QCD-like theories at next-to-next-to-leading order with $N_F=2$ non-degenerate fermions

QCD-like theories with $N_F=2$ fermion flavours in real and pseudoreal representations are studied within Chiral Perturbation Theory. For the pseudoreal symmetry-breaking pattern $SU(4)\mathrm{/}Sp(4)$, the reduced NLO Lagrangian is derived. The NLO and NNLO corrections to the pion masses, decay constants, and vacuum condensates are calculated for non-degenerate fermion masses, extending previous results obtained for degenerate masses and for the non-degenerate case at NLO. Using the available spectroscopic and scattering lattice data for the $Sp(N_c=4)$ gauge theory with two fermion flavours, fits of the NLO low-energy constants are performed at NNLO precision. It is found that higher-order corrections are important for reproducing lattice observables and have a significant impact on the phenomenology of strongly interacting pionic dark matter, particularly in the regime of big $M_{\pi }\mathrm{/}{F}_{\pi }$.