Small- x physics 1- High-energy scattering in pQCD: the BFKL equation

1. Small-x physics1- High-energy scattering in pQCD: the BFKL equation Cyrille Marquet Columbia University

2. for , what is the behavior of the scattering amplitude ? s t Outstanding problems in pQCD • What is the high-energy limit of hadronic scattering ? • What is the wave function of a high-energy hadron ? spin indices momenta wave function color indices

3. Outline of the first lecture • Hadronic scattering in the high-energy limitleading logarithms and kT factorization in perturbative QCD • The wave function of a high-energy hadronthe dipole picture and the BFKL equation • The BFKL equation at leading orderconformal invariance and solutions of the equation • BFKL at next-to-leading orderpotential problems and all-order resummations • How to go beyond the BFKL approachthe ideas that led to the Color Glass Condensate picture

4. The scattering amplitude

5. + z • after the collision final-state particles rapidity pseudo rapidity High-energy scattering light-cone variables • before the collision • during the collision the momentum transfer is mainly transverse

6. initial momenta and final state parametrized by and • consider 2 to 2 scattering with (Regge limit): • next-order diagram: the new final-state gluon yields the factor this contribution goes as and is as large as the zeroth order in  using the perturbative expansion is not the right approach 2 to 2 scattering at high energy the exchanged particle has a very small longidudinal momentum: the final-state particles are separated by a large rapidity interval:

7. the leading-logarithmic approximation only gluons contribute in the LLA, and the coupling doesn’t run n-th order this is schematic, but the actual summation of the leading logs by BFKL confirms this power-law growth with energy in practice, NLL corrections are large Summing large logarithms • the relevant perturbative expansion in the high-energy limit: : leading-logarithmic approximation (LLA), sums : next-to-leading logarithmic approximation (NLLA), sums . . . Balitsky, Fadin, Kuraev, Lipatov

8. the unintegrated gluon density and obey the same BFKL equation, with different initial condition is related to the hadron’s wave function which we will study in the following kT factorization is also proved at NLL but there are many complications Fadin et al. (2005-2006) kT factorization • from parton-parton scattering to hadron-hadron scattering impact factors no Y dependence Green function, this is what resums the powers ofαSY

9. the solution of this linear equation the saddle point atgives the high-energy behavior initial condition obtained from the impact factor The BFKL equation • for the unintegrated gluon distribution real-virtual cancellation when comes from real gluon emission comes from virtual corrections we will derive this equation with a wave function calculation

10. The hadronic wave function

11. the wave functions can be computed following Feynman rules a bit different from those of standard covariant perturbation theory Brodsky and Lepage • the simplest light-cone wave function ~ energy denominator like in quantum mechanics The wave function of a hadron • light-cone perturbation theory (in light-cone gauge ) a quantum superposition of states - the unintegrated gluon distribution is given by - the partons in the wave function are on-shell - their virtuality is reflected by the non-conservation of momentum in the x- direction

12. the key to the simplicity of this approach : the mixed space the wave function depends only on the dipole size r because of momentum conservation The wave function of a dipole • replace the gluon cascade by a dipole cascade Mueller’s idea to compute the evolution of the unintegrated gluon distribution  simple derivation of the BFKL equation this dipole picture of a hadron is used a lot in small-x physics

13. The wavefunction • in momentum space this selects the leading logarithm using the wave function in the limit, one gets • in mixed space we have to compute

14. x = two-dipole wave function amplitude probability for dipole splitting original dipole wave function interpretation: • the evolution of the hadron wave function is that of a dipole cascade dipole splitting probability the evolution the density of dipoles in the hadron wave function is the same as the evolution of the unintegrated gluon distribution Dipole cascade in position space • the zero-th order wave function factorizes !

15. or equivalently • back to momentum space the BFKL equation for the unintegrated gluon distribution can be recovered using The BFKL equation • for the dipole density no-splitting probability splitting into a dipole of size r using and :

16. Solving the leading logarithmic (LL) BFKL equation

17. eigenfunctions: labeled by one discrete index n and one continuous ν • eigenvalues: Conformal invariance Lipatov (1986) • the BFKL kernel is conformal invariant: under the conformal transformation it becomes note the complex notation:

18. the saddle point at high energies is at high energies, one can neglect non-zero conformal spins after Fourier transforming to momentum space, one recovers the solution given earlier for the unintegrated gluon distribution BFKL solutions • a linear superposition of eigenfunctions discrete index called conformal spin specified by the initial condition continuous index ~ Mellin transformation

19. BFKL at next-to-leadinglogarithmic (NLL) accuracy

20. with The NLL-BFKL Green function it took about 10 years to compute the NLL Green function Fadin and Lipatov (1998), Ciafaloni (1998) up to running coupling effects, the eigenfunctions are unchanged the eigenvalues are

21. On the NLO impact factors the NLO impact factors are very difficult to compute for deep inelastic scattering it took about 10 years to compute the photon impact factors Bartels et al. for jet production in hadron-hadron collisions the impact factors are known but after 5 years there is still no numerical result Bartels, Colferai and Vacca (2002) for vector meson production the impact factors are known the first complete NLL-BFKL calculation was for Ivanov and Papa (2006) but the results are very unstable when varying the renormalization scheme  impossible to make reliable predictions

22. different resummation schemes there are different proposal to add the relevant higher-order corrections Ciafaloni, Colferai, Salam, Stasto, Altarelli, Ball, Forte, Brodsky, Lipatov, Fadin, … (1999-now) there are equivalent at NLL accuracy and produce similar numerical results Salam’s schemes are the only ones used so far for phenomenological studies because they are easy to implement All-order resummations • truncating the BFKL perturbative series generates singularities Salam (1998), Ciafaloni, Colferai and Salam (1999) has spurious singularities in Mellin (γ) space, they lead to unphysical results, this is an artefact of the truncation of the perturbative series to produce meaningful NLL-BFKL results, one has to add the higher order corrections which are responsible for the canceling the singularities

23. regularization implicit equation Strategy: there is some arbitrary: different schemes S3, S4, … in practice, each value k1k2 leads to a different effective kernel the S3 kernel (extended to p≠ 0) : with expanding in powers of , one recovers Salam’s resummation schemes in momentum space, the poles of correspond to the known so-called DGLAP limits k1 >> k2 and k1 << k2 this gives information/constraints on what add to the next-leading kernel

24. Resummed NLL BFKL • the resummed NLL-BFKL Green function now running coupling (with symmetric scale) effective kernel values of at the saddle point the power-law growth of scattering amplitudes with energy is slowed down compared to the LLA result the growth with rapidity of the gluon density in the hadronic wave function is also slower

25. Going beyond the BFKL approach

26. The problem with BFKL • the growth of scattering amplitudes with energy this leads to unitarity violations, for instance for the total cross-section, the Froissart bound cannot be verified at high energies what did we do wrong ? use a perturbative treatment when we shouldn’t have • the growth of gluon density with increasing rapidity even if this initial condition is a fully perturbative wave function (no gluons with small ) the BFKL evolution populates the non perturbative region this so-called infrared diffusion invalidates the perturbative treatment

27. the color glass condensate (CGC) in this approach, the hadronic wave function is described by classical fields the BFKL growth is due to the approximation that gluons in the wave function evolve independently when the gluon density is large enough, gluon recombination becomes important the idea of the CGC is to take into account this effect via strong classical fields the CGC sums both and Proposals to go beyond BFKL summing terms isn’t enough, high-density effects are missing to deal with this many body problem, one needs effective degrees of freedom • the modified leading logarithmic approximation (MLLA) in this approach, hadronic scattering is described by the exchange of quasi-particles called Reggeized gluons (or Reggeons) Bartels, Ewertz, Lipatov, Vacca the BFKL approximation corresponds to the exchange of two Reggeons (a Pomeron), the idea of the MLLA is to include multiple exchanges

28. the saturation regime: for with the numerous small-x gluons can be described by large color fields which can be treated as classical fields • an effective theory to describe the saturation regime of QCD higher-x gluons act as static color sources for these fields McLerran and Venugopalan (1994) The saturation phenomenon • gluon recombination in the hadronic wave function gluon density per unit area recombination cross-section recombinations important when gluon kinematics this regime is non-linear, yet weakly coupled magnitude of Qs x dependence

29. The Color Glass Condensate  short-lived fluctuations lifetime of the fluctuations ~ separation between high-x partons ≡ static sources and low-x partons ≡ dynamical fields effective wave function for the dressed hadron when computing the unintegrated gluon distribution we recover the BFKL equation in the low-density regime what I will cover: how the wave function evolves with x how do we “measure” it with well-understood probes what I will not cover: how this formalism is applied to heavy ion collisions

30. Outline of the second lecture • The evolution of the CGC wave functionthe JIMWLK equation and the Balitsky hierarchy • A mean-field approximation: the BK equationsolutions: QCD traveling waves the saturation scale and geometric scaling • Beyond the mean field approximationstochastic evolution and diffusive scaling • Computing observables in the CGC frameworksolving evolution equation vs using dipole models