5 QCD on the Lattice
185
Normally one would expect that the chiral Ward identities ensure that the axial
current does not get renormalized. However, this no longer applies if the discretization conflicts with the symmetries of the classical action. This is clearly the case
for Wilson fermions, which break chiral symmetry, such that the resulting shortdistance corrections must be absorbed into a renormalization factor Z A . Similar
considerations apply to the vector current: if the discretization does not preserve
chiral symmetry, current conservation is only guaranteed if the vector current is
suitably renormalized by a factor Z V , which must be considered even in the massless
theory. Unlike the case of the renormalization factor of the pseudoscalar density, Z A
and Z V are scale-independent, i.e. they only depend on the bare coupling g 0 . From
the above discussion it is obvious that perturbative estimates of Z A and Z V are
inadequate in order to compute hadronic matrix elements of the axial and vectors
currents with controlled errors. A non-perturbative determination of Z A and Z V can
be achieved by imposing the chiral Ward identities as a renormalization condition.
Two widely used intermediate schemes, namely the Schrödinger functional (SF)
and the Regularization independent momentum subtraction (RI/MOM) schemes are
briefly reviewed in the following. We strongly recommend that the reader consult
the original articles (Refs. [72–75] for the SF, and [76] for RI/MOM) for further
details.
5.5.2 Finite Volume Scheme: The Schrödinger Functional
The Schrödinger functional is based on the formulation of QCD in a finite volume
of size L 3 ·T —regardless of whether space-time is discretized or not—with suitable
boundary conditions. Assuming that lattice regularization is employed, one imposes
periodic boundary conditions on the fields in all spatial directions, while Dirichlet
boundary conditions are imposed at Euclidean times x 0 = 0 and x 0 = T . In order
to make this more precise, let C and C denote classical configurations of the gauge
potential. For the link variables at the temporal boundaries one then imposes
U k (x)| x 0 =0 = e
aC ,
U k (x)| x 0 =T = e
aC
.
(5.121)
In other words, the links assume prescribed values at the temporal boundaries, but
remain unconstrained in the bulk (see Fig. 5.13).
Quark fields are easily incorporated into the formalism. Since the Dirac equation
is first order, only two components of a full Dirac spinor can be fixed at the
boundaries. By defining the projection operator P ± =
1
2 (1 ± γ 0 ), one requires that
the quark fields at the boundaries satisfy
P + ψ(x)| x 0 =0 = ρ( x), P − ψ(x)| x 0 =T = ρ
( x),
¯
ψ(x)P −
x 0 =0
= ¯
ρ( x),
¯
ψ(x)P +
x 0 =T
= ¯
ρ
( x),
(5.122)
185
Normally one would expect that the chiral Ward identities ensure that the axial
current does not get renormalized. However, this no longer applies if the discretization conflicts with the symmetries of the classical action. This is clearly the case
for Wilson fermions, which break chiral symmetry, such that the resulting shortdistance corrections must be absorbed into a renormalization factor Z A . Similar
considerations apply to the vector current: if the discretization does not preserve
chiral symmetry, current conservation is only guaranteed if the vector current is
suitably renormalized by a factor Z V , which must be considered even in the massless
theory. Unlike the case of the renormalization factor of the pseudoscalar density, Z A
and Z V are scale-independent, i.e. they only depend on the bare coupling g 0 . From
the above discussion it is obvious that perturbative estimates of Z A and Z V are
inadequate in order to compute hadronic matrix elements of the axial and vectors
currents with controlled errors. A non-perturbative determination of Z A and Z V can
be achieved by imposing the chiral Ward identities as a renormalization condition.
Two widely used intermediate schemes, namely the Schrödinger functional (SF)
and the Regularization independent momentum subtraction (RI/MOM) schemes are
briefly reviewed in the following. We strongly recommend that the reader consult
the original articles (Refs. [72–75] for the SF, and [76] for RI/MOM) for further
details.
5.5.2 Finite Volume Scheme: The Schrödinger Functional
The Schrödinger functional is based on the formulation of QCD in a finite volume
of size L 3 ·T —regardless of whether space-time is discretized or not—with suitable
boundary conditions. Assuming that lattice regularization is employed, one imposes
periodic boundary conditions on the fields in all spatial directions, while Dirichlet
boundary conditions are imposed at Euclidean times x 0 = 0 and x 0 = T . In order
to make this more precise, let C and C denote classical configurations of the gauge
potential. For the link variables at the temporal boundaries one then imposes
U k (x)| x 0 =0 = e
aC ,
U k (x)| x 0 =T = e
aC
.
(5.121)
In other words, the links assume prescribed values at the temporal boundaries, but
remain unconstrained in the bulk (see Fig. 5.13).
Quark fields are easily incorporated into the formalism. Since the Dirac equation
is first order, only two components of a full Dirac spinor can be fixed at the
boundaries. By defining the projection operator P ± =
1
2 (1 ± γ 0 ), one requires that
the quark fields at the boundaries satisfy
P + ψ(x)| x 0 =0 = ρ( x), P − ψ(x)| x 0 =T = ρ
( x),
¯
ψ(x)P −
x 0 =0
= ¯
ρ( x),
¯
ψ(x)P +
x 0 =T
= ¯
ρ
( x),
(5.122)