306
20. Introduction to the Phenomenology of Weak Interactions
requires a double quark transition from suu to udd). All known data on such
decays can be fit with a value sin θ C ⋍ 0.22 for the Cabibbo angle θ C . This
relatively small angle is therefore a measure of the suppression of |ΔS| = 1
processes relative to ΔS = 0 ones.
The Cabibbo current can be written in a more compact form by introducing the ‘mixed’ field
d ˆ ′
ˆ
≡ cos θ C d + sin θ C s. ˆ
(20.97)
Then
(1 − γ 5 )
uγ
μ
ˆ j
μ (u, d, s) = ¯ ˆ
d ˆ ′ .
(20.98)
Cab
2
In 1970 Glashow, Iliopuolos and Maiani (GIM) (1970) drew attention to
a theoretical problem with the interaction (20.92) if used in second order.
Now it is, of course, the case that this interaction is not renormalizable, as
noted previously for the purely leptonic one (20.40), since G F has dimensions
of an inverse mass squared. As we saw in section 11.7, this means that oneloop diagrams will typically diverge quadratically, so that the contribution
of such a second-order process will be of order (G F .G F Λ
2 ) where Λ is a cutoff, compared to the first-order amplitude G F . Recalling from (20.46) that
G F ∼ 10
−5 GeV
−2 , we see that for Λ ∼ 10 GeV such a correction could
be significant if accurate enough data exists. GIM pointed out, in particular,
that some second-order processes could be found which violated the (hitherto)
well-established phenomenological selection rules, such as the |ΔS| = 1 and
ΔS = ΔQ rules already discussed. For example, there could be ΔS = 2
amplitudes contributing to the K L − K S mass difference (see Renton 1990,
section 9.1.6, for example), as well as contributions to unobserved decay modes
such as
K
+
→ π
+ + ν + ¯
ν
(20.99)
which has a neutral lepton pair in association with a strangeness change for
the hadron. In fact, experiment placed very tight limits on the rate for (20.99)
– and still does: the branching fraction is (1.7 ± 1.1) × 10
−10 (Nakamura et
al. 2010). This seemed to imply a surprisingly low value of the cut-off, say
∼ 3 GeV (Mohapatra et al. 1968).
Partly in order to address this problem, and partly as a revival of an
earlier lepton-quark symmetry proposal (Bjorken and Glashow 1964), GIM
introduced a fourth quark, now called c (the charm quark) with charge
2 e.
3
Note that in 1970 the τ -lepton had not been discovered, so only two lepton
family pairs (ν e , e), (ν μ , μ) were known; this fourth quark therefore did restore
the balance, via the two quark family pairs (u,d), (c,s). In particular, a
second quark current could now be hypothesized, involving the (c,s) pair. GIM
postulated that the c-quark was coupled to the ‘orthogonal’ d-s combination
(cf (20.97))
′
ˆ
s ˆ = − sin θ C d + cos θ C s. ˆ
(20.100)
20. Introduction to the Phenomenology of Weak Interactions
requires a double quark transition from suu to udd). All known data on such
decays can be fit with a value sin θ C ⋍ 0.22 for the Cabibbo angle θ C . This
relatively small angle is therefore a measure of the suppression of |ΔS| = 1
processes relative to ΔS = 0 ones.
The Cabibbo current can be written in a more compact form by introducing the ‘mixed’ field
d ˆ ′
ˆ
≡ cos θ C d + sin θ C s. ˆ
(20.97)
Then
(1 − γ 5 )
uγ
μ
ˆ j
μ (u, d, s) = ¯ ˆ
d ˆ ′ .
(20.98)
Cab
2
In 1970 Glashow, Iliopuolos and Maiani (GIM) (1970) drew attention to
a theoretical problem with the interaction (20.92) if used in second order.
Now it is, of course, the case that this interaction is not renormalizable, as
noted previously for the purely leptonic one (20.40), since G F has dimensions
of an inverse mass squared. As we saw in section 11.7, this means that oneloop diagrams will typically diverge quadratically, so that the contribution
of such a second-order process will be of order (G F .G F Λ
2 ) where Λ is a cutoff, compared to the first-order amplitude G F . Recalling from (20.46) that
G F ∼ 10
−5 GeV
−2 , we see that for Λ ∼ 10 GeV such a correction could
be significant if accurate enough data exists. GIM pointed out, in particular,
that some second-order processes could be found which violated the (hitherto)
well-established phenomenological selection rules, such as the |ΔS| = 1 and
ΔS = ΔQ rules already discussed. For example, there could be ΔS = 2
amplitudes contributing to the K L − K S mass difference (see Renton 1990,
section 9.1.6, for example), as well as contributions to unobserved decay modes
such as
K
+
→ π
+ + ν + ¯
ν
(20.99)
which has a neutral lepton pair in association with a strangeness change for
the hadron. In fact, experiment placed very tight limits on the rate for (20.99)
– and still does: the branching fraction is (1.7 ± 1.1) × 10
−10 (Nakamura et
al. 2010). This seemed to imply a surprisingly low value of the cut-off, say
∼ 3 GeV (Mohapatra et al. 1968).
Partly in order to address this problem, and partly as a revival of an
earlier lepton-quark symmetry proposal (Bjorken and Glashow 1964), GIM
introduced a fourth quark, now called c (the charm quark) with charge
2 e.
3
Note that in 1970 the τ -lepton had not been discovered, so only two lepton
family pairs (ν e , e), (ν μ , μ) were known; this fourth quark therefore did restore
the balance, via the two quark family pairs (u,d), (c,s). In particular, a
second quark current could now be hypothesized, involving the (c,s) pair. GIM
postulated that the c-quark was coupled to the ‘orthogonal’ d-s combination
(cf (20.97))
′
ˆ
s ˆ = − sin θ C d + cos θ C s. ˆ
(20.100)