda}^{(ad)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\lambda(x)+\bar{\theta}\,(0)\equiv\lambda(x)+\bar{\theta}\,[s_{ad}\,\lambda(x)],$
$\displaystyle\rho(x)\;\;\longrightarrow\;\;\tilde{\rho}^{(ad)}(x,\bar{\theta})$
$\displaystyle=$
$\displaystyle\rho(x)+\bar{\theta}\,(0)\equiv\rho(x)+\bar{\theta}\,[s_{ad}\,\rho(x)],$
$\displaystyle{\cal B}(x)\;\;\longrightarrow\;\;\tilde{\cal
B}^{(ad)}(x,\bar{\theta})$ $\displaystyle=$ $\displaystyle{\cal
B}(x)+\bar{\theta}\,(m\,\lambda)\equiv{\cal B}(x)+\bar{\theta}\,[s_{ad}\,{\cal
B}(x)],$ $\displaystyle{\bar{\cal
B}}(x)\;\;\longrightarrow\;\;{\tilde{\bar{\cal B}}}^{(ad)}(x,\bar{\theta})$
$\displaystyle=$ $\displaystyle{\bar{\cal
B}}(x)+\bar{\theta}\,(0)\equiv{\bar{\cal
B}}(x)+\bar{\theta}\,[s_{ad}\,{{\bar{\cal B}}}(x)].$ (50)

The above final expansions explicitly show that we have already derived the
anti-co-BRST symmetry transformations $(s_{ad})$ as the coefficients of
$\bar{\theta}$. In other words, we note that
$\partial_{\bar{\theta}}\,\Omega^{(ad)}(x,\bar{\theta})=s_{ad}\,\omega(x)$
where $\Omega^{(ad)}(x,\bar{\theta})$ is the generic superfield that stands
for all the anti-chiral superfields which are present on the l.h.s. of Eqs.
(47) as well as (50) and $\omega(x)$ is the generic field which corresponds to
the ordinary basic and auxiliary fields of ${\cal L}_{(\bar{B},{\bar{\cal
B})}}$ that are the first terms on the r.h.s. of Eqs. (47) as well as (50).
This observation also implies that we have interconnection between the
nilpotency ($s_{ad}^{2}=0$) of $s_{ad}$ and nilpotency
($\partial_{\bar{\theta}}^{2}=0$) of the translational generator
$\partial_{\bar{\theta}}$ along $\bar{\theta}$-direction of the anti-chiral
super sub-manifold as:
$s_{ad}^{2}=0\Leftrightarrow\partial_{\bar{\theta}}^{2}=0$.

##  5 Invariance of the Lagrangian Densities:
ACSA to BRST Formalism

In this section, we establish the existence of the CF-type restrictions [cf.
Eq. (21)] within the framework ACSA by capturing the symmetry invariance of
the Lagrangian densities ${\cal L}_{(B,{{\cal B})}}$ and ${\cal
L}_{(\bar{B},\bar{\cal B})}$. This exercise also proves the equivalence of the
coupled Lagrangian densities ${\cal L}_{(B,{\cal B})}$ and ${\cal
L}_{(\bar{B},{\bar{\cal B})}}$ w.r.t. the (anti-)BRST and (anti-)co-BRST
symmetry transformations in the space of fields where the CF-type restrictions
[cf. Eq. (21)] are satisfied. Our present section is divided into two parts.
In sub-section 5.1, we discuss the (anti-) BRST invariance and the CF-type
restrictions (associated with the nilpotent (anti-)BRST symmetries). Our sub-
section 5.2 is devoted to the discussion of the (anti-)co-BRST invariance and
derivation of the CF-type restrictions (associated with these nilpotent
symmetries).

###  5.1 (Anti-)BRST Invariance and CF-Type Restrictions

In this sub-section, we discuss the (anti-)BRST invariance and derivation of
the proper CF-type restrictions within the framework of ACSA to BRST
formalism. Toward this objective in mind, we perform the following
generalization of the ordinary Lagrangian density: ${\cal L}_{(B,\cal
B)}\longrightarrow\tilde{\cal L}^{(ac)}_{(B,\cal B)}(x,\bar{\theta}),$ namely;

$\displaystyle\tilde{\cal L}^{(ac)}_{(B,\cal B)}(x,\bar{\theta})$
$\displaystyle=$ $\displaystyle\frac{1}{2}{\cal B}_{\mu}(x)\;{\cal
B}^{\mu}(x)-{\cal
B}^{\mu}(x)\Big{(}\frac{1}{2}\varepsilon_{\mu\nu\eta\kappa}\,\partial^{\nu}{\tilde{B}}^{\eta\kappa(b)}(x,\bar{\theta})$
$\displaystyle-$
$\displaystyle\frac{1}{2}\,\partial_{\mu}\tilde{\varphi}(x)+m\tilde{\phi}_{\mu}(x)\Big{)}-\frac{m^{2}}{4}\,{\tilde{B}}^{\mu\nu{(b)}}(x,\bar{\theta})\,{\tilde{B}}_{\mu\nu}^{(b)}(x,\bar{\theta})$
$\displaystyle-$
$\displaystyle\frac{1}{2}\,\partial^{\mu}{\tilde{\Phi}}^{\nu(b)}(x,\bar{\theta})\,\Big{(}\partial_{\mu}{\tilde{\Phi}}_{\nu}^{(b)}(x,\bar{\theta})-\partial_{\nu}{\tilde{\Phi}}_{\mu}^{(b)}(x,\bar{\theta})\Big{)}$
$\displaystyle+$ $\displaystyle
m\,\tilde{B}^{\mu\nu(b)}(x,\bar{\theta})\,\partial_{\mu}{\tilde{\Phi}}_{\nu}^{(b)}(x,\bar{\theta})+\frac{1}{4}\,{\tilde{\Phi}}^{\mu\nu}(x)\,{\tilde{\Phi}}_{\mu\nu}(x)$
$\displaystyle+$
$\displaystyle\frac{m}{2}\,\varepsilon^{\mu\nu\eta\kappa}{\tilde{B}}^{(b)}_{\mu\nu}(x,\bar{\theta})\partial_{\eta}{\tilde{\phi}}_{\kappa}(x)-\frac{1}{2}\,B^{\mu}(x)\,B_{\mu}(x)$
(51) $\displaystyle+$ $\displaystyle
B^{\mu}(x)\;\left(\partial^{\nu}{\tilde{B}}_{\nu\mu}^{(b)}(x,\bar{\theta})-\frac{1}{2}\,\partial_{\mu}\tilde{\Phi}^{(b)}(x,\bar{\theta})+m{\tilde{\Phi}}_{\mu}^{(b)}(x,\bar{\theta})\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\,B(x)\,B(x)+B(x)\,\Big{(}\partial_{\mu}{\tilde{\Phi}}^{\mu(b)}(x,\bar{\theta})+\frac{m}{2}\,\tilde{\Phi}^{(b)}(x,\bar{\theta})\Big{)}$
$\displaystyle-$ $\displaystyle\frac{1}{2}\,{\cal B}(x)\,{\cal B}(x)-{\cal
B}(x)\Big{(}\partial_{\mu}\tilde{\phi}^{\mu}(x)+\frac{m}{2}\,\tilde{\varphi}(x)\Big{)}$
$\displaystyle+$ $\displaystyle\Big{(}\partial_{\mu}{\tilde{\bar{\cal
F}}}^{(b)}(x,\bar{\theta})-m\,{\tilde{\bar{\cal
F}}}_{\mu}^{(b)}(x,\bar{\theta})\,\Big{)}$
$\displaystyle\Big{(}\partial^{\mu}\,{\tilde{\cal
F}}^{(b)}(x,\bar{\theta})-m\,{\tilde{\cal F}}^{\mu(b)}(x,\bar{\theta})\Big{)}$
$\displaystyle-$ $\displaystyle\Big{(}\partial_{\mu}{\tilde{\bar{\cal
F}}}_{\nu}^{(b)}(x,\bar{\theta})-\partial_{\nu}{\tilde{\bar{\cal
F}}}_{\mu}^{(b)}(x,\bar{\theta})\Big{)}\Big{(}\partial^{\mu}{\tilde{\cal
F}}^{\nu(b)}(x,\bar{\theta})\Big{)}$ $\displaystyle-$
$\displaystyle\frac{1}{2}\,\partial_{\mu}{\tilde{\bar{\beta}}}^{(b)}(x,\bar{\theta})\,\partial^{\mu}\beta(x)+\frac{1}{2}\,m^{2}\,\tilde{\bar{\beta}}^{(b)}(x,\bar{\theta})\,\beta(x)$
$\displaystyle-$
$\displaystyle\frac{1}{2}\left(\partial_{\mu}{\tilde{\bar{\cal
F}}}^{\mu(b)}(x,\bar{\theta})+m\,{\tilde{\bar{\cal
F}}}^{(b)}(x,\bar{\theta})+\frac{1}{4}\,\rho(x)\right)\lambda(x)$
$\displaystyle-$ $\displaystyle\frac{1}{2}\left(\partial_{\mu}{\tilde{\cal
F}}^{\mu(b)}(x,\bar{\theta})+m\,{\tilde{\cal
F}}^{(b)}(x,\bar{\theta})-\frac{1}{4}\,\lambda(x)\right)\rho(x),$

where the superscript $(ac)$ on the super Lagrangian density denotes that we
have taken into account the anti-chiral superfields in the anti-chiral super
Lagrangian density $[\tilde{\cal L}_{(B,{\cal B})}^{(ac)}(x,\bar{\theta})]$
which incorporates a combination of the ordinary fields and anti-chiral
superfields with superscript $(b)$ that have been derived earlier in Eq. (34).
We would like to point out that the ordinary fields are those which are
trivially BRST invariant [cf. Eq. (13)]. Hence, they are independent of the
Grassmannian variable. It is now straightforward to check that we have the
following expression when we apply $\partial_{\bar{\theta}}$ on (51), namely;

$\displaystyle\frac{\partial}{\partial\bar{\theta}}\Big{[}\tilde{\cal
L}^{(ac)}_{(B,\cal B)}\Big{]}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\,\tilde{\phi}_{\nu}\big{(}\partial_{\eta}C_{\kappa}\big{)}+B_{\nu}\big{(}\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu}\big{)}$
(52) $\displaystyle+$
$\displaystyle\frac{1}{2}\,B^{\mu}\,\lambda-B\big{(}\partial^{\mu}C-mC^{\mu}\big{)}-\frac{1}{2}\,\big{(}\partial^{\mu}\beta\big{)}\,\rho\bigg{]}\equiv
s_{b}\,{\cal L}_{(B,{\cal B})},$

which demonstrates that the r.h.s. is a total spacetime derivative that has
been derived earlier in Eq. (15) due to the BRST transformation of the
Lagrangian density ${\cal L}_{(B,{\cal B})}$ in the ordinary space.

To capture the anti-BRST invariance of ${\cal L}_{(\bar{B},\bar{\cal B})}$
within the framework of ACSA, we perform the generalization: ${\cal
L}_{(\bar{B},\bar{\cal B)}}\longrightarrow\tilde{\cal
L}^{(c)}_{(\bar{B},\bar{\cal B)}}(x,\theta)$ as follows

$\displaystyle\tilde{\cal L}^{(c)}_{(\bar{B},\bar{\cal B)}}(x,\theta)$
$\displaystyle=$ $\displaystyle\frac{1}{2}\bar{\cal B}_{\mu}(x)\;\bar{\cal
B}^{\mu}(x)+\bar{\cal
B}^{\mu}(x)\Big{(}\frac{1}{2}\varepsilon_{\mu\nu\eta\kappa}\,\partial^{\nu}{\tilde{B}}^{\eta\kappa(ab)}(x,\theta)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\,\partial_{\mu}\tilde{\varphi}(x)+m\;\tilde{\phi}_{\mu}(x)\Big{)}-\frac{m^{2}}{4}\,{\tilde{B}}^{{\mu\nu}(ab)}(x,\theta)\,{\tilde{B}}_{\mu\nu}^{(ab)}(x,\theta)$
$\displaystyle-$
$\displaystyle\frac{1}{2}\,\partial^{\mu}{\tilde{\Phi}}^{\nu(ab)}(x,\theta)\,\Big{(}\partial_{\mu}{{\tilde{\Phi}}_{\nu}}^{(ab)}(x,\theta)-\partial_{\nu}{\tilde{\Phi}}_{\mu}^{(ab)}(x,\theta)\Big{)}$
$\displaystyle+$ $\displaystyle
m\,{\tilde{B}}^{\mu\nu(ab)}(x,\theta)\,\partial_{\mu}{\tilde{\Phi}}_{\nu}^{(ab)}(x,\theta)+\frac{1}{4}\,{\tilde{\Phi}}^{\mu\nu}(x){\tilde{\Phi}}_{\mu\nu}(x)$
(53) $\displaystyle+$
$\displaystyle\frac{m}{2}\,\varepsilon^{\mu\nu\eta\kappa}{\tilde{B}}^{(ab)}_{\mu\nu}(x,\theta)\partial_{\eta}{{\tilde{\phi}}}_{\kappa}(x)-\frac{1}{2}\,\bar{B}^{\mu}(x)\,\bar{B}_{\mu}(x)$
$\displaystyle-$
$\displaystyle\bar{B}^{\mu}(x)\,\left(\partial^{\nu}{\tilde{B}}_{\nu\mu}^{(ab)}(x,\theta)+\frac{1}{2}\,\partial_{\mu}\tilde{\Phi}^{(ab)}(x,\theta)+m\;{\tilde{\Phi}}_{\mu}^{(ab)}(x,\theta)\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\,\bar{B}(x)\,\bar{B}(x))-\bar{B}(x)\,\left(\partial_{\mu}{\tilde{\Phi}}^{\mu(ab)}(x,\theta)-\frac{m}{2}\,\tilde{\Phi}^{(ab)}(x,\theta)\right)$
$\displaystyle-$ $\displaystyle\frac{1}{2}\,\bar{\cal B}(x)\,\bar{\cal
B}(x)+\bar{\cal
B}(x)\left(\partial_{\mu}\tilde{\phi}^{\mu}(x)-\frac{m}{2}\,\tilde{\varphi}(x)\right)$
$\displaystyle+$ $\displaystyle\Big{(}\partial_{\mu}{\tilde{\bar{\cal
F}}}^{(ab)}(x,\theta)-m\,{\tilde{\bar{\cal
F}}}_{\mu}^{(ab)}(x,\theta)\,\Big{)}$
$\displaystyle\Big{(}\partial^{\mu}\,{\tilde{\cal
F}}^{(ab)}(x,\theta)-m\,{\tilde{\cal F}}^{\mu(ab)}(x,\theta)\Big{)}$
$\displaystyle-$ $\displaystyle\Big{(}\partial_{\mu}{\tilde{\bar{\cal
F}}}_{\nu}^{(ab)}(x,\theta)-\partial_{\nu}{\tilde{\bar{\cal
F}}}_{\mu}^{(ab)}(x,\theta)\Big{)}\Big{(}\partial^{\mu}{\tilde{\cal
F}}^{\nu(ab)}(x,\theta)\Big{)}$ $\displaystyle-$
$\displaystyle\frac{1}{2}\,\partial_{\mu}{\bar{\beta}}(x)\,\partial^{\mu}\beta^{(ab)}(x,\theta)+\frac{1}{2}\,m^{2}\,\bar{\beta}(x)\,\beta^{(ab)}(x,\theta)$
$\displaystyle-$
$\displaystyle\frac{1}{2}\left(\partial_{\mu}{\tilde{\bar{\cal
F}}}^{\mu(ab)}(x,\theta)+m\,{\tilde{\bar{\cal
F}}}^{(ab)}(x,\theta)+\frac{1}{4}\,\rho(x)\right)\lambda(x)$ $\displaystyle-$
$\displaystyle\frac{1}{2}\left(\partial_{\mu}{\tilde{\cal
F}}^{\mu(ab)}(x,\theta)+m\,{\tilde{\cal
F}}^{(ab)}(x,\theta)-\frac{1}{4}\,\lambda(x)\right)\rho(x),$

where we have obtained a chiral super Lagrangian density from the ordinary
Lagrangian density ${\cal L}_{(\bar{B},\bar{\cal B})}$ [that is characterized
by a superscript $(c)$]. This chiral super Lagrangian density is made up of
the ordinary fields as well as the chiral superfields with superscript $(ab)$
that have been obtained in Eq. (40). Now we are in the position to apply a
derivative (${\partial_{\theta}}$) w.r.t. the Grassmannian variable $\theta$
on (53). This operation leads to the following:

$\displaystyle\frac{\partial}{\partial\theta}\Big{[}\tilde{\cal
L}^{(c)}_{{(\bar{B},\bar{\cal B)}}}\Big{]}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\,\tilde{\phi}_{\nu}\big{(}\partial_{\eta}\bar{C}_{\kappa}\big{)}-\bar{B}_{\nu}\big{(}\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu}\big{)}$
(54) $\displaystyle+$
$\displaystyle\frac{1}{2}\,\bar{B}^{\mu}\,\rho+\bar{B}\;\big{(}\partial^{\mu}\bar{C}-m\;\bar{C}^{\mu}\big{)}-\frac{1}{2}\,\big{(}\partial^{\mu}\bar{\beta}\big{)}\,\lambda\big{]}\equiv
s_{ab}\;{\cal L}_{(\bar{B},{\bar{\cal B}})}.$

Keeping in mind the mapping $\partial_{\theta}\longleftrightarrow s_{ab}$
[4-6], it is straightforward to note that we have captured the perfect anti-
BRST invariance of the Lagrangian density ${\cal L}_{(\bar{B},{\bar{\cal
B}})}$ [as we have: $\partial_{\theta}\tilde{\cal
L}^{(c)}_{(\bar{B},{\bar{\cal B}})}\equiv s_{ab}{\cal L}_{(\bar{B},{\bar{\cal
B}})}$].

To establish the existence of the CF-type restrictions [cf. Eq. (21)] and
equivalence of the Lagrangian densities ${\cal L}_{(B,{{\cal B}})}$ and ${\cal
L}_{(\bar{B},{\bar{\cal B}})}$ w.r.t. the nilpotent symmetries $s_{(a)b}$, we
generalize the Lagrangian density ${\cal L}_{(B,{{\cal B}})}$ to its chiral
counterpart: ${\cal L}_{(B,\cal B)}\longrightarrow\tilde{\cal
L}^{(c)}_{(B,\cal B)}$ as

$\displaystyle\tilde{\cal L}^{(c)}_{(B,\cal B)}(x,\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\cal B}_{\mu}(x)\;{\cal B}^{\mu}(x)-{\cal
B}^{\mu}(x)\Big{(}\frac{1}{2}\varepsilon_{\mu\nu\eta\kappa}\,\partial^{\nu}{\tilde{B}}^{\eta\kappa(ab)}(x,\theta)$
$\displaystyle-$
$\displaystyle\frac{1}{2}\,\partial_{\mu}\tilde{\varphi}(x)+m\;\tilde{\phi}_{\mu}(x)\Big{)}-\frac{m^{2}}{4}\,{\tilde{B}}^{\mu\nu(ab)}(x,\theta)\,{\tilde{B}}_{\mu\nu}^{(ab)}(x,\theta)$
$\displaystyle-$
$\displaystyle\frac{1}{2}\,\partial^{\mu}{\tilde{\Phi}}^{\nu(ab)}(x,\theta)\,\Big{(}\partial_{\mu}{{\tilde{\Phi}}_{\nu}}^{(ab)}(x,\theta)-\partial_{\nu}{{\tilde{\Phi}}}_{\mu}^{(ab)}(x,\theta)\Big{)}$
(55) $\displaystyle+$
$\displaystyle{m}\,{\tilde{B}}^{\mu\nu(ab)}(x,\theta)\,\partial_{\mu}{{\tilde{\Phi}}}_{\nu}^{(ab)}(x,\theta)+\frac{1}{4}\,\tilde{\Phi}^{\mu\nu}(x)\tilde{\Phi}_{\mu\nu}(x)$
$\displaystyle+$
$\displaystyle\frac{m}{2}\,\varepsilon^{\mu\nu\eta\kappa}{\tilde{B}}^{(ab)}_{\mu\nu}(x,\theta)\partial_{\eta}{{\tilde{\phi}}}_{\kappa}(x)-\frac{1}{2}\,B^{\mu(ab)}(x,\theta)\,B_{\mu}^{(ab)}(x,\theta)$
$\displaystyle+$ $\displaystyle
B^{\mu(ab)}(x,\theta)\,\Big{(}\partial^{\nu}{\tilde{B}}_{\nu\mu}^{(ab)}(x,\theta)-\frac{1}{2}\,\partial_{\mu}\tilde{\Phi}^{(ab)}(x,\theta)$
$\displaystyle+$ $\displaystyle
m\;{\tilde{\Phi}}_{\mu}^{(ab)}(x,\theta)\Big{)}+\frac{1}{2}\,B^{(ab)}(x,\theta)\,B^{(ab)}(x,\theta)$
$\displaystyle+$ $\displaystyle
B^{(ab)}(x,\theta)\,\left(\partial_{\mu}{\tilde{\Phi}}^{\mu(ab)}(x,\theta)+\frac{m}{2}\,\tilde{\Phi}^{(ab)}(x,\theta)\right)$
$\displaystyle-$ $\displaystyle\frac{1}{2}\,{\cal B}(x)\,{\cal B}(x)-{\cal
B}(x)\left(\partial_{\mu}\tilde{\phi}^{\mu}(x)+\frac{m}{2}\,\tilde{\varphi}(x)\right)$
$\displaystyle+$ $\displaystyle\Big{(}\partial_{\mu}{\tilde{\bar{\cal
F}}}^{(ab)}(x,\theta)-m\,{\tilde{\bar{\cal
F}}}_{\mu}^{(ab)}(x,\theta)\,\Big{)}$
$\displaystyle\Big{(}\partial^{\mu}\,{\tilde{\cal
F}}^{(ab)}(x,\theta)-m\,{\tilde{\cal F}}^{\mu(ab)}(x,\theta)\Big{)}$
$\displaystyle-$ $\displaystyle\Big{(}\partial_{\mu}{\tilde{\bar{\cal
F}}}_{\nu}^{(ab)}(x,\theta)-\partial_{\nu}{\tilde{\bar{\cal
F}}}_{\mu}^{(ab)}(x,\theta)\Big{)}\Big{(}\partial^{\mu}{\tilde{\cal
F}}^{\nu(ab)}(x,\theta)\Big{)}$ $\displaystyle-$
$\displaystyle\frac{1}{2}\,\partial_{\mu}{\bar{\beta}}(x)\,\partial^{\mu}\tilde{\beta}^{(ab)}(x,\theta)+\frac{1}{2}\,m^{2}\,\bar{\beta}(x)\,\tilde{\beta}^{(ab)}(x,\theta)$
$\displaystyle-$
$\displaystyle\frac{1}{2}\left(\partial_{\mu}{\tilde{\bar{\cal
F}}}^{\mu(ab)}(x,\theta)+m\,{\tilde{\bar{\cal
F}}}^{(ab)}(x,\theta)+\frac{1}{4}\,\rho(x)\right)\lambda(x)$ $\displaystyle-$
$\displaystyle\frac{1}{2}\left(\partial_{\mu}{\tilde{\cal
F}}^{\mu(ab)}(x,\theta)+m\,{\tilde{\cal
F}}^{(ab)}(x,\theta)-\frac{1}{4}\,\lambda(x)\right)\rho(x),$

where the superfields with the superscript $(ab)$ have been obtained earlier
[cf. Eq.(40)] after the applications of anti-BRST invariant restrictions and
superscript $(c)$ on ${\cal L}_{(B,{{\cal B}})}$ denotes that we have taken
the chiral generalization of the Lagrangian density ${\cal L}_{(B,{{\cal
B}})}$. It will be noted that there are superfields in Eq. (55) as the
ordinary fields because they are anti-BRST invariant fields. It is now
elementary to check that we have the following when $\partial_{\theta}$
operates on $\tilde{\cal L}^{(c)}_{(B,\cal B)}$, namely;

$\displaystyle\frac{\partial}{\partial\theta}\Big{[}\tilde{\cal
L}^{(c)}_{(B,\cal B)}\Big{]}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}\bigg{[}m\,\varepsilon^{\mu\nu\eta\kappa}\,\tilde{\phi}_{\nu}\big{(}\partial_{\eta}\bar{C}_{\kappa}\big{)}+\Big{(}\partial_{\nu}B^{\nu\mu}+\frac{1}{2}\,\bar{B}^{\mu}+m\;\phi^{\mu}\Big{)}\rho$
(56) $\displaystyle+$ $\displaystyle
B_{\nu}\big{(}\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu}\big{)}-B\big{(}\partial^{\mu}\bar{C}-m\;\bar{C}^{\mu}\big{)}-\frac{1}{2}\,\big{(}\partial^{\mu}\bar{\beta}\big{)}\,\lambda\bigg{]}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\,\big{[}B_{\mu}+\bar{B}_{\mu}+\partial_{\mu}\varphi\big{]}\big{(}\partial^{\mu}\rho\big{)}+\partial_{\mu}\big{[}B_{\nu}+\bar{B}_{\nu}+\partial_{\nu}\varphi\big{]}$
$\displaystyle\big{(}\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu}\big{)}+m\big{[}B_{\mu}+\bar{B}_{\mu}+\partial_{\mu}\varphi\big{]}$
$\displaystyle\big{(}\partial^{\mu}\bar{C}-m\;\bar{C}^{\mu}\big{)}-\frac{m}{2}\,\big{[}B+\bar{B}+m\;\varphi\big{]}\rho$
$\displaystyle-$
$\displaystyle\partial_{\mu}\big{[}B+\bar{B}+m\;\varphi\big{]}\big{(}\partial^{\mu}\bar{C}-m\;\bar{C}^{\mu}\big{)}\equiv
s_{ab}\;{\cal L}_{(B,\cal B)}.$

The above equation establishes (keeping in our mind
$\partial_{\theta}\leftrightarrow s_{ab}$) that when the anti-BRST symmetry
operates on the Lagrangian density ${\cal L}_{(B,{\cal B})}$ in the ordinary
spacetime, we obtain the variation of ${\cal L}_{(B,{\cal B})}$ such that it
transforms to a total spacetime derivative plus terms that vanish off in the
space of fields where the CF-type restrictions [cf. Eq. (21)] are satisfied.
Hence, the Lagrangian density ${\cal L}_{(B,{\cal B})}$ respects both the BRST
and anti-BRST symmetry transformations together provided we consider the whole
theory on the sub-space of fields (defined on the flat 4D Minkowskian
spacetime manifold) on which the CF-type restrictions [cf. Eq. (21)] are fully
satisfied together.

Now we capture the BRST invariance of the Lagrangian density ${\cal
L}_{(\bar{B},\bar{\cal B})}$ within the framework of ACSA. Toward this goal in
mind, we generalize the ordinary 4D Lagrangian density ${\cal
L}_{(\bar{B},\bar{\cal B})}$ onto the (4, 1)-dimensional anti-chiral
supermanifold as the anti-chiral super Lagrangian density $\tilde{\cal
L}^{(ac)}_{(\bar{B},\bar{\cal B})}(x,\bar{\theta})$. The explicit and lucid
form of $\tilde{\cal L}^{(ac)}_{(\bar{B},\bar{\cal B})}(x,\bar{\theta})$ is

$\displaystyle\tilde{\cal L}^{(ac)}_{(\bar{B},\bar{\cal B)}}(x,\bar{\theta})$
$\displaystyle=$ $\displaystyle\frac{1}{2}\bar{\cal B}_{\mu}(x)\;\bar{\cal
B}^{\mu}(x)+\bar{\cal
B}^{\mu}(x)\Big{(}\frac{1}{2}\varepsilon_{\mu\nu\eta\kappa}\,\partial^{\nu}{\tilde{B}}^{\eta\kappa(b)}(x,\bar{\theta})$
(57) $\displaystyle+$
$\displaystyle\frac{1}{2}\,\partial_{\mu}\tilde{\varphi}(x)+m\;\tilde{\phi}_{\mu}(x)\Big{)}-\frac{m^{2}}{4}\,{\tilde{B}}^{\mu\nu(b)}(x,\bar{\theta})\,{\tilde{B}}_{\mu\nu}^{(b)}(x,\bar{\theta})$
$\displaystyle-$
$\displaystyle\frac{1}{2}\,\partial^{\mu}{{\tilde{\Phi}}}^{\nu(b)}(x,\bar{\theta})\,\Big{(}\partial_{\mu}{{\tilde{\Phi}}}_{\nu}^{(b)}(x,\bar{\theta})-\partial_{\nu}{{\tilde{\Phi}}}_{\mu}^{(b)}(x,\bar{\theta})\Big{)}$
$\displaystyle+$
$\displaystyle{m}\,{\tilde{B}}^{\mu\nu(b)}(x,\bar{\theta})\,\partial_{\mu}{{\tilde{\Phi}}}_{\nu}^{(b)}(x,\bar{\theta})+\frac{1}{4}\,\tilde{\Phi}^{\mu\nu}(x)\tilde{\Phi}_{\mu\nu}(x)$
$\displaystyle+$