Mesons and Glueballs September 23, 2009 By Hanna Renkema

Mesons and Glueballs

September 23, 2009

By Hanna Renkema

Overview

• 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( ),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

1,1 sSsS

Quantum numbers and symmetries



– 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

1,1 sSsS







1,1 sSsS

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







1,1 sSsS

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).

• 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

one is and the other

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

2

1 2

1

1322





– Flavor quantum numbers (u,d,s), SU(3)f


1,1 sSsS

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:

1833

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)







1,1 sSsS

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

• 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 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: 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 ++

q q

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

Documents

Mesons and Glueballs September 23, 2009 By Hanna Renkema