Chromomagnetic condensation and perturbative confinement induced by imaginary rotation in SU(2) Yang-Mills Theory

We perturbatively investigate the rotation effect on the Polyakov loop potential in SU(2) gauge theroy within a chromomagnetic background. It is observed that the imaginary rotation spontaneously induces both confinement and chromomagnetic condensation at high temperatures, thereby provides a perturbative window to explore non-perturbative dynamics. Compared to the case without including the induced chromomagnetic field, the perturbative confinement transition becomes first-order, with a temperature-dependent phase boundary that asymptotically approaches $\tildeΩ_c = π/\sqrt{3}$ at high temperatures. This leads to a significantly enriched $\tildeΩ$-$T$ phase diagram characterized by an expanded deconfined region. For real angular velocities, we find that the chromomagnetic condensate decreases with increasing rotation, and that the coupling between rotation, spin, and the chromomagnetic background leads to a cusp in the Polyakov loop potential, suggesting that the underlying dynamics could be more intricate.

💡 Research Summary

In this paper the authors investigate how an imaginary angular velocity (℧ = −i ω) influences the deconfinement transition in pure SU(2) Yang‑Mills theory when a constant chromomagnetic background field H is present. By treating the chromomagnetic field as the Savvidy‑type vacuum and working at one‑loop order, they derive the effective potential for the Polyakov loop variable ϕ (related to the traced Wilson line) in a rotating Euclidean spacetime. The rotation is introduced through a modified covariant derivative that adds the term −i ℧ L_z − s ℧ to the temporal derivative, where s = ±1 denotes the spin orientation of the charged W‑boson modes. This shift translates into a modification of the Matsubara frequencies by (l − s) ℧, where l is the orbital angular momentum quantum number.

The spectrum of the charged gluon modes in the chromomagnetic field remains the familiar Landau‑level structure, E² = k_z² + (2m + 2s + 1) g H, but the rotation introduces an additional “chemical‑potential‑like” shift. At the rotation axis (r = 0) the orbital quantum number freezes (δ_{l,0}) and only the spin states contribute. Consequently the effective potential separates into two parts:

1. V_H, the chromomagnetic‑favoring contribution, contains the classical Savvidy term (g H)² ln(g H/Λ²) and the contribution of the lowest Landau level (LLL). Because the LLL term depends on the combination (ϕ + ℧), the Z₂ center symmetry ϕ ↔ 2π − ϕ is explicitly broken when ℧ ≠ 0.

2. V_nonH, the chromomagnetic‑suppressing contribution, comes from all higher Landau levels (m ≥ 1). In the limit H → 0 this part reproduces the known Ω‑dependent Polyakov‑loop potential of pure Yang‑Mills theory.

The total dimensionless potential β⁴ V(ϕ, H) is minimized with respect to both ϕ and H, yielding coupled gap equations ∂V/∂ϕ = 0 and ∂V/∂H = 0. Numerical evaluation is performed by truncating the Landau‑level sum at O(H⁻¹) and using a hybrid scheme (integral representation for the LLL term, series representation for the higher‑level terms). A lower cutoff β √(g H) = 0.01 ensures convergence while effectively treating H ≈ 0 as the chromomagnetic‑suppressing limit.

The main findings are:

• Imaginary rotation induces chromomagnetic condensation: For any non‑zero ℧, the minimization yields a non‑zero H even at high temperature, a phenomenon absent in the Ω = 0 case.

• Perturbative confinement becomes first‑order: When ℧ exceeds a critical value, the Polyakov‑loop potential develops two degenerate minima (ϕ ≈ 0 and ϕ ≈ π) separated by a barrier, signalling a first‑order deconfinement transition.

• Temperature‑dependent phase boundary: The critical line Ω_c(T) approaches the asymptotic value Ω_c = π/√3 as T → ∞. At lower temperatures the line bends, producing an enlarged deconfined region in the Ω–T plane compared with the Ω = 0 diagram.

• Real rotation (ω) suppresses the condensate: Analytically continuing Ω → i ω and imposing the causality bound ω < Ω_max, the authors find that V_H diminishes with increasing ω, while V_nonH remains sizable. Consequently the chromomagnetic condensate H(ω) decreases monotonically.

• Cusp in the Polyakov‑loop potential: The interplay of rotation, spin, and the background field generates a non‑analytic point (cusp) in V(ϕ) at ϕ = π − ℧ (mod 2π). This reflects a sudden change in the occupation of spin‑aligned versus spin‑anti‑aligned modes and hints at richer dynamics in rotating quark‑gluon plasma.

The paper concludes that an imaginary angular velocity provides a unique perturbative window into non‑perturbative phenomena such as confinement and chromomagnetic condensation. The Ω‑dependent phase diagram, the change of transition order, and the behavior under real rotation suggest several avenues for future work, including extensions to SU(3), inclusion of dynamical quarks, and exploration of combined magnetic‑rotational effects on the QCD phase diagram.