Dependence of Kπ ratio on temperature and chemical potential as a signature of quark-gluon plasma

PhysicsLetters B 278 (1992) 357-362 North-Holland

P HYSIC $ LETT ERS B

Dependence of K/n ratio on temperature and chemical potential as a signature of quark-gluon plasma Saeed U d d i n a and C.P. Singh ~,b a Departmentof Physics, Banaras Hindu University, Varanasi221 005, India b lnternationalCentrefor TheoreticalPhysics, 1-34100 Trieste, Italy Received 30 May 1991

We suggestthat the variation of the ratio nK+/n~. with the thermodynamicalintensive variables, e.g., baryon chemical potential and temperature shouldbe studied in high energycollisionsof two heavynucleiand a changein the variation of this ratio can be used to distinguish between the quark-gluon plasma and the hadronic contributions.

1. Introduction

It is widely believed that the ultrarelativistic nuclear collisions can lead to the formation of a new state of matter called quark-gluon plasma ( Q G P ) where coloured quarks and gluons get deconfined and behave almost as free particles [ 1-12]. Therefore it is crucial to find an unambiguous signature of such a phase transition. Recently we have suggested [ 9 ] that the variation of the ratio n ~ . / n K - and no/no with baryon density (or rapidity) a n d / o r with temperature can serve as a potential probe for detecting the QGP formation because it discriminates clearly the signals arising from the QGP and the resulting hadronic phases. In this paper we propose a similar signature by studying the variations of nK+/n~÷ with baryon chemical potential/~n a n d / o r with temperature T. Our analysis reveals that this ratio increases rapidly as/tB increases if the hadronic matter undergoes a phase transition. However, if the matter does not suffer a phase transition, this ratio decreases very slowly. This clearly depicts a marked difference in the behaviour of the matter with and without a QGP formation. The abundance of strangeness in ultrarelativistic nuclear collisions has long been believed to provide a signal for the production of quark-gluon plasma. In particular, it has been suggested that the relative or absolute abundance of strange particles, e.g., the ra-

tio nidn~, can give a signal for testing the QGP formation [ 12-17 ]. Recently, the data [ 18-20 ] from the Brookhaven National Laboratory AGS machine from the reaction of Si on Au at an incident energy of 14.5 GeV/nucleon reveal nK+/n~÷ ratio as about 20% and nK-/n~- as 5%. The expected ratio for both nK÷/n~÷ and nK-/n~- is about 5% from p - p or p-~ collisions around the same energy. Thus the factor of 4 enhancement in the ratio nK+/n~+ has given us an optimism about the formation of the quark-gluon plasma. Recently Xia and Ko [ 16,17 ] have used an expanding fireball model to determine nK÷/n~÷ as 23% and nK-/n~- as 5%, respectively, and assumed the baryon-rich fireball to be in thermal but not in chemical equilibrium. They thus conclusively support the formation of the QGP in the above experiment. However, the prediction of a large value of nK/n~ as a signature of QGP has theoretically been quite controversial until today. The controversies exist on how to properly compare the two phases and whether proper consideration was given to the conservation laws for baryon number, strangeness and entropy etc. in the dynamical evolution of the system. In particular, assuming the entropy conservation during hadronization after QGP has been formed, it was suggested [21,22] that the entropy carried by gluons can go into extra pions resulting in a small nidn~ ratio. In such a confusing situation we suggest in this paper that instead of concentrating on

0370-2693/92/$ 05.00 © 1992Elsevier SciencePublishers B.V. All rights reserved.

357

Volume 278, number 3

PHYSICS LETTERSB

the absolute abundance of the ratio nK/n,~ in the relativistic heavy ion collisions in comparison to the hadronic collisions, we should study the variation of the ratio nK+/n,~+ with baryon chemical potential/tB, and/or temperature T. It can thus provide a potential signature for the QGP formation in the ultrarelativistic collisions of two nuclei because these have a marked different behaviour for a chemically equilibrated hadronic gas and the QGP. Thus this signal discriminates the QGP contribution from the hadronic background contribution.

26 March 1992

For bosons q= - 1 and for fermions r/= + 1. Here m is the particle mass, the fugacity 2 = exp (/z/T) with /z as the chemical potential and g the degeneracy factor. The average multiplicity of particles of type i in the grand canonical system can be calculated as 01nz (2)

/V-----'~i ~ i

Using the Boltzmann approximation for massless light quarks q we can write l n Z q = ~2V f d k k 3 2 q e x p ( - k / T ) .

2. nx/n,, ratio from Q G P

Thus the light quark density can be calculated as

Once a QGP is formed, the gluonic production rate dominates the process of strange quark pair production and the time needed to saturate the phase space is almost comparable and even smaller than the estimates of plasma lifetime based on free hydrodynamic expansion [ 13 ]. We assume that the QGP system expands and cools so rapidly that the strange quarks and antiquarks do not annihilate to any significant extent and thus the hadronization of the plasma does not upset the relative abundance of strangeness. Thus our assumption implies that the strangeness abundance freezes out and does not annihilate. Using this simple hadronization scheme we can calculate the nK+/n~+ ratio in a Q G P as follows: / = / = / and similarly

(nK- >/ = / = / . Here nq (n~) indicates the density of light quarks (antiquarks) and ns (n~) the density of strange (antistrange) quarks. Since strange-antistrange quarks are produced in pairs only, therefore < ns > = < n~ >. For calculating the number density of particles we make use of the statistical mechanics formalism and write the partition function for a grand canonical ensemble as follows [ 11 ]:

= -12 ~ T 3 exp(flq/T) ,

(3)

where the quark chemical potential/.~q is related to the baryon chemical potential aB as aq = ~/% The light antiquark density n~ can be obtained by simply replacing/Zq by - / t q in eq. (3). Using ( l ) and (2), we can obtain the strange (antistrange) quark density as follows: 2

( ns >=
(4)

From eqs. (3) and (4), we can get the ratio nK+/n,~+ for the QGP system as follows:

m 72 K2(ms/T) exp(/tq/T), nK +/ n~ += ~1 -T-

(5)

where ms is the strange quark mass and 1';2 is the modified Bessel function of second kind. It is obvious that for nK-/n,~- we have to substitute -/~q for /tq in order to incorporate the antiquark density. It can be stressed here that in the above analysis we have assumed that thermal kaons emerging from the Q G P suffer only elastic scattering and thus the expansion of the system does not alter the number densities of K + and K - , respectively.

3. nK/n,, ratio from hadronic gas In Z =

gV 61t2T

× ~ d k k 4 {2 - l exp[(k2+m2)l/2/T] +t/} - t (k2+m2)l/2 J 358

(1)

For the hadronic gas we assume that the total partition function is additive for strange and nonstrange parts if we consider a chemically equilibrated hadron

Volume 278, number 3

PHYSICS LETTERSB

gas as a mixture of various noninteracting resonances. We thus write

26 March 1992

Therefore, the effect of increasing PB will be to enhance the ratio nK+/n,~÷.

In Z = In Z . . . . ~range+ In Z strange. The abundance of the strange particles can be obtained by considering In Z strange only. We can define [14] the grand canonical partition function for strange particles with s = 1 in the Boltzmann approximation as In Z strange

=C[2W(XK) + 6W(XK*) ] [,~.q,~.sI +~-q l~-s] +C[2W(xa)+6W(xz)][22q2s+2~z2~ -1 ] ,

(6)

where C = VT3/2/[ 2,

x=m/T,

W ( x ) =X2KE(X) ,

nK/n~

The nK+/n,~+ ratio from the hadron gas is strongly affected by the contributions from the resonance production and their decay after freeze-out. These decays produce mostly additional pions. In order to make the calculation more realistic, we show the effects of A(1236), p(765) and K*(892) production and decay. The equilibrium density of the A resonance can be obtained from the partition function given in eq. ( 1 ) in the Boltzmann approximation:

gaT 3 n,x = ~ (ma/T)EKz(ma/T) e x p ( # B / T ) ,

2q=2~/3=exp(#q/T).

Since the hadronic matter is with a zero net strangeness, we can thus derive the expression for the strange quark fugacity as ;t~ =y3.q,

4. Effect of resonances on

(7)

where

2 ([W(xx)+3W(xK')l+'~ff'[W(xA)+3W(xz)]'~'/2 Using eqs. (2) and (6) we get the number densities of K + and K - mesons as

(10)

where ga is the degeneracy factor for the A resonance. From eq. (10) we find that the A number density is large at large baryon chemical potential/~B and, therefore, the number of decay pions also increases with #B- Assuming that the A resonance decays after freeze-out, we can calculate the correction to the ratio nx+/n,~+ from decay n + arising due to A + +--,pn + and A + - , n n + decays. Pions can also appear due to p-meson decay. The equilibrium density of the p meson can be obtained as 2

nK÷ =

nK_

=

K(mK/T)2K2(mK/T)

,

(8a)

~.r~(m~/r)2g~(mK/r)

.

(8b)

Here the fugacities are defined according to the quark contents of the hadron species, i.e., 2K= 2 q 2 ~ - l = l / y and 2~=2~-~2s=~,. The thermal pion density can be written [ 12 ] as

3T3 ( m ~ 2 n~ = -f~n2 \--~- J [K2 ( m J T) + ½K2 ( 2 m J T) + ~K2( 3m,dT) ] .

nt>=~(~-~)2= K2(mplT) 2

,

(11)

where go is the degeneracy factor of the p resonance. The main decay channels contributing to the n + abundance are p°--,n+n- and p + ~ n + n °. Similarly K ÷ and n ÷ can also arise from K*(892) decay. The number density for nK* can be calculated from eqs. (2) and (6) as follows: 3T3 2 nK*-- ---~'-2K*(mK*/T) K2(mK./T) •

(12)

(9)

The quantity y as is obvious from eq. (7) decreases with increasing gB at fixed temperature T. We, therefore, expect that nK+ should increase and nK- should decrease, respectively with increasing #a at fixed T.

5. Results and discussion

In fig. 1 we have shown the contributions for the ratio nK+/n,~+ from the hadronic gas and the Q G P 359

Volume 278, number 3

..... .....

3.0

+

PHYSICS LETTERS B

experimental data o f Abbott et al. [20] can be assumed to correspond to similar baryon rich matter with T a n d / t 8 falling in the above region and give an nK+/n,~+ ratio 0.24+0.05. Therefore, our calculations do not favour the possibility o f Q G P formation in the above experiment. In fig. 2 we have illustrated the variation of the ratio nK+/n~+ with temperature T at fixed values of Pn = 200, 600 and 900 MeV, respectively. We find that in the hadronic gas the ratio nK+/n,~+ increases with temperature. Although the increase is slow particularly at small/zB, its increase becomes more rapid as /zs increases. On the other hand it is seen that if the matter undergoes a phase transition and a Q G P is formed this ratio decreases very rapidly with increasing temperature particularly for larger values of/zs. The fall of the ratio is less rapid and becomes almost constant at small/zB. In fig. 2, we have also shown the result o f our calculation for the ratio nK÷/n=+from the hadronic gas when K*, p and A resonance corrections have been incorporated. We again find a significant modification in our results since the values are not only reduced in magnitude but the ratio nK+/n,~+ becomes almost constant with respect to temperature.

QGP Hodronicgas with resononces Hodronic gas without resononces

2.5

,K~./ . / /

T-200 MeV

o# 1.5

T=2oo,ev

\

1.0 .

.

.

.

.

.

.

.

0.5 ----

-,L~T= 160 M e V

- - -- = = ------.:?.~

,;o ,do ,;o lab (MeV)

7;o "

nK./n,,,

Fig. 1. Variation of the ratio with baryon chemical potential Pa for various values of temperature T. The solid curve represents the contribution of QGP. The strange quark mass M~ has been taken as 150 MeV. The dash-dotted curve gives the contribution of hadronic gas without resonances. The dashed curves are for the hadronic gas with K*, p and A resonance decay correction.

and we have demonstrated the variation of this ratio with the baryon chemical potential/~a at a fixed temperature T. In our calculation ~tB is related with the baryon density o f the evolving system and hence it is also related with the rapidity variable. Similarly the other thermodynamical variable used here is the temperature T and it is related with the initial energy density of the matter formed. We see that the ratio nK+/n~+ increases with increasing Ps for QGP. It is almost comparable in magnitude and increases for the hadronic matter without any resonance decay corrections. In fig. 1, we have also shown the ratio nK+/n~+ from the hadronic gas when K*, p and A resonance production and decay corrections have been incorporated. The behaviour obtained for hadronic gas as shown in fig. 1 is quite similar to that obtained by Cleymans et al. [ 23 ]. The contrast in the behaviour becomes more transparent because the hadronic ratio nK+/n~+ now slowly decreases as/tB increases whereas in the Q G P case, this ratio rapidly increases. From fig. 1 we find that the ratio nK+/n~+ from the Q G P phase is 0.5-1.0 at 160 < T < 240 MeV. However, its value for the hadronic gas involving resonances lies approximately between 0.26-0.30 up to/2a ~ 150-500 MeV. The BNL 360

26 March 1992

- - Q G P ..... Hodronic ..... Hodronic

gas with resonances gas wit hout resonances

3.0

I

.,....~ ,~

2.5

PB . 9 0 0 M e V

+

t-2. I

#

_..-."- NB = 6 0 0 M e V

1.! 1.0

..I.1./*/

~

UB=6OOMeV . ....- UB = 2 0 0 H E Y , .,.-. ~"

0.5

~

,.,.. ~ " ~ ~. ~" ~'=~= ................... I

160

I

200

T (MeV)

UB = 200MeV

r_..__lala=2OOMeV _ L _ - - IJ~=6OOMeV 1--- pe=gOOMeV I

I

240

280 -

Fig. 2. Dependence of the n~./n,,, ratio on Tfor fixed values of /ta = 200, 600 and 900 MeV. The solid curves represent the QGP contribution and dash-dotted curves give the contribution of hadronic gas without resonances. The dashed curves are for the hadronic gas with K*, p and A and resonance decay correction.

Volume 278, number 3

PHYSICS LETTERS B

We hope that the rapidly increasing behavior of nK+/n~ + with respect to the baryon chemical potential Ira at a fixed value of temperature when the QGP is formed but its slightly decreasing behaviour for the hadronic gas phase without a phase transition to QGP can provide a clear and unambiguous signal for detecting this phase transition. Similarly the rapid decrease of nK+/n~+ with increasing temperature at /ta ~ 900 MeV for the Q G P case and its slightly increasing or almost constant values for the hadronic gas can provide a confirmatory test for Q G P detection. The reason for this entirely different behaviour can be understood as follows. In QGP, the ratio nK÷/n,~÷ represents effectively the ratio of antistrange (strange) quark density to the light antiquark density. Since strange-antistrange quarks are produced in pairs to conserve strangeness, they maintain zero strange chemical potential i.e., #s= 0 and their density is controlled by the temperature T alone. However, as temperature increases, the equilibrium density of massless light antiquark increases more rapidly than that of the massive s or g quark whose current quark mass is ms = 150 MeV. Similarly at large quark chemical potential/lq ( = ~/IB), the light antiquark density is suppressed because of the Pauli blocking [12]. Consequently, the ratio nK÷/n,~+ arising from the quark matter is quite large at large #a and decreases rapidly with temperature. The hadronic gas behaviour is, however, slightly complicated. Here the strange baryons play an important role by providing a background chemical potential to the strange mesons. When the temperature increases at a given baryon chemical potential/~a the density of all particles including that of kaons and pions in a hadronic gas increases. However, as strange baryons become more abundant the conservation of strangeness requires that more K + ( = q g ) mesons be present. Moreover, at sufficiently large/ts, the strange baryons A, E, etc. are more abundant than A, E. This further results in a large K ÷ density and it increases with temperature. Moreover, the contribution from resonance production and decay strongly modifies the resuit. We find that their effect is strongly temperature and/ta dependent. This shows that at large T and/za, the pions are mostly decay pions. However, our resuits indicate that this can be utilized to separate the information on the Q G P behaviour from that of the hadronic background.

26 March 1992

Recently it has been pointed out be Cleymans et al. [ 23 ] that if the expansion from an initial plasma to the final hadronic state takes place as equilibrium process, then at freeze-out we have an equilibrated hadronic gas. The observable particle ratios in this case are just those of an equilibrated hadronic gas. In this paper, we have avoided this by assuming a specific nonequilibrium expansion of Q G P and we postulate the freeze-out of the strange quark sector already in the initial plasma state. Thus we have ignored the role played by gluons in the subsequent fragmentation process as qcl pair creation. In fig. 3, we have shown the variation of the fraction of the initial energy density carried by gluons Re= eg/(e~+ eq + e~) with baryon chemical potential #~. Here eg denotes the energy density of gluons in QGP, eq (e~) is the energy density carried by light quarks (antiquarks), respectively. We find that R, decreases very slowly as #B increases. If we assume that energy density carded by gluons goes mostly into pions after hadronization, then the behaviour of the ratio nK +/n~ ÷ as shown for QGP in fig. 1 will not change at all and thus the proposed signature will still survive. We have also shown in fig. 3 the entropy ratio R,=sg(Sg+Sq+S~) carded by gluons where sg is the entropy density carried by gluons, Sq (Sq) is that of light quarks (antiquarks). In conclusion, our suggestions on the signals of the

T= 240 Mev

0.7

--0.6

Froction of 9[uon entropy d e n s i t y Froction of gtuon energy density

0.5 0,4 ¢,r

0,3 0.2 0.1 ,

,

~oo 200

'

3 o 400 soo 6 o 700 ~ue (MeY)

e o 9 o looo

Fig. 3. Variation of the fraction of initial energy density carried by gluons as measured by the ratio R = ~ / ( eg+ eq+ eq) with baryon chemical potential shown by a solid line. Similarly variation of fraction of entropy density in the gluon sector of QGP with/t. is shown by a dashed curve.

361

Volume 278, number 3

PHYSICS LETTERS B

Q G P f o r m a t i o n can be put as follows. The variations o f the ratios nK+/nK-, n,/n,o (as discussed in refs. [ 8,9 ] ) a n d nK÷/n~ + either with t e m p e r a t u r e or with baryon chemical potential/tB can serve as unique signature for detecting the Q G P formation because their increasing b e h a v i o u r in one a n d the decreasing behaviour in the other can discriminate clearly the signals arising from the Q G P and the resultant hadronic phase. However, the picture presented here is a simple one. In nuclear collisions whether or not the Q G P is formed, the h a d r o n i z a t i o n is still a c o m p l i c a t e d p r o b l e m which is not under control due to its essential nonperturbative nature. The simple use of a grand canonical partition for the calculation o f the equilibrium ratio nl,:+/n,~+ m a y also be questionable. We have ignored the role o f fragmentation and recombination processes involving gluons which produce additional qcl and sg quark pairs during h a d r o n i z a t i o n and thus the nK+/n~+ ratio m a y be strongly affected. However, it is obvious from fig. 3 that the p r o p o s e d signal will still survive even after incorporation o f this correction in our model. Thus in the absence o f any unambiguous signal our suggestions will help in the detection o f QGP.

Acknowledgement S.U. is grateful to the University G r a n t s C o m m i s sion for providing a Senior Research Fellowship. One o f us (C.P.S.) is grateful to S A R E C for an award of Associate Membership and to Professor Abdus Salam for the hospitality at ICTP. He is also grateful to Pro-

362

26 March 1992

fessor H. Satz and J. Schukraft for m a n y stimulating discussions and hospitality at CERN.

References [ 1] E.V. Shuryak, Phys. Rep. 61 (1979) 71. [2] L.D. McLerran and T. Toimela, Phys. Rev. D 31 (1985) 545. [ 3 ] T. Biro and J. Zimanyi, Nucl. Phys. A 395 ( 1983 ) 525. [4] L. Van Hove, Phys. Lett. B 118 (1982) 138. [5] M. Jacob and J. Rafelski, Phys. Lett. B 190 (1987) 173. [6] J. Rafelski, Phys. Rep. 88 (1982) 331. [7] A. Shor, Phys. Rev. Lett. 54 (1985) 1122. [8] C.P. Singh, Phys. Rev. Lett. 56 (1986) 1750; Phys. Lett. B 188 (1987) 369. [9] C.P. Singh and S. Uddin, Phys. Rev. D 41 (1990) 870. [ 10] C.P. Singh and S. Uddin, Prog. Theor. Phys. 83 (1990) 77. [ I 1] J. Cleymans, R.V. Gavai and E. Suhonen, Phys. Rep. 130 (1986) 217. [12] P. Koch, B. Muller and J. Rafelski, Phys. Rep. 142 (1986) 167. [ 13 ] J. Rafelski and B. Muller, Phys. Rev. Lett. 48 ( 1982 ) 1066; 56 (1986) 2334(E). [14] P. Koch, J. Rafelski and W. Greiner, Phys. Lett. B 123 (1983) 151. [ 15] J. Rafelski, Nucl. Phys. A 418 (418 (1984) 215. [ 16] C.M. Ko and L. Xia, Phys. Rev. C 38 (1988) 179. [17] L. Xia and C.M. Ko, Phys. Lett. B 222 (1989) 343. [ 18] S. Nagamiya, Nucl. Phys. A 488 (1988) 3c. [ 19] J. Maike and G.S.F. Stephans, Z. Phys. C 38 (1988) 135. [20] T. Abbott et al., Brookhaven National Laboratory report BNL-42194 (1988). [21 ] K. Redlich, Z. Phys. C 27 (1985) 633. [22 ] J. Kapusta and A. Mekjian, Phys. Rev. D 33 ( 1986 ) 1304. [23 ] J. Cleymans, H. Satz, E. Suhonen and D.W. Von Oertzen, Phys. Lett. B 242 (1990) 111; J. Cleymans et al., Nucl. Phys. A 525 ( 1991 ) 205c.