I'm reading Zee book on quantum field theory.(adsbygoogle = window.adsbygoogle || []).push({});

He wants to explain that pion is the goldstone boson arising from the spontaneous simmetry breaking of the chiral symmetry.

So he

start with the weak decay

[itex]\pi^{-} \rightarrow \bar{\nu} + e^{-} [/itex]

and write this equation

[itex] <0|J^{\mu}_{5}|k>=f k^{\mu}[/itex] (1),

where [itex] k [/itex] is the momentum of the pion.

Then, of course, if you act with [itex] k_{\mu} [/itex] on the left, you get

[itex] k_{\mu }<0|J^{\mu}_{5}|k>=f m _{\pi} [/itex] (2)

and we see that if we consider the pion as a massless spin 0 particle, it is a good candidate for a goldstone boson associated with the spontaneous breaking of the chiral symmetry, because it then follows that

[itex] \partial_{\mu} J_{5}^{\mu} = 0 [/itex]

My questions are:

-why in (1) he wrote only the axial current and he doesn't write something like

[itex] <0|J^{\mu}-J^{\mu}_{5}|k> [/itex]

?

-why the right hand side of (1) is just a vector while the left hand side is a pseudovector?

Thank you!

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# Pion as a goldstone boson

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