2540 km: Bimagic baseline for neutrino oscillation parameters
We show that a source-to-detector distance of 2540 km, motivated recently 2540-umasankar () for a narrow band superbeam, offers multiple advantages for a low energy neutrino factory with a detector that can identify muon charge. At this baseline, for any neutrino hierarchy, the wrong-sign muon signal is almost independent of CP violation and in certain energy ranges. This allows the identification of the hierarchy in a clean way. In addition, part of the muon spectrum is also sensitive to the CP violating phase and , so that the same setup can be used to probe these parameters as well.
Introduction.— The data from ongoing neutrino experiments confirm that neutrinos have distinct masses and the three neutrino flavors mix among themselves. While the mass squared difference and the magnitude of , as well as two of the mixing angles, and , are well measured, three parameters of the leptonic mixing matrix still remain elusive: the mixing angle , the sign of , and the CP phase latest-fit (). The determination of hierarchy (NH/normal: , IH/inverted: ), in particular, would be crucial in identifying the mechanism of neutrino mass generation muchun ().
If the actual value of is not much below the current bound of , it may be measured at detectors at a distance of km from a reactor/accelerator. In order to determine the hierarchy, however, the most efficient avenue is to have the neutrinos travel through Earth for thousands of km before detection. Here, the difference between Earth matter effects in the two hierarchies can help in distinguishing them. This can be achieved, for instance, by using the decay of accelerated muons – or – as a source (“neutrino factory” (NF)) and a detector that can detect muons and identify them as right-sign (the same sign as the source) or wrong-sign.
The wrong-sign muon signal is hailed as the “golden channel” since it is sensitive to all the three parameters: , the sign of , and . However, the dependence on also introduces large uncertainties, making the unambiguous determination of the true parameters difficult degpapers (); magic (). A potential way out is to have the detector at km (“the magic baseline” magic (); magic2 ()) from the source, where the effect of CP violation vanishes for both hierarchies. However, this very feature makes it impossible to measure the CP phase at this baseline. Moreover, such a long baseline requires an extremely well-collimated muon source, else the flux at the detector is highly reduced.
It is therefore desirable to look for a shorter baseline that will still give a wrong-sign muon signal independent of the CP phase for one of the hierarchies, albeit only in a part of the spectrum. The remaining part of the spectrum would still be sensitive to the CP phase and can be used to detect CP violation for the same hierarchy.
In the context of a superbeam, it was recently pointed out 2540-umasankar () that the baseline of 2540 km satisfies the above condition for IH at a neutrino energy of GeV and a narrow band neutrino beam was therefore deemed desirable. In this Letter, we point out that this baseline also satisfies the desired condition for NH, at the energy 1.9 GeV. The two energies at which the desired condition is satisfied are termed as magic energies, and the baseline is referred to as “bimagic”. The bimagic property, first realized in this work, makes it more desirable to have a broadband beam covering the range 1–4 GeV. We use the – channel in a low energy neutrino factory (LENF) with a muon energy of 5 GeV lownufac (), as opposed to the – channel used for superbeams 2540-umasankar (). The detection of muons is easier compared to that of electrons. Moreover with muon charge identification, NFs do not have beam contamination problems, thus enabling sensitivity to smaller values. Thus, the bimagic nature in conjunction with a LENF helps in an efficient identification of hierarchy, nonzero and CP violation, even with a single polarity of decaying muons, as we shall motivate and demonstrate in this Letter. It is remarkable that the distance 2540 km also happens to be close to the distance between Brookhaven and Homestake bnlhomestake (), as well as that between CERN and Pyhasalmi mine pyhasalmi (), which is one of the proposed sites for the LENA detector.
The bimagic baseline.— The source beam from a neutrino factory that accelerates consists of and . Charged current interactions at the detector can give muons in two ways: the original that survive as give (right-sign muons) while the original that oscillate to give (wrong-sign muons). The oscillation probability , relevant for the wrong-sign muon signal, can be written in the constant matter density approximation as akhmedov-prob ()
keeping terms up to second order in and . Here , . Also,
where is the Fermi constant and is the electron number density. For neutrinos, the signs of and are positive for normal hierarchy and negative for inverted hierarchy. picks up an extra negative sign for anti-neutrinos. The last term in Eq. (1) clearly mixes the dependence on hierarchy and , leading to a degeneracy between them degpapers (), which can be overcome if one manages to have either or . The first condition is achieved at the magic baseline ( km) for all and for both the hierarchies. The second condition, on the other hand, is sensitive to hierarchy. This sensitivity can be maximized if one has for one of the hierarchies and for the other. In such a situation, only the term in Eq. (1) survives for the hierarchy for which , making independent of both and . At the same time, for the other hierarchy the first term in Eq. (1) enhances the number of events as well as sensitivity, and the third term enhances the sensitivity to .
If we demand “IH-noCP” (no sensitivity to CP phase in IH), these conditions imply
where are integers, . These two conditions are exactly satisfied at a particular baseline and energy, given by
Note that the relevant is independent of any oscillation parameters. A viable solution for these set of equations (with and ) is km, g/cc and GeV, as was first pointed out in 2540-umasankar (). On the other hand, one may demand “NH-noCP” (no sensitivity to CP phase in NH), which leads to the conditions
with integers, and . These lead to the same condition on as in Eq. (4a) except for an overall negative sign, while continues to be given by Eq. (4b). These conditions are also satisfied at km (for and ) at GeV. The magic energies and would be suitable for a neutrino factory with a parent muon energy of GeV.
Eqs. (4a, 4b) indicate that many combinations of and are possible for a given baseline. Indeed, the 2540 km baseline also satisfies IH-noCP at GeV () and NH-noCP at GeV (). However the flux at these energies would be small, so we do not consider these in this Letter.
Fig. 1 shows the probability for . In this and all other plots, we have solved the exact neutrino propagation equation numerically using the Preliminary Reference Earth Model prem (). Clearly the IH-noCP and NH-noCP conditions are satisfied at the energies and , respectively. At , the probabilities for NH and IH are distinct, hence a measurement of the neutrino spectrum around this energy would be a clean way of distinguishing between the hierarchies. The oscillatory nature of for non-zero vis-a-vis the monotonic behavior for helps in the discovery of a nonzero . Finally, the significant widths of the bands (near for NH, and near for IH) imply sensitivity to .
The simplified forms of probabilities at the magic energies offer insights into the CP sensitivity at this baseline. At , we have
while at , we have
Near the magic energies, where the CP sensitivity is the highest, the values giving the highest and the lowest probabilities would be and , respectively.
Experimental setup and numerical simulation.— We use a magnetized totally active scintillator detector (TASD) which is generally used in the context of a LENF lownufac (). We use a 25 kt detector with a energy threshold of 1 GeV. We choose a typical Neutrino factory setup with 5 GeV parent muon energy and useful muon decays per year huber (); lenf-improve (). We consider the running with only one polarity of the parent muon, so that we have a neutrino flux consisting of and . We assume a muon detection efficiency of 94% for energies above 1 GeV, 10% energy resolution for the whole energy range up to 5 GeV and a background level of for the and channels. Detection of or is not considered in this study, which seems to have a very small effect when the initial flux is as large as above lenf-improve (). A 2.5% normalization error and 0.01% calibration error, both for signal and background, have also been taken into account throughout this study. The detector characteristics have been simulated by GLoBES globes ().
The top panel in Fig. 2 shows the energy spectra of wrong-sign muon events. For illustration, in addition to , we choose which would give the maximum dependence near the magic energies, as indicated by Eqs. (6) and (7). It is clear from this figure that there is considerable sensitivity to CP phase near GeV for NH (IH). It may be noted from the figure that the CP sensitivity for IH is actually better at slightly higher energies than . This is because the spectrum at the source as well as the cross section of at the detector are strongly increasing functions of energy around GeV, and push the peak in the IH spectrum to higher energies.
In order to illustrate the effectiveness of the magic energies, we show in the bottom panel of Fig. 2 the total number of events in two bins near the magic energies – (3.0–3.6 GeV) and (1.7–2.1 GeV) – as functions of . Clearly, the bin itself is enough to identify the hierarchy as long as . If the actual hierarchy is NH, this bin is also sensitive to . The sensitivity to may be estimated by comparing the error bars at different values. If the actual hierarchy is IH, one needs the events data from the energy bin in order to discern and . The actual identification of hierarchy and the measurement of and is done by using the complete wrong-sign events spectrum as well as the right-sign events spectrum. We present below the results of this analysis.
Mass hierarchy determination.— In Fig. 3 we quantify the hierarchy sensitivity of the bimagic neutrino factory setup. The experimental data are generated with the chosen true hierarchy. The true values of and are plotted along the axes while the true values of the other parameters are set to the values quoted in Fig. 1. For each pair of –, we obtain by marginalizing over other parameters. We have taken 4% error on each of and , and 5% error on each of and , for calculating the priors. has been varied over . A 2% error has also been considered on the earth matter profile and marginalized over.
The contours in Fig. 3 suggest that if the true hierarchy is NH, then for favorable values of , an exposure of muonskt may determine the hierarchy at 3 even for as small as . If the true hierarchy is IH then that can be established at 3 for . This sensitivity is better than that indicated by the superbeam studies at this baseline bnlhomestake (); 2540-umasankar ().
An optimized LENF setup with a baseline of 1300 km has been recently proposed lenf-improve (). However, the relatively small baseline does not allow matter effects to develop sufficiently, and one does not have the advantage of the magic energies. So the sensitivity of this setup to the hierarchy is rather limited. Indeed if the true hierarchy is NH, the bimagic baseline will rule out IH at 3 for values almost an order of magnitude smaller than the expected reach of the 1300 km setup with the same exposure. If IH is the true hierarchy, the performance of both the setups is almost the same. Thus the bimagic baseline is a more optimal setup as far as the hierarchy is concerned.
and measurement.— The top panel of Fig. 4 shows that the exposure of muonskt will be able to discover a nonzero to 3 as long as for either hierarchy and for any value. For NH and , the discovery of is possible even for as low as .
The bottom panel of Fig. 4 shows the discovery reach with this setup. It shows that the exposure allows the discovery of nonzero for NH for as low as , as long as . This is the value at which we expect the highest deviation in the events spectrum from , as indicated by Eqs. (6) and (7). For IH, the results are about one order of magnitude worse than those for NH.
The discovery potential for and at the bimagic baseline is comparable to that of the 1300 km setup if the true hierarchy is NH, while it is not as good if the true hierarchy is IH. However, note that this is valid if only are available at the source. With both polarities available, the bimagic baseline would be almost as good as the 1300 km setup for and , and will have a better sensitivity to the hierarchy. Indeed, once the hierarchy is identified – for which the bimagic baseline performs better – running the bimagic setup with () as the source beam for NH (IH) would offer a sensitivity similar to the 1300 km setup. Thus, overall the bimagic baseline seems like an optimal one to probe the three most important unknown parameters of the leptonic mixing matrix: and the sign of .
Conclusion.— We have shown the “bimagic” nature of the 2540 km baseline: at this baseline with judicious choice of energies, the dependence of the wrong-sign muon signal on and can be made to vanish for either hierarchy. This energy turns out to be around 3.3 GeV for IH and 1.9 GeV for NH. This helps in an efficient identification of hierarchy even at very low , when one uses a neutrino factory with parent muon energy GeV as a source. On the other hand the sensitivity to and is maximum at 3.3 GeV for NH and 1.9 GeV for IH, allowing the determination of these parameters as well with the same beam-baseline setup. To exploit these features, a broadband beam of a neutrino factory is more effective as compared to a narrow band beam.
Acknowledgments.— We thank P. Ghoshal, J. Kopp, S. Uma Sankar and W. Winter for useful discussions. S.R. would like to thank Prof. Yuval Grossman for support and hospitality.
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