Classical Gluon Fields Rainer J Fries Texas AampM University NFQCD Kyoto December 5 2013 Classical Gluon Fields and MV Model Analytic Solutions for Early Times Phenomenology Beyond BoostInvariance ID: 296584
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Slide1
Flow and Energy Momentum Tensor From Classical Gluon Fields
Rainer J. FriesTexas A&M University
NFQCD Kyoto, December 5, 2013Slide2
Classical Gluon Fields and MV ModelAnalytic Solutions for Early Times
PhenomenologyBeyond Boost-InvarianceWork in collaboration mainly withGuangyao Chen (Texas A&M)
Sener Ozonder (INT/Seattle, Minnesota)Rainer Fries2
NFQCD 2013
OverviewSlide3
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
3
NFQCD 2013
The “Standard Model” of URHICs
[B.
Schenke
et al., PRL108 (2012); C. Gale et al. , PRL 110 (2013)]Slide4
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
4
NFQCD 2013
MV Model: Classical YM Dynamics
[L.
McLerran
, R. Venugopalan]Slide5
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
5
NFQCD 2013
MV Model: Classical YM Dynamics
[A.
Kovner
, L.
McLerran
, H.
Weigert
]Slide6
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
6
NFQCD 2013Glasma
in the Forward Light Cone
[
Krasnitz
,
Venugopalan
]
[T.
Lappi
]
…Slide7
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
7NFQCD 2013Fields: Before CollisionSlide8
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 Fries8
NFQCD 2013
Fields: At Collision
E
0
B
0
[L.
McLerran
, T.
Lappi
, 2006]
[RJF, J.I.
Kapusta
, Y. Li, 2006]Slide9
NFQCD 2013
9Rainer Fries
Fields: Into the Forward Light Cone
[
Guangyao
Chen, RJF]
Once the
non-
abelian
longitudinal fields E
0
, B
0
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 E
0
, B
0
decrease in both z
and t away from the light coneFirst abelian
theory:Gauss’ Law at fixed time t
Long. flux imbalance compensated by transverse fluxGauss: rapidity-odd radial fieldAmpere/Faraday as function of t: Decreasing long. flux induces transverse fieldAmpere/Faraday: rapidity-even curling
field
Full classical QCD:
[G. Chen, RJF, PLB 723 (2013)]Slide10
Transverse fields for randomly seeded A
1, A2 fields (abelian case).
= 0: Closed field lines around longitudinal flux maxima/minima 0: Sources/sinks for transverse fields appear
Rainer Fries
10
NFQCD 2013
Initial Transverse Field: VisualizationSlide11
Rainer Fries
11NFQCD 2013
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
R
ecursive solution for gluon field:
0
th
order = boundary conditionsSlide12
Rainer Fries
12NFQCD 2013
Analytic Solution: Small Time Expansion Convergence for weak field limit: recover analytic solution for all times.
Convergence for strong fields: convergence radius ~ 1/Q
s for averaged quantities like energy density.Slide13
Initial ( = 0)
structure of the energy-momentum tensor from purely londitudinal fields
Rainer Fries13
NFQCD 2013
Energy Momentum Tensor
Transverse pressure
P
T
=
0
Longitudinal pressure
P
L
= –
0
Slide14
NFQCD 2013
14Rainer Fries
Energy Momentum TensorFlow emerges from pressure at order 1:
Transverse
Poynting vector gives transverse flow.
Like hydrodynamic flow, determined by gradient of
transverse pressure
P
T
=
0
; even in rapidity.
Non-hydro like; odd in rapidity ??
[RJF, J.I.
Kapusta
, Y. Li, (2006)]
[G. Chen, RJF, PLB 723 (2013)]Slide15
NFQCD 2013
15Rainer Fries
Energy Momentum TensorCorrections to energy density and pressure due to flow at second order in timeExample: energy density
Depletion/increase of energy density due to transverse flow
Due to longitudinal flowSlide16
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
16
NFQCD 2013
Transverse Flow: VisualizationSlide17
Rainer Fries
17NFQCD 2013
Modelling Color ChargesSo 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.
[G. Chen, RJF, PLB 723 (2013)]
[G. Chen et al., in preparation]Slide18
NFQCD 2013
18Rainer Fries
Averaged Density and FlowEnergy density ~ product of nuclear gluon distributions ~ product of color source densities“Hydro” flow:
“Odd“ flow term:
With
we
have
Order
2
terms …
[T.
Lappi
, 2006]
[RJF,
Kapusta
, Li, 2006]
[
Fujii
, Fukushima, Hidaka, 2009]
[G. Chen, RJF, PLB 723 (2013)]
[G. Chen et al., in preparation]Slide19
NFQCD 2013
19Rainer Fries
Higher Orders in TimeGenerally: 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]Slide20
Ratio of longitudinal and transverse pressure. Pocket formula for homogeneous nuclei (no transverse gradients):
Rainer Fries
20NFQCD 2013Comparison with Numerical Solutions
[F.
Gelis
, T.
Epelbaum
,
arxiv:1307.2214]
Slide21
Ratio of longitudinal and transverse pressure. Pocket formula for homogeneous nuclei (no transverse gradients):
Rainer Fries
21NFQCD 2013Comparison with Numerical Solutions
[F.
Gelis
, T.
Epelbaum
,
arxiv:1307.2214]
Slide22
NFQCD 2013
22Rainer Fries
Check: Event-By-Event PictureExample: numerical sampling of charges for “odd” vector i in
Au+Au (b
=4 fm).
Averaging over events: recover analytic result.
PRELIMINARY
Analytic expectation value
Energy density
N
=1Slide23
Rainer Fries
23NFQCD 2013
Flow Phenomenology: b 0
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 @
~ 0.1Slide24
NFQCD 2013
24
Rainer FriesPhenomenology: 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
v
1
:
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)]
…Slide25
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
25
NFQCD 2013
Phenomenology: A
BSlide26
CGC fingerprints? Need to evolve systems to larger times.Examples:
Forward-backward asymmetries of typeHere p+Pb:
Rainer Fries26
NFQCD 2013
Phenomenology: A B
Directed flow asymmetry
Radial flow asymmetrySlide27
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
27
NFQCD 2013
Matching to Hydrodynamics
p
T
LSlide28
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:
R
otation and odd flow terms readily translate into hydrodynamics fields.
Rainer Fries
28
NFQCD 2013
Matching to HydrodynamicsSlide29
Intricate 3+1 D global structure emergesExample: nodal plane of longitudinal flow, i.e.
vz = 0 in x-y-
spaceFurther analysis: Need to run viscous 3+1-D hydro with large viscous corrections.Work in progress.
Rainer Fries
29
NFQCD 2013
Initial Event ShapeSlide30
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
30
NFQCD 2013
Classical QCD Beyond Boost Invariance
[S.
Ozonder
, RJF,
arxiv:1311.3390]
[
C.S. Lam, G.
Mahlon
, PRD 62 (2000)
]Slide31
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
31
NFQCD 2013
Summary