MODERN COSMOLOGY

The SM of particle physics 191

5.2.2 Fermion masses

Fermions are spinors with respect to the Lorentz groupSU( 2 )⊗SU( 2 ).Weyl
spinors are two-component spinors which transform under the Lorentz group:

χL as(^12 , 0 ) (5.10)

ηR as( 0 ,^12 ) (5.11)

and therefore are said to be left-handed and right-handed respectively.
A fermion mass term must be invariant under the Lorentz group. We have
two possibilities:

(i) A Majorana mass term couples just one spinor with itself:

χαχβαβ or η ̇αη

β ̇

α ̇β ̇. (5.12)

It is not invariant under any local or global symmetry under which the field

transforms not trivially;

(ii) A Dirac mass term involves two different spinorsχLandηR:

χαη ̄βαβ or χ ̄α ̇ηβ ̇α ̇β ̇. (5.13)

This can be present even if the fields carry quantum numbers.

In the SM Majorana masses are forbidden by the gauge symmetry; in fact
we have that, for example,

eLeL⇒Q= 0

νLνL⇒SU( 2 )L=

andSU( 2 )Lforbids Dirac mass terms:

eLeR⇒SU( 2 )L=. (5.14)

Therefore no direct mass term can be present for fermions in the SM.
However, when the gauge symmetry breaks spontaneously the Yukawa
couplings provide Dirac mass terms to fermions which read:

LMmat=+

1

√

2

λeve ̄LeR+

1

√

2

λuvu ̄LuR+

1

√

2

λdvd ̄LdR+h.c. (5.15)

with masses:

me=

1

√

2

λev mu=

1

√

2

λuv md=

1

√

2

λdv. (5.16)

We note that neutrinos are massless and so remain at any order in
perturbation theory: