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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)]
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
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
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
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