10.3 Solutions 583
(h) Massmx≥(
mπ+mμ+mυ)
/c^2 =( 135 + 106 + 0 )/c^2 =241 MeV/c^2
(i)Xis identified asK+-meson.10.77 The initial state^16 O∗has odd parity, while the parity of^12 C is even and so
also that ofα.Ifαis emitted withl=0 the parity of the final state is even.
Clearly the decay goes through a weak interaction because parity is violated
and the observed width is consistent with a weak decay. On the other hand
the width of the electro-magnetic decay^16 O∗→^16 O+γ,is3× 10 −^3 eV.
10.78ττ=B×τμ
(
mμ
mτ) 5
=^17100.^7 × 2. 197 × 10 −^6 ×
( 105. 658
1784) 5
= 0. 28 × 10 −^12 sa value
which is in excellent agreement with the experimental value of 0. 3 × 10 −^12 s10.79 Pions haveT=1. A system of two pions can exist in the isospin stateI=0,
1, and 2 state. Therefore, isospin is conserved in all the three cases.
Now pions obey Bose statistics and soΨtotalis symmetrical. AsJ = 0
for pions, the spatial part of pions system must be symmetrical. The intrinsic
parity of each pion being negative, does not contribute to the parity of the
state. The parity of the state is mainly determined by the L-value;p=(−1)l.
Therefore, allowed states correspond toL= 0 , 2 , 4 ,...It follows that par-
ticles withJp= 0 +, 2 +...can decay to two pions, so that only possible
decay isf^0 →π++π−; the other two particlesω^0 andη^0 actually decay
into three pions.
10.80 The mass ofη– meson is 549 Mev/c^2 while the mass of four pions is
558 MeV/c^2 if all the four pions are charged and 540 MeV/c^2 if uncharged.
In the first state the decay cannot occur because of energy conservation and
in the second case the decay will be heavily suppressed because of a small
phase space on account of small Q-value. Apart from this the only possible
pionic decay is the 3-body mode, the strong decay into two pions or four
pions being forbidden by parity conservation.
10.81 The leptonic decays are assumed to proceed via exchange of a single virtual
photon, Fig. 10.11
Fig. 10.11Leptonic decay of
vector meson
Apart from numerical factors the partial widthΓ(V→l+l−) is propor-
tional to the squared sum of the charges of the quarks in the meson, that is
Γ∝∑
|aiQi|^2 , where aithe amplitudes from all the quarks in the meson
are superposed. The approach is similar to the Rutherford scattering where
the cross-sections are assumed to be proportional to the sum of squares of
the charges of quarks. The quark wave functions and the quantity∣
∣∑aiQi∣
∣^2
are tabulated below