Explicit chiral symmetry breaking in the Nambu-Jona-Lasinio model

N~lear Physics AS47 ( i 992 ) 612--632 No, nh~Hol|,and

NUCLEAR PHYSICS A

Explicit chiral symmetry breaking in the Nambu-Jona-Lasinio model

C. Schiiren '~, E. Ruiz Arriola 'l'b and K. Goeke ~ ~ laslitut liir Theorelische Physik Ii, Ruhr-Uni~ersitiit Bochum, D.4630 Bochum, Germany

~' Departamento de Fisica Moderna, Universidad de Granada, E-18071 Granada, Spain

Received 28 November 1991 {Revised 7 April 1992)

Abstract: We consider a chirally symmetric bosonization of the SU(2) Nambu-Jona-Lasinio model within the Pauli-Villars regularization scheme. Special attention is paid to the way in which chiral symmetry is broken explicitly. The parameters of the model are fixed in the light of chirai perturbation theory by performing a covariant derivative expansion in the presence of external fields. As a by-product we obtain the corresponding low-energy parameters and pion radii as well as some threshold parameters for pion-pion scattering. The nucleon is obtained in terms of the solitonic solutions of the action in the sector with baryon number equal to one. It is found that for a constituent quark mass M --- 350 MeV most of the calculated vacuum and pion properties agree reasonably well with the experimental ones and coincide with the region where localized so|irons with the right size exist. For this value, however, the scalar and vector pion radii turn out to be very small. A unique determination of the sigma term is proposed, obtaining a value of ~r{0) = 41.3 MeV. The scalar nucleon form factor is evaluated in the Breit frame. The extrapolation to the Cheng-Dashen point leads to tr(2m 2) - tr(0) = 7.4 MeV.

1. Introduction

In the past years the old Nambu and Jona-Lasinio (NJL) model ~) has received increasing attention as an effective chiral model for low- and intermediate-energy hadronic physics :). Although schematic in nature, it has proved to be a useful framework to study the consequences of spontaneous chiral symmetry breaking on a semi-quantitative level. As a chiral model, it is the simplest one which can be written in terms of quark degrees of freedom only and, as pointed out by various authors, this model seems to be related to QCD in some short-wavelength approximation 3-7). In the vacuum and meson sectors this model has been studied in great detail both in pure fermionic language 8-L~) and by means of a partial bosonization procedure ~4-,7). Furthermore, its bosonized version suits very nicely in the soliton approach to baryon structure pioneered by Skyrme ~s) and followed by a number of ~!uthors ~9). In contrast to other soliton models, the NJL model does not make a priori any assumption about the validity of the valence-quark picture, i.e. whether

0375-9474/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

C. Schiiren et al. / Chiral symmetry breaking 613

the baryon charge is carried by the valence quarks or by the mesons. The existence of localized solitonic solutions without relying on the heat kernel or gradient expansion has been suggested 2o-2,) and demonstrated by various authors for Baryon number one 22-24) and higher 25). All calculations done so far indicate that a valence picture of deeply bound quarks emerges as far as various baryonic properties are concerned 26-29). This is usually achieved for constituent quark masses of M--- 350-400 MeV. As with any effective model, NJL has its specific drawbacks, namely the absence of renormalizability and confinement. The former makes use of a UV cut-off mandatory which cannot be removed by renormalization. This cut-off (---1 GeV) is usually interpreted as the typical scale for low-energy phenomena. In general one knows that the vacuum sector depends strongly 3o) dnd the soliton sector weakly 3'-32) on the particular regularization scheme employed. The lack of confinement is not expected to be relevant provided the valence quarks are deeply bound.

Many of the calculations done in this model, at least in the soliton sector 22-32), suffer from a few drawbacks which are mainly related to the explicit breaking of chirai symmetry. First, the bosonization used in those works is not chirally invariant, i.e. the functional integral is multiplied I~y a field-indeper~dent factor that breaks chiral symmetry explicitly. Thus the bosonizing fields do not have simple chiral transformation properties. In the second place, the pion weak-decay constant is computed in the soft-pion limit but fixed to the experimental value of 93 MeV. This is not fully consistent. As it has been shown in the framework of chiral perturbation theory (ChPT), the extrapolation to the soft-pion limit leads to a value of 88 MeV. Finally, the sigma term has been incorrectly evaluated thereby leading to results which may differ by a factor of two.

The purpose of the present work is to reconsider the SU(2) NJL model at the one-quark-loop level and to treat in detail the problems mentioned above. First, we consider a chiral symmetric bosonization of the model. Secondly, we perform a covariant gradient expansion up to fourth order in the presence of external fields and obtain an effective lagrangian which resembles the tree-level lagrangian of ChPT for mesons. From there we evaluate some pionic properties. After that, we consider the solitonic sector and propose a correct way of extracting the sigma term. Finally, the scalar form factor of the nucleon is computed in the Breit frame. It is clear that we do not expect dramatic changes with respect to previous works except perhaps in those observables related to the explicit breaking of chiral symmetry.

Chiral perturbation theory (ChPT)33) imposes an important constraint on the structure of chirai models which try to mimic QCD at low energies. This is equivalent to an effective theory containing only pseudoscalars in the presence of external fields, expanded in a power series of the external momentum p2 of the pseudoscalar and of the current quark masses Mo. In the lowest non-trivial order O(p2 Mo) the corresponding lagrangian is characterized by two quantities: the pion mass m~ and the pion decay constant f,~. The next to leading order O(p 4, p2Mo, M2o) includes in

614 c Schiiren er al. / Chiral symmetry breaking

the SU{2) case seven dimensionless parameters which can be related to experiment and have been analyzed in detail and determined by Gasser and Leutwyler ~3) by looking at different processes involving pions, photons and ieptons. A direct calculation of these coefficients from QCD itself goes beyond the present computational capability, although there have been many attempts to do so within certain approxi- mations or in specific models 34-a3).

It is therefore interesting to study systematically the corresponding low-energy parameters at the one-quark-loop level in comparison with the results of ChPT, and therefore to decide the validity of NJL at least up to fourth order in the chiral expansion. In addition, one would like also to check whether the solitonic properties are compatible with the obtained low-energy parameters, i.e. whether one can or cannot reproduce beth for the same values of the constituent quark mass.

In order to evaluate the low-energy parameters within the NJL model one ought to be careful with the regularization scheme used. One needs a scheme which preserves gauge invariance and at the same time reproduces the quark condensate and the current quark mass. To this purpose we will suggest the Pauli-Viilars regularization*~) which fulfills both requirements and even has simple analytical properties.

The paper is organized as follows. In sect. 2 we quickly review the model together with the chiraUy symmetric bosonization and the regularization method employed. In sect. 3 we concentrate on the vacuum sector indicating how to extract various properties from the corresponding effective potential. In sect. 4 we consider a covariant gradient expansion of the regularized quark determinant in the presence of external fields and compare it with the outcome of ChPT. This allows us to fix the parameters of the model. In addition, we compute some pionic properties such as scalar and vector radii and thresho!d parameters for ¢rrr scattering. Sect. 5 is devoted to the soliton sector and compute among other observables the sigma term in a unique way. |n sect. 6 the scalar form factor of the nucleon and in particular the extrapolation to the Cheng-Dashen point are computed. In sect. 7 we present our numerical results and in sect. 8 we come to the conclusions.

2. Bosonization and regularization

The SU(2)R®SU(2)L®U(|)v invariant NJL generating functional in euclidean space and in the presence of external sources reads*

Z[ S, P, V,A ]= f D(t Dq

× exp ( - ( SNJL[ #, q ] "~- Tr[ #Sq d_ #i,~5 pq --I-- # V~t y~t q .~_ ~A~, y~" ysq ]) },

* We use the same conventions used in refs. 16.t7).

(l)

C. Schiiren et al. / Chiral symmetry breaking

where the NJL action is given by

SNj~ = f d4x{~(iy- 0 - Mo)q + [G[(qq)2 + (~iys,q)2]} J

615

(2)

and Tr= IV, ~ d4x try tr,. The field q(x) represents a Dirac spinor with IV,. colors and two (u and d) flavors. Mo is the average up and down quark mass. In what follows explicit isospin breaking will be neglected, i.e. we set mu = md.

Using well-known techniques 46) this generating functional can be bosonized by multiplying it with a chirally invariant factor. Integrating out the quarks we get the following generating functional,

z[s, P, v, A] = I _ _ ~ e f f l" 44 Dd/exp ( oNj , t~ , S, P, V, A]), (3)

where the corresponding effective action is given by

l S~,L[J~, & P, V , A ! = - T r l o g ( i D + M,,)+~-~Tr (J~+~) (4)

and the Dirac operator defined as

iD = iy. [0- i( V + ysA)] + S + iysP + ~ (5)

has been introduced. Here ~ff = t r+ iys r . 7r. Notice that contrary to previous works 2>3_,) we consider the explicit chiral symmetry breaking included in the fermion determinant rather than in the bosonizing term. In practice one can go from one prescription to another by shifting the path integration variable d/l-, .14- Mo. However, this shift does not preserve chiral invariance and therefore ~ does not transform well chirally. Moreover, this would give a wrong result for some of the low-energy parameters. In what follows we leave out the path integration over boson fields and consider only classical configurations, therefore neglecting any effect coming from bosonic fluctuations. For brevity we will include the current quark mass in the external scalar field. We will also consider the real part in euclidean space of the effective action only'. This action is divergent and has to be regularized. In the Pauli-Villars 44) regularization scheme (PV) we have [see also ref. 4s)]

, + l_~_Tr(~+~)" R e S = - ~ N " Y ~ c ' T r l ° g ( D ÷ D + A ; ) 4G i

(6)

Here c~ (Co = 1) are coefficients to be chosen so as to make the action finite and A, (Ao = O) stand for the PV regulators. It is interesting to note that all our calculations will be always done with this action.

* We do not consider here the imaginary part of the action which is responsible for anomalous

processes.

616 C Schiiren et aL / Chiml symmetry breaking

3. Effective potential and vacuum sector

It is now straightforward to compute from eq. (6) the corresponding effective potential by considering translationally invariant configurations. (In the vacuum is constant.) That leads to

V = _4Nc I d4k 1 (2~)~ E c, log [k 2 + (~+ MoY +,,~ + A ~] + T6 (,~ +,,~). (7)

Working out the integral it turns out that both logarithmic and quadratic divergencies can be rendered finite with just two conditions:

X e~=0, E c,a~=0. (8) i i

Hence the minimal number of PV regulators fulfilling these conditions is two, say A~ and An. For simplicity we consider the limiting case A~ = A2 = A. Then the coefficients c~ can be eliminated by making use of the convenient identity (see appendix A)

2 lim E cJ(A ~) =f(0) - f ( A 2) + a 2f,(A 2)

l t " 1 2 - ~ A i=0 (9)

with A the Pauli-Villars cut-off. The effective potential as a function of or has two minima at non-vanishing values of the or-field, which are tilted by the inclusion of a finite Mo. |~he minimization of the effective potential leads in the chiral limit to or, = M and ,rv= 0. One can then obtain the value of the quark condensate and vacuum energy density,

(au + dd)o = ( ,q)o = - S M 3 L 2 ,

- e = 2 M 4 I o - 8 - ~ 2 ~ c,A~41og A2+M2], (10)

with the integrals I, defined below. To go away from the chiral limit we take the expectation value of the scalar field in the vacuum and the Pauli-Viilars regulators to be zeroth order in Mo. This can be achieved by making 1/G depending on Mo to all necessary orders as follows from the minimization condition

OV I I d4k M+Mo 1 =-8N,. (2rr)4 ~ Ci k 2 + ( M r -F- M o ) 2 + A 2 (11)

which suggests for 1/G the following expansion,

+ o + . . . 0 2 M 4

(12)


with

Go I =8M21_2 ,

G-~ ! = 8M2( I_~, + 21o) ,

G ~ = 8 M Z ( I o + 2 1 2 ) .

The dimensionless integrals

f d4 k lo=Nc (2¢r)"

d4k 1 (27r)4~ ei k2 + A 2 M 2

• i +

N,. /A,' +1 Iog(A i

1 C'(k2 + A~ + M2) 2

NO" c~ log (A, + Me),

(4,tr):

1 2 = 6 N , . M 2 f ~ d4k l

(2~)4 ~ ¢i (k2+ A2+ M2)3 N,. M 2

-" (4w)2 ~ Ci A2_{_ M 2 , i

/4 = 12NcM 4 f d4k 1 Nc M 4

c'tk + + C' tA , + W J

have been introduced.

+ M2),

(14)

4. Low-energy expansion and meson sector

Taking the bosonized effective action (6), one can study the different mesonic correlation functions. It can be proven 47) that a saddle point evaluation of the path integral over the bosonic fields in the presence of external fields leads to a full Bethe-Salpeter treatment very similar to that of refs. ~1-~3). In this sense the bosonization procedure does not represent in our view any particular restriction, at least up to one quark loop. An approximate way of doing so is to perform a gradient expansion of the bosonized NJL action in the presence of external fields up to fourth order. Of course, this expansion breaks down for high momenta and it should be applied for low momenta only. We take the convenient parametrization d / = M(1 + 8d~)(s + iyer- p) with (s, pi) a unitary four-vector s z +p2 = 1 and 34, the classical fluctuation in the scalar field. Following ref. 33) the derivative 0~ and the external vector and axial fields are counted as first order and the current quark mass and the external scalar and pseudoscalar fields as second order. We also take the fluctuation in the scalar field to be second order. The method we use is based upon the average procedure of ref. 48) as expressed by the equation

, c, T r l o g ( D + D + A ~ ) - 8 4 ( O ) d4k

(2¢r)4 ~ . c, Trlog ( D + [ k ] D [ k ] + 32) , (15)

where D[k] = D + iy. k. A straightforward though tedious calculation leads to the

618 C, Schiiren et aL / Chirai symmetry breaking

etfective lagrangian

" V M-" 5 ~ = 2M ° i , .~, U t W ' U + 8 l..~ 2M -~ ~ Y XT U

_--_/ M'\ " T ,' ) 2__ ~__~)-(xT - I ]4 (v~uTv~u)2 ' 4 +g/4(V~U ~' U 81-, _ U) 2

+ 41,, M__~'- (V, xTV . U)-~I,_( U rFu''F.,,U) - 312(V~, U TF~'' 'V,,U) A-"

M-" , , t , x t UtV~

+ 6 6 4 M - " (! , , - i ._)V~UtV~U+Io 1 _ x T u -(~$#b)28M41o, (16)

where A" = 21 2/loM'-. For an easier compar ison we have in t roduced the no ta t ion of re/'. 33) with U a = ( s , p , ) a uni tary four-vector u T u = UAU a= 1 and also X =

2B,(S, P,). The covariant derivative and the field strength tensor are defined

V~ U a = ~ ,. U a + A A n U l~, (V ~, V ,. - V ,.V ~, ) U a = _~FAnun, (17)

respectively. At the mean field level, the scalar f luctuation can be integrated out by

means of the classical equat ion of motion

4M26d> = ( l - 1 2 / I , , ) v ~ u t v ~ u + ( I - - 4 ~ ' 2 ) xTu , : (18)

which indicates that 8& is indeed of second order. Thus, fur ther terms in ~$d> will

not contr ibute at the mean field level up to fourth order. Finally, this lagrangian

can be brought to the Gasser and Leutwyler form by making use of the classical

equat ions of mot ion for the pion field. Up to fourth order they read

V 2 U g - ( u T v 2 U) U A = X A - ( u T x ) U A. (19)

The classical lagrangian is given then by

1 ~ = t , f 2 v u u T v u u + f 2 x T U + , [6,/~,xTx + ~$,/~, tr F~,,.F ~'"

-" 32rr- - "

+ v, f, (v,, u T v " u )-" + y: v,, u v"u)2 + v., ]3ix u)2

+ y 4 f 4 ( V , . x T v " U ) + y_~( U r F ' " F , . , . U ) + y . I ' ~ , ( V , . , U t F " " V . U ) ] , (20)

where the numerical factors

2 8, = 2 , 82 - " / , ~/! = ~, "}/2 = 3 ,

! ! 3'4 = 2, Y5 = - ~, Y6 = -3

I I 3

(21)


have been introduced. At this level of approximation the constants in the lagrangian are given by

f2 = 4M21o,

2 B o M = A 2 = m 2 M M 2 Mo 2 2 G o f

21 -2 / Io ,

m

Ii

12 = N,.J4, ~, = - 8 -~- J2 + ~ ,

-

4 " 15=2N"J2" 1 6 : 4 N , . J 2 ,

h2=8 - 3 N,.J2 ,

Mr2- g2,, = Yi ci ( A 2 + M2), , (22)

i

where the lowest-order pion mass m has been introduced. Combining this with the expression for the chiral condensate (10) we get the current algebra result (~lq)Mo =

f 2 m 2 as it should be. Notice that i7=0 as a consequence of explicit isospin w

conservation mu = md. Notice also that in contrast to ChPT the coefficients h,, h 2

are finite and calculable. The iagrangian above resembles only the tree-level lagrangian of ChPT, and hence does not include neither the tadpole nor the unitarity correction. Therefore there is an additive ambiguity in the identification of the low-energy coefficients. In ref. 43) it has been proposed that this problem may be overcome by computing pion loops within the model itself. There it was found that the correction induced by them is about 4. We win not consider pion loops here and prefer to compare to experiment directly. In a sense this is equivalent to refit the low-energy parameters at the tree level. It should be clear that in doing so we cannot perform better than ChPT itself, but rather provide an understanding of how the low-energy coefficients arise in a model with dynamical quarks. In the present work we evaluate some pion properties as well as threshold parameters for # o n - # o n scattering. Their explicit expressions are given in appendix B.

The parameters f and m have been estimated in ref. 33) to be

f - 8 8 MeV, m = 140.1 MeV. (23)

For definiteness we will assume these values throughout this paper. We will see that variations in f can be effectively taken into account by changing the constituent

quark mass M.

620 C Schiiren el al. / Chiral symmetry breaking

& SoHton so|uriah and baryon sector

Following refs.-'"°~) the nucleon appears naturally in our approach as a bound state of three valence quarks in a self-consistent solitonic background. We consider the classical static solutions of the action:

SN, L[ U, S, P, V, A] = - T r log (iD+ M o ) + - ~ Tr [J/f+J/f]. (24)

Unfortunately, the low-energy approximation (16) of this action does not support solitonic solutions. In fact, one has to consider the total action in its full non-locality. Setting the external sources to zero, considering time-independent hedgehog fields on the chirai circle [U = M exp (iT5¢" £0(r)] and subtracting the vacuum energy we get the following expression for the soliton energy with baryon number equal to one in the PV regulafization scheme:

°IR(e°, A)], (25) E = N : l , . ~ , , e , , a , - } N , . ~. A ) - l e , .

where the reguladzation function R(e, A ) = ~ i ci(1 + A~/e2) I/2 and the eigenvalues of the Dirac hamiltonian satisfying the equation

[ t~ " V + Mfl[c°s O( r) + iysr " £ sin O( r)] + M°fl ] q~(x) = e~q~ ( (26)

have been used. The sums are performed over the whole Dirac spectrum. In practice, this sum is restricted by the regularization function (see fig. 1) since it cuts the higher part of the spectrum. Notice that in contrast to previous works 22-32) our non-linear constraint is chirally symmetric. In order to guarantee B = 1 we define r/va~ = 1 if e,~> 0 and ri,,..,~ = 0 if ev,,~ < 0. Since our bosonization is chirally symmetric, the

-o 1.00

u ~n 0 .80

0 .60 <

t~ 0 . 4 0

i f 0.20 @

0.00 I 0

M = 3 9 6 Me V

' ' ' ' I I ' I ' I ' L I I I I I , , I . . L

1 2 3 4 5

e n e r g y e ( s c a l e d )

Fig. 1. The Pauli-Vil lars regularized contr ibution to the sum of eq. (25) as a funct ion of the energy eigenvalues for a constituent quark mass M = 396 MeV. The scaling is done with respect to M.


chiral circle constraint is chirally invariant. Varying with respect to the chiral angle 0(r) we obtain the following equation of motion:

O(r) =. [ d~ [E~ (l~,,IR(e,~))'el~(x)iys~'. ~q~(x)] + ~valqval(X)i~5 T" Xqval(X)] tan d:~ [Eo~ (lealR(ea))'qa(x)qa(x) 4" "qvaIFlval(X)qval(X)]

(27)

where ~ d~ stands for angular integration. The set of self-consistent equations (26) and (27) are solved numerically by the iteration method described in detail in refs. 22-23) by putting the system in a spherical box. After 15-20 iterations convergence is achieved. Once a solution is obtained the corresponding energy can be computed by using eq. (25). In addition one can evaluate other observables. In the present work we consider only a few, such as the isoscalar-electric mean square radius (r2) !=°, the axial coupling constant gA and the sigma commutator 2,~N [see e.g. ref. 27) for more details].

6. Sigma term and scalar form factor of the nucleon

For on-shell nucleons the scalar form factor is given by the matrix element

or(q2)figp,, s)ugp, s)= Mo(N<p', s)lO(O)q(O)lN(p, s)), <28)

where u(p, s) and u(p', s) are free nucleon Dirac spinors normalized to one and q = p ' - p represents the four-momentum transfer. We choose the Breit frame to make our calculations,

p=(E,½q), p '=(E,-½q), E=x/MN+~q 2 , q2=_q2, t29)

i.e. we have no energy transfer and the momentum transfer becomes space-like. In this frame we have

= Mo<N(p, s)lO(x)q(x)lNgp, s)>e , <30)

where we have made use of translation invariance. The nucleon state with definite momentum can be obtained by making a boost from the system at rest. Making the static approximation and neglecting recoil corrections we get the following final

expression,

d3xjo( qr)( N[(t(x)q(x)iN) . (31)

The value of the scalar form factor at zero momentum transfer is

or(0)= f dax Mo(N[Ft(x)q(x)[N). (32) 3

622 C Schiiren el al. / Chiral symmetry breaking

The scalar form factor cannot be evaluated at the Cheng-Dashen point q2= 2m 2 directly since we are in the Breit frame. However, one can prolongate analytically (q-~ iq) the formula (3 | ) to the time-like region,

o'(q") = ~ d3x ho(qrj(Nl~l(x)q(xJlN) , (33)

up to the next singularity. Here we have used the fact that jo(iqr)= ho(qr). Using

this we can evaluate the quantity

a,, = o-(2m~)- o'(0). (34)

The numerical value has been estimated by means of ChPT 5o.5~) and more recently by dispersion relations ~-') giving 4.7 and 15.2 MeV, respectively.

[n the present model after bosonization and integration of the quarks we have the following formula for the scalar density,

0 ( Nl~l( x )q( x )l N) = ~,,,,~l,,,,( x )q,~,,( x ) +.E ~ [[e,,IR(e., a ) ](t. ( x )q,, ( x )

0 0 - 0 0 - ) ] q . ( ). x)q,,(x • cge,,

At zero momentum transfer we have

(35)

OE o-(0) = Mo OMo" (36)

which is nothing but the Feynman-Helimann theorem. This relation has been used a few times already in this context ~n,33) and will prove to be a useful check of our numerical calculation.

7. Numerical results and discussion

7.1. FIXING OF THE PARAMETERS

As already mentioned the minimal number of Pauli-Villars regulators fulfilling the conditions (6) is two, say A~ and A2. For simplicity we consider the limiting case A~= A2 = A. We fix the parameters in the following way: First we fix the pion weak-decay constant and the pion mass in the soft-pion limit to the values f = 88 MeV and m = 140.1 MeV. For a given constituent quark mass M we adjust the cut-off A in order to obtain a properly normalized pion kinetic energy [first two terms of eq. (20)). The rest of the parameters listed in eq. (22) and the solitonic solutions as well as their propelties can then be evaluated unambiguously. The resulting cut-off together with the integrals J_,,-/4 can be seen in table 1. In the range of masses considered the value of the cut-off is A---0.8 GeV, a quite reasonable value for a low-energy l 'adronic scale. The values of J2 and J4 give an idea of the finite cut-off


TABLt- I

For different values of the constituent quark mass M different parameters are presented. The cut-off A is always adjusted to reproduce the pion decay constant in the soft-pion limit f = 88 MeV

M(MeV) 220 264 308 352 396 440

A (MeV) 1015 862 808 793 800 816 A (MeV) 890 834 835 860 897 941

-/2 0.91 0.82 0.73 0.65 0.57 0.50 • /4 0.99 0.98 0.94 0.90 0.85 0.80 /| -0.60 -2.16 -3.23 -3.85 -4.11 -4.13 ]'2 2.98 2.93 2.83 2.70 2.57 2.39 T 3 5.37 2.20 0.60 -0.23 -0.66 -0.87 /4 3.59 1.92 1.03 0.53 0.24 0.07 ]'S 5.45 4.93 4.38 3.87 3.42 3.02 /6 10.9 9.86 8.77 7.74 6.83 6.04 /~| 3.7 1.86 0.88 0.35 0.05 -0. I I /~2 42.3 27.70 19.21 i 3.93 10.47 8. ! 0

effects since in the infinite cut-off limit A >> M they should be identically equal to one. We see that finite cut-off effects become more important for increasing M.

7.2. VACUUM SECTOR

Our results for the vacuum observables are shown in table 2. For the vacuum parameters, i.e. the quark condensate (tY/q), the current quark mass Mo and the vacuum energy density e, we observe a smooth dependence on the constituent quark mass around 300 MeV and a rather good agreement with the generally accepted values 33.53.54). As we see the relative deviation of the quark condensate from the

chiral limit becomes less important as M is increased.

7.3. MESON SECTOR

The low-energy parameters r~ , . . . , 16,/~1, h-, show a strong dependence on the constituent quark mass. Besides the last two coefficients, which cannot be related directly to experiment, they have been estimated in ref. 33). One should, however, not compare them directly to those analyzed by Gasser and Leutwyler since their analysis includes the effect of pion loops. We have calculated the tree-level contribution of various pion properties such as the pion mass and decay constant, scalar and isoscalar radii and some threshold parameters for pion-pion scattering. Their numerical values are presented in table 2. As it can be seen, soft-pion corrections to the pion mass and pion decay constant become smaller for increasing M. Actually, our values are in agreement with ChPT for M---250 MeV.

624 C Schiiren eta,. / Chiml symmetry breaking

TABLE 2

For different values of the constituent quark mass M different vacuum, pion and nucleon properties are presented. The cut-off A is always adjusted to reproduce the pion decay constant in the soft-pion limit f = 88 MeV. The experimental values for the vacuum parameters are taken from refs. 33.s3.54). The experimental values for u-~r threshold parameters are from ref. 5s) and are given in appropriate units of

the pion mass. The scalar radius is taken from the dispersive analysis of ref. 60). - - - i ill i w i

M ~MeV) 220 264 308 352 396 440

(_<~))~I~ (MeV) 303 273 259 253 250 249 283 ± 31 Mo (MeV) 7.0 8.1 8.8 9.1 9.2 9.1 7 + 2 (_~)9/4 (MeV) 123 136 150 162 174 186 150-240 <(Iq)/((lq)o 1.08 1.04 1.02 1.01 1.01 1.00 - -

a~ 0.178 0.181 0.182 0.181 0.181 0.180 0.26±0.05 b~ 0.24 0.22 0.21 0.20 0.20 0.20 0.25 ± 0.03 ~) -0.045 -0.046 -0.048 -0.049 -0.049 -0.050 -0.028 + 0.012 b~ -0.067 -0.091 -0.104 -0.111 -0.114 -0.115 -0.082 +0.008 a ~ 0.036 0.037 0.038 0.038 0.038 0.038 0.038 ± 0.002 a°( 10 -~) 16.3 13.7 ! ! .6 10.0 8.8 7.8 17.3 ± 3 a~.{ |0 ) 3.4 I. i - 0 . 6 - i . 7 - 2 . 2 -2 .5 1.3±3

m~ (MeV) 137.1 138.9 139.8 140.2 140.5 140.5 139.6 J~. { MeV) 93 91 90 , 89 88 88 93 (r')~" (|'m ~) 0.72 0.38 0.20 0.10 0.05 0.01 0.60 + 0.03

" ~ f f

( r ' )v (fro 2) 0.35 0.31 0.28 0.25 0.22 0.19 0.44±0.03 y 0.91 0.82 0.73 0.65 0.57 0.50 0.44+0.12

EN (MeV) ~ ~ ~ 1223 1214 i 198 938 e~,~l (MeV) ~ ~ ~ 677 587 506 (r2)i =o (fro") ~ ~ - - 0.62 0.50 0.44 0.62 o'{0) (MeV) ~ ~ ~ 41.3 40.7 39.2 "-45 g~ ~ - - - - 0.75 0.72 0.71 1.23

TABLE 3

Results for 7r:r threshold parameters in appropriate units of the pion mass. NJL SU(2) represents this work. NJL SU(3) refers to ref. s6) and includes vector and axial couplings as well as flavour mixing.

ChPT are the results of ref. 33). The experimental values are from ref. 55).

a ° b ° a2o bo 2 a[ a ° a 2 . . . . . . . . . .

NJL SU(2) 0.18 0.22 -0.046 -0.091 0.037 13.7 1.1 NJL SU(3) 0.26 0.29 -0.062 -0.12 0.047 13.0 -0 .8 ChPT 0.20 0.24 -0.042 -0.075 0.037 input input Exp. 0.26 0.25 -0.028 -0.082 0.038 17.3 1.3

+0.015 +0.03 ±0.012 +0.008 ±0.002 +3 +-3


We see that the experimental threshold properties 55) are best reproduced for M = 260 MeV. Higher masses do not give, however, a bad result. For comparison, we present the results of a SU(3) calculation with vector and axial couplings in table 3 with our result for M = 264 MeV. As we see the generalization of the mL:del to include vector and axial couplings leads to a substantial improvement in a°o.

The situation with the radii is drastically different (see table 2). They exhibit a strong dependence on the constituent quark mass and become very small at masses higher than 300 MeV. One might conclude from here that lower masses than usual ones are preferred. In fact, after completion of this work we became aware of ref. 57). There the scalar form factor of the pion was analyzed and 13,/'4 extracted in the Hartree-Fock approximation. These authors do not apply, however, our low-energy expansion and can therefore go to higher momenta. Despite of this we can reproduce, whenever comparable, their results for the same values of parameters.

7.4. BARYON SECTOR

For constituent masses bigger than 320 MeV the interaction gets strong enough allowing for the existence of self-consistent solitonic solutions, with bound valence quarks. Our results are presented in table 2. The calculated solitonic properties show a weak dependence in terms of the constituent quark mass. The soliton mass comes out to be 20% larger than the nucleon mass. As it has been suggested in re£ 58) this problem can be solved by removing zero-point energies. The axial-coupling constant gA extracted from the tail of the pion field turns out to be 30% smaller than its experimental value. This is a common problem in soliton models [also in the Skyrme model ~8.~9)] which has not yet been understood. The corresponding pion-nucleon coupling constant computed through the Goldberger=Treiman relation turns out to be g~NN ~" 10, a somewhat small value. The isoscalar nucleon radius (r e) t=o= (r2)p_ (r2)" exhibits a stronger dependence on the constituent quark mass and its experimental value can be reproduced for M ~ 352. As we can see in table 2 this situation corresponds to a valence picture of deeply bound quarks (Eva I - - 0.6M). This circum- stance both discards a purely bosonic configuration a , d allows to neglect confinement effects. These conclusions are in agreement with the findings of several authors using other regularization schemes 22-32).

On the other hand, it is interesting to note that for M = 352 MeV the pion scalar and vector radii are much too small. Had we tried to reproduce them we would not have found any soliton since the binding force becomes too small. In fact this is a direct consequence of the absence of confinement within the model. This result represents a clear limitation of the NJL model in its present form. However, as it has been shown in ref. 59) the inclusion of a vector-isovector type coupling increases the binding and allows the existence of solitons for lower constituent quark masses. Unfortunately, this calculation does not preserve chiral symmetry. In this sense it

626 C Schiiren et al. / Chiral symmetry breaking

......... ', 2 ~ b

. . . . . . . .

" ~ [ i ' [

f

,%2

Fig. 2. Dependence of the self-consistent energy as a function of the current quark mass Mo for a constituent quark mass M = 352 MeV. The derivative at Me = 9.2 MeV is proportional to the sigma term.

wea|d be very interesting to reanalyze this point in a model which includes vector and axial couplings. Such a calculation is now in progress.

7.5. SIGMA TERM A N D SCALAR FORM FACTOR OF THE N U C L E O N

Using eq. (36) we compute the derivative numerically from the functional dependence of the nucleon mass on Mo. Such dependence is exhibited in fig. 2. We observe a rather smooth dependerice over the regime considered. We would like to stress the main differences with the previous works _,2-32): First, we take f = 88 MeV; second, we perform a chirally symmetric bosonization; third, our calculation of the sigma term is unambiguous. The result for the sigma term computed in this way is given in fig. 3 and compared with the corresponding result evaluated within the

> 50

4 0

30

2 0 , , , , I , , i I , , , , I I I , I I | I I i I

3 5 0 4 0 0 4 5 0 5 0 0 5 5 0 6 0 0

c o n s t i t u e n t quark mass [ M e V ]

Fig. 3. o-(0) as a function of the constituent quark mass M. The solid line represents this work. The dashed line refers to the non-chirai bosonization of ref. 32).

C. Schiiren et aL / Chiral symmetry breaking 627

usual scheme. We see that the difference decreases as the constituent quark mass increases, eventually coinciding for mass higher than 600 MeV. However, in the physically relevant regime the difference is 20%. The value that we obtain lies around 41.3 MeV which is closer to the recent determination of ref. s,). The analysis of these authors indicates, however, that half of the difference between 25 MeV (lowest order in the chiral expansion) and 45 MeV (empirical value extrapolated to zero momentum transfer) stems from the strange quarks. Since ours is a pure SU(2) calculation one might infer that the value 45 MeV is a bit too large. In this sense it would be highly desirable to extend the present calculation to the SU(3) case and hence to obtain an estimate of the strange content of the nucleon.

The result for the ,,;calar form factor is presented in fig. 4 for M = 400 MeV. As we have mentioned ~dready, due to the space-like kinematics of the Breit frame, extrapolation to the t:Lme-like region requires analytical continuation up to the next singularity. To analy;,e this point we show in fig. 5 the scalar density together with the corresponding valence- and sea-quark contributions separated. We see that this function drops off rather quickly in the surface of the soliton. In this region one can apply a derivative expansion giving

9 , -~ e-2mr

Mo(Ni( l (x )q(x) lN) 32,n.2g-Am- r----T-_-i-''', (37)

which is an exponer~tiai tail. Clearly, analytical continuation can be safely applied up to q = 2m. From the calculation we find

A,, = cr(2m2)- or(0) = 7.4 MeV. (38)

•0 I l I i I I i I i I i i i i I i i ' | i i i i " i i "

4O

:Z 3O

20 -

10 - ,1

o , , , , I , , , , I , , , , I . . . . I . . . . I , , , -2 0 2 4 6 8 10

qZ (mZ)

Fig. 4. Scalar form factor of the nucleon as a function of the three-momentum transfer q2 in units of the pion mass. The Cheng-Dashen point corresponds to the value q2=-2m~.


-1

' ' ' 1 . . . . I . . . . I ' ' '

\

_ \ \

S,:alar Density

-a , , , . ! , , , , I , , , , ! . , , , 0 0,5 1 1.5 2

r(fm)

Fig, 5, Sea!at densities as a function of the distance in fm. The dot-dashed line represents the valence- quark contribution. The dotted line is the sea-quark contribution. The solid line is the total scalar density.

For large distances the total density goes to the chiral condensate.

This value lies above the ChPT 50,5~) at the one-loop level and is half the result

obtained recently by means of dispersion relations 52). In fact it is known that the

scalar form factor is dominated by the strong final-state interaction of pions in the

isoscalar s-wave channel. In the NJL model this final-state interaction is described

by means of the scalar field or at least in the mean-field approximation. It would be interesting to see whether a generalized model of the NJL type is able to describe this final-state interaction properly.

8. Conclusions

In this work the effects of explicit chirai symmetry breaking have been analyzed

in a bosonized version of the SU(2) NJL model. This has been done by means of

a chirally symmetric bosonization. We have studied various hadronic properties in

the vacuum, meson and baryon sectors. The main results of our study are:

(i) Vacuum parameters compare well with the usually admitted values.

(ii) Pion scalar and vector radii require unusual low constituent quark masses M --- 250 MeV.

(iii) Pion-pion threshold parameters are reasonably well reproduced within a wide range of constituent quark masses.

(iv) Solitons are found for masses bigger than 320 MeV. The isoscalar nucleon

radius can be obtained for 352 MeV. The axial coupling constant turns out to be too small as it is usual in soliton models.

C. Schiiren et aL / Chiral symmetry breaking 629

(v) The scalar form factor of the nucleon has been evaluated at zero momentum transfer and at the Cheng-Dashen point giving o-(0)--41.3 MeV and o-(2m~)- 48.7 MeV, respectively. These values have to be compared with o-(0)---45 MeV and o-(2m~)--- 60 MeV.

Ours is not intended to be a fundamental calculation since the model has clear limitations: no renormalizability and absence of confinement. In addition we have considered the SU(2) case only and no effect of vector and axial couplings. In this sense, it would be very interesting to study both generalizations of the model. Such generalizations are now underway. On the other hand, such a simple model allows to describe a wide variety of hadronic properties with a unique action containing only one adjustable parameter: the constituent quark mass. In addition, the predic- tions of the model agree with experiment within 30%. In our opinion this makes the NJL model a useful tool to study some low- and intermediate-energy hadronic properties.

One of us (E.R.A.) is very much indebted to G. Ecker (U. Wien) for numerous discussions and a critical reading of a previous version of the manuscript. Thanks are also due to F. Cornet (U. Grmada) for clarifying some points and J. Gasser (U. Bern) for suggesting the calculation of the scalar form factor. We thank L. L. Salcedo (U. Granada) for reading the final version of the manuscript.

This work has been partially supported by the KFA Jiilich (COSY-Project), the Bundesministerium fiir Forschung und Technologie, Bonn (Contract 06-BO-702), the DGICYT under contract PB90-0873 and by the Junta de Andalucia (Spain).

Appendix A

REGULARIZATION SCHEME

Since we have two conditions to satisfy, the minimal number of PV regulators fulfilling those is two. If one solves the system of equations we obtain the following general formula,

A 2 A 2 rtA2 x ~,, c , f(A2)=f(O)+A2_ A2f(A2) /,,'2--- ,, 2a',_/,2 2,. (A.1)

One can also consider for further simpiicity the one cut-off limit (A1 - '> A2 = A ). The f~dowing formula holds:

~, c cf(A~)=f(O)-f(A2)+ A2f'(A 2) (A.2) i

with A the Pauli-Villars cut-off. We obtain the following formulas:

(A2 ) - E c, log (A ,2- + M = log l

i

A 2

A 2 + M 2 '

630 C Schiiren et aL / Chiral symmetry breaking

v c , ( A ; + M : l i o g ( A - i + M ~ ) = A a + ( M 2 - A 2 ) l o g 1+

E c~ M,)2 = 1 - 1 + n (A~+ ( A 2 + M 2 ) , A 2 + 2 ,

( A:( a-',": ~.c~(l+A~/e2)' /2=l - 1+- 7 +2--~e_~ 1+-e-52) • (A.3)

i

One can make contact to the proper-time formalism by rewriting the expression of the regularized action as fellows,

Re S = +~N,. f, orb(r), --r Tr e - * ° ' ° (A.4)

with

d~(r) = E c, e - ~ . (A.5)

In many cases the proper-time function has been taken to a simple step function,

d~(~-) = 0 ( r - ~ 1 ) (A.6)

with Art the proper-time cut-off. In the PV case this function becomes a smoothed step function. |n the general case the following formulas hold,

fq 'L d7 rM-" " --~ d~(r) e = - Y ~ c , ( A ; + M " ) i o g ( A 2 + M 2 ) ,

D T - i

b T

f '- d r r ''-! d~(r) e --'~!" b

= - ~ c, log (A ~ + M 2), i

1 = r ( n ) Ei Ci (A~'+" M 2 ) '' " (A.7)

Appendix B

LOW-ENERGY PION OBSERVABLES

The calculation of the pertinent two, three and four point functions follows closely re|: 33). The soft-pion corrections to the quark condensate, the pion mass and decay constant read

m-" q (qq)=(qq)o 14 321r2f 2 (4/~,- I-,)j, (B.1)

( m - ~ = m 2 \1 32-~-T2f2 , (B.2)

f,.~ = f ( l 16-~2f2 . (B.2)

C. Schiiren et at / Chiral symmetry breaking 631

With the lagrangian of eq. (20) one obtains the following results for pion-pion scattering lengths.

S-wave: 7m2[ a ° - 3~-~--f2 1-t

P-wave:

5 m2 ] 84~2 f2 ( ]'! + 2]'2-~]'3) ,

bO - m 2 [ 1 m E ] 47rf2 l+12¢r-~fi f2 (2]',+3[2) ,

m2 [ 1m2 ] a~= 16--~-f2 1 12--r 2 ~ ( ] - ~ + 2 ~ ) ,

b~- m2[ ]

8 --fz 1 12 r ;

m2[,m2 ] a1-24-~f 2 1 12-13. 2 ~ (]'~- ~) ,

D-wave:

m 4

a~= 14401r3f 4 (l~ + 4 ~ ) ,

m 4

b' -]'l ; ,-288~r-,~4(.j _ + r~)

., m 4

a_; = 1440¢r3f 4 ( ]'i + !"2).

The mean square scalar and vector pion radii are given by

3 r. 2 7r ~. 2f" ( r ) s - 8 " " 1 (m2/32¢Pf")[ , '

(r2)~ - - 87rz~.f_ ]'6.

The amplitude associated with the decay 7r--> e vy is determined by

~='(r~- ~).

(B.3)

(B.4)

(B.5)

(B.6)

(B.7)

(B.8)

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Explicit chiral symmetry breaking in the Nambu-Jona-Lasinio model