Initial particle production in high-energy nucleus-nucleus collisions

Initial particle production in high-energy nucleus-nucleus collisions François Gelis CEA / DSM / SPhT François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 1

Typical e+e- or pp collision Run : even t 2513 : 61702 Da t e 910910 T ime 85656 C t r k (N= 37 Sump= 65. 7 ) Eca l (N= 55 SumE= 44. 8 ) Hca l (N=19 SumE= 8. 6 ) Ebeam 45. 613 Ev i s 90. 2 Emi s s 1. 1 V t x ( - 0. 09, 0. 13, - 0. 20 ) Muon (N= 2 ) Sec V t x (N= 3 ) Fde t (N= 0 SumE= 0. 0 ) Bz=4. 028 Bunch l e t 1 / 1 Th r us t =0. 6788 Ap l an=0. 0381 Ob l a t =0. 4249 Sphe r =0. 6273 Y Z X 100. cm. 2 5 10 20 GeV Cen t r e o f s c r een i s ( 0. 0000, 0. 0000, 0. 0000 ) François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 2

Why is QCD predictive there? Why is QCD predictive despite the fact that hadrons are non-perturbative bound states? Factorization : When we add one extra loop to the hard part, all the divergences that remain after having summed over degenerate final states can be absorbed into the evolution of the parton distributions Universality : the structure of these divergences is independent of the nature of the colliding hadrons and of the process under consideration universal distributions François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 3

Typical nucleus-nucleus collision Very high multiplicity (1000s of particles) 99% of the multiplicity below p 3 GeV François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 4

Goals Describe nucleus-nucleus collisions within QCD Generalize the concept of parton distribution Due to the high density of partons, we expect to need higher correlations (beyond the usual parton distributions, which are 2-point correlation functions) If divergences show up in loop corrections, one should be able to factor them out into the evolution of these distributions These distributions should be universal The non-perturbative information is relegated into the initial condition for this evolution Find which observables are infrared safe François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 5

Stages of a nucleus-nucleus collision z = -t t z = t z (beam axis) François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 6

Stages of a nucleus-nucleus collision t z (beam axis) François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 6

Stages of a nucleus-nucleus collision t strong fields z (beam axis) classical EOMs François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 6

Stages of a nucleus-nucleus collision t gluons & quarks out of eq. kinetic theory strong fields z (beam axis) classical EOMs François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 6

Stages of a nucleus-nucleus collision t gluons & quarks in eq. gluons & quarks out of eq. hydrodynamics kinetic theory strong fields z (beam axis) classical EOMs François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 6

Stages of a nucleus-nucleus collision t hadrons in eq. gluons & quarks in eq. gluons & quarks out of eq. hydrodynamics kinetic theory strong fields z (beam axis) classical EOMs François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 6

Stages of a nucleus-nucleus collision t freeze out hadrons in eq. gluons & quarks in eq. gluons & quarks out of eq. hydrodynamics kinetic theory strong fields z (beam axis) classical EOMs François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 6

Stages of a nucleus-nucleus collision t freeze out hadrons in eq. gluons & quarks in eq. gluons & quarks out of eq. hydrodynamics kinetic theory strong fields z (beam axis) classical EOMs calculate the initial production of semi-hard particles prepare the stage for kinetic theory François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 6

Outline and bookkeeping at leading order and resummations FG, Venugopalan, hep-ph/0601209, 0605246 FG, Jeon, Venugopalan, in preparation + work in progress with Lappi, Venugopalan and Fukushima, McLerran François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 7

Nucleon at rest Nucleon at high energy Degrees of freedom Hadronic collisions François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 8

Nucleon at rest Nucleon at rest Nucleon at high energy Degrees of freedom Hadronic collisions Very complicated non-perturbative object... Contains 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 The only role of short lived fluctuations is to renormalize the masses and couplings 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 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 9

Nucleon at high energy Nucleon at rest Nucleon at high energy Degrees of freedom Hadronic collisions Dilation of all internal time-scales of the 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 Many fluctuations live long enough to be seen by the probe. The nucleon appears denser at high energy. Pre-existing fluctuations are totally frozen over the time-scale of the probe, and act as static sources of new partons In a nucleus, soft gluons (long wavelength) belonging to different nucleons overlap in the longitudinal direction coherent effects (known as saturation) François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 10

Degrees of freedom and their interplay Nucleon at rest Nucleon at high energy Degrees of freedom Hadronic collisions McLerran, Venugopalan (1994), Iancu, Leonidov, McLerran (2001) Soft modes have a large occupation number they are described by a classical color field A µ that obeys Yang-Mills s equation: [D ν, F νµ ] a = J µ a The source term J µ a comes from the faster partons. The hard modes, slowed down by time dilation, are described as frozen color sources ρ a. Hence : J µ a = δ µ+ δ(x )ρ a ( x ) (x (t z)/ 2) The color sources ρ a are random, and described by a distribution functional W Y [ρ], with Y the rapidity that separates soft and hard. Evolution equation (JIMWLK) : W Y [ρ] Y = H[ρ] W Y [ρ] François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 11

Hadronic collisions Nucleon at rest Nucleon at high energy Degrees of freedom Hadronic collisions In order to study the collisions of two hadrons at leading order, solve the classical Yang-Mills equations in the presence of the following current : J µ δ µ+ δ(x ) ρ 1 ( x ) + δ µ δ(x + ) ρ 2 ( x ) Compute the observable O of interest in the background field created by a configuration of the sources ρ 1, ρ 2. Note : the sources are of order 1/g very non-linear problem Average over the sources ρ 1, ρ 2 Z ˆDρ1 ˆDρ2 O Y = WYbeam [ρ ˆρ2 Y 1 WY +Ybeam O[ρ1, ρ 2 Note : the boundary conditions depend on the observable François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 12

Main difficulties Power counting Bookkeeping François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 13

Main difficulties Main difficulties Power counting Bookkeeping Dilute regime : one source in each projectile interact François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 14

Main difficulties Main difficulties Power counting Bookkeeping Dilute regime : one source in each projectile interact Dense regime : non linearities are important François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 14

Main difficulties Main difficulties Power counting Bookkeeping Dilute regime : one source in each projectile interact Dense regime : non linearities are important Many gluons can be produced from the same diagram There can be many simultaneous disconnected diagrams François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 14

Main difficulties Main difficulties Power counting Bookkeeping Dilute regime : one source in each projectile interact Dense regime : non linearities are important Many gluons can be produced from the same diagram There can be many simultaneous disconnected diagrams Some of them may not produce anything (vacuum diagrams) François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 14

Main difficulties Main difficulties Power counting Bookkeeping Dilute regime : one source in each projectile interact Dense regime : non linearities are important Many gluons can be produced from the same diagram There can be many simultaneous disconnected diagrams Some of them may not produce anything (vacuum diagrams) All these diagrams can have loops (not at LO though) François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 14

Power counting Main difficulties Power counting Bookkeeping In the saturated regime, the sources are of order 1/g The order of each disconnected diagram is given by : 1 g 2 g# produced gluons 2(# loops) g The total order of a graph is the product of the orders of its disconnected subdiagrams quite messy... François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 15

Bookkeeping Main difficulties Power counting Bookkeeping François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 16

Bookkeeping Main difficulties Power counting Bookkeeping Consider squared amplitudes (including interference terms) rather than the amplitudes themselves François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 16

Bookkeeping Main difficulties Power counting Bookkeeping Consider squared amplitudes (including interference terms) rather than the amplitudes themselves See them as cuts through vacuum diagrams Consider only the simply connected ones, thanks to : X all the vacuum diagrams «= exp j X simply connected ff vacuum diagrams Simpler power counting for connected vacuum diagrams : 1 loops) g2(# g2 François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 16

Bookkeeping Main difficulties Power counting Bookkeeping There is an operator D that acts on a pair of vacuum diagrams by removing two sources and attaching a cut propagator instead : D can also act directly on single diagram, if it is already cut François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 17

Bookkeeping Main difficulties Power counting Bookkeeping There is an operator D that acts on a pair of vacuum diagrams by removing two sources and attaching a cut propagator instead : D can also act directly on single diagram, if it is already cut By repeated action of D, on can generate all the diagrams with an arbitrary number of cuts Thanks to this operator, one can write : P n = 1 n! Dn e iv e iv, iv = ( ) connected uncut vacuum diagrams ( ) all the cut = e D e iv e iv vacuum diagrams François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 17

First moment Gluon production at LO Boost invariance François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 18

First moment of the distribution First moment Gluon production at LO Boost invariance It is easy to express the average multiplicity as : N = { n P n = D e D e iv iv } e n N is obtained by the action of D on the sum of all the cut vacuum diagrams. There are two kind of terms : D picks two sources in two distinct connected cut diagrams D picks two sources in the same connected cut diagram François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 19

Gluon multiplicity at LO First moment Gluon production at LO Boost invariance At LO, only tree diagrams contribute the second type of topologies can be neglected (it starts at 1-loop) In each blob, we must sum over all the tree diagrams, and over all the possible cuts : tree N LO = trees cuts tree A major simplification comes from the following property : + = retarded propagator The sum of all the tree diagrams constructed with retarded propagators is the retarded solution of Yang-Mills equations : [D µ, F µν ] = J ν with A µ (x 0 = ) = 0 François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 20

Gluon multiplicity at LO First moment Gluon production at LO Boost invariance Krasnitz, Nara, Venugopalan (1999 2001), Lappi (2003) dn LO dy d 2 p = 1 16π 3 Z x,y X e ip (x y) x y ǫ µ λ ǫν λ A µ (x)a ν (y) At LO : Yang-Mills equations with null retarded boundary conditions λ François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 21

Gluon multiplicity at LO First moment Gluon production at LO Boost invariance Krasnitz, Nara, Venugopalan (1999 2001), Lappi (2003) dn LO dy d 2 p = 1 16π 3 Z x,y X e ip (x y) x y ǫ µ λ ǫν λ A µ (x)a ν (y) At LO : Yang-Mills equations with null retarded boundary conditions initial value problem on the light-cone λ François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 21

Gluon multiplicity at LO First moment Gluon production at LO Boost invariance 2 k T )dn/d 2 1/(πR 10-1 KNV I 10-2 10-3 10-4 KNV II Lappi 10-5 10-6 10-7 0 1 2 3 4 5 6 k T /Λ s Lattice artefacts at large momentum (they do not affect much the overall number of gluons) Important softening at small k compared to pqcd (saturation) François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 22

Initial conditions and boost invariance First moment Gluon production at LO Boost invariance Gauge condition : x + A + x A + = 0 8 < A i (x) = α i (τ, η, x ) : A ± (x) = ± x ± β(τ,η, x ) τ = const η = const Initial values at τ = 0 + : α i (0 +, η, x ) and β(0 +, η, x ) do not depend on the rapidity η α i and β remain independent of η at all times (invariance under boosts in the z direction) numerical resolution performed in 1 + 2 dimensions François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 23

Local anisotropy After some time, the gluons have a longitudinal velocity tied to their space-time rapidity by v z = tanh (η) : First moment Gluon production at LO Boost invariance François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 24

Local anisotropy After some time, the gluons have a longitudinal velocity tied to their space-time rapidity by v z = tanh (η) : First moment Gluon production at LO Boost invariance at late times : if particles fly freely, only one longitudinal velocity can exist at a given η : v z = tanh(η) François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 24

1-loop contributions to N Initial state factorization Unstable modes and summation of leading divergences François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 25

1-loop contributions to N 1-loop diagrams for N 1-loop contributions to N Initial state factorization Unstable modes tree 1-loop tree François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 26

1-loop contributions to N 1-loop diagrams for N 1-loop contributions to N Initial state factorization Unstable modes tree 1-loop tree This can be rewritten as a perturbation of the initial value problem encountered at LO : François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 26

1-loop contributions to N The to N can be written in terms of the generator T( x) of translations of the initial condition at the point x of the light-cone : 1-loop contributions to N Initial state factorization Unstable modes δn 1 = [ ] δa( x) T( x) N LO x light cone δn 2 = x, y light cone 1 [ 2 λ ] ζ λ ( x)ζ λ( y) T( x)t( y) N LO François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 27

Divergences 1-loop contributions to N Initial state factorization Unstable modes If taken at face value, the are plagued by two kinds of divergences : The two coefficients δa( x) and λ ζ λ( x)ζ λ ( y) are infinite, because of an unbounded integration over a rapidity variable At late times, T( x)a(τ, y) diverges exponentially, T( x)a(τ, y) τ + e µτ because of an instability of the classical solution of Yang-Mills equations against non-boost invariant perturbations (Romatschke, Venugopalan (2005)) We can decompose the 1-loop correction δn as [ [ ] [ ] δn = δn ]finite + δn + δn divergent coefficients unstable modes François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 28

Initial state factorization Anatomy of the full calculation : W Y beam -Y[ρ 1 ] 1-loop contributions to N Initial state factorization Unstable modes N[ A in(ρ 1, ρ 2 ) ] W Y beam +Y[ρ 2 ] François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 29

Initial state factorization Anatomy of the full calculation : W Y beam -Y[ρ 1 ] 1-loop contributions to N Initial state factorization Unstable modes N[ A in(ρ 1, ρ 2 ) ] + δ N W Y beam +Y[ρ 2 ] When the observable N[A in (ρ 1, ρ 2 )] is corrected by an extra gluon, one gets divergences of the form α s dy in δn one would like to be able to absorb these divergences into the Y dependence of the source densities W Y [ρ 1,2 ] François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 29

Initial state factorization Anatomy of the full calculation : W Y beam -Y 0 [ρ 1 ] Y + Y beam 1-loop contributions to N Initial state factorization Unstable modes N[ A in(ρ 1, ρ 2 ) ] + δ N W Y beam +Y 0 [ρ 2] Y 0 Y 0 - Y beam When the observable N[A in (ρ 1, ρ 2 )] is corrected by an extra gluon, one gets divergences of the form α s dy in δn one would like to be able to absorb these divergences into the Y dependence of the source densities W Y [ρ 1,2 ] Equivalently, if one puts some arbitrary frontier Y 0 between the observable and the source distributions, the dependence on Y 0 should cancel between the various factors François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 29

Initial state factorization 1-loop contributions to N Initial state factorization Unstable modes The two kind of divergences don t mix, because the divergent part of the coefficients is boost invariant. Given their structure, the divergent coefficients seem related to the evolution of the sources in the initial state In order to prove the factorization of these divergences in the initial state distributions of sources, one needs to establish : h δn i divergent coefficients = h (Y 0 Y ) H [ρ 1 ] + (Y Y 0) H [ρ 2 ] i N LO where H[ρ] is the Hamiltonian that governs the rapidity dependence of the source distribution W Y [ρ] : W Y [ρ] Y = H[ρ] W Y [ρ] FG, Lappi, Venugopalan (work in progress) François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 30

Initial state factorization Why is it plausible? 1-loop contributions to N Initial state factorization Unstable modes Reminder : [ ] δn divergent coefficients = { + 1 2 x x, y [ ] δa( x) T( x) div [ λ } ] ζ λ ( x)ζ λ( y) T( x)t( y) div N LO Compare with the evolution Hamiltonian : δ H[ρ] = σ( x ) δρ( x ) + 1 δ 2 χ( x, y 2 ) δρ( x )δρ( y ) x x, y The coefficients σ and χ in the Hamiltonian are well known. There is a well defined calculation that will tell us if it works... François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 31

Unstable modes 1-loop contributions to N Initial state factorization Unstable modes The coefficient δa( x) is boost invariant. Hence, the divergences due to the unstable modes all come from the quadratic term in δn : h i δn unstable modes = 8 >< 1 >: 2 with Z x, y 9 >= Σ( x, y) T ( x)t ( y) >; N [A LO in(ρ 1, ρ 2 )] Σ( x, y) Z λ ζ λ ( x)ζ λ( y) When summed to all orders, these divergences exponentiate : h i δn unstable modes = e 1 2 R x, y Σ( x, y) T ( x)t ( y) N LO [A in (ρ 1, ρ 2 )] François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 32

Unstable modes 1-loop contributions to N Initial state factorization Unstable modes This can be arranged in a more intuitive way : h i δn unstable modes = = Z ˆDξ e 1 2 R x, y ξ( x)ξ( y) Σ( x, y) {z } ei Z fluct [ξ] Z ˆDξ Zfluct[ξ] N [A LO in(ρ 1, ρ 2 )+ξ] R x ξ( x) T ( x) N LO [A in (ρ 1, ρ 2 )] summing these divergences simply requires to add Gaussian fluctuations to the initial condition for the classical problem Interpretation : Despite the fact that the fields are coupled to strong sources, the classical approximation alone is not good enough, because the classical solution has unstable modes that can be triggered by the quantum fluctuations François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 33

Unstable modes 1-loop contributions to N Initial state factorization Unstable modes François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 34

Unstable modes 1-loop contributions to N Initial state factorization Unstable modes Combining everything, one should write dn dy d 2 p = [Dρ1 ] [Dρ 2 ] WYbeam Y [ρ 1 ] W Ybeam+Y [ρ 2 ] [Dξ ] Zfluct [ξ] dn[a in (ρ 1, ρ 2 )+ξ] dy d 2 p This formula resums the divergences that occur at Leading Order François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 34

François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 35

When the parton densities in the projectiles are large, the study of particle production becomes rather involved Many disconnected diagrams contribute Vacuum diagrams play a crucial role At Leading Order, the inclusive gluon spectrum can be calculated from a classical solution with retarded boundary conditions However, the bring divergences : Large logs of 1/x Unstable modes of the classical solution Resummation of the leading divergences to all orders : Evolution with Y of the distribution of sources Quantum fluctuations on top of initial condition for the classical solution in the forward light-cone The devil is in the details, and many need to be checked... François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 36

Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory Extra bits François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 37

Parton evolution Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory assume that the projectile is big, e.g. a nucleus, and has many valence quarks (only two are represented) on the contrary, consider a small probe, with few partons at low energy, only valence quarks are present in the hadron wave function François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 38

Parton evolution Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory when energy increases, new partons are emitted the emission probability is α dx s x α sln( 1 x ), with x the longitudinal momentum fraction of the gluon at small-x (i.e. high energy), these logs need to be resummed François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 39

Parton evolution Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory 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 (BFKL) François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 40

Parton evolution Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory eventually, the partons start overlapping in phase-space 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 François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 41

Saturation criterion Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory Gribov, Levin, Ryskin (1983) Number of gluons per unit area: Recombination cross-section: ρ xg A(x, Q 2 ) πr 2 A 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 sxg A (x, Q 2 s) πr 2 A A 1/3 1 x 0.3 At saturation, the phase-space density is: dn g d 2 x d 2 p ρ Q 2 1 α s François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 42

Saturation domain log(x -1 ) Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory log(q 2 ) Λ QCD François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 43

Definition One can encode the information about all the probabilities P n in a generating function defined as : F(z) P n z n n=0 Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory From the expression of P n in terms of the operator D, we can write : F(z) = e zd e iv e iv Reminder : e D e iv e iv is the sum of all the cut vacuum diagrams The cuts are produced by the action of D Therefore, F(z) is the sum of all the cut vacuum diagrams in which each cut line is weighted by a factor z François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 44

What would it be good for? Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory Let us pretend that we know the generating function F(z). We could get the probability distribution as follows : P n = 1 2π Z 2π 0 dθ e inθ F(e iθ ) Note : this is trivial to evaluate numerically by a FFT : 1 0.01 F 1 (z) F 2 (z) 1e-04 1e-06 P n 1e-08 1e-10 1e-12 1e-14 0 500 1000 1500 2000 2500 n François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 45

F(z) at Leading Order We have : F (z) = D { e zd e iv e iv } By the same arguments as in the case of N, we get : Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory F (z) F(z) = + The major difference is that the sum of cut graphs that must be evaluated have a factor z attached to each cut line At tree level (LO), we can write F (z)/f(z) in terms of solutions of the classical Yang-Mills equations, but these solutions are not retarded anymore, because : + z retarded propagator François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 46

F(z) at Leading Order Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory The derivative F /F has an expression which is formally identical to that of n gluons, F (z) F(z) = Z d 3 p (2π) 3 2E p Z x,y e ip (x y) x y X λ ǫ µ λ ǫν λ A (+) µ (x)a ( ) ν (y), with A (±) µ (x) two solutions of the Yang-Mills equations If one decomposes these fields into plane-waves, A (ε) µ (x) Z d 3 p n f (ε) (2π) 3 + 2E (x0, p)e ip x + f (ε) (x0, p)e ip xo p the boundary conditions are : f (+) + (, p) = f( ) (, p) = 0 f ( ) + (+, p) = z f(+) + (+, p), f(+) (+, p) = z f( ) (+, p) There are boundary conditions both at x 0 = and x 0 = + not an initial value problem hard ++... François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 47

Remarks on factorization As we ve seen before, the fact that the calculation of the first moment N can be formulated as an initial value problem is probably going to be crucial for proving factorization Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory If this is indeed the case, then factorization does not hold for the generating function F(z), or equivalently for the individual probabilities P n Preliminary studies seem to indicate that the computation of higher moments of the distribution can also be reformulated as initial value problems the moments would be factorizable as well François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 48

Quark production FG, Kajantie, Lappi (2004, 2005) E p d n quarks d 3 p = 1 16π 3 Z x,y e ip (x y) / x / y ψ(x)ψ(y) Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory Dirac equation in the classical color field : François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 49

Spectra for various quark masses Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory dn/dyd 2 q T [arbitrary units] 0 5 10 4 1 10 5 2 10 5 m = 60 MeV m = 300 MeV m = 600 MeV m = 1.5 GeV m = 3 GeV 0 1 2 3 4 ^q [GeV] François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 50

Exclusive processes So far, we have considered only inclusive quantities i.e. the P n are defined as probabilities of producing particles anywhere in phase-space Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory What about events where a part of the phase-space remains unoccupied? e.g. rapidity gaps empty region Y François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 51

Main issues 1. How do we calculate the probabilities Pn excl with an excluded region Ω in the phase-space? Can one calculate the total gap probability P gap = n P n excl? Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory 2. What is the appropriate distribution of sources W excl [ρ] to Y describe a projectile that has not broken up? 3. How does it evolve with rapidity? See : Hentschinski, Weigert, Schafer (2005) 4. Are there some factorization theorems, and for which quantities do they hold? François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 52

Exclusive probabilities Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory The probabilities Pn excl [Ω], for producing n particles but none in the region Ω can also be constructed from the vacuum diagrams, as follows : Pn excl [Ω] = 1 n! Dn eiv e iv Ω where D Ω is an operator that removes two sources and links the corresponding points by a cut (on-shell) line, for which the integration is performed only in the region Ω One can define a generating function, F Ω (z) n P excl n [Ω] z n, whose derivative is given by the same diagram topologies as the derivative of the generating function for inclusive probabilities François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 53

Exclusive probabilities Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory Differences with the inclusive case : In the diagrams that contribute to F Ω (z)/f Ω(z), the cut propagators are restricted to the region Ω of the phase-space at leading order, this only affects the boundary conditions for the classical fields in terms of which one can write F Ω (z)/f Ω (z) not more difficult than the inclusive case Contrary to the inclusive case where we know that F(1) = 0 the integration constant needed to go from F Ω (z)/f Ω(z) to F Ω (z) is non-trivial. This is due to the fact that the sum of all the exclusive probabilities is smaller than unity F Ω (1) is in fact the probability of not having particles in the region Ω i.e. the gap probability François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 54

Survival probability Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory We can write : F Ω (z) = F Ω (1) exp z 1 dτ F (τ) Ω F Ω (τ) the prefactor F Ω (1) will appear in all the exclusive probabilities This prefactor is nothing but the famous survival probability for a rapidity gap One can in principle calculate it by the general techniques developed for calculating inclusive probabilities : F Ω (1) = F incl 1 Ω (0) Note : it is incorrect to say that a certain process with a gap can be calculated by multiplying the probability of this process without the gap by the survival probability François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 55

Factorization? Except for the case of Deep Inelastic Scattering, nothing is known regarding factorization for exclusive processes in a high density environment Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory For the overall framework to be consistent, one should have factorization between the gap probability, F Ω (1), and the source density studied in Hentschinski, Weigert, Schafer (2005) (and the ordinary W Y [ρ] on the other side) The total gap probability is the most inclusive among the exclusive quantities one may think of. For what quantities if any does factorization work? François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 56

Link with Kinetic Theory Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory The distribution n(τ 0, x, p) of gluons is evaluated from the previous formula at a finite time τ 0 After the time τ 0, it evolves according to a Boltzmann equation : v X n + g X A p n = S + C[n] The source term S comes from the fact that there are still time-dependent fields in the system, that keep producing particles This procedure makes sense only if the final result is insensitive to the value of τ 0, within a reasonable range. This independence comes from the fact that, for field amplitudes 1 A g 1, the system can be described in a dual way : as fields governed by classical equations of motion as on-shell particles governed by a kinetic equation François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 57

Importance of fluctuations Extra bits Parton saturation Generating function Quark production Exclusive processes Link with Kinetic Theory With a background field, the collision term of the Boltzmann equation allows 1 2 splittings In the classical calculation, the average over the fluctuations of the initial condition also amounts to including gluon splittings these fluctuations may be crucial for the independence with respect to the transition time τ 0 (so that the two descriptions work equally well between τ 0 and τ 0) τ 0 τ 0 François Gelis 2006 From RHIC to LHC: Achievements and Opportunities, INT, Seattle, October 2006 - p. 58