189
16.5. Lattice QCD
where c n
(r) and d
(r) are dimensionless constants independent of the lattice spacing a, but dependent on the particular Y
(r) , and α V (d
(r) /a) is the running
QCD coupling in the V-scheme, with N f = 3 light quark flavours. The per(r)
turbative coefficients c n for the various Y ’s were computed using Feynman
diagrams, for n ≤ 3, for the same quark and gluon actions which were used to
create the sets of gluon field configurations employed in the numerical evaluation of the Y ’s. The renormalization scale d
(r) /a varies for each short-distance
quantity, being chosen according to the Lepage-Mackenzie (1993) prescription
(or in some cases a more robust procedure due to Hornbostel, Lepage and
Morningstar (2003)).
There were 22 Y
(r) ’s, each of which was analyzed separately, fitting the
expansion (16.152) to the 12 values of that Y calculated using the 12 gluon
configurations. In the simplest terms, the result of each such fit would be the
value of α V at a particular scale, which was chosen to be α V (7.5 GeV). The
values required at the scales α V (d
(r) /a i ) were found by numerically integrating the evolution equation (at four-loop order) for α V ; here a i is the lattice
spacing for each configuration (there were 6 different spacings). In fact, the
fitting was more sophisticated, including further parameters related to various corrections; the interested reader can consult Davies et al. (2008) for the
details. Having obtained α V (7.5 GeV), this was then converted to the MS
scheme, using the relation (Brodsky, Lepage and Mackenzie 1983)
α V (μ) = α (e
−5/6 μ).
(16.153)
MS
Finally, the resultant α MS was evolved to M Z
2 . The value (16.149) is the final
result after performing a weighted average over the 22 separate determinations. A full discussion of the error estimate, which includes finite lattice
spacing, finite lattice volume, and chiral extrapolation uncertainties, is given
in Davies et al. (2008).
16.5.4 Hadron masses
For our last example of a precise lattice QCD calculation, it is appropriate to
consider the mass spectrum of light hadrons. After all, protons and neutrons
account for nearly all the mass of ordinary matter, and 95% of their mass is
the result of QCD interactions. It has long been a fundamental challenge to
predict hadron masses accurately from QCD.
As one example of such calculations, we show in figure 16.12 the light
hadron spectrum of QCD as reported by D¨ urr et al. (2008). Horizontal lines
and bands are the experimental values (which have been isospin-averaged)
with their decay widths. The solid circles are the predicted values. Vertical
error bars represent combined statistical and systematic error estimates. The
masses of the π, K and Ξ have no error bars, because they have been used to
set the values of m u = m d , m s and the overall scale, respectively. Once again,
the agreement with experiment is very impressive.
16.5. Lattice QCD
where c n
(r) and d
(r) are dimensionless constants independent of the lattice spacing a, but dependent on the particular Y
(r) , and α V (d
(r) /a) is the running
QCD coupling in the V-scheme, with N f = 3 light quark flavours. The per(r)
turbative coefficients c n for the various Y ’s were computed using Feynman
diagrams, for n ≤ 3, for the same quark and gluon actions which were used to
create the sets of gluon field configurations employed in the numerical evaluation of the Y ’s. The renormalization scale d
(r) /a varies for each short-distance
quantity, being chosen according to the Lepage-Mackenzie (1993) prescription
(or in some cases a more robust procedure due to Hornbostel, Lepage and
Morningstar (2003)).
There were 22 Y
(r) ’s, each of which was analyzed separately, fitting the
expansion (16.152) to the 12 values of that Y calculated using the 12 gluon
configurations. In the simplest terms, the result of each such fit would be the
value of α V at a particular scale, which was chosen to be α V (7.5 GeV). The
values required at the scales α V (d
(r) /a i ) were found by numerically integrating the evolution equation (at four-loop order) for α V ; here a i is the lattice
spacing for each configuration (there were 6 different spacings). In fact, the
fitting was more sophisticated, including further parameters related to various corrections; the interested reader can consult Davies et al. (2008) for the
details. Having obtained α V (7.5 GeV), this was then converted to the MS
scheme, using the relation (Brodsky, Lepage and Mackenzie 1983)
α V (μ) = α (e
−5/6 μ).
(16.153)
MS
Finally, the resultant α MS was evolved to M Z
2 . The value (16.149) is the final
result after performing a weighted average over the 22 separate determinations. A full discussion of the error estimate, which includes finite lattice
spacing, finite lattice volume, and chiral extrapolation uncertainties, is given
in Davies et al. (2008).
16.5.4 Hadron masses
For our last example of a precise lattice QCD calculation, it is appropriate to
consider the mass spectrum of light hadrons. After all, protons and neutrons
account for nearly all the mass of ordinary matter, and 95% of their mass is
the result of QCD interactions. It has long been a fundamental challenge to
predict hadron masses accurately from QCD.
As one example of such calculations, we show in figure 16.12 the light
hadron spectrum of QCD as reported by D¨ urr et al. (2008). Horizontal lines
and bands are the experimental values (which have been isospin-averaged)
with their decay widths. The solid circles are the predicted values. Vertical
error bars represent combined statistical and systematic error estimates. The
masses of the π, K and Ξ have no error bars, because they have been used to
set the values of m u = m d , m s and the overall scale, respectively. Once again,
the agreement with experiment is very impressive.