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November 7, 2013 223
The last term in Eq.(9.15) appears to deviate significantly from the spinorial
structure of the first term, which coincides with the Fermi model. Hoewever,
notice that
u(k 1 )(1 + γ 5 )/Qu(p) = u(k 1 )(1 + γ 5 )(/p − /k 1 )u(p)
= u(k 1 ) ( − /k 1 (1 − γ 5 ) + (1 + γ 5 )/p ) u(p)
= m µ u(k 1 )(1 + γ 5 )u(p) (9.17)
upon application of the Dirac equation to the external spinors ; and since, in
the same way,
u(q)(1 + γ 5 )/Qv(k 2 ) = m e u(q)(1 − γ 5 )v(k 2 ) , (9.18)
the second term in Eq.(9.15) is actually suppressed by a factor (m e m µ )/m W 2 ,
which is small if m W is sufficiently large 5 . Neglecting this term, we see that the
Fermi-model amplitude is recovered with the single replacement of the coupling
constant G F / √ 2 by g W 2 /(Q 2 − m W 2 ). Now, the maximum value that Q 2 can
take in this process is m µ 2 , which is attained in the improbable case that the
muon neutrino emerges with zero momentum from the decay. If, therefore, we
assume that m W is large compared to m µ , we see that the successes of the Fermi
model in describing muon decay will be completely reproduced provided 6
g W
2
m W
2 = G F
√
2
, (9.19)
which we may also write in purely dimensionless terms as
( )
gW
c √¯h
= 1 (
mW c 2 )
. (9.20)
2 1/4 Λ W
9.2.2 The cross section for µ − ν µ → e − ν e revisited
We can now study the modification that the IVB hypothesis makes in the cross
section for the process µ − ν µ → e − ν e , where the Fermi model fails. In this case
the total invariant mass is (assumed to be) much larger than the W mass, so
that the modified prediction can immediately be seen to be
σ(µ − ν µ → e − ν e ) = 2¯h2 g W
4
3π
s
(s − m W2 ) 2 = ¯h2 G 2 F s
3π
(
mW
2
s − m W
2
) 2
, (9.21)
and this cross section does decrease as 1/s for large s.
Of course, the unitarity limit (9.13) still has to be observed, which puts an
upper limit 7 on the useful values of m W :
m W c 2 ≤ (72π 2 ) 1/4 Λ W ≈ 1.5 TeV . (9.22)
5 In fact, for the actual values of the masses the suppression factor is about 10 −7 .
6 We disregard the overall sign difference between the two forms as Q 2 /m W 2 → 0.
7 This value is close to the value of √ s at which unitairy breaks down in the unmodified
Fermi model, see Eq.(9.13). This is not a coincidence. Whatever we do to the electroweak
interactions, 1.5 TeV appears to be the energy régime where things get tricky.