Appearance and disappearance signals at a beta-Beam and a Super-Beam facility

In this letter we present the study of the eightfold degeneracy in the $(\theta_{13},\delta)$ measurement including both appearance and disappearance channels. We analyse, for definiteness, the case of a standard low-$\gamma$ $\beta$-Beam and a 4 MWatt SPL Super-Beam facility, both aiming at a UNO-like Mton water Cerenkov detector located at the Frejus laboratory, $L = 130$ km. In the $\beta$-Beam case, the \nue disappearance channel does not improve the $(\theta_{13},\delta)$ measurement when a realistic (i.e. $\ge$ 2%) systematic error is included. In the Super-Beam case, the \numu disappearance channel could, instead, be quite useful in reducing the impact of the eightfold degeneracy in the $(\theta_{13},\delta)$ measurement, especially once the error on the atmospheric mass difference is fully taken into account in the fit.


Introduction
After more than 30 years of successful neutrino oscillation experiments [1] two parameters still remain undetermined in the three-family Pontecorvo-Maki-Nakagawa-Sakata [2] mixing matrix: the mixing angle θ 13 , for which only a upper limit has been set [3], and the CPviolating phase δ that is still completely unknown. The full understanding of the leptonic mixing matrix constitutes, together with the discrimination of the Dirac/Majorana character and the measure of the neutrino absolute mass scale, the main neutrino-physics goal for the next decade(s).
It is well known that the best way to simultaneously measure (θ 13 , δ) is the (golden) ν e → ν µ appearance channel [4] (and its T and CP conjugate ones). Unfortunately this measure is, in general, severely affected by the presence of degeneracies. When a measurement of the two unknown parameters is performed using a beam able to produce both neutrinos and antineutrinos, the following four systems of equations must be solved: 13 ,δ;s atm ,s oct ) = N l + (θ 13 ,δ; ±s atm , ±s oct ) , N l − (θ 13 ,δ;s atm ,s oct ) = N l − (θ 13 ,δ; ±s atm , ±s oct ) , where s atm = sign(∆m 2 atm ) and s oct = sign(tan 2θ 23 ) are two discrete unknowns, the sign of the atmospheric mass difference and the θ 23 -octant. The r.h.s. of this equation implies that four different models (each of them with a definite (s atm , s oct ) choice) must be used to fit the data on the l.h.s. The eight solutions form what is known as the eightfold degeneracy [5]- [8]. Various methods have been considered to get rid of degeneracies (using spectral analysis [5], combination of experiments [9] and/or different channels [10]). In principle, the eightfold degeneracy can be completely solved if a sufficient number of independent informations is added. At the cost, of course, of increasing the number of detectors and/or beams and consequently the budget needs.
In this letter we try to understand if the effect of degeneracies can be reduced using informations from both the appearance and the disappearance channels at a given experiment. We consider, as reference, the proposal for two CERN-based facilities, the standard 1 low-γ β-Beam [11] and the Super-Beam based on the 4 MWatt SPL 2.2 GeV proton driver [15]. Both beams are directed from CERN toward the underground Fréjus laboratory, where it has been proposed to locate a 1 Mton UNO-like [16] waterČerenkov detector with a 440 kton fiducial mass. The considered baseline is L = 130 km. To be at the first peak in the leading oscillation probability term, the average neutrino energy for both beams has been chosen of the order of a few hundreds MeV. Of course, many other similar setups could be considered, instead the "standard" ones adopted in this letter. Anyway our considerations are quite general and will hold for any comparable low-γ β-Beam and Super-Beam setup.
Needless to say that a similar analysis can be performed in any experiment where disappearance and appearance channels are simultaneously available. At the Neutrino Factory, for example (see [17,18]), theν µ disappearance channel can be certainly used together with the appearance channel ν e → ν µ , whereas the ν e disappearance channel is extremely difficult to exploit (due to the need to measure the electron charge to distinguish ν e → ν e fromν µ →ν e ).
The eightfold degeneracy for these two facilities has been comprehensively studied in [19] and we refer to that paper for all the technical details regarding the used cross-sections, efficiencies and backgrounds. The results of [19] show that to run the two facilities simultaneously does not help in solving the degeneracies, mainly because the two beams, running on the same baseline and with approximately the same energy, are not complementary at all. The only effect is to increase the statistics by roughly a factor two and to reduce some of the systematics, but leaving practically unaffected the main systematic error that it's due to the definition of the fiducial volume of a Mton water detector. In this sense no real synergy is achieved adding these two experiments (if not for using the same detector, thus halving the corresponding costs). For this reason, in the following we will analyse the performance (in the appearance and disappearance channels) of the two facilities separately. 2

β-Beam Appearance and Disappearance Channels
The considered β-Beam setup consists of aν e -beam produced by the decay of 6 He ions boosted at γ = 60 and of a ν e -beam produced in the decay of 18 Ne ions boosted at γ = 100. The γ-ratio has been chosen to store both ions simultaneously into the decay ring. A flux of 2.9 × 10 18 6 He decays/year and 1.1 × 10 18 18 Ne decays/year, as discussed in [20], will be assumed. The average neutrino energies of the ν e ,ν e beams corresponding to this configuration are 0.37 GeV and 0.23 GeV, respectively. Although the boosting factor has been chosen to maximize the oscillation probability at L = 130 km, a severe drawback of this option is the impossibility to use energy resolution, due to nuclear effects.  Table 1 Disappearance event rates for a 10 years run at the considered β-Beam with a 440 kton detector at L = 130 km, for different values of θ 13 and of the sign of the atmospheric mass difference, s atm . Appearance event rates have been quoted in ref. [19]. A 5% systematic error is assumed and backgrounds are computed as in ref. [19]. Continuous, dotted, dashed and dot-dashed lines stand for the intrinsic, sign, octant and mixed degeneracies, respectively.
The measurement of (θ 13 , δ) at this facility has been already actively discussed in the literature [17,21]. In particular, a complete analysis of the eightfold degeneracy was done in [19]. In Fig. 1 we plot our results for three different CP phases,δ = 0 • (left plot) andδ = 45 • , −90 • (right plot), and for two different mixing anglesθ 13 = 2 • and 8 • . The input (θ 13 ,δ) value used in the fit is always shown as a filled black box. As in [19], we use the following reference values for the atmospheric and solar parameters: ∆m 2 atm = ∆m 2 23 = 2.5 × 10 −3 eV 2 , θ 12 = 33 • and ∆m 2 sol = ∆m 2 12 = 8.2 × 10 −5 eV 2 [22]. The atmospheric mixing angle, θ 23 , has been fixed at θ 23 = 40 • , thus inducing the so-called octant degeneracies 3 . The 90% CL contours for each of the degenerate solutions are depicted in the plot: continuous lines stand for s atm =s atm , s oct =s oct (the true solution and the intrinsic clone); dotted lines stand for s atm = −s atm , s oct =s oct (the sign clone); dashed lines stand for s atm =s atm , s oct = −s oct (the octant clone); dot-dashed lines stand for s atm = −s atm , s oct = −s oct (the mixed clone). These plots are obtained assuming a 5% systematic error. Backgrounds have been computed as in [19].
In Fig. 1 it can be seen the dramatic impact that degeneracies have in the precision of the measure of (θ 13 , δ): (1) the knowledge of θ 13 is worsened by, roughly, a factor four (two) for large 4 (small) values of θ 13 , as four possible separate solution-regions appear. The presence of degeneracies has a small impact on the ultimate θ 13 sensitivity; (2) the knowledge of δ is worsened in a significant way in presence of the clones, almost spanning half of the parameter space for small values of θ 13 . These facts are well understood. From the appearance channel of a counting experiment, like the standard β-Beam , with a baseline of hundreds of km (i.e practically in vacuum) there are not enough independent informations to cancel any of the degeneracies. We can also rephrase this fact in the following "statistical" way: the clones have always the same χ 2 of the true solution, making impossible any discrimination between true solution and the degeneracies.
It has been claimed [25] that the ν e disappearance channel at a reactor experiment can help Super-Beam experiments in solving part of the eightfold degeneracy. Indeed, the ν e disappearance probability does not depend on the CP violating phase δ and the atmospheric θ 23 mixing angle. Thus, the θ 13 measurement is not affected by θ 13 − δ correlations nor by the octant and mixed ambiguities. The ν e → ν e matter oscillation probability, expanded at second order in the small parameters θ 13 and (∆m 2 sol L/E) reads [26]: where ∆ atm = ∆m 2 atm /2E, ∆ sol = ∆m 2 sol /2E and B ∓ = |A ∓ ∆ atm | with ∓ for neutrinos (antineutrinos), respectively. The dependence on the sign of the atmospheric mass difference arises from the first non-trivial term of eq. (2) and from higher order terms, O(θ 2 13 × ∆m 2 12 L/E). As a consequence, the s atm dependence is relevant for large values of θ 13 , only.
In Tab. 1 we summarize the relevant numbers for the β-Beam disappearance analysis. In the appearance analysis having a realistic description of backgrounds and efficiencies is of fundamental importance for providing the correct sensitivity on (θ 13 , δ). Conversely, their relevance is much smaller in the disappearance measure, that is limited primarily by systematic errors. Lacking a complete realistic description of systematics we decided to present in Fig. 2 the disappearance measure for three different systematic errors, namely 0% ("theoreticalunrealistic" scenario), 2% ("optimistic" scenario) and 5% ("pessimistic" scenario). We decided to show the 0% systematic line, as we think it is "theoretically" important to have an idea of the ultimate reach of this experiment. The 2% and 5% lines will cover the optimistic and pessimistic feelings about future experimental improvements in understanding a Mton water detector.
In Fig. 2 the 90% CL contours in the (θ 13 , ∆m 2 atm ) plane using the ν e disappearance channel are shown for the input valuesθ 13 = 2 • , 8 • and ∆m 2 atm = 2.5 × 10 −3 eV 2 . The left plot (θ 13 = 2 • ) represents, in practice, the θ 13 sensitivity reach of the β-Beam disappearance the explicit derivation. channel.If systematic errors cannot be controlled better than the 5% level, the β-Beam disappearance channel alone does not improve significantly the present bound on θ 13 . The "theoretical" sensitivity (sys=0%) is around θ 13 = 4 • , while if a systematics of 2% is achieved the disappearance channel alone could test θ 13 down to 6 • . In Fig. 2(right) we show our results for a large value of θ 13 ,θ 13 = 8 • . With a systematics of 2% the mixing angle can be measured in the disappearance channel alone with an error of ±2 • . Again if the 5% systematics is assumed no improvement on the present bound is obtained.
As it is clear from Fig. 2, the ν e disappearance channel is only slightly sensitive to the sign clone, as the full and dashed lines are almost superimposed for every value of θ 13 and ∆m 2 atm . The ν e disappearance channel is an almost "clone-free" environment for the β-Beam, as it is for reactor experiments. However, even in the case of an optimistic (but non-zero) 2% systematic error, no improvement is obtained adding the disappearance channel informations to the results of Fig. 1 for the appearance channel. The resulting 90% CL contours practically coincide with the previous ones, and for this reason we do not consider to present them in a separate figure. The θ 13 indetermination coming from the clone presence in the appearance channel is smaller than the disappearance error itself. Only considering an unrealistic 0% systematics the disappearance channel starts to be useful to eliminate some of the clones.
Summarizing, the β-Beam has two available oscillation channels: the ν e → ν e disappearance and the ν e → ν µ appearance (and their CP-conjugates). The appearance channel can measure (θ 13 , δ), but being the considered β-Beam a pure counting experiment this measurement is severely affected by degeneracies. The disappearance channel does not provide any further useful informations once realistic systematic errors are taken into account.

Super-Beam Appearance and Disappearance Channels
The considered Super-Beam setup is a conventional neutrino beam based on the 4 MWatt SPL 2.2 GeV proton driver that has been proposed at CERN, described in ref. [15]. The average neutrino energies of the ν µ ,ν µ beams corresponding to this configuration are 0.27 GeV and 0.25 GeV, respectively. The possibility to measure (θ 13 , δ) with a Super-Beam has been already widely discussed in the literature [27]. A complete analysis of the eightfold degeneracy at this facility has been done in [19].
In Fig. 3 we plot our results for three different CP phases,δ = 0 • (left plot) andδ = 45 • , −90 • (right plot), and for two different mixing angles,θ 13 = 2 • and 8 • . The input (θ 13 ,δ) value used in the fit is shown as a filled black box. As in the previous section we use the following reference values for the atmospheric and solar parameters: θ 23 = 40 • , ∆m 2 atm = 2.5 × 10 −3 eV 2 , θ 12 = 33 • and ∆m 2 sol = 8.2 × 10 −5 eV 2 . The 90% CL contours for each of the degenerate solutions are depicted in the plot and explained in the caption. These plots are obtained assuming a 5% systematic error. Backgrounds have been computed as in [19]. As it appears from comparison of Fig. 3 with Fig. 1, the "figures of merit" of a standard β-Beam and the SPL Super-Beam are very similar. Also the Super-Beam appearance channel is severely affected by proliferation of clones. The precision in measuring (θ 13 , δ) is practically identical in the two cases. This is well explained by the comparable statistics in the golden channel (ν e → ν µ vs ν µ → ν e ) and an almost equal (L/E) ratio for the two experiments. For this reason there could be no real synergy between this two setups (i.e. they are not complementary) and the only effect in summing these two experiments, concerning the (θ 13 , δ) measure, is to double the statistics.
Nevertheless, the great advantage of the Super-Beam facility compared with the standard-γ β-Beam one is the possibility to measure directly the atmospheric parameters using the ν µ disappearance channel [28] reducing, in particular, the atmospheric mass difference error to less than 10%.
In Tab. 2 we summarize the relevant numbers used in the Super-Beam disappearance analysis. In Fig. 4 we show the measure of (θ 23 , ∆m 2 atm ) at the SPL Super-Beam with a 2% systematic error, in the case (left) of non-maximal atmospheric mixing, θ 23 = 40 • , and (right) maximal atmospheric mixing, θ 23 = 45 • . In both cases, the input value for the atmospheric mass difference has been fixed to ∆m 2 atm = 2.5 × 10 −3 eV 2 . The continuous contour represents the fit to the right choice of the sign of the atmospheric mass difference (i.e. s atm =s atm ) whereas the dotted contour represents the fit to the wrong choice of s atm (i.e. s atm = −s atm ). Since we plot the results in the full θ 23 ∈ [35 • − 55 • ] parameter space, the octant and mixed clones in the left plot are automatically taken into account and do not appear  Table 2 Disappearance event rates for a 2 + 8 years run at the 4 MWatt SPL Super-Beam with a 440 kton detector at L = 130 km, for different values of θ 13 , δ and of the sign of the atmospheric mass difference, s atm . Appearance event rates have been presented in ref. [19].
as separate regions. Notice, however, that being the contours for θ 23 ≤ 45 • and θ 23 ≥ 45 • slightly different for θ 13 = 0 • , if we were to plot the contours in the (sin 2 2θ 23 , ∆m 2 atm ) plane a fourfold degeneracy would be manifest. In the right plot only the sign clone is present, being θ 23 = 45 • . Two comments are in order: first, the sign ambiguity implies that the errors on the atmospheric mass difference are roughly doubled with respect to what expected in the absence of degeneracies; second, the left plot is significantly worse than the right plot. If θ 23 is not maximal, the errors on the atmospheric parameters (θ 23 , ∆m 2 atm ) can be much larger than expected.
The presence of degeneracies in the ν µ disappearance channel can be easily understood looking at the the ν µ → ν µ vacuum oscillation probability expanded to the second order in the small parameters θ 13 and (∆m 2 sol L/E), [26]: whereJ = cos θ 13 sin 2θ 12 sin 2θ 13 sin 2θ 23 . The first non-trivial term is the dominant (atmospheric) contribution. It does not depend on the solar parameters and it reduces to the usual two-family approximation when θ 13 = 0 • . The last term is the subleading solar contribution, suppressed by two powers of the solar mass difference. This term is independent (in this approximation) from θ 13 . Eventually, the term in the second line is the interference between the atmospheric and the solar contributions: it is small but not negligible compared to the first term, being suppressed by only one power of the solar mass difference. This term encodes both a θ 13 -dependence (through theJ coefficient) and a small CP-conserving δ-dependence (suppressed by one power of ∆ sol and one power of θ 13 ). Changing the sign of the atmospheric mass difference makes the interference term change sign, also, mimicking an increase A 5% (2%) systematic error is assumed for the appearance (disappearance) channel, and backgrounds are computed as in ref. [19]. Continuous, dotted, dashed and dot-dashed lines stand for the intrinsic, sign, octant and mixed degeneracies, respectively. of ∆m 2 atm . Notice, finally, that the three non-trivial terms in eq. (3) are not symmetric for θ 23 → π/2 − θ 23 . However, the non-symmetric dependence on θ 23 is suppressed by at least two powers of θ 13 , ∆ sol or their combination, making the asymmetry extremely small.
In Fig. 5 we present the simultaneous measurement of (θ 13 , δ) using both the appearance (with a 5% systematic error) and the disappearance (with a 2% systematic error) channels at the Super-Beam. As it can be noticed, contrary to the β-Beam case, the disappearance channel in the Super-Beam fit introduces significant changes. Notably enough, the sign clone has disappeared in any case considered. This is not a surprise as these fits are performed at a fixed ∆m 2 atm : since in the disappearance channel the sign clone manifests itself at a larger value of ∆m 2 atm (see Fig. 4), in the combination with the appearance channel the tension between the two suffices to remove the unwanted clone in the (θ 13 , δ) plane. Notice, moreover, that in some cases the octant clone is considerably reduced or even solved, due to the octant-asymmetric contributions in the disappearance probability, eq. (3). Nonetheless, this does not mean that thanks to the combination of the appearance and the disappearance channels we are indeed able to measure the sign of the atmospheric mass difference, s atm . The mixed clones are generally still present for large values of θ 13 , thus preventing us from measuring s atm , if the θ 23 -octant is not known at the time the experiment takes place.
It is clear that these results should be confirmed by a complete multi-dimensional analysis that is actually underway [29]. As a first step, in Fig. 6(left) we show the projection on the (θ 13 , δ) plane of the Super-Beam appearance three-dimensional fit in the parameters (θ 13 , δ, ∆m 2 atm ) for the following input values:θ 13 = 8 • ,δ = 45 • and ∆m 2 atm = 2.5 × 10 −3 eV 2 . We let ∆m 2 atm varying freely in the range ∆m 2 atm ∈ [2.0 − 3.0] × 10 −3 eV 2 , obtained from Fig. 4. In Fig. 6(right) we show the same three-dimensional fit, adding this time both the Super-Beam appearance and disappearance channels. It can be seen that in a complete three-parameters analysis small remnants of the sign clones are still present, contrary to the case of the two-parameters analysis of Fig. 5. It is, however, still true that the degeneracy structure gets strongly reduced 5 . touch δ = 0 • , 180 • at θ 13 =θ 13 , also 6 . The outcome of this procedure is finally plotted, representing the region in the (θ 13 , δ)-parameter space for which a CP-violating signal is observed at 3σ. The novelty of this plot, with respect to Fig. 6 of [20] and to Fig. 13 of [13], is that we fully take into account the impact of the parameter degeneracies to derive the CP-violation discovery power of these facilities. Moreover, we present the results for the whole allowed range in δ, δ ∈ [−180 • , 180 • ]. This is particularly appropriate, since only an approximate symmetry is observed for |δ| ≥ π/2 and |δ| ≤ π/2 and no symmetry at all between positive and negative δ. For both facilities, we have applied a 2% systematic error on the disappearance channel (i.e., ν µ → ν µ and ν e → ν e ) and a 5% systematic error on the appearance channel (i.e., ν e → ν µ and ν µ → ν e ).
First of all, notice that the Super-Beam CP discovery potential in θ 13 is symmetric in δ (for both sectors, we observe an ultimate sensitivity of sin 2 θ 13 ≃ 6 − 8 × 10 −4 ). This is not the case for the β-Beam, where for positive δ the facility outperforms the Super-Beam and for negative δ is outperformed by it. We know, however, that this asymmetric behaviour of the β-Beam for positive and negative δ is a statistical mirage caused by the low background in the appearance antineutrino sample and the high background in the appearance neutrino one (see [19]). A proper statistical treatment should be performed, following [31], to get rid of this asymmetry in the small sin 2 θ 13 case: the treatment, however, is extremely time consuming and we do not consider meaningful applying it here. Regarding the discovery potential to δ, notice how for large θ 13 the Super-Beam is generally performing better than the β-Beam, in particular for negative δ. This can be understood comparing the right plots in Fig. 1 and Fig. 5: forθ 13 = 8 • ,δ = −90 • we can see that the 90% CL contours for the Super-Beam are much smaller than for the β-Beam. For positive δ the difference is not so relevant, since although the Super-Beam contours forθ 13 = 8 • ,δ = 45 • are certainly smaller than the β-Beam ones, it can be seen that the spread in δ is similar.
Eventually, we stress that for the Super-Beam a small improvement in the discovery potential in δ is achieved combining appearance and disappearance channels. On the contrary, practically no effect is observed in the β-Beam case.

Conclusions
Summarizing, in this letter we have tried to understand the impact of a disappearance measurement on the (θ 13 , δ) eightfold degeneracy for two specific facilities, the 4 MWatt SPL Super-Beam and the standard low-γ β-Beam proposed at CERN. We presented a complete analysis of degenerations in the ν e and ν µ disappearance channels: the ν e disappearance is affected by a twofold degeneracy, since the ν e probability depends on s atm only, eq. (2); the ν µ disappearance is affected by a fourfold degeneracy, depending on both s atm and s oct , eq. (3).
The standard low-γ β-Beam setup looks somewhat limited, when compared with facilities with many channels to exploit, such as the Neutrino Factory: indeed, the golden ν e → ν µ appearance channel is severely affected by degeneracies (being the neutrino energy too low to use energy resolution techniques) and the combination with the ν e → ν e disappearance, potentially of interest, is in practice useless once a realistic systematic error is taken into account. The β-Beam idea should be certainly pursued further, using for example higher γ options. For neutrino energies around 1 GeV is, in fact, possible to take advantage of energy resolution [32] and, for energies higher than 4-5 GeV, the silver channel ν e → ν τ becomes available. In both cases, the different informations can be used to reduce the parameter space degeneracies and solve some of the clones, consequently improving our knowledge on θ 13 and δ beyond the Super-Beam reach.
The SPL Super-Beam appearance channel, ν µ → ν e , is also severely affected by degeneracies (being a counting experiment, as the β-Beam). However, in this case the complementarity between the appearance and disappearance channels, ν µ → ν e and ν µ → ν µ , can be fully exploited even when a realistic systematic error is taken into account. In particular, the sign ambiguity can be strongly reduced (Fig. 6), a consequence of the fact that the disappearance sign clone is located at a different ∆m 2 atm for different choices of s atm , Fig. 4. Notice, finally, that the ν µ disappearance channel is interesting on its own for a precise measurement of the atmospheric oscillation parameters, (θ 23 , ∆m 2 atm ), Fig. 4.
It is clear that, in the case where only one of the two facilities were to be built at CERN, the 4 MWatt SPL Super-Beam would represent a more interesting choice than the standard low-γ β-Beam to study the leptonic mixing matrix. This is because the two experiments have a similar discovery potential to (θ 13 , δ) and Super-Beam can add useful informations from the disappearance ν µ channel. Anyway, it is now evident that the Super-Beam race against T2K is going to be lost. So one should concentrate on different beam technologies, β-Beam being one of the options. We believe that a higher-γ β-Beam should be considered instead of the standard one, in such a way that, adding spectral information, a measure of (θ 13 , δ) preciser than at the Super-Beam could be obtained.