In relativistic heavy ion collisions a high energy density matter Quark-Gluon Plasma (QGP) may be formed.

Various signals have been proposed which probe this new phase of matter.

In particular photons (and dileptons) measure the properties of this plasma in a relatively clean manner, since such electro-magnetically interacting particles do not take part in strong interactions and only weakly interact with the system.

Photons are emitted from every stage of the collision, from pre-equilibrium stage, from quark matter, and from hadronic matter. The thermal photon emission from the QGP and the hadronic phase are obtained by integrating the rates of emission over the space-time history of the fireball.

E dN d3p=∫[{…}exp (-p.u/T )] d4x

Where,

p=( pT coshY, px , py , pTsinhY ) =>4-momentum of the photons

u=T(cosh , Vx(x,y) ,Vy(x,y) , sinh) =>4-velocity of the flow field

d4x= d dx dy d =>4-volume element

Also Y = rapidity (=0 as mid-rapidity region) =tanh-1(z/t) => space-time rapidity

= [t2 – z2]1/2 =>proper time

T=1/[1 VT2]1/2 => Lorentz factor

VT= [Vx2 + Vy

2 ]1/2 =>radial flow velocity

And pT= [px2 + py 2 ]1/2 =>transverse momentum

Where, px=pTcos and py=pTsin

Taking the azimuthal angle for transverse momenta as (in the reaction plane ) we can write :

p.u =T [ pTcosh(Y-)–pxvx–pyvy]

z =0 plane

Impact parameter b

x

y

Reaction plane

Overlap region

y

x

Spatial asymmetry

Momentum asymmetry

High pressure

Low pressure

Initial spatial

anisotropy

Final momentum

anisotropy

INPUT

OUTPUT

ELLIPTIC FLOW or AZIMUTHAL FLOW is

nothing but the system,s response to the initial spatial anisotropy.

The azimuthal asymmetry in the distribution of thermal photons , defined in terms of the coefficientsvn :

dN(b)/d2pTdy =dN(b)/2pTdpTdy[1+2v2(pT,b)cos(2)

+ 2v4(pT,b)cos(4)+ ………..]

Initial parameters for Au+Au collision at 200 AGeV

Initial time = 0.2 fmInitial entropy density (s0) = 351 fm-3

Initial baryon number density (n0) = 0.40 fm-3

Impact parameter b = 7 fmFreeze-out energy density (f )= 0.075 GeV/fm3

Initial energy density ~ *NWN +(1-)*NBC

where, 0.25 NWN (x,y,b) = Number of wounded nucleons NBC (x,y,b) = Number of binary collisions

Transverse momentum pT (GeV)

dN

/d2 p

Td

y (1

/GeV

2 )b=0.0fmb=3.0 fmb=5.4 fmb=7.0 fmb=8.3 fmb=9.4 fmb=10.4 fm

The yield is maximum for central collision ( b=0 fm) and it goes down as we move towards peripheral collisions with increase of impact parameter b.

b=10.4fm

b=9.4 fm

b=8.3 fm

b=7.0 fm

b=5.4 fm

b=3.0 fm

b=0.0 fm

b =7 fm

QGP

Hadronic

Total

At high values of transverse momenta (pT) the yield from the QGP dominates over the yield from hadrons. As pT gets lower, the yield from the hadrons makes a significant

contribution to the total emission of thermal photons.

Transverse momentum pT (GeV)

dN

/d2 p

T d

y (1

/Ge

V2 )

Constant energy density contours for =q, h and

f , along y(x=0) (solid curves)

and x (y=0) (dashed curves).

Where,

q = 1.6 GeV/fm3

h = 0.444 GeV/fm3

f = 0.075 GeV/fm3

The difference between the solid and the dashed curves get pushed to larger x or y as the energy density decreases or the time increases. Both the curves are identical for 0 impact parameter .

nucl-th/0511079

nucl-th/0511079 –Rupa Chatterjee,

Evan S. Frodermann, Ulrich Heinz

and Dinesh K.Srivastava .

t (f

m)

Fluid velocity along the constant energy density contour for =q(for

QGP phase),=h (for mixed phase) and =f (for hadronic phase ) for x (y=0) (dashed curve) and y (x=0) (solid curve) .

Velocities ( zero at initial time t0 ) which start differing over extended volume by the end of the QGP phase, become almost identical at the end of hadronic phase.

nucl-th/0511079 –Rupa Chatterjee, Evan S. Frodermann, Ulrich Heinzand Dinesh K.Srivastava .

.

v2(QM) is small at higher pT (due to absence of transverse flow at early times) and gradually build up as pT (and temperature) decreases . After reaching a peak value (around 1.5 GeV) v2

decreases again as pT 0.

Elliptic flow of hadronic photons is much larger than v2(QM). The hadronic contribution to the photon spectrum is increasingly suppressed below the QGP contribution once pT exceeds 1.5- 2.0 GeV.

Hence the overall photonic v2 is larger than pure QM contribution and also decreases for large pT , approaching the v2 of the QM photons in -

spite of the larger elliptic flow of the hadronic photons .

•v2 for thermal photons from 200 AGeV Au+Au collision is shown by the red curve.

.Quark and hadronic contributions to v2 are shown separately. •v2 for pion is also shown in comparison with hadronic v2. • pion v2 tracks the hadronic v2.

nucl-th/0511079

Impact parameter dependence of the elliptic flow

Impact parameter are chosen to roughly correspond tocollision centralities of 0-10%(b=3 fm), 10-20%(b=5.4 fm), 20-30%(b=7 fm), 30-40% (b=8.3 fm), 40-50% (b=9.4 fm), and 50-60% (b=10.4) .

nucl-th/0511079

Initial entropy density dependence of theelliptic flow is shown.

Initial entropy densities are takenIn steps of 100, 200,..,1000 fm-3.

With rise in initial entropy density the flow also increases.Higher v2 at LHC!

so=100 fm-3

so=351 fm-3

so=1000 fm-3

Transverse momentum pT (GeV)

Elli

pti

c fl

ow

co

-eff

icie

nt

v 2

We have presented a first calculation of elliptic flow of thermal photons, emitted from relativistic collisions of heavy ions.

The azimuthal flow revealed by the thermal photons shows a rich structure and sensitivity to the evolution of the expansion dynamics as well as the rates of emission from the hadronic and quark matter .

v2 of thermal photons from hadronic matter tracks the v2 of pions, while v2 of thermal photons from the quark matter tracks the v2 of quarks.

Summary & Conclusions

v2shows a strong dependence on the impact parameter of the collisions as the relative contributions of the hadronic and quark

matter depend on it.

My collaborators for this work:-

Evan S. Frodermann and Ulrich Heinz The Ohio State University, USA

&Dinesh K. Srivastava, VECC Kolkata