Hydrodynamical Description of the QGP Using Energy-momentum - PowerPoint PPT Presentation

In collaboration with A. Ayala (ICN-UNAM), I. Dominguez (FFM-UAS), M.E. Tejeda (DF-USON) Hydrodynamical Description of the QGP Using Energy-momentum In-medium Deposition By An Extended Source J. Maldonado (M.Sc. student at Universidad de Sonora)

I address this questions about the QGP medium’s response using linearized viscous hydrodynamics. Deposition could be on the sides of the path traveled, or along the direction of the traveling parton. This is work in progress and I will briefly explain the beginning of my work. 1 Introduction How does a parton travel through the QGP and deposits energy-momentum? What can we learn from the energy-momentum deposition into the medium?

2 (1) (2) 0 0 By equilibrium conditions, local conservation laws are presented, The medium 4-velocity, Equilibrium Medium g u µ = ( 1 , u ) , u = ϵ 0 ( 1 + c 2 s ) with g the momentum density related with the perturbation. ϵ 0 and c s static background’s energy density and sound velocity. = − Pg µν + ( ϵ + P ) u µ u ν ∂ µ Θ µν Θ µν = 0 ,

3 the perturbation made by the parton. medium’s energy-momentum tensor can be written as [1] (5) 0 (4) (3) Assuming that the disturbance introduced by the parton is small, the 0 Introducing a disturbance Θ µν = Θ µν + δ Θ µν where Θ µν is the equilibrium energy-momentum tensor and δ Θ µν is Considering at first order in shear ( η ) and bulk ( ξ ) viscosity, δ Θ 00 = δϵ, δ Θ 0 i = g δ Θ ij = δ ij c 2 4 Γ s ( ∂ i g j + ∂ j g i − 2 3 δ ij ∇ · g ) − ξδ ij ∇ · g s δϵ − 3 with Γ s ≡ 4 η/ 3 ϵ 0 ( 1 + c 2 s ) , the sound attenuation length.

4 in terms of the source components, (9) (6) The hydrodynamics equations are, (7) (8) Substituting the tensor components, and using the Fourier transform for these equations, the energy and momentum density are written Introducing a disturbance ∂ 0 δϵ + ∇ · g = J 0 ∂ 0 g i + ∂ j δ Θ ij = J i with J µ the source of the disturbance. δϵ = i k · J ( k , ω ) + J 0 ( k , ω )( i ω − Γ s k 2 ) ω 2 − c 2 s + i Γ s ω k 2 [ ω k 2 k · J ( k , ω ) + c 2 s J 0 ( k , ω ) i J T ] g L = i g T = , ω 2 − c 2 s k 2 + i Γ s ω k 2 ω + i 3 4 Γ s k 2

5 dx in [1] (11) 2 v dx 2 v 2 v (10) The last integration is rewritten using dimensionless quantities, The parton can be represented by a localized source, as it’s chosen Its Fourier’s transform, Localized disturbance source ( dE ) J ν ( x , t ) = v ν δ ( x − v t ) dE / dx is the parton’s energy loss per unit length and it’s taken as a constant along the parton’s path. v ν = ( 1 , v ) is the parton’s velocity. 2 π ( dE ) J ν ( x , t ) = v ν δ ( k · v − ω ) ( 2 π ) 4 ( 3 Γ s ) ( 3 Γ s ) ( 3 Γ s ) − 1 ξ ≡ k T , α ≡ z , β ≡

6 Taking as parameters α and β , the plots were performed from an alpha minimum value of 0 . 5, to 6, and from − 6 to 6 for β . Figure 1: ( ⃗ g T ) z , ( ⃗ g T ) y ( ⃗ g L ) z , ( ⃗ g T ) y and δϵ quantities for a localized source with α min = 0 . 5

7 (12) (13) 4 v 2 k 4 2 v 2 dx 1 An extended source was proposed, inspired by [2]. The main idea is x , the Fourier space, the comparison with the previously localized source, 1 dx An extended source ( dE ) ) 3 e − ( ⃗ ⃗ vt ) 2 x − J ν ( ⃗ v ν x , t ) = 2 σ 2 ( √ 2 πσ Parton’s velocity ⃗ v makes a constant angle with the position vector ⃗ represented by ⃗ x · ⃗ v = | ⃗ x || ⃗ v | cos γ . Again, the source is transform to ) √ ( dE 2 πσ [( ) ⃗ sin2 γ k 2 ] ⃗ − σ 2 J ν ( ⃗ 1 + ω 2 − 8 sin2 γ ω + v ν e v · k , ω ) = sin2 γ v sin 3 γ ( 2 π ) 4

8 1 (14) 0 dx 2 11 It is feasible to calculate ( ⃗ g T ) z performing the integration over ω and k z using a contour integral that contains at least one of the function poles. The final expression was rewritten using the variables α , β and ξ , ) 3 ∫ ∞ ( dE ) σ ( 2 v ( ⃗ d ξξ 2 J 0 ( βξ ) e − αξ g T ) z = 2 sin 2 γ 3 Γ s ( 2 π ) Figure 2: From left to right, ( ⃗ g T ) z for an extended source with α min = 0 . 1, α min = 0 . 5, α min = 1 . 0

It’s possible to obtain similar plots for the other energy-momentum components. The final integration could be performed using a Monte-Carlo integration method, and then, to compare with the integration already done for a localized source. expecting in the delta approximation, it would be convenient to consider non-linear terms in the hydrodynamics equations. This is work in progress and eventually we want to generate initial conditions with energy and momentum maps that can be used as input on numerical simulations in different hydrodynamical set-ups. 9 Final Remarks If we want to consider a σ value that is far away from the one we are

A. Ayala, I. Dominguez, and M. E. Tejeda-Yeomans. Phys. Rev. C , 88, 2013. B. Betz, J. Noronha, G. Torrieri, M. Gyulassy, I. Mishustin, and D. H. Rischke. Phys. Rev. C , 79, 2009. References i Head shock versus mach cone: Azimuthal correlations from 2 → 3 parton processes in relativistic heavy-ion collisions. Conical correlations, bragg peaks, and transverse flow deflections in jet tomography.