The high-energy limit of DIS and DDIS cross-sections in QCD Cyrille Marquet Service de Physique Théorique CEA/Saclay based on Y. Hatta, E. Iancu, C.M., G. Soyez and D. Triantafyllopoulos, hep-ph/

Contents Introduction kinematics and notations The Good and Walker picture realized in pQCD - eigenstates of the QCD S-matrix at high energy: color dipoles - the virtual photon expressed in the dipole basis - inclusive and diffractive cross-sections in terms of dipole amplitudes High-energy QCD predictions for DIS and DDIS - the geometric scaling regime - the diffusive scaling regime - consequences for the inclusive and diffractive cross-sections Conclusions towards the LHC

Kinematics and notations photon virtuality Q 2 = - (k-k’) 2 > 0  *p collision energy W 2 = (k-k’+p) 2 the  *p total cross-section:  DIS (x, Q 2 ) high-energy limit: x << 1 Bjorken x:  pp’ diffractive mass of the final state: M X 2 = (p-p’+q) 2 some events are diffractive x pom = x/  rapidity gap:  = ln(1/x pom ) high-energy limit: x pom <<1 the diffractive cross-section:  DDIS ( , x pom, Q 2 ) k k’ size resolution 1/Q Q >>  QCD

The Good and Walker picture realized in perturbative QCD

Description of diffractive events hadronic projectile before the collision after the collision: hadronic particules target which remains intact picture in the target rest frame:  : eigenstates of the interaction which can only scatter elastically the final state is then some multi-particule state elastic scattering amplitude: Good and Walker (1960)

Eigenstates of the QCD S-matrix Eigenstates? The interaction should conserve their spin, polarisation, color, momentum … Degrees of freedom of perturbative QCD: quarks and gluons In general, we don’t know, and In the high-energy limit, eigenstates are simple, provided a Fourier transform of the transverse momenta: color singlet combinations of quarks and gluons. For instance: colorless quark-antiquark pair: colorless quark-antiquark-gluon triplet: x, y, z : transverse coordinates In the large-Nc limit, further simplification: the eigenstates are only made of dipoles other quantum numbers aren’t explicitely written, the eigenvalues depend only on the transverse coordinates

The virtual photon in the dipole basis (I) For the virtual photon in DIS and DDIS, we do know them: the wavefunction of the virtual photon is computable from perturbation theory. For an arbitrary hadronic projectile, we don’t know wavefunction computed from QED at lowest order in  em k : quark transverse momentum k -k x y For instance, the component: with transverse coordinates: x : quark transverse coordinate y : antiquark transverse coordinate To describe higher-mass final states, we need to include states with more dipoles

y z x The virtual photon in the dipole basis (II) With the component: from QCD at order g S x y z1z1 z N-1 … In principle, all are computable from perturbation theory Large-Nc limit: we include an arbitrary number of gluons in the photon wave function N-1 gluons emitted at transverse coordinates  N dipoles

Formula for  DIS (x, Q 2 ) Via the optical theorem:  DIS (x, Q 2 ) = 2 Im[A el (x, Q 2 )] : Y 0 specifies the frame in which the cross-section has been calculated, therefore  DIS is independent of Y 0 Y = ln(1/x): the total rapidity average over the target wave function Two consequences: - One can use the simplest expression (obtained for Y 0 = 0) with - One can derive an hierarchy of evolution equations for the dipole amplitudes see talk by G. Soyez yesterday and previous talk by E. Iancu Y Y0Y0 arbitrary

Formula for  DDIS ( , x pom, Q 2 ) For the diffractive cross-section one obtains: ln(1/  ) the are easily computable with a Monte Carlo but we need to solve the hierarchy (or the equivalent Langevin equation) Salam (1995) see yesterday’s talk by G. Soyez in the structure function session now we need all the  = ln(1/x pom ) new factorization formula

Recovering known results - The diffractive cross-section for  close to 1: Y - The diffractive cross-section for  smaller 1, but not too small (with only the and components): - When making the assumption, we also recover the evolution equation for Kovchegov and Levin (2000) same for total and diffractive cross-section (and also exclusif processes, jet production…)

High-energy QCD predictions for the DIS and DDIS cross-sections

The dipole scattering amplitudes The dipole amplitudes,, … should be obtained from the recently derived Pomeron-loop equation This is a Langevin equation, for an event-by-event dipole amplitude, the physical amplitudes, … are them obtained fromafter proper averaging Although one cannot solve this equation yet, some properties of the solutions have been obtained, exploiting the similarities between the Pomeron-loop equation and the s-FKPP equation well-known in statistical physics Mueller and Shoshi (2004); Iancu and Triantafyllopoulos (2005); Mueller, Shoshi and Wong (2005) As explained in the previous talk by E. Iancu: Iancu, Mueller and Munier (2005)C.M., R. Peschanski and G. Soyez, hep-ph/

The different high-energy regimes in an intermediate energy regime, it predicts geometric scaling: HERA it seems that HERA is probing the geometric scaling regime at higher energies, a new scaling takes over : diffusive scaling due to the fluctuations which have been amplified as the energy increased In the diffusive scaling regime, saturation is the relevant physics up to momenta much higher than the saturation scale

Stasto, Golec-Biernat and Kwiecinski (2001) The geometric scaling regime this is seen in the data with  0.3 saturation models with the features of this regime fit well F 2 data and they give predictions which describe accurately a number of observables at HERA (F 2 D, F L, DVCS, vector mesons) and RHIC (nuclear modification factor in d-Au) Golec-Biernat and Wüsthoff (1999) Bartels, Golec-Biernat and Kowalski (2002) Iancu, Itakura and Munier (2003) see yesterday’s talk by H. Kowalski

The diffusive scaling regime D : diffusion coefficient Blue curves: different realizations of Red curve: the physical amplitude : average speed Y1Y1 Y 2 >Y 1 The dipole amplitude is a function of one variable: We even know the functional form for : the smallest size controls the correlators

Consequences on the observables geometric scaling regime:  DIS dominated by relatively hard sizes  DDIS dominated by semi-hard sizes total diff integrand total diff geometric scaling diffusive scaling dipole size r dipole size both  DIS and  DIS are dominated by hard sizes diffusive scaling regime: yet saturation is the relevant physics

Some analytic estimates Analytical estimates for  DIS (x, Q 2 ) in the diffusive scaling regime: valid for And for the diffractive cross-section (integrated over  at fixed x) withand Physical consequences: up to momenta Q² much bigger than the saturation scale the cross-sections are dominated by small dipole sizes there is no Pomeron (power-like) increase the diffractive cross-section is dominated by the scattering of the quark-antiquark component

QCD in the high-energy limit provides a realization of the Good and Walker picture, and allows to make accurate QCD predictions for diffractive observables From the recently-derived Pomeron-loop equation, a new picture emerges for DIS. In an intermediate energy regime: geometric scaling - inclusive and diffractive experimental data indicate that HERA probes this regime At higher energies: diffusive scaling - up to values of Q² much higher than the saturation scale, saturation is the relevant physics - cross-sections are dominated by rare events, in which the photon hits a black spot, that he sees dense (at saturation) at the scale Q² - the features expected when are extended up to much higher Q² Towards the LHC - the energy there may be high enough to see the diffusive scaling regime - we want to say more before LHC starts: determination of and D ?  work in progress Conclusions and Outlook