5
1.2. The fermions of the Standard Model
would account for their interaction (see section 1.3.2). In particular, its mass
(105.7 MeV) was nicely within the range predicted by Yukawa. However, experiments by Conversi et al. (1947) established that the muon could not be
Yukawa’s quantum since it did not interact strongly; it was therefore a lepton.
The μ
− seems to behave in exactly the same way as the electron, interacting
only electromagnetically and weakly, with interaction strengths identical to
those of an electron.
In 1975 Perl et al. (1975) discovered yet another ‘replicant’ electron, the
τ
− with a mass of 1.78 GeV. Once again, the weak and electromagnetic interactions of the τ
− (τ
+ ) are identical to those of the e
− (e
+ ).
At this stage one might well wonder whether we are faced with a ‘lepton
spectroscopy’, of which the e
− , μ
− and τ
− are but the first three states. Yet
this seems not to be the correct interpretation. First, no other such states have
(so far) been seen. Second, all these leptons have the same spin (
1 ), which
2
is certainly quite unlike any conventional excitation spectrum. And third,
no γ-transitions are observed to occur between the states, though this would
normally be expected. For example, the branching fraction for the process
−
μ
−
→ e + γ
(not observed)
(1.1)
is currently quoted as less than 1.2 × 10
−11 at the 90% confidence level
(Nakamura et al. 2010). Similarly there are (much less stringent) limits on
τ
−
→ μ
− + γ and τ
−
→ e
− + γ.
If the e
− and μ
− states in (1.1) were, in fact, the ground and first excited
states of some composite system, the decay process (1.1) would be expected
to occur as an electromagnetic transition, with a relatively high probability
because of the large energy release. Yet the experimental upper limit on the
rate is very tiny. In the absence of any mechanism to explain this, one systematizes the situation, empirically, by postulating the existence of a selection
rule forbidding the decay (1.1). In taking this step, it is important to realize that ‘absolute forbidden-ness’ can never be established experimentally: all
that can be done is to place a (very small) upper limit on the branching fraction to the ‘forbidden’ channel, as here. The possibility will always remain
open that future, more sensitive, experiments will reveal that some processes,
assumed to be forbidden, are in fact simply extremely rare.
Of course, such a proposed selection rule would have no physical content if
it only applied to the one process (1.1); but it turns out to be generally true,
applying not only to the electromagnetic interaction of the charged leptons,
but to their weak interactions also. The upshot is that we can consistently
account for observations (and non-observations) involving e’s, μ’s and τ ’s by
assigning to each a new additive quantum number (called ‘lepton flavour’)
which is assumed to be conserved. Thus we have electron flavour L e such that
L e (e
− ) = 1 and L e (e
+ ) = −1; muon flavour L μ such that L μ (μ
− ) = 1 and
L μ (μ
+ ) = −1; and tau flavour L τ such that L τ (τ
− ) = 1 and L τ (τ
+ ) = −1.
Each is postulated to be conserved in all leptonic processes. So (1.1) is then
1.2. The fermions of the Standard Model
would account for their interaction (see section 1.3.2). In particular, its mass
(105.7 MeV) was nicely within the range predicted by Yukawa. However, experiments by Conversi et al. (1947) established that the muon could not be
Yukawa’s quantum since it did not interact strongly; it was therefore a lepton.
The μ
− seems to behave in exactly the same way as the electron, interacting
only electromagnetically and weakly, with interaction strengths identical to
those of an electron.
In 1975 Perl et al. (1975) discovered yet another ‘replicant’ electron, the
τ
− with a mass of 1.78 GeV. Once again, the weak and electromagnetic interactions of the τ
− (τ
+ ) are identical to those of the e
− (e
+ ).
At this stage one might well wonder whether we are faced with a ‘lepton
spectroscopy’, of which the e
− , μ
− and τ
− are but the first three states. Yet
this seems not to be the correct interpretation. First, no other such states have
(so far) been seen. Second, all these leptons have the same spin (
1 ), which
2
is certainly quite unlike any conventional excitation spectrum. And third,
no γ-transitions are observed to occur between the states, though this would
normally be expected. For example, the branching fraction for the process
−
μ
−
→ e + γ
(not observed)
(1.1)
is currently quoted as less than 1.2 × 10
−11 at the 90% confidence level
(Nakamura et al. 2010). Similarly there are (much less stringent) limits on
τ
−
→ μ
− + γ and τ
−
→ e
− + γ.
If the e
− and μ
− states in (1.1) were, in fact, the ground and first excited
states of some composite system, the decay process (1.1) would be expected
to occur as an electromagnetic transition, with a relatively high probability
because of the large energy release. Yet the experimental upper limit on the
rate is very tiny. In the absence of any mechanism to explain this, one systematizes the situation, empirically, by postulating the existence of a selection
rule forbidding the decay (1.1). In taking this step, it is important to realize that ‘absolute forbidden-ness’ can never be established experimentally: all
that can be done is to place a (very small) upper limit on the branching fraction to the ‘forbidden’ channel, as here. The possibility will always remain
open that future, more sensitive, experiments will reveal that some processes,
assumed to be forbidden, are in fact simply extremely rare.
Of course, such a proposed selection rule would have no physical content if
it only applied to the one process (1.1); but it turns out to be generally true,
applying not only to the electromagnetic interaction of the charged leptons,
but to their weak interactions also. The upshot is that we can consistently
account for observations (and non-observations) involving e’s, μ’s and τ ’s by
assigning to each a new additive quantum number (called ‘lepton flavour’)
which is assumed to be conserved. Thus we have electron flavour L e such that
L e (e
− ) = 1 and L e (e
+ ) = −1; muon flavour L μ such that L μ (μ
− ) = 1 and
L μ (μ
+ ) = −1; and tau flavour L τ such that L τ (τ
− ) = 1 and L τ (τ
+ ) = −1.
Each is postulated to be conserved in all leptonic processes. So (1.1) is then