mike lisa ohio state university

The multiplicity evolution of spectra at RHIC

How different (really) are p+p and Au+Au collisions?

Mike LisaOhio State University

Z. Chajecki & MAL, arXiv:0803.0022 [nucl-th]; Z. Chajecki & MAL arXiv:0807.3569 [nucl-th]

Outline

• 2 Prime Discoveries in (heavy ion) collisions at RHIC: flow & RAA

• Observed multiplicity evolution of spectra at RHIC (& conclusions derived)

• EMCIC* effects

• soft-sector implications

• hard-sector speculations

• Summary

mike lisa - WPCF - 11-14 Sept 2008 - Krakow Poland

* EMCIC = “Energy and Momentum Induced Correlation/Constraint”2


Perfect Press Releases

• Perfect or not, creation of a bulk system at RHIC is established – soft sector: flow• This system is very color dense and largely opaque to partons traversing it – hard sector: RAA

? Are these statements unique to A+A collisions?

blah blahthe quick brown/..fox...jumped..ove... th lazydog /// whatever one wants to say here s.....is just fine with mw. It’s not mattering at all. This is just a bunch of squiglly, unreadable text on this sllide I hope nobody can read itanyways since it is all nonsense. Not like that distinguishes it very much from much of my other writing, of course. But what the hell... OKlet’s just finish this lnbe and we’re done

ature of EoS unde estigation ; agreement wi

data might be accidental ; viscous hydrodynamics under

development ; assumptionof thermalization in question

sensitivity to modeling ofinitial state, underintense study

3

Spectra signals from hard & soft sectors

• Jet “quenching” RAA

–comparing “big” to “small” systems indispensible–clear evidence for suppression of fragmentation daughters

mike lisa - WPCF - 11-14 Sept 2008 - Krakow Polandmike lisa - WPCF - Krakow - 11 Sept 2008

€

RAA ≡dNdpT AA

TAA × dNdpT pp

€

RCP ≡TP × dN

dpT C

TC × dNdpT P

4




• Soft sector: evidence for collectivity– spectra–azimuthal correlations (v2)– femtoscopy (best probe…)

–no reference to small system required?


Spectra

v2

HBT

5







Central Au+Au collisions

T~105 MeV

<β> ~ 0.55 c• consistent with femtoscopy• Insensitive to space


T~105 MeV


Spectra

v2

HBT

Can one have bulk collective behaviour in smaller systems?Can one have bulk collective

behaviour in smaller systems?

6








T~105 MeV



T~105 MeV


Spectra

v2

HBT



mike lisa - WPCF - Krakow - 11 Sept 2008

E735 Collaboration, PRD48 1931 (1993)also PLB 2002consistent with an expanding shell model.

NA22 Collaboration Z. Phys. C 71, 405–414 (1996)(hadron-hadron collisions)[based on shape of C(q)…]Our data do not confirm the expectation from the string type model… A good description of our data is, however, achieved in the framework of the hydrodynamical expanding source model.

W. Kittel Acta Phys.Polon. B32 (2001) 3927 [Review article]… and suggests the existence of an important “collective flow”, even in the system of particles produced in e+e− annihilation!

A 1/√m T scaling first observed in heavy-ion collisions is now also observed in Z fragmentation and may suggest a “transverse flow” even there!

OPAL Collaboration, Eur.Phys.J.C52:787-803,2007; arXiv:0708.1122 [hep-ex]

R2tside , R2tout and, less markedly, R2long decrease with increasing kt . The presence of correlations between the particle production points and their momenta is an indication that the pion source is not static, but rather expands during the particle emission process.

7








T~105 MeV



T~105 MeV


Spectra

v2

HBT



mike lisa - WPCF - Krakow - 11 Sept 2008

E735 Collaboration, PRD48 1931 (1993)also PLB 2002consistent with an expanding shell model.

NA22 Collaboration Z. Phys. C 71, 405–414 (1996)(hadron-hadron collisions)[based on shape of C(q)…]Our data do not confirm the expectation from the string type model… A good description of our data is, however, achieved in the framework of the hydrodynamical expanding source model.

W. Kittel Acta Phys.Polon. B32 (2001) 3927 [Review article]… and suggests the existence of an important “collective flow”, even in the system of particles produced in e+e− annihilation!

A 1/√m T scaling first observed in heavy-ion collisions is now also observed in Z fragmentation and may suggest a “transverse flow” even there!

OPAL Collaboration, Eur.Phys.J.C52:787-803,2007; arXiv:0708.1122 [hep-ex]

R2tside , R2tout and, less markedly, R2long decrease with increasing kt . The presence of correlations between the particle production points and their momenta is an indication that the pion source is not static, but rather expands during the particle emission process.

RHIC: “comparison machine”Vary size. All else fixed.• spectra• femtoscopy (Z. Chajecki)

RHIC: “comparison machine”Vary size. All else fixed.• spectra• femtoscopy (Z. Chajecki)

8

STAR PRL 92 112301 (2004)

Au+Au 0-5%

Au+Au 60-70%

p+p minbias

Multiplicity evolution of pT spectra

9mike lisa - WPCF - 11-14 Sept 2008 - Krakow Poland


Blast-wave fit to spectra:•much less explosive flow, (& higher “temperature”) in low-multiplicity collisions (as per prejudice)

10


Blast-wave fit to spectra:•much less explosive flow, (& higher “temperature”) in low-multiplicity collisions (as per prejudice)

•BUT: correlation studies (e.g. Chajęcki) clear finite-number effects. •Would that affect spectra?

11

energy-momentum conservation in n-body states

€

f α( ) =d

dαM

2⋅Rn( )

where

M = matrix element describing interaction

(M =1 → all spectra given by phasespace)

spectrum of kinematic quantity (angle, momentum) given by

€

Rn = δ 4 P − p j

j=1

n

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ δ pi

2 − mi2

( )d4pi

i=1

n

∏4n

∫

where

P = total 4 - momentum of n - particle system

pi = 4 - momentum of particle i

mi = mass of particle i

n-body Phasespace factor Rn

€

δ pi2 − mi

2( )d

4pi =r p i

2

E i

dr p i ⋅d cosθ i( ) ⋅dφi

statistics: “density of states”

larger particle momentum more available states

P conservation

€

δ 4 P − p j

j=1

n

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Induces “trivial” correlations(i.e. even for M=1)

mike lisa - WPCF - 11-14 Sept 2008 - Krakow Poland 12

Example of use of total phase space integral

• In absence of “physics” in M : (i.e. phase-space dominated)

€

Γ pp → πππ( )Γ pp → ππππ( )

=R3 1.876;π ,π ,π( )

R4 1.876;π ,π ,π ,π( )

€

In limit where "α "="event" = collection of momenta r p i

"spectrum of events" = f α( ) =d

dαRn

→ Probevent α ∝d3n

dpi3

i=1

n

∏Rn

• single-particle spectrum (e.g. pT):

• “spectrum of events”:

F. James, CERN REPORT 68-15 (1968)

€

W pi( ) = d 3 pi ⋅S n pi( )Rn

Hagedorn


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

€

˜ f ( pi) = 2E i

dN

d3 pi

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

€

˜ f c (p1,..., pk ) ≡ ˜ f ( pi)i=1

k

∏ ⎛ ⎝ ⎜ ⎞

⎠ ⎟⋅

d3 pi

2E i

˜ f (pi)i= k +1

N

∏ ⎛

⎝ ⎜

⎞

⎠ ⎟∫ δ 4 pi

i=1

N

∑ − P ⎛

⎝ ⎜

⎞

⎠ ⎟

d3 pi

2E i

˜ f (pi)i=1

N

∏ ⎛

⎝ ⎜

⎞

⎠ ⎟∫ δ 4 pi

i=1

N

∑ − P ⎛

⎝ ⎜

⎞

⎠ ⎟

= ˜ f ( pi)i=1

k

∏ ⎛ ⎝ ⎜ ⎞

⎠ ⎟⋅

d4 piδ(pi2 − mi

2) ˜ f (pi)i= k +1

N

∏ ⎛ ⎝ ⎜ ⎞

⎠ ⎟∫ δ 4 pi

i=1

N

∑ − P ⎛

⎝ ⎜

⎞

⎠ ⎟

d4 piδ(pi2 − mi

2) ˜ f (pi)i=1

N

∏ ⎛ ⎝ ⎜ ⎞

⎠ ⎟∫ δ 4 pi

i=1

N

∑ − P ⎛

⎝ ⎜

⎞

⎠ ⎟

k-particle distribution (k<N) with P.S. restriction

no othercorrelations

what wemeasure


Simplification for “large” N-k (1)


Denominator is “just” a constant normalization (indep pi)

€

pi

i= k +1

N

∑ − P + pi

i=1

k

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

€

= ˜ f (pi)i=1

k

∏ ⎛ ⎝ ⎜ ⎞

⎠ ⎟⋅

d4 piδ( pi2 − mi

2) ˜ f ( pi)i= k +1

N

∏ ⎛ ⎝ ⎜ ⎞

⎠ ⎟∫ δ 4 pi

i=1

N

∑ − P ⎛

⎝ ⎜

⎞

⎠ ⎟

d4 piδ( pi2 − mi

2) ˜ f ( pi)i=1

N

∏ ⎛ ⎝ ⎜ ⎞

⎠ ⎟∫ δ 4 pi

i=1

N

∑ − P ⎛

⎝ ⎜

⎞

⎠ ⎟

€

pi

i= k +1

N

∑ − P − pi

i=1

k

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

Numerator is the probability distribution of a sum of many (N-k) uncorrelated vectors

(i.e. the probability that they will add up to P-Σi=1kpi)

If (N-k) big Multivariate Central Limit Theorem

15

Using central limit theorem (“large* N-k”)

€

˜ f c(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

€

pμ2 ≡ d3p ⋅pμ

2 ⋅ ˜ f p( )unmeasuredparent distrib

{∫ ≠ d3p ⋅pμ2 ⋅ ˜ f c p( )

measured{∫N.B.

relevant later

–Danielewicz et al, PRC38 120 (1988)–Borghini, Dinh, & Ollitraut PRC62 034902 (2000)–Borghini Eur. Phys. J. C30:381-385, (2003)–Chajecki & MAL, arXiv:0803.0022 [nucl-th], sub PRC * “large”: N > ~10 18

k-particle correlation function

€

C(p1,...,pk ) ≡˜ f c(p1,...,pk )

˜ f c(p1)....̃ f c(pk )

=

N

N − k

⎛

⎝ ⎜

⎞

⎠ ⎟2

N

N −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

+E i − E( )

i=1

k

∑ ⎛ ⎝ ⎜ ⎞

⎠ ⎟2

E 2 − E2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟i=1

k

∑

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

exp −1

2(N −1)

px,i2

px2

+py,i

2

py2

+pz,i

2

pz2

+E i − E( )

2

E 2 − E2

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

i=1

k

∑ ⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

Dependence on “parent” distrib f vanishes,except for energy/momentum means and RMS

2-particle correlation function (1st term in 1/N expansion)

€

C(p1,p2) ≅1−1

N2

r p T,1 ⋅

r p T,2

pT2

+pz,1 ⋅pz,2

pz2

+E1 − E( ) ⋅ E 2 − E( )

E 2 − E2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


Effects on single-particle distribution

€

˜ f c(pi) = ˜ f (pi)N

N −1

⎛

⎝ ⎜

⎞

⎠ ⎟2

exp −pi,μ − pμ( )

2

2(N −1)σ μ2

μ = 0

3

∑ ⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

= ˜ f (pi)N

N −1

⎛

⎝ ⎜

⎞

⎠ ⎟2

exp −1

2(N −1)

px,i2

px2

+py,i

2

py2

+pz,i

2

pz2

+E i − E( )

2

E 2 − E2

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

in this case, the index i is only keepingtrack of particle type, reallymike lisa - WPCF - 11-14 Sept 2008 - Krakow Poland

Finite-particle “distortion”•1st, 2nd moments of parent <E>, <E2>, <pT

2>•multiplicity N

20

“the system”… a nontrivial concept


€

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 primary particles (including unmeasured γ’s etc)

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

3. “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

1.jets: “of the system” or only stealing energy from the system which then redistributes?

2.“relevant system” ≠ “whole system”, total energy-momentum will vary somewhat event-to-event – see definition above

• <E2> etc: averages of the parent distribution

€

pμ2 ≡ d3p ⋅pμ

2 ⋅ ˜ f p( )unmeasuredparent distrib

{∫ ≠ d3p ⋅pμ2 ⋅ ˜ f c p( )

measured{∫

“the system”… a nontrivial concept

€

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

• What to expect?


€

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

What drives the evolution?

€

˜ 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

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

measured

“matrix element”

STAR PRL 92 112301 (2004)

Au+Au 0-5%

Au+Au 60-70%

p+p minbias

What drives the evolution?

€

˜ 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

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

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

measured

“matrix element”

€

same ˜ f p( ) , pT2 , E , E2

€

˜ f cp+ p pi( )

˜ f cA+A pi( )

=N

N −1

⎛

⎝ ⎜

⎞

⎠ ⎟2

exp −1

2(N −1)

2 pT ,i2

pT2

+pz,i

2

pz2

+Ei − E( )

2

E2 − E2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

where N is for minbias p + p, and N A+A ≈ ∞

Then we would measure:


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

€

˜ f cp + p pT ,i( )

˜ f cA +A pT ,i( )

= K ⋅N

N −1

⎛

⎝ ⎜

⎞

⎠ ⎟3 / 2

exp −1

2(N −1)

2pT ,i2

pT2

+E i − E( )

2

E 2 − E2

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

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

K ~ unity. driven by conservation of discrete quantum #s (strangeness, etc) 26

By popular demand…

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

Apparent changes in β, T with dN/dη caused by EMCICs*

* EMCIC = Energy & Momentum Conservation Induced Constraint 27

Kinematic scales of “the system”

€

˜ f cp + p pT ,i( )

˜ f cA +A pT ,i( )

= K ⋅N

N −1

⎛

⎝ ⎜

⎞

⎠ ⎟3 / 2

exp −1

2(N −1)

2 pT ,i2

pT2

+E i − E( )

2

E 2 − E2

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

€

non - rel ultra - rel if T = .15 ÷ .35 What we find

pT2 2mT 8T 2 0.045 ÷ 0.98 GeV/c( )

20.12 GeV/c( )

2

E 2 154 T 2 + m2 12T 2 0.10 ÷1.5 GeV2 0.43 GeV2

E 32 T + m 3T 0.36 −1 GeV 0.61 GeV

28

Multiplicity evolution of spectra - within p+p (soft sector)

What if the only difference betweenmultiplicity-selected p+p collisions was N?

€

same ˜ f p( ) , pT2 , E , E2

€

˜ f cN1 pT ,i( )

˜ f cN 2 pT ,i( )

=N2 −1( )N1

N1 −1( )N2

⎛

⎝ ⎜

⎞

⎠ ⎟

3/2

exp1

2 N2 −1( )−

1

2 N1 −1( )

⎛

⎝ ⎜

⎞

⎠ ⎟2 pT ,i

2

pT2

+Ei − E( )

2

E2 − E2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

…Then we would measure:

pion mass assumedmike lisa - WPCF - 11-14 Sept 2008 - Krakow Poland

STAR, PRD74 032006 (2006)

29

Multiplicity evolution of spectra - within p+p (soft sector)

€

˜ f cN1 pT ,i( )

˜ f cN 2 pT ,i( )

=N2 −1( )N1

N1 −1( )N2

⎛

⎝ ⎜

⎞

⎠ ⎟

3/2

exp1

2 N2 −1( )−

1

2 N1 −1( )

⎛

⎝ ⎜

⎞

⎠ ⎟2 pT ,i

2

pT2

+Ei − E( )

2

E2 − E2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

pion mass assumedmike lisa - WPCF - 11-14 Sept 2008 - Krakow Poland

STAR, PRD74 032006 (2006)

unidentified particles cannot concludethat EMCICs is the whole story…

…but it appears to be most of it.

unidentified particles cannot concludethat EMCICs is the whole story…

…but it appears to be most of it.

Account for “trivial” before inferring more “profound” messagesAccount for “trivial” before inferring more “profound” messages

30

Summary• E&M conservation induces phasespace constraints w/ explicit N dependence

–obviously should not be ignored in (crucial!) N-dependent comparisons

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

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

• Soft sector:

–differences observed in pT spectra consistent with EMCIC “distortion” of unchanging parent distribution


Can we look at higher pT ?

Calculation shown: GENBOD program of Fred James (CERN 1968)• correlations due only to EMCICs• intrinsic “algorithmic” parent distribution

Not quantitatively…1.CLT approximation becomes unreliable for E > ~2÷3<E>c

2.also: jets – are they“of the system” ?• statistical energy-sharing questionable

€

RN−1 p1( )

CLT approx

€

Ec

=1 GeVBut do it anyways, just to see…

32

Au+Au collisions


High multiplicity collisions suppressed at high pT, relative to low multiplicity:•NOT explained by phasespace restrictions (EMCICs)•parent (physics) drives the change

33

p+p collisions


High multiplicity collisions enhanced at high pT, relative to low multiplicity:•but not as enhanced as if only EMCICs were at play: Rpp

34

p+p collisions


High multiplicity collisions enhanced at high pT, relative to low multiplicity:•but not as enhanced as if only EMCICs were at play: Rpp

•naively (!!) “correct” for EMCICs R’pp

• high-pT suppression in p+p ???

Calculation way too quantitatively (conceptually?) unreliable at this pT

Conclude only that “trivial” effects might be large here, should be considered in addition to more profound physics conclusions

Calculation way too quantitatively (conceptually?) unreliable at this pT

Conclude only that “trivial” effects might be large here, should be considered in addition to more profound physics conclusions

35





• Soft sector (~quantitative):


• Hard sector (qualitative):

–EMCICs do not explain N-dependence of spectra (RAA) – parent changes

–EMCIC effects in p+p big, may modify physics conclusions






• Soft sector (~quantitative):


• Hard sector (qualitative):

–EMCICs do not explain N-dependence of spectra (RAA) – parent changes

–EMCIC effects in p+p big, may modify physics conclusions• “jet quenching” in p+p ??


In terms of our “big physics” at RHIC,How different (really) are p+p and A+A collisions?•flow in p+p? – spectral signal same as A+A•jet quenching in p+p? much less clear

In terms of our “big physics” at RHIC,How different (really) are p+p and A+A collisions?•flow in p+p? – spectral signal same as A+A•jet quenching in p+p? much less clear

mike lisa ohio state university

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