Rewriting neutral gauge bosons (EW) for vacuum with Y=-1

[Valeriia Lukashenko]

Homework Statement

So, my textbook proposes a to check what will change in mass and mass eigenvectors of Z and photon in terms of $W_{3}$ and $B_{\mu}$ fields in Higgs mechanism for EW if we choose a vacuum hypercharge to be -1 and compare results to SM (where we know that photon is massless and Z is not)

Homework Equations

Mass term for $W_{3}$ and $B_{\mu}$:
$$(-gW_{3}+g^{'}Y_{vac}B_{\mu})^2=(W_3, B_{\mu})\Bigg(\begin{matrix}
g^2 & -gg^{'} Y_{vac} \\
-gg^{'} Y_{vac} & g^{'2}
\end{matrix}\Bigg)\Bigg(\begin{matrix}W_3\\
B_{\mu}
\end{matrix}\Bigg)$$

The Attempt at a Solution

Let's find eigenvalues and eigenvectors of $$
\Bigg(\begin{matrix}
g^2 & -gg^{'} Y_{vac} \\
-gg^{'} Y_{vac} & g^{'2}
\end{matrix}\Bigg)
$$
for $Y_{vac}=-1$
$$
det\Bigg(\begin{matrix}
g^2-\lambda & gg^{'}\\
gg^{'} & g^{'2}-\lambda
\end{matrix}\Bigg)=0
$$
$$(g^2-\lambda)(g^{'2}-\lambda)-(gg^{'})^2=(gg^{'})^2-\lambda g^2 -\lambda g^{'2} + \lambda^2=
\lambda(\lambda-(g^2+g^{'2}))=0$$

There are 2 eigenvalues: $\lambda=0$ and $\lambda=(g^2+g^{'2})$
For $\lambda=0$ the eigenvector is:
$$g^2x+gg'y=0$$
$$gx=g'y$$
$$x=g', y=g$$
+ normilize it:
$$
\frac{1}{\sqrt{g^2+g^{'2}}}\Bigg(\begin{matrix}
g'\\
g
\end{matrix}\Bigg)=0
$$For $\lambda=(g^2+g^{'2})$ following the same procedure:
$$g^2x-gg'y=g^2x+g^{'2}x$$
$$-gy=g'x$$
$$x=g, y=-g'$$
+ normilize it:
$$
\frac{1}{\sqrt{g^2+g^{'2}}}\Bigg(\begin{matrix}
g\\
-g'
\end{matrix}\Bigg)=0
$$

Eigenvectors describe mass. We have two bosons, one of them is massless (photon) other is not (Z-boson).

In terms $W_3$ and $B_{\mu}$ we can write:
$$A'_{\mu}(photon)=\frac{1}{\sqrt{g^2+g^{'2}}}(g^{'}W_3+gB_{\mu})$$
$$Z'_{\mu}=\frac{1}{\sqrt{g^2+g^{'2}}}(gW_3-g^{'}B_{\mu})$$

BUT!
$$(-g W_{3}-g^{'}B_{\mu})^2 \neq (g^2+g^{'2}) Z'^2_{\mu} +0 \cdot A'^2_{\mu}$$

We need a mass of $Z$ to be equal 0 and photon to survive, because $$A'^{2}_{\mu}=(g^{'}W_3+g B_{\mu})^2=(-g W_{3}-g^{'}B_{\mu})^2$$

For $Y_{vac}=1$ everything works fine, but in this case not! It kind of surprises me, because choice of vacuum hypercharge is described as arbitrary in our lecture notes (however, they say that it becomes clear why it is better to choose +1 if you look at photon coupling with electric charge).

So, my question is: am I missing something in math or everything is fine and -1 hypercharge for vacuum doesn't work already in this case?

 

[mark726]

Your solution looks correct to me. It seems that choosing a vacuum hypercharge of -1 does not work in this case because it does not give a massless photon and a massive Z-boson as we expect in the Standard Model. This may indicate that there is some deeper reason behind why we choose a vacuum hypercharge of +1 in the Standard Model, and this is something that may become clearer as you continue your studies in this topic.