Indirect Detection Signatures for the Origin of Asymmetric Dark Matter

We study the decay signatures of Asymmetric Dark Matter (ADM) via higher dimension operators which are responsible for generating the primordial dark matter (DM) asymmetry. Since the signatures are sensitive both to the nature of the higher dimension operator generating the DM asymmetry and to the sign of the baryon or lepton number that the DM carries, indirect detection may provide a window into the nature of the mechanism which generates the DM asymmetry. We consider in particular dimension-6 fermionic operators of the form ${\cal O}_{ADM} = X {\cal O}_{B-L}/M^2$, where ${\cal O}_{B-L} = u^c d^c d^c,~\ell \ell e^c,~q \ell d^c$ (or operators related through a Hermitian conjugate) with the scale $M$ around or just below the GUT scale. We derive constraints on ADM particles both in the natural mass range (around a few GeV), as well as in the range between 100 GeV to 10 TeV. For light ADM, we focus on constraints from both the low energy gamma ray data and proton/anti-proton fluxes. For heavy ADM, we consider $\gamma$-rays and proton/anti-proton fluxes, and we fit $e^+/e^-$ data from AMS-02 and H.E.S.S. (neglecting the Fermi charged particle fluxes which disagree with AMS-02 below 100 GeV). We show that, although the best fit regions from electron/positron measurement are still in tension with other channels on account of the H.E.S.S. measurement at high energies, compared to an ordinary symmetric dark matter scenario, the decay of DM with a primordial asymmetry reduces the tension. Better measurement of the flux at high energy will be necessary to draw a definite conclusion about the viability of decaying DM as source for the signals.

While the idea that DM may carry a particle asymmetry has existed in the literature for a long time [1][2][3][4][5][6][7][8], it has only been relatively recently that robust classes of models based on higher dimension operators were introduced [9].
The ADM operators communicate an asymmetry between the DM and visible sectors, and have the advantage that they naturally decouple at low energies, leading to conserved baryon and DM asymmetries separately in the two sectors late in the Universe. These operators take on the form where O B−L has dimension m and O X has dimension n. By sharing a primordial asymmetry between the two sectors, the models naturally realize the relationship n X − nX ∼ n b − nb.
Since the observed baryon to DM energy density is ρ DM /ρ b ∼ 5, this implies the natural mass scale of ADM is ∼ 5 GeV. 1 For a review and list of references of DM models employing higher dimension operators, see [10].
As outlined in [10], for such higher-dimension ADM models, there are two basic categories of models. In the first class a primordial matter anti-matter asymmetry is shared between the DM and visible sectors via interactions that are mediated by heavy particles that become integrated out as the temperature of the Universe drops [12][13][14][15][16]. Such scenarios give rise to DM particles whose relic abundance carries the same baryon or lepton number as visible particles.
Examples of operators which may transfer an asymmetry between sectors are where L is the chiral supermultiplet of a SM lepton doublet, H is the Higgs doublet, U c , D c are right-handed anti-quarks, E c is a right-handed charged anti-lepton, and Q is a quark doublet. In the context of supersymmetry, these operators are R-parity violating, and having the simplest interaction with the DM X, the simplest ADM interactions take the form where now we have explicitly included a flavor index i, j, k on the generic scale of the operator M.
In the context of supersymmetry, the ADM particle is stabilized by R-parity. On the other hand, the analogue fermionic operators, of the form 2 may also share a primordial between the two sectors. To distinguish from superpotential multiplets in SUSY, we use lower case letters for the SM fermionic fields in the Lagrangian, and to label the operator conveniently, we use the SM part of the operator as a subscript. For example, we label can be obtained by the Lagrangian L ⊃ y i Xℓ i Φ + y ′ jk Φ † ℓ j e c k , where i, j, k = 1, 2, 3 for 3 generations and Φ is a heavy scalar field in fundamental representation of SU(2) W . If y i = y ′ jk for all i, j and k, we obtain a universal flavor structure for Xℓ i ℓ j e c k M 2 ijk . 3 2 We do not include other choices of Lorentz structures for these 4-fermion interactions since they do not make a substantial difference in the indirect detection signals. 3 We emphasize that one can UV complete this operator in another way, i.e. L ⊃ y 1,i Xe c i Φ + y 2,jk Φ † ℓ j ℓ k . In this case, Φ is a heavy scalar field but a singlet in SU (2) W . Since ℓ j and ℓ k have to contract by an anti-symmetric tensor in the SU (2) W basis, they must be in different generations. A similar subtlety also occurs for the While these operators induce an asymmetry in the two sectors, they also cause the fermionic X to decay. If its abundance has not been cosmologically depleted in the early Universe, and M is a high scale, the decay lifetime can be long. Assuming the heavy mediator is a scalar field, i.e. in the form of the effective operators in Eq. (4), the decay lifetime is approximately Here C color , C f lavor and C SU (2) W indicate the constants introduced from color, flavor and weak isospin combinations in the final states.
Observations of the DM decay products in high energy gamma rays and in charged particles (electrons, positrons and anti-protons) thus will constrain M. As we will show, if M 10 13 GeV, these lifetimes are on the order of current constraints, and their decay may be detectable both in photons and in charged cosmic ray byproducts. Similar decay signatures have also been studied in many other contexts. (Please see [33] and the references therein for a review.) As pointed out in [34,35], current constraints from indirect detection implies a suppression scale around the GUT scale if weak scale DM decays through dimension 6 operators. Most studies, however, have mainly focused on symmetric DM. In this paper, we focus on the asymmetric DM scenario, and, as we will see, the sign of the effective DM baryon or lepton number substantially affects the results. Refs. [35][36][37][38][39] also studied scenarios where DM particles decay asymmetrically. In these studies, however, the operators which induce DM decay may not be those which are responsible for generating the asymmetry in DM sector as in ordinary ADM models. In Ref. [40], the authors briefly mentioned the possibility of ADM decay induced by the operators in ordinary ADM models, though they were mainly focused on the neutrino fluxes induced from other operators. In addition, the studies mentioned above only focused on a few specific decay channels, while we carry out a comprehensive study of ADM decay through various operators.
The goal of this paper is two-fold. First, we aim to study the constraints from photons in the galactic center and diffuse extra galactic background on the scale M in Eq. (4) from fermionic ADM, assuming the fermionic ADM composes all (or most of) the DM. We do this both for ADM in its natural mass window (from a few GeV up to approximately 20 GeV), and for ADM with a heavier mass near the weak scale. Second, we study models of ADM that may generate part or all of the charged cosmic ray signals observed by PAMELA and AMS-02, consistent with the flux of anti-protons in the Universe.
There are many ADM models where the DM mass is much heavier than a few GeV. In this case a mechanism must be present to reduce the DM number density relative to the baryon number density. This can be achieved, for example, by inducing DM/anti-DM oscillations that wash out the asymmetry so that subsequent annihilations can reduce the DM number density. In this case the DM is not asymmetric from an indirect detection point of view. It is not difficult, however, to build a model where the DM is electroweak scale while retaining its asymmetry throughout the history of the Universe. One straightforward way to achieve this is to assume a non-zero primordial baryon/lepton (B/L) number in a parent particle (such as the state integrated out to generate the operators Eq. (1)) which subsequently decays with different branching fractions to the DM and the visible sectors. Such a scenario is discussed in [41][42][43].
As long as the DM and SM sectors are never in thermal equilibrium after decay of the heavy particles, the DM mass can be tuned to any value by changing the primordial asymmetry. In addition, the asymmetry can be diluted later in the Universe through a DM-number violating process (such as annihilation) which washes out the asymmetry; we present such a model in The outline of this paper is as follows. We first discuss the details of the operators we study and specify the flavor structure for each operator in Sec. II. Then, in Sec. III, we provide details of the gamma ray flux calculation, for both the galactic and diffuse extra-galactic gamma rays.
In Sec. IV, we focus on the light ADM scenario and present constraints on the ADM decay lifetime. Finally, in Sec. V, we consider the heavy ADM scenario, discussing in detail charged cosmic ray fluxes and finding the best-fit region for electron/positron fluxes, before concluding.

II. OPERATORS FOR ASYMMETRIC DARK MATTER DECAY
There are many signatures that can arise from DM decay through the operators in Eq. (4).
It is the purpose of this section to motivate the particular choices of flavor structures in these operators that we study below. We do not consider the XℓH operator, which is marginal and will lead to rapid DM decay.
As discussed in the introduction, in most ADM models, the mass of the DM particle is naturally 1 ∼ 20 GeV. The DM may, however, be heavier. Besides the possibility of a primordial asymmetry in the heavy particles which induce the asymmetry in DM/SM sectors through decay [41][42][43], we provide an alternative option in Appendix A. There we build a toy model of thermal ADM where the DM is heavier, which occurs if some X-violating interaction (mediating annihilations) is in thermal equilibrium when the temperature T ∼ m X . In this case, the DM number density is suppressed by a Boltzmann factor e −m X /T f o , where T f o is the temperature at which freeze-out of the X-violating interactions occurs. Since we focus on the phenomenology of ADM decay, we treat the DM mass as a free parameter, and we divide our discussion into two parts. We will first focus on the natural mass range of ADM models, i.e. 3 GeV < m DM < 20 GeV. Then we study the case that 100 GeV m X 10 TeV. We emphasize that this latter case, while motivated by models of ADM, may arise in many GUT-inspired models, such as those explored in [34,35,44].
In ADM models, the DM effectively carries non-zero baryon or lepton number, which may be positive or negative in sign. The gamma ray spectra are indifferent to the sign of the baryon or lepton number of the DM, but it is crucial for the charged cosmic ray measurements. We will consider both cases in our study. we study one more decay channel for O LLE , DM → e ± + τ ∓ + ν(ν).
In addition to the flavor structure of operators, each class of operators has several variations.
As mentioned above, the Lorentz structure of the four Fermi interaction is not important for the indirect detection signals, so that we focus on the contraction integrating out the scalar particle which generates the four Fermi interaction in Eq. (4). Further, one can change the operators by taking charge conjugation on part of the operator. For example, with a small change of the field content to preserve gauge symmetry, O QLD we may have not only However, such changes leave the indirect detection signals essentially unchanged, so that we do not study this variation of the operators further. 4 Operator light ADM heavy ADM ℓℓe c flavor universal e + + e − + ν or ν + ν +ν Finally, one can also change the SU(2) W field content of the operator. For example, O QLD can be changed to The new operator eliminates the hard neutrino, and only a charged lepton appears in the final state. This change impacts both the gamma ray flux and the electron/positron flux. We will take this operator as an example to illustrate the differences induced by this modification.
We summarize the combinations of operators we consider in Table I.

III. PHOTONS FROM DARK MATTER DECAY
Photons can be produced in many ways in DM decay processes. Charged particles in the final state can produce photons through bremsstrahlung. If there are colored particles in the final states, hadronization produces π 0 s, which will decay to photons. Since these photons are produced directly from the primary decay process, they are generically energetic. We will call photons from either bremsstrahlung or hadronic decays FSRγ. The other important source of photons is Inverse-Compton (IC) scattering between energetic electrons/positrons and galactic ambient light, which is mainly CMB photons and starlight. Since the galactic ambient light has very low energy, these IC photons are generically much softer than FSR photons.
In this section, for completeness, we overview the gamma ray spectra from these sources.
We first focus on the gamma ray spectrum from the DM halo in our galaxy, then we will discuss the diffuse gamma ray background. We summarize the data and statistical procedure we used in our analysis.
A. Photon Flux from DM Decay

Galactic DM Halo
The galactic DM halo provides a promising place to look for the gamma ray flux produced through DM decay processes, where the FERMI collaboration has released the sky map of the gamma ray measurement up to a few hundred GeV [45,46]. Electrons/positrons propagating in the galaxy scatter with starlight, as well as infrared and CMB photons to produce Inverse-Compton photon. The spectrum from IC scattering, especially in the inner galaxy, depends strongly on details of the galaxy, such as starlight spectrum and distribution. To avoid introducing large uncertainties, we do not consider the IC spectrum and only focus on the FSRγ for the galactic halo constraints.
The flux of photons from DM decay in our galaxy can be written as where the integral is along the line of sight, dNγ dE is the gamma ray spectrum from ADM decay, and ρ DM (r) is the DM profile in our galaxy. We choose an NF W profile, with r s = 24.42 kpc and ρ s = 0.184 GeV/cm 3 . To get the gamma ray spectrum from DM decay, i.e. dNγ dE , we use MadGraph to generate parton level events, and use PYTHIA to shower and hadronize the events.

Extra-galactic γ-ray
In addition to the galactic halo, the gamma ray flux from the decay or annihilation of DM particles in the early Universe can propagate to the Earth and contribute as a diffuse extragalactic gamma ray background. The measurement of the diffuse extra-galactic gamma ray spectrum is provided by FERMI in [47], and provides a particularly important constraint on DM decay. The ratio of extra-galactic gamma ray flux from DM decay, Φ exG−γ , to the galactic halo gamma ray flux, Φ halo , can be estimated as, where ρ cosmo is the average DM energy density in the Universe, R cosmo is the size of the Universe, ρ ⊙ is the local DM energy density and R ⊙ is the distance from the solar system to the galactic center. Due to this numerical coincidence, the constraints from the diffuse extra-galactic gamma ray flux are comparable to the constraints from the galactic halo.
There are again two dominant contributions to extra-galactic gamma rays, one from FSRγ and the other from scattering between hard electron/positrons produced from the decay and the soft photon background. Unlike in the galaxy, the IC scattering is dominated by scattering off CMB photons. Since the uncertainty is rather small in this case, we will include the IC contribution to the diffuse extra-galactic gamma ray flux.
For the photons produced directly from DM decay, the spectrum can be calculated by properly redshifting the photon injection spectrum at any redshift z. High energy photons can be absorbed in a cosmological length. The dominant absorption is caused by the scattering with CMB photons. This is only important, however, for extremely high energy photons. For the energy range we consider in this paper, i.e. E γ smaller than few T eV , the absorption is negligible. Given an injection spectrum from DM decays at redshift a = 1/(1 + z), i.e. dN γ,FSR dEγ (a) , the flux of photons is We take Ω m + Ω Λ ≃ 1 and Ω DM ρ c ≃ 1.3 × 10 −6 GeV/cm 3 , when calculating the gamma ray flux from prompt photons.
To estimate the gamma ray flux from the IC scattering between high energy electrons/positrons and CMB photons, we closely follow the procedure of [48]. When the DM mass is small, the IC contribution to the extra-galactic gamma spectrum is negligible. However, when the DM is very heavy, e.g. O(TeV), the IC contribution is dominant.
We will see this explicitly when we discuss the heavy ADM scenario.

B. Data and Statistical Methodology
For the galactic gamma ray spectrum, the FERMI collaboration provides two sets of measurements which we use. One is focused on the low energy regime, ranging from 0.2 GeV to 100 GeV [45]. In this measurement, the gamma ray spectrum is provided on different patches on the sky. We choose the patch of the full sky without the galactic plane, i.e. 0 • ≤ l ≤ 180 • and 8 • ≤ b ≤ 90 • . When the DM mass is small, the low energy measurement is the most sensitive probe. The other measurement from the FERMI collaboration is in the high energy energy data [46], and the Fermi diffuse extragalactic gamma ray data [47]. We also show the ADM decay spectra through the O LLE operator assuming a flavor universal structure, shown as red curves.
For the extragalactic gamma ray flux, when the DM mass is large, both FSR and IC contributions are important. The decay lifetime is chosen so that gamma ray from DM decay does not exceed any bin by 1 − σ.
regime, from 4.8 GeV to 264 GeV [46]. The region of coverage is the full sky minus the galactic plane while keeping galaxy center, i.e. (|b| > 10 • )|(l ≤ 10 • )|(l ≥ 350 • ). This will be more useful for constraining the heavy ADM decay scenario. For the diffuse extra-galactic gamma ray spectrum, we take the most recent published measurement from FERMI [47]. In Fig. 1, we overlay all the data sets we use for our gamma ray analysis.
In this paper we provide the most conservative constraints on the ADM decay scenario from gamma ray spectra. We require the flux from ADM decay does not exceed the central value plus twice the error bar in any bin, without any assumption about the background flux. One could improve the constraints by subtracting the astrophysical background, gaining perhaps a factor of a few on the constraints. This, however, induces larger systematic uncertainties from the background. For this reason, we focus on the most conservative analysis.

IV. CONSTRAINTS ON LIGHT DECAYING DM
We begin with gamma ray constraints on ADM particles with mass in the natural window,  Table I. This choice aims to illustrate the effects of final state quark kinematics including the b-quark threshold effect.
As discussed in Sec. III A, one can derive conservative constraints by requiring the spectrum from DM decay not saturating the data in any bin at 95% C.L.. In Fig. 2, we present our results; the left panel shows the constraint from the low energy gamma ray data in the halo, while the right shows the constraint from the extragalactic spectrum. To demonstrate how signals compare to data, we plot the observed spectra overlaid with the spectra from ADM impacts signatures through the sign of the baryon or lepton number that the DM carries, which in turn determines the nature of the decay products. Since the signatures depend on the B/L sign, we will consider both cases. In addition, as usual, the flavor structure of the operators affects the signatures substantially; as summarized in Table I, we will take two extremal cases in this section -DM decaying to the first generation only, or to the third generation only; other flavor combinations fall between these two choices. In addition, for the O LLE operator we make another flavor choice, decay to e + τ − ν, that highlights the asymmetric nature of the decay. When DM is a symmetric relic, generically, one expects the same spectra of electrons and positrons in the final state. 5 However, this is not necessarily the case for ADM -since there may be no hard electrons in the final state, the positron ratio from DM decay alone can be as high as 1. This special feature of the ADM scenario helps to reduce, for example, the tension between the AMS-02 anomaly and measured electron/positron total flux. Further, since there are no hard electrons in the final state, the number of hard photons from FSR, as well as the photons from IC processes, is also reduced.
We have already discussed in Sec. III the methods that we use for constraining ADM decay with photons. Thus in Sec. V A, we will focus on the electron/positron flux and proton/antiproton flux, where we provide details on the data we use and the statistics we apply. In Sec.
V B, we present our results by combining all channels for indirect detection, both gamma and charged cosmic rays.

A. Charged Cosmic Rays
We consider first the fit of ADM decay models to AMS-02 data, and then examine the constraints from the anti-proton flux.

e+/e-Fit from AMS-02
Electron/positron fluxes provide potentially powerful probes of DM properties. In 2008, PAMELA [49] published their measurements of the electron/positron fluxes, showing that the positron fraction rises at energies above 10 GeV. Recently AMS-02 [50] confirmed PAMELA's result but with smaller uncertainties and extending to higher energies. Since ADM decays to quarks and leptons through the operators in Eq. (1), it is interesting to see how well the 5 There are some special cases where even symmetric dark matter decay can induce asymmetric electron/positron spectra. One example is assuming DM is a Majorana fermion with several different decay channels. If there is a non-trivial CP-violating phase, then the electron/positron spectra in the final states can be different from each other. This scenario is realized in [16], though not aimed at inducing DM decay. electron/positron flux can be fit by these operators.
To obtain the electron/positron fluxes received near the Earth, we use GALPROP to calculate the propagation [51]. We run the 2D mode of the code, which calculates the propagation equations on (r, z) grid. We use the same DM distribution profile applied in previous studies, i.e. Eq. (8), and we choose the propagation parameters in a conventional way. The diffusion constant K(E) is taken to be 5.8 × 10 28 (E/4 GeV) 0.33 cm 2 /s, and the root-mean-square of the magnetic field is modeled by an exponential disk, where B 0 = 5 µG, r B = 10 kpc and z B = 2 kpc.
To estimate how well electron/positron fluxes constrain the decay lifetime, we carry out a χ 2 fit including an astrophysical background, which we take to be [52,53]  where e = E 1 GeV . To treat the background uncertainties, we allow variation in both overall normalization and index of the power law. More precisely, we take where 0 < A ± < +∞ and −0.05 < P ± < 0.05. We find the best-fit of the AMS-02 data with these 6 parameters, A ± , P ± , m DM and τ . We will present the 3-sigma best fit region in the (m DM − τ ) plane.
To illustrate how well one can fit AMS-02 electron/positron data, we choose several benchmark points and show the comparison between the fit and the data. For positron ratios, we extend curves beyond current energy range to show how various models behave as more AMS-02 data is accumulated. Complete results for different ADM operators will be shown below, in Sec. V B. results [50]. For positron ratios, we extend curves beyond the current energy range, to show how AMS-02 data might appear at higher energies.

Constraints from p+/p-Fluxes
For operators we are considering, DM decay products may include quarks so that modifications of the proton/anti-proton fluxes are possible. The best data for the proton flux is from AMS-02 [54], while PAMELA provides the most updated results for the anti-proton flux and anti-proton/proton ratio [55,56]. For proton/anti-proton fluxes, the data agrees well with the astrophysical expectation, so that we use this data to constrain the decay lifetime for each oper- To compute the anti-proton flux as a constraint on heavy ADM decay, we applied GALPROP to calculate the propagation of the proton/anti-proton flux, where the parameters are the same as in Sec. V A 1. Due to large uncertainties in the solar modulation of the flux, we focus on proton/anti-proton fluxes whose energy is larger than 1 GeV. To model the astrophysical background of proton and anti-proton fluxes, we fit the proton/anti-proton fluxes as sum of polynomials. Similar to the electron/positron cases, we allow small variations in both the overall normalization and the index of the power law, 0 < A ± < +∞ and −0.05 < P ± < 0.05.
For each DM mass, we find the values of A ± , P ± and τ which best fit the data. Then we constrain the DM decay lifetime at the 2σ level with respect to the best fit point. We show a benchmark O QLD model point which is constrained at the 2σ level in Fig. 4.

B. Combination of All Channels of Indirect Detection
In the previous section, we addressed each indirect detection channel carefully. Now we combine all channels for each operator to determine in detail which regimes of parameter space  all constraints. We also showed that whether ADM carries positive or negative B(L) number strongly impacts the signatures, providing a possible handle to probe the asymmetry generating mechanism of ADM.

VI. CONCLUSIONS
In this paper we have studied signatures for decaying ADM through a higher dimension operator. While most models of ADM in the literature have assumed that the ADM is absolutely stable (e.g. through a Z 2 symmetry or through R-parity), the apparent stability of the DM may simply be due to a very high suppression scale of the higher dimension operator. These same higher dimension operators, as shown in Eq. (1), are responsible for the asymmetry generation in the DM sector. Thus one may be able to connect indirect detection signatures to the ADM mechanism. In addition, the asymmetry in the DM sector gives unique signatures that allow one to prove through indirect detection the sign of the B/L number carried by the DM.
We focused on four Fermi interactions, where a suppression scale M for the operator just below the GUT scale is sufficient to be consistent with all constraints. We considered both ADM in its natural mass window around 10 GeV, as well as heavier ADM with mass between 100 GeV and 10 TeV. In the former case, only low-energy FERMI data, both from the galactic halo as well as the isotropic flux, constrain the lifetime; generally the constrained lifetime translates to a constraint on the suppression scale of around 10 14 GeV. For heavier ADM, we fit the AMS-02 data to the models and consider constraints from high energy FERMI data as well as the anti-proton flux in PAMELA. In this case, a suppression scale of around 10 15 − 10 16 GeV is appropriate for fitting the AMS-02 data. We were able to demonstrate the effect of the sign of the ADM B/L asymmetry on the signatures.
Determining the nature of the DM is a complex multi-faceted problem. Further determining how the DM density is set, for example through a cosmic asymmetry, is an even greater challenge. Astrophysical objects, such as stars and neutron stars can also be crucial probes, though they give no hint as to how the asymmetry was generated in the first place in the DM sector. (See [10] and the references therein for review.) For ADM communicating with the SM through higher dimension operators, if the suppression scale of the operator is between 1 TeV and 10 4 TeV, collider and flavor signatures are relevant for probing ADM, as explored in [57]. For a much higher suppression scale, around the GUT scale, however, one may worry that determining the nature of the ADM mechanism becomes essentially impossible. Here we have shown that indirect detection in these cases may provide a handle, lending one more tool in the hunt for the DM.
where X L and X c R are two Weyl spinors components of X. X carries one unit of baryon/lepton number, depending on how X couples to SM sector. φ is a complex scalar field which carries two units of B/L number. We assume m X is much larger than m φ .
If we assume that the transfer of the SM baryon or lepton number to the DM sector decouples at a high temperature, the baryon or lepton number in the DM sector is locked. The details are highly model dependent, but the ratio of the primordial asymmetries in the two sectors is O(1).
When the temperature drops below the transfer decoupling temperature, the interaction within the DM sector is still active. Due to B conservation, there is no 2-to-2 process (if we restrict ourselves to marginal operators for the annihilation) 7 capable of transferring baryon number from X to φ. One has to rely on a 2-to-3 process, i.e. X +X → 2φ+φ * . The scattering cross section for this process is which controls the abundance of X in the DM sector. We label the temperature when this 2to-3 process freezes out as T X,φ . This is the freeze-out temperature of the chemical equilibrium between X and φ. We assume that the freeze-out temperature for kinetic equilibrium is much lower than T X,φ . Thus both X and φ are thermal, and their number densities are described by a Boltzmann distribution at T X,φ . This is a reasonable assumption, because one needs a large annihilation cross section to deplete the symmetric component of ADM.
If T X,φ is larger than m X , both X and φ are relativistic. The asymmetries of number densities in X and φ depend on the chemical potentials as Since the chemical potentials for X and φ only differ by a factor of 2, the asymmetries carried by these two particles are still comparable to each other. Thus the DM mass cannot be too 7 If we instead allow the annihilation to proceed through higher dimension operators (for example through an interaction XX ′ φ, where X ′ is exchanged in the t-channel and is heavier than X), 2-to-2 annihilation XX → φφ may proceed, though suppressed by the mass scale of the particle (X ′ here) being integrated out. The essential dynamics of the models we consider below is unchanged, though some numbers will be modified.
large to obtain the correct DM density.
If instead T X,φ < m X , X is non-relativistic while φ is relativistic. For non-relativistic particles, the chemical potential is related to the number density difference as Given the fact that m X > T X,φ > m φ , we have Assuming the symmetric component of X is annihilated completely and φ's are too light to contribute significantly to the DM energy density, then we need a ∼ 10 −3 to obtain the correct relic abundance for TeV mass of X. This implies m X /T X,φ ∼ 10 from Eq. (A5).
To determine the required cross section, we compare the interaction rate with Hubble, The cross section of X + X → 2φ + φ * is calculated as Eq. (A2). For T < m X , n X = g X ( m X T 2π ) 3/2 exp[−(m X − µ X )/T ], H = 1.66 √ g * T 2 /M pl . Taking m X = 5 TeV as an example, to satisfy Eq. (A6), one needs y 2 λ 2 ∼ 10 −4 , which is a reasonable choice of parameters with y ∼ η ∼ 0.1.