18.3. Effective Lagrangians
247
mass values lighter than can presently be simulated. For example, Noaki et al.
(2008) have reported the results of such a calculation, using 2 light dynamical
quark flavours, in the overlap fermion formalism (Neuberger 1998a, 1998b),
which preserves chiral symmetry at finite lattice spacing. Their pion masses
ranged from 290 MeV to 750 MeV, and they compared their results with the
predictions of ChPT at one-loop (Gasser and Leutwyler 1984) and two-loop
(Colangelo et al. 2001). They found good fits to the ChPT formulae, and
extracted quark masses (in the MS scheme at the scale 2 GeV) of about 4.5
¯
MeV; they also found |‹0| ˆˆ
f f |0› ∼ (235 GeV)
3 , in the MS scheme at 2 GeV
scale. Studies by this and other groups are continuing, with 3 light flavours,
lighter pion masses, and other lattice fermion formalisms.
18.3.3 Extension to SU(3) f L ×SU(3) f R
To the extent that the strange quark is also ‘light’ on hadronic scales, the
QCD Lagrangian has the larger symmetry of SU(3) f L ×SU(3) f R , which breaks
spontaneously so as to preserve the flavour symmetry SU(3) f , and produce
¯
an SU(3) octet of pseudoscalar Goldstone bosons: π
± , π
0 , K
± , K
0 , K
0 and η 8
(see figure 12.4). The effective Lagrangian approach to the dynamics of the
Goldstone fields can be easily extended to chiral SU(3). One simply replaces
ˆ
ˆ
U = exp(iτ ·
ˆ
π ˆ /f π ) by V = exp(iλ · φ/f π ) where
( 1
)
ˆ
8
π ˆ +
∑
√
0
2
√ 1 η ˆ
π ˆ
+
K
+
8
1
6
ˆ
|
|
√
λ a φ a = (
π ˆ
−
− √ 1
ˆ
π ˆ
0 + √ 1 η ˆ
0
8
K
2
2
6
) . (18.114)
a=1
ˆ
K
−
¯
K
0
− √ 2 η ˆ 8
6
One easily verifies that the kinetic terms in
f
2
L ˆ 2 =
π
ˆ
Tr∂ μ V ∂
μ ˆ
V
(18.115)
4
have the correct normalization, using Trλ a λ b = 2δ ab . The 3-flavour quark
mass term is now
¯
− ˆ
f exp[−iλ · ˆ
ˆ
ˆ
φγ 5 /(2f π )] m 3 exp[−iλ · φγ 5 /(2f π )] f
(18.116)
where
(
)
m u 0
0
m 3 = ( 0 m d 0 ) .
(18.117)
0
0 m s
The axial SU(3) symmetry breaking vev is
¯
‹ ˆ ˆ
0|f i f j |0› = −f
2
π Bδ ij (i, j = 1, 2, 3)
(18.118)
and the meson mass term is
B
− Tr
2
{(λ · ˆ
φ)
2
m 3 }.
(18.119)
247
mass values lighter than can presently be simulated. For example, Noaki et al.
(2008) have reported the results of such a calculation, using 2 light dynamical
quark flavours, in the overlap fermion formalism (Neuberger 1998a, 1998b),
which preserves chiral symmetry at finite lattice spacing. Their pion masses
ranged from 290 MeV to 750 MeV, and they compared their results with the
predictions of ChPT at one-loop (Gasser and Leutwyler 1984) and two-loop
(Colangelo et al. 2001). They found good fits to the ChPT formulae, and
extracted quark masses (in the MS scheme at the scale 2 GeV) of about 4.5
¯
MeV; they also found |‹0| ˆˆ
f f |0› ∼ (235 GeV)
3 , in the MS scheme at 2 GeV
scale. Studies by this and other groups are continuing, with 3 light flavours,
lighter pion masses, and other lattice fermion formalisms.
18.3.3 Extension to SU(3) f L ×SU(3) f R
To the extent that the strange quark is also ‘light’ on hadronic scales, the
QCD Lagrangian has the larger symmetry of SU(3) f L ×SU(3) f R , which breaks
spontaneously so as to preserve the flavour symmetry SU(3) f , and produce
¯
an SU(3) octet of pseudoscalar Goldstone bosons: π
± , π
0 , K
± , K
0 , K
0 and η 8
(see figure 12.4). The effective Lagrangian approach to the dynamics of the
Goldstone fields can be easily extended to chiral SU(3). One simply replaces
ˆ
ˆ
U = exp(iτ ·
ˆ
π ˆ /f π ) by V = exp(iλ · φ/f π ) where
( 1
)
ˆ
8
π ˆ +
∑
√
0
2
√ 1 η ˆ
π ˆ
+
K
+
8
1
6
ˆ
|
|
√
λ a φ a = (
π ˆ
−
− √ 1
ˆ
π ˆ
0 + √ 1 η ˆ
0
8
K
2
2
6
) . (18.114)
a=1
ˆ
K
−
¯
K
0
− √ 2 η ˆ 8
6
One easily verifies that the kinetic terms in
f
2
L ˆ 2 =
π
ˆ
Tr∂ μ V ∂
μ ˆ
V
(18.115)
4
have the correct normalization, using Trλ a λ b = 2δ ab . The 3-flavour quark
mass term is now
¯
− ˆ
f exp[−iλ · ˆ
ˆ
ˆ
φγ 5 /(2f π )] m 3 exp[−iλ · φγ 5 /(2f π )] f
(18.116)
where
(
)
m u 0
0
m 3 = ( 0 m d 0 ) .
(18.117)
0
0 m s
The axial SU(3) symmetry breaking vev is
¯
‹ ˆ ˆ
0|f i f j |0› = −f
2
π Bδ ij (i, j = 1, 2, 3)
(18.118)
and the meson mass term is
B
− Tr
2
{(λ · ˆ
φ)
2
m 3 }.
(18.119)