Abstract
Building on an older method used to derive non-decoupling effects of a heavy Higgs boson in the Standard Model, we describe a general procedure to integrate out heavy fields in the path integral. The derivation of the corresponding effective Lagrangian including the one-loop contributions of the heavy particle(s) is particularly transparent, flexible, and algorithmic. The background-field formalism allows for a clear separation of tree-level and one-loop effects involving the heavy fields. Using expansion by regions the one-loop effects are further split into contributions from large and small momentum modes. The former are contained in Wilson coefficients of effective operators, the latter are reproduced by one-loop diagrams involving effective tree-level couplings. The method is illustrated by calculating potential non-decoupling effects of a heavy Higgs boson in a singlet Higgs extension of the Standard Model. In particular, we work in a field basis corresponding to mass eigenstates and properly take into account non-vanishing mixing between the two Higgs fields of the model. We also show that a proper choice of renormalization scheme for the non-standard sector of the underlying full theory is crucial for the construction of a consistent effective field theory.
Highlights
Particular strengths of the method are: (i) a clear separation of tree-level and loop effects of the heavy fields; (ii) the possibility to fix the gauge in intermediate steps of the calculation and to restore gauge invariance of the effective Lagrangian at the end; (iii) transparency in the sense that at each stage of the calculation it is possible to identify the origin of all contributions to the effective Lagrangian in terms of Feynman diagrams; (iv) flexibility due to the fact that no ansatz is made for the effective Lagrangian
In this article we have described a general procedure to integrate out heavy fields directly in the path integral and to derive an effective Lagrangian at the one-loop level
The method is based on the background-field formalism, which implies a natural separation of tree-level and loop effects of the heavy fields, and on the expansion by regions, which further separates loop effects into contributions from large and small momentum modes
Summary
The method described in the following is a further development of the method introduced in Refs. [14,15], where a heavy Higgs field was integrated out in an SU(2) gauge theory and the SM, respectively, directly in the path integral. [29,39]) within a procedure to calculate the different parts in the effective Lagrangian, but we consider our formulation in terms of heavy and light modes of background and quantum fields and their different treatments in the path integral conceptually more transparent. They can conveniently be removed from the effective Lagrangian by applying their equations of motions (EOMs) in the large-mass expansion This procedure can be viewed as a saddle-point approximation in the path integral over the light modes of the heavy quantum field combined with a large-mass expansion. Parts (i) and (ii) combine to a single effective Lagrangian at lowest order in the coupling constants, which can be used to evaluate tree-level amplitudes at different orders in the large-mass expansion and one-loop contributions resulting from insertions of effective vertices in loop diagrams (reproducing the soft momentum regions of loops in the full theory). In a decoupling scenario this means that the final effective Lagrangian differs from the SM only by operators with dimensions higher than four
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