Symmetries of the \(\sigma\) model
The Lagrangian of the \(\sigma\) model: \[\begin{aligned} \mathcal{L} &= \frac{1}{2} \big[ (\partial_\mu \sigma)(\partial^\mu \sigma) + (\partial_\mu \boldsymbol{\pi}) \cdot (\partial^\mu \boldsymbol{\pi}) \big] + i \overline{\Psi} \cancel{\partial} \Psi + g \overline{\Psi} (\sigma + i \boldsymbol{\tau} \cdot \boldsymbol{\pi} \gamma_5) \Psi\\ &- \frac{m^2}{2} (\sigma^2 + \boldsymbol{\pi}^2) - \frac{\lambda}{4} (\sigma^2 + \boldsymbol{\pi}^2)^2, \end{aligned}\] where
\(\sigma(x)\) is a scalar field,
\(\boldsymbol{\pi}(x)\) is a three-component scalar field,
\(\Psi(x)\) is a doublet of fermion fields,
\(\boldsymbol{\tau}\) are the Pauli matrices.
Symmetry: Prove that the Lagrangian \(\mathcal{L}\) has the following symmetry: \[\sigma(x) \to \sigma(x),\] \[\boldsymbol{\pi}(x) \to \boldsymbol{\pi}(x) - \boldsymbol{\alpha} \times \boldsymbol{\pi}(x),\] \[\Psi(x) \to \Psi(x) + i \frac{\boldsymbol{\alpha} \cdot \boldsymbol{\tau}}{2} \Psi(x),\] where \(\boldsymbol{\alpha}\) is a 3-vector of infinitesimal parameters of the transformation. Find the corresponding Noether current.