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to obtain inflationary potential. A simple calculation of the virtual effects of the heavy scalars and fermions on the light scalar potential, which can be the inflaton, can be ob- tained by matching the one-loop corrected effective potential for the full theory. This gives the following result following Coleman-Weinberg [360] $^{28}$: 

<img src='content_image/329478.jpg'>

where V$_{inf}$ is now the renormalized potential, Λ( φ ) is the renormalization mass scale. The sum extends over all helicity states i , F$_{i}$ is the fermion number, and M ( φ ) is the mass of the i-th state. 

As an example, let us consider a chiral Lagrangian for N fermions invariant under Z$_{N}$ symmetry [361]. In general, Z$_{N}$ symmetry can be defined for an Abelian and non-Abelian gauge group, which keeps the action, S , invariant under the symmetry transformation: S −→ e 2 πij/N × S , where j = 1 , 2 , · · · , N . The relevant Lagrangian is: 

<img src='content_image/329479.jpg'>

where ψ$_{(}$$_{R, L}$$_{)}$ = (1 ± γ $^{5}$) ψ/ 2, m$_{0}$ is an explicit breaking term, and the scalar VEV responsible for generating ϵ via some Yukawa interaction between scalar and N fermions is given by: 〈 φ 〉 = e iφ/f $_{f/}$√$_{2. Under the}$ Z$_{N}$ discrete symmetry: ψ$_{j}$ → ψ$_{j}$$_{+1}$$^{, ψ$_{N}$}$− 1 → ψ$_{0}$, φ → φ + 2 πjf/N . The induced one-loop potential can be calculated from Eq. (98), with a cut-off: λ < f . The potential is: 

where M 2 j = m 2 $_{0}$ϵ 2 + 2 m$_{0}$ϵ cos ( φ/f + 2 πj/N ). The scalar field, φ , is a simple example of pseudo-Nambu Goldstone Boson (pNGB), which can protect its potential naturally due to symmetries $^{29}$. For N = 2 the pNGB mass is m$_{φ}$ ∼ m$_{0}$ϵ/f . 

<img src='content_image/329480.jpg'>

