Constraint on the solar $\Delta m^2$ using 4,000 days of short baseline reactor neutrino data

TL;DR

This paper uses 4,000 days of short baseline reactor neutrino data from Daya Bay and RENO to constrain the solar $ m^2$, providing an upper limit that complements existing measurements and will improve with more data.

Contribution

It introduces a method to constrain solar  m^2 using publicly available short baseline reactor neutrino data, achieving a new upper limit and demonstrating potential for future improvements.

Findings

Combined Daya Bay and RENO data set an upper limit of 18  10^{-5} eV^2 on solar  m^2 at 95% C.L.

The constraint is slightly more than twice the KamLAND value.

Results are statistically limited but will improve with additional data.

Abstract

There is a well known 2$\sigma$ tension in the measurements of the solar $\Delta m^2$ between KamLAND and SNO/Super-KamioKANDE. Precise determination of the solar $\Delta m^2$ is especially important in connection with current and future long baseline CP violation measurements. Reference \cite{Seo:2018rrb} points out that currently running short baseline reactor neutrino experiments, Daya Bay and RENO, can also constrain solar $\Delta m^2$ value as demonstrated by a GLoBES simulation with a limited systematic uncertainty consideration. In this work, the publicly available data, from Daya Bay (1,958 days) and RENO (2,200 days) are used to constrain the solar $\Delta m^2$. Verification of our method through $\Delta m^2_{ee}$ and $\sin^2 \theta_{13}$ measurements is discussed in Appendix A. Using this verified method, reasonable constraints on the solar $\Delta m^2$ are obtained using above Daya Bay and RENO data, both individually and combined. We find that the combined data of Daya Bay and RENO set an upper limit on the solar $\Delta m^2$ of 18 $\times 10^{-5}$ eV$^2$ at the 95% C.L., including both systematic and statistical uncertainties. This constraint is slightly more than twice the KamLAND value. As this combined result is still statistics limited, even though driven by Daya Bay data, the constraint will improve with the additional running of this experiment.

Tables

Table 1. Table 1: Live days (not operational days), effective baseline distance ( L eff subscript 𝐿 eff L_{\rm eff} ), observed IBD and background events for Daya Bay and RENO used in this work. For Daya Bay there are two near detectors ( N ​ 1 𝑁 1 N1 and N ​ 2 𝑁 2 N2 ) in different sites.

		Daya Bay	RENO
Live days	Near ($N1,N2$)	(1,637.12 , 1,647.64)	1,807.88
	Far	1,692.69	2,193.04
$L_{\mathrm{eff}}$ (m)	Near ($N1,N2$)	(562.2 , 594.2)	430.4
	Far	1637	1445.4
Total # of IBD events	Near ($N1,N2$)	(1,763,939 , 1,651,088)	833,433
	Far	486,873	98,292
Total # of background events	Near ($N1,N2$)	(19,056 , 13,634)	17,229
	Far	2,230	4,912

Table 2. Table 2: Relative systematic uncertainties used in this work for Daya Bay and RENO, taken from Adey:2018zwh ; Bak:2018ydk respectively.

	Daya Bay	RENO
Source	Uncertainty %
Detection efficiency	0.13	0.21
Energy scale	0.2	0.15
Li-He background	30	5-8
Fast neutron background	13-17	–
Accidental background	1	–

Equations

$$\Delta m_{21}^2=5.1_{-1.0}^{+1.3}\times 10^{-5}{\mathrm{eV}}^2,$$

$$\Delta m_{21}^2=7.50_{-0.20}^{+0.20}\times 10^{-5}{\mathrm{eV}}^2,$$

$$P({\bar{\nu }}_{\mu }\to {\bar{\nu }}_e)-P({\nu }_{\mu }\to {\nu }_e)\approx \pi J\left(\frac{\Delta m_{21}^2}{\Delta m_{31}^2}\right)$$

$$N({\nu }_{\mu }\to {\nu }_e)=37\mathrm{and}N({\bar{\nu }}_{\mu }\to {\bar{\nu }}_e)=4$$

$$P({\bar{\nu }}_e\to {\bar{\nu }}_e)=1-P_{12}-P_{13}\mathrm{with}$$

$$P_{12}$$

$$P_{13}$$

$$P_{13}\approx {\sin }^{2}2{\theta }_{13}{\sin }^2{\Delta }_{ee}$$

$$P_{12}$$

$$P_{13}$$

$$P_{12}$$

$${\chi }^2$$

$${\left(\frac{\Delta m_{ee,\mathrm{prior}}^2-\Delta m_{ee}^2}{\sigma }\right)}^2$$

$$\Delta m_{ee}^2=2.45\pm 0.15\times 10^{-3}{\mathrm{eV}}^2$$

$$X_i^d$$

$$X_i^d({ϵ}^d,{\eta }^d)$$

$$B_i^d(b_{\mathrm{LH}}^d,b_{\mathrm{acc}}^d,b_{\mathrm{n}}^d)$$

Full text

Neutrino Physics, Reactor Experiments

††preprint: FERMILAB-PUB-19-190-T

I Introduction

Evidence that neutrinos are massive and mix is well established by a significant number of experiments. In this paper, we are interested in the mass squared difference, $\Delta m^{2}_{21}$; the mass squared difference of the two mass eigenstates that have the greatest fraction of electron neutrino, $\nu_{1}$ and $\nu_{2}$. This mass splitting is responsible for the neutrino flavor transformations that occur inside the Sun (hence the name the solar mass squared difference), and for the antineutrino oscillations observed at an L/E $\sim$ 15 km/MeV.

In this paper, we use publicly available data to follow up a recent paper Seo:2018rrb , that Daya Bay An:2015rpe and RENO RENO:2015ksa , the short baseline ($\sim$1.5 km) reactor antineutrino experiments currently running , have enough data already collected to constrain $\Delta m^{2}_{21}$.

The combined constraint by Daya Bay and RENO, gives an important consistency check of the standard three neutrino paradigm as well as adding addition information to the size of $\Delta m^{2}_{21}$. The $\sim$2$\sigma$ tension between the combined Super-Kamiokande (SK) Abe:2010hy & Sudbury Neutrino Observatory (SNO) Aharmim:2011vm solar neutrino measurements and KamLAND Gando:2010aa reactor experiment ($L/E\sim$ 50 km/MeV) is not directly addressed by this constraint. However such a combined Daya Bay plus RENO constraint is at a different $L/E$ range ($\sim$ 0.5 km/MeV) than the above mentioned measurements as well as JUNO An:2015jdp . Moreover, the ratio of $\Delta m^{2}_{21}$ to $\Delta m^{2}_{31}$, at an $L/E\sim$ 0.5 km/MeV, is required for the precision measurement of leptonic CP violation parameter, by NOvA Ayres:2004js , T2K Abe:2011ks and future Long Baseline (LBL) experiments.

Currently there are two measurements of the solar mass squared difference, $\Delta m^{2}_{21}$. One measurement comes from a combined measurement by SNO and SK using the the observation of a day-night asymmetry by SK and the non-observation of the low energy up turn of the 8B neutrino survival probability by SNO and SK. This combined result is

[TABLE]

from SNO and SK. Similar results are obtained by Nu-Fit Esteban:2018azc . The other measurement is from KamLAND, the long baseline reactor anti-neutrino experiment, see Gando:2010aa , at

[TABLE]

If CPT invariance is a good symmetry of nature then the $\Delta m_{21}^{2}$ measured from solar neutrinos and reactor anti-neutrinos is required to give the same value. Currently this important parameter for neutrino physics suffers from a 2$\sigma$ level tension. This tension could come from new physics, some error in the analysis of one or more of the experiments or a statistical fluctuation.

Moreover, the ratio of $\Delta m^{2}_{21}$ to $\Delta m^{2}_{31}$ is required for the determination of the CP phase, $\delta$, in the long baseline neutrino111In the rest of this paper, when referring to neutrinos, we mean both neutrinos and/or anti-neutrinos. oscillation experiments (NOvA, DUNE Acciarri:2015uup , T2K, T2HK Abe:2015zbg , T2HKK Abe:2016ero ) as the size of the CP violation is proportional to $\Delta m^{2}_{21}$ to $\Delta m^{2}_{31}$, as well as the Jarlskog invariant. At $L/E\sim$ 500 km/GeV=0.5 km/MeV, the first oscillation peak in vacuum, for $\nu_{\mu}\rightarrow\nu_{e}$

[TABLE]

where the Jarlskog invariant, J, is

$J=\sin 2\theta_{12}\sin 2\theta_{13}\cos\theta_{13}\sin 2\theta_{23}\sin\delta\,\approx 0.3\sin\delta$.

In the bi-event plane for T2K, see Fig 44 of Abe:2017vif ,

[TABLE]

is outside the allowed region (by about 1 $\sigma$). This can be well accommodated by a $\Delta m^{2}_{21}$ value, approximately twice the KamLAND value. Again, this is probably a statistical fluctuation but with only the KamLAND precision measurement of $\Delta m^{2}_{21}$, other possibilities are still viable.

The future medium baseline, $L/E\sim$ 15 km/MeV, reactor experiment JUNO will measure to better than 1% precision $\Delta m^{2}_{21}$ and $\sin^{2}\theta_{12}$, see An:2015jdp . JUNO experiment is currently under construction and their precision measurements of $\Delta m^{2}_{21}$ and $\sin^{2}\theta_{12}$ will not be available until approximately 5 years from now. Later next decade, the proposed experiments Hyper-K & DUNE will also give us precision measurements of $\Delta m^{2}_{21}$ using 8B solar neutrinos, see Abe:2018uyc and Beacom:2018xyz respectively.

In section II, we briefly discuss in detail the effects of increasing $\Delta m_{21}^{2}$ on the $\bar{\nu}_{e}$ survival probability. Then in section III Daya Bay and RENO data sets used in this work are discussed followed by section IV, V, and VI for methods and systematic uncertainties, results, and conclusion, respectively. In Appendix A it is described the verification of the method used in this work by comparing $\Delta m^{2}_{ee}$ vs $\sin^{2}2\theta_{13}$ measurements. In Appendix B we describe expected events and how pull parameters are inserted. In Appendix C the effects of fixing or floating the value of $\Delta m^{2}_{ee}$ are discussed.

II Survival Probability

In vacuum, the electron antineutrino survival probability is

[TABLE]

where the kinematic phases are given by $\Delta_{jk}\equiv\Delta m^{2}_{jk}L/(4E)$ and $\theta_{13}\approx 8^{\circ}$ and $\theta_{12}\approx 33^{\circ}$ are the reactor and solar mixing angles respectively. The $P_{12}$ term is associated with the solar oscillation scale of 15 km/MeV and the $P_{13}$ term is associated with the atmospheric oscillation scale of 0.5 km/MeV. To excellent fractional precision222The fractional precision is better than 0.05% for L/E $<$ 1 km/MeV. Also, in this L/E range, the exact $P_{13}$ is very insensitive to mass ordering provided the value of $|\Delta m^{2}_{ee}|$ is the same for both mass orderings. , the $P_{13}$ term can be approximated by

[TABLE]

where $\Delta m^{2}_{ee}\equiv\cos^{2}\theta_{12}\Delta m^{2}_{31}+\sin^{2}\theta_{12}\Delta m^{2}_{32}$ Nunokawa:2005nx ; Parke:2016joa , interpreted as the $\nu_{e}$ average of $\Delta m^{2}_{31}$ and $\Delta m^{2}_{32}$.

Using the fit values given in Esteban:2018azc , and an $L/E$ range around the first oscillation minimum ($L/E\sim 0.5\,{\rm km/MeV}$), $P_{12}$ and $P_{13}$ is well approximated by:

[TABLE]

The $P_{12}$ term is almost negligible for all $L/E<1\,{\rm km/MeV}$, if $\Delta m^{2}_{21}=7.5\times 10^{-5}\,{\rm eV^{2}}$. For Daya Bay and RENO this covers the full $L/E$ range.

Suppose that $\Delta m^{2}_{21}$ is 3 times larger than KamLAND value, i.e. $22.5\times 10^{-5}\,{\rm eV^{2}}$, then

[TABLE]

Now $P_{12}$ is now no longer tiny compared to $P_{13}$ at $L/E=0.5\,{\rm km/MeV}$, oscillation minimum, and as $L/E$ gets larger than 0.5 km/MeV, $P_{12}$ gets bigger, whereas $P_{13}$ is getting smaller. At an $L/E=1\,{\rm km/MeV}$, $P_{12}$ would be approximately equal to $\sin^{2}2\theta_{13}$ (0.08) for this value of $\Delta m^{2}_{21}$. It is this quadratic rise in $P_{12}$ as $\Delta m^{2}_{21}$ increases that we exploit to place an upper limit on $\Delta m^{2}_{21}$. For further details on the survival probability as $\Delta m^{2}_{21}$ increases see Seo:2018rrb .

III Daya Bay and RENO Data Sets

In this work, 1,958 days of Daya Bay data Adey:2018zwh and 2,200 days of RENO data Bak:2018ydk are used, where Daya Bay has about five times more inverse beta decay (IBD) events than RENO in their far detectors. Daya Bay data including background estimation, energy response function, and systematic uncertainties are taken from the supplementary material in Adey:2018zwh . RENO data and background estimation are extracted from FIG.1 in Bak:2018ydk and systematic uncertainties are also taken from Bak:2018ydk . Table 1 shows summary of the basic parameters, i.e., $L_{\rm eff}$, IBD rate, and background rate, for near and far detectors of Daya Bay and RENO used in this analysis. Note that there are two near detectors in different sites for Daya Bay.

IV Methods and Systematic Uncertainties

Best fit values on $\Delta m^{2}_{21}$ and $\sin^{2}2\theta_{13}$ are obtained by finding minimum $\chi^{2}$ values between data and predictions for all possible combination of the two parameters. Far-to-near ratio method is employed in this $\chi^{2}$ analysis to avoid the spectral shape anomaly around 5 MeV region Seo:2014xei as well as to reduce systematic uncertainties.

The $\chi^{2}$ formalism as written below contains a covariance matrix (Vstat,ij) to include statistical uncertainty and pull parameters ($\xi_{\alpha}$) to include systematic uncertainties.

[TABLE]

where, $D^{F/N}_{i}\equiv\frac{O^{F}_{i}-B^{F}_{i}}{O^{N}_{i}-B^{N}_{i}}$, $P^{F/N}_{i}\equiv\frac{X^{F}_{i}}{X^{N}_{i}}$, and $F(N)$ and $i$ ($j$) represent the Far (Near) detector and $i^{th}$ ($j^{th}$) prompt energy bin, respectively. Being $O$ the observed number of IBD candidate events, $B$ the estimated background number of events and $X$ the expected number of events for a given $\Delta m^{2}_{21}$ and $\sin^{2}2\theta_{13}$ pair. A total of 26 energy bins ($N_{bins}$) is used for RENO from 1.2 to 8.4 MeV. The same number of energy bins are used for Daya Bay from 0.7 to 12 MeV but two near detectors are taken into account in the $\chi^{2}$ formalism by replacing $N_{bins}$ to $2N_{bins}$ where for $1\leqslant i\leqslant\rm N_{bins}$, $F=$EH3 and $N=$EH1, and for ${\rm N_{bins}}+1\leqslant i\leqslant 2{\rm N_{bins}}$, $F=$EH3 and $N=$EH2.

For both Daya Bay and RENO, systematic uncertainties on the relative detection efficiency, relative energy scale and the main background contributions are taken into account as summarized in Table 2.

Besides the systematic uncertainties, additional systematic paddings (fudge factors) are added in our work to match Daya Bay and RENO results on $\theta_{13}$ and $\Delta m^{2}_{ee}$ measurements. For Daya Bay a 1.3 fudge factor to the relative energy scale and Li-He background uncertainties is added. Whereas in RENO a 1.4 fudge factor is added to the relative detection efficiency uncertainty. More details on the validation of our method and expected event description can be found in Appendices A and B. The RENO predictions are computed using the Daya Bay detector response function and the relative far-to-near normalization is computed comparing our total number of expected events with the total number of expected events in the RENO Far detector. In order to match the best fit values of $\theta_{13}$ and $\Delta m^{2}_{ee}$ a 0.984 fudge factor is added to this normalization of a total event rate for RENO.

V Results

A 2-dimensional scan over $\Delta m^{2}_{21}$ and $\sin^{2}2\theta_{13}$ is performed to find the best fit value pair at the minimum value of $\chi^{2}$ described earlier, where in the oscillation probability, the parameter $\theta_{12}$ is fixed333A discussion on the effects of varying $\theta_{12}$ in this analysis can be found in Seo:2018rrb . at $\sin^{2}\theta_{12}=0.310$. The $\Delta m^{2}_{ee}$ parameter is constrained with a pull parameter, allowing it to vary within a $2\sigma$ range of a prior $\Delta m^{2}_{ee}$ value with a penalizing term

[TABLE]

The prior $\Delta m^{2}_{ee}$ value and its uncertainty are taken to be

[TABLE]

which is inferred from the combined measurement on $\Delta m^{2}_{\mu\mu}$ by current long baseline neutrino experiments in Esteban:2018azc through $\Delta m^{2}_{ee}\simeq\Delta m^{2}_{\mu\mu}\pm\cos 2\theta_{12}\Delta m^{2}_{21}$, see Nunokawa:2005nx , where the $+/-$ comes from the unknown mass ordering (NO/IO) and ignoring terms proportional to $\sin\theta_{13}\Delta m^{2}_{21}$. The unknown mass ordering is treated as an additional uncertainty (4%) to $\Delta m^{2}_{\mu\mu}$ uncertainty (4%) for the $\Delta m^{2}_{ee}$ uncertainty which, therefore, becomes about 6%.

The best fit, 1, 2, and 3 $\sigma$ allowed regions of $\Delta m^{2}_{21}$ vs $\sin^{2}2\theta_{13}$ are shown in Fig. 1 with (solid lines) and without (dashed lines) systematic uncertainties for Daya Bay and RENO. Daya Bay result is better than RENO due to about five time more statistics.

To obtain the best result on solar $\Delta m^{2}$ measurement, a combined analysis of both Daya Bay and RENO data sets is performed. Figure 2 left plot shows the best fit, 1, 2, and 3 $\sigma$ allowed regions of $\Delta m^{2}_{21}$ vs $\sin^{2}2\theta_{13}$ of the combined analysis, and as expected the result is slightly improved by combining the two data sets.

Figure 2 right plot shows the $\chi^{2}$ projection over $\Delta m^{2}_{21}$, obtained by minimizing over $\sin^{2}2\theta_{13}$. The upper bounds on $\Delta m^{2}_{21}$, including systematic uncertainties, are 12.3, 18.3 and 22.3 $\times 10^{-5}$ eV2 at 1, 2 and 3 $\sigma$ CL, respectively. Current upper bounds are limited by statistics.

In Fig. 3, we give the constraints on the three parameter fit, $\Delta m^{2}_{21}$, $\Delta m^{2}_{ee}$ and $\sin^{2}2\theta_{13}$, without imposing any constrain on $\Delta m^{2}_{ee}$, using the combined Daya Bay and RENO data sets. Both statistical and systematic uncertainties are included in this plot. As before $\theta_{12}$ is fixed at $\sin^{2}\theta_{12}=0.310$, see Seo:2018rrb for discussion on allowing $\sin 2\theta_{12}$ to also vary.

Results with $\Delta m^{2}_{ee}$ fixed or free are obtained for each experiment and for when the data from both experiments are combined. These are described and given in Appendix C. It was found that the effect of free $\Delta m^{2}_{ee}$ is bigger than that of systematic uncertainty, but our representing results are based on constrained $\Delta m^{2}_{ee}$ since it is a reasonably well measured oscillation parameter using LBL experiments .

VI Conclusion

Using the currently available public data from Daya Bay (1,958 days) and RENO (2,200 days), we have provided additional information on the solar $\Delta m^{2}$. A reasonable upper bound is obtained from a combined analysis of the Daya Bay and RENO data as 18 $\times 10^{-5}$ eV2 at 95% CL , where $\Delta m^{2}_{ee}$ was constrained using a pull parameter with input information from LBL experiments. Our combined analysis result is currently limited by statistics and, as expected, Daya Bay data drives the combined analysis results. Our analysis method was validated by reproducing the $\Delta m^{2}_{ee}$ and $\sin^{2}\theta_{13}$ contours for each experiment as discussed in Appendix A.

Given that the previous measurements by KamLAND and SK/SNO of the solar $\Delta m^{2}$ are in a 2$\sigma$ tension and the importance of solar $\Delta m^{2}$ for the determination of CP violation in LBL experiments, it is crucial that we understand the value of the solar $\Delta m^{2}$ better. It is expected by circa 2025 that the JUNO experiment will provide additional, important information on the value of the of solar $\Delta m^{2}$.

Acknowledgements.

We are grateful to Thomas Schwetz for fruitful discussions. This work (SHS) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Ministry of Science and ICT (MSIT) (No. 2017R1A2B4012757 and IBS-R016-D1-2019-b01). This manuscript has been authored (SJP) by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. This project (SJP) has received funding/support from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 690575 & No 674896.

Appendix A VALIDATION OF OUR ANALYSES

Using the data and the $\chi^{2}$ formalism described in section III and IV, our method reproduces the contours in the $\Delta m^{2}_{ee}$ vs $\sin^{2}2\theta_{13}$ from the the Daya Bay and RENO collaborations as it is shown in figures 4, 5. The Day Bay and RENO collaboration contours are taken from the supplementary material of Adey:2018zwh and from FIG. 3 of Bak:2018ydk , respectively.

The agreement between our results and Daya Bay as well as RENO for the measurements of $\Delta m^{2}_{ee}$ vs $\sin^{2}2\theta_{13}$ is an excellent validation of the methods and numbers used in our analysis. Therefore, our constraint on $\Delta m^{2}_{21}$, using the publicly available data of Daya Bay and RENO, has a firm basis.

Appendix B NUMBER OF EXPECTED EVENTS AND PULL PARAMETERS IN $\chi^{2}$

The expected numbers of signal events in a detector $d$ in a prompt energy bin $i$, $X_{i}^{d}$, is computed as follows up to a common input (e.g. reactor power, total number of protons) which cancels when taking ratios in the $\chi^{2}$ computation.

[TABLE]

where, the indices $i$, $r$, $d$, and iso refers to the $i^{th}$ energy bin, $r^{th}$ reactor, $d^{th}$ detector, and a fissionable isotope (${}^{235}\rm U$, ${}^{239}\rm Pu$, ${}^{238}\rm U$, or ${}^{241}\rm Pu$), respectively, and $a^{d}$ is the detector efficiency. $L_{rd}$ is the baseline between the reactor $r$ and the detector $d$. $E_{\nu}$ and $E^{\text{rec}}$ are the neutrino true energy and the reconstructed energy, both related by the detector response function $R(E^{\text{rec}},E_{\nu})$. The $\sigma(E_{\nu})$ is the IBD cross section computed performing the integral in ${\rm d}\!\cos\theta$ of the differential cross section in Vogel:1999zy and the $f^{\text{iso}}$ is the averaged fission fraction444Ideally we would have the information on the fission factions as a function of time in each reactor, but since we do not have this information we take the same averaged values for all the detectors. This means that any systematic uncertainty on the flux predictions will cancel when taking ratios of the expected events in different experimental sites. and the $\phi^{\text{iso}}(E_{\nu})$ is the Huber-Mueller flux prediction Huber:2011wv ; Mueller:2011nm . $P^{rd}_{\bar{\nu}_{e}\rightarrow\bar{\nu}_{e}}(E_{\nu})$ is the oscillation probability from reactor r to detector d in the three neutrino oscillation paradigm.

The pull parameters accounting for detection efficiency ($\epsilon^{d}$) and relative energy scale ($\eta^{d}$) are included in the number of expected events as follows

[TABLE]

For RENO, the efficiency pull parameter is included in the ratio.

The background pull parameters are included in background events $B_{i}^{d}$ used in $D^{F/N}_{i}\equiv\frac{O^{F}_{i}-B^{F}_{i}}{O^{N}_{i}-B^{N}_{i}}$ as follows

[TABLE]

where $B_{i}^{d}$ ($B_{{\rm LH},i}^{d}$, $B_{{\rm acc},i}^{d}$ and $B_{{\rm n},i}^{d}$) represents the number of total (Li-He, accidental and fast neutron) background events in the $i^{th}$ prompt energy bin in the $d^{th}$ detector, and the small $b$ represents the corresponding pull parameter.

Appendix C FIXED VS FREE $\Delta m^{2}_{ee}$

For the results in the main body of our paper we constrained $\Delta m^{2}_{ee}$ treating it as a pull parameter using LBL experiments input. In this Appendix we show the impact of $\Delta m^{2}_{ee}$ fixed and set free. A 2-dimensional scan over $\Delta m^{2}_{21}$ and $\sin^{2}2\theta_{13}$ is performed to find the best fit value pair at the minimum value of $\chi^{2}$ described earlier, where in the oscillation probability $\theta_{12}$ is fixed as $\sin^{2}\theta_{12}=0.310$ but $\Delta m^{2}_{ee}$ is set free within the range of $[1.55,3.55]\times 10^{-3}$ eV2. Results with a fixed $\Delta m^{2}_{ee}=2.45\times 10^{-3}{\rm eV}^{2}$ are also obtained and compared to those with $\Delta m^{2}_{ee}$ set free. Figure 6, left and middle panels, shows the results of $\Delta m^{2}_{ee}$ fixed and free for Daya Bay and RENO. It is observed that the effect of floating $\Delta m^{2}_{ee}$ is bigger than adding systematic uncertainty for both Daya Bay and RENO. For floating $\Delta m^{2}_{ee}$ case, the corresponding $\Delta m^{2}_{ee}$ values for the minimum $\chi^{2}$ are found to be $2.50\times 10^{-3}$eV2 ($2.68\times 10^{-3}$ eV2) for Daya Bay (RENO) and it is within 1 $\sigma$ uncertainty of each of their measurements. Figure 6, right panels shows the results with combined analysis. For floating $\Delta m^{2}_{ee}$ case, the corresponding $\Delta m^{2}_{ee}$ value for the minimum $\chi^{2}$ is found to be $2.54\times 10^{-3}$eV2 and it is within 1 $\sigma$ uncertainty of the Daya Bay best fit value, i.e., $[2.52\pm 0.07]\times 10^{-3}$.