Scalar Thermal Field Theory for a Rotating Plasma

Referee: The v→1 enhancement narrative and bound Ω<1/R are derived under the Dirichlet choice J_m(αR)=0; no argument is given that this models a black-hole corona, rotating star, or lab plasma. Different boundary conditions can change v_max and the v→1 behavior. The paper should (i) demonstrate robustness under a representative class of BCs or (ii) restrict §6 claims to hard-wall systems.

Authors: We agree that the convergence bound Ω<1/R and the enhancement towards v→1 are derived using the Dirichlet spectrum J_m(αR)=0, which was adopted as a computationally convenient regulator. The structural feature on which Ω<1/R relies is that the lowest radial root j_{m,1} grows linearly in m for large m with j_{m,1}/m→1; this asymptotic is shared by Neumann and Robin conditions on a cylinder of radius R, so the bound Ω<1/R and the qualitative v→1 growth survive in this representative class. MIT-bag-type or soft-confining boundaries change the prefactor in the spectrum and hence the precise critical velocity, but not the existence of a finite v_max<1 set by causality on the boundary. We will (a) add an explicit paragraph in §3.1 stating that the bound Ω<1/R follows from the universal large-m asymptotics shared by a class of BCs, (b) qualify the statements in §6 to make clear that the precise numerical enhancement depends on the wall model, and (c) note that the qualitative conclusion (sizable enhancement at moderate v, divergence of the boundary-confined idealization as v→1) is robust, while the strict v→1 limit should be interpreted as the unattainable causal limit of the rotating-wall idealization. revision: yes

Referee: The large-volume limit holds v=ΩR fixed, forcing Ω→0 as R→∞, yet §6 quotes v≃1 for black-hole coronas. Rigid rotation of an unbounded thermal state at finite Ω is ill-defined (light surface r=1/Ω must lie inside). The setup being modeled — finite cylinder with rotating wall vs. bulk rigid rotation with light cylinder excluded — should be clarified.

Authors: The referee is correct that the order of limits taken corresponds physically to a finite cylinder co-rotating with the plasma, with R kept finite (or the limit R→∞ taken with v=ΩR fixed, i.e. Ω→0), rather than bulk rigid rotation of an unbounded medium at finite Ω. This is precisely the well-known Vilenkin/Iyer setup, in which the light cylinder coincides with the boundary at v=1 and is excluded from the system. We will add a clarifying paragraph at the end of §3.1 (and a forward reference in §6) emphasizing that: (i) Ω is the angular velocity of the bounding cylinder, with the plasma in rigid co-rotation inside r≤R; (ii) the causal limit r=1/Ω is realized as v→1 from below and is never crossed in the integration domain; (iii) phenomenological identifications with black-hole coronas should be read as order-of-magnitude indications, with R interpreted as the characteristic transverse size of the rotating region and Ω as a local angular velocity, not as a global rigid-rotation parameter of an infinite medium. We thank the referee for pointing out that this physical picture was implicit but not stated. revision: yes

Referee: The divergences of ⟨ρ_E⟩ and ⟨J_z⟩ as v→1 (Fig. 2) appear dominated by modes with α≈|y|, i.e. peaked near the wall. The paper should display the radial profile of the energy and angular-momentum densities to clarify whether the enhancement is bulk or shell-localized. It is also unclear whether ⟨ρ_E⟩ in (3.43) is volume-averaged or local at r.

Authors: This is a fair observation. The quantities ⟨ρ_E⟩, ⟨J_z⟩ in Eqs. (3.43)–(3.45) are obtained from logZ via β- and τ-derivatives and therefore correspond to volume-integrated densities divided by the cylinder volume (i.e. volume-averaged densities), not local densities at fixed r. This is reflected in the absence of any explicit r-dependence in the integrands. The referee's intuition that the v→1 growth is dominated by modes with α≈|y| (large m, lowest-n) is correct, and these modes are radially concentrated near r∼R; hence the enhancement is, in this idealized hard-wall setup, a near-boundary effect rather than a bulk effect. We will (a) state explicitly below Eq. (3.43) that ⟨ρ_E⟩ etc. are volume-averaged, (b) add a short discussion of the radial profile constructed from |J_m(αr)|² weighted by the relevant occupation numbers, and (c) include a representative radial-profile plot showing the shell-like localization as v approaches 1. This sharpens the physical interpretation considerably and we are grateful for the suggestion. revision: yes

Referee: The argument that for μ_a=0 the −m²Ω² contribution is compensated by α²≥(m/R)² is tied to the Dirichlet spectrum. The conclusion that the real part of the Euclidean action is bounded from below is therefore conditional on this BC choice and should be stated explicitly to avoid over-interpretation as a generic sign-problem-free formulation of bulk rigid rotation.

Authors: We agree. The bound α≥|m|/R that we use in §5 to argue α²−m²Ω²>0 is indeed inherited from the same Dirichlet spectrum used in §3.1, and the resulting boundedness of the real part of the Euclidean action is not a generic statement about bulk rigid rotation. We will revise the paragraph following Eq. (5.12) to make this conditionality explicit, stating that (i) the positivity argument relies on the existence of a confining boundary together with v=ΩR<1, (ii) it therefore applies to lattice formulations on a finite cylinder co-rotating with the plasma, and (iii) it does not by itself establish the absence of a sign problem for an attempted formulation of bulk rigid rotation in infinite volume at finite Ω, which is in any case ill-defined for the reasons noted in the referee's second comment. We will also add a corresponding caveat to the summary in §7. revision: yes

Referee: In Eq. (6.7) and footnote 8, the vanishing of the decay-type (1+f)(1+f) and (1+f)f contributions is justified by a brief energy–momentum-conservation argument. In a rotating system, with f_B(ω−mΩ) and large m, ω−mΩ small modes are populated; the kinematic exclusion of the decay channel deserves a more careful argument.

Authors: The referee raises a legitimate concern that we did not spell out carefully. The relevant point is that the kinematic constraint enforced by the radial integral I_{q,q1,q2}, namely that α, α_1, α_2 form the sides of a triangle (Eq. (6.8) and Ref. [33]), together with conservation of p (translation along z) and m (rotational symmetry around z), is equivalent in the rotating case to ordinary 4-momentum conservation among the three on-shell modes labeled by {ω,α,p,m}. The shift ω→ω−mΩ enters only through the Bose factors and not through the triangle/conservation conditions, because Ω couples to the conserved m, not to the dispersion relation of an individual mode. Hence the decay S→hS in the rotating frame is kinematically forbidden by exactly the same boost argument as in the non-rotating case, regardless of how populated the large-m, small (ω−mΩ) modes are: large occupation does not change the on-shell kinematics, only the statistical weight of the (kinematically allowed) coalescence channel. We will expand footnote 8 into a short paragraph that (i) separates the kinematic content (triangle condition, on-shellness) from the statistical content (Bose factors with ω−mΩ), and (ii) explicitly verifies that the decay channels vanish term by term in the rotating case. revision: yes