Date post: | 01-Jan-2016 |
Category: |
Documents |
Upload: | britton-reeves |
View: | 215 times |
Download: | 0 times |
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
• 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,c,t,b), SU(3)f
– Color quantum numbers (r,b,g), SU(3)c
1,1 sSsS
Quantum numbers and symmetries
• 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,c,t,b), SU(3)f
– Color quantum numbers (r,b,g), SU(3)c
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
Quantum numbers and symmetries
• 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,c,t,b), SU(3)f
– Color quantum numbers (r,b,g), SU(3)c
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
Quantum numbers and symmetries
• 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
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)
Quantum numbers and symmetries
• 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,c,t,b), SU(3)f
– Color quantum numbers (r,b,g), SU(3)c
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
• 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