Not the SIMPlest Miracle Martin Hansen  , ∗ Kasper Langæble  , † and Francesco Sannino ‡  CP 3 -Origins and the Danish IAS, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark. We investigate the phenomenological viability of a recently proposed class of composite dark mat- ter models where the relic density is determined by 3 → 2 number-changing processes in the dark sector. Here the pions of the strongly interacting field theory constitute the dark matter particles. By performing a consistent next-to-leading and next-to-next-to-leading order chiral perturbative inves- tigation we demonstrate that the leading order analysis cannot be used to draw conclusions about the viability of the model. We further show that higher order corrections substantially increase the tension with phenomenological constraints challenging the viability of the simplest realisation of the strongly interacting massive particle (SIMP) paradigm. Preprint: CP 3 -Origins-2015-025 DNRF90, DIAS-2015-25 I. INTRODUCTION Dark Matter accounts for circa 85% of the matter in the universe, but besides from its cosmological abun- dance, very little is known about its nature. In a wide class of models, the relic abundance is generated via a thermal freeze-out in the early universe. Typi- cally 2 → 2 annihilation processes into e.g. standard model particles keep dark matter in thermal equilib- rium with the standard model bath until the annihila- tion processes drop below the Hubble expansion rate. After this point in time the abundance of dark matter is essentially constant throughout the universe. This constitutes the weakly interacting massive particle (WIMP) paradigm. In a recent paper [1] the authors revived an alter- native mechanism [2, 3] for achieving the observed dark matter relic density. Instead of using 2 → 2 annihilation processes they assume that a dominant 3 → 2 number-changing process occurs in the dark sector involving strongly interacting massive parti- cles (SIMPs). This process reduces the number of dark particles at the cost of heating up the sector. However, the presence of hot dark matter is problem- atic for structure formation, which means that at the time of freeze-out, the dark matter particles must to be in thermal equilibrium with the standard model ones. This, in turn, requires small couplings between the dark and the standard model sectors. In this way the energy from the dark sector can be transferred to the standard model via scattering processes. The coupling between the two sectors allows for direct and indirect detection, while the large self- interactions can play a role in structure formation, by solving the core vs. cusp problem [4]. Compared to the WIMP paradigm, where the dark matter particles typically are believed to be around the TeV scale, this model can yield dark matter particles with masses around a few 100 MeVs. This is an interesting alter- native to the WIMP paradigm given the fact that cur- ∗ Electronic address: hansen@cp3.dias.sdu.dk † Electronic address: langaeble@cp3.dias.sdu.dk ‡ Electronic address: sannino@cp3.dias.sdu.dk rent experiments are putting substantial constraints on this old paradigm. These constraints can, how- ever, be alleviated or even be completely offset within the recently proposed safe dark matter paradigm [5]. A follow-up paper [6] introduced a realization of the SIMP mechanism based on an underlying strongly coupled sector described via chiral pertur- bation theory. In this set-up the pions constitute the dark matter particles and a key role is played by the time-honoured Wess-Zumino-Witten (WZW) term [7–9]. The WZW term is non-vanishing in the- ories where the coset space of the symmetry break- ing pattern has a non-trivial fifth homotopy group. This topological term introduces a 5-point pion inter- action, making it an ideal candidate for the 3 → 2 annihilation process. In QCD, for example, the term describes the annihilation of two kaons into three pi- ons. In this paper we shall only be concerned with the symmetry breaking pattern SU(2N f ) → Sp(2N f ) for N f = 2. The simplest realization of this breaking pat- tern comes from an underlying Sp(2) = SU(2) gauge group, but in general it can be realized for any Sp(N c ) gauge group. The actual pattern of chiral symmetry breaking depends on the number of flavours, colors and matter representation. A comprehensive study of the conformal window for Sp(N c ) gauge groups can be found in [10]. Lattice simulations have fur- ther demonstrated that such an underlying dynamics truly leads to the expected pattern of chiral symme- try [11] with the spectrum of the composite spin-one resonances first computed in [11–13]. The computations performed in [6] make use of the first non-vanishing order in the chiral expansion for the 3 → 2 and 2 → 2 processes. For the 3 → 2 annihi- lation process it is one order higher than the related 2 → 2 self-interaction process. We will demonstrate that it is important to analyse the physical results via a consistent next-to-leading (NLO) and next-to-next-to- leading order (NNLO) chiral perturbative treatment. We will, in fact, show that the leading order analysis is phenomenologically unreliable because it is outside the range of convergence. The important higher order corrections substantially increase the range of conver- gence of the theory, and therefore the phenomeno- logical reach. Within our controllable analysis we discover that higher order corrections increase the arXiv:1507.01590v1 [hep-ph] 6 Jul 2015