Correlations and uctuations in pA and AA collisions M. · PDF fileCorrelations and uctuations in pA and AA collisions ... pair NN correlations inclusion feasible; ﬃ to estimate uctuations

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Correlations and fluctuations in pA and AA collisions

M. Alvioli

CNRConsiglio Nazionale delle Ricerche

(IRPI - Perugia, Italy)

M. Alvioli 1 ECT⋆, June ’15

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OUTLINE

1Monte Carlo Glauber approach for pA and AA

1.a Glauber vs. Monte Carlo Glauber

1.b Nuclear configurations including NN correlations

1.c Npart and Geometry fluctuations in pA and AA collisions

1.d Recent updates

2. Beyond the Glauber approach

2.a NN interaction strenght fluctuations

2.b Determination of centrality

2.c Perspectives


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1.a - Glauber multiple scattering pA and AA scattering

Glauber approach: quantum mechanics of high-energy many-body scat-tering =⇒ frozen approximation; straight line trajectories, no transverse mo-mentum exchange.

z

Incoming Hadron

Target A

s

Inputs:⋆ (charge) densities of nuclei⋆ energy-dependent Nucleon--Nucleon (NN) cross sections

Side view Beam-line view

Projectile B Target A

s–b

s–b

s sb

b

z

B

A

for given energy andAA impact parameter b:

−→ interacting−→ spectators

−→ elastically scattered


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1.a - Glauber: semi-analytic description

• continuous density distributions of nuclei, ρ(r); r = (b, z)

• probability of NN interaction from thickness function T (b) =∫dzρ(b, z)

• binomial distribution for n binary collisions in AA:

Pn(b) =

(A2

n

)[TAA(b)σ

inNN

]n [1− TAA(b)σ

inNN

]A2−n

e.g., total AA inelastic cross section (TAA convolution of two Ts):

σinAA =∫db

∫ ∏A∗Bi dsiTAA(si)

1−

∏Aj∏Ak σ(b− sjsk)

• optical limit : assuming uncorrelated scattering centers, A*B integrationsover transverse coordinates are reduced to one integration:

σin,optAA =

∫db

1−

[1− σinNNTAA(b)

]A2

• accurate for most needs. Fine details of density are lost; pair NN correlationsinclusion feasible; difficult to estimate fluctuations

4 ECT⋆, June ’15

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1.a - Monte Carlo Glauber (MCG) description

• event-by-event simulation: details of density distributions by randomly gen-erated nucleons positions : in average give the nucleus density.

•MCG introduces of Npart and Ncoll, not directly measurable, but containa lot of information about the fluctuating collision geometry .

In particular:

→ Npart experimentally related to energy in ZDCs ⇔ centrality

→ charged particle multiplicity scales with Npart, Ncoll ⇔ centrality

→ participant distribution shape determines elliptic flow of low pT parti-cles

• quantitative description of geometry reduce systematic errors

•MCG is a starting point for models that requireproduction points for individual subprocesses

• shadowing naturally built-in with MGC

M. Alvioli 5

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1.a - Monte Carlo Glauber (MCG) description: fluctuations

partN0 50 100 150 200 250 300

def

ault

par

t /

NX par

tN

0.8

0.9

1

1.1

1.2

partN0 50 100 150 200 250 300

def

ault

par

t /

NX par

tN

0.8

0.9

1

1.1

1.2

= 39 mbinelNNσ

= 45 mbinelnnσ

Woods-Saxon: R = 6.65 fm, d = 0.55 fmWoods-Saxon: R = 6.25 fm, d = 0.53 fm

= 0.4 fm)hard

Nucleons with hard core (r

Different simulation of ZDC energyDifferent BBC fluctuations

Gray disk NN overlap functionGaussian NN overlap functionVariation in 2d BBC-ZDC centrality selectionBBC trigger eff. (91.4 + 2.5) % (more central)BBC trigger eff. (91.4 - 3.0) % (less central)Sys. error of exp. centr. sel. (more central)Sys. error of exp. centr. sel. (less central)

collN0 200 400 600 800 1000

def

ault

coll

/ N

X coll

N

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

collN0 200 400 600 800 1000

def

ault

coll

/ N

X coll

N

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4Miller, Reygers, Sanders, Steinberg,Ann. Rev. Nucl. Part. Sci. 57 (2007)

effects of different sourcesof fluctuations and

parameter dependencieswithin MGC

and detector simulation

We focus onfluctuations due to:

• inclusion of NN correlations inpreparing nuclear configurations

• no black-disk approximation for NN−→ P (|b− bj|)

• initial nucleon positions −→ initialgeometry

• fluctuation of the NN cross sec-tion−→ average number of partic-ipants −→ different impact param-eter dependence


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1.b - A Monte Carlo generator for nucleon configurations

• Configurations generated accordingto the independent particle modelcontain overlapping nucleons

• Ad-hoc hard-core rejection methods avoids overlapping nucleons but is notlinked to realistic correlations and do not reproduce two-body density

•We developed a Metropolis codewhich includes realistic NNcorrelations functions ina way which is consistent withthe input one-body density

•We also have a two-body densitywhich compares to the realistic one

0 2 4 6 8 10 120.00

0.04

0.08

0.12

0.16

0 1 2 3 4

(r) [fm

-3]

r [fm]

16O 40Ca 208Pb

16O

(r)

[fm-3]

r [fm]

M. Alvioli 7

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•We used |Ψ|2 as a Metropolis weight function

Ψ(r1, ..., rA) =A∏i<j

f (rij) Φ(r1, ..., rA)

where Φ is given by the independent particle model.0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

Step Gaussian

f(r)

r [fm]

⇓

0 1 2 3 4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

C(r)

r [fm]

Monte Carlo Step Gaussian Realistic

Variational calculation0 1 2 3 4 5

-0.04-0.020.000.020.040.060.080.10 f(r)

4 - 5 - S 6 - S

16O - AV8'

r [fm]

1 - fc (/10) 2 - 3 - =⇒

⇑

•We use realistic correlation functions from variational calculation


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1.b - Correlations signatures in coordinate space densities

• realistic relative two-body density ρ(r) =∫dR ρ(2)

(r = r1 − r2,R = (r1 + r2)/2

)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

CN 0,

1 ρre

l0,

1 (r

) [fm

-3]

r [fm]

αt

h

α*

0 1 2 3 4r (rm)

0.00

0.02

0.04

0.06

0.08

ρ0 1,0(

r)/R

1,0

(fm

-3)

1S0 VBS

4He

6Li

16O

2H (unpolarized)

0 1 2 3 4 5 6 80.0

0.1

0.2

0.3

0.4

0.5

12C 16O 40Ca

(2) (r)

[fm

-3]

r [fm]

Feldemeier et al, Forest et al, Alvioli et al, Phys. Rev. C72;Phys. Rev. C84, 054003 (2011) Phys. Rev. C54 (1996) 646-667 Phys. Rev. Lett. 100 (2008)

•MC algorithm to include correlations in heavy nuclei

0 1 2 3 4 5 6 7 80.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(2) (r

) [fm

-3]

r [fm]

Mean Field Correlations

16O

(a)

0 1 2 3 4 5 6 7 8 9 100.000.25

0.500.751.00

1.251.50

1.752.00

(b)

r [fm]


40Ca

0 2 4 6 8 10 12 140

2

4

6

8

10

12

14

r [fm]


208Pb

(c)

0

2

4

6

8

10

12

14

16

18

20

0 2 4 6 8 10

238U

r [fm]

UncorrelatedCentral Corr.Tensor Corr.

M. Alvioli, H.-J. Drescher, M. Strikman, Phys. Lett. B680 (2009) 225M. Alvioli 9 ECT⋆, June ’15

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1.b - Probabiltity distribution functions in pA collisions

• probability of interaction with nucleon i: P (b, bi) = 1 − [1 − Γ(b− bi)]2

• black disk approximation replaced bythe Glauber profile Γ(b):

Γ(b) =σtotNN4πB e

−b2/(2B)

• probability of interaction with Nnucleons, vs impact parameter b →

0 2 4 6 8 10 120.0

0.1

0.2

0.3

0.4

√s = 5000 GeV

P12P9

P6

P3

P N(b

)

b [fm]

208Pb

P1

given by:

PN (b) =∑Ni1,...,iN

P (b, bi1) · · · · · P (b, biN )∏A−Nj =i1,...,iN

[1 − P (b, bj)

](M. Alvioli, H.-J. Drescher, M. Strikman Phys. Lett. B680 (2009))


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1.c - fluctuations in pA collisions

• average number of single and double collisions:

⟨N⟩ =∑N N PN (b)

⟨N(N − 1)⟩ =∑N

(N2 − N

)PN (b)

• dispersion: D(b) =⟨N2⟩− [⟨N⟩]2

⟨N⟩ −→0.8

0.9

1.0

0 2 4 6 8 100.7

0.8

0.9

1.0

0.7

0.8

0.9

1.0

(<N

2 > - <

N>2 ) /

<N

> Poissonian Uncorrelated Gaussian

40Ca

b [fm]

208Pb

16O

• Poissonian result is obtained using

PN (b) =(AN

) (σinNNT (b)

)N[1− σinNNT (b)

]A−N(M. Alvioli, H.-J. Drescher, M. Strikman Phys. Lett. B680 (2009))


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1.c - Fluctuations of the geometry of participant matter

• non-spherical distribution of participantnucleons causes pressure gradients

• the system undergoes thermalizationand hydrodynamical evolution, whichcan be performed event-by-event undera few assumptions

• The transformation of spatialanysotropies into momentum spaceis investigated thorugh elliptic flow v2and higher order flows

−→ transverse plane sketch of participantnucleons •x

y


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1.c - Sources of event-by-event fluctuations

-15 -10 -5 0 5 10x (fm)

-10

-5

0

5

10

y (f

m) RP

Φ2

Ψ2

EP

PP

• non-uniform initial densities; modifiedby inclusion of NN correlations

• spectators # and # (ZDCs, centrality)

• reaction (RP), event (EP) plane

• moments of participant matter distri-bution and

ϵn = − ⟨w(r) cosn(ϕ− ψn)⟩⟨w(r)⟩

• dispersion of the moments: ∆ϵn =

√∑(ϵin−⟨ϵi⟩)2N

• NN int. probability: P (bij) = 1−∣∣1− Γ(bij)

∣∣2; Γ(bij) =σtotNN4πB e

−b2ij/2B

• fluctuations of NN cross sections (second part)


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0 100 200 300 4000.3

0.4

0.5

0.6

(bij) Profile

No NN Correlations Central Correlations Full Corr. (2b only) Full Corr. (3b chains)

2 /

2

Npart

0 50 100 150 200 250 300 350 4000.0

0.1

0.2

0.3

0.4

0.5

0.6

*/<*>

NW

d=0 d=0.9 correlations, Ref. [1]


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%M. Alvioli 15 ECT⋆, June ’15

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1.d - Latest updates of nuclear configurations - I

•Nucleus deformation – for 238U we use a modified WS profile:

ρ(r) =ρ0

1 + e(r−R0)/a−→ ρ(r, θ) =

ρ0

1 + e(r−R0−R0β2Y20(θ)−R0β4Y40(θ))/a

Y20(θ) =1

4 r2

√5

π

(2z2 − x2 − y2

)Y40(θ) =

1

16 r4

√9

π

(35z4 − 30z2r2 + 3r4

)(P. Filip, R. Lednicky, H. Masui, N. Xu Phys. Lett. C80 (2009))

• deformation effect on dispersion of moments:

0.5

0.52

0.54

0.56

0.58

0.6

0.62

0 50 100 150 200 250 300 350 400 450 500

Dis

pers

ion

of D

ipol

e A

sym

met

ry: ∆

ε 1

Npart

Effect of correlations deformation on ∆ε1

No Corr., SphericalCentral Corr., Spherical

3b Induced, SphericalNo Corr., Deformed

Central Corr, Deformed3b Induced, Deformed

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0 50 100 150 200 250 300 350 400 450 500

Dis

pers

ion

of E

ccen

tric

ity: ∆

ε 2

Npart





0.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0 50 100 150 200 250 300 350 400 450 500D

ispe

rsio

n of

Tri

angu

lari

ty: ∆

ε 3Npart





18

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1.d - Latest updates of nuclear configurations - II

•Neutron skin – for 208Pb we use different p/n profiles:

ρ(r) =ρ(p,n)0

1 + e(r−Rp,n0 )/ap,n

(ρp0, R

p0, a

p0) = (”82”, 6.680fm, 0.447fm)

(ρn0 , Rn0 , a

n0 ) = (”126”, 6.700fm, 0.550fm)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10

ρp (r)

/ ρn (r

)

r [fm]

PRL 112 (2014) 242502UncorrelatedCentral Corr.

• additional tool for determination of centrality:

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Centrality

0-10

%

10-2

0%

20-3

0%

30-4

0%

40-5

0%

50-6

0%

60-7

0%

70-8

0%

80-9

0%

90-1

00%effe

ctiv

epr

oton

-to-

neut

ron

ratio

p+Pb Pb+Pb

0.8

0.85

0.9

0.95

1.0

1.05

1.1

1.15

1.2

0.8

0.85

0.9

0.95

1.0

1.05

1.1

1.15

1.2

Centrality

0-5%

5-10

%

10-2

0%

20-4

0%

40-8

0%

ATLAS Pb-PbOptical Glauber +Neutron skin +NLO pQCD

(

W+

W−

)

Ck/(

W+

W−

)

0−80%

10-5

10-4

10-3

10-2

10-1

100

0 2 4 6 8 10 12 14

Pp (b)

/ Pn (b

)b [fm]

Un-correlatedCentral Correlations

H. Paukkunen, PLB745 (2015) Alvioli, Strikman19

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2 - Beyond Glauber approach (also Mark’s talk tomorrow)

−→ Glauber model: in rescatter-ing diagrams the proton cannotpropagate in intermediate states

−→ Gribov-Glauber model: theproton can access a set of inter-mediate state as in pN diffraction;relevant at high energies (Einc >>10 GeV)

X is a set of intermediate states that stay frozen during pA interaction


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2.a - NN interaction Fluctuations in high-energy pA scattering

G. Baym, B. Blattel, L. Frankfurt, M. Strikman, Phys.Rev. D47 (1993)

Heiselberg, Baym, Blattel, Frankfurt, Strikman, Phys.Rev.Lett. 70 (1993)


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2.a - Color Fluctuations in high-energy pA scattering

• we implement the Gribov-Glauber dynamics - interaction with n nucle-ons gives different effective cross-section - in Monte Carlo studies with theprobability of the incoming proton to interact with cross section σ with thetarget nucleons, P (σ)

P (σ) = γσ

σ + σ0e−σ/(σ0−1)2

Ω2∫dσ P (σ) = 1 ,

∫dσ σ P (σ) = σtot

1

σ2tot

∫dσ (σ − σtot)

2P (σ) = ωσ

proposed by V. Guzey, M. Strikman, Phys. Lett. B633 (2006)

used in MCG: M. Alvioli, M. Strikman, Phys. Lett. B722 (2013);M. Alvioli, V. Guzey, L. Frankfurt, M. Strikman, PRC90 (2014);

Alvioli, Cole, Frankfurt, Perepelitsa, Strikman, arXiv:1409.7381 [hep-ph]M. Alvioli 22 ECT⋆, June ’15

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2.a - Color Fluctuations: probability of N interactions at b

• fluctuations of the number of wounded nucleons Ncoll for given impactparameter b =⇒ smearing of centrality

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 2 4 6 8 10

PN

(b)

b [fm]

LHC - N=1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2 4 6 8 10

PN

(b)

b [fm]

N = <N> - 0.5 <N> = 3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2 4 6 8 10

PN

(b)

b [fm]

N = <N> = 7

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2 4 6 8 10

PN

(b)

b [fm]

N = <N> + 0.5 <N> = 10

GlauberGribov-Glauber P(σ)

• we find enhancement of the probability of events with large N = NcollM. Alvioli, M. Strikman, Phys. Lett. B722 (2013)


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2.a - Color Fluctuations: probability of N interactions

• fluctuations of the number of wounded nucleons Ncoll for given impactparameter b =⇒ smearing of centrality

• PN =∫dbPN (b); N = Ncoll

10-1210-1010-810-610-410-2100

0 10 20 30 40 50

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 5 10 15 20 25 30

P N -

inte

grat

ed o

ver

b

Ncoll

Glauber, σtot=93.0 mbGlauber + CF, ωσ=0.2Glauber + CF, ωσ=0.1

M. Alvioli, M. Strikman, Phys. Lett. B722 (2013)M. Alvioli, V. Guzey, L. Frankfurt, M. Strikman, Phys. Rev. C90 (2014)


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2.b - Color Fluctuations: Ncoll and b dependence

•We use ATLAS (ATLAS-CONF-2013-096 ) model for ∆ET in pp colli-sions with a convolution to obtain the pA model

=⇒

0

0.05

0.1

0.15

0.2

0 5 10 15 20 25 30

P(ν)

ν

90-100%

60-90%

40-60%

30-40%

20-30%

10-20%

0-10%

P(ν)

10-3

10-2

10-1

0 5 10 15 20 25 30

Alvioli, Cole, Frankfurt, Perepelitsa, Strikman, arXiv:1409.7381 [hep-ph]

• ATLAS and CMS found deviations from the Glauber model (Ncoll tail)

• we derive a non-trivial relation between bins in ∆ET and Ncoll and thusdetermine P (Ncoll) dependence on centrality (ν = Ncoll)


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2.b - Soft processes and hard processes (Mark’s talk)

•We have developed a model to characterize events with one hard scatteringand the remaining soft scatterings, as a function of Ncoll

• The hard event (HT) is triggered in a probabilistic way, using the gluondistributions in the transverse plain Fg(ρ) = exp(−ρ2/B2)/πB2

•We have coupled the MC Glauber-Gribov averaging (< ... >) for the N-1soft interactions with 2-d integral over the position of the hard scattering

ρi

b

θ x

ρ

b i

iθ

M. Alvioli, L. Frankfurt, V. Guzey, M. Strikman, Phys. Rev. C90 (2014)

FNcoll =< σHT

∫db

A∏i=1

dρiFg(ρ)A∑i=1

Fg(ρi) phard(Ncoll) >


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2.c - Perspectives: modeling Double Partonic Interactions

• we can extend the formalism developed for one hard +(A-1) soft interac-tions to additional hard interactions

i

b

θ x

b

θ ρy

ρ

ρρ

i

j

θj

1

2

i

• we can easily describe events where two hard interactions are from onetarget nucleon or two different nucleons, useful to study MPI and partoniccorrelations, i.e. as in

M. Strikman, D. Treleani, Phys. Rev. Lett. 88 (2002)


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my final message:

use our MC-generated configurations available at:http://www.phys.psu.edu/~malvioli/eventgenerator

described in:M. Alvioli, H.-J. Drescher, M. Strikman, Phys. Lett. B 680 (2009)


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Additional Slides


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2.a - Glauber-Gribov: probability of N interactions

in the Gribov-Glauber model we have, for the diffraction cross section:

σhAin =

∫db

[1 − (1 − x)A

]=

A∑Ncoll=1

(−1)Ncoll+1A!

(A − Ncoll)!Ncoll!

∫dbxNcoll ,

where x =1

AσhNin T (b) ,

∫dbT (b)

and we can rewrite

σhAin =

A∑Ncoll=1

σNcoll ,A!


∫dbxNcoll (1 − x)A−Ncoll

hence the probability of collisions with Ncoll nucleons and < Ncoll > are:

PNcoll =σNcollσhAin

, < Ncoll >=

A∑Ncoll

NcollσNcollσhAin

= Aσin

σhAin

L. Bertocchi, D. Treleani, J. Phys. G 3 (1977)


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2.a - Color fluctuations: probability of N interactions

we use the distribution P (σ) to implementi Gribov-Glauber in Monte Carlo:

σhAin =

∫dσinP (σin)

∫db

[1 − (1 − x)A

]where

x =1

AσhNin T (b) ,

∫dbT (b)

and

σNcoll =

∫dσinP (σin)

A!


∫dbxNcoll (1 − x)A−Ncoll

and we have for the probability of collisions with Ncoll nucleons:

PNcoll =σNcollσhAin

G. Baym, B. Blattel, L. Frankfurt, M. Strikman, Phys.Rev. D47 (1993)

Heiselberg, Baym, Blattel, Frankfurt, Strikman, Phys.rev.Lett. 70 (1993)


Documents

Correlations and uctuations in pA and AA collisions M. · PDF fileCorrelations and uctuations in pA and AA collisions ... pair NN correlations inclusion feasible; ﬃ to estimate uctuations