619
HOT HADRONIC MATTER Johann Rafelski Institut flir Theoretische Physik der Johann Wolfgang Goethe-Universitat Frankfurt am Main, Germany
A description of highly excited hadronic matter is presented . I assume the kinetic and chemical equilibrium and develop a thermody namic description valid for high temperatures and different chemical hadronic compositions. Two physically different domains are describ ed : firstly the hadronic gas phase in which individual hadrons co exist as separate entities, though sometimes assembled in large hadronic clusters. In the second domain individual hadrons dissolve into one large hadronic cluster consisting of quarks and gluons - this is the quark-gluon phase. Observation of enhancement in the partial cross-section at subthreshold energies in nuclear collisionsA is discussed as one of possible characteristic phenomena of this phase . 620
1. Introduction A theory of hot hadronic matter is being developed1) based on the present knowledge about strong interactions . Physically most impor tant is the phase transition in which baryons and mesons are dis 2) solved into the quark-gluon plasma • Essential quantity controll ing the occurence of this transition is the energy density of ha dronic matter . Hence the heat and pions created in central high 3) energy heavy ion collisions or low energy p annihilations on nuclei lower the necessary nuclear density to about 1 -3 times normal nu cleon density at temperatures of the order of the pion mass. For lack of time and space I can only discuss the nuclear collisions here ; for a discussion of p annihilations the reader may consult ref. (3) . The theoretical descriptions of both phase s are entirely diffe rent: In the hadronic gas phase1) the hadron-nucleon cross section being dominated by the formation of hadronic resonances, the essen tial physical input is the particle mass spectrum, here derived in the statistical bootstrap model. In the dense domain of the matter it is essential to take into account the finite size and the clustering of individual quark bags . In the quark-gluon plasma , we have to deal with a many body gauge field theory at moderate inter action strength2) . An insight into the behavi.our of the plasma can be gathered studying the Fermi-Bose interacti.ng quantum gases in a large quark bag4) . In order to test our understanding of hot hadronic matter in ki netic and chemical equilibrium and to show the existence of the plasma phase different experiments are neces£>ary . As an initial £>tep I consider the temperature£> of particles emitted in a hypothetical i,5) central fireball created in high energy nuclear collisions . Suitable care must be taken in an experiment to eliminate the con tributions of projectile and target nuclei - here we do not know the model dependent internal excitation . Among the results I dis cuss here a substantial entropy production in the explosion of the fireball is due primarily to the production of new particles. Under certain conditions I also anticipate that strangeness would equili brate - the relative yields of (anti) strange particles can be used as a measure of the size of the reaction zone to be confronted with correlation experiments . If the phase transition occurs at
moderate energy densities (500 MeV/fm3) , then the relative A to p 621
yields can be used as a measure of the relative antistrange quark abundance in the quark phase for heavy ions energies between 2 and 7l 5 GeV/N . Other phenomena will be studied at higher energies : prompt photons , leptons and eventually heavy flavours (charm etc) . It is important to appreciate that all this depends sensitively on the existence of cooperative phenomena in which many nucleons from the projectile and target participate . This behaviour was not so apparent in hadron-nucleus collisions . However , we note that in a first nucleon-nucleon collision the first participants are slowed down and other nucleons can run into the reaction zone . Some of the reaction products acquire significant transverse momen �a . Thuci the assumption of kinetic equilibrium may be correct in a clightly restricted sense: the center of the energy and momentum distribu tion will be at a certain mean value - while the low and the impor tant high energy tails will be underpopulated . The chemical equili brium in which the relative population of different particle states is also governed by the statistical distribution is more difficult to achieve . It is controlled by partial reaction cross sections which are much smaller than total cross sections. Hence for example at low energies (e.g. 1-2 GeV/N .kinetic energy) or large impact parameters strangeness may be decoupled from the presumed equili brium state of hadronic matter . I would like to emphasize that only in very high energetic cen tral collisions of heavy nuclei we will be able to study the pro perties of the quark-gluon plasma at high temperatures . Is this worth the effort? Let us consider as an example the production of + heavy quark flavours in e e- annihilations and in nuclear coJ lisions : while the first kind of experiment is almost certainly much clearer , a complex state like8l ccss could only be produced when high quark density is reached at high plasma temperature T�1 GeV . [This pro
cess may be seen as an analogy to the production of a in pp-ISR collisions .] Another very fundamental aspect of experiments with high energy nuclei is the exploration of the phase transition be tween hadron gas and quark plasma states of hadronic matter and the determination of the critical energy density . 2, Hot Hadronic Gas Phase The basic assumption will be the conceptual validity of the quark 4) bag model - what I propose below is practically thermodynamics of finite size bags9) interacting through creation or destruction of new bag states. I will neglect quantum statistics - this approxima- 622
tion turns out to be permissible when the temperatures of individual hadrons are above 40-50 MeV. From the partition function Z(V,µ,T) all physically interesting quantities , such as energy , pressure , entropy can be desired as they are simply first derivatives of Here Vis the total reaction vo z. lume , µ the baryonic chemical potential and T = 1/B the temperature .
We will also use often the baryon fugacity \ = exp (µ/T) . \ or resp. µ are introduced to allow the conservation of the baryon number . The partition function can be expressed in terms of the hadronic mass - spectrum T(p2 ,b) : T(m2 ,b)dm2 is the number of hadronic resonances of baryon number bin the mass intervall (m2 ,m2 + dm2). We obtain 1 ) : 2ti ln d ( S, \) (1a) z (2'TT ) 3H 36
+= b ( A ) H \ e-SPT(p2 ,b)d0p (1b) cp s I l: f b=-= The bootstrap constant H has been introduced here mainly for dimen
sional convenience: the dimensionless quantity will be derived from the statistical bootstrap model . ti is the available volume in which hadrons are free to move after subtraction of their proper volumina :
ti=V - l:V . (2) i h ,J. Here the sum over all individual hadronic volumina V . in V is h 4) taken. We know that in the bag model of massless relaf vistic quarks we have � Vh,i = Mi/4B (3) and hence the sum in Eq . (4) may be easily carried out :
ti = V -
----'>E + oo 4B (6) 1 s /4B p + p 1 The factor (1+s /4B) - represents the part of the van der Waals p effect introduced by the finite size of individual hadrons . In order to obtain a complete quantitative description of the ha dronic phase we must derive the bootstrap function ¢, Eq . (1b) . By requiring that in the limit 6 + O the hadronic state as described by Eq . (1) is again just another state . In order to obtain a quantitative description of the hadronic phase we must derive an expression for the bootstrap function + ¢(13,;\) = 2¢-e The input function ¢ is defined in terms of the basic hadronic states [q q and qq q] as m � ( rr ) 8 cos h (µ/T) m K1( (8) ¢ = 2rrHT 3m K1 +. N [ Tr T T ] } We recall that Eq . ( 7 ) has a real solution > ., � 500 :::!.. 0 100 150 0 50 150 200 0 50 100 20 T (MeV) T (MeV) �: The critical relationship between temperature T and a) chemical potential µ; b) baryon density Curves 1-5 of constant (shown) projectile kineticv. energy per baryon indicate the hypothetical evolution of central fireballs at selected kinetic energies; v0 = .56 4B/mN corresponds to normal nuclear baryon density v0 .14/fm 3 for the old value of = the bag constant: B 1/4 = 145 MeV . large number of mesons generated by high T is sufficient to induce the change to the quark plasma . As a consequence of the Van der Waals effect discussed the energy density along the phase boundary is equal to 4B while pressure vanishes. With B 170 MeV) 110 MeV/fm3 we ll:: ( 4 expect the phase transition at about 0.5 GeV/fm3 • To test these ideas a5out the hadronic gas phase the temperatures (slope parameters) of inclusive particle cross sections expected in relativistic heavy ion collisions have been computed. These calcula- tions require that : 1) small impact parameter collisions are identified , 2) only particles from the central fireball are counted . Under these circumstances the initial kinetic energy of the projec tile nucleus per nucleon , Ek,p' defines the available excitation energy per participating baryon in the fireball (9) The equations of state distribute this energy among kinetic and chemical degrees of freedom [collective motion and its energy is negle�ted]. The temperature-density relationship of the exploding fireball is shown for the LBL-DUBNA energies .in Fig . 1b. Averaging the temperatures of emitted particles along these cooling curves 625 gives the results shown in Fig. 2. We record that the nucleon tem perature TN is significantly higher from that of pions . [Hagedorn is presently finetuning the parameters H and B and a ±10 MeV change in shown results is anticipated]. The substantial rise of the tem perature with the kinetic energy shown in Fig. 2 is in good agree 12l ment with experiment (BEVELAC-ISR) . ------ - -- - - 150 ... - ... -- ,.... ,. · - · - ·-·-·- - -·- -· -· - -·-·- ,"""" -·-· � JI' ,, ;: .... �," - ! /' ;·" 100 ��-,... 10 100 1000 �: The dependence of temperatures of nucleons and pions on initial projectile kinetic energy . T is the highest temperature of the hadronic gg§ phase. In Fig . 3 I show another interesting result: Along the cooling lines of constant energy per baryon , Fig. 1 the (specific) entropy per baryon rises significantly from the (high) value computed at the critical line . This is due to the onset of strong pionization of the available energy . The entropy at the critical line is found recalling that when pressure vanishes (S/b) / = E/b-µ / E/bT 0 (10) P=O T /P=O high E We thus keep in mind that hadronic fireballs do not expand adiaba tically and that substantial cooperative entropy production is ex pected. We will return to this point again below. 3. The Quark-Gluon Plasma When hadrons have coalesced into a large quark bag at the critical curve , we must change the theoretical model underlying the descrip tion of hot hadronic matter. The new central assumption , valid strictly only at very high energies, is the weakness of the quark quark interaction: only in this case a description of interacting 2) quantum Fermi-Bose gases may be successful. The quark masses in the relativistic quark-bag model are small: mq� 10-30 MeV , while 626 S/b 10 5 2 Fig . 3: Specific entropy per baryon along cooling lines for given projectile kinetic energy as function of T. The boundary is the specific entropy at the critical line . gluons are massless. [These masses should not be confounded with the nonrelativistic quark model masses M which correspond to the kine nr tic energies of the bag model: M 2.04 11c/8. 400 MeV. ] Hence in nr � bag:=- the region of interest to us of µ and T, the quark chemical potential 1 µ µ > m . The factor 1 /3 arises in view of the quark baryon num- q = 3 q ber 1/3. As long as this condition is satisfied , the Fermi-Bose gas 13) with interaction O(a ) can be integrated ana1ytically and we find : s Quarks : ( 11a) T ln Z q Gluons : 8V 15 T ln z 1 - a (rrT) (11b) g s 4 45rr2 4rr ) Vacuum: ( T ln Z -BV (11c) V = where z z z z is the total partition function . = q g v Here g 2·2·3 = 12 is the number of distinct quark modes with spin , isospin and colour . a is the QCD colour coupling constant [a �.5 s s for space-like q 2 and a � . 2 for time-like q 2 - but we ignored the s q 2 dependence of a ]. s In Eq. (11) we show separately the contributions of quarks and gluons ; the vacuum term is a phenomenological supplement at this 6 27 stage of the discussion and has been chosen so as the bag energy density is inside the region of the plasma and that an inside point B ing pressure P -B acts on the surface of the bag region . Contribu = tions of heavy and strange quark flavours have been neglected . The vacuum contribution , (1 1c) is as postulated in the quark Eq . bag model. However, at finite temperatures additional difficulty arises not shown by the perturbative expression , Eq . (11) . We re call that the vacuum structure term originates presumably in the absence of the true vacuum gluonic structure from the region of space containing quarks . Therefore should be calcu lated non-per B turbatively together with the glue term , Eq . (11 b) to yield some T-dependent quantity . In particular , this gauge pressure should va nish at some high temperature T 1. 11 , when the structure of cr�� SB 4 1 4 ) the true vacuum is de stroyed . Only above T the expression (11b ) cr is valid - wh ile (11c ) is only correct for T-+ 0. For T T Eq . < cr there is only a partial restoration of symmetry and not all 8·2 gluonic degrees of freedom can be exc ited . Another aspect of this point is that gluons may not be able to exist as independent par ticles in the plasma region and are rapidly absorbed on the surface by the true vacuum. This glue dissipation may be the origin of a 15l high instability of gluonic states . In order to estimate the boundary of the quark-gluon plasma phase in the µ-T plane , c.f. Fig . 1a we search for the line of zero pressure PV = T ln Z, as given by Eq . (11), though omitting the little important , but obscure glue term at relatively small tempe ratures. The result is very similar to the bootstrap line , Fig . 1a, with T0:::: at O: now it is the quark-antiquark pair pressure B1/4 µ that balances the vacuum pressure . This perhaps not accidental coincidence of the critical lines leads to the conclusion that both phases described here are directly adjacent to each other . Ad justung slightly H, and a(T) we can achieve exact coincidence B of the critical lines along which in both phases the energy density is and pressure vanishes. However, we can have a discontinuity 4 B of the baryon density : As in nuclear collisions baryon number should be conserved, the hadronic volume would be discontinuous . Of course this will not be the case - instead we have a Van der Waals transi tion shown schematically in Fig . On the hadronic gas side we have 4: to construct a new state consisting of a mixture between plasma and hadronic cluster . We will not enter here further into the discussion of this subject. 628 p T fixed I ' I ', I ', V1 Vm V2 V-1/V Fig. Pressure vs Volume , 4: ( ·1 /v) qualitatively with the Maxwell construction Finally let us note that as in the hadronic gas phase we can con struct in the µ-T plane curves of given available energy per baryon . In a qualitative picture , Fig . 5, the most important aspects are illustrated : there is a highest temperature that can be reached : T "' Furthermore , at hadronization we have very small chemi max ls/A. cal potentials ·for ISR energies - hence we find from Eq. (10) that the expected specific entropy in a nucleus·-nucleus collision at ISR could reach ... 100. In other words , following Boltzmann , we find that there are 2 10 0 different final states available to each 15 GeV baryon . We conclude that this enormous amount of entropy must be produced in the nonadiabatic explosions of highly compressed quark plasma . 4. Strangeness in Nuclear Collisions Unlike p-p collisions , strangeness may be close to kinetic equili brium in nuclear collisions : it is the large reaction volume with typical lengths exceeding the Compton wavelength of the strange mass 6) that is of great importance here . Let me explain why we should not expect the usual kinetic equilibrium result in p-p collisions : in general when strangeness-antistrangeness pair is mode. in hadro nic reactions at given temperature T, the expected number of pairs is (12a) where m � 280 MeV is the relativistic strange quark mass . Instead , s when we consider production of particles in a hot star we find µ rr-condensotes = Nuclear Motter 1 GeV Quark - Gluon Plasma T0� 160- 200 MeV Fig. 5: Cooling lines in the µ-T plane in the quark-gluon plasma (qualitatively) The fact that a pair is produced seems not to matter here . The cru cial question is where the transition from case a) to b) occurs . The 3 quantitative result is that in the hadronic volume V = 4TI/3 (1 fm) h the pair mass matters , case a) , while already for V 6 -8 V we are ::$ h in the limit b) - assuming a typical T�150 MeV . The suppression of the kaon yield in p-p collisions as compared with thermodynamic models has so found an explanation . It is expected that in nuclear collisions sufficiently large hadronic volumes are found . Further 6) more , I have found from this study that there is no information about the hadronic aggregate state that can be extracted from the total strangeness yields . However , it is the antistrangeness that may be very usefull in the search for quark matter : below the AA threshold [at 4-5 GeV taking account of Fermi-motion in both nuclei] the antistrangeness that must balance the strangeness produced in hadronic reactions will generally be found only in kaons , except if quark plasma state were formed . In that case the kinematic limit is of no importance and it is the relative abundance of antiquacks in the plasma that will con trol the antibaryon yields . Consider for example the A/N ratio as a measure of the relative and or abundances: on theoretical s u d grounds I expect to be as abundant as and taken together , s u d supposing the typical T and µ values at 4-5 GeV/N . Hence if quark 630 plasma is formed at these energies , A/N ratio could become an important observable . 16l A preliminary search for A at 2.1 GeV/N has been negative . As the energy density reached at 2.1 GeV/N - projectile kinetic energy is perhaps only 250 MeV/fm , this result fulfills expectations and sets a lower limit on the value of the critical energy density . It is quite conceivable [say 50%-50%] that at 4-·5 GeV/N in some collisions the quark plasma state is formed . With direct production A still strongly suppressed I am awaiting impatiently future results on A/N ratios at these energies . 5. Conclusions In order to obtain a theoretical description of the hadronic gas and quark plasma phases I have used some 'common ' knowledge and plausible interpretations of the currently available experimental observations . In particular , the strongly attractive hadronic interactions are included through the rich , exponentially growing hadronic ma ss spectrum T(m 2 ,b) while the introduction of the finite volume of each hadron is responsible for an effective shortrange repulsion . Aside from these manifestations of strong interactions , I only satisfy the usual conservation laws of energy , momentum and baryon number . I neglect quantum statistics since quantitative study has revealed that this is allowed above T = 50 MeV . But particle production is allowed , which introduces a quantum physical aspect into the other wise 'classical ' theory of Boltzmann particles. The study of the properties of hadronic matter has just begun and it is too early to say if the features of strong interactions that have been included in these considerations are the most relevant ones . It is important to observe that the currently predicted pion and nucleon mean transverse momenta and temperatures show the required substantial rise , see Fig. 2, as required by the experimental results 13) available at E /A = 2 GeV [BEVALAC ] and at 1000 GeV [ISR] . k,lab Further comparisons involving , in particular , particle mu ltiplicities and strangeness production discussed above are under consideration . I wish to emphasize the internal consistency of the two-fold approach . With the proper interpretation the description of hot ha dronic matter leads us , in a straight forward fashion , to the postu late of a phase transition to the quark-gluon plasma . This new phase is treated by a quite different method ; in addition to the standard many-body theory of weakly interacting particles at finite temperature and density, we also introduce the phenomenological vacuum pressure 631 and energy density B. Perhaps the most interesting and far reaching aspect of this work is the realization that the transition to quark matter will occur at much lower baryon density for highly excited hadronic matter in the ground state (T = O) . The precise baryon density of the phase tran sition depends somewhat on the phenomenological value of the bag constant; we estimate it to be at about 2-4 at T 150 MeV. The Vo = detailed study of the different aspects of this phase transition , as well as of possible characteristic signatures of quark matter, must still be carried out . I have given here only a very preliminary re port on the status of my present understanding . The occurence of the quark plasma phase will certainly be an oberservable phenomenon . Here I have discussed a measurement of the A/p relative yield at about 4-5 GeV/N kinetic energy nuclear collisions . In the quark plasma phase we expect a significant en hancement of production which will be most likely visible in the A A/p relative rate . Thinking ahead a decade from now , I can foresee colliding nuclear beams with energies of the order of 100 GeV/N : The anticipated temperatures of several GeV in the quark plasma may lead to the formation of very exotic heavy flavour states involving c and b quarks at the same time . Many fruitful discussions with the GSI/LBL Relativistic Heavy Ion group stimulated many of the ideas presented here . I would like to thank R. Bock and R. Stock for their hospitality at GSI. Parts of this work were performed in collaboration with R. Hagedorn , B. Mliller, H.-Th . Elze , and M. Danos . I thank also the organizers of the Rencontre de Moriond for the opportunity to present these ideas and for the fine organization of the meeting . References 1) R. Hagedorn and J. Rafelski , manuscript in preparation for Phy sics Reports. See also 'From Hadron Gas to Quark Matter ', CERN preprints TH 2947 and TH 2969 , to appear in the Proceedings of the 'International Symposium on Statistical Mechanics of Quarks and Hadrons ', Bielefeld , Germany , August 1980, H. Satz , editor , North Holland Publishing Company . 2) The many-body theory for QCD at finite temperatures has been dis cussed by : B.A. Freedman and L.D. McLerran , Phys . Rev . £.!§_ (1977) 1169; S.A. Chin , Phys . Lett . 78B (1978) 552; P.D. Morley and M.B. Kislinger , Phys . Rep . 51 (1979) 63; J.I. Kapusta , Nucl. Phys . B148 (1979) 461 ; E.V. Shuryak , Phys . Lett . 81B (1979) 65 and Phys.Lett . � (1980) 71 ; 632 O.K. Kalashnikov and V.V. Klimov , Phys . Lett . 88B (1979) 328 3) J. Rafelski , H.-Th . Elze , and R. Hagedorn , 'Hot Hadronic Matter in p-Annihilation on Nuclei ', CERN preprint TH 2912, in Proceed ings of the 5th European p-Sympo sium , June 1980 , Bressanone , Italy . CLEUP , Edts ., Padova 1980. See also J. Rafelski , Phys . Lett . 2..!l?_ ( 1980) 2 8 4) For review see , for example , K. Johnson , 'The MIT Bag Model ', Acta Phys . Polon . B6 (1975) 865 5) R. Hagedorn and J. Rafelski , Phys . Lett . 97B (1980) 136 6) J. Rafelski and M. Danos , Phys . Lett . 2.::� (1980) 279 7) J. Rafelski , 'Extreme States of Nuclear Matter ', Frankfurt Preprint UFTP 52 (1981 ) in Proceedings of the Workshop on 'Future Relativistic Heavy Ion Experiments ', R. Bock and R. Stock , eds . Darmstadt 1981 . 8) It has been suggested in the lecture by P. Richard at this con ference that this would be a stable state . 9) We record the first attempt by J. Baacke , Ac ta Phys. Pol B8 (1977) 625 to develop a thermodynamic description of a gas of quark bags . 10) R. Hagedorn , I. Montvay , and J. Rafelsk.i , 'Hadronic Matter at Extreme Energy Density ', Proceedings of Erice Workshop , eds . N. Cabibbo and L. Sertorio , Plenum Pcess (New York 1980) , p. 49 11) J. Yellin , Nucl . Phys . B52 (1973) 583 ; see also E. Schroder , Zs. flir Math . und Physik 15 (1870) 361 12) For LBL experiments see e.g. S. Nagamiya, 'Heavy Ion Collisions at Relativistic Energies ', LBL preprint 9494 (1979) in Proceedings of the Symposium on Heavy-Ion Physics , Brookhaven National Labora tory , Upton , N.Y. , July 15-20 , 1979. The ISR inclusive pp results are summarized in G. Giacomelli and M. Jacob , Phys . Lett . C55 (1979) 1 13) H.-Th . Elze , Greiner, and J. Rafelski , J. Phys . G6 (1980) L149 w. and in Ref . 3 above , H.-Th . Elze et al ., to be published . 14) In the model of magnetic Savvidy vacuum the restoration of the perturbative vacuum has first been shown by B. Mliller and J. Rafelski , 'Temperature Dependence of the Bag Constant and the Effective Lagrangian for Gauge Fields at Finite Temperatures' , CERN preprint TH 2928, Phys . Lett . B in print . See also J. Kapusta , Phys . Rev . D in print and Dittrich and V. Schanbacher , Phys . w. Lett . B in print . 15) This view is at odds with the one presented by S. Lindenbaum at this conference . 16) R. Stock , private communication .