1. AND THE RISE AND FALL OF THE RIDGE
QGP phase
quark and gluon degrees of freedom
hadronization
kinetic
freeze-out
lumpy initial energy density
distributions and correlations of produced particles
DENSITY FLUCTUATIONS AND TWO-PARTICLE CORRELATIONS
⎯⎯⎯⎯⎯
AND THE RISE AND FALL OF THE RIDGE
Paul Sorensen
Brookhaven National LabOratory
P. S., B. Bolliet, A. Mocsy, Y. Pandit, N. Pruthi, arXiv:

2. Initial Gluon Density x=10-5 x=10-3
In CGC for example, an effective field theory for QCD, this is a transverse projection of the gluon density
Gluons are localized around valence charges in the nuclei
x=10-5
x=10-3
H. Kowalski, T. Lappi and R. Venugopalan, Phys.Rev.Lett. 100:022303
The initial state is inhomogenous: a likely source of final-state correlations
S. Voloshin, Phys.Lett.B632: ,2006
A.P. Mishra, R. K. Mohapatra, P. S. Saumia, A. M. Srivastava, Phys. Rev. C77: , 2008
C.Pruneau, S. Gavin, S. Voloshin, Nucl. Phys. A802: ,2008

3. Initial to Final State
distributions and correlations of produced particles
kinetic
freeze-out
hadronization
lumpy initial
density
Do initial spatial correlations manifest as hotspots in final momentum correlations?

4. Analogy with the Early Universe
The Universe
RHIC
QGP phase
quark and gluon degrees of freedom
hadronization
kinetic
freeze-out
lumpy initial energy density
distributions and correlations of produced particles
Credit: NASA
WMAP
HIC

5. Analogy with the Early Universe
The Universe
RHIC
QGP phase
quark and gluon degrees of freedom
hadronization
kinetic
freeze-out
lumpy initial energy density
distributions and correlations of produced particles
Credit: NASA
WMAP
RHIC

6. Analogy with the Early Universe
The Universe
RHIC
QGP phase
quark and gluon degrees of freedom
hadronization
kinetic
freeze-out
lumpy initial energy density
distributions and correlations of produced particles
Credit: NASA
WMAP
WMAP
RHIC

7. Analogy with the Early Universe
The Universe
RHIC
QGP phase
quark and gluon degrees of freedom
hadronization
kinetic
freeze-out
lumpy initial energy density
distributions and correlations of produced particles
Credit: NASA
Δρ/√ρref Δφ
WMAP
RHIC

8. Analogy with the Early Universe
The Universe
RHIC
QGP phase
quark and gluon degrees of freedom
hadronization
kinetic
freeze-out
lumpy initial energy density
distributions and correlations of produced particles
Credit: NASA
Δρ/√ρref Δφ
WMAP
RHIC

9. Relationship to Viscous Effects
How much of the initial inhomogeneity is transferred to the final state?
Spherical harmonic expansion of CMB
sum l
Higher harmonics probe smaller length-scales.
lmfp
lmfp
Fourier expansion
of HIC
n=2
n=3
n=4
n=10
n=15
Efficiency of conversion depends on relation of various length scales like lmfp to the scale probed at n

10. Length Scale: Pixel Size
ℓmfp
Mean free path sets the resolution scale (1/n) or pixel size 10

11. Length Scale: Pixel Size11

12. Length Scale: Pixel Size12

13. Length Scale: Pixel Size13

14. Length Scale: Pixel Size14

15. Length Scale: Pixel Size
Higher harmonics probe finer detail. Is there any detail there? 15

16. Characterizing the Initial Eccentricity
Broniowski, Bozek, & Rybczynski: Phys. Rev. C76: , 2007
PHOBOS: Phys. Rev. C77: ,2008
Alver and Roland: Phys. Rev. C81: , 2010
The eccentricity calculation is approximately a random walk.
Each participant represents a step
The distribution of eccentricity will end up as a 2-D gaussian.
The shift in x is the standard eccentricity
The number of steps determines the width
For odd n, the shift is zero
But participant eccentricity considers the length of the eccentricity vector which is positive definite, even for n=1,3,5,7…

17. Eccentricity and Length Scales
We can introduce a length scale by smearing out the participants
Monte Carlo Glauber
rpart
P. S., B. Bolliet, A. Mocsy, Y. Pandit, N. Pruthi, arXiv:
Smearing out the participants washes out the higher harmonics
Ideal case only retains curvature due to correlations in the Glauber Model

18. Eccentricity and Two-particle Correlations
If vn2 = cn〈ε2n,part〉, a Gaussian will appear in 2-particle correlations vs Δφ
Width in Δφ is inversely related to width in n and the length scale 
P. S., B. Bolliet, A. Mocsy, Y. Pandit, N. Pruthi, arXiv:
Gaussian width depends on the variation of the transfer function cn with n
Shift to center of mass 〈x〉=〈y〉=0 means 〈ε1,part〉≈0. This leads to a negative cos(Δφ) term which is related to the near-side amplitude

19. Observed 2-Particle Correlations
STAR Preliminary
peripheral
central
200 GeV Au+Au Collisions
ρ12 is the density of particle pairs
ρ1ρ2 is the product of single particle densities
Correlations show non-trivial evolution from p+p to most central Au+Au
Are these related to the initial density fluctuations? What about jets, resonance decays, cluster formation, several possible sources…

20. Centrality Dependence: Rise and Fall
STAR Preliminary A1
σηΔ
σφΔ AD
STAR Preliminary
Both collision energies show a rise and fall for A1 and AD

21. The Away-side Amplitude
Shift to center of mass 〈x〉=〈y〉=0 means 〈ε1,part〉≈0.
This leads to a negative cos(Δφ) term which is related to the near-side amplitude: data exhibit just this trend.

22. Estimate of the Amplitude
1) take conversion efficiency from c=(v2/ε2)2
2) take initial eccentricity from Monte-Carlo Glauber
3) convert <cos(3Δφ)> into equivalent Gaussian Amplitude
prediction for “minijet” amplitude from density fluctuations

23. Are 2-particle Correlations Dominated by the Initial State Density Fluctuations?
Model based on ansatz that initial density is converted into 2 particle correlations: conversion efficiency increases with density
initial eccentricities
Amplitude estimate agrees reasonably well with the data.
Exhibits a rise and fall: where does that feature come from?

24. Rise and Fall and the Almond Shape
See D. Teaney, L. Yan, arXiv: v1
Triangular fluctuations increase in non-symmetric overlaps
fluctuations for lower harmonics are highly non-statistical and depend on elliptic geometry
excess from ellipse
eccentricity
scaled eccentricity
statistical
P. S., B. Bolliet, A. Mocsy, Y. Pandit, N. Pruthi, arXiv:
Linking the final-state correlations to initial density fluctuations
When the collision becomes spherical, the coupling subsides
This leads to the rise and fall: a feature unique to this explanation

25. Rise and Fall and the Almond Shape
Triangular fluctuations are driven by edge effects: one nucleon near the edge of nucleus A can impinge on many nucleons in the center of nucleus B. Effect depends on the Woods-Saxon diffuseness parameter.
a = 0.535: standard 197Au Woods-Saxon parameter
a = 0: sharp edge
ρ(r) r

26. Rise and Fall and the Almond Shape
See D. Teaney, L. Yan, arXiv: v1
Lowest order fluctuations couple to higher orders: In this sketch a rightward shift couples with the ellipse to produce a triangular fluctuation
How I learned to stop worrying (about minijets) and love vn
ellipse couples most strongly to nearby harmonics
excess from ellipse
eccentricity
scaled eccentricity
statistical
P. S., B. Bolliet, A. Mocsy, Y. Pandit, N. Pruthi, arXiv:
Linking the final-state correlations to initial density fluctuations
When the collision becomes spherical, the coupling subsides
This leads to the rise and fall: a feature unique to this explanation

27. LHC Predictions
A1 will be several times larger at the LHC: driven by increased multiplicity and flow
ALICE: Phys. Rev. Lett. 105, (2010)
STAR: Phys. Rev. C 72, (2005)
Fit function Drescher, et. al. Phys. Rev. C 76:024905, 2007
P. S., B. Bolliet, A. Mocsy, Y. Pandit, N. Pruthi, arXiv:
Prediction: rise and fall of the ridge will be present at all energies: it’s a feature of the overlap geometry

28. LHC Predictions
A1 will be several times larger at the LHC: driven by increased multiplicity and flow
A. Timmins, QM2011
Prediction confirmed by ALICE at QM 2011

29. All The Terms Rise and Fall Viscous Effects STAR Preliminary A1 σφΔ
σηΔ
σφΔ
Viscous Effects AD
STAR Preliminary

30. All The Terms Rise and Fall CM Shift Viscous Effects
STAR Preliminary
Rise and Fall
CM Shift A1
σηΔ
σφΔ
Viscous Effects AD
STAR Preliminary
What can we learn from the longitudinal dependence

31. Longitudinal Dependence
Wide
50-60%
STAR Preliminary
STAR Preliminary
STAR Preliminary
Wide Gaussian
10-20%
Narrow Gaussian
STAR Preliminary
σ≈2 is indistinguishable from linear within acceptance
Initial state density correlations may drop with Δy: interesting physics σΔy~1/αs?
Fit with a wide and a narrow peak. Wide peak amplitude first drops with 1/N but then deviates from trend near Npart=50. Above that it follows an Npartε23,part trend
Dusling, Gelis, Lappi & Venugopalan, Nucl. Phys. A 836, 159 (2010)
Petersen, Greiner, Bhattacharya & Bass, arXiv:

32. Conclusions
High precision correlations data reveal information similar to CMB analysis:
Data consistent with correlations in the initial overlap geometry converted into momentum space
Fine structure is washed out and large scale structure persists
Non central collisions induce larger fluctuations explaining the centrality dependence of the ridge:
A clear demonstration of the role of density fluctuations in the development of the ridge
Heavy Ion Collisions act as a femto-scope, revealing the structure in the initial conditions!
Longitudinal dependence still needs comprehensive study

33. Density Fluctuations➙2p Correlations
NexSPheRIO: J.Takahashi, B.M.Tavares, W.L.Qian, F.Grassi, Y.Hama, T.Kodama and N.X
The gluon density at small x seen by a di-quark
IPsat GCG
A partial reference list:
S. Voloshin, Phys.Lett.B632: ,2006
A.P. Mishra, R. K. Mohapatra, P. S. Saumia, A. M. Srivastava, Phys. Rev. C77: , 2008
C.Pruneau, S. Gavin, S. Voloshin, Nucl. Phys. A802: ,2008
P. Sorensen, arXiv: WWND Proc.
A. Dumitru, F. Gelis, L. McLerran, R. Venugopalan, T. Lappi, Nucl.Phys.A810:91-108,2008; S. Gavin, L. McLerran, G. Moschelli, Phys.Rev.C79:051902,2009
J.Takahashi, B.M.Tavares, W.L.Qian, F.Grassi, Y.Hama, T.Kodama and N.Xu
P. Sorensen, J. Phys. G37: , 2010
B. Alver, G. Roland Phys. Rev. C81:054905, 2010
B. Alver, C. Gombeaud, M. Luzum. J-Y. Ollitrault, Phys. Rev. C82: , 2010
H. Petersen, G-Y. Qin, S. Bass, B. Muller, Phys. Rev. C 82: , 2010
A. Mocsy, P. Sorensen, arXiv: [hep-ph]
G-Y. Qin, H. Petersen, S. Bass, B. Muller, arXiv: [nucl-ex]
D. Teaney, L. Yan, arXiv: [nucl-th]

34. The Widths
Length scales like rpart, lmfp, cτfs, the acoustic horizon, width of thermal broadening all suppress higher harmonics and broaden the Δφ width
A full dynamic model will be needed to disentagle various effects

35. Relationship of the cos(Δφ) Term to the Gaussian
no C.M. shift
n=1 restored
-cos(Δφ) term should have the same centrality dependence as the near- side peak (as seen in data)
-cos(Δφ) comes from the Gaussian shape of 〈ε2n,part〉 with the n=1 term removed (momentum conservation)

36. Calculating the -cos(Δφ) Term
For a Gaussian, we can calculate the magnitude of the n=1 term that is removed by shifting to the C.O.M.
This will be the magnitude of the –cos term

37. Further Predictions: Ridge at High pT
Any mechanism causing space-momentum correlations (eg. quenching, flow) will give rise to higher vn terms
The jet picks up v2, v3, v4… from quenching (jet tomography)
A low or intermediate pT particle picks up v2, v3, v4 from flow
The ridge structure arises from
where the drop of vn with n is sensitive to the length-scales relevant in the different kinematic ranges (q-hat, viscosity…)
This does not require the jet to be from the same flux tube as the associated particle, only that vnquench and vnflow are each sensitive to the geometry
The width can be quite narrow