Weinberg Angle and Electroweak Mixing

1. Recapping the Neutral Boson Mass Term

Let's recall the part of the Lagrangian responsible for the masses of the neutral gauge bosons, which we derived from the term $(D_{\mu }H{)}^{†}(D^{\mu }H)$ by substituting the Higgs VEV, $H=\frac{1}{\sqrt{2}}\left(\begin{array}{c}0\\ v\end{array}\right)$:

Expanding this and writing it in a matrix form suitable for a Lagrangian density (which has the form $\frac{1}{2}{X}^T{M}^{2}X$), we have:

As we noted, the off-diagonal $-g{g}^{\prime }$ terms tell us that $W_{\mu }^3$ and $B_{\mu }$ are not mass eigenstates. A particle with a definite mass cannot be a mixture of other states that gets transformed by propagation. To find the fields that correspond to physical particles with definite masses, we must diagonalize this mass-squared matrix.

2. Diagonalization and the Physical Fields

The physical states are the eigenvectors of the mass-squared matrix:

Last time, we found the eigenvalues to be:

• ${\lambda }_1=M_Z^2=\frac{v^2}{4}(g^2+g^{\prime 2})$ (The massive boson)
• ${\lambda }_2=M_{\gamma }^2=0$ (The massless boson)

Now, let's find the corresponding eigenvectors.

For the massless eigenvalue (${\lambda }_2=0$):
We need to find the vector $\left(\begin{array}{c}w\\ b\end{array}\right)$ such that $M^2\left(\begin{array}{c}w\\ b\end{array}\right)=0$.

The first row gives the equation $g^{2}w-g{g}^{\prime }b=0$, which simplifies to $gw=g^{\prime }b$.
The un-normalized eigenvector is therefore proportional to $\left(\begin{array}{c}{g}^{\prime }\\ g\end{array}\right)$.

For the massive eigenvalue (${\lambda }_1=M_Z^2$):
The eigenvector will be orthogonal to the first one. An orthogonal vector to $\left(\begin{array}{c}{g}^{\prime }\\ g\end{array}\right)$ is $\left(\begin{array}{c}g\\ -g^{\prime }\end{array}\right)$. Let's verify this is indeed an eigenvector for $M_Z^2$:

It works perfectly.

The physical fields are the normalized eigenvectors. We define the photon field $A_{\mu }$ as the normalized massless state and the Z boson field $Z_{\mu }$ as the normalized massive state:

This is the explicit mixing of the original gauge fields into the physical particles we observe.

3. The Weinberg Angle

This transformation from the $(W^3,B)$ basis to the $(Z,A)$ basis is a rotation. We can parameterize this rotation with an angle, which we call the Weinberg angle (or weak mixing angle), ${\theta }_W$.

We define the angle such that:

With these definitions, our new fields become:

This expresses the physical fields as a clean rotation of the original interaction fields.

The following reading provides a very clear and standard derivation of this result.

4. Recovering Electromagnetism

We have identified the massless field $A_{\mu }$ as the photon. If this is true, it must couple to the familiar electric charge Q via the electromagnetic coupling e. Let's see if this is the case.

The interaction part of the general covariant derivative for a field with weak isospin $T^3$ and hypercharge $Y$ is:

To see how this works in the physical basis, we must invert our rotation to express $W^3$ and $B$ in terms of $Z$ and $A$:

Substituting these into the covariant derivative gives:

Now, let's group the terms for the $A_{\mu }$ and $Z_{\mu }$ fields separately.

Coupling to $A_{\mu }$:

Using our definitions $\sin {\theta }_W=g^{\prime }/\sqrt{...}$ and $\cos {\theta }_W=g/\sqrt{...}$, we find a remarkable equality:

So, $g\sin {\theta }_W=g^{\prime }\cos {\theta }_W$. Let's define this common coupling as the electric charge coupling, $e$.

The coupling term for $A_{\mu }$ now becomes:

In the last lesson, we identified the unbroken generator as $Q=T^3+Y$, which we call the electric charge. So, the coupling is simply $-ieQ{A}_{\mu }$. This is precisely the interaction term for electromagnetism! The field $A_{\mu }$ is indeed the photon, and we have unified the electroweak couplings $g$ and $g^{\prime }$ with the electric charge coupling $e$.

This is a profound result, showing how electromagnetism emerges from a more fundamental electroweak structure.

For your interest, given your background in computational physics, the dissertation "Shift on W boson mass" contains Python code in Appendix E to numerically diagonalize the $4\times 4$ mass matrix for all $(W^1,W^2,W^3,B)$ fields. This provides a nice practical verification of the analytical diagonalization we have performed here for the neutral sector.

Conclusion

Key Takeaways:

• The physical neutral gauge bosons, the photon ($A_{\mu }$) and the $Z_{\mu }$ boson, are orthogonal mixtures of the original $SU(2{)}_L$ field $W_{\mu }^3$ and $U(1{)}_Y$ field $B_{\mu }$.
• This mixing is a rotation described by the Weinberg angle, ${\theta }_W$, which is determined by the ratio of the gauge couplings: $\tan {\theta }_W=g^{\prime }/g$.
• The photon corresponds to the massless eigenvector of the neutral mass matrix and couples to the unbroken $U(1{)}_{EM}$ charge, $Q=T^3+Y$.
• This framework unifies the fundamental forces, providing a relationship between the weak couplings ($g,g^{\prime }$) and the electromagnetic coupling $e$: $e=g\sin {\theta }_W=g^{\prime }\cos {\theta }_W$.

We have now fully established how the gauge bosons of the electroweak theory acquire their physical identities and masses. The picture is almost complete. The final piece of the puzzle is understanding how the matter fields themselves get their mass.

Preview of the Next Lesson:

In our next lesson, we will complete the story of mass generation in the Standard Model by introducing Yukawa couplings to the Higgs field and showing how they generate fermion masses after spontaneous symmetry breaking. This will explain why quarks and leptons, like the electron, have mass.