Drozdov in [@Bordag2003:prd] computed the vacuum energy due to purely fermionic fluctuations on a Nielsen-Olesen vortex. Together with other colleague we performed similar calculations in the purely bosonic planar AHM in References [@Alonso2004:prd] and [@Alonso2005:prd]. We used the spectral heat kernel/zeta function regularization procedure to control the divergences, both ultraviolet and infrared, arising in the computation of vacuum energies caused by one-loop fluctuations of BPS vortices, as well as those associated to tadpole and self-energy graphs. Invented by Hawking [@Hawking1977:cmp] and Dowker et al [@Dowker1976:prd] to describe quantum fields in curved space-times this method was used for the first time in the analysis of quantum fluctuations of kinks and solitons by van Nieuwenhuizen et al in [@Bordag2002:prd] within a $N=2$ supersymmetric framework. We took profit of these ideas to calculate the quantum corrections to the masses of several types of topological kinks in scalar field models with different number of fields in References [@Alonso2002:npb; @Alonso2002:npb2; @Alonso2004:npb].

The vortex Casimir energy is the main ingredient in the formula giving the vortex mass (2D) or string tension (3D) quantum corrections. It is formally given by the trace, both in the matrix and the $L^2$-functional sense, of the square root of the matrix 