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The imaginary-$\theta$ dependence of the SU($N$) spectrum

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Pith's one-line read The O($\theta^2$) coefficients of the SU(3) glueball mass and string tension are negative, and the data at N=3 and N=6 follow 1/N$^2$ scaling.

desk verdict First continuum-limit numbers for theta^2 dependence of the SU(3) glueball mass and string tension, with a plausible but not airtight large-N check. read the letter →

arxiv 2411.14022 v2 pith:XMH55LRX submitted 2024-11-21 hep-lat hep-th

Claudio Bonanno , Claudio Bonati , Mario Papace , Davide Vadacchino This is my paper · ORCID
classification hep-lathep-th
keywords thetadependenceYang-Millstheoryglueballmassstringtensionimaginaryanalyticcontinuationlarge-Nlimitlatticegauge
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 reports the first continuum-limit determination of the leading $\theta$-dependence of the lightest glueball mass and the string tension in SU(3) Yang–Mills theory: both decrease as $\theta$ is turned on, with coefficients $m_2 = -0.0083(23)$ and $s_2 = -0.0258(14)$. It also gives the first quantitative large-$N$ evidence, combining N=3 with N=6 data, that these coefficients scale as $1/N^2$, with combined estimates $\bar{m}_2 \simeq -0.075(20)$ and $\bar{s}_2 \simeq -0.23(1)$. The method is imaginary $\theta$: simulating at imaginary values of the topological-angle parameter keeps the lattice action real, and analytic continuation back to real $\theta$ yields the Taylor coefficients. The central reason to care is that these quantities enter the Witten–Veneziano mechanism, axion phenomenology, and any lattice calculation performed at fixed topological charge, where the mass shift is proportional to $m_2$.

What carries the argument

The carrying device is the imaginary-$\theta$ method combined with a Taylor expansion around $\theta=0$. At imaginary $\theta$ the action is real, so standard Monte Carlo applies; the lattice parameter $\theta_L$ is related to the physical one by $\theta = i Z_Q \theta_L$, with $Z_Q$ the renormalization constant obtained by cooling. Assuming analyticity around $\theta=0$, the measured dependence on $\theta_L$ is fit to a quadratic, giving $Z_Q^2 m_2$ and $Z_Q^2 s_2$. Parallel Tempering on Boundary Conditions (PTBC) swaps replicas with differing boundary conditions to prevent topological freezing and keep topology ergodic. The spectrum itself comes from a variational basis of blocked and smeared operators, with masses extracted from GEVP-correlation-function plateaus, and the string tension from the torelon mass via the Lüscher correction term.

What would settle it

Compute $m_2$ and $s_2$ directly at real $\theta$ using an independent method that does not rely on analytic continuation (for example, reweighting at small real $\theta$ or the fixed-sector formula extrapolated to $\theta=0$); if the result differed from these coefficients by more than the quoted errors, the analyticity assumption would be falsified. Alternatively, measuring the $\theta^4$ coefficient and finding it comparable to the $\theta^2$ term at the simulated imaginary values would indicate that the quadratic truncation is contaminated.

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Extended reading notes

Core claim

For pure SU(N) Yang–Mills in four dimensions, the paper establishes that the $\theta$-dependence of the spectrum is, at $\mathcal{O}(\theta^2)$, non-trivial and negative for both the mass gap and the string tension. In the continuum limit for N=3 it obtains $m_G/\sqrt{\sigma}$ at $\theta=0$ equal to $3.398(25)$, and $m_2 = -0.0083(23)$, $s_2 = -0.0258(14)$, using the parametrization $m(\theta)=m_0[1+m_2 \theta^2+\ldots]$ and $\sigma(\theta)=\sigma[1+s_2 \theta^2+\ldots]$. For N=6 the data, on two fine lattice spacings, are compatible with the large-N expectation $m_2 = \bar{m}_2/N^2 + \mathcal{O}(1/N^4)$, $s_2 = \bar{s}_2/N^2 + \mathcal{O}(1/N^4)$, yielding the large-N estimates $\bar{m}_2 \simeq -0.075(20)$ and $\bar{s}_2 \simeq -0.23(1)$. The ratio $s_2/m_2 \simeq 3.07(82)$ agrees with a holographic prediction of 4, while the ratio $T_c/m_G$ is found to be $\theta$-dependent already at leading order, in contrast with that same holographic model.

Load-bearing premise

The whole analytic-continuation program assumes the spectrum is analytic in $\theta$ around zero, so the curvature measured at imaginary $\theta$ equals the real-$\theta$ Taylor coefficient; if non-analytic behavior or higher-order terms intervene, the quoted numbers would not be the real-$\theta$ coefficients.

Editorial extensions

If this is right

  • If confirmed, the negative sign and magnitude of $m_2$ fix the leading $\theta$-dependence of the glueball mass and string tension, with direct consequences for axion cosmology and for any observable that probes the vacuum angle.
  • The $1/N^2$ scaling means the $\theta$-dependence of the spectrum vanishes in the planar limit at fixed $\theta$, consistent with confinement in large-N gauge theories, and it sharpens predictions for N=2 and N=4.
  • The combination $t_2 = R + s_2/2$ gives a non-zero $\theta$-dependence of $T_c/\sqrt{\sigma}$, which can be compared with holographic and other model predictions for the deconfinement transition.
  • The value of $m_2$ controls the systematic error of lattice spectra computed at fixed topological charge; the paper estimates this error is below 0.1% for N=3 at typical volumes and shrinks with N.
  • The ratio $s_2/m_2 \simeq 3.07(82)$ is consistent with the holographic prediction of 4, but the ratio $R/m_2$ disagrees, so the full pattern of $\theta$-dependence in the spectrum discriminates among holographic models.

Reading between the lines

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

  • The near-equality $m_2 \approx s_2/2$ at N=3 means the ratio $m_G/\sqrt{\sigma}$ is almost $\theta$-independent at leading order; a natural extension is to test whether this approximate cancellation persists at N=6 and higher N, which would suggest an underlying non-renormalization in this ratio.
  • The large-N scaling seen here could be combined with the known $1/N^2$ behavior of the deconfinement temperature to build a unified large-N description, where all dimensionless ratios in the confining sector have fixed leading $\theta$-dependence.
  • Because the N=6 data are limited to two fine lattices, running at an additional lattice spacing or at N=4,5 would sharpen the $1/N^2$ fit and test the subleading $\mathcal{O}(1/N^4)$ corrections.
  • The same imaginary-$\theta$ plus PTBC framework could be extended to the excited glueball spectrum and to the tensions of $k$-strings, not just the fundamental one, to see whether the negative $\theta^2$ coefficient is universal across the spectrum.
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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

1 major / 5 minor

Summary. This proceedings contribution reports a lattice computation of the O(theta^2) coefficients of the mass gap and string tension of SU(N) Yang-Mills for N=3 and N=6, using the imaginary-theta method with Parallel Tempering on Boundary Conditions. For N=3 the authors obtain continuum-extrapolated values m2 = -0.0083(23) and s2 = -0.0258(14), together with m0++/sqrt(sigma) = 3.398(25), in agreement with previous determinations. For N=6, data at two lattice spacings are used to test the expected large-N scaling of N^2 m2 and N^2 s2, yielding the estimates m2bar = -0.075(20) and s2bar = -0.23(1). The paper also compares the ratio s2/m2 with a holographic prediction and discusses implications for the theta-dependence of Tc/sqrt(sigma) and for fixed-topology systematic errors.

Significance. If correct, these are the first continuum-extrapolated determinations of the theta-dependence of the glueball mass and string tension in SU(3) Yang-Mills, and a first quantitative large-N check. The use of PTBC to mitigate topological freezing at fine lattice spacings is a useful methodological step, and the cross-check against the known m0++/sqrt(sigma) value is a genuine strength. The paper is honest about the absence of an N=6 continuum limit, but the large-N estimates that follow from the two N=6 points need stronger qualification, as discussed in the major comment.

major comments (1)
  1. [Section 3, Eqs. (18)-(19), Fig. 3] The quoted large-N estimates m2bar = -0.075(20) and s2bar = -0.23(1) rest on N=6 data at only two finite lattice spacings with no continuum extrapolation. The manuscript itself states that 'we cannot perform a continuum limit of these data alone', yet the errors quoted in Eqs. (19) do not include any contribution from the unknown O(a^2) discretization effects. The statement that the data are 'perfectly compatible' with N^2 scaling does not by itself determine the continuum large-N coefficients, and the numerical estimates in (19) are not fully supported. Please either estimate the discretization systematic (for example, from the spread between the two N=6 points or from a combined N=3/N=6 fit with a shared O(a^2) term) or explicitly present (19) as a preliminary estimate with statistical errors only.
minor comments (5)
  1. [Eqs. (11)-(14)] The notation O(theta^2) after the explicit 1 + m2 theta^2 term should be O(theta^4) (and analogously O(theta_L^4) in Eqs. (13)-(14)); as written, the error term is indistinguishable from the kept term.
  2. [Eq. (21)] The derivative in the definitions of m2 and s2 should be the second derivative with respect to theta, d^2/dtheta^2; the current notation d/dtheta^2 is ambiguous and, read literally, introduces a factor of two relative to the parameterization in Eqs. (11)-(12).
  3. [Section 3, Fig. 2] Please provide or cite the specific ensemble table, Z_Q values, and fit ansatz used in the continuum extrapolations; without these, the quoted continuum values in Eqs. (16)-(17) cannot be verified from this proceedings alone.
  4. [Eq. (29)] The ratio s2/m2 is written twice, 's2/m2 = s2/m2 = 4'; the first equality is tautological and should be removed or corrected.
  5. [References, [58]] The companion paper [58] is cited only in the introduction; it would be helpful to cite it again where the numerical results are presented, for example after Eq. (17) and in the caption of Fig. 3, so that readers know where to find the full error budget and numerical tables.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the coefficients m2 and s2 are measured quantities from lattice fits, with external benchmarks and honest statements of assumptions.

full rationale

The central results m2 and s2 are not assumed inputs but are extracted by fitting the measured lattice dependence of am_G and a^2 sigma on the imaginary theta_L to the Taylor forms in Eqs. (13)-(14), after dividing by the independently determined renormalization factor Z_Q obtained from cooling. The continuum limit for N=3 is a standard extrapolation of those extracted coefficients, and the check of m0++/sqrt(sigma) against Refs. [103] is an external benchmark. The holographic comparison with Ref. [108] is also an external prediction, not an input. The large-N estimates in Eq. (19) are not derived from a fit to a claimed scaling law; the paper explicitly states that the N=6 data alone cannot be continuum-extrapolated and that Eq. (18) is an expected scaling used to check compatibility. This is a stated modeling limitation rather than a circular step. Self-citations, including the companion paper [58], serve as provenance or methodological references (PTBC, imaginary-theta method) and do not carry the derivation by themselves. The analyticity assumption around theta=0 is explicitly stated as an assumption, not disguised as a derived result. No equation reduces by construction to an input, and no fitted parameter is renamed as a prediction.

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

The paper introduces no ad hoc model constants; the numbers it reports are the measured coefficients themselves. The load-bearing assumptions are analyticity in theta, truncation of the Taylor expansion, and the large-N 1/N^2 scaling used to quote the N=6 estimates. No invented entities are postulated.

free parameters (4)
  • m2 for SU(3) = -0.0083(23)
    Fitted theta_L dependence of the glueball mass plus continuum extrapolation; this is the central output.
  • s2 for SU(3) = -0.0258(14)
    Fitted theta_L dependence of the string tension plus continuum extrapolation; this is the central output.
  • large-N coefficient N^2 m2 = -0.075(20)
    Quoted using the expected 1/N^2 scaling in Eq. (18) from N=3 and N=6 data.
  • large-N coefficient N^2 s2 = -0.23(1)
    Quoted using the same expected 1/N^2 scaling from N=3 and N=6 data.
assumptions (4)
  • domain assumption The mass gap and string tension are analytic in theta around theta=0, allowing analytic continuation from imaginary theta_L.
    Section 2 states 'Assuming analyticity around theta=0 it is possible to use analytic continuation'; this is the load-bearing premise for the whole method.
  • domain assumption The Taylor expansion truncated at O(theta^2) is valid over the fitted theta_L range.
    Eqs. (11)-(14) expand to order theta^2; the paper cites fit-range stability as evidence but does not quantify O(theta^4) contamination separately.
  • domain assumption The large-N scaling m2 = m2bar/N^2 + O(1/N^4) and s2 = s2bar/N^2 + O(1/N^4) holds for N=3 and N=6.
    Eq. (18) states the expected scaling; the large-N estimates in Eqs. (19) and (22) use it as an input.
  • domain assumption The clover topological charge is multiplicatively renormalized to the integer charge by Z_Q determined via cooling.
    Section 2, Eqs. (4)-(6); the theta_L to theta mapping depends on this renormalization.

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Cite this review

Pith. "Pith review of The imaginary-$\theta$ dependence of the SU($N$) spectrum." pith.science (2026). https://pith.science/paper/XMH55LRX

@misc{pith2026241114022,
  author       = {Pith},
  title        = {Pith review of: The imaginary-$\theta$ dependence of the SU($N$) spectrum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XMH55LRX}},
  note         = {Machine review of arXiv:2411.14022}
}
abstract

In this talk we will report on a study of the $\theta$-dependence of the string tension and of the mass gap of four-dimensional SU($N$) Yang--Mills theories. The spectrum at $N=3$ and $N=6$ was obtained on the lattice at various imaginary values of the $\theta$-parameter, using Parallel Tempering on Boundary Conditions to avoid topological freezing at fine lattice spacings. The coefficient of the $\mathcal{O}(\theta^2)$ term in the Taylor expansion of the spectrum around $\theta=0$ could be obtained in the continuum limit for $N=3$, and on two fairly fine lattices for $N=6$.

Figures

Figures reproduced from arXiv: 2411.14022 by the authors.

Figure 1
Figure 1. Results for 𝑁 = 3 with 𝛽 = 6.40 (finest lattice spacing explored). 3. Results We determined the mass gap of the theory (i.e., the mass of the lightest glueball state) and the string tension in lattice units for several values of 𝛽 and 𝜃L. The 𝜃 dependence was parameterized by Taylor expanding in up to the next-to-leading order, around 𝜃 = 0, 𝑎𝑚G (𝜃) = 𝑎𝑚0 ++ 1 + 𝑚2𝜃 2 + O (𝜃 2 ) [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 2
Figure 2. Continuum limit of our 𝑁 = 3 results for 𝑚2 and 𝑠2. We also computed the continuum limit of 𝑚0 ++ / √ 𝜎, which is in perfect agreement with the previous result of [103]. 0 2 4 6 a 2σ ×10−2 −6 −5 −4 −3 −2 −1 0 1 2 N 2 m 2 ×10−1 N = 3 N = 6 0 2 4 6 a 2σ ×10−2 −5 −4 −3 −2 −1 N 2 s 2 ×10−1 N = 3 N = 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Continuum scaling of 𝑁 2𝑚2 and 𝑁 2 𝑠2 for 𝑁 = 3 and 𝑁 = 6. Concerning 𝑁 = 6, we obtained results only for two fairly fine lattice spacings, thus we cannot perform a continuum limit of these data alone. However, our data allowed for a first quantitative check of the following expected large-𝑁 scaling: 𝑚2 = 𝑚2 𝑁2 + O  1 𝑁4  , 𝑠2 = 𝑠2 𝑁2 + O  1 𝑁4  . (18) Our 𝑁 = 3 and 6 results are perfectly compatible with this e… view at source ↗

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

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Reference graph

Works this paper leans on

110 extracted references · 8 canonical work pages · cited by 1 Pith paper

  1. [58]

    Bonanno, C

    C. Bonanno, C. Bonati, M. Papace and D. Vadacchino,The𝜃-dependence of the Yang-Mills spectrum from analytic continuation, JHEP 05(2024) 163 [2402.03096]

    arXiv 2024

  2. [1]

    Witten,Current Algebra Theorems for the𝑈(1) Goldstone Boson,Nucl

    E. Witten,Current Algebra Theorems for the𝑈(1) Goldstone Boson,Nucl. Phys. B156 (1979) 269

    1979

  3. [2]

    Veneziano,𝑈(1) Without Instantons,Nucl

    G. Veneziano,𝑈(1) Without Instantons,Nucl. Phys. B159 (1979) 213

    1979

  4. [3]

    Kawarabayashi and N

    K. Kawarabayashi and N. Ohta,The Problem of𝜂 in the Large𝑁 Limit: Effective Lagrangian Approach, Nucl. Phys. B175 (1980) 477

    1980

  5. [4]

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

    E. Witten,Large𝑁 Chiral Dynamics, Annals Phys.128 (1980) 363. 8 The imaginary-𝜃 dependence of the SU(𝑁) spectrum Davide Vadacchino

    1980

  6. [5]

    Coleman,Aspects of Symmetry: Selected Erice Lectures

    S. Coleman,Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, Cambridge, U.K., 1985

    1985

  7. [6]

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

    1981

  8. [7]

    Schäfer and E

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

    arXiv 1998

Show all 110 references
  1. [8]

    D’Adda, M

    A. D’Adda, M. Lüscher and P. Di Vecchia,A 1/𝑁 Expandable Series of Nonlinear Sigma Models with Instantons,Nucl. Phys. B146 (1978) 63

    1978

  2. [9]

    Witten,Instantons, the Quark Model, and the1/𝑁 Expansion, Nucl

    E. Witten,Instantons, the Quark Model, and the1/𝑁 Expansion, Nucl. Phys. B149 (1979) 285

    1979

  3. [10]

    T. G. Kovacs, E. T. Tomboulis and Z. Schram,Topology on the lattice: 2-d Yang-Mills theories with a theta term, Nucl. Phys. B454 (1995) 45 [hep-th/9505005]

    1995 arXiv

  4. [11]

    Bonati and P

    C. Bonati and P. Rossi,Topological susceptibility of two-dimensional𝑈(𝑁) gauge theories, Phys. Rev. D99(2019) 054503 [1901.09830]

    2019 arXiv

  5. [12]

    Bonati and P

    C. Bonati and P. Rossi,Topological effects in continuum two-dimensional𝑈(𝑁) gauge theories,Phys. Rev. D100(2019) 054502 [1908.07476]

    2019 arXiv

  6. [13]

    Jackiw,Introduction to the Yang-Mills Quantum Theory,Rev

    R. Jackiw,Introduction to the Yang-Mills Quantum Theory,Rev. Mod. Phys.52(1980) 661

    1980

  7. [14]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg,Theta, Time Reversal, and Temperature, JHEP 05(2017) 091 [1703.00501]

    2017 arXiv

  8. [15]

    Bonati and M

    C. Bonati and M. D’Elia,Topological critical slowing down: variations on a toy model, Phys. Rev. E98(2018) 013308 [1709.10034]

    2018 arXiv

  9. [16]

    Di Vecchia and G

    P. Di Vecchia and G. Veneziano,Chiral Dynamics in the Large𝑁 Limit,Nucl. Phys. B171 (1980) 253

    1980

  10. [17]

    Grilli di Cortona, E

    G. Grilli di Cortona, E. Hardy, J. Pardo Vega and G. Villadoro,The QCD axion, precisely, JHEP 01(2016) 034 [1511.02867]

    2016 arXiv

  11. [18]

    F.-K.GuoandU.-G.Meißner, CumulantsoftheQCDtopologicalchargedistribution ,Phys. Lett. B749 (2015) 278 [1506.05487]

    2015 arXiv

  12. [19]

    Lu, M.-L

    Z.-Y. Lu, M.-L. Du, F.-K. Guo, U.-G. Meißner and T. Vonk,QCD𝜃-vacuum energy and axion properties,JHEP 05 (2020) 001 [2003.01625]

    2020 arXiv

  13. [20]

    N. R. Acharya, F.-K. Guo, M. Mai and U.-G. Meißner,𝜃-dependence of the lightest meson resonances in QCD, Phys. Rev. D92 (2015) 054023 [1507.08570]

    2015 arXiv

  14. [21]

    Boccaletti and D

    A. Boccaletti and D. Nogradi,The semi-classical approximation at high temperature revisited,JHEP 03(2020) 045 [2001.03383]. 9 The imaginary-𝜃 dependence of the SU(𝑁) spectrum Davide Vadacchino

    2020 arXiv

  15. [22]

    Bonati, M

    C. Bonati, M. D’Elia, M. Mariti, G. Martinelli, M. Mesiti, F. Negro et al.,Axion phenomenology and𝜃-dependence from𝑁 𝑓 = 2+ 1 lattice QCD,JHEP 03 (2016) 155 [1512.06746]

    2016 arXiv

  16. [23]

    Petreczky, H.-P

    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]

    2016 arXiv

  17. [24]

    Frison, R

    J. Frison, R. Kitano, H. Matsufuru, S. Mori and N. Yamada,Topological susceptibility at high temperature on the lattice,JHEP 09 (2016) 021 [1606.07175]

    2016 arXiv

  18. [25]

    Borsanyi et al.,Calculation of the axion mass based on high-temperature lattice quantum chromodynamics,Nature 539 (2016) 69 [1606.07494]

    S. Borsanyi et al.,Calculation of the axion mass based on high-temperature lattice quantum chromodynamics,Nature 539 (2016) 69 [1606.07494]

    2016 arXiv

  19. [26]

    Bonati, M

    C. Bonati, M. D’Elia, G. Martinelli, F. Negro, F. Sanfilippo and A. Todaro,Topology in full QCDathightemperature: amulticanonicalapproach ,JHEP 11(2018)170[ 1807.07954]

    2018 arXiv

  20. [27]

    F.Burger,E.-M.Ilgenfritz,M.P.LombardoandA.Trunin, Chiralobservablesandtopology in hot QCD with two families of quarks, Phys. Rev. D98(2018) 094501 [1805.06001]

    2018 arXiv

  21. [28]

    Chen, T.-W

    TWQCD collaboration, Y.-C. Chen, T.-W. Chiu and T.-H. Hsieh,Topological susceptibility in finite temperature QCD with physical (u/d,s,c) domain-wall quarks, Phys. Rev. D106 (2022) 074501 [2204.01556]

    2022 arXiv

  22. [29]

    Athenodorou, C

    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,JHEP 10 (2022) 197 [2208.08921]

    2022 arXiv

  23. [30]

    Alles, M

    B. Alles, M. D’Elia and A. Di Giacomo,Topological susceptibility at zero and finite T in SU(3) Yang-Mills theory, Nucl. Phys. B494 (1997) 281 [hep-lat/9605013]

    1997 arXiv

  24. [31]

    Alles, M

    B. Alles, M. D’Elia and A. Di Giacomo,Topology at zero and finite𝑇 in𝑆𝑈(2) Yang–Mills theory,Phys. Lett. B412 (1997) 119 [hep-lat/9706016]

    1997 arXiv

  25. [32]

    Del Debbio, L

    L. Del Debbio, L. Giusti and C. Pica,Topological susceptibility in the𝑆𝑈(3) gauge theory, Phys. Rev. Lett.94(2005) 032003 [hep-th/0407052]

    2005 arXiv

  26. [33]

    Del Debbio, H

    L. Del Debbio, H. Panagopoulos and E. Vicari,𝜃 dependence of𝑆𝑈(𝑁) gauge theories, JHEP 08 (2002) 044 [hep-th/0204125]

    2002 arXiv

  27. [34]

    D’Elia,Field theoretical approach to the study of theta dependence in Yang-Mills theories on the lattice, Nucl

    M. D’Elia,Field theoretical approach to the study of theta dependence in Yang-Mills theories on the lattice, Nucl. Phys. B661 (2003) 139 [hep-lat/0302007]

    2003 arXiv

  28. [35]

    Lucini, M

    B. Lucini, M. Teper and U. Wenger,Topology of𝑆𝑈(𝑁) gauge theories at𝑇≃ 0 and 𝑇≃𝑇𝑐, Nucl. Phys. B715 (2005) 461 [hep-lat/0401028]

    2005 arXiv

  29. [36]

    Giusti, S

    L. Giusti, S. Petrarca and B. Taglienti,𝜃 dependence of the vacuum energy in the𝑆𝑈(3) gauge theory from the lattice, Phys. Rev. D76 (2007) 094510 [0705.2352]. 10 The imaginary-𝜃 dependence of the SU(𝑁) spectrum Davide Vadacchino

    2007 arXiv

  30. [37]

    Vicari and H

    E. Vicari and H. Panagopoulos,𝜃 dependence of𝑆𝑈(𝑁) gauge theories in the presence of a topological term, Phys. Rept.470 (2009) 93 [0803.1593]

    2009 arXiv

  31. [38]

    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

  32. [39]

    Bonati, M

    C. Bonati, M. D’Elia, H. Panagopoulos and E. Vicari,Change of𝜃 Dependence in4𝐷 𝑆𝑈(𝑁) Gauge Theories Across the Deconfinement Transition,Phys. Rev. Lett.110 (2013) 252003 [1301.7640]

    2013 arXiv

  33. [40]

    M. Cè, C. Consonni, G. P. Engel and L. Giusti,Non-Gaussianities in the topological charge distributionofthe 𝑆𝑈(3) Yang–Millstheory,Phys.Rev.D 92(2015)074502[ 1506.06052]

    2015 arXiv

  34. [41]

    M. Cè, M. Garcia Vera, L. Giusti and S. Schaefer,The topological susceptibility in the large-𝑁 limit of SU(𝑁) Yang-Mills theory,Phys. Lett. B762 (2016) 232 [1607.05939]

    2016 arXiv

  35. [42]

    Berkowitz, M

    E. Berkowitz, M. I. Buchoff and E. Rinaldi,Lattice QCD input for axion cosmology, Phys. Rev. D92(2015) 034507 [1505.07455]

    2015 arXiv

  36. [43]

    Borsanyi, M

    S. Borsanyi, M. Dierigl, Z. Fodor, S. Katz, S. Mages, D. Nogradi et al.,Axion cosmology, lattice QCD and the dilute instanton gas, Phys. Lett. B752 (2016) 175 [1508.06917]

    2016 arXiv

  37. [44]

    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

  38. [45]

    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

  39. [46]

    Bonati, M

    C. Bonati, M. Cardinali and M. D’Elia,𝜃 dependence in trace deformed𝑆𝑈(3) Yang-Mills theory: a lattice study,Phys. Rev. D98(2018) 054508 [1807.06558]

    2018 arXiv

  40. [47]

    Bonati, M

    C. Bonati, M. Cardinali, M. D’Elia and F. Mazziotti,𝜃-dependence and center symmetry in Yang-Mills theories, Phys. Rev. D101 (2020) 034508 [1912.02662]

    2020 arXiv

  41. [48]

    Campostrini and P

    M. Campostrini and P. Rossi,1/𝑁 expansion of the topological susceptibility in the𝐶𝑃 𝑁−1 models, Phys. Lett. B272(1991) 305

    1991

  42. [49]

    Del Debbio, G

    L. Del Debbio, G. M. Manca, H. Panagopoulos, A. Skouroupathis and E. Vicari, 𝜃-dependence of the spectrum of𝑆𝑈(𝑁) gauge theories,JHEP 06 (2006) 005 [hep-th/0603041]

    2006 arXiv

  43. [50]

    Rossi,Effective Lagrangian of𝐶𝑃 𝑁−1 models in the large𝑁 limit,Phys

    P. Rossi,Effective Lagrangian of𝐶𝑃 𝑁−1 models in the large𝑁 limit,Phys. Rev. DD94 (2016) 045013 [1606.07252]

    2016 arXiv

  44. [51]

    Vicari,Monte Carlo simulation of lattice𝐶𝑃 𝑁−1 models at large𝑁, Phys

    E. Vicari,Monte Carlo simulation of lattice𝐶𝑃 𝑁−1 models at large𝑁, Phys. Lett. B309 (1993) 139 [hep-lat/9209025]

    1993 arXiv

  45. [52]

    Bonanno, C

    C. Bonanno, C. Bonati and M. D’Elia,Topological properties of𝐶𝑃 𝑁−1 models in the large-𝑁 limit,JHEP 01 (2019) 003 [1807.11357]. 11 The imaginary-𝜃 dependence of the SU(𝑁) spectrum Davide Vadacchino

    2019 arXiv

  46. [53]

    Berni, C

    M. Berni, C. Bonanno and M. D’Elia,Large-𝑁 expansion and𝜃-dependence of2𝑑𝐶𝑃 𝑁−1 models beyond the leading order,Phys. Rev. D100(2019) 114509 [1911.03384]

    2019 arXiv

  47. [54]

    Berni, C

    M. Berni, C. Bonanno and M. D’Elia,𝜃-dependence in the small-𝑁 limit of2𝑑𝐶𝑃 𝑁−1 models, Phys. Rev. D102 (2020) 114519 [2009.14056]

    2020 arXiv

  48. [55]

    Bonanno, M

    C. Bonanno, M. D’Elia and F. Margari,Topological susceptibility of the 2D CP1 or O(3) nonlinear𝜎 model: Is it divergent or not?,Phys. Rev. D107(2023) 014515 [2208.00185]

    2023 arXiv

  49. [56]

    Nogradi,An ideal toy model for confining, walking and conformal gauge theories: the O(3) sigma model with theta-term, JHEP 05(2012) 089 [1202.4616]

    D. Nogradi,An ideal toy model for confining, walking and conformal gauge theories: the O(3) sigma model with theta-term, JHEP 05(2012) 089 [1202.4616]

    2012 arXiv

  50. [57]

    Alles, M

    B. Alles, M. Giordano and A. Papa,Behavior near𝜃=𝜋 of the mass gap in the two-dimensional O(3) non-linear sigma model, Phys. Rev. B90 (2014) 184421 [1409.1704]

    2014 arXiv

  51. [59]

    Bhanot and F

    G. Bhanot and F. David,The Phases of the𝑂(3)𝜎 Model for Imaginary𝜃, Nucl. Phys. B 251 (1985) 127

    1985

  52. [60]

    Azcoiti, G

    V. Azcoiti, G. Di Carlo, A. Galante and V. Laliena,New proposal for numerical simulations of theta vacuum - like systems, Phys. Rev. Lett.89(2002) 141601 [hep-lat/0203017]

    2002 arXiv

  53. [61]

    Alles and A

    B. Alles and A. Papa,Mass gap in the2𝐷𝑂(3) non-linear sigma model with a𝜃=𝜋 term, Phys. Rev. D77 (2008) 056008 [0711.1496]

    2008 arXiv

  54. [62]

    Imachi, M

    M. Imachi, M. Kambayashi, Y. Shinno and H. Yoneyama,The𝜃-term,𝐶𝑃 𝑁−1 model and the inversion approach in the imaginary-𝜃 method, Prog. Theor. Phys.116 (2006) 181

    2006

  55. [63]

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

    arXiv

  56. [64]

    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

  57. [65]

    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

  58. [66]

    Bonanno, C

    C. Bonanno, C. Bonati and M. D’Elia,Large-𝑁 𝑆𝑈(𝑁) Yang-Mills theories with milder topological freezing,JHEP 03 (2021) 111 [2012.14000]

    2021 arXiv

  59. [67]

    Bonanno, M

    C. Bonanno, M. D’Elia and L. Verzichelli,The𝜃-dependence of the𝑆𝑈(𝑁) critical temperature at large𝑁, JHEP 02(2024) 156 [2312.12202]

    2024 arXiv

  60. [68]

    Campostrini, A

    M. Campostrini, A. Di Giacomo and H. Panagopoulos,The Topological Susceptibility on the Lattice, Phys. Lett. B212(1988) 206. 12 The imaginary-𝜃 dependence of the SU(𝑁) spectrum Davide Vadacchino

    1988

  61. [69]

    Berg,Dislocations andTopological Background inthe Lattice𝑂(3)𝜎 Model,Phys

    B. Berg,Dislocations andTopological Background inthe Lattice𝑂(3)𝜎 Model,Phys. Lett. B 104(1981) 475

    1981

  62. [70]

    Iwasaki and T

    Y. Iwasaki and T. Yoshie,Instantons and Topological Charge in Lattice Gauge Theory, Phys. Lett. B131(1983) 159

    1983

  63. [71]

    S. Itoh, Y. Iwasaki and T. Yoshie,Stability of Instantons on the Lattice and the Renormalized Trajectory, Phys. Lett. B147 (1984) 141

    1984

  64. [72]

    Teper,Instantons in the Quantized𝑆𝑈(2) Vacuum: A Lattice Monte Carlo Investigation, Phys

    M. Teper,Instantons in the Quantized𝑆𝑈(2) Vacuum: A Lattice Monte Carlo Investigation, Phys. Lett. B162(1985) 357

    1985

  65. [73]

    Ilgenfritz, M

    E.-M. Ilgenfritz, M. Laursen, G. Schierholz, M. Müller-Preussker and H. Schiller,First Evidence for the Existence of Instantons in the Quantized𝑆𝑈(2) Lattice Vacuum,Nucl. Phys. B268 (1986) 693

    1986

  66. [74]

    Campostrini, A

    M. Campostrini, A. Di Giacomo, H. Panagopoulos and E. Vicari,Topological Charge, Renormalization and Cooling on the Lattice,Nucl. Phys. B329(1990) 683

    1990

  67. [75]

    Alles, L

    B. Alles, L. Cosmai, M. D’Elia and A. Papa,Topology in2𝐷𝐶𝑃 𝑁−1 models on the lattice: A Critical comparison of different cooling techniques,Phys. Rev. D62(2000) 094507 [hep-lat/0001027]

    2000 arXiv

  68. [76]

    APEcollaboration,M.Albaneseetal., GlueballMassesandStringTensioninLatticeQCD , Phys. Lett. B192 (1987) 163

    1987

  69. [77]

    Morningstar and M

    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

  70. [78]

    Lüscher,Trivializing maps, the Wilson flow and the HMC algorithm, Commun

    M. Lüscher,Trivializing maps, the Wilson flow and the HMC algorithm, Commun. Math. Phys. 293(2010) 899 [0907.5491]

    2010 arXiv

  71. [79]

    Lüscher,Properties and uses of the Wilson flow in lattice QCD, JHEP 08(2010) 071 [1006.4518]

    M. Lüscher,Properties and uses of the Wilson flow in lattice QCD, JHEP 08(2010) 071 [1006.4518]

    2010 arXiv

  72. [80]

    Luscher and P

    M. Luscher and P. Weisz,Perturbative analysis of the gradient flow in non-abelian gauge theories,JHEP 02(2011) 051 [1101.0963]

    2011 arXiv

  73. [81]

    Bonati and M

    C. Bonati and M. D’Elia,Comparison of the gradient flow with cooling in𝑆𝑈(3) pure gauge theory,Phys. Rev. DD89(2014) 105005 [1401.2441]

    2014 arXiv

  74. [82]

    Alexandrou, A

    C. Alexandrou, A. Athenodorou and K. Jansen,Topological charge using cooling and the gradient flow, Phys. Rev. D92(2015) 125014 [1509.04259]

    2015 arXiv

  75. [83]

    Creutz,Overrelaxation and Monte Carlo Simulation,Phys

    M. Creutz,Overrelaxation and Monte Carlo Simulation,Phys. Rev. D36(1987) 515

    1987

  76. [84]

    Creutz,Monte Carlo Study of Quantized𝑆𝑈(2) Gauge Theory,Phys

    M. Creutz,Monte Carlo Study of Quantized𝑆𝑈(2) Gauge Theory,Phys. Rev. D21 (1980) 2308. 13 The imaginary-𝜃 dependence of the SU(𝑁) spectrum Davide Vadacchino

    1980

  77. [85]

    A. D. Kennedy and B. J. Pendleton,Improved Heat Bath Method for Monte Carlo Calculations in Lattice Gauge Theories, Phys. Lett. B156 (1985) 393

    1985

  78. [86]

    Hasenbusch,Fighting topological freezing in the two-dimensional𝐶𝑃 𝑁−1 model,Phys

    M. Hasenbusch,Fighting topological freezing in the two-dimensional𝐶𝑃 𝑁−1 model,Phys. Rev. D96(2017) 054504 [1706.04443]

    2017 arXiv

  79. [87]

    Alles, G

    B. Alles, G. Boyd, M. D’Elia, A. Di Giacomo and E. Vicari,Hybrid Monte Carlo and topological modes of full QCD, Phys. Lett. B389 (1996) 107 [hep-lat/9607049]

    1996 arXiv

  80. [88]

    Del Debbio, G

    L. Del Debbio, G. M. Manca and E. Vicari,Critical slowing down of topological modes, Phys. Lett. B594(2004) 315 [hep-lat/0403001]

    2004 arXiv

  81. [89]

    Schaefer, R

    ALPHAcollaboration, S. Schaefer, R. Sommer and F. Virotta,Critical slowing down and error analysis in lattice QCD simulations,Nucl. Phys. B845 (2011) 93 [1009.5228]

    2011 arXiv

  82. [90]

    Bonanno,Lattice determination of the topological susceptibility slope𝜒′ of 2𝑑 CP𝑁−1 models at large𝑁,Phys

    C. Bonanno,Lattice determination of the topological susceptibility slope𝜒′ of 2𝑑 CP𝑁−1 models at large𝑁,Phys. Rev. D107 (2023) 014514 [2212.02330]

    2023 arXiv

  83. [91]

    Bonanno, G

    C. Bonanno, G. Clemente, M. D’Elia, L. Maio and L. Parente,Full QCD with milder topological freezing,JHEP 08 (2024) 236 [2404.14151]

    2024 arXiv

  84. [92]

    Bonanno, M

    C. Bonanno, M. D’Elia, B. Lucini and D. Vadacchino,Towards glueball masses of large-N SU(N) pure-gauge theories without topological freezing,Phys. Lett. B833 (2022) 137281 [2205.06190]

    2022 arXiv

  85. [93]

    J. L. Dasilva Golán, C. Bonanno, M. D’Elia, M. García Pérez and A. Giorgieri,The twisted gradient flow strong coupling with parallel tempering on boundary conditions, PoS LATTICE2023(2024) 354 [2312.09212]

    2024 arXiv

  86. [94]

    Bonanno, J

    C. Bonanno, J. L. Dasilva Golán, M. D’Elia, M. García Pérez and A. Giorgieri,The SU(3) twisted gradient flow strong coupling without topological freezing,Eur. Phys. J. C84 (2024) 916 [2403.13607]

    2024 arXiv

  87. [95]

    Berg and A

    B. Berg and A. Billoire,Glueball spectroscopy in4𝑑 su(3) lattice gauge theory (i), Nuclear Physics B221 (1983) 109

    1983

  88. [96]

    Teper,An improved method for lattice glueball calculations,Physics Letters B183 (1987) 345

    M. Teper,An improved method for lattice glueball calculations,Physics Letters B183 (1987) 345

    1987

  89. [97]

    M. J. Teper,SU(𝑁) gauge theories in(2+ 1)-dimensions,Phys. Rev. D59(1999) 014512 [hep-lat/9804008]

    1999 arXiv

  90. [98]

    Lucini and M

    B. Lucini and M. Teper,𝑆𝑈(𝑁) gauge theories in four-dimensions: Exploring the approach to𝑁 =∞, JHEP 06(2001) 050 [hep-lat/0103027]

    2001 arXiv

  91. [99]

    Lucini, M

    B. Lucini, M. Teper and U. Wenger,Glueballs and𝑘-strings in𝑆𝑈(𝑁) gauge theories: Calculations with improved operators, JHEP 06(2004) 012 [hep-lat/0404008]. 14 The imaginary-𝜃 dependence of the SU(𝑁) spectrum Davide Vadacchino

    2004 arXiv

  92. [100]

    Blossier, M

    B. Blossier, M. Della Morte, G. von Hippel, T. Mendes and R. Sommer,On the generalized eigenvalue method for energies and matrix elements in lattice field theory, JHEP 04(2009) 094 [0902.1265]

    2009 arXiv

  93. [101]

    Lucini, A

    B. Lucini, A. Rago and E. Rinaldi,Glueball masses in the large-𝑁 limit,JHEP 08 (2010) 119 [1007.3879]

    2010 arXiv

  94. [102]

    Bennett, J

    E. Bennett, J. Holligan, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini et al.,Glueballs and strings in𝑆𝑝(2𝑁) Yang-Mills theories, Phys. Rev. D103 (2021) 054509 [2010.15781]

    2021 arXiv

  95. [103]

    Athenodorou and M

    A. Athenodorou and M. Teper,The glueball spectrum of SU(3) gauge theory in 3 + 1 dimensions,JHEP 11 (2020) 172 [2007.06422]

    2020 arXiv

  96. [104]

    Athenodorou and M

    A. Athenodorou and M. Teper,SU(N) gauge theories in 3+1 dimensions: glueball spectrum, string tensions and topology, JHEP 12(2021) 082 [2106.00364]

    2021 arXiv

  97. [105]

    de Forcrand, G

    P. de Forcrand, G. Schierholz, H. Schneider and M. Teper,The String and Its Tension in SU(3) Lattice Gauge Theory: Towards Definitive Results,Phys. Lett. B160 (1985) 137

    1985

  98. [106]

    Vadacchino,A review on Glueball hunting, 2305.04869

    D. Vadacchino,A review on Glueball hunting, 2305.04869

    arXiv

  99. [107]

    Yamada, M

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

    arXiv

  100. [108]

    Bigazzi, A

    F. Bigazzi, A. L. Cotrone and R. Sisca,Notes on𝜃 Dependence in Holographic Yang–Mills, JHEP 08(2015) 090 [1506.03826]

    2015 arXiv

  101. [109]

    Brower, S

    R. Brower, S. Chandrasekharan, J. W. Negele and U. J. Wiese,QCD at fixed topology,Phys. Lett. B560 (2003) 64 [hep-lat/0302005]

    2003 arXiv

  102. [110]

    S. Aoki, H. Fukaya, S. Hashimoto and T. Onogi,Finite volume QCD at fixed topological charge,Phys. Rev. D76(2007) 054508 [0707.0396]. 15

    2007 arXiv

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