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W_L-W_R Mixing in the Left-Right Symmetric Model

  1. Apr 21, 2015 #1
    1. The problem statement, all variables and given/known data

    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
    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. Relevant 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) +
    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?

    Attached Files:

  2. jcsd
  3. Apr 22, 2015 #2
    Found the solution! I think this can be deleted. Thanks!
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