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Study of symmetries in finite temperature $N_f=2$ QCD with M\"obius Domain Wall Fermions

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that in two-flavor QCD the axial U(1)_A symmetry, which is broken by the quantum anomaly, is effectively restored already at the chiral crossover temperature T_c ~ 165 MeV, below the 1.2-1.3 T_c threshold reported by…

desk verdict Useful preliminary JLQCD update pointing to early U(1)_A restoration, but the key near-T_c claim lacks shown finite-volume checks and error bars. read the letter →

arxiv 2412.06574 v3 pith:OFFNE5C5 submitted 2024-12-09 hep-lat hep-th

David Ward , Sinya Aoki , Yasumichi Aoki , Hidenori Fukaya , Shoji Hashimoto , Issaku Kanamori , Takashi Kaneko , Jishnu Goswami
show 1 more author
Yu Zhang
This is my paper · ORCID
classification hep-lathep-th
keywords QCDchiralsymmetryaxialU(1)anomalyscreeningmassesMöbiusdomain-wallfermionsfinitetemperaturelatticeSU(2)_CScrossovertwo-flavor
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that in two-flavor QCD the axial U(1)_A symmetry—the one broken by the quantum anomaly—is effectively restored already at the chiral crossover temperature T_c ~ 165 MeV, rather than at the 1.2-1.3 T_c reported by earlier lattice studies. If true, the chiral transition would take place with both SU(2)_L times SU(2)_R and U(1)_A restored, which changes the expected order and universality class of the transition. The evidence comes from screening masses extracted from spatial two-point correlators in simulations with Möbius domain-wall fermions whose residual quark mass is below 0.1 MeV, so chiral symmetry is much better preserved than in Wilson or staggered fermion actions. The paper also reports that the emergent chiral-spin symmetry SU(2)_CS is not seen up to 330 MeV, so that approximate high-temperature symmetry does not set in near T_c.

What carries the argument

The central tool is the screening mass extracted from the z-axis spatial two-point meson correlator, C_Gamma(z) ~ exp(-M_Gamma z), fitted to A cosh(m(z-L/2)). Symmetry restoration is diagnosed by mass differences between channels connected by the relevant transformation: V-A for SU(2)_L times SU(2)_R, X_t - T_t and PS-S for U(1)_A, and X-A for SU(2)_CS. The Möbius domain-wall action keeps the residual quark mass below 0.1 MeV, so the lattice breaks chiral symmetry far less than Wilson or staggered fermion actions, which is what makes the U(1)_A comparison at T_c credible. The free-quark propagator at T to infinity is used to derive the SU(2)_CS structure, showing why that symmetry is expected only at high temperature.

What would settle it

Compare the V-A and X_t-T_t screening mass differences between the small (L=36, L=32) and large (L=48, L=40) volumes at T=147 and 165 MeV: if either difference is significantly nonzero on the larger volumes, the claimed restoration at T_c is a finite-volume artifact.

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Extended reading notes

Core claim

The central claim is that effective U(1)_A restoration in N_f=2 QCD happens at or very near the chiral crossover temperature, not at the higher temperatures previously reported. Using the screening mass differences between symmetry-partner channels, the paper sees SU(2)_L times SU(2)_R broken at 147 MeV and consistent with zero at 165 MeV and above, and it sees the U(1)_A partner difference X_t - T_t vanish near 165 MeV as well, with the PS-S difference supporting the same conclusion. The paper contrasts this with the 1.2-1.3 T_c threshold from earlier staggered-fermion and domain-wall studies, and attributes part of the discrepancy to an isospin-breaking artifact in staggered fermions that opens an unphysical two-pion decay channel for the scalar. It also finds that the X-A difference, which would signal emergent SU(2)_CS symmetry, does not converge to zero up to 330 MeV.

Load-bearing premise

The near-T_c conclusion rests on the assumption that the L=36 and L=32 lattices are large enough that long-range correlations do not shift the screening masses; the paper itself flags that these volumes may not control those effects and does not show the comparison with the larger L=48 and L=40 lattices.

Editorial extensions

If this is right

  • If U(1)_A is restored at T_c, analyses of the chiral transition should treat the effective symmetry as SU(2)_L times SU(2)_R times U(1)_A rather than a theory with a residual axial anomaly, changing the expected critical behavior.
  • The scalar screening mass in a theory with preserved isospin is heavy and cannot decay to two pions, so earlier staggered-fermion results that saw a light scalar were seeing a lattice artifact rather than the physical channel.
  • The screening spectrum at 0.9 T_c already matches zero-temperature hadron masses, implying that the chiral condensate is close to its vacuum value just below the crossover.
  • The absence of SU(2)_CS pairing up to 330 MeV means the chiral-spin symmetry, if it exists, emerges only at higher temperatures than the range studied here.

Reading between the lines

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

  • If the larger-volume ensembles confirm the T=165 MeV result, the longstanding question of whether the two-flavor transition is second-order O(4) or first-order should be revisited with U(1)_A restored, since the effective-action arguments that link the anomaly to the transition order change.
  • A direct test within the same framework is to measure the low-lying Dirac eigenvalue density: effective U(1)_A restoration should appear as a suppression of near-zero modes, and the paper's screening-mass claim predicts that suppression already at T approximately 165 MeV.
  • The scalar-channel discrepancy suggests that part of the difference between restoration temperatures in the literature may be an isospin-breaking artifact; a staggered simulation that preserves isospin would isolate that effect.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This proceedings contribution from the JLQCD Collaboration studies the restoration of chiral and axial symmetries in N_f=2 QCD at finite temperature using Möbius domain-wall fermions with a small residual mass. Screening masses are extracted from spatial two-point meson correlators via a standard cosh fit, and the mass differences between symmetry-related channels (V-A for SU(2)_L x SU(2)_R, X-T and PS-S for U(1)_A, A-X for SU(2)_CS) are plotted versus temperature for four quark masses. New ensembles at T=147 MeV and T=165 MeV are added to earlier higher-temperature data. The paper claims that SU(2)_L x SU(2)_R is restored at T_c ~ 165 MeV, that U(1)_A is effectively restored already at T ~ 165 MeV (below the 1.2-1.3 T_c threshold reported elsewhere), and that no convincing emergence of SU(2)_CS is seen in the studied temperature range.

Significance. If the central claim holds, that U(1)_A is effectively restored at or very near the chiral crossover temperature T_c ~ 165 MeV in two-flavor QCD, it would be a noteworthy result: it would place the U(1)_A restoration below the 1.2-1.3 T_c threshold reported by HotQCD and other groups, potentially affecting expectations for the order and universality of the chiral transition. The paper has clear strengths: chiral symmetry is well preserved (residual mass <0.1 MeV in the main text), several quark masses are studied, the screening-mass analysis uses a standard cosh fit, and the comparison against HotQCD data (including the explanation of the scalar-channel artifact in staggered fermions) is informative. The exploratory treatment of SU(2)_CS is also a useful addition. However, the quantitative support for the central claim is incomplete: no error bars are shown in the key figures, and the finite-volume check that the authors themselves state is needed for the two new near-T_c ensembles is not presented.

major comments (3)
  1. [Section 4 (lattice extents) and Section 5] The claim that SU(2)_L x SU(2)_R and U(1)_A are restored at T ~ 165 MeV rests on the screening-mass differences shown for the T=147 and T=165 MeV ensembles. Section 4 states that L=36 (T=147 MeV) and L=32 (T=165 MeV) "may not be enough to control the long range correlation effects" and that L=48 and L=40 lattices have been generated, but no comparison of screening masses or mass differences between the two volumes is shown anywhere in the proceedings. If the larger volumes shift Delta M_{V-A}, Delta M_{X-T} or Delta M_{PS-S} by more than the (unshown) statistical errors, the conclusion that U(1)_A is restored at or near T_c would not follow. The volume cross-check, or an explicit quantitative estimate of the finite-volume systematic, is load-bearing and should be presented before the central claim is made.
  2. [Figures 2-4 and Section 5] None of the figures showing Delta M_{V-A}, Delta M_{X-T} and Delta M_{A-X} include statistical or systematic error bars, yet the text draws quantitative conclusions from these plots, e.g. that Delta M_{V-A} "remains zero" above 165 MeV and that Delta M_{X-T} signals U(1)_A restoration at 165 MeV. Without uncertainties, a reader cannot distinguish a genuine symmetry signal from a fluctuation. At minimum, the statistical errors from the cosh fits in Eq. (7) should be displayed, and the fit ranges and chi-squared per degree of freedom should be reported for the quoted screening masses.
  3. [Section 4 and Section 5, Figure 3] The U(1)_A claim is supported by two mass differences, Delta M_{X-T} and Delta M_{PS-S}. Section 4 states that for the S channel, and "in particular the lowest two temperatures in our study", fit values are omitted because the correlator and effective mass are too unstable; these are exactly the T=147 and 165 MeV ensembles relevant to the near-T_c claim. Thus the PS-S evidence for U(1)_A restoration is absent at the decisive point, and the quantitative basis for the T~165 MeV statement reduces to one channel. In addition, the text in Section 5 says U(1)_A is restored at T~165 MeV, while the Figure 3 caption attributes the suppression to T=189 MeV; this discrepancy should be resolved explicitly.
minor comments (5)
  1. [Abstract and Section 4] The abstract states that the residual mass is "~1 MeV or less," while the introduction (Section 1) says "~0.1 MeV" and Section 4 states "<0.1 MeV"; these values should be reconciled.
  2. [Figure 1 and Section 5] The comparison with HotQCD shaded bands in Figure 1 uses N_f=2+1 data while the simulation is N_f=2; the difference in flavor content and in the light-quark mass (2.6 MeV versus the physical average) should be stated in the caption or text to avoid over-interpreting small differences.
  3. [Eq. (5)] The phrase "lowest Matsubara modes with M=+/- pi T" is confusing; the symbol M is otherwise used for the screening mass, and the text should instead write p_0 = +/- pi T or define the notation explicitly.
  4. [Table 1] There are several typographical issues: "Psuedo Scalar" should be "Pseudo Scalar," and the table layout makes the symmetry correspondences hard to read; please format the table entries cleanly.
  5. [Acknowledgments] The name of the Yukawa Institute is misspelled as "Yuakawa Institute" in the acknowledgments.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the symmetry-restoration claims are driven by newly measured screening masses, with external HotQCD benchmarks and an independent free-quark derivation for SU(2)_CS.

full rationale

The paper's central results are the screening masses extracted from spatial two-point correlators via a standard cosh fit (Eq. 7); the symmetry-restoration statements compare measured mass differences (Delta M_{V-A}, Delta M_{X-T}, Delta M_{PS-S}) to zero and to the HotQCD bands [22]. The quark masses are simulation inputs, not fitted parameters, and the conclusion that Delta M approaches zero is not imposed by the fitting procedure. The SU(2)_CS discussion is derived from the free-quark propagator in Section 3 (Eqs. 3-6), independently of the lattice data. Self-citations (e.g. [5-8,15-17,26]) appear in background statements and in comparisons with previously reported thresholds, but the load-bearing evidence for the new T approximately 165 MeV claim is the new ensemble data with Mobius domain-wall fermions, and the threshold comparison includes external references [22-25,27] as well. The finite-volume concern about L=36/32 lattices is a correctness and systematics risk rather than a circularity: the paper openly states that these volumes 'may not be enough to control the long range correlation effects' and reports that larger lattices were generated, but no circular definition or fitted-parameter-as-prediction is present. Therefore no circular step is identifiable.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No new parameters are fitted to the target conclusions; the quark masses are simulation inputs. The central claims rest on standard lattice QCD assumptions, chiefly control of finite-volume and chiral-symmetry-breaking systematics.

assumptions (5)
  • domain assumption Euclidean spatial correlators at large z are dominated by the lowest screening mass with cosh(z-L/2) form (Eq. 7).
    Standard spectral representation; needed to extract M_Gamma from fits to C(z)=A cosh(m[z-L/2]).
  • domain assumption Möbius domain-wall fermion action with small residual mass preserves SU(2)_L x SU(2)_R sufficiently for these measurements.
    The whole analysis depends on good chiral symmetry; the abstract quotes residual mass '~1 MeV or less', while Section 4 states '<0.1 MeV'.
  • domain assumption The lattice scale is set by a^{-1}=2.463 GeV at beta=4.30 from prior determinations.
    Temperature values T=1/(a N_t) rely on this scale; no error is quoted for the scale.
  • domain assumption The high-T derivation of SU(2)_CS (Eqs. 3-5) keeps only the lowest Matsubara modes and neglects gluonic interactions.
    Used to motivate the SU(2)_CS analysis; it is an approximate free-quark result, not a full interacting derivation.
  • domain assumption Finite-size effects at L=32/36 are small enough to extract reliable screening masses near T_c.
    The paper itself states these volumes 'may not be enough to control the long range correlation effects', and larger lattices are not compared.

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Cite this review

Pith. "Pith review of Study of symmetries in finite temperature $N_f=2$ QCD with M\"obius Domain Wall Fermions." pith.science (2026). https://pith.science/paper/OFFNE5C5

@misc{pith2026241206574,
  author       = {Pith},
  title        = {Pith review of: Study of symmetries in finite temperature $N_f=2$ QCD with M\"obius Domain Wall Fermions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OFFNE5C5}},
  note         = {Machine review of arXiv:2412.06574}
}
abstract

We report on the ongoing study of symmetry of $N_f=2$ QCD around the critical temperature. Our simulations of $N_f = 2$ QCD employ the M\"obius domain-wall fermion action with residual mass $\sim 1\mbox{MeV}$ or less, maintaining a good chiral symmetry. Using the screening masses from the two point spatial correlators we compare the mass difference between channels connected through various symmetry transformations. Our analysis focuses on restoration of the $SU(2)_L\times SU(2)_R$ as well as anomalously broken axial $U(1)_A$. We also present additional study of a potential $SU(2)_{CS}$ symmetry which may emerge at sufficiently high temperatures.

Figures

Figures reproduced from arXiv: 2412.06574 by the authors.

Figure 1
Figure 1. For the lightest mass quark ensembles the 𝑁𝑓 = 2 screening mass data plotted with respect to temperature, shown as points. The shaded bands are 𝑁𝑓 = 2 + 1 data from HotQCD in [22]. where Σ𝑘 =        𝛾𝑘 𝑖𝛾𝑘𝛾5 𝛾5        , which is the so-called 𝑆𝑈(2)𝐶𝑆 chiral-spin symmetry, explored in [7, 9, 12, 13, 18]. For the free quark case shown above, the chiral-spin symmetry is approximate and emerges in the high… view at source ↗
Figure 2
Figure 2. The screening mass difference plot for 𝑆𝑈(2)𝐿 × 𝑆𝑈(2)𝑅 with respect to temperature for the range of masses in this study. 5. Summary of Preliminary Results [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. The screening mass difference plot for 𝑈(1)𝐴 with respect to temperature, the axial anomaly appears to be suppressed at 𝑇 = 189 MeV. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The screening mass difference plot for 𝑆𝑈(2)𝐶𝑆 for various temperatures and masses, while it looks close to being restored at 𝑇 = 330 MeV it is ∼ 40 MeV and most likely approximate if the symmetry is emergent at this temperature. sion of this work in its later stages d…

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Symmetry of screening masses of mesons in two-flavor lattice QCD at high temperatures

    hep-lat 2025-01 accept novelty 6.0 of 10

    Using Möbius domain-wall fermions in two-flavor QCD, the authors find that vector/axial-vector and tensor/axial-tensor screening masses become degenerate above Tc, while the chiral-spin A-X splitting stays around 40 M...

  2. $U(1)_A$ Breaking in Hot QCD in the Chiral Limit

    hep-lat 2025-02 conditional novelty 5.0 of 10

    A free instanton gas random matrix model predicts a singular Dirac spectral peak and a nonvanishing pion-minus-delta susceptibility in the chiral limit of two-flavor hot QCD.

Reference graph

Works this paper leans on

31 extracted references · 7 canonical work pages · cited by 2 Pith papers

  1. [1]

    S. Aoki, H. Fukaya and Y. Taniguchi,Chiral symmetry restoration, eigenvalue density of Dirac operator and axial U(1) anomaly at finite temperature,Phys. Rev. D86(2012) 114512 [1209.2061]. 7 Study of symmetries in finite temperature𝑁𝑓 = 2 QCD with Möbius Domain Wall FermionsDavid Ward

    arXiv 2012

  2. [2]

    JLQCD:collaboration, Chiral susceptibility and axial U(1) anomaly near the (pseudo-)critical temperature, PoS LATTICE2023(2024) 184 [2401.06459]

    arXiv 2024

  3. [3]

    Cossu, S

    G. Cossu, S. Aoki, H. Fukaya, S. Hashimoto, T. Kaneko, H. Matsufuru et al.,Finite temperature study of the axial U(1) symmetry on the lattice with overlap fermion formulation, Phys. Rev. D87(2013) 114514 [1304.6145]

    arXiv 2013

  4. [4]

    Pisarski and F

    R.D. Pisarski and F. Wilczek,Remarks on the Chiral Phase Transition in Chromodynamics, Phys. Rev. D29(1984) 338

    1984

  5. [5]

    JLQCDcollaboration,AxialU(1)symmetry, topology, andDiracspectraathightemperature in𝑁𝑓 = 2 lattice QCD, PoS CD2018(2019) 085 [1908.11684]

    arXiv 2019

  6. [6]

    Symmetries of the light hadron spectrum in high temperature QCD

    C. Rohrhofer, Y. Aoki, G. Cossu, H. Fukaya, C. Gattringer, L.Y. Glozman et al.,Symmetries of the Light Hadron Spectrum in High Temperature QCD, PoS LATTICE2019(2020) 227 [1912.00678]

    work page Pith review arXiv 2020

  7. [7]

    Rohrhofer, Y

    C. Rohrhofer, Y. Aoki, G. Cossu, H. Fukaya, C. Gattringer, L.Y. Glozman et al.,Symmetries of spatial meson correlators in high temperature QCD, 1902.03191

    arXiv 1902

  8. [8]

    JLQCD collaboration,Study of the axial𝑈(1) anomaly at high temperature with lattice chiral fermions, Phys. Rev. D103 (2021) 074506 [2011.01499]

    arXiv 2021

Show all 31 references
  1. [9]

    Glozman,𝑆𝑈(2𝑁𝐹) hidden symmetry of QCD, 1511.05857

    L.Y. Glozman,𝑆𝑈(2𝑁𝐹) hidden symmetry of QCD, 1511.05857

    arXiv

  2. [10]

    L.Y.Glozman, ChiralspinsymmetryandQCDathightemperature ,Eur.Phys.J.A 54(2018) 117 [1712.05168]

    2018 arXiv

  3. [11]

    Glozman, O

    L.Y. Glozman, O. Philipsen and R.D. Pisarski,Chiral spin symmetry and the QCD phase diagram, Eur. Phys. J. A58(2022) 247 [2204.05083]

    2022 arXiv

  4. [12]

    Glozman,𝑆𝑈(2𝑁𝐹) symmetry of QCD at high temperature and its implications,Acta Phys

    L.Y. Glozman,𝑆𝑈(2𝑁𝐹) symmetry of QCD at high temperature and its implications,Acta Phys. Polon. Supp.10 (2017) 583 [1610.00275]

    2017 arXiv

  5. [13]

    Rohrhofer, Y

    C. Rohrhofer, Y. Aoki, G. Cossu, H. Fukaya, L. Glozman, S. Hashimoto et al.,Observation of approximate SU(2)𝐶𝑆 and SU(2𝑛𝑓) symmetries in high temperature lattice QCD,Nucl. Phys. A982(2019) 207

    2019

  6. [14]

    Rohrhofer,Symmetries of QCD at high temperature, Ph.D

    C. Rohrhofer,Symmetries of QCD at high temperature, Ph.D. thesis, Graz U., 2018

    2018

  7. [15]

    JLQCD collaboration,Axial U(1) symmetry and mesonic correlators at high temperature in 𝑁𝑓 = 2 lattice QCD,PoS LATTICE2019(2020) 178 [2001.07962]

    2020 arXiv

  8. [16]

    Zhang, Y

    Y. Zhang, Y. Aoki, S. Hashimoto, I. Kanamori, T. Kaneko and Y. Nakamura,Finite temperature QCD phase transition with 3 flavors of Mobius domain wall fermions,PoS LATTICE2022(2023) 197 [2212.10021]

    2023 arXiv

  9. [17]

    D. Ward, S. Aoki, Y. Aoki, H. Fukaya, S. Hashimoto, I. Kanamori et al.,Study of Chiral Symmetry and𝑈(1)𝐴 using Spatial Correlators for𝑁𝑓 = 2 + 1 QCD at finite temperature with Domain Wall Fermions,PoS LATTICE2023(2024) 182 [2401.07514]. 8 Study of symmetries in finite temperat...

    2024 arXiv

  10. [18]

    D. Bala, O. Kaczmarek, P. Lowdon, O. Philipsen and T. Ueding,Pseudo-scalar meson spectral properties in the chiral crossover region of QCD, 2310.13476

    arXiv

  11. [19]

    Chiu,Symmetries of meson correlators in high-temperature QCD with physical (u/d,s,c) domain-wall quarks, Phys

    T.-W. Chiu,Symmetries of meson correlators in high-temperature QCD with physical (u/d,s,c) domain-wall quarks, Phys. Rev. D107 (2023) 114501 [2302.06073]

    2023 arXiv

  12. [20]

    Chiu,Symmetries of spatial correlators of light and heavy mesons in high temperature lattice QCD, Phys

    T.-W. Chiu,Symmetries of spatial correlators of light and heavy mesons in high temperature lattice QCD, Phys. Rev. D110 (2024) 014502 [2404.15932]

    2024 arXiv

  13. [21]

    Dalla Brida, L

    M. Dalla Brida, L. Giusti, T. Harris, D. Laudicina and M. Pepe,Non-perturbative thermal QCD at all temperatures: the case of mesonic screening masses, JHEP 04(2022) 034 [2112.05427]

    2022 arXiv

  14. [22]

    Bazavov et al.,Meson screening masses in (2+1)-flavor QCD,Phys

    A. Bazavov et al.,Meson screening masses in (2+1)-flavor QCD,Phys. Rev. D100 (2019) 094510 [1908.09552]

    2019 arXiv

  15. [23]

    Buchoff et al.,QCD chiral transition, U(1)A symmetry and the dirac spectrum using domain wall fermions,Phys

    M.I. Buchoff et al.,QCD chiral transition, U(1)A symmetry and the dirac spectrum using domain wall fermions,Phys. Rev. D89 (2014) 054514 [1309.4149]

    2014 arXiv

  16. [24]

    Cheng et al.,Meson screening masses from lattice QCD with two light and the strange quark, Eur

    M. Cheng et al.,Meson screening masses from lattice QCD with two light and the strange quark, Eur. Phys. J. C71(2011) 1564 [1010.1216]

    2011 arXiv

  17. [25]

    HotQCD collaboration,The chiral transition and𝑈(1)𝐴 symmetry restoration from lattice QCD using Domain Wall Fermions,Phys. Rev. D86(2012) 094503 [1205.3535]

    2012 arXiv

  18. [26]

    Tomiya, G

    A. Tomiya, G. Cossu, S. Aoki, H. Fukaya, S. Hashimoto, T. Kaneko et al.,Evidence of effective axial U(1) symmetry restoration at high temperature QCD,Phys. Rev. D96(2017) 034509 [1612.01908]

    2017 arXiv

  19. [27]

    Brandt, A

    B.B. Brandt, A. Francis, H.B. Meyer, O. Philipsen, D. Robaina and H. Wittig,On the strength of the𝑈𝐴(1) anomaly at the chiral phase transition in𝑁𝑓 = 2 QCD, JHEP 12 (2016) 158 [1608.06882]

    2016 arXiv

  20. [28]

    Boyle, A

    P. Boyle, A. Yamaguchi, G. Cossu and A. Portelli,Grid: A next generation data parallel C++ QCD library, 1512.03487

    arXiv

  21. [29]

    Meyer, P

    N. Meyer, P. Georg, S. Solbrig and T. Wettig,Grid on QPACE 4,PoS LATTICE2021(2022) 068 [2112.01852]

    2022 arXiv

  22. [30]

    S. Ueda, S. Aoki, T. Aoyama, K. Kanaya, H. Matsufuru, S. Motoki et al.,Development of an object oriented lattice QCD code ’Bridge++’,J. Phys. Conf. Ser.523 (2014) 012046

    2014

  23. [31]

    Akahoshi, S

    Y. Akahoshi, S. Aoki, T. Aoyama, I. Kanamori, K. Kanaya, H. Matsufuru et al.,General purpose lattice QCD code set Bridge++ 2.0 for high performance computing, J. Phys. Conf. Ser. 2207(2022) 012053 [2111.04457]. 9

    2022 arXiv

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