5 QCD on the Lattice
187
which are shown schematically in the middle and right panels of Fig. 5.13. In the
above expressions, the Pauli matrices act on the first two flavour components of the
fields.
The specific boundary conditions of the Schrödinger functional ensure that the
Dirac operator has a minimum eigenvalue proportional to 1/T in the massless case
[73]. As a consequence, renormalization conditions can be imposed at vanishing
quark mass. If the aspect ratio T /L is set to some fixed value, the spatial length L is
the only scale in the theory, and thus the masses and couplings in the SF scheme run
with the box size. The recursive finite-size scaling study described below can then
be used to map out the scale dependence of running quantities non-perturbatively
from low to high energies. It is important to realize that in this way the relevant
scale for the RG running (the box size L) is decoupled from the regularization scale
(the lattice cutoff a). It is this features which ensures that the running of masses and
couplings can be obtained in the continuum limit.
Let us now return to our earlier example of the renormalization of quark masses.
The transition from lattice regularization and the associated hadronic scheme to the
SF scheme is achieved by computing the scale-dependent renormalization factor
which links the pseudoscalar density in the intermediate scheme to the bare one, i.e.
(¯ sγ 5 u) SF (μ 0 ) = Z P (g 0 , aμ 0 ) (¯ sγ 5 u) lat (a).
(5.126)
A renormalization condition that defines Z P can be formulated in terms of SF
correlation functions:
Z P (g 0 , aμ 0 ) = c
√
f 1
f P (x 0 )
x 0 =T /2
,
μ 0 = 1/L max ,
(5.127)
where the constant c must be chosen such that Z P = 1 in the free theory. In order
to determine the RG running of the quark mass non-perturbatively one can perform
a sequence of finite-size scaling steps, as illustrated in Fig. 5.14. To this end one
simulates pairs of lattices with box lengths L and 2L, at fixed lattice spacing a. The
ratio of Z P evaluated for each box size yields the ratio ¯
m SF (L)/ ¯
m SF (2L) (upper
horizontal step in Fig. 5.14), which amounts to the change in the quark mass when
the volume is scaled by a factor 2. In a subsequent step, the physical volume can be
doubled once more, which gives ¯
m SF (2L)/ ¯
m SF (4L). The important point to realize
is that the lattice spacing can be adjusted for a given physical box size. In this way
the number of lattice sites can be kept at a manageable level, while the physical
volume is gradually scaled over several orders of magnitude, as indicated by the
zig-zag pattern in Fig. 5.14. Furthermore, each horizontal step can be performed for
several lattice resolutions, so that the continuum limit can be taken. By contrast,
if one attempted to scale the physical volume for fixed lattice spacing, one would,
after only a few iterations, end up with systems so large that they would not fit into
any computer’s memory.
187
which are shown schematically in the middle and right panels of Fig. 5.13. In the
above expressions, the Pauli matrices act on the first two flavour components of the
fields.
The specific boundary conditions of the Schrödinger functional ensure that the
Dirac operator has a minimum eigenvalue proportional to 1/T in the massless case
[73]. As a consequence, renormalization conditions can be imposed at vanishing
quark mass. If the aspect ratio T /L is set to some fixed value, the spatial length L is
the only scale in the theory, and thus the masses and couplings in the SF scheme run
with the box size. The recursive finite-size scaling study described below can then
be used to map out the scale dependence of running quantities non-perturbatively
from low to high energies. It is important to realize that in this way the relevant
scale for the RG running (the box size L) is decoupled from the regularization scale
(the lattice cutoff a). It is this features which ensures that the running of masses and
couplings can be obtained in the continuum limit.
Let us now return to our earlier example of the renormalization of quark masses.
The transition from lattice regularization and the associated hadronic scheme to the
SF scheme is achieved by computing the scale-dependent renormalization factor
which links the pseudoscalar density in the intermediate scheme to the bare one, i.e.
(¯ sγ 5 u) SF (μ 0 ) = Z P (g 0 , aμ 0 ) (¯ sγ 5 u) lat (a).
(5.126)
A renormalization condition that defines Z P can be formulated in terms of SF
correlation functions:
Z P (g 0 , aμ 0 ) = c
√
f 1
f P (x 0 )
x 0 =T /2
,
μ 0 = 1/L max ,
(5.127)
where the constant c must be chosen such that Z P = 1 in the free theory. In order
to determine the RG running of the quark mass non-perturbatively one can perform
a sequence of finite-size scaling steps, as illustrated in Fig. 5.14. To this end one
simulates pairs of lattices with box lengths L and 2L, at fixed lattice spacing a. The
ratio of Z P evaluated for each box size yields the ratio ¯
m SF (L)/ ¯
m SF (2L) (upper
horizontal step in Fig. 5.14), which amounts to the change in the quark mass when
the volume is scaled by a factor 2. In a subsequent step, the physical volume can be
doubled once more, which gives ¯
m SF (2L)/ ¯
m SF (4L). The important point to realize
is that the lattice spacing can be adjusted for a given physical box size. In this way
the number of lattice sites can be kept at a manageable level, while the physical
volume is gradually scaled over several orders of magnitude, as indicated by the
zig-zag pattern in Fig. 5.14. Furthermore, each horizontal step can be performed for
several lattice resolutions, so that the continuum limit can be taken. By contrast,
if one attempted to scale the physical volume for fixed lattice spacing, one would,
after only a few iterations, end up with systems so large that they would not fit into
any computer’s memory.