main goals in this paper is to compute one-loop vortex mass shifts. In References [@Alonso2004:prd; @Alonso2005:prd; @Alonso2008:npb], see also the reviews [@Alonso2006:hepth; @Mateos2009:pos], we performed these calculations by applying the spectral zeta function regularization procedure to the second-order small vortex fluctuation operator ${\cal H}^+$ both in the Abelian Higgs model and in Semilocal Abelian gauge systems. The scheme developed by our group was based in the standard Gilkey-de Witt heat kernel asymptotic expansion. An important obstacle found in developing this program is that the Gilkey-de Witt approach is well established only for operators with strictly positive spectrum and the operator ${\cal H}^+$ exhibits zero modes. In the papers [@Alonso2012:epjc; @Alonso2014:jhep] two of us improved on the Gilkey-de Witt expansion by showing how to generalize the method to cope with the existence of zero modes. Application of the generalized Gilkey-de Witt heat kernel asymptotic expansion to the computation of one-loop kink mass shifts showed a remarkably better precision and unveiled the appearance of forces between kinks of pure quantum nature.

In this Section, having in the back of the mind computations of BPS vortex mass shifts, we shall generalize the standard Gilkey-de Witt heat kernel expansion to operators with zero modes 