Low energy effective theories of composite dark matter with real representations

We consider pseudo Nambu-Goldstone bosons arising from Dirac fermions transforming in real representations of a confining gauge group as dark matter candidates. We consider a special case of two Dirac fermions and couple the resulting dark sector to the Standard Model using a vector mediator. Within this construction, we develop a consistent low energy effective theory, with special attention to Wess-Zumino-Witten term given the topologically non-trivial coset space. We furthermore include the heavier spin-0 flavour singlet state and the spin-1 vector meson multiplet, by using the Hidden Local Symmetry Lagrangian for the latter. Although we concentrate on special case of two flavours, our results are generic and can be applied to a wider variety of theories featuring real representations. We apply our formalism and comment on the effect of the flavour singlet for dark matter phenomenology. Finally, we also comment on generalisation of our formalism for higher representations and provide potential consequences of discrete symmetry breaking.