\nonumber
&&s_{d}\,C\,=\,-\frac{i}{2}(\dot\phi\,-\,\dot v\,+\,v'\,-\,\phi'),\qquad
s_{d}\,(\dot \phi\,-\,\dot v\,+\,v'\,-\,\phi')\,=\,0,\end{aligned}$$ because the Lagrangian density (8) transforms to the total “time" derivatives as $$\begin{aligned}
&&s_{ad}\,{\cal L}_{B}^{(s)}\,=\,\frac{\partial}{\partial t}\,\Big[\frac {\dot C}{2}\,
(\dot \phi\,-\,\dot v\,+\,v'\,-\,\phi')\Big],\\ \nonumber
&&s_{d}\,{\cal L}_{B}^{(s)}\,=\,\frac{\partial}{\partial t}\,\Big[\frac {\dot {\bar C}}{2}\,
(\dot \phi\,-\,\dot v\,+\,v'\,-\,\phi')\Big].\end{aligned}$$ Thus, the action integral $S\,=\,\int d^{2}x\,{\cal L}_{B}^{(s)}$ remains invariant under $s_{(a)d}$ for the physical fields that vanish off at $t = \pm \infty $.

The decisive features of the (anti-)dual BRST \[i.e. (anti-)co-BRST\] symmetry transformations are the observations that (i) they are nilpotent of order two (i.e. $ s_{(a)d}^2\,=\,0$) which demonstrates their fermionic nature, (ii) these nilpotent symmetries are also absolutely anticommuting $(s_{d}\,s_{ad}\,+\,s_{ad}\,s_{d}\,=\,0)$ in nature that shows the linear independent of $s_{d}$ and $s_{ad}$, (iii) the gauge-fixing terms, owing their origin to the co-exterior derivative (see, e.g. \[16-20\]), remain invariant under the (anti-)dual BRST symmetry transformations $(s_{(a)d})$. Thus, the nomenclature (anti-)co-BRST symmetries is appropriate for these symmetries. This observation should be contrasted with the (anti-)BRST symmetries where the total kinetic term, owing its origin to the exterior derivative, remains invariant \[16-20\].

Nilpotent (Anti-)co-BRST Symmetries: Superfield Approach to the Abelian 1-Form Gauge Theories
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We briefly discuss here the derivation of the (anti-)co-BRST symmetries of our 1-form gauge theories by exploiting the geometrical superfield 