Flow and Energy Momentum Tensor From

Classical Gluon Fields

Rainer J. FriesTexas A&M University

NFQCD Kyoto, December 5, 2013

2NFQCD 2013

Classical Gluon Fields and MV Model

Analytic Solutions for Early Times

Phenomenology

Beyond Boost-Invariance

Work in collaboration mainly with Guangyao Chen (Texas A&M) Sener Ozonder (INT/Seattle, Minnesota)

Rainer Fries

Overview

3NFQCD 2013

Bulk evolution Local thermal equilibrium with small (?) dissipative corrections after time

0.2-1 fm/c. Expansion and cooling via viscous hydrodynamics. Hadronic phase: Hydro or Transport.

Pre-equilibrium: color glass condensate (CGC); successful predictions of eccentricities and fluctuations of the energy density flow observables

IP-Glasma Poor constraints on initial flow and other variables. p+A?

Rainer Fries

The “Standard Model” of URHICs

[B. Schenke et al., PRL108 (2012); C. Gale et al. , PRL 110 (2013)]

4NFQCD 2013

Nuclei/hadrons at asymptotically high energy: Saturated gluon density ~ Qs

-2 scale Qs >> QCD, classical fields.

Single nucleus: solve Yang-Mills equations for gluon field A().

Source = light cone current J (given by SU(3) charge distribution ). Calculate observables O( ) from the gluon field A(). from random Gaussian color fluctuations of a color-neutral nucleus.

Rainer Fries

MV Model: Classical YM Dynamics

JFD ,

[L. McLerran, R. Venugopalan]

5NFQCD 2013

Two nuclei: intersecting currents J1, J2 (given by 1, 2), calculate gluon field A( 1, 2) from YM.

Equations of motion

Boundary conditions

Rainer Fries

MV Model: Classical YM Dynamics

iA1iA2

[A. Kovner, L. McLerran, H. Weigert]

0,,,1

0,,1

0,,1

2

33

jijii

ii

ii

FDADAigA

AAigAD

ADDA

xAA

xAxAii ,

,

xAxAigxA

xAxAxA

ii

iii

21

21

,2

,0

,0

6NFQCD 2013

Numerical Solutions of Classical YM in the forward light cone.

Quantum corrections, Instabilities, Thermalization, …

Here: Analytic solution of classical theory. Analyze pressure and flow for small times.

Rainer Fries

Glasma in the Forward Light Cone

[Krasnitz, Venugopalan][T. Lappi]…

7NFQCD 2013

Before the collision: color glass = pulse of strictly transverse (color) electric and magnetic fields, mutually orthogonal, with random color orientations, in each nucleus.

Rainer Fries

Fields: Before Collision

ii AxF 11

ii AxF 22

8NFQCD 2013

Before the collision: color glass = pulse of strictly transverse (color) electric and magnetic fields, mutually orthogonal, with random color orientations, in each nucleus.

Immediately after overlap (forward light cone, 0): strong longitudinal electric & magnetic fields. Non-abelian effect.

Rainer Fries

Fields: At Collision

0B0E

E0

B0

jiij

ii

AAigF

AAigF

21210

210

,

,

[L. McLerran, T. Lappi, 2006][RJF, J.I. Kapusta, Y. Li, 2006]

9NFQCD 2013Rainer Fries

Fields: Into the Forward Light Cone

00

00

,sinh, cosh2

, cosh,sinh2

BDEDB

BDEDE

ijiji

jijii

[Guangyao Chen, RJF]

Once the non-abelian longitudinal fields E0, B0 are seeded, the first step of further evolution can be understood in terms of the QCD versions of Ampere’s, Faraday’s and Gauss’ Law. Longitudinal fields E0, B0 decrease in both z and t away from

the light cone First abelian theory: Gauss’ Law at fixed time t

Long. flux imbalance compensated by transverse flux Gauss: rapidity-odd radial field

Ampere/Faraday as function of t: Decreasing long. flux induces transverse field Ampere/Faraday: rapidity-even curling field

Full classical QCD:[G. Chen, RJF, PLB 723 (2013)]

10NFQCD 2013

Transverse fields for randomly seeded A1, A2 fields (abelian case).

= 0: Closed field lines around longitudinal flux maxima/minima

0: Sources/sinks for transverse fields appear

Rainer Fries

Initial Transverse Field: Visualization) d(backgroun 0BE i ) d(backgroun 0EB i

0

1

11NFQCD 2013Rainer Fries

Analytic Solution: Small Time Expansion

[RJF, J. Kapusta, Y. Li, 2006][Fujii, Fukushima, Hidaka, 2009]

Here: analytic solution using small-time expansion for gauge field

Recursive solution for gluon field:

0th order = boundary conditions

xAxA

xAxA

in

n

ni

nn

n

0

0

,

,

422

2

,,,1

,,2

1

nmlkm

ilk

nlk

jil

jk

in

nmlkm

il

ikn

ADAigFDn

A

ADDnn

A

xAxAigxA

xAxAxA

ii

iii

210

210

,2

12NFQCD 2013Rainer Fries

Analytic Solution: Small Time Expansion Convergence for weak field limit: recover analytic solution

for all times.

Convergence for strong fields: convergence radius ~ 1/Qs for averaged quantities like energy density.

kJAA

kJk

AA

ii00

LO

10LO

,

2,

kk

kk

ini

n

nn

Akn

A

Aknn

A

02

2

02

2

!!1

2!!2

13NFQCD 2013

Initial ( = 0) structure of the energy-momentum tensor from purely londitudinal fields

Rainer Fries

Energy Momentum Tensor

0

0

0

0

)0(f

T

Transverse pressure PT = 0

Longitudinal pressure PL = –0

ε 0=12 (𝐸0

2+𝐵02 )

14NFQCD 2013Rainer Fries

Energy Momentum Tensor

Flow emerges from pressure at order 1:

Transverse Poynting vector gives transverse flow.

20

22112

2220

222

11220

11

2221120

f

coshsinhcoshsinhcoshsinhsinhcoshcoshsinhsinhcosh

sinhcoshsinhcosh

OOOO

OOOO

T

0000

0

,,2

2

BEDEBD jjiji

ii

BES

Like hydrodynamic flow, determined by gradient of transverse pressure PT = 0; even in rapidity.

Non-hydro like; odd in rapidity ??

[RJF, J.I. Kapusta, Y. Li, (2006)][G. Chen, RJF, PLB 723 (2013)]

sinh,, sinh2

cosh,,cosh2

0000odd

0000even

ijjiji

iiii

EDBBDES

BDBEDES

15NFQCD 2013Rainer Fries

Energy Momentum Tensor

Corrections to energy density and pressure due to flow at second order in time

Example: energy density

42

000 2cosh2 2sinh2

8 OT iiii

Depletion/increase of energy density due to transverse flow

Due to longitudinal flow

16NFQCD 2013

Transverse Poynting vector for randomly seeded A1, A2 fields (abelian case).

= 0: “Hydro-like” flow from large to small energy density

0: Quenching/amplification of flow due to the underlying field structure.

Rainer Fries

Transverse Flow: Visualization) d(backgroun 0

flow) odd (no0 1

17NFQCD 2013Rainer Fries

Modelling Color Charges So far color charge densities 1, 2 fixed. MV: Gaussian distribution around color-neutral average

Sample distribution to obtain event-by-event observables. Next: analytic calculation of expectation values (as function of

average color charge densities 1, 2).

Transverse flow comes from gradients in nuclear profiles. Original MV model = const. Here: relaxed condition, constant on length scales 1/Qs ,

allow variations on larger length scales 1/m where m << Qs.

xdxxxxNgxx

x

iiTTiab

ijc

bj

ai

ai

1

0

212

2112

2

21 xx

Tji

Ti

T mm xxx 22212 [G. Chen, RJF, PLB 723 (2013)][G. Chen et al., in preparation]

18NFQCD 2013Rainer Fries

Averaged Density and Flow

Energy density ~ product of nuclear gluon distributions ~ product of color source densities

“Hydro” flow:

“Odd“ flow term:

With we have

Order 2 terms …

2

22

21

26

0 ln8

1mQNNg cc

2

22

212

26

ln64

1mQNNg icci

[T. Lappi, 2006][RJF, Kapusta, Li, 2006] [Fujii, Fukushima, Hidaka, 2009]

2

22

21122

26

ln64

1mQNNg iicci

[G. Chen, RJF, PLB 723 (2013)][G. Chen et al., in preparation]

jiijii AAigBAAigE 210210 , , , 0000 EBBE ii

19NFQCD 2013Rainer Fries

Higher Orders in Time

Generally: powers of go with factors of Q or factors of transverse gradients

Number of terms with gradients in 1, 2 rises rapidly for higher orders in time.

For the case of near homogeneous charge densities when gradients can be neglected the expressions are fairly simple.

Example: ratio of longitudinal and transverse pressure at midrapidity

[G. Chen et al., in preparation]

20NFQCD 2013

Ratio of longitudinal and transverse pressure. Pocket formula for homogeneous nuclei (no transverse gradients):

Rainer Fries

Comparison with Numerical Solutions

[F. Gelis, T. Epelbaum, arxiv:1307.2214]

𝑃𝜀0

𝑂 (𝜏4 )

21NFQCD 2013

Ratio of longitudinal and transverse pressure. Pocket formula for homogeneous nuclei (no transverse gradients):

Rainer Fries

Comparison with Numerical Solutions

𝑃𝜀0

[F. Gelis, T. Epelbaum, arxiv:1307.2214]

𝑂 (𝜏4 )

22NFQCD 2013Rainer Fries

Check: Event-By-Event Picture Example: numerical sampling of charges for “odd” vector i in

Au+Au (b=4 fm).

Averaging over events: recover analytic result.

1N 500N

PRELIMINARY

Analytic expectation valueEnergy density N=1

23NFQCD 2013Rainer Fries

Flow Phenomenology: b 0

0

0

TTV

ii

10

0 Background

Odd flow needs asymmetry between sources. Here: finite impact parameter

Flow field for Au+Au collision, b = 4 fm.

Radial flow following gradients in the fireball at = 0. In addition: directed flow away from = 0. Fireball is rotating, exhibits angular momentum. |V| ~ 0.1 at the surface @ ~ 0.1

00T

xT 0

24NFQCD 2013Rainer Fries

Phenomenology: b 0 Angular momentum is natural: some

old models have it, most modern hydro calculations don’t. Some exceptions

Do we determine flow incorrectly when we miss the rotation?

Directed flow v1: Compatible with hydro with suitable

initial conditions.

[Gosset, Kapusta, Westfall (1978)]

[Liang, Wang (2005)]

MV only, integrated over transverse plane, no hydro

[L. Csernai et al. PRC 84 (2011)] …

25NFQCD 2013

Odd flow needs asymmetry between sources. Here: asymmetric nuclei.

Flow field for Cu+Au collisions:

Odd flow increases expansion in the wake of the larger nucleus, suppresses flow on the other side.

b0 & A B: non-trivial flow pattern characteristic signatures from classical fields? Rainer Fries

Phenomenology: A B

0

0

TTV

ii

fm 2b

0b

0b

26NFQCD 2013

CGC fingerprints? Need to evolve systems to larger times.

Examples: Forward-backward asymmetries of type

Here p+Pb:

Rainer Fries

Phenomenology: A B

⟨𝑇0 𝑥 ⟩⟨𝑇 00 ⟩

( =−2 ) −⟨𝑇0 𝑥 ⟩⟨𝑇 00 ⟩

( =+2 )

Directed flow asymmetry

Radial flow asymmetry

27NFQCD 2013

No equilibration here; see other talks at this workshop. Pragmatic solution: extrapolate from both sides (r() =

interpolating fct.)

Here: fast equilibration assumption:

Matching: enforce (and other conservation laws). Analytic solution possible for matching to ideal hydro.

4 equations + EOS to determine 5 fields in ideal hydro. Up to second order in time:

Odd flow drops out: we are missing angular momentum!

Rainer Fries

Matching to Hydrodynamics

rTrTT 1plf

pT

L0 T

0r

28NFQCD 2013

Instantaneous matching to viscous hydrodynamics using in addition

Mathematically equivalent to imposing smoothness condition on all components of T.

Leads to the same procedure used by Schenke et al. Numerical solution for hydro fields:

Rotation and odd flow terms readily translate into hydrodynamics fields.

Rainer Fries

Matching to Hydrodynamics

TxTxMM 0

0yvz 001 y 0yvx

29NFQCD 2013

Intricate 3+1 D global structure emerges Example: nodal plane of longitudinal flow, i.e. vz = 0 in x-y-

space

Further analysis: Need to run viscous 3+1-D hydro with large viscous corrections. Work in progress.

Rainer Fries

Initial Event Shape

30NFQCD 2013

Real nuclei are slightly off the light cone.

Classical gluon distributions calculated by Lam and Mahlon.

Nuclear collisions off the light cone?

Approximations for RA/ << 1/Qs: Because of d/d = 0 the energy

density just after nuclear overlap is a linear superposition of all energy density created during nuclear overlap.

New (transverse) field components Lorentz suppressed only count longitudinal fields in initial energy density.

Then use Lam-Mahlon gluon distributions in “old” formula for 0.

Rainer Fries

Classical QCD Beyond Boost Invariance

[S. Ozonder, RJF, arxiv:1311.3390]

[C.S. Lam, G. Mahlon, PRD 62 (2000)]

31NFQCD 2013

We can calculate the fields and energy momentum tensor in the clQCD approximation for < 1/Qs analytically.

Simple expressions for homogeneous nuclei. Transverse energy flow shows interesting and unique (?)

features: directed flow, A+B asymmetries, etc. Interesting features of the energy flow are naturally translated

into their counterparts in hydrodynamic fields in a simple matching procedure.

Future phenomenology: hydrodynamics with large dissipative corrections.

3 Questions: Are there corrections to observables we think we know well (like elliptic

flow)? Are there phenomena that we completely miss with too simplistic state-of-

the art initial conditions? Are there flow signatures unique to CGC initial conditions that we can

observe?Rainer Fries

Summary