Spectrum of the QCD flux tube in 3d SU(2) lattice gauge theory
Evidence from the lattice suggests that formation of a flux tube between a pair in the QCD vacuum leads to quark confinement. For large separations between the quarks, it is conjectured that the flux tube has a behavior similar to an oscillating bosonic string, supported by lattice data for the groundstate potential. We measure the excited states of the flux tube in 3d SU(2) gauge theory with three different couplings inside the scaling region. We compare our results to predictions of effective string theories.
Simulations over the last few years have accumulated strong evidence that gluonic dynamics indeed leads to the formation of a flux tube between test quark and antiquark () in the vacuum of Yang-Mills theory bali (); ymstring (). This implies a linearly rising potential between quark and antiquark in the QCD vacuum and thus leads to quark confinement. At large separations, this flux tube is expected to behave like a string. Open bosonic string descriptions of the dynamics of this flux tube have been attempted for a long time God (). Using the Nambu-Goto (NG) action, first Alvarez Alvarez () (in the limit ) and later Arvis Arvis () obtained the energy states of the flux tube as
where is the string tension, the quark-antiquark separation and the number of space-time dimensions. A closed string description was proposed by Polchinski and Strominger (PS) PS () where they suggested how effective string theory with vanishing conformal anomaly could be formulated in arbitrary dimensions. The spectrum of the string using the PS prescription has been computed in drum (); pmd (); unpub (). To order it has been found that the spectrum is universal (depends only on the number of space-time dimensions and the string tension), and to this order it coincides with the NG spectrum. For a calculation of the closed string spectrum on the lattice see teper (). Using symmetry properties of the Polyakov loop correlation function, Lüscher and Weisz (LW) showed that any effective string picture posseses an open-closed duality property lweisz (). They also showed that demanding this duality constrains the possible string spectra and in particular forbids terms in the effective string action. In the LW formulation too, the spectrum is consistent with the NG spectrum. In 3-dimensions it is exactly the same and in 4-dimensions one undetermined parameter remains which can of course assume the NG value. The NG partition function itself respects open-closed duality. Predictions from these theories can be tested by comparing them to results coming from simulations of pure Yang-Mills theories.
In this Letter we will present a new scheme for measuring observables in simulations of pure Yang-Mills theories. Using this scheme we will accurately measure observables in 3-dimensional lattice gauge theory and try to extract the excitation spectrum of the flux tube. For earlier studies of the QCD string spectra using different schemes in both 3- and 4-dimensions for several gauge theories, see Kuti () and the references therein.
To probe the properties of the flux tube formed between test quark and antiquark, the two observables in pure Yang-Mills theories are the Polyakov loop correlation function and spatio-temporal Wilson loops. While Polyakov loop correlation functions project very strongly onto the ground state, Wilson loops are much more sensitive to the excited states of the flux tube and in addition allows one to couple more strongly to a particular state while suppressing the others, by appropriate choice of the spatial parts of the loop. It is therefore the observable of choice if one wants to study the excitation spectrum of the flux tube.
ii.1 Extraction of the excited states
In 3-dimensions, the states of the oscillating string can be classified by charge-conjugation and parity properties Pushan2 (); dipl (). Choosing the spatial parts of the Wilson loops as shown in figure 1, the projectors are given by the superpositions
where stands for even or odd states under and . Although in each of the channels, there are an infinite number of states, we are going to look only at the ground states and will label these channels as respectively. The Wilson loop projecting onto a channel has the spectral representation
where are the energies of the states in a given channel. Using Wilson loops with different temporal extents, one obtains the energies and energy differences at leading order as
where . In the following, the values obtained from the LHS of eqns.(4) and (5) will be called naïve values and be denoted by and . The quantities and will be called corrected values. They, along with and , are obtained as fit parameters. The fits are done by taking all possible combinations of and and are discussed in more detail in Section D.
ii.2 Noise reduction techniques
String like behavior of the flux tube is expected to occur at large separations. The Wilson loops we measure must therefore extend to large enough values. Also we see from equations (4) and (5) that to reduce the contaminations due to other states we must either go to large values of or tune to small values by appropriate choice of the basis states. The latter method has been followed in Juge () where asymmetric lattices with small temporal extents were used. The former method requires accurate measurements of expectation values of large Wilson loops, made possible recently by the multilevel algorithm, proposed in Weisz1 (). This method has been used in Weisz2 (); PundD () to accurately measure the ground state properties of the flux tube. Here we will try to combine both methods by using a slight variant of the multilevel algorithm.
In the multilevel algorithm, intermediate averages are computed for the temporal links by updating certain sub-lattices with the sources on the spatial links of the boundaries of these sub-lattices. The boundaries are held fixed during the sub-lattice updates. We now put the sources on space-like surfaces in the middle of a sub-lattice of thickness where is the lattice spacing. Two temporal links are attached to the two ends of the source which terminates on the space-like surfaces that are fixed during the sub-lattice updates (see fig.2). The advantage is that the sub-lattice updates reduces fluctuations of the sources as well as fluctuations of the temporal links. Moreover, in addition to using multihit on the temporal links, we can now use multihit on the spatial links as well. For further details see ownPos (); dipl () and references therein. Adopting the notation in Weisz1 (), we now define the expectation value of the Wilson loop by
A similar method was used in forcrand () to measure the static potential and observe string breaking in the adjoint representation.
In most parts of the calculations, the operators (figure 1 top; see also Pushan2 (); ownPos ()) are good enough for a reliable signal. However, for the second excited state beyond the error reduction was not sufficient and we had to use another set of operators , (figure 1 bottom) with a stronger coupling to that particular excited state. We will henceforth refer to the two sets as set and set , respectively. To check the error reduction with the new set of operators we performed 460 measurements at with spatial extents using the simulation parameters shown in table 1. The resulting relative errors of are also tabulated there. The values for the naïve energies using both sets are plotted in figure 3. For comparison we also plot , which was calculated from , in the same region. It is clearly seen that not only do the naïve values of set already coincide with the corrected values of set , they also have much smaller error bars. Set therefore significantly improves the signal for the state .
There are several parameters for the algorithm, that have to be tuned properly in order to achieve the maximal error reduction. For the temporal links we follow the same procedure as outlined in Pushan2 (). In addition we now have another parameter which is the number of updates () for the sub-lattices containing the spatial operators. We found it beneficial for the excited states to have quite a large number of such sub-lattice updates. Since there are only two such sub-lattices for every loop this does not increase the cost of the simulation very much. Table 2 contains the resulting optimized run parameters and we refer to dipl () for details of the optimization.
ii.3 Simulation parameters and lattice scales
|10.0||8.6602(8)||0.05812(1)||A||8||0.465||6||48000||3000||1272, 1272, 1296|
|10||0.581||4||3000||2352, 2544, 2568|
|14||0.814||6||6000||6384, 6216, 6480|
|18||1.046||4||12000||8664, 8304, 8472|
Our simulations were done with three different couplings, and (chosen so as to lie in the scaling region) in 3-dimensional lattice gauge theory, using usual heatbath sweeps heatb (), combined with three overrelaxation sweeps. The scale was set by the Sommer parameter sommer (), which has been computed for these values with high accuracy in PundD (). For each beta value we calculated Wilson loops with four different temporal extents and used cubic lattices.
Our simulation parameters, lattice scales, along with the parameters for the multilevel algorithm are also tabulated in table 2.
ii.4 Error analysis and control of the fits
For estimating the error of all the energy values and differences, we used the usual binned jackknife method, with 44, 24, 40 and 40 bins for the lattices , , and , respectively. We also checked that the errors did not vary by more than a few percent with bin size.
For the energy difference at and we used the results obtained from set for while the values for were obtained from set . Since these two values come from independent simulations, we added the individual errors in quadrature to obtain error estimates for .
The remaining issue is the control of the fits (4) and (5), and this was done in two ways. For the energies , we expect to be smaller then the ratio of the degeneracies 111At the order where we are comparing the data with the theory, the degeneracies are the same as in the free theory Weisz2 () of the energy states considered and should be of the order of the energy gap to the next level in the channel. Similar conclusions hold for the parameters of the energy differences Pushan2 (); dipl () as well. If the resulting fit parameters were far of from these expectations we did not trust the fits. As a second check we plot the expected corrections
obtained with averaged parameters and against the differences , for all possible combinations of and . We show an example for these fits (, , and ) in figure 5. If for some of the values there was a big discrepancy to the expectation the fit was also regarded as unreliable (for more details and systematics of the fits see dipl () and Pushan2 ()).
ii.5 Finite volume effects
Finite volume effects are possible due to finite spatio-temporal extents of the lattices. To control the effects due to finite temporal extents, we use four different extents to make sure that any systematic error in our energy values is much lower than our statistical errors.
The dominant corrections due to finite spatial extents are around-the-world glueball exchanges of the form , where is the glueball mass and is the spatial extent of the lattice. These are relevant for large values of . While glueball masses for 3-dimensional SU(2) lattice gauge theory has been studied in cmichael (); miketeper (), the mixing coefficient is not known. To get an estimate for this coefficient, we did runs at different lattice volumes at (for which has been measured in miketeper ()), varying by a factor two. Since we did not see any finite size effects in the energies, we concluded, that in the range considered, is a number of order one.
Using interpolation, we obtain from the data in miketeper (), the glueball masses at = 7.5, 10.0 and 12.5 to be 0.856, 0.671 and 0.536 respectively in lattice units. Since our data at large , with systematic errors under control is only at , we concentrate on that value. The only lattice where this correction matters is the lattice on which we measure Wilson loops with and between . On this lattice, the correction ranges between and for between . The error bars for the wilson loop corresponding to the second excited state are of similar magnitude for and 20 while for the first excited state, has a similar error bar. We therefore expect that there can be some correction due to the glueball for these energy values while for all other values, the corrections are well below our statistical errors.
Iii Simulation Results
In this section we discuss the results of our simulations, that are tabulated in table 3. We compare to the full NG spectrum, eq.(1) and to leading order (LO) and next to leading order (NLO) models which are obtained by truncating the expansion of the square root of (1) in at leading order and next to leading order respectively. The LW and PS type string theories give identical results to NLO order. The curves in the plots were drawn using the string tension at . All the results have been rescaled so that they are visible on a single plot.
iii.1 Energy states
|7.5 set||PundD ()||10.0 set||10.0 set||PundD ()||12.5 set||12.5 set||PundD ()|
We use the groundstate to determine the string tension and fix an additive constant , appearing in the potential, by fitting to the form Arvis ():
The results of the fits are shown in table 4, where also the values from PundD () are shown for comparison. We see that all results agree well within error bars. Also and from both operator sets agree with each other.
In figure 5 the results for the total energies are shown against , together with the predictions of the NG spectrum. The energies are rescaled such that . The results for the groundstate are completely in agreement with the NG predictions. In this case only the curve of the full spectrum is shown since deviations at different orders of the expansion are not visible on this scale.
For the excited states we were only able to obtain corrected results over a limited range of , except for at where we have results over the full range.
We see that the corrected values for follow the Nambu curves quite well. In the region where we are able to obtain corrected energies for all values of , we see good agreement between the different values. At smaller values of where the difference between the NG, LO and NLO curves are visible, the data seems to favor the NG curve.
iii.2 Energy differences
We now turn to the energy differences which are more sensitive to subleading properties of the flux tube. In addition they have the advantage that the constant does not contribute. The results for the energy differences and are shown in figure 6.
We were able to obtain corrected values using eq.(5) only for at . That data set seems to follow the NG curves quite nicely. For , our best estimates are the naïve values from eq.(5) with and . Nevertheless we expect very little systematic effects in these results as the physical temporal extents for these loops are 1fm. Again we see that the NG curve is favored by the data.
For at and 12.5, our best estimates come from the difference with the errors being calculated in quadrature. Even though this gives larger error bars, especially at , we see that both data sets are consistent with values.
For at and 12.5, the physical temporal extents of the loops are not large enough that the higher order corrections, unaccounted for in equations (4) and (5), are negligible. Nevertheless, at , where it was possible at least partly to take into account the corrections, we see the trend of the data is to follow the NG curve. For , where even that was not possible, the data contains systematic errors and we have not plotted it.
In this Letter we have looked at a variant of the multi-level error reduction scheme suitable for studying the excited states of the QCD flux tube. Using this scheme we have looked at the excited states of the flux tube at three different couplings for separations between 0.5 and 1.7 fm.
Compared to measurements with the older method Pushan2 (), we have seen very significant improvements due to the new measurement process. It has been possible to increase the range of by more than 50%. Since the error in grows roughly proportional to , this would be very hard to achieve by increasing computing power alone. Compared to Pushan2 (), the errors on and have been reduced by a factor between 2 and 3 in the overlapping range of . This would also be very hard to achieve by increasing statistics alone. Both of these observations clearly demonstrate the superiority of the new measurement process over the older one. Moreover using the older method we did not have a signal for the second excited state whereas we do have a reasonable measurement now at least for one value.
Our results seem to indicate that in this scheme, with our basis, upto the values of we have considered, one needs temporal extents of about 1 fm to make sure that the corrections due to the higher states are well under control. Wherever this criterion has been met, we have seen that the data follows the NG curve.
The corrected data sets for different values seem to fall on top of each other indicating that there is very little effect due to finite lattice spacing. As such it seems to be much more important to go to larger temporal extents than finer lattices.
Unfortunately our data is still not good enough to distinguish between NG, LW and PS type string theories as that would require a sensitivity at the level of . Deviations from the NG predictions at higher orders have recently been reported in gliozzi () for gauge duals of random percolation problems.
This was our first attempt to show how to combine the two powerful techniques of multi-level error reduction and sophisticated sources for Wilson loops. Using this we have been able to go to much larger values of and than was possible before. For ruling out different string models, the error bars will have to be reduced by at least an order of magnitude. That at the moment looks like a project for the future.
The simulations were distributed over three different computational facilities, namely the computing resources of the Westfälische Wilhelms-Universität Münster (organized with the condor system condor ()), where this study was started, the teraflop Linux cluster KABRU at IMSc, Chennai and the Linux cluster LC2 at the ZDV of the Johannes Gutenberg-Universität Mainz. We are indebted to the institutes for these facilities. We would also like to thank P. Weisz and H. Wittig for useful discussions on finite size effects for Wilson loops. B.B. is funded by the DFG via the SFB 443.
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