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When trying to unificate $SU(2)_L$ and $U(1)_{em}$ one introduces and extended Gauge group with the direct product $SU(2)_L \times U(1)_Y$, where are introduced the concepts of hypercharge $Y(\psi)$, that should be determined for different particles, and a new coupling costant $g'$ to $U(1)_Y$ interaction.

The neutral current lagrangian $\mathcal{L}_n$ is introduced coupling the gauge fields $W_{\mu}^{(3)}$ (related to the third generator of $SU(2)_L$) and $B_{\mu}$ (the only generator of $U(1)_Y$) to the relative currents. The physical relevant gauge bosons are obtained as linear combination of these two, and are related by a rotation $$ B_{\mu}=\cos\theta A_{\mu}-\sin\theta Z_{\mu} $$ $$ W_{\mu}^{(3)}=\sin\theta A_{\mu}+\cos\theta Z_{\mu} $$

$\mathcal{L}_n$ is written with explicit dependece from $Z_{\mu},A_{\mu}$, and we want to isolate the terms proportional to $A_{\mu}$ to be able to see the limit in which the e.m. current is found.

Here is the question:

Formally in the new expression of $\mathcal{L}_n$=$A_{\mu}[...]+Z_{\mu}[...]$, in the first squared parenthesis the terms $Y(\psi_l)$ and $g'$ comes always together as a product $g'Y(\psi_l)$. It is said that because of that we can choose the value of the hypercharge of the first spinor arbitrally (es. $Y(\psi_L)=-1$), and any "error" or compensation can be corrected rescaling the $g'$ factor accordingly. I have problems understanding this passage, as being $g'$ the coupling constant of the $U(1)_Y$ part of the interaction, I think it can be measured, and then its value can't be rescaled arbitrally. If instead $g'$ can't be measured I have a problem understanding what really is its meaning in the $U(2)_L \times U(1)_Y$ model.

A similar question has been asked about the constant $k_B$ and the temprature $T$ here, Is the Boltzmann constant really that important? . So I was wandering if there is a similar argument as the one presented in the accepted answer for the $g'Y(\psi)$ case. Thanks a lot for the help.

I didn't wrote the explicit calculation as the aim of the first part is more of a background, anyway if neccessary I can write the explicit passages.

SuperCiocia
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Ratman
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  • Who gave you the idea that g' can be measured independently (presumably of θ)? How on earth would you measure it? Due diligence. – Cosmas Zachos Oct 13 '20 at 14:02
  • @CosmasZachos, Thanks for the reference, I forgot to write that even every resource on topic was appreciated, as I am just starting with weak interactions theory – Ratman Oct 13 '20 at 14:21
  • WP reminds you that, for all practical purposes, $g'\equiv e/\cos \theta$. I suspect if you stick to a good textbook on the subject all such questions will dissolve... – Cosmas Zachos Oct 13 '20 at 14:40
  • Actually, maybe my problem is more trivial, It isn't a problem related to the measure of g'. The arbitrariness of the first Y is just up to choosing what we consider to be the unit hypercharge, Anyway I'll think more about this – Ratman Oct 13 '20 at 17:02
  • The normalization of the Hypercharge generator is a matter of (Physics-universal) convention. – Cosmas Zachos Oct 13 '20 at 17:58

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