5 QCD on the Lattice
169
of this problem was given in [39], but so far no firm conclusion has been reached.
Nevertheless, the probability measure Eq. (5.90) and the “rooting trick” it is based
on, have been employed in large-scale simulations (see, e.g. Ref. [40]).
Discretizations based on twisted mass QCD have also been proposed as a
numerically more efficient quark action. Here, the twisted mass parameter μ q
protects the operator against arbitrarily small eigenvalues. The smallest mass in the
pion channel that has been reached with this formulation was as low as 300 MeV
[41]. This corresponds to a physical quark mass of about m s /5, which may be
sufficient to enter the regime where the quark mass behaviour of observables can
be described analytically using Chiral Perturbation Theory.
Owing to several major algorithmic improvements, simulations based on the
Wilson-Dirac operator can now be performed much more efficiently. Without going
into much detail, we simply state that most of the gain is due to the use of suitably
chosen factorizations of the Wilson-Dirac operator into its low- and high-frequency
parts. The various factors are then “better conditioned”. In particular, fluctuations in
the condition number can be controlled via a separate and optimized treatment of
the low-energy part. In this way the step size τ can be increased whilst keeping
a reasonably high acceptance rate for fixed total trajectory length τ . Algorithmic
implementations of factorization range from Hasenbusch’s “mass preconditioning” [42, 43], Lüscher’s domain decomposition technique based on the Schwarz
Alternating Procedure (DD-HMC algorithm) [44], to factorizations based on mass
preconditioning combined with rational approximations of the contributions from
multiple pseudo-fermion fields [45]. Thanks to these developments, it appears that
the spectral properties of the Wilson-Dirac operator are no longer an obstacle to
the efficient simulation of lattice QCD with light dynamical quarks. At the same
time, large-scale simulations employing the recent algorithmic improvements are
only just starting.
5.3 Hadron Spectroscopy
The determination of the spectrum of hadrons, i.e. mesons, baryons, glueballs,
and possibly “exotic” hadronic states, starting from the underlying gauge theory
of quarks and gluons has traditionally been one of the main applications of
lattice QCD. The rôle of lattice calculations in this context is twofold: first, the
determination of the experimentally known values of hadron masses from first
principles represents a stringent test of QCD. Second, lattice calculations can make
predictions for the masses of undiscovered or poorly established states. For instance,
lattice results have been instrumental in the search for glueball candidates, and have
also contributed significantly to the debate on the existence of pentaquarks.
The principles of hadronic mass calculations have already been outlined at the
end of Sect. 5.2.3: After defining a suitable interpolating operator with the quantum
numbers of the desired hadronic channel, one computes its Euclidean two-point
function. The mass (energy) of the ground state in that channel is then extracted
169
of this problem was given in [39], but so far no firm conclusion has been reached.
Nevertheless, the probability measure Eq. (5.90) and the “rooting trick” it is based
on, have been employed in large-scale simulations (see, e.g. Ref. [40]).
Discretizations based on twisted mass QCD have also been proposed as a
numerically more efficient quark action. Here, the twisted mass parameter μ q
protects the operator against arbitrarily small eigenvalues. The smallest mass in the
pion channel that has been reached with this formulation was as low as 300 MeV
[41]. This corresponds to a physical quark mass of about m s /5, which may be
sufficient to enter the regime where the quark mass behaviour of observables can
be described analytically using Chiral Perturbation Theory.
Owing to several major algorithmic improvements, simulations based on the
Wilson-Dirac operator can now be performed much more efficiently. Without going
into much detail, we simply state that most of the gain is due to the use of suitably
chosen factorizations of the Wilson-Dirac operator into its low- and high-frequency
parts. The various factors are then “better conditioned”. In particular, fluctuations in
the condition number can be controlled via a separate and optimized treatment of
the low-energy part. In this way the step size τ can be increased whilst keeping
a reasonably high acceptance rate for fixed total trajectory length τ . Algorithmic
implementations of factorization range from Hasenbusch’s “mass preconditioning” [42, 43], Lüscher’s domain decomposition technique based on the Schwarz
Alternating Procedure (DD-HMC algorithm) [44], to factorizations based on mass
preconditioning combined with rational approximations of the contributions from
multiple pseudo-fermion fields [45]. Thanks to these developments, it appears that
the spectral properties of the Wilson-Dirac operator are no longer an obstacle to
the efficient simulation of lattice QCD with light dynamical quarks. At the same
time, large-scale simulations employing the recent algorithmic improvements are
only just starting.
5.3 Hadron Spectroscopy
The determination of the spectrum of hadrons, i.e. mesons, baryons, glueballs,
and possibly “exotic” hadronic states, starting from the underlying gauge theory
of quarks and gluons has traditionally been one of the main applications of
lattice QCD. The rôle of lattice calculations in this context is twofold: first, the
determination of the experimentally known values of hadron masses from first
principles represents a stringent test of QCD. Second, lattice calculations can make
predictions for the masses of undiscovered or poorly established states. For instance,
lattice results have been instrumental in the search for glueball candidates, and have
also contributed significantly to the debate on the existence of pentaquarks.
The principles of hadronic mass calculations have already been outlined at the
end of Sect. 5.2.3: After defining a suitable interpolating operator with the quantum
numbers of the desired hadronic channel, one computes its Euclidean two-point
function. The mass (energy) of the ground state in that channel is then extracted