F. Scardina University of Catania INFN-LNS

Heavy Flavor in Medium Momentum Evolution: Langevin vs

Boltzmann

V. GrecoS. K. DasS. PlumariV. MinissaleJ. Bellone

Heavy Quark Physics in Heavy-Ion collisions: Experiments, Phenomenology and TheoryTrento 16-20/03/2015

Outline

Fokker Planck approach

Boltzmann approach

Comparison in a static medium

Comparison in realistic HIC (RAA and

V2)

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

fpBp

fpApt

fij

ji

i

The Fokker-Planck equation

The interaction is encoded in the drag and diffusion coefficents

MHQ >> QDC (Mcharm1.3 GeV; Mbottom 4.2 GeV)

MHQ >> T

MHQ >> gT≈K (momentum trasferred)

pfk,pkpfk,kpkdC 322

fpp

kkfp

kpfkpkpfkkpji

ji

2

2

1.),(,

Fokker Planck equation

If k<< P

22Ct,p,xfxEP

t

22Ct,pft

The Fokker Planck eq can be derived from the Bolzmann-Equation

kq,kpq,prelv)q(f)(

qdg)k,p(

3

3

2

w(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

fpBp

fpApt

fij

ji

i

Where Bij can be divided in a longitudinal and in a transverse component B||, BT

Langevin Equation

kjkjj

j

dpptCdtdtpdp

dtE

pdxj

),(

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

G is the deterministic friction (drag) force

Cjk is a stochastic force in terms of independent

Gaussian-normal distributed random variable ρ=(ρx,ρy,ρz)

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

l

ijlkji p

CCpA

ii

cpqqpi

ppppqfqpqp

ME

pd

E

qd

E

qd

EA

)(2

1

2222222

1

44

2

3

3

3

3

3

3

Drag and diffusion coefficients from pQCDThe infrared singularity is regularized introducing a Debye-screaning-mass mD

Dmtt

11

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

p

D

pET

DA

1

We have studied the impact on RAA and v2 of evaluating the drag and the diffusion from pQCD or using the different form of the FDT

1) A and D (from pQCD )

2) D (pQCD) ;A (FDT)

3) A (pQCD) ; D (FDT) AETD

BT and B|| =D We assume

Impact on RAA and v2 of the different form of FDT

Au+Au (200 GeV) b=8.0

If B||=BT =D we can get the same RAA of the pure pQCD case reducing the drag and the diffusion by 20%

1) A and D (pQCD)

2) D (pQCD) ;A (FDT)

3) A (pQCD) ; D (FDT)

mD=gT

BBp

n

p

B

pp

E

TB

pA ||2

||||

1111 4) BT and B|| (pQCD) ; A FDT

5) B||=D (pQCD) ; A (pQCD)

ET

DA

The long time solution is recovered relating the Drag and Diffusion coefficent by means of the fluctuation dissipation relation

Different form of FDT

p

D

pET

DA

1

We have studied the impact on RAA and v2 of evaluating the drag and the diffusion from pQCD or using the different form of the FDT 1) A and D (pQCD)

2) D (pQCD) ;A (FDT)

3) A pQCD ; D (FDT) AETD

Impact on RAA and v2 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 pT dependence of RAA and v2 is quite different

4) BT and B|| (pQCD) ; A FDT

5) B||=D (pQCD) ; A (pQCD)

mD=gT

Free-streaming

22Cp,XfXMXM)p,x(fp p

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)]

x

tv

NN

NP kq,kpq,prel

gHQ

coll322

Transport theory

Collision integral (stochastic algorithm)

Evaluation of Drag and diffussion

ii

cpqqpi

pppp)q(fqpqp

ME

pd

E

qd

E

qd

EA

44

2

3

3

3

3

3

3

2

1

2222222

1

jiij ppppB )(2

1

Boltzmann approach

M -> M -> Ai, Bij

Langevin approach

20

2222

16

1

sMsc

gcgc MdtMs

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 momentum evolves according to<pi>=p0

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

b/c=2.55mb/mcFor a very inclusive quantity BM and LV give same result

T=400 MeV

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, S. Plumari andV. Greco PRC90 044901 (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 PRC90 044901 (2014)]

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

Dmtt

11

[S. K. Das , F. Scardina, V. Greco PRC90 044901 (2014)]

Boltzmann vs Langevin (Charm)

The smaller <k> the better Langevin approximation works

mD=0.4 GeV mD=0.83 GeV (gT)

mD=1.6 GeV

RAA and v2 Boltzmann vs Langevin (RHIC)

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)

Au+Au@200AGeV, b=8 fm

mD=1.6 GeV

40%

[F. ScardinaJ.Phys.Conf.Ser. 535 (2014) 012019]

RAA and v2 Boltzmann vs Langevin (RHIC) Au+Au@200AGeV, b=8 fm

mD=gT

Fixed same RAA(pT) [one has to multiply A and D by 0.65]

Energy loss of a single HQLangevin Boltzmann

T=400 MeV Mc/T≈3 Mb/T≈10

Charm

Bottom

T=400 MeV

Mc/T≈3

Mb/T≈10

[F. ScardinaJ.Phys.Conf.Ser. 535 (2014) 012019]

Back to Back correlation

Back to back correlation observable could be sensitive to such a detail

Langevin Boltzmann

Initial d(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)

P0=10 GeV

Boltzmann

Langevin vs Boltzmann angular correlation (static medium)

Striking difference also at Df =0:

- The evolution of the yield from 2 to 5 GeV is about 50 times different

P0=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

P0=10 GeV

LV vs BM c-c angular correlation in HIC

RHIC Au+Au 200 AGeV b=8.0

mD=0.4 GeV mD=1.6 GeV

For the two cases we fix the same RAA

pT cut [0,2] GeV[2,4][4,6][6,10]

LV

BM

RHIC Au+Au 200 AGeV b=8

mD=0.4 GeV mD=1.6 GeV

For the two cases we fix the same RAA

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] there is one

order of magnitude of difference in the yield for the mD=1.6 case

pT 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=8.0mD=1.6

pT 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-c b-b

mD=1.6 GeV

RHIC 200 AGeV vs LHC 2.76 AGeVWe found that the broadening of dN/df is created mainly at initial times

Mc/Tinit≈3.5

mD=0.4 GeVRHIC

Mc/Tinit≈2.5

LHC

pT cut [0,2] GeV[2,4][4,6][6,10]

LV

BM

LHC Pb+Pb 2.76 TeV (charm) b=9.0

mD=0.4 GeV

For mD=1.6 we observe the partonic wind effect (enhancement of the azimuthal correlations in the region of Df=0) with the BM and not with the LVBecause of the radial flow the cc pair are pushed in the same direction toward smaller opening angles

mD=1.6 GeV

LV vs BM c-c angular correlation in HIC (LHC 2.76 ATeV)

[ X. Zhu et al PRL 100, 152301 (2008)]

pT 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 mD=1.6 (more isotropic cross section) not for mD=0.4

[M. Nahrgang et al PRC 90, 024907 (2014)]

mD=1.6 GeV

pT cut [0,2] GeV[2,4][4,6][6,10]

mD=1.6 mD=0.4

c-c

Angular correlation sensitive to the microscopical dynamics

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 collisional DE) -BM vs LV -The broadening of dN/df 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

kpfk,kp HQ pfk,p HQGain term Loss term

kq,kpq,prelv)q(f)(

qdg)k,p(

3

3

2

w(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(fPfq

gpxt

N

3

3

33

3

2

2

Transport theory Collision integral (stochastic

algorithm)

collision rate per unit phase space for this pair

kq,kpq,prel

gHQcoll vqx

N

Px

Nqg

Pxt

N

33

3

33

3

3

3

33

3 22

2

2

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

x

tv

NN

NP kq,kpq,prel

gHQ

coll322

Δ3x

Langevin Equation

kjkjj

j

dpptCdtdtpdp

dtE

pdxj

),(

)( ,, 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

Boltzmann vs Langevin (Bottom)