Constraints on the Solar $\Delta m^2$ using Daya Bay&RENO

We demonstrate that the currently running short baseline reactor experiments, especially Daya Bay, can put a significant upper bound on $\Delta m^2_{21}$. This novel approach to determining $\Delta m^2_{21}$ can be performed with the current data of both Daya Bay \&RENO and provides additional information on $\Delta m^2_{21}$ in a different $L/E$ range ($\sim$ 0.5 km/MeV) for an important consistency check on the 3 flavor massive neutrino paradigm. Upper limits by Daya Bay and RENO and a possible lower limit from Daya Bay, before the end of 2020, will be the only new information on this important quantity until the medium baseline reactor experiment, JUNO, gives a very precise measurement in the middle of the next decade. In this study $\theta_{12}$ value is fixed since its impact on the $\Delta m^2_{21}$ measurement is relatively small as discussed in the Appendix.


I. INTRODUCTION
The fact that neutrinos have mass and mix is now well established by a large number of experiments. In this paper we concentrated on the mass difference squared between the two mass eigenstates that have the most electron neutrino, ν 1 and ν 2 . The splitting between these two neutrinos, ∆m 2 21 ≡ m 2 2 −m 2 1 , is responsible for the (anti-) neutrino oscillations observed at an L/E = 15 km/MeV and for the neutrino flavor transformations inside the Sun, hence the name the solar mass squared difference.
In this paper, we demonstate that the currently running short baseline (∼1.5 km) reactor anti-neutrino experiments, Daya Bay [1] and RENO [2] both have enough data already collected to constrain ∆m 2 21 to be less than 3 times the KamLAND central value (7.5 × 10 −5 eV 2 ). By the end of the running time of these experiments, they will be able to constrain this parameter to less than twice the KamLAND value. Setting a lower limit maybe possible for the Daya Bay experiment with improvements on their systematic uncertainties. Upper, and maybe lower, limits from Daya Bay and RENO, will add independent information to our knowledge of ∆m 2 21 and provide an important consistency check of the 3 flavor massive neutrino paradigm. While not capable of directly addressing the ∼2σ tension between KamLAND [3] reactor experiment (L/E ∼ 50 km/MeV) and the combined Super KamiokANDE [4] & Sudbury Neutrino Observatory [5] solar neutrino measurements of ∆m 2 21 , measurements of ∆m 2 21 by Daya Bay and RENO are at a different L/E range (∼ 0.5 km/MeV) than previous measurements. Furthermore, the ratio of ∆m 2 21 to ∆m 2 31 , at L/E ∼ 0.5 km/MeV, is needed by the long baseline ν e appearance experiments for the precision measurement of leptonic CP violation.
Currently the best measurement of the solar mass squared difference, ∆m 2 21 , is from the long baseline reactor anti-neutrino experiment, KamLAND, which has determined see [3]. The only other measurement of ∆m 2 21 comes from a combined measurement using the solar neutrino experiments principle Super KamiokaNDE (SK) and Sudbury Neutrino Observatory (SNO). This combined measurement is from SNO [5]. Similar results can be found in SK [4] and Nu-Fit [6]. This solar neutrino determination of ∆m 2 21 comes from the non-observation of the low energy up turn of the 8 B neutrino survival probability by both SNO and SK and the observation of a day-night asymmetry by SK. CPT invariance implies that the ∆m 2 21 measured in reactor anti-neutrinos and solar neutrinos should be identical. However, at the 2σ level there is some tension between these two determinations of this important quantity. This tension could arise from a statistical fluctuation, some error in the analysis of one or more of the experiments or new physics.
The future medium baseline reactor experiment JUNO (L/E ∼ 15 km/MeV) will measure ∆m 2 21 and sin 2 θ 12 with better than 1% precision, [13]. However, this experiment is under construction and the precision measurements of the solar neutrino oscillation parameters will not be available until approximately 5 years from now. In more than a decade from now, the DUNE & HyperK proposed experiments will make a precise measurement of ∆m 2 21 using solar neutrinos, see [14] and [15] respectively.
In section II, we discuss in detail the effects of changing ∆m 2 21 on the oscillation probability. Then in section III we explain and give the results of a simulation of both Daya Bay and RENO using 3000 live days of data with and without systematic uncertainties followed by a conclusion.
Using typical fit values and considering a L/E range around the first oscillation minimum (L/E = 0.5 km/MeV), we can approximate P 13 and P 12 as follows: For ∆m 2 21 = 7.5 × 10 −5 eV 2 , the P 12 term is essentially negligible for all L/E < 1 km/MeV. This encompasses the L/E range of all current short baseline experiments.
However, consider the case that ∆m 2 21 is 3 times larger than this value, i.e. 22.5 × 10 −5 eV 2 , then P 12 is now no longer negligible compared to P 13 at oscillation minimum (L/E = 0.5 km/MeV) and P 12 gets larger for L/E > 0.5 km/MeV whereas P 13 is getting smaller. In fact, at L/E = 1 km/MeV, P 12 would be as large as sin 2 2θ 13 (0.08) for this value of ∆m 2 21 . Therefore the short baseline reactor experiments can constrain ∆m 2 21 to be less than 2 to 3 times the current best fit value depending on the experiment, Daya Bay or RENO, run time and the confidence level. Setting a lower bound on ∆m 2 21 will be challenging for these experiments due to systematic uncertainties. As data above L/E ∼ 0.5 km/MeV is important for this constrain, the Double Chooz experiment, which has no data with L/E > 0.5 km/MeV, is not considered.
To keep the disappearance probability the same as we vary ∆m 2 21 , at these small L/E, we must keep the quantity in [· · · ] in the above equation unchanged. If we also keep the position of the first minima fixed by holding ∆m 2 ee fixed (see eq. (8) to leading order in s 2 13 . So as we vary ∆m 2 21 from Kam-LAND value of 7.5 × 10 −5 eV 2 , we must also change s 2 13 from 0.021 so as to keep the combination in eq. (11) unchanged.
In Fig. 1, we show the electron anti-neutrino disappearance probability as function of L/E, keeping the quantity given in eq. (11) fixed, as we vary ∆m 2 21 in multiples of 7.5×10 −5 eV 2 . Note that if ∆m 2 21 > 3×10 −4 eV 2 then there is no minimum 3 around L/E ≈ 0.5 km/MeV. The red points with error bars, represents the statistical uncertainties for a detector 1.6 km from a single reactor core which has 9 × 10 5 events. Clearly, an experimental setup with this number of events in the far detector, 1.6 km from a reactor core, will be able to set an upper limit smaller than 3 times the KamLAND central value for ∆m 2 21 assuming systematic uncertainties are no larger than the statistical uncertainties. A lower limit on ∆m 2 21 will be challenging.
In the rest of this paper, we report on a simulation of the setups for Daya Bay and RENO experiments, to estimate the constraints these experiments can place on ∆m 2 21 .

III. SIMULATIONS FOR DAYA BAY AND RENO USING GLOBES
Our sensitivity study on ∆m 2 21 for the short baseline reactor experiments, Daya Bay and RENO, is performed 3 For ∆ 21 < 1, so that sin ∆ 21 ≈ ∆ 21 , one can find the minima by finding ∆ee such that,  is varied in multiples (labels =(0, .., 6, 10)) of the KamLAND value of 7.5 × 10 −5 eV 2 . θ13 is also varied, see eq. (11), to keep the same disappearance probability for L/E < 0.2 km/MeV. The red points with error bars, are the statistical uncertainties only, for a detector at 1.6 km from a reactor core with an exposure such that there are 900k events in this detector assuming the KamLAND value for ∆m 2 21 . This number of events corresponds to 3,000 days of Daya Bay data, see Table  I . This figure demonstrates that the Daya Bay experiment can put an upper limits on ∆m 2 21 of approximately 2 times the KamLAND central value or smaller, assuming the systematic uncertainties are smaller than the statistical uncertainties shown here. The dotted line is the two term approximation to the disappearance probability, see eq. (9).
using GLoBES [18]. In this study 3000 live days of data are assumed for both experiments and systematic uncertainties are taken into account as described in [19] for Daya Bay and [20] for RENO. Table I lists the effective baselines, L eff , and the number of observed IBD ν e events per day used.
To find the best fit values of ∆m 2 21 and sin 2 (2θ 13 ), a χ 2 formalism with pull parameters is constructed using the far-to-near ratio method to cancel out correlated systematic uncertainties. The χ 2 is given by where, is the observed far-to-near ratio of IBD ν e events in the i-th E ν bin, . Red, blue, and green lines represent 1σ, 2σ, and 3σ (2 dof) allowed regions, respectively. The point "×" is the input for the simulation given by eq. 13. In the bottom row, we also show the 1σ uncertainty band on ∆m 2 21 from KamLAND (cyan) and SNO/SK (yellow), see eq. 1 and 2.
is the expected far-to-near ratio of IBD ν e events for a given ∆m 2 21 and θ 13 pair, • f, , s, and b d are pull parameters for systematic uncertainties of neutrino flux (σ r flux ), detection efficiency (σ eff ), energy scale (σ scale ), and background (σ d bkg ), respectively. The indices r and d represent r-th reactor and d-th detector, respectively. Both Daya Bay and RENO have six reactors. For Daya Bay, two near detector sets (N 1 and N 2 ) are used in the last pull term of the χ 2 due to their differences in the baselines, backgrounds, and systematic uncertainties [19]. As a cross check of our simulations we have reasonably well reproduced the ∆m 2 ee vs. sin 2 2θ 13 sensitivity curves for both experiments.
True values used in the signal simulation are sin 2 θ 12 = 0.304, ∆m 2 21 = 7.65 × 10 −5 eV 2 , sin 2 (2θ 13 ) = 0.085, ∆m 2 31 = 2.50 × 10 −3 eV 2 . (13) To minimize the χ 2 , expected values for different pairs of ∆m 2 21 and sin 2 (2θ 13 ) are compared to the simulated signal ν e data from 1.8 to 8 MeV with 31 energy bins. Figure 2 shows the results of our simulation for contour plots of ∆m 2 21 vs. sin 2 (2θ 13 ) sensitivities using 3000 live days of data for RENO and Daya Bay, respectively, without (top) and with (bottom) systematic uncertainties. Adding systematic uncertainties effects RENO less than Daya Bay, because after 3,000 days of data taking, Daya Bay has ≈ 5 times more events in the far detector(s) than RENO, see Table I. Clearly, both of these experiments can constrain ∆m 2 21 to be less than two to three times the KamLAND central value, i.e. ∆m 2 21 < 15−22×10 −5 eV 2 . Setting a lower limit on ∆m 2 21 maybe possible with Daya Bay if modest improvements in their systematic uncertainties, over those used for this simulation, can be achieved. We encourage both Daya Bay and RENO to perform a measurement of ∆m 2 21 using their more precise information on their experiments.

IV. CONCLUSION
We have argued that Daya Bay and RENO can add to the information of the solar mass squared difference, ∆m 2 21 , now. A simulation study for these experiments was performed with and without systematic uncertainties using GLoBES. We have found that ∆m 2 21 can be reasonably well constrained by Daya Bay 3000 live days of data to be less than twice the KamLAND central value. Without systematic uncertainties Daya Bay can exclude ∆m 2 21 = 0 with 1σ confidence level but when current systematic uncertainties are included only an upper bound can be set. Until JUNO measures ∆m 2 21 with great precision early next decade, we expect the ∆m 2 21 measurement by Daya Bay can play an important role for the leptonic CP violation measurement by T2K and NOvA and provides an important consistency check on the 3 flavor massive neutrino paradigm. A truly realistic simulation and a true measurement of ∆m 2 21 can only be performed by the short baseline reactor experiments, Daya Bay and RENO.