Neutrino flux prediction at MiniBooNE

The booster neutrino experiment (MiniBooNE) searches for ${\nu }_{\mu }\to {\nu }_e$ oscillations using the $\mathcal{O}(1\mathrm{GeV})$ neutrino beam produced by the booster synchrotron at the Fermi National Accelerator Laboratory). The booster delivers protons with 8 GeV kinetic energy ($8.89\mathrm{GeV}/c$ momentum) to a beryllium target, producing neutrinos from the decay of secondary particles in the beam line. We describe the Monte Carlo simulation methods used to estimate the flux of neutrinos from the beam line incident on the MiniBooNE detector for both polarities of the focusing horn. The simulation uses the Geant4 framework for propagating particles, accounting for electromagnetic processes and hadronic interactions in the beam line materials, as well as the decay of particles. The absolute double differential cross sections of pion and kaon production in the simulation have been tuned to match external measurements, as have the hadronic cross sections for nucleons and pions. The statistical precision of the flux predictions is enhanced through reweighting and resampling techniques. Systematic errors in the flux estimation have been determined by varying parameters within their uncertainties, accounting for correlations where appropriate.

Figure 1

Overall layout of the BNB. The primary proton beam, extracted from the booster, enters the target hall from the left. Upon exiting the target hall, particles encounter a 50-meter-long decay region, terminating in the beam stop on the right.

Figure 2

The MiniBooNE target assembly. The top shows an exploded view of the components, while the bottom shows the assembled configuration. The proton beam enters from the left in both figures, striking the finned beryllium slugs. Dimensions are in inches.

Figure 3

Left: Neutrino event times relative to the nearest RF bucket (measured by the RWM) corrected for expected time-of-flight. Right: An oscilloscope trace showing the coincidence of the beam delivery with the horn pulse. The top trace (labeled “2” on the left) is a discriminated signal from the resistive wall monitor (RWM), indicating the arrival of the beam pulse. The bottom trace (labeled “1” on the left) is the horn pulse. The horizontal divisions are $20\mu \mathrm{s}$ each.

Figure 4

The MiniBooNE pulsed horn system. The outer conductor (gray) is transparent to show the inner conductor components running along the center (dark green and blue). The target assembly is inserted into the inner conductor from the left side. In neutrino-focusing mode, the (positive) current flows from left-to-right along the inner conductor, returning along the outer conductor. The plumbing associated with the water cooling system is also shown.

Figure 5

Measurements of the azimuthal magnetic field within the horn. The points show the measured magnetic field, while the line shows the expected $1/R$ dependence. The black lines indicate the minimum and maximum radii of the inner conductor.

Figure 6

The MiniBooNE horn as simulated in the Geant4 beam Monte Carlo.

Figure 7

Total and elastic hadron-proton cross sections (Top: proton/neutron, Bottom: ${\pi }^{+}/{\pi }^{-}$) compiled by the Particle Data Group and the COMPAS collaboration [21], with the parametrizations used in the Glauber model calculations.

Figure 8

Measured values of $\alpha$ the real-to-imaginary ratio of the forward scattering amplitude for $p-p$ and $n-p$ scattering (left) and ${\pi }^{+}-p$ and ${\pi }^{-}-p$ scattering (right) with parametrizations. The parametrizations used in the Glauber model calculation are shown.

Figure 9

$\beta$ parameters for $p-p$ (left) and $n-p$ (right) scattering obtained from fits to the data with the parametrizations used in the Glauber model calculation.

Figure 10

Compilation of measured $\beta$ parameters for ${\pi }^{+}-p$ elastic scattering (left) and ${\pi }^{-}-p$ elastic scattering (right) versus incident pion momentum with the parametrizations used in the Glauber model calculation. The measured values of $\beta$ include the Lasinski compilation [31] as well as our own fits to the $q^2$ distributions measured in Refs. [22, 24, 30].

Figure 11

Total hadronic cross sections calculated using the Glauber model and the optical theorem for beryllium (left) and aluminum (right) targets. The calculated results for neutrons (protons) are shown as triangles (circles). The parametrization used in the flux prediction is shown as the solid line, while the default Geant4 parametrization is shown as a dashed line. The measured values of ${\sigma }_{\mathrm{TOT}}$ for $n$-Be/Al from Refs. [62, 63, 64, 65, 66] are shown as squares.

Figure 12

Total hadronic cross sections for ${\pi }^{\pm }$-Be calculated using the Glauber model (points) for ${\pi }^{+}$ (left) and ${\pi }^{-}$ (right). The Breit-Wigner parametrization based on the Carroll data [32] on the $\Delta (1232)$ resonance is shown as a dotted line, while the parametrization of the Glauber model points is shown as a solid line. The Geant4 default model is shown as a dashed line.

Figure 13

Total hadronic cross sections for ${\pi }^{\pm }$-Al calculated using the Glauber model (points) for ${\pi }^{+}$ (left) and ${\pi }^{-}$ (right). The Breit-Wigner parametrization based on the Carroll data [32] on the $\Delta (1232)$ resonance is shown as a dotted line, while the parametrization of the Glauber model points is shown as a solid line. The Geant4 default model is shown as a dashed line.

Figure 14

$p$-Be (left) and $p$-Al (right) inelastic cross sections measured from Gachurin et al. [33] ($1–4\mathrm{GeV}/c$) and Bobchenko et al. [34] ($5–9\mathrm{GeV}/c$). The solid line is the parametrization used in the flux prediction, while the dashed line shows the Geant4 default parametrization.

Figure 15

Inelastic cross sections for ${\pi }^{+}$-Be (top left), ${\pi }^{-}$-Be (top right), ${\pi }^{+}$-Al (bottom left) and ${\pi }^{-}$-Al (bottom right) as measured in Refs. [35] (squares), [36] (triangles), and [34] (circles). The solid lines are the parametrizations used in the flux prediction, while the dashed lines are the default Geant4 parametrizations.

Figure 16

Calculated values of ${\sigma }_{\mathrm{QEL}}$ for $p$-Be (left) and $p$-Al (right) interactions along with the parametrization used in the flux prediction.

Figure 17

Quasielastic cross sections for ${\pi }^{+}$-Be (top left), ${\pi }^{-}$-Be (top right), ${\pi }^{+}$-Al (bottom left) and ${\pi }^{-}$-Al (bottom right) as measured in Refs. [35] (squares) and calculated using the shadowed scattering model (circles). The solid black lines are the parametrizations used in the flux prediction.

Figure 18

Left: Production angle $\theta$ vs momentum $p$ for the ${\pi }^{+}$ in the flux simulation that contribute to the ${\nu }_{\mu }$ flux at the MiniBooNE detector. The color scale indicates the relative cross section for ${\pi }^{+}$ production in each bin of angle and momentum. The black box marks the kinematic range covered by the HARP measurements [38]. Right: Transverse momentum $p_T$ vs the Feynman-scaling variable $x_F$ for the $K^{+}$ in the flux simulation that contribute to the neutrino flux at the MiniBooNE detector (black squares). The colored points indicate the kinematic regions measured by $p$-Be $K^{+}$ production measurements [4].

Figure 19

Comparison of HARP ${\pi }^{+}$ production cross section data [38] (circles) versus $p_{\pi }$ in bins of ${\theta }_{\pi }$ from $8.89\mathrm{GeV}/c$ $p$-Be interactions and best-fit SW model (solid lines). The dashed lines represent the uncertainty band resulting from varying the parameters within their correlated uncertainties, as described in the text.

Figure 20

Comparison of E910 ${\pi }^{+}$ production cross section data [39] (circles) versus $p_{\pi }$ in bins of ${\theta }_{\pi }$ from $6.4\mathrm{GeV}/c$ $p$-Be interactions and best-fit SW model (solid lines). The dashed lines represent the uncertainty band for 68% confidence level for 7 fit parameters.

Figure 21

Comparison of E910 ${\pi }^{+}$ production cross section data [39] (circles) versus $p_{\pi }$ in bins of ${\theta }_{\pi }$ from $12.3\mathrm{GeV}/c$ $p$-Be interactions and best-fit SW model (solid lines). The dashed lines represent the uncertainty band resulting from varying the parameters within their correlated uncertainties, as described in the text.

Figure 22

Comparison of HARP ${\pi }^{-}$ production cross section data [47] (circles) versus $p_{\pi }$ in bins of ${\theta }_{\pi }$ from $8.89\mathrm{GeV}/c$ $p$-Be interactions and best-fit SW model (solid lines). The dashed lines represent the uncertainty band resulting from varying the parameters within their correlated uncertainties, as described in the text.

Figure 23

Comparison of E910 ${\pi }^{-}$ production cross section data [39] (circles) versus $p_{\pi }$ in bins of ${\theta }_{\pi }$ from $6.4\mathrm{GeV}/c$ $p$-Be interactions and best-fit SW model (solid lines). The dashed lines represent the uncertainty band resulting from varying the parameters within their correlated uncertainties, as described in the text.

Figure 24

Comparison of E910 ${\pi }^{-}$ production cross section data [39] (circles) versus $p_{\pi }$ in bins of ${\theta }_{\pi }$ from $12.3\mathrm{GeV}/c$ $p$-Be interactions and best-fit SW model (solid line). The dashed lines represent the uncertainty band resulting from varying the parameters within their correlated uncertainties.

Figure 25

Comparison of $K^{+}$ invariant cross section data (points) as a function of $p_{K^{+}}$, the $K^{+}$ momentum, in bins of ${\Theta }_{K^{+}}$, the $K^{+}$ production angle (in radians), with the Feynman-scaling based parametrization with best-fit parameters shown as a solid line. The scaling has been used to relate the measurements at different primary beam momenta to the $8.89\mathrm{GeV}/c$ primary momentum in the BNB. Normalization factors from the best-fit are applied to the data. The dashed lines represent the uncertainty band resulting from varying the parameters within their correlated uncertainties. The uncertainty bands include the factor 1.5 error inflation to set ${\chi }^2/\mathrm{DOF}=1$.

Figure 26

Invariant pion production cross section from HARP and E910 versus $p_{\pi }$ in bins of ${\theta }_{\pi }$. The E910 measurements are rescaled to $p_B=8.89\mathrm{GeV}/c$.

Figure 27

Total predicted flux at the MiniBooNE detector by neutrino species with horn in neutrino mode.

Figure 28

Total predicted flux at the MiniBooNE detector by neutrino species with horn in anti-neutrino mode.

Figure 29

Predicted ${\nu }_{\mu }$ (top) and ${\overline{\nu }}_{\mu }$ (bottom) fluxes at the MiniBooNE detector by parent meson species with horn in neutrino mode. The black line is the total predicted flux, while all the subcomponents apart from the dashed black are from nucleon-induced meson production of the indicated decay chains. The dashed black histogram includes all other contributions, primarily from meson decay chains initiated by meson-nucleus interactions.

Figure 30

Predicted ${\nu }_e$ (top) and ${\overline{\nu }}_e$ (bottom) flux at the MiniBooNE detector by parent meson species with horn in neutrino mode. The black line is the total predicted flux, while all the subcomponents apart from the dashed black are from nucleon-induced meson production of the indicated decay chains. The dashed black histogram includes all other contributions, primarily from meson decay chains initiated by meson-nucleus interactions.

Figure 31

Predicted ${\nu }_{\mu }$ (top) and ${\overline{\nu }}_{\mu }$ (bottom) fluxes at the MiniBooNE detector by parent meson species with horn in anti-neutrino mode. The black line is the total predicted flux, while all the subcomponents apart from the dashed black are from nucleon-induced meson production. The dashed black histogram includes all other contributions, primarily from meson decay chains initiated by meson-nucleus interactions.

Figure 32

Predicted ${\nu }_e$ (top) and ${\overline{\nu }}_e$ (bottom) fluxes at the MiniBooNE detector by parent meson species with horn in anti-neutrino mode. The black line is the total predicted flux, while all the subcomponents apart from the dashed black are from nucleon-induced meson production. The dashed black histogram includes all other contributions, primarily from meson decay chains initiated by meson-nucleus interactions.

Figure 33

Left: The fractional uncertainties in the ${\pi }^{+}\to {\nu }_{\mu }$ and ${\pi }^{+}\to {\mu }^{+}\to {\nu }_e$ flux with the horn in neutrino mode due to uncertainties in the ${\pi }^{+}$ production in $p$-Be interactions. Right: Same for the $K^{+}\to {\nu }_{\mu }$ and $K^{+}\to {\nu }_e$ flux from uncertainties in the $K^{+}$ production in $p$-Be interactions.

Figure 34

Left: The fractional uncertainties in the neutrino flux ${\pi }^{-}\to {\overline{\nu }}_{\mu }$ and ${\pi }^{-}\to {\mu }^{-}\to {\overline{\nu }}_e$ flux with the horn in anti-neutrino mode due to uncertainties in the ${\pi }^{-}$ production in $p$-Be interactions. Right: Same for the $K^{+}\to {\nu }_{\mu }$ and $K^{+}\to {\nu }_e$ flux from uncertainties in the $K^{+}$ production in $p$-Be interactions.

Figure 35

Left: Correlation matrix for variations in the ${\pi }^{+}\to {\nu }_{\mu }$ flux due to uncertainties in ${\pi }^{+}$ production in $p$-Be interactions with the horn in neutrino mode. Center/Right: Same for variations in the ${\pi }^{+}\to {\mu }^{+}\to {\nu }_e$ flux (center) and correlations between the ${\pi }^{+}\to {\nu }_{\mu }$ flux and the ${\pi }^{+}\to {\mu }^{+}\to {\nu }_e$ flux (right). The color scale on each plot ranges from 0 to 1.

Figure 36

Left: Correlation matrix for variations in the $K^{+}\to {\nu }_{\mu }$ flux due to uncertainties in $K^{+}$ production in $p$-Be interactions with the horn in neutrino mode. Center/Right: Same for variations in the $K^{+}\to {\nu }_e$ flux (center) and correlations between the $K^{+}\to {\nu }_{\mu }$ flux and the $K^{+}\to {\nu }_e$ flux (right). The color scale on each plot ranges from 0 to 1.

Figure 37

Left: Correlation matrix for variations in the ${\pi }^{-}\to {\overline{\nu }}_{\mu }$ flux due to uncertainties in ${\pi }^{-}$ production in $p$-Be interactions with the horn in anti-neutrino mode. Center/Right: Same for variations in the ${\pi }^{-}\to {\mu }^{-}\to {\overline{\nu }}_e$ flux (center) and correlations between the ${\pi }^{-}\to {\overline{\nu }}_{\mu }$ flux and the ${\pi }^{-}\to {\mu }^{-}\to {\overline{\nu }}_e$ flux (right). The color scale on each plot ranges from 0 to 1.

Figure 38

Left: Correlation matrix for variations in the $K^{+}\to {\nu }_{\mu }$ flux due to uncertainties in $K^{+}$ production in $p$-Be interactions with the horn in anti-neutrino mode. Center/Right: Same for variations in the $K^{+}\to {\nu }_e$ flux (center) and correlations between the $K^{+}\to {\nu }_{\mu }$ flux and the $K^{+}\to {\nu }_e$ flux (right). The color scale on each plot ranges from 0 to 1.

Figure 39

Observed $q^2$ distribution from Bellettini et al. [59] for $20\mathrm{GeV}/c$ $p$-Be scattering. The solid line represents the fit to two exponential distributions with the dashed and dotted lines showing the elastic and quasielastic contributions, respectively.

Figure 40

Top: Change in the ${\nu }_{\mu }$ flux from ${\pi }^{+}$ due to the dominant sources of systematic uncertainty apart from particle production. Left: Ratio of flux from increased horn current (circles), increased nucleon-nucleus quasielastic cross section (squares) and increased pion-nucleus quasielastic cross section (upward triangles) to the default flux. Right: Ratio of flux from decreased horn current (circles), decreased nucleon-nucleus quasielastic cross section (squares), decreased pion-nucleus quasielastic cross section (upward triangles), and turning off the skin effect (downward trianges), to the default flux. Bottom: Same for predicted ${\overline{\nu }}_{\mu }$ flux from ${\pi }^{-}$ in antineutrino mode.

Figure 41

Comparison of the observed (points) and predicted (histogram) energy distribution in ${\nu }_{\mu }$ CCQE events selected in the MiniBooNE data. A normalization factor of 1.21 has been applied to the predicted distribution as described in the text. The error bars on the predicted distribution are the estimated uncertainties in the shape of the spectrum once the normalization has been fixed to match the data.