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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(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 • • • • 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