5 QCD on the Lattice
189
1
10
100
1000
/
)
(
SF scheme, = 2
N f
3 - loop
0.5
0.4
0.3
0.2
0.1
0
Fig. 5.15 Running of α s (left panel) [77] and quark mass in units of the RGI mass M (right panel)
[78] in the SF scheme. The results from simulations (full circles) are compared to the integration
of the perturbative RG equations
5.5.3 Regularization-Independent Momentum Subtraction
Scheme
An alternative choice of intermediate renormalization scheme is based on imposing
renormalization conditions in terms of Green’s functions of external quark states in
momentum space, evaluated in a fixed gauge (e.g. Landau gauge) [76]. The external
quark fields are off-shell, and their virtualities are identified with the momentum
scale. Here we summarize the basic steps in this procedure by considering a quark
bilinear non-singlet operator O = ¯
ψ 1 2 , where denotes a generic Dirac
structure, e.g. = γ 5 in the case of the pseudoscalar density. The corresponding
renormalization factor Z is fixed by requiring that a suitably chosen renormalized
vertex function (p) be equal to its tree-level counterpart:
(p)
p 2 =μ 2 = Z Z
−1
ψ (p)
p 2 =μ 2
= (p).
(5.128)
This condition defines Z up to quark field renormalization. Such a prescription
can be formulated in any chosen regularization, which is why the method is said
to define a regularization-independent momentum subtraction (RI/MOM) scheme.
However, Z does depend on the external states and the gauge.
In order to connect to our previous example of the renormalization of quark
fields, we consider the pseudoscalar density for concreteness: = γ 5 = “P ”. In
this case, P,0 = γ 5 ⊗ 1 colour , and Eq. (5.128) can be cast into the form
Z
MOM
P
(g 0 , aμ) Z
−1
ψ (g 0 , ap)
1
12
Tr { P (p)γ 5 }
p 2 =μ 2
= 1,
(5.129)
where the trace is taken over Dirac and colour indices.
189
1
10
100
1000
/
)
(
SF scheme, = 2
N f
3 - loop
0.5
0.4
0.3
0.2
0.1
0
Fig. 5.15 Running of α s (left panel) [77] and quark mass in units of the RGI mass M (right panel)
[78] in the SF scheme. The results from simulations (full circles) are compared to the integration
of the perturbative RG equations
5.5.3 Regularization-Independent Momentum Subtraction
Scheme
An alternative choice of intermediate renormalization scheme is based on imposing
renormalization conditions in terms of Green’s functions of external quark states in
momentum space, evaluated in a fixed gauge (e.g. Landau gauge) [76]. The external
quark fields are off-shell, and their virtualities are identified with the momentum
scale. Here we summarize the basic steps in this procedure by considering a quark
bilinear non-singlet operator O = ¯
ψ 1 2 , where denotes a generic Dirac
structure, e.g. = γ 5 in the case of the pseudoscalar density. The corresponding
renormalization factor Z is fixed by requiring that a suitably chosen renormalized
vertex function (p) be equal to its tree-level counterpart:
(p)
p 2 =μ 2 = Z Z
−1
ψ (p)
p 2 =μ 2
= (p).
(5.128)
This condition defines Z up to quark field renormalization. Such a prescription
can be formulated in any chosen regularization, which is why the method is said
to define a regularization-independent momentum subtraction (RI/MOM) scheme.
However, Z does depend on the external states and the gauge.
In order to connect to our previous example of the renormalization of quark
fields, we consider the pseudoscalar density for concreteness: = γ 5 = “P ”. In
this case, P,0 = γ 5 ⊗ 1 colour , and Eq. (5.128) can be cast into the form
Z
MOM
P
(g 0 , aμ) Z
−1
ψ (g 0 , ap)
1
12
Tr { P (p)γ 5 }
p 2 =μ 2
= 1,
(5.129)
where the trace is taken over Dirac and colour indices.