Uncertainties in atmospheric neutrino fluxes

An evaluation of the principal uncertainties in the computation of neutrino fluxes produced in cosmic ray showers in the atmosphere is presented. The neutrino flux predictions are needed for comparison with experiment to perform neutrino oscillation studies. The paper concentrates on the main limitations which are due to hadron production uncertainties. It also treats primary cosmic ray flux uncertainties, which are at a lower level. The absolute neutrino fluxes are found to have errors of around 15% in the neutrino energy region important for contained events underground. Large cancellations of these errors occur when ratios of fluxes are considered, in particular, the ${\nu }_{\mu }/{\overline{\nu }}_{\mu }$ ratio below $E_{\nu }=1\mathrm{GeV}$, the $({\nu }_{\mu }+{\overline{\nu }}_{\mu })/({\nu }_e+{\overline{\nu }}_e)$ ratio below $E_{\nu }=10\mathrm{GeV}$ and the up/down ratios above $E_{\nu }=1\mathrm{GeV}$ are at the 1% level. A detailed breakdown of the origin of these errors and cancellations is presented.

Figure 1

Summary of measurements of single particle production yields as a function of primary energy and secondary energy. The bands for each experiment represent the range of primary and secondary particle energies where measurements exist for at least one value of $p_T$. The boxes on the plot show the contribution of the phase space to the generation of contained underground neutrino events as computed by the simulation. The outer (red) and inner (black) boxes indicate the extremes of geomagnetic field effects for high and low geomagnetic latitude, respectively.

Figure 2

Uncertainties assigned to the production rate of charged pions (left) and charged kaons (right) as a function of $x_{\mathrm{lab}}$. The uncertainties are shown for various ranges of incident particle energy $E_{\mathrm{i}}$ for interactions of protons on light nuclei.

Figure 3

Uncertainty sources for hadron production. The uncertainties which are applied are fully correlated within each region shown and completely uncorrelated between regions. The letters used to label each region are used on subsequent figures. The levels of uncertainties applied are shown in Fig. 2.

Figure 4

Comparison of proton flux measurements to the GSHL parametrization with parameters given in Table  as a function of energy. The lines show the uncertainties on the fluxes used in this paper.

Figure 5

Comparison of helium flux measurements to the GSHL parametrization with parameters given in Table  as a function of energy per nucleon. The lines show the uncertainties on the fluxes used in this paper.

Figure 6

Fluxes, flux ratios and uncertainties for muon neutrino/antineutrinos plotted as a function of neutrino energy. The muon neutrino and antineutrino fluxes are shown by (red) circular points, the upper line is ${\nu }_{\mu }$ and the lower line is ${\overline{\nu }}_{\mu }$ plotted against the scale on the right axis, the error bars represent the uncertainty from hadron production and primary fluxes. All of the following curves are plotted against the scale on the left axis. The uncertainties (in percent) in the fluxes are shown again with the (green) lines with no points for ${\nu }_{\mu }$ (lower line) and ${\overline{\nu }}_{\mu }$ (upper line). The ratio of ${\nu }_{\mu }$ flux to ${\overline{\nu }}_{\mu }$ flux is plotted by the (blue) line with triangles and error bars indicating the estimated uncertainty in the ratio and the uncertainties in the ratio are shown in percent with the (black) line with square points. The large cancellation in the uncertainties in the ratio is apparent.

Figure 7

Uncertainties in neutrino-type ratios as a function of neutrino energy. ${\nu }_{\mu }/{\overline{\nu }}_e$ is shown with (black) lines with squares, ${\nu }_e/{\overline{\nu }}_e$ with (red) lines with circles and $({\nu }_{\mu }+{\overline{\nu }}_{\mu })/({\nu }_e+{\overline{\nu }}_e)$ with (green) lines with triangles.

Figure 8

Uncertainties in neutrino-type ratios as a function of zenith angle for (a) 0.3–3 GeV, (b) 3–30 GeV and (c) above 30 GeV neutrino energy ranges. ${\nu }_{\mu }/{\overline{\nu }}_{\mu }$ is shown with (black) lines with squares, ${\nu }_e/{\overline{\nu }}_e$ with (red) lines with circles and $({\nu }_{\mu }+{\overline{\nu }}_{\mu })/({\nu }_e+{\overline{\nu }}_e)$ with (green) lines with triangles.

Figure 9

Uncertainties in directional ratios as a function of neutrino energy.

Figure 10

Breakdown of flux uncertainties (shown here for angle averaged muon neutrinos) with different regions of hadron production as a function of neutrino energy. The capital letters in the key correspond with the hadron production uncertainty zones in Fig. 3 (green for $\pi$, blue for K) and the one labeled “Chg” represents the pion charge ratio uncertainty. The lower case (red) letters in the key correspond to variation of the flux parameters in Table . See text for more information. The topmost thick (black) line with no points is the total error on the flux and the lower line of the same style is the total hadronic error (i.e. excluding the uncertainty from the primary flux).

Figure 11

Breakdown of uncertainties in flavor ratios with different regions of hadron production, shown as a function of neutrino energy. (a) $({\nu }_{\mu }+{\overline{\nu }}_{\mu })/({\nu }_e+{\overline{\nu }}_e)$ ratio (b) ${\nu }_{\mu }/{\overline{\nu }}_{\mu }$ ratio and (c) ${\nu }_e/{\overline{\nu }}_e$. Key as in Fig. 10.

Figure 12

Breakdown of uncertainties in up/down ratio for muon neutrinos, shown as a function of neutrino energy, key as in Fig. 10.

Figure 13

Breakdown of uncertainties in up/horizontal ratio, key as in Fig. 10.

Figure 14

Stability of the uncertainty estimates as a function of different choices of uncertainty sources ((1), (2), (3)), detector location (Soudan) and base hadron production generator.

Figure 15

Uncertainty sources used in the alternative 2 hadron production weighting scheme. The sources 01–25 allow more independence to different values of $x_{\mathrm{lab}}$ than the standard scheme and have full correlation between pions and kaons in the same region of $(E_{\mathrm{i}},x_{\mathrm{lab}})$.

Figure 16

Uncertainty sources used in the alternative 3 hadron production weighting scheme. The sources 01–23 allow more independence to different values of $E_{\mathrm{i}}$ than the standard scheme and have full correlation between pions and kaons in the same region of $(E_{\mathrm{i}},x_{\mathrm{lab}})$.

Figure 17

Increase in muon neutrino to electron neutrino flux uncertainty when comparing fluxes from different parts of the solar cycle at Soudan.

Figure 18

Line with open squares shows the standard uncertainty in the up/horizontal ${\nu }_{\mu }$ flux ratio and line with closed circles the uncertainty with the effect of the Earth’s magnetic field removed.