\quad s_{ad}\,C\,=\,0,\quad s_{ad}\,\bar C
\,=\,i\,(E\,-\,m\,\tilde \phi), \\ \nonumber
&&s_{ad}\,E\,=\,\square\, C,\quad s_{ad}\,(\partial \cdot A\,+\,m\,\phi)\,=\,0,\quad 
s_{ad}\,\phi\,=\,0, \quad s_{ad}\,\tilde \phi\,=\,-m\,C, \\\nonumber
&&s_{d}\,A_{\mu}\,=\,-\varepsilon_{\mu\nu}\,\partial^\nu\,\bar C, \quad s_{d}\,\bar C\,=\,0,
\quad s_{d}\,C\,=\,-\,i\,(E\,-\,m\,\tilde \phi), \\ \nonumber
&&s_{d}\,E\,=\,\square\, \bar C,\quad s_{d}\,(\partial \cdot A\,+\,m\,\phi)\,=\,0,\quad 
s_{d}\,\phi\,=\,0, \quad s_{d}\,\tilde \phi\,=\,-m\,\bar C,\end{aligned}$$ leave the action integral invariant because the Lagrangian density transforms to the total spacetime derivatives (see, e.g. \[17\],\[18\] for details). It is to be noted that the total gauge-fixing term remains invariant under $s_{(a)d}$ \[i.e. $\,s_{(a)d}\,(\partial \cdot A + m \phi) = 0$\].

Another modified version of the 2$D$ Abelian 1-form model is the bosonized version of anomalous Abelian 1-form gauge theory which is described by the following (anti-)BRST invariant Lagrangian density (see, e.g., \[19\] for details) $$\begin{aligned}
\mathcal{L}_{B}^{(a)}\,&=&\,-\,\frac{1}{4}\,F^{\mu\nu}\,F_{\mu\nu}\,
+\,\frac{1}{2}\,\partial_{\mu}\phi\,\partial^{\mu}\phi\,
+\,\frac{a}{2}\,A_{\mu}\,A^{\mu}\\ \nonumber
&+&\,(\eta^{\mu\nu}\,-\,\varepsilon^{\mu\nu})\,\partial_{\mu}\phi\,A_{\nu}\,+\,
\sigma \,[(a-1)\,(\partial \cdot A)\,+\,\varepsilon^{\mu\nu}\,\partial_{\mu} \,A_{\nu}] \\ \nonumber
&+&\frac{(a-1)}{2}\,\partial_{\mu}\,\sigma\,\partial^{\mu}\,\sigma\,+\,B\,(\partial \cdot A)\,+\,
\frac{B^2}{2} \,+\,i\,\partial_{\mu}\,\bar C\,\partial^{\mu}\,C,\end{aligned}$$ where, as explained earlier, the 2-form curvature $F_{\mu\nu}$ has only electric field as its existing component and $a$ is the ambiguity parameter in the regularization of the fermionic determinant when the 2$D$ chiral Schwinger model (with electric charge $e = 1$) is bosonized in terms of the scalar field $\phi$. We have introduced an extra 2$D$ bosonic field $\sigma(x)$ in the theory to convert the 