Physics Letters B 269 ( 1991 ) 17-22 North-Holland PHYSICS LETTERS B

Color singlet projection in the Nambu-Jona-Lasinio model at finite temperature

K. K u s a k a Institute of Theoretical Physics, University of Regensburg, W-8400 Regensburg, FRG

Received 6 May 1991 ; revised manuscript received 9 August 1991

We study the baryon number susceptibility in the Nambu-Jona-Lasinio (NJL) model within the mean field approximation. We find that the naive extension of the NJL model to the finite temperature situation cannot reproduce the correct low tempera- ture behavior of the baryon number susceptibility. We propose the color singlet projection in order to remove the colored quark states from the thermodynamic potential. We find that the color singlet projection considerably improves the low temperature behavior of the susceptibility.

Recently the proper t ies o fhad ron ic mat ter at finite tempera ture and finite densi ty have a t t racted consid- erable interest. These extreme envi ronments are con- s idered to be realized in the early universe [ 1 ] and in the ul trarelat ivist ic heavy-ion collisions [2] . Nu- merical QCD simulat ions on the latt ice have shown that chiral symmetry, which is spontaneously broken at zero tempera ture and zero density, is restored with a t ransi t ion tempera ture in the range 100-200 MeV [3] . This restorat ion of chiral symmet ry is also ex- tensively studied in the effective lagrangian approach.

The low tempera ture expansion analysis based on the non-l inear sigma model also suggests the restora- t ion of chiral symmet ry at T # 0. The finite tempera- ture correct ions to the quark condensate < 01 qq] 0 >, which is the order pa ramete r of the chiral phase tran- sition, have been calculated [4] , with the result

<O[4qlO>~

=<Olqq lO>o(1 N ~ - I T2 ) Nf 12./" 2 + . . . . ( l )

in the ehiral l imit (mq-~O), where f~= 93 MeV is the pion decay constant and Nr is the number of light quarks.

The Nambu-Jona-Las in io (NJL) model [ 5 ] is also

¢~ Work supported in part by Deutsche Forschungsgemein- schaft, grant We 655/9-3.

considered as an approximate low energy effective la- grangian of QCD, because this model exhibits the dy- namical generat ion of quark masses and the emerg- ence of composi te Golds tone pseudoscalar mesons [5,6]. Indeed, if we regularize the quark loop inte- gral suitably, the NJL model leads to the chiral me- son lagrangian by bosonization techniques [6,7]. The finite tempera ture and the densi ty effects are also s tudied in the NJL model by several authors [8,9]. However, the low tempera ture behavior of the quark condensate predicted by the NJL model differs from that of the chiral lagrangian approach. Up to the one loop calculation, the finite tempera ture correct ion to the quark condensate in the NJL model is

<OlqqlO>-r

( T ) 3/2 = <Orqq lO>o[1-4NrN~.GM 2 \ ~ )

× e x p ( - M J T ) + . . . ] , /

(2)

in the chiral limit. Here G is the four-quark coupling constant of d imension ( length) 2, No= 3 is the num- ber of colors, and Mq is the const i tuent quark mass at T = 0, which is dynamical ly generated. The thermal correct ion in the NJL model is evident ly smaller than that of the chiral meson lagrangian approach (see eq.

0370-2693/9 t/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved. 17

Volume 269, number 1,2 PHYSICS LETTERS B 24 October 1991

( 1 ) ) due to the suppression factor e x p ( - M u / T ) in (2).

This difference comes from the modes which con- tribute to the thermal average of the Boltzmann fac- tor. In the chiral lagrangian approach, the thermal average is dominated by the massless pseudoscalar meson (pion) states. On the other hand, only con- stituent quark states are taken into account in the one loop calculation of the NJL model. Since the low temperature thermodynamics is governed by the lightest particle of the system, clearly the pion loop contribution is more important than that of constit- uent quarks. Thus we cannot neglect the thermal pion loop contribution, even though it is a higher loop contribution in the NJL model. However, the calcu- lations beyond the one loop in the NJL model are complicated, so that we will consider another quan- tity which is almost independent of the pion loop at low temperature.

The baryon number susceptibility 7,e is one such quantity. It measures the response of the baryon number density p~ with respect to a change of the chemical potential ~tz~. In other words, Z8 indicates how easy it is to create baryonic excitations. Z~ is re- lated with the quark number susceptibility Zu, which is the derivative of the quark number with respect to the quark chemical potential J~q as

O(pq/N~) 1 ZB= O(N~jtu) - N 2 Zu. (3)

Because the pion loop has no explicit dependence on the chemical potential/~8, the susceptibility ZB is not strongly affected by the pion loops at low tempera- ture. Note that the susceptibility ZB does not vanish even at zero chemical potential limit (/tRY0).

ZB has been measured in a numerical QCD lattice simulation [ 10 ], with the result that Z~ is very small in the low-T chiral symmetry broken phase and be- comes finite in the high-T chiral symmetric phase. This suppression in the broken phase can be easily understood. Consider the thermal baryon gas. If T is sufficiently low, we can neglect heavy baryons and their interactions, so that we can easily get the low-T expansion of the susceptibility from the Fermi distri- bution function of the nucleons:

2 -- . 1 / 2

Z~'d=8 ~ " \ ~ - ~ 7 " " J zlr ,~rMN e x p ( - M N / T ) + .... (4)

Since the baryon masses are large (MN~ 1 GeV) in the chiral symmetry broken phase, the exponential factor exp ( - MN/T) suppresses the susceptibility.

However, the susceptibility in the NJL model has a different suppression factor,

2 1 / 2

x ~ J L _ 4 N r M ~ ( T ~ - N¢ 27c \2~Mq] exp ( -Mq/T)+ . . . , (5)

because 4NrN¢ constituent quarks carry the baryon number l/Nc. Since Mq~MN/Nc, the difference be- tween X~ ad and X NJL is numerically significant. Note that the inclusion of the pion loop cannot remove this discrepancy. We should obviously remove the quasi- free constituent quark contribution from the thermal average, since confinement does not permit such states. In order to get the correct low-T behavior of the susceptibility ,~ we propose here the color singlet projection [ l 1,12].

Let us consider the partition function

Z = t r e x p ( - f l H ) = ~ ( n l e x p ( - f l H ) l n ) , (6) n

where H is the hamiltonian of the NJL model and In) denotes arbitrary multi-quark (anti-quark) states. In order to eliminate free constituent quarks, we shall restrict the sum in (6) to color singlet states. Such a constraint can be introduced by making use of the color singlet projection operator P [ 1 1,12 ]. The projected partition function is

Z pr°J = tr P exp ( - fill)

= f [dg] ~ { n l e x p ( - f l H ) U(g) l n ) , (7) n

where U(g) is the color rotation operator generated by the SU (No) group element g and [ dg] is the prop- erly normalized Haar measure of the SU (N~) group. Since the color symmetry in the NJL model is simply a global SU(N~) and the hamiltonian H commutes with all the SU (No) generators we can replace U(g) in (7) by the diagonal rotation operator [ 12 ]. Let Q~ represent the diagonal generators of the SU(N~) group. Then we have

( n l exp ( - f l H) U(g) In) pl

= ~ (nlexp(- f lH) exp(iO.Q.)ln). (8) n

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Volume 269, number 1,2 PHYSICS LETTERS B 24 October 1991

Here 0~ are the parameters characterizing the group element g such that

U(g)=U(g' ')exp(iO~Q~) U(g'), (9)

where g' is an element ofSU (N~) which diagonalizes U(g). Furthermore, we can choose the diagonal gen- erator such that only the ath diagonal component is non-vanishing and all others are zero. With this rep- resentation, the ath colored quark is defined as the column vector whose non-vanishing component is the ath row. Now, the color singlet projection operator P is given by the quark field in the diagonal color rep- resentation as follows:

P = ~ [ d 0 o ] e x p ( i ~.=, Id3x~ .yo0 .~ '~ ) , (10)

where [d0.] is the measure of this representation. The explicit form if this measure is well known from lat- tice gauge theory [ 1 3 ] :

x~ i dO. [d0~] = 1-I 2n a=l

× [A (exp (i01), exp(i02) ..... exp(i0~.~) )12

X gpcr(Z Oa)' (11)

where A(z,, z2 ..... zx) is the Nth order Vandermonde determinant and 6per is the periodic cLfunction with period 2n. This d-function ensures that the group ele- ment is SU (N~).

Now, we can evaluate the projected partition func- tion Z p~°~ with the following functional integral:

/proj= ~ [dgl [d~'] [dq/]

× exp[i f d4x ( 51~jL + ygT)'o ~'+ x~. 7o -~

(12)

where we have followed the convention of Dolan and Jackiw [ 1 4 ]'

-i/>' 1 ~d4x~ ~ Ckvofd3x, with fl=~.

0 I'

Furthermore, we set the spatial volume V finite in or- der to integrate over 0, without ambiguities. We will take an infinite volume limit after the 0, integration.

In this letter, we will consider the simplest NJL lagrangian

~NJL = gT(i~-- mo)~'+ ½G[ ( ~ , ) 2 + (~ i~,s~,)2 ] . (13)

Introducing the auxiliary fields a and n for the qT~, and ~'iysr~ bilinear combinations, respectively, we can integrate over the quark field with the anti-peri- odic boundary condition, as usual. Note that the ex- tra color rotation term acts as an imaginary chemical potential. Therefore, we can easily obtain the mean field partition function Z °r°j just replacing the chem- ical potential in the ordinary NJL calculation by the complex "effective chemical po ten t i a l " / l - i0,/fl for the ath colored quark. We then have

ZPr°J=exp(-flV~--~ (0") 2)

× ~[d0~] exp[tr(flE

+ln {1 +exp[ - f l ( E + / z ) ] exp(i0.)}

+ln{1 + e x p [ - f l ( E - / ~ ) ] exp( - i0~)}) ] , (14)

where E is the energy of the constituent quark, E=xfpZ+(mo+(a)) 2, and the trace means the summation over the spin, flavor, color and integra- tion over momentum p.

Since we set the space-time volume V finite, we can rewrite the quark determinant contribution in the product form

x l q l q [2 cosh HE+ 2 cos(0. +ifl/~) ], (15)

where the products over l a n d s refer to flavor and spin states, respectively and we have replaced the pe- riodic c~-function by the summation over the Fourier components exp (im~ 0,). Utilizing the properties of the Vandermonde determinant, an orthogonal poly- nomial method [ 15 ] provides us with the following formula [ 16,17]:

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Volume 269, number 1,2

7r N t" d 0 a [ I ~ - I A ( e x p ( i 0 ~ ) , exp(i02), ..., exp(i0Nc)) I 2

a=l

×f(0o)

=N! det [ l ,_j l i j= ~,2,....N, (16)

where,/'(0) is an arbitrary function of 0 and L, is given by the following integral:

Im= i~f(O) exp(imO). - r e

Applying this formula, we have the mean field ther- modynamic potential

f2 p'°j = - In Z p~°j

1 - 2G ( a ) 2

l J" d3p - ~ ( ~ ) 3 1 n , . = ~ _ ~ dettlm+i_,]/,a=L2....,N~, (17)

where we have taken the infinite volume limit ( V-. ~ ) and

i d0 Ira= ~ l ~ I [2coshflE+2cos(O+iflg)]

X e x p ( i m 0 ) . (18)

Note that I,n can be calculated exactly and vanishes for I m l > 2 N r which is the maximum number o f quarks with the same color and momentum. Thus the sum over m in (17) is finite. Here we show the ex- plicit form of ( 17 ) in the case of Arc = 3 and Nf= 2:

• 1 I d3p E n° r ° '= 2---G ( o . ) 2 _ 12 (-~g)3

1 [" d3p ln{l + 16 e x p ( - 2 f l E )

+ 2 0 exp[ - 3 f l ( E - #) ] + 2 0 exp[ - 3 f l (E+ lz) ]

+. . .} , ( 1 9 )

where we have shown only the first few terms in the logarithm for simplicity. The first and the second terms on the right hand side of (19) represent the

PHYSICS LETTERS B 24 October 1991

zero temperature energy density. They are identical to those obtained with the ordinary NJL model, so that the phenomenologically successful predictions of the NJL model at zero temperature are not modified by the color singlet projection. Due to the projection, thermal contributions are written as a sum of Boltz- mann factors for the color singlet states. The second term in the logarithm of eq. (19) can be interpreted as the Boltzmann factor of one meson (q~) states whose energy is 2E. The third and the fourth terms are also interpreted as one baryon (qqq) and one anti- baryon (c]qc]) states with energies 3E (in general NEE). The dots in the logarithm indicates states of many mesons and baryons. As we have mentioned above, the number of these terms are finite. We can therefore calculate g2 pr°j exactly, and we will actually use the exact form in the following numerical calcu- lations. Note that the chemical potential # always ap- pears with the factor No=3 due to the color singlet projection. Of course, these color singlet states are not real bound states, but just combinations o f the con- stituent quarks. However, we expect that this pro- jected thermodynamic potential simulates the con- fined hadrons well with the exception of the pseudoscalar Goldstone bosons, which are due to higher loop contributions as we have discussed above.

Now, let us consider the finite temperature behav- ior of physical observables. Hereafter, we will con- centrate on the zero chemical potential l imit /2- .0. First, we would like to discuss the quark condensate (01 c]ql 0 ) . Although we cannot get any reliable low- T behavior of ( 01 c]q] 0 ) without thermal pion con- tributions, it is worthwhile to see how the color sin- glet projection modifies the low-T corrections to the quark condensate in the NJL model. Minimizing the thermodynamic potential (19) with (or), we obtain the gap equation which determines the quark con- densate. In fig. 1, we show the quark condensates (01 c]ql 0 ) as functions o f the temperature T. For the choice o f parameters we follow the work of Hatsuda and Kunihiro [ 8 ] : three-momentum cutoff A = 631 MeV, current quark mass rno--5.5 MeV and four- quark coupling constant G=0 .428 fm -2. The solid line denotes the projected condensate ( 01 qql 0 ) p~oj and the dashed line the original non-projected one, ( 01 qql 0 ) NJL. AS the thermodynamic potential (19) suggests, the chiral restoration temperature of the projected model is higher than that of the non-pro-

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Volume 269, number 1,2 PHYSICS LETTERS B 24 October 1991

12 <01~ql0>~

< 01qql0>o 1.0

08

06

04

02

t i 100 200 300 t.oo

T [HeV1

Fig. 1. The quark condensates (Ol~lqlO)yl(OlqqlO)o as func- tions of temperature T. The projected condensate is denoted by the solid line and the non-projected condensate is denoted by the dashed line.

jected NJL mode. However, we should of course remember that we have dropped the pion loop con- tributions which will dominate the low-T thermody- namics. The pion loop finite-T corrections to the quark condensate are actually more important than those of the quark loop, because of the small pion mass. The inclusion of the pion loop terms in the NJL model should give the low-Tcorrections of the quark condensate as in ( 1 ).

Next, consider the baryon number susceptibility XB, which is not affected by the pion loop correction, at least, in the low-T region. Expanding the logarithm in (19), we can easily extract the low-Tbehavior of the susceptibility Z/~:

2 1 / 2 • 40 (3Mq) ( T

Z7;°~-27 ~ \ 6 ~ } exp(-3M~/T)

+ .... (20)

Comparing (20) with (4), we find that 3Mq plays the role of the baryon mass MN, SO that the projected susceptibility Z~ ~°j has the correct suppression factor e x p ( - 3 M ~ / T ) ~ e x p ( - - M N / T ) , and this low-T be- havior is not modified by the pion loop effects. How- ever, (20) has a factor 40 _~, instead of 8, which is the number of nucleon and anti-nucleon states. Since the "baryon" and "anti-baryon" states in the projected NJL model correspond to the ones in the SU(2)~pm×SU(2)n .. . . quark model in which the nucleon and the delta are degenerate, the 40 degen-

erate states of baryons and anti-baryons contribute to the baryon number susceptibility at low T. Another extra factor ~7 comes from the fact that the Boltz- mann factor for the projected "baryon" state is not exp[-flxfp2+(3Mq) =] but e x p ( - 3 f l ~ ) . Scaling the momentum p-,p' = 3p, we can write the Boltzman n factor as exp [ - flx/p '2 + (3Mq) 2 ] and this scaling changes the phase space d3p~d3p ' /27. In fig. 2, we show the numerical calculation of the baryon number susceptibilities as functions of temperature T. The solid line denotes the projected susceptibility X~ r°j, whereas the dashed line gives the non-projected one, Z~ JL. We also show the susceptibility of the free nucleon gas X~ ad by the dotted line. All the suscepti- bilities vanish in the zero temperature limit. When the temperature increases, X~ JL starts to grow around T~ 50 MeV but X~ r°j and X~f d stay almost zero up to T~ 80 MeV. This difference is a consequence of the difference of the modes which carry the baryon num- ber. The deviation between X~ r°j and X~ ad in the low- T region comes from the different numbers of de- grees of freedom and the difference in their masses. While the nucleon mass My is 940 MeV, the mass of the three constituent quark state is 3Mq= 1005 MeV in this model. We conclude that the color singlet pro- jection improves the low-Tbehavior of the suscepti- bility. But the high-T ( T> 200 MeV) behavior of the free nucleon susceptibility differs from the one of the projected or unprojected NJL model. XhB ad diverges at high T, but X~ r°j and Z~ JL do not. This is because the

6000 l MeV~] / . i ~-- ~ ~--.~.~..

4500 /

xNJL / ."'/""

3oo0 i I / xhcld

,,,,, / XpcoJ 1500 / ~

i 0 100 200 300 400

T IHeVl Fig. 2. The baryon number susceptibilit ies ZB as functions o f the temperature T. The projected susceptibil ity X~ r°j is denoted by the solid l ine and the non-proJected one Z~ JL by the dashed line. The dotted line denotes the susceptibility of the free nucleon gas X~ a~.

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Volume 269, number 1,2 PHYSICS LETTERS B 24 October 1991

N J L m o d e l has a cutoff , so that the suscept ib i l i ty in

the N J L mode l does not d iverge . We cons ide r this

d i sc repancy as meaningless , s ince ne i the r the N J L

mode l nor the free nucleon gas are reliable in the high-

T region.

In summary , we have inves t iga ted the baryon

n u m b e r suscept ib i l i ty XR in the N J L m o d e l wi th color

singlet pro jec t ion . Since the suscept ib i l i ty z , at low T

is expec ted to be a lmos t i n d e p e n d e n t o f the m e s o n

( p i o n ) loop effects, we can ob ta in the rel iable T-de-

pendence o f Z . wi th in the one - loop a p p r o x i m a t i o n ,

at least in the l o w - T region. F r o m the l o w - T b e h a v i o r

o f Z~ in the N J L mode l , we conc lude that the co lor

singlet p ro jec t ion is a useful and pract ical dev ice to

r e m o v e the ar t i f ic ia l t he rma l qua rk states f r o m the

the rmal average in the " N J L - l i k e " quark models ,

which have no con f inemen t .

The au tho r thanks W. Weise for helpful discus-

s ions and read ing o f the manusc r ip t . T h a n k s are also

due to H. Yabu and M. Tak izawa for helpful discus-

sions.

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