Confining Solution of the Dyson-Schwinger Equations in Coulomb Gauge

D. Epple; H. Reinhardt; W. Schleifenbaum

REVIEW

Confining Solution of the Dyson-Schwinger Equations in Coulomb Gauge

D. Epple, H. Reinhardt, W. Schleifenbaum

High Energy Physics - Theory · Submitted 21 Dec 2006 · 13 references

Pith's verdict: A competent numerical companion to the authors' own infrared analysis: it finds the β=1 branch in a full momentum-space solve and gets a linear Coulomb potential, but the exponent extraction has no stability analysis and the potential is algebraically the same statement, not a second result.

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arXiv IDhep-th/0612241 (version 2)

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CitationD. Epple, H. Reinhardt, W. Schleifenbaum. "Confining Solution of the Dyson-Schwinger Equations in Coulomb Gauge." arXiv:hep-th/0612241 (2006). https://pith.science/paper/QW2NZR3K

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Editor's note

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Bottom line: this is the numerical follow-up to the authors' own infrared analysis [5], and it delivers what it promises — the β=1 branch shows up in a full momentum-space solve, and the Coulomb potential comes out linear. If you work in the Coulomb-gauge Hamiltonian program, this is the reference for that solution. If you don't, it won't move your picture of confinement, and the paper doesn't claim it will (abstract:s1 is careful to say the existence was previously observed analytically). That framing is honest and I'd give them credit for it. The method section is genuinely followable: Chebyshev on a log grid, Gauss-Legendre quadrature, relaxation, and asymptotic forms whose exponents and coefficients are fitted rather than inserted. That last choice is the right one, and it's the part of the paper I'd point a student to. The soft spots are real but contained. First, the fitted exponent has no stability study: β₁ and β₂ sit about 25% apart (Eq. 18), the fit runs over a finite window with a finite IR cutoff ε, and nothing shows where α lands when the window or ε moves. The sentence in §III that the method 'provides a check on the analytical results rather than imposing the latter' is asserted, not demonstrated. Second, V(r)=σr is not independent evidence — with d(q)~Bq^β, β=1 makes d²/q² constant and linearity follows algebraically. Fig. 3 right is a consistency plot; the abstract reads slightly as though it were a second result. Third, the shift from [3]'s β₁ to β₂=1 is credited to an 'improved numerical method' whose actual change is never described, which matters precisely because the earlier code landed on the other branch. The 'less than one percent' linearity error is also asserted without a budget. On the reader's weakest-assumption note: the horizon condition is imported from Zwanziger [6], a published result, and used within its stated role. That's a scope assumption, not the load-bearing flaw, and I'd not hold it against them. Send it out. Ask the referees for a fit-window and ε stability table and one sentence on what changed since [3]; with those, it's a clean specialist publication.

Referee report

Significance

If the exponent extraction holds up, this is a useful result: it would establish that the Coulomb-gauge Hamiltonian DSEs admit the confining branch numerically, not only in the infrared power-law analysis, and it would connect the variational approach directly to a linearly rising Coulomb potential and to a non-perturbative running coupling with an infrared fixed point. The methodological description is a genuine strength: the Chebyshev expansion on a logarithmic grid, the Gauss-Legendre quadrature with logarithmic mapping of the radial integrals, the relaxation iteration, and above all the decision to fit the infrared exponents and ultraviolet coefficients as free parameters rather than imposing the analytic forms (Eqs. (22)-(25)) are described in enough detail to be reproduced, and the paper is candid that β₂=1 was predicted in ref. [5]. The value of the paper therefore turns on whether the numerical selection of the branch is demonstrated quantitatively rather than asserted.

Major objections (4)

[§III, Eqs. (22)-(23) and Fig. 1] The central claim of the paper is that the numerics select the β₂=1 branch, yet no numerical value of the fitted infrared exponent is quoted anywhere in the text. The reader is told only that ω(k) and χ(k) 'obey the same infrared divergent behaviour, in agreement with the infrared analysis', and is left to judge from a log-log plot. Since the two analytic branches differ by only about 25% (β₁≈0.796 versus β₂=1, Eq. (18)), a fitted number with an uncertainty is essential: please report the extracted α and β from the fits of Eqs. (22)-(23), together with the fit window and the estimated error, and state explicitly by how many standard deviations the result excludes β₁≈0.796.
[§III, numerical method] The sentence 'this numerical method provides a check on the analytical results (18) rather than imposing the latter' is the load-bearing methodological claim, but it is asserted rather than demonstrated. No stability analysis is presented for the extracted exponent against the infrared cutoff ε, the ultraviolet cutoff Λ, the Chebyshev order, the relaxation parameter r, or the iteration seed. Given that the two branches are separated by only 25%, a power-law fit over a finite window with a finite ε could plausibly land between them, and the reader cannot tell whether it does. A table showing the fitted exponent as a function of fit range and ε would settle this. Relatedly, the shift from the β₁≈0.796 result of ref. [3] to β₂=1 here is credited to an 'improved numerical method' whose specific change is never described; please state what changed, so that it is clear whether the improvement removed a bias or introduced one.
[§III, Eq. (20) and Fig. 3 (right panel)] The linearly rising potential is presented as a second, independent piece of evidence for the confining branch, but it is not independent: with d(q)∼B q^β and β=1, the integrand d²(q)/q² is scale invariant and V(r)∝r follows algebraically from the same exponent. The potential plot and the exponent determination are one claim, not two, and the text should say so. In addition, the strict linearity depends on f(q→0)=const, which the paper states is an input from the one-loop perturbative treatment of the Coulomb form factor rather than a self-consistent output; any residual infrared scale dependence in f would convert the linear term into r times a logarithm. The sensitivity of the linearity to the treatment of f should be stated and, if feasible, tested.
[§III, Fig. 2 and renormalization constants] The paper asserts that the infrared exponents and ultraviolet behaviour are independent of the renormalization constants once the horizon condition is adopted, but Fig. 2 shows ω(k) and d(k) only for three values of ω̄(1), and the exponent extraction is displayed only for ω̄(1)=1.6. Since ω̄(1) is an unfixed parameter of the calculation, the robustness of β=1 across it is load-bearing for the central claim. Please report the fitted exponent for the other values shown (15.6 and 3.1), or state on what basis the independence is claimed.

Minor objections (7)

[Eq. (20)] The Fourier factor in the integrand is written e^{ik·r} while the integration variable is q; it should read e^{iq·r}.
[§III, Fig. 3 (right panel)] The text quotes 'an estimated error of less than one percent' for the linearity of V(r) without saying how this estimate was obtained; please state the procedure, for example the deviation from the linear fit over the fitted range.
[§III, Fig. 1 (left panel)] The left panel plots |χ(k)|, which hides the sign change of χ. Since the zero of χ at k=μ is a renormalization choice (χ̄(1)=0) and the sign structure is used in constructing the fit nodes, a plot showing χ itself, or at least marking the zero, would be clearer.
[Eq. (13)] The bracketing of the factor [ω(q)−χ(q)]^n − [ω(k)−χ(k)]^n over ω(q) is ambiguous as typeset; please check the displayed formula.
[§III, axes and scale] The figures are given in units of σ^{1/2}, but the fitted value of σ and the corresponding physical scale (for instance μ in GeV) are never quoted. Providing them would let the reader compare directly with the lattice values of ref. [12] and with the factor 1.5-3 quoted in the text.
[§IV and ref. [10]] The concluding statement that the intermediate-momentum behaviour will be tested by the 't Hooft loop rests on ref. [10], listed as 'to be published'. Either summarize the relevant result here or soften the forward reference.
[Abstract] The abstract describes a 'confining solution' without mentioning that the horizon condition d^{-1}(0)=0 is imposed as an input. Since the confining properties follow from that condition, a brief qualifier in the abstract would be appropriate.

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The paper

The paper claims

The Coulomb-gauge equations of Yang-Mills theory admit a second, confining solution whose static quark potential rises strictly linearly.

The reading

This paper asks whether the integral equations that follow from minimizing the vacuum energy of Yang-Mills theory in Coulomb gauge have more than one consistent infrared solution, and whether any of them actually confines. An earlier numerical treatment found only the branch with infrared exponent $beta approx 0.796$; a separate analytic study had argued a second branch exists at $beta = 1$. The authors improve the numerical machinery enough to land on that second branch, and report that it produces a static quark potential rising strictly linearly with separation, together with a running coupling that approaches a finite infrared fixed point. The point of interest is that confinement here is a property of the same coupled equations that determine the gluon and ghost propagators, rather than a separate ingredient bolted on afterwards.

What carries the argument

Three ingredients carry the argument. First, the horizon condition $d^{-1}(mu=0)=0$, imported from Zwanziger, forces the ghost exponent $beta$ positive and is what makes the gluon energy infrared divergent. Second, the sum rule $2beta = alpha + 1$, which follows from the non-renormalization of the ghost-gluon vertex, ties the gluon exponent $alpha$ to the ghost exponent $beta$; combined with the gap equation's infrared limit $lim_{kto 0}(omega(k)-chi(k)) = c$, it is what allows $beta_2=1$ to produce a strictly linear potential rather than a merely rising one. Third, the numerical scheme: Chebyshev expansion on a logarithmic momentum grid, double-logarithmic representation for divergent functions, analytic power-law and logarithmic asymptotics used as fit forms whose exponents and coefficients are still determined by nonlinear least squares, Gauss-Legendre quadrature after a logarithmic change of radial variable, and relaxed iteration with $r approx 0.1$. The design point is that the asymptotics are used as fitting templates, not imposed, so the numerically extracted exponents serve as a check on the analytic predictions.

Extended reading notes

Core claim

The central claim is that the one-loop Dyson-Schwinger equations of the Hamiltonian (variational) approach to Yang-Mills theory in Coulomb gauge possess a second numerical solution, with infrared exponent $beta_2 = 1$, in addition to the previously found $beta_1 approx 0.796$. The authors solve the coupled equations for the ghost form factor $d(k)$, the gluon energy $omega(k)$ and the scalar curvature $chi(k)$ over the whole momentum range, imposing the horizon condition $d^{-1}(0)=0$ and the renormalization choices $bar{chi}(1)=0$, $c=0$, with $bar{omega}(1)$ left as a free parameter (their displayed solution uses $bar{omega}(1)=1.6$). In the infrared, $omega$ and $chi$ are found to share the same divergent power law, consistent with the analytic expectation, and $omega(k)-chi(k)$ tends to a constant. Feeding the solution into the Coulomb term of the Hamiltonian for two static colour charges gives a potential that is linear in $r$ to within an estimated error under one percent, and the ghost-gluon running coupling $alpha(k) = frac{2}{3pi},k,d^2(k),omega^{-1}(k)$ is monotonic with an infrared fixed point matching the analytically predicted $alpha(0) = 16pi/(3N_c)$.

Load-bearing premise

The horizon condition $d^{-1}(0)=0$ is imposed as an input from outside rather than derived from the variational principle, and without it the ghost propagator is not enhanced, the exponent is not forced positive, and the confining solution does not follow.

Editorial extensions

If this is right

The variational Hamiltonian approach in Coulomb gauge reproduces a linearly rising static quark potential from its own gap equations, so confinement need not be added as an external assumption about the infrared behaviour of the propagators.
The ghost-gluon running coupling has a finite infrared limit $alpha(0)=16pi/(3N_c)$, so the coupling does not diverge in the infrared in this scheme, which constrains how the beta function can behave at low momentum.
The Coulomb string tension extracted from the linear fit can be compared with lattice determinations, which the authors note are a factor of roughly 1.5 to 3 larger than the Wilson-loop string tension, in line with the analytic statement that the Coulomb tension is an upper bound.
Because the infrared exponents and ultraviolet behaviour are reported to be insensitive to the remaining renormalization constants once the horizon condition is adopted, the qualitative confining picture is claimed to be robust against those choices, with changes confined to intermediate momenta.

Reading between the lines

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

If the $beta=1$ branch is a genuine feature of the untruncated equations rather than of the one-loop truncation and the fitted asymptotic templates, then the earlier numerical search that found only $beta approx 0.796$ was limited by its discretization and not by the physics; a convergence study in grid density and fit range would settle which it is.
The reported insensitivity to $bar{omega}(1)$ suggests a one-parameter family of confining solutions; whether the Coulomb string tension varies across that family is a concrete question the paper's own figures could answer but does not quantify.
The linearity claim is a statement about the Coulomb potential alone, so it is testable against lattice Coulomb-gauge measurements of the same quantity at intermediate separations, where the variational truncation is least controlled.
Since the horizon condition is what drives the whole chain, a useful extension would be to solve the same truncated system with $d^{-1}(0)$ held at small nonzero values and watch the exponent and the potential's slope move, turning an input assumption into a dial.

Original abstract

The Dyson-Schwinger equations arising from minimizing the vacuum energy density in the Hamiltonian approach to Yang-Mills theory in Coulomb gauge are solved numerically. A new solution is presented which gives rise to a strictly linearly rising static quark potential and whose existence was previously observed in the infrared analysis of the Dyson-Schwinger equations. For the new solution we also present the static quark potential and calculate the running coupling constant from the ghost-gluon vertex.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.19 · compares with prior results 0.64 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The Dyson-Schwinger equations arising from minimizing the vacuum energy density in the Hamiltonian approach to Yang-Mills theory in Coulomb gauge are solved numerically. A new solution is presented which gives rise to a strictly linearly rising static quark potential and whose existence was previously observed in the infrared analysis of the Dyson-Schwinger equations. For the new solution we also present the static quark potential and calculate the running coupling constant from the ghost-gluon vertex.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Dyson-Schwinger equations arising from minimizing the vacuum energy density in the Hamiltonian approach to Yang-Mills theory in Coulomb gauge are solved numerically.	method 0.82	0.92	0.08	asserted 1.00
s1	A new solution is presented which gives rise to a strictly linearly rising static quark potential and whose existence was previously observed in the infrared analysis of the Dyson-Schwinger equations.	result 0.72	0.74	0.75	asserted 1.00
s2	For the new solution we also present the static quark potential and calculate the running coupling constant from the ghost-gluon vertex.	contribution 0.51	0.94	0.15	asserted 1.00

What kind of paper the classifier reports this is: simulation study 0.92 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
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Figures

**Figure 1.** FIG. 1: Left panel: the gluon energy

**Figure 2.** shows the dependency of ω(k) and d(k) upon the renormalization constant ω(µ). We find that this constant does not change the infrared exponents and the ultraviolet behaviour but does have impact on the prefactors A, B of the power laws for ω(k), χ(k) and d(k). Also, there are some quantitative changes in the intermediate momentum range

**Figure 3.** FIG. 3: (left) The running coupling

Figures reproduced from arXiv:hep-th/0612241 by the authors.

References

Works cited (13)

[5]

W. Schleifenbaum, M. Leder and H. Reinhardt, Phys. Rev. D 73 (2006) 125019 [arXiv:hep-th/0605115]. 8

View on Pith Pith review arXiv 2006
[3]

C. Feuchter and H. Reinhardt, Phys. Rev. D 70 (2004) 105021 [arXiv:hep-th/0408236]

View on Pith Pith review arXiv 2004
[6]

D. Zwanziger, Nucl. Phys. B 485 (1997) 185 [arXiv:hep-th/9603203]

View on Pith Pith review arXiv 1997
[1]

N. H. Christ and T. D. Lee, Phys. Rev. D 22 (1980) 939 [Phys. Scripta 23 (1981) 970]

View on Pith 1980
[2]

A. P. Szczepaniak and E. S. Swanson, Phys. Rev. D 65 (2001) 025012 [arXiv:hep-ph/0107078]; A. P. Szczepaniak, Phys. Rev. D 69 (2004) 074031 [arXiv:hep-ph/0306030]

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[4]

H. Reinhardt and C. Feuchter, Phys. Rev. D 71 (2005) 105002 [arXiv:hep-th/0408237]

View on Pith Pith review arXiv 2005
[7]

J. C. Taylor, Nucl. Phys. B 33 (1971) 436; A. Cucchieri, T. Mendes and A. Mihara, JHEP 0412 (2004) 012 [arXiv:hep-lat/0408034]; W. Schleifenbaum, A. Maas, J. Wambach and R. Alkofer, Phys. Rev. D 72 (2005) 014017 [arXiv:hep-ph/0411052]

View on Pith Pith review arXiv 1971
[8]

C. S. Fischer and D. Zwanziger, Phys. Rev. D 72 (2005) 054005 [arXiv:hep-ph/0504244]

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D. Atkinson and J. C. R. Bloch, Phys. Rev. D 58 (1998) 094036 [arXiv:hep-ph/9712459]; J. C. R. Bloch, arXiv:hep-ph/0208074; C. S. Fischer and R. Al kofer, Phys. Lett. B 536 (2002) 177 [arXiv:hep-ph/0202202]; A. Maas, Comput. Phys. Commun. 175 (2006) 167 [arXiv:hep-ph/0504110]

View on Pith Pith review arXiv 1998
[10]

H. Reinhardt and D. Epple, to be published

View on Pith
[11]

H. Reinhardt, D. Epple and W. Schleifenbaum, arXiv:hep -th/0610324

View on Pith
[12]

A. Cucchieri and D. Zwanziger, Nucl. Phys. Proc. Suppl. 119 (2003) 727 [arXiv:hep-lat/0209068]; J. Greensite and S. Olejnik, Phy s. Rev. D 67 (2003) 094503 [arXiv:hep-lat/0302018]; K. Langfeld and L. Moyaerts, Phy s. Rev. D 70 (2004) 074507 [arXiv:hep-lat/0406024]

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[13]

D. Zwanziger, Phys. Rev. Lett. 90 (2003) 102001 [arXiv:hep-lat/0209105]. 9

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Forward citations

Cited by 7 Pith papers

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Correlation functions of Landau gauge Yang-Mills theory
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Subcritical solution of the Yang-Mills Schroedinger equation in the Coulomb gauge
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The Coulomb-gauge Yang-Mills Dyson-Schwinger system admits a self-consistent solution only below a critical coupling, because the Coulomb form factor equation has no infrared-critical solution.
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hep-ph 2007-07 conditional novelty 5.0 of 10

The authors identify the fifth dimension of anti-de Sitter space with a light-front separation variable, yielding analytic pion wavefunctions and form factors that match space-like data with one fitted parameter.

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A holographic model identifies the fifth AdS coordinate with the light-front impact separation of quarks, producing analytic hadron wavefunctions that fit spectra and form factors with one or two fitted scales.
The QCD Running Coupling
hep-ph 2016-04 accept novelty 2.0 of 10

A comprehensive review of the QCD running coupling across perturbative and nonperturbative regimes, closing with a proposal that effective charges defined directly from observables give the most unified definition of alpha_s.

How later papers used it

49 later papers cite it

Evidence: some. 16 of the citing papers share no author with it, and 2 of those carry a confident label from Pith's citation classifier.

Pith read 18 of the 124 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 papers by the same authors are listed below: 2 used its data or values.

Uses its data or values 2 of 5 papers with this label · 2 by the same authors

By the same authors

Replaced the running coupling, which this paper calculated from the ghost-gluon vertex, by its infrared value.

Quark Sector of the QCD Groundstate in Coulomb Gauge (2013) (same authors) · 0.87 · paraphrased: the licence does not allow quoting
Noted that the ghost form factor this paper obtained for the alpha=1 solution can be fitted by another work's form.

Vertex functions of Coulomb gauge Yang--Mills theory (2014) (same authors) · 0.92 · paraphrased: the licence does not allow quoting

How the literature uses it

47 arXiv papers cite this in 121 sentences

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Papers citing it · 79 sentences without a confident label

Builds on it 2 papers

In Ref. [ref] the variational Hamiltonian approach to Yang–Mills theory in Coulomb gauge [ref] was extended to full QCD by including the quark-gluon coupling explicitly in the vacuum wave functional.

Variational and Dyson--Schwinger Equations of Hamiltonian Quantum Chromodynamics · Introduction · 0.83
The unperturbed basis constructed from the eigenstates of the unperturbed Hamiltonian, which up to the colour index of the gauge field coincides with the Hamiltonian for QED (without fermions), was generalized to multiple quasi-gluon states on top of the non-perturbative vacuum wave functional used in the variational approach [ref] .

Perturbation theory in the Hamiltonian approach to Yang-Mills theory in Coulomb gauge · Summary and Conclusions · 0.84

Uses its method or tool 3 papers

[ref] and [ref] .

The Yang-Mills Vacuum in Coulomb Gauge in D=2+1 Dimensions · Numerical results · 0.91
In the variational approach developed in Refs. [ref] one attempts to solve the functional Schrödinger equation [equation] for the QCD vacuum state $|⟩$ by means of the variational principle with suitable trial Ansätze for the vacuum wave functional $[A]=⟨A|⟩$.

Dynamical Quark Mass Generation in QCD\textsubscript{3} within the Hamiltonian approach in Coulomb gauge · Vacuum wave functional and quark propagator · 0.86
Technically, the system is solved in the same way as described in [ref] , except for the following differences.

Subcritical solution of the Yang-Mills Schroedinger equation in the Coulomb gauge · Full subcritical solutions · 0.98

Uses its data or values 5 papers

(Within our approximation, this is the only place where the actual form of the vacuum wave functional enters.) The exact analytical expressions are unknown, but we have reliable data from recent variational calculations [ref] , which are also in decent agreement with high precision lattice measurements [ref] .

On the temporal Wilson loop in the Hamiltonian approach in Coulomb gauge · Yang-Mills theory in $(3+1)$ dimensions · 0.94
Fig. 1 shows the result for the dielectric function ([ref]) obtained from the solutions to the Dyson-Schwinger equations reported in [ref] .

Dielectric function of the QCD vacuum · The dielectric function · 0.91
All numerical plots displayed in this section were calculated by D.Epple [ref] .

Nonperturbative aspects of Yang-Mills theory · Full numerical solutions · 0.85

Compares with it 2 papers

In the momentum regime where reliable lattice data are available, there is a reasonable agreement between lattice data [ref] , [ref] and the continuum results [ref] , [ref] , [ref] .

Dielectric function of the QCD vacuum · The dielectric function · 0.92
We can compare these results with the solutions of DSEs obtained in the variational approach in Refs. [ref] .

Color Coulomb Potential in Yang-Mills Theory from Hamiltonian Flows · Introduction · 1.00

Cites it as support 4 papers

Solutions of the Dyson-Schwinger equations for the three-gluon and four-gluon couplings [ref] and phenomenological studies [ref] of QCD couplings based on physical observables such as $τ$ decay [ref] and the Bjorken sum rule [ref] , show that the QCD $β$ function vanishes and $α_s(Q^2)$ become constant at small virtuality; i.e., effective charges develop an “infrared fixed point." Recent lattice simulations [ref] and nonperturbative analyses [ref] have also indicated an infrared fixed point for QCD.

AdS/CFT and Light-Front QCD · Introduction · 1.00
Our variational approach has given a quite decent description of the infrared sector of Yang-Mills theory as, for example, a linearly rising non-Abelian Coulomb potential [ref] , an infrared diverging gluon energy (expressing confinement) [ref] in accord with lattice data [ref] , an infrared finite running coupling constant [ref] , a perimeter law for the 't Hooft loop [ref] , an area law for the Wilson loop [ref] and a dielectric function of the Yang-Mills vacuum in accord with the bag model picture [ref] .

Quark Sector of the QCD Groundstate in Coulomb Gauge · Introduction · 0.97
It is precisely this picture which is supported by the variational calculation in the Hamiltonian approach to Yang-Mills theory in Coulomb gauge developed in Refs. [ref] .

A covariant variational approach to Yang-Mills Theory · Gaussian trial measure · 1.00

Cites it as background 21 papers

Recently progress has been made in variationally solving the Yang-Mills Schrödinger equation in Coulomb gauge [ref] .

The 't Hooft loop in the Hamiltonian approach to Yang-Mills theory in Coulomb gauge · Introduction · 0.82
The Coulomb form factor $f(k)$ measures the ratio of the vev of its square to the square of vevs, [equation] Finally we also consider the expectation value of the curvature [ref] defined by [equation] The DS equations for these correlations functions were independently studied in [ref] and [ref] under different approximation schemes.

Subcritical solution of the Yang-Mills Schroedinger equation in the Coulomb gauge · Variational Coulomb QCD · 0.96
Recently, there has been much activity in the Hamiltonian approach to Yang–Mills theory in Coulomb gauge [ref] .

Perturbation theory in the Hamiltonian approach to Yang-Mills theory in Coulomb gauge · Introduction · 0.91

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [5]

W. Schleifenbaum, M. Leder and H. Reinhardt, Phys. Rev. D 73 (2006) 125019 [arXiv:hep-th/0605115]. 8

View on Pith Pith review arXiv 2006

[2] [3]

C. Feuchter and H. Reinhardt, Phys. Rev. D 70 (2004) 105021 [arXiv:hep-th/0408236]

View on Pith Pith review arXiv 2004

[3] [6]

D. Zwanziger, Nucl. Phys. B 485 (1997) 185 [arXiv:hep-th/9603203]

View on Pith Pith review arXiv 1997

[4] [1]

N. H. Christ and T. D. Lee, Phys. Rev. D 22 (1980) 939 [Phys. Scripta 23 (1981) 970]

View on Pith 1980

[5] [2]

A. P. Szczepaniak and E. S. Swanson, Phys. Rev. D 65 (2001) 025012 [arXiv:hep-ph/0107078]; A. P. Szczepaniak, Phys. Rev. D 69 (2004) 074031 [arXiv:hep-ph/0306030]

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