where ϵ denotes the number of external lines, L loops and V$_{n}$ vertices with d$_{n}$ derivatives. The factor E/ 4 πM$_{P}$ ≪ 1 ensures that successive insertions of interactions to be smaller than preceding ones. One particular application is integrating out a particle of mass m ≫ M . In the infla- tionary context, M ∼ H$_{inf}$ ∼ v $^{2}$/M$_{P}$ for the above potential Eq. (96), and λ$_{n}$ ∼ O (1). Expanding the above amplitude Eq. (97) for arbitrary derivatives and comparing the coef- ficients with Eq. (95), one finds v $^{4}$/ ( E $^{2}$M2 $_{P}$) ≃ v $^{4}$/ ( m $^{2}$M2 $_{P}$) ≃ H 2 $_{inf}$/m 2 ≪ 1. In addition, the generic loop factor ( m/ 4 πM$_{P}$ ) $^{2}$, the d$_{n}$ ≥ 4 interactions are suppressed by at least two powers of m $^{2}$/M2 $_{P}$, and λ$_{n}$ are additionally suppressed by powers of H 2 $_{inf}$/m $^{2}$. Only the d$_{n}$ = 2 interactions remain unsuppressed beyond the basic loop factor if λ$_{n}$ < ∼ O (1). On the other hand, if there are interactions in the scalar potential that are unsuppressed by powers of M$_{P}$ , such as if λ$_{n}$ ≃ ( M$_{P}$/v ) $^{N$_{n}$}$λ$_{n}$ , i.e. higher powers of φ and h$_{µν}$ , then loops involving these interactions can modify the inflaton potential [290, 353] $^{26}$. One such known example is in the case of Higgs inflation [86], where the presence of ξH † HR (interaction vertex involves d$_{n}$ = 2 derivatives) term helps flattening the Higgs potential $^{27}$. A successful inflation requires ξ ∼ 5 × 10 4 √$_{λ}$$_{, where}$ λ is the Higgs self- coupling. The unitarity bound on interactions such as Higgs-Higgs scattering, graviton- Higgs scattering (through graviton exchange) leads to E < E$_{max}$ ≈ M$_{P}$/ξ , which constraints the validity range for an effective field theory treatment for the Higgs inflation to be within a narrow range √$_{λ}$ ≪ H$_{inf}$/M ≪ 1, where H$_{inf}$ ∼ √$_{λM}$$_{P}$$_{/ξ}$$_{.}$ Any Higgs coupling to the matter field will induce corrections to the Higgs potential and alter the predictions for inflation, see [290]. In other extreme limit, when a light particle is running in the loop, i.e. m ≪ H$_{inf}$ and m ≪ ˙ φ/φ , the above analysis when applied to 2-point correlations, yields a well known result, 〈 φ $^{2}$〉 ∼ H 2 $_{inf}$/ 4 π 2 [290]. The one-loop effective potential, which can be generated when the heavy fields are ex- plicitly integrated out, serves as an useful tool to lift a generic flat direction, which is helpful 26 There are two types of operators; (i) relevant operators , which are proportional to positive powers of M , and (ii) marginal operators , which grow logarithmically with M , have been considered in studying the trans-Planckian physics, see for instance [294, 359]. 27 In an Einstein frame the Higgs potential with canonical kinetic terms is given by Eq. (215), see section IV D. 45