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    d 3 p= ∫ [{…}exp (-p  . u  /T )] d 4 x Where, p  =( p T cosh Y, p x, p y, p T sinh Y ) =>4-momentum of the photons  u  =  T ( cosh , V x (x,y), V y (x,y), sinh  ) =>4-velocity of the flow field d 4 x =  d  d x d y d  =>4-volume element

Also Y = rapidity (=0 as mid-rapidity region)  = tanh -1 (z/t) => space-time rapidity   = [t 2 – z 2 ] 1/2 => proper time  T =1/[1 V T 2 ] 1/2 => Lorentz factor V T = [V x 2 + V y 2 ] 1/2 => radial flow velocity And p T = [p x 2 + p y 2 ] 1/2 => transverse momentum Where, p x =p T cos  and p y =p T sin  Taking the azimuthal angle for transverse momenta as    (in the reaction plane ) we can write : p . u  =  T [ p T cosh(Y-  )–p x v x –p y v y ]

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 coefficients v n : dN(b) / d 2 p T dy =dN(b) / 2  p T dp T dy[1+2v 2 (p T,b)cos(2  ) + 2v 4 (p T,b)cos(4  )+ ………..]

Initial parameters for Au+Au collision at 200 AGeV Initial time   = 0.2 fm Initial entropy density (s 0 ) = 351 fm -3 Initial baryon number density (n 0 ) = 0.40 fm - 3 Impact parameter b = 7 fm Freeze-out energy density (  f )= GeV/fm 3 Initial energy density ~  *N WN +(1-  )*N BC where,  0.25 N WN (x,y,b) = Number of wounded nucleons N BC (x,y,b) = Number of binary collisions

Transverse momentum p T ( GeV ) dN/d 2 p T dy (1/GeV 2 ) b=0.0fm b=3.0 fm b=5.4 fm b=7.0 fm b=8.3 fm b=9.4 fm b=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 (p T ) the yield from the QGP dominates over the yield from hadrons. As p T gets lower, the yield from the hadrons makes a significant contribution to the total emission of thermal photons. Transverse momentum p T (GeV) dN/d 2 p T dy (1/GeV 2 )

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/fm 3   h = GeV/fm 3  f = GeV/fm 3 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/ nucl-th/ –Rupa Chatterjee, Evan S. Frodermann, Ulrich Heinz and Dinesh K.Srivastava. t (fm)

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 t 0 ) which start differing over extended volume by the end of the QGP phase, become almost identical at the end of hadronic phase. nucl-th/ –Rupa Chatterjee, Evan S. Frodermann, Ulrich Heinz and Dinesh K.Srivastava..

v 2 (QM) is small at higher p T (due to absence of transverse flow at early times) and gradually build up as p T (and temperature) decreases. After reaching a peak value (around 1.5 GeV) v 2 decreases again as p T  0. Elliptic flow of hadronic photons is much larger than v 2 (QM). The hadronic contribution to the photon spectrum is increasingly suppressed below the QGP contribution once p T exceeds GeV. Hence the overall photonic v 2 is larger than pure QM contribution and also decreases for large p T, approaching the v 2 of the QM photons in - spite of the larger elliptic flow of the hadronic photons.

v 2 for thermal photons from 200 AGeV Au+Au collision is shown by the red curve.. Quark and hadronic contributions to v 2 are shown separately. v 2 for pion is also shown in comparison with hadronic v 2. pion v 2 tracks the hadronic v 2. nucl-th/

Impact parameter dependence of the elliptic flow Impact parameter are chosen to roughly correspond to collision 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/

Initial entropy density dependence of the elliptic flow is shown. Initial entropy densities are taken In steps of 100, 200,..,1000 fm -3. With rise in initial entropy density the flow also increases. Higher v 2 at LHC! s o =100 fm -3 s o =351 fm -3 s o =1000 fm -3 Transverse momentum p T (GeV) Elliptic flow co-efficient 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. v 2 of thermal photons from hadronic matter tracks the v 2 of pions, while v 2 of thermal photons from the quark matter tracks the v 2 of quarks. Summary & Conclusions

v 2 shows 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