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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.

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Presentation on theme: "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."— Presentation transcript:

1 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/ hep-ph/

2 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) ) Motivation with stationnary stationnary ? Separate short distance (indiv collisions) and large distance (collective answer; polarization) For partons produced in medium, proper field needs time (   /m D ) to reach its full amplitude and fully polarize the medium Naive expectation: No full dissipation until t ret   /m D. Retardation effect (bad for single-e puzzle) 2

3 Physical situation Heavy quark Q: Avoid some complications due to collinear divergences Q V1V1 V2V2 Q Situation under study (exploratory) and hypothesis "Usual" QGP: no quasi bound state, at rest, static, « confined » over some domain L plasma Assume high temperature  g<<1  ordering 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 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) Energy loss within Linear Response Theory (I) With : long. (transv.) projection of the Q current on HTL gluon propagator Substract the vacuum value 2. Solution for the induced field: 4

5 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:  = L/v Only imaginary part contributes Energy loss within Linear Response Theory (II) Energy loss = - Work on particle 5

6 In stationnary regime: Only the Im(  ) contributes (ok); collective modes do not get excited (no Cherenkov radiation in HTL). x=  /k Pole long collect. mode. Vanishes at T=0 Space-like cut imaginary part = 0 Non vanishing imaginary part (Landau damping) Time-like Pole trans collect. mode.  constrain from stationary current (Sp-like) Energy loss within Linear Response Theory (III) 6

7 Energy loss within Linear Response Theory (IV) with 4. Make the maths… UV divergent due to lack of granularity  apply some appropriate cut off : 7

8 (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 V1V1 V2V2 Q Non stationary case Several effects (all included in the work done on the Q): 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 8

9 Physical situation Assume factorization between hard production process (at t=0) and subsequent evolution Q V1V1 V2V2 Q "Renormalization" w.r.t. Vacuum -  E vac = E vac (t=0)- E vac (t=L/v)= Quantity considered. Cures most of UV diverg. Factorization -  E med = E med (t=0)- E med (t=L/v)= Same -  E (med w.r.t. vac) = -  E med +  E vac = E vac (t=L/v) - E med (t=L/v) = Same as for stationary; includes all effects Operationally: Th. nucl. modif factor : 9

10 Energy loss for finite path length L (I)  current complex Retarded prescription !!! Poles Coll modes Space-like cut Time-like Pole from current created at t=0. Connects with all other singularities after contour integration 10

11 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

12 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:and d L ret 12

13 Fractional energy loss of MINUS 10-15% Large L ret  induced force acting on Q is delayed, but… -d  /E   2 C R  S m D /M Q  The "retardation" function d: results Exclusively from transverse field 13

14 Spectral functions Talking to the rest of the world… … one needs to separate various contributions. 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  E coll ? Rest 14

15 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

16 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 t ret   /m D 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 Time In work: both and  16

17 Physical situation In "true" life, asymptotic medium is vacuum and this difference = 0 Up to now, we assumed L plasma =   asymptotic states differ. Difference is evaluated to be  C R  S m D /M Q /  2 = - d  /2 (the PP in the spectral function). Correct definition of E loss within our model Mechanical Work (considered up to now) -  E correct = E vac (t=  ) - E med (t=  ) = [E vac, self field + E vac, mech (t=  ) ]- [E med, self field + E med, mech (t=  ) ] = [E vac, self field - E med, self field ] + [E vac, mech (t=  ) - E med, mech (t=  ) ] Energy difference of asymptotic dressed quarks of given velocity -  E correct = -  E previous - d  /2 The other  scaling part in retardation 17

18 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/ ) Moreover, does not seem to scale like  m D .. Original naive argument does not apply… WHY ? TM effect + self energy Good news for collisional energy loss as a relevant mechanism in URHIC. 18

19 Transition radiation (gossiaux, peigne et al; hep-ph/ ) 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 (  L ret ) Transition radiation does not compensate TM effect. All Z.O.L effects included -  E 0 (L p )/E   2/3 C R  S m D /M Q  3-5% For L p > L ret Order of magnitude compatible with Djordjevic (nucl-th/ ) 19

20 Conclusions and Remarks 1.No retardation of collisional energy loss…within HTL (Im(  T )   2 -k 2 ) Is it always the case ? 2.-  E 0 (L p )/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  E 0 (L p ) and the L 2 dependance of  E 1 rad can cope with the offset seen in R AA (  L p ) ? If not, maybe come back to point 1. 4.Errare humanum est, perseverare diabolicum est 20

21 Retardation effect for heavy parton collisional energy loss in QGP. What if the heavy quark is not produced at t = -  ? Q World IWorld II PP’

22 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

23 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 : for v  1. IR divergent

24 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) :  n  tr  P  Im(  )] Quite general relation : includes the soft collisions (k

25 CEL in stationnary regime: far collisions (II). For k>q*: through kinetic expressions For k

26 CEL in stationnary regime: far collisions (III). Poor man’s prescription: take with some UV regulator k cut : Results at the logarithmic accuracy: 0

27 Talking to the rest of the world … one needs to separate various contributions.  finite constant (indep of v) when L  . Dipole separation.

28 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)):

29 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) : PP’ 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

30 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

31 Vacuum vs Infinite medium Vacuum P=10 GeV  =1GeV Medium P=10 GeV  =0.5GeV 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 m D Vacuum

32 Infinite medium vs Finite (L=5fm) medium P=10 GeV  =0.5GeV Infinite  =1GeV 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.

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