# Effective field theory approach to Casimir interactions on soft matter surfaces

We utilize an effective field theory approach to calculate Casimir interactions between objects bound to thermally fluctuating fluid surfaces or interfaces. This approach circumvents the complicated constraints imposed by such objects on the functional integration measure by reverting to a point particle representation. To capture the finite-size effects, we augment the Hamiltonian by a term Δ that encapsulates the particles' response to external fields. Δ is systematically expanded in a series of terms, each of which scales homogeneously in the two power counting parameters: λ≡*R*/*r* , the ratio of the typical object size (*R*) to the typical distance between them (*r*), andδ≡*k*B*T*/*k*, where *k* is the modulus characterizing the surface energy. The coefficients of the terms in Δ correspond to generalized polarizabilities and thus the formalism applies to rigid as well as deformable objects. We first illustrate and verify our approach by re-deriving known pair forces between circular objects bound to films or membranes. To demonstrate its efficiency and versatility, we then derive a number of new results, among them the triplet interactions present in these systems, a higher-order correction to the film interaction, and general scaling laws for the leading-order interaction valid for objects of arbitrary shape and internal flexibility.