New perspective of QCD at high energy New perspective of QCD at high energy - introduction to Color Glass Condensate - Kazunori Itakura Service de Physique Theorique CEA/Saclay, France

Plan of the lectures 1. Introduction and overview 2. Kinematics and Evolution equations Bjorken x and virtuality Q 2 in DIS, DGLAP and BFKL equations 3. Growth of the gluon distribution and unitarity violation Solution to the BFKL equation, High energy behaviour of cross section 4. Color Glass Condensate McLerran-Venugopalan model, Quantum evolution and JIMWLK equation, the Balitsky-Kovchegov equation 5. The Balitsky-Kovchegov equation Solutions (analytical and numerical), saturation scale, geometric scaling, absence of infrared diffusion 6. Recent progress in phenomenology HERA DIS, RHIC (Au-Au, d-Au) 7. Recent progress in theory Traveling wave, Structure of JIMWLK equation, Evolution equation for n-point functions, attempts beyond the BK equation (fluctuation) I II III IV

Lecture I

1. Introduction and overview

Introduction Questions/problems which we would like to answer/understand: What is the high energy limit of QCD? Can we see it in experiments?  What is the high energy limit of “scattering” involving hadrons?  How can we treat it from first principles (QCD)?  Is the energy in the present experiments enough large to see it? Is there an unexplored regime which is still approachable in weak coupling techniques?  In fact, most of the “success stories” in QCD is due to identifying new perturbative regimes. What’s next?  Can we define a perturbative regime in relation to the high energy limit of scattering? What is the dynamical information of the nucleons?  Static properties such as nucleon’s constituents, mass, radius, charge etc. do not help to describe the actual scattering. We need information about, say, higher Fock components, which becomes relevant at high energy. This is the dynamical aspects of the nucleon!

References: review and lecture notes -- Iancu & Venugopalan, hep-ph/ , published in “QGP3” “The Color Glass Condensate and High Energy Scattering in QCD” -- Iancu, Leonidov & McLerran, hep-ph/ Cargese lectures “The Colour Glass Condensate: An introduction” (actually this is not an introduction) -- Al Mueller, hep-ph/ Cargese lectures “Parton Saturation - An Overview” For japanese reading (only introductory discussion), -- Itakura, JPS membership journal (February 2004) “Color Glass Condensate – a universal picture of hadrons and nuclei” Textbooks -- Devenish & Cooper-Sarkar, Oxford University Press, April 2004 “Deep Inelastic Scattering” For basic understanding before CGC -- Forshaw & Ross, Cambridge University Press, 1997 “Quantum Chromodynamics and the Pomeron” -- Ellis, Stirling, & Webber, Cambridge University Press, “QCD and Collider Physics”

Overview (I) Phase diagram ancient < 1970 old ~ 1980 new > 2002 Regge theory 1/x Q2Q2 saturation parton gas Extended scaling regime color glass condensate -Better understanding of the saturation line -new phase = extended scaling regime  -proton scatt. Gribov,Levin,Ryskin already considered gluon saturation

Overview (II) Universality As the scattering energy is increased, hadrons and nuclei eventually become the color glass condensate, irrespective of the details of the system. This is due to the multiple production of gluons, while the species properties are carried by the valence quarks.  Universal fixed point of the evolution. Indeed, in the saturation regime, the gluon distributions of a nucleon and a nucleus are expressed in the same functional form. The only difference is the magnitude of the saturation scale. High energy limit of QCD is the Color Glass Condensate.

Overview (III) Important Experimental Results 1993: Beginning of excitement in hard small x physics (not the soft Pomeron physics) Steep rise of F 2 at small x evidence for BFKL ?  LO, NLO(98), resumed NLO (99)

Overview (III) Important Experimental Results Hadronic cross section at high energy (pp, total) ln s, ln 2 s (Froissart bound), or s   (Pomeron) ?? Most recent PDG  consistent with ln 2 s. include cosmic ray pp data of AKENO & Fly’s eye Seems to saturate the Froissart bound…. How can the usual soft Pomeron description be modified so that the unitarity bound is satisfied?? The same question should be asked to the hard Pomeron, too. S 1/ GeV

Overview (III) Important Experimental Results 2001: Discovery of geometric scaling in DIS (ep) 2002: eA Need saturation to understand this phenomena total  * p cross section

Overview (III) Important Experimental Results 2004: high pt suppression at forward rapidity in dAu at RHIC Another evidence of CGC??

“Shattered Glass” by D. Appell Scientific American, April 2004 (Nikkei Science, June 2004, p8)

2. Kinematics and evolution equations equations Light Cone variables momentum q  = (q 0, q 1, q 2, q 3 )  q + = (q 0 + q 3 )/ longitudinal momentum q - = (q 0 – q 3 )/ LC energy q T =(q 1,q 2 ) transverse momentum √ 2 √ 2 Infinite momentum frame = a frame in which the target proton is moving very fast (in z direction). p  =(p 0,p 1,p 2,p 3 )~(p+M 2 /2p, 0, 0, p)~(p, 0, 0, p). p + = (p 0 + p 3 )/ ~ p very large p - = (p 0 - p 3 )/ ~ 0 p T =(p 1,p 2 ) = 0 √ 2 √ 2 √ 2

DIS Kinematics (I) Two basic kinematical variables in deep inelastic scattering electron(k) + proton(p)  electron(k’) + X Lorentz invariant quantities qq pp Q 2 = - q 2 > 0 : virtuality of photon x = Q 2 /2p q : Bjorken variable.

DIS Kinematics (II) qq pp 1) Infinite momentum frame p  =(p+m 2 /2p,0,0,p)~(p,0,0,p), where p is large. Take q  = (q 0,q 1,q 2,0),(Breit frame) so that q 0 ~p  q  /p  0 as p  infinity. Thus, one finds Q 2 ~ q T 2 2) On-shell condition for the struck quark (p q  + q  ) 2 =2p q (  - x)+  2 M p 2 =0 Ignoring the proton mass  x =  : x is the fraction of momentum carried by the parton to the total nucleon momentum. 0 < x < 1. In IMF, this becomes just the longitudinal momentum fraction. p + = p, p - ~ 0. Q 2 = - q 2 : transverse resolution, transverse size of measured partons x = Q 2 /2p q : fraction of longitudinal momentum of a parton Physical meaning of Q 2 and x p q  =  p  qq. √ 2.

DIS Kinematics (III) Q 2 = q T 2 : transverse resolution x =p + /P + : longitudinal momentum fraction transverse longitudinal

Structure functions of a proton F 1 and F 2 structure functions (neglecting the proton mass) from Lorentz decomposition of hadronic tensor W  The Bjorken limit: Q 2, =p q  infinity with x fixed x =Q 2 /2 F i (x,Q 2 )  F i (x) Bjorken Scaling -- A proton is made of point-like objects (otherwise Q 2 dependent) -- naïve parton model: a proton is an incoherent collection of partons whose distribution is given by the probability q(x)dx  with x  being a fraction of longitudinal momentum carried by a parton. F 2 (x)= 2 x F 1 (x)=  q e q 2 x q(x) q(x) quark distribution function -- there is a weak violation of Bjorken scaling  log Q 2 dependence (QCD effect!) y=q p/k p...

Structure functions and DGLAP equation One gluon emission gives the logarithmic Q 2 dependence  Change of the resolution scale Q 2  evolution equation for the parton distribution functions DGLAP equation (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) “Splitting function” P ij (x): a probability of finding a parton of type “i” in a parton of type “j”. The equation for the change of “transverse resolution”

DGLAP equation at small x Splitting functions at leading order O(  S 0 ) At small x, only P gq and P gg are relevant.  Gluon dominant at small x! The double log approximation (DLA) of DGLAP is easily solved. -- increase of gluon distribution at small x

BFKL evolution Evolution with respect to x or rapidity y = ln 1/x Resum all the contributions (  S ln 1/ x ) n (n>0) in gluon distribution xg(x,Q 2 ) even if they are not accompanied by powers of ln Q 2. A 0 ~  S ln 1/x ~ (  S ln 1/x ) n x 1 >> x 2 >>….>> x n >> x strong ordering in x  gluon number ~  C n (1/n!) (  S ln 1/x ) n ~ exp{   S ln 1/x } n

BFKL equation Linear equation for the unintegrated gluon distribution f(x,k) recursive equation Relation to the gluon distribution

Summary for lecture I DIS process is described by two independent kinematical variables x and Q 2 High energy = small x Change of energy  multiple gluon production  BFKL equation at high energy [(  S ln 1/x) is not small]