- #1
ChrisVer
Gold Member
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I'm having problem in deriving 23.6.11 from Weinberg's-Quantum Theory of fields...
We have: [itex] \psi_f \rightarrow \exp (i a_f \gamma_5) \psi_f[/itex], f denoting the flavor.
Then for the mass term lagrangian he writes:
[itex] L_m = - \frac{1}{2} \sum_f M_f \bar{\psi}_f (1+ \gamma_5) \psi_f - \frac{1}{2} \sum_f M^*_f \bar{\psi}_f (1- \gamma_5) \psi_f[/itex]
With [itex]M_f[/itex] the mass parameters. He says that by making a transformation of the fields as above, the mass parameter will be redefined:
[itex]M_f \rightarrow M_f \exp (2i a_f)[/itex]
However I think he is missing a [itex]\gamma_5[/itex]?
Because the first for example term:
\begin{multline}
\\
-\frac{1}{2} \sum_f M_f\psi^\dagger_f e^{-i \gamma_5 a_f}\gamma_0 (1+ \gamma_5) e^{i \gamma_5 a_f} \psi_f=\\
\approx -\frac{1}{2} \sum_f M_f\psi^\dagger_f (1-i \gamma_5 a_f)\gamma_0 (1+ \gamma_5) (1+i \gamma_5 a_f) \psi_f=\\
=-\frac{1}{2} \sum_f \psi^\dagger_f \gamma_0 M_f (1+i \gamma_5 a_f) (1+i \gamma_5 a_f) (1+ \gamma_5)\psi_f=\\
=-\frac{1}{2} \sum_f \bar{\psi}_f M_f (1+i 2 \gamma_5 a_f) (1+ \gamma_5)\psi_f
\end{multline}which leads in the redifinition of M:
[itex]M_f \rightarrow M_f \exp (2i a_f \gamma_5)[/itex]
Any help?
We have: [itex] \psi_f \rightarrow \exp (i a_f \gamma_5) \psi_f[/itex], f denoting the flavor.
Then for the mass term lagrangian he writes:
[itex] L_m = - \frac{1}{2} \sum_f M_f \bar{\psi}_f (1+ \gamma_5) \psi_f - \frac{1}{2} \sum_f M^*_f \bar{\psi}_f (1- \gamma_5) \psi_f[/itex]
With [itex]M_f[/itex] the mass parameters. He says that by making a transformation of the fields as above, the mass parameter will be redefined:
[itex]M_f \rightarrow M_f \exp (2i a_f)[/itex]
However I think he is missing a [itex]\gamma_5[/itex]?
Because the first for example term:
\begin{multline}
\\
-\frac{1}{2} \sum_f M_f\psi^\dagger_f e^{-i \gamma_5 a_f}\gamma_0 (1+ \gamma_5) e^{i \gamma_5 a_f} \psi_f=\\
\approx -\frac{1}{2} \sum_f M_f\psi^\dagger_f (1-i \gamma_5 a_f)\gamma_0 (1+ \gamma_5) (1+i \gamma_5 a_f) \psi_f=\\
=-\frac{1}{2} \sum_f \psi^\dagger_f \gamma_0 M_f (1+i \gamma_5 a_f) (1+i \gamma_5 a_f) (1+ \gamma_5)\psi_f=\\
=-\frac{1}{2} \sum_f \bar{\psi}_f M_f (1+i 2 \gamma_5 a_f) (1+ \gamma_5)\psi_f
\end{multline}which leads in the redifinition of M:
[itex]M_f \rightarrow M_f \exp (2i a_f \gamma_5)[/itex]
Any help?