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Mesons and Glueballs
September 23, 2009
By Hanna Renkema
Overview
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Conventional mesons
Quantum numbers and symmetries
Quark model classification
Glueballs
Glueball spectrum
Glueball candidates
Decay of glueballs
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(S s   1, S s  1),baryon number B,
charge Q, hypercharge Y=S+B
– JPC
– Isospin, SU(2) symmetry
– Flavor quantum numbers (u,d,s), SU(3)f
– Color quantum numbers (r,b,g), SU(3)c

Quantum numbers and symmetries
• Mesons (like all hadrons) are identified by their quantum
numbers.
– Strangeness S( S s   1, S s   1 ),baryon number B,
charge Q, hypercharge Y=S+B
– 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
Quantum numbers and symmetries
• Mesons (like all hadrons) are identified by their quantum
numbers.
– Strangeness S( S s   1, S s   1 ),baryon number B,
charge Q, hypercharge Y=S+B
– 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
JPC
• J: total angular momentum, it is given by:
|L-S| ≤ J ≤ L+S, 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.
PΨ(x)= PΨ(-x)
PP Ψ(x)= PPΨ(-x)= P2 Ψ(x)
so P=±1
Quarks have P=+1, 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)L+1
JPC
• C: charge parity is the behavior of a state under charge conjugation.
Charge conjugation changes a particle into it’s 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)L+S
Quantum numbers and symmetries
• Mesons (like all hadrons) are identified by their quantum
numbers.
– Strangeness S(S s   1, S s   1 ),baryon number B,
charge Q, hypercharge Y=S+B
– 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
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
SU(2) symmetry
• Four configurations are expected from SU(2). 2  2  3 1
• A meson in SU(2) will have I=1, so Iz=+1,0,-1. Three pions were
found: π+, π0,π• If we take two particles with isospin up or down:
1:↑↓ 2:↑↓ they can combine as follows
↑↑ with Iz=+1, ↓↓ with Iz=-1
and two possible linear combinations of ↑↓, ↓↑ with both Iz=0
1
  
one is 12   
and the other
2
There are 2 states with Iz=0, one is π0 the other is η
• SU(2) for u and d quarks, can be extended to SU(3)f for u,d and s
quarks
Quantum numbers and symmetries
• Mesons (like all hadrons) are identified by their quantum
numbers.
– Strangeness S(S s   1, S s   1 ),baryon number B,
charge Q, hypercharge Y=S+B
– JPC
– Isospin, SU(2) symmetry
– Flavor quantum numbers (u,d,s), SU(3)f
– Color quantum numbers (r,b,g), SU(3)c
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
3  3  8 1
Quantum numbers of u,d and s:
SU(3)f symmetry
• Two triplets in SU(3)
combine into octets and
singlets
• In SU(2) two states for
Iz=0 were obtained. In a
similar manner we can
obtain three Iz=0 states in
SU(3)
Quantum numbers and symmetries
• Mesons (like all hadrons) are identified by their quantum
numbers.
– Strangeness S(S s   1, S s   1 ),baryon number B,
charge Q, hypercharge Y=S+B
– 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
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
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
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
Glueball spectrum
• What are the possible glueball
states?
• Use: J=(|L-S| ≤ J ≤ L+S,
P=(-1)L and C=+1 for two gluon
states, C=-1 for three gluon
states
• e.g. take L=0, S=0:
J=0 P=+1 C= +1
give states: 0++
• Masses obtained form LQCD
Mass spectrum of glueballs
in SU(3) theory
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
The lightest glueball
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0++ scalar particle is considered to be the lightest state
Mass: 1 ~2 GeV
Candidates: I=0 f0(1370), f0(1500), f0(1710)
Glueball must be identified by its decay products
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
J=0, prefer to decay into s-quarks.
– f0(1710) decay more frequent into kaons (s
composition) than into pions (u, d compositions)
Chiral suppression
Chiral suppression
• If 0++ decays into a quark and an antiquark, we go from a state with
J=L=S=0 to a state which must also have J=L=S=0
• Chiral symmetry requires q and q 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: J=L=0, but S=1
• 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 ++
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