The early stages of Heavy Ion Collisions

Transcription

1 The early stages of Heavy Ion Collisions SEWM, Swansea University, July 2012 t z François Gelis IPhT, Saclay

2 Outline 2 1 Initial state factorization 2 Final state evolution In collaboration with : K. Dusling (NCSU) T. Epelbaum (IPhT) T. Lappi (Jyväskylä U.) R. Venugopalan (BNL)

3 Stages of a nucleus-nucleus collision 3 t freeze out hadrons gluons & quarks in eq. gluons & quarks out of eq. strong fields z kinetic theory ideal hydro viscous hydro classical dynamics

4 Stages of a nucleus-nucleus collision 3 t freeze out hadrons gluons & quarks in eq. gluons & quarks out of eq. strong fields z kinetic theory ideal hydro viscous hydro classical dynamics Goals : evolution from t = to the start of hydrodynamics first principles description in QCD weakly coupled (but strongly interacting...)

5 What do we need to know about nuclei? 4 Nucleus at rest At low energy : valence quarks

6 What do we need to know about nuclei? 4 Slightly boosted nucleus At low energy : valence quarks At higher energy : Lorenz contraction of longitudinal sizes Time dilation slowing down of the internal dynamics Gluons start becoming important

7 What do we need to know about nuclei? High energy nucleus At low energy : valence quarks At higher energy : Lorenz contraction of longitudinal sizes Time dilation B slowing down of the internal dynamics Gluons start becoming important At very high energy : gluons dominate 4

8 Kinematics 5 Low typical final state transverse momentum p 1 GeV Incoming partons have low momentum fractions x p /E x 10 2 at RHIC (E = 200 GeV) x at the LHC (E = TeV)

9 Nucleon parton distributions 6 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

10 Nucleon parton distributions 6 H1 and ZEUS xf xg 10 xs Large x : dilute regime 2 2 Q = 10 GeV 1 xu v xd v HERAPDF1.0 exp. uncert. model uncert. parametrization uncert x

11 Nucleon parton distributions 6 xf H1 and ZEUS xg 2 2 Q = 10 GeV 10 xs Small x : dense regime, gluon saturation 1 xu v xd v HERAPDF1.0 exp. uncert. model uncert. parametrization uncert x

12 Saturation domain 7

13 Saturation domain 7 Saturation criterion [Gribov, Levin, Ryskin (1983)] α s Q }{{ 2 A } 2/3 xg(x, Q 2 ) }{{} σ gg g surface density 1 Q 2 Q 2 s α sxg(x, Q 2 s) A 1/3 x 0.3 }{{ A 2/3 } saturation momentum

14 Color Glass Condensate = effective theory of small x gluons 8 [McLerran, Venugopalan (1994), Jalilian-Marian, Kovner, Leonidov, Weigert (1997), 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

15 Initial state factorization [FG, Lappi, Venugopalan (2008)]

16 Renormalization group evolution, JIMWLK equation 10 fields Λ - 0 sources P - k - The cutoff between the sources and the fields is not physical, and should not enter in observables Loop corrections contain logs of the cutoff (physically, due to soft gluon radiation) In inclusive Deep Inelastic Scattering: cancelled by letting the distribution of the sources depend on the cutoff ( ) = H ρ, W[ρ] (JIMWLK equation) Λ W[ρ] Λ δ δρ What about nucleus-nucleus collisions? Are the logs still universal?

17 Handwaving argument for factorization 11 τ coll E -1 The duration of the collision is very short: τ coll E 1

18 Handwaving argument for factorization 11 τ coll E -1 The duration of the collision is very short: τ coll E 1 Soft radiation takes a long time it must happen (long) before the collision

19 Handwaving argument for factorization 11 τ coll E -1 space-like interval The duration of the collision is very short: τ coll E 1 Soft radiation 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

20 Power counting 12 In the saturated regime: J µ g 1 g 2 g # of external legs 2 (# of loops) g No dependence on the number of sources J µ infinite number of graphs at each order

21 Leading Order 13 All observables can be expressed in terms of classical solutions of Yang-Mills equations : [D µ, F µν ] = J ν 1 + J ν 2 Boundary conditions for inclusive observables : retarded, with A 0 at x 0 = Example : inclusive spectra at LO dn 1 d 3 p d 4 xd 4 y e ip (x y) x y A(x) A(y) LO dn n d 3 p 1 d 3 = dn 1 dn 1 p n d 3 p 1 d 3 p n LO LO LO

22 Next to Leading Order 14 Relation between LO and NLO [ 1 [ O NLO = ak T ] [ 2 u a k T ] + v k u,v u [ αt ] u ] O LO T is the generator of the shifts of the initial field : [ exp u [ αt ] u ] [class. field at τ {}}{ O A τ( A }{{} init ) ] = O [ A τ( A init + α ) ] }{{} init. field shifted init. field a k, α are calculable analytically Valid for all inclusive observables, e.g. spectra, the energy-momentum tensor, etc...

23 Initial state logarithms 15 In the CGC, upper cutoff on the loop momentum : k ± < Λ, to avoid double counting with the sources J ν 1,2 logarithms of the cutoff Central result for factorization 1 2 k u,v [ ak T ] [ u a k T ] v + u [ αt ] u = = log ( Λ +) H 1 + log ( Λ ) H 2 + terms w/o logs H 1,2 = JIMWLK Hamiltonians of the two nuclei No mixing between the logs of the two nuclei Since the LO NLO relationship is the same for all inclusive observables, these logs have a universal structure

24 Factorization of the logarithms 16 The JIMWLK Hamiltonian H is self-adjoint : [Dρ] W ( H O ) = [Dρ] ( H W ) O Inclusive observables at Leading Log accuracy [Dρ1 ] [ ] [ ] O Leading Log = Dρ 2 W1 ρ1 W2 ρ2 OLO [ρ 1, ρ 2 ] }{{} fixed ρ 1,2 Logs absorbed into the scale evolution of W 1,2 Λ W Λ = H W (JIMWLK equation) Universality : the same W s for all inclusive observables

25 Final state evolution [Dusling, Epelbaum, FG, Venugopalan (2010)] [Dusling, FG, Venugopalan (2011)] [Epelbaum, FG (2011)]

26 Energy momentum tensor at LO 18 Q S -1

27 Energy momentum tensor at LO 18 T µν for longitudinal E and B Q S -1 T µν (τ = 0+ ) = diag (ɛ, ɛ, ɛ, ɛ) LO far from ideal hydrodynamics

28 Weibel instabilities for small perturbations 19 [Mrowczynski (1988), Romatschke, Strickland (2003), Arnold, Lenaghan, Moore (2003), Rebhan, Romatschke, Strickland (2005), Arnold, Lenaghan, Moore, Yaffe (2005), Romatschke, Rebhan (2006), Bodeker, Rummukainen (2007),...] max τ 2 T ηη / g 4 µ 3 L 2 1e-05 1e-06 1e-07 1e-08 1e-09 1e-10 1e-11 1e-12 c 0 +c 1 Exp(0.427 Sqrt(g 2 µ τ)) c 0 +c 1 Exp( g 2 µ τ) [Romatschke, Venugopalan (2005)] 1e g 2 µ τ

29 Weibel instabilities for small perturbations 19 [Mrowczynski (1988), Romatschke, Strickland (2003), Arnold, Lenaghan, Moore (2003), Rebhan, Romatschke, Strickland (2005), Arnold, Lenaghan, Moore, Yaffe (2005), Romatschke, Rebhan (2006), Bodeker, Rummukainen (2007),...] max τ 2 T ηη / g 4 µ 3 L 2 1e-05 c 0 +c 1 Exp(0.427 Sqrt(g 2 µ τ)) For some k s, the field fluctuations 1e-06 c 0 +c 1 Exp( g 2 a k diverge µ τ) like exp µτ when τ + 1e-07 Some components of T 1e-08 µν have secular divergences when evaluated at fixed loop order 1e-09 1e-10 When a k A g 1, the power counting breaks down and additional contributions must be resummed : 1e-11 1e-12 g e [Romatschke, µτ 1 at Venugopalan τ max µ (2005)] 1 log 2 (g 1 ) 1e g 2 µ τ

30 Improved power counting 20 Loop g 2, T e µτ T µν (x) 1 loop : (ge µτ ) 2 u Γ 2 (u,v) v

31 Improved power counting 20 Loop g 2, T e µτ T µν (x) 1 loop : (ge µτ ) 2 2 disconnected loops : (ge µτ ) 4

32 Improved power counting 20 Loop g 2, T e µτ T µν (x) 1 loop : (ge µτ ) 2 2 disconnected loops : (ge µτ ) 4 Γ 3 (u,v,w) 2 nested loops : g(ge µτ ) 3 subleading Leading terms at τ max All disconnected loops to all orders exponentiation of the 1-loop result

33 Resummation of the leading secular terms 21 T µν = exp resummed [ 1 2 u,v ] [a k T] u [a kt] v + [αt] u k }{{} u G(u,v) = T µν + T µν + T µν LO NLO NNLO }{{}}{{ + } partially in full T µν [A LO init]

34 Resummation of the leading secular terms 21 T µν = exp resummed = [ 1 2 ] [a k T] u [a kt] v + [αt] u k }{{} u G(u,v) ] χ(u)g 1 (u, v)χ(v) u,v [ [Dχ] exp 1 2 u,v T µν [A LO init] T µν LO [A init + χ + α] The evolution remains classical, but we must average over a Gaussian ensemble of initial conditions The 2-point correlation G(u, v) of the fluctuations is known Fluctuations 1/2 quantum per mode

35 Analogous scalar toy model 22 φ 4 field theory coupled to a source L = 1 2 ( αφ) 2 g2 4! φ4 + Jφ Strong external source: J Q3 g In 3+1-dim, g is dimensionless, and the only scale in the problem is Q This theory has unstable modes (parametric resonance) Two setups have been studied : Fixed volume system Longitudinally expanding system

36 Pathologies in fixed order calculations 23 Tree time P LO ε LO Oscillating pressure at LO : no equation of state

37 Pathologies in fixed order calculations 23 Tree + 1-loop time P LO+NLO ε LO+NLO Oscillating pressure at LO : no equation of state Small NLO correction to the energy density (protected by energy conservation) Secular divergence in the NLO correction to the pressure

38 Spectrum of fluctuations 24 Initial conditions d 3 k [ ] φ(t 0, x) = ϕ(t 0, x) + (2π) 3 c k e iλ kt 0 V k (x) + c.c. 2k ϕ(t 0, x) = classical background field c k = complex random Gaussian number: ck = 0, ck c l = (2π) 3 k δ(k l) λ k, V k determined by: [ ] 2 g2 ϕ 2 (t 0, x) V k (x) = λ 2 kv k (x)

39 Resummed energy momentum tensor p x +p y +p z time ε No secular divergence in the resummed pressure The pressure relaxes to the equilibrium equation of state

40 Time evolution of the occupation number 26 f k Bose-Einstein with µ=0.54,t=1.31 T/(ω k -µ)-1/2 with µ=0.54,t=1.31 const / k 5/3 t = k Resonant peak at early times Late times : classical equilibrium with a chemical potential µ m + excess at k = 0 : Bose-Einstein condensation?

41 Bose-Einstein condensation t = f k k t = T/(ω k -µ)-1/2 Start with the same energy density, but an empty zero mode Very quickly, the zero mode becomes highly occupied Same distribution as before at late times

42 Volume dependence L = 20 L = 30 L = f(0) / V time f(k) = 1 e β(ω k µ) 1 + n 0δ(k) = f(0) V = L 3

43 Evolution of the condensate g 2 = lattice, 256 configurations, V(φ) = g 2 φ 4 /4! 10 2 f time Formation time almost independent of the coupling Condensate lifetime much longer than its formation time Smaller amplitude and faster decay at large coupling

44 Longitudinal expansion: spectrum of fluctuations 30 Initial conditions d 2 k dν [ ] φ(τ 0, η, x ) = ϕ(τ 0, x ) + c (2π) 3 νk H (2) iν (λ kτ 0 ) V k (x ) e iνη + c.c. ϕ(τ 0, x ) = classical background field (boost invariant) c νk = complex random Gaussian number: cνk = 0, cνk c σl = (2π) 3 δ(k l )δ(ν σ) λ k, V k determined by: [ ] 2 g2 ϕ 2 (τ 0, x ) V k (x ) = λ 2 kv k (x )

45 Longitudinal expansion: initial time 31 The EoM is singular when τ 0 : one must start at τ τφ + 1 τ τφ 1 τ 2 2 ηφ 2 φ + g2 6 φ3 = 0 The spectrum of fluctuations of the initial field also depends on τ 0, in such a way that the end result is independent of τ P T (τ 0 =0.01) 10 3 ε(τ 0 =0.01) P T (τ 0 =0.1) ε(τ 0 =0.1) τ

46 Longitudinal expansion : occupation number 32 τ = 0.1 [ ] ν k Note : ν is the Fourier conjugate of the rapidity η Initially, only the ν = 0 modes are occupied

47 Longitudinal expansion : occupation number 32 τ = 10 [ ] ν k Note : ν is the Fourier conjugate of the rapidity η Initially, only the ν = 0 modes are occupied

48 Longitudinal expansion : occupation number 32 τ = 50 [ ] ν k Note : ν is the Fourier conjugate of the rapidity η Initially, only the ν = 0 modes are occupied Rapid expansion of the distribution in ν (ν max τ 2/3 for a thermalized system)

49 Longitudinal expansion : occupation number 32 τ = 100 [ ] ν k Note : ν is the Fourier conjugate of the rapidity η Initially, only the ν = 0 modes are occupied Rapid expansion of the distribution in ν (ν max τ 2/3 for a thermalized system)

50 Longitudinal expansion : occupation number 32 τ = 150 [ ] ν k Note : ν is the Fourier conjugate of the rapidity η Initially, only the ν = 0 modes are occupied Rapid expansion of the distribution in ν (ν max τ 2/3 for a thermalized system)

51 Longitudinal expansion : occupation number 32 τ = 200 [ ] ν k Note : ν is the Fourier conjugate of the rapidity η Initially, only the ν = 0 modes are occupied Rapid expansion of the distribution in ν (ν max τ 2/3 for a thermalized system)

52 Longitudinal expansion : occupation number 32 τ = 300 [ ] ν k Note : ν is the Fourier conjugate of the rapidity η Initially, only the ν = 0 modes are occupied Rapid expansion of the distribution in ν (ν max τ 2/3 for a thermalized system)

53 Longitudinal expansion : equation of state τ -4/3 τ -1 2P T + P L ε τ After a short time, one has 2P T + P L ɛ Change of behavior of the energy density: τ 1 τ 4/3. Since one has τ ɛ + (ɛ + P L )/τ = 0, this suggests that P L gets close to ɛ/3

54 Longitudinal expansion : isotropization P T + P L ε 10 0 P T P L τ At early times, P L drops much faster than P T (redshifting of the longitudinal momenta due to the expansion) Drastic change of behavior when the expansion rate becomes smaller than the growth rate of the unstability Eventually, isotropic pressure tensor : P L P T

55 Summary and Outlook

56 Summary 36 Factorization of high energy logarithms in AA collisions universal distributions, also applicable in pa or DIS controls the rapidity dependence of correlations limited to inclusive observables Resummation of secular terms in the final state evolution stabilizes the NLO calculation leads to the equilibrium equation of state isotropization even with longitudinal expansion full thermalization on longer time-scales Bose-Einstein condensation if overoccupied initial state (so far, all numerical studies done for a toy scalar model) What s next? QCD generalizable to Yang-Mills theory gauge invariant seamless integration with the CGC description of AA collisions computationally expensive ( [scalar case] 3 (N 2 c 1))

57 Extra

58 Dense-dilute collisions 38 p q g -4 g -2 1 g 2 p q p q g 4 g 1 g -1 pa AA ρ 1

59 Dense-dilute collisions 38 p q Expected g -4 complications g -2 ( ) 2 ( p 1 g 2 ρ 2 g 4 ρ 2 ρ ρ g 2 g 4 More diagrams to consider even at Leading Order More terms in the evolution Hamiltonian if ρ g: p q q ) 4 g 1 g -1 pa AA ρ 1

60 Exclusive processes 39

61 Exclusive processes 39 Example : differential probability to produce 1 particle at LO dp 1 d 3 = F[0] d 4 xd 4 y e ip (x y) x y A + (x)a (y) p LO The vacuum-vacuum graphs do not cancel in exclusive quantities : F[0] 1 (in fact, F[0] = exp( c/g 2 ) 1) z=0 A + and A are classical solutions of the Yang-Mills equations, but with non-retarded boundary conditions

62 Thermalization in Yang-Mills theory 40 Recent analytical work : Kurkela, Moore (2011) Going from scalars to gauge fields : More fields per site (3 Lorentz components 8 colors) More complicated spectrum of initial conditions Expansion : UV overflow on a fixed grid in η

63 BEC and dilepton production 41

64 BEC and dilepton production 41 Two topologies for virtual photons at LO Connected ω M inv Q s k Q s Disconnected ω M inv Q s k Q s excess of dileptons with k M inv

65 Links to Quantum Chaos 42 Quantum Chaos : how does the chaos at the classical level manifests itself in quantum mechanics? Berry s conjecture [M.V. Berry (1977)] High lying eigenstates of such systems have nearly random wavefunctions. The corresponding Wigner distribution is almost uniform on the energy surface Srednicki s eigenstate thermalization hypothesis [M. Srednicki (1994)] For sufficiently inclusive measurements, these high lying eigenstates look thermal. If the system starts in a coherent state, decoherence is the main mechanism to thermalization

66 Phenomenology [Dumitru, FG, McLerran, Venugopalan (2008)] [Dusling, FG, Lappi, Venugopalan (2010)]

67 2-particle correlations in AA collisions 44 [STAR Collaboration, RHIC] Long range rapidity correlation Narrow correlation in azimuthal angle Narrow jet-like correlation near y = ϕ = 0

68 Probing early times with rapidity correlations 45 A t B detection ( 1 m/c) freeze out ( 10 fm/c) latest correlation z By causality, long range rapidity correlations are sensitive to the dynamics of the system at early times : τ correlation τ freeze out e y /2

69 Color field at early time 46 [(g 2 µ) 4 /g 2 ] [Lappi, McLerran (2006)] 2 B z 2 E z 2 B T 2 E T g 2 µτ

70 Color field at early time 46 [(g 2 µ) 4 /g 2 ] [Lappi, McLerran (2006)] B z 2 E z 2 Q S -1 B T 2 E T The field lines form tubes of transverse size Q 1 Rapidity correlation length : η α 1 s g 2 µτ s

71 2-hadron correlations from color flux tubes 47 η-independent fields lead to long range correlations :

72 2-hadron correlations from color flux tubes 47 η-independent fields lead to long range correlations : Q S -1 R Particles emitted by different flux tubes are not correlated (RQ s ) 2 sets the strength of the correlation

73 2-hadron correlations from color flux tubes 47 η-independent fields lead to long range correlations : Q S -1 R Particles emitted by different flux tubes are not correlated (RQ s ) 2 sets the strength of the correlation At early times, the correlation is flat in ϕ

74 2-hadron correlations from color flux tubes 47 η-independent fields lead to long range correlations : v r Particles emitted by different flux tubes are not correlated (RQ s ) 2 sets the strength of the correlation At early times, the correlation is flat in ϕ The collimation in ϕ is produced later by radial flow

75 Centrality dependence 48 Amplitude radial boost model STAR preliminary N part Main effect : increase of the radial flow velocity with the centrality of the collision

76 Rapidity dependence 49 1/N trig dn ch /d η Au+Au 0-30% (PHOBOS) p+p (PYTHIA) y trig = 0 y trig =0.75 y trig =1.5 p T trig = 2.5 GeV p T assoc = 350 MeV η

77 Rapidity dependence 49 1/N trig dn ch /d η Au+Au 0-30% (PHOBOS) 1.2 Prediction at LHC energy p+p (PYTHIA) y trig = y trig =0.75 Pb+Pb at LHC y trig = trig p T = GeV assoc 0.6 p T = 350 MeV d 2 N/(d 2 p T d 2 q T dy p dy q ) [GeV -4 ] y q - y p η p T = 2 GeV q T = 2 GeV y p = 0 y p = -1.5 y p = -3 y p = -4.5

78 Backup

79 Boundary conditions 51 A +(x)= A (x)= y y [ ] [ ] G 0 ++(x, y) J(y) V (A +(y)) G 0 + (x, y) J(y) V (A (y)) [ ] [ ] G 0 +(x, y) J(y) V (A +(y)) G 0 (x, y) J(y) V (A (y)) G 0 + (p) = 2π z(p) θ( p 0 )δ(p 2 ) A + and A are solutions of the classical EoM Decompose the fields in Fourier modes : d 3 p [ A ɛ(x) a (+) (2π) 3 ɛ (x 0, p) e ip x + a ( ) ɛ (x 0, p) e +ip x] 2E p Boundary conditions x 0 = : a (+) + ( p) = a ( ) ( p) = 0 x 0 = + : a (+) ( p) = z(p) a (+) + ( p), a ( ) + ( p) = z(p) a ( ) ( p)

80 Moyal bracket 52 Wigner transform of the commutator Explicit expression : {{A, B}} = 2 h ) ( h A(Q, P) sin 2 ( Q P P Q ) B(Q, P) Quantum deformation of the Poisson bracket : {{A, B}} = {A, B} + O( h 2 )