Homotopy Transfer and Effective Field Theory I: Tree-level

TL;DR

This paper develops an algebraic framework using homotopy transfer and $L_$-algebras to describe the process of integrating out degrees of freedom in field theories at tree level, with potential extensions to loop level.

Contribution

It introduces explicit formulas for homotopy transfer in $L_$-algebras, linking algebraic structures to effective field theory procedures at tree level.

Findings

Provides explicit formulas for homotopy transfer in $L_$-algebras.

Shows how to integrate out degrees of freedom algebraically at tree level.

Lays groundwork for extending the framework to loop level in future work.

Abstract

We use the dictionary between general field theories and strongly homotopy algebras to provide an algebraic formulation of the procedure of integrating out of degrees of freedom in terms of homotopy transfer. This includes more general effective theories in which some massive modes are kept while other modes of a comparable mass scale are integrated out, as first explored by Sen in the context of closed string field theory. We treat $L_\infty$-algebras both in terms of a nilpotent coderivation and, on the dual space, in terms of a nilpotent derivation (corresponding to the BRST charge of the field theory) and provide explicit formulas for homotopy transfer. These are then shown to govern the integrating out of degrees of freedom at tree level, while the generalization to loop level will be explored in a sequel to this paper.