182
16. Lattice Field Theory, and the Renormalization Group Revisited
that corrections to continuum theory results stemming from finite lattice spacing could be diminished systematically by the use of lattice actions that also
include suitable irrelevant terms. This procedure is routinely adopted in accurate lattice calculations with ‘Symanzik-improved’ actions.
One further word should be said about terms such as ‘m
2 φ
2 ’ (which arise
in the Higgs sector of the Standard Model, for instance). As we have seen,
m
2 scales by m
′ 2 = m
2 f
2 , which is a rapid growth with f . If we imagine
starting at a very high scale, such as 10
15 TeV and flowing down to 1 TeV,
then the ‘initial’ value of m will have to be very finely ‘tuned’ in order to end
up with a mass of order 1 TeV. Thus, in this picture, it seems unnatural to
have scalar particles with masses much less than the physical cut-off scale,
unless some symmetry principle ‘protects’ their light masses. We shall return
to this problem in section 22.8.1.
We now return to lattice QCD, with a brief survey of some of the impressive
results now being obtained numerically.
16.5
Lattice QCD
16.5.1 Introduction, and the continuum limit
Let us begin by considering some numbers. The lattice must be large enough
so that the spatial dimension R of the object we wish to describe – say the size
of a hadron – fits comfortably inside it, otherwise the result will be subject
to ‘finite size effects’ as the hypercube side length L is varied. We also need
R ≫ a, or else the granularity of the lattice resolution will become apparent.
Further, as indicated earlier, we expect the mass m (which is of order R
−1 )
to be very much less than a
−1 . Thus ideally we need
a ≪ R ∼ 1/m ≪ L = N a
(16.133)
so that N must be large. For example, if N = 64 and a ∼ 0.1fm the condition
(16.133) would be reasonably satisfied by a light hadron mass. But remember
that each field at each lattice point is an independent degree of freedom: dealing with integrals such as (16.87) presents a formidable numerical challenge.
Ignoring any statistical inaccuracy, the results will depend on the parameters g L and N , where g L is the bare lattice gauge coupling (we assume for
simplicity that the quarks are massless). Despite the fact that g L is dimensionless, we shall now see that its value actually controls the physical size of
the lattice spacing a, as a result of renormalization effects. The computed
mass of a hadron M , say, must be related to the only quantity with mass
dimension, a
−1 , by a relation of the form
1
M = f (g L ).
(16.134)
a
Thus in approaching the continuum limit a → 0, we shall also have to change
16. Lattice Field Theory, and the Renormalization Group Revisited
that corrections to continuum theory results stemming from finite lattice spacing could be diminished systematically by the use of lattice actions that also
include suitable irrelevant terms. This procedure is routinely adopted in accurate lattice calculations with ‘Symanzik-improved’ actions.
One further word should be said about terms such as ‘m
2 φ
2 ’ (which arise
in the Higgs sector of the Standard Model, for instance). As we have seen,
m
2 scales by m
′ 2 = m
2 f
2 , which is a rapid growth with f . If we imagine
starting at a very high scale, such as 10
15 TeV and flowing down to 1 TeV,
then the ‘initial’ value of m will have to be very finely ‘tuned’ in order to end
up with a mass of order 1 TeV. Thus, in this picture, it seems unnatural to
have scalar particles with masses much less than the physical cut-off scale,
unless some symmetry principle ‘protects’ their light masses. We shall return
to this problem in section 22.8.1.
We now return to lattice QCD, with a brief survey of some of the impressive
results now being obtained numerically.
16.5
Lattice QCD
16.5.1 Introduction, and the continuum limit
Let us begin by considering some numbers. The lattice must be large enough
so that the spatial dimension R of the object we wish to describe – say the size
of a hadron – fits comfortably inside it, otherwise the result will be subject
to ‘finite size effects’ as the hypercube side length L is varied. We also need
R ≫ a, or else the granularity of the lattice resolution will become apparent.
Further, as indicated earlier, we expect the mass m (which is of order R
−1 )
to be very much less than a
−1 . Thus ideally we need
a ≪ R ∼ 1/m ≪ L = N a
(16.133)
so that N must be large. For example, if N = 64 and a ∼ 0.1fm the condition
(16.133) would be reasonably satisfied by a light hadron mass. But remember
that each field at each lattice point is an independent degree of freedom: dealing with integrals such as (16.87) presents a formidable numerical challenge.
Ignoring any statistical inaccuracy, the results will depend on the parameters g L and N , where g L is the bare lattice gauge coupling (we assume for
simplicity that the quarks are massless). Despite the fact that g L is dimensionless, we shall now see that its value actually controls the physical size of
the lattice spacing a, as a result of renormalization effects. The computed
mass of a hadron M , say, must be related to the only quantity with mass
dimension, a
−1 , by a relation of the form
1
M = f (g L ).
(16.134)
a
Thus in approaching the continuum limit a → 0, we shall also have to change