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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?

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

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

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  • $\begingroup$ 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. $\endgroup$ Dec 15, 2019 at 23:22
  • $\begingroup$ If $H$ is a triplet field you need to use the $3 \times 3$ representation of $\mathrm{SU(2)}$ and its associated generators. $\endgroup$
    – DavidH
    Dec 16, 2019 at 1:00
  • $\begingroup$ @DavidH, what about the Yukawa coupling between the triplet H and the left-handed isospin doublet? $\endgroup$
    – MadMax
    Dec 16, 2019 at 19:10

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