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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.

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 !

PHSD !

June 30

PHSD !

June 30, I fried to Frankfurt

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


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


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

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

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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