Observational constraints on dark matter decaying via gravity portals

Global symmetry can guarantee the stability of dark matter particles (DMP). However, nonminimal coupling between dark matter (DM) and gravity can destroy the global symmetry of DMP, which in turn leads to their decay. Under the framework of nonminimal coupling between DM and gravity, it is worth exploring to what extent the symmetry of DMP is broken. It is suggested that the total amount of decay products of DMP cannot exceed current observational constraints. Along these lines, the data obtained with satellites such as Fermi-LAT and AMS-02 can limit the strength of the global symmetry breaking of DMP. Since the mass of the DMP may be in the GeV--TeV range, we determine a reasonable parameter range for the coupling strength between DM and gravity in this range. We find that when the mass of the DMP is around the electroweak scale (246 GeV), we can exclude values of the nonminimal coupling parameter $\xi$ greater than $1.5\times10^{-10}$. We also show that the larger the mass of the DMP, the stronger this restriction. Our results give the strongest constraints to date of the possible coupling strength under current observations.


Introduction
Observations of the rotation curves of galaxies, the Bullet Cluster, gravitationally lensed galaxy clusters, type Ia supernova, baryonic acoustic oscillations and anisotropies in the cosmic microwave background have all implied the existence of DM [1]. The Standard Model of particle physics describes electromagnetism, weak and strong nuclear forces successfully [2], however it does not currently accommodate the existence of any DMP.
Among the various properties of DMP, we are extremely concerned with their stability, because stability may be closely related to whether we can observe their decay or annihilation products with current satellites [3]. The stability of electrons is guaranteed by electric charge conservation, while the stability of neutrinos is guaranteed by Lorentz symmetry. Similarly, current observations have suggested that DM is stable and may be composed of particles. It is usually assumed that DMP have global symmetry in Minkowski space-time, such as the hypothetical Z 2 symmetry [4] [5]. But every particle is subject to gravitational interactions. In reality, there is no Minkowski space-time, and gravity does not necessarily couple to DM minimally. In the minimal coupling regime, matter distribution decides the geometry of space-time, and gravity and matter do not transform each other. However, if gravity couples to DM nonminimally, the global symmetry of DMP can be broken [6] [7]. Consequently the stability of DMP is no longer preserved under the influence of gravity [8] [9] [10] [11], implying that DMP could decay via nonminimal coupling due to gravity.
Catà et al. [12] give such a model of DM with global symmetry breaking induced by nonminimal coupling to gravity. There were also other attempts to study nonminimal coupling regime. For example, the Higgs field may have nonminimal coupling with gravity in Higgs inflation [13]. If the mass of the DMP is less than 270 MeV, such a particle could concurrently acting as an inflaton [14]. There are also nonminimal coupling models of DM and gravity where the global symmetry is not destroyed by gravity [15] [16]. As both an inflaton and DMP, the nonminimal coupling between a complex scalar field and gravity has also been used to explain the electroweak phase transition [17] [18]. The possible decay of inert doublet DM and fermionic DM through nonminimal coupling with gravity has also been studied [19].
Many observations and experiments could set limits on the strength of global symmetry breaking of DMP. Currently, there are many types of experiments and observational methods being used to search for DMP. Direct detection methods rely on monitoring nucleon recoil induced by interactions with DMP distributed around the Earth [20]. Indirect detection methods search for photons, neutrinos and/or cosmic rays produced by DMP using satellites and Earth-based instrumentation [21]. The Large Hadron Collider (LHC) also serves as complementary experiment in the search for DM. Cosmological studies have also provided constraints on DM. If nonminimal coupling to gravity breaks the global symmetry of DMP, DM would not be stable, and it would consequently decay into observable particles such as cosmic rays [22], neutrinos [23] or cosmic gamma-rays [24]. So while no conclusive particle signal has yet been attributed to DM [25], current observations can still be used to set limitations on the stability of DMP.
Using chiral perturbation theory, Catà et al. [26] provided the allowed parameter space of light DMP less massive than 1 GeV, in which the decay products were described by a sharp feature photon spectrum. These authors obtained the strongest restriction to date using Fermi-LAT gamma-ray observations. However, the mass range of weakly interacting massive particles (WIMP) and super WIMPs proposed based on the gauge hierarchy problem as well as hidden DM based on the gauge hierarchy problem and new flavor physics is expected to be GeV-TeV [27]. And, if the mass of the DMP is in the GeV-TeV range, more decay channels will be opened and the decay properties of DMP will be quite diverse. Assuming that the lifetime of a DMP is longer than the age of the universe, Catà et al. [12] provided a rough restriction of the nonminimal coupling coefficient between the DM field and gravity around the GeV-TeV range.
In case of DM decay, constraints obtained via direct-detection methods are not as strong as those obtained via indirect-detection methods. As indirect-detection methods, satellites such as Fermi-LAT [28], Alpha Magnetic Spectrometer (AMS) [29] and Dark Matter Particle Explorer (DAMPE) [30] have the ability to obtain sensitive observations of high-energy photons and cosmic rays. However, DAMPE is unable to distinguish between positrons and electrons, thus in this current work we only consider positron data obtained by AMS-02 [29] and photon data obtained by Fermi-LAT [28] to yield the strongest indirect restrictions of the GeV-TeV range to date.
According to the work of Catà et al. [19], the first step is to construct the problem of the action of a system in a Jordan frame. When using a Feynman diagram to calculate the specific decay channel, one can choose to calculate it in either a Jordan or an Einstein frame. For example, Ren et al. [31] used quantum field theory method to calculate Higgs inflation both in a Jordan frame and an Einstein frame. They obtained the same result using both, which reflects the equivalence of the Jordan frame and the Einstein frame in these scenarios. After framing the problem in a Jordan frame, we calculate the spectra of photons and positrons arising from the decay of DMP in the GeV-TeV range where WIMPs, super WIMPs and hidden DM mass may likely be. Finally, we obtain constraints on nonminimal coupling constant ξ, which reflects the strength of the global symmetry breaking of DMP by comparing our theoretical spectra to observations made by Fermi-LAT and AMS-02.
The structure of this paper is as follows. In Section 2, we describe the calculation of the decay spectrum induced by global symmetry breaking. In Section 3, we compare the expected spectrum with that which can be observed from decaying DM in the Solar System with observations, and give constraints on the nonminimal coupling strength between DM and gravity. The discussion and conclusions are presented in Section 4.

Decay spectrum induced by global symmetry breaking
Catà et al. [12] considered that DM has a nonminimal coupling with gravity and whose global symmetry is broken in curved space-time. In a Jordan Frame, the action S of system can be written as (2.1): where Ω 2 = 1 + 2ξM κ 2 ϕ, R is the Ricci scalar, ξ is the coupling constant, and M is a parameter with dimension one so that ξ is dimensionless. For convenience, we fix is the DM potential, X donates a Standard Model particle, ϕ is the scalar singlet DM field, σ bc with D µ the gauge covariant derivative and e νc the vierbein, T F = − 1 4 g µν g λρ F a µλ F a νρ is the kinetic term of a spin-one particle, is the kinetic term of the Higgs boson, L Y is the Yukawa interaction term, and V H is the Higgs potential.
Using a conformal transformation, as shown in Eq. (2.2): one can acquire action in an Einstein Frame, which is shown as Eq. (2.3): In the table, f i represents a fermion and index i includes all fermion flavours, Y µ represents a spin-one particle, a f ij and b f ij can be obtained from the expansion ofT f . W µ represents the W boson and Z µ represents the Z boson, h represents the Higgs boson, v = 246.2 GeV is the Higgs vacuum expectation value, m Yµ represents the mass of the spin-one particle, m f i represents the mass of the fermion, m h represents mass of the Higgs boson. The second column lists the decay channels. For example, ϕ →f i , f i represents the channel through which DM ϕ decays into a pair of fermions.  [32] provided a detailed procedure to calculate decay rates. These authors gave expressions for differential decay rates, e.g. Eq. (2.6), relativistically invariant three-body phase space, e.g. Eq. (2.7), and relativistically invariant four-body phase space, e.g. Eq. (2.10).
For the convenience of description, in the following, we mark the three product particles arising from three-body decay as particle 1, particle 2 and particle 3. We also use nomenclature for the rest frame of particle i and particle j as F ij .
The expression of the differential decay rate is: where Γ is the decay rate of ϕ in its rest frame, m ϕ is mass of the DMP,M is the invariant matrix element, Φ (n) is the n-body phase space, and p i is the four momentum of terminal particle i. We also use the definitions p ij = p i + p j , m 2 ij = p 2 ij , so that the element of three body phase space dΦ (3) can be written as: where(| p * 1 |, Ω * 1 ) is the three momentum of particle 1 in F 12 , and Ω 3 is the angle of particle 3 in the rest frame of the decaying particle. The symbol * always donates a quantity in F 12 .
The relationship between E 3 and m 12 is: where m 3 and E 3 are the mass and energy of particle 3, respectively. The energy spectrum of particle 3 per decay in a channel with final state l can be calculated following Eq. (2.9): Using the vertex rules given in Table 1, and following Eqs. (2.6), (2.7), (2.8) and (2.9), we numerically calculated the decay rate Γ and energy spectrum dρ l /dE 3 . According to translatable symmetry, dρ l /dE 1 and dρ l /dE 2 were also calculated, where E 1 is the energy of particle 1, E 2 is the energy of particle 2.
As for four-body decay, there are three channels: ϕ → W + , W − , h, h; ϕ → Z, Z, h, h and ϕ → h, h, h, h. We will consider ϕ → W + , W − , h, h here to illustrate our method of calculation. In order to demonstrate the calculation of Γ and dρ l /dE 1 clearly, we regard the W + boson as particle 1 and the W − boson as particle 2, while the remaining two Higgs bosons are particles 3 and 4. As before, we still denote the rest frame of particles i and j as F ij .
The element of four-body phase space dΦ (4) can be written as: where(| p 12 |, Ω 12 ) is the three momentum of p 12 , and ( p * * 3 , Ω * * 3 ) is the three momentum of particle 3 in F 34 . The symbol * * always donates a quantity in F 34 . Using Eqs. (2.6) and (2.10), we numerically calculated Γ and dρ l /(dm 12 dm 34 ), where dρ l = dΓ l /Γ l . Then we applied Lorentz transformations to | p * 1 | and E * 1 . We find that the isotropic spectrum of particle 1 with momentum | p * 1 | in F 12 has a spectrum described by Eq. (2.11) in the rest frame of ϕ: where β ij is the velocity of F ij relative to the decaying DMP, γ ij = (1 − β 2 ij ) −1/2 , E ± ≡ γ 12 E * 1 ± γ 12 β 12 | p * 1 | and Θ(x) the Heaviside function. The energy spectrum of particle 1 produced per decay in the channel with final state l can be described by Eq. (2.12): As before, according to translatable symmetry, dρ l /dE 2 , dρ l /dE 3 and dρ l /dE 4 can also be calculated, where E 2 , E 3 and E 4 represent the energy of particles 2, 3 and 4 respectively.
So far, we have obtained many spectra of stable and unstable particles, such as of the Higgs boson, Z boson and neutrino. All these spectra are called primary spectra. For comparison with observations, we calculated the spectra of stable particles, called secondary spectra, such as those of photons, neutrinos and cosmic rays. Cirelli et al. [33] use the Herwig and Pythia codes to generate the secondary spectra f E (E γ,e + ) of photons and positrons induced by a primary state particle with given energy E, where E γ,e + represents energy of the photon or positron. Then, the secondary photon or positron energy spectrum produced per decay in a channel with final state l represented by dρ l /dE γ,e + was numerically calculated as: where s includes all final state particles in the channel with final state l. In the threebody decay case, s runs from 1 to 3, while in the four-body decay case s runs from 1 to 4.
Finally, following the procedure provided by Cirelli et al. [33], we calculated the spectra that could be detected by satellites. We used the Navarro-Frenk-White (NFW) DM distribution model and MED (Median) propagation model to calculate the positron fluxes. The predict flux of each photon was calculated by using the NFW DM distribution model and minUV (minimal universe background) propagation model.

Results
Based on the procedure outlined in Section 2, the photon flux and the positron flux arising from DMP decay, and which would be detected in the Solar System, were calculated. Figures 1 and 2 show two specific examples of our calculated photon and positron fluxes, respectively. In Fig 1, we considered the case of ξ = 2.5 × 10 −11 , and plotted three cases of photon flux predicted by the decay of dark particles with masses of 50 GeV, 246 GeV and 5 TeV. The isotropic diffuse gamma-ray background observed by Fermi-LAT is also shown for reference. The photon flux produced by the 50 GeV DMP is at least four orders of magnitude smaller than isotropic diffuse gamma-ray background. The predicted photon flux from a decay DMP of the typical energy of the electroweak scale (246 GeV) is at least 1.5 orders of magnitude smaller than the observations. This means that the parameter combinations (ξ = 2.5 × 10 −11 , m ϕ = 50 GeV) and (ξ = 2.5 × 10 −11 , m ϕ = 246 GeV) are not excluded by the most sensitive gamma-ray observations obtained to date. For the photons produced by the 5 TeV DMP decay, the photon flux is at least 3.5 orders of magnitude larger than the isotropic diffuse gamma-ray background measured by Fermi-LAT. The parameter combination (ξ = 2.5 × 10 −11 , m ϕ = 5 TeV) is excluded by Fermi-LAT's observations.
In Fig. 2, one can see that the positron flux from the decay of a DMP with a mass of 50 GeV is at least four orders of magnitude smaller than the observations. As this positron flux does not exceed the flux measured by AMS-02, the parameter combination (ξ = 2.5 × 10 −11 , m ϕ = 50 GeV) is not excluded. The situation is the same for the 246 GeV DMP, where the predicted positron flux is at least two orders of magnitude smaller than the detected cosmic ray positron flux. Thus, the parameter combination (ξ = 2.5 × 10 −11 , m ϕ = 246 GeV) is also allowed by the observations of AMS-02. When assuming that the positron flux was produced by a 5 TeV DMP decay, the positron flux was about three orders of magnitude larger than that measured by AMS-02. This means that the parameter combination (ξ = 2.5 × 10 −11 , m ϕ = 5 TeV) can be excluded by the AMS-02 data.
The two variables of nonminimal coupling constant of DM and gravity ξ and the mass of DM m ϕ determine the exclusion range of their two-dimensional parameter space, as shown in Fig. 3. The shadowed area above the dashed line is the excluded region of parameter space (ξ, m ϕ ) as constrained by Fermi-LAT. The shadowed area above the dotted line is the parameter space (ξ, m ϕ ) excluded by AMS-02. In addition, we also plotted the line for Λ EW = 246 GeV, which represents the typical energy of the electroweak scale. It is clear that when the mass of the DMP is around the electroweak scale, a nonminimal coupling parameter ξ greater than 1.5 × 10 −10 can be excluded.

Discussion and Conclusions
Global symmetry can guarantee the stability of DMP. However, nonminimal coupling between DM and gravity can destroy their global symmetry, hence leading to their decay.
In this study we set constraints on the symmetry breaking strength of DMP using the most sensitive observations of photons and cosmic rays respectively made by Fermi-LAT and AMS-02. The results show that the larger the mass of the DMP, the stronger the limitation of the indirect detection. This behaviour is attributed to the fact that a DMP with a larger mass has more decay channels and hence a larger phase space.
Different to the previous work by [26], the mass range of DMP considered in our study is around the GeV-TeV range. Near this scale, the decay channels are more abundant and the phase space is larger, and consequently it is more restricted by observations. In the work by Catà et al. [26], the nonminimal coupling strength between DM and gravity larger than 10 −6 could be excluded assuming that the mass of the DMP is 1 GeV. As was expected, in our work a nonminimal coupling parameter ξ greater than 1.5 × 10 −10 can be excluded assuming that the mass of the DMP is around the electroweak scale (246 GeV). We note that our results are consistent with the work by Catà et al. [26]. However, the mass region analyzed here contains all possible DM decay channels, so the analysis of the extent to what the global symmetry of DM is broken under the influence of gravity is more comprehensive.