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Heavy Flavor in Medium: Langevin vs. Boltzmann

This study compares the Langevin and Boltzmann approaches in describing the evolution of heavy quarks in a medium, focusing on the impact on RAA and v2. The effects of drag and diffusion coefficients evaluated from pQCD and different forms of the fluctuation-dissipation relation are investigated.

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Heavy Flavor in Medium: Langevin vs. Boltzmann

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  1. HeavyFlavor in Medium MomentumEvolution: Langevin vs Boltzmann F. Scardina University of Catania INFN-LNS 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

  2. Outline • Fokker Planck approach • Boltzmann approach • Comparison in a static medium • Comparison in realistic HIC (RAA and V2) • cc angular correlation

  3. Description of HQ propagation in the QGP • MHQ >> QDC (Mcharm1.3 GeV; Mbottom  4.2 GeV) • MHQ >> T • MHQ >> gT≈K(momentum trasferred) Brownian Motion Many soft scatterings are necessary to change significantly the momentum and the trajectory of Heavy quarks The Fokker-Planckequation The interactionisencoded in the drag and diffusioncoefficents

  4. Fokker Planck equation The FokkerPlanckeq can bederivedfrom the Bolzmann-Equation Integratingovermomenta w(p,k) is the transition rate for collisions of HQ with heath bath changing the HQ momentum from p to p-k Ifk<< P WhereBij can bedivided in a longitudinal and in a transversecomponent B||, BT

  5. Langevin Equation The Fokker-Planck equation is equivalent to an ordinary stochastic differential equation • is the deterministicfriction (drag) force • Cjkis a stochasticforce in termsofindependent • Gaussian-normaldistributedrandomvariableρ=(ρx,ρy,ρz) the covariance matrix and  arerelated to the diffusion matrix and to the drag coefficient by

  6. Drag and diffusion coefficients from pQCD For 2->2 collsionprocess The infraredsingularityisregularizedintroducing a Debye-screaning-massmD

  7. Different form of FDT The long time solution is recovered relating the Drag and Diffusion coefficent by means of the fluctuation dissipation relation Wehavestudied the impact on RAA and v2ofevaluating the drag and the diffusionfrompQCD or using the differentformof the FDT We assume BT andB|| =D 1) A and D (from pQCD ) 2) D (pQCD) ;A (FDT) 3) A (pQCD) ; D (FDT)

  8. Impact on RAA and v2 of the different form of FDT Au+Au (200 GeV) b=8.0 1) A and D (pQCD) mD=gT 2) D (pQCD) ;A (FDT) 3) A (pQCD) ; D (FDT) If B||=BT =Dwe can get the same RAA of the pure pQCD case reducing the drag and the diffusion by 20%

  9. Different form of FDT The long time solution is recovered relating the Drag and Diffusion coefficent by means of the fluctuation dissipation relation Wehavestudied the impact on RAA and v2ofevaluating the drag and the diffusionfrompQCD or using the differentformof the FDT 1) A and D (pQCD) 2) D (pQCD) ;A (FDT) 3) A pQCD ; D (FDT) 4) BT andB|| (pQCD) ; A FDT 5) B||=D (pQCD) ; A (pQCD)

  10. Impact on RAA and v2 of the different form of FDT Au+Au (200 GeV) b=8.0 4) BT andB|| (pQCD) ; A FDT mD=gT 5) B||=D (pQCD) ; A (pQCD) If B||is evaluated from pQCD one has to reduce the drag and the diffusion by 55% but the pT dependence of RAA and v2 is quite different

  11. Transport theory Free-streaming Mean Field Collisions Describes the evolution of the one body distribution function f(x,p) Itisvalidtostudy the evolutionofboth bulk and Heavyquarks 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) [Greco, et al. PLB670(09)], [ Z. Xu, et al PRC71(04)] [Scardina,et al PLB724(13)],[Scardina,et al PLB727(13)]

  12. Transport theory • Collision integral (stochastic algorithm)

  13. Evaluation of Drag and diffussion Langevinapproach Boltzmannapproach M -> Ai, Bij M -> 

  14. Mean momentum evolution in a static medium Weconsiderasinitialdistribution in p-space a (p-1.1GeV) forboth C and B withpx=(1/3)p T=400 MeV Eachcomponentofaveragemomentumevolvesaccordingto <pi>=p0iexp(-t) where 1/  is the relaxationtimetoequilibrium () For a very inclusive quantity BM and LV givesameresult b/c=2.55mb/mc

  15. Boltzmann vsLangevin (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. PlumariandV. Greco PRC90 044901 (2014)] We studied the effect of the mass and of the momentum transferred on the approximation involved in the F-P

  16. Boltzmann vsLangevin (Bottom) In bottom case LangevinapproximationgivesresultssimilartoBoltzmann The Larger M the BetterLangevinapproximationworks [S. K. Das , F. Scardina, V. Greco PRC90 044901 (2014)]

  17. Boltzmann vsLangevin (Charm) • simulatingdifferentaveragemomentum transfer Angulardependenceof Mometum transfer vs P Decreasing mD makes the  more anisotropic Smaller average momentum transfer [S. K. Das , F. Scardina, V. Greco PRC90 044901 (2014)]

  18. Boltzmann vsLangevin (Charm) mD=0.4 GeV mD=0.83 GeV (gT) mD=1.6 GeV The smaller <k> the betterLangevinapproximationworks

  19. RAAand v2 Boltzmann vsLangevin (RHIC) Au+Au@200AGeV, b=8 fm 40% mD=1.6 GeV • Fixed same RAA(pT) [one has to reduce A and D] v2(pT) 35% higher (mD=1.6 GeV) • - dependence on the specific scattering matrix (isotropic case -> larger effect) [F. ScardinaJ.Phys.Conf.Ser. 535 (2014) 012019]

  20. RAAand v2 Boltzmann vsLangevin (RHIC) Au+Au@200AGeV, b=8 fm mD=gT Fixed same RAA(pT) [one has to multiply A and D by 0.65]

  21. Energy loss of a single HQ T=400 MeV Langevin Boltzmann Charm Mc/T≈3 Bottom Mb/T≈10 T=400 MeV Mc/T≈3 Mb/T≈10 [F. ScardinaJ.Phys.Conf.Ser. 535 (2014) 012019]

  22. Back to Back correlation Back to back correlationobservablecouldbe sensitive tosuch a detail Langevin Boltzmann Initiald(p=10) can betoughtas a Near side charm withmomentumequal 10 Finaldistribution can betoughtas the momentumprobabiltydistribution tofindanAway side charm Boltzmannimplies a largermomentum spread

  23. Langevinvs Boltzmann angular correlation (static medium) P0=10 GeV

  24. Langevinvs Boltzmann angular correlation (static medium) t= 5 fm/c Langevin Boltzmann The larger spread ofmomentumwith the Boltzmannimplies a large spread in the angulardistributionsof the Away side charm Striking difference also at Df =0: - The evolution of the yield from 2 to 5 GeV is about 50 times different P0=10 GeV

  25. LV vs BM c-c angular correlation in HIC P0=10 GeV

  26. LV vs BM c-c angular correlation in HIC RHIC Au+Au 200 AGeV b=8.0 For the twocases wefix the same RAA pT cut [0,2] GeV [2,4] [4,6] [6,10] LV BM mD=1.6 GeV mD=0.4 GeV

  27. LV vs BM c-c angular correlation in HIC ( RHIC 200 AGeV) RHIC Au+Au 200 AGeVb=8 For the twocases wefix the same RAA pT cut [0,2] GeV [2,4] [4,6] [6,10] LV BM mD=0.4 GeV mD=1.6 GeV • There is not the large difference in the yield we found in the static case • because now RAA is fixed • However if (Df-π)>1 for pt cut [2,4] and [4,6] thereisoneorderof • magnitudeofdifference in the yieldfor the mD=1.6 case

  28. LV vs BM c-c and b-b angular correlation in HIC ( RHIC 200 AGeV) RHIC Au+Au 200 AGeV b=8.0 mD=1.6 pT cut [0,2] GeV [2,4] [4,6] [6,10] LV BM c-c b-b

  29. RHIC 200 AGeVvs LHC 2.76 AGeV Wefoundthat the broadeningofdN/dfiscreatedmainly at initialtimes pT cut [0,2] GeV [2,4] [4,6] [6,10] LV BM mD=0.4 GeV mD=1.6 GeV RHIC Mc/Tinit≈3.5 LHC Mc/Tinit≈2.5

  30. LV vs BM c-c angular correlation in HIC (LHC 2.76 ATeV) LHC Pb+Pb 2.76 TeV (charm) b=9.0 pT cut [0,2] GeV [2,4] [4,6] [6,10] LV BM mD=0.4 GeV mD=1.6 GeV For mD=1.6 weobserve the partonicwindeffect (enhancementof the azimuthalcorrelations in the regionofDf=0) with the BM and notwith the LV Becauseof the radial flow the cc pair are pushed in the same direction towardsmaller opening angles [ X. Zhuet al PRL 100, 152301 (2008)]

  31. Partonic wind with BM LHC Pb+Pb 2.76 TeV (charm) b=9.0 pT cut [0,2] GeV [2,4] [4,6] [6,10] mD=1.6 GeV mD=1.6 mD=0.4 c-c Angularcorrelation sensitive to the microscopicaldynamics Partonicwind: LHC (largerradial flow) and not at RHIC mD=1.6 (more isotropic cross section) notfor mD=0.4 [M. Nahrganget al PRC 90, 024907 (2014)]

  32. Summary • The more one looks at differential observables RAA->V2->dNcc/dDf • the more the differences between the BM and F-P approach increase • Very similar dynamics between FP and LV for Bottom at least for • RAA and V2 • Boltzmann is more efficient in building up the v2 related to HQ • Angular correlation depends on: • -microscopic details of the Dynamics (mD, radiative or collisionalDE) • -BM vs LV • -The broadening of dN/df Is mainly created at initial times • -BM vs LV differences larger at high collision energy

  33. Asymptoticsolution

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

  35. Transport theory • Collision integral (stochastic algorithm) • Assuming two particle • In a volume 3x in space • momenta in the range • (P,P+3P) ; (q,q+3q) collision rate per unit phase space for this pair Exact solution Δ3x

  36. Langevin Equation The Fokker-Planck equation is equivalent to an ordinary stochastic differential equation •  is the deterministicfriction (drag) force • Cijis a stochasticforce in termsofindependent • Gaussian-normaldistributedrandomvariable Interpretation of the momentum argument of the covariance matrix. =0 the pre-point Ito =1/2 mid-point Stratonovic-Fisk  =1 the post-point Ito (or H¨anggi-Klimontovich)

  37. Boltzmann vsLangevin (Bottom)

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