Coulomb-gauge gaugeon correlators as a string-tension and confinement probe in lattice gauge theories with dynamical fermions

Marinari, Paciello, Parisi and Taglienti (Phys. Lett. B 298 (1993) 400) proposed to measure the string tension and to test confinement from Coulomb-gauge correlation functions $C_n(d)$ of timelike Wilson lines of length $n$ ("gaugeons") projected to zero transverse momentum, predicted to decay as $e^{-\sigma n d}$ in a confining phase (the source's own notation writes this $e^{-\sigma^2 n d}$, its symbol for the string tension being $\sigma^2$) and argued — unlike Wilson loops or Polyakov-loop correlators — to retain a confinement signal when dynamical fermions screen the static potential. Their numerical demonstration covered only pure-gauge SU(3) at a single coupling ($\beta=5.8$), leaving open the dynamical-fermion validation, operator improvement for the large-distance asymptotics, continuum extrapolation of systematic effects, and Gribov-copy robustness. Later work adopted the same observable (the Coulomb-gauge timelike-line correlator $G(R,T)$), confirmed in quenched theory that it interpolates between an overconfining instantaneous color-Coulomb string tension ($\sigma_{\rm coul}/\sigma\approx 2$–$4$) and the physical one, and found it robust against Gribov copies — but every published measurement remains in pure gauge or in gauge-Higgs models with scalar matter as a screening proxy, and analytic abelian-model checks give cautionary results. The dynamical-fermion validation, the method's central selling point, has not been performed, and what confinement signal the correlator carries in full QCD remains open.

Background

In lattice gauge theory, spacetime is discretized and the gauge field is represented by group-valued link variables $U_\mu(x)$ on the edges of a hypercubic lattice. A defining phenomenon of non-abelian gauge theories such as QCD is confinement: the energy of the color-electric field between two static color sources grows linearly with their separation $R$, $V(R)\simeq\sigma R$, and the coefficient $\sigma$ is the string tension. Two standard lattice diagnostics of confinement and of $\sigma$ are Wilson loops (traced products of link variables around closed curves, expected to decay with the enclosed area in a confining theory) and correlators of Polyakov loops (Wilson lines closed through the periodic time direction, whose large-distance decay is governed by $\sigma$). Both fail once the theory contains dynamical fermions — quark fields included in the Monte Carlo ensemble ("sea quarks"), as opposed to the quenched approximation in which gauge fields are sampled without them: light quarks screen the sources, the flux tube breaks ("string breaking"), Wilson loops cross over to perimeter-law decay and Polyakov correlators acquire a nonzero connected part, so neither observable measures confinement in full QCD.

The source paper (Marinari, Paciello, Parisi and Taglienti, 1993) proposed a third diagnostic. One fixes the gauge field to Coulomb gauge — on the lattice, maximizing $\sum\mathrm{Re}\,\mathrm{Tr}\,U_i$ over spatial links, so that spatial gauge fields are as close as possible to the identity on each Euclidean time slice. This does not exhaust the gauge freedom: time-dependent but space-independent gauge transformations still preserve the condition, leaving a global color $SU(3)$ symmetry on each time slice (the remnant symmetry). Following an old observation of Ferrari and Picasso (1971) that in Coulomb gauge the photon can be regarded as the Goldstone boson of the gauge symmetry, the authors argued that in a Coulomb-like (deconfined) phase this remnant symmetry breaks spontaneously and the correlator of timelike links tends to a constant at large distance, while in a confining phase the symmetry stays unbroken and the correlator decays exponentially. The elementary operator is the "gaugeon" of length $n$, $G_n(\mathbf{x},t_0)=\prod_{t=t_0}^{t_0+n}U_0(\mathbf{x},t)$ — a straight timelike Wilson line — projected to zero transverse momentum, and the measured quantity is the equal-time, zero-transverse-momentum correlator

$$C_n(d)\propto\sum_{x_\alpha}\big\langle G_n(x_\delta,t_0)\,G_n^\dagger(x_\delta+d,t_0)\big\rangle .$$

The prediction is $C_n(d)\sim e^{-\sigma n d}$ at large $n$ and $d$ in the confining phase (the source denotes the string tension by $\sigma^2$, so its formula reads $e^{-\sigma^2nd}$). Crucially, global color is not spontaneously broken even with dynamical quarks, so the authors argued that the gaugeon correlator — unlike Wilson loops and Polyakov correlators — should keep a confinement signal in the full theory. Their numerical demonstration was confined to pure-gauge SU(3) at $\beta=5.8$ on $10^3\times 20$ and $10^3\times 6$ lattices: the effective decay rates followed the predicted form, with $\sigma^2(n)=\sigma^2(\infty)+c_1/n$ fitting the data well, but the largest accessible effective-mass estimate (from separations 3 and 4) still overshot the asymptotic string tension by about 50%.

Later work placed the proposal in a sharper framework. The gaugeon correlator is the Coulomb-gauge timelike-line correlator $G(R,T)$, whose $T\to\infty$ limit yields the static potential and whose equal-time limit yields the instantaneous color-Coulomb potential, i.e. the instantaneous part of the time-time gluon propagator in Coulomb gauge. Zwanziger proved the variational inequality $V(R)\le V_{\rm coul}(R)$: if the physical potential is confining, the instantaneous Coulomb potential must confine too. Quenched lattice studies found that it does, but with a slope (the "Coulomb string tension" $\sigma_{\rm coul}$) two to four times the physical $\sigma$, and with the slope $\sigma(T)$ extracted from $G(R,T)$ decreasing monotonically in $T$ from $\sigma_{\rm coul}$ toward $\sigma$ — precisely the $1/n$ decrease seen empirically in the source. A further practical concern for any gauge-fixed observable is Gribov copies: after gauge fixing, many gauge-equivalent configurations ("copies") can satisfy the gauge condition, and non-gauge-invariant quantities could depend on which copy is selected. What the later literature did not do is test any of this on ensembles with dynamical sea quarks: every published measurement of these Coulomb-gauge correlators and potentials is in quenched pure-gauge theory, or in gauge-Higgs models with scalar matter serving as a proxy for screening.

Problem Statement

Validate the Coulomb-gauge gaugeon correlator method — equivalently the timelike-line correlator $G(R,T)$ in Coulomb gauge — as a measurement of the string tension and as a confinement criterion, in fermion-coupled as well as pure lattice gauge theories. Concretely: on gauge-field ensembles that include dynamical fermions (full lattice QCD), after fixing Coulomb gauge, measure

$$C_n(d)\quad\text{or}\quad G(R,T)=\Big\langle\tfrac{1}{N}\mathrm{Tr}\,L^\dagger(\mathbf{x},T)\,L(\mathbf{x}+\mathbf{R},T)\Big\rangle,\qquad L(\mathbf{x},T)=\prod_{t=1}^{T}U_0(\mathbf{x},t),$$

at zero transverse momentum as in the source, and determine: (i) whether $C_n(d)$ retains the predicted exponential decay in $d$ with a rate $f(n)$ that grows linearly in $n$ at large $n$; (ii) which slope the method returns in a theory whose physical static potential is screened by string breaking — in particular whether the instantaneous color-Coulomb part extracted at small $T$ remains linearly rising with sea quarks present, how it compares with the quenched ratios $\sigma_{\rm coul}/\sigma\approx2$–$4$ and with Zwanziger's inequality $V(R)\le V_{\rm coul}(R)$, and what confinement information the decay constant carries; and (iii) the systematic controls the source flagged as prerequisites for validating the method: operator improvement (smearing or variational bases) for reaching the large-distance asymptotic regime, the magnitude and continuum-limit behavior of the difference between gauge-invariant and Coulomb-gauge-fixed determinations as a function of $\beta$ and lattice spacing, and Gribov-copy dependence across couplings, volumes and gauge-fixing procedures. Items in (iii) have been substantially settled for the quenched theory by subsequent work, so the question that remains open is whether (i)–(ii) hold with dynamical fermions, the (iii) controls serving as prerequisites for that determination.

References

1. E. Marinari, M. L. Paciello, G. Parisi, B. Taglienti, "The string tension in gauge theories. A suggestion for a new measurement method", Phys. Lett. B 298 (1993) 400-404, DOI 10.1016/0370-2693(93)91840-J, arXiv:hep-lat/9210021, https://arxiv.org/abs/hep-lat/9210021
2. R. Ferrari, L. E. Picasso, "Spontaneous breakdown in quantum electrodynamics", Nucl. Phys. B 31 (1971) 316-330, DOI 10.1016/0550-3213(71)90235-5, https://doi.org/10.1016/0550-3213(71)90235-5
3. T. H. Hansson, I. Zahed, "String tension in QED from Coulomb gauge correlators", arXiv:hep-th/9303179 (1993, unpublished preprint), https://arxiv.org/abs/hep-th/9303179
4. E. Marinari, M. L. Paciello, B. Taglienti, "The string tension in gauge theories", Int. J. Mod. Phys. A 10 (1995) 4265-4310, DOI 10.1142/S0217751X95001984, arXiv:hep-lat/9503027, https://arxiv.org/abs/hep-lat/9503027
5. D. Zwanziger, "No confinement without Coulomb confinement", Phys. Rev. Lett. 90 (2003) 102001, DOI 10.1103/PhysRevLett.90.102001, arXiv:hep-lat/0209105, https://arxiv.org/abs/hep-lat/0209105
6. J. Greensite, Š. Olejník, "Coulomb energy, vortices, and confinement", Phys. Rev. D 67 (2003) 094503, DOI 10.1103/PhysRevD.67.094503, arXiv:hep-lat/0302018, https://arxiv.org/abs/hep-lat/0302018
7. J. Greensite, Š. Olejník, "Adventures in Coulomb gauge", in: Color Confinement and Hadrons in Quantum Chromodynamics (World Scientific, 2004) pp. 152-163, DOI 10.1142/9789812702845_0011, arXiv:hep-lat/0309172, https://arxiv.org/abs/hep-lat/0309172
8. J. Greensite, Š. Olejník, D. Zwanziger, "Coulomb energy, remnant symmetry, and the phases of non-Abelian gauge theories", Phys. Rev. D 69 (2004) 074506, DOI 10.1103/PhysRevD.69.074506, arXiv:hep-lat/0401003, https://arxiv.org/abs/hep-lat/0401003
9. A. Nakamura, T. Saito, "QCD color interactions between two quarks", Phys. Lett. B 621 (2005) 171-175, DOI 10.1016/j.physletb.2005.06.053, arXiv:hep-lat/0512043, https://arxiv.org/abs/hep-lat/0512043
10. A. Nakamura, T. Saito, "Color confinement in Coulomb gauge QCD", Prog. Theor. Phys. 115 (2006) 189-200, DOI 10.1143/PTP.115.189, arXiv:hep-lat/0512042, https://arxiv.org/abs/hep-lat/0512042
11. Y. Nakagawa, A. Nakamura, T. Saito, H. Toki, D. Zwanziger, "Properties of color-Coulomb string tension", Phys. Rev. D 73 (2006) 094504, DOI 10.1103/PhysRevD.73.094504, arXiv:hep-lat/0603010, https://arxiv.org/abs/hep-lat/0603010
12. T. Heinzl, K. Langfeld, M. Lavelle, D. McMullan, "Coulomb gauge Gribov copies and the confining potential", Phys. Rev. D 76 (2007) 114510, DOI 10.1103/PhysRevD.76.114510, arXiv:0705.2718, https://arxiv.org/abs/0705.2718
13. G. Burgio, M. Quandt, H. Reinhardt, H. Vogt, "Coulomb versus physical string tension on the lattice", Phys. Rev. D 92 (2015) 034518, DOI 10.1103/PhysRevD.92.034518, arXiv:1503.09064, https://arxiv.org/abs/1503.09064
14. J. Greensite, A. P. Szczepaniak, "Coulomb string tension, asymptotic string tension, and the gluon chain", Phys. Rev. D 91 (2015) 034503, DOI 10.1103/PhysRevD.91.034503, arXiv:1410.3525, https://arxiv.org/abs/1410.3525
15. G. Burgio, M. Quandt, H. Reinhardt, H. Vogt, "Gribov horizon and Gribov copies effect in lattice Coulomb gauge", Phys. Rev. D 95 (2017) 014503, DOI 10.1103/PhysRevD.95.014503, arXiv:1608.05795, https://arxiv.org/abs/1608.05795
16. J. Greensite, E. Owen, "Unphysical properties of the static quark-antiquark four-point correlator in Landau gauge", Phys. Rev. D 99 (2019) 014506, DOI 10.1103/PhysRevD.99.014506, arXiv:1811.01028, https://arxiv.org/abs/1811.01028
17. R. Pasechnik, M. Šumbera, "Different Faces of Confinement", Universe 7 (2021) 330, DOI 10.3390/universe7090330, arXiv:2109.07600, https://arxiv.org/abs/2109.07600
18. S. M. Dawid, W. A. Smith, A. Rodas, R. J. Perry, C. Fernández-Ramírez, E. S. Swanson, A. P. Szczepaniak, "Coulomb confinement in the Hamiltonian limit", Phys. Rev. D 110 (2024) 094509, DOI 10.1103/PhysRevD.110.094509, arXiv:2408.09007, https://arxiv.org/abs/2408.09007