58
G. Altarelli and S. Forte
charge Q = +2/3 and one neutrino with Q = 0 and, with t 3 = −1/2, three colours
of down quarks with charge Q = −1/3 and one l − with Q = −1. Thus we obtain
tr{t 3 QQ} = 1/2 . 3 . 4/9 − 1/2 . 3 . 1/9 − 1/2 . 1 = 0. This impressive cancellation
suggests an interplay among weak isospin, charge and colour quantum numbers
which appears as a miracle from the point of view of the low energy theory but is in
fact understandable from the point of view of the high energy theory. For example,
in Grand Unified Theories (GUTs) (for reviews, see, for example, [31]) there are
similar relations where charge quantization and colour are related: in the five of
SU(5) we have the content (d, d, d, e + , ¯
ν) and the charge generator has a vanishing
trace in each SU(5) representation (the condition of unit determinant, represented by
the letter S in the SU(5) group name, translates into zero trace for the generators).
Thus the charge of d quarks is −1/3 of the positron charge because there are three
colours. A whole family fits perfectly in one 16 of SO(10) which is anomaly free.
So GUTs can naturally explain the cancellation of the chiral anomaly.
An important implication of chiral anomalies together with the topological
properties of the vacuum in non abelian gauge theories is that the conservation of the
charges associated to baryon (B) and lepton (L) numbers is broken by the anomaly
[32], so that B and L conservation is actually violated in the standard electroweak
theory (but B-L remains conserved). B and L are conserved to all orders in the
perturbative expansion but the violation occurs via non perturbative instanton effects
[33] (the amplitude is proportional to the typical non perturbative factor exp −c/g 2 ,
with c a constant and g the SU (2) gauge coupling). The corresponding effect is
totally negligible at zero temperature T , but becomes relevant at temperatures close
to the electroweak symmetry breaking scale, precisely at T ∼ o(T eV ). The non
conservation of B+L and the conservation of B−L near the weak scale plays a role
in the theory of baryogenesis that quantitatively aims at explaining the observed
matter antimatter asymmetry in the Universe (for a recent review, see, for example,
[34]; see also Chap. 9).
3.9 QED Tests: Lepton Anomalous Magnetic Moments
The most precise tests of the electroweak theory apply to the QED sector. Here
we discuss some recent developments. The anomalous magnetic moments of the
electron and of the muon are among the most precise measurements in the whole
of physics. The magnetic moment
μ and the spin
S are related by
μ = −ge
S/2m,
where g is the gyromagnetic ratio (g = 2 for a pointlike Dirac particle). The quantity
a = (g − 2)/2 measures the anomalous magnetic moment of the particle. Recently
there have been new precise measurements of a e and a μ for the electron [35] and
the muon [36]:
a
exp
e
= 11596521808.5(7.6)
. 10
−13 ,
a
exp
μ = 11659208.0(6.3)
. 10
−10 .
(3.89)
G. Altarelli and S. Forte
charge Q = +2/3 and one neutrino with Q = 0 and, with t 3 = −1/2, three colours
of down quarks with charge Q = −1/3 and one l − with Q = −1. Thus we obtain
tr{t 3 QQ} = 1/2 . 3 . 4/9 − 1/2 . 3 . 1/9 − 1/2 . 1 = 0. This impressive cancellation
suggests an interplay among weak isospin, charge and colour quantum numbers
which appears as a miracle from the point of view of the low energy theory but is in
fact understandable from the point of view of the high energy theory. For example,
in Grand Unified Theories (GUTs) (for reviews, see, for example, [31]) there are
similar relations where charge quantization and colour are related: in the five of
SU(5) we have the content (d, d, d, e + , ¯
ν) and the charge generator has a vanishing
trace in each SU(5) representation (the condition of unit determinant, represented by
the letter S in the SU(5) group name, translates into zero trace for the generators).
Thus the charge of d quarks is −1/3 of the positron charge because there are three
colours. A whole family fits perfectly in one 16 of SO(10) which is anomaly free.
So GUTs can naturally explain the cancellation of the chiral anomaly.
An important implication of chiral anomalies together with the topological
properties of the vacuum in non abelian gauge theories is that the conservation of the
charges associated to baryon (B) and lepton (L) numbers is broken by the anomaly
[32], so that B and L conservation is actually violated in the standard electroweak
theory (but B-L remains conserved). B and L are conserved to all orders in the
perturbative expansion but the violation occurs via non perturbative instanton effects
[33] (the amplitude is proportional to the typical non perturbative factor exp −c/g 2 ,
with c a constant and g the SU (2) gauge coupling). The corresponding effect is
totally negligible at zero temperature T , but becomes relevant at temperatures close
to the electroweak symmetry breaking scale, precisely at T ∼ o(T eV ). The non
conservation of B+L and the conservation of B−L near the weak scale plays a role
in the theory of baryogenesis that quantitatively aims at explaining the observed
matter antimatter asymmetry in the Universe (for a recent review, see, for example,
[34]; see also Chap. 9).
3.9 QED Tests: Lepton Anomalous Magnetic Moments
The most precise tests of the electroweak theory apply to the QED sector. Here
we discuss some recent developments. The anomalous magnetic moments of the
electron and of the muon are among the most precise measurements in the whole
of physics. The magnetic moment
μ and the spin
S are related by
μ = −ge
S/2m,
where g is the gyromagnetic ratio (g = 2 for a pointlike Dirac particle). The quantity
a = (g − 2)/2 measures the anomalous magnetic moment of the particle. Recently
there have been new precise measurements of a e and a μ for the electron [35] and
the muon [36]:
a
exp
e
= 11596521808.5(7.6)
. 10
−13 ,
a
exp
μ = 11659208.0(6.3)
. 10
−10 .
(3.89)