Small- x physics 3- Saturation phenomenology at hadron colliders

1. Small-x physics3- Saturation phenomenologyat hadron colliders Cyrille Marquet Columbia University

2. Outline of the third lecture • The hadronic wave functionsummary of what we have learned • The saturation modelsfrom GBW to the latest ones • Deep inelastic scattering (DIS)the cleanest way to probe the CGC/saturationallows to fix the model parameters • Diffractive DIS and other DIS processesthese observables are predicted • Forward particle production in pA collisionsand the success of the CGC picture at RHIC

3. The hadronic/nuclearwave function

4. The hadron wave function in QCD • one can distinguish three regimes S (kT ) << 1 perturbative regime, dilute system of partons: hard QCD (leading-twist approximation) weakly-coupled regime, dense system of partons (gluons) non linear QCD the saturation regime non-perturbative regime: soft QCD relevant for instance for the total cross-section in hadron-hadron collisions relevant for instance for top quark production not relevant to experiments until the mid 90’s with HERA and RHIC: recent gain of interest for saturation physics

5. for instance, the total cross-section in DIS parton density partonic cross-section The dilute regime as kT increases, the hadron gets more dilute the dilute (leading-twist) regime: 1/kT ~ parton transverse size transverse view of the hadron leading-twist regime hadron = a dilute system of partons which interact incoherently Dokshitzer Gribov Lipatov Altarelli Parisi

6. the saturation regime: hadron = a dense system of partons whichinteract coherently the saturation regime of QCD:the weakly-coupled regime that describes the collective behaviorof quarks and gluons inside a high-energy hadron The saturation regime as xdecreases, the hadron gets denser the separation between the dilute and dense regimes is caracterized by a momentum scale: the saturation scaleQs(x) Balitsky Fadin Kuraev Lipatov

7. N = 1 N << 1 the saturation scale: the amplitude is invariant along any line parallel to the saturation line traveling wave solutions  geometric scaling Geometric scaling from BK • what we learned about the transition to saturation: the dipole scattering amplitude

8. When is saturation relevant ? in processes that are sensitive to the small-x part of the hadron wavefunction • deep inelastic scattering at small xBj : • particle production at forward rapidities y : at HERA, xBj ~10-4 for Q² = 10 GeV² in DIS small xcorresponds to high energy saturation relevant for inclusive, diffractive, exclusive events at RHIC, x2 ~10-4 for pT ² = 10 GeV² pT , y in particle production, small xcorresponds to high energy and forward rapidities saturation relevant for the production of jets, pions, heavy flavors, photons

9. The dipole models

10. improvement for small dipole sizes Bartels, Golec-Biernat and Kowalski (2002) obtained by including DGLAP-like geometric scaling violations standard leading-twist gluon distribution this is also what is obtained in the MV model for the CGC wave function, the behavior is recovered The GBW parametrization • the original model for the dipole scattering amplitude Golec-Biernat and Wusthoff (1998) it features geometric scaling: the saturation scale: the parameters: fitted on F2 data λconsistent with BK + running coupling main problem: the Fourier transform behaves badly at large momenta:

11. improvement with the inclusion of heavy quarks Soyez (2007) matching point size of scaling violations quark masses fixed numbers: the parameters: originally, this was fixed at the leading-log value The IIM parametrization • a BK-inspired model with geometric scaling violations Iancu, Itakura and Munier (2004) α and β such that N and its derivative are continuous at the saturation scale: main problem: the Fourier transform features oscillations

12. the t-CGC model C.M., Peschanski and Soyez (2007) the idea is to Fourier transform where is directly related to the measured momentum transfer the hadron-size parameter is always of order Impact parameter dependence the impact parameter dependence is not crucial for F2, it only affects the normalization however for exclusive processes it must be included • the IPsat model Kowalski and Teaney (2003) same as before impact parameter profile • the b-CGC model Kowalski, Motyka and Watt (2006) IIM model with the saturation scale is replaced by

13. the DHJ version Dumitru, Hayashigaki and Jalilian-Marian (2006) KKT modified to better account for geometric scaling violations • the BUW version Boer, Utermann and Wessels (2008) KKT modified to feature exact geometric scaling in practice is always replaced by before the Fourier transformation The KKT parametrization • build to be used as an unintegrated gluon distribution Kovchegov, Kharzeev and Tuchin (2004) the idea is to modify the saturation exponent

14. Deep inelastic scattering (DIS)

15. k’ k p size resolution 1/Q Kinematics of DIS lh center-of-mass energyS = (k+p)2*h center-of-mass energyW2 = (k-k’+p)2photon virtualityQ2 = - (k-k’)2 > 0 x ~ momentum fraction of the struck parton y~ W²/S • the measured cross-section experimental data are often shown in terms of

16. as usual we go to the mixed space where the interaction with the CGC is diagonal x: quark transverse coordinate y: antiquark transverse coordinate in DIS we need the overlap function The virtual photon wave functions • computable from perturbation theory wave function computed from QED at lowest order in em

17. then at small x, the dipole cross section is comparable to that of a pion, even though r ~ 1/Q << 1/QCD up to deviations due to quark masses the geometric scaling implies The dipole factorization • the virtual photon overlap functions • scattering off the CGC we already computed the dipole-CGC scattering amplitude average over the CGC wave function

18. HERA data and geometric scaling Soyez (2007) Stasto, Golec-Biernat and Kwiecinski (2001) geometric scaling seen in the data, but scaling violations are essential for a good fit IIM fit (~250 points)

19. Diffractive DIS

20. k’ k’ some events are diffractive k k when the hadron remains intact p p p’ rapidity gap diffractive mass MX2 = (p-p’+k-k’)2 momentum transfert = (p-p’)2 < 0 momentum fraction of the exchanged object (Pomeron) with respect to the hadron • the measured cross-section Inclusive diffraction in DIS

21. Fourier transform to MX2 comes from overlap of wavefunctions Fourier transform to t dipole amplitudes The dipole picture the diffractive final state is decomposed into contributions • the contribution double differential cross-section (proportional to the structure function) for a given photon polarization: geometric scaling implies

22. diffraction directly sensitive to saturation contribution of the different r regions in the hard regime DIS dominated by relatively hard sizes DDIS dominated by semi-hard sizes Hard diffraction and saturation • the total cross sections recall the dipole scattering amplitude in DIS in DDIS dipole size r

23. Comparison with HERA data with proton tagging e p  e X p H1 FPS data (2006) ZEUS LPS data (2004) without proton tagging e p  e X Y H1 LRG data (2006) MY < 1.6 GeV ZEUS FPC data (2005) MY < 2.3 GeV parameter-free predictions with IIM model (~450 points) C.M. (2007)

24. geometric scaling C.M. and Schoeffel (2006) Important features • the βdependence contributions of the different final statesto the diffractive structure function: tot =F2D at small  : quark-antiquark-gluon at intermediate  : quark-antiquark (T) at large  : quark-antiquark (L)

25. nuclear effects Kowalski, Lappi, C.M. and Venugopalan (2008) enhancement at large  suppression at small  Hard diffraction off nuclei • the dipole-nucleus cross-section Kowalski and Teaney (2003)  averaged with the Woods-Saxon distribution position of the nucleons • the Woods-Saxon averaging in diffraction, averaging at the level of the amplitude corresponds to a final state where the nucleus is intact averaging at the cross-section level allows the breakup of the nucleus into nucleons

26. rho J/Psi • success of the dipole models t-CGC b-CGC appears to work well also but no given predictions for DVCS are available Exclusive vector meson production • sensitive to impact parameter the overlap function: instead of lots of data from HERA measurements:

27. Forward particle productionin pA collisions

28. Forward particle production • forward rapidities probe small values of x kT , y transverse momentum kT, rapidity y > 0 values of x probed in the process: the large-x hadron should be described by standard leading-twist parton distributions the small-x hadron/nucleus should be described by CGC-averaged correlators the cross-section: single gluon production probes only the unintegrated gluon distribution (2-point function)

29. how the CGC is being probed if the emitted particle is a quark, involves xA xA xd xp if the emitted particle is a gluon, involves RHIC vs LHC • typical values of x being probed at forward rapidities (y~3) RHIC deuteron dominated by valence quarks nucleus dominated by early CGC evolution LHC the proton description shouldinclude both quarks and gluons on the nucleus side, the CGC picture would be better tested RHIC LHC

30. the other Wilson lines and (coming from the interaction of non-mesured partons) cancel when summing all the diagrams h h • the gluon production cross-section the transverse momentum spectrum is obtained from a Fourier transformation of the dipole size r very close to the unintegrated gluon distribution introduced earlier Inclusive gluon production • effectively described by a gluonic dipole gg dipole scattering amplitude: adjoint Wilson line with this derivation is for dipole-CGC scattering but the result valid for any dilute projectile q : gluon transverse momentum yq : gluon rapidity

31. the suppression of RdA y the suppression of RdA was predicted xA decreases (y increases) in the absence of nuclear effects, meaning if the gluons in the nucleus interact incoherently like in A protons A CGC prediction • the unintegrated gluon distribution in the geometric scaling regime is peaked around QS(Y) the infrared diffusion problem of the BFKL solutions has been cured by saturation

32. quantitative agreement Dumitru, Hayashigaki and Jalilian-Marian (2006) for the pT – spectrum with the DHJ model shows the importance of both evolutions: xA (CGC) and xd (DGLAP) shows the dominance of the valence quarks RdA and forward pion spectrum RdA • first comparison to data Kharzeev, Kovchegov and Tuchin (2004) qualitative agreement with KKT parametrization

33. some results for azimuthal correlations k2 is varied from 1.5 to 3 GeV as k2 decreases, it gets closer to QSand the correlation in azimuthal angle is suppressed obtained by solving BK, not from model C.M. (2007) 2-particle correlations in pA • inclusive two-particle production at forward rapidities in order to probe small x final state : probes 2-, 4- and 6- point functions one can test more information about the CGC compared to single particle production

34. What is going on now in this field • Link with the MLLA ?we would like to understand the differences between the picturessimilar objects have already been identified (triple Pomeron vertex) • Higher order correctionsrunning coupling corrections to BK are known,but not the full non linear equation at next-to-leading log • Heavy ion collisionswhat is the system at the time ~1/Qs after the collisioncrucial for the rest of the space-time evolution • Calculations for RHIC/LHCtotal multiplicities, jets, pions, heavy flavors, photons, dileptons