Chiral thermodynamics in the 1N expansion

Physics Letters B 311 (1993) 213-218 North-Holland

PHYSICS LETTERS B

Chiral thermodynamics in the 1/N expansion H . M e y e r - O r t m a n n s , H . J . P i r n e r i a n d B.-J. S c h a e f e r

InstitutJ~r TheoretischePhysik, UniversitdtHeidelberg,Heidelberg,FRG Received 3 April 1993; revised manuscript received 13 May 1993 Editor: P.V. Landshoff

We calculate the partition function of the O (4) nonlinear a-model with explicit symmetry breaking to lowest order in 1/N(N= 4 ). Fixing the parameters at temperature T= 0 to reproduce the observed pion mass m~ and the pion decay constant f~ we calculate the pressure, the energy density and entropy density as a function of the temperature. With PCAC we also determine the order parameter ((lq(T)) ofchiral symmetry breaking. At low temperatures T< 100 MeV our results coincide with those obtained in chiral perturbation theory in the three loop approximation.

Relativistic n u c l e u s - n u c l e u s collisions with CM

energiesEcm~ 20 G e V / n u c l e o n create large systems o f sizes R > 20 fm at freeze-out with > 104 pions [ i ]. It is natural to try statistical m e t h o d s to describe such hadronic fireballs. A good starting point m a y be to use e q u i l i b r i u m t h e r m o d y n a m i c s with pions when one is interested in the later stages o f these collisions, where the t e m p e r a t u r e is below possible phase transitions [2]. The low energy interaction o f pions is fully d e t e r m i n e d by chiral s y m m e t r y [3 ]. It can be parametrized by the nonlinear sigma model O ( 4 ) [ 4 ] with one scalar ( 6 ) and three pseudoscalar n-fields, which are constrained by the c o n d i t i o n 6 2 + n 2 = f 2, w h e r e f , is the pion decay constant. The experimental challenge is to measure the equation o f state o f p i o n s from the inclusive pion spectra. The theoretical task is to calculate this equation o f state. F o r this purpose we need reliable techniques to treat field theories at finite temperatures. One can then extrapolate from the m e a s u r e d physics at T = 0 to the yet unknown physics at high temperatures. A very accurate t r e a t m e n t o f the soft m o d e s with lowest mass is essential at low temperatures. The m e t h o d we are proposing treats the effective potential in terms o f a selfconsistent field, which in lowest o r d e r 1/N represents an effective mass for the unconstrained (6,

7t) fields. A generalization to theories without the expansion p a r a m e t e r 1 / N is discussed in ref. [ 5 ]. The partition function ~ for the 0 ( 4 ) nonlinear 6-model is given in terms of the multiplet (no, n) = (6, n ) with n z = 6 2 q - n 2 as

~=~ ~n(x) 1-['~(n2(x)-f~) x

p Xexp(-f

d ~ f d3x[½(~un)2-Cno]). 0

(1)

V

At zero temperature T = 1//~= 0 the parameters o f the m o d e l are well known. The pion decay constant f~ equals to 93 MeV. The classical v a c u u m expectation value o f no is d e t e r m i n e d as ( n o ) =f~ by m i n i m i z i n g the v a c u u m energy. Expanding the d e p e n d e n t field n o = x / / ~ - n 2 to leading order in n 2, one obtains the mass o f the pion as m 2 =c/f~. The basic idea o f our m e t h o d is to postpone the t r e a t m e n t o f the nonlinear n - d y n a m i c s to the end. Therefore, we eliminate the nonlinear constraint r t z = 6 z + n 2 = f 2 n by the introduction of an auxiliary field 2 (x) via an integral along the imaginary axis [ 6 ].

J Supported by the Bundesministerium f'tir Forschung und Technologie BMFT contract 06HD729. 0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

213

Volume 311, number 1,2,3,4

PHYSICS LETTERSB

a + i oe

,8

a--i~

0

that the topology of a graph determines its order in 1/N. Thus a saddle point approximation is most convenient, if one is interested in a large N expansion. The lowest order saddle point automatically gives the leading contribution to a 1/N expansion. Since in our case N = 4, we have to be aware of subleading corrections, which may not be negligible. In the present paper we consider only the leading terms. For constant 2 the saddle point approximation leads to a partition function of a free relativistic Bose gas with N = 4 components and effective masses

j ga(x) n(x)exp(-fd

V After shifting the zeroth component no to no ~=(~o,~) =

no- ~2,n ,

we obtain a Gaussian action for the O(N) multiplet field h, when we evaluate eq. (2) in a saddle point approximation. The resulting partition function is given as

m~fr=22,

(5)

Se = e x p [ - fl V

(

B

.--f

29 July 1993

× UT(m~fr, A)+Uo(

exo[-I d.

2me~,A)-2f]-~

(6)

0

×fd3x(½(Ou~)2+2~2-2f~-~-2)].

(3)

V Here we have dropped the X-integration and chosen 2(x) =2=const. Upon Gaussian integration over the four ( N = 4 ) ~-fields, it follows from eq. (3) that

UT( m~rr, A ) A

i f d3k =47j ° (-~)31n[1-exp(-flx/~+m~rr)],

and Uo the zero point energy

0

Uo(m~fr, A)=4 f (2rc)3 d3k ½x/kx/kx/kx/kx/kx/kmZrr x~

(7)

.4 0 'n4ff~

(4)

V The optimal choice of 2 will be determined later, cf. eq. (16). Evaluating eq. (2) in the saddle point approximation corresponds to the leading order in a 1/Nexpansion. It may not be obvious that the intrinsic, small parameter of fluctuations around the saddle point in 2 is 1/v@, as we have introduced the auxiliary field for eliminating the nonlinear constraint. But we know from the work of Coleman et al. [ 7 ] that the introduction of an auxiliary field in O(N) field theories with scalar fields ¢ amounts to a new diagrammatic expansion with new vertices and propagators for 2and 0-fields. This expansion has the advantage over the original Feynman graph expansion for q~-fields 214

Here UT denotes the contribution from thermal fluctuations

p

× I d3x(NTr log(-02+ 22)-2f 2-~2)1.

.

A

= 4 4n 2 (4mefr

~(

A2"~ 3 l+

mZefr]

-g + ln(\meff A--d-+ X/1 + ~2~2 A2 )3 } .

(8)

We regularize the k-integrations in eqs. (7) and (8) with a cut-offA, since we do not believe our x-effective theory to be correct for momenta beyond A. At momenta k> A the compositeness of the pions manifests itself in resonance excitations and/or higher derivative couplings of the pion states, which are neglected. For the numerical calculations we take cutoff values A = 700 MeV, 800 MeV, 1000 MeV.

Volume 31 l, n u m b e r 1,2,3,4

PHYSICS LETTERS B

After regularization we adopt the following renorrealization procedure. We define a renormalized potential at arbitrary Taccording to u r e n ( m eft 2

-

A)=UT(m2eff, A).4_ Uo ( m eff 2 ,A )

( Uo(m2, A)+(m2rf-m~)~--2SS-20Vol

Om~ff lm~]

29 July 1993

~(A, T)=expI-flV X{urentm 2

.

(9)

The two subtractions guarantee the two renormalization conditions at T = 0 2 2 m e f r ( / 1 o ) --=--m . ,

(10)

=f~,

(11)

c2

¢.~*

where 2*(T) extremizes lnff at a given temperature T # 0. The saddle point equation for/l* is solved numerically, since in the interesting temperature range the relevant parameter mcff/T= x ~ / T c a n have all values 0 ~ behaves as a function of temperature. In fig. 1 we present the result for

where/1o extremizes the exponent of Se at T = O, i.e. 0/~

--

uren

2

2

(rn,ff(/1),A)-2f ~ - ~

X=~o= 0 .

(12)

Eq. (12) reduces to 0Uo ~=~o 02

0m~ff 0Uo 02 Om~ffm~- f ~ +

c2

~0

~

(13)

i.e. c

/1o'-2f,

(14)

in agreement with eqs. (5) and (10). It can also be easily checked that the expectation value of ( no > remains at its classical valuef, at T = 0

1 0In Se =f~. - flV Oc I~=~o

(15)

It is well known that the nonlinear sigma model is not renormalizable in four dimensions. Therefore higher order divergences can only be compensated by higher order derivative terms in the original action. The coefficients of these higher order terms have to be determined by experiment. We do not include such terms (cf. ref. [2] ). A strong cut-off dependence of our resulting thermodynamic functions signals the necessary extra experimental input. In further work [8] we will extend the calculation to the linear ormodel ( S U ( 3 ) ® S U ( 3 ) ) which contains higher masses and strange mesons. The thermodynamic observables at finite T are obtained from the partition function

(no(T) > (no(T=0))

0 In Se(T) /0 In S e ( T = 0 )

0c

/

0c

In quark language this ratio corresponds to the ratio of the quark condensate < @ ( T ) > at finite temperature over the quark condensate at T = 0 , since the symmetry breaking term of the 0 ( 4 ) Lagrangian ~ss=cno equals the symmetry breaking term f s s = - 2 m @ in the QCD Lagrangian. The result for ( q q ( T ) ) / ( @ ( T = 0 ) > is rather insensitive to the cut-off. It agrees well with the three loop calculation of ref. [ 2 ]. Chiral symmetry is only very gradually restored. At low temperature the zrTrinteraction is weak and (qq> does not change very much. In fig. 2 we show the energy density u ( T ) / T 4, the pressure P / T 4 and the entropy density s ( T ) / T 3 a s function of the temperature T u(T)=

1 0 V0flln ~ ,

s(T)=

VOTk, fl lnSe '

(18)

P(T)=-

~vln ~=Ts-u.

(19)

(17)

)

The energy density and pressure are defined to be zero at T = 0 . It is interesting to recall the results of the thermodynamical functions for a massless pion gas: u = ~ r 2 T 4, s=~Tr2T 3 and P=]u. One sees in fig. 2 that around T = 100 MeV the values of u, s, P for the interacting pion gas coincide with the values of a 215

Volume 311, number 1,2,3,4

PHYSICS LETTERS B i

i '

'[' ']'

1.0

29 July 1993

'

I

. . . .

I

'

'

'

0.9 A = 700 MeV
"6

0.8

Io" V

A = lOOO MeV ~

A

~

G

-

Io-,

\\

o.7

V

÷

\ ~

x

0.6

0

'

'

, , I

. . . .

I . . . . 100

I

. . . .

150 T e m p e r a t u r e [ldeV]

50

I , 200

,

,

Fig. 1. The order parameter for chiral symmetry breaking ( Clq( T ) ) / (Clq( T= 0 ) ) for two different cut-offs A = 700 MeV (solid line) and A = 1000 MeV (dashed line). For comparison with the three loop results of chiral perturbation theory we include a few selected points (crosses) from a curve of Gerber and Leutwyler [ 2 ].

1.50 . . . .

L.25

I

. . . .

_

l

. . . .

I

s(T)/T a

[ .00

0.75

0.50

--

/~//

/

/

//'/ 0,25

0.00

50 tO0 T e m p e r a t u r e [MeV]

150

Fig. 2. The entropy density s~ T x, the energy density u~ T 4 and the pressure P~ T 4 of the interacting pion gas (solid lines) and of a free pion gas (dashed lines). The cut-off is A = 700 MeV. 216

Volume 31 I, number 1,2,3,4

PHYSICS LETTERS B

. . . .

I

. . . .

[

/~

1.0 /

~

. . . .

29 July 1993

I

t --

. . . .

I

'

'

'

~ A = tO00 MeV _

~

\

-, A

=

800

MeV

0.8

1

e

170f 0

0.6

0.4

f ~ ] JAt=

0.2

0.0

¢ ¢-

0

i

50

100 150 Temperature [MeV]

200

Fig. 3. The energy density u~ T 4 as a function of temperature for A = 700 MeV, A = 800 MeV and A = 1000 MeV. massless three c o m p o n e n t gas. At low t e m p e r a t u r e s the finite pion mass leads to an exponential suppression o f the excitation o f pionic modes. F o r c o m p a r i s o n we also show in fig. 2 the t h e r m o d y n a m i c functions o f the noninteracting pion gas with mass m~. Because o f the attractive interaction which becomes i m p o r t a n t at higher t e m p e r a t u r e s T > 9 0 MeV, the energy density and the pressure are larger for the interacting system. Describing the interacting pions as a system o f quasiparticles with energies oJ* ( k ) , one obtains for the energy density u(T) =3

" d3k co* ( k ) (2zr) 3 exp [floJ* ( k ) ] - 1

(20)

One sees that with a lower effective energy 09* ( k ) the density o f states increases. This effect o v e r c o m p e n sates the effect o f a r e d u c e d oJ*(k) in the n u m e r a t o r o f the integral for the energy density u ( T ) . The energy density is rather sensitive to the choice o f the cut-offA. Because o f the increase o f rn~fr with T the average thermal m o m e n t a also strongly increase with temperature. A l r e a d y at T = 1 0 0 - 1 2 0 M e V different cut-offs A p r o d u c e different curves u ( T ) / T 4 for the energy density as shown in fig. 3. The energy density u ( T ) / T 4 changes by + 10%, i f A is increased from 700 M e V to 1000 MeV. At high

temperatures u ( T ) / T 4 decreases as a function of temperature, because the cut-off does not allow the phase space for the thermal pion modes to grow oc T 4. In reality other modes like K,/~, p, etc. produced by nn-interactions b e c o m e relevant a r o u n d these temperatures. It will be especially interesting to study the pion p r o p a g a t o r at finite temperature. In the quasiclassical approximation the four independent (a, ~ ) modes have an effective mass meff=22 . Due to q u a n t u m corrections the pion will develop a different screening and d y n a m i c a l mass. It has been speculated that the pion propagator leads to a dispersion relation with a roton like behavior [ 9 ] similarly to 4He. This will be the subject o f further studies [8 ].

2 *

References [ 1] H.R. Schmidt and J. Schukraft, The physics ofultrarelativistic heavy ion collisions, Darmstadt, GSI-CERN preprint GSI92-19 (March 1992). 12 ] P. Gerber and H. Leutwyler, Nucl. Phys. B 321 (1989) 387. [3] B.W. Lee, in: Chiral dynamics (Gordon and Breach, New York ); S. Weinberg, Phys. Rev. Lett. 18 (1967) 188. [4] B.W. Lee and T.H. Nieh, Phys. Rev. 166 (1968) 1507. 217

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PHYSICS LETTERS B

[5] G. Amelino-Camelia and S.-Y. Pi, Self-consistent improvement of the finite temperature effective potential, Boston University preprint BUHEP-92-26 (November 1992). [6]A.M. Polyakov, in: Gauge fields and strings (Harwood Academic, New York) pp. 125ff.

218

29 July 1993

[7] S. Coleman, R. Jackiw and H,D. Politzer, PHys. Rev. D 10 (1974) 2491. [8] D. Metzger, H. Meyer-Ortmanns, H.J. Pirner and B.-J. Schaefer, work in progress. [9] E.V. Shuryak, Phys. Rev. D 42 (1990) 1764,