354
11. Loops and Renormalization II: QED
theory, only a finite number of diagrams are ultraviolet divergent, to all orders
in perturbation theory.
It is clear that some kind of opposite situation must obtain when the
coupling constant dimensionality is negative; for then, as the order of the perturbation theory increases, the negative powers of M in the coupling constant
factors must be compensated by positive powers of k in the numerators of
loop integrals. Hence the divergence will tend to get worse at each successive
order. A famous example of such a theory is Fermi’s original theory of β-decay
(Fermi 1934a, b), referred to in section 1.3.5, in which the interaction density
has the ‘four-fermion’ form
¯ ˆ
¯ ˆ
G F ψ p (x)ψ ˆ n (x)ψ e (x)ψ ˆ νe (x)
(11.96)
where G F is the ‘Fermi constant’. To find the dimensionality of G F , we first
¯
establish that of the fermion field by considering a mass term mψ ˆ ψ ˆ , for example. The integral of this over d
3
x gives one term in the Hamiltonian, which has
¯ ¯
dimension M . We deduce that [ψ ˆ ] = 2
3 , since [d
3
x] = −3. Hence [ψ ˆ ψ ˆ ψ ˆ ψ ˆ ] = 6,
and so [G F ] = −2. The coupling constant G F in (11.96) therefore has a negative mass dimension, just like the coefficient K/m in (11.80). Indeed, the
four-fermion theory is also non-renormalizable.
Must such a theory be rejected? Let us briefly sketch the consequences of
an interaction of the form (11.96), but slightly simpler, namely
¯
¯
G F ψ ˆ
n (x)ψ ˆ n (x)ψ ˆ
νe (x)ψ ˆ νe (x)
(11.97)
where, for the present purposes, the neutron is regarded as point-like. Consider, for example, the scattering process ν e + n → ν e + n. To lowest order
in G F , this is given by the tree diagram – or ‘contact term’ – of figure 11.11,
which contributes a constant −iG F to the invariant amplitude for the process,
disregarding the spinor factors for the moment. A one-loop O(G
2 ) correction
F
is shown in figure 11.12. Inspection of figure 11.12 shows that this is an s[2]
channel process (recall section 6.3.3): let us call the amplitude −iG F G (s),
l
where one G F factor has been extracted, so that the correction can be com[2]
[2]
pared with the tree amplitude and G (s) is dimensionless. Then G (s) is
l
l
given by
∫ d
4 k
i
i
[2]
G (s) = −iG F
.
(11.98)
l
(2π) 4 k / − m νe (p νe + p n − k /) − m n
As expected, the negative mass dimension of G F leaves fewer k-factors in the
denominator of the loop integral. Indeed, manipulations exactly like those
we used in the case of Σ
[2] shows that G
[2] (s) has a quadratic divergence,
l
[2]
and that dG /ds has a logarithmic divergence. The extra denominators
l
[2]
associated with second and higher derivatives of G (s) are sufficient to make
l
these integrals finite.
11. Loops and Renormalization II: QED
theory, only a finite number of diagrams are ultraviolet divergent, to all orders
in perturbation theory.
It is clear that some kind of opposite situation must obtain when the
coupling constant dimensionality is negative; for then, as the order of the perturbation theory increases, the negative powers of M in the coupling constant
factors must be compensated by positive powers of k in the numerators of
loop integrals. Hence the divergence will tend to get worse at each successive
order. A famous example of such a theory is Fermi’s original theory of β-decay
(Fermi 1934a, b), referred to in section 1.3.5, in which the interaction density
has the ‘four-fermion’ form
¯ ˆ
¯ ˆ
G F ψ p (x)ψ ˆ n (x)ψ e (x)ψ ˆ νe (x)
(11.96)
where G F is the ‘Fermi constant’. To find the dimensionality of G F , we first
¯
establish that of the fermion field by considering a mass term mψ ˆ ψ ˆ , for example. The integral of this over d
3
x gives one term in the Hamiltonian, which has
¯ ¯
dimension M . We deduce that [ψ ˆ ] = 2
3 , since [d
3
x] = −3. Hence [ψ ˆ ψ ˆ ψ ˆ ψ ˆ ] = 6,
and so [G F ] = −2. The coupling constant G F in (11.96) therefore has a negative mass dimension, just like the coefficient K/m in (11.80). Indeed, the
four-fermion theory is also non-renormalizable.
Must such a theory be rejected? Let us briefly sketch the consequences of
an interaction of the form (11.96), but slightly simpler, namely
¯
¯
G F ψ ˆ
n (x)ψ ˆ n (x)ψ ˆ
νe (x)ψ ˆ νe (x)
(11.97)
where, for the present purposes, the neutron is regarded as point-like. Consider, for example, the scattering process ν e + n → ν e + n. To lowest order
in G F , this is given by the tree diagram – or ‘contact term’ – of figure 11.11,
which contributes a constant −iG F to the invariant amplitude for the process,
disregarding the spinor factors for the moment. A one-loop O(G
2 ) correction
F
is shown in figure 11.12. Inspection of figure 11.12 shows that this is an s[2]
channel process (recall section 6.3.3): let us call the amplitude −iG F G (s),
l
where one G F factor has been extracted, so that the correction can be com[2]
[2]
pared with the tree amplitude and G (s) is dimensionless. Then G (s) is
l
l
given by
∫ d
4 k
i
i
[2]
G (s) = −iG F
.
(11.98)
l
(2π) 4 k / − m νe (p νe + p n − k /) − m n
As expected, the negative mass dimension of G F leaves fewer k-factors in the
denominator of the loop integral. Indeed, manipulations exactly like those
we used in the case of Σ
[2] shows that G
[2] (s) has a quadratic divergence,
l
[2]
and that dG /ds has a logarithmic divergence. The extra denominators
l
[2]
associated with second and higher derivatives of G (s) are sufficient to make
l
these integrals finite.