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Event by Event Analysis of the Anisotropy ofthe Low Energy Direct Photons
inRelativistic Heavy Ion Collisions
Classical Bremsstrahlung Revisited
T. Kodama1,3, T. Koide1, Y. Nara2,3 and H. Stöcker3
1 Federal University of Rio de Janeiro2 Akita International University
3 FIAS, Frankfurt
Direct Photons in Heavy Ion CollisionsThe electromagnetic probes carry the information from the early stage ofthe collision dynamics. Many theoretical Works since the very early stage ofthe heavy ion program.
J. Kapsta (1977); J. D. Bjorken and L. McLerran (1985); J. Thiel, T. Lippert, N. Grün (1989);
V. Koch, B. Blättel, W. Cassing, U. Mosel (1990); P. A. Ruuskanen, (1992), A. Dumitru, L.
McLerran, H.Stoecker (1993); D. Srivastava (1994); N. Arbex, U. Ornik, M. Plumer, R.Weiner
(1995); A. Dumitru, J. A. Marhuhn, D. H. Rischke(1995), T. Hirano, S. Muroya, M. Namiki
(1995); J. Alam, S. Raha and B. Sinha(1996). S. Jeon, A. Chikanian, J. Kapusta, S. M. H.
Wong (1999); U. Eichmann, C. Ernst, L.M. Satarov. W. Greiner (2000); R. Chatterjee, H.
Holopainen, T. Renk, K. J. Eskola (2011); A. K. Chaudhuri and B. Sinha (2011), T.S. Biro, M.
Gyulassi, Z. Schram (2012); G. Basar, D. Kharzeev, V. Skokov,........
And more recently, using the state-of-art hydro or transport theories,
C. Shen, Ulrich Heinz, J-F. Paquet, I. Kozlov, C. Gale, C. Shen, G. S. Denicol, M. Luzum, B.
Schenke, S. Jeon, Y. Hidaka, S. Lin, R. D. Pisarski, D. Satow, V. V. Skokov, G. Vujanovic, O.
Linnyk, E. L. Bratkovskaya, W. Cassing, C. Greiner, M. Greif, S. Endres, H. v. Hess, J. Weil,
M. Bleicher ..........
Early days model of photon emission
Coherent Bremsstrahlung Emission due toDeceleration of Incident Nuclei
J. Kapsta, Phys. Rev. C15, 1580 (1977), J. D. Bjorken and L. McLerran, Phys. Rev. D31, 63 (1985), J. Thiel et. al., Nucl. Phys. A504, 864 (1989).V. Koch et. al., Phys. Lett. B236, 135 (1990), T. Lippert et. al., Int. J. Mod. Phys. A29, 5249 (1991), A. Dumitru et. al., Phys. Lett. B318, 583 (1993), U. Eichmann and W. Greiner, J. Phys. G23, L65 (1997), S. Jeon et. al., Phys. Rev. C58, 1666 (1998), J. Kapusta and S. M. H. Wong, Phys. Rev. C59, 3317 (1999), U. Eichmann, C. Ernst, L.M. Satarov and W. Greiner, Phys.Rev. C62, 044902 (2000) ..
Decelation of incidente nuclei
the velocity. t’ is the emission time, defined by
is the trajectory of the point charge, and is /t d dt
Classical Electromagnetic Radiation by an accelerated point charge is given by (Liénard-Wiechert Potential),
1 1 ( ')
, 1 ,4 '1 ''
Te d tE x t nn
dtt nx t
, , ,B x t n E x t
where
' ( ')t t x t
t
and .( ')
( ')
x tn
x t
Continuum Sources (Integrate over moving charges)
31, 1 , ,
4
T
t x
eE x t nn d J
r
, , ,B x t n E x t
For ,r x
with 1
.n xr
How to calculate the photon spectrum?
Equivalent Photon Method - Jackson´s book
How to calculate the photon spectrum?
Equivalent Photon Method - Jackson´s book
Or introduce the Wigner function,
where is the wave function. ( , )x t
3 /( , ; ) †( / 2, ) ( / 2, )ip u
Wf x p t d u e x u t x u t
Identify the Classical Electromagnetic fields as thevector type wavefunction similar to the Dirac form*.
L. Silberstein, Ann. d. Phys. 22, 579 (1907); 24, 783 (1907); H. Bateman, Cambridge 1915, Dover, New York,1955). I. Bialynicki-Birula, Act. Phys. Pol. A86 97 (1994). P. Holland, Proc. R. Soc. A461, 359 (2005).
( , ) / 2,x t E iB
, and
then the Maxwell equations are equivalent to
0 0 0 0 0 1 0 1 01 1 1
0 0 1 , 0 0 0 , 1 0 0
0 1 0 1 0 0 0 0 0i i i
with generators of O(3) for spin 1
ti j
0,ti
The time evolution after “freeze-out” ( , 0j
The energy density and Poyinting Vector are expressed as
†
†
,
,P i
) we need only
Then the corresponding Wigner function (scalar) decomposes into the electric and magnetic contributions,
( , ; ) ( , ; ) ( , ; )W E Bf x p t f x p t f x p t
where the product is defined as . † † †
, ,i i
i x y z
323
3( , ; ) ( ) ,W FO
d nd r f x p t p
d p
From the definition, we know that
where tFO is the freezeout time. But it is instructive tocalculate the spectrum as the energy that enters into thedetector from the large distance behavior of the Wigner function as;
3 ( , ; ),Detector Wr D
E d r f x p t
where D is the domain of the detector positioned at a solid angle d2W.
0
x
q
f
D
( , ; )Wf x p t
Minimal Model as Extreme Coherence
• Each set of participant protons of the twoincidente nuclei are considered as point-likecharges with the effective charge Zeff .
• These two effective charges stop instantaneously as the two nuclei colide.
Minimal Model as Extreme Coherence
• Each set of participant protons of the twoincidente nuclei are considered as point-likecharges with the effective charge Zeff .
• These two effective charges stop instantaneously as the two nuclei colide.
x
y
z
2d
0
x
2
1
q
f
1
0
( ) 0 ,
( )
d
t
tV t
q
2
0
( ) 0
( )
d
t
tV t
q
0
1 3
2 2
( )1
( , ) ( ),4
( )
eff
x d zeV Z
E x t xy t rr
x d y
0
1 2
1( , ) ( ) ( ),
40
eff
yeV Z
B x t x d t rr
1 1 2 2
0 0
( , ), ( , ) ( , ), ( , )
, ,
E x t B x t E x t B x t
d V d V
where 2 2 2( )r x d y z
Taking the trajectories as
we get for the fields created by the trajectory-1 as
and for the trajectory-2, weget simply by the following replacement,
(0) (0)
, , ,
(0)
,
( , ; ) ( , ; ) ( , ; )
2cos (2 ) ( , ; )
E B E B E B
E B
f x p t f x d p t f x d p t
p d f x p t
From these, we have
where(0)
, , 0( , ; ) ( , ; )E B E B d
f x p t f x p t
3 /( , ; ) ( / 2, ) ( / 2, ) ,ip q
Ef x p t d u e E x q t E x q t and
and analogously for fB .
which reduces for large distance (i.e., r,t >> 1/p, d) to
2 2
/// 2 / 2 ( ) ( )t x q t x q t x q
for arbitrary vector with , and is the parallelcomponent of
q q d
xq//q
with respect to .
These integrals contain the product of two delta functions
2 2/ 2 / 2t x q t x q
Finally we get
2 2 2 2
0 2 2
1( , ; ) 4 1 cosW EM eff p xf x p t Z V p d
p r W W
The delta function (the last term) means that at a large(macroscopic) distance, the observational directioncoincides with that of the momentum detected.
where RD is the position of the detector D. Photon number distribution is
The energy spectrum is given by the integral within thevolume of the detecter D, which is equivalente to
32 2
3 3
1( , ; )
(2 )x D W D
D
ddt d R f R p t
dp
W
W
2 23
0
2 2 2 2
11 cos 2
2 cosh
EM eff
T
Z Vd Np d
dp dy p y
Angle integrated pT spectrum is given by (V0 ~ 1)
23
3
02
0
0.37 10 1 22
eff
T T Ty
Zd NJ pd
p dp dy p
10
10-1
10-3
10-5
10-7
d2N
/2
pTd
pTd
y (
GeV
/c)-2
pT Spectrum
0 1,
80,
1
eff
V
Z
d fm
10
10-1
10-3
10-5
10-7
d2N
/2
pTd
pTd
y (
GeV
/c)-2
0 1,
80,
1
eff
V
Z
d fm
pT Spectrum
0
/2
3/2
x
y
f
Angular distribution
Anisotropy Parameter v2
22
0
(2 )
1 (2 )
T
T
J p dv
J p d
1d fm
Large increase ofv2 for lower pT !
(b ~ 2fm)
If we suppose incoherence occurs between electro-magnetic fields generated by the incidente nuclei as exponetially with pT, we may expect*
22 2 /
0
(2 )
1 2 / (2 )T
T
p p
eff T
J p dv
e Z J p d
* T.S. Biró, Z. Szendi and Z. Schram, Euro. Phys.J A50, 62 (2014)
Curiously,...
PHENIX dataCentrality 20-40% arXiv:1509.07758
0 1, 80,
1 ,
0.2 ,
effV Z
d fm
p GeV
T. Koide and T. K, J. Phs. G 43 (2016) 095103.
Can the Classical Bremsstrahlung ofElectromagnetic Fields be useful to understand
Event by Event Initial Geometry
Event by Event inhomogeneity and fluctuations
Baryon Stopping
?
Can the Classical Bremsstrahlung ofElectromagnetic Fields be useful to understand
Event by Event Initial Geometry
Event by Event inhomogeneity and fluctuations
Baryon Stopping
?
More realistic simulations !
Can the Classical Bremsstrahlung ofElectromagnetic Fields be useful to understand
Event by Event Initial Geometry
Event by Event inhomogeneity and fluctuations
Baryon Stopping
?
More realistic simulations !
Can the Classical Bremsstrahlung ofElectromagnetic Fields be useful to understand
Event by Event Initial Geometry
Event by Event inhomogeneity and fluctuations
Baryon Stopping
?
More realistic simulations !
PHSD !
June 30
PHSD !
June 30, I fried to Frankfurt
PHSD !
Yasushi Horst
PHSD !
Yasushi Horst
PHSD -> JAM
the time evolution of electric charge current density of a few events for
Au+Au b=2fm Sqs = 5GeV
x
z
t=1.8fmExample of the time evolution in y=0 plane
Au+Au, SQS=5GeV, b=2fm
Ch
arge
cu
rren
t d
ensi
tyC
har
ge d
ensi
ty
t=2.2fmExample of the time evolution of current desity
Ch
arge
cu
rren
t d
ensi
tyC
har
ge d
ensi
ty
t=2.6fmExample of the time evolution of current desity
Ch
arge
den
sity
Ch
arge
cu
rren
t d
ensi
ty
t=4.2fmExample of the time evolution of current desity
Ch
arge
den
sity
Ch
arge
cu
rren
t d
ensi
ty
-8 -6 -4 -2 0 2 4 6 8
-4
-2
0
2
4
6
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
-8 -6 -4 -2 0 2 4 6 8
-4
-2
0
2
4
6
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
t=6.0fmExample of the time evolution of current desity
Ch
arge
den
sity
Ch
arge
cu
rren
t d
ensi
ty
-8 -6 -4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
-8 -6 -4 -2 0 2 4 6 8
-6
-4
-2
0
2
4
6
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
t=8.0fm
Ch
arge
den
sity
Ch
arge
cu
rren
t d
ensi
ty
-10 -8 -6 -4 -2 0 2 4 6 8 10
-8
-6
-4
-2
0
2
4
6
8
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-10 -8 -6 -4 -2 0 2 4 6 8 10
-8
-6
-4
-2
0
2
4
6
8
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Ch
arge
cu
rren
t d
ensi
ty
-8 -6 -4 -2 0 2 4 6 8 10
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8 10
-6
-4
-2
0
2
4
6
-8 -6 -4 -2 0 2 4 6 8 10
-4
-2
0
2
4
-8 -6 -4 -2 0 2 4 6 8 10
-2
0
2
0
x
q
f
D3
2
d N
d d W
At large distance,
3 3
2
2 2
1, ,
T
d N d NA n
d p dy d d
W
3, 1 ,4
i ni r Te JA n e nn d d e
,zJ Mostly is dominant.
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-14
-13.4
-12.8
-12.2
-11.6
-11
-10.4
-9.8
-9.2
-8.6
-8
-7.4
-6.8
-6.2
-5.6
-5
-4.4
-3.8
-3.2
-2.6
-2
-1.4
-0.8
-0.2
0.4
T=2.0 fmT=1.8 fmT=1.6 fm
T=1.4 fmT=1.2 fmT=1.0 fm
x-y profile of , /z zd dJ dt
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-14
-13.4
-12.8
-12.2
-11.6
-11
-10.4
-9.8
-9.2
-8.6
-8
-7.4
-6.8
-6.2
-5.6
-5
-4.4
-3.8
-3.2
-2.6
-2
-1.4
-0.8
-0.2
0.4
T=2.0 fmT=1.8 fmT=1.6 fm
T=1.4 fmT=1.2 fmT=1.0 fm
x-y profile of , /z zd dJ dt
(y=0)
Change of v2 vector as function of energy
2
2
2
cos(2 )
sin(2 )
V Cosv
V Sin
f
f
Change of v2 vector as function of energy
2
2
2
cos(2 )
sin(2 )
V Cosv
V Sin
f
f
2
2
2
cos(2 )
sin(2 )
V Cosv
V Sin
f
f
Similar behavior in V3 ...
Azimuthal Angular distribution at y=0
Azimuthal Angular distribution at y=1.0
Zenith angle distribution (forward)
1sin ,
cosh yq
tan sin .yq
Rapidity Distributions 4 events
10
10-1
10-3
10-5
10-7
d2N
/2
pTd
pTd
y (
GeV
/c)-2
0 1,
80,
1
eff
V
Z
d fm
pT Spectrum
Toy Model
pT Spectrum
pT Spectrum
SQS=10GeV b=2fm V2 aligned
Summary
The classical electromagnetic fields radiated by the decelerationprocess of relativistic heavy ions were re-examined.
A toy model and more sophisticated JAM simulations show a similar prediction for the v2 anisotropy parameter, indicating thatthe photons from the charge decelation (or stopping) processreflects very well the initial state in the EbyE basis.
If the increase of v2 in PHENIX data for lower pT really tellssomething, we may think of some coherent (collective) deceleration mechanism of the incident charges.
Indication of large EbyE fluctuations and formation of hot spots in JAM simulation.
If this this is the case, the classical EM radiation may offer a veryinteresting approach to determine the collision geometry, baryonstopping, etc..... But,
Experimental difficulties due to critically low multiplicity (~ 102-103). A further study is necessary to examine more in detail usingdifferent models of the initial condition.
Some Basic Questions
How to deal with the string dynamics and charge current ?
ˆ,J
Strings J Strings
How does the coarse-graining scale influence ?
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