NLO observables for QCD-like theories and application to pion dark matter

QCD-like theories are of interest in various areas of beyond-Standard-Model phenomenology, including composite Higgs models or pionic dark matter. The effective field theories provide a framework for describing the dynamics of such strongly coupled gauge theories at low energies. In this work, we present next-to-leading order (NLO) expressions for masses, condensates, decay constants, and scattering amplitudes in the chiral expansion of QCD-like theories with $N_F=2$ fermions with non-degenerate masses in both real and pseudoreal representations of the gauge group. We further apply the NLO formulas to fit existing lattice spectroscopic and scattering data for $Sp(N_c=4)$ gauge theory with $N_F=2$ fermions in fundamental representation, extracting the NLO low-energy constants of the theory. Using these fits, we refine the NLO formulas describing the $2\to 2$ pion self-interactions and confirm that the NLO contributions play a crucial role in determining the viable parameter space of pion dark matter scenarios like the strongly interacting massive particles (SIMP).