content_image/340846.jpg content_image/340847.jpg content_image/340848.jpg
Similar couplings are used to generate mass terms for the charged leptons, the neutrino has no right handed partner, it remains massless within SM. 

In order to obtain neutrino masses, one would have to introduce right handed neutrinos with a Yukawa coupling, i.e. L ∼ h ( LH ) ν$_{R}$ . From the VEV of the Higgs the neutrinos will obtain masses ∝ hv . In order to get neutrino masses in the interesting range m$_{ν}$ ∼ 10 − 1 eV, for solar and atmospheric neutrino mixing, the Yukawa coupling has to be very tiny, i.e. h ∼ 10 − $^{12}$. The origin of neutrino masses can also arise from higher dimensional lepton number violating operator [347–350]: 

<img src='content_image/340846.jpg'>

When the Higgs gets a VEV, these gives rise to Majorana masses for the neutrinos of order 

<img src='content_image/340847.jpg'>

In order to get neutrino masses in the interesting range we require M ∼ 10 14 GeV, remark- ably close to the GUT scale. 

## B. Radiative corrections in an effective field theory 

An effective field theory is a powerful tool to study the loop corrections of the scalars coupled to gravity [288, 290, 351–357]. The most general action for N dimensionless scalar fields, θ $^{i}$, and the metric g$_{µν}$ can be written in terms of derivative expansion, with terms involving up to two derivatives, see [288, 290, 353, 354]: 

<img src='content_image/340848.jpg'>

where M ≪ M$_{P}$ is regarded as the lightest mass scale of particles which would be integrated out, the coefficient functions, V ( θ ), G$_{ij}$ ( θ ) , A ( θ ) , B ( θ ) , C ( θ ) , E ( θ ) , F ( θ ) are dimensionless, and R 3 collectively represents all possible independent invariants constructed from three Riemann tensors, or two Riemann tensors and two of its covariant derivatives; R ( ∂θ ) 2 de- notes all possible invariants involving one power of the Riemann tensor and two derivatives acting on θ $^{i}$. 

