Astrophysical background and dark matter implication based on latest AMS-02 data
The cosmic ray(CR) positrons and antiprotons are often regarded as the products of collisions of CR nucleons with the interstellar medium. However this conclusion is challenged by recent experimental data. In this work, we choose the latest AMS-02 data to analyze the astrophysical background of CR positrons and antiprotons based on the GALPROP code for CR propagation and QGSJET-II-04 for hadronic CR interactions. The results show that in low energies the flux of CR antiprotons is consistent with AMS-02 data, and the over-predicted flux of CR positrons is well reduced in a diffusion model combining the re-acceleration and convection terms. Using this model, the calculated flux of CR protons is consistent with AMS-02 data with the hardening feature above 330 GeV. Based on this model, using the total fluxes of CR electrons and positrons from AMS-02, interpretation of dark matter annihilation on the positron excess are also analyzed.
In the origin of Galactic CRs, the primary particles, such as nucleons and electrons, are commonly regarded as the injection of Supernova relics(SNRs), in which, CRs are accelerated by the diffusive shock[1, 2, 3, 4, 5]. In this mechanism, the injection spectra of CRs are a power law below Knee, which are verified by the experimental data. The other particles called as the secondary CRs, are mainly produced in the collision between the primary particles and the interstellar medium in the Galaxy. In the analysis of the secondary CRs, the collision cross-sections between them are often calculated with the hadronic interaction models. Since the spectra of the primary CRs are a single power law, the spectral features of the secondary CRs, deriving the origin of CRs, are mainly relevant to the hadronic interaction models. Besides the hadronic interaction models, the exclusive cross-section of the produced particles are also calculated with the empirical parameterizations of the accelerator data. In practice, these calculations are often performed using Monte Carlo(MC) event generators, and combining the accelerator data, which are developed into the parameterization model package, such as FLUKA[7, 8], QGSJET-II-04, EPOS-LHC, etc. In this paper, QGSJET-II-04 is chosen to calculate the secondary antiproton product.
When the charged particles of CRs propagate in the Galaxy, they may be accelerated via the interaction with the turbulent interstellar magnetic field, which is also called as re-acceleration in contrast to DSA. On the meantime, CRs may commit the energy loss when propagating in the Galactic winds, which are blowing outwards from the Galactic disc. That is called as convection in CR propagation models. Besides the Galactic diffusion of CRs, these interactions alter the original structure of the spectra of CRs. As a result, the measured spectra of CRs are different from the injection of the sources. Thus, the uncertainties of the secondary particle spectra are also from the propagation models of CRs. In the conventional model, CR production and propagation are governed by the same mechanism at energies below eV, and CR propagation is often described by the diffusion equation. Thus, based on the propagation models and the hadronic interaction models, the astrophysical spectra of the secondary particles are predicted with the experimental data of the primary particles of CRs.
Recently, AMS-02 reported their observed results of cosmic rays. Below TeV, the spectra of CR protons can be described by the high precision data. CR positrons, antiprotons and B/C have also precise measurements. In the previous analysis, we have performed a global analysis of CR propagation parameters with the only AMS-02 data. The propagation parameters have been shown to be well determined by the only CR proton and B/C data of AMS-02, which is the first strategy of fitting parameters different from the two ratios of CRs, such as Be/Be and B/C. The constrained parameters of the proton and B/C case have very narrow range at the 95% confidence level(CL) and are also less than the Be/Be and B/C case in the same CL. Based on the propagation parameter models constrained by the measured data of CRs, the astrophysical spectra of the secondary particles are naturally predicted in a given CL. Since the astrophysical spectra of the secondary particles, such as CR positrons and antiprotons, already have the measured data with high precision, the predicted spectra of them may be used to verify the propagation model and explore the origin of the experimental data. The positron excess has been discovered by PAMELA, Fermi-LAT and AMS-02 in the past years, which indicated that above 10 GeV the CR positron fluxes of experimental data are greater than the astrophysical fluxes predicted by the CR propagation models and the hadronic interaction models. The phenomenological implications can be found in refs.[20, 21, 22, 16, 23]
Below 10 GeV the astrophysical fluxes of CR positrons and antiprotons analyzed based on the current models are actually both inconsistent with the latest data of AMS-02, i.e. the fluxes of CR positrons are greater than the measured data and the fluxes of antiprotons are less. In the paper, the fluxes of the CR positrons and antiprotons ions are predicted from a global bayesian analysis by using GALPROP package, which are inconsistent with the experimental data below 10 TeV. It also indicated that the under-predicted antiprotons may result from a general feature of the re-acceleration models. In GALPROP package, the default of the hadronic interaction model is from the parameterizations of Tan & Ng and Duperray et al., which is called as a Conventional model. In ref., authors have compared the antiproton yields from the Conventional model, QGSJET-II-04 and EPOS-LHC. The result shows below 10 GeV the Conventional model have less contribution to the antiproton yields than the others. And in the ranges from 10 GeV to 100 GeV, Conventional model is favored, but EPOS-LHC model has more contribution to the antiproton yields than Conventional and QGSJET-II-04 model.Thus, QGSJET-II-04 is moderate model to describe the hadronic interaction from the low to high energy, which is chosen as the hadronic interaction model to enhance the flux of CR antiprotons in the low energy. In the GALPROP package, the codes relevant to producing CR antiprotons are replaced with QGSJET-II-04 model by us. The modified version of GALRPOP is used in our analysis.
Recently, the uncertainties of the antiproton flux have been analyzed sufficiently in ref.[26, 27], which involves the cross-section of interaction between nucleons, the propagation of CRs, solar modulation and the spectral slope of the primary CRs. As a result, there are some of those parameters that may predict the consistent flux of antiproton with AMS-02 data. But, in ref., the figures and chi-squares concerned are not found, which may indicate whether these parameters could also lead to a well prediction for the flux of protons and B/C. The similar analysis is also found in ref.. In the analyses[29, 30, 31, 32], the under-predicted antiprotons were considered as the possible evidence of a new primary component of CR antiprotons. Another possible excess can be found at high energies around 300 GeV.
In this paper, the analysis strategy of CR positrons and antiprotons is the following. The calculations of the cross-section relevant to producing antiprotons between the interstellar medium and the primary CRs, are performed by Monte Carlo (MC) generator QGSJET-II-04. That has been parameterized as the Z-factors in ref.. In order to compare the difference between the Conventional and QGSJET-II-04 models, the Z-factors are also recalculated for CR antiprotons and positrons using CRMC v1.6.0 package[35, 36]. In the source terms of propagation equation, the injection spectra of the primary CRs are characterized by a continuous function in the referred rigidity of CRs, which analytically express a simple power law with the same indices below/above the referred rigidity. That is also used in the analysis of the features of the measured spectra of CRs by Voyager[37, 38]. In the propagation models, the re-acceleration and convection of the Galactic CRs are both taken into account. The propagation equation of CRs is solved numerically using GALPROP package. The effect of the solar modulation of CRs is considered using the model of force-field approximation.
Based on the above strategy, in the re-acceleration diffusion model, the under-predicted flux of CR antiprotons, has been intensified to be consistent with the AMS-02 data, while the predicted flux of CR positrons below 10 GeV is not automatically consistent with the AMS-02 data. Such a consistency results from the recalculation of antiproton production in the nucleon collision by using MC generator QGSJET-II-04 parameterized as Z-factors in ref.[40, 34]. But, for CR positrons the QGSJET-II-04 does not depress the flux of them. That situation is also found in the other model, such as FLUKA[7, 8]. In the strategy of the exclusive potential of the solar modulation for CR positrons, the over-predicted flux of CR positrons below 10 GeV cannot be reduced. The propagation parameters relevant to the consistent flux of CR positrons at the larger CL, favored by the experimental data of CR protons and B/C ratio, do not be also found. In the strategy of combining the convection and re-acceleration diffusion models, the over-predicted flux of CR positrons can be depressed. In the meantime the flux of CR antiprotons are also consistent with the AMS-02 data. However, in the compatible analysis with ACE data (Be/Be), the constrained the propagation parameters with the CR protons, antiprotons, positron and B/C from AMS-02 experiment do not well predict the values of Be/Be compatible with ACE data(/DOF=38/4). In the fact, not too bad, if the experimental data of CR protons, Be/Be and B/C are only considered to constrain the propagation parameters, the predicted values of them are consistent completely with the experimental data (/DOF for the three kind of experimental data). Thus, it implied that the predicted fluxes of secondary antiprotons and positrons have some tensions with the predicted value of Be/Be. The further exploration will be coped with in the next paper. The following sections will illustrate the details of the analysis concerned.
This paper is organized as follows. In section 2, we outline the framework for the calculation of the propagation of the cosmic-ray particles and the cross-section of interaction between nucleons. In section 3, we describe the data selection and the strategy of the data fitting in a number of propagation models. The numerical results are presented in section 4. Our conclusions are given in section 5.
2 Cosmic ray propagation and hadronic interaction model
2.1 Cosmic ray propagation equation and parameters
In the conventional model, CR production and propagation are governed by the same mechanism at energies below eV. CR propagation is often described by the diffusion equation:
where is the number density per unit of total particle momentum, which is related to the phase space density as . is the spatial diffusion coefficient parametrized as
where is the rigidity of the CR particles, and is the index below (above) a reference rigidity . The parameter is a normalization constant and is the ratio of the velocity of the CR particles to the speed of light. is the convection velocity related to the drift of CR particles from the Galactic disc due to the Galactic wind. The diffusion in the momentum space is described by the re-acceleration parameter related to the Alfvn speed , i.e. the velocity of turbulences in the hydrodynamical plasma, whose level is characterized as [11, 42]:
where or is the index of the spatial diffusion coefficient. , and are the momentum loss rate, the time scales for fragmentation and the time scales for radioactive decay, respectively. The momentum loss rate of CR electrons is not the same as CR nucleons, and the relevant expressions are found in the APPENDIX C of paper .
z is the height perpendicular to the Galactic disc and also appears in the next equation (5).
The source 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):
where , and the parameter is normalized with the propagated flux of CR protons. is the relative abundance of the Ath nucleon. The reference rigidity is described as the breaks of injection spectrum. is 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 is replaced by the following,
and are the spectral indices below/above the referred rigidity, determines the smoothness of the spectral change in the left and right sides of the referred rigidity, when is , as a broken power law, the expression is same as the one in the equation (5). In this work, is taken as a free parameter to analyze the smoothness of the injection spectra of CR nucleons.
2.2 the secondary particle production in the interstellar medium and the analytical expression of Z-factors
is the interstellar H and He density, is density of No. CR nucleon, and is cross-section between the No. 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, the equation(7) is often used to calculate the flux of secondary CRs. Recently, these calculations are improved by using Z-factors[40, 34], where they took a numerical calculation instead of the equation(7). The concerned details may be found in the papers[40, 34]. 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 CRMC v1.6.0 package. The details of CRMC are found in the paper. In the recalculation of Z-factors, the collision between the particles is more than 8,780,000 times.
For the Z-factor expression of CR antiproton case, the equation(7) is modified as:
Here, are the power indice of the interstellar spectra of No. CR nucleon. the -factor is expressed via the inclusive spectra of antiprotons with , as follows
In Table (1) of paper, 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’s 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
Where, the dimensionless quantity denotes kinetic energy over GeV per nucleon. The above spectral indice is replaced by . and are the nucleon number of the th interstellar gas and the th CR nucleon respectively. a=0.5 when and a=1.0 when . In the expression of , the three parameters , and are served for the broken power law spectra of the primary CRs.
With the equation (10), is simply rewritten as,
Where, is a discrete function expressed as
Based on the analytic expression (11), the source term of the secondary antiproton is modified as,
is the maximum nucleon number of the chosen particle of CRs, which mainly contribute to the production of CR antiprotons.
|() in this paper Eq.(11)||() in Ref.|
In order to check the values of the expression (11), the calculated values in an example =2.4 are listed in Table 1. If the standard deviation is 0.02, the total over data points calculated by the differences between the values of the expression (11) and the reference is near 1.0 for all cases (). Thus, the errors may be accepted.
In this paper, the CR propagation equation (2.1) is solved by GALPROP v54 package, which is based on a Crank-Nicholson implicit second-order scheme. 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 and half-height . The calculation option of the tertiary antiprotons is turn on in GALPROP package. 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 the 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 denotes the force field intensity. In this paper, is a free parameter and in some fitting, the difference of that between the experimental data is also taken for granted.
3 Data selection and fitting schemes
3.1 The propagation parameters and the experimental data selections
In our previous paper, it was found that the propagation parameters: half-height , diffusion parameters and , Alfvn speed , and power indices: below(above) a reference rigidity of CR protons, can be determined by the AMS-02 data: proton flux (P) and the ratio of Boron to Carbon flux (B/C). Thus, based on these parameters constrained from AMS-02 data, the fluxes of the secondary CRs may be predicted. In this paper, in order to constrain the fluxes of CR antiprotons and positrons, besides AMS-02 data of P and B/C, we shall also include the latest released AMS-02 data of CR antiprotons and positrons. It is already known that in the re-acceleration diffusion model, the flux of CR positrons below 10 GeV is over-predicted, which perhaps implies that the other effects of CRs propagation and interaction etc. would be considered. These effects concern the solar modulation and the convection, i.e. Galactic winds blowing outwards from the Galactic disc, which is denoted by and in the context.
In order to explore the difference between the Conventional and QGSJET-II-4 model, the Z-factors are recalculated for the secondary antiproton and positron based on a broken power law spectrum of the primary particles. The relevant figures are shown in Figure 1. In the left of Figure 1, it is seen that the Z-factors relevant to QGSJET-II-4 model are greater than the Conventional model in the low energies. For CR positrons, the Z-factors relevant to QGSJET-II-4 model are just slightly less than the Conventional model, which is seen in the right of Figure 1. Thus, QGSJET-II-4 model is not used to produce CR positrons in the paper.
In the chis-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, the sensitive intervals, which are sensitive to the interaction model (for antiproton) and source (for positron), are necessary to be chosen. For CR positrons, the energy range of the fitting to CR positron data of AMS-02 is limited in 0.6 GeV - 6.0 GeV. Above the 6.0 GeV, the positron excess begins to appear and increases with energy raising in the CR positron data of 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 the 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 the figure 1. Thus, the energy range of CR antiproton flux is restricted in 2.0GeV - 6.0 GeV. In these energy ranges, for CR antiprotons and positrons, the fluxes of them fitting to the experimental data mean that the dominant yields are derived from the hadronic interaction between the primary particles and interstellar gas.
In order to explore the energy range sensitive to the propagation parameters, the chis-square fitting for all of CR antiproton data from AMS-02 are done as a test. The result shows the overlarge convention velocity (3050 km/s) is favored to match the minimal chis-square for CR antiprotons (/DOF=94/57). It implies that the other sources need be considered to contribute to CR antiprotons in the high energy 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.
Besides the smoothness parameter in the Equation (6), there are in general 12 fitting parameters: (Normalization of CR flux), , , and , which are determined from fitting four groups of AMS-02 data: Proton, B/C, antiproton and positron. Be/Bedata from ACE experiment are also used to analyze the compatibility with AMS-02 data in the constrained propagation parameters.
With the 12 parameters and four groups of AMS-02 data, the parameter models are constructed to analyze the constrained flux of CR antiprotons and positrons from AMS-02 data. The results are presented in Tables 3 and 4. In Table 3 and 4, DCR denotes for a diffusion model including the convection and re-acceleration effects, DR for a diffusion model involving only the re-acceleration effect, and DC for a diffusion model containing only the convection effect. In the DR model, the solar modulation potential is taken with different values for CR positrons and CR nucleons. In the DCR model, some hadronic models are improved to replace the equation (11). In order to compare the recalculated flux of CR antiprotons, a DCR model denoted as DCR model is included, which is relevant to the conventional hadronic model. In the DCR model, Alfvn speed is not different between CR positrons and CR nucleons. The DCR model is constructed to verify whether re-acceleration has the different effect between CR positrons and CR nucleons.
In order to check the value of compatible with ACE data, the propagation parameters are re-fitted by including the ACE data based on DCR model. In DCR model, the five groups of the experimental data: CR protons, antiprotons, positrons, B/C and Be/Be are included to fit the best-fit parameters. In DCR 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 for CR protons, B/C and Be/Be.
3.2 The fitting schemes of dark matter implications
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 CR antiprotons, whose fluxes are calculated with the above best-fit parameters from the constraints of AMS-02 data. The fluxes of CR electrons are divided into the primary and secondary ones. In ref., by using the re-acceleration diffusion model of CRs, the flux of primary electrons are predicted by the difference between CR electron and positron of the AMS-02 data. In this paper, the propagation parameters are chosen from DCR Model in Table 3 and the parameters relevant to the primary electron are directly fitted with the constraints of AMS-02 data. The best-fit is 21.74/72. The concerned parameters are given in Table 2. As seen in the table, above GeV there are three spectral indices to describe the spectra of the primary electrons, which indicates that the primary electrons have a complex feature connected with the other origin. That has been discussed in the paper.
In the fitting of annihilation cross-section of dark matter, the annihilation channels of dark matter contributing to CR electrons, positrons and antiprotons involve the following particle states:
hh and for CR positrons and electrons
and 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 as the mass-fixed best-fit and also the more tensive in the annihilation cross-section limits. A global minimum in any values of dark matter mass is relevant to the minimal chi-square in the entire fitting.
The annihilation spectra of Majorana dark matter particles via these channels are calculated using the numerical package PYTHIA v8.175. In the paper, it is known that for the dark matter particles with a mass in the TeV range, electroweak corrections are important in particular for annihilation/decay in leptonic channels. In the paper, electroweak corrections are always turned on for the calculation of the annihilation spectra of dark matter. The analysis from dark matter decays is found in the paper.
Through the global -fit using the MINUIT package, the best-fit values of the parameters and spectrum of CRs are derived from the minimized . In Table 3, the best-fit parameters of each model are listed. In Table 4, the corresponding relations of the models and their concerned experimental data are presented, which shows the best-fit values for the models with the relevant experiments.
4.1 The propagation parameter model constrained by AMS-02 data
In Table 3, the parameters of all models are listed. These parameters are the best-fit values in the fitting by using the MINUIT package. The best-fit values for are much less than 1 in all the models. In fact, approaches to vanish in the DCR and DC models, which indicates that the injection spectra of CR nucleons become very sharp near the referred rigidity. With the comparison of between the models and the effective judgment of the propagation parameters, it is known that DCR model is favored by AMS-02 experiment and may well predict the flux of CR protons, antiprotons, positrons and B/C consistent with AMS-02 data. For the other models, in the following paragraphs the details of exclusion by AMS-02 data will be described.
In Table 4, the best-fit relevant to the three types of model DCR, DR and DC are given. With the estimation of the total over the point numbers of the experimental data(/DOF), DC, DR and DCR models are excluded obviously from the favored propagation parameter models by the latest data from AMS-02. The strong constraint on these models is derived from the latest released B/C data, which gets higher precision and has a more complex feature of spectra than the other experimental data. From the values in Table 4 and 3, it is also seen that in DR model the predicted flux of CR positrons and B/C have remarkable deviation from AMS-02 data, though the solar modulations are separately considered for CR positrons and nucleons. In the DCR model, the hadronic interaction model is Conventional model and that tension also does not been eliminated. In DCR model, with an alternative interaction model, i.e. QGSJET-II-4, the predicted fluxes of CR antiprotons and positrons are well fitted to the AMS-02 data in the same time. It implies that the hadronic interaction model is a key to relax the predicted tension between CR positrons and antiprotons. As seen in Figure 1, for CR antiprotons, Z-factors relevant to QGSJET-II-4 model are greater than the Conventional model in the low energies. But for CR positrons, Z-factors relevant to QGSJET-II-4 model are slightly less than the Conventional model.
In DCR model, Alfvn speed has the two different values relevant to CR positrons and CR nucleons, which are found in Table 3. Comparing between the DCR and DCR models in Table 4, it is found that the differences of re-acceleration effect does not apparently 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 and the diffusion coefficient D are converged into the large values.
In order to explore the correlations between the propagation parameters in DCR model, MCMC sampling is done using CosmoMC package, which call the functions of GALPROP in the sampling steps. Exception the smoothness parameter , there are 11 propagation parameters to be used. In the sampling result, the numbers of the sampling steps in the 28 chains amount to more than 428000 after burn-in. The details of the sampling methods are found in the previous paper. For the contour drawing, the covariance matrix of 11 parameters is calculated using the sampling data. Based on the inverse of the covariance matrix, with the best-fit values of the propagation parameters in DCR model, the correlation contours between any two parameters are drawn in Figure 2 and 3.
As seen in these contours, the Alfvn speed , the Galactic wind velocity and the diffusion coefficient D are completely the position correlations between the two parameters exception for the combination between the Alfvn speed and the diffusion coefficient D. Thus, it is clear that they are converged into the large values in the context.
In order to check the value of compatible with ACE data, the propagation parameters are re-fitted by including the ACE data based on DCR model. The best-fit flux ratio from the refitted parameters and the predicted values from the chosen parameters of DCR model are drawn in Figure 4. For ISOMAX data, in DCR and DCR models the predicted values are both consistent. For ACE data, as seen in Figure 4, is compatible with ACE data in DCR model, but deviate remarkably from ACE data in DCR model. In 4 for Be/Be from ACE data /DOF is much greater than 2, but for the others from AMS-02 /DOF are less than 2. Thus, as a result, there is a tension between Be/Be and CR antiprotons and positrons, which is being expected to be released in the further exploration. Since CR protons and B/C data from AMS-02 do not favor DR and DC model, Be/Be data does not help to constrain the propagation parameters. In the propagation parameters constrained by CR protons, antiprotons, positrons, B/C and Be/Be data, the large convective velocity, which appears in DCR model, is limited and do not help to predict the value of Be/Be compatible with ACE data. That situation is derived from the high precision data from AMS-02, which is dominant in chi-square fitting and depresses the constraint from ACE data with low accuracy.
It is known that the spectra of CR protons from the measurement of AMS-02 experiment have two breaks of power index with the kinetic energy increasing from 0.5 GeV to 2 TeV. 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 as CR hardening. From a comparison of for the CR protons in Table 4, it is indicated that the flux of CR protons with the multiple power indices may be well predicted and derived from the re-acceleration and convection processes considered here. Though the DCR model does not help to improve the prediction of the fluxes of CR positrons and antiprotons, the large and D promote the relevant to CR protons decreasing from 22.2 to 14.12. The experimental data of CR protons with high precision from AMS-02 has 72 points. The 14.12 means the flux of CR protons is better predicted in the DCR model, which justifies the convection effect for CR propagation.
On the left of the first row of Figure 5, the fluxes of CR protons are drawn. It is apparently seen that the fluxes of CR protons above 330 GeV have different trends changing with different models. As a result of the comparison among the models, DCR model can well predict the hardening flux of CR protons and the flux at the 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 5, 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 5, the fluxes of CR antiprotons are plotted at the left of the second row. From the low energy to high one, except for the DC and DCR models, the DCR and DR models lead to a consistent prediction for the flux of CR antiprotons, though the tail data of AMS-02 show a bulge, which cannot completely be fitted. At the low energies, the best-fit flux of CR antiprotons prevents the excess interpretation in many existing papers. In the DCR model, which is relevant to a conventional hadronic interaction model, the excess flux of CR antiprotons is remarkable in the low energies. In some papers, the access flux was interpreted as the contribution from dark matter annihilation. Nevertheless, at the high energies, the bulge relative to the predicted flux of CR antiprotons still exists and needs a further interpretation. We shall make a detailed analysis in next paragraphs.
On the right of the second row in Figure 5, the predicted fluxes of CR positrons are drawn. In DC, DR and DCR models there are the fully inconsistent prediction with AMS-02 data. In contrast, in the DCR model the flux of CR positrons may be well fitted to the AMS-02 data at the low energies. And the maximal energy range is up to about 10 GeV. Above 10 GeV, the CR positron access becomes significant and begins to increase with the energy raising. It implies that new sources, which are different from the secondary particle production, need to be considered. In the next paragraphs, a possible dark matter interpretation of positron excess will be discussed.
4.2 dark matter interpretations for CR antiproton and positron excesses based on AMS-02 data
In Figure 6, all of the figures are drawn with the data calculated using 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 the previous paper. That situation is relevant to the flux of CR antiprotons fitting to AMS-02 data at the low energies and weaker than AMS-02 data at the high energies. It is known that the dark matter interpretation of the positron excess take the large annihilation cross-sections, which does not match weak excess of the CR antiprotons consistent with the experiment data. As the astrophysical background flux of CR antiprotons in DCR model is less than the other models, the large annihilation cross-sections of dark matter are predicted. That tension between the excess interpretations of CR antiprotons and positrons is relieved in a certain extent, which may give some hints about symmetry between the quark and lepton particles annihilated from dark matter.
In Figure 6, it is seen that in the whole mass of dark matter, the annihilation cross-sections are greater than the predicted values from Fermi-LAT gamma-ray data of dwarf spheroidal satellite galaxies of the Milky Way. That is also different completely from the other papers, which present the annihilation cross-sections from CR antiproton excess consistent with gamma-ray data of dwarf spheroidal satellite galaxies. That situation is obviously from the low estimation of the astrophysical background of CR antiprotons.
In the second row of Figure 6, the best-fit masses of dark matter are found near 600 GeV. In the different channels of dark matter annihilation the predicted best-fit mass only is from 300 GeV to 700 GeV. Dark matter interpretation of CR antiproton excess has the weak effect to discriminate the dark matter annihilation channels.
Based on the change of with dark mater mass shown on the left figure of the second row, the masses of dark matter are chosen by referring to the minimal . And the fluxes of CR antiprotons from the total contribution of astrophysical background and dark matter annihilation are drawn on the left of the first row. 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 contributing to CRs. 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 general, the analysis of positron excess is based on the total fluxes of CR electrons and positrons, which is different from the CR positron flux or fraction data of AMS-02 used in the previous papers. The relevant compatibility of the large annihilation cross section with the thermal relic density is discussed in terms of self-interacting dark matter [57, 58, 21, 22, 59, 60, 61, 61] and in connection with collider physics [62, 63]. Being similar to the CR antiprotons, the astrophysical backgrounds of the total electrons and positrons are also calculated in DCR model. The injection spectra of the primary electrons are best-fit to the data of AMS-02 and the relevant power indices are found in Table 2. Apparently, based on the redefined background of CR electrons, the total fluxes of CR electrons and positrons also have the excess to CR background. On the right figure of first row, the flux of background is well consistent with the AMS-02 data at the low energies, but at the high energies, the experimental data of AMS-02 are manifestly in access of the background.
In the last row of Figure 6, the mass-fixed best-fit annihilation cross-sections of dark matter contributing to CR electrons and positrons via many channels: , , etc., are plotted. As seen in the figures, the mass-fixed best-fit values relevant to the lepton channels are different from the other channels with mass of dark matter increasing, and the trend of the differences between them is clear. The cross masses of dark matter between the channels are near 10 TeV, where the same annihilation cross-sections appear. In the second row of Figure 6, the best-fit masses of dark matter are found easily from the change of with the mass of dark matter increasing. Except for and final states, the are less than the twice of the experimental data points, and the predicted masses of dark matter are from 400 GeV to 4 TeV. The different channels of dark matter annihilation may be discriminated by the best-fit masses of dark matter. That is obviously different from the excess interpretation CR antiprotons. The annihilation of heaviest dark matter is via the final state and the lightest one is relevant to . 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 6. As seen in the figure, in the high energies, the predicted fluxes of CR electrons and positrons are consistent with AMS-02 data and the best fluxes fitting to AMS-02 data are relevant to the final state.
In the last row of Figure 6, the differences of annihilation cross-sections of dark matter for all the annihilation channels are remarkable between the constraints from Fermi-LAT and AMS-02 data. The issues of the large annihilation cross-sections still exist. In details, though the limit lines of annihilation cross-sections relevant to and channels are lower than the other channels, the annihilation cross-sections still one order of magnitude greater than the ones from Fermi-LAT gamma ray data of dwarf spheroidal satellite galaxies. The further exploration may be found in the papers[62, 63].
In this paper, as the excess is relevant to the experimental measurement of CR electrons and positrons with ignoring the charge polarity, the interpretation of the excess is easy to be promoted at higher energies for the upcoming data of DAMPE satellite experiment.
In the re-acceleration diffusion model, the propagation parameters were well constrained only with CR protons and B/C data of AMS-02 in the previous paper. However, based on those parameter models, the predicted fluxes of CR positrons and antiprotons at the low energies deviate significantly from the experimental data. For the latest AMS-02 data, the deviation is more remarkable. In the past analyses by many groups, the situation could not be improved completely. The analysis in ref. indicated that the under-predicted antiprotons may result from a general feature of the re-acceleration models. In our present considerations, the CR antiproton productions have been recalculated using QGSJET-II-4 model. As a result, in the re-acceleration diffusion model, the under-predicted flux of CR antiprotons can be enhanced to fit consistently the AMS-02 data. Even so, the predicted flux of CR positrons and B/C deviate apparently from AMS-02 data in this model.
In the conventional model, the potential of the solar modulation is taken to be the same for the CR positrons and nucleons. In re-acceleration diffusion model, it has been shown that the prediction of CR positron flux cannot be improved even if the solar modulation is made to be different for the CR positrons and nucleons. For the CR positrons, in re-acceleration diffusion model the over-predicted flux could not be depressed by QGSJET-II-4 model, which is explained from the Z-factor comparison in Figure 1. In this paper, using the diffusion model with the convection and re-acceleration terms, the flux of CR positrons is predicted consistently.
In the further exploration on the tension between fluxes of CR positrons and nucleons predicted by propagation models, it is found that the diffusion model including the convection and re-acceleration effects need be used for the propagation of CRs, and the secondary antiproton product is calculated using QGSJET-II-4 model. Based on those conditions, such a tension is relaxed completely, and the predicted fluxes of CR positrons and antiprotons are both consistent with the latest AMS-02 data. In addition, the flux of CR protons with the hardening feature above 330 GeV is well predicted consistently.
In the DCR model, the predicted value of is compatible with ACE data only in the propagation parameters constraint only by CR protons and B/C data from AMS-02 without CR antiproton and positron data. If the five groups of the experimental data: CR protons, antiprotons, positrons, B/C and Be10/Be9 are included to fit the best-fit parameters, the chi-square over DOF for Be10/Be9 is much greater than 2, but the others are less than 2. Thus, as a result, there is a tension between Be/Be and CR antiprotons and positrons, which is being expected to be released in the further exploration. Since CR protons and B/C data from AMS-02 do not favor DR and DC model, Be/Be data from ACE does not help to constrain the propagation parameters.
Based on the predicted background of CR positrons and antiprotons, the contribution to CRs from dark matter is analyzed. For CR antiprotons, the slight bulge in the tail data of AMS-02 does not strongly constrain the interpretations of dark matter contributing to CRs. The annihilation cross-sections of dark matter contributing to CR antiprotons are calculated with the latest AMS-02 data. The result indicate that the limit lines of the annihilation cross-sections of dark matter, constrained by AMS-02 data, are higher apparently than the ones from Fermi-LAT gamma ray data of dwarf spheroidal satellite galaxies. The best-fit masses of dark matter are found near 600 GeV. In the different channels of dark matter annihilation the predicted best-fit mass only is from 300 GeV to 700 GeV. dark matter interpretation of CR antiproton excess has the weak effect to discriminate the dark matter annihilation channels.
Our analysis on the positron excess is based on the total fluxes of CR electrons and positrons, which is different from the CR positron flux or fraction data of AMS-02 adopted in the published papers. The injection spectra of the primary electrons are best-fit to the data of AMS-02. As a result, based on the redefined the astrophysical background of CR electrons, the total fluxes of CR electrons and positrons are manifestly in excess. For the excess of the total fluxes of CR electrons and positrons, the differences of annihilation cross-sections of dark matter for all the annihilation channels are remarkable between the constraints from Fermi-LAT and AMS-02 data. The issues of the large annihilation cross-sections still exist. Except for and final states, in the other channels, the best-fit masses of dark matter are from 400 GeV to 4 TeV. The annihilation of heaviest dark matter is through the final state and the lightest one is via the channel. The best flux fitting to AMS-02 data is relevant to the final state.
For the issue of positron excess, in the paper the excess is relevant to the experimental measurements of CR electrons and positrons with ignoring the charge polarity. The interpretation of the excess can easily be extended to analyze the upcoming data of DAMPE satellite experiment in a high-energy region.
Y. L. Wu is grateful to S. Ting for insightful discussions. We thank P. Zuccon, A. Kounine, A. Oliva, and S. Haino for helpful discussions on the details of the AMS-02 detector. This work is supported in part by the National Basic Research Program of China (973 Program) under Grant No. 2010CB833000; and the National Nature Science Foundation of China (NSFC) under Grants No. 11335012, No. 11475237 and No. 11121064, and also by the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No. XDB23030100 as well as the CAS Center for Excellence in Particle Physics (CCEPP).The numerical calculations were done using the HPC Cluster of SKLTP/ITP-CAS. We thank the referees of this paper for many suggestions, which are very useful to improve many parts in the paper.
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