The exact operator solutions of two-dimensional anomaly-free chiral abelian
gauge theories are obtained. We show that anomaly-cancellation conditions arise
as consistency requirements of these solutions. For a certain class of flavour
symmetries, fermion condensates are constructed. These are shown to violate the
fermion-number conservation rule. The models are extended to include massive
fermions. We propose a bosonised lagrangian for the massive theory and verify
that it complies with the Gupta-Bleuler condition.Comment: 14 pages, latex fil

Abdalla, E.

English

arXiv

The exact operator solutions of two-dimensional anomaly-free chiral abelian gauge theories are obtained. We show that anomaly-cancellation conditions arise as consistency requirements of these solutions. For a certain class of flavour symmetries, fermion condensates are constructed. These are shown to violate the fermion-number conservation rule. The models are extended to include massive fermions. We propose a bosonised lagrangian for the massive theory and verify that it complies with the Gupta-Bleuler condition.The exact operator solutions of two-dimensional anomaly-free chiral abelian gauge theories are obtained. We show that anomaly-cancellation conditions arise as consistency requirements of these solutions. For a certain class of flavour symmetries, fermion condensates are constructed. These are shown to violate the fermion-number conservation rule. The models are extended to include massive fermions. We propose a bosonised lagrangian for the massive theory and verify that it complies with the Gupta-Bleuler condition

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CERN-TH/96-329November 1996Operator solutions of two-dimensional chiralgauge theoriesE. Abdallaa;b1aCERN, Theory DivisionbICTP, High-Energy Divisionb elcio@ictp.trieste.itAbstractThe exact operator solutions of two-dimensional anomaly-free chi-ral abelian gauge theories are obtained. We show that anomaly-cancellation conditions arise as consistency requirements of these so-lutions. For a certain class of flavour symmetries, fermion condensatesare constructed. These are shown to violate the fermion-number con-servation rule. The models are extended to include massive fermions.We propose a bosonised lagrangian for the massive theory and verifythat it complies with the Gupta-Bleuler condition.CERN-TH/96-329November 19961Permanent address: Instituto de Fsica-USP, C.P. 66.318, S. Paulo, Brazil.11 IntroductionTwo-dimensional gauge theories have provided us with valuable in-formations since the pioneering work of Schwinger on two-dimensionalquantum electrodynamics [1]. The subject evolved rapidly with thediscovery of the operator solution of this model [2]. It was subse-quently realised that this solution is of a key importance for the under-standing of fundamental physical problems [2, 3] such as connementand the structure of the -vacuum.Any operator solution of this kind can only be of physical relevanceif it respects gauge symmetries. In chiral gauge theories, the gaugeeld couples to a non-conserved current and the gauge symmetry isbroken. In Salam-Weinberg model, dierent multiplets of right andleft chiralities are introduced in order to restore the gauge symmetry.The gauge anomalies only cancel for a certain combination of mul-tiplets. This anomaly-cancellation condition lead to the discovery ofthe top quark.In this paper, we obtain the operator solution of a general two-dimensional quantum theory with SUR(n)SUL(q) symmetry. As inSalam-Weinberg model, the anomalies cancel between dierent mul-tiplets. We start by deriving the anomaly-cancellation condition inSection 2. In Section 3, we write the gauge and the fermion eldsin terms of bosonic variables. The full operator solutions are thenfound by requiring that these elds obey Dirac and Maxwell equa-tions. These solutions are similar to the operator solutions of theflavoured Schwinger model [4]. Massive excitations are obtained fromthe massive mesons. These are, in turn, constructed from the gaugeeld and their masses are generated by a dynamical Higgs mecha-nism. The massless operators build the  vacua which are labelledby two angles 1 and 2. Out of these solutions, condensates withnon-vanishing fermion numbers are also constructed. We calculatethe two-point functions of these condensates. The non-vanishing re-sult shows that the fermion-number conservation rule is violated. InSection 4, we study the massive theory. We propose an ansatz for thebosonised massive lagrangian, using the principle of form invariance.This lagrangian depends on free parameters. In order to x them,we identify the combinations of the elds that are required to remainmassless by the BRST condition. This then insures the validity of ourlagrangian. The last section is devoted to conclusions and comments.22 The massless modelThe interaction of massless right and left chiral fermions ( and) of charges e and e0 with a U(1) gauge eld (A) is described by theLagrangian density 2L = −14FF +nXj=1 jγ(i@ + eA1 + γ52) j+qXl=1lγ(i@ + e0A1− γ52)l; (1)where the space-time indices are ;  = f0; 1g. The coupling of thegauge eld to the non-conserved currents,J+ =e2Xj jγ+ j = e +− − (2)andJ− =e02Xllγ−l = e0+++ (3)renders this model anomalous. The breakdown of the gauge symmetryis made manifest in the anomaly equation which expresses the non-vanishing of the total divergence of the chiral currents, i.e.,@J =14(ne2 − qe02)F : (4)A well-known method of dealing with an anomalous model is tointroduce extra degrees of freedom [5] into the theory. In this proce-dure, the chiral theory is replaced by a gauge-invariant theory whichincorporates the additional modes. The original anomalous theorycan be recovered by a suitable choice of gauge [4]. However, it isbelieved that a reliable physical theory cannot contain gauge anoma-lies. The gauge symmetries can be restored by imposing the anomalycancellation condition on the theory. Specically, such a cancellationis achieved by requiring that the coecient of the anomaly equationvanishes, i.e.,ne2 = qe02) (ee0)2 =qn: (5)2In our notation [4], γ0 =( 0 11 0, γ1 =( 0 −11 0, γ5 =( 1 00 −1, γγ5 =γ , 01 = +1 and ~p = p .3We shall study this anomaly-free quantum model and solve it in termsof elementary quantum scalar elds.3 Quantum solution of the masslesstheoryIn order to obtain the complete operator solution to the model(1), we need to solve the equations of motion and to insure that thephysical Hilbert space is consistent with Maxwell equations.The Dirac equations of motion, obtained from the Lagrangian (1)is 3i@ + ~e =A1 + γ52 = 0; (6)where  is  i or l,  = 1 and ~e = e; e0. The most general expressionfor the gauge eld isA = −(γ ~@ + @); (7)where  and  are any scalar elds. Consequently, the eld strengthtakes the form,F = γ2: (8)Similarly, the fermions can be expressed in terms of bosonic elds ’f ,f ,  and  as follows4: +;f = (2)1=2Kf expi4 + ip(− ’f +Zx1dy1 _’f (x0; y1));(9) −;f = (2)1=2Kf exp−i4 + ip(’f +Zx1dy1 _’f (x0; y1)) expi(eγ + e); (10)+;f 0 = (−2)1=2Kf expi4 + ip(− f +Zx1dy1 _f 0(x0; y1))3Note that the Dirac equations are similar to those obtained for flavoured Schwingermodel.4The elds  and  do not appear in the expressions for  + or − which are the freechiral components.4 exp− i(e0γ0 + e000); (11)−;f 0 = (2)1=2Kf exp−i4 + ip(− f 0 +Zx1dy1 _f 0(x0; y1));(12)where the Klein-factors Kf insure the correctness of the commuta-tion relations [4]. The gauge and the fermion elds satisfy the Diracequations provided γ = γ0,  = 0 and the elds f , ’f and  aremassless.In addition to the Dirac equations, Maxwell equations,@F + eXj γ1 + γ52 + e0Xlγ1− γ52 = 0; (13)should also be satised 5. Using eqs. (9-12), and after some manipu-lations [4], we obtain the currents,J − = γ− = −1p@−’f −e@− −eγ@−; (14)J + = fγ+ f =1p@+’f ; (15)J− = −1p@−f ; (16)J+ =1p@+f +e0@+ +e0γ@+: (17)On inserting these currents into the Maxwell equations (13), we obtainfor  = −,γ@+2 +e0p@+Xff +e0q2@+ + eq02 γ@+ = 0; (18)and for  = +,−γ@−2−ep@−Xf’f −1e2n@− −γne2@− = 0: (19)The massless terms (i.e. the terms containing massless elds) in theabove two equations have to vanish separately. This is only possible5In fact, Maxwell equations can only be satised on the physical Hilbert space.5if the space of states is chosen in such a way that the longitudinalcurrentsL+ = @+[e0pXff +qe02] (20)andL− = @−[epXf’f +qe2] (21)vanish weakly, i.e., the physical subspace is dened to be that gener-ated by states which obeyhphysjLjphysi = 0: (22)In other words, the Gupta-Bleuler (or BRST) condition must be sat-ised.In contrast to the elds , f and ’f , which are massless, thebosonic eld  obeys a massive equation. For this eld, the contra-dictory equations of motion,(γ2+γqe02) = 0 (23)and(γ2+γne2) = 0; (24)only coincide if the anomaly-cancellation condition (5) is imposed onthe charges e and e0. Consequently, the mass of the  eld is 6m2 =ne2=qe02: (25)The parameters γ and  can now be computed. We require the van-ishing of the norm of L and impose canonical quantisation conditionon , taking care of the negative metric character of the commutator7. We then obtainne2[4−ne22] = 0)  =1ern=1e0rq(26)6This corresponds to the dynamical Higgs mechanism.7The elds f and ’f are also canonically quantised, but with positive metric.6Similarly, by imposing the canonical commutation relation on thegauge eld A18 and on the scalar eld , we obtainγ =1ern=1e0rq: (27)Next, we construct the physical operators. We dene ’ = 1pnPf ’fand  = 1pqPf 0 f 0 . The longitudinal currents (20, 21) simplify toL+ = e0rq@+( + ) (28)andL− = ern@−(’+ ): (29)The negative metric part of the theory can now be isolated by re-expressing the physical operators in terms of the combined elds ’+and +. The eld  − depends on ’ and  and the eld + on  and. However, physical quantities can only depend on the combinations’+ or +, due to the physical state condition (22). Also, the eldswith opposite chiralities  + and − should remain invariant. Thus weneed a chiral gauge transformation. Taking these considerations intoaccount, we dene the following transformations 9: − ! eipnRx1_ − ;  + !  +;+ ! e−ipqRx1_+ ; − ! −;A− ! A− −1ern@−;A+ ! A+ +1ern@+: (30)The physical elds are now given by, −f = (2)1=2Kfeipn exp ip[’f −1pn’+Z( _’f −1pn_’)] exp irn[ + ’+Z( _ + _’)] (31)8The gauge eld is A1 = γ@0 and the momentum conjugate to it is 1 = F01 =γ2 = −γ ne.9The partition function remains invariant as a result of the ne tuning of the chargese and e0.7and+f 0 = (2)1=2Kf 0e−ipq exp ip[− f 0 +1pq +Z( _f 0 −1pq)] exp−irq[ +  −Z( _ + _)]; (32)which depend on the required combination of elds. The last expo-nentials appearing in the expressions for the physical elds, i.e.,(x) = exp irn[ + ’+Z( _ + _’)] = ei1 (33)and(x) = exp−irq[ +  −Z( _ + _)] = ei2 ; (34)act as constant operators on the physical Hilbert space.10 This givesrise to the angles 1 and 2. Thus, the vacuum has a structure similarto that of the Schwinger model{it is degenerate and is labelled by theangles 1 and 2.For general values of n and q, the solutions are similar to thoseof the flavoured Schwinger model. However, for rational values ofpn=q a special situation arises. One can construct condensates in thefollowing way:f1;fpn0f01;f0pq0=  f1    fpn0f 01   f 0pq0; (35)where n0 and q0 are such thatpn0 andpq0 are integers and obeypn0=q0 =pn=q. It is also possible to dene the duals~f1;fn0−pn0f 01;f0q0−pq0=  f1    fn0−pn0f 01   f 0q0−pq0; (36)The {~-operators have non-vanishing two-point functionshffgff 0g(x)~f ~fgf ~f 0g(y)i which break the fermion-number conservation10This can be shown by using the two-point-functions relations,h’(x)’(y)i = h(x)(y)i = −h(x)(y)i :8law[6] 11. Let us take the example of the SU(4)R  U(1)L theory,which contains particles of the type(x)ij =  i j(x) (37)and where the dual coincides with the original condensate, after apermutation of indices. We are interested in computing the correlatorsfor long distances. In that case, we can drop the contribution of themassive eld, and we are left withh12(x)34(0)i  he−i2p(’(+)1 +’(+)2 −’(+)3 −’(+)4 )(x)e−i2p(’(+)1 +’(+)2 −’(+)3 −’(+)4 )(0)i;= e−144ln(−x) = [−x−]−1 : (38)where the label-+ means the combination ’(+) = ’+Rx1 dy1 _’(x0; y1).Notice the use of the infrared regularization of the exponential of themassless eld in two dimensions, crucial in the present computation.In general, such two-point correlators are expected to vanish. Herethey do not, in view of instanton eect as already announced in [6].These excitations can be created in a world wherepn=q takes onrational values. Moreover, these condensates only have a −1 powerdecay (that is as a fermion) for very particular models, such as theexample SU(4)U(1) above, discussed in [6]. If we take the exampleof a model SU(9)U(1), we have for the corresponding correlator theexpressionh123(x)~456789(0)i  [−x−]−1 : (39)For a general symmetry SU(n2)SU(q2), with both n and q not unit,we obtainhff 0(x)~ ~f ~f 0(0)i  [−x−]−1[−x+]−1 : (40)Analogously, for a symmetry SU(n2) U(1) a correlatorh(x1)    (xn)i Yi<j[x−i − x−j ]−1 (41)where all indices are dierent does not vanish either.11Such two-point functions can be computed by known techniques of two-dimensionalquantum eld theory [4].94 The massive theoryUnlike the massless theory, the massive theory is not exactly solv-able. However, to obtain a bosonised massive lagrangian, the massterm of the original lagrangian can be treated perturbatively. Massterms of the bosonised lagrangian are constructed by using the eldsappearing in the massless bosonised theory. In addition, one shouldallow for possible renormalisation constants. This procedure is knownas the principle of form invariance [8].The massive theory is described by the lagrangianLm = L+ Lmass ; (42)where L is given by (1) andLmass = mXf y+f −f +m0Xf 0y+f−f + c:c: : (43)The combinations  y+ − or y+− in the mass terms are determinedby the charge superselection rules. The massive lagrangian can bebosonised by using the formulae (9-12), obtained in the massless case.We nd,L(0)mass = mXfcos[2p’f +rn( + )]+ m0f 0cos[2pf +rq( + )]: (44)The bosonised massive lagrangian obtained from a simple extensionof the massless lagrangian is, however, incomplete. The equations ofmotion are inconsistent and the physical operators fail to satisfy theBRST conditions.Nevertheless, expression (44) gives us an insight into the possibleform of the correct lagrangian. It shows that all the elds in themassive theory are interacting. We also use the principle of forminvariance, discussed above, to propose the ansatzLmass = mXfcos[’f +  + γ +  + ’+ ]+ m0Xf 0cos[0’f 0 + 0 + γ0 + 0 + 0’+ 0]: (45)10The equations of motion obtained from the total lagrangian whichincludes the above mass terms are,(2+ 2) + mγXfsin[’f +  + γ + ’+ ]+ m0γ0Xf 0sin[0f 0 + 0 + γ0 + 0’+ 0] = 0;(46)2’f + mXfsin[’f +  + γ + ’+ ]+ mpnX~fsin[’ ~f +  + γ + +’+ ]+ m00X~f 0sin[0 ~f 0 + 0 + γ0 + 0 + 0’] = 0;(47)2f + m00Xfsin[0’f + 0 + γ0 + 0 + 0’]+ m00pqX~f 0sin[0 ~f 0 + 0 + γ0 + 0 + 0’]+ mX~fsin[’ ~f +  + γ + ’+ ] = 0; (48)−2 + mXfsin[’f +  + γ + ’+ ]+ m00Xf 0sin[0f 0 + 0 + γ0 + 0’+ 0] = 0:(49)The equations of motion of the elds which are relevant for the BRSTconditions follow from these;2(’+ ) = −m(pn+  − )Xfsin[’f +  + γ + ’+ ]− m0(0 − 0)Xf 0sin[0f 0 + 0 + γ0 + 0’+ 0]; (50)2( + ) = −m0(0pq+ 0 − 0)Xf 0sin[0f + 0 + γ0 + 0’+ 0]− m(− )Xfsin[’f +  + γ + ’+ ]: (51)The above combined operators are required, by the Gupta-Bleulercondition, to be free massless elds (see eqs. (28) and (29)). Therefore,11we obtain =pn+  =  (52)and0 =0pq+ 0 = 0: (53)The dimensions of the mass operators,dim (   ) =2 + 2=pn+ 2 + γ2 + 2 − 24;=2 + 2 + γ2 + 2=pn4; (54)dim () =02 + 02 + γ02 + 200=pn4; (55)determine the range of the allowed values of the parameters γ andγ0. If, as in the massive Schwinger model, we require the absence ofThirring interactions and x the parameter  to be 2p, the massoperator will have dimension greater than one [4]. On evaluating theoperator-product expansion of the canonical fermion elds, we alsond that 0 =  = 0. Under such conditions, we obtaindim (   ) =2 + γ24! 1 +γ24; (56)dim () ! 1 +γ024: (57)The mass perturbation expansion is only well-dened if the dimensionsof these mass operators do not exceed 2. This is required for renor-malizability. Therefore, we conclude that, γ  2p and γ0  2p.5 Conclusion and CommentsWe have constructed the full operator solutions of chiral gaugetheories and obtained the anomaly-cancellation conditions by requir-ing the consistency of these solutions. The solutions together withthe structure of the -vacuum are similar to their counterparts in theSchwinger model. It would be interesting to see if these theories alsohave phase structures similar to the screening and conning phases ofthe Schwinger model.12For symmetry groups SU(n)R  SU(q)L withpn=q a rationalnumber, we have constructed condensates. For long distances thesecondensates behave like fermions. We have shown that they violatefermion-number conservation law. It remains to be seen if this phe-nomenon can be generally and satisfactorily explained in terms ofinstantons, as discussed in the case of symmetry SU(4)  U(1). Inthe bosonised theory, the fermion number is a topological numberand the instantons can interplay between dierent vacua. Thus, thenon-conservation of the fermion number can be traced back to instan-ton eects. The generalization of these results to a nonabelian gaugegroup would be welcome,[7] but the nonabelian bosonisation formulasare more involved, and do not permit a straightforward solution.We have formulated a bosonised massive lagrangian by using theprinciple of form invariance, the BRST condition and mass pertur-bation techniques. However, it remains a challenging open problemto nd a bosonisation formula which represents massive fermions asoperator functions of bosonic variables.AcknowledgementsI wish to thank R. Mohayaee, J. Narayanan, H. Neuberger and K.D.Rothe for discussions and suggestions, which considerably improvedthis paper.References[1] J. Schwinger, Phys. Rev. 128 (1962) 2425.[2] J. Lowenstein and J.A. Swieca, Annals of Phys. 68 (1971) 172.[3] A. Casher, J. Kogut and L. Susskind, Phys. Rev D10 (1974)732.[4] E. Abdalla, M. C. B. Abdalla and K. D. Rothe, \Non-perturbative methods in two dimensional quantum eld theory",World Scientic, 1991.13[5] K. Harada and I. Tsutsui, Phys. Lett. B183 (1987) 311.[6] J. Narayanan and H. Neuberg, RU-96-84 Preprint, hep-lat/9609031.[7] E. Abdalla and M. C. B. Abdalla, Phys. Rep. 265 (1996) 253.[8] S. Coleman, R. Jackiw and L. Susskind, Ann. Phys. 13 (1975)267.14

Operator solutions of two-dimensional chiral gauge theories

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