Four-neutrino analysis of 1.5km-baseline reactor antineutrino oscillations

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 1.5km-baseline detector. Here, we present four-neutrino analysis of the results announced by RENO and Daya Bay, which performed the definitive measurements of $\theta_{13}$ based on the disappearance of reactor antineutrinos at km-order baselines. Our results using 3+1 scheme include the exclusion curve of $\Delta m^2_{41}$ vs. $\theta_{14}$ and the adjustment of $\theta_{13}$ due to correlation with $\theta_{14}$. The value of $\theta_{13}$ obtained by RENO and Daya Bay with a three-neutrino oscillation analysis is included in the $1\sigma$ interval of $\theta_{13}$ allowed by our four-neutrino analysis.


I. 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 multi-detector 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], 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 sin 2 2θ 13 . For instance, the best-fit values given in the analysis of Fogli et al. [9] are ∆m 2 21 = 7.5 × 10 −5 eV 2 , sin 2 θ 12 = 3.2 × 10 −1 , ∆m 2 32 = 2.4 × 10 −3 eV 2 , sin 2 θ 13 = 2.8 × 10 −2 , and sin 2 θ 23 = 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][10][11], we focus on θ 13 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], Gallium solar neutrino experiments [14] and some reactor experiments [15] 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 [17]. There are also other works with similar motivations [18].
After realizing the impact of the large size of θ 13 , both reactor neutrino experiments have continued and updated the far-to-near ratios and sin 2 2θ 13 . 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 sin 2 2θ 13 = 0.100 replaced the former best fit of sin 2 2θ 13 = 0.113 [19].
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 ∆m 2 14 within the range of O(0.001eV 2 ) to O(0.1eV 2 ), where ∆m 2 14 oscillations might have appeared in the superposition with ∆m 2 13 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 ∆m 2 14 ∼ O(1eV 2 ) [20,21]. This article is organized as follows: In Section II, 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 ∆m 2 41 , when reactor neutrinos in the energy range of 1.8 to 8 MeV are detected after travel along a km-order baseline. In Section III, 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 sin 2 2θ 13 due to the coexistence with sterile neutrinos. Broad ranges of ∆m 2 41 and sin 2 2θ 14 remain. In the conclusion, the exclusion bounds of sin 2 2θ 14 and the best fit of sin 2 2θ 13 are summarized, and the consistency between rate-only analysis and shape analysis is discussed.

II. FOUR NEUTRINO ANALYSIS OF EVENT RATES IN MULTI DETECTORS
The four-neutrino extension of unitary transformations from mass basis to flavor basis is given in terms of six angles and three Dirac phases: 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 where the PMNS type of a 3-by-3 matrix U PMNS with three rows, , is imbedded. The CP phases δ 2 and δ 3 introduced in Eq.
(1) are omitted for simplicity, since they do not affect the electron antineutrino survival probability at the reactor neutrino oscillation.
The survival probability ofν e produced from reactors is where ∆m 2 ij denotes the mass-squared difference (m 2 i − m 2 j ). It can be expressed in terms of combined ∆m 2 ij -driven oscillations as where ∆m 2 32 ≈ ∆m 2 31 and ∆m 2 42 ≈ ∆m 2 41 . The size of m 4 relative to m 3 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 averages accessible energies of the neutrinos emerging from the reactors. The measured probability of survival 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 where E e ≈ E ν − (M n − M p ) [21,22]. The flux distribution φ(E) from the four isotopes (U 235 , Pu 239 , U 238 , Pu 241 ) at the reactors is expressed by the following exponential of a fifth where f 0 = +4.57491 × 10, f 1 = −1.73774 × 10 −1 , f 2 = −9.10302 × 10 −2 , f 3 = −1.67220 × curve corresponds to the oscillation due to ∆m 2 41 , while the second bump that appears near 1500m corresponds to the oscillation due to ∆m 2 31 . RENO and Daya Bay were designed to observe the ∆m 2 31 -driven oscillations at far detector(FD) according to three-neutrino analysis, while additional detector(s) at a closer baseline perform 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 sin 2 2θ 13 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 θ 13 . 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 Fig.   1, it can be shown that the magnitude of ∆m 2 41 can affect not only the normalization factor but also the ratio between the FD and ND. 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) [24]. 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) [19]. Their measurements are marked in Fig.1. In three-neutrino analysis, the far-to-near ratios give the ∆m 2 31 -oscillation amplitude sin 2 2θ 13 = 0.089 ± 0.010(stat) ± 0.005(syst) and sin 2 2θ 13 = 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-toexpected ratio are understood as a combination of ∆m 2 31 oscillations and ∆m 2 41 oscillations as shown in Fig.1. For a given value of ∆m 2 41 , 0.01eV 2 or 0.01eV 2 , the combination of sin 2 2θ 14 and sin 2 2θ 13 is described in Fig.2. In the case of RENO, the P (∆m 2 41 = 0.01eV 2 ) and P (∆m 2 41 = 0.1eV 2 ) 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 sin 2 2θ 13 − sin 2 2θ 14 space using rate-only analysis. The corresponding analysis for Daya Bay is shown together in Fig.2. The value of sin 2 2θ 13 is in good agreement with the results released by the two experiments.
In rate-only analysis, the ratio of the observed to the expected number of events at FD in Eq.(10) is just the survival at FD, since the denominator in Eq. (12) is eliminated. Thus, R coincides with R far in Fig. 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 Fig.3 are obtained by the definition of the ratio R given in Eq. (10) and Eq. (11) per 0.25MeV bin, as the energy varies from 1.8MeV to 12.8MeV. The data dots and error bars were updated by including additional data from March to October in 2012 officially announced at Neutrino Telescope 2013 [19]. The ratio in Eq.(12) is compared with theoretical curves overlaid on the data points. The theoretical curves are described by where P (L) is given in Eq. (6). In Fig. 4, the best fit of (∆m 2 31 , sin 2 2θ 13 ) is presented when θ 14 = 0. The point A (0.00283eV 2 , 0.09) indicates the χ 2 minimum where sin 2 2θ 13 and ∆m 2 31 are parameters, while the point B (0.00232eV 2 , 0.10) is the minimum where ∆m 2 31 is 0.00232 which RENO and Daya Bay used for the fixed value. Hereafter, two cases depending on ∆m 2 31 are discussed: One is for ∆m 2 31 = 0.00283eV 2 marked by A and the other is for ∆m 2 31 = 0.00232eV 2 marked by B. According to the analysis performed with θ 14 = 0, the red curves for the two values of ∆ 2 31 are overlaid on the spectral data in Fig.  3. Fig. 5 shows interpretation of the spectral shape in terms of four-neutrino oscillation. In case (B) where the value of ∆m 2 31 is the same as the one that RENO and Daya Bay took for it, the best fit of sin 2 2θ 13 is 0.118 in company with non-zero sin 2 2θ 14 . The best fit sin 2 2θ 13 = 0.100 with the restriction sin 2 2θ 14 = 0 is still within 1σ region of four-neutrino analysis. Also in case (B) which is specified by a rather large ∆m 2 31 compared to the value taken by RENO and Daya Bay or the value suggested by global analyses, the best fit sin 2 2θ 13 = 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 Fig. 3 is analyzed in this rough estimation.

IV. 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 ∆m 2 31 = 0.00232eV 2 , clues about the fourth neutrino can be perceived only if the order of ∆m 2 41 is not much larger than that of ∆m 2 31 . Therefore, this work examined the possibility of a kind of sterile neutrino in the range of mass-squared differences below 0.1eV 2 , considering the two announced results of RENO and Daya Bay. Anomalies of reactor antineutrino oscillations have been considered for the range, 0.1eV 2 < ∆m 2 14 < 1eV 2 . 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 [25].
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 sin 2 2θ 13 = 0.100. We compared the spectral shape with theoretical curves of the superpositions of ∆m 2 41 oscillations and ∆m 2 31 oscillations. In summary, sin 2 2θ 14 > 0.2 is excluded at 3σ CL. When ∆m 2 31 = 0.00232eV 2 is fixed, the best-fit in four-neutrino parameters is (∆m 2 41 , sin 2 2θ 14 ) = (0.0078eV 2 , 0.054). When we search the fit of ∆m 2 31 along with other parameters of four-neutrino analysis, the best value is obtained ∆m 2 31 = 0.00283eV 2 with sin 2 2θ 13 = 0.090 from the shape in three-neutrino analysis. When the parameters are extended to four-neutrino scheme, the best fit is (∆m 2 41 , sin 2 2θ 14 ) = (0.039eV 2 , 0.049). As shown in Fig.6, the three-neutrino analysis of RENO (sin 2 2θ 13 , sin 2 2θ 14 ) = (0.100, 0.0) is also included within 1σ CL in four-neutrino analysis. Thus, it is not yet known whether the superposition with ∆m 2 41 oscillations is preferred to the single ∆m 2 31 oscillations at RENO detectors. Fig. 7 shows that the rate-only analysis and the spectral shape analysis are in good agreement within their 1σ CL range.