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An effective action for a scalar field expanded about a classical solution θ $^{i}$( x ) = ϑ $^{i}$( x ) + φ $^{i}$( x ) /M$_{P}$, g$_{µν}$ ( x ) = ˆ g$_{µν}$ ( x ) + h$_{µν}$ ( x ) /M$_{P}$ can be given by [290]: 

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where ˆ L eff is the classical Lagrangian density evaluated at the background configuration, φ and h$_{µν}$ are the small perturbations around the background scalar field and the metric. The sum over n runs over the labels for a complete set of interactions, O $_{n}$, each of which involves N$_{n}$ = N ( φ ) n + N ( h ) n ≥ 2 powers of the fields φ i (for N fields) and h$_{µν}$ (with N$_{n}$ ̸ = 1). The parameter d$_{n}$ counts the number of derivatives appearing in O $_{n}$, and c$_{n}$/M d$_{n}$ is a dimensionless quantity, where M ≪ M$_{P}$ denotes the scale at which heavy degrees of freedom have been integrated out. The overall prefactor, M $^{2}$M2 $_{p}$, is to keep the kinetic terms canonical for individual fields. 

In terms of dimensionless couplings, λ$_{n}$ , the scalar part of the potential can be expanded as $^{25}$: 2 4 

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Note that the natural scale for the scalar masses under the above assumptions is m ≃ v $^{2}$/M$_{P}$ . The quartic coupling constant, λ$_{4}$ ( v/M$_{P}$ ) $^{4}$, is similarly Planck suppressed. For the purpose of inflation, the scale of the scalar potential will be governed by V ∼ v 4 ≪ M $^{4}$. Such small masses and couplings are needed to keep the successes of inflation. 

In general, in order to study the evolution of couplings, c$_{n}$ , and how the two scales M and M$_{P}$ appear in a given problem, one has to study the 1 − particle irreducible (1PI) graphs perturbatively in the interaction part of the Lagrangian density. To this end, one can obtain the leading order expression for the amplitude of such graphs, by assuming that E denotes the largest of physical scales that appear explicitly in the propagators or vertices of the calculation. One can neglect any other smaller scales compared with E when estimating the size of a particular Feynman graph [288, 290, 354]. 

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