Physics Letters B 285 (1992) 119-125 North-Holland P H Y$1C $ k E TT E R $ g

Consistent treatment of the bosonized Nambu-Jona-Lasinio model

V6ronique Bernard, A.A. Osipov '

Division de Physique ThOorique, Centre de Recherches NuclOaires et UniversitO Louis Pasteur de Strasbourg, B.P. 20Cr, F-67037 Strasbourg Cedex 2, France

and

UIf-G. MeiBner 2 Universitiit Bern, Institut J~r Theoretische Physik, Sidlerstrasse 5, CH-3012 Bern, Switzerland

Received 16 April 1992

We suggest a systematic method for evaluating any mesonic N-point function in the bosonized Nambu-Jona-Lasinio model. We introduce a transformation of the bosonic fields which allows to develop the Hartree-Fock method in the framework of the bosonized NJL model. We obtain the usual picture of one-loop quark graphs with the external meson legs. However, the meson- quark couplings are momentum-dependent and the finite pieces of the quark loop integrals can be calculated exactly and in harmony with the low-energy theorems of current algebra. As an example, the nn-scattering amplitude is considered.

I. Introduction

The Nambu- Jona -Las in io (NJL) model is an extremely useful theoret ical laboratory for analyzing low-en- ergy processes involving the pseudoscalar and scalar mesons [ 1 ]. One of its advantages is the inherent mecha- nism of chiral symmetry violat ion with explicit symmetry breaking by the current quark masses and dynamical breaking by the qua rk -an t i qua rk condensates. This model is related to QCD in the short wavelength approxi- mat ion [ 2-5 ] and has been s tudied both in pure fermionic language [ 6-9 ] and by means of bosonizat ion pro- cedures [ 10-13 ]. In this paper we explore the l inear bosonized version of the U (2) X U (2) NJL model [ 10 ].

There is an essential difference between our approach and the usual t rea tment of bosonized NJL models. The s tandard approach is based on the der ivat ive expansion of the lagrangian. In this paper we develop the Har t r ee - Fock ( H F ) method in the f ramework of the bosonized theory ~. Instead of the usual der ivat ive expansion we obtain a quark one-loop expansion with increasing number of external mesonic fields. The first approaches to use the H F method for analyzing the bosonized NJL model date back to the seventies and can be found in ref. [ 14 ]. However, these authors systematical ly considered only the divergent pieces of the quark loop integrals, losing as a result impor tan t informat ion from the finite parts. In our paper we construct a special t ransforma- t ion of the bosonic fields which allows to take into account the full loop contr ibut ions without destroying the

¢r Work supported in part by Deutsche Forschungsgemeinschaft under contract no. Me864/2-2 and by Schweizerischer Nationalfonds. Permanent address: Joint Institute for Nuclear Research, Laboratory of Nuclear Problems, 141 980 Dubna, Moscow Region, Russia.

2 Heisenberg fellow. ~ In what follows, we will only consider the Hartree-approximation. The exchange contributions can easily be generated by Fierz-

transformation of the lagrangian.

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved. 1 19

Volume 285, number 1,2 PHYSICS LETTERS B 2 July 1992

current algebra predictions. In this sense we exactly realize the HF program in the framework of the bosonized NJL approach.

Particular attention will be concentrated on the consequences of explicit chiral symmetry breaking. This ques- tion was the subject of ref. [ 15 ]. Using the usual derivative expansion the authors of that paper were able to evaluate the corrections of order p4 [ 15 ]. Here we show that one can obtain the general expressions for any mesonic amplitude of the model. The N-point functions calculated by this method can easily be expanded in powers ofp 2, p4, p 6 ... (which allows to investigate how fast such an expansion converges). In momentum space this leads to the picture of the momentum dependent meson-quark couplings in harmony with the constraints from chiral symmetry. As an example we consider the ztzr scattering amplitude and obtain the explicit symmetry breaking corrections to the scattering lengths and slope parameters for the first S-, P- and D-waves.

We would like to stress that our approach is a bosonized version of the HF method which is successfully applied in purely fermionic NJL models [ 6 ]. In this sense we extend it to the framework of bosonized models, that, as we hope, simplifies certain calculations.

2. Bosonized NJL model for scalar and pseudoscalar mesons in the Hartree-Fock approximation

Consider the NJL lagrangian with a local four-quark interaction,

L~(q) =~](i~- rh)q+ ½G[ (qraq)2 + (c]iy5 ra q) 2 ] , ( 1 )

where ~]= (a, aT) are coloured current quark fields with current mass rh, %= (to, ri), %=1, r, ( i= 1, 2, 3) are the Pauli matrices of the flavour group SU (2)f. The current mass term explicitly violates the U (2) X U (2) chiral symmetry of the lagrangian ( 1 ). Introducing meson fields in the standard way, the lagrangian takes the form

1 5e(q, 6, 77)=~(i~- r h + 8 + i 7 5 ~ ) q - ~ (6 2 + ~ 2 ) . (2)

Here 8=8 ,%, 2 = 2 , z , . The vacuum expectation value of the field 8o turns out to be non-zero ( ( 8 o ) S0) . To obtain the physical field 60 with (~o) = 0 one performs a field shift leading to a new quark mass rn to be iden- tified with the mass of the constituent quarks

8 o - r h = ~ o - m , 8 i=# , , (3)

where m is determined from the gap equation (see eq. (5) below). Let us integrate in the generating functional associated with the lagrangian (2) over the quark fields. Evalu-

ating the resulting quark determinant by a loop expansion (HF approximation) one obtains

1 5a(#, 2) = - i Tr ln[1 + ( i ~ - m ) - 1 (~+ iys~) ]A -- ~ (d] q-~]) • (4)

The index A indicates that a regularization of the divergent loop integrals is introduced. We use here the usual cut-off A characterizing the scale of chiral symmetry breaking.

Consider the first terms of this expansion. From the requirement for the terms linear in # to vanish we get a modified gap equation

m-rh=8rnGl~ . (5)

The integral It is equal to

, ,=_ iNc f d4q O(A2+q 2) _ 3 [A2_m21 n ( A ~ ] ] (6) (2/l") 4 m2--q 2 (4;,r) 2 l + m - / j '

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with Arc the number of colours. The second order terms in pseudoscalar and scalar fields lead to the self-energies

H~b(p2)=~ab[ (8I, -G-~)+p2g-2 (p2) ] , H~b(pZ)=6ab[ (81, - G - ' ) + (p2-4m2)g-2(p2) ] . (7)

The function g(p2 ) is determined by the integral

( 42 /2[ (A2+3m2/l g- 2(p2)=412(p)---- 412( O ) + 8zc2m2\A2+m2 ] 1+ l ~ - 2 m Z \ ~ - ~ j j + O ( p 6 ) , (8)

where I2 (p) is defined in eq. ( 18 ) below. If one takes into account only the first term of the expansion (8) one gets ~2 from (7) the usual lagrangian of the free meson fields (z~a, Oa) with the masses M 2 = ( G - l - 8 1 1 ) / 4•2(0) =rh/4mGl2(O), M 2 = M ~ + 4 m 2. This is the standard method [ I0,1 1,14]. To include the momentum- dependent corrections one has to consider all the terms of expression (8). In this case one should perform the following field transformations

fG(p)=Z;l /2g(p2)zG(p) , 6~(p)=Zgl/2g(pZ)ao(p). (9)

The new bosonic fields (zt~, ~a) have the self-energies

n 2 o" ~ - - Hab(P )=dabZ;l[p2--m2(p2) ] , H,,,(p-)=O, hZ~ '[p2-m~(p2) ] . (10)

Note that one should use all the terms of the function (8) for the transformation (9). I f we had included some of them in a momentum-dependent mass in the case of the pseudoscalar meson we would have obtained the wrong conclusion that the mass of pion does not vanish when the current quark mass rh goes to zero. The p2. dependent masses are equal to

m2(p 2) = (G - i - 8 1 1 )g2(p2) = rh ( m G ) - ,gZ(p2), m2(p2) =m2(p2) + 4 m 2 " ( 11 )

The constants Z , and Z~ are determined by the requirement that the inverse z~, and a~ propagators Hgb(p2), Hgb(p 2) satisfy the normalization conditions

n 2 2 2 Hab(P ) = P - -m , +O( (p2--m2) 2) ~ 2 , Hab(p )=p2--mZ+O((pZ--m2)2) (12)

around the physical mass points p2= m ~ and p 2= m 2, respectively. The restrictions (12) lead to the values

p2=m2 m~--4m 2 OI2(p) v 2 .... m 2 ~ O12(p) Z o = I + i2(m2) (13) Z,~= 1 + i2(m2) Op 2 , 0p 2 ,

2 2 2 2 where m2=m=g (mo)/g ( m 2 ) + 4 m 2. The on-mass shell evaluation of the pion decay amplitude (0[A~(0)[~zb(p))=if=6~bP ~, where A g ( 0 ) =

0(0)7*% r~q(0) is the axial-vector current, shows that there is a simple relation between the transformation function g(p2) at the point p2 = m ] and the weak pion decay constant f=:

g( m2)f= =rnZ; 1/2 . (14)

In the chiral limit it is easy to obtain from (14) the well-known Goldberger-Treiman relation (at the quark level )

g~qq(m~)=Z; l /2g (m~)=7 l + ~ + O ( r h ~ ) , (15)

with Jt defined in eq. (20). To order p2, the Goldberger-Treiman relation is exact. It is clear now that one can evaluate any meson vertex using the lagrangian (4) with the new wavefunctions

~2 After appropriate normalization of the meson fields which is fixed by the standard form of the kinetic term.

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of the mesonic fields (9). It will depend on the transferred momenta and the parameters of the model: A, m, G. This picture corresponds to known calculations in the framework of the pure fermionic NJL model [6 ]. Now we shall show that doing this way one obtains the known current algebra results.

3. Low-energy fr~ scattering

The celebrated current algebra theorem [ 16 ] that states the form of the low-energy nn scattering amplitude is a good testing ground for our method. According to this theorem the expansion of the nn scattering amplitude A (s, t, u) in the external momenta starts from the terms

s - r h ~ +O(p4) ' (16) A(s,t, u)= f2

It is known that in the linear bosonized NJL model Weinberg's prediction arises as a tree level result from the box and a-exchange graphs if one extracts only the divergent pieces of the quark loops [ 17 ]. Here we show that in our approach where the finite parts of the quark loops are taken into account the contributions of order O(s, m] ) from the finite pieces of the loop integrals are cancelled.

There is another reason for our calculations. The nn scattering amplitude is relatively well known from the other explorations and experiment. It is interesting to compare the results of our approach with the known ones. Of course, this model does not include the very important p meson exchange graph, which plays a prominent role in the P- and D-waves. This will be done elsewhere when the vector and axial-vector mesons will be taken into account. But even without p mesons one can hope to obtain reasonable values for the S- and P-wave char- acteristics, because in that case the main contribution comes from the Weinberg amplitude.

Consider the box and a-exchange graph contributions, which in our case follow to be

AU°X(s, t, u ) = -4g4(m])Z~Z[412(q, +qz)+4(s-2mZ)13(q,, q~ + q 2 ) + (2m4-us)I4(q,, q, +q2, q4)

+ (2m4-ts)I4(q~, q~ +q2, q3) - ( 2m4 -ut)14(q,, q, -q3, q4)] ,

4mZZ;2 [412(ql +q2) +2(s-2m])I3(q~, q, +q2) ]2. (17) A~(s, t, u)=4g4(m~)g2(s) m~(s) -s

We have written here the renormalization factors at each quark meson vertex and the full contributions of the quark loops. The following notations are used:

f d4q (-iNc)O(AZ+q 2) I2(k) = J (2n) 4 (q2-mZ)[(q+k)2-m2] '

I d4 q (-iNc)O(A2+q 2) 13(kl,k2)= (2n) 4 ( q 2 _ m 2 ) [ ( q + k l ) 2 _ m 2 ] [ ( q + k 2 ) 2 m2],

I d4q (-iNc)O(A2+q2) . (18) I4(k~,k2, k3)= (2n) 4 ( q z _ m 2 ) [ ( q + k l ) 2 mZ][(q+k2)2_mZ][(q+k3)2 m2]

The kinematical variables are s= (ql +q2) 2, t = (q l - -q3) 2, U= (q l - -q4) 2 where q~, q2 (q3, q4) are the momenta of the initial (final) pions.

The full amplitude can be computed straightforwardly:

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Z;2 ( s - -mZ(s ) A (s, t, u) - 412(m~ ) 4 mZ(s ) -~s [I2(ql +q2) + (s--2m])I3(ql, q, +q2)] + (us-2m4)I4(ql, q, +q2, q4)

(s-2m2~) z 13(q,, q, +q2) ] + (ts-2ml)I4(q,, ql +q2, q3)+ (2m 4 - blt)14(ql, ql --q3, q4) + 4 m 2 m2(s ) --s -17, ~-~2) J" (19)

It is clear, that expanding this expression around the point s = t= u = m ~ = 0 (soft pion limit) one obtains the result (16) to leading order. These calculations show that corrections from higher derivatives which naturally arise in the NJL model in the HF approximation do not disturb the current algebra predictions. Moreover, using our approach one can exactly calculate all these corrections.

4. The p4 approximation and beyond

Before proceeding, we must fix the parameters of the model. For a direct comparison with the empirical numbers and chiral perturbation theory (zPT) results, we have used the values for the pion decay constant, f~, and the pion mass, m,, close to their physical values, f,_~ 93 MeV and m~-~ 139 MeV, respectively. We further- more demand that the pion decay constant in the chiral l imi t , f and the leading term in the quark mass expan- sion of the pion mass, rh,, are close to the zPT result f~_ 88 MeV, rh,~ 140 MeV [ 18 ]. Within the purely fer- mionic NJL model [ 19 ], such a set of parameters has already been determined. For the case at hand, we used G=7.74 GeV -2, A= 1.0 GeV and rh= 5.5 MeV. With these parameters, we findf~=93.1 MeV, f=88 .6 MeV, m~= 139 MeV, rh~= 141.5 MeV and < qq) J/3= _ 248 MeV, consistent with the empirical and zPT constraints. These results are obtained from the fully self-consistent solution within the HF approximation and do not in- volve any expansion in powers o f p 2. We should stress that for these set of parameters the constituent quark mass is rather low, m = 242 MeV (see ref. [ 19 ] for a more detailed discussion).

For later comparison, let us consider the pa approximation to the full nn-amplitude A (s, t, u ) (eq. ( 19 ) ). One obtains

s-rh] A(s,t ,u)= f2 +A(4)(s,t,u)+O(p6),

At4)(s,t,u)_ (s -2rh2)2 ( rhZJ, 9rh 4J~ "~ [s(u+t)-ut-2rh4lJ2 (2rhf) 2 1 - (nf) ~ + (~ f )4 ] + 2(2nf2)2 , (20)

with

{ A2 ,2 A 2_t_ 3~t 2 Jl = \A2_t_r~/2 J , J2=Jl A2.t_~/2 •

In this expression, we have used the chiral expansion for the pion decay constant, the pion mass and the constit- uent quark mass to orderp 4. To show the power of our approach, let us give these expansions up to and including the terms of order p6:

4rh2 (2-~--~-i] - 32rh4Ll+ ( - ~ \ ~ 1 5 ~ J _ ] J '

{ rh~( rhZJ, "] 5rh4[ rh2J, (13rh2j, 69A2+97rh2"~l ~ m=.=m~ l - ~ m = I ( ~ / ) 2 1 + ~ I+2(-i--~7\i~= 25(A=+m=)]Aj ,

(21a)

(21b)

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m = 6 6 [ 1 + 662 3664 [ 662j, '~1 466~ 3 - ~ - 4 ~ 1 2/r2f 2j.] , (21c)

L

(~q> = <~q>o [ i + 4662662 rh66 326643664 ( I 2rc2f 2//_] . 66 2J' -'~l (21d)

It is interest ing to see how rapidly the expansion given in (21 a ) - ( 21 c ) converge to the full result. In table 1, we give the values for the terms to order p 2, p4 and p6, in that order. In all cases, the p4 approx imat ion is within less than 1% of the total result and the p6 approx ima t ion even within less than 0.2%. The quark condensate also given in table 1 and evaluated from (21d) shows a less rap id convergence. We wish to point out, however, that in QCD the VEV of the scalar quark densi ty is only well-defined in the chiral limit. In the NJL model, it is f inite also for finite quark masses. This behav iour of < ~q) reflects itself in the devia t ion from the G e l l - M a n n - O a k e s - Renne t relation, m 2 f 2 = - 2 6 6 < 0 q ) , which is exact at order p 2. For the full result, it is fulfilled within 0.3%, but for the p4 (p6) approx imat ion , we f ind a devia t ion of 17% (14%). Similar results are found in ref. [ 15] for comparable parameters ( to order p4) ~3. The results for the n n scattering lengths and effective ranges for the S-, P- and D-waves are given in table 2 in compar ison to the xPT results of Gasser and Leutwyler [20] and the empir ical values from Peterson [21,22]. For the S- and P-waves, the scattering lengths and the effective ranges only vary by less than 7% between the p4 approx imat ion and the fully self-consistent result. We find a rather good agreement with the EPT predict ions. In the D-waves, the total result is comparable with the empir ical data, the p4 approx imat ion gives somewhat too large values. We should stress, however, that in the P- and D-waves some cont r ibut ion from the p-meson is expected. We are presently investigating a more complete model includ- ing also (axia l ) vector mesons along the same lines. Our results for the p4 approx imat ion are s imilar to the ones found in ref. [ 15 ] (for s imilar pa ramete rs ) . We would like to stress again that we have not only considered the p4 approximat ion , but also the fully self-consistent result, which allows to est imate the effects of yet higher order terms in the m o m e n t u m expansion. Finally, a compar ison with the results obta ined in the fermionic NJL model [23] should only be done after inclusion o f the vector mesons.

~3 Note that in that paper the VEV of qq should be divided by a factor two compared with the usual definitions.

Table 1 Chiral expansion of the pion decay constant (f~), the pion mass (m~), the constituent quark mass (m) and the vacuum expectation value ( <qq> ) to order p 2, p4, p6 in comparison to the self-consistent total result. All numbers are given in MeV.

Order f,~ m,~ m _ < qq) I/3

O(p 2) 88.6 141.5 221.2 242.7 O(p 4 ) 93.8 138.4 243.9 261.4 O (p6) 93.0 139.1 241.4 258.5

total 93.1 139.0 241.8 248.1

Table 2 nn scattering threshold parameters. We show the NJL results for the p4 approximation and the total self-consistent solution. For compar- ison, the •PT results of ref. [20] and the empirical data [21,22] are given. All scattering lengths and effective ranges are given in appropriate units of the pion mass.

a ° b ° a~ b~ 2a°o-5a 2 al a ° a~

O(p 4) 0.188 0.265 -0.041 -0.076 0.58 0.034 21.2×10 -4 8.2x 10 4 total 0.190 0 .268 -0.044 -0.079 0.60 0.034 16.7× 10 -4 3.2× 10 -4 zPT 0.20 0.24 - 0.045 - 0.075 0.61 0.037 input input experiment 0.26_+0.05 0.25_+0.03 -0.028_+0.012 -0.082+_0.008 0.66_+0.05 0.038_+0.002 (17_+3))<10 4 (1.3_+3)×10-4

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5. Summary and conclusions

In this paper we have developed the systemat ic me thod to deal with the chiral expans ion ( in powers of exter- nal m o m e n t a a n d quark masses) wi th in the f ramework of the bosonized NJL model . The m a i n idea resides in the cons t ruc t ion of new bosonic var iables for the meson fields which have the usual kinet ic te rms bu t non- t r iv ia l m o m e n t u m d e p e n d e n t qua rk -meson couplings. F r o m the ma themat i ca l po in t o f view we exactly realize the

H a r t r e e - F o c k program us ing the language of a bosonized theory. In this sense we essential ly improve the pre- v ious a t t empts [10,11,14] a n d connec t the bosonized vers ion of the NJL model with the known fe rmionic

(Be the -Sa lpe te r ) approach [ 6 ]. Our m e t h o d allows to calculate the correct ions of order p4, p6, ... to any mesonic N-poin t funct ion . These

calcula t ions are s t ra ightforward a n d allow one to systematical ly s tudy higher order corrections. Fur the rmore , the m e t h o d is in h a r m o n y with current algebra, i.e. the leading terms genera ted are no th ing bu t the ones de- m a n d e d by chiral symmetry . As an example, we have cons idered the classical p rob lem of ~z~r scattering. We have recovered the Weinberg ampl i t ude to lowest order and calculated the p4 app rox im a t ion as well as the fully self- consis tent ampl i tude . In the lowest part ial waves, the p4 expans ion is a very good approx imat ion . We have also shown that the chiral expans ions for the p ion decay constant , the p ion mass and the cons t i tuen t quark mass are rapidly converging (i.e. the p6 a p p r o x i m a t i o n is wi th in 0.2% of the full resul t) . We are present ly inves t iga t ing the same p rob lem in the ex tended vers ion of the NJ L lagrangian inc lud ing also vector and axial-vector mesons.

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