]>NUPHB14562S05503213(19)30030610.1016/j.nuclphysb.2019.02.002The Author(s)Quantum Field Theory and Statistical SystemsFig. 1Comparison of the three point correlation function Γ3f at β = 0.41 with the fit using Yl and Y laws.Fig. 1Fig. 2Twopoint (left) and threepoint (right) correlation functions in the fundamental representation versus β. The green solid lines represent the strong coupling expansions, given respectively in Eqs. (31) and (32). (For interpretation of the colours in the figure(s), the reader is referred to the web version of this article.)Fig. 2Fig. 3Twopoint (left) and threepoint (right) correlation functions in the adjoint representation versus β. The solid green lines represent the strong coupling expansions, given respectively in Eqs. (33) and (34). The round (square) symbols refer to simulations in the standard (dual) formulation.Fig. 3Fig. 4Scatter plots of the complex magnetization Mf at β = 0.1,0.42,0.44 on a 32 × 32 lattice.Fig. 4Fig. 5Fits of the βpc(L) values determined from the magnetization susceptibility χL(Mf) (left) and of the peak value of the magnetic susceptibility (right) versus the lattice size L. The solid red lines give the result of the fits with the scaling functions in Eqs. (41) and (42), respectively.Fig. 5Fig. 6Γ3f versus σΔ (left) and σY (right), for the isosceles geometry. In both cases the value of σww is used for σ.Fig. 6Fig. 7Comparison of the threepoint correlation Γ3f with the fit using the Δ law (left) and the Y law (right) for β = 0.423 isosceles triangles with the angles less than 2π/3 and Y < 35.Fig. 7Fig. 8σqq‾eff versus R, σwweff versus R and σqqqeff versus Δ (left) and Y (right) at β = 0.41 on a 128 × 128 lattice, for the isosceles geometry.Fig. 8Fig. 9Same as Fig. 8 at β = 0.42.Fig. 9Fig. 10Same as Fig. 8 at β = 0.423.Fig. 10Fig. 11Same as Fig. 10 (right) with data points for triangles having an angle larger than 2π/3 explicitly marked as σqqq(Y → Λ).Fig. 11Fig. 12σww versus β. The solid green line gives the result of the fit with the analytic form A(βc−β)ν.Fig. 12Fig. 13Γ2ad versus R at β = 0.41 (left) and β = 0.42 (right). The solid green line gives the result of the fit with the function in Eq. (53).Fig. 13Fig. 14Γ3ad versus the height of triangles with base b = 4 and β = 0.41 (left) and β = 0.42 (right). The solid green line gives the result of the fit with the function in Eq. (54).Fig. 14Fig. 15Γ2f versus R at β = 0.43 (left) and β = 0.44 (right). The solid green line gives the result of the fit with the function in Eq. (55).Fig. 15Fig. 16Γ3ad versus the height of triangles with base b = 4 at β = 0.43 (left) and β = 0.44 (right). The solid green line gives the result of the fit with the function in Eq. (56).Fig. 16Fig. 17Γ2ad versus R at β = 0.43 (left) and β = 0.44 (right). The solid green line gives the result of the fit with the function in Eq. (57).Fig. 17Fig. 18Γ3ad versus the height of triangles with base b = 4 at β = 0.43 (left) and β = 0.44 (right). The solid green line gives the result of the fit with the function in Eq. (58).Fig. 18Table 1Bestfit parameters η and σqq‾, obtained from fits of the Monte Carlo values for σqq‾eff(R) with the function σqq‾+η/2ln[R/(R−2)] on a 128 × 128 lattice. The second and third columns give the minimum and maximum values of the distance R considered in the fit. The last column gives the determination of σww.Table 1βRminRmaxησqq‾χr2σww
0.414200.6242(36)0.3309(12)0.600.3436(49)
6200.659 (14)0.3245(28)0.35
8200.735 (50)0.3144(69)0.28
10200.61 (18)0.327 (19)0.31

0.4124200.6274(33)0.29939(10)0.700.3101(32)
6200.601 (14)0.3044 (27)0.49
8200.595 (54)0.3051 (72)0.59
10200.62 (18)0.303 (19)0.74

0.4144260.6439(34)0.2620 (10)0.750.2733(16)
6260.620 (13)0.2663 (25)0.60
8260.600 (44)0.2687 (57)0.66
10260.60 (0.12)0.269 (13)0.75

0.4154220.6480(25)0.2434(16)0.580.2501(20)
6220.6330(93)0.2460(17)0.47
8220.611 (28)0.2487(38)0.50
10220.560 (83)0.2537(84)0.55

0.4164260.6553(43)0.2242(13)1.630.2356(14)
6260.623 (13)0.2296(24)1.04
8260.640 (61)0.2305(53)1.16
10260.57 (10)0.235 (11)1.29

0.4174280.6649(31)0.20343(94)1.190.2144(10)
6280.6349(69)0.2084 (12)0.43
8280.627 (21)0.2093 (27)0.47
10280.598 (52)0.2121 (86)0.51

0.4184280.6747(28)0.18187(83)1.120.1927(13)
6280.6515(75)0.1857 (13)0.61
8280.647 (22)0.1863 (28)0.67
10280.598 (49)0.1909 (50)0.66

0.4194360.6835(35)0.16037(97)1.620.1700(21)
6360.668 (10)0.1628 (18)1.48
8360.667 (27)0.1630 (34)1.59
10360.690 (58)0.1609 (57)1.69

0.424360.6938(38)0.13758(53)0.690.1476(13)
6360.6841(54)0.13900(90)0.59
8360.670 (13)0.1405 (15)0.58
10360.652 (26)0.1421 (25)0.59

0.4234560.7756(16)0.05295(40)1.330.05809(56)
6560.7887(37)0.05143(52)0.87
8560.7841(68)0.05185(73)0.89
10560.781 (11)0.0521(90)0.92

0.4244520.8844(44)0.02306(68)3.840.01612(30)
6520.9089(44)0.02073(53)1.30
8520.9132(71)0.02040(69)1.33
10520.903 (11)0.02106(90)1.31
Table 2Parameters extracted from the fits of the Γ2ad and Γ3ad at some given β < βc. For each value of β the first line contains the result of the fit of Γ2ad to (53), and the next lines contain the result of the fit of the values of Γ3ad obtained for the isosceles triangles with fixed base b to (54).Table 2βbMAσηχr2
0.410.557288(39)0.743(12)0.284(34)0.57(10)0.11
20.55102(39)0.8567(90)0.308(16)0.419(53)0.094
40.55637(15)0.777(22)0.348(21)0.278(80)0.21
60.55716(10)0.742(61)0.330(33)0.32(15)0.27
80.557269(75)0.65(11)0.354(43)0.17(22)0.25

0.4150.645366(83)0.792(18)0.235(26)0.499(79)0.023
20.63984(75)0.878(13)0.244(22)0.400(73)0.047
40.64379(29)0.825(30)0.247(24)0.376(94)0.051
60.64501(21)0.791(67)0.255(30)0.34(14)0.036
80.64526(17)0.77(13)0.250(40)0.36(22)0.062

0.420.79511(12)0.870(16)0.134(16)0.531(53)0.0078
20.79391(79)0.907(11)0.143(13)0.452(48)0.048
40.79353(24)0.880(24)0.140(14)0.450(59)0.064
60.79433(30)0.885(50)0.136(16)0.474(83)0.040
80.79487(26)0.90(10)0.132(23)0.51(14)0.040
Table 3Parameters extracted from the fits of Γ2f and Γ3f at β > βc. For each value of β the first line contains the result of the fit of Γ2f to (55), and the next lines contain result of the fit of the values of Γ3f obtained for isosceles triangles with fixed base b to (56).Table 3βbMαmηχr2
0.4251.15180(36)0.4186(45)0.1031(76)0.527(29)0.028
21.17711(55)0.2368(36)0.1089(80)0.510(34)0.051
41.16585(43)0.2600(80)0.1055(96)0.531(48)0.028
61.16034(48)0.287(17)0.100(11)0.572(70)0.0093
81.15729(53)0.314(31)0.095(14)0.61(10)0.0036

0.431.4678468(36)0.18067(16)0.3663(25)0.7054(77)0.00012
21.474944(26)0.11717(43)0.3905(62)0.574(21)0.0029
41.469304(13)0.1292(12)0.3754(66)0.610(26)0.0019
61.468205(12)0.1490(32)0.3541(85)0.728(40)0.0014
81.467957(11)0.1714(88)0.348(13)0.808(71)0.0012

0.4351.582348(16)0.1364(26)0.560(77)0.71(23)0.0039
21.586050(80)0.0908(24)0.568(44)0.58(14)0.012
41.582808(45)0.0997(67)0.542(49)0.67(18)0.0077
61.582422(37)0.122(20)0.512(61)0.87(27)0.0047
81.582353(30)0.194(61)0.436(73)1.37(37)0.0052

0.441.6581470(11)0.11482(22)0.722(13)0.687(37)0.0020
21.660580(28)0.0744(10)0.789(40)0.34(12)0.0077
41.658353(11)0.0754(49)0.780(63)0.34(23)0.0088
61.6581876(83)0.069(18)0.89(13)0 ± 0.550.0085
81.6581549(54)0.090(53)0.80(17)0.36(91)0.0042
Table 4Parameters extracted from the fits of Γ2ad and Γ3ad at β > βc. For each value of β the first line contains the result of the fit of Γ2ad to (57), and the next lines contain result of the fit of the values of Γ3ad obtained for isosceles triangles with fixed base b to (58).Table 4βbMαmηχr2
0.4251.52849(31)0.04314(83)0.086(16)1.097(63)0.0062
21.54873(77)0.03227(73)0.100(15)1.008(59)0.022
41.53420(47)0.0348(16)0.093(17)1.043(81)0.0087
61.53086(50)0.0385(35)0.084(20)1.11(12)0.0046
81.52966(53)0.0422(64)0.079(23)1.17(16)0.0049

0.431.9422411(77)0.0385(23)0.335(17)1.079(93)0.019
21.95648(20)0.02654(42)0.369(19)0.891(64)0.018
41.94449(13)0.0287(14)0.345(32)0.97(13)0.016
61.94279(11)0.0343(38)0.316(40)1.15(19)0.021
81.94237(10)0.0458(89)0.275(44)1.43(24)0.017

0.4352.159588(31)0.03716(58)0.546(69)0.96(21)0.026
22.17020(25)0.02495(70)0.530(61)0.89(20)0.051
42.16073(12)0.0270(35)0.51(11)0.96(42)0.033
62.15976(76)0.033(12)0.49(16)1.16(69)0.023
82.159588(60)0.063(28)0.38(12)1.87(67)0.028

0.442.3244414(44)0.03673(14)0.727(26)0.840(76)0.00080
22.33305(39)0.0234(13)0.74(14)0.64(43)0.011
42.32509(12)0.0236(25)0.729(67)0.65(25)0.017
62.324522(75)0.0245(64)0.801(92)0.45(38)0.0087
82.324454(56)0.060(48)0.56(21)1.7(29)0.015
Threequark potentials in an SU(3) effective Polyakov loop modelO.Borisenkoaoleg@bitp.kiev.uaV.Chelnokovb1Volodymyr.Chelnokov@lnf.infn.itE.Mendicellicemanuelemendicelli@hotmail.itA.Papabcpapa@fis.unical.itaBogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, 03143 Kiev, UkraineBogolyubov Institute for Theoretical PhysicsNational Academy of Sciences of UkraineKiev03143UkrainebIstituto Nazionale di Fisica Nucleare, Gruppo collegato di Cosenza, I87036 Arcavacata di Rende, Cosenza, ItalyIstituto Nazionale di Fisica NucleareGruppo collegato di CosenzaArcavacata di RendeCosenzaI87036ItalycDipartimento di Fisica, Università della Calabria, I87036 Arcavacata di Rende, Cosenza, ItalyDipartimento di FisicaUniversità della CalabriaArcavacata di RendeCosenzaI87036Italy1On leave from Bogolyubov Institute for Theoretical Physics, Kiev.Editor: Hubert SaleurAbstractThreequark potentials are studied in great details in the threedimensional SU(3) 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–antiquark 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 colours N. The threequark potential is tested against the expected Δ and Y laws and the 3q string tension entering these laws is compared to the conventional qq¯ string tension. As a byproduct of this investigation, essential features of the critical behaviour across the deconfinement transition are elucidated.1IntroductionThe interest in studying the interquark potential for a threequark 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 [3–17]. New results are somewhat contradictory, which could be reasonably explained by the difficulty of accurate measurements of the threequark potential. But from these discussions spanning many years, two main Änsatze emerged to describe the threequark potential: the Δ law and the Y law. Denoting by r1, r2, r3 the sides of the triangle having the quarks at its vertices, the Δ law is defined by(1)V3=12σqq‾(r1+r2+r3)=σqq‾Δ, which describes a potential linearly rising with half the perimeter of the triangle. The Y law describes the threequark potential as linearly rising with the minimal total length of the flux lines connecting the three quarks,(2)V3=σqq‾Y, where Y is the sum of the distances of the three quarks from the FermatTorricelli point F, which is the point such that this sum is the least possible. When all inner angles of the triangle are less than 2π/3, one has(3)Y=r12+r22+r32+43A2, where A is the area of the triangle; if one of the angles is larger than 2π/3, we have instead(4)Y=min(r1+r2,r1+r3,r2+r3)≡Λ, which gives rise to the Λ law,(5)V3=σqq‾Λ. Some earlier [4,8,9] and the most recent studies [13,16] in the SU(3) pure gauge theory seem to support the Yansatz, while other simulations [3,5–7] prefer the Δansatz, at least for not too large triangles. An even more complicated picture emerged after simulations of the simpler, Z(3) Potts model in twodimensions, 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 Δ law and the Y law when the size of triangles grows (see, however, [11] where this scenario has been criticized). Also, the paper [10] proposes a new ansatz in which both the Y law and the Λ law are present.In this paper we are going to study an SU(3) 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 [10], only its twodimensional version. Our primary goal is to get some analytical predictions for the threepoint correlation function of the Polyakov loops and compare them with numerical simulations. For that we use the SU(3) spins both in the fundamental and adjoint representations. The main tool of our analytical investigation is the largeN expansion. Within this expansion we demonstrate that the fundamental threepoint correlator is described by a sum of the Y and Λ laws. The Δ contribution is not present. In turn, the connected part of the adjoint threepoint correlation follows the Δ 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 twodimensional Z(3) spin model.This paper is organized as follows. In the next section we introduce our notations, define SU(N) Polyakov loop model and its dual. Certain analytical predictions for two and threepoint correlation functions are obtained in the strong coupling expansion and in the largeN limit. Moreover, we check the restoration of the rotational symmetry for the 3quark 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 finitesize scaling analysis. Section 4 presents the results of Monte Carlo simulations for the fundamental and adjoint two and threepoint 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.2The model and theoretical expectations2.1Partition and correlation functionsWe work on a 2d Euclidean lattice Λ=L2, with sites x=(x1,x2), xn∈[0,L−1], and denote by en the unit vector in the nth direction. Periodic boundary conditions (BC) are imposed in all directions. Let W(x)∈SU(N), and TrW be the character of SU(N) in the fundamental representation. Consider the following partition function on Λ, which describes the interaction of nonAbelian spins:(6)ZΛ(β,N)=∫∏xdW(x)exp[β∑x,nReTrW(x)TrW⁎(x+en)]. The trace of an SU(N) matrix can be parameterized with the help of N angles, e.g. by taking W=diag(eiω1,⋯,eiωN), subject to the constraint ∏keiωk=1. In this parameterization the action has the form(7)ReTrW(x)TrW⁎(x+en)=∑i,j=1Ncos[ωi(x)−ωj(x+en)]. The invariant measure for SU(N) is given by(8)∫dW=∫02πD(ω)D⁎(ω)δ(∑kωk)∏k=1Ndωk2π, where(9)D(ω)=∏k<l(eiωk−eiωl) and δ(x) is the periodic deltafunction. Due to this constraint, the SU(N) model is invariant only under the global discrete shift ωk(x)→ωk(x)+2πnN for all k and x. This is just the global Z(N) symmetry.The main subjects of this work are the two and threepoint correlation functions for the SU(3) model. In the fundamental representation these correlations are given by(10)Γ2f(β,R)=〈TrW(0)TrW⁎(R)〉,(11)Γ3f(β,{xi})=〈TrW(x1)TrW(x2)TrW(x3)〉, while in the adjoint representation the correlations are written as(12)Γ2ad(β,R)=〈χad(W(0))χad(W(R))〉,(13)Γ3ad(β,{xi})=〈χad(W(x1))χad(W(x2))χad(W(x3))〉, where we use the relation χad(W)=TrWTrW⁎−1.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 3d lattice gauge theory (LGT) at finite temperature (see, e.g., [18] and references therein). Namely, the integration over the spatial gauge links on the anisotropic (d+1)dimensional lattice with two couplings βs and βt≡β in the limit βs=0 and for β sufficiently small leads to the ddimensional spin model (6). It describes the deconfinement phase transition of the pure gauge theory, which is of second order for SU(3) if d=2. It is widely assumed that the phase transition is in the universality class of the twodimensional Z(3) (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 ν=5/6. Another important critical index η, which is a characteristic of the massless phase, equals 4/15 exactly at the critical point. Thus, on the basis of the universality arguments [19] we expect the same values for these indices also in the effective SU(3) Polyakov loop model. More on the critical behaviour of threedimensional SU(N) LGTs can be found in Refs. [20,21].The model (6) cannot be solved exactly at any finite N and D>1. Therefore, to get some analytical predictions for the behaviour of the threepoint correlation functions we consider the largeN limit of the model. This limit can in turn be solved exactly by using the dual representation which we are going to describe shortly.2.2Dual representationIn some applications the dual formulation of the Polyakov loop model (6) can be useful. Such formulation for the SU(3) model has been derived in [22]. Here we use the dual representation obtained by some of us in [23,24]. This form of dual theory is valid for all N and can be used both for numerical simulations and for the study of the largeN limit of the theory. For the 2d theory the partition function (6) on the dual lattice takes the form(14)ZΛ(β,N)=∑{r(x)}=−∞∞∑{q(l)}=0∞∑{k(l)}=−∞∞∏pQN(s(p),s¯(p))×∏l[(β2)r(x)−r(x+en)+k(l)N+2q(l)(q(l)+r(x)−r(x+en)+k(l)N)!q(l)!], where QN(s,s¯) results from the invariant integration over the SU(N) measure,(15)QN(s,s¯)=∑λ⊢min(s,s¯)d(λ)d(λ+kN). Here d(λ) is the dimension of the irreducible representation λ of the permutation group Ss, s−s¯=kN and(16)s(p)=12k(p)N+∑l∈p(q(l)+12r(l)),(17)k(p)=k(l1)+k(l2)−k(l3)−k(l4),li∈p,r(l)=r(x)−r(x+en)+k(l)N. λ is enumerated by the partition λ=(λ1,λ2,⋯,λl(λ)) of s, i.e. ∑i=1l(λ)λi=s, where l(λ) is the length of the partition and λ1≥λ2⋯λl(λ)>0. The sum in (15) is taken over all λ’s such that l(λ)≤N and the convention λ+qN≡(λ1+q,⋯,λN+q) has been adopted. For the exact expressions of the different correlation functions we refer the reader to the paper [24].2.3LargeN solutionUsing the dual representation (14), one can construct an exact solution of the model in the largeN limit [25] and even estimate the first nontrivial corrections specific for the SU(N) 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(18)〈∏x(TrW(x))η(x)(TrW⁎(x))η¯(x)〉=Z(η,η¯)Z(0,0),(19)Z(η,η¯)=∫−∞∞∏xdα(x)dσ(x)(α(x)+iσ(x))η(x)(α(x)−iσ(x))η¯(x)e−∑x,x′Gx,x′(α(x)α(x′)+σ(x)σ(x′))∏x(1+2N!Re(α(x)−iσ(x))N). The Gaussian part describes the solution in the largeN limit, while the product over x in the second line presents the first correction due to SU(N). Gx,x′ is the massive twodimensional 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 threepoint correlations in that limit. Different results are obtained in the small and large β regions separated by the deconfinement phase transition. If we take N=3, then for the confinement phase we get(20)Γ2f(β,R)∼G(β,R),(21)Γ3f(β,{xi})∼∑y∏i=13G(β,xi−y),(22)Γ2ad(β,R)∼G(β,R)2+Mad(β)2,(23)Γ3ad(β,{xi})∼∏i=13G(β,xi−xi+1)+Mad(β)∑i=13G(β,xi−xi+1)2+Mad(β)3 and in the deconfinement phase we get(24)Γ2f(β,R)∼Mf(β)2exp[α(β)G(β,R)],(25)Γ3f(β,{xi})∼Mf(β)3exp[α(β)∑i=13G(β,xi−xi+1)],(26)Γ2ad(β,R)∼(Mad(β)+1)2exp[4α(β)G(β,R)]−2Mad(β)−1,(27)Γ3ad(β,{xi})∼(Mad(β)+1)3exp[4α(β)∑i=13G(β,xi−xi+1)]−∑i=13(Mad(β)+1)2exp[4α(β)G(β,xi−xi+1)]+3Mad(β)+2. In the equations above the Green function in the thermodynamical limit is given by(28)G(β,x)=∫02πdω1dω2(2π)2ei∑nωnxnm(β)+2−∑n=12cosωn∼e−m(β)RR, where R2=x12+x22 and the functional dependence of the mass m on β is different in the confined and deconfined phases. In the confinement phase the mass m(β) coincides with the qq¯ string tension, while in the deconfinement phase this quantity has the meaning of screening mass. Mf(β) and Mad(β) define the fundamental and adjoint magnetizations at a given β, correspondingly. They also depend on the considered phase. For example, Mf(β)=0 in the confined phase. α(β) is another Rindependent quantity which appears due to Gaussian integration around the largeN solution. All four quantities—m(β), α(β), Mf(β), Mad(β)—are known exactly in the largeN expansion.2.43q potentialIn what follows our strategy relies on the assumption that the largeN formulae (20)–(27) remain qualitatively valid (up to one correction explained below) at finite N, in particular for N=3. We expect that the most essential difference between the largeN limit and the N=3 case exhibits itself in the vicinity of the critical point. Indeed, both our solution [25] and the meanfield solution of Ref. [26] reveal the existence of a third order phase transition at large N. Meanwhile, as described above, the SU(3) Polyakov loop model belongs to the universality class of the twodimensional Z(3) model. It means, in particular that the critical behaviour of two and threepoint correlation functions is described by a different set of the critical indices ν and η. Therefore, we shall use (20)–(27) as fitting functions, where the quantities m(β), α(β), Mf(β), Mad(β) are unknown parameters to be found from fits of numerical data. In most cases, we use the asymptotic expansion for the Green function G(β,x) given on the righthand side of Eq. (28). As we explained above, only the critical indices appearing in these quantities can vary with N. Also, we introduce here another quantity, namely the index η, in order to describe the power dependence of the correlation function, R−η, on the distance. This could again be important in the vicinity of the critical point. In general, this introduces a correction to the potential of the form(29)VCoulomb=ηlnD,D=R,Y,Δ,Λ, and is interpreted as the Coulomb part of the full potential in the twodimensional theory.Since the asymptotic behaviour of G(β,x) is known, it follows that we actually know the largedistance behaviour of all two and threepoint functions listed above, but Γ3f(β,{xi}). The behaviour of the latter can be analyzed by the saddlepoint method when at least one side of the triangle is large enough. We find two types of the behaviour:1.All inner angles of the triangle are less than 2π/3. The threepoint fundamental correlation function is given by the sum of two terms corresponding to Y and Λ laws(30)Γ3f(β,{xi})≈Ae−σqqqYY+Be−σqqqΛΛ, where A,B are constants and σqqq=σqq¯. This behaviour resembles the behaviour of the threepoint correlation function in the Z(3) spin model [10].2.One of the angles is larger than 2π/3. In this case the asymptotics is described by the above formula with A=0. Thus, only the Λ law is present. This again agrees with the Z(3) spin model. Let us also emphasize that we could not find the Δ law contribution in our largeN approach. Nevertheless, we attempt to fit numerical data both to Y and Δ laws in the following. Finally, let us stress that the connected part of the threepoint adjoint correlation follows the Δ law in the confinement phase, as is seen from Eq. (23).2.5Strong coupling expansionWhen β is sufficiently small, one can use the conventional strong coupling expansion to demonstrate the exponential decay of the fundamental two and threepoint correlation functions. Instead, adjoint correlations stay constant over large distance. To check our codes we have calculated the leading orders of the strongcoupling expansion for the twopoint correlator at distance R=1 and for the threepoint correlator in the isoscelestriangle geometry T with base b=2 and height h=1. The results read(31)Γ2f(β,1)=12β+18β2+98β3+385384β4+O(β5),(32)Γ3f(β,T)=18β3+12β4+145128β5+298β6+O(β7),(33)Γ2ad(β,1)=14β2+16β3+178β4+O(β5),(34)Γ3ad(β,T)=2716β6+487192β7+O(β8). For arbitrary isosceles triangle T with base b and height h one obtains(35)Γ3f(β,T)∼βh+b≡βYl. On a cubic lattice Yl=h+b is the minimal sum of the lattice distances from the triangle vertices to an arbitrary lattice point. Then, according to Eq. (2),(36)V3=σqq‾Yl,σqq‾≈lnβ in the strong coupling region on the finite lattice. Thus, strictly speaking the strong coupling expansion predicts not an exact Y law, as it is often stated in the literature, but rather a Yl law. In general, Yl>Y and we expect that the rotational symmetry will be restored quickly with β and the triangle sides increasing. This should result in the restoration of the genuine Y law. To demonstrate that such a restoration really takes place, we have studied the threepoint correlation function for triangles with 2≤b≤10 and 6≤h≤14 at β=0.41. The fact that the rotational symmetry is already restored at this value of β is shown on Fig. 1, where we compare numerical data with the fits of the form e−σqqqD/Dη for D=Yl and D=Y. Clearly, D=Y describes data better than D=Yl.3Details of numerical simulationsTo 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 ωi(x) of the SU(3) spins, described in more detail in [21]. 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 Z(3) reflections of the clusters. An alternative approach is the simulation of the dual model (14), using the heatbath update for the link variables k, q and the dual site variables r. In this case, we can measure only observables invariant under the global Z(3) symmetry.In both approaches we measured two and threepoint correlation functions in the fundamental and adjoint representations, taking for the two point correlation function pairs of points separated by R in one of the two lattice directions, with R=2,4,…,L/2. For the threepoint correlation functions two geometries were studied: isosceles triangles with base b and height h, and rightangled triangles with the catheti (of lengths a1 and a2) along the two lattice directions. In both cases, b and h, and a1 and a2, took independent values in the set {2,4,…,L/2}.In addition to the two and threepoint correlations (10)–(13), the magnetizations and their susceptibilities were measured:(37)Mf=〈χf(Wx)〉=〈TrWx〉,(38)Mad=〈χad(Wx)〉=〈TrWxTrWx⁎−1〉,(39)χL(Mf)=L2(〈(χf(Wx))2〉−〈χf(Wx)〉2),(40)χL(Mad)=L2(〈(χad(Wx))2〉−〈χad(Wx)〉2). The xdependent values are averaged over all sites of the lattice.For each simulation we performed 104 thermalization updates, and then made measurements every ten whole lattice updates (sweeps), collecting a statistics of 105–106. 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 β, while giving much larger fluctuations than the standard one when β is close to its critical value. What is more important—at larger values of β 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 crosscheck purposes.3.1Comparison with strong couplingTo test our algorithms we performed a set of simulations at small values of β (β<0.15), and compared the obtained values of Γ2f, Γ3f, Γ2ad and Γ3ad 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 twopoint correlation, both in the fundamental and adjoint representation, is in good agreement with the strong coupling expansion. For the threepoint 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 β=0.1. Since the results of the two simulation codes agree in the region around βc, where most of our simulations were carried out, we are confident in the reliability of our measurements.3.2Critical behaviourA clear indication of the twophase structure of the model is provided by the scatter plots of the complex magnetization at different values of β, shown in Fig. 4.To precisely locate the βc at which the phase transition occurs, we have studied the magnetization susceptibility for different lattice sizes L, extracting the value of βpc(L) from a fit of the peak of the susceptibility with a Lorentzian function. The obtained values of βpc(L) have been fitted with the scaling law for a second order transition (see the left panel of Fig. 5)(41)βpc(L)=βc+AL1/ν with the following resulting parameters:A=−0.0675(62),βc=0.4242748(39),ν=0.835(17),χr2=1.18. The value for the critical index ν is in agreement with the critical index ν=5/6 of the twodimensional threestate Potts model, to whose universality class our model is believed to belong. A direct extraction of the critical exponent ν, performed in the subsection 4.3, gives a compatible result, which is sensitive to the choice of the region of β 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)(42)χL(M)(βpc)(L)=BLγ/ν. We foundB=0.0282(27),γ/ν=1.737(17),χr2=0.30. The obtained value for γ/ν is in agreement with the hyperscaling relation 2−η=γ/ν for the threestate Potts model, which gives η≈0.263. The expected value of η is 4/15. These findings support the Z(3) universality class of the present Polyakov loop spin model.4Correlation functions in confinement phase4.1Extraction of σqq‾ from Γ2fThe potential parameter σqq‾ is extracted from the measurements of the observable Γ2f. Following Eq. (20) and the explanation in subsection 2.4, we expect(43)Γ2f(R)=Ae−σqq‾RRη. One can extract σqq‾ from the following ratio:(44)σqq‾eff(R)≡−12ln[Γ2f(R)Γ2f(R−2)]=σqq‾+η2ln[RR−2]. We have compared our Monte Carlo data for σqq‾eff(R) with the formula σqq‾+η/2ln[R/(R−2)]. The interval of β values that we considered for the extraction of σqq‾ was [0.41,0.42], since for β<0.41 the twopoint correlation drops too fast to be of any use. The values of σqq‾eff obtained in the selected range of β do not show any significant difference when moving from a 64×64 to a 128×128 lattice, thus making unnecessary to perform simulation on even larger lattices.As an alternative method for extracting σqq‾, we measured the wall–wall correlation function,(45)Γ2ww(R)=〈1L3∑x,y1,y2=1LTrW(x,y1)TrW†(x+R,y2)〉, which is known to obey the exponential decay law, with no power corrections,(46)Γ2ww(R)=Ae−σqq‾R.Introducing, similarly to (44),(47)σwweff(R)≡−12ln[Γ2ww(R)Γ2ww(R−2)], we found that σwweff(R) exhibits a long plateau at each considered β value; we took as plateau value σww the value of σwweff(R) at the smallest value of R after which all values of σwweff(R) agree within statistical uncertainties. Results for the 128×128 lattice are summarized in Table 1: we can see that results of σqq‾ obtained from the fitting of σqq‾eff(R) according to (44) are in good agreement with the plateau values of σww. We ascribe the difference between the σqq‾ values obtained for different choices of Rmin to possible systematic effects arising from the difficulty to extract the parameters of the exponential decay corrected by a power law and to our treating of the correlation function errors as independent values. These systematic errors seem also to result in a value of σww being in most cases slightly higher than the estimates of σqq‾.4.2Extraction of σqqq from Γ3fFirst we studied the dependence of Γ3f on the geometry, considering Δ and Y laws. In Fig. 6 we see that if we consider σ to be proportional to σww we get a reasonable collapse for all β values except the largest one (β=0.424), which might be too close to the critical point for our lattice size, L=128. Still this does not allow us to discriminate between the two laws.We turned therefore to fits with the two laws of the threepoint correlation function for small (σwwY<2) triangles, excluding those having an angle larger than 2π/3. The results of these fits for β=0.423 are shown in Fig. 7. In this case the fit with the Δ law gives A=2.319(14),σ=0.0631(4),η=0.503(4),χr2=28.6, while the fit with the Y law gives A=2.502(17),σ=0.0573(3),η=0.506(5),χr2=4.67. For all values of β the Y law performed better than the Δ law (giving smaller χr2), the difference getting more and more clear for larger values of β.To extract σqqq we followed a procedure similar to the one used for σqq‾. We supposed, according to (21) and (30), the decay law(48)Γ3f(R)=Be−σqqqRRη. In this subsection R is a distance parameter depending on the geometry, that could be Δ or Y, respectively for the Δ and Y laws, given in Eqs. (1) and (2). In this proposed fitting function we excluded the second term, corresponding to the Λ law, since, for the size of triangles considered in the fitting procedure, we could hardly distinguish it from the first one, corresponding to the Y law.22Recently, a new method was suggested to reveal the distinction between the Δ and Y laws, based on the use of hyperspherical threebody variables [17].We calculated the following ratio(49)σqqqeff=−1R1−R2ln[Γ3f(R1)Γ3f(R2)] for the pairs of triangles (b+2,h+2) and (b,h) for the isosceles geometry and (a1+2,a2+2) and (a1,a2) for the rightangled geometry, where R1 and R2 are the distance parameters of the two triangles. The ratio is equal to(50)σqqqeff=−1R1−R2ln[Γ3f(R1)Γ3f(R2)]=σqqq+ηR1−R2ln[R1R2], where, for sufficiently large distances we can assume R1∼R2∼R and get(51)σqqqeff=−1R1−R2ln[Γ3f(R1)Γ3f(R2)]≃R≫1σqqq+ηR.It turned out that for part of the triangle pairs σqqqeff 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 σqqqeff was possible, strongly depends on the value of β; for example, for the isosceles geometry we had 274 triangle pairs for β=0.41, 558 for β=0.42 and 783 for β=0.423.After extracting σqqqeff, we plotted it directly against the halfperimeter Δ and against the sum Y of the distances of the triangle vertices from the FermatTorricelli point. We overlapped these plots with the plots of σqq‾eff and σwweff versus R (Figs. 8–10). We see that on the plots for the Δ law the values of σqqqeff fail to collapse into a single line, while the collapse is much better for the Y law for all β values we studied. The residual spread of the points can be at least partially explained by different triangle pairs having different R1−R2 values, which are not distinguished on these plots. Another observation that supports the Y law is that the collapse line for the σqqqeff closely matches the line of σqq‾eff, which suggests that not only the sigma values entering the two and the threepoint correlation are the same if we consider the Y law, but also that the parameters η are similar in these cases.It is worth mentioning that in our study the results of the extraction of σqqqeff are compared for triangles that have strongly different geometries: there are triangles that have similar Y 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 2π/3 from the study of σqqqeff, despite the fact that for them the FermatTorricelli point coincides with one of the vertices, thus leading to a different dependence of Y on h and b. The fact that even these “extreme” triangles obey the Y 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 σqqq(Y→Λ) implies that for such triangles the FermatTorricelli point coincides with one of the vertices turning the Y law into the Λ law. As can be seen from Figs. 8–10 these differences in geometry give negligible corrections to the values of σqqqeff up to β=0.423, providing us with an additional point in support of the Y law.4.3Extraction of critical index ν from the scaling of the twopoint string tensionWe used the values of the string tension obtained from the wall–wall correlation function close to the critical point to extract the critical index ν.The values of σww, as well as the result of the fit in the region 0.422<β<βc with the scaling function(52)σ=A(βc−β)ν,A=12.5(2.8),βc=0.424255(34),ν=0.806(37),χr2≈3.69, are shown in the Fig. 12. The value of the critical index ν obtained in this way is compatible with both the critical index ν=5/6 of the threestate 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 ν is quite sensitive to the inclusion of the points outside this region.4.4Adjoint correlations in the confinement phaseWe have performed measurements of the two and threepoint correlation functions in the adjoint representation, defined in Eqs. (12) and (13), at some values of β below the critical one.Following formulae (22) and (23) and replacing in them the massive Green function G(β,R) with its asymptotic behaviour, we got the following models:(53)Γ2ad(R)=Mad2+Aexp[−2σR]R2η.(54)Γ3ad({xi})=A3∏i=13exp[−σxi−xi+1]xi−xi+1η=X+MadA2∑i=13exp[−2σxi−xi+1]xi−xi+12η+Mad3. Note that Γ3ad({xi}) exhibits the Δlaw decay after subtracting terms proportional to (powers of) the magnetization.The results of the fitting of the adjoint correlations to the models in Eqs. (53) and (54) are given in Table 2 (see also Figs. 13 and 14).Unusually low χr2 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 η value is inaccurate, since it describes shortrange 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 η and σ which is very close to −1.Despite that, the fact that the parameters σ and M show some degree of stability when going from the description of the twopoint correlation to the threepoint one, and also the compatibility of the results for σ with the σqq‾ and σww 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 Mad extracted from the fits do agree with direct measurements of this quantity as defined in Eq. (38).5Correlation functions in deconfinement phaseUsing the same approach adopted in the subsection 4.4 for the description of adjoint correlations in the confinement phase, we get the following models for the correlations in the deconfinement phase from Eqs. (24)–(27):(55)Γ2f(R)=Mf2exp[αexp[−mR]Rη],(56)Γ3f(β,{xi})=Mf3exp[α∑i=13exp[−mxi−xi+1]xi−xi+1η],(57)Γ2ad(R)=(Mad+1)2exp[4αexp[−mR]Rη]−2Mad−1,(58)Γ3ad({xi})=(Mad+1)3exp[4α∑i=13exp[−mxi−xi+1]xi−xi+1η]−∑i=13(Mad+1)2exp[4αexp[−mxi−xi+1]xi−xi+1η]+3Mad+2, where m can be interpreted as the chromoelectric screening mass.We used these formulae to describe the correlation functions measured above βc. The remarks given at the end of subsection 4.4 apply here as well. The results of the fits are gathered in Table 3 for the fundamental correlations (see also Figs. 15 and 16) and in Table 4 for the adjoint ones (see also Figs. 17 and 18). We see that the values of M and m extracted from the fits for twopoint and threepoint correlations are compatible. Moreover, the m values extracted from the fits of the fundamental and adjoint correlations are also compatible between themselves (the same should not apply to the M values, since they represent the average magnetization in two different representations). This supports the validity of the formulae (55)–(58) for the description of the correlation functions in the deconfinement phase.Also in this case, we found that the values of Mad extracted from the fits do agree with direct measurements of this quantity as defined in Eq. (38).In the deconfinement phase the value of m becomes the inverse correlation length for the connected part of the correlation, as can be seen from the Taylor expansion of the outer exponent in Eq. (55). When we approach the critical point from above the value of m should vanish as(59)m=Adec(β−βc)ν. The ratio Adec/A, where A is the amplitude for the scaling of σ in the deconfinement phase given by Eq. (52), equals 2.657 for 2d Z(3) universality class [27]. Using this universal ratio and the parameters obtained in subsection 4.3, we have calculated the prediction that Eq. (59) gives for the values of m. It turns out that for β=0.425 the predicted value m=0.100(35) is in good agreement with the values of m in Tables 3, 4, while for larger values of β this agreement becomes worse (m=0.52(15) for β=0.43). This might be explained by the scaling holding only in a narrow region around βc, similarly to Fig. 12.6SummaryIn this study we performed an extensive analysis of a twodimensional effective SU(3) Polyakov loop model. Differently from other approaches of the same kind, in our effective model the basic degrees of freedom are traces of SU(3) matrices and not Z(3) spins. The partition function is therefore integrated with a groupinvariant measure. The motivation for this choice is that it can help to catch some important features of the (2+1)dimensional SU(3) lattice gauge theory at finite temperature that escape approaches based on the centre degrees of freedom. For example, one cannot define adjoint correlation functions in Z(3) models. Our main goal was to examine the behaviour of the threequark potential below the critical point and to distinguish between possible scenarios for the threepoint correlation function decay: the Δ law and the Y law. Considering the triangle whose vertices are the positions of the three sources, the Δ potential depends on the perimeter of this triangle, while the Y potential depends on the sum Y of the distances of its vertices from the FermatTorricelli point. Other studies accomplished in the paper include investigation of the two and threepoint correlation functions in the adjoint representation both below and above critical point and calculation of critical indices in the vicinity of the deconfinement phase transition.Our main findings can be summarized as follows:•Similarly to the pure gauge SU(3) LGT and Z(3) spin model, the leading contribution to the threepoint fundamental correlation function in the strong coupling region is described by the Yl law, as explained in section 2.5. Exact Y law is restored as soon as the rotational symmetry is also restored.•From the study of the largeN limit of the model we also obtained the general form for the two and threepoint correlations in fundamental and adjoint representations both above and below the critical point. The analytical results suggest that the fundamental threepoint correlation behaves as in Eq. (30), i.e. it is described by a combination of Y and Λ laws if all angles of the triangle are less than 2π/3 or by only the Λ law if any of the angles is larger than 2π/3. We have not found analytical support in favour of Δ law (one possibility is that this law is suppressed in the largeN limit). The comparison with the results of numerical simulations shows that these forms can indeed be used to describe the behaviour of the corresponding correlations.•The critical behaviour across the deconfinement transition supports the universality conjecture that this model is in the universality class of the twodimensional Z(3) spin model. In particular, we have determined the critical indices ν and η from finite size scaling (the index ν has been also evaluated directly from the twopoint correlation). Their values agree well with the values in the Potts model.•The fact that the assumption of Y law gives much better collapse of the effective string tension to a single curve, when different locations of the three sources are taken with the same value of Y, indicates that in this effective model the Y law is preferred. We have found the agreement of the effective string tensions for the two and threepoint correlation functions. Moreover, all string tensions appearing in the twoquark potential and in the threequark potential with Y or Δ law agree up to uncertainties. This result is also supported by the analytic study of the model in the largeN limit. Since our results are supportive of the Y law even for small triangles, we do not observe a smooth crossover from Δ to Y law as conjectured in [6,10]. For triangles with one of the angles larger than 2π/3 the threepoint fundamental correlation function follows the Λ law.•In the deconfinement phase the screening masses for fundamental two and threepoint correlations also coincide up to numerical errors.•Adjoint correlations share a similar pattern in the deconfinement phase, namely the corresponding screening masses are consistent with each other for two and threepoint correlators. Moreover, they seem to coincide in the deconfined phase with the fundamental screening masses.•In the confined phase the connected part of the adjoint twopoint correlation equals the square of the fundamental one after subtraction of magnetization. The connected part of the adjoint threepoint correlation is consistent with the Δ law below the critical point and agrees with the largeN predictions.This work can be straightforwardly extended to the case of three dimensions, which is certainly more relevant from the physical point of view, though being technically more involved. Another possible extension is to consider the threequark system in different colour channels, both in the confined and in the deconfined phase (see, e.g., Ref. [28]).AcknowledgementsWe gratefully acknowledge useful discussions with Leonardo Cosmai. Numerical simulations have been performed on the ReCaS Data Center of INFNCosenza. V.C. acknowledges financial support from the INFN HPC_HTC and NPQCD projects. A.P. acknowledges financial support from the INFN NPQCD project. O.B. also thanks INFN (Fondo FAI) for financial support.References[1]R.SommerJ.WosiekPhys. Lett. B1491984497[2]R.SommerJ.WosiekNucl. Phys. B2671986531[3]G.BaliPhys. 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