F. Scardina University of Catania INFN-LNS

Heavy Flavor in Medium Momentum Evolution: Langevin vs

Boltzmann

V. GrecoS. K. DasS. Plumari

The 30th Winter Workshop on Nuclear Dynamics

Outline Heavy Flavors

Transport approach

Fokker Planck approach

Comparison in a static medium

Comparison in HIC

Conclusions and future developmentsThe 30th Winter Workshop on Nuclear Dynamics

Introduction Heavy Quarks MHQ >> QDC (Mcharm1.3 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

c

K+

lepton–D0

c

D0

π+

K–

RAA of Heavy QuarksNuclear Modification factor:

dydp/Nddydp/Nd

N)p(R

Tpp

TAA

collTAA 2

21

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

RAA at LHC

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

Simultaneous description of RAA and v2 is a tough challenge for all models

RAA and Elliptic flow

Description of HQ propagation in the QGPBrownian

Motion

Described by the Fokker Planck equation

Description of HQ propagation in the QGPBrownian

Motion

Described by the Fokker Planck equationFokker Plank[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. Alberico et al. Eur. Phys. J. C,71,1666(2011)][T. Lang, H. van Hees, J. Steinheimer and 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]

Free-streaming

22Cp,XfXMXM)p,x(fp p

Mean Field Collisions

22CfFfvtf

pr

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)]

pfk,pkpfk,kpkdC HQHQ 322

xtv

NNNP kq,kpq,prel

gHQ

coll322

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)]

kq,kpq,prelv)q(f)(qdg)k,p(

3

3

2(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 -> gcDominant contribution

The infrared singularity is regularized introducing a Debye-screaning-mass mD

Dmtt

11Tm sD 4

[B. L. Combridge, Nucl. Phys. B151, 429 (1979)] [B. Svetitsky, Phys. Rev. D 37, 2484 (1988) ]

pfk,pkpfk,kpkdC 322

fpp

kkfp

kpfkpkpfkkpji

ji

2

21.),(,

Fokker Planck equation

If k<< P

B-E 22Ct,p,xfxEP

t

22Ct,pft

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

fpp

kkfp

kpfkpkpfkkpji

ji

2

21.),(,

fpBpfpAptf

ijj

ii

i3

i k)k(pkdA ,

ji3

ij kk)k(pkdB ,

where we have defined the kernels → Drag Coefficient

→ Diffusion Coefficient

pf)k,p(pp

kkfp

kkdCji

jii

i

2

322 2

1

[B. Svetitsky PRD 37(1987)2484]

Where Bij can be divided in a longitudinal and in a transverse component B0 , B1

Langevin Equation

kjkjj

j

dpptCdtdtpdp

dtEp

dxj

),(

jkki tttt )()()(

The Fokker-Planck equation is equivalent to an ordinary stochastic differential equation is the deterministic friction (drag) force

Cij is a stochastic force in terms of independent Gaussian-normal distributed random variable ρ=(ρx,ρy,ρz)

0 )t(i

)exp()(P22

1 23

the covariance matrix and are related to the diffusion matrix and to the drag coefficient by

l

ijlkji pC

CpA

ρ obey the relations:

For Collision Process the Ai and Bij can be calculated as following :

Evaluation of Drag and diffussion

ii

cpqqpi

pppp)q(fqpqp

MEpd

Eqd

Eqd

EA

44

2

3

3

3

3

3

3

2

12222222

1

jiij ppppB )(21

Boltzmann approach

M -> M -> Ai, Bij

Langevin approach

20

2222

161

sMs

c

gcgc MdtMs

Charm evolution in a static mediumSimulations in which a particle ensemble in a box evolves dynamicallyBulk composed only by gluons in thermal equilibrium at T=400 MeV ->mD=gt=0.83 GeV

C and C initially are distributed: uniformily in r-space, while in p-space

Mcharm=1.3 GeV

[M. Cacciari, P. Nason and R. Vogt, PRL95 (2005) 122001]

Charm evolution in a static mediumSimulations in which a particle ensemble in a box evolves dynamicallyBulk composed only by gluons in thermal equilibrium at T=400 MeV

Due to collisions charm approaches to thermal equilibrium with the bulk

fm

mD=gt=0.83 GeVC and C initially are

distributed: uniformily in r-space, while in p-space[M. Cacciari, P. Nason and R. Vogt, PRL95 (2005) 122001]

MCharm=1.3 GeV

Bottom evolution in a static medium

MBottom=4.2 GeV

Boltzmann vs Langevin (Charm)BoltzmannLangevin

pddN

pddN

33

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]

Boltzmann vs Langevin (Charm)

Mometum transfer vs P

Boltzmann vs Langevin (Charm)• simulating different average momentum transfer

Decreasing mD makes the more anisotropic

Smaller average momentum transfer

Angular dependence of Mometum transfer vs P

Boltzmann vs Langevin (Charm)

The smaller <k> 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)

Langevin Boltzmann GeVp

pddN

initial

103

• 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)

GeVppd

dN

initial

103

Langevin Boltzmann

T=400 MeV Mc/T≈3 Mb/T≈10

[S. K. Das , F. Scardina, V. Greco arXiV:1312.6857]

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≈10T=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 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

RAA at RHIC centrality 20-30 %

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 RAA thus a larger suppression.One can get very similar RAA for both the approaches just reducing the diffusion coefficent

RAA at RHIC for different <k>

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

v2 at RHIC centrality 20-30 %

Boltzmann is more efficient in producing v2 for fixed RAA

Also for v2 the smaller averege transfered momentum the better Langevin works

RAA LHC centrality 30-50%

One can get very similar RAA for both the approaches just reducing the diffusion coefficent The smaller averege transfered momentum the better Langevin works

V2 at LHC centrality 30-50%

Boltzmann is more efficient in producing v2 for fixed RAA

Also for v2 the smaller averege transfered momentum the better Langevin works

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

Transport theory Collision integral

HQ with momentum

p+k

HQ with momentum

p

HQ with momentum

p-k

kpfk,kp HQ pfk,p HQGain term Loss term

kq,kpq,prelv)q(f)(qdg)k,p(

3

3

2(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

pfk,pkpfk,kpkdC HQHQ 322

0t03 x

Exact solution

kq,kpq,prelgHQ

coll v)q(fPfqgpxt

N

3

3

33

3

22

Transport theory Collision integral (stochastic

algorithm)

collision rate per unit phase space for this pair

kq,kpq,prel

gHQcoll vqxN

PxNqg

PxtN

33

3

33

3

3

3

33

3 222

2

Assuming two particle• In a volume 3x in space• momenta in the range (P,P+3P) ; (q,q+3q)

xtv

NNNP kq,kpq,prel

gHQ

coll322

Δ3x

Langevin Equation

kjkjj

j

dpptCdtdtpdp

dtEp

dxj

),(

)( ,, zyx )exp()(P22

1 23

jkki tttt )()()(

The Fokker-Planck equation is equivalent to an ordinary stochastic differential equation

is the deterministic friction (drag) force Cij 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.

0 )t(i

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

l

ijlkji pC

CpA

Langevin Equation

For a process in which B0=B1=D

The long-time solution of the Fokker Planck equation does not reproduces the equilibrium distribution (we are away from thermalization around 35-40 % 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 mediumWe consider as initial distribution in p-space a (p-1.1GeV) for both C and B with px=(1/3)p

Each component of average momentuma evolves according to<pi>=p0

iexp(-t) where 1/ is the relaxation time to equilibrium ()

b/c=2.55mb/mc