We study the phase transition properties of the nonlinear O(N) σ model in two dimensions when O(N) gauge interactions are included. With nonzero gauge coupling, this theory exhibits a first-order phase transition in the large-N limit. The broken-symmetry phase is stabilized by the Higgs mechanism and Goldstone bosons do not appear. © 1976 The American Physical Society

Bander, M

Bardeen, WA

We study the phase transition properties of the nonlinear O(N) σ model in two dimensions when O(N) gauge interactions are included. With nonzero gauge coupling, this theory exhibits a first-order phase transition in the large-N limit. The broken-symmetry phase is stabilized by the Higgs mechanism and Goldstone bosons do not appear

Bardeen, William A

Bander, Myron

eScholarship - University of California

UC IrvineUC Irvine Previously Published WorksTitlePhase transition in an O(N) gauge model in two dimensionsPermalinkhttps://escholarship.org/uc/item/1td81326JournalPhysical Review D, 14(8)ISSN2470-0010AuthorsBardeen, William ABander, MyronPublication Date1976-10-15DOI10.1103/physrevd.14.2117Copyright InformationThis work is made available under the terms of a Creative Commons Attribution License, availalbe at https://creativecommons.org/licenses/by/4.0/ Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaPHYSICAL REVIEW 0 VOLUME 14, NUMBER 8 15 OCTOBER 1976Phase transition in an O(N) gauge model in two dimensions*William A. BardeenFermi National Accelerator Laboratory, Batavia, Illinois 60510Myron BanderUniversity of California, Irvine, California 92664(Received 19 May 1976)We study the phase transition properties of the nonlinear O(N) cr model in two dimensions when O(N) gaugeinteractions are included. With nonzero gauge coupling, this theory exhibits a first-order phase transition inthe large-N limit. The broken-symmetry phase is stabilized by the Higgs mechanism and Goldstone bosons do notappear.Recent].y„Bardeen and Pearson have proposeda transverse-lattice formulation of quantum chro-modynamics. ' In this theory, color confinement isobtained when the vacuum is invariant under thetransverse gauge symmetry. A quark-gluon phase,where color is not confined, can result if this sym-metry is spontaneously broken. In this paper wewill study some aspects of the mechanisms whichare responsible for generating such a phase trans-sition.The independent gauge degrees of freedom or thetransverse-lattice theory are associated with linkson the transverse lattice. The longitudinal dynam-ics of a single transverse link consists of a non-linear SU(3) xSU(3) o model with SU(3) x SU(3) gaugeinteractions. For a single transverse link the lon-gitudinal dynamics is two-dimensional. As thetransverse gauge symmetry is a local symmetryon the lattice, the phase-transition properties ofa single transverse link are relevant to the phase-transition properties of the full gauge theory.We shall study a somewhat simpler though an-alogous version of the single-link problem. Themodel consists of a two-dimensional nonlinearO(N) o model with O(N) gauge interactions. Thistheory has the advantage that the model may besystematically studied in the large-N limit. Thenonlinear cr model in 2 and 2+& dimensions has re-cently been extensively studied by Brezin and Zinn-Justin' and by Bardeen, Lee, and Shrock. ' In thistheory, spontaneous symmetry breaking can occurin 2+a dimensions but only the symmetric phasecan exist in 2 dimensions. The basic result of ourpaper is that the broken-symmetry phase can bestabilized when gauge interactions are introduced.The existence of nontrivial phase-transition prop-erties in two dimensions makes this theory inter-esting in its own right.The nonlinear O(N) o model with gauge interac-tions is described by the Lagrangian~ =-.'(f~„y)'--,'(G„„,)', (I)where P is an N-component scalar field with theconstraint P'= f,'. The gauge fields may be writtenas an antisymmetric tensor (A,.&, i,j =I, . . . , N).The covariant derivative is defined by(D„4)( = s„4;+g&„~;4'gand the Yang-Mills field strength is given byG„„,= B„A„,. —BQ„;,+gA„;Q„„+gA„,Q„,~.In two dimensions this theory is renorma]. izablewith respect to the dimensionless coupling constantI/f, ', and super-renormalizable with respect to thegauge coupling constant g'.This theory may be studied directly using themethods discussed in Ref. 4. Since the nonlineartheory is renormalizable, we must be careful topreserve the symmetry structure of the theory inour calculation. Dimensional regularization is notparticularly convenient in this case as we wouldconfront the necessity of including contributionsfrom the transverse gauge fields. Instead, wechoose to regularize the theory by considering thelinear 0 model in precisely two dimensions. Thenonlinear theory is recovered as a limit of the lin-ear theory. 'The linearized theory is described by the Lagran-gian~ =-.'(I~g)'--.'~.(f.'- y')'- —,'(G„,„)', (4)where the constraint P =f,' has been relaxed. Thenonlinear model is obtained by taking the limit Ap~ with f,' and g' fixed. Since the nonlinear theo-ry is renormalizable, only a logarithmic depen-dence on Xp can occur and is absorbed by the re-normalization of f,'. In the large-N limit no suchlogarithms appear and the limit may be taken with-out the renormalization involving Xp.14 21172118 WILLIAM A. BARDEEN AND MYRON BANDER 14The theory is most easily studied in the light-cone gauge, A „=0, A, =(1/W)(Aa+A, ). The gaugefields A„, are dependent and may be eliminated infavor of a "Coulomb" interaction. In this gauge,the Lagrangian of Eq. (4) becomes-" =-.'(8„4)' —-'~.(f.'- 4')'4g-'(4(a 4g)( 8-') '(4;8 4g),where8 g = (aa + 8 g)1. We wish to consider the possibility of spontan-eous symmetry breaking of the O(N) symmetrywhere the field, P, acquires a, nonzero vacuum ex-pectation value. The direction of the symmetrybreaking may be rotated to any fixed axis. Thereremains an O(N —1) symmetry. We rewrite thefield, P, as afield, o, which can have a vacuumexpectation value, and fields [w;, i = 1, . . . , (N 1)]-which have zero vacuum expectation value. We mayprobe the broken phase by adding a Lagrange mul-tiplier, J, to force the o field to have a given ex-pectation value, f. A spontaneously broken sym-metry phase exists if fe 0 for J =0 and if this phasehas the lowest vacuum energy. The Lagrangian ofEq. (5) may be rewritten as2 = —,' (ao)'+ —,' (8 rr)' ——,'A.,(f,' —o' —rr')' +J(o f)—',g'(oa -)rr(- 8') '(oa rr, ),'g'(rr, a—rr))( .8') '-(rr,.a rr,.).t+g' g ((8,)(- 8 ') '( )&.'C+ O(l/N),I'~ (P) =P +Rafa'-X.a(o'+ w$a+g'((8 o)(-a ') '(oa )&,+g+ O(1/N).The vacuum expectation value of the o-field equa-tion of motion may be used to determine the valueof the Lagrange multiplier, J', such that (o), =f.We obtain the expression0 =J +X,f,'(rr&, —X,(o(o'+ rr')&,+g' Q((a rr,.)(-a ') '(&,.8 o)&,.The various vacuum expectation values in Eqs.(7), (8), and (9) may be evaluated in leading orderN by using the full propagators for the 0 and zfields. We obtain the following results:(9)The large-N limit of this theory is obtained byletting N-~ while holding XaN, g'N, f,'/N, andf'/N fixed. In this limit, a Hartree calculation ofthe o and g propagators becomes exact, with thez mass being determined self-consistently.The propagators may be computed using the La-grangian of Eq. (6):I" '(P) =P2+ rafa' -Xa(3o'+ rr )a(a) (o&.=f,(b) (v,&, =0,2(c) (o &a =(o)'+, , d'k(k' m') ' =f '+-(4rr) ' ln27r '.' mQ2(d) (rr ), =(N 1),~d'k(k' m—,') '—=(N 1)(4rr) 'ln—(e) ((8 o)( 8') '(oa—)), = —(o&'—,dk(k'- ') '(k +P )'(k -P ) '27r'.2 2 y g 2 2 y 2k 3P(2 )dk(k —ma )(2 )2dk(k ™a)(k P )2 (10)=-f —(4rr) ln, +—,maA'(f)g((a rr, )(-a ') '(rr, a )), = —(N 1)—2, dk(k-'-m„') '(k +P )'(k -P ) '=(N 1)(4rr) 'ln, +-rr rA'(g) (o'), =f '+ Sf(4rr) ' lnm QA'(h) ( '),o=rrf(N-1)(4rr)-'ln(i) Q ((8 rr, )(—8 ') '(rr;8 o)&a=f —(N-1)(4rr) 'ln, +A (N 1)—7r14 PHASE TRANSITION IN AN O(N) GAUGE MODEL IN TWO. . . 2119IIII(o)FIG. I. Diagrams summed in Hartree calculation: (a) tadpole, (b) cactus, and (c) rainbow.The divergent self-energy integrals have been corn-puted using a Wick rotation and a cutoff, A'.Using the results of Eq. (10), we may computeall the large-N contributions to Eqs. (7), (8), and(9). The propagators are given byr. (I ') =Z' -m. ',f ' =m „'(g',A—(A. o(g )m = —Xofo +RON(4v) 'ln m. '+g'N(4v) 'lnA gNm. '(14)r„'(P')=I'-m, ',whereAm, =-A,f, 3+3.,f . +X,N(4n) 'ln mr'+g N(4v) 'lnAm~ =-Xofo'+Aof '+RON(4w. ) 'ln mr'+g'f'+g'N(4v) 'lnmg' rThe Lagrange multiplier is determined by the equa-tionA2J =-Xofo f+Xof +f+N(4v) 'ln m. 'A'+fg'N(4v) 'ln mr' rexp —i dxV = exp i ', dxgwhere gistheLagrangian of Eq. (6). Instead ofcomputing V directly, we shall evaluate 8&V usingthe result of Eq. (10). This ensures a. systematictreatment of all divergences. We obtain—a, V= -Z+(s,g(o -f), = -Z. (16)This expression may be integrated by consideringa similar derivative of the mass equation for m, ',Eq. (11). We obta, in[1+(A, ,+g')N(4v) '(m „') ']8&m „'= 2f(X, +g').Now we wish to compute the dependence of thevacuum energy density, V, on the order parameter,f. The vacuum energy may be computed from=f(m, '-g'f '). (12) (17)The Hartree calculation sums consistently, inleading order N, all tadpole graphs [Fig. 1(a)], allcactus graphs [Fig. 1(b)], and all rainbow graphs[Fig. 1(c)]. All other graphs are nonleading in thelarge-N limit.The m mass, m„', must be determined self-con-sistently from Eq. (11) and Eq. (12). In the caseJ= 0, there are two possible solutions correspond-ing to the symmetric phase, f= 0, and a spontan-eously broken phase, ft 0. In the symmetric phasewe have f=0,When Eqs. (16), (12), and (17) are combined, thederivative of the vacuum energy becomesB~V=J=f(m~ —g f )=-,'[m„'(X, +g') '+N(4v) ']s~m, '- g'f'.(18)Equation (18) may be directly integrated to yieldthe vacuum energy:V=-,'m, '(X, +g') '+ —,'N(4v) 'm, ' ——,'gf'+ V„A'm„= —Af+A, N(4v) 'ln.mm'A N+g'N(4v) 'lnm. 'In the broken-symmetry phase we have(13)(19)where Vo can depend on A. o, g', and f,' but not on f.By using our results of Eq. (11), Eq. (12), andEq. (19) we may study the phase properties of thelinear O(N) gauge theory in two dimensions. Wesummarize the results in Eq. (20):WILLIAM A. BARDEEN AND MYRON BANDER 14A'm = —X,f,'+3k.,f'+X,N(47t) 'ln mr'A gN+g N(47t) 'ln m.theory may be obtained by taking the nonlinear lim-it of Eq. (21) and Eq. (22). The symmetric phaseis determined by=0,Am, = X,f,—'+X,f'+A, N(4z. ) 'ln mr'+g'f'+g'N(4w) 'ln A g Nm. '(20)Afo =N(4z) 'ln2= 2m Q mV =-,'N(4w)-'m, '.(23)J =f(m, ' gf ),-'V= —'m 4(A. +g ) '+ —'N(4n') 'm ' —,'g'f . —'The two distinct phases occur when J =0. The sym-metric phase is determined by=0A'm„= —Xofo +RON(4m) 'ln m. 'This phase is characterized by a full restorationof the O(N) symmetry. As discussed in Ref. 4, theo. particle forms as a bound state of the nonlineardegrees of freedom. Since the full O(N) Coulombinteraction is still operative, only O(N)-singletbound states of o and 7r exist in the physical spec-trum. ' The broken-symmetry phase is determinedby22=m~+g N(4w) lnm, '2= 2m Q m Q 7(21)2 A2f,'=,' +N(4w) 'ln mr" (24)V=~, '(A. ,+g') '+-', N(4z) 'm, '.The broken phase is determined byf =m~A—(Xo/g )m~' = —Xofo +A, ON(4v) 'ln mr'A+g'N(4n) 'lnm. 'm, ' =m, '+ (K, g')f'—= 2(X,/g')m, ',V=~, '(X, +g') '+ ,'N(4m) 'm—„'—~,'/g'.Since we are using the linear O(N) model to reg-ularize the nonlinear O(N) model, we will not dis-cuss the linear theory but proceed to a discussionof the nonlinear theory. We note that all express-sions we have have used are unrenormalized. Theonly divergent renormalization necessary in thelarge-N limit is a logarithmic divergence in f, .The nonlinear O(N) gauge theory is obtained bytaking the limit A. ,-~ with all other parametersheld fixed. One might worry that such a limitmight reorder the large-N expansion. However,we have noted that in two dimensions the nonlineartheory is renormalizable with only logarithmic di-vergences. Hence, the large-N limit cannot bemodified by powers of N and no reordering canoccur.The phase-transition properties of the nonlinear4V=-', N(4~)-'m„'--,' + .In this phase the o is not formed as a bound state.Only the O(N- 1) Coulomb interactions are opera-tive with the remaining Coulomb interactions beingscreened. The physical states are O(N- 1)-singletbound states of z. The broken-symmetry phaseavoids Coleman's theorem' as the Higgs mechan-ism does not permit the existence of Goldstonebo sons.We must now discuss the stability of the twophases to see the range of couplings where ea,chphase can exist and the character of the phasetransition. We introduce a renormalized couplingconstant, h, through the relation=f '- N(4~)-'inwhere M' is a normalization scale, and we definethe fine-structure constant, o. =g'N/4m. Equations(23) and (34) become the following.(a) symmetric Phase1 M—=ln- —, or m '=M'e ' ",(25)V=-,'N(4~)-'m „'.(b) broken symmetry phase-1 m„' M2+lnp2 0V=-,'N(4v)-'(m, ' ——,'m, '/o. ).PHASE TRANSITION IN AN 0 (N) GAUGE MODEL IN TWO. . . 2121.55.0.2rrlbV2.0I.O—.20 .I .2Irn S.5 .6.2 .5FIG. 2. x masses in symmetric phase (I) and broken-symmetry phases (0, II/ as a function of mass in sym-metric phase for (a) n =1, (b) ~ =2, and (c) ~ =~.The symmetric phase can exist for all values ofthe coupling constants. The broken- symmetryphase can exist only if the coupling constant, h, issufficiently small. Since the o-model coupling con-stant is directly related to the mass in the sym-metric phase, we consider the phase propertiesof the theory expressed in terms of this mass. Wedenote the w masses in the symmetric and brokenphases by m, ' and m~', respectively. As mentionedabove we use m, ' to parametrize theory in bothphases. In Fig. 2 we plot the m masses in eachphase for different values of the fine-structure con-stant. In Fig. 3 we plot the vacuum energy for afixed value of the fine-structure constant. In Fig.4, w'e plot the phase-transition line in coupling-constant space.By examining Fig. 2, we see that only the sym-metric phase (I) can exist for m, &n/8 For m, '.& n/e, the broken-symmetry phase can also existwith two possible branches (II, III). As we de-crease the gauge coupling constant (a-b- c), theregion where the broken phase can exist is re-stricted to smaller values of m, ' or, equivalently,smaller values of the coupling constant h. Whenthe gauge coupling constant vanishes (n =0), onlythe symmetric phase can exist, in agreement withthe discussions of Ref. 3 and Ref. 4.Although two broken-symmetry phases can existFIG. 3. Vacuum energy as a function of mass in sym-metric phase for symmetric phase Q and broken-symmetry phases (II, III).for m, '& n/e -=0.368m, the phase transition doesnot occur until m, 's 0.321m, as seen in Fig. 3. Forsmall m, ' (s 0.321m), the stable phase is the larger-~mass broken-symmetry phase (III). The vac-uum energy is, of course, continuous through thephase transition. However, both m ' and j' arediscontinuous, which indicates that it is a first-or-der phase transition. In Fig. 4 we plot the phase-transition line in coupling-constant space whereregion A, is the symmetric phase and region B isthe broken-symmetry phase.FIG. 4. Phase diagram in terms of coupling constantsfor symmetric phase (A) and broken-symmetry phase (8).2122 WILLIAM A. BARDEEN AND MYRON BANDER 14In this paper we have studied the phase-transitionproperties of a nonlinear O(N) o model with gaugeinteractions. When studied in the large-N limit thetheory exhibits a first-order phase transition. Thesymmetric phase is characterized by the generationof a bound-state o particle degenerate with the m'' sand by all physical states being O(N) singlet boundstates of cr and z' s. The broken-symmetry phaseis characterized by a residual O(N —1) symmetry.The v bound states are not formed and all physicalstates are O(N -1) singlet bound states. The Higgsmechanism avoids the necessity of Goldstone bo-sons and stabilizes the broken-symmetry phase intwo dimensions.These results indicate that the longitudinal dy-namics of the gluon fields in the transverse-latticetheory of Bardeen and Pearson' is rich enough tosupport a phase transition in the quark-gluon theo-ry, as would be expected in 4+& dimensions or ifthe number of quarks were to be sufficiently largeas to destabilize confinement phase. The phase-transition properties of the full gauge theory are ofcourse much more complex than the simple modelstudied in this paper. We do think that the resultsof this paper shed some light on the mechanismswhich operate in a gauge theory.Note added. After this paper was completed, wereceived a report by J. S. Kang [Phys. Rev. D14, 1587 (1976)], who studies the linear 0(N) gaugemodel. His calculations include the leading-N con-tribution from the meson self-interaction but areto first order in the gauge coupling. Our resultsrepresent the full leading-N calculation of the prop-erties of the linear and nonlinear O(N) gauge theo-ries in two dimensions.W. A. B. would like to thank R. Shrock, R. Pear-son, and B. W. Lee for useful discussions.*Operated by Universities Research Association Inc.under contract with the Energy Hesearch and Develop-ment Administration. .William A. Bardeen and Robert B. Pearson, Phys.Rev. D 14, 547 (1976).2Alternative lattice formulations of this theory havebeen proposed by K. Wilson [Phys. Rev. D 10, 2445(1974)] and by J. Kogut and L. Susskind [ibid. 11, 399(1975)].E. Brezin and J. Zinn-Justin, Phys. Rev. Lett. 36, 691(1976) and CEN Saclay report (unpublished); see alsoA. Polyakov, Phys. Lett. 59B, 79 (1975) and A. Migdal,Landau Institute Report, 1975 (unpublished).4W. A. Bardeen, B.W. Lee, and R. E. Shrock, Phys.Rev. D 14, 985 (1976).5Without gauge couplings, the linear model has beenstudied in the large-N limit by S. Coleman, R. Jackiw,and H. D. Politer [Phys. Rev. D 10, 2491 (1974)],R. G. Root [ibid. 10, 3322 (1974)]; and L. F. Abbott etal. [ibid. 13, 2212 (1976)l.6To establish this intuitive result rigorously, we wouldhave to study a Bethe-Salpeter equation analogous tothat proposed by G. 't Hooft [Nucl. Phys. B75, 461(1964)] and further elaborated upon by C. G. Callan,N. Coote, and D. Gross [Phys. Rev. D 13, 1649 (1976)]and by M. E. Einhorn [Fermilab Report No. Fermilab-Pub-76/22- THY (unpublished)].S. Coleman, Commun. Math. Phys. 31, 259 (1973).

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Phase transition in an O(N) gauge model in two dimensions

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Phase transition in an O(N) gauge model in two dimensions

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