5 QCD on the Lattice
155
Equation (5.55) requires some further explanation:
•
denotes the representation of in the (effective) theory, where the quark fields
have been integrated out and only the link variables remain in the functional
integral measure;
• D lat denotes a generic, massive lattice Dirac operator. For instance, for Wilson
quarks one has D lat = D w +m 0 . For simplicity we have displayed the expression
for QCD with N f flavours of equal mass m 0 , which accounts for the power N f .
In the case of non-degenerate quarks {det D lat } N f must be replaced by a product
of determinants, in which each factor represents the contribution from a single
flavour:
• The lattice formulation has given a well-defined meaning to the measure D[U ].
The integration over the gauge degrees of freedom reduces to a finite-dimensional
integration over the gauge group, based on the invariant group (Haar) measure.
The numerical evaluation of via Monte Carlo integration proceeds as follows.
One starts by generating a set of gauge configurations using a computer program.
One configuration in the set represents the collection of all link variables on a given
lattice, i.e.
U μ (x) |x ∈ E , μ = 0, . . . , 3
,
(5.56)
for which we shall use the shorthand {U μ (x)} below. A collection of an infinite
number of configurations is called an ensemble. The statistical weight, W , of an
individual configuration is given by
W = {det D lat }
N f e
−S G [U ] .
(5.57)
In other words, the composition of the ensemble is determined by a probability
distribution, which is given by the negative exponentiated classical action in the integrand of the Euclidean functional integral. Owing to the weight factor, the integrand
of the functional integral will be strongly peaked around those configurations for
which W is large. This particular feature makes the expectation value amenable to a
Monte Carlo treatment. The key idea is to replace the ensemble by a finite sample of
N cfg gauge configurations, which is dominated by those configurations for which W
is large. Provided that one can construct a suitable algorithm, the sample will then
consist predominantly of those configurations which give a large contribution to the
Euclidean functional integral and thus . Such a procedure is called importance
sampling.
Technically, the sample is produced by generating a sequence of configurations
via a Markov process:
U μ (x)
1
−→
U μ (x)
2
−→ . . . −→
U μ (x)
N cfg
.
(5.58)
155
Equation (5.55) requires some further explanation:
•
denotes the representation of in the (effective) theory, where the quark fields
have been integrated out and only the link variables remain in the functional
integral measure;
• D lat denotes a generic, massive lattice Dirac operator. For instance, for Wilson
quarks one has D lat = D w +m 0 . For simplicity we have displayed the expression
for QCD with N f flavours of equal mass m 0 , which accounts for the power N f .
In the case of non-degenerate quarks {det D lat } N f must be replaced by a product
of determinants, in which each factor represents the contribution from a single
flavour:
• The lattice formulation has given a well-defined meaning to the measure D[U ].
The integration over the gauge degrees of freedom reduces to a finite-dimensional
integration over the gauge group, based on the invariant group (Haar) measure.
The numerical evaluation of via Monte Carlo integration proceeds as follows.
One starts by generating a set of gauge configurations using a computer program.
One configuration in the set represents the collection of all link variables on a given
lattice, i.e.
U μ (x) |x ∈ E , μ = 0, . . . , 3
,
(5.56)
for which we shall use the shorthand {U μ (x)} below. A collection of an infinite
number of configurations is called an ensemble. The statistical weight, W , of an
individual configuration is given by
W = {det D lat }
N f e
−S G [U ] .
(5.57)
In other words, the composition of the ensemble is determined by a probability
distribution, which is given by the negative exponentiated classical action in the integrand of the Euclidean functional integral. Owing to the weight factor, the integrand
of the functional integral will be strongly peaked around those configurations for
which W is large. This particular feature makes the expectation value amenable to a
Monte Carlo treatment. The key idea is to replace the ensemble by a finite sample of
N cfg gauge configurations, which is dominated by those configurations for which W
is large. Provided that one can construct a suitable algorithm, the sample will then
consist predominantly of those configurations which give a large contribution to the
Euclidean functional integral and thus . Such a procedure is called importance
sampling.
Technically, the sample is produced by generating a sequence of configurations
via a Markov process:
U μ (x)
1
−→
U μ (x)
2
−→ . . . −→
U μ (x)
N cfg
.
(5.58)