1. Radiative energy loss Marco van Leeuwen, Utrecht University

2. 2 Medium-induced radiation If <  f, multiple scatterings add coherently Zapp, QM09 L c =  f,max propagating parton radiated gluon Landau-Pomeranchuk-Migdal effect Formation time important Radiation sees length ~  f at once Energy loss depends on density: and nature of scattering centers (scattering cross section) Transport coefficient

3. 3 A simple model `known’ from e + e - known pQCDxPDF extract Parton spectrum Fragmentation (function) Energy loss distribution This is where the information about the medium is P(  E) combines geometry with the intrinsic process – Unavoidable for many observables Notes: This formula is the simplest ansatz – Independent fragmentation after E-loss assumed Jet,  -jet measurements ‘fix’ E, removing one of the convolutions We will explore this model during the week; was ‘state of the art’ 3-5 years ago

4. 4 Two extreme scenarios p+p Au+Au pTpT 1/N bin d 2 N/d 2 p T Scenario I P(  E) =  (  E 0 ) ‘Energy loss’ Shifts spectrum to left Scenario II P(  E) = a  (0) + b  (E) ‘Absorption’ Downward shift (or how P(  E) says it all) P(  E) encodes the full energy loss process R AA not sensitive to energy loss distribution, details of mechanism

5. 5 What can we learn from R AA ? This is a cartoon! Hadronic, not partonic energy loss No quark-gluon difference Energy loss not probabilistic P(  E) Ball-park numbers:  E/E ≈ 0.2, or  E ≈ 2 GeV for central collisions at RHIC  0 spectra Nuclear modification factor PHENIX, PRD 76, 051106, arXiv:0801.4020 Note: slope of ‘input’ spectrum changes with p T : use experimental reach to exploit this

6. 6 Four formalisms Hard Thermal Loops (AMY) –Dynamical (HTL) medium –Single gluon spectrum: BDMPS-Z like path integral –No vacuum radiation Multiple soft scattering (BDMPS-Z, ASW-MS) –Static scattering centers –Gaussian approximation for momentum kicks –Full LPM interference and vacuum radiation Opacity expansion ((D)GLV, ASW-SH) –Static scattering centers, Yukawa potential –Expansion in opacity L/ (N=1, interference between two centers default) –Interference with vacuum radiation Higher Twist (Guo, Wang, Majumder) –Medium characterised by higher twist matrix elements –Radiation kernel similar to GLV –Vacuum radiation in DGLAP evolution Multiple gluon emission Fokker-Planck rate equations Poisson ansatz (independent emission) DGLAP evolution See also: arXiv:1106.1106

7. 7 Kinematic variables Gluon(s) L: medium length T, q: Density E: incoming parton energy/momentum  : (total) momentum of radiated gluon (=  E) k T : transverse momentum of radiated gluon p: outgoing parton momentum = E-  E = E- 

8. 8 Large angle radiation Emitted gluon distribution Opacity expansion k T < k Calculated gluon spectrum extends to large k  at small k Outside kinematic limits GLV, ASW, HT cut this off ‘by hand’ Estimate uncertainty by varying cut; sizeable effect Gluon momentum k Gluon perp momentum k 

9. 9 Limitations of soft collinear approach Soft: Collinear: Need to extend results to full phase space to calculate observables (especially at RHIC) Soft approximation not problematic: For large E, most radiation is soft Also:  > E  full absorption Cannot enforce collinear limit: Small ,   k T always a part of phase space with large angles Calculations are done in soft collinear approximation:

10. 10 Opacity expansions GLV and ASW-SH x + ~ x E in soft collinear limit, but not at large angles Different large angle cut-offs: k T <  = x E E k T <  = 2 x + E Blue: k Tmax = xE Red: k Tmax = 2x(1-x)E Single-gluon spectrum Blue: m g = 0 Red: m g =  /√2 Horowitz and Cole, PRC81, 024909 Single-gluon spectrum Different definitions of x: ASW: GLV: Factor ~2 uncertainty from large-angle cut-off Horowitz and Cole, PRC81, 024909

11. 11 Opacity expansion vs multiple soft Salgado, Wiedemann, PRD68, 014008 Different limits: SH (N=1 OE): interference between neighboring scattering centers MS: ‘all orders in opacity’, gaussian scattering approximation Quantitative differences sizable OE and MS related via path integral formalism Two differences at the same time

12. 12 AMY, BDMPS, and ASW-MS Single-gluon kernel from AMY based on scattering rate: BMPS-Z use harmonic oscillator: BDMPS-Z: Salgado, Wiedemann, PRD68, 014008 Finite-L effects: Vacuum-medium interference + large-angle cut-off

13. 13 AMY and BDMPS Using based on AMY-HTL scattering potential L=2 fm Single gluon spectraL=5 fm Single gluon spectra AMY: no large angle cut-off + sizeable difference at large  at L=2 fm

14. 14 L-dependence; regions of validity? Emission rate vs  (=L) Caron-Huot, Gale, arXiv:1006.2379 AMY, small L, no L 2, boundary effect Full = numerical solution of Zakharov path integral = ‘best we know’ GLV N=1 Too much radiation at large L (no interference between scatt centers) H.O = ASW/BDMPS like (harmonic oscillator) Too little radiation at small L (ignores ‘hard tail’ of scatt potential) E = 16 GeV k = 3 GeV T = 200 MeV

15. 15 HT and GLV Single-gluon kernel GLV and HT similar HT: kernel diverges for k T  0  √(E/L) GLV: HT: GLV similar structure, phase factor  However HT assumes k T >> q T, so no explicit integral over q T GLV HT gives more radiation than GLV

16. 16 Single gluon spectra Same temperature @Same temperature: AMY > OE > ASW-MS L = 2 fm L = 5 fm Size of difference depends on L, but hierarchy stays

17. 17 Multiple gluon emission Poisson convolution example Average number of gluons: Poisson fluctuations: Total probability: (assumed)

18. 18 Outgoing quark spectra Same temperature: T = 300 MeV @Same T: suppression AMY > OE > ASW-MS Note importance of P 0

19. 19 Outgoing quark spectra Same suppression: R 7 = 0.25 At R 7 = 0.25: P 0 small for ASW-MS P 0 = 0 for AMY by definition

20. 20 Model inputs: medium density Multiple soft scattering (ASW-MS): Quenching weights (AliQuenchingWeights) Inputs: N.B: keep track of factors hc = 197.327 MeVfm Quenching weights: Opacity expansion (DGLV, ASW-SH): Gluon spectra +Poisson Ansatz for multiple gluon radiation

21. 21 Medium properties Some pocket formulas gluon gas, Baier scheme: HTL: See also: arXiv:1106.1106

22. 22 Geometry Density profile Profile at  ~  form known Density along parton path Longitudinal expansion dilutes medium  Important effect Space-time evolution is taken into account in modeling

23. 23 Geometry II L-moments of density along path:

24. 24 ‘Analytic’ calculations vs MC Event Generators Analytic calculations (this lecture) –Easy to include interference –So far: only soft-collinear approx –Energy-momentum conservation ad-hoc Monte Carlo event generators (modified parton showers) –Energy-momentum conservation exact –May be able to introduce recoil (dynamic scatt centers) –More difficult to introduce interference –Examples: JEWEL, qPYTHIA, YaJEM

25. 25 AliQuenchingWeights Based on Salgado, Wiedemann, hep-ph/0302184 AliQuenchingWeights::InitMult() Initialises multiple soft scattering Quenching Weights AliQuenchingWeights::CalcMult(ipart, R, x, cont, disc) Multiple soft scattering Quenching Weights input: ipart 0=gluon, 1=quark x =  /  c return: cont:  dI/d  disc: P(0)

26. 26 Extra slides

27. 27 Thoughts about black-white scenario At RHIC, we might have effectively a ‘black-white scenario’ –Large mean E-loss –Limited kinematic range Different at LHC? –Mean E-loss not much larger, kinematic range is? –Or unavoidable: steeply falling spectra In addition: the more monochromatic the probe, the more differential sensitivity  -jet, jet-reco promising! Or: Hitting the wall with P(  E)

28. 28 Opacity expansion Opacity expansion (DGLV, ASW-SH): Gluon spectra Poisson Ansatz: