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Physics Letters B 626 (2005) 161–166

www.elsevier.com/locate/physlet

Reconsidering gauge-Higgs continuity

Michael Grady

Department of Physics, State University of New York at Fredonia, Fredonia, NY 14063, USA

Received 2 August 2005; accepted 2 September 2005

Available online 13 September 2005

Editor: H. Georgi

Abstract

The 3d Z(2) lattice gauge-Higgs theory is cast in a partial axial gauge leaving a residual Z(2) symmetry, globadirections and local in one. It is shown both analytically and numerically that this symmetry breaks spontaneouslHiggs phase and is unbroken in the confinement phase. Therefore they must be separated everywhere by a phase tcontradiction to a theorem by Fradkin and Shenker. It relied on a fully fixed unitary gauge, which prohibits this phase trexplicitly. Thus the unfixed gauge theory is not, in this case, equivalent to the unitary-gauge version. 2005 Elsevier B.V. All rights reserved.

PACS: 11.15.Ha; 11.30.Qc; 64.60.Cn; 05.50.+q

Keywords: Lattice gauge; Phase transitions; Gauge fixing

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This Letter presents evidence for a hidden symetry breaking in 3d Z(2) lattice gauge-Higgs theomade visible by a partial axial gauge fixing. The symetry which breaks is simply a subgroup of the orinal gauge symmetry, which is fixed enough to invadate Elitzur’s theorem[1] but leaves unbrokenN “lay-ered” Z(2) symmetries on theN3 lattice, one for eachplane perpendicular to the third axis. In this gauthe average Higgs field on each layer and the avethird-dimension-pointing link (link-magnetization) oeach layer become order parameters for this resi

E-mail address: grady@fredonia.edu(M. Grady).

0370-2693/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2005.09.001

exact Z(2)N symmetry. It is found that in the Coulomphase, the link magnetization takes on a non-zeropectation value, but the Higgs field does not. InHiggs phase they both have non-zero expectationues, which can be shown analytically. In the confiment phase it can be shown analytically that bothpectation values vanish and the symmetry is unbrothere. Therefore, due to the symmetry difference, thphases must be separated by a phase transition.is a surprising result because a well-known theorby Fradkin and Shenker (FS)[2] states that the Higgand confinement phases are analytically connecThis phase-continuity has played an important roleattempts to understand the confinement mechan

.

162 M. Grady / Physics Letters B 626 (2005) 161–166

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The FS proof relies crucially on using a completefixed unitary gauge, in which all of the Higgs fieldare set to unity. Normally, any fixed gauge is consered equivalent to the unfixed gauge theory, withpartition function differing only by an infinite constant. However, if a portion of the gauge symmeitself breaks spontaneously (e.g., global gauge smetry or the layered symmetry considered above) tthey will not be equivalent, because of the loss ofgodicity at a phase transition. The unfixed theory wdiffer from the unitary-gauge theory by adifferent infi-nite constant in the symmetry-broken phase than inunbroken phase. Therefore, the FS proof, valid inunitary gauge, is not necessarily valid for the unfixor partially-fixed theories. In fact the presence ofphase transition mentioned above shows it is invaThe change in ergodicity at the phase transition pduces a sudden change in entropy which can beas the source of non-analyticity.

The plan of this Letter is as follows. First, thpreviously-known phase structure of the theoryreviewed including the FS analyticity region athe Monte Carlo results of Jongeward, Stack, aJayaprakash[3]. Then it is shown that in partial axial gauge along theβ = 0 line, the theory is equivalent to the one-dimensional Ising model (hereβ isthe gauge coupling parameter). This places a pof non-analyticity at (β = 0, βH = ∞), well withinthe FS analyticity region. The full theory is self-duso if there is only one phase transition it must ocalong the self-dual line. Thus the FS analyticityguments can be safely used away from it. Theyused to show that the order parameters are identiczero in the strong coupling region (confined phaand non-zero in the dual region (Higgs phase). Tis sufficient to prove the two regions are everywhseparated by a phase boundary. Monte Carlo simtions are then used to explore the low-β region whereno phase transition was seen in Ref.[3]. Using thenew order parameters a clear transition is seen neaself-dual line, which is most likely weakly first-ordeFinally the situation in the Coulomb phase, wheresymmetry is also broken is discussed. The fact thatgauge theory apparently “breaks itself” (i.e., breathe layered partially-global gauge symmetry) hasteresting implications for continuum gauge theories

The theory under consideration is a 3d classicaltice theory with action

Fig. 1. Previously known phase diagram of 3d Z(2) gauge-Himodel. Dashed lines are second-order transitions, solidfirst-order as found in Ref.[3]. Open circles trace the self-dual linStriped region is schematic representation of FS analyticity re(exact numerical bounds not given).

S = β∑

�x,k,l

(1− U�kl(�x)

)

+ βH

∑

�x,k

(1− φ(�x)Uk(�x)φ(�x + k̂)

).

The Uk are gauge fields lying on the links of the latice, k = 1 to 3, and theφ are Higgs fields living onsites.U�kl is the standard plaquette variable. Theytake values±1. There is a Z(2) local gauge symmetbecause the action is invariant if both the Higgs fiand all links attached to a given site are multiplied±1. The phase diagram, as determined in Ref.[3], isshown inFig. 1. The theory on the right-hand bounary is just the 3d Ising model and on the lower bouary the 3d Ising lattice gauge theory, its dual[4]. Thesecond-order transitions on the edges continue intophase diagram and meet at the self-dual line, whis given byβH = −1

2 ln(tanhβ). Either before or athis meeting the transitions become first-order, athe single joined transition continues up the self-dline where the first-order transition was seen to dispear, aroundβ = 0.62 in an apparent critical point. Ntransition was seen beyond this. If a phase transisimply ends in a critical point, it cannot be symmtry breaking. However, the attached Ising transitis symmetry-breaking. Indeed, it is argued below tthe entire transition is symmetry breaking, with brokand unbroken phases everywhere separated by a ptransition. This transition most likely continues up t

M. Grady / Physics Letters B 626 (2005) 161–166 163

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self-dual line (unless it splits again, but that seemslikely).

The results of this Letter are mostly for the patial axial gauge. As usual, we fix all 1-direction linkto unity, except for thex1 = 0 plane. Onx1 = 0 all2-direction links are fixed to unity except for whex2 = 0. In the normal maximal-tree axial gauge,line of third-direction links atx1 = x2 = 0 are alsofixed, however we will leave that line unfixed, givina residual Z(2) symmetry on eachx3 = const plane.Because discrete symmetries can break spontanein two dimensions, this should be enough gaugeing to evade Elitzur’s theorem, and allow one to viwhether or not the remaining partially-global symmtry breaks or not. Consider first, the theory onβ = 0 line in this gauge. For this argument one canlax even further and fix only the 1-direction links. Onthe Higgs interaction remains. Along the 1-directiodue to the fixing of the gauge field, one has a 1d Isinteraction. The sideways 2- and 3-direction intertions do not affect the Higgs fields, because whatethe Higgs field configuration, one choice of sidewalink will always be frustrated and the other will alwabe satisfied. Because the sideways links are otherunrestricted (due to the lack of a gauge interactiowhen they are integrated out one has an equal cobution to every Higgs configuration. Therefore theinteractions do not affect the Higgs field probabiliand the system is equivalent to a set of 1d Ising mels. The 1d Ising model is unbroken at non-zero teperature, but broken atT = 0 (βH = ∞). The freeenergy has an essential singularity here. This is csistent with the requirement that if there is only oline of singularities, it must be the self-dual line, whihas (β = 0, βH = ∞) as its endpoint. This singularitis within the FS analyticity zone. In the unitary gauin which the FS proof is given, all Higgs fields are sto unity, so the considered phase transition cannotplace because the symmetry that is breaking is expitly broken. This would seem to indicate that the prois only valid in the unitary gauge.

Assuming the only possible singularities in tlow-β region are along the self-dual line, it seemsafe to assume the FS analyticity arguments are vaway from that line. Therefore, a convergent strocoupling expansion exists in the neighborhood(0,0), and a convergent weak coupling expansionists in the opposite dual region (∞, ∞). Consider

y

calculating〈φ〉 in the strong coupling expansion. Hethe exponential in the partition function is expandin a power series, resulting in a terms involving tfield φi and various powers of terms from the actioThe only terms which survive the partition-functiosummation are those that have all even powersfields. However, it is easily seen that for〈φi〉 therewill always be an odd power of one or moreφ fields.Therefore〈φ〉 = 0 everywhere the strong coupling epansion converges, i.e., at least in a finite patch aro(0,0). Turning to the upper right-hand corner of tphase diagram, theβ = ∞ line in the axial gauge isjust the 3d Ising model, which definitely has〈φ〉 �= 0in the highβH (Higgs) phase. A function identicallzero in one portion of the plane cannot become nzero without passing through a singularity. Therefobecause an exact symmetry is being broken hereHiggs and confinement phases must be everywseparated by a phase transition.

In the symmetry-broken region, the infinite lattiloses ergodicity and the partition function is restricto a single symmetry-broken sector. In the unbroregion the system is ergodic. In the unfixed or ptially fixed gauges, the gauge-copy factor multiplyithe partition function suddenly changes at the trsition, giving an extra term ofN ln(2) to βF in thebroken phase (hereF is the free energy). Althougthis does not, in the thermodynamic limit, changespecific free energy,f = F/N3, it can change one oits derivatives. This is because the change occursan increasingly smaller beta-range as the latticebecomes infinite. This range is of order theβ-shift,�β ∝ N−1/ν . If one computes higher derivativesf with respect toβ through finite differences, theone will eventually find one that diverges asN → ∞.In the unitary gauge, only one gauge copy is keptboth sides of the transition, so the non-analyticityprevented from forming. This can be seen as duexplicitly breaking a symmetry that would otherwibreak spontaneously. Thus one can see that the ungauge is not valid in the case that part of the gasymmetry itself breaks spontaneously. In this casemust be careful not to fix those symmetries that wnaturally break spontaneously, if one wishes a thewhich is physically equivalent to the unfixed theoThe safest fixed gauge is therefore one which ofixes to the point of invalidating Elitzur’s theorem bno further. Because a discrete symmetry cannot b

164 M. Grady / Physics Letters B 626 (2005) 161–166

n

of possible

Fig. 2. Histograms of Higgs field expectation value on lattice layers perpendicular to third direction on a 443 lattice. Histogram centered ozero is forβ = 0.48. Histogram with peaks away from zero is forβ = 0.585.βH is held fixed at− 1

2 ln(tanh(0.5)) for whichβ = 0.5 would bethe self-dual point. All runs shown here and later are hot starts. Bin size and spacing is carefully chosen to have an equal numberdiscrete values in each (8 in this case).

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spontaneously at non-zero temperature in less thandimensions, it would seem that our partially-fixed aial gauge that fixes independently on planes meetscriterion, but some additional minor thinning of thgauge fixing might be possible.

Monte Carlo simulations were performed to illutrate the behavior of these new order parametersthe self-dual line in the region previously thoughtbe continuous. A clear symmetry breaking is searound the self-dual pointsβ = 0.5, βH = 0.385968,and β = 0.25, βH = 0.703415, which are both weinto the region seen as continuous in Ref.[3]. TheHiggs coupling was held fixed andβ varied. For thefirst of these, at gauge-couplings below and incling β = 0.48, the Higgs field order parameter (averaHiggs field on 2d layers) shows a Gaussian distribuaround zero, consistent with an unbroken symmeAboveβ = 0.5 it becomes non-Gaussian (as measuby Binder cumulant), first flattening and eventuapicking up a clear peak away from zero aroundβ =0.58 (Fig. 2). At the intermediate coupling of 0.55a triple peak structure is seen, suggestive of a fiorder phase transition (Fig. 3). One would need datfrom larger lattices to be certain of this identificatiof the order. For the self-dual point atβ = 0.25, sim-ilar behavior is seen (Gaussian below 0.23 and hig

Fig. 3. Histogram of layered Higgs-field expectation valueβ = 0.55 with sameβH .

peaked away from origin above 0.33). The layelink magnetization order parameter behaves similato the layered Higgs field at these couplings (Fig. 4),but at a givenβ is slightly less strongly broken. Simulations were done with generally 150 000 sweepster 20 000–100 000 equilibration sweeps. Hot and cstarts agreed reasonably well up toβ = 0.585, exceptthat cold-start distributions were still asymmetric.β = 0.6 metastabilities apparently prevent agreemeven with 100 000 equilibration sweeps. This stro

M. Grady / Physics Letters B 626 (2005) 161–166 165

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Fig. 4. Histogram of layered link expectation value for sameβ ’s asFig. 1.

metastability is also highly suggestive of a first-ordtransition. A simulated annealing procedure or clter algorithm may be necessary to study the stronbroken region accurately. Monte Carlo simulatiowere also used to verify that the layered link orderrameter undergoes a symmetry breaking in the pgauge theory (βH = 0) at the location of the knowphase transition between Coulomb and confinemphases.

These transitions are unusual in that there areN

separate symmetries breaking instead of the usualThusN tunneling events must take place for the lattto go from the completely broken to the completeunbroken state. These can be watched by monitothe order parameters for all of the layers. (For the pgauge theory, the link order parameters are not setive to a global Z(2) gauge transformation, so thereonly N −1 broken symmetries, however the link ordparameter also breaks the Z(2) Polyakov loop symtry in the third direction, giving a total ofN .) The totallatent heat of the entire transition is also split intoN

small jumps, making it difficult to measure and son top of random fluctuations. Thus without the ordparameter for guidance, the transition may be hardistinguish from a crossover. The effective volume tenters the finite size scaling is for most quantities oN2, the layer volume, as opposed to the total syssizeN3. These features conspire to make this sysa surprisingly difficult one to study using the MonCarlo method with finite-size scaling. Further detaconcerning the finite-lattice scaling behavior willgiven elsewhere.

.

In this Letter it has been shown, both analyticaand through Monte Carlo simulations, that the 3d Zgauge-Higgs system has a drastically different behior in a partially-fixed axial gauge than it doesfully-fixed unitary gauge. This is because the partiaglobal Z(2) N -fold layered symmetry on anN3 lat-tice breaks spontaneously in the Higgs (and Coulophases, which is prevented from happening throexplicit breaking in the unitary gauge. ThereforeFradkin–Shenker proof of analyticity in a region ecompassing both the Higgs and confinement phaperformed in the unitary gauge, breaks down inmore relaxed gauge, and by implication, in the unfixcase. The unitary gauge misses the singularity induby the ergodicity change that occurs when the absymmetry-breaking takes place. In the partially-fixaxial gauge it was shown that the Higgs and Confiment phases are everywhere separated by a phasesition. The above arguments also hold in four dimsions and for continuous groups. (Here only in fodimensions due to the Mermin–Wagner theorem, s3d layers are needed to have a layered phase-transat non-zero coupling for a continuous group.)

The same order parameters (Higgs and link mnetizations) could be studied in the unfixed theory,defining a covariant Higgs field and covariant linkthe following way. Simply multiply the Higgs fieldat a given site by all of the links on the gauge-fixitree from that site to the root for that layer. For tlink, do this to both ends. These will be equal to tisolated Higgs field or link themselves in the partaxial gauge. Then simply average these covariantjects over each layer to obtain order parameters eqalent to our layered Higgs and link magnetizatioAssuming no further symmetry breakings take plathe unfixed and partial axial gauge simulations wbe equivalent. However, computing the non-localder parameters given above in the unfixed case wbe numerically prohibitive. The idea that the knowphase transitions could be extended if non-local orparameters were considered was discussed in Ref[5].

It is also very interesting to consider implicationsthe layered symmetry for the Coulomb phase, whwas not the focus of this Letter. In the Coulomb phait is easy to show that the Higgs order parametezero (due to the 3d Ising transition on RHS ofFig. 1).However Monte Carlo simulations show that the laered link magnetization order parameter is non-z

166 M. Grady / Physics Letters B 626 (2005) 161–166

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21

here, becoming zero at the confinement transitionthis holds for continuous groups in 4d, then for U(1)four dimensions the layered 3d global 1d local gausymmetry is spontaneously broken at weak couplThat is, the gauge symmetry is partially broken bygauge field itself. This leads to a rather different pture in which gauge theories are much like magnspin systems. The physical gauge fields (photons inU(1) case) may be spin-wave like Goldstone collecexcitations of the broken vacuum. Despite the vtor order parameter, Lorentz symmetry is effectivnot broken because a combined Lorentz and gaugvariance remains unbroken. Whether this realizasimply results in a new point of view or has physicimplications will require further study. Its also possble that certain gauge-fixings are not equivalent incontinuum theory as well, because they might infere with the formation of Goldstone modes. It is a

very interesting to consider the new order parameintroduced here in the non-Abelian case. A prelimnary investigation of the 4d SU(2) lattice gauge theshows the layered symmetry to be broken at weak cpling there as well[6], which suggests the pure SU(theory may itself have a Coulomb phase, contraryusual expectations.

References

[1] S. Elitzur, Phys. Rev. D 12 (1975) 3978.[2] E. Fradkin, S.H. Shenker, Phys. Rev. D 19 (1979) 3682.[3] G.A. Jongeward, J.D. Stack, C. Jayaprakash, Phys. Rev. D

(1980) 3360.[4] F.J. Wegner, J. Math. Phys. 12 (1971) 2259.[5] K. Szlachanyi, Commun. Math. Phys. 108 (1987) 319.[6] M. Grady, hep-lat/0409163.