Color Glass Condensate

Transcription

1 Color Glass Condensate Schleching, Germany, February 2014 François Gelis IPhT, Saclay

2 Heavy Ion Collisions

3 From atoms to nuclei, to quarks and gluons m : atom (99.98% of the mass is in the nucleus) François Gelis Color Glass Condensate 1/140 Schleching, February 2014

4 From atoms to nuclei, to quarks and gluons < m : quarks + gluons François Gelis Color Glass Condensate 1/140 Schleching, February 2014

5 Quarks and gluons Strong interactions : Quantum Chromo-Dynamics Matter : quarks ; Interaction carriers : gluons a j i g (t a ) ij a c b g (T a ) bc i, j : quark colors ; a, b, c : gluon colors (t a ) ij : 3 3 SU(3) matrix ; (T a ) bc : 8 8 SU(3) matrix Lagrangian L = 1 4 F2 + f ψ f (i/d m f )ψ f Free parameters : quark masses m f, scale Λ QCD François Gelis Color Glass Condensate 2/140 Schleching, February 2014

6 Asymptotic freedom Running coupling : α s = g 2 /4π α s(e) = 2πN c (11N c 2N f ) log(e/λ QCD ) α S Durham 4-Jet Rate OPAL (preliminary) JADE ALEPH JADE Preliminary α S (M Z )=0.1182± s [ GeV ] François Gelis Color Glass Condensate 3/140 Schleching, February 2014

7 Color confinement The quark-antiquark potential increases linearly with distance François Gelis Color Glass Condensate 4/140 Schleching, February 2014

8 Color confinement In nature, we do not see free quarks and gluons (the closest we have to actual quarks and gluons are jets) Instead, we see hadrons (quark-gluon bound states): François Gelis Color Glass Condensate 5/140 Schleching, February 2014

9 Debye screening at high density r V(r) = exp( - r / r debye ) r In a dense environment, color charges are screened by their neighbours The Coulomb potential decreases exponentially beyond the Debye radius r debye Bound states larger than r debye cannot survive François Gelis Color Glass Condensate 6/140 Schleching, February 2014

10 Deconfinement transition 5 4 p/t 4 p SB /T flavour 2+1 flavour 2 flavour pure gauge T [MeV] Fast increase of the pressure : at T 270 MeV, if there are only gluons at T MeV, depending on the number of light quarks François Gelis Color Glass Condensate 7/140 Schleching, February 2014

11 QCD phase diagram Temperature Quark Gluon plasma hadronic phase Nuclei Neutron stars Color superconductor Net Baryon Density François Gelis Color Glass Condensate 8/140 Schleching, February 2014

12 QGP in the early universe Temperature Early Universe Quark Gluon plasma hadronic phase Nuclei Neutron stars Color superconductor Net Baryon Density François Gelis Color Glass Condensate 9/140 Schleching, February 2014

13 QGP in the early universe François Gelis Color Glass Condensate 10/140 Schleching, February 2014

14 Heavy ion collisions Temperature Heavy Ion Collision Quark Gluon plasma hadronic phase Nuclei Neutron stars Color superconductor Net Baryon Density François Gelis Color Glass Condensate 11/140 Schleching, February 2014

15 Experimental facilities : RHIC and LHC François Gelis Color Glass Condensate 12/140 Schleching, February 2014

16 Heavy ion collision at the LHC François Gelis Color Glass Condensate 13/140 Schleching, February 2014

17 t z What can be said about hadronic and nuclear collisions in terms of the underlying quarks and gluons degrees of freedom? François Gelis Color Glass Condensate 14/140 Schleching, February 2014

18 Stages of a nucleus-nucleus collision t freeze out hadrons gluons & quarks in eq. gluons & quarks out of eq. strong fields z kinetic theory ideal hydro viscous hydro classical dynamics François Gelis Color Glass Condensate 15/140 Schleching, February 2014

19 Stages of a nucleus-nucleus collision t Lecture I freeze out hadrons in eq. gluons & quarks in eq. gluons & quarks out of eq. strong fields z hydrodynamics viscous hydro classical fields Lecture I : Parton model, Gluon saturation François Gelis Color Glass Condensate 16/140 Schleching, February 2014

20 Stages of a nucleus-nucleus collision t Lecture II Lecture I freeze out hadrons in eq. gluons & quarks in eq. gluons & quarks out of eq. strong fields z hydrodynamics viscous hydro classical fields Lecture I : Parton model, Gluon saturation Lecture II : Color Glass Condensate, Factorization François Gelis Color Glass Condensate 16/140 Schleching, February 2014

21 Stages of a nucleus-nucleus collision t Lecture III Lecture II Lecture I freeze out hadrons in eq. gluons & quarks in eq. gluons & quarks out of eq. strong fields z hydrodynamics viscous hydro classical fields Lecture I : Parton model, Gluon saturation Lecture II : Color Glass Condensate, Factorization Lecture III : Instabilities, Thermalization François Gelis Color Glass Condensate 16/140 Schleching, February 2014

22 Parton model

23 Hadronic spectrum In nature, we do not see free quarks and gluons (the closest we have to actual quarks and gluons are jets) Instead, we see hadrons (quark-gluon bound states): The hadron spectrum is uniquely given by Λ QCD, m f But this dependence is non-perturbative (it can now be obtained fairly accurately by lattice simulations) François Gelis Color Glass Condensate 17/140 Schleching, February 2014

24 Nuclear spectrum But nuclear spectroscopy is out of reach of lattice QCD, even for the lightest nuclei François Gelis Color Glass Condensate 18/140 Schleching, February 2014

25 Do we need to know all this in order to describe hadronic/nuclear collisions in Quantum Chromodynamics? François Gelis Color Glass Condensate 19/140 Schleching, February 2014

26 Do we need to know all this in order to describe hadronic/nuclear collisions in Quantum Chromodynamics? NO! François Gelis Color Glass Condensate 19/140 Schleching, February 2014

27 A nucleon at rest is a very complicated object... Contains valence quarks + fluctuations at all space-time scales smaller than its own size Only the fluctuations that are longer lived than the external probe participate in the interaction process Interactions are very complicated if the constituents of the nucleon have a non trivial dynamics over time-scales comparable to those of the probe François Gelis Color Glass Condensate 20/140 Schleching, February 2014

28 Dilation of all internal time-scales for a high energy nucleon Interactions among constituents now take place over time-scales that are longer than the characteristic time-scale of the probe the constituents behave as if they were free the reaction sees a snapshot of the nucleon internals Many fluctuations live long enough to be seen by the probe. The nucleon appears denser at high energy (it contains more gluons) François Gelis Color Glass Condensate 20/140 Schleching, February 2014

29 What do we need in order to describe a hadronic collision? Provide a snapshot of the two projectiles Flavor and color of each parton Transverse position and momentum Since these properties are not know event-by-event, one should aim at a probabilistic description of the parton content of the projectiles François Gelis Color Glass Condensate 21/140 Schleching, February 2014

30 Why is this non trivial? In quantum mechanics, the transition probability from some hadronic states to the final state is expressed as : transition probability from hadrons to X Amplitudes h 1 h 2 X 2 The parton model assumes that we may be able to write it as : transition probability from hadrons to X partons {q,g} probability to find {q, g} in {h 1, h 2 } Amplitudes {q, g} X 2 This property is known as factorization. It can be justified in QCD, and it is a consequence of the separation between the timescale of confinement and the collision timescale François Gelis Color Glass Condensate 22/140 Schleching, February 2014

31 Deep Inelastic Scattering

32 Introduction to DIS Basic idea : smash a well known probe on a nucleon or nucleus in order to try to figure out what is inside... Photons are very well suited for that purpose because their interactions are well understood Deep Inelastic Scattering : collision between an electron and a nucleon or nucleus, by exchange of a virtual photon Variant : collision with a neutrino, by exchange of Z 0, W ± François Gelis Color Glass Condensate 23/140 Schleching, February 2014

33 Kinematical variables k P k q θ X Note : the virtual photon is space-like: q 2 0 Other invariants of the reaction : ν P q s (P + k) 2 M 2 (P + q) 2 = m 2 + 2ν + q2 X N One uses commonly : Q 2 q 2 and x Q 2 /2ν In general M 2 m2, and we have : 0 x 1 X N (x = 1 corresponds to the case of elastic scattering) François Gelis Color Glass Condensate 24/140 Schleching, February 2014

34 DIS cross-section The inclusive cross-section can be written as : E dσ e N 1 e 2 d 3 k = 32π 3 (s m 2 ) q 4π 4 Lµν W µν N where W µν is the hadronic tensor, defined as: 4π W µν [dφ X ](2π) 4 δ(p + q P X ) states X N(P) J ν(0) X X J µ(0) N(P) spin 4π W µν = d 4 y e iq y N(P) Jν (y) J µ (0) N(P) spin W µν contains all the informations about the properties of the nucleon under consideration that are relevant to the interaction with the photon François Gelis Color Glass Condensate 25/140 Schleching, February 2014

35 Structure functions For interactions with a photon : ( W µν = F 1 g µν qµqν q 2 ) + F 2 ν ( ) ( ) P q P q P µ q µ P q 2 ν q ν q 2 DIS cross-section in the nucleon rest frame : [ ] dσ e N de dω = α 2 em 2 sin 2 (θ/2)f 4m N E 2 sin m2 N (θ/2) ν cos2 (θ/2)f 2 where Ω is the solid angle of the scattered electron Note: F 1 is proportional to the interaction cross-section between the nucleon and a transverse photon François Gelis Color Glass Condensate 26/140 Schleching, February 2014

36 Bjorken scaling Bjorken scaling : F 2 depends very weakly on Q SLAC 0.5 F x François Gelis Color Glass Condensate 27/140 Schleching, February 2014

37 Longitudinal structure function F L F 2 2xF 1 is quite smaller than F 2 : F L vs. F 2 for Q 2 = 20 GeV 2 F L F e x François Gelis Color Glass Condensate 28/140 Schleching, February 2014

38 Analogy with the e- mu- cross-section In terms of F 1 and F 2, the DIS cross-section reads: [ ] dσ e N de dω = α 2 em 2F 4m N E 2 sin 4 θ 1 sin 2 θ 2 + m2 N ν F 2 cos 2 θ 2 2 Compare with the e µ cross-section: [ ] dσ e µ de dω = α2 emδ(1 x) sin 2 θ 4m µe 2 sin 4 θ 2 + m2 µ θ ν cos2 2 2 If the constituents of the nucleon that interact in the DIS process were spin 1/2 point-like particles, we would have: 2F 1 = m N m c δ(1 x c), F 2 = mc m N δ(1 x c) where m c is some effective mass for the constituent (comparable to m N because it is trapped inside the nucleon) and x c Q 2 /2q p c with p µ c the momentum of the constituent François Gelis Color Glass Condensate 29/140 Schleching, February 2014

39 Analogy with the e- mu- cross-section If p µ c = x F P µ, then x c = x/x F, and: 2F 1 δ(x x F ), F 2 δ(x x F ) The structure functions F 1 and F 2 would therefore not depend on Q 2, but only on x Conclusion : Bjorken scaling could be explained if the constituents of the nucleon that are probed in DIS are spin 1/2 point-like particles The variable x measured in DIS would have to be identified with the fraction of momentum carried by the struck constituent François Gelis Color Glass Condensate 30/140 Schleching, February 2014

40 Pre-QCD parton model The historical parton model describes the nucleon as a collection of point-like fermions, called partons A parton of type i, carrying the fraction x F of the nucleon momentum, gives the following contribution to the hadronic tensor : 4π W µν d 4 p i = (2π) 2π 4 δ(p 2 ) (2π) 4 δ(x F P + q p ) x F P J µ (0) p p J ν (0) xf P spin François Gelis Color Glass Condensate 31/140 Schleching, February 2014

41 Pre-QCD parton model The historical parton model describes the nucleon as a collection of point-like fermions, called partons A parton of type i, carrying the fraction x F of the nucleon momentum, gives the following contribution to the hadronic tensor : 4π W µν d 4 p i = (2π) 2π 4 δ(p 2 ) (2π) 4 δ(x F P + q p ) x F P J µ (0) p p J ν (0) xf P spin For the scattering on a spin 1/2 elementary constituent, one has: 4π W µν i = 2π x F δ(x F x) e 2 i [ (g µν qµ q ν q 2 ) + 2x F P q ( ) ( )] P µ q µ P q P ν q ν P q q 2 q 2 François Gelis Color Glass Condensate 31/140 Schleching, February 2014

42 Pre-QCD parton model If there are f i (x F )dx F partons of type i with a momentum fraction between x F and x F + dx F, we have W µν = i 1 0 dx F x F f i (x F ) W µν i, F 1 = 1 2 e 2 i f i (x), F 2 = 2xF 1 i Callan-Gross relation : F 2 = 2xF 1 Consequence of spin 1/2 point-like partons Exercise : for spin 0 partons, show that Caveats and puzzles : W µν i (2x F P µ + q µ )(2x F P ν + q ν ) and F 1 = 0 The parton model assumes that partons are free inside the nucleon. How does this work in a strongly bound state? One would like to have a field theoretical description of what is going on, including the effect of interactions, quantum fluctuations... François Gelis Color Glass Condensate 32/140 Schleching, February 2014

43 What have we learned from QCD? Asymptotic freedom + time dilation in a high energy hadron explain why the partons appear as almost free at large Q 2 QCD loop corrections lead to violations of Bjorken scaling, that are visible as a Q 2 dependence of the structure functions. Physically, 1/Q is the spatial resolution at which the hadron is probed Parton distributions are non-perturbative in QCD, but their Q 2 and x dependence are governed by equations that are perturbative (DGLAP, BFKL) One can prove that the parton distributions are universal, i.e. are the same in all inclusive processes François Gelis Color Glass Condensate 33/140 Schleching, February 2014

44 DIS results for F 2 and DGLAP fit at NLO : François Gelis Color Glass Condensate 34/140 Schleching, February 2014

45 NNLO parton distributions and possible caveats 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 François Gelis Color Glass Condensate 35/140 Schleching, February 2014

46 NNLO parton distributions and possible caveats xf H1 and ZEUS xg 2 2 Q = 10 GeV 10 xs Large x : dilute, dominated by single parton scattering 1 xu v xd v HERAPDF1.0 exp. uncert. model uncert. parametrization uncert x François Gelis Color Glass Condensate 35/140 Schleching, February 2014

47 NNLO parton distributions and possible caveats xf H1 and ZEUS xg 2 2 Q = 10 GeV 10 xs Small x : dense, multi-parton interactions become likely 1 xu v xd v HERAPDF1.0 exp. uncert. model uncert. parametrization uncert x François Gelis Color Glass Condensate 35/140 Schleching, February 2014

48 Small x data displayed differently... (Geometrical scaling) Small x data (x 10 2 ) displayed against τ log(x 0.32 Q 2 ) : 10 1 σ γ* p (mb) H1 ZEUS E665 NMC τ François Gelis Color Glass Condensate 36/140 Schleching, February 2014

49 Eikonal Scattering

50 DIS off a highly boosted target Note : cross-sections are Lorentz invariant, but the microscopic interpretation may be frame dependent Some useful insight can be gained with a frame in which the target proton has a large P 3 momentum All the proton internal time scales are Lorentz dilated : its constituents appear frozen to the incoming virtual photon (they behave as if they have a mass P 3 ) In the limit P 3, the interactions with such a constituent are equivalent to the interactions with its radiated field E a, B a François Gelis Color Glass Condensate 37/140 Schleching, February 2014

51 Setup Consider the scattering amplitude off an external potential : S βα β out αin = βin U(+, ) αin where U(+, ) is the evolution operator from t = to t = + [ ] U(+, ) = T exp i d 4 x L int(φ in(x)) Note : L int contains the self-interactions of the fields and their interactions with the external potential We want to calculate its high energy limit (eikonal limit): S ( ) βα lim βin e iωk 3 U(+, ) e +iωk3 αin ω + }{{} boosted state where K 3 is the generator of boosts in the +z direction François Gelis Color Glass Condensate 38/140 Schleching, February 2014

52 Eikonal scattering in a nutshell In a scattering at high energy, the collision time goes to zero as E 1 With scalar interactions, this implies a decrease of the scattering amplitude as E 1 With vectorial interactions, this decrease is compensated by the growth of the components J 0,3 of the vector current the eikonal approximation gives the finite limit of the scattering amplitude in the case of vectorial interactions when E + François Gelis Color Glass Condensate 39/140 Schleching, February 2014

53 Light-cone coordinates Light-cone coordinates are defined by choosing a privileged axis (generally the z axis) along which particles have a large momentum. Then, for any 4-vector a µ, one defines : a + a0 + a 3 2, a a0 a 3 2 a 1,2 unchanged. Notation : a (a 1, a 2 ) Some useful formulas : x y = x + y + x y + x y d 4 x = dx + dx d 2 x = Notation : + x, x + François Gelis Color Glass Condensate 40/140 Schleching, February 2014

54 Light-cone coordinates The Dalembertian is bilinear in the derivatives +, The metric tensor is non diagonal : g µν = g µν = x _ t x + x + = 0 x _ = 0 z François Gelis Color Glass Condensate 41/140 Schleching, February 2014

55 Action of K (K = K 3 ) e iωk P e iωk = e ω P e iωk P + e iωk = e +ω P + e iωk P j e iωk = P j Simple rescaling of the various operators. This suggests that the light-cone framework is simpler in order to study processes involving highly boosted particles These relations play an essential role in the eikonal approximation François Gelis Color Glass Condensate 42/140 Schleching, February 2014

56 Eikonal limit Consider an external vector potential, that couples via e A µ (x)j µ (x) (J µ is the Noether current associated to some conserved charge.) Assume that the external potential is non-zero only in a finite range in x +, x + [ L, +L] Action of K on states and operators e iωk p in = (e ω p +, p ) in e iωk a in (q)eiωk = a in (eω q +, e ω q, q ) e iωk φ in (x)e iωk = φ in (e ω x +, e ω x, x ) François Gelis Color Glass Condensate 43/140 Schleching, February 2014

57 Eikonal limit Split the S matrix U(+, ) into three factors : U(+, ) = U(+, +L) U(+L, L) U( L, ) Upon application of K, this becomes : e iωk U(+, )e iωk = e iωk U(+, +L)e iωk e iωk U(+L, L)e iωk e iωk U( L, )e iωk The external potential A µ (x) is unaffected by K Action of K on J µ (x) e iωk J i (x)e iωk = J i (e ω x +, e ω x, x ) e iωk J (x)e iωk = e ω J (e ω x +, e ω x, x ) e iωk J + (x)e iωk = e ω J + (e ω x +, e ω x, x ) François Gelis Color Glass Condensate 44/140 Schleching, February 2014

58 Eikonal limit The factors U(+, +L) and U( L, ) do not contain the external potential. In order to deal with these factors, it is sufficient to change variables : e ω x + x +, e ω x x. This leads to : lim ω + eiωk U(+, +L)e iωk = U 0 (+, 0) lim ω + eiωk U( L, )e iωk = U 0 (0, ) where U 0 is the same as U, but with the self-interactions only François Gelis Color Glass Condensate 45/140 Schleching, February 2014

59 Eikonal limit Therefore, in the limit ω +, we have : [ lim ω + eiωk U(+L, L)e iωk = exp ie with ] d 2 x χ( x )ρ( x ) χ( x ) dx + A (x +, 0, x ) ρ( x ) dx J + (0, x, x ) The high-energy limit of the scattering amplitude is : S ( ) βα = [ β U0 in (+, 0) exp ie χ( x )ρ( x ) ]U 0 (0, ) αin x Only the component of the vector potential matters The self-interactions and the interactions with the external potential are factorized parton model This is an exact result in the limit ω + François Gelis Color Glass Condensate 46/140 Schleching, February 2014

60 Eikonal limit For each intermediate state δ in {k + i, k i }, define the corresponding light-cone wave function by : Ψ δα ({k + i, x i }) i d 2 ki (2π) 2 e i ki x i δin U(0, ) αin Each charged particle going through the external field acquires a phase proportional to its charge (antiparticles get an opposite phase) : Ψ δα ({k + i, x i }) Ψ δα ({k + i, x i }) i U i ( x ) U i ( x ) T + exp [ig i dx + A a (x +, 0, x )t a] François Gelis Color Glass Condensate 47/140 Schleching, February 2014

61 Eikonal limit The number and the nature of the particles is unchanged under the action of the eikonal operator. In terms of the transverse coordinates, we simply have γin e ig ρχ [ ] δin = δnn 4πk + i δ(k + i k + i )δ( x i x i )U Ri ( x i ) i where U R ( x ) is a Wilson line operator, in the representation R appropriate for the particle going through the target Therefore, the high energy scattering amplitude can be written as : S ( ) βα = [ [ ] dφ i ]Ψ δβ ({k+ i, x i }) U Ri ( x i ) Ψ δα ({k + i, x i }) δ i δ i δ François Gelis Color Glass Condensate 48/140 Schleching, February 2014

62 DIS in the Eikonal limit

63 Differential DIS cross-section Differential photon-target cross-section (γ T qq + X) : d 3 k d 3 p 1 dσ γ T = (2π) 2 2E k (2π) 3 2E p 2q 2πδ(q k p ) M µ (q k, p)m ν (q k, p) ɛ µ(q) ɛ ν(q), k, p : momenta of the quark and antiquark q : momentum of the virtual photon ɛ µ(q) : polarization vector François Gelis Color Glass Condensate 49/140 Schleching, February 2014

64 Total inclusive cross-section If we integrate out the final quark and antiquark, we get : with σ γ T = 1 0 dz d 2 r ψ(q z, r ) 2 σ dipole ( r ) σ dipole( r ) 2 d 2 X Tr 1 U( X N + r c 2 )U ( X r 2 ) and ψ for the light-cone wave function for a photon that splits into a quark-antiquark intermediate state. François Gelis Color Glass Condensate 50/140 Schleching, February 2014

65 Dipole cross-section Computing F 2 requires to know the dipole amplitude T( x, y ) Y 1 N c Tr 1 U( x )U ( y ) as a function of dipole size and rapidity This object is often presented in the form of the dipole cross-section : σ dip ( r, Y) 2 d 2 b T( b r 2, b + r 2 ) Y François Gelis Color Glass Condensate 51/140 Schleching, February 2014

66 Golec-Biernat Wusthoff model GBW modeled the dipole cross-section as a Gaussian, with an energy dependence entirely contained in Q s [ σ dip ( r, Y) = σ 0 1 e Qs(Y)2 r 2 /4] Q 2 s(y) = Q 2 0 e λ(y Y 0) The exponential form in σ dip is inspired of Glauber scattering The fit parameters are σ 0, Q 0, λ and possibly an effective quark mass in the photon wave-function Quite good for all small-x HERA data, with some problems at large Q 2 François Gelis Color Glass Condensate 52/140 Schleching, February 2014

67 State of the art : AAMQS model GBW model used only as initial condition Evolution with running coupling BK equation Comparison with HERA data Fit with only light quarks Q =0.85 GeV Data Q =2.0 GeV Theory 1 σ r 0.5 Fit including heavy quarks Q =0.85 GeV Data Q =2.0 GeV Theory 1 σ r Q =4.5 GeV 2 2 Q =8.5 GeV Q =4.5 GeV 2 2 Q =8.5 GeV σ r 0.5 σ r Q =10.0 GeV 2 2 Q =12.0 GeV Q =10.0 GeV 2 2 Q =12.0 GeV σ r 0.5 σ r Q =15.0 GeV 2 2 Q =28.0 GeV Q =15.0 GeV 2 2 Q =28.0 GeV σ r 0.5 σ r σ r Q =35 GeV Q =45 GeV σ r Q =35 GeV Q =45 GeV x x x x 2 10 François Gelis Color Glass Condensate 53/140 Schleching, February 2014

68 Summary of Lecture I

69 DIS results for F 2 (DGLAP equation at NLO) François Gelis Color Glass Condensate 54/140 Schleching, February 2014

70 Small x data displayed differently... (Geometrical scaling) Small x data (x 10 2 ) displayed against τ log(x 0.32 Q 2 ) : 10 1 σ γ* p (mb) H1 ZEUS E665 NMC τ François Gelis Color Glass Condensate 55/140 Schleching, February 2014

71 DIS cross-section in the eikonal limit σ γ T (x, Q 2 ) = 1 0 dz d 2 r ψ(q 2 z, r ) 2 σ dipole (x, r ) Golec-Biernat Wusthoff model : σ dip(x, r ) = σ 0 [ 1 e Q2 s (x)r2 /4] Q 2 s(x) = Q 2 0 (x/x 0 ) λ State of the art : AAMQS model The GBW model is used as input at x Smaller x s are obtained from the Balitsky-Kovchegov equation σ r σ r σ r σ r Q =0.85 GeV Data Theory Q =4.5 GeV 2 2 Q =10.0 GeV 2 2 Q =15.0 GeV Fit with only light quarks 2 2 Q =2.0 GeV 2 2 Q =8.5 GeV 2 2 Q =12.0 GeV 2 2 Q =28.0 GeV σ r Q =35 GeV x Q =45 GeV x 2 10 François Gelis Color Glass Condensate 56/140 Schleching, February 2014

72 BFKL equation

73 Scattering of a dipole Take a virtual photon as initial and final state. At lowest order, the scattering amplitude can be written as : Ψ (0) ( x, y ) 2 tr [ ] U( x )U ( y ) It turns out that 1-loop corrections to this contribution are enhanced by α s log(p + ), which can be large when the quark or antiquark has a large p + In the gauge A + = 0, the emission of a gluon of momentum k by a quark can be written as : = 2gt a ɛ λ k k 2 François Gelis Color Glass Condensate 57/140 Schleching, February 2014

74 Scattering of a dipole The following diagrams must be evaluated : + h.c. When connecting two gluons, one must use : ɛ i λ ɛ j λ = gij λ François Gelis Color Glass Condensate 58/140 Schleching, February 2014

75 Virtual corrections Consider first the loop corrections inside the wavefunction of the incoming or outgoing dipole Example = Ψ (0) ( x, y ) 2 tr [ t a t a U( x )U ( y ) ] 2α s dk + k + d 2 z (2π) 2 ( x z ) ( x z ) ( x z ) 2 ( x z ) 2 Reminder : t a t a = (N 2 c 1)/2N c (denoted C F ) François Gelis Color Glass Condensate 59/140 Schleching, February 2014

76 Virtual corrections The sum of all virtual corrections is : C F αs dk + d 2 ( x y z ) 2 π 2 k + ( x z ) 2 ( y z ) 2 Ψ (0) ( x, y ) 2 [ ] tr U( x )U ( y ) The integral over k + is divergent. It should have an upper bound at p + : p + dk + k + = ln(p+ ) = Y When Y is large, α s Y may not be small,and these corrections should be resummed François Gelis Color Glass Condensate 60/140 Schleching, February 2014

77 Real corrections There are also real corrections, for which the state that interacts with the target has an extra gluon Example = Ψ (0) ( x, y ) 2 tr [ t a U( x )t b U ( y ) ] 4α s dk + k + d 2 z (2π) 2 Ũab( z ) ( x z ) ( x z ) ( x z ) 2 ( x z ) 2 (Ũab( z ) is a Wilson line in the adjoint representation) In order to simplify the color structure, use : + the SU(N c ) Fierz identity : t a Ũ ab ( z ) = U( z )t b U ( z ) t b ijt b kl = 1 2 δ ilδ jk 1 2N c δ ij δkl François Gelis Color Glass Condensate 61/140 Schleching, February 2014

78 Evolution equation Denote : S( x, y ) 1 N c tr [ U( x )U ( y ) ] The full LO + NLO scattering amplitude reads : [ Ψ N (0) c ( x, y ) 2 S( x, y ) αsncy d 2 ( x y z ) 2 2π 2 ( x z ) 2 ( y z ) 2 { } ] S( x, y ) S( x, z )S( z, y ) S( x, y ) Y = αsnc d 2 ( x y z ) 2 2π 2 ( x z ) 2 ( y z ) 2 { } S( x, y ) S( x, z )S( z, y ) François Gelis Color Glass Condensate 62/140 Schleching, February 2014

79 BFKL equation Kuraev, Lipatov, Fadin (1977), Balitsky, Lipatov (1978) Write S( x, y ) 1 T( x, y ) and assume that we are in the dilute regime, so that the scattering amplitude T is small Drop the terms that are non-linear in T BFKL equation in coordinate space T( x, y ) Y = αsnc d 2 z ( x y ) 2 2π 2 ( x z ) 2 ( y z ) 2 { } T( x, z ) + T( z, y ) T( x, y ) François Gelis Color Glass Condensate 63/140 Schleching, February 2014

80 Gluon Saturation

81 Unitarity problem The mapping T α s N c T has a positive eigenvalue ω z Solutions of the BFKL equation grow exponentially as exp(ωy) when Y + violation of unitarity... In perturbation theory, the forward scattering amplitude between a small dipole and a target made of gluons reads : T( x, y ) x y 2 xg(x, x y 2 ) where Y ln(1/x) Therefore, the exponential behavior of T is related to the increase of the gluon distribution at small x T e ωy xg(x, Q 2 ) 1 x ω François Gelis Color Glass Condensate 64/140 Schleching, February 2014

82 Parton evolution under boosts at low energy, only valence quarks are present in the hadron wave function François Gelis Color Glass Condensate 65/140 Schleching, February 2014

83 Parton evolution under boosts when energy increases, new partons are emitted the emission probability is α dx s x momentum fraction of the gluon α sln( 1 x ), with x the longitudinal at small-x (i.e. high energy), these logs need to be resummed François Gelis Color Glass Condensate 65/140 Schleching, February 2014

84 Parton evolution under boosts as long as the density of constituents remains small, the evolution is linear: the number of partons produced at a given step is proportional to the number of partons at the previous step François Gelis Color Glass Condensate 65/140 Schleching, February 2014

85 Parton evolution under boosts eventually, the partons start overlapping in phase-space François Gelis Color Glass Condensate 65/140 Schleching, February 2014

86 Parton evolution under boosts parton recombination becomes favorable after this point, the evolution is non-linear: the number of partons created at a given step depends non-linearly on the number of partons present previously Balitsky (1996), Kovchegov (1996,2000) Jalilian-Marian, Kovner, Leonidov, Weigert (1997,1999) Iancu, Leonidov, McLerran (2001) François Gelis Color Glass Condensate 65/140 Schleching, February 2014

87 Saturation domain François Gelis Color Glass Condensate 66/140 Schleching, February 2014

88 Saturation domain Saturation criterion [Gribov, Levin, Ryskin (1983)] α s Q }{{ 2 A } 2/3 xg(x, Q 2 ) }{{} σ gg g surface density Q 2 Q 2 s α sxg(x, Q 2 s) A 2/3 }{{} saturation momentum 1 A 1/3 x 0.3 François Gelis Color Glass Condensate 66/140 Schleching, February 2014

89 Balitsky-Kovchegov equation

90 Non-linear evolution equation The first evolution equation we derived has the non-linear effects due to recombination : T( x, y ) = αsnc d 2 ( x y z ) 2 Y 2π 2 ( x z ) 2 ( y z ) 2 { } T( x, z ) + T( z, y ) T( x, y ) T( x, z )T( z, y ) (Balitsky-Kovchegov equation) The r.h.s. vanishes when T reaches 1, and the growth stops. The non-linear term lets both dipoles interact after the splitting of the original dipole Both T = 0 and T = 1 are fixed points of this equation T = ɛ : r.h.s.> 0 T = 0 is unstable T = 1 ɛ : r.h.s.> 0 T = 1 is stable François Gelis Color Glass Condensate 67/140 Schleching, February 2014

91 Caveats So far, we have studied the scattering amplitude between a color dipole and a god given patch of color field. This is too crude to describe any realistic situation One can describe Deep Inelastic Scattering as an interaction between a dipole and the target, but for that we need to improve the treatment of the target An experimentally measured cross-section is an average over many collisions, and the target fields fluctuate event-by-event : T T François Gelis Color Glass Condensate 68/140 Schleching, February 2014

92 Balitsky hierarchy Because of this average over the target configurations, the evolution equation we have derived should be written as : T( x, y ) = αsnc d 2 ( x y z ) 2 Y 2π 2 ( x z ) 2 ( y z ) 2 { } T( x, z ) + T( z, y ) T( x, y ) T( x, z )T( z, y ) As one can see, the equation is no longer a closed equation, since the equation for T depends on a new object, T T One can derive an evolution equation for T T. Its right hand side contains objects with six Wilson lines There is in fact an infinite hierarchy of nested evolution equations, whose generic structure is (UU ) n = Y (UU ) n (UU ) n+1 François Gelis Color Glass Condensate 69/140 Schleching, February 2014

93 BK equation as a mean-field approximation In the large N c approximation, the equations of the Balitsky hierarchy can be rewritten in terms of the dipole operator T tr(uu ) only. But they still contain averages like T n In order to truncate the hierarchy of equations, one may assume a mean field approximation T T T T This approximation gives for T the same evolution equation as the one we had for a fixed configuration of the target (Balitsky-Kovchegov equation) François Gelis Color Glass Condensate 70/140 Schleching, February 2014

94 Geometrical Scaling from BK evolution

95 Analogy with reaction-diffusion processes Munier, Peschanski (2003,2004) Assume translation and rotation invariance, and define : N(Y, k ) 2π d 2 x e i k x T(0, x ) Y x 2 From the Balitsky-Kovchegov equation for T Y, we obtain the following equation for N : with N(Y, k ) Y [ ] = αsnc χ( L )N(Y, k ) N 2 (Y, k ) π L ln(k 2 /k 2 0) χ(γ) 2ψ(1) ψ(γ) ψ(1 γ) d ln Γ(z) ψ(z) dz François Gelis Color Glass Condensate 71/140 Schleching, February 2014

96 Analogy with reaction-diffusion processes Expand the function χ(γ) to second order (diffusion approximation) around its minimum γ = 1/2 Introduce new variables : t Y z L + αsnc 2π χ (1/2) Y The equation for N becomes : tn = 2 zn + N N 2 (known as the Fisher-Kolmogorov-Petrov-Piscounov equation) François Gelis Color Glass Condensate 72/140 Schleching, February 2014

97 Analogy with reaction-diffusion processes Interpretation : this equation is typical for all the diffusive systems subject to a reaction A A + A 2 zn : diffusion term (the quantity under consideration can hop from a site to the neighboring sites) +N : gain term corresponding to A A + A N 2 : loss term corresponding to A + A A Note : this equation has two fixed points : N = 0 : N = 1 : unstable stable The stable fixed point at N = 1 exists only if one keeps the loss term. One would not have it from the BFKL equation François Gelis Color Glass Condensate 73/140 Schleching, February 2014

98 Traveling waves Assume an initial condition N(t 0, z) that goes smoothly from 1 at z = to 0 at z = +, and behaves like exp( βz) when z 1 N(t,z) z The solution of the F-KPP equation is known to behave like a traveling wave at asymptotic times : N(t, z) with v(t) 2t 3 ln(t)/2 N(z v(t)) t + François Gelis Color Glass Condensate 74/140 Schleching, February 2014

99 Traveling waves Assume an initial condition N(t 0, z) that goes smoothly from 1 at z = to 0 at z = +, and behaves like exp( βz) when z 1 N(t,z) z The solution of the F-KPP equation is known to behave like a traveling wave at asymptotic times : N(t, z) with v(t) 2t 3 ln(t)/2 N(z v(t)) t + François Gelis Color Glass Condensate 74/140 Schleching, February 2014

100 Traveling waves Assume an initial condition N(t 0, z) that goes smoothly from 1 at z = to 0 at z = +, and behaves like exp( βz) when z 1 N(t,z) z The solution of the F-KPP equation is known to behave like a traveling wave at asymptotic times : N(t, z) with v(t) 2t 3 ln(t)/2 N(z v(t)) t + François Gelis Color Glass Condensate 74/140 Schleching, February 2014

101 Traveling waves Assume an initial condition N(t 0, z) that goes smoothly from 1 at z = to 0 at z = +, and behaves like exp( βz) when z 1 N(t,z) z The solution of the F-KPP equation is known to behave like a traveling wave at asymptotic times : N(t, z) with v(t) 2t 3 ln(t)/2 N(z v(t)) t + François Gelis Color Glass Condensate 74/140 Schleching, February 2014

102 Geometrical scaling in DIS Iancu, Itakura, McLerran (2002) Mueller, Triantafyllopoulos (2002) Munier, Peschanski (2003) Going back to the original variables, one gets : N(Y, k ) = N (k /Q s(y)) with Q 2 s(y) = k 2 0 Y 3 2(1 γ) e αsχ ( 1 2 )( 1 2 γ)y Going from N(Y, k ) to T(0, x ) Y, we obtain : T(0, x ) Y = T(Q s(y)x ) François Gelis Color Glass Condensate 75/140 Schleching, February 2014

103 Geometrical scaling in DIS Reminder : γ p cross-section expressed in terms of T: 1 σ γ p(y, Q 2 ) = 2 σ 0 d 2 x dz ψ(z, x, Q 2 ) 2 T(0, x ) Y 0 If one neglects the quark masses in ψ, the scaling property of T Y imply that σ γ p depends only on the ratio Q 2 /Q 2 s(y), rather than on Q 2 and Y separately François Gelis Color Glass Condensate 76/140 Schleching, February 2014

104 F 2 (x, Q 2 ) σ γ p(log(q 2 /Q 2 s(x))) 10 1 σ γ* p (mb) H1 ZEUS E665 NMC τ François Gelis Color Glass Condensate 77/140 Schleching, February 2014

105 Color Glass Condensate

106 DIS and other elementary reactions Reactions involving a hadron or nucleus and an elementary projectile (γ, q or g) are fairly straightforward to study Deep Inelastic Scattering is the archetype of all these processes These processes play a role in the study of proton-nucleus collisions, where the proton is described as a dilute beam of quarks and gluons François Gelis Color Glass Condensate 78/140 Schleching, February 2014

107 DIS and other elementary reactions Reactions involving a hadron or nucleus and an elementary projectile (γ, q or g) are fairly straightforward to study qa qx These processes play a role in the study of proton-nucleus collisions, where the proton is described as a dilute beam of quarks and gluons François Gelis Color Glass Condensate 78/140 Schleching, February 2014

108 DIS and other elementary reactions Reactions involving a hadron or nucleus and an elementary projectile (γ, q or g) are fairly straightforward to study ga gx These processes play a role in the study of proton-nucleus collisions, where the proton is described as a dilute beam of quarks and gluons François Gelis Color Glass Condensate 78/140 Schleching, February 2014

109 DIS and other elementary reactions Reactions involving a hadron or nucleus and an elementary projectile (γ, q or g) are fairly straightforward to study qa qγx These processes play a role in the study of proton-nucleus collisions, where the proton is described as a dilute beam of quarks and gluons François Gelis Color Glass Condensate 78/140 Schleching, February 2014

110 DIS and other elementary reactions Reactions involving a hadron or nucleus and an elementary projectile (γ, q or g) are fairly straightforward to study qa qgx These processes play a role in the study of proton-nucleus collisions, where the proton is described as a dilute beam of quarks and gluons François Gelis Color Glass Condensate 78/140 Schleching, February 2014

111 DIS and other elementary reactions Reactions involving a hadron or nucleus and an elementary projectile (γ, q or g) are fairly straightforward to study ga qqx These processes play a role in the study of proton-nucleus collisions, where the proton is described as a dilute beam of quarks and gluons François Gelis Color Glass Condensate 78/140 Schleching, February 2014

112 Nucleus-nucleus collisions? What about collisions where the two projectiles are equally dense? It would be nice to have a formalism that allows one to treat the two projectiles on the same footing :? Conjecture (for the Leading Order): Before the collision, the two projectiles are described by their own color field, like in DIS After the collision (i.e. in the forward light-cone), there is a color field that obeys Yang-Mills equations, and whose boundary condition on the light-cone is given by the fields of the incoming nuclei François Gelis Color Glass Condensate 79/140 Schleching, February 2014

113 Nucleus-nucleus collisions? Can we set up a framework where this can be justified? Note : this new formalism should lead to the same results for DIS, not something completely different... Can we include all the multiple scattering corrections? How do we compute observables for two saturated objects? Can we compute and resum all the large logs of 1/x 1,2? François Gelis Color Glass Condensate 80/140 Schleching, February 2014

114 Introduction to the Color Glass Condensate The BK equation, can be viewed as a projectile-centric description of a collision process. The rapidity evolution comes from the dressing of the projectile as it is boosted One may see the Color Glass Condensate as a description of the same physics from the point of view of the target In this target-centric description, the projectile does not change, but the color fields of the target depend on the rapidity François Gelis Color Glass Condensate 81/140 Schleching, February 2014

115 Degrees of freedom and their interplay McLerran, Venugopalan (1994) Iancu, Leonidov, McLerran (2001) 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) may evolve during the collision treated as standard gauge fields eikonal coupling to the current J µ : J µ A µ S = 1 F µν F µν 4 }{{} S YM + J µ A µ }{{} fast partons François Gelis Color Glass Condensate 82/140 Schleching, February 2014

116 Semantics McLerran (2000) Color : pretty much obvious... Glass : the system has degrees of freedom whose timescale is much larger than the typical timescales for interaction processes. Moreover, these degrees of freedom are stochastic variables, like in spin glasses for instance Condensate : the soft degrees of freedom are as densely packed as they can (the density remains finite, of order α 1 s, due to the interactions between gluons) François Gelis Color Glass Condensate 83/140 Schleching, February 2014

117 Target average The averaged dipole amplitude T studied in the Balitsky-Kovchegov approach can be written as : [ T( x, y ) = [Dρ] W Y [ρ] 1 1 ] tr(u( x )U ( y N )) c The Y dependence of the expectation value T comes from the Y dependence of W Y [ρ] François Gelis Color Glass Condensate 84/140 Schleching, February 2014

118 JIMWLK evolution equation Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner W Y [ρ] Y = 1 δ 2 x, y δρ a( x ) χ δ ab( x, y ) δρ b ( y ) }{{} H (JIMWLK Hamiltonian) W Y [ρ] with χ ab ( x, y ) αs d 2 ( x z ) ( y z z ) 4π 3 ( x z ) 2 ( y z ) 2 [( )( )] 1 Ũ ( x )Ũ( z ) 1 Ũ ( z )Ũ( y ) ab Ũ is a Wilson line in the adjoint representation, that exponentiates the gauge field A + such that 2 A + = ρ François Gelis Color Glass Condensate 85/140 Schleching, February 2014

119 JIMWLK evolution equation Sketch of a derivation : exploit the frame independence in order to write : O Y = [Dρ] W 0 [ρ] O Y [ρ] = [Dρ] W Y [ρ] O 0 [ρ] }{{}}{{} Balitsky-Kovchegov description CGC description Universality : the evolution of W Y [ρ] does not depend on the observable one is considering François Gelis Color Glass Condensate 86/140 Schleching, February 2014

120 Initial condition : McLerran Venugopalan model The JIMWLK equation must be completed by an initial condition, given at some moderate x 0 As with DGLAP, the initial condition is non-perturbative The McLerran-Venugopalan model is often used as an initial condition at moderate x 0 for a large nucleus : partons distributed randomly z many partons in a small tube no correlations at different x The MV model assumes that the density of color charges ρ( x ) has a Gaussian distribution : [ ] W x0 [ρ] = exp d 2 ρ a( x )ρ a( x ) x 2µ 2 ( x ) François Gelis Color Glass Condensate 87/140 Schleching, February 2014

121 CGC applied to DIS

122 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 : François Gelis Color Glass Condensate 88/140 Schleching, February 2014

123 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 François Gelis Color Glass Condensate 89/140 Schleching, February 2014

124 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 Λ - 1 Λ - 0 sources P - k - At NLO, the q q dipole must be corrected by a gluon, e.g. : François Gelis Color Glass Condensate 90/140 Schleching, February 2014

125 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 François Gelis Color Glass Condensate 91/140 Schleching, February 2014

126 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 1 [ρ]: fields Λ - 1 Λ - 0 sources P - k - T LO [ ( Λ ) ] W Λ ln 1 Λ H W Λ 0 1 (JIMWLK equation for a small change in the cutoff) François Gelis Color Glass Condensate 92/140 Schleching, February 2014

127 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 - François Gelis Color Glass Condensate 93/140 Schleching, February 2014

128 Summary of Lecture II

129 Projectile-centric description : Balitsky-Kovchegov equation T(x, y ) 1 1 N c tr (U(x )U (y )) T Y αs preserves unitarity [ ] T TT }{{} T T dynamical geometrical scaling input: model for T at the initial Y 0 : Golec-Biernat-Wusthof, McLerran-Venugopalan,... François Gelis Color Glass Condensate 94/140 Schleching, February 2014

130 Projectile-centric description : Balitsky-Kovchegov equation T(x, y ) 1 1 N c tr (U(x )U (y )) T Y αs preserves unitarity [ ] T TT }{{} T T dynamical geometrical scaling input: model for T at the initial Y 0 : Golec-Biernat-Wusthof, McLerran-Venugopalan,... basis of the hybrid description in hadron-hadron reactions : projectile 1 : dilute parton beam projectile 2 : saturated François Gelis Color Glass Condensate 94/140 Schleching, February 2014

131 Target-centric description : Color Glass Condensate Color source distribution ρ(x ) in the target Color field A µ given by Yang-Mills equations : [D µ, F µν ] = δ ν ρ Observable O evaluated on this field configuration Expectation value obtained by averaging over ρ : O = [Dρ(x )] W Y [ρ] O[ρ] Rapidity evolution : W Y Y = H W (JIMWLK equation) Y François Gelis Color Glass Condensate 95/140 Schleching, February 2014

132 CGC description of A-A collisions at LO

133 CGC and Nucleus-Nucleus collisions? L = 1 4 F µνf µν + (J µ 1 + Jµ 2 )A µ }{{} J µ Given the sources ρ 1,2 in each projectile, how do we calculate observables? Is there some kind of perturbative expansion? Loop corrections and factorization? François Gelis Color Glass Condensate 96/140 Schleching, February 2014

134 Main difficulty : bookkeeping Dilute regime : one parton in each projectile interact François Gelis Color Glass Condensate 97/140 Schleching, February 2014

135 Main difficulty : bookkeeping Dilute regime : one parton in each projectile interact Dense regime : multiparton processes become crucial (+ pileup of many partonic scatterings in each AA collision) François Gelis Color Glass Condensate 97/140 Schleching, February 2014

136 Power counting François Gelis Color Glass Condensate 98/140 Schleching, February 2014

137 Power counting In the saturated regime, the sources are of order 1/g (because ρρ occupation number 1/αs ) The order of each connected subdiagram is 1 g 2 g# produced gluons 2(# loops) g François Gelis Color Glass Condensate 98/140 Schleching, February 2014

138 Power counting Example : gluon spectrum : dn 1 d 3 p = 1 [ ] g 2 c 0 + c 1 g 2 + c 2 g 4 + The coefficients c 0, c 1, are themselves series that resum all orders in (gρ 1,2 ) n. For instance, c 0 = c 0,n (gρ 1,2 ) n n=0 At Leading Order, we want to calculate the full c 0 /g 2 contribution François Gelis Color Glass Condensate 99/140 Schleching, February 2014

139 Inclusive gluon spectra at LO The gluon spectrum at LO is given by : dn 1 = 1 e ip (x y) x dyd 2 p 16π 3 y x,y LO λ ɛ µ λ ɛν λ A µ(x)a ν(y) where A µ(x) is the classical solution such that lim x 0 A µ(x) = 0 Classical Yang-Mills equations [D µ, F µν ] = J ν 1 + J ν 2 Inclusive multigluon spectra at Leading Order dn n d 3 p 1 d 3 = dn 1 p n d 3 dn 1 p 1 d 3 p n LO LO LO François Gelis Color Glass Condensate 100/140 Schleching, February 2014

140 Retarded classical fields This sum of trees obeys : A + U (A) = J, lim x 0 A(x) = 0 Perturbative expansion (illustrated here for U(A) A 3 ) : Built with retarded propagators François Gelis Color Glass Condensate 101/140 Schleching, February 2014

141 Retarded classical fields This sum of trees obeys : A + U (A) = J, lim x 0 A(x) = 0 Perturbative expansion (illustrated here for U(A) A 3 ) : Built with retarded propagators François Gelis Color Glass Condensate 101/140 Schleching, February 2014

142 Retarded classical fields This sum of trees obeys : A + U (A) = J, lim x 0 A(x) = 0 Perturbative expansion (illustrated here for U(A) A 3 ) : Built with retarded propagators François Gelis Color Glass Condensate 101/140 Schleching, February 2014

143 Retarded classical fields This sum of trees obeys : A + U (A) = J, lim x 0 A(x) = 0 Perturbative expansion (illustrated here for U(A) A 3 ) : Built with retarded propagators Classical fields resum the full series of tree diagrams François Gelis Color Glass Condensate 101/140 Schleching, February 2014

144 Space-time evolution of the classical field in AA collisions Sources located on the light-cone: J µ = δ µ+ ρ 1 (x, x ) +δ µ ρ }{{} 2 (x +, x ) }{{} δ(x ) δ(x + ) t Region 0 : A µ = z Regions 1,2 : A µ depends only on ρ 1 or ρ 2 (known analytically) 0 Region 3 : A µ = radiated field after the collision, only known numerically François Gelis Color Glass Condensate 102/140 Schleching, February 2014

145 Space-time evolution of the classical field in AA collisions Sources located on the light-cone: J µ = δ µ+ ρ 1 (x, x ) +δ µ ρ }{{} 2 (x +, x ) }{{} δ(x ) δ(x + ) t Region 0 : A µ = 0 easy hard! easy z Regions 1,2 : A µ depends only on ρ 1 or ρ 2 (known analytically) trivial Region 3 : A µ = radiated field after the collision, only known numerically François Gelis Color Glass Condensate 102/140 Schleching, February 2014

146 Single gluon spectrum at LO k T 2 )dn/d 2 1/(πR 10-1 KNV I 10-2 KNV II Lappi k T /Λ s Lattice artifacts at large momentum (they do not affect much the overall number of gluons) Important softening at small k compared to pqcd (saturation) François Gelis Color Glass Condensate 103/140 Schleching, February 2014