Small-x physics 2- The saturation regime of QCD and the Color Glass Condensate Cyrille Marquet Columbia University

The saturation regime of QCD the BFKL approximation summing terms leads to a growth of the gluon density in the hadronic wave function but high-density effects are missing when the gluon density is large enough, gluon recombination is important gluon recombination in the hadronic wave function gluon density per unit area recombination cross-section gluon kinematics recombinations important when the saturation regime: for with magnitude of Qs x dependence this regime is non-linear, yet weakly coupled the idea of the CGC is to take into account this effect via strong classical fields

The Color Glass Condensate effective description of a high-energy hadron/nucleus the CGC is moving in the direction and the gauge is the current is solving Yang-Mills equations gives  short-lived fluctuations separation between the long-lived high-x partons and the short-lived low-x gluons CGC wave function in practice we deal with CGC averages such as we are interested in the evolution of with x, which sums both the BFKL equation is recovered in the low-density regime

Outline of the second lecture The evolution of the CGC wave function the JIMWLK equation and the Balitsky hierarchy A mean-field approximation: the BK equation solutions: QCD traveling waves the saturation scale and geometric scaling Beyond the mean field approximation stochastic evolution and diffusive scaling Computing observables in the CGC framework solving evolution equation vs using dipole models

The evolution of the CGC wave function: the B-JIMWLK equations

The JIMWLK equation a functional equation for the x evolution of : Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner a functional equation for the x evolution of : with the JIMWLK equation gives the evolution of the wave function for small enough x the equivalent Balitsky equations are obtained by considering the scattering of simple test projectiles (dipoles) off the CGC the Wilson lines sum powers of adjoint representation

Scattering off the CGC eigenstates ? eigenvalues their interaction should conserve their spin, polarisation, color, momentum … in the high-energy limit, the eigenstates are simple when the partons transverse momenta are Fourier transformed to transverse coordinates in the large-Nc limit, further simplification: the eigenstates are only made of dipoles x, y : transverse coordinates other quantum numbers aren’t explicitely written, the eigenvalues depend only on the transverse coordinates eigenvalues scattering amplitude off the (dense) target CGC the interaction with the CGC conserves transverse position

Dipoles as test projectiles dipoles are ideal projectiles to probe small distances the dipole: u : quark space transverse coordinate v : antiquark space transverse coordinate scattering of the quark: dependence kept implicit in the following JIMWLK equation → evolution equation for the dipole correlators

The Balitsky hierarchy equations for dipoles scattering of the CGC Balitsky (1996) the BFKL kernel an hierarchy of equation involving correlators with more and more dipoles BFKL saturation general structure: we will now derive the first equation of the hierarchy solving the B-JIMWLK equations gives , … which can then be used to compute observables in the large Nc limit, the hierarchy is restricted to dipoles

Recall the dipole wave function the valence component (using the mixed space): such that only perturbative sizes r << 1/ QCD are included x : quark space transverse coordinate y : antiquark space transverse coordinate other degrees of freedom are not explicitely written from QCD at order gS color index the dipole rapidity: the dipole dressed with one gluon

Elastic scattering of the dipole the dipole scattering is described by Wilson lines scattering of the quark scattering of the antiquark for the qqg component: let’s compute the elastic scattering amplitude Ael(Y ) in the dipole-CGC collision where is the total rapidity Y = YCGC + Yd in the frame in which and

Frame independence of Ael(Y ) in the frame in which and using the following identity to get rid of the adjoint Wilson line = 1/2 -1/(2Nc) one obtains our two expressions for Ael(Y ) should be identical, this requirement in the limit Y - YCGC  dY  0 gives the first Balitsky equation frame independence:

A mean field approximation: the Balitsky-Kovchegov equation

The BK equation obtain by neglecting correlations the BK equation is a closed equation for obtained by assuming solutions: qualitative behavior r = dipole size let’s consider impact-parameter independent solutions at small Y, is small, and the quadratic term can be neglected, the equation reduces then to the linear BFKL equation and rises exponentially with Y as gets close to 1 (the stable fixed point of the equation), the non-linear term becomes important, and , saturates at 1 with increasing Y, the unitarization scale get bigger

Coordinate vs momentum space let’s go to momentum space  due to conformal invariance, the linear part of the equation is the same for and coordinate space momentum space dipole scattering amplitude unintegrated gluon distribution linear BFKL equation linear BFKL equation genuine saturation: no real saturation: both equation are in the same universality class as the F-KPP equation

The F-KPP equation same features as the BK equation Fisher, Kolmogorov, Petrovsky, Piscounov time derivative space derivative linear term meaning exponential growth non linear term meaning saturation dictionary F-KPP → BK when expanding to second order, the equations are the same (in momentum space) in spite of small difference, same universality class the precise forms of the space derivatives and of the non-linear term doesn’t matter these equations belong to the same universality class F-KPP and BK asympotic (in t or Y) solutions are the same: traveling waves

Traveling wave solutions what is a traveling wave position: the speed of the wave is determined only by the linear term of the equation saturation region wave traveling at speed v, independently of the initial condition provided it is steep enough there the initial condition has not been erased yet BK solutions: same rapidity evolution quantitative features can be derived the formation of the traveling wave

Recall the BFKL solutions a superposition of waves with speeds initial condition in Mellin space the minimal speed is obtained for and its value is this will be the speed of the asymptotic traveling wave the saturation exponent for the traveling wave to form, the initial condition must feature

QCD traveling waves the initial condition is steep enough Munier and Peschanski (2004) in QCD asymptotic solutions of BK same features for except in the saturation region Y0 Y >Y0 the saturation scale  with then called geometric scaling

Numerical solutions numerical simulations confirm the results for the scattering amplitude for the saturation scale sub-asymptotic corrections (≡ geometric scaling violations) are also known:

Running coupling corrections running coupling corrections to the BK equation taken into account by the substitution Kovchegov Weigert Balitsky consequences similar to those first obtained by the simpler substitution running coupling corrections slow down the increase of Qs with energy also confirmed by numerical simulations, however this asymptotic regime is reached for larger rapidities

Beyond the mean field approximation

Recall the Balitsky hierarchy equations for dipole correlators it describes the so-called dense regime where the BK equation is a good approximation general structure at large Nc the equation for is fine, its dilute limit is the BFKL equation, however the other equations are incomplete, they do not agree with Mueller’s dipole model in the limit incorrect results for have consequences for still there is something missing describing properly the dilute regime is also important the B-JIMWLK equations do not describe it properly

A new hierarchy of equations the hierarchy must be completed Iancu and Triantafyllopoulos (2005) fluctuations (included to recover Mueller’s dipole model in the dilute limit) important when BFKL saturation important when this QCD evolution is equivalent to a stochastic process reformulation into a Langevin problem average over the realisations of the stochastic process the correlators can all be obtained from a single stochastic quantity which obeys a Langevin equation CGC wave function

The stochastic F-KPP equation the QCD Langevin equation deterministic part of the equation: BK ! noise high-energy QCD evolution = stochastic process in the universality class of reaction-diffusion processes, of the sF-KPP equation Iancu, Mueller and Munier (2005) the reduction to one dimension introduces the noise strength parameter  noise r = dipole size the stochastic F-KPP equation

A stochastic saturation scale the stochastic evolution the Langevin evolution generates, at each rapidity, a stochastic ensemble of solutions for the average speed of the waves: the dispersion coefficient: D analytic expressions for v and D are known only for very small equivalent to a stochastic saturation scale Qs (for ) Qs is a stochastic variable distributed according to a Gaussian probability law: corrections to the Gaussian law for improbable fluctuations also known, picture confirmed by numerical simulations C.M., Soyez and Xiao average saturation scale: the probability distribution of Qs

A new scaling law the average dipole scattering amplitude : the diffusion is negligible and one recovers geometric scaling : the diffusion is important and new regime: diffusive scaling very high energies

The diffusive scaling regime important properties in the diffusive scaling regime: the amplitudes are dominated by events that feature the hardest fluctuation of in average the scattering is weak, yet saturation is the relevant physics running coupling corrections Dumitru et al. push the diffusive scaling regime up to asymptotically high energies conclusion at the moment using the mean field approximation is good enough for phenomenology

Computing observables in the CGC framework

The CGC averages the Wilson line correlators in the CGC framework, any cross-section is determined by colorless combinations of Wilson lines, averaged over the CGC wave function the energy evolution of cross-sections is encoded in the evolution of this wave function is mainly non-perturbative, but its evolution is known  Balitsky/BK equation for the correlators this is the most common average, (for instance it determines deep inelastic scattering) its Fourier transform is the unintegrated gluon distribution, it obeys the BK equation the 2-point function or dipole amplitude for 2-particle correlations more complicated correlators for instance

The strategy what should be done in principle what is done in practice - express the cross-sections in terms of correlators of Wilson lines the same correlators enter in the formulation of very different processes the more exclusive the final state is, the more complicated the correlators are - use B-JIMWLK or BK equation to compute the correlators use appropriate initial conditions include non-perturbative effects such as the impact parameter dependence running coupling effects should also be included many processes have been studied in the CGC framework using dipole models what is done in practice solving evolution equations with running coupling is not the favored approach instead one uses parametrizations with features inspired from theoretical results it makes it easier to deal with impact parameter dependence of the dipole amplitude famous dipole models: GBW, IIM, IPsat

Outline of the third lecture The hadron wave function summary of what we have learned The saturation models from GBW to the latest ones Deep inelastic scattering (DIS) the cleanest way to probe the CGC/saturation allows to fix the model parameters Diffractive DIS and other DIS processes these observables are predicted Forward particle production in pA collisions and the success of the CGC picture at RHIC