Astrophysical Background and Dark Matter Implication Based on Latest AMS-02 Data

In the conventional model, CR production and propagation are governed by the same mechanism at energies below 10¹⁷ eV. CR propagation is often described by the diffusion equation (Berezinskii et al. 1990)

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial \psi }{\partial t} & = & {\rm{\nabla }}({D}_{{xx}}{\rm{\nabla }}\psi -{{\boldsymbol{V}}}_{c}\psi )+\displaystyle \frac{\partial }{\partial p}{p}^{2}{D}_{{pp}}\displaystyle \frac{\partial }{\partial p}\displaystyle \frac{1}{{p}^{2}}\psi \\ & & -\,\displaystyle \frac{\partial }{\partial p}\left[\dot{p}\psi -\displaystyle \frac{p}{3}({\rm{\nabla }}\cdot {{\boldsymbol{V}}}_{c})\psi \right]\\ & & -\,\displaystyle \frac{1}{{\tau }_{f}}\psi -\displaystyle \frac{1}{{\tau }_{r}}\psi +q({\boldsymbol{r}},p),\end{array}\end{eqnarray} \tag{ 1 }$$

where $\psi ({\boldsymbol{r}},p,t)$ is the number density per unit of total particle momentum, which is related to the phase-space density $f({\boldsymbol{r}},p,t)$ as $\psi ({\boldsymbol{r}},p,t)=4\pi {p}^{2}f({\boldsymbol{r}},p,t)$. D_xx is the spatial diffusion coefficient parameterized as

$$\begin{eqnarray}&&{D}_{{xx}}=\beta {D}_{0}{\left(\displaystyle \frac{\rho }{{\rho }_{0}}\right)}^{{\delta }_{\mathrm{1,2}}},\end{eqnarray} \tag{ 2 }$$

where $\rho =p/({Ze})$ is the rigidity of the CR particles and ${\delta }_{1(2)}$ is the index below (above) a reference rigidity ${\rho }_{0}$. The parameter D₀ is a normalization constant, and $\beta =v/c$ is the ratio of the velocity v of the CR particles to the speed of light c. V_c is the convection velocity related to the drift of CR particles from the Galactic disk due to the Galactic wind. The diffusion in the momentum space is described by the reacceleration parameter D_pp related to the Alfvèn speed V_a, i.e., the velocity of turbulences in the hydrodynamical plasma, whose level is characterized as ω (Berezinskii et al. 1990; Seo & Ptuskin 1994):

$$\begin{eqnarray}&&{D}_{{pp}}=\displaystyle \frac{4{V}_{a}^{2}{p}^{2}}{3{D}_{{xx}}{\delta }_{i}\left(4-{\delta }_{i}^{2}\right)\left(4-{\delta }_{i}\right)\omega },\end{eqnarray} \tag{ 3 }$$

where ${\delta }_{i}={\delta }_{1}$ or ${\delta }_{2}$ is the index of the spatial diffusion coefficient. $\dot{p}$, ${\tau }_{f}$, and ${\tau }_{r}$ are the momentum-loss rate, the timescales for fragmentation, and the timescales for radioactive decay, respectively. The momentum-loss rate of CR electrons is not the same as that of CR nucleons, and the relevant expressions are found in Appendix C of Strong & Moskalenko (1998).

The convection term in Equation (1) is used to describe the Galactic wind blowing outward from the Galactic disk, and in the GALPROP package (Strong et al. 2000), the wind velocity V_c is expressed as (Strong & Moskalenko 1998)

$$\begin{eqnarray}&&{V}_{c}={V}_{0}+z\displaystyle \frac{{{dV}}_{c}}{{dz}},\end{eqnarray} \tag{ 4 }$$

where z is the height perpendicular to the Galactic disk and also appears in Equation (5).

The source $q({\boldsymbol{r}},p)$ of the primary particles is often described as a broken power-law spectrum multiplied by the assumed spatial distribution described in the cylindrical coordinate (R, z) (Strong & Moskalenko 1998):

$$\begin{eqnarray}\begin{array}{rcl}{q}_{A}(R,z) & = & {q}_{0}{c}_{A}{\left(\displaystyle \frac{\rho }{{\rho }_{\mathrm{br}}}\right)}^{{\gamma }_{s}}{\left(\displaystyle \frac{R}{{R}_{\odot }}\right)}^{\eta }\\ & & \times \,\exp \left[-\xi \displaystyle \frac{R-{R}_{\odot }}{{R}_{\odot }}-\displaystyle \frac{| z| }{0.2\,\mathrm{kpc}}\right],\end{array}\end{eqnarray} \tag{ 5 }$$

where η = 0.5, ξ = 1.0, and the parameter q₀ is normalized with the propagated flux of CR protons. c_A is the relative abundance of the Ath nucleon. The reference rigidity ρ_br is described as the breaks of injection spectrum. ${\gamma }_{s}$ are the power indices below (above) a reference rigidity.

In this paper, the injection spectra of the primary CRs are described by the expression of the continuous functions in the referred rigidity of CRs, which may express a simple power law with the same indices below/above the referred rigidity. The spectral index difference between CR species is not considered in the paper. The expression ${\left(\tfrac{\rho }{{\rho }_{\mathrm{br}}}\right)}^{{\gamma }_{s}}$ is replaced by the following:

$$\begin{eqnarray}&&{\left(\displaystyle \frac{\rho }{{\rho }_{\mathrm{br}}}\right)}^{-{\gamma }_{1}}{\left(1+{\left(\displaystyle \frac{\rho }{{\rho }_{\mathrm{br}}}\right)}^{\tfrac{-{\gamma }_{1}-(-{\gamma }_{2})}{\alpha }}\right)}^{-\alpha },\end{eqnarray} \tag{ 6 }$$

where ${\gamma }_{1}$ and ${\gamma }_{2}$ are the spectral indices below/above the referred rigidity, and α determines the smoothness of the spectral change on the left and right sides of the referred rigidity; when α is 0, as a broken power law, the expression is the same as the one in Equation (5). In this work, α is taken as a free parameter to analyze the smoothness of the injection spectra of CR nucleons.

The flux of secondary particles is derived from the primary particle's spectra, spatial distribution, and interaction with the interstellar medium. The calculation methods of the secondary particle fluxes are referred to in the papers of Strong & Moskalenko (1998), Moskalenko et al. (2001), and Kelner et al. (2009).

CR antiprotons and positrons are produced in the collision with the interstellar medium, and in the propagation Equation (1) their source terms are described as follows (Moskalenko & Strong 1998):

$$\begin{eqnarray}\begin{array}{rcl}{q}_{\bar{p},{e}^{\pm }}(p) & = & \displaystyle \frac{c}{4\pi }\displaystyle \frac{{dn}(p)}{{dt}}=\displaystyle \frac{c}{4\pi }\\ & & \times \,\displaystyle \sum _{i={\rm{H}},\mathrm{He}}{n}_{i}\displaystyle \sum _{j}\displaystyle \int {{dp}}^{{\prime} }\beta {n}_{j}({p}^{{\prime} })\displaystyle \frac{d{\sigma }_{{ij}}({E}_{\mathrm{tot}},{p}^{{\prime} })}{{{dE}}_{\mathrm{tot}}},\end{array}\end{eqnarray} \tag{ 7 }$$

where n_i is the interstellar H and He density, ${n}_{j}({p}^{{\prime} })$ is the density of the jth CR nucleon, and ${\sigma }_{{ij}}$ is the cross section between the jth CR nucleon and interstellar H and He.

With the cross sections between the nucleon's collision, which are taken from the theoretical models or the collision experimental data, Equation (7) is often used to calculate the flux of secondary CRs. Recently, these calculations have been improved by using Z-factors (Kachelriess et al. 2014, 2015), where they took a numerical calculation instead of Equation (7). The concerned details may be found in Kachelriess et al. (2014, 2015). As the calculated Z-factors are relative to a simple power-law spectrum, in this paper a broken power-law spectrum is chosen to recalculate the Z-factors using the CRMC package (Pierog et al. 2015; Baus 2015). The details of the CRMC package (Pierog et al. 2015; Baus 2015) are found in Baus (2015). In the recalculation of Z-factors, the collision between the particles is more than 8,780,000 times.

For the Z-factor expression of the CR antiproton case, Equation (7) is modified as

$$\begin{eqnarray}&&{q}_{\bar{p}}^{{ij}}({E}_{\bar{p}})={n}_{i}\,{I}_{j}({E}_{\bar{p}})\,{Z}_{\bar{p}}^{{ij}}({E}_{\bar{p}},{\alpha }_{j}).\end{eqnarray} \tag{ 8 }$$

Here ${\alpha }_{j}$ are the power indices of the interstellar spectra ${I}_{j}({E}_{\bar{p}})\,\propto {E}_{\bar{p}}^{-{\alpha }_{j}}$ of the jth CR nucleon. The Z-factor ${Z}_{\bar{p}}^{{ij}}$ is expressed via the inclusive spectra of antiprotons $d{\sigma }^{{ij}\to \bar{p}}(E,{z}_{\bar{p}})/{{dz}}_{\bar{p}}$ with ${z}_{\bar{p}}\,={E}_{\bar{p}}/E$, as follows:

$$\begin{eqnarray}&&{Z}_{\bar{p}}^{{ij}}({E}_{\bar{p}},{\alpha }_{j})={\int }_{0}^{1}{dz}\,{z}^{\alpha -1}\,\displaystyle \frac{d{\sigma }^{{ij}\to \bar{p}}({E}_{\bar{p}}/z,z)}{{dz}}.\end{eqnarray} \tag{ 9 }$$

In Table 1 of Kachelriess et al. (2015), Z-factors are calculated with the modified QGSJET-II-4 model, whose values are listed discretely in the limited ranges of the energy and the spectral indices. In the calculation of secondary particle spectra, these numbers are not used conveniently. In this paper, these numbers are interpolated and fitted to an analytic expression, which is written in a good approximation as continuous functions

$$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{j}({\epsilon }_{\bar{p}},{\gamma }_{j}) & = & {\gamma }_{j}^{-6}{\mathrm{ln}}^{0.8}[\,0.16{\epsilon }_{\bar{p}}(10-\mathrm{ln}\,{A}_{j})+1\,]\\ & & \times {\left(1+\displaystyle \frac{1}{{\epsilon }_{\bar{p}}}+\displaystyle \frac{a}{{\epsilon }_{\bar{p}}^{3}}\right)}^{-2},\\ {\gamma }_{j} & = & \displaystyle \frac{{\gamma }_{2,j}}{1+\tfrac{1}{12}{\left(\tfrac{{\rho }_{\mathrm{br}}}{{E}_{\bar{p}}}\right)}^{{\gamma }_{2,j}-{\gamma }_{1,j}}}.\end{array}\end{eqnarray} \tag{ 10 }$$

Here the dimensionless quantity ${\epsilon }_{\bar{p}}=\tfrac{{E}_{\bar{p}}}{{\rm{GeV}}/{\rm{n}}}$ denotes kinetic energy ${E}_{\bar{p}}$ over GeV per nucleon. The above spectral index ${\alpha }_{j}$ is replaced by γ_j. Aⁱ_gas and A_j are the nucleon number of the ith interstellar gas and the jth CR nucleon, respectively. a = 0.5 when ${A}_{j}\leqslant 4$ and a = 1.0 when ${A}_{j}\gt 4$. In the expression of ${\gamma }_{j}$, the three parameters ${\gamma }_{1,j}$, ρ_br, and ${\gamma }_{2,j}$ are served for the broken power-law spectra of the primary CRs.

Table 1.  The Z-factors from Expression (11) and Kachelriess et al. (2015)

${E}_{\bar{p}}$	(${A}_{\mathrm{gas}}^{i},{A}_{j}$) in This Paper, Equation (11)				(${A}_{\mathrm{gas}}^{i},{A}_{j}$) in Kachelriess et al. (2015)
	(P,P)	(P,He)	(He,P)	(He,CNO)	(P,P)	(P,He)	(He,P)	(He,CNO)
1	0.008	0.028	0.025	0.189	0.00772	0.0248	0.0277	0.196
10	0.094	0.360	0.310	3.617	0.1	0.35	0.339	3.24
100	0.178	0.694	0.587	7.106	0.187	0.715	0.612	7.15
1000	0.244	0.960	0.805	9.902	0.248	0.978	0.787	9.81
10000	0.304	1.199	1.002	12.428	0.307	1.2	0.959	12

Note. As an example, γ_j is chosen as 2.4 without the break for the injection nucleons A_j; the nucleon numbers are from the values in Table 1 of Kachelriess et al. (2015). As an example, A_j is 14 for CNO. The target nucleons Aⁱ_gas are P and He.

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With Equation (10), ${Z}_{\bar{p}}^{{ij}}({\epsilon }_{\bar{p}},{\gamma }_{j})$ is simply rewritten as

$$\begin{eqnarray}&&{Z}_{\bar{p}}^{{ij}}({\epsilon }_{\bar{p}},{\gamma }_{j})={C}^{{ij}}{A}_{\mathrm{gas}}^{i}{A}_{j}{\sigma }_{j}({\epsilon }_{\bar{p}}).\end{eqnarray} \tag{ 11 }$$

Here C^ij is a discrete function expressed as

$$\begin{eqnarray}{C}^{{ij}}=\left\{\begin{array}{ll}9.44, & {A}_{\mathrm{gas}}^{i}=1\\ 8.97, & {A}_{\mathrm{gas}}^{i}=4,{A}_{j}=1\\ 7.08, & {A}_{\mathrm{gas}}^{i}=4,{A}_{j}\gt 1.\end{array}\right.\end{eqnarray} \tag{ 12 }$$

Based on the analytic expression (11), the source term of the secondary antiproton is modified as

$$\begin{eqnarray}&&{q}_{\bar{p}}({E}_{\bar{p}})=\displaystyle \sum _{i={\rm{H}},\mathrm{He}}{n}_{i}\displaystyle \sum _{j=1}^{{A}_{\max }}\,{I}_{j}({E}_{\bar{p}})\,{Z}_{\bar{p}}^{{ij}}({E}_{\bar{p}},{\gamma }_{j}),\end{eqnarray} \tag{ 13 }$$

where A_max is the maximum nucleon number of the chosen particle of CRs, which mainly contribute to the production of CR antiprotons. In order to check the values of expression (11), the calculated values in the example of γ_j = 2.4 are listed in Table 1. If the standard deviation σ is 0.02, the total χ² over data points calculated by the differences between the values of expression (11) and Kachelriess et al. (2015) is near 1.0 for all cases (${A}_{\mathrm{gas}}^{i},{A}_{j}$). Thus, the errors may be accepted.

In this paper, the CR propagation Equation (1) is solved by the GALPROP package (Strong et al. 2000), which is based on a Crank–Nicholson implicit second-order scheme (Strong & Moskalenko 1998). In order to solve the equation, a cylindrically symmetric geometry is assumed, and the spatial boundary conditions assume that the density of CR particles vanishes at the boundaries of radius R_h and half-height Z_h. The calculation option of the tertiary antiprotons is turned on in the GALPROP package (Strong et al. 2000). As the flux of the tertiary antiprotons is less than the secondary ones, the production of the tertiary antiprotons does not enhance the total fluxes of CR antiprotons to match the AMS-02 data.

At the top of the atmosphere of Earth, CR particles are affected by the solar winds and the heliospheric magnetic field. The force-field approximation is used to describe that effect, and the solar modulation potential ϕ denotes the force-field intensity (Gleeson & Axford 1968). In this paper, ϕ is a free parameter, and in some fittings, the difference of that from the experimental data is also taken for granted.

In order to compare the secondary particle production between the conventional and QGSJET-II-4 models, the Z-factors are recalculated for the secondary antiproton and positron based on a broken power-law spectrum of the primary particles, which is different from the a simple power law referred to in Kachelriess et al. (2015). The calculated Z-factors for the secondary antiproton and positron are drawn in the left and right panels of Figure 1. As seen in the left panel of Figure 1, for the secondary antiproton antiproton, the calculated Z-factors with a break in QGSJET-II-4 are greater than those of the conventional model at low energies. The Z-factors relevant to a simple power-law spectrum of the primary particles in QGSJET-II-4 are close to the ones in the conventional model. In the right panel of Figure 1, for the secondary positron, the calculated Z-factors in QGSJET-II-4 are slightly less than those in the conventional model from 1 to 10 GeV. Thus, we only replace the hadronic interaction model for the antiproton production with the QGSJET-II-04 model in the GALPROP package (Strong et al. 2000).

In the propagation models, the propagation parameter Alfvèn speed V_A expresses the reacceleration of CRs, and the convection velocity V_c and dV_c/dZ_h describe the convection of CRs. There are reacceleration parameters in both the DCR and DR models. In order to analyze the exclusive effectivity of the convection, we remove the reacceleration in the DCR model to make up the diffusion model with the convection (DC) model, which only includes the convection velocity. In the DC, DR, and DCR models, the propagation parameters in common are the following:

• 1.  
  Half-height Z_h, diffusion parameter D₀, and $\delta (\delta ={\delta }_{\mathrm{1,2}})$.
• 2.  
  Spectral index: ${\gamma }_{1,2}^{p}$ below (above) a reference rigidity ρ_br and the spectral smoothness α for the source expression of primary nucleons.

In the DCR model, Alfvèn speed V_A is not different between CR positrons and nucleons. The DCR${}_{V}$ model is constructed to verify whether reacceleration has a different effect between CR positrons and nucleons. The DCR₀ model is relevant to the conventional hadronic model, which is the default in the GALPROP package (Strong et al. 2000). CR antiproton production in the conventional and QGSJET-II-4 models can be compared with the DCR₀ and DCR models.

In a previous paper (Jin et al. 2015a), it was found that the propagation parameters can be constrained by the AMS-02 data: proton flux (P) and the ratio of boron to carbon flux (B/C). In this paper, CR protons and B/C from AMS-02 are chosen as the basic data to constrain the propagation parameters. With the best-fit propagation parameters, the fluxes of CRs are predicted by the solution of the propagation equation. Though the fluxes of CR antiprotons and positrons are also predicted, the measured data from AMS-02 for CR antiprotons and positrons are included to shorten the confidence interval of the constrained parameters by CR proton and B/C data. In order to check the predicted value of ${{\rm{Be}}}^{10}/{{\rm{Be}}}^{9}$ compatible with ACE data, the ACE data are also included to constrain the propagation parameters. In the DCR${}_{1}$ model, there are the five groups of the experimental data: CR protons, antiprotons, positrons, B/C, and Be¹⁰/Be⁹. In the DCR${}_{2}$ model, the fluxes of CR antiprotons and positrons are not predicted and only calculated with the chosen parameters, which are the scanned values relevant to ${\chi }^{2}/{\rm{N}}\lt 2$ for CR protons, B/C, and Be¹⁰/Be⁹ from the DCR${}_{1}$ model. As the data from ACE and AMS-02 are obtained in different time periods, the potentials of solar modulation of CRs for ACE and AMS-02 data should be different. DCR_ϕ is the updated DCR${}_{1}$ model. In the DCR_ϕ model, the potentials of solar modulation of CRs for ACE and AMS-02 data are divided into two groups: ϕ and ${\phi }_{\sec }$, which are the free parameters as well as the propagation parameters. The potential ${\phi }_{\sec }$ is used for ACE data: Be¹⁰/Be⁹. The potentials ϕ are used for AMS-02 data: CR protons, antiprotons, positrons, and B/C.

In the chi-square fitting of combining the secondary antiproton and positron data, in order to avoid the propagation parameters deviating the determined ranges by CR protons and B/C data, it is necessary to fix the sensitive intervals of CR energy in the given ranges. For CR positrons, the energy range of the fitting to CR positron data of AMS-02 is limited to 0.6–6.0 GeV. Above 6.0 GeV, the positron excess begins to appear in the CR positron data of the AMS-02 experiment. For CR antiprotons, the cross point of the Z-factors between the conventional model and QGSJET-II-4 is 6 GeV, which is found in Figure 1. From the comparison between the fitting line and MC points, below 2 GeV, the uncertainties of the calculated z-factors are large, which is found in Figure 1. The energy range of CR antiproton flux is restricted to 2.0–6.0 GeV. In the limited energy ranges, the fluxes of CR positrons and antiprotons are just closest to AMS-02 data, which is found in Figure 5. Thus, the secondary antiproton and positron data have only a slight constraint on the global chi-square fitting. In order to check the energy range sensitive to the propagation parameters, the chi-square fittings for all of the CR antiproton data from AMS-02 are done as a test. The result shows that the overlarge convention velocity (3050 km s⁻¹) is favored to match the minimal chi-square for CR antiprotons (χ²/N = 94/57). Above 100 GeV the predicted flux of CR antiprotons is closest to AMS-02 data and includes the part of the excess. This is shown in the right panel of Figure 4. The relevant tag in the legend is DCR${}_{\bar{P}}$. It implies that the other sources need to be considered to contribute to CR antiprotons so as to tune the propagation parameters to the reasonable bounds. For the estimation of the astrophysical background, the flux of CR antiprotons should be predicted in the energy range sensitive to the propagation parameters.

In the DCR model, as the energy range of the fitting to CR positron and antiproton data of AMS-02 is limited in the given values and the calculated fluxes of CR positrons and antiprotons are closest to the experimental data in the given C.L. regions, the best-fit parameters come mainly from the constraint of CR protons and B/C. This is found in Figure 5. The experimental data of CR positrons and antiprotons have a slight constraint on the best-fit values of propagation parameters. Thus, the fluxes of CR positrons and antiprotons are well predicted by the DCR model, which is used for the interpretations of the positron excess above 6 GeV and the antiproton excess above several tens of GeV.

For the dark matter implications of the AMS-02 data, the prediction of mass and annihilation cross section of dark matter are based on the background of the total CR electrons and antiprotons, whose fluxes are calculated with the best-fit parameters in the DCR model. The parameters of the source term of CR electrons are fixed in the best-fit values, which are listed in Table 2. As seen in the table, above 1 GeV there are three spectral indices to describe the spectra of the primary electrons, which means that the primary electrons have a complex feature connected with the other origin. That has been discussed in Chen et al. (2015a).

Table 2.  Best-fit Values of the Parameters with the Constraints of AMS-02 Data

Ne	${\gamma }_{1}^{e}$	${\rho }_{\mathrm{br}1}^{e}$	${\gamma }_{2}^{e}$	${\rho }_{\mathrm{br}2}^{e}$	${\gamma }_{3}^{e}$
0.4167	0.5718	1.4623	2.6743	93.32	2.4768

Note. The unit of normalization N_e is ${10}^{-9}\,{{\rm{MeV}}}^{-1}{{\rm{cm}}}^{-2}{{\rm{sr}}}^{-1}{{\rm{s}}}^{-1}$, and the referred energy of CR electrons is 34.5 GeV. The breaks of injection electron spectra are ${\rho }_{{br}1,2}^{e}$, whose unit is GV. ${\gamma }_{1,2,3}^{e}$ are spectral indices below/above the breaks, respectively.

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In the analysis of the annihilation cross section of the dark matter, the annihilation channels of the dark matter contributing to CR electrons, positrons, and antiprotons involve the following particle states:

• 1.  
  $2\mu ,4\mu ,2\tau ,4\tau ,{W}^{+}{W}^{-},b\bar{b},{ZZ},q\bar{q},$ hh, and $t\bar{t}$ for CR positrons and electrons;
• 2.  
  ${W}^{+}{W}^{-},b\bar{b},{ZZ},{hh},q\bar{q}$, and $t\bar{t}$ for CR antiprotons.

In the fitting strategy, the limits on annihilation cross section are calculated with the best-fit parameters based on the given mass of dark matter, which is called the mass-fixed best fit. A global minimum in any values of dark matter mass is relevant to the minimal chi-square.

The annihilation spectra of Majorana dark matter particles via these channels are calculated using the numerical package PYTHIA (Sjöstrand et al. 2008). Ciafaloni et al. (2010) indicate that for the dark matter particles with the mass in the TeV scale the electroweak corrections are important in particular for annihilation/decay in leptonic channels. In this work, the electroweak corrections are always turned on for the calculation of the annihilation spectra of the dark matter. The analysis from dark matter decays is found in Mambrini et al. (2015).

Through the global χ² fit using the MINUIT package (James 1994), all the best-fit values of the propagation parameters are derived from the minimized χ². In Table 3, the best-fit parameters of each model are listed. In Table 4, the corresponding relations between the models and their concerned experimental data are presented. It is also shown that the best-fit χ² values are relevant to the experimental data in the given propagation model.

Table 3.  Best-fit Parameters of Models DCR, DR, and DC

Para.	DCR	DR	DC	DCR${}_{V}$	DCR0	DCR1	DCR2	DCR${}_{\phi }$
α	0.0	0.238	8.0E-3	0.0	0.0	0.0	0.0	0.0
$\phi ({\phi }_{\sec })$	701.93	597.4(817.6)	394.8	603.5	601.2	664.8	463.2	701.73 (67.95)
${V}_{A}({V}_{A}^{{e}^{+}})$	83.12	44.46		156.37 (178.43)	68.44	83.2	43.9	83.10
Vc0	496.75		32.21	4673.1	435.8	483.9	114.2	496.62
$\tfrac{{{dV}}_{c}}{{dz}}$	90.99		2.57	26.2	61.78	75.5	44.7	90.92
Zh	2.695	3.7	4.0	4.013	4.1	3.0	3.0	2.7
${D}_{0}/{Z}_{h}$	2.726	1.636	1.164	21.166	1.891	2.946	1.222	2.727
δ	0.305	0.321	0.393	0.144	0.366	0.291	0.403	0.305
Np	4.609	4.533	4.53	4.564	4.649	4.583	4.571	4.609
ρbr	6.6	10.153	8.41	5.032	4.873	6.6	6.6	6.6
${\gamma }_{1}^{p}$	1.8398	1.772	1.748	1.654	1.586	1.808	1.538	1.8395
${\gamma }_{2}^{p}$	2.4125	2.447	2.427	2.411	2.398	2.408	2.386	2.4125

Note. The unit of N_p is ${10}^{-9}\,{{\rm{MeV}}}^{-1}\,{{\rm{cm}}}^{-2}\,{{\rm{sr}}}^{-1}{{\rm{s}}}^{-1}$. ϕ, ρ_br, Z_h, ${D}_{0}/{Z}_{h}$, v_A, and V_c0 are in units of MV, GV, kpc, ${10}^{28}\,{{\rm{cm}}}^{2}\,{{\rm{s}}}^{-1}/{\rm{kpc}}$, km ${{\rm{s}}}^{-1}$, and km ${{\rm{s}}}^{-1}$, respectively. In the DR model, the bracketed ${\phi }_{\sec }$ or ${V}_{A}^{{e}^{+}}$ is only used for CR positrons. In the DCR_ϕ model, the bracketed ${\phi }_{\sec }$ is only used for Be¹⁰/Be⁹ data from ACE. The rest of the propagation parameters are referred to the example 01 of GALPROP WebRun (Vladimirov et al. 2011).

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Table 4.  Best-fit χ² Relevant to the Models DCR, DR, and DC

Models	${\chi }_{{\rm{P,N=72}}}^{2}$	${\chi }_{{\rm{B/C,N=67}}}^{2}$	${\chi }_{\bar{{\rm{P}}},N=10}^{2}$	${\chi }_{{e}^{+},N=16}^{2}$	${\chi }_{{\rm{ACE}},N=4}^{2}$	χ2	χ2/N
DCR	22.2	50.16	12.94	13.01		98.31	0.596
DR	47.02	288.31	8.74	117.27		461.33	2.8
DC	424.6	414	594.5	1037.33		2470.4	14.97
DCRV	14.12	77.77	17.51	14.32		123.72	0.75
DCR0	49.17	91.42	382.62	185.58		708.79	4.3
DCR1	28.10	50.49	12.05	21.13	38.76	150.54	0.89
DCR2	123.57	109.26	356.98	1923.1	5.95	2518.9	14.9
DCRϕ	22.20	50.148	12.93	13.03	2.93	101.24	0.601

Note. The subscript N of χ² denotes the number of the experiment data points, for instance, the number of AMS-02 proton data points is N = 72. The total χ² and its value over the total data points of the chosen experiment for each model are presented in the last two columns.

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In Table 3, the parameters are the best-fit values in the χ² fitting by the MINUIT package (James 1994). The best-fit values for α are much less than 1 in all the models. In fact, α nearly vanishes in the DCR and DC models, which indicates that the injection spectra of CR nucleons become very sharp near the referred rigidity. In Table 4, the best-fit χ² relevant to the three types of models DCR, DR, and DC are shown. With the comparison of χ² between the models, it is shown that the DCR model is more favored by AMS-02 data and the other models have some flaw or large deviation from AMS-02 data. As a whole, AMS-02 data have higher precision than the other data and are strongly constrained on the propagation models and the hadronic interaction models. For instance, the latest released B/C data have not only higher precision but also a more complex feature of spectra than the other experimental data. ${\chi }^{2}/N=288.31/67,414/67$ relevant to B/C data of AMS-02 in the DR and DC models are much greater than 2. Thus, only based on the B/C data of AMS-02, the conventional reacceleration diffusion model (DR) and the convention diffusion model (DC) are excluded. In the following several paragraphs, the DCR${}_{x}$ models are deeply explained.

First, there is a comparison between DCR and DCR₀. In the DCR₀ model, the hadronic interaction model is the conventional model. As the secondary particles, at low energies the flux of CR antiprotons is not favored by AMS-02. In the right panel of Figure 4, the pink dotted–dashed line relevant to the DCR₀ model is below the experimental data line of AMS-02. At low energies ${\chi }^{2}/N=382.62/10$ relevant to CR antiproton data of AMS-02 is very large. ${\chi }^{2}/N=185.58/16$ relevant to CR positrons is also brought to the worst value. In the DCR model, with an alternative interaction model, i.e., QGSJET-II-4 for the antiproton production, the predicted fluxes of CR antiprotons and positrons are well fitted to the AMS-02 data, though for CR positrons the hadronic interaction model is still the conventional model. This implies that the hadronic interaction model is key to relaxing the predicted tension between CR positrons and antiprotons. As seen in Figure 1, for CR antiprotons, Z-factors relevant to the QGSJET-II-4 model are greater than those of the conventional model at low energies. But for CR positrons, Z-factors relevant to the QGSJET-II-4 model are almost the same as those of the conventional model.

Second, in order to check the values of ${{\rm{Be}}}^{10}/{{\rm{Be}}}^{9}$ compatible with ACE data, the propagation parameters are refitted by including the ACE data in the DCR${}_{1}$ and DCR${}_{2}$ models. The predicted values of ${{\rm{Be}}}^{10}/{{\rm{Be}}}^{9}$ in the DCR${}_{1}$ and DCR${}_{2}$ models are drawn in the left panel of Figure 4. For ISOMAX data (Hams et al. 2004), in the DCR${}_{1}$ and DCR${}_{2}$ models the predicted values are both consistent. For ACE data (Yanasak et al. 2001), the predicted values of ${{\rm{Be}}}^{10}/{{\rm{Be}}}^{9}$ are consistent with ACE data in the DCR${}_{2}$ model. But in the DCR${}_{1}$ model the relevant values (${\chi }^{2}/N=38.76/4$) deviate remarkably from ACE data. In the DCR${}_{2}$ model only CR proton and B/C data of AMS-02 are included to perform the chi-square fitting. This means that the propagation parameters constrained by CR proton and B/C data of AMS-02 are compatible with ACE data. However, in the DCR${}_{1}$ model, CR antiprotons and positrons are both included. In the chi-square fitting, five groups of experimental data are chosen: CR protons, CR antiprotons, CR positrons, and B/C from AMS-02 and ${{\rm{Be}}}^{10}/{{\rm{Be}}}^{9}$ from ACE. Except for ACE data, the four groups of experimental data from AMS-02 favor completely the predicted values by best-fit propagation parameters, and all of the relevant χ²/N are less than 2. This means that in the DCR model the prediction has a tension between Be¹⁰/Be⁹ from ACE and CR antiprotons and positrons from AMS-02.

In order to explore the tensions of the propagation parameter constraint from AMS-02 and ACE data, the fluxes of CR positrons and antiprotons and values of Be¹⁰/Be⁹ are calculated with the marginal values of the propagation parameters at the 99% C.L. region. The relevant contours are found in Figures 2 and 3. In the propagation parameters, dV_c/dz is relevant to the most upper/lower limits in all the contours. The values of Be¹⁰/Be⁹ decrease with dV_c/dz dropping. When dV_c/dz arrives at the lower limit of the 99% C.L. region, the values of Be¹⁰/Be⁹ are only near the ACE data. When the value of dV_c/dz drops to 25 km kpc⁻¹ s⁻¹, the values of Be¹⁰/Be⁹ arrive at the error bars of ACE data. The relevant χ²/N is less than 2. However, the fluxes of CR positrons and antiprotons increase with dV_c/dz dropping. When dV_c/dz arrives at the lower limit of the 99% C.L. region, the fluxes of CR positrons and antiprotons deviate from AMS-02 data obviously. Thus, it is not easy to release the tensions between AMS-02 and ACE data only by the constrained propagation parameters. In the DCR_ϕ model, the potentials of the solar modulation of CRs for ACE and AMS-02 data are divided into two groups of values. The values of the best-fit potentials and the propagation parameters constrained by ACE and AMS-02 data are shown in the last column of Table 3. The potentials 67.95 and 701.73 MV of the solar modulation of CRs for ACE and AMS-02 data, respectively, have a remarkable difference. The values of constrained propagation parameters are very close to the values in the DCR model, which is shown in the first column of Table 3. The χ²/N relevant to the predicted fluxes of CRs and ratios of Be¹⁰/Be⁹ and B/C are shown in the last row of Table 4. Except the χ²/N relevant to CR antiprotons, which is less than 2, all other χ²/N are less than 1. Specially, χ²/N relevant to Be¹⁰/Be⁹ is 2.93/4. The predicted ratios of Be¹⁰/Be⁹ by the best-fit parameters are drawn in Figure 4. Except for Be¹⁰/Be⁹, the χ²/N relevant to CRs from AMS-02 is almost the same as the values in the first row of Table 4 for the DCR model. Thus, the tension between ACE and AMS-02 data can be relieved by the different potentials of the solar modulation in the DCR model.

In order to check the low-energy behavior consistency between the total beryllium flux and the ratios of Be¹⁰/Be⁹ about the tension, we performed a chi-square fitting to search the best-fit values ϕ of the solar modulation potential for the two groups of data: Be¹⁰/Be⁹ and the total beryllium flux. We add the normalized factor k multiplied by the predicted fluxes of all the beryllium isotopes. The factor k expresses the uncertainties of the absolute fluxes of CRs and the spallation cross section of the secondary particle production of CRs. When ϕ is at 67.95 MV, the best-fit value of the normalized factor k is 0.277 and the χ²/N is 25.24/6 for beryllium flux data from ACE. This means that the normalized factor does not improve exclusively the prediction of the total beryllium flux. The solar modulation potential has a remarkable influence on the gradient of the total beryllium flux. When ϕ is also free, that prediction is improved. The best-fit values of ϕ and k are 165.53 MV and 0.3785, respectively. The relevant χ²/N are 5.717/4 and 6.128/6 for Be¹⁰/Be⁹ and the total beryllium flux data. That means that the the total beryllium flux is also well predicted, together with the Be¹⁰/Be⁹. The ϕ of 165.53 MV is consistent with the solar minimum periods mentioned in Cholis et al. (2016). In Figure 4, the relevant curves are drawn.

In the DCR_V model, Alfvèn speed V_A has the two different values relevant to CR positrons and CR nucleons, which are found in Table 3. By comparing χ² between the DCR and DCR_V models in Table 4, it is found that the differences of the reacceleration effect do not improve the fluxes of CR positrons and CR nucleons to fit to AMS-02 data. At the meantime, as a bad result, the Galactic wind velocity V_c and the diffusion coefficient D₀ are converged into the large values. In the chi-square fitting, the propagation parameters are not converged. This means that the Alfvèn speed V_A is not sensitive to the difference between CR positrons and CR nucleons.

In order to explore the correlations between the propagation parameters in the DCR model, Markov Chain Monte Carlo sampling is done using the CosmoMC package (Lewis & Bridle 2011), which calls the functions of the GALPROP package (Strong et al. 2000) in the sampling steps. Except for the smoothness parameter α, there are 11 propagation parameters to be used. In the sampling results, the sampling steps in the 28 chains amount to more than 428,000 after burn-in. The details of the sampling methods are found in Jin et al. (2015a). For the contour drawing, the covariance matrix of 11 parameters is calculated with the sampling data. Based on the inverse of the covariance matrix, with the best-fit values of the propagation parameters in the DCR model, the correlation contours between any two parameters are drawn in Figures 2 and 3. As seen in these contours, the Alfvèn speed V_A, the Galactic wind velocity V_c, and the diffusion coefficient $D{}_{0}/{Z}_{h}$ are completely the positive correlations between the two parameters. These correlations have been presented in the DCR_V model. The Galactic wind velocity V_c and the diffusion coefficient D₀ are converged into the large values together. The well-known correlations of the diffusion coefficient Z_h, $D{}_{0}/{Z}_{h}$, and δ are also clearly shown. In Figure 2 and 3, the regions enclosing the $68 \% ,95 \%$, and 99% C.L. are shown for 11 propagation parameters. As seen for the regions in the given C.L., the confidence interval of the propagation parameters constrained by AMS-02 data is very narrow. This means that these correlations do not impact the convergence of the propagation parameters in the chi-square fitting for the DCR model. In Figure 5, the contours of the fluxes of CR protons, antiprotons, and positrons and the ratio of boron to carbon flux are predicted from the propagation parameter regions of the 99% C.L.. As seen in Figure 5, all the best-fit lines are in the middle of the contours.

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It is known that the spectra of CR protons from the measurement of the AMS-02 experiment have two breaks with the kinetic energy increasing from 0.5 GeV to 2 TeV (Aguilar et al. 2015a). The second break of the CR proton spectra means that the absolute index of CR proton spectra begins to decrease above 330 GeV, which is often called the CR hardening. From a comparison of χ² for the CR protons in Table 4, it is seen that in the DCR model the flux of CR protons with the multiple power indices is well predicted. Though the DCR_V model does not help to improve the prediction of the fluxes of CR positrons and antiprotons, the large V_c and D₀ promote the relevant χ² decreasing from 22.2 to 14.12. The experimental data of CR protons with high precision from AMS-02 have 72 points. The ${\chi }^{2}=14.12$ means that the flux of CR protons is better predicted in the DCR_V model, which justifies the convection effect for CR propagation.

On the left of the first row of Figure 6, the fluxes of CR protons are drawn. It is apparently seen that the flux of CR protons above 330 GeV has a different trend change with different models. As a result of the comparison among the models, the DCR model can well predict the hardening flux of CR protons. And the flux at low energies is also well consistent with the AMS-02 data, which distinguishes from the DR model. On the right of the first row in Figure 6, the ratio of boron to carbon flux in the DC model is inconsistent with the AMS-02 data below 2 GeV, which indicates that the DC model cannot give a significant prediction.

In Figure 6, the fluxes of CR antiprotons are plotted at the left of the second row. From the low energy to the high one, except for the DC and DCR₀ models, the consistent prediction for the flux of CR antiprotons is found in the DCR and DR models, though the tail data of AMS-02 show a bulge. That cannot completely be fitted. At low energies, the best-fit flux of CR antiprotons does not have the excess. In the DCR₀ model, which is relevant to a conventional hadronic interaction model, the excess flux of CR antiprotons is remarkable at low energies. At high energies, the bulge relative to the predicted flux of CR antiprotons still exists and needs a further interpretation. As the fluxes of CR helium are one order of magnitude less than the fluxes of CR protons, though the spectrum of CR helium has a hardening feature, the flux of CR antiprotons from the contribution of CR helium is very little. In Figure 4, the difference of best-fit fluxes of CR antiprotons between the DCR and DCR_He models cannot be distinguished. In this paper, the excess is interpreted as the contribution from dark matter annihilation.

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On the right of the second row in Figure 6, the predicted flux of CR positrons is drawn. In the DC, DR, and DCR₀ models, there is a fully inconsistent prediction with AMS-02 data. In contrast, in the DCR model the fluxes of CR positrons are well fitted to the AMS-02 data at low energies. And the maximal energy range is up to about 10 GeV. Above 10 GeV, CR positron excess becomes significant and begins to increase with rising energy. This implies that new sources different from the secondary particle production need to be considered. The dark matter interpretation of positron excess appears in the next paragraphs.

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In Figure 7, all of the figures are drawn with the calculated data in the DCR model. On the left of the last row, the mass-fixed best-fit annihilation cross sections of dark matter contributing to CR antiprotons are drawn. It is seen that the annihilation cross sections of dark matter are different completely from those of Jin et al. (2015b). That situation is relevant to the flux of CR antiprotons fitting to AMS-02 data at low energies and weaker than AMS-02 data at high energies. It is known that the dark matter interpretation of the positron excess takes the large annihilation cross sections. But in this paper, the large annihilation cross section does not match the weak excess of CR antiprotons. The tension between the excess interpretations of CR antiprotons and positrons is perhaps relieved to a certain extent. For instance, the symmetry between the quark and lepton particles annihilated from dark matter needs the new hyperthesis. In Figure 7, it is also seen that all of the annihilation cross sections are greater than the predicted values from Fermi-LAT gamma-ray data of dwarf spheroidal satellite galaxies in the Milky Way.

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Based on the change of χ² with the dark mater mass shown in the left of the second row in Figure 7, the best-fit mass of dark matter is found by referring to the minimal χ². It is seen that the best-fit mass of dark matter is found to be near 600 GeV. As a whole, in all of the channels of the dark matter annihilation the predicted best-fit masses are only from 300 to 700 GeV. Meanwhile, the dark matter interpretation of the antiproton excess has the weak effect of discriminating the dark matter annihilation channels. In the left of the first row of Figure 7 the predicted fluxes of CR antiprotons from the total contribution of astrophysical background and dark matter annihilation are drawn. As the annihilation spectra of dark matter contributing to CR antiprotons are not sharp, as well as CR electrons, the bulge in the tail data of AMS-02 does not strongly constrain the interpretations of dark matter. As a result, for all the channels the predicted values do not match the fluxes of the last second point from AMS-02 data.

In this paper, the analysis of positron excess is based on the total fluxes of CR electrons and positrons, which is different from CR positron flux or the fraction data of AMS-02 used in the previous papers. In the right of Figure 7, the relevant figures are drawn from the first to the third rows. Being similar to the CR antiprotons, the astrophysical backgrounds of the total electrons and positrons are also calculated in the DCR model. In first row of Figure 7, the calculated flux of the CR electron background is well consistent with the AMS-02 data at low energies, but at high energies the experimental data of AMS-02 are manifestly in excess of the calculated background.

In the last row of Figure 7, the mass-fixed best-fit annihilation cross sections of dark matter contributing to CR electrons and positrons via many channels (2μ, ${W}^{+}{W}^{-}$, etc.) are plotted. As seen in the panels, the mass-fixed best-fit values relevant to the lepton channels are different with mass of dark matter increasing. The cross mass of dark matter between the channels is near 10 TeV, where the same annihilation cross sections appear. As a whole, annihilation cross sections of dark matter still have large values. The relevant compatibility of the large annihilation cross section with the thermal relic density is discussed in terms of self-interacting dark matter (Liu et al. 2011, 2013; Dev et al. 2013; Chen et al. 2015b; Li et al. 2014; Li & Zhou 2014; Liang et al. 2016) and in connection with collider physics (Huang et al. 2015, 2016).

In the second row of Figure 7, the best-fit mass of dark matter is found from the change of χ² with the mass of dark matter increasing. Except for 2 and 4 μ final states, the χ² are less than the twice the experimental data points, and the predicted mass of dark matter is from 400 GeV to 4 TeV. The different channels of dark matter annihilation may be discriminated by the best-fit mass of dark matter. That is obviously different from the interpretation of the antiproton excess. The annihilation of heaviest dark matter is via the $t\bar{t}$ final state, and the lightest one is relevant to 2τ. The total fluxes of CR electrons and positrons from the astrophysical background and the dark matter annihilation are drawn in the first row of Figure 7. As seen in the figure, at high energies the predicted fluxes of CR electrons and positrons are consistent with AMS-02 data. The best-fit fluxes are relevant to the ZZ final state. As seen in the last row of Figure 7, for all the annihilation channels the annihilation cross sections of dark matter are still one order of magnitude greater than from Fermi-LAT gamma-ray data of dwarf spheroidal satellite galaxies. The issues of the large annihilation cross sections still exist. The relevant exploration is found in Huang et al. (2015, 2016).