Spin-one dark matter and gamma ray signals from the galactic center

In this work we study the production of gamma rays in the annihilation of dark matter with a $(1,0)\oplus(0,1)$ space-time structure and compare our results with data from several experiments. Considering next to leading order standard model effects, we find that the velocity averaged cross section for the annihilation of dark matter into $\gamma\gamma$ is well below present upper limits, except for Higgs resonant exchange in which case $M\simeq M_{H}/2$. A proper description of the galactic center gamma ray excess (GRE) is obtained only if photons are produced in Higgs resonant processes. In this case, results are highly sensitive to the precise value of $M$ and the scalar Higgs coupling $g_{s}$. We refine our previous calculation of the dark matter relic density in this framework to take into account resonant effects. Consistency of the GRE and dark matter relic density requires $M=M_{H}/2$ and $g_{s}=1.268\times 10^{-3}$. Upper bounds for the annihilation of dark matter into $\bar{b}b$, $\tau^{+}\tau^{-}$ as well as the limits on the spin-independent dark matter-nucleon cross-section $\sigma_{p}$ are satisfied for these values. The cross-section for the annihilation into $\gamma\gamma$ and $\mu^{+}\mu^{-}$ are of the order of the present upper bounds, and a clear signal should be seen in these channels if the corresponding limits are lowered by one order of magnitude

The understanding of the nature of dark matter is presently one of the major challenges in particle physics, astrophysics and cosmology. Dark matter amounts for 26% of the content of the universe and there is compelling evidence for its existence from the measurements of several independent observables, among which we have galaxy rotation curves, cosmic microwave background and dark matter relic density (for a recent review with a comprehensive list of references see [1]).
The measured dark matter relic density [2] can be obtained from its thermal decoupling from the primordial plasma. This requires σv r ≈ 10 −9 GeV −2 ≈ 10 −26 cm 3 /seg, i.e. of the order of the weak scale cross sections, and a mass of the order of a hundred GeV for the so called weakly interactive massive particle (WIMP). This result can be understood in terms of the exchange of a massive particle between dark matter and standard model (SM) fields, with couplings of the order of the weak interactions, which points to the unification route, i.e. to identify dark matter with particles arising in formalisms unifying the three interactions of the SM or including gravity with a wide variety of models yielding candidates with different space-time structure. Recently, a thorough study of the possibilities for scalar, fermionic and vector dark matter was done in [3], concluding that little room is left by available data for WIMPs with these space-time structures.
The SM uses only a very restricted set of the Homogeneous Lorentz Group (HLG) irreducible representations (irreps) . Indeed, from the isomorphism of the HLG with SU (2) ⊗ SU (2), the irreps of the HLG can be labelled by two SU (2) quantum numbers (a, b). The SM uses only the scalar irrep, (0, 0), for the Higgs, the spin 1/2 chiral representations ( 1 2 , 0) and (0, 1 2 ), for quarks and leptons and the vector irrep, ( 1 2 , 1 2 ), for the gauge fields. Quantum field theory proposals for physics beyond the SM in general use these very same representations for their field content, except for supergravity which includes the gravitino transforming in the (1, 1 2 )⊕( 1 2 , 1) representation and the graviton transforming in the (1, 1) representation.
Recently, in Ref. [4] we proposed an alternative (1, 0) ⊕ (0, 1) space-time structure for dark matter, in a formalism that generalizes the structure of spin 1/2 Dirac theory to spin-one matter particles. The corresponding quantum field theory was developed in [5] and it is based in the parity-based construction of a covariant basis for (j, 0) ⊕ (0, j) fields done in [6]. Spin-one matter fields are described by a six-component spinor and can be endowed with a vector gauge structure (the kinetic term is not chiral thus chiral gauge interactions are not permitted). The interaction of spin-one dark matter (SODM) with SM fields is constructed under the effective field theory philosophy and a basic principle: SM fields are singlets of the dark gauge group and viceversa. Considering for simplicity the dark gauge group as U (1) D , the leading interacting terms in the effective field theory are [4] L int =ψ(g s 1+ig p χ)ψφφ+g tψ M µν ψB µν +L self int , (1) where g s , g p and g t are low energy constants. The selfinteraction terms L self int are given in [5] and are not relevant for the purposes of this work. After spontaneous symmetry breaking, this Lagrangian yields a spin portal (photon and Z 0 coupling to higher multipoles of dark matter) and a Higgs portal with a scalar and a parityviolating pseudo-scalar interaction.
Light dark matter (M < M Z /2) turns out to be inconsistent with the measured relic density and the generated invisible widths of the Z 0 and Higgs boson. The upper bounds for the termal average σv r forDD →bb, τ + τ − extracted by the FermiLAT-DES collaboration in the analysis of photon signals coming from Milky Way dwarf satellite galaxies [7] are well satisfied. On the direct de-arXiv:1911.01604v1 [hep-ph] 5 Nov 2019 tection side, the appropriate description of XENON1T measurement of σ p [8] requires spin portal coupling g t ≤ 10 −5 for M ≈ 100 GeV to g t ≤ 10 −3 for M ≈ 1 T eV . Less stringent upper bounds are obtained for the Higgs portal coupling g s ≤ 10 −2 and no significant constraint is obtained for g p [4].
In this formalism, the annihilation of dark matter into final states containing photons occurs at tree level. On the experimental side, gamma ray physics started a new era with the launch of PAMELA [9], AMS-02 [10], and the Large Area Telescope of the Fermi experiment (Fer-miLAT) [11], and strong upper limits have been obtained by these collaborations for the annihilation of dark matter into SM particles, which are complementary to searches at higher energies in experiments like HESS [12].
The simplest process for photon production is the direct annihilation of spin-one dark matter into two gamma rays which yields cuasi-monochromatic photons with an energy centered at ω = M and a width ∆ω = M v 2 r /8 ≈ 10 −6 GeV (for M ≈ 100 GeV ), thus difficult to resolve. The intensity of this line is related to the corresponding cross section and there are stringent upper bounds on this observable by FermiLAT [13] and HESS [12] collaborations.
The leading contributions for SODM annihilation into two photons are shown in Fig. 1. The stringent bounds for the coupling g t imposed by the XENON1T results [4] lead us to consider also next to leading order SM contributions mediated by the Higgs boson. The Hγγ and HγZ (needed below) three-point functions have been calculated in the literature [14][15] [16]. Considering only the parts contributing to our processes, the SM yields the following effective interactions where G γγ , G Zγ are the corresponding form factors. We will follow a phenomenological approach here, and normalizing the form factors as M H , will extract the couplings from the measured branch- [2] to obtain g γγ = 1.91 × 10 −3 , g Zγ = 3.30 × 10 −3 . These couplings correspond to the form factors for on-shell momentum. As it will be shown below, only the resonant processes produce sizable contributions for most of the channels relevant in this work, hence this approximation is justified for our purposes.
A straightforward calculation to leading order in v 2 where v = 246 GeV stands for the Higgs vacuum expectation value.
In Fig. 2 we show the upper bounds obtained by Fermi-LAT [13] and our results as a function of the dark matter mass for coupling values consistent with the XENON1T results for direct detection of dark matter (see [4]). The Higgs width has not been measured yet and we use the SM prediction Γ SM H 4 M eV in the numerics. It is clear from this plot that the process is dominated by the Higgs exchange and the cross section is well below the FermiLAT upper bounds, except for a tiny energy region around the resonant Higgs contribution. Another important observable is the excess over the known gamma ray sources in the FermiLAT data claimed by several groups [17] [27], whose main results are summarized in a recent analysis by the FermiLAT collaboration [28]. Taking into account all the sources of uncertainties, it is shown in [28] that a GRE excess in a region around 3 GeV indeed exists, but a broad band of possible values for the corresponding flux as a function of the photon energy is permitted by these uncertainties. Although the GRE can be explained by little known astrophysical sources [17] [29] [30] [31] [32] [33], the annihilation of dark matter into final states containing photons remains as an attractive possibility and we work out here the results for dark matter with a (1, 0) ⊕ (0, 1) structure.
The gamma-ray differential intensity from the annihilation of (non-self-conjugated) dark matter is The sum runs over all annihilation channels containing at least one photon in the final state; σv r i is the velocity averaged cross section corresponding to the i channel and B i is the number of photons produced in this process. The term in the first parentheses contains all the information from the dark matter interactions and the second parentheses contains the so called J-factor which depends only on the DM distribution and observation geometry.
A mechanism proposed to explain the GRE is the delayed emission of secondary photons from Bremstrahlung or Inverse Compton Scattering of electrons produced in dark matter annihilation, which propagates over the galactic center and scatter photons from the Cosmic Microwave Background (CMB) or starlight [34]. The required σv r ee is of the order of the thermal cross section. For SODM, this cross section turns out to be very small but under certain conditions the annihilation into muons can reach the required values. The analogous contribution of muons was studied in [35] and it was shown there that this mechanism can account for the GRE if the annihilation of dark matter into muons is σv µµ ≈ 1.7 × 10 −26 cm 3 /seg for M ≈ 60 GeV . The cross-section for the annihilation of SODM into muons is given by [4] σv This cross-section is small (of the order of 10 −32 cm 3 /seg or lower) except at the Higgs reso- 4M 2 ). These contributions dominate the related three-body final state processes obtained considering the resonance decay into two particles, which have continuous photon spectrum. Again, these three-body processes are induced by the coupling g t constrained to small values by the XENON1T results [4], hence we consider all possible three-body contributions, depicted in Fig. 4, which can be classified into initial state radiation (dominated by the resonance), internal radiation and final state radiation transitions. It is well known that initial state radiation yield spectrums with shape similar to the GRE when there are resonant effects involved in the process. In this case the resonant effects translates into wider peaks in the photon spectrum (see eg. [37] for these effects in colliders at low energies ). A calculation of the first two diagrams in Fig. 4 yields γ and Z exchange of order O(v 2 r ) and initial state radiation contributions are dominated by the Higgs exchange. For this contribution we obtain where N c = 1(3) for leptons (quarks) and the sum runs over all kinematically allowed SM fermions. The corresponding results for the observable reported by the experimental collaborations is shown in Fig. 5. As expected, for M > M H /2 the photon spectrum has a bump at energies corresponding to di-fermion invariant mass close to the Higgs resonance. The location of this bump depends on the dark matter mass and for M = 64 GeV it coincides with the GRE bump at 3 GeV . However, from this plot it is clear that initial state radiation yields a very small photon flux compared to the measured GRE.
Final state radiation is given by the next two diagrams in Fig. 4. The γ and Z exchange are O(v 2 r ) and the leading contributions are given by the diagrams with the Higgs exchange which yields where Q f stands for the fermion charge in units of e > 0. The corresponding photon spectrum as a function of the dark matter mass is shown in Fig.6. These contributions are also resonant but in this case the resonant effects occur exactly at From this plot it is clear that results for ω 2 dN/dω of the order of the GRE are obtained only for Higgs exchange in the resonance region. Since dark matter is non-relativistic, this requires M M H /2. The internal radiation is given by the last two diagrams of Fig. 4. For the sequential decay with γ and Z 0 intermediate states we obtain where There is also an enhancement at the Higgs resonance in these processes and a double-resonant effect in the case of the Z * intermediate state. The last diagram in Fig. 4 involve non-perturbative QCD effects. We calculated these contributions finding them negligible even at the Higgs resonance. There are also contributions with the sequential decaysDD → γ * , Z 0 * → γH → γf f not shown in Fig. 4 which are not resonant and are also very small. Our final results are shown in Fig. 7. The spin portal and parity-violating contributions given by g t and g p are negligible and the spectrum depends only on the parityconserving Higgs-DM interaction coupling g s . Here the secondary photon contributions calculated in [35] are scaled by a factor σv r µµ /2 σv r C µµ , where σv r µµ is the result of our model and σv r C µµ is the cross-section used in [35]. The factor 2 is necessary because the SODM particle is not self-conjugate. Additionally, since the contributions for these secondary photons are reported in units of GeV /cm 2 seg sr, we multiply their result by the solid angle Ω 0 = 0.121 sr corresponding to |l| < 10 • and |b| < 10 • which is the ROI studied in [28]. Finally, the J-factors over the whole sky are J = 1.79 GeV 4 /cm 2 seg and J = 2.51 GeV 4 /cm 2 seg for Navarro-Frenk-White (NFW) [38] and generalized NFW (gNFW) [39] [40] profiles, respectively. We use the gNFW profile with γ = 1.25 and since the ROI is |l| < 10 • and |b| < 10 • , we multiply our result by Ω 0 /4π. The result M = M H /2 arising from our calculation of the GRE, lead us to check calculation of other observables in [4] for this specific value. The XENON1T upper bound for σ p at M = M H /2 is σ p ≤ 9.86 × 10 −47 cm 2 and this upper bound requires g s ≤ 5.12 × 10 −3 or g t ≤ 1.7 × 10 −5 for the spin portal. No significant constraint is obtained for g p whose contributions are v 2 r -suppressed. Our results for the dark matter relic density in [4] must be refined if M = M H /2, since we are at the Higgs resonance and it has been shown previously in a general analysis that the naive calculation of the dark matter relic density using the non-relativistic expansion can fail in the presence of resonances [41]. The scope of this failure depends on the specific values of the resonance mass and width, but it can be important well beyond the resonance region. Our model allows to test how important it can be for the Higgs resonance. The spin portal contributions generated by the g t coupling turn out to be very small and the pseudoscalar Higgs portal induced by g p violates parity, thus in the following we consider only the scalar Higgs portal contributions given by g s .
The calculation of the relic density requires to consider the full thermal average cross-section σv r (x) for the annihilation of dark matter into standard model states. For masses M M H /2 dark matter annihilates only intof f , γγ and Z 0 γ. For the scalar Higgs portal, the cross section for the annihilation into light fermions is given by Eq.(29) of [4]. A straightforward calculation ofD D → γγ, Z 0 γ induced by Higgs exchange yields We use these results to calculate numerically the complete thermal average cross section. Our results compared with those obtained with the non-relativistic expansion are shown in Fig. 8, where it can be seen that even for values of M far from the resonance there are important differences and, at the resonance, these differences are dramatic and extend to the highly non relativistic regime. We use the complete function σv r (x) to solve the freezing condition and the Boltzmann equation for the relic density following the conventional procedure used in [4]. The freezing temperature is still around x f ≈ 25. The corresponding values of the coupling g s and the dark matter mass M consistent with the measured relic density are shown in Fig. 9 Fig. 7, relic density requires g s = 1.268 × 10 −3 . For these values we get σv r µµ = 2.2 × 10 −26 cm 3 /seg which is of the order of the upper bounds obtained in [42] for this observable using the AMS02 data on the positron fraction in primary cosmic rays [43]. Summarizing, in this work we study the production of gamma rays in the annihilation of dark matter with a (1, 0) ⊕ (0, 1) space time structure. We calculate the annihilation into two photons and show that σv r γγ is well below the experimental upper bounds except at the Higgs resonance in whose case M M H /2. Next we show that the explanation proposed in [35] for the gamma ray excess (GRE) from the galactic center as delayed photon emission in inverse Compton scattering of muons from cosmic microwave background and starlight, with muons produced in dark matter annihilation, works only if muons are produced in a Higgs resonant process, in whose case M = M H /2. We study the production of prompt photons finding initial state radiation negligible, whereas final state and internal radiation are sizable only at the Higgs resonance. We refine our previous calculation of the dark matter relic density in [4] to take into account the resonant Higgs effects and next to leading order standard model effects. Consistency of dark matter relic density and GRE requires M = 62.545 GeV and g s = 1.268 × 10 −3 . For these values, we find a consistent description of the measured relic density [2], the GRE [28], XENON1T results on σ p [8], upper limits on the annihilation of dark matter into two photons from FermiLAT [13] and HESS [12], and upper limits by Fermilat-DES on dark matter annihilation intobb, τ + τ − in the analysis of photon signals from Milky Way dwarf spheroidal satellite galaxies [7]. The cross-sections for the annihilation of dark matter into µ + µ − and γγ are of the order of the present upper bounds. It would be important to focus experimental effort in lowering these limits to further test the consistent picture arising from the calculations in this work.