VOLUME 62, NUMBER 13 PHYSICAL REVIEW LETTERS 27 MARCH 1989

Four-Fermion Theory is Renormalizable in 2+ I Dimensions

Baruch Rosenstein and Brian J. WarrTheory GroupD, epartment of Physics, University of'Texas, Austin, Texas 78712

Seon H. ParkCenter for Relativity, Department of Physics, University of Texas, Austin, Texas 78712

(Received 13 October 1988)

The four-fermion interaction in 2+1 dimensions is nonrenormalizable in weak-coupling perturbationtheory. We show that this model is in fact renormalizable order by order in the 1//Vf expansion, and wefind in this theory a finite ultraviolet fixed point.

PACS numbers: 11.10.Gh, 11.10.Ef

It is widely believed that in quantum field theory therequirement of renormalizability is very strict for space-time dimensions d & 2, so that the possible list of contin-uum models is very short. The argument here is basedon a simple "power-counting" analysis, which can bejustified to all orders in the weak-coupling expansion'.There is only a finite number of polynomials in the ele-mentary fields with total "mass dimension" ~ d, andthese are the so-called "relevant" and "marginal" opera-tors. The "irrelevant" operators with dimension & dhave their couplings driven to zero as the ultravioletcutoA A is taken to infinity, and moreover, these cou-plings have no effect at all on the form of the renormal-ized Green's functions. Two famous examples of non-renormalizable interactions (in weak coupling) are"four-fermion" models in d=3, 4, and Einsteinian gravi-ty.

In this Letter we show that outside weak-coupling per-turbation theory new possibilities emerge. Specifically,we prove that the four-fermion interaction in d =3,defined by the Euclidean Lagrangian

2

/klffl/fj + ( l/fj l/fj )2%y

is in fact renormalizable. In (1) the index j runs from 1

to Ny and denotes the "flavor" quantum number. Usinga systematic I/N~ expansion we prove that order by or-der in I//Vj all the ultraviolet divergences can be ab-sorbed in the couplings already present in (1). More-over, the I/Wj. expansion provides here a reliable approx-imation scheme for a whole host of interesting, "nonper-turbative" phenomena.

The model has been studied comprehensively in theframework of the I/jvj. expansion in d=2 by Gross andNeveu, and the reliability of this scheme was estab-lished by comparing results with the known, exact 5 ma-trix. In d=2 the operator (l/fl/f) is already relevant bypower counting, and its coupling is asymptotically free.%'e find that in d=3 the theory is not asymptoticallyfree, but rather possesses a finite ultraviolet fixed point.This fixed point resembles the conjectured nontrivialfixed point in QED in four dimensions widely discussedrecently, and we hope that the theory may provide agood "laboratory" for studying such behavior.

The theory does, however, share some features with itsd=2 analog, namely, that the discrete chiral symmetry

y @5' is dynamically broken, and that the interactionbetween the fermions is su%ciently attractive to lead tothe appearance of weakly bound "mesons. "

Let us briefly sketch the derivation of the I/jVj expan-sion. The expansion can be defined purely in terms ofthe ordinary Feynman diagrams, but it is convenient torewrite the Lagrangian (1) using a scalar auxiliary fieldcr(x):

WyJ (l/f, I/f, a) l/fj kl/fj +G'l//jl/fj2

a' (2)2g

The Lagrangian (2) is equivalent to (1) by the exactequation of motion for a(x). The discrete chiral symme-try is now written as

Y 3'& Y Y YX»

We define a functional integral ZA(g, g) by coupling insources for the fermionic fields and regularizing with amomentum cutofi A:

rZA(g, g) =& Dl/fDl/fDaexp —JI d x[Xf, (l/f, l/f, a) —

gl/f—

l/foal

Since the exponent is purely quadratic in the fermion fields we can integrate directly to obtain

2

Z~(g, g) =„DcrexP — —jVj-J d x —Njtrln(&A+cr)+& d xd y g„(kf,+a) y'(y'2gw

We calculate (5) by the method of steepest descents. The first step is the solution of the stationary condition (the so-

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VOLUME 62, NUMBER 13 PHYSICAL REVIEW LETTERS 27 MARCH 1989

G;; (p) M:—0 exists for all values of the large-Nf bare couplinggA, and a "nontrivial" branch exists in the range

D(p )

where

2 ~ 2gcrit —gA (7a)

FIG. 1. Feynman rules for the 1/1Vf expansion. gcrit g ~ ~ 3(2R') p p(7b)

called "gap equation") which defines the vacuum expec-tation value M of the field cr(x) (Ref. 10):

M ~ d p 10= — —„ trg,' " (2rr) ' ip, +M

This equation has two branches. The "trivial" branch

Now let us consider the theory on the nontrivialbranch. M is a physical quantity, in fact in the Nf~limit it is the pole mass of the fermions, and therefore itshould be independent of A. [Thus for A))M the cou-pling is tuned to be very close, or equal, to the criticalvalue (7b).] The Feynman rules are now given by Fig. 1,where the fermion and auxiliary field propagators aregiven, respectively, by

G„(p) = (ig+M) (ga)

D(p') = 1

Nf1

" d, + ~, tr

gA" (2rt) ' (ig, +M) [i(g—p), +M]

(gb)

and in this algorithm ZA(g, g) is given by the set of allFeynman diagrams with external fermion "legs," exceptthose containing "bubbles" and "tadpoles" as subgraphs(see Fig. 2). The point is that these have already beensummed over in Eqs. (6) and (8).

The necessary fact in the renormalizability proof isthat the reciprocal of the cr propagator, D '(p ), isfinite as A ~ for fixed

~M

~

~ 0, and gA obeying (6).Thus the propagator is not driven to zero, and is given by

D(p', A = ) =(p +4M )arctan[(p ) ' /2M]

Dp =3 —E~ —E (10)

and so the only primitively A-divergent graphs are thosein Fig. 3. Notice that, for example, a four-fermion

This means that in "tree approximation, " which is theleading order in 1/Nf, the connected Green's functionsare finite and nonzero as A

We observe that for large momenta D (p )—1/(p ) 'i, so its "ultraviolet scaling" is the same as forthe fermion propagator. Thus the "superficial degree ofdivergence, "

D&, of a Feynman diagram with E~ exter-nal fermion legs and E external cr legs is

graph is not superficially A divergent, so the "one-loop"graph in Fig. 4 is A finite.

Now from Fig. 3 we may expect to need countertermsfor the renormalization of the form yy, pity, cryy, cr,

a, and o. . Importantly, this list does pot contain a ki-netic term (B„cr), since in Fig. 3(c) the A divergence ismomentum independent.

In fact, we will only need counterterms pity, cryy,and a, which are precisely the original operators in theLagrangian (2). To see this, consider the case M=O.Figures 3(d) and 3(e) vanish identically, as does themomentum-independent divergence in Fig. 3(a), and thisis a consequence of the chiral symmetry. Now, to cir-cumvent a discussion concerning infrared divergences,we carry out the systematic renormalization using the"method of eA'ective Lagrangians. " A small adaptionof the arguments in Ref. 2 shows that the "anomalousdimensions" of operators built out of y and o. are pertur-batively small in 1/Nf, so the "naive" power-countinganalysis in (10) is correct. ' Moreover, the chiral sym-metry in the M=0 case can be preserved order by order,so indeed here the counterterms yy, o, and o. are for-bidden. Now it is easy to discuss the case M&0, sincethis is generated by a finite shift in the cr field. As is

simply proved in this method ' this shifting has no

FIG. 2. Illegal subgraphs. FIG. 3. Primitive divergences.

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VOLUME 62, NUMBER 13 PHYSICAL REVIEW LETTERS 27 MARcH 1989

eA'ect at all on the ultraviolet structure, ' and this completes the proof.Using the Feynman rules, Fig. 1, the four-point fermionic Green s function, in leading order, is

Gij kl (p 1 ~p2, p3,p4) =(4) + crossed

=2K [(p i+p2) 'i 'j'

[(pt+p2) +4M jarctan[[(pt+pq) l' /2Mj

We see from (11) there are no tachyons, so the theory is

consistent. We therefore conclude that for gA in therange (7a) the four-fermion theory is not free and the

chiral symmetry (3) is broken dynamically. Note thereis no "dimensional transmutation, " which takes place in

d =2, since the coupling constant g~ is not dimensionless.

Let us consider the "deep Euclidean" region of thefour-point function (11). We define a dimensionless cou-pling X(E) which expresses the strength of interaction

between the fermions, by ) (E)=NAG (E—). Here thefour-point function is taken at the symmetric point,(pt+p2) =(pt+p3) =(pt+p4) =E, and i =j=k

27' . 7rm2 =Msin sin

Nf —2 Nf —2(14)

2 Z'l JsAij kl (s, t i u ) = ~ij ~kl~f (s —4M') [ln

t(Js +2M)/(Js

There is an s-channel pole at s =4M coincident withthe two-body threshold. The situation closely resemblesthat in d=2, in which the exact solution for the ampli-tude is known. There the pole moves away from thethreshold into the "bound-state region" in order 1/Ng,and the exact mass of this meson is

—2M)t

—itro(s —4M')](13)

tphenomena in d =3 and also the attempt to quantizegravity.

The authors thank Dr. Lee Brekke for discussions andProfessor Lee Smolin for bringing Ref. 5 to our atten-tion. This research was supported in part by the RobertA. Welch Foundation and NSF Grants No. PHY8605978 and No. PHY 8404931.

! =l. From (11) we obtain

E 21).(E) =2n (12)~~+4M2 arctan(E/2M)

We observe that for large momenta X, (E) has a finitelimit ) (~) =4, which is therefore a nontrivial ultravioletfixed point. This feature recalls the running of the elec-tric charge in the "nonperturbative phase" of QED.This result is in sharp contrast with the d=2 Gross-Neveu model, which is asymptotically free.

Finally, let us investigate the behavior of the 2 2scattering amplitude, found by analytic continuation ofthe four-point function (11):

In d=3 we also expect the pole to move away from thethreshold in higher orders; however, an exact solution isnot available.

To summarize, we proved the renormalizability of the"scalar-scalar" four-fermion theory (1) in d=3. Theproof is easily extended to all four-fermion interac-tions. ' The theory possesses a finite ultraviolet fixedpoint and at least one mesonic bound state. In con-clusion, it is clear that weak-coupling nonrenormalizabletheories can nevertheless be renormalizable. This maybe interesting in two areas of physics, namely, critical

FIG. 4. Finite four-fermion coupling in one loop.

'C. Itzykson and J.-B. Zuber, Quantum Field Theory(McGraw-Hill, New York, 1980).

~K. G. Wilson and J. G. Kogut, Phys. Rep. 12, 75 (1974); J.Polchinski, Nucl. Phys. B231, 269 (1984).

There have been attempts to construct a "nonperturbative"continuum limit for such theories, both analytically and on thelattice. See J. Glimm and A. JaA'e, Quantum Physics(Springer-Verlag, New York, 1987); B. de Witt, in "Proceed-ings of the International Meeting on Geometrical and Algebra-ic Aspects of Nonlinear Field Theories" (University of Salerno,Amalfi, Italy, to be published).

4In this notation the Dirac matrices are 4&&4 and Hermitian.5The renormalizability of the (2+ 1)-dimensional models

was noted in the following references. However, these refer-ences carry out the 1/jv expansion incorrectly, and do not ob-tain Eq. (11) for the four-point function. D. J. Gross, in

Methods in Field Theory, Proceedings of the Les HouchesSummer School, Session XXVIII, edited by R. Balian and J.

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VOr VME62, NUMBER 13 PHYSICAL REVIEW LETTERS 27 MARcH 1989

Zinn-Justin (North-Holland, Amsterdam, 1976); G. Parisi,Nucl. Phys. 8100, 368 (1975); K. Shizuya, Phys. Rev. D 21,2327 (1980).

6D. J. Gross and A. Neveu, Phys. Rev. D 10, 3235 (1974).7A. B. Zamalodchikov and A. B. Zamalodchikov, Ann. Phys.

(N. Y.) 120, 253 (1979).~J. G. Kogut, E. Dagotto, and A. Kocic, Phys. Rev. Lett. 60,

772 (1988); E. Dagotto and J. G. Kogut, Nucl. Phys. 8295,125 (1988); A. Kocic, E. Dagotto, and J. G. Kogut, Universityof' Illinois Report No. ILL-TH-88-22 (to be published).

9S. Coleman, Aspects of Symmetry: Selected Frice Lectures (Cambridge Univ. Press, Cambridge, 1985).

'oln (6) and subsequently we consider only constant-fieldsolutions to the gap equation. There will also exist finite-actionnonconstant solutions, which must be accounted for to establishthe equivalence of Lagrangians (I) and (2). We assume thatthese solutions do not spoil the renormalizability of the model.A full discussion of the fermion model in 1+1 dimensions is

given by R. I". Dashen, B. Hasslacher, and A. Neveu, Phys.

Rev. D 12, 2443 (1975), and for analogous models by A. T.Vainshtein, V. T. Zakharov, V. A. Novikov, and M. A. Shif-man, Fiz. Elem. Chastits At. Yadra 17, 472 (1986) [Sov. J.Part. Nucl. 17, 204 (1986)].

''Now we can see what happens if we choose the trivialbranch, with gA outside the range (7a). From (8b) we see thatD '(p ) blows up linearly with A, and the connected Green'sfunctions vanish as an inverse power of A. (This also happensin the weak-coupling expansion. ) The theory defined here is

perfectly consistent: it is a theory of free massless fermions.The trivial branch with gA inside the range (7a) has higherfree energy than the coexistant nontrivial branch, and is madeinconsistent by the presence of tachyons. This exhausts thepossibilities from the gap equation (6).

'28. Rosenstein, B. J. Warr, and S. H. Park (to be pub-lished).

'38. J. Warr, Ann. Phys. (N. Y.) 183, I (1988).'4G. 't Hooft and M. Veltman, CERN Report No. 73-9 (un-

published).

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