Partons and waves Alejandro Ayala*, Isabel Domínguez and Maria Elena Tejeda-Yeomans (*) Instituto de Ciencias Nucleares, UNAM ayala@nucleares.unam.mx December,
Seminal work
Outline. Azimuthal angular correlations. Mach cones?. Head shock vs. Mach cones. vs. processes. Small viscosity Hydro: energy-momentum deposited in medium 6. Cooper-Frye: parton distribution from energy-momentum deposited in medium 7. Correlations 8. Conclusions
Azimuthal angular correlations from RHIC (STAR) φ) dn/d( φ) dn/d( /N /NTRIGGER Trigger.. d+au FTPC-Au -% p+p min. bias Au+Au Central - C( Φ) = φ (radians) dn d η N trigger d Φd η Peak suppressed at φ = ±π rad Away-side parton absorbed by medium due to energy loss Nucl. Phys. A 77, -8 () [STAR].
Azimuthal angular correlations from RHIC (PHENIX). (a) -.- GeV/c (b) - - GeV/c Au + Au -% p + p.. a ab Y = (/N )dn /d φ jet_ind. (c). - - GeV/c (d) - - GeV/c. SR HR SR. (e) - - GeV/c (f) - - GeV/c...6 (g).. NR - - GeV/c (h) φ (rad) - - GeV/c. As the p T difference between leading and associate particles increases Excess of particles at φ π/ y φ π/ rad Phys. Rev. C 78, 9 (8) [PHENIX].
6 Mach cones? Difficult to produce by a fast moving parton for the conditions in HIC (low viscosity, large parton velocity. I. Bouras, A. El, O. Fochler, F. Reining, F. Senzel, J. Uphoff, C. Wes, Z. Xu, C. Greiner, arxiv:7.7v [hep-ph]
7 Head shock (wakes) vs. Mach cones (shock waves) A fast moving parton produces a wake rather than a shock wave for small η/s.
8 processes: double hump only through shock waves Single parton moving in plasma with momentum p T deposits energy as a shock wave.
9 processes: double hump possible with wakes from two away-side particles Two partons moving in plasma with momentum p T +p T = p T deposit energy, each as a wake, rather than a shock wave.
p T,a p T,a p = p + p T,t T,a T,a p T suppressed by α s compared to but enhanced since on average (for same deposited energy) two partons come with lower momentum than one in the away side
Linearized, small viscosity hydro Total energy-momentum of medium T µν form initial T µν perturbed by fast parton T µν = T µν +δt µν Small deviations from equilibrium hydro equations µ T µν = µ δt µν = J ν Relativistic (small viscosity) fluid energy-momentum tensor components in terms of energy density δǫ and momentum density g transferred by fast parton to the medium δt δǫ δt i g i δt ij c sδǫδ ij Γ s( i g j + j g i gδij )
Linearized, small viscosity hydro Speed of sound: c s = /. Sound attenuation length: Γ s = η/st. Equations more easily solved in Fourier space δǫ = ikj L +J (iω Γ s k ) ω c s k +iγ s ωk g L = g T = J L (J ˆk) ˆk J T J J L iωj L +icsˆkj ω csk +iγ s ωk ij T ω + iγ sk
Energy-momentum deposited by partons in medium away-side partons in processes absorbed by medium deposit energy and momentum. Model the current they produce by: J ν (x) = de dx δ(x ut)uν U ν γ(,u), u is the corresponding parton velocity average energy-loss per unit length: de dx R. B. Neufeld, Thorsten Renk. Phys. Rev. C. 8. 9 ()
Energy-momentum deposited by partons in medium Density of particles produced at central rapidity (Cooper-Frye): dn p f T (y = ) = dy dφ pt i dp T p T (π) dσ µ p µ (f(x,p ) f ) Constant time freeze-out hyper surface: dσ µ p µ = d r. Equilibrium distribution (Boltzmann): f = e p T/T. Distribution generated by deposited energy-momentum: ( )( ) f(x,p ) f pt δǫ T ǫ ǫ + gy(x )sinφ+g z(x )cosφ ǫ (+cs ) e p T/T Angle between g and ẑ: φ Initial medium s energy density and temperature: ǫ, T
Energy-momentum deposited by fast particle in medium (α, β) I gtz... -. -. β - - - - - α 6 (α, β) I gty -. -. -.6 -.8 -. -. -. -.6 -.8 β - - - - - α 6 (α, β) I glz.8.6....8.6.. β - - - - - α 6...8.7 (α, β) I gly.8.6.. -. β - - - - - α 6 (α, β) I δ.6..... β - - - - - α 6
6 Particle distribution around parton direction of motion depends on the relative strength of g y (the coefficient of sinφ) and g z (the coefficient of cosφ). sin φ cos φ. -. - -. G + G cos φ.. y sin φ z (G y > G z ) G y sin φ (G z > G y ) + G cos φ z I α max = 6 g T,z g L,z g T,y g L,y δ g z g y G y sin φ (G z >> G y ) + G cos φ z -...6.8 α min -. - -... φ [rad]
] 7 Surface emission of leading particle together with two away-side partons moving through medium ( processes) - [(GeV/c) T ]..8 partons p T,l -.6 [(GeV/c) /N dn/dp.. T.8 partons p T,a.6 /N dn/dp.. 6 8 p [GeV/c] T 6 p [GeV/c] T
- 8 Azimuthal correlations compared to PHENIX data. Systematics of double hump correlation signal decreasing as the momentum of the away-side hadron gets closer to the momentum of the leading hadron is well reproduced. (a) PHENIX Au-Au Model < p T,l. < p T,a < GeV/c < GeV/c. ] - [(rad) /N dn/d φ...6. (b) (c) < p T,l < p T,a x. < GeV/c < GeV/c - < p T,l < p T,a < GeV/c < GeV/c..6.. (d) < p T,l < p T,a x. < GeV/c < GeV/c - φ [rad]
9 CONCLUSIONS vs. processes in-medium. Lower production rate compensated by higher momentum space population. Energy-momentum deposited in medium described by low viscosity hydro. Particles produced along direction of motion of partons (Wakes instead of Mach cones). Surface emission of leading particle together with away side partons depositing energy momentum within medium reproduce after hadronization systematics of two hadron correlations.