REVISTA MEXICANA DE FíSICA 45 SUI'LE~IENTO 2, 116-119

The expanding pion Iiquid and the pion spectra

Alejandro AyalaInstituto de Ciencias Nucleares, Universidad Nacional Autónoma de México

Apartado posta/70-543, 045/0 México, D.F, Mexico

Recibido el 28 de enero de 1999; aceptado el I de junio de 1999

OCTUBRE 1999

Wc compute lhe pian inclusive morncnlum distribution in hcavy-ion collisions fOf encrgics al thc Altcrnating Gradient Synchrotron (AGS),assllllling thcrmnl equilibrium ano accounling ror dcnsity ami expansion clfcCIS al frcczc out. We compare to uala on mid-rapidity chargedpions produccd in central Au + Au collisions and tind a vcry gooo agreemcnl. Thc shapc of the distribution al low transvcrsc mass iscxplaincd in part as <ln effeel arising from the high mean pian dcnsity achicvcd in thcsc rcactions. The differcncc bctween the positive andncg<ltive pian distributions in the sume region is attributed in pan lo the differcnt averagc yicJds 01'cach kind ol' charged pions.

Kev\I'onls: Scattering heavy ion; gold; 7r production; 71" charged partidc; 7r momcntum spcctrum: rapidity; mas s trnnsverse.

En este trabajo se calcula la distribución inclusiva de piones producidos en colisiones de iones pesndos relativistas a energías del Altert/a(;llgGnulü,1lt SYllc/¡rotmll (AGS). bajo la suposición de que los piones se producen en equilibrio térmico y tomando en cuenta los efectosde densidad y expansión hidrodinámica al momento del desacoplo. El cálculo se compnra con datos de piones producidos en colisioncscentrales de Au + Au. en la región central de rapidez. y la comparación resulta muy huena. La curvalUra de la distribución en la región devalores bajos de la masa transversa se explica en parte como consecuencia de la alta densidad pionica alcanzada en la reacción. La diferenciaentre las distribuciones de piones positivos y negativos en la misma región se ntrihuye en parte n que el número promedio total producido dec(lda especie de pion cargado es diferente.

Dcscriptores: Dispersión iones pesados; oro; 71" producción: 1T partícula cargada; 1T espectro de momento; rapidez: masn transversa.

PACS: 25.75.-q

1. Inlroduction

A great deal 01' experimental er1'ort has been devoled in re-ccnl ycars 10 lhe protluction of a highly cornprcssed slatc 01'matlcf in high-cnergy collisions bctween heavy ions. Sornenf lhe most abundantly produced particles in these reactionsare pions. An undcrstanding 01' pion production in lhis envi-fOllmcnt has long been searched. particularly in view 01' one01' Ihe lIlos1 remarkable propertie5 exhibiled by lheir spec-tra 11J. c01l11l10nly referrcd to as an enhancemenl of either lhelow 01' lhe high lransvcrse momentum regions in lhe inclusivesingle pion dislribution. as compared to p-p collisiolls. Thisproperty is concomitant with lhe difficulty lo describe lhe in-variant transverse mass distribution with a single exponentialfUJ1ction [2]. An understanding 01' pion spectra should thusprovide useful information about the dynamics and evo1utionnI' the kind of matler formed in lhese rcactions.

Several possibilities have been suggesled lo explain thepeculiar shape 01' the pion dislribution, ror example. the dif-ferent contributions to lhe pion yield coming from the dccayof ~ resonances produced during different stages of (he c01-lision [3J amI Ihe superposilion of primary pions anu pinnscoming from resonance decay, mainly ñ,'s [4]. The impor-tance of transverse llow in describing Ihe spectra has alsobeen strcssed [5]. The proper trcatment ofCoulomb final stateinleractions has also been pointed out. More recentiy, Wongand MOSlafa [6] nOliced Ihal when parlicles feel Ihe cffecIs 01'a houndary during the evolution from the first stages 01' the

collision lo Iheir final free slrcaming. Iheir momenlum dis-tribution is affecled due to lhe discrelizalion of energy levelsand Ihe correspondingly differenl density 01' slales introducedby Ihe finile size of Ihe syslem jusI befare freele oul. Thisidea is hascd on Ihe concept 01' a pion liquid. first discussedhy Shuryak [7] and was fUrlher developed in ReL 8 wilh lheinlrodllction of a flnite chemical potcnlial associaled to lhemean pion multiplicil)' per event in central collisions.

The existing calculations hased on the ef1'ects 01' a bound-ary resorl to lhe assumplion oí' lhermal cquilibration. Theyare successflll in reproducing !he concave shape 01' the dis-trihution at high transverse momcnlum but fail to describe itsoverall fal! off. This failure could have heen anlicipated since,as poinled out in Ref. 9, thennodynamics is lanlamount ofhydrodynamics alld a simple lhermal spcctrum should be C<Jr-

rectcd hy the Doppler shift resulting fmm collective expan-sion. In this paper we describe the formalism required lo ac-counl silllultaneously 1'01'cxpansion and bounuary effecls ina phenornenulogical calculalion of pion spectra in rclativisticheavy-ion collisions. We apply this forrnalism 10 compute lheinvariant transverse rnolTlenlum distrihution for mid.rapiditypions at AGS energics. We flnd a very good agrecrncnt withdata on cenlral Au+Au reaclions at 11.6 A GeV/c.

2. A pion liquid

Wc slarl by noticing that ir lhe average pion scparation d everhecomes srnaller than lhe average range of lhe pian strong

THE EXPANDlt\G PION LIQUID AND TlIE P10;-JSPECTRA 117

3. Thc mOmC/ltlll1l distriblltio/l

In lhe ahsel1ce 01'expansiono the solution has been presentedin Rcf. H. JI involves solving the Klein-Gordon equation toflnd lhe slalionary wavc functions satisfying

Au+Au reaclions 121. In this case, a more symmctric geomc-lry bclwccn transvcrsc and longitudinal dircctions secms ap-propriate. \Ve lhus consider a scenario in which the system ofpiolls 01'a given specics is both in lhermal and hydrodynam-ical cquilibrium ano is confined wilhin a sphere 01' radius R(t1rchall) as viewcd frolll lhe center 01' Illass of Ihe collidingnuclei al the time 01'decoupling. As discllssed in Re!'. 11, lhistime need not he Ihe same over Ihe entire reaction volume.Ncverthcless. in lhe spirit of the flrehall model of Ref. 12,we cO[lsider lhat decoupling takes place over a constanl timesurface in space-lime. This asslllllplion should be essentiallycorrect ir the freeze oul inlerval is shorl compared to the sys-lem's life lime.

and f¡nite at lhe origino Tu incluJe lhe effects 01' hydrody-namical 110w,we observe Ihat lhe presence of an ordcred 010-tion, represented hy a fonr-veloeity I¡cld u'> = ')(1')[1, v(r)],alllollnts to a redistrihulion 01'momentum in each 01'Ihe fluidcclls, as vicwcd from a given reference frame (the ccnter01' mass in our case). The tendcncy of mancr to occupy alarger volume is compensated by tlle distribution 01'momentain each cell hccoming narrower (121. The disaiburion in theceH becoJ1les also cenlcred arollnd Ihe rnomentum associatedwilh lhe velocity nf lhe lluid elcmcnl. Consequel1lly, lhe Iher-mal speclrum in cach cell should he describcd on top 01'thiscollective 110w,Ihal is, rcferred from the collectivc fluid's ele-meol IllOlllentum. To describe [his hehavior we make the suh-slilution 01'lhe lllomenlull1 operalor pi' hy pll - 11/,/1/1, wherc111 is lile pion mass. The lerm 1111111 represents the colleclivemOlllentum of Ihe given pion Huid elemen1. Thc correspond-ing cquation hecollles

(I)

(2)1/'(i"1 = R, t) = O,

[D' ., .,]DI' - \7- + In- 1/)('",1)= O,

suhjccl 10 lhe condition

inlcraction ,¡.~("" 1.4 fm) during the evollllion 01' Ihe colli-SiOll,lhe pioll dispersion curve can be modificd and thc col.kctive propcrlics 01' the pion system could rescmblc moreIhose 01'a I¡quid than those 01'a gas [7]. An important con-seqllcncc is Ihe de\/clopmCnl 01' a surface tension thal actsas a rcl1ccting boundary. The E.802/R66 collaboralion hasreported that a baryon density of aboul eighl times normalnuclear densily is achicveJ in central Au+Au collisions all1.G A GcV/c. A large fraclion of this densily is due lo pi-(lIlS. Prom Fig. 3 01' Ref. 2, one can read lhal the tolal nUIIl-hcr 01'chargcd pions one 1I11itamulld central rapidity in Ihiskind of reaclions is about 200. Assuming that lhe numhcr 01'ncutral pions in (he same rapidity interval is half thc lotalIlumbcr 01'charged ones, the lotal pion yield in central col-lisions at lllid.rapidity is dNrr/dy "" 300. Since Ihe averagepion scparation is inverscly proportional lo one Ihird 01' Ihepion densily ", = (l/Alo)dN,/dy, where A(- G4 fm') islhe trans\/crse arca of the reaction and toe "" I fm) is a typicalformalion lime [10], d "" 0.6 1'01< ds and Ihus the condilion10 regard lhe pion systcm as a Iiquid is me1.

Pions lhat move towards lhe boundary 01'lhe syslelll fcellhe altractive potential hchind them and are reflected nack.The retlectioll properties inlroduced by lhe boundary shoulddepend on lhe 1110menlumcarricd by the givcn particle hut themain fcature introduccd by lhe reflecting surfacc is lhal it al-lows very lilllc wavc functioll Icakage and lo a good approxi-Illation, the pion wavc functions vanish oulside the boundary.\Vhen, as a cOllsequence oftlle expansion ofthe initially COIll.pressed ami hot syslem, Ihe pion average scparation beco meslarger than Ihe range of slrong interactions. the systcm be.comes a free gas hut the lransilion between Ihe liquid and lhegas phases is very rapid ami the pion mOlllcntulll distrihuliol1should he dctennined hy lhe distribution jusI befare frecleout.

Thus ir Ihe syslem of pions can he cOl1sidcrcd as COI1-tiflel! ami their wave rllnCliol1sas salisfying a given condilionat Ihe boundary just heforc frccze out Ihen the energy statesform a discrcte se1. The shapc 01'the voluI1Ie within (he COI1-IIning hOllndary deserves sorne atlenlion. For reactions witha large degrec 01' transparency, Bjorkcl1 Iike geomelry, witha prcdominanlly longitudinal elongation, sccms belter slliled.However, for AGS encrgies. a signifkant amount 01'stoppinghas been reponed hy lhe E-802/866 collahoration in central

{D" ., }-[i

DI-111')(1')]- + [-;\7-111)(r)v(r)]- +11/' ~'(•.,t)=O (3 )

alld we look for stationary solutions subjcct lO Ihe same con-dilion in Eq. (2) and also flnite al the origino Here wc considera parametriwlion of the three velocity veclor v( 1') Ihat scaleswith the dislance fram the cenlcr al' the fireball

(5)l'

v(r) = 13y/. (4)

Iignorc any asymlllelry between transverse ami longitudinalexpansiono Ü < ¡3 < 1, reprcsents lhe surfacc firehall \/eloc-ily. The corresponding explicit exprcssion far ¡(r) is

I-1(1') = JI - (pI" / R' .

\Ve identify Ihis velocity with the transversc Ilow velocity antl Eq. (3) ",ilh the gaJ1lma faclor givcn hy Eq. (5) can nnly be

Re\'. Mex. Fí.\". -15S2(1999) 116-119

lIS ALEJANDRO AYALA

L20_6 0.8mt - m (GeV)

0.40.2

R=8fmT = 120MeVN = 160~ ~ 0.5

0.1

o

1000 r--,--r---,--,---,--,---,-,

lOO~

lO ~

~~

~'~4L:

(6)

solvcd l1ulllcrically. In ordcr lo provide an analytical solution,""'e appruximatc lhe fUlletinn l' by lhe flrst tcrms DI' its Taylorcxpansion

This approximation is valid for nol too large valucs 01' /3. JnIhis case, Eq. (3) oecomcs an equation for particlcs movingin a sphcrical harmonic wcll with a r¡gid boundary.

The stationary statcs are

! (. t) - ~e~iE ••lteim¡Jr2 /(2R)y (i')e-n;'110Z /2rlIpnlm' ), - ~E lm'y':'wnl

R = 8fmT = 120MeVN = 115fi ~ 0.5

[(1+3/2) é;,1 3.2.,]

Xl F¡ - -4 2 ,l + :),Ctlll' ,(7)2 Ctni ~

whcrc 1 F¡ is a conllucnt hypcrgcomctric function ano lím'is a sphcrical harmonic. Thc quantitics A1I1 are rhe normal-izalion constants and are found from lhe condition

;-;

.l(ll,.~';'I""(r,t)gt</J"I,,,,(r.t) = 1. (x)

The paramctcrs 0111 ami [ni are rclatcd 10 lhe encrgy cigcn-valucs Enl by

I ( ) "'/ .,O'~¡/ ::::::m Enl - In ,B- R- 1

JOOO

10

01

() 0_2 004 06 0.8mI - m (GeV)

12

(9) FIGURE 1. Dislribution of .'lo = El - Eo, whcrc Eo is the grounostate and El is lhe Ilrst excitcd statcs.

Enl are given as the solutions to

Conseqllenl!y, the lt10lt1entllm dislrihlltiol1 IS obtaineJ byweighing lhe cOlltrihulioll fmm cach slale wilh the statisli-cal Bose-Einslcin factor and aJding up the contrihution fromal! nf tlle states

The normalized contrihution to the 1ll0lllentum Jislrihutionfmm the cncrgy statc wilh quantum numbers 11, l, mi is givenin terms nI' the absolule value squared of the Fourier Irans-I"rm ,,1 Eq. (7).

4')1!lm'(P)::::=:./ d~I'C-ip.r1j'nlm,(r). ([2)

Since lhe prohlclll has an azimuthal symmclry, the wavefllllClioll in 1ll0lt1cntum space docs not dcpend 011 the quan-IUlll nUlllher 1/1' ami is a function on!y ofthe 1ll0lllcntuIll mag-niludc.

( 15)N =" _._(2_' +_1)_L.- (,(l~,,[-II)/T _ 1n,l

where lhe chemical potential Ji is computeJ fmm

for a givell 1l1l111ber01"parlicles N.

\Ve now procecd lo describe the mid-rapidity pion dataon cenlral Au+Au reactions al 11.6 A GeV/c 12]. RatherIhan perfonning an exhaustive search in lhe wholc parame-ler space anJ in order to tesl ,he plausibility of this type 01'deseriplioll, here we fix Ihe valuc nf the parameters ¡nvolvedto reasonahle anJ more or)ess acceptcJ values. A more com-plele analysis rcquires tile propcr trcalmenl of Coulomh cf-feets, \Ve are eurrenlly wnrking in (his dircction. Wc take [1:31T::::=: 120 MeV, ji::::=: O.':> (colTcsponJing to an average collec-tivc expansion velocily (1') ~ 0.'1 e). For lhe mean negativepion lll11ltiplicily \Ve take Nrr- ::::=: 160 ]14j. \Ve consiJer al¡rcball raJius R ::::=: S fm. Figure! a shO\vs lhe thcoretical dis-triblltion eompared to data. In mder to compare with the dif-I"erenlia! cross section rcporte~l in Ref. 2 which is normalizedlo a subset 01" lió (centra! rapiJily) negative pions, Ihe curvehas heen lllultiplicd by a constant h'- ::::=: 0.56 that minimizesthe \ 2 when we compare to data aboye HlI - 111 = 0.4 GeV,Ihe rcgion where Coulomb ellects should start hecoming ¡esssignilicanl. Ahove 1111-111 = 0.'1 GeV lhe agrccIllent hetwcendala and lheory is very good. Below 111( - 111 ::::=: 0.4 GeV Ihe

( 10)

(11 )

( 13)

( 14)JIX " 9"1(1')(pp = L ('(1:;,,1-/1)/"1' _ 1

11,1

. 2E,,1 ( )1'f/)nlm' (p) ::::=: (2rr)3 I'I/'n/m' P

F [(1+3/2)_ é;', , ~. 'R'] =0.I 1 ,) 4 ,2 ,. + 2' ünl_ (1,,1

\l,'here

Hev. Me.\". ¡:¡.\....• 5 S2 (1999) 116-119

TIIE EXI'ANDlNG 1'10:-; L1QUIIJ ANIl THE PION SI'ECTRA 119

curve follow5 (he shapc of lhe data poinls hu( these lasl areslill ahove Ihe ealculation. This eould he a good fealure sineeolle knows Ihat lhe long-rangc Coulornh cffects should pushIhe dislrihution for low momentum ncgativc pions upwards.givcll thal thcir Coulornh intcraction with Ihe overall posilivcchargc is altractivc.

\Ve now use these paramctcrs lo descrihe data on posilivcpions. Fig. Ih shows lhe thcorctical distribution ca1culatcdfor T = 120 MeY, (3 = 0.5, R = 8 fm hUI a lolal posi-tive pioll Illultiplicity NTr+ = 1151H], compared lo dala. Inordcr lo compare with lhe diffcrcntial eross seclion rep0rledin Rcf. 2 which is normalizcd 10 a suhsct of 94 (central ra-pidity) positivc pions, Ihe curve has heen mulliplicJ hy theconstanl .,V+ = 0.59 that minirnizes lhe .\2 when we com-pare lo dala ahoye mt - m = 0..1 Gev. The agreemenL he-lweell daLa and Lheory is also very gooJ for lhe rcgion ahoye11I, - 11I = 0.4 Gev. Howevcr, Lhe raise of the curve below/1/1 - 1// = 0.4 GeV is less sleep than for the negative pioncase. This is casy to understand since rol' positive pions thetlcnsity is lower than 1'01'lhe negativc ones. Also in the sameregioll. the theoretical curve is marginally oelow data and lheCoulomb dístortion will push it evcn more bclow. \Ve spec.ulate that lhis signals that data prcfer a slightly lower valucnf Ihe radius hut again, this can only be confirrned after lhcpropcr inelusion 01'Coulomh effeeLs.

Finally. we IIlcntinn that the nonnalizalion constants Ar:t.are inlrmluccL! bccause lhe tolal experimental yields in Reí".2are nOfmalized nnly to a subset 01'charged pions in Ihe rapid-iLyinLerval I~yl < 1 around central rapidity, whereas in (hiswork. we need LOcon sidel' the full set of protluced pions sincein our scherne, all of lhem feel the presence 01'the boundary,irrespective nf lheir momcnlUm.

1. 11. Strühclc l'/ al. (NA35 Collahoralion), Z. ¡'hys. e 3M (l9XS)SI); R. Alhrccht et ai. (WASO Collahor.ltion). Z Ph)'.s. e 47(IYYO) 367; A 5YO(1995) 259c ; J. Barre"e /" al. (E814 Col-Iaboration). Phys, Let!. B .151 (1995) 93.

2. 1.. AlJlc e/ al. (E-802J866 Collahoration), Phys. Re\: e 57(I'N8) R466.

3. B.-A. Li and W. 8auer, Phys.l.(,tt. B 25~ (1991) 335.

'1. R. Brocrnann er al., "h)'5. Re\'. Lett. 5.1 (19S4) 2012; J, Soll-frank. P. Koch and U. Heinz, Php.I.A.'t!. B 252 (1990) 256; G.E.Brown, J. Stachei nnd G.,\1. Welke. Phys, /.l,t!.1l253 (1991) 19.

;,. See fOl" ex.amplc T.W. Atwaler. P.S. rrcier, amI J.1. Kapusla./'!lr.\'. f.d(. n 199 (1987) 30; K.S. Lec and LJ. Ilcinz, Z. Ph\'J. e~-'(IY89)~25. .

G. ('.- Y. \Vnng. Phys. Rev. e 4N (1993) 902; M.G.-H. MO.l.lnfnandc.-Y. \Vong:, Phy.s. Re~~e 51 (1995) 2135.

j. I~.V. Shuryak. Phys. Rev. [) 42 (1990) 1764.S. A. Ayala and A. Srnerzi. Phys. Ll'tr. U40S (1997) 20.9. I..IJ. Landi.lu. LlV. Akad. Nauk SSSR Ser. riz. 17 (1953) 51;

Also in enffee/n/ papers 01L.D. Lmu/alt, (Pergarnon Press andGordon and Breach, New York, 19(5) p. 569; E.Y. Shuryak and<.>.V.Zhirov, Phys. Letr. S9B (191:'0) 253.

4. Conclusions

In conclusion, \ve have shown Ihat a very good description 01'mid.rapidity, charged pion spectra can he achieved by a phe-nOlllenological calculation whose key ingredient is the proper¡rcatmenl 01' the large pion del1sily produced in cenLral, rela-tivislic Au+Au reacLiolls. This large density allows the con-sideration 01' the pion systelll as contlned during the earlyslages 01' the collision, heforc Jccoupling. Such a schcmehas consequenccs on hoth cnds 01' the spectra. At low trans-verse mass, Lheconvex shape 01'the distribution is due to thelarge value nf lhe chemical potenlial associated with the meanpion mulliplicity per evenl. Al high transverse momentum,Lheconvex shape 01"the dislrihulion is due lO the higher den-siLy01'states as compared Loa calculation without boundary.Another importanl elelllent is the inclusion 01'col1ccLivellow.\Ve finJ Lhatan average llow velocity 01'(v) ~ 0.4 c togetherwilh ti temperalUrc T = 120 MeV docs a very good joo de-scrihing dala above m, - 111 = 0.4 GeV when Ihe t1rebal-I's radius is ahoul R = 8 fm. A more conclusivc statemcntcan be maJe only afler \••..e include a proper trcalment of theCoulomb correcLions 1151. Perhaps more importantly is lhefacI that lhe ditlercnt rates al which lhe posilivc and nega-live pion distributions rise at low \'alues 01'mi - 111 can hcunderstood in pan as an ellecL rclated to the correspondinglydilferent mcasureJ yic1ds. ll1aking Lhe negative pion suhsys-Lcmdcnscr LhatLhepositive pion (lne.

Acknnwlcdgmcnts

SUppor! for lhis work has heen received in par! hy CONA-CyT México under granl No. 127212-E.

111.JD. Bjorkcn, Ph,.,. R"". /) 27 (IY83) I~O.

11. D. Kusne/ov nnd G. Bcrtsch, Ph.v.\". Rel'. e 40 (1989) 2075.

12. P.J Sic,"cns and J.O RaslIlllsscn, "h\'J. Rev. Lell. ~2 (1979)8811.

13. As discussed by S. ESlIl1li. S. Chapman, H. van Hecke, andN. Xu. Phys. Rl'l'. e 55 (1997) R2163; there exists nn anlicor.relation bclwecn lhe Iransverse Ilow velocity and Ihe [reezc outtcmpcra(llfe in such a way that higher tcmpcralures irnply lowcrcxp:lllsioll veJocities and vice versa. 11is likcly that for AGS en-rrgies not too high Icmpcraturcs are rcached and thus the corn-hination of T and ti tllat we considero

1,t. F. Vidchaek el al. (E-S02JH6() Collaboration). Proc. Qllark Mat-ler 95, NI/el. Phys. A590 (1995) 249c. Thc E-802J866 Collaho-ralion has reponed a total nUlllhcr of negatively and positivelycharged pions on Ihe order of N,,_ :::' IGO and X

ff+ :::: 115.

15. A lllelhoo to incorporate Coulomh corrections in the descrip-lion 01' charged plon spcclra, Illat cfln he applicable in (he COtl-lexl 01' Ihis work, has recently hern proposed. see A. Ayala amI1. Kapusta.I'/IYs. Rl'I'. e S(, (1997) 407; A. Ayala. S. kon. andJ. Kapllsla. I'hys. Rel'. e 5t) (1999) 3324.

R('I~Akr. FÍ.\'. 45S2(1999) 116-119