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REVIEW 4 major objections 5 minor 27 references

Numerical evidence for a CP broken deconfined phase at $\theta =\pi$ in 4D SU(2) Yang-Mills theory through simulations at imaginary $\theta$

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Lattice evidence shows CP is spontaneously broken at θ=π at low temperature and restored above T_dec(π), leaving a CP-broken deconfined phase.

desk verdict The CP-restoration claim is an artifact of switching fit ansatz at T_c; the paper's technical setup is good, but the central conclusion does not follow from the data. read the letter →

arxiv 2502.09115 v2 pith:4IS7CA3I submitted 2025-02-13 hep-lat hep-phhep-th

Mitsuaki Hirasawa , Masazumi Honda , Akira Matsumoto , Jun Nishimura , Atis Yosprakob This is my paper · ORCID
classification hep-lathep-phhep-th MSC 81T1381T2581T80 PACS 11.15.Ha11.30.Er
keywords SU(2)Yang-Millstheta=piCPsymmetrybreakingimaginarythetaanalyticcontinuationdeconfinementtemperaturetopologicalchargelatticegaugetheory
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

At θ=π, four-dimensional SU(2) Yang-Mills theory has an exact CP symmetry, and the question is whether the vacuum spontaneously breaks it. This paper reports lattice evidence that it does: simulations at imaginary θ, continued analytically to real θ, give a non-vanishing topological-charge density at θ=π at low temperature, which is the CP-breaking order parameter. The CP-restoring temperature is found to be close to the deconfining temperature at θ=0, while the deconfining temperature at θ=π is clearly lower. The paper concludes that a deconfined but CP-broken phase exists, a situation allowed by the anomaly-matching argument but different from the large-N prediction.

What carries the argument

The load-bearing device is analytic continuation from imaginary θ: with θ = iθ̃ the topological term becomes a real weight $e^{{θ̃Q}}$, the topological charge density ⟨Q⟩_{iθ̃}/V is measured and fitted at each temperature to one of two holomorphic ansätze — the odd polynomial g(iθ̃)=χ0θ̃ − a3θ̃^3 + a5θ̃^5 or the $\sinh$-series h(iθ̃)=(χ0−2b2−3b3)sinhθ̃ + b2 sinh2θ̃ + b3 sinh3θ̃ — whose first coefficients are fixed by the topological susceptibility χ0 at θ=0. Continuing the chosen ansatz to θ=π gives the CP order parameter; a non-vanishing value signals spontaneous breaking. An essential auxiliary ingredient is the dynamical stout smearing of the links used in the topological-charge definition, which makes the charge take near-integer values, a prerequisite for the CP symmetry at θ=π to exist on the lattice. The deconfining temperature is obtained separately from the peak of the Polyakov-loop susceptibility, extrapolated to infinite volume and fitted as a quadratic in (θ/π)^2.

What would settle it

Repeat the measurement at T ≈ 0.9 T_c at larger imaginary θ̃ and refit with a higher-order polynomial (adding a θ̃^7 term): if the analytically continued value of i⟨Q⟩/V at θ=π changes by more than its statistical error, the truncation is not reliable. Alternatively, a direct real-θ lattice computation of the CP order parameter and the Polyakov-loop susceptibility in the window 0.79 T_c < T < 1.0 T_c — e.g., with a subvolume method — would settle whether a deconfined CP-broken phase actually exists.

Watch

Extended reading notes

Core claim

The central claim is that in 4D SU(2) Yang-Mills at θ=π the inequality T_CP > T_dec(π) holds: CP is spontaneously broken in the confined phase and remains broken across the deconfining transition, being restored only at a temperature T_CP ~ T_dec(0) > T_dec(π). The evidence is obtained by simulating at imaginary θ (where the action is real), measuring the topological-charge density and the Polyakov-loop susceptibility, and continuing the fitted holomorphic forms to real θ. A non-vanishing i⟨Q⟩/V at θ=π at low temperature, which vanishes smoothly at T between 1.0 and 1.01 T_c, is taken as the CP order parameter, while the deconfining-temperature fit T_dec(θ)/T_c = c0 − c2(θ/π)^2 with c0 = 1.0183(16), c2 = 0.225(12) gives T_dec(π) < T_c. This establishes a window of temperatures in which the plasma is deconfined yet CP-broken, consistent with the 't Hooft anomaly-matching condition and unlike the large-N limit where the two transitions coincide.

Load-bearing premise

The conclusion turns on assuming that the topological-charge density, as a function of θ, is smooth and well approximated all the way to θ=π by the particular polynomial or sinh fit chosen at each temperature, even though the fits are made only at imaginary θ and the choice of ansatz is made after inspecting the data.

Editorial extensions

If this is right

  • SU(2) Yang-Mills at θ=π would exhibit three distinct regimes: a confined CP-broken phase at low T, a deconfined CP-broken phase for T_dec(π) < T < T_CP, and a deconfined CP-restored phase above T_CP.
  • The strict inequality T_CP > T_dec(π) contrasts with the large-N result T_CP = T_dec(π), so the ratio of the two critical temperatures becomes an N-dependent quantity.
  • The deconfining line bends downward in θ, with T_dec(π)/T_c ≈ 0.79 from the quadratic fit, so the transition temperature at the CP-symmetric point is significantly suppressed.
  • The imaginary-θ method with dynamically smeared topological charge provides a practical route to θ=π observables in other gauge groups, such as SU(3), where the order parameter and the deconfining line can be mapped the same way.

Reading between the lines

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

  • If the phase exists, the 't Hooft anomaly between the Z_2 center symmetry and CP in the deconfined regime is matched by CP breaking, so the topological susceptibility should develop a discontinuity in ⟨Q⟩/V exactly at θ=π through that window.
  • The switch of ansatz between g and h at T_c suggests the truncated functional forms are not controlled near θ=π; a single Padé or higher-order fit across all temperatures, or data at larger θ̃, would test whether the inferred T_CP is an artifact of truncation.
  • One nearby extension is to scan temperatures between 0.79 T_c and 1.0 T_c to look for a possible first-order line or critical endpoint where the CP-restoration line meets the deconfinement line.
  • A similar imaginary-θ analysis in SU(3) Yang-Mills could reveal whether the strict inequality and the CP-broken deconfined window are special to N=2 or persist at N=3, where the deconfinement transition is first order.
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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

4 major / 5 minor

Summary. The paper studies 4D SU(2) Yang-Mills theory at θ=π using simulations at imaginary θ. The topological charge density ⟨Q⟩ at imaginary θ is measured on lattices with L_s=16,20,24 and fixed L_t=5, using a dynamically stout-smeared topological charge in the action. The data are fit to two holomorphic forms, g(θ̃)=χ0θ̃−a3θ̃^3+a5θ̃^5 for T≤T_c and h(θ̃)=(χ0−2b2−3b3)sinh θ̃+b2 sinh 2θ̃+b3 sinh 3θ̃ for T>T_c, and then analytically continued to real θ. The Polyakov-loop susceptibility is used to extract T_dec(θ), with the imaginary-θ results fit to T_dec(θ)/T_c = c0 − c2(θ/π)^2. The authors conclude that ⟨Q⟩ is nonzero at θ=π below T_c, vanishes near T_c, and that T_dec(π)<T_c, so that T_CP > T_dec(π) and a CP-broken deconfined phase exists for SU(2), unlike the large-N case.

Significance. If the conclusion is correct, this is an important result: it would provide numerical evidence for a CP-broken deconfined phase in 4D SU(2) Yang-Mills and for the inequality T_CP > T_dec(π) implied by 't Hooft anomaly matching, in contrast to the large-N prediction. The use of dynamical stout smearing to define a near-integer topological charge in the action is a useful methodological step, and the paper includes infinite-volume extrapolations and transparent fit forms. These strengths, however, do not by themselves establish the central claim, because the CP-restoration signal is built into the chosen high-temperature ansatz and the analytic continuation is long and unvalidated. The paper is best read as a promising preliminary study whose central quantitative claim requires substantial additional support.

major comments (4)
  1. [§4.1, Eq. (4.2)] The high-temperature fit form h(θ̃) is a finite sum of sinh(nθ̃) terms. After analytic continuation to real θ it becomes a finite sine series h(θ)=(χ0−2b2−3b3)sin θ+b2 sin 2θ+b3 sin 3θ, which vanishes identically at θ=π. Therefore the statement that CP is restored above T_c is imposed by the ansatz, not extracted from the data. The switch from g to h is made after inspecting the data, as the text states: 'we plot g(θ) for T ≤ T_c and h(θ) for T > T_c since g(π)<0 is not consistent with DIGA at higher temperature'. This post-hoc selection makes the disappearance of the gap near T_c uninformative unless the authors show that the imaginary-θ data themselves select between the two forms and that adding higher-order terms (θ^7, sinh 4θ̃, etc.) does not change the endpoint at θ=π.
  2. [§4.1, Figs. 2 and 3] The low-temperature polynomial g(θ̃)=χ0θ̃−a3θ̃^3+a5θ̃^5 is fitted to imaginary-θ data over roughly θ̃/π ≤ 0.7 and then continued to θ=π, which is a long extrapolation. The value at θ=π is therefore not a measured order parameter but the endpoint of a fitted polynomial. No stability check is reported for including a θ^7 term or for using an alternative extrapolant (e.g., a Padé or a sine series). Since the claimed CP-broken phase hinges entirely on this endpoint, the authors should demonstrate that the conclusion is stable under reasonable variations of the fit form and fit range.
  3. [§4.1 and §5] The CP-restoration temperature is reported only qualitatively: the gap 'disappears at some T within 1.0 ≲ T/T_c ≲ 1.01' and T_CP ∼ T_c, with no error bar and no interpolation. The central inequality (1.2) requires a quantitative comparison between T_CP and T_dec(π), so an estimate of T_CP with a statistical and systematic uncertainty is needed. In addition, the analysis is performed at a single lattice spacing N_t=5, so a continuum extrapolation or at least a second N_t value is required before the claim can be considered numerical evidence rather than a lattice-artifact.
  4. [§4.2, Eq. (4.4)] The deconfining temperature is obtained by fitting T_dec(θ̃)/T_c to c0 − c2(θ̃/π)^2 and extending this quadratic to θ=π. The conclusion T_dec(π)<T_c depends on this functional form; higher-order terms in (θ/π)^2 or a different extrapolant could shift T_dec(π). The agreement with Ref. [27] is encouraging, but the authors should report a stability check (e.g., adding a quartic term) and propagate the corresponding uncertainty into the relation T_CP > T_dec(π).
minor comments (5)
  1. [§4.1, Figure 2 (Right)] At T=T_c the figure shows both g(θ) and h(θ), while the text says g is used for T≤T_c and h for T>T_c. Please clarify which form is used at T=T_c and how this choice affects the apparent gap at θ=π.
  2. [§5] There is a typo: 'deconfinig' should be 'deconfining'.
  3. [§2, Eq. (2.3)] The order parameter is defined as the ε→0 limit of ⟨Q⟩/(V) at θ=π−ε, but the paper later reports values of ⟨Q⟩ at θ=π obtained by analytic continuation. Please explain the relation between the two, especially since the analytic continuation gives a single branch rather than the two degenerate CP-related branches.
  4. [§3] Simulation details needed for reproducibility and for assessing error bars are not given: number of configurations, thermalization, autocorrelation times, HMC trajectory length, and the ranges of θ̃ simulated. Please provide these.
  5. [§3, Eq. (3.5)] The topological-charge rescaling factor w is fitted by minimizing the cost function (3.5). The resulting w is used in defining Q in the action, and the systematic uncertainty in w is not propagated into the final ⟨Q⟩ values. Please state w for each ensemble and quantify the effect of its uncertainty.

Circularity Check

2 steps flagged · score 6.0 of 10

The CP-restoration temperature T_CP is set by the ansatz switch, not by data: the high-temperature fit h(θ) is a sine series that vanishes identically at θ=π.

  1. self definitional [Section 4.1, eqs. (4.1)-(4.2) and the paragraph after Fig. 3]
    "Here we plot g(θ) for T ≤ Tc and h(θ) for T > Tc since g(π) < 0 is not consistent with DIGA at higher temperature, which is presumably due to the truncation of the polynomial expansion."

    Eq. (4.2) is h(iθ̃)=(χ0−2b2−3b3)sinh θ̃+b2 sinh 2θ̃+b3 sinh 3θ̃; analytically continued to real θ this is (χ0−2b2−3b3)sin θ+b2 sin 2θ+b3 sin 3θ, and every term vanishes at θ=π. Hence ⟨Q⟩_{θ=π}=0 for all T>T_c regardless of the fitted coefficients b2,b3. The decision to use h for T>T_c—made because g(π)<0 is deemed inconsistent with DIGA—therefore installs CP restoration above T_c by construction. T_CP∼T_c is the temperature at which the ansatz is switched, not a measured vanishing of an order parameter.

  2. fitted input called prediction [Section 1 (strategy) and Section 4.1, eq. (4.1)]
    "we first calculate the expectation value of the topological charge at imaginary θ and fit the results to an appropriate holomorphic function. Then ⟨Q⟩θ at real θ is obtained through analytic continuation of the fitting function. A non-vanishing ⟨Q⟩θ at θ=π signals spontaneous breaking of CP symmetry."

    The CP-breaking signal is the value at θ=π of the fitted polynomial g(iθ̃)=χ0θ̃−a3θ̃³+a5θ̃⁵, whose coefficients are determined by imaginary-θ data at small θ̃ (Fig. 2 shows fits up to θ̃/π≈0.8). Extrapolating this quintic to θ=π is what produces the nonzero 'gap'; the alternative sine-series h would give zero there. Thus the low-temperature CP-broken phase is also an output of the chosen analytic-continuation ansatz rather than a measured order parameter, making the inferred T_CP and the inequality T_CP>T_dec(π) dependent on the ansatz choice.

full rationale

The lattice work itself is careful (stout smearing, infinite-volume extrapolation, Polyakov-loop susceptibility), and the T_dec(θ) quadratic fit and the w-rescaling are standard fittable ingredients rather than circular loads. The self-citations [6,7,25] are not load-bearing: [6] is a theoretical prediction by an overlapping author that the numerics purport to test, not to prove, and [25] is the companion paper providing technical details. However, the central CP claim is circular in the specific sense above: for T>T_c the authors choose a sine-series ansatz whose analytic continuation is identically zero at θ=π, so the 'restoration' is a property of the ansatz, not of the data; for T≤T_c the nonzero gap is the extrapolated value of a polynomial. No stability check against adding θ^7 or sinh(4θ̃) terms is reported, and the analysis is at a single lattice spacing N_t=5, so the high-temperature zero is not a falsifiable measurement. Score 6: one central 'prediction' reduces by construction, while substantial independent lattice input remains.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central conclusion rests on fitting imaginary-theta data to assumed analytic forms, on a lattice-artifact rescaling of the topological charge, and on the absence of a continuum limit. These are the quantities the reader must accept on the authors' say-so.

free parameters (4)
  • w = minimizer of F(w) = <1-cos(2*pi*w*Q[U_tilde])>
    Section 3, eq (3.5). Rescales the topological charge to make it near-integer; affects the theta term and hence the location of CP restoration.
  • a3, a5 = not listed in the proceedings
    Section 4.1, eq (4.1). Coefficients of the polynomial fit for T <= T_c; the sign of g(pi) depends on a5.
  • b2, b3 = not listed
    Section 4.1, eq (4.2). Coefficients of the sinh-series fit for T > T_c.
  • c0, c2 = c0 = 1.0183(16), c2 = 0.225(12)
    Section 4.2, eq (4.4). Parameters of the quadratic fit controlling T_dec(pi) prediction.
assumptions (5)
  • domain assumption The topological-charge density is holomorphic in theta and the truncated ansatze (4.1) or (4.2) represent the true theta-dependence up to theta = pi.
    Section 4.1: the CP-breaking order parameter at theta=pi is obtained by analytically continuing these fits. The unstated justification is that higher-order terms and non-perturbative effects are negligible.
  • domain assumption The w-rescaled topological charge is integer-valued on the lattice, so the theta-periodicity and CP symmetry at theta=pi survive lattice discretization.
    Section 3: the cost function (3.5) is minimized to choose w; the claim that peaks are 'shifted due to lattice artifacts' and that rescaling makes them 'closer to integers' is an ansatz, not a proven property.
  • domain assumption Finite lattice-spacing effects do not change the qualitative phase structure; results at N_t = 5 represent continuum physics.
    The paper fixes N_t = 5 and changes temperature via beta; no continuum extrapolation is performed, so the conclusion silently assumes negligible cutoff effects.
  • ad hoc to paper The deconfining transition temperature varies with theta according to T_dec(theta)/T_c = c0 - c2 (theta/pi)^2 over the full range to theta = pi.
    Section 4.2, eq (4.4): a quadratic ansatz is fitted to imaginary-theta data and extrapolated to theta=pi; nothing in the theory fixes this form.
  • domain assumption The Polyakov-loop susceptibility peak, after infinite-volume extrapolation, locates the deconfining transition at imaginary theta.
    Section 4.2, eq (4.3): the critical temperature is read from the Lorentzian peak; this assumes the transition is second order and the susceptibility shape is Lorentzian.

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Cite this review

Pith. "Pith review of Numerical evidence for a CP broken deconfined phase at $\theta =\pi$ in 4D SU(2) Yang-Mills theory through simulations at imaginary $\theta$." pith.science (2026). https://pith.science/paper/4IS7CA3I

@misc{pith2026250209115,
  author       = {Pith},
  title        = {Pith review of: Numerical evidence for a CP broken deconfined phase at $\theta =\pi$ in 4D SU(2) Yang-Mills theory through simulations at imaginary $\theta$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4IS7CA3I}},
  note         = {Machine review of arXiv:2502.09115}
}
abstract

We investigate the possibility of the spontaneous breaking of CP symmetry in 4D SU(2) Yang-Mills at $\theta=\pi$, which has recently attracted much attention in the context of the higher-form symmetry and the 't Hooft anomaly matching condition. Here we provide a numerical evidence that the CP symmetry is indeed spontaneously broken at low temperature and it gets restored above the deconfining temperature at $\theta=\pi$, which is consistent with the anomaly matching condition and yet differs from the situation predicted in the large-$N$ limit. We avoid the severe sign problem by performing simulations at imaginary $\theta$. We obtain the critical temperature of the CP restoration and that of deconfinement at $\theta=\pi$ by analytic continuation, which leads to the above conclusion.

Figures

Figures reproduced from arXiv: 2502.09115 by the authors.

Figure 1
Figure 1. The histogram of the topological charge after the stout smearing for 𝜌 = 0, 0.05, 0.09, and 0.15 with the fixed number of smearing steps 𝑁𝜌 = 40. The lattice volume is 𝑉L = 203 × 5, and the temperature is 𝑇 = 1.2𝑇c, where 𝑇c is the deconfining temperature at 𝜃 = 0 in the continuum limit. To study the 𝜃 dependence of the topological charge, we measure ⟨𝑄⟩𝑖 𝜃˜ at various values of 𝜃˜, and then we perform the infinite … view at source ↗
Figure 2
Figure 2. (Left) ⟨𝑄⟩ 𝜃 /𝑉s is plotted against 𝜃˜/𝜋 for 𝑉s = 163 , 203 , 243 and ∞ at 𝑇 = 𝑇c. (Right) ⟨𝑄⟩ 𝜃 /𝑉s after the infinite volume limit is fitted to the forms (4.1) and (4.2). 0.0 0.2 0.4 0.6 0.8 1.0 / 0.000 0.001 0.002 0.003 0.004 0.005 i Q /Vs T/Tc 0.90 0.96 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.10 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Our prediction for lim𝑉s→∞ 𝑖 ⟨𝑄⟩ 𝜃 /𝑉s is plotted against 𝜃/𝜋 at various temperature within 0.9 ≤ 𝑇/𝑇c ≤ 1.1. The gap at 𝜃 = 𝜋 disappears at some 𝑇 within 1.0 ≲ 𝑇/𝑇c ≲ 1.01. which is presumably due to the truncation of the polynomial expansion. Based on this analysis, we conclude that 𝑇CP ∼ 𝑇c. 4.2 𝜃 dependence of the deconfining temperature 𝑇dec (𝜃) In order to determine the critical temperature of the deconfing ph… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Susceptibility of Polyakov loop 𝜒P is plotted against 𝑇/𝑇c at 𝜃˜ = 𝜋/2 for 𝐿s = 16, 𝐿t = 5. The curve represents the fit to the Lorentz function (4.3). 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 ( / )2 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Td e c( )/ Tc [PITH_FULL_IMAGE:figur…
Figure 5
Figure 5. Figure 5: The deconfining temperature after the infinite volume extrapolation is plotted against (𝜃/𝜋) 2 . The line with error band represents the fit to the function (4.4). function 𝜒P(𝑇) = 𝐴 (𝑇 − 𝑇peak) 2 + 𝑤2 , (4.3) where 𝐴, 𝑇peak and 𝑤 are the fitting parameters. By extrapo…

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Works this paper leans on

27 extracted references · 10 canonical work pages

  1. [25]

    Hirasawa, M

    M. Hirasawa, M. Honda, A. Matsumoto, J. Nishimura and A. Yosprakob,Evidence of a CP broken deconfined phase in 4D SU(2) Yang-Mills theory at𝜃=𝜋 from imaginary𝜃 simulations, 2412.03683

    arXiv

  2. [27]

    Yamada, M

    N. Yamada, M. Yamazaki and R. Kitano,𝜃 dependence of𝑇𝑐 in SU(2) Yang-Mills theory, 2411.00375. 10

    arXiv

  3. [1]

    Gross, R.D

    D.J. Gross, R.D. Pisarski and L.G. Yaffe,QCD and Instantons at Finite Temperature, Rev. Mod. Phys.53(1981) 43. 8 Numerical evidence for a CP broken deconfined phase at𝜃=𝜋 in 4D SU(2) Mitsuaki Hirasawa

    work page 1981

  4. [2]

    Weiss,The Effective Potential for the Order Parameter of Gauge Theories at Finite Temperature,Phys

    N. Weiss,The Effective Potential for the Order Parameter of Gauge Theories at Finite Temperature,Phys. Rev. D24(1981) 475

    work page 1981

  5. [3]

    Witten,Large N Chiral Dynamics,Annals Phys.128 (1980) 363

    E. Witten,Large N Chiral Dynamics,Annals Phys.128 (1980) 363

    work page 1980

  6. [4]

    Witten,Theta dependence in the large N limit of four-dimensional gauge theories, Phys

    E. Witten,Theta dependence in the large N limit of four-dimensional gauge theories, Phys. Rev. Lett.81(1998) 2862 [hep-th/9807109]

    arXiv 1998

  7. [5]

    Bigazzi, A.L

    F. Bigazzi, A.L. Cotrone and R. Sisca,Notes on theta dependence in holographic yang-mills, JHEP08 (2015) 090 [1506.03826]

    arXiv 2015

  8. [6]

    S. Chen, K. Fukushima, H. Nishimura and Y. Tanizaki,Deconfinement andCP breaking at 𝜃=𝜋 in Yang-Mills theories and a novel phase for SU(2), Phys. Rev. D102 (2020) 034020 [2006.01487]

    arXiv 2020

Show all 27 references
  1. [7]

    Matsumoto, K

    A. Matsumoto, K. Hatakeyama, M. Hirasawa, M. Honda, Y. Ito, J. Nishimura et al.,A new technique for solving the freezing problem in the complex Langevin simulation of 4D SU(2) gauge theory with a theta term, PoSLATTICE2021(2022) 087 [2112.01805]

    2022 arXiv

  2. [8]

    F.K. Guo, R. Horsley, U.G. Meissner, Y. Nakamura, H. Perlt, P.E.L. Rakow et al.,The electric dipole moment of the neutron from 2+1 flavor lattice QCD, Phys. Rev. Lett.115 (2015) 062001 [1502.02295]

    2015 arXiv

  3. [9]

    S. Aoki, R. Horsley, T. Izubuchi, Y. Nakamura, D. Pleiter, P.E.L. Rakow et al.,The Electric dipole moment of the nucleon from simulations at imaginary vacuum angle theta, 0808.1428

    arXiv

  4. [10]

    Panagopoulos and E

    H. Panagopoulos and E. Vicari,The 4D SU(3) gauge theory with an imaginary𝜃 term,JHEP 11 (2011) 119 [1109.6815]

    2011 arXiv

  5. [11]

    D’Elia and F

    M. D’Elia and F. Negro,𝜃 dependence of the deconfinement temperature in Yang-Mills theories, Phys. Rev. Lett.109 (2012) 072001 [1205.0538]

    2012 arXiv

  6. [12]

    D’Elia and F

    M. D’Elia and F. Negro,Phase diagram of Yang-Mills theories in the presence of a𝜃 term, Phys. Rev. D88(2013) 034503 [1306.2919]

    2013 arXiv

  7. [13]

    Bonati, M

    C. Bonati, M. D’Elia and A. Scapellato,𝜃 dependence in𝑆𝑈(3) Yang-Mills theory from analytic continuation,Phys. Rev. D93(2016) 025028 [1512.01544]

    2016 arXiv

  8. [14]

    Bonati, M

    C. Bonati, M. D’Elia, P. Rossi and E. Vicari,𝜃 dependence of 4D𝑆𝑈(𝑁) gauge theories in the large-𝑁 limit,Phys. Rev. D94(2016) 085017 [1607.06360]

    2016 arXiv

  9. [15]

    Bonanno, M

    C. Bonanno, M. D’Elia and L. Verzichelli,The𝜃-dependence of the SU(N) critical temperature at large N, JHEP02(2024) 156 [2312.12202]

    2024 arXiv

  10. [16]

    Bonanno, C

    C. Bonanno, C. Bonati, M. Papace and D. Vadacchino,The𝜃-dependence of the Yang-Mills spectrum from analytic continuation, JHEP05(2024) 163 [2402.03096]. 9 Numerical evidence for a CP broken deconfined phase at𝜃=𝜋 in 4D SU(2) Mitsuaki Hirasawa

    2024 arXiv

  11. [17]

    Morningstar and M.J

    C. Morningstar and M.J. Peardon,Analytic smearing of SU(3) link variables in lattice QCD, Phys. Rev. D69(2004) 054501 [hep-lat/0311018]

    2004 arXiv

  12. [18]

    Unsal,Theta dependence, sign problems and topological interference,Phys

    M. Unsal,Theta dependence, sign problems and topological interference,Phys. Rev. D86 (2012) 105012 [1201.6426]

    2012 arXiv

  13. [19]

    Poppitz, T

    E. Poppitz, T. Schäfer and M. Ünsal,Universal mechanism of (semi-classical) deconfinement and theta-dependence for all simple groups,JHEP03 (2013) 087 [1212.1238]

    2013 arXiv

  14. [20]

    Anber,Θ dependence of the deconfining phase transition in pure𝑆𝑈(𝑁𝑐) Yang-Mills theories, Phys

    M.M. Anber,Θ dependence of the deconfining phase transition in pure𝑆𝑈(𝑁𝑐) Yang-Mills theories, Phys. Rev. D88(2013) 085003 [1302.2641]

    2013 arXiv

  15. [21]

    Otake and N

    N. Otake and N. Yamada,𝜃 dependence of T𝑐 in 4d SU(3) Yang-Mills theory with histogram method and the Lee-Yang zeros in the large N limit, JHEP06(2022) 044 [2202.05605]

    2022 arXiv

  16. [22]

    Borsanyi, Z

    S. Borsanyi, Z. Fodor, D.A. Godzieba, R. Kara, P. Parotto, D. Sexty et al.,Topological features of the deconfinement transition, Phys. Rev. D107 (2023) 054514 [2212.08684]

    2023 arXiv

  17. [23]

    Di Vecchia, K

    P. Di Vecchia, K. Fabricius, G.C. Rossi and G. Veneziano,Preliminary Evidence for U(1)-A Breaking in QCD from Lattice Calculations, Nucl. Phys. B192 (1981) 392

    1981

  18. [24]

    Del Debbio, H

    L. Del Debbio, H. Panagopoulos and E. Vicari,Theta dependence of SU(N) gauge theories, JHEP08 (2002) 044 [hep-th/0204125]

    2002 arXiv

  19. [26]

    Engels, F

    J. Engels, F. Karsch and K. Redlich,Scaling properties of the energy density in SU(2) lattice gauge theory,Nucl. Phys. B435(1995) 295 [hep-lat/9408009]

    1995 arXiv

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