Wing Sheung Chan

20 The Standard Model and lepton flavour violation acquire their masses from two types of mass terms in the Lagrangian, namely the Dirac mass term and the Majorana mass term. A Dirac mass term has the basic structure of m ¯ ψψ in the Lagrangian. Since it is an interaction of a field with its conjugation, the mass term conserves charges, flavours and number of particles. Any fermions could have a Dirac mass term. In the SM, the Yukawa interaction of the Higgs field and charged fermion fields is a Dirac mass term. A Majorana mass term has the basic structure of mψ T ψ , without the conjugation as in a Dirac mass term. As apparent, any quantum numbers that ψ carries are not conserved with the Majorana mass term. If the conservation of charge is not to be violated, only particles that are antiparticles of themselves could have Majorana mass. Fermions with such a property are called Majorana fermions. In the SM, only neutrinos could possibly be Majorana fermions. If neutrinos only have Dirac mass like the other fermions in the SM, then the Yukawa coupling of the neutrinos must be many orders of magnitude smaller than that of the other SM fermions. Hence, there is a fine-tuning problem in such a model: there lacks a natural explanation for the smallness of the neutrino-Higgs Yukawa coupling. However, the problem could be avoided if neutrinos are indeed Majorana fermions and there exist some mechanisms other than the SM Yukawa interactions that give mass to the neutrinos. One type of commonly studied models are models with heavy right-handed neutrinos [38] . Right-handed neutrinos can couple with the left-handed neutrinos via the SM-like Yukawa coupling, giving a Dirac mass term. On top of that, if neutrinos are Majorana fermions, the right-handed neutrinos may also have a Majorana mass term. The relevant terms in the Lagrangian after the electroweak symmetry breaking can be expressed as: L ν = − 1 2 m D ¯ ν L N R − 1 2 M R N T R N R + h.c. , (1.27) where N R denotes the conjectured right-handed neutrinos. This could also be represented using a mass (block) matrix in the basis of { ν L , N R } : M ν = 0 m D m T D M R ! . (1.28) The null matrix at the top left represents that there is no Majorana mass term for the left-handed neutrinos at tree level. This is important since such a term will clearly violate the isospin SU(2) symmetry for left-handed leptons. The off-diagonal terms are the Dirac masses, which have an origin similar to the charged fermion masses in the SM. It is natural to assume that these masses are also at the electroweak scale. The bottom-right Majorana mass term for right-handed neutrinos could have an origin from BSM physics at a much higher energy scale. This could be, for example, the SU(5), SO(10) or E 6 model of a Grand Unification Theory. The values of M R will then naturally be at the scale where New Physics appears.