伶人传a r X i v :h e p -l a t /9701005v 1 9 J a n 1997Preprint HLRZ 84/96
海赋江城Gauge-ball spectrum
of the four-dimensional pure U(1)gauge
theory
J.Cox,W.Franzki,J.Jers´a k Institut f¨u r Theoretische Physik E,RWTH Aachen,Germany C.B.Lang Institut f¨u r Theoretische Physik,Karl-Franzens-Universit¨a t Graz,Austria T.Neuhaus Niels Bohr Institute,Univ.of Copenhagen,Denmark P.W.Stephenson DESY,Zeuthen,Germany
Preprint submitted to Elvier Preprint 1February 2008
1Introduction
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Since studies of lattice gauge theories began it has been conjectured that the pure U(1)gauge theory in four dimensions(4D)might posss two contin-uum limits at the pha transition between the confinement and the Coulomb phas[1,2],provided the transition is of2nd order.The continuum limit de-pends on the pha in which it is approached.In the confinement pha various gauge-balls(GB)wit
hfinite mass,analogous to the glue-balls of pure QCD, as well as afinite string tensionσ,are expected.In the Coulomb pha with a massless photon dilute magnetic monopoles might exist,whereas in the con-finement pha they conden.The phenomena were clearly obrved for afinite lattice cutoff1/a long ago[3,4].However,the continuum limit has remained elusive for the subquent13years.
The main reason for this delay has been the two-state signal at the pha transition obrved onfinite lattices with periodic boundary conditions[5,6]. However,it has recently been demonstrated that if,instead of such toroidal lattices,tho with trivial homotopy group are ud,the two-state signal dis-appears for the Wilson action[7],extended Wilson action(defined below)at γ≤0[8,9],and the Villain action[10].Furthermore,in high precision simu-lations on specially constructed homogeneous spherical lattices,thefinite size scaling(FSS)behaviour turned out to be well described by the leading term of the critical behaviour with a non-Gaussian value of the correlation length critical exponent,ν=0.365(8)[8–10].
Within the intrinsic uncertainties of numerical evidence this demonstrates that the pha transition is of cond order in some parameter region and the cor-responding continuum limit is governed by a non-Gaussianfixed point.The expected continuum limit is thus presumably a nontrivial quantumfield t
heory which is not asymptotically free.Until now no such nonperturbatively defined continuum quantumfield theory has been established in four dimensions.Fur-ther investigation of this theory is therefore of high theoretical interest.
The u of spherical lattices has been crucial for the study of the model by FSS methods,becau this method combines scaling andfinite size phenom-ena.Thus the latter should not be distorted by topological effects.However, once the critical properties are established,there is no urgent need to u such complicated lattices for investigations of the spectrum of the model.The topology offinite lattices of volume V ought to be irrelevant for the thermo-dynamic limit,V→∞,if that is taken before the continuum limit,when the lattice constant a→0.The restrictions impod by the two-state signal on toroidal lattices can be avoided by using large V and choosing the coupling parameters outside the metastability region on the lattices.In fact,fairly preci values ofν=0.28−0.42have been obtained by this approach in the
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past.
祷告In this work we therefore return to the4D toroidal lattices and adapt the latest methods of glue-ball m
easurements in QCD on such lattices to the U(1)gauge group.Since the most recent studies have indicated universality of the scaling behaviour[8–10],we have chon an action reducing the two-state signal while keeping the autocorrelation time reasonably short:the extended Wilson action at the double-charge couplingγ=−0.2.
We have investigated in both phas on large lattices the correlation functions in20channels with the zero momentum and in some channels with the smallest nonzero momentum as well.We have also determined the static potential from the same runs.
In the confinement pha,we have obtained mass m j(inver correlation lengths)in most channels.Within the considered parameter range thefinite size effects are negligible.Also the string tensionσhas been determined from the values of the potential at the largest parations.We investigate the scaling behaviour of the heir vanishing in lattice units,whenβ, the standard Wilson coupling,approaches its critical valueβc.
Our main result for the confinement pha in the vicinity of the critical point βc is the evidence for two groups of GB mass with distinctly different scaling behaviour when the pha transition is approached,τ=|β−βc|→0.Most
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of the GB mass,and approximately also
we also give the result for the ratio of both exponents,
νng
Coulomb pha.The conclusions and a long list of interesting open questions are given in c.8.The appendix contains tables with more detailed data.
2The pure U(1)lattice gauge theory
2.1Action and pha diagram
We consider pure compact U(1)lattice gauge theory with an extended Wilson action,
S=− P[βcos(ΘP)+γcos(2ΘP)].(5) Here,ΘP∈[0,2π)is the plaquette he argument of the product of U(1)link variables around a plaquette P,andβandγare the single and the double charge reprentation couplings,respectively.TakingΘP=a2gFµν, where a is the lattice spacing,andβ+4γ=1/g2,one obtains for weak coupling g the usual continuum action S=1
on pha transition lines where the change between the1st and2nd order takes place.More is required:in a parameter space enlarged by an external coupling to the order parameter three critical lines should emerge from the TCP[17,6].This results in TCP’s being associated with tricritical exponents different from the critical ones on the emerging critical lines.Up to now there has been no evidence that the pointγ0in the pure compact U(1)gauge theory is really a TCP,becau the other critical lines are not known.The problem is infinding the suitably enlarged coupling space while only nonlocal order parameters distinguishing between the confinement and Coulomb phas,like the string tension or the photon mass,exist.This problem may be specific for gauge theories.It makes an interpretation of the two obrved scaling laws in terms of critical and tricritical scaling behaviour uncertain.
Various investigations of the scaling behaviour of the model by analytic means [18],thefinite size scaling analysis[16,19]and the Monte Carlo renormalization group(MCRG)method[11,12,20]found the correlation length exponentνto be in the rangeν≃0.28−0.42.In some MCRG studies the existence of a cond value of this exponent,consistent withν=0.5,was obrved[11,12] suggesting that one of the exponents is tricritical and the other critical.In ref.[11]the tricritical value was suggested to beν=0.5,whereas in refs.[12] the assignment was opposite,the tricritical exponent being the non-Gau
ssian one.Recently,evidence has been provided,that in the coupling space enlarged by monopole coupling,a cond order pha transition with non-Gaussian critical exponents may be obrved[21].
Becau of the doubts as to whether a continuum limit can be obtained,cal-culations of the spectrum of the model are sporadic.The massless photon in the Coulomb pha was obrved in various studies[4,22–25]and the prence of veral gauge-balls in both phas was indicated in[4].In[23,24]the1−−gauge-ball,the“massive photon”in the confinement pha was found and in[23]its scaling investigated and found to be consistent withν=0.33.The monopole mass was measured in the Coulomb pha in a dual formulation of the theory[26].To our knowledge its scaling behaviour has not yet been studied.
2.3Overview of our measurements
代写年终总结Motivated by the recent progress in understanding the continuum limit,in the prent work we investigate the spectrum and its scaling behaviour.Our choice ofγ=−0.2is a compromi between the requirements of minimizing both the two-state signal and the autocorrelation time.
In the simulations we have ud a vectorized three-hit Metropolis algorithm
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