170
16. Lattice Field Theory, and the Renormalization Group Revisited
Now we may write
[
]
∏ N f
∑
detM f (U ) = exp
ln detM f (U ) ,
(16.92)
f=1
f
so that the effect of N f fermions is to contribute an additional term
∑
S eff (U ) = −
ln det[M f (U )]
(16.93)
f
to the gluonic action. But although formally correct, this fermionic contribution is computationally very time-consuming to include. Until the mid-1990s
it could not be done, and instead calculations were made using the quenched
approximation, in which the determinant is set equal to a constant independent of the link variables U . This is equivalent to the neglect of closed fermion
loops in a Feynman graph approach, i.e. no vacuum polarization insertions
on virtual gluon lines. Vacuum polarization amplitudes typically behave as
q
2 /m
2
f for q
2
≪ m
2
f , where q is the momentum flowing into the loop (see equation (11.39), for example, in the case of QED). The quenched approximation
is therefore poorer for the light quarks u, d and s.
By the later 1990s it was possible to include the determinant provided the
quark masses were not too small: the computation slowed down seriously for
light quark masses. So calculations were done for unphysically large values of
m u , m d and m s , and the results extrapolated towards the physical values.
Beginning in the early 2000s, however, more precise calculations with substantially lighter quark masses became possible, using the staggered fermion
formulation discussed in section 16.2.2. It will be recalled that this saves a
factor of four in the number of degrees of freedom. But there is still the remaining problem of the four unwanted additional ‘tastes’. If these tastes are
degenerate, as they would be in the continuum limit, then we can use the
simple trick of replacing S eff (U ) by
1 S eff (U ), which means that we take the
4
fourth root of the staggered fermion determinant. The true physical (nondegenerate) quark flavour multiplicity still remains, of course, and we arrive
at
S
= − ln det{M
(U )M
(U )M
(U )}
1/4
eff,stag.
stag. u
stag. d
stag. s
.
(16.94)
Unfortunately, things are not so simple away from the continuum limit, at
finite lattice spacing a. Bernard, Golterman and Shamir (2006) pointed out
that the quantity
{detM
(U )}
1/4
stag.
(16.95)
cannot be represented by a local single-taste theory except in the continuum
limit: at finite a, it represents a non-local single-taste action. Locality is a
very fundamental property of all successful quantum field theories, and its
recovery from (16.95) in the limit a → 0 is not obvious. We refer to Sharpe
(2006) for a full discussion, and further references. Meanwhile, as we shall see
in section 16.6, some of the currently (in 2011) most accurate published results
in lattice QCD are using staggered fermions with the ‘rooting’ procedure.
16. Lattice Field Theory, and the Renormalization Group Revisited
Now we may write
[
]
∏ N f
∑
detM f (U ) = exp
ln detM f (U ) ,
(16.92)
f=1
f
so that the effect of N f fermions is to contribute an additional term
∑
S eff (U ) = −
ln det[M f (U )]
(16.93)
f
to the gluonic action. But although formally correct, this fermionic contribution is computationally very time-consuming to include. Until the mid-1990s
it could not be done, and instead calculations were made using the quenched
approximation, in which the determinant is set equal to a constant independent of the link variables U . This is equivalent to the neglect of closed fermion
loops in a Feynman graph approach, i.e. no vacuum polarization insertions
on virtual gluon lines. Vacuum polarization amplitudes typically behave as
q
2 /m
2
f for q
2
≪ m
2
f , where q is the momentum flowing into the loop (see equation (11.39), for example, in the case of QED). The quenched approximation
is therefore poorer for the light quarks u, d and s.
By the later 1990s it was possible to include the determinant provided the
quark masses were not too small: the computation slowed down seriously for
light quark masses. So calculations were done for unphysically large values of
m u , m d and m s , and the results extrapolated towards the physical values.
Beginning in the early 2000s, however, more precise calculations with substantially lighter quark masses became possible, using the staggered fermion
formulation discussed in section 16.2.2. It will be recalled that this saves a
factor of four in the number of degrees of freedom. But there is still the remaining problem of the four unwanted additional ‘tastes’. If these tastes are
degenerate, as they would be in the continuum limit, then we can use the
simple trick of replacing S eff (U ) by
1 S eff (U ), which means that we take the
4
fourth root of the staggered fermion determinant. The true physical (nondegenerate) quark flavour multiplicity still remains, of course, and we arrive
at
S
= − ln det{M
(U )M
(U )M
(U )}
1/4
eff,stag.
stag. u
stag. d
stag. s
.
(16.94)
Unfortunately, things are not so simple away from the continuum limit, at
finite lattice spacing a. Bernard, Golterman and Shamir (2006) pointed out
that the quantity
{detM
(U )}
1/4
stag.
(16.95)
cannot be represented by a local single-taste theory except in the continuum
limit: at finite a, it represents a non-local single-taste action. Locality is a
very fundamental property of all successful quantum field theories, and its
recovery from (16.95) in the limit a → 0 is not obvious. We refer to Sharpe
(2006) for a full discussion, and further references. Meanwhile, as we shall see
in section 16.6, some of the currently (in 2011) most accurate published results
in lattice QCD are using staggered fermions with the ‘rooting’ procedure.