rpp2010-rev-structure-functions

rpp2010-rev-structure-functions - 16. Structure functions 1...

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16. Structure functions 1 16. STRUCTURE FUNCTIONS Updated July 2009 by B. Foster (University of Oxford), A.D. Martin (University of Durham), and M.G. Vincter (Carleton University). 16.1. Deep inelastic scattering High-energy lepton-nucleon scattering (deep inelastic scattering) plays a key role in determining the partonic structure of the proton. The process `N ` 0 X is illustrated in Fig. 16.1. The ±lled circle in this ±gure represents the internal structure of the proton which can be expressed in terms of structure functions. k k q P, M W Figure 16.1: Kinematic quantities for the description of deep inelastic scattering. The quantities k and k 0 are the four-momenta of the incoming and outgoing leptons, P is the four-momentum of a nucleon with mass M ,and W is the mass of the recoiling system X . The exchanged particle is a γ , W ± ,o r Z ; it transfers four-momentum q = k k 0 to the nucleon. Invariant quantities: ν = q · P M = E E 0 is the lepton’s energy loss in the nucleon rest frame (in earlier literature sometimes ν = q · P ). Here, E and E 0 are the initial and ±nal lepton energies in the nucleon rest frame. Q 2 = q 2 =2( EE 0 −→ k · k 0 ) m 2 ` m 2 ` 0 where m ` ( m ` 0 ) is the initial (±nal) lepton mass. If 0 sin 2 ( θ/ 2) À m 2 ` , m 2 ` 0 ,then 4 0 sin 2 ( 2), where θ is the lepton’s scattering angle with respect to the lepton beam direction. x = Q 2 2 where, in the parton model, x is the fraction of the nucleon’s momentum carried by the struck quark. y = q · P k · P = ν E is the fraction of the lepton’s energy lost in the nucleon rest frame. W 2 =( P + q ) 2 = M 2 +2 Q 2 is the mass squared of the system X recoiling against the scattered lepton. s k + P ) 2 = Q 2 xy + M 2 + m 2 ` is the center-of-mass energy squared of the lepton-nucleon system. K. Nakamura et al. (Particle Data Group), JP G 37 , 075021 (2010) (http://pdg.lbl.gov) December 17, 2010 12:57
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2 16. Structure functions The process in Fig. 16.1 is called deep ( Q 2 À M 2 ) inelastic ( W 2 À M 2 ) scattering (DIS). In what follows, the masses of the initial and scattered leptons, m ` and m ` 0 ,a re neglected. 16.1.1. DIS cross sections : d 2 σ dx dy = x ( s M 2 ) d 2 σ dx dQ 2 = 2 πMν E 0 d 2 σ d Ω Nrest dE 0 . (16 . 1) In lowest-order perturbation theory, the cross section for the scattering of polarized leptons on polarized nucleons can be expressed in terms of the products of leptonic and hadronic tensors associated with the coupling of the exchanged bosons at the upper and lower vertices in Fig. 16.1 (see Refs. 1–4) d 2 σ dxdy = 2 πyα 2 Q 4 X j η j L μν j W j μν . (16 . 2) For neutral-current processes, the summation is over j = γ,Z and γZ representing photon and Z exchange and the interference between them, whereas for charged-current interactions there is only W exchange, j = W . (For transverse nucleon polarization, there is a dependence on the azimuthal angle of the scattered lepton.) L μν is the lepton tensor associated with the coupling of the exchange boson to the leptons. For incoming leptons of charge e = ± 1 and helicity λ = ± 1, L γ μν =2 ³ k μ k 0 ν + k 0 μ k ν k · k 0 g μν iλε μναβ k α k 0 β ´ , L μν =( g e V + eλg e A
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rpp2010-rev-structure-functions - 16. Structure functions 1...

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