3 The Standard Model of Electroweak Interactions
59
Fig. 3.10 The hadronic
contributions to the
anomalous magnetic moment:
vacuum polarization (left)
and light by light scattering
(right)
The theoretical calculations in general contain a pure QED part plus the sum of
hadronic and weak contribution terms:
a = a
QED
+ a
hadronic
+ a
weak
=
i
C i (
α
π
)
i
+ a
hadronic
+ a
weak .
(3.90)
The QED part has been computed analytically for i = 1, 2, 3, while for i = 4
there is a numerical calculation with an error (see, for example, [38] and refs
therein). Some terms for i = 5 have also been estimated for the muon case. The
hadronic contribution is from vacuum polarization insertions and from light by light
scattering diagrams (see Fig. 3.10). The weak contribution is from W or Z exchange.
For the electron case the weak contribution is essentially negligible and the
hadronic term (a hadronic
e
∼ (16.71 ± 0.19) . 10 −13 ) does not introduce an important
uncertainty. As a result this measurement can be used to obtain the most precise
determination of the fine structure constant [37]:
α
−1
∼ 137.035999710(96) ,
(3.91)
with an uncertainty about 10 times smaller than the previous determination.
However, very recently a theoretical error in the α 4 terms was corrected [39]. As a
result the value of α −1 in Eq. (3.91) is shifted by −6.41180(73) 10 −7 (about 7 σ ’s).
This change has a minor impact in the following discussion of the muon (g − 2).
In the muon case the experimental precision is less by about three orders of
magnitude, but the sensitivity to new physics effects is typically increased by a
factor (m μ /m e ) 2 ∼ 4 . 10 4 (one mass factor arises because the effective operator
needs a chirality flip and the second one is because, by definition, one must factor
out the Bohr magneton e/2m). From the theory side, the QED term (using the value
of α from a e in Eq. (3.91)), and the weak contribution are affected by small errors
and are given by (all theory number are taken here from the review [40])
a
QED
μ
= (116584718.09 ± 1.6)
. 10
−11 ,
a
weak
μ
= (154 ± 2.2)
. 10
−11
(3.92)
The dominant ambiguities arise from the hadronic term. The lowest order (LO)
vacuum polarization contribution can be evaluated from the measured cross sections
in e + e − → hadrons at low energy via dispersion relations (the largest contribution
is from the ππ final state), with the result a LO.
μ 10 −11 = 6909 ± 44. The higher
59
Fig. 3.10 The hadronic
contributions to the
anomalous magnetic moment:
vacuum polarization (left)
and light by light scattering
(right)
The theoretical calculations in general contain a pure QED part plus the sum of
hadronic and weak contribution terms:
a = a
QED
+ a
hadronic
+ a
weak
=
i
C i (
α
π
)
i
+ a
hadronic
+ a
weak .
(3.90)
The QED part has been computed analytically for i = 1, 2, 3, while for i = 4
there is a numerical calculation with an error (see, for example, [38] and refs
therein). Some terms for i = 5 have also been estimated for the muon case. The
hadronic contribution is from vacuum polarization insertions and from light by light
scattering diagrams (see Fig. 3.10). The weak contribution is from W or Z exchange.
For the electron case the weak contribution is essentially negligible and the
hadronic term (a hadronic
e
∼ (16.71 ± 0.19) . 10 −13 ) does not introduce an important
uncertainty. As a result this measurement can be used to obtain the most precise
determination of the fine structure constant [37]:
α
−1
∼ 137.035999710(96) ,
(3.91)
with an uncertainty about 10 times smaller than the previous determination.
However, very recently a theoretical error in the α 4 terms was corrected [39]. As a
result the value of α −1 in Eq. (3.91) is shifted by −6.41180(73) 10 −7 (about 7 σ ’s).
This change has a minor impact in the following discussion of the muon (g − 2).
In the muon case the experimental precision is less by about three orders of
magnitude, but the sensitivity to new physics effects is typically increased by a
factor (m μ /m e ) 2 ∼ 4 . 10 4 (one mass factor arises because the effective operator
needs a chirality flip and the second one is because, by definition, one must factor
out the Bohr magneton e/2m). From the theory side, the QED term (using the value
of α from a e in Eq. (3.91)), and the weak contribution are affected by small errors
and are given by (all theory number are taken here from the review [40])
a
QED
μ
= (116584718.09 ± 1.6)
. 10
−11 ,
a
weak
μ
= (154 ± 2.2)
. 10
−11
(3.92)
The dominant ambiguities arise from the hadronic term. The lowest order (LO)
vacuum polarization contribution can be evaluated from the measured cross sections
in e + e − → hadrons at low energy via dispersion relations (the largest contribution
is from the ππ final state), with the result a LO.
μ 10 −11 = 6909 ± 44. The higher