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8 Probing the Proton: Electron - Proton Scattering

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8 Probing the Proton: Electron - Proton Scattering 8 Probing the Proton: Electron - proton scattering Scattering of charged leptons by protons is an electromagnetic interaction. Electron beams have been used to probe the structure of the proton (and neutron) since the 1960s, with the most recent results coming from a high energy electron-proton collider called HERA at DESY in Hamburg. These experiments provide direct evidence for the composite nature of protons and neutrons, and measure the distributions of the quarks and gluons inside the nucleon. The results of e−p e−p scattering depend strongly on the wavelength λ = E/c. → At very low electron energies λ>>rp, where rp is the radius of the proton, the • scattering is equivalent to that from a point-like spin-less object At low electron energies λ rp the scattering is equivalent to that from a extended • charged object ∼ At high electron energies λ<rp: the wavelength is sufficiently short to resolve • sub-structure. Scattering from constituent quarks At very high electron energies λ<< rp: the proton appears to be a sea of quarks • and gluons. 8.1 Form Factors Extended object - like the proton - have a matter density ρ(r), normalised to unit volume: d3rρ (r ) = 1. The Fourier Transform of ρ(r) is the form factor, F (q): F (q )= d3r exp iq r ρ(r ) F (0) = 1 (8.1) { · } ⇒ Cross section from extended objects are modified by the form factor: dσ dσ F (q ) 2 (8.2) dΩ extended ≈ dΩ point like| | − For e−p e−p scattering we need form factors are required: F to describe the distri- → 1 bution of the electric charge F2 to describing the recoil of the proton 8.2 Elastic Scattering The elastic scattering of a pointlike spin-1/2 electron by a pointlike spin 1/2 target is described in the relativistic limit pe = Ee by the Mott formula: dσ α2 θ q2 θ = cos2 sin2 (8.3) dΩ 4p2 sin4 θ 2 − 2m2 2 point e 2 p 51 In this formula θ and pe are in the Lab frame. The no recoil limit corresponds to a very massive target m2 q2. p In the non-relativistic limit p m this reduces to Rutherford scattering: e e dσ α2 = (8.4) dΩ 4m2v4 sin4 θ NR e e 2 e−(p1) e−(p3) ieγµ q ieKµ − p(p2) p(p4) 2m ν = Q2 = q2 (8.5) p − Note that ν>0 by energy conservation, so Q2 > 0 and the mass squared of the virtual photon is negative, q2 < 0! 8.3 Form Factors Deviations from the point-like Mott scattering are described in the no recoil limit by a form factor F (q2), related to the finite size of a charge distribution inside the proton: dσ dσ = F (q2) 2 (8.6) dΩ dΩ | | point At low q2 the distances probed are large compared to the size of the proton, so the scattering still appears point-like with F (0) = 1. As q2 gets larger the electron probes deeper into the proton, and F (q2) is found to decrease. Mathematically, the form factor is the Fourier transform of the charge distribution inside the proton: F (q2)= ρ(x ) exp(iq x )d3x (8.7) · The charge distribution ρ(x ) is assumed to be spherically symmetric, and normalised to one. It has a mean square radius <r2 >: ρ(r)d3r =1 <r2 >= r2ρ(r)d3r (8.8) The matrix element for elastic scattering is written in terms of electron and proton currents: 2 e µ (e−p e−p)= (¯u γ u )(¯u K u ) (8.9) M → (p p )2 3 1 4 µ 2 1 − 3 52 where the proton is treated as an extended structure, with a current operator Kµ that is more complex than the point-like γµ: µ µ 2 iκp 2 µν ν 2 K = γ F1(q )+ F2(q )σ qν + q F3(q ) (8.10) 2mp In this general form there are three form factors F1, F2 and F3 which are functions of 2 µ q . However, electromagnetic current conservation δµ(¯u4K u2) = 0 implies that F3 = 0. F1 is the electrostatic form factor, while F2 is associated with the recoil of the proton. The anomalous magnetic moment of the proton is defined by: e(1 + κp) µp = κp =1.79 (8.11) 2mp The differential cross section for elastic electron-proton scattering is 2 2 2 2 dσ α E3 2 κ q 2 2 θ q 2 2 θ = F F cos (F1 + κF2) sin (8.12) dΩ 4E2 sin4 θ E 1 − 4m2 2 2 − 2m2 2 lab 1 2 1 p p 2 where the two form factors F1 and F2 are functions of q which parameterise the struc- ture of the proton. They have to be determined by experiment. For a point-like spin 1/2 particle, F1 = 1, κ = 0, and the above equation reduces to the Mott scattering result. It is common to use linear combinations of the form factors: κq2 GE = F1 + 2 F2 GM = F1 + κF2 (8.13) 4mp which are referred to as the electric and magnetic form factors, respectively. The differential cross section can be rewritten as: dσ α2 E G2 + τG2 θ θ = 3 E M cos2 +2τG2 sin2 (8.14) dΩ 4E2 sin4 θ E 1+τ 2 M 2 lab 1 2 1 2 2 where we have used the abbreviation τ = Q /4mp. The experimental data on the form factors as a function of q2 are in good agreement with a “dipole” fit: G β2 2 G = M = β =0.84GeV (8.15) E µ β2 + Q2 p This corresponds to an exponential charge distribution: ρ(r)=ρ exp( r/r )1/r2 =0.71GeV2 <r2 >=0.81fm2 (8.16) 0 − 0 0 The whole of the above discussion can be repeated for electron-neutron scattering, with similar form factors for the neutron. 53 Note that even though the neutron has zero total charge, it has an anomalous magnetic moment: eκn µn = κn = 1.91 (8.17) 2mn − The anomalous magnetic moments are themselves evidence that the proton and neutron are not pointlike Dirac fermions. 54 8.4 Deep Inelastic Scattering During inelastic scattering the proton can break up into its constituent quarks which then form a hadronic jet. At high q2 this is known as deep inelastic scattering (DIS). e−(p1) e−(p3) ieγµ q X p(p2) The invariant mass, W , of the final state hadronic jet is: 2 2 2 W = mp +2mpν + q (8.18) Since W = m the four-momentum and energy transfer, q2 and ν, are two independent p variables in DIS, and it is necessary to measure E1, E3 and θ in the Lab frame to determine the full kinematics. It is useful to introduce two dimensionless variables, a parton energy x, and a rapid- ity y, which replace ν and q2 Q2 q2 p q ν x = = − y = 2 · = (8.19) 2m ν 2m ν p p E p p 2 · 1 1 It is an exercise to show that the allowed kinematic ranges of these variables are 0 x 1 and 0 y 1. ≤ ≤ ≤ ≤ 8.5 Structure Functions The matrix element squared for DIS can be factorised into lepton and hadron currents: e4 2 = Lµν(W ) (8.20) |M| q2 e hadron µν µν where the hadronic part Whadron describes the inelastic breakup of the proton. As in elastic scattering, there are two independent form factors in Whadron, which are known 2 2 as structure functions W1(ν,Q ) and W2(ν,Q ). In the Lab frame, the doubly differential cross section for deep inelastic scattering is: 2 dσ α 2 2 θ 2 2 θ = W2(ν,Q )cos +2W1(ν,Q ) sin (8.21) dE dΩ 4E2 sin4 θ 2 2 3 lab 1 2 55 Bjorken predicted that at high energy the structure functions should exhibit a property called scaling: m W (ν,Q2) F (x) νW (ν,Q2) F (x) (8.22) p 1 → 1 2 → 2 where F1(x) and F2(x) are now functions of x alone. Note that the structure functions F1(x) and F2(x) in DIS are different from the elastic 2 2 form factors F1(q ) and F2(q )! The experimental data for F2 are shown in figure 8.1. For intermediate regions of x and Q2 scaling holds, but at high Q2 and low x there is a significant amount of scaling violation which we will discuss later. 2 Figure 8.1: Structure function F2 for large Q and small x, as measured at HERA using collisions between 30 GeV electrons and 830 GeV protons. 56 8.6 The Parton Model The parton model was proposed by Feynman in 1969, to describe deep inelastic scat- tering in terms of point-like constituents inside the nucleon known as partons with an effective mass m<mp. Nowadays partons are identified as being quarks or gluons. e−(p1) e−(p3) q m m The parton model restores the elastic scattering relationship between q2 and ν, with m replacing mp: q2 ν + = 0 (8.23) 2m and the cross section for elastic electron-parton scattering is: dσ Z2α2 E θ q2 θ = 3 cos2 sin2 (8.24) dΩ 4E2 sin4 θ E 2 − 2m2 2 lab 1 2 1 where Z is the charge of the parton (+2/3 or 1/3 for quarks). − Effectively the DIS structure functions for scattering offa single parton, have become delta functions: Q2 Q2 Q2 2W = δ(ν ) W = δ(ν ) (8.25) 1 2m2 − 2m 2 − 2m It can be seen that the parton energy variable: Q2 m x = = (8.26) 2mpν mp is the fraction of the proton rest mass carried by the parton.
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