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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
Tr⇥W (x)W †(x̄)
⇤�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
Tr⇥W (x)W †(x̄)
⇤ 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
Tr⇥W (x)W †(x̄)
⇤· · ·
�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
< 185trkoffline N≤150
(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
< 220trkoffline N≤185
(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
< 260trkoffline N≤220
(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