Proposition of FSR Photon Suppression Employing a Two-Positron Decay Dark Matter Model to Explain Positron Anomaly in Cosmic Rays

The origin of an anomalous excess of high-energy (about 100 GeV and higher) positrons in cosmic rays is one of the rare problems in this field, which is proposed to be solved with dark matter (DM). Attempts to solve this problem are faced with the issue of having to satisfy the data on cosmic positrons and cosmic gamma radiation, which inevitably accompanies positron production, such as FSR (final state radiation), simultaneously. We have been trying to come up with a solution by means of two approaches: making assumptions (*) about the spatial distribution of the dark matter and (**) about the physics of its interactions. This work is some small final step of a big investigation regarding the search for gamma suppression by employing the second approach, and a model with a doubly charged particle decaying into two positrons (X++ $\rightarrow$ e+ e+) is suggested as the most prospective one from those considered before.

Here, we are trying to reduce the fine-tuning problem in the framework of a DM explanation of PA.This explanation faces the issue of disagreement with data on cosmic gamma radiation, first of all, the so-called isotropic gamma-ray background (IGRB) obtained by Fermi-LAT [11]; an illustration is provided in Figure 1.The authors are aware that Figure 1 does not contain the latest data from the AMS experiment [25].However, the choice for this particular figure was made nonetheless in order to portray the issue, which only intensifies in case of an increase in energy range as in new AMS data.Any positrons (electrons) e + e − produced by annihilation or decay of DM particles induce prompt photons (mainly final state radiation (FSR)) and photons due to the interaction of e + e − with medium photons (mainly due to inverse Compton (IC) scattering on starlight).As one can see from Figure 1, the main problem arises due to, basically, FSR photons and occurs at high energies.
Generally, in spite of this work initially being connected to the problems of dark matter and positron anomaly, it does not exclude its independent importance concerning the issue of FSR suppression.

MATTER
It is possible to propose two approaches for solving the problem of disagreement with gamma-ray data in a DM explanation of PA.First, one is due to a spatial distribution of DM components, including DM clumps and other structures, such as dark disk.The second approach is related to the physical properties of DM particles that govern the decay/annihilation process.
Our group proposed the so-called "dark disk model" [11,[26][27][28][29][30][31] in the framework of the first approach in order to explain positron anomaly in AMS-02 data.The idea is the following.The contradiction is caused by a finite traveling length of high-energy positrons because of the energy losses they suffer and the existence of a magnetic field around the galactic disk, which makes their trajectory tangled and does not allow positrons born outside of it to reach the Earth.However, gammas are unaffected by these and, therefore, arrive from virtually all the distances.This enables one to artificially decrease the amount of gamma flux while keeping the amount of positrons unchanged by "cutting off" an area of space outside the magnetic disk.In fact, there can be one minor "active" component of DM that gives a positron signal and a major passive one, which forms a halo of the galaxy.
It was shown in [10] that the implementation of this particular model greatly reduces the contradiction with IGRB data.
In the framework of the second approach, different attempts were undertaken to find a physical model of DM (Lagrangian) to provide the suppression of gamma-ray output.
However, the focus here lies on doubly charged DM particles.
Earlier, DM models based on technicolor [32,33], where doubly charged techniparticles, in the composition of electrically neutral dark atoms, decay into two positrons (X ++ → e + e + ), were considered.Details of a technicolor DM model can be found in the Appendix A.
Additionally, different DM models with doubly charged particles, based on various standard model extensions [34][35][36][37][38], were discussed and elaborated to solve the contradiction of the results of an underground experiment DAMA with the results of other similar experiments.We do not fix any specific physical model here, but it is believed that such DM candidate containing a doubly charged particle, hidden inside compact neutral dark atoms,1 may avoid different constraints of underground, (above)ground, and space experiments (see, e.g.[39]), including those based on the observation of white dwarfs [40].As to the positron anomaly, a model with the decay X ++ → e + e + has a simple advantage as compared with the more traditional one X 0 → e + e − , since there are twice as many positrons per one FSR photon.
In this short letter, we follow the second approach related to the physical properties of DM, which account for decay with positron production.More specifically, our aim was to point out that the DM model with a doubly charged unstable particle has one more advantage in the context of positron anomaly solution.This additional advantage is associated with two identical particles in the final state [10].Such a system (e + e + ) does not have classical dipole radiation since it has a zero electric dipole moment.The so-called "single photon theorem" (or "radiation zeros") [41] claims partial suppression of identically charged particle radiation, thus restoring a correspondence between classical and quantum descriptions to some extent.

III. MODELS USED
Following the theoretical simplification of the model in [10], two models of a decaying dark matter particle were considered.
• A model with a decay of a scalar DM particle into two positrons according to Lagrangian with an accompanying decay of a DM particle into two positrons and an FSR photon • and a more conventional model, to be compared with, with a decay of a scalar DM particle into an electron and a positron according to Lagrangian respectively accompanied by a decay Ψ represents the positron/electron wave function, index C stands for charge conjugation, a = b = 1 was used in this work during calculations, and γ 5 is the Dirac matrix.
Photon (FSR) suppression is of interest to us, since it is necessary to eliminate the contradiction with the excess of IGRB during the decay of DM particles.This implies that the ratio of the width of the three-body to two-body decay should be minimal: Here, we denoted this ratio as Br, which is (since Γ(X → e + e ± γ) ≪ Γ(X → e + e ± )) close to the branching ratio.

IV. RESULTS
Processes ( 1),( 3),( 4),( 6) were simulated by making use of the CompHEP [42][43][44] and MadGraph [45] MC generators.Numerical results were obtained for the mass of X being equal to 1000 GeV.For the presentation of the results, the relation ( 7) is used in differential form for photon energy spectra in both model cases (e + e − and e + e + ).
where Γ i and Br i are the widths of the respective processes and their ratio (according to (7)), and E is the FSR photon energy.
The results for these two types of processes are shown in Figure 2. As one can note, the X → e + e − γ mode has a more smooth drop in photon energy, especially at the upper kinematic limit.This can finally be observed in Figure 3, where the ratio of these two spectra is shown.This is the main result, which shows the essential suppression of FSR photons in the model with the decay X ++ → e + e + γ as compared with X 0 → e + e − γ with the growth of photon energy, as was necessary for the resolution of contradiction between the DM explanation of PA and data on IGRB.
This behavior of spectra ratio has a qualitative explanation.The highest FSR photon energy corresponds to the situation when two charged leptons move with the maximum possible energy in the direction opposite to that of the photon (lepton and photon momenta are related as follows: ⃗ p e1 = ⃗ p e2 = −⃗ p γ /2).However, two positrons cannot be born with identical momenta because of the Pauli exclusion principle.FIG.3: The ratio R(E) of photon energy spectra from the two processes X → e + e + γ and X → e + e − γ.

V. CONCLUSIONS
In this work, an overview of prerequisites for solving the problem of DM explanation of positron anomaly in CR was conveyed.Such an explanation faces discrepancy with data on cosmic gamma rays.The result of this note is a suggestion of the model, which provides the suppression of FSR photons in comparison with the traditional case.The model suggested is based on a decaying doubly charged DM particle X ++ → e + e + .This displays the suppression of the FSR photon yield, relatively to positrons, for two reasons: first, we have twice as many positrons per photon as compared with the more conventional case X 0 → e + e − ; the second, which is the main result of this note, is the effect of the suppression of FSR photons due to an identity of final charged fermions.The latter leads to an additional essential suppression of FSR.This suppression takes place in the classical case since two same charged particles do not have an electric dipole momentum and, therefore, radiation.In the quantum case, the so-called single-photon theorem tells a similar thing in an implicit way.We have shown here an effect in a specific model example that is yet to be applied to concrete astrophysical and cosmological problems.We do not show here how this suppression helps in explaining the PA problem further.This requires a separate comprehensive study.In any case, such an effect will facilitate its solution, and this is what we pay attention to.
particles).In the second case, on the other hand, an excess of ζ and a smaller excess of U U are assumed.In this case, the main component of DM will consist of the states Heζ (SIMP) and a small component will consist of the states U U ζ (WIMP).
In both cases, it is assumed that U U is the lightest technibaryon, and ζ is the lightest technilepton.The assumption of the smallness of the WIMP component is due to the results of underground experiments on the direct search for DM.The constraint obtained from the underground experiments requires that the relative fraction of the WIMP component has to be at the level of ∼ 10 −6 [32].This value of the WIMP fraction and the corresponding values of initial excesses between particles and antiparticles can be obtained on the base of the mechanism of sphaleron transitions in the early universe and can be associated with an excess of baryons and leptons [32].It is important to note that the U U state has both charge +2 and spin 0, which is important for our final goal.
This is an example of the model, which can incorporate an unstable doubly charged DM candidate.It does not exhaust all the physical frameworks in which such candidates may be possible.
FIG. 1: A comparison of the data of CR experiments and the predicted results of the model for decaying DM: (left) cosmic positron fraction, red line indicates the theoretical prediction, black line is the expected background, and data points are from AMS-02; (right) IGRB, red line corresponds to the total expected contribution of photons of the same DM model as in the left plot, and the data points are of the Fermi-LAT.Figures were taken from[11] [11]

FIG. 2 :
FIG. 2: Energy distribution of photons dBr e + e ± γ (E) dE from DM particle decay through e + e + mode (left) and e + e − one (right).Dotted lines show errors.Cyan color represents errors.