Regularization of Functional Determinants of Radial Operators via Heat Kernel Coefficients

TL;DR

This work develops two complementary regularization strategies for functional determinants of radial operators in quantum field theory: a large angular momentum (LAM) approach tied to heat kernel coefficients, and a full heat kernel coefficient (HKC) subtraction with IR regulation. By systematically identifying angular-momentum-space representations of heat-kernel traces and constructing compact reference series up to $a_{\max}=6$, the authors achieve UV regularization of determinants up to $D=13$ dimensions and demonstrate improved accuracy over prior WKB-based methods. They also provide a robust dimensional-regularization conversion, ensuring consistency between $\zeta$-function and DR schemes. Applications to Fubini-Lipatov instantons and a generic quartic potential illustrate substantial computational gains and practical utility for vacuum decay rate prefactors in higher-dimensional settings.

Abstract

We propose an efficient regularization method for functional determinants of radial operators using heat kernel coefficients. Our key finding is a systematic way to identify heat kernel coefficients in the angular momentum space. We explicitly obtain the formulas up to sixth order in the heat kernel expansion, which suffice to regularize up to 13-dimensional functional determinants. We find that the heat kernel coefficients accurately approximate the large angular momentum dependence of functional determinants, and make numerical computations more efficient. In the limit of a large angular momentum, our formulas reduce to the Wentzel-Kramers-Brillouin formulas in previous studies, but are extended to higher orders. All the results are available in both the zeta function regularization and the dimensional regularization.

Figures (7)

• Figure 1: Contours for the integral and the analytic structure of the integrand. The poles of $\odv{}{k}\ln u_\nu(k)$ are at $k=k_i=\sqrt{\omega_{\nu i}}$. The wavy line indicates the branch cut due to $k^{-2s}$.
• Figure 2: Convergent region of $S^{\rm HKC}(0;s;\varepsilon)$. In the white region, the momentum integral is absolutely convergent. The shaded regions are IR/UV divergent and are defined only through analytic continuation of either $s$ or $\varepsilon$. The red dashed line corresponds to the zeta function regularization and the blue solid line corresponds to the dimensional regularization.
• Figure 3: The potential (left) and the bounce (right) for $\hat{m}^2=0.2$ (blue) and $\hat{m}^2=0.4$ (orange) with $v=1$. The false vacuum is at $\phi=0$ and the true vacuum is at $\phi=1$.
• Figure 4: Absolute value of $d_\nu\ln R_\nu^{\rm fin}$ using the large angular momentum subtraction (left) and the heat kernel coefficient subtraction with $z=\hat{m}^2$ (right). The parameters are $\hat{m}^2=0.2$ and $v=1$, corresponding to a thick-wall bounce. Markers indicate $a_{\rm max}=1$ (blue circle), $2$ (orange square), $3$ (green diamond), $4$ (red triangle), and $5$ (purple inverted triangle). The dashed line shows the original $\ln R_\nu$ before subtraction. The two solid lines correspond to the Feynman diagrammatic subtraction for $a_{\max}=1$ and $2$ (from above).
• Figure 5: The same figure as in Fig. \ref{['fig:numerical-thick']}, but for $\hat{m}^2=0.4$ and $v=1$, which corresponds to a thin-wall bounce.