Subatomic Physics

11.10. Weak Currents in Nuclear Physics 357
spin-parity J π =0 + . Parity and angular momentum selection rules then severely
restrict the matrix elements.
Using Eq. (11.59) for writing down the weak current of hadrons, and taking into
consideration that in nuclear beta decay there is is not enough energy available for
transformations involving quarks other than u and d:
J h w (nuclear physics) = VudJw. (11.61)
Denoting the wave functions of the initial and final nuclear states by ψ 0 + α and ψ 0 + β
and writing the weak current Jw in the same form as J e w, Eq. (11.44), J h w becomes
J h w (0+ −→ 0 + )=cVudψ ∗ 0 + β (V−A)ψ 0 + α.
With Eqs. (11.25) and (11.44), the matrix element of Hw then becomes
�
d 3 xψ ∗ e +(V −A)ψ¯νe · ψ ∗ 0 + β (V−A)ψ0 + α.
〈β|Hw|α〉 = 1 √ 2 GF Vud
The positron and the neutrino are leptons, and they do not interact hadronically
with the nucleus. After emission, they can therefore be described by plane waves,
like free particles:
� �
� �
ipe · x
ipν · x
ψe + = ue exp , ψ¯ν = u¯ν exp . (11.62)
�
�
Here the spin wave functions ue and u¯ν are no longer functions of x. (The plane
wave for the electron is slightly distorted by the Coulomb field of the nucleus. This
distortion results in a small correction that has been discussed in Section 11.2 and
is given by the function F introduced there.) The energies of the leptons are less
than a few MeV, the reduced wavelengths λ = �/p are long compared to the nuclear
radius, and the lepton wave functions can be replaced by their values at the origin,
ue and u¯ν. The matrix element then becomes
〈β|Hw|α〉 = 1 √ GF Vudu
2 ∗ e (V−A)u¯ν
�
·
d 3 xψ ∗ 0 + β (V−A)ψ 0 + α. (11.63)
Parity and angular momentum conservation simplify this expression. Consider parity
first. (21) Under P , the nuclear wave functions ψ 0 + α and ψ 0 + β remain unchanged.
According to Eqs. (11.42) and (11.43), V and A0 change sign. Consequently, the
corresponding integrands are odd under P and the integrals vanish. The term
involving A also vanishes because the wave functions are scalars under rotation,
whereas A behaves like a vector. The average of a vector over a spherical surface
21 At first sight, the parity argument seems inappropriate, because the weak interaction does
not conserve parity. However, the parity of the initial and the final nuclear states is given by the
hadronic interaction, which, due to the non-relativistic nature of the motion of the hadrons, does
conserve parity. The argument is therefore correct.