Coulomb-Gauge Correlator Approach in QCD

• The Coulomb-gauge correlator approach is a noncovariant QCD framework that fixes the gauge with ∇·A=0 to analyze equal-time propagators and reveal confinement mechanisms.
• It employs both Hamiltonian and first-order formulations to derive static observables such as the non-Abelian Coulomb potential, gluon gap equations, and ghost propagator relations.
• The method informs applications ranging from lattice confinement diagnostics and chiral symmetry breaking to finite-temperature transitions and hadron spectroscopy.

The Coulomb-gauge correlator approach denotes a family of noncovariant QCD and Yang–Mills constructions in which one fixes the gauge by $\nabla\!\cdot\!A=0$ and studies static or equal-time correlators—ghost and gluon propagators, Coulomb kernels, temporal-link correlators, Wilson-line correlators, or gauge-fixed bilinears—to extract confinement, static interquark energies, chiral symmetry breaking, and, in more recent work, parton observables (1111.7279, Greensite et al., 2018). Its characteristic feature is that Gauss’ law makes the non-Abelian Coulomb interaction explicit, so that the Faddeev–Popov operator, the ghost sector, and the residual gauge symmetry directly control the long-distance behavior of the relevant correlators (Reinhardt et al., 2016). The same feature also makes interpretation delicate: depending on the observable, one may be dealing with a gauge-fixed overlap, an expectation value of the Coulomb kernel, an instantaneous upper bound on the static potential, or a genuinely gauge-invariant Wilson-loop quantity (Iritani et al., 2010).

1. Formal setting and basic objects

In the Hamiltonian formulation, one starts in Weyl gauge $A_0=0$, imposes Coulomb gauge $\nabla\!\cdot\!\mathbf A=0$, and resolves Gauss’ law explicitly. The gauge-fixed Yang–Mills Hamiltonian then takes the form

$$H = \frac{1}{2} \int d^3 x \left( J^{-1}[A]\, \Pi^a(x)\cdot J[A]\,\Pi^a(x) + B^a(x)\cdot B^a(x) \right) + H_C ,$$

with Faddeev–Popov determinant

$J[A]=\Det(-\hat D\cdot \partial),$

and Coulomb term

$$H_C = \frac{1}{2}\int d^3x\, d^3y\; J[A]^{-1}\rho^a(x)J[A] \left[ (-\hat D\cdot \partial)^{-1}(-\partial^2)(-\hat D\cdot \partial)^{-1} \right]^{ab}(x,y)\rho^b(y).$$

The color charge density is

$$\rho^a(x)= -f^{abc}A^b(x)\cdot \Pi^c(x) + \rho_m^a(x).$$

The scalar product contains the same determinant,

$$\langle \phi \vert \ldots \vert \psi\rangle = \int \mathcal D A \; J[A]\; \phi^*[A]\ldots \psi[A],$$

so the ghost sector is built into both the Hamiltonian and the measure (Reinhardt et al., 2016).

A complementary first-order functional formulation rewrites Coulomb-gauge QCD as a ghost-free but nonlocal theory after integrating out temporal and longitudinal components. In that language, Gauss’ law generates both the nonlocal Coulomb kernel

$$\hat F_x^{ab} = \left[-\vec\nabla_x\!\cdot\!\vec D_x^{ac}\right]^{-1} (-\nabla_x^2) \left[-\vec\nabla_x\!\cdot\!\vec D_x^{cb}\right]^{-1}$$

and the global color-neutrality constraint

$\delta\!\left(\int d^3x\,\hat\rho^a(x)\right),$

so total color charge must vanish (1111.7279). This makes static and equal-time correlators the natural observables. The central ones are the equal-time gluon propagator, the ghost propagator, the curvature or ghost loop, and the vacuum expectation value of the Coulomb kernel. In the Gaussian Hamiltonian ansatz the static gluon propagator is essentially

$A_0=0$0

while the ghost propagator is

$A_0=0$1

with ghost form factor

$A_0=0$2

and the non-Abelian Coulomb potential is

$A_0=0$3

(Reinhardt et al., 2017).

This setup already indicates the distinctive logic of the approach. Equal-time Green’s functions are not auxiliary quantities; they are the objects from which confinement, color screening, and static energies are read off. A recurring theme in the literature is that the physically robust information is often carried by static propagators or color-singlet combinations, whereas full colored propagators retain charge-constraint constants and infrared divergences that are not themselves observable (1111.7279).

2. Yang–Mills correlators and confinement

The pure-glue variational program is organized around the vacuum wave functional

$A_0=0$4

or, in non-Gaussian extensions,

$A_0=0$5

Minimization of the vacuum energy yields the gluon gap equation

$A_0=0$6

where the curvature

$A_0=0$7

is determined by the ghost sector (Reinhardt et al., 2016, Reinhardt et al., 2017).

The ghost Dyson–Schwinger equation requires an infrared boundary condition, and the central Coulomb-gauge choice is the horizon condition

$A_0=0$8

With the infrared ansatz

$A_0=0$9

one finds the sum rule

$\nabla\!\cdot\!\mathbf A=0$0

with solutions quoted in the Hamiltonian literature as $\nabla\!\cdot\!\mathbf A=0$1 in $\nabla\!\cdot\!\mathbf A=0$2 and $\nabla\!\cdot\!\mathbf A=0$3 or $\nabla\!\cdot\!\mathbf A=0$4 in $\nabla\!\cdot\!\mathbf A=0$5 (Reinhardt et al., 2017). The physical interpretation is sharpened by the identity

$\nabla\!\cdot\!\mathbf A=0$6

which identifies the inverse ghost dressing with the dielectric function of the Yang–Mills vacuum. Under the horizon condition, $\nabla\!\cdot\!\mathbf A=0$7, so the vacuum behaves as a perfect color dielectric medium, i.e. as a dual superconductor (Reinhardt et al., 2016).

The infrared behavior of the gluon sector is correspondingly confining. The equal-time gluon propagator is well fitted by Gribov’s formula

$\nabla\!\cdot\!\mathbf A=0$8

with $\nabla\!\cdot\!\mathbf A=0$9 in the review literature. This gives $$H = \frac{1}{2} \int d^3 x \left( J^{-1}[A]\, \Pi^a(x)\cdot J[A]\,\Pi^a(x) + B^a(x)\cdot B^a(x) \right) + H_C ,$$0 in the ultraviolet and $$H = \frac{1}{2} \int d^3 x \left( J^{-1}[A]\, \Pi^a(x)\cdot J[A]\,\Pi^a(x) + B^a(x)\cdot B^a(x) \right) + H_C ,$$1 in the infrared, so low-momentum gluons become infinitely costly excitations (Reinhardt et al., 2016). The Coulomb kernel then generates a linearly rising non-Abelian Coulomb potential, with Coulomb string tension determined by

$$H = \frac{1}{2} \int d^3 x \left( J^{-1}[A]\, \Pi^a(x)\cdot J[A]\,\Pi^a(x) + B^a(x)\cdot B^a(x) \right) + H_C ,$$2

The Coulomb string tension is an upper bound on the Wilsonian string tension and is typically larger by a factor quoted as $$H = \frac{1}{2} \int d^3 x \left( J^{-1}[A]\, \Pi^a(x)\cdot J[A]\,\Pi^a(x) + B^a(x)\cdot B^a(x) \right) + H_C ,$$3 in the lattice-informed Hamiltonian surveys (Reinhardt et al., 2017).

A particularly explicit continuum realization appears in the heavy-quark limit. Expanding QCD in $$H = \frac{1}{2} \int d^3 x \left( J^{-1}[A]\, \Pi^a(x)\cdot J[A]\,\Pi^a(x) + B^a(x)\cdot B^a(x) \right) + H_C ,$$4 and truncating the Yang–Mills sector to dressed two-point functions only, the rainbow gap equation and ladder Bethe–Salpeter equation become exact within the truncation. If the temporal gluon dressing behaves as

$$H = \frac{1}{2} \int d^3 x \left( J^{-1}[A]\, \Pi^a(x)\cdot J[A]\,\Pi^a(x) + B^a(x)\cdot B^a(x) \right) + H_C ,$$5

then the heavy-quark potential becomes

$$H = \frac{1}{2} \int d^3 x \left( J^{-1}[A]\, \Pi^a(x)\cdot J[A]\,\Pi^a(x) + B^a(x)\cdot B^a(x) \right) + H_C ,$$6

so

$$H = \frac{1}{2} \int d^3 x \left( J^{-1}[A]\, \Pi^a(x)\cdot J[A]\,\Pi^a(x) + B^a(x)\cdot B^a(x) \right) + H_C ,$$7

In the same framework, only color-singlet $$H = \frac{1}{2} \int d^3 x \left( J^{-1}[A]\, \Pi^a(x)\cdot J[A]\,\Pi^a(x) + B^a(x)\cdot B^a(x) \right) + H_C ,$$8 states have finite confined bound-state poles, while nonsinglet channels retain uncanceled infrared divergences; in the $$H = \frac{1}{2} \int d^3 x \left( J^{-1}[A]\, \Pi^a(x)\cdot J[A]\,\Pi^a(x) + B^a(x)\cdot B^a(x) \right) + H_C ,$$9 sector, a finite antisymmetric bound state occurs only for $J[A]=\Det(-\hat D\cdot \partial),$0 (Popovici et al., 2010). This makes precise the often qualitative statement that Coulomb-gauge confinement is encoded in the temporal correlator.

Lattice tests of Coulomb-gauge confinement diagnostics support this picture, but with an important numerical caveat. In $J[A]=\Det(-\hat D\cdot \partial),$1-dimensional $J[A]=\Det(-\hat D\cdot \partial),$2 Yang–Mills theory, a proposed vacuum wave functional reproduces the Coulomb-gauge ghost propagator almost perfectly and yields a linearly rising color-Coulomb potential. The latter is, however, highly sensitive to rare configurations with very small lowest Faddeev–Popov eigenvalue $J[A]=\Det(-\hat D\cdot \partial),$3, so agreement between recursion and Monte Carlo ensembles becomes satisfactory only after matched cuts such as $J[A]=\Det(-\hat D\cdot \partial),$4 are imposed (Greensite et al., 2010). This suggests that the Coulomb-gauge correlator approach is simultaneously physically informative and numerically delicate in observables containing two inverse Faddeev–Popov operators.

The same confinement mechanism is tied in the Hamiltonian reviews to center vortices. Removing center vortices from lattice ensembles makes the ghost form factor infrared flat and removes the Coulomb string tension, while removing only spatial vortices already removes the Coulomb string tension. This is used to argue that the Coulomb string tension is tied to the spatial string tension and that the Gribov-horizon and center-vortex pictures are not competing explanations but overlapping ones (Reinhardt et al., 2016).

3. Quark sector and chiral symmetry breaking

In full QCD, the variational Coulomb-gauge program supplements the gluonic vacuum by a generalized Slater determinant,

$J[A]=\Det(-\hat D\cdot \partial),$5

with kernel

$J[A]=\Det(-\hat D\cdot \partial),$6

The scalar kernel $J[A]=\Det(-\hat D\cdot \partial),$7 generates quark–antiquark pairing, while $J[A]=\Det(-\hat D\cdot \partial),$8 and $J[A]=\Det(-\hat D\cdot \partial),$9 encode quark–gluon correlations in two distinct Dirac structures (Campagnari et al., 2016).

In momentum space, the quark mass function is defined exactly by

$$H_C = \frac{1}{2}\int d^3x\, d^3y\; J[A]^{-1}\rho^a(x)J[A] \left[ (-\hat D\cdot \partial)^{-1}(-\partial^2)(-\hat D\cdot \partial)^{-1} \right]^{ab}(x,y)\rho^b(y).$$0

Variation with respect to $$H_C = \frac{1}{2}\int d^3x\, d^3y\; J[A]^{-1}\rho^a(x)J[A] \left[ (-\hat D\cdot \partial)^{-1}(-\partial^2)(-\hat D\cdot \partial)^{-1} \right]^{ab}(x,y)\rho^b(y).$$1 and $$H_C = \frac{1}{2}\int d^3x\, d^3y\; J[A]^{-1}\rho^a(x)J[A] \left[ (-\hat D\cdot \partial)^{-1}(-\partial^2)(-\hat D\cdot \partial)^{-1} \right]^{ab}(x,y)\rho^b(y).$$2 yields algebraic kernels,

$$H_C = \frac{1}{2}\int d^3x\, d^3y\; J[A]^{-1}\rho^a(x)J[A] \left[ (-\hat D\cdot \partial)^{-1}(-\partial^2)(-\hat D\cdot \partial)^{-1} \right]^{ab}(x,y)\rho^b(y).$$3

$$H_C = \frac{1}{2}\int d^3x\, d^3y\; J[A]^{-1}\rho^a(x)J[A] \left[ (-\hat D\cdot \partial)^{-1}(-\partial^2)(-\hat D\cdot \partial)^{-1} \right]^{ab}(x,y)\rho^b(y).$$4

where

$$H_C = \frac{1}{2}\int d^3x\, d^3y\; J[A]^{-1}\rho^a(x)J[A] \left[ (-\hat D\cdot \partial)^{-1}(-\partial^2)(-\hat D\cdot \partial)^{-1} \right]^{ab}(x,y)\rho^b(y).$$5

The crucial technical result is that the quark gap equation is ultraviolet finite only when both Dirac structures are retained and the ultraviolet Coulomb term is included with the correct sign (Campagnari et al., 2016).

The cancellation pattern is explicit. The UV-divergent contributions induced by $$H_C = \frac{1}{2}\int d^3x\, d^3y\; J[A]^{-1}\rho^a(x)J[A] \left[ (-\hat D\cdot \partial)^{-1}(-\partial^2)(-\hat D\cdot \partial)^{-1} \right]^{ab}(x,y)\rho^b(y).$$6 and $$H_C = \frac{1}{2}\int d^3x\, d^3y\; J[A]^{-1}\rho^a(x)J[A] \left[ (-\hat D\cdot \partial)^{-1}(-\partial^2)(-\hat D\cdot \partial)^{-1} \right]^{ab}(x,y)\rho^b(y).$$7 are

$$H_C = \frac{1}{2}\int d^3x\, d^3y\; J[A]^{-1}\rho^a(x)J[A] \left[ (-\hat D\cdot \partial)^{-1}(-\partial^2)(-\hat D\cdot \partial)^{-1} \right]^{ab}(x,y)\rho^b(y).$$8

and

$$H_C = \frac{1}{2}\int d^3x\, d^3y\; J[A]^{-1}\rho^a(x)J[A] \left[ (-\hat D\cdot \partial)^{-1}(-\partial^2)(-\hat D\cdot \partial)^{-1} \right]^{ab}(x,y)\rho^b(y).$$9

while the Coulomb term contributes

$$\rho^a(x)= -f^{abc}A^b(x)\cdot \Pi^c(x) + \rho_m^a(x).$$0

The linear and logarithmic divergences cancel in the sum (Campagnari et al., 2016). A separate one-loop Hamiltonian perturbative analysis of the static quark and gluon propagators shows that the Hamiltonian equal-time propagators agree with the equal-time limits of the four-dimensional Coulomb-gauge propagators from the functional-integral formalism, and it reproduces the standard one-loop $$\rho^a(x)= -f^{abc}A^b(x)\cdot \Pi^c(x) + \rho_m^a(x).$$1, $$\rho^a(x)= -f^{abc}A^b(x)\cdot \Pi^c(x) + \rho_m^a(x).$$2, and quark contribution to the QCD $$\rho^a(x)= -f^{abc}A^b(x)\cdot \Pi^c(x) + \rho_m^a(x).$$3-function (Campagnari et al., 2014). This places the equal-time quark correlator program on a perturbatively consistent footing.

The quark sector also sharpens the physical role of the Coulomb kernel. Using

$$\rho^a(x)= -f^{abc}A^b(x)\cdot \Pi^c(x) + \rho_m^a(x).$$4

with the choice

$$\rho^a(x)= -f^{abc}A^b(x)\cdot \Pi^c(x) + \rho_m^a(x).$$5

the variational calculation reproduces the phenomenological condensate

$$\rho^a(x)= -f^{abc}A^b(x)\cdot \Pi^c(x) + \rho_m^a(x).$$6

for

$$\rho^a(x)= -f^{abc}A^b(x)\cdot \Pi^c(x) + \rho_m^a(x).$$7

Without coupling to transverse gluons, one obtains only

$$\rho^a(x)= -f^{abc}A^b(x)\cdot \Pi^c(x) + \rho_m^a(x).$$8

while the infrared mass stays nearly unchanged:

$$\rho^a(x)= -f^{abc}A^b(x)\cdot \Pi^c(x) + \rho_m^a(x).$$9

for $$\langle \phi \vert \ldots \vert \psi\rangle = \int \mathcal D A \; J[A]\; \phi^*[A]\ldots \psi[A],$$0 and $$\langle \phi \vert \ldots \vert \psi\rangle = \int \mathcal D A \; J[A]\; \phi^*[A]\ldots \psi[A],$$1 for $$\langle \phi \vert \ldots \vert \psi\rangle = \int \mathcal D A \; J[A]\; \phi^*[A]\ldots \psi[A],$$2. Most importantly, if the linearly rising Coulomb part is removed,

$$\langle \phi \vert \ldots \vert \psi\rangle = \int \mathcal D A \; J[A]\; \phi^*[A]\ldots \psi[A],$$3

the only solution is

$$\langle \phi \vert \ldots \vert \psi\rangle = \int \mathcal D A \; J[A]\; \phi^*[A]\ldots \psi[A],$$4

so there is no spontaneous chiral symmetry breaking (Campagnari et al., 2016). In this formulation, confinement and chiral symmetry breaking are therefore linked by the same infrared Coulomb correlator.

A broader spectroscopy-oriented use of the same Hamiltonian language appears in work on excited baryons. There the running quark mass $$\langle \phi \vert \ldots \vert \psi\rangle = \int \mathcal D A \; J[A]\; \phi^*[A]\ldots \psi[A],$$5 enters the chiral charge explicitly, and in highly excited states one uses the weak expansion

$$\langle \phi \vert \ldots \vert \psi\rangle = \int \mathcal D A \; J[A]\; \phi^*[A]\ldots \psi[A],$$6

to argue for parity doubling when $$\langle \phi \vert \ldots \vert \psi\rangle = \int \mathcal D A \; J[A]\; \phi^*[A]\ldots \psi[A],$$7. In that setting the parity splitting is parameterized as

$$\langle \phi \vert \ldots \vert \psi\rangle = \int \mathcal D A \; J[A]\; \phi^*[A]\ldots \psi[A],$$8

with inferred running mass

$$\langle \phi \vert \ldots \vert \psi\rangle = \int \mathcal D A \; J[A]\; \phi^*[A]\ldots \psi[A],$$9

(Llanes-Estrada, 2010). This does not define a correlator formalism by itself, but it shows how equal-time Coulomb-gauge dressing functions feed directly into hadron spectroscopy.

4. Static-source correlators and interquark energies

One concrete meaning of the Coulomb-gauge correlator approach is the use of gauge-fixed static-source correlators built from timelike Wilson lines. For a static quark at $$\hat F_x^{ab} = \left[-\vec\nabla_x\!\cdot\!\vec D_x^{ac}\right]^{-1} (-\nabla_x^2) \left[-\vec\nabla_x\!\cdot\!\vec D_x^{cb}\right]^{-1}$$0 and antiquark at $$\hat F_x^{ab} = \left[-\vec\nabla_x\!\cdot\!\vec D_x^{ac}\right]^{-1} (-\nabla_x^2) \left[-\vec\nabla_x\!\cdot\!\vec D_x^{cb}\right]^{-1}$$1, separated by

$$\hat F_x^{ab} = \left[-\vec\nabla_x\!\cdot\!\vec D_x^{ac}\right]^{-1} (-\nabla_x^2) \left[-\vec\nabla_x\!\cdot\!\vec D_x^{cb}\right]^{-1}$$2

the connected correlator studied on the lattice is

$$\hat F_x^{ab} = \left[-\vec\nabla_x\!\cdot\!\vec D_x^{ac}\right]^{-1} (-\nabla_x^2) \left[-\vec\nabla_x\!\cdot\!\vec D_x^{cb}\right]^{-1}$$3

with

$$\hat F_x^{ab} = \left[-\vec\nabla_x\!\cdot\!\vec D_x^{ac}\right]^{-1} (-\nabla_x^2) \left[-\vec\nabla_x\!\cdot\!\vec D_x^{cb}\right]^{-1}$$4

In Coulomb gauge, the remnant symmetry $$\hat F_x^{ab} = \left[-\vec\nabla_x\!\cdot\!\vec D_x^{ac}\right]^{-1} (-\nabla_x^2) \left[-\vec\nabla_x\!\cdot\!\vec D_x^{cb}\right]^{-1}$$5 implies

$$\hat F_x^{ab} = \left[-\vec\nabla_x\!\cdot\!\vec D_x^{ac}\right]^{-1} (-\nabla_x^2) \left[-\vec\nabla_x\!\cdot\!\vec D_x^{cb}\right]^{-1}$$6

so the disconnected subtraction vanishes; in Landau gauge the weaker remnant symmetry $$\hat F_x^{ab} = \left[-\vec\nabla_x\!\cdot\!\vec D_x^{ac}\right]^{-1} (-\nabla_x^2) \left[-\vec\nabla_x\!\cdot\!\vec D_x^{cb}\right]^{-1}$$7 leaves a nonzero disconnected term that must be subtracted (Greensite et al., 2018).

The operational assumption is a spectral decomposition

$$\hat F_x^{ab} = \left[-\vec\nabla_x\!\cdot\!\vec D_x^{ac}\right]^{-1} (-\nabla_x^2) \left[-\vec\nabla_x\!\cdot\!\vec D_x^{cb}\right]^{-1}$$8

so that at large $$\hat F_x^{ab} = \left[-\vec\nabla_x\!\cdot\!\vec D_x^{ac}\right]^{-1} (-\nabla_x^2) \left[-\vec\nabla_x\!\cdot\!\vec D_x^{cb}\right]^{-1}$$9 the dominant-state energy $\delta\!\left(\int d^3x\,\hat\rho^a(x)\right),$0 may be read off from the Euclidean-time decay. In the lattice study of $\delta\!\left(\int d^3x\,\hat\rho^a(x)\right),$1 pure gauge theory, the Coulomb-gauge correlator at $\delta\!\left(\int d^3x\,\hat\rho^a(x)\right),$2 on $\delta\!\left(\int d^3x\,\hat\rho^a(x)\right),$3 was fitted, with periodic images included, to

$\delta\!\left(\int d^3x\,\hat\rho^a(x)\right),$4

for $\delta\!\left(\int d^3x\,\hat\rho^a(x)\right),$5 and $\delta\!\left(\int d^3x\,\hat\rho^a(x)\right),$6. The extracted effective string tension was described by

$\delta\!\left(\int d^3x\,\hat\rho^a(x)\right),$7

so that

$\delta\!\left(\int d^3x\,\hat\rho^a(x)\right),$8

in lattice units, in agreement within errors with the known asymptotic $\delta\!\left(\int d^3x\,\hat\rho^a(x)\right),$9 string tension at that coupling (Greensite et al., 2018). The same study emphasizes that this quantity was not identified with the instantaneous color-Coulomb potential in the usual Zwanziger sense, nor with a gauge-invariant Wilson-loop potential. Rather, it was treated as a gauge-fixed operator overlap: the lowest-energy state in the Coulomb-gauge Hamiltonian with nonzero overlap onto the chosen static $A_0=0$00 operator.

The value of the Coulomb-gauge result becomes clearer in the side-by-side comparison with Landau gauge. At $A_0=0$01 on a $A_0=0$02 lattice, the Landau-gauge connected correlator becomes negative for $A_0=0$03 already at $A_0=0$04, and similar positivity violation appears in all displayed $A_0=0$05 plots. At $A_0=0$06 on $A_0=0$07, an apparent intermediate-$A_0=0$08 linear potential survives for finite $A_0=0$09, but the fitted string tension behaves as

$A_0=0$10

and on $A_0=0$11 as

$A_0=0$12

so that $A_0=0$13 as $A_0=0$14 (Greensite et al., 2018). The large-$A_0=0$15 negativity, the strong finite-size dependence, and the collapse of the slope at large Euclidean time are interpreted there as evidence that unphysical states dominate the Landau-gauge correlator. This is one of the sharpest cautionary results in the subject: not every gauge-fixed static correlator is physically interpretable in the same way.

A related lattice program studies the equal-time correlator of temporal links in the generalized Landau gauge

$A_0=0$16

interpolating between Landau gauge ($A_0=0$17) and Coulomb gauge ($A_0=0$18). The equal-time correlator

$A_0=0$19

defines the “instantaneous potential”

$A_0=0$20

and the finite-time terminated-Polyakov correlator

$A_0=0$21

defines

$A_0=0$22

In Landau gauge the instantaneous potential has no linear term, whereas in Coulomb gauge it is well described for $A_0=0$23 fm by

$A_0=0$24

with

$A_0=0$25

so the instantaneous Coulomb-gauge potential is overconfining (Iritani et al., 2010). This was interpreted there through the usual inequality

$A_0=0$26

and by the statement that in Coulomb gauge “the lowest energy state is considered to be a gluon-chain state.”

The same interpolating-gauge study found a special near-Coulomb gauge,

$A_0=0$27

where the instantaneous slope approximately matches the physical one:

$A_0=0$28

Around this point the finite-time potential is unusually stable in $A_0=0$29 (Iritani et al., 2010). This suggests that Coulomb gauge is not the only useful gauge-fixed setting for temporal-link correlators, but it also reinforces a common misconception that the strict Coulomb-gauge equal-time potential is automatically the physical static potential. In the lattice literature it is instead treated as a confining but overconfining instantaneous interaction, while the physically relevant large-$A_0=0$30 quantity requires further projection.

5. Wilson loops, perturbative consistency, and the definition of Coulomb gauge

The relation between Coulomb-gauge correlators and the gauge-invariant Wilson potential is subtle. In the Hamiltonian approach, the temporal Wilson loop may be rewritten using a unitary transformation that shifts the momentum operator by an induced electric field. In QED and in $A_0=0$31-dimensional Yang–Mills theory this reproduces the exact result, while in $A_0=0$32 dimensions one uses variational Coulomb-gauge input for the equal-time gluon propagator $A_0=0$33 and Coulomb kernel expectation value $A_0=0$34. Within the approximations of Jensen’s inequality, “Abelization,” factorization, and neglect of certain commutators, one obtains a Wilson loop dominated at large distance by

$A_0=0$35

so that

$A_0=0$36

The same analysis stresses that this is likely too strong quantitatively, since lattice simulations typically find

$A_0=0$37

and the missing physics was identified with screening by dynamical gluons (Quandt et al., 2013). A closely related Hamiltonian treatment reaches the same qualitative conclusion: Coulomb-gauge equal-time correlators encode the confining backbone, but the true Wilsonian potential requires additional screening dynamics beyond the simplest factorized correlator treatment (Reinhardt et al., 2011).

At the perturbative level, the Coulomb-gauge correlator approach faces a more basic problem: strict Coulomb gauge has energy divergences. A controlled definition is therefore obtained by introducing the interpolating gauge

$A_0=0$38

with Coulomb gauge recovered in the limit $A_0=0$39. In the corresponding Hamiltonian formulation the ghost term is

$A_0=0$40

and the propagators involve the regulated denominator

$A_0=0$41

The gauge is renormalizable for arbitrary $A_0=0$42, but renormalization requires not only multiplicative constants but also field/source mixings, and $A_0=0$43 itself renormalizes according to

$A_0=0$44

An all-orders structural relation emphasized in this framework is

$A_0=0$45

between coupling and ghost renormalization in the given conventions (Andrasi et al., 2021). The conceptual lesson is that Coulomb gauge should be understood as a regulated limit of a BRST-controlled interpolating theory, not as a naively gauge-fixed standalone perturbative system.

The effective-action literature shows why this is necessary. At two loops, individual Coulomb-gauge graphs contain ill-defined energy integrals, but suitable graph combinations reorganize them into the convergent structure

$A_0=0$46

whose integral is finite. This construction was first established for transverse external gluons and later extended, for the two-gluon function, to longitudinal external spatial fields so that the BRST identities involving $A_0=0$47 can be formulated off shell (Andrasi et al., 2014). At three loops, the new issue is the insertion of UV-divergent renormalized subgraphs inside energy-divergent Coulomb-gauge skeletons. Explicit examples with quark-loop subgraphs show that these too can be organized into finite graph sets when manipulated in the interpolating gauge and only then sent to the Coulomb limit (Andrasi et al., 2018). No all-order theorem is supplied, but the result supports a perturbatively consistent notion of Coulomb-gauge Green’s functions as sums of regulated graph combinations rather than as naive individual diagrams.

This entire line of work also modifies the interpretation of the Christ–Lee terms in the Hamiltonian. At two loops the energy-divergent graph sums reproduce the known $A_0=0$48 Christ–Lee operator structure, but at three loops the radiative corrections generated by the same consistency program are not generally instantaneous and therefore do not reduce to simple new local Hamiltonian terms (Andrasi et al., 2018). The implication is that even in Hamiltonian language, multiloop Coulomb-gauge correlators are more intricate than a purely instantaneous picture might suggest.

6. Extensions to finite temperature, hadron structure, and spectroscopy

Once the equal-time Coulomb-gauge correlators are known, the same framework extends to thermodynamics. A particularly effective Hamiltonian method introduces finite temperature by compactifying a spatial dimension,

$A_0=0$49

so that the partition function is governed by the ground-state energy on the compactified space. With a constant background field along the compact direction, the effective potential is

$A_0=0$50

where

$A_0=0$51

Using the zero-temperature variational solution, the survey literature reports that the deconfinement transition is second order for $A_0=0$52 and first order for $A_0=0$53, with critical temperatures

$A_0=0$54

respectively. With quarks included, the transition becomes a crossover, and the pseudo-critical temperatures extracted from the dual and chiral condensates are

$A_0=0$55

for deconfinement and chiral restoration, respectively (Reinhardt et al., 2017).

A much newer extension applies the Coulomb-gauge correlator idea to lattice parton physics. Instead of the gauge-invariant quasi-PDF operator with a Wilson line,

$A_0=0$56

one fixes to Coulomb gauge and studies the equal-time bilinear

$A_0=0$57

from which the quasi-distribution

$A_0=0$58

is constructed. Because there is no Wilson line, the linear power divergence is absent, and the renormalization becomes

$A_0=0$59

with $A_0=0$60 independent of $A_0=0$61. In the hybrid scheme used there,

$A_0=0$62

for the Coulomb-gauge correlator, whereas those quantities are nontrivial in the gauge-invariant case (Gao et al., 2023).

The same work verifies the LaMET factorization formula

$A_0=0$63

at one loop and presents an exploratory pion valence-PDF calculation. The Coulomb-gauge and gauge-invariant extractions differ at the quasi-PDF stage but agree within errors after matching, especially for moderate $A_0=0$64, and the Coulomb-gauge observable shows preserved $A_0=0$65 rotational symmetry for off-axis momenta together with improved long-range precision (Gao et al., 2023). This suggests that the correlator approach is not limited to confinement diagnostics or static potentials; it can also serve as a gauge-fixed alternative to Wilson-line observables in lattice hadron structure.

A broader, less formal extension uses Coulomb gauge as a spectroscopy tool because it is formulated in terms of physical transverse gluons and quarks alone. In that setting, state counting is transparent, parity doubling in highly excited baryons is linked to the momentum dependence of the running quark mass, and heavy-hadron decays are interpreted through a Franck–Condon-like mapping of internal heavy-quark momentum to open-flavor decay products (Llanes-Estrada, 2010). This suggests that the Coulomb-gauge correlator approach is best viewed not as a single method but as a wider program: once gauge fixing has isolated the relevant equal-time structures, those structures can be used to study static forces, confinement mechanisms, thermal transitions, hadron structure, and parts of the excited spectrum within a common noncovariant framework.