Gauge and Lorentz invariance for Lagrangians

[spaghetti3451]

Consider the following Lagrangian:

$YHLN_{1}^{c} + Y^{c}H^{\dagger}L^{c}N_{1} + \text {h.c.},$

where $L=(N_{0}, E')$ and $L^{c} = (E^{'c}, N_{0}^{c})$ are a pair of $SU (2)$ doublets and $N_{1}$ and $N_{1}^{c}$ are a pair of neutral Majorana fermions.

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1. In the first term, $H$ and $L$ are column vectors. How do you multiply two column vectors in the first term of the Lagrangian?

2. Are $L$ and $L^{c}$ $4$-component spinors? Are $N_{1}$ and $N_{1}^{c}$ also $4$-component spinors? How do the components of the vectors and spinors in the first term multiply?

 

[nrqed]

Consider the following Lagrangian:

$YHLN_{1}^{c} + Y^{c}H^{\dagger}L^{c}N_{1} + \text {h.c.},$

where $L=(N_{0}, E')$ and $L^{c} = (E^{'c}, N_{0}^{c})$ are a pair of $SU (2)$ doublets and $N_{1}$ and $N_{1}^{c}$ are a pair of neutral Majorana fermions.

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1. In the first term, $H$ and $L$ are column vectors. How do you multiply two column vectors in the first term of the Lagrangian?

2. Are $L$ and $L^{c}$ $4$-component spinors? Are $N_{1}$ and $N_{1}^{c}$ also $4$-component spinors? How do the components of the vectors and spinors in the first term multiply?

Can you give a reference for this expression? Is it a supersymmetric system? I am not sure what you mean by "H and L are column vectors", it could mean SU(2) doublets or it could mean spinors. In either case, there is something that is not working so the original reference would be helpful.