- #1

CAF123

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## Homework Statement

Show that if a transformation ##\Phi \rightarrow \Phi + \alpha \partial \Phi/ \partial \alpha## is not a symmetry of the Lagrangian, then the Noether current is no longer conserved, but rather ##\partial_{\mu}J^{\mu} = \partial L/ \partial \alpha##. Use this result to show that for a massive Dirac fermion the conservation of the chiral current is softly broken by the mass term: ##\partial_{\mu} j^{5,\mu} = -2m\bar{\psi} \gamma^5 \psi##

## Homework Equations

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$$\delta L = \frac{\partial L}{\partial \Phi} \delta \Phi + \frac{\partial L}{\partial(\partial_{\mu} \Phi)} \delta (\partial_{\mu}\Phi)$$

## The Attempt at a Solution

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I understand ##\Phi \rightarrow \Phi + \alpha \partial \Phi/ \partial \alpha## to not be a symmetry of the Lagrangian to mean that when this transformation is imposed on the fields the lagrangian is not invariant.

Can rewrite the equation in relevant equations as $$\frac{\partial L}{\partial \Phi} \delta \Phi + \frac{\partial L}{\partial(\partial_{\mu} \Phi)} \partial_{\mu} (\delta \Phi) = \alpha \left( \frac{\partial L}{\partial \Phi} + \frac{\partial L}{\partial( \partial_{\mu}\Phi)} \partial_{\mu} \right) \frac{\partial \Phi}{\partial \alpha} = \alpha \left( \frac{\partial L}{\partial \alpha} + \frac{\partial L}{\partial (\partial_{\mu} \Phi)} \partial_{\mu} \frac{\partial \Phi}{\partial \alpha}\right)$$

The question doesn't state whether if the piece of the lagrangian after the transformation not making it invariant is a total derivative. If it was, then I could use the equations of motion since the action wouldn't change.

Many thanks for any tips to proceed!