16.5. Lattice QCD
187
16.5.3 Calculation of α(M 2
Z )
Our second example of a precision lattice calculation with dynamical quarks is
the determination of α s (M
2
Z ) by Davies et al. (2008) (HPQCD Collaboration).
The reported value is
α s (M
2
Z ) = 0.1183(8).
(16.149)
The accuracy of this result is extremely impressive, and it implies that this
determination is an important ingredient in the world average value quoted
in (15.62). It is worth sketching some of the elements that went into this
landmark calculation.
The work used 12 gluon configurations from the MILC collaboration (Aubin
et al. 2004), and built on a joint effort by several groups (see Davies et al.
(HPQCD, UKQCD, MILC, and Fermilab collaborations) 2004). Vacuum polarization effects from all three light quarks u, d and s were included, using a
Symanzik-improved staggered-quark discretization, with rooting. The effects
of c and b quarks were incorporated using perturbation theory. The strange
quark mass was physical, while the u and d quark mass (set to be the same)
was three times too large, but small enough for chiral perturbation theory
(see chapter 18) to be reliable for extrapolating to the physical mass.
There were 5 parameters: m u = m d , m s , m c , m b and the bare QCD coupling g L (or equivalently the lattice spacing a). The mass parameters were
tuned to reproduce experimentally measured values of m
2 , 2m
2
2
π
K −m π , m D and
m Υ respectively. The lattice spacing was adjusted to make the Υ − Υ
′ mass
difference agree with experiment (Gray et al. 2005). With the free parameters
all determined, the simulation accurately reproduced QCD, and predictions
for physical quantities could proceed. En passant, we show in figure 16.11 results obtained (Davies et al. 2004), divided by experimental results, for nine
different quantities, with and without quark vacuum polarization (left and
right panels respectively). The values on the left deviate from experiment by
as much as 10% − 15%; those on the right agree with experiment to within
systematic and statistical errors of 3% or less.
To extract a value of the coupling constant, the general strategy is to
calculate (with the tuned simulation) a non-perturbative numerical value for
a short-distance quantity, for which perturbation theory should be reliable.
Then, by comparing the numerically computed value to the known perturbative expansion, a value of the coupling constant can be found.
In this case, the quantities calculated were vacuum expectation values of
small Wilson loop operators W mn (and related quantities) where
∫
1
W mn ≡ ‹0|Re Tr P exp[−ig L
A
]
3
· dx
nm
|0›,
(16.150)
where P denotes path ordering, A μ = λ/2 · A μ is the QCD (matrix-valued)
vector potential, and the integral is over a closed ma × na rectangular path,
not necessarily planar. The 1 × 1 Wilson loop is just the vev of the simple
plaquette operator U ❗ of section 16.2.3.
187
16.5.3 Calculation of α(M 2
Z )
Our second example of a precision lattice calculation with dynamical quarks is
the determination of α s (M
2
Z ) by Davies et al. (2008) (HPQCD Collaboration).
The reported value is
α s (M
2
Z ) = 0.1183(8).
(16.149)
The accuracy of this result is extremely impressive, and it implies that this
determination is an important ingredient in the world average value quoted
in (15.62). It is worth sketching some of the elements that went into this
landmark calculation.
The work used 12 gluon configurations from the MILC collaboration (Aubin
et al. 2004), and built on a joint effort by several groups (see Davies et al.
(HPQCD, UKQCD, MILC, and Fermilab collaborations) 2004). Vacuum polarization effects from all three light quarks u, d and s were included, using a
Symanzik-improved staggered-quark discretization, with rooting. The effects
of c and b quarks were incorporated using perturbation theory. The strange
quark mass was physical, while the u and d quark mass (set to be the same)
was three times too large, but small enough for chiral perturbation theory
(see chapter 18) to be reliable for extrapolating to the physical mass.
There were 5 parameters: m u = m d , m s , m c , m b and the bare QCD coupling g L (or equivalently the lattice spacing a). The mass parameters were
tuned to reproduce experimentally measured values of m
2 , 2m
2
2
π
K −m π , m D and
m Υ respectively. The lattice spacing was adjusted to make the Υ − Υ
′ mass
difference agree with experiment (Gray et al. 2005). With the free parameters
all determined, the simulation accurately reproduced QCD, and predictions
for physical quantities could proceed. En passant, we show in figure 16.11 results obtained (Davies et al. 2004), divided by experimental results, for nine
different quantities, with and without quark vacuum polarization (left and
right panels respectively). The values on the left deviate from experiment by
as much as 10% − 15%; those on the right agree with experiment to within
systematic and statistical errors of 3% or less.
To extract a value of the coupling constant, the general strategy is to
calculate (with the tuned simulation) a non-perturbative numerical value for
a short-distance quantity, for which perturbation theory should be reliable.
Then, by comparing the numerically computed value to the known perturbative expansion, a value of the coupling constant can be found.
In this case, the quantities calculated were vacuum expectation values of
small Wilson loop operators W mn (and related quantities) where
∫
1
W mn ≡ ‹0|Re Tr P exp[−ig L
A
]
3
· dx
nm
|0›,
(16.150)
where P denotes path ordering, A μ = λ/2 · A μ is the QCD (matrix-valued)
vector potential, and the integral is over a closed ma × na rectangular path,
not necessarily planar. The 1 × 1 Wilson loop is just the vev of the simple
plaquette operator U ❗ of section 16.2.3.