LowEnergy Baryons and the MIT Bag Model

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Low-Energy Baryons and the MIT Bag Model
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Introduction

• Static (additive) quark model is based on sum
  rules
• SU(n) hadronic multiplets are obtained in
  additive quark model
• Hadrons are constructed in the Fock space
• Flavor SU(3) is not a perfect symmetry because
  quarks have not the same mass
• The meson mass formula
• Color SU(3) gauge group
• A quark field operator

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Magnetic Moments

• The magnetic dipole moment of a spin-1/2 point
  particle of charge q and
• mass m is

• Quark dynamics is investigated in lepton-nucleon
  scattering
• Momentum transfer
• Relativistic description because typical
  excitation energies of hadrons lie in the range
  300-500 MeV

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MIT Bag Model

• hadrons are described as collective excitations
  of quark and gluon fields
• the choice of the lowest energy state (that is,
  vacuum)
• - "true QCD vacuum"
• - "perturbative vacuum "
• two boundary conditions
• -
• -

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• free Dirac equation
• Corresponding symmetrized energy-momentum tensor
  is
• The 4-momentum vector is constant

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Interaction of quarks

• Quarks interact by exchange of gluons
• Gluon field tensor is

• A separate quark cant appear color Coulomb
  field is analogous
• to EM. Coulomb field

Figure 2
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• Equation for colored gluon field
  is analogous to
• Maxwell equation
• where
• continuity equation
• where
• the color on the gluons inside the bag can be
  neglected
• we get quark confinement and the total color
  charge neutrality of the bag

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• Boundary conditions for gluons

-for
-for
Figure 4
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Solution for quarks in a rigid spherical bag

• Solution of a free Dirac equation for massless
  quark, S1/2 state (l0, s1/2)
• where
• and (R is the bag radius,
  n1,2,...8) are eigenfrequencies of spherical
  cavity
• From the boundary conditions for quarks
• -infinite number of solutions
  ,etc.
• Quarks in the lowest state ground states
  of hadrons

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• Masses of hadrons
• Total energy of hadron EH
• The condition PDB is equivalent to
• We can choose that, for example, mass of the
  proton determines B (n3, Mp938 MeV)
• Masses of all other hadrons can be determined
  from this value of B, but it is an
  oversimplification
• - fluctuations in the bag center of mass
• - gluon field contribution
• - energy of interaction of quarks and
  gluons
• - "zero-point" energy
• - the mass of strange quark differs from
  zero

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• Parameters can be determined from masses of ?
  (m?1.236 GeV), proton (mp0.938 GeV), ?
  (m?0.783 GeV) and O- (mO1.672 GeV).

Table 1
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Figure 5
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• The pion mass comes out too large hybrid
  chiral-bag model
• Approximate chiral symmetry
• spontaneously broken
  Goldstone bosons bag is surrounded with a
  pion cloud

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Magnetic moments

• Magnetic moment operator
• Magnetic moment of a proton
• We can calculate, with R1 fm and 2.04
• Gyromagnetic ratios

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• Table 2 Magnetic moments of some hadrons

• Table 4 Some hadron sizes as measured by the
  charge radius

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Experimental Investigation of the Baryon structure

• Elastic lepton-nucleon (e.g. e-p) scattering-low
  energies, around GeV
  
  
  
• Proton EM current
• Measurable elastic form factors
• -electric
• -magnetic
• In the MIT bag model
• the static MIT value for in the
  limit q 0

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• The axial vector current
• - in the MIT bag model in the q 0
  limit
• - HA is two orders of magnitude below the
  experimental value
• pionic degree of freedom
• Experimental curve - dipole formula for
  (Q2-q2, Breit frame)
• mean quadratic radius of the
  proton
• 2) Inelastic scattering few tens of GeV
• - but proton can decay

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• 3) Deep inelastic regime hundreds of GeV
• - scattering at point particles-partons
  -quarks -valence quarks
• 
  -sea of quarks
• 
  -gluons

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Puzzle of the proton spin

• Expected value of the fraction of the proton spin
  carried by quarks

• Experimentally, in pp collisions

• Explanation gluon and antiquark polarization
  is large

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• Figure 6 Illustration of the proton spin
  structure
• 
  
• 
  up quark
• 
  down quark
• 
  strange quark
• 
  antiquark
• 
  gluon
• 
  spin ½
• 
  spin 1