Spectral methods for coupled channels with a mass gap

Abstract

We develop a method to compute the vacuum polarization energy for coupled scalar fields with different masses scattering off a background potential in one space dimension. As an example we consider the vacuum polarization energy of a kinklike soliton built from two real scalar fields with different mass parameters.

Vacuum polarization energy of the Shifman–Voloshin soliton

“…While the classical energy does not vary with these parameters, the vacuum polarization energy diverges logarithmically to negative infinity as a → 1 unless the two masses are equal. We have employed (generalized) spectral methods [8,9] for this computation but note that our results are consistent with those obtained in a heat kernel expansion [10][11][12]. However, those studies did not identify the importance and origin of this divergence, 3 which implies that for any finite value of λ, we can find a 1 such that the total energy is negative, and hence the quantum corrections destabilize the solitons.…”

“…We thus find that the various known analytical solutions are not independent but are related by a single parameter. If these solitons were independent, a third zero mode for the small amplitude fluctuations about the soliton along the direction in field space connecting the solutions would have emerged, but only two have been observed [8]. Stated otherwise, the solitons are parameterized by two continuous parameters [6]: the center of the soliton, which we set to zero, and the amplitude of the χ field, which we parameterize by a. Varying these parameters produces the two observed zero modes.…”

Quantum Corrections to Soliton Energies

“…In Ref. 49 it has been shown that these two seemingly contradictory relations can be consistently combined as…”

“…This prescription leads to additional singularities, but they occur only for momenta with negative imaginary parts, and it passes numerous consistency checks. 49 Hence it is straightforwardly permissible to analytically continue within the upper half plane with k = it and…”

Quantum instabilities of solitons

“…To compute the VPE this requires further extensions of the spectral methods that are described and applied in Refs. [5,15]. Some of the numerical results of that analysis are shown in the right panel of figure 3.…”