24
1. The Particles and Forces of the Standard Model
FIGURE 1.6
Z
0 -exchange process.
Some initial idea of how this works in the ‘electroweak’ case may be gained
2
by considering the amplitude for figure 1.5(a) in the low −q limit. In a
simplified version analogous to (1.29) which ignores the spin of the W and of
the leptons, this amplitude is
2
− M
2
g
2 /(q
W )
(1.30)
where g is a ‘weak charge’ associated with W-emission and absorption. In
actual β-decay, the square of the 4-momentum transfer q
2 is tiny compared to
2
M
2 , so that (1.30) becomes independent of q and takes the constant value
W
−g
2 /M
2 . This corresponds, in configuration space, to a point-like interaction
W
(the Fourier transform of a delta function is a constant). Just such a pointlike interaction, shown in figure 1.7, had been postulated by Fermi (1934a, b)
in the first theory of β-decay: it is a ‘four-fermion’ interaction with strength
G F . The value of G F can be determined from measured β-decay rates. The
dimensions of G F turn out to be energy × volume, so that G F /(ħc)
3 has
dimension (energy
−2 ). In our units ħ = c = 1, the numerical value of G F is
G F ∼ (300 GeV)
−2 .
(1.31)
If we identify this constant with g
2 /M
2 we obtain
W
g
2
∼ M
2
(1.32)
W /(300 GeV)
2
∼ 0.064
2
a value quite similar to that of the electromagnetic charge e as determined
2
from e = 4πα ∼ 0.09. Though this is qualitatively correct, we shall see
in volume 2 that the actual relation, in the electroweak theory, between the
weak and electromagnetic coupling strengths is somewhat more complicated
than the simple equality ‘g = e’. (Note that a corresponding connection with
Fermi’s theory was also made by Yukawa!)
1. The Particles and Forces of the Standard Model
FIGURE 1.6
Z
0 -exchange process.
Some initial idea of how this works in the ‘electroweak’ case may be gained
2
by considering the amplitude for figure 1.5(a) in the low −q limit. In a
simplified version analogous to (1.29) which ignores the spin of the W and of
the leptons, this amplitude is
2
− M
2
g
2 /(q
W )
(1.30)
where g is a ‘weak charge’ associated with W-emission and absorption. In
actual β-decay, the square of the 4-momentum transfer q
2 is tiny compared to
2
M
2 , so that (1.30) becomes independent of q and takes the constant value
W
−g
2 /M
2 . This corresponds, in configuration space, to a point-like interaction
W
(the Fourier transform of a delta function is a constant). Just such a pointlike interaction, shown in figure 1.7, had been postulated by Fermi (1934a, b)
in the first theory of β-decay: it is a ‘four-fermion’ interaction with strength
G F . The value of G F can be determined from measured β-decay rates. The
dimensions of G F turn out to be energy × volume, so that G F /(ħc)
3 has
dimension (energy
−2 ). In our units ħ = c = 1, the numerical value of G F is
G F ∼ (300 GeV)
−2 .
(1.31)
If we identify this constant with g
2 /M
2 we obtain
W
g
2
∼ M
2
(1.32)
W /(300 GeV)
2
∼ 0.064
2
a value quite similar to that of the electromagnetic charge e as determined
2
from e = 4πα ∼ 0.09. Though this is qualitatively correct, we shall see
in volume 2 that the actual relation, in the electroweak theory, between the
weak and electromagnetic coupling strengths is somewhat more complicated
than the simple equality ‘g = e’. (Note that a corresponding connection with
Fermi’s theory was also made by Yukawa!)