The influence of the nuclear Coulomb field on transverse spectra of $\pi^+$
and $\pi^-$ measured in $Pb+Pb$ reactions at 158 A GeV has been investigated.
Pion trajectories are calculated in the field of an expanding fireball. The
observed enhancement of the $\pi^-/\pi^+$ ratio at small momenta depends on the
temperature and transverse expansion velocity of the source, the rapidity
distribution of the net positive charge, and mainly the time of the freeze-out.Comment: 11 pages including 2 figure

D. Röhrich

H. Bøggild

H. Heiselberg

H. Sorge

H. W. Barz

I. G. Bearden

J. Bächler

J. J. Gaardhøje

J. P. Bondorf

P. Braun-Munzinger

S. V. Afanasiev

S. V. Afanasiev et al.

T. Alber

W. Benenson et al.

English

arXiv

The influence of the nuclear Coulomb field on transverse spectra of  λ + and λ- measured in Pb + Pb reactions at 158 A GeV has been investigated. Pion trajectories are calculated in the field of an expanding fireball. The observed enhancement of the λ-/λ+ ratio at small momenta depends on the temperature and transverse expansion velocity of the source, the rapidity distribution of the net positive charge, and mainly the time of the freeze-out

Bondorf, J. P.

Gaardhoje, J. J.

Heiselberg, H.

Barz, Hanns-Werner

Qucosa

Freeze-Out Time in Ultrarelativistic Heavy Ion Collisions from Coulomb Effects in Transverse Pion Spectra

Qucosa – Hemholtz-Zentrum Dresden-Rossendorf

FORSCHUNGSZENTRUM ROSSENDORF e.v. F Z R  FZR-168 February 1997 Preprint H. W. Barz, J. P. Bondorf, J. J. Gaardhflje, und H. Heiselberg Freeze-Out Time in Ultrarelativistic Heavy Ion Collisions from Coulomb Effects in Transverse Pion Spectra Forschungszentrum Rossendorf e.V. Postfach 51 01 19 D-01314 Dresden Bundesrepublik Deutschland Telefon (0351) 260 2263 Telefax (0351) 260 3700 E-Mail barz@fz-rossendorf.de Freeze-Out Time in Ultrarelativistic Heavy Ion Collisions from Coulomb Effects in Transverse Pion Spectra H.W. Barzl, J.P. BondorfL, J.J. Gaardh@je2, and H. Heiselberg3 Institut für Kern- und Hadronenphysik, FZ Rossendorf, Pf 510119, 01314 Dresden, Germany 2Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark 3NORDITA, Blegdamsvej 17, 2100 Copenhagen, Denmark Abstract: The influence of the nuclear Coulomb field on transverse spectra of T+ and X- measured in P b  + P b  reactions at 158 A GeV has been investigated. Pion tra- jectories are calculated in the field of an expanding fireball. The observed en- hancement of the T- /x+  ratio at  small momenta depends on the temperature and transverse expansion velocity of the source, the rapidity distribution of the net positive charge, and mainly the time of the freeze-out. PACS numbers: 25.75.+r, 25.70.Pq In heavy ion collisions at ultrarelativistic energies, nuclear systems with a total net positive charge of around Z = 160 can now be created and studied. It has recently been observed [I] in central Pb+Pb collisions a t  CERK-SPS energies (158 A GeV) that the ratio of negative pions to positive pions is significantly increased at  lotv transverse masses (mLNm, + 50MeV) for central rapidities. A possible cause for this effect is the Coulomb interaction between the produced pions and the positive charge from the reaction partners. Such effects are already known in reactions at intermediate energies [2]. If the two incoming ions pass through each other without stopping, the com- bined electromagnetic field of the two moving ions has only a minimal effect on the motion of the pions that are produced around midrapidity. On the average, a charged pion experiences at time rf and transverse distance R a momentum change of Ze2R/(y,rf)*, where Z is the combined charge of the target and projectile nuclei and y, is the Lorentz factor. This effect is inversely propor- tional to the bombarding energy. On the other hand, if the colliding ions are strongly decelerated (large nuclear stopping) the total charge stays together for a sufficient amount of time to accelerate or decelerate the produced charged pions significantly. Under such conditions distortions of the pion spectra can occur leading to  non-uniform pion yield ratios. In this letter we report on a study within a dynamical model of the enhance- ment of the T-/T+ ratio as a function of the properties of the participant fireball and investigate the sensitivity to dynamical features such as the total participant charge, the transverse expansion velocity and the temporal and spatial extent of the source. In relativistic heavy ion eollisions a t  SPS energies most of the pions are pro- duced rather late in the collision history. They are formed either directly from strings or in the decay of resonances. In the dense medium they undergo sev- eraf. coHisions so that the pionic ensemble Comes close to thermal equilibrium, as carr be deduced from properties of measured transverse spectra. Due to the long range nature of the Coulomb force the details of the production mechanism are not 1-ery important for our purpose. We thus consider the system at a proper time rf sufficiently large that the violent stage of the collision has ceased, i.e. close to freeze-out. Already at this time a certain charge separation has occurred during the equilibration phase due to the inhomogeneous Coulomb field. How- ever, the main Coulomb acceleration effect occurs after freeze-out during the free motion in the field of the source. At this time the pions already hat-e a spectrum very similar to that measured in experiment, since only the Coulomb field leads to further significant modification of the spectrum shape, particularly for slowly moving particles. We assume that the pions can be described by a thermal distribution (char- acterized by a temperature T) that is superimposed on a collective outward Aow of the matter characterized by a 4-velocity field u p  This field describes slices which move with rapidity y' in longitudinal z-direction (i-e. along the beam axis). We assume that the reaction Zone has an axially symmetric cylindrical shape and that the the transverse 4-velocity u~ scales linearly with the radial distance r l  from the axis. The transverse velocity describes the motion of a characteristic radius R. Assuming that the rapidity slices follow a Gaussian distribution centered a t  the center-of-mass rapidity y„ the pion source distribution at freeze-out reads d 6 N  d y d p l d z d r l  (Y' - 6 ( z -  T - ~  sinh y ' ) ~ ( % )  " J 'Y' e x ~  [- 2Ay2 R where Y, e, and -4" denote the pion momentum and charge and the ekectrornag- netic 4-potential of the source, respectively. The distribution is a funetion oi" the rapidity y,  the transverse momentum pl ,  and tfie longitudinal and transt-erce extensions z and r l .  In the spirit of the Bjorken picture the longitudinal dis- tance z is related to the rapidity y' of the cells via an effective freeze-out time T~ although in reality there will be a ,distribution of freeze-out times. The width h y  is taken from experiment. For the transverse density profile F(rl /R)  we find that two different profiles of the transverse pion distribution, a sharp cut-off at radius F ( r l )  0(1 - r I / R )  and a Gaussian distribution exp ( -2r t /R2)  both with the Same mean Square radius, give similar pion ratios within a few percent. Due to the Coulomb potential Ap the shapes of the distributions are different for positively and negatively charged pions. The subsequent motion of the pions in the expanding electromagnetic field created by the net charge is described by the 4-velocity wp which satisfies the equation of motion in the electromagnetic potential Alb where T denotes the proper time and rn is the pion mass. The final spectra are obtained by sampling over a set of different initial conditions using eq. ( 2 ) .  The potential Ap is obtained from the current distribution of the (positive) net charge. Most of this charge is carried by the baryons which are not fully stopped [3, 41 and, thus, may have a wider rapidity distribution than the pions which are preferentially produced at  midrapidity. In the case of symmetric reactions tve ."- - parametrize the charge distribution as 1 f c h ( ~ )  = Zcent (Y - Y d 2  ) + exp (- (Y + y d 2  &Ach 2 A ~ &  )I describing the participant charge as ttvo distributions of width Agch centered a t  rapidities f yl in the center of mass System. For non-central collisions the rest charges z t ,  of the target and projectile continue to move with their original rapidities yt and y,. The participant Zone further expands due to the collective transverse flow (1) and the random thermal motion. We model the time evolution of the charge dis- tribution by a Monte-Carlo sample of charged test particles. These test particles move on straight trajectories with initial conditions given by eq. (2) but now with the longitudinal distribution (4) instead of the single Gaussian distribution. The potential Ap is then calculated by summing up the retarded potentials of the test particles. Due to the large retardation effects an essential part of the elec- tromagnetic potential is generated by the charge distribution prior to freeze-out. We describe this situation between 0 < T < 70 by the hydrodynamic Aoiv (2) of the matter. During this period the transverse flow is assumed to increase linearly with time. The potential depends somewhat on the particle composition of the charged Zone since the thermal velocity of the particles depends on their masses. We have included an excess of negative pions over positive pions compensated by the positive kaon net charge [5] .  In Fig. la-C we compare the result of the calculations to recent data from the NA44 experiment for P b  + P b  collisions at E = 158AGeV and for S + P b  and S + S at 200AGeVA [1], all for central collisions and rapidities 3.3 < y < 4.0. The calculations take the actual NA44 detector acceptance [I] into account. In addition, we also plot calculations corresponding to a detector located esactly a t  midrapidity. Since absolute ~ - / n +  ratios are not available, we normalize our calculated ratios to unity in the region 200MeV < mt - m, < 400MeV, as is done for the experimental results. Lead data were taken with a trigger selecting 15% of the total interaction cross section. In a sharp cut-off model this implies an average charge of Zmnt = 122 in the fireball. In the calculations rve fix the charge of the participant zone to this value. We transform the transverse area into a circle of B„, = 5.8 fm. At freeze-out we take a radius of R = R(rf) = Rgeom + Prf/2 assurnirig the linear increase of the transverse velocity. The rvidths of the pion rapidity distributions was chosen to be Ay = 1.3 corresponding to measurernents in ,945, S + Pb [6] and mith measured transverse energy spectra in Pb + Pb collisions['7]. 1Ve use a temperature of T = 120kIeV and ß= 0.62 corresponding to a mean transt-erse espansion velocity (ß) = 0.42. These parameters are well compatible with the values obtained from a systematic analysis of measured transverse T, K and p spectra from Pb + Pb data [8], where a definite anticorrelation between extracted temperature and transverse velocity was established. Recent d N / d y  vs y data from the NA44 [3] and NA49[4] experiments provide us with the rapidity distribution of protons which can be characterized by the parameters AyCh = 0.84 and yl = 1.1, implying a rapidity loss of 1.7. Using these values a good description of the experimental data (see Fig. la)  was obtained using a freeze-out time of 7 fm/c. Figs. l b  and l c  show a comparison for lighter systems. Here we use a higher temperature of 130 MeV and a smaller velocity (ß) = 0.29 [6, 91. A fit to the S + S data [6] indicates a smaller rapidity loss of 1.3, leading to 4ych = 1.25, yl = 1.7. For the asymmetric S + Pb reaction a good description of the data is obtained (see Fig. lb) by assuming that 30 protons from the Pb target and all protons from the projectile participate in the fireball decelerated to their respective rapidities yl in eq. (4). The efFect of varying the various parameters is exhibited in Fig 2. The mag- nitude of the enhancement clearly scales with the total participant charge and diminishes if the charge is distributed over a wider rapidity range (Fig. 2a). If the system expands fast, the slower pions are overtaken by the expanding po- tential and experience a smaller net charge. This can be caused by increasing collective flow (Fig. 2b) or by increasing thermal motion (Fig. 2c) and results in a decrease of Coulomb effects. A stronger efFect is observed when the formation time rj increases, See Fig. 2d, reflecting a more diluted charge distribution. The profile of the rapidity distribution of the net charge also plays a role. Further, we remark that the pion ratio decreases slightly from 1.60 to 1.55 if the net charges of kaons and pions are not taken into account. Hanbury-Brown and Twiss (HBT) analysis provides two independent com- parisons of the freeze-out time. In a longitudinally expanding system with Bjorken scaling the freeze-out time is related to the longitudinal HBT radius by rf = Rl d m ~ / ~ .  From the NA44 data Rl = 5.5fm [10] we estimate rf = 8fm/c. From transverse HBT radii we obtain R ' =  2R, 10fm. Com- paring to R = Rp, + Prf /2 we deduce rf = 8fm/c; this value should, however, be corrected for transverse flow which affects the HBT radii. We conclude that both freeze-out times extracted from the longitudinal and transverse HBT radii are compatible with the one we find from r-/n+. We also remark that RQMD calculations predict substantially larger values of rf =I5 fm/c [ll]. A simple estimate of the Coulomb effect can be made from the retarded elec- tric field resulting from a net charge distribution, diVch/dy, that is approsimately constant in rapidity Such a longitudinally streaming charge distribution gener- ates in transverse direction an electric field E E 2e(dNch/dy)rl/(tR2). Here, t is the time after collision and R is the transverse size of the charge distribution. In average this field leads to a momentum change of the n" by for small transverse momenta. The factor f is of order the inverse of the average proton velocity but also depends on the freeze-out time and the transverse expan- sion [12]. Assuming a thermal transverse momentum distribution at freeze-out, dN/dp: exp(-ml/T), as has been used the spectra [8], we obtain for small transverse momenta after correcting for the Coulomb effect of Eq. (5) This approximate result shows that the linear decreasc of the ratio takcs placc orr a scale of m l  - m = T. The enhaneement for Pb+Pb collisionc a t  midrapidities can be estimated using diVch/dy 35 [4]? R 10 fm and f = 2 to be 4 3  0.6. This approximate vafue and Eq. (6) are in qualitative agseenient with the pion ratio in Fig. 1. TVe have also calculated the kaon ratio K-/KS in P b  + Pb collisions using the Same parameter set as used for the pions (Fig. ld). As the kaon is heavier we expect the corresponding K-/K+ ratio to be smaller by a factor of rnK/m, - 3 accordingly to Eq. (5). In Summary, we find that the observed n-/71-+ ratios can be qualitatively explained as resulting from the Coulomb acceleration of pions by the positive participant charge in a rapidly expanding system. The quantitative agreement with experimental data requires a relatively short freeze-out time of about 7 fm/c assuming a mean transverse flow velocity of (P )  = 0.42 and a temperature of 120 MeV. Such values are compatible with analyses of transverse pion, kaon and pro- ton spectra. Larger freeze-out times require less amount of flow. Consequently, the measured pionic charge ratios provide independent information on the freeze- out time scale. The freeze-out time obtained for central Pb+Pb coilisions is compatible with freeze-out times and the source sizes extracted in HBT analyses. Finally, we note that the magnitude of the Coulomb effects in pion spectra are sensitive to the degree of stopping and the resulting distribution of the positive charge. At RHIC and LHC energies the stopping and in particular the net charge at midrapidity are expected to be smaller, and consequently we predict (see Eq. (6) and Fig. 2a) correspondingly smaller Coulomb effects. We acknowledge fruitful discussions with Dr. Nu Xu (Los Alamos). The support of the German BMBF and the Danish Natural Science Research Council is appreciated. References [I] H. Boggild et al., N444 collaboration., Phys. Lett. B 372, 343 (1996) [2] W. Benenson et al., Phys. Rev. Lett. 43, 683 (1979); K. G. Libbrecht and S.E. Koonin, Phys. Rev. Lett. 43, 1581 (1979) [3] I. G. Bearden et al., NA44 collab., Phys. Lett. B388 (1996) 431 [4] T. Wienold for NA49 collab., Quark matter '96, Proc. of the 12th Int. Conf. on Ultra-Relativistic Nucleus-Nucleus Collisions, Heidelberg, Germany, hIay 20-24, 1996 [5] A 10 % n- excess is assumed, J. Stachel, NA49, priv. comm. [6] D. Röhrich et al., NA35 collaboration, Nucl. Phys. A566, 35c (1994); J. Bächler et al., Phys. Rev. Lett. 72, 1419 (1994) [7] T. Alber et al., Phys. Rev. Lett. 75, 3814 (1995) [8] I. G. Bearden et al., NA44 collaboration, submitted to Phys. Rev. Lett. [9] P. Braun-Munzinger et al., Phys. Lett. B 365, 1 (1996) [I01 A. Franz for NA44 collab., Quark matter '96, ibid. as ref. [4] [ l l ]  H. Sorge, Phys. Lett. B373, 16 (1996) 1121 H.W. Barz et al., to be published. I I I I 8 8 - (0) charge - l ( b )  radial flow - - L -- 8 8 - - -  <P>= 0.0 - <ß> = 0.42 I I I I - (C) temperoture -T (d) break-up t ime L s T s * s * 8 8 - - -  T= 80 MeV - - -  T,' 3 fm/c - - T= 120 MeV - T ~ =  7 fm/c - 0 100 200 0 100 200 m,-mK (MeV) ml-m, (MeV) Fig. 2. Comparison of the sensitivity of the calculations to  variation of the main parameters. Solid lines represent the best fit calculations for the P b  + P b  reaction shown in Fig. I. Variation of : (a) total participant charge Zent and width of rapidity distribution, (b) average transverse expansion velocity (P) ,  (C) temperature a t  freeze-out, (d) freeze-out time rj. I I I I I I - (a) charge - (b) radial f low . -- - D I I - - -  <P>= 0.0 - - <ß> = 0.42 0 100 200 0 100 200 ml-m, (MeV) m,-m, (MeV) I 1 ( C )  temperature 1 T - s I Fig. 2. Comparison of the sensitivity of the calculations to variation of the main parameters. Solid lines represent thc best fit calculations for the Pb + P b  reaction shown in Fig. 1. Variation of : (a) total participant chargc ZcenL and width of rapidity distribution, (b) average traiisverse expansioii velocity (P ) ,  (C) temperature at freeze-out, (d) frcczc-out time T/. I I -T (d) break-up time D - D - - -  T= 80 MeV * ,  - T,,= 7 fm/c 

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Freeze-out time in ultrarelativistic heavy ion collisions from Coulomb effects in transverse pion spectra

HSSS - Hochschulschriftenserver der SLUB

Freeze-Out Time in Ultrarelativistic Heavy Ion Collisions from Coulomb
  Effects in Transverse Pion Spectra

Freeze-Out Time in Ultrarelativistic Heavy Ion Collisions from Coulomb Effects in Transverse Pion Spectra

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