5 QCD on the Lattice
245
weak and the strong interactions, i.e.
a
SM
μ = a
QED
μ
+ a
weak
μ
+ a
strong
μ
.
(5.257)
While QED effects account for about 99.994% of the absolute value of a SM
μ , its
total uncertainty is completely dominated by the contribution from a
strong
μ
. Since the
latter is mostly due to hadronic effects that are intrinsically non-perturbative, it is
clear that special attention must be paid to their reliable evaluation.
The most important quantum corrections to a SM
μ arising from strong interaction
physics are the leading hadronic vacuum polarization (HVP) and hadronic light-bylight scattering (HLbL) contributions. The HVP contribution, a
hvp
μ , which arises at
order α 2 (where α is the fine structure constant), can be expressed in terms of a dispersion integral of the cross section ratio R(s) = σ (e + e − → hadrons)/σ (e + e − →
μ + μ − ), multiplied by a known kernel function. At small values of the centre-ofmass energy s, the dispersion integral is evaluated using experimental data for the
R-ratio R(s) as input [476–480]. For instance, the recent analysis of Ref. [479],
which is based on the available data for e + e − → hadrons, produced an estimate of
a
hvp
μ = (693.1 ± 3.4) · 10 −10 . While the total error is at the level of 0.5%, it is clear
that experimental uncertainties enter the SM prediction for a μ in this approach.
The HLbL contribution has been quantified mostly using hadronic models,
although efforts are under way to formulate and apply a dispersive or data-driven
framework to treat some of the dominant sub-processes [481–491]. The current SM
estimate a SM
μ is based on model calculations such as the “Glasgow consensus”, i.e.
a hlbl
μ
= (105 ± 26) · 10 −11 [492]. Other studies, which have produced consistent
results, can be found in Refs. [474, 478, 493].
Given the importance of a μ for testing the limits of the SM, it is crucial to verify
the current estimates of a
hvp
μ and a hlbl
μ and possibly reduce their overall errors using
an ab initio approach such as lattice QCD. Given that two new experiments (E989 at
Fermilab and E34 at J-PARC) are set to improve the precision of the measurement of
a μ by a factor four, the importance of reliably estimating the hadronic contributions
has become even higher. In order to make an impact, lattice QCD must be able to
constrain a
hvp
μ with sub-percent accuracy, while an estimate of a hlbl
μ
at the level
of 10% would already be a major step forward. Both tasks, however, present a
considerable challenge to lattice QCD. The current status of lattice calculations of
a
hvp
μ and a hlbl
μ was reviewed extensively in Ref. [494], which can be consulted for
details. Here we present merely an overview of the main issues and a guide to the
literature.
The hadronic vacuum polarization contribution, a
hvp
μ , is accessible in lattice QCD
via different integral representations involving the correlator of the electromagnetic
current. The first possibility is to consider a convolution integral over Euclidean
momenta Q 2 of the subtracted vacuum polarization function [500, 501]. The second
possibility is the so-called time-momentum representation defined in Ref. [502], in
which the product of the spatially summed vector correlator G(x 0 ) and a kernel
245
weak and the strong interactions, i.e.
a
SM
μ = a
QED
μ
+ a
weak
μ
+ a
strong
μ
.
(5.257)
While QED effects account for about 99.994% of the absolute value of a SM
μ , its
total uncertainty is completely dominated by the contribution from a
strong
μ
. Since the
latter is mostly due to hadronic effects that are intrinsically non-perturbative, it is
clear that special attention must be paid to their reliable evaluation.
The most important quantum corrections to a SM
μ arising from strong interaction
physics are the leading hadronic vacuum polarization (HVP) and hadronic light-bylight scattering (HLbL) contributions. The HVP contribution, a
hvp
μ , which arises at
order α 2 (where α is the fine structure constant), can be expressed in terms of a dispersion integral of the cross section ratio R(s) = σ (e + e − → hadrons)/σ (e + e − →
μ + μ − ), multiplied by a known kernel function. At small values of the centre-ofmass energy s, the dispersion integral is evaluated using experimental data for the
R-ratio R(s) as input [476–480]. For instance, the recent analysis of Ref. [479],
which is based on the available data for e + e − → hadrons, produced an estimate of
a
hvp
μ = (693.1 ± 3.4) · 10 −10 . While the total error is at the level of 0.5%, it is clear
that experimental uncertainties enter the SM prediction for a μ in this approach.
The HLbL contribution has been quantified mostly using hadronic models,
although efforts are under way to formulate and apply a dispersive or data-driven
framework to treat some of the dominant sub-processes [481–491]. The current SM
estimate a SM
μ is based on model calculations such as the “Glasgow consensus”, i.e.
a hlbl
μ
= (105 ± 26) · 10 −11 [492]. Other studies, which have produced consistent
results, can be found in Refs. [474, 478, 493].
Given the importance of a μ for testing the limits of the SM, it is crucial to verify
the current estimates of a
hvp
μ and a hlbl
μ and possibly reduce their overall errors using
an ab initio approach such as lattice QCD. Given that two new experiments (E989 at
Fermilab and E34 at J-PARC) are set to improve the precision of the measurement of
a μ by a factor four, the importance of reliably estimating the hadronic contributions
has become even higher. In order to make an impact, lattice QCD must be able to
constrain a
hvp
μ with sub-percent accuracy, while an estimate of a hlbl
μ
at the level
of 10% would already be a major step forward. Both tasks, however, present a
considerable challenge to lattice QCD. The current status of lattice calculations of
a
hvp
μ and a hlbl
μ was reviewed extensively in Ref. [494], which can be consulted for
details. Here we present merely an overview of the main issues and a guide to the
literature.
The hadronic vacuum polarization contribution, a
hvp
μ , is accessible in lattice QCD
via different integral representations involving the correlator of the electromagnetic
current. The first possibility is to consider a convolution integral over Euclidean
momenta Q 2 of the subtracted vacuum polarization function [500, 501]. The second
possibility is the so-called time-momentum representation defined in Ref. [502], in
which the product of the spatially summed vector correlator G(x 0 ) and a kernel