Pomeron loop equations and phenomenological consequences

1. Pomeron loop equations and phenomenological consequences Cyrille Marquet RIKEN BNL Research Center ECT* workshop, January 2007

2. Contents • The B-JIMWLK equations- scattering off a dense target • The dipole model equations- scattering off a dilute target • The Pomeron loop equations- combining dense and dilute evolution- stochasticity in the QCD evolution • Phenomenological consequences- diffusive scaling- implications for deep inelastic scattering- implications for particle production

3. Introduction x : parton longitudinal momentum fraction kT: parton transverse momentum transverse view of the hadron Regime of interest: weak coupling regime effective coupling  dense system of partons mainly gluons (small-x gluons) high-energy scattering processes are sensitive to the small-x gluons the dilute/dense separation is caracterized by the saturation scale Qs(x)

4. The B-JIMWLK equationsscattering off a dense target

5. : smallest value of longitudinal impulsion is called the hadron rapidity To describe a hadron dressed with many small-x gluons, we use an effective theory:  effective wavefunction for the dressed hadron α: large color fields created by the small-x gluons Effective description of the hadron McLerran and Venugopalan (1994) the numerous small-x gluons are responsible for a large color field which can be treated as a classical field light-cone gauge

6. the Wilson lines sum powers ofαgS ~ 1 adjoint representation The JIMWLK equation Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner a functional equation for the rapidity evolution of the JIMWLK equation gives evolution of the hadron wavefunction for large enough Y study the high-energy scattering of simple projectiles (dipoles) off this dense hadron

7. Dipoles as test projectiles the dipole: u: quark space transverse coordinate v: antiquark space transverse coordinate scattering amplitude off the dense target scattering of the quark: JIMWLK equation → evolution equation for the dipole correlators

8. An hierarchy of equations for dipoles scattering off a dense target Balitsky (1996) an hierarchy of equation involving correlators with more and more dipoles in the large Nc limit, the hierarchy is restricted to dipoles general structure: BFKL saturation

9. is not invariant, it transform into the Wilson lines sum powers of gSδ/δρ ~ 1 also approach with effective action  study the dilute regime ρ ~ gS Balitsky (2005), Hatta, Iancu, McLerran, Stasto and Triantafyllopoulos (2006) Something is missing frame invariance requires that His invariant under the following transformation (dense-dilute duality) color field color charge Kovner and Lublinsky (2005)

10. The dipole model equationsscattering off a dilute target

11.  no splitting splitting this transforms the functional equation for into a master equation for the probabilities The dipole model in the large Nc limit, the emission cascade of soft gluons is a dipole cascade: N-1 gluons emitted at transverse coordinates  N dipoles ansatz for the wavefuntion of a dilute hadron : ~ dipole creation operator Iancu and Mueller (2004) Mueller, Shoshi and Wong (2005) C.M., Mueller, Shoshi and Wong (2006) Hatta, Iancu, McLerran and Stasto (2006)

12. at lowest order with respect to αS : dipole-dipole cross-section from the master equation for the probabilities, one obtains the equation for the dipole correlators Scattering of projectile dipoles high-energy scattering of dipoles off this dilute hadron obtained from T [α] after inverting 

13. u v x z y A new hierarchy of equations for dipoles scattering off a dilute target I denote The equation for T(n) reads k = 1  the BFKL equation k > 1  fluctuation terms

14. Structure of the fluctuation term general structure: BFKL fluctuation, important when except for n = 1, there is more than BFKL analogous to recent toy models : Kovner and Lublinsky (2006) Blaizot, Iancu and Triantafyllopoulos (2006) Iancu, de Santana Amaral, Soyez and Triantafyllopoulos (2006) differences to understand work in progress previous hierarchy of Iancu and Triantafyllopoulos: Iancu and Triantafyllopoulos (2005) obtained requiring that the target dipoles scatter only once

15. The Pomeron-loop equationscombining dense and dilute evolution

16. fluctuation, important when BFKL saturation, important when for instance the truncated hierarchy can be reformulated into a Langevin equation for a stochastic dipole amplitude and the correlators are obtained by averaging the realizations: Iancu and Triantafyllopoulos (2005) A stochastic evolution by combining the evolution equations of the dense and dilute regimes, (counting the BFKL term only once), one gets the QCD evolution is equivalent to a stochastic process

17. solutions of the deterministic part of the equation: traveling waves the saturation scale: The sF-KPP equation high-energy QCD evolution = stochastic process in the universality class of reaction-diffusion processes, of the sF-KPP equation Iancu, Mueller and Munier (2005) noise r = dipole size the reduction to one dimension introduces the noise strength parameter 

18. (for ) corrections to the Gaussian law for improbable fluctuations also known confirmed by exact results in the strong noise limit and numerical results for arbitrary values of the noise strength C. M., Soyez and Xiao (2006) C. M., Peschanski and Soyez (2006) Soyez (2005) A stochastic saturation scale The noise term introduce a stochastic saturation scale the saturation scale is a stochastic variable distributed according to a Gaussian probability law: average saturation scale v : average speed of the waves D: dispersion coefficient

19. 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 A new scaling law the average dipole scattering amplitude: : the diffusion is negligible and with we obtain geometric scaling : the diffusion is important and new regime: diffusive scaling

20. Phenomenological consequencesdiffusive scaling

21. photon virtuality Q2 = - (k-k’)2 >> QCD *p collision energy W2 = (k-k’+p)2 k’ k p size resolution 1/Q Geometric scaling and DIS data Stasto, Golec-Biernat and Kwiecinski (2001) 2 this is seen in the data with

22. at higher energies, a new scaling law: diffusive scaling no Pomeron (power-like) increase HERA In the diffusive scaling regime, saturation is the relevant physics up to momenta much higher than the saturation scale High-energy DIS Y. Hatta, E. Iancu, C.M., G. Soyez and D. Triantafyllopoulos (2006) an intermediate energy regime: geometric scaling it seems that HERA is probing the geometric scaling regime

23. geometric scaling regime: diffusive scaling geometric scaling DIS dominated by relatively hard sizes DDIS dominated by semi-hard sizes diffusive scaling regime: both DIS and DDIS are dominated by hard sizes yet saturation is the relevant physics the photon hits black spots Consequences for the observables Y. Hatta, E. Iancu, C.M., G. Soyez and D. Triantafyllopoulos (2006) dipole size r

24. the transverse momentum spectrum is obtained from a Fourier transform of the dipole size r: result valid for any dilute projectile unintegrated gluon distribution Inclusive gluon production scattering amplitude with gluon production is effectively described by a gluonic dipole (gg): adjoint Wilson line the other Wilson lines (coming from the interaction of non-mesured partons) cancel h h q : gluon transverse momentum yq : gluon rapidity

25. Y in the geometric scaling regime is peaked around k ~ QS(Y) Forward particle production important in view of the LHC: large kT , small values of x kT , y particle production at forward rapidities y (in hadron-hadron and heavy-ion collisions): in forward particle production, the transverse momentum spectrum is obtained from the unintegrated gluon distribution of the small-x hadron

26. Y Consequences in particle production E. Iancu, C.M. and G. Soyez (2006) In the diffusive scaling regime, flattens with increasing Y Is diffusive scalingwithin the LHC energy range? hard to tell: theoretically, we have a poor knowledge of the coefficient D Consequences for RpA (~ ratio of gluon distribution) : Kozlov, Shoshi and Xiao (2006)

27. Conclusions • Scattering off a dense targetB-JIMWLK equations • Scattering off a dilute targetdipole model equations • Pomeron loop equationscombining the dense and dilute regimeshigh-energy QCD evolution  stochastic processthis implies: geometric scaling at intermediate energiesdiffusive scaling at higher energies • Phenomenological consequencesnew scaling laws in DIS and particle production for large momenta and small xof strong interest in view of the LHC