Method for Determination of $|U_{e3}|$ in Neutrino Oscillation Appearance Experiments

We point out that determination of the MNS matrix element |U_{e3}| = s_{13} in long-baseline \nu_{\mu} \to \nu_e neutrino oscillation experiments suffers from large intrinsic uncertainty due to the unknown CP violating phase \delta and sign of \Delta m^2_{13}. We propose a new strategy for accurate determination of $\theta_{13}$; tune the beam energy at the oscillation maximum and do the measurement both in neutrino and antineutrino channels. We show that it automatically resolves the problem of parameter ambiguities which involves \delta, \theta_{13}, and the sign of \Delta m^2_{13}.


I. INTRODUCTION
With the accumulating evidences for neutrino oscillation in the atmospheric [1], the solar [2] and the accelerator neutrino experiments [3], it is now one of the most important subjects in particle physics to explore the full structure of neutrino masses and the lepton flavor mixing. In particular, it is the challenging task to explore the relatively unknown (1)(2)(3) sector of the MNS matrix [4], namely, θ 13 , the sign of ∆m 2 13 and the CP violating phase δ. The only available informations to date are the upper bound on θ 13 from the reactor experiments [5], and an indication for positive sign of ∆m 2 13 by neutrinos from supernova 1987A [6]. Throughout this paper, we use the standard notation of the three flavor MNS matrix, in particular U e3 = s 13 e −iδ , and define the neutrino mass-squared difference as ∆m 2 ij ≡ m 2 j − m 2 i . The long baseline ν µ → ν e neutrino oscillation experiment is one of the most promising way of measuring θ 13 . In particular, it is expected that the JHF-Kamioka project which utilizes low energy superbeam can go down to the sensitivity sin 2 2θ 13 ≃ 6 × 10 −3 [7]. A similar sensitivity is expected for the proposed CERN → Frejus experiment [8]. Although a far better sensitivity is expected to be achieved in neutrino factories [9], it is likely that the low energy conventional superbeam experiments are the ones which can start much earlier.
Therefore, it is of great importance to examine how accurately θ 13 can be determined in this type of experiments.
In this paper, we point out that determination of sin 2 2θ 13 by using only neutrino channel suffers from large intrinsic uncertainty of ± (30-70) % level due to the unknown CP violating phase δ and the undetermined sign of ∆m 2 13 . It should be noted that the intrinsic uncertainty exists on top of the usual experimental (statistical and systematic) errors. To overcome the problem of the intrinsic uncertainty, we suggest a new strategy for determination of θ 13 by doing appearance experiments utilizing both antineutrino and neutrino beams. Our proposal is a very simple one at least at the conceptual level; tune the beam energy to the oscillation maximum and run the appearance experiments in bothν µ →ν e and ν µ → ν e channels.
We will show that it not only solves the problem of intrinsic uncertainty mentioned above but also resolves the (δ − θ 13 ) two-fold ambiguity discussed in Ref. [10]. Furthermore, it does not suffer from possible ambiguity due to the unknown sign of ∆m 2 13 , the problem first addressed in Refs. [11,12]. 1 We are aware that there are combined ambiguities to be resolved (even ignoring experimental uncertainties) to determine a complete set of parameters including δ, θ 13 , and the sign of ∆m 2 13 , which are as large as four-fold [14]. We take experimentalists' approach to the ambiguity problem and try to resolve them one by one, rather than developing mathematical framework for the simultaneous solutions. The most important issue here is again to accurately determine θ 13 , because then all the combined ambiguities will be automatically resolved, as we will show below. Let us clarify how large uncertainty is expected for determination of θ 13 due to our ignorance of δ in the ν µ → ν e appearance experiment. To achieve intuitive understanding of the issue we use the CP trajectory diagram introduced in previous papers [11,12]. Plotted in Fig. 1 are the CP trajectory diagrams in bi-probability space spanned by P (ν) ≡ P (ν µ → ν e ) and P (ν) ≡ P (ν µ →ν e ) averaged over Gaussian distribution (see next paragraph) with three values of θ 13 , sin 2 2θ 13 = 0.05 and 0.02 for ∆m 2 23 > 0 case and sin 2 2θ 13 = 0.064 for ∆m 2 23 < 0 case. Since we assume |∆m 2 23 | ≫ |∆m 2 12 | the sign of ∆m 2 23 is identical with that of ∆m 2 13 . (The fourth one with sin 2 2θ 13 = 0.04 is for our later use.) The values of sin 2 2θ 13 for the second and the third trajectories are chosen so that the maximum (minimum) value of P (ν) of the second (third) trajectory coincides with about 1.1 %, the minimum value of P (ν) of the first trajectory. The remaining mixing parameters are taken as the best fit value of the Super-Kamiokande (SK) and the K2K experiments [15], |∆m 2 23 | ≡ ∆m 2 atm = 3 × 10 −3 eV 2 , and the typical ones for the large mixing angle (LMA) MSW solar neutrino solution as given in the caption of Fig. 1.
While we focus in this paper on the JHF experiment with baseline length of 295 km, JAERI-Kamioka distance, many of the qualitative features of our results remains valid also for the CERN-Frejus experiment. Throughout this paper we take the neutrino energy 1 Our new strategy and these results were announced in the 8th Tokutei-RCCN workshop [13]. distribution of Gaussian form with width of 20 % of the peak energy. Of course, it does not represent in any quantitatively accurate manner the effects of realistic beam energy spread and the energy dependent cross sections. But we feel that it is sufficient to make the point of this paper clear, illuminating our new strategy toward accurate determination of θ 13 .
Suppose that a measurement of appearance events gives us the value of the oscillation probability P (ν) ≃ 1.1 %. Then, it is obvious from Fig. 1 that a full range of values of sin 2 2θ 13 from 0.02 to 0.064 are allowed (even if we ignore experimental errors) due to our ignorance to the CP violating phase δ and the sign of ∆m 2 13 . 2 If we know that the sign is positive, for example, the uncertainty region would be limited to 0.02-0.05, which is still large.
Let us estimate in a syatematic way the uncertainty in the determination of θ 13 due to the CP violating phase δ. To do this we rely on perturbative formulae of the oscillation probabilities P (ν) and P (ν) which are valid to first order in the matter effect [18]. With relatively short baseline ∼ 300 km or less the first-order formula gives reasonably accurate results. Ignoring O(sin 3 2θ 13 ) terms the formula can be written with use of the notation (L and E denote baseline length and neutrino energy, respectively) in the form P (ν/ν) = P ± sin 2 2θ 13 + 2Q sin 2θ 13 cos 2 It may be worth to remark the following: Low energy neutrino oscillation experiments with superbeams are primarily motivated as a result of the search for the place where CP violating effects are comparatively large and easiest to measure [16]. See e.g., [17] for works preceding to [16]. Unfortunately, this large effect of δ is the very origin of the above mentioned large intrinsic uncertainty in determination of θ 13 .
where a = √ 2G F N e denotes the index of refraction in matter with G F being the Fermi constant and N e a constant electron number density in the earth. The ± signs in P ± refer to the neutrino and the antineutrino channels, respectively.
The maximum and the minimum of P (ν) for given mixing paramters, neutrino energy and baseline is obtained at cos δ + ∆ ij 2 = ±1. Then, the allowed region of sin 2θ 13 for a given value of P (ν), assuming blindness to the sign of ∆m 2 13 , is given by In Fig. 2 presented is the allowed region of sin 2 2θ 13 for a given value of measured oscillation probability P (ν). The size of the intrinsic uncertainty must be compared with the statistical and the systematic errors which are expected in the actual experiments. A detailed estimation of the experimental uncertainties is performed for the JHF experiment by Obayashi [19] assuming the off-axis beam (OA2) [7] and running of 5 years. The results strongly depend upon θ 13 .
We implemented these errors in Fig. 2b which is drawn with the similar energy as the peak energy of OA2 beam (∼ 780 MeV). We should note, however, an important difference between Fig. 2 and the plot in [19]; the abscissa of Fig. 2 is the Gaussian averaged probability, whereas the corresponding axis of the plot in [19] is the number of events. Therefore, we tentatively determined the location of errors in Fig. 2 so that the center of the error bars coincide with the center of the allowed band of sin 2 2θ 13 . Keeping this difference in mind, we still feel it informative for the readers to display the expected experimental uncertainties in Fig. 2b for comparison.
Therefore, the intrinsic uncertainty due to δ and undetermined sign of ∆m 2 13 is larger than the expected experimental errors in most of the sensitivity region for θ 13 in the experiment.
We note that the experimental errors are dominated by the statistical one in phase I of the JHF-SK neutrino project and hence it should be improved by a factor of ∼ 10 in two years of running in the phase II with a megaton water Cherenkov detector [7]. Thus, the intrinsic uncertainty completely dominates over the experimental ones if one stays only on the neutrino channel. It is tempting to think about seeking better resolution by adding more informations. A natural candidate for such possibilities in this line of thought is to do additional appearance experimentν µ →ν e using antineutrino beam. While it strengthens constraints, it does not completely solve the uncertainty problem even if we ignore the experimental errors. It is due to the inherent two-fold ambiguity which exists in simultaneous determination of δ and θ 13 as has been pointed out by Burguet-Castell et al. [10]. While their discussion anticipates applications to neutrino factory, the issue of the two-fold ambiguity is in fact even more relevant to our case because of the large effect of δ as we saw in the previous section.
The existence of the two-fold (θ 13 − δ) ambiguity is easy to recognize by using the CP trajectory diagram. We show in Fig. 1 by a dash-dotted curve another trajectory drawn with sin 2 2θ 13 = 0.04 which has two intersection points with the solid curve trajectory with sin 2 2θ 13 = 0.05. Suppose that measurements of neutrino and antineutrino oscillation probabilities P (ν) and P (ν) had resulted into either one of the two intersection points.
Then, it is clear that we have two solutions, for positive ∆m 2 13 , (sin 2 2θ 13 , δ) = (0.04, 0.65π) and (0.05, 0.35π) for the upper intersection point, and (sin 2 2θ 13 , δ) = (0.04, 1.4π) and (0.05, 1.7π) for the lower intersection point. Similar two-fold (θ 13 − δ) ambiguity also exists for negative ∆m 2 13 which however is not shown in Fig. 1. In other word, we can draw two different CP trajectories which pass through a point determined by given values of P (ν) and P (ν). This is the simple pictorial understanding of the (θ 13 − δ) two-fold ambiguity which is uncovered and analyzed in detail in [10]. We will show in the next two sections that the ambiguity is automatically resolved by our proposal.

IV. NEW STRATEGY FOR DETERMINATION OF θ 13
We now present our new strategy for determination of θ 13 which avoids the problem of the large intrinsic uncertainty. It is intuitively obvious from the CP trajectory diagram displayed in Fig. 1 that if one can tune the experimental parameters so that its radial thickness (which measures the cos δ term in Eq. (1)) vanishes then the two-fold ambiguity is completely resolved. It occurs if we tune the beam energy at the oscillation maximum so that ∆ 13 = π as is clear from Eq. (1).
We explain below in more detail how it occurs and then discuss by what kind of quantity θ 13 is determined. In the following discussion we assume that the mixing parameters We note that the oscillation probabilities (1) can be written as , and C ± = P ± sin 2 2θ 13 in the present approximation. It is easy to show from this expression that CP trajectory diagram is elliptic in the approximation that we are working [11]. (In fact, it is the case for all the known perturbative formulae.) Given (5) it is simple to observe that the CP trajectory is a straight line at the oscillation maximum, A = 0; the equation obeyed by the oscillation probabilities is given as P (ν) + P (ν) = C + + C − . Moreover, the first order matter effect cancels in C + + C − , leaving the vacuum peace of P ± . Therefore, the slope of the straight-line CP trajectory is the same as that in vacuum, and the matter effects affects only on the maximum and the minimum points of the straight line in P (ν) and P (ν) coordinates. Thus, once a set of values of P (ν) and P (ν) is given by the experiments, one can determine C + + C − as the segment of the "CP straight line" in the diagram, and hence sin 2 2θ 13 to which C + + C − is proportional.
Thus, measurement of P (ν) and P (ν) at the oscillation maximum implies determination of θ 13 without suffering from any uncertainties due to unknown value of δ and the sign of ∆m 2 13 . In Fig. 3  MeV if we sit on the oscillation maximum. It arises because the contributions from higher and lower energy parts around the peak energy do not completely cancel because of the extra 1/E factor in the cos δ term for symmetric Gaussian beam width. Thus, we need slightly higher energy to have the thinnest trajectory. It should be noted, however, that the feature highly depends upon the specific beam shape, and will also be affected by the fact that the cross section has an extra approximately linear E dependence.
The slightly different slope of the straight-line trajectories of positive and negative ∆m 2 13 indicates the higher order matter effect. This effect must be (and can be) taken into account when one try to determine θ 13 following the method proposed above.

AMBIGUITIES
We now show that the (θ 13 − δ) ambiguity is automatically resolved by tuning neutrino energy at the oscillation maximum. It must be the case because two straight-line trajectories with the same slope do not have intersection points. For our purpose, it suffices to work with oscillation probability at a fixed monochromatic beam energy because averaging over a finite width complicates the formalism and may obscure the essence of the problem. It can be shown [10] that the difference between the true (θ 13 ) and the false (θ ′ 13 ) solutions of θ 13 for a given set of P (ν) and P (ν) is given under the small θ 13 approximation by where z = P − + P + P − − P + tan ∆ 13 2 (7) Hence, the difference vanishes at the oscillation maximum, ∆ 13 = π, which means z → ∞.
It should be emphasised that our strategy of tuning beam energy at the oscillation maximum is not affected by the ambiguity correlated with the sign of ∆m 2 13 which is discussed in Ref. [11]. It is because the matter effect split the straight-line CP trajectories of positive and negative ∆m 2 13 toward the direction of the line itself in first order of the matter effect. The possible correction comes from higher order matter effect which is small in the relatively short baseline of the JHF (as well as the CERN → Frejus) experiment, as shown in Fig. 3.
The effect can be easily taken care of in the actual determination of θ 13 .

VI. CONCLUDING REMARKS
In this paper, we proposed a new strategy for accurate determination of θ 13 without suffering from the intrinsic ambiguity due to unknown value of δ. That is, tune the beam energy at the thinnest CP trajectory and do the measurement both in neutrino and antineutrino channels. We have shown that our new strategy completely resolves the ambiguities in the determination of θ 13 due to δ and due to the sign of ∆m 2 13 within the experimental accuracy attainable in such experiments.
One of the proposal which could be extracted from the strategy described in this paper is a possibility of havingν µ beam as early as possible. It would be the promising option for the case of relatively large sin 2 2θ 13 , say, within a factor of 2-3 smaller than the CHOOZ bound. In this case, the ν µ → ν e appearance events can be easily established in a few years of running of next generation neutrino oscillation experiments. Then, the uncertainties in determination of θ 13 would be greatly decreased by switching toν µ beam rather than just running with the ν µ beam.
What would be the implication of our strategy to the determination of δ? The tuning of beam energy at thinnest trajectory in fact also provides a good way of measuring δ. 3 The ambiguity (δ → π − δ), however, is unresolved and it would necessitate supplementary measurement either by using "fattest" trajectory configuration [11], or by second detector with different baseline distance [10]. We should emphasize that once θ 13 is measured accurately there is no more intrinsic ambiguities in determination of δ. We have explicitly shown that (δ −θ 13 ) ambiguity is resolved. The only ambiguity which would survive (from the viewpoint of determination of δ) would be the accidental one that arises in a correlated way (δ − sign of ∆m 2 13 ), which is nothing but the remnant of (δ → π − δ) degeneracy in vacuum [11]. But it is also resolved by either one of the two second measurements mentioned above.
Nore added: While this paper was being written, we became aware of the paper by Barger et al. [21] whose results partially overlaps with ours. However, most of the ambiguities discussed in the paper will be gone once θ 13 is determined accurately, as we noted above.

ACKNOWLEDGMENTS
We thank Takashi Kobayashi and Yoshihisa Obayashi for valuable informative correspondences on low energy neutrino beams, detector backgrounds, and θ 13 sensitivity in the JHF experiment. This work was supported by the Brazilian funding agency Fundação de Amparoà Pesquisa do Estado de São Paulo (FAPESP), and by the Grant-in-Aid for Scientific Research in Priority Areas No. 12047222, Japan Ministry of Education, Culture, Sports, Science, and Technology. 3 Tuning of beam energy at the oscillation maximum itself has been proposed before for differing reasons from ours. First of all, it is preferred experimentally because it maximizes disappearance of ν µ as well as the number of electron appearance events [7]. The tuning of beam energy to the oscillation maximum for measurement of CP violating phase δ was proposed by Konaka for the purpose of having maximal CP-odd (sin δ) term at the energy [20,7]. The beam profile, the mixing parameters and the matter density are taken as in Fig. 1.