Mesons and Glueballs

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Mesons and Glueballs

• September 23, 2009
• By Hanna Renkema

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Overview

• Conventional mesons
• Quantum numbers and symmetries
• Quark model classification
• Glueballs
• Glueball spectrum
• Glueball candidates
• Decay of glueballs

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Conventional mesons

• They consist of a quark and an antiquark. Mesons
  have integer spin.
• Mesons (like all hadrons) are identified by their
  quantum numbers.
• Strangeness S( ),baryon number B, charge Q,
  hypercharge YSB
• JPC
• Isospin, SU(2) symmetry
• Flavor quantum numbers (u,d,s), SU(3)f
• Color quantum numbers (r,b,g), SU(3)c

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Quantum numbers and symmetries

• Mesons (like all hadrons) are identified by their
  quantum numbers.
• Strangeness S( ),baryon number B, charge Q,
  hypercharge YSB
• JPC
• Isospin, SU(2) symmetry
• Flavor quantum numbers (u,d,s,c,t,b), SU(3)f
• Color quantum numbers (r,b,g), SU(3)c

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Quantum numbers and symmetries

• Mesons (like all hadrons) are identified by their
  quantum numbers.
• Strangeness S( ),baryon number B, charge Q,
  hypercharge YSB
• JPC
• Isospin, SU(2) symmetry
• Flavor quantum numbers (u,d,s,c,t,b), SU(3)f
• Color quantum numbers (r,b,g), SU(3)c

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JPC

• J total angular momentum, it is given by
• L-S J LS, integer steps.
• L is the orbital angular momentum and S the
  intrinsic spin.
• P parity defines how a state behaves under
  spatial inversion.
• P is the parity operator, P is the eigenvalue
  of the state.
• P?(x) P?(-x)
• PP ?(x) PP?(-x) P2 ?(x) so P1
• Quarks have P1, antiquarks have P-1 this will
  give a meson with P-1. But if the meson has an
  orbital angular momentum, another minus sign is
  obtained from the Ylm of the state.
• So parity of mesons P(-1)L1

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JPC

• C charge parity is the behavior of a state under
  charge conjugation.
• Charge conjugation changes a particle into its
  antiparticle
• Only for neutral systems we can define the
  eigenvalues of the state,like we did for parity
• with
• For other systems things get more complicated
• Charge parity of mesons C(-1)LS

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Quantum numbers and symmetries

• Mesons (like all hadrons) are identified by their
  quantum numbers.
• Strangeness S( ),baryon number B, charge Q,
  hypercharge YSB
• JPC
• Isospin, SU(2) symmetry
• Flavor quantum numbers (u,d,s,c,t,b), SU(3)f
• Color quantum numbers (r,b,g), SU(3)c

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Isospin and SU(2) symmetry

• Isospin (I) indicates different states for a
  particle with the same mass and the same
  interaction strength
• The projection on the z-axis is Iz
• u and d quarks are 2 different states of a
  particle with I ½, but with different Iz. Resp.
  ½ and - ½
• c.p. electron with S ½ with up and down states
  with Sz ½ and Sz -½
• Isospin symmetry is the invariance under SU(2)
  transformations

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SU(2) symmetry

• Four configurations are expected from SU(2).
• A meson in SU(2) will have I1, so Iz1,0,-1.
  Three pions were found p, p0,p-
• If we take two particles with isospin up or
  down 1?? 2?? they can combine as follows
• ?? with Iz1, ?? with Iz-1
• and two possible linear combinations of ??, ??
  with both Iz0
• one is and the other
• There are 2 states with Iz0, one is p0 the
  other is ?
• SU(2) for u and d quarks, can be extended to
  SU(3)f for u,d and s quarks

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Quantum numbers and symmetries

• Mesons (like all hadrons) are identified by their
  quantum numbers.
• Strangeness S( ),baryon number B, charge Q,
  hypercharge YSB
• JPC
• Isospin, SU(2) symmetry
• Flavor quantum numbers (u,d,s), SU(3)f
• Color quantum numbers (r,b,g), SU(3)c

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Flavor quantum numbers and SU(3)f symmetry

• From the six existing flavors, u, d and s and
  their anti particles will be considered
• According to SU(3)f this gives nine combinations
• Quantum numbers of u,d and s

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SU(3)f symmetry

• Two triplets in SU(3) combine into octets and
  singlets
• In SU(2) two states for Iz0 were obtained. In a
  similar manner we can obtain three Iz0 states in
  SU(3)

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Quantum numbers and symmetries

• Mesons (like all hadrons) are identified by their
  quantum numbers.
• Strangeness S( ),baryon number B, charge Q,
  hypercharge YSB
• JPC
• Isospin, SU(2) symmetry
• Flavor quantum numbers (u,d,s,c,t,b), SU(3)f
• Color quantum numbers (r,b,g), SU(3)c

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Color quantum numbers and SU(3)c symmetry

• Three color charges exist red, green and blue
• These quantum numbers are grouped in the SU(3)
  color symmetry group
• Only colorless states appear, because SU(3)c is
  an exact symmetry

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Quark model classification

• f and f are mixtures of wave functions of the
  octet and singlet
• There are 3 states isoscalar states identified by
  experiment f0(1370),f0(1500) and f0(1710)
• Uncertainty about the f0 states

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Glueballs

• Glueballs are particles consisting purely of
  gluons
• QED Photons do not interact with other photons,
  because they are charge less.
• QCD Gluons interact with each other, because
  they carry color charge
• The existence of glueballs would prove QCD

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Glueball spectrum

• What are the possible glueball states?
• Use J(L-S J LS,
• P(-1)L and C1 for two gluon states, C-1 for
  three gluon states
• e.g. take L0, S0J0 P1 C 1
• give states 0
• Masses obtained form LQCD

• Mass spectrum of glueballs
• in SU(3) theory

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LQCD

• Define Hamiltonian on a lattice
• To all lattice points correspond to a wave
  function
• Lattice is varied within the boundaries given by
  the quantum numbers
• Energy can be minimized

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The lightest glueball

• 0 scalar particle is considered to be the
  lightest state
• Mass 1 2 GeV
• Candidates I0 f0(1370), f0(1500), f0(1710)
• Glueball must be identified by its decay products

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Decay of glueballs

• Interaction of gluons is thought to be
  flavor-blind. No preference for u,d or s
  interactions.
• f0(1500) decays with the same frequency to u,d
  and s states
• From chiral suppression, it follows that
  glueballs with J0, prefer to decay into
  s-quarks.
• f0(1710) decay more frequent into kaons (s
  composition) than into pions (u, d compositions)

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Chiral suppression
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Chiral suppression

• If 0 decays into a quark and an antiquark, we
  go from a state with JLS0 to a state which
  must also have JLS0
• Chiral symmetry requires and to have
  equal chirality (they are not equal to their
  mirror image)
• As a concequence the spins are in the same
  directions and they sum up. We have obtained
  state with JL0, but S1
• Chiral symmetry is broken for massive particles.
  This allows unequal chirality.
• Heavy quarks break chiral symmetry more and will
  occur more in the decay of a glueball in state 0
  

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Conclusion

• By using quantum numbers quark states can be
  identified
• More states are found by experiment than the
  states existing in the quark model
• Which state the glueball must be is unclear,
  depending on the considered theory