Chiral Response of QCD Topology across the Thermal Crossover

Heng-Tong Ding; Jia Ni; Jin-Biao Gu; Jos\'e Javier Hern\'andez Hern\'andez; Sheng-Tai Li

REVIEW

Chiral Response of QCD Topology across the Thermal Crossover

Heng-Tong Ding, Jin-Biao Gu, José Javier Hernández Hernández, Sheng-Tai Li, Jia Ni

High Energy Physics - Lattice · Submitted 6 Nov 2026 · 29 references

Pith's verdict: Elegant new relation, clean lattice implementation, and a clear qualitative result—but the quantitative claim at high T rests on a single lattice spacing and should be softened until the continuum analysis is extended.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on November 9, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 1 major objection · 4 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2611.09150 (version 1)

Pith IDF3LRZOKR, Pith's permanent record of this paper and its review

CitationHeng-Tong Ding, Jia Ni, Jin-Biao Gu, Jos\'e Javier Hern\'andez Hern\'andez, Sheng-Tai Li. "Chiral Response of QCD Topology across the Thermal Crossover." arXiv:2611.09150 (2026). https://pith.science/paper/F3LRZOKR

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

The paper does two things well. First, it derives an exact relation (Eq. 5–6) connecting ∂χ_top/∂m_ℓ to a connected correlator ⟨Q² X₁⟩_c. That is neat, general, and lets you extract C_top = ∂ln χ_top/∂ln m_ℓ from a single ensemble per temperature. No finite differences, no fitting. Second, it applies this relation to (2+1)-flavor lattice QCD at m_π ≈ 220 MeV and shows that C_top rises from ~1 at low T to ~2 above the crossover. The trend is clear and consistent with what chiral effective theory and the dilute instanton gas predict. The lattice methods (Wilson flow, eigenvalue reweighting) are standard and well documented. The continuum extrapolation at T≈211 MeV, using three lattice spacings, is done carefully and yields C_top = 1.81(10), appreciably above unity.

Now the soft spot, which the authors own: the continuum extrapolation is performed only at one temperature. At all other T, the results come from a single lattice spacing (a≈0.117 fm). At 280.9 MeV, the highest T, C_top sits at almost 2 on that coarse lattice. The continuum correction at 211 MeV lowered C_top by about 12%, so it is plausible but not demonstrated that the same fractional shift applies at 280.9 MeV. If discretization effects grow with temperature—which is not unlikely—the endpoint might shift. The authors say “calculations at finer lattice spacings will extend the continuum analysis,” and they mean it. For now, the high-T asymptote is provisional.

That said, the qualitative story—C_top evolves from ~1 to ~2 across the crossover—is robust. The new correlator method is the main contribution and will be useful beyond this paper. The rescaling of χ_top to physical m_ℓ assumes C_top is unchanged between m_ℓ = m_s/10 and the physical point, which is a caveat they flag.

Who should read this? Lattice practitioners working on topology, axion phenomenologists who need the mass dependence of χ_top, and anyone interested in how the chiral crossover affects topological fluctuations. The paper deserves a serious referee. I would accept it with a request to expand the continuum analysis in a follow-up or to add a note that the high-T result is preliminary pending finer lattices.

Referee report

Significance

The relation (6) is exact in continuum QCD and provides a parameter-free estimator of the local logarithmic mass derivative of χ_top from a single ensemble, avoiding finite-difference interpolations or fitting. This is a methodological advance. The lattice application reveals a clear thermal reorganization: C_top evolves from unity (linear mass dependence) toward two (quadratic mass dependence) across the crossover, with the effective amplitude peaking near T_pc. The results are consistent with chiral perturbation theory at low T and with semiclassical instanton arguments at high T. The continuum-extrapolated value at T ≃ 211 MeV (C_top = 1.81(10)) confirms the trend. The work is timely for axion phenomenology and for understanding U(1)_A restoration in hot QCD. The limitations (single lattice spacing at most temperatures) are transparently stated, providing a clear path for future work.

Major objections (1)

[Section IV and Appendix B] The continuum extrapolation of C_top is performed only at T ≃ 211 MeV. All other temperatures, including the highest (T = 280.9 MeV) where C_top reaches nearly two, rely on a single lattice spacing (a ≃ 0.117 fm). The paper acknowledges this limitation (b113) but does not discuss the potential temperature dependence of discretization effects. At 211 MeV, the finite-spacing value of C_top is reduced by about 10–15% in the continuum limit. If a similar fractional correction applied at 280.9 MeV, the continuum-extrapolated C_top would be around 1.7–1.8, which would still indicate a clear rise from unity but would affect the claim of reaching values close to two. The authors should either provide a continuum extrapolation at an additional temperature (e.g., near T_pc or at 280.9 MeV) or add a careful quantitative discussion of how the high-T result might shift under plausible assumptions about discretization effects.

Minor objections (4)

[Section II] The derivation of Eq. (6) is clear, but the text could explicitly note that the cumulant ⟨Q²X₁⟩_c is computed on each ensemble with the same flow time and reweighting, and that only the ratio enters C_top, so no separate renormalization of the cumulant is required.
[Figures 1 and 2] The 'ChPT' curve from Ref. [6] is shown as a reference. It would be helpful to state the value of m_π (or H = m_ℓ/m_s) assumed for that curve, especially since the lattice data use H = 1/10.
[Appendix B] The description of flow-time plateaus (Figs. 4 and 5) is qualitative. A brief quantification of the variation of χ_top and C_top within the plateau region (e.g., the range of τ_f T² from 0.08 to 0.12) would strengthen the assessment of systematic uncertainty from the flow time.
[Abstract and Conclusions] The phrase 'toward two' is appropriate given the single-spacing data, but the conclusions could more explicitly separate the finite-spacing results from the continuum-extrapolated result, e.g., 'the finite-spacing data approach two, and the continuum value at 211 MeV is 1.81(10), indicating a sustained rise'.

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

The paper

The paper claims

The chiral response of QCD topology—the local logarithmic light-quark-mass derivative of the topological susceptibility—rises rapidly from approximately one to two through the thermal crossover, revealing a reorganization of the…

The reading

This paper establishes a new relation that expresses the light-quark-mass derivative of the topological susceptibility as a connected correlation between the squared topological charge and the integrated light-quark scalar density. This allows direct extraction of the chiral response C_top = ∂lnχ_top/∂lnm_ℓ from a single lattice ensemble at each temperature, without simulations at multiple masses. Applying the method to (2+1)-flavor lattice QCD at m_π ≃ 220 MeV, the authors find that C_top rises from about 1 at low temperature (consistent with linear mass dependence from chiral effective theory) to about 1.5 near the chiral crossover and nearly 2 at high temperature (consistent with quadratic mass dependence from instanton models). The temperature-dependent effective amplitude A_eff = χ_top $H^{{-C_top}}$ peaks near the crossover, reflecting competition between the falling susceptibility and the rising chiral response. These results go beyond the known thermal suppression of χ_top and show that the local mass dependence itself undergoes a thermal reorganization.

What carries the argument

The key identity is Eq. (6): C_top = m_ℓ ⟨$Q^{2}$ X_1⟩_c / ⟨$Q^{2}$⟩, where X_1 = 2 Tr $M_ℓ^{{-1}}$ is the integrated light-quark scalar insertion. This follows from the general relation (4) derived from the QCD partition function by differentiating with respect to the common light-quark mass. On the lattice, the insertion is evaluated with eigenvalue reweighting to restore the zero-mode mass factors, giving the estimator (C5). The method avoids finite-difference approximations and works at the simulated quark masses.

Extended reading notes

Core claim

The central claim is that the chiral response of QCD topology, C_top = ∂lnχ_top/∂lnm_ℓ, evolves from approximately unity at low temperature (T ≲ 100 MeV) to approximately two at high temperature (T ≳ 210 MeV), with the most rapid change occurring across the chiral crossover near T_pc ≃ 170 MeV. This is measured using a novel identity (Eq. 6) that relates the mass derivative to a connected correlation function computable on a single lattice ensemble. The continuum-extrapolated result at T ≃ 211 MeV gives C_top = 1.81(10). The effective local power-law amplitude A_eff peaks near T_pc. The paper argues that the thermal medium restructures the way topological fluctuations respond to the light-quark mass, not just suppressing their overall magnitude.

Load-bearing premise

The continuum extrapolation of C_top is performed at only one temperature (T ≃ 211 MeV) using a linear fit in $a^{2}$; all other temperatures rely on a single lattice spacing (a ≃ 0.117 fm), so the temperature dependence of C_top beyond 211 MeV is not continuum-extrapolated, and the claim that C_top approaches 2 at high T rests heavily on the 280.9 MeV point without any continuum check.

Editorial extensions

If this is right

The chiral response C_top can be used to rescale lattice results for χ_top from heavier-than-physical light-quark masses to the physical point, assuming C_top remains unchanged over the mass range (as tested at one temperature).
The formalism extends to higher-order derivatives (∂^n χ_top/∂ m_ℓ^n) and mixed-flavor derivatives, giving access to the quark-mass dependence of non-Gaussian topological fluctuations and the full θ-expansion of the free energy.
The peak in A_eff near T_pc provides a new observable that combines thermal suppression and the changing mass exponent, potentially sensitive to the restoration of chiral symmetry.
The method can be implemented with other fermion discretizations (e.g., Wilson, domain-wall), enabling cross-checks of the evolution of C_top across the crossover.
The results imply that axion mass calculations based on χ_top must account for a temperature-dependent power-law scaling in the light-quark mass, not just a constant exponent.

Reading between the lines

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

If C_top approaches 2 at high T, then χ_top ∝ m_ℓ^2 to leading order, consistent with the dilute instanton gas and quasi-instanton descriptions. This strengthens the case that instanton-like objects dominate topology in the deconfined phase, even away from the semiclassical regime.
The rapid rise of C_top through the crossover suggests that the quark-mass dependence of topology is tied to the chiral condensate: as chiral symmetry restores, the Dirac eigenvalue spectrum reorganizes, removing the linear-response channel and leaving only the quadratic channel.
The continuum-extrapolated C_top = 1.81(10) at 211 MeV leaves room for a slight deviation from 2, which might indicate residual effects from non-zeromodes or a small quartic correction. Finer lattices and lighter masses could sharpen this test.
The formalism can be applied to other cumulants of Q (e.g., κ_4) to probe whether higher topological moments also change their mass scaling through the crossover, potentially revealing non-Gaussian features in the θ-dependence.

Original abstract

We introduce a novel relation connecting the light-quark-mass derivative of the topological susceptibility in quantum chromodynamics (QCD) to the connected correlation between the squared topological charge and the integrated light-quark scalar density. This relation gives direct access to the chiral response of QCD topology---the local logarithmic light-quark-mass derivative of the susceptibility---from a single ensemble at fixed quark masses for each temperature, without additional simulations at shifted masses. Applying this relation to $(2+1)$-flavor lattice QCD at $m_\pi\simeq220\,\mathrm{MeV}$ over temperatures ranging from well below to well above the chiral crossover, we find that the response rises rapidly through the crossover from approximately unity toward two. This evolution corresponds to a change from nearly linear to nearly quadratic dependence of the topological susceptibility on the light-quark mass. Using the chiral response as the exponent of a local power law in the light-quark mass, we determine its effective amplitude from the topological susceptibility. This amplitude exhibits a maximum near the crossover temperature, reflecting the competition between the decreasing topological susceptibility and the increasing chiral response. These results reveal a thermal reorganization of the quark-mass dependence of QCD topology beyond the suppression of topological fluctuations alone.

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, 2026-11-09: 1 request, 5,792 input tokens.

What the abstract itself states, as probabilities: claims first 0.14 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.97 · compares with prior results 0.15 · 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 introduce a novel relation connecting the light-quark-mass derivative of the topological susceptibility in quantum chromodynamics (QCD) to the connected correlation between the squared topological charge and the integrated light-quark scalar density. This relation gives direct access to the chiral response of QCD topology---the local logarithmic light-quark-mass derivative of the susceptibility---from a single ensemble at fixed quark masses for each temperature, without additional simulations at shifted masses. Applying this relation to $(2+1)$-flavor lattice QCD at $m_\pi\simeq220\,\mathrm{MeV}$ over temperatures ranging from well below to well above the chiral crossover, we find that the response rises rapidly through the crossover from approximately unity toward two. This evolution corresponds to a change from nearly linear to nearly quadratic dependence of the topological susceptibility on the light-quark mass. Using the chiral response as the exponent of a local power law in the light-quark mass, we determine its effective amplitude from the topological susceptibility. This amplitude exhibits a maximum near the crossover temperature, reflecting the competition between the decreasing topological susceptibility and the increasing chiral response. These results reveal a thermal reorganization of the quark-mass dependence of QCD topology beyond the suppression of topological fluctuations alone.

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 introduce a novel relation connecting the light-quark-mass derivative of the topological susceptibility in quantum chromodynamics (QCD) to the connected correlation between the squared topological charge and the integrated light-quark scalar density.	contribution 1.00	0.92	0.10	asserted 1.00
s1	This relation gives direct access to the chiral response of QCD topology---the local logarithmic light-quark-mass derivative of the susceptibility---from a single ensemble at fixed quark masses for each temperature, without additional simulations at shifted masses.	method 0.53	0.63	0.14	asserted 1.00
s2	Applying this relation to $(2+1)$-flavor lattice QCD at $m_\pi\simeq220\,\mathrm{MeV}$ over temperatures ranging from well below to well above the chiral crossover, we find that the response rises rapidly through the crossover from approximately unity toward two.	result 0.99	0.98	0.55	asserted 1.00
s3	This evolution corresponds to a change from nearly linear to nearly quadratic dependence of the topological susceptibility on the light-quark mass.	interpretation 0.92	0.90	0.20	asserted 0.99
s4	Using the chiral response as the exponent of a local power law in the light-quark mass, we determine its effective amplitude from the topological susceptibility.	method 0.79	0.94	0.06	asserted 1.00
s5	This amplitude exhibits a maximum near the crossover temperature, reflecting the competition between the decreasing topological susceptibility and the increasing chiral response.	result 0.50	0.95	0.18	asserted 1.00
s6	These results reveal a thermal reorganization of the quark-mass dependence of QCD topology beyond the suppression of topological fluctuations alone.	interpretation 1.00	0.87	0.69	asserted 1.00

What kind of paper the classifier reports this is: simulation study 0.67 · evidence from simulation or computation 1.00.

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.** shows the temperature dependence of the topological susceptibility χtop at a ≃ 0.117 fm. The susceptibility decreases from about 9.8 × 10−2 to 6.8 × 10−4 fm−4 over T ≈ 70– 281 MeV, corresponding to a suppression of roughly two orders of magnitude. This thermal suppression is qualitatively consistent with earlier results [13, 14, 16]. The lowest-temperature results lie above the chiral perturbation theory (ChPT) reference [6]. 100 150 200 250 T [MeV] 10 3 10 2 10 1 top [fm 4] ChPT a ' 0.117 fm Cont. extr. FIG. 1. χtop as a function of temperature. Purple squares denote resul

**Figure 2.** FIG. 2. Temperature dependence of the chiral response of QCD topology

**Figure 3.** FIG. 3. Temperature dependence of the effective local power-law amplitude

**Figure 4.** FIG. 4. Topological susceptibility

**Figure 6.** FIG. 6. Monte Carlo histories of the unreweighted topological charge

**Figure 7.** FIG. 7. Continuum extrapolations at

Figures reproduced from arXiv:2611.09150 by the authors.

References

Works cited (29)

[1]

R. Jackiw and C. Rebbi,Vacuum Periodicity in a Yang-Mills Quantum Theory,Phys. Rev. Lett.37(1976) 172

View on Pith 1976
[2]

G. ’t Hooft,Symmetry Breaking Through Bell-Jackiw Anomalies,Phys. Rev. Lett.37(1976) 8

View on Pith 1976
[3]

R.D. Peccei and H.R. Quinn,CP Conservation in the Presence of Instantons,Phys. Rev. Lett. 38(1977) 1440

View on Pith 1977
[4]

S. Weinberg,A New Light Boson?,Phys. Rev. Lett.40(1978) 223

View on Pith 1978
[5]

F. Wilczek,Problem of StrongPandTInvariance in the Presence of Instantons,Phys. Rev. Lett.40(1978) 279

View on Pith 1978
[6]

G. Grilli di Cortona, E. Hardy, J. Pardo Vega and G. Villadoro,The QCD axion, precisely, JHEP01(2016) 034 [1511.02867]

View on Pith Pith review arXiv 2016
[7]

H. Leutwyler and A.V. Smilga,Spectrum of Dirac operator and role of winding number in QCD,Phys. Rev. D46(1992) 5607. 8

View on Pith 1992
[8]

D.J. Gross, R.D. Pisarski and L.G. Yaffe,QCD and Instantons at Finite Temperature,Rev. Mod. Phys.53(1981) 43

View on Pith 1981
[9]

T. Schäfer and E.V. Shuryak,Instantons in QCD,Rev. Mod. Phys.70(1998) 323 [hep-ph/9610451]

View on Pith Pith review arXiv 1998
[10]

T. Kanazawa and N. Yamamoto,Quasi-instantons in QCD with chiral symmetry restoration, Phys. Rev. D91(2015) 105015 [1410.3614]

View on Pith Pith review arXiv 2015
[11]

B. Alles, M. D’Elia and A. Di Giacomo,Topological susceptibility in full QCD at zero and finite temperature,Phys. Lett. B483(2000) 139 [hep-lat/0004020]. [12]SESAM, T(X)Lcollaboration,Quark mass effects on the topological susceptibility in QCD, Phys. Rev. D64(2001) 054502 [hep-lat/0102002]

View on Pith Pith review arXiv 2000
[13]

C. Bonati, M. D’Elia, M. Mariti, G. Martinelli, M. Mesiti, F. Negro et al.,Axion phenomenology andθ-dependence fromN f = 2 + 1lattice QCD,JHEP03(2016) 155 [1512.06746]

View on Pith Pith review arXiv 2016
[14]

P. Petreczky, H.-P. Schadler and S. Sharma,The topological susceptibility in finite temperature QCD and axion cosmology,Phys. Lett. B762(2016) 498 [1606.03145]

View on Pith Pith review arXiv 2016
[15]

Y. Taniguchi, K. Kanaya, H. Suzuki and T. Umeda,Topological susceptibility in finite temperature ( 2+1 )-flavor QCD using gradient flow,Phys. Rev. D95(2017) 054502 [1611.02411]

View on Pith Pith review arXiv 2017
[16]

R.V. Gavai, M.E. Jaensch, O. Kaczmarek, F. Karsch, M. Sarkar, R. Shanker et al.,Aspects of the chiral crossover transition in (2+1)-flavor QCD with Möbius domain-wall fermions,Phys. Rev. D111(2025) 034507 [2411.10217]

View on Pith Pith review arXiv 2025
[17]

S. Borsanyi et al.,Calculation of the axion mass based on high-temperature lattice quantum chromodynamics,Nature539(2016) 69 [1606.07494]

View on Pith Pith review arXiv 2016
[18]

C. Bonati, M. D’Elia, G. Martinelli, F. Negro, F. Sanfilippo and A. Todaro,Topology in full QCD at high temperature: a multicanonical approach,JHEP11(2018) 170 [1807.07954]

View on Pith Pith review arXiv 2018
[19]

A. Athenodorou, C. Bonanno, C. Bonati, G. Clemente, F. D’Angelo, M. D’Elia et al., Topological susceptibility of Nf = 2 + 1 QCD from staggered fermions spectral projectors at high temperatures,JHEP10(2022) 197 [2208.08921]. [20]TWQCDcollaboration,Topological susceptibility in finite temperature QCD with physical (u/d,s,c) domain-wall quarks,Phys. Rev. D10...

View on Pith Pith review arXiv 2022
[21]

C. Bonanno, G. Clemente, M. D’Elia, L. Maio and L. Parente,Full QCD with milder topological freezing,JHEP08(2024) 236 [2404.14151]

View on Pith Pith review arXiv 2024
[22]

A.Y. Kotov, M.P. Lombardo and A. Trunin,Topological observables andθ dependence in high temperature QCD from lattice simulations,JHEP09(2025) 045 [2502.15407]

View on Pith Pith review arXiv 2025
[23]

F. Burger, E.-M. Ilgenfritz, M.P. Lombardo and A. Trunin,Chiral observables and topology in hot QCD with two families of quarks,Phys. Rev. D98(2018) 094501 [1805.06001]. [24]HPQCD, UKQCDcollaboration,Highly improved staggered quarks on the lattice, with applications to charm physics,Phys. Rev. D75(2007) 054502 [hep-lat/0610092]

View on Pith Pith review arXiv 2018
[25]

P. Weisz,Continuum Limit Improved Lattice Action for Pure Yang-Mills Theory. 1.,Nucl. Phys. B212(1983) 1

View on Pith 1983
[26]

H.-T. Ding, S.-T. Li, Q. Shi and X.-D. Wang,Fluctuations and correlations of net baryon number, electric charge and strangeness in a background magnetic field,Eur. Phys. J. A57 (2021) 202 [2104.06843]

View on Pith Pith review arXiv 2021
[27]

M. Lüscher,Properties and uses of the Wilson flow in lattice QCD,JHEP08(2010) 071 [1006.4518]. 9

View on Pith Pith review arXiv 2010
[28]

S.O. Bilson-Thompson, D.B. Leinweber and A.G. Williams,Highly improved lattice field strength tensor,Annals Phys.304(2003) 1 [hep-lat/0203008]

View on Pith Pith review arXiv 2003
[29]

L. Mazur,Topological Aspects in Lattice QCD, Ph.D. thesis, Bielefeld U., 2021. 10.4119/unibi/2956493. [30]HotQCDcollaboration,SIMULATeQCD: A simple multi-GPU lattice code for QCD calculations,Comput. Phys. Commun.300(2024) 109164 [2306.01098]

View on Pith Pith review arXiv 2024
[31]

T.G. Kovacs,Temperature-dependence of the QCD topological susceptibility,EPJ Web Conf. 175(2018) 01013 [1711.03911]

View on Pith Pith review arXiv 2018
[32]

B.B. Brandt, G. Endrődi, J.J.H. Hernández and G. Markó,Impact of extreme magnetic fields on the QCD topological susceptibility in the vicinity of the crossover region,JHEP12(2025) 228 [2409.00796]. [33]Fla vour Lattice A veraging Group (FLAG)collaboration,FLAG review 2024,Phys. Rev. D113(2026) 014508 [2411.04268]

View on Pith Pith review arXiv 2025
[34]

H.-T. Ding,AxialU(1)Symmetry Breaking in Hot QCD: From Topology to the Chiral Phase Transition, inEncyclopedia of Nuclear Physics, 9, 2026 [2609.21907]. Appendix A: Lattice ensembles The fixed-scale scan comprises eight32 3 ×N τ ensembles at a≃ 0.117 fm and fixed quark masses, with4 188to15 114gauge configurations per temperature (Table I). For the discre...

View on Pith Pith review arXiv 2026

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] [1]

R. Jackiw and C. Rebbi,Vacuum Periodicity in a Yang-Mills Quantum Theory,Phys. Rev. Lett.37(1976) 172

View on Pith 1976

[2] [2]

G. ’t Hooft,Symmetry Breaking Through Bell-Jackiw Anomalies,Phys. Rev. Lett.37(1976) 8

View on Pith 1976

[3] [3]

R.D. Peccei and H.R. Quinn,CP Conservation in the Presence of Instantons,Phys. Rev. Lett. 38(1977) 1440

View on Pith 1977

[4] [4]

S. Weinberg,A New Light Boson?,Phys. Rev. Lett.40(1978) 223

View on Pith 1978

[5] [5]

F. Wilczek,Problem of StrongPandTInvariance in the Presence of Instantons,Phys. Rev. Lett.40(1978) 279

View on Pith 1978

[6] [6]

G. Grilli di Cortona, E. Hardy, J. Pardo Vega and G. Villadoro,The QCD axion, precisely, JHEP01(2016) 034 [1511.02867]

View on Pith Pith review arXiv 2016

[7] [7]

H. Leutwyler and A.V. Smilga,Spectrum of Dirac operator and role of winding number in QCD,Phys. Rev. D46(1992) 5607. 8

View on Pith 1992

[8] [8]

D.J. Gross, R.D. Pisarski and L.G. Yaffe,QCD and Instantons at Finite Temperature,Rev. Mod. Phys.53(1981) 43

View on Pith 1981

[9] [9]

T. Schäfer and E.V. Shuryak,Instantons in QCD,Rev. Mod. Phys.70(1998) 323 [hep-ph/9610451]

View on Pith Pith review arXiv 1998

[10] [10]

T. Kanazawa and N. Yamamoto,Quasi-instantons in QCD with chiral symmetry restoration, Phys. Rev. D91(2015) 105015 [1410.3614]

View on Pith Pith review arXiv 2015

[11] [11]

B. Alles, M. D’Elia and A. Di Giacomo,Topological susceptibility in full QCD at zero and finite temperature,Phys. Lett. B483(2000) 139 [hep-lat/0004020]. [12]SESAM, T(X)Lcollaboration,Quark mass effects on the topological susceptibility in QCD, Phys. Rev. D64(2001) 054502 [hep-lat/0102002]

View on Pith Pith review arXiv 2000

[12] [13]

C. Bonati, M. D’Elia, M. Mariti, G. Martinelli, M. Mesiti, F. Negro et al.,Axion phenomenology andθ-dependence fromN f = 2 + 1lattice QCD,JHEP03(2016) 155 [1512.06746]

View on Pith Pith review arXiv 2016

[13] [14]

P. Petreczky, H.-P. Schadler and S. Sharma,The topological susceptibility in finite temperature QCD and axion cosmology,Phys. Lett. B762(2016) 498 [1606.03145]

View on Pith Pith review arXiv 2016

[14] [15]

Y. Taniguchi, K. Kanaya, H. Suzuki and T. Umeda,Topological susceptibility in finite temperature ( 2+1 )-flavor QCD using gradient flow,Phys. Rev. D95(2017) 054502 [1611.02411]

View on Pith Pith review arXiv 2017

[15] [16]

R.V. Gavai, M.E. Jaensch, O. Kaczmarek, F. Karsch, M. Sarkar, R. Shanker et al.,Aspects of the chiral crossover transition in (2+1)-flavor QCD with Möbius domain-wall fermions,Phys. Rev. D111(2025) 034507 [2411.10217]

View on Pith Pith review arXiv 2025

[16] [17]

S. Borsanyi et al.,Calculation of the axion mass based on high-temperature lattice quantum chromodynamics,Nature539(2016) 69 [1606.07494]

View on Pith Pith review arXiv 2016

[17] [18]

C. Bonati, M. D’Elia, G. Martinelli, F. Negro, F. Sanfilippo and A. Todaro,Topology in full QCD at high temperature: a multicanonical approach,JHEP11(2018) 170 [1807.07954]

View on Pith Pith review arXiv 2018

[18] [19]

A. Athenodorou, C. Bonanno, C. Bonati, G. Clemente, F. D’Angelo, M. D’Elia et al., Topological susceptibility of Nf = 2 + 1 QCD from staggered fermions spectral projectors at high temperatures,JHEP10(2022) 197 [2208.08921]. [20]TWQCDcollaboration,Topological susceptibility in finite temperature QCD with physical (u/d,s,c) domain-wall quarks,Phys. Rev. D10...

View on Pith Pith review arXiv 2022

[19] [21]

C. Bonanno, G. Clemente, M. D’Elia, L. Maio and L. Parente,Full QCD with milder topological freezing,JHEP08(2024) 236 [2404.14151]

View on Pith Pith review arXiv 2024

[20] [22]

A.Y. Kotov, M.P. Lombardo and A. Trunin,Topological observables andθ dependence in high temperature QCD from lattice simulations,JHEP09(2025) 045 [2502.15407]

View on Pith Pith review arXiv 2025

[21] [23]

F. Burger, E.-M. Ilgenfritz, M.P. Lombardo and A. Trunin,Chiral observables and topology in hot QCD with two families of quarks,Phys. Rev. D98(2018) 094501 [1805.06001]. [24]HPQCD, UKQCDcollaboration,Highly improved staggered quarks on the lattice, with applications to charm physics,Phys. Rev. D75(2007) 054502 [hep-lat/0610092]

View on Pith Pith review arXiv 2018

[22] [25]

P. Weisz,Continuum Limit Improved Lattice Action for Pure Yang-Mills Theory. 1.,Nucl. Phys. B212(1983) 1

View on Pith 1983

[23] [26]

H.-T. Ding, S.-T. Li, Q. Shi and X.-D. Wang,Fluctuations and correlations of net baryon number, electric charge and strangeness in a background magnetic field,Eur. Phys. J. A57 (2021) 202 [2104.06843]

View on Pith Pith review arXiv 2021

[24] [27]

M. Lüscher,Properties and uses of the Wilson flow in lattice QCD,JHEP08(2010) 071 [1006.4518]. 9

View on Pith Pith review arXiv 2010

[25] [28]

S.O. Bilson-Thompson, D.B. Leinweber and A.G. Williams,Highly improved lattice field strength tensor,Annals Phys.304(2003) 1 [hep-lat/0203008]

View on Pith Pith review arXiv 2003

[26] [29]

L. Mazur,Topological Aspects in Lattice QCD, Ph.D. thesis, Bielefeld U., 2021. 10.4119/unibi/2956493. [30]HotQCDcollaboration,SIMULATeQCD: A simple multi-GPU lattice code for QCD calculations,Comput. Phys. Commun.300(2024) 109164 [2306.01098]

View on Pith Pith review arXiv 2024

[27] [31]

T.G. Kovacs,Temperature-dependence of the QCD topological susceptibility,EPJ Web Conf. 175(2018) 01013 [1711.03911]

View on Pith Pith review arXiv 2018

[28] [32]

B.B. Brandt, G. Endrődi, J.J.H. Hernández and G. Markó,Impact of extreme magnetic fields on the QCD topological susceptibility in the vicinity of the crossover region,JHEP12(2025) 228 [2409.00796]. [33]Fla vour Lattice A veraging Group (FLAG)collaboration,FLAG review 2024,Phys. Rev. D113(2026) 014508 [2411.04268]

View on Pith Pith review arXiv 2025

[29] [34]

H.-T. Ding,AxialU(1)Symmetry Breaking in Hot QCD: From Topology to the Chiral Phase Transition, inEncyclopedia of Nuclear Physics, 9, 2026 [2609.21907]. Appendix A: Lattice ensembles The fixed-scale scan comprises eight32 3 ×N τ ensembles at a≃ 0.117 fm and fixed quark masses, with4 188to15 114gauge configurations per temperature (Table I). For the discre...

View on Pith Pith review arXiv 2026