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H. Wittig
First numerical investigations via Monte Carlo simulations, focusing in particular
on the confinement mechanism in pure Yang–Mills theory, were carried out around
1980. The following years saw already several valiant attempts to study QCD
numerically, yet it was realized that the available computer power was grossly
inadequate to incorporate the effects of dynamical quarks. It was then that the socalled “quenched approximation” of QCD was proposed as a first step to solving
full QCD numerically. This approximation rests on the ad hoc assumption that
the dominant non-perturbative effects are mediated by the gluon field. Hadronic
observables can then be computed on a pure gauge background with far less
numerical effort compared to the real situation where quarks have a feedback on the
gluon field. The main focus of activity during the 1980s was on bosonic theories:
numerical simulations were used to compute the glueball spectrum in pure Yang–
Mills theory. Another important result during this period concerned φ 4 -theory and
the implications of its supposed “triviality” for the Higgs-Yukawa sector of the
Standard Model. Using a combination of analytic and numerical techniques, the
triviality of φ 4 theory could be rigorously established.
Except for a brief spell of activity around the turn of the decade to simulate
QCD with dynamical fermions, most projects in the 1990s were devoted to explore
quenched QCD. Having recognized that the available computers and the efficiency
of known algorithms were by far not sufficient to perform “realistic” simulations
of QCD with controlled errors, lattice physicists resorted to exploring the quenched
approximation and its limitations for a number of phenomenologically interesting
quantities. Although the systematic error that arises by neglecting dynamical quarks
could not be quantified reliably, many important quantities, such as quark and
hadron masses, the strong coupling constant and weak hadronic matrix elements,
were computed for the first time. One of the icons of that period was surely a
plot of the masses of the lightest hadrons in the continuum limit of quenched
QCD, produced by the CP-PACS Collaboration: their results indicated that the
quenched approximation works surprisingly well (at least for these quantities), since
the computed spectrum agreed with experimental determinations at the level of
10%. Simultaneously, a number of sophisticated techniques have been developed
during the 1990s, thereby helping to control systematic effects, mainly pertaining
to the influence of lattice artefacts, as well as the renormalization of local operators
in the lattice regularized theory and their relation to continuum schemes such as
MS. Perhaps the most significant development at the end of the 1990s was the
clarification of the issue of chiral symmetry and lattice regularization. Following
this work it is now understood under which conditions the lattice formulation is
compatible with chiral symmetry. The importance of this development extends far
beyond QCD and implies new prospects for the non-perturbative study of chiral
gauge theories.
Since 2000 the focus has decidedly shifted from the quenched approximation
to serious attempts to simulate QCD with dynamical quarks, thereby tackling the
biggest remaining systematic uncertainty. Progress in this area has not just been
determined by the vast increase in computer power since the very first Monte Carlo
simulations, but rather by the development of new algorithmic ideas, combined
H. Wittig
First numerical investigations via Monte Carlo simulations, focusing in particular
on the confinement mechanism in pure Yang–Mills theory, were carried out around
1980. The following years saw already several valiant attempts to study QCD
numerically, yet it was realized that the available computer power was grossly
inadequate to incorporate the effects of dynamical quarks. It was then that the socalled “quenched approximation” of QCD was proposed as a first step to solving
full QCD numerically. This approximation rests on the ad hoc assumption that
the dominant non-perturbative effects are mediated by the gluon field. Hadronic
observables can then be computed on a pure gauge background with far less
numerical effort compared to the real situation where quarks have a feedback on the
gluon field. The main focus of activity during the 1980s was on bosonic theories:
numerical simulations were used to compute the glueball spectrum in pure Yang–
Mills theory. Another important result during this period concerned φ 4 -theory and
the implications of its supposed “triviality” for the Higgs-Yukawa sector of the
Standard Model. Using a combination of analytic and numerical techniques, the
triviality of φ 4 theory could be rigorously established.
Except for a brief spell of activity around the turn of the decade to simulate
QCD with dynamical fermions, most projects in the 1990s were devoted to explore
quenched QCD. Having recognized that the available computers and the efficiency
of known algorithms were by far not sufficient to perform “realistic” simulations
of QCD with controlled errors, lattice physicists resorted to exploring the quenched
approximation and its limitations for a number of phenomenologically interesting
quantities. Although the systematic error that arises by neglecting dynamical quarks
could not be quantified reliably, many important quantities, such as quark and
hadron masses, the strong coupling constant and weak hadronic matrix elements,
were computed for the first time. One of the icons of that period was surely a
plot of the masses of the lightest hadrons in the continuum limit of quenched
QCD, produced by the CP-PACS Collaboration: their results indicated that the
quenched approximation works surprisingly well (at least for these quantities), since
the computed spectrum agreed with experimental determinations at the level of
10%. Simultaneously, a number of sophisticated techniques have been developed
during the 1990s, thereby helping to control systematic effects, mainly pertaining
to the influence of lattice artefacts, as well as the renormalization of local operators
in the lattice regularized theory and their relation to continuum schemes such as
MS. Perhaps the most significant development at the end of the 1990s was the
clarification of the issue of chiral symmetry and lattice regularization. Following
this work it is now understood under which conditions the lattice formulation is
compatible with chiral symmetry. The importance of this development extends far
beyond QCD and implies new prospects for the non-perturbative study of chiral
gauge theories.
Since 2000 the focus has decidedly shifted from the quenched approximation
to serious attempts to simulate QCD with dynamical quarks, thereby tackling the
biggest remaining systematic uncertainty. Progress in this area has not just been
determined by the vast increase in computer power since the very first Monte Carlo
simulations, but rather by the development of new algorithmic ideas, combined