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
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