Exploring the Dirac Fields in Srednicki (Ch88)

[LAHLH]

Hi,

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

The covariant derivaties are thus

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

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

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

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

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

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

 

[AndyW]

Hi,

You are correct in your interpretation of the first two terms in the kinematic Lagrangian. The third term, however, can also be written in terms of the Dirac fields. This is because the Majorana field N can be expressed as a linear combination of the Dirac fields \varepsilon and \bar{\varepsilon}. In particular, we have:

N = N_L + N_R = \left(\begin{array}{c} \nu \\ \nu^{\dagger} \end{array}\right) + \left(\begin{array}{c} 0 \\ \nu \end{array}\right) = \left(\begin{array}{c} \nu \\ \nu \end{array}\right) = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)\left(\begin{array}{c} \nu \\ \nu \end{array}\right) = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)\varepsilon

Therefore, the third term in the kinematic Lagrangian can be written as i\bar{N_R}\gamma^{\mu}D_{\mu}N_R = i\bar{\varepsilon}\gamma^{\mu}D_{\mu}\varepsilon.