A full imaginary turn
A theoretical preprint reports an exact relation in hot QCD: increasing the dimensionless imaginary angular velocity Ω̃I by 2π is equivalent, at the density-operator level, to shifting the quark angle θq by π. In the notation of the calculation, ρ(Ω̃I + 2π, θq) = ρ(Ω̃I, θq + π). At (Ω̃I, θq) = (2π, 0), the density operator therefore equals the one at (0, π), the Roberge-Weiss point. The work is identified as arXiv:2608.25849v1 [hep-ph], dated 26 Aug 2026.
That full-turn correspondence carries the stated Roberge-Weiss phase structure to the line Ω̃I = 2π: charge-conjugation symmetry, or C symmetry, is broken for temperatures above T_RW and restored below it. At physical quark masses, the endpoint has the three-dimensional Ising classification. These are results of a theoretical calculation, not measurements from participants or an observational dataset.
What the rotation axis reveals
To examine changes along the rotation axis, the one-loop analysis fixes θq = 0, defines t = Ω̃I/π, and numerically minimizes over the entire stated Weyl alcove. The study examines theoretical parameter cases in the thermodynamic limit; it has no participant sample or observational cohort.
For massless three-flavor QCD, the one-loop calculation reports the second-order C-breaking point at t_C ≈ 1.4021. The signal Im L turns on continuously there and follows square-root behavior above the point. The result describes a smooth onset of the symmetry-breaking signal, rather than a jump at this threshold.
A separate first-order lock-in point appears at t_lock ≈ 1.4471. At lock-in, two distant global minima share the same free energy, and the global minimum jumps from x = 2/9 to x = 1/3. The thresholds therefore describe different changes on the same axis: a continuous onset followed by a discontinuous change in the preferred minimum.
In the stated 2 + 1-flavor case, with massless up and down quarks and m_s/T = 0.30, the finite-mass check gives Ω̃I,C/π ≈ 1.4013 and Ω̃I,lock/π ≈ 1.4483. Each differs from its corresponding massless value by less than 0.1%. This comparison is specific to that parameter choice.
The ambiguity behind the massless result
The massless approximation has a built-in ambiguity. Its one-loop potential contains a degenerate circle of minima that includes both C-invariant and C-noninvariant directions, so the calculation cannot select a unique realization of C symmetry. A finite strange-quark mass lifts this accidental degeneracy in full QCD. In pure SU(3) Yang-Mills, there is no quark mass to deform it, and the one-loop treatment does not determine how the analogous degeneracy is lifted.
The calculation also tracks the imaginary quark density n_I, providing a second C-odd thermodynamic response. For 2 + 1-flavor QCD with m_s/T = 0.30, the two n_I branches have equal magnitude and opposite sign, rise continuously from zero at Ω̃I,C, and jump at Ω̃I,lock. It therefore follows the same continuous-then-discontinuous sequence as the on-axis calculation.
The boundary of the result
The exact correspondence and the phase analysis have different scopes. The density-operator relation is system-wide, whereas the one-loop C-breaking analysis is restricted to the rotation axis. Off-axis radial C-breaking remains unresolved.
The work identifies the mixed-periodicity relation, 4π periodicity, the mirror relation, and the density-operator and observable mappings as exact relations for direct lattice testing with dynamical quarks. No new lattice measurements are supplied, so the transition points remain one-loop theoretical predictions rather than empirical measurements. Higher-order and nonperturbative effects may shift those one-loop thresholds, and corrections beyond one loop may lift the massless degeneracy.
Paper data and sources
Original title: Imaginary Rotation Breaks Charge Conjugation in Hot QCD
Authors: Rin Takada
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text