# Measuring Dirac CP-violating phase with intermediate energy beta beam facility

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## Abstract

Taking the established nonzero value of \(\theta _{13}\), we study the possibility of extracting the Dirac CP-violating phase by a beta beam facility with a boost factor \(100<\gamma <450\). We compare the performance of different setups with different baselines, boost factors, and detector technologies. We find that an antineutrino beam from \(^6\)He decay with a baseline of \(L=1300\) km has a very promising CP-discovery potential using a 500 kton water Cherenkov detector. Fortunately this baseline corresponds to the distance between FermiLAB to Sanford underground research facility in South Dakota.

## Keywords

Lepton Sector Inverted Hierarchy Oscillation Probability Hadronic Shower Boost Factor## 1 Introduction

The developments in neutrino physics in recent 15 years have been overwhelmingly fast. Nonzero neutrino mass has been established and five out of nine neutrino mass parameters have been measured with remarkable precision. In 2012, finally, the relatively small mixing angle, \(\theta _{13}\) was measured [1, 2, 3]. This nonzero value of \(\theta _{13}\) opens up the possibility of having CP-violating effects in the neutrino oscillations; i.e., \(P(\nu _\alpha \rightarrow \nu _\beta )\ne P({\bar{\nu }}_\alpha \rightarrow {\bar{\nu }}_\beta )\). With this nonzero value of \(\theta _{13}\), the quest for measuring the Dirac CP-violating phase, \(\delta _D\), has been gathering momentum. A well-studied way to extract \(\delta _D\) is the precision measurement and comparison of \(P(\nu _\mu \rightarrow \nu _\mathrm{e})\) and \(P({\bar{\nu }}_\mu \rightarrow {\bar{\nu }}_\mathrm{e})\) by superbeam and neutrino factory facilities [4]. However, this is not the only way. In fact by studying the energy dependence of just one appearance mode e.g., \(P(\nu _\mu \rightarrow \nu _\mathrm{e})\), the value of \(\delta _{CP}\) can be extracted [5]. In [6], a novel method for extracting \(\delta _D\) (or more precisely \(\cos \delta _D\)) was suggested that was based on reconstructing the unitary triangle in the lepton sector. The idea of reconstructing the unitary triangle in the lepton sector has been later on studied in [7, 8, 9, 10, 11, 12, 13].

The flux at the detector scales as \(\gamma ^2\) and the scattering cross section of neutrinos increases by increasing the energy (i.e., increasing \(\gamma \)). As a result, for a given baseline, the statistics increases with \(\gamma \). Based on this observation, most attention in recent years has been given to \(\gamma >300\). However, one should bear in mind that for \(\gamma <300\), there is the advantage of using very large water Cherenkov (WC) detectors. In this energy range, the neutrino interaction will be dominantly quasi-elastic and its scattering cross section is known with high precision.

In the literature, the CP-discovery potential of a beta beam setup from CERN to Frejus with \(\gamma <150\) and \(L=130\) km has been investigated [27]. Moreover, varying the values of \(\gamma \) and baselines, it has been shown that for \(150<\gamma <300\) with 500 kton WC detector [28, 29, 30] or iron calorimeter [31], there is a very good chance of CP-discovery. Reference [22] explores the \(\theta _{13}\)–\(\delta _D\)-discovery reach with a 300 kton WC and 50 kton LAr detector at Deep Underground Science and Engineering Laboratory (DUSEL), taking maximum boosts possible at Tevatron. Now that the value of \(\theta _{13}\) is measured and found to be sizeable, reconsidering \(\gamma <300\) setup is imperative. In vacuum, the dependence of the oscillation probability on \(L\) and \(\gamma \) would be through \(L/\gamma \). However, for setups under consideration, because of the matter effects, the dependence on \(L\) and \( \gamma \) is more sophisticated so the dependence on \(E\) and \(L\) has to be investigated separately. In particular, while Ref. [28] focuses on \(L/\gamma =2.6\) km, we have found that for \(L/\gamma >2.6\) km, there is a very good discovery potential. The present paper is devoted to such a study.

In Sect. 2, we describe the inputs and how we carry out the analysis. We outline the characteristics of the beam and the detector as well as the sources of the background and the systematic errors. In Sect. 3, we present our results and analyze them. In Sect. 4, we summarize our conclusions and propose an optimal setup for the \(\delta _D\) measurement.

## 2 The inputs for our analysis

Using the GloBES software [32, 33], we investigate the CP-discovery potential of a beta beam setup with various baselines and beam boost factors, \(\gamma \). For the central values of the neutrino parameters, we have taken the latest values in Ref.[34]. The hierarchy can be determined by other experiments such as PINGU [35, 36, 37, 38, 39, 40, 41] or combining PINGU and DAYA Bay II results [42], so we assume that this hierarchy is known by the time that such a beta beam setup is ready. We study both normal and inverted hierarchies. T2K and Nova can also solve the octant degeneracy and determine whether \(\theta _{23}<45^\circ \) or \(\theta _{23}>45^\circ \) [43]. The data already excludes the \(\theta _{23}>45^\circ \) solution at 1\(\sigma \) CL. For the uncertainty of the mixing parameters, we take the values that will be achievable by forthcoming experiments. Namely, we take the following uncertainties: 0.4 % for \(\theta _{12}\) [44], 1.8 % for \(\theta _{13}\) [45], 2 % for \(\theta _{23}\) [46], 0.2 % for \(\Delta m_{12}^{2}\) [44] and 0.7 % for \(\Delta m_{13}^{2}\) [35, 36, 37, 38, 39, 40]. As predefined by GLoBES, we use the matter profiles in [47, 48]. We consider 5 % error for matter density. The uncertainties are treated by the so-called pull-method [32, 33]. While the effects of uncertainty in matter density is more important for larger baselines, the uncertainties of neutrino parameters affect the results from smaller baselines more.

While the disappearance probabilities (i.e., \(P(\nu _\mathrm{e} \rightarrow \nu _\mathrm{e})\) or \(P({\bar{\nu }}_\mathrm{e}\rightarrow {\bar{\nu }}_\mathrm{e})\)) are not sensitive to \(\delta _D\), the appearance probabilities (i.e., \(P(\nu _\mathrm{e} \rightarrow \nu _\mu )\) or \(P({\bar{\nu }}_\mathrm{e}\rightarrow {\bar{\nu }}_\mu )\)) contain information on \(\delta _D\). In our analysis, we, however, employ both appearance and disappearance modes. In principle, the disappearance mode can help to reduce the effect of uncertainty in other parameters but we have found that the effect of turning off the disappearance mode on the \(\delta _D\) measurement is less than 1 %. To derive the value of \(\delta _D\), the detector has to distinguish \(\nu _\mu \) from \(\nu _\mathrm{e}\). We focus on a 500 kton water Cherenkov (WC) detector and compare its performance with a 50 kton totaly active scintillator detector (TASD).

In the energies of our interest with \(\gamma <300\), the main interaction mode is Charged Current (CC) quasi-elastic mode with a non-negligible contribution from inelastic charged current interaction which produce one or more pions along with the charged lepton. In principle, the quasi-elastic CC events can be distinguished from the inelastic CC ones by counting the number of Cherenkov rings. However, to distinguish the two interactions with a WC detector will be challenging. We take the signal to be composed of both quasi-elastic and inelastic charged current events and conservatively assume that the WC detector cannot distinguish between the two.

In the case of QE interaction by measuring the energy and the direction of the final charged lepton, the energy of the initial neutrino can be reconstructed up to an uncertainty of 0.085 GeV, caused by the Fermi motion of the nucleons inside the nucleus. However, in the inelastic interaction, a fraction of the initial energy is carried by pions, so the energy of the initial neutrino cannot be reconstructed by measuring the energy and the direction of the final lepton alone. A WC detector cannot measure the energy deposited in hadronic showers, so with this technique the reconstruction of the energy spectrum will be possible only for the QE interactions. Following the technique in [28, 50] we take an unknown normalization for QE events and use its spectrum as a basis for energy reconstruction. Of course, with this method energy reconstruction cannot be carried out on an event by event basis and information on the spectrum will only be statistical. TASD can measure the energy deposited in hadronic showers, too. As a result, energy reconstruction by TASD can be possible on an event by event basis.

As shown in [51], the background from atmospheric neutrinos can be neglected and the main source of background for both TASD and WC detectors are neutral current interactions of the beam neutrinos. In our analysis, for the cross sections of the quasi-elastic, inelastic and neutral current interactions we employ the results of [52, 53]. Recently the MiniBooNE collaboration has measured the antineutrino cross section in the energy range of our interest [54] with remarkable precision. In the near future, the measurement of the cross section will become even more precise. Unless otherwise stated, we assume 4 years of data taking.

For the treatment of the efficiencies and backgrounds we implement the same methods as used in [28, 50]. While for the purpose of this paper the methods used in [28, 50] are adequate, we would like to note that a more complete discussion of reconstruction of events in the large WC detectors can be found in [51]. To be more specific, similarly to [28] we assume the following characteristics for the WC detector performance. We take a signal efficiency of 55 % for neutrinos and of 75 % for antineutrinos. We take the uncertainty in the normalization of the total signal to be 2.5 % but as we mentioned above, we take a free normalization for QE events. We assume a background rejection of 0.3 % for neutrinos and 0.25 % for antineutrinos. The normalization uncertainty of the background is taken to be 5 %. For both background and signal, the calibration error is 0.0001. For the energy reconstruction of the background, we use the migration matrices tabulated for the GLoBES package [55, 56]. We consider the energy range between 0.2 and 3 GeV and divide it into 28 bins. The energy resolution for QE CC interactions is assumed to be of the form \(0.085+0.05\sqrt{E/\hbox {GeV}}\) GeV for both muon and electron neutrino detection. The first term originates from the Fermi motion of the nucleons inside nuclei and the second term reflects the error in measuring the energy of the final charged lepton [22].

As in Ref.[28], we assume the following features for TASD: A signal efficiency of 80 % for \(\nu _\mu \) and \({\bar{\nu }}_\mu \) and of 20 % for \(\nu _\mathrm{e}\) and \({\bar{\nu }}_\mathrm{e}\); background rejection of 0.1 %; a signal normalization uncertainty of 2.5 %; normalization uncertainty of 5 %; a calibration error of 0.0001. The energy resolution is given by \(0.03\sqrt{E/\mathrm{{GeV}}}\) GeV for muon (anti)neutrinos and \(0.06 \sqrt{E/\mathrm{{GeV}}}\) GeV for electron (anti)neutrinos. The energy range is taken to be 0.5–3.5 GeV and is divided into 20 bins. We have studied the dependence of our results on the number of bins. It seems that the results do not change by increasing the number of bins to 30.

## 3 Results and the interpretation

The oscillatory behavior of the curves is driven by the 13-splitting and has a frequency given by \(\sim \Delta m_{31}^2/(2E)\). Such a behavior can be understood by the following consideration on Eqs. (1) and (2): While \(\lambda _2-\lambda _1\) is driven by \((\Delta m_{21}^2)_\mathrm{{eff}}\) and slowly varies with \(L\), \(\lambda _3-\lambda _2\) is driven by \((\Delta m_{32}^2)_\mathrm{{eff}}\) and varies rapidly. For the values of \(L\) with \(\lambda _3-\lambda _2=2n\pi \), the sensitivity is lost. This consideration explains the oscillatory behavior of Fig. 1. Notice, however, that this consideration holds for a given \(E_\nu \). If the energy spectrum is wide, the effect will smear out. In other words, if the number of energy bins from which information on \(\delta _D\) can be deduced (i.e., the bins for which the number of events without oscillation is sizeable and the quasi-elastic interactions dominate) is relatively large, missing information in few of these bins for which \((\lambda _3-\lambda _2)\rightarrow 0\) will not affect much the precision in the determination of \(\delta _D\). In the opposite case, when at all such bins \((\lambda _3 -\lambda _2)\rightarrow 0\), the precision in \(\delta _D\) will be dramatically deteriorated. Increasing the boost factor increases both the peak energy and the energy width. Thus, we expect for higher \(\gamma \) that the oscillatory behavior is to be smeared out. Figure 1 confirms this expectation. In the case of antineutrinos, the information on \(\delta _D\) can be deduced from a larger range of the spectrum mainly because of the shape of the spectrum at the source and the fact that for antineutrinos, the QE interactions dominate over the inelastic interaction for a wider energy range compared to the case of neutrino [52, 53]. As a result, the modulation driven by \(\Delta m_{31}^2\) is less severe for antineutrinos. As seen in the lower panels of Fig. 1, the antineutrino beam with \(\gamma =300\) and a WC detector can achieve an impressive precision of better than 20\(^\circ \) for baselines over 500 km.

For \(L=1300\) km, (corresponding to the baseline of the LBNE setup from the FermiLAB to Sanford underground research facility in South Dakota [57]), we also observe that \(\gamma =300\) with WC detector is promising and can outperform the \(\gamma =450\) setup with TASD detector. Figure 5 shows \(\Delta \delta _D\) versus \(\gamma \) for \(L=1300\) km and \(L=2300\) km. The latter corresponds to the baseline for the LBNO setup from CERN to Finland [58]. The plots show that the setup with \(\gamma =200{-}300\) and \(L=1300\) km can measure \(\delta _D=90^\circ \) with a remarkable precision and also have an outstanding coverage of the \(\delta _D\) range. At this baseline, increasing \(\gamma \) from 200 to 300 does not much improve the sensitivity to \(\delta _D\).

For the 130 km setup, the oscillation probabilities can be approximately written as \(P_{\bar{e}\bar{\mu }}\simeq |i s_{12}^mc_{23}\sin (\lambda _2-\lambda _1)+s_{13}^ms_{23} \hbox {e}^{i\delta _D}(\hbox {e}^{i \lambda _3}-1)|^2\) and \(P_{e\mu }\simeq |ic_{12}^mc_{23}\sin (\lambda _2-\lambda _1)+s_{13}^ms_{23} \hbox {e}^{-i\delta _D}(\hbox {e}^{i \lambda _3}-1)|^2\). Since we are far from the 31-resonance, \(s_{13}^m\) is not very different from \(s_{13}\) and is approximately the same for normal and inverted hierarchies. As a result, replacing \(\delta _D\rightarrow \pi -\delta _D\) and \(\Delta m_{13}^2\rightarrow -\Delta m_{13}^2\) (i.e., \(\lambda _3 \rightarrow -\lambda _3\)), the oscillation probability does not change. That is why in Fig. 6 the \( \Delta \delta _D\) plots for normal and inverted hierarchies are practically the same. If we take a value other than \(90^\circ \) as the true value of \(\delta _D\), we will not have such a symmetry. However, as seen in Figs. 5 and 6, the general behavior for normal and inverted hierarchies are similar. With the present SPS setup, CERN cannot enhance \(\gamma \) over 150 for the \(^6\)He ions [29]. On the other hand, from Fig. 6, we observe that with \(L=130\) km, there is no point in seeking higher values of \(\gamma \). In fact, at \(\gamma =150\), the fraction of CP-violating \(\delta _D\) parameter for which CP-violation can be established is slightly higher than that for \(\gamma >250\).

From comparing Figs. 5 and 6, we observe that the best performance can be achieved by an \(L=1300\) km setup and antineutrino run. For example, while with the CERN to Frejus setup, \(\delta _D=90^\circ \) can be measured with only an uncertainty of \(\Delta \delta _D=35^\circ \), with a 1,300 km setup, the uncertainty can be lowered down to \(\Delta \delta _D=15^\circ \). Notice that for these setups, the same detector (500 kton WC) is assumed. Although with longer baselines the flux decreases, instead \(\lambda _2-\lambda _1\) in Eq. (2) becomes larger, so a moderate precision in the \(P_{{\bar{e}}{\bar{\mu }}}\) measurement will suffice to extract \(\delta _D\). For measuring \(\delta _D=90^\circ \), the \(L=2300\) km setup with the \({\bar{\nu }}_\mathrm{e}\) run seems to be competitive with the \(L=1300\) km setup; however, the fraction of \(\delta _D\) to be established by the \(L=1300\) km setup is considerably higher. Among the setups that we have considered the \(L=1300\) km setup with a 500 kton WC and the \({\bar{\nu }}\) run seems to be the most promising one. In Fig. 5, we observe that for \(200<\gamma <300\), the curves corresponding to the \({\bar{\nu }}_\mathrm{e}\) run are almost flat.

## 4 Conclusions

Measuring the CP-violating phase by a beta beam facility has been extensively studied in the literature. Most of the recent studies have focused on relatively high energy beams with \(\gamma >300\). The reason is that for a given baseline, the number of detected neutrinos increases approximately as \(\gamma ^3\). However, for lower energy beta beam, a large volume WC detectors [30] can be employed that can compensate for the decrease of flux and cross section. Moreover, with the relatively large value of \(\theta _{13}\) chosen by Nature, having enough statistics will not be the most serious challenge for measuring the CP-violating phase. Considering these facts, we have explored the CP-discovery reach of an intermediate energy beta beam for various baselines and different neutrino vs. antineutrino combinations using the GLoBES software [32, 33]. We have discussed the precision with which \(\delta _D\) can be measured, assuming that by the time that the required facilities are ready the hierarchy is also determined. Our results do not depend much on which mass ordering is chosen.

We have found that a setup with only an antineutrino run with \(200<\gamma <300\) and a baseline of \(L=1,300\) km has an excellent discovery potential. Four years run of such a setup with \(5.8\times 10^{18}\) \(^6\)He decays per year can establish CP-violation at 95 % CL for more than 85 % of the \(\delta _D\) parameter range. If \(\delta _D=90^\circ \), this setup can determine it with impressive precision \(\delta _D=90^\circ \pm 8^\circ \) for an inverted hierarchy and \(\delta _D=90^\circ \pm 7^\circ \) for a normal hierarchy at 1\(\sigma \) CL. Such a baseline corresponds to the distance between FermiLAB to Sanford underground research facility in South Dakota. A baseline of \(L=1300\) km seems to be close to the optimal distance to measure the Dirac CP-violating phase. We have found that for this baseline a setup with intermediate values of \(\gamma \) in the range \(200<\gamma <300\) with a 500 kton WC detector can outperform that with \(\gamma =450\) and 50 kton TASD.

For very long baselines with \(L>1000\) km, a pure antineutrino source from \(^6\)He enjoys a better performance than a mixed neutrino antineutrino run. On the other hand for shorter baselines, a balanced neutrino–antineutrino mode gives better results. We have specifically discussed the CERN to Frejus setup with \(L=130\) km baseline. We have found that with 2 years of neutrino mode from \(2.2 \times 10^{18}\) decays of \(^{18}\)Ne per year combined with 2 years of antineutrino mode from \(5.8 \times 10^{18}\) decays of \(^6\)He per year both with \(\gamma =150\) (the largest boost that can be obtained for \(^6\)He with the present SPS accelerator at CERN [29]), the CP-violation can be established for about 80 % of the \(\delta _D\) parameter range. With such a setup and runtime, if the true value of \(\delta _D\) is equal to \(90^\circ \), it can be measured as \(\delta _D=90^\circ \pm 18^\circ \) at 1\(\sigma \) CL. By increasing \(\gamma \) to higher values, the precision in the \(\delta _D\) measurements slightly improves, however, still with a similar detector and antineutrino run, the performance of \(L=1,300\) km can be better.

Our conclusion is that a beta beam facility with \(200<\gamma <300\), a baseline of \(L\simeq 1300\) km, and 500 kton WC running in the antineutrino mode from \(^6\)He decay is an optimal option for establishing CP-violation in the lepton sector and the measurement of \(\delta _D\). The location of source and detector might be, respectively, FermiLAB and Sanford underground laboratory in South Dakota.

## Notes

### Acknowledgments

The authors would like to thank A. Yu. Smirnov for encouragement and very useful comments. They also thank F. Terranova and W. Winter for fruitful comments. P. B. acknowledges H. Mosadeq for technical help in running the computer codes. They also acknowledges partial support from the European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442). Y.F. thanks the staff of Izmir technical institute (IZTECH) where a part of this work was carried out for generous support and hospitality. The authors also thank the anonymous referee for useful remarks.

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