Deriving the charged pion decay matrix element

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?

1. Starting from $$\mathcal{L}_{\rm weak}$$ given eq. (28.29) on p. 570 of the book of Schwartz, one integrates out the $$W$$ boson arriving at the low-energy Lagrangian (4-Fermi-interaction)

$$\mathcal{L}_{4F} = -\frac{G_F}{\sqrt{2}} J^\mu J_\mu^\dagger$$,

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).

1. For the decay $$\pi^+ \to \mu^+ \nu_\mu$$ one has to compute the S-matrix element

$$\langle \mu^+(q_1) \nu_\mu(q_2)| S |\pi^+(p)\rangle = \frac{iG_F}{\sqrt{2}}\int d^4 x \, \langle\mu^+(q_1) \nu_\mu(q_2)| J^\mu(x) J_\mu^\dagger(x)| \pi^+(p)\rangle$$.

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

$$\langle \mu^+(q_1) \nu_\mu(q_2) | J^\mu(x) J_\mu^\dagger(x) | \pi^+(p) \rangle=\langle \mu^+(q_1) \nu_\mu(q_2) | \ell^\mu(x)|0\rangle \, \langle 0 |h_\mu^\dagger(x)|\pi^+(p)\rangle$$.

1. Employing the relation $$O(x) = e^{iP\cdot x} O(0) e^{-iP\cdot x}$$, where $$P^\mu$$ is the energy-momentum operator generating space-time translations on an arbitrary field operator $$O$$, the leptonic and the hadronic matrix elements can be further simplified as

$$\langle \mu^+(q_1) \nu_\mu(q_2)| \ell^\mu(x) |0\rangle= e^{i(q_1+q_2)\cdot x} \langle\mu^+(q_1) \nu_\mu(q_2)|\ell^\mu(0) | 0\rangle$$ 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

$$\int d^4 x \, e^{i(q_1+q_2-p)\cdot x}= (2\pi)^4 \delta^{(4)}(q_1+q_2-p)$$.

1. 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.

2. 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.

1. Sometimes you find diagrammatic illustrations of semileptonic or nonleptonic processes, where a $$u$$-quark line and $$\bar{d}$$-quark line with many gluons interchanged represent the $$\pi^+$$. These kind of diagrams should be taken with a grain of salt, as hadronic matrix elements are intrinsically nonperturbative (nonperturbative QCD in the present case encoded in the pion decay constant $$F_\pi$$). Chiral perturbation theory, the low-energy limit of QCD, allows a consistent treatment of low-energy QCD in terms of asymptotic fields (pions, eta, kaons) fully exploiting the restrictions imposed by spontaneous chiral symmetry breaking and other symmetries. In this case, Feynman diagrams (with pions, etc.) do indeed make sense.

The decay $$\pi^+ \to \mu^+\nu_\mu$$ is a bit of a "centaur": The pion couples to the hadronic current and disappears into the vacuum, a strong interaction; while the hadronic current weakly converts to a $$W^+$$ which then weakly converts to the antimuon-neutrino pair.

You are done with (perturbative) Feynman diagrams, here: the W has been integrated out to produce the 4-Fermi contact interaction (28.31), as described in your text, leaving only $$G_F$$ as its only trace! You are really asking about notation of a plugin. This is the language of "current algebra", or "current-current interaction", framing the field before the advent of the standard model, and ingeniously bypassing our ignorance about the nonperturbative chiral symmetry breaking, connecting quarks to hadrons, (pseudoscalars here), a subtle process.

As your text indicates, (28.30) leads to the "horse half" $$J_\mu^{5a}=F_\pi\partial_\mu \pi^a$$, the low-energy hadronic equivalent of (28.20), the high-energy "human" half--the quark axial current $$J_\mu^{5a}=\bar q \tau^a\gamma_\mu \gamma^5 q$$ fitting into the L weak currents of (28.31), $$F_\pi\partial_\mu \pi^a \longleftrightarrow \bar q \tau^a\gamma_\mu \gamma^5 q~~.$$

So, the only pieces of (28.32) contributing to the decay $$\pi^+ \to \mu^+\nu_\mu$$ summarizing all effective low-energy weak interactions in (28.31) is, really, $$J_\mu^{L+} \approx - \frac{F_\pi}{\sqrt 2}\partial_\mu \pi^+ + \overline{\psi_{\mu}}\gamma^\mu (1 - \gamma^5) \psi_{\nu_\mu} + \cdots \tag{28.32'}$$ where, like Matt, I have relegated all useless pieces to the ellipses (...).$$^\natural$$ The first term kills a $$\pi^+$$ and the second kills a $$\mu^+$$ and a muon neutrino. The h.c. $$J_\mu^{L-}$$ multiplying it creates these states, so the cross term destroys a $$\pi^+$$ and creates a $$\mu^+$$ and a neutrino.

Taking matrix elements $$\langle \mu^+ \nu_\mu | \frac{G_F}{\sqrt 2} J^{L+}\cdot J^{L-} | \pi^+\rangle$$ yields $$\mathcal{M} (\pi^+ \to \mu^+\nu_\mu) = \frac{G_F}{\sqrt{2}} F_\pi p_\mu \langle \mu^+ \nu_\mu | \bar{\psi}_{\nu_{\mu}} \gamma^\mu (1- \gamma^5)\psi_\mu|0\rangle. \tag{28.33}$$

$$^\natural$$ I seem to disagree with Matt's perverse (28.32) with the L projector in the quark instead of the leptonic piece, ... puzzling, probably designed to keep signs straight... (28.33) equates an amplitude to an operator?? They might be fixed like lots of typos in subsequent editions. Worry about the concepts involved, not the accuracy of his normalizations.