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.