Small-x physics 3- Saturation phenomenology at hadron colliders Cyrille Marquet Columbia University

Outline of the third lecture The hadronic 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

The hadronic/nuclear wave function

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

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 for instance, the total cross-section in DIS partonic cross-section parton density

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

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

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 x corresponds 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 x corresponds to high energy and forward rapidities saturation relevant for the production of jets, pions, heavy flavors, photons

The dipole models

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 λ consistent with BK + running coupling main problem: the Fourier transform behaves badly at large momenta: 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 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 matching point size of scaling violations quark masses Soyez (2007) improvement with the inclusion of heavy quarks the parameters: fixed numbers: originally, this was fixed at the leading-log value

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 the t-CGC model the hadron-size parameter is always of order C.M., Peschanski and Soyez (2007) the idea is to Fourier transform where is directly related to the measured momentum transfer

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 the DHJ version the BUW version KKT modified to feature exact geometric scaling Dumitru, Hayashigaki and Jalilian-Marian (2006) Boer, Utermann and Wessels (2008) in practice is always replaced by before the Fourier transformation KKT modified to better account for geometric scaling violations

Deep inelastic scattering (DIS)

Kinematics of DIS size resolution 1/Q k k’ p lh center-of-mass energy S = (k+p)2 *h center-of-mass energy W2 = (k-k’+p)2 photon virtuality Q2 = - (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

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

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 then up to deviations due to quark masses the geometric scaling implies at small x, the dipole cross section is comparable to that of a pion, even though r ~ 1/Q << 1/QCD

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)

Diffractive DIS

Inclusive diffraction in DIS k k’ p k k’ p p’ when the hadron remains intact rapidity gap some events are diffractive momentum fraction of the exchanged object (Pomeron) with respect to the hadron diffractive mass MX2 = (p-p’+k-k’)2 the measured cross-section momentum transfer t = (p-p’)2 < 0

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

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

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)

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

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 Kowalski, Lappi, C.M. and Venugopalan (2008) nuclear effects enhancement at large  suppression at small  averaging at the cross-section level allows the breakup of the nucleus into nucleons

Exclusive vector meson production sensitive to impact parameter the overlap function: instead of lots of data from HERA 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 measurements:

Forward particle production in pA collisions

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)

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

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

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

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

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 as k2 decreases, it gets closer to QS and the correlation in azimuthal angle is suppressed some results for azimuthal correlations obtained by solving BK, not from model k2 is varied from 1.5 to 3 GeV C.M. (2007)

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