Four-Neutrino Analysis of 1.5 km Baseline Reactor Antineutrino Oscillations

Abstract

The masses of sterile neutrinos are not yet known, and depending on the orders of magnitudes, their existence may explain reactor anomalies or the spectral shape of reactor neutrino events at 1.5 km baseline detector. Here, we present four-neutrino analysis of the results announced by RENO and Daya Bay, which performed the definitive measurements of 𝜃₁₃ based on the disappearance of reactor antineutrinos at km order baselines. Our results using 3 + 1 scheme include the exclusion curve of Δ𝑚²₄₁ versus 𝜃₁₄ and the adjustment of 𝜃₁₃ due to correlation with 𝜃₁₄. The value of 𝜃₁₃ obtained by RENO and Daya Bay with a three-neutrino oscillation analysis is included in the 1𝜎 interval of 𝜃₁₃ allowed by our four-neutrino analysis.

1. Introduction

Understanding of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [1] is now moving to another stage, due to the determination of the last angle by multidetector observation of reactor neutrinos at Daya Bay [2] and RENO [3], whose success was strongly expected from a series of oscillation experiments, (T2K [4], MINOS [5], and Double Chooz [6, 7]), which all contributed to the forefront of neutrino physics [8]. A number of 3𝜈 global analyses [9, 10] have presented the best fit and the allowed ranges of masses and mixing parameters at 90% confidence level (CL) by crediting RENO and Daya Bay for the definitive measurements of sin²2𝜃₁₃. For instance, the best-fit values given in the analysis of Fogli et al. [9] are Δ𝑚²₂₁=7.5×10⁻⁵eV², sin²𝜃₁₂=3.2×10⁻¹, Δ𝑚²₃₂=2.4×10⁻³eV², sin²𝜃₁₃=2.8×10⁻², and sin²𝜃₂₃=4.8×10⁻¹ for normal hierarchy. While all three mixing angles are now known to be different from zero, the values of the CP violating phases are completely unknown. Although there are a number of global analysis which presented consistent values of masses and mixing parameters [9–11], we focus on 𝜃₁₃ and its associated factors obtained by RENO and Daya Bay.

Although the three-neutrino framework is well established phenomenologically, we do not rule out the existence of new kinds of neutrinos, which are inactive so-called sterile neutrinos. Over the past several years, the anomalies observed in LSND [12], MiniBooNE [13–15], Gallium solar neutrino experiments [16, 17], and some reactor experiments [18] have been partly reconciled by the oscillations between active and sterile neutrinos. In a previous work, we also examined whether the oscillation between sterile neutrinos and active neutrinos is plausible, especially when analyzing the first results released from Daya Bay and RENO [19]. There are also other works with similar motivations [20–25].

After realizing the impact of the large size of 𝜃₁₃, both reactor neutrino experiments have continued and updated the far-to-near ratios and sin²2𝜃₁₃. Daya Bay improved their measurements and explained the details of the analysis. RENO announced an update with an extension until October 2012 and modified their results as follows. The ratio of the observed to the expected number of neutrino events at the far detector 𝑅=0.929 replaced the former value of 𝑅=0.920, and sin²2𝜃₁₃=0.100 replaced the former best fit of sin²2𝜃₁₃=0.113 [26]. The spectral shape was also modified. Again, we examine the oscillation between a sterile neutrino and active neutrinos in order to determine whether four-neutrino oscillations are preferred to three-neutrino oscillations. This work is focused on Δ𝑚²₁₄ within the range of 𝒪(0.001eV²) to 𝒪(0.1eV²), where Δ𝑚²₁₄ oscillations might have appeared in the superposition with Δ𝑚²₁₃ oscillations at far detectors of 𝒪(1.5km) baselines. Since the mass of the fourth neutrino is unknown, it is worth verifying its existence at all available orders of magnitude which are accessible from different baseline sizes. For instance, the near detector at RENO can search reactor antineutrino anomalies with Δ𝑚²₁₄∼𝒪(1eV²) [27, 28].

This paper is organized as follows. In Section 2, the survival probability of electron antineutrinos is presented in four-neutrino oscillation scheme. We exhibit the dependence of the oscillating aspects on the order of Δ𝑚²₄₁, when reactor neutrinos in the energy range of 1.8 to 8 MeV are detected after travel along a km order baseline. In Section 3, the curves of the four-neutrino oscillations are compared with the spectral shape of data through October 2012 to search for any clues of sterile neutrinos and to see the changes in sin²2𝜃₁₃ due to the coexistence with sterile neutrinos. Broad ranges of Δ𝑚²₄₁ and sin²2𝜃₁₄ remain. In the conclusion, the exclusion bounds of sin²2𝜃₁₄ and the best fit of sin²2𝜃₁₃ are summarized, and the consistency between rate-only analysis and shape analysis is discussed.

2. Four-Neutrino Analysis of Event Rates in Multidetectors

The four-neutrino extension of unitary transformations from mass basis to flavor basis is given in terms of six angles and three Dirac phases: 𝑈_F=𝑅₃₄𝜃₃₄𝑅₂₄𝜃₂₄,𝛿₂𝑅₁₄𝜃₁₄⋅𝑅₂₃𝜃₂₃𝑅₁₃𝜃₁₃,𝛿₁𝑅₁₂𝜃₁₂,𝛿₃, (1) where 𝑅_𝑖𝑗(𝜃_𝑖𝑗) denotes the rotation of the 𝑖𝑗 block by an angle of 𝜃_𝑖𝑗. When a 3+1 model is assumed as the minimal extension, the 4-by-4 𝑈_F is given by 𝑈_F=𝑐₁₄00𝑠₁₄−𝑠₁₄𝑠₂₄𝑐₂₄0𝑐₁₄𝑠₂₄−𝑐₂₄𝑠₁₄𝑠₃₄−𝑠₂₄𝑠₃₄𝑐₃₄𝑐₁₄𝑐₂₄𝑠₃₄−𝑐₂₄𝑐₃₄𝑠₁₄−𝑠₂₄𝑐₃₄−𝑠₃₄𝑐₁₄𝑐₂₄𝑐₃₄×𝑈_𝑒1𝑈_𝑒2𝑈_𝑒30𝑈_𝜇1𝑈_𝜇2𝑈_𝜇30𝑈_𝜏1𝑈_𝜏2𝑈_𝜏30=𝑐0001₁₄𝑈_𝑒1𝑐₁₄𝑈_𝑒2𝑐₁₄𝑈_𝑒3𝑠₁₄⋯⋯⋯𝑐₁₄𝑠₂₄⋯⋯⋯𝑐₁₄𝑐₂₄𝑠₃₄⋯⋯⋯𝑐₁₄𝑐₂₄𝑐₃₄, (2) where the PMNS type of a 3-by-3 matrix 𝑈_PMNS with three rows, (𝑈_𝑒1𝑈_𝑒2𝑈_𝑒3),(𝑈_𝜇1𝑈_𝜇2𝑈_𝜇3), and (𝑈_𝜏1𝑈_𝜏2𝑈_𝜏3), is imbedded. The CP phases 𝛿₂ and 𝛿₃ introduced in (1) are omitted for simplicity, since they do not affect the electron antineutrino survival probability at the reactor neutrino oscillation.

The survival probability of 𝜈_𝑒 produced from reactors is 𝑃_Th𝜈_𝑒⟶𝜈_𝑒=⁴_𝑗=1𝑈_𝑒𝑖²exp𝑖Δ𝑚²_𝑗1𝐿2𝐸_𝜈²=1−_𝑖<𝑗4𝑈_𝑒𝑖²𝑈_𝑒𝑗²sin²Δ𝑚²_𝑖𝑗𝐿4𝐸_𝜈, (3) where Δ𝑚²_𝑖𝑗 denotes the mass-squared difference (𝑚²_𝑖−𝑚²_𝑗). It can be expressed in terms of combined Δ𝑚²_𝑖𝑗-driven oscillations as 𝑃_Th𝜈_𝑒⟶𝜈_𝑒=1−𝑐⁴₁₄𝑐⁴₁₃sin²2𝜃₁₂sin²1.27Δ𝑚²₂₁𝐿𝐸−𝑐⁴₁₄sin²2𝜃₁₃sin²1.27Δ𝑚²₃₁𝐿𝐸−𝑐²₁₃sin²2𝜃₁₄sin²1.27Δ𝑚²₄₁𝐿𝐸−𝑠²₁₃sin²2𝜃₁₄sin²1.27Δ𝑚²₄₃𝐿𝐸, (4) where Δ𝑚²₃₂≈Δ𝑚²₃₁ and Δ𝑚²₄₂≈Δ𝑚²₄₁. The size of 𝑚₄ relative to 𝑚₃ is not yet constrained. The above 𝑃_Th is understood only within a theoretical framework, since the energy of the detected neutrinos is not unique but is continuously distributed over a certain range. So, the observed quantity is established with a distribution of neutrino energy spectrum and an energy-dependent cross section. Analyses of neutrino oscillation average accessible energies of the neutrinos emerging from the reactors. The measured probability of survival is ⟨𝑃⟩=∫𝑃_Th(𝐸)𝜎_tot(𝐸)𝜙(𝐸)𝑑𝐸∫𝜎_tot(𝐸)𝜙(𝐸)𝑑𝐸, (5) where 𝜎_tot(𝐸) is the total cross-section of inverse beta decay (IBD), and 𝜙(𝐸) is the neutrino flux distribution from the reactor. The total cross section of IBD is given as 𝜎_tot(𝐸)𝐸=0.0952^𝑒𝐸²_𝑒−𝑚²_𝑒1MeV²×10⁻⁴²cm², (6) where 𝐸_𝑒≈𝐸_𝜈−(𝑀_𝑛−𝑀_𝑝) [28, 29]. The flux distribution 𝜙(𝐸) from the four isotopes (U²³⁵,Pu²³⁹,U²³⁸,andPu²⁴¹) at the reactors is expressed by the following exponential of a fifth order polynomials of 𝐸_𝜈: 𝜙𝐸_𝜈=exp⁵_𝑖=0𝑓_𝑖𝐸^𝑖_𝜈, (7) where 𝑓₀=+4.57491×10, 𝑓₁=−1.73774×10⁻¹, 𝑓₂=−9.10302×10⁻², 𝑓₃=−1.67220×10⁻⁵, 𝑓₄=+1.72704×10⁻⁵, and 𝑓₅=−1.01048×10⁻⁷ are obtained by fitting the total flux of the four isotopes with the fission ratio expected at the middle of the reactor burn up period [30].

The curves in Figure 1 show ⟨𝑃⟩ as 𝐿 increases in a logarithmic manner, where the three patterns of probabilities are shown according to the order of Δ𝑚²₄₁. The first bump in each curve corresponds to the oscillation due to Δ𝑚²₄₁, while the second bump that appears near 1500m corresponds to the oscillation due to Δ𝑚²₃₁. RENO and Daya Bay were designed to observe the Δ𝑚²₃₁-driven oscillations at far detector (FD) according to three-neutrino analysis, while additional detector(s) at a closer baseline performs the detection of neutrinos in the same condition. The comparison of the number of neutrino events at FD to the number of events at the near detector (ND) is an effective strategy to determine the disappearance of antineutrinos from reactors. That is, the sin²2𝜃₁₃ is evaluated by the slope of the curve between ND and FD, while their absolute values of event numbers do not affect the estimation of the angle 𝜃₁₃. Both experiments used the normalization to adjust the data to satisfy the boundary condition which is that there is no oscillation effect before the ND. From Figure 1, it can be shown that the magnitude of Δ𝑚²₄₁ can affect not only the normalization factor but also the ratio between the FD and ND.

Figure 1 

The dependency of ⟨𝑃⟩ in (5) on Δ𝑚241 is presented, when Δ𝑚231=2.32×10−3eV2 as taken in RENO and Daya Bay. Typical shapes of ⟨𝑃⟩ versus distance are drawn for comparison with the measured ratios in the two experiments. The amplitudes of Δ𝑚231 oscillation and Δ𝑚241 oscillation are given by sin22𝜃13=0.10 and sin22𝜃14=0.10, respectively, as an example.

The six baselines of the near detector (ND) and the far detector (FD) of RENO are 𝐿_near(meters)={660,445,302,340,520,746} and 𝐿_far(meters)={1560,1460,1400,1380,1410,1480}, while their flux-weighted averages 𝐿_near and 𝐿_far are 407.3 m and 1443 m, respectively. The baselines of Daya Bay, named EH1, EH2, and EH3, have lengths of EH1=494m, EH2=554m, and EH3=1628m, respectively, so that, conventionally, EH1 and EH2 are regarded as near detectors while EH3 is regarded as a far detector.

After the first release of results, Daya Bay and RENO updated the far-to-near ratio of neutrino events with additional data. Daya Bay reported a ratio of 𝑅=0.944±0.007(stat)±0.003(syst) with 𝑅(EH1)=0.987±0.004(stat)±0.003(syst) [31]. RENO also reported an update with additional data from March to October in 2012, where 𝑅(FD)=0.929±0.006(stat)±0.009(syst) [26]. Their measurements are marked in Figure 1. In three-neutrino analysis, the far-to-near ratios give the Δ𝑚²₃₁-oscillation amplitude sin²2𝜃₁₃=0.089±0.010(stat)±0.005(syst) and sin²2𝜃₁₃=0.100 ± 0.010(stat)±0.015(syst) in Daya Bay and RENO, respectively. On the other hand, the far-to-near ratio and the measured-to-expected ratio are understood as a combination of Δ𝑚²₃₁ oscillations and Δ𝑚²₄₁ oscillations as shown in Figure 1. For a given value of Δ𝑚²₄₁, 0.01eV² or 0.01eV², the combination of sin²2𝜃₁₄ and sin²2𝜃₁₃ is described in Figure 2. In the case of RENO, the ⟨𝑃(Δ𝑚²₄₁=0.01eV²)⟩ and ⟨𝑃(Δ𝑚²₄₁=0.1eV²)⟩ curves which pass the error bars at ND and FD are drawn as blue- (gray-) shaded areas. The area where the two shaded areas, ND and FD, overlap is the allowed region in sin²2𝜃₁₃−sin²2𝜃₁₄ space using rate-only analysis. The corresponding analysis for Daya Bay is shown together in Figure 2. The value of sin²2𝜃₁₃ is in good agreement with the results released by the two experiments.

Figure 2 

Four-neutrino analysis for the observed to expected ratios at both ND and FD of RENO. The shaded areas indicate the available combination of sin22𝜃13 and sin22𝜃14 which is compatible with the ranges of 𝑅Near and 𝑅Far released by RENO and Daya Bay. Blue (gray) areas are obtained with Δ𝑚241=0.01(0.1)eV2. The 4𝜈 analysis of the far-to-near ratio at Daya Bay is also given for comparison.

3. Four-Neutrino Analysis of Updated Spectral Shape in RENO

One of RENO’s results was the ratio of the observed to the expected number of antineutrinos in the far detector, 𝑅=0.929±0.011 (see [26]), where the observed is simply the number of events at FD. On the other hand, the expected number of events at FD can be obtained using several adjustments of the number of events at ND: []𝑅≡ObservedatFD≡[]ExpectedatFDNo.ofeventsatFD[No.ofeventsatND]^∗, (8)(9) where the number of events at each detector is normalized. The normalization of the neutrino fluxes at ND and FD requires an adjustment between the two individual detectors which includes corrections due to DAQ live time, detection efficiency, background rate, and the distance to each detector. The numbers of events at FD and ND in (9) have already been normalized by these correction factors, and so we have 𝑅_far=0.929±0.017 and 𝑅_near=0.990±0.025 as shown in Figure 2. The normalization guarantees 𝑅=1 at the center of the reactors. RENO removes the oscillation effect at ND when evaluating the expected number of events at FD by dividing the denominator of (9) by 0.990 which is taken from 𝑅_near. Now, []𝑅=No.ofeventsatFD[].No.ofeventsatND/0.990 (10)

In rate-only analysis, the ratio of the observed to the expected number of events at FD in (8) is just the survival at FD, since the denominator in (10) is eliminated. Thus, 𝑅 coincides with 𝑅_far in Figure 2.

In spectral shape analysis, however, the denominator cannot be neglected, since the oscillation effect at ND differs depending on the neutrino energy. The data points in Figure 3 are obtained by the definition of the ratio 𝑅 given in (8) and (9) per 0.25 MeV bin, as the energy varies from 1.8 MeV to 12.8 MeV. The data dots and error bars were updated by including additional data from March to October in 2012 officially announced at Neutrino Telescope 2013 [26]. The ratio in (10) is compared with theoretical curves overlaid on the data points. The theoretical curves are described by ⟨𝑃(FD)⟩⟨𝑃(ND)⟩(0.990)⁻¹, (11) where ⟨𝑃(L)⟩ is given in (4). In Figure 4, the best fit of (Δ𝑚²₃₁,sin²2𝜃₁₃) is presented when 𝜃₁₄=0. The point A(0.00283eV²,0.09) indicates the 𝜒² minimum where sin²2𝜃₁₃ and Δ𝑚²₃₁ are parameters, while the point B(0.00232eV²,0.10) is the minimum where Δ𝑚²₃₁ is 0.00232 which RENO and Daya Bay used for the fixed value. Hereafter, two cases depending on Δ𝑚²₃₁ are discussed: one is for Δ𝑚²₃₁=0.00283eV² marked by A and the other is for Δ𝑚²₃₁=0.00232eV² marked by B. According to the analysis performed with 𝜃₁₄=0, the red curves for the two values of Δ²₃₁ are overlaid on the spectral data in Figure 3. Figure 5 shows interpretation of the spectral shape in terms of four-neutrino oscillation. For given values of Δ²₃₁ and sin²2𝜃₁₃, the 1, 2, 3𝜎 CL exclusion curves are obtained by convolution with the energy resolution of RENO detectors (5.9/√𝐸+1.1)%. Using the best fits (Δ𝑚²₄₁,sin²2𝜃₁₄)=(0.039eV²,0.05) for (A) and (Δ𝑚²₄₁,sin²2𝜃₁₄)=(0.0078eV²,0.033) for (B), the blue curves are also added on the spectral data in Figure 3. To see a distinct different aspect due to the magnitude of Δ𝑚²₄₁, we choose 𝑚²₄₁=0.0078eV² for the best fit of (B), avoiding 𝑚²₄₁=0.039eV² which is the same as the 𝑚²₄₁ for (A).

Figure 3 

The data and error bars are the reproduction of updated RENO for an extended period until October 2012. The red fits are drawn with sin22𝜃14=0 for Δ𝑚231=2.83×10−3eV2 and for Δ𝑚231=2.32×10−3eV2 obtained from the analysis in Figure 4. For each case, the blue fits are overlaid which are the superposition with Δ𝑚241=0.039eV2 oscillation of amplitude sin22𝜃14=0.050, and the superposition with Δ𝑚241=0.0078eV2 oscillation of amplitude sin22𝜃14=0.054, respectively, obtained from the analysis in Figure 5.

Figure 4 

Three-neutrino analysis of the spectral shape updated until October 2012. The best fit in (Δ𝑚231,sin22𝜃13) is (0.00283, 0.09) denoted by A. When Δ𝑚231=0.00232eV2 is fixed as taken by RENO, the best fit of sin2𝜃13 is 0.100. The 1𝜎,2𝜎, and 3𝜎 exclusion curves are drawn by solid lines.

Figure 5 

The 1𝜎,2𝜎, and 3𝜎 exclusion curves are drawn by solid lines. Apparently, a broad range of sin22𝜃14 apparently remains not excluded. Only Δ𝑚241>Δ𝑚231 is considered and sin22𝜃14>0.2 is excluded at 3𝜎 CL. For (A) specified by Δ𝑚231=2.83×10−3eV2, the minimum 𝜒2min/dof=0.48 is at (Δ𝑚241,sin22𝜃14)=(0.039eV2,0.050), while, for (B) specified by Δ𝑚231=2.32×10−3eV2, two minima 𝜒2min/dof=1.06 and 0.85 are located at (Δ𝑚241,sin22𝜃14)=(0.0078eV2,0.033) and (0.039eV2,0.049), respectively.

The solid curves in Figure 6 explain 1𝜎,2𝜎, and 3𝜎 CL of Δ𝜒² in parameters (sin²2𝜃₁₃,sin²2𝜃₁₄), of which the best-fits are found at (0.092, 0.049) for case (A) of Δ𝑚²₃₁=0.00283eV² and at (0.118, 0.054) for case (B) of Δ𝑚²₃₁=0.00232eV². In case (B) where the value of Δ𝑚²₃₁ is the same as the one that RENO and Daya Bay took for it, the best fit of sin²2𝜃₁₃ is 0.118 in company with nonzero sin²2𝜃₁₄. The best fit sin²2𝜃₁₃=0.100 with the restriction sin²2𝜃₁₄=0 is still within 1𝜎 region of four-neutrino analysis. Also in case (B) which is specified by a rather large Δ𝑚²₃₁ compared to the value taken by RENO and Daya Bay or the value suggested by global analyses, the best fit sin²2𝜃₁₃=0.090 of three-neutrino analysis is placed in the region of 1𝜎 CL. This implies no preference between three-neutrino and four-neutrino schemes when the shape in Figure 3 is analyzed in this rough estimation.

Figure 6 

The 1𝜎,2𝜎, and 3𝜎 fit of combination of the sin22𝜃13 and sin22𝜃14 for chosen values of (Δ𝑚231 and Δ𝑚241). For (A), the 𝜒2min/dof of (0.092,0.049) is 0.51. For (B), the 𝜒2min/dof of (0.118,0.049) is 0.96. In both cases, the best fit of sin22𝜃14=0 is included in 1𝜎 region.

4. Conclusion

If a fourth type of neutrino has a mass not much larger than the other three masses, the results of reactor neutrino oscillations like RENO, Daya Bay, and Double Chooz can be affected by the fourth state. For detectors established for oscillations driven by Δ𝑚²₃₁=0.00232eV², clues about the fourth neutrino can be perceived only if the order of Δ𝑚²₄₁ is not much larger than that of Δ𝑚²₃₁. Therefore, this work examined the possibility of a kind of sterile neutrino in the range of mass-squared differences below 0.1eV², considering the two announced results of RENO and Daya Bay. Anomalies of reactor antineutrino oscillations have been considered for the range, 0.1eV²<Δ𝑚²₁₄<1eV². Thus, it is worth analyzing the absolute flux at the near detector and the ratio of the far-to-near flux on a common basis [32].

RENO announced an update of rate-only analysis and the spectral shape of neutrino events [neutrino telescope], including an observed-to-expected ratio 𝑅=0.929 and an oscillation amplitude of sin²2𝜃₁₃=0.100. We compared the spectral shape with theoretical curves of the superpositions of Δ𝑚²₄₁ oscillations and Δ𝑚²₃₁ oscillations. In summary, sin²2𝜃₁₄>0.2 is excluded at 3𝜎 CL. When Δ𝑚²₃₁=0.00232eV² is fixed, the best-fit in four-neutrino parameters is (Δ𝑚²₄₁,sin²2𝜃₁₄)=(0.0078eV², 0.054). When we search the fit of Δ𝑚²₃₁ along with other parameters of four-neutrino analysis, the best value is obtained Δ𝑚²₃₁=0.00283eV² with sin²2𝜃₁₃=0.090 from the shape in three-neutrino analysis. When the parameters are extended to four-neutrino scheme, the best fit is (Δ𝑚²₄₁,sin²2𝜃₁₄) = (0.039eV²,0.049). As shown in Figure 6, the three-neutrino analysis of RENO (sin²2𝜃₁₃,sin²2𝜃₁₄)=(0.100,0.0) is also included within 1𝜎 CL in four-neutrino analysis. Thus, it is not yet known whether the superposition with Δ𝑚²₄₁ oscillations is preferred to the single Δ𝑚²₃₁ oscillations at RENO detectors. Figure 7 shows that the rate-only analysis and the spectral shape analysis are in good agreement within their 1𝜎 CL range.

Figure 7 

Comparison of rate-only analysis and spectral shape analysis: in both (A) and (B), the regions allowed by rate-only analysis are overlapped in 1𝜎 range of shape analysis, while they do not include the 𝜒2-minimum best fits.

Acknowledgments

This research was supported by a Chung-Ang University Research Scholarship Grant in 2013 and the National Research Foundation of Korea (NRF) Grants funded by the Korea Government of the Ministry of Education, Science and Technology (MEST) (2011-0014686).