Help with a formula in Weinberg

[Jim Kata]

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.

He writes:

$$iL_e = - \left( {\begin{array}{*{20}c}<br /> {\bar \upsilon _e } \\<br /> {\bar e} \\<br /> <br /> \end{array} } \right)\sum\limits_\alpha {\gamma _\mu A^\mu _\alpha } t_\alpha \left( {\begin{array}{*{20}c}<br /> {\upsilon _e } \\<br /> e \\<br /> <br /> \end{array} } \right)$$

but I think he meant to write

$$iL_e = - \left( {\begin{array}{*{20}c}<br /> {\bar \upsilon _e } \\<br /> {\bar e} \\<br /> <br /> \end{array} } \right)\sum\limits_\alpha {\gamma _\mu \left( {A^\mu _\alpha t_{\alpha L} + B^\mu y} \right)} \left( {\begin{array}{*{20}c}<br /> {\upsilon _e } \\<br /> e \\<br /> <br /> \end{array} } \right)$$

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

Namely $$e_L = \frac{1}{2}\left( {1 + \gamma _5 } \right)$$ and $$e_R = \frac{1}<br /> {2}\left( {1 - \gamma _5 } \right)$$

but whatever

From this I was able to derive next line

Namely

$$\begin{gathered}<br /> - \left( {\begin{array}{*{20}c}<br /> {\bar \upsilon _e } \\<br /> {\bar e} \\<br /> <br /> \end{array} } \right)[\frac{1}<br /> {{\sqrt 2 }}\gamma _\mu W^\mu \left( {t_{1L} - it_{2L} } \right) + \frac{1}<br /> {{\sqrt 2 }}\gamma _\mu W^{*\mu } \left( {t_{1L} + it_{2L} } \right) \hfill \\<br /> + \gamma _\mu Z^\mu \left( {t_{3L} \cos \theta _W + y\sin \theta _W } \right) + \gamma _\mu A^\mu \left( { - t_{3L} \sin \theta _W + y\cos \theta _W } \right)]\left( {\begin{array}{*{20}c}<br /> {\upsilon _e } \\<br /> e \\<br /> <br /> \end{array} } \right) \hfill \\ <br /> \end{gathered}$$

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

$$\begin{gathered}<br /> - \frac{g}<br /> {{\sqrt 2 }}\left( {\bar e\gamma _\mu W^\mu \left( {\frac{{1 + \gamma _5 }}<br /> {2}} \right)\upsilon _e } \right) - \frac{g}<br /> {{\sqrt 2 }}\left( {\bar \upsilon _e \gamma _\mu W^{*\mu } \left( {\frac{{1 + \gamma _5 }}<br /> {2}} \right)e} \right) \hfill \\<br /> + \frac{1}<br /> {2}\sqrt {g^2 + g'^2 } \bar \upsilon _e \gamma _\mu Z^\mu \left( {\frac{{1 + \gamma _5 }}<br /> {2}} \right)\upsilon _e - \frac{1}<br /> {2}\frac{{\left( {g^2 - g'^2 } \right)}}<br /> {{\sqrt {g^2 + g'^2 } }}\bar e\gamma _\mu Z^\mu \left( {\frac{{1 + \gamma _5 }}<br /> {2}} \right)e \hfill \\<br /> - g'\sin \theta _W \bar e\gamma _\mu Z^\mu \left( {\frac{{1 - \gamma _5 }}<br /> {2}} \right)e - \left( {\begin{array}{*{20}c}<br /> {\bar \upsilon _e } \\<br /> {\bar e} \\<br /> <br /> \end{array} } \right)\gamma _\mu A^\mu \left( { - t_{3L} \sin \theta _W + y\cos \theta _W } \right)\left( {\begin{array}{*{20}c}<br /> {\upsilon _e } \\<br /> e \\<br /> <br /> \end{array} } \right) \hfill \\ <br /> \end{gathered}$$

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

$$- g'\bar e\gamma _\mu Z^\mu \left( {\frac{{1 - \gamma _5 }}<br /> {2}} \right)e + e\left( {\bar e\gamma _\mu A^\mu e} \right)$$

from
$$- \left( {\begin{array}{*{20}c}<br /> {\bar \upsilon _e } \\<br /> {\bar e} \\<br /> <br /> \end{array} } \right)[\gamma _\mu Z^\mu \left( {\frac{{1 - \gamma _5 }}<br /> {2}} \right)\sin \theta _W + \gamma _\mu \left( { - t_{3L} \sin \theta _W + y\cos \theta _W } \right)\left( {\begin{array}{*{20}c}<br /> {\upsilon _e } \\<br /> e \\<br /> <br /> \end{array} } \right)$$

even with

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

 

[Jim Kata]

Anybody? Help!

 

[mjsd]

trying to learn EW theory from Weinberg?... not a good choice!
sorry don't have a copy of Weinberg at hand

 

[Jim Kata]

Plain and simple, I think his formula is wrong. He got his signs backwards and he is missing a
$$\sin \theta _W$$ term in

$$- g'\bar e\gamma _\mu Z^\mu \left( {\frac{{1 - \gamma _5 }}<br /> {2}} \right)e$$

It should read:

$$- g'\sin \theta_W \bar e\gamma _\mu Z^\mu \left( {\frac{{1 - \gamma _5 }}<br /> {2}} \right)e$$

 

[Demystifier]

trying to learn EW theory from Weinberg?... not a good choice!

The funny thing is that Weinberg is the guy who discovered EW theory. Moreover, he received the Nobel prize for it.

 

[mjsd]

The funny thing is that Weinberg is the guy who discovered EW theory. Moreover, he received the Nobel prize for it.

And the most amusing part is that that book assumes that you know "the entire thing" before you start reading... I guess that's what make him the Nobel laureate. you need to be ahead of your time

 

[Demystifier]

And the most amusing part is that that book assumes that you know "the entire thing" before you start reading... I guess that's what make him the Nobel laureate. you need to be ahead of your time

I have a general rule: If you want to learn the basics of something, never ask the best experts for that to explain it to you!

 

[kdv]

Anybody? Help!

Jim, have you resolved the issue to your satisfaction? Do you agree with him now or are you now convinced there is a mistake? I am asking because I was considering double checking this.

 

[Jim Kata]

Jim, have you resolved the issue to your satisfaction? Do you agree with him now or are you now convinced there is a mistake? I am asking because I was considering double checking this.

I think there is a mistake, take a look