Pith. sign in

REVIEW 4 major objections 4 minor 35 references

Effective string description of the reconfined phase in the trace deformed $\mathrm{SU}(2)$ Yang-Mills theory in (2+1) dimensions

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read In the reconfined phase of trace-deformed $\mathrm{SU}(2)$ Yang-Mills in (2+1) dimensions, the flux-tube ground-state energy follows the Polchinski-Yang rigid-string solution rather than the Nambu-Goto prediction.

desk verdict First flux-tube data in the reconfined phase suggest the Polchinski-Yang rigid string, but the evidence is underreported and the claimed regime is only marginally approached. read the letter →

arxiv 2501.13684 v1 pith:OYKKQQZN submitted 2025-01-23 hep-lat hep-th

Caludio Bonati , Michele Caselle , Alessio Negro , Dario Panfalone , Lorenzo Verzichelli This is my paper · ORCID
classification hep-lathep-th
keywords tracedeformationreconfinementrigidstringPolchinski-YangsolutioneffectivetheoryfluxtubelatticegaugeSU(2)Yang-Mills
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 studies the flux tube between two Polyakov loops in a modified version of $\mathrm{SU}(2)$ Yang-Mills theory in three spacetime dimensions, where an extra trace-deformation term keeps the Polyakov loop's expectation value at zero even at high temperature. It claims that in this reconfined phase the ground-state energy $E_0(N_t)$ of the flux tube is described by the Polchinski-Yang rigid-string solution, in which the curvature-squared rigidity term dominates the string action, and not by the Nambu-Goto prediction that holds in ordinary confinement. If correct, this would show that the confining mechanism in the reconfined phase has a genuinely different effective string description, and that the trace-deformed model provides a numerical laboratory where a normally unphysical regime of the rigid string can be realised and studied.

What carries the argument

The central object is the ground-state energy $E_0(N_t)$ of the effective string stretched between two Polyakov loops, extracted from the large-$R$ behaviour of the correlator using the modified Bessel form of the string free energy. The action that carries the argument is the rigid-string action $S_R = \int d^2\xi \sqrt{g}\,(\sigma + \gamma_2\, \mathcal{K}^2 + \cdots)$, where $\mathcal{K}$ is the extrinsic curvature of the world-sheet; in the regime where the rigidity term dominates and the Nambu-Goto quadratic term is a perturbation, the Polchinski-Yang solution gives $E_0 = w\lambda$, and this two-parameter formula is what the numerical data are compared with. The single fitted string tension $\sigma$ is cross-checked with the known zero-temperature scale-setting relation.

What would settle it

Compute $E_0(N_t)$ at values where $N_t^2\sigma$ is close to 1, for instance the largest $N_t$ in each set of Table 1 or a slightly smaller deformation parameter $h$ just inside the reconfined phase; if the data drift toward the Nambu-Goto prediction instead of the Polchinski-Yang curve, the claimed rigid-string regime is not realised. A direct independent value of $\gamma_2$ obtained from the flux-tube width would also settle whether the two-parameter fit is forced or physically meaningful.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that the reconfined-phase data for $E_0/\sqrt{\sigma}$ as a function of $T/T_c$ fall on the Polchinski-Yang curve rather than on the Nambu-Goto curve. The ground-state energy is extracted from a fit of the Polyakov-loop correlator at two values of the bare coupling $\beta$, several compactification sizes $N_t$, and two values of the deformation parameter $h$, and it agrees with the two-parameter rigid-string formula with $\sigma$ and $\gamma_2$ as the only free parameters. The paper presents this agreement as evidence that the reconfined phase sits in the rigid regime $\gamma_2 \gg N_t^2\sigma$, $N_t^2\sigma \ll 1$, where the extrinsic-curvature term dominates and the Nambu-Goto term is a small perturbation.

Load-bearing premise

The whole result depends on the simulated trace-deformed theories actually lying in the regime where the curvature-squared rigidity term dominates the string action and the quadratic tension term is only a small correction; the paper asserts this regime is realised but does not verify the required inequalities directly, and for the simulated lattice sizes the product $N_t^2\sigma$ is only about 0.3 to 0.9, so the regime is approached only marginally.

Editorial extensions

If this is right

  • The reconfined phase is described by a different effective string theory from the ordinary confining phase: the rigidity term, not the Nambu-Goto term, dominates, so the two confining mechanisms are distinct.
  • The two-parameter Polchinski-Yang formula accounts for $E_0(N_t)$ at both lattice spacings and all values of $h$ studied, supporting the claim that the trace-deformed model realises the normally unphysical rigid-string regime on the lattice.
  • The reconfinement transition in the trace-deformed phase diagram can be read as the transition between the Nambu-Goto regime and the Polchinski-Yang rigid-string regime.
  • If the rigid-string description is correct, the flux-tube width and shape in the reconfined phase should follow the rigid-string predictions as well, a check the authors state they will address in a forthcoming paper.

Reading between the lines

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

  • Editorial inference: the same two-parameter Polchinski-Yang test could be applied to trace-deformed $\mathrm{SU}(3)$ Yang-Mills, whose reconfined phase is already known to share other properties of ordinary confinement, to see whether the rigid-string regime is generic or specific to $\mathrm{SU}(2)$.
  • Editorial inference: since $N_t^2\sigma$ is only marginally small in the simulated range, the good fit could be partly a two-parameter description of a crossover; an independent determination of $\gamma_2$ from the flux-tube width would sharpen the claim that the rigid-string regime is truly approached.
  • Editorial inference: the trace-deformed model offers a controlled lattice setting for the high-temperature rigid-string regime that was originally studied to mimic large-$N$ QCD behaviour, so the agreement found here may connect to that earlier motivation.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies the confining flux tube in the trace-deformed SU(2) Yang-Mills theory in (2+1) dimensions, in the reconfined phase where the Polyakov loop vanishes. Using Monte Carlo simulations of the Polyakov loop correlator, the authors extract the ground state energy E0(N_t) of the effective string for five datasets (two beta values and several h values). They show that E0/sqrt(sigma) versus T/Tc is clearly different from the Nambu-Goto prediction of the ordinary confining phase, and they claim that the data are described by the Polchinski-Yang rigid-string solution (Eqs. 12-14) with a large rigidity term. The paper also reports evidence for a first-order reconfinement transition from the finite-size scaling of the Polyakov loop susceptibility.

Significance. If the claimed agreement is established, this would be an interesting example of a gauge theory whose flux tube is described by the Polchinski-Yang regime of the rigid string, and it would sharpen the distinction between ordinary and center-symmetric confining mechanisms. The numerical data and the qualitative comparison to Nambu-Goto are valuable, and the use of Eq. (4) for scale setting is sensible. However, the central quantitative evidence is incomplete: the fitted values of sigma and gamma_2, their uncertainties, and the goodness of fit are not reported, and the regime inequalities gamma_2 >> N_t^2 sigma and N_t^2 sigma << 1 are not demonstrated. The qualitative difference from Nambu-Goto is clear, but the specific Polchinski-Yang identification is not yet established beyond a two-parameter fit to the same data.

major comments (4)
  1. [Sec. 4.3, Eqs. (12)-(14), Figs. 2-3] The paper's central claim is that the data agree with the Polchinski-Yang solution, but Sec. 4.3 reports only that 'fitting these two parameters with our data we find the green dash-dotted line'; the fitted values of sigma and gamma_2, their uncertainties, and the chi^2/dof are not given. Since both parameters are fitted to the very same E0(N_t) values that the curve is then said to match, the visual agreement is not yet distinguishable from a two-parameter interpolation. Please report the fit results (including correlated errors and the covariance between sigma and gamma_2) for each dataset or for a combined fit, and state the exact fitting range and statistical treatment.
  2. [Sec. 4.3, regime conditions] The identification with the Polchinski-Yang vacuum is controlled by the inequalities gamma_2 >> N_t^2 sigma and N_t^2 sigma << 1, which the paper asserts but does not verify. Using Eq. (4) with beta = 23.3805 and beta = 27.4745, N_t^2 sigma ranges from about 0.28 to 0.68 and from about 0.30 to 0.90, respectively, so the second inequality is only marginally satisfied for the larger N_t values, and the first inequality cannot be checked because gamma_2 is not reported. Please provide numerical evidence for the regime, or alternatively qualify the claim as a test of the Polchinski-Yang formula outside its strict asymptotic regime.
  3. [Sec. 4.1, Eq. (8), Table 2] The extraction of E0 is done by fitting the correlator with a single K0 term over R in [15,23], and the text states that a good chi^2 is always found, but no chi^2 values or fit-range variations are reported for any ensemble. Since the later Polchinski-Yang claim depends on these E0 values, please provide the chi^2/dof for each fit, show stability under changing Rmin and Rmax, and discuss the possible contamination from higher string states or boundary effects.
  4. [Sec. 4.3, Ref. [17]] The paper repeatedly refers to a forthcoming publication for further details of the analysis; however, the present manuscript should be self-contained on the points that support its central claim. The missing fit parameters and regime verification should be included here, or the claims should be explicitly presented as preliminary.
minor comments (4)
  1. [Introduction and Sec. 2] Please correct the typos 'resarch', 'origianl', 'the the', and 'in the in the' that appear in the Introduction and Section 2.
  2. [Eq. (13)] The square-root nesting in Eq. (13) is hard to parse; please check the formula against Ref. [34] and define the spacetime dimension d explicitly (for this paper d=3, so d-2=1).
  3. [Figs. 2 and 3] The horizontal axes use T/Tc, where Tc is extrapolated from Ref. [20] for the two beta values; the uncertainties in this extrapolation are not discussed and may affect the comparison. Please state the Tc values used and their errors.
  4. [Table 2] Table 2 reports E0 for only one of the five datasets; to facilitate comparison, the corresponding E0 values for the other datasets could be given in an appendix or as auxiliary material.

Circularity Check

1 steps flagged · score 6.0 of 10

The 'remarkable agreement' with the Polchinski-Yang solution is produced by fitting both σ and γ2 to the same E0(Nt) data, and the asserted large-rigidity regime is not demonstrated.

  1. fitted input called prediction [Sec. 4.3, Eqs. (12)-(14); Abstract]
    "This expression depends on only two degrees of freedom: σ and γ2. Fitting these two parameters with our data we find the is the green dash-dotted line of fig.2, which agrees remarkably well with the results of the simulations."

    The curve displayed as a 'prediction' of the rigid string is obtained by fitting the two free parameters σ and γ2 of Eqs. (12)-(14) to the very same E0(Nt) data points it is then said to match. The agreement in Figs. 2 and 3 is therefore an interpolation, not an independent check. The fitted values, uncertainties, and chi^2 are not reported, and the abstract's wording 'predictions of the so called rigid string' presents this fit as if it were a prediction. Without the fitted parameters one cannot even test whether the fitted σ is consistent with the independent zero-temperature string tension of Eq. (4), or whether the fitted γ2 satisfies the assumed regime γ2 >> N_t^2 σ.

full rationale

The only load-bearing step that qualifies as circular is in Sec. 4.3: the Polchinski-Yang curves shown as 'predictions' in Figs. 2 and 3 are obtained by fitting σ and γ2 to the same E0(Nt) values that the curves are then said to reproduce. This fits the pattern of a fitted input being called a prediction, so the central 'agreement' is partly constructed rather than independently verified. The Polchinski-Yang formula itself is an external 1992 result, so the functional form is not derived in this paper and retains independent theoretical content; however, the paper's central claim that the reconfined flux tube is described by the rigid-string vacuum rests on this two-parameter fit. The fitted values, uncertainties, and chi^2 are not reported, and details are deferred to an in-preparation self-citation [17], making the claim difficult to falsify. Separately, the regime condition N_t^2 σ << 1 is asserted but not demonstrated; using Eq. (4) with the simulated parameters gives N_t^2 σ as large as about 0.9, so the largest points are not deep in the claimed Polchinski-Yang asymptotic regime. This is a correctness and transparency concern rather than an additional circular step. No load-bearing argument reduces to a self-citation chain: the comparison with the ordinary confining phase uses external data, and the cited rigid-string solution is independent of the present fitted values. Overall, the central 'prediction' reduces to a fit, but the external functional form keeps the circularity partial, giving a score of 6.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the Polchinski-Yang formula for the rigid string, on the assumption that the trace deformed theory sits in the large-rigidity regime, and on the standard single-state EST representation of the Polyakov correlator. The two fitted parameters sigma and gamma_2 carry much of the weight, and the paper does not report their values or errors.

free parameters (2)
  • sigma (string tension) = not reported
    Appears in the Polchinski-Yang formula Eq. (14). In Sec. 4.3 both sigma and gamma_2 are fitted to the same E0 data; no fitted value or uncertainty is given. An independent zero-temperature value exists in Ref. [7], but the paper does not state whether it was used as an input.
  • gamma_2 (rigidity coefficient) = not reported
    Coefficient of the extrinsic curvature squared term in the rigid string action Eq. (9). It is fitted together with sigma to the measured E0(N_t) values, and its value and uncertainty are not reported.
assumptions (4)
  • standard math The Polchinski-Yang solution (Eqs. 12-14) gives the ground state energy of the rigid string in the regime where the rigidity term dominates.
    Invoked in Sec. 4.3 and used to fit the data; treated as correct without re-derivation.
  • domain assumption The trace-deformed action Eq. (1) with the selected h values realizes a reconfined phase with a confining flux tube describable by an effective string.
    Assumed from prior trace-deformation literature [3-5,18] and from the first-order transition seen in Fig. 1; not independently established in this paper.
  • ad hoc to paper The simulated parameters lie in the Polchinski-Yang regime, gamma_2 >> N_t^2 sigma and N_t^2 sigma << 1, and no other higher-order terms contribute.
    Stated as a claim in Sec. 4.3 without a direct check of the inequalities; for the simulated N_t values, N_t^2 sigma is of order 0.3 to 0.9, so the regime is only marginally satisfied.
  • domain assumption The Polyakov loop correlator is described by the single-state form Eq. (8) for R in [15,23].
    Standard EST prediction; validated by the quoted 'good chi2' but no quantitative chi2 values are shown.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Effective string description of the reconfined phase in the trace deformed $\mathrm{SU}(2)$ Yang-Mills theory in (2+1) dimensions." pith.science (2026). https://pith.science/paper/OYKKQQZN

@misc{pith2026250113684,
  author       = {Pith},
  title        = {Pith review of: Effective string description of the reconfined phase in the trace deformed $\mathrmSU(2)$ Yang-Mills theory in (2+1) dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OYKKQQZN}},
  note         = {Machine review of arXiv:2501.13684}
}
abstract

We study the behaviour of the flux tube in the reconfined phase of the trace deformed $\mathrm{SU}(2)$ Yang-Mills theory in (2 + 1) dimensions. In this phase the Polyakov loop has a vanishing expectation value (and center symmetry is recovered) even at high temperatures. We study, by means of numerical simulations, the confining potential between two Polyakov loops. We show that its behaviour is very different from that of usual confining gauge models and shows a remarkable agreement with the predictions of the so called "rigid string" in the limit in which the rigidity term (i.e. a term proportional to the square of the extrinsic curvature of the string) is very large and is the dominant contribution in the action.

Figures

Figures reproduced from arXiv: 2501.13684 by the authors.

Figure 1
Figure 1. Behaviour of the susceptibility for the SU(2) model in (2+1) dimensions with 𝑁𝑡 = 10 and 𝛽 = 23.3805, for different values of the lattice extent 𝑁𝑠. For this value of 𝛽 the deconfinement transition of the ordinary SU(2) gauge theory is located at 𝑁𝑡,𝑐 = 15, so with 𝑁𝑡 = 10 we are deeply in the deconfined phase. 3. The effective string description of the flux tube. A general introduction to the subject can be found i… view at source ↗
Figure 2
Figure 2. Values of the adimensional ratio 𝐸0/ √ 𝜎 as a function of 𝑇/𝑇𝑐. The blue open circles are taken from [16] and correspond to the ordinary confining SU(2) model, while the green open squares are the data discussed in the present paper for the reconfined phase. The continuous line is the Nambu-Goto prediction, the blue dashed line describes the Nambu-Goto prediction plus the term proportional to K4 discussed in [16] wh… view at source ↗
Figure 3
Figure 3. Data for all values of 𝛽 and ℎ that we studied, compared with the Polchinski-Yang solution. Acknowledgements This work has been supported by the Italian PRIN “Progetti di Ricerca di Rilevante Interesse Nazionale – Bando 2022” prot. 2022TJFCYB, and by the “Simons Collaboration on Confinement and QCD Strings” funded by the Simons Foundation. The simulations were run on CINECA computers. C. Bonati acknowledges support … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 10 canonical work pages

  1. [7]

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

    arXiv 1999

  2. [1]

    Unsal and L

    M. Unsal and L. G. Yaffe,Center-stabilized Yang-Mills theory: Confinement and large N volume independence, Phys. Rev. D78 (2008) 065035 [0803.0344]

    arXiv 2008

  3. [2]

    J. C. Myers and M. C. Ogilvie,New phases of SU(3) and SU(4) at finite temperature, Phys. Rev.D77 (2008) 125030 [0707.1869]

    arXiv 2008

  4. [3]

    Athenodorou, M

    A. Athenodorou, M. Cardinali and M. D’Elia,Spectrum of trace deformed Yang-Mills theories, Phys. Rev. D104 (2021) 074510 [2010.03618]

    arXiv 2021

  5. [4]

    Bonati, M

    C. Bonati, M. Cardinali, M. D’Elia, M. Giordano and F. Mazziotti,Reconfinement, localization and thermal monopoles in𝑆𝑈(3) trace-deformed Yang-Mills theory,Phys. Rev. D 103 (2021) 034506 [2012.13246]

    arXiv 2021

  6. [5]

    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]

    arXiv 2018

  7. [6]

    Ambjørn, P

    J. Ambjørn, P. Olesen and C. Peterson,Observation of a string in three-dimensional SU(2) lattice gauge theory,Phys. Lett.B142(1984) 410

    work page 1984

  8. [8]

    Caselle, M

    M. Caselle, M. Pepe and A. Rago,Static quark potential and effective string corrections in the (2+1)-d SU(2) Yang-Mills theory, JHEP10(2004) 005 [hep-lat/0406008]

    arXiv 2004

Show all 35 references
  1. [9]

    Caselle, A

    M. Caselle, A. Feo, M. Panero and R. Pellegrini,Universal signatures of the effective string in finite temperature lattice gauge theories, JHEP04(2011) 020 [1102.0723]

    2011 arXiv

  2. [10]

    Bringoltz and M

    B. Bringoltz and M. Teper,A Precise calculation of the fundamental string tension in SU(N) gauge theories in 2+1 dimensions, Phys. Lett.B645(2007) 383 [hep-th/0611286]

    2007 arXiv

  3. [11]

    B. B. Brandt,Probing boundary-corrections to Nambu-Goto open string energy levels in 3d SU(2) gauge theory, JHEP02(2011) 040 [1010.3625]

    2011 arXiv

  4. [12]

    B. B. Brandt,Spectrum of the open QCD flux tube and its effective string description I: 3d static potential in SU(N = 2, 3),JHEP 07(2017) 008 [1705.03828]

    2017 arXiv

  5. [13]

    B. B. Brandt,Spectrum of the open QCD flux tube and its effective string description, PoS Confinement2018 (2018) 039 [1811.11779]

    2018 arXiv

  6. [14]

    B. B. Brandt,Revisiting the flux tube spectrum of 3d SU(2) lattice gauge theory, Indian Journal of Physics95(2021) 1613 [2102.06413]

    2021 arXiv

  7. [15]

    Bonati, M

    C. Bonati, M. Caselle and S. Morlacchi,The Unreasonable effectiveness of effective string theory: The case of the 3D SU(2) Higgs model,Phys. Rev. D104(2021) 054501 [2106.08784]. 9 EST description of reconfinement. Michele Caselle

    2021 arXiv

  8. [16]

    Caristo, M

    F. Caristo, M. Caselle, N. Magnoli, A. Nada, M. Panero and A. Smecca,Fine corrections in the effective string describing SU(2) Yang-Mills theory in three dimensions,JHEP03 (2022) 115 [2109.06212]

    2022 arXiv

  9. [17]

    Bonati, M

    C. Bonati, M. Caselle, A. Negro, D. Panfalone and L. Verzichelli,In preparation,

  10. [18]

    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

  11. [19]

    K. G. Wilson,Confinement of Quarks,Phys. Rev.D10 (1974) 2445

    1974

  12. [20]

    Edwards and L

    S. Edwards and L. von Smekal,SU(2) lattice gauge theory in 2+1 dimensions: Critical couplings from twisted boundary conditions and universality, Phys. Lett. B681 (2009) 484 [0908.4030]

    2009 arXiv

  13. [21]

    Aharony and Z

    O. Aharony and Z. Komargodski,The Effective Theory of Long Strings,JHEP05(2013) 118 [1302.6257]

    2013 arXiv

  14. [22]

    B. B. Brandt and M. Meineri,Effective string description of confining flux tubes,Int. J. Mod. Phys.A31(2016) 1643001 [1603.06969]

    2016 arXiv

  15. [23]

    Caselle,Effective String Description of the Confining Flux Tube at Finite Temperature, Universe7 (2021) 170 [2104.10486]

    M. Caselle,Effective String Description of the Confining Flux Tube at Finite Temperature, Universe7 (2021) 170 [2104.10486]

    2021 arXiv

  16. [24]

    Lüscher and P

    M. Lüscher and P. Weisz,String excitation energies in SU(N) gauge theories beyond the free-string approximation,JHEP07 (2004) 014 [hep-th/0406205]

    2004 arXiv

  17. [25]

    Dubovsky, R

    S. Dubovsky, R. Flauger and V. Gorbenko,Effective String Theory Revisited, JHEP09 (2012) 044 [1203.1054]

    2012 arXiv

  18. [26]

    Caselle, N

    M. Caselle, N. Magnoli, A. Nada, M. Panero, D. Panfalone and L. Verzichelli,Confining strings in three-dimensional gauge theories beyond the Nambu-Got¯o approximation, JHEP 08(2024) 198 [2407.10678]

    2024 arXiv

  19. [27]

    Peliti and S

    L. Peliti and S. Leibler,Effects of Thermal Fluctuations on Systems with Small Surface Tension, Phys. Rev. Lett.54(1985) 1690

    1985

  20. [28]

    Helfrich,Effect of thermal undulations on the rigidity of fluid membranes and interfaces, J

    W. Helfrich,Effect of thermal undulations on the rigidity of fluid membranes and interfaces, J. Phys. France46 (1985) 1263

    1985

  21. [29]

    Förster,On the scale dependence, due to thermal fluctuations, of the elastic properties of membranes, Phys

    D. Förster,On the scale dependence, due to thermal fluctuations, of the elastic properties of membranes, Phys. Lett.A114(1986) 115

    1986

  22. [30]

    A. M. Polyakov,Fine Structure of Strings, Nucl. Phys.B268(1986) 406

    1986

  23. [31]

    Kleinert,The Membrane Properties of Condensing Strings, Phys

    H. Kleinert,The Membrane Properties of Condensing Strings, Phys. Lett.B174(1986) 335

    1986

  24. [32]

    Braaten, R

    E. Braaten, R. D. Pisarski and S.-M. Tse,The Static Potential for Smooth Strings, Phys. Rev. Lett.58 (1987) 93. 10 EST description of reconfinement. Michele Caselle

    1987

  25. [33]

    Germán and H

    G. Germán and H. Kleinert,Perturbative Two Loop Quark Potential of Stiff Strings in Any Dimension, Phys. Rev.D40(1989) 1108

    1989

  26. [34]

    Polchinski and Z

    J. Polchinski and Z. Yang,High temperature partition function of the rigid string, Phys. Rev. D 46(1992) 3667 [hep-th/9205043]

    1992 arXiv

  27. [35]

    Ambjørn, Y

    J. Ambjørn, Y. Makeenko and A. Sedrakyan,Effective QCD string beyond the Nambu-Goto action,Phys. Rev. D89(2014) 106010 [1403.0893]. 11

    2014 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.