1000 Solved Problems in Modern Physics

578 10 Particle Physics – II

10.61 Neithere+nore−is an eigen state ofC. However, the systeme+e−in a
definite (l, s) state is an eigen state ofC. According to the generalized Pauli
Principle under the total exchange of particles consisting of changingQ,r
andslabels, the total functionΨmust change its sign.
Space exchange gives a factor (−1)las this involves parity operation.
Spin exchange gives a factor (−1)S+^1.
Charge exchange gives a factorC.
The condition becomes

(−1)l(−1)S+^1 C=−1orC=(−1)l+S

where S is the total spin.

We shall now show howC-invariance restricts the number of photons into

which the positrinium annihilates. Let n be the number of photons in the final

state. Conservation ofC-parity gives

(−1)l+S=(−1)n

Two cases arise

(i) Singlet s state^1 S 0 ;l=S=0 (para – positronium)

e+e−→ 2 γallowed with lifetime 1. 25 × 10 −^7 s.

→ 3 γforbidden

(ii) Triplet s state^3 S 1 ;l= 0 ,S=1 (ortho – positronium)

e+e−→ 3 γallowed with lifetime 1. 5 × 10 −^7 s.

→ 2 γforbidden

Note that annihilation into a single photon is not possible as it would violate

the conservation of linear momentum.

10.62 (a) It is forbidden as electromagnetic interaction becauseΔS
=0 and also
forbidden as weak interaction because there is no strangeness changing
current
(b) It is allowed as an electromagnetic process becauseΔS=0 (bothΣ^0 and
ΛhaveS=−1.Sdoes not apply toe+, ande−)

10.3.5 WeakInteractions..................................

10.63 The decay rateWis given by the inverse of lifetime multiplied by the branch-
ing fractionB, that isW=B/τ

W(D+)=

0. 19

10. 6 × 10 −^13

= 1. 8 × 1011 s−^1

W(D^0 )=

0. 08

4. 2 × 10 −^13

= 1. 9 × 1011 s−^1