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

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

Introduction Heavy Quarks M HQ >>  QDC (M charm  1.3 GeV; M bottom  4.2 GeV) M HQ >> T M HQ >> T They are produced in the early stage They interact with the bulk and can be used They interact with the bulk and can be used to study the properties of the medium to study the properties of the medium There are two way to detect charm: There are two way to detect charm: 1) Single non-photonic electron 1) Single non-photonic electron 2) D and B mesons 2) D and B mesons c K+K+ lepton – D0D0 c D0D0 π+π+ K–K–

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

R AA at LHC Again at LHC energy heavy flavor suppression is similar to light flavor especially at high p T

Simultaneous description of R AA and v 2 is a tough challenge for all models R AA and Elliptic flow

Description of HQ propagation in the QGP Brownian Motion Described by the Fokker Planck equation

Description of HQ propagation in the QGP Brownian Motion Described by the Fokker Planck equation Fokker Plank [H. van Hees, V. Greco and R. Rapp, Phys. Rev. C73, (2006) ] [Min He, Rainer J. Fries, and Ralf Rap PRC ] [G. D. Moore, D Teaney, Phys. Rev. C 71, (2005)] [S. Cao, S. A. Bass Phys. Rev. C 84, (2011) ] [Majumda, T. Bhattacharyya, J. Alam and S. K. Das, Phys. Rev. C,84, (2012)] [Y. Akamatsu, T. Hatsuda and T. Hirano, Phys. Rev.C 79, (2009)] [W. M. Alberico et al. Eur. Phys. J. C,71,1666(2011)] [T. Lang, H. van Hees, J. Steinheimer and M. Bleicher arXiv: [hep-ph]] [C. Young, B. Schenke, S. Jeon and C. Gale PRC 86, (2012)] Boltzmann [J. Uphoff, O. Fochler, Z. Xu and C. Greiner PRC ; PLB 717] [S. K. Das, F. Scardina, V. Greco arXiV: ] [B. Zhang, L. W. Chen and C. M. Ko, Phys. Rev. C 72 (2005) ] [P. B. Gossiaux, J. B. Aichelin PRC ] [D. Molnar, Eur. Phys. J. C 49(2007) 181]

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

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

Cross Section gc -> gc Dominant contribution The infrared singularity is regularized introducing a Debye-screaning-mass m D [B. L. Combridge, Nucl. Phys. B151, 429 (1979)] [B. Svetitsky, Phys. Rev. D 37, 2484 (1988) ]

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

Fokker Planck equation where we have defined the kernels → Drag Coefficient → Diffusion Coefficient [B. Svetitsky PRD 37(1987)2484] Where B ij can be divided in a longitudinal and in a transverse component B 0, B 1

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 ρ=(ρ x,ρ y,ρ z ) the covariance matrix and  are related to the diffusion matrix and to the drag coefficient by ρ obey the relations:

For Collision Process the A i and B ij can be calculated as following : Evaluation of Drag and diffussion Boltzmann approach M ->  M -> A i, B ij Langevin approach

Charm evolution in a static medium Simulations in which a particle ensemble in a box evolves dynamically Bulk composed only by gluons in thermal equilibrium at T=400 MeV ->m D =gt=0.83 GeV C and C initially are distributed: uniformily in r-space, while in p-space M charm =1.3 GeV [M. Cacciari, P. Nason and R. Vogt, PRL95 (2005) ]

Charm evolution in a static medium Simulations in which a particle ensemble in a box evolves dynamically Bulk composed only by gluons in thermal equilibrium at T=400 MeV Due to collisions charm approaches to thermal equilibrium with the bulk fm m D =gt=0.83 GeV C and C initially are distributed: uniformily in r-space, while in p-space [M. Cacciari, P. Nason and R. Vogt, PRL95 (2005) ] M Charm =1.3 GeV

Bottom evolution in a static medium M Bottom =4.2 GeV

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, V. Greco arXiV: ]

Boltzmann vs Langevin (Charm) Mometum transfer vs P

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

Boltzmann vs Langevin (Charm) The smaller the better Langevin approximation works

Boltzmann vs Langevin (Bottom) In bottom case Langevin approximation gives results similar to Boltzmann The Larger M the Better Langevin approximation works

Momentum evolution starting from a  (Charm) LangevinBoltzmann Clearly appears the shift of the average momentum with t due to the drag force The gaussian nature of diffusion force reflect itself in the gaussian form of p-distribution Boltzmann approach can throw particle at low p instead Langevin can not A part of dynamic evolution involving large momentum transfer is discarded with Langevin approach

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

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

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

Back to Back correlation The larger spread of momentum with the Boltzmann implicates a large spread in the angular distributions of the Away side charm

R AA at RHIC centrality % Langevin: Describes the propagation of HQ in a background which evolution is described by the Boltzmann equation Boltzmann : Describes the evolution of the bulk as well as the propagation of HQ The Langevin approach indicates a smaller R AA thus a larger suppression. One can get very similar R AA for both the approaches just reducing the diffusion coefficent

R AA at RHIC for different R AA at RHIC for different The Langevin approach indicates a smaller R AA thus a larger suppression. One can get very similar R AA for both the approaches just reducing the diffusion coefficent The smaller averege transfered momentum the better Langevin works

v 2 at RHIC centrality % Boltzmann is more efficient in producing v 2 for fixed R AA Also for v 2 the smaller averege transfered momentum the better Langevin works

R AA LHC centrality 30-50% One can get very similar R AA for both the approaches just reducing the diffusion coefficent The smaller averege transfered momentum the better Langevin works

V 2 at LHC centrality 30-50% Boltzmann is more efficient in producing v 2 for fixed R AA Also for v 2 the smaller averege transfered momentum the better Langevin works

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

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.

Langevin process defined like this is equivalent to the Fokker-Planck equation: the covariance matrix and  are related to the diffusion matrix and to the drag coefficent by Langevin Equation For a process in which B 0 =B 1 =D

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

D=Constant A= D/ET from FDT The long time solution is recovered relating the Drag and Diffusion coefficent by mean of the fluctaution dissipation relation Imposing the simple relativistic dissipation-fluctuation relations Charm propagation with the langevin eq

D(E) Charm propagation with the langevin eq Imposing the full relativistic dissipation-fluctuation relations

Boltzmann vs Langevin (Bottom)

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 momentuma evolves according to =p 0 i exp(-  t) where 1/  is the relaxation time to equilibrium (  )  b /  c =2.55  m b /m c