The bottomonium triplet has PDG 2024 masses , , : spin-triplet ( )

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Symmetry, groups and the quark model · Symmetry, groups and the quark model

Symmetry and Flavour, Ch Hadron spectroscopy: quarkonium, orbital excitations and exotic states

Problem

The

1P

bottomonium triplet has PDG 2024 masses

\chi_{b0}(1P)=9859.4\,\mathrm{MeV}

\chi_{b1}(1P)=9892.8\,\mathrm{MeV}

\chi_{b2}(1P)=9912.2\,\mathrm{MeV}

: spin-triplet (

S=1

), orbital P-wave (

L=1

) states with

J=0,1,2

respectively. ((a)) Write the

n\,{}^{2S+1}L_J

label and

J^{PC}

for each state, using

P=(-1)^{L+1}

C=(-1)^{L+S}

. ((b)) Confirm that the mass ordering

\chi_{b0}<\chi_{b1}<\chi_{b2}

follows the same spin-orbit pattern established for the

\chi_{cJ}

triplet, where

\langle\mathbf{L}\cdot\mathbf{S}\rangle=-2,-1,+1

for

J=0,1,2

and the spin-orbit coefficient is positive. ((c)) State which of the three states can decay to two photons, using

C

-parity and the Landau-Yang theorem.

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