In this section, we describe the model of a pseudo-scalar dark matter. The reader interested in alternative scenarios can find in the appendix the formulas in the case of fermionic dark matter. The model was originally built with success to interpret the recent monochromatic signal observed in different clusters of galaxies [36]. In this model, a scalar or pseudo-scalar particle is by definition a self-interacting particle. The Higgs boson, the unique observed spin 0 particle until now, is a self-interacting particle through its quartic coupling. Several other self-interacting candidates have been proposed in the literature, but usually these were spin 1/2 particles. However, in this case, it becomes necessary to invoke specific processes (like Sommerfeld enhancement, or strong interaction) to compensate for the dimensionality of the 4-fermion couplings. In the case of a scalar or pseudo-scalar dark matter ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}, the self-interaction term λ4|ϕ|4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\lambda }{4} |\phi |^4$$\end{document} is always allowed by a global U(1) invariance and induces then necessarily self-interacting processes. Moreover, in the framework of spontaneous symmetry breaking, a strong correlation exists between the vacuum expectation value (vev) of ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}, its mass, and the quartic coupling λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, rendering the construction very predictive.

The general renormalizable potential for a scalar complex field |Φ|2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Phi |^2$$\end{document} respecting a global U(1) symmetry is31VΦ=-μ2|Φ|2+λ4|Φ|4,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {V}_{\Phi }=-\mu ^2|\Phi |^2+\frac{\lambda }{4}|\Phi |^4, \end{aligned}$$\end{document}where μ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu ^2$$\end{document} is the bare mass of Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} and λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is the quartic coupling of Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document}.

After a spontaneous breaking of the symmetry, it is straightforward to re-express the potential as a function of the fundamental components of Φ=v+s+ia2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi = v + \frac{s + i a}{\sqrt{2}}$$\end{document} with v=⟨Φ⟩=2λμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v= \langle \Phi \rangle = \sqrt{\frac{2}{\lambda }} \mu $$\end{document}. Absorbing the unphysical constants, we obtain2VΦ=ms22s2+λ22mss3+λ22msa2s+λ16s4+λ16a4+λ8a2s2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {V}_{\Phi }= & {} \frac{m_s^2}{2} s^2 + \frac{\sqrt{\lambda }}{2 \sqrt{2}} m_s s^3 + \frac{\sqrt{\lambda }}{2 \sqrt{2}} m_s a^2 s \nonumber \\&+ \frac{\lambda }{16} s^4 + \frac{\lambda }{16} a^4 + \frac{\lambda }{8} a^2 s^2, \end{aligned}$$\end{document}with the scalar mass ms=2μ=λv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_s=\sqrt{2} \mu = \sqrt{\lambda } v$$\end{document}. It is important to notice that if our U(1) symmetry was exact (prior to developing a VEV), the pseudo-scalar dark matter mass ma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_a$$\end{document} would remain massless to all orders in perturbation theory. In the following, we will assume that the U(1) symmetry is broken by non-perturbative effects down to a discrete ZN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_N$$\end{document} symmetry. It is actually standard in string theory that all symmetries are gauged symmetries in the UV.4 Thus a non-zero dark matter mass ma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_a$$\end{document} being much lighter than ms\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_s$$\end{document} is expected.Fig. 1

Feynman diagrams for dark matter self-interacting cross section

In our model, we have four diagrams contributing to the self-interacting cross section as depicted in Fig. 1. Once the scalar part of Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} develops a VEV it becomes possible to re-express the total cross section as3σaama=λ2ma32πms41-4ma2ms22≃λ2ma32πms4,(ms≫ma).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\sigma _{aa}}{m_a}= & {} \frac{\lambda ^2 m_a}{32 \pi m_s^4 \left( 1-4 \frac{m_a^2}{m_s^2}\right) ^2} \nonumber \\\simeq & {} \frac{\lambda ^2 m_a}{32 \pi m_s^4},\quad (m_s \gg m_a). \end{aligned}$$\end{document}It is interesting to note that the cross section is of the form σaa∝ma2/ms4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{aa} \propto m_a^2/m_s^4$$\end{document} and then null for ma=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_a=0$$\end{document}, whereas if one takes into account only the quartic vertex aaaa, it should naively be proportional to 1/ma2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/m_a^2$$\end{document} and could potentially diverge. The mechanism canceling the divergences is in fact similar to the Higgs contribution occurring in the WW scattering in the Standard Model. This can easily be understood as ma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_a$$\end{document} can be considered as the pseudo-Goldstone boson generated by the breaking of the global U(1) symmetry. This fundamental feature would not have been observed in the framework of an effective approach if one introduces a dimensional coupling of the form μ~saa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mu }s aa $$\end{document}, μ~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\mu }$$\end{document} being a free mass parameter. It is thus the dynamical structure of the construction which defines precisely its self-coupling constants. Another interesting point is that, for a MeV scale mediator s, one does not need to invoke very large values of λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} to obtain a self-interacting cross section compatible with recent analysis. For instance, in the case of ma=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_a = 3 $$\end{document} keV and ms=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_s=1$$\end{document} MeV, one obtains σaa/ma≃7λ2cm2/g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{aa}/m_a \simeq 7 \lambda ^2 ~\mathrm {cm^2 /g}$$\end{document}, which is of the order of the measured limit (σ/m≲1cm2/g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma /m\lesssim 1~\mathrm {cm^2/g}$$\end{document}) for a reasonable value of λ≃1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \simeq 1$$\end{document}, much below the perturbativity limit, without invoking velocity enhancement.

Concerning the coupling to the photons, we consider the coupling which can be written as4Lsγγ=sΛFμνFμν,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {L}_{s\gamma \gamma }=\frac{s}{\Lambda }F_{\mu \nu }F^{\mu \nu }, \end{aligned}$$\end{document}with Fμν=∂μAν-∂νAμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{\mu \nu } = \partial _\mu A_\nu - \partial _\nu A_\mu $$\end{document} being the electromagnetic field strength. The scale Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} can be interpreted in a UV completion since it can be determined by a set of new heavy charged particles running in triangular loops. The mass scale of new charged particles is assumed to be heavier than 300 GeV to respect the LEP constraint, depending on the number of charged fermions. Several experiments restrict Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} from the Horizontal Branch (HB) stars processes [74–76] to the LEP [77] or beam dump experiment constraints [78, 79]. We will review them in detail in the next section, but roughly speaking, the coupling of a scalar to photons is extremely suppressed (Λ≳1010GeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \gtrsim 10^{10}~\mathrm {GeV}$$\end{document}) for ms≲300\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_s \lesssim 300$$\end{document} keV, largely due to the HB limits. For ms≳300\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_s \gtrsim 300$$\end{document} keV, a window opens, allowing values of Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} as low as 10 GeV. In a UV complete model, such low values of Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} can be understood if the number of fermions running in the loop is relatively important (of the order of 10).Fig. 2

Feynman diagrams for dark matter annihilation into two photons. The second diagram can be generated by higher dimensional operators (see the text for details)

Building a complete ultraviolet model is far beyond the scope of this work, but we can give some hints for further developments. Indeed, even if the Lagrangian Eq. (4) breaks explicitly the U(1) symmetry, we can have a look at higher dimensional operators which can generate such a term after the breaking of the U(1) symmetry. The simplest dimension 6 operator can be written as8LΦγγ=|Φ|2Λ~2FμνFμν,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {L}_{\Phi \gamma \gamma }=\frac{|\Phi |^2}{\tilde{\Lambda }^2}F_{\mu \nu }F^{\mu \nu }, \end{aligned}$$\end{document}with Λ~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Lambda }$$\end{document} being a different cut-off scale from Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} introduced in Eq. (4). After the symmetry breaking, one obtains the interaction terms9LΦγγ⊃2λmssΛ~2+a22Λ~2FμνFμν.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {L}_{\Phi \gamma \gamma }\supset \left( \sqrt{\frac{2}{\lambda }}\frac{m_ss}{\tilde{\Lambda }^2}+\frac{a^2}{2\tilde{\Lambda }^2}\right) F_{\mu \nu }F^{\mu \nu }. \end{aligned}$$\end{document}One can then deduce from Eq. (9) the relation between Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} and Λ~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\Lambda }$$\end{document}: Λ=λ2Λ~2ms\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda =\sqrt{\frac{\lambda }{2}}\frac{\tilde{\Lambda }^2}{m_s}$$\end{document}. The effective model built from the Lagrangian generates the second term in Eq. (9). This contact interaction contributes also to the annihilation cross section aa→γγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$aa\rightarrow \gamma \gamma $$\end{document}. Including this new contribution, the total cross section is then given by10σvγγ=32ma6πΛ~4(4ma2-ms2)2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma {v}_{\gamma \gamma }=\frac{32m_a^6}{\pi \tilde{\Lambda }^4(4m_a^2-m_s^2)^2}. \end{aligned}$$\end{document}However, in the rest of our work we will continue to consider the dimension-5 coupling approach sΛFμνFμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{s}{\Lambda } F^{\mu \nu } F_{\mu \nu }$$\end{document} because a complete dimension-6 operator analysis would require a much more careful study of all the possible operators involved in the processes.

The status of the (non-)observation of self-interacting dark matter has become somewhat quite confusing recently, due to the release of (seemingly) contradictory results. Indeed, some authors of Ref. [66] using the new Hubble Space Telescope imaging, claimed to have observed that the dark matter halo of at least one of the central galaxies belonging to the cluster Abell 3827 is spatially offset from its stars. The offset, of the order of 1.62 kpc, could be interpreted as evidence of self-interacting dark matter with a ratio of cross section over mass of σ/m≃1.7×10-4cm2/g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma /m \simeq 1.7 \times 10^{-4} ~\mathrm {cm^2/g}$$\end{document}.6 In the meantime, using a different kinematical approach from [66], the authors of [67] obtained a value of σ/m≃(1.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma /m \simeq (1.5$$\end{document}–3)cm2/g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3)~\mathrm {cm^2/g}$$\end{document}, resulting in tension with the upper bounds set by other astrophysical objects such as the “Bullet Cluster” (1E 0657-56), which are typically of the order of σ/m≲1cm2/g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma /m \lesssim 1 ~\mathrm {cm^2/g}$$\end{document} [66, 69, 70, 80–82]. The main difference between the two analyses came from some approximations concerning the evolution time and from the gravitational back-reaction of the halo on its stars during the separation process due the drag forces. The authors of Ref. [67] have already addressed this issue some time ago in [83]. They clearly distinguished the contact interaction or rare interaction (our case) from the long-range force (involving Sommerfeld enhancement) or frequent self-interactions through the position of the peak of the dark matter distribution compared to the position of the stars/galaxies after the interaction.

In this work, we decided to take the two values proposed by the two groups as benchmark points, to show the correlation between an indirect signal (monochromatic photon in our case) and the self-interaction, once one has built an explicit microscopic model. Some recent phenomenological constructions explaining these observations can be found in [84] for a model with a gauged Z3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_3$$\end{document} discrete symmetry; a very nice interpretation in the framework of a Higgs portal (freeze-in mechanism) in [85, 86] whereas other authors introduced a dark photon sector [87, 88] or a strong interacting sector [89].

When a light (pseudo-)scalar interacts with photons, the helium burning period of HB stars is shortened due to non-standard energy loss since the light (pseudo-)scalar is produced in the stellar interior by photons within thermal distribution [74, 75]. This effect gives a strong constraint on the coupling between the light (pseudo-)scalar and photons for the (pseudo-)scalar mass lighter than 300keV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$300~\mathrm {keV}$$\end{document}. The detailed analysis has been done in Ref. [76] and we used their result in our study.7

The interaction between the (pseudo-)scalar and photons is also constrained by the mono-photon search at leptonic collider experiments. Its signature is e+e-→γ+missingenergy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^+e^-\rightarrow \gamma + \mathrm{missing\,energy}$$\end{document}. The collider bound has been compiled in Ref. [74], taking into account the anomalous single photon (ASP) experiment [90]. In addition, the improved Large Electron-Positron Collider (LEP) limits based on the data of ALEPH, OPAL, L3, and DELPHI have been published in Ref. [77].

The computation of the relic abundance of dark matter in our framework has already been studied in detail in [36]. We will not repeat the analysis in this work, but we recall its main point. Adding interactions to the neutrino sector through s as a mediator can fulfill perfectly the relic abundance of dark matter measured by PLANCK while in the meantime generating naturally a massive neutrino sector respecting the recent cosmological bounds on neutrino masses (mν≲1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\nu \lesssim 1$$\end{document} eV; see [91] for a review on the subject). The presence of a dark bath between the neutrino and dark matter allows it in the parameter range of our work. However, keeping in mind this elegant possibility, our aim is to study the properties of self-interacting dark matter at the present time and the correlation between different observations in present large scale structures, independently on hypotheses concerning the thermal history generating the correct amount of dark matter abundance.

Recent claims for a detection of X-ray line observed in galaxies and galaxy clusters like Perseus, by the XMM-Newton observatory [33–35] increased the interest in light dark matter scenarios. Keeping in mind that the status is still in debate (see the thermal atomic transition interpretation in [61] for instance), it is nevertheless interesting to apply our analysis in this concrete example to check if such a signal can be compatible with the limits derived from the recent self-interaction measurements.

The flux generated by the annihilation of dark matter in the Perseus cluster for instance can be computed from the luminosity of the cluster L [36]11L=∫0RPe4πr2nDM2(r)⟨σv⟩γγdr=∫0RPe4πr2ρDM(r)ma2⟨σv⟩γγdr,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} L= & {} \int _{0}^{R_{Pe}} 4 \pi r^2 n^2_\mathrm {DM}(r) \langle \sigma v \rangle _{\gamma \gamma }\mathrm{d}r\nonumber \\= & {} \int _{0}^{R_{Pe}} 4 \pi r^2 \left( \frac{\rho _\mathrm {DM}(r)}{m_a} \right) ^2 \langle \sigma v \rangle _{\gamma \gamma }\mathrm{d}r, \end{aligned}$$\end{document}with the Perseus radius RPe\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{Pe}$$\end{document}, the number density of dark matter nDM(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_\mathrm {DM}(r)$$\end{document}, the dark matter profile ρDM(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _\mathrm {DM}(r)$$\end{document} and the thermally averaged cross section ⟨σv⟩γγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle \sigma {v}\rangle _{\gamma \gamma }$$\end{document}. At a first approximation, one can consider a mean density of dark matter in the cluster as in Ref. [92]. The Perseus observation involved the mass of MPe=1.49×1014M⊙\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{Pe}= 1.49 \times 10^{14} M_{\odot }$$\end{document} in the region of RPe=0.25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{Pe}=0.25$$\end{document} Mpc at the distance of DPe=78\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{Pe}=78$$\end{document} Mpc from the solar system. One can then estimate12nDM≃1.49×1014M⊙ma/4πRPe33=1.9×10-37GeV3(ma=3.5keV)=2.5×104cm-3.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n_\mathrm {DM}\simeq & {} \frac{1.49 \times 10^{14}M_{\odot } }{m_a}\!\bigg /\frac{4 \pi R_{Pe}^3}{3}\nonumber \\= & {} 1.9\times 10^{-37}~\mathrm {GeV^3}~~~(m_a=3.5~\mathrm {keV})\nonumber \\= & {} 2.5 \times 10^4~\mathrm {cm^{-3}}.\quad \end{aligned}$$\end{document}Combining Eqs. (11) and (12), one can then compute the luminosity in the Perseus cluster in the “mean” approximation ⟨L⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle L\rangle $$\end{document},13⟨L⟩≃1.2×10553.5keVma2×⟨σv⟩γγ10-26cm3s-1photon/s.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \langle L\rangle\simeq & {} 1.2 \times 10^{55} \left( \frac{3.5~\mathrm {keV}}{m_a} \right) ^2\nonumber \\&\times \left( \frac{\langle \sigma v \rangle _{\gamma \gamma }}{10^{-26} \mathrm {cm^3\,s^{-1}}} \right) ~\mathrm {photon/s} . \end{aligned}$$\end{document}This estimation would be reasonable since dark matter in our model does not have a cusp profile such as NFW or Einasto, but a cored profile due to the large self-interacting cross section of dark matter. One can then deduce the flux ϕγγ=L/(4πDPe2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{\gamma \gamma } = L/(4 \pi D_{Pe}^2)$$\end{document} that one should observe on earth,14ϕγγ=1.6×10-53.5keVma2×⟨σv⟩γγ10-32cm3s-1cm-2s-1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \phi _{\gamma \gamma }= & {} 1.6 \times 10^{-5} \left( \frac{3.5 ~ \mathrm {keV}}{m_a} \right) ^2\nonumber \\&\times \left( \frac{\langle \sigma v \rangle _{\gamma \gamma }}{10^{-32} \mathrm {cm^3\,s^{-1}}} \right) ~\mathrm {cm^{-2}\, s^{-1}}. \end{aligned}$$\end{document}According to the authors of Refs. [33–35], one can identify the monochromatic signal arising from Andromeda galaxy (M31) or Perseus cluster with the flux ϕγγ=5.2-2.13+3.70×10-5cm-2s-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{\gamma \gamma }=5.2_{-2.13}^{+3.70} \times 10^{-5}~\mathrm {cm^{-2}\,s^{-1}}$$\end{document} at 3.56 keV including the cluster core.8 We will parametrize our uncertainty from the dark matter distribution in the source by the classical “astrophysical” parameter Jastro≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_\mathrm{astro} \ge 1$$\end{document} with Jastro=L/⟨L⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_\mathrm{astro} = L/\langle L\rangle $$\end{document}, L being the effective luminosity for a steeper profile than the mean one we considered above.

Finally, extending the analysis by taking into account also other observations like M31, we will impose in our analysis a conservative annihilation cross section which is required to reproduce the X-ray line estimated as15⟨σv⟩γγ≃1Jastro(2×10-33-8.5×10-33)cm3s-1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \langle \sigma v \rangle _{\gamma \gamma } \simeq \frac{1}{J_\mathrm{astro}}(2 \times 10^{-33} {-} 8.5 \times 10^{-33}) \ \mathrm {cm^3\,s^{-1}}. \end{aligned}$$\end{document}

We show in Figs. 3 and 4 the combined analysis, including the HB stars, ASP and LEP constraints [76, 77] for two different values of the self-interacting cross section, σaa/ma=1.7×10-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{aa}/m_a=1.7 \times 10^{-4}$$\end{document} and 1.5cm2/g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.5 ~\mathrm {cm^2/g}$$\end{document} corresponding to the values derived in [66, 67], respectively. In Fig. 3 the analysis is made by taking into account the current limits from different observations, whereas in Fig. 4 we fixed the annihilation cross section to fit the 3.5 keV line observation by XMM-Newton, Eq. (15).

We would like to insist that our aim is not to affirm that these two observations are the signatures of dark matter, but that combining these two physical measurements one can deduce a very strong constraint and/or information on Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}, especially if one uses the current limits from different experiments looking at the sky from the keV to the MeV energy range. To illustrate our purpose, we can extract lower bounds for the scale of the Beyond the Standard Model (BSM) Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} from the observations of the satellites HEAO-1, INTEGRAL, COMPTEL, EGRET, and FERMI [93]. For our analysis, we required that the photon flux coming from the dark matter annihilation does not exceed the observed central value plus twice the error bar where the NFW dark matter profile is assumed [94]. This is depicted in Fig. 3 where we plot the limit we obtained on Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} for σaa/ma=1.7×10-4cm2/g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{aa}/m_a = 1.7 \times 10^{-4} ~ \mathrm {cm^2/g}$$\end{document} and σaa/ma=1.5cm2/g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{aa}/m_a = 1.5~\mathrm {cm^2/g}$$\end{document} assuming the mass ratio ms/ma=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_s/m_a=10$$\end{document}. We notice that the limits are quite stronger than the ones obtained by LEP, especially for a large self-interaction cross section.

As one can see from Fig. 4, it is interesting to note that there exists a band of parameter space, for ms≃\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_s \simeq $$\end{document}1–10 MeV and Λ≃10-1000\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \simeq 10{-}1000$$\end{document} TeV where one can explain the observed 3.5 keV line from the Perseus cluster for a self-interaction cross section of the order of magnitude of the one claimed to have been recently observed and still being largely compatible with accelerator searches.Fig. 3

Limits on Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} obtained from different observatories and satellites with the mass ratio fixed to ms/ma=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_s/m_a=10$$\end{document} where the white region is allowed and the colored region is excluded. The lower bounds of Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} have been obtained from the data of satellites HEAO-1 (red), INTEGRAL (green), COMPTEL (blue), EGRET (brown), and FERMI (dark-yellow). The HB bound (violet) and perturbativity bound (gray) for λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} are also shown

If the 3.5keV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3.5~\mathrm {keV}$$\end{document} X-ray line excess discussed above is interpreted as a dark matter signal, a same excess should be observed from the other galaxies such as the Milky Way, M31 and dwarf spheroidal galaxies in addition to the Perseus and Centaurus clusters. However, such an excess has not been observed for the Milky Way [62], M31 [63], stacked galaxies [64], and stacked dwarf galaxies [65]. For completeness, keeping open all possible interpretations of the 3.5 keV line signal, we present the result of these analyses in Fig. 5.

This inconsistency can be managed in some models. The first example is the scenario of decaying dark matter into an axion-like particle [95]. In this model, dark matter decays into an axion-like particle with the energy 3.5keV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3.5~\mathrm {keV}$$\end{document}. The axion-like particle produced in the process can be converted into photon via the astrophysical magnetic field around the galaxy clusters. Since the X-ray flux from dark matter depends on the strength of the magnetic field of each galaxy, the non-detection of the X-ray excess in some galaxies can be consistent.

Another type of interpretation concerns the possibility of an exciting dark matter  [96]. In this scenario, dark matter χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi $$\end{document} with the mass of the order of 10 GeV possesses an excited state χ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi ^*$$\end{document}. The excited state can be produced by up-scattering process χχ→χ∗χ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi \chi \rightarrow \chi ^*\chi ^*$$\end{document} in the center of the cluster and converting the kinetic energy of dark matter. Then the excited state decays into the ground state and photon χ∗→χγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi ^*\rightarrow \chi \gamma $$\end{document}. One can reproduce the X-ray line excess with the mass difference of 3.5keV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3.5~\mathrm {keV}$$\end{document}. Moreover, since the up-scattering process can occur in more massive and hotter environments such as clusters, non-detection of the X-ray line excess in smaller galaxies would be reasonable.Fig. 5

Same as Fig. 4 if one considers the non-observation of the 3.5 keV line in the Milky Way or dwarf galaxies. The region below the red-full line is excluded by the analyses of the Milky Way in [62], whereas the zone below the green-dashed line is excluded by the non-observation of the line in stacked dwarf galaxies [65]

Such a light keV–MeV dark matter particle is clearly out of the reach of any present direct detection technology. However, recently, the authors of Ref. [97] proposed a new class of superconducting detectors which are sensitive to O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}$$\end{document}(meV) electron recoils from dark matter–electron scattering. Such devices could detect dark matter as light as 10 keV which is exactly the mass range of interest for our model. The idea is to observe the dark matter scattering off free electrons in a superconducting metal. Indeed, in a superconductor, the free electrons are bound into Cooper pairs, which typically have a meV scale (or less) binding energy, which is the typical energy transported by a O(10)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(10)$$\end{document} keV dark matter with a local velocity ≃300km/s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\simeq 300~\mathrm {km/s}$$\end{document}. Assuming that ge\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_e$$\end{document} is the coupling of the electron to the scalar mediator s, one can straightforwardly compute the scattering cross section with an electron σDDe\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^e_{\mathrm {DD}}$$\end{document}:16σDDe=λ2ge22πms4μea2ms24ma2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma ^e_{\mathrm {DD}}= \frac{\lambda ^2g_e^2}{2\pi m_s^4}\mu _{ea}^2\left( \frac{m_s^2}{4m_a^2}\right) , \end{aligned}$$\end{document}where μea≡mame/ma+me\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{ea}\equiv m_am_e/\left( m_a+m_e\right) $$\end{document} is the reduced mass.

In Fig. 6, we show the 95 % expected sensitivity reached after 1 kg·\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cdot $$\end{document}year exposure, corresponding to the cross section required to obtain 3.6 signal events [98] supposing a detector sensitivity to recoil energies between 1 meV and 1 eV [97]. One can see that even for quite low values of the coupling ge\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_e$$\end{document}, the prospect of discovery of self-interacting dark matter through this new proposal is quite promising.