235
18.2. Pion decay and the Goldberger–Treiman relation
SU(2) L ×SU(2) R since they treat the L and R parts differently. Indeed from
(18.37) and (18.38) we find
′
¯
¯ ′
¯
q ˆ L q ˆ R → q ˆ L q
= q ˆ L (1 + i(∊ − η) · τ /2)(1 − i(∊ + η) · τ /2)ˆ q R (18.43)
R
¯
¯
= q ˆ L q ˆ R − iη · q ˆ L τ q ˆ R
(18.44)
and
¯
¯
¯
q ˆ R q ˆ L → q ˆ R q ˆ L + iη · q ˆ R τ q ˆ L .
(18.45)
¯
Equations (18.44) and (18.45) confirm that the term q ˆq ˆ in (18.40) is invariant
under the isospin part of SU(2) L ×SU(2) R (since ∊ is not involved), but not
invariant under the axial isospin transformations parametrized by η. The
¯ ˆ q term explicitly breaks the third component of isospin (resembling an
qτ 3 ˆ
electromagnetic effect), but its magnitude may be expected to be smaller
than that of the ¯ q ˆq ˆ term, being proportional to the difference of the masses,
rather than their sum. This suggests that the vacuum will ‘align’ in such a
way as to preserve isospin, but break axial isospin.
18.2 Pion decay and the Goldberger–Treiman relation
We now discuss some of the rather surprising phenomenological implications of
spontaneously broken chiral symmetry – specifically, the spontaneous breaking of the axial isospin symmetry. We start by ignoring any ‘explicit’ quark
μ
ˆ
masses, so that the axial isospin current is conserved, ∂ μ j i,5 = 0. From sections
17.4 and 17.5 (suitably generalized) we know that this current has non-zero
matrix elements between the vacuum and a ‘Goldstone’ state, which in our
case is the pion. We therefore set (cf (17.94))
μ
‹0| ˆ j (x)|π j , p› = −ip
μ f π e
−ip·x δ ij
(18.46)
i,5
where f π is a constant with dimensions of mass, and which we expect to be
related to a symmetry breaking vev. This is just what we shall find in section
ˆ
2
μ
18.3.1. Note that (18.46) is consistent with ∂ μ j i,5 = 0 if p = 0, i.e. if the
pion is massless.
We treat f π as a phenomenological parameter. Its value can be determined
from the rate for the decay π
+
→ μ
+ ν μ by the following reasoning. In chapter
20 we shall learn that the effective weak Hamiltonian density for this low
energy strangeness non-changing semileptonic transition is
G F
¯
H ˆ W (x) = √ V ud ψ ˆ
d (x)γ
μ (1 − γ 5 )ψ ˆ u (x)
2
¯
×[ψ ˆ νe (x)γ μ (1 − γ 5 )ψ ˆ e (x) + ψ ˆ
νμ (x)γ μ (1 − γ 5 )ψ ˆ μ (x)] (18.47)
18.2. Pion decay and the Goldberger–Treiman relation
SU(2) L ×SU(2) R since they treat the L and R parts differently. Indeed from
(18.37) and (18.38) we find
′
¯
¯ ′
¯
q ˆ L q ˆ R → q ˆ L q
= q ˆ L (1 + i(∊ − η) · τ /2)(1 − i(∊ + η) · τ /2)ˆ q R (18.43)
R
¯
¯
= q ˆ L q ˆ R − iη · q ˆ L τ q ˆ R
(18.44)
and
¯
¯
¯
q ˆ R q ˆ L → q ˆ R q ˆ L + iη · q ˆ R τ q ˆ L .
(18.45)
¯
Equations (18.44) and (18.45) confirm that the term q ˆq ˆ in (18.40) is invariant
under the isospin part of SU(2) L ×SU(2) R (since ∊ is not involved), but not
invariant under the axial isospin transformations parametrized by η. The
¯ ˆ q term explicitly breaks the third component of isospin (resembling an
qτ 3 ˆ
electromagnetic effect), but its magnitude may be expected to be smaller
than that of the ¯ q ˆq ˆ term, being proportional to the difference of the masses,
rather than their sum. This suggests that the vacuum will ‘align’ in such a
way as to preserve isospin, but break axial isospin.
18.2 Pion decay and the Goldberger–Treiman relation
We now discuss some of the rather surprising phenomenological implications of
spontaneously broken chiral symmetry – specifically, the spontaneous breaking of the axial isospin symmetry. We start by ignoring any ‘explicit’ quark
μ
ˆ
masses, so that the axial isospin current is conserved, ∂ μ j i,5 = 0. From sections
17.4 and 17.5 (suitably generalized) we know that this current has non-zero
matrix elements between the vacuum and a ‘Goldstone’ state, which in our
case is the pion. We therefore set (cf (17.94))
μ
‹0| ˆ j (x)|π j , p› = −ip
μ f π e
−ip·x δ ij
(18.46)
i,5
where f π is a constant with dimensions of mass, and which we expect to be
related to a symmetry breaking vev. This is just what we shall find in section
ˆ
2
μ
18.3.1. Note that (18.46) is consistent with ∂ μ j i,5 = 0 if p = 0, i.e. if the
pion is massless.
We treat f π as a phenomenological parameter. Its value can be determined
from the rate for the decay π
+
→ μ
+ ν μ by the following reasoning. In chapter
20 we shall learn that the effective weak Hamiltonian density for this low
energy strangeness non-changing semileptonic transition is
G F
¯
H ˆ W (x) = √ V ud ψ ˆ
d (x)γ
μ (1 − γ 5 )ψ ˆ u (x)
2
¯
×[ψ ˆ νe (x)γ μ (1 − γ 5 )ψ ˆ e (x) + ψ ˆ
νμ (x)γ μ (1 − γ 5 )ψ ˆ μ (x)] (18.47)