Three-quark potentials are studied in great details in the three-dimensional pure gauge theory at finite temperature, for the cases of static sources in the fundamental and adjoint representations. For this purpose, the corresponding Polyakov loop model in its simplest version is adopted. The potentials in question, as well as the conventional quark–anti-quark potentials, are calculated numerically both in the confinement and deconfinement phases. Results are compared to available analytical predictions at strong coupling and in the limit of large number of colors . The three-quark potential is tested against the expected and laws and the string tension entering these laws is compared to the conventional string tension. As a byproduct of this investigation, essential features of the critical behaviour across the deconfinement transition are elucidated.
Three-quark potentials in an effective Polyakov loop model
O. Borisenko, V. Chelnokov, E. Mendicelli, A. Papa
Bogolyubov Institute for Theoretical Physics,
National Academy of Sciences of Ukraine,
03143 Kiev, Ukraine
Istituto Nazionale di Fisica Nucleare, Gruppo collegato di Cosenza,
I-87036 Arcavacata di Rende, Cosenza, Italy
Dipartimento di Fisica, Università della Calabria,
I-87036 Arcavacata di Rende, Cosenza, Italy
On leave from Bogolyubov Institute for Theoretical Physics, Kiev
The interest in studying the interquark potential for a three-quark system is not a recent issue at all. It has instead a long history due to its importance in the spectroscopy of baryons. The first studies date back to the mid ‘80s [1, 2] and after more than a decade a new turn of research has started around the year 2000 which continues till now [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. New results are somewhat contradictory, which could be reasonably explained by the difficulty of accurate measurements of the three-quark potential. But from these discussions spanning many years, two main Änsatze emerged to describe the three-quark potential: the law and the law. Denoting by , , the sides of the triangle having the quarks at its vertices, the law is defined by
which describes a potential linearly rising with half the perimeter of the triangle. The law describes the three-quark potential as linearly rising with the minimal total length of the flux lines connecting the three quarks,
where is the sum of the distances of the three quarks from the Fermat-Torricelli point F, which is the point such that this sum is the least possible. When all inner angles of the triangle are less than , one has
where is the area of the triangle; if one of the angles is larger than , we have instead
which gives rise to the law,
Some earlier [4, 8, 9] and the most recent studies [12, 13] in the pure gauge theory seem to support the -ansatz, while other simulations [3, 5, 6, 7] prefer the -ansatz, at least for not too large triangles. An even more complicated picture emerged after simulations of the simpler, Potts model in two-dimensions, which is believed to capture the most essential features of the gauge model [7, 10]. Namely, it was conjectured that there might be a smooth crossover between the law and the law when the size of triangles grows (see, however,  where this scenario has been criticized). Also, the paper  proposes a new ansatz in which both the law and the law are present.
In this paper we are going to study an spin model which is an effective model for Polyakov loops and can be derived from the original gauge theory in the strong coupling region. For simplicity, we consider, following , only its two-dimensional version. Our primary goal is to get some analytical predictions for the three-point correlation function of the Polyakov loops and compare them with numerical simulations. For that we use the spins both in the fundamental and adjoint representations. The main tool of our analytical investigation is the large- expansion. Within this expansion we demonstrate that the fundamental three-point correlator is described by a sum of the and laws. The contribution is not present. In turn, the connected part of the adjoint three-point correlation follows the law in the confinement phase. In addition, we study the critical region of the model and confirm that it belongs to the universality class of the two-dimensional spin model.
This paper is organized as follows. In the next section we introduce our notations, define Polyakov loop model and its dual. Certain analytical predictions for two- and three-point correlation functions are obtained in the strong coupling expansion and in the large- limit. Moreover, we check the restoration of the rotational symmetry for the 3-quark system. Section 3 outlines some details of our numerical simulations. Here we compare numerical data with the strong coupling expansion and study the critical behaviour of the model using the finite-size scaling analysis. Section 4 presents the results of Monte Carlo simulations for the fundamental and adjoint two- and three-point correlations in the confinement region. Results for the same quantities above critical temperature are described in Section 5. In Section 6 we summarize our results.
2 The model and theoretical expectations
2.1 Partition and correlation functions
We work on a Euclidean lattice , with sites , , and denote by the unit vector in the -th direction. Periodic boundary conditions (BC) are imposed in all directions. Let , and be the character of in the fundamental representation. Consider the following partition function on , which describes the interaction of non-Abelian spins:
The trace of an matrix can be parameterized with the help of angles, e.g. by taking , subject to the constraint . In this parameterization the action has the form
The invariant measure for is given by
and is the periodic delta-function. Due to this constraint, the model is invariant only under the global discrete shift for all and . This is just the global symmetry. The partition function (6) can be regarded as the simplest effective model for the Polyakov loops which can be derived in the strong coupling region of lattice gauge theory (LGT) at finite temperature (see, e.g.,  and references therein).
The main subjects of this work are the two- and three-point correlation functions for the model. In the fundamental representation these correlations are given by
while in the adjoint representation the correlations are written as
where we use the relation .
This spin model is one of the simplest versions of Polyakov loop models. It can be derived from the finite-temperature pure gauge LGT in the strong coupling approximation. Namely, the integration over the spatial gauge links on the anisotropic -dimensional lattice with two couplings and in the limit and for sufficiently small leads to the -dimensional spin model (6). It describes the deconfinement phase transition of the pure gauge theory, which is of second order for if . It is widely assumed that the phase transition is in the universality class of the two-dimensional (Potts) model. The inverse correlation length (mass gap) is the string tension of the gauge theory. The correlation length diverges when approaching the critical point with the critical index . Another important critical index , which is a characteristic of the massless phase, equals exactly at the critical point. Thus, on the basis of the universality arguments  we expect the same values for these indices also in the effective Polyakov loop model. More on the critical behaviour of three-dimensional LGTs can be found in Refs.[16, 17].
The model (6) cannot be solved exactly at any finite and . Therefore, to get some analytical predictions for the behaviour of the three-point correlation functions we consider the large- limit of the model. This limit can in turn be solved exactly by using the dual representation which we are going to describe shortly.
2.2 Dual representation
In some applications the dual formulation of the Polyakov loop model (6) can be useful. Such formulation for the model has been derived in . Here we use the dual representation obtained by some of us in [19, 20]. This form of dual theory is valid for all and can be used both for numerical simulations and for the study of the large- limit of the theory. For the theory the partition function (6) on the dual lattice takes the form
where results from the invariant integration over the measure,
Here is the dimension of the irreducible representation of the permutation group , and
is enumerated by the partition of , i.e. , where is the length of the partition and . The sum in (15) is taken over all ’s such that and the convention has been adopted. For the exact expressions of the different correlation functions we refer the reader to the paper .
2.3 Large- solution
Using the dual representation (14), one can construct an exact solution of the model in the large- limit  and even estimate the first non-trivial corrections specific for the group. As an example, we give here the expression for the most general correlation function and for the partition function in the confinement region in the presence of sources
The Gaussian part describes the solution in the large- limit, while the product over in the second line presents the first correction due to . is the massive two-dimensional Green function for the scalar field.
This solution, together with a similar one in the deconfinement phase, enables one to calculate both fundamental and adjoint two- and three-point correlations in that limit. Different results are obtained in the small and large regions separated by the deconfinement phase transition. If we take , then for the confinement phase we get
and in the deconfinement phase we get
In the equations above the Green function in the thermodynamical limit is given by
where and the functional dependence of the mass on is different in the confined and deconfined phases. In the confinement phase the mass coincides with the string tension, while in the deconfinement phase this quantity has the meaning of screening mass. and define the fundamental and adjoint magnetizations at a given , correspondingly. They also depend on the considered phase. For example, in the confined phase. is another -independent quantity which appears due to Gaussian integration around the large- solution. All four quantities - , , , - are known exactly in the large- expansion.
In what follows our strategy relies on the assumption that the large- formulae (20)-(27) remain qualitatively valid (up to one correction explained below) at finite , in particular for . Therefore, we shall use these formulae as the fitting functions, where the quantities , , , are unknown parameters to be found from fits of numerical data. In most cases, we use the asymptotic expansion for the Green function given on the right-hand side of Eq. (28). Here, we introduce another quantity, namely the index , in order to describe the power dependence of the correlation function, , on the distance. This could be important in the vicinity of the critical point. In general, this introduces a correction to the potential of the form
and is interpreted as the Coulomb part of the full potential in the two-dimensional theory.
Since the asymptotic behaviour of is known, it follows that we actually know the large-distance behaviour of all two- and three-point functions listed above, but . The behaviour of the latter can be analyzed by the saddle-point method when at least one side of the triangle is large enough. We find two types of the behaviour:
All inner angles of the triangle are less than . The three-point fundamental correlation function is given by the sum of two terms corresponding to and laws
where are constants and . This behaviour resembles the behaviour of the three-point correlation function in the spin model .
One of the angles is larger than . In this case the asymptotics is described by the above formula with . Thus, only the law is present. This again agrees with the spin model.
Let us also emphasize that we could not find the law contribution in our large- approach. Nevertheless, we attempt to fit numerical data both to and laws in the following. Finally, let us stress that the connected part of the three-point adjoint correlation follows the law in the confinement phase, as is seen from Eq. (23).
2.5 Strong coupling expansion
When is sufficiently small, one can use the conventional strong coupling expansion to demonstrate the exponential decay of the fundamental two- and three-point correlation functions. Instead, adjoint correlations stay constant over large distance. To check our codes we have calculated the leading orders of the strong-coupling expansion for the two-point correlator at distance and for the three-point correlator in the isosceles-triangle geometry with base and height . The results read
For arbitrary isosceles triangle with base and height one obtains
On a cubic lattice is the minimal sum of the lattice distances from the triangle vertices to an arbitrary lattice point. Then, according to Eq. (2),
in the strong coupling region on the finite lattice. Thus, strictly speaking the strong coupling expansion predicts not an exact law, as it is often stated in the literature, but rather a law. In general, and we expect that the rotational symmetry will be restored quickly with and the triangle sides increasing. This should result in the restoration of the genuine law.
To demonstrate that such a restoration really takes place, we have studied the three-point correlation function for triangles with and at . The fact that the rotational symmetry is already restored at this value of is shown on Fig. 1, where we compare numerical data with the fits of the form for and . Clearly, describes data better than .
3 Details of numerical simulations
To calculate the correlation functions from numerical simulations we used two different approaches. The first is the simulation of the model in terms of the eigenvalues of the spins, described in more detail in . In this approach (denoted as standard in the following), an updating sweep consisted in the combination of a local Metropolis update of each lattice site, followed by two updates by the Wolff algorithm, consisting in reflections of the clusters. An alternative approach is the simulation of the dual model (14), using the heatbath update for the link variables , and the dual site variables . In this case, we can measure only observables invariant under the global symmetry.
In both approaches we measured two- and three-point correlation functions in the fundamental and adjoint representations, taking for the two point correlation function pairs of points separated by in one of the two lattice directions, with . For the three-point correlation functions two geometries were studied: isosceles triangles with base and height , and right-angled triangles with the catheti (of lengths and ) along the two lattice directions. In both cases, and , and and , took independent values in the set .
The -dependent values are averaged over all sites of the lattice.
For each simulation we performed thermalization updates, and then made measurements every ten whole lattice updates (sweeps), collecting a statistics of – . To estimate statistical errors a jackknife analysis was performed at different blocking over bins with size varying from 500 to 10000.
A comparison of the two simulation methods showed that the dual code performs better at small values of , while giving much larger fluctuations than the standard one when is close to its critical value. What is more important – at larger values of for the fundamental correlation function the fluctuations rapidly increase with the distance between the points. Due to this, most of the results presented here have been obtained in the standard approach, and the dual code was used only for cross-check purposes.
3.1 Comparison with strong coupling
To test our algorithms we performed a set of simulations at small values of (), and compared the obtained values of , , and with the corresponding determinations in the strong coupling expansion (Eqs. (31)-(34)). The results of the comparison are shown in Fig. 2 for the correlations in the fundamental representation, and in Fig. 3 for the correlations in the adjoint representation. It can be seen that the two-point correlation, both in the fundamental and adjoint representation, is in good agreement with the strong coupling expansion. For the three-point correlation, due to its small absolute value, statistical errors in the standard simulation are too large to make any statement about agreement with the strong coupling prediction. The results for the adjoint correlation from the dual code are compatible with the strong coupling expansion up to . Since the results of the two simulation codes agree in the region around , where most of our simulations were carried out, we are confident in the reliability of our measurements.
3.2 Critical behaviour
A clear indication of the two-phase structure of the model is provided by the scatter plots of the complex magnetization at different values of , shown in Fig. 4.
To precisely locate the at which the phase transition occurs, we have studied the magnetization susceptibility for different lattice sizes , extracting the value of from a fit of the peak of the susceptibility with a Lorentzian function. The obtained values of have been fitted with the scaling law for a second order transition (see the left panel of Fig. 5)
with the following resulting parameters:
The value for the critical index is in agreement with the critical index of the two-dimensional three-state Potts model, to whose universality class our model is believed to belong. A direct extraction of the critical exponent , performed in the subsection 4.3, gives a compatible result, which is sensitive to the choice of the region of values where critical scaling is supposed to hold.
As a second check of the order of the phase transition and of the universality class, we studied the dependence of the peak value of the magnetic susceptibility for different lattice sizes using the scaling law (see the right panel of Fig. 5)
The obtained value for is in agreement with the hyperscaling relation for the three-state Potts model, which gives . The expected value of is 4/15. These findings support the universality class of the present Polyakov loop spin model.
4 Correlation functions in confinement phase
4.1 Extraction of from
The potential parameter is extracted from the measurements of the observable . Following Eq. (20) and the explanation in subsection 2.4, we expect
One can extract from the following ratio:
We have compared our Monte Carlo data for with the formula . The interval of values that we considered for the extraction of was , since for the two-point correlation drops too fast to be of any use. The values of obtained in the selected range of do not show any significant difference when moving from a to a lattice, thus making unnecessary to perform simulation on even larger lattices.
As an alternative method for extracting , we measured the wall-wall correlation function,
which is known to obey the exponential decay law, with no power corrections,
Introducing, similarly to (44),
we found that exhibits a long plateau at each considered value; we took as plateau value the value of at the smallest value of after which all values of agree within statistical uncertainties. Results for the lattice are summarized in Table 1: we can see that results of obtained from the fitting of according to (44) are in good agreement with the plateau values of .
|10||20||0.61 (18)||0.327 (19)||0.31|
|6||20||0.601 (14)||0.3044 (27)||0.49|
|8||20||0.595 (54)||0.3051 (72)||0.59|
|10||20||0.62 (18)||0.303 (19)||0.74|
|6||26||0.620 (13)||2663 (25)||0.60|
|8||26||0.600 (44)||0.2687 (57)||0.66|
|10||26||0.60 (0.12)||0.269 (13)||0.75|
|10||26||0.57 (10)||0.235 (11)||1.29|
|8||28||0.627 (21)||0.2093 (27)||0.47|
|10||28||0.598 (52)||0.2121 (86)||0.51|
|8||28||0.647 (22)||0.1863 (28)||0.67|
|10||28||0.598 (49)||0.1909 (50)||0.66|
|6||36||0.668 (10)||0.1628 (18)||1.48|
|8||36||0.667 (27)||0.1630 (34)||1.59|
|10||36||0.690 (58)||0.1609 (57)||1.69|
|8||36||0.670 (13)||0.1405 (15)||0.58|
|10||36||0.652 (26)||0.1421 (25)||0.59|
4.2 Extraction of from
First we studied the dependence of on the geometry, considering and laws. In Fig. 6 we see that if we consider to be proportional to we get a reasonable collapse for all values except the largest one (), which might be too close to the critical point for our lattice size, . Still this does not allow us to discriminate between the two laws.
We turned therefore to fits with the two laws of the three-point correlation function for small () triangles, excluding those having an angle larger than . The results of these fits for are shown in Fig. 7. In this case the fit with the law gives , while the fit with the law gives . For all values of the law performed better than the law (giving smaller ), the difference getting more and more clear for larger values of .
In this subsection is a distance parameter depending on the geometry, that could be or , respectively for the and laws, given in Eqs. (1) and (2). In this proposed fitting function we excluded the second term, corresponding to the law, since, for the size of triangles considered in the fitting procedure, we could hardly distinguish it from the first one, corresponding to the -law.
We calculated the following ratio
for the pairs of triangles and for the isosceles geometry and and for the right-angled geometry, where and are the distance parameters of the two triangles. The ratio is equal to
where, for sufficiently large distances we can assume and get
It turned out that for part of the triangle pairs and its jackknife error estimate could not be extracted reliably, due to one of the correlation functions being too close to zero. We removed from the study all the triangle pairs in which for at least one of the triangles, at least one of the jackknife samples gave negative correlation. The actual number of the triangle pairs for which the extraction of was possible, strongly depends on the value of ; for example, for the isosceles geometry we had 274 triangle pairs for , 558 for and 783 for .
After extracting , we plotted it directly against the half-perimeter and against the sum of the distances of the triangle vertices from the Fermat-Torricelli point. We overlapped these plots with the plots of and versus (Figs. 8-10). We see that on the plots for the law the values of fail to collapse into a single line, while the collapse is much better for the law for all values we studied. The residual spread of the points can be at least partially explained by different triangle pairs having different values, which are not distinguished on these plots. Another observation that supports the law is that the collapse line for the closely matches the line of , which suggests that not only the sigma values entering the two- and the three-point correlation are the same if we consider the law, but also that the parameters are similar in these cases.
It is worth mentioning that in our study the results of the extraction of are compared for triangles that have strongly different geometries: there are triangles that have similar distance, but some of them can have a small base and a large height, while others can have small heights and large bases. In particular, we did not exclude triangles with angles larger than from the study of , despite the fact that for them the Fermat-Torricelli point coincides with one of the vertices, thus leading to a different dependence of on and . The fact that even these “extreme” triangles obey the law is explicitly demonstrated in Fig. 11 (instead, the most outlying data points turn out to be the ones with smallest triangle base), where the caption implies that for such triangles the Fermat-Torricelli point coincides with one of the vertices turning the law into the law. As can be seen from Figs. 8-10 these differences in geometry give negligible corrections to the values of up to , providing us with an additional point in support of the law.
4.3 Extraction of critical index from the scaling of the two-point string tension
We used the values of the string tension obtained from the wall-wall correlation function close to the critical point to extract the critical index .
The values of , as well as the result of the fit in the region with the scaling function
are shown in the Fig. 12. The value of the critical index obtained in this way is compatible with both the critical index of the three-state Potts model and with our previous estimate in (41). We note, however, that the scaling region in this case is extremely narrow, and the value of is quite sensitive to the inclusion of the points outside this region.
4.4 Adjoint correlations in the confinement phase
Note that exhibits the -law decay after subtracting terms proportional to (powers of) the magnetization.
Unusually low values in the fits arise due to treating the measurements of the correlations at different distances as independent despite being obtained from the same set of measurements. This fact makes the error estimates of the fit parameters unreliable. Also, the estimation of the value is inaccurate, since it describes short-range corrections to the exponential decay, on which only a few points in the fitting range have impact. This fact is especially visible from the covariance between and which is very close to .
Despite that, the fact that the parameters and show some degree of stability when going from the description of the two-point correlation to the three-point one, and also the compatibility of the results for with the and values given in Table 1, support the validity of the suggested descriptions for the adjoint correlations.
We would like to stress that the values of extracted from the fits do agree with direct measurements of this quantity as defined in Eq. (38).