The Development of Particle Physics. Dr. Vitaly Kudryavtsev E45, Tel.:

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1 The Development of Particle Physics Dr. Vitaly Kudryavtsev E45, Tel.:

2 The structure of the nucleon Electron - nucleon elastic scattering Rutherford, Mott cross-sections Nucleon form-factors Rosenbluth cross-section Size of the nucleon Electron - nucleon inelastic scattering Structure functions Key ideas: scaling and partons Partons and quarks Neutrino - nucleon inelastic scattering Weak structure functions Neutrino scattering cross-sections

3 Elastic and inelastic scattering N N γ γ N X Electron - nucleon scattering can be described as an exchange of a virtual photon. Three cases can be considered: 1. Momentum carried by photon is low, its wavelength is long compared with the size of the nucleon. It will not be able to resolve any nucleon structure but will see the nucleon as a point.. Photon momentum is higher, its wavelength is comparable to the nucleon size. The photon begins to resolve the finite size of the nucleon. 3. With very high momentum transfer the photon wavelength is much shorter than the nucleon size and the photon can resolve the internal structure of the nucleon.

4 e (k,e) N (P,M) Elastic electron - nucleon scattering e(k',e') θ γ (q) W k and P are the four-momenta The interaction is assumed to be due to the exchange of a single virtual photon. In the case of elastic scattering the final state W is the scattered nucleon. Each coupling of the photon (vertex) gives a factor of e in the scattering amplitude. In total two vertices: e / (4πε 0 ħc) = α 1 / 137. Assumptions to derive cross-sections of elastic scattering: Relativistic electrons (E>>m e ) With electron spin taken into account Spinless electrons Scattering on a static point charge dσ dω = α dσ 4E sin 4 θ dω = α cos θ 4E sin 4 θ Rutherford formula Mott formula

5 Elastic cross-section Scattering on a static point charge with infinite mass: E'= E Scattering on a target with finite mass: E'= q = 4 E sin θ q = 4 EE ' sin θ E 1+ E M sin θ Elastic scattering of an electron by a point-like Dirac particle of mass M: dσ dω = α cos θ 4E sin 4 θ ' E 1 q M tan * θ ( ) +, E ' Reduces to the Mott cross-section with the increase of target mass

6 Form factors and Rosenbluth formula In the case of elastic scattering from a target with a particular charge distribution, ρ(r), the scattering amplitude is modified by a form factor: F(q ) = d 3 re i! q! r ρ(r) The Mott cross-section should be multiplied by F(q ). For a unit charge target: d 3 rρ(r) =1 and the form factor is equal to unity for zero momentum transfer.

7 Form factors and Rosenbluth formula The differential cross-section for elastic electron-proton scattering can be calculated including form factors using current algebra. The result is known as Rosenbluth formula: dσ dω = α cos θ 4E sin 4 θ E' -' E F 1 + κ Q /) 4M F.( *, + Q F + M 1 + κf ( ) tan θ where Q =-q (Q is positive), κ is the anomalous magnetic moment of nucleon in units of nuclear magneton, e!/(mc). For proton κ = 1.79, for neutron κ = Dirac magnetic moment µ = e!j / (Mc). F 1 (q ) is the charge form factor; F (q ) is the magnetic moment form factor. For a proton, F 1 (0) = F (0) = 1. For point-like Dirac particle (like electron) F 1 (q ) = 1, κ F (q ) = 0. For a neutron, F 1 (0) =

8 Measurement of the elastic electron-proton scattering McAllister and Hofstadter MeV and 36 MeV electron beam from linear accelerator at Stanford Hydrogen and helium (gas) targets Magnetic spectrometer Measurements of scattering angle and energy

9 Elastic profiles E'= E 1+ E M sin θ Typical elastic profiles obtained in the scattering experiment with hydrogen gas The elastic profiles were used to obtain cross-section of scattering - the cross-section at a particular scattering angle is proportional to the number of observed events at this angle (area under the curve)

10 Energy estimates Energy of scattered electrons as a function of scattering angle in the laboratory frame Energy was measured as peak value of the elastic profile (from spectrometer data) Theoretical curve was obtained using relativistic kinematics

11 Cross-section of elastic scattering a - Mott curve for spinless point-like proton b - Rosenbluth curve for a point-like proton with the Dirac magnetic moment (without anomalous magnetic moment): F 1 (q ) = 1, κf (q ) = 0 c - Rosenbluth curve with contribution from anomalous magnetic moment for point-like proton F 1 (q ) = 1, κf (q ) = κ The deviation of experimental data from curve (c) were interpreted as an effect from proton form-factors - finite size proton

12 The size of the proton The experiment was not sensitive enough to the q dependence of the form factors. All that could be determined was a mean square radius of the nucleon. The best fit was achieved for <r > 1/ = (0.70 ±0.4) cm. Another experiment with 36 MeV electrons showed <r > 1/ = (0.78±0.0) cm. Combined result - <r > 1/ = (0.74±0.4) cm. F(q ) 1 (1+ q / M V ) M V 0.9 GeV

13 Inelastic scattering When the scattering is not elastic (new particles are produced) the energy and direction of the scattered electron are independent variables, unlike the elastic scattering situation. W is the mass squared of the produced hadronic system From the measurement of the direction θ (solid angle element dω) and the energy E ' of the scattered electron, the four momentum transfer Q =-q can be calculated. The differential cross-section is determined as a function of E ' and Q. e (k,e) N (P,M) k ' = k q e(k',e') θ γ (q) ν = E E ' q = 4 EE ' sin θ W = (P + q) = P + P q + q = M + Mν Q W

14 Electron - proton inelastic scattering Bloom et al. (SLAC-MIT group) in 1969 performed an experiment with high-energy electron beams (7-18 GeV). Scattering of electrons from a hydrogen target at 6 0 and Only electrons are detected in the final state - inclusive approach. The data showed peaks when the mass W of the produced hadronic system corresponded to the mass of the known resonances.

15 Inelastic scattering cross-section Similar to the electron-proton elastic scattering, the differential cross-section of electron-proton inelastic scattering can be written in a general form: dσ = α cos θ dωde ' 4 E sin 4 θ [ W (ν,q ) + W 1 (ν,q )tan θ ] The cross-section is double differential because θ and E ' are independent variables. The expression contains Mott cross-section as a factor and is analogous to the Rosenbluth formula. It isolates the unknown shape of the nucleon target in two structure functions W 1 and W, which are the functions of two independent variables ν and q. The structure functions correspond to the two possible polarisation states of the virtual photon: longitudinal and transverse. Longitudinal polarisation exists only because photon is virtual and has a mass. For elastic scattering, (P+q) =M and the two variables ν and Q are related by Q =Mν.

16 Scaling ν = E E ' q = 4 EE ' sin θ To determine W 1 and W separately it is necessary to measure the differential cross-section at two values of θ and E ' that correspond to the same values of ν and Q. This is possible by varying the incident energy E. SLAC result: the ratio of σ /σ Mott depends only weakly on Q for high values of W. For small scattering angles σ /σ Mott W. Thus, the structure function W does not depend on Q.

17 Scaling Instead, at high values of W the function νw depends on the single variable ω = Mν / Q (at present the variable x=1/ω is widely used). This is the so-called "scaling" behaviour of the cross-section (structure function). It was first proposed by Bjorken in W 1, (ν,q ) W 1, (x) when ν,q.

18 Scaling Recent data on the scaling behaviour of structure functions obtained with electron-proton and muonproton scattering Scaling is better seen at large x=q / Mν. These are electromagnetic structure functions which describe electromagnetic content of the proton.

19 Parton model The parton model was first put forward by Richard Feynman. The basic idea: the nucleon is made up of point-like constituents - partons. Partons share the total momentum of the nucleon by taking up variable fractions of its momentum, x. The probability for a parton to carry momentum fraction x, f(x), does not depend on the process or nucleon energy but is intrinsic property of a nucleon. Natural question: are partons the quarks? Yes, but not only the three valence quarks that describe the composition of the nucleon. There are also sea quarks - virtual quark-antiquark pairs emerging briefly from vacuum borrowing energy due to Heisenberg's uncertainty principle. There are also gluons - quanta of the strong force of quark interactions. Quarks interact by exchange of gluons. Gluons glue the quarks together.

20 Parton model The distribution functions for various quarks are given as u(x), d(x), s(x) etc. Momenta of quarks (and gluons) are added to give proton momentum, so there is a constraint: x[ u(x) + u(x) + d(x) + d(x) + s(x) + s(x) +...] dx =1 The nucleon quantum number should come out correctly. For proton: dx[ u(x) u(x) ] = dx[ d(x) d(x) ] =1 dx[ s(x) s(x) ] = 0 This replaces the statement that proton is composed of two u-quarks and one d-quark.

21 Parton model and scaling Feynman's parton model gives explanation of the Bjorken scaling. If the quarks or partons are treated as real particles, then p = p - E = -m, p f = (p i + q) = (xp + q) = -m 0, x P + q + xp q = 0 If x P = x M << q, then x = -q /P q = Q /Mν, where invariant scalar product P q is evaluated in the laboratory system, in which the energy transfer is ν and the nucleon is at rest. Thus, x = 1/ω - the variable used in the experiment at SLAC to find scaling behaviour of the cross-section. Cross-section of inelastic scattering can be rewritten with other variables s=me, x = Q /Mν and y = ν /E, and dimensionless functions F 1 and F (must not be confused with elastic form factors). Traditionally F 1 = MW 1 F = νw and the cross-section is written as:

22 Structure functions in parton model dσ dxdy = 4πα s where & 1 Q 4 ' ( [ 1+ ( 1 y ) ]xf 1 + ( 1 y) ( F xf 1 ) M E xyf F 1 = MW 1 = 1 " 4 9 u(x) d(x) u(x) + 1 $ d(x) +... # 9 % # F = νw = x 4 9 u(x) d(x) u(x) + 1 % d(x) +... $ 9 & The factors 4/9 and 1/9 arise from the squares of the quark charges. The absence of Q dependence in F 1 and F is the manifestation of scaling. The functions u(x), d(x) etc. show the probability for quarks (partons) to carry a fraction x of nucleon momentum. F = xf 1 - Callan-Gross relation: partons are point-like Dirac particles. ) * +

23 Inelastic scattering in parton model Scaling and parton model: incoherent scattering of electron off the individual parton. Other partons do not feel the scattering. This is the consequence of high q. Consequence of scaling and parton model: partons are free from mutual interactions over the space-time distances of the electron-parton interaction. But: no partons or quarks were observed in the final state, only hadrons. Somehow scattered and unscattered partons should recombine to form hadrons. Collision occurs in two independent stages. First stage: one parton is scattered. The collision time is determined by the ratio: τ 1 λ / c! / ν. If the wavelength of the virtual photon is small, then the collision time is smaller than the time of interparton interaction: τ d / c, where d is the distance between partons. Second stage: partons (quarks) recombine with virtual quark-antiquark pairs to form hadrons in final state. The time for this is τ! / M >> τ 1 if ν >> M.

24 Inelastic neutrino-nucleon scattering Parton model is used to make predictions for deep inelastic neutrino-nucleon scattering. Neutrino beams from pion and kaon decays, dominated by muon neutrinos are used to study this process. ν µ + nucleon µ + X ν µ + nucleon µ + + X Since parity is not conserved in weak interactions, there are more structure functions for weak processes, like neutrino scattering, than for electromagnetic processes, like electron scattering. Again the variables x = Q /Mν and y = ν /E can be used.

25 Weak structure functions General form for the neutrino-nucleon deep inelastic scattering crosssection, neglecting lepton masses, Cabibbo angle and corrections of the order of M/E: dσ ν,ν dxdy = G F ME π, & ( 1 y)f νn + y xf νn 1 y y ) / νn. ( + xf ' * 0 The functions F 1, F and F 3 are the functions of Q and ν. In the scaling limit they are the functions of x only.

26 Quark distribution functions Cross-section can be expressed in terms of quark distributions. For neutrino-proton scattering the struck quark must gain charge (lepton loses charge) and only d and anti-u quarks contribute, while for antineutrinoproton scattering anti-d and u quarks contribute : dσ ν dxdy = G F ME π x[ d(x) + ( 1 y) u(x) ] dσ ν dxdy = G F ME π x[ d(x) + ( 1 y) u(x) ] For the scattering on isoscalar target (equal mixture of protons and neutrons - u and d quarks) the cross-sections are (replacing d quark distribution in neutrons with u quark distribution in protons): dσ ν dxdy = G F ME π x[ q(x) + ( 1 y) q(x) ] dσ ν dxdy = G F ME π x[ q(x) + ( 1 y) q(x) ] where q(x) = u(x) + d(x) q(x) = u(x) + d(x)

27 Structure functions as parton distributions In terms of parton distributions the structure functions can be expressed as: [ ] F 3 (x) = [ q(x) q(x) ] F (x) = xf 1 (x) F (x) = x q(x) + q(x) The crucial point of the parton model: quark distributions should not depend on the interaction process but should be intrinsic property of the nucleon. For an isoscalar target the electromagnetic structure function is: F = 5 18 x(u + d + u + d) + 1 x(s + s) 9 which is 5/18 times the corresponding structure function for weak interactions, if we neglect strange quark contribution.

28 Predictions for total cross-sections The integrated cross-sections are expressed in terms of Q xq(x)dx Q xq(x)dx the momentum fractions carried by the quarks and the antiquarks. ( ) σ ν = MEG F ( Q+ 1 3 Q) σ ν = MEG F Q+ 1 3 Q π π Since more momentum is carried by quarks than by antiquarks, we expect σ ν σ ν 1 3 σ ν E =1.56 ( Q+ 1 Q 3 )10 38 cm /GeV

29 Neutrino-nucleon cross-sections CERN Proton Synchrotron (PS) Gargamelle - heavy liquid (freon) bubble chamber. Muons - charged particle signal without hadrons. Energy of neutrino = total liberated energy (track curvature, range measurements, energy deposition). Neutrino (antineutrino) - nucleon cross-sections have been measured as a function of neutrino energy σ ν E = (0.74 ± 0.0) cm /GeV σ ν E = (0.8± 0.01) cm /GeV

30 Quasi-elastic cross-sections of neutrinos Quasi-elastic processes: ν µ + n µ + p ν µ + p µ + + n

31 Ratio of neutrino to antineutrino crosssections Ratio of antineutrinonucleon to neutrinonucleon cross-sections has been determined as 0.37±0.0 in good agreement with parton model predictions

32 Rise of mean q with energy Mean q was found to be linear function in neutrino (antineutrino) energy.

33 Neutrino cross-section (recent data) Compilation of recent data on neutrino-nucleon and antineutrino-nucleon total cross-sections (from the Review of Particle Physics)

34 Structure functions Compilation of data from neutrino and muon scattering experiments. Note the factor 18/5 for electromagnetic structure functions to allow the comparison with the neutrino data (5/18 - average charge squared of the light quarks). The third distribution shown is: q ν (x) = x[ u(x) + d(x) + s(x) ] from the Review of Particle Properties, Phys. Lett., 170B (1986) 79.

35 Scaling behaviour of the structure functions Compilation of the data on structure functions in deep inelastic neutrino scattering (from F. Dydak. Proc. of the 1983 Intern. Lepton/Photon Symposium, 1983, p. 634)

36 Summary and conclusions The elastic electron-proton scattering showed that nucleon is not a point-like particle and allowed determination of its size. The inelastic electron-proton scattering helped to measure electromagnetic structure functions of nucleons. The scaling found in inelastic scattering experiments was successfully explained within the framework of parton model (quark distributions in nucleons). The inelastic neutrino-nucleon scattering provided another test for the parton model and the measurements of neutrino interaction cross-section. The electromagnetic and weak structure functions, and the parton distribution functions were measured in deep inelastic scattering experiments with electrons, muons and neutrinos.

37 References 1. R. W. McAllister and R. Hofstadter. "Elastic scattering of 188-MeV electrons from the proton and alpha particles". Phys. Rev., 10 (1956) E. D. Bloom et al. "High energy inelastic e-p scattering at 6 o and 10 o." Phys. Rev. Lett., 3 (1969) M. Breidenbach et al. "Observed behavior of highly inelastic electron proton scattering". Phys. Rev. Lett., 3 (1969) T. Eichten et al. "Measurement of the neutrino-nucleon and antineutrino-nucleon total cross-sections." Phys. Lett., 46B (1973) 74.