Chromomagnetic Condensate in Finite-Temperature SU(2) Yang-Mills Theory under Imaginary Rotation

We investigate the finite-temperature SU(2) Savvidy model under an imaginary angular velocity. Employing the background-field method, we derive the one-loop effective potential and analyze both its real and imaginary parts. We demonstrate that imaginary rotation modifies the chromomagnetic condensate and the Polyakov loop, and can partially suppress the Nielsen-Olesen instability of the chromomagnetic background. Moreover, a high-temperature expansion shows that imaginary rotation strengthens the effective coupling and that the chromomagnetic field induces a negative contribution to the moment of inertia.

💡 Research Summary

The paper investigates the finite‑temperature SU(2) Yang‑Mills theory in the presence of a constant chromomagnetic background (the Savvidy vacuum) while introducing an imaginary angular velocity Ω_I. The authors adopt the background‑field method, splitting the gauge field into a classical background (\bar A_\mu^a) (containing both a uniform chromomagnetic field H aligned with the rotation axis and a constant Polyakov‑loop variable φ) and quantum fluctuations A_\mu^a. The Euclidean metric is modified to include Ω_I, and a vierbein formalism is used to keep the kinetic operators well‑defined.

The one‑loop effective potential V is derived by evaluating the functional determinant of the quadratic fluctuation operator. The spectrum consists of Landau levels λ, spin s = ±1, and Matsubara frequencies ω_n = 2π n T. The imaginary rotation enters as a spin‑dependent chemical potential, shifting the Matsubara frequencies to ω_n ± Ω_I. This shift modifies the Nielsen‑Olesen tachyonic mode (λ = 0, s = –1) that normally renders the Savvidy vacuum unstable. The authors show that the instability is avoided if the inequality ((\omega_n + g φ – Ω_I)^2 ≥ g H) holds for all n, i.e. a sufficiently large Ω_I can partially suppress the tachyonic contribution.

For the non‑tachyonic sector the authors employ Schwinger proper‑time representation to rewrite the logarithm, perform the k_z integration, and sum over λ and s. The resulting expression can be written in terms of Jacobi theta functions, allowing a compact high‑temperature form. The tachyonic sector is treated separately by splitting the longitudinal momentum integral at |k_z| = √(gH). The region |k_z| > √(gH) contributes only a real part, while |k_z| < √(gH) yields both real and imaginary pieces. The imaginary part is extracted using the principal‑branch identity (\Im \ln(1 – e^{i x}) = \frac12 ( \text{mod}(x,2π) – π )).

The full effective potential is the sum of the zero‑temperature vacuum term (including the standard (-i (gH)^2/8π) Nielsen‑Olesen imaginary piece), the finite‑temperature non‑tachyonic contribution V_T^{nt}, and the tachyonic contributions V_T^{>}, Re V_T^{<}, Im V_T^{<}. In the limit of vanishing Polyakov loop and rotation (φ = Ω_I = 0) with a finite H, the imaginary part reduces to (- (gH)^{3/2} / (2π T)), reproducing known results. In the opposite limit H → 0, only the non‑tachyonic sector survives, and the effective potential reduces to the well‑known Gross‑Pisarski‑Yaffe (GPY) or Weiss potential, expressed through polylogarithms of e^{±iβ(gφ ± Ω_I)}.

A small‑Ω_I expansion at high temperature reveals that the effective coupling constant is enhanced by the imaginary rotation, i.e. the one‑loop β‑function coefficient is effectively reduced. Moreover, the chromomagnetic field contributes a negative term to the moment of inertia, proportional to (- (gH)^{3/2} / (2π T)), indicating a “negative inertia” effect that aligns with recent lattice observations of a negative moment of inertia in rotating gauge theories.

Numerical evaluations of the real part of the potential as a function of β g φ for several values of β√(gH) at Ω_I = π/2 show that the minimum shifts away from φ = 0, demonstrating that imaginary rotation can induce a non‑trivial Polyakov‑loop condensate even when the background magnetic field is present. The imaginary part, originating from the tachyonic mode, diminishes as Ω_I grows, confirming the partial stabilization mechanism.

In summary, the paper provides a comprehensive analytical treatment of the Savvidy vacuum under imaginary rotation, elucidates how Ω_I modifies both the chromomagnetic condensate and the Polyakov loop, and shows that the tachyonic Nielsen‑Olesen instability can be mitigated. The high‑temperature analysis further connects imaginary rotation to an effective strengthening of the gauge coupling and to a negative contribution to the rotational inertia, offering valuable insights for future studies of rotating QCD matter, both in continuum approaches and lattice simulations.