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Observing Long Colour Flux Tubes in SU(2) Lattice Gauge Theory

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G.S. Bali, K. Schilling, C. Schlichter

High Energy Physics - Lattice · Submitted 7 Sep 1994 · 40 references

Pith's verdict: A solid, methodologically careful lattice study that genuinely images the flux tube; the abstract's '2 fm' is a bit more generous than the cleanest geometric evidence supports.

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CitationC. Schlichter, G.S. Bali, K. Schilling. "Observing Long Colour Flux Tubes in SU(2) Lattice Gauge Theory." arXiv:hep-lat/9409005 (1994). https://pith.science/paper/XZ47BNKE
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The paper

The paper claims

A lattice SU(2) simulation traces the confining colour flux tube between two static quarks out to separations of about two femtometres, and finds its transverse width levelling off near one femtometre.

The reading

The paper asks a geometric question about confinement: when a static quark and antiquark are held apart, does the colour field between them actually take the shape of a tube, or does the linearly rising potential come from something with no tube-like structure? Working in pure $SU(2)$ lattice gauge theory on volumes up to $48^3$ (times) $64$ with lattice spacings down to about $0.04$ fm, the authors measure the action and energy density of the chromoelectric and chromomagnetic field in the presence of the pair, using smeared Wilson loops to prepare the ground state and analytically integrated timelike links to reduce noise. They report that the flux distribution persists as a tube over separations as large as $2$ fm, that its transverse profile scales between two lattice spacings, and that the integrated densities satisfy Michael's sum rules, which tie the local field density to the derivative of the static potential with respect to the coupling. If this holds, the dual-superconductor picture of confinement acquires direct geometric support at distances far beyond where effective string models had previously been tested, and the confining field can be treated as an object with a measurable width rather than as a bookkeeping device for the potential.

What carries the argument

The load-bearing object is the correlator between a smeared Wilson loop of spatial extent $R$ and temporal extent $T$ and a single plaquette inserted at a distance $S$ from the loop's central time slice, written $<plaq(S)>_W$ and normalised by subtracting the vacuum plaquette expectation. Two techniques make it usable at large $R$. The first is an iterative spatial smearing of the parallel transporter in the quark-antiquark creation operator, with $150$ iterations and $alpha=2$, which raises the ground-state overlap to between $0.72$ and $0.98$ depending on separation. The second is an analytic integration of the timelike links of the Wilson loop against the neighbouring staples, exploiting the fact that $SU(2)$ integrals of this form give modified Bessel function ratios; the authors estimate this reduces the computer time needed for a given error by a factor of about $17$ at $T=6$. The spectral decomposition of the correlator, organised by the string-motivated level spacings $pi T/R$ and $2pi T/R$, supplies the fit form used to extrapolate $T$ to infinity. A separate argument, based on a $Z_2$ centre transformation of spatial links crossing a plane, shows that mirror-source contributions with odd winding are projected out, which is what allows tubes of length up to three quarters of the spatial extent to be measured without periodic contamination.

Extended reading notes

Core claim

The central claim is that string formation is established over physical distances as large as $2$ fm in quenched $SU(2)$ gauge theory. The evidence is threefold. First, the action density is traced as a tube along up to thirty lattice sites at $R=30$ on the $48^3$ (times) $64$ lattice, which corresponds to roughly $1.7$ fm. Second, a differential diagnostic is introduced: the change in action density under stretching the sources from $R-2$ to $R$ becomes confined to the centre plane between the quarks for $R$ around $9$ and above, whereas at smaller separations a measurable flow of action into the centre plane from neighbouring planes remains, as a dipole picture would predict. Third, the numerically integrated centre-plane action density, rescaled by the measured anomalous dimension, agrees with the force obtained from the static potential for $R+1$ of $6$ and above, which is the differential form of Michael's action sum rule. As a by-product the authors extract the static potential with high statistics and fit it with three- and four-parameter forms, obtaining string tension values at three couplings and an interpolating determination of the lattice $\beta$ function over $2.4$ to $2.85$.

Load-bearing premise

The field distributions at infinite temporal extent are obtained by fitting a specific spectral form, with excitation gaps taken as multiples of $pi/R$ from string models, to data whose ground-state overlap is only about $0.72$ at the largest separation.

Editorial extensions

If this is right

  • The confining field between static sources can be treated as an extended object with a measurable transverse profile, not merely as the derivative of a linearly rising potential, at separations up to $2$ fm.
  • The transverse mean-square width of the action density rises with separation and then saturates at a roughly constant level between $1$ and $2$ fm, with a half width near $0.7$ fm, which bounds the ultraviolet cutoff of any effective string description to be above about $1.3$ GeV.
  • Michael's action sum rule, which relates the integrated local action density to $-\partial V/\partial \ln \beta$, holds against the data over the range where the differential form is applicable, providing an independent consistency check on the extracted field distributions.
  • The same measurement machinery, smearing plus link integration, extends to multi-source problems such as three static quarks or two quark-antiquark pairs, and to studies of confinement in gauges where monopoles are made explicit.

Reading between the lines

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

  • If the tube width genuinely saturates rather than growing logarithmically, that would be in tension with the Gaussian-fluctuation picture of a quantum string, which predicts a logarithmic growth; the authors' data are consistent with either, so a wider range of $R$ at fixed lattice spacing would discriminate between a constant and a log.
  • The centre-group projection argument that suppresses mirror sources is general and would apply to any Wilson-loop-based measurement of local operators in $SU(2)$, which suggests the same trick could rescue finite-volume studies at smaller spatial extents than the $1.3$ fm used here.
  • Because the energy density is argued to be better described by perturbation theory than the action density, the two densities should be reported separately in any future flux-tube study rather than combined into a single action-like quantity.
Original abstract

We present results of a high statistics study of the chromo field distribution between static quarks in SU(2) gauge theory on lattices of volumes 16^4, 32^4, and 48^3*64, with physical extent ranging from 1.3 fm up to 2.7 fm at beta=2.5, beta=2.635, and beta=2.74. We establish string formation over physical distances as large as 2 fm. The results are tested against Michael's sum rules. A detailed investigation of the transverse action and energy flux tube profiles is provided. As a by-product, we obtain the static lattice potential in unpreceded accuracy.

The compiled record

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We present results of a high statistics study of the chromo field distribution between static quarks in SU(2) gauge theory on lattices of volumes 16^4, 32^4, and 48^3*64, with physical extent ranging from 1.3 fm up to 2.7 fm at beta=2.5, beta=2.635, and beta=2.74. We establish string formation over physical distances as large as 2 fm. The results are tested against Michael's sum rules. A detailed investigation of the transverse action and energy flux tube profiles is provided. As a by-product, we obtain the static lattice potential in unpreceded accuracy.

background contribution method result interpretation main claim

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We present results of a high statistics study of the chromo field distribution between static quarks in SU(2) gauge theory on lattices of volumes 16^4, 32^4, and 48^3*64, with physical extent ranging from 1.3 fm up to 2.7 fm at beta=2.5, beta=2.635, and beta=2.74. method 0.61 0.95 0.07 asserted 0.97
s1 We establish string formation over physical distances as large as 2 fm. result 1.00 0.96 0.73 asserted 1.00
s2 The results are tested against Michael's sum rules. method 0.92 0.80 0.04 asserted 0.91
s3 A detailed investigation of the transverse action and energy flux tube profiles is provided. contribution 0.96 0.94 0.05 asserted 0.96
s4 As a by-product, we obtain the static lattice potential in unpreceded accuracy. result 0.99 0.96 0.09 asserted 1.00

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

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Figures
Figure 1. Visualization of a smearing iteration. In the present simulation an iterative procedure (with niter iteration steps) has been applied [20, 21]: each spatial link Ui(n), occurring in the transporter, is substituted by a “fat” link, Ui(n) −→ N  αUi(n) + X j6=i Uj (n)Ui(n + ˆj)U † j (n +ˆi)   (13) with the appropriate normalization constant N and free parameter α. One such it￾eration step is visualized in
Figure 2. The ground state overlaps C0(R) versus R at β = 2.635. In addition to the overlaps of smeared operators (diamonds), finite T approximants, C0(R, T) for the overlaps from unsmeared (on-axis) operators are plotted. The dashed line denotes the extrapolated large T limit for the unsmeared overlaps and should be a valid approxima￾tion to the large R behaviour. for large R and T and arrive at the estimate (with constants V0 and e obtained from the potential analysis below), C0(R, T) = De−(V0−e/T)R or C0(R) = De−V0R . (18) This parametrization of the unsmeared overlaps turns out to…
Figure 3. Comparison between continuum and lattice dipole fields in the center plane between two sources, separated by R = 12. The ordinate, n, is the distance from the QQ¯ axis. The values have been multiplied by a factor (4π) 2/(CF g 2 ). Crosses and the solid line correspond to the infinite volume results. Squares and the dashed line are obtained at finite volume, R/LS = 12/32. In the continuum limit one finds the expression (from Eq. 90), ǫ (c) R (0, n⊥) = g 2CF 1 (4π) 2 R2 (R2/4 + n 2 ⊥) 3 , (32) for the energy density distribution in the central transverse plane. In Figs. 3
Figure 5. V (R, T) as a function of T for R = 20, R = 10, R = 5 and R = 2, respectively, at β = 2.635. The plateau values obtained at T = 5 are indicated by the horizontal dashed lines. To prepare the stage for the colour flux investigations, we had to calculate the static QQ¯ potential. It will render the string tension, which serves to fix the physical scale and relates the results, obtained at different β values to each other. 4.1.1 Data The potential data has been obtained by the method described in Section 2.3. Finite T approximants V (R, T) and C0(R, T) to the ground state potential…
Figure 6. The potential, measured on the four lattices, scaled in units of the string tension. The solid line refers to the string picture expectation V (R) = KR−π/(12R). R Path V (R) C0(R) R Path V (R) C0(R) 1.00 1 .3356( 2) .979(1) 4.90 5 .6671(10) .927(3) 1.41 2 .4292( 4) .975(1) 5.00 1 .6706( 9) .930(3) 1.73 4 .4735( 5) .970(2) 5.20 4 .6817(15) .928(5) 2.00 1 .4844( 4) .963(1) 5.66 2 .7012(14) .921(4) 2.24 3 .5137( 4) .963(2) 6.00 1 .7137(11) .915(3) 2.45 5 .5330( 5) .960(2) 6.00 6 .7157(13) .916(4) 2.83 2 .5585( 7) .954(2) 6.71 3 .7418(13) .902(4) 3.00 1 .5660( 6) .956(2) 6.93 4…
Figure 7. Λ−1 MS(a) √ κ versus the lattice spacing a √ κ, both measured in units of the string tension. The estimates for the Λ values have been obtained from the perturbative two loop formula by use of the bare coupling (upper values) and the βE scheme (lower values). Linear fits to the data are indicated. A cut-off parameter ΛE(a) from the effective coupling βE = 3 4 (1 − h✷i) −1 , introduced in Ref. [30], has also been computed. This cut-off parameter is translated into the MS scheme by the relation ΛMS(a) = 11.51ΛE(a). As can be seen, the slope is substantially reduced but asymptotic…
Figure 8. The energy density distribution at β = 2.5, R = 8 (r ≈ .7 fm) in units of the string tension. We are now in the position to present a survey on the flux distributions and watch the formation of flux tubes with increasing distance between the static sources. In Figs. 8 and 9, we display the situation at β = 2.5 and Ra = 8a ≈ .7 fm, for the energy and action densities, respectively, in units of the string tension. Notice, that the mesh is not equidistant in the perpendicular direction because the off-axis separations n⊥ ∝ (1, 1) are included. We confirm the earlier observation [3]…
Figure 9. The action density distribution at β = 2.5, R = 8 (r ≈ .7 fm) in units of the string tension. vertical axis of
Figure 10. The action density distribution at β = 2.635, R = 12 (r ≈ .7 fm) in units of the string tension. Relative to
Figure 12. The action density distributions for quark separations R = 20 at β = 2.5 and R = 30 at β = 2.635, corresponding to r ≈ 1.7 fm. The z axis of the first plot is expanded by the ratio of the two B˜(β) values in respect to the second plot to account for the anomalous dimension. tain the corresponding continuum expressions. This is beyond the scope of the present paper, where we are mainly interested in non-perturbative large distance effects. 4.2.3 Finite size effects Lattice results for the heavy quark potential and colour flux distributions are subject to finite size effects (FSE).…
Figure 13. Comparison of longitudinal action density profiles at r = .7 fm between the β = 2.5 data (R = 8) and the β = 2.635 data (R = 12) at n⊥a = .17 fm in units of the string tension. The vertical axis has been multiplied by B˜(β). One source is placed at position (n1, n⊥) = 0. The second source is located at n1 √ K ≈ 1.6 (outside the visible range). simulation we are able to confirm this observation by comparing the 164 and 324 potential data at β = 2.5. In addition one might worry about the impact of mirror sources, due to the toroidal structure of the lattice: if one places sources at t
Figure 14. Comparison of transverse action density profiles at the center plane of two sources, separated by r = .7 fm between the β = 2.5 data (R = 8) and the β = 2.635 data (R = 12) in units of the string tension. The vertical axis has been multiplied by B˜(β). However, in weak coupling, perturbation theory yields W(R, T) = W(LS −R, LT −T) in the non zero momentum sector. At least to the lowest order (O(g 2/(L 3 SLT ))) the zero modes do not obey this behaviour [33]. Their influence might become even more important to higher orders, especially in the infra red regime of large R. The na¨ıve
Figure 15. Differences between the longitudinal action density profiles measured on 164 and 324 lattices at β = 2.5 for R = 8 and n⊥ = 2. -0.002 -0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 -6 -4 -2 0 2 4 6 V=16^4 V=32^4 difference
Figure 16. Differences between the transverse action density profiles measured on 164 and 324 lattices at β = 2.5 for R = 8 and n1 = 0. 32
Figure 17. Comparison between smeared and unsmeared finite T energy density ap￾proximants in the center (n1 = n⊥ = 0) between two sources, separated by R = 4 at β = 2.74. The horizontal axis is the time T. 4.2.4 T-stability In the two preceding subsections limitations in the lattice geometry have been discussed to substantiate the relevance of our lattice results to continuum physics. Here, we address the reliability of our ground state results in view of the (necessarily) limited temporal extent of the lattice operators. We will explain in some detail how the (T → ∞) results shown above have…
Figure 18. Finite T approximants to the electric plaquette expectation value E6(0, 3) in presence of two quarks (diamonds), separated by R = 6 at β = 2.5 at the position n1 = 0, n⊥ = 3 in lattice units. Two-, three- and four-parameter fits are indicated, together with the corresponding extrapolated asymptotic values (rightmost points with error bars). 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0 1 2 3 4 5 6 ex B^2 2-par 3-par 4-par
Figure 20. The central transverse energy density profile at R = 2, β = 2.5 together with fit curves of methods (1) and (4). R method c1 c2 δ A χ 2 /NDF 2 1 — 0.043 (1) 1 0.0223(2) 16.0 2 2 0.0711( 8) — 1 0.0356(4) 17.4 2 3 0.0296( 8) 0.0252(5) 1 0.0280(5) 15.6 2 4 0.0570(10) — 0.90(1) 0.0354(8) 1.6 2 6 0.0310( 3) — 0.65(1) 0.0368(5) 7.4 4 1 — 0.082( 2) 2 0.0142( 3) 1.2 4 2 0.162(3) — 2 0.0205( 4) 5.5 4 3 0.017(3) 0.074( 2) 2 0.0150( 6) 1.2 4 4 0.065(7) — 1.55(5) 0.0136(17) 0.9 4 5 0.001(2) 0.077( 8) 1.09(1) 0.0134(14) 1.3 4 6 0.023(2) — 1.01(3) 0.011 ( 1) 1.6 4 7 0.011(4) 0.040(10) 1.03(6)…
Figure 21. The central transverse energy density profile at R = 4, β = 2.5 together with a fit to the lattice expression, fl (method (1)). R method c1 c2 δ A χ 2/NDF 2 1 — 0.0271( 5) 1 0.0143( 3) 3.7 2 2 0.0446( 8) — 1 0.0223( 4) 1.3 2 3 0.0383( 8) 0.0039( 5) 1 0.0211( 5) 1.3 2 6 0.0230( 9) — 0.83(3) 0.0160(10) 1.6 4 1 — 0.052( 1) 2 0.0093(2) 1.0 4 2 0.102( 2) — 2 0.0128(3) 3.3 4 3 0.019( 2) 0.043( 1) 2 0.0101(4) 0.9 4 4 0.045( 5) — 1.59(5) 0.009 (1) 0.5 4 5 0.040(10) 0.010(10) 1.59(6) 0.010 (3) 0.5 4 6 0.016( 2) — 1.03(3) 0.008 (1) 0.9 4 7 0.009( 3) 0.025( 8) 1.06(5) 0.007 (2) 0.5 6 1 —…
Figure 23. The central transverse action density profile at R = 8, β = 2.5 together with fit curves of methods (4) and (6) (dipole and Gauss). 1. c2fl(n⊥), 2. c1fd(n⊥) with δ = R/2, 3. c1fd(n⊥) + c2fl(n⊥) with δ = R/2, 4. c1fd(n⊥; δ), 5. c1fd(n⊥; δ) + c2fl(n⊥), 6. c1fg(n⊥; δ), and 7. c1fg(n⊥; δ) + c2fl(n⊥). The stable fit results are collected in Tabs. 8–11 which also contain the integrated area, A = c1 2δ 2  + 2c2 R2  . (62) The formula is only exact in the infinite volume limit. The second term has been corrected by numerical computations of the corresponding lattice sums. In c
Figure 25. Differences, ∆σR, between the action density distributions at R and R−2 for R = 6, 8, and 10 (β = 2.5). The sources have been aligned. The labelling corresponds to lattice units. 46
Figure 26. The QQ¯ force, obtained from the potential at β = 2.5, compared to the integrated center plane action density, scaled by the anomalous dimension B˜ in lattice units. In addition to the numerically integrated data (num), the dipole and Gauss fit results from Section 5.1 are displayed. Within the string regime, Ra > .75 fm, one can write a differential version of the sum rules: F( √ R2 − 1) = 1 2 (V (R − 1) − V (R + 1)) (63) = − a 3 2 X n2,n3 (ǫR−1(0, n2, n3) + ǫR+1(0, n2, n3)) + O(β −1 ) (64) ≈ −aπ Z xcut 0 dx x (ǫR−1(0, x) + ǫR+1(0, x)) . (65) For the second equality we
Figure 27. The width of the flux tube, δ, against R at β = 2.5 (in lattice units). The dashed line corresponds to δ = R/2. In addition to the numerically integrated results from the energy and action density distributions, fit results to the action density from Tab. 10, and results from the geometric method (with γ=1) are displayed. Ref. [3] and is consistent with our fit results (Tabs. 8 and 10). An alternative approach to study the functional dependence, δ(R), is to constrain the center plane analysis to the results of the differential action sum rule. In addition, we apply a geometric…
Figure 28. The width of the flux tube, δ, against R in units of the string tension, obtained by use of the geometric method (γ = 1) for the energy and action density distributions at β = 2.5 and β = 2.635. The vertical line indicates the lower limit of applicability of the geometric method. The dashed-dotted curve is the string picture expectation (Eq. 37) for R0 √ K = 1/3, δ0 q K/γ = 0.5. A is the area below the curve. It can be fixed by the sum rules with high accuracy. For the action density we obtain A from A = B˜−1 (β)(V (R) − V (R − 1)) . (69) Note, that here we have taken an asymmetr
Figure 29. The half widths of the energy and action flux tubes at β = 2.5 and β = 2.635, ρ, against R in units of the string tension. The line corresponds to the small R dipole expectation. of the interpolating curve. We observe nice scaling between both β values. We also confirm the width of the energy flux tube to be smaller than the width of the action flux tube for distances below .5 fm. Both densities increase till r ≈ 1.1 fm. The action density saturates at the level ρ ≈ .7 fm. We conclude that the data beyond 1 fm is in agreement with a constant but does not contradict the expected…

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References
How later papers used it

61 later papers cite it

Evidence: substantial. 61 of the citing papers share no author with it, and 6 of those carry a confident label from Pith's citation classifier.

Pith read 16 of the 81 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

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45 arXiv papers cite this in 56 sentences

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Papers citing it · 39 sentences without a confident label

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Cites it as background 11 papers
  • Extensive foundational studies on flux tubes in QCD have established a robust framework, encompassing theoretical analyses and lattice QCD simulations [ref] .

    Machine Learning Insights into Quark-Antiquark Interactions: Probing Field Distributions and String Tension in QCD · Introduction · 0.98

  • It is known since long that, in lattice numerical simulations, tubelike structures emerge by analyzing the chromoelectric fields between static quarks [ref] .

    Flux tubes in QCD with (2+1) HISQ fermions · Introduction · 0.82

  • A large amount of numerical investigations of flux tubes has accumulated in SU(2) and SU(3) Yang-Mills theories [ref] mostly aimed at studying the shape of the chromoelectric field on the transverse plane at the midpoint of the line connecting the static quark and antiquark, given that the other two components of the chromoelectric field and all the three components of the chromomagnetic field are suppressed in that plane.

    The flux tube profile in full QCD · Introduction · 0.98

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