Calculation of atmospheric neutrino flux using the interaction model calibrated with atmospheric muon data

Using the “modified DPMJET-III” model explained in the previous paper [T. Sanuki et al., preceding Article, Phys. Rev. D 75, 043005 (2007).], we calculate the atmospheric neutrino flux. The calculation scheme is almost the same as HKKM04 [M. Honda, T. Kajita, K. Kasahara, and S. Midorikawa, Phys. Rev. D 70, 043008 (2004).], but the usage of the “virtual detector” is improved to reduce the error due to it. Then we study the uncertainty of the calculated atmospheric neutrino flux summarizing the uncertainties of individual components of the simulation. The uncertainty of $K$-production in the interaction model is estimated using other interaction models: FLUKA’97 and FRITIOF 7.02, and modifying them so that they also reproduce the atmospheric muon flux data correctly. The uncertainties of the flux ratio and zenith angle dependence of the atmospheric neutrino flux are also studied.

Figure 1

Left panel: Ratio of all direction averaged flux with a smaller virtual detector (${\Phi }_{\nu }({\theta }_d=5°)$) to that with the larger virtual detector (${\Phi }_{\nu }({\theta }_d=10°)$) used in HKKM04. Right panel: Zenith angle (${\theta }_z$) dependence of the ratio at $E_{\nu }=0.1\mathrm{GeV}$ in azimuthal average.

Figure 2

Left: Atmospheric neutrino fluxes for vertical directions calculated with the virtual detectors with different radii. Right: Ratio of fluxes calculated with virtual detectors with ${\theta }_d=10$, 5, and 2.5° to the flux estimated with Eq. (5).

Figure 3

The comparison of the atmospheric neutrino fluxes calculated with modified and original DPMJET-III in ratio. The denominator is the original DPMJET-III.

Figure 4

The comparison of all direction average of the atmospheric neutrino fluxes with other calculations [8, 19, 20, 24]; (a) the absolute values of each kind of neutrinos and (b) the ratio of them.

Figure 5

$Z$-factors for ${\pi }^{+}+{\pi }^{-}(Z_{\pi })$ and $K^{+}+K^{-}(Z_K)$, and their ratio in each interaction model as the function of projectile energy. Solid lines are for DPMJET-III, dashed lines for FLUKA’97, and dotted lines for FRITIOF 7.02.

Figure 6

Left: The comparison of muon flux data and the calculations. The observed data are shown as the ratios to the calculations with modified FLUKA. The dashed line shows the ratio of the calculation with the original FLUKA’97 to that with modified FLUKA. Right: Comparison of observed muon charge ratio with the calculations. The solid line shows the calculation with modified FLUKA and the dashed line the calculation with original FLUKA’97.

Figure 7

Left: The comparison of muon flux data and the calculations. The observed data are shown as the ratios to the calculations with modified FRITIOF. The dashed line shows the ratio of the calculation with the original FRITIOF 7.02 to that with modified FRITIOF. Right: Comparison of observed muon charge ratio with the calculations. The solid line shows the calculation with modified FRITIOF and the dashed line the calculation with original FRITIOF 7.02.

Figure 8

Left: The comparison of muon flux data and calculations. The observed data are shown as the ratios to the calculations in modified primary flux scheme. The dashed line shows the ratio of calculation in original HKKM04 scheme to that in modified primary flux scheme. Right: The comparison of muon charge ratio between observed data and the calculated in modified primary flux scheme. The solid line shows the calculation with modified primary flux scheme and the dashed line the calculation in original HKKM04 scheme. Note, in the modified primary flux scheme, a modification is applied to DPMJET-III to reproduce the observed muon charge ratio. For the details, see the text.

Figure 9

$Z$-factors for ${\pi }^{+}+{\pi }^{-}(Z_{\pi })$ and $K^{+}+K^{-}(Z_K)$, and their ratio in the modified interaction models as the function of projectile energy. Solid lines are for DPMJET-III, dashed lines for FLUKA’97, and dotted lines for FRITIOF 7.02. We also plotted the values of original DPMJET-III multiplying $(E_{\mathrm{proj}}/100\mathrm{GeV}{)}^{0.05}$ above 100 GeV, to compare the modified primary flux scheme in terms of the $Z$-factor.

Figure 10

Left panel: comparison of the atmospheric neutrino fluxes calculated in the different schemes. The atmospheric neutrino fluxes are averaged over all directions and the ratios to that of modified DPMJET-III are shown. The solid lines are for the modified DPMJET-III, the dashed lines for the modified FLUKA, the dotted lines for the modified FRITIOF, and the dash-dot lines for the modified primary flux. Right panel: comparison of neutrino ratios, ${\nu }_{\mu }/{\overline{\nu }}_{\mu }$, ${\nu }_e/{\overline{\nu }}_e$, and $({\nu }_{\mu }+{\overline{\nu }}_{\mu })/({\nu }_e+{\overline{\nu }}_e)$. The neutrino ratios are calculated for all direction averaged flux in each calculation scheme, and the ratio to the modified DPMJET-III shown. In addition, we show the ratios calculated with a slightly different atmospheric model with $\epsilon =\pm 0.05$ in Eq. (7).

Figure 11

The uncertainty of each error source for atmospheric neutrino flux and their sum with Eq. (8). Note, Eq. (9) loses its validity in the shaded region. The total error for $\lesssim 1\mathrm{GeV}$ is estimated differently from Eq. (8), as stated in the text. Note the statistical and systematic error are not shown in the figure.

Figure 12

The muon fluxes at balloon altitudes at Fort Sumner (Sept. 2001). Left: comparison calculated and observed muon fluxes. The dashed lines show $\pm 10\%$ deviation from the calculated values. Right: $p_{\mu }/p_{\nu }$ ratio for the same momentum of parent $\pi$’s in average. The dashed lines are the same quantities but for the muons at the ground level.