1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted