collectivity from the initial statea. dumitru and j. jalilian-marian, phys. rev. lett., 89:022301,...

Collectivityfromtheinitialstate:Four-particlecorrelationsinproton-nucleuscollisions

AworkinprogresswithMarkMace(StonyBrook/Brookhaven)RajuVenugopalan(Brookhaven)

Contents

•  Background/Motivation

•  Particleproductioninproton-nucleuscollisions

•  Beyondthedipole:n-pointcorrelatorsintheMVmodel

• Results

BACKGROUND/MOTIVATION

Background1:CollectivityinsmallsystemsV.Khachatryanetal.(CMSCollaboration)JHEP,1009:091,2010.

V.Khachatryanetal.(CMSCollaboration)Phys.Rev.Lett.,116(17):172302,2016.

K.Dusling,W.Li,andB.SchenkeNovelcollectivephenomenainhigh-energyproton–protonandproton–nucleuscollisionsInt.J.Mod.Phys.,E25(01):1630002,2016.[arXiv:1509.07939]

BackgroundII:Glasma-graphs

η∆-4

-20

24

φ∆0

24

φ∆

dη∆d

pair

N2 dtrgN1

1.301.351.40

CMS Preliminary 110≥ = 7 TeV, N spp

MotivationI:Four-particlecumulants

ATLASCollaborationPhys.Lett.B725(2013)60-78.

CMSCollaborationPhys.Lett.,B724(2013)213–240.

offlinetrkN

50 100 150 200

{4}

2c

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

-310×

bin width of 2offlinetrk N bin width of 5offlinetrk N bin width of 30offlinetrk N

= 5.02 TeVNNs(a) Data, pPb

gen-levelchN

50 100 150 200

{4}

2c

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

-310×

bin width of 2gen-levelch N bin width of 5gen-levelch N bin width of 30gen-levelch N

= 5.02 TeVNNs(b) HIJING Gen-level, pPb

MotivationII:Thecolordomainmodel

A.KovnerandM.Lublinsky,Phys.Rev.,D84:094011,2011.A.DumitruandA.V.Giannini,Nucl.Phys.,A933:212–228,2015.A.Dumitru,L.McLerran,andV.Skokov,Phys.Lett.,B743:134–137,2015.A.DumitruandV.Skokov,Phys.Rev.,D91(7):074006,2015.T.Lappi,B.Schenke,S.Schlichting,andR.Venugopalan.JHEP,01:061,2016.

g2

Nc

⌦Eai (b1)E

bj (b2)

↵=

1

(N2c � 1)�ab�ijQ

2s�(b1 � b2)

g2

Nc

⌦Eai (b1)E

bj (b2)

↵=

1

(N2c � 1)�abQ2s�(b1 � b2)

✓�ij + 2A

âiâj �

1

2�ij

�◆

•  GaussiancorrelationofE-dieldswillproduceparticlesisotropically

•  Considersub-classofeventswithaprededinedorientation:

PARTICLEPRODUCTIONINPROTON-NUCLEUSCOLLISIONS

Thehybridframework•  Considereikonalscatteringofquarkfromadensetarget

•  Differentialcross-sectionforquarkscatteringonanucleus

A.DumitruandJ.Jalilian-Marian,Phys.Rev.Lett.,89:022301,2002.J.D.Bjorken,J.B.Kogut,andD.E.Soper,Phys.Rev.,D3:1382,1971.

dN

d

2p

' 1⇡B

p

Z

xx̄

e

�(x2+x̄2)/2Bp⌧

1

N

c

Tr⇥W (x)W †(x̄)

⇤�e

ip·(x�x̄)

p+ � p�

p? ⇠ ⇤QCD p? ⇠ Qs

A�

W [A](x) = P expig

Zdz

+A

�a (z

+, x)

�

•  DipolecorrelatorisevaluatedintheMVmodelwherebyaveragingoverthecolordieldofthenucleus

•  resultsin

Multiple-scatteringintheCGC

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

x?[GeV]

Q2s = 1.0 GeV2

r =p

2/Q2s

r = 1/⇤QCD

g

2⌦A

�a

(x)A�b

(y)↵= �abL

xy

L

xy

= �g4µ

2

16⇡|x� y|2 ln 1

⇤ |x� y|

D(x, x̄) =

1

N

c

Tr

⇥W (x)W

†(x̄)

⇤= exp(C

F

L

xx̄

)

N(x?)

•  Singlequarkscattering:

•  Twoquarkscattering:

•  Fourquarkscattering:

Multi-particleproduction

dN

d

2p

' 1⇡B

p

Z

xx̄

e

�(x2+x̄2)/2Bp⌧

1

N

c


⇤�e

ip·(x�x̄)

d

2N

d

2p1d

2p2

' 1(⇡B

p

)2

Z

xx̄yȳ

e

�(x2+x̄2)/2Bpe

�(y2+ȳ2)/2Bpe

ip1·(x�x̄)e

ip2·(y�ȳ)

⇥⌧

1

N

c


⇤ 1N

c

Tr⇥W (y)W †(ȳ)

⇤�

d

4N ⇠

Z ⌦Tr

⇥W (w)W †(w̄)

⇤Tr

⇥W (x)W †(x̄)

⇤Tr

⇥W (y)W †(ȳ)

⇤Tr

⇥W (z)W †(z̄)

⇤↵

BEYONDTHEDIPOLEn-pointcorrelatorsintheMVmodel

Multiple-scatteringintheCGC•  Startwiththedipolescatteringmatrix:

•  Expandoutthelastsliceinrapidity:

•  Takealltwo-pointfunctions:

•  Andre-exponentiate:

W (x) ⌘ P expig

Zdz

+A

�a (z

+, x)

�' V (x)

⇥1 + igA

�a (⇠, x)T

a+ · · ·

⇤

hD(x, x̄)iW

= exp (C

F

L

xx̄

)

hD(x, x̄)iW =1

Nc

⌦Tr

⇥W (x)W †(x̄)

⇤↵

�abLxx̄

�

ab

2Tr

⇥V (x)V †(x̄)

⇤

hD(x, x̄)iW ' hD(x, x̄)iV + g2⌦A

�a (x)A

�b (x̄)

↵ 1Nc

Tr⇥V (x)T aT bV †(x̄)

⇤

•  AsbeforeexpandoutallWilsonlines:

•  Fourpointfunction:

•  Eightpointfunction:

KovnerandU.A.Wiedemann.Phys.Rev.,D64:114002,2001.H.Fujii..Nucl.Phys.,A709:236–250,2002.J.P.Blaizot,F.Gelis,andR.Venugopalan.Nucl.Phys.,A743:57–91,2004.F.Dominguez,C.Marquet,andB.Wu,Nucl.Phys.,A823:99–119,2009.

W (x) ⌘ P expig

Zdz

+A

�a (z

+, x)

�' V (x)

⇥1 + igA

�a (⇠, x)T

a+ · · ·

⇤

hDxx̄

Dyȳ

iW

' ↵xx̄yȳ

hDxx̄

Dyȳ

iV

+ �xyx̄ȳ

hQxȳyx̄

iV

✓hD

xx̄

Dyȳ

ihQ

xȳyx̄

i

◆

W

=

✓↵xx̄yȳ

�xyx̄ȳ

�xyȳx̄

↵xȳyx̄

◆✓hD

xx̄

Dyȳ

ihQ

xȳyx̄

i

◆

V

hDww̄

Dxx̄

Dyȳ

Dzz̄

iW

' ↵ww̄xx̄yȳzz̄

hDww̄

Dxx̄

Dyȳ

Dzz̄

iV

+�wxw̄x̄

hQxw̄wx̄

Dyȳ

Dzz̄

iV

+ �wyw̄ȳ

hQyw̄wȳ

Dxx̄

Dzz̄

iV

+�wzw̄z̄

hQzw̄wz̄

Dxx̄

Dyȳ

iV

+ �xyx̄ȳ

hQyx̄xȳ

Dww̄

Dzz̄

iV

+�xzx̄z̄

hQzx̄xz̄

Dww̄

Dyȳ

iV

+ �yzȳz̄

hQzȳyz̄

Dww̄

Dxx̄

iV

CorrelatorofeightWilsonlines

1 5 10 15 20 24

1

5

10

15

20

24

1 5 10 15 20 24

1

5

10

15

20

24

2

66666666666666666666666666666666666666664

hDww̄

Dxx̄

Dyȳ

Dzz̄

ihQ

ww̄xx̄

Dyz̄

Dzȳ

ihQ

ww̄xȳ

Dyx̄

Dzz̄

ihQ

ww̄xȳ

Dyz̄

Dzx̄

ihQ

ww̄xz̄

Dyx̄

Dzȳ

ihQ

ww̄xz̄

Dyȳ

Dzx̄

ihQ

wx̄xw̄

Dyȳ

Dzz̄

ihQ

wx̄xw̄

Qyz̄zȳ

ihQ

wx̄xȳ

Qyw̄zz̄

ihQ

wx̄xȳ

Qyz̄zw̄

ihS

wx̄xz̄yw̄

Dzȳ

ihS

wx̄xz̄yȳ

Dzw̄

ihS

wȳxw̄yx̄

Dzz̄

ihS

wȳxw̄yz̄

Dzx̄

ihS

wȳxx̄yw̄

Dzz̄

ihS

wȳxx̄yz̄

Dzw̄

ihS

wȳxz̄yw̄

Dzx̄

ihS

wȳxz̄yx̄

Dzw̄

ihO

wz̄xw̄yx̄zȳ

ihO

wz̄xw̄yȳzx̄

ihO

wz̄xx̄yw̄zȳ

ihO

wz̄xx̄yȳzw̄

ihO

wz̄xȳyw̄zx̄

ihO

wz̄xȳyx̄zw̄

i

3

77777777777777777777777777777777777777775

(47)

[A] ⌘⇥↵ww̄xx̄yȳzz̄

⇤(48)

[B] ⌘⇥�wxw̄x̄

�wyw̄ȳ

�wzw̄z̄

�xyx̄ȳ

�xzx̄z̄

�yzȳz̄

⇤(49)

[C] ⌘

2

666664

�wxx̄w̄

�wyȳw̄

�wzz̄w̄

�xyȳx̄

�xzz̄x̄

�yzz̄ȳ

3

777775(50)

[D] ⌘

2

666664

↵wx̄xw̄yȳzz̄

0 0 0 0 0

0 ↵wȳyw̄xx̄zz̄

0 0 0 0

0 0 ↵wz̄zw̄xx̄yȳ

0 0 0

0 0 0 ↵xȳyx̄ww̄zz̄

0 0

0 0 0 0 ↵xz̄zx̄ww̄yȳ

0

0 0 0 0 0 ↵yz̄zȳww̄xx̄

3

777775(51)

7

2

66666666666666666666666666666666666666664

hDww̄

Dxx̄

Dyȳ

Dzz̄

ihQ

ww̄xx̄

Dyz̄

Dzȳ

ihQ

ww̄xȳ

Dyx̄

Dzz̄

ihQ

ww̄xȳ

Dyz̄

Dzx̄

ihQ

ww̄xz̄

Dyx̄

Dzȳ

ihQ

ww̄xz̄

Dyȳ

Dzx̄

ihQ

wx̄xw̄

Dyȳ

Dzz̄

ihQ

wx̄xw̄

Qyz̄zȳ

ihQ

wx̄xȳ

Qyw̄zz̄

ihQ

wx̄xȳ

Qyz̄zw̄

ihS

wx̄xz̄yw̄

Dzȳ

ihS

wx̄xz̄yȳ

Dzw̄

ihS

wȳxw̄yx̄

Dzz̄

ihS

wȳxw̄yz̄

Dzx̄

ihS

wȳxx̄yw̄

Dzz̄

ihS

wȳxx̄yz̄

Dzw̄

ihS

wȳxz̄yw̄

Dzx̄

ihS

wȳxz̄yx̄

Dzw̄

ihO

wz̄xw̄yx̄zȳ

ihO

wz̄xw̄yȳzx̄

ihO

wz̄xx̄yw̄zȳ

ihO

wz̄xx̄yȳzw̄

ihO

wz̄xȳyw̄zx̄

ihO

wz̄xȳyx̄zw̄

i

3

77777777777777777777777777777777777777775

(47)

[A] ⌘⇥↵ww̄xx̄yȳzz̄

⇤(48)

[B] ⌘⇥�wxw̄x̄

�wyw̄ȳ

�wzw̄z̄

�xyx̄ȳ

�xzx̄z̄

�yzȳz̄

⇤(49)

[C] ⌘

2

666664

�wxx̄w̄

�wyȳw̄

�wzz̄w̄

�xyȳx̄

�xzz̄x̄

�yzz̄ȳ

3

777775(50)

[D] ⌘

2

666664

↵wx̄xw̄yȳzz̄

0 0 0 0 0

0 ↵wȳyw̄xx̄zz̄

0 0 0 0

0 0 ↵wz̄zw̄xx̄yȳ

0 0 0

0 0 0 ↵xȳyx̄ww̄zz̄

0 0

0 0 0 0 ↵xz̄zx̄ww̄yȳ

0

0 0 0 0 0 ↵yz̄zȳww̄xx̄

3

777775(51)

7

=

VW

RESULTS

Quantitiesofinterest

•  Particlespectra:•  Two-particlecumulants:•  Four-particlecumulants:

n{2} ⌘Z

d2p1d2p2 cos [n (�p1 � �p2)]

d2N

d2p1d2p2

cn{4} =n{4}0{4}

� 2✓n{2}0{0}

◆2, vn{4} = (�cn{4})1/4

cn{2} =n{2}0{2}

, vn{2} =p

cn{2}

n{4} ⌘Z

d2p1d2p2d

2p3d2p4 cos [n (�p1 + �p2 � �p3 � �p4)]

d4N

d2p1d2p2d2p3d2p4

Q2s ⇠ 2 GeV2Bp = 4 GeV�2

d

n

N

d

2p · · · ⇠

Ze

�(x2+x̄2)/2Bp···⌧

1

N

c


⇤· · ·

�e

ip·(x�x̄)···

Integratedellipticdlow

T.Lappi,Phys.Lett.,B744:315–319,2015.T.Lappi,B.Schenke,S.Schlichting,andR.Venugopalan.JHEP,01:061,2016.

Integratedellipticdlow

v2{2}

v2{4}

offlinetrkN

0 100 200 300

2v0.00

0.05

0.10 |>2}η∆{2, |2v2}η∆{2, |2v {4}2v

= 5.02 TeVNNsCMS pPb

< 5 GeV/cT

ATLAS, 0.3 < p

< 3 GeV/cT

0.3 < p

, 50-100% sub.|>2}η∆{2, |2v{4}2v

Differentialdlow

(GeV/c)T

p2 4

2v

0.0

0.1

0.2

0.3 = 2.76 TeVNNsCMS PbPb

< 150trkoffline N≤120

|>2}η∆{2, |2v2}η∆{2, |2v

{4}2v

(GeV/c)T

p2 4

2v0.0

0.1

0.2

0.3 = 5.02 TeVNNsCMS pPb >80 GeVPbT EΣATLAS,

|>2}η∆{2, |2v{4}2v

|>0.8}η∆{2, |2vALICE, 0-20%

(GeV/c)T

p2 4

2v

0.0

0.1

0.2

0.3


(GeV/c)T

p2 4

2v

0.0

0.1

0.2

0.3 (GeV/c)T

p2 4

2v

0.0

0.1

0.2

0.3


(GeV/c)T

p2 4

2v

0.0

0.1

0.2

0.3 (GeV/c)T

p2 4

2v

0.0

0.1

0.2

0.3


(GeV/c)T

p2 4

2v

0.0

0.1

0.2

0.3

18≥ part = 4.4 TeV, NNNspPb Hydro v2{2}

v2{4}

Symmetriccumulants

A random event having v2 > hv2i will be more likely to have a v4 > hv4i.

SC(m,n) =⌦v2mv

2n

↵�⌦v2m

↵ ⌦v2n

↵

Furtherthoughts…

Conclusions

• DevelopedthemachinerytocomputecorrelatorsofsixandeightfundamentalWilsonlinesatdiniteNc• Modulomanycaveatsqualitativefeaturesofthemeasuredfour-particlecorrelationsareunderstoodbynon-linearclassicaldields

BACKUP

Symmetriccumulants

BackgroundII:Collectivity

A.Adareetal.(PHENIXCollaboration)Phys.Rev.Lett.115,142301(2015)

Schenke,VenugopalanNucl.Phys.A931(2014)1039-1044

J.L.Nagle,etal.Phys.Rev.Lett.113,112301

Bożek,BroniowskiPhysicsLettersB747(2015)135–138

collectivity from the initial statea. dumitru and j. jalilian-marian, phys. rev. lett., 89:022301,...

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