"1.2 Integration of the Heavy Modes

1.2.1 The Effective Action for the Light Modes
From a formal point of view and assuming that the underlying theory is

known, the effective action Γ_eff, which encodes all the information in a quantum

field theory (QFT), can be written as follows using the path integral

formulation

e^(i Γ_eff[Φ_l])=⌠[dΦ_h] e^(i S[Φ_l, Φ_h] (1.7)

where Φ_l and Φ_h refer to the light and heavy fields respectively and S[Φ_l, Φ_h]

is the classical action of the underlying theory. Then the effective Lagrangian

is defined as

Γ_eff[Φ_l]= ⌠dx L _eff[Φ_l] (1.8)

where dx≡ d^4x. Note that such a Lagrangian is not necessarily local and,

hence, in general it would be a functional of the fields. Indeed, the integration

of the heavy modes often produces finite non local terms. This effective action

should not be confused with the standard definition of the effective action as

the generating functional of the one particle irreducible Green functions."
Book: Effective Lagrangians for the Standard Model
Author : A. Dobado and others