{"id":"99f68195-2c49-4597-880b-211e57d5edbd","arxiv_id":"2411.15085","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The authors show that the inhomogeneous confinement/deconfinement phase in rotating gluon plasma is caused by the quadratic magnetovortical coupling of angular velocity to chromomagnetic gluon fields.","lead":"Lattice simulations of rotating gluon plasma show that the mixed confined/deconfined phase is driven by the quadratic \"magnetovortical\" coupling between rotation and the chromomagnetic field, not by the usual linear mechanical coupling. The result challenges the naive Tolman-Ehrenfest picture of how temperature and rotation shape the QCD phase diagram.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Real-rotation Tc formula (7.2) rests on analytic continuation of a fit to imaginary data; the paper's own S2-only test shows polynomial continuation fails beyond u^2≈0.2, leaving quantitative real-rotation phase structure unestablished.","rationale":"The paper is a careful lattice study with a strong operational decomposition of the action into S1 and S2, and it provides direct real-rotation evidence for the S2-only regime. The local thermalization approximation is validated against the full inhomogeneous simulation for imaginary rotation up to u_I^2~0.5, and the systematic error analysis in Appendices C-D is thorough. The most load-bearing assumption is that the analytic continuation from imaginary to real rotation is quantitatively reliable for the full action. This assumption is load-bearing because the central quantitative claim and the TE-violation argument rest on Eq. (7.2); it is also the least secure because the full real-rotation action has a sign problem and is never simulated. The paper's own Fig. 12 demonstrates that, in the S2-only case, the naive polynomial continuation fails beyond u^2≈0.2, requiring the rational function (6.3). This is a concrete, acknowledged limitation that directly affects the confidence in the real-rotation phase boundary. The reader's weakest assumption identifies exactly this point, and I agree. The verdict should remain CONDITIONAL: the qualitative finding (S2 dominance) is well supported, but the quantitative real-rotation formula should be regarded as an extrapolation subject to the (6.2)/(6.3) ambiguity.","tokens_in":77,"tokens_out":9698,"duration_ms":151839,"concrete_test":"In the local thermalization approximation, simulate the full action (6.1) with λ1=λ2=1 at imaginary rotation over an extended range, e.g., 0<u_I^2<1.5, at the finest lattice available (Nt=6 or 8), and fit Tc(u_I) with the polynomial (6.2) and the rational function (6.3). If the rational fit is preferred and its pole lies at 1/b2>1, the continuation to real rotation up to ΩR≈0.7 is reliable. If the pole enters the physical region u^2<1, the continuation fails and the phase diagram in Fig. 7 must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 7's central real-rotation prediction, Tc(r)/Tc0 = 1 + κ2(Ωr)^2 (Eq. 7.2), is obtained from the imaginary-rotation fit (4.6) via the substitution Ω_I^2 → -Ω^2. This is a Taylor extrapolation in z=Ω_I^2 to negative z. The direct evidence for the continuation is limited to the S2-only action (Re2 regime) in the local thermalization approximation, where Fig. 12 shows that the polynomial (6.2), fitted to imaginary rotation, fails to describe real-rotation data beyond u^2≈0.2 and must be replaced by the rational function (6.3) with a pole at 1/b2≈1.44. The paper explicitly notes that the difference between (6.2) and (6.3) for u^2>0 is a systematic uncertainty for the full action, yet the phase diagram in Fig. 7 and the summary equation (7.2) are built on the simple linear-in-Ω^2 form without propagating this uncertainty. Since the full action with S1 (linear in Ω_I) continues to a complex action at real rotation and is never simulated, the quantitative validity of Eq. (7.2) for the full action is not established. The qualitative inverted phase structure (deconfinement at center) is robust because it is directly seen in the S2-only real simulation, but the specific Tc(r) formula and the claimed quantitative disagreement with the Tolman-Ehrenfest prediction are contingent on an untested extrapolation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies SU(3) Yang–Mills theory in a rigidly rotating reference frame on the lattice, using imaginary angular velocity to circumvent the sign problem. It confirms the existence of a mixed inhomogeneous phase, introduces a local (pseudo)critical temperature Tc(r) determined from the Polyakov-loop susceptibility, and parametrizes it by a quartic fit in r/R with coefficients that are linearly fitted in the squared imaginary velocity. The action is decomposed into a linear mechanical coupling S1 and a quadratic magnetovortical coupling S2, and each is simulated separately; the authors find that S2 dominates the formation of the mixed phase, while S1 plays a subleading role. A local-thermalization approximation is introduced, and in the sign-problem-free S2-only (Re2) regime the paper directly simulates real rotation, showing that the polynomial analytic continuation of the imaginary-rotation fit fails for u^2 ≳ 0.2 whereas a rational function works. The main conclusion is that the real-rotation phase structure has deconfinement at the center and confinement at the periphery, in contradiction with a naive Tolman–Ehrenfest expectation.","tokens_in":32293,"tokens_out":6394,"duration_ms":59504,"significance":"The paper's operator decomposition into mechanical and magnetovortical couplings is a conceptually valuable step, and the direct real-rotation simulation of the S2-only theory (Re2 regime) is a clever and important methodological advance because it provides a sign-problem-free window into real rotation. The numerical work is careful: boundary conditions are cross-checked (OBC vs PBC), finite-volume and lattice-spacing effects are analyzed in Appendices C and D, and continuum extrapolations are performed. If the central claim holds, it sharpens our understanding of why the inhomogeneous phase in vortical gluon plasma is inverted relative to the simple Tolman–Ehrenfest picture. However, the quantitative real-rotation prediction rests on an analytic continuation that the paper's own data show is not quantitatively reliable beyond |u^2| ≈ 0.2 in the directly tested sector, so the significance of the specific formula (7.2) is currently partly aspirational.","major_comments":[{"comment":"The central real-rotation prediction Tc(r)/Tc0 = 1 + κ2(Ωr)^2 in Eq. (7.2) is obtained by substituting Ω_I^2 → −Ω^2 into the imaginary-rotation fit Eq. (4.6). However, the paper's own test of this continuation in the sign-problem-free Re2 sector (Fig. 12) shows that the polynomial continuation (6.2) agrees with real-rotation data only for u^2 ≲ 0.2, while the full data are described by the rational function (6.3). The paper explicitly states that the difference between (6.2) and (6.3) for u^2 > 0 is a systematic uncertainty of the analytic continuation, yet Eq. (7.2) and the phase diagram in Fig. 7 are presented without propagating this uncertainty. Since the full action with the S1 term is never simulated at real Ω, the quantitative validity of Eq. (7.2) for the full theory is not established; the qualitative inverted phase structure is supported by the direct Re2 simulation, but the central quantitative formula should be reframed as an extrapolation whose systematic error is quantified.","section":"§6.2, Eq. (7.2)"},{"comment":"The text claims that because Eq. (4.6) is quadratic in Ω_I, the phase diagram for real frequency at T = Tc0 + ΔT has the same shape as for imaginary frequency at T = Tc0 − ΔT. This is not correct when κ4 ≠ 0: substituting Ω_I^2 → −Ω^2 flips the sign of the κ4 term in Eq. (4.6). For fixed ΩR and x = r/R, the imaginary-rotation function is 1 − κ2(ΩR)^2 x^2 + κ4(ΩR)^2 x^4, while the real-rotation function is 1 + κ2(ΩR)^2 x^2 − κ4(ΩR)^2 x^4. These are different shapes, especially near r/R ≳ 0.8 where κ4 is most important; Fig. 7 should either present the two cases separately or justify explicitly why the sign change of the κ4 term can be neglected.","section":"§4.3, Fig. 7"},{"comment":"The summary states that 'The coefficient κ2 in Eqs. (7.1) and (7.2) is given in (4.5)', but Eq. (4.5) is the earlier value κ2 = 0.902(33) obtained in Ref. [41] via a quadratic fit, whereas the new continuum-extrapolated value obtained in this paper from the quartic fit is κ2 = 1.051(29) in Eq. (4.4). Since Eq. (7.2) is the central quantitative prediction, the summary must use the new value and clearly state which fit it is taken from; the current text is internally inconsistent.","section":"§7, Eqs. (7.1)–(7.2)"}],"minor_comments":[{"comment":"Eq. (7.1) drops the κ4 term and presents 1 − κ2(Ω_I r)^2 as 'well described' by the results, but Section 4.2 states that the quadratic fit is valid only for r/R ≲ 0.5; the summary should specify the restricted radial domain of Eq. (7.1).","section":"§7, Eq. (7.1)"},{"comment":"The local-thermalization approximation relies on a correlation length ζ, but the text does not specify which correlation length (e.g., the Polyakov-loop correlation length) is meant; please define ζ explicitly.","section":"§6.1"},{"comment":"The caption uses the abbreviations 'fit' and 'a.c.' (analytic continuation) without defining them; please spell these out so that the figure is self-contained.","section":"Fig. 12 caption"},{"comment":"The text says that the transition region 'has a regular circle form'; based on Fig. 1 the boundary appears approximately circular, so the wording should be softened to 'approximately circular' to avoid overstatement.","section":"§4.1"},{"comment":"Reference [38] (Yang and Huang, 'QCD on Rotating Lattice with Staggered Fermions') is cited only by arXiv number without journal or publication status; please update the citation if the paper has been published.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper overlaps substantially with the authors' earlier publication [41] and their previous work on the negative moment of inertia [44–47]; the genuinely new elements are the S1/S2 operator decomposition and the direct Re2 simulation. The editor may wish to ask the authors to more clearly delineate which results are new to this paper. The central analytic-continuation caveat is acknowledged by the authors but not propagated into the main quantitative claim, which is the main reason I recommend major revision rather than minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know before you read it. First, the paper's main claim—that the mixed inhomogeneous phase in rotating gluon plasma is produced by the quadratic magnetovortical coupling S2, not the linear mechanical coupling S1—is well supported and is the real contribution. Second, the quantitative real-rotation formula Tc(r)/Tc0 = 1 + κ2(Ωr)^2 is an analytic continuation of a fit to imaginary-rotation data; the paper's own S2-only real-rotation test shows that continuation fails beyond u^2 ≈ 0.2. The qualitative phase inversion (deconfinement at the center, confinement in the periphery) is robust, but the specific curve is not.\n\nThe operator decomposition (5.4) is clean and gauge invariant, and the comparison of the Im1, Im2, Im12, and Re2 regimes is convincing. The local thermalization approximation is introduced and validated, and it gives a nice physical picture: the co-rotating metric anisotropizes the couplings and shifts Tc away from the naive Tolman-Ehrenfest expectation. The lattice work is careful—boundary conditions, lattice spacing, and finite volume are checked in Appendices C and D.\n\nThe soft spot is the corner cut to get to real rotation. The full action at real rotation has a sign problem and is never simulated. Instead, they continue the fitted formula (4.6) with Ω_I^2 → -Ω^2. Their own Re2 simulation (S2 only) shows the polynomial (6.2) fitted to imaginary data fails for real u^2 > 0.2, and the rational function (6.3) with a pole at 1/b2 ≈ 1.44 describes the data better. They call this a systematic uncertainty for the full action, but then Fig. 7 and Eq. (7.2) use only the simple linear form. So the quantitative real-rotation phase structure is not established. The paper is openly aware of this; the problem is that the abstract and conclusions present the quantitative TE violation as established. No code or data are released, which makes independent verification of the fit parameters harder than it should be.\n\nWhat survives is qualitative and important. S1 alone gives the opposite phase structure, so the linear coupling is genuinely subleading. S2 dominates. The sign-problem-free S2-only real simulation directly shows the inverted radial pattern. That is enough to make the paper worth reading for anyone working on rotation in QCD or heavy-ion phenomenology.\n\nMy recommendation: send it to peer review, conditional acceptance. Ask the authors to either provide a controlled estimate of the S1 contribution to the real-rotation action, or soften Eq. (7.2) and the TE-violation claim to what the data actually support. The mechanism and the decomposition are solid.","headline":"A solid lattice study pinning the mixed phase on the quadratic magnetovortical coupling, but the real-rotation Tc formula rests on an analytic continuation whose quantitative reach is untested.","tokens_in":32856,"tokens_out":4876,"would_cite":true,"duration_ms":43659,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The spatial structure of the mixed phase in rotating gluon plasma is governed by the quadratic magnetovortical coupling, with the linear mechanical coupling subleading.","keywords":["rotating quark-gluon plasma","lattice SU(3) gluodynamics","mixed inhomogeneous phase","magnetovortical coupling","Tolman-Ehrenfest law","analytic continuation","local critical temperature","Polyakov loop"],"falsifier":"Simulate the full rotating SU(3) action at a real angular velocity with a sign-problem-free method (for instance complex Langevin or Lefschetz-thimble sampling) at $\\Omega R \\approx 0.5$--$0.7$ and measure the radial Polyakov-loop profile: the paper predicts deconfinement at the center and confinement at the rim with $T_c(r)/T_{c0} = 1 + \\kappa_2(\\Omega r)^2$, whereas the Tolman-Ehrenfest expectation places deconfinement at the rim, so a rim-deconfined profile would settle the question against the claim.","tokens_in":31720,"feed_emoji":"🌀","tokens_out":14580,"duration_ms":119660,"temperature":0.7,"pith_summary":"This paper tries to establish why hot gluon matter under rigid rotation forms a mixed phase with deconfinement at the center and confinement at the periphery, and to identify which piece of the action creates that arrangement. The authors decompose the rotating gluon action into a linear coupling of the angular velocity to the gluons' mechanical angular momentum and a quadratic \"magnetovortical\" coupling to the chromomagnetic field, and their lattice simulations show that the quadratic term mimics the full theory while the linear term alone produces the opposite arrangement. On this basis they argue that the co-rotating metric's anisotropy—not the Tolman-Ehrenfest redshift—controls the radial dependence of the local critical temperature, giving $T_c(r)/T_{c0} = 1 + \\kappa_2(\\Omega r)^2$ for real rotation. The result matters for the vortical quark-gluon plasma of non-central heavy-ion collisions, where the same inverted phase ordering is expected to appear.","feed_headline":"Magnetic coupling, not angular momentum, shapes rotating gluon plasma","feed_subtitle":"Lattice results: deconfinement at the center, confinement at the rim, contrary to the Tolman-Ehrenfest law.","key_machinery":"The machinery is the decomposition of the Euclidean rotating-gluon action $S(\\Omega_I) = S_0 + S_1 \\Omega_I + S_2 \\Omega_I^2$ in cylindrical co-rotating coordinates, and in particular the magnetovortical term $S_2$, which couples the square of the angular velocity to the squared components of the chromomagnetic field. Its work in the argument is to act as a switchable knob: the factors $\\lambda_1$ and $\\lambda_2$ in $S = S_0 + \\lambda_1 S_1 \\Omega_I + \\lambda_2 S_2 \\Omega_I^2$ define four regimes (Im1, Im2, Im12, Re2) whose comparison isolates the contribution of each coupling. The companion piece is the local-thermalization approximation, which freezes the $\\Omega r$-dependent coefficients of the action at a fixed radius $r_0$ and reduces the rotating system to a homogeneous, anisotropic action with couplings $\\beta$ and $\\tilde{\\beta} = (1 - \\Omega^2 r_0^2)\\beta$, showing that the metric-induced asymmetry alone shifts the local critical temperature with radius.","core_discovery":"On the paper's own terms, the central discovery is that the inhomogeneous mixed phase of rotating SU(3) Yang-Mills plasma is driven by the quadratic magnetovortical term $S_2 = (1/(2g^2))\\int d^4x\\, r^2[(F^a_{\\hat{\\varphi} z})^2 + (F^a_{r\\hat{\\varphi}})^2]$ in the co-rotating action, while the linear mechanical term $S_1$, which couples rotation to gluon angular momentum, is subleading. Switching the two terms on and off through the couplings $\\lambda_1$ and $\\lambda_2$ in $S = S_0 + \\lambda_1 S_1 \\Omega_I + \\lambda_2 S_2 \\Omega_I^2$, the simulations show that the full-action regime (Im12) matches the $S_2$-only regime (Im2), whereas the $S_1$-only regime (Im1) produces the opposite phase ordering and a much weaker radial dependence of $T_c(r)$. Analytic continuation $\\Omega_I^2 \\to -\\Omega^2$ turns the fitted local critical temperature into $T_c(r)/T_{c0} = 1 + \\kappa_2(\\Omega r)^2$ for real rotation, with $\\kappa_2$ of order unity, which puts deconfinement at the axis and confinement at the rim. The paper further identifies the physical origin of this behavior in the anisotropy that the curved co-rotating metric induces between chromoelectric and chromomagnetic couplings ($\\tilde{\\beta} = (1 - \\Omega^2 r_0^2)\\beta$ in the local-thermalization approximation), an effect that a straightforward Tolman-Ehrenfest treatment misses.","pith_inferences":["If $\\kappa_2$ is universal as the paper suggests, the same quadratic radial law should appear in QCD with dynamical quarks; because quarks couple to rotation linearly, the gluonic $S_2$ term should still set the shape of the phase boundary.","The pole-like rational fit ($b_2 \\approx 0.9$ in the local-thermalization approximation) hints at a limiting angular velocity below the causality bound where the deconfining temperature would diverge; a dedicated scan at $u^2$ between 0.5 and 1 could distinguish a physical singularity from a fitting artifact.","The local-thermalization approximation, if it holds at higher velocities, offers a computationally cheap way to map the rotating phase diagram: instead of simulating large inhomogeneous lattices, one simulates homogeneous anisotropic actions over a grid of radii."],"forward_implications":["For real rotation, the mixed deconfinement/confinement phase exists only above the non-rotating critical temperature, with the phase-boundary radius fixed by $T_{c0} + \\Delta T = T_{c0}(1 + \\kappa_2(\\Omega r)^2)$.","Models of rotating quark-gluon plasma that retain only the standard linear $\\Omega \\cdot J$ coupling will produce the wrong spatial phase ordering; the chromomagnetic sector must be included.","The Tolman-Ehrenfest law, applied naively to the rotating gluon plasma, predicts the inverse arrangement and is therefore not the controlling factor for vortical gluon matter.","Because the $S_2$-only action is free of the sign problem, real-rotation simulations of the magnetic sector are feasible and support the analytic continuation between imaginary and real angular frequencies within the studied accuracy.","The same quadratic magnetovortical coupling underlies the negative moment of inertia below the supervortical temperature, so the phase-structure anomaly and the mechanical-inertia anomaly share one microscopic origin."],"supporting_citations":[{"why":"introduces lattice QCD in rotating frames and the imaginary-rotation discretization that the present simulations build on.","marker":"[32]"},{"why":"establishes the rotating SU(3) lattice setup, boundary-condition checks, and the increase of the transition temperature with rotation that this paper refines.","marker":"[34]"},{"why":"provides the first lattice observation of the mixed inhomogeneous phase and the quadratic fit ($\\kappa_2 = 0.902$) that this paper reanalyzes and explains.","marker":"[41]"},{"why":"supplies the earlier suggestion that the linear and quadratic couplings shift the critical temperature in opposite directions, which is tested directly here.","marker":"[44]"},{"why":"gives the decomposition of the gluon moment of inertia into mechanical and chromomagnetic parts and the negative-Barnett-effect interpretation that motivates the $S_2$-dominance claim.","marker":"[47]"},{"why":"is the low-dimensional model prediction of an inhomogeneous phase following the Tolman-Ehrenfest law, whose phase ordering the present simulations contradict.","marker":"[39]"},{"why":"states the Tolman-Ehrenfest local-temperature law whose naive application the paper argues is violated by the lattice results.","marker":"[59, 60]"},{"why":"fixes the causality condition $v_I < 1/\\sqrt{2}$ that delimits the imaginary-rotation domain used for analytic continuation.","marker":"[63]"},{"why":"shows that asymmetric lattice couplings alter lattice spacings and the critical temperature, supporting the anisotropy mechanism proposed in Section 6.","marker":"[64]"}],"fun_headline_variants":["Rotating gluon plasma: magnetovortical coupling wins","Quadratic magnetovortical term drives gluon phase split","Vortical gluon plasma shaped by magnetic coupling","Magnetovortical effects, not angular momentum, split gluon plasma","Anisotropy explains gluon plasma phase separation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The real-rotation phase structure rests on one unverified step: the formula fitted at an imaginary rotation speed is assumed to stay correct when the speed is made real by flipping the sign of its square, since the full real-speed action has a complex weight that ordinary Monte Carlo cannot handle and was never simulated; if that step fails beyond small speeds, the central deconfinement-at-the-center claim is not established.","fun_headline_variants_meta":{"raw":{"variants":["Rotating gluon plasma: magnetovortical coupling wins","Quadratic magnetovortical term drives gluon phase split","Vortical gluon plasma shaped by magnetic coupling","Magnetovortical effects, not angular momentum, split gluon plasma","Anisotropy explains gluon plasma phase separation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000258,"raw_usage":{"total_tokens":1685,"prompt_tokens":1151,"completion_tokens":534,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":767,"completion_tokens_details":{"reasoning_tokens":454}},"tokens_in":767,"tokens_out":534,"duration_ms":5280,"temperature":1.0,"reasoning_tokens":454,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:30:06.525401+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the full rotating SU(3) action at a real angular velocity with a sign-problem-free method (for instance complex Langevin or Lefschetz-thimble sampling) at $\\Omega R \\approx 0.5$--$0.7$ and measure the radial Polyakov-loop profile: the paper predicts deconfinement at the center and confinement at the rim with $T_c(r)/T_{c0} = 1 + \\kappa_2(\\Omega r)^2$, whereas the Tolman-Ehrenfest expectation places deconfinement at the rim, so a rim-deconfined profile would settle the question against the claim.","supporting_citations":[{"cited_title":"Karsch, SU(N) Gauge Theory Couplings on Asymmetric Lattices , Nucl","cited_arxiv_id":null,"evidence_quote":"shows that asymmetric lattice couplings alter lattice spacings and the critical temperature, supporting the anisotropy mechanism proposed in Section 6."}],"review_version":1}