Neutral currents and tests of three-neutrino unitarity in long-baseline experiments

We examine a strategy for using neutral current measurements in long-baseline neutrino oscillation experiments to put limits on the existence of more than three light, active neutrinos. We determine the relative contributions of statistics, cross section uncertainties, event misidentification and other systematic errors to the overall uncertainty of these measurements. As specific case studies, we make simulations of beams and detectors that are like the K2K, T2K, and MINOS experiments. We find that the neutral current cross section uncertainty and contamination of the neutral current signal by charge current events allow a sensitivity for determining the presence of sterile neutinos at the 0.10--0.15 level in probablility.


Introduction
In recent years a series of exciting experimental results have shown that neutrinos have finite masses and mixings. For a recent review of the status see Ref. [1]. Solar neutrino and atmospheric neutrino results indicate that all three known neutrino flavors (e, µ, τ ) participate in neutrino mixing, and hence neutrino oscillations. Consequently, the standard framework to describe the experimental results and analyse neutrino oscillation data is that of three-flavor mixing in which the three flavor eigenstates are related to three mass eigenstates by a 3 × 3 mixing matrix [2]. The positive signal forν µ →ν e oscillations from the LSND experiment [3] challenges the three-flavor mixing paradigm [4]. However, the neutrino oscillation interpretation of the LSND observations is yet to be confirmed. Independent of whether or not the LSND results are confirmed by MiniBooNE [5], the three-flavor mixing framework deserves further experimental scrutiny in the coming years. Much of the focus on future experiments so far has been directed to the determination of the 3-mixing angles and the CP-violating phase with long-baseline or oscillation experiments [6] and reactor experiments [7]. Of interest in this paper is the measurement of the neutral current, which could allow tests of the unitarity of the 3×3 mixing matrix and thus indirectly probe the existence of sterile neutrinos.
In a three-flavor neutrino model, the sum of the oscillation probabilities y=e,µ,τ P (ν x → ν y ) is unity. If there are more than three light neutrinos, we know from measurements of the invisible width of the Z [8] that the additional neutrinos must be sterile. If additional light neutrinos mix with the three known flavors we can expect a non-zero oscillation probability to sterile neutrinos, P (ν x → ν s ) = 0. To test the three-flavor neutrino-mixing paradigm it is important to search for a sterile neutrino component within the neutrino flux from natural and manmade sources. Since sterile neutrinos have no strong or electroweak interactions, they cannot be detected directly. However, neutral current (NC) measurements allow y=e,µ,τ P (ν x → ν y ) to be determined which, by probability conservation, is equal to 1 − P (ν x → ν s ).
Therefore, in principle a NC measurement alone is sufficient to determine P (ν x → ν s ).
However, in a realistic detector misidentifications of CC and NC events, together with systematic uncertainties on the relevent neutrino interaction cross sections, compli-cate the analysis.
In this paper we study the use of NC measurements to determine limits on the sterile neutrino content in long-baseline neutrino oscillation experiments. First we consider the sensitivity to the sterile content that might be obtained in a K2K-like [9], T2K-like [10], and MINOS-like[11] experiment with a "perfect" detector and "perfect" beam if there are no systematic uncertainties. We then consider the impact on the sensitivity of event misidentification and systematic uncertainties. Our study is based on a simple simulation of the long-baseline neutrino beams, neutrino interactions [12], and detector responses. We present our results versus event rates and the size of the cross section uncertainty in order to show the dependence on these quantities.
2 Using NC data to determine sterile content

Formalism
The present and proposed long-baseline neutrino oscillation experiments exploit conventional neutrino beams that are produced by the decays of charged pions in a long channel. This produces a beam which is initially almost entirely ν µ . Kaons and muons decaying in the channel introduce a small (typically ∼ 1%) ν e component in the neutrino beam. As the neutrino beam travels towards a distant detector its flavor content will evolve. In our analysis we will consider three active neutrinos (ν e , ν µ , ν τ ) and one sterile neutrino (ν s ), with oscillation probabilities P (ν µ → ν x ) ≡ P µx , x = e, µ, τ, s. We begin by considering an oscillation experiment that has an initially pure ν µ beam with well known neutrino spectrum and flux, and a detector with perfect identification of the produced events. Then the event rates at the far detector will be where N 0 µ is the predicted number of ν µ CC interactions in the detector in the absence of oscillations and the σ x denote the interaction cross sections (σ e , σ µ , σ τ ) for (ν e , ν µ , ν τ ) CC interactions and σ NC for ν x NC interactions.
In an ideal experiment N NC determines P µs and N CC determines P µµ . In practice the presence of ν e and ν τ CC events complicates the analysis if these events are not distinguished from NC events. In that circumstance the NC events provide a measure of 1 − P µs + ǫ e P µe + ǫ τ P µτ , where the factors ǫ e and ǫ τ reflect the contaminations.
Probability conservation (P µe + P µµ + P µτ = 1) can be used to eliminate P µτ or P µe , but not both. If the beam energy is below the threshold for τ production or the probability P µe is small and can be neglected, then P µs can still be determined. However, for a realistic detector with particle misidentifications and/or a ν e component at the far detector that cannot be neglected, the problem of determining P µs can be complex but still solvable, as we shall discuss.
In general, let the probability that an event of type x (NC, ν e CC, ν µ CC, or ν τ CC) be identified in the detector as an event of type y be given by ζ xy , where x, y = (NC, e, µ, τ ) (note that ζ xx is the efficiency for detecting an event of type x). If N 0 µ is the predicted number of ν µ CC interactions in the detector in the absence of oscillations, then after including oscillations, detector efficiencies and misidentifications, and integrating over the energy dependence, the number of measured events of type y will be: where the interaction cross sections for ν e CC, ν µ CC, ν τ CC, and NC events are given by σ e , σ µ , σ τ and σ N C , respectively.

Ignore ν e 's
We consider first the situation in which the ν e component in the beam at the far detector is so small that ν e CC interactions can be neglected. In this case we let P µe → 0 (it is known to be small, at the 5% level or less from the CHOOZ experiment [13]).
It is convenient to define the following two ratios, where and f x,y ≡ ζ xy σ x /ζ yy σ y is a normalized misidentification factor that gives the fraction of events identified as being of type y that are really events of type x. Measuring R N C and R µ is sufficient for deducing P µs (and P µµ ). The analysis depends on whether or not we are above the ν τ CC interaction threshold, i.e., whether or not there are ν τ CC events produced in the detector.

Below τ threshold
For neutrino energies below the τ threshold σ τ = 0 and f τ,j = 0. In this case we can invert Eqs. 6 and 7 to obtain Adding uncertainties in quadrature we get where in the limit of Gaussian statistical uncertainties and The first term in each δR j is the usual statistical uncertainty, the second comes from the normalization uncertainty (flux and cross section).
Note that the normalized mis-identification factors f µ,N C and f N C,µ will be sensitive to the neutrino energy spectrum and the detector technology, and therefore must be evaluated for each experimental setup. Most of the mis-identification terms are we can ignore all terms of order f 2 , (see Table 1) we have which measures the significance of the deviation of P µs from zero.
For a perfect detector that can identify each event correctly, f x,y = δ xy . In this This ratio depends only on P µs , the experimental statistics, and the systematic uncertainty on the NC measurement. Thus, Eq. 17 defines the maximum sensitivity that is in principle achievable for a given N 0 N C and ǫ N C .

Above τ threshold
If the neutrino energy is above the τ threshold and there is not a clean signature for ν τ CC events, we can still deduce P µµ and P µs by using the identity P µµ + P µτ + P µs = 1 to eliminate P µτ in Eqs. 6 and 7 (we are still assuming P µe = 0), which gives If no other process contaminates the ν µ CC events (i.e., f j,µ = 0 as appears to be the case for a MINOS-like experiment; see Sec. 3), then For a perfect detector, P µs /δP µs is again given by Eq. 17.

Do not ignore ν e 's
If the ν e CC interaction rate in the far detector is not negligible (which could be the case if sin 2 2θ 13 is near its upper bound and we want to push the uncertainty in the measurement of P µs down to the few per cent level), then we need three measurements to be able to solve for all of the probabilities. The potential measurables are and, if we are above the ν τ CC interaction threshold,

Below τ threshold
Below the ν τ CC threshold energy the three measurements must be R µ , R N C , and R e . Then f τ,j = 0, the P µτ terms drop out, and we can invert Eqs. 21-23 to obtain where calculation of the δP 's is straightforward; each R term has a statistical and systematic uncertainty given by Eq. 14.
Note that for an idealized detector in which no other processes significantly contaminate ν e CC events (i.e., f j,e ≃ 0) and ν e CC events do not contaminate ν µ CC events (i.e., f e,µ ≃ 0), then P µe = R e . Since P µe is small (of order 0.1 or less, as indicated by current oscillation limits), eliminating terms of order f 2 and f P µe in this case will recover the situation where we ignored ν e (i.e., Eq. 16).

Above τ threshold, no τ measurement
For energies above the ν τ CC interaction threshold the P µτ terms do not drop out of Eqs. 21-23. If we do not have the means to measure ν τ CC events but can measure ν e CC events, then we can use probability conservation to eliminate P µτ , giving The general solution for the probabilities is somewhat messy, but if we assume that no other processes contaminate the ν µ CC signal (i.e., f j,µ ≃ 0) and the ν e CC events do not contaminate the other signals (f e,j ≃ 0), (see Sec. 3), then P µµ = R µ and we can invert Eqs. 28 and 30 to obtain The calculation of δP µs is straightforward.
If no other processes contaminate the ν e CC signal (i.e., f j,e → 0), then P µe = R e and we obtain

Above τ threshold with a τ measurement
If R τ is also measured, in addition to R e , then there are four measurements (R µ , R N C , R e and R τ ), but there are only three independent quantities (since P µµ + P µe + P µτ + P µs = 1). One possible approach would be to assume that P µs is independent of the other probabilities and use these four measurements to test probability conservation.
We do not pursue this option here. Instead, we use probability conservation to eliminate one of the probabilities and use three of the four measurements to determine P µs (the fourth measurement could be used to check probability conservation afterwards).
Since P (ν µ → ν τ ) is most likely much larger than P (ν µ → ν e ) in the L/E regime we are considering, we use R τ as the third measurement (along with R µ and R N C ).
Then the appropriate formulas for the measurables R N C , R µ and R τ can be found by the interchange τ ↔ e in Eqs. 28-30.

Detector simulations
We wish to explore how well in principle a neutrino three-flavor unitarity test can be performed with a given muon-neutrino beam as a function of dataset size, and study which systematic uncertainties are likely to be important, and their impact.
We consider first a "perfect" experiment in which the sensitivity of the unitarity test is determined only by the statistical uncertainties, calculated using a parameterization of the known beam flux and spectrum, together with a simulation of neutrino interactions in the detector. An event simulation is used to determine the relevent detection efficiencies and misidentification factors. We use the NEUGEN Monte Carlo code [12] to simulate neutrino interactions in the detector. Events are classified as ν µ CC, ν e CC, or NC. In practice the requirements used to identify events of a given type will depend upon the detector technology. For example, for a water cherenkov detector in our simple analysis we will define a ν e CC event candidate as an event with an electron candidate above threshold. An electron candidate is either a real electron or a π 0 with an energy exceeding 1 GeV (in which case the two daughter photons from the high energy π 0 produce cherenkov rings that overlap in the detector and cannot be distinguished from a single electromagnetically showering particle).
A NC event candidate would be an event containing a π 0 candidate but no muon candidate, where a π 0 candidate has two e-like rings above threshold (which come from a π 0 with energy less than 1 GeV). The definition of CC and NC events can of course be varied, and then tuned to give favorable values for the signal efficiencies and mis-identification factors. Examples are shown in Table 1.

K2K-like and T2K-like Experiments
To identify the most important systematic uncertainties it is useful to compare the sensitivity of our "perfect experiment" with that of a realistic experiment. We begin with the K2K experiment. K2K uses a beam from the KEK laboratory in Japan. The neutrinos in the KEK beam have a mean energy of 1.3 GeV [9], and the neutrinos travel 250 km to the Super-K water Cerenkov detector. A new experiment T2K is being planned that will exploit a more intense neutrino source that is presently under construction at Tokai, Japan. T2K will also use the Super-K detector, but with a slightly longer baseline (300 km) and narrow-band beam with an axis displaced slightly from pointing directly at the far detector (an "off-axis" beam will depend upon the sizes of these systematics. We use the following parameterization of a Super-K-like detector response: (a) A threshold of 197 MeV/c for the detection and measurement of muons [14], and 100 MeV/c for electrons and π 0 's. These thresholds approximate those used for the atmospheric neutrino analysis of Super-K [15,14].
In addition, we use a parametrization of the spectra for the K2K and T2K neutrino beams.
In our analysis we will use only simulated events with visible energy greater than 0.1 GeV. For our "basic" signals we define a ν µ CC event candidate as an event with a single muon-like ring, a ν e CC event candidate as an event with a single e-like ring, and a NC event candidate as an event with two e-like rings, which are assumed to be two  photons from a single π 0 decay. Given these definitions, the detector efficiencies and mis-identification factors determined from our simulations are listed in Table 1. As shown in the table, the efficiencies ζ jj are of order one-half, and there is no significant contamination of one signal by another due to mis-identification. Also shown are the results of a more aggressive signal definition, where a simulated event with an odd number of e-like rings is labeled as a ν e CC event candidate, and the remaining events (those with an even number of e-like rings) are labeled as ν µ CC event candidates if they have one or more µ-like rings or NC if they do not. In this more aggressive scenario no events are discarded, i.e., all events were used for one of the targeted signals. Although some of the misidentification factors are slightly larger for the aggressive scenario, overall they are not greatly changed, while there is a significant improvement in the efficiencies for the CC events.
To investigate whether our analysis is sensitive to the assumed details of the neutrino spectrum we have repeated the calculation of efficiencies and misidentification factors for a K2K-like experiment with a beam that has the same average energy and beam spread as the K2K beam, but with a Gaussian energy spectrum (no long highenergy tail). For the Gaussian beam, the misidentification factors involving ν e were greatly reduced (since backgrounds from the high-energy tail are now suppressed), but f µ,N C and f N C,µ were only slightly affected. Since f µ,N C is the dominant f factor for a K2K-like experiment, we conclude that our results are not very sensitive to the detailed beam spectrum we assume.
We now consider a T2K-like experiment, where we have used a beam spectrum that corresponds to a detector 2 degrees off-axis. The resulting mis-identification factors for a T2K-like experiment are shown in Table 1. All of the misidentification factors are reduced except for f µ,N C , which is now 0.25. Therefore, in both the K2Klike and T2K-like experiments, the most important contamination is ν µ CC events being mis-identified as NC events.

A MINOS-like experiment
The MINOS experiment is a long-baseline oscillation exeriment that will use a neutrino beam from the Fermilab Main Injector and an iron-scintillator sampling calorimeter 730 km away in Minnesota. MINOS is expected to begin data taking early in 2005.
With a beam energy that is about a factor of three higher than the KEK beam, and a detector that is very different from the water cerenkov detector used by K2K and T2K, the efficiencies and misidentification factors for MINOS will be very different than those for the experiments in Japan. To compute the numbers given in Table 1 we have once again used a parametrization of the neutrino beam spectrum, the NEU-GEN Monte Carlo Program to simulate neutrino interactions in an iron detector, and a simple parametrization of the response of a MINOS-like detector. In particular we assume: (a) An energy threshold of 50 MeV for the detection and measurement of electrons, and charged and neutral pions, and a threshold of 1 GeV for the identification and measurement of muons. Note that the MINOS detector is expected to be able to determine the charge and measure the momenta of muons from 0.5 GeV/c to 100 GeV/c, and to distinguish ν µ CC events from NC events if the muons have momenta exceeding about 1 GeV/c [16].
(b) Energy resolutions given by for electrons and π 0 's, for charged pions, and ∆p rms p = 0.05 , for muons. Note that in practice the muon energy resolution for the MINOS experiment is expected to be somewhat better (worse) than described by Eq. 37 if the muon ranges out (does not range out) in the detector. We found that ∆p rms /p values as high as 0.10 do not appreciably change our results.
As shown in the table, for a MINOS-like experiment there is a very large contamination of the NC channel by ν µ CC events, and mis-identification of ν τ CC events as NC events is also significant. The efficiency for identifying NC events is about one-half, similar to the K2K-like and T2K-like experiments 1 .
1 Although to first order for MINOS all events with an electron or photon candidate will be classified as NC events, there are three independent probabilities, and it is necessary to extract a separate ν e signal, in addition to ν µ and NC signals, to be able to solve for all of the probabilities.
Hence we must try to select genuine ν e interactions from the large NC background. In the table we also show mis-identification factors when all non-µ events are classified as NC events, which could be used when P µe is very small.

A perfect detector
We first find the sensitivity of the NC unitarity test for a perfect detector, i.e., a detector that can categorize each event correctly as CC muon or NC, with no misidentification and 100% efficiency. The figure of merit for a perfect detector is given by Eq. 17 with ζ N CN C = 1. We show the 3σ sensitivity for P µs (i.e., the minimum value of P µs for which P µs = 3δP µs ) versus N 0 µ (the number of CC muons expected in the detector with no oscillations) for several values of the NC systematic error in Fig. 1 (the dotted curves). At low statistics the sensitivity is very poor, and for high statistics the sensitivity approaches the asymptotic limit of 3ǫ N C /(1 + 3ǫ N C ), where ǫ N C ≡ δN 0 N C /N 0 N C is the fractional NC normalization uncertainty.

More realistic K2K-like and T2K-like experiments
Next we find the NC sensitivity for the K2K-like detector described in Sec. 3.1 for the case P µe ≃ 0. We generated 400,000 neutrino events using the NEUGEN simulator, from which the normalized mis-identification factors f x,y were calculated. For a given set of probabilities P xy , the values of R N C and R µ were calculated, and the corresponding measured value of P µs was determined from Eq. 11. The uncertainty on P µs was calculated using Eq. 14, assuming the uncertainties δR N C and δR µ are uncorrelated and add in quadrature. The 3σ sensitivity for P µs is shown in Fig. 1 (solid curves) for various values of ǫ N C for the case P µs = 1 − P µµ (all ν µ oscillating to ν s ; we will consider cases with nonzero P µτ later). For both low and high statistics the K2K-like 3σ sensitivity can be approximated by which can be derived from Eq. 16, where factors quadratic in the f x,y are ignored.
Since f µ,N C ≃ 0.08 for our K2K-like experiment, the NC sensitivity is at most about 1.08 worse than that of the perfect detector for large numbers of events where the statistical uncertainty becomes negligible compared to the systematic uncertainty. At low statistics the efficiency becomes important and the K2K-like performance will be more than 1.08 worse than a perfect detector.
The K2K-like curves in Fig. 1 are plotted for the simple K2K signals in Table 1.
The corresponding curves for the more aggressive K2K-like signals are very similar to the simple case; the improved efficiencies are partially compensated for by the slightly higher value of f µ,N C . Thus the result is fairly insensitive to the exact signal criteria used.
We next consider the effects of nonzero P µτ . If we assume P µτ = 1 − P µs (i.e., P µµ = P µe = 0), the curves are very close to those of the perfect detector, since the dominant mis-identification term f µ,N C does not contribute to R N C when P µµ = 0.
If both P µµ and P µτ are both nonzero (with P µe ≃ 0), the results will lie somewhere between the curves for K2K-like and the perfect detector.
Finally, we consider nonzero P µe , in which case R e must also be measured and P µs is determined using Eq. 27. As discussed in Sec. 2.3.1, if P µe is of order 0.1 or less (as indicated by oscillation bounds such as from the CHOOZ reactor), and if the misidentification factors are also of order 0.1 or less, then this case reduces to that where the ν e are ignored. We have verified this numerically for the K2K-like misidentification factors in Table 1.
In summary, the sensitivity of the K2K-like detector to the NC signal is only slightly worse than that of a perfect detector, with the dominant loss of sensitivity coming from the mis-identification of CC muon events as NC. For comparison, in Fig. 1 we have also shown sensitivity curves for the T2K-like experiment with a Gaussian beam spread. Since f µ,N C = 0.25 in this case, the sensitivity is about a factor of 1.25/1.08 = 1.16 worse than for K2K.

The MINOS-like detector
For the MINOS-like case, we generated 320,000 neutrino events using the NEUGEN simulator, and calculated the corresponding mis-identification factors. The 3σ sensitivity for P µs was calculated as described above for the case P µs = 1 − P µµ ; the results are shown in Fig. 1. At low statistics, the MINOS-like experiment does better than the K2K-like and T2K-like experiments because of the higher NC efficiency, but at high statistics it does worse because of the larger mis-identification factors.

Exclusion limit when P µs = 0
If a 3σ signal for P µs is not observed, then an exclusion limit (upper bound) for P µs can then be obtained. The 90% C.L. exclusion limit for P µs is shown in Fig. 2 for a perfect detector (dotted curves), K2K-like with basic signals (solid curves), and T2Klike with basic signals (dashed curves). To model realistic oscillation probabilities we have assumed a three-neutrino model assuming the parameters δm 2 31 = 2.0×10 −3 eV 2 , sin 2 2θ 23 = 1.0, and sin 2 2θ 13 = 0.1. At high statistics the relative values of the exclusion limits are approximately proportional to (1 + f µ,N C ), similar to the 3σ sensitivity levels calculated previously. As was the case for the 3σ sensitivity, the MINOS-like detector does better than the K2K-like and T2K-like detectors at low statistics, due to the higher NC efficiency, but not as well at high statistics due to larger misidentification factors.

Summary
At low statistics ( 1000 events), experiments with a larger NC efficiency, such as our MINOS-like example, tend to have better sensitivity to the sterile oscillation probability P µs . At high statistics, the sensitivity in the cases we considered is primarily limited by the systematic uncertainty in the NC rate, ǫ N C , and the contamination of the NC signal from CC µ events, f µ,N C (and NC contamination from CC τ events, f τ,N C , above τ threshold). The best anticipated ǫ N C is of order a few per cent, so the best 3σ sensitivity and 90% C.L. exclusion limits that can be expected for the sterile oscillation probability will be of order 0.10-0.15 (0.2-0.3 for the oscillation amplitude). The lowest contamination rates are realized for the K2K-like and T2Klike cases. Significant improvements in these sterile probability sensitivies or limits can only be achieved by lowering the uncertainty in NC cross sections or improving the event selection criteria, both of which could prove to be challenging but very worthwhile.   Other assumptions are the same as in Fig. 1.