5 QCD on the Lattice
233
Simulations with light dynamical quarks, whose masses correspond to the physical
value of the pion mass, have become the state of the art, and the effects of
dynamical strange and charm quarks are now routinely included as well. In fact,
lattice calculations of certain observables have reached (or are aiming for) a level
of precision where the effects of the breaking of isospin symmetry can no longer
be ignored. This necessitates that lattice QCD must account not only for the effects
of unequal u and d quark masses but also for corrections due to electromagnetism,
owing to the different electric charges of up- and down-type quarks.
In this context it is interesting to quote a remark by Ken Wilson, made at the
1989 International Conference on Lattice Field Theory [242]: “I still believe that an
extraordinary increase in computing power (10 8 is I think not enough) and equally
powerful algorithmic advances will be necessary before a full interaction with
experiment takes place.” Given that, in 1989, the most powerful supercomputers
could sustain 10 GFlops (i.e. 10 10 floating point operations per second), Wilson’s
estimate was tantamount to requiring ExaFlops capabilities (10 18 Flops) for lattice
QCD to make an impact, a performance figure that has only been reached very
recently by less than a handful of machines. The enormous progress that the field
of lattice QCD has already seen over the past decade proves that Wilson’s view was
far too pessimistic. 21 For instance, results from lattice calculations for the decay
constants and form factors of mesons and baryons containing heavy quarks are vital
input for global analyses of observables in flavour physics, designed to constrain the
elements of the Cabibbo–Kobayashi–Maskawa matrix. Furthermore, lattice QCD
yields precise values for the masses of the light (u, d, s) quarks [244].
An impressive testimony to the importance of lattice QCD for the entire field
of particle physics is the regular report provided by the Flavour Lattice Averaging
Group (FLAG). Since its inception in 2007, FLAG has been charting the progress
in lattice QCD, by collecting results for a range of phenomenologically relevant
quantities. Taking inspiration from the Particle Data Group, FLAG assesses the
quality of individual calculations and produces world averages by combining those
results that satisfy a defined set of requirements regarding the overall control over
systematic effects. Three editions of the FLAG report, published in 2010 [245],
2013 [246] and 2016 [247], have appeared until now, and a fourth one has been
published in 2019 [248]. In fact, the current status of lattice calculations of many
observables that have been reviewed in the first edition of this article can be found
in these comprehensive reports.
This short review is organized as follows. In Sects. 5.9.2 and 5.9.3 we give
an update of lattice calculations applied to hadron spectroscopy, weak hadronic
matrix elements and the determination of Standard Model parameters such as quark
masses and the strong coupling constant. These quantities were covered extensively
in the original edition of [241]. Then, in Sect. 5.9.4 we extend the discussion to
the determination of quantities that describe structural and other properties of the
nucleon, such as form factors and the axial charge. Finally, in Sect. 5.9.5 we discuss
lattice calculations of the hadronic contributions to the muon anomalous magnetic
21 Even Wilson himself acknowledged, at least partially, that this was the case [243].
233
Simulations with light dynamical quarks, whose masses correspond to the physical
value of the pion mass, have become the state of the art, and the effects of
dynamical strange and charm quarks are now routinely included as well. In fact,
lattice calculations of certain observables have reached (or are aiming for) a level
of precision where the effects of the breaking of isospin symmetry can no longer
be ignored. This necessitates that lattice QCD must account not only for the effects
of unequal u and d quark masses but also for corrections due to electromagnetism,
owing to the different electric charges of up- and down-type quarks.
In this context it is interesting to quote a remark by Ken Wilson, made at the
1989 International Conference on Lattice Field Theory [242]: “I still believe that an
extraordinary increase in computing power (10 8 is I think not enough) and equally
powerful algorithmic advances will be necessary before a full interaction with
experiment takes place.” Given that, in 1989, the most powerful supercomputers
could sustain 10 GFlops (i.e. 10 10 floating point operations per second), Wilson’s
estimate was tantamount to requiring ExaFlops capabilities (10 18 Flops) for lattice
QCD to make an impact, a performance figure that has only been reached very
recently by less than a handful of machines. The enormous progress that the field
of lattice QCD has already seen over the past decade proves that Wilson’s view was
far too pessimistic. 21 For instance, results from lattice calculations for the decay
constants and form factors of mesons and baryons containing heavy quarks are vital
input for global analyses of observables in flavour physics, designed to constrain the
elements of the Cabibbo–Kobayashi–Maskawa matrix. Furthermore, lattice QCD
yields precise values for the masses of the light (u, d, s) quarks [244].
An impressive testimony to the importance of lattice QCD for the entire field
of particle physics is the regular report provided by the Flavour Lattice Averaging
Group (FLAG). Since its inception in 2007, FLAG has been charting the progress
in lattice QCD, by collecting results for a range of phenomenologically relevant
quantities. Taking inspiration from the Particle Data Group, FLAG assesses the
quality of individual calculations and produces world averages by combining those
results that satisfy a defined set of requirements regarding the overall control over
systematic effects. Three editions of the FLAG report, published in 2010 [245],
2013 [246] and 2016 [247], have appeared until now, and a fourth one has been
published in 2019 [248]. In fact, the current status of lattice calculations of many
observables that have been reviewed in the first edition of this article can be found
in these comprehensive reports.
This short review is organized as follows. In Sects. 5.9.2 and 5.9.3 we give
an update of lattice calculations applied to hadron spectroscopy, weak hadronic
matrix elements and the determination of Standard Model parameters such as quark
masses and the strong coupling constant. These quantities were covered extensively
in the original edition of [241]. Then, in Sect. 5.9.4 we extend the discussion to
the determination of quantities that describe structural and other properties of the
nucleon, such as form factors and the axial charge. Finally, in Sect. 5.9.5 we discuss
lattice calculations of the hadronic contributions to the muon anomalous magnetic
21 Even Wilson himself acknowledged, at least partially, that this was the case [243].