The triplet ( ) consists of states with orbital angular momentum and total quark spin

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

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

Problem

The

\chi_{cJ}

triplet (

J=0,1,2

) consists of

^3P_J

c\bar c

states with orbital angular momentum

L=1

and total quark spin

S=1

, so

\mathbf J = \mathbf L + \mathbf S

. ((a)) Using

\mathbf L\cdot\mathbf S = \tfrac12[J(J+1)-L(L+1)-S(S+1)]

, find

\langle\mathbf L\cdot\mathbf S\rangle

for

J=0,1,2

, and hence the ratio

[M(\chi_{c1})-M(\chi_{c0})] : [M(\chi_{c2})-M(\chi_{c1})]

predicted by a splitting law of the pure spin-orbit form

\Delta E = A\langle\mathbf L\cdot\mathbf S\rangle

with a single constant

A

. ((b)) The PDG 2024 masses are

M(\chi_{c0})=3414.7\,\mathrm{MeV}

M(\chi_{c1})=3510.7\,\mathrm{MeV}

M(\chi_{c2})=3556.2\,\mathrm{MeV}

. Compute the observed ratio of the two splittings and compare it with the prediction of part (a). ((c)) The

c\bar c

2S-1S

spacing is

589.2\,\mathrm{MeV}

(from

\psi(2S)=3686.1\,\mathrm{MeV}

J/\psi=3096.9\,\mathrm{MeV}

), a scale fixed by the Coulomb-plus-linear Cornell potential, and the charmonium velocity satisfies

v^2\approx(M_{2S}-M_{1S})/m_c\approx0.4

for

m_c\approx1.5\,\mathrm{GeV}

. Compute the ratio of the total

P

-wave splitting

M(\chi_{c2})-M(\chi_{c0})

to the

2S-1S

spacing, and say whether this ratio is of the order expected for a relativistic (

v^2

-suppressed) correction to the Cornell-potential spectrum.