Alright, I'm trying to follow Weinberg's derivation of the matter part of the Lagrangian for electroweak theory, and I am all confused. This is equation (21.3.20) in volume II.(adsbygoogle = window.adsbygoogle || []).push({});

He writes:

[tex]

iL_e = - \left( {\begin{array}{*{20}c}

{\bar \upsilon _e } \\

{\bar e} \\

\end{array} } \right)\sum\limits_\alpha {\gamma _\mu A^\mu _\alpha } t_\alpha \left( {\begin{array}{*{20}c}

{\upsilon _e } \\

e \\

\end{array} } \right)

[/tex]

but I think he meant to write

[tex]

iL_e = - \left( {\begin{array}{*{20}c}

{\bar \upsilon _e } \\

{\bar e} \\

\end{array} } \right)\sum\limits_\alpha {\gamma _\mu \left( {A^\mu _\alpha t_{\alpha L} + B^\mu y} \right)} \left( {\begin{array}{*{20}c}

{\upsilon _e } \\

e \\

\end{array} } \right)

[/tex]

He defines his left handed and right handed parts different than Itzykson and Zuber

Namely [tex]e_L = \frac{1}{2}\left( {1 + \gamma _5 } \right)[/tex] and [tex]

e_R = \frac{1}

{2}\left( {1 - \gamma _5 } \right)

[/tex]

but whatever

From this I was able to derive next line

Namely

[tex]

\begin{gathered}

- \left( {\begin{array}{*{20}c}

{\bar \upsilon _e } \\

{\bar e} \\

\end{array} } \right)[\frac{1}

{{\sqrt 2 }}\gamma _\mu W^\mu \left( {t_{1L} - it_{2L} } \right) + \frac{1}

{{\sqrt 2 }}\gamma _\mu W^{*\mu } \left( {t_{1L} + it_{2L} } \right) \hfill \\

+ \gamma _\mu Z^\mu \left( {t_{3L} \cos \theta _W + y\sin \theta _W } \right) + \gamma _\mu A^\mu \left( { - t_{3L} \sin \theta _W + y\cos \theta _W } \right)]\left( {\begin{array}{*{20}c}

{\upsilon _e } \\

e \\

\end{array} } \right) \hfill \\

\end{gathered}

[/tex]

but when I tried to go to the final line of his derivation I got an opposite sign and some different stuff, namely I got

[tex]

\begin{gathered}

- \frac{g}

{{\sqrt 2 }}\left( {\bar e\gamma _\mu W^\mu \left( {\frac{{1 + \gamma _5 }}

{2}} \right)\upsilon _e } \right) - \frac{g}

{{\sqrt 2 }}\left( {\bar \upsilon _e \gamma _\mu W^{*\mu } \left( {\frac{{1 + \gamma _5 }}

{2}} \right)e} \right) \hfill \\

+ \frac{1}

{2}\sqrt {g^2 + g'^2 } \bar \upsilon _e \gamma _\mu Z^\mu \left( {\frac{{1 + \gamma _5 }}

{2}} \right)\upsilon _e - \frac{1}

{2}\frac{{\left( {g^2 - g'^2 } \right)}}

{{\sqrt {g^2 + g'^2 } }}\bar e\gamma _\mu Z^\mu \left( {\frac{{1 + \gamma _5 }}

{2}} \right)e \hfill \\

- g'\sin \theta _W \bar e\gamma _\mu Z^\mu \left( {\frac{{1 - \gamma _5 }}

{2}} \right)e - \left( {\begin{array}{*{20}c}

{\bar \upsilon _e } \\

{\bar e} \\

\end{array} } \right)\gamma _\mu A^\mu \left( { - t_{3L} \sin \theta _W + y\cos \theta _W } \right)\left( {\begin{array}{*{20}c}

{\upsilon _e } \\

e \\

\end{array} } \right) \hfill \\

\end{gathered}

[/tex]

The first four terms I got are similar to his, but with a different sign, and I understand he uses a Gell Mann Nishijima equation to get last part, but how do you get

[tex]

- g'\bar e\gamma _\mu Z^\mu \left( {\frac{{1 - \gamma _5 }}

{2}} \right)e + e\left( {\bar e\gamma _\mu A^\mu e} \right)

[/tex]

from

[tex]

- \left( {\begin{array}{*{20}c}

{\bar \upsilon _e } \\

{\bar e} \\

\end{array} } \right)[\gamma _\mu Z^\mu \left( {\frac{{1 - \gamma _5 }}

{2}} \right)\sin \theta _W + \gamma _\mu \left( { - t_{3L} \sin \theta _W + y\cos \theta _W } \right)\left( {\begin{array}{*{20}c}

{\upsilon _e } \\

e \\

\end{array} } \right)

[/tex]

even with

[tex]

q = - \sin \theta _W t_3 + \cos \theta _W y

[/tex]

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# Help with a formula in Weinberg

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