Post on 22-Feb-2016
description
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