J.M. Burward-Hoy Lawrence Livermore National Laboratory for the PHENIX Collaboration

Source Parameters from Identified Hadron Spectra and HBT Radii for Au-Au Collisions

at sNN = 200 GeV in PHENIXJ.M. Burward-Hoy

Lawrence Livermore National Laboratoryfor the

PHENIX Collaboration

Motivation• The objective is to measure the characteristics of the particle emitting source from both spectra and HBT radii simultaneously.• An interpretation of the data assuming relativistic hydrodynamic expansion is presented.• The study uses PHENIX Preliminary data as presented by T. Chujo and A. Enokizono.

Jane M. Burward-Hoy QM2002: Particle Yields I 2

Single Kp SpectraPHENIX Time of Flight

HBT AnalysisPHENIX Electromagnetic Calorimeter

momentum resolution p/p ~ 1% 1% p

TOF resolution 120 ps EMCal resolution 450 ps

Detecting , K, p in PHENIX


• Model by Wiedemann, Scotto, and Heinz , Phys. Rev. C 53, 918 (1996)E. Schnedermann, J. Sollfrank, and U. Heinz, Phys. Rev. C 48, 2462 (1993)

• Fluid elements each in local thermal equilibrium move in space-time with hydrodynamic expansion.– No temperature gradients

• Boost invariance along collision axis z.• Infinite extent along rapidity y.• Cylindrical symmetry with radius r.

• Particle emission – hyperbola of constant proper time 0

• Short emission duration t < 1 fm/c

z

A Simple Model for the Source

t 220 zt

z

r


Transverse Kinetic Energy Spectra5% Au-Au at sNN = 200 GeV

• Kinetic energy spectra broaden with increasing mass, from to K to p (they are not parallel).


Mean Transverse Momentum

• Mean pt increases with Npart and m0, indicative of radial expansion.• Relative increase from peripheral to central greater for (anti)p than for ,

K. • Systematic uncertainties: 10%, K 15%, and (anti-)p 14%

Open symbols: sNN = 130 GeV

<pT>

(GeV

/c)

Npart


Reproducing the Shape of the Single Particle Spectra

parametersnormalization Afreeze-out temperature Tfo

surface velocity T

t

mt

1/m

t dN

/dm

t

TfoA

Minimize contributions from hard processes (mt-m0) < 1 GeV

Exclude resonance region pT < 0.5 GeV/c

Linear flow profile () = T <T > = 2T/3

S. Esumi, S. Chapman, H. van Hecke, and N. Xu, Phys. Rev. C 55, R2163 (1997)

t()

1

f()

Radial position on freeze-out surface = r/R

Particle density distribution f() is independent of

Shape of spectra important


Fitting the Transverse Momentum Spectra

• Simultaneous fit in range (mt -m0 ) < 1 GeV is shown.• The top 5 centralities are scaled for visual clarity.• Similar fits for negative particles.


Fitting the Transverse Momentum Spectra

• Simultaneous fit in range (mt -m0 ) < 1 GeV is shown.• The top 5 centralities are scaled for visual clarity.• Similar fits for positive particles.


In each centrality, the first 20 n- contour levels are shown.

From the most peripheral to the most central data, the single particle spectra are fit simultaneously for all pions, kaons, and protons.

For All Centralities2 Contours in Parameter Space Tfo and T

PHENIX Preliminary

PHENIX Preliminary


A close-up …Most Central and Most Peripheral

For the 5% spectra T = 0.7 ± 0.2 syst. Tfo = (110 23 syst.) MeV

For the most peripheral spectra:T = 0.46 ± 0.02 stat. 0.2 syst. Tfo = 135 3 stat. 23 syst. MeV


The Parameters Tfo and T vs. Npart

• Expansion parameters in each centrality

• Overall systematic uncertainty is shown.

• A trend with increasing Npart is observed:– Tfo and T

• Saturates at mid-central


Expansion from the kT Dependence of HBT Radii

f = 0f = 0.3f = 0.6f = 0.9

parametersgeometric radius Rfreeze-out temperature Tfo

flow rapidity at surface T

freeze-out proper time 0

T = 150 MeV, R = 3 fm, 0 = 3 fm/c

Use analytical forms for the radii from Wiedemann, Scotto, and HeinzRef: PRC 53 (No. 2), Feb. 1996

Rs (fm)

Ro (fm) RL (fm)

Rs (fm)

RL (fm)Ro (fm)

++PHENIX Preliminary


Fitting the kT Dependence of HBT Radii

2 contours in parameter space Tfo and T

• The contours are not closed• In this region of parameter space, the minimum 2 value is found • These contours are the n- values relative to this minimum


HBT Radii and Single Particle Spectra

• Spectra and RL are consistent within 2.5 .

• Ro and Rs disagree with the spectra.

• Rs prefers high flow and low temperatures T > 1.0 and Tfo < 50 MeV

• Ro prefers T > 1.4 and Tfo > 100 MeV

• (R-contours not closed)

PHENIX Preliminary


Using spectra information to constrain HBT fits…

• 10% central positive pion HBT radii (similar result for negative pion data).• Systematic uncertainty in the data is 8.2% for Rs, 16.1% for Ro, 8.3% for RL.

From the spectra (systematic errors):T = 0.7 ± 0.2 syst. Tfo = 110 23 syst. MeV

++

0 = 132 fm/cR = 9.6±0.2 fm

PHENIX PreliminaryRs (fm) Ro (fm) RL (fm)


Using spectra information to constrain HBT fits…

• 10% central negative pion HBT radii.• Systematic uncertainty in the data is 8.2% for Rs, 16.1% for Ro, 8.3% for RL.

From the spectra (systematic errors):T = 0.7 ± 0.2 syst. Tfo = 110 23 syst. MeV

--

0 = 132 fm/cR = 9.7±0.2 fm

PHENIX PreliminaryRs (fm) Ro (fm) RL (fm)


Conclusions• Expansion measured from

spectra depends on Npart.– Saturates to constant for

most central.• Used simple profiles for

the expansion and particle density distribution– Linear velocity profile– Flat particle density

• Within this hydro model, no common source parameters could be found for spectra and all HBT radii simultaneously.

For the most peripheral spectra:T = 0.5 0.2 syst. (< T> = 0.3 ± 0.2 syst. )Tfo = 135 23 MeV

… to the 5% spectra T = 0.7 ± 0.2 syst. (< T> = 0.5 ± 0.2 syst.)Tfo = 110 23 MeV

Rs prefers T > 1.0 and Tfo < 50 MeV

Ro prefers T > 1.4 and Tfo > 100 MeV


Particle Yields per Npart pair vs. Npart

• Yield per pair Npart shown on a log scale for visual clarity only.• Linear dependence on Npart.• Relative increase from peripheral to central greater for K than for , (anti-)p. • Systematic uncertainties shown as lines.

Open symbols: sNN = 130 GeV

dN

/dy/

0.5N

part

Npart

+

K+

p

positive negative-

K-

p

PHENIX Preliminary


Year-1 Mean transverse momentum

• Mean pt with Npart , m0 radial flow• Relative increase from peripheral to central same for , K, (anti)p• (Anti)proton significant from pp collisions

20+/- 5 % increase

20+/- 5 % increase

Open symbols: pp collisions


Year-1:Fitting the Single Particle Spectra

Exclude resonances by fitting pt > 0.5 GeV/c

The resonance region decreases T by ~20 MeV. This is no surprise! Sollfrank and Heinz also observed this in their study of S+S collisions at CERN energies.

NA44 also had a lower pt cut-off for pions in Pb+Pb collisions.

Simultaneous fit (mt -m0 ) < 1 GeV (see arrows)

PHENIX Preliminary

PHENIX Preliminary

PHENIX Preliminary


Year-1: Single Particle Spectra

PHENIX Preliminary


Year-1: An example of a fit to ++. . .

PHENIX Preliminary

The 2 is better for the higher order fit. . .


HBT Radii and Single Particle Spectra

• HBT radii suggest a lower temperature and higher flow velocity

• Use best fit of singles and convert to

• Singles and HBT radii are within 2

f

TfoPHENIX Preliminary


From Hydrodynamics:Radial Flow Velocity Profiles

Plot courtesy of P. Kolb

t()

2 fm/c20 fm/c

5 fm/c

At each “snapshot”in time during theexpansion, thereis a distributionof velocities that vary with theradial position r

Velocity profileat ~ 20 fm/c “freeze-out”hypersurface


Hydrodynamics-based parameterization

parametersnormalization Afreeze-out temperature Tfo

surface velocity t

t

mt

1/m

t dN

/dm

t

TfoA

1/mt dN/dmt = A f() d mT K1( mT /Tfo cosh ) I0( pT /Tfo sinh )

minimize contributions from hard processes fit mt-m0<1 GeV

linear velocity profile t() = t surf. velocity t ave. velocity <t > = 2/3 t boost () = atanh( t() )

integration variable radius r= r/R

definite integral from 0 to 1particle density distribution f() ~ const

t()

1

f()


Two Approaches to Calculating HBT Radii. . .

(After assuming something about the source function. . .)A numerical approach is to

– numerically determine C(K,q) from S(K,q)– C(K,q) ~ 1 + exp[ -qs

2Rs2(K) – q0

2Ro2(K) – ql

2Rl2(K)-2qlqoRlo

2(K)]

– there is an “exact” calculation of these radii (full integrations)– there are lower-order and higher-order approximations (from

series expansion of Bessel functions).• The lowest-order form for Rs was used in Phenix PRL. (A similar

expression is used by NA49). • The higher-order approximation is very good when compared to the

exact calculation for Rs and RL.

What I’m doing


Integration over is done exactly.– Boost invariance (vL = z/t). Space-

time rapidity equals flow rapidity – Infinitely long in y.– In LCMS, y and L = 0.– Integrals expressed in terms of the

modified Bessel functions:

For HBT radii, approximations are used in integration over x and y.

– Saddle point integration using “approximate” saddle point

– Series expansion of Bessel functions

• Assume mT/T>1

nT

mdT

mKT

T

T

Tn coshcosh

0 coshexp

cosh

Important Assumptions Used. . .

TmnA T

n 21

0,An

Btanh 1

ηf

R, yx nn

sinh,cosh

11

21

zt

nvv

L

LT

TkB T

As is also assumed in calculating particle spectra


Calculating the HBT RadiiLinear flow rapidity profile

Defined weight function Fn Rx

,0xηRx

fntt

nft

~

f = 0f = 0.3f = 0.6f = 0.9

Constants are determined up to order 3 from Bessel function expansion

p

nn n

p

nsn n

l

p

nn n

p

nsn n

s

Fe

nRFexR

Fe

nRFeyR

0

0

2

20

22

0

0

22

2

~

xZnRnRmT

F nnso

n

Tn

~21

parametersgeometric radius Rfreeze-out temperature Tflow rapidity at surface f

freeze-out proper time 0

Rs

T = 150 MeV, R = 3 fm, 0 = 3 fm/c


The Analytical Evaluation of the HBT Radii

p

nn n

p

nsn n

l

p

nn n

p

nsn n

s

Fe

nRFexR

Fe

nRFeyR

0

0

2

20

22

0

0

22

2

~

xRnR

xRnR

nss

noo

~111

~111

222

222

Linear flow rapidity profile

xnnns

tnno

BA

Bx

BAx

22 2

2

2

22 22

1~

1

~1

xxλR

Rx

Rx,0xη

Rx

nno

n

fntt

nft

22

2~

~

,128105,

815,1,0~

,1024105,

12815,

83,1

~

2

~sinhcosh

21exp~

21

2

2

e

e

xZnRnRmT

F

RxBAnxZ

n

n

nnso

n

Tn

nttnnn

Up to order 3 in Bessel function expansion

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J.M. Burward-Hoy Lawrence Livermore National Laboratory for the PHENIX Collaboration