Methods to Determine Neutrino Flux at Low Energies:Investigation of the Low $\nu$ Method

We investigate the"low-$\nu$"method (developed by the CCFR/NUTEV collaborations) to determine the neutrino flux in a wide band neutrino beam at very low energies, a region of interest to neutrino oscillations experiments. Events with low hadronic final state energy $\nu<\nu_{cut}$ (of 1, 2 and 5 GeV) were used by the MINOS collaboration to determine the neutrino flux in their measurements of neutrino ($\nu_\mu$) and antineutrino ($\nub_\mu$) total cross sections. The lowest $\nu_\mu$ energy for which the method was used in MINOS is 3.5 GeV, and the lowest $\nub_\mu$ energy is 6 GeV. At these energies, the cross sections are dominated by inelastic processes. We investigate the application of the method to determine the neutrino flux for $\nu_\mu$, $\nub_\mu$ energies as low as 0.7 GeV where the cross sections are dominated by quasielastic scattering and $\Delta$(1232) resonance production. We find that the method can be extended to low energies by using $\nu_{cut}$ values of 0.25 and 0.50 GeV, which is feasible in fully active neutrino detectors such as MINERvA.


Introduction
A detailed understanding of neutrino (ν µ ) and antineutrino (ν µ ) interaction cross sections for various final states is required for the next generation neutrino oscillations experiments. The relevant neutrino energy region of interest for the large neutrino detectors such as T2K [1], MINOS [2,3], and NOVA[4] is 0.5 < E ν < 3 GeV.
The MINERvA[5] experiment at the NUMI wide band beam at Fermilab uses a fine grain fully active scintillator target-detector to investigate neutrino (ν µ ) and antineutrino (ν µ ) cross sections for energies above 0.5 GeV. These measurements require a reliable determination of the flux as a function of ν µ ,ν µ energy.
Previous neutrino experiments in wide band beams used five methods for the determination of flux as a function of energy.
1. Modeling the distribution of pions and kaons produced by incident proton beam in the target. Then, tracking the pions and kaons though the Horn focussing magnetic fields, and modeling the decays of pions and kaons in the decay pipe. 2. Measuring the muon flux that exits the decay pipe and relating it to the neutrino flux. 3. Monitoring Inverse muon decay events ( ν µ +e → µ − + ν e ) in the detector. 4. Monitoring neutrino-electron scattering events (ν µ + e → ν µ + e) in the detector.
5. The "low-ν" method for the determination of the energy dependence of the relative neutrino and antineutrino flux.
Here, "low-ν" [6,7] refers to events with low energy transfer to the target nucleon in the scattering processes ν µ + N → µ − + X andν µ + N → µ + + X. This energy transfer manifests itself as the energy (ν = E had ) of the final state hadrons (X) in the laboratory frame.
There are inherent difficulties in the each of those techniques: 1. In method 1, the differential cross sections for the production of pions and kaons by protons incident on a thick nuclear target must be known very well. In addition, the magnetic field of the horn focussing magnets must be modeled reliably. 2. In method 2, the response of the muon detectors at the end of the decay pipe must be very well understood (for absolute calibration of the neutrino flux). The response of the muon detectors is sensitive to δ rays. In addition, since the energy of the muons is not measured, it is difficult to determine the energy dependence of the neutrino flux. 3. In method 3, the threshold for the reaction ν µ + e → µ − + ν e is about 12 GeV. Therefore, this method can only be used at higher energies. Unfortunately, this method cannot be used for the determination of the flux for antineutrinos. Inverse muon decay was used by NOMAD to constrain their neutrino flux at high energies. 4. In method 4, only the sum of the fluxes for neutrinos and antineutrinos can be measured. This is because calorimetric detectors such as MINERvA cannot determine the charge of final state electron in ν µ +e → ν µ +e events. Both methods 4 and 5 are statistically limited. In addition, in both methods, the total final state energy in the events is not fully reconstructed since there is a neutrino in the final state. This places a limitation on the determination of the energy dependence of the neutrino and antineutrino fluxes. Despite of these limitations, these two additional methods are valuable as important consistency checks.
Consequently, having another independent technique such as the "low-ν" method is extremely valuable.
The "low-ν" method was initially developed by the CCFR/NUTEV [6,7] collaboration. At high energy, the method was used to determine the relative neutrino flux as a function of neutrino energy (E ν ). The method relies on the observation that the charged current differential cross section, dσ ν,ν dν in the limit ν → 0, only depends on the structure function F 2 , and therefore is independent of energy.
The "low-ν" method was used by the CCFR/NuTeV collaborations to measure the energy dependence of σ ν /E and σν/E for charged current interactions for energies higher than 30 GeV for an iron target. The absolute level of the charged cross sections is normalized to previous measurements of σ ν /E in a high energy narrow band neutrino beam.
Most recently, the method was extended to lower energies by the MINOS [3] collaboration. The lowest neutrino energy for which this method was used in MINOS is 3.5 GeV for neutrinos and 6 GeV for antineutrinos.
The absolute normalization used by MINOS is to the world average value of charged current σ ν /E measurements for an isoscalar target for neutrino energies between 30 to 50 GeV. The average value used by MINOS is σ ν /E 30−50 = 0.675 × 10 −38 cm 2 /GeV per nucleon. The antineutrino sample is not independently normalized but is related to the neutrinos by using the same normalization factor.
In this communication we investigate the application of the technique to much lower neutrino energies (E ν > 0.5 GeV). Neutrino interactions in this energy range are currently being studied at MINERvA. 2 The "low-ν" method at high energies If we neglect terms which are proportional to the muon mass, the differential cross section d 2 σ ν,ν dxdy for charged current scattering of ν µ (ν µ ) with an incident energy E ν , muon final energy E µ and scattering angle θ can be written in terms of the structure functions F 1 = M W 1 (x, Q 2 ), F 2 = νW 2 (x, Q 2 ) and F 3 = νW 3 (x, Q 2 ): where G F is the Fermi weak coupling constant, M is the proton mass, y = ν/E ν , ν = E had = E ν − E µ , Q 2 = 4E ν E µ sin 2 (θ/2) is the square of four momentum transfer, and x = Q 2 /(2M ν) is the Bjorken scaling variable. The plus sign in front of the xF 3 term is for neutrinos and the minus is for antineutrinos.
Here, R(x, Q 2 ) is defined as the ratio of the longitudinal and transverse structure functions (σ L /σ T ). It is related to the other structure functions by, where F L is called the longitudinal structure function, Other useful relations are: W 1 (x, Q 2 ) = W 2 (x, Q 2 ) 1 + ν 2 /Q 2 1 + R(x, Q 2 ) .
Integrating over x, the differential dependence on ν can be written in the simplified form The coefficients A, B, and C depend on integrals over structure functions, where In the limit ν/E ν → 0, the A term dominates and the B and C terms are very small. The MINOS collaboration used the number of "low-ν" events (with ν < ν cut ) in the detector to determine the relative flux of neutrinos and antineutrinos as a function of E ν . In the MINOS analysis, the relative flux is determined using events with ν < 1 GeV for ν µ energies in the range 3 < E ν < 9 GeV, and forν µ in the range 5 < E ν < 9 GeV. Events with ν < 2 GeV are used for ν µ andν µ events in the range 9 < E ν < 18 GeV, and events with ν < 5 GeV are used for E ν > 18 GeV.
MINOS divides the number of "low-ν" events with y < y cut = ν cut /E ν by correction term f C to account for the energy dependence from the B and C terms. Here As seen in equation 6, the negative contribution of F 2 (x) in B partially cancels the positive contribution of xF 3 (x) for ν µ 's. Forν µ 's both contributions are negative. There are additional small corrections that are applied to equation 6 to correct for differences in F 2 between neutrinos and antineutrinos.
In practice, a neutrino interaction generator model [8] is used to compute f C from The corrections factors f C used by MINOS for ν µ andν µ are shown in Fig. 1. The measured "low-ν" sample is corrected for detector smearing and acceptance by multiplying the number of observed "low-ν" events in the data in each energy bin by which is obtained from a Monte Carlo detector simulation. N GEN and N REC are the number of generated and reconstructed events below ν cut in each reconstructed energy bin, respectively. In the first pass the initial input flux from a beam model is used. It is then replaced by the extracted "low-ν" flux and the procedure is reiterated to account for the effect of the flux model on the acceptance corrections. (The change in the extracted flux is found to be negligible). The "low-ν" sample is further corrected for radiative effects using Ref. [9]. The absolute level of the flux is set by normalizing the cross section in data to a nominal world average charged current cross section at some high energy. As mentioned earlier, in MINOS the normalization is set to the average of previous σ total /E measurements for neutrino energies between 30 and 50 GeV.
There are three criteria for the effectiveness of the "low-ν" method.
1. The number of "low-ν" events that are used in the determination of the flux should not be a large fraction of the total number of neutrino events in each energy bin. 2. The systematic uncertainty in the energy dependent correction factor f C should be small. 3. The number of "low-ν" events that are used in the determination of the flux should be sufficiently large to have flux sample with small statistical errors.
The first two criteria require a ν cut which is as low as possible. The third requires a ν cut which is as large as possible.
The MINOS collaboration uses the criteria that the fractional contribution of events with ν < ν cut to the total charged current cross section should be less than 60%. MINOS uses events with ν < 1 GeV for determination of the flux at their lowest ν µ ,ν µ energies.
The fraction of events with ν < 1 GeV is less than 60% for ν µ interactions with E ν > 3 GeV and forν µ interactions with Eν > 5 GeV. Therefore, to determine the flux for E ν < 3 GeV and Eν < 5 GeV we need to use a ν cut which is smaller than 1 GeV.
We investigate ν cut = 0.25 GeV to be used for E ν,ν > 0.7 GeV, and ν cut = 0.5 GeV to be used for E ν,ν > 1.4 GeV. These samples can be cross calibrated against the ν cut = 1 GeV sample in the range E ν > 3 GeV for neutrinos and Eν > 5 GeV for antineutrinos. Similarly, they can be calibrated against the ν cut = 2 GeV and ν cut = 5 GeV samples in the range E ν,ν > 9 GeV and E ν,ν > 18 GeV, respectively.
3 The "low-ν" method at low energies In the few GeV region, there are several types of neutrino interaction processes as defined by the final state invariant mass W . These include quasielastic (QE) reactions (W < 1.07 GeV), production of the ∆(1232) resonance (1.1 < W < 1.4 GeV), coherent pion production, production of higher mass resonances (1.4 < W < 2.0 GeV) and the inelastic continuum (W > 2.0 GeV ). Fig. 2 shows the kinematic region in Q 2 (in GeV 2 and ν = E had (in GeV) for E ν < 4 GeV. Fig. 3 shows the kinematic region in Q 2 and ν for E ν < 1.5 GeV. In this paper we focus on ν <0.25 GeV region (shaded area in Fig. 2 and Fig. 3) for the lowest neutrino energies. In addition, we investigate the ν <0.50 GeV region as an additional check.
For E ν = 3 GeV, about 1/3 of the total charged current cross section originates from QE scattering, 1/3 from resonance production and 1/3 from inelastic scattering.
As seen in Fig. 3 the ν < 0.25 GeV sample is dominated almost entirely by QE events with Q 2 < 0.45 GeV 2 .
The ν < 0.5 GeV sample includes both QE events with Q 2 < 0.95 GeV 2 and also ∆(1232) resonance events with Q 2 < 0.3 GeV 2 . Both samples include a very small fraction of events originating from coherent pion production (as discussed in Appendix II).
In the very "low-ν" region it is more convenient to write the expression for the charged current differential cross sections as follows [10,11]: where S cos = G 2 2π cos 2 θ C = 80 × 10 −40 cm 2 /GeV 2 . In the scattering process, there are additional small contributions from strangeness and charm non-conserving processes. In the discussion below we do not show these terms explicitly, but charm and strangeness changing contributions are assumed to be included in the analysis. (The strangeness changing valence quark contributions are proportional to G 2 2π sin 2 θ C ). Each of the structure functions has a vector and axial component (except for W 3 which originates from axialvector interference). The vector part of W 4 and W 5 are well known since they are related to the vector part of W 2 and W 1 by the following expressions [10]: At 'low-ν' and very high energy the charged current cross section is only a function of W 2 . If we integrate the cross section from ν min ≈ 0 up to ν= ν cut (where ν cut is small), we can write the expression for the cross section in terms of W 2 only, and energy dependent corrections ratios to the W 2 component: Top panel: Neutrino partial charged cross sections per nucleon for "low-ν" events determined from the GENIE Monte Carlo [12]. Also shown are the measurements of MINOS on iron (per nucleon corrected for the excess number of neutrons). Bottom panel: The fraction of "low-ν" neutrino events in the GENIE [12] Monte Carlo as compared with the measurements of MINOS (color online).
The technique does not depend on the modeling of W 2 because the σ W2 cross section is the same at all energies. All energy dependent corrections are expressed in terms of ratios to σ W2 . In quark parton language, the uncertainty in f 1 is related to the uncertainty in the longitudinal structure function at low Q 2 and the uncertainty in f 3 is related to the uncertainty in level of antiquarks in the nucleon at low Q 2 . For QE scattering and resonance production the structure functions are expressed in terms of form factors.

Partial charged current cross sections
The top panel of Fig. 4 shows the partial neutrino charged current cross section per nucleon for "low-ν" events (for ν cuts of 0.25, 0.5, 1, 2 and 5 GeV) as a function of energy as determined by the GENIE [12] Monte Carlo for a carbon target. The top panel of Fig. 5 shows the corresponding partial charged current cross sections for antineutrinos.
Also shown are the measurements of the partial charged current cross sections on iron from the MINOS collaboration (for ν cuts of 1, 2 and 5 GeV). The MINOS cross sections for iron have been corrected for the excess number of neutrons in iron. Note that the nuclear corrections to the structure functions in iron nucleus are larger than in carbon. Therefore, the partial cross sections on carbon and on iron may not be the same.
At high energies (as shown in Fig. 4 and 5) the partial cross sections for a fixed ν cut are independent of energy and are approximately equal for neutrinos and antineutrino. The fact that these partial charged current cross section are relatively independent of energy is the basis for the "low-ν" method.
The bottom panels of figures 4 and 5 show the fraction of "low-ν" events predicted by the GENIE Monte Carlo as compared with the measurements in MINOS. In order to use the technique at low energies the fractions must be smaller than 0.6. Therefore, at the lowest energies we must use ν cuts of 0.25 and 0.50 GeV.
MINOS is a sampling target calorimeter which has poor resolution at low hadron energy. Therefore, "low-ν" samples with ν < 0.25 GeV and ν < 0.5 GeV cannot be defined reliably. On the other hand, since the MINERvA detector is a fully active target calorimeter, "low-ν" samples with ν < 0.25 GeV and ν < 0.5 GeV can be used.

Absolute normalization
Since the neutrino energy range for MINERvA is limited to lower energies, we propose that the MINERvA charged current cross section measurements be normalized to the cross section in the energy range between 10 to 20 GeV (e.g. at a mean energy of 15.1 GeV). The absolute level of the charged current cross section at this energy range has been measured by both the MINOS and NOMAD collaborations.
The MINOS total cross section measurement for an isoscalar iron target at a neutrino energy of 15.1 GEV is per nucleon in iron. Here the total error of 0.02 is the combined statistical, systematic and normalization errors of 0.008 ± 0.012 ± 0.015, respectively.
The NOMAD cross section measurement for an isoscalar carbon target at a neutrino energy of 15.1 GEV is σ N OM AD ν /E = 0.698 ± 0.025 × 10 −38 cm 2 /GeV per nucleon in carbon.
The MINOS total cross section measurement for an isoscalar iron target at an antineutrino energy of 15.1 GEV is per nucleon in iron. Here, the total error of 0.012 is the combined statistical, systematic and normalization errors of 0.007 ± 0.007 ± 0.006, respectively. Alternatively, it may be possible for MINERvA to normalize to the partial cross sections measured by MINOS for ν < 1 GeV and ν < 2 GeV at 15.1 GeV. These partial cross sections (which were used by MINOS to determine their relative flux) are relatively constant between 10 and 20 GeV. However, the MINOS partial cross sections are measured on iron. The MINERvA target is solid scintillator (i.e. carbon), and the partial cross sections for iron and carbon can be different. For a neutrino energy of 15.1 GeV MINOS measured the following isoscalar partial cross sections on iron (per nucleon): 4 Using "low-ν" events with ν < 0.25 GeV As seen in Fig. 3 the ν < 0.25 GeV region is dominated by QE events. This is illustrated in Fig. 6 which shows the relative contributions of QE and non-QE processes to ν < 0.25 GeV cross section as a function of energy (as determined from the GENIE Monte Carlo). The ν < 0.25 GeV cross sections for ν µ are shown on the top panel, and the ν < 0.25 GeV cross sections forν µ are shown on the bottom panel. The QE contribution is shown in red, the contribution from pion production process (e.g. ∆, inelastic and coherent pion) is shown in blue and the total is shown in black. Most of the events are QE and the contribution from pion production processes is negligible.
As mentioned earlier, the technique does not rely on the modeling of W 2 , or the modeling of nuclear effects (e.g. Fermi motion smearing) on W 2 . This is because the cross section σ W2 (including nuclear effects) is the same at all neutrino energies.
The uncertainty in the flux extracted from the event sample with ν < 0.25 GeV is determined by how well we can model the relative contributions of W 1 and W 3 for the case of QE scattering on bound nucleons, or equivalently the relative contributions of f 1 and f 3 to f C . Here f 1 and f 3 are proportional to the ratios W1 W2 and W3 W2 . Since the ratios W1 W2 and W3 W2 for QE scattering on free nucleons are very well known, the uncertainty in f C originates primarily from modeling the nuclear corrections to W1 W2 and W3

W2
for nucleons bound in a nuclear target.
Band from Bosted--Mamyan fit to electron sca3ering data Parametriza8on Fig. 7. The transverse enhancement ratio [14] (RT ) as a function of Q 2 . Here, RT is ratio of the integrated transverse response function for QE electron scattering on nucleons bound in carbon divided by the integrated response function for independent nucleons. The black points are extracted from Carlson et al [16], and the blue bands are extracted from a fit [19] to QE data from the JUPITER [18] experiment (Jlab experiment E04-001). The curve is a fit to the data of the form RT = 1 + AQ 2 e −Q 2 /B . The dashed lines are the upper and lower error bands (color online).
. and Here, G p E (Q 2 ), G n E (Q 2 ), G p M (Q 2 ) and G n M (Q 2 ) are the electric and magnetic nucleon form factors, which are mea-sured in electron scattering experiments. Note that: Therefore, for QE ν µ ,ν µ scattering only G V M contributes to the transverse virtual boson absorption cross section.

Transverse enhancement QE scattering from nuclei
Studies of QE electron scattering on nuclear targets [16] indicate that only the longitudinal part of the QE cross section can be described in terms of a universal response function of independent nucleons bound in a nuclear potential (and free nucleon form factors). In contrast, a significant additional enhancement with respect to the model is observed in the transverse part of the QE cross section.
The enhancement in the transverse QE cross section has been attributed [16] to meson exchange currents (MEC) in a nucleus. Within models of meson exchange currents [16] the enhancement is primarily in the transverse part of the QE cross section, while the enhancement in the longitudinal QE cross section is small (in agreement with the electron scattering experimental data).
The conserved vector current hypothesis (CVC) implies that the corresponding vector structure function for the QE cross section in ν µ ,ν µ scattering can be expressed in terms of the structure functions measured in electron scattering on nuclear targets. Therefore, there should also be a transverse enhancement in neutrino scattering. In models of meson exchange currents the enhancement in the axial part of ν µ ,ν µ QE cross section on nuclear targets is also expected small.
The transverse enhancement observed in electron scattering is a function of both Q 2 and ν. However, a simple way to account for the integrated transverse enhancement [14] from nuclear effects is to assume that G p M (Q 2 ) and G n M (Q 2 ) are enhanced in a nuclear targets by factor √ R T L . Bodek, Budd and Christy [14] have used electron scattering data [16,18,19] to parametrize R T L as follows: with A = 6.0 and B = 0.34 GeV 2 . The electron scattering data indicates that the transverse enhancement is maximal near Q 2 =0.3 GeV 2 and is small for Q 2 greater than 1.5 GeV 2 . The upper error band is given by A = 6.7 and B = 0.35 GeV 2 , and the lower error band is given by A = 5.3 and B = 0.33 GeV 2 .
In modeling ν µ ,ν µ QE scattering on nuclear targets we use BBBA2007 25 parameterization [17] of the free nucleon electromagnetic form factors , and a dipole axial form factor with M A = 1.014 GeV. We apply the transverse enhancement correction to G p M (Q 2 ) and G n M (Q 2 ). We also apply Pauli blocking corrections to the differential QE cross section as parametrized by Paschos and Yu [11]. We refer to this model as the Transverse Enhancement (TE) model. This is the nominal model that is used in this paper.
We also compare calculations based on the nominal TE model to two other models. The first model is the independent nucleon model with Pauli blocking with M A = 1.014 GeV, without transverse enhancement. We refer to this model as the "Independent Nucleon (MA=1.014)" model. This model, which is used by the NOMAD [22] collaboration, is very close to the model which is currently implemented in the GENIE Monte Carlo (the GE-NIE default value is M A = 0.99 GeV). The second model is the independent nucleon model with Pauli blocking, M A = 1.3 GeV, without transverse enhancement. This model is used by the MiniBooNE Collaboration [21]. We refer to this model as the "Larger M A (M A =1.3) model". We use the difference between the three models as a conservative systematic error on the flux extracted from the ν samples. Fig. 8 shows a comparison of predictions of various model predictions for the ν µ ,ν µ total QE cross section sections to experimental data on nuclear targets. Shown are "Independent Nucleon (MA=1.014)" model, the "Larger M A (M A =1.3) model", and the TE model (with upper and lower error bands). Also shown are the predictions of the "QE+np-nh RPA" MEC model of Martini et al. [20] The data points are the QE cross section measurements of MiniBooNE [21] (gray stars) and NOMAD [22] (purple circles). Note that there is an overall ≈ 10% systematic error in the experimental QE cross sections because of uncertainties in the determination of the neutrino and antineutrino fluxes in each of the two experiments.
In this paper we use the error band in the transverse enhancement parameters as a lower limit on systematic error in the modeling. We use the "Independent Nucleon (MA=1.014)" and the "Larger M A (M A =1.3) model" as conservative upper limits on the errors in the modeling. Fig. 9 shows the contribution of the various components (σ W2 , σ 2 , σ 1 , σ 3 , σ 4 , σ 5 ) to the total QE cross section (as defined by Eq. 14) as a function of incident energy. These contributions are calculated using the TE model. The top panel shows the contribution of the various components for the neutrino QE cross section, and the bottom panel shows the contribution of the various components for the antineutrino QE cross section.

Neutrino and antineutrino total cross sections
The MINOS collaboration uses the criteria that the fraction of "low-ν" events that are used for the determination of the relative neutrino flux in an energy bin should be less than 60% of the total number of charged current events. In order to test for this fraction, we need to use a parameterization to estimate the energy dependence of the neutrino and antineutrino charged current total cross sections. Fig. 10 and 11 show the ν µ andν µ total charged current cross sections measured on isoscalar nuclear targets by the MINOS [3] (iron), NOMAD [22](carbon), and Serpukov [23] (Serp96, aluminum) experiments. The total cross sections per nucleon (divided by neutrino energy) are shown in units of 10 −38 cm 2 /GeV (with statistical, systematic and normalization errors combined in quadrature). The ratio of theν µ and ν µ total charged current cross sections is shown in the bottom panel of Fig. 11. The cross sections reported by the MINOS collaboration were measured using a neutrino flux extracted from "low-ν" samples with ν less than 1, 3, and 5 GeV.
Also shown in Fig. 10 are low energy cross sections measured by at BNL [24] (BNL82). Since the BNL82 cross sections were measured on a deuterium target we apply a correction to account for nuclear effects. The BNL82 points shown in the figure were increased by the difference of the predictions of the TE model for the QE cross section (which is expected to describe the cross section on a heavy nuclear target) and the "Independent Nucleon (MA=1.014)" model (which is expected to describe the QE cross sections on deuterium).
In our investigation of the "low-ν" technique, we use a parameterization to estimate the total ν µ ,ν µ charged current cross sections. The parameterization, which is shown as the thick red line in Fig. 10, is given by  The above form is constrained to yield the average world cross section measurements in the 30 to 50 GeV region of of 0.675 10 −38 cm 2 /GeV, and 0.329 10 −38 cm 2 /GeV for ν µ andν µ , respectively.
We only use this parameterization to estimate the fractional contribution of "low-ν" events to the total cross section to determine the region where it is less than 60%. When improved total cross section measurements become available (e.g. from MINERvA), this parameterization can be updated to include the new data.
4.4 Results with ν < 0.25 GeV  section. This sample is dominated by QE ν µ N → µ − P events. The partial cross section as a function of energy for neutrinos is shown in the top panel and the partial cross section for antineutrinos is shown in the bottom panel. The partial cross section (per nucleon) is calculated on a carbon target using the TE model.
The uncertainty in the relative values of the ν < 0.25 GeV partial cross section as a function of energy determines the uncertainty in the determination of the relative fluxes. Here f C (E) is the ratio of the partial cross section to the value of the partial cross section at E = ∞. Fig. 13(a) (top) shows the correction factor f C for the ν < 0.25 GeV sample for neutrinos as a function energy. The error bands in f C (originating from the uncertainty in the transverse enhancement) are shown as the dotted lines, and represent the lower limit on errors. Also shown on the figure is the negative contribution from the kinematic correction f 2 (which is well known), and the contributions of f 1 , f 3 , f 4 and f 5 . Here the contribution of f 4 and f 5 is negligible. For the case of neutrino scattering, the positive contributions of f 1 and f 3 partially cancel the negative contribution of f 2 . Fig. 13(b) (bottom) shows the fractional contribution of the ν < 0.25 GeV sample to the total neutrino charged current cross section. This fraction is less than 60% for ν µ energies above 0.70 GeV.

Uncertainty in the f C correction factors
It has been traditional to use the value and error in the effective M A extracted from neutrino scattering data as an estimate of various uncertainties. Typically, the difference between results with M A = 1.014 GeV and M A = 1.3 GeV are used an upper limit on the error.
We find that the values of the f C correction factor are insensitive to M A . This is because at small Q 2 , both ratios f 1 , and f 3 are insensitive to M A . Specifically, both are insensitive to M A because the change in F A at small Q 2 is small. Since f C is insensitive to large variations in M A one may naively surmise that the error in f C is small.
However, we find that the difference between the values f C calculated with and without transverse enhancement is larger than the error estimate extracted from the uncertainty in M A . This is because , which depends on the magnitude of the transverse enhancement at small Q 2 . Fig. 15 shows the errors in f C from the uncertainty in the TE parameters. The error originating from uncer-tainties in the TE parameters is also very small (less than 0.005).
We obtain a more conservative estimate of the systematic error in f C originating from uncertainties in the modeling the QE cross section by taking the difference between f C calculated with and without transverse enhancement. At the lowest energy of 0.7 GeV, this difference is -0.05 for ν µ . Since at 0.7 GeV f ν C ≈ 1.3 this corresponds to a maximum error in the determination of the ν µ flux of 3.8%.
Forν µ the difference between f C calculated with and without transverse enhancement at an energy of 1.0 GeV is +0.03. Since at 1.0 GeV fν C ≈ 0.6 this corresponds a maximum error in the determination of theν µ flux of 5%.

Comparison to GENIE andf C:ν<0.25 (15.1 GeV)
We have used a sample of events generated by the GENIE Monte Carlo. Our studies are done at the generated level and therefore do not depend on the detector parameters or energy resolutions of any specific experiment.
We extract the energy dependence of the ν < 0.25 GeV cross section from the GENIE MC sample using the following expression: where the superscript M C refers to events generated by the GENIE Monte Carlo.
Here, N M C (E) is the number of events generated by the Monte Carlo with neutrino energy E, and N M C ν<0.25 (E) is the subset of these events with ν < 0.25 GeV.
As mentioned earlier, we propose that the neutrino cross sections at low energy be measured relative to the neutrino cross section at 15.1 GeV. For any cross section model we can define the normalized quantityf C:ν<0. 25 Fig. 16. The top panel shows the comparison for neutrinos and the bottom panel shows the comparison for antineutrinos. Our calculation for the TE model is shown in black. Our calculation assuming no transverse enhancement and M A = 1.014 GeV is shown in red. As mentioned earlier, the values for M A = 1.014 GeV (red line) and M A = 1.3 GeV (blue line) are very close to each other. The GENIE prediction (which has no transverse enhancement and uses M A = 0.99 GeV) is close to the red curve as expected. The GENIE predictions include a contribution from coherent pion production. As shown in Appendix II, for the ν < 0.25 GeV sample, the contribution from coherent pion production is less than 0.1% for neutrinos and less than 0.6% for antineutrinos.

Conclusions of the studies with ν <0.25 GeV
In conclusion, we find that the method works very well for ν < 0.25 GeV. If one takes the average of all the models, a conservative upper limit of the model uncertainty in the relative flux extracted from the ν < 0.25 GeV sample is 1.9% for ν µ energies above 0.7 GeV and 2.5% forν µ energies above 1.0 GeV. The GENIE Monte Carlo is in reasonable agreement with the models and therefore can be used to obtain a first order neutrino flux.
A study of the Q 2 distributions of QE events in MIN-ERvA can be used to constrain the Q 2 dependence of the QE differential cross sections and thus reduce the model dependence in the determination of the relative flux to a negligible level. A GENIE Monte Carlo which is tuned to agree with the new data can be used to extend the technique to lower energies. 5 Using "low-ν" events with ν < 0.5 GeV The ν < 0.5 GeV ν µ andν µ samples have close to twice the number of events as the ν < 0.25 GeV samples. These samples for scattering are also dominated by QE events, but include a significant fraction (about 1/3) of events in which a single pion is produced in the final state. As seen in Fig. 3, the ν < 0.5 GeV samples are composed of QE events with Q 2 < 0.9 GeV 2 , and ∆(1232) events with Q 2 < 0.3 GeV 2 . Fig. 17 shows the ν < 0.5 GeV partial charged current cross sections as a function of energy. The partial cross sections extracted from the GENIE Monte Carlo are shown as black points with MC statistical errors. The ν < 0.5 GeV partial cross section for ν µ scattering is shown on the top panel, and the ν < 0.5 GeV partial cross section forν µ scattering is shown on the bottom panel. The QE contribution to the ν < 0.5 GeV partial cross section is shown in red, and the contribution from pion production processes (∆, inelastic and coherent pion production) is shown in blue.
As seen in Fig. 17, the pion production contribution to the ν < 0.5 GeV partial cross section is relatively constant with energy, while the QE contribution has some energy dependence. Therefore, the energy dependence of the sum of the two contributions to the ν < 0.5 GeV partial cross

GENIE Total GENIE QE GENIE Non-QE (Resonance, Coherent and DIS)
<0.5) Components for Antineutrino on Carbon ν ( σ Fig. 17. The ν < 0.5 GeV partial charged current cross section as a function of energy from the GENIE Monte Carlo. The QE contribution is shown in red, the contribution from pion production process (∆, inelastic and coherent pion production) is shown in blue, and the total ν < 0.5 GeV GeV partial cross section is shown in black. The ν < 0.5 GeV GeV partial cross section for νµ is shown in the top panel, and the ν < 0.5 GeV partial cross section forνµ is shown in the bottom panel. (color online).
section requires modeling of the relative magnitude of QE and pion production processes (specifically at low Q 2 ). As shown in Fig. 18 and 19, the consistency among the experimental measurements of pion production cross sections in the region of the ∆(1232) resonance is about 20% (depending on the neutrino energy and the nuclear target). We use this variation to get an estimate of the model uncertainty in the determination of the neutrino flux from the ν < 0.5 GeV samples. This uncertainty can be greatly reduced when more precise measurements of the QE and pion production cross sections become available (e.g. from MINERvA).
We define the cross section for ν µ P → µ − ∆ ++ as the integrated cross section for W < 1.4 GeV for the following single final state: We define the cross section forν µ N → µ + ∆ − as the integrated cross section for W < 1.4 GeV for the following single final state:ν µ N → µ + N π − Therefore, our definition includes the sum of the contributions of the resonant cross section and the non-resonant continuum.
The structure functions (form factors) for the reactions ν µ P → µ − ∆ ++ andν µ N → µ + ∆ − defined above are the same (except that for antineutrinos the structure function W 3 changes sign). It has been experimentally determined [32] that ν µ P cross section for W < 1.4 GeV is dominated by the resonant ∆ ++ production process. Similarly, the W < 1.4 GeV cross section forν µ N is dominated by the resonant ∆ − production process.
As discussed in the Appendix, we parametrize the ∆ ++ and ∆ − production cross sections in terms of form factors as given by Paschos and Lalakulich [10], with the form factors of Paschos and Schalla [10]. In order to obtain predic- tions for the W < 1.4 GeV region, we divide all theoretical ∆ production cross sections by a factor of 1.2 (because 20% of the resonant cross section is above W = 1.4 GeV). We vary two of the parameters in the model, specifically M ∆ A and C A 5 to obtain a band that span the experimental data. We extract M ∆ A from the measured Q 2 distributions and use C A 5 to set the overall normalization. The top panel in Fig. 18 shows a summary of cross section measurements for ν µ P → µ − ∆ ++ on free nucleons (hydrogen or deuterium targets.) Shown are bubble chamber measurements at low energy from Argonne (ANL73 [25], ANL79 [26], ANL82 [27]) and measurement at low energy from Brookhaven (BNL86 [28]). Also shown are measurements at higher energies from the Fermilab bubble chamber (FNAL78 [29], FNAL81 [30]) and high energy data from CERN (BEBC80 [31], BEBC80 [32]). The bottom panel in Fig. 18 shows the BEBC90 [32] cross section measurements forν µ N → µ + ∆ − on free nucleons (deu- The top panel in Fig. 19 shows a summary of cross section measurements for ν µ P → µ − ∆ ++ data on nuclear targets. Shown are the measurements of Gargamelle78 [33] (Propane), SKAT88 [34] (Freon), and SKAT89 [35] (Freon). The bottom panel shows measurements ofν µ N → µ + ∆ − cross sections on nuclear targets from Gargamelle78 [36] (Propane) and SKAT89 [35] (Freon).
Aside from Pauli suppression and final state interaction, the structure functions (form factors) for the processes in Fig. 18 and 19 are the same. The black (Paschos-2011), red (FIT-A1) and blue (FIT-A2) curves shown in Fig. 18 and 19 use the free nucleon form factors (but include the Pauli suppression for the case of nuclear targets). The calculations do not include the effect of final state interaction for the nuclear targets. The three curves (Paschos-2011, FIT-A1 and FIT-A2) conservatively span all the available ∆ ++ and ∆ − production cross sections on hydrogen, deuterium and nuclear targets, as shown in Fig. 18 and 19. The cross sections for the production of ∆ ++ and ∆ − on nuclear targets predicted by GENIE are near the upper bound of our three parameterizations. Additional details are given in the Appendix.
We define the cross section for ν µ N → µ − ∆ + as the sum of the integrated cross sections for W < 1.4 GeV for the following two final states: We define the cross section forν µ P → µ + ∆ 0 as the sum of the integrated cross sections for W < 1.4 GeV for the following two final states: Therefore, our definition includes the sum of the contributions of the resonant cross section and non-resonant continuum.
The structure functions (form factors) for the reactions ν µ N → µ − ∆ + andν µ P → µ + ∆ 0 defined above are the same (except that for antineutrinos the structure function W 3 changes sign). Because of Clebsch-Gordan coefficients [10] the resonant cross section for ∆ + production in ν µ N collisions is a third of the resonant cross section for ∆ ++ production in ν µ P collisions. Similarly, the resonant cross section for ∆ 0 production inν µ P collisions is a third of the cross section for resonant production of ∆ − inν µ N collisions.
However, unlike the case for ν µ P (∆ ++ ) andν µ N (∆ − ), where the cross sections are dominated by the resonant process, there is a significant contribution from the non-resonant continuum to the W < 1.4 GeV cross section in ν µ N andν µ P collisions.
The top panel of Fig. 20 shows the ν µ N → µ − ∆ + cross sections (W < 1.4 GeV) measured on free nucleons (deuterium). Shown are measurements from ANL79 [26], ANL82 [27], and BEBC90 [32]. The predictions from the GENIE MC are shown as black points with MC statistical errors. In order to describe the data (which has a large  The cross sections on nuclear targets are expected to be somewhat lower than the cross sections on free nucleons (which are shown in Fig. 20). Here, FIT-B includes the effect of Pauli suppression (but not final state interaction). The GENIE MC cross sections for the production of ∆ + and ∆ 0 on nuclear targets are lower than the fit. Additional details are given in the Appendix.

Comparisons of W < 1.4 GeV cross sections on carbon
A more relevant comparison is to determine how well the GENIE Monte Carlo describes the sum of the proton and neutron cross sections on carbon, since it is the total number of ν < 0.5 GeV events on carbon that are used in the determination of the neutrino flux. Fig. 22 shows the predictions from the GENIE MC for total ∆ production cross section for W < 1.4 GeV on carbon (per nucleon). The neutrino cross sections for ν µ C → µ − (∆ ++ or ∆ + ) are shown in the top panel, and the antineutrino cross sectionsν µ C → (µ + ∆ 0 or ∆ − ) are shown in the bottom panel. The cross sections which are predicted by GENIE are compared to our three parameterizations. (Paschos-2011, FIT-A1 and FIT-A2 for ∆ ++ and ∆ − , and FIT-B for ∆ + and ∆ 0 ). The GENIE cross section predictions for the total ∆ production cross sections on carbon (which use the Rein and Seghal model [37]  for resonance production) fall near the lower bound of our three parameterizations of the experimental data.
As described below, the uncertainties in the measurements of the ∆ production cross sections do not place a serious limitation on the flux extractions using the low ν method.
The top panel of Fig. 23 shows the total correction factor f C (E) for the ν < 0.5 GeV sample (defined as f C:ν<0.5 (E)) for neutrino running. Also shown are the various contributions to f C:ν<0.5 (E) including the kinematic correction to W 2 (f 2 ), and the contributions from , W 4 (f 4 ), and W 5 (f 5 ). The bottom panel shows the fractional contribution of ν < 0.5 GeV events to the charged current neutrino total cross section. Using our nominal model (TE model for QE scattering and the Paschos 2011 model for ∆ production) we find that the fraction of ν < 0.5 GeV events is less than 60% for ν µ energies above 1.2 GeV.
The top panel of Fig. 24 shows the total correction factor f C:ν<0.5 (E) for antineutrino running. Also shown are the various contributions to f C:ν<0.5 including the kinematic correction to W 2 (f 2 ), and the contributions from , W 4 (f 4 ), and W 5 (f 5 ). The bottom panel shows the fractional contribution of ν < 0.5 GeV events to the charged current antineutrino total cross section. Using our nominal model (TE model for QE scattering and the Paschos 2011 model for ∆ production) we find that the fraction of ν < 0.5 GeV events is less than 60% forν µ energies above 2 GeV.
As for the ν < 0.25 sample, we propose that the neutrino and antineutrino cross sections at low energy be measured relative to the cross sections at 15.1 GeV. Therefore, we define normalized quantityf C:ν<0.5 (15.1)(E) for the ν < 0.5 sample as: which is equivalent tō Here, f C:ν<0.5 (E=15.1 GeV)=1.0113 (for ν) and 0.9507 (forν). These values can be used to convert betweenf C:ν<0.5 (E) and f C:ν<0.5 (E). Fig. 25 shows comparisons of our calculated values of the normalizedf C:ν<0.5 (15.1)(E) (shown as the solid black line) to values extracted from the GENIE MC. The GE-NIE predictions include a contribution from coherent pion production. As shown in Appendix II, for the ν < 0.50 GeV sample, the contribution from coherent pion production is less than 0.7% for neutrinos and less than 3% for antineutrino Our values are calculated from our nominal model which uses the TE model for QE scattering and the Paschos 2011 model for ∆ production. Neutrinos are shown on the top panel and antineutrinos are shown on the bottom panel (color online). Figure 26 shows the error band in the correction factor f C:ν<0.5 (15.1)(E) for neutrinos (top panel) and antineutrinos (bottom panel). The error band is defined as the differences between our nominal model and other model assumptions. For neutrinos with energies greater than 1.2 GeV, the error inf C (15.1) is less than 0.03, which corresponds to a 2.6% upper limit on the model uncertainty in the neutrino flux extracted from the ν < 0.5 GeV sample. For antineutrinos with energies greater than 2 GeV the error inf C (15.1) is less than 0.01 (which corresponds to a 1.4% upper limit on the model uncertainty in the antineutrino flux extracted from the ν < 0.5 GeV sample).
In order to go to lower neutrino and antineutrino energies we need to use the ν < 0.25 GeV sample. The model uncertainty in the relative flux extracted from the ν < 0.25 GeV sample is 1.9% for ν µ energies above 0.7 GeV and 2.5% forν µ energies above 1.0 GeV. With improved determination of QE and ∆ production cross sections (e.g. in MINERvA), the model uncertainties can be further reduced, and the method may be extended to lower energies.

Resolution, acceptance and radiative corrections
The ν < 0.25 GeV events are primarily QE events with Q 2 < 2M × 0.25 ≈ 0.45 GeV 2 . We can select either all events with ν < 0.25 GeV or only QE events with Q 2 < 0.5 GeV The ratio of the number of reconstructed events with ν < 0.25 GeV (or Q 2 < 0.5 GeV 2 ) in data and MC as a function of energy is proportional to the ratio of the true flux to the simulated flux in the MC. This ratio provides a measure of the relative neutrino flux as a function of energy. A complete Monte Carlo should include the small contributions from coherent pion production (shown in Appendix II), strange particle production such as QE production of hyperons [38], and radiative corrections [39,40]. The effects of experimental resolution and acceptance should also be simulated.
At present the GENIE Monte Carlo includes coherent pion production, but does not include the QE production of hyperons, nor radiative effects.
If the GENIE Monte Carlo is used, then one may wish to weight the rate of QE events (as a function of Q 2 ) by the ratio of events expected in the TE model to the number of events predicted by the model which is implemented in GENIE (i.e. the "Independent Nucleon" model with M A = 0.99 GeV). In addition, QE production of hyperons and radiative effects need to be added.

Conclusions
We find that the model uncertainties in using the "low-ν" event samples with ν < 0.25 and ν < 0.5 GeV are well under control (less than 3%). Therefore, the "low-ν" technique can be used at low energies (0.7 GeV for neutrinos and 1 GeV for antineutrinos). Once data from MINErVA on QE scattering and resonance production becomes available, the model uncertainties can be made even smaller, and the technique may be extended to even lower energies.
Since the model uncertainties are under control, the dominant systematic error originates from how well the detector response is understood, Specifically, the mis-reconstruction of high ν events as "low-ν" events must be modeled reliably. This is because at high energies (as shown in Fig. 2) mis-reconstruction of the hadron energy of high ν events can increase the number of "low-ν" events, while at low energies there are fewer high ν events that can be mis-reconstructed at low ν.
The dominant uncertainty in the method comes from the calibration and resolution smearing in the measurement of the hadronic energy. This was the dominant error when this method was used in MINOS because of the poor resolution of the MINOS target calorimeter at low hadronic energy.
As mentioned in the introduction, the standard method for the determination of the neutrino flux requires the modeling of pion production as well as the complicated magnetic focusing elements. The determination of the flux for the Fermilab NUMI beam with the standard method is limited at present by the uncertainties in pion production cross sections. The resulting error in the flux is about 5% at low energies (1-2 GeV) and 10%-15% at the higher energies (10-20 GeV). Therefore, having the "low ν" method which yields the relative neutrino flux as a function of energy very useful. In principle, the uncertainties in the standard method can be improved with better measurements of pion and kaon production cross sections. Plans for such future measurements at the CERN Laboratory are currently under discussion.
A second method, which requires the measurement of the muon rate downstream of the decay pipe, can not determine the energy dependence of the flux. It mostly constrains the overall level of the flux. At present, the uncertainties in the overall calibration of the muon chambers yields an uncertainty in the flux of about 10%. . A third method uses inverse muon decay ν µ + e → µ − +ν e events in the detector. The threshold for this reaction is about 12 GeV. Therefore, this method can only be used at higher energies. Inverse muon decay was used by NOMAD to constrain their neutrino flux at high energies. In addition to being statistically limited, the final state energy of inverse muon decay events is not fully measured since there is a neutrino in the final state. This places a limitation on the determination of the energy dependence of the neutrino fluxes. This method cannot be used for the determination of the flux for antineutrinos.
A fourth method uses the neutral current reaction ν µ + e → ν µ + e. In addition to being statistically limited, the final state energy in ν µ + e → ν µ + e events is not fully measured since there is a neutrino in the final state. This places a limitation on the determination of the energy dependence of the fluxes. In this method only the sum of the fluxes for neutrinos and antineutrinos is measured because calorimetric detectors such as MINERvA cannot determine the charge of final state electrons. 8 Appendix I: ∆ production cross sections 8

.1 ∆ production form factors
For the vector contribution we use the formulae for the structure functions W 1 , W 2 , W 3 , W 4 , and W 5 on free nucleons from Lalakulich and Paschos [10]. We neglect the effect of Fermi motion. The form factors that we use are taken from Paschos and Schalla [10]. Specifically, the vector form factors are with M V = 0.84 GeV, which have been extracted from electroproduction data.
For the vector-axial interference W 3 (Q 2 , ν) Paschos and Schalla use the form factor C A 5 (Q 2 ) where Here, we define C A 5 = C A 5 (0) Paschos and Schalla use low energy π + p → ∆ ++ where the non-resonant background is smallest. With M A =1.05 GeV they extract value of C A 5 (0) = 1.08 from the data. Since this value is close to 1.20 predicted by the Goldberger-Treiman relation, they chose to use C A 5 =1.2.
Paschos and Schalla mention that several recent articles also calculate C A 5 (0) by fitting experimental data [41,42,43,44,45,46,47] with values varying from 0.87 up to 1.20. Models with a resonant background [41,42] prefer the power value, while the other articles [43,44,45,46,47] prefer values closer to 1.20. The reasons for the differences is the treatment of the non-resonant background, the form of the axial form factor that is used, and the exact kinematics at small Q 2 .
For ∆ ++ and ∆ − we define the Pachos-2011 parameterization using above form factors with C A 5 =1.2 (extracted through PCAC), M A =1.05 GeV, and the vector form factors described above. As mentioned earlier, FIT-A1 and FIT-A2 use the same form but with different values of C A 5 and M A . For ∆ + and ∆ 0 production our Fit-B uses the same form factors multiplied by a factor of 1/ (3) (as expected from Clebsch-Gordan coefficients [10]). However, in order to account for the large non-resonance background, we use different values C A 5 and M A .

Various parameterizations
The low energy and high energy data for neutrino and antineutrino production of the ∆(1232) resonance are not entirely consistent. Therefore, we use range of parameterization to span the systematic error in our modeling of ∆ production cross sections. The form factors for ν µ P → µ − ∆ ++ andν µ N → µ + ∆ − should be the same. The dσ/dQ 2 differential cross sections (W < 1.4 GeV) for ν µ P → µ − ∆ ++ measured at high energies are shown in the top panel of Fig. 27  2) is from fits to lower energy ν µ P → µ − ∆ ++ data (BNL and Argonne). The red curve labeled FIT-A1 (M A =1.93, C A 5 = 0.62) is a fit to the BEBC90ν µ N → µ + ∆ − data . The blue curve labeled FIT-A2 (M A =1.75, C A 5 = 0.49) is a fit to the BEBC90 ν µ P → µ − ∆ ++ data. The variation among the three curves is taken as a systematic error.
The bottom panel of Fig. 28 shows values of dσ/dQ 2 differential cross sections for ν µ N → µ − ∆ + (for W < 1.4 GeV) measured by BEBC90 on free nucleons on deuterium. This reaction has different form factors then ν µ P → µ − ∆ ++ . The green curve labeled FIT-B (M A =1.62, C A 5 = 1.27) represents a fit to the BEBC90 ν µ N data.
We use the above models with the addition of Pauli suppression in order to model the differential cross sections on nuclear targets. Fig. 29 shows our three ν µ P/ν µ N dσ/dQ 2 cross sections models (for W < 1.4 GeV) with Pauli suppression for nuclear targets at an energy of 40.5 GeV. The cross sections for for ν µ P → µ − ∆ ++ are shown on the top panel and the cross sections forν µ N → µ + ∆ − are shown  on the bottom panel. These two reactions should be described by the same form factors. The black curve labeled Paschos-2011(M ∆ A =1.05, C A 5 = 1.2) is from fits to lower energy ν µ P free nucleon data (BNL and Argonne). The red curve labeled FIT-A1 (M A =1.93, C A 5 = 0.62) is from a fit to the BEBC90ν µ N free nucleon data. The blue curve labeled FIT-A2 (M A =1.75, C A 5 = 0.49) is from a fit to the BEBC90 ν µ P free nucleon data. The variation among the three curves is taken as a systematic error.    For the ν < 0.25 GeV sample, the contribution from coherent pion production is less than 0.1% for neutrinos and less than 0.6% for antineutrinos Fig. 32 shows the fraction of events from coherent pion production in the ν < 0.50 GeV event sample (calculated with GENIE) as a function of neutrino energy. Neutrinos are shown on the top panel and antineutrinos are shown on the bottom panel. For the ν < 0.50 GeV sample, the contribution from coherent pion production is less than 0.7% for neutrinos and less than 3% for antineutrinos