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] $^{28}$: 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: 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 . 28 A supersymmetric (SUSY) generalization of the Coleman-Weinberg formula will be presented in Sec. III E 1. 29 Technically natural small mass scales are protected by symmetries, such that when the small mass vanishes it can not be generated in any order of perturbation theory. 46 V$_{eff}$ ( φ ) = V$_{inf}$ ( φ ) + Δ V Δ V = 1 64 π 2 ∑ i ( − ) $^{F$_{i}$}$M$_{i}$ ( φ ) $^{4}$ln M$_{i}$ ( φ ) 2 Λ( φ ) 2 , (98) L = (1 / 2) ∂$_{µ}$φ∂ $^{µ}$φ +$^{N}$ − 1 ∑ j =0 ¯ ψ$_{j}$iγ $^{µ}$∂$_{µ}$ ψ$_{j}$ + [ m$_{0}$ + ϵe i ( φ/f +2 πj/N $^{)}$] ¯ ψ$_{j L}$ψ$_{j R}$ + h.c. , (99) Δ V = − N − 1 ∑ j =0 M 4 j 16 π ln ( M 2 j Λ 2 ) , (100)