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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 electrons and 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 the high energy HERA electron-proton collider 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. 8.1 Electron - Proton Scattering The results of e−p e−p scattering depends strongly on the wavelength λ = hc/E. → 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 is from constituent quarks. At very high electron energies λ rp: the proton appears to be a sea of quarks • and gluons. 8.2 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 two form factors are required: F to describe the distribution → 1 of the electric charge, F2 to describe the recoil of the proton. 8.3 Elastic Electron-Proton Scattering Elastic electron-proton scattering is illustrated in figure 8.3. As the proton is a com- posite object the vertex factor is modified by Kµ (compared to γµ): 2 e µ (e−p e−p)= (¯u γ u )(¯u K u ) (8.3) M → (p p )2 3 1 4 µ 2 1 − 3 51 e−(p1) e−(p3) ieγµ q ieKµ − p(p2) p(p4) Figure 8.1: Elastic electron proton scattering. As the proton is a composite object the vertex factor is Kµ. µ µ 2 iκp 2 µν K = γ F1(q )+ F2(q )σ qν (8.4) 2mp F1 is the electrostatic form factor, while F2 is associated with the recoil of the proton (as described above). F1 and F2 parameterise the structure of the proton, and are functions of the momentum transferred by the photon (q2). 8.4 Elastic Scattering The elastic scattering in the relativistic limit pe = Ee by the Mott formula: dσ α2 θ q2 θ = cos2 sin2 (8.5) dΩ 4p2 sin4 θ 2 − 2m2 2 point e 2 p For θ and pe are in the Lab frame. In the non-relativistic limit p m this reduces to Rutherford scattering: e e dσ α2 = (8.6) dΩ 4m2v4 sin4 θ NR e e 2 8.4.1 Q2 and ν We define a two new quantities: Q2 and ν, which are useful in describing scattering. 2 2 2 p2 q Q q =(p1 p3) > 0 ν · (8.7) ≡− − ≡ Mp Note that ν>0 so Q2 > 0 and the mass squared of the virtual photon is negative, q2 < 0! 52 8.4.2 Higher Energy Scattering At higher energy, we have to account for the recoil of the proton. The differential cross section for electron-proton scattering becomes: 2 2 2 2 dσ α E3 2 κ q 2 2 θ q 2 2 θ = F F cos (F1 + κF2) sin (8.8) dΩ 4E2 sin4 θ E 1 − 4m2 2 2 − 2m2 2 lab 1 2 1 p p 1 For a point-like spin- 2 particle, F1 = 1, κ = 0, and the above equation reduces to the Mott scattering result, equation (8.5). It is common to use linear combinations of the form factors: κq2 GE = F1 + 2 F2 GM = F1 + κF2 (8.9) 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 (8.10) 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 correspond to an exponential charge distribution: ρ(r)=ρ exp( r/r )1/r2 =0.71 GeV2 r2 =0.81 fm2 (8.11) 0 − 0 0 The proton is an extended object with an rms radius of 0.81 fm. The whole of the above discussion can be repeated for electron-neutron scattering, with similar form factors for the neutron. 8.5 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) 53 Figure 8.2: Differential cross section for E1 =10 GeV and θ =6◦, as a function of the hadronic jet mass mX (labeled W in plot). The peaks represent elastic scattering, and then the excitation of baryonic resonances at higher mass. Above 2 GeV there is a continuum of non-resonant scattering. This data was taken by Friedman, Kendall and Taylor at SLAC in the 1960 and 1970’s and was accepted as the first experimental proof of quarks. Friedman, Kendall and Taylor won the Nobel prize in 1990 for this work. We introduce a third variable, known as Bjorken x: Q2 x (8.12) ≡ 2p q 2 · where again q2 is the four-momentum transferred by the photon. The invariant mass, MX , of the final state hadronic jet is: 2 2 2 2 2 MX = p4 =(q + p2) = mp +2p2q + q (8.13) The system X will be a baryon. The proton is the lightest baryon, therefore MX >mp. Since M = m q2 and ν, are two independent variables in DIS, and it is necessary to X p measure E1, E3 and θ in the Lab frame to determine the full kinematics. Futhermore Q2 =2p q + M 2 M 2 Q2 < 2p q (8.14) 2 · p − X ⇒ 2 · Implying the allowed range of x is 0 to 1. 0 <x<1 represents inelastic scattering, x = 1 represents elastic scattering. 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. 54 Figure 8.3: The structure function F2 as measured at HERA using collisions between 30 GeV electrons and 830 GeV protons. 55 e−(p1) e−(p3) q m m The parton model restores the elastic scattering relationship between q2 and ν, with m (parton mass) replacing mp: q2 ν + = 0 (8.15) 2m and the cross section for elastic electron-parton scattering is: 2 2 d σ α 1 2 2 θ 2 2 2 θ = F2(x, Q )cos + F1(x, Q ) sin (8.16) dE dΩ 4E2 sin4 θ ν 2 M 2 3 1 2 The structure functions are sums over the charged partons in the proton: 2 2xF1(x)=F2(x)= xeqq(x) (8.17) q where eq is the charge of the parton (+2/3 or 1/3 for quarks). Essentially the structure functions are just describing the parton content− of the proton! 8.7 Parton Distribution Functions We introduce the parton distribution functions, fi(x), defined as the probability that to find a parton in the proton the carries energy between x and x + dx. The structure functions can then be written: 1 F (x)= e2f (x) (8.18) 1 2 q i i 2 F2(x)= xeqfi(x) (8.19) i Where the sum is over all the the different flavour of partons in the proton, and eq = 1/3, +2/3 is the charge of the parton. The partons in the protons are: − The valance quarks, uud. • The sea quarks, which come in quark anti-quark pairs: u¯u, dd¯, s¯s, c¯c, bb.¯ • Gluons, g. • 56 The parton distribution functions must describe a proton with total fractional momen- tum x = 1: 1 dx x[u(x) + ¯u(x) + d(x)+d(¯ x) + s(x) + ¯s(x)+...] = 1 (8.20) 0 1 ¯ 1 dx[d(x)+d(x)] = 1 0 dx[u(x) + ¯u(x)] = 2 (8.21) 0 1 1 dx[s(x) + ¯s(x)] = 0 0 dx[c(x) + ¯c(x)] = 0 (8.22) 0 8.8 Structure Functions and Quarks in the Nucleon Scattering experiments can be used to measure the structure functions q(x), these are shown in figure 8.6. The structure function F2(x) (equation 8.19) for a proton, consisting of just uud valance quarks is: 4 1 F p(x)= xe2q(x)= xu(x)+ xd(x) (8.23) 2 q 9 9 q=u,d If we integrate the area the structure function F2(x) we measure the total fraction carried by the valence and sea quarks: 1 4 1 1 4 1 F p(x)dx = f + f F n(x)dx = f + f (8.24) 2 9 u 9 d 2 9 d 9 u 0 0 Where fu (fd) is the fraction of momentum carried by the up (down) quarks. Using isospin symmetry (see chapter 9) the neutron can be described by d u: ↔ 1 4 1 F n(x)dx = f + f (8.25) 2 9 d 9 u 0 Experimental measurement give: 1 1 p n F2 (x)dx =0.18 F2 (x)dx =0.12 (8.26) 0 0 Implying fu =0.36 fd =0.18 (8.27) The quarks constitute only 54% of the proton rest mass! The remainder is carried by gluons which must also be considered as partons.
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