156
16. Lattice Field Theory, and the Renormalization Group Revisited
to the fermion action in this one-dimensional case, where r is dimensionless.
Evidently this is a second difference, and it would correspond to the term
∫
1
− ra d
3
¯
xdτ ψ(x)(∂
2
τ + ∇
2 )ψ(x)
(16.22)
2
in the four-dimensional continuum action. Note the presence of the lattice
spacing ‘a’ in (16.22), which ensures its disappearance as a → 0. The higher¯
derivative term ψ(∂
2
τ + ∇
2 )ψ has mass dimension 5, and therefore requires a
coupling constant with mass dimension -1, i.e. a length in units ħ = c = 1;
it is, in fact, a non-renormalizable term. However, if we recall the discussion of section 11.8 in volume 1, we would expect it to be suppressed at low
momenta much less than the cut-off π/a. Hence it is natural to see a coupling proportional to a appearing in (16.22). (We shall see in section 16.5.3
how renormalization group ideas provide a different perspective on such nonrenormalizable interactions, classifying them as ‘irrelevant’).
How does the extra term (16.21) help the doubling problem? One easily
finds that it changes the (one-dimensional) inverse propagator to
[
]
E sin(k ν )
γ
1 a
r
i 1
+ m + (1 − cos(k ν1 a)).
(16.23)
a
a
By considering the expansion of the cosine near k ν1 ≈ 0 it can be seen that
the second term disappears in the continuum limit, as expected. However,
for k ν1 ≈ π/a it gives a large term of order
1 which adds to the mass m,
a
effectively banishing the ‘doubled’ state to a very high mass, far from the
physical spectrum.
Unfortunately there is a price to pay. The problem is that, as we learned in
section 12.3.2, the QCD lagrangian has an exact chiral symmetry for massless
quarks. To the extent that m u and m d (and m s , but less so) are small on
a hadronic scale such as Λ MS , we expect chiral symmetry to have important
physical consequences. These will indeed be explored in chapter 18. For the
moment, we note merely that it is important for lattice-based QCD calculations to be able to deal correctly with the light quarks. Now we cannot
simply choose the bare Lagrangian mass parameters to be small, and leave it
at that. In any interacting theory, renormalization effects will cause shifts in
these masses. In a chirally symmetric theory, or one which is chirally symmetric as a fermion mass goes to zero, such a mass shift is proportional to the
fermion mass itself; in particular it does not simply add to the mass. We drew
attention to this fact in the case of the electron mass renormalization in QED,
in section 11.2. So in chirally symmetric theories, mass renormalizations are
‘protected’, in this sense. But the modification (16.21), while avoiding physical fermion doublers, breaks chiral symmetry badly. This can easily be seen
by noting (see (12.154) for example) that the crucial property required for
chiral symmetry to hold is
γ 5 D+ Dγ 5 = 0,
(16.24)
/
/
16. Lattice Field Theory, and the Renormalization Group Revisited
to the fermion action in this one-dimensional case, where r is dimensionless.
Evidently this is a second difference, and it would correspond to the term
∫
1
− ra d
3
¯
xdτ ψ(x)(∂
2
τ + ∇
2 )ψ(x)
(16.22)
2
in the four-dimensional continuum action. Note the presence of the lattice
spacing ‘a’ in (16.22), which ensures its disappearance as a → 0. The higher¯
derivative term ψ(∂
2
τ + ∇
2 )ψ has mass dimension 5, and therefore requires a
coupling constant with mass dimension -1, i.e. a length in units ħ = c = 1;
it is, in fact, a non-renormalizable term. However, if we recall the discussion of section 11.8 in volume 1, we would expect it to be suppressed at low
momenta much less than the cut-off π/a. Hence it is natural to see a coupling proportional to a appearing in (16.22). (We shall see in section 16.5.3
how renormalization group ideas provide a different perspective on such nonrenormalizable interactions, classifying them as ‘irrelevant’).
How does the extra term (16.21) help the doubling problem? One easily
finds that it changes the (one-dimensional) inverse propagator to
[
]
E sin(k ν )
γ
1 a
r
i 1
+ m + (1 − cos(k ν1 a)).
(16.23)
a
a
By considering the expansion of the cosine near k ν1 ≈ 0 it can be seen that
the second term disappears in the continuum limit, as expected. However,
for k ν1 ≈ π/a it gives a large term of order
1 which adds to the mass m,
a
effectively banishing the ‘doubled’ state to a very high mass, far from the
physical spectrum.
Unfortunately there is a price to pay. The problem is that, as we learned in
section 12.3.2, the QCD lagrangian has an exact chiral symmetry for massless
quarks. To the extent that m u and m d (and m s , but less so) are small on
a hadronic scale such as Λ MS , we expect chiral symmetry to have important
physical consequences. These will indeed be explored in chapter 18. For the
moment, we note merely that it is important for lattice-based QCD calculations to be able to deal correctly with the light quarks. Now we cannot
simply choose the bare Lagrangian mass parameters to be small, and leave it
at that. In any interacting theory, renormalization effects will cause shifts in
these masses. In a chirally symmetric theory, or one which is chirally symmetric as a fermion mass goes to zero, such a mass shift is proportional to the
fermion mass itself; in particular it does not simply add to the mass. We drew
attention to this fact in the case of the electron mass renormalization in QED,
in section 11.2. So in chirally symmetric theories, mass renormalizations are
‘protected’, in this sense. But the modification (16.21), while avoiding physical fermion doublers, breaks chiral symmetry badly. This can easily be seen
by noting (see (12.154) for example) that the crucial property required for
chiral symmetry to hold is
γ 5 D+ Dγ 5 = 0,
(16.24)
/
/