mass parameter $m$) which is described by the following (anti-)BRST invariant Lagrangian density (see, e.g. \[17, 18\] for details) $$\begin{aligned}
\mathcal L_{B}^{(p)} \,&=&\,\frac {1}{2} (E\,-\,m\,\tilde \phi)^2\,+\,m\,E\,\tilde\phi\,-\,
\frac {1}{2}\,\partial_{\mu}\,\tilde \phi\,\partial^{\mu}\,\tilde \phi\,+\,\frac {m^2}{2}\,A_{\mu}\,A^{\mu}\,
+\,\frac{1}{2}\,\partial_{\mu}\,\phi\,\partial^{\mu}\,\phi \\ \nonumber
&-&\,m\,A_{\mu}\,\partial^{\mu}\,\phi\,-\,\frac{1}{2}\,(\partial \cdot A\,+\,m\,\phi)^2
\,-\,i\,\partial_{\mu}\,\bar C\,\partial^{\mu}\,C\,+\,i\,m^2\,\bar C\,C,\end{aligned}$$ where the 1-form $A^{(1)}\,=\,dx^{\mu}\,A_{\mu}$ defines the 2$D$ gauge potential $A_{\mu}$ and the corresponding curvature tensor $F_{\mu\nu}$ is defined from the 2-form $F^{(2)}\,=\,dA^{(1)}\,=\,
[(dx^{\mu}\wedge dx^{\nu}/2!)\,F_{\mu\nu}]$ where $d\,=\,dx^{\mu}\partial_{\mu}$ (with $d^2\,=\,0$) is the exterior derivative. In 2$D$, the curvature tensor $F_{\mu\nu}$ contains only [*one*]{} independent component which is nothing but the electric field $E$. The latter turns out to be a [*pseudoscalar*]{} in two (1 + 1)-dimensions of spacetime. In the above, we have a pair $(\phi, \tilde \phi)$ of fields which is constructed by a scalar Stueckelberg field $\phi$ and a pseudoscalar field $\tilde \phi$. The latter has been introduced in the theory on the physical as well as mathematical grounds \[17,18\]. The fermionic $(C^2\,=\,\bar C^2\,=\,0, \,\,C\,\bar C\,+\,\bar C\,C\,=\,0)$ fields are the (anti-)ghost fields $(\bar C)\,C$ which are required to maintain the unitarity in the theory. It can be readily checked that the following nilpotent ($ s_{(a)d}^2 = 0$) and absolutely anticommuting $(s_{d} s_{ad}\,+\,s_{ad} s_{d}\,=\,0)$ (anti-)dual BRST symmetry transformations $(s_{(a)d})$ $$\begin{aligned}
&&s_{ad}\,A_{\mu}\,=\,-\varepsilon_{\mu\nu}\,\partial^\nu\,C, 