F. Scardina University of Catania INFN-LNS Heavy Flavor in Medium Momentum Evolution: Langevin vs Boltzmann V. Greco S. K. Das S. Plumari V. Minissale J. Bellone Heavy Quark Physics in Heavy-Ion collisions: Experiments, Phenomenology and Theory Trento 16-20/03/2015

Outline  Fokker Planck approach  Boltzmann approach  Comparison in a static medium  Comparison in realistic HIC (R AA and V 2 )  cc angular correlation

Description of HQ propagation in the QGP Brownian Motion Many soft scatterings are necessary to change significantly the momentum and the trajectory of Heavy quarks The Fokker-Planck equation The interaction is encoded in the drag and diffusion coefficents M HQ >>  QDC (M charm  1.3 GeV; M bottom  4.2 GeV) M HQ >> T M HQ >> T M HQ >> gT≈K (momentum trasferred)

Fokker Planck equation If  k  <<  P  The Fokker Planck eq can be derived from the Bolzmann-Equation  (p,k) is the transition rate for collisions of HQ with heath bath changing the HQ momentum from p to p-k Integrating over momenta Where B ij can be divided in a longitudinal and in a transverse component B ||, B T

Langevin Equation The Fokker-Planck equation is equivalent to an ordinary stochastic differential equation  is the deterministic friction (drag) force C jk is a stochastic force in terms of independent Gaussian-normal distributed random variable ρ=(ρ x,ρ y,ρ z ) the covariance matrix and  are related to the diffusion matrix and to the drag coefficient by

Drag and diffusion coefficients from pQCD The infrared singularity is regularized introducing a Debye-screaning-mass m D For 2->2 collsion process

The long time solution is recovered relating the Drag and Diffusion coefficent by means of the fluctuation dissipation relation Different form of FDT We have studied the impact on R AA and v 2 of evaluating the drag and the diffusion from pQCD or using the different form of the FDT 1) A and D (from pQCD ) 1) A and D (from pQCD ) 2) D (pQCD) ;A (FDT) 2) D (pQCD) ;A (FDT) 3) A (pQCD) ; D (FDT) 3) A (pQCD) ; D (FDT) B T and B || =D We assume

Impact on R AA and v 2 of the different form of FDT Au+Au (200 GeV) b=8.0 If B || =B T =D we can get the same R AA of the pure pQCD case reducing the drag and the diffusion by 20% 1) A and D (pQCD) 1) A and D (pQCD) 2) D (pQCD) ;A (FDT) 2) D (pQCD) ;A (FDT) 3) A (pQCD) ; D (FDT) 3) A (pQCD) ; D (FDT) m D =gT

4) B T and B || (pQCD) ; A FDT 4) B T and B || (pQCD) ; A FDT 5) B || =D (pQCD) ; A (pQCD) 5) B || =D (pQCD) ; A (pQCD) The long time solution is recovered relating the Drag and Diffusion coefficent by means of the fluctuation dissipation relation Different form of FDT We have studied the impact on R AA and v 2 of evaluating the drag and the diffusion from pQCD or using the different form of the FDT 1) A and D (pQCD) 1) A and D (pQCD) 2) D (pQCD) ;A (FDT) 2) D (pQCD) ;A (FDT) 3) A pQCD ; D (FDT) 3) A pQCD ; D (FDT)

Impact on R AA and v 2 of the different form of FDT Au+Au (200 GeV) b=8.0 If B || is evaluated from pQCD one has to reduce the drag and the diffusion by 55% but the p T dependence of R AA and v 2 is quite different 4) B T and B || (pQCD) ; A FDT 4) B T and B || (pQCD) ; A FDT 5) B || =D (pQCD) ; A (pQCD) 5) B || =D (pQCD) ; A (pQCD) m D =gT

Free- streaming Mean Field Collisions Transport theory To solve numerically the B-E we divide the space into a 3-D lattice and we use the standard test particle method to sample f(x,p) Describes the evolution of the one body distribution function f(x,p) It is valid to study the evolution of both bulk and Heavy quarks [Greco, et al. PLB670(09)], [ Z. Xu, et al PRC71(04)] [Scardina,et al PLB724(13)],[Scardina,et al PLB727(13)]

Transport theory Collision integral (stochastic algorithm)

Evaluation of Drag and diffussion Boltzmann approach M ->  M -> A i, B ij Langevin approach

Mean momentum evolution in a static medium We consider as initial distribution in p-space a  (p-1.1GeV) for both C and B with p x =(1/3)p Each component of average momentum evolves according to =p 0 i exp(-  t) where 1/  is the relaxation time to equilibrium (  )  b /  c =2.55  m b /m c For a very inclusive quantity BM and LV give same result T=400 MeV

Boltzmann vs Langevin (Charm) We have plotted the results as a ratio between LV and BM at different time to quantify how much the ratio differs from 1 [S. K. Das, F. Scardina, S. Plumari andV. Greco PRC (2014)] We studied the effect of the mass and of the momentum transferred on the approximation involved in the F-P

Boltzmann vs Langevin (Bottom) In bottom case Langevin approximation gives results similar to Boltzmann The Larger M the Better Langevin approximation works [S. K. Das, F. Scardina, V. Greco PRC (2014)]

Boltzmann vs Langevin (Charm) simulating different average momentum transfer Decreasing m D makes the  more anisotropic Smaller average momentum transfer Angular dependence of  Mometum transfer vs P [S. K. Das, F. Scardina, V. Greco PRC (2014)]

Boltzmann vs Langevin (Charm) The smaller the better Langevin approximation works m D =0.4 GeV m D =0.83 GeV (gT) m D =1.6 GeV

R AA and v 2 Boltzmann vs Langevin (RHIC) Fixed same R AA (p T ) [one has to reduce A and D   v 2 (p T ) 35% higher (m D =1.6 GeV) - dependence on the specific scattering matrix ( isotropic case -> larger effect ) m D =1.6 GeV 40% [F. ScardinaJ.Phys.Conf.Ser. 535 (2014) ]

R AA and v 2 Boltzmann vs Langevin (RHIC) Fixed same R AA (p T ) [one has to multiply A and D by 0.65]

Energy loss of a single HQ LangevinBoltzmann T=400 MeV Mc/T≈3 Mb/T≈10 CharmCharm BottomBottom T=400 MeV M c /T≈3 M b /T≈10 [F. ScardinaJ.Phys.Conf.Ser. 535 (2014) ]

Back to Back correlation Back to back correlation observable could be sensitive to such a detail Langevin Boltzmann Initial  (p=10) can be tought as a Near side charm with momentum equal 10 Final distribution can be tought as the momentum probabilty distribution to find an Away side charm Boltzmann implies a larger momentum spread

Langevin vs Boltzmann angular correlation (static medium) P 0 =10 GeV

Boltzmann Langevin vs Boltzmann angular correlation (static medium) Striking difference also at  =0: - The evolution of the yield from 2 to 5 GeV is about 50 times different P 0 =10 GeV t= 5 fm/c The larger spread of momentum with the Boltzmann implies a large spread in the angular distributions of the Away side charm Langevin

LV vs BM c-c angular correlation in HIC P 0 =10 GeV

LV vs BM c-c angular correlation in HIC RHIC Au+Au 200 AGeV b=8.0 m D =0.4 GeV m D =1.6 GeV For the two cases we fix the same R AA p T cut [0,2] GeV [2,4] [4,6] [6,10] LV BM

RHIC Au+Au 200 AGeV b=8 m D =0.4 GeV m D =1.6 GeV For the two cases we fix the same R AA There is not the large difference in the yield we found in the static case because now R AA is fixed However if (  -π)>1 for pt cut [2,4] and [4,6] there is one order of magnitude of difference in the yield for the mD=1.6 case p T cut [0,2] GeV [2,4] [4,6] [6,10] LV BM LV vs BM c-c angular correlation in HIC ( RHIC 200 AGeV)

RHIC Au+Au 200 AGeV b=8.0 m D =1.6 p T cut [0,2] GeV [2,4] [4,6] [6,10] LV BM LV vs BM c-c and b-b angular correlation in HIC ( RHIC 200 AGeV) c-cb-b

m D =1.6 GeV RHIC 200 AGeV vs LHC 2.76 AGeV We found that the broadening of dN/d  is created mainly at initial times M c /T init ≈3.5 m D =0.4 GeV RHIC M c /T init ≈2.5 LHC p T cut [0,2] GeV [2,4] [4,6] [6,10] LV BM

LHC Pb+Pb 2.76 TeV (charm) b=9.0 m D =0.4 GeV For m D =1.6 we observe the partonic wind effect (enhancement of the azimuthal correlations in the region of  =0) with the BM and not with the LV Because of the radial flow the cc pair are pushed in the same direction toward smaller opening angles m D =1.6 GeV LV vs BM c-c angular correlation in HIC (LHC 2.76 ATeV) [ X. Zhu et al PRL 100, (2008)] p T cut [0,2] GeV [2,4] [4,6] [6,10] LV BM

Partonic wind with BM LHC Pb+Pb 2.76 TeV (charm) b=9.0 Partonic wind: LHC (larger radial flow) and not at RHIC m D =1.6 (more isotropic cross section) not for m D =0.4 [M. Nahrgang et al PRC 90, (2014)] m D =1.6 GeV p T cut [0,2] GeV [2,4] [4,6] [6,10] m D =1.6 m D =0.4 c-c Angular correlation sensitive to the microscopical dynamics

Summary The more one looks at differential observables R AA ->V 2 -> dN cc /d  the more the differences between the BM and F-P approach increase Very similar dynamics between FP and LV for Bottom at least for R AA and V 2 Boltzmann is more efficient in building up the v 2 related to HQ Angular correlation depends on: - microscopic details of the Dynamics (m D, radiative or collisional  E) - BM vs LV -The broadening of dN/d  Is mainly created at initial times -BM vs LV differences larger at high collision energy

Asymptotic solution

Transport theory Collision integral HQ with momentum p+k HQ with momentum p HQ with momentum p-k Gain term Loss term  (p,k) is the transition rate for collisions of HQ with heath bath changing the HQ momentum from p to p-k Element of momentum space with momentum p

Exact solution Transport theory Collision integral (stochastic algorithm) collision rate per unit phase space for this pair Assuming two particle In a volume  3 x in space In a volume  3 x in space momenta in the range momenta in the range (P,P+  3 P) ; (q,q+  3 q) (P,P+  3 P) ; (q,q+  3 q) Δ3xΔ3x

Langevin Equation The Fokker-Planck equation is equivalent to an ordinary stochastic differential equation  is the deterministic friction (drag) force C ij is a stochastic force in terms of independent Gaussian-normal distributed random variable  =0 the pre-point Ito  =1/2 mid-point Stratonovic-Fisk  =1 the post-point Ito (or H¨anggi-Klimontovich) Interpretation of the momentum argument of the covariance matrix.

Boltzmann vs Langevin (Bottom)