QCD-2004 Lesson 2 :Perturbative QCD II 1)Preliminaries: Basic quantities in field theory 2)Preliminaries: COLOUR 3) The QCD Lagrangian and Feynman rules 4) Asymptotic freedom from e + e - -> hadrons 5) Deep Inelastic Scattering Guido Martinelli Bejing 2004

NO DEPENDENCE ON THE CUTOFF, NON INFRARED DIVERGENCE

Deep Inelastic Scattering DIS Guido Martinelli Bejing 2004 hadronic system with invariant mass W and momentum p X l(k) l=e, ,  (q) q=k-k’ k’ proton,neutron of momentum p

pXpX l(k) l=e, ,  (q) q=k-k’ k’ p Bjorken dimensionless variables q Kinematics

pXpX l(k) l=e, ,  (q) q=k-k’ k’ p Structure Functions

Scaling limit CROSS SECTION pXpX l(k) l=e, ,  (q) q=k-k’ k’ p

Naive Parton Model For electromagnetic scattering processes: fragments  (q) + q(p i ) -> q(p f ) by neglecting parton virtuality and transverse momenta pipi pfpf strucked quark

Naive Parton Model pipi pfpf Parton cross-section: From which we find: longitudinal cross-section

THE LONGITUDINAL STRUCTURE FUNCTION (CROSS-SECTION) IS ZERO FOR HELICITY CONSERVATION: p i =(Q/2,Q/2,0,0) p f =(Q/2,-Q/2,0,0) q=(0,-Q,0,0) massless spin 1/2 partons = helicity longitudinally polarized photon spinless partons would give F transverse =0

Parton Model:Useful Relations and Flavour Sum Rules strange quarks in the proton? proton = uud + qq pairs u gluon s s photon Gottfried Sum Rule

Neutrino Cross Section pipi pfpf W y dd From neutrino-antineutrino cross-section we can distinguish quarks from antiquarks

Parton Model and QCD q +  q´ for simplicity let us consider first only the non-singlet case, namely q +  q´ + g

Parton Model and QCD  is a cutoff necessary to regularize collinear divergences Effective quark distribution

Classic Interpretation p = z P z´=(x/z)p = x P  dW is the probability of finding a quark with a fraction x/z of its ``parent” quark and a given k 2 T <<Q 2 The total probability (up to non leading logarithms) is

22 THE EFFECTIVE NUMBER OF QUARKS WITH THE APPROPRIATE X VARIES WITH Q 2 z1 z2 z3 x  2 )

22 Q2Q2 THE EFFECTIVE NUMBER OF QUARKS WITH THE APPROPRIATE X VARIES WITH Q 2 z1 z2 z3 x Q2)Q2)

t=ln(Q 2 /  2 ) Mellin Transform Differential equation Solution

It will be shown later as q(n,t 0 ) can be related to hadronic matrix elements of local operators which can computed in lattice QCD

GLUON CONTRIBUTION TO THE STRUCTURE FUNCTIONS THE GLUON DISTRIBUTION IS DIFFICULT TO MEASURE BECAUSE IT ENTERS ONLY AT ORDER

z x/z SPLITTING FUNCTIONS

By B. Foster (Bristol U.), A.D. Martin (Durham U.), M.G. Vincter (Alberta U.),. On Page of the Review of Particle Properties, please cite the entire review Phys.Lett.B592: 1,2004.

By B. Foster (Bristol U.), A.D. Martin (Durham U.), M.G. Vincter (Alberta U.),. On Page of the Review of Particle Properties, please cite the entire review Phys.Lett.B592: 1,2004.

NEXT-TO-LEADING CORRECTIONS TO THE STRUCTURE FUNCTIONS IN THE NAÏVE PARTON MODEL F 3 (x) = q(x) - q(x) ˜ q V (x) IN THE LEADING LOG IMPROVED PARTON MODEL F 3 (x Q 2 ) = q(x,Q 2 ) - q(x, Q 2 ) ˜ q V (x, Q 2 ) Gluon contribution Next-to-leading correction

NON UNIVERSAL REGULARIZATION PRESCRIPTION DEPENDENT CANNOT HAVE A PHYSICAL MEANING, HOWEVER What matters is the combination: regularization independent process dependent

NLL EVOLUTION LET US DEFINE BY ABSORBING THE ENTIRE NLL CORRECTION IN THE DEFINITION OF THEN

The Operator Product Expansion pipi ,W,W  d d  X

The Operator Product Expansion  The term at x 0 < 0 does not contribute because cannot satisfy the 4-momentum  -function

Neglecting the light quark mass (up to a factor i): the covariant derivative corresponds to momenta of order  QCD the covariant derivative corresponds to large momenta of order q >> M N,  QCD Thus, a part a trivial Lorentz structure, we have to compute

Short Distance Expansion x -> 0 Local operator ô x0 Higher twist Suppressed as

Local operators and Mellin Transforms of the Structure Functions Renormalization scale DEFINE:

Moment of the Structure Functions and Operators Total momentum conservation Current conservation (Adler Sum Rule)