# Z_N symmetry in SU(N) gauge theories

aa r X i v : . [ h e p - l a t ] F e b IMSc/2021/02/02 Z N symmetry in SU ( N ) gauge theories 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 Z N symmetry in SU ( N ) gauge theories in the presence of matter ﬁelds in the funda-mental representation. The Polyakov loop dependence of the free energy is calculated by restrictingthe lattice partition function integration to matter ﬁelds which are uniform in spatial directionsand gauge ﬁelds with vanishing spatial components. We show that in the limit of large number oftemporal sites the explicit breaking of Z N symmetry vanishes, driven by dominance of the densityof states. We argue that the spatial links as well as the spatial modes of the matter ﬁelds determinethe boundaries separating regions where Z N symmetry is realised from the rest. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] . INTRODUCTION Gauge theories such as quantum chromodynamics (QCD), the standard model (SM) etc.are crucial for understanding evolutions of the early universe as well as the quark-gluonplasma (QGP) formed in relativistic heavy-ion collisions. Studies of phases and phase tran-sitions in these theories are important as the main aim of experimental observations is tolook for their signatures. One of the phase transitions common to all these theories is theconﬁnement deconﬁnement transition(s) (CD) at ﬁnite temperatures. In the pure gaugelimit of these theories the CD transition is described by the center Z N ∈ SU ( N ) symmetryand the Polyakov loop which plays the role of an order parameter [1–9]. Above the criticaltemperature, the Polyakov loop acquires a non-zero thermal average value which leads tothe spontaneous symmetry breaking (SSB) of the Z N symmetry [4, 7, 9, 10]. The SSB leadsto N degenerate states in the deconﬁned phase and global topological defects such as stringsand domain walls in physical space [11–15].The Z N symmetry arises from the fact that the allowed gauge transformations in Euclideanspace are periodic up to a factor z ∈ Z N . These gauge transformations are responsiblefor the Z N symmetry in pure gauge theory. In turn the Z N symmetry plays an importantrole on the nature of the CD transition [16]. The presence of matter ﬁelds in the abovegauge theories spoil the Z N symmetry. The requirement that the matter ﬁelds satisfy eitherperiodic or anti-periodic boundary conditions in the temporal direction forces the gaugetransformations to be periodic [8, 12, 17–20]. However gauge ﬁeld conﬁgurations relatedby Z N gauge transformations can contribute to the partition function, though not equally.Since the matter ﬁelds cannot be subjected to these gauge transformations the diﬀerentconﬁgurations will have diﬀerent actions. This situation appears similar to Ising model inthe presence of external ﬁeld [21]. In the present context, the gauge-matter ﬁeld interactionplays the role of explicit Z N symmetry breaking term [8, 21].There are several studies of explicit breaking of Z N symmetry due to matter ﬁelds over theyears [8, 12, 15, 17, 22–24]. Perturbative one loop calculations show that the Z N symmetryexplicit breaking increases with decrease in mass of the matter ﬁelds [12]. The explicit break-ing is found to increase with temperature. Recently there are extensions of loop calculations2o higher order with similar trend [25]. The non-perturbative studies, which are mostlyaround the CD transition regions, show decrease in explicit breaking with the number oftemporal lattice points ( N τ ) [19, 20, 26]. These studies show a trend that in the continuumlimit likely there will be reemergence of Z N symmetry.The exact calculation of the partition function to validate the reemergence of Z N symme-try is almost impossible. An explanation of this analytically, is highly desirable even withsome simpliﬁcations. In this paper, we attempt an exact calculation of the lattice partitionfunction after restricting the spatial gauge ﬁelds to zero and matter ﬁelds uniform in thespatial directions. With these restrictions only the terms of the action which break the Z N gauge transformations remain and the problem eﬀectively reduces to a 1-dimensional model.The matter ﬁelds can then be integrated out exactly after a suitable gauge choice for thetemporal links. The integrations can be carried out for any arbitrary value of N τ . Fromthese calculations the free energy for the given gauge ﬁeld background or Polyakov loopis obtained. The results show that the explicit symmetry breaking vanishes in the limit oflarge N τ even when the relevant couplings are ﬁnite and the action breaks the Z N symmetry.These results suggest that for the parameters for which the Z N symmetry realised, the freeenergy is dominated by the density of states. Recent studies of Z + Higgs have shown thatthe histogram of the explicit breaking term exhibits Z symmetry [27]. The modes/aspectsof the ﬁelds neglected here will aﬀect the Z N symmetry in parts of the phase diagram bydriving the system away from the point where the density of states dominate the thermody-namics. This is observed in Monte Carlo simulations of the partition function with the fullaction [27].The paper is organised as follows. In section II, we discuss the Z N symmetry in the presenceof matter ﬁelds in the fundamental representation. In section III, we calculate the partitionfunction for SU ( N )+ Higgs followed by the calculations for SU ( N )+ fermions in section IV.The discussions and conclusions are presented in section V.3 I. Z N SYMMETRY IN SU ( N ) GAUGE THEORIES

The action for a minimally coupled SU ( N ) gauge theory of fermions and bosons in Euclidean space is given by S = Z V d x Z β dτ (cid:20) (cid:8) T r ( F µν F µν ) + | D µ Φ | + m b Φ † Φ (cid:9) + ¯Ψ( /D + m f )Ψ (cid:21) (1) F µν = ∂ µ A ν − ∂ ν A µ + g [ A µ , A ν ] , D µ Φ = ( ∂ µ + igA µ )Φ , /D Ψ = ( /∂ + ig /A )Ψ . Here A µ , Φ and Ψ are the gauge, Higgs and the fermion ﬁelds respectively. Φ and Ψ arein the fundamental representation. g is the gauge coupling strength, m b ( m f ) is the mass of Φ(Ψ) ﬁelds and β is the inverse of temperature, i.e β = 1 /T . The corresponding partitionfunction takes the form Z = Z [ DA ][ D Φ][ D Φ † ][ D Ψ][ D ¯Ψ] Exp[ − S] . (2)The ﬁelds contributing to the partition function satisfy the following temporal boundaryconditions, A µ ( τ = 0) = A µ ( τ = β ) , Φ( τ = 0) = Φ( τ = β ) , Ψ( τ = 0) = − Ψ( τ = β ) . (3)These ﬁelds transform under a gauge transformation V ∈ SU ( N ) as A µ −→ V A µ V − + 1 g ( ∂ µ V ) V − , Φ −→ V Φ , Ψ −→ V Ψ . (4)In absence of the matter ﬁelds one can consider V which is not necessarily periodic, i.e V ( τ = 0) = zV ( τ = β ) , where z is an element of the center Z N of SU ( N ) . While the puregauge action is invariant under this transformation, the Polyakov loop, L ( ~x ) = 1 N Tr (cid:20) P (cid:26) Exp (cid:18) − ig Z β A dτ (cid:19)(cid:27)(cid:21) , (5)transforms as L −→ zL . This transformation property of the Polyakov loop is crucial forit playing the role of an order parameter for the CD phase transition and also SSB of the Z N symmetry in the deconﬁned phase. In the presence of the matter ﬁelds ( Φ , Ψ ), the4oundary conditions in Eq.3, restrict the gauge transformations to be periodic in τ . Sincethe non-periodic gauge transformations can not act on the matter ﬁelds, two conﬁgurationswith Polyakov loops L ( ~x ) and zL ( ~x ) do not necessarily contribute equally to the partitionfunction. Hence the Z N symmetry is explicitly broken. If L ( ~x ) belongs to the identity sectorof Z N then the conﬁguration corresponding to zL ( ~x ) will have higher action.Symmetry in the action automatically leads to symmetry in the free energy. However explicitbreaking at the level of action does not necessarily mean the same is true at the level offree energy. This is because the free energy diﬀerence between L ( ~x ) and zL ( ~x ) can only bedecided after the matter ﬁelds are integrated out. Integrating the matter ﬁeld is very diﬃculttask in four dimensions. Therefore in the following we consider SU ( N ) gauge theory of Higgsand fermions separately by restricting to the spatial gauge ﬁelds and spatial variations ofthe matter ﬁelds to zero. III. GAUGED − d CHAIN OF SU ( N )+ HIGGS

The SU ( N )+ Higgs part of Eq.1 on the Euclidean lattice can be written as [20], S = β g X p (cid:20) −

12 ( U p + U † p ) (cid:21) − b X n,µ (Φ † n U n,µ Φ n +ˆ µ + h.c ) + a X n Φ † n Φ n . (6) β g is the gauge coupling constant, a = and the coupling b = ( m b + 8) − , the Higgs mass m b is expressed in lattice units [20]. For unit spatial links and Φ uniform in the spatialdirections, the action reduces to, S = a N τ X i =1 Φ † i Φ i − b N τ X i =1 (Φ † i U i Φ i +1 + h.c. ) , (7)apart from an overall spatial volume factor. The pure gauge part is dropped as the eﬀectof matter ﬁelds on the Z N symmetry in the back ground of temporal gauge links is beingconsidered. For convenience the subscripts of the ﬁeld variables have been replaced by i . N τ denotes the number of temporal sites. Φ satisﬁes periodic boundary condition, i.e Φ N τ +1 = Φ . We consider a gauge choice in which U i = I for i = 1 , , ..., N τ − and U N τ = U . The Polyakov loop is L = T r ( U ) /N . In order to derive the free energy V ( L ) ,5nly the Φ i ﬁelds in the partition function Z L are to be integrated out, Z L = Z N τ Y i =1 d Φ † i d Φ i Exp[ − S ] . (8)For convenience the action is written as S = S + S as in the following, S = a Φ † Φ − b (cid:16) Φ † N τ U Φ + h.c. (cid:17) , S = a N τ X i =2 Φ † i Φ i − b N τ − X i =1 (Φ † i Φ i +1 + h.c. ) . (9)At ﬁrst, the ﬁelds Φ to Φ N τ − are integrated out sequentially, i.e, Z = Z N τ − Y i =2 d Φ † i d Φ i Exp[ − S ] (10)Afterwards the remaining integration of Φ and Φ N τ can be carried out to obtain the partitionfunction, Z L = Z d Φ † d Φ d Φ † N τ d Φ N τ ( Z ×

Exp[ − S ]) . (11)The integration of Φ and Φ N τ requires evaluation of determinant of a matrix of size N × N .The integrations of Φ to Φ N τ − greatly simplify the problem, otherwise one would have todeal with evaluating of matrix whose size depends on N τ .In the integration of Z in Eq.10, due to the gauge choice mentioned above the diﬀerentcomponents as well as the real and imaginary parts of Φ i ’s do not mix. Therefore it can bewritten as, Z = N Y r =1 I (Φ N τ ,r ) , (12)where Φ Nτ ,r is the r-th component of Φ N τ and I (Φ N τ ,r ) is obtained by integrating out r-thcomponent of Φ to Φ N τ − . Denoting the r-th component by φ we can write I ( φ N τ ) = Z N τ − Y i =2 dφ i Exp[ − S ′ ] , (13)where S ′ = a N τ − X i =2 φ i − b N τ − X i =1 φ i φ i +1 . (14)6he integration I , in Eq.13, can be also be written as, I ( φ N τ ) = Z N τ − Y i =3 dφ i e − S ′ Z dφ Exp (cid:2) − aφ + 2 φ ( bφ + bφ ) (cid:3) (15) S ′ is obtained by taking out terms from S ′ which are dependent on φ . After φ is integratedout, I ( φ N τ ) = Z N τ − Y i =3 dφ i e − S ′ r πa Exp (cid:20) a ( bφ + bφ ) (cid:21) (16)which can also be written as I ( φ N τ ) = Z N τ − Y i =4 dφ i e − S ′ × r πa Z dφ Exp (cid:20) − (cid:18) a − b a (cid:19) φ + 2 φ (cid:18) b a φ + bφ (cid:19) + b a φ (cid:21) . (17)Here again S ′ is S ′ without terms containing φ and φ . Given the forms of I ( φ N τ ) in Eq.15and Eq.17 one easily write down the would be form of I ( φ N τ ) after integration of φ k − as, I ( φ N τ ) = Z dφ k +1 ....dφ N τ − e − S ′ k +1 × I k Z dφ k Exp (cid:2) − A k φ k + 2 φ k ( B k φ + bφ k +1 ) + E k φ (cid:3) . (18)Carrying out the φ k integration will result in, I ( φ N τ ) = Z dφ k +2 ....dφ N τ − e − S ′ k +2 × I k +1 Z dφ k +1 Exp (cid:2) − A k +1 φ k +1 + 2 φ k +1 ( B k φ + bφ k +2 ) + E k +1 φ (cid:3) . (19)From equations 18 and 19, one can read oﬀ the following recursion relations, I k +1 = r πA k , A k +1 = a − b A k , B k +1 = bB k A k , E k +1 = E k + B k A k , (20)with I = 1 , A = a , B = b and E = 0 . Using these recursion relations we can workoutthe integration, I ( φ N τ ) completely. Using I ( φ N τ ) ’s one can write the partition function as, Z L = Q Z d Φ † d Φ d Φ † N τ d Φ N τ xp h − A N τ Φ † N τ Φ N τ − C N τ Φ † Φ + (cid:16) Φ † N τ (B N τ I + bU)Φ + H . C . (cid:17)i , (21)where Q = N τ Y k =2 I n k , n = 2 N. (22) n corresponds to the number of components of the Φ ﬁeld. The coeﬃcient C N τ = a − E N τ .After the integration of the remaining ﬁelds Φ and Φ N τ the partition function takes theform, Z L = Q s π Det ( M ) . (23) M is (4 N × N ) given by, A N τ B N τ + bUB N τ + bU † C N τ The exact form of

Det ( M ) for arbitrary N is diﬃcult to ﬁnd. However the sequentialintegration has greatly simpliﬁed the problem. For arbitrary N τ we need to deal with amatrix of ﬁnite size. In the following we consider N = 2 and evaluate Z L explicitly for anarbitrary U , which can be parametrised as, U = α + iα.σ, α = ( α , α , α ) , (24)where σ i ’s are the Pauli matrices. The corresponding matrix M is given by, A N τ B bα − bα bα A N τ − bα B − bα − bα A N τ bα bα B − bα A N τ − bα bα bα B B − bα bα − bα C N τ bα B bα bα C N τ − bα − bα B bα C N τ bα − bα − bα B C N τ ,where B = − ( bα + B N τ ) . The determinant of M is given by, DetM = (cid:0) B N τ − A N τ C N τ + 2 bB N τ α + b (cid:1) (25)8 rotation of the Polyakov loop changes α → − α . So in the determinant the explicitsymmetry breaking of Z is bB N τ α . It is observed that B N τ rapidly decreases, vanishing inthe larger N τ limit restoring the Z symmetry. Even for higher N one can see the realisationof Z N symmetry as the oﬀ diagonal elements of the matrix M turn out to be just U and U † due to vanishing of B N τ . Eﬀecting a Z N transformation, ie U → zU , the factor z in U willcancel with z ∗ in U † leaving the determinant unchanged.In the following we consider the eﬀects of staggered fermion ﬁelds on the Z N symmetry. IV. GAUGED − d CHAIN OF SU ( N )+ FERMIONS

The lattice action for SU ( N ) staggered fermions is given by [28] S = β g X p (cid:20) −

12 ( U p + U † p ) (cid:21) + 2 m f X n ¯Ψ n Ψ n + X n,µ η n,µ h ¯Ψ n U n,µ Ψ n + µ − ¯Ψ n U † n − µ,µ Ψ n − µ i (26)Here the fermion mass as well as the ﬁelds are expressed in lattice unit. The analog of Eq.7in this case turns out to be, S = 2 m f N τ X i =1 ¯Ψ i Ψ i + N τ − X i =1 (cid:0) ¯Ψ i Ψ i +1 − ¯Ψ i +1 Ψ i (cid:1) − ¯Ψ N τ U Ψ + ¯Ψ U † Ψ N τ , (27)The change in the sign of the last two terms is due to the anti-periodicity of Ψ . Here wehave considered the KS phase η to be +1 [29, 30], however the results/conclusions do notdepend on η . As in the previous section we work in the gauge in which all temporal linksexcept the last one are set to identity. The last link is denoted by U . The correspondingPolyakov loop is L = T r ( U ) /N . To ﬁnd out the free energy V ( L ) we need to integrate outonly the fermion ﬁelds. For convenience we write S = S + S where S = 2 m f ¯Ψ Ψ − ¯Ψ N τ U Ψ + ¯Ψ U † Ψ N τ , (28) S = 2 m f N τ X i =2 ¯Ψ i Ψ i + N τ − X i =1 (cid:0) ¯Ψ i Ψ i +1 − ¯Ψ i +1 Ψ i (cid:1) . (29)9nitially we integrate the ﬁelds Ψ , ¯Ψ to Ψ N τ − , ¯Ψ N τ − sequentially just as in the previoussection. Afterwards Ψ , ¯Ψ and Ψ N τ , ¯Ψ N τ are integrated out to obtain the partition function, Z L = Z d ¯Ψ d ¯Ψ N τ d Ψ d Ψ N τ Exp[ − S ] Z , (30)where Z is given by Z = Z N τ − Y i =2 d ¯Ψ i d Ψ i Exp[ − S ] . (31)Since S is diagonal in colour space we consider a particular colour of Ψ i and denote it by ψ i . For this choice the relevant integral is, I ψ = Z N τ − Y i =2 d ¯ ψ i dψ i Exp[ − S ,ψ ] . (32)After integrating ψ , ¯ ψ and ψ , ¯ ψ the integral takes the form, I ψ = Z N τ − Y i =4 d ¯ ψ i dψ i Exp[ − S ,ψ ] × (cid:2) m f − m f ¯ ψ ψ − m f ¯ ψ ψ + ¯ ψ ψ − ¯ ψ ψ + ¯ ψ ψ ¯ ψ ψ (cid:3) (33) S ,ψ is obtained by dropping terms which depend on ψ , ¯ ψ and ψ , ¯ ψ . The sequentialintegration ψ up to ψ N τ − and their conjugates leads to I ψ = A N τ − B N τ ¯ ψ ψ − C N τ ¯ ψ N τ ψ N τ + ¯ ψ N τ ψ + D N τ ¯ ψ ψ N τ + E N τ ¯ ψ N τ ψ N τ ¯ ψ ψ (34)where the coeﬃcients A N τ to E N τ can be obtained by recursion as A k +1 = 2 m f A k + C k , B k +1 = 2 m f B k + E k , C k +1 = A k , D k +1 = ( − k , E k +1 = B k , (35)with A = (1 + 4 m f ) , B = 2 m f , C = 2 m f , E = 1 , (36)Taking the I integrals into account we can write the partition function as Z L = Z d ¯ ψ dψ d ¯ ψ N τ dψ N τ Exp (cid:2) ¯ ψ N τ U ψ − ¯ ψ U † ψ N τ (cid:3) × r (cid:0) − m f ¯ ψ r ψ r − m f ¯ ψ rN τ ψ rN τ + 4 m f ¯ ψ r ψ ¯ ψ rN τ ψ rN τ (cid:1) × (cid:0) A N τ − B N τ ¯ ψ r ψ r − C N τ ¯ ψ rN τ ψ rN τ + ¯ ψ rN τ ψ r + D N τ ¯ ψ r ψ rN τ + E N τ ¯ ψ rN τ ψ rN τ ¯ ψ r ψ r (cid:1) . (37)Note that ψ ri denotes the colour r of the ﬁeld Ψ i at the temporal site i . This expression canbe simpliﬁed as, Z L = Z d ¯ ψ dψ d ¯ ψ N τ dψ N τ Exp (cid:2) ¯ ψ N τ U ψ − ¯ ψ U † ψ N τ (cid:3) × Y r (cid:16) ˜ A − ˜ B ¯ ψ r ψ r − ˜ C ¯ ψ rN τ ψ rN τ + ¯ ψ rN τ ψ r + ˜ D ¯ ψ r ψ rN τ + ˜ E ¯ ψ rN τ ψ rN τ ¯ ψ r ψ r (cid:17) , (38)where ˜ A = A N τ , ˜ B = (2 m f A N τ + B N τ ) , ˜ C = (2 m f A N τ + C N τ ) , ˜ D = D N τ and ˜ E = E N τ + 2 m f C N τ + 2 m f B N τ + 4 m f A N τ . The superscript r denotes the colour of the fermionﬁeld. For N = 2 , integration of the rest of the ﬁelds in Eq.38 leads to, Z L = ˜ E + 2 ˜ E ˜ A | U | + ˜ A + 2 ˜ B ˜ C | U | + 2(1 − ˜ DRe ( U ))+( ˜ E + ˜ A )(1 − ˜ D ) tr ( U ) . (39)As one can see the Z explicit breaking term is linear in ˜ E + ˜ A . For non zero m f , in thefree energy V ( L ) the ﬁrst four terms of Z L dominate over ˜ E + ˜ A . The dominance onlygrows with N τ , hence in the limit of large N τ the Z symmetry is recovered. For higher N it is diﬃcult to evaluate Z L for a general U . To proceed further we assume the U to be U rs = λ r δ rs . After the exponential in Eq.38 is written as a polynomial, Z L = Z d ¯ ψ dψ d ¯ ψ N τ dψ N τ Y r (cid:0) λ r ¯ ψ rN τ ψ r − λ ∗ r ¯ ψ r ψ rN τ + ¯ ψ rN τ ψ rN τ ¯ ψ r ψ r (cid:1) × (cid:16) ˜ A − ˜ B ¯ ψ r ψ r − ˜ C ¯ ψ rN τ ψ rN τ + ¯ ψ rN τ ψ r + ˜ D ¯ ψ r ψ rN τ + ˜ E ¯ ψ rN τ ψ rN τ ¯ ψ r ψ r (cid:17) (40) = Z d ¯ ψ dψ d ¯ ψ N τ dψ N τ × Y r ( A − B ¯ ψ r ψ r − C ¯ ψ rN τ ψ rN τ + F r ¯ ψ rN τ ψ r + D r ¯ ψ r ψ rN τ + E r ¯ ψ rN τ ψ rN τ ¯ ψ r ψ r ) , (41)where A = ˜ A , B = ˜ B , C = ˜ C , D r = ˜ D − λ ∗ r ˜ A , E r = ˜ E − λ r ˜ D + λ ∗ r + ˜ A and F r = (1 + λ r ˜ A ) .After the ﬁelds are integrated out we get the following result for the partition function, Z L = Y r E r (42)11he corresponding free energy is V ( L ) = − T X r ( log (cid:16) ˜E + ˜A (cid:17) + log − λ r ˜D − λ ∗ r ˜E + ˜A !) . (43)For non-zero m f , the second term vanishes in the limit of large N τ . Hence, in this limit, thefree energy V ( L ) is independent of L leading to realisation of the Z N symmetry. The form of U considered above includes λ r = λ for all r , with λ = Exp(i2 π n / N) with n = 0 , , , ...N − .We mention here that for this case one would have expected the explicit breaking of Z N tobe maximal. V. DISCUSSIONS AND CONCLUSIONS

In this paper we report on the explicit breaking of Z N symmetry in the presence of bosonsand fermions. We show that analytical treatment of the problem is possible by simplifyingthe functional integral where the spatial links are set to unity and matter ﬁelds uniform inspatial direction. In this simpliﬁcation most of the terms of the original action drop out ex-cept for the ones which break the Z N symmetry. Also the problem reduces to 1-dimensionalchain of gauged bosons/fermions making analytical calculations possible. To derive thefree energy V ( L ) for the Polyakov loop L , the partition function is evaluated for a givenbackground of temporal gauge links. The calculations become simple in the gauge where wecan set all the links except the last one to unity. Then the matter ﬁelds are integrated outsequentially except for the two ﬁelds connected to the last gauge link. The integration of thelast two ﬁelds result in determinant of a ﬁnite sized matrix for arbitrary N τ . The resultingfree energy V ( L ) turns out to be Z N invariant in the limit of larger N τ . This symmetry willalso be observed in the expectation value of the Polyakov loop, which can be obtained byrestricting the integration of U to a given Z N sector. Note that V ( L ) being calculated hereis the contribution coming from the interaction of gauge ﬁelds with the matter ﬁelds. Forthermodynamic studies the contribution of the pure gauge part must be taken into accountbefore an integration over U is taken, which we plan to do in future.The present calculations leave out the eﬀect of the spatial links and non-zero spatial modes ofthe matter ﬁelds. These modes are responsible for the Higgs and the chiral transitions, which12re entropy driven. It is expected that in the phase diagram where the action dominates overthe entropy the Z N symmetry will be explicitly broken. Recent Monte Carlo simulationsin the presence of Higgs show that this is indeed the case. The Z N symmetry is explicitlybroken in the Higgs broken phase even for large N τ . ACKNOWLEDGMENTS

We thank A. P. Balachandran, S. Datta and S. Sharma for valuable discussions and sugges-tions.

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