5 QCD on the Lattice
205
quark masses are input parameters in the simulations, lattice QCD allows to map out
the quark mass dependence of the masses of Goldstone bosons and thus determine
the LEC B 0 . We shall see below that B 0 is related to the quark condensate ¯
which can be considered as the order parameter for spontaneous chiral symmetry
breaking. Furthermore, as we have already discussed in Sect. 5.5.6, absolute values
of quark masses are accessible via lattice QCD.
We end our brief introduction to ChPT with the derivation of a few relations
which will be useful for our discussion of chiral symmetry breaking below. In
particular, we shall derive the leading-order mass formulae such as Eq. (5.80) and
establish a link between the quark condensate and B 0 . To this end we expand the
field U in the chiral Lagrangian L
(2)
eff in powers of the Goldstone boson fields.
Assuming exact isospin symmetry, m u = m d , one finds at lowest order in φ a :
L
(2)
eff =
1
2
8
a=1
∂ μ φ a ∂ μ φ a + . . .
+
1
2
(m u + m d )B 0
3
a=1
φ
2
a +
1
2
( ˆ
m + m s )B 0
7
a=4
φ
2
a
+
1
3
( ˆ
m + 2m s )B 0 φ
2
8 + . . . .
(5.173)
After identifying φ 1 , φ 2 , φ 3 with the pions, φ 4 , . . . , φ 7 with the kaons, and φ 8 ≡ η,
one derives the leading-order relations between the quark masses and the masses of
the Goldstone bosons, viz.
m
2
π = 2B 0 ˆ
m, m
2
K = B 0 ( ˆ
m + m s ), m
2
η =
2
3 B 0 ( ˆ
m + 2m s ).
(5.174)
Thus, the relation for a generic pseudoscalar Goldstone boson made up of quarks
with masses m 1 and m 2 is precisely what was already shown in Eq. (5.80). We note
that from Eq. (5.174) one easily derives the Gell-Mann–Okubo mass relation, i.e.
3m
2
η + m
2
π − 4m
2
K = 0,
(5.175)
which is satisfied experimentally within a few percent. Furthermore, Eq. (5.174)
yields the ratio m s / ˆ
m at lowest order, viz.
m s
ˆ
m
=
2m 2
K − m 2
π
m 2
π
24,
(5.176)
which is already close to the estimate at next-to-leading order of m s / ˆ
m = 24.4 ±
1.5 [38], quoted in Sect. 5.5.6.
For the discussion of spontaneous symmetry breaking, it is useful to establish
a connection between the quark condensate in QCD,
¯
, and the LECs which
205
quark masses are input parameters in the simulations, lattice QCD allows to map out
the quark mass dependence of the masses of Goldstone bosons and thus determine
the LEC B 0 . We shall see below that B 0 is related to the quark condensate ¯
which can be considered as the order parameter for spontaneous chiral symmetry
breaking. Furthermore, as we have already discussed in Sect. 5.5.6, absolute values
of quark masses are accessible via lattice QCD.
We end our brief introduction to ChPT with the derivation of a few relations
which will be useful for our discussion of chiral symmetry breaking below. In
particular, we shall derive the leading-order mass formulae such as Eq. (5.80) and
establish a link between the quark condensate and B 0 . To this end we expand the
field U in the chiral Lagrangian L
(2)
eff in powers of the Goldstone boson fields.
Assuming exact isospin symmetry, m u = m d , one finds at lowest order in φ a :
L
(2)
eff =
1
2
8
a=1
∂ μ φ a ∂ μ φ a + . . .
+
1
2
(m u + m d )B 0
3
a=1
φ
2
a +
1
2
( ˆ
m + m s )B 0
7
a=4
φ
2
a
+
1
3
( ˆ
m + 2m s )B 0 φ
2
8 + . . . .
(5.173)
After identifying φ 1 , φ 2 , φ 3 with the pions, φ 4 , . . . , φ 7 with the kaons, and φ 8 ≡ η,
one derives the leading-order relations between the quark masses and the masses of
the Goldstone bosons, viz.
m
2
π = 2B 0 ˆ
m, m
2
K = B 0 ( ˆ
m + m s ), m
2
η =
2
3 B 0 ( ˆ
m + 2m s ).
(5.174)
Thus, the relation for a generic pseudoscalar Goldstone boson made up of quarks
with masses m 1 and m 2 is precisely what was already shown in Eq. (5.80). We note
that from Eq. (5.174) one easily derives the Gell-Mann–Okubo mass relation, i.e.
3m
2
η + m
2
π − 4m
2
K = 0,
(5.175)
which is satisfied experimentally within a few percent. Furthermore, Eq. (5.174)
yields the ratio m s / ˆ
m at lowest order, viz.
m s
ˆ
m
=
2m 2
K − m 2
π
m 2
π
24,
(5.176)
which is already close to the estimate at next-to-leading order of m s / ˆ
m = 24.4 ±
1.5 [38], quoted in Sect. 5.5.6.
For the discussion of spontaneous symmetry breaking, it is useful to establish
a connection between the quark condensate in QCD,
¯
, and the LECs which