Factorization in high energy nucleus-nucleus collisions

Transcription

1 Factorization in high energy nucleus-nucleus collisions ISMD, Kielce, September 2012 François Gelis IPhT, Saclay 1 / 30

2 Outline 1 Color Glass Condensate 2 Factorization in Deep Inelastic Scattering 3 Factorization in nucleus-nucleus collisions 4 Extensions 2 / 30

3 1 Color Glass Condensate Small-x gluons matter Gluon saturation Saturation domain Color Glass Condensate 2 / 30

4 Longitudinal momentum fraction in AA collisions The partons that are relevant for the process under consideration carry the longitudinal momentum fractions: x 1,2 = P s e ±Y P : transverse momentum Y : rapidity s : collision energy 3 / 30

5 Longitudinal momentum fraction in AA collisions 99% of the produced particles below p 2 GeV x 10 2 at RHIC ( s = 200 GeV) x at the LHC ( s = 5.5 TeV) partons at small x are the most important 4 / 30

6 Growth of the gluon distribution at small x Parton distributions at small x H1 and ZEUS xf 10 xg xs 2 2 Q = 10 GeV 1 xu v xd v HERAPDF exp. uncert. model uncert. parametrization uncert x Gluons dominate at any x / 30

7 1 Color Glass Condensate Small-x gluons matter Gluon saturation Saturation domain Color Glass Condensate 5 / 30

8 Multiple scatterings and gluon recombination? Main difficulty: How to treat collisions involving a large number of partons? 6 / 30

9 Multiple scatterings and gluon recombination Dilute regime : one parton in each projectile interact large Q 2, no small-x effects single parton distributions + DGLAP evolution 6 / 30

10 Multiple scatterings and gluon recombination Dense regime : multiparton processes become crucial gluon recombinations are important (saturation) multi-parton distributions + JIMWLK evolution new techniques are required (Color Glass Condensate): L = 1 4 F 2 + J A (gluons only, field A for k + < Λ, classical source J for k + > Λ) 6 / 30

11 1 Color Glass Condensate Small-x gluons matter Gluon saturation Saturation domain Color Glass Condensate 6 / 30

12 Criterion for gluon recombination Gribov, Levin, Ryskin (1983) Number of gluons per unit area : ρ xg A (x, Q2 ) πr 2 A Recombination cross-section : σ gg g α s Q 2 Recombination happens if ρσ gg g 1, i.e. Q 2 Q 2 s, with : Q 2 s α s xg A (x, Q 2 s ) πr 2 A A 1/3 1 x / 30

13 Saturation domain Saturation scale as a function of x and A 8 / 30

14 1 Color Glass Condensate Small-x gluons matter Gluon saturation Saturation domain Color Glass Condensate 8 / 30

15 CGC: Degrees of freedom CGC = effective theory of small x gluons The fast partons (k + > Λ + ) are frozen by time dilation described as static color sources on the light-cone : J µ = δ µ+ ρ(x, x ) (0 < x < 1/Λ + ) The color sources ρ are random, and described by a probability distribution W Λ +[ρ] Slow partons (k + < Λ + ) cannot be considered static over the time-scales of the collision process must be treated as standard gauge fields eikonal coupling to the current J µ : A µ J µ 9 / 30

16 CGC: renormalization group evolution Independence w.r.t Λ + evolution equation (JIMWLK) : J i a( u ) W Λ + ln(λ + ) = H W Λ + H = 1 d 2 u J ( u ) J( u ) π d 2 x 2π ( x u ) i ( x u ) 2 [ Ω( x ) Ω( u ) ] ab b x Ω is a Wilson line in the color field of the nucleus Originally derived for the RG evolution of observables that involve a single nucleus (e.g. Deep Inelastic Scattering) Can this be generalized to observables in nucleus-nucleus collisions? 10 / 30

17 Why factorization should work: causality τ coll E -1 The duration of the collision is very short: τ coll E 1 11 / 30

18 Why factorization should work: causality τ coll E -1 The duration of the collision is very short: τ coll E 1 The logarithms we want to resum arise from the radiation of soft gluons, which takes a long time it must happen (long) before the collision 11 / 30

19 Why factorization should work: causality τ coll E -1 space-like interval The duration of the collision is very short: τ coll E 1 The logarithms we want to resum arise from the radiation of soft gluons, which takes a long time it must happen (long) before the collision The projectiles are not in causal contact before the impact the logarithms are intrinsic properties of the projectiles, independent of the measured observable 11 / 30

20 2 Factorization in Deep Inelastic Scattering Leading Order Next to Leading Order Leading Log resummation 11 / 30

21 Inclusive DIS at Leading Order CGC effective theory with cutoff at the scale Λ 0 : fields Λ - 0 sources P - k - At Leading Order, DIS can be seen as the interaction between the target and a q q fluctuation of the virtual photon : 12 / 30

22 Inclusive DIS at Leading Order Forward dipole amplitude at leading order: T LO ( x, y ) = 1 1 tr(u( x )U ( y N )) c }{{} Wilson lines 1/xP U( x ) = P exp ig dz + A (z +, x ) [D µ,f µν ] = δ ν ρ(x +, x ) at LO, the scattering amplitude on a saturated target is entirely given by classical fields Note: the q q pair couples only to the sources up to the longitudinal coordinate z + (xp ) 1. The other sources are too slow to be seen by the probe 13 / 30

23 2 Factorization in Deep Inelastic Scattering Leading Order Next to Leading Order Leading Log resummation 13 / 30

24 Inclusive DIS at NLO Consider now quantum corrections to the previous result, restricted to modes with Λ 1 < k < Λ 0 (the upper bound prevents double-counting with the sources): fields sources k - Λ1 - Λ0 - P - At NLO, the q q dipole must be corrected by a gluon, e.g. : 14 / 30

25 Inclusive DIS at NLO fields Λ - 1 Λ - 0 sources P - k - δt NLO T LO At leading log accuracy, the contribution of the quantum modes in that strip is : δt NLO ( x, y ) = ln ( Λ ) 0 Λ H T LO ( x, y ) 1 15 / 30

26 Inclusive DIS at NLO These NLO corrections can be absorbed in the LO result, T LO +δt NLO Λ 0 = T LO Λ 1 provided one defines a new effective theory with a lower cutoff Λ 1 and an extended distribution of sources W Λ [ρ]: 1 fields Λ - 1 Λ - 0 sources P - k - W Λ 1 T LO [ ( Λ ) ] 0 1+ln Λ H W Λ 0 1 (JIMWLK equation for a small change in the cutoff) 16 / 30

27 2 Factorization in Deep Inelastic Scattering Leading Order Next to Leading Order Leading Log resummation 16 / 30

28 Inclusive DIS at Leading Log Iterate the previous process to integrate out all the slow field modes at leading log accuracy: Inclusive DIS at Leading Log accuracy 1 σ γ T = dz d 2 r ψ(q z, r ) 2 σ dipole (x, r ) 0 [Dρ σ dipole (x, r ) 2 d 2 ] X WxP [ρ] T LO ( x, y ) One does not need to evolve down to Λ 0: the DIS amplitude becomes independent of Λ when Λ xp fields sources Λ - xp - Λ1 - Λ0 - P - k - 17 / 30

29 3 Factorization in nucleus-nucleus collisions Power counting Leading Order Next to Leading Order Initial state factorization 17 / 30

30 Power counting CGC effective theory with cutoff at the scale Λ + 0 : fields Λ + 0 sources P + k + S = 1 F µν F µν 4 }{{} S YM + (J µ 1 + Jµ 2 ) A µ }{{} fast partons Expansion in g 2 in the saturated regime: dn g 1 [ ] dyd 2 p g 2 c 0 + c 1 g 2 + c 2 g / 30

31 3 Factorization in nucleus-nucleus collisions Power counting Leading Order Next to Leading Order Initial state factorization 18 / 30

32 Leading Order dn g dyd 2 p LO = trees Inclusive gluon spectrum at LO : dn g d 4 xd 4 y e ip (x y) A µ (x)a ν (y) dyd 2 p LO [ Dµ,F µν] = J ν 1 + J ν 2 }{{} Yang Mills equation, lim t Aµ (t, x) = 0 19 / 30

33 3 Factorization in nucleus-nucleus collisions Power counting Leading Order Next to Leading Order Initial state factorization 19 / 30

34 Next to Leading Order [FG, Venugopalan (2006)] dn g dyd 2 p NLO = trees Inclusive gluon spectrum at NLO : [ dn g 1 [ = ak Ì ] [ dyd 2 p 2 u a k Ì ] + v k NLO u, v Σ u Σ [ αì ] u ] dn g dyd 2 p LO Σ = initial Cauchy surface, Ì δ/δa init does not include virtual quarks loops a k and α are calculable analytically 20 / 30

35 Shift operator Ì Definition Equations of motion for a field and a small perturbation A+V (A) = J [ +V (A)] a = 0 Linear relationship between A and a : a(x) = [ aì ] u A(x) u Σ Diagrammatic interpretation : A(x) J A init Σ 21 / 30

36 Shift operator Ì Definition Equations of motion for a field and a small perturbation A+V (A) = J [ +V (A)] a = 0 Linear relationship between A and a : a(x) = [ aì ] u A(x) u Σ Diagrammatic interpretation : a(x) J a init Σ 21 / 30

37 3 Factorization in nucleus-nucleus collisions Power counting Leading Order Next to Leading Order Initial state factorization 21 / 30

38 Leading Logs [FG, Lappi, Venugopalan (2008)] Logs of Λ + and Λ 1 2 k u, v Σ [ ak Ì ] [ u a k Ì ] v + u Σ [ α Ì ] u = = ln ( Λ +) H 1 + ln ( Λ ) H 2 + terms w/o logs H 1,2 = JIMWLK Hamiltonian No mixing between the logs of Λ + and Λ Ensures the factorizability of these logs into JIMWLK-evolved distributions W[ρ 1,2 ] Roughly speaking, the mapping is: [ ak Ì ] u d 2 x 2π ( x u ) i ( x u ) 2 [ Ω( x ) Ω( u ) ] ab b x 22 / 30

39 Factorization of the Leading Logs of 1/x One can factorize all the powers of α s log(1/x 1,2 ) Single inclusive gluon spectrum at Leading Log accuracy dng [Dρ1 ] [ ] [ ] dn g [ρ 1,2 ] = Dρ dyd 2 p 2 W1 ρ1 W2 ρ2 dyd 2 p LLog }{{ } for fixed ρ 1,2 The factor dn g /dyd 2 p under the integral does not depend on y: the rapidity dependence comes entirely from the distributions W 1,2 This factorization establishes a link to other reactions (such as DIS) in the saturated regime 23 / 30

40 Multi-gluon correlations at Leading Log The previous factorization can be extended to multi-particle inclusive spectra : dn n g = dy 1 d 2 p 1 dy n d 2 p n LLog [Dρ1 ] [ ] [ ] dn g [ρ 1,2 ] = Dρ 2 W1 ρ1 W2 ρ2 dn g[ρ 1,2 ] dy 1 d 2 p 1 dy n d 2 p n Note: at Leading Log accuracy, all the rapidity correlations come from the evolution of the distributions W[ρ 1,2 ] they are a property of the pre-collision initial state This formula predicts long range ( y α 1 s ) correlations in rapidity 24 / 30

41 4 Extensions Quark production Beyond Leading Log 24 / 30

42 Quark spectrum at LO [FG, Kajantie, Lappi (2005)] p dn Q dyd 2 p LO = trees q Inclusive quark spectrum at LO : dn Q d 4 xd 4 y e ip (x y) ψ q(x)ψ dyd 2 q (y) p LO [i/ g/a m] ψ q (x) = 0 }{{} Dirac equation, lim t ψ q(t, x) = v(q)e iq x Note : for the quarks, LO = 1-loop, NLO = 2-loops 25 / 30

43 Quark spectrum at NLO [FG, Laidet (in preparation)] p dn Q dyd 2 p NLO = trees q Inclusive quark spectrum at NLO : dn Q dyd 2 p NLO = [ 1 2 k u, v Σ + [ ak Ì+b k Ì ψ ] u Σ u [ αì+β Ì ψ ] [ a k Ì+b k Ì ψ ] u ] dn Q dyd 2 p LO v Ì ψ δ/δψ init 26 / 30

44 Quark spectrum Leading Logs Reminder : [ ak Ì ] u d 2 x 2π where [ b x ] A Conjecture : [ bk Ì ψ ]u d 2 x 2π u i x i [ Ω(x (u x ) 2 ) Ω(u ) ] [ ] ab b x A acts only on Wilson lines contained in color fields u i x i [ Ω(x (u x ) 2 ) Ω(u ) ] [ ] ab b x ψ Then: [ b ] x + [ b ] x = b x, with no restriction on where the A ψ derivative acts (color field or spinor) we would recover the JIMWLK Hamiltonian, and factorization for quark production would ensue 27 / 30

45 4 Extensions Quark production Beyond Leading Log 27 / 30

46 Gluons at NNLO } {{ } LO } {{ } NLO } {{ } NNLO LO and NLO are known All the lines are dressed by the background classical field 28 / 30

47 Gluon spectrum at Next-to-Leading Log : sketch of the strategy i. Express the NNLO correction in terms of retarded objects ii. Conjecture : it can be related to the LO by a formula of the form : [ dn g = α dyd 2 1 Ì u + 1 α 2 Ì u Ì v p ! NNLO u, v, w Σ u Σ α 3 Ì u Ì v Ì w + 1 4! u, v Σ u, v, w, x Σ α 4 Ì u Ì v Ì w Ì x ] dn g dyd 2 p LO iii. Extract the logarithms of the cutoffs from the coefficients α 1 4 iv. Subtract two iterations of the Leading Log JIMWLK evolution Note : the JIMWLK Hamiltonian at next-to-leading log probably involves terms with higher derivatives (up to 4) 29 / 30

48 Summary Collisions of hadrons/nuclei at high energy require some knowledge about the multigluon Fock states of the projectiles. An effective description of these states is provided by the color glass condensate The LO is given by classical fields Higher orders contains logs of the energy For inclusive gluonic quantities, the leading logs can be factorized into two factors that describe the color content of the projectiles Most likely, this is also true for the inclusive quark spectrum Next to leading log may be tractable? 30 / 30