to some extent, dimensional regularization technique with Chowla-Selberg lattice summation [@Milton2010].

Integrals in Eq.  can be evaluated within dimensional regularization scheme. The first integral can be recast with the substitution $q^2=\frac{\alpha_{\sigma,n}^2}{R^2}+k^2$, as $I_1=2\int_{\alpha_{\sigma,n/R}}^\infty \frac{dq}{2\pi}q^2(q^2-\frac{\alpha_{\sigma,n}^2}{R^2})^{-1/2}$, which for the sake of dimensional regularization, can be further recast as $I_1=2\int_{\alpha_{\sigma,n/R}}^\infty \frac{dq}{2\pi}q^{-2s}(q^2-\frac{\alpha_{\sigma,n}^2}{R^2})^{-d}$ where $s=-1$ and $d=1/2$. It is easy to check that, this integral diverges for $s=-1$ and $d=1/2$. This divergence can be avoided within dimensional regularization scheme by substituting $d=1/2+\epsilon$ (where $\epsilon\rightarrow0$) and $s=-1+\delta$ (where $\delta\rightarrow0$), and by subsequently performing analytic continuation of gamma and zeta functions for $\Re[s]+d\le1/2$. In a similar way, we can regularize the second integral in Eq. . Thus within the dimensional regularization scheme, Eq.  can be recast as $$\begin{aligned}
\label{eqn:13o0}
E=L\hbar c[D^{(-2s)}+N^{(-2s)}], \end{aligned}$$ where keeping $s\rightarrow-1$ unaltered we put $d=1/2$ after evaluating both the integrals in Eq. , as, $$\begin{aligned}
\label{eqn:13o1}
D^{(-2s)}=\frac{1}{4\sqrt{\pi}}\sum_{\sigma=-\infty,n=1}^{\infty,\infty}\left(\frac{\alpha^2_{\sigma,n}}{R^2}\right)^{-s}\frac{\Gamma(s)}{\Gamma(\frac{1+2s}{2})},\end{aligned}$$ and $$\begin{aligned}
\label{eqn:13o2}
N^{(-2s)}=\frac{1}{4\sqrt{\pi}}\sum_{\sigma=-\infty,n=0}^{\infty,\infty}\left(\frac{\beta^2_{\sigma,n}}{R^2}\right)^{-s}\frac{\Gamma(s)}{\Gamma(\frac{1+2s}{2})}.\end{aligned}$$ Here ‘D’ and ‘N’ respectively stand for Dirichlet and Neumann boundary conditions. Regularization of $D^{(-2s)}$ in Eq.  and that of $N^{(-2s)}$ in Eq. , with the use of Chowla-Selberg lattice-sum formula [@Chowla-Selberg], needs exact values of $\alpha_{\sigma,n}$ and $\beta_{\sigma,n}$ which although appear in closed forms for triangular-cylinder, square-cylinder, *etc*, unfortunately do not 