248
H. Wittig
The method has yet to produce explicit estimates for a hlbl
μ . A variant was proposed
by RBC/UKQCD in Ref. [525]. Another project of the Mainz group has focussed on
the forward light-by-light scattering amplitude, which can be linked via the optical
theorem and dispersive sum rules to models of the cross section for the process
γ ∗ γ ∗ → hadrons [526, 527]. The results provide an important test for model
estimates of a hlbl
μ .
Finally, lattice QCD calculations can also be used to directly test model estimates
of the expected dominant contribution to a hlbl
μ from the pion pole, which requires
knowledge of the transition form factor for π 0 → γ ∗ γ ∗ . The calculation of Ref.
[528], which was performed in two-flavour QCD, gives
(a
hlbl
μ )
π 0 = (65.0 ± 8.3) · 10
−11
(5.259)
which is in very good agreement with model estimates [491]. It will be interesting
to extend this calculation by including the corresponding contributions of the η and
η mesons.
This brief survey demonstrates that lattice QCD contributes in many different
and complementary ways to constrain the hadronic contributions to the muon g − 2
more precisely.
5.9.6 Concluding Remarks
In this short review we have charted the progress of lattice QCD calculations over
more than a decade, i.e. since the publication of the original review article. Back in
2007, lattice QCD was on the verge of providing estimates for hadronic observables
from first principles, which were of immediate phenomenological relevance. In the
meantime, lattice QCD has become an indispensable tool in particle and hadron
physics: In addition to to providing accurate estimates of SM parameters and input
quantities for analyses in flavour physics, lattice QCD is now also making inroads
into field such as nucleon structure and precision observables. This underlines the
important role of lattice calculations for exploring the limits of the SM and searches
for new physics.
Furthermore, studying hadronic interactions, i.e. the physics of resonances and
multi-hadron systems, has become a major activity in lattice QCD and also serves
as a basis for the understanding of light nuclei from first principles. Other important
applications of the lattice formulation that have not been covered in this article
are studies of matter under extreme conditions. Indeed, many features of the
QCD phase diagram and properties of the quark-gluon plasma that are otherwise
inaccessible can nowadays be obtained reliably from lattice calculations. Perhaps
the most significant development since Ken Wilson’s 1989 remark, quoted in the
introduction, is the fact that there is now a vigorous interaction between lattice QCD
and experiment.
H. Wittig
The method has yet to produce explicit estimates for a hlbl
μ . A variant was proposed
by RBC/UKQCD in Ref. [525]. Another project of the Mainz group has focussed on
the forward light-by-light scattering amplitude, which can be linked via the optical
theorem and dispersive sum rules to models of the cross section for the process
γ ∗ γ ∗ → hadrons [526, 527]. The results provide an important test for model
estimates of a hlbl
μ .
Finally, lattice QCD calculations can also be used to directly test model estimates
of the expected dominant contribution to a hlbl
μ from the pion pole, which requires
knowledge of the transition form factor for π 0 → γ ∗ γ ∗ . The calculation of Ref.
[528], which was performed in two-flavour QCD, gives
(a
hlbl
μ )
π 0 = (65.0 ± 8.3) · 10
−11
(5.259)
which is in very good agreement with model estimates [491]. It will be interesting
to extend this calculation by including the corresponding contributions of the η and
η mesons.
This brief survey demonstrates that lattice QCD contributes in many different
and complementary ways to constrain the hadronic contributions to the muon g − 2
more precisely.
5.9.6 Concluding Remarks
In this short review we have charted the progress of lattice QCD calculations over
more than a decade, i.e. since the publication of the original review article. Back in
2007, lattice QCD was on the verge of providing estimates for hadronic observables
from first principles, which were of immediate phenomenological relevance. In the
meantime, lattice QCD has become an indispensable tool in particle and hadron
physics: In addition to to providing accurate estimates of SM parameters and input
quantities for analyses in flavour physics, lattice QCD is now also making inroads
into field such as nucleon structure and precision observables. This underlines the
important role of lattice calculations for exploring the limits of the SM and searches
for new physics.
Furthermore, studying hadronic interactions, i.e. the physics of resonances and
multi-hadron systems, has become a major activity in lattice QCD and also serves
as a basis for the understanding of light nuclei from first principles. Other important
applications of the lattice formulation that have not been covered in this article
are studies of matter under extreme conditions. Indeed, many features of the
QCD phase diagram and properties of the quark-gluon plasma that are otherwise
inaccessible can nowadays be obtained reliably from lattice calculations. Perhaps
the most significant development since Ken Wilson’s 1989 remark, quoted in the
introduction, is the fact that there is now a vigorous interaction between lattice QCD
and experiment.