W_L-W_R Mixing in the Left-Right Symmetric Model

[andres0l1]

Homework Statement

The question is how to get the mass term for the W Gauge bosons within the Left Right Symmetric Model (LRSM). My main struggle is with the Left-Right Mixing part.
So the LRSM has the gauge group $SU(2)_L\times SU(2)_R \times U(1)_{B-L}$ and with an additional discrete symmetry, we can make the left and right gauge couplings to be equal $g_L=g_R=g$. Now, the Higgs sector of the theory has the following Vacuum Expectation Values (VEV):
$$
\ev{\phi}=\left(\begin{array}{c c }
k_1 & 0 \\
0 & k_2
\end{array}\right), \,\, \ev{\Delta_{L,R}}=\left(\begin{array}{c c }
0 & 0 \\
\kappa_{L,R} & 0
\end{array}\right),
$$
Where the $\phi$ is a doublet under both $SU(2)_L$ and $SU(2)_R$, the $\Delta_L$ is a triplet under $SU(2)_L$ and transform in the trivial representation of $SU(2)_R$ and vice versa for the $\Delta_R$ Higgs. Therefore, $\tau^a=\frac{\sigma^a}{2}$ for the $\phi$ terms.
What is the mass term for the W bosons? in particular the mixing term?

2. Homework Equations

$$
D_\mu\phi=\partial_\mu\phi + ig_LW_{L,\mu}^a \tau^a\phi + ig_RW_{R,\mu}^a \tau^a+ ig'(B-L)B_\mu\phi
$$
$$
D_\mu\Delta_L=\partial_\mu + ig_LW_{L,\mu}^a \tau^a\Delta_L + ig'(B-L)B_\mu\Delta_L
$$
$$
D_\mu\Delta_R=\partial_\mu + ig_RW_{R,\mu}^a \tau^a\Delta_R + ig'(B-L)B_\mu\Delta_R
$$
The kinetic term in the lagrangian which gives mass to the gauge bosons is
$$
\mathcal{L}=Tr[(D^\mu\phi)^\dagger(D_\mu\phi) + (D^\mu\Delta_L)^\dagger(D_\mu\Delta_L) +
(D^\mu\Delta_R)^\dagger(D_\mu\Delta_R)]
$$
where the Tr is over the 2x2 matrices.

3. The Attempt at a Solution

So if we only focus on the $W^1,W^2$ terms which are the ones that give mass to the charged W bosons. I get pairs of the type $W_L,W_L$ and $W_RW_R$
The mixing will be given by the interacion with the $\phi$ field so let us only focus on it.
Within it, the mixing terms are of the form

$$W_L^aW_R^b\sigma^a\sigma^b +W^a_RW^b_L\sigma^a\sigma^b$$.

Using the properties of the sigma matrices, this is reduced to the unitary matrix multiply by $2W_L^1W_R^1 + W_L^2W_R^2 + W_L^3W_R^3$ which, when multiply by the $\phi$ at its VEV's gives

$$(k_1^2 +k_2^2)2[W_L^1W_R^1 + W_L^2W_R^2 + W_L^3W_R^3]$$

When I should be getting something of the type

$$(k_1k_2)2[W_L^1W_R^1 + W_L^2W_R^2] + 2(k_1^2+k_2^2)W_L^3W_R^3$$

What am I doing wrong?

 

[andres0l1]

Found the solution! I think this can be deleted. Thanks!