PHYSICAL REVIEW D 66, 127701 ~2002! Affine Toda model coupled to matter and the string tension in 2D QCD Harold Blas* Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, SP, Brazil ~Received 8 September 2002; published 13 December 2002! The sl(2) affine Toda model coupled to matter is shown to describe various features, such as the spectrum and string tension, of the low-energy effective Lagrangian of two-dimensional QCD~one flavor andN colors!. The corresponding string tension is computed when the dynamical quarks are in thefundamentalrepresentation of SU(N) and in theadjoint representation ofSU(2). DOI: 10.1103/PhysRevD.66.127701 PACS number~s!: 11.10.Kk, 11.15.Kc, 11.30.Na, 12.38.Aw o n- n n e o i u rn f rg s d t a a in re ls e e e a - - - c- er- e- tum - ve ing e bi- It has been conjectured that the low-energy action of tw dimensional QCD (QCD2) (e@mq , mq quark mass ande gauge coupling! might be related to massive two dime sional integrable models, thus leading to the exact solutio the strong coupled QCD2 @1#. Although some hints toward a integrable structure in QCD2 have been encountered th problem remains open@2#. In recent papers by Armoniet al. @3# it was proved that bosonized QCD2 @1# exhibits a screening nature~vanishing of the string tension! when the dynamical quarks have n mass both in the case when the source and the dynam fermions belong to the same representation of the ga group and in the case when the representation of the exte charge is smaller than the representation of the massless mions. The string tension also vanishes when the test cha are in the adjoint representation and the dynamical one the fundamental representation. Confinement is restore the nonstandard matter content case~e.g., dynamical adjon matter and fundamental probe charge! when a small mass (mq!e) is given to the quarks, as initially argued in@4#. Similar phenomena occur in QED2 @2#. Integer charges can screen fractional charges when the dynamical electrons massless. The confinement phase is restored when the namical electrons are massive and when the external ch is not an integer multiple of the dynamical charge. The str tension in QCD2 is @3# s5mqmR( i @12cos 4pl i kext/kdyn#, ~1! wheremR;e „m f und5@exp(g)/(2p)3/2#e, g is the Euler num- ber…, l i are the isospin eigenvalues of the dynamical rep sentation,kext andkdyn are the affine current algebra leve of the external and dynamical representations, respectiv R5 fundamental and adjoint representations. A possible g eralization of Eq. ~1! to representations to which th bosonization techniques are applicable, among them the tisymmetric and symmetric representations, is in@3#. In addition, thesl(n) affine Toda model~ATM ! coupled to matter~Dirac! fields @5–11# constitutes an excellent labo ratory to test ideas about confinement@6#, the role of solitons in quantum field theories@5#, duality transformations inter changing solitons and particles@5,9#, as well as the reduction *Electronic address: blas@ift.unesp.br 0556-2821/2002/66~12!/127701~4!/$20.00 66 1277 - of cal ge al er- es in in re dy- rge g - ly. n- n- processes of the~two-loop! Wess-Zumino-Novikov-Witten ~WZNW! theory from which the ATM are derivable @7,10,11#. We show that thesl(2) ATM model describes the low energy spectrum of QCD2 ~one flavor andN colors in the fundamental andN52 in the adjoint representations, respe tively!. The exact computation of the string tension is p formed. A key role will be played by the equivalence b tween the Noether and topological currents at the quan level @6#. The Lagrangian of the ATM model is@5–7# ~1/k!L52 1 4 ]mw]mw1 i c̄gm]mc2mcc̄ e2iw g5c, ~2! wherek5k/2p (kPZ), w is a real field,mc is a mass pa- rameter, andc is a Dirac spinor. Notice thatc̄[c̃Tg0. We shall takec̃5ecc* @6#, whereec is a real dimensionless constant. The conformal version~CATM! of Eq. ~2! has been constructed in@11#. The integrability properties and the re duction processes, WZNW→CATM→ATM→ sine-Gordon ~SG! 1 free field, have been considered@5,7,10,6#. The sl(n) ATM exhibits a generalized sine-Gordon–massi Thirring correspondence@9#. Moreover, Eq. ~2! exhibits mass generation despite chiral symmetry@12# and confine- ment of fermions in a self-generated potential@6,13#. The Lagrangian is invariant underw→w1np, thus the topological charge,Qtopol.[*dx j0, j m5(1/p)emn]nw, can assume nontrivial values. A reduction is performed impos the constraint@6,7,5# ~1/2p! emn]nw5~1/p!c̄gmc, ~3! whereJm5c̄gmc is theU(1) Noether current. Equation ~3! implies c†c;]xw, thus the Dirac field is confined to live in regions where the fieldw is not constant. The 1(2)-soliton~s! solution~s! for w andc are of the sine- Gordon~SG! and massive Thirring~MT! types, respectively; they satisfy Eq.~3! for uecu51, and so are solutions of th reduced model@6#. Similar results hold insl(n) ATM @8,9#. Introduce a new boson field representation of fermion linears as @14# :c̄(16g5)c:52 (cm/p) :e(6 iA4pf): , :c̄gmc:52 (1/Ap) emn]nf,wherec5 1 2 exp(g) andm is an infrared regulator. Define the fieldsF andr as F5 2 b ~Apf1w!, r5 A2/p b ~2Apecf1pw!. ~4! ©2002 The American Physical Society01-1 fr y s r r’’ a - te b- t ten- n- - in in e ass BRIEF REPORTS PHYSICAL REVIEW D66, 127701 ~2002! Then the Lagrangian~2! becomes@6# Lbos52 e 2ec ~]mr!21e 1 2 ~]mF!21 m2 b2 ~cosb F!m ~5! where b25u4p28ecu/uecku, m25c mcm/pu4p28ecu, and e5sgn(4p28ec). Imposinge51, ec,p/2 we get a unitary sine-Gordon theory and a decoupled massless field. The bosonized version of the constraints~3! is ^C8uAu4p28ecu/2p ]mruC&50, ~6! where theuC& ’s are the space of states of the theory. The low-energy spectrum of QCD2 has been studied b means of Abelian@15# and non-Abelian bosonizations@1,16#. In this limit the baryons of QCD2 are sine-Gordon soliton @1#. In the largeN limit approach~weake and smallmq) the SG theory also emerges@17#. The low-energy limit of QCD2 (Nf51) with quarks in the fundamentalrepresentation ofSU(N) is described by the SG theory with ~see the Appendix! @1# b5A4p/N (N .1). Now, let us introduce in Eq.~5! a new mass paramete m8 by renormal ordering @2# (cosb F)m 5(m8/m)b2/4p(cosb F)m8 , then one has Lbos 52 (1/2ec) (]mr)21 1 2 (]mF)212(m8)2(cosb F)m8 where ~m8!25@ ukecuc ~mc/2p! ~m!(N21)/N#2N/(2N21), ukecu5 N Usgn~ec! 4N uku 61U . ~7! From Eq.~7! and the QCD2 parameterm8 ~A7! one can make the identificationsm5e/A2p; mc;mq . An exact re- lationship betweenmc and mq will be found below. In the largeN limit @16,17#, (m8)2;N e mq . On the other hand, QCD2 (Nf51) with quarks in the adjoint representation ofSU(2) is described by the SG theory with b254p ~see the Appendix!. This allows us to make the identifications from Eqs.~5! and ~A8!: mc;mq , m;S(;e). Let us study the question of confinement of the ‘‘colo degrees of freedom associated to the fieldc ~see the Appen- dix! in the ATM model by computing the string tension. In semi-classical analysis@2,3#, we put a pair of classical exter nal probe ‘‘color’’ chargesq and2q at L and2L described by the static potentialQc5a@Q(x1L)2Q(x2L)# (a is a yet unknown factor!, in the ‘‘color’’ space directionText 3 5diag(l1 ,l2 , . . . ,l l ,0,0, . . . ,0), $l i% being the ‘‘isospin’’ components of the representationR under aSU(2) sub- group. Then comparing the vacuum expectation value~VEV! of the Hamiltonian given earlier in the presence of an ex nal (qext)(2qext) source with the relevant one in the a sence of such a source, we define the string tension in limit L→` @2# s5^H&2^H0&, whereH0 ~H! is the Hamil- tonian in the absence~presence! of the probe charges. 12770 ee r- he Let us examine the ‘‘mass’’ term in Eq.~2! with the SU(N) ‘‘color’’ sector of external fields (ca)ext in the fun- damental representation coupled to the Toda fieldw. In ATM type theories the fermions confine in a self-generated po tial @13#, thus coupling the (cext)’s to w implies some kind of self-coupling in view of the equivalence~3!. From Eq. ~A3! we write (cL †a)ext(cR b)ext5cm/2p(eiQcText 3 )ab ; then the mass term ~A2! becomes kmcecc̄ae2iw g5ca 5(m2/2b2)Tr(eiQcText 3 e2iw1e2 iQcText 3 e22iw). Defining the analogue ofF in Eq. ~4! as FQc i 52/b(l iQc1w), and from Eq.~4! replacingw in terms of the fieldsF andr, the mass term in Eq.~5! can be written as m2 b2 ( i 51 l cosb FQc i 5 m2 b2 ( i 51 l FcosS l iQc2 8 sgn~ec! buku F 1 8 bukecu Ap/2r D G m . ~8! The fieldsF andr in Eq. ~4! inherit from w the symme- tries r→r1(A2p/b)pn and F→F1(2p/b)n (nPZ); then the theory~5! has a degenerate vacuau0n&. In order to computes we need the VEV of the fields, thus we conce trate our attention on one of these vacua, sayu0o&F ^ u0o&r . Therefore, in accordance with the constraint~6! we shall setr50. In Eq. ~8! setting r50 ~ATM → SG reduction! and a 50 ~absence of external charges! we must recover the inter action term (m2/b2)cosb F ~without the sum ini ), so we requireb56@28 sgn(ec)/buku#, where the6 signs encode the F→6F symmetry of the SG theory. Then one gets uku55 6sgn~ec!4N, dyn. quarks in the f undamental rep. of SU~N! 6sgn~ec!4, dyn. quarks in the ad joint rep. of SU~2!. ~9! From Eqs.~7! and ~9! for the fundamentalrepresentation one hasukecu5N/2, uku54N, uecu5p/4'0.78, therefore mc/4p5mq . The limit ukecu/4N→ 1 8 as k, N→`, is the semi-classical limit of the SG theory (b→0). In theadjoint case one has to compare the coefficients of the ‘‘cos’’ term Eq. ~5! with its QCD2 analogue~A8!, thus foruku54 one has egmcmuecu/p25mqS→mc;mq , m;S(;e). In order to describe the chirally rotated mass term QCD2 ~A5! we seta54p(kext /kdyn) @the caseSU(2) re- quires 2p instead of 4p] @3#. Actually, this is the first order term in the (e2/Mq 2) expansion when the external prob charge is viewed as a dynamical field with very large m Mq ~see more details in@3#!. Then the energy VEV in the limit L→` is ^H&52m2/b2( i^cos@4pli(kext/kdyn)1bF#&. Thens5^H&2^H0& becomes 1-2 . ca in r c- int ure and ces r e ts, rk for ge- d l n - n BRIEF REPORTS PHYSICAL REVIEW D66, 127701 ~2002! s5 m2 b2 ( i 51 l F S 12cos 4pl i kext kdyn D ^cosbF& 1sin 4pl i kext kdyn ^sinbF&G . ~10! Thus the values of̂cosbF& and ^sinbF& in the SG theory are needed. TheexactVEV of type ^eiaF& @Re(a)^A2p/b# in the SG theory has recently been proposed@18#. The au- thors studiedLSG5 1 2 (]F)222mocosbF, assuming the nor- malization ^cosbF~x!cosbF~0!&mo505 1/2uxub 2/2p . ~11! From @18# we quote the expectation value fora5b 5A4p/N (N.1) @19#: ^exp~ ibF!&5C~N!mo 1/(2N21) , C~N![ 2N 2N21 16 sinS p 2N21D S pGS 12 1 2ND GS 1 2ND D 2N/(2N21) 3 1 FGS N 2N21D G2 S GS 4N23 4N22D 4Ap D (122N)/N 3F 4 Ap sinS p 4N22DGS 1 4N22D G 1/N . ~12! To use this exact result we have to relatemo andm8. This is done comparing Eq.~11! with ^@cosbF(x)#m8@cosbF(0)#m8& 5cosh@b2D(m8,uxu)#;cosh@b2/2p (2g2ln(m8uxu/2))#, for small m8uxu. We have (m8)25c2/(2N21)(mo 2N/(2N21)). Then the string tension~10! becomes sR55 2~m8!2 ~c2/(2N21)! C~N!( i 51 l S 12cos 4pl i kext kdyn D , R5 f undamental rep. o f SU~N!, ~mqS!( i 51 l S 12cos 2pl i kext kdyn D , R5ad joint rep. o f SU~2!, ~13! whereS in R5ad j. is the fermion condensate~see the Ap- pendix!. We propose Eq.~13! as the exact QCD2 string ten- sion in the limite/mq→`. Some comments are in order. ~i! The string tension~1! reproduces qualitatively Eq ~13!. Equation~1! has been derived using a semi-classi average for the bosonized fields in Eq.~A5! (^g&51) @3#. ~ii ! In the largeN limit for R5fund., Eq.~13! takes the form s52Ncmq(e/A2p)( i@12cos 4pli(kext/kdyn)#, which 12770 l has the samemq ande dependence as Eq.~1!, except for a 2pN factor @20#. Note that when the dynamical matter is the fundamental (kdyn51) the string tension vanishes fo any external matter. In theR5ad joint of SU(2) case (kdyn52) and external charges in the fundamentalkext51, Eq. ~13! reproduces the result of@4# up to a factor 2. Con- siderlW f und5( 1 2 ,2 1 2 ,0,0, . . . ,0), andlW ad j5(1,0,21). The sl(n) ATM models may be relevant in the constru tion of the low-energy effective theories of multiflavor QCD2 with the dynamical fermions in the fundamental and adjo representations, thus providing an extension of the pict described above. Notice that in these models the Noether topological currents@generalizations of Eq.~3!# and the gen- eralized sine-Gordon/massive Thirring models equivalen ~see Refs.@8,9#! take place. A work in this direction is unde current investigation and will appear elsewhere. I thank A. Armoni for his valuable comments about th manuscript which allowed the clarification of certain poin Professor L.A. Ferreira for collaboration in a previous wo which motivated the present investigation, R. Casana useful conversations, Professor A. Accioly for encoura ment and FAPESP-Brazil for financial support. APPENDIX: THE EXTERNAL COLOR CHARGES The equivalence~3! for multisolitons describes,w5wN @Qtopol5N sgn(ec)# and C N-solitons of the SG and MT type, respectively. Asymptotically one can write 1 2p emn]nwN' ( a51 N 1 p c̄agmca , ~A1! where theca’s are the solutions for the individual localize lowest energy fermion states. In fact, Eq.~A1! encodes the classical SG/MT correspondence@21#. Thus, the ATM model can accommodateN-fermion confined states with interna ‘‘color’’ index a @13#. If we considerN free Dirac fermions ca we will have aSU(N)3U(1) symmetry with currents Jm a 5c̄gmTac andJm5c̄agmca where theTa’s are the gen- erators of SU(N) in the fundamental representatio @Tr(TaTb)5 1 2 dab#. TheU(1) currentJm was bosonized ear lier. In order to gain insight into the QCD2 origin of the ca fields @22# let us write the mass term in the multifermio sector of ATM theory as c̄ae2iw g5ca5cL †acR ae 2iw1cR †acL ae22iw. ~A2! The non-Abelian bosonization@14# allows us to write Ja5 2 i 2p Tr~]2hh†Ta!, J̄a5 i 2p Tr~h†]1hTa!, cL †acR b5M ~hb a!M~eiA4pf!M , ~A3! whereh is a SU(N) matrix field andx65t6x. Then Eq. ~A2! becomes 1-3 e ize d s ’ - : um g the n BRIEF REPORTS PHYSICAL REVIEW D66, 127701 ~2002! M ~Tr h eibF1Tr h†e2 ibF!M , ~A4! whereF52/b(Apf1w) from Eq. ~4! has been used. Th ATM mass term in the multifermion sector, Eq.~A4!, must be compared to the corresponding term in the boson QCD2 in order to identify the fields related to the flavor an color degrees of freedom. The bosonized QCD2 action (Nf51) with a chirally ro- tated mass term in thefundamentaland adjoint representa- tions can be schematically represented by@3# S5SWZW@g#1Skinetic@Am#2 ikdyn 4p E d2xTr A1 a g]2g† 1 1 2 mqmRE dx2Tr~gei4p(kext /kdyn)Tdyn 3 1e2 i4p(kext /kdyn)Tdyn 3 g†!, ~A5! where g is an N3N unitary matrix @(N221)3(N221) orthogonal# for the fundamental ~adjoint! representation and Am is the gauge field~the gaugeA250 was used!. When the quarks transform in theadjoint rep. the WZW and the interaction terms must be multiplied by1 2 becauseg is real and represents Majorana fermions (kdyn51 for the fundamental andkdyn5N for the adjoint reps., respectively!, mR is to be fixed, andSWZW@g#5 1/8*pd2xTr(]mg]mg21) 11/12p*d3ye i jkTr(g21] ig)(g21] jg)(g21]kg). For quarks in thefundamentalrepresentation we setg 5heibF (b5A4p/N), h,eSU(N); then the mass term is 1 2 m m f und*d2xTr~hei4p(kext /kdyn)Tdyn 3 eibF 1e2 i4p(kext /kdyn)Tdyn 3 h†e2 ibF!. ~A6! et s. , 12770 d In the strong coupling limit (e/mq→`) the heavy fields can be ignored (h51) after normal ordering at the mas scalee/A2p. Then Eq.~A5! becomes the SG model@set kext50 in Eq. ~A5!, i.e. absence of external charges# @1#, Se f f5*d2x@ 1 2 (]mF)212(m8)2(cosbF)m8#, where ~m8!25@N c mq~e/A2p!(N21)/N#2N/(2N21). ~A7! From Eqs.~A4! and ~A6! one concludes that the ‘‘color’ degreesca (h matrix! confined inside the SG solitons corre spond to the heavy fields of QCD2 which decouple from the light field F at low-energies. For quarks in the adjoint, one has @23# gab 52 Tr(TauTbu21), whereu is a unitaryN3N matrix. For N52 and u5eiApFnW •sW (nW 251; sa , Pauli matrices!, the mass term of Eq.~A5! (kext50) reproduces exactly cosA4pF, the remaining terms in Eq.~A5! are the kinetic and derivative interaction terms for the fieldsnW andF. The kinetic terms do not contribute to the change of the vacu energy in the presence of the external source@3#, and the interaction terms will not contribute in the strong couplin limit. Actually, the change in the vacuum energy is due to mass term@3#. We have the SG model withb254p @19#, Se f f5*d2x@ 1 2 ~]mF!212mqmad j~cosbF!mad j #. ~A8! When N52, instantons bring about a bilinear fermio condensate ~for small mq) @23#: 2mad j^cosA4pF& 5S(;e). ive tion @1# Y. Frishman and J. Sonnenschein, Phys. Rep.223, 309~1993!. @2# E. Abdalla, M.C.B. Abdalla, and K.D. Rothe,Non-perturbative Methods in Two-dimensional Quantum Field Theory, 2nd ed. ~World Scientific, Singapore, 2001!. @3# A. Armoni, Y. Frishman, and J. Sonnenschein, Phys. Rev. L 80, 430 ~1998!; Int. J. Mod. Phys. A14, 2475~1999!. @4# D.J. Grosset al., Nucl. Phys.B461, 109 ~1996!. @5# H. 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Phys.B493, 571 ~1997!. @19# According to the SG/MT correspondence the caseb5A4p is the free fermion point in the SG theory, i.e., the free mass fermions of the MT model. @20# Use sin(x)/x'1, G(x)'(1/x)2g1O(x), at x50. @21# S.J. Orfanidis, Phys. Rev. D14, 472 ~1976!. @22# In the case of dynamical fermions in the adjoint representa of SU(2) it is possible to constructSU(2) fund. rep. from the adj. rep.of SU(2) via bosonization@4#. @23# A. Smilga, Phys. Rev. D49, 6836~1994!; 54, 7757~1996!. 1-4