The Development of Particle Physics - vitaly · Dr. Vitaly Kudryavtsev The Development of Particle Physics Lectures 7-8 The structure of the nucleon • Electron - nucleon elastic

The Development of Particle Physics

Dr. Vitaly Kudryavtsev E45, Tel.: 0114 2224531

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Dr. Vitaly Kudryavtsev The Development of Particle Physics Lectures 7-8

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


Elastic and inelastic scattering

γ

N N

N X

γ

Electron - nucleon scattering can be described as anexchange of a virtual photon. Three cases can be considered:1. Momentum carried by photon is low, its wavelength islong compared with the size of the nucleon. It will not be able to resolve any nucleon structure but will see thenucleon as a point. 2. Photon momentum is higher, its wavelength is comparable to the nucleon size. The photon begins to resolvethe finite size of the nucleon.3. With very high momentum transfer the photon wavelengthis much shorter than the nucleon size and the photon canresolve the internal structure of the nucleon.


Elastic electron - nucleon scattering

The interaction is assumed to be due to theexchange 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 factorof e in the scattering amplitude.In total two vertices: e2 / (4πε0ħc) = α ≈ 1 / 137.

•  Relativistic electrons (E>>me)•  Spinless electrons•  Scattering on a static point charge

€

dσdΩ

=α 2

4E 2 sin4 θ2Rutherford formula

e (k,E)

N (P,M)

e(k',E')

γ (q)

W

θ

Assumptions to derive cross-sections of elastic scattering:

•  With electron spin taken into account

€

dσdΩ

=α 2 cos2 θ24E 2 sin4 θ2

Mott formula

k and P are the four-momenta


Elastic cross-section

Scattering on a static point charge withinfinite mass:

Scattering on a target withfinite mass:

€

E '= E

€

q2 = −4E 2 sin2 θ2

€

q2 = −4EE ' sin2 θ2

€

E '= E1+ 2E

M sin2 θ2

Elastic scattering of an electron by a point-like Dirac particle of mass M:

€

dσdΩ


E '

E1−

q2

2M 2 tan2 θ2

'

( ) *

+ , Reduces to the Mott cross-sectionwith the increase of target mass


Form factors and Rosenbluth formula

In the case of elastic scattering from a target with a particular chargedistribution, ρ(r), the scattering amplitude is modified by a form factor:

€

F(q2) = d3rei! q •! r ρ(r)∫

The Mott cross-section should be multiplied by |F(q2)|2.For a unit charge target:

€

d3rρ(r)∫ =1and the form factor is equal to unity for zero momentum transfer.


Form factors and Rosenbluth formula

The differential cross-section for elastic electron-proton scattering can becalculated including form factors using current algebra. The result is knownas Rosenbluth formula:

€

dσdΩ


E 'E

F12 +

κ 2Q2

4M 2 F22

'

( )

*

+ , +

Q2

2M 2 F1 +κF2( )2 tan2 θ2-

. /

0

1 2

where Q2=-q2 (Q2 is positive), κ is the anomalous magnetic moment of nucleonin units of nuclear magneton, . For proton κ = 1.79, for neutronκ = -1.91. Dirac magnetic moment .F1 (q2) is the charge form factor; F2 (q2) is the magnetic moment form factor.For a proton, F1 (0) = F2 (0) = 1.For point-like Dirac particle (like electron) F1 (q2) = 1, κ F2 (q2) = 0. For a neutron, F1 (0) = 0.

€

e! /(2Mc)µ = e!J / (Mc)


Measurement of the elastic electron-proton scattering

•  McAllister and Hofstadter 1956 - 188 MeV and 236 MeV electron beam from linear accelerator at Stanford

•  Hydrogen and helium (gas) targets

•  Magnetic spectrometer •  Measurements of

scattering angle and energy


Elastic profiles

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 numberof observed events at this angle (area under the curve)

€

E '= E1+ 2E

M sin2 θ2


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


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): F1 (q2) = 1, κF2 (q2) = 0

•  c - Rosenbluth curve with contribution from anomalous magnetic moment for point-like proton F1 (q2) = 1, κF2 (q2) = κ

•  The deviation of experimental data from curve (c) were interpreted as an effect from proton form-factors - finite size proton


The size of the proton

•  The experiment was not sensitive enough to the q2 dependence of the form factors.

•  All that could be determined was a mean square radius of the nucleon.

•  The best fit was achieved for <r2>1/2 = (0.70 ±0.24)×10-13 cm.

•  Another experiment with 236 MeV electrons showed <r2>1/2 = (0.78±0.20)×10-13 cm.

•  Combined result - <r2>1/2 = (0.74±0.24)×10-13 cm.

€

F(q2 )∝ 1(1+ q2 /MV

2 )2

MV2 ≈ 0.9 GeV


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.

•  W2 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 Q2=-q2 can be calculated.

•  The differential cross-section is determined as a function of E' and Q2.

€

q2 = −4EE ' sin2 θ2

e (k,E)

N (P,M)

e(k',E')

γ (q)W

θ

€

k ' = k − q

€

W 2 = (P + q)2 =

P 2 + 2P⋅ q + q2 =

M 2 + 2Mν −Q2

€

ν = E − E '


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 60 and 100.

•  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.


Inelastic scattering cross-section

•  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 W1 and W2, which are the functions of two independent variables ν and q2. 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)2=M2 and the two variables ν and Q2 are related by Q2=2Mν .

Similar to the electron-proton elastic scattering, the differential cross-sectionof electron-proton inelastic scattering can be written in a general form:

€

dσdΩdE ' =

α 2 cos2 θ24E 2 sin4 θ2

W2(ν,q2) + 2W1(ν,q

2)tan2 θ2[ ]


Scaling

•  To determine W1 and W2 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 Q2.

•  This is possible by varying the incident energy E.

•  SLAC result: the ratio of σ /σMott depends only weakly on Q2 for high values of W.

•  For small scattering angles σ /σMott ≈ W2 . Thus, the structure function W2 does not depend on Q2.

€

ν = E − E '

€

q2 = −4EE ' sin2 θ2


Scaling

•  Instead, at high values of W the function νW2 depends on the single variable ω = 2Mν / Q2 (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 1967.

•  W1,2(ν,q2) → W1,2(x) when ν,q2 → ∞.


Scaling

•  Recent data on the scaling behaviour of structure functions obtained with electron-proton and muon-proton scattering

•  Scaling is better seen at large x=Q2 / 2Mν .

•  These are electromagnetic structure functions which describe electromagnetic content of the proton.


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.


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)[ ]∫ = 2

€

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.


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 p2 = p2 - E2 = -m2, pf

2 = (pi + q)2 = (xP + q)2 = -m2 ≈ 0 ,

x2P2 + q2 + 2xP•q = 0 If x2P2 = x2M2 << q2, then x = -q2/2P•q = Q2/2Mν , 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=2ME, x = Q2/2Mν and y = ν /E, and dimensionless functions F1 and F2 (must not be confused with elastic form factors). Traditionally F1 = MW1 F2 = νW2 and the cross-section is written as:


Structure functions in parton model

€

dσdxdy

=4πα2sQ4

121+ 1− y( )2[ ]2xF1 + 1− y( ) F2 −2xF1( ) − M

2ExyF2

& ' (

) * +

where

€

F1 = MW1 =1249u(x) +

19d(x) +

49u(x) +

19d(x) + ..."

# $ %

€

F2 = νW2 = x49u(x) +

19d(x) +

49u(x) +

19d(x) + ...#

$ % &

The factors 4/9 and 1/9 arise from the squares of the quark charges. The absence of Q2 dependence in F1 and F2 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. F2 = 2xF1 - Callan-Gross relation: partons are point-like Dirac particles.


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 q2.

•  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: τ1 ≈ 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 τ2 ≈ / M >> τ1 if ν >> M.

€

!

€

!


Inelastic neutrino-nucleon scattering

•  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 = Q2/2Mν and y = ν /E can be used.

€

νµ + nucleon → µ− + X

•  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.

€

νµ + nucleon → µ+ + X


Weak structure functions

General form for the neutrino-nucleon deep inelastic scattering cross-section, neglecting lepton masses, Cabibbo angle and corrections of the order of M/E:

€

dσν ,ν

dxdy=

GF2 MEπ

1− y( )F2νN + y 2xF1νN ∓ y − y 2

2&

' (

)

* + xF3

νN,

- .

/

0 1

The functions F1 , F2 and F3 are the functions of Q2 and ν. In the scaling limit they are the functions of x only.


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 antineutrino-proton scattering anti-d and u quarks contribute :

•  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):

where

€

dσν

dxdy=2GF

2MEπ

x d(x) + 1− y( )2u(x)[ ]

€

dσν

dxdy=2GF

2MEπ

x d(x) + 1− y( )2u(x)[ ]

€

dσν

dxdy=2GF

2MEπ

x q(x) + 1− y( )2q(x)[ ]

€

dσν

dxdy=2GF

2MEπ

x q(x) + 1− y( )2q(x)[ ]

€

q(x) = u(x) + d(x)

€

q(x) = u(x) + d(x)


Structure functions as parton distributions

•  In terms of parton distributions the structure functions can be expressed as:

•  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:

which is 5/18 times the corresponding structure function for weak interactions, if we neglect strange quark contribution.

€

F2(x) = x q(x) + q(x)[ ]

€

F3(x) = q(x) −q(x)[ ]

€

F2(x) = 2xF1(x)

€

F2 =518

x(u + d + u + d) +19x(s + s)


Predictions for total cross-sections

•  The integrated cross-sections are expressed in terms of

the momentum fractions carried by the quarks and the antiquarks. •  Since more momentum is carried by quarks than by antiquarks, we expect

€

Q ≡ xq(x)dx∫

€

Q ≡ xq(x)dx∫

€

σν =MEGF

2

πQ+ 1

3Q( )

€

σν =MEGF

2

πQ+ 1

3Q( )

€

σν

σν

≈13

€

σν

E=1.56 Q+ 1

3Q( )10−38cm2 /GeV


Neutrino-nucleon cross-sections

•  CERN Proton Synchrotron (PS) - 1973. •  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

σν

E= (0.74±0.02)×10−38cm2 /GeV

σν

E= (0.28±0.01)×10−38cm2 /GeV


Quasi-elastic cross-sections of neutrinos

Quasi-elastic processes:

€

νµ + n→ µ− + p

€

νµ + p→ µ+ + n


Ratio of neutrino to antineutrino cross-sections

Ratio of antineutrino-nucleon to neutrino-nucleon cross-sections has been determined as 0.37±0.02 in good agreement with parton model predictions


Rise of mean q2 with energy

Mean q2 was found to be linear function in neutrino (antineutrino) energy.


Neutrino cross-section (recent data)

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


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:

€

qν(x) = x u(x) + d(x) + 2s(x)[ ]

from the Review of Particle Properties, Phys. Lett., 170B (1986) 79.


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)


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.


References

1.  R. W. McAllister and R. Hofstadter. "Elastic scattering of 188-MeV electrons from the proton and alpha particles". Phys. Rev., 102 (1956) 851.

2.  E. D. Bloom et al. "High energy inelastic e-p scattering at 6o and 10o." Phys. Rev. Lett., 23 (1969) 930.

3.  M. Breidenbach et al. "Observed behavior of highly inelastic electron proton scattering". Phys. Rev. Lett., 23 (1969) 935.

4.  T. Eichten et al. "Measurement of the neutrino-nucleon and antineutrino-nucleon total cross-sections." Phys. Lett., 46B (1973) 274.

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The Development of Particle Physics - vitaly · Dr. Vitaly Kudryavtsev The Development of Particle Physics Lectures 7-8 The structure of the nucleon • Electron - nucleon elastic