Charmonium masses (PDG 2024): ( ) , ( ) ; the spacing is for and for

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

Charmonium masses (PDG 2024):

J/\psi

(

1\,{}^{3}S_1

)

=3096.9\,\mathrm{MeV}

\chi_{c2}(1P)

(

1\,{}^{3}P_2

)

=3556.2\,\mathrm{MeV}

; the

2S-1S

spacing is

589\,\mathrm{MeV}

for

c\bar c

and

563\,\mathrm{MeV}

for

b\bar b

. Constituent quark masses are

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

m_b\approx4.7\,\mathrm{GeV}

. The leading Regge trajectory of the light

\rho

a_2

\rho_3

a_4

family has slope

0.92\,\mathrm{GeV^{-2}}

. ((a)) For a Coulomb potential the level spacing scales as

\Delta E\propto m\alpha^2

and for a linear potential as

\Delta E\propto(\kappa^2/m)^{1/3}

, at fixed coupling and string tension. Using

m_c

and

m_b

above, compute the ratio (bottomonium/charmonium) of

2S-1S

spacings that each model predicts on its own, and compare both with the observed ratio

563/589

. ((b)) The

J/\psi

(

L=0

S=1

J=1

) and

\chi_{c2}(1P)

(

L=1

S=1

J=2

) form the charmonium analogue of the spin-aligned (

S=1

J=L+1

)

\rho

a_2

family. Using only these two charmonium points, compute the two-point Regge slope

\Delta J/\Delta M^2

for this family, and compare it with the light-quark slope of

0.92\,\mathrm{GeV^{-2}}

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