2 Gauge Theories and the Standard Model
15
of slow pions that make the exact separation of the two jets impossible. In some
cases a third well separated jet of hadrons is also observed: these events correspond
to the radiation of an energetic gluon from the parent quark-antiquark pair.
In the EW sector the SM inherits the phenomenological successes of the old
(V − A) ⊗ (V − A) four-fermion low-energy description of weak interactions, and
provides a well-defined and consistent theoretical framework including weak interactions and quantum electrodynamics in a unified picture. The weak interactions
derive their name from their intensity. At low energy the strength of the effective
four-fermion interaction of charged currents is determined by the Fermi coupling
constant G F . For example, the effective interaction for muon decay is given by
L eff = (G F /
√
2)
¯
ν μ γ α (1 − γ 5 )μ
¯
eγ
α (1 − γ 5 )ν e
,
(2.2)
with [7]
G F = 1.16639(1) × 10
−5 GeV
−2 .
(2.3)
In natural units ¯
h = c = 1, G F has dimensions of (mass) −2 . As a result, the intensity
of weak interactions at low energy is characterized by G F E 2 , where E is the energy
scale for a given process (E ≈ m μ for muon decay). Since
G F E
2
= G F m
2
p (E/m p )
2
10
−5 (E/m p )
2 ,
(2.4)
where m p is the proton mass, the weak interactions are indeed weak at low energies
(up to energies of order a few ten’s of GeV). Effective four fermion couplings for
neutral current interactions have comparable intensity and energy behaviour. The
quadratic increase with energy cannot continue for ever, because it would lead to a
violation of unitarity. In fact, at large energies the propagator effects can no longer
be neglected, and the current–current interaction is resolved into current–W gauge
boson vertices connected by a W propagator. The strength of the weak interactions
at high energies is then measured by g W , the W − −μ–ν μ coupling, or, even better,
by α W = g 2
W /4π analogous to the fine-structure constant α of QED (in Chap. 3,
g W is simply denoted by g or g 2 ). In the standard EW theory, we have
α W =
√
2 G F m
2
W /π ∼ = 1/30 .
(2.5)
That is, at high energies the weak interactions are no longer so weak.
The range r W of weak interactions is very short: it is only with the experimental
discovery of the W and Z gauge bosons that it could be demonstrated that r W is
non-vanishing. Now we know that
r W =
¯
h
m W c
2.5 × 10
−16 cm,
(2.6)
15
of slow pions that make the exact separation of the two jets impossible. In some
cases a third well separated jet of hadrons is also observed: these events correspond
to the radiation of an energetic gluon from the parent quark-antiquark pair.
In the EW sector the SM inherits the phenomenological successes of the old
(V − A) ⊗ (V − A) four-fermion low-energy description of weak interactions, and
provides a well-defined and consistent theoretical framework including weak interactions and quantum electrodynamics in a unified picture. The weak interactions
derive their name from their intensity. At low energy the strength of the effective
four-fermion interaction of charged currents is determined by the Fermi coupling
constant G F . For example, the effective interaction for muon decay is given by
L eff = (G F /
√
2)
¯
ν μ γ α (1 − γ 5 )μ
¯
eγ
α (1 − γ 5 )ν e
,
(2.2)
with [7]
G F = 1.16639(1) × 10
−5 GeV
−2 .
(2.3)
In natural units ¯
h = c = 1, G F has dimensions of (mass) −2 . As a result, the intensity
of weak interactions at low energy is characterized by G F E 2 , where E is the energy
scale for a given process (E ≈ m μ for muon decay). Since
G F E
2
= G F m
2
p (E/m p )
2
10
−5 (E/m p )
2 ,
(2.4)
where m p is the proton mass, the weak interactions are indeed weak at low energies
(up to energies of order a few ten’s of GeV). Effective four fermion couplings for
neutral current interactions have comparable intensity and energy behaviour. The
quadratic increase with energy cannot continue for ever, because it would lead to a
violation of unitarity. In fact, at large energies the propagator effects can no longer
be neglected, and the current–current interaction is resolved into current–W gauge
boson vertices connected by a W propagator. The strength of the weak interactions
at high energies is then measured by g W , the W − −μ–ν μ coupling, or, even better,
by α W = g 2
W /4π analogous to the fine-structure constant α of QED (in Chap. 3,
g W is simply denoted by g or g 2 ). In the standard EW theory, we have
α W =
√
2 G F m
2
W /π ∼ = 1/30 .
(2.5)
That is, at high energies the weak interactions are no longer so weak.
The range r W of weak interactions is very short: it is only with the experimental
discovery of the W and Z gauge bosons that it could be demonstrated that r W is
non-vanishing. Now we know that
r W =
¯
h
m W c
2.5 × 10
−16 cm,
(2.6)