How two neutrino superbeam experiments do better than one

We examine the use of two superbeam neutrino oscillation experiments with baselines $\lsim 1000$ km to resolve parameter degeneracies inherent in the three-neutrino analysis of such experiments. We find that with appropriate choices of neutrino energies and baselines two experiments with different baselines can provide a much better determination of the neutrino mass ordering than a single experiment alone. Two baselines are especially beneficial when the mass scale for solar neutrino oscillations $\delta m^2_{\rm sol}$ is $\gsim 5\times10^{-5}$ eV$^2$. We also examine CP violation sensitivity and the resolution of other parameter degeneracies. We find that the combined data of superbeam experiments with baselines of 295 and 900 km can provide sensitivity to both the neutrino mass ordering and CP violation for $\sin^22\theta_{13}$ down to 0.03 for $|\delta m^2_{\rm atm}| \simeq 3\times10^{-3}$ eV$^2$. It would be highly advantageous to have a 10% determination of $|\delta m^2_{\rm atm}|$ before the beam energies and baselines are finalized, although if $|\delta m^2_{\rm atm}|$ is not that well known, the neutrino energies and baselines can be chosen to give fairly good sensitivity for a range of $|\delta m^2_{\rm atm}|$.


I. INTRODUCTION
Atmospheric neutrino data from Super-Kamiokande provides strong evidence that ν µ 's created in the atmosphere oscillate to ν τ with mass-squared difference |δm 2 atm | ∼ 3×10 −3 eV 2 and almost maximal amplitude [1]. Furthermore, the recent solar neutrino data from the Sudbury Neutrino Observatory (SNO) establishes that electron neutrinos change flavor as they travel from the Sun to the Earth: the neutral-current measurement is consistent with the solar neutrino flux predicted in the Standard Solar Model [2], while the charged-current measurement shows a depletion of the electron neutrino component relative to the total flux [3]. Global fits to solar neutrino data give a strong preference for the Large Mixing Angle (LMA) solution to the solar neutrino puzzle, with δm 2 sol ∼ 5×10 −5 eV 2 and amplitude close to 0.8 [3,4].
The combined atmospheric and solar data may be explained by oscillations of three neutrinos, that are described by two mass-squared differences, three mixing angles and a CP violating phase. The atmospheric and solar data roughly determine δm 2 atm , δm 2 sol and the corresponding mixing angles. The LMA solar solution will be tested decisively (and δm 2 sol measured accurately) by the KamLAND reactor neutrino experiment [5,6]. More precise measurements of the other oscillation parameters may be performed in long-baseline neutrino experiments. The low energy beam at MINOS [7] plus experiments with ICARUS [8] and OPERA [9] will allow an accurate determination of the atmospheric neutrino parameters and may provide the first evidence for oscillations of ν µ → ν e at the atmospheric mass scale [10]. It will take a new generation of long-baseline experiments to further probe ν µ → ν e appearance and to measure the leptonic CP phase. Matter effects are the only means to determine sgn(δm 2 atm ); once sgn(δm 2 atm ) is known, the level of intrinsic CP violation may be measured. Matter effects and intrinsic CP violation both vanish in the limit that the mixing angle responsible for ν µ → ν e oscillations of atmospheric neutrinos is zero.
It is now well-known that there are three two-fold parameter degeneracies that can occur in the measurement of the oscillation amplitude for ν µ → ν e appearance, the ordering of the neutrino masses, and the CP phase [11]. With only one ν and oneν measurement, these degeneracies can lead to eight possible solutions for the oscillation parameters; in most cases, CP violating (CP V ) and CP conserving (CP C) solutions can equally explain the same data. Studies have been done on how a superbeam [11][12][13][14][15][16], neutrino factory [16][17][18], superbeam plus neutrino factory [19], or two superbeams with one at a very long baseline [20,21] could be used to resolve one or more of these ambiguities.
In this paper we show that by combining the results of two superbeam experiments with different medium baselines, < ∼ 1000 km, the ambiguity associated with the sign of δm 2 atm can be resolved, even when it cannot be resolved by the two experiments taken separately. Furthermore, the ability to determine sgn(δm 2 atm ) from the combined data is found to not be greatly sensitive to the size of δm 2 sol , unlike the situation where data from only a single baseline is used. If both experiments are at or near the peak of the oscillation, a good compromise is obtained between the sensitivities for resolving sgn(δm 2 atm ) and for establishing the existence of CP violation. If |δm 2 atm | is not known accurately, the neutrino energies and baselines can be chosen to give fairly good sensitivity to the sign of δm 2 atm and to CP violation for a range of |δm 2 atm |. The organization of our paper is as follows. In Sec. II we discuss the parameter de-generacies that can occur in the analysis of long-baseline oscillation data. In Sec. III we analyze how two long-baseline superbeam experiments can break degeneracies, determine the neutrino parameters, and establish the existence of CP violation in the neutrino sector, if it is present. A summary is presented in Sec. IV.

II. PARAMETER DEGENERACIES
We work in the three-neutrino scenario using the parametrization for the neutrino mixing matrix of Ref. [11]. If we assume that ν 3 is the neutrino eigenstate that is separated from the other two, then δm 2 31 = δm 2 atm and the sign of δm 2 31 can be either positive or negative, corresponding to the mass of ν 3 being either larger or smaller, respectively, than the other two masses. The solar oscillations are regulated by δm 2 21 = δm 2 sol , and thus |δm 2 21 | ≪ |δm 2 31 |. If we accept the likely conclusion that the solar solution is LMA [3,4], then δm 2 21 > 0 and we can restrict θ 12 to the range [0, π/4]. It is known from reactor neutrino data that θ 13 is small, with sin 2 2θ 13 ≤ 0.1 at the 95% C.L. [22]. Thus a set of parameters that unambiguously spans the space is δm 2 31 (magnitude and sign), δm 2 21 , sin 2 2θ 12 , sin θ 23 , and sin 2 2θ 13 ; only the θ 23 angle can be below or above π/4.
For the oscillation probabilities for ν µ → ν e andν µ →ν e we use approximate expressions given in Ref. [11], in which the probabilities are expanded in terms of the small parameters θ 13 and δm 2 21 [23,24], which reproduces well the exact oscillation probabilities for E ν > ∼ 0.5 GeV, θ 13 < ∼ 9 • , and L < ∼ 4000 km [11]. In all of our calculations we use the average electron density along the neutrino path, assuming the Preliminary Reference Earth Model [25]. Our calculational methods are described in Ref. [12].
We expect that |δm 2 31 | and sin 2 2θ 23 will be measured to an accuracy of ≃ 10% at 3σ from ν µ → ν µ survival in long-baseline experiments [7-10], while δm 2 21 will be measured to an accuracy of ≃ 10% at 2σ and sin 2 2θ 12 will be measured to an accuracy of ±0.1 at 2σ in experiments with reactor neutrinos [6]. The remaining parameters (θ 13 , the CP phase δ, and the sign of δm 2 31 ) must be determined from long-baseline appearance experiments, principally using the modes ν µ → ν e andν µ →ν e with conventional neutrino beams, or ν e → ν µ and ν e →ν µ at neutrino factories. However, there are three parameter degeneracies that can occur in such an analysis: (i) the (δ, θ 13 ) ambiguity [17], (ii) the sgn(δm 2 31 ) ambiguity [13], and (iii) the (θ 23 , π/2 − θ 23 ) ambiguity [11] (see Ref. [11] for a complete discussion of these three parameter degeneracies). In each degeneracy, two different sets of values for δ and θ 13 can give the same measured rates for both ν andν appearance and disappearance. For each type of degeneracy the values of θ 13 for the two equivalent solutions can be quite different, and the two values of δ may have different CP properties, e.g., one can be CP conserving and the other CP violating.
A judicious choice of L and E ν can reduce the impact of the degeneracies. For example, if L/E ν is chosen such that ∆ ≡ |δm 2 31 |L/(4E ν ) = π/2 (the peak of the oscillation in vacuum), then the cos δ terms in the average appearance probabilities vanish, even after matter effects are included [11]. Then since it is sin δ that is being measured, the (δ, θ 13 ) ambiguity is reduced to a simple (δ, π − δ) ambiguity, CP V solutions are no longer mixed with CP C solutions, and θ 13 is in principle determined (for a given sgn(δm 2 31 ) and θ 23 ). If L is chosen to be long enough ( > ∼ 1000 km), then the predictions for δm 2 31 > 0 and δm 2 31 < 0 no longer overlap if θ 13 > ∼ a few degrees, and the sgn(δm 2 31 ) ambiguity is removed; our previous studies indicated that for δm 2 21 = 5 × 10 −5 eV 2 this happens at L > ∼ 1300 km if sin 2 2θ 13 > 0.01 [11] (before experimental uncertainties are considered). However, the persistence of the sgn(δm 2 31 ) ambiguity is highly dependent on the size of the solar oscillation mass scale, because large values of δm 2 21 cause the predictions for δm 2 31 > 0 and δm 2 31 < 0 to overlap much more severely than when δm 2 21 is smaller. Also, existing neutrino baselines are no longer than 735 km. In this paper we explore the possibility that two experiments with medium baselines ( < ∼ 1000 km) can determine sgn(δm 2 31 ), even when data from one of the baselines alone cannot. We then address the sensitivity for establishing CP violation.

A. Description of the experiments and method
For our analysis we take one baseline to be 295 km, the distance for the proposed experiment from the Japan Hadron facility (JHF) to the Super-Kamiokande detector at Kamioka. For the neutrino spectrum of this experiment we use their 2 • off-axis beam with average neutrino energy of 0.7 GeV [26]. For the second experiment we assume an off-axis beam in which the beam axis points at a site 735 km from the source (appropriate for a beamline from NuMI at Fermilab to Soudan, or from CERN to Gran Sasso). For the off-axis spectra of the NuMI experiment we use the results presented in Ref. [27], which provides neutrino spectra for 39 different off-axis angles ranging from 0.32 • to 1.76 • .
Using the off-axis components of the beam has the advantage of a lower background [15,28,29] due to reduced ν e contamination and a smaller high-energy tail. Off-axis beams also offer flexibility in the choice of L and E ν . For example, for a beam nominally aimed at a ground-level site a distance L 0 from the source, the distance to a ground-level detector with off-axis angle θ OA can lie anywhere in the range where sin θ = L 0 /(2R e ), and R e = 6371 km is the radius of the Earth. Then for L 2 0 ≪ R 2 e the possible range of distances for an off-axis detector at approximately ground level is The neutrino energy and neutrino flux Φ ν decrease with increasing off-axis angle as where γ = E π /m π is boost factor of the decaying pion. Thus a wide range of L and E ν can be achieved with a single fixed beam, although the event rate will drop with increasing off-axis angle because the flux decreases and the neutrino cross section is smaller at smaller E ν (thereby putting a limit on the usable range of L and E ν ). For the first experiment at L 1 = 295 km, we assume that the neutrino spectrum is chosen so that the cos δ terms in the ν andν oscillation probabilities vanish (after averaging over the neutrino spectrum), using the best existing experimental value for δm 2 31 . The JHF 2 • off-axis beam [30] satisfies this condition for δm 2 31 = 3 × 10 −3 eV 2 . This spectrum choice reduces the (δ, θ 13 ) ambiguity to a simple (δ, π − δ) ambiguity, as described in Sec. II. For the second experiment we allow L 2 and θ OA to vary within the restrictions of Eq. (2). This flexibility can be fully utilized if a deep underground site is not required; the short duration of the beam operation (an 8.6 µs pulse with a 1.9 s cycle time [31]) may enable a sufficient reduction in the cosmic ray neutrino background. We assume that the proton drivers at the neutrino sources have been upgraded from their initial designs (from 0.8 to 4.0 MW for JHF [30] and from 0.4 to 1.6 MW for FNAL [32]), so that they are both true neutrino superbeams. We assume two years running with neutrinos and six years with antineutrinos at JHF, and two years with neutrinos and five years with antineutrinos at FNAL; these running times give approximately equivalent numbers of charged-current events for neutrinos and antineutrinos at the two facilities, in the absence of oscillations. For detectors, we assume a 22.5 kt detector in the JHF beam (such as the current Super-K detector) and a 20 kt detector in the FNAL beam (which was proposed in Ref. [15]). Larger detectors such as Hyper-Kamiokande or UNO would allow shorter beam exposures or higher precision studies. In all of our calculations, we assume |δm 2 31 | = 3 × 10 −3 eV 2 , θ 23 = π/4, δm 2 21 = 5 × 10 −5 eV 2 , and sin 2 2θ 12 = 0.8, unless noted otherwise.
We first consider the minimum value of sin 2 2θ 13 for which the signal in the neutrino appearance channel can be seen above background at the 3σ level (the discovery reach), varying over a range of allowed values for θ OA and L 2 in the second experiment. The discovery reach depends on the value of δ and the sign of δm 2 31 ; the best (when δm 2 31 > 0) and worst (when δm 2 31 < 0) cases in the ν channel (after varying over δ) are shown in Fig. 1. In theν channel, the best case occurs for δm 2 31 < 0 and the worst for δm 2 31 > 0. In our calculations we assume a background that is 0.5% of the unoscillated charged-current rate (see Ref. [15]), and that the systematic error is 5% of the background. However, we note that our general conclusions are not significantly affected by reasonable changes in these experimental uncertainty assumptions. Detector positions where there is no cos δ dependence in the rates are denoted by boxes. The best reach is sin 2 2θ 13 ≃ 0.003, which occurs for θ OA ≃ 0.5-0.9 • . In the worst case scenario the reach degrades to sin 2 2θ 13 ≃ 0.01.
The measurement of P andP at L 1 allows a determination of sin 2 2θ 13 and sin δ, modulo the possible uncertainty caused by the sign of δm 2 31 , assuming for the moment that θ 23 = π/4, so there is no (θ 23 , π/2 − θ 23 ) ambiguity. The question we next consider is whether an additional measurement of P andP at L 2 can determine sgn(δm 2 31 ), measure CP violation, and distinguish δ from π − δ. We define the χ 2 of neutrino parameters (δ ′ , θ ′ 13 ) relative to the parameters (δ, θ 13 ) as where N i and N ′ i are the event rates for the parameters (δ, θ 13 ) and (δ ′ , θ ′ 13 ), respectively, δN i is the uncertainty in N i , and i is summed over the measurements being used in the analysis (ν andν at L 1 and ν andν at L 2 ). For δN i we assume that the statistical error for the signal plus background can be added in quadrature with the systematic error. For a two-parameter system (δ and θ 13 unknown), two sets of parameters can be resolved at the 2σ (3σ) level if χ 2 > 6.17 (11.83). To determine if measurements at L 1 and L 2 can distinguish one set of oscillation parameters with one sign of δm 2 31 from all other possible sets of oscillation parameters with the opposite sign of δm 2 31 , we sample the (δ, θ 13 ) space for the opposite sgn(δm 2 31 ) using a fine grid with 1 • spacing in δ and approximately 2% increments in sin 2 2θ 13 . If the χ 2 between the original set of oscillation parameters and all of those with the opposite sgn(δm 2 31 ) is greater than 6.17 (11.83), then sgn(δm 2 31 ) is distinguished at the 2σ (3σ) level for that parameter set. Figure 2 shows contours (in the space of possible L 2 and θ OA for the second experiment) for the minimum value of sin 2 2θ 13 (the sin 2 2θ 13 reach) for distinguishing sgn(δm 2 31 ) at the 3σ level when ν andν data from L 1 and L 2 are combined. As in Fig. 1, the boxes indicate the detector positions where the cos δ terms in the average probabilities vanish. The best reach of about sin 2 2θ 13 ≃ 0.03 can be realized for θ OA ≃ 0.7-1.0 • and L 2 values near the maximum allowed by Eq. 2 (≃ 875-950 km). Table I shows the sensitivity for determining sgn(δm 2 31 ) for different combinations of detector size and proton driver power in the two experiments. The table shows that once enough statistics are obtained at JHF (with a 22.5 kt detector and a 4 MW source), combined JHF and NuMI data significantly improve the sin 2 2θ 13 reach for determining sgn(δm 2 31 ) at 3σ (by nearly a factor of two compared to data from a 1.6 MW NuMI alone).  The ability to distinguish the sign of δm 2 31 is greatly affected by the size of the solar mass scale δm 2 21 , because the predictions for δm 2 31 > 0 and δm 2 31 < 0 overlap more for larger values of δm 2 21 . In Fig. 3a we show the region in (δ, sin 2 2θ 13 ) space for which parameters with δm 2 31 > 0 can be distinguished from all parameters with δm 2 31 < 0 at the 3σ level for several possible values of δm 2 21 , using combined data from L 1 = 295 km and L 2 = 890 km, with θ OA = 0.74 • for the second experiment. With this configuration the cos δ terms in the average probabilities vanish for both experiments and nearly maximal reach for distinguishing sgn(δm 2 31 ) is achieved. A similar plot using only data at L 2 = 890 km and θ OA = 0.74 • is shown in Fig. 3b. We do not show a corresponding plot for L 1 = 295 km because the shorter baseline severely inhibits the determination of sgn(δm 2 31 ). A comparison of the two figures shows that for δ = 270 • (where the δm 2 31 > 0 predictions have the least overlap with any of those for δm 2 31 < 0) the sensitivity to sgn(δm 2 31 ) is not significantly improved by adding the data at L 1 . However, at δ = 90 • the ability to distinguish sgn(δm 2 31 ) is much less affected by the value of δm 2 21 when the data at L 1 is included. With data only at L 2 , sgn(δm 2 31 ) can be determined for sin 2 2θ 13 = 0.1 when δ = 90 • only for δm 2 21 < ∼ 8 × 10 −5 eV 2 , while with data at L 1 and L 2 it can be determined for sin 2 2θ 13 as low as 0.04 for δm 2 21 as high as 2 × 10 −4 eV 2 . The corresponding results for δm 2 31 < 0 are approximately given by reflecting the curves in Fig. 3 about δ = 180 • .
We conclude that combining measurements of ν µ → ν e andν µ →ν e from two superbeam experiments at different L results in a much more sensitive test of the sign of δm 2 31 than one experiment alone, especially for larger values of the solar mass scale δm 2 21 . The ability to determine sgn(δm 2 31 ) is also affected by the value of θ 23 . We found that the sin 2 2θ 13 reach for determining sgn(δm 2 31 ) at 3σ varied from 0.02 to 0.04 for sin 2 2θ 23 = 0.90 (compared to 0.03 when θ 23 = π/4), depending on whether δm 2 31 is positive or negative, and whether θ 23 < π/4 or θ 23 > π/4. The sgn(δm 2 31 ) sensitivities for different possibilities are shown in Table II. An important goal of long-baseline experiments is to determine whether or not CP is violated in the leptonic sector. In order to unambiguously establish the existence of CP violation, one must be able to differentiate between (δ, θ 13 ) and all possible (δ ′ , θ ′ 13 ), where δ ′ = 0 • or 180 • and θ ′ 13 can take on any value. For our CP violation analysis we vary sin 2 2θ ′ 13 in 2% increments, as was done in the previous section when testing the sgn(δm 2 13 ) sensitivity. Figure 4 shows contours of sin 2 2θ 13 reach for distinguishing δ = 90 • from the CP conserving values δ = 0 • and 180 • at 3σ (with the same sgn(δm 2 31 )), plotted in the (θ OA , L 2 ) plane, assuming ν andν data at both L 1 and L 2 are combined. The CP reach in sin 2 2θ 13 can go as low as 0.01 for θ OA ≃ 0.5 to 0.9 • . Results for δ = 270 • are similar to those for δ = 90 • . Figure 5 shows the minimum value of sin 2 2θ 13 for which δ can be distinguished from all CP conserving parameter sets with δ = 0 • and 180 • , including those with the opposite sgn(δm 2 31 ), at the 3σ level when θ OA = 0.74 • and L 2 = 890 km, for several different values of δm 2 21 . Figure 5a shows the reaches if data from JHF and NuMI are combined, while Fig. 5b shows the reaches if data from NuMI only are used. For most values of δ, when δm 2 21 is higher the CP effect is increased, and hence CP violation can be detected for smaller values of θ 13 . However, there is a possibility that a CP V solution with one sgn(δm 2 31 ) may not be as easily distinguishable from a CP C solution with the opposite sgn(δm 2 31 ); this occurs, e.g., in Fig. 5a for δm 2 21 = 1 × 10 −4 eV 2 , where the predictions for (δ = 45 • and 135 • , δm 2 31 > 0) are close to those for (δ = 0 • and 180 • , δm 2 31 < 0); in this case the CP reach for those values of δ is about the same for δm 2 21 = 1 × 10 −4 eV 2 and δm 2 21 = 5 × 10 −5 eV 2 . We note that if data from only JHF are used (and assuming sin 2 2θ 13 ≤ 0.1) no value of the CP phase can be distinguished at 3σ from the CP conserving solutions when δm 2 21 < ∼ 8 × 10 −5 eV 2 , principally because the intrinsic CP violation due to δ and the CP violation due to matter have similar magnitudes and it is hard to disentangle the two effects. For larger values of δm 2 21 , the intrinsic CP effects are larger and CP violation can be established; e.g., if δm 2 21 = 1 × 10 −4 (2 × 10 −4 ) eV 2 , maximal CP violation (δ = 90 • or 270 • ) can be distinguished from CP conservation at 3σ for sin 2 2θ 13 > ∼ 0.006 (0.001). Therefore, when δm 2 21 = 1 × 10 −4 eV 2 , most of the CP sensitivity of the combined JHF plus NuMI data results from the JHF data; for δm 2 21 = 2 × 10 −4 eV 2 the two experiments contribute about equally to the CP sensitivity.
The boxes in Figs. 2 and 4 indicate the values of L 2 and θ OA for which the cos δ terms in the average probabilities vanish for the second experiment. As indicated in the figures, these detector positions are good for both distinguishing sgn(δm 2 31 ) (see Fig. 2) and for establishing the existence of CP violation (see Fig. 4), especially for larger values of L 2 . A good compromise occurs at θ OA ≃ 0.74 • with L 2 ≃ 890 km. In Ref. [15] it was shown that similar values for θ OA and L 2 using the NuMI off-axis beam gave a favorable figure-ofmerit for the signal to background ratio; our analysis shows that such an off-axis angle and baseline is also very good for distinguishing sgn(δm 2 31 ) and establishing CP violation, when combined with superbeam data at L 1 = 295 km. D. Resolving the (δ, π − δ) ambiguity If L 2 ≃ 890 km and θ OA ≃ 0.74 • are chosen for the location of the second experiment, as suggested in the previous section, then both the first and second experiments are effectively measuring sin δ, and it is impossible to resolve the (δ, π − δ) ambiguity. Different values of L 2 and θ OA would be needed to distinguish δ from π − δ. Figure 6 shows contours (in the space of possible L 2 and θ OA ) for the minimum value of sin 2 2θ 13 needed to distinguish δ = 0 • from δ = 180 • at the 2σ level using ν andν data from L 1 and L 2 (it is not possible to distinguish δ = 0 • from δ = 180 • at the 3σ level for any value of sin 2 2θ 13 ≤ 0.1). Two choices are possible: one with θ OA < ∼ 0.3-0.5 • and L 2 ≃ 650-775 km, and another near θ OA ≃ 1.0 • with L 2 ≃ 950 km. The former choice does not do well in distingishing sgn(δm 2 31 ), while the latter choice is nearly optimal for sgn(δm 2 31 ) sensitivity but significantly worse for CP violation sensitivity. Thus the ability to also resolve the (δ, π − δ) ambiguity is rather poor, and comes at the expense of CP V sensitivity.
E. Resolving the (θ 23 , π/2 − θ 23 ) ambiguity If θ 23 = π/4, there is an additional ambiguity between θ 23 and π/2 − θ 23 . This ambiguity gives two solutions for sin 2 2θ 13 whose ratio differs by a factor of approximately tan 2 θ 23 , which can be as large as 2 if sin 2 2θ 23 = 0.9 [11]. Assuming L 1 = 295 km for the first experiment, we could not find any experimental configuration of L 2 and θ OA for the second experiment that could resolve the (θ 23 , π/2 − θ 23 ) ambiguity for sin 2 2θ 13 ≤ 0.1 at even the 1σ level for the entire range of detector sizes and source powers listed in Table I. Therefore we conclude that superbeams are not effective at resolving the (θ 23 , π/2 − θ 23 ) ambiguity using ν e andν e appearance data. Since the approximate oscillation probability for ν e → ν τ is given by the interchanges sin θ 23 ↔ cos θ 23 and δ → −δ in the expression for the ν µ → ν e probability, a neutrino factory combined with detectors having tau neutrino detection capability provides a means for resolving the (θ 23 , π/2 − θ 23 ) ambiguity [11]. Another possibility is to measure survival ofν e 's from a reactor, which to leading order is sensitive to sin 2 2θ 13 but not θ 23 [33,34]. The foregoing analysis assumed |δm 2 31 | = 3 × 10 −3 eV 2 . If the true value differs from this, then to sit on the peak (where the cos δ terms vanish) requires tuning the beam energy and baseline according to the measured value of |δm 2 31 |. JHF has the capability of varying the average E ν from 0.4 GeV to 1.0 GeV, which would correspond to realizing the peak condition for |δm 2 31 | = 1.6-4.0 × 10 −3 eV 2 [30]. In principle, NuMI can vary both L 2 and θ OA to be on the peak. If |δm 2 31 | < 3 × 10 −3 eV 2 , then the best sensitivity to sgn(δm 2 31 ) is obtained for larger θ OA and longer distances (the larger angle makes E ν smaller while the longer distance enhances the matter effect), and the sensitivity is reduced (since the matter effect is smaller for smaller δm 2 31 ). The CP violation sensitivity is also reduced, although not as significantly. For larger values of |δm 2 31 | the sensitivity to sgn(δm 2 31 ) is better, with CP violation sensitivity about the same.
The tuning of the experiments to the peak (where the cos δ terms in the average probabilities vanish) requires knowledge of |δm 2 31 | before the experimental design is finalized. The values of |δm 2 31 | and θ 23 will be well-measured in the survival channel ν µ → ν µ measurements that would run somewhat before or concurrently with the appearance measurements being discussed here, but of course this information may not be available when the configurations for the off-axis experiments are chosen. If |δm 2 31 | is known to 10% at 3σ (the expected sensitivity of MINOS), then the sensitivities to sgn(δm 2 31 ) and CP violation are not greatly affected by baselines that are slightly off-peak. If the baselines and neutrino energies for the superbeam experiments must be chosen before a 10% measurement of |δm 2 31 | can be made, a loss of sensitivity to sgn(δm 2 31 ) could result by not being on the peak. For example, if the experiments are designed for |δm 2 31 | = 3 × 10 −3 eV 2 but in fact |δm 2 31 | = 2.5 × 10 −3 eV 2 , the 3σ sgn(δm 2 31 ) reach is less (sin 2 2θ 13 = 0.04, compared to 0.03 for |δm 2 31 | = 3×10 −3 eV 2 ). If |δm 2 31 | is actually 2×10 −3 eV 2 , the 3σ sgn(δm 2 31 ) reach extends only down to sin 2 2θ 13 ≃ 0.075, just a little below the CHOOZ bound.
Since the sgn(δm 2 31 ) determination has the worst reach in sin 2 2θ 13 (compared to the discovery reach and the CP V sensitivity), and since not knowing sgn(δm 2 31 ) can induce a CP V /CP C ambiguity, the measurement of sgn(δm 2 31 ) is crucial. If |δm 2 31 | is not known precisely, then the exact peak position is not known, and an off-axis angle and baseline should be chosen that will give a reasonable reach for sgn(δm 2 31 ) over as much of the allowed range of |δm 2 31 | as possible. For example, θ OA = 0.85 • -0.90 • and L ≃ 930 km gives a sgn(δm 2 31 ) reach that is fairly good for the range |δm 2 31 | = 2 × 10 −3 eV 2 to 4 × 10 −3 eV 2 . The reach for sgn(δm 2 31 ) is farthest from optimal at the extremes (sin 2 2θ 13 = 0.06 versus the best reach of 0.05 when |δm 2 31 | = 2 × 10 −3 eV 2 and 0.03 versus the best reach of 0.02 when |δm 2 31 | = 4 × 10 −3 eV 2 ). But the CP V reach remains at least as good as the sgn(δm 2 31 ) reach for this range of |δm 2 31 |.

IV. SUMMARY
We summarize the important points of our paper as follows: (i) Two superbeam experiments at different baselines, each measuring ν µ → ν e andν µ → ν e appearance, are significantly better at resolving the sgn(δm 2 31 ) ambiguity than one experiment alone. Using beams from a 4.0 MW JHF with a 22.5 kt detector 2 • off axis at 295 km and a 1.6 MW NuMI with a 20 kt detector 0.7-1.0 • off axis at 875-950 km, sgn(δm 2 31 ) can be determined for sin 2 2θ 13 > ∼ 0.03 if δm 2 31 = 3 × 10 −3 eV 2 . Sensitivities for other beam powers and detector sizes are given in Table I. (ii) For the most favorable cases, a higher value for the solar oscillation scale δm 2 21 does not greatly change the sensitivity to sgn(δm 2 31 ) when ν andν data from two different baselines are combined (unlike the single baseline case, where the ability to determine sgn(δm 2 31 ) is significantly worse for δm 2 21 > ∼ 5 × 10 −5 eV 2 ).
(iii) Running both experiments at the oscillation peaks, such that the cos δ terms in the average probabilities vanish, provides good sensitivity to both sgn(δm 2 31 ) and to CP violation. On the other hand, the ability to resolve the (δ, π − δ) ambiguity is lost, and the (θ 23 , π/2 − θ 23 ) ambiguity is not resolved for any experimental arrangement considered. However, the (δ, π −δ) and (θ 23 , π/2−θ 23 ) ambiguities do not substantially affect the ability to determine whether or not CP is violated (although the latter ambiguity could affect the inferred value of θ 13 by as much as a factor of 2).
(iv) Since running at or near the oscillation peaks is favorable, knowledge of |δm 2 31 | to about 10% (from MINOS) before these experiments are run would be advantageous. If |δm 2 31 | is not known that precisely in advance, then the detector off-axis angle and baseline can still be chosen to give fairly good (though not optimal) sensitivities to sgn(δm 2 31 ) and CP violation.
We conclude that superbeam experiments at different baselines may greatly improve the prospects for determining the neutrino mass ordering in the three-neutrino model. Since a good compromise between determining sgn(δm 2 31 ) and establishing the existence of CP violation is obtained when both experiments are tuned so that the cos δ terms in the average probabilities approximately vanish, knowledge of |δm 2 31 | would be helpful for the optimal design for the experiments. Contours of (a) best-case (when δm 2 31 > 0), and (b) worst-case (when δm 2 31 < 0), sin 2 2θ 13 3σ discovery reach in the (θ OA , L 2 ) plane, for the ν channel at NuMI, where θ OA is the off-axis angle and L 2 is the baseline of the NuMI detector. For the other neutrino parameters we assume |δm 2 31 | = 3 × 10 −3 eV 2 , θ 23 = π/4, δm 2 21 = 5 × 10 −5 eV 2 , and sin 2 2θ 12 = 0.8. The boxes indicate detector positions for which the cos δ terms in the average oscillation probabilities vanish. For theν channel the results are similar, except that the best case occurs for δm 2 31 < 0 and the worst case for δm 2 31 > 0. Contours of sin 2 2θ 13 reach for resolving the sign of δm 2 31 at the 3σ level in the (θ OA , L 2 ) plane when data from JHF and NuMI are used. The JHF detector is assumed to have baseline L 1 = 295 km. Other parameters and notation are the same as in Fig. 1.  6. Contours of sin 2 2θ 13 reach for distinguishing δ = 0 • from δ = 180 • (the (δ, π − δ) ambiguity) at the 2σ level, plotted in the (θ OA , L 2 ) plane, when data from JHF and NuMI are combined. Other parameters and notation are the same as in Fig. 1.