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Writing lepton sector kinematic term in terms of N_L, E_L projections of Dirac fields

  1. Feb 10, 2012 #1

    In srednicki (ch88) he starts off considering the electron and associated neutrino, by introducing the left handed Weyl fields [itex]l, \bar{e} [/itex] in the representations (2,-1/2), (1,+1) of SU(2)XU(1).

    The covariant derivaties are thus

    [itex] (D_{\mu}l)_i=\partial_{\mu}l_i-ig_2A_{\mu}^a(T^a)_i^jl_j-ig_1(-1/2)B_{\mu}l_i[/itex] and [itex]D_{\mu}\bar{e}=\partial_{\mu}\bar{e}-ig_1(+1)B_{\mu}\bar{e} [/itex] where the T's are SU(2) gens and Y=-1/2I for l, and Y=+1 for the [itex]\bar{e} [/itex]. He then relabels the SU(2) components of [itex]l=\left( \begin{array}{c} \nu\\ e\end{array} \right) [/itex] and defining the Dirac field [itex]\varepsilon=\left( \begin{array}{c} e\\ \bar{e}^{\dagger}\end{array} \right) [/itex] and finally the Majorana feld for the neutrino:[itex]N=\left( \begin{array}{c}
    \nu\\ \nu^{\dagger}\end{array} \right) [/itex].

    Now the kinematic term in the Lagrangian starts life in terms of the Weyl fields looking like:
    [itex] L_{\text{kin}}=i\nu^{\dagger}\bar{\sigma}^{\mu}(D_{\mu}\nu)+ie^{\dagger}\bar{\sigma}^{\mu}(D_{\mu}e)+i\bar{e}^{\dagger}\bar{\sigma}^{\mu}(D_{\mu}\bar{e}) [/itex]

    My question is how exactly do I rewrite this in terms of the Dirac fields?

    I believe if you define [itex] N_L:=P_L N=\left( \begin{array}{c} \nu\\ 0\end{array} \right) [/itex] then the first term can be expressed as [itex]i\bar{N_L}\gamma^{\mu}D_{\mu}N_L [/itex]. This is the case because [itex] \bar{N_L}=(0, \nu^{\dagger}) [/itex] so we get:

    [tex] i\bar{N_L}\gamma^{\mu}D_{\mu}N_L =i(0, \nu^{\dagger})\left( \begin{array}{cc} 0 & \sigma^{\mu}\\ \bar{\sigma}^{\mu}&0\end{array} \right)\left( \begin{array}{c} D_{\mu}\nu\\ 0\end{array} \right) =i\nu^{\dagger}\bar{\sigma}^{\mu}(D_{\mu}\nu)[/tex]

    The second term of the kinematic Lagrangian is similarly [itex] i\bar{\varepsilon_L}\gamma^{\mu}D_{\mu}\varepsilon_L [/itex] as far as I can tell, but then the third term does not seem to fall into such an expression?
  2. jcsd
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