239
18.3. Effective Lagrangians
the Goldberger–Treiman (1958) relation. Taking M = 939 MeV, g A = 1.26
and f π = 92 MeV one obtains g πNN ≈ 12.9, which is only 5% below the
experimental value of this effective pion-nucleon coupling constant.
We can repeat the argument leading to the G-T relation but retaining
μ
2
ˆ
2
m / 0. Equation (18.46) tells us that ∂ μ /(m f π ) behaves like a properly
=
j
π
i,5
π
normalized pion field, at least when operating on a near mass-shell pion state.
μ
ˆ
This means that the one-nucleon matrix element of ∂ μ j i,5 is (cf (18.59))
τ i
2
2g πNN u ¯(p
′ )γ 5 u(p)
i
m π f π ,
(18.62)
2
q 2 − m 2
π
while from (18.55) it is given by
i¯ u(p
′ )[−2M γ 5 F 1
5 (q
2 ) + q
2 γ 5 F 3
5 (q
2 )]
τ i u(p).
(18.63)
2
Hence
2
2g πNN m π f π
2 F
5
−2M F 1
5 (q
2 ) + q 3 (q
2 ) =
.
(18.64)
q 2 − m 2
π
Also, in place of (18.60) we now have
F 3
5 (q
2 ) =
1
2g πNN f π .
(18.65)
q 2 − m 2
π
2
2
Equations (18.64) and (18.65) are consistent for q = m if
π
2
2
F 1
5 (q = m π ) = g πNN f π /M.
(18.66)
F
5
2
2
1 (q
2 ) varies only slowly from q
2 = 0 to q = m π , since it contains no rapidly
varying pion pole contribution, and so we recover the G-T relation again.
Amplitudes involving two Goldstone pions can be calculated by an extension of these techniques. However, a much more efficient method is available,
through the use of effective Lagrangians, which capture the low energy dynamics of the Goldstone modes.
18.3 Effective Lagrangians
18.3.1 The linear and non-linear σ-models
We begin by considering the linear σ-model, which has the same Lagrangian
as the one considered in section 17.6,
L ˆ Φ = (∂ μ φ ˆ † )(∂
μ φ ˆ ) + μ
2 φ ˆ † φ ˆ −
λ (φ ˆ † φ ˆ )
4 ,
(18.67)
4
18.3. Effective Lagrangians
the Goldberger–Treiman (1958) relation. Taking M = 939 MeV, g A = 1.26
and f π = 92 MeV one obtains g πNN ≈ 12.9, which is only 5% below the
experimental value of this effective pion-nucleon coupling constant.
We can repeat the argument leading to the G-T relation but retaining
μ
2
ˆ
2
m / 0. Equation (18.46) tells us that ∂ μ /(m f π ) behaves like a properly
=
j
π
i,5
π
normalized pion field, at least when operating on a near mass-shell pion state.
μ
ˆ
This means that the one-nucleon matrix element of ∂ μ j i,5 is (cf (18.59))
τ i
2
2g πNN u ¯(p
′ )γ 5 u(p)
i
m π f π ,
(18.62)
2
q 2 − m 2
π
while from (18.55) it is given by
i¯ u(p
′ )[−2M γ 5 F 1
5 (q
2 ) + q
2 γ 5 F 3
5 (q
2 )]
τ i u(p).
(18.63)
2
Hence
2
2g πNN m π f π
2 F
5
−2M F 1
5 (q
2 ) + q 3 (q
2 ) =
.
(18.64)
q 2 − m 2
π
Also, in place of (18.60) we now have
F 3
5 (q
2 ) =
1
2g πNN f π .
(18.65)
q 2 − m 2
π
2
2
Equations (18.64) and (18.65) are consistent for q = m if
π
2
2
F 1
5 (q = m π ) = g πNN f π /M.
(18.66)
F
5
2
2
1 (q
2 ) varies only slowly from q
2 = 0 to q = m π , since it contains no rapidly
varying pion pole contribution, and so we recover the G-T relation again.
Amplitudes involving two Goldstone pions can be calculated by an extension of these techniques. However, a much more efficient method is available,
through the use of effective Lagrangians, which capture the low energy dynamics of the Goldstone modes.
18.3 Effective Lagrangians
18.3.1 The linear and non-linear σ-models
We begin by considering the linear σ-model, which has the same Lagrangian
as the one considered in section 17.6,
L ˆ Φ = (∂ μ φ ˆ † )(∂
μ φ ˆ ) + μ
2 φ ˆ † φ ˆ −
λ (φ ˆ † φ ˆ )
4 ,
(18.67)
4