Equation (28.33) in Matthew Schwartz's QFT text book $$ \mathcal{M}(\pi^+ \rightarrow \mu^+ \nu_\mu) = \frac{G_F}{\sqrt{2}}F_\pi p^\mu \bar{\psi}_{\nu_\mu}\gamma^\mu(1-\gamma^5)\psi_\mu $$ where $F_\pi$ is pion decay constant. But, how can QCD pions interact with SM neutrinos and leptons?

Axial current associated with axial part of chiral symmetry excites vacuum to generate QCD pions$(\langle0|J^{a5}_\mu|\pi^a\rangle=i\frac{f_\pi p_\mu}{\sqrt{2}})$. Since chiral symmetry is among 3 flavors of quarks, this axial current consists only of quarks. But Matthew Schartz's textbook seems to identify axial current of leptons with axial currents arising from QCD chiral symmetry by introducing Equation (28.32) $$ \mathcal{L}_{4F} = \frac{G_F}{\sqrt{2}}J_\mu^LJ_\mu^L $$ where $$ J_\mu^L = \bar{\psi}_u\gamma^\mu(1-\gamma^5)\psi_d + \bar{\psi}_{\nu_\mu}\gamma^\mu\gamma^5\psi_\mu+\cdots $$ where $\psi_\mu, \psi_{\mu_\nu}$ refers to muon and muon neutrino fields.

If subgroup of chiral symmetry is gauged with electroweak gauge bosons corresponding to electroweak symmetry $SU(2)_L\times U(1)_Y$ and is spontaneously broken by QCD vacuum, then 3 pions that are associated Goldstone bosons should be eaten by gauge bosons and become unphysical. Does that pions are physical mean electroweak symmetry $SU(2)_L\times U(1)_Y$ is not broken by QCD vacuum and therefore these are not subgroup of chiral symmetry?

Equation (28.33) in Matthew Schwartz's QFT text book $$ \mathcal{M}(\pi^+ \rightarrow \mu^+ \nu_\mu) = \frac{G_F}{\sqrt{2}}F_\pi p^\mu \bar{\psi}_{\nu_\mu}\gamma^\mu(1-\gamma^5)\psi_\mu \tag{28.33} $$ where $F_\pi$ is pion decay constant. But, how can QCD pions interact with SM neutrinos and leptons?

Axial current associated with axial part of chiral symmetry excites vacuum to generate QCD pions $$\langle0|J^{a5}_\mu|\pi^a\rangle=i\frac{f_\pi p_\mu}{\sqrt{2}}.\tag{28.30'}$$ Since chiral symmetry is among 3 flavors of quarks, this axial current consists only of quarks. But Matthew Schartz's textbook seems to identify axial current of leptons with axial currents arising from QCD chiral symmetry by introducing Equation (28.32) $$ \mathcal{L}_{4F} = \frac{G_F}{\sqrt{2}}J_\mu^LJ_\mu^L \tag{28.31} $$ where $$ J_\mu^L = \bar{\psi}_u\gamma^\mu(1-\gamma^5)\psi_d + \bar{\psi}_{\nu_\mu}\gamma^\mu\gamma^5\psi_\mu+\cdots \tag{28.32} $$ where $\psi_\mu, \psi_{\mu_\nu}$ refers to muon and muon neutrino fields.

If subgroup of chiral symmetry is gauged with electroweak gauge bosons corresponding to electroweak symmetry $SU(2)_L\times U(1)_Y$ and is spontaneously broken by QCD vacuum, then 3 pions that are associated Goldstone bosons should be eaten by gauge bosons and become unphysical. Does that pions are physical mean electroweak symmetry $SU(2)_L\times U(1)_Y$ is not broken by QCD vacuum and therefore these are not subgroup of chiral symmetry?

Equation (28.33) in Matthew Schwartz's QFT text book $$ \mathcal{M}(\pi^+ \rightarrow \mu^+ \nu_\mu) = \frac{G_F}{\sqrt{2}}F_\pi p^\mu \bar{\psi}_{\nu_\mu}\gamma^\mu(1-\gamma^5)\psi_\mu $$ where $F_\pi$ is pion decay constant. But, how can QCD pions interact with SM neutrinos and leptons?

Axial current associated axial part of chiral symmetry excites vacuum to generate QCD pions$(\langle0|J^{a5}_\mu|\pi^a\rangle=i\frac{f_\pi p_\mu}{\sqrt{2}})$. Since chiral symmetry is among 3 flavors of quarks, this axial current consists only of quarks. But Matthew Schartz's textbook seems to identify axial current of leptons with axial currents arising from QCD chiral symmetry by introducing Equation (28.32) $$ \mathcal{L}_{4F} = \frac{G_F}{\sqrt{2}}J_\mu^LJ_\mu^L $$ where $$ J_\mu^L = \bar{\psi}_u\gamma^\mu(1-\gamma^5)\psi_d + \bar{\psi}_{\nu_\mu}\gamma^\mu\gamma^5\psi_\mu+\cdots $$ where $\psi_\mu, \psi_{\mu_\nu}$ refers to muon and muon neutrino fields.

If subgroup of chiral symmetry is gauged with electroweak gauge bosons, corresponding to electroweak symmetry $SU(2)_L\times U(1)_Y$ and is spontaneously broken by QCD vacuum, then 3 pions that are associated Goldstone bosons, should be eaten by gauge bosons and become unphysical. Does that pions are physical mean electroweak symmetry $SU(2)_L\times U(1)_Y$ is not broken by QCD vacuum and therefore these are not subgroup of chiral symmetry?

Equation (28.33) in Matthew Schwartz's QFT text book $$ \mathcal{M}(\pi^+ \rightarrow \mu^+ \nu_\mu) = \frac{G_F}{\sqrt{2}}F_\pi p^\mu \bar{\psi}_{\nu_\mu}\gamma^\mu(1-\gamma^5)\psi_\mu $$ where $F_\pi$ is pion decay constant. But, how can QCD pions interact with SM neutrinos and leptons?

Axial current associated with axial part of chiral symmetry excites vacuum to generate QCD pions$(\langle0|J^{a5}_\mu|\pi^a\rangle=i\frac{f_\pi p_\mu}{\sqrt{2}})$. Since chiral symmetry is among 3 flavors of quarks, this axial current consists only of quarks. But Matthew Schartz's textbook seems to identify axial current of leptons with axial currents arising from QCD chiral symmetry by introducing Equation (28.32) $$ \mathcal{L}_{4F} = \frac{G_F}{\sqrt{2}}J_\mu^LJ_\mu^L $$ where $$ J_\mu^L = \bar{\psi}_u\gamma^\mu(1-\gamma^5)\psi_d + \bar{\psi}_{\nu_\mu}\gamma^\mu\gamma^5\psi_\mu+\cdots $$ where $\psi_\mu, \psi_{\mu_\nu}$ refers to muon and muon neutrino fields.

If subgroup of chiral symmetry is gauged with electroweak gauge bosons corresponding to electroweak symmetry $SU(2)_L\times U(1)_Y$ and is spontaneously broken by QCD vacuum, then 3 pions that are associated Goldstone bosons should be eaten by gauge bosons and become unphysical. Does that pions are physical mean electroweak symmetry $SU(2)_L\times U(1)_Y$ is not broken by QCD vacuum and therefore these are not subgroup of chiral symmetry?

Equation (28.34) in Matthew Schwartz's QFT text book $$ \Gamma(\pi^+ \rightarrow \mu^+ \nu_\mu) = \frac{G_F^2F_\pi^2}{4\pi}m_\pi m_\mu^2\left(1-\frac{m_\mu^2}{m_\pi^2}\right)^2 $$ where $F_\pi$ is pion decay constant. But, how can QCD pions interact with SM neutrinos and leptons?

Axial current associated axial part of chiral symmetry excites vacuum to generate QCD pions$(\langle0|J^{a5}_\mu|\pi^a\rangle=i\frac{f_\pi p_\mu}{\sqrt{2}})$. Since chiral symmetry is among 3 flavors of quarks, this axial current consists only of quarks. But Matthew Schartz's textbook seems to identify axial current of leptons with axial currents arising from QCD chiral symmetry by introducing Equation (28.32) $$ \mathcal{L}_{4F} = \frac{G_F}{\sqrt{2}}J_\mu^LJ_\mu^L $$ where $$ J_\mu^L = \bar{\psi}_u\gamma^\mu(1-\gamma^5)\psi_d + \bar{\psi}_{\nu_\mu}\gamma^\mu\gamma^5\psi_\mu+\cdots $$ where $\psi_\mu, \psi_{\mu_\nu}$ refers to muon and muon neutrino fields.

If subgroup of chiral symmetry is gauged with electroweak gauge bosons, corresponding to electroweak symmetry $SU(2)_L\times U(1)_Y$ and is spontaneously broken by QCD vacuum, then 3 pions that are associated Goldstone bosons, should be eaten by gauge bosons and become unphysical. Does that pions are physical mean electroweak symmetry $SU(2)_L\times U(1)_Y$ is not broken by QCD vacuum and therefore these are not subgroup of chiral symmetry?

Equation (28.33) in Matthew Schwartz's QFT text book $$ \mathcal{M}(\pi^+ \rightarrow \mu^+ \nu_\mu) = \frac{G_F}{\sqrt{2}}F_\pi p^\mu \bar{\psi}_{\nu_\mu}\gamma^\mu(1-\gamma^5)\psi_\mu $$ where $F_\pi$ is pion decay constant. But, how can QCD pions interact with SM neutrinos and leptons?

Axial current associated axial part of chiral symmetry excites vacuum to generate QCD pions$(\langle0|J^{a5}_\mu|\pi^a\rangle=i\frac{f_\pi p_\mu}{\sqrt{2}})$. Since chiral symmetry is among 3 flavors of quarks, this axial current consists only of quarks. But Matthew Schartz's textbook seems to identify axial current of leptons with axial currents arising from QCD chiral symmetry by introducing Equation (28.32) $$ \mathcal{L}_{4F} = \frac{G_F}{\sqrt{2}}J_\mu^LJ_\mu^L $$ where $$ J_\mu^L = \bar{\psi}_u\gamma^\mu(1-\gamma^5)\psi_d + \bar{\psi}_{\nu_\mu}\gamma^\mu\gamma^5\psi_\mu+\cdots $$ where $\psi_\mu, \psi_{\mu_\nu}$ refers to muon and muon neutrino fields.

If subgroup of chiral symmetry is gauged with electroweak gauge bosons, corresponding to electroweak symmetry $SU(2)_L\times U(1)_Y$ and is spontaneously broken by QCD vacuum, then 3 pions that are associated Goldstone bosons, should be eaten by gauge bosons and become unphysical. Does that pions are physical mean electroweak symmetry $SU(2)_L\times U(1)_Y$ is not broken by QCD vacuum and therefore these are not subgroup of chiral symmetry?