For a quark-antiquark pair with relative orbital angular momentum and total spin

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 Parity, charge conjugation and time reversal as operators, and the CPT theorem

Problem

For a quark-antiquark pair with relative orbital angular momentum

L

and total spin

S\in\{0,1\}

, the parity and charge-conjugation eigenvalues are

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

and

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

, and the total angular momentum

J

is obtained by coupling

\mathbf L

and

\mathbf S

. ((a)) Show that the product

CP

depends only on

S

, not on

L

, and give its value for

S=0

and for

S=1

. ((b)) Using this result, show that any

q\bar q

state with

P=+1

C=-1

must have

S=0

and odd

J

. ((c)) Hence show that

0^{+-}

and

2^{+-}

can never be built from a quark-antiquark pair, while

1^{+-}

(realised by the

b_1(1235)

and

h_1(1170)

) is allowed.

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