Regularized stress tensor of vector fields in de Sitter space

TL;DR

This work regularizes the vacuum stress tensor of the Stueckelberg vector field in de Sitter space by applying a minimal subtraction-based adiabatic scheme that assigns 0th order to the transverse part and 2nd order to the longitudinal, temporal, and gauge-fixing parts. The resulting total regularized tensor is UV/IR finite, covariantly conserved, and maximally symmetric, with a positive energy density that can be interpreted as a cosmological constant driving inflation; in the massless limit it vanishes, in agreement with Maxwell+GF theory. Crucially, the authors show that the conventional 4th-order regularization leads to unphysical negative energy density and spurious trace anomalies, while the componentwise 0th/2nd-order scheme avoids these issues. The Stueckelberg tensor consistently reduces to Proca (and to Maxwell+GF in the massless limit), underscoring the importance of matching regularization order to the field and background. Overall, the results clarify regularization choices for vector fields in curved spacetimes and their cosmological implications, including the interpretation of vacuum energy as a cosmological constant.

Abstract

We study the Stueckelberg field in de Sitter space, which is a massive vector field with the gauge fixing (GF) term $\frac{1}{2ζ} (A^μ\,_{;\, μ})^2$. We obtain the vacuum stress tensor, which consists of the transverse, longitudinal, temporal, and GF parts, and each contains various UV divergences. By the minimal subtraction rule, we regularize each part of the stress tensor to its pertinent adiabatic order. The transverse stress tensor is regularized to the 0th adiabatic order, the longitudinal, temporal, and GF stress tensors are regularized to the 2nd adiabatic order. The resulting total regularized vacuum stress tensor is convergent and maximally-symmetric, has a positive energy density, and respects the covariant conservation, and thus can be identified as the cosmological constant that drives the de Sitter inflation. Under the Lorenz condition $A^μ\,_{;\, μ}=0$, the regularized Stueckelberg stress tensor reduces to the regularized Proca stress tensor that contains only the transverse and longitudinal modes. In the massless limit, the regularized Stueckelberg stress tensor becomes zero, and is the same as that of the Maxwell field with the GF term, and no trace anomaly exists. If the order of adiabatic regularization were lower than our prescription, some divergences would remain. If the order were higher, say, under the conventional 4th-order regularization, more terms than necessary would be subtracted off, leading to an unphysical negative energy density and the trace anomaly simultaneously.

Figures (10)

• Figure 1: The unregularized spectral stress tensor. The unit of $\rho_k$ and $p_k$ are $H^4/2\pi^2$. (a) the transverse; (b) the longitudinal; (c) the temporal; (d) the GF. Red: the spectral energy density, Blue: the spectral pressure. For illustration, $\frac{m^2}{H^2}=0.1$, $\zeta=20$, and $|\tau|=1$ are taken in Figure 1 through Figure 6, except otherwise specified.
• Figure 2: The unregularized total $\rho_k$ (red) and $p_k$ (blue) are UV divergent, $\propto k^4$ at high $k$.
• Figure 3: The regularized transverse spectral energy density (red) and spectral pressure (blue). (a) the 0th-order $\rho_{k~reg}^{TR(0)}$ is UV convergent and positive. (b) the 2nd-order $\rho_{k~reg}^{TR(2)} <0$. (c) the 4th-order $\rho_{k~reg}^{TR(4)}$ shows a negative segment.
• Figure 4: The regularized longitudinal spectral energy density (red) and spectral pressure (blue). (a) the 0th-order, still divergent. (b) the 2nd-order $\rho_{k~reg}^{L(2)}$ is convergent and positive. (c) the 4th-order $\rho_{k~reg}^{L(4)}$ is negative.
• Figure 5: The regularized temporal spectral energy density (red) and spectral pressure (blue). (a) the 0th-order. (b) the 2nd-order. (c) the 4th-order.