Ivan Vitev
A Light-front Wave-function Approach
to the In-medium Modification of
Heavy-quark Fragmentation Functions
Ivan Vitev, Nuclear Theory, T-2 , LANL
”Heavy Quark Physics in Nucleus-Nucleus Collisions” Workshop
UCLA, Los Angeles, 2009
-mesons, -mesonsD B
Time evolution
c
u
Ivan Vitev
2
Outline of the Talk
Jet tomography of the QGP
• Jet quenching for light hadrons, QGP tomography
• The heavy quark puzzle at RHIC. A space-time picture of hadronization
Collisional dissociation of hadrons in dense QCD matter • Dissociation: new approach to D- and B-mesons suppression in the QGP • Light-front quantization and light-front wave-functions
• Possibilities to calculate parton distribution functions and fragmentation
functions
• Evaluating the medium modification of heavy quark fragmentation
Phenomenological results
• Heavy hadron cross sections and correlations
• Solving the rate equations and relative meson suppression in Cu+Cu
• Results for decay electrons and caveats
Summary and outlook
Talk based upon: R.Sharma, I.Vitev, in preparation A.Adil, I.Vitev, Phys. Lett. B649 (2007)
Ivan Vitev
3
I.V., Phys.Lett.B 639 (2006)
Light Hadron vs Heavy Meson Quenching
• Predictions of this formalism tested vs
particle momentum, C.M. energy, centrality
• Nuclear modification factor
T
NN
T
AA
coll
TAAdpdd
dpdd
NpR
/
/1),(
2
2
><=
Ivan Vitev
4
S. Wicks et al., Nucl.Phys.A (2007)
• Radiative Energy Loss using (D)GLV
(both c + b)
• Radiative + Collisional + Geometry (both c + b) (overestimated)
• Deviation by a factor of two
• Is it accidental or is it symptomatic?
Non-Photonic Electron / Heavy Flavor Quenching
• Single electron measurements (presumably from heavy quarks) may be
problematic 1
1
(1 ) (1 )
2
2 22
2 2
22
2
,g
n n
gm x M
m
xE
k k k
x Mx
p Ek k
+
+
+
=+
+
+
… …
M.Djordjevic, M.Gyulassy, Nucl.Phys.A (2004)
Proceed to A+A collisions
Ivan Vitev
5
STAR Collab., preliminary (2009)
Non-Photonic Electron / Heavy Flavor Quenching
• Is there a mechanism where D suppression = B suppression
arises naturally?
Another way to look at the same problem
“Problem” is that
B is more suppressed than D
“Problem” is that
D is more suppressed than Light
Ivan Vitev
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The Space-Time Picture of Hadronization
In mesoscopic systems one has to account for the space-time evolution
form
=0
E
m=
0
A. Bialas, M.Gyulassy, Nucl.Phys.B 291, (1987)
J.D. Bjorken, Lect.NotesPhys.56, (1987)
• Inside-outside cascade
• Outside-inside cascade
0~ 1 fm
- Correctly accounts for the leading energy and
mass dependence. Lack of control over t0
- Correctly points at the reduction of tform at large
values of x. Specific for Lund string fragmentation.
Mass dependence obscured
p + p
A + AB. Kopeliovich JETP, (1984)
Ivan Vitev
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Conceptually Different Approach to D / B
• Fragmentation and dissociation of hadrons from heavy quarks inside the QGP
• Problem: treated in the same way as light quarks
D B
20 fm 1.5 fm 0.4 fm form ( 10 )
Tp GeV=
Parton
Hadron
p+
zp+
(1 )z p+
~ QCDk
B
D
QGP extent
2 2 2
form
(0.2 . ) 2 (1 )1
(1 ) (1 )
/(1 )
h q
Q
GeV fm z z py
p k z m z z M
y
+
+
+
= =+
= +
2
, ,02
q
q
Mp p
p
+
+=
2 2
, ,2
hh
k mp zp k
zp
+
+
+=
2
(1 ) , ,2(1 )
g
kp z p k
z p
+
+=+
C.Y.Wong, Phys.Rev.C 72, (2005) and others
Ivan Vitev
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Collisional Dissociation of D / B Mesons
• An alternative
-mesons, -mesonsD B
Time evolution
c
u
A. Adil, I. Vitev, Phys. Lett. B649, 139 (2007)
Simultaneous fragmentation and
dissociation call for solving a system
of coupled equations
f Q ( pT,t = 0) =
d
dyd 2 pT
( y, pT
)
f H ( pT,t = 0) = 0
• Initial
conditions
• Example: radioactive decay chain
dNi
dt= i 1Ni 1 iNi
F.Dominguez, C.Marquet, B.Wu (2008)
Ivan Vitev
Light Front Quantization 9
S.Brodsky, H.C.Pauli, S.Pinsky, Phys. Rep. (1998)
• Advantages of light front quantization: simple vacuum, the only state with
p+=0
• Full set of operators, commuting: M 2= 2p+ p p2 , p+ , p
S2 , Sz
Ivan Vitev
QCD on the Light Front 10
Commutation relations and normalization of states
• States:
a (x ) =dp+
2p+
d 2 p
2( )3 aa (p+ )u (p)e ip x
+ b†a (p+ )v (p)e+ ip x( )|x+ =0
Aa (x ) =dp+
2p+
d 2 p
2( )3 da (p+ ) (p)e ip x
+ d†a (p+ ) * (p)e+ ip x( )|x+ =0
a (x ) =dp+
2p+
d 2 p
2( )3 ba (p+ )v (p)e ip x
+ a†a (p+ )u (p)e+ ip x( )|x+ =0
• Quarks
• Anti-quarks
• Gluons
The free theory
a 'a ' (p+ ' ),a†a (p+ ){ } = 2p+ 2( )
3 3 p+ p+ '( ) aa ' '
b 'a ' (p+ ' ),b†a (p+ ){ } = 2p+ 2( )
3 3 p+ p+ '( ) aa ' '
b 'a ' (p+ ' ),b†a (p+ ) = 2p+ 2( )
3 3 p+ p+ '( ) aa ' '
n, p+
n{ }, n{ } an{ } = ... a ,i†a
i, j ,k n
(p+
i )...b , j†a (p+
j )...d ,k†a (p+
k ) 0
• Implicit: quark flavor, (anti)symmetrization
• Normalization trivially obtained from above
Ivan Vitev
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Light Front Wave Functions
Composite hadron creation operator:
The normalization then becomes
• Expansion in Fock components
P+ , P ,S 2 ,Sz=
n=2,3 i=1
n dxi
2xi
d 2ki
2( )3 n
x{ }i, k
i{ },i{ } a
i{ }( ) xi
i=1
n
1 ki
i=1
n
...ai
† a (xiP+
+ ki)...
i, j ,k n
bj
† a (xjP+
+ kj)... d
k
† a (xkP+
+ kk)... 0
P
+ ', P
',S
2 '',S
z
'P
+, P ,S
2,S
z= 2P
+2( )
33
P+
P+ '( ) s
zs
z'
1=1
2 2( )3
n=2,3 i=1
n dxi
2xi
d2k
i
2( )3 n
x{ }i, k
i{ },i{ } a
i{ }( )2
xi
i=1
n
1 ki
i=1
n
aH† sz (P+ )
Baryon
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From Low to High Fock Components
• Perturbative generation of the
higher Fock states
At the QCD vertexes: conserve color, momentum, flavor, …
2
22( )
2
s
a bca
ddzP
dzdP =
• The lowest lying Fock state (non-perturbative) – the most
important Correct quantum #s carry over to higher states
Ivan Vitev
13Calculating the Meson Wave Function
• Relativistic Dirac equation
M. Avila, Phys. Rev. D49 (1994)
V =1
r, =
4
3 s S = br
dG
dr= V '+ S '+ m( )F
k +1
r
b
2MG
dF
dr=
k 1
r
b
2MF + V ' S ' m( )G
Reduces to:
• Radial density: (r) ~ (F2+G2 )
3S1
Reduces to:
1S0
=l +1, j = l +1 / 2
l, j = l 1 / 2
Coulomb Linear
( k ,x)2
Expk 2
+ 4mQ
2 (1 x) + 4mq
2 (x)
4 2x(1 x)
D0 ,D0 ,D_D+ ,Ds ... The*, .... Same for B
S ' = S3
2
VS
MQ
m
MQ
V
V ' = S1
2
S2
MQ
m
MQ
S
Boost with large P+ - end up at
the same longitudinal rapidity
Ivan Vitev
Parton Distribution Functions 14
q /P (x) =dy
2e ixP+ y P a (y ,0)
+
2a (0,0) P
q /P (x) =dy
2e ixP+ y Tr P
+
2a (y ,0) a (0,0) P
Factorization
• Light cone gauge A+=0 , 0<x<1
• We have a technique of calculating the PDFs
in any hadron
R. Sharma, I. Vitev, in progress
q / H(x) =
1
2 2( )3
n=2,3 i=1
n dxi
2xi
d 2ki
2( )3 n
x{ }i, k
i{ },i{ } a
i{ }( )2
xi
i=1
n
1 ki
i=1
n
xq
x( )
q / H(x) =
1
2 2( )3
n=2,3 i=1
n dxi
2xi
d 2ki
2( )3 n
x{ }i, k
i{ },i{ } a
i{ }( )2
xi
i=1
n
1 ki
i=1
n
xq
x( )
P H
J. Collins, D. Soper, Nucl.Phys.B194 (1982)
Ivan Vitev
Fragmentation Functions 15
DH /q (z) = zdy
2eiP
+ /z y 1
3Trcolor
1
2TrDirac
+
20 aa (y ,0)aH
† (P+ )aH (P+ ) a (0,0) 0
Factorization
• Light cone gauge A+=0 , 0<z<1
• Kinematically FFs at tree level do not exist
except for exclusive processes
DH / q
(z)1
20
lnlnQ2 / 2
ln mQ
/2 2
1
2 2( )3
dy0
1
Pqq
( y)n=2,3 i=1
n dxi
2xi
d 2ki
2( )3
nx{ }
i, k
i{ },i{ } a
i{ }( )2
xi
i=1
n
1 ki
i=1
n
xq
y / z( )
DH /q (z) = zdy
2eiP
+ /z y 1
3Trcolor
1
2TrDirac 0 aa (y ,0)
+
2aH† (P+ )aH (P
+ ) a (0,0) 0
pQ+
pH+ pQ
+
pQ+ ' pH
+
J. Collins, D. Soper, Nucl.Phys.B194 (1982)
y
1 y
DH / q
(z)1
20
lnlnQ2 / 2
ln mQ
/2 2
1
2 2( )3
dy0
1
Pqq
( y)n=2,3 i=1
n dxi
2xi
d 2ki
2( )3
nx{ }
i, k
i{ },i{ } a
i{ }( )2
xi
i=1
n
1 ki
i=1
n
xq
y / z( )
Remarkable connection between PDFs and FFs
Ivan Vitev
Modification Fragmentation Functions 16
DH /q (z) = zdy
2eiP
+ /z y 1
3Trcolor
1
2TrDirac
+
20 aa (y ,0)aH
† (P+ )aH (P+ ) a (0,0) 0
Start from the definition
G.Nayak (2008)
1. Fragmentation of the partons, just from the QGP
phase space density
2. Thermal modification
DH /q (z) = zdy
2eiP
+ /z y 1
3Trcolor
1
2TrDirac
+
20 aa (y ,0)aH
† (P+ )aH (P+ ) a (0,0) 0
New solution for the wave function. As a
function of time
3. Pure coalescence from the QGP Need to work out the Fiertz decomposition
factors
4. Corrections to the hard fragmentation Need to work also the pQCD vs thermal
rates
Ivan Vitev
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Medium Dissociation of Heavy Mesons
• Heavy meson acoplanarity:
K2
= 2 2μ2L
q
Initial distribution:
Resum using GLV the multiple scattering in
impact parameter (B,b) space
• Broadening (separation) the q q-bar pair:
2 2
0
12 2 2 2 ( )
( )
L
q q
Ll dl
lμ μ
f ( k , x)2=
eK 2
4 μ2
4 μ2Norm2 x(1 x) 2
μ2+ x(1 x) 2
ek2
4( μ2 + x(1 x ) 2 )em12 (1 x )+m2
2 x
x(1 x ) 2
i ( k , x)2=
2 (K ) Norm2ek2
4 x(1 x ) 2
em12 (1 x )+m2
2 x
x(1 x ) 2
f ( k , x) = a M ( k , x) + (1 a)qq dissociated
( k , x)
?
K
k
Ivan Vitev
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Heavy Quark Production and Correlations
D
, ,k
D
D
• Possibility for novel studies of heavy
quark-triggered (D and B) jets: hadron
composition of associated yields
• Fast convergence of the perturbative
series
Ivan Vitev
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Heavy Meson Dissociation at RHIC and LHC
Coupled rate equations
• The asymptotic solution in the QGP -
sensitive to t0~0.6 fm and expansion dynamics
• Features of energy loss
• B-mesons as suppressed as D-mesons at pT~ 10 GeV at the LHC
1
/2
0
1
/2
0
( , ) ( , )
( / , )
( / , )
( , )
( / , )
( , )
( / , )
1
1 1 + ( )
1
1 1
(
+ ( )
, ) ( , )
form T
diss T
diss T
for
t t
Q H
t t
m
H
H H
Q
T T
H
Q
T T
T
Q
T
T
Q
f
f p
p t
p x t
p t
p z
dx xx
dz
f p
t f
t f p t
p x t
f p z tzt
z
t
D
p
=
=
( )1, 1x z< <
Unique feature
A. Adil, I. Vitev, Phys. Lett. B649, 139 (2007)
Ivan Vitev
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Quenching of Non-Photonic Electrons
• B-mesons are included. They give a
major contribution to (e++e-)
• Similar to light , however, different
physics mechanism
2
2
coll
/( )
/
e
AA T
e
pp T
e
AA T
d dyd p
d d dR
yp
N p
±
±
±
=
• Full semi-leptonic decays of C- and B-
mesons and baryons included. PDG branching fractions and kinematics.
PYTHIA event generator
0
Note on applicability
(e++e-) to 25 GeV
D-, B-mesons to RAA (D) = RAA (B)
Ivan Vitev
Electron Suppression in Cu+Cu 21
• Of more recent relevance are the
results in Cu+Cu. Calculated for several centralities, both mesons and
electrons
J.Bielchick (2008)
• Main caveat: cylindrical geometry. While this is not very important
for radiative e-loss it is more important for dissociation (short distance). Expect stronger centrality dependence.
Ivan Vitev
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Conclusions
Heavy Quarks / Hadrons
• Time dependence of fragmentation/hadronization in the spotlight
• The heavy quark puzzle at RHIC may require different solution than the
interaction strength
Collisional dissociation of hadrons in dense QCD matter • Begin to understand from QCD the FFs and PDFs beyond global
analysis. Shifts the problem to the wave-function. HQ tractable
• Derived the theoretical results. Identified the sources of medium
modification of FFs
Phenomenological results
• Gave results in Cu+Cu directly comparable to previous Au+Au
calculations. Both mesons and electrons
• Found comparable suppression of light and heavy
To do list
• Carry the numerical implementation of the modification of the FFs
• Update calculations in Cu+Cu, estimate geometry effects
• Compare fragmentation/coalescence contributions calculated in this
approach
Ivan Vitev
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Effects of Partial Chiral Symmetry Restoration
Kaon
• Scale of chiral symmetry restoration
Phi meson
L. Holt, K.Haglin, J. Phys. G31, S245 (2005)
m 4 f
Mass shifts
Width broadening
Manifestation for baryons
- Includes approximately strange quarks
SU(N )L SU(N )R SU(N )L+R L = i N /D N + 0 N N +Lg
Ivan Vitev
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Motivation / Estimates
• Mass of heavy resonances: Falls in the right region to ensure early
formation
Haglin, K. (2004)
coll = 2RA / o = 1 / pT
Ivan Vitev
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In-medium Lifetime
Phi meson
• Spectral function: medium broadening
Haglin, K. (2004)
Phi meson
Ivan Vitev
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Cross Sections / Fragmentation Distributions
Single inclusive
• Spectral function: medium broadening
Vitev, I. et al (2006)
Double inclusive
Ivan Vitev
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I.V., M.Gyulassy, Phys.Rev.Lett. 89 (2002)
F.Karsch, Nucl.Phys.A698 (2002)
SPS RHIC LHC
0 0
' / ( ') ' ( ')
0 0
( ')
( )
r r
abs absdr r dr r r
I r I e I e= =
Determining the properties of the QGP: ,T
Jet Tomography
Ivan Vitev
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Conclusions
The physics of jet (inclusive particle) quenching• The stopping power of QCD matter drives hard physics
in p+A and A+A collisions
• Theoretical advances have been made in understanding coherent non-
Abelian bremsstrahlung
• Rich phenomenology of jet quenching has been developed
• Jet tomography stringly suggests deconfinement in HIC
Collisional dissociation of heavy mesons in the QGP
• Time scales are important in mesoscopic systems
• Derived the dissociation rate of heavy mesons in the QGP
• Provided possible solution to the heavy quark puzzle
• Successful description of non-photonic e-suppression from RHIC
New opportunities for heavy ion physics will emerge at the LHC
• Bridging the gap between high energy and nuclear physics
• First real measurement of the QGP-induced modification of jet properties
• Rich phenomenology of topological jet observables
• Significantly larger discriminating power for theoretical models
Ivan Vitev
31
S. Wicks et al., (2005)
• Diffusion coefficient D and eventually
• Existence of heavy heavy
resonances near Tc in the QGP
Non-Photonic Electron / Heavy Flavor Quenching
( , )( , )( ,( , ) )ii
i i
jiA pf p t
p f p tB pt p p
t t= +
Langevin simulation of heavy quark
diffusion
N. Armesto et al., (2006)
H. van Hees, R. Rapp, (2005) G. Moore, D.Teaney (2005)
Radiative and collisional energy loss
/ s• Ratio:
• Opacity of the QGP
Ecoll . / Erad .
L / g
Ivan Vitev
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The Path Forward
• An interesting idea valid physics explanation
A.Adil, I.Vitev, Phys. Lett. B (2006) W.Horowitz, M. Gyulassy, (2007)
• To understand heavy flavor modification in the QGP we need direct and
separate measurements of D- and B-mesons, excellent statistics
Measurable at RHIC Measurable at the LHC
RAAc (pT )
RAAb (pT )
= 1
Meson dissociation
String theoryAdS/CFT
PQCD,Transport
PT [GeV]10-15 50-100 Never
Ivan Vitev
33
Heavy Flavor Elliptic Flow and Suppression
Understand the structure of mesons
light cone wave functionsc
A. Adil, I. Vitev, Phys.Lett.B (2007)
Sensitive to the opacity of the QGP
and its formation time 0
D. Molnar (2004)
Test coalescence model fits to the v2 of
light hadrons via heavy flavor