25
1.3. Particle interactions in the Standard Model
FIGURE 1.7
Point-like four-fermion interaction.
We can now understand the ‘weakness’ of the weak interactions from an2
other viewpoint. For q ≪ M
2 , the ratio of the electromagnetic amplitude
W
(1.29) to the weak amplitude (1.30) is of order q
2 /M
2 , given that e ∼ g.
W
Thus despite having an intrinsic strength similar to that of electromagnetism,
weak interactions will appear very weak at low energies such that q
W .
2
≪ M
2
At energies approaching M W , however, weak interactions will grow in im2
portance relative to electromagnetic ones and, when q ≫ M
2 , weak and
W
electromagnetic interactions will contribute roughly equally.
‘Similar’ coupling strengths are still not ‘unified’, however. True unification only occurs after a more subtle effect has been included, which goes
beyond the one-quantum exchange mechanism. This is the variation or ‘running’ of the coupling strengths as a function of energy (or distance), caused
by higher-order processes in perturbation theory. This will be discussed more
fully in chapter 11 for QED, and in volume 2 for the other gauge couplings.
It turns out that the possibility of unification depends crucially on an important difference between the weak interaction quanta W
± (to take the present
example) and the photons of QED, which has not been apparent in the simple
β-decay processes considered so far. The W’s are themselves ‘weakly charged’,
acting as both carriers and sources of the weak force field, and they therefore
interact directly amongst themselves even in the absence of other matter.
By contrast, photons are electromagnetically neutral and have no direct selfinteractions. In theories where the gauge quanta self-interact, the coupling
strength decreases as the energy increases, while for QED it increases. It is
this differing ‘evolution’ that tends to bring the strengths together, ultimately.
Even granted similar coupling strengths and the fact that both are 4-vector
fields, the idea of any electroweak unification appears to founder immediately
on the markedly different ranges of the two forces or, equivalently, of the
masses of the mediating quanta (m γ = 0, M W ∼ 80 GeV!). This difficulty
becomes even more pointed when we recall that, as previously mentioned,
the masslessness of the photon is related to gauge invariance in electrodynamics: how then can there be any similar kind of gauge symmetry for weak
interactions, given the distinctly non-zero masses of the mediating quanta?
Nevertheless, in one of the great triumphs of 20th century theoretical physics,
it is possible to see the two theories as essentially similar gauge theories, the
1.3. Particle interactions in the Standard Model
FIGURE 1.7
Point-like four-fermion interaction.
We can now understand the ‘weakness’ of the weak interactions from an2
other viewpoint. For q ≪ M
2 , the ratio of the electromagnetic amplitude
W
(1.29) to the weak amplitude (1.30) is of order q
2 /M
2 , given that e ∼ g.
W
Thus despite having an intrinsic strength similar to that of electromagnetism,
weak interactions will appear very weak at low energies such that q
W .
2
≪ M
2
At energies approaching M W , however, weak interactions will grow in im2
portance relative to electromagnetic ones and, when q ≫ M
2 , weak and
W
electromagnetic interactions will contribute roughly equally.
‘Similar’ coupling strengths are still not ‘unified’, however. True unification only occurs after a more subtle effect has been included, which goes
beyond the one-quantum exchange mechanism. This is the variation or ‘running’ of the coupling strengths as a function of energy (or distance), caused
by higher-order processes in perturbation theory. This will be discussed more
fully in chapter 11 for QED, and in volume 2 for the other gauge couplings.
It turns out that the possibility of unification depends crucially on an important difference between the weak interaction quanta W
± (to take the present
example) and the photons of QED, which has not been apparent in the simple
β-decay processes considered so far. The W’s are themselves ‘weakly charged’,
acting as both carriers and sources of the weak force field, and they therefore
interact directly amongst themselves even in the absence of other matter.
By contrast, photons are electromagnetically neutral and have no direct selfinteractions. In theories where the gauge quanta self-interact, the coupling
strength decreases as the energy increases, while for QED it increases. It is
this differing ‘evolution’ that tends to bring the strengths together, ultimately.
Even granted similar coupling strengths and the fact that both are 4-vector
fields, the idea of any electroweak unification appears to founder immediately
on the markedly different ranges of the two forces or, equivalently, of the
masses of the mediating quanta (m γ = 0, M W ∼ 80 GeV!). This difficulty
becomes even more pointed when we recall that, as previously mentioned,
the masslessness of the photon is related to gauge invariance in electrodynamics: how then can there be any similar kind of gauge symmetry for weak
interactions, given the distinctly non-zero masses of the mediating quanta?
Nevertheless, in one of the great triumphs of 20th century theoretical physics,
it is possible to see the two theories as essentially similar gauge theories, the