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 –J PC –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( ),baryon number B, charge Q, hypercharge Y=S+B –J PC –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( ),baryon number B, charge Q, hypercharge Y=S+B –J PC –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

J PC 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)= P 2 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 Y lm of the state. So parity of mesons: P=(-1) L+1

J PC 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 –J PC –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 I z u and d quarks are 2 different states of a particle with I= ½, but with different I z. Resp. ½ and - ½ c.p. electron with S= ½ with up and down states with S z = ½ and S z = -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 I z =+1,0,-1. Three pions were found: π +, π 0,π - If we take two particles with isospin up or down: 1:↑↓ 2:↑↓ they can combine as follows ↑↑ with I z =+1,↓↓ with I z =-1 and two possible linear combinations of ↑↓, ↓↑ with both I z =0 one is and the other There are 2 states with I z =0, one is π 0 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( ),baryon number B, charge Q, hypercharge Y=S+B –J PC –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 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 I z =0 were obtained. In a similar manner we can obtain three I z =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 –J PC –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: f 0 (1370),f 0 (1500) and f 0 (1710) Uncertainty about the f 0 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=0f 0 (1370), f 0 (1500), f 0 (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. –f 0 (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. –f 0 (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 ++

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