Simplified dark matter models in the light of AMS-02 antiproton data

In this work we perform an analysis of the recent AMS-02 antiproton flux and the antiproton-to-proton ratio in the framework of simplified dark matter models. To predict the AMS-02 observables we adopt the propagation and injection parameters determined by the observed fluxes of nuclei. We assume that the dark matter particle is a Dirac fermionic dark matter, with leptophobic pseudoscalar or axialvector mediator that couples only to Standard Model quarks and dark matter particles. We find that the AMS-02 observations are consistent with the dark matter framework within the uncertainties. The antiproton data prefer a dark matter (mediator) mass in the 700 GeV-5 TeV region for the annihilation with pseudoscalar mediator and greater than 700 GeV (200 GeV-1 TeV) for the annihilation with axialvector mediator, respectively, at about 68% confidence level. The AMS-02 data require an effective dark matter annihilation cross section in the region of 1×10−25−1×10−24 (1×10−25−4×10−24) cm3/sforthesimplifiedmodelwithpseudoscalar (axialvector) mediator. The constraints from the LHC and Fermi-LAT are also discussed.


Introduction
Charged cosmic rays connect information about galactic astrophysics with that about possibly new fundamental particle physics.Explaining the precise measurements of cosmic ray spectra requires the detailed knowing the propagation and injection of cosmic rays and the microscopic properties of the fundamental particle such as dark matter.The recent observations of cosmic ray nuclei by AMS-02, e.g.proton [1], antiproton [2], Helium [3], etc., provide updated understanding the propagation/source injection parameters and leptophobic dark matter models.These measurements gain attentions of both astrophysicists and particle physicists [4][5][6][7][8][9][10][11].
The propagation parameters can be determined by fitting the secondary-to-primary ratio of cosmic ray nuclei, such as the Boron-to-Carbon ratio (B/C), and the ratio of secondary nuclei, such as the Beryllium isotope ratio 10 Be/ 9 Be.The observed proton flux can further fix the unified injection parameters of all nuclei.Based on these obtained parameters, one can derive an up-to-date astrophysical background for the secondary production of antiprotons so as to study the extra sources like dark matter.A self-consistent way to take into account the dark matter source is to propagate the antiproton spectrum induced by dark matter annihilation through the Galaxy and calculate the antiproton flux under the exact same set of the above astrophysical parameters.This procedure ensures a consistent astrophysical treatment of cosmic rays from the standard astrophysical source and dark matter [12].
In this work, we examine the constraint of AMS-02 data of antiproton flux and antiprotonto-proton ratio on leptophobic simplified models of dark matter.This hypothesis is widely adopted in the analysis of dark matter search at the Large Hadron Collider (LHC), satellites in the sky and underground direct detection experiments [13][14][15][16][17][18].It uses minimal and general theoretical assumptions with only two parameters, i.e. the dark matter mass and the mediator mass, and the simultaneous presence of various annihilation channels provides the dark matter models with considerable flexibility.We specifically consider a Dirac fermionic dark matter, with pseudoscalar and axialvector mediators that couple only to quarks and dark matter particles.The resulted dark matter annihilations are not velocity suppressed [19].Meanwhile the dark matter-nucleon elastic scattering cross sections are spin-dependent (SD) thus do not receive stringent constraint from direct detection.We also derive the AMS-02 preferred region in the parameter space of the dark matter models.This paper is organized as follows.In Sec. 2 we describe the propagation equation and injection spectra for cosmic ray nuclei.The values of corresponding parameters are also given.In Sec. 3, we describe the simplified dark matter models we use.Our numerical results are given in Sec. 4. Finally, in Sec. 5 we summarize our conclusions.

Propagation and Injection of Cosmic Rays
Cosmic rays in the Galaxy are categorized into primary and secondary types [20][21][22][23].The interstellar mediums (ISM) are accelerated to produce primary cosmic rays.The produced primary cosmic ray protons and nuclei collide with the ISM and then produce secondary cosmic rays.The cosmic ray propagation within the galaxy is described by the following transport equation [24] where ψ( r, t, p) is the density of cosmic rays per unit of total particle momentum p. V is the convection velocity and τ f (τ r ) is the time scale for fragmentation (radioactive decay).
The spatial diffusion coefficient is usually written in this form with R and β being the rigidity and particle velocity divided by light speed respectively.The diffusion coefficient in momentum space, i.e.D pp , is dependent on the square of the Alfven velocity v A .z 0 is the height of the cylindrical diffusion halo.The above key propagation parameters can be constrained by fitting the latest ratios of nuclei, that is the Boron-to-Carbon ratio (B/C) and the Beryllium ratio ( 10 Be/ 9 Be).We adopt the diffusion reacceleration model and the values of propagation parameters shown in Table 1, determined by the B/C and 10 Be/ 9 Be data [6].In Eq. (2.1), the source term of cosmic ray species i can be described by the product of the spatial distribution and the injection spectrum function For the spatial distribution of the primary cosmic rays we use the following supernova remnants distribution where r = 8.5 kpc is the distance between the Sun and the Galactic center, the height of the Galactic disk is z s = 0.2 kpc, and the two parameters a and b are chosen to be 1.25 and 3.56, respectively [25].We assume the following power law with one break for the injection spectrum of various nuclei The corresponding injection parameters in Eq. (2.5), i.e. rigidity break R p br and power law indexes ν 1 , ν 2 , can be determined by fitting the latest AMS-02 proton data [1].We adopt injection parameters obtained by performing such a fit in Ref. [6].The values of these injection parameters are shown in Table 1, together with the Fisk potential φ i (i = p, p) for solar modulation effect.propagation value nucleon injection value solar modulation value D 0 (10 28 cm 2 s −1 ) 7.09 Parameters of propagation, nucleon injection and solar modulation and their values adopted in our numerical analysis.The proton flux is normalized to A p at 100 GeV in the units of

The Simplified Dark Matter Models
In this section, we describe the simplified dark matter models restricted by the AMS-02 data of antiproton flux and antiproton-to-proton ratio.We assume that dark matter is composed of Dirac fermionic particles, which we denote by χ.The dark matter particles couple to the Standard Model (SM) quarks through a pseudoscalar mediator S or an axialvector mediator V .The corresponding interactions are as follows [16] where v 0 = 246 GeV.Following the general choices in the analysis of dark matter searches in literatures, we take g S DM = g S q = 1 and g A DM = 1, g A q = 1 4 in the calculations below.Under the above assumptions the dark matter models are described by two parameters, i.e. the dark matter mass m χ and the mediator mass m S or m V .The scan range for these parameters is Induced by the interactions in Eqs.(3.1) and (3.2), the pairs of dark matter particle χ can either annihilate into SM quark pairs via the mediator particle in s channel χχ → S/V → qq, or annihilate into the mediator pairs in t channel followed by mediators decaying to SM quarks χχ → SS/V V → qq q q .The resulting cosmic ray spectra can thus be categorized into 2-body spectrum and 4-body spectrum, respectively.
The source term arising from dark matter annihilation contributing to the cosmic ray species i is given by where σ ann v is the total velocity averaged dark matter annihilation cross section of all kinematically allowed channels.dN i /dE is the total energy spectrum of cosmic ray particle i produced in the annihilation, that is the sum of 2-body spectrum and 4-body spectrum For the 2-body spectrum, one has where σ ann v q = σ ann v( χχ → S/V → q q), σ ann v g = σ ann v( χχ → S → gg) for the pseudoscalar mediator case and σ ann v g = 0 for the axialvector mediator case.dN q i /dE and dN g i /dE are the cosmic ray spectra given by dark matter direct annihilating into quark pairs qq and gluons gg, respectively.The 4-body spectrum is where σ ann v Med = σ ann v( χχ → SS/V V ), Γ Med→q q = Γ S/V →q q and the total decay width of the mediator is Γ Med = Γ S/V .Γ Med→gg = Γ S→gg for the pseudoscalar mediator case and Γ Med→gg = 0 for the axialvector mediator case.d N q i /dE and d N g i /dE are the cosmic ray spectra in the lab frame given by the spectrum from the mediator decay in its rest frame, denoted by dN q i /dE 0 and dN g i /dE 0 , after a Lorentz boost [26,27]: where ) As a result of the non-trivial involvement of the mediator, σ ann v and dN i /dE are dependent on both the dark matter mass and the mediator mass.AMS-02 data thus play an important role in constraining these two parameters.We show the σ ann v i / σ ann v as a function of m χ in Fig. 1.For pseudoscalar mediator case, we find that χχ → gg channel is dominant for small dark matter mass region.After t t channel is open, χχ → qq channel turns to be dominant.χχ → SS channel is always very small as it is a process through p wave.For axialvector mediator case, χχ → qq is dominant before χχ → V V is forbidden and after χχ → tt is open.In Fig. 2 we show the resulted total antiproton spectrum x 2 dN i /dE as a function of x = E/m χ .

Results
As discussed in Sec. 2, the propagation and injection parameters of cosmic rays are determined by fitting the B/C and 10 Be/ 9 Be data and cosmic ray proton data from AMS-02, respectively.The parameters in Table 1 thus imply prediction for cosmic ray measurements inferred from standard astrophysical sources.One can investigate the constraint on extra sources, such as dark matter, based on this fiducial astrophysical background.
For each group of dark matter mass and mediator mass, we use PPPC4DMID [29] to generate the antiproton spectrum in Eqs.(3.5) and (3.6), and calculate the dark matter annihilation cross sections following the formulas in Appendix.These dark matter model dependent variables are then passed into the public code Galprop v54 [30][31][32][33][34] to ensure that near Earth cosmic ray fluxes from dark matter annihilation and background spectra are obtained in a consistent way.
The calculated cosmic ray fluxes, together with the measured data points, are put into a composite likelihood function, defined as Here f th i are the theoretical predictions and f exp i are the corresponding central value of the experimental data.The uncertainty σ i combines the theoretical and experimental uncertainties in quadrature.We stipulate a 50% uncertainty of the theoretical prediction of antiproton flux and antiproton-to-proton ratio according to the estimates in Refs.[6,[35][36][37].This uncertainty takes into account, amongst other, the uncertainty related to the fixed propagation parameters.The sum in Eq. (4.1) runs over all the AMS-02 antiproton cosmic ray spectral data points: the antiproton flux (57 points) and antiproton-proton ratio (57 points).
As the dark matter-nucleon elastic scattering cross sections induced by the simplified models we consider are spin-dependent, the most stringent constraints come from collider search and indirect detection of dark matter [38][39][40][41].LHC performed dark matter search using events with large missing transverse momentum plus energetic jets [38] and dijet events [39,40] at 13 TeV collisions.Their exclusion limits can be directly presented in the plane of dark matter mass vs. mediator mass for simplified model with a pseudoscalar mediator or an axialvector mediator.Moreover, Fermi Large Area Telescope (LAT) searched for gamma ray emission from Milky Way satellite galaxies using 6 years of data.They recently released the observed constraints on the dark matter annihilation cross section for pure b b channel [42].We can convert the Fermi-LAT limit into a bound on our dark matter we claim the corresponding set of m χ , m Med is excluded.FIGs. 3 and 4 show our main results: AMS-02 cosmic ray flux data are consistent with the dark matter hypothesis within the uncertainties.The two plots in each figure display the antiproton cosmic ray: antiproton flux and antiproton-to-proton ratio.AMS-02 central value measurements are shown by red dots and error bars in black indicate measurement uncertainties.The green solid curves are obtained using the parameters shown in Table 1 and display the predicted background flux originating from standard astrophysical sources.The blue solid lines show the predictions of the total cosmic ray flux with dark matter contribution that fit the AMS-02 data best and are the sum of the background flux (green curve) and the dark matter contribution at the best fit point (purple curve).A series of salmon colored vertical bars indicate the theoretical uncertainty of the dark matter prediction given by the 2σ confidence region of dark matter model parameters.The plots show that adding a dark matter contribution to the background flux yields a better fit to the AMS-02 data.In the left frame of Fig. 5 we show the regions of the mass parameter space preferred by the AMS-02 data and the LHC limit for the pseudoscalar mediator case.Solid circles and squares denote the estimated 1σ and 2σ confidence regions, respectively.We find the AMS-02 antiproton data favor region 700 GeV m χ 5 TeV at about 1σ confidence level.The LHC excludes a part of the 2σ confidence region with m χ 170 GeV and 300 GeV m S 420 GeV.
The right frame of Fig. 5 shows that the AMS-02 data require an effective dark matter annihilation cross section in the region of 1 × 10 −25 -1 × 10 −24 (5 × 10 −27 -2 × 10 −24 ) cm 3 /s at about 1(2)σ C.L. The LHC excludes a part of the region below thermal relic cross section, denoted by green dots.The Fermi-LAT bound becomes rather weak after t t channel is open and thus does not constrain the AMS-02 favored region.
In the left frame of Fig. 6, for the axialvector mediator case, we can see that the AMS-02 antiproton data favor region m χ 700 GeV and 200 GeV m V 1 TeV at about 1σ confidence level.The region with m χ 1 TeV and m V 500 GeV can evade the LHC limit.

Conclusions
In this work we examine the plausibility of leptophobic dark matter annihilation contributing to the recent AMS-02 data, i.e. the antiproton flux and antiproton-to-proton ratio.Besides the standard astrophysical cosmic ray flux prediction we include a dark matter component.Our choice of the dark matter model is two simplified models of a Dirac fermionic dark matter, with leptophobic pseudoscalar and axialvector mediators that couple only to SM quarks and dark matter particles.The fluxes from standard astrophysical sources and dark matter annihilation are propagated through the Galaxy using the same set of diffusion parameters.The propagation and injection parameters are determined by fitting the latest AMS-02 cosmic ray fluxes of nuclei.
We have shown that not only AMS-02 observations are consistent with the dark matter hypothesis within the uncertainties, but also including a dark matter contribution to the background flux gives a better fit to the data.We also estimated the most plausible parameter regions of the dark matter parameter space in light of AMS-02 data.The observation of antiproton prefers a dark matter (mediator) mass in the 700 GeV-5 TeV (5 GeV-10 TeV) region for the annihilation with pseudoscalar mediator and in the 700 GeV-10 TeV (200 GeV-1 TeV) region for the annihilation with axialvector mediator, respectively, at about 68% confidence level.The AMS-02 data require an effective dark matter annihilation cross section in the region of 1 × 10 −25 -1 × 10 −24 (1 × 10 −25 -4 × 10 −24 ) cm 3 /s for the simplified model with pseudoscalar (axialvector) mediator.The LHC excludes a part of the region below thermal relic cross section for the pseudoscalar mediator model and the region with axialvector mediator mass greater than 500 GeV.The Fermi-LAT bound does not constrain the AMS-02 favored region.

9 )
with = m Med /m χ and x = E/m χ ≤ 0.5.The expressions of dark matter annihilation cross sections and mediator decay widths in Eqs.(3.5) and (3.6) are collected in Appendix.

Figure 1 .
Figure 1.The annihilation cross section fractions σ ann v i / σ ann v as a function of m χ for the pseudoscalar mediator case (left) and the axialvector mediator case (right).The mediator mass is fixed to be 100 GeV.

Figure 2 .
Figure 2. Total antiproton spectrum x 2 dN i /dE as a function of x = E/m χ for the pseudoscalar mediator case (left) and the axialvector mediator case (right).

Figure 3 .
Figure 3. Antiproton flux (left) and antiproton-to-proton ratio (right) observed by AMS-02 (red dots and dark error bars) in the simplified dark matter model with a pseudoscalar mediator.The blue solid line shows the prediction of the total cosmic ray flux with dark matter parameter values that best fit the AMS-02 data.The total predicted flux is the sum of the background flux (green curve) and the dark matter contribution (purple curve).Salmon dots indicate the 2σ confidence region of the prediction.

Figure 4 .
Figure 4. Antiproton flux (left) and antiproton-to-proton ratio (right) observed by AMS-02 (red dots and dark error bars) in the simplified dark matter model with an axialvector mediator.

Figure 5 .
Figure 5. Left: the AMS-02 favored region of masses (m χ vs. m S ) in the simplified dark matter model with a pseudoscalar mediator we consider.The solid circles and squares estimate 1σ and 2σ confidence regions, respectively.The best fit point is indicated by a triangle.The green curve is the LHC exclusion limit [38].Right: the AMS-02 favored region of cross sections (σv vs. m χ ).The green points are excluded by LHC search.The red curve is the converted upper bound from Fermi-LAT, i.e. the right hand side of Eq. (4.2).The black dashed curve corresponds to the thermal cross section [43].