The Spectrum of Darkonium in the Sun

Dark matter (DM) is approximately five times more abundant than baryonic matter in the Universe. Although DM can be in the form of more conventional compact objects like primordial black holes Carr:2009jm , there is also the possibility that DM might be in the form of particles. No Standard Model (SM) particle can play the role of DM. Therefore, in the case DM is in the form of particles, it must be related to physics beyond the SM. The simplest example of such a realization is the Weakly Interacting Massive Particle (WIMP) paradigm. In that case, WIMPs are produced in the early Universe and occasionally annihilate to SM particles. If the annihilation cross section is appropriate, annihilations are sufficient to produce the DM abundance we observe today. Such a scenario can potentially be tested. Annihilation of WIMPs in the Sun and the center of the Galaxy to SM particles could create observable signals. Similarly, this scenario allows production of WIMPs in collider experiments as long as there is enough energy to produce them or create nuclear recoils when WIMPs scatter off nuclei in underground detectors.

A minimal extension of the SM that could facilitate the above characteristics can be realized by adding a new $U(1)$ gauge symmetry which breaks spontaneously providing the dark photon with a mass Kobzarev:1966qya ; Okun:1982xi . In this scenario, the DM particle is charged under the $U(1)$ symmetry and since it is possible to have a small kinetic mixing between the dark photon and the SM photon under very generic grounds Holdom:1985ag , the dark and bright sectors are linked. As we mentioned above, a particular way of testing this scenario is by searching for signals of DM annihilation in the Sun. DM particles can be trapped in the Sun simply by interacting with nuclei or electrons as they cross it. Trapped DM particles may thermalize with the interior of the Sun and sink to the center where they can meet each other and annihilate to SM particles. In particular, annihilation to neutrinos (which can easily escape from the Sun) may potentially lead to detectable signals in Earth-based detectors. The capture and annihilation process in the Earth and the Sun as well as the produced neutrino spectrum from DM annihilation has been studied extensively in the past Press:1985ug ; Gould:1987ir ; Gould:1987ju ; Silk:1985ax ; Freese:1985qw ; Krauss:1985aaa ; Griest:1986yu . Within the context of dark photons, similar studies have also been made regarding indirect signals from the Sun and the Earth or direct detection Pospelov:2007mp ; Schuster:2009au ; Schuster:2009fc ; Meade:2009mu ; An:2013yfc ; An:2013yua ; Fradette:2014sza ; An:2014twa ; Feng:2015hja ; Feng:2016ijc .

In this paper we investigate fermionic DM that self-interacts via a light dark photon mediator. In particular, we are interested in the region of the parameter space, where DM captured by the Sun has a substantial probability of recombining to positronium-like bound states of DM and anti-DM, which we from now on call darkonium. Darkonium states can subsequently decay to either two dark photons in the case of para-darkonium (where the spin of DM and anti-DM are opposite) or to three dark photons in the case of ortho-darkonium (where the spin of DM and anti-DM are aligned). We assume that dark photons are linked to SM via kinetic mixing with the ordinary photons. At the end of the day, the spectrum of the produced dark photons in the Sun has four components: i) monochromatic dark photons of energy $m_{X}$ (the mass of DM) which come from direct DM annihilation ii) monochromatic dark photons of energy $m_{X}-\Delta/2$ where $\Delta$ is the binding energy of the para-darkonium iii) dark photons that span energy up to $m_{X}-\Delta/2$ from the decay of ortho-darkonium states and iv) monochromatic photons of energy $\Delta$ produced from the recombination of DM to darkonium. We investigate under what conditions the aforementioned dark photons leave the Sun without decaying to SM particles. Those that exit the Sun intact, will decay sooner or later producing an electron-positron pair due to the kinetic mixing. Based on that, we find the precise positron spectrum that potentially could be observed by AMS-02 and we show how one can set constraints on these models. Interestingly, we find that there is parameter space where recombination dark photons could be detected earlier than the annihilation ones.

We should also mention at this point that the possibility of forming darkonium can also have important consequences in the physics of the early Universe. For instance, if DM is thermally produced by a freeze-out of DM annihilation into mediators, the effect of darkonium bound states is to delay the freeze-out until later times vonHarling:2014kha . Additionally, the effect of the bound states in indirect Galactic searches and on the Galactic structure has been investigated in An:2016gad ; Feng:2009mn . The indirect detection signal in models with light dark photons in the center of the Earth and the Sun have been recently investigated in Feng:2015hja ; Feng:2016ijc albeit neglecting DM bound states. As we will show, in the region of the parameter space where darkonium can form, the spectrum of positrons is dominated by the decay pattern of the bound states, and direct annihilation contributes a subleading part in the full signal.

The paper is structured as follows: In section II we review the DM model and the necessary formalism to calculate the solar DM and darkonium populations. In section III we demarcate the allowed parameter space in which darknonium can lead to a significant indirect detection signal. In section IV we review the kinematics and geometry of the positron signal from a decaying mediator emitted from the Sun. In section V we present the positron spectra and compare them with the flux observed by AMS-02. In section VI we make our concluding remarks.

Throughout the paper we use natural units, i.e. $\hbar=c=k_{\text{B}}=1$.

We start by introducing a generic DM model that can accommodate the formation of darkonium bound states. The DM model includes a Dirac fermion $X$ (and its antiparticle $\bar{X}$) which constitutes the bulk of the DM relic abundance. The DM candidate interacts through a dark $U(1)$ gauge symmetry. The associated dark photon $\phi$ interacts with the SM by kinetic mixing with the SM photon. The Lagrangian density of the model reads

$$\mathcal{L}=\bar{X}\left(i\gamma^{\mu}D_{\mu}-m_{X}\right)X-\frac{1}{4}\Phi_{\mu\nu}\Phi^{\mu\nu}+\frac{m_{\phi}^{2}}{2}\phi_{\mu}\phi^{\mu}-\frac{\epsilon}{2}\Phi_{\mu\nu}F^{\mu\nu},$$

(1)

where $m_{X}$ is the DM mass, $m_{\phi}$ the mediator mass, $\epsilon$ the kinetic mixing parameter and $D_{\mu}=\partial_{\mu}-ig_{X}\phi_{\mu}$ is the covariant derivative. $F_{\mu\nu}$ is the electromagnetic field strength tensor and $\Phi_{\mu\nu}=\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu}$ is the dark $U(1)$ field strength tensor. We further define $\alpha=e^{2}/(4\pi)$ and $\alpha_{X}=g_{X}^{2}/(4\pi)$ as the electromagnetic and dark fine structure constants. The kinetic mixing term can arise through integrating out a heavy particle that is charged under both the dark and electromagnetic gauge groups Holdom:1985ag ; Collie:1998ty , leading naturally to a small $\epsilon$. For the purposes of this paper we will treat $\epsilon$ as a free, yet small, parameter.

The above generic model allows for self-interactions of DM of Yukawa type since the dark photon has a mass (either through a Higgs- or Stueckelberg-mechanism) of the form

$$V=\pm\alpha_{X}\frac{e^{-m_{\phi}r}}{r},$$

(2)

$r$ being the distance between two DM particles. The potential is repulsive ($+$ sign) for $XX$ or $\bar{X}\bar{X}$ interactions, while $X\bar{X}$ interactions are attractive ($-$ sign) and can lead to formation of bound states. In fact DM self-interactions are welcome since they can solve a range of problems arising in the collisionless cold DM paradigm such as the too big to fail problem BoylanKolchin:2011de , the missing satellites problem Klypin:1999uc ; Moore:1999nt ; Kauffmann:1993gv ; Liu:2010tn ; Tollerud:2011wt ; Strigari:2011ps and the core-cusp problem Moore:1994yx ; Flores:1994gz ; Navarro:1996gj . Numerical simulations suggest that generally speaking DM self-interactions alleviate the aforementioned problems within approximately the range of $\sigma/m_{X}$ ($\sigma$ being the DM-DM cross section) $0.1-1\,\text{cm}^{2}/\text{g}$ Rocha:2012jg ; Peter:2012jh .

The model has four free parameters: $\epsilon$, $m_{\phi}$, $m_{\chi}$ and $\alpha_{X}$. In this paper, we assume that the DM relic abundance $\Omega_{X}\simeq 0.23$ is produced though the thermal freeze-out of $X+\bar{X}\to\phi+\phi$, i.e. the DM density is fixed once the annihilation rate drops below the Hubble expansion rate, $\Gamma(X+\bar{X}\to\phi+\phi)\lesssim H$. We use this criterion to fix the value of $\alpha_{X}$ as a function of $m_{X}$. The relation between $\alpha_{X}$ and $m_{X}$ has been calculated in detail in vonHarling:2014kha including the effects of both Sommerfeld enhancement and darkonium recombination¹¹1We note that the recombination cross sections used in vonHarling:2014kha differs from that used in An:2016gad . The coupling $\alpha_{X}$ fixed by the DM relic abundance in vonHarling:2014kha differs slightly from the result of direct annihilation with Sommerfeld enhancement, while An:2016gad claims that the effect of recombination on freeze-out is negligible all together. For the purposes of removing $\alpha_{X}$ as a free parameter we adopt the coupling found by vonHarling:2014kha ..

The decay rate of the dark photon into SM fermions is given by Feng:2016ijc

$$\Gamma(\phi\to f\bar{f})=\frac{\epsilon^{2}q_{f}^{2}\alpha(m_{\phi}^{2}+2m_{f}^{2})}{3m_{\phi}^{2}}\sqrt{1-\frac{4m_{f}^{2}}{m_{\phi}^{2}}},$$

(3)

where $q_{f}$ is the electric charge of the fermion (in units of $e$) and $m_{f}$ is the fermion mass. The decay length into positron electron pairs is

$$L=\text{Br}(\phi\to e^{+}e^{-})\left(\frac{1.1\cdot 10^{-9}}{\epsilon}\right)^{2}\left(\frac{m_{X}/m_{\phi}}{1000}\right)\left(\frac{100\text{MeV}}{m_{\phi}}\right)R_{\odot},$$

(4)

where $R_{\odot}=7.0\cdot 10^{8}$ m is the radius of the Sun, and the branching ratio to electron-positron pairs $\text{Br}(\phi\to e^{+}e^{-})\simeq 1$, if the dark photon cannot decay to heavier particles, e.g. to muons $m_{\phi}<2m_{\mu}=211$ MeV. The branching ratios for heavier dark photons has been calculated in Buschmann:2015awa .

In this section we map the allowed and interesting region of the parameter space. The parameter space allowed by observations, which simultaneously permits bound state formation, is quite complex. It is beyond the scope of this paper to make a detailed analysis and scan over all possible parameter sets. Instead we list the most important constraints within the interesting region of darkonium formation. The available regions in parameter space is demarcated by the requirements listed below. Bullets 1-5 are regions of interest (not constraints), while bullets 6-8 contain observational constraints.

1. 1.

   Bound state formation: Darkonium bound states can in principle exist if $1/m_{\phi}$ is longer than the Bohr length, $a_{0}=2/(\alpha_{X}m_{X})$. However, a stronger requirement arises by demanding the darknonium to be able to form in the first place by emission of an on-shell dark photon. This condition can be written as $m_{\phi}<\Delta=\alpha_{X}^{2}m_{X}/4$.

2. 2.

   Electron-positron decay: We require that the dark photon can decay into an electron-positron pair, i.e. $m_{\phi}>2m_{e}$. Within the model we consider, this inequality is always fulfilled if the observational constraints 6-8 are obeyed. We would also like the branching into electron-positrons to be close to one, and therefore keep $m_{\phi}<2m_{\mu}$.

3. 3.

   Scattering unitarity: To keep our model self-consistent we need to impose unitarity of DM self-scattering. This limits the coupling from above to be $\alpha_{X}\lesssim 0.54$. Correspondingly, by assuming a thermal relic DM abundance limits the DM mass below $m_{X}\lesssim 139$ TeV vonHarling:2014kha .

4. 4.

   Efficient recombination: It is useful to note the Coulombic regime where the mediator mass is smaller than the transferred momentum $m_{\phi}<\mu v$ but larger than the kinetic energy of the scattering, $m_{\phi}>\mu v^{2}/2$. If the lower limit is respected it is a good approximation to use Coulombic energy levels An:2016gad . The region can be written as $2m_{\phi}/m_{X}<v<2\sqrt{m_{X}/m_{\phi}}$. Within this range the recombination cross section is much larger than the direct annihilation cross section and approximately given by the analytic expression of Eq. 17. In the Sun the velocity of a thermalized DM particle is

   $$v_{\text{th}}=\sqrt{\frac{2T_{\odot}}{m_{X}}}\approx 5.1\times 10^{-5}\sqrt{\frac{\text{TeV}}{m_{X}}}.$$

   (30)

   Taking the relative velocity of colliding DM particles to be $v=\sqrt{2}v_{\text{th}}$, the lower limit translates to the following inequality on the mediator mass

   $$m_{\phi}<\frac{v_{\text{th}}m_{X}}{\sqrt{2}}=36\text{MeV}\sqrt{\frac{m_{X}}{\text{TeV}}}.$$

   (31)

   We stress that satisfying this inequality only ensures that the analytic expression in Eq. 17 is well described when $n_{\text{max}}\gg 1$. For somewhat heavier mediators the recombination rate may still be larger than the direct annihilation rate, although Eq. 17 breaks down. As already discussed, even if the recombination rate is far smaller, the indirect signal from recombination photons may be important nonetheless.

5. 5.

   Solar steady state: For the solar signal to be maximal, the Sun’s age must be longer than the time it takes to reach the darkonium steady state population, i.e. $\tau_{o/p}<\tau_{\odot}\approx 4.6$ Gyr, where $\tau_{o/p}=\tau_{X}$ for the parameters we will consider, and is thus given by Eq. 27.

6. 6.

   Self-interaction constraints: $N$-body simulations suggests that DM self-interactions may flatten the core of dwarf spheroidal galaxies, alleviating tension with observations. Self-interactions stronger than roughly $0.1-10\text{cm}^{2}/\text{g}$ will however reduce the central densities too much. Furthermore, at higher velocity dispersions, the ellipticity of the Milky Way is threatened if the cross section is stronger than roughly $0.1-1\text{cm}^{2}/\text{g}$ Tulin:2013teo . The scattering cross section when $m_{X}v/\alpha_{X}\gg 1$ is well described by the classical momentum transfer cross sections defined by $\int d\Omega(1-\cos\theta)d\sigma/d\Omega$ Khrapak1 ; Khrapak:2014xqa

   $$\sigma_{\text{att}}=\begin{dcases}\tfrac{4\pi}{m_{\phi}^{2}}\beta^{2}\log\left(1+\beta^{-1}\right)&\beta\lesssim 10^{-1}\\ \tfrac{8\pi}{m_{\phi}^{2}}\beta^{2}\left(1+1.5\beta^{1.65}\right)^{-1}&10^{-1}\lesssim\beta\lesssim 10^{3}\\ \tfrac{\pi}{m_{\phi}^{2}}\left(1+\log\beta-\tfrac{1}{2\log\beta}\right)^{2}&\beta\gtrsim 10^{3}\end{dcases},$$

   (32)

   in the case of attractive interactions and

   $$\sigma_{\text{rep}}=\begin{dcases}\tfrac{2\pi}{m_{\phi}^{2}}\beta^{2}\log\left(1+\beta^{-1}\right)&\beta\lesssim 1\\ \tfrac{\pi}{m_{\phi}^{2}}\left(\log 2\beta-\log\log 2\beta\right)^{2}&\beta\gtrsim 1\end{dcases},$$

   (33)

   for repulsive ones, where $\beta=2\alpha_{X}m_{\phi}/(vm_{X})$. The typical $v$ for dwarf galaxies is of the order of $v_{0}\sim 10$ km/s, while galactic velocity dispersions are significantly larger at the order of $v_{0}\sim 200$ km/s in the case of the Milky Way. Since symmetric DM with a vector mediator interacts attractively between $X\bar{X}$ pairs and repulsively between $XX$ or $\bar{X}\bar{X}$, the total cross section per mass is $\langle\sigma\rangle/m_{X}=\langle\sigma_{\text{att}}+\sigma_{\text{rep}}\rangle/(2m_{X})$. Here $\langle\cdot\rangle=\int d^{3}ve^{-\tfrac{1}{2}v^{2}/v_{0}^{2}}/(2\pi v_{0}^{2})^{3/2}$ is the velocity average. The DM self-interactions are mapped in Fig. 2.

   

7. 7.

   Direct detection: Direct detection places an upper bound on $\epsilon$ for a particular choice of $m_{X}$ and $m_{\phi}$. Currently the strongest limits are placed by the LUX-experiment’s 2013 results Akerib:2013tjd . Following the procedure of DelNobile:2015uua , we find the exclusion contours at 90% confidence level in the $\epsilon$ versus $m_{\phi}$ parameter space summarized in Fig. 3. For heavy mediator masses, $\epsilon$ is less constrained. Notice, that the bounds become roughly constant when $m_{\phi}<q\approx\sqrt{2}\mu_{N}v_{\odot}$, where $q$ is the typical value of the transferred momentum in a nuclear recoil, with $\mu_{N}$ the reduced DM-nucleus mass and $v_{\odot}$ the Sun’s velocity in the galactic frame.

8. 8.

   Cosmology: If the dark photon is abundant in the early universe and decays into SM particles, predictions of the Big Bang nucleosynthesis (BBN) may be adversely affected. To avoid this possibility we can demand that the mediator decays before BBN begins, i.e. the decay rate of the dark photon is $\Gamma_{\phi}\approx\alpha_{\text{em}}m_{\phi}\epsilon^{2}/3>1\,\text{s}^{-1}$.

   For non-thermal DM the cosmic microwave background (CMB) constrains the dark matter coupling $\alpha_{X}<0.17(m_{X}/\text{TeV})^{1.61}$ from the imprints annihilation products would leave in the CMB. This upper limit is a fit of the results in Slatyer:2015jla obtained by Feng:2016ijc . The couplings we consider are always smaller than this upper limit.

To illustrate the behaviour of the signal, we choose three dark matter masses as benchmark points. For each benchmark the mediator mass is varied in the region compatible with efficient bound state formation in the Sun. The benchmark points are chosen to be:

1. (B1)

   For our first benchmark we choose the (nearly) lightest DM mass where recombination can take place with DM self-interaction within the range solving the collisionless DM problems (see Fig. 2): $m_{X}=400$ GeV and $\alpha_{X}=1.1\cdot 10^{-2}$. The mediator mass is chosen to be $m_{\phi}=10$ MeV, slightly below the binding energy of the ground state which is $\Delta\approx 12$ MeV. The smallest kinetic mixing parameter for which the mediator decay is faster than one second is $\epsilon_{\text{min}}\approx 1.6\cdot 10^{-10}$, while the maximum allowed by direct detection is $\epsilon_{\text{max}}\approx 3.3\cdot 10^{-10}$.

2. (B2)

   Our second benchmark point corresponds to the point where the energy threshold of AMS-02 ($\sim 0.5$ GeV) is close to the maximum positron energy from a decaying recombination photon. This corresponds to $m_{X}=2$ TeV and $\alpha_{X}=4\cdot 10^{-2}$, for which the ground state binding energy is $\Delta=0.8$ GeV. We choose three values of the mediator mass: (a) $m_{\phi}=5$ MeV, (b) $m_{\phi}=15$ MeV and (c) $m_{\phi}=70$ MeV.

3. (B3)

   For the last benchmark point we choose the heaviest DM particle allowed by unitarity of scattering. We adopt the value reported in vonHarling:2014kha which is $m_{X}=139$ TeV and $\alpha_{X}=0.54$. Again we choose three values of the mediator mass: (a) $m_{\phi}=2$ MeV, (b) $m_{\phi}=20$ MeV and (c) $m_{\phi}=100$ MeV. The value of $n_{\text{max}}$ is very large for this benchmark. However, Eq. 17 is well satisfied within the range of the chosen parameters of this benchmark and therefore we use it for the recombination cross section in this case.

In table 2 of appendix B we summarize key quantities for each benchmark.

In this section we review the formalism necessary to extract the positron spectrum from DM annihilating in the Sun. It is well known that the received gamma-ray flux emerging from the decay of a monochromatic (single energy) particle to two photons has a characteristic box-like spectrum (see e.g. Ibarra:2012dw ), i.e the gamma-ray flux is independent of energy. Alternatively, if the mediator is a Dirac fermion with chiral interactions, the spectrum form a triangle/trapezoid Ibarra:2016fco . In our scenario dark photons are produced through either direct annihilation of DM particles or via recombination and subsequent darkonium decay. An addittional population is produced because a dark photon is emitted each time a recombination takes place inside the Sun. Dark photons created through direct annihilations, $p$-darkonium state decays and recombination are monochromatic, whereas dark photons originating from $o$-darkonium state decays are distributed according to Eq. 19. Decays of monochromatic dark photons to positron-electron pairs produce a box-like spectrum similar to the one of the gamma-rays.

In principle, the energy of recombination photons depends on the excitation of the formed bound state. As already discussed, we make the simplifying approximation that all recombination photons have the ground state energy. As a further simplification, we neglect all dark photons, which may be emitted in transitions between darkonium states. For light mediators ($<2m_{\mu}$) the branching into electrons is almost 100%. The signal is for the most part heavily boosted, and therefore clearly directed towards the Sun. For highly energetic positrons, the bending in the magnetic field of the Sun will be small. Since the lightest DM particle we consider has a mass of $400$ GeV, the typical energy of positrons will be quite large. The positrons resulting from decays of recombination photons will in general be less energetic and thus experience a stronger effect. For simplicity we neglect all effects from the solar magnetic field.

We now want to extract the flux of positrons in a solid angle directed towards the Sun. When the mediator is more boosted than the positrons there exists a maximum detector angle $\theta_{\text{d}}^{\text{max}}$, beyond which no positrons reach the Earth; see figure 4 for definitions of lengths and angles. The flux is described by the integral

$$\phi_{+}=-\int_{0}^{\theta_{\text{d}}^{\text{max}}}d\theta_{\text{d}}\sin\theta_{\text{d}}\frac{dN_{+}}{dAdtd\cos\theta_{\text{d}}dE_{\text{d}}},$$

(34)

where $\theta_{\text{d}}^{\text{max}}=\theta_{\text{l}}(E_{\text{d}})$ is the maximal detector angle for a specific detector energy $E_{\text{d}}$ (it is easy to understand why $\theta_{\text{d}}^{\text{max}}=\theta_{\text{l}}$ since two angles in a triangle sum to less than $\pi$, i.e. $\pi-\theta_{\text{l}}+\theta_{\text{d}}<\pi$). When comparing the flux with AMS-02 we smear out the signal in the regions where the maximum angle is smaller than experimental resolution, i.e. $\theta_{\text{d}}^{\text{max}}<\theta_{\text{AMS}}(E_{\text{d}})$, where $\theta_{\text{AMS}}(E_{\text{d}})=\sqrt{(5.8^{\circ})^{2}/(E_{\text{d}}/\text{GeV})+(0.23^{\circ})^{2}}$ is the angular resolution of AMS-02 Gallucci:2015fma . In order to determine the integrand in Eq. 34 we estimate the flux of positrons with energy $E_{\text{d}}$ arriving at a detector close to the Earth integrating over the volume $2\pi R^{2}d\cos\theta_{\text{d}}dR$

$$\frac{dN_{+}}{dAdtdE_{\text{d}}}=\int_{0}^{\infty}dR\int_{0}^{\infty}dE_{\phi}\frac{dN_{\phi}}{dVdtdE_{\phi}}\frac{1}{2\pi R^{2}}\frac{d\Gamma}{d\cos\theta_{\text{l}}}(2\pi R^{2}d\cos\theta_{\text{d}})\delta(E_{\text{d}}-E_{+})\Theta(r-R_{\odot}),$$

(35)

where $E_{+}$ is the positron energy, $d\Gamma/d\cos\theta_{\text{l}}$ is the fraction of positrons emitted at a particular angle, such that it reaches the detector with energy $E_{+}$. Rearranging Eq. 35 gives the integrand of Eq. 34 ⁴⁴4We note that our formula disagrees with Eq. 21 of Ajello:2011dq .

$$\frac{dN_{+}}{dAdtd\cos\theta_{\text{d}}dE_{\text{d}}}=\int_{0}^{\infty}dR\int_{0}^{\infty}dE_{\phi}\frac{dN_{\phi}}{dVdtdE_{\phi}}\frac{d\Gamma}{d\cos\theta_{\text{l}}}\delta(E_{\text{d}}-E_{+})\Theta(r-R_{\odot}).$$

(36)

The last term is a Heaviside function that excludes positrons created inside the Sun. The first term in the integrand is the number of mediator decays per time, volume and energy $E_{\phi}$ at a distance $r$ from the center of the Sun. It can be written as

$$\frac{dN_{\phi}}{dVdtdE_{\phi}}=\frac{e^{-r/L}}{L}\frac{\Gamma_{\phi}(E_{\phi})}{4\pi r^{2}},$$

(37)

where $L$ is the decay length of $\phi$ in Eq. 4 and $\Gamma_{\phi}(E_{\phi})$ is the rate of emitted mediators per energy $E_{\phi}$. The form of $\Gamma_{\phi}$ changes depending on the specific DM model. If only direct annihilation is taken into account the rate is just $\Gamma_{\phi}(E_{\phi})=C_{\text{cap}}2\delta(E_{\phi}-m_{X})$. In our case $\Gamma_{\phi}$ is significantly more involved

$$\Gamma_{\phi}(E_{\phi})=C_{\text{cap}}\left(k_{o}\frac{dN_{\phi}^{(o)}}{dE_{\phi}}+k_{p}\frac{dN_{\phi}^{(p)}}{dE_{\phi}}+k_{\text{ann}}\frac{dN_{\phi}^{(\text{ann})}}{dE_{\phi}}+(k_{o}+k_{p})\frac{dN_{\phi}^{(\text{rec})}}{dE_{\phi}}\right),$$

(38)

where $k_{i}$ is the fraction of DM particles that are converted to photons through either direct annihilation or decay of the $o$- and $p$-darkonium states. By inspection of the Boltzmann equations Eqs. 21a-21c and using the steady state values of Eq. 29, we get $k_{o}=3k_{p}=3C_{\text{rec}}/(4(C_{\text{rec}}+C_{\text{ann}}))$ and $k_{\text{ann}}=C_{\text{ann}}/(C_{\text{rec}}+C_{\text{ann}})$. $dN_{\phi}^{(i)}/dE_{\phi}$ are given by Eqs. 16, 18, 19 and 20. The factor $1/4\pi r^{2}$ in Eq. 37 assumes that the mediators are radiated isotropically from the Sun. Eq. 36 can be simplified significantly by noting that $\phi$ decays isotropically in its center of mass (cm) frame, and that the energy of a positron in the lab frame is only a function of the decay angle in the cm frame $\theta_{\text{cm}}$

$$E_{+}=\frac{\gamma_{\phi}m_{\phi}}{2}(1+v_{+}v_{\phi}\cos\theta_{\text{cm}}).$$

(39)

Here $v_{\phi}=(1-m_{\phi}^{2}/m_{X}^{2})^{1/2}$, $v_{+}=(1-4m_{e}^{2}/m_{\phi}^{2})^{1/2}$ and $\gamma_{\phi}=(1-v_{\phi}^{2})^{-1/2}$. Each angle $\theta_{\text{cm}}$ corresponds to a particular angle in the lab frame

$$\cos\theta_{\text{l}}=\frac{\gamma_{\phi}\left(\cos\theta_{\mathrm{cm}}+\alpha\right)}{\sqrt{\gamma_{\phi}^{2}\left(\cos\theta_{\mathrm{cm}}+\alpha\right)^{2}+\sin^{2}\theta_{\mathrm{cm}}}}\,,$$

(40)

where $\alpha=v_{\phi}/v_{+}$. Combining Eq. 39 and 40 we can write $\theta_{l}(E_{+})$. When $\alpha\geq 1$, there is a maximum angle $\theta_{\text{l}}^{\text{max}}\leq\pi/2$, such that for each $\theta_{\text{l}}<\theta_{\text{l}}^{\text{max}}$ there are two corresponding angles in the center of mass frame and thus two positron energies. We note, that the width of the positron energy spectrum in the lab frame is given by $\Delta E_{+}=\gamma_{\phi}m_{\phi}v_{+}v_{\phi}$. We can now rewrite the last terms of the integrand by applying the chain rule and the identity $\delta[f(x)]=\delta(x-x_{0})/|f^{\prime}(x_{0})|$

	$\displaystyle\frac{d\Gamma}{d\cos\theta_{\text{l}}}\delta(E_{\text{d}}-E_{+})$	$\displaystyle=\frac{d\Gamma}{d\cos\theta_{\text{cm}}}\left|\frac{d\cos\theta_{\text{cm}}}{d\cos\theta_{\text{l}}}\right|\delta(E_{\text{d}}-E_{+})$
		$\displaystyle=\frac{d\Gamma}{d\cos\theta_{\text{cm}}}\left|\frac{dE_{+}}{d\cos\theta_{\text{cm}}}\frac{d\cos\theta_{\text{l}}}{dR}\right|^{-1}\delta(R-R_{0})$
		$\displaystyle=-\frac{1}{\gamma_{\phi}m_{\phi}v_{+}v_{\phi}}\frac{r^{3}}{\sin^{2}\theta_{\text{d}}D_{\odot}^{2}}\delta(R-R_{0}),$		(41)

where the absolute value in the first line ensures that a positive number of particles are emitted in the interval $d\cos\theta_{\text{l}}$, $D_{\odot}=1$ a.u. $=1.50\cdot 10^{11}$ m is the distance to the Sun, $d\Gamma/d\cos\theta_{\text{cm}}=-1/2$ from isotropy and $dE_{+}/d\cos\theta_{\text{cm}}=\gamma_{\phi}m_{\phi}v_{+}v_{\phi}/2$. In the last line we use $d\cos\theta_{\text{l}}/dR=-\sin^{2}\theta_{\text{d}}D_{\odot}^{2}/r^{3}$ which follows from the geometric identity

$$\cos\theta_{\text{l}}=\frac{D_{\odot}\cos\theta_{\text{d}}-R}{r},$$

(42)

with $r^{2}=R^{2}+D_{\odot}^{2}-2RD_{\odot}\cos\theta_{\text{d}}$. When evaluating the $R$-integral in Eq. 36 it is useful to know $R_{0}$ in Eq. 41 which for a particular detector energy (or equivalently a particular $\theta_{\text{l}}$) can be found from Eq. 42

$$R_{0}=\left(\cos\theta_{\text{d}}-\sin\theta_{\text{d}}\cot\theta_{\text{l}}\right)D_{\odot}.$$

(43)

For a fixed detector angle, the distance $r$ is only a function of $R$. For evaluating the angular integral it is also useful to note the quantity $r_{0}=r(R_{0})$ which is

$$r_{0}=\sin\theta_{\text{d}}\csc\theta_{\text{l}}D_{\odot}.$$

(44)

After trivially integrating over $R$ (by using the delta function) Eq. 36 can be written as

$$\frac{dN_{+}}{dAdtd\cos\theta_{\text{d}}dE_{\text{d}}}=-\int_{0}^{\infty}dE_{\phi}\frac{\Gamma_{\phi}(E_{\phi})\csc\theta_{\text{l}}}{4\pi E_{\phi}v_{+}v_{\phi}D_{\odot}L}\,\csc\theta_{\text{d}}e^{-\sin\theta_{\text{d}}\csc\theta_{\text{l}}D_{\odot}/L}\Theta(\sin\theta_{\text{d}}\csc\theta_{\text{l}}D_{\odot}-R_{\odot}),$$

(45)

where $E_{\phi}=\gamma_{\phi}m_{\phi}$ is the energy of the mediator and $\Gamma_{\phi}(E_{\phi})$ is the rate Eq. 38 . In the case where $\alpha>1$, $\theta_{\text{d}}^{\text{max}}=\theta_{\text{l}}(E_{\text{d}})<\theta_{\text{AMS}}(E_{\text{d}})\ll 1$, $\phi_{+}$ becomes approximately

$$\phi_{+}=\int_{0}^{\infty}dE_{\phi}\frac{\Gamma_{\phi}(E_{\phi})}{4\pi E_{\phi}v_{+}v_{\phi}D_{\odot}^{2}}\,\left(e^{-R_{\odot}/L}-e^{-D_{\odot}/L}\right)\,,$$

(46)

where the last term is the fraction of dark photons with energy $E_{\phi}$, which decays between the surface of the Sun and the Earth. The three processes; annihilation, recombination and para-decay contribute to $\Gamma_{\phi}(E_{\phi})$ in the form of delta-functions. In these cases, when Eq. 46 is valid, the signal is independent on $E_{d}$. Hence, when the mediator is more boosted than the electron-positron pair it decays into, the corresponding spectrum is approximately a box. However, for $\alpha<1$, $\cos\theta_{\text{l}}$ takes values in its full range, and for energies $E_{\text{d}}$ where $\theta_{\text{l}}(E_{\text{d}})>\theta_{\text{AMS}}(E_{\text{d}})$, the signal is spread over more than one angular bin of AMS. In this case, we should also subtract positrons emitted behind the Sun. The width of the spectrum is given by $\Delta E_{+}=\gamma_{\phi}m_{\phi}v_{+}v_{\phi}$. The height of the signal scales linearly with the capture rate, and inversely with the width of the spectrum. Furthermore, Eq. 46 shows the dependence on the decay length $L$. For $L\ll R_{\odot}$ all the decays happen within the Sun, whereas for $L\gg D_{\odot}$ all decays occur past the Earth. From Eq. 4 and 11, we get that $L\propto\epsilon^{-2}$, $C_{\text{cap}}\propto\epsilon^{2}$, and conclude that for each pair $\gamma_{\phi},m_{\phi}$ there is an optimum value of $\epsilon$ that maximizes the detected positron flux. When combining the positron spectrum from mediators emitted in different processes, the signal will not scale uniformly with $\epsilon$.