Article Dilaton Effective Field Theory Thomas Appelquist, 1 , James Ingoldby 2 , and Maurizio Piai 3   Citation: Appelquist, T.; Ingoldby, J.; Piai, M. Dilaton Effective Field Theory . Preprints 2021, 1, 0. https://doi.org/ Received: Accepted: Published: Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. 1 Department of Physics, Sloane Laboratory, Yale University, New Haven, Connecticut 06520, USA 2 Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151, Trieste, Italy 3 Department of Physics, Faculty of Science and Engineering, Swansea University (Singleton Park Campus), Singleton Park, SA2 8PP Swansea, Wales, United Kingdom Abstract: We review and extend recent studies of dilaton effective field theory (dEFT) which provide a framework for the description of the Higgs boson as a composite structure. We first describe the dEFT as applied to lattice data for a class of gauge theories with near-conformal infrared behavior. It includes the dilaton associated with the spontaneous breaking of (approximate) scale invariance, and a set of pseudo-Nambu-Goldstone bosons (pNGBs) associated with the spontaneous breaking of an (approximate) internal global symmetry. The theory contains two small symmetry-breaking parameters. We display the leading-order (LO) Lagrangian, and review its fit to lattice data for the SU(3) gauge theory with N f = 8 Dirac fermions in the fundamental representation. We then develop power-counting rules to identify the corrections emerging at next-to-leading order (NLO) in the dEFT action. We list the NLO operators that appear and provide estimates for the coefficients. We comment on implications for composite-Higgs model building. Keywords: lattice gauge theory, physics beyond the standard model 1. Introduction It has long been thought that the dilaton, the neutral Nambu-Goldstone boson (NGB) arising from the spontaneous breaking of scale invariance, might play a role in fundamental physics—see, e.g., Refs [1,2]. The idea is intriguing yet elusive. If an approximate symmetry under scale transformations sets in over some energy range, and if the forces are such that this symmetry is not respected by the ground state (vacuum) of the system, then the appearance of an approximate, light dilaton would seem natural. It is easy to realize this possibility at the classical level, there being no better example than the Higgs potential of the standard model (SM) with its minimum at a vacuum expectation value (VEV) v W > 0 of the Higgs field. The Higgs particle becomes lighter as the self–coupling strength is reduced with fixed v W . In this limit, the explicit breaking of scale invariance by the Higgs mass becomes smaller, and the breaking dominantly spontaneous, due to the VEV v W . The Higgs particle can then be viewed as an approximate dilaton at the classical level. The dilaton idea becomes more subtle at the quantum level. At either level, it makes sense only if the explicit breaking is relatively small, so that there is an approximate scale invariance (dilatation symmetry) to be broken spontaneously. In a quantum field theory, explicit breaking arises not only from the fixed dimensionful parameters in the Lagrangian, but also through the renormalization process. For example, the quantum corrections to the Higgs potential can lead to large contributions to the Higgs mass, requiring fine tuning to maintain its lightness. Yet, its interpretation as a dilaton has striking implications for the standard model as well as its extensions [3]. In a gauge theory like quantum chromodynamics (QCD), renormalization leads to a confinement scale Λ, explicitly breaking dilatation symmetry. Approximate scale invariance sets in only at higher energies, while the vacuum structure and composite-particle spectrum are determined at scales of order Λ itself. There is no reason to expect the appearance of arXiv:2209.14867v3 [hep-ph] 1 Dec 2022