578
10 Particle Physics – II
10.61 Neither e
+ nor e
− is an eigen state of C. However, the system e
+ e
− in a
definite (l, s) state is an eigen state of C. According to the generalized Pauli
Principle under the total exchange of particles consisting of changing Q, r
and s labels, the total function Ψ must change its sign.
Space exchange gives a factor (−1)
l as this involves parity operation.
Spin exchange gives a factor (−1)
S+1 .
Charge exchange gives a factor C.
The condition becomes
(−1)
l (−1)
S+1 C = −1 or C = (−1)
l+S
where S is the total spin.
We shall now show how C-invariance restricts the number of photons into
which the positrinium annihilates. Let n be the number of photons in the final
state. Conservation of C-parity gives
(−1)
l+S
= (−1)
n
Two cases arise
(i) Singlet s state
1 S 0 ; l = S = 0 (para – positronium)
e
+ e
−
→ 2γ allowed with lifetime 1.25 × 10
−7 s.
→ 3γ forbidden
(ii) Triplet s state
3 S 1 ; l = 0, S = 1 (ortho – positronium)
e
+ e
−
→ 3γ allowed with lifetime 1.5 × 10
−7 s.
→ 2γ forbidden
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 because ΔS = 0 and also
forbidden as weak interaction because there is no strangeness changing
current
(b) It is allowed as an electromagnetic process because ΔS = 0 (both Σ
0 and
Λ have S = −1. S does not apply to e
+ , and e
− )
10.3.5 Weak Interactions
10.63 The decay rate W is given by the inverse of lifetime multiplied by the branching fraction B, that is W = B/τ
W (D
+ ) =
0.19
10.6 × 10 −13 = 1.8 × 10
11 s
−1
W (D
0 ) =
0.08
4.2 × 10 −13 = 1.9 × 10
11 s
−1
10 Particle Physics – II
10.61 Neither e
+ nor e
− is an eigen state of C. However, the system e
+ e
− in a
definite (l, s) state is an eigen state of C. According to the generalized Pauli
Principle under the total exchange of particles consisting of changing Q, r
and s labels, the total function Ψ must change its sign.
Space exchange gives a factor (−1)
l as this involves parity operation.
Spin exchange gives a factor (−1)
S+1 .
Charge exchange gives a factor C.
The condition becomes
(−1)
l (−1)
S+1 C = −1 or C = (−1)
l+S
where S is the total spin.
We shall now show how C-invariance restricts the number of photons into
which the positrinium annihilates. Let n be the number of photons in the final
state. Conservation of C-parity gives
(−1)
l+S
= (−1)
n
Two cases arise
(i) Singlet s state
1 S 0 ; l = S = 0 (para – positronium)
e
+ e
−
→ 2γ allowed with lifetime 1.25 × 10
−7 s.
→ 3γ forbidden
(ii) Triplet s state
3 S 1 ; l = 0, S = 1 (ortho – positronium)
e
+ e
−
→ 3γ allowed with lifetime 1.5 × 10
−7 s.
→ 2γ forbidden
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 because ΔS = 0 and also
forbidden as weak interaction because there is no strangeness changing
current
(b) It is allowed as an electromagnetic process because ΔS = 0 (both Σ
0 and
Λ have S = −1. S does not apply to e
+ , and e
− )
10.3.5 Weak Interactions
10.63 The decay rate W is given by the inverse of lifetime multiplied by the branching fraction B, that is W = B/τ
W (D
+ ) =
0.19
10.6 × 10 −13 = 1.8 × 10
11 s
−1
W (D
0 ) =
0.08
4.2 × 10 −13 = 1.9 × 10
11 s
−1