Power counting of the pion-dilaton effective field theory

TL;DR

This paper analyzes different power counting schemes in the pion-dilaton effective field theory, confirming the validity of one approach and critiquing another, to improve understanding of light scalar mesons near the conformal window.

Contribution

It compares proposed power counting methods for the dilaton EFT, validating the systematic approach previously derived and identifying issues with an alternative phenomenological scheme.

Findings

One power counting scheme fails to produce a systematic expansion.

The other scheme matches the previously derived systematic power counting.

The $ riangle$-potential aligns with the invalid power counting's tree-level potential.

Abstract

Confining QCD-like theories close to the conformal window have a ``walking'' coupling. This is believed to lead to a light singlet scalar meson in the low-energy spectrum, a dilaton, which is the pseudo Nambu--Goldstone boson for the approximate scale symmetry. Extending chiral perturbation theory to include the dilaton requires a new small parameter to control the dilaton mass and its interactions. In our previous work we derived a systematic power counting for the dilaton couplings by matching the effective low-energy theory to the underlying theory using mild assumptions. In this paper we examine two alternative power countings which were proposed in the literature based on a phenomenological picture for the conformal transition. We find that one of these power countings fails, in fact, to generate a systematic expansion; the other coincides with the power counting we derived. We also point out that the so-called $\Delta$-potential coincides with the tree-level potential of the former, invalid, power counting.