Perturbation theory of rotating scalar fields and vacuum insensitivity to rotation

TL;DR

This work develops a finite-temperature perturbation theory for a rotating scalar field with a $\lambda\phi^4$ interaction by employing a Fourier-Bessel basis to address radial non-invariance. The authors derive the noninteracting partition function $\ln Z_0$ in cylindrical coordinates, showing that rotation alters finite-temperature thermodynamics but leaves zero-temperature physics invariant, modulo the causality-imposed finite size $\Omega R \le 1$. They then compute the leading interaction correction $\ln Z_1$, highlighting a radial four-Bessel integral that captures the rotational modification of the interaction vertex, and demonstrate that the zero-temperature, finite-$\Omega$ part cancels, reproducing the Cartesian limit $\ln Z_1 \to -\frac{\lambda}{48} V T^3$ as $R\to\infty$. The one-loop self-energy $\Pi_1$ exhibits off-diagonal radial structure and converges to the Cartesian form in the large-$R$ limit, with rotation-updated, temperature-dependent behavior. Collectively, the Feynman rules reveal that the vacuum is perturbatively insensitive to rotation up to $\mathcal{O}(\lambda^2)$, while nonperturbative effects may still modify the vacuum structure, motivating extensions to other theories and effective models in rotating backgrounds.

Abstract

We formulate the finite-temperature perturbation theory of interacting scalar fields under external rotation. Because of the translational non-invariance in the radial direction, Green's functions are described using the Fourier-Bessel basis, instead of the conventional Fourier basis. We derive the leading-order correction to the partition function and the one-loop self-energy. The Feynman rules obtained in our perturbation theory shows that due to the finite-size effect required by the causality constraint, the zero-temperature thermodynamics in the perturbation theory is unaffected by rotation, similarly to that in the noninteracting theory.

Figures (4)

• Figure 1: Contour $C_l$ for $l> 0$ (left) and $l<0$ (right).
• Figure 2: Diagonal elements of the self-energy for $\Omega = 0$.
• Figure 3: Off-diagonal elements of the self-energy for $\Omega = 0$.
• Figure 4: $\Omega$ and $T$ dependence of the diagonal self-energy.