Energy loss of heavy quark produced in a finite size medium (QGP).

Pol-Bernard Gossiaux

SUBATECH (Nantes)

work done with J. Aichelin, A.C. Brandt, S. Peigné & Th. Gousset

hep-ph/0509185hep-ph/0608061

• Taking collisional energy loss for partons of moderate energy could help to solve the single electron puzzle.

• One generally assumes

Figure taken from Mustafa et al (Phys.Rev. C72 (2005)

014905 )

Motivation

with stationnarydx

EdEcoll )(L dx

dE(L)E collcoll

stationnary ?

• Separate short distance (indiv collisions) and large distance (collective answer; polarization)

• For partons produced in medium, proper field needs time ( /mD) to reach its full amplitude and fully polarize the medium

Naive expectation: No full dissipation until tret

/mD. Retardation effect (bad for single-e puzzle)2

Physical situation

Heavy quark Q: Avoid some complications due to collinear divergencesQ

V1

V2Q

Situation under study (exploratory) and hypothesis

"Usual" QGP: no quasi bound state, at rest, static, « confined » over some domain Lplasma

Assume high temperature g<<1 ordering

TggTr

Tr

MD 2

1 1 1 1

QGP: viewed as continuous medium acting collectively on Q ; interaction described by lin. resp. th. ( solve Maxwell's

equations)

Untouched !

Distance between two constituants in QGP

3

) ( 4 )( 22TLTTLL jj

iEk-E

1. Start from Maxwell equations in Fourier space (,k):

With L and T : dielectric functions of the plasma

(Polarization tensor taken in H.T.L approximation)

22 )/(1 )( ; )/(1 )( kk,kkk, TTLL

Energy loss within Linear Response Theory (I)

With : long. (transv.) projection of the Q current on k

)( TL jj

HTL gluon propagatorSubstract the vacuum value

ind

2

ind

222

2

4i 4 ),(

TTL

LT

T

L

L

ind jjkj

kj

kk

ikE

2. Solution for the induced field:

4

3. Evaluate the work done on the heavy quark by the induced field, up to t=L/v, where L is the path length:

Special case of stationary current: )1,( with)( ),( 3 vVtvxVqxtj aa

)( 2 )( 2 )( vkVqVKVqKj aaa

= L/v

Only imaginary part contributes

tvqedtKjk

Kjk

kdkd iLE atViK

aT

T

aL

L

a

L/v

0ind

,22,

2

2

3

3

)( )( )( 4

)(

)()( )( 2

)(

ind

222

2

2

2

2

3

vkKvk

Kvk

kdkdvi

LCLE

TT

L

LSR

Energy loss within Linear Response Theory (II)

Energy loss = - Work on particle

5

In stationnary regime: Only the Im() contributes (ok); collective modes do not get excited (no Cherenkov radiation in HTL).

)()( )( ind

222

2

2

2

vkKvk

Kvk

kT

T

L

L

x=/k

Polelong collect.

mode.Vanishes at

T=0

Space-like

cut

imaginary part = 0

Non vanishing imaginary part

(Landau damping)

Time-likeTime-like

Poletrans collect. mode.

constrain from stationary

current (Sp-like)

Energy loss within Linear Response Theory (III)

6

)()( )( 2

)(

ind

222

2

2

2

2

3

vkKvk

Kvk

kdkdvi

LCLE

TT

L

LSR

Energy loss within Linear Response Theory (IV)

, min max ETMETk with

4. Make the maths… UV divergent due to lack of granularity apply some appropriate cut off : max

k

7

• (delayed ?) coll energy loss ( heat released in medium)

• initial Bremstrallung

• dipole separation

• induced radiation: not considered (Zero Opacity Limit)

Physical situation

Non stationnary conserved current :Q

V1

V2Q

Non stationary case

Several effects (all included in the work done on the Q):

j

N.B.: Only one parton contributes to and appears in the expression of field, but the system is in fact color neutral.

Also in vacuum "renormalization" needed

With and)1,( 11 vV )01,( 2

V

8

Physical situation

Assume factorization between hard production process (at t=0) and subsequent evolution Q

V1

V2Q

"Renormalization" w.r.t. Vacuum

-Evac= Evac(t=0)- Evac(t=L/v)=

Quantity considered. Cures most of UV diverg.

Factorization

-Emed= Emed(t=0)- Emed(t=L/v)=

Same

tVEj vac

vL

dd 1

/

0

tVEj med

vL

dd 1

/

0

-E(med w.r.t. vac) = -Emed +Evac

= Evac(t=L/v) - Emed(t=L/v)

=tVEj ind

vL

dd 1

/

0

Same as for stationary; includes all effects

Operationally: Th. nucl. modif factor :)(

))(( ),(Ed

LEEdLERvac

vacAA

9

Energy loss for finite path length L (I)

)( 2 )( 2 )( vkVqVKVqKj aaa

edtvqKjk

Kjk

kdkd iLE tViKaaT

T

aL

L

a

1L/v

0ind

,22,

2

2

3

3

)( )( 4

)(

0but 0 1 VKVK

currentcomplex

Retarded prescription !!!

PolesColl modes

Space-like

cut

Time-like

Pole from current created at t=0. Connects with all other singularities after contour integration

10

Energy loss for finite path length L (II)

Contour integration in lower half-(complex) plane :

when L

and generates the stationnary E

loss -E add and remove

With the spectral functions :

Residue

Magnitude of the cut

Inholds the modifications ./. stationary case

-E(L) = -E(L) + d(L)

11

The "retardation" function d

Retardation length

Or reduced (induced) E loss ?Ex: dipole energy

Retardation of produced heat ?

Brings a 1/k additional factor instead of L UV convergent !

Asymptotics: )( dLdx

dEE col and )( - dx

dEdL colret

d

Lret

12

Lfm0.5

0.51

1.52

2.5EGeV

E

p20GeV

5 10 15 20Lfm

0.5

0.5

1

1.5

2EGeV

E

p10GeV

5 10 15 20

Fractional energy loss of MINUS 10-15%

Large Lret induced force acting on Q is delayed, but…

-d/E 2 CRS mD/MQ

The "retardation" function d: results

Exclusively from transverse field

13

Spectral functions

Talking to the rest of the world…

… one needs to separate various contributions.

Lfm0.5

0.5

1

1.5

2EGeV

E

p10GeV

5 10 15 20

TM effect

d, TM = 1/2 x d

From gluonic pole : PP + i Radiation; Ter-Mikayelian effect Djordjevic & Gyulassy (03)

TM

For :Genuine retardation of Ecoll ?

Rest

Rest

14

Intermediate conclusions

• Even after substraction of radiative effects, it seems that

collisional energy loss suffers large retardation.

• Flaw: Transition radiation not included (when Q leaves QGP).

Corrected in the meanwhile; why should it affects the coll E loss ?

• Bad news for single electron puzzle… TOO BAD to be true !?

• Other Flaw ?

15

Indeed: calculation of Work done on the charge is correct but inappropriate

• Initially: the charge is bare.

• Part of the work done on the charge by charge is used to build its

self field, i.e. to dress oneself. This takes a time tret /mD

• Asymptotically, one has to take into account the (self) field

contribution in order to perform energy balance

• Beware of talking of En. Loss without specifying "of what" !!!

Growing self field

Bare quark

Grown self field

Dressed quark

: Genuine En. Loss

: "Internal" En. Transfer

TimeIn work: both and

16

Physical situation

• In "true" life, asymptotic medium is vacuum and this difference = 0• Up to now, we assumed Lplasma= asymptotic states differ. Difference is evaluated to be CRS mD/MQ/2 = - d/2 (the PP in the spectral function).

Correct definition of E loss within our model

Mechanical Work (considered up to now)

-Ecorrect = Evac(t=) - Emed(t=)

= [Evac, self field + Evac, mech (t=) ]-

[Emed, self field + Emed, mech (t=) ] =

[Evac, self field - Emed, self field ] +

[Evac, mech (t=) - Emed, mech (t=) ]Energy difference of asymptotic dressed quarks of given velocity

-Ecorrect = -Eprevious - d/2 The other scaling part in retardation

17

Intermediate conclusions (revisited)

• With this correct definition of energy loss, there seems to be little room for retardation of collisional energy loss (see also as Djordjevic nucl-th/0603066)

• Moreover, does not seem to scale like mD.. Original naive argument does not apply… WHY ?

Lfm0.5

0.5

1

1.5

2EGeV

E

p10GeV

5 10 15 20 Lfm0.5

0.51

1.52

2.5EGeV

E

p20GeV

5 10 15 20

TM effect + self energy

• Good news for collisional energy loss as a relevant mechanism in URHIC.

18

Transition radiation (gossiaux, peigne et al; hep-ph/0608061)

Provided one takes good care of asymptotic states, the semi-classical approach can incorporate a lot of physical effects consistently… Taking into account transition radiation:

Retardation of "in medium" emission (Lret)

Transition radiation does not compensate TM effect.

All Z.O.L effects included

-E0(Lp)/E 2/3 CRS mD/MQ 3-5% For Lp > Lret

Order of magnitude compatible with Djordjevic (nucl-th/0512089)19

Conclusions and Remarks

1. No retardation of collisional energy loss…within HTL (Im(T) 2-k2) Is it always the case ?

2. -E0(Lp)/E sometimes neglected in MC simulation of E loss with the argument that "transition radiation compensates TM effect". We confirm that this is not true, strictly speaking.

3. Do E0(Lp) and the L2 dependance of E1rad can cope with

the offset seen in RAA(Lp) ? If not, maybe come back to point 1.

4. Errare humanum est, perseverare diabolicum est

20

Retardation effect for heavy parton collisional energy

loss in QGP.

What if the heavy quark is not produced at t = - ?

Q

World I World II

P P’<P Q

Untouched !

Physically reasonable, but HOW BIG ?

CEL in stationnary regime.

Heavy quark probes the medium via virtual gluon of momentum k

Zone I: k>q* : hard; close collisions; individual; incoherent.

Zone II: k<q*: soft; far collisions; collective; coherent; macroscopic.

Assume pQCD regime at high temperature: g<<1 and ordering

TggTr

qTr

MD 2

1 1 *

1 1 1

Debye radius

Av. dist r

Q

Dr

I

II

CEL in stationnary regime: close collisions.

Bjorken (82) evaluated this contribution using kinetic theory.

Reaction rate :

n(p) (1-

n(p’)) (P+p-P’-q’)Q

q

P

p p’

P’

{P’,p,p’}

From rate to energy loss :

E'EE'E'

EE'xE d

v

d),(d

dd for v1.

IR divergent

*ln

32 2

D qETm

CEL in stationnary regime: far collisions.

Will embed most of the retardation effect.

First evaluated in stationnary regime by Braaten & Thoma (82)

following Weldon (83) : ntrPIm()]

Quite general relation : includes the soft collisions (k<q*) as well, provided one takes the full gluon propagator :

P

p

Q

p’

P’

k

Im(hard)=

CEL in stationnary regime: far collisions (II).

For k>q*: through kinetic expressions

For k<q*: evaluate the self energy with HTL ressumation for consistent with collective polarizationImaginary time formalism.

Legitimate as both contributions are separately gauge invariant.

P

p

Q

p’

P’

k

CEL in stationnary regime: far collisions (III).

Poor man’s prescription: take with some UV regulator kcut:

Results at the logarithmic accuracy:

0<v<1: v1:

UV div

*

ln 11 ln

11 32 2

22D Mq

ETvv

vv

vm

dxdEhard

*ln

32 2

D qETm

dxdEhard

3/

*ln 11 ln

11 32

D2

22D

mq

vv

vv

vm

dxdEsoft

3/*ln

32

D

2D

mqm

dxdEsoft

3/ln

32

D

2D

mETm

dxdE

dxdEsoft

3/

ln 11 ln

11 32

D2

22D

mMET

vv

vv

vm

dxdE

, min cut ETMETk Method followed by Thoma & Gyulassy (91)

Talking to the rest of the world

… one needs to separate various contributions.

finite constant (indep of v) when L . Dipole separation.

Retardation effect… and radiation.

PP[1/(-s(k))]

+ i(-s(k))

• Clear signature of some radiation (cf. identification of Cherenkov radiation in stationnary Collisional E loss)

• Does not scale like L Not Cherenkov (although emitted by the time dependent medium polarization), but ‘initial’ Brehmstrallung .

• E incorporates the difference of energies radiated in the medium (neglecting radiative rescattering) and in the vacuum.

Space-like

cut

Time-like

Pole from current created at t=0

(-s(k)):

In-vacuum radiation

‘‘Radiates like crazy’’, but what should matter is the difference with the ‘‘In-medium’’ radiation.

P

p

p’

P’

k but also (even at lower order) :

P P’

k

Radiative contribution W(L) to the energy loss :

N.B.: For v=1, this is just the Z.O.L. result of GLV for x<<1

In-medium radiation

Radiation of collective modes, described at the

level of the HTL ressumation

P

p

p’

P’

k

Long: suppressed at large k

Vacuum vs Infinite medium

0 0.5 1 1.5 2rad0.01

0.02

0.05

0.1

0.2

0.5

1

2

dWdk dcos

Vacuum

P=10 GeV=1GeV

Medium

0 0.5 1 1.5 2rad0.01

0.02

0.05

0.1

0.2

0.5

1

2

dWdk dcos

P=10 GeV=0.5GeV

Medium

0 0.5 1 1.5 2rad0.01

0.02

0.05

0.1

0.2

0.5

1

2

dWdk dcos

Medium

P=10 GeV=2GeV

• In all cases, the integrated In-medium radiation is weaker than in the vacuum (finite mass, residues < 1)

• Interplay of trans. and long. radiation for of the order of mD

Vacuum

0 0.25 0.5 0.75 1 1.25 1.51.75 2min0.01

0.02

0.05

0.1

0.2

0.5

1

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2min0.01

0.02

0.05

0.1

0.2

0.5

1

Infinite medium vs Finite (L=5fm) medium

P=10 GeV=0.5GeV

Infinite

=1GeV

Infinite

0 0.25 0.50.75 1 1.251.5 1.75 2min0.01

0.02

0.05

0.1

0.2

0.5

1 Infinite

=2GeV

• Depletion at small angle

• Gluon formed:

averages to ½

Provides a criteria for formation time.

• Possible consequences on the understanding of far away hadrons.