198
H. Wittig
masses are indeed small compared to typical hadronic scales, such as the mass of
the nucleon, for instance. Thus, the magnitude of χ is identified with a value close
to 1 GeV. In ChPT, quantities like hadron masses, decay rates or cross sections
are computed through an expansion in powers of quark masses (and 4-momenta)
about the chiral limit. The inclusion of the charm quark into the formalism is rather
useless, since the masses if the lightest charmed pseudoscalar mesons are far greater
than χ ≈ 1 GeV.
The top quark can be safely ignored in this context, since its lifetime is an order
of magnitude shorter than typical QCD processes. As a consequence, the top quark
does not undergo any hadronization effects (for instance, “toponium”, i.e. t ¯
t bound
states have never been observed), but rather decays weakly into a W -boson and a
b-quark.
The mass of the b-quark is rather large (and to some extent this is also true
for the charm quark), so that one may attempt to determine their values from
perturbative expansions in α s of some mass-dependent quantity. By contrast, in
the light quark sector non-perturbative effects such as spontaneous chiral symmetry
breaking dominate. As far as the determination of the masses of the u, d, s quarks
is concerned, ChPT is of limited value, since only ratios of quark masses can
be predicted, but not their absolute values. The reason is that although the light
quark masses appear as parameters of ChPT, their values cannot be fixed by
chiral symmetry (see Sect. 5.6.1 for more details). The absolute normalization must
therefore be provided by non-perturbative methods such as lattice simulations or
QCD sum rules.
Below we will focus on attempts to compute the values of the light quark
masses in units of some hadronic quantity. As indicated in Sect. 5.5.1, this entails
the knowledge of the renormalization factor that links lattice regularization to the
chosen continuum scheme. Lattice simulations have maximum impact in the light
quark sector, owing to the dominance of non-perturbative effects, which is in fact
signified by the large uncertainties quoted for the values of the u, d and s quark
masses in the particle data book [61].
The general procedure for the determination of light quark masses in lattice QCD
starts from the PCAC relation, Eq. (5.116). Assuming exact isospin symmetry, m u =
m d , one can consider a generic light flavour with mass m ≡ ˆ
m =
1
2 (m u + m d ).
In order to determine, say, the combination ˆ
m + m s , one must define a particular
hadronic renormalization scheme, by specifying the lattice scale and the hadronic
quantity that fixes the value of ˆ
m + m s . Furthermore, the renormalization factor
which connects hadronic and continuum schemes must be known. Equation (5.116)
can then be rewritten such that it yields the sum of RG-invariant quark masses ˆ
M +
M s in units of the quantity Q which sets the lattice spacing:
ˆ
M + M s
Q
= Z M ×
f bare
PS Q
G
bare
PS
m PS =m K
×
m 2
K
Q 2
exp
+ O(a
p ).
(5.156)
H. Wittig
masses are indeed small compared to typical hadronic scales, such as the mass of
the nucleon, for instance. Thus, the magnitude of χ is identified with a value close
to 1 GeV. In ChPT, quantities like hadron masses, decay rates or cross sections
are computed through an expansion in powers of quark masses (and 4-momenta)
about the chiral limit. The inclusion of the charm quark into the formalism is rather
useless, since the masses if the lightest charmed pseudoscalar mesons are far greater
than χ ≈ 1 GeV.
The top quark can be safely ignored in this context, since its lifetime is an order
of magnitude shorter than typical QCD processes. As a consequence, the top quark
does not undergo any hadronization effects (for instance, “toponium”, i.e. t ¯
t bound
states have never been observed), but rather decays weakly into a W -boson and a
b-quark.
The mass of the b-quark is rather large (and to some extent this is also true
for the charm quark), so that one may attempt to determine their values from
perturbative expansions in α s of some mass-dependent quantity. By contrast, in
the light quark sector non-perturbative effects such as spontaneous chiral symmetry
breaking dominate. As far as the determination of the masses of the u, d, s quarks
is concerned, ChPT is of limited value, since only ratios of quark masses can
be predicted, but not their absolute values. The reason is that although the light
quark masses appear as parameters of ChPT, their values cannot be fixed by
chiral symmetry (see Sect. 5.6.1 for more details). The absolute normalization must
therefore be provided by non-perturbative methods such as lattice simulations or
QCD sum rules.
Below we will focus on attempts to compute the values of the light quark
masses in units of some hadronic quantity. As indicated in Sect. 5.5.1, this entails
the knowledge of the renormalization factor that links lattice regularization to the
chosen continuum scheme. Lattice simulations have maximum impact in the light
quark sector, owing to the dominance of non-perturbative effects, which is in fact
signified by the large uncertainties quoted for the values of the u, d and s quark
masses in the particle data book [61].
The general procedure for the determination of light quark masses in lattice QCD
starts from the PCAC relation, Eq. (5.116). Assuming exact isospin symmetry, m u =
m d , one can consider a generic light flavour with mass m ≡ ˆ
m =
1
2 (m u + m d ).
In order to determine, say, the combination ˆ
m + m s , one must define a particular
hadronic renormalization scheme, by specifying the lattice scale and the hadronic
quantity that fixes the value of ˆ
m + m s . Furthermore, the renormalization factor
which connects hadronic and continuum schemes must be known. Equation (5.116)
can then be rewritten such that it yields the sum of RG-invariant quark masses ˆ
M +
M s in units of the quantity Q which sets the lattice spacing:
ˆ
M + M s
Q
= Z M ×
f bare
PS Q
G
bare
PS
m PS =m K
×
m 2
K
Q 2
exp
+ O(a
p ).
(5.156)