From M Schwarz's QFT (p 570), Goldstone's theorem indicates that pions are created from the vacuum by the chiral ${\rm SU}(2)$ current $J_\mu^{5a}$,
$$
\langle
\Omega|
J_\mu^{5a}(x)|\pi^b(p)
\rangle
=
ip_\mu F_\pi e^{ip\cdot x} \delta^{ab}. \tag{28.30}
$$
We also know that for charged pion decay we can use the 4-Fermi interaction,
$$
\mathcal{L}_{4F}
=
\frac{G_F}{\sqrt{2}}
J_\mu^L J_\mu^L , \tag{28.31}
$$
where
$$
J_\mu^L
=
\bar{\psi}_u\gamma^\mu (1 – \gamma^5) \psi_d + \bar{\psi}_{\nu_\mu}\gamma^\mu\gamma^5 \psi_\mu
+
\cdots \tag{28.32}
$$
Schwarz proceeds to say that the matrix element for $\pi^+ \to \mu^+\nu_\mu$ is
$$
\mathcal{M}
(\pi^+ \to \mu^+\nu_\mu)
=
\frac{G_F}{\sqrt{2}}
F_\pi p^\mu \bar{\psi}_{\nu_{\mu}}
\gamma^\mu(1 – \gamma^5)\psi_\mu, \tag{28.33}
$$
as derived from the previous equations.
How is this result obtained? Quite often, I have seen matrix elements written in terms of currents. Is there a way to understand this different approach given an understanding of Feynman rules?
Best Answer
$\mathcal{L}_{4F} = -\frac{G_F}{\sqrt{2}} J^\mu J_\mu^\dagger$,
where $\frac{G_F}{\sqrt{2}}=\frac{g^2}{8M_W^2}$ and $J^\mu = V_{ud} \, \bar{\psi}_u \gamma^\mu (1-\gamma_5) \psi_d+\bar{\psi}_{\nu_\mu} \gamma^\mu (1-\gamma_5) \psi_\mu + \ldots$, which shows that the second current in eq. (28.31) on p. 271 in the book of Schwartz should be replaced by its adjoint (otherwise the Lagrangian would fail to be hermitean). The missing minus sign and the wrong index structure (one of the $\mu$'s should be upstairs) in eq. (28.31) are minor points. Note that the Cabibbo-Kobayashi-Maskawa matrix element $V_{ud}$ in the current was neglected on purpose by Schwartz as he sets $V_{ij}=\delta_{ij}$ for simplicity (see the remark in the third line after eq. (28.29) in his book).
$\langle \mu^+(q_1) \nu_\mu(q_2)| S |\pi^+(p)\rangle = \frac{iG_F}{\sqrt{2}}\int d^4 x \, \langle\mu^+(q_1) \nu_\mu(q_2)| J^\mu(x) J_\mu^\dagger(x)| \pi^+(p)\rangle$.
As the V-A current $J^\mu = h^\mu + \ell^\mu$ is the sum of the hadronic part $h^\mu = V_{ud} \bar{\psi}_u \gamma^\mu (1-\gamma_5) \psi_d+\ldots$ and the
leptonic part $\ell^\mu=\bar{\psi}_{\nu_\mu} \gamma^\mu (1-\gamma_5)\psi_\mu + \ldots$, the above matrix element can be factorized as
$\langle \mu^+(q_1) \nu_\mu(q_2) | J^\mu(x) J_\mu^\dagger(x) | \pi^+(p) \rangle=\langle \mu^+(q_1) \nu_\mu(q_2) | \ell^\mu(x)|0\rangle \, \langle 0 |h_\mu^\dagger(x)|\pi^+(p)\rangle $.
$\langle \mu^+(q_1) \nu_\mu(q_2)| \ell^\mu(x) |0\rangle= e^{i(q_1+q_2)\cdot x} \langle\mu^+(q_1) \nu_\mu(q_2)|\ell^\mu(0) | 0\rangle$ and $\langle 0 |h_\mu^\dagger(x) |\pi^+(p)\rangle = e^{-ip\cdot x} \langle 0 | h_\mu^\dagger(0) | \pi^+(p)\rangle$ .
Inserting these two matrix elements in the formula for the S-matrix element, the x-integration can be carried out, expressing energy-momentum conservation via
$\int d^4 x \, e^{i(q_1+q_2-p)\cdot x}= (2\pi)^4 \delta^{(4)}(q_1+q_2-p)$.
The computation of the leptonic matrix element $\langle \mu^+(q_1) \nu_\mu(q_2) | \ell^\mu(0) | 0\rangle$ is trivial. The hadronic matrix element $\langle 0 | h_\mu^\dagger(0) | \mu^+(p)\rangle$ can be related to eq. (28.30) in the book of Schwartz by noting that $\pi^+ = (\pi^1 +i\pi^2)/\sqrt{2}$ and the fact that only the axial vector part contributes.
In this way, one should finally arrive at the invariant matrix element
$\mathcal{M}(\pi^+ \to \mu^+ \nu_\mu) = \frac{G_F}{\sqrt{2}} V_{ud}^\ast F_\pi p^\mu \bar{u}(\nu_\mu;q_2) \gamma_\mu (1-\gamma_5)v(\mu;q_1)$.
Note again the small inconsistencies in the corresponding eq. (28.33) of the book.