186
16. Lattice Field Theory, and the Renormalization Group Revisited
dynamical quarks
3 on a 16
3
× 32 lattice. As usual, one dimensionful quantity
has to be fixed in order to set the scale. In the present case this has been
done via the scale parameter r 0 of Sommer (1994), defined by
dV
2
r
= 1.65 .
(16.144)
0
|
|
|
|
dr r=r0
Applying (16.144) to the Cornell (Eichten et al. 1980) or Richardson (1979)
phenomenological potentials gives r 0 ⋍ 0.49 fm, conveniently in the range
which is well-determined by c¯ c and b ¯
b data. The data are well described by
the expression
A
V (r) = V 0 + σr − ,
(16.145)
r
where in accordance with (16.144)
(1.65 − A)
σ =
,
(16.146)
2
r 0
and where V 0 has been chosen such that V (r 0 ) = 0. Thus (16.145) becomes
(
)
(
)
r
r 0
r 0 V (r) = (1.65 − A)
− 1 − A
− 1 .
(16.147)
r 0
r
This is – up to a constant – exactly the functional form mentioned in chapter
√
1, equation (1.33). The quantity σ (there called b) is referred to as the
‘string tension’, and has a value of about 465 MeV in the present calculations.
Phenomenological models suggest a value of around 440 MeV (Eichten et al.
1980). The parameter A is found to have a value of about 0.3. In lowestorder perturbation theory, and in the continuum limit, A would be given by
one-gluon exchange as
4
A = α s (μ)
(16.148)
3
where μ is some energy scale. This would give α s ⋍ 0.22, a reasonable value
for μ ⋍ 3 GeV. Interestingly, the form (16.147) is predicted by the ‘universal
bosonic string model’ (L¨ uscher et al. 1980, L¨ uscher 1981), in which A has the
‘universal’ value
π
⋍ 0.26.
12
The existence of the linearly rising term with σ > 0 is a signal for confinement, since – if the potential maintained this form – it would cost an infinite
amount of energy to separate a quark and an antiquark. But at some point,
enough energy will be stored in the ‘string’ to create a q¯ q pair from the vacuum: the string then breaks, and the two q¯ q pairs form mesons. There is no
evidence for string breaking in figure 16.10, but we must note that the largest
distance probed is only about 1.3 fm.
3 Comparison with matched data in the quenched approximation revealed very little
difference, in this case.
16. Lattice Field Theory, and the Renormalization Group Revisited
dynamical quarks
3 on a 16
3
× 32 lattice. As usual, one dimensionful quantity
has to be fixed in order to set the scale. In the present case this has been
done via the scale parameter r 0 of Sommer (1994), defined by
dV
2
r
= 1.65 .
(16.144)
0
|
|
|
|
dr r=r0
Applying (16.144) to the Cornell (Eichten et al. 1980) or Richardson (1979)
phenomenological potentials gives r 0 ⋍ 0.49 fm, conveniently in the range
which is well-determined by c¯ c and b ¯
b data. The data are well described by
the expression
A
V (r) = V 0 + σr − ,
(16.145)
r
where in accordance with (16.144)
(1.65 − A)
σ =
,
(16.146)
2
r 0
and where V 0 has been chosen such that V (r 0 ) = 0. Thus (16.145) becomes
(
)
(
)
r
r 0
r 0 V (r) = (1.65 − A)
− 1 − A
− 1 .
(16.147)
r 0
r
This is – up to a constant – exactly the functional form mentioned in chapter
√
1, equation (1.33). The quantity σ (there called b) is referred to as the
‘string tension’, and has a value of about 465 MeV in the present calculations.
Phenomenological models suggest a value of around 440 MeV (Eichten et al.
1980). The parameter A is found to have a value of about 0.3. In lowestorder perturbation theory, and in the continuum limit, A would be given by
one-gluon exchange as
4
A = α s (μ)
(16.148)
3
where μ is some energy scale. This would give α s ⋍ 0.22, a reasonable value
for μ ⋍ 3 GeV. Interestingly, the form (16.147) is predicted by the ‘universal
bosonic string model’ (L¨ uscher et al. 1980, L¨ uscher 1981), in which A has the
‘universal’ value
π
⋍ 0.26.
12
The existence of the linearly rising term with σ > 0 is a signal for confinement, since – if the potential maintained this form – it would cost an infinite
amount of energy to separate a quark and an antiquark. But at some point,
enough energy will be stored in the ‘string’ to create a q¯ q pair from the vacuum: the string then breaks, and the two q¯ q pairs form mesons. There is no
evidence for string breaking in figure 16.10, but we must note that the largest
distance probed is only about 1.3 fm.
3 Comparison with matched data in the quenched approximation revealed very little
difference, in this case.