# Confinement-Deconfinement transition and Z_2 symmetry in Z_2+Higgs theory

IIMSc/2021/02/01

Conﬁnement-Deconﬁnement transition and Z symmetry in Z + Higgs theory

Minati Biswal, ∗ Sanatan Digal,

2, 3, † Vinod Mamale,

2, 3, ‡ and Sabiar Shaikh

2, 3, § Indian Institute of Science Education and Research, Mohali 140306, India The Institute of Mathematical Sciences, Chennai 600113, India Homi Bhabha National Institute, Training School Complex,Anushakti Nagar, Mumbai 400085, India

Abstract

We study the Polyakov loop and the Z symmetry in the lattice Z +Higgs theory in 4D Euclideanspace using Monte Carlo simulations. The results show that this symmetry is realised in the Higgssymmetric phase for large number of temporal lattice sites. To understand the dependence on thenumber of temporal sites, we consider a one dimensional model by keeping terms of the originalaction corresponding to a single spatial site. In this approximation the partition function can becalculated exactly as a function of the Polyakov loop. The resulting free energy is found to havethe Z symmetry in the limit of large temporal sites. We argue that this is due to Z invarianceas well as dominance of the distribution or density of states corresponding to the action. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - l a t ] F e b . INTRODUCTION Z N symmetry plays an important role in the conﬁnement-deconﬁnement (CD) transitionin pure SU ( N ) gauge theories [1–3]. In these theories, at ﬁnite temperature, the allowedgauge transformations are classiﬁed by the centre of the gauge group, i.e Z N . Under these Z N gauge transformations, i.e Z N symmetry, the Polyakov loop ( L ) transforms like mag-netisation, in spin models [4]. In the conﬁnement and deconﬁnement phases the Polyakovloop acquires vanishing and non-zero thermal average values respectively, hence plays therole of an order parameter for the conﬁnement-deconﬁnement (CD) transition, [5–9]. In thedeconﬁned phase, the Z N symmetry is spontaneously broken which leads to N -degeneratestates [10–12].The Z N symmetry of pure SU ( N ) gauge theory is spoiled when matter ﬁelds are in-cluded. Gauge transformations which are not periodic in temporal directions can not act onthe matter ﬁelds. These may act only on the gauge ﬁelds but in the process the action doesnot remain invariant. There are many studies on the eﬀect of matter ﬁelds on this symme-try. Perturbative loop calculations of the Polyakov loop eﬀective potential show that thissymmetry is explicitly broken by matter ﬁelds in the fundamental representation [13–15].The mean-ﬁeld approximations of lattice partition functions in the strong coupling limit alsoshow the explicit breaking of the Z N symmetry [16, 17]. Non-perturbative studies of CD transition in 2 − colour QCD show a sharp transition suggesting small explicit breaking of Z symmetry [18].Recent non-perturbative Monte Carlo simulations of SU (2)+Higgs theory show that thestrength of Z explicit breaking depends on the Higgs condensate [19]. These studies ﬁndthat the CD transition exhibits critical behaviour in the Higgs symmetric phase for largenumber of temporal sites ( N τ ) [19]. The distributions of the Polyakov loop are found to be Z symmetric, albeit within statistical errors, suggesting the realisation of Z symmetry inthe Higgs symmetric phase. In reference [20], it was argued that the emergence of Z symme-try is due to enhancement of the conﬁguration/ensemble space with N τ . This enhancementmakes it possible that the change in the Euclidean action due to Z “rotation” of gaugelinks can be compensated by appropriately changing the Higgs ﬁeld. This was numericallytested by updating the Higgs ﬁeld using Monte Carlo steps after Z rotating the gauge ﬁelds.2he non-invariance of the action under Z gauge transformation which are not periodicin temporal directions does not necessarily imply the explicit breaking of Z symmetry. Thepresence of Z symmetry or it’s explicit breaking can only be inferred from the free energyof the Polyakov loop. In the free energy or the partition function calculations, two factorsplay an important role. The distribution of the action also known as the density of states( DoS ) and the Boltzmann factor. The latter clearly does not respect the Z symmetry. Sothe realisation of the Z symmetry must come from the DoS and it’s dominance over theBoltzmann factor. Computing the

DoS in SU ( N )+Higgs theory is a diﬃcult task as theconﬁguration space is inﬁnite. In this situation, the Z +Higgs theory in four dimensionsprovides a suitable alternative. Since the ﬁeld variables take values ±

1, it is possible tocalculate the

DoS with some simpliﬁcations.The Z +Higgs theory has been extensively studied in literature [21–27]. The phasediagram of this theory is found to be similar to that of SU ( N )+Higgs theories in 3 and4 − dimensions [28, 29]. This theory has no continuum limit and the temporal direction of SU ( N ) gauge theories has no analog in this case. For the purpose of understanding therole of DoS on the Z symmetry it is suﬃcient to consider the Higgs ﬁeld to be periodicin any one of the four dimensions. This will aﬀect the gauge transformations, which arenot periodic in this direction but preserve the invariance of the pure gauge action, exactlythe same way matter ﬁelds aﬀect the corresponding gauge transformations in SU ( N ) gaugetheories. The Polyakov loop along this “temporal direction” behaves as an order parameterfor the CD transition.In this paper, we consider one of the four dimensions to deﬁne the “temporal direction”and the Polyakov loop. The Z symmetry of this observable and the nature of CD transitionis then studied by varying the number of lattice points, N τ , along the temporal direction.The computations are mostly done on the Higgs symmetric side of the Higgs transition line.Our results show that the Z symmetry is realised for large N τ . Also the behaviour of the CD transition is found to be similar to the pure gauge case apart from the location of thecritical point. To understand the role of N τ a 0 + 1 dimensional model is considered bykeeping temporal component of the gauge Higgs interaction corresponding to a given spatialcoordinate. The reason for this choice is the fact that only the temporal component of thegauge Higgs interaction is sensitive to the Z gauge transformations. For the one dimensionalmodel the Polyakov loop can take values ±

1. For each of these cases the free energy can3e calculated exactly. The free energy calculations show the emergence of Z symmetry inthe large N τ limit for arbitrary interaction coupling. Further the Monte Carlo results forthe distribution of the interaction term is reproduced well by 0 + 1 dimensional DoS with asimple Boltzmann factor, though with a diﬀerent value of the interaction strength coupling.The

DoS for both values of the Polyakov loop is sharply peaked at zero. Z symmetry isclearly observed near the peak, the diﬀerences appear when the action takes the limitingvalues. Since the peak height grows with N τ , the DoS will dominate the thermodynamicsin the N τ → ∞ , restoring the Z symmetry.This paper is organised as follows. In section II, we discuss the Z symmetry in Z +Higgstheory. This is followed by numerical simulations of CD transition and the Z symmetryin pure gauge theory and in the presence of Higgs in section III. In section IV, we derivethe free energy of the Polyakov loop in a 0 + 1 dimensional model, and relate the resultsto 4 − dimensional Monte Carlo simulations. In section V, discussions and conclusions arepresented. II. Z SYMMETRY IN Z + HIGGS GAUGE THEORY.

The action for the Z +Higgs theory in four dimensional Euclidean lattice ( N s × N τ ) is givenby, S = − β g (cid:88) P U P − κ (cid:88) n, ˆ µ Φ n +ˆ µ U n, ˆ µ Φ n . (1)Here n = ( n , n , n , n ) represents a point on the lattice with 1 ≤ n , n , n ≤ N s and1 ≤ n ≤ N τ . As mentioned above we assume that the fourth direction is the temporaldirection. U n, ˆ µ represents the gauge links in ˆ µ direction between the lattice point n and n + ˆ µ . The Higgs ﬁeld Φ n lives at the site n . Both U n, ˆ µ and Φ n take values ± β g is thegauge coupling and κ is the gauge Higgs interaction strength. Figure. 1 shows a schematiclayout of the gauge links and Higgs variable on the lattice.4 U FIG. 1. Position of gauge links U and Higgs ﬁelds Φ on lattice

The plaquette U P which is path ordered product of the links along an elementary square onthe µ − ν plane, i.e U P = U n, ˆ µ U n +ˆ µ, ˆ ν U n +ˆ ν, ˆ µ U n, ˆ ν . (2)Figure. 2 shows the sketch of an elementary plaquette. The pure gauge part of the action,n n+ˆ µ n+ˆ µ +ˆ ν n+ˆ ν U n, ˆ µ U n +ˆ µ, ˆ ν U n +ˆ ν, ˆ µ U n, ˆ ν FIG. 2. Sketch of an elementary plaquette U P ﬁrst term in Eq.1, is invariant under the Z gauge transformations, U n, ˆ µ → V n U n, ˆ µ V − n +ˆ µ (3)where V n = ± ∈ Z . The V n ’s satisfy the following boundary condition, V ( (cid:126)n, n = 1) = zV ( (cid:126)n, n = N τ ) . (4) z = ± ∈ Z . So the gauge transformations can be classiﬁed by the group Z . For z = − L ( (cid:126)n ) = N τ (cid:89) n =1 U ( (cid:126)n,n ) , ˆ4 (5)transforms non-trivially under Z gauge transformations [9]. It is easy to see that thePolyakov loop transforms as, L ( (cid:126)n ) → zL ( (cid:126)n ) . (6)This transformation property of the Polyakov loop under Z (or Z N in general) gauge trans-formation is similar to that of magnetisation in the Ising model. The partition function inthe pure gauge case ( κ = 0) is given by, Z = (cid:90) DU e − S . (7)Since the action for κ = 0 is invariant under Z gauge transformations, any conﬁgurationand it’s gauge rotated counterpart will contribute equally to the partition function. There-fore the distribution of the Polyakov loop exhibits Z symmetry in this case. Equivalentlythe free energy of the Polyakov loop will have Z symmetry.The presence of the Higgs ﬁeld changes the space of allowed gauge transformations. Thereason being that the Higgs ﬁeld is required to be periodic in the temporal direction. Undera gauge transformation, Φ n transforms as,Φ n → V n Φ n . (8)Now the periodic boundary condition of Φ would be spoiled if non-periodic gauge transfor-mations, characterised by z = − Z counterpart in which only the gauge links are Z rotated. Obviously these pair ofconﬁgurations will not contribute equally to the partition function for κ (cid:54) = 0. So accordingto the Boltzmann factor, (cid:80) (cid:126)n L ( (cid:126)n ) and − (cid:80) (cid:126)n L ( (cid:126)n ) are non degenerate. This situation issimilar to the presence of an external ﬁeld in the Ising model. However, the status of Z symmetry in the free energy can be answered only after integrating out the Higgs ﬁeld fora given L ( (cid:126)n ) and it’s Z rotated conﬁgurations.The Polyakov loop and Ising spins are similar in how they transform under respective6ransformations. However there is an important diﬀerence between them. This becomesclear when one compares L ( (cid:126)n ) and an Ising spin at a spatial point (cid:126)n = { n , n , n } . Agiven value of L ( (cid:126)n ) is associated with an entropy factor. This is because there are manydiﬀerent combinations of U ( (cid:126)n,n ) , ˆ4 and Φ (cid:126)n,n are possible for a given value of L ( (cid:126)n ). Largerthe N τ , larger is the corresponding entropy. This aspect of the Polyakov loop needs to betaken into account to understand the explicit breaking or realisation of Z ( Z N ) symmetry,which is done in section IV. In the following section, we discuss the algorithm of the MonteCarlo simulations [30], present simulation results for the phase diagram in the β g − κ plane,distribution of the Polyakov loop and CD transition in the Higgs symmetric phase etc. III. NUMERICAL TECHNIQUE AND MONTE CARLO SIMULATION RESULTS.

In the Monte Carlo simulations, the Metropolis algorithm is used for sampling the sta-tistically signiﬁcant conﬁgurations [31]. To update a particular gauge link U n,µ , we considerthe change in the action by ﬂipping it. If the action decreases then the ﬂipped gauge linkis accepted for the new conﬁguration. If the action increases by ∆ S then the new link isaccepted with probability Exp ( − ∆ S ). The same procedure is adopted for Φ n . The processof updating is carried out over all n and µ in multiple sweeps. Conﬁgurations separated by10 sweeps are used in our analysis, which brings down the autocorrelation between successiveconﬁgurations to an acceptable level. For this simulations, N τ = 4 −

24 and N s = 16 − N s /N τ = 4 lattices have been considered [32].The pure gauge simulations are initially performed to understand the nature of CD transition and Z symmetry of the Polyakov loop. The simulations were repeated in thepresence of Φ to study its eﬀects. The pure gauge transition has been studied previously inthe mean-ﬁeld approximations [21], which ﬁnds the transition is ﬁrst order in four dimen-sions. Also using duality transformations it can be shown that the critical β g ∼ . κ = 0 [22]. These results are supported by Monte Carlo simulations of smaller lattices [23].The simulations carried out in this work are also consistent with these results. In ﬁgure. 3the average of the Polyakov loop is plotted vs β g for N τ = 4 ,

8. There is a range in β g forwhich clearly separated peaks in the distribution of the Polyakov loop has been observed.We take average of the Polyakov loop values corresponding to each peak separately. There-7ore we have two points in the ﬁgure for a given β g . The two peaks also suggest that thetransition is ﬁrst order. For larger lattice sizes the range of β g over which two states areobserved increases [33]. This is expected as strength of ﬂuctuations relatively decrease withvolume (when correlation length is smaller than the spatial size of the system), making itdiﬃcult for the ﬁeld to climb over the barrier and cross to the other side. < | L | > β g κ =0, 16 x4 < | L | > β g κ =0, 32 x8 FIG. 3. The average of the Polyakov loop vs β g for N τ = 4 and 8. The eﬀect of the Φ ﬁeld on the CD transition and Z symmetry is expected to dependon κ . To relate these two aspects of pure gauge theory to the phases of the Higgs ﬁeld,simulations were performed to obtain the Higgs transition line. For a given β g , κ > κ c corresponds to the Higgs broken phase. In this phase the action term dominates. For κ < κ c the ﬂuctuations of the Higgs rather than the action dominate the thermodynamicproperties. This situation is similar to the Ising model at high temperatures. In Fig.4 theHiggs transition line is plotted in the β g − κ plane. The location of the phase boundaryis obtained by studying the κ dependence of the interaction term and it’s ﬂuctuations fordiﬀerent values of β g . In our simulations the Higgs transition is found to be ﬁrst order forintermediate range of β and crossover for both small and large β , as observed in previousstudies [28, 29]. For large β g critical κ c remains ﬂat and increases with β g in the small β g range. As expected the critical values ( β c , κ c ) depend on N τ , however for N τ values used inour simulations the dependence is very mild.In the Higgs broken phase, i.e large κ , the interaction term dominates over the entropy.The action takes the largest value when all the temporal links are +1. So it is expectedthat in the Higgs phase Z symmetry is badly broken, also observed in our simulations. Inthe Higgs symmetric phase, it is the ﬂuctuations of Higgs in other words the distribution8 κ β g x 8 FIG. 4. Phase diagram of the interaction term dominate. In this phase there is a possibility for realisation of Z symmetry. In ﬁgure. 5 we show CD transition in the Higgs symmetric phase ( κ = . κ = 0 results also have been included. The CD transition is ﬁrst order even inthe presence of Φ, though the transition point shifts to lower values of β g . < | L | > β g κ =0,16 x4 κ =0.13,16 x4 < | L | > β g κ =0, 32 x8 κ =0.13, 32 x8 FIG. 5. The average of the Polyakov loop vs β g for N τ = 4 and 8. To check the N τ dependence of the Z symmetry at κ = .

13, the distribution of Polyakovloop is computed both in the conﬁned and the deconﬁned phases for N τ = 2 , , and 8. Inthe deconﬁned phase, L < Z rotated and then compared with L > N τ = 2 the histograms clearly showthere is no Z symmetry. In the deconﬁnement side there is no Z symmetry as the twoPolyakov loop sectors do not overlap. For N τ = 3 the two peaks corresponding to the two9 H ( L ) L κ =0, 12 x2 κ =0.13,12 x2 FIG. 6. Histogram of L in the conﬁnedphase. H ( L ) L κ =0.13,L > 0,12 x2 κ =0.13,L < 0,12 x2 FIG. 7. Histogram of L in the deconﬁnedphase. H ( L ) L κ =0, 12 x3 κ =0.13,12 x3 FIG. 8. Histogram of L in the conﬁnedphase. H ( L ) L κ =0.13, L > 0, 12 x3 κ =0.13, L < 0, 12 x3 FIG. 9. Histogram of L in the deconﬁnedphase. H ( L ) L κ =0, 32 x8 κ =0.13, 32 x8 FIG. 10. Histogram of L in the conﬁnedphase. H ( L ) L κ =0.13, L > 0, 32 x8 κ =0.13, L < 0, 32 x8 FIG. 11. Histogram of L in the deconﬁnedphase. sectors are approaching towards each other. For N τ = 8, the histogram of Polyakov loop fortwo Z sectors agree well with each other. 10 < sk > κ L > 0, 16 x 4L < 0, 16 x 4 FIG. 12. sk average vs κ for β g = 0 .

435 on16 × × -5 × -4 × -4 × -4 x 4 χ sk κ L > 0L < 0

FIG. 13. sk ﬂuctuation vs κ for β g = 0 . × < sk > κ L > 0, 64 x 16L < 0, 64 x 16 FIG. 14. sk average vs κ for β g = 0 .

435 on64 ×

16 lattice × -7 × -7 × -6 × -6 × -6 x16 χ sk κ L > 0L < 0

FIG. 15. sk ﬂuctuation vs κ for β g = 0 . ×

16 lattice

The κ dependence of the Z symmetry is studied by computing the thermal averageof the temporal part of the interaction, i.e sk = (cid:80) n Φ n U n, ˆ4 Φ † n +ˆ4 and the correspondingsusceptibility χ sk . These simulations are carried out in the deconﬁned phase, as there aretwo Z states corresponding to each sector of the Polyakov loop. The results for ( (cid:104) sk (cid:105) , χ sk )are shown in ﬁgures. 12-15. For all N τ values the diﬀerence in ( (cid:104) sk (cid:105) , χ sk ) for these twosectors is vanishingly small for small enough κ . For larger N τ , the kappa value at whichthe two polyakov loop sectors diﬀer signiﬁcantly in sk and χ sk is higher. For the largestconsidered, N τ = 24 , the two sectors agree in ( (cid:104) sk (cid:105) , χ sk ) up to the Higgs crossover point κ < κ c . When Higgs transition is ﬁrst order the Z symmetry is observed in the Higgs sym-metric phase even for κ > κ c . Note that for κ > κ c the Higgs symmetric phase is meta-stable.It is clear from our 3 + 1 dimensional simulations that the Z symmetry is realised inthe Higgs symmetric phase for large N τ . To understand these results, a 0 + 1 dimensional11odel with temporal component of the interaction term corresponding to a single spatialcoordinates has been considered.. IV. THE PARTITION FUNCTION AND DENSITY OF STATES IN

DIMEN-SIONS

The temporal component of the gauge Higgs interaction corresponding to a particularspatial site can be written as, S D = − κsk , sk = N τ (cid:88) n =1 Φ n U n Φ n +1 . (9) n denotes the temporal lattice site, i.e 1 ≤ n ≤ N τ . Φ n satisﬁes the periodic boundary con-dition Φ N τ +1 = Φ . Since the action will not be invariant if a z = − U i ’s, the action breaks the Z symmetry explicitly. For this model the Polyakovloop can take values ±

1. To see the N τ dependence of the Z symmetry we calculate thefree energy V ( L, N τ ). To simplify the calculations we set U i = 1, for i = 1 , , ...N τ − U N τ = L . All other conﬁgurations of U i corresponding to a given value of L are gaugeequivalent. Now the partition function for L = 1 is nothing but that of the one dimensionalIsing chain. For L = − N τ and Φ isanti-ferromagnetic. For each choice of L the partition function can be calculated exactly,i.e, Z ( L = 1) = λ N τ + λ N τ , Z ( L = −

1) = λ N τ − λ N τ , (10)where λ = e κ + e − κ and λ = e κ − e − κ . The corresponding free energies in large N τ aregiven by, V ( L = 1) = V ( L = −

1) = N τ log( λ ) . (11)This results show that there is Z symmetry in 0 + 1 dimensions in the limit of N τ → ∞ .As noted previously the restoration of the Z symmetry must come from the Z symmetryof the entropy or the DoS . For L = 1 the sequence of allowed value of sk is { N τ , N τ − , ...... ≥ − N τ } . On the other hand for L = − { N τ − , N τ − , ........ ≥ − N τ } . The DoS or ρ ( sk ) for N τ = 4 , ,

12 and 16 are shown in ﬁgures.16-19.For small N τ there are clear diﬀerence for L = ±

1. The diﬀerence persists for the largest as12 τ =4 ρ ( sk ) sk L > 0L < 0

FIG. 16. ρ ( sk ) for κ = 0 in 0+1 τ =8 ρ ( sk ) sk L > 0L < 0

FIG. 17. ρ ( sk ) for κ = 0 in 0+1 τ =12 ρ ( sk ) sk L > 0L < 0

FIG. 18. ρ ( sk ) for κ = 0 in 0+1 × × × × -15 -10 -5 0 5 10 15N τ =16 ρ ( sk ) sk L > 0L < 0

FIG. 19. ρ ( sk ) for κ = 0 in 0+1 well as smallest values of sk . For large N τ , ρ ( sk )’s for both L = ± sk = 0, with √ N τ as standard deviation. The logarithm of the peakhight is given by (cid:39) log N τ ! − N τ / N τ even. For N τ = 2 n + 1 the samecan be approximated by log N τ ! − log( n + n ) + log2. The thermodynamics in the N τ → ∞ limit will be dominated by peak height and distribution of ρ ( sk ) around the peak, which is Z symmetric, for all ﬁnite κ . Interestingly this situation is similar to one dimensional Isingchain where entropy dominates for any non-zero ﬁnite temperature.In order to take into account the eﬀect of nearest neighbour coupling along the spatialdirection we consider 1 + 1 dimensional model with N s = 2 and vary N τ . In this case thePolyakov loop can take value L = 0 , ±

2. The exact calculation of ρ ( sk ) get increasinglydiﬃcult with N τ . One can however consider generating conﬁgurations randomly by givingequal probability for each allowed value of a given variable. The results for the distributionof the total action for N τ = 4 and N τ = 16 are shown in Figs.20-21. As one can see that forhigher N τ , ρ ( sk ) around the peak sk = 0 do not depend on L .13 .0 × × × × -20 -10 0 10 20N s =2N τ =4 ρ ( sk ) sk L=2L=0L=-2

FIG. 20. ρ ( sk ) for κ = 0 in 0+1 -40 -20 0 20 40N s =2N τ =16 ρ ( sk ) sk L=2L=0L=-2

FIG. 21. ρ ( sk ) for κ = 0 in 0+1 × × × -15 -10 -5 0 5 10 15 20k=0.1 β g =0.435 H ( sk ) sk L > 0,64 x16L < 0,64 x16 FIG. 22. H ( sk ) for κ = 0 . , β g = 0 .

435 for3 + 1 dimension × × × -15 -10 -5 0 5 10 15 20 H ( sk ) sk L > 0,64 x16L < 0,64 x16L > 0,N τ =16L < 0,N τ =16 FIG. 23. H ( sk ) ﬁtted with 0 + 1 density ofstates with a Boltzmann factor To ﬁnd out how well the ρ ( sk ) describe the Monte Carlo simulations of the 4 D partitionfunction, the thermal average of the distribution function H ( sk ) of sk has been computed.For each conﬁguration H ( sk ) is given by the number of spatial sites with a given value of sk . Note that the distribution of sk takes into account the Boltzmann factor which shiftsthe peak of ρ ( sk ) to the right. The ﬁgure. 22 shows the distribution H ( sk ) for N τ = 16at κ = 0 . β g = 0 . κ and β g , the system is found to be in thedeconﬁned and Higgs symmetric phase. The thermal average of the Polaykov loop for thetwo sectors are found to be (cid:104) L (cid:105) = 0 . ± .

002 and − . ± . (cid:104) L (cid:105) (cid:54) = 1there is a smaller but ﬁnite fraction of spatial site where the Polyakov loop takes oppositevalue. This results in the lower envelope in H ( sk ). The results clearly show that H ( sk ) forboth the Polyakov loop sectors can be approximately described by single function in otherwords the presence of Z symmetry.In ﬁgure. 23, we try to ﬁt the 3 + 1 dimensional simulation result with 0 + 1 dimensional14 oS by including an extra Boltzmann factor, i.e exp( κ (cid:48) sk ). The resulting ﬁt agree very wellwith H ( sk ). We expect that the 0+1 results can describe the 3+1 Monte Carlo simulationsin most of the phase diagram except for critical points. Note here, H ( sk ) values correspondto κ = 0 .

1, however to ﬁt

DoS one needs a κ value which is higher. This is due to thefact that in 3 + 1 dimensions sk at a given spatial point interacts with sk at the nearestneighbour sites. Considering a mean-ﬁeld approximation one can compute the free energydiﬀerence between L = 1 and L = − κ = κ (cid:48) for the 3 + 1 dimensional system at κ = 0 . − . V. CONCLUSIONS

In this paper the CD transition and Z symmetry are studied in Z +Higgs theory for 3 + 1dimensional Eulidean space. The results show that for large N τ the Z symmetry is realizedin the Higgs symmetric phase within statistical errors. To understand the mechanism ofemergence of the Z symmetry a simpliﬁed one dimension model of Z +Higgs is consid-ered by keeping only the temporal interaction terms at a given spatial site. The partitionfunction and the corresponding free energy for each of the two Polyakov loop sectors isexactly calculated. It is shown that the free energy diﬀerence between the two Polyakovloop sectors vanishes in the large N τ limit, which leads to Z symmetry purely due todominance of entropy. The DoS for ﬁnite N τ are calculated exactly where the asymmetrybetween the diﬀerent Polyakov loop sectors rapidly decreases with N τ . The eﬀect of nearestneighbour interaction along the spatial directions in a simple model shows the persistenceof Z symmetry in the DoS . Further it is shown that the 3 + 1 Monte Carlo simulationscan be reproduced using the

DoS of the one dimensional model.

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