content_image/353884.jpg content_image/353885.jpg content_image/353886.jpg content_image/353887.jpg
Let us follow the discussion of [582], where they begin by considering the simplest model which involves a pNGB θ which comes from the breaking of a global SO (2) symmetry. After integrating out the “radial” degree of freedom and pushing the cutoff of this non-linear sigma model to the point where the interactions become strongly coupled, namely Λ ∼ 4 πf . The inflaton, Φ, can be parameterized as 

Let us consider the tree-level potential to be: 

where σ T = ( σ$_{1}$ σ$_{2}$ ) and τ$_{1}$ is the first Pauli matrix. Let us consider a simple situation when g$_{1}$, g$_{2}$ = g ̸ = 0. From expanding out the Φs in the potential, one finds: 

<img src='content_image/353884.jpg'>

<img src='content_image/353885.jpg'>

<img src='content_image/353886.jpg'>

Now computing the one-loop corrections to the mass of θ , the authors of Ref. [582] obtained that there is no one-loop quadratic divergent contribution to a θ mass. This is because θ only couples to the combination σ$_{1}$σ$_{2}$ making it impossible to close a loop with only one vertex. There is a logarithmic divergence at one loop proportional to g$_{1}$g$_{2}$ = g 2 

<img src='content_image/353887.jpg'>

The value of Λ could be as large as M$_{P}$ or below, but the corrections to the potential is only logarithmic dependent. The pNGB inflaton could also originate in SUSY inflation models and in extra dimensional models [582, 584, 585]. 

Some of the above mentioned challenges can be addressed if inflation is explicitly em- bedded within an observable sector. One such example of inflaton is the SM Higgs in a non-SUSY context. 

