$\rm SU(2)$ isospin invariance of the nucleon-pion system in infinitesimal transformation for Noether current calculation

Question

I'm reading "Dynamics of the Standard Model, Cambridge monographs on particle physics, nuclear physics and cosmology".

In page 10 the authors achieve an expression for the Lagrangian which is the following: $$\mathcal{L}'(\hat{\psi},\hat{\pi})=\mathcal{L}(\hat{\psi},\hat{\pi})+\frac{1}{2}\bar{\psi}\gamma^\mu\boldsymbol{\tau}\cdot\partial_\mu\boldsymbol{\alpha}\psi-\epsilon^{ijk}(\partial_\mu\pi^i)\pi^j\partial^\mu\alpha_k$$

I can obtain the first two terms, the one I'm having trouble getting is the last one. The pion transforms as $\hat\pi^i=\pi^i-\epsilon^{ijk}\pi^j\alpha^k(x)$ and the Lagrangian is: $$\mathcal{L}=\bar\psi(i {\partial\!\!\!/} -\boldsymbol{m})\psi+\frac{1}{2}[\partial_\mu\boldsymbol\pi\partial^\mu\boldsymbol\pi -m_\pi^2\boldsymbol{\pi}\cdot\boldsymbol{\pi}]+ig\bar\psi\boldsymbol\tau\cdot\boldsymbol\tau\gamma_5\psi-\frac{\lambda}{4}(\boldsymbol\pi\cdot\boldsymbol\pi)^2$$

I'm having trouble with the term: $$\partial_\mu\boldsymbol\pi\partial^\mu\boldsymbol\pi$$ which is the one where one obtains $$-\epsilon^{ijk}(\partial_\mu\pi^i)\pi^j\partial^\mu\alpha_k.$$

I obtain that term, but some extra terms that I don't know how to cancel. I have done the following:

$$(\partial_\mu\pi\partial^\mu\pi) {'}=(\partial_\mu\pi^i-\epsilon^{ijk}(\partial_\mu\pi^j)\alpha^k(x))(\partial^\mu\pi^i-\epsilon^{ijk}(\partial^\mu\pi^j)\alpha^k(x))=\\ \partial_\mu\pi^i\partial^\mu\pi^i-\partial_\mu\pi^i\epsilon^{ijk}(\partial^\mu\pi^j)\alpha^k(x)-\partial_\mu\pi^i\epsilon^{ijk}\pi^j(\partial^\mu\alpha^k(x))-\epsilon^{ijk}(\partial_\mu\pi^j)\alpha^k(x)(\partial^\mu\pi^i)+\epsilon^{ijk}\epsilon^{ijk}(\partial_\mu\pi^i)\alpha^{k^2}(x)(\partial^\mu\pi^j)+\epsilon^{ijk}\epsilon^{ijk}(\partial_\mu\pi^j)\alpha^k(x)\pi^j\partial^\mu\alpha^k(x)-\epsilon^{ijk}\pi^j(\partial_\mu\alpha^k(x))\partial^\mu\pi^i+\epsilon^{ijk}\epsilon^{ijk}\pi^j\partial_\mu\alpha^k(x)\partial^\mu\pi^j\alpha^k(x)+\epsilon^{ijk}\epsilon^{ijk}\pi^j\pi^j\partial_\mu\alpha^k(x)\partial^\mu\alpha^k(x)$$

How do I cancel the extra terms? Should I cancel every term that depends on $\alpha^2$?

Answer 1

Terms like $$\partial_\mu\pi^i\epsilon^{ijk}(\partial^\mu\pi^j)\alpha^k(x)\\ \epsilon^{ijk}(\partial_\mu\pi^j)\alpha^k(x)(\partial^\mu\pi^i)$$

vanish since $\epsilon^{ijk}$ is totally antisymmetric and $\partial_\mu\pi^i\partial^\mu\pi^j$ is symmetric in the two indices.

Terms depending on $\alpha^2$ should be cancelled since the expansion should be done to first order.