# Glashow-Weinberg-Salam (GWS) Theory for Gauge Boson Masses

I am supposed to find the mass of the $$W^\pm$$ and $$Z$$ bosons from the following covariant derivative:

$$D_\mu = \partial\mu - \frac{ig_1}{2}B^\mu -ig_2t^iW^{i\mu}$$

and from

$$\mathcal{L}_{kin. Higgs}=(D_\mu\Phi^\dagger)(D_\mu\Phi)+\frac{1}{2}(D_\mu H)^\dagger (D_\mu H)$$

where:

$$B_\mu$$ is the $$U(1)_Y$$ gauge boson;

$$g_1$$ is the hypercharge coupling;

$$g_2$$ is the weak coupling;

$$W^{i\mu}$$ are the $$SU(2)_L$$ gauge bosons;

$$t^i$$ the generators of $$SU(2)_L$$ in the appropriate representation.

I know I must calculate the first half of the lagrangian equation, according to the Glashow-Weinberg-Salam Theory of Weak Interactions (Peskin & Schroeder 20.2), but I am not certain how to do so.

I think the $$t^i$$ (the generators of $$SU(2)$$) can be represented as $$i\sigma_i$$, where $$\sigma$$ is a Pauli matrix but if this is the case I don't know how to introduce the Pauli Matrices into the equation. Am I supposed to only introduce one of Pauli matrices? Am I supposed to introduce one at the time and solve the equation three different times?

## 1 Answer

The covariant derivative you wrote down contains a sum over the $$\mathrm{SU(2)}$$ indices, so you need to use all Pauli matrices:

$$W_\mu^i \sigma^i = \begin{pmatrix} W_\mu^3 & W_\mu^1 - i W_\mu^2 \\ W_\mu^1 + i W_\mu^2 & -W_\mu^3 \end{pmatrix}$$

• Thank you that was great help. But how would I use that for H which is a triplet (3x1) as I can't multiply a 2x2 by a 3x1. – user7077252 Dec 15 '19 at 23:22
• If $H$ is a triplet field you need to use the $3 \times 3$ representation of $\mathrm{SU(2)}$ and its associated generators. – DavidH Dec 16 '19 at 1:00
• @DavidH, what about the Yukawa coupling between the triplet H and the left-handed isospin doublet? – MadMax Dec 16 '19 at 19:10