Abstract
We derive fermionic Green's functions in the background of the Euclidean solitons describing false vacuum decay in a prototypal Higgs-Yukawa theory. In combination with appropriate counterterms for the masses, couplings and wave-function normalization, these can be used to calculate radiative corrections to the soliton solutions and transition rates that fully account for the inhomogeneous background provided by the nucleated bubble. We apply this approach to the archetypal example of transitions between the quasi-degenerate vacua of a massive scalar field with a quartic self-interaction. The effect of fermion loops is compared with those from additional scalar fields, and the loop effects accounting for the spacetime inhomogeneity of the tunneling configuration are compared with those where gradients are neglected. We find that scalar loops lead to an enhancement of the decay rate, whereas fermion loops lead to a suppression. These effects get relatively amplified by a perturbatively small factor when gradients are accounted for. In addition, we observe that the radiative corrections to the solitonic field profiles are smoother when the gradients are included. The method presented here for computing fermionic radiative corrections should be applicable beyond the archetypal example of vacuum decay. In particular, we work out methods that are suitable for calculations in the thin-wall limit, as well as others that take account of the full spherical symmetry of the solution. For the latter case, we construct the Green's functions based on spin hyperspherical harmonics, which are eigenfunctions of the appropriate angular momentum operators that commute with the Dirac operator in the solitonic background.
Highlights
The possible metastability of the electroweak vacuum is among the most important features of the Standard Model (SM) and may point to its embedding in the framework of more fundamental theories [1,2,3,4,5]
In the thin-wall approximation, we have found that it is possible to reduce the functional determinant of the Dirac operator to functional determinants of two scalar operators of second order in derivatives, which simplifies explicit calculations
We have used this procedure to calculate the radiative corrections to the bubble profile and the decay rate numerically in the thin-wall approximation
Summary
The possible metastability of the electroweak vacuum is among the most important features of the Standard Model (SM) and may point to its embedding in the framework of more fundamental theories [1,2,3,4,5]. One-loop radiative corrections due to fluctuations about the bounce, which fully account for the inhomogeneity of the background, have been calculated using the Gel’fand-Yaglom theorem [37,38,39,40] While the latter is a powerful way of evaluating functional determinants, it has shortcomings: If the quantum-corrected bounce cannot be obtained by improving a classical solution via perturbation theory, further elaboration is necessary. Compared to calculations in homogeneous backgrounds, the reduced symmetry makes it harder, to advance calculations to high orders Problems such as the perturbative improvement of bounce solutions and decay rates [48,49], and finding bounces in radiatively generated [45] or classically scale invariant potentials [41] can be addressed this way.
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