zbigniew chaj ę cki, mike lisa ohio state university

Z. Ch. - QM 2009, Knoxville, Tennessee, Mar 31, 2009 1

Zbigniew Chajęcki,

Mike Lisa

Ohio State University

Do p+p Collisions Flow at Do p+p Collisions Flow at RHIC?RHIC? Understanding One- and Two-particle Understanding One- and Two-particle

Distributions, Multiplicity Evolution, and Conservation Distributions, Multiplicity Evolution, and Conservation LawsLaws

Z. Ch. & M. Lisa, PRC 78 064903 (2008)Z. Ch. & M. Lisa, PRC 79 034908 (2009)Z. Ch., arXiv:0901.4078 [nucl-ex]Z. Ch. & M. Lisa, to be published


Outline & MotivationOutline & Motivation p+p as a reference to heavy ion collisions

Effect of the phase-space constraints due to energy and momentum conservation

Re-examining multiplicity-evolution of pT spectra, considering evolution of available phase space

postulate of unchanging parent distribution

Consistent treatment of the phase-space constraints and bulk in femtoscopy and spectra

[hard sector]

Heavy ion collisions as a reference to p+p?

Summary

[soft sector]


Small vs BigSmall vs BigLarge system (Au+Au)Small system (p+p)

STAR, PRL93 (2004) 252301

Hard sector : p+p apparently different than Au+Au Soft sector : Is p+p a clear reference to Au+Au?

STAR PRL 92 112301 (2004)

p+p

Au+Au


Phase-Space varies with Phase-Space varies with multiplicitymultiplicity

Phase-space constraints

Extreme case, N=3, easily calculable with Dalitz plot

What about the effect for higher number of particles?

Dalitz plot for a three-body final state. (p at 3 GeV), PDG 2008Pn ∝ SnRn

Phase-space factor: Hagedorn/FermiPhase-space factor: Hagedorn/Fermi

€

Sn - dynamics

Rn - kinematics


Correlations arising (only) from Correlations arising (only) from conservation laws (PS constraints)conservation laws (PS constraints)

€

˜ f ( pi) = 2E i

dN

d3 pi

single-particle “parent” distributionw/o P.S. restriction

%fc (p1,..., pk ) : %f (pi )i=1

k∏( )⋅ d4 piδ(pi2 −mi

2 ) %f (pi )i=k+1

N∏( )∫ δ 4 pii=1

N

∑ −P⎛⎝⎜

⎞⎠⎟

≅ %f (pi )i=1

k∏( ) N

N −k⎛⎝⎜

⎞⎠⎟

2

exp −pi,μ − pμ( )

i=1

k

∑⎛⎝⎜⎞⎠⎟

2

2(N −k)σ μ2

μ=0

3

∑

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

k-particle distribution (k<N)

no othercorrelations

what wemeasure

with P.S. restriction

CLT approximation works best for N>10 & Ei < 23<E>


Phase-space effect on k-particle Phase-space effect on k-particle distributiondistribution

%fc (p1,..., pk ) = %f (pi )i=1

k∏( ) N

N −k⎛⎝⎜

⎞⎠⎟

2

exp −pi,μ − pμ( )

i=1

k

∑⎛⎝⎜⎞⎠⎟

2

2(N −k)σ μ2

μ=0

3

∑

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

where

σ μ2 = pμ

2 − pμ

2

pμ =0 for μ =1,2,3

k-particle distribution in N-particle system (in CMS frame)

pμ2 ≡ d3p⋅pμ

2 ⋅ %f p( )unmeasuredparent distrib

{∫ ≠ d3p⋅pμ2 ⋅%fc p( )

measured{∫

–Danielewicz et al, PRC38 120 (1988)–Borghini, Dinh, & Ollitraut PRC62 034902 (2000)–Borghini, Eur. Phys. J. C30:381-385, (2003)–Chajecki & Lisa, PRC 78 064903 (2008), PRC 79 034908 (2009)

“distortion” due to PS constraints


N=5

N=40

Z.C

h, M.Lisa, P

RC

79 034908 (2009)

1-particle PS effect1-particle PS effect

Phase-space effect on kinematic Phase-space effect on kinematic observablesobservables

%fc (pi ) =%f (pi )×

×N

N −1⎛⎝⎜

⎞⎠⎟

2

exp −1

2(N −1)2pT ,i

2

pT2

+pz,i2

pz2

+Ei − E( )

2

E2 − E 2

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

Finite-particle constrains

C(p1, p2 ) ≅1−1N

2rpT ,1 ⋅

rpT ,2

pT2

+pz,1 ⋅pz,2

pz2

+E1 − E( )⋅E2 − E( )

E2 − E 2

⎛

⎝⎜

⎞

⎠⎟

NA49 pions Borghini et al, PRC 66 014901 (2002)- also, Danielewicz, PLBB157:146 (1985)


CF (GenBod)

EMCICs

Z. Ch, M. Lisa, PRC 78 064903 (2008)

N. Borghini, PRC75:021904 (2007)


C(p1,..., pk ) ≡%fc(p1,..., pk)

%fc(p1)....%fc(pk)

=

NN −k

⎛⎝⎜

⎞⎠⎟2

NN −1

⎛⎝⎜

⎞⎠⎟2k

exp −1

2(N −k)

px,ii=1

k∑( )2

px2 +

py,ii=1

k∑( )2

py2

+pz,ii=1

k∑( )2

pz2 +

Ei − E( )i=1

k∑( )2

E2 − E 2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟i=1

k

∑⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

exp −1

2(N −1)px,i2

px2 +

py,i2

py2

+pz,i2

pz2 +

Ei − E( )2

E2 − E 2

⎛

⎝⎜

⎞

⎠⎟

i=1

k

∑⎛

⎝⎜⎜

⎞

⎠⎟⎟


1-particle phase-space 1-particle phase-space effect effect

€

˜ f c( pi ) = ˜ f (pi )N

N −1

⎛

⎝ ⎜

⎞

⎠ ⎟2

exp −1

2(N −1)

2 pT ,i2

pT2

+pz ,i

2

pz2

+Ei − E( )

2

E2 − E2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

“distortion” of single-particle spectra

What if the only difference between p+p and A+A collisions was N?

measured

“matrix element”

€

same ˜ f p( ) , pT2 , E , E2

STAR PRL 92 112301 (2004)

Au+Au 0-5%

Au+Au 60-70%

p+p minbias

STAR PRL 92 112301 (2004)

Then we would measure:

€

˜ f cpp pT ,i( )

˜ f cAA pT ,i( )

=NAA −1( )N pp

N pp −1( )NAA

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

exp1

2 NAA −1( )−

1

2 N pp −1( )

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟2pT ,i

2

pT2

+E i − E( )

2

E 2 − E2

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟


Multiplicity evolution of spectra - p+p to A+A (soft Multiplicity evolution of spectra - p+p to A+A (soft sector)sector)

N evolution of spectra dominated by PS “distortion”

p+p system samples same parent distribution, but under stronger PS constraints

N evolution of spectra dominated by PS “distortion”

p+p system samples same parent distribution, but under stronger PS constraints

€

˜ f cpp pT ,i( )

˜ f cAA pT ,i( )

∝ exp1

2 NAA −1( )−

1

2 N pp −1( )

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟2 pT ,i

2

pT2

+E i − E( )

2

E 2 − E2

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

STAR PRL 92 112301 (2004)


Kinematic scales of “the Kinematic scales of “the system”system”

postulate of same parent consistent with all spectra• magnitude• pT dependence (shape)

• mass dependence

postulate of same parent consistent with all spectra• magnitude• pT dependence (shape)

• mass dependence

Fit results for p+p consistent with expectations from Maxwell-Boltzman equation, Blast-wave, Pythia, …


By By popular popular demanddemand

Almost universal “flow” & “temperature”parameters in a BlastWave fit

Apparent changes in β, T with dN/dη caused by finite phase-space effect

p+p

STAR PRL 92 112301 (2004)

Blast-Wave Model: F. Retiere, M. Lisa,

PRC70:044907,2004.


Blast-wave in p+p@200GeV: Blast-wave in p+p@200GeV: simultaneous description of spectra, simultaneous description of spectra,

HBTHBT

€

T =105.5 MeV

ρ 0 = 0.934 β = 0.535( )

R = 2.19 fm

τ = 2.25 fm/c

Δτ ~ 0.15

determined entirelyby spectra

STAR PreliminarySee M. Lisa’s poster (STAR)


Fits to pion CF in p+p by STARFits to pion CF in p+p by STAR

C(p1, p2 ) =a 1+ λ ⋅ Kcoul (Qinv) 1+ exp −Rout2 Qout

2 −Rside2 Qside

2 −Rlong2 Qlong

2( )( )−1⎡⎣

⎤⎦{ } ×

1−2⋅

rp1,T ⋅

rp2,T

N pT2

−p1,Z ⋅p2,Z

N pz2

−E1 − E( )⋅E2 − E( )

N E2 − E 2( )

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

N =14

pT2 =0.17 (GeV / c)2

pz2 =0.32 (GeV / c)2

E =0.68 GeV

E2 =0.50 GeV2HBT

exp CF

HBT+ “conservation”

STAR preliminary

kT = [0.35,0.45] GeV/c

Use parameters obtained from the fit to STAR femtoscopic correlation function and use them to “correct” spectra

%fc (pi ) =%f(pi )N

N −1⎛⎝⎜

⎞⎠⎟

2

exp −1

2(N −1)2pT ,i

2

pT2

+pz,i2

pz2

+Ei − E( )

2

E2 − E 2

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟



Combined fit: consistent flow-based Combined fit: consistent flow-based descriptiondescription

Blast-Wave Model: F. Retiere, M. Lisa, PRC70:044907,2004.

€

T =106 ± 3 MeV

β = 0.48 ± 0.03

R = 2.09 ± 0.04 fm

τ 0 = 2.25 ± 0.05 fm/c

Δτ = 0.1± 0.2 fm/c



Combined fit: consistent flow-based Combined fit: consistent flow-based descriptiondescription

“raw” (ignoring PS effets)

“raw” (ignoring PS effects)

PS effects fixed by correlationsJoint spectra/HBT BW fit

PS effects free adjustedto spectra & fit to spectra

PS effects fixed by correlationsJoint spectra/HBT BW fit

p+p collisions show same flow signals as A+A collisions

p+p collisions show same flow signals as A+A collisions


SummarySummary

Energy and momentum conservation induces phase-space constraint that has explicit multiplicity dependence– should not be ignored in (crucial!) N-dependent comparisons

– significant effect on 2- (and 3-) particle correlations [c.f. Ollitrault, Borghini, Voloshin…]

– …and single-particle spectra (often neglected because no “red flags”)

Femtoscopy & Spectra– in H.I.C., well understood, detailed fingerprint of flow

– RHIC – first opportunity for direct comparison with p+p

– accounting for finite phase-space effects identical flow signals in p+p

• Has A+A become the reference system for p+p in non-perturbative sector???

?

HBT

exp CF

HBT+ “conservation”

STAR PRL 92 112301 (2004)

STAR Preliminary


““the system”… a nontrivial the system”… a nontrivial conceptconcept

€

N, E , E 2 , pT2 , pZ

2

Characteristic scales of relevant system in which limited energy-momentum is shared

• Not known a priori• should track measured quantities, but not be identical to them

1. N includes all primary particles (including unmeasured γ’s etc)

2. secondary decay (resonances, fragmentation) smears connection b/t <E2> and measured one

3. <E2> etc: averages of the parent distribution

4. “relevant system” almost certainly not the “whole” (4π) system• e.g. beam fragmentation probably not relevant to system emitting at midrapidity

• characteristic physical processes (strings etc): Δy ~ 1÷2

• jets: “of the system” ??• or just stealing energy from “the system?”

1.if “relevant system” ≠ “whole system”, then total energy-momentum will fluctuate e-by-e

€

pμ2 ≡ d3p ⋅pμ

2 ⋅ ˜ f p( )unmeasuredparent distrib

{∫ ≠ d3p ⋅pμ2 ⋅ ˜ f c p( )

measured{∫


Consistency check ….Consistency check ….

€

N, E , E 2 , pT2 , pZ

2

Characteristic scales of relevant system in which limited energy-momentum is shared

€

Blastwave, T =100 MeV ρ 0 = 0.9

pT2

π= 0.240 GeV2 pT π

= 0.405 GeV( )

mT π= 0.435 GeV

mT2

π= 0.259 GeV2

€

Maxwell - Boltzmann parent d3N

d3 p~ e−E /T

non - rel ultra - rel if T = .15 ÷ .35

pT2 2mT 8T 2 0.045 ÷ 0.98 GeV/c( )

2

E 2 154 T 2 + m2 12T 2 0.10 ÷1.5 GeV2

E 32 T + m 3T 0.36 −1 GeV


Spectra

v2

HBT

€

mT ≈ T + mβ flow2

Heavy Ion Collisions : Explosive flow revealed through Heavy Ion Collisions : Explosive flow revealed through specific fingerprints specific fingerprints on soft-sector observableson soft-sector observables

calculable in hydrodynamics or toy “blast wave” models

slow

fast


Femtoscopy - direct evidence of Femtoscopy - direct evidence of flowflow

Spectra

v2

HBT

Flow-dominated “Blast-wave”toy models capture main characteristicse.g. PRC70 044907 (2004)

KR

(fm

)

mT (GeV/c)

STAR PRL 91 262301 (2003)

space-momentum substructure mapped in detail


Implication: A+A is just a collection Implication: A+A is just a collection of flowing p+p?of flowing p+p?

• No! Quite the opposite.– femtoscopically

• A+A looks like a big BlastWave• not superposition of small

BlastWaves• A+A has thermalized globally

–spectra• superposition of spectra from p+p

has same shape as a spectrum from p+p!

• relaxation of P.S. constraints indicates A+A has thermalized globally

• rather, p+p looks like a “little A+A”


EMCIC fit to STAR p+p dataEMCIC fit to STAR p+p data

STAR preliminary

kT = [0.15,0.25] GeV/c kT = [0.25,0.35] GeV/c

kT = [0.35,0.45] GeV/c kT = [0.45,0.60] GeV/c


Average matrix element - Average matrix element - factorizationfactorization

W ′p1( )d3 ′p1 ∝ d3 ′p1 L δ ′p12 −m2( )dp01 δ ′pi

2 −m2( )d4 ′pii=2

n

∏ ×∫∫

δ 4 ′pj −p1 −p2j=1

n

∑⎛

⎝⎜⎞

⎠⎟S ′p1K ′pn |p1, p2( )

≡d3 ′p1 ⋅Sn ′p1( )RF

Probability for an n-particle final state:

Single-particle spectrum

R. Hagedorn, Relativistic Kinematics 1963

dynamics kinematics


……STAR PRC 2007


World Systematics : R(pWorld Systematics : R(pTT/m/mTT) in small ) in small systemssystems

**

€

pT = 2 / 3 ⋅r p

STAR preliminary

€

*RT ≈ RO ≈ RSfrom STAR talk at WWND 2009non-STAR data taken from Z. Ch. arXiv:0901.4078 [nucl-ex]

STAR preliminary


Non-femto correlationsNon-femto correlations

CLEO PRD32 (1985) 2294

NA22, Z. Phys. C71 (1996) 405

Qx<0.04 GeV/cOPAL, Eur. Phys. J. C52 (2007) 787-803

Qx<0.2 GeV/cNA23, Z. Phys. C43 (1989) 341

E766, PRD 49 (1994) 4373M

ultip

licity

incr

ease

s


Significant non-femto correlations, but little effect on Significant non-femto correlations, but little effect on “message”“message”

STAR preliminary Ratio of (AuAu, CuCu, dAu) HBT radii by pp

rather, “suggestion”: explosive flow in p+p?

Z. Ch. - QM 2009, Knoxville, Tennessee, Mar 31, 2009 28arXiv:0810.4979 submitted

looks to me like the spectrum evolves…arXiv:0809.4737

PLB 612 (2005) 181 99 (2007) 112301

folks use this onefor p+p data

… and this onefor Au+Au data(looks better than the one to the left!)

these ones arerecently submittedpapers that replotthe data from the above

these ones arerecently submittedpapers that replotthe data from the above

jetty starting ~ mT-m=1.5 pT=2.3

STAR ϕ spectra


phi agrees as well/poorly phi agrees as well/poorly as pi/K/p from our paperas pi/K/p from our paper

Z. Chajecki & M. Lisa PRC 79 034908 (2009)

as discussed in our paper, EMCICsalone is not enough to explain behaviorbeyond ~ 1GeV/c

using same parameters as in our paper,multiplicity-dependence of phi is described,as well, and up to same pT, as pi/k/p


Phase-space effects in Phase-space effects in PYTHIAPYTHIA

correlation function from PYTHIA

It’s likely that there are also other correlations in PYTHIA than just due to E&M correlations


b

zbigniew chaj ę cki, mike lisa ohio state university

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