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F. Scardina University of Catania INFN-LNS

Heavy Flavor in Medium Momentum Evolution : Langevin vs Boltzmann. F. Scardina University of Catania INFN-LNS. V. Greco S. K. Das S. Plumari. The 30 th Winter Workshop on Nuclear Dynamics. Outline. Heavy Flavors Transport approach Fokker Planck approach

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F. Scardina University of Catania INFN-LNS

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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 The 30th Winter Workshop on Nuclear Dynamics

  2. Outline • Heavy Flavors • Transport approach • Fokker Planck approach • Comparison in a static medium • Comparison in HIC • Conclusions and future developments The 30th Winter Workshop on Nuclear Dynamics

  3. Introduction Heavy Quarks • MHQ >> QDC (Mcharm1.3 GeV; Mbottom  4.2 GeV) • MHQ >> T • They are produced in the early stage • They interact with the bulk and can be used • to study the properties of the medium • There are two way to detect charm: • 1) Single non-photonic electron • 2) D and B mesons D0 K+ lepton– c c D0 K– π+

  4. RAA of Heavy Quarks Nuclear Modification factor: Decrease with increasing partonic interaction RHIC [PHENIX:PRL98(2007)172301] In spite of the larger mass at RHIC energy heavy flavor suppression is not so different from light flavor

  5. RAA at LHC Again at LHC energy heavy flavor suppression is similar to light flavor especially at high pT

  6. RAA and Elliptic flow Simultaneous description of RAA and v2 is a tough challenge for all models

  7. Description of HQ propagation in the QGP Brownian Motion Describedby the FokkerPlanckequation

  8. Description of HQ propagation in the QGP Brownian Motion Describedby the FokkerPlanckequation • FokkerPlank • [H. van Hees, V. Greco and R. Rapp, Phys. Rev. C73, 034913 (2006) ] • [Min He, Rainer J. Fries, and Ralf Rap PRC86 014903] • [G. D. Moore, D Teaney, Phys. Rev. C 71, 064904(2005)] • [S. Cao, S. A. Bass Phys. Rev. C 84, 064902 (2011) ] • [Majumda, T. Bhattacharyya, J. Alam and S. K. Das, Phys. Rev. C,84, 044901(2012)] • [Y. Akamatsu, T. Hatsuda and T. Hirano, Phys. Rev.C 79, 054907 (2009)] • [W. M. Albericoet al. Eur. Phys. J. C,71,1666(2011)] • [T. Lang, H. van Hees, J. Steinheimerand M. Bleicher arXiv:1208.1643 [hep-ph]] • [C. Young , B. Schenke , S. Jeon and C. Gale PRC 86, 034905 (2012)] • Boltzmann • [J. Uphoff, O. Fochler, Z. Xu and C. Greiner PRC 84 024908; PLB 717] • [S. K. Das , F. Scardina, V. Greco arXiV:1312.6857] • [B. Zhang, L. W. Chen and C. M. Ko, Phys. Rev. C 72 (2005) 024906] • [P. B. Gossiaux, J. B. Aichelin PRC 78 014904] • [D. Molnar, Eur. Phys. J. C 49(2007) 181]

  9. Transport theory Free-streaming Mean Field Collisions Classic Boltzmann equation 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) [ Z. Xhu, etal… PRC71(04)],[Ferini, et al. PLB670(09)], [Scardina,et al PLB724(13)],[Ruggieri,et al PLB727(13)]

  10. Transport theory • Collision integral (stochastic algorithm) w(p,k) is the transition rate for collisions of HQ with heath bath changing the HQ momentum from p to p-k • Collision integral (stochastic algorithm) [ Z. Xhu, etal… PRC71(04)],[Ferini, et al. PLB670(09)], [Scardina,et al PLB724(13)],[Ruggieri,et al PLB727(13)]

  11. Cross Section gc -> gc Dominantcontribution The infraredsingularityisregularizedintroducing a Debye-screaning-massmD [B. L. Combridge, Nucl. Phys. B151, 429 (1979)] [B. Svetitsky, Phys. Rev. D 37, 2484 (1988) ]

  12. Fokker Planck equation HQ interactions are conveniently encoded in transport coefficients that are related to elastic scattering matrix elements on light partons. The FokkerPlanckeq can bederivedfrom the B-E B-E Ifk<< P

  13. Fokker Planck equation where we have defined the kernels → Drag Coefficient → Diffusion Coefficient WhereBij can bedivided in a longitudinal and in a transversecomponent B0 , B1 [B. Svetitsky PRD 37(1987)2484]

  14. 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-normaldistributedrandomvariableρ=(ρx,ρy,ρz) ρobey the relations: the covariance matrix and  arerelated to the diffusion matrix and to the drag coefficient by

  15. Evaluation of Drag and diffussion Boltzmannapproach Langevinapproach For Collision Process the Ai and Bij can be calculated as following : M -> Ai, Bij M -> 

  16. Charm evolution in a static medium Simulations in which a particle ensemble in a box evolves dynamically Bulk composedonlybygluons in thermalequilibriumat T=400 MeV ->mD=gt=0.83 GeV C and Cinitially are distributed: uniformily in r-space, while in p-space Mcharm=1.3 GeV [M. Cacciari, P. Nason and R. Vogt, PRL95 (2005) 122001]

  17. Charm evolution in a static medium Simulations in which a particle ensemble in a box evolves dynamically Bulk composedonlybygluons in thermalequilibriumat T=400 MeV mD=gt=0.83 GeV C and Cinitially are distributed: uniformily in r-space, while in p-space MCharm=1.3 GeV fm [M. Cacciari, P. Nason and R. Vogt, PRL95 (2005) 122001] Due to collisions charm approaches to thermal equilibrium with the bulk

  18. Bottom evolution in a static medium MBottom=4.2 GeV

  19. 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, V. Greco arXiV:1312.6857]

  20. Boltzmann vsLangevin (Charm) Mometum transfer vs P

  21. Boltzmann vsLangevin (Charm) • simulatingdifferentaveragemomentum transfer Angulardependenceof Mometum transfer vs P Decreasing mD makes the  more anisotropic Smaller average momentum transfer

  22. Boltzmann vsLangevin (Charm) The smaller <k> the betterLangevinapproximationworks

  23. Boltzmann vsLangevin (Bottom) In bottom case LangevinapproximationgivesresultssimilartoBoltzmann The Larger M the BetterLangevinapproximationworks

  24. Momentum evolution starting from a  (Charm) Langevin Boltzmann • Clearlyappears the shiftof the average • momentumwith t due to the drag force • Boltzmannapproach can throwparticle • at low p insteadLangevin can not • The gaussian nature ofdiffusion • forcereflectitself in the gaussian • formofp-distribution • A part ofdynamicevolution • involvinglargemomentum transfer • isdiscardedwithLangevinapproach

  25. Momentum evolution starting from a  (Bottom) Langevin Boltzmann T=400 MeV Mc/T≈3 Mb/T≈10 [S. K. Das , F. Scardina, V. Greco arXiV:1312.6857]

  26. Momentum evolution for charm vs Temperature Boltzmann Mc/T>5 Boltzmann Mc/T≈3 T= 400 MeV T= 200 MeV Charm Charm T=400 MeV Mc/T≈3 Mb/T≈10 T=200 MeV Mc/T≈6 T= 400 MeV • At T=200 MeV start to see a symmetric shape also for charm Bottom

  27. 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

  28. Back to Back correlation Back to Back correlation The larger spread ofmomentumwith the Boltzmannimplicates a large spread in the angulardistributionsof the Away side charm

  29. RAA at RHIC centrality 20-30 % Boltzmann : Describes the evolutionof the bulk aswellas the propagationof HQ Langevin:Describes the propagationof HQ in a background whichevolutionisdescribedby the Boltzmannequation The Langevin approach indicates a smaller RAA thus a larger suppression. One can getverysimilar RAAforboth the approaches just reducing the diffusioncoefficent

  30. RAA at RHIC for different <k> The Langevin approach indicates a smaller RAA thus a larger suppression. One can getverysimilar RAAforboth the approaches just reducing the diffusioncoefficent The smalleraveregetransferedmomentum the betterLangevinworks

  31. v2 at RHIC centrality 20-30 % Alsofor v2 the smalleraveregetransferedmomentum the betterLangevinworks Boltzmann is more efficient in producing v2 for fixed RAA

  32. RAA LHC centrality 30-50% One can getverysimilar RAAforboth the approaches just reducing the diffusioncoefficent The smalleraveregetransferedmomentum the betterLangevinworks

  33. V2 at LHC centrality 30-50% Alsofor v2 the smalleraveregetransferedmomentum the betterLangevinworks Boltzmann is more efficient in producing v2 for fixed RAA

  34. Conclusions and perspective • The Langevin with drag and diffusion evaluated within pQCD overestimates the • interaction for charm , especially for large value of the average • transferred momentum. However reducing the drag and diffusion we can get • the same RAA . Instead for the Bottom case LV and BM gives similar results • Looking at more differential observable like v2 or angular correlation • Boltzmann and Langevin give results quite different • Boltzmann seems more efficient in producing v2 for the same fixed RAA • Boltzmann implicates a much larger angular spread in the angular distribution • of the Away side charm • 2->3 collisions • Hadronization mechanism (coalescence and fragmentation) The 30th Winter Workshop on Nuclear Dynamics

  35. 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

  36. 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

  37. 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)

  38. Langevin Equation Langevin process defined like this is equivalent to the Fokker-Planck equation: the covariance matrix and  arerelated to the diffusion matrix and to the drag coefficent by For a process in which B0=B1=D

  39. Charm propagation with the langevineq We solve LangevinEquation in a box in the identical environment of the B-E Bulk composed only by gluon in Thermal equilibrium at T= 400 MeV. The long-time solution of the Fokker Planck equation does not reproduces the equilibrium distribution (we are away from thermalizationaround 35-40 % at intermediate pt ). This is however a well-know issue related to the Fokker Planck

  40. Charm propagation with the langevineq The long time solution is recovered relating the Drag and Diffusion coefficent by mean of the fluctaution dissipation relation • Imposing the simplerelativisticdissipation-fluctuation relations D=Constant A= D/ET from FDT

  41. Charm propagation with the langevineq • Imposing the full relativisticdissipation-fluctuation relations D(E)

  42. Boltzmann vsLangevin (Bottom)

  43. mean momentum evolution in a static medium Weconsiderasinitialdistribution in p-space a (p-1.1GeV) forboth C and B withpx=(1/3)p Eachcomponentofaveragemomentumaevolvesaccordingto <pi>=p0iexp(-t) where 1/  is the relaxationtimetoequilibrium () b/c=2.55mb/mc

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