246
18. Chiral Symmetry Breaking
A systematic exposition of ChPT at the one-loop level was given by Gasser
and Leutwyler (1984). Bijnens et al. (1996) carried the π − π calculation to
two-loop order; see also Colangelo et al. (2001).
It is clear that there must be some relation between the masses of the u
and d quarks (in the SU(2) flavour case) and the pion mass, since the latter
must vanish in the limit m u = m d = 0. To see this connection, we consider
the quark mass term in the 2-flavour QCD Lagrangian, which is
(
)
m u 0
− ¯ q ˆm 2 q, ˆ m 2 =
0 m d
.
(18.108)
Let us now redefine the quark fields (compare (17.107) and (18.17)) by
q ˆ = exp[−iτ · π ˆ γ 5 /(2f π )] f. ˆ
(18.109)
This transformation is a perfectly good parametrization of the Goldstone fields
associated with the axial symmetry (18.17), and effectively removes them from
the new fermion fields f ˆ . The quark mass term now becomes
¯
−f ˆ exp[−iτ · π ˆ γ 5 /(2f π )] m 2 exp[−iτ · π ˆ /(2f π )]f .
ˆ
(18.110)
We now make the assumption that the axial SU(2) is spontaneously broken
¯ ˆ
in QCD, by imposing a non-zero vev on the symmetry-breaking operator f f ˆ :
¯
‹0|f ˆ
i f ˆ j |0› = −f π
2 Bδ ij (i, j = 1, 2).
(18.111)
Expanding (18.110) up to second order in the pion fields, retaining just the
expectation value of the fermion bilinear
4 , we find a mass term
1
π
2
B(m u + m d ) ˆ ,
(18.112)
− 2
from which the relation (Gasser and Leutwyler 1982)
(m u + m d ) ¯
2
m = −
‹0|f ˆ f ˆ |0›
(18.113)
π
f 2
π
¯
¯
¯
f ˆ ˆ
ˆ ˆ
ˆ ˆ
follows, where f represents either f u f u or f d f d . From (18.113) we can see
that the square of the pion mass is proportional to the average u-d quark mass
(provided of course that B does not accidentally vanish), and goes to zero as
¯ ˆ
they do; ‹0|f f ˆ |0› is the ‘chiral condensate’ (cf figure 18.1).
Lattice QCD (see chapter 16) can be used to test equation (18.113), since
simulations can be done for a range of quark masses, and the relation between
m and m u, d can be checked. Conversely, ChPT can assist lattice QCD
π
calculations by guiding the extrapolation of the calculated results to quark
4 A formal justification of this step is provided by Weinberg (1996), section 19.6.
2
18. Chiral Symmetry Breaking
A systematic exposition of ChPT at the one-loop level was given by Gasser
and Leutwyler (1984). Bijnens et al. (1996) carried the π − π calculation to
two-loop order; see also Colangelo et al. (2001).
It is clear that there must be some relation between the masses of the u
and d quarks (in the SU(2) flavour case) and the pion mass, since the latter
must vanish in the limit m u = m d = 0. To see this connection, we consider
the quark mass term in the 2-flavour QCD Lagrangian, which is
(
)
m u 0
− ¯ q ˆm 2 q, ˆ m 2 =
0 m d
.
(18.108)
Let us now redefine the quark fields (compare (17.107) and (18.17)) by
q ˆ = exp[−iτ · π ˆ γ 5 /(2f π )] f. ˆ
(18.109)
This transformation is a perfectly good parametrization of the Goldstone fields
associated with the axial symmetry (18.17), and effectively removes them from
the new fermion fields f ˆ . The quark mass term now becomes
¯
−f ˆ exp[−iτ · π ˆ γ 5 /(2f π )] m 2 exp[−iτ · π ˆ /(2f π )]f .
ˆ
(18.110)
We now make the assumption that the axial SU(2) is spontaneously broken
¯ ˆ
in QCD, by imposing a non-zero vev on the symmetry-breaking operator f f ˆ :
¯
‹0|f ˆ
i f ˆ j |0› = −f π
2 Bδ ij (i, j = 1, 2).
(18.111)
Expanding (18.110) up to second order in the pion fields, retaining just the
expectation value of the fermion bilinear
4 , we find a mass term
1
π
2
B(m u + m d ) ˆ ,
(18.112)
− 2
from which the relation (Gasser and Leutwyler 1982)
(m u + m d ) ¯
2
m = −
‹0|f ˆ f ˆ |0›
(18.113)
π
f 2
π
¯
¯
¯
f ˆ ˆ
ˆ ˆ
ˆ ˆ
follows, where f represents either f u f u or f d f d . From (18.113) we can see
that the square of the pion mass is proportional to the average u-d quark mass
(provided of course that B does not accidentally vanish), and goes to zero as
¯ ˆ
they do; ‹0|f f ˆ |0› is the ‘chiral condensate’ (cf figure 18.1).
Lattice QCD (see chapter 16) can be used to test equation (18.113), since
simulations can be done for a range of quark masses, and the relation between
m and m u, d can be checked. Conversely, ChPT can assist lattice QCD
π
calculations by guiding the extrapolation of the calculated results to quark
4 A formal justification of this step is provided by Weinberg (1996), section 19.6.
2