# Four-Neutrino Analysis of 1.5km Baseline Reactor Antineutrino Oscillations

Authors: Sin Kyu Kang; Yeong-Duk Kim; Young-Ju Ko; Kim Siyeon
Journal: Advances in High Energy Physics
 (2013)

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## 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 ofθ13 based on the disappearance of reactor antineutrinos at km order baselines. Our results using 3 + 1 scheme include the exclusion curve of Δm412 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.


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## Body

## 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 Δm212=7.5×10-5eV2, sin2θ12=3.2×10-1, Δm322=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 ofθ13, 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 Δm142 within the range of 𝒪(0.001eV2) to 𝒪(0.1eV2), where Δm142 oscillations might have appeared in the superposition with Δm132 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 Δm142~𝒪(1eV2) [27, 28].This paper is organized as follows. In Section2, 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 Δm412, 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 Δm412 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:(1)U~F=R34(θ34)R24(θ24,δ2)R14(θ14)·R23(θ23)R13(θ13,δ1)R12(θ12,δ3),
where Rij(θij) denotes the rotation of the ij block by an angle of θij. When a 3+1 model is assumed as the minimal extension, the 4-by-4 U~F is given by
(2)U~F=(c1400s14-s14s24c240c14s24-c24s14s34-s24s34c34c14c24s34-c24c34s14-s24c34-s34c14c24c34)×(Ue1Ue2Ue30Uμ1Uμ2Uμ30Uτ1Uτ2Uτ300001)=(c14Ue1c14Ue2c14Ue3s14⋯⋯⋯c14s24⋯⋯⋯c14c24s34⋯⋯⋯c14c24c34),
where the PMNS type of a 3-by-3 matrix UPMNS with three rows, (Ue1Ue2Ue3),(Uμ1Uμ2Uμ3), and (Uτ1Uτ2Uτ3), 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 ofν-e produced from reactors is
(3)PTh(ν-e⟶ν-e)=|∑j=14‍|U~ei|2expiΔmj12L2Eν|2=1-∑i<j‍4|U~ei|2|U~ej|2sin2(Δmij2L4Eν),
where Δmij2 denotes the mass-squared difference (mi2-mj2). It can be expressed in terms of combined Δmij2-driven oscillations as
(4)PTh(ν-e⟶ν-e)=1-c144c134sin22θ12sin2(1.27Δm212LE)-c144sin22θ13sin2(1.27Δm312LE)-c132sin22θ14sin2(1.27Δm412LE)-s132sin22θ14sin2(1.27Δm432LE),
where Δm322≈Δm312 and Δm422≈Δm412. The size of m4 relative to m3 is not yet constrained. The above PTh 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
(5)〈P〉=∫PTh(E)σtot(E)ϕ(E)dE∫σtot(E)ϕ(E)dE,
where σtot(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
(6)σtot(E)=0.0952(EeEe2-me21MeV2)×10-42cm2,
where Ee≈Eν-(Mn-Mp) [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 Eν:
(7)ϕ(Eν)=exp(∑i=05‍fiEνi),
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 Figure1 show 〈P〉 as L increases in a logarithmic manner, where the three patterns of probabilities are shown according to the order of Δm412. The first bump in each curve corresponds to the oscillation due to Δm412, while the second bump that appears near 1500m corresponds to the oscillation due to Δm312. RENO and Daya Bay were designed to observe the Δm312-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 Δm412 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 Δm412 is presented, when Δm312=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 Δm312 oscillation and Δm412 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 areLnear(meters)={660,445,302,340,520,746} and Lfar(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 ofR=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 Δm312-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 Δm312 oscillations and Δm412 oscillations as shown in Figure 1. For a given value of Δm412, 0.01eV2 or 0.01eV2, the combination of sin22θ14 and sin22θ13 is described in Figure 2. In the case of RENO, the 〈P(Δm412=0.01eV2)〉 and 〈P(Δm412=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 ofsin22θ13 and sin22θ14 which is compatible with the ranges of RNear and RFar released by RENO and Daya Bay. Blue (gray) areas are obtained with Δm412=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:
(8)R≡[ObservedatFD][ExpectedatFD](9)≡[No.ofeventsatFD][No.ofeventsatND]*,
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 Rfar=0.929±0.017 and Rnear=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 Rnear. Now,
(10)R=[No.ofeventsatFD][No.ofeventsatND]/0.990.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 Rfar 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 Figure3 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
(11)〈P(FD)〉〈P(ND)〉(0.990)-1,
where 〈P(L)〉 is given in (4). In Figure 4, the best fit of (Δm312,sin22θ13) is presented when θ14=0. The point A(0.00283eV2,0.09) indicates the χ2 minimum where sin22θ13 and Δm312 are parameters, while the point B(0.00232eV2,0.10) is the minimum where Δm312 is 0.00232 which RENO and Daya Bay used for the fixed value. Hereafter, two cases depending on Δm312 are discussed: one is for Δm312=0.00283eV2 marked by A and the other is for Δm312=0.00232eV2 marked by B. According to the analysis performed with θ14=0, the red curves for the two values of Δ312 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 Δ312 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 (Δm412,sin22θ14)=(0.039eV2,0.05) for (A) and (Δm412,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 Δm412, we choose m412=0.0078eV2 for the best fit of (B), avoiding m412=0.039eV2 which is the same as the m412 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 withsin22θ14=0 for Δm312=2.83×10-3eV2 and for Δm312=2.32×10-3eV2 obtained from the analysis in Figure 4. For each case, the blue fits are overlaid which are the superposition with Δm412=0.039eV2 oscillation of amplitude sin22θ14=0.050, and the superposition with Δm412=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(Δm312,sin22θ13) is (0.00283, 0.09) denoted by A. When Δm312=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
The1σ,2σ, and 3σ exclusion curves are drawn by solid lines. Apparently, a broad range of sin22θ14 apparently remains not excluded. Only Δm412>Δm312 is considered and sin22θ14>0.2 is excluded at 3σ CL. For (A) specified by Δm312=2.83×10-3eV2, the minimum χmin2/dof=0.48 is at (Δm412,sin22θ14)=(0.039eV2,0.050), while, for (B) specified by Δm312=2.32×10-3eV2, two minima χmin2/dof=1.06 and 0.85 are located at (Δm412,sin22θ14)=(0.0078eV2,0.033) and (0.039eV2,0.049), respectively.The solid curves in Figure6 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 Δm312=0.00283eV2 and at (0.118, 0.054) for case (B) of Δm312=0.00232eV2. In case (B) where the value of Δm312 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 Δm312 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
The1σ,2σ, and 3σ fit of combination of the sin22θ13 and sin22θ14 for chosen values of (Δm312 and Δm412). For (A), the χmin2/dof of (0.092,0.049) is 0.51. For (B), the χmin2/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 byΔm312=0.00232eV2, clues about the fourth neutrino can be perceived only if the order of Δm412 is not much larger than that of Δm312. 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<Δm142<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 ratioR=0.929 and an oscillation amplitude of sin22θ13=0.100. We compared the spectral shape with theoretical curves of the superpositions of Δm412 oscillations and Δm312 oscillations. In summary, sin22θ14>0.2 is excluded at 3σ CL. When Δm312=0.00232eV2 is fixed, the best-fit in four-neutrino parameters is (Δm412,sin22θ14)=(0.0078eV2, 0.054). When we search the fit of Δm312 along with other parameters of four-neutrino analysis, the best value is obtained Δm312=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 (Δm412,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 Δm412 oscillations is preferred to the single Δm312 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 in1σ range of shape analysis, while they do not include the χ2-minimum best fits.



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