arX

iv:n

ucl-

ex/0

3060

33v2

26

Sep

2003

Charged Pion Production in 2 to 8 AGeV Central Au+Au Collisions

J.L. Klay,1 N.N. Ajitanand,2 J.M. Alexander,2 M.G. Anderson,1 D. Best,3 F.P. Brady,1 T. Case,3

W. Caskey,1 D. Cebra,1 J.L. Chance,1 P. Chung,2 B. Cole,4 K. Crowe,3 A.C. Das,5 J.E. Draper,1

M.L. Gilkes,6 S. Gushue,7 M. Heffner,1 A.S. Hirsch,6 E.L. Hjort,6 L. Huo,8 M. Justice,9 M. Kaplan,10

D. Keane,9 J.C. Kintner,11 D. Krofcheck,12 R.A. Lacey,2 J. Lauret,2 C. Law,2 M.A. Lisa,5 H. Liu,9

Y.M. Liu,8 R. McGrath,2 Z. Milosevich,10 G. Odyniec,3 D.L. Olson,3 S.Y. Panitkin,9 C. Pinkenburg,2

N.T. Porile,6 G. Rai,3 H.G. Ritter,3 J.L. Romero,1 R. Scharenberg,6 B. Srivastava,6 N.T.B. Stone,3

T.J.M. Symons,3 S. Wang,9 R. Wells,5 J. Whitfield,10 T. Wienold,3 R. Witt,9 L. Wood,1 and W.N. Zhang8

(E895 Collaboration)1University of California, Davis, California 95616

2Department of Chemistry and Physics, SUNY, Stony Brook, New York 11794-34003Lawrence Berkeley National Laboratory, Berkeley, California 94720

4Columbia University, New York, New York 100275The Ohio State University, Columbus, Ohio 43210

6Purdue University, West Lafayette, Indiana 47907-13967Brookhaven National Laboratory, Upton, New York 11973

8Harbin Institute of Technology, Harbin, 150001 People’s Republic of China9Kent State University, Kent, Ohio 44242

10Carnegie Mellon University, Pittsburgh, Pennsylvania 1521311St. Mary’s College of California, Moraga, California 94575

12University of Auckland, Auckland, New Zealand

(Dated: November 3, 2018)

Momentum spectra of charged pions over nearly full rapidity coverage from target to beam rapidityhave been extracted from 0-5% most central Au+Au collisions in the beam energy range from 2to 8 AGeV by the E895 Experiment. Using a thermal parameterization to fit the transverse massspectra, rapidity density distributions are extracted. The observed spectra are compared withpredictions from the RQMD v2.3 cascade model and also to a thermal model including longitudinalflow. The total 4π yields of the charged pions are used to infer an initial state entropy produced inthe collisions.

PACS numbers: 25.75.-q, 25.60.Gc

I. INTRODUCTION

One of the primary goals of the study of heavy ion col-lisions at relativistic energies is the improvement of ourunderstanding of the bulk properties of nuclear/hadronicmatter at high temperatures and densities. As a first stepin understanding these properties, one should carefullycharacterize the particle species that make up the bulkof the matter. For the energy regime of the Bevalac/SIS(0.2 to 1.2 AGeV), collisions between heavy nuclei causea compression of the nuclear matter, resulting in a disas-sembly into the constituent neutrons and protons, whichare emitted either individually or bound within variouslight composite fragments (d, t, 3He, 4He)[1, 2].

At the top energy of AGS Au+Au collisions of 10.8AGeV, the most copiously produced charged particles arethe lightest of the mesons, the pions[3, 4]. The matterhas evolved from a heated and compressed gas of nucleonsinto a hot dense gas of hadrons, predominantly pi mesons.Thus the energy region from 2 to 8 AGeV represents atransition regime. Studies of nuclear stopping suggestthe maximum density achieved increases from three timesnormal nuclear density at 1 AGeV[5, 6, 7] to eight timesnormal nuclear density at 10 AGeV[8]. Measurements ofthe proton directed and elliptic flow indicate that hydro-

dynamic flow has saturated across this energy range andcannot account for the increased energy available[9, 10].This additional available energy goes primarily into pionproduction. By studying the pion production across thisregime we are able to observe nuclear matter in transi-tion.

Early on in the study of heavy ion collisions, an en-hanced yield of pions was seen as a possible signatureof a transition to a deconfined state of matter. How-ever, in the early studies of pion yields in the 1 AGeVenergy range at the Bevalac[11, 12, 13], it was ob-served that the measured pion production cross sectionswere smaller than predicted at the time. This obser-vation led to the conjecture, which was later experi-mentally demonstrated, that the excess kinetic energywas converted into hydrodynamical flow effects. Thestrong radial flow observed at this energy implies a sig-nificant expansion and cooling, which limits the freeze-out pion multiplicities[14, 15]. More recently, more de-tailed mesurements of the pion yields from 1 AGeVAu+Au collisions have become available from the SISexperiments[16, 17, 18, 19]. These results demonstrate aroughly two to one ratio of π− over π+ and a strong non-thermal low-pt enhancement. Both features are strongindicators that pions at this energy are produced almost

2

exclusively through the Delta resonance[20]. In full en-ergy AGS collisions (Au+Au at 10.8 AgeV), pion pro-duction has been studied at midrapidity[3] and at targetrapidity[21]. Although there is still an asymmetry be-tween π− and π+ production and there is also still evi-dence of a low-pt enhancement in the momentum spectra,these effects are much less significant than at SIS. Thebroad rapidity coverage for both pions and protons atthese energies has also been used to study the develop-ment of longitudinal flow[22].This paper will detail the development of pion produc-

tion across the beam energy range from 2 to 8 AGeV. Therole of the Delta resonance production mechanism will beexplored through observations of the overall pion ratiosand through RQMD simulations. The rapidity densitydistributions will be used with previously published pro-ton rapidity distributions[23] to explore the effects of col-lective longitudinal flow. The overall pion multiplicitieswill be used to establish a low energy baseline for a re-cent pion multiplicity based QGP search at the CERNSPS[24].

II. DATA COLLECTION AND ANALYSIS

The data were taken at the Brookhaven National Lab-oratory Alternating Gradient Synchrotron (AGS) by theE895 Experiment using the EOS TPC [25] during a seriesof runs in 1996. This article presents charged pion trans-verse mass and rapidity density spectra from Au+Aucollisions at nominal beam energies of 2, 4, 6, and 8GeV/nucleon (AGeV). (After correcting for energy lossbefore the target, the actual beam energies of collisionsat 2 and 4 AGeV were found to be 1.85 and 3.91 AGeV,respectively. No corrections were necessary for 6 and8 AGeV.) The EOS TPC provides nearly 4π solid an-gle coverage, which makes global characterization of thecollision events possible. Charged particle momenta arereconstructed from the helical trajectories of tracks re-constructed from the ionization trails left by particlespassing through the TPC, which was situated inside theMulti-Particle Spectrometer (MPS) magnet. Data pre-sented for 2 AGeV collisions were taken in a 0.75 Teslafield, while 4, 6, and 8 AGeV collisions were taken in a 1Tesla field.A primary track multiplicity for each event is obtained

by rejecting those tracks which do not pass within 2.5cm of the reconstructed event vertex in the target. Themultiplicity distribution from a minimum trigger-biasedsample of events is used to discriminate event centralityclasses by assuming a monotonic relationship between theimpact parameter and multiplicity [26]. The data pre-sented in this paper are selected on the top 5% most cen-tral events; an ensemble of approximately 20,000 eventsat each energy for this centrality selection were used toobtain the spectra.Identification of particle species is determined via mul-

tiple sampling (up to 128 samples) of the ionization in

P10 (10% methane, 90% argon) drift gas. The averageionization energy loss, 〈dE/dx〉, is computed for eachtrack from the available samples via a truncated meanto reduce the influence of the large energy tail of the dis-tribution. The 20% highest dE/dx samples are discardedfrom the calculation of the average.

Fig. 1 shows a typical particle identification map with〈dE/dx〉 plotted versus reconstructed track rigidity (=p/Z) at 6 AGeV. The various particle species are iden-tified by their separation into bands. The pion spectrawere obtained by fitting projections of the 〈dE/dx〉 innarrow bins ofmt−m0 and rapidity, using an assumptionof the pion mass and charge to calculate the (mt−m0,y)coordinates for a given measured particle (px,py,pz).

The single particle 〈dE/dx〉 projections are often de-scribed by a Gaussian distribution centered on the meanvalue predicted by a Bethe-Bloch formulation. However,this assumes that the calculation of 〈dE/dx〉 for eachtrack was obtained in an identical fashion for all tracksin the distribution. In fact, the truncated mean methodused in this analysis introduces a skewing toward larger〈dE/dx〉 which comes from combining tracks of differ-ent number of samples, Nhits. One way to avoid thisskewing is to divide the data into bins of Nhits, whichreduces the effect. However the reduction in statisticsfor each bin leads to increased uncertainty in the deter-mined yields. Therefore, for this analysis, a model ofthe Nhits-integrated single particle 〈dE/dx〉 distributionshapes was used with the predicted mean values from aBethe-Bloch parameterization of the 〈dE/dx〉 as a func-tion of βγ to extract the total yields of pions from eachprojection. The model is represented by a correlated sumof Gaussian distributions: a main Gaussian for the bulkof the distribution plus a smaller, offset “shoulder” Gaus-sian for the high-〈dE/dx〉 tail. The parameters of themodel were studied as a function of beam energy, parti-cle type and (mt − m0,y) bin in regions where the sin-gle particle distributions can be separately characterized.Tight constraints on the model parameters were appliedin order to extrapolate the model into regions where mul-tiple particle distributions partially overlap. Thus therelative yields of different particles were deconvoluted ineach (mt − m0,y) bin: 0.1 unit rapidity slices over thefull rapidity range from target to beam rapidity, withthe midrapidity bin covering the range -0.05 < ycm <0.05, and in 25 MeV/c2 bins in mt-m0 in the range 0< mt-m0 < 1.0 GeV/c2. Particles of the wrong massand/or charge contaminating the pion sample in a given(mt −m0,y) bin were discarded.

Fig. 2 shows an example of a single (mt −m0,y) sliceat midrapidity for 6 AGeV and 0.125 <mt −m0< 0.150GeV/c2. The total yields are obtained by integrating thearea under the fitted distributions for each particle. Op-positely charged particles are projected separately. Theinset shows the projection of negatively charged particles,which are well-characterized by this method of particleidentification over the entire range of transverse mass andrapidity. However, as the rapidity and transverse mass

3

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-3 -2 -1 0 1 2 3plab/Z (GeV/c)

log 10

(<dE

/dx>

) (a

rbitr

ary

units

)

6 AGeV

π- π+

pd He

FIG. 1: 〈dE/dx〉 as a function of rigidity at 6 AGeV. A pa-rameterized Bethe-Bloch prediction is used to fix the meanenergy loss for each particle species as a function of rigid-ity. The narrow slices illustrate the location of the 〈dE/dx〉projections shown in Fig. 2.

increase, the positively charged pions suffer increasingcontamination due to the overlap with the positive kaonsand protons. The range of mt − m0 and rapidity overwhich the π+ yields are extracted is therefore more lim-ited.

Electron contamination was estimated by studyingtheir yields in the regions of phase space where theyare relatively cleanly identified (for plab < 150 MeV/cand 300 < plab < 500 MeV/c) and interpolating betweenthese limits. It was found that electrons contribute ap-proximately 10% to the observed yield of pions for labmomenta 150 < plab < 250 MeV/c. Since the electronyields fall significantly as a function of their own trans-verse momentum, the contamination predominantly ef-fects the lowest pion mt − m0 bins. The errors on thequoted pion yields account for possible systematic uncer-tainties in the determination of the electron contamina-tion.

Observed kaon yields from the same beam energyrange, measured by E866/E917 [27, 28], folded with theEOS TPC detector response allows us to extend the reachof the π+ fitting out to plab ∼ 1.2 GeV/c. For lab mo-menta above 1.2 GeV/c, however, the π+ become hope-lessly entangled with the protons, which are approachingminimum ionizing 〈dE/dx〉. This cut-off manifests atdifferent mt − m0, depending on the rapidity and bom-barding energy.

Detector response to the final-state collision products

0

100

200

300

400

500

600

0 0.02 0.04 0.06 0.08 0.1 0.12x 10

-3

5654 π+

448 e+

283 K+1852 p

<dE/dx>

Cou

nts 6 AGeV-0.05 < y-yc.m. < 0.05

0.125 < mt-mπ < 0.150 GeV/c2

<ptot> = 0.552 GeV/c

0

100

200

300

400

500

600

0 0.05 0.1x 10

-3

7485 π-

80 K-

515 e-

<dE/dx>

Cou

nts

FIG. 2: Projections of the positive and negative (inset) parti-cle 〈dE/dx〉 at mid-rapidity and 0.125 < mt − m0 < 0.150Gev/c2, in the rigidity ranges indicated on Fig. 1. The(mt −m0,y) ranges were calculated assuming the pion mass.The function used to fit the data is the two-Gaussian model(described in the text) of the single particle 〈dE/dx〉 distri-butions, one for each particle type. The integrals under thepion peaks are the total yield for the fitted (mt −m0,y) binand the contributions from all other particles are discarded.

has been extensively studied using the GEANT 3.21 sim-ulation package. Small (maximum of 4 per event) sam-ples of pions with momentum distributions approximat-ing the real data are embedded into full data events.These particles are tagged and propagated through thedata reconstruction chain to determine the effects of de-tector acceptance, tracking efficiency, and momentumresolution.Since the beam actually passes through the sensitive

volume of the detector, there is no explicit low-pt mea-surement cut-off. However, forward-focusing causes in-creased tracking losses at low pt and forward rapiditiesdue to higher track densities and track merging. Losses atbackward rapidities and high mt −m0 are dominated bythe geometric acceptance of the detector, while at moreforward rapidities, the high mt − m0 tracks are stifferand therefore suffer from worsening momentum resolu-tion. Fig. 3 shows the overall detection efficiency ob-tained from simulations as a function of mt − m0 andrapidity for pions at each beam energy. The contoursindicate steps of ∼15%. The raw data are corrected forthese effects to extract the total yields of positive andnegative pions as a function of mt −m0 and rapidity ateach beam energy.Due to the large acceptance of our device, we can test

4

Rapidity (y-yc.m.)

mt-m

π (G

eV/c

2 )

2 AGeV 4 AGeV

6 AGeV

15%75%

90%8 AGeV

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

-1 0 1 -1 0 1

FIG. 3: EOS TPC total “efficiency” (≡ 1/correction, includ-ing effects of acceptance, tracking efficiency and momentumresolution) for pions produced in 2, 4, 6 and 8 AGeV 0-5% central Au+Au collisions. The efficiencies are obtainedfrom GEANT simulations of the detector response to MonteCarlo tracks embedded into real data events and propagatedthrough the reconstruction chain. The contours, shown instep of ∼15%, include the effects of geometric acceptance,tracking efficiency and momentum resolution.

the systematic uncertainties on the yields by comparingforward and backward rapidity corrected spectra. Over-lapping acceptances at 2, 4, 6 and 8 GeV allow us toconclude that rapidity bins corresponding to ylab < 0.5are most affected by the systematic uncertainties in ourcorrections. The underestimated corrections in this re-gion of phase space are not surprising given that nei-ther detector performance nor detector simulations wereoptimized for target rapidity in the lab frame. Heretracks cross the fewest pad rows and have the small-est radii of curvature. Mid-rapidity yields at all fourbeam energies were checked against published resultsfrom E866/E917[29, 30]. The average level of agreementbetween our spectra and E866/E917 over all mt−m0 andbeam energies is observed to be approximately 7%. Theminimum systematic uncertainties at all rapidities for theE895 data presented here are estimated to be 5%, whilefor the most backward rapidity bins, the uncertainties athigh mt −m0 can become as large as 50%. These errorsare included on the spectra reported in this article.

III. RESULTS

Figs. 4 and 5 show the fully corrected, invariant yieldsof charged pions per event from 0-5% central Au+Au col-lisions at 2,4,6 and 8 AGeV. Mid-rapidity is shown un-scaled as black circles, while each bin forward/backwardof mid-rapidity is scaled down by a successive factor of 10.Forward rapidities are indicated as open circles and back-ward rapidities as open triangles. The reported error barsinclude both statistical and systematic uncertainties.The approximately exponential decay of the particle

yields as a function of transverse mass has been observedin high energy particle and heavy ion experiments overa wide range of conditions. In order to extract the full4π yields of pions (including those from resonance feed-down) produced in the beam energy range studied here,a simple parameterization of the pion mt − m0 spec-tra which reproduces the observed shapes of the spectraover the full range of mt − m0, from 0 to 1 GeV/c2 isused. This parameterization is the sum of two indepen-dent Maxwell-Boltzmann thermal functions, each term ofwhich can be expressed in terms of the measured coordi-nates, (mt −m0,y), integrated over azimuth as

1

2πmt

d2N

dmtdy= A(y)mte

−(mt−m0)/Teff (y) (1)

where the amplitude A and inverse slope parameter, Teff

are parameters which can be extracted from fits to thedata at each rapidity slice. Integration of either of thesetwo distributions over mt produces its contribution to thetotal number of particles per unit of rapidity in the givenrapidity slice[31].

dN

dy(y) = 2πA(y)(m2

0Teff (y) + 2m0T2eff (y) + 2T 3

eff(y))

(2)Note that A(y) in Eq. (1) can be re-written in terms ofdN/dy(y), using Eq. (2).The low-pt enhancement observed in the pion yields

has been attributed to feed-down from late-stage reso-nance decays [32, 33, 34], which tend to populate lowerpion pt in the frame of the collision due to the natureof the decay kinematics. In particular, at the beamenergies presented here, the delta resonances, (such asthe ∆(1232)) are the predominant mechanism for pionproduction, due to the very large cross-section for pion-nucleon interactions. The observed asymmetry in thespectral shapes of positive and negative pions at lowmt − m0 has been described elsewhere by a final-stateCoulomb interaction of the pions with the nuclear fire-ball [35, 36, 37, 38].The two-slope model which has been applied to the

pion spectra was chosen to provide the simplest phe-nomenological description of the data. Since this pa-rameterization does reasonably well at describing the ob-served shapes of the spectra over all mt −m0, it is usedto extract the 4π yields of pions in these collisions with

5

2 AGeV

π-

|y-yc.m.| < 0.05(unscaled)

y-yc.m. < 0y-yc.m. > 0

4 AGeV 6 AGeV 8 AGeV

mt-mπ (GeV/c2)

(1/2

πmt)

d2 N/d

mtd

y [(

GeV

/c2 )-2

]

10-15

10-12

10-9

10-6

10-3

1

103

0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0

FIG. 4: Invariant yield per event as a function of mt −m0 for π− at 2, 4, 6, and 8 AGeV. Midrapidity is shown unscaled, whilethe 0.1 unit forward/backward rapidity slices are scaled down by successive factors of 10. The functions plotted with the dataare two-slope Boltzmann parameterizations described in the text.

minimal extrapolation.

1

2πmt

d2N

dmtdy=

dN/dy1mte−(mt−m0)/T1

2πT1 (m20 + 2m0T1 + 2T 2

1 )+

dN/dy2mte−(mt−m0)/T2

2πT2 (m20 + 2m0T2 + 2T 2

2 )(3)

The four parameter fit includes two independently fitinverse slope parameters (T1(y), T2(y)), and two inde-pendently fit yield parameters (dN/dy1(y), dN/dy2(y)),which reasonably describe the low-(mt −m0) and high-(mt − m0) portions of the spectrum at each rapidity.These fits are shown as solid lines in Figs. 4 and 5.The data at more forward rapidities, where in partic-ular the π+ spectra are limited to the range mt −m0 <200 MeV/c, are not included in the two-slope parameter-ization fits.

More detailed examples of these two-slope fits at mid-rapidity are shown in Fig. 6. Figs. 7-10 show the four fitparameters as a function of rapidity from the π− spectraand the resulting total dN/dy. The parameters plotted inthe top row are from fits in which all four parameters areallowed to be free. All of them show an approximatelyGaussian dependence on rapidity, which is demonstratedby the solid line in each panel. The bottom row of eachfigure demonstrates the result of a second fit of the spec-tra in which the high-(mt −m0) inverse slope parameter

is constrained to the Gaussian (solid line) value. Thisprocedure smooths out the covariance between the in-dividual yield parameters but has a negligible effect ontotal dN/dy (rightmost panel).

If the source of pions were a static thermal sourceof temperature T0 and zero chemical potential, the ex-pected shape of Teff in Figs. 7-11 would be T0/ cosh(y).(Of course, there would then be needed only a single termin Eq. (3).) We have shown in Ref. [23] that there issignificant longitudinal flow - i.e., the source is not static.As the main purpose of the fits in Figs. 4 and 5 is for in-tegration of the (mt −m0,y) spectra to obtain 4π yields,a Gaussian fit of the inverse slope parameters in Figs.7-11, rather than 1/cosh(y), was used.

Fig. 11 shows the π+ fit parameters at each beam en-ergy. The limited range in mt −m0 of the π+ spectra iscaused by the 〈dE/dx〉 overlap with the protons, whichmakes it impossible to get good spectral data at all rapid-ity slices. In order to make reasonable estimates of the4π positive pion yields, the high-(mt −m0) positive pioninverse slope parameters were assumed to be the same asthose of the negative pions[16]. In the estimation of theπ+ dN/dy, the high-(mt−m0) inverse slopes were there-fore fixed to the negative pion values. Experimentally,the mt − m0 negative pion inverse slope parameters do

6

2 AGeV

π+

|y-yc.m.| < 0.05(unscaled)

y-yc.m. < 0y-yc.m. > 0

4 AGeV 6 AGeV 8 AGeV

mt-mπ (GeV/c2)

(1/2

πmt)

d2 N/d

mtd

y [(

GeV

/c2 )-2

]

10-15

10-12

10-9

10-6

10-3

1

103

0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0

FIG. 5: Invariant yield per event as a function of mt −m0 for π+ at 2, 4, 6, and 8 AGeV. Midrapidity is shown unscaled, whilethe 0.1 unit forward/backward rapidity slices are scaled down by successive factors of 10. The functions plotted with the dataare two-slope Boltzmann parameterizations described in the text.

reasonably reproduce the observed high-(mt −m0) posi-tive pion inverse slopes in the rapidity regions where theycan be measured. Where themt−m0 spectra are severelytruncated, the positive pion yields were obtained fromsingle-slope fits. The reported systematic uncertaintiesin these rapidity slices are correspondingly larger to ac-count for this missed yield.

Fig. 12 shows the beam energy dependence of the ra-pidity distributions of negative and positive pions ex-tracted from the fits. These distributions are well-described by Gaussians, which are used to obtain thetotal 4π yields by integrating the fitted distributions overall rapidity.

〈π〉 =∫ +∞

−∞

dN

dydy =

√2πwQ (4)

where Q is the value of dN/dy at y-yCM = 0 and w isthe width. The fit parameters and their uncertainties arelisted in Tables I and II. Both the widths and the over-all yields of pions increase as a function of beam energy.However, there is a more significant increase in the ob-served pion yields between 2 and 4 AGeV than between4 and 6 AGeV or 6 and 8 AGeV.

Ebeam Qπ− 〈π−〉 wπ−

2 AGeV 21.3 ± 0.1 ± 1.3 36.1 ± 0.3 ± 2.0 0.675 ± 0.006 ± 0.005

4 AGeV 39.0 ± 0.1 ± 2.1 76.0 ± 0.2 ± 4.2 0.780 ± 0.003 ± 0.002

6 AGeV 50.8 ± 0.1 ± 2.7 104.0 ± 0.2 ± 5.4 0.817 ± 0.002 ± 0.001

8 AGeV 61.1 ± 0.1 ± 3.3 130.7 ± 0.4 ± 7.9 0.854 ± 0.003 ± 0.008

TABLE I: dN/dy fit parameters for negative pions at eachbeam energy with statistical and systematic uncertainties re-ported separately. Qπ− is the amplitude of the distributionat y-yCM = 0, 〈π−〉 is the 4-π yield (total area) and wπ− isthe width.

IV. DISCUSSION

In this section, some of the characteristics of the ob-served pion spectra obtained from E895 are discussed.The pion spectral shapes are compared to the predictionsof a microscopic transport model, the dN/dy distribu-tions are evaluated in the context of collective dynamics,and the overall yields are used to infer the initial stateentropy density obtained in these collisions.

It is interesting to note the asymmetry between thepositive and negative pion yields at each beam energy.Although the ratio of negative to positive pions decreases

7

0 0.5 0 0.5 0 0.5 0 0.5

2 AGeV 4 AGeV 6 AGeV 8 AGeV

|y-yc.m.| < 0.05

π-

π+

mt-mπ (GeV/c2)

(1/2

πmt)

d2 N/d

mtd

y [(

GeV

/c2 )-2

]

1.010-2

10-1

1

10

102

10-1

1

10

102

FIG. 6: Pion transverse mass spectra at mid-rapidity for 2,4, 6, and 8 AGeV (uppermost curves on Figs. 4 and 5). Two-slope thermal fits are shown superposed on the data alongwith the contributions to the total from the two separate in-verse slope terms in Eq. (3).

0

50

100

150

0

10

20

30

0

50

100

150

-1 0 1 -1 0 10

10

20

30

-1 0 1

T1 T2 dN/dy1 dN/dy2 dN/dy1+dN/dy2

(Fixed)

Rapidity (y-yc.m.)

T (

MeV

/c2 )

dN/dy

2 AGeV π-

FIG. 7: Rapidity dependence of the fit parameters from 2AGeV π− mt −m0 spectra fits with a two-slope model. Theleftmost panels show the inverse slope parameters. T1 (indi-cated by closed triangles) dominates the low-(mt−m0) rangeof the spectra, whereas T2, (indicated by closed squares) dom-inates the high-(mt − m0) end of the spectra. The middlepanels show the two dN/dy fit parameters, (dN/dy1 in opentriangles and dN/dy2 in open squares) and the rightmost pan-els show the total dN/dy, which is the sum of dN/dy1 anddN/dy2. The top row reports the fit results when all fourparameters are allowed to be independent in the fit. The bot-tom row shows the results for the remaining three parameterswhen the high-(mt−m0) inverse slope is fixed to the Gaussianestimation. Note that the total dN/dy is fairly insensitive tothe interplay among the parameters.

0

50

100

150

0

10

20

30

40

0

50

100

150

-1 0 1 -1 0 10

10

20

30

40

-1 0 1

T1 T2 dN/dy1 dN/dy2 dN/dy1+dN/dy2

(Fixed)

Rapidity (y-yc.m.)

T (

MeV

/c2 )

dN/dy

4 AGeV π-

FIG. 8: Rapidity dependence of the fit parameters from 4AGeV π− mt −m0 spectra fits with a two-slope model. Seecaption of Fig. 7 for details.

0

50

100

150

0

20

40

60

0

50

100

150

-1 0 1 -1 0 10

20

40

60

-1 0 1

T1 T2 dN/dy1 dN/dy2 dN/dy1+dN/dy2

(Fixed)

Rapidity (y-yc.m.)

T (

MeV

/c2 )

dN/dy

6 AGeV π-

FIG. 9: Rapidity dependence of the fit parameters from 6AGeV π− mt −m0 spectra fits with a two-slope model. Seecaption of Fig. 7 for details.

Ebeam Qπ+ 〈π+〉 wπ+

2 AGeV 11.5 ± 0.3 ± 1.2 19.2 ± 1.3 +2.0−1.0 0.668 ± 0.053 ± 0.011

4 AGeV 27.7 ± 0.3 +1.2−1.4 46.3 ± 0.8 +4.5

−3.3 0.667 ± 0.015 ± 0.025

6 AGeV 38.4 ± 0.3 +2.4−1.8 75.7 ± 1.1 +3.4

−2.9 0.787 ± 0.014 ± 0.016

8 AGeV 46.2 ± 0.4 +2.7−3.4 95.9 ± 1.1 +6.1

−5.9 0.828 ± 0.013 ± 0.005

TABLE II: dN/dy fit parameters for positive pions at eachbeam energy with statistical and systematic uncertainties re-ported separately. Qπ+ is the amplitude of the distributionat y-yCM = 0, 〈π+〉 is the 4-π yield (total area) and wπ+ isthe width.

8

0

50

100

150

0

20

40

60

0

50

100

150

-1 0 1 -1 0 10

20

40

60

-1 0 1

T1 T2 dN/dy1 dN/dy2 dN/dy1+dN/dy2

(Fixed)

Rapidity (y-yc.m.)

T (

MeV

/c2 )

dN/dy

8 AGeV π-

FIG. 10: Rapidity dependence of the fit parameters from 8AGeV π− mt −m0 spectra fits with a two-slope model. Seecaption of Fig. 7 for details.

0

50

100

0

5

10

15

(Fixed)

2 AGeV

0

50

100

150

0

10

20

30

(Fixed)

4 AGeV

0

50

100

150

0

10

20

30

40

(Fixed)

6 AGeV

0

50

100

150

-1 0 1 -1 0 10

20

40

60

-1 0 1

(Fixed)

Rapidity (y-yc.m.)

T (

MeV

/c2 )

dN/dy

π+

8 AGeV

T1 T2 dN/dy1 dN/dy2 dN/dy1+dN/dy2

FIG. 11: Rapidity dependence of the fit parameters for eachbeam energy from the π+ mt − m0 spectra fits with a two-slope model. (See caption of Fig. 7 for more details.) Thehigh-(mt − m0) inverse slope parameters have been fixed tothe π− parameterized values.

over the studied beam energy range, from 1.88 at 2 AGeVto 1.36 at 8 AGeV (see Tables I and II), it does not reachthe asymptotic value of ∼ 1.0 observed at the top CERNSPS energy[24, 39]. The neutron excess in Au+Au col-lisions (118+118 neutrons compared to 79+79 protons)combined with the pion branching ratios suggests that1.95 negative pions for every positive pion will be pro-

0

10

20

30

40

50

60

70

-1 0 10

10

20

30

40

50

60

70

-1 0 1Rapidity (y-yc.m.)

dN/d

y

π- π+

8 AGeV6 AGeV4 AGeV2 AGeV

FIG. 12: Yield per event of π− (left-hand panel) and π+

(right-hand panel) in 0-5% central Au+Au collisions at 2, 4,6, and 8 AGeV integrated over mt−m0 as a function of rapid-ity. A Gaussian parameterization is applied to extract the 4πyields. Reported uncertainties are statistical plus systematic.

duced [13]. If the isospin fractions are folded with the ob-served cross-sections for NN → NNπ from experimentalmeasurements[40], the expected ratio of 〈π−〉:〈π+〉 at 1AGeV is ∼ 1.91:1. The 2 AGeV data, with

√sNN not far

from the ∆(1232) production threshold, are quite near,though a little lower than this predicted ratio. As theenergy increases, the number of directly produced pionpairs (π−π+) is expected to lower the ratio, asymptoti-cally approaching 1.0, which is the trend we observe. At8 AGeV the negative pion excess is only approximately36%, compared to 88% at 2 AGeV.

A. RQMD Comparisons

RQMD (Relativistic Quantum Molecular Dynamics)v2.3 [41] is a microscopic transport model which attemptsto simulate heavy ion collisions by propagating individ-ually all particles through the six dimensions of phasespace in the fireball. Interaction probabilities are ap-proximated by using published interaction cross-sectionsof free hadrons and the relative phase space proximityof pairs of particles at each time step of the reaction.Inelastic collisions may produce new particles, such aspions, which are also propagated through phase spacealong with the nucleons. The reaction ends when thephase space density reaches a low enough threshold suchthat the probability of further interactions is small - thefreeze-out point. This model has been reasonably suc-cessful in describing many final state observables experi-mentally measured in the beam energy range studied forthis analysis [42].In this and other cascade models, final-state particle

distributions are frozen at the end of the reaction. Postfreeze-out effects, such as the Coulomb interaction of thepions with the nuclear fireball, which will be discussedin detail using results from this analysis elsewhere [38],are not included. However, RQMD combined with an

9

afterburner to simulate final state Coulomb interactions[43] and to permit the decay of residual resonances hasbeen successful at describing asymmetries in observedpion yields at low pt.

RQMD version 2.3, with the nucleon mean field set-ting turned on, was used to generate pion distributionsfrom central (b ≤ 3 fm) Au+Au collisions for compari-son with the data obtained by E895 for this paper. Oneof the output parameters from RQMD records the na-ture of the last collision, “lastcl”, of each particle beforefreeze-out. Particles whose lastcl was a thermal rescat-tering are labelled “thermal”. Particles whose lastcl wasthe decay of a resonance have the parent particle listedexplicitly. However, high mass resonances which are notDeltas or vector mesons are combined in a single cate-gory labelled “himass” in the user’s notes for the code.Particles whose lastcl was String or Rope fragmentationare indicated separately. Figs. 13 and 14 show the na-ture of the last collision for π− and π+ for a centralityselection of b ≤ 3 fm at 2 AGeV and 8 AGeV, respec-tively. The distributions are normalized to the per eventyield of pions. In each case, the Delta resonances are thelargest single contributor to the last pion interactions be-fore freeze-out.

The dependence of the pion spectral shapes on feed-down from Delta resonances is evident in Fig. 15, whichshows the mt − m0 spectra of pions near mid-rapidity(|y| < 0.3) from RQMD. Pions whose last collision wasa Delta are plotted separately from the pions with allother last interactions. The spectral shapes are stronglyaffected by a Delta lastcl. Plotted in Fig. 16 are theratios of the pion yields from Deltas and all other lastinteractions to the total yields as a function of mt −m0.In all cases the Delta pions contribute more significantlyto the total yield at low mt −m0 than at high mt −m0,but the influence of the Delta contribution to the spectradiminishes as the beam energy increases. The total mid-rapidity pion spectra are fit with the same model as wasused to describe the data, Eq. (3), with the fit parame-ters shown in Tables III and IV. At both high and lowmt−m0, the RQMD π− and π+ appear to have commoninverse slope parameters and there is no obvious evolu-tion with beam energy. The average value at low mt−m0

(T1) is ∼72 MeV, while the high mt −m0 (T2) averageis ∼140 MeV. The magnitude of the low mt−m0 RQMDinverse slope parameters is much larger than what is ob-served in the data at all beam energies. At high mt−m0,the RQMD inverse slope parameters are much closer tothe observed values, except at 2 AGeV, where the datashow T2 ∼ 120 MeV at mid-rapidity.

The rapidity density distributions of charged pions pre-dicted by RQMD are shown in Fig. 17 alongside the E895data from Figs. 7-11. The contributions to the totaldN/dy from Delta resonance decay pions and all otherpions are also shown. As the beam energy increases, thecontribution to the total from Delta resonance decay pi-ons becomes less important, which is consistent with thetrends seen in the data.

Ebeam (AGeV) dN/dy1 T1 (MeV/c2) dN/dy2 T2 (MeV/c2)

2 17.4 73 1.8 146

4 21.3 73 10.9 131

6 26.6 74 15.5 146

8 31.7 77 20.1 155

TABLE III: Mid-rapidity RQMD π− fit parameters, Eq. (3).

Ebeam (AGeV) dN/dy1 T1 (MeV/c2) dN/dy2 T2 (MeV/c2)

2 10.2 73 1.6 134

4 14.4 72 8.5 131

6 15.5 64 17.7 131

8 21.4 70 21.3 144

TABLE IV: Mid-rapidity RQMD π+ fit parameters, Eq. (3).

The

rmal

Stri

ng

Him

ass

Rop

e ∆ π K n,p η Λ Σ Ξ Ω φ ρ

Mis

c

2 AGeV π-

2 AGeV π+

RQMD v2.3: Nature of last collision before freezeout

(1/N

even

ts)

dNπ/

d la

stcl

10-310-210-1

110

100

10-310-210-1

110

FIG. 13: RQMD nature of pion last collision, “lastcl”, beforefreeze-out at 2 AGeV for impact parameter, b ≤ 3 fm.

The

rmal

Stri

ng

Him

ass

Rop

e ∆ π K n,p η Λ Σ Ξ Ω φ ρ

Mis

c8 AGeV π-

8 AGeV π+

RQMD v2.3: Nature of last collision before freezeout

(1/N

even

ts)

dNπ/

d la

stcl

10-310-210-1

110

100

10-310-210-1

110

FIG. 14: RQMD nature of pion last collision, “lastcl”, beforefreeze-out at 8 AGeV for impact parameter, b ≤ 3 fm.

10

0 0.5 0 0.5 0 0.5 0 0.5

2 AGeVRQMD v2.3

4 AGeV 6 AGeV 8 AGeV

All∆Other

π-

π+

mt-mπ (GeV/c2)

(1/2

πmt)

d2 N/d

mtd

y [(

GeV

/c2 )-2

]

1.0

10-2

10-1

1

10

100

10-2

10-1

1

10

100

FIG. 15: RQMD charged pion spectra at mid-rapidity forcollisions with impact parameter, b ≤ 3 fm. The contributionto the spectra from pions whose last interaction was a Deltadecay and the sum of all other processes are shown separately.

00.20.40.60.8

1

00.20.40.60.8

1

0 0.5 0 0.5 0 0.5 0 0.5

2 AGeV 4 AGeV 6 AGeVRQMD v2.3

8 AGeV

∆Other

mt-mπ (GeV/c2)

Tot

al Y

ield

Rat

io

1.0

FIG. 16: Ratios of pion yield for different production mech-anisms to the total yield as a function of mt − m0 at mid-rapidity from RQMD collisions with impact parameter, b ≤3 fm. The contribution to the spectra from pions whose lastinteraction was a Delta decay and from the sum of all otherprocesses are shown separately.

B. Longitudinal flow

An important question in heavy ion collisions is the de-gree of collectivity of the produced particles, which canarise from the build-up of pressure in the hot, dense col-lision zone. RQMD has no such hydrodynamic effect ex-plicitly included. However, individual particle thermalrescattering, the microscopic analog to pressure, can alsodrive collective flow, and has been observed in RQMDcalculations at RHIC energies [44]. Rapidity broaden-

0

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5

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010203040

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30

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60

010203040

0

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-1 0 1 -1 0 10

20

40

-1 0 1 -1 0 1

2 AGeVRQMD v2.3

4 AGeV

6 AGeV

8 AGeV

All

∆

Other

π-

dN/dy1 dN/dy2 dN/dy1+dN/dy2

π+

Rapidity (y-yc.m.)

dN/d

y

FIG. 17: Rapidity density of charged pions from RQMD cal-culations for collisions with impact parameter, b ≤ 3 fm. Thecontribution from Delta resonance decay pions and the sum ofall other processes are also plotted separately to show the rel-ative contribution to the pion yields from Delta decays. Alsoplotted are the data from Figs. 7-11.

ing along the beam axis has been observed in heavy ioncollisions at many beam energies for a wide range of sys-tems and particle species [22, 23, 31, 45]. This broad-ening has been interpreted as arising from the collec-tive motion of the system after collision in the longitu-dinal direction (longitudinal flow). The observed rapid-ity distributions are compared with the expectation fora stationary thermal source, and with a longitudinallyboost-invariant superposition of multiple boosted indi-vidual isotropic, locally thermalized sources in a givenrapidity interval. Each locally thermalized source is mod-elled by the mt-integrated Maxwell-Boltzmann distribu-tion, Eq. (1), with the rapidity dependence of the energy,E = mt cosh(y) explicitly included, and T is true tem-perature:

dNth

dy(y) = BT 3(

m2

T 2+

m

T

2

cosh y+

2

cosh2 y)e(−

mT cosh y).

(5)The distributions are integrated over source element

rapidity to extract the maximum longitudinal flow, ηmax

dN

dy=

∫ ηmax

ηmin

dηdNth

dy(y − η) (6)

βL = tanh(ηmax).

where ηmax = - ηmin, from symmetry about the centerof mass, and βL is the maximum longitudinal velocity inunits of c. An average longitudinal flow velocity can bedefined as 〈βL〉 = tanh(ηmax/2).Ref. [23] presents longitudinal flow velocities extracted

from the proton rapidity densities measured by E895 for

11

the same event selection as the pions presented in thisanalysis and compares the results to values extractedfrom a wide array of experiments over the beam energyrange from 1 to 160 AGeV. Since the protons are presentbefore the collision, at higher beam energies they maybe strongly affected by nuclear transparency, which canalso broaden the rapidity distributions. Therefore, theprotons alone cannot be used to determine the absolutemagnitude of the collective motion, if it is present.In order to determine the degree of collectivity, it is

important to compare multiple particle species from thesame collisions simultaneously. This has been done withcentral Si+Al collisions at 14.6 AGeV for protons, pi-ons, kaons and lambda hyperons [45] and in central S+Scollisions at 200 AGeV for pions, kaons and lambda hy-perons [31]. A common collective flow velocity was ableto reasonably describe all of the observed rapidity dis-tributions, except the protons in S+S collisions at 200AGeV, where apparent nuclear transparency is more pro-nounced. A similar simultaneous description is possiblehere, by combining the proton rapidity distributions from[23] with the present pion rapidity distributions.The rapidity densities of pions and the protons from

Ref. [23] are shown in Figure 18. Stationary thermalsource emission functions, the sum of two of Eq. (5) forthe pions using the two inverse slope parameters fromthe pion transverse mass spectra fits at mid-rapidity, areshown as dashed lines. We have not measured the truesystem temperature, T, since T1 and T2 parametrizethe temperature and known radial flow, resonance andCoulomb effects in the pion spectra. Consequently, T2

overestimates the system temperature and T1 may un-derestimate (due to competing effects of radial flow, res-onance feed-down and the Coulomb interaction with thefireball). Therefore the plotted distributions (dashedlines) are probably wider than the true thermal distribu-tions would be, and yet are still too narrow to reproducethe observed rapidity spectra. The inclusion of longitudi-nal flow remedies this. For the protons, a single Eq. (5)is used for the thermal rapidity distribution, as in Ref.[23].The solid curves are the emission functions including

longitudinal flow, Eq. (6), with the velocities fixed tothe values extracted from the protons (〈βL〉 = 0.28, 0.42,0.48, 0.50 at 2, 4, 6 and 8 AGeV, respectively). The longi-tudinally expanding source clearly better reproduces themeasured pion distributions at all beam energies than dothe dashed curves in Fig. 18. Consistency among parti-cles of different masses supports a hydrodynamical inter-pretation of the rapidity density broadening; the systemis expanding like a fluid with a common longitudinal flowvelocity.

C. Entropy production

It was suggested many years ago by Fermi [46] andlater Landau [47] that pion production may be used to

0

10

20

30

020406080

01020304050

020406080

0

20

40

60

020406080

0

20

40

60

-1 0 1 -1 0 10

20406080

-1 0 1

dN/d

y

Rapidity (y-yc.m.)

2 AGeV

4 AGeV

6 AGeV

8 AGeV

π- π+ p

FIG. 18: π−, π+ and proton rapidity density distributionsat 2,4,6, and 8 AGeV. The expectation for isotropic emissionfrom a stationary thermal source is indicated with dashedlines, whereas the solid curves represent the form includinglongitudinal flow from the proton ηmax values given in Ref.[23]. In all cases the thermal model is too narrow. The protondata and the longitudinal expansion velocity come from Ref.[23].

estimate the amount of entropy produced in high energyparticle collisions. Later, Van Hove [48] extended thisidea to heavy ion collisions and proposed that this maybe a way to distinguish events in which a Quark GluonPlasma (QGP) is formed. In a QGP the color degrees offreedom of the liberated partons introduce a significantnumber of new energy states unavailable in a hadron gas.By studying pion production over a broad range of colli-sion systems and energies, discontinuities in the observedmultiplicities might indicate the onset of QGP formation.In 1995, Gazdzicki [49] took the available data from

heavy ion collisions and showed that there is an increasein the observed entropy produced at the SPS (NA35 Ex-periment with S+S collisions at 200 AGeV [50]) com-pared with AGS energies. The low energy heavy ion datafollow the trend for p+p collisions, while at the SPS thereis an apparent factor of 3 increase in the effective num-ber of degrees of freedom [51]. The entropy productionanalysis of Pb+Pb collisions at 158 AGeV (NA49) [52]supported this observation.This model assumes that the entropy is produced at

the early stage of the collision when the incident matteris in a highly excited state. The thermalized, stronglyinteracting matter is assumed to expand adiabatically tothe freeze-out point, preserving the early stage entropy.Since the majority of produced particles are pions, to

12

first order, the mean pion (boson) multiplicity shouldbe nearly proportional to the entropy. The ratio of themean pion multiplicity to the mean number of participat-ing nucleons, 〈π〉/〈Npart〉, provides a simple estimate ofthe entropy density. 〈Npart〉 for a nucleus-nucleus colli-sion can be estimated using a Glauber model calculationof the mean free path of the nucleons through the nu-clei as they collide at a given impact parameter. Forthe present analysis, 〈NP 〉 was estimated using RQMD.The nucleons are distributed according to a Woods-Saxonnuclear density profile and the impact parameter of thesimulated collision is used with the nucleon-nucleon in-teraction cross-sections[53] to determine the number ofparticipants. The top 5% of collisions (determined byintegrating the 〈NP 〉 distribution from a set of minimumbias RQMD events at each beam energy) correspond to〈NP 〉 = 364, 366, 365, 363 at 2,4,6, and 8 AGeV, respec-tively. The estimated uncertainty on these values is ± 5participants.

Following Ref.[24], the entropy densities for each beamenergy, here approximated as 〈π〉/〈Npart〉, with 〈π〉 =1.5(〈π−〉 + 〈π+〉) to account for the neutral pions, areplotted in Fig. 19 as a function of the Fermi energy vari-

able, F ≡ (√sNN−2mN )3/4√sNN

1/4 . E895 data are indicated by

stars and the values are tabulated in Table V. NA49results for 40, 80 and 158 AGeV Pb+Pb collisions fromthe CERN SPS were obtained from Ref.[24]. The numberof participants from [24], calculated using the Fritiof[54]model, are also listed in Table V.

The linear dependence of the entropy per participantnucleon as a function of F in the proton-proton(anti-proton) data is not evident for the full range of the heavyion collision data. At and below AGS energies, the heavyion data lie below the p+p data, and appear to be approx-imately linear with F. In Ref. [24], the SPS results com-bined with RHIC results at much higher energies from thePHOBOS Collaboration show a linear trend with a slopethat is approximately 1.3 times larger than at the lowerenergies. There appears to be a transition in the regionbetween the AGS and top SPS energies. The third-orderpolynomial fit to the heavy ion data shown on Fig. 19may indicate that a smooth trend with increasing F canaccurately describe the excitation function without theneed for a discontinuous jump, such as one might expectfrom a first-order phase transition. Two more runs atthe SPS with beam energies of 20 AGeV and 30 AGeVmay be able to improve the resolution in this importanttransition region.

V. SUMMARY

Transverse mass and rapidity spectra of charged pi-ons in 2-8 AGeV 0-5% central Au+Au collisions havebeen measured by the E895 experiment. The transversemass spectra exhibit a low-pt enhancement which can belargely ascribed to the feed-down from late stage reso-

√sNN (GeV) F (GeV1/2) 〈Np〉 〈π〉/〈Np〉2.630 0.644 364 0.2279 ± 0.0159 +0.0124

−0.0165

3.279 0.965 366 0.5012 ± 0.0111 +0.0307−0.0357

3.838 1.190 365 0.7385 ± 0.0148 +0.0341−0.0362

4.289 1.351 363 0.9364 ± 0.0170 +0.0570−0.0579

8.830 2.452 349 2.6433 ± 0.0858

12.280 3.099 349 3.9542 ± 0.0869

17.260 3.821 362 5.2127 ± 0.1823

TABLE V: Tabulated mean number of pions per participantand Fermi energy variable for 2, 4, 6, and 8 AGeV Au+Aucollisions and 40, 80, and 158 AGeV Pb+Pb collisions fromRef.[24]. Statistical and systematic uncertainties are reportedseparately.

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

F(GeV1/2)

<π>/

<Np>

E895

p+pBevalac+DubnaAGSNA49

FIG. 19: Mean pion multiplicity per participant vs. F, theFermi energy variable. The solid symbols represent data fromheavy ion collisions, whereas the open symbols come fromp+p interactions. The E895 data points are indicated bystars. The third-order polynomial fit to the heavy ion datashow that a smooth trend with F is possible, although thelow energy and high energy data (including RHIC results) canbe well-described by separate linear parameterizations whoseslopes differ by a factor of approximately 1.3[24].

nance decays. Differences in the π+ and π− spectra atlow mt −m0 are not reproduced by RQMD, which doesnot include final state interactions such as the Coulombinteraction with the nuclear source. The inverse slopeparameters increase as a function of beam energy andappear to be charge independent at high mt −m0. Themeasured rapidity distributions show excellent forward-backward rapidity symmetry and are well described by amodel which includes collective longitudinal flow, with a

13

velocity that is common to both pions and protons emit-ted in these collisions. The 4π yields of pions, obtained byintegrating the rapidity distributions, have been used toinfer an initial state entropy which increases with beamenergy and is consistent with a smooth non-linear trendas a function of F from the 2 AGeV Au+Au collisions to158 AGeV Pb+Pb collisions at the SPS.This work was supported in part by the U.S.

Department of Energy under grants DE-FG02-87ER40331.A008, DE-FG02-89ER40531, DE-FG02-88ER40408, DE-FG02-87ER40324, and contractDE-AC03-76SF00098; by the US National Science Foun-dation under Grants No. PHY-98-04672, PHY-9722653,PHY-96-05207, PHY-9601271, and PHY-9225096; andby the University of Auckland Research Committee,NZ/USA Cooperative Science Programme CSP 95/33.

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