20.2. Parity violation in weak interactions, and V-A theory
291
Equations (20.27) and (20.28) are very important. In particular, equation
(20.27) implies that in the limit of zero mass m (and hence E → |p|), only
the negative helicity u-spinor will enter. More quantitatively, using
√
√
E − |p
√
|
E 2 p 2
m
=
−
≈
for m
E,
(20.29)
E + |p|
(E + |p|)
2E
≪
we can say that positive helicity components of all fermions are suppressed in
V-A matrix elements, relative to the negative helicity components, by factors
of order (m/E). Bearing in mind that the helicity operator σ · p/|p| is a
pseudoscalar, this ‘unequal’ treatment for λ = +1 and λ = −1 components is,
of course, precisely related to the parity violation built in to the V-A structure.
A similar analysis may be done for the v-spinors. They satisfy ( p+m)v = 0
and the normalization vv ¯ = −2m. We must however remember the ‘small
subtlety’ to do with the labelling of v-spinors, discussed in section 3.4.3: the
2-component spinors χ in v(p, λ = +1) actually satisfy σ pχ = p χ ,
−
· −
−| | −
and similarly the χ + ’s in v(p, λ = −1) satisfy σ · pχ + = |p|χ + . We then find
(problem 20.3) the results
( √
)
E
p χ
v(p, λ = +1) =
− √ − | | −
(20.30)
E + |p|χ −
and
( √
)
χ
v(λ = 1) =
√ E + |p|
−
+
.
(20.31)
− E − |p|χ +
Once again, the action of P L removes the top two components, leaving the result that, in the massless limit, only the λ = +1 state survives. Recalling the
‘hole theory’ interpretation of section 3.4.3, this would mean that the positive
helicity components of all antifermions dominate in V-A interactions, negative
helicity components being suppressed by factors of order m/E. The proportionality of the negative helicity amplitude to the mass of the antifermion is
of course exactly as noted for π
+
→ μ
+ ν μ decay in section 18.2.
We should emphasize that although the above results, stated in italics,
were derived in the convenient representation (20.14) for the Dirac matrices,
they actually hold independently of any choice of representation. This can be
shown by using general helicity projection operators.
In Pauli’s original letter, he suggested that the mass of the neutrino might
be of the same order as the electron mass. Immediately after the discovery of
parity violation, it was realized that the result could be elegantly explained
by the assumption that the neutrinos were strictly massless particles (Landau
1957, Lee and Yang 1957 and Salam 1957). In this case, u and v spinors
satisfy the same equation p (u or v) = 0, which reduces via (20.21) (in the
/
/
291
Equations (20.27) and (20.28) are very important. In particular, equation
(20.27) implies that in the limit of zero mass m (and hence E → |p|), only
the negative helicity u-spinor will enter. More quantitatively, using
√
√
E − |p
√
|
E 2 p 2
m
=
−
≈
for m
E,
(20.29)
E + |p|
(E + |p|)
2E
≪
we can say that positive helicity components of all fermions are suppressed in
V-A matrix elements, relative to the negative helicity components, by factors
of order (m/E). Bearing in mind that the helicity operator σ · p/|p| is a
pseudoscalar, this ‘unequal’ treatment for λ = +1 and λ = −1 components is,
of course, precisely related to the parity violation built in to the V-A structure.
A similar analysis may be done for the v-spinors. They satisfy ( p+m)v = 0
and the normalization vv ¯ = −2m. We must however remember the ‘small
subtlety’ to do with the labelling of v-spinors, discussed in section 3.4.3: the
2-component spinors χ in v(p, λ = +1) actually satisfy σ pχ = p χ ,
−
· −
−| | −
and similarly the χ + ’s in v(p, λ = −1) satisfy σ · pχ + = |p|χ + . We then find
(problem 20.3) the results
( √
)
E
p χ
v(p, λ = +1) =
− √ − | | −
(20.30)
E + |p|χ −
and
( √
)
χ
v(λ = 1) =
√ E + |p|
−
+
.
(20.31)
− E − |p|χ +
Once again, the action of P L removes the top two components, leaving the result that, in the massless limit, only the λ = +1 state survives. Recalling the
‘hole theory’ interpretation of section 3.4.3, this would mean that the positive
helicity components of all antifermions dominate in V-A interactions, negative
helicity components being suppressed by factors of order m/E. The proportionality of the negative helicity amplitude to the mass of the antifermion is
of course exactly as noted for π
+
→ μ
+ ν μ decay in section 18.2.
We should emphasize that although the above results, stated in italics,
were derived in the convenient representation (20.14) for the Dirac matrices,
they actually hold independently of any choice of representation. This can be
shown by using general helicity projection operators.
In Pauli’s original letter, he suggested that the mass of the neutrino might
be of the same order as the electron mass. Immediately after the discovery of
parity violation, it was realized that the result could be elegantly explained
by the assumption that the neutrinos were strictly massless particles (Landau
1957, Lee and Yang 1957 and Salam 1957). In this case, u and v spinors
satisfy the same equation p (u or v) = 0, which reduces via (20.21) (in the
/
/