The nonperturbative color field in the SU(3) flux tube

Using lattice Monte Carlo simulations of pure SU(3) gauge theory, we determine the spatial distribution of all components of the color fields created by a static quark and antiquark. We identify the components of the measured chromoelectric field transverse to the line connecting the quark-antiquark pair with the transverse components of an effective Coulomb-like field $\vec{E}^C $ associated with the quark sources. Subtracting $\vec{E}^C$ from the total simulated chromoelectric field $\vec{E}$ yields a non-perturbative, primarily longitudinal chromoelectric field $\vec{E}^{NP}$, which we identify as the confining field. This is the first time that such a separation has been effected, creating a new tool to study the development of the QCD flux tube as the quark and antiquark move apart.

Introduction.-Quantumchromodynamics (QCD), the theory of the strong interactions describing the dynamics of quarks and gluons, has yet to provide a theoretical explanation of the experimentally established phenomenon of confinement, i.e., the confinement of quarks and gluons inside hadrons.Several mechanisms of confinement have been proposed (for a review, see Refs.[1,2]), each with its own merits and limitations, but a comprehensive picture is still missing.In particular, it is not yet clear which feature of QCD is responsible for the area-law behavior of Wilson loops that implies a linear confining potential between a static quark and antiquark at large distances.Results from numerical simulations have shown this linear potential for quark-antiquark distances 0.5 fm, and up to distances of about 1.4 fm in presence of dynamical quarks, where the string breaking should take place [3][4][5].
Monte Carlo simulations of QCD on a space-time lattice provide a powerful tool with which to investigate the nonperturbative properties of QCD.A wealth of numerical analyses of SU (2) and SU(3) Yang-Mills theories  have found that the dominant color field generated by a static quark-antiquark pair is the component of the chromoelectric field along the line connecting the pair.(See, in particular, the SU(2) studies of Ref. [15].)This longitudinal field results in tube-like structures (flux tubes) that naturally give rise to a long-distance linear potential between the quark and antiquark [29][30][31][32].
The aim of this paper is to obtain further insight into the color field distributions generated by a static quark-antiquark pair.To do this, we first perform a series of new simulations in pure SU(3) gauge theory, measuring all six components of the color field on all transverse planes passing through the line between the quarks.We find that the chromomagnetic field is everywhere much smaller than the chromoelectric field.We then fit the measured transverse components of the chromoelectric field to an effective Coulomb-like field generated by sources at the positions of the quarks.A nonperturbative, mostly longitudinal, chromoelectric field is then obtained by subtracting the effective Coulomb-like field from the total chromoelectric field.To our knowledge, this is the first time such a separation has been carried out.Analysis of this nonperturbative chromoelectric field provides new understanding of the structure of confining flux tubes.
After the definition of the lattice observables used to extract the field strength tensor of the static quark-antiquark sources and the description of our lattice setup, we illustrate the subtraction procedure to extract the nonperturbative field and present the numerical results of our analysis.Theoretical background and lattice observables.-Thefield configurations generated by a static quark-antiquark pair can be probed by calculating on the lattice the vacuum expectation value of the following connected correlation function [10,11,33,34]: Here U P = U µν (x) is the plaquette in the (µ, ν) plane, connected to the Wilson loop W by a Schwinger line L, and N is the number of colors (see Fig. 1).The correlation function defined in Eq. ( 1) measures the field strength, since in the naive continuum limit [11] ρ conn where q q denotes the average in the presence of a static q q pair and 0 is the vacuum average, which is expected to vanish.This leads to the following definition of the quarkantiquark field strength tensor: In the particular case when the Wilson loop W lies in the plane with μ = 4 and ν = 1 (see Fig. 1 (left)) and the plaquette U P is placed in the planes 41 , 42 , 43 , 23 , 31 , 12 , we get, respectively, the color field components E x , E y , E z , B x , B y , B z , at the spatial point corresponding to the position of the center of the plaquette, up to a sign depending on the orientation of the plaquette.For symmetry reasons, at spatial points connected by rotations around the axis on which the sources are located (the 1or x-axis in the given example) the color fields take the same value.
As far as the color structure of the field F µν is concerned, we observe that the Wilson loop connected to the plaquette is the source of a color field which points, on average, in an unknown direction n a in color space determined by the Wilson loop itself.We thus measure the average projection of the color field onto that direction.The role of the Schwinger lines entering the definition ( 1) is to realize the color parallel transport between the source loop and the "probe" plaquette.Therefore, the color fields F µν appearing in Eq. ( 3), should be understood as n a F a µν , This relation is a necessary consequence of the gauge-invariance of the operator defined in (1) and of its linear dependence on the color field in the continuum limit.An explicit verification of the latter property was exhibited in Ref. [35] (see Fig. 3 there).
Lattice setup and numerical results.-Weperformed all simulations in pure gauge SU(3), with the standard Wilson action as the lattice discretization.We made use of the publicly available MILC code [36], suitably modified in order to introduce the relevant observables.A summary of the runs performed is given in Table I.To allow for thermalization we typically discarded a few thousand sweeps.The error analysis was performed by the jackknife method over bins at different blocking levels.We set the physical scale for the lattice spacing by using the value √ σ = 420 MeV for the string tension, and the parameterization [37] for a √ σ that gave an accurate fit in a high-statistics simulation for all β in the range 5.6 ≤ β ≤ 6.5.
The correspondences between β and the distance d shown in Table I were obtained from this parameterization.
The connected correlator defined in Eq. ( 1) suffers from large fluctuations at the scale of the lattice spacing, which are responsible for a bad signal-to-noise ratio.To extract the physical information carried by fluctuations at the physical scale (and, therefore, at large distances in lattice units) we smoothed out configurations by the smearing procedure.Our setup consisted of (just) one step of HYP smearing [38] on the temporal links, with smearing parameters (α 1 , α 2 , α 3 ) = (1.0,0.5, 0.5), and N APE steps of APE smearing [39] on the spatial links, with smearing parameter α APE = 0.25.Here α APE is the ratio between the weight of one staple and the weight of the original link.
As noted in the Introduction, we have determined the six components of the color field on all possible two-dimensional planes transverse to the line joining the color sources which are allowed by the lattice discretization, by Monte Carlo evaluations of the expectation value of the operator ρ conn W, µν over smeared ensembles.Such comparison was carried out for three values of the distance d between the static sources, at values of β lying inside the continuum scaling region, as determined in Ref. [27].We find that the chromomagnetic field is everywhere much smaller than the longitudinal chromoelectric field and is compatible with zero within statistical errors.Here, we plot in Fig. 2 the components of the simulated chromoelectric field E at β = 6.240 as a function of its longitudinal displacement from one of the quarks, x l , and its transverse distance from the axis, x t .
While the transverse components of the chromoelectric field are also smaller than the longitudinal components, they are larger than the statistical errors in a region wide enough that we can match them to the transverse components of an effective Coulomb-like field E C ( r) produced by two static sources.
Remarkably, apart from the field very close to the quarks this matching can be carried out with a single fitting parameter, the value of their effective charges Q and −Q.We interpret E C ( r) as the effective field of the quark sources.To the extent that we can fit the transverse components of the simulated field E to those of E C ( r) with an appropriate choice of Q, the nonperturbative difference E N P between the simulated chromoelectric field E and 0 0.5 1 FIG. 2. Surface and contour plots for the three components of the chromoelectric field at β = 6.240 and d = 1.142 fm.All plotted quantities are in physical units.0 0.5 1 (5) will be purely longitudinal.We then identify E N P as the confining field of the QCD flux tube.
Our goal is to carry out lattice simulations for a sufficient range of quark-antiquark distances to begin to see the formation and development of the proposed flux tube field, Evaluation of the nonperturbative color field.-Toextract the longitudinal component of the confining field E N P , ( 5), from lattice simulations, we must first determine the effective charge of the sources, Q, by fitting the transverse components of the simulated field to those of an effective Coulomb field E C ( r): where r Q and r −Q are the positions of the two static color sources and R 0 is the effective radius of the color source, introduced to, at least partially, explain the decrease of the field close to the sources.Due to the axial symmetry around the line connecting the static charges1 , without loss in generality we may consider the color field distributions in the x y plane.Then x ≡ x l , y ≡ x t .
Remarkably, we find that with an appropriate choice of Q the y-component of the simulated chromoelectric field, E y , is approximately equal to the y-component of the Coulomb field, E C y , at least not too close to the quarks (i.e., at distances greater than 1-2 lattice spacings).In making the fit we must take into account that the color fields are probed by a plaquette, so that the measured field value should be assigned to the center of the plaquette.This also means that the z-component of the field is probed at a distance of 1/2 lattice spacing from the x y plane, where the z-component of the Coulomb field E C z is nonzero and can be matched with the measured value E z for the same value of Q.
In Table II, we list the values of the effective charge Q obtained from lattice measurements of E z and E y at three values of d, the quark-antiquark separation.The uncertainties on the .
quoted Q values result from the comparison among Coulomb fits of E y and E z at the values of x l , for which we were able to get meaningful results for the fit.The stability of Q under change of the fitting strategy, its dependence on the values of x l included in the fit and the global assessment of the systematic uncertainties will be presented in a forthcoming extended version of this work.The values of R 0 in physical units grow with the growth of the lattice step a, while in lattice units they show more stability.This suggests that the effective size of a color charge in our case is mostly explained by lattice discretization artifacts and the smearing procedure, and is not a physical quantity.
Once the charge Q determining the Coulomb field has been fixed from the fits of the transverse fields E y and E z , we subtract the longitudinal component of the Coulomb field from the measured longitudinal chromoelectric field E x to determine E N P according to Eq. (5).In this way we obtain the nonperturbative structure of the flux tube.To the best of our knowledge, this is the first time that a confining part of the measured longitudinal chromoelectric field has been extracted making use only of lattice data.
In Fig. 3 we display the shape of E N P x as a function of the longitudinal and transverse displacements x l , x t at β = 6.240.As expected, the nonperturbative longitudinal chromoelectric fields is almost uniform along the flux tube, at least for distances not too close to the static color sources.This feature is better seen in Fig. 4, where cross sections of the nonperturbative longitudinal field are shown, for all possible values of x l , except too close to the sources, in the same lattice setup as for Fig. 3.The shape of the nonperturbative longitudinal field is basically the same along the axis connecting them.Though figures refer only to the case of β = 6.240 and d = 1.142 fm, the scenario is the same for the other two lattice setups listed in Table I.
In Table III  x l = 3.5 a x l = 4.5 a x l = 5.5 a x l = 6.5 a x l = 7. x (x l , x t = 0) assumes these same values at all points x l on the axis for all distances larger than approximately 0.1 − 0.2 fm from the quark sources.(25) .
The value of the chromoelectric field at the position of the quarks is equal to the force on the quarks, which in turn is equal to the string tension [40].Then, according to this relation, along with the value σ = (420 MeV) 2 ≈ 0.1765 GeV 2 and the lattice data in Table III, the the chromoelectric field into the perturbative and nonperturbative components is done in a model-independent way.We have used the lattice data to investigate their structure and to provide a new picture of the confining field.We have analyzed lattice data taken for three quark-antiquark distances using this procedure, and we plan to carry out simulations for smaller quark-antiquark distances.More lattice data, particularly close to the quarks, must be taken to determine the extent to which the separation of the simulated fields can be carried out, and to check the consistency of our picture.
To the best of our knowledge this separation between perturbative and nonperturbative components has not been carried out previously.It provides a new tool with which to probe the chromoelectric field surrounding the quarks.Our procedure can be straightforwardly extended to the case of QCD with dynamical fermions with physical masses and at nonzero temperature and baryon density.Any successful theoretical mechanism of color confinement should be able to explain the findings of the present work.

FIG. 1 .
FIG. 1.(a) The connected correlator given in Eq. (1) between the plaquette U P and the Wilson loop (subtraction in ρ conn W, µν not explicitly drawn).(b) The longitudinal chromoelectric field E x (x t ) with respect to the position of the static sources (represented by the white and black circles), for a given value of the transverse distance x t .

3 .
FIG. 3. Surface and contour plots for the three components of the nonperturbative chromoelectric field, E N P ≡ E − E C , at β = 6.240 and d = 1.142 fm.All plotted quantities are in physical units.

FIG. 4 .
FIG. 4. Transverse cross sections of the nonperturbative field E N P x (x t ) at β = 6.240, d = 1.142 fm, for several values of x l .

TABLE I .
Summary of the runs performed in the SU(3) pure gauge theory (measurements are taken every 100 upgrades of the lattice configuration).

TABLE II .
Values of the fit parameters extracted from Coulomb fits of the transverse components of the chromoelectric field.
below we compare the values of the measured longitudinal chromoelectric

TABLE III .
Values of the longitudinal chromoelectric fields at (d/2, 0), the midpoint between the sources and transverse distance zero, for several values of distance d.E x (d/2, 0) is the unsubtracted