PHYSICAL REVIEW C VOLUME 49, NUMBER 6 JUNE 1994

EfFective 1/N, expansion in the Nambu —Jona-Lasinio model

E. Quack and S. P. KlevanskyInstitut j('6'r Theoretische Physik, Philosophens)eg i9, D 691-20 Hei delberg, Germany

(Received 23 November 1993)

The 1/N expansion is developed for the Nambu —Jona-Lasinio model, a commonly used low-

energy model of +CD. The next-to-leading order in this expansion that represents pion and sigmameson exchange in the gap equation is evaluated perturbatively. These contributions are seen to bedifFering in sign, a feature that is due to their opposite parities. Physically one can interpret thismeson exchange as giving rise to screening (x) or antiscreening (6r) of the bare four-point interaction.As a consequence of the higher isospin degeneracy of the pion, the contributions from it are foundto dominate. In total, we and that the resulting corrections to the usual Hartree approximation areof the order of 10%, justifying a poateriori the perturbative approach.

PACS number(s): 24.85.+p, 12.39.Fe

The need for evaluating low-momentum transfer pro-cesses, corresponding to the nonperturbative QCDregime, has lead to a renaissance of effective quark andmeson theories. One of the most popular ones is theNambu —Jona-Lasinio (NJL) model [1]. It incorporatesthe spontaneous breaking of chiral symmetry, which playsa crucial role at the low energies, and it gives rise to thetransition &om current to constituent quarks in a naturalfashion.

The model, which can be be formulated at Bnite tem-perature and density as well, leads to a successful de-scription of the static meson properties, such as masses,sizes, form factors, etc. [2,3]. Usually calculations arecarried out either in the Hartree or in the Hartree-Fockapproximation. It is, however, of interest to go beyondthese mean Beld calculations. Some attempts have al-ready been made in this direction [4,5], that incorporatein some fashion the terms of higher order. In performingsuch calculations, it is not clear, however, what a suit-able expansion parameter should be, since the theory isone in which the coupling is large. Therefore, the inclu-sion of higher-order terms in [5], for example, accordingto interaction strength, seems unjusti6able. In this pa-per, we suggest an expansion in the inverse number ofcolors 1/N„as an ordering scheme for the contributingdiagrams. We proceed as follows: after briefly review-

ing the commonly used lowest-order or Hartree approx-imation, we develop the 1/N, expansion in a diagram-matic fashion. We classify 6rst the higher-order termsthat contribute to the self-energy. We then discuss a di-agrammatic method of obtaining the associated effectiveinteraction. A correct identification of the polarizationkom this is in principle necessary to ensure that the low-

energy theorems are retained, such as that the Goldstonemode be recovered at every order. In the Bnal sectionof this paper, we present a numerical calculation of thedynamically generated mass to next-to-leading order in1/N that is done perturbatively. This corresponds to in-troducing corrections to the self-energy of the quark dueto the "meson cloud" that surrounds it, and which mayact to screen or antiscreen the bare four-point interac-tion. We 6nd that, while the dressing due to m mesonsleads to an increase in the dynamically generated quark

mass, the u mesons act in the opposite fashion. Thisis brought about by the fact that the pion and sigmacarry opposite parities in entering the expression for theself-energy or gap equation. The net effect indicates thatthe pions dominate, which can be understood as. due totheir higher isospin degeneracy. This level of approxima-tion xnade justifies the approach of Ref. [4], who howeverhave only examined the effect of x mesons.

We 6nd that the resulting corrections to the usualHartree approximation are of the order of 10%, whichin turn justi6es the perturbative approach. Since, how-

ever, in this case, the calculation of the next order is notself-consistent, the Goldstone mode cannot be recovered.

The NJL model in the two flavor version is defined bythe Lagrangian

CNJL —4(iP —mp)C + 6[(@@)'+ (Cits r@) ], (1)

where @ = (&) and a is the vector of the isospin matrices.The current quark masses are assumed to be degeneratein isospin, YAp = fA = mp. For vanishing mp, the La-grangian has the chiral symmetry SU(2)L, SU(2) Jt. Ingeneral, the equation of motion for the quark propagatoris given as

(i() —mo]s(2:, x') —/ dys(z, y)s(y, z') = 6 (z —z').

(2)The solution of this equation for S depends on a knowl-edge of the self-energy Z, while the calculation of Z inturn depends on a knowledge of S. Thus the problem isone of self-consistency, and for an arbitrary Lagrangianto arbitrary order, this is in general not possible to solve.However, by now it is well known that in the NJL modelin the Hartree (or Hartree-Fock) approximation, at leastfor an infinite system, Eq. (2) can be solved exactly. Inthis case, the self-energy is simply a constant so that, inmomentum space, S is given as

S (p) =P —m

with the constituent quark mass m = mp + ZH. Thefact that the theory does not incorporate con6nement

49 3283

3284 E. QUACK AND S. P. KLEVANSKY

of the quarks allows one to identify the quark massesdirectly with the corresponding poles of the quark prop-agator S(p). The model requires a regularization cutoffparameter A, which can be motivated as a crude incor-poration of asymptotic freedom at large Q . The NJLmodel is thus, in what concerns its range of applicability,complementary to that of perturbative @CD.

In order to establish concepts and notation, we brieHyreview the usual calculation of the masses of constituentquarks and mesons in the lowest order approximation.This corresponds to the coupling of the quark to thescalar quark density, or the quark condensate, (QQ). TheGreen function for this process is shown in the upper partof Fig. 1. From the self-energy insertion, one obtains theHartree gap equation which reads

d4k .Z„= 2G Tr),iS(k)r„2ir 4 (4)

m = mp + m8iGN, NyIr (k ),

where I'I, C (I, ipsq ), and the trace is carried out overthe color, Havor, and Dirac indices, Tr = tr, trftr~. Onperforming the traces, one has

2 qq——g qqg(x)ips7. qrQ(x)

for the pion triplet and

~~qq = g~qq&(x)+&(x) (8)

for the u meson.On the quark level, the mesonic modes are described

as polarization excitations with the appropriate quan-tum numbers [6]. The momentum-dependent polariza-tion function has to be evaluated again in some approx-imation. The most commonly used one is the randomphase approximation (RPA), which amounts to summingover single virtual quark-antiquark loops as is shown inFig. 2. The identi6cation of the resulting expression forthe scattering diagram in constructing an effective inter-action with the single meson exchange yields the mesonmasses. In general, the dispersion relation becomes

1 —2GIIM(p )i = 0,

Corresponding to the SU(2)-flavor quarks, one mayconsider the pseudoscalar m triplet and the scalar a me-son. The minimal coupling of these fields to the quarkspinors reads

with

Ii(k ) =(2~)4 pz —m2 '

with M referring to the 7t or 0 channel, as appropriate.%Kith some formal algebraic manipulations, one may ex-press these dispersion relations as [2]

where Iq(k ) has to be suitably regularized. Using, forexample, a standard O(4) cutoff, one recovers the usualform [1)

and

1 —2GII„,(k ) = + 4iGN, Nfk I2(k ) (10)

GA'N. N, m'm=mo+ ' ~m 1 — ln +1 . (6)

2irz A2 (m2

Using appropriate values for the parameters mo, G, andA, the resulting solution describes constituent quarkswith m mN/3, where mN is the nucleon mass. Thus,the original ehiral symmetry is spontaneously broken.Since, in this case, the Goldstone theorem implies theexistence of a corresponding massless boson, mesons ex-ist in the theory and must also be included in a consistentway.

1 —2GII, (k ) = +4iGN, Nf(k —4m )Iz(k ), (ll)m

with Iz(k2) defined as

d4I2(k') =

(2ir)4 [(p+ k)z —mz][p2 —mz]'

This can be evaluated most simply on introducing theFeynman form for the inverse product

1dx

ab o [ax+ b(1 —x)]2'

One 6nds directly that

A2I,(k') =, dx —, , +ln 1+

(14)

I I I

+ I I + I + a ~ ~

FIG. 1. Quark propagators in lowest order. Above: Inself-consistent form; below: In terms of the free quark prop-agator. Thick lines indicate the propagator of constituentquarks, thin lines of current quarks. For clarity, the pointlikeinteraction is indicated by a dashed line.

FIG. 2. Meson polarization modes in the RPA approxi-mation. The boxes indicate the effective meson-quark-quarkcoupling, the double dashed line stands for a meson exchange.

49 EFFECTIVE 1/N, EXPANSION IN THE NAMBU —JONA-. . . 3285

where E = m —k z(l —x). From the relations (10)and (11), it follows that

mp 1

m 4GN, Ny iI2 (m2 )'

while

mp 1m =4m

m 4GN, NyiI2(m )'

so that, approximately,m' - 4m' + m', (17)

given that I2 is a slowly varying function of its argu-ment. In this form, the physics of these results becomestransparent. Since the pion mass vanishes in the chirallimit mp —+ 0, the pion is the Goldstone boson corre-sponding to the spontaneous chiral symmetry breaking.For mo g 0, the vr gains its small mass via the explicitsymmetry breaking.

For the o meson, no such restriction applies, and itsmass lies above 2m. Thus, besides the decay channelo. m qq, the physical channel o ~ xx is open. Due tothe large phase space available, the cr appears at most asa broad resonance in the ~sr channel.

In this manner, one obtains in the Hartree and randomphase approximations a realistic and consistent descrip-tion of the low-energy quark and mesonic sector. Wenow introduce the 1/N, expansion graphically as a toolto classify systematically these approximations accordingto their orders in this expansion. This allows us to checkthe consistency of the approximation. One may see sim-ply in the diagrammatic formulation, that the Hartreeand random phase approximations are complete in theirrespective orders, and are consistent with each other.

The 1/N, expansion is an approximation scheme,where the product of the coupling constant a and somefermionic degree of &eedom N is kept fixed while N —+oo. Originally, it was proposed for @CD with N = N, [7].In this case, for N m oo and fixed nN„ the nonabeliangluon contributions are suppressed against the fermionicones.

We now carry over this procedure to the NJL modeland consider an expansion in the (diagonal) color de-gree of freedom N, . Let g = GA2Ny/vr2 with the massscale given by the cutoff parameter A. With this cou-pling constant, gN, = 0(1). Since every fermion loopcontributes via the trace with a factor N, a diagramwith k fermion loops and l interaction lines is of the or-der g N," = (gN, ) N," = (1/N, ) ". Thus, the Hartreeterm of the self-energy, containing one loop and one in-teraction, is the only contribution of (0[1/ N] )s= 0(1)in the 1/N, expansion, as has been emphasized alreadyby [8]. On the other hand, keeping the Fock term in thecombined Hartree-Fock approximation amounts to keep-ing only a single term of the next order as well, which isnot consistent in a 1/N, expansion. One must examinethe full series in the self-energy and extract the furtherterms of the same order.

In order to connect the self-energy with the effective in-teraction scattering diagrams, we first represent the self-

energy contributions in terms of the bare propagatorsSs(p) = Q —mo] . This is visualized in the lower partof Fig. 1 by iterating the corresponding lower-order dia-grams, as enforced by the self-consistency condition. Thescattering diagrams can now be obtained by a functionalderivative of the self-energy with respect to the currentquark propagator So(p),

bZ 'bSp

'

see e.g. [9]. Here, the improper diagrams that arise onlyrenormalize the external lines and thus do not changethe form of the effective interaction. The self-consistencyin the gap equation then, in turn, implies that thepropagators occurring in the scattering diagrams are re-stored to the "full" propagators of the constituent quarks,S(p) = [yl

—m], which are indicated by thick lines in thediagrams. In terms of diagrams, this functional deriva-tive corresponds to cutting one of the quark "bubbles"in the self-energy diagrams. Omitting this trace over thequark loop reduces the order of the resulting scatteringdiagram by one. Thus, starting from an approximationto the self-energy to a given order, the order of the cor-responding set of scattering diagrams is

0(U) = 0(E) —1. (19)Starting from the lowest order in 1/N, for the self-energy,with the Hartree approximation, this method immedi-ately gives the sum over quark loops indicated in Fig. 2.The RPA, being of 0(1/N, ), is therefore complete andconsistent with the Hartree approximation.

In this way, we obtain an instrument to classify dia-grams systematically according to their order, and to ob-tain a consistent description of the mesonic sector, i.e., acomplete set of scattering diagrams of the appropriateorder n can be obtained from a given order n —1 of theself-energy.

In order for the 1/N, expansion to be a sensible ap-proximation scheme, however, it remains to be shownthat the higher-order contributions to the self-energy de-crease with increasing order in the expansion. We willexamine this for the next order in 1/N, .

In a straightforward manner, one would on first sighttry to carry out the same calculation as before in thenext order. That is to say, one identifies the contribu-tions to the quark self-energy in next-to-leading order(NLO), 0(1/N, ). The resulting self-consistent gap equa-tion is shown in Fig. 3, where at each vertex, one has tosum over the matrices I'g C (1,ipse}. It amounts to con-sidering corrections for Z beyond the mean field. Notethat the 1/N, term that contains an interaction occur-ring inside the tadpole diagram does not require a newfundamental diagram, but is included implicitly in Fig. 3via recurrence.

The resulting full gap equation for ZNI, Q from Fig. 3,1S

d4aZNr, Q(p) = mo + 2iGN, Ny

(2z)4TrS(k)

d4I S(a)(2z )4 1 —2GII -g (p —k)

(20)

3286 E. QUACK AND S. P. KLEVANSKY 49

I I I

+ L + I

I~II I II

!

1I I

+ ' ' +

I

I I

I + ~ ~ ~

FIG. 3. Contributions to the quark propagator in the nextorder of the 1/N, expansion in a self-consistent fashion.

I

I

, 02-

~

L

r~ ~ ~

+ ' — +

I I I I

+ ' + ' ' + ' + ~ ~

II / I

+ I I + I I + I I

I I

+ + ~ ~ ~

I

I + + I + ~ ~ ~

I

I + I I~ ~ ~

PIG. 4. Contributions to the quark self-energy, expressedwith the current quark propagator So(p) = [IIi

—mo] . Thesecontributions are implied in the gap equation, Fig. 3, via theself-consistency condition.

where the combination of vertex matrices I' must corre-spond to the correct quantum numbers of the respectivepolarization state Ph. ysically this corresponds to dress-ing the quark line with pseudoscalar and scalar mesonicstates.

In order to identify the corresponding terms of the scat-tering diagrams, one again writes the gap equation interms of the bare propagator So, as shown in Fig. 4. Byapplying Eq. (18) and again resumming the diagrams,one identi6es the scattering diagrams contributing to thisorder, some of which are displayed in Fig. 5. In additionto the RPA series of lowest order, one obtains furtherclasses of diagrams such as vertex corrections and boxdiagrams. Additional polarization diagrams can simplybe read Rom this series. Since parity is conserved onlybetween the outer vertices, additional parity mixing canoccur in the diagrams of higher order.

What is required now in an exact calculation is a fullyself-consistent solution to Eq. (2) for S and Eq. (20) forZNQQ together with an exact summation of the associ-ated polarization modes, as extracted from Fig. 5. Thislooks forbidding, and we found no way of easily sum-

FIG. S. Examples of scattering diagrams which describemeson exchange in O([1/N, ] ). They are obtained by a func-tional derivative of the corresponding self-energy diagramsextracted from Fig. 4 with respect to the current quark prop-agator.

ming up all the contributions to the polarization modes.Therefore, we choose the alternative path of formulatinga perturbation theory to calculate corrections over andabove the self-consistent Hartree approximation; see e.g. ,Sec. 7 of Ref. [10], and also [4], which take a similar ap-proach for the m on physical grounds. Doing this, one canreplace the full Green function S occurring in Eq. (20)by the mean field S of Eq. (3), with m = mH. Then thefirst integral in Eq. (20) can simply be identified as mH,while the higher-order corrections occurring in Eq. (20)are treated in the pole approximation using the physicalvr and 0 meson masses. This amounts to calculating thehigher-order corrections to the quark self-energy with-out describing the meson sector explicitly on the quarklevel. It corresponds to the physical picture of dressingthe quark line with a meson cloud that is created by thevr or o.. This can also be viewed as a calculation of thescreening (antiscreening) of the bare four-point interac-tion by the 7r(o). This concept is discussed at the endof this paper in more detail, when a numerical calcula-tion is performed. We note that once we choose to takesuch a perturbative approach, the consistency check ofdescribing the pion properly as a Goldstone boson can-not be carried out any more. For the approach to besensible, the higher-order contributions must actually besmall in order to be tractable on a perturbative basis.A posteriori, we will show that this is indeed well justi-6ed.

This perturbative approach is shown graphically inI"ig. 6. The contributions of next-to-leading order termsin 1/N, as shown in the lower part of Fig. 3, can besummed up and are expressed in the pole approxima-tion, i.e., 2iG/[1 —2GIIM] = —i(gM~~)2/(k2 —m2M) withthe mesons M = vr and 0.

The resulting quark self-energy reads

ZNi, o ——mH + K (JI, mH) + Z (p, mH), (21)

where mH is the solution of the self-consistent gap Eq.(5) on the Hartree level. It enters into the expression forthe mesonic self-energy contributions, E and Z, viaS(k). For example,

49 El'I'PCTIVE 1/N, EXPANSION IN THE NAMBU-JONA-. . . 3287

and

1 . d k p. k 1Z. v(p') = ——3ig'p2 ~qq (2~)4 (p I,)2

(26)

rg

+ IS W + W

p —k

and correspondingly for the contribution of the 0 meson.Both integrals in (25) and (26) may be expressed moresimply on introducing the Feynrnan form of Eq. (13).Explicitly, one has

2= 3 2Z s(p)= — g m16'

dx2 2

—ln 1+—2

FIG. 6. Perturbation theory to evaluate the quark propa-gator in next order of 1/N, . Above: Self-consistently in lowestorder; middle: Perturbative corrections to the quark propa-gator from the efFective coupling to m and cr mesons; below:Notation for the momenta.

d4k .Z~(p) = i ipsvig~qqiS(k)ipsvig~qq2x 4

while

2= 3 2Z-,v(p ) = —16,g.„x dzz

2 2—ln 1+

(27)

(2S)

X(p —k)2 —m' ' (22)

Written in full, this becomes

d4k 1 ~ —P+ mlZ (p) = —3ig2

(2x) (p —k) —m~

k —m

(23)

with E'2 = p2x2 —(p2 —m2)z+m2(1 —z). Correspondinglyfor the 0-meson mass, we obtain similar relations, withZ = sZ (E -+ E,mrs m —m~).

The dispersion relation of the quarks, po(p), and itseffective mass are determined &om the pole of the quarkpropagator. For this, we write

~Nr, (p) = P —ZNto(p)

where the four-dimensional integral requires a regula-tor. The expression for Z can be obtained by replacingZ = sZ (E ~ E,mH -+ —mH). The factor of 3 dif-ference between the two expressions is due to the isospinmultiplicity of the pion, while the sign change in mH isdue to parity. We assume also that g~qq ——g~qq.

ZM can in general be decomposed into two terms,

in the form

~NL'(p) = ~(p') 0 —Z(p') j

with

Z( 2) ZH+Z, (p')+Z, (p')1 —Z,v(p') —Z,v(p')

(29)

(30)

Z-(p) = Z .(p')+HZ. , (p') (24)

containing two scalar functions Z s(p2) and Z v(p2).We may determine these functions via the inverse rela-tions

and a renormalization constant Z(p2) = ] —Z v(p2)—Z,v(p ). To determine the pole, we write the dispersionrelation (p —Z)(p+ Z) = p2 —Z2 = 0 as

p' = po —V' = Z'(p') .

and

to be

Z s(p)= —tr Z (p)

Z-, v(p') = 4, t"WZ-(p)j

1 1

(2n-) (p-k) -m k -m

Here, Z(p2) is the effective mass. This implicit equationfor the function po(p) is directly solved by po —pconst. On physical grounds, this is to be expected sincewe are considering the self-energy of a &ee, i.e., only self-interacting, quark in vacuum.

For the quantitative evaluation of these next-to-leadingorder corrections we use the parameter set A = 1.103GeV, G = 2.98 GeV, and mo ——9 MeV. For these pa-rameter values, the value of the quark condensate densityis {@@) —(261 MeV)s, and the pion mass and decayconstant are m = 137 MeV and f = 92.0 MeV. Theyalso determine the values for the mesonic sector to be m

3288 E. QUACK AND S. P. KLEVANSKY

1.0 l I I I

(

I I I I

t

1 I I I

(

I

Hartree

0.8 —— — H + 7I

0.6

CO

04

0

0.00

I I I

0.2

p„-(c,ev')0.4 0.6

FIG. 7. Quark dispersion relation in the effective mesonmodel. Shown is po(p ) which includes the mesonic correc-tions. The 7r and the cr contribution are seen to compensateeach other in part.

= 461 MeV and g zq——g ~z

——3.69 [2].The result of the numerical solution of Eq. (31) is

shown in Fig. 7 in the form of the dispersion relationpo2(p2). To lowest order, the gap equation on the Hartreelevel yields the quark mass mH ——220 MeV. This is theinput to the mesonic corrections. As can be deduced fromFig. 7, the contributions of the m and u corrections tothe self-energy are opposite in sign, and therefore canceleach other in part. The contribution &om the vr mesonacts to increase the dynamically generated quark mass,

while the cr meson contributes to decrease this quantity.The fact that the two contributions to the quark massare opposite in sign is brought about by the fact thatthe ~ and 0 mesons enter with opposite parities into thegap equation. Further, it is seen that the effect due tothe x meson is dominating, which is a consequence ofits higher isospin degeneracy. One may note too thatan effective Hartree calculation with increased couplingstrength would produce the same effect. Alternatively,regarding the dynamically generated quark mass as a fun-damental quantity that is "fixed," would require that adecreased coupling strength be used to maintain its value.We therefore attribute the term "screening" to the 7t me-son and conversely "antiscreening" to the ~. The neteffect necessary to maintain a fixed quark mass thus re-quires a lowered or screened interaction strength. In ourcalculation, in which we retain our original parameterset, and calculate the resulting quark mass including thehigher-order corrections, we find mNj. o ——256 MeV. Thiscorresponds to a correction of 16% as compared to theleading order.

Since the contribution of the next-order corrections tothe quark self-energy is only of O(10%%up), the perturba-tive approach used is justified. One sees that the meanGeld calculations represent the dominant part of the con-tributions to the calculated quantities. Futher, the I/N,analysis justifies the inclusion of screening corrections viaselective infinite partial summations, rather than an ap-proach in which higher-order terms in the interactionsstrength are included.

It remains to perform a fully self-consistent expansionfor the next-to-leading corrections, as well as to inves-

tigate this for the SU(3) version of the NJL model aswell.

It is a pleasure to thank J. Hiifner and R. H. I em-mer for interesting and enlightening discussions. Thiswork was supported in part by the DFG, Project No.Hu 233/4-2.

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