The High Energy Neutrino Nuisance at a Medium Baseline Reactor Experiment

10 years from now medium baseline reactor experiments will attempt to determine the neutrino mass hierarchy from the differences (RL+PV) between the extrema of the Fourier transformed neutrino spectra. Recently Qian et al. have claimed that this goal may be impeded by the strong dependence of the difference parameter RL+PV on the reactor neutrino flux and on slight variations of Delta M^2_32. We demonstrate that this effect results from a spurious dependence of the difference parameter on the very high energy (8+ MeV) tail of the reactor neutrino spectrum. This dependence is spurious because the high energy tail depends upon decays of exotic isotopes and is insensitive to the mass hierarchy. An energy-dependent weight in the Fourier transform not only eliminates this spurious dependence but in fact increases the chance of correctly determining the hierarchy.

This year the Daya Bay [1,2] and RENO [3] experiments have demonstrated beyond any reasonable doubt that θ 13 is as much as an order of magnitude larger than had been suspected several years ago. This large value of θ 13 implies that 1-3 neutrino oscillations may be observed at medium baselines, which we define to be 40-80 km. The medium baseline neutrino spectrum may then be used to determine the neutrino mass hierarchy [4]. Such experiments are now not only practical but indeed they will be performed within the next decade [5,6,7].
However at these baselines, due to a degeneracy in the high energy neutrino spectrum [8,9,10], a determination of the hierarchy requires a measurement of 1-3 oscillations at low neutrino energies E. As a result of the finite energy resolution of the detector and various interference effects [10] these low energy peaks are difficult to identify individually at medium baselines L. Nonetheless if the nonlinear energy response of a detector is well understood then one may measure the sum of the peaks by studying the k ∼ |∆M 2 31 |/2 region of the L/E-Fourier transform of the neutrino spectrum [11]. The most popular variables for such a determination are the fractional difference RL between the deepest minima of the Fourier cosine transform and the difference P V between the deepest minimum and the highest peak of the Fourier sine transform [12,13]. Although these two variables are somewhat degenerate, an improvement may be obtained by considering their sum RL + P V .
A serious obstruction to this analysis, and thus to plans to measure the neutrino mass hierarchy at medium baselines, has been described in Ref. [9]. The authors observed that the combination RL + P V is very sensitive to the choice of model of the reactor neutrino flux Φ(E) and to variations of |∆M 2 32 | which are smaller than the precision to which this mass difference has been determined by MINOS [14]. While the observed shift appears to depend upon both |∆M 2 32 | and the hierarchy, from Fig. 4 of Ref. [9] it can be seen that the shift depends only upon the effective mass difference [15,10] ∆M 2 eff = cos 2 (θ 12 )|∆M 2 31 | + sin 2 (θ 12 )|∆M 2 32 |.
The neutrino flux from reactors is known poorly. The theoretical normalization has recently increased by about 3% [16] and the 6+ MeV flux has increased by an additional 3% [17].
The flux beyond about 8 MeV is not known at all due to its strong dependence upon decays of exotic isotopes [16]. Even worse, all of these theoretical fluxes are about 6% above the observed fluxes at very short [18] and 1 km [19] baselines. Thus the large sensitivity of RL + P V upon the poorly known fluxes and ∆M 2 eff appreciably reduces the probability that a medium baseline reactor experiment can correctly determine the neutrino mass hierarchy.
We will now explain the cause of this strong dependence. As the dependence of RL, P V and RL + P V upon these parameters is virtually indistinguishable, for brevity we will consider only which is the fractional difference between two minima R and L of the Fourier cosine transform of the neutrino spectrum where the tree level neutrino inverse β decay cross section is [20] σ and the electron neutrino survival probability is P ee = sin 4 (θ 13 ) + cos 4 (θ 12 )cos 4 (θ 13 ) + sin 4 (θ 12 )cos 4 (θ 13 ) + 1 2 (P 12 + P 13 + P 23 ) Following Ref. [12], a 3%/ √ E energy resolution is included by convoluting the observed energy spectrum with We use the neutrino mass matrix parameters of Ref. [5].
As was demonstrated in Ref. [10], the minima whose difference defines RL lie just on either side of k = |∆M 2 31 |/2. These minima arise from the Fourier transform of P 13 which is independent of the hierarchy, but the contribution of P 23 provides a perturbation which makes the right (left) minimum deeper for the normal (inverted) hierarchy.
The problem observed in Ref. [9] is that, depending upon the reactor flux model used, the transform of the unoscillated reactor flux Φ(E)σ(E)E 2 /L 3 itself may contribute peaks near k = |∆M 2 31 |/2 which interfere with those of P 13 + P 23 and so affect RL. While the cosine transform of the unoscillated flux Φ(E)σ(E)E 2 /L 3 is independent of the neutrino mass splittings, the locations of the peaks of P 13 + P 23 are proportional to ∆M 2 eff . This means that the relative phase between the Fourier transform of the unoscillated spectrum and that of P 13 + P 23 depends on the precise value of ∆M 2 eff . As a result the oscillations in the Fourier transform of Φ(E)σ(E) lead to an ∆M 2 eff -dependence in the quantity RL just of the kind observed in Ref. [9] using old reactor flux models. In fact, using the 235 U flux from Ref. [21], the 239 Pu and 241 Pu fluxes from [22] and the Gaussian approximated 238 U flux from Ref. [23] with the isotope ratios of Ref. [12] we find an oscillation in the unoscillated spectrum term in Eq. (3). Using this old model of the reactor flux, in Fig. 1 we compare the Fourier transform of the unoscillated term with that of the P 13 + P 23 term, which is sensitive to the hierarchy. One can see that the unoscillated term is periodic with the same wavelength as was observed in Fig. 4 of Ref. [9], and thus the interference between these two terms oscillates as ∆M 2 eff varies, shifting the P 13 + P 23 peaks and so reproducing the effect reported in that note.
Ref. [9] concludes that this strong dependence of RL upon the reactor flux means that a precise knowledge of this flux is desirable to determine the neutrino mass hierarchy at a short baseline experiment. Our conclusions differ. This difference arises from the observation that, as can be seen in Fig. 2, near k = |∆M 2 31 |/2 the oscillations of the cosine transform of the reactor spectrum are governed by the highest energy neutrinos (8+ MeV) which, according to Refs. [8,10], are not sensitive to the hierarchy. On the other hand in Fig. 2 we see that the effect is present in the case of both old and new fluxes if the spectrum is cut off at 8.5 MeV while it is negligible if the spectrum is cut off at 12.8 MeV. However since the quadratic and quintic fits to these fluxes are only reliable below 6.5 MeV and marginally reliable below 8 MeV, there is no reason to trust a naive extrapolation to 12.8 MeV.
As RL depends strongly on the spectrum between 8.5 and 12.8 MeV, which in turn is independent of the hierarchy, this high energy spectrum provides a nuisance parameter for the determination of the hierarchy using RL. The solution suggested in Ref. [9] is to determine the spectrum precisely, however so few neutrinos are observed in this range that such a determination would be difficult, indeed the spectrum is not understood at the required precision even at the energies with high fluxes [18]. Even if such a measurement were possible, then RL would still be likely to depend upon ∆M 2 eff with a higher sensitivity than the mass determination at MINOS and probably at NOνA, making a determination of the hierarchy at a medium baseline more challenging.
Our solution is to replace RL and P V with indicators that are insensitive to the high energy neutrino spectrum, by providing an energy-dependent weight w(E) on the neutrino spectrum in the Fourier transform. As we saw in Fig. 2, a simple cutoff in the Fourier transform will amplify the spurious dependence. The weight needs to cut off the high energies gradually, with derivative scales much longer than |∆M 2 31 |, so as to not itself introduce spurious peaks in the critical part of the Fourier transforms. One such choice of weight which we have found works quite well is a Gaussian The same weight serves well in both the sine transform and also the nonlinear transforms of inverted (red dashed curve) hierarchies. One can see that the solid, blue unoscillated curve is very close to zero. We have checked that this curve is essentially independent of the cutoff and so the reactor spectrum no longer affects RL. Comparing with Fig. 1 one can see that the difference RL between the depths of the minima is even greater in this weighted case, allowing for a better determination of the hierarchy than was possible with an unweighted Fourier transform.
Ref. [10] which determine the hierarchy more reliably than RL+P V at baselines below about 55 km [24]. As can be seen by comparing the unweighted and weighted cosine transforms in Figs. 1 and 3, not only does the weighting procedure preserve RL, but it actually increases the difference in the peak sizes between the normal and inverted hierarchies. Thus this solution to the dependence upon the high energy neutrino tail not only removes the spurious dependence, for any high energy reactor spectrum, but it increases the chance of success of the determination of the hierarchy. In simulations [24] we will show that when the weight function is optimized with a neural network this increase is of order 2%.