Isospin invariance and the vacuum polarization energy of cosmic strings

We corroborate the previously applied spectral approach to compute the vacuum polarization energy of string configurations in models similar to the standard model of particle physics. The central observation underlying this corroboration is the existence of a particular global isospin transformation of the string configuration. Under this transformation the single particle energies of the quantum fluctuations are invariant, while the inevitable implementation of regularization and renormalization requires operations that are not invariant. We verify numerically that all such variances eventually cancel, and that the vacuum polarization energy obtained in the spectral approach is indeed gauge invariant.

Figure 1

Partial sums that enter the VPE, Eq. (19), for the string profiles with $w_G=w_H=4.82$, ${\xi }_1=0.3\pi$, and ${\xi }_2=0.25\pi$. The left panel shows the fermion part for four different values of $w_{\xi }$. The right panel shows the total integrand for three values of $w_{\xi }$ relative to $w_{\xi }=2.0$. The double-dashed lines that start at $\tau =2$ in the right panel are simple power decays which serve to guide the eye on the large $\tau$ behavior. Note the different scales in the two panels.

Figure 2

The fermion part of the momentum integrand in Eq. (19) (similar to the left panel of Fig. 1). The selected width parameters are $w_H=w_G=3.5$.

Figure 3

The vacuum polarization energy for different background profiles with the two isospin angles swapped. In the right panel we zoom in by omitting narrow profiles that suffer from the Landau ghost problem [18, 33, 34]. Details of the profiles are listed in Appendix pp2.

Figure 4

Same as Fig. 3 for a second pair of isospin angles.