- #1
Valeriia Lukashenko
- 8
- 1
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?