184
16. Lattice Field Theory, and the Renormalization Group Revisited
In a similar way, integrating (16.137) using (16.138) gives, in (16.134),
[
(
)]
1
2π
M = constant ×
exp
a
−
(16.141)
β 2
0 g L
= constant × Λ L .
(16.142)
Equation (16.141) is known as asymptotic scaling: it predicts how any physical
mass, expressed in lattice units a
−1 , should vary as a function of g L . The
form (16.142) is remarkable, as it implies that all calculated masses must be
proportional, in the continuum limit a → 0, to the same universal scale factor
Λ L .
How are masses calculated on the lattice? The principle is very similar to
the way in which the ground state was selected out as τ i → −∞, τ f → +∞ in
(16.78). Consider a correlation function for a scalar field, for simplicity:
C(τ ) = ‹Ω|φ(x = 0, τ)φ(0)|Ω
∑
2
›
=
|‹Ω|φ(0)|n›| e
−Enτ .
(16.143)
n
As τ → ∞, the term with the minimum value of E n , namely E n = M φ , will
survive; M φ can be measured from a fit to the exponential fall-off as a function
of τ .
The behaviour predicted by (16.141) and (16.142) can be tested in actual
calculations. A quantity such as the ρ meson mass is calculated via a correlation function of the form (16.143), the result being expressed in terms of a
certain number of lattice units a
−1 at a certain value of g L . By comparison
with the known ρ mass, a
−1 can be converted to GeV. Then the calculation
is repeated for a different g L value and the new a
−1 (GeV) extracted. A
plot of ln[a
−1 (GeV)] versus 1/g
2
L should then give a straight line with slope
2π/β 0 and intercept ln Λ L . Figure 16.9 shows such a plot, taken from Ellis
et al. (1996), from which it appears that the calculations are indeed being
performed close to the continuum limit. The value of Λ L has been adjusted
to fit the numerical data, and has the value Λ L = 1.74 MeV in this case. This
may seem alarmingly far from the kind of value expected for Λ QCD , but we
must remember that the renormalization schemes involved in the two cases are
quite different. In fact, we may expect Λ QCD ≈ 50Λ L (Montvay and Munster
(1994), section 5.1.6).
16.5.2 The static q¯ q potential
The calculations of m ρ represented in figure 16.9 were done in the quenched approximation. As a first example of a calculation with dynamical (unquenched)
fermions we show in figure 16.10 a lattice calculation of the static qq ¯ potential
(Allton et al. 2002, UKQCD Collaboration ), using two degenerate flavours of
16. Lattice Field Theory, and the Renormalization Group Revisited
In a similar way, integrating (16.137) using (16.138) gives, in (16.134),
[
(
)]
1
2π
M = constant ×
exp
a
−
(16.141)
β 2
0 g L
= constant × Λ L .
(16.142)
Equation (16.141) is known as asymptotic scaling: it predicts how any physical
mass, expressed in lattice units a
−1 , should vary as a function of g L . The
form (16.142) is remarkable, as it implies that all calculated masses must be
proportional, in the continuum limit a → 0, to the same universal scale factor
Λ L .
How are masses calculated on the lattice? The principle is very similar to
the way in which the ground state was selected out as τ i → −∞, τ f → +∞ in
(16.78). Consider a correlation function for a scalar field, for simplicity:
C(τ ) = ‹Ω|φ(x = 0, τ)φ(0)|Ω
∑
2
›
=
|‹Ω|φ(0)|n›| e
−Enτ .
(16.143)
n
As τ → ∞, the term with the minimum value of E n , namely E n = M φ , will
survive; M φ can be measured from a fit to the exponential fall-off as a function
of τ .
The behaviour predicted by (16.141) and (16.142) can be tested in actual
calculations. A quantity such as the ρ meson mass is calculated via a correlation function of the form (16.143), the result being expressed in terms of a
certain number of lattice units a
−1 at a certain value of g L . By comparison
with the known ρ mass, a
−1 can be converted to GeV. Then the calculation
is repeated for a different g L value and the new a
−1 (GeV) extracted. A
plot of ln[a
−1 (GeV)] versus 1/g
2
L should then give a straight line with slope
2π/β 0 and intercept ln Λ L . Figure 16.9 shows such a plot, taken from Ellis
et al. (1996), from which it appears that the calculations are indeed being
performed close to the continuum limit. The value of Λ L has been adjusted
to fit the numerical data, and has the value Λ L = 1.74 MeV in this case. This
may seem alarmingly far from the kind of value expected for Λ QCD , but we
must remember that the renormalization schemes involved in the two cases are
quite different. In fact, we may expect Λ QCD ≈ 50Λ L (Montvay and Munster
(1994), section 5.1.6).
16.5.2 The static q¯ q potential
The calculations of m ρ represented in figure 16.9 were done in the quenched approximation. As a first example of a calculation with dynamical (unquenched)
fermions we show in figure 16.10 a lattice calculation of the static qq ¯ potential
(Allton et al. 2002, UKQCD Collaboration ), using two degenerate flavours of