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I have some points not so clear for me about the fermions mass terms in SM; first, why

## \bar{\psi}\psi = \bar{\psi}_L \psi_R + \bar{\psi}_R \psi_L ##, that since

## \bar{\psi} = \bar{\psi}_R + \bar{\psi}_L ## and

## \psi = \psi_R + \psi_L ##

Where are such terms: ## \bar{\psi}_R \psi_R , \bar{\psi}_L \psi_L ##

Second: Why in SM the Majorana mass terms has not taken into account, that since term as:

## m~ \bar{\psi}_L \psi_R ## is not ## SU(2)_L ## invariant, a term as

## m~ \bar{\psi}_L \bar{\psi}^c_L, ~ m ~\bar{\psi}_R \bar{\psi}^c_R ## are gauge invariant. I mean could not Majorana mass terms replace the Higgs mechanism ?

Last: I read in literatures that ## m~ \bar{\psi}_L \psi_R ##, is not a re-normalizable term, but I don't see why , if ## \psi ## has dimensions of mass equals 3/2, then the the mass dimension of this term is only 4 !

Thanks,

S.