Volume 244, number 3,4 PHYSICS LETTERS B 26 July 1990

Meson properties at finite density in an extended Nambu-Jona-Lasinio model

A. Hosaka Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, USA

Received 8 March 1990; revised manuscript received 11 May 1990

Properties of various mesons such as n, g, p, al and to at finite density are studied in an extended Nambu-Jona-Lasinio model. The bosonization technique is applied as an alternative convenient method rather than solving the Bethe-Salpeter equations. We recover many of the previous results obtained by Bernard and Meissner. In addition, a possible density dependence of the param- eter a is pointed out.

Hadron properties in baryonic matter (for example, in a nucleus) may change from those in free space [ 1 ]. In particular, possible changes of meson features including masses up to about 1 GeV are very important, since, for example, they manifest themselves in some fundamental hadronic phenomena such as the nucleon-nucleon interaction [ 2 ]. Furthermore, in the large-Nc limit of QCD, baryon properties are themselves accounted for in the same framework of effective meson theory [ 3,4 ].

To study such aspects of meson properties, it is good to start with a microscopic theory which has the desired properties of QCD. The Nambu-Jona-Las in io (NJL) model [5 ] is one theory suited to this purpose. As dis- cussed by a number of authors, the model predicts various meson properties pretty well in free space, including not only chiral mesons (n and a ) but also vector mesons (p, a~ and co) [6 -8] . It is then natural to apply the model to situations which are different from free space. Previously, properties of chiral mesons (n and ~) at finite density were investigated by Bernard, Meissner and Zahed in the NJL model [9 ]. Later, Bernard and Meissner extended their analysis to the case of vector mesons [ 10 ]. They solved the Bethe-Salpeter equations in the relevant mesonic channels and studied various meson masses and quark-meson coupling constants. Their results are consistent with the chiral symmetry restoration expected at finite density.

The purpose of this paper is to present an alternative convenient way to study meson properties at finite density. Here, instead of solving the Bethe-Salpeter equations, the quark field is integrated out by introducing meson degrees of freedom in the relevant qft channels, which is usually called bosonization. To calculate the fermion determinant, we apply the loop expansion method [ 1 1 ]. The density dependence is then easily treated by introducing a lower momentum cutoff kv in the quark loop integrals. Although the present approach may not be as rigorous as solving the Bethe-Salpeter equations for discussing the results quantitatively, one of its nice features is that various (symmetry) relations among the meson parameters such as the KSRF relation [ 12 ] are presented in a manifest form. In particular, the parameter a is related to the p- and a~-meson masses [8 ], which is then followed by a possible density dependence of a.

Let us start with the following NJL model lagrangian extended by including vector meson channels p and al:

L(~ , g2) = ~ , + ½G, [ (yv¢)2 + (~ys~r~,) 2 ] - ½G2 [ (Y?Pu~'g/)2 + (~2yuYs~r¢) / ] , ( 1 )

where G~ and Gz are dimensional coupling constants for four-point quark interactions. Here we have neglected the current quark masses and shall concentrate on the exact chiral limit. A major effect of finite current quark

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Volume 244, number 3,4 PHYSICS LETTERS B 26 July 1990

masses is to smooth various singularities at the transition point. The partition functional generated by the la- grangian ( 1 ),

W=f[d~dlp]exp(i fd4xL(~/ ,~ ' ) ) ,

can be transformed into the following form by introducing auxiliary fields a, re, Vu and A u [ 1 1,13 ]:

W=f[d~udtpdad~dVdA]exp(ifd4xL'(gt ,~,a, zr, V,A)),

with

1 1 L' (~u, qT, a, 7r, V, A) = ~ [ i ~ - (a+irr75 + V~y~'+AuTU75)]~u- ~ (a2+n 2) + 2~2 ( V~ +A2u). (2)

Here the isospin matrix notation is adopted for zt, V u and A~:n=v .n etc. The quark fields gt and ~ are now integrated out to yield

W=~ [dadredVdA]exp(ild4xLerf(a,~,V,A)) ,

where the effective lagrangian Lcrf(a, re, V, A ) takes the form

L e ~ = - i t r l n 1 i~-m(S+inys+Vuy~+AuYUT') - ~ ( a 2 + ~ 2 ) + ~ ( V ~

~ i r ( i _ ~ m )n 1 1 2 2 .=, n t (s+i~,, + V S + A f ~ , ) ~ ( ~ + ~ ) + ~ (V,,+Au). (3)

Here the a field is separated into its vacuum expectation value m and its quantum fluctuation s. The parameter m is interpreted as a constituent quark mass.

We calculate the series (3) up to fourth order (n = 4) and take terms including second derivatives or less. Multi-meson vertices including more than four mesons and higher derivative terms are neglected, since they are not very important at low energies. The leading terms thus calculated include all the divergent terms, both quadratic and logarithmic divergences. In the NJL model which is not renormalizable, these terms must be regularized as will be shown later.

After suitable field renormalizations, we lind the following e-model lagrangian with 0 and a~ mesons intro- duced by the massive Yang-Mills construction [4,7,8 ]:

a Lerf= a--S-i- Itr (D~, U DuU*) - ~tr( V~. +A~.) + ¼m~tr( V~ +A 2)

- ~ 2 ( a + n - f ~ ) + a - ~ / 2 - ~ (a +re 2) . (4)

Here we adopt the standard notation for various terms. The chiral field U is parametrized by the linear chirai fields a and n such that U= a+ iT-it. The covariant derivative on the chiral field U takes the form

Du U= O u U- ½ig[ V~, U] + ½ig{Au, U}. (5)

The non-abelian field tensors are

Vu~=OuV~-O.V,,-½ig[V~, , V.]-½ig[Au,A~] , (6)

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Au~ = GA. - GA~ - ~ig[ v u, A,] - ½ig[A u, V~I . (7)

In these equations, various mesonic parameters are expressed in terms of the model parameters G1, G2 and quark loop integrals Io and 12:

2_ 3 1 g = \ ~ o , ] , rap-- ~ o -- I 2 - m Z -F ~ 2 ' (8,9)

a = ( 1 m2 , - 1 ( a - I .~,/2 )~= 1 ( a ~2. m E + e m 2 J ' f ~ = 2 k a I ° ) m, ~ \ a ~ - l , / (10,11,12)

The quark loop integrals are defined by A

Io = - i N c ~ d4p 1 Ncf d3p 1 (27~) 4 (pZ-m2)2 - 4 (2n) 3 (p2q_m2)3/2, (13)

A I 2 = i N c f d4p 1 Nc I d3p 1

(2z0 4 ( p 2 - m 2 ) - 2 (2n)3(p2+m2)1/2 , (14)

where the color number Nc arises from the trace in color space. These integrals are divergent and the three- momentum cutoffA must be introduced. This is appropriate in the present study, since the density dependence is conveniently expressed in the rest frame of matter. Various relations in eqs. ( 8 ) - (12) are very similar to those obtained in the heat kernel expansion method [7 ]. We note, however, the difference &the p-meson mass due to the appearance of the factor/2 q- m 2 in (9).

Other interesting physical quantities are the masses of o and a~, and the strength of several decay processes. From eq. (4) the o and al meson masses are given by

ma 2, = m 2 + 6 m 2, (15)

mZ~ =4m 2 (for p<p¢)

= 2(~GL --I2 ) ( forp>pc) . (16)

The meson decays we are interested in here are o - 27t, p-, 27t and a,-~ pTt decays. The strengths for these decays calculated by first order Born diagrams are denoted ;t . . . . ;tp~ and 2a,p~, respectively. To evaluate 2p~ and ~alp'/g properly, one has to take into account the presence of the n-a~ mixing term: [ a / ( a - 1 ) ]gf~0un.A ~. After suita- ble redefinition of the physical a, field [ 7,8 ], one finds

a 2

g Z ( ~ _ _ l ) f ~ , ~,prc~:=(1 2~)g , J.alpn=g2f~ (17,18,19) ,~o~ = ~- - .

The second term of eq. (18 ) arises after the diagonalization and violates the relation 2p~ = g. In ref. [ 4 ] non- minimal terms were added to recover this relation. Here we do not argue this subtle point in the model (4) but only study the qualitative behavior of these parameters as functions of density.

We can also study the properties of the co meson by adding a third quartic term - G3/2 (qT~u~')2 in ( 1 ). Here we are interested in the co-meson mass rn,o and the co--, 3n decay constant go. We find

m~ ~/o (3--I2 -- m2+ ~G3) Nc = , g o , = - - ~ g . (20,21)

The parameter go, is calculated from the fourth order diagram including one co and three n-vertices. The term contains three derivatives and is convergent unlike other terms in (4) [ 14]. The o~-37t coupling is important

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not only in describing the relevant anomalous process but also in accounting for the stability of the soliton solutions ofeq. (4).

Before studying these meson parameters at finite density, we determine the model parameters G1, G2, G3 and A, assuming that they are independent of density (see discussions in ref. [ 10 ] ). We impose the following con- ditions in free space (at zero density):

( a )mo= m,o= 770 MeV, (b) f~=93 MeV, (c) a=2 , (d) the KSRF relation [ 12]

m~ = x/~gf~. (22)

We find G~ = (960 MeV) -2, G2=G3= (740 MeV) 2 and A= 1300 MeV. We note that the conditions fixing the vector meson masses are the renormalization of the coupling constants G2 and G3 [ 11 ]. Furthermore, the last term of (4) is eliminated for p~<pc by the gap equation

1 -812 . (23)

Gj

Finite-density effects are taken into account by introducing a lower momentum cutoff kv in (13) and (14). A convenient way to see this is to use a quark propagator with the chemical potential/t = ~ [ 15 ]. Then

fo --'fkF" The Fermi momentum kv is related to the nuclear the integrals Io and I2 are modified according to A A matter density p by kv= (3n2p)1/3. The normal nuclear matter density is po = 0.17 fm -3. Using these prescrip- tions we study the density dependence of the meson parameters.

The constituent quark mass m is determined by the gap equation (23). This is an important order parameter ofchiral symmetry, since it is related to the quark condensate (qT~u) by (~Tq/) = -4mi2 . At low densities, there is a non-trivial solution to the gap equation which provides non-zero values for m and ( ~ / ) . Chiral symmetry is thus broken spontaneously. As the density is increased, these parameters decrease and become zero at and beyond the critical density Pc ~ 3.3po. A similar behavior is also found for the pion decay constant f~. These results are in qualitative agreement with those obtained by Bernard, Meissner and Zahed [ 9,10 ].

In fig. 1 we plot the meson masses as functions ofp. Again these results are qualitatively consistent with the previous findings [9,10 ]. The masses of the p and n are relatively stable for p <Pc (although the p-meson mass increases slowly as the density is increased). Beyond the critical density the n- and ~-masses and the p and a~ masses get degenerate, as expected from chiral symmetry. We also plot the parameter a in fig. 1, which relates the p- and a~-meson masses as seen from (10) and ( 15 ). We find that the parameter a is strongly density depen- dent. Since the p-meson mass is almost density independent, the KSRF relation (22) implies that the parameter a must be density dependent so as to compensate the density dependence off~. This is an interesting observation, since the parameter a has been usually assumed to be density independent [ 16 ].

The coupling constant g and the decay strengths 2 . . . . 2 ,~ and )'a,on as functions ofp are plotted in fig. 2. The gauge coupling constant g is almost independent of density (so is the coupling constant go), while the physical p--,2x decay strength 2~= has a weak density dependence. They coincide at P>Pc, since the difference between them depends on 1/a [eq. (18) ]. The parameters 2 ,~ and 2a,o~ decrease as the density is increased. This im- plies that the couplings of ~xx and of a lpx are suppressed at finite density. A similar behavior of 2o~ at finite temperature was discussed by Hatsuda and Kunihiro [ 17 ].

In summary, we have studied various mesonic parameters at finite density in the NJL model. In the present approach of bosonization, various relations among parameters are presented in a manifest manner at finite density as well as at zero density. Then the mesonic parameters vary if the density is increased, while preserving these relations. As a consequence, for example, the parameter a can be density dependent with the KSRF rela- tion unchanged.

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I I | I I

[ -12

~ . - 1 0

E ~ /a I - 6

- Y - . . I d ° E 2

P( Pc, ) 0 1 I I I ~ l '0

o 1 2 3 t 4 Pc

8 - I I I

g

2 - X%pTr

Oo I I ~ 1 2

8O > (i) (.9

60 "-"

- ~.

--40 ~<~ !- 2o

0

Fig. 1. Physical meson masses for n,o,p and a~, and the parameter a as functions of the density p. The density is normalized by the normal nuclear matter density Po ~ 0.17 fm -3. The to-meson mass m,~ has the same value as the p-meson mass mp.

Fig. 2. The gauge coupling constant g and meson decay strength 2,=~, A ~ and J-a,p~ as functions of the density p.

T h e a u t h o r t h a n k s R. A m a d o a n d M. O k a for d i s cus s ions a n d r e a d i n g o f t he m a n u s c r i p t . T h a n k s are a lso due

to N. Wa le t for he lp fu l d i scuss ions . T h i s work is s u p p o r t e d in pa r t by t he N a t i o n a l Sc ience F o u n d a t i o n .

R e f e r e n c e s

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