# Potential between adjoint sources in arbitrary representations

###### Abstract

The potential between sources in arbitrary representations of the gauge group is studied on an anisotropic lattice in a spherical model approximation. It is shown analytically that for half-integer and in the confinement phase the potential rises linearly , whereas for integer and half-integer it risesinfinitely which means a strong suppression of the combination of such states . For integer and the potential shows Debay screening and Coulomb behavior in the deconfinement phase . It is also shown, that when and is in agreement with the mean field theory prediction, and for which agrees with MC experiment. String tension model-computed for sources invariant under center group transformations demonstrates Casimir scaling in the intermediate distance regime and turns into zero at large distances.

## 1 Introduction.

Statistical QCD evaluated on the lattice by means of computer simulation is perhaps the only case in statistical physics where the critical behavior can be calculated from the first principles of dynamics [1, 2, 3]. The power of the approach is best seen in the thermodynamics of pure gauge theory, i.e., for the systems consisting of gluons only [1, 4, 5]. The main features considered are the deconfinement transition and the properties of the hot gluon plasma [6]. Lattice gauge theory (LGT) provides a powerful tool to compute nonperturbative properties of strong interactions, such as a potential between pairs of infinitely heavy sources.

Its known from Monte Carlo simulations that probes in representations insensitive to the centre gauge group transformation yield a screened potential, while those sensitive to this transformations provide a linearly rising one (where - string tension, - distance between sources) [47, 8, 9]. As indicated in [10], the potential between sources transforming as arbitrary irreducible representations of the gauge group provides a good laboratory for testing the dynamics behind screening and confining mechanisms. In particular, one can measure the relevant distance scales, i.e. those that separate “short distances” (essentially perturbative physics), from the “long distances”. The dynamics of the intermediate region has been found quite rich in many aspects. Indeed, some very simple numerical simulations have shown that adjoint sources, for example, may feel a linearly rising potential at intermediate distances [11, 12, 13, 14, 15]. Such sources appear to “deconfine” at precisely the same critical temperature at which fundamental sources deconfine [16, 17]. The string tension is representation-dependent and appears to be roughly proportional to the eigenvalue of the quadratic Casimir operator [11, 12, 18, 13].

As the distance between color sources with zero ”n-ality” increases they are eventually screened by gluons. This asymptotic regime extends from the color-screening length to infinity, and in the case of gauge group the string tensions become equal for half-integer -s and turn into zero for integer ones. In particular, the string between quarks in an adjoint representation must break at some distance that presumably depends on the mass of ”gluelumps” (i.e. on the energy of a gluon bound to a massive adjoint quark) [19].

¿From universality hypothesis it follows, in particular, that all higher representations sensitive to the center of the gauge group should be equivalent order parameters with the same critical behavior as the fundamental representation. However different behavior was found for the higher representations at lattice gauge theory in dimensions [16, 17].

The purpose of this work is an analytical investigation of the potential between sources in arbitrary representations of the gauge group near the critical point on an anisotropic lattice in a spherical model approximation.

The application of spherical model in statistical physics has a long history since the time it had been introduced to investigate critical phenomena in the ferromagnet [20] and until recently (see, e.g., [21, 22]). Stanley [23] established the correspondence between the spherical and the Heisenberg models.

Although this model is of no direct experimental relevance, it may provide useful insight since many physical quantities of interest can be exactly evaluated with its help. In this context, the spherical model is quite a useful tool in providing explicit checks of general concepts in critical phenomena, see [24, 25, 26, 27]. Recently [21] it was successfully used for studying the transitions between a paramagnetic, a ferromagnetic and an ordered incommensurate phase (Lifshitz point). Models of this kind were investigated extensively (see recent review in [28]). The spherical model predicts reasonable values for critical exponents. Moreover, a ’basic’ set of exponent relations is also satisfied by spherical model for [29].

## 2 LGT on anisotropic lattice.

The simplest way to increase the accuracy in numerical study of LGT at large distances and high temperatures is to use anisotropic lattices in which the temporal spacing is much smaller than that in the spatial directions (, the Hamiltonian limit). This approach becomes a very popular technique at the present time since it allows Monte Carlo calculations to be carried out with reasonable computational resources and reap many of the benefits of fine lattices, while still using cheap coarse spatial ones. There are convincing arguments [30, 31] that in the anisotropic Wilson gauge action no coefficients are to be tuned in order to restore space-time exchange symmetry up to O() errors. On such lattices heavy quarks will not suffer large lattice artifacts as long as their masses are small in the units of .

To compute the partition function on a lattice of a size ( is the temporal extent, is the spatial extent of a lattice) after [32] (see also [33, 34]) we use the anisotropic lattice

(1) |

where is electric and magnetic part of the gluodynamics action and

(2) |

with

(3) |

In the weak coupling region [35, 36] and in this case does not differ essentially from . Moreover, in the “naive” limit

(4) |

and one must put exactly . However, in LGT the effective values of the integration variables are far enough from the “naive” limit (especially in the deconfinement region). The deviation of from is regarded as ’quantum corrections’ or ’renormalisation’. The behavior of in the vicinity has been studied both perturbatively [37] and by non-perturbative methods [38, 39, 40, 41, 42, 43]. Since we cannot exclude, that in the area of the asymptotically large the behavior of may be essentially different, we will not specify the dependence of on , but only assume that may be made arbitrary large by increasing at fixed .

It is well known [44, 45] (see a review in [46]) that there is a variety of ways to get (in accordance with a universality hypothesis [32]) the effective action

(5) |

expressed in terms of the Polyakov lines in representation of group (we omit index for fundamental representation). The crucial assumption for obtaining is to discard the magnetic part () of action. Then with

(6) |

As a rule it can be done in a strong coupling approximation [44, 45]. However, as it is pointed out in [32] at high ’temperatures’ and for the couplings one gets and .Hence we may assume that for finite the magnetic part of action may be neglected [34] compared to the electric one () in a Hamiltonian limit even at low temperatures . Such an assumption doesn’t look as harmless as in a strong coupling case. Indeed, there are serious reasons to believe that the magnetic part of the action may be of crucial importance for creating the confining forces [47, 48, 49], even at high temperatures [50]. Therefore, strictly speaking, without a magnetic part may be considered in a weak coupling area only as a specific (’toy’) model.

Summarizing for the partition function with static sources we may eventually write

(7) |

## 3 gauge model

In a simple case of gauge group one may write

(8) |

so by introducing new variables: and the partition function can be rewritten as

(9) |

To compute the partition function we use the spherical model
approximation^{1}^{1}1We want to stress, that the spherical model
approximation does not break the global - invariance of
the model.:

(10) | ||||

where constant is chosen so that the integration path is placed at the right side of all singularities of the integrand to ensure the legitimacy of interchanging the integration order over and . Therefore we can rewrite as

(11) |

where

(12) |

It is clear that integration over can be done trivially

(13) |

Integration over may be fulfilled after Fourier
transformation^{2}^{2}2Discrete variables for can be
considered as continuous:

(14) |

which diagonalizes bilinear form

(15) |

where

(16) |

Now we can do the integration over and write for partition function

(17) |

where

(18) |

and

(19) |

with

(20) |

and - modified Bessel function.

Taking into account that

(21) |

we obtain for the uniform sources

(22) |

with

(23) |

To calculate (22) we use the steepest descent method that in the limit gives an accuracy of order . The saddle point is determined by and considering that one may write [29] for .

(24) |

we get

(25) |

Hence for the solution of for is given by

(26) |

where the critical coupling is defined as the solution of at the lowest value of (and which evidently gives In a vicinity of the critical point one may write for

(27) |

It is easy to see, that in the deconfinement region the solution of with is absent. The only way to regain the solution in such region is to tend , because in this case the term must be preserved even in a limit Now the solution

(28) |

sticks to the point so in this region we can write

(29) |

and

(30) |

Hence in the deconfinement region but steadily approaches zero when in accordance with the well known fact, that - gluodynamics undergoes second order phase transition.

For the two point correlation function for we get

(31) |

where and string tension (in lattice units) is given by

(32) |

The expression for correlation function as is just the Fourier transform of the propagatorwhere –dimensional momentum space. In the perturbation theory expression can be attributed to the lattice one-particle exchange. A basic non-perturbative feature of the high temperature plasma phase of QCD is the occurrence of the gluon screening mass [51], so parameter , defined by can be interpreted as screening mass as well. The expression is not specific for the spherical model and appears in many popular approaches, for example in the matrix model (see e.g., [52]) and at the perturbation expansion () (see [53, 54]). is the lattice laplacian in

A more detailed case is considered in the Appendix. Here we present only the final results :

Tab. I Pair correlation for states in higher representations

We work in the parameter area and where and so we preserve only the terms with lower powers of and .

It is easy to see that in all cases the correlation function

(33) |

exponentially decreases in the confinement phase and can be used for the computation of which plays a role of string tension when and may be interpreted as a screening mass for

## 4 Conclusions.

As it can be seen from TABLE 1, our model gives linearly rising free energy in the confinement phase for half-integer and . The potential between states with integerand half-integer (or vice versa) turns into infinity which means strong suppression of such combination of states. For integer and we get the screening potential: . In the deconfinement phase we get . We would like to stress here that in the deconfinement phase for integer and the screening mass is twice as much than in confinement one.

As it was predicted by the mean field theory [55]

(34) |

This result was confirmed in MC simulations [56] and as it can be seen from TABLE 1 roughly agrees with such data when . However, as it was pointed out in [57] in the vicinity of critical point the relation does not agree with MC experiment. TABLE also shows, that our result

(35) |

obtained in such area agrees with the data given in [57] and with theoretical predictions developed both in framework flux picture in a strong coupling approximation and within field theory () on the basis of universality arguments [57].

Finally, there is the issue of the Casimir scaling (whose importance were emphasized in [11, 12]) of higher-representation string tensions in the intermediate distance regime. For values far from asymptotic region we can write and is independent either from or from and so it follows from and : where

(36) |

Therefore, the adjoint sources may ’feel’ a linearly rising potential at intermediate distances, and appear to “deconfine” at precisely the same critical point at which the fundamental sources really deconfine. This qualitatively agrees with Casimir scaling observed in such parameter region [11, 12]. It is easy to see, that at very large distance scales Casimir scaling breaks down and color screening is set.

As it was pointed long ago, sources trivially transformed under centre gauge group transformations can not be used as an order parameter. An average value of a pair of such sources also is not a good laboratory for critical phenomena studies, because the magnetization differs from zero and masks the Debay term. Therefore with so and are statistically independent at asymptotically large distances and long range order is absent either for or for On the other hand, the fact that the correlation function exponentially decreases for all and can be used for determination of string tension even in the cases when or are invariant under transformations.

We also wish to note that computing the Binder cumulant , defined as [58]:

(37) |

we get for a case gauge group

(38) |

Therefore, in a vicinity of a critical point we get because in the confinement region and in deconfinement region, if we are close enough to the critical point. Vanishing of means that the distribution of is Gaussian, therefore the spherical model must give reliable results in both regions.

## 5 Appendix

The - order correlations may be easily computed after Fouirier transformation of

(39) |

and we get in a spherical model approximation

(40) |

with

(41) |

Therefore after integration over

(42) |

and taking into account

we obtain inverting Fourier transformation

(43) |

with

(44) |

where is matrix is defined by the condition . In particular one can write

(45) | ||||

where the dispersion is given by

(46) |

The average value of the character in an arbitrary
irreducible representation may be computed as^{3}^{3}3Here refers to an integer part of and are
Hermitian polinomials of order .

(47) |

or

(48) |

that for meaning immediately gives

(49) |

In the ’deep’ deconfinement area or

(50) |

The pair correlation

(51) | ||||

where matrix is given by

(52) |

one may do computation expanding over powers and taking into account simple relation we get . In particular for magnetization

(53) |

In the confinement area for integer for

(54) |

In particular for we obtain for

(55) |

where is quadratic Casimir operator.

For half-integer

(56) |

with

(57) |

and

(58) |

Therefore taking into account

(59) |

we obtain

(60) | ||||

In particular we get

(61) |

and

(62) | ||||

therefore, correlation function for characters in the adjoint representation may be written as

(63) |

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