Noether Theorem and Nilpotency Property of the (Anti-)BRST Charges in the BRST
Formalism: A Brief Review

A. K. Rao(a), A. Tripathi(a), B. Chauhan(a), R. P. Malik(a,b)
(a) Department of Physics, 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: In some of the physically interesting gauge systems, we show that
the application of the Noether theorem does not lead to the deduction of the
Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST charges that obey precisely the
off-shell nilpotency property despite the fact that these charges are $(i)$
derived by using the off-shell nilpotent (anti-)BRST symmetry transformations,
$(ii)$ found to be the generators of the above continuous symmetry
transformations, and $(iii)$ conserved w.r.t. the time-evolution due to the
Euler-Lagrange equations of motion derived from the Lagrangians/Lagrangian
densities (that describe the dynamics of these suitably chosen physical
systems). We propose a systematic method for the derivation of the off-shell
nilpotent (anti-)BRST charges from the corresponding non-nilpotent Noether
(anti-)BRST charges. To corroborate the sanctity and preciseness of our
proposal, we take into account the examples of $(i)$ the one (0 +
1)-dimensional (1D) system of a massive spinning (i.e. SUSY) relativistic
particle, $(ii)$ the D-dimensional non-Abelian 1-form gauge theory, and
$(iii)$ the Abelian 2-form and the St${\ddot{u}}$ckelberg-modified version of
the massive Abelian 3-form gauge theories in any arbitrary D-dimension of
spacetime. Our present endeavor is a brief review where some decisive
proposals have been made and a few novel results have been obtained as far as
the nilpotency property is concerned.

PACS numbers: 11.15.-q, 12.20.-m, 03.70.+k

Keywords: Noether theorem in BRST approach; continuous (anti-)BRST symmetries;
conserved Noether currents; conserved (anti-)BRST charges; nilpotency property

##  1 Introduction

It is an undeniable truth that the symmetries of all kinds (e.g. global,
local, discrete, continuous, spacetime, internal, etc.) have played a decisive
role in the realm of theoretical physics as they have provided a set of deep
insights into various working aspects of the physical systems of interest. The
basic principles behind the local gauge symmetries and diffeomorphism
symmetries provide the precise theoretical descriptions of the standard model
of particle physics and theory of gravity (i.e. general theory of relativity
and (super)string theories). The Becchi-Rouet-Stora-Tyutin (BRST) formalism
[1-4] is applied fruitfully to the gauge theories as well as the
diffeomorphism invariant theories. Some of the salient features of BRST
formalism are ($i$) it covariantly quantizes the gauge theories which are
characterized by the existence of the first-class constraints on them in the
terminology of Dirac’s prescription for the classifications of constraints
(see, e.g. [5, 6] for details), ($ii$) it is consistent with the Dirac
quantization scheme because the physicality criteria with the nilpotent and
conserved (anti-)BRST charges imply that the physical states, in the total
quantum Hilbert space, are those that are annihilated by the operator form of
the first-class constraints of the gauge theories (see, e.g. [7-9] for
details), ($iii$) it maintains the unitarity and quantum gauge (i.e. BRST)
invariance at any arbitrary order of perturbative computations of a given
physical process that is allowed by an interacting gauge theory, and ($iv$) it
has deep connections with some of the key ideas behind differential geometry
and its (anti-)BRST transformations resemble that of the ${\cal N}=2$
supersymmetry transformations. There is a decisive difference, however,
between the key properties associated with the (anti-)BRST symmetries and the
${\cal N}=2$ supersymmetries in the sense that the nilpotent BRST and anti-
BRST symmetries are absolutely anticommuting in nature but the nilpotent
${\cal N}=2$ supersymmetries are not. To sum-up, we note that the application
of the BRST formalism is physically very useful and mathematically its horizon
is quite wide.

The purpose of our present endeavor is related with the derivations of the
conserved (anti-)BRST charges by exploiting the theoretical potential of
Noether’s theorem and a thorough study of their nilpotency property. We take
into account diverse examples of physically interesting models of gauge
theories and demonstrate that the Noether theorem does not always lead to the
derivation of conserved and nilpotent (anti-)BRST charges***It is found that
the Noether conserved (anti-)BRST charges are the generators of the off-shell
nilpotent (anti-)BRST symmetry transformations from which they are derived by
using the Noether theorem.. One has to apply specific set of theoretical
tricks and techniques to obtain the nilpotent versions of the (anti-)BRST
charges from the Noether conserved (anti-)BRST charges which are found to be
non-nilpotent. In simple examples, we show that one equation of motion is good
enough to convert the non-nilpotent Noether conserved (anti-)BRST charges into
the conserved and nilpotent (anti-)BRST charges. For instance, in the case of
a 1D massive spinning (i.e. SUSY) relativistic particle (see, e.g. [10-13] and
references therein), the Euler-Lagrange (EL) equations of motion (EoM) w.r.t.
the “gauge” and “supergauge” variables are sufficient to convert a non-
nilpotent set of (anti-)BRST charges into nilpotent of order two (cf. Sec. 2).
However, in the case of a D-dimensional non-Abelian 1-form theory, the Gauss
divergence theorem and EL-EoM w.r.t. the gauge field are needed to obtain the
nilpotent (anti-)BRST charges from the conserved and non-nilpotent Noether
(anti-)BRST charges. These simple examples, we have purposely chosen so that
it becomes clear that the celebrated Noether theorem does not lead to the
derivation of nilpotent (anti-)BRST charges where $(i)$ the non-trivial
(anti-)BRST invariant Curci-Ferrari (CF) type restrictions exist, and ($ii$) a
set of coupled (but equivalent) Lagrangians/Lagrangian densities respect the
off-shell nilpotent (anti-)BRST symmetry transformations. It is worthwhile to
mention here that the limiting cases of the above two examples are the free
scalar relativistic particle and D-dimensional Abelian 1-form gauge theory
where there is existence of a single Lagrangian/Lagrangian density. In these
cases, the Noether conserved (anti-)BRST charges are off-shell nilpotent
automatically because the CF-type restriction is trivial. In fact, it is
observed that the trivial CF-type restriction of the scalar relativistic
particle is the limiting case of the non-trivial CF-type restriction of the
spinning relativistic particle and the trivial CF-type restriction of the
Abelian 1-form gauge theory is the limiting case of the non-trivial CF-
condition of non-Abelian 1-form gauge theory.

One of the central issue we address in our present investigation is the cases
of the BRST approach to higher $p$-form ($p=2,3,...$) gauge theories†††The
higher $p$-form ($p=2,3,...$) gauge fields (and corresponding theories) are
important because such fields appear in the quantum excitations of the
(super)string theories (see, e.g. [14] for details). where there is always
existence of ($i$) a set of coupled (but equivalent) Lagrangian densities, and
($ii$) a set of (anti-)BRST invariant CF-type restrictions. In such cases, the
Noether theorem always leads to the derivation of conserved (anti-)BRST
charges which are found to be non-nilpotent. In fact, the expressions for
these charges are quite complicated and the EL-EoMs are too many because of
the presence of too many fields in the theory (cf. Sec. 5 below for details).
We make a systematic proposal which enables us to obtain the off-shell
nilpotent (anti-) BRST charges from the conserved Noether (anti-)BRST charges
which are found to be non-nilpotent. One of the key ingredients of our
proposal is the observation that one has to start with the EL-EoM w.r.t. gauge
field of the $p$-form gauge theories where, most of the time, the Gauss
divergence theorem is required to be applied (before we exploit the potential
and power of the EL-EoM w.r.t. the gauge fields). To be precise, all the
D-dimensional ($D\geq 2$) higher $p$-form gauge theories require the
application of the Gauss divergence theorem before we exploit the potential of
EL-EoM w.r.t. the gauge field. After this, it is the interplay amongst ($i$)
the application of the EL-EoMs, ($ii$) use of the (anti-) BRST
transformations, and ($iii$) the requirement of Gauss’s divergence theorem
that lead to the derivation of the nilpotent versions of the (anti-)BRST
charges from the conserved Noether (anti-)BRST charges which are found to be
non-nilpotent (cf. Secs. 4, 5, 6).

The theoretical contents of our present endeavor are organized a follows. In
Sec. 2, we exploit the theoretical potential and power of Noether’s theorem in
the context of a gauge system of 1D spinning relativistic particle to deduce
the explicit expressions for the conserved (anti-)BRST charges $Q_{(a)b}$ and
establish that they are not off-shell nilpotent. We focus, in our Sec. 3, on
the D-dimensional non-Abelian 1-form gauge theory (without any interaction
with matter fields) and demonstrate that the Noether conserved (anti-)BRST
charges, once again, are non-nilpotent to begin with. We pinpoint the specific
EL-EoMs that have to be used to make these (anti-)BRST charges off-shell
nilpotent. Our Sec. 4 is devoted to the discussion of Noether’s theorem in the
context of D-dimensional (anti-) BRST invariant Abelian 2-form theory and
discuss the nitty-gritty details of the nilpotency property. The theoretical
content of our Sec. 5 is concerned with the (anti-)BRST invariant coupled (but
equivalent) Lagrangian densities of the (anti-)BRST invariant modified massive
Abelian 3-form gauge theory and our discussion is centered around the property
of the off-shell nilpotency of the (anti-)BRST charges. Finally, in Sec. 6, we
make some concluding remarks and comment on the future prospects of our
present investigation.

In our Appendix A, we discuss the St${\ddot{u}}$ckelberg-modified
D-dimensional massive Abelian 2-form theory and deduce the off-shell nilpotent
versions of the (anti-)BRST charges [$Q_{(a)b}^{(1)}$] from the non-nilpotent
Noether conserved (anti-)BRST charges [$Q_{(a)b}$].

Convention and Notations for the 1D Massive Spinning (SUSY) Relativistic
Particle and D-dimensional Abelian 2-Form as well as 3-Form Theories: We
follow the convention of the left-derivatives w.r.t. all the fermionic
variables/fields of our theory in the computations of the canonical conjugate
momenta and the Noether conserved currents. The flat metric tensor
$\eta_{\mu\nu}=$ diag $(+1,-1,-1,...)$ is chosen for the D-dimensional flat
Minkowskian space so that the dot product between two non-null vectors
$P_{\mu}$ and $Q_{\mu}$ is denoted by $P\cdot
Q=\eta_{\mu\nu}\,P^{\mu}\,Q^{\nu}=P_{0}\,Q_{0}-P_{i}\,Q_{i}$ where the Greek
indices $\mu,\nu,\,\lambda,...=0,1,2,...D-1$ correspond to the time and space
directions and the Latin indices $i,\,j,\,k...=1,\,2,\,...D-1$ stand for the
space directions only. Through out the whole body of our text, the nilpotent
(anti-)BRST symmetry transformations carry the symbol $s_{(a)b}$ and the
corresponding conserved (anti-) BRST charges are denoted by $Q_{(a)b}$ for all
kinds of the Abelian systems that have been chosen for our present discussion.
For our discussions on the D-dimensional non-Abelian gauge theory, we shall
adopt different convention in Sec. 3.

##  2 Preliminary: (Anti-)BRST Charges and Nilpotency for a Massive Spinning
Relativistic Particle

We begin with the following coupled (but equivalent) (anti-)BRST invariant
(see, e.g. [12, 13]) Lagrangians that describe the dynamics of a one (0 +
1)-dimensional massive spinning (i.e. supersymmetric) relativistic particle in
the D-dimensional target space, namely:

$\displaystyle L_{b}$ $\displaystyle=$ $\displaystyle
L_{f}+b^{2}+b\,({\dot{e}}+2\,{\bar{\beta}}\,\beta)-i\,\dot{\bar{c}}\,{\dot{c}}+{\bar{\beta}}^{2}\,{\beta}^{2}+2\,i\,\chi\,(\beta\,\dot{\bar{c}}-\bar{\beta}\,\dot{c})-2\,e\,(\bar{\beta}\,\dot{\beta}+\gamma\,\chi)$
(1) $\displaystyle+$ $\displaystyle
2\,\gamma\,(\beta\,\bar{c}-\bar{\beta}\,c)+m\,(\bar{\beta}\,\dot{\beta}-\dot{\bar{\beta}}\,\beta+\gamma\,\chi)-\dot{\gamma}\,\psi_{5},$
$\displaystyle L_{\bar{b}}$ $\displaystyle=$ $\displaystyle
L_{f}+{\bar{b}}^{2}-\bar{b}\,({\dot{e}}-2\,{\bar{\beta}}\,\beta)-i\,\dot{\bar{c}}\,{\dot{c}}+{\bar{\beta}}^{2}\,{\beta}^{2}+2\,i\,\chi\,(\beta\,\dot{\bar{c}}-\bar{\beta}\,\dot{c})+2\,e\,(\dot{\bar{\beta}}\,\beta-\gamma\,\chi)$
(2) $\displaystyle+$ $\displaystyle
2\,\gamma\,(\beta\,\bar{c}-\bar{\beta}\,c)+m\,(\bar{\beta}\,\dot{\beta}-\dot{\bar{\beta}}\,\beta+\gamma\,\chi)-\dot{\gamma}\,\psi_{5},$

where $L_{f}$ is the first-order Lagrangian for our system [10]

$\displaystyle
L_{f}=p_{\mu}\,\dot{x}^{\mu}+\frac{i}{2}(\psi_{\mu}\,\dot{\psi}^{\mu}-\psi_{5}\,\dot{\psi}_{5})-\frac{e}{2}\;(p^{2}-m^{2})+i\,\chi\,(p_{\mu}\,\psi^{\mu}-m\,\psi_{5}).$
(3)

In the above, the target space canonical conjugate quantities
$\big{(}x_{\mu}(\tau),p^{\mu}(\tau)\big{)}$ are the bosonic coordinates
$(x_{\mu})$ and canonical momenta $(p^{\mu})$, respectively, with the Greek
indices $\mu=0,1,2,...,D-1$ corresponding to the D-dimensional flat
Minkowskian target space. The trajectory of the spinning particle is
parameterized by $\tau$ and the generalized velocities:
${\dot{x}}_{\mu}=({d\,x^{\mu}}/{d\,\tau}),\,{\dot{\psi}}_{\mu}=({d\,\psi^{\mu}}/{d\,\tau})$
are defined w.r.t. it. The pair of fermionic
($\psi_{\mu}^{2}=0,\,\psi_{\mu}\,\psi_{\nu}+\psi_{\nu}\,\psi_{\mu}=0,\,\psi_{5}^{2}=0,\,\psi_{5}\,\psi_{\mu}+\psi_{\mu}\,\psi_{5}=0,$
etc.) variables $(\psi_{\mu},\psi_{5})$ are introduced in the theory to (i)
maintain the SUSY gauge symmetry transformations, and (ii) incorporate the
mass-shell condition $p^{2}-m^{2}=0$ where $m$ is the rest mass of the
particle. The variable $e\,(\tau)$ and $\chi(\tau)$ are the superpartners of
each-other where $e\,(\tau)$ is the einbein variable and fermionic
$(\chi^{2}=0)$ variable $\chi\,(\tau)$ is its superpartner and both of them
behave as the “gauge” and “supergauge” variables. We have incorporated a pair
of variables $(b,\bar{b})$ as the bosonic Nakanishi-Lautrup type auxiliary
variables which participate in defining the CF-type restriction‡‡‡The
(anti-)BRST symmetry transformations (4) and (5) are absolutely anticommuting
(i.e. $\\{s_{b},\,s_{ab}\\}=0$) in nature on the submanifold of the Hilbert
space of quantum variables which is defined by the (anti-) BRST invariant
(i.e. $s_{(a)b}\,[b+\bar{b}+2\,\bar{\beta}\,\beta]=0$) CF-type restriction:
$b+\bar{b}+2\,\bar{\beta}\,\beta=0$.: $b+\bar{b}+2\,\bar{\beta}\,\beta=0$
where the bosonic (anti-)ghost variables $(\bar{\beta})\,\beta$ are the
counterparts of the fermionic
$(c^{2}=0,\,{\bar{c}}^{2}=0,\,c\,\bar{c}+\bar{c}\,c=0)$ (anti-)ghost variables
$(\bar{c})\,c$ in our supersymmetric (i.e. spinning) system of a 1D
diffeomorphism invariant theory [10, 12, 13]. We require an additional
auxiliary variable $(\gamma)$ that is fermionic $(\gamma^{2}=0)$ in nature as
it anticommutes with all the other fermionic variables of our theory (i.e.
$\gamma\,\chi+\chi\,\gamma=0,\,\gamma\,\psi_{5}+\psi_{5}\,\gamma=0,\,\gamma\,\psi_{\mu}+\psi_{\mu}\,\gamma=0,\,\gamma\,c+c\,\gamma=0,\,\gamma\,\bar{c}+\bar{c}\,\gamma=0$,
etc.).

The above Lagrangians $L_{b}$ and $L_{\bar{b}}$ are coupled (but equivalent)
on the submanifold of the quantum variables where the CF-type restriction:
$b+\bar{b}+2\,\bar{\beta}\,\beta=0$ is satisfied [12, 13]. Furthermore, it is
straightforward to check that under the following off-shell nilpotent
$[s_{(a)b}^{2}=0]$ (anti-)BRST symmetry transformations $[s_{(a)b}]$, namely;

$\displaystyle
s_{ab}\;x_{\mu}={\bar{c}}\;p_{\mu}+\bar{\beta}\;\psi_{\mu},\quad\qquad
s_{ab}\;e=\dot{\bar{c}}+2\;\bar{\beta}\;\chi,\;\quad\qquad
s_{ab}\;\psi_{\mu}=i\;\bar{\beta}\;p_{\mu},$ $\displaystyle
s_{ab}\;\bar{c}=-i\;{\bar{\beta}}^{2},\quad s_{ab}\;c=i\;\bar{b},\quad
s_{ab}\;\bar{\beta}=0,\;\quad s_{ab}\;\beta=-i\;\gamma,\quad
s_{ab}\;p_{\mu}=0,$ $\displaystyle s_{ab}\;\gamma=0,\quad
s_{ab}\;\bar{b}=0,\quad s_{ab}\;\chi=i\;\dot{\bar{\beta}},\quad
s_{ab}\;b=2\;i\;\bar{\beta}\;\gamma,\quad s_{ab}\,\psi_{5}=i\,\bar{\beta}\,m,$
(4) $\displaystyle s_{b}\;x_{\mu}=c\;p_{\mu}+\beta\;\psi_{\mu},\quad\qquad
s_{b}\;e=\dot{c}+2\;\beta\;\chi,\quad\qquad
s_{b}\;\psi_{\mu}=i\;\beta\;p_{\mu},$ $\displaystyle
s_{b}\;c=-i\;\beta^{2},\;\quad s_{b}\;{\bar{c}}=i\;b,\;\quad
s_{b}\;\beta=0,\;\quad s_{b}\;\bar{\beta}=i\;\gamma,\;\quad s_{b}\;p_{\mu}=0,$
$\displaystyle s_{b}\;\gamma=0,\quad s_{b}\;b=0,\quad
s_{b}\;\chi=i\;\dot{\beta},\qquad s_{b}\;\bar{b}=-2\;i\;\beta\;\gamma,\quad
s_{b}\,\psi_{5}=i\,\beta\,m,$ (5)

the Lagrangians $L_{b}$ and $L_{\bar{b}}$ transform§§§The off-shell nilpotent
$[s_{(a)b}^{2}=0]$ (anti-)BRST symmetry transformations $s_{(a)b}$ [cf. Eqs.
(4), (5)] are said to be perfect symmetry transformations for the Lagrangians
$L_{(\bar{b})b}$, respectively, because of our observations in (6) and (7)
where no EL-EoMs and/or CF-type restriction(s) are used for their validity. to
the total derivatives as:

$\displaystyle
s_{ab}\,L_{\bar{b}}=\frac{d}{d\,\tau}\,\Big{[}\frac{\bar{c}}{2}\,(p^{2}+m^{2})+\frac{\bar{\beta}}{2}\,(p_{\mu}\,\psi^{\mu}+m\,\psi_{5})-\bar{b}\,(\dot{\bar{c}}+2\,\bar{\beta}\,\chi)\Big{]},$
(6) $\displaystyle
s_{b}\,L_{b}=\frac{d}{d\,\tau}\,\Big{[}\frac{c}{2}\,(p^{2}+m^{2})+\frac{\beta}{2}\,(p_{\mu}\,\psi^{\mu}+m\,\psi_{5})+b\,(\dot{c}+2\,\beta\,\chi)\Big{]}.$
(7)

As a consequence, the action integrals
$S_{1}=\int^{\infty}_{-\infty}d\,\tau\,L_{b}$ and
$S_{2}=\int^{\infty}_{-\infty}d\,\tau\,L_{\bar{b}}$ remain invariant for the
physically well-defined variables that vanish-off as
$\tau\rightarrow\pm\,\infty$. The application of Noether’s theorem yields the
following explicit expressions for the conserved (anti-)BRST charges
[$Q_{(a)b}$] for our 1D SUSY system of a relativistic spinning particle,
namely;

$\displaystyle Q_{ab}$ $\displaystyle=$
$\displaystyle\frac{\bar{c}}{2}\,(p^{2}-m^{2})+{\bar{\beta}}\,(p_{\mu}\,\psi^{\mu}-m\,\psi_{5})-\bar{b}\,\dot{\bar{c}}-2\,\bar{b}\,\bar{\beta}\,\chi-i\,m\,\bar{\beta}\gamma\,-\bar{\beta}{{}^{2}}\,\dot{c}-2\,\beta\,\bar{\beta}{{}^{2}}\,\chi,$
(8) $\displaystyle Q_{b}$ $\displaystyle=$
$\displaystyle\frac{c}{2}\,(p^{2}-m^{2})+{\beta}\,(p_{\mu}\,\psi^{\mu}-m\,\psi_{5})+b\,\dot{c}+2\,b\,\beta\,\chi-i\,m\,\beta\gamma\,+\beta{{}^{2}}\,\dot{\bar{c}}+2\,\bar{\beta}\,\beta{{}^{2}}\,\chi,$
(9)

where we have used the following explicit relationships

$\displaystyle Q_{ab}$ $\displaystyle=$
$\displaystyle(s_{ab}\,x_{\mu})\,\frac{\partial\,L_{\bar{b}}}{\partial\,{\dot{x}}_{\mu}}+(s_{ab}\,\psi_{\mu})\,\frac{\partial\,L_{\bar{b}}}{\partial\,{\dot{\psi}}_{\mu}}+(s_{ab}\,e)\,\frac{\partial\,L_{\bar{b}}}{\partial\,{\dot{e}}}+(s_{ab}\,c)\,\frac{\partial\,L_{\bar{b}}}{\partial\,{\dot{c}}}+(s_{ab}\,\bar{c})\,\frac{\partial\,L_{\bar{b}}}{\partial\,\dot{\bar{c}}}$
(10) $\displaystyle+$
$\displaystyle(s_{ab}\,\beta)\,\frac{\partial\,L_{\bar{b}}}{\partial\,{\dot{\beta}}}+(s_{ab}\,\psi_{5})\,\frac{\partial\,L_{\bar{b}}}{\partial\,\dot{\psi_{5}}}-X,$
$\displaystyle Q_{b}$ $\displaystyle=$
$\displaystyle(s_{b}\,x_{\mu})\,\frac{\partial\,L_{b}}{\partial\,{\dot{x}}_{\mu}}+(s_{b}\,\psi_{\mu})\,\frac{\partial\,L_{b}}{\partial\,{\dot{\psi}}_{\mu}}+(s_{b}\,e)\,\frac{\partial\,L_{b}}{\partial\,{\dot{e}}}+(s_{b}\,c)\,\frac{\partial\,L_{b}}{\partial\,{\dot{c}}}+(s_{b}\,\bar{c})\,\frac{\partial\,L_{b}}{\partial\,\dot{\bar{c}}}$
(11) $\displaystyle+$
$\displaystyle(s_{b}\,\bar{\beta})\,\frac{\partial\,L_{b}}{\partial\,{\dot{\bar{\beta}}}}+(s_{b}\,\psi_{5})\,\frac{\partial\,L_{b}}{\partial\,\dot{\psi_{5}}}-Y,$

for the derivations of conserved (anti-)BRST charges $Q_{(a)b}$.

The following points are pertinent as far as the equations (10) and (11) are
concerned. First, even though some of the auxiliary variables (e.g.
$b,\,\bar{b},\,\chi$) transform [cf. Eqs. (4), (5)] under the (anti-)BRST
symmetry transformations, we have not used their contributions because their
“time” derivative is not present in the Lagrangians $L_{b}$ and $L_{\bar{b}}$.
Second, we have not taken into account the contributions of $\beta$ and
$\bar{\beta}$ in (11) and (10), respectively, because $s_{ab}\,\bar{\beta}=0$
and $s_{b}\,\bar{\beta}=0$. Third, the expressions X and Y are the quantities
that are present in the square brackets of (6) and (7). Fourth, the direct
applications of the EL-EoMs ensure that $Q_{(a)b}$ are conserved quantities
[13]. Finally, the conserved charges $Q_{(a)b}$ are the generators for the
continuous (anti-)BRST symmetry transformations (4) and (5) because it can be
checked that the following is true, namely;

$\displaystyle s_{r}\,\Phi=-\,i\,[\Phi,Q_{r}]_{\pm},\qquad r=b,\,ab,$ (12)

where the $(\pm)$ signs (as the subscript) on the square bracket, on the
r.h.s., denote that the square bracket is an (anti)commutator for the given
variable $\Phi$ being fermionic/bosonic in nature. Here $\Phi$ denotes the
generic variable of $L_{b}$ and $L_{\bar{b}}$. In other words, we have:
$\Phi=x_{\mu},\,\phi_{\mu},\,\psi_{\mu},\,\psi_{5},\,b,\,\bar{b},\,e,\,\chi,\,c,\,\bar{c},\,\beta,\,\bar{\beta},\,\gamma$.
Using the principle behind the continuous symmetries and their corresponding
generators [cf. Eq. (12)], we have the following

$\displaystyle s_{b}\,Q_{b}=-\,i\,\\{Q_{b},\,Q_{b}\\},\qquad
s_{ab}\,Q_{ab}=-\,i\,\\{Q_{ab},\,Q_{ab}\\},$ (13)

where the l.h.s. of both the entries can be directly computed by using the
(anti-)BRST symmetry transformations [cf. Eqs. (4), (5)] and the explicit
expressions for the Noether conserved (anti-)BRST charges (8) and (9). It is
interesting to point out that the explicit computations of $s_{b}\,Q_{b}$ and
$s_{ab}\,Q_{ab}$ are as follows:

$\displaystyle
s_{b}\,Q_{b}=i\,\beta^{2}\,\Big{[}\dot{b}+\frac{1}{2}\,(p^{2}-m^{2})+2\,\gamma\,\chi+2\,\bar{\beta}\dot{\beta}\Big{]}\equiv-\,i\,\\{Q_{b},\,Q_{b}\\},$
$\displaystyle
s_{ab}\,Q_{ab}=i\,\bar{\beta}^{2}\,\Big{[}\frac{1}{2}\,(p^{2}-m^{2})-\,\dot{\bar{b}}+2\,\gamma\,\chi-2\,\beta\,\dot{\bar{\beta}}\Big{]}\equiv-\,i\,\\{Q_{ab},\,Q_{ab}\\}.$
(14)

It is obvious, from the above expressions, that the (anti-)BRST charges
$Q_{(a)b}$ are not off-shell nilpotent. However, these non-nilpotent conserved
charges can be made off-shell nilpotent if $(i)$ we use the EL-EoMs w.r.t. the
“gauge” and “supergauge” variables $e\,(\tau)$ and $\chi\,(\tau)$,
respectively, which are, in some sense, superpartners of each-other, $(ii)$ we
apply the principle that the off-shell nilpotent continuous (anti-)BRST
transformations are generated by the conserved (anti-)BRST charges, and
$(iii)$ we apply the (anti-)BRST symmetry transformations at appropriate
places.

We propose here a systematic method to obtain the off-shell nilpotent version
$Q_{b}^{(1)}$ of the BRST charge $Q_{b}$. Our aim would be to obtain
$Q_{b}^{(1)}$ from the non-nilpotent Noether BRST charge from $Q_{b}$ such
that $s_{b}\,Q_{b}^{(1)}=-\,i\,\\{Q_{b}^{(1)},\,Q_{b}^{(1)}\\}=0$. In other
words, the l.h.s. (i.e. $s_{b}\,Q_{b}^{(1)}$) should be precisely equal to
zero. Towards this goal in mind, first of all, we focus on the EL-EoMs that
emerge out from $L_{b}$ w.r.t. $e\,(\tau)$ and $\chi\,(\tau)$ which are the
“gauge” and “supergauge” variables. These, in their useful form, are as
follows:

$\displaystyle\frac{1}{2}\,(p^{2}-m^{2})=-\,\dot{b}-2\,(\bar{\beta}\,\dot{\beta}+\gamma\,\chi),$
$\displaystyle(p_{\mu}\,\psi^{\mu}-\,m\,\psi_{5})=-\,i\,m\,\gamma+2\,i\,e\,\gamma-2\,(\beta\,\dot{\bar{c}}-\,\bar{\beta}\,\dot{c}).$
(15)

In the second step, the substitutions of EL-EoMs w.r.t. “gauge” and
“supergauge” variables in the appropriate terms of Noether conserved BRST
charge $Q_{b}$. For instance, the substitutions of (15) lead to the
modifications of the following terms:

$\displaystyle\frac{c}{2}\,(p^{2}-m^{2})=-\,\dot{b}\,c-2\,c\,(\bar{\beta}\,\dot{\beta}+\gamma\,\chi),$
$\displaystyle\beta\,(p_{\mu}\,\psi^{\mu}-\,m\,\psi_{5})=-\,i\,m\,\beta\,\gamma+2\,i\,e\,\beta\,\gamma-2\,\beta\,(\beta\,\dot{\bar{c}}-\,\bar{\beta}\,\dot{c}).$
(16)

In the third step, we observe whether the above “modified” terms add, subtract
and/or cancel out with some of the terms of $Q_{b}$. For instance, we note
that, in our present case, only the term which is added in (16) is
“$-\,i\,m\,\beta\,\gamma$” from $Q_{b}$. The total sum of the expressions in
(16) and this term is the following explicit expression:

$\displaystyle-\,\dot{b}\,c-2\,c\,(\bar{\beta}\,\dot{\beta}+\gamma\,\chi)-\,2\,i\,m\,\beta\,\gamma+2\,i\,e\,\beta\,\gamma-\,2\,\beta\,(\beta\,\dot{\bar{c}}-\bar{\beta}\dot{c}).$
(17)

In the fourth step, we apply the BRST transformations on (17) which yields:

$\displaystyle-\,i\,\dot{b}\,\beta^{2}+2\,i\,\beta^{2}\,\chi\,\gamma-2\,i\,\beta^{2}\,\dot{\beta}\,\bar{\beta}.$
(18)

In our fifth step, we keenly observe whether some of terms of the Noether
conserved charge $Q_{b}$ should be modified so that the terms of (18) cancel
out precisely when we apply the BRST symmetry transformations on them. In our
present case, we observe luckily that

$\displaystyle
s_{b}\,[\beta^{2}\,\dot{\bar{c}}+2\,\beta^{2}\,\bar{\beta}\,\chi]=i\,\dot{b}\,\beta^{2}+2\,i\,\beta^{2}\,\gamma\,\chi+2\,i\,\beta^{2}\,\dot{\beta}\,\bar{\beta},$
(19)

which cancels out whatever we have obtained in (18). In the final step, we
apply the BRST symmetry transformations on the left-over terms of the Noether
conserved charge $Q_{b}$. It turns out that we have the following

$\displaystyle
s_{b}\,[b\,\dot{c}+2\,b\,\beta\,\chi]=-\,2\,i\,b\,\beta\,\dot{\beta}+2\,i\,b\,\beta\,\dot{\beta}=0.$
(20)

It is pertinent to point out that all the terms that cancel out due to the
application of the BRST symmetry transformations should be present in the off-
shell nilpotent version of the (anti-)BRST charges $Q_{(a)b}^{(1)}$. For
instance, all the terms of (17) and the terms, on the l.h.s. of (19) and (20)
in the square bracket, will be present¶¶¶This is due to the fact that, as
pointed out earlier, we wish to obtain $Q_{b}^{(1)}$ such that
$s_{b}\,Q_{b}^{(1)}=0$. in $Q_{b}^{(1)}$. Ultimately, in our present case, we
obtain the following off-shell nilpotent version of the BRST charge:

$\displaystyle Q_{b}\longrightarrow
Q_{b}^{(1)}=b\,\dot{c}-\dot{b}\,c+2\,\beta\,[i\,e\,\gamma+\bar{\beta}\,\dot{c}-i\,m\,\gamma+b\,\chi+\beta\,\bar{\beta}\,\chi]-\,\beta^{2}\dot{\bar{c}}-2\,c\,[\bar{\beta}\,\dot{\beta}+\gamma\,\chi].$
(21)

At this stage, it is straightforward to note that the following observation is
true, namely;

$\displaystyle
s_{b}\,Q_{b}^{(1)}=-\,i\,\\{Q_{b}^{(1)},\,Q_{b}^{(1)}\\},\quad\Longrightarrow\quad[Q_{b}^{(1)}]^{2}=0.$
(22)

In other words, we point out that the off-shell nilpotent version of the
conserved BRST charge [$Q_{b}^{(1)}$] is obtained from the non-nilpotent
Noether conserved charge by using the EL-EoMs w.r.t. the “gauge” variable
$e\,(\tau)$ and “supergauge” variable $\chi\,(\tau)$ and the application of
the (anti-)BRST symmetry transformations at appropriate places.

Against the backdrop of the above paragraph, we note that, to obtain the off-
shell nilpotent version of the conserved anti-BRST charge, we use the
following EL-EoMs

$\displaystyle\frac{1}{2}\,(p^{2}-m^{2})=\dot{\bar{b}}+2\,(\dot{\bar{\beta}}\,\beta-\gamma\,\chi),$
$\displaystyle(p_{\mu}\,\psi^{\mu}-\,m\,\psi_{5})=2\,i\,e\,\gamma-\,i\,m\,\gamma-2\,(\beta\,\dot{\bar{c}}-\,\bar{\beta}\,\dot{c}).$
(23)

that are derived from the perfectly anti-BRST invariant $L_{\bar{b}}$. We
follow exactly the same steps as in the case of BRST charge $Q_{b}$ to obtain
the off-shell nilpotent version of the anti-BRST $Q_{ab}^{(1)}$. In fact, the
substitution of (23) into the expression for $Q_{ab}$ [cf. Eq. (8)] at
appropriate places leads to the following expression for $Q_{ab}^{(1)}$,
namely;

$\displaystyle Q_{ab}\longrightarrow Q_{ab}^{(1)}$ $\displaystyle=$
$\displaystyle\dot{\bar{b}}\,\bar{c}-\bar{b}\,\dot{\bar{c}}+2\,\bar{\beta}\,[i\,e\,\gamma-\beta\,\dot{\bar{c}}-\bar{b}\,\chi-i\,m\,\gamma-\beta\,\bar{\beta}\,\chi]$
(24) $\displaystyle+$ $\displaystyle
2\,\bar{c}\,(\bar{\beta}\,\dot{\beta}+\gamma\,\chi)+\bar{\beta}^{2}\,\dot{c}.$

It is now straightforward to note that we have the following:

$\displaystyle
s_{ab}\,Q_{ab}^{(1)}=-\,i\,\\{Q_{ab}^{(1)},\,Q_{ab}^{(1)}\\},\quad\Longrightarrow\quad[Q_{ab}^{(1)}]^{2}=0.$
(25)

The above observation is nothing but the proof of the off-shell nilpotency of
the anti-BRST charge $Q_{ab}^{(1)}$ where the l.h.s. is computed explicitly by
using (4) and (24).

We end this section with the following remarks. First of all, we note that it
is the EL-EoMs w.r.t. the “gauge” and “supergauge” variables that have been
used and these have been singled out from the rest of the EL-EoM. This
observation is one of the key ingredients of our proposal followed by the
steps that have been discussed from Eq. (15) to Eq. (20). Second, in our
present simple case of a 1D spinning relativistic particle, only a single step
is good enough to enable us to obtain an off-shell nilpotent set of
(anti-)BRST conserved charges. However, we shall see that, in the context of
Abelian 2-form and 3-form gauge theories defined in any arbitrary dimension of
spacetime, more steps will be required to obtain the off-shell nilpotent set
of conserved (anti-)BRST charges from the non-nilpotent forms of the
(anti)BRST charges (that are derived directly from the applications of
Noether’s theorem). Third, it is interesting to mention that, in the limiting
case of the spinning relativistic particle when $\beta=\bar{\beta}=\gamma=0$,
we obtain the (anti-)BRST charges for the scalar relativistic particle from
(8) and (9) as

$\displaystyle Q_{ab}\longrightarrow
Q_{ab}^{(sr)}=\frac{\bar{c}}{2}\,(p^{2}-m^{2})-\bar{b}\,\dot{\bar{c}},\qquad
Q_{b}\longrightarrow Q_{b}^{(sr)}=\frac{c}{2}\,(p^{2}-m^{2})+b\,\dot{c},$ (26)

which are off-shell nilpotent of order two because $s_{b}\,Q_{b}^{(sr)}=0$ and
$s_{ab}\,Q_{ab}^{(sr)}=0$ due to: $s_{b}\,c=0,\,s_{b}\,b=0,\,s_{b}\,p_{\mu}=0$
and $s_{ab}\,\bar{c}=0,\,s_{ab}\,\bar{b}=0,\,s_{ab}\,p_{\mu}=0$ which are true
for the scalar relativistic particle. In the above equation, the superscript
$(sr)$ on the charges stand for the conserved and off-shell nilpotent
(anti-)BRST charges for the scalar relativistic particle. Finally, it can be
explicitly checked that the modified versions of the (anti-)BRST charges
$Q_{(a)b}^{(1)}$ are also conserved quantities if we use the proper EL-EoMs
that are derived from the coupled (but equivalent) Lagrangians $L_{b}$ and
$L_{\bar{b}}$ of our 1D massive SUSY gauge theory.

##  3 (Anti-)BRST Charges and Nilpotency: Arbitrary Dimensional non-Abelian
1-Form Gauge Theory

In this section, we show that the Noether conserved (anti-)BRST charges for
the D-dimensional non-Abelian 1-form gauge theory are non-nilpotent. However,
following our proposal, we can obtain the appropriate forms of the conserved
and off-shell nilpotent expressions for the (anti-)BRST charges for our non-
Abelian theory (without any interactions with the matter fields). We begin
with the following coupled (but equivalent) Lagrangian densities (see, e.g.
[9]) in the Curci-Ferrari gauge (see, e.g. [15, 16])

$\displaystyle{\cal L}_{B}=-\,\frac{1}{4}\,F^{\mu\nu}\cdot
F_{\mu\nu}+B\,\cdot(\partial_{\mu}\,A^{\mu})+\frac{1}{2}\,(B\cdot
B+\bar{B}\cdot\bar{B})-\,i\,\partial_{\mu}\bar{C}\cdot D^{\mu}\,C,$
$\displaystyle{\cal L}_{\bar{B}}=-\,\frac{1}{4}\,F^{\mu\nu}\cdot
F_{\mu\nu}-\bar{B}\,\cdot(\partial_{\mu}\,A^{\mu})+\frac{1}{2}\,(B\cdot
B+\bar{B}\cdot\bar{B})-\,i\,D^{\mu}\,\bar{C}\cdot\partial_{\mu}C,$ (27)

where the field strength tensor $F_{\mu\nu}\equiv
F_{\mu\nu}^{a}\,T^{a}\,(a=1,2,...,N^{2}-1)$ has been derived from the non-
Abelian 2-form: $F^{(2)}=d\,A^{(1)}+i\,A^{(1)}\wedge A^{(1)}$ where the 1-form
$A^{(1)}=dx^{\mu}\,A_{\mu}\equiv d\,x^{\mu}\,A_{\mu}^{q}\,T^{a}$ defines the
non-Abelian gauge field ($A_{\mu}^{a}$) so that we have:
$F_{\mu\nu}^{a}=\partial_{\mu}\,A_{\nu}^{a}-\partial_{\nu}\,A_{\mu}^{a}+i\,f^{abc}\,A_{\mu}^{b}\,A_{\nu}^{c}$.
For the $SU(N)$ Lie algebraic space, we have the Lie algebra:
$[T^{a},\,T^{b}]=f^{abc}\,T^{c}$ that is satisfied by the $SU(N)$ generators
$T^{a}\,(a=1,2,...,N^{2}-1)$ where $f^{abc}$ are the structure constants that
can be chosen [17] to be totally antisymmetric in all the indices for the
semi-simple Lie group $SU(N)$. In this section, we adopt the dot and cross
products only in the $SU(N)$ Lie algebraic space where we have: $S\cdot
T=S^{a}\,T^{a},\;(S\times T)^{a}=f^{abc}\,S^{b}\,T^{c}$ for the two non-null
vectors $S^{a}$ and $T^{a}$ (in this space and $a,b,c,...=1,2,...N^{2}-1$). We
have also taken into account the summation convention where the repeated
indices are summed over and $D_{\mu}C=\partial_{\mu}C+i\,(A_{\mu}\times C)$
and $D_{\mu}\bar{C}=\partial_{\mu}\bar{C}+i\,(A_{\mu}\times\bar{C})$ are the
covariant derivatives in the adjoint representation of the $SU(N)$ Lie
algebra.

The above coupled (but equivalent) Lagrangian densities (27) respect the
following off-shell nilpotent [$s_{(a)b}^{2}=0$] (anti-)BRST symmetry
transformations [$s_{(a)b}$]

$\displaystyle s_{ab}\;A_{\mu}=D_{\mu}\bar{C},\qquad
s_{ab}\;\bar{C}=-\frac{i}{2}\,(\bar{C}\times\bar{C}),\qquad
s_{ab}\;C=i{\bar{B}},\qquad s_{ab}\;\bar{B}=0,$ $\displaystyle
s_{ab}\;F_{\mu\nu}=i\,(F_{\mu\nu}\times\bar{C}),\qquad
s_{ab}(\partial_{\mu}A^{\mu})=\partial_{\mu}D^{\mu}\bar{C},\qquad
s_{ab}\;B=i\,(B\times\bar{C}),$ $\displaystyle s_{b}\;A_{\mu}=D_{\mu}C,\qquad
s_{b}\;C=-\frac{i}{2}\;(C\times C),\qquad s_{b}\;\bar{C}\;=i\,B,\;\qquad
s_{b}\;B=0,$ $\displaystyle s_{b}\;\bar{B}=i\,(\bar{B}\times C),\qquad
s_{b}\;(\partial_{\mu}A^{\mu})=\partial_{\mu}D^{\mu}C,\qquad
s_{b}\;F_{\mu\nu}=i\,(F_{\mu\nu}\times C),$ (28)

because of the following observations

$\displaystyle s_{b}{\cal L}_{B}$ $\displaystyle=$
$\displaystyle\partial_{\mu}(B\cdot D^{\mu}C),\quad s_{ab}{\cal
L}_{\bar{B}}=-\;\partial_{\mu}{(\bar{B}\cdot D^{\mu}\bar{C})},$ $\displaystyle
s_{b}{\cal L}_{\bar{B}}\;$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,[{\\{B+(C\times\bar{C})\\}}\cdot\partial^{\mu}C\,]-{\\{B+\bar{B}+(C\times\bar{C})\\}}\cdot
D_{\mu}\partial^{\mu}C,$ $\displaystyle s_{ab}{\cal L}_{B}$ $\displaystyle=$
$\displaystyle-\;\partial_{\mu}\,[{\\{\bar{B}+(C\times\bar{C})\\}\cdot\partial^{\mu}\bar{C}}\,]+\;\\{(B+\bar{B}+(C\times{\bar{C}})\\}\cdot
D_{\mu}\partial^{\mu}\bar{C},$ (29)

which establish that the action integrals $S_{1}=\int d^{D}x\,{\cal
L}_{B},S_{2}=\int d^{D}x\,{\cal L}_{\bar{B}}$ remain invariant
($s_{b}\,S_{1}=0,\;s_{ab}\,S_{2}=0$) under the BRST and anti-BRST symmetry
transformations, respectively, because the physical fields vanish-off as
$x\longrightarrow\pm\infty$ due to Gauss’s divergence theorem. If we confine
our whole discussion on the submanifold of the total quantum Hilbert space of
fields where the CF-condition: $B+\bar{B}+(C\times\bar{C})=0$ is respected
[18], we note that both the Lagrangian densities respect both the nilpotent
symmetries. In other words, we have: $s_{b}{\cal
L}_{\bar{B}}=-\;\partial_{\mu}{[\bar{B}\cdot\partial^{\mu}C]},\;s_{ab}{\cal
L}_{B}=\partial_{\mu}[B\cdot\partial^{\mu}\bar{C}]$ on the above submanifold
of Hilbert space of quantum fields. We christen the transformations:
$s_{b}{\cal L}_{B}=\partial_{\mu}[B\cdot D^{\mu}C],\;s_{ab}{\cal
L}_{\bar{B}}=-\;\partial_{\mu}{[\bar{B}\cdot D^{\mu}\bar{C}]}$ as perfect
symmetry transformations because we do not use any EL-EoMs and/or CF-type
condition for their proof.

In addition to the equivalence of the coupled Lagrangian densities (27) from
the point of view of the (anti-)BRST symmetry considerations, we note that the
absolute anticommutativity (i.e. $\\{s_{b},\,s_{ab}\\}=0$) property of the
(anti-)BRST symmetry transformations is satisfied if and only if we invoke the
sanctity of the CF-condition: $B+\bar{B}+(C\times\bar{C})=0$. This becomes
obvious when we observe that the following are true, namely;

$\displaystyle\\{s_{b},s_{ab}\\}\,A_{\mu}$ $\displaystyle=$ $\displaystyle
i\,D_{\mu}\,\big{[}B+\bar{B}+(C\times\bar{C})\big{]},$
$\displaystyle\\{s_{b},s_{ab}\\}\,F_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\,F_{\mu\nu}\times\big{[}B+\bar{B}+(C\times\bar{C})\big{]}.$
(30)

Thus, it is clear that the absolute anticommutativity properties:
$\\{s_{b},s_{ab}\\}\,A_{\mu}=0$ and $\\{s_{b},s_{ab}\\}\,F_{\mu\nu}=0$ are
true if and only if $B+\bar{B}+(C\times\bar{C})=0$. We further note that
$\\{s_{b},s_{ab}\\}\,\Phi=0$ where $\Phi=C,\,\bar{C},\,B,\,\bar{B}$ is the
generic field of the theory (besides $A_{\mu}$ and $F_{\mu\nu}$). Thus, the
absolute anticommutativity property (i.e. $\\{s_{b},s_{ab}\\}\,\Phi=0$) is
automatically satisfied for the fields: $B,\,\bar{B},\,C,\,\bar{C}$ due to the
off-shell nilpotent (anti-)BRST symmetry transformations (28). It is very
interesting to point out that the straightforward equivalence $({\cal
L}_{B}={\cal L}_{\bar{B}})$ of both the Lagrangian densities (27) of our
theory leads to

$\displaystyle(\partial_{\mu}\,A^{\mu})\cdot\big{[}B+\bar{B}+(C\times\bar{C})\big{]}=0,$
(31)

modulo a total spacetime derivative. The above observation establishes the
fact that both the Lagrangian densities ${\cal L}_{B}$ and ${\cal
L}_{\bar{B}}$ of equation (27) are coupled in the sense that the Nakanishi-
Lautrup auxiliary fields $B$ and $\bar{B}$ are not free but these specific
fields are restricted to obey $B+\bar{B}+(C\times\bar{C})=0$ (which is nothing
but the CF-condition [18]). This condition, for the $SU(N)$ non-Abelian gauge
theory, is physically sacrosanct because it is an (anti-)BRST invariant (i.e.
$s_{(a)b}\,[B+\bar{B}+(C\times\bar{C})]=0$) quantity which can be verified by
using the (anti-)BRST symmetry transformations (28).

The perfect symmetry invariance of the Lagrangian densities ${\cal L}_{B}$ and
${\cal L}_{\bar{B}}$ under the infinitesimal, continuous and off-shell
nilpotent $[s_{(a)b}^{2}=0]$ BRST and anti-BRST symmetry transformations,
respectively, leads to the derivation of the following expressions for the
conserved Noether currents

$\displaystyle J_{(B)}^{\mu}=(s_{b}\,A_{\nu})\,\frac{\partial\,{\cal
L}_{B}}{\partial\,(\partial_{\mu}\,A_{\nu})}+(s_{b}\,\bar{C})\,\frac{\partial\,{\cal
L}_{B}}{\partial\,(\partial_{\mu}\,\bar{C})}+(s_{b}\,C)\,\frac{\partial\,{\cal
L}_{B}}{\partial\,(\partial_{\mu}\,C)}-\,B\cdot D^{\mu}\,C,$ $\displaystyle
J_{(\bar{B})}^{\mu}=(s_{ab}\,A_{\nu})\,\frac{\partial\,{\cal
L}_{\bar{B}}}{\partial\,(\partial_{\mu}\,A_{\nu})}+(s_{ab}\,C)\,\frac{\partial\,{\cal
L}_{\bar{B}}}{\partial\,(\partial_{\mu}\,C)}+(s_{ab}\,\bar{C})\,\frac{\partial\,{\cal
L}_{\bar{B}}}{\partial\,(\partial_{\mu}\,\bar{C})}+\,\bar{B}\cdot
D^{\mu}\,\bar{C},$ (32)

where we have followed the convention of the left derivative w.r.t. all the
fermionic fields. The explicit form of the (anti-)BRST Noether currents are:

$\displaystyle J_{(\bar{B})}^{\mu}=-\,F^{\mu\nu}\cdot
D_{\nu}\,\bar{C}-\,\bar{B}\cdot
D^{\mu}\,\bar{C}-\,\frac{1}{2}\,(\bar{C}\times\bar{C})\cdot\partial^{\mu}\,C,$
$\displaystyle J_{(B)}^{\mu}=B\cdot D^{\mu}\,C-\,F^{\mu\nu}\cdot
D_{\nu}\,C+\frac{1}{2}\,\partial^{\mu}\,\bar{C}\cdot(C\times C).$ (33)

The conservation law $\partial_{\mu}\,J_{(r)}^{\mu}=0$ (with $r=\bar{B},B$)
can be proven by exploiting the power and potential of the EL-EoMs. For the
proof of $\partial_{\mu}\,J_{(\bar{B})}^{\mu}=0$, we have to use the following
EL-EOMs w.r.t. the gauge field $A_{\mu}$ and (anti-)ghost fields
$(\bar{C})\,C$, namely;

$\displaystyle
D_{\mu}\,F^{\mu\nu}+\partial^{\nu}\,\bar{B}+(\bar{C}\times\partial^{\nu}\,C)=0,\qquad\partial_{\mu}\,(D^{\mu}\,\bar{C})=0,\qquad
D_{\mu}\,(\partial^{\mu}\,C)=0,$ (34)

that are derived from ${\cal L}_{\bar{B}}$. In exactly similar fashion, for
the proof of $\partial_{\mu}\,J_{(B)}^{\mu}=0$, we utilize the following EL-
EoMs w.r.t. $A_{\mu},\,C$ and $\bar{C}$, namely;

$\displaystyle
D_{\mu}\,F^{\mu\nu}-\partial^{\nu}\,B-(\partial^{\nu}\,\bar{C}\times
C)=0,\qquad\partial_{\mu}\,(D^{\mu}\,C)=0,\qquad
D_{\mu}\,(\partial^{\mu}\,\bar{C})=0,$ (35)

which are derived from the Lagrangian density ${\cal L}_{B}$. Ultimately, we
claim that the Noether currents (33) are conserved and they lead to the
derivation of conserved (anti-)BRST charges for our D-dimensional non-Abelian
1-form gauge theory. Following the sacrosanct prescription of Noether theorem,
we derive the expressions for the conserved (anti-)BRST charges
($Q_{(\bar{B})B}=\int d^{D-1}x\,J_{(\bar{B})B}^{0})$ as follows:

$\displaystyle Q_{\bar{B}}=-\,\int d^{D-1}x\,\Big{[}F^{0i}\cdot
D_{i}\,\bar{C}+\bar{B}\cdot
D_{0}\,\bar{C}+\,\frac{1}{2}\,(\bar{C}\times\bar{C})\cdot\dot{C}\Big{]},$
$\displaystyle Q_{B}=\int d^{D-1}x\,\Big{[}B\cdot D_{0}\,C-\,F^{0i}\cdot
D_{i}\,C+\frac{1}{2}\,\dot{\bar{C}}\cdot(C\times C)\Big{]}.$ (36)

A few comments, at this juncture, are in order. First of all, it can be
checked that $\partial_{0}\,Q_{\bar{B}}=0$ and $\partial_{0}\,Q_{B}=0$ where
we have to use the EL-EoMs from ${\cal L}_{B}$ and ${\cal L}_{\bar{B}}$, and
(ii) the above conserved (anti-)BRST charges are the generators of all the
symmetry transformations (28) provided we use the canonical (anti-)commutators
by deriving the explicit expressions for the canonical conjugate momenta from
${\cal L}_{B}$ and ${\cal L}_{\bar{B}}$. Using the principle behind the
continuous symmetry transformations and their generators as the Noether
conserved charges, we note that the expressions (36) lead to the following

$\displaystyle s_{ab}\,Q_{\bar{B}}=-\,\int
d^{D-1}x\,\Big{[}i\,(F^{0i}\times\bar{C})\cdot
D_{i}\,\bar{C}-\,\frac{i}{2}\,(\bar{C}\times\bar{C})\cdot\dot{\bar{B}}\Big{]}\neq
0,$ $\displaystyle s_{b}\,Q_{B}=\int d^{D-1}x\,\Big{[}-\,i\,(F^{0i}\times
C)\cdot D_{i}\,C+\frac{i}{2}\,\dot{B}\cdot(C\times C)\Big{]}\neq 0,$ (37)

which are not equal to zero. In other words, we note that:
$s_{ab}\,Q_{\bar{B}}=-\,i\,\\{Q_{\bar{B}},\,Q_{\bar{B}}\\}\neq 0$ and
$s_{b}\,Q_{B}=-\,i\,\\{Q_{B},\,Q_{B}\\}\neq 0$. Hence, the expressions for the
Noether (anti-)BRST charges (36) are not off-shell nilpotent (i.e.
$Q_{\bar{B}}^{2}\neq 0,\,Q_{B}^{2}\neq 0$) of order two.

Following the proposal mentioned in the context of 1D massive spinning
relativistic particle, in the first step, we have to find out the EL-EOMs
w.r.t. the non-Abelian gauge field and substitute it in the non-nilpotent
Noether conserved charges $Q_{B}$ and $Q_{\bar{B}}$. In our present case, it
can be done only after the application of the Gauss divergence theorem so that
we have the following for the first term in $Q_{\bar{B}}$ and the second term
$Q_{B}$, namely;

$\displaystyle\int
d^{D-1}\,x\,\Big{[}-F^{0i}\cdot(\partial_{i}\,{\bar{C}}+i\,A_{i}\times{\bar{C}})\Big{]}$
$\displaystyle\equiv$ $\displaystyle+\int
d^{D-1}\,x\,(\partial_{i}\,F^{0i})\cdot{\bar{C}}$ (38) $\displaystyle-$
$\displaystyle i\,\int
d^{D-1}\,x\,\Big{[}F^{0i}\cdot(A_{i}\times{\bar{C}})\Big{]},$
$\displaystyle\int
d^{D-1}\,x\,\Big{[}-F^{0i}\cdot(\partial_{i}\,{C}+i\,A_{i}\times{C})\Big{]}$
$\displaystyle=$ $\displaystyle+\int
d^{D-1}\,x\,(\partial_{i}\,F^{0i})\cdot{C}$ (39) $\displaystyle-$
$\displaystyle i\,\int d^{D-1}\,x\,\Big{[}F^{0i}\cdot(A_{i}\times{C})\Big{]},$

where we can use the following EL-EoM w.r.t the gauge field, namely;

$\displaystyle(\partial_{i}\,F^{0i})\cdot{\bar{C}}$ $\displaystyle=$
$\displaystyle\dot{\bar{B}}\cdot{\bar{C}}+({\bar{C}}\times{\dot{C}})\cdot{\bar{C}}-i\,(A_{i}\times
F^{0i})\cdot{\bar{C}},$ $\displaystyle(\partial_{i}\,F^{0i})\cdot{C}$
$\displaystyle=$
$\displaystyle-{\dot{B}}\cdot{C}+(\dot{\bar{C}}\times{C})\cdot{C}-i\,(A_{i}\times
F^{0i})\cdot{C}.$ (40)

In the second step, we have to find out if there are addition, subtraction
and/or cancellations with the rest of the terms of the conserved Noether
charges $Q_{\bar{B}}$ and $Q_{B}$ in (36). At this stage, taking the help of:
$+({\bar{C}}\times{\dot{C}})\cdot{\bar{C}}=+({\bar{C}}\times{\bar{C}})\cdot{\dot{C}}$
and $-(\dot{\bar{C}}\times{C})\cdot{C}=-\dot{\bar{C}}\cdot({C}\times{C})$, we
have the following expressions for $Q_{(\bar{B})B}^{(1)}$ from the expressions
for $Q_{\bar{B}}$ and $Q_{B}$, namely:

$\displaystyle Q_{\bar{B}}\rightarrow Q_{\bar{B}}^{(1)}$ $\displaystyle=$
$\displaystyle\int
d^{D-1}\,x\,\Big{[}\dot{\bar{B}}\cdot{\bar{C}}-{\bar{B}}\cdot
D_{0}\,{\bar{C}}+\frac{1}{2}\,({\bar{C}}\times{\bar{C}})\cdot{\dot{C}}\Big{]},$
$\displaystyle Q_{B}\rightarrow Q_{B}^{(1)}$ $\displaystyle=$
$\displaystyle\int d^{D-1}\,x\,\Big{[}{B}\cdot
D_{0}\,{C}-\dot{B}\cdot{C}-\frac{1}{2}\,\dot{\bar{C}}\cdot({C}\times{C})\Big{]},$
(41)

where ($i$) a cancellation has taken place between $\big{[}-i\,(A_{i}\times
F^{0i})\cdot{\bar{C}}\big{]}$ and
$\big{[}-i\,F^{0i}\cdot(A_{i}\times{\bar{C}})\big{]}$ in the expression for
$Q_{\bar{B}}$, ($ii$) in the expression for $Q_{B}$, there has been a
cancellation between $\big{[}-i\,(A_{i}\times F^{0i})\cdot{C}\big{]}$ and
$\big{[}-i\,F^{0i}\cdot(A_{i}\times{C})\big{]}$, ($iii$) the pure ghost terms
have been added to yield
$\big{[}+\frac{1}{2}\,(\bar{C}\times\bar{C})\cdot\dot{C}\big{]}$ in
$Q_{\bar{B}}$ and $-\frac{1}{2}\,\dot{\bar{C}}\cdot(C\times C)$ in the
expression for $Q_{B}$, and ($iv$) the final contributions, after the
applications of the Gauss divergence theorem [cf. Eqs. (38), (39)] and the
equation of motion (40), are as follows

$\displaystyle\dot{\bar{B}}\cdot\bar{C}+\frac{1}{2}\,(\bar{C}\times\bar{C})\cdot\dot{C},\qquad\qquad-\dot{B}\cdot
C-\frac{1}{2}\,\dot{\bar{C}}\cdot(C\times C),$ (42)

in $Q_{\bar{B}}$ and $Q_{B}$, respectively. In the third step, we have to
apply the anti-BRST and BRST symmetry transformations on (42), respectively.
It turns out that we have the following:

$\displaystyle
s_{ab}\Big{[}\dot{\bar{B}}\cdot\bar{C}+\frac{1}{2}\,(\bar{C}\times\bar{C})\cdot\dot{C}\Big{]}=0,\qquad\qquad
s_{b}\Big{[}-\dot{B}\cdot C-\frac{1}{2}\,\dot{\bar{C}}\cdot(C\times
C)\Big{]}=0.$ (43)

Thus, our all the relevant steps terminate here and we have the final
expression for the (anti-)BRST charges as quoted in (41) because it is
elementary to check that the left-over terms of (36) are (anti-)BRST
invariant: $s_{ab}\,(\bar{B}\,\cdot D_{0}\,\bar{C}\,)=0,\;s_{b}\,(B\,\cdot
D_{0}\,C\,)=0$. It is straightforward now to observe that the above
expressions for the conserved (anti-)BRST charges are off-shell nilpotent
[$(Q_{(\bar{B})B}^{(1)})^{2}=0$] of order two. To corroborate this statement,
we have the following observations related with the modified version of
$Q_{(\bar{B})B}^{(1)}$

$\displaystyle
s_{ab}\,Q_{\bar{B}}^{(1)}=-\,i\,\\{Q_{\bar{B}}^{(1)},\,Q_{\bar{B}}^{(1)}\\}=0\quad\Longrightarrow\quad[Q_{\bar{B}}^{(1)}]^{2}=0,$
$\displaystyle
s_{b}\,Q_{B}^{(1)}=-\,i\,\\{Q_{B}^{(1)},\,Q_{B}^{(1)}\\}=0\quad\Longrightarrow\quad[Q_{B}^{(1)}]^{2}=0,$
(44)

where the l.h.s. of the above equation can be computed explicitly by taking
the help of (28) and (34). Thus, the (anti-)BRST charges
$Q_{(\bar{B})B}^{(1)}$ are off-shell nilpotent.

We end this section with the following useful remarks. First, we observe that
the Noether conserved (anti-)BRST charges $Q_{\bar{B}}$ and $Q_{B}$ are not
off-shell nilpotent of order two as was the case with the massive spinning
(i.e. SUSY) relativistic particle. Second, to obtain the conserved and off-
shell nilpotent expressions for the (anti-)BRST charges $Q_{\bar{B}}^{(1)}$
and $Q_{B}^{(1)}$, we have, first of all, taken the help of Gauss’s divergence
theorem and, then, used only the equation of motion for the gauge field from
the coupled Lagrangian densities ${\cal L}_{\bar{B}}$ and ${\cal L}_{B}$. This
observation is similar to our observation in the context of the 1D massive
spinning relativistic particle (modulo Gauss’s divergence theorem). Finally,
we observe that the Abelian limit (i.e. $f^{abc}=0$ plus no dot and/or cross
products plus no covariant derivative, etc.) of the Noether conserved charges
$Q_{\bar{B}}$ and $Q_{B}$ from (31) are:

$\displaystyle Q_{\bar{B}}\longrightarrow Q_{ab}=-\,\int
d^{D-1}x\,\Big{[}F^{0i}\,\partial_{i}\,\bar{C}-B\,\dot{\bar{C}}\Big{]},$
$\displaystyle Q_{B}\longrightarrow Q_{b}=\int
d^{D-1}x\,\Big{[}F^{0i}\,\partial_{i}\,C-B\,\dot{C}\Big{]},$ (45)

where $Q_{(a)b}$ are the (anti-)BRST charges for the free Abelian 1-form gauge
theory. It is elementary exercise to note that the above expressions for the
charges are off-shell nilpotent
($s_{b}\,Q_{b}=-\,i\,\\{Q_{b},\,Q_{b}\\}=0,\;s_{ab}\,Q_{ab}=-\,i\,\\{Q_{ab},\,Q_{ab}\\}=0$)
where we have to apply the analogues of the (anti-)BRST transformations (28)
on $Q_{(a)b}$ for the Abelian 1-form theory theory which are:
$s_{b}\,F_{0i}=0,\,s_{b}\,C=0,\,s_{b}\,B=0$ and
$s_{ab}\,F_{0i}=0,\,s_{ab}\,\bar{C}=0,\,s_{ab}\,\bar{B}=0$.

##  4 (Anti-)BRST Charges and Nilpotency: Arbitrary Dimensional Abelian 2-Form
Gauge Theory

We begin with the (anti-)BRST invariant coupled (but equivalent) Lagrangian
densities for the free D-dimensional Abelian 2-form gauge theory as follows
(see, e.g. [8, 19] for details)

$\displaystyle{\cal L}_{B}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\,H^{\mu\nu\kappa}\,H_{\mu\nu\kappa}+B^{\mu}\,\Bigl{(}\partial^{\nu}B_{\nu\mu}-\partial_{\mu}\phi\Bigr{)}+B\cdot
B+\partial_{\mu}{\bar{\beta}}\,\partial^{\mu}\beta$ (46) $\displaystyle+$
$\displaystyle(\partial_{\mu}{\bar{C}}_{\nu}-\partial_{\nu}{\bar{C}}_{\mu})\,(\partial^{\mu}C^{\nu})+(\partial\cdot
C-\lambda)\,\rho+(\partial\cdot{\bar{C}}+\rho)\,\lambda,$ $\displaystyle{\cal
L}_{\bar{B}}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\,H^{\mu\nu\kappa}\,H_{\mu\nu\kappa}+{\bar{B}}^{\mu}\,\Bigl{(}\partial^{\nu}\,B_{\nu\mu}+\partial_{\mu}\phi\Bigr{)}+{\bar{B}}\cdot{\bar{B}}+\partial_{\mu}\,{\bar{\beta}}\,\partial^{\mu}\beta$
(47) $\displaystyle+$
$\displaystyle(\partial_{\mu}{\bar{C}}_{\nu}-\partial_{\nu}{\bar{C}}_{\mu})\,(\partial^{\mu}C^{\nu})+(\partial\cdot
C-\lambda)\,\rho+(\partial\cdot{\bar{C}}+\rho)\,\lambda,$

where the totally antisymmetric tensor
$H_{\mu\nu\lambda}=\partial_{\mu}\,B_{\nu\lambda}+\partial_{\nu}\,B_{\lambda\mu}+\partial_{\lambda}\,B_{\mu\nu}$
is derived from the Abelian 3-form
$H^{(3)}=d\,B^{(2)}\equiv\big{[}{(d\,x^{\mu}\wedge\,d\,x^{\nu}\wedge\,d\,x^{\lambda})}/{3!}\big{]}\,H_{\mu\nu\lambda}$.
Here the Abelian 2-form
$B^{(2)}=\big{[}{(d\,x^{\mu}\wedge\,d\,x^{\nu})}/{2!}\big{]}\,B_{\mu\nu}$ is
antisymmetric $(B_{\mu\nu}=-\,B_{\nu\mu})$ tensor gauge field and
$d=d\,x^{\mu}\,\partial_{\mu}$ (with $d^{2}=0$) is the exterior derivative.
The gauge-fixing term for the gauge field has its origin in the co-exterior
derivative of differential geometry (see, e.g. [20-23]) as it is
straightforward to check that
$\delta\,B^{(2)}=-*\,d\,*\,B^{(2)}\equiv(\partial^{\nu}\,B_{\nu\mu})\,d\,x^{\mu}$
where $*$ is the Hodge duality operator on the flat D-dimesnional spectime
manifold. A derivative on a scalar field $(\phi)$ has been incorporated into
the gauge-fixing term on dimensional ground. The Nakanishi-Lautrup type
auxiliary vector fields $B_{\mu}$ and ${\bar{B}}_{\mu}$ are restricted to obey
the CF-type restriction: $B_{\mu}-{\bar{B}}_{\mu}-\partial_{\mu}\,\phi=0$
(see, e.g. [24]). Here the fermionic
($C_{\mu}^{2}=0,\,{\bar{C}}_{\mu}^{2}=0,\,C_{\mu}\,C_{\nu}+C_{\nu}\,C_{\mu}=0,\,{\bar{C}}_{\mu}\,{\bar{C}}_{\nu}+{\bar{C}}_{\nu}\,{\bar{C}}_{\mu}=0,\,C_{\mu}\,{\bar{C}}_{\nu}+{\bar{C}}_{\nu}\,C_{\mu}=0$,
etc.) vector (anti-)ghost fields $({\bar{C}}_{\mu})\,C_{\mu}$ carry the ghost
numbers $(-1)+1$, respectively, and the bosonic (anti-)ghost fields
$(\bar{\beta})\,\beta$ are endowed with the ghost numbers $(-2)+2$,
respectively. The auxiliary (anti-)ghost fields $(\rho)\,\lambda$ are
fermionic ($\rho^{2}=\lambda^{2}=0,\,\rho\,\lambda+\lambda\,\rho=0$, etc.) in
nature and they also carry the ghost numbers $(-1)+1$, respectively, due to
the fact that $\lambda=\frac{1}{2}\,(\partial\cdot C)$ and
$\rho=-\frac{1}{2}\,(\partial\cdot{\bar{C}})$. The (anti-)ghost fields are
invoked to maintain the unitarity in the theory.

The above coupled Lagrangian densities ${\cal L}_{B}$ and ${\cal L}_{\bar{B}}$
respect the following perfect off-shell nilpotent [$s_{(a)b}^{2}=0$]
(anti-)BRST symmetry transformations [$s_{(a)b}$], namely;

$\displaystyle
s_{ab}B_{\mu\nu}=-(\partial_{\mu}{\bar{C}}_{\nu}-\partial_{\nu}{\bar{C}}_{\mu}),\quad
s_{ab}{\bar{C}}_{\mu}=-\partial_{\mu}{\bar{\beta}},\qquad
s_{ab}C_{\mu}={\bar{B}}_{\mu},$ $\displaystyle s_{ab}\phi=\rho,\quad
s_{ab}\beta=-\lambda,\;\;s_{ab}B_{\mu}=\partial_{\mu}\rho,\quad
s_{ab}\bigl{[}\rho,\lambda,{\bar{\beta}},{\bar{B}}_{\mu},H_{\mu\nu\kappa}\bigr{]}=0,$
$\displaystyle
s_{b}B_{\mu\nu}=-(\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}),\qquad
s_{b}C_{\mu}=-\partial_{\mu}\beta,\qquad s_{b}\bar{C}_{\mu}=-B_{\mu},$
$\displaystyle s_{b}\phi=\lambda,\quad s_{b}\bar{\beta}=-\rho,\quad
s_{b}\bar{B}_{\mu}=-\,\partial_{\mu}\lambda,\quad
s_{b}\bigl{[}\rho,\lambda,\beta,B_{\mu},H_{\mu\nu\kappa}\bigr{]}=0,$ (48)

due to our observations that:

$\displaystyle s_{ab}{\cal
L}_{\bar{B}}=-\partial_{\mu}\Bigr{[}(\partial^{\mu}{\bar{C}}^{\nu}-\partial^{\nu}{\bar{C}}^{\mu}){\bar{B}}_{\nu}-\rho\;{\bar{B}}^{\mu}+\lambda\partial^{\mu}{\bar{\beta}}\Bigl{]},$
$\displaystyle s_{b}{\cal
L}_{B}=-\partial_{\mu}\Bigr{[}(\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu})B_{\nu}+\rho\;\partial^{\mu}\beta+\lambda
B^{\mu}\Bigl{]}.$ (49)

Thus, it is crystal clear that the action integrals $S_{1}=\int d^{D}x\,{\cal
L}_{B}$ and $S_{2}=\int d^{D}x\,{\cal L}_{\bar{B}}$ remain invariant (i.e.
$s_{b}S_{1}=0,\;s_{ab}S_{2}=0$) for the physical fields that vanish-off as
$x\longrightarrow\pm\infty$ due to Gauss’s divergence theorem. It should be
noted that the above (anti-)BRST symmetry transformations are absolutely
anticommuting only on the submanifold of the Hilbert space of quantum fields
where the CF-type restriction: $B_{\mu}-\bar{B}_{\mu}-\partial_{\mu}\phi=0$ is
satisfied. This statement can be corroborated by the following observation

$\displaystyle\\{s_{b},s_{ab}\\}\,B_{\mu\nu}=\partial_{\mu}(B_{\nu}-\bar{B}_{\nu})-\partial_{\nu}(B_{\mu}-\bar{B}_{\mu}),$
(50)

which establishes that $\\{s_{b},s_{ab}\\}\,B_{\mu\nu}=0$ if and only if we
invoke the sanctity of CF-type restriction
$B_{\mu}-\bar{B}_{\mu}-\partial_{\mu}\phi=0$. It can be checked that the
absolute anticommutativity property $\\{s_{b},s_{ab}\\}\,\Psi=0$ is satisfied
automatically if we use (48) for the generic field
$\Psi=C_{\mu},\,\bar{C}_{\mu},\,\phi,\,\lambda,\,\rho,\,\beta,\bar{\beta},\,B_{\mu},\,\bar{B}_{\mu}$
of ${\cal L}_{B}$ and ${\cal L}_{\bar{B}}$.

The above infinitesimal, continuous and off-shell nilpotent (anti-)BRST
symmetry transformations lead to the derivations of the Noether conserved
currents

$\displaystyle
J^{\mu}_{(ab)}=\rho\,{\bar{B}}^{\mu}-(\partial^{\mu}{\bar{C}}^{\nu}-\partial^{\nu}{\bar{C}}^{\mu})\,{\bar{B}}_{\nu}-(\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu})\,\partial_{\nu}{\bar{\beta}}-\lambda\;\partial^{\mu}{\bar{\beta}}-H^{\mu\nu\kappa}\;(\partial_{\nu}\bar{C}_{\kappa}),$
$\displaystyle
J^{\mu}_{(b)}=(\partial^{\mu}{\bar{C}}^{\nu}-\partial^{\nu}{\bar{C}}^{\mu})\;\partial_{\nu}\beta-(\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu})\,B_{\nu}-\lambda\;B^{\mu}-\rho\;\partial^{\mu}\beta-H^{\mu\nu\kappa}\;(\partial_{\nu}C_{\kappa}),$
(51)

where it is quite straightforward to check (see, e.g. [19] for details) that
the conservation law ($\partial_{\mu}J^{\mu}_{(r)},\;r=ab,\,b$) is true
provided we use the EL-EoMs from the coupled (but equivalent) (anti-)BRST
invariant Lagrangian densities ${\cal L}_{B}$ and ${\cal L}_{\bar{B}}$,
respectively. The conserved Noether (anti-)BRST charges
$Q_{r}=\int\,d^{D-1}\,x\,J^{0}_{(r)}\;(r=ab,\,b)$ are as follows

$\displaystyle
Q_{ab}={\displaystyle\int}d^{D-1}x\,\Bigr{[}\rho\,{\bar{B}}^{0}-(\partial^{0}{\bar{C}}^{i}-\partial^{i}{\bar{C}}^{0})\,{\bar{B}}_{i}-(\partial^{0}C^{i}-\partial^{i}C^{0})\,\partial_{i}{\bar{\beta}}-\lambda\;\partial^{0}{\bar{\beta}}-H^{0ij}\;(\partial_{i}\bar{C}_{j})\Bigl{]},$
$\displaystyle
Q_{b}={\displaystyle\int}d^{D-1}x\,\Bigr{[}(\partial^{0}{\bar{C}}^{i}-\partial^{i}{\bar{C}}^{0})\,\partial_{i}\beta-(\partial^{0}C^{i}-\partial^{i}C^{0})B_{i}-\lambda\;B^{0}-\rho\;\partial^{0}\beta-H^{0ij}\;(\partial_{i}C_{j})\Bigl{]},$
(52)

which are the generators for the infinitesimal, continuous and off-shell
nilpotent [$s_{(a)b}^{2}=0$] (anti-)BRST symmetry transformations (48). At
this stage, it is worthwhile to point out that the following observations are
true, namely;

$\displaystyle
s_{ab}\,Q_{ab}=-\,i\,\\{Q_{ab},Q_{ab}\\}={\displaystyle\int}\;d^{D-1}x\Bigr{[}-\,(\partial^{0}\bar{B}^{i}-\partial^{i}\bar{B}^{0})\,\partial_{i}\bar{\beta}\Bigl{]}\;\neq
0,$ $\displaystyle
s_{b}\,Q_{b}=-\,i\,\\{Q_{b},Q_{b}\\}={\displaystyle\int}\;d^{D-1}x\Bigr{[}-\,(\partial^{0}B^{i}-\partial^{i}B^{0})\,\partial_{i}\beta\Bigl{]}\;\neq
0,$ (53)

when we apply the principle behind the continuous symmetry transformations and
their generators as the conserved Noether charges. The above observations
establish that the Noether conserved charges $Q_{b}$ and $Q_{ab}$ are not off-
shell nilpotent (i.e. $Q_{b}^{2}\neq 0,\;Q_{ab}^{2}\neq 0$).

At this juncture, we follow our proposal to obtain the off-shell nilpotent
versions [$Q_{(a)b}^{(1)}$] of the (anti-)BRST conserved charges from the
conserved Noether (anti-)BRST charge $Q_{(a)b}$ [cf. Eq. (52)] which are found
to be non-nilpotent [cf. Eq. (53)]. Our objective is to prove the validity of
$s_{(a)b}Q_{(a)b}^{(1)}=-\,i\,\\{Q_{(a)b}^{(1)},\,Q_{(a)b}^{(1)}\\}=0$ by
computing precisely the l.h.s. (i.e. $s_{(a)b}Q_{(a)b}^{(1)}$). In the first
step, we have to use the equation of motion w.r.t. the gauge field. Towards
this goal in mind, first of all, we note that the last terms of $Q_{b}$ and
$Q_{ab}$ can be re-expressed as follows

$\displaystyle-\,\int d^{D-1}x\;H^{0ij}\,\partial_{i}C_{j}=-\,\int
d^{D-1}x\;\partial_{i}\,[H^{0ij}\,C_{j}]+\int
d^{D-1}x\;(\partial_{i}H^{0ij})\,C_{j},$ $\displaystyle-\,\int
d^{D-1}x\;H^{0ij}\,\partial_{i}\bar{C}_{j}=-\,\int
d^{D-1}x\;\partial_{i}\,[H^{0ij}\,\bar{C}_{j}]+\int
d^{D-1}x\;(\partial_{i}H^{0ij})\,\bar{C}_{j},$ (54)

where the first-term will be zero due to Gauss’s divergence theorem for the
physical fields (which vanish-off as $x\longrightarrow\mp\,\infty$) and we can
apply the first step of our proposal where the equations of motion w.r.t. the
gauge field from ${\cal L}_{B}$ and ${\cal L}_{\bar{B}}$ are as follows:

$\displaystyle\partial_{i}H^{0ij}=(\partial^{0}B^{j}-\partial^{j}B^{0}),\qquad\partial_{i}H^{0ij}=(\partial^{0}\bar{B}^{j}-\partial^{j}\bar{B}^{0}).$
(55)

Thus, the last terms of $Q_{ab}$ and $Q_{b}$ are as follows:

$\displaystyle\int d^{D-1}x\,(\partial_{i}H^{0ij})\,\bar{C}_{j}=\int
d^{D-1}x\,[(\partial^{0}\bar{B}^{i}-\partial^{i}\bar{B}^{0})\,\bar{C}_{i}],$
$\displaystyle\int d^{D-1}x\,(\partial_{i}H^{0ij})\,C_{j}=\int
d^{D-1}x\,[(\partial^{0}B^{i}-\partial^{i}B^{0})\,C_{i}].$ (56)

We remark here that the terms in (56) here emerged out from the EL-EoMs w.r.t.
the gauge field and they are sacrosanct. As a consequence, they will be
present in the off-shell nilpotent version of $Q_{(a)b}^{(1)}$. In the second
step, we apply the (anti-)BRST symmetry transformations on the above terms
(modulo integration) which lead to the following:

$\displaystyle
s_{ab}\,[(\partial^{0}\bar{B}^{i}-\partial^{i}\bar{B}^{0})\,\bar{C}_{i}]=-\,(\partial^{0}\bar{B}^{i}-\partial^{i}\bar{B}^{0})\,\partial_{i}\bar{\beta},$
$\displaystyle
s_{b}\,[(\partial^{0}B^{i}-\partial^{i}B^{0})\,C_{i}]=-\,(\partial^{0}B^{i}-\partial^{i}B^{0})\,\partial_{i}\beta.$
(57)

In the third step, we have to modify∥∥∥It will be noted that, in the cases of
the 1D massive spinning relativistic particle and D-dimensional non-Abelian
1-form gauge theory, the original term of the Noether conserved charges are
good enough to ensure that:
$s_{b}\,Q_{b}^{(1)}=0,\,s_{ab}\,Q_{ab}^{(1)}=0,\,s_{b}\,Q_{B}^{(1)}=0$ and
$s_{ab}\,Q_{\bar{B}}^{(1)}=0$.  appropriate terms of $Q_{ab}$ and $Q_{b}$ so
that (57) cancels out when we apply the (anti-)BRST symmetry transformations
on them. Towards this goal in mind, first of all, we focus on the BRST charge
$Q_{b}$ and explain clearly the third step of our proposal. It can be seen
that the first term of $Q_{b}$ can be re-written as

$\displaystyle(\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0})\,\partial_{i}\beta=2\,(\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0})\,\partial_{i}\beta-(\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0})\,\partial_{i}\beta.$
(58)

It is clear that if we apply the BRST symmetry transformations on the second
term of the above equation, it will serve our purpose and cancel out the
second entry in (57). At this stage, we comment that the second term of (58)
and (56) will be present in the off-shell nilpotent version ($Q_{b}^{(1)}$) of
the BRST charge. This is due to the fact that our central aim is to show that
$s_{b}Q_{b}^{(1)}=0$. Now we focus on the first term of (57) which can be re-
written in an appropriate form:

$\displaystyle
2\,(\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0})\;\partial_{i}\beta=\partial_{i}\,[2\;(\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0})\,\beta]-2\;\partial_{i}\,[\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0}]\,\beta.$
(59)

The above terms are inside the integration w.r.t. $d^{D-1}x$. Hence, the first
term of the above equation will vanish due to the Gauss divergence theorem.
The second term can be written, using the following equations of motion,
derived from ${\cal L}_{B}$, as:

$\displaystyle\Box\,\bar{C}_{\mu}-\partial_{\mu}\,(\partial\cdot\bar{C})-\partial_{\mu}\rho=0,$
$\displaystyle\Longrightarrow-\,2\,\partial_{i}\,[\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0}]\,\beta=2\,\beta\,\partial^{0}\rho\equiv
2\,\beta\,\dot{\rho}.$ (60)

In the fourth step, we have to apply the BRST symmetry transformations on
$(2\,\beta\,\dot{\rho})$ which turns out to be zero. We comment that this term
will be present in the off-shell nilpotent version $(Q_{b}^{(1)})$ of the BRST
charge due to our objective to show that $s_{b}\,Q_{b}^{(1)}=0$. Hence, all
the steps of our proposal terminate at this stage. As a consequences, we have
the following off-shell nilpotent version of the BRST charge:

$\displaystyle Q_{b}\longrightarrow Q_{b}^{(1)}$ $\displaystyle=$
$\displaystyle{\displaystyle\int}\;d^{D-1}x\;\Bigr{[}(\partial^{0}B^{i}-\partial^{i}B^{0})\,C_{i}-(\partial^{0}{\bar{C}}^{i}-\partial^{i}{\bar{C}}^{0})\;\partial_{i}\beta$
(61) $\displaystyle+$ $\displaystyle
2\,\beta\,\dot{\rho}-(\partial^{0}C^{i}-\partial^{i}C^{0})B_{i}-\lambda
B^{0}-\rho\,\dot{\beta}\Bigl{]},$

where we have taken into account the appropriate term from (56), the second
term from (58) and the r.h.s. of (60). It is straightforward now to check that

$\displaystyle
s_{b}\,Q_{b}^{(1)}=-\,i\,\\{Q_{b}^{(1)},\;Q_{b}^{(1)}\\}=0\quad\Longrightarrow\quad[Q_{b}^{(1)}]^{2}=0,$
(62)

which proves the off-shell nilpotency of the modified version [i.e.
$Q_{b}^{(1)}$] of the non-nilpotency Noether conserved charges $Q_{b}$. It is
worthwhile to mention that $s_{b}\,[\lambda\,B^{0}+\rho\,\dot{\beta}]=0$.
Hence, these terms [i.e. $-\,(\lambda\,B^{0}+\rho\,\dot{\beta})$] remain
intact and they are present in the expression for $Q_{b}^{(1)}$. We follow
exactly the above prescription of our proposal in the case of the non-
nilpotent Noether anti-BRST charge $Q_{ab}$ to obtain its modified off-shell
nilpotent version as

$\displaystyle Q_{ab}\longrightarrow Q_{ab}^{(1)}$ $\displaystyle=$
$\displaystyle{\displaystyle\int}\;d^{D-1}x\Bigr{[}(\partial^{0}\bar{B}^{i}-\partial^{i}\bar{B}^{0})\,\bar{C}_{i}+(\partial^{0}{C}^{i}-\partial^{i}{C}^{0})\,\partial_{i}\bar{\beta}$
(63) $\displaystyle-$
$\displaystyle(\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0})\,\bar{B}_{i}+\rho\,\bar{B}^{0}-2\,\bar{\beta}\,\dot{\lambda}-\lambda\,\dot{\bar{\beta}}\Bigl{]},$

which satisfies the following relationship:

$\displaystyle
s_{ab}\,Q_{ab}^{(1)}=-\,i\,\\{Q_{ab}^{(1)},\;Q_{ab}^{(1)}\\}=0\quad\Longrightarrow\quad[Q_{ab}^{(1)}]^{2}=0.$
(64)

The above observations establish the sanctity of our proposal that enables us
to obtain precisely the off-shell nilpotent versions of the (anti-)BRST
charges [$Q_{(a)b}^{(1)}$] from the conserved Noether (anti-)BRST charges
[$Q_{(a)b}$] which are non-nilpotent.

We end this section with the following remarks. First, in the case of the 1D
massive spinning particle, only the EL-EoMs w.r.t. the “gauge” and
“supergauge” variables were good enough to convert the non-nilpotent Noether
(anti-)BRST charges into the off-shell nilpotent versions of the (anti-)BRST
conserved charges. Second, to obtain the off-shell nilpotent versions of the
(anti-)BRST conserved charges in the context of the D-dimensional non-Abelian
1-form theory, we required Gauss’s divergence theorem plus the EL-EoMs with
respect to the gauge field. Third, in the context of the D-dimensional Abelian
2-form theory, we invoked twice the Gauss divergence theorem and the EL-EoMs
w.r.t. the gauge field and fermionic (anti-)ghost fields
$(\bar{C}_{\mu})\,C_{\mu}$ to obtain the off-shell nilpotent versions of the
(anti-)BRST charges from the conserved Noether non-nilpotent (anti-)BRST
charges. In all the above examples, we have also used the strength of the
(anti-)BRST symmetry transformations, at appropriate places, so that we could
obtain the off-shell nilpotent versions of the (anti-)BRST charges from the
Noether conserved charges which are non-nilpotent.

##  5 (Anti-)BRST Charges and Nilpotency: Arbitrary Dimensional
St${\ddot{u}}$ckelberg-Modified Massive Abelian 3-Form Gauge Theory

We begin our present section with the (anti-)BRST invariant coupled (but
equivalent) Lagrangian densities******The Lagrangian densities ${\cal L}_{B}$
and ${\cal L}_{\bar{B}}$ are coupled and equivalent due to the existence of
six (anti-)BRST invariant CF-type restrictions on our theory which have been
systematically derived using the augmented version of superfield approach to
BRST formalism in [25]. for the St${\ddot{u}}$ckelberg-modified massive
Abelian 3-form gauge theory (see, e.g. [25] for details)

$\displaystyle{\cal L}_{B}$ $\displaystyle=$ $\displaystyle{\cal
L}_{S}+(\partial_{\mu}A^{\mu\nu\lambda})B_{\nu\lambda}-\frac{1}{2}B_{\mu\nu}\,B^{\mu\nu}+\frac{1}{2}\,B^{\mu\nu}\,\Big{[}\partial_{\mu}\,\phi_{\nu}-\partial_{\nu}\,\phi_{\mu}\mp\,m\,\Phi_{\mu\nu}\Big{]}$
(65) $\displaystyle-$
$\displaystyle\,(\partial_{\mu}\,\Phi^{\mu\nu})\,B_{\nu}-\frac{1}{2}\,B^{\mu}\,B_{\mu}+\frac{1}{2}\,B^{\mu}\,\Big{[}\pm\,m\phi_{\mu}-\partial_{\mu}\,\phi\Big{]}+\frac{m^{2}}{2}\,\bar{C}_{\mu\nu}\,C^{\mu\nu}$
$\displaystyle+$
$\displaystyle(\partial_{\mu}\bar{C}_{\nu\lambda}+\partial_{\nu}\bar{C}_{\lambda\mu}+\partial_{\lambda}\bar{C}_{\mu\nu})(\partial^{\mu}C^{\nu\lambda})\pm
m\,(\partial_{\mu}\,\bar{C}^{\mu\nu})\,C_{\nu}\pm\,m\,\bar{C}^{\nu}\,(\partial^{\mu}\,C_{\mu\nu})$
$\displaystyle+$
$\displaystyle(\partial_{\mu}\,\bar{C}_{\nu}-\partial_{\nu}\,\bar{C}_{\mu})\,(\partial^{\mu}\
C^{\nu})-\frac{1}{2}\,\Big{[}\pm
m\,\bar{\beta}^{\mu}-\partial^{\mu}\,\bar{\beta}\Big{]}\,\Big{[}\pm
m\,\beta_{\mu}-\partial_{\mu}\,\beta\Big{]}$ $\displaystyle-$
$\displaystyle\,(\partial_{\mu}\,\bar{\beta}_{\nu}-\partial_{\nu}\,\bar{\beta}_{\mu})\,(\partial^{\mu}\,\beta^{\nu})-\partial_{\mu}\bar{C}_{2}\partial^{\mu}C_{2}-\,m^{2}\,\bar{C}_{2}\,C_{2}+[(\partial\cdot\bar{\beta})\mp
m\,\bar{\beta}]\,B$ $\displaystyle-$ $\displaystyle[(\partial\cdot\phi)\mp
m\,\phi]\,B_{1}-[(\partial\cdot\beta)\mp
m\,\beta]\,B_{2}+\Big{[}\partial_{\nu}\bar{C}^{\nu\mu}+\partial^{\mu}\,\bar{C}_{1}\mp\frac{m}{2}\,\bar{C}^{\mu}\Big{]}\,f_{\mu}$
$\displaystyle-$
$\displaystyle\,2\,F^{\mu}\,f_{\mu}-2\,F\,f-\,\Big{[}\partial_{\nu}C^{\nu\mu}+\partial^{\mu}\,C_{1}\mp\frac{m}{2}\,C^{\mu}\Big{]}\,F_{\mu}+\Big{[}\frac{1}{2}\,(\partial\cdot
C)\mp m\,C_{1}\Big{]}\,F$ $\displaystyle-$
$\displaystyle\Big{[}\frac{1}{2}\,(\partial\cdot\bar{C})\mp
m\,\bar{C}_{1}\Big{]}\,f-\,B\,B_{2}-\,\frac{1}{2}\,B_{1}^{2},$
$\displaystyle{\cal L}_{\bar{B}}$ $\displaystyle=$ $\displaystyle{\cal
L}_{S}-(\partial_{\mu}A^{\mu\nu\lambda})\bar{B}_{\nu\lambda}-\frac{1}{2}\bar{B}_{\mu\nu}\,\bar{B}^{\mu\nu}+\frac{1}{2}\,\bar{B}^{\mu\nu}\,\Big{[}\partial_{\mu}\,\phi_{\nu}-\partial_{\mu}\,\phi_{\nu}\pm\,m\,\Phi_{\mu\nu}\Big{]}$
$\displaystyle+$
$\displaystyle(\partial_{\mu}\,\Phi^{\mu\nu})\,\bar{B}_{\nu}-\frac{1}{2}\,\bar{B}^{\mu}\,\bar{B}_{\mu}+\frac{1}{2}\,\bar{B}^{\mu}\,\Big{[}\pm\,m\phi_{\mu}-\partial_{\mu}\,\phi\Big{]}+\frac{m^{2}}{2}\,\bar{C}_{\mu\nu}\,C^{\mu\nu}$
$\displaystyle+$
$\displaystyle(\partial_{\mu}\bar{C}_{\nu\lambda}+\partial_{\nu}\bar{C}_{\lambda\mu}+\partial_{\lambda}\bar{C}_{\mu\nu})(\partial^{\mu}C^{\nu\lambda})\pm
m\,(\partial_{\mu}\,\bar{C}^{\mu\nu})\,C_{\nu}\pm\,m\,\bar{C}^{\nu}\,(\partial^{\mu}\,C_{\mu\nu})$
$\displaystyle+$
$\displaystyle(\partial_{\mu}\,\bar{C}_{\nu}-\partial_{\nu}\,\bar{C}_{\mu})\,(\partial^{\mu}\
C^{\nu})-\frac{1}{2}\,\Big{[}\pm
m\,\bar{\beta}^{\mu}-\partial^{\mu}\,\bar{\beta}\Big{]}\,\Big{[}\pm
m\,\beta_{\mu}-\partial_{\mu}\,\beta\Big{]}$ $\displaystyle-$
$\displaystyle\,(\partial_{\mu}\,\bar{\beta}_{\nu}-\partial_{\nu}\,\bar{\beta}_{\mu})\,(\partial^{\mu}\,\beta^{\nu})-\partial_{\mu}\bar{C}_{2}\partial^{\mu}C_{2}-\,m^{2}\,\bar{C}_{2}\,C_{2}+[(\partial\cdot\bar{\beta})\mp
m\,\bar{\beta}]\,B$ (66) $\displaystyle-$
$\displaystyle[(\partial\cdot\phi)\mp m\,\phi]\,B_{1}-[(\partial\cdot\beta)\mp
m\,\beta]\,B_{2}+\Big{[}\partial_{\nu}C^{\nu\mu}-\partial^{\mu}\,C_{1}\mp\frac{m}{2}\,C^{\mu}\Big{]}\,\bar{f}_{\mu}$
$\displaystyle+$
$\displaystyle\,2\,\bar{F}^{\mu}\,\bar{f}_{\mu}+2\,\bar{F}\,\bar{f}-\,\Big{[}\partial_{\nu}\bar{C}^{\nu\mu}-\partial^{\mu}\,\bar{C}_{1}\mp\frac{m}{2}\,\bar{C}^{\mu}\Big{]}\,\bar{F}_{\mu}+\Big{[}\frac{1}{2}\,(\partial\cdot\bar{C})\pm
m\,\bar{C}_{1}\Big{]}\,\bar{F}$ $\displaystyle-$
$\displaystyle\Big{[}\frac{1}{2}\,(\partial\cdot C)\pm
m\,C_{1}\Big{]}\,\bar{f}-\,B\,B_{2}-\,\frac{1}{2}\,B_{1}^{2},$

where ${\cal L}_{S}$ is the St${\ddot{u}}$ckelberg-modified classical
Lagrangian density that incorporates into it the totally antisymmetric tensor
gauge field $(A_{\mu\nu\lambda})$ and antisymmetric
$(\Phi_{\mu\nu}=-\,\Phi_{\nu\mu})$ St${\ddot{u}}$ckelberg field
$(\Phi_{\mu\nu})$ along with their kinetic terms as follow [25]:

$\displaystyle{\cal
L}_{S}=\frac{1}{24}\,H^{\mu\nu\lambda\zeta}\,H_{\mu\nu\lambda\zeta}-\frac{m^{2}}{6}\,A^{\mu\nu\lambda}\,A_{\mu\nu\lambda}\pm\frac{m}{3}\,A^{\mu\nu\lambda}\,\Sigma_{\mu\nu\lambda}-\frac{1}{6}\,\Sigma^{\mu\nu\lambda}\,\Sigma_{\mu\nu\lambda}.$
(67)

In the above, the kinetic term with the tensor field $H_{\mu\nu\lambda\zeta}$
for the gauge field $A_{\mu\nu\lambda}$ owes its origin to the exterior
derivative $d=d\,x^{\mu}\,\partial_{\mu}\,(\mu=0,1...D-1)$ because the Abelian
4-form is:
$H^{(4)}=d\,A^{(3)}=\big{[}{(d\,x^{\mu}\wedge\,d\,x^{\nu}\wedge\,d\,x^{\lambda}\,\wedge\,d\,x^{\xi})}/{4!}\big{]}\,H_{\mu\nu\lambda\xi}$
where the Abelian 3-form totally antisymmetric gauge field is defined through:
$A^{(3)}=\big{[}{(d\,x^{\mu}\wedge\,d\,x^{\nu}\wedge\,d\,x^{\lambda})}/{3!}\big{]}\,A_{\mu\nu\lambda}$.
In addition, the Abelian 3-form
$\Sigma^{(3)}=\big{[}{(d\,x^{\mu}\wedge\,d\,x^{\nu}\wedge\,d\,x^{\lambda})}/{3!}\big{]}\,\Sigma_{\mu\nu\lambda}$
is defined through the Abelian St${\ddot{u}}$ckelberg 2-form
$\Phi^{(2)}=\big{[}{(d\,x^{\mu}\wedge\,d\,x^{\nu})}/{2!}\big{]}\,\Phi_{\mu\nu}$
as: $\Sigma^{(3)}=d\,\Phi^{(2)}$ which implies that
$\Sigma_{\mu\nu\lambda}=\partial_{\mu}\Phi_{\nu\lambda}+\partial_{\nu}\Phi_{\lambda\mu}+\partial_{\lambda}\Phi_{\mu\nu}$.
It is also evident that $H_{\mu\nu\lambda\xi}$ is equal to

$\displaystyle
H_{\mu\nu\lambda\zeta}=\partial_{\mu}\,A_{\nu\lambda\zeta}-\partial_{\nu}\,A_{\lambda\zeta\mu}+\partial_{\lambda}\,A_{\zeta\mu\nu}-\partial_{\zeta}\,A_{\mu\nu\lambda},$
(68)

which is invoked in the definition of the kinetic term for the gauge field
$A_{\mu\nu\lambda}$.

In the (anti-)BRST invariant Lagrangian densities ${\cal L}_{B}$ and ${\cal
L}_{\bar{B}}$, we have the bosonic auxiliary fields
$(B_{\mu\nu},\,{\bar{B}}_{\mu\nu},\,B_{\mu},\,{\bar{B}}_{\mu}\,B_{1},\,B_{2},\,B)$
and a fermionic set of auxiliary fields is
$(F_{\mu},\,{\bar{F}}_{\mu},\,f_{\mu},\,{\bar{f}}_{\mu},\,F,\,{\bar{F}},\,f,\,{\bar{f}})$
out of which the two bosonic auxiliary fields $(B,\,B_{2})$ carry the ghost
numbers $(-2,\,+2)$, respectively, and a set of fermionic auxiliary fields
$(F_{\mu},\,{\bar{f}}_{\mu},\,{\bar{f}},\,F)$ carry the ghost number $(-1)$.
On the other hand, a fermionic set of auxiliary fields
$({\bar{F}}_{\mu},\,f_{\mu},\,f,\,\bar{F})$ is endowed with ghost number
$(+1)$. To maintain the unitarity in the theory, we need the fermionic set of
ghost fields
$(C_{\mu\nu},\,{\bar{C}}_{\mu\nu},\,C_{\mu},\,{\bar{C}}_{\mu},\,{\bar{C}}_{2},\,C_{2},\,{\bar{C}}_{1},\,C_{1})$
as well as the bosonic set of (anti-)ghost fields
$({\bar{\beta}}_{\mu},\,\beta_{\mu},\,\bar{\beta},\,\beta)$ where the latter
(anti-)ghost fields carry the ghost numbers $(-2,\,+2)$. To be precise, the
set of bosonic anti-ghost fields $({\bar{\beta}}_{\mu},\,\bar{\beta})$ and the
ghost fields $(\beta_{\mu},\,\beta)$ are endowed with $(-2)$ and $(+2)$ ghost
numbers, respectively. The fermionic set of (anti-)ghost fields
$({\bar{C}}_{2})\,C_{2}$ have the ghost numbers $(-3)+3$, respectively, and
all the rest of the fermionic (anti-)ghost fields:
$({\bar{C}}_{\mu},\,{\bar{C}}_{1})$ and $(C_{\mu},\,C_{1})$ carry the ghost
numbers $(-1)$ and $(+1)$, respectively. We have the bosonic vector and scalar
fields $(\phi_{\mu},\,\phi)$, too, in our theory which appear in the gauge-
fixing terms.

The following off-shell nilpotent $[s_{(a)b}^{2}=0]$ anti-BRST transformations
$[s_{(a)b}]$

$\displaystyle
s_{ab}A_{\mu\nu\lambda}=\partial_{\mu}\bar{C}_{\nu\lambda}+\partial_{\nu}\bar{C}_{\lambda\mu}+\partial_{\lambda}\bar{C}_{\mu\nu},\quad
s_{ab}\bar{C}_{\mu\nu}=\partial_{\mu}\bar{\beta}_{\nu}-\partial_{\nu}\bar{\beta}_{\mu},$
$\displaystyle
s_{ab}B_{\mu\nu}=-\,(\partial_{\mu}F_{\nu}-\partial_{\nu}F_{\mu})\equiv(\partial_{\mu}\bar{f}_{\nu}-\partial_{\nu}\bar{f}_{\mu}),\quad
s_{ab}C_{\mu\nu}=\bar{B}_{\mu\nu},$ $\displaystyle
s_{ab}\,\Phi_{\mu\nu}=\pm\,m\,\bar{C}_{\mu\nu}-\,(\partial_{\mu}\,\bar{C}_{\nu}-\partial_{\nu}\,\bar{C}_{\mu}),\quad
s_{ab}\,F_{\mu}=-\,\partial_{\mu}\,B_{2},$ $\displaystyle
s_{ab}\,\bar{C}_{\mu}=\pm\,m\,\bar{\beta}_{\mu}-\partial_{\mu}\,\bar{\beta},\qquad
s_{ab}\phi_{\mu}=\bar{f}_{\mu},\quad
s_{ab}\bar{\beta}_{\mu}=\partial_{\mu}\,\bar{C}_{2},$ $\displaystyle
s_{ab}\,B_{\mu}=\pm\,m\,\bar{f}_{\mu}-\,\partial_{\mu}\,\bar{f},\quad\quad
s_{ab}\beta_{\mu}=\bar{F}_{\mu},\quad s_{ab}f_{\mu}=\partial_{\mu}B_{1},$
$\displaystyle s_{ab}\,C_{\mu}=\bar{B}_{\mu},\qquad
s_{ab}\,f=\pm\,m\,B_{1},\qquad s_{ab}\,\phi=\bar{f},$ $\displaystyle
s_{ab}\,\bar{\beta}=\pm\,m\,\bar{C}_{2},\qquad s_{ab}\,\beta=\bar{F},\qquad
s_{ab}\,F=\mp\,m\,B_{2}$ $\displaystyle s_{ab}\bar{C}_{1}=-B_{2},\qquad
s_{ab}C_{1}=B_{1},\qquad s_{ab}C_{2}=B,$ $\displaystyle
s_{ab}\,[H_{\mu\nu\lambda\zeta},\,\bar{B}_{\mu\nu},\,\bar{B}_{\mu},\;\bar{F}_{\mu},\,\bar{f}_{\mu},\,\bar{F},\,\bar{f},\,B,\,B_{1},\,B_{2},\,\bar{C}_{2}]=0,$
(69)

are the symmetry transformations for the action integral
$S_{1}=\int\,d^{D-1}\,x\,{\cal L}_{\bar{B}}$ because the Lagrangian density
${\cal L}_{\bar{B}}$ transforms to a total spacetime derivative under the
infinitesimal, continuous and off-shell nilpotent $(s_{ab}^{2}=0)$ anti-BRST
symmetry transformations $s_{ab}$:

$\displaystyle s_{ab}\,{\cal L}_{\bar{B}}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,\Big{[}\bar{B}^{\mu\nu}\,\bar{f}_{\nu}-(\partial^{\mu}\,\bar{C}^{\nu\lambda}+\partial^{\nu}\,\bar{C}^{\lambda\mu}+\partial^{\lambda}\,\bar{C}^{\mu\nu})\,\bar{B}_{\nu\lambda}+B\,\partial^{\mu}\,\bar{C}_{2}$
(70) $\displaystyle-$ $\displaystyle
B_{2}\,\bar{F}^{\mu}-B_{1}\,\bar{f}^{\mu}-(\partial^{\mu}\,\bar{\beta}^{\nu}-\partial^{\nu}\,\bar{\beta}^{\mu})\,\bar{F}_{\nu}+\frac{1}{2}\,(\pm\,m\,\bar{\beta}^{\mu}-\partial^{\mu}\,\bar{\beta})\,\bar{F}$
$\displaystyle-$
$\displaystyle(\partial^{\mu}\,\bar{C}^{\nu}-\partial^{\nu}\,\bar{C}^{\mu})\,\bar{B}_{\nu}\mp\,m\,\bar{B}^{\mu\nu}\,\bar{C}_{\nu}-\,\frac{1}{2}\,\bar{B}^{\mu}\,\bar{f}$
$\displaystyle\pm$ $\displaystyle m\,C^{\mu\nu}\,(\pm
m\,\bar{\beta}_{\nu}-\partial_{\nu}\,\bar{\beta})\pm\,m\,(\partial^{\mu}\,\bar{\beta}^{\nu}-\partial^{\nu}\,\bar{\beta}^{\mu})\,C_{\nu}\Big{]}.$

On the other hand, under the infinitesimal, continuous and off-shell nilpotent
$(s_{b}^{2}=0)$ BRST symmetry transformations $(s_{b})$

$\displaystyle
s_{b}A_{\mu\nu\lambda}=\partial_{\mu}C_{\nu\lambda}+\partial_{\nu}C_{\lambda\mu}+\partial_{\lambda}C_{\mu\nu},\quad
s_{b}C_{\mu\nu}=\partial_{\mu}\beta_{\nu}-\partial_{\nu}\beta_{\mu},$
$\displaystyle
s_{b}\bar{B}_{\mu\nu}=-\,(\partial_{\mu}\bar{F}_{\nu}-\partial_{\nu}\bar{F}_{\mu})\equiv(\partial_{\mu}\,f_{\nu}-\partial_{\nu}\,f_{\mu}),\quad
s_{b}\bar{C}_{\mu\nu}=B_{\mu\nu},$ $\displaystyle
s_{b}\,\Phi_{\mu\nu}=\pm\,m\,C_{\mu\nu}-\,(\partial_{\mu}\,C_{\nu}-\partial_{\nu}\,C_{\mu}),\qquad\quad
s_{b}\bar{F}_{\mu}=-\partial_{\mu}B,$ $\displaystyle
s_{b}\,C_{\mu}=\pm\,m\,\beta_{\mu}-\partial_{\mu}\,\beta,\qquad
s_{b}\phi_{\mu}=f_{\mu},\qquad s_{b}\beta_{\mu}=\partial_{\mu}C_{2},$
$\displaystyle s_{b}\,\bar{B}_{\mu}=\pm\,m\,f_{\mu}-\partial_{\mu}f,\qquad
s_{b}\bar{f}_{\mu}=-\,\partial_{\mu}B_{1},\quad
s_{b}\bar{\beta}_{\mu}=F_{\mu},$ $\displaystyle
s_{b}\,\bar{C}_{\mu}=B_{\mu},\qquad s_{b}\bar{C}_{2}=B_{2},\qquad
s_{b}\bar{C}_{1}=-\,B_{1},$ $\displaystyle s_{b}\,\phi=f,\qquad
s_{b}\,\beta=\pm\,m\,C_{2},\qquad s_{b}\,\bar{\beta}=F,$ $\displaystyle
s_{b}\,\bar{F}=\mp\,m\,B,\qquad s_{b}\,\bar{f}=\mp\,m\,B_{1},\qquad
s_{b}\,C_{1}=-B,$ $\displaystyle
s_{b}\,[H_{\mu\nu\lambda\zeta},\,B_{\mu\nu},\,B_{\mu},\,f_{\mu},\,F_{\mu},\,F,\,f,\,B,\,B_{1},\,B_{2},\,C_{2}]=0,$
(71)

the Lagrangian density ${\cal L}_{B}$ transforms to a total spacetime
derivative

$\displaystyle s_{b}\,{\cal L}_{B}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,\Big{[}(\partial^{\mu}\,C^{\nu\lambda}+\partial^{\nu}\,C^{\lambda\mu}+\partial^{\lambda}\,C^{\mu\nu})\,B_{\nu\lambda}+B^{\mu\nu}\,f_{\nu}-B_{2}\,\partial^{\mu}\,C_{2}$
(72) $\displaystyle-$ $\displaystyle
B_{1}\,f^{\mu}+B\,F^{\mu}-(\partial^{\mu}\,\beta^{\nu}-\partial^{\nu}\,\beta^{\mu})\,F_{\nu}+\frac{1}{2}\,(\pm\,m\,\beta^{\mu}-\partial^{\mu}\,\beta)\,F$
$\displaystyle+$
$\displaystyle(\partial^{\mu}\,C^{\nu}-\partial^{\nu}\,C^{\mu})\,B_{\nu}\pm\,m\,B^{\mu\nu}\,C_{\nu}-\,\frac{1}{2}\,B^{\mu}\,f$
$\displaystyle\mp$ $\displaystyle m\,\bar{C}^{\mu\nu}\,\big{(}\pm
m\,\beta_{\nu}-\partial_{\nu}\,\beta\big{)}\mp\,m\,(\partial^{\mu}\,\beta^{\nu}-\partial^{\nu}\,\beta^{\mu})\,\bar{C}_{\nu}\Big{]},$

which establishes the fact that the action integral
$S_{2}=\int\,d^{D-1}\,x\,{\cal L}_{B}$ respects the above infinitesimal,
continuous and nilpotent BRST symmetry transformations ($s_{b}$).

According to Noether’s theorem, the above observations imply that the BRST
conserved currents can be derived, by exploiting the standard theoretical
formula, as [26]:

$\displaystyle J^{\mu}_{(b)}$ $\displaystyle=$ $\displaystyle
H^{\mu\nu\lambda\zeta}\,(\partial_{\nu}\,C_{\lambda\zeta})\pm\frac{m}{2}\,[\pm
m\,\bar{\beta}^{\mu}-\partial^{\mu}\,\bar{\beta}]\,C_{2}\pm
m\,\bar{C}^{\mu\nu}\,(\pm
m\,\beta_{\nu}-\partial_{\nu}\,\beta)+(\partial^{\mu}\,C^{\nu\lambda}$ (73)
$\displaystyle+$
$\displaystyle\partial^{\nu}\,C^{\lambda\mu}+\partial^{\lambda}\,C^{\mu\nu})\,B_{\nu\lambda}-(\partial^{\mu}\,\bar{C}^{\nu\lambda}+\partial^{\nu}\,\bar{C}^{\lambda\mu}+\partial^{\lambda}\,\bar{C}^{\mu\nu})\,(\partial_{\nu}\,\beta_{\lambda}-\partial_{\lambda}\,\beta_{\nu})\mp
m\,C^{\mu\nu}\,B_{\nu}$ $\displaystyle-$ $\displaystyle B_{1}\,f^{\mu}+[\pm
m\,A^{\mu\nu\lambda}-\,\Sigma^{\mu\nu\lambda}]\,[\pm
m\,C_{\nu\lambda}-(\partial_{\nu}\,C_{\lambda}-\partial_{\lambda}\,C_{\nu})]\,-B_{2}\,\partial^{\mu}\,C_{2}+B^{\mu\nu}\,f_{\nu}$
$\displaystyle-$
$\displaystyle\,(\partial^{\mu}\,\bar{\beta}^{\nu}-\partial^{\nu}\,\bar{\beta}^{\mu})\,(\partial_{\nu}\,C_{2})-\,(\partial^{\mu}\,\beta^{\nu}-\partial^{\nu}\,\beta^{\mu})\,F_{\nu}+(\partial^{\mu}\,C^{\nu}-\partial^{\nu}\,C^{\mu})\,B_{\nu}+B\,F^{\mu}$
$\displaystyle-$
$\displaystyle\,\frac{1}{2}\,B^{\mu}\,f+\,\frac{1}{2}\,\big{(}\pm
m\,\beta^{\mu}-\partial^{\mu}\,\beta\big{)}\,F-\,(\partial^{\mu}\,\bar{C}^{\nu}-\partial^{\nu}\,\bar{C}^{\mu})\,(\pm
m\,\beta_{\nu}-\partial_{\nu}\,\beta).$

In exactly similar fashion, we obtain the precise expression for the conserved
$(\partial_{\mu}\,J^{\mu}_{(ab)}=0)$ anti-BRST Noether current
$J^{\mu}_{(ab)}$ [26]:

$\displaystyle J^{\mu}_{(ab)}$ $\displaystyle=$ $\displaystyle
H^{\mu\nu\lambda\zeta}\,(\partial_{\nu}\,\bar{C}_{\lambda\zeta})\pm\frac{m}{2}\,[\pm
m\,\beta^{\mu}-\partial^{\mu}\,\beta]\,\bar{C}_{2}\mp m\,C^{\mu\nu}\,(\pm
m\,\bar{\beta}_{\nu}-\partial_{\nu}\,\bar{\beta})-(\partial^{\mu}\,\bar{C}^{\nu\lambda}$
(74) $\displaystyle+$
$\displaystyle\partial^{\nu}\,\bar{C}^{\lambda\mu}+\partial^{\lambda}\,\bar{C}^{\mu\nu})\,\bar{B}_{\nu\lambda}+\,(\partial^{\mu}\,C^{\nu\lambda}+\partial^{\nu}\,C^{\lambda\mu}+\partial^{\lambda}\,C^{\mu\nu})\,(\partial_{\nu}\,\bar{\beta}_{\lambda}-\partial_{\lambda}\,\bar{\beta}_{\nu})\pm
m\,\bar{C}^{\mu\nu}\,\bar{B}_{\nu}$ $\displaystyle-$
$\displaystyle\bar{f}^{\mu}\,B_{1}+[\pm
m\,A^{\mu\nu\lambda}-\,\Sigma^{\mu\nu\lambda}]\,[\pm
m\,\bar{C}_{\nu\lambda}-(\partial_{\nu}\,\bar{C}_{\lambda}-\partial_{\lambda}\,\bar{C}_{\nu})]+B\,\partial^{\mu}\,\bar{C}_{2}+\bar{B}^{\mu\nu}\,\bar{f}_{\nu}$
$\displaystyle-$
$\displaystyle\,(\partial^{\mu}\,\beta^{\nu}-\partial^{\nu}\,\beta^{\mu})\,(\partial_{\nu}\,\bar{C}_{2})-\,(\partial^{\mu}\,\bar{\beta}^{\nu}-\partial^{\nu}\,\bar{\beta}^{\mu})\,\bar{F}_{\nu}-\,(\partial^{\mu}\,\bar{C}^{\nu}-\partial^{\nu}\,\bar{C}^{\mu})\,\bar{B}_{\nu}-B_{2}\,\bar{F}^{\mu}$
$\displaystyle-$
$\displaystyle\,\frac{1}{2}\,\bar{B}^{\mu}\,\bar{f}+\frac{1}{2}\,\big{(}\pm
m\,\bar{\beta}^{\mu}-\partial^{\mu}\,\bar{\beta}\big{)}\,\bar{F}+(\partial^{\mu}\,C^{\nu}-\partial^{\nu}\,C^{\mu})\,(\pm
m\,\bar{\beta}_{\nu}-\partial_{\nu}\,\bar{\beta}).$

The conservation law $(\partial_{\mu}\,J^{\mu}_{(a)b}=0)$ can be checked by
using the EL-EoMs that have been derived in our earlier work [26]. In these
proofs, the algebra is a bit involved but it is quite straightforward to check
that $\partial_{\mu}\,J^{\mu}_{(r)}=0$ with $r=b,\,ab$.

The conserved Noether charges $Q_{(a)b}=\int\,d^{D-1}\,x\,J^{0}_{(a)b}$ (that
emerge out from the conserved currents) $J^{\mu}_{(a)b}$ are as follows [26]:

$\displaystyle Q_{ab}$ $\displaystyle=$ $\displaystyle\int
d^{D-1}x\,J^{0}_{(ab)}\equiv\int
d^{D-1}x\,\Big{[}H^{0ijk}\,(\partial_{i}\,\bar{C}_{jk})\pm\frac{m}{2}\,[\pm
m\,\beta^{0}-\partial^{0}\,\beta]\,\bar{C}_{2}$ (75) $\displaystyle-$
$\displaystyle[\,\pm\,m\,C^{0i}\,-(\partial^{0}\,C^{i}-\partial^{i}\,C^{0})]\,(\,\pm\,m\,\bar{\beta}_{i}-\partial_{i}\,\bar{\beta})+[\,\pm\,m\,\bar{C}^{0i}\,-\,(\partial^{0}\,\bar{C}^{i}-\partial^{i}\,\bar{C}^{0})]\,\bar{B}_{i}$
$\displaystyle-$
$\displaystyle(\partial^{0}\,\bar{C}^{ij}+\partial^{i}\,\bar{C}^{j0}+\partial^{j}\,\bar{C}^{0i})\,\bar{B}_{ij}+\,(\partial^{0}\,C^{ij}+\partial^{i}\,C^{j0}+\partial^{j}\,C^{0i})\,(\partial_{i}\,\bar{\beta}_{j}-\partial_{j}\,\bar{\beta}_{i})$
$\displaystyle-$ $\displaystyle\bar{f}^{0}\,B_{1}+[\pm
m\,A^{0ij}-\,\Sigma^{0ij}]\,[\pm
m\,\bar{C}_{ij}-(\partial_{i}\,\bar{C}_{j}-\partial_{j}\,\bar{C}_{i})]+B\,\partial^{0}\,\bar{C}_{2}-\,\frac{1}{2}\,\bar{B}^{0}\,\bar{f}-B_{2}\,\bar{F}^{0}$
$\displaystyle+$
$\displaystyle\bar{B}^{0i}\,\bar{f}_{i}-\,(\partial^{0}\,\beta^{i}-\partial^{i}\,\beta^{0})\,(\partial_{i}\,\bar{C}_{2})-\,(\partial^{0}\,\bar{\beta}^{i}-\partial^{i}\,\bar{\beta}^{0})\,\bar{F}_{i}+\frac{1}{2}\,\big{(}\pm
m\,\bar{\beta}^{0}-\partial^{0}\,\bar{\beta}\big{)}\,\bar{F}\Big{]},$
$\displaystyle Q_{b}$ $\displaystyle=$ $\displaystyle\int
d^{D-1}x\,J^{0}_{(b)}\equiv\int
d^{D-1}x\,\Big{[}H^{0ijk}\,(\partial_{i}\,C_{jk})\pm\frac{m}{2}\,[\pm
m\,\bar{\beta}^{0}-\partial^{0}\,\bar{\beta}]\,C_{2}$ (76) $\displaystyle+$
$\displaystyle[\,\pm\,m\,\bar{C}^{0i}\,-(\partial^{0}\,\bar{C}^{i}-\partial^{i}\,\bar{C}^{0})]\,(\,\pm\,m\,\beta_{i}-\partial_{i}\,\beta)-[\pm\,m\,C^{0i}\,-(\partial^{0}\,C^{i}-\partial^{i}\,C^{0})]\,B_{i}$
$\displaystyle+$
$\displaystyle(\partial^{0}\,C^{ij}+\partial^{i}\,C^{j0}+\partial^{j}\,C^{0i})\,B_{ij}-(\partial^{0}\,\bar{C}^{ij}+\partial^{i}\,\bar{C}^{j0}+\partial^{j}\,\bar{C}^{0i})\,(\partial_{i}\,\beta_{j}-\partial_{j}\,\beta_{i})$
$\displaystyle-$ $\displaystyle B_{1}\,f^{0}+[\pm
m\,A^{0ij}-\,\Sigma^{0ij}]\,[\pm
m\,C_{ij}-(\partial_{i}\,C_{j}-\partial_{j}\,C_{i})]\,-B_{2}\,\partial^{0}\,C_{2}-\,\frac{1}{2}\,B^{0}\,f+B^{0i}\,f_{i}$
$\displaystyle-$
$\displaystyle\,(\partial^{0}\,\bar{\beta}^{i}-\partial^{i}\,\bar{\beta}^{0})\,(\partial_{i}\,C_{2})-\,(\partial^{0}\,\beta^{i}-\partial^{i}\,\beta^{0})\,F_{i}+B\,F^{0}+\,\frac{1}{2}\,\big{(}\pm
m\,\beta^{0}-\partial^{0}\,\beta\big{)}\,F\Big{]}.$

These conserved charges are the generators of all the (anti-)BRST symmetry
transformations. However, they are not off-shell nilpotent of order two.
Exploiting the principle behind the continuous symmetry transformations and
their generators, we obtain the following

$\displaystyle s_{b}\,Q_{b}=-\,i\,\\{Q_{b},\,Q_{b}\\}$ $\displaystyle=$
$\displaystyle\int
d^{D-1}x\,\Big{[}\pm\frac{m}{2}\,\Big{(}\pm\,m\,F^{0}-\partial^{0}F\Big{)}\,C_{2}-(\partial^{0}F^{i}-\partial^{i}F^{0})\,(\partial_{i}C_{2})$
$\displaystyle-$
$\displaystyle(\partial^{0}\,B^{ij}+\partial^{i}\,B^{j0}+\partial^{j}\,B^{0i})\,(\partial_{i}\beta_{j}-\partial_{j}\beta_{i})$
$\displaystyle+$
$\displaystyle\Big{[}\pm\,m\,B^{0i}-(\partial^{0}B^{i}-\partial^{i}B^{0})]\,(\pm\,m\,\beta_{i}-\partial_{i}\beta)\Big{]}\neq
0,$ $\displaystyle s_{ab}\,Q_{ab}=-\,i\,\\{Q_{ab},\,Q_{ab}\\}$
$\displaystyle=$ $\displaystyle\int
d^{D-1}x\,\Big{[}\pm\frac{m}{2}\,\Big{(}\pm\,m\,\bar{F}^{0}-\partial^{0}\bar{F}\Big{)}\,\bar{C}_{2}-(\partial^{0}\bar{F}^{i}-\partial^{i}\bar{F}^{0})\,(\partial_{i}\bar{C}_{2})$
(77) $\displaystyle+$
$\displaystyle(\partial^{0}\,\bar{B}^{ij}+\partial^{i}\,\bar{B}^{j0}+\partial^{j}\,\bar{B}^{0i})\,(\partial_{i}\bar{\beta}_{j}-\partial_{j}\bar{\beta}_{i})$
$\displaystyle-$
$\displaystyle\Big{[}\pm\,m\,\bar{B}^{0i}-(\partial^{0}\bar{B}^{i}-\partial^{i}\bar{B}^{0})]\,(\pm\,m\,\bar{\beta}_{i}-\partial_{i}\bar{\beta})\Big{]}\neq
0,$

where the l.h.s. of the above equations have been explicitly computed by using
the (anti-) BRST symmetry transformations [cf. Eqs. (69), (71)] and the
expressions for the Noether conserved charges $Q_{(a)b}$ [cf. Eqs. (75),
(76)]. The above observations establish that the conserved Noether (anti-)BRST
charges $Q_{(a)b}$ are not off-shell nilpotent of order two.

We now follow our step-by-step proposal to obtain the off-shell nilpotent
expressions for the (anti-)BRST charges $Q_{(a)b}^{(1)}$ where the off-shell
nilpotency is proven by using the principle behind the continuous symmetry
transformations and their generators as:
$s_{b}\,Q_{b}^{(1)}=-\,i\,\\{Q_{b}^{(1)},\,Q_{b}^{(1)}\\}=0$ and
$s_{ab}\,Q_{ab}^{(1)}=-\,i\,\\{Q_{ab}^{(1)},\,Q_{ab}^{(1)}\\}=0$. First of
all, we focus on the BRST charge $Q_{b}$. The first step is to use the
equation of motion w.r.t. the gauge field. Towards this goal in mind, we note
that the first term of $Q_{b}$ can be re-expressed as:

$\displaystyle\int
d^{D-1}\,x\,\Big{[}H^{0ijk}\,\partial_{i}\,C_{jk}\Big{]}=\int
d^{D-1}\,x\,\Big{[}\partial_{i}\,\big{\\{}H^{0ijk}\,C_{jk}\big{\\}}\Big{]}-\int
d^{D-1}\,x\,\Big{[}(\partial_{i}\,H^{0ijk})\,C_{jk}\Big{]}.$ (78)

The first term on the r.h.s. of the above equation vanishes for the physical
fields due to celebrated Gauss’s divergence theorem. In the second term, we
apply the following equation of motion w.r.t. the gauge field
$A_{\mu\nu\lambda}$, namely;

$\displaystyle\partial_{\rho}\,H^{\rho\mu\nu\lambda}+m^{2}\,A^{\mu\nu\lambda}\mp
m\,\Sigma^{\mu\nu\lambda}+(\partial^{\mu}\,B^{\nu\lambda}+\partial^{\nu}\,B^{\lambda\mu}+\partial^{\lambda}\,B^{\mu\nu})=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\Longrightarrow-\,(\partial_{i}\,H^{0ijk})\,C_{jk}=\mp\,m\Big{[}\pm\,mA^{0ij}-\Sigma^{0ij}\Big{]}\,C_{ij}-(\partial^{0}\,B^{ij}+\partial^{i}\,B^{j0}+\partial^{j}\,B^{oi})\,C_{ij}.$
(79)

In the second step, we look for the terms of $Q_{b}$ that cancel out with some
of the terms that emerge out after the first step. In this context, we note
that such an appropriate term is

$\displaystyle+\Big{[}\pm\,m\,A^{0ij}-\Sigma^{0ij}\Big{]}\,\Big{[}\pm\,m\,{C}_{ij}-(\partial_{i}\,{C}_{j}-\partial_{j}\,{C}_{i})\Big{]}.$
(80)

The sum of (79) and (80) yields the following:

$\displaystyle-$
$\displaystyle\,(\partial_{i}\,H^{0ijk})\,C_{jk}+\Big{[}\pm\,m\,A^{0ij}-\Sigma^{0ij}\Big{]}\,\Big{[}\pm\,m\,{C}_{ij}-(\partial_{i}\,{C}_{j}-\partial_{j}\,{C}_{i})\Big{]}$
(81) $\displaystyle\equiv$
$\displaystyle-(\partial^{0}\,B^{ij}+\partial^{i}\,B^{j0}+\partial^{j}\,B^{oi})\,C_{ij}-2\,\Big{[}\pm\,m\,A^{0ij}-\Sigma^{0ij}\Big{]}\,(\partial_{i}\,C_{j}).$

We comment here that, as pointed out earlier, one of the key ingredients of
our proposal is the use of EL-EoM w.r.t. the gauge field. This step opens up a
decisive door for (i) our further applications of Gauss’s divergence theorem,
(ii) use of the appropriate EL-EoM, and (iii) the application of the BRST
symmetry transformations ($s_{b}$) at appropriate places. In the third step,
we focus on the second term of the r.h.s. which can be written as:

$\displaystyle-2\,\partial_{i}\,\Big{[}(\pm\,m\,A^{0ij}-\Sigma^{0ij})\,C_{j}\Big{]}+2\,\partial_{i}\,\big{[}(\pm\,m\,A^{0ij}-\Sigma^{0ij})\big{]}\,C_{j}.$
(82)

The terms in (82) are inside the integration. Thus, the first term of (82)
will vanish due to Gauss’s divergence theorem for physical fields. Using the
following equation of motion w.r.t. the St${\ddot{u}}$ckelberg field
$\Phi_{\mu\nu}$ from the Lagrangian density ${\cal L}_{B}$, we obtain:

$\displaystyle\partial_{\mu}\,\Sigma^{\mu\nu\lambda}\mp\,m\,(\partial_{\mu}\,A^{\mu\nu\,\lambda})+\frac{1}{2}\,(\partial^{\nu}\,B^{\lambda}-\partial^{\lambda}\,B^{\nu})\mp\,\frac{m}{2}\,B^{\nu\lambda}=0$
$\displaystyle\Longrightarrow\;\;2\,\partial_{i}\,\Big{[}\pm\,m\,A^{0ij}-\Sigma^{0ij}\Big{]}\,C_{j}=\pm\,m\,B^{0i}\,C_{i}-(\partial^{0}\,B^{i}-\partial^{i}\,B^{0})\,C_{i},$
(83)

where the l.h.s. is nothing but the second term of (82). Using (83), we can
re-write the whole sum of (81) as follows:

$\displaystyle+\Big{[}\pm\,m\,B^{0i}-(\partial^{0}\,B^{i}-\partial^{i}\,B^{0})\Big{]}\,C_{i}-(\partial^{0}\,B^{ij}+\partial^{i}\,B^{j0}+\partial^{j}\,B^{oi})\,C_{ij}.$
(84)

Within the framework of our proposal, the terms in (84) will be present in the
off-shell nilpotent form of the BRST charge ($Q_{b}^{(1)}$) because they have
come out from the use of EL-EoMs and, hence, will remain intact. This is also
required due to the fact that, as pointed out earlier, we wish to obtain
$s_{b}\,Q_{b}^{(1)}=0$ which will automatically imply that
$[Q_{b}^{(1)}]^{2}=0$. In the fourth step, we apply the BRST symmetry
transformation $(s_{b})$ on (84) which leads to the following explicit
expression:

$\displaystyle+\Big{[}\pm\,m\,B^{0i}-(\partial^{0}\,B^{i}-\partial^{i}\,B^{0})\Big{]}\,(\pm\,m\,\beta_{i}-\partial_{i}\,\beta)$
$\displaystyle-(\partial^{0}\,B^{ij}+\partial^{i}\,B^{j0}+\partial^{j}\,B^{oi})\,(\partial_{i}\,\beta_{j}-\partial_{j}\,\beta_{i}).$
(85)

In the fifth step, we modify some of the terms of $Q_{b}$ so that when $s_{b}$
acts on a part of them, there is precise cancellation between the ensuing
result and (85). In this context, we modify the following terms from the
explicit expression for $Q_{b}$ [cf. Eq. (76)], namely;

$\displaystyle-$
$\displaystyle(\partial^{0}\,{\bar{C}}^{ij}+\partial^{i}\,{\bar{C}}^{j0}+\partial^{j}\,{\bar{C}}^{oi})\,(\partial_{i}\,\beta_{j}-\partial_{j}\,\beta_{i})=+(\partial^{0}\,{\bar{C}}^{ij}+\partial^{i}\,{\bar{C}}^{j0}+\partial^{j}\,{\bar{C}}^{oi})\,(\partial_{i}\,\beta_{j}-\partial_{j}\,\beta_{i})$
(86) $\displaystyle-$ $\displaystyle
2\,(\partial^{0}\,{\bar{C}}^{ij}+\partial^{i}\,{\bar{C}}^{j0}+\partial^{j}\,{\bar{C}}^{oi})\,(\partial_{i}\,\beta_{j}-\partial_{j}\,\beta_{i}),$
$\displaystyle+$
$\displaystyle\Big{[}\pm\,m\,{\bar{C}}^{0i}-(\partial^{0}\,{\bar{C}}^{i}-\partial^{i}\,{\bar{C}}^{0})\Big{]}\,(\pm\,m\,\beta_{i}-\partial_{i}\,\beta)=2\,\Big{[}\pm\,m\,{\bar{C}}^{0i}-(\partial^{0}\,{\bar{C}}^{i}$
$\displaystyle-$
$\displaystyle\partial^{i}\,{\bar{C}}^{0})\Big{]}\,(\pm\,m\,\beta_{i}-\partial_{i}\,\beta)-\Big{[}\pm\,m\,{\bar{C}}^{0i}-(\partial^{0}\,{\bar{C}}^{i}-\partial^{i}\,{\bar{C}}^{0})\Big{]}\,(\pm\,m\,\beta_{i}-\partial_{i}\,\beta).$

In the sixth step, we note that, in the above modified expressions, when we
apply $s_{b}$ on a part of (86) as explicitly written below, namely;

$\displaystyle
s_{b}\,\Big{[}(\partial^{0}\,{\bar{C}}^{ij}+\partial^{i}\,{\bar{C}}^{j0}+\partial^{j}\,{\bar{C}}^{0i})\,(\partial_{i}\,\beta_{j}-\partial_{j}\,\beta_{i})$
$\displaystyle-\big{\\{}\pm\,m\,{\bar{C}}^{0i}-(\partial^{0}\,{\bar{C}}^{i}-\partial^{i}\,{\bar{C}}^{0})\big{\\}}\,(\pm\,m\,\beta_{i}-\partial_{i}\,\beta)\Big{]},$
(87)

we obtain the desired result

$\displaystyle(\partial^{0}\,{B}^{ij}+\partial^{i}\,{B}^{j0}+\partial^{j}\,{B}^{0i})\,(\partial_{i}\,\beta_{j}-\partial_{j}\,\beta_{i})$
$\displaystyle-\big{\\{}\pm\,m\,{B}^{0i}-(\partial^{0}\,{B}^{i}-\partial^{i}\,{B}^{0})\big{\\}}\,(\pm\,m\,\beta_{i}-\partial_{i}\,\beta)\Big{]},$
(88)

which precisely cancels out with whatever we have obtained in (85). At this
juncture, we point out that, in the off-shell nilpotent version of the BRST
charge $Q_{b}^{(1)}$, the terms in the square bracket of (87), will be always
[along with (84) (as pointed out earlier)] present. We now focus on the left-
over terms of (86) which can be re-expressed as follows:

$\displaystyle\pm\,2\,m\,\big{\\{}\pm\,m\,{\bar{C}}^{0i}-(\partial^{0}\,{\bar{C}}^{i}-\partial^{i}\,{\bar{C}}^{0})\big{\\}}\,\beta_{i}$
$\displaystyle-\;2\,\big{\\{}\pm\,m\,{\bar{C}}^{0i}-(\partial^{0}\,{\bar{C}}^{i}-\partial^{i}\,{\bar{C}}^{0})\big{\\}}\,\partial_{i}\,\beta$
$\displaystyle-\;4\,\big{[}\partial^{0}\,{\bar{C}}^{ij}+\partial^{i}\,{\bar{C}}^{j0}+\partial^{j}\,{\bar{C}}^{0i}\big{]}\,(\partial_{i}\,\beta_{j}).$
(89)

All the above terms are inside the integration. Hence, we can apply the
following algebraic tricks on the last two terms of (89) to obtain the
following

$\displaystyle-2\,\big{\\{}\pm\,m\,{\bar{C}}^{0i}-(\partial^{0}\,{\bar{C}}^{i}-\partial^{i}\,{\bar{C}}^{0})\big{\\}}\,\partial_{i}\,\beta$
$\displaystyle=$
$\displaystyle\partial_{i}\,\Big{[}-2\,\big{\\{}\pm\,m\,{\bar{C}}^{0i}-(\partial^{0}\,{\bar{C}}^{i}-\partial^{i}\,{\bar{C}}^{0})\big{\\}}\,\beta\Big{]}$
$\displaystyle+$ $\displaystyle
2\,\partial_{i}\,\Big{[}\pm\,m\,{\bar{C}}^{0i}-(\partial^{0}\,{\bar{C}}^{i}-\partial^{i}\,{\bar{C}}^{0})\Big{]}\beta$
$\displaystyle-4\,\big{[}\partial^{0}\,{\bar{C}}^{ij}+\partial^{i}\,{\bar{C}}^{j0}+\partial^{j}\,{\bar{C}}^{0i}\big{]}\,\partial_{i}\,\beta_{j}$
$\displaystyle=$
$\displaystyle\partial_{i}\,\Big{[}-4\,\big{\\{}\partial^{0}\,{\bar{C}}^{ij}+\partial^{i}\,{\bar{C}}^{j0}+\partial^{j}\,{\bar{C}}^{0i}\big{\\}}\,\beta_{j}\Big{]}$
(90) $\displaystyle+$ $\displaystyle
4\,\partial_{i}\,\Big{[}\partial^{0}\,{\bar{C}}^{ij}+\partial^{i}\,{\bar{C}}^{j0}+\partial^{j}\,{\bar{C}}^{0i}\Big{]}\,\beta_{j},$

so that the total space derivative terms vanish due to Gauss’s divergence
theorem for the physical fields and we are left with the following from the
above equation (90):

$\displaystyle
2\,\partial_{i}\,\Big{[}\pm\,m\,{\bar{C}}^{0i}-(\partial^{0}\,{\bar{C}}^{i}-\partial^{i}\,{\bar{C}}^{0})\Big{]}\beta+4\,\partial_{i}\,\Big{[}\partial^{0}\,{\bar{C}}^{ij}+\partial^{i}\,{\bar{C}}^{j0}+\partial^{j}\,{\bar{C}}^{0i}\Big{]}\,\beta_{j}.$
(91)

In the above, we apply the following equation of motion:

$\displaystyle\partial_{\mu}\,[\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu}]-\frac{1}{2}\,\partial^{\nu}F\,\mp\,m\,(\partial_{\mu}\bar{C}^{\mu\nu})\,\pm\,\frac{m}{2}\,F^{\nu}=0,$
$\displaystyle\Longrightarrow\;\;2\,\partial_{i}\,[\,\pm\,m\,\bar{C}^{0i}-(\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0})]\,\beta=-\,(\pm\,m\,F^{0}-\partial^{0}F)\,\beta,$
(92)

which provides the concise value of the first term of (91). Exactly, in a
similar fashion, the application of the following equation of motion

$\displaystyle\partial_{\mu}\,[\partial^{\mu}\bar{C}^{\nu\lambda}+\partial^{\nu}\bar{C}^{\lambda\mu}+\partial^{\lambda}\bar{C}^{\mu\nu}]\pm\,\frac{m}{2}\,(\partial^{\nu}\,\bar{C}^{\lambda}-\partial^{\lambda}\,\bar{C}^{\nu})$
$\displaystyle+\frac{1}{2}\,(\partial^{\nu}\,F^{\lambda}-\partial^{\lambda}\,F^{\nu})-\,\frac{m^{2}}{2}\,\bar{C}^{\nu\lambda}=0,$
$\displaystyle\Longrightarrow\;\;4\,\partial_{i}\,[\partial^{0}\bar{C}^{ij}+\partial^{i}\bar{C}^{j0}+\partial^{j}\bar{C}^{0i}]\,\beta_{j}=2\,(\partial^{0}F^{i}-\partial^{i}F^{0})\,\beta_{i}$
$\displaystyle~{}~{}~{}~{}~{}~{}\,\mp\,2\,m\,\big{[}\,\pm\,m\,\bar{C}^{0i}-(\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0})\big{]}\,\beta_{i},$
(93)

leads to the alternative (but appropriate) form of the second term of (91).
The sum of the r.h.s. of (92), (93) and the left-over term (i.e. the first
term) of (89) is equal to:

$\displaystyle
2\,(\partial^{0}F^{i}-\partial^{i}F^{0})\,\beta_{i}-(\pm\,m\,F^{0}-\partial^{0}F)\,\beta.$
(94)

At this stage, we take the seventh step and apply the BRST symmetry
transformations $(s_{b})$ on (94) which yields the following explicit
expression, namely;

$\displaystyle-\,2\,(\partial^{0}F^{i}-\partial^{i}F^{0})\,(\partial_{i}\,C_{2})+(\pm\,m\,F^{0}-\partial^{0}F)\,(\pm\,m\,C_{2}).$
(95)

We emphasize that the term in (94) will be present in the off-shell nilpotent
version of the BRST charge $(Q_{b}^{(1)})$. We take now the eighth step, and
modify some of the terms of $Q_{b}$ so that when we apply the BRST
transformations $(s_{b})$ on a part of them, the resulting expressions must
cancel out (95) in a precise manner. Towards this goal in mind, we modify the
following appropriate terms of the original expression for $Q_{b}$ [cf. Eq.
(76)]:

$\displaystyle-$
$\displaystyle(\partial^{0}\,{\bar{\beta}}^{i}-\partial^{i}\,{\bar{\beta}}^{0})\,\partial_{i}\,C_{2}\pm\frac{1}{2}\,m\,(\pm\,m\,{\bar{\beta}}^{0}-\partial^{0}\,\bar{\beta})\,C_{2}$
(96) $\displaystyle\equiv$ $\displaystyle
2\,(\partial^{0}\,{\bar{\beta}}^{i}-\partial^{i}\,{\bar{\beta}}^{0})\,\partial_{i}\,C_{2}\mp
m\,(\pm\,m\,{\bar{\beta}}^{0}-\partial^{0}\,\bar{\beta})\,C_{2}$
$\displaystyle\pm$
$\displaystyle\frac{3}{2}\,(\pm\,m\,{\bar{\beta}}^{0}-\partial^{0}\,\bar{\beta})\,C_{2}-3\,(\partial^{0}\,{\bar{\beta}}^{i}-\partial^{i}\,{\bar{\beta}}^{0})\,\partial_{i}\,C_{2}.$

It is evident that if we apply the BRST symmetry transformations $(s_{b})$ on
the first two terms of (96), the resulting expressions will cancel out the
terms that are written in (95). Hence, along with (94), the above first two
terms will be present in the off-shell nilpotent version of the BRST charge
$(Q_{b}^{(1)})$ so that our central objective $s_{b}\,Q_{b}^{(1)}=0$ can be
fulfilled. At this juncture, we focus on the last term of (96) which can be
re-written as

$\displaystyle-3\,(\partial^{0}\,{\bar{\beta}}^{i}-\partial^{i}\,{\bar{\beta}}^{0})\,\partial_{i}\,C_{2}=\partial_{i}\,\big{[}-3\,(\partial^{0}\,{\bar{\beta}}^{i}-\partial^{i}\,{\bar{\beta}}^{0})\,C_{2}\big{]}+3\,\partial_{i}\,\big{[}(\partial^{0}\,{\bar{\beta}}^{i}-\partial^{i}\,{\bar{\beta}}^{0})\big{]}\,C_{2}.$
(97)

The above terms are inside the integral. Hence, the first term on the r.h.s.
will vanish due to Gauss’s divergence theorem and only the second term on the
r.h.s. of (97) will survive. Now, we apply the following EL-EoM

$\displaystyle\Box\,{\bar{\beta}}_{\mu}-\partial_{\mu}\,(\partial\cdot\bar{\beta})+\partial_{\mu}\,B_{2}-\frac{m^{2}}{2}\,{\bar{\beta}}_{\mu}\pm\frac{m}{2}\,\partial_{\mu}\,\bar{\beta}=0$
$\displaystyle\Longrightarrow$ $\displaystyle
3\,\partial_{i}\,\big{[}\partial^{0}\,{\bar{\beta}}^{i}-\partial^{i}\,{\bar{\beta}}^{0}\big{]}\,C_{2}=3\,{\dot{B}}_{2}\,C_{2}\mp\frac{3}{2}\,m\,(\pm\,m\,{\bar{\beta}}^{0}-\partial^{0}\,\bar{\beta})\,C_{2},$
(98)

which will replace the last term of (96). The substitution of (98) into (96)
yields the following concise and beautiful result, namely;

$\displaystyle\pm\,\frac{3}{2}\,m\,(\pm\,m\,{\bar{\beta}}^{0}-\partial^{0}\,\bar{\beta})\,C_{2}+3\,\partial_{i}\,\big{[}\partial^{0}\,{\bar{\beta}}^{i}-\partial^{i}\,{\bar{\beta}}^{0}\big{]}\,C_{2}=3\,{\dot{B}}_{2}\,C_{2}.$
(99)

If we apply further the BRST symmetry transformation $(s_{b})$ on (99), it
turns out to be zero. Here, all our steps terminate (according to our
proposal). It is self-evident that the r.h.s. of (99) will be part in
$Q_{b}^{(1)}$. Ultimately, the off-shell nilpotent version $Q_{b}^{(1)}$, from
the non-nilpotent Noether conserved charge $Q_{b}$, is as follows

$\displaystyle Q_{b}\longrightarrow Q_{b}^{(1)}$ $\displaystyle=$
$\displaystyle\int
d^{D-1}x\,\Big{[}(\partial^{0}C^{ij}+\partial^{i}C^{j0}+\partial^{j}C^{0i})\,B_{ij}-\,[\,\pm\,m\,C^{0i}-(\partial^{0}C^{i}-\partial^{i}C^{0})]\,B_{i}$
(100) $\displaystyle-$
$\displaystyle(\partial^{0}B^{ij}+\partial^{i}B^{j0}+\partial^{j}B^{0i})\,C_{ij}+[\,\pm\,m\,B^{0i}-(\partial^{0}B^{i}-\partial^{i}B^{0})]\,C_{i}$
$\displaystyle-$
$\displaystyle[\,\pm\,m\,\bar{C}^{0i}-(\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0})\,(\pm\,m\,\beta_{i}-\partial_{i}\beta)]+2\,(\partial^{0}\bar{\beta}^{i}-\partial^{i}\bar{\beta}^{0})\,\partial_{i}C_{2}$
$\displaystyle\mp$
$\displaystyle\,m\,(\pm\,m\,\bar{\beta}^{0}-\partial^{0}\bar{\beta})\,C_{2}+(\partial^{0}\bar{C}^{ij}+\partial^{i}\bar{C}^{j0}+\partial^{j}\bar{C}^{0i})\,(\partial_{i}\beta_{j}-\partial_{j}\beta_{i})$
$\displaystyle+$ $\displaystyle
2\,(\partial^{0}F^{i}-\partial^{i}F^{0})\,\beta_{i}-(\pm\,m\,F^{0}-\partial^{0}F)\,\beta+3\,\dot{B}_{2}\,C_{2}-B_{2}\,\dot{C}_{2}+B\,F^{0}$
$\displaystyle-$ $\displaystyle
B_{1}\,f^{0}-\frac{1}{2}B^{0}f+B^{0i}\,f_{i}+\frac{1}{2}\,(\pm\,m\,\beta^{0}-\partial^{0}\beta)F-(\partial^{0}\beta^{i}-\partial^{i}\beta^{0})\,F_{i}\Big{]}.$

It will be noted that we have not touched several terms of the Noether
conserved charge $Q_{b}$ [cf. Eq. (76)] because these terms are BRST
invariant. For instance, we have the following explicit observations:

$\displaystyle
s_{b}\,\Big{[}(\partial^{0}C^{ij}+\partial^{i}C^{j0}+\partial^{j}C^{0i})\,B_{ij}\Big{]}=0,\quad
s_{b}\,\Big{[}\big{\\{}\pm\,m\,C^{0i}-(\partial^{0}C^{i}-\partial^{i}C^{0})\big{\\}}\,B_{i}\Big{]}=0,$
$\displaystyle s_{b}\Big{[}(\pm\,m\,F^{0}-\partial^{0}F)\,\beta\Big{]}=0,\quad
s_{b}\,(B_{1}\,f^{0})=0,\quad s_{b}\,(B^{0}\,f)=0,\quad
s_{b}\,(B^{0i}\,f_{i})=0,$ $\displaystyle
s_{b}\,\Big{[}(\pm\,m\,\beta^{0}-\partial^{0}\beta)\,F\Big{]}=0,\quad
s_{b}\,\Big{[}(\partial^{0}\beta^{i}-\partial^{i}\beta^{0})\,F_{i}\Big{]}=0,\quad
s_{b}\,(\dot{B}_{2}\,C_{2})=0,$ $\displaystyle
s_{b}\,(B_{2}\,\dot{C}_{2})=0,\quad s_{b}\,(B\,F^{0})=0.$ (101)

It is straightforward to check that the following is true, namely;

$\displaystyle
s_{b}\,Q_{b}^{(1)}=-\,i\,\\{Q_{b}^{(1)},\,Q_{b}^{(1)}\\}=0\;\;\Longrightarrow\;\;[Q_{b}^{(1)}]^{2}=0,$
(102)

where the l.h.s. is computed directly by using the BRST symmetry
transformations (71) on the expression for the modified version of the BRST
charge $Q_{b}^{(1)}$.

We follow the prescriptions and proposal outlined above to compute the exact
expression for the off-shell nilpotent version of the anti-BRST charge
$Q_{ab}^{(1)}$ from the non-nilpotent conserved Noether anti-BRST charge
$Q_{ab}$ as follows:

$\displaystyle Q_{ab}\longrightarrow Q_{ab}^{(1)}$ $\displaystyle=$
$\displaystyle\int\,d^{D-1}\,x\,\Big{[}(\partial^{0}\bar{B}^{ij}+\partial^{i}\bar{B}^{j0}+\partial^{j}\bar{B}^{0i})\,\bar{C}_{ij}-\,[\,\pm\,m\,\bar{B}^{0i}-(\partial^{0}\bar{B}^{i}-\partial^{i}\bar{B}^{0})]\,\bar{C}_{i}$
(103) $\displaystyle-$
$\displaystyle(\partial^{0}\bar{C}^{ij}+\partial^{i}\bar{C}^{j0}+\partial^{j}\bar{C}^{0i})\,\bar{B}_{ij}+[\,\pm\,m\,\bar{C}^{0i}-(\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0})]\,\bar{B}_{i}$
$\displaystyle+$
$\displaystyle[\,\pm\,m\,C^{0i}-(\partial^{0}C^{i}-\partial^{i}C^{0})]\,(\pm\,m\,\bar{\beta}_{i}-\partial_{i}\bar{\beta})+2\,(\partial^{0}\beta^{i}-\partial^{i}\beta^{0})\,\partial_{i}\bar{C}_{2}$
$\displaystyle\mp$ $\displaystyle
m\,(\pm\,m\,\beta^{0}-\partial^{0}\beta)\,\bar{C}_{2}-(\partial^{0}C^{ij}+\partial^{i}C^{j0}+\partial^{j}C^{0i})\,(\partial_{i}\bar{\beta}_{j}-\partial_{j}\bar{\beta}_{i})$
$\displaystyle+$ $\displaystyle
2\,(\partial^{0}\bar{F}^{i}-\partial^{i}\bar{F}^{0})\,\bar{\beta}_{i}-(\pm\,m\,\bar{F}^{0}-\partial^{0}\bar{F})\,\bar{\beta}-3\,\dot{B}\,\bar{C}_{2}+B\,\dot{\bar{C}}_{2}-B_{2}\,\bar{F}^{0}$
$\displaystyle-$ $\displaystyle
B_{1}\,\bar{f}^{0}-\frac{1}{2}\,\bar{B}^{0}\bar{f}+\bar{B}^{0i}\,\bar{f}_{i}+\frac{1}{2}\,(\pm\,m\,\bar{\beta}^{0}-\partial^{0}\bar{\beta})\,\bar{F}-(\partial^{0}\bar{\beta}^{i}-\partial^{i}\bar{\beta}^{0})\bar{F}_{i}\Big{]}.$

It is now straightforward to check that:

$\displaystyle
s_{ab}Q_{ab}^{(1)}=-\,i\,\\{Q_{ab}^{(1)},\,Q_{ab}^{(1)}\\}=0\;\;\Longrightarrow\;\;[Q_{ab}^{(1)}]^{2}=0.$
(104)

The above observation proves that we have derived the off-shell nilpotent
version $[Q_{ab}^{(1)}]$ of the non-nilpotent Noether conserved charge
$Q_{ab}$ in a precise and logical manner. Our final results are the
expressions for $Q_{(a)b}^{(1)}$ in (104) and (100).

##  6 Conclusions

In our present investigation, for a few physically interesting gauge systems,
we have shown that wherever there is existence of the coupled (but equivalent)
Lagrangians/Lagrangian densities due to the presence of the (anti-)BRST
invariant CF-type restriction(s), we observe that the Noether theorem does not
lead to the derivation of the off-shell nilpotent versions of the conserved
(anti-)BRST charges $[Q_{(a)b}]$ within the framework of BRST formalism. These
Noether conserved charges are found to be the generators for the
infinitesimal, continuous and off-shell nilpotent (anti-)BRST symmetry
transformations from which they are derived by exploiting the theoretical
strength of Noether’s theorem. However, they are found to be non-nilpotent in
the sense that $s_{b}\,Q_{b}=-i\,\\{Q_{b},\,Q_{b}\\}\neq 0$ and
$s_{ab}\,Q_{ab}=-i\,\\{Q_{ab},\,Q_{ab}\\}\neq 0$ which can be verified by
applying the (anti-)BRST transformations directly on the Noether conserved
charges $Q_{(a)b}$. In other words, by directly computing the explicit
expressions for $s_{b}\,Q_{b}$ and $s_{ab}\,Q_{ab}$, which are the l.h.s. of:
$s_{b}\,Q_{b}=-i\,\\{Q_{b},\,Q_{b}\\}$ and
$s_{ab}\,Q_{ab}=-i\,\\{Q_{ab},\,Q_{ab}\\}$, we show that these charges are not
nilpotent [see, e.g. Eq. (77)]. However, a close and careful look at them
demonstrate that $s_{b}\,Q_{b}$ and $s_{ab}\,Q_{ab}$ are proportional to the
EL-EoM. For instance, we are sure that, in the simple case of a BRST-invariant
1D massive spinning relativistic particle, the r.h.s. of (14) are zero due to
the EL-EoMs (15) and (23) that are derived from $L_{b}$ and $L_{\bar{b}}$,
respectively.

In the derivations of the off-shell nilpotent versions of the (anti-)BRST
charges $[Q_{(a)b}^{(1)}]$, the crucial roles are played by (i) the Gauss
divergence theorem, (ii) the appropriate EL-EoMs from the appropriate
Lagrangian/Lagrangian density of a set of coupled (but equivalent)
Lagrangians/Lagrangian densities, and (iii) the application of the (anti-)BRST
symmetry transformations at appropriate places (cf. Sec. 5 for details). We
lay emphasis on the fact that, for the D-dimensional (D $\geq$ 2) higher
$p$-form ($p=1,2,3,...$) gauge theories, it is imperative to first exploit the
Gauss divergence theorem so that we can use the EL-EoMs w.r.t. the gauge
field. For the 1D case of a spinning (i.e. SUSY) relativistic particle, we
have shown that the Gauss divergence theorem, for obvious reasons, is not
required at all and we directly use the EL-EoMs, right in the beginning,
w.r.t. the “gauge” and “supergauge” variables. This is not the case with the
rest of the examples considered in our present investigation which are
connected with the D-dimensional (D $\geq$ 2) (non-)Abelian $p$-form
$(p=1,2,3,...)$ massless and massive gauge theories.

The sequence of our proposal, to obtain the off-shell nilpotent versions of
the (anti-) BRST charges from the non-nilpotent Noether (anti-)BRST charges,
is as follows. In the first step, we apply the Gauss divergence theorem and
take the help of EL-EoM w.r.t. the gauge field. This is the crucial and key
first step of our proposal. In the next (i.e. second) step, we observe
carefully whether there are any addition, subtraction and/or cancellation of
the resulting terms (from the first step) with any of the terms of the non-
nilpotent Noether conserved charges. The existing terms, after these two
steps, will always be present in the off-shell nilpotent version of the
(anti-)BRST charges $Q_{(a)b}^{(1)}$. After the above two steps, we apply the
(anti-)BRST symmetry transformations on the existing terms. In the third step,
we modify some of the appropriate terms of the non-nilpotent Noether conserved
(anti-)BRST charges and see to it that a part of these modified terms cancel
precisely with the terms that have appeared after the application of the
nilpotent (anti-)BRST transformations on the existing terms (after the first
two steps of our proposal). The parts which participate in the above
cancellation are also always present in the off-shell nilpotent versions of
(anti-)BRST charges $[Q_{(a)b}^{(1)}]$. After this step, it is the interplay
amongst the Gauss divergence theorem, appropriate††††††It is worthwhile to
point out here that the number of fields and corresponding equations of motion
become too many even in the case of the modified version of a massive Abelian
3-form gauge theory (see. e.g. [26, 27] for details) and it becomes very
difficult to know which equation of motion will be picked-up, from amongst
these total number of EL-EoMs, to render the non-nilpotent versions of the
Noether conserved charges into the off-shell nilpotent versions of the
conserved charges. Moreover, the expressions for the conserved Noether charges
themselves become very complicated and cumbersome (see, e.g. [26] for details)
when we discuss the modified massive higher $p$-form ($p=2,3,...$) gauge
theories. EL-EoMs and application of the nilpotent (anti-) BRST symmetry
transformations at appropriate places that lead to the derivation of the
precise forms of the off-shell nilpotent versions‡‡‡‡‡‡There are two ways by
which $s_{b}\,Q_{b}^{(1)}=0$ and $s_{ab}\,Q_{ab}^{(1)}=0$ can be proven
because of the integration present in the expressions for $Q_{(a)b}^{(1)}$.
The first option is a clear-cut proof that the total integrand turns out to be
zero when we apply $s_{(a)b}$ on it. The second option is the case when the
integrand transforms to a total space derivative under the applications of
$s_{(a)b}$. In our present endeavor, it is the first option that has been
chosen where we have proven that $s_{b}\,Q_{b}^{(1)}=0$ and
$s_{ab}\,Q_{ab}^{(1)}=0$. of the (anti-)BRST charges $[Q_{(a)b}^{(1)}]$ from
the non-nilpotent Noether conserved (anti-)BRST charges (cf. Secs. 4 and 5 for
details).

Before we end this section by pointing out our future directions of
investigation in the next paragraph, it is worthwhile to point out the
physical significance of the conserved and off-shell nilpotent (anti-)BRST
charges in the context of a given gauge theory which is endowed with a set of
non-trivial CF-type restrictions(s). The physicality condition
($Q_{B}^{(1)}\mid phys>\,=0$, cf. Sec. 3) ensures that the operator form of
the first-class constraints of the given gauge theory annihilate the physical
state which is consistent with the requirements of the Dirac quantization
condition for a theory endowed with constraints (see, e.g. [5, 6] for
details). As has been shown by Weinberg [19], in the Fock-space, the gauge
transformed states differ from their original counterparts by the BRST-exact
states. Hence, if we have off-shell nilpotent BRST charge, their difference
becomes trivial when we invoke the physicality condition on the states w.r.t.
the BRST charge. This argument has been extended in our earlier research works
(see, e.g. [28, 29]) where we have been able to prove that the 2D
(non-)Abelian 1-form and 4D Abelian 2-form gauge theories are the tractable
field-theoretic models for the Hodge theory. In the case of the 2D
(non-)Abelian 1-form theories, we have been able to show that the BRST and co-
BRST symmetry transformations “gauge away” both the d.o.f. of the gauge fields
and these theories become a new kind of topological field theory (see, e.g.
[30] for details).

We end this section with the final comment that our proposal is very general
and it can be applied to any physical system where (i) the (anti-)BRST
invariant non-trivial CF-type restrictions exist, and (ii) the coupled (but
equivalent) Lagrangians/Lagrangian densities describe the dynamics of the
above physical systems within the framework of BRST formalism. The systematic
application of our proposal sheds light on the appropriate directions that
should be followed in order to obtain the off-shell nilpotent versions of the
(anti-)BRST charges from the non-nilpotent versions of the Noether conserved
(anti-)BRST charges . We plan to extend our present ideas in the context of
more challenging problems of physical interest [31] in the future.

Acknowledgments

One of us (AKR) thankfully acknowledges the financial support from the
Institution of Eminence (IoE) Research Incentive Grant of PFMS Scheme No.
3254-World Class Institutions to which Banaras Hindu University, Varanasi,
belongs. The present investigation has been carried out under the above
Research Incentive Grant which has been launched by the Govt. of India. All
the authors express their deep sense of gratitude to all the Reviewers for
their fruitful comments which have made our presentation more accurate and
beautiful.

Conflicts of Interest
The authors declare that there are no conflicts of interest.

Data Availability

No data were used to support this study.

Appendix A: On the Massive Abelian 2-Form Theory

To supplement our discussions in Sec. 4, in this Appendix, we concisely
mention the off-shell nilpotency property of the non-nilpotent Noether
conserved (anti-)BRST charges for the St$\ddot{u}$ckelberg-modified massive
Abelian 2-form gauge theory within the framework of BRST formalism. For this
purpose, we begin with the generalized (i.e. St$\ddot{u}$ckelberg-modified)
versions of the (anti-)BRST invariant Lagrangian densities of equations (46)
and (47) for the D-dimensional massive Abelian 2-form theory (see, e.g. [32,
27] for details) as

${\cal
L}_{b}=\frac{1}{12}\,H_{\mu\nu\eta}H^{\mu\nu\eta}-\frac{1}{4}\,m^{2}\,B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}\,\Phi_{\mu\nu}\Phi^{\mu\nu}+\frac{1}{2}\,m\,B_{\mu\nu}\Phi^{\mu\nu}-B^{2}$
$~{}~{}~{}~{}-B\left(\partial\cdot\phi+m\,\varphi\right)+B^{\mu}B_{\mu}+B^{\mu}\left(\partial^{\nu}B_{\nu\mu}-\partial_{\mu}\varphi+m\phi_{\mu}\right)-m^{2}\,\bar{\beta}\beta$
$~{}~{}~{}~{}~{}~{}~{}~{}+\left(\partial_{\mu}\bar{C}_{\nu}-\partial_{\nu}\bar{C}_{\mu}\right)\left(\partial^{\mu}C^{\nu}\right)-\left(\partial_{\mu}\bar{C}-m\bar{C}_{\mu}\right)\left(\partial^{\mu}C-mC^{\mu}\right)+\partial_{\mu}\bar{\beta}\,\partial^{\mu}\beta$
$+\left(\partial\cdot\bar{C}+\rho+m\,\bar{C}\right)\lambda+\left(\partial\cdot
C-\lambda+m\,C\right)\rho,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$None$ ${\cal
L}_{\bar{b}}=\frac{1}{12}\,H_{\mu\nu\eta}H^{\mu\nu\eta}-\frac{1}{4}\,m^{2}\,B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}\,\Phi_{\mu\nu}\Phi^{\mu\nu}+\frac{1}{2}\,m\,B_{\mu\nu}\Phi^{\mu\nu}-{\bar{B}}^{2}$
$~{}~{}~{}~{}+\bar{B}\left(\partial\cdot\phi-m\,\varphi\right)+\bar{B}_{\mu}\bar{B}^{\mu}+\bar{B}^{\mu}\left(\partial^{\nu}B_{\nu\mu}+\partial_{\mu}\varphi+m\phi_{\mu}\right)-m^{2}\,\bar{\beta}\beta$
$~{}~{}~{}~{}~{}~{}~{}~{}+\left(\partial_{\mu}\bar{C}_{\nu}-\partial_{\nu}\bar{C}_{\mu}\right)(\partial^{\mu}C^{\nu})-\left(\partial_{\mu}\bar{C}-m\bar{C}_{\mu}\right)\left(\partial^{\mu}C-mC^{\mu}\right)+\partial_{\mu}\bar{\beta}\,\partial^{\mu}\beta$
$+\left(\partial\cdot\bar{C}+\rho+m\,\bar{C}\right)\lambda+\left(\partial\cdot
C-\lambda+m\,C\right)\rho,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$None$

where $\Phi_{\mu\nu}=\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu}$ is the
field-strength tensor for the vector St$\ddot{u}$ckelberg field $\phi_{\mu}$.
There are additional fermionic (anti-)ghost fields
($\bar{C}_{\mu},\,C_{\mu},\,\bar{C},\,C$) and bosonic (anti-)ghost fields
$(\bar{\beta})\beta$ in our theory along with an additional scalar field
$\varphi$ with ghost numbers $(-\,1)+1,(-\,2)+2$ and zero, respectively. There
are a couple of additional Nakanishi-Lautrup type fields $(\bar{B})B$, too. It
will be noted that the massive Abelian 2-form field has the rest mass $m$
which happens to be the mass of the rest of the fields of our theory [32]. The
above Lagrangian densities ${\cal L}_{\bar{b}}$ and ${\cal L}_{b}$ respect the
following off-shell nilpotent $[s_{(a)b}^{2}=0]$ (anti-)BRST symmetry
transformations $[s_{(a)b}]$, namely;

$~{}~{}~{}~{}s_{ab}B_{\mu\nu}=-\,(\partial_{\mu}\bar{C}_{\nu}-\partial_{\nu}\bar{C}_{\mu}),\qquad
s_{ab}\bar{C}_{\mu}=-\,\partial_{\mu}\bar{\beta},\qquad
s_{ab}\phi_{\mu}=\partial_{\mu}\bar{C}-m\,\bar{C}_{\mu},$
$~{}~{}~{}s_{ab}C_{\mu}=\bar{B}_{\mu},\quad s_{ab}\beta=-\,\lambda,\quad
s_{ab}\bar{C}=-\,m\,\bar{\beta},\quad s_{ab}C=\bar{B},\quad
s_{ab}B=-\,m\,\rho,$ $s_{ab}\varphi=\rho,\qquad
s_{ab}B_{\mu}=\partial_{\mu}\rho,\qquad
s_{ab}[\bar{B},\rho,\lambda,\bar{\beta},\bar{B}_{\mu},H_{\mu\nu\kappa}]=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$s_{b}B_{\mu\nu}=-\,(\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}),\qquad
s_{b}C_{\mu}=-\,\partial_{\mu}\beta,\qquad
s_{b}\phi_{\mu}=\partial_{\mu}C-\,m\,C_{\mu},$
$s_{b}\bar{C}_{\mu}=-\,B_{\mu},\quad s_{b}\bar{\beta}=-\,\rho,\quad
s_{b}C=-\,m\,\beta,\quad s_{b}\bar{C}=B,\quad s_{b}\bar{B}=-\,m\,\lambda,$
$s_{b}\varphi=\lambda,\qquad
s_{b}\bar{B}_{\mu}=-\,\partial_{\mu}\lambda,\qquad
s_{b}[B,\rho,\lambda,\beta,B_{\mu},H_{\mu\nu\kappa}]=0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$None$

The above (anti-)BRST symmetries are perfect symmetries for the Lagrangian
densities ${\cal L}_{\bar{b}}$ and ${\cal L}_{b}$, respectively, because we
observe that:

$s_{ab}{\cal
L}_{\bar{b}}=\partial_{\mu}\Big{[}(\partial^{\mu}\bar{C}+m\,\bar{C}^{\mu})\,\bar{B}-(\partial^{\mu}\bar{C}^{\nu}-\partial^{\nu}\bar{C}^{\mu})\,\bar{B}_{\nu}-\lambda\,\partial^{\mu}\bar{\beta}+\rho\,\bar{B}^{\mu}\Big{]},$
$s_{b}{\cal
L}_{b}=-\,\partial_{\mu}\,\Big{[}(\partial^{\mu}C-\,m\,C^{\mu})\,B+(\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu})\,B_{\nu}+\rho\,\partial^{\mu}\beta+\lambda\,B^{\mu}\Big{]}.$
$None$

The above results establish that the action integrals $S_{1}=\int
d^{D}\,x\,{\cal L}_{\bar{b}}$ and $S_{2}=\int d^{D}\,x\,{\cal L}_{b}$ remain
invariant under the (anti-)BRST symmetry transformations $[s_{(a)b}]$,
respectively, due to Gauss’s divergence theorem.

There are a few comments in order here . First, we note that the kinetic term
for the gauge field $(B_{\mu\nu})$, owing its origin to the exterior
derivative (i.e. $H^{(3)}=d\,B^{(2)}$), remains invariant under the
(anti-)BRST symmetry transformations $[s_{(a)b}]$. Second, we differ in some
of our notations, a few terms in the coupled (but equivalent) Lagrangian
densities and signs of the symmetry transformations from the earlier works
[32, 27]. Finally, we note that the (anti-)BRST symmetry transformations
$[s_{(a)b}]$ are off-shell nilpotent $[s_{(a)b}^{2}=0]$ of order two and
absolutely anticommuting (i.e. $\\{s_{b},\,s_{ab}\\}=0$) in nature, namely;

$~{}~{}~{}~{}\\{s_{b},\,s_{ab}\\}B_{\mu\nu}=\partial_{\mu}(B_{\nu}-\bar{B}_{\nu})-\partial_{\nu}(B_{\mu}-\bar{B}_{\mu}),$
$\\{s_{b},\,s_{ab}\\}\Phi_{\mu}=\partial_{\mu}(B+\bar{B})-m\,(B_{\mu}-\bar{B}_{\mu}),$
$None$

provided we invoke the (anti-)BRST invariant

$s_{(a)b}\big{[}B_{\mu}-\bar{B}_{\mu}-\partial_{\mu}\varphi\big{]}=0,\qquad
s_{(a)b}\big{[}B+\bar{B}+m\,\varphi\big{]}=0,$ $None$

CF-type restrictions of our theory for the proof of the absolute
anticommutativity in (A.5).

The Noether conserved (anti-)BRST charges have been computed in [32]. We quote
here these expressions (with appropriate and correct signs) as (see, e.g. [32]
for details):

$Q_{ab}=\int
d^{D-1}x\Big{[}-H^{0ij}\,\big{(}\partial_{i}\bar{C}_{j})-\big{(}\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0}\big{)}\,\bar{B}_{i}+\big{(}\partial^{0}\bar{C}-m\,\bar{C}^{0}\big{)}\,\bar{B}$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\big{(}\partial_{i}\bar{C}-m\,\bar{C}_{i}\big{)}\,\big{(}\Phi^{0i}-m\,B^{0i}\big{)}+m\,\big{(}\partial^{0}C-m\,C^{0}\big{)}\,\bar{\beta}$
$-\big{(}\partial^{0}C^{i}-\partial^{i}C^{0}\big{)}(\partial_{i}\bar{\beta})-\lambda\,\partial^{0}\bar{\beta}+\rho\,\bar{B}^{0}\Big{]},$
$Q_{b}=\int
d^{D-1}x\Big{[}-H^{0ij}\,\big{(}\partial_{i}C_{j})-\big{(}\partial^{0}C^{i}-\partial^{i}C^{0}\big{)}\,B_{i}-\big{(}\partial^{0}C-m\,C^{0}\big{)}\
B$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\big{(}\Phi^{0i}-m\,B^{0i}\big{)}\,\big{(}\partial_{i}C-m\,C_{i}\big{)}-m\,\big{(}\partial^{0}\bar{C}-m\,\bar{C}^{0}\big{)}\,\beta$
$~{}~{}~{}+\big{(}\partial^{0}\bar{C}^{i}-\partial^{i}\bar{C}^{0}\big{)}(\partial_{i}\beta)-\rho\,\partial^{0}\beta-\lambda\,B^{0}\Big{]}.$
$None$

These Noether conserved charges are the generators for all the (anti-)BRST
symmetry transformations (A.3) which have been explicitly quoted in (A.3). At
this stage, we observe that the above Noether conserved (anti-)BRST charges
are non-nilpotent. This can be checked explicitly by applying the BRST
symmetry transformations ($s_{b}$) on $Q_{b}$ and anti-BRST symmetry
transformation ($s_{ab})$ on $Q_{ab}$ as illustrated below:

$s_{b}Q_{b}=-\,i\,\\{Q_{b},\,Q_{b}\\}=-\,\int
d^{D-1}x\,\Big{[}m\,(\partial^{0}B+m\,B^{0})\,\beta+(\partial^{0}B^{i}-\partial^{i}B^{0})\,\partial_{i}\beta\Big{]}\neq
0,~{}~{}~{}~{}~{}~{}~{}~{}$ $s_{ab}Q_{ab}=-\,i\,\\{Q_{ab},\,Q_{ab}\\}=\int
d^{D-1}x\,\Big{[}m\,(\partial^{0}\bar{B}-m\,\bar{B}^{0})\,\bar{\beta}-(\partial^{0}\bar{B}^{i}-\partial^{i}\bar{B}^{0})\,\partial_{i}\bar{\beta}\Big{]}\neq
0.$ $None$

Thus, we note that the Noether (anti-)BRST charges (i) are conserved
quantities, (ii) are the generators [28] for the appropriate (anti-)BRST
symmetry transformations, and (iii) are found to be not nilpotent of order two
(i.e. $Q_{b}^{2}\neq 0,\;Q_{ab}^{2}\neq 0$).

In what follows, we follow a systematic method to convert the non-nilpotent
Noether conserved (anti-)BRST charges (A.7) into their off-shell nilpotent
versions. Since we have demonstrated our method in the context of the BRST
charge for the St$\ddot{u}$ckelberg-modified massive Abelian 3-form theory
(cf. Sec. 5), we derive the off-shell nilpotent version [$Q_{ab}^{(1)}$] of
the no-nilpotent Noether anti-BRST $Q_{ab}$ following our proposal. First of
all, we apply the Gauss divergence theorem whch turns the first term of
$Q_{ab}$ [cf. Eq. (A.7)] as

$(\partial_{i}H^{0ij})\,\bar{C}_{j}+m\,(\Phi^{0i}-m\,B^{0i})\,\bar{C}_{i}=(\partial^{0}\bar{B}^{i}-\partial^{i}\bar{B}^{0})\,\bar{C}_{i},$
$None$

where we have used the following EL-EoM that is derived from ${\cal
L}_{\bar{b}}$

$\partial_{\mu}H^{\mu\nu\lambda}+(\partial^{\nu}\bar{B}^{\lambda}-\partial^{\nu}\bar{B}^{\lambda})-m\,(\Phi^{\nu\lambda}-m\,B^{\nu\lambda})=0.$
$None$

We claim that the r.h.s. of (A.9) will be present in the off-shell nilpotent
version of the anti-BRST charge [$Q_{ab}^{(1)}$].

Now let us focus on the remaining part of the fourth term inside the
integration. We note that, using the Gauss divergence theorem, we obtain

$-\,(\Phi^{0i}-m\,B^{0i})\,\partial_{i}\,\bar{C}\equiv\partial_{i}\,[(\Phi^{0i}-m\,B^{0i})\,\bar{C}].$
$None$

At this stage, we use the following equation of motion in the above:

$\partial_{\mu}\,\Phi^{\mu\nu}-\partial^{\nu}\,\bar{B}-m\,(\partial_{\mu}\,B^{\mu\nu})+m\,\bar{B}^{\nu}=0.$
$None$

The appropriate substitution leads to the following

$\partial_{i}\,[(\Phi^{0i}-m\,B^{0i})\,\bar{C}]=-\,\dot{\bar{B}}\,\bar{C}+m\,\bar{B}^{0}\,\bar{C}\equiv-\,(\dot{\bar{B}}-m\,\bar{B}^{0})\,\bar{C}.$
$None$

Ultimately, the sum of the first term and fourth term [i.e. sum of (A.10) and
(A.13)] leads to the following two terms:

$+\,(\partial^{0}\,\bar{B}^{i}-\partial^{i}\,\bar{B}^{0})\,\bar{C}_{i}-(\dot{\bar{B}}-m\,\bar{B}^{0})\,\bar{C}.$
$None$

These two terms will stay in the off-shell nilpotent version of the anti-BRST
charge [$Q_{ab}^{(1)}$]. If we apply an anti-BRST symmetry transformations
$s_{ab}$ on (A.14), we obtain the following

$s_{ab}\,[(\partial^{0}\,\bar{B}^{i}-\partial^{i}\,\bar{B}^{0})\,\bar{C}_{i}-(\dot{\bar{B}}-m\,\bar{B}^{0})\,\bar{C}]=-\,(\partial^{0}\,\bar{B}^{i}-\partial^{i}\,\bar{B}^{0})\,\partial_{i}\,\bar{\beta}+m\,(\dot{\bar{B}}-m\,\bar{B}^{0})\,\bar{\beta}.$
$None$

We have to look, according to our proposal, whether the above terms in (A.15)
can cancel and/or can be added and/or subtracted with any terms of $Q_{ab}$
(i.e. Noether charge). We find that the following modifications of the fifth
and sixth terms, namely;

$+m\,(\partial^{0}\,C-m\,C^{0})\,\bar{\beta}-(\partial^{0}\,C^{i}-\partial^{i}\,C^{0})\,\partial_{i}\,\bar{\beta}\equiv
2\,m\,(\partial^{0}\,C-m\,C^{0})\,\bar{\beta},$
$-2\,(\partial^{0}\,C^{i}-\partial^{i}\,C^{0})\,\partial_{i}\,\bar{\beta}-\,m\,(\partial^{0}\,C-m\,C^{0})\,\bar{\beta}+(\partial^{0}\,C^{i}-\partial^{i}\,C^{0})\,\partial_{i}\,\bar{\beta},$
$None$

will enable us to observe that the anti-BRST symmetry transformations $s_{ab}$
on the last two terms of the above equation, namely;

$s_{ab}[(\partial^{0}C^{i}-\partial^{i}C^{0})\,\partial_{i}\bar{\beta}-m\,(\partial^{0}C-m\,C^{0})\,\bar{\beta}]=(\partial^{0}\bar{B}^{i}-\partial^{i}\bar{B}^{0})\,\partial_{i}\bar{\beta}-m\,(\dot{\bar{B}}-m\,\bar{B}^{0})\,\bar{\beta},$
$None$

cancels out with the ones that have been obtained in (A.15). Hence, we note
that, in addition to the terms in (A.14), the terms in the square bracket of
(A.17) will also stay in the off-shell nilpotent version [$Q_{ab}^{(1)}$] of
the anti-BRST charge.

We concentrate now on the left-over terms of (A.16) which are as follows

$2\,m\,(\partial^{0}C-m\,C^{0})\,\bar{\beta}-2\,(\partial^{0}C^{i}-\partial^{i}C^{0})\,\partial_{i}\bar{\beta}.$
$None$

Using the Gauss divergence theorem, we note that the last term can be written
as

$2\,\partial_{i}(\partial^{0}C^{i}-\partial^{i}C^{0})\,\bar{\beta},$ $None$

because both the terms of (A.18) are inside the integration. At this stage, we
use the following EL-EoM derived from ${\cal L}_{\bar{b}}$, namely;

$\partial_{\mu}(\partial^{\mu}C^{\nu}-\partial^{\nu}C^{\mu})+\partial^{\nu}\lambda-m\,(\partial^{\nu}C-m\,C^{\nu})=0,$
$None$

which leads to the following for the choice $\nu=0$, namely;

$\partial_{i}(\partial^{0}C^{i}-\partial^{i}C^{0})\,\bar{\beta}=2\,\dot{\lambda}\,\bar{\beta}-2\,m\,(\dot{C}-m\,C^{0})\,\bar{\beta}.$
$None$

The above relationship plays an important and decisive role. For instance, the
substitution of the above into equation (A.18) leads to

$2\,m\,(\dot{C}-m\,\,C^{0})\,\bar{\beta}+2\,\partial_{i}\,(\partial^{0}C^{i}-\partial^{i}C^{0})\,\bar{\beta}=2\,\dot{\lambda}\,\bar{\beta}.$
$None$

We find that if we apply an anti-BRST symmetry transformation $(s_{ab})$ on
the r.h.s. of (A.22), it turns out to be zero. Thus, the theoretical tricks of
our proposal terminate here. Hence, the term ($2\,\dot{\lambda}\,\bar{\beta}$)