572
10 Particle Physics – II
While the p–p system is a pure I = 1 state the p-n system is a linear superposition with equal statistical weight of I = 1 and I = 0 states. Therefore if
isospin is conserved only the I = 1 state can contribute since the final state
is a pure I = 1 state. The expected ratio of cross-sections at a given energy is
σ
+
σ 0 =
1
1
√
2
2 = 2
The observed ratio 3.15/1.5 = 2.1 is consistent with the expected ratio of
2.0.
10.43 Pions obey Bose statistics and the system of pions must be symmetric upon
interchange of space and isospin coordinates of any two pions. For each pion
T = 1 and for a system of two pions, total isospin, I = 2, 1, 0. The corresponding states are
|2, ±2 , |2, ±1 , |2, 0
(Symmetric)
|1, ±1 , |1, 0
(Anti symmetric)
|0, 0
(Symmetric)
Now ψ(total) = ψ (space)ψ (isospin)
Thus for the symmetric states like |2, ±1 or |0, 0 , ψ (space) must be
symmetric because of Bose symmetry. But for two pions the interchange of
the spatial coordinates introduces a factor (−1)
l so that only even l states
are allowed. Thus, (π
0
, π
0 ), (π
+
, π
+ ), (π
−
, π
− ) in (a), (b) and (f) will
be found in the states L = 0, 2, . . . However, the states |1, ±1 , |1, 0 are
antisymmetric and this requires, L = odd, that is L = 1, 3, . . . (c) being
antisymmetric can have odd values for L, that is L = 1, 3, . . . (d) and (e)
can exist in I = 1 regardless of the L values because of two particles in
these two cases are fermion and boson, so that the previous considerations
are inconsequential.
10.3.3 Quarks
10.44 High energy collisions can cause the quarks within hadrons, or newly created
quark–antiquark pairs, to fly apart from each other with very high energies.
As the distance from the origin increases the colour force also increases,
and it is energetically more favourable for the quarks to be fragmented,
that is transformed into a jet of hadrons, mostly pions. Gluons may also
radiate a separate jet. However, the quarks can not be dislodged as free
particles.
10 Particle Physics – II
While the p–p system is a pure I = 1 state the p-n system is a linear superposition with equal statistical weight of I = 1 and I = 0 states. Therefore if
isospin is conserved only the I = 1 state can contribute since the final state
is a pure I = 1 state. The expected ratio of cross-sections at a given energy is
σ
+
σ 0 =
1
1
√
2
2 = 2
The observed ratio 3.15/1.5 = 2.1 is consistent with the expected ratio of
2.0.
10.43 Pions obey Bose statistics and the system of pions must be symmetric upon
interchange of space and isospin coordinates of any two pions. For each pion
T = 1 and for a system of two pions, total isospin, I = 2, 1, 0. The corresponding states are
|2, ±2 , |2, ±1 , |2, 0
(Symmetric)
|1, ±1 , |1, 0
(Anti symmetric)
|0, 0
(Symmetric)
Now ψ(total) = ψ (space)ψ (isospin)
Thus for the symmetric states like |2, ±1 or |0, 0 , ψ (space) must be
symmetric because of Bose symmetry. But for two pions the interchange of
the spatial coordinates introduces a factor (−1)
l so that only even l states
are allowed. Thus, (π
0
, π
0 ), (π
+
, π
+ ), (π
−
, π
− ) in (a), (b) and (f) will
be found in the states L = 0, 2, . . . However, the states |1, ±1 , |1, 0 are
antisymmetric and this requires, L = odd, that is L = 1, 3, . . . (c) being
antisymmetric can have odd values for L, that is L = 1, 3, . . . (d) and (e)
can exist in I = 1 regardless of the L values because of two particles in
these two cases are fermion and boson, so that the previous considerations
are inconsequential.
10.3.3 Quarks
10.44 High energy collisions can cause the quarks within hadrons, or newly created
quark–antiquark pairs, to fly apart from each other with very high energies.
As the distance from the origin increases the colour force also increases,
and it is energetically more favourable for the quarks to be fragmented,
that is transformed into a jet of hadrons, mostly pions. Gluons may also
radiate a separate jet. However, the quarks can not be dislodged as free
particles.