Recoilless resonant absorption of a monochromatic neutrino beam for measuring Δm231 and θ13

We discuss, in the context of precision measurement of Δ m231 and θ13, physics capabilities enabled by the recoilless resonant absorption of a monochromatic antineutrino beam enhanced by the Mössbauer effect recently proposed by Raghavan. Under the assumption of a small relative systematic error of the level of a few tenths of a per cent between measurements at different detector locations, we give analytical and numerical estimates of the sensitivities to Δ m231 and sin22θ13. The accuracies of their determination are enormous; the fractional uncertainty in Δ m231 achievable by ten point measurement is 0.6% (2.4%) for sin22θ13 = 0.05, and the uncertainty of sin22θ13 is 0.002 (0.008) both at 1σ confidence level (CL) with the optimistic (pessimistic) assumption of systematic error of 0.2% (1%). The former opens a new possibility of determining the neutrino mass hierarchy by comparing the measured value of Δ m231 with that from accelerator experiments, while the latter will help to resolve the θ23 octant degeneracy.

and θ 13 , physics capabilities enabled by the recoilless resonant absorption of a monochromatic antineutrino beam enhanced by the Mössbauer effect recently proposed by Raghavan. Under the assumption of a small relative systematic error of the level of a few tenths of a per cent between measurements at different detector locations, we give analytical and numerical estimates of the sensitivities to m 2 31 and sin 2 2θ 13 . The accuracies of their determination are enormous; the fractional uncertainty in m 2 31 achievable by ten point measurement is 0.6% (2.4%) for sin 2 2θ 13 = 0.05, and the uncertainty of sin 2 2θ 13 is 0.002 (0.008) both at 1σ confidence level (CL) with the optimistic (pessimistic) assumption of systematic error of 0.2% (1%). The former opens a new possibility of determining the neutrino mass hierarchy by comparing the measured value of m 2 31 with that from accelerator experiments, while the latter will help to resolve the θ 23 octant degeneracy. 3 Author to whom any correspondence should be addressed.

Introduction
Recently, the intriguing possibility was suggested by Raghavan [1,2] that the resonant absorption reaction [3]ν e + 3 He + orbital e − → 3 H (1) with simultaneous capture of an atomic orbital electron can be dramatically enhanced. The key idea is to use a monochromaticν e beam with the energy 18.6 keV from the inverse reaction 3 H →ν e + 3 He + orbital e − , by which the resonance condition is automatically satisfied.
(See [4,5] for earlier suggestions.) He then suggested an experiment to measure θ 13 by utilizing the ultra low-energy monochromaticν e beam. Though similar to the reactor θ 13 experiments [6]- [8], the typical baseline length is of order 10 m due to the much lower energy of the beam by a factor of 150, making it achievable in the laboratories. The mechanism, in principle, would work with a more generic setting in which 3 H and 3 He in (1) are replaced by nuclei A(Z) and A(Z + 1). The author of [1,2] then went on to an even bolder proposal of enhancement by a factor of ∼10 11 by embedding both 3 H and 3 He into solids [4,5] by which the broadening of the beam due to nuclear recoil is severely suppressed by a mechanism similar to the Mössbauer effect [9]. Then, the event rate of the θ 13 experiment is enhanced by the same factor, allowing an extremely high counting rate. Thanks to the ultimate energy resolution of E ν /E ν 2 × 10 −17 enabled by the recoilless mechanism, he was able to propose a table top experiment to measure the gravitational red shift of neutrinos, a neutrino analogue of the Pound-Rebka experiment for photons [10].

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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT In this paper, we examine possible physics potentials of the θ 13 experiment proposed in [1,2]. The characteristic feature of the experiment, which clearly marks the difference from the reactor θ 13 measurement, is the use of a monochromatic beam apart from the shorter baseline by a factor of 150. Then, the most interesting question is how accurately m 2 31 can be determined. Note that even without recoilless setting, the beam energy width is of the order of E ν /E ν ∼ 10 −5 , and it can be ignored for all practical purposes. It is also interesting to explore the accuracy of the θ 13 measurement. In addition to possible extremely high statistics, the baseline as short as ∼10 m should allow us to utilize the setting of a continuously movable detector, which was once proposed in a reactor θ 13 experiment [11] but that did not survive in the (semi-) final proposal. The method will greatly help to reduce the experimental systematic uncertainties of the measurement.
We will show in our analysis that the accuracies one can achieve for m 2 31 and θ 13 determination by recoilless resonant absorption are enormous. At sin 2 2θ 13 = 0.05, for example, the fractional uncertainty in m 2 31 determination is 0.6% (2.4%) and the uncertainty of sin 2 2θ 13 is 0.002 (0.008) both at 1σ CL under an optimistic (pessimistic) assumption of systematic error of 0.2% (1%).
What is the scientific merit of such precision measurement of m 2 31 and θ 13 ? With a 1% level precision of m 2 31 , the method for determining neutrino mass hierarchy by comparing between the two effective m 2 measured in reactor and accelerator (or atmospheric) disappearance measurements [12,13] would work, opening another door for determining the neutrino mass hierarchy. It is also proposed [7] that the θ 23 octant degeneracy can be resolved by combining reactor measurements of θ 13 with accelerator disappearance (appearance) measurements of sin 2 2θ 23 (s 2 23 sin 2 2θ 13 ). 4 (See [16,19] for earlier qualitative suggestions.) The results of the recent quantitative analysis [20], however, indicate that the resolving power of the method is limited at small θ 13 primarily because of the uncertainties in reactor measurement of θ 13 . Therefore, the highly accurate measurement of m 2 31 and θ 13 which is enabled by using the resonant absorption reaction should help resolve the mass hierarchy and the θ 23 degeneracies.
In section 2, we discuss 'conceptual design' of the possible experiments. In section 3, we define the statistical method for our analysis. In section 4, we present a numerical analysis of the sensitivities of θ 13 and m 2 31 measurement. In section 5, we complement the numerical estimate in section 4 by giving an analytic estimate of the sensitivities. In section 6, we give some remarks on the implications of our results. In the appendix, we give a general formula for the inverse of the error matrix.
where the neutrino mass squared difference is defined as m 2 ji ≡ m 2 j − m 2 i with neutrino masses 5 m i (i = 1-3) and L is the distance from source to detector. With E ν = 18.6 keV, the first oscillation maximum (minimum in P(ν e →ν e )) is reached at the baseline distance While the current value of m 2 31 which comes from the atmospheric [22] and the accelerator [23] measurement has large uncertainties, it should be possible to narrow down the value, thanks to the ongoing and the forthcoming disappearance measurement by MINOS [24] and T2K [25] experiments. Furthermore, the experiment considered in this paper is powerful enough to determine both quantities accurately at the same time, if detector locations are appropriately chosen.
The whole discussion of the θ 13 experiment must be preceded by the test measurement at ∼10 cm or so to verify the principle, namely to demonstrate that the mechanism of resonant enhancement proposed in [1,2] is indeed at work. At the same time, the flux times cross-section must be measured to check the consistency of the Monte Carlo estimate. Then, one can go on to the measurement of θ 13 and m 2 31 , and possibly other quantities. Because of the expected high statistics of the experiment it is natural to think about using spectrum information. In the case of a monochromatic beam, this amounts to considering measurements at several different detector locations.
Let us estimate the event rate. Although the precise rate is hard to estimate, the numbers displayed below will give the reader a feeling on what would be the timescale for the experiment. Theν e flux from 3 H source with strength S MCi due to bound state beta decay is given by where the ratio of bound state beta decay to free space decay is taken to be 4.7 × 10 −3 based on [26]. The rate of the resonant absorption reaction can be computed by using cross-section σ res and number of target atoms N T as R = N T fν e σ res . Without the Mössbauer enhancement, the cross-section is estimated to be σ res 10 −42 cm 2 [1,2] based on [3]. Then, the rate with target mass M T without neutrino oscillation is given by 5 When we speak about discriminating the neutrino mass hierarchy by comparing the two 'large' m 2 measured inν e disappearance and ν µ disappearance channels, one has to be careful about the definition of m 2 which enters into the survival probabilities [12]. While keeping this point in mind, we do not try to elaborate the expressions of m 2 this paper by just writing it as m 2 31 in theν e disappearance channel which may be interpreted as m 2 eff | e in [12].

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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT An improved estimate in [2] entailed a factor of 10 11 enhancement of the cross-section by the Mössbauer effect after the source and the target are embedded into solids. Assuming the enhancement factor, σ res 5 × 10 −32 cm 2 and the rate becomes Therefore, one obtains about 1.2 × 10 6 events per day for a 1 MCi source and 100 g 3 He target at a baseline distance L = 10 m. If the enhancement factor is not reached, the running time for collecting the same number of events becomes longer accordingly. Thus, once the 3 He (and much easier 3 H) implementation into solid is achieved, the event rate is sufficient. The real issue for high sensitivity measurement of m 2 31 and sin 2 2θ 13 is whether the produced 3 H can be counted directly without waiting for decay back to 3 He by emitting an electron. This is because the long lifetime of 12.33 years [27] of 3 H makes it impossible to identify in which period the decayed 3 H was produced, resulting in errors of the event rate at each detector location. Possibilities of real-time counting and direct counting by extracting 3 H atoms are mentioned in [2]. In this paper, we assume that at least one such method works, and it offers the opportunity of direct counting of events. Note that the detection efficiency need not be high because of huge number of events. What is important is the time-stable counting rate which allows relative systematic errors between measurements at different detector locations small enough.

Statistical method for analysis
In this section, we define the statistical procedure for our analysis to estimate the sensitivities of m 2 31 and sin 2 2θ 13 to be carried out in the following sections. We aim at illuminating general properties of the χ 2 under the assumption of the small uncorrelated systematic errors compared to the correlated ones.

Definition of χ 2 and characteristic properties of errors
We consider measurement at n different distances L = L i (i = 1, 2, . . . , n) from the source. Then, the appropriate form of χ 2 which is suited for analytic study [28] and is simply denoted as χ 2 hereafter, is as follows: where N obs is the number of events computed with the values of parameters given by nature, and N exp is the one computed with a certain trial set of parameters. σ c is the systematic error common to measurement at n different distances, the correlated error, whereas σ usys,i indicate errors that cannot be attributed to σ c , the uncorrelated errors. Examples of the former and latter errors are as follows: 1. σ c (correlated error): uncertainties in number of target 3 He atoms, errors in counting the number of produced tritium nuclei, errors in calculating resonant absorption cross-section, errors in estimating the efficiency of counting tritium nuclei. Since we consider a moving detector setting the list of the possible uncorrelated systematic errors is quite limited. If a near detector with an identical structure to a movable far detector exists the error can, in principle, be vanishingly small. One may think of errors of the order of 0.1-0.3%. This is because the flux times cross-section can be monitored in real time by a near detector. In fact, similar values for uncorrelated systematic error are adopted in sensitivity estimates of some of the reactor θ 13 experiments such as the Braidwood, the Daya Bay, and the Angra projects [29]- [31]. In near future experiments, somewhat larger values are taken, 0.6% in the Double-Chooz project [32] and 0.35% in KASKA [33].
On the other hand, it may not be so easy to control the correlated systematic error σ c . The number of 3 H nuclei may be measured when they are implemented into solid. The number of target nuclei times the resonant absorption cross-section may be measured in a research and development stage with a near detector. Therefore, we suspect that the largest error may come from uncertainty in the counting rate of the produced 3 H nuclei. Of course, a reliable estimate of systematic errors σ c and σ usys requires specification of the site to estimate the background caused by n 3 H reaction etc. But it can be experimentally measured by the source on and off procedure, as pointed out in [1]. Lacking definitive numbers for σ c at the moment, we use a tentative value σ c = 10% throughout our analysis. We have checked that the results barely change even if we use the more conservative number σ c = 20%.
If direct counting of 3 H atoms does not work, we may have to expect much larger systematic errors, because one has to extract the event rate at each detector location only by fitting the decay curve. In this case, determination of baseline-dependent event rates would be more and more difficult for a larger number of detector locations. Probably a better strategy without direct counting would be to place multiple identical detectors (or of the same structure) at appropriate baseline distances. Even in this case, it is quite possible that the uncorrelated systematic error σ usys can be controlled to 1% level, as expected in a variety of reactor θ 13 experiments [8].

Approximate form of χ 2 with hierarchy in errors
By eliminating α through minimization the χ 2 can be written as where x is defined as Using the general formula given in the appendix, V −1 is given by only through this combination. Therefore, the particular case that will be taken in the next section, in fact, includes many cases with different event number but with the same σ u .
Under the approximation σ 2 ui σ 2 c , V −1 simplifies; The remarkable feature of (11) is the 'scaling behaviour' in which χ 2 is independent of the correlated error σ c , and the sensitivity to sin 2 2θ 13 and m 2 31 can be made higher as the uncorrelated systematic errors as well as the statistical error become smaller. It may be counterintuitive because the leading term of the error matrix V is of order σ 2 c . (See the appendix.) It is due to the singular nature of the leading order matrix, as noted in [34].

Estimation of sensitivities of m 2 31 and θ 13
We now examine the sensitivities of m 2 31 and sin 2 2θ 13 achievable by the recoilless resonant absorption of monochromaticν e enhanced by the Mössbauer effect. The numerical estimate of the sensitivities in this section will be followed by that by the analytic method in section 5.
The setting of the movable detector and the expected high statistics of the experiment make it possible to consider the situation that an equal number of events are taken at each detector location. Of course, the farther the detector from the source, the longer an exposure will take. In the following analysis, the number of events is assumed to be 10 6 at each detector location. Given the rate in (6), and assuming that direct counting works, it is obtainable in ∼10 days for a 100 g 3 He target even if the detector is located at the second oscillation maximum, L = 3L OM . On the other hand, the number of events 10 6 is sufficient for our purpose because it is unlikely that the uncorrelated systematic errors can be made much smaller than 0.1%.
We take a 'common-sense approach'to determine the locations of the detectors and postpone the discussion of the optimization problem. We examine the following four types of Run: Run I, IIA, IIB, and III, for the measurement.
In the following two subsections 4.1 and 4.2, we examine the cases of the optimistic (σ usys = 0.2%) and the pessimistic (σ usys = 1%) systematic errors. We stress here that the analyses 8 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT we will present there contain much more general cases. For example, because of the scaling behaviour discussed in the previous section, the case with N = 10 6 and σ usys = 0.2% is equivalent to N = 2 × 10 5 and σ usys = 0.0%. Similarly, the case with N = 10 6 and σ usys = 1% is equivalent to N = 1.33 × 10 4 and σ usys = 0.5%. In subsection 4.3, we give an estimate of the sensitivities using a tentative setting which may be possible without direct counting of 3 H atoms.

Case of optimistic systematic error
We focus in this subsection on the case of optimistic systematic error, from which one may obtain some feeling for the ultimate sensitivities achievable by the present method with the four Run options described above. As we mentioned earlier, the correlated systematic error σ c is taken to be a tentative value of 10% throughout our analysis. The uncorrelated systematic error σ usys , which is assumed to be equal for all detector locations, is taken to be 0.2% in this subsection.
In figure 1, we show in sin 2 2θ 13 − m 2 31 plane the expected allowed region by Run I, IIA, IIB, and III with number of events 10 6 in each location. Throughout the analysis, the true value of m 2 31 is assumed to be m 2 31 = 2.5 × 10 −3 eV 2 . The input values of sin 2 2θ 13 are taken as 0.1 and 0.01 in the left-and the right-hand panels in figure 1, respectively. Throughout the numerical analyses in this paper, the other oscillation parameters are taken as: m 2 21 = 7.9 × 10 −5 eV 2 , sin 2 θ 12 = 0.31 and sin 2 θ 23 = 0.5. In each panel, the red-solid, the green-dashed, and the blue-dotted lines are for 1σ (68.27%), 2σ (95.45%) and 3σ (99.73%) CL for 2 DOF, respectively.
To complement figure 1, we give in table 1 the expected sensitivities to m 2 31 at 1σ and 3σCL (the latter in parentheses) for 1 DOF for Run I-III. They are obtained by optimizing sin 2 2θ 13 in the fit. For relatively large θ 13 , sin 2 2θ 13 0.05 the expected sensitivities to m 2 31 are enormous. For sin 2 2θ 13 = 0.05, the sensitivities are already less than 1% in Run IIA, and it is about 0.6% in Run IIB both at 1σCL. The scaling behaviour mentioned at the end of the previous section is roughly satisfied, as indicated in table 1. (See section 5 for more detailed discussions.) For a small value of θ 13 , sin 2 2θ 13 = 0.01, the sensitivities to m 2 31 are much worse, as shown in table 1. They are about 6% in Run IIA, and 3% in Run IIB both at 1σCL. If Run III is carried out it can go down to 2%.
In table 2, the expected sensitivities to sin 2 2θ 13 at 1σ and 3σCL (the latter in parentheses) for 1 DOF are given. The sensitivities to sin 2 2θ 13 can be better characterized by δ(sin 2 2θ 13 ), not its fraction to sin 2 2θ 13 , as will be understood in our analytic treatment in section 5. By Run I one can already achieve an accuracy of δ(sin 2 2θ 13 ) 0.003, and Run IIA or IIB reach δ(sin 2 2θ 13 ) 0.002. The effect of measurement at multiple detector locations on improvement of the sensitivity is relatively minor in the case of sensitivities to sin 2 2θ 13 . This is in sharp contrast to the case of m 2 31 . To show the sensitivity limit on θ 13 achievable by the present method, we present in figure 2 the excluded regions in sin 2 2θ 13 − m 2 31 space, assuming the case of no depletion ofν e flux. The four panels in figure 2 correspond to Run I, IIA, IIB, and III. In each panel, the red-solid, the green-dashed, and the blue-dotted lines are for 1σ (68.27%), 2σ (95.45%) and 3σ (99.73%) CL for 1 DOF, respectively. The sensitivities indicated in figure 2 are quite impressive, which reach sin 2 2θ 13 0.006 at 2σCL even in Run I, and sin 2 2θ 13 0.004 at the same CL in Run IIB. As expected the improvement by adding more detector locations is relatively minor.

Case of pessimistic systematic error
It might be possible that we end up with the error of ∼1% due to, e. g., time dependence of the source even though the method of movable detector with direct counting of 3 H works. In figure 3, we present the similar allowed region in sin 2 2θ 13 − m 2 31 space obtained by the same Run I, IIA, IIB, and III with the same number of events of 10 6 in each location but with a pessimistic systematic error of σ usys = 1%. At large θ 13 , sin 2 2θ 13 = 0.1, we still have reasonable sensitivities to m 2 31 . For Run IIB and III, for example, the sensitivities are about 1-2% level for 2 DOF. At sin 2 2θ 13 = 0.01, however, the sensitivity to m 2 31 is lost except for the one at 1σCL in Run III. It indicates that the value of θ 13 is close to the sensitivity limit, and hence we do not include the figure for it though we did for the case of optimistic error, figure 2.
For more detailed information on sensitivities with the pessimistic systematic error of σ usys = 1%, we give in tables 3 and 4 the sensitivities at 1σ and 3σ CL (the latter in parentheses) for 1 DOF to m 2 31 and sin 2 2θ 13 , respectively. The column without number represents that no limit is obtained, in a similar way as seen in the right-hand panels of figure 3. At relatively large θ 13 , sin 2 2θ 13 = 0.1 and 0.05, the sensitivities to m 2 31 remain good, 1.2 and 2.4% at 1σ CL for Run IIB. But, the sensitivity quickly drops and is about 15% in the same Run for sin 2 2θ 13 = 0.01. The sensitivity to sin 2 2θ 13 is still reasonable, δ(sin 2 2θ 13 ) 0.008 at 1σCL for Run IIB even at sin 2 2θ 13 = 0.01. For disappearance measurement of P(ν e →ν e ), sin 2 2θ 13 = 0.01 is too small a value for a pessimistic systematic error of σ usys = 1% to retain the sensitivity to m 2 31 . Therefore, reduction of the uncorrelated systematic error is a mandatory requirement in this method for accurate measurement of m 2 31 at small θ 13 .

Case without direct detection of 3 H
Suppose that the direct detection of 3 H in the target is not possible. Then we may have to take the option of multiple detectors with the same structure, giving up the idea of the movable detector. In this case, most probably, we have to accept a pessimistic value of the uncorrelated systematic error of 1-3%. This will cause two important changes in designing the experiment. (i) Number of events that can be accumulated in a reasonable timescale would be smaller by a factor of ∼10 than the case of direct detection. (ii) Number of detectors that can be prepared by keeping their identity to suppress the uncorrelated systematic errors may be limited. Therefore, three detector settings, for example, (in addition to a near detector which monitors the flux) would be more practical.
To understand the performance of such a reduced setting with larger errors, we have carried out a similar χ 2 analysis as done in the previous subsections. We take three detector settings with tentatively determined baselines L = 1 9 L OM , L OM and 3L OM , and assume 10 5 events in each detector. We call the setting Run 0. The three cases of uncorrelated systematic errors, 1, 2 and 3%, are examined. In table 5, the expected fractional uncertainty δ( m 2 )/ m 2 31 (0) and sin 2 2θ 13 for Run 0 are presented. With 1% of the uncorrelated systematic errors, while a sensitivity comparable to Run I is reached for sin 2 2θ 13 , uncertainty of m 2 31 is larger by a factor of 2 Table 3. The expected fractional uncertainty δ( m 2 )/ m 2 31 (0) in % for the pessimistic systematic error of σ usys = 1% reachable by Runs I-III defined in the text. The uncertainties are given at 1σ (68.27%) CL for 1 DOF, and the numbers in parentheses are the ones at 3σ (99.73%) CL for 1 DOF. In the left, middle and right columns, the input value of θ 13 are taken as sin 2 2θ 13 = 0.1, 0.05 and 0.01, respectively. The column without number represents that no limit is obtained.   For the cases of uncorrelated systematic errors of 2 and 3%, the uncertainties of sin 2 2θ 13 get worse by a factor of 2 and 3, respectively. The behaviour of sensitivities to m 2 31 are more complicated and no numbers are obtained for uncertainties at 3σ level for most cases. We note that loss of the sensitivities mainly comes from fewer number of detectors with larger systematic errors, but not from an order of magnitude smaller number of events.
The results obtained above indicate that direct counting, either real-time counting or an efficient extraction, of the produced 3 H is mandatory to make the type of experiments under discussion useful.

Analytic estimation of the sensitivities
We complement our numerical analysis of the sensitivities in the previous section by presenting analytical treatment of the uncertainties in the m 2 31 and θ 13 determination. In particular, Table 5. The expected fractional uncertainty δ( m 2 )/ m 2 31 (0) in % and δ(sin 2 2θ 13 ) for the pessimistic systematic error of σ usys = 1, 2 and 3% reachable by Run 0 with three detectors as defined in the text. The uncertainties are given at 1σ (68.27%) CL for 1 DOF, and the numbers in parentheses are the ones at 3σ (99.73%) CL for 1 DOF. In the left, middle, and right columns, the input value of θ 13 are taken as sin 2 2θ 13 = 0.1, 0.05 and 0.01, respectively. The column without number represents that no limit is obtained. In section 3, we have argued, assuming feasible direct counting of 3 H atoms, that the hierarchy of errors is very likely to hold.
We restrict ourselves, consistent with the numerical analysis done in the previous section, to the case that an equal number of events are taken in each baseline, N obs i = N obs , which may be translated into N exp i = N exp . We also assume, for simplicity, the case of equal uncorrelated systematic error in each detector location, σ 2 ui ≡ σ 2 usys,i + 1/N exp i = σ 2 u . Under these assumptions, V −1 has a simple form where H n×n is an n × n matrix whose elements are all unity, H i,j = 1 for any i and j. It indicates again the independence of the χ 2 on the correlated error σ 2 c and the 'scaling behaviour' with respect to the uncorrelated systematic error. Then, the χ 2 simplifies:

Optimal baselines and sensitivities for two detector locations
Let us start by examining sensitivities for the case of two detector locations. Because of a simple setting with monochromaticν e beam, we can give explicit expression of χ 2 in terms of small deviation of the parameters from the true (nature's) values. For this purpose, we note that the number of events is given by where fν e denotes the neutrino flux, σ res the absorption cross-section, N T the number of target nuclei, T the running time, and P ee is a short-hand notation for P(ν e →ν e ). In the setting in this section, T is adjusted such that an equal number of events is collected at each location of the detector. We recall that N obs denotes the event number computed with the true value of the parameters, m 2 31 = m 2 (0) and sin 2 2θ 13 = sin 2 2θ 13 (0), whereas N exp denotes the event number computed with possible small deviations δ( m 2 ) and δ(sin 2 2θ 13 ) from the true values of the parameters. Then, ith component of x vector in (9) is given to first order in the deviation as where P (0) ee (L i ) ≡ P ee (θ 13 (0), m 2 31 (0), L i ) for which we have used the two-flavour expression (2). For simplicity we restrict our discussion in this section to the analysis with single DOF. This is a natural setting for estimating ultimate sensitivities; When we discuss sensitivity of m 2 31 we optimize χ 2 in terms of sin 2 2θ 13 , and vice versa. Or, one can think of the situation that, in determination of m 2 31 , sin 2 2θ 13 is accurately determined by other ways, e.g., long-baseline accelerator experiments. Under the approximation σ 2 u σ 2 c and using V −1 in (12), χ 2 is given for small deviations of θ 13 and m 2 31 as follows: Now, we can address the problem of optimal baseline and estimate the sensitivities of sin 2 2θ 13 and m 2 31 under the approximations stated above. Since sin 2 2θ 13 0.1 [35], it may be a reasonable approximation to set P (0) ee (L i ) = 1 in the denominator, as we do in the rest of the section.
Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT 5.1.1. Optimal setting and sensitivity to sin 2 2θ 13 . To maximize (16) one should take L 1 as short as possible, and L 2 at the oscillation maximum, the well-known feature in the reactor θ 13 experiments. Thus, we take L 1 = 0 and L 2 = L OM which makes the square parenthesis in (16) unity. Then, one can obtain the sensitivity at N CL σCL for two detector locations as For σ u = 0.2%, δ(sin 2 2θ 13 ) = 2.8 × 10 −3 (8.5 × 10 −3 ) at 1σ (3σ) CL. For σ u = 1%, δ(sin 2 2θ 13 ) = 0.014 (0.043) at 1σ (3σ) CL, which is not so far from the sensitivities quoted in the literatures of the reactor θ 13 experiments. 2 31 . The optimal baseline setting is quite different for m 2 31 . We first recall a property of the function x sin x; it has the first maximum x sin x = 1.82 at x = 2.02 (L = 0.64L OM ), has the first minimum x sin x = −4.81 at x = 4.91 (L = 1.56L OM ), and then the second maximum x sin x = 7.92 at x = 7.98 (L = 2.54L OM ), and so on. For simplicity, we restrict ourselves to x 3π, which means L 3L OM so that the running time does not blow up. Then, the optimal setting is L 1 = 0.64L OM and L 2 = 1.56L OM if we restrict to L 2L OM , and L 1 = 1.56L OM and L 2 = 2.54L OM if we allow baseline until L 3L OM . Despite the factor of 4 different baseline lengths, we still assume equal numbers of events at L = 0.64L OM and L = 2.54L OM , which implies 16 times longer exposure time at the latter distance.

Optimal setting and sensitivity to m
The χ 2 is given approximately by The coefficient c is 5.5 for L 2L OM and 20.3 for L 3L OM . Then, we obtain the sensitivity at N CL σCL for two detector locations as

The problem of n detector locations reduces to the two-location case
We first show that the problem of optimal setting of n detector locations reduces to the case of two locations under the assumption of equal number of events in each location. To indicate the essential point let us first consider a simplified χ 2 of the form χ 2 = n i,j=1 (x i − x j ) 2 and 0 x i 1, which is the essential part of χ 2 for sin 2 2θ 13 , the equation (16). In the case of Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT two locations the configuration which maximizes the χ 2 (n = 2) is x 1 = 0 and x 2 = 1 and χ 2 max (n = 2) = 1. It is not difficult to observe that in the case of n locations the configuration which maximizes χ 2 (n) is 1. Even n : n = 2M; x = 0 appears M times, and x = 1 appears M times, χ 2 (n) max = M 2 . 2. Odd n : n = 2M + 1; x = 0 appears M+1 times, and x = 1 appears M times, or vice versa, Thus, the problem of optimal setting with equal number of events at each detector location is reduced to the two-location case.
For χ 2 for m 2 31 in (17), the situation is slightly different because the function |x sin x| increases without limit as x becomes large. Therefore, mathematically speaking, one can obtain better and better accuracies as one goes to longer and longer distances in our setting of equal number of events at any detector location. 6 But, since we want to remain to a reasonable running time, we have restricted our discussions to baselines limited by L 2L OM or L 3L OM in the two-location case, the restriction which is kept throughout this section. Then, one can show that the same result follows for χ 2 for m 2 31 , (17). Namely, in the case of n locations, the highest sensitivity is achieved at the same baselines L 1 and L 2 of the two location case; L 1 in [ n 2 ] ([ n 2 ] + 1 for odd n) times, and L 2 in [ n 2 ] times, where [ ] implies Gauss' symbol. The maximal value of χ 2 is, therefore, given by It can be translated into the uncertainties at N CL CL as δ(sin 2 2θ 13 )(n) = 2 n δ(sin 2 2θ 13 )(n = 2), for even n. For odd n, n in (22) must be replaced by (n 2 − 1)/n. δ(sin 2 2θ 13 )(n = 2) and (δ( m 2 31 )/ m 2 31 (0))(n = 2) are given respectively by (18) and (20). Therefore, the sensitivity gradually improves as number of runs becomes larger. 7 At the end of this subsection, we want to note the following: the reason why we did not take these sets of optimal distances for θ 13 and m 2 31 obtained in this subsection in the numerical analyses in section 4 is that the sensitivity to m 2 31 is lost if we tune the setting optimal for sin 2 2θ 13 , and vice versa. The reason for this is easy to understand; at baselines L 1 and L 2 which maximizes χ 2 θ (χ 2 m 2 ), χ 2 m 2 (χ 2 θ ) vanishes (approximately vanishes), because sin x 1 = sin x 2 = 0 at x 1 = 0 and x 2 = π (sin 2 x 1 2 sin 2 x 2 2 1 2 at x 1 π 2 and x 2 3π 2 ). Nonetheless, we will see, 6 This explains at least partly the reason why the sensitivities to m 2 31 and sin 2 2θ 13 differ in dependence on distance from the source to a detector, as indicated in figure 3 in [36]. 7 It is a well-known feature in the multiple detector setting in the reactor θ 13 experiments in which one obtains better sensitivity as in (18) if the two identical detectors, the near and the far, are each divided into small detectors in the same way, if the uncorrelated systematic error σ usys is made to be equal with that of the original large detector and if the statistical errors are negligible even for divided detectors. This point was emphasized by Yasuda [37].
Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Table 6. The analytically estimated fractional uncertainties of m 2 31 , δ( m 2 )/ m 2 31 (0) in % are given for the optimistic systematic error of σ usys = 0.2% and for the pessimistic one (in parentheses) of σ usys = 1%. The uncertainties are given at 1σ (68.27%) CL for 1 DOF for the cases of 5, 10, and 20 detector locations. The upper three rows are for the case of restricted baselines, L 2L OM , whereas the lower three rows are for the cases of somewhat relaxed baseline setting, L 3L OM . In the left, middle, and right columns, the input value of θ 13 are taken as sin 2 2θ 13 = 0.1, 0.05, and 0.01, respectively. in the following subsections, that the sensitivities analytically estimated with optimal baseline distances and the numerically calculated ones with baselines taken by 'common sense' agree reasonably well with each other.

Analytic estimation of the sensitivities; sin 2 2θ 13
Let us examine the case of σ usys = 0.2% and the number of events N = 10 6 which was considered in our numerical analysis in section 4. Then, σ u = 0.22%. In the case of five locations, n = 5, δ(sin 2 2θ 13 ) = 2.0 × 10 −3 (6.1 × 10 −3 ) at 1σ (3σ) CL. Similarly for ten and 20 locations δ(sin 2 2θ 13 ) = 1.4 × 10 −3 (4.2 × 10 −3 ) and 0.99 × 10 −3 (3.0 × 10 −3 ), respectively, at 1σ (3σ) CL. They compare well with the numbers in table 2 though the latter are obtained with notso-tuned baseline settings. Notice that δ(sin 2 2θ 13 ) is independent of θ 13 under the present approximation of small deviation from the best fit. With σ usys = 1% the corresponding sensitivities are δ(sin 2 2θ 13 ) = 9.2 × 10 −3 (2.8 × 10 −2 ), 6.4 × 10 −3 (1.9 × 10 −2 ), and 4.5 × 10 −3 (1.3 × 10 −2 ) for five, ten and 20 locations, respectively, at 1σ (3σ) CL. They are again roughly consistent with the ones in table 4. In table 6, we give the fractional uncertainties of m 2 31 , δ( m 2 )/ m 2 31 (0) in % at 1σ CL for the optimistic and the pessimistic systematic errors of σ usys = 0.2 and 1% (the latter in parenthesis), respectively, obtained by using the equations (20) and (22). We do not show errors at 3σ CL because it is obtained simply by multiplying 3. Overall, the analytically estimated uncertainties are in reasonable agreement with those obtained by the numerical analysis in section 4. Notice that one has to compare the sensitivities of Run I and IIA with the case of severer restriction L 2L OM , and the ones of Run IIB and III with the case of milder restriction L 3L OM , because distances beyond 2L OM are involved in the latter runs. The fact that our analytical estimates of the errors are smaller than the numerical ones by ∼30% or so, apart from approximations involved, is consistent with that the latter are based on non-optimal baseline distances. It also implies that the baseline setting chosen by the 'common sense' used in the numerical analysis in section 4 is not so far from the optimal one, indicating that the sensitivities are rather stable against changes of baseline setting.
Based on the numerical and the analytical estimate of the uncertainties of m 2 31 determination, we conclude that sensitivities of less than 1% at 90% CL required for resolution of mass hierarchy proposed in [12,13] are in reach if the uncorrelated systematic error of σ usys = 0.2% is realized, and if θ 13 is relatively large, sin 2 2θ 13 0.05 in Run IIB. We have confined ourselves to the problem of optimal setting of distances under the constraint of equal number of events at each location. To minimize running time for a given sensitivity, we have to address the problem of optimal detector locations and exposure times for a given total running time. This is left for a future study.

Concluding remarks
In this paper, we have explored the potential of high sensitivity measurement of m 2 31 and θ 13 which is enabled by using the resonant absorption of a monochromaticν e beam enhanced by the Mössbauer effect. With baseline distances of ∼10 m, the movable detector setting is certainly possible. Assuming that the direct detection of produced 3 H atom either by real-time counting or extraction of 3 H atoms works, we have argued that the uncorrelated systematic error can be as small as 0.1-0.3%, if the near detector has the same structure as the far one. This will allow us to determine m 2 31 to the accuracies m 2 31 of 0.3(sin 2 2θ 13 /0.1) −1 % at 1σ CL (Run IIB, σ usys = 0.2%). The error of sin 2 2θ 13 is also small, sin 2 2θ 13 = 1.8 × 10 −3 almost independently of θ 13 with the same setting. The accuracy of the θ 13 measurement even in Run I, if the systematic error of 0.2% is reached, is comparable with that of the next generation accelerator ν e appearance experiments [25,38]. It may exceed the accelerator sensitivity if Run IIB is performed. If the systematic error is of 1% level, the sensitivity to θ 13 is comparable to the first-stage reactor θ 13 experiments even in Run IIB.
What is the scientific merit of the precision measurement of m 2 31 and θ 13 ? As we have already mentioned in section 1, the precision measurement of m 2 31 and θ 13 will have a great impact, at least, on two of the unknowns in lepton flavour mixing: the neutrino mass hierarchy and resolving the θ 23 octant degeneracy. It would be very interesting to carry out quantitative analyses of such possibilities. 8 We emphasize that such physics capabilities can only be made possible by direct counting of the produced 3 H atoms which makes a movable detector feasible. We hope that these exciting possibilities will stimulate further development of the experimental technology towards that goal.
What are the additional capabilities of the resonant absorption of the monochromaticν e beam? With 18.6 keV of neutrino energy, the solar oscillation maximum would be reached at L solarOM = 290( m 2 21 /8 × 10 −5 eV 2 ) −1 m. Then, it would be worthwhile to explore the possibility of precision measurement of θ 12 and m 2 21 , as was done for the reactor experiments [36]. But in the present case, neither geo-neutrinos norν e flux from nearby reactors (if any) contaminate the measurement. In particular, a possible movable or multiple baseline set up should allow improvement of accuracy of m 2 21 determination. Detection of CP violating effect due to δ in theν e disappearance measurement requires going down by a factor of O(10 −6 ) compared to CP conserving terms [40], Unfortunately, this would not be within reach despite great potential sensitivities achievable by the resonant absorption reaction.
Finally, the setup of multiple baseline lengths of ∼10 m allows a precision test of the pure vacuum oscillation hypothesis by observing a sine curve slightly modified by the solar m 2 21 oscillation. It will constrain various possible sub-leading effects such as de-coherence or new neutrino interactions to great precision.