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)

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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

><=

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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

6

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)

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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

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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

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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

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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

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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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Formation Time of Resonances

LHC

• Spectral function: medium broadening

RHIC

Ivan Vitev

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Triggered Measurements

• Additional handle on formation times

Haglin, K. (2004)

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

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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

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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