Solving the Conformal Bootstrap Four-Fermion Interactions at 2d4
a r X i v :h e p -t h /9302029v 1 9 F e
b 1993
UBCTP-92-34
SOL VING THE CONFORMAL BOOTSTRAP:FOUR-FERMION INTERACTIONS AT 2 Wei Chen ?,Yuri Makeenko ?and Gordon W.Semeno?? ? Department of Physics,University of British Columbia Vancouver,British Columbia,Canada V6T 1Z1 ? Institute of Theoretical and Experimental Physics 117259Moscow,Russia PACS No.:11.10.Gh Abstract We formulate the conformal bootstrap approach to four–fermion theory at its strong coupling ?xed point in dimensions 2 to order 1/N 2. Conformal symmetry describes the critical behavior of quantum?eld theories or statistical systems at points of second order phase transition.It is widely used in two dimensions where it provides exact solutions to a variety of non–trivial interacting ?eld theories.In this Letter we shall demonstrate that it can also be useful in higher dimensions.If a?eld theory has a?xed point of the renormalization group?ow,the dynamics at that point is conformally invariant.In any spacetime dimensions,confor-mal symmetry determines the form of the two and three-point Green functions of the ?eld theory up to a few constants,the scaling dimension of the?eld operators and the value of the coupling constant at the?xed point.This information,when combined with Dyson–Schwinger equations can provide a powerful tool for the analysis of?eld theories. In fact,in a?eld theory with only three-point coupling constants,the Dyson–Schwinger equations which relate two and three point functions could in principle be used to determine all anomalous dimensions in terms of the coupling constants,which are then left to be determined by higher order equations.Unfortunately,this direct approach is plagued by technical di?culties involving indeterminate expressions.This problem was partially solved by Parisi[1]who showed how to use the information of the lower order Dyson–Schwinger equations to derive bootstrap equations for the e?ective coupling constants.The resulting equations can not be solved exactly,but can be used to?nd interesting approximate solutions of a?eld theory. In this Letter,we shall consider the example of a fermionic?eld theory with a four-fermion interaction,?λ2 In this Letter we shall assume thatλ?exists and analyze the resulting conformal ?eld theory.We?nd a solution of the bootstrap equations in the three and?ve vertex approximation.In particular,we show that these give an easy way to?nd the anomalous dimensions ofψandˉψψin the large N limit to order1/N2. Under rescaling of the coordinate,x→ρx,the?elds transform as ψ′(ρx)=ρ?lψ(x),ˉψ′(ρx)=ρ?lˉψ(x),φ′(ρx)=ρ?bφ(x),(1) with l and b the scaling dimensions of the fermion(anti–fermions)and the boson, respectively.Under a special conformal transformation parameterized with a constant vector tμ, xμ→x′μ= xμ+tμx2 ?x′ |1/d=1+2t·x+t2x2,they transform as ψ(x)→ψ′(x′)=σl?1/2 x (1+?t?x)ψ(x),(3) ˉψ(x)→ˉψ′(x′)=σl?1/2 x ˉψ(x)(1+?x?t),(4) φ(x)→φ′(x′)=σb xφ(x),(5) where?x=γμxμ(γμare the Dirac Matrices). These,together with translation and Lorentz invariance,form the global?nite dimensional conformal group.Conformal invariance determines the form of two-point correlation functions.In momentum space, G(p)=1 p2 (p2)b?h+1.(6) A convenient normalization has been chosen.(This can always be done using a?nite rescaling ofφandψ.)Moreover,conformal invariance?xes the three point vertex up to a dimensionless constant factor,the e?ective couplingλ, Γ(p1,p2)=λ N(γ) πh ?k+?p 1 [(k+p2)2]b/2+1/2 1 Γ(τ) ;and γ=l+b/2?h,(8) which is de?ned as the index of the vertex.By dimensional analysis,it is easy to check that the (anomalous)dimension of the vertex Γin momentum units is ?2γ.Our task is to determine the scaling dimensions l and b ,which describe the critical behavior of the model,and the critical coupling constant λ=λ?.We shall use the bootstrap equations.These are derived from the Dyson-Schwinger equations for the fermion and boson self-energies and for the Yukawa vertex.We present the one for the vertex ?rst.Graphically,the bootstrap equation of the Yukawa vertex is [3][1] Γ= u ???e e e ?? ? ?? ? ??e e e e e e e e k p 1p 2+ u [p 2]γ .(9) The r.h.s.of the bootstrap vertex equation is an in?nite series.By dimensional analysis,each term is proportional to1 [p2]γ ,(10) where the function f(2n+1)(l,l,b)is given by the diagrams with2n+1conformal vertices.The arguments l,l,and b refer to the scaling dimensions of the external fermion,anti-fermion,and boson,respectively.The bootstrap equation for the vertex is 1=λ2?f(l,l,b;λ?),(11) f(l,l,b;λ?)= ∞ n=1 λ2(n?1) ? f(2n+1)(l,l,b).(12) We call f(l,l,b;λ?)the vertex function. There are two similar Dyson-Schwinger equations for the fermion and boson self-energies,respectively, Σ= &% '$u resolve this ambiguity was suggested by Parisi[1].It involves taking derivatives of each side by external momenta and by the scale dimensions of the operators.The result is the bootstrap equations for the vertex function.In the present model,we obtain[4] 1=?λ4? N2(b)N(l?b/2) ?f(l′,l,b;λ?) (4π)hΓ(h) N(γ)?N2(b/2)N(d?b) ?b′/2 b′=b,(14) where N(τ)is de?ned after(7)and?N(τ)≡Γ(h?τ+1/2) [p2]γ ?N(b/2)?N(l) It is easy to check that,at the lowest order in γ,(18)coincides with (9). To solve the set of conformal bootstrap equations,we start with calculating the three-vertex correction to the vertex https://www.sodocs.net/doc/2510060902.html,ing the conformal propagators and vertex,and setting the momentum carried by the external boson zero (i.e.p 1=p 2),the three-vertex diagram in Fig.1is ?λ3 d d k [k 2]h ?l +1/2Γ(k,k ) ?k [(k ?p )2]h ?b =? λ3 (4π)h [ ?N (b/2)?N (l )N (b )N (γ) . (19) Above,we have used (18)for Γ(p,k ),as the integration over the region of large internal momentum dominates.Then we have f 3(l,l,b )=? 1 γ[1? γ N Tr 1Γ(2h ?1)sin(πh )h ?1 γ(1) ψ, (22) which reproduce the known results,with (λ2?) (1) =?1 πΓ(h ) ,(23)γ(1)= 1 πΓ2(h ) . (24) Moreover,to the next to leading order,as we shall see below,the vertex function f takes a same form as that in (20).This implies to the order ?f (l ′,l,b ;λ?) ?b ′/2 b ′=b . (25) Then,with no need of further details of the vertex function f (l,l,b ),by using the boostrap equations (13)and (14),one can determine the fermion anomalous dimenion to order 1/N 2.Dividing one of them by the other,we have 1=? 1 N (b )N (d ?b ) . (26) Expand the r.h.s.of the above equation over γψand γφ(both ～1/N +...),we obtain γψ= γ(1)ψ[1 + γ(1) ψ h ?1 +ψ(2h ?1)?ψ(1)+πctg (hπ))+...],(27) where ψ(x )=Γ′(x )/Γ(x ),and γ(1) ψand γ(1) φare given in (21)and (22). To obtain the anomalous dimension of φ=ˉψψto order 1/N 2,one needs the vertex function f to the same order.Besides the second term in the three-vertex correction (20),it involves the leading order of the ?ve-vertex correction,and also the leading order of the seven-vertex diagrams with a fermion loop. Notice that only one ?ve-vertex diagram,depicted in Fig.1,contributes to f 5(l,l,b ).(Another diagram which also contains ?ve conformal vertices has a subdiagram with a fermion loop attached to three boson legs and is zero by parity symmetry).The Feynman integral (setting p 1=p 2)is d d q (2π)d Γ(p,k )G (k )Γ(k,k +q ?p )G (k +q ?p )Γ(k +q ?p,k +q ?p ) G (k +q ?p )Γ(k +q ?p,q )G (q )Γ(q,p )D (k ?p )D (k ?q ).(28) Performing the integral,to the leading logarithmic term,we obtain f 5(l,l,b )=? 1 γ+.... (29) To order 1/N 2,the seven-vertex diagrams with a fermion loop attached to four boson lines contribute,as a fermion loop carries a factor N .This sort of seven-vertex diagrams are given in Fig.3. u u u u u ????e e e e ?? e e u u u u u ????e e e e ?? e e u u u u u ????e e e e ?? e e Fig.3The diagrams of seven-vertex correction with a fermion loop.Each fermion loop carries a factor N. For f7in the second order,to the leading logarithmic term again,we have N Tr1Γ(2?h) γ[1? 3Γ3(h/3)Γ(h)(2h/3?1) (4π)3h(h?1)2Γ(2h?2)1 h?1 ? Γ3(h/3)Γ(h) h?1+ λ2?(1) (4π)2hΓ(2h?2)(h?1)2( 2h?1 2Γ3(2h/3) (1+ 1 2h ))+...]. (34) Then?nally, γφ=2(γ?γψ),(35) whereγandγψare given in(34)and(27). In particular,when d=2h=3and Tr1=2,we have γψ=2 9π2N +...],(36) γφ=?16 36π2N +...].(37) Theψanomalous dimension(γψ)has been computed to this order by conventional diagrammatic methods in[6].That calculation involves much more labor than the present one.Also,the present calculation gives the anomalous dimension of the composite operatorφ=ˉψψwhich has not been computed elsewhere.Note that the result forγψis not the same as in[6].(The numerator of the second term quoted in[6]has122instead of148.)We presently do not understand the source of this discrepancy. Here,we see the power of the bootstrap method in performing approximate cal-culations,such as the large N expansion to higher orders.It would be interesting to apply this approach to other?eld theories,such as gauge theory or theories with vector-like vertices.It would also be interesting to search for other,non–perturbative solutions in the present four–fermion,or inλφ4theory[7]. Note added:After this work was submitted for publication,J A Gracey informed the authors that the number122was a misprint in[6],his result would be112,instead. In a recent publication[8]he also calculated the exponentγφto order1/N2. References [1]G.Parisi,Lett.Nuovo Cim.4(1972)777. [2]C.de Calan,P.A.Faria Da Veiga,J.Magnen and R.Seneor,Phys.Rev.Lett.66 (1991)3233. [3]A.A.Migdal,Phys.Lett.37B(1971)98;386. [4]W.Chen,Yu.M.Makeenko,and G.W.Semeno?,Four-Fermi Theory and the Confor- mal Bootstrap,UBCTP92-30. [5]S.Ferrara,G.Gatto,A.Grillo and G.Parisi,Nuovo Cim.19A(1974)667. [6]J.A.Gracey,Int.J.Mod.Phys.A6(1991)395,2755(E). Interaction,ITEP-44(1979). [7]Yu.M.Makeenko,Conformal Bootstrap forΦ4 (4) [8]J.A.Gracey,Phys.Lett.B297(1992)293. 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