A.N.Makhlin, Yu.M.Sinyukov HYDRODYNAMICS OF HADRON MATTER UNDER PION IHTERPEROMETRIC MICROSCOPE

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1 f i ITP-87-64B A.N.Makhlin, Yu.M.Sinyukov HYDRODYNAMICS OF HADRON MATTER UNDER PION IHTERPEROMETRIC MICROSCOPE ii

2 УДК ; А.Н.Махлин, Ю.М.Синюкв Изучение гидрдинамики адрннй материи метдм пиннг интерферметрическг микрскпа Предлжен метд бнаружения гидрдинамическг движения в адрннм веществе, ржденнм в ультрярелятивитских ядерных рр и рр-суцярениях. Метд снван на анализе бяеэйнштейнвских пмнных крреляций. Он пзвляет измерит;, время и темп расширения, а также температуру замраживания для вещества, ржденнг в выскэнергетических адрнных и ядерных сударениях. A.H.Makhlin, Yu.M.Sinyukov Hydrodynamics of Hadron Matter under Pion Interferometric Microscope A method which makes it possible to find out the hydrodynamical motion of hadron matter is proposed. The method is based on the analysis of Bose-Einstein pion correlations. It makeu it possible to measure the time, the expansion rate and the freezeout temperature for the matter produced in the central region of high-energy hadronic and nuclear collisions Институт теретическй физики АН УССР Александр Натанвич йахлин Юрий Михайлвич Синюкв Изучение гидрдинамики адрннй материи метдм пиннг интерферметрическг микрскпа Утвержден к печати ученым светм Института теретическй физики АН УССР Редактр А.А.Храбрва Техн.редактр Л.П.Львва Зак. В19 Фрмат.60x84/16.Уч.-изд.л. 0,7 Пписан к печати г. Тираж 200.Цена 4 кп. Плиграфический участк Института теретическй физики АН УССР

3 Academy of Sciences of the Ukrainian SSR Institute for Theoretical Physics Preprint ITP-87-64E A.N. Makhlln, YU. II. Sinyukov HYHtODYNAMICS OP HADRON MATTER UNDER PION INTERPEROME'JHIC MICROSCOPE Kiev

4 -2- A study of the two-particle correlations of identical particles ia now widely used to determine the size and shape of the multiparticle generation region in hadronio, nuclear and e + e~collisions. The main point of the method is that the width of a two-particle correlator, which ia considered as a function of the momentum difference Д D, is inversely proportional to the emission region dimension CL along a direction parallel* to д"р [i-4j However, this approach does not take into account the possibility of an internal relative motion of radiation sources in a system. In this paper we shall demonstrate how the hydrodynamical motion anticipated for ultrarelativistio nuclei- and pp, pp-collisions [5»6j, can be detected by the pion interferometric method. 1. We base on the pion interferometry theory for a hydrodynamical stage of multiparticle processes, which was developed in our previous papers Г7,в1. The one-particle pion inclusive spectrum in hydrodynamical theory looks like = f (1) where pv is the secondary particle 4-momentum, 6T T ie the total cross-section, c : "t -~ь с ( )1в the critical hypersurface in the space-time where a hadronic fluid achieves the freeze out temperature T a and decays into secondary particles, $»(2JT) X ^ *and U^* is the hydrodynamical 4-velocity of hadronic matter at the is *y stage. In this picture the fluid elements can be considers! as radiation sources. The sources have approximatly identical spectra in its own rest systems [9], but they move with different velocities Ufa) and radiate in different times 't=~t a C<#)' The radiation duration time AT (the decay time of a fluid element) is negligible [5J«The two-particle inclusive JT JT cross-section of identical pions in hydrodynamical theory has the form [ 7J:.6

5 where the correlator K(p^ft^ arises from the interference nf identical pions O ) We note that the pion interferometry method for moving sources was etudied independently in papers [10] for special case of a scaling expansion. But the inside-outside cascade model, used in the papers, is not suitable for the hydrodynamical theory of multiparticle production. This leads to differences in initial formula (3) and, of course, in numerical results. Hydrodynamical approach deale with real local thermodynamical equilibrium for pion gaa at the decay hypersurface 2«[5l. This initial suggestion leads to the correlator (3) in the unambiguout way[7j. The explicit calculations of correlator (3) are easily performed in terms of the auxiliary function so that л л J J,, _ <4> (5) 2. In the hydrodynamical theory of multiparticle production the one-dimensional expansion of matter along the collision axis ^(l'=x is assumed to be an initial approximation. So the speedograph transformation may be used [ 5j: To t - e"" 2 [ where it = ^ь(т/т у=аг11) & is the hydrodynamic rapidity of the fluid element moving along the Л-axis,..I e is some reference point of temperature,% (3t,4^is the hydrodynamics! potential. The explicit forms of the potentials are well known for Landau and scallng-models [5»11^«The decay region S e is limited by longitudinal coordinates X w<x =-Х Л1,- п< = а/д» or rapidities ц =-ч, =Y i n c.m.s. Let us pass to rapidity variables of ae-

6 condary pione Jf s t Jf^ ; t - -52k, б-- -4-,*w5=wj^, a= ^ > 0- ^ -art^ Here р г - is the momentum projection on the collision axis, "p x^ are the transversal components. When we analyse the two-particle correlator, we suppose l^=t> zl Using variables (6), (7) we can then write down 11 where Ф(«р and ^T(^^ are the known functions [э'. Sf=JT/2 is the transversal area of the hydrodynamical tube at the final stage. If only the particles with tkj^? 7^ are selected, the Bose- Einsteln factor J- in (8) can be replaced by a Boltzman exponent. Under the above assumption the function Ъ (р*, (^can be evaluated by the saddle-point method in the central rapidifcy region 6 г ' ^-^. The equation which defines the saddle-point is If cjb=.o the obvious solution of Eq.(1O) allows us to evaluate the one-particle spectra (1), (5) and to find the pion rapidity distribution whereк = ($% Ь / ГГ г )НКя/Чс) Й» i-3 for the pion triplet, d is the gradient of 4-velocity longitudinal component of the fluid element, which moves with rapidity Э. The solution of Eq.(1O) at ol ^ О can be obtained by the

7 -5- succesive approximation detailed in Ref. [_8l. If we limit ourselves to the correlation measurements in the region ~tlfa 06 < < {. and neglect the terms -^tltc6 in the pion correlator, the hydrodynamical correlator in the central rapidity region for pione with equal transversal masses 1Л> ±1 ~tft ex has the form: +^T c^)uv] (12) The factor ^-i at jv^»"^- The factor У 4 4 in the general case when the directions of the momenta ~b Xx and ~^3X are not fixed to be exactly parallel and when we take into account that the interference is not complete for various reasons. Aa distinct from the standard models Г 1 4-J* where the relative motion of sources is absent, dimension of a system is not present in expression (12). If one determines it formally according to the usual interferometric method via the correlator width GL O (connected with the rapidity oc 0 according to (7)): & e^~i/# e there are then two effective lengths depending on the hydrodynamical regime. If we deal with the moderate velocity gradients of a hadron fluid, dtv/cu= =!/ *<.<T c» tne effective length H -k*tlfc/* 2 is the longitudinal size of a fluid element forming the one-particle spectral density at the points Pj,p A as /2 [в]. In this case, the correlator (12) has an exponential behaviour. If the correlator behaviour is oscillatory, the velocity gradient is large,dw/elx >Т С., and the effective length 0, в^-сьц= l/t/^is the distance betv/een the fluid elements which contribute to the one-particle spectral densities at the points A and Р л» The distance Q, и exceeds the size CL T of the elements themselves 10> H > d T.The typical regimes of the correlator (&, ) behavior for t^e Landau-model [5} and the sea].ing-model [б, 11,12 in Pp'-collisiona ( T c - t^-д-, У" х с: i/ubj.) are plotted in Pig.1 for P=0. The factor jj-i.the effective dimensions depending upon the hydrodynamioal velocity gradient and the heat broadening of the hydrodynamical spectrum агэ of the order Q>tj^- 1r5fm, while the whole lengths of decaying system are within fm. So the energy density calculated by the formula 6 = F/^^^v.ould be overestimated if the system possessing a developed hydrodynamical motion is mistaken for a system without internal motion.

8 -6-3. We perform the qualitative experimental test enabling us to find out what variant of matter evolution takes place in high energy hadronic or nuclear collisions. a). The formation of an intermediate massive cluster (fireball); the absence of a developed hydrodynamical motion. If the rest spherical cluster of radius Г decays, the correlator looks like [.1,81: where j^is the spherical Beauel function. The first zero at ^p! = = лр of Eq.(i3) enables one to determine the clusters size» г*» Ц.Н«/Дрв If the prolate shape rest system radiates, the correlator is approximated by a 3-dimensional Gaussian form [~4^. Its widths determine the longitudinal and the transversal ^x effective sizes of emitting objects ^ 1.(14) At the present time the interferometric analysis of the dimensions and shape of an omission region is based on models of thie type. The common property of the models (13), (14) is that the L(Q,, *} -correlator in momentum space depends only in the momentum difference 2. Q. along the direction of interest, but not on the moment urn вши 2 P. The width of the- same correlator RAc^e"} exprestsed in the rapidity variables decreases according to Kq.(7) when the rapidity вши 20 increases (see Fig.2) b) The emergence of a global hydrodynamic regime for the matter evolution in high multiplicity fluctuations of p^ -collisions and ultra-relativistic heavy ion collisions. In accordance with the basic results of the hydrodynamical approach [5,6,11,12~, the function T(0} connected with oneparticle rapidity spectrum via Eq.(7) ia a constant in the всаling-model or slowly decreases as in the Landau-model when the

9 -7- Э increases (in cm.s.). If the plateau in the central region of the rapidity destribution is observed, the width 0 of the hydrodynamics! correlator R(ol,9) in the rapidity variables doea not change when the detected-particles rapidity sum A 6 increases in cm. e. (see Fig. 2). The same correlator R,(u,P) in momentum variables undergoes a broadening with increasing momentum sum 2 P, due to (7) (see Pig.3) This gives rise to the imitation of decreasing in the sources size d,, ~ i/6j o rp)according to (15) when the momentum sum inoreases. Thus the rapidity difference is a natural variable for hydro-dynamical correlators, and the momentum difference is natural variable for the interference of the radiation from a rest пм dia. If the rapidity plateau is absent (the Landau-model leads to a Gaussian-type falling of rapidity distribution), the hydrodynaraical correlator gains a broadening additional to (15) (see Pig.2,3). We als< note that the effective hydrodynamical length has specific dependence on the transversal mass of detecting particles, Q. e jj i/vw-j» f r all hydrodynamical models. If the tests indicate the existence of a global hydrodynamical regime, one can perform detailed analyses of the regime using the correlator (12). We shall demonstrate the analyses for the scaling regime of matter evolution forming In nucleus-nucleus collisions [6,12]. For the scaling-model the decay isotherm has the form Х г х*, and T -^(dil/dx) means the proper time of the system expansion. The longitudinal velocity destribution has the form -O = &/4~. In hadronic and nuclear collisions the initial condition and, therefore, the parameters of hydrodynamical flow change from one collision event to another. So it is nececoary to express the parameters in terms of observables and to select the events with the same conditions. The plateau height in the central region H = H(O^ (in c.m.s.), will be taken as one of the main observable quantities identifying the physical picture. The height of plateau H and the total pion multiplicity in the central region Г^ are connected with the hyd. odynamical parameters of the scaling-model [8,12^ (for the Landau-пю-

10 del see [в]): -8- L where S o is the initial entropy density,t o =(0,5 4 1)fm is the initial time of the hydrodynaraioal stage formation. We note that the measurement of only the plateau height H in ultrarelativistic nuclear collisions does not make it poasible to reliably determine X, since only 40# variations of the transverse ^adius V ± and the temperature T c at the final stage of matter evolution lead to a change ±n t by an order of magnitude due to the presence of the factor i^ N 0 (%)±n (11). The proper time of the expansion IT and the freeze-out temperature T c can be determined directly from the correlation data. The correlator behaviour at ehoeen M»,:>>Te (г/»^»л^) depends on the two parameterst and T c (and the common normalizing multiplier Д ). It can be determined by the fitting of correlation data by formula (12) when the plateau height H/0) is fixed. We would remind if the plateau is absent in events with a fixed value of H(0^, the ~(G) is the inversall gradient of the longitudinal component of the hydrodynamical 4- velocity u,' l '(^&t the final stage. If the values f and T c are determined at fixed Htf^one can find the transversal area Sj. according to formula (16) == /fa/(t c ) In conclusion, we shall demonstrate the possibilities of correlation analysis to elucidate the character of the hadronic matter evolution in ultrarelativistic nucleus-nucleus collisions. We are here concerned with the transition of a one-dimensional nydrodynamical regime into a three-dimensional one and the occurence of a mixed phase as a manifestation of the first-order phase transition. If f *" f in central collisions, the intensive transversal motion has no time to develop and the transversal area at the decay moment is equal to the nuclear area S^ S A. The oscillation factor dominates in correlator (12) because ТГ is small. If at the same time T c MeV, this is evidence of the standard hydrod.,namicel picture [ 51. The QCD-degrees of freedom do not manifest themselves. If T c > 140 Me7, this can indicate the specific mechanism of concentration freezing i3j» The mechanism appears, because the relaxat on processes which guarantee

11 -9- the settlement of local thermodynamical equilibriuir' have no time to compensate the changes connected with t^e fast rate of system expansion. The picture corresponds to the hard equation of state at all phases of hydrodynamical evolution. If the mixed phase is realized, the proper time T becomes long [12]. The greater part of the matter moves by ineartia during this phase, because the pressure gradient is absent* If the initial energy density does not exceed the energy density in a mixed phase the transversal expansion of the greater part of the matter is negligible. The decay of system occurs, in fact, just after the hadronic phase is formed (during time й,х^т л ), because all the parameters of the system are closed to its decay values. In this case the correlation analysis must lead tot:»r^and S^e-S»^. If one finds outt"»qat < э 1 "* > 2> д, It would be evidence of the active hydrodynamicbl regime forming alread; in the quark-gluon phase. The transversal motion developed preserves by inertia in the mixed phase (simple estimations give S x a (3T4)S^). The described method of determining the freeze-out temperature and the transversal area from the correlation data on longitudinal momenta enables us to separate the contributions to the transversal momentum spectrum from heat radiation and transversal hydrodynamical motion. We are indebted to V.L.Lyuboshitz, M.I.Podgoretsky M.I., G.M.Zinovjev for fruitful discussions. REFERENCES 1. Goldhaber G., Goldhnbor S., Lee W., Pais Л. Influence of Вве- Einstein statistics on the annihilation processes. - Phys. Rev., 1960,.120, p Кпылв Г.И., Пдгрецкий М.И. Крреляции тждественных частиц, испускаемых высквэбужденными ядрами. - ЯФ, 1973, 18, 3, с Щуряк Э.В. Крреляция тждественных пинв в реакциях мнжест-г веннг рждения. - ЯФ, 1973, 18, 6, с Асевзп Т. et al. Evidence for a directional dependence of Bose-Einstein correlations at the CERN ISR. - Phys. Lett., 1987, 187B, N 3,4, p Ландау Л.Д. О мнжественнм бразвании частиц при стлкнвении быстрых частиц. - Изв.АН СССР. Сер.физ., 1953,17,»1,

12 -10- o Bjorken J. Highly relativistic nucleus-nucleus collisions» the central rapidity region. - Phys. Rev., 1983, D27, p Makhlin Yu.M., Sinyukov Yu.M. The pion interferometry theory for the hydrodynamic stage of multiple processes. Preprint ITP-86-27E, 1986 ( Sov. J. ijucl. Phys., 1$«7, 46, II 8 ). 8. Averchencov V.A., Makhlin A.N., Sinyukov Уц.М. Study of collective motion in hadron matter by a pion interferometry me thod. Preprint ITP E, 1986;(Sov. J.Nucl.Phys., 1987, 46, N 10). 9. Gorenstein M.I., Sinyukov Yu.M. Local anisotropy effects in the hydrodynamical stage of multiple processes. - Phys.Lett., 1985, 142B, N 5,6, p Kolehmainen K., Gyulassy M. Pion interferometry of ultrarelativistic hadron collisions. - Phys.Lett., 1986, 180B, 14 3, p Pratt S. Piori interf erometry of quark-gluon plasma. - Phys. Rev., 1986, D3_3, N 5, p. 13H Gorenstein M.I., Sinyukov Yu.M., Zhdanov V.I. On scaling solution in the hydrodynamical theory of multiple production.- JETF(USSR), 1978, 7_i, N 3, p Gersdorff H., von, blclerran L., Kataja M., Ruuskanen P.V. Studies of the hydrodynamic evolution of matter produced In fluctuations in pp collisions and in ultrarelativistic nuclear collisions. - Phys. Rev., 1986, D34, p «Rosental I.L., Tarasov U.A.On transversal momenta of the secondary particles in hydrodynamical theory of multiple production. - JETF(USSR), 1985, 8, p Received May 7, 1987.

13 R P в О N/H - 5 ш в 0.44 Gev Q (Gev/o) Pig.1. The Q-dependence of the correlator H(Q t P) atp a & for the scaling-model (8), plotted by a continuous line and the Landau-mode3 (L) plotted by dashes at different plateau heights H(0) in pp-collisions. When H=1, j=i7fm, <Z fc «Hfm,tf.^ fm; when H=3 <Z S -50fm, <2 Ы «42fm, ^2, 5fm; H=5 CL S =BOfin, <2^

14 «Н 1 О см 1 1 А V 1 1 ю Ь II i V 1 СО О Г" х О II 1 J > & *** * в i. i; с V * * - v» О ф - «л с1 *~ 4» й й и а 01 ^«elati tt endence Р. ш Ф ш Pig. 2 (dots gdel 4» В ClUl rest в с а а f-4 43 А к СО

15 Pig..3. "Hie correlator R(Q,P) at different values P for 3-, L- and C-models. The hydrodynamical 3-, L- correlators at different values P are approximated by the cluster model with different cluster radii Z : Z t =4,3 fm,? г = a 1,8 fm, Zj я 0,8 fm. At P=0 the curves of S-, L-, C-r:odels are indistinguishable in the figure scale. Q- (Gev/c)

16 Препринты Института терет:гиткпи фиг.нкн ЛИ УССР рассылаются научным рганизациям и тдельным ученым па спине цлапмнг бмена Наш адрес: , Киев-130 ИТФ АН УССР Инфрмацинный тдел The preprints of the Institute for Theoretical Physics are distributed to scientific institutions and individual scientists on the mutual exchange basis. Our address: Information Department Institute for Theoretical Physics , Kiev-130, USSR