Rotational effects on the chiral crossover transition in QCD matter

Arghya Chatterjee; Kshitish Kumar Pradhan; Nandita Padhan; Raghunath Sahoo

REVIEW

Rotational effects on the chiral crossover transition in QCD matter

Nandita Padhan, Kshitish Kumar Pradhan, Arghya Chatterjee, Raghunath Sahoo

High Energy Physics - Phenomenology · Submitted 14 Sep 2026 · 69 references

Pith's verdict: A clean HRG calculation of rotational curvature coefficients—the half-maximum criterion is the Achilles heel, but the paper is transparent about it.

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arXiv ID2609.15513 (version 1)

Pith ID2BDGW2BZ, Pith's permanent record of this paper and its review

CitationArghya Chatterjee, Kshitish Kumar Pradhan, Nandita Padhan, Raghunath Sahoo. "Rotational effects on the chiral crossover transition in QCD matter." arXiv:2609.15513 (2026). https://pith.science/paper/2BDGW2BZ

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Claim paper

Why Pith says this

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Editor's note

read the letter

The main thing: this paper does exactly what it says—computes the chiral condensate in a rotating HRG and extracts curvature coefficients for the Tc(ω) line. That's new. Not a breakthrough, but it fills a gap in the model literature. The derivation is sound within the HRG assumptions, and the paper is clearly written. The comparison with lattice QCD and other models is properly contextualized.

What it does well: it systematically extends the existing HRG treatment of the chiral condensate to include rotation. The formalism is well-documented. The results are presented clearly, and the curvature coefficients are fitted with error estimates. The paper also checks the combined effect with baryon chemical potential and spatial dependence.

Soft spots: the half-maximum criterion for defining Tc is imported from lattice QCD and is not independently justified for ω>0. The paper itself notes that the HRG has no inflection point, so they rely on this external definition. The stress-test concern is real: if rotation changes the shape of Δ_l^R(T), the half-maximum point might not track the physical crossover. The paper does not test sensitivity to the threshold fraction, nor does it cross-check with another observable. This makes the curvature coefficients potentially an artifact of the criterion. However, the paper is transparent about this, and the authors do not overclaim—they present it as a model prediction. A referee should ask for a sensitivity test (e.g., 45% or 55% of maximum). Another minor point: the cylindrical boundary condition with R=6 fm leads to discretization of transverse momentum; the dependence on R is not explored. The paper fixes Rω≤1 and uses one R value; varying R would test robustness.

Who it's for: anyone working on the QCD phase diagram under rotation, particularly those comparing model predictions. It's a useful reference point for the HRG approach, but not a definitive result.

Recommendation: send to peer review. The paper is competent and honest. The concern about the half-maximum criterion is manageable with a sensitivity check. I'd suggest the authors add a test varying the threshold and discuss the robustness of the coefficients.

Referee report

Significance

If the results are robust, this work offers a systematic HRG-based investigation of the chiral crossover in rotating QCD matter, providing curvature coefficients that could be compared with future lattice QCD or effective model calculations. The inclusion of both baryon chemical potential and radial dependence adds to the comprehensiveness. However, the significance is tempered by the central reliance on a definition of the pseudocritical temperature that is not independently validated within the model for rotating systems. The HRG model's limited range of validity (T ≲ 160 MeV) and its inability to reproduce the saturation of the condensate at higher T also restrict the impact. The results are consistent with some effective model predictions (e.g., NJL, PNJL, holography) but conflict with recent lattice QCD findings that indicate an increase in Tc with ω, so the paper does not resolve the existing discrepancy.

Major objections (3)

[Sec. III, Fig. 2 and Eq. (10)] The definition of the pseudocritical temperature as the temperature where Δ_l^R reaches half its vacuum value is imported from lattice QCD (Refs. [55,66]) and calibrated only in the non-rotating limit. The paper acknowledges (near Fig. 1) that no inflection point exists in the HRG chiral susceptibility, making this external criterion necessary. However, there is no test of whether this half-maximum criterion remains a faithful proxy for the chiral crossover at finite ω. If rotation changes the shape of Δ_l^R(T) (e.g., its derivative or curvature near the crossover), the half-maximum point could shift differently than the true pseudocritical temperature. The extracted curvature coefficients k2^ω and k4^ω would then reflect a definitional artifact. The authors should perform a sensitivity analysis by varying the threshold fraction (e.g., 0.4 and 0.6) and report whether the curvature coefficients are stable. Without this, the central claim is not adequately supported.
[Sec. II C] The rotating HRG framework assumes a cylindrical boundary with fixed radius R = 6 fm and imposes Rω ≤ 1. The results, including the curvature coefficients, may depend on the choice of R. For example, a larger R would lower the first quantized transverse momentum and could affect the condensate at a given ω. The authors should either test the sensitivity of Tc(ω) to R (e.g., R = 5, 7 fm) or provide a physical justification for the chosen value. Similarly, the cutoff in the hadron mass spectrum at 2.6 GeV is mentioned but not assessed; contributions from heavier states, especially those with higher spins, could be non-negligible at large ω due to the factor e^{(l+s)ω/T}.
[Fig. 1 (left panel)] The left panel of Fig. 1 compares the renormalized chiral condensate from the HRG model with lattice QCD results from Ref. [44]. The authors note that the HRG curve does not exhibit the saturation behavior seen in lattice data at T ≳ 200 MeV and remains systematically lower. This discrepancy indicates that the HRG model is not reliable for temperatures above approximately 160–180 MeV. However, the pseudocritical temperature extracted from the half-maximum criterion for ω=0 is 162.45 MeV, which is at the upper edge of this range. For finite ω, the extracted Tc values can be as low as ~130 MeV (from Fig. 3, at ω=0.025 and μ_B=0.4 GeV), which is within the HRG's range of applicability. The authors should explicitly discuss the temperature range where the HRG model is trustworthy for this analysis and provide an uncertainty estimate from the model's truncation.

Minor objections (5)

[Sec. I, introduction] The introduction states that 'the curvature coefficients of the chiral crossover transition line in the T–μ_B plane within the HRG framework have been found to be consistent with lQCD results [49]' but does not cite specific values. Providing a brief comparison would help contextualize the new ω curvature coefficients.
[Eq. (6), Eq. (7), Eq. (9)] The notation for the exponential factor in the distribution functions differs: Eq. (6) uses 'exp[(Ei−μi)/T]−1', Eq. (7) uses 'exp[(Ei−μi)/T]±1', and Eq. (9) uses 'exp[(ε_{l,i}−μ_i)/T]±1'. While correct, it would be clearer to explicitly state that the '±' corresponds to bosons (−) and fermions (+), as done in Eq. (1). Also, in Eq. (9), the denominator 'exp[(εl,i−μi)/T]±1' should include the ± sign as in the text.
[Sec. III, Fig. 2] The fit in Fig. 2 uses the ansatz Eq. (10) truncated at O(ω^4). The data points appear to extend to ω/Tc(0) ≈ 0.14, which is near the boundary Rω=1. It would be informative to include the reduced χ^2 of the fit and an indication of whether including an O(ω^6) term significantly improves the fit. Also, error bars on the data points from the half-maximum extraction (e.g., from temperature resolution) should be specified.
[Sec. III, Fig. 4] The radial dependence of Tc is shown for r from 1 to 6 fm. At ω=0.025 GeV and r=6 fm, the product ωr = 0.15, which is less than 1 in natural units (since c=1). However, the cylindrical boundary condition imposes R=6 fm and Rω≤1, so at ω=0.025 GeV, Rω=0.15, which is fine. But the authors should clarify whether the radial profiles are computed at fixed ω (global rotation) or whether the local angular velocity is assumed to be constant. The text implies the latter, but a brief comment would help.
[Sec. IV, summary] The summary states that the study provides 'a first systematic HRG-based investigation of the rotational dependence of the chiral crossover.' This claim should be tempered by the acknowledgment that the definition of Tc is imported and not internally derived. Also, the abstract uses 'pseudocritical' and pseudo-critical interchangeably; one form should be chosen for consistency.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Rotation suppresses the QCD chiral crossover temperature in a hadron resonance gas model, with leading curvature coefficient k₂^ω = 10.578 ± 0.118.

The reading

The paper investigates how uniform rotation affects the chiral crossover transition of QCD matter, using a hadron resonance gas (HRG) model extended to a rotating frame. By coupling the total angular momentum of hadronic states to the angular velocity, the authors compute the renormalized chiral condensate and extract the pseudocritical temperature via the half-maximum criterion. They find that the pseudocritical temperature systematically decreases with increasing angular velocity, quantified by leading and next-to-leading curvature coefficients. The suppression becomes stronger when both rotation and baryon chemical potential are applied, and the pseudocritical temperature also drops with radial distance from the rotation axis. These results align with predictions from effective models and holography but stand in tension with recent lattice QCD findings that report an increase of the transition temperature with rotation.

What carries the argument

The central object is the renormalized chiral condensate Δ_l^R, computed in a rotating HRG model. Rotation enters through a shift of the single-particle energy ε_ℓ,i = √(p_r² + p_z² + m_i²) − (ℓ + s)ω, where ℓ is the orbital angular momentum about the rotation axis and s the spin projection. The pressure sum over ℓ and radial modes (quantized by Bessel zeros) yields the condensate derivative. The half-maximum of Δ_l^R – the temperature where the condensate drops to half its vacuum value – defines the pseudocritical temperature; this criterion is imported from lattice QCD analyses.

Extended reading notes

Core claim

Within the rotating hadron resonance gas framework, the chiral crossover pseudocritical temperature decreases monotonically with angular velocity ω. For ω up to 0.015 GeV, the suppression reaches about 10% relative to the non-rotating value Tc(0) = 162.45 MeV. The dependence is parameterized as Tc(ω)/Tc(0) = 1 − κ₂^ω (ω/Tc(0))² − κ₄^ω (ω/Tc(0))⁴, with fitted curvature coefficients κ₂^ω = 10.578 ± 0.118 and κ₄^ω = −68.145 ± 6.416. Additionally, the combined effect of ω and baryon chemical potential μ_B produces a stronger reduction than either alone, and the pseudocritical temperature decreases with increasing radial distance from the rotation axis.

Load-bearing premise

The pseudocritical temperature is defined as the temperature where the renormalized chiral condensate drops to half its vacuum value – a criterion imported from lattice QCD at zero rotation – and this criterion is assumed to remain valid for rotating systems without independent verification.

Editorial extensions

If this is right

If rotation indeed suppresses the chiral crossover, the QCD phase diagram must be extended to a three-dimensional space T–μ_B–ω, and heavy-ion collisions with large vorticity (e.g., non-central events) may sample a colder transition region near the rotation axis.
The curvature coefficients κ₂^ω and κ₄^ω provide a benchmark for lattice QCD calculations and other effective models, allowing quantitative comparisons of the rotational response of the chiral transition.
The spatial dependence of Tc suggests that a rotating fireball undergoes chiral restoration at different radii at the same global temperature, potentially imprinting radial correlations in hadron yields or flow observables.
If the suppression is genuine, the freeze-out temperature in rotating systems (extracted via thermal fits) should be lower than in non-rotating systems, consistent with earlier HRG predictions for freeze-out.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The half-maximum criterion for Tc is borrowed from lattice QCD at zero rotation; its validity in a rotating system has not been independently tested. A different criterion (e.g., peak of the chiral susceptibility) might yield a different functional dependence on ω and could change the curvature coefficients.
The tension between this HRG prediction (Tc drops with ω) and recent lattice results (Tc rises with ω) may reflect the truncation of the HRG model at low temperature or the treatment of the rotational boundary conditions. A direct lattice calculation of the chiral condensate in a rotating system with the same quark masses could break the deadlock.
If the suppression is confirmed, it implies that the chiral transition temperature in the early universe (which may have been rotating) was lower than commonly assumed, affecting predictions for primordial magnetic field generation or baryogenesis scenarios.

Original abstract

We analyze the impact of rotation on the chiral crossover transition of QCD matter within the framework of a hadron resonance gas model. By extending the conventional formulation of the renormalized chiral condensate to a rotating hadronic medium, we find that rotation significantly modifies the chiral condensate and systematically suppresses the pseudocritical temperature as the angular velocity increases. The pseudocritical line in the $T-\omega$ plane is examined, and the rotational dependence of the pseudocritical temperature is quantified through leading- and next-to-leading-order rotational curvature coefficients. We further investigate the combined effects of angular velocity and baryon chemical potential and find that their interplay leads to a stronger suppression of the pseudocritical temperature. Additionally, the spatial dependence of the chiral crossover transition is examined through the radial variation of the pseudocritical temperature. We find that the pseudocritical temperature decreases with increasing radial distance, indicating that the chiral crossover sets in at progressively lower temperatures as one goes away from the rotation axis.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,042 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.12 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We analyze the impact of rotation on the chiral crossover transition of QCD matter within the framework of a hadron resonance gas model. By extending the conventional formulation of the renormalized chiral condensate to a rotating hadronic medium, we find that rotation significantly modifies the chiral condensate and systematically suppresses the pseudocritical temperature as the angular velocity increases. The pseudocritical line in the $T-\omega$ plane is examined, and the rotational dependence of the pseudocritical temperature is quantified through leading- and next-to-leading-order rotational curvature coefficients. We further investigate the combined effects of angular velocity and baryon chemical potential and find that their interplay leads to a stronger suppression of the pseudocritical temperature. Additionally, the spatial dependence of the chiral crossover transition is examined through the radial variation of the pseudocritical temperature. We find that the pseudocritical temperature decreases with increasing radial distance, indicating that the chiral crossover sets in at progressively lower temperatures as one goes away from the rotation axis.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We analyze the impact of rotation on the chiral crossover transition of QCD matter within the framework of a hadron resonance gas model.	contribution 0.91	0.90	0.05	asserted 0.70
s1	By extending the conventional formulation of the renormalized chiral condensate to a rotating hadronic medium, we find that rotation significantly modifies the chiral condensate and systematically suppresses the pseudocritical temperature as the angular velocity increases.	result 0.97	0.96	0.77	asserted 1.00
s2	The pseudocritical line in the $T-\omega$ plane is examined, and the rotational dependence of the pseudocritical temperature is quantified through leading- and next-to-leading-order rotational curvature coefficients.	contribution 0.74	0.94	0.12	asserted 0.98
s3	We further investigate the combined effects of angular velocity and baryon chemical potential and find that their interplay leads to a stronger suppression of the pseudocritical temperature.	result 0.98	0.96	0.15	asserted 1.00
s4	Additionally, the spatial dependence of the chiral crossover transition is examined through the radial variation of the pseudocritical temperature.	contribution 0.97	0.92	0.05	asserted 0.86
s5	We find that the pseudocritical temperature decreases with increasing radial distance, indicating that the chiral crossover sets in at progressively lower temperatures as one goes away from the rotation axis.	result 0.98	0.96	0.21	asserted 0.83

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 0.80.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: (Colour Online) Temperature dependence of the renormalised chiral condensate

**Figure 2.** FIG. 2: (Colour online) The pseudocritical line as a function

**Figure 3.** FIG. 3: (Colour online) The pseudocritical temperature as a

**Figure 4.** FIG. 4: (Colour Online) Radial dependence of the pseudo

Figures reproduced from arXiv:2609.15513 by the authors.

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [44]

S. Borsanyiet al.[Wuppertal-Budapest], JHEP09, 073 (2010)

View on Pith 2010

[2] [1]

A. I. Bochkarev and M. E. Shaposhnikov, Phys. Lett. B 145, 276 (1984)

View on Pith 1984

[3] [2]

C. M. Ko, Z. G. Wu, L. H. Xia and G. E. Brown, Phys. Rev. Lett.66, 2577 (1991)

View on Pith 1991

[4] [3]

C. M. Ko and B. H. Sa, Phys. Lett. B258, 6 (1991)

View on Pith 1991

[5] [4]

A. Bazavovet al.[HotQCD], Phys. Lett. B795, 15 (2019)

View on Pith 2019

[6] [5]

W. T. Deng and X. G. Huang, Phys. Rev. C85, 044907 (2012)

View on Pith 2012

[7] [6]

K. Fukushima and Y. Hidaka, Phys. Rev. Lett.117, 102301 (2016)

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