Measuring the CP-violating phase by a long base-line neutrino experiment with Hyper-Kamiokande

We study the sensitivity of a long-base-line (LBL) experiment with neutrino beams from the High Intensity Proton Accelerator (HIPA), that delivers 10^{21} POT per year, and a proposed 1Mt water-Cherenkov detector, Hyper-Kamiokande (HK) 295km away from the HIPA, to the CP phase (delta_{M N S}) of the three-flavor lepton mixing matrix. We examine a combination of the nu_mu narrow-band beam (NBB) at two different energies, vev{p_pi}=2, 3GeV, and the bar{nu}_mu NBB at vev{p_pi}=2GeV. By allocating one year each for the two nu_mu beams and four years for the bar{nu}_mu beam, we can efficiently measure the nu_mu to nu_e and bar{nu}_mu to bar{nu}_e transition probabilities, as well as the nu_mu and bar{nu}_mu survival probabilities. CP violation in the lepton sector can be established at 4sigma (3sigma) level if the MSW large-mixing-angle scenario of the solar-neutrino deficit is realized, |\dmns| or |delta{M N S}-180^{circ}|>30^{circ}, and if 4|U_{e3}|^2 (1-|U_{e3}|^2) equiv sin^2 2 theta_{CHOOZ}>0.03 (0.01). The phase delta_{M N S} is more difficult to constrain by this experiment if there is little CP violation, delta_{M N S} sim 0^{circ} or 180^{circ}, which can be distinguished at 1sigma level if sin^2 2 theta_{CHOOZ}>~ 0.01.

Neutrino oscillation experiment is one of the most attractive experiments in the first quarter of 21st century. Many experiments will measure precisely the model parameters in the neutrino oscillations. In this article, we discuss the sensitivity of a long-base-line (LBL) experiment with conventional neutrino beams to measure the CP phase in the lepton sector.
The Super-Kamiokande (SK) collaboration showed that the ν µ created in the atmosphere oscillates into ν τ with almost maximal mixing [1]. The SNO collaboration reported that the ν e 's from the sun oscillate into the other active neutrinos [2]. A consistent picture in the three active-neutrino framework is emerging.
In the three-neutrino-model, neutrino oscillations depend on two mass-squared differences, three mixing angles and one CP violating phase of the lepton-flavor mixing (Maki-Nakagawa-Sakata (MNS) [3]) matrix. These parameters are constrained by the solar and atmospheric neutrino observations. One of the mixing angles and one of the mass-squared differences are constrained by the atmospheric-neutrino observation, which we may label [4] as sin 2 θ ATM and δm 2 ATM , respectively. The K2K experiment, the ongoing LBL neutrino oscillation experiment from KEK to SK, constrains the same parameters [5]. Their findings are consistent with the maximal mixing, sin 2 2θ ATM ∼ 1 (sin 2 θ ATM ∼ 0.5) and δm 2 ATM ∼ (2 ∼ 4) × 10 −3 (eV 2 ). The solar-neutrino observations constrain another mixing angle and the other mass-squared difference, sin 2 2θ SOL and δm 2 SOL , respectively. Four possible solutions to the solar-neutrino deficit problem [6] are found: the MSW [7,8] largemixing-angle (LMA) solution, the MSW small-mixing-angle (SMA) solution, the vacuum oscillation (VO) solution [9], and the MSW low-δm 2 (LOW) solution. The SK collaboration [6] and the SNO collaboration [2] suggested that the MSW LMA solution is the most favorable solution among them, for which sin 2 2θ SOL = 0.7 ∼ 0.9 and δm 2 SOL = (3 ∼ 15) × 10 −5 eV 2 . For the third mixing angle, only the upper bound is obtained from the reactor neutrino experiments. CHOOZ [10] and Palo Verde [11] found sin 2 2θ RCT < 0.1 for δm 2 ATM ∼ 3×10 −3 eV 2 . No constraint on the CP phase (δ MNS ) has been reported.
Several future LBL neutrino-oscillation experiments [12]- [15] have been proposed to confirm the results of these experiments and to measure the neutrino oscillation parameters more precisely. One of those experiments proposed in Japan makes use of the beam from High Intensity Proton Accelerator (HIPA) [16] and SK as the detector [15]. The facility HIPA [16] has a 50 GeV proton accelerator to be completed by the year 2007 in the site of JAERI (Japan Atomic Energy Research Institute), as a joint project of KEK and JAERI. The proton beam of HIPA will deliver neutrino beams of sub-GeV to several GeV range, whose intensity will be two orders of magnitudes higher than that of the KEK PS beam for the K2K experiment. The HIPA-to-SK experiment with L=295 km baseline length and E ν ≃ 1 GeV will measure δm 2 ATM at about 3 % accuracy and sin 2 θ ATM at about 1 % accuracy from the ν µ survival rate, while ν µ -to-ν e oscillation can be discovered if sin 2 2θ RCT = 4|U 2 e3 |(1 −|U 2 e3 |) ∼ > 0.006 [15]. As a sequel to the HIPA-to-SK LBL experiment, prospects of using the HIPA beam for a very long base-line (VLBL) experiments with the base-line length of a few thousand km have been studied [4,17,18]. Use of narrow-band high-energy neutrino beams ( E ν = 3 ∼ 6GeV) and a 100kton-level waterČerenkov detector [17] will allow us to distinguish the neutrino mass hierarchy (the sign of m 2 3 − m 2 1 ), if sin 2 2θ RCT ∼ > 0.03 [4]. If the LMA solution of the solar neutrino deficit is chosen by the nature, we can further constrain the allowed region of the δ MNS and sin 2 2θ RCT [4]. However, because ν µ → ν e appearance is strongly suppressed by the matter effect at such high energies, the measurement is not sensitive to the CP violating effects, ∼ sin δ MNS . In this paper, we study the capability of an LBL experiment between HIPA and Hyper-Kamiokande (HK), a megaton-level waterČerenkov detector being proposed to be built at the Kamioka site [19]. Here a combination of the shorter distance (L = 295km) and low ν-energy ( E ν ∼ 1GeV) makes the matter effect small, and the comparison of ν µ → ν e and ν µ → ν e appearance experiments is expected to have sensitivity to the CP violation effects proportional to sin δ MNS . The MNS matrix of the three-neutrino model is defined as where α = e, µ, τ are the lepton-flavor indices and ν i (i = 1, 2, 3) denotes the neutrino masseigenstates. The 3 × 3 MNS matrix, V MNS , has three mixing angles and three phases in general for Majorana neutrinos. In the above parameterization, the two Majorana phases reside in the diagonal phase matrix P, and the matrix U, which has three mixing angles and one phase, can be parameterized in the same way as the CKM matrix [20]. Because the present neutrino oscillation experiments constrain directly the elements, U e2 , U e3 , and U µ3 , we find it most convenient to adopt the parameterization [21] where these three matrix elements in the upper-right corner of the U matrix are chosen as the independent parameters. Without losing generality, we can take U e2 and U µ3 to be real and non-negative while U e3 is a complex number. All the other matrix elements of the U are then determined by the unitary conditions [21].
The probability of finding the flavor-eigenstate β at base-line length L in the vacuum from the original flavor-eigenstate α is given by where satisfy ∆ 12 + ∆ 23 + ∆ 31 = (δm 2 12 + δm 2 23 + δm 2 31 )(L/2E ν ) = 0. The two independent mass-squared differences are identified with the two "measured" ones, as follows; With the above identification, the MNS matrix elements are constrained by the observed survival probabilities, P ν µ →ν µ from the atmospheric neutrinos [22], P ν e →ν e from the reactor anti-neutrinos [10,11], and P ν e →ν e from the solar neutrinos [6]. The four independent parameters of the MNS matrix are then related to the observed oscillation amplitudes as The CP phase of the MNS matrix, δ MNS , is not constrained. The solution eq.(5c) follows from our convention [4], U e1 > U e2 , which defines the mass-eigenstate ν 1 . In this convention, there are four mass hierarchy cases corresponding to the sign of δm 2 ij ; I (δm 2 13 > δm 2 12 > 0), II (δm 2 13 > 0 > δm 2 12 ), III (δm 2 12 > 0 > δm 2 13 ), and IV (0 > δm 2 12 > δm 2 13 ) [4]. If the MSW effect is relevant for the solar neutrino oscillation, then the neutrino mass hierarchy cases II and IV are not favored. When sin 2 2θ ATM = 1, there is an additional twofold ambiguity in the determination of U µ3 in eq.(5b). In order to avoid the ambiguity, we adopt the U µ3 element itself, or equivalently sin 2 θ ATM defined in eq.(5b), as an independent parameter of the MNS matrix. Summing up, we parametrize the three-flavor neutrino oscillation parameters in terms of the 5 observed (constrained) parameters δm 2 ATM , δm 2 SOL , sin 2 θ ATM , sin 2 2θ SOL , sin 2 2θ RCT and one CP-violating phase δ MNS , for four hierarchy cases.
Neutrino-flavor oscillation inside of the matter is governed by the Schrödinger equation where H 0 is the Hamiltonian in the vacuum and a is the matter effect term [7] Here n e is the electron density of the matter, E ν is the neutrino energy, G F is the Fermi constant, and ρ is the matter density. In our analysis, we assume for brevity that the density of the earth's crust relevant for the LBL experiment, between HIPA and HK is a constant, ρ = 3, with an overall uncertainty of ∆ρ = 0.1; The Hamiltonian is diagonalized as by the MNS matrix in the matter U. The neutrino-flavor oscillation probabilities in the takes the same form as those in the vacuum, with ∆ ij = (λ j − λ i )L/2E ν , if the matter density can be approximated by a constant throughout the base-line. Because the effective matter potential for anti-neutrinos has the opposite sign with the same magnitude, the total Hamiltonian H governing the anti-neutrino oscillation in the matter is obtained from H as follows [4], We make the following simple treatments in estimating the signals and the backgrounds in our analysis.
• We assume a 1 Mega-ton waterČerenkov detector, which is capable of distinguishing between e ± CC events and µ ± CC events, but cannot distinguish their charges.
• We do not require capability of the detector to reconstruct the neutrino energy.
Although the waterČerenkov detector has the capability of measuring the energy of the produced µ and e as well as a part of hadronic activities, we do not make use of these information in this analysis. We only use the total numbers of the produced µ ± and e ± events from ν µ or ν µ narrow-band-beams (NBB). The NBBs from HIPA deliver 10 21 protons on target (POT) in a typical 1 year operation, corresponding to about 100 days of operation with the design intensity [16]. Details of the NBB's used for this study are available from the web-page [23].
much in the range E ν ≃ 0.6 ∼ 1.2 GeV, our results do not depend strongly on the true value of the δm 2 ATM : as long as it stays in the range (2 ∼ 5) × 10 −3 eV 2 [4]. The signals in this analysis are the numbers of ν µ and ν e CC events from NBB (2,3GeV) and those of the ν µ and ν e CC events from NBB(2GeV). These are calculated as for l = e or µ, where M is the mass of detector (1Mega-ton), N A = 6.017 × 10 23 is the Avogadro number, Φ ν µ (E ν ; p π ) and Φ ν µ (E ν ; p π ) are the flux of ν µ in NBB( p π GeV) and ν µ in NBB( p π GeV), respectively. The flux is negligibly small at E ν > 10GeV for the NBB's used in our analysis. The cross sections are obtained by assuming a pure water target [24].
Typical numbers of expected CC signals are tabulated in Table 1 for the parameter sets * : ρ = 3 g/cm 3 .
(13d) * Recently KamLAND collaboration confirmed that only the LMA solution of the solar-neutrino deficit problem is consistent with the data[27]. The allowed region of δm 2 SOL is found to be either (6 − 9) or (13 − 19) × 10 −5 eV 2 , slightly below or above our input value. The conclusions of this paper remain valid no matter which region its true value is.
The numbers in the Table 1 are for 1 Mt·year exposure with 10 21 POT per year for 0.77 MW operation of HIPA at L = 295 km. From Table 1, we learn that the transition events, N e and N e , are sufficiently large to have the potential of distinguishing the CP conserved cases, δ MNS = 0 • and 180 • , from the CP violating cases of δ MNS = 90 • and 270 • , even if sin 2 2θ RCT = 0.01. We also find that the survival events, N µ and N µ , barely depend on the CP phase. The ratio N µ (2GeV)/N µ (2GeV) is approximately σ CC νµ /σ CC νµ ≃ 2.9, because both the flux and the survival rates are approximately the same for ν µ and ν µ [4]. From the comparison of N ℓ (2GeV) and N ℓ (3GeV), we find that N µ (3GeV)/N µ (2GeV) ∼ 3 because of the rise in the cross section (∼ 1.5) and the increase in the survival rate (∼ 2). The ν e appearance signal N e increases only slightly at higher energies because a slight decrease in the transition probability cancels partially the effect of the rising cross section. Most notably, we find that the difference between the predictions of δ MNS = 0 • and 180 • cases is significantly larger for N e (ν µ ; p π = 3GeV) than that for N e (ν µ ; p π = 2GeV).
The above results can be seen clearly in Fig.1, where we show the expected number of ν e CC events N e for NBB(2GeV) with 4Mton·year plotted against those of the ν e CC event N e for NBB(2GeV) (left) and for NBB(3GeV) (right), both with 1Mton·year. The CP-phase dependence of the predictions are shown as closed circles for the parameters of eq.(13) at sin 2 2θ RCT = 0.06, 0.04, 0.02, and 0.01. Comparable numbers of ν e CC events (N e ) and ν e CC events (N e ) are expected by giving 4 times more ν µ than ν µ beams. At each sin 2 2θ RCT the ν µ → ν e events are expected to be smaller at δ MNS = 90 • (solid-squares) than at δ MNS = 270 • (open-squares). The trend is opposite for the ν µ → ν e events, and thus anti-correlation allows us to distinguish the two cases clearly. On the other hand, the expected number at δ MNS = 0 • (solid-circles) and that at δ MNS = 180 • (open-circles) do not differ much for NBB(2GeV) and NBB(2GeV). We find that NBB(3GeV) predicts significant differences between the two CP-invariant cases without loosing event numbers.
In this report, we assume 1Mton·year exposure each with NBB(2GeV) and NBB(3GeV) and 4Mton·year exposure of NBB(2GeV), and examine the capability of HIPA-to-Hyper-Kamiokande experiments to measure the CP phase, δ MNS , under the following simplified treatments of the backgrounds and systematic errors.
The first 3 terms in the r.h.s. are calculated as where Φ ν α and Φ ν α stands, respectively, for the secondary ν α and ν α flux of the primarily ν µ NBB. The last term in eq.(14a) for the e-like events gives the contribution of the NC events where produced π 0 's mimic the electron shower in the HK. By using the estimations from the K2K experiments [5], we use N e,e (NC; p π ) = P e/N C ν α =ν e ,ν e ,ν µ ,ν µ with where the NC event numbers are calculated as in eq.(15) by replacing σ CC ν ℓ by σ NC ν ℓ . The 10% error in the misidentification probability of 0.25% is accounted for as a systematic error [15]. The τ -lepton contribution is found to be negligibly small for the NBB's considered in this analysis. The background for the ν µ enriched beam NBB(2GeV) are evaluated in the same way.
Summing up, the event numbers for each energy neutrino and anti-neutrino NBB's are calculated from the sum : Most importantly, we do not require the capability of the HK detector to distinguish charges of electrons and muons. In Table 2   and NBB(2GeV), we find that the fraction of the secondary-beam contributions is much larger for the ν µ -beam than that for the ν µ -beam. This is essentially because ν ℓ CC cross section is about a factor of three smaller than the ν ℓ CC cross section at E ν ∼ 1GeV. In Fig.2, we show the expected ν µ → ν e (ν µ → ν e ) signal and background event numbers for the parameters of eq.(13) for sin 2 2θ RCT = 0.01 ∼ 0.06. The solid-circles show the number of expected signal events for δ MNS = n × 10 • (n = 1 ∼ 36). The numbers of signal events are largest at around δ MNS = 270 • for NBB(2, 3GeV), while those for NBB(2GeV) are largest at around 90 • , as is expected from the CP phase dependence of N e and N e shown in Fig.1. The open-triangle denotes ν e → ν e CC events, which give the largest background for the experiments with NBB(2, 3GeV), and the second largest background for NBB(2GeV). The open-square denotes ν e → ν e CC events that gives the largest background for NBB(2GeV), but is negligible for NBB (2,3GeV). The open-diamond denotes the background from the NC events, where π 0 's are miss-identified as electrons. They give the second largest background for NBB(2, 3GeV). Backgrounds from ν µ → ν e transition events for NBB(2, 3GeV) and those from ν µ → ν e transition events for NBB(2GeV) are shown by open-circle. These transition backgrounds depend on the CP phase and they tend to cancel the δ MNS dependence of the signals, but their magnitudes are small. The background level starts dominating the signal at sin 2 2θ RCT ∼ < 0.02.
The background numbers for the µ-like signals are found to be negligibly small (∼ 10 −2 ) for NBB (2,3GeV). Those for NBB(2GeV) are found to be about 21% of the signal almost independent of sin 2 2θ RCT . In Both cases, the major background comes from the secondary ν µ (ν µ ) survival events.
Our analysis proceeds as follows. For a given set of the model parameters, we calculate the expected numbers of all the signal and background events for each NBB( p π ) and NBB( p π ), by assuming 100% detection efficiencies for simplicity. The resulting numbers of µ-like and e-like events are then denoted by N true µ ( p π ) and N true e ( p π ) for NBB( p π ), and N µ true ( p π ) and N e true ( p π ) for NBB( p π ).
We account for the following two effects as major parts of the systematic uncertainty in this analysis. One is the uncertainty in the total flux of each neutrino beam, for which we assign the uncertainty, independently for ν α = ν e , ν µ , ν e , ν µ and for NBB(2GeV), NBB(3GeV), and NBB(2GeV).
Although it is likely that correlation exists among the flux uncertainties, we ignore possible effects of correlations in this analysis. By using the above flux factors, theoretical predictions for the event numbers, N f it l ( p π ) and N f it l ( p π ), are calculated as N f it ℓ ( p π ) = f ν e ( p π )N ℓ (ν e , p π ) + f ν µ ( p π )N ℓ (ν µ , p π ) +f ν e ( p π )N ℓ (ν e , p π ) + f ν µ ( p π )N ℓ (ν µ , p π ) +δ ℓ,e P e/N C where the last terms proportional to δ ℓ,e are counted only for ℓ = e. As the second major systematic error, we allocate 3.3% overall uncertainty in the matter density along the base-line, eq.(8). The fit functions are hence calculated for an arbitrary set of the 6 model parameters, the 12 flux normalization factors, and the matter density ρ.
The χ 2 function of the fit in this analysis can now be expressed as where the summation is over NBB(2GeV), NBB(3GeV) and NBB(2GeV). Even though we have only one NBB in our analysis, we retain the summation symbol in eq.(21) for the sake of clarity. The last two terms are added because KamLAND experiment [25] will measure δm 2 SOL at 10% level and the solar neutrino experiments constrain sin 2 2θ SOL with the 1σ error of about 0.06 for the LMA parameters of eq.(13). The individual error for each N µ ( p π ) (N µ ( p π )) is statistical only, whereas the error for each N e ( p π ) (N e ( p π )) is a sum of the statistical errors and the systematic error coming from the 10% uncertainty in the e/π 0 misidentification probability of eq.(17), σ e ( p π ) = N true e ( p π ) + 0.1N true e,e (NC; p π ) 2 .
The errors for the NBB(2GeV) case are calculated similarly as above.
We show in Fig.3  by a solid-circle for sin 2 2θ true RCT = 0.06, and by a solid-square for sin 2 2θ true RCT = 0.01. The regions where χ 2 min <1, 4, and 9 are depicted by solid, dashed, and dotted boundaries, respectively. All the 6 parameters, the matter density ρ f it , and the 12 flux normalization factors are allowed to vary freely in the fit.
From the top-right and bottom-right figures for δ true MNS = 90 • and 270 • respectively, we learn that δ MNS can be constrained to ±30 • (±60 • ) at the 1σ (3σ) level, even if sin 2 2θ true RCT = 0.01. This is because N e + N e constrain sin 2 2θ RCT and N e /N e distinguishes between δ MNS = 90 • and 270 • in Fig.3, whereas the remaining parameters (δm 2 ATM and sin 2 θ ATM ) are constrained by the ν µ and ν µ survival data, N µ and N µ . The accuracy of the δ MNS measurement does not decrease significantly for sin 2 2θ true RCT = 0.01 despite the large background level, because the δ MNS -dependence of the signal exceeds significantly the 3% uncertainty of the background level from the flux normalization factors in eq. (19). We find that the CP violation signal can be distinguished from the CP-conserving cases (δ MNS = 0 • or 180 • ) at 4σ (3σ) level for all δ MNS values in the region |δ MNS |, |δ MNS − 180 • | > 30 • if sin 2 2θ true RCT ∼ > 0.03 (0.01), for the LMA parameters of eq.(13) and for the systematic errors assumed in this analysis.
The situation is quite different for the CP-conserving cases of δ true MNS = 0 • or 180 • shown in the left-hand side of Fig.3. δ MNS can be constrained to better than ±7 • (11 • ) accuracy at 1σ level for sin 2 2θ RCT ∼ > 0.06 (0.01), but the two cases cannot be distinguished at 2σ level. This is mainly because of the similarity of N e /N e between δ MNS = 0 • and 180 • in Fig.1. The difference between the two cases is larger for NBB(3GeV). If we remove the NBB(3GeV) data from the fit, we find that the two cases cannot be distinguished even at 1σ level. This two-fold ambiguity between δ MNS and 180 • − δ MNS is found in general for all δ MNS , because the difference in the predictions can be adjusted by a shift in the fitted sin 2 2θ RCT value; see Fig.1. As a demonstration of the effect of using two NBB's, NBB(2GeV) and NBB(3GeV), in the analysis, we show in Fig.4 the fit results when the data are generated by using NBB(2GeV) and NBB(2GeV) only, each at 2 Mton·year and 4 Mton·year, respectively. It is clearly seen from the figures that the 'mirror' solution at δ MNS = 180 • (0 • ) can fit the data as well as the 'true' solution at δ MNS = 0 • (180 • ). Essentially the same results are obtained when we replace NBB(2GeV) by NBB(3GeV) in the above analysis. It is only by combining the two NBB's that we can distinguish the two solutions as shown in Fig.3. We find less significant difference from the results of Fig.3 when the input δ MNS value is 90 • or 270 • .
It is remarkable that the 1σ error of δ MNS is as large as 30 • for δ true MNS = 90 • and 270 • while it is less than 10 • for δ true MNS = 0 • and 180 • . This is simply because the δ MNS dependence of the ν µ -to-ν e (and also ν µ -to-ν e ) oscillation probability is roughly proportional to sin δ MNS , in the vicinity of the first dip of the ν µ -to-ν µ survival probability.
We close this article by pointing out that the low-energy LBL experiment like HIPAto-HK cannot distinguish between the neutrino-mass hierarchy cases (between I and III) because of the small matter effect at low energies. If we repeat the analysis by using the same input data but assuming the hierarchy III in the analysis, we obtain another excellent fit to all the data where the fitted model parameters are slightly shifted from their true (input) values. VLBL experiments at higher energies at L > 1000km [4] are needed to determine the mass hierarchy.