We derive an effective Lagrangian for meson fields. This is done in the light-cone gauge for two-dimensional large-N_c QCD by using the bilocal auxiliary field method. The auxiliary fields are bilocal on light-cone space and their Fourier transformation determines the parton momentum distribution. As the first test of our method, the 't Hooft equation is derived from the effective Lagrangian

Kuroki, S I

Morikawa, K

Morozumi, T

arXiv

We derive an effective Lagrangian for meson fields. This is done in the light-cone gauge for two-dimensional large-N_c QCD by using the bilocal auxiliary field method. The auxiliary fields are bilocal on light-cone space and their Fourier transformation determines the parton momentum distribution. As the first test of our method, the 't Hooft equation is derived from the effective Lagrangian.We derive an effective Lagrangian for meson fields. This is done in the light-cone gauge for two-dimensional large-N_c QCD by using the bilocal auxiliary field method. The auxiliary fields are bilocal on light-cone space and their Fourier transformation determines the parton momentum distribution. As the first test of our method, the 't Hooft equation is derived from the effective Lagrangian

Kuroki, Shin-Ichiro

Morikawa, Keiichi

Morozumi, Takuya

CERN Document Server

CERN-TH/2001-070HUPD-0102Large-Nc meson theoryShin-Ichiro Kuroki 1, Keiichi Morikawa 2 and Takuya Morozumi 31 Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama,Higashi-Hiroshima, Japan2 Research Center for Nanodevices and Systems, Hiroshima University,1-4-2 Kagamiyama, Higashi-Hiroshima, Japan3 CERN, Geneva, SwitzerlandAbstractWe derive an eective Lagrangian for meson elds. This is done in the light-conegauge for two-dimensional large-Nc QCD by using the bilocal auxiliary eld method. Theauxiliary elds are bilocal on light-cone space and their Fourier transformation determinesthe parton momentum distribution. As the rst test of our method, the ’t Hooft equationis derived from the eective Lagrangian.1E-mail address: kuroki@theo.phys.sci.hiroshima-u.ac.jp2E-mail address: morikawa@sxsys.hiroshima-u.ac.jp3On leave of absence from Hiroshima University. E-mail address: takuya.morozumi@cern.ch1 IntroductionQCD successfully describes the dynamics of quarks and gluons in the perturbative regime.However the dynamics of mesons and baryons are not yet understood from the rst prin-ciple. Although we believe that the meson is the bound state of quarks and the numericalsimulation based on lattice QCD is successful, it still is a dicult task to clarify this con-nection analytically. Large-Nc QCD has played an important role on our understanding ofQCD [1{3]. Especially in two dimensions, that is, in the ’t Hooft model, a bound-stateequation is obtained. This is the well-known ’t Hooft equation [1, 2], and it is derived froma ladder Bethe{Salpeter equation. This equation can be solved by several methods, e.g.,the discretized light-cone quantization [4, 5], the variational method [6], and the Multhopptechnique [7]. These calculations give us various information about the meson wave functionand the bound state mass. Our problem is how we can understand this equation in the La-grangian level and, more, how we can deduce the Lagrangian corresponding to this equationfrom QCD.In this paper, we derive an eective Lagrangian of large-Nc QCD2. The derivation ofthe Lagrangian is based on the bilocal auxiliary eld method [8{10], and then the eectiveLagrangian contains a meson eld as a bilocal eld. At rst, we take the light-cone gaugeA+ = 0. The remaining gauge eld A− induces non-local four-fermi interaction with light-cone distance r− at equal light-cone time x+. In this interaction term, we introduce the bilocalauxiliary elds using the path-integral, and this eld corresponds to the meson. The Fouriertransformation of the bilocal auxiliary elds determines the parton momentum distributionfunction. From the eective Lagrangian, we obtain the ’t Hooft equation for the bilocalauxiliary elds.The paper is organized as follows. In section 2, we briefly review QCD2 with the light-cone gauge. In section 3, we derive the large-Nc eective Lagrangian, and we decomposethe auxiliary elds into the vacuum expectation value and its fluctuation eld. This vacuumexpectation value of the auxiliary elds is obtained by the eective potential. In section 4,we discuss the eective potential and the vacuum expectation value, which is obtained fromthe gap equation and determines the dynamical quark mass. In section 5, we derive the ’tHooft equation from our eective Lagrangian. We conclude the paper in section 6, and wesummarize our discussion.2 SU(Nc) QCDAn SU(Nc) gauge theory is dened by the LagrangianL = qi(iγD −mi) qi −14FaFa ; (1)where i denotes the flavour (i = 1;    ;Nf ) and a denotes the adjoint representation of colour(a = 1;    ;N2c − 1). Fa is the eld-strength tensor Fa = @Aa − @Aa − gfabcAbAc ,and the covariant derivative is dened as D = @ + igtaAa. The fermion eld qi is thefundamental representation of SU(Nc), and in two dimensions this eld is a two-componentspinor. The light-cone coordinate is dened by x+ = x0 + x3; x− = x0 − x3. We employ aspecic representation γ = 1  i2. In this representation, the spinor is given by qT =(q−; q+), where the lower component (upper component) q+ (q−) is a projection of  =114γγ. The light-cone gauge is given by A+ = 0. In this gauge there are no ghosts, and theremaining gauge eld is only A−. Then the Lagrangian is given byL = qiy+ i@− qi+ + qiy− i@+ qi− −miqiy+ qi− + qiy− qi+− gqiy+ taA−a qi+ +18(@+A−a2: (2)Integrating over A−, this gauge eld induces a non-local four-fermi interaction:L = qiy+ i@− qi+ + qiy− i@+ qi− −miqiy+ qi− + qiy− qi++ 2g2qiy+ ta qi+ 1@+2qjy+ ta qj+; (3)where the inverse operator 1@+2is dened by [11, 12]1@+2q+(x−) =18Z 1−1dy−jx− − y−jq+(y−): (4)The equation of motion for qiy+ is given as qi− =mii@+ qi+, and then the eld qi− is the constrainedeld because at any x+ it is determined by qi+. The action is given in terms of the one-component spinor qi+ as:S =12Zdx−dx+ qiy+ i@− − m2ii@+!qi++g28Zdx−dx+dy−qiy+(x−; x+) ta qi+(x−; x+)jx− − y−jqjy+ (y−; x+) ta qj+(y−; x+): (5)The non-local four-fermi interaction with linear potential induces the bound state of quarkand antiquark.3 The large-Nc effective action with the auxiliary fieldsWe expand the Lagrangian by 1=Nc, with g2Nc xed. We discuss the theory in the leadingorder of 1=Nc expansion. At rst we rearrange the non-local four-fermi interaction by usingthe identity (ta)(tb)γ = − 12Nc γ + 12γ . In the leading order only the colour singletpart of the quark pair contributes. Then the eective action in the large-Nc limit is given bySLargeNc =12Zdx−dx+qyi+ i@− − m2i@+!qi+−g216Zdx−dx+dy−qyi+(x)qj+(y)jx− − y−jqyj+ (y)qi+(x); (6)where the quark pair in the interaction term is a colour singlet, and this quark pair is non-local in the light-cone space, that is qyi+(x)qj+(y)  qyi+(x−; x+) qj+(y−; x+). Here we ignorenext-to-leading terms. The partition function of this theory is written asZLargeNc =Z Yi[dqyi+ ][dqi+] expniSLargeNco: (7)2In order to treat the singlet of quark pair as a dynamical variable, we introduce bilocalauxiliary elds. After integrating over quark elds, the bilocal auxiliary eld corresponds tothe singlet of the quark pair. At rst we multiply the partition function by a constant factorof -functions:ZLargeNc =Z Yi;j[dqyi+ ][dqi+][dsij ] expniSLargeNcoYi;jsij − qyj+ (x)qi+(y); (8)where we introduce the auxiliary elds sij, which correspond to qyj+ (x)qi+(y). These -functions are dened with the additional bilocal elds :Yi;jsij(y; x) − qyj+ (x)qi+(y)/Z Yi;j[dij ] expi2Zdx−dx+dy− ji(x−; y−;x+)(sij(y; x)− qyi+(x)qj+(y))o: (9)The partition function with the bilocal auxiliary elds is given byZLargeNc =Z Yi;j[dqyi+ ][dqi+][dsij ][dij ] expniSLargeNc + iSAuxo; (10)where the action SAux is written asSAux =12Zdx−dx+dy− ji(x−; y−;x+)sij(y; x)− qyj+ (x)qi+(y): (11)In order to consider the correspondence between the bilocal elds ij and quarks qi+, varyingZLargeNc by sij(x; y), we get the relationhij(x−; y−;x+)i = hg28jx− − y−jqyj+ (y)qi+(x)i: (12)This equation tells us that the auxiliary elds are of order g2 and that these elds correspondto quark pairs with gluons; they would then be identied as mesonic dynamical variables. Thiswill be mentioned later. Integrating over auxiliary elds sij and quarks qi+, the Lagrangianis given bySLargeNc = −iNc ln Det M(x; y)+Zdx−dx+dy−ij(y−; x−;x+)1g2jx− − y−jji(x−; y−;x+); (13)where the matrix M is dened byM(x; y) "ji i@− − m2i@+!2(x− y)− ji(x; y)#; (14)withij(x; y)  ij(x−; y−;x+)(x+ − y+): (15)3The vacuum expectation value of these bilocal auxiliary elds ij is obtained by the eectivepotential. From the charge conservation, the diagonal part of ij takes non-zero vacuumexpectation value, and the other is zero:hij(x−; y−;x+)i =(vi (x− − y−) if i = j;0 if i 6= j: (16)The bilocal auxiliary elds are expanded into the vacuum expectation values and the fluctu-ation,ij(x−; y−;x+) −! vi (x− − y−) ij + ij(x−; y−;x+): (17)Finally we arrive at the large-Nc eective action:SLargeNc = S(0) + S(2); (18)whereS(0) = −iNcTrLn S−1F (y − z)+Zdx−dx+dy−vi(y− − x−)vi(x− − y−)g2jx− − y−j ; (19)S(2) =iNc2Tr[SF (z − u)(u; x)]2+Zdx−dx+dy−ij(y−; x−;x+)ji(x−; y−;x+)g2jx− − y−j : (20)Here the subscript of S denotes the degree of . The rst-degree action S(1) vanishes if weuse the gap equation of vyi , which will be discussed in the next section. In this Lagrangian,the matrices S−1F and  are dened byS−1F (x− y) = diag  i@− − m2ii@+!2(x− y)− vi (x− − y−)(x+ − y+)!; (21)and(x; y) =ij(x; y): (22)Here we ignore the higher-order terms of the 1=Nc expansion in Eq. (18).4 The effective potential and the gap equationIn this section, we discuss the behaviour of the vacuum expectation value of the bilocalauxiliary eld by considering the leading order S(0) solely. We give the momentum spacedescription of S(0). Then we can easily extract the space-time volume from the eectiveaction. The rst term in Eq. (19) isTrLn S−1F (y − x)=Zdx+dx−dy+dy− 2(y − x)Zd2p(4)2eip(y−x)ln detS−1F (p)= 2(0)Zdp+dp−Xiln p− − m2ip+− vi(p+)!; (23)4where S−1F (p) and v(p+) are the Fourier transformation of Eq. (21) and vi(x−):S−1F (p) = diag p− − m2ip+− vi(p+)!; (24)vi(x−) =Zdp+4e−i2p+x−vi(p+): (25)The second term in Eq. (19) isZdx+dx−dy−vi(y− − x−)vi(x− − y−)g2jx− − y−j= 4(0)Zdx−dy−vi(y− − x−)vi(x− − y−)g2jx− − y−j= 2(0)Zdp+dk+vi(p+)G(p+ − k+)vi(k+); (26)where we dene the kernel G:1g2jx−j =Zdk+4e−i2k+x−G(k+): (27)The eective potential is dened by S(0) = −2(0)Veff , thenVeff = iNcXiZdp+dp−ln p− − m2ip+− vi(p+)!−Zdp+dk+vi(p+)G(p+ − k+)vi(k+): (28)The behaviour of the vacuum expectation value vi(p+) is determined by the gap equation,which is given by varying the eective potential with respect to vi(p+):Veffvi(p+)= −iNcZdp−1p− − m2ip+ − vi(p+) + i sgn(p+)−2Zdk+G(p+ − k+)vi(k+)= 0: (29)Multiplying these equations by the inverse of G(p+), the gap equation is written asvi(p+) =iNcg242Zdk+dk−P(p+ − k+)21k− − m2ik+ − vi(k+) + i sgn(k+): (30)The symbol P means the principal value of (p+ − k+)−2. This equation is equivalent tothe Schwinger{Dyson equation. The vacuum expectation value gives the self-energy of thequark and the dynamical quark mass. Performing the integration in Eq. (30), we arrive atvi(p+) = −Ncg22 1p+ . However we will not use this explicit expression in the derivation of the’t Hooft equation discussed in the next section.5qq P−i,j,Pp +i, p(a) (b) (c)Figure 1: Feynman rule: (a) quark propagator SiF (p) =p− − m2ip+ − vi(p+) + i sgn(p+)−1,which is the diagonal part of SF (p), (b) gluon propagator, with vertices G−1(p+) = −8g2 Pp+2 ; and (c)bilocal eld ij(q+; q+ − P+ : P−).5 The ’t Hooft equationOur bilocal auxiliary eld is supposed to be the mesonic eld in the large-Nc theory. Intwo dimensional large-Nc QCD, there is a well-known bound-state equation called the ’tHooft equation. If our bilocal auxiliary eld is a bound state, it should satisfy this equation.This equation was rst derived in [1], and the mesonic eld was introduced as a solutionof the ladder Bethe{Salpeter equation. As we will show, our eective theory contains thebilocal eld as a meson, which satises the ’t Hooft equation. In this section we derive the’t Hooft equation from our eective theory. At rst we give the denition of the Fouriertransformation, which, for ij(x−; y−;x+), is dened byij(x−; y−;x+) =ZdP−4dp+4dk+4e−i2P−x+e−i2(p+x−−k+y−) ij(p+; k+;P−); (31)or in the variables p+ ! q+ and k+ ! q+ − P+,ij(x−; y−;x+) =ZdP−4dq+4dP+4e−i2P−x+e−i2(q+(x−−y−)+P+y−)ij(q+; q+ − P+;P−); (32)where this P+ is the total momentum and q+ is the momentum carried by the quark (seeFig. 1). The quark propagator is given bySF (p) =Zdx+dx−eipxSF (x)= diag0@ 1p− − m2ip+ − vi(p+) + i sgn(p+)1A : (33)The Feynman rule is shown in Fig. 1. The momentum-space description of the action (20)is given byS(2) =iNc2trZdP−dP+dq−dq+(4)4SF (q+; q−) (q+; q+ − P+;P−)SF (q+ − P+; q− − P−) (q+ − P+; q+;−P−)+trZdP−dP+dq0+dq+(4)4(q0+; q0+ − P+;P−)G(q0+ − q+) (q+ − P+; q+;−P−): (34)6= − +S (2) Tr TrFigure 2: Diagrammatic representation of the eective action (34).This action is depicted in Fig. 2. The Bethe{Salpeter equation is derived by varying the S(2)with respect to the bilocal eld ji(q+ − P+; q+;−P−)(4)4S(2)ji(q+ − P+; q+;−P−)= iNcZdq−SiF (q+; q−)ij(q+; q+ − P+;P−)SjF (q+ − P+; q− − P−)+2Zdq0+G(q+ − q0+)ij(q0+; q0+ − P+;P−)= 0: (35)This equation is depicted in Fig. 3. We multiply the Bethe{Salpeter equation by the kernelG−1, which is written as a simple form:ij(q+; q+ − P+;P−)=iNcg242Zdk−dk+P(q+ − k+)2 1k− − m2ik+ − vi(k+) + i sgn(k+)ij(k+; k+ − P+;P−) 1k− − P− − m2jk+−P+ − vj(k+ − P+) + i sgn(k+ − P+): (36)In Eq. (36) the variable k− is only included in the quark propagators, and we can easilyperform the integration over k−. If the condition sgn(k+) = −sgn(k+ − P+) is satised, theintegration over k− gives the nite contribution. If P+ is positive4, the integration region ofk+ is restricted as follows:0 < k+ < P+: (37)Then the Bethe{Salpeter equation is written asij(q+; q+ − P+;P−)= −Ncg22Z P+0dk+P(q+ − k+)2ij(k+; k+ − P+;P−)P− − (m2ik+ +m2jP+−k+ )− vi(k+) + vj(k+ − P+): (38)We dene the eld  ij by the eld ij and two quark propagators:4In spite of the sign of P+, we arrive at the same expression Eq.(38). In the light-cone quantization, P+is positive, because of the dispersion relation and the positive-definite light-cone energy.7=qi,q P−j,qi,q P−j,P PFigure 3: Diagrammatic representation of Eq. (35). ij(q+; q+ − P+;P−)  ij(q+; q+ − P+;P−)P− − (m2iq+ +m2jP+−q+ )− vi(q+) + vj(q+ − P+): (39)This eld  ij is depicted in Fig. 4. Using the eld  ij , the Bethe{Salpeter equation isrewritten as "P− − m2iq++m2jP+ − q+!− vi(q+) + vj(q+ − P+)# ij(q+; q+ − P+;P−)= −Ncg22PZ P+0dk+ ij(k+; k+ − P+;P−)(q+ − k+)2 : (40)qi,q P−j,PFigure 4: Diagrammatic representationof  ij(q+; q+ − P+;P−).The bound-state massM and the momentum frac-tion variable x are dened asM2  P+P−; x  q+P+: (41)The eld  ij is rewritten in terms of the momen-tum fraction x, ij(xP+) =  ij(q+; q+ − P+;P−): (42)Using these variables, the Bethe{Salpeter equa-tion is written as"M2 − m2ix+m2j1− x!− P+ vi(q+) + P+ vj(q+ − P+)# ij(xP+)= −Ncg22PZ 10dy ij(yP+)(x− y)2 : (43)The dierence between the two vacuum expectation values is easily calculated by using thegap equation Eq. (30),vi(q+)− vj(q+ − P+) = Ncg22Z P+0dk+P(q+ − k+)2 : (44)8At last, the ’t Hooft equation is derived:"M2 − m2ix+m2j1− x! # ij(xP+) = −Ncg22PZ 10dy ij(yP+)−  ij(xP+)(x− y)2 : (45)In our large-Nc eective theory, this equation is derived using a bilocal auxiliary eld. Thisshows that the auxiliary eld is a collective eld, corresponding to the bound state.6 SummaryWe construct the large-Nc QCD2 eective theory of the bilocal auxiliary elds. In this theorywe introduce the bilocal auxiliary eld as a colour singlet, which is bilocal in the light-conespace and local in the light-cone time. This non-locality means the distance between theconstituent quarks. From the consideration of the ’t Hooft equation, we identify the bilocaleld as the parton momentum distribution function, or the meson eld. Original derivation ofthe ’t Hooft equation was based on the ladder Bethe{Salpeter equation, and the meson eldis introduced as the solution of this equation. From our point of view, there is a meson theorythat is reduced from large-Nc QCD2, and from this theory we derive the ’t Hooft equation.In our approach the vacuum expectation value of the bilocal eld gives the dynamical quarkmass, and then it corresponds to the self-energy of the quark. We may suppose the resultof the gap equation contradicts the triviality of the light-cone vacuum. In this respect, ourauxiliary eld is bilocal and indeed has a vacuum expectation value. The result implies thatthe light-cone vacuum is not trivial.AcknowledgementsWe like to thank S. Fukae for collaboration at the early stage. We also like to thank V.Gusynin, K. Harada, M. Harada, M. Hirata and Y. Yamamoto for discussions. T. M. likesto thank the PQCD working group members for their encouragement. The work of T. M. issupported by Grand-in Aid for Special Project Research (Physics of CP violation) under theGrant NO. 12014211 and by Fellowship for Japanese scholar and researcher in abroad fromthe Ministry of Education, Science, and Culture of Japan.References[1] G. ’t Hooft, Nucl. Phys. B 75 (1974) 461.[2] S. Coleman, 1=N , in Aspects of Symmetry, Cambridge University Press, Cambridge,1985.[3] A.V. Manohar, hep-ph/9802419.[4] S. J. Brodsky, H.C. Pauli and S. S. Pinsky, Phys. Rep. 301 (1998) 299.[5] T. Eller, H.C. Pauli and S. J. Brodsky,Phys. Rev. D 35 (1987) 1493.[6] K. Harada, T. Heinzl and C. Stern, Phys. Rev. D 57 (1998) 2460.9[7] B. Grinstein and R. Lebed, Phys. Rev. D 57 (1998) 1366.[8] H. Kleinert, Phys. Lett. B 62 (1976) 429.[9] T. Kugo, Phys. Lett. B 76 (1978) 625.[10] T. Morozumi and H. So, Prog. Theor. Phys. 77 (1987) 1434.[11] S. Coleman, Ann. Phys. (NY) 101 (1976) 239.[12] M. B. Einhorn, Phys. Rev. D 14 (1976) 3451.10

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