242
18. Chiral Symmetry Breaking
The first of these represents a four-pion contact interaction with amplitude
−iλ/2,
(18.86)
while the second contributes an s-exchange graph in the t-channel with amplitude
(−iλv/2)
2
i
,
(18.87)
q 2 − 2μ 2
′
′
where q is the 4-momentum transfer q = p − p + = p 0 − p 0 . The sum of these
+
is
2
q
−iλ/2
,
(18.88)
q 2 − 2μ 2
which reduces to iq
2 /v
2 for q ≈ 0. Thus, despite the apparent constant 4boson piece (18.86), the total amplitude in fact vanishes as q
2
→ 0, due to a
cancellation.
This cancellation is not an accident. It is generally true that Goldstone
fields enter only via their derivatives, which bring factors of momenta into the
amplitudes. We drew attention to this following equation (17.85), and the
same is true of the θ ˆ fields in (17.107). This suggests that it is both possible,
and more efficient, to recast L ˆ s into a form in which only the derivatives of
the Goldstone fields enter. Equation (17.107) indicates how to do this: we
define new pion fields (but call them the same) by
ˆ
+ S ˆ ) ˆ ˆ
Σ = (v
U , U = exp(iτ · π ˆ /v),
(18.89)
where S ˆ is invariant under SU(2) L × SU(2) R , and where U ˆ transforms by
†
ˆ
ˆ
U → U L U U R .
(18.90)
Now Σ ˆ † ˆ
+ ˆ
Σ = (v S)
2 , and the Goldstone modes have been transformed away
from the potential terms in L ˆ Σ , reappearing in the derivative terms instead.
ˆ
We write the transformed Lagrangian as L S where
1
1
1
λ
ˆ S − μ
2 S ˆ 2
+ ˆ
ˆ U
† ) −
L ˆ S = ∂ μ S∂
μ ˆ
+ (v S)
2 Tr(∂ μ U∂
μ ˆ
λvS ˆ 3 − S ˆ 4 , (18.91)
2
4
4
16
where we have used
U ˆ † ∂
μ ˆ + ∂
μ U ˆ † ˆ
u
U = 0,
(18.92)
which follows from the unitary condition U ˆ † U ˆ = 1.
ˆ
When ∂ μ U is expanded in powers of π ˆ , we recover a kinetic energy piece
1 ∂ μ π ˆ · ∂
μ
π ˆ ,
(18.93)
2
and all other terms involve derivatives of π ˆ . In particular, the term with