Structure of Hadrons

1
Structure of Hadrons
2
Internal structure

• The Dirac equation describes point-particles.
• The measured magnetic moment of an electron
  coincides with the prediction by Dirac (g2). The
  electron is therefore a point-particle.
• The magnetic moment of a proton is g5.6. A
  neutron has a magnetic moment of g-3.8. These
  are not point-particles.
• Scattering by a charge cloud
• Probing a charge distribution with electrons

q

• Whereby q the four-momentum carried by the photon
• Structure of the charge cloud parameterized by
  (unknown) function F(q). Deduce the structure of
  the target from F(q).
• Take as first example scattering of unpolarized
  electrons from static, spinless charge
  distribution Ze?(x), with

3
Mott scattering

• For static charge, it is found that F(q) is just
  the Fourier transform of the charge distribution
• This can be obtained by considering a static
  electromagnetic field
• The scattering amplitude becomes
• With the integral equal to
• Including summing final and averaging initial
  spins
• While the reference cross section for a
  structureless target is the Mott
  scattering(which simplifies to the Rutherford
  scattering cross section in the non-relativistic
  limit)
• The normalization of the formfactor F(q) is
• Hence experimental determination of the function
  F(q) gives insight in the charge distribution of
  the charge cloud.

4
Elastic proton scattering

• Apply these ideas to get the structure of the
  proton
• The proton has a magnetic moment
• Relativistic treatment including recoiling proton
• Start by taking the result from e-? scattering
• Neglect electron mass
• Keep the muon mass
• Calculate kinematics in laboratory frame
• replace m? by Mmp
• Remember that lowest order amplitude is given by

And the currents
5
Proton form factors

• Parameterize the (unknown) proton vertex by
  unknown form factors
• And we obtain (the Rosenbluth formula)
• All ignorance of the proton is parameterized with
  the form factors.
• If the charge in the proton is spread out
  isotropic then
• K1(0)1 Limit to get Mott scattering back.
  Proton is a point charge for large wave-lengths
• K1(?)0 Photon wave is too short and flies
  through proton undisturbed
• Hence K1(q) is a monotonically decreasing
  function of q
• Experimentally K(q) can be measured by
  determining d?/d? as function of ? and q

K1/2
K2
6
Proton electric charge

• The size of the proton can be determined via the
  formfactors
• The charge distribution of the proton has an
  exponential form as function of the radius (size)

Electric form factor
fit to the data points
For an exponential charge distribution it follows
The scale m2, with a value of 0.71 GeV, is an
empiric quantity that characterizes the
distribution of electric charge in the proton
7
Inelastic scattering

• Inelastic scattering ep?eX
• When increasing value of q, at some point the
  proton breaks up in a number of particles X
• A second degree of freedom comes in to play
  elasticity(or the mass of the system X)
• Description in terms of a current J?u??u is no
  longer adequate!
• The final state is not a single fermion described
  by a spinor u!
• Describing the cross section analogue to
  e?-scattering
• With the differential cross section equal to
• In practice we sum over the hadronic final state,
  and observe the scattered electron only

8
Inelastic e-p scattering

• With the integral over the final state
• Hence we obtain for the cross section
• Note that E is an independentvariable for
  inelastic scattering thehadron final state can
  soak up a lot ofenergy
• W is a second rank tensor in Lorentz indices. The
  most general parameterization has 5 terms
• Parity conservation conserved current at the
  hadron vertexq?W??q?W??0 leaves only two

Inelastic scattering
Elastic scattering
9
Inelastic e-p scattering

• The structure functions W1 and W2 depend on two
  independent variables
• One can use E and ? of the final electron
• Or Q2-q2 and x, with
• Note that the elastic form factors K depend on
  one dimensional variable, e.g. q2.
• You can get back Rosenbluth formula by the
  identification
• I.e. scattering with x1.
• So far so good! But we have learned not much. We
  only have parameterized our ignorance of the
  electron-proton inelastic scattering by two
  functions, W1 and W2.
• What is the physics interpretation of these
  functions?

10
Deep inelastic scattering

• In the sixties Bjorken predicted that the q2
  dependence of the structure functions fades away
  at large values of Q2q2.
• Sign that there are structureless particles
  inside a complex system like the proton and that
  the scattering of a proton becomes scattering
  between the photon and a free Dirac particle
• Formfactors K(q2) are dimensionless
• There is an intrinsics scale m0.71 GeV that
  determines the size of the proton. The form
  factor becomes dependent on the ratio Q2/m2.
• Structure functions F(x,Q2) are also
  dimensionless (with proton break-up)
• If the scattering takes place with
  point-particles inside the proton, there is no
  scale for these interactions. Hence the
  structure function cannot depend on Q2
• Dramatic prediction for scattering of quarks
  gluons inside the proton The structure
  functions are independent of Q2 (scaling of
  the structure functions)

Deep-Inelastic-Scattering
11
Partons and Bjorken scaling

• The scale-indepence of the structure functions
  has spectacularly been confirmed at Stanford e-p
  collisions (1972)
• Scattering takes placeon point-like
  objects(partons) inside the proton
• Quarks and gluons becomereality. They no longer
  areonly mathematical tricksto order the
  particle zoo
• With this picture in mind, how can we interpret
  the structure functions F1 and F2?
• Various types of point particles make up the
  proton, each with charge ei.
• Introduce the parton momentum distribution
  fi(x)

X momentum fraction of the proton carried
by the struck quark
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Parton model
xP
k

• The parton model can be summarized as
• All hadrons consists of partons. In QCD the
  partons are identified with the quarks and gluons
• The scattering between the electron and proton
  proceeds via the electron-parton scattering
• The parton is point-like, and hence has no form
  factor
• The partons stay on-shell during the elastic
  interaction
• Exchange between the struck quark and the
  spectator partons can be neglected
• The scattering is incoherent between various
  partons in the proton

Before hard interaction. The proton is Lorentz
contracted.
Hard interaction between electron andone parton.
The others are spectators.
k
After the hard collision the hadronization takes
place
13
Partons and Bjorken scaling

• The kinematics can be summarized as
• The parton distribution function fi(x) gives the
  probability that quark i carries a momentum
  fraction of the proton x
• The number of partons i in the interval (x,xdx)
  is given by fi(x)dx
• Kinematics can be described by a number of
  variables

1/Q wavelength of the exchanged photon
Fraction of the proton momentum carried by the
struck quark
In the rest-system of the proton fraction of
energy transfer of the electron
Center-of-mass energy
Invariant mass of the hadronic system
14
Cross sections for partons

• Back in business We can treat the DIS
  effectively as electron-quark scattering
• Write cross section in terms of Q2 and x (and
  yQ2/sx)
• First start with the Mott cross section on a
  spinless target
• Since quark has spin ½, the cross section becomes
  (no derivation here!)
• If we denote the number of up-quarks in the
  proton in interval (x,xdx) by u(x)
• Sum over all quarks in the proton to get

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Parton model, Callan Gross

• Compare this result with the cross section
  obtained via the tensor W??
• Also written as function of x and Q2
• Hence for the parton model with partons having
  s½ this predicts (Callan Gross relation)
• With F1 and F2 independent of Q2 (scaling)
• And the quark distribution functions

Callan Gross relation, 2xF1F2,has been verified
experimentallyin 1979. It shows the quarks
havespin s½
16
Parton distributions

• The parton distributions are normalized as
• The proton and neutron structure function can be
  written as
• Parton model predict the behavior of neutron
  scattering by interchange of u(x) with d(x)
• Since the proton and neutron are members of the
  same isospin doublet
• We describe the parton distribution as a sum of
  valence and sea quarks

17
Parton distribution functions

• The quantum numbers of the proton must correspond
  to the uud combination of valence quarks
• This gives rise to a sum rule, theGottfried
  sum rule
• Where are the gluons?? Indeed the integral over
  all parton distribution functions is not unity!
• The difference being the momentum carried by
  gluons (44)

18
Parton distribution functions

• So we know that the proton consists of partons.
  What else?
• Well, we know the quarks are held together by
  gluons, g(x,Q2)
• Gluons carry part of the proton momentum
• Gluons split themselves in quark-anti-quark pairs
• The proton is a very dynamic system
• Described by QCD

19
Scaling violations

• We argued that the structure functions F1,2 are
  independent of the scale Q2
• Dramatic experimental confirmation at SLAC ?
  parton model
• However, the dynamics of QCD start to play a role
• The parton distribution function change slowly
  when increasing the resolution, or Q2 value, as
  the result of gluon radiation.
• Resolution virtual photon wave ?1/?Q2
• A number of reactions happen
• A parton can split up into two (g?gg, g?qq, q?qg)
• The resulting partons each have smaller value for
  x than the parent partons
• And hence
• At large values of x the parton distribution gets
  smaller when increasing Q2
• At small values of x the parton distribution get
  larger when increasing Q2

Scaling violations as the result of gluon
bremsstrahlung
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HERA
HERA is the worlds first and only elektron-proton
collider
Situated in HAMBURG. Has been an accelerator lab
since 1965. Several accelerators DESY,DORIS,
PETRA, HERA Active
e-p collisions with cm energy of 320 GeV
Circumference 6335 m
Protons (H- atoms) are accelerated in a LINAC to
500 MeV, stripped of electrons, injected into
DESY and accelerated to 7 GeV, then into PETRA
accelerated up to 40 GeV and finally in HERA up
to 920 GeV. Electrons follow similar route but
only to 27.5 GeV
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ZEUS
One of the two experiments at HERAto register
e-p collisions (other is H1)
e 27.5 GeV
Proton 920 GeV
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Structure functions again

• The structure of the proton depends strongly on
  how you look at it Predicted by perturbative QCD

• F2 is the structure function of the proton in
  the parton model it is given by
• q(x,Q2) are the quark or parton densities
• The number of partons we see depend on the scale
  Q2

Low Q2
Q2
x
High Q2
23
Structure evolution
The QCD evolution in pictures
Gluondistribution
F2 structure function
Parton Splitting
(the negative contributions not shown)
24
ZEUS data F2(x,Q2) as function of x
At small values of xthe sea quarks
dominate. At large values of Q2this
dominationis complete The QCD
predictions(line) only availableabove Q24
GeV2where perturbation theory becomes valid
25
ZEUS data F2(x,Q2) as function of Q2
At small values of xscaling violations are most
strong. At large values of xapproximate
scalingis valid.
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DGLAP-evolution

• Evolution in Q2 described by perturbative QCD
• Dokshitzer-Gribov-Lipatov-Altarelli-Parisi
  integro-differential equations
• With the splitting functions
• For a variation in unit logQ2,
  is the probability of finding a parton i inside
  parton j with a fraction y of the parent momentum

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Instead of a static proton built from three
quarks.
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e versus e- proton DIS

• Small differences seen in the cross section.
• Described by a new structure function, F3

Due to Z exchange
Only visible at large Q2
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Charged current DIS

• And even more clearly when we look at charged
  current, i.e. when the electron becomes a
  neutrino by exchanging a W-boson.