Cosmic ray spectrum of protons plus helium nuclei between 6 and 158 TeV from HAWC data

A measurement with high statistics of the differential energy spectrum of light elements in cosmic rays, in particular, of primary H plus He nuclei, is reported. The spectrum is presented in the energy range from 6 to 158 TeV per nucleus. Data was collected with the High Altitude Water Cherenkov (HAWC) Observatory between June 2015 and June 2019. The analysis was based on a Bayesian unfolding procedure, which was applied on a subsample of vertical HAWC data that was enriched to 82% of events induced by light nuclei. To achieve the mass separation, a cut on the lateral age of air shower data was set guided by predictions of CORSIKA/QGSJET-II-04 simulations. The measured spectrum is consistent with a broken power-law spectrum and shows a kneelike feature at around $E={24.0}_{-3.1}^{+3.6}\mathrm{TeV}$, with a spectral index $\gamma =-2.51\pm 0.02$ before the break and with $\gamma =-2.83\pm 0.02$ above it. The feature has a statistical significance of $4.1\sigma$. Within systematic uncertainties, the significance of the spectral break is $0.8\sigma$.

Figure 1

The lateral effective charge distribution of an EAS event measured with HAWC on June 2, 2019. The estimated energy, zenith angle, and azimuth are ${\log }_{10}(E_{\mathrm{rec}}/\mathrm{GeV})=5.05$, $\theta =1.04°$, and $\varphi =202.24°$, respectively. The gray dots represent the measured $Q_{\mathrm{eff}}$ per PMT in $PE$ (photoelectron) units. The vertical errors are the systematic uncertainties. The result of the fit with Eq. (1) is shown with a red line. The corresponding fit parameters are shown; the number of degrees of freedom is 1018.

Figure 2

Nominal composition model [21] used for the analysis in this work. The model was obtained from fits to the AMS-2 [42, 43], CREAM I-II [44, 45], and PAMELA [46] cosmic ray data. The black bold line represents the all-particle energy spectrum, the thin continuous line and the short-dashed line, the fitted spectra of H and He nuclei. The sum of the C and O energy spectra is indicated by the dashed-dotted line, and the combination of the spectra of Ne, Mg, and Si primaries, by the long-dashed line. The dotted line correspond to the fit spectrum of Fe nuclei.

Figure 3

The mean bias (black circles) and resolution (red open squares) of the primary energy of cosmic ray induced EAS as a function of the estimated energy in HAWC according to MC predictions with QGSJET-II-04. The plots were obtained for events with $\theta <16.7°$ and using the all-particle spectrum described in our nominal composition model. The selection cuts discussed in the paper were also applied. The energy bias is defined as $\mathrm{\Delta }{\log }_{10}(E_{\mathrm{rec}})={\log }_{10}(E_{\mathrm{rec}})-{\log }_{10}(E)$, where $E_{\mathrm{rec}}$ and $E$ are the reconstructed and true primary energies of the EAS. The energy resolution is defined as the standard deviation of the $\mathrm{\Delta }{\log }_{10}(E_{\mathrm{rec}})$ distribution.

Figure 4

Predictions of the QGSJET-II-04 model for the energy dependence of the mean lateral age in vertical air showers initiated by four cosmic ray species at HAWC. From top to bottom, the MC points correspond to Fe (solid triangles), C (hollowed triangles), He (hollowed circles), and H (solid circles) primaries, respectively. For clarity, not all the elemental nuclei simulated in this work were included in the plot. HAWC data has also been added to the figure. They are shown with black squares. The $s_{\mathrm{He}-\mathrm{C}}$ cut employed to extract the enriched subsample of light nuclei is plotted using a dashed line in red.

Figure 5

The average lateral age as a function of the estimated energy for MC simulations and HAWC data. The curves are shown with their respective $1\sigma$ statistical errors: for MC, error bands are used, while for measured data, vertical error bars. Mean results for HAWC data are shown with open black squares. Meanwhile, expected values for proton and iron primaries are represented by circles (lower curve) and triangles (upper curve), respectively. They were obtained for EAS with $\theta <16.7°$ using QGSJET-II-04.

Figure 6

The raw energy distribution of the subsample of HAWC data enriched to events initiated by light primaries (dark gray) compared with the distribution previous to the employment of the age cut (light gray). The plots are not corrected for energy bin resolution effects.

Figure 7

The ratio between the event rates for measured data and MC simulations using QGSJET-II-04 and different cosmic ray composition models (cf. Appendix pp1) after applying the shower age cut. The ratios are plotted against the reconstructed primary energy. The composition models used for each curve are the nominal one (data points), ATIC-2 (dashed line), Polygonato (long dashed line), JACEE (dotted line), and MUBEE (dashed-dotted line). Statistical errors are displayed as vertical error bars for the case of the nominal model.

Figure 8

The efficiency of the shower age cut for HAWC data and MC simulations versus the reconstructed primary energy. The efficiency is estimated from its effect on the event samples selected with the criteria of Sec. 4. The shown curves represent the age cut efficiencies for HAWC data (black circles) and QGSJET-II-04 simulations with the nominal (open squares), ATIC-2 (dashed line), Polygonato (long dashed line), JACEE (dotted line), and MUBEE (dashed-dotted line) composition models of cosmic rays. The vertical error bars represent statistical errors.

Figure 9

The response matrix $P(E_{\mathrm{rec}}|E)$ for the subset of EAS enriched to events induced by protons and helium nuclei as estimated from MC simulations using our nominal composition model.

Figure 10

Left: the correction factor applied to the energy spectrum of the light mass group of cosmic rays for the contamination of heavy nuclei vs the true primary energy $E$. Right: the effective area for light primaries (black data points) compared to the effective area for protons (upper dotted line) and helium (lower dashed line) primaries and plotted against the true primary energy $E$. The gray bands represent the statistical uncertainties in both panels. In these plots, we used our MC simulations with the nominal cosmic ray composition model.

Figure 11

The reconstructed cosmic ray energy spectrum for protons plus helium primaries in the present analysis with HAWC (black points). The gray error band and the error bars represent the systematic and statistical uncertainties, respectively. The all-particle energy spectrum for cosmic rays measured by HAWC and presented in [21] is also shown (open squares). Statistical errors smaller than the data points are not shown.

Figure 12

The fits to the HAWC energy spectrum for H plus He nuclei (black circles) in the range ${\log }_{10}(E/\mathrm{GeV})=[3.8,5.2]$. The black dotted line shows the fit with the single power-law function of Eq. (8) and the red dashed line, the fit with the broken power-law expression (9). Only statistical uncertainties are shown, which are represented by vertical error bars. At low energies, the diameter of the data points is larger than the error bars.

Figure 13

The spectrum for $\mathrm{H}+\mathrm{He}$ cosmic ray nuclei as measured by HAWC (black circles) and calibrated with the post-LHC hadronic interaction model QGSJET-II-04 in comparison with similar measurements of the spectrum from direct and indirect experiments. In particular, the spectra from the direct cosmic ray detectors ATIC-2 (squares) [22], CREAM I-III (diamonds) [23], NUCLEON (downward solid triangles) [20], JACEE (upward triangles) [65], and DAMPE (crosses) [66] are presented. Indirect measurements are also shown from the EAS observatories ARGO-YBJ (downward hollowed triangles) [67], Tibet AS-gamma (open circles) [68] and EAS-TOP (hollowed star) [69]. The gray band around the HAWC data points represents systematic uncertainties. The statistical uncertainties of the HAWC measurements are shown with vertical error bars. The magnitude of the systematic uncertainty in the spectrum after varying the energy scale within systematic errors $\delta E=\pm 16\%$ is shown in the upper right corner of the plot with arrows.

Figure 14

The systematic (continuous line) and statistical (dotted line) relative uncertainties of the cosmic ray spectrum for protons plus helium nuclei measured with HAWC, which is shown in Fig. 11.

Figure 15

The total systematic uncertainties of the energy scale as a function of the primary energy $E$.

Figure 16

The intensity of the light cosmic ray nuclei obtained in this work divided by the all-particle spectrum of cosmic rays obtained with HAWC in [21] plotted as a function of the primary energy. The vertical error bars represent the total uncertainty.

Figure 17

Left: the energy spectra for light (gray, $\mathrm{H}+\mathrm{He}$) and heavy (black, $Z\ge 3$) cosmic ray primaries in the 100 GeV – 1 PeV regime as predicted with the different composition models used in this work. The nominal one (continuous line) was taken from [21]. The Polygonato model (long-dashed line) was obtained from [89], and the other models, from fits with broken power-law formulas to data from ATIC-2 [90] (short-dashed line), JACEE [65] (dotted line), and MUBEE [91] (dashed-dotted line). Right: the relative abundance of the light component of cosmic rays in the interval 100 GeV – 1 PeV as predicted with the above composition models. The same line style conventions used on the left panel are used here.

Figure 18

Left: expected values of the mean age parameter of vertical EAS for HAWC according to EPOS-LHC versus the reconstructed energy in comparison with the measured data (black open squares). For clarity, not all the elemental nuclei simulated in this work were included in the plot. Model predictions are only shown for H (circles) and Fe (triangles) primaries. Vertical lines in each data point represent the errors on the mean. The $s_{\mathrm{He}\text{−}\mathrm{C}}$ cut derived from EPOS-LHC simulations is shown with the segmented red line. Right: the energy spectrum of protons plus helium cosmic ray nuclei as measured with HAWC and reconstructed within the framework of the QGSJET-II-04 (black circles) and EPOS-LHC (hollowed red circles) hadronic interaction models. Error bars represent the statistical uncertainties. They are larger for the reconstruction with the EPOS-LHC model due to the smaller size of the MC sample used in this case within the unfolding procedure.

Figure 19

Energy dependence of the relative statistical and systematic uncertainties of the energy spectrum presented in this work. The total error is calculated as the sum in quadrature of the statistical and systematic uncertainties. For a description of each uncertainty source, see Appendix pp2. On the plot of the right, systematic uncertainties due to smoothing in unfolding procedure, seed for the unfolding method and unfolding algorithm were added in quadrature and are shown as a single contribution with a dotted line.

Figure 20

The total systematic uncertainties on the energy scale against the primary energy. The results are shown along with the contributions from the different systematic sources listed in Appendix pp2.

Figure 21

Left: the cosmic ray energy spectrum for protons plus helium primaries according to the MUBEE composition model as reconstructed with the unfolding technique described in this work (black data points). The gray error band and the vertical error bars represent the systematic and statistical uncertainties, respectively. The systematic error due to the hadronic interaction model was not included. The true energy spectrum is shown with a continuous line, while the result of the fit with a power-law formula like Eq. (8) to the reconstructed spectrum is shown with a short-dashed line. The true spectral index of the spectrum is $\gamma =-2.67$, while the fitted one is $\gamma =-2.66\pm 0.01$. Right: the reconstructed energy spectrum of light primaries (black circles) for the same model as before, but with the difference that the true spectrum of the light component (continuous line) has a break at $E_c=24\mathrm{TeV}$ with $\mathrm{\Delta }{\gamma }_c=-0.32$. The fit with the broken power-law function of Eq. (9) (represented with a short-dashed line) gave a smaller value for the change of spectral index ($\mathrm{\Delta }\gamma =-0.30\pm 0.06$) due to the contamination of heavy primaries, which is larger in this model than in the nominal one at high energies.

Figure 22

Predictions of the QGSJET-II-04 model for the energy dependence of the mean lateral age in vertical air showers initiated by four cosmic ray species at HAWC. From top to bottom, the MC points correspond to Fe (solid triangles), C (hollowed triangles), He (hollowed circles), and H (solid circles) primaries, respectively. For clarity, not all the elemental nuclei simulated in this work were included in the plot. The standard cut $s_{\mathrm{He}-\mathrm{C}}$ employed to extract the enriched subsample of light nuclei used for the main analysis of the paper is plotted using a gray dotted line, while the optimized age cut based on the maximization of the $Q$ factor and used in one of the systematic checks of Appendix pp3 is shown with a dashed black line.

Figure 23

The corrected effective area for light primaries estimated with Eq. (6) for HAWC using QGSJET-II-04 simulations and the optimized age cut based on the maximization of the $Q$ factor (black data points). For comparison, also the corrected effective area for light primaries obtained after applying the standard age cut $s_{\mathrm{He}\text{−}\mathrm{C}}$ is displayed (gray open squares). Statistical uncertainties are smaller than the size of the data points.

Figure 24

The energy spectrum of protons plus helium cosmic ray nuclei as measured with HAWC and reconstructed within the framework of QGSJET-II-04 using a subset of data selected with the standard age cut of Fig. 4 (gray open squares) and another experimental subsample obtained by applying the optimized cut of Fig. 22 (black circles). Vertical error bars represent the statistical uncertainties.