A Global Fit Determination of Effective $\Delta m_{31}^2$ from Baseline Dependence of Reactor $\bar{\nu}_e$ Disappearance

Recently, three reactor neutrino experiments, Daya Bay, Double Chooz and RENO have directly measured the neutrino mixing angle $\theta_{13}$. In this paper, another important oscillation parameter, effective $\Delta m_{31}^2$ (= $\Delta \tilde{m}_{31}^2$) is measured using baseline dependence of the reactor neutrino disappearance. A global fit is applied to publicly available data and $\Delta \tilde{m}_{31}^2 = 2.95^{+0.42}_{-0.61} \times 10^{-3}$ eV$^2$, $\sin^22\theta_{13} = 0.099^{+0.016}_{-0.012}$ are obtained by setting both parameters free. This result is complementary to $\Delta tilde{m}_{31}^2$ to be measured by spectrum shape analysis. The measured $\Delta \tilde{m}_{31}^2$ is consistent with $\Delta \tilde{m}_{32}^2$ measured by $\nu_{\mu}$ disappearance in MINOS, T2K and atmospheric neutrino experiments within errors. The minimum $\chi^2$ is small, which means the results from the three reactor neutrino experiments are consistent with each other.

Another effective mass squared difference ∆m 2 31 can be measured by energy spectrum distortion and baseline dependence of the reactor-θ 13 experiments. This paper is to measure ∆m 2 31 by baseline dependence of the reactor neutrino-θ 13 experiments.
In reactor-θ 13 experiments, usually the neutrino disappearance is analysed by a two flavor neutrino * Electronic address: thiago@awa.tohoku.ac.jp † Electronic address: furuta@awa.tohoku.ac.jp ‡ Electronic address: suekane@awa.tohoku.ac.jp § Electronic address: matsubara@hepmail.phys.se.tmu.ac.jp oscillation formula; P (ν e →ν e ) = 1 − sin 2 2θ 13 sin 2 ∆m 2 where L is baseline which is ∼ 1 km and E ν is neutrino energy, which is around a few MeV. ∆m 2 31 is a weighted average of the two mass square differences, |∆m 2 31 | and |∆m 2 32 | of the standard parametrization, with c ij and s ij representing cos θ ij and sin θ ij , respectively [7]. In the analyses of reactor-θ 13 experiments so far published, sin 2 2θ 13 is extracted assuming ∆m 2 31 = ∆m 2 32 , which is measured by MINOS experiment [8]. ∆m 2 32 can be expressed as, ∆m 2 32 = (s 2 12 + s 13 t 23 sin 2θ 12 cos δ) ∆m 2 31 + (c 2 12 − s 13 t 23 sin 2θ 12 cos δ) ∆m 2 32 , where t ij = tan θ ij [7]. Since there is a relation in the standard three neutrino flavor scheme, the difference between ∆m 2 31 and ∆m 2 32 is expressed as follows, 2(∆m 2 31 − ∆m 2 32 ) ∆m 2 31 + ∆m 2 32 ∼ ± (1 − s 13 t 23 tan 2θ 12 cos δ) where the overall sign depends on mass hierarchy, and the ±0.3 term comes from the ambiguity of cos δ. The difference is much smaller than the current precisions of measurements and can be treated practically equivalent. A precision better than 1% is necessary to distinguish the mass hierarchy. However, if ∆m 2 31 and ∆m 2 32 are separately measured and if they turn out to be significantly different, it means the standard three flavour neutrino scheme is wrong. Thus it is important to measure ∆m 2 31 independently from ∆m 2 32 to test the standard three flavour neutrino oscillation.
The E dependence and L dependence analyses to extract ∆m 2 31 use independent information, namely energy distortion and normalization and thus are complementary. Some of the authors demonstrated ∆m 2 31 measurement using L dependence of deficit value of each reactor-θ 13 experiment in 2012 [9,10]. In this paper the analysis is significantly improved by applying a detailed global fit making use of the publicly available information of the three reactor neutrino experiments.
In next section we re-analyze the published data of each experiment and compare with the results written in the papers in order to demonstrate our analysis produces identical result. Section-III discusses about possible correlations between the experiments. In section-IV, most recent Double Chooz, Daya Bay and RENO results [2,4,11] are combined and ∆m 2 31 is extracted. Finally, a summary of this study is presented in section-V.

II.
REACTOR NEUTRINO DATA Details of each experiment and their data are presented in this section and they are re-analysed by the authors in order to demonstrate that the analysis methods used in this work are consistent with the publications from the experimental groups. The χ 2 used in this section will be used to form global χ 2 function in section-IV.

A. Daya Bay
The Daya Bay (DB) reactor neutrino experiment consists of three experimental halls (EH), containing one or more antineutrino detectors (AD). The AD array sees 6 reactors clustered into 3 pairs: Daya Bay (DB1, DB2), Ling Ao (L1, L2) and Ling Ao-II (L3, L4) power stations. Fig.-1 shows the relative locations of reactors and AD and table-I shows the distance between each combination of reactor and detector. All reactors are functionally identical pressurized water reactors with maximum thermal power of 2.9 GW [3].
In DB publication, the χ 2 is defined as where M d are the measured neutrino candidate events of the d-th AD with background subtracted, B d is the corresponding background, T d is the prediction from neutrino flux, Monte Carlo simulation (MC) and neutrino oscillation. ω d r is the fraction of neutrino event contribution of the r-th reactor to the dth AD determined by baselines and reactor fluxes. The uncorrelated reactor uncertainty is σ r . σ d is the uncorrelated detection uncertainty, and σ b d is the background uncertainty, with the corresponding pullterms (α r , ǫ d , η d ). An absolute normalization factor a is determined from the fit to the data.
The values of ω d r are not shown in Daya Bay publications and was estimated using where w r is the thermal power of each reactor and L rd is the baseline of r-th reactor to d-th detector. In this analysis, the value of p r is considered 1/6 since all reactors have same nominal thermal power. The calculated ω d r is shown in table-II. All the others terms are shown in table-III.
By using equation-(6) and the data from tables-II and -III, we were able to reproduce Daya Bay's result, where T d was multiplied by the value of the deficit probability (P def dr ), defined as: with ∆m 2 being measured in eV 2 , L dr in meters and E in MeV. n ν (E) is the expected energy spectrum of the observed neutrinos which is calculated by n ν (E) = S(E)σ IBD (E). S(E) is the energy spectrum of the reactor neutrinos, which is a sum of the energy spectrum of neutrinos from the four fissile elements: where S i (E ν ) is reactor neutrino spectrum per fission from fissile element i and β i is a fraction of fission rate of fissile element i. For equilibrium light water reactors, β i are similar and we use the values of Bugey paper [12], namely 235 U : 238 U : 239 Pu : 241 Pu = 0.538 : 0.078 : 0.328 : 0.056. In this study, S i (E) is approximated as an exponential of a polynomial function which is defined in [13], σ IBD is the cross section of the inverse process of neutron β-decay (IBD), that can be precisely calculated from the neutron lifetime [14]. The energy dependence of the IBD cross section is,  7), affect the final result. Dependence on the burn-up values is less than 0.001, as it was determined by replacing the burn-up assumption with that of the Chooz reactors at the beginning and end of the reactor cycle. Extreme assumptions on equation-(7) (one or two reactors off for the whole data period, for example) had an effect of less than 0.002 on the central value, with no change on the sensitivity. Moreover, the good agreement between the χ 2 distributions, shows that the assumptions are reasonable.
FIG. 2: χ 2 distribution with respect to sin 2 2θ13 by fixing ∆m 2 as ∆m 2 32 for Daya Bay data. The black curve is the χ 2 distribution shown in their paper [11] with central value and 1σ uncertainty of 0.089 ± 0.011, while the red curve shows the χ 2 distribution calculated in this analysis with central value and 1σ uncertainty of 0.090 +0.011 −0.010 .

B. RENO
The Reactor Experiment for Neutrino Oscillation (RENO) is located in South Korea and has two identical detectors, one near (ND) and one far (FD) from an array of six commercial nuclear reactors, as shown in fig.-3.

FIG. 3:
Relative locations of detectors and reactors of RENO. Scale is approximate.
Together with the distances of each detector reactor pair, the contribution of each reactor flux to each detector for the period of their first analysis is available [15] and are summarized in table-IV. The RENO χ 2 is defined as: where N d obs is the number of observed IBD candidates in each detector after background subtraction and N d,r exp is the number of expected neutrino events, including detection efficiency, neutrino oscillations and contribution from the r−th reactor to each detector determined from baseline distances and reactor fluxes. A global normalization n is taken free and determined from the fit to the data. The uncorrelated reactor uncertainty is σ r , σ ξ d is the uncorrelated detection uncertainty, and σ b d is the background uncertainty, and the corresponding pull parameters are (f r , ξ d , b d ). The values of these variables are shown in the table-V.
The expected number of events for both detectors are not present in the RENO paper, but the ratio between data and expectation is shown. This ratio and the quantities of table-IV were used to calculate the expectation value (N d,r exp ). Using the data in the table-V, equation-(8) and the MINOS ∆m 2 32 , we obtained sin 2 2θ 13 = 0.111 ± 0.024 which is in good agreement with their published value of sin 2 2θ 13RE = 0.113 ± 0.023. The χ 2 distributions are also very similar as shown in fig.-4. FIG. 4: χ 2 distribution with respect to sin 2 2θ13 by fixing ∆m 2 as ∆m 2 32 for RENO data. The black curve shows the χ 2 distribution shown in their paper [4] with central value and 1 σ uncertainty of 0.113±0.023, while the red curve shows the χ 2 distribution calculated in this analysis with central value and 1 σ uncertainty of 0.111 ± 0.024.

C. Double Chooz
The Double Chooz (DC) experiment uses the two Chooz B reactors with thermal power of 4.25 GW th each. Currently, the experiment is using only the far detector, since its near detector is not complete yet. The Bugey-4 measurement [12] is used as a reference of the absolute neutrino flux in the analysis, and the relative location of the far detector and reactors are shown in fig.-5, where the distances from the detector to each reactor are 998.1 and 1114.6 meters [16]. The Double Chooz collaboration published a rate plus shape analysis result [2].
An effect of the shape analysis in this case is an evaluation of main backgrounds of 9 Li and fast-neutron from the energy spectrum beyond the reactor neutrino energy range. Since information of detailed energy spectrum, which is necessary to reproduce the analysis, are not publicly available, we do not consider here the shape analysis but restrict only to the rate analysis.
After the second publication on the sin 2 2θ 13 measurement, the Double Chooz group published a result of the direct measurement of backgrounds by making use of 7.53 days reactor-OFF period [17]. We used these data in addition to the background evaluation inputs written in [2] to improve the background estimation instead of the energy spectrum analysis. The relative neutrino-flux uncertainty for reactor-OFF period is much larger than reactor-ON period. The dominant uncertainty comes from long-life isotopes whose abundance are not well known. It has negligible contribution in reactor-ON period [17]. Therefore, we regard the error correlation on neutrino flux between the reactor ON and OFF periods to be uncorrelated. We performed a similar χ 2 analysis as Daya Bay and RENO cases, assuming that the detector and background related uncertainties of [17] and [2] are fully correlated.
where N obs is the number of the observed neutrino event candidates. The subscript "i" represents reactor-ON and OFF period. N exp is the number of expected neutrino events, including detection efficiency and oscillation effects, and B is the total expected number of background events. The σ r , σ d , and σ b are the reactor, detection and background uncertainties, respectively. The corresponding pull parameters are (α, ǫ, b). Using the parameters shown in table-VI, we obtained sin 2 2θ 13 = 0.131 ± 0.048 which is consistent with the result of the DC publication, sin 2 2θ 13 = 0.109± 0.039, although the background evaluation methods are different using different data sets. We also did a rate only analysis of the Double Chooz data, which result agreed with the published one.

III. CORRELATION EVALUATION OF SYSTEMATIC UNCERTAINTIES
In reactor neutrino experiments, the expected number of observed events (N exp ) is defined by: where L is the reactor-detector baseline, N p is the number of targets in the detector, ε is the detector efficiency, P th is the reactor thermal power, E f is the mean energy released per fission, and σ f is the crosssection per fission defined as: For each experiment, L, N p , ε, and P th terms are determined independently. Therefore they can be assumed to be uncorrelated. On the other hand, E f and σ f terms are taken from the same references and the uncertainties of these terms are correlated between the experiments. From the Bugey and Chooz experimental results, the total uncertainty on spectrum prediction is 2.7%, where a 2% correlation is expected between the experiments as treated in [18]. Fully correlated signal prediction uncertainties between experiments, which come from neutrino flux and detection efficiency, can be cancelled by overall normalization factors used in the analyses of the Daya Bay and RENO. It allows us only to take into account remaining uncertainties between detectors or periods for each experiment. Daya Bay and RENO treat the remaining uncertainties as uncorrelated in their publications.

IV. COMBINED ANALYSIS
As explained before, the main method of this work is to combine all the data of the current neutrino reactor experiments in a single χ 2 function. Then we look for the minimum χ 2 value, calculate the ∆χ 2 distribution, and determine the confidence level regions. The χ 2 function used for such analysis was chosen so as to use the data from tables-I to -VI as well as the correlation as described in section-III. The definition of our global χ 2 is, with the χ 2 of each experiment defined as in section-II. Therefore, this function has 32 pull terms: 18 for Daya Bay (6 reactors, 6 detectors and 6 backgrounds), 10 for RENO (6 reactors, 2 detectors and 2 backgrounds) and 4 for Double Chooz (2 reactors, 1 detector and 1 background). It also contains the two overall normalization factors, one for Daya Bay and the other for RENO data set.
For all combinations of ∆m 2 and sin 2 2θ, the χ 2 G is minimized with respect to the pull terms. measured by accelerator experiments [8,19], confirming the standard three flavor neutrino oscillation within the error. The sin 2 2θ 13 obtained here is independent from ∆m 2 32 . The small χ 2 min /DoF means the data from the three reactor neutrino experiments are consistent with each other.
All the pull terms output were within 1 σ from the input value, and the normalization factors obtained from the fit to the data, were both less than 1%.
In fig.-8  and sin 2 2θ13, and by the minimization of the pull terms. For higher values of ∆m 2 31 (bigger than 10 −2 eV 2 ) some valleys are present, although they are about more than ten times less sensitive than the minimum χ 2 . probability is calculated using the parameters output which give the best fit. The Double Chooz has a large effect on this ∆m 2 31 determination because it locates at a baseline where the slope of the oscillation is large. In the near future, when the near detector of the Double Chooz experiment starts operation, the accuracy of this ∆m 2 31 measurement is expected to improve much. The data points are below the ∆m 2 32 because they are calculated using the parameters returned by the best fit solution. Generally, a detector sees several reactors. The horizontal axis is a weighted baseline L and the horizontal bar in each data point shows the standard deviation of the distribution of the baselines, which is defined by σL = k is the reactor index and L k and P k are the baseline and thermal power of the reactor k.
Complementary to this study, we demonstrated a similar, but simpler and robust measurement of the effective ∆m 2 31 from the baseline dependence of the disappearance probabilities of the three reactor-θ 13 experiments [9,10]. The result obtained on that work of ∆m 2 31 = 2.99 +1.13 −1.58 × 10 −3 eV 2 , is compatible with the value obtained in this paper. In addition, a similar ∆χ 2 distribution is presented in [20, fig.-4]. However, the central value of ∆m 2 31 could not be compared since only the distribution is presented.

V. SUMMARY
In this work, a global fit of the data from all the current reactor-θ 13 experiments was performed to measure ∆m 2 31 . The combination of the data from Daya Bay, RENO and Double Chooz resulted in ∆m 2 31 = 2.95 +0.42 −0.61 × 10 −3 eV 2 . This is consistent with ∆m 2 32 and it confirms that the experiments are observing standard three flavor neutrino oscillations within the error. The mixing angle obtained this analysis is sin 2 2θ 13 = 0.099 +0.016 −0.012 . The small χ 2 min /DoF value indicates that the data from the three reactor experiments are consistent with each other. This analysis uses independent information from the energy spectrum distortion and it is possible to improve the accuracy of ∆m 2 31 combining with results from energy spectrum analysis. It will be important to perform this kind of analysis to improve ∆m 2 31 accuracy and to check the consistency of the results from the reactor-θ 13 experiments. Tables   TABLE I: Daya Bay: Baselines, in meters, between each  detector and core [11, tab.-2].   DB1  DB2  L1  L2  L3  L4  AD1  362  372  903  817  1354  1265  AD2  358  368  903  817  1354  1266  AD3  1332  1358  468  490  558  499  AD4  1920  1894  1533  1534  1551  1525  AD5  1918  1892  1535  1535  1555  1528  AD6  1925  1900  1539  1539 1556 1530