157
16.2. Discretization
where / D is the SU(3) c -covariant Dirac derivative. Any addition to D / which
is proportional to the unit 4 × 4 matrix will violate (16.24), and hence break
chiral symmetry. The Lagrangian mass m itself is of this form, and it breaks
chiral symmetry, but ‘softly’ – i.e. in a way that disappears as m goes to zero
(thereby preserving the symmetry in this limit). The Wilson addition (16.21)
also breaks chiral symmetry, but it remains there even as m → 0: it is a ‘hard’
breaking.
This means that in the theory with the Wilson modification (i.e. with
‘Wilson fermions’) fermion mass renormalization will not be protected by the
chiral symmetry, so that large additive renormalizations are possible. This
will require repeated fine-tunings of the bare mass parameters, to bring them
down to the desired small values. And it turns out that this seriously lengthens
the computing time.
Another approach (‘staggered fermions’) was suggested by Kogut and
Susskind (1975), Banks et al. (1976), and Susskind (1977). This essentially
involves distributing the 4 spin degrees of freedom of the Dirac field across
different lattice sites (we shall not need the details). At each site there is now
a one-component fermion, with the colour degrees of freedom, which speeds
the calculations. The 16-fold ‘doubling’ degeneracy can be re-arranged as
four different tastes of 4-component fermions, while retaining enough chiral
symmetry to forbid additive mass renormalizations.
Since the different components of the staggered Dirac field now live on
different sites, they will experience slightly different gauge field interactions.
(These are of course local in the continuum limit, but the point remains true
after discretization, as we shall see in the following section.) These interactions
will mix fields of different tastes, causing new problems, but they can be
suppressed by adding further terms to the action. There is still the 4-fold
degeneracy to get rid of, but a trick is available for that, as we shall explain
in section 16.3.
One might wonder if a lattice theory with fermions could be formulated
such that it both avoids doublers and preserves chiral symmetry. For quite
a long time it was believed that this was not possible – a conclusion which
was essentially the content of the Nielsen–Ninomaya theorem (Nielsen and
Ninomaya 1981a, b, c). But more recently a way was found to formulate chiral
gauge theories with fermions satisfactorily on the lattice at finite spacing a.
The key is to replace the condition (16.24) by the Ginsparg–Wilson (1982)
relation
γ 5 /
/
= /
/
(16.25)
D+ Dγ 5 a Dγ 5 D .
This relation implies (L¨ uscher 1998) that the associated action has an exact
symmetry, with infinitesimal variations proportional to
(
)
1
δψ = γ 5 1 − a /
ψ
D
(16.26)
2 )
(
1
δψ = ψ 1 − a /
γ 5 .
(16.27)
D
2
16.2. Discretization
where / D is the SU(3) c -covariant Dirac derivative. Any addition to D / which
is proportional to the unit 4 × 4 matrix will violate (16.24), and hence break
chiral symmetry. The Lagrangian mass m itself is of this form, and it breaks
chiral symmetry, but ‘softly’ – i.e. in a way that disappears as m goes to zero
(thereby preserving the symmetry in this limit). The Wilson addition (16.21)
also breaks chiral symmetry, but it remains there even as m → 0: it is a ‘hard’
breaking.
This means that in the theory with the Wilson modification (i.e. with
‘Wilson fermions’) fermion mass renormalization will not be protected by the
chiral symmetry, so that large additive renormalizations are possible. This
will require repeated fine-tunings of the bare mass parameters, to bring them
down to the desired small values. And it turns out that this seriously lengthens
the computing time.
Another approach (‘staggered fermions’) was suggested by Kogut and
Susskind (1975), Banks et al. (1976), and Susskind (1977). This essentially
involves distributing the 4 spin degrees of freedom of the Dirac field across
different lattice sites (we shall not need the details). At each site there is now
a one-component fermion, with the colour degrees of freedom, which speeds
the calculations. The 16-fold ‘doubling’ degeneracy can be re-arranged as
four different tastes of 4-component fermions, while retaining enough chiral
symmetry to forbid additive mass renormalizations.
Since the different components of the staggered Dirac field now live on
different sites, they will experience slightly different gauge field interactions.
(These are of course local in the continuum limit, but the point remains true
after discretization, as we shall see in the following section.) These interactions
will mix fields of different tastes, causing new problems, but they can be
suppressed by adding further terms to the action. There is still the 4-fold
degeneracy to get rid of, but a trick is available for that, as we shall explain
in section 16.3.
One might wonder if a lattice theory with fermions could be formulated
such that it both avoids doublers and preserves chiral symmetry. For quite
a long time it was believed that this was not possible – a conclusion which
was essentially the content of the Nielsen–Ninomaya theorem (Nielsen and
Ninomaya 1981a, b, c). But more recently a way was found to formulate chiral
gauge theories with fermions satisfactorily on the lattice at finite spacing a.
The key is to replace the condition (16.24) by the Ginsparg–Wilson (1982)
relation
γ 5 /
/
= /
/
(16.25)
D+ Dγ 5 a Dγ 5 D .
This relation implies (L¨ uscher 1998) that the associated action has an exact
symmetry, with infinitesimal variations proportional to
(
)
1
δψ = γ 5 1 − a /
ψ
D
(16.26)
2 )
(
1
δψ = ψ 1 − a /
γ 5 .
(16.27)
D
2