Charm production in AA collisions

Transcription

1 Charm production in AA collisions François Gelis and CEA/Saclay François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 1

2 François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 2

3 Heavy quark production Standard collinear factorization François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 3

4 Heavy quark production Standard collinear factorization...or higher twist effects? François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 3

5 In-medium J/psi suppression Debye screening prevents the formation of quarkonium states Matsui, Satz (1986) the heavy quarks pick a light quark instead and form a D meson Heavy quark potential, screening masses, and spectral functions (?) calculable on the lattice Relevant observable : [J/ψ] / [Open charm] François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 4

6 ...or QQbar recombination? Many QQ pairs are produced in each AA collision Braun-Munzinger, Stachel (2000) Thews, Schroedter, Rafelski (2001) A Q from one pair can recombine with a Q from another pair Avoids the conclusion of the Matsui-Satz scenario, provided that the average distance between heavy quarks is smaller than the Debye screening length Leads to an enhancement of J/ψ formation François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 5

7 Outline QQ production at small x François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 6

8 Fixed order Resummations François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 7

9 Fixed order calculations Fixed order Resummations LO [O(α 2 s)] : François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 8

10 Fixed order calculations Fixed order Resummations LO [O(α 2 s)] : NLO [O(α 3 s)] : Nason, Dawson, Ellis (1988) NLO almost as large as LO + rather large scale dependence NNLO not known yet for this process François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 8

11 Fixed order calculations Fixed order Resummations Plain LO+NLO has been problematic for a long time: B production at CDF vs NLO-pQCD, as of 2001 data / theory somewhat embarassing for pqcd... François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 9

12 Resummation of logarithms Fixed order Resummations The coefficients of the perturbative expansion may be enhanced by logarithms dσ ij QQ = X c n αs n, c n = n=2 n 2 X k=0 c (n 2 k) n ˆln Q n 2 k where Q might be large enough so that α s ln Q 1 Logs that are independent of the observable : Threshold logs: Q = bs/4m 2 Q 1 Small-x logs: Q = bs/m 2 Q Logs that depend on the details of the observable : Single Q spectrum at large momentum: Q = p (Q)/m Q QQ spectrum at low pair momentum: Q = m Q /p (QQ) QQ spectrum in a back-to-back configuration: Q = 1 φ QQ /π François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 10

13 Resummation of logarithms Fixed order Resummations Including these logarithms amounts to taking into account extra radiation in the final state Rearrangement of the perturbative expansion: dσ = α 2 s X n=0 α n s X i=0 r (n) i [α s ln Q] i + O `Q 1 n = 0 : Leading Log (LL) n = 1 : Next-to-Leading Log (NLL) Two different implementations : FONLL : NLO fixed order + analytic resummation of leading logs Cacciari, Greco, Nason (1998) MC@NLO : NLO fixed order + resummation of logs via a parton shower Frixione, Webber (2002) François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 11

14 Present data vs theory situation Fixed order Resummations Resummations + better fragmentation functions: better agreement with data : B production at Tevatron II Note : the data has gone down as well... François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 12

15 Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 13

16 Relevant x range at the LHC Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions x coverage for c c production at the LHC : central rapidity François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 14

17 Relevant x range at the LHC x coverage for c c production at the LHC : forward rapidity Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions very small values of x reached in one of the projectiles François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 15

18 Kt-factorization Included diagrams : Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions QQ cross-section in k -factorized form : Z dσ pp QQ = dφ Q dφ Q δ( k1 + k 2 p (QQ)) k 2 1 k 2 ϕ p (x 1, k 1 )ϕ p (x 2, k 2 ) M 2 2 Pros : Includes intrinsic k and of resums logs of 1/x Some NLO and NNLO diagrams are already included This formalism can be generalized to include saturation Cons : The incoming gluons are off-shell difficult calculations Only a subset of the NLO terms is included Factorization proven only to Leading Log François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 16

19 Color glass condensate Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions McLerran, Venugopalan (1994) Iancu, Leonidov, McLerran (2001) Small x modes have a large occupation number they are described by a classical color field Large x modes are described by frozen color sources ρ a The classical field obeys Yang-Mills equations: [D ν, F νµ ] a = δ µ+ δ(x )ρ a ( x ) The color sources ρ a are random, and their distribution is described by a functional W x0 [ρ], where x 0 is the separation between small x et large x. W x0 [ρ] changes with x 0 according to the JIMWLK equation. Observables are calculated in the presence of the classical field, and then averaged over the configurations of the sources ρ a : O = Z [Dρ a ] W x0 [ρ a ] O[ρ a ] François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 17

20 Quark production in the CGC Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions get W x1 [ρ 1 ] for the first projectile François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 18

21 Quark production in the CGC Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions get W x2 [ρ 2 ] for the second projectile François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 19

22 Quark production in the CGC Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions solve the Yang-Mills equations for the sources ρ 1, ρ 2 François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 20

23 Quark production in the CGC Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions compute the quark propagator in the classical field François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 21

24 Quark production in the CGC Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions Baltz, FG, McLerran, Peshier (2001) FG, Venugopalan (2003) Blaizot, FG, Venugopalan (2004) The single inclusive quark spectrum can be expressed in terms of the retarded quark propagator : E p dn Q d 3 p Z d 3 q (2π) 3 2E q u( p)t R (p, q)v( q) 2 The calculation can be carried out analytically only at lowest order in one of the two sources i.e. ρ 1 g 1 ρ 2 0 François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 22

25 Cronin effect for single quarks RpA at moderate x : Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions 2 Q s 2 = 4 GeV R pa gluons s quarks c quarks b quarks q (GeV) François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 23

26 Cronin effect for single quarks RpA at small x : Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions 2 Q s 2 = 4 GeV R pa gluons c quarks b quarks q (GeV) François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 24

27 Formulation for AA collisions Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions FG, Kajantie, Lappi (2004, 2005) To all orders in both sources, the gauge field is known only numerically. It is possible to reformulate the problem of quark production in a way which is suitable for a numerical approach Alternate representation of the retarded amplitude: Z u( q)t R (p, q)v( p) = lim τ τ + dηd 2 x e ip x u ( p) e ηγ0 γ 3 ψ q (t, x) where ψ q (t, x) obeys Dirac s equation with retarded boundary conditions : (i/ x g/a(x) m)ψ q (x) = 0, ψ q (t, x) v( q)e iq x t François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 25

28 Classical color field Space-time structure of the classical color field: Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions t 4 Region 1: A µ = 0 Region 2: A ± = 0, 2 3 z A i = i g U 1 i U 1 Region 3: A ± = 0, 1 A i = i g U 2 i U 2 Region 4: A µ 0 Notes: U 1,2 ( x ) = exp( ig 1 ρ 2 1,2 ) In the region 4, A µ is known only numerically Krasnitz, Venugopalan (2000,2001), Lappi (2003) François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 26

29 Quark propagation Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions Propagation through region 4: t τ f τ i z h τ ψ p (τ, η, x ) = 1 2τ γ0 γ 3 ( η + iga η ) τ +γ 0 γ ( + iga i ) iγ 0 m ψ p (τ, η, x ) The initial condition is known analytically at τ i 0 + François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 27

30 Time dependence g 2 µ = 2 GeV, (*) g 2 µ = 1 GeV : Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions dn / dy m = 60 MeV m = 300 MeV m = 600 MeV m = 1.5 GeV m = 300 MeV * τ [fm] François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 28

31 Spectra for various quark masses g 2 µ = 2 GeV, τ = 0.25 fm : Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions dn/dyd 2 q T [arbitrary units] m = 60 MeV m = 300 MeV m = 600 MeV m = 1.5 GeV m = 3 GeV ^q [GeV] François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 29

32 Qs dependence of dn/dy Number of quarks at τ = 0.25 fm : Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions dn / dy m = 300 MeV g 2 µ [GeV] François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 30

33 Shortcomings Relevant x range Kt-factorization Color glass condensate High density effects for pa AA collisions Factorization of logs of 1/x plausible not proven Numerical evaluation done on a coarse lattice due to limited computational ressources large lattice artifacts at large momentum/mass Instead of dn Q /d 3 p, one should calculate f Q (t, x, p) (easy to fix) François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 31

34 Dynamical evolution QQbar recombination Dynamical evolution François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 32

35 QQbar recombination Dynamical evolution QQbar recombination What has been said so far is correct if there is only a few QQ pairs in the system At LHC energies, pqcd predicts that hundreds of cc pairs are being produced in a central PbPb collision Q and Q that have been produced uncorrelated may encounter and form a quarkonium state Model independent estimates : Prob(J/ψ) N c /N u,d,s N c c /N ch N J/ψ Nc c/n 2 ch Since Nc c 2 grows faster with energy than N ch, this mechanism of J/ψ production will eventually be dominant Two different implementations : Statistical hadronization Kinetic models François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 33

36 Statistical hadronization Dynamical evolution QQbar recombination Braun-Munzinger, Stachel (2000) Early attemps to include charm in thermal fits underpredicted the yield of charmed hadrons However, the ratio σ ψ /σ J/ψ measured at SPS goes to its thermal value when N part is large One assumes that the number of c, c quarks is determined by early hard collisions (no thermal production/annihilation) Hadronization is assumed to follow thermal distributions, modified by an enhancement factor γ c (one power of γ c per c or c quark in the hadron). Conservation of charm : N direct c c = 1 2 γ cv X i (n th (D i ) + n th (Λ i )) + γ 2 cv X i n th (ψ i ) + Then : N D = γ c V n th (D) and N J/ψ = γ 2 cv n th (J/ψ) François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 34

37 Statistical hadronization LHC : J/ψ yield per participant Dynamical evolution QQbar recombination (dn/dy) cc (b=0) = 25 (dn/dy) cc (b=0) = (dn/dy) cc (b=0) = (dn/dy) J/ψ / N part N part François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 35

38 Statistical hadronization LHC : J/ψ yield per c c pair 3.0 Dynamical evolution QQbar recombination (dn/dy) J/ψ / (dn/dy) cc (dn/dy) cc (b=0) = 25 (dn/dy) cc (b=0) = (dn/dy) cc (b=0) = N part this behavior with centrality is the opposite of what one expects in the Matsui-Satz scenario François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 36

39 Kinetic formation Dynamical evolution QQbar recombination Thews, Schroedter, Rafelski (2001) Dominant in-medium J/ψ breakup process : g + J/ψ c c The reverse process c c J/ψ + g should also occur, with a probability that increases like the square of the density of charmed quarks Kinetic equation : dn J/ψ dτ = λ F N c N c V (τ) λ D ρ gn J/ψ V (τ) : τ-dependent volume (expansion plays against recombination) ρ g : gluon density λ F,D : formation and dissociation rates (λ = σv rel ) h R N J/ψ (τ) = ǫ(τ) N J/ψ (τ i ) + Nc c 2 τ dτ τ i Solution : with ǫ(τ) = exp( R τ τ i dτ ρ g λ D ) λ F V (τ)ǫ(τ) i François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 37

40 Kinetic formation LHC : J/ψ yield per c c pair Dynamical evolution QQbar recombination N J/ψ / Initial Charm N cc (b=0) =100, y = 4 N cc (b=0) =150, y = 4 N cc (b=0) =200, y = 4 N cc (b=0) =100, y = 7 N cc (b=0) =150, y = 7 N cc (b=0) =200, y = 7 N cc (b=0) =100, Thermal N cc (b=0) =150, Thermal N cc (b=0) =200, Thermal N part very sensitive to the distribution of initial charm See Gossiaux, Guiho, Aichelin (2004) for a Fokker-Plank description of the time evolution of the c, c distributions François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 38

41 Bound states in a medium Hadronization in vaccum in-medium suppression lattice results Bound states in a dense medium François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 39

42 Quarkonium production in vacuum Bound states in a medium Hadronization in vaccum in-medium suppression lattice results More difficult than the inclusive fragmentation c D + X LO is clearly insufficient in order to get the p distribution of J/ψ or Υ, since by construction p (QQ) = 0 at this order Several approaches : Color Singlet Model (CSM) Non-Relativistic QCD (NRQCD) [aka Color Octet Model] Color Evaporation Model (CEM) Comover Enhancement Scenario (CES) François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 40

43 Charmonium suppression in the QGP Bound states in a medium Hadronization in vaccum in-medium suppression lattice results Matsui, Satz (1986), Kharzeev, Satz (1994), and many others... If the Debye screening radius is smaller than the size of quarkonium state, the binding of the Q and Q is destroyed by the surrounding light quarks and gluons The Q and Q drift in the QGP, and cannot find each other again At hadronization time, they pick up a light quark and form D or B mesons A suppression of the ratio [J/ψ] / [Open charm] could be a signature of the QGP Not as simple though : there is also a suppression in pa collisions. One should therefore look for anomalous suppression effects François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 41

44 Normal nuclear suppression Bound states in a medium Hadronization in vaccum in-medium suppression lattice results Parameterization of the J/ψ absorption in cold nuclear matter : σ abs ( s) = σ abs ( s s 0 ) s 0 «/2 σ abs ( s 0 = 17.3 GeV) = 5 ± 0.5 mb, Quarkonium survival probability in an AB collision : Z S( b) = " exp (A 1) d 2 sdz A dz B ρ A ( s, z A )ρ B ( b s, z B ) # " Z z A dzρ A σ abs exp (B 1) Z z B dzρ B σ abs # François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 42

45 Normal nuclear suppression Impact parameter dependence of the survival probability : Bound states in a medium Hadronization in vaccum in-medium suppression lattice results J/ψ nuclear absorption at LHC Ar+Ar Pb+Pb b [fm] François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 43

46 Normal nuclear suppression N part dependence of the survival probability : Bound states in a medium Hadronization in vaccum in-medium suppression lattice results J/ψ nuclear absorption in Pb-Pb collisions SPS RHIC LHC N part François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 44

47 Lattice results Bound states in a medium Hadronization in vaccum in-medium suppression lattice results Reminder : only Euclidean quantities can be calculated directly in lattice Monte-Carlo simulations : Minkowkian : e is[aµ ] Euclidean : e S[Aµ ] Potential between pairs of heavy quarks in a QGP Can be fed into a non-relativistic Shöedinger equation in order to compute the binding energy of the bound states Extraction of the QQ spectral functions from lattice data Fairly new method, still in developement Results in qualitative agreement with the previous one These issues are totally unexplored at finite µ B François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 45

48 Heavy quark potential Bound states in a medium Hadronization in vaccum in-medium suppression lattice results The averaged free-energy is obtained from Polyakov loops : e F(r,T)/T = 1 9 D E trl( r) trl ( 0), L( r) = YN τ i=1 U 0 ( r, τ) It can be divided into a color singlet and a color octet parts : e F(r,T)/T = 1 9 e F 1(r,T)/T e F 8(r,T)/T e F 8(r,T)/T = 1 8 e F 1(r,T)/T = 1 3 D E trl( r)l ( 0) D E trl( r) trl ( 0) 1 D E trl( r)l ( 0) 24 In principle, one needs to transform that into the potential energy U : F = U TS, S = F T François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 46

49 Heavy quark potential Results for T/T c = 1.5 : Bound states in a medium Hadronization in vaccum in-medium suppression lattice results 600 U 1 [MeV] F 1 [MeV] r [fm] François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 47

50 Heavy quark potential T -dependence of the potential above T c : Bound states in a medium Hadronization in vaccum in-medium suppression lattice results 400 U 1 (r,t) [MeV] r [fm] 1.95T c 2.60T c 4.50T c 7.50T c François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 48

51 Heavy quark potential Bound states in a medium Hadronization in vaccum in-medium suppression lattice results What do we do with that? Shröedinger equation for QQ bound states :» 2m Q U1 (r, T) ψ i = M i (T)ψ i m Q Non-relativistic Assumes 2-body interactions only Dissociation temperatures : state J/ψ χ c ψ Υ χ b Υ T d /T c the quarkonium states do not get immediately dissolved above the critical temperature François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 49

52 Heavy quark spectral functions Bound states in a medium Hadronization in vaccum in-medium suppression lattice results Method for extracting spectral functions : G H (τ, p) = G H (τ, p) = Z Z 0 dω ρ H (ω, p T) D E d 3 x e i p x J H (τ, x)j H (0, 0) cosh(ω(τ 1/2T)) sinh(ω/2t), J H = ψ Γ H ψ state χ 0 c η c J/ψ χ 1 c Γ H 1 γ 5 γ µ γ µ γ 5 ρ H (ω, p) has a sharp peak for stable states in the corresponding channel (broad peak for an unstable state) Main problem : G H (τ, p) is known at a finite number of τ s the inversion of the spectral integral in order to obtain the function ρ H is a mathematically ill-defined problem François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 50

53 Heavy quark spectral functions Bound states in a medium Hadronization in vaccum in-medium suppression lattice results Maximum Entropy Method : Many more degrees of freedom in ρ H (ω, p) than data points a χ 2 -fit would have flat directions... Most of the multiple solutions would have unphysical features: non-positivity, not smooth, incorrect large ω behavior Idea : add a convex term F to the χ 2 so that there is a unique minimum χ 2 χ 2 + αf[ρ H ] MEM : F[ρ H ] = Z 0 dω [ρ H (ω) ρ 0 (ω) ρ H (ω) ln(ρ H (ω)/ρ 0 (ω))] ensures the positivity of ρ H for α, the solution wants to be identical to the prior ρ 0 use with extreme caution because you may only get what you bring... François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 51

54 Heavy quark spectral function J/ψ spectral function below T c : Bound states in a medium Hadronization in vaccum in-medium suppression lattice results σ(ω)/ω 2 0.9T c 1.5T c ω[gev] The second and third peaks (the fat ones...) are lattice artifacts. Shouldn t we worry about them contaminating the physical peak? François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 52

55 Heavy quark spectral function J/ψ spectral function above T c : Bound states in a medium Hadronization in vaccum in-medium suppression lattice results 0.8 σ(ω)/ω T c 3T c ω[gev] The J/ψ peak starts going down for T above 2T c good qualitative agreement with the method based on the heavy quark potential François Gelis 2008 Initial conditions in Heavy Ion Collisions, Goa, September p. 53