Determination of mass hierarchy with medium baseline reactor neutrino experiments
I discuss the sensitivity of future medium baseline reactor antineutrino experiments on the neutrino mass hierarchy. By using the standard analysis, we find that the sensitivity depends strongly on the baseline length and the energy resolution , where and parameterize the statistical and systematic uncertainties, respectively. The optimal length is found to be km, the larger resolution the shorter optimal . For a 5 kton detector (with 12% weight fraction of free proton) placed at km away from a reactor, an experiment would determine the mass hierarchy with on average after 5 (15) or more years of running with the energy resolution. This type of experiment can also measure the relevant mixing parameters with the accuracy of .
Now that a large has been measured at Daya Bay An et al. (2012, 2013) and RENO Ahn et al. (2012) experiments accurately, neutrino physics enters a new era. One of the next challenges is determination of the mass hierarchy.
Among many ideas proposed, the medium baseline reactor antineutrino experiment Petcov and Piai (2002); Choubey et al. (2003); Learned et al. (2008); Zhan et al. (2008); Batygov et al. (2008); Zhan et al. (2009); Ghoshal and Petcov (2011) has stimulated various re-evaluations of its physics potential and sensitivity recently. Some works utilize the Fourier transform technique Ciuffoli et al. (2012a, b); Qian et al. (2012a), first discussed in refs. Learned et al. (2008); Batygov et al. (2008); Zhan et al. (2008), to distinguish the mass hierarchy. The main advantage of this technique is that the mass hierarchy can be determined without precise knowledge of the reactor antineutrino spectrum, the absolute value of the large mass-squared difference , and the energy scale of a detector. Although interesting and attractive, this technique is somewhat subtle to incorporate the uncertainties of the mixing parameters and to estimate its sensitivity to the mass hierarchy.
On the other hand, some works adopt the analysis Ghoshal and Petcov (2011, 2012); Qian et al. (2012a); Li et al. (2013) and new measure based on Bayesian approach Qian et al. (2012b). These methods utilize all available information from experiments, and it is straightforward to incorporate the uncertainties to evaluate the sensitivity, providing robust and complementary results to the Fourier technique.
In this proceedings, we analyze the sensitivity of medium baseline reactor antineutrino experiments to the mass hierarchy for the baseline length of – km and the energy resolution in the range and with the analysis. The optimal baseline length and the expected statistical uncertainties of the neutrino parameters, and , are also estimated.
Ii Reactor antineutrino flux
In this section, we briefly discuss the evaluation of how many electron antineutrinos, , would be detected at a far detector with a medium baseline length from a reactor.
In a nuclear reactor, antineutrinos are mainly produced via beta decay of the fission products of the four radio-active isotopes, and , in the fuel. The flux of antineutrinos with energy (MeV) at a reactor of thermal power is then expressed as Zhan et al. (2008)
where and are the relative fission contribution and the released energy per fission of the isotope , respectively Huber and Schwetz (2004). is the number of antineutrinos produced per fission Vogel and Engel (1989).
This rate is then modulated by neutrino oscillation. The survival probability is expressed as
where is the neutrino mixing-matrix element relating the electron neutrino to the mass eigenstate . The variables and are the mass and energy of the corresponding mass eigenstate, while represent the neutrino mixing angles. The oscillation phases are defined as
with a baseline length . The plus or minus sign in the last term of eq. (2) corresponds to the mass hierarchy; the plus sign is for normal hierarchy (NH), and the minus sign for inverted hierarchy (IH). Note that this last term is the only source of the mass hierarchy difference. We have neglected the matter effect because it is small enough for the energy range and the baseline lengths we concern in this study Hagiwara et al. (2011).
Similar as the current reactor experiments, such as Daya Bay An et al. (2012, 2013), RENO Ahn et al. (2012) and Double Chooz Abe et al. (2012), future medium-baseline reactor-antineutrino experiments can also use free protons as targets to detect electron antineutrinos via the inverse neutron-beta-decay (IBD) process, producing a neutron and a positron. The threshold neutrino energy of this process is
The produced positron then interacts with scintillator, converting its kinetic energy to photons. Eventually, the positron annihilates with an electron in the detector and emits two 0.5 MeV photons. The energies of photons are accumulated as the visible energy, , which is the sum of the positron’s total energy and an electron’s rest energy,
The observed antineutrino distribution by a detector with free protons after an exposure time can then be expressed as
where is the cross section of the IBD process Vogel and Beacom (1999), is the detector response function with the energy resolution , and is the observed visible energy by the detector. In this study, we take the normalized gaussian function as the response function, i.e.,
The detector energy resolution, , is parameterized into two parts,
The first term in the square root represents the statistical uncertainty, and the second one gives the systematic uncertainty Ere ().
Iii The sensitivity to the mass hierarchy
After obtaining the energy distribution of reactor antineutrinos, we would like to estimate the sensitivity of determining the mass hierarchy using the standard analysis Choubey et al. (2003); Batygov et al. (2008); Ghoshal and Petcov (2011, 2012); Qian et al. (2012a); Li et al. (2013).
To set the stage, we introduce the function as
The first term summarizes the prior knowledge on fitting parameters. In reactor antineutrino experiments, these are the mixing angles, and , and the two mass-square differences, and . In this study we also consider the event-number normalization factor , assuming the uncertainty. Their contributions look like,
The input values and their uncertainties are listed in Table 1.
The second term in (9) represents the statistical fluctuation. When we introduce binning w.r.t. , it looks like
with the summation running over all the bins. Here, is the event number for the bin when the hierarchy is NH (IH), while is the theoretical prediction of the event number either with right or wrong mass hierarchy, calculated as a function of the four model parameters and the normalization factor , which are all varied under the constraints of (10). In this study we prepare the data by using eq. (6) with the input values of the five parameters for each mass hierarchy.
In the limit of infinitely many events, the bin size can be reduced to zero, and the sum (11) can be replaced by an integral. Although a finite bin size is required for actual experiments, we adopt this zero-bin-size limit as a measure of the maximum sensitivity.
We then define as
where is the minimum of , which is obviously zero in our approximation of neglecting statistical fluctuations in data, . When wrong mass hierarchy is assumed in the fit, the minimum of , , will deviate from zero, and the wrong mass hierarchy can be rejected with the significance .
In this section, we discuss the sensitivity to the mass hierarchy, the optimal length and the statistical uncertainties of the neutrino parameters. All our results are obtained by assuming a reactor of 20 thermal power, a far detector of 5 kton fiducial volume with 12% weight fraction of free proton and 5 years exposure time.
Figures 1 shows energy distributions for km (left) and 50 km (right), in which the exact measurement is assumed in the upper panel, whereas in the lower panel the energy resolution of with in eq. (8) is assumed. The dashed blue curve corresponds to the NH case, and the dashed red curve to the IH case, while the solid curve is obtained using the parameter values fitted to the NH data with the “wrong” IH assumption. The red arrow points out the energy at which the difference due to mass hierarchy vanishes (the degeneracy point).
At km, the solid curve almost coincides with the dashed blue one even with the exact energy measurement, implying that it is almost impossible to distinguish the mass hierarchy by experiments at km. This is because the small phase shift between the NH and IH predictions can be absorbed by a small shift in by a fraction of its present uncertainty, .
The situation changes when the second peak of the mass-hierarchy dependent term appears in the energy range. The mass hierarchy difference can no longer be absorbed by a shift in since the relative phase difference between the NH and IH oscillations changes across the degeneracy point. There is no way to make the differences on the both sides compensated, resulting in the distinct mismatch between the dashed blue curve (the NH data) and the solid curve (the best fit under the IH assumption) as shown in the upper panel of the right plot in Fig. 1, where the antineutrino energy is exactly measured.
Once the finite energy resolution is introduced, the phase difference in the lower energy side of the degeneracy point is significantly smeared out as it oscillates faster w.r.t. at the low energy. Hence it is easier for one oscillation period to be covered by a sizable Gaussian profile of the detector response function. The remaining difference in the higher energy side can then be absorbed by a small shift in , resulting in an excellent fit (solid curve) to the NH data (blue dashed curve) in the lower panel of the right plot in Fig. 1, shown for energy resolution.
Solid curves are for NH, while dashed curves are for IH. The results clearly show that the mass hierarchy can be determined by those experiments only if the energy resolution of the detector is or better, and that the optimal baseline length (as shown by the cross symbol) is around 50 km for that resolution. The small for the baseline length km and km is due to a shift in and low statistics, respectively. For the and cases stays almost zero at all .
The right plot in Fig. 2 shows the value as a function of the baseline length for different values with . The curves from the top to the bottom are obtained for and , respectively. The effect of the systematic uncertainty is significant as discussed in ref. Qian et al. (2012a), reducing the peak value of from 11.0 () to 9.7 (), 8.4 () and 6.9 () for NH. The optimal shortens from 51 km for to 47 km for .
In addition, the neutrino parameters, and , can be measured accurately with statistical uncertainties shown in Fig. 3.
with the energy resolution of at km; the percentage values in the parentheses denote the relative accuracy of the measurement. Those uncertainties are almost independent of the mass hierarchy and of the energy resolution, with the only exception of the uncertainty for which the larger resolution results in the larger uncertainty. and are measured most accurately around km.
V Discussions and Conclusion
In this proceedings we have investigated the sensitivity of medium-baseline reactor-electron-antineutrino oscillation experiments for determining the neutrino mass hierarchy by performing the standard analysis.
We have carefully studied the impacts of the energy resolution and find that the sensitivity strongly depends on it. The optimal baseline length is found to depend slightly on the energy resolution, preferring the length slightly shorter than 50 km for the energy resolution of and . At the optimal baseline length, the energy resolution better than the level is needed to determine the neutrino mass hierarchy pattern. For a 5 kton detector (with 12% weight fraction of free proton) placed at km away from a reactor, an experiment would determine the mass hierarchy with on average after five or more years of running if the energy resolution of is achieved, while a factor of three larger or longer experiment is needed to achieve the same goal for the energy resolution of .
It is also found that this experiment can measure the neutrino parameters, , and , very accurately as shown in (13) for an experiment of 5kt (12% free-proton weight fraction)5yrs at km.
Acknowledgements.This proceedings is based on the work in collaboration with Shao-Feng Ge, Kaoru Hagiwara and Naotoshi Okamura. We wish to thank Jun Cao, Jarah Evslin, Soo-Bong Kim, Serguey Petcov, Xin Qian, Yifang Wang and Xinmin Zhang for valuable discussions on reactor neutrino experiments. This work was in part supported by Korea Neutrino Research Center (KNRC) through National Research Foundation of Korea Grant.
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