Estimation of shear viscosity based on transverse momentumcorrelations

Monika Sharma for the STAR CollaborationDepartment of Physics & Astronomy, Wayne State University, Detroit, MI, USA

Abstract

Event anisotropy measurements at RHIC suggest the strongly interacting matter created in heavyion collisions flows with very little shear viscosity. Precise determination of “shear viscosity-to-entropy” ratio is currently a subject of extensive study [1]. We present preliminary resultsof measurements of the evolution of transverse momentum correlation function with collisioncentrality of Au + Au interactions at

√sNN = 200 GeV. We compare two differential correlation

functions, namely inclusive [2] and a differential version of the correlation measure C̃ introducedby Gavin et.al. [1, 3]. These observables can be used for the experimental study of the shearviscosity per unit entropy.

Key words: azimuthal correlations, QGP, Heavy Ion Collisions 25.75.Gz, 25.75.Ld, 24.60.Ky,24.60.-k

1. Introduction

Measurements of elliptic flow at RHIC (Relativistic Heavy Ion Collider) indicate based oncomparisons with ideal hydrodynamics calculations that the quark gluon plasma produced inheavy ion collisions is a nearly perfect liquid [4]. A measure of fluidity is provided by the ratioof shear viscosity to entropy density (η/s). Calculations based on Super-symmetric gauge theo-ries [5] and uncertainty principle [6] suggest a lower bound, η/s ≥ 1/4π. Elliptic flow has beenthe basic experimental probe for the estimation of η/s. Based on recent measurements of ellipticflow and comparison with hydro models, the estimated range is 1 < 4πη/s < 5. This suggeststhat the matter produced in Au + Au collisions is indeed a low viscosity medium [7].

In this paper, we present preliminary results of an alternative technique to determine themedium viscosity. The technique, proposed by Gavin et. al. [1], relies on measurements of thecollision centrality evolution of transverse momentum two-particle correlation functions. Thisη/s is estimated based on the longitudinal broadening of the correlations with increasing colli-sion centrality. The broadening arises from longitudinal diffusion of momentum currents. It isquantitatively determined by the magnitude of the kinematic viscosity, ν = η

T s (where “T” standsfor temperature), and the lifetime of the colliding system. We use differential extensions of theintegral correlation observable C̃ proposed by Gavin et. al [1].

Nuclear Physics A 830 (2009) 813c–816c

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www.elsevier.com/locate/nuclphysa

doi:10.1016/j.nuclphysa.2009.10.074

We present measurements of C̃ and inclusive (ρΔp1Δp2

2 ) as a function of the relative pseudora-pidity and azimuthal angles of the measured particles. The observable C̃ is defined as

C̃ =

⟨nα(η1,ϕ1)∑

i=1

nα(η2,ϕ2)∑i� j=1

pα,i (η1, ϕ1) pα, j (η2, ϕ2)

⟩

〈nα (η1, ϕ1) nα (η2, ϕ2)〉 −

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⟨nα(η1,ϕ1)∑

i=1pα,i (η1, ϕ1)

⟩

〈nα (η1, ϕ1)〉

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⟨nα(η2,ϕ2)∑

j=1pα, j (η2, ϕ2)

⟩

〈nα (η2, ϕ2)〉

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(1)

and inclusive is defined as

ρΔp1Δp2

2 (Δη,Δϕ) =

⟨nα(η1,ϕ1)∑

i=1

nα(η2,ϕ2)∑j�i=1

(pα,i (η1, ϕ1) − 〈p (η1, ϕ1)〉) (pα, j (η2, ϕ2) − 〈p (η2, ϕ2)〉

)⟩

〈nα (η1, ϕ1) nα (η2, ϕ2)〉 (2)

nα(ηi, φi) represents the number of particles detected in an event α at pseudorapidity ηi andazimuthal angle φi. pT,α,i(ηi, φi) stands for the transverse momentum of the ith particle in an eventα. 〈pT (η1, ϕ1)〉 is the average of the particle transverse momentum at ηi and φi over the wholeevent ensemble.

2. Analysis

This analysis is based on data recorded using the solenoidal tracker at RHIC (STAR) detectorduring the 2004 data RHIC run at Brookhaven National Laboratory. Au + Au collisions at

√sNN

= 200 GeV were acquired with minimum bias triggers [8]. This analysis is restricted to chargedparticle tracks from the STAR-Time Projection Chamber (TPC) in the momentum range 0.2< pT < 2.0 GeV/c within the pseudorapidity acceptance of |η| <1.0. A nominal cut of distanceof closest approach (DCA < 3.0 cm) was applied in order to limit the selected tracks to primarycharged particle tracks only. An event was accepted for analysis if its collision vertex lay within|z| <25 cm, where z stands for the maximum distance along the beam axis from the center ofthe TPC. The results reported here are based on 10 million minbias events. We define centralitybased on primary tracks within |η| <1.0. Centrality bins are calculated as a fraction of the totalmultiplicity distribution.

3. Results

Figures 1(a, c) and (b, d) show a comparison of inclusive and C̃ correlations functions for70-80% & 0-5% centrality, respectively, in Au + Au collisions at

√sNN = 200 GeV. The correla-

tion function is plotted as a function of the particles’ relative pseudorapidity, Δη, and azimuthalangles, Δφ, using 31 and 36 bins, respectively. The two observables exhibit a ridge-like structurein the most central collisions (0-5%) which is narrow in azimuth (near Δη = 0) and extendedover particles’ relative pseudorapidity, Δη. In peripheral collisions both C̃ and inclusive fea-ture a near-side peak centered at Δϕ=0, and a broad away-side (Δφ ∼ π) ridge. The near sidepeak broadens progressively with centrality reaching a maximum in the most central collisionswhile the away-side amplitude progressively decreases from peripheral to central collisions. Theinclusive exhibits a single near-side peak structure whereas C̃ features a dip near Δφ ≈ Δη ≈ 0.The cause of this dip is under investigation. We assume in this analysis that the broadening of thecorrelation function C̃ in Δη is solely due to viscous diffusion effects and proceed to determine

M. Sharma / Nuclear Physics A 830 (2009) 813c–816c814c

φΔ-1

0 1 23 4

5ηΔ

-2.0-1.5

-1.0-0.5

0.00.5

1.01.5

2.0

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

(a)STAR Preliminary

φΔ-1

0 1 23 4

5ηΔ

-2.0-1.5

-1.0-0.5

0.00.5

1.01.5

2.0-0.050.000.050.100.150.200.250.300.35

-310×

(b)

φΔ-1 0 1 2 3 4 5

ηΔ

-2.0-1.5

-1.0-0.5

0.00.5

1.01.5

2.0-0.004-0.0020.0000.0020.0040.0060.0080.0100.012

(c)STAR Preliminary

φΔ-1 0 1 2 3 4 5

ηΔ

-2.0-1.5

-1.0-0.5

0.00.5

1.01.5

2.0

-0.003

-0.002

-0.001

0.000

0.001

0.002

(d)

Figure 1: (Color online) (a, b) Correlation function, inclusive (ρΔpT,1ΔpT,22 (Δη,Δφ)) shown for 70-80 & 0-5% centrality,

(c, d) C̃ shown for 70-80 & 0-5% centrality in Au+ Au collisions at√

s=200 GeV. These observables are plotted in unitsof (GeV/c)2 , and the relative azimuthal angle Δφ in radians.

the evolution of the Δη width with collision centrality. This is accomplished by fitting the Δηprojections of C̃ in the range |Δφ| <1.0 radians. Figures 2(a, c) and (b, d) show Δη projectionsfor |Δφ| <1.0 radians for inclusive and C̃ correlations functions for peripheral (70-80%) and cen-tral (0-5%) collisions, respectively. We parameterize the projections with a 5-component model.A wide Gaussian approximates the overall shape of the correlation function.

C̃ (b, aw, σw, an, σn) = b + aw exp(−Δη2/2σ2

w

)+ an exp

(−Δη2/2σ2

n

)(3)

where an and aw are the amplitude of the narrow and wide Gaussians, respectively. Similaryσn and σw are the widths of the narrow and wide Gaussians. “b” stands for baseline in Eq. 3.Widths obtained for peripheral (σw,70−80%) and central (σw,0−5%) collisions for C̃ are 0.53±0.01,1.3±0.4, respectively. Assuming the shear viscosity dominates the broadening of the correlationfunction for increasing system life times, the following expression provides an estimate of theviscosity:

σ2c − σ2

p = 4υ(τ−1

f ,p − τ−1f ,c

)(4)

where τ−1f ,p and τ−1

f ,c stand for the freeze-out time estimates in peripheral and central collisions. σc

and σp represent the width of the correlation functions in the central and peripheral collisions.

4. Summary

We presented measurements of differential transverse momentum correlation functions inorder to estimate the value of η/s based on the model by Gavin et.al. [1]. The determination

M. Sharma / Nuclear Physics A 830 (2009) 813c–816c 815c

Entries 16304

/ ndf 2χ 3.641 / 26

constW 0.0000350± 0.0007655

constN 0.0000452± 0.0002537

sigmaW 0.017± 0.361

sigmaN 0.01375± 0.07695

offset 0.0000125± 0.0002815

ηΔ-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

2 p

Δ1

pΔ 2ρ

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

Entries 16304

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constW 0.0000350± 0.0007655

constN 0.0000452± 0.0002537

sigmaW 0.017± 0.361

sigmaN 0.01375± 0.07695

offset 0.0000125± 0.0002815

STAR Preliminary

(a)

Entries 25856

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constW 2.231e-06± 3.125e-05

constN 3.32e-06± 4.15e-05

sigmaW 0.05± 0.52

sigmaN 0.00700± 0.06533

offset 1.754e-06± 4.713e-05

ηΔ-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

2 p

Δ1

pΔ 2ρ

0.02

0.04

0.06

0.08

0.10

0.12

-310×Entries 25856

/ ndf 2χ 160.7 / 26

constW 2.231e-06± 3.125e-05

constN 3.32e-06± 4.15e-05

sigmaW 0.05± 0.52

sigmaN 0.00700± 0.06533

offset 1.754e-06± 4.713e-05

(b)

Entries 111396

/ ndf 2χ 75.7 / 26

constW 0.000059± 0.006862

constN 0.000101± -0.002712

sigmaW 0.0064± 0.5304

sigmaN 0.00279± 0.07353

offset 4.635e-05± -5.423e-05

ηΔ-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

C~

0

0.001

0.002

0.003

0.004

0.005

0.006

Entries 111396

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constW 0.000059± 0.006862

constN 0.000101± -0.002712

sigmaW 0.0064± 0.5304

sigmaN 0.00279± 0.07353

offset 4.635e-05± -5.423e-05

(c)

STAR Preliminary

Entries 6662

/ ndf 2χ 21.12 / 26

constW 0.0002440± 0.0009388

constN 0.0000633± -0.0006818

sigmaW 0.361± 1.283

sigmaN 0.0251± 0.1856

offset 0.0002876± -0.0003287

ηΔ-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

C~

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-310×Entries 6662

/ ndf 2χ 21.12 / 26

constW 0.0002440± 0.0009388

constN 0.0000633± -0.0006818

sigmaW 0.361± 1.283

sigmaN 0.0251± 0.1856

offset 0.0002876± -0.0003287

(d)

Figure 2: (Color online) Projection of |Δφ| <1.0 radians on the Δη axis (a, b) inclusive, ρΔp1Δp22 (Δη,Δφ) correlations

function for 70-80% & 0-5% centralities, respectively, (c, d) C̃ shown for same centralities in Au + Au collisions at√s=200 GeV. These observables are plotted in units of (GeV/c)2 .

of η/s will be sensitive to the freeze-out time estimate of the peripheral collisions posited byGavin et.al. [1]. However, STAR measurements [9] indicate a larger freeze-out time in peripheralcollisions than that used in Ref. [1].

References

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el al. ibid. 184; B. B Back et al. (PHOBOS Collaboration), ibid. 28; J. Adams et al. (STAR Collaboration), ibid.102.

[5] P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. 87 (2001) 081601.[6] P. Danielewicz and M. Gyulassy, Phys. Rev. D 31 (1985) 53.[7] D. Teaney, Phys. Rev. C 68, (2003) 034913; P. Romatschke and U. Romatschke, Phys. Rev. Lett. 99 (2007) 172301;

H. Song and U. W. Heinz, Phys. Rev. C 77 (2008) 064901; H. J. Drescher, A. Dumitru, C. Gombeaud and J. Y.Ollitrault, Phys. Rev. C 76 (2007) 024905.

[8] M. Anderson et. al. Nucl. Instrum. Meth. A 499 (2003) 624;[9] J. Adams et. al. (STAR Collaboration) Phys. Rev. C 71 (2005) 044906.

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