Probing neutrino quantum decoherence at reactor experiments

We explore how well reactor antineutrino experiments can constrain or measure the loss of quantum coherence in neutrino oscillations. We assume that decoherence effects are encoded in the size of the neutrino wave-packet, $\sigma$. We find that the current experiments Daya Bay and the Reactor Experiment for Neutrino Oscillation (RENO) already constrain $\sigma>8.9\times 10^{-5}$ nm and estimate that future data from the Jiangmen Underground Neutrino Observatory (JUNO) would be sensitive to $\sigma<2.3\times 10^{-3}$ nm. If the effects of loss of coherence are within the sensitivity of JUNO, we expect $\sigma$ to be measured with good precision. The discovery of nontrivial decoherence effects in JUNO would indicate that our understanding of the coherence of neutrino sources is, at least, incomplete.

is much longer than the "atmospheric" oscillation length. Nonetheless, the energy and position resolutions are such that "atmospheric" oscillations are visible, rendering JUNO uniquely well suited to probe decoherence effects.
In Sec. II, we introduce neutrino oscillations and discuss the formalism we will use to describe and constrain decoherence, concentrating on how it modifies the neutrino oscillation probabilities at reactor experiments. In Sec. III, we analyze data from the ongoing Reactor Experiment for Neutrino Oscillation (RENO) and the Daya Bay reactor neutrino experiment, and discuss bounds on the wave-packet width, introduced in Sec. II. In Sec. IV, we discuss the sensitivity of JUNO. We summarize our results in Sec. V, and offer some concluding remarks.

II. NEUTRINO OSCILLATIONS, INCLUDING DECOHERENCE
Nuclear reactors produce an intense flux of electron antineutrinos with energies roughly in the [1 − 8] MeV range. These are detected some distance away from the source via inverse beta-decay, which allows one to measure the neutrino energy on an event-by-event basis with good precision. If the flux of electron antineutrinos is, somehow, known, reactor neutrino oscillation experiments can measure the survival probability of electron antineutrinos, P (ν e →ν e ), as a function of energy and baseline.
It is straight forward to compute P (ν e →ν e ). Here we include, rather generally, the effects of decoherence among the mass eigenstates. For a fixed neutrino energy E and baseline L, the density matrix ρ jk , j, k = 1, 2, 3, of the antineutrino state produced in the nuclear reactor, in the mass basis, can be written as where ∆m 2 jk = m 2 j − m 2 k , and ξ jk (L, E) = ξ kj (L, E) quantifies the loss of coherence as a function of the neutrino energy and the baseline. In the absence of decoherence, ξ jk = 0. The survival probability, including decoherence effects, is simply the ee element of the density matrix and reads P dec (ν e → ν e ) = j,k It is trivial to see that we recover the standard expression for the electron antineutrino disappearance when all ξ jk → 0. Throughout, we will use the standard PDG parameterization of the leptonic mixing matrix where |U e1 | 2 = cos 2 θ 12 cos 2 θ 13 , |U e2 | 2 = sin 2 θ 12 cos 2 θ 13 , and |U e3 | 2 = sin 2 θ 13 , and, unless otherwise noted, we assume that the true values of the relevant neutrino oscillation parameters are ∆m 2 31 = 2.5 × 10 −3 eV 2 , ∆m 2 21 = 7.55 × 10 −5 eV 2 , sin 2 θ 13 = 0.0216, sin 2 θ 12 = 0.32, in agreement with the best-fit values obtained from the world's neutrino data [10]. We assume the neutrino massordering is normal (∆m 2 31 > 0) and assume this information is known. We will comment on the consequences of this assumption when relevant.
Different physical effects lead to decoherence [11][12][13][14][15][16][17][18][19]. Here, we will concentrate on decoherence effects that grow as the baseline grows and parameterize the decoherence parameters as [3,13,16]  and further parameterize the coherence lengths as [3,13,16] Concretely, as discussed in [3,13,16], σ is the width of the neutrino wave-packet and depends on the properties of the neutrino source and of the detector. The physics that leads to this type of decoherence is the fact that the different neutrino mass eigenstates propagate with different speeds and, given enough time, the wave-packets ultimately separate. From a more pragmatic point of view, here σ is the single parameter that characterizes the effects of decoherence, and has dimensions of length. Decoherence effects vanish as the coherence lengths become very long: σ → ∞, and we highlight that the different coherence lengths are inversely proportional to the associated neutrino mass-squared differences. Here, constraining neutrino decoherence assuming the data are consistent with a perfectly coherent beam is equivalent to placing a lower bound on σ.
Decoherence effects in reactor experiments grow with the baseline and decrease with the neutrino energy. Fig. 1 depicts the expectedν e →ν e oscillation probability for typical reactor neutrino energies and the JUNO average baseline L = 52.5 km, assuming the oscillation parameters are the ones in Eqs. (5). The green, solid curve corresponds to standard oscillations with no decoherence effects while the red and black dashed ones are the expected disappearance probabilities in the presence of decoherence effects with σ = 2 × 10 −4 nm and σ = 2 × 10 −3 nm, respectively. Note that the fast oscillations "disappear" first and that the effect is more pronounced at smaller neutrino energies.

III. CURRENT CONSTRAINTS FROM RENO AND DAYA BAY
RENO and Daya Bay are reactor neutrino experiments in South Korea and China, respectively, that measure the flux of antineutrinos from nuclear reactors at L ∼ 100 m and L ∼ 1 km, using information from both the near and far detectors to measure P (ν e → ν e ). Given the typical reactor neutrino energies and the 1 km baselines, these experiments are sensitive to ∆m 2 31 and sin 2 θ 13 but insensitive to the "solar" parameters ∆m 2 21 and sin 2 θ 12 . This effective-two-flavor approximation also applies to the decoherence effect since L coh 12 L coh 13 L coh 23 . Hence, at the relevant energies and baselines, is an excellent description of electron antineutrino disappearance at RENO and Daya Bay. * For the same reasons, data from RENO and Daya Bay are insensitive to the neutrino mass-ordering and the results presented here do not depend on our assumption that the neutrino mass-ordering is normal. RENO uses a power plant with six nuclear reactors as neutrino sources and consists of two identical detectors at two different locations. Daya Bay makes use of six nuclear reactors located at two nearby sites. In the case of Daya Bay, there are eight identical detectors located at three different experimental halls; two experimental halls contain two detectors each that serve as near detectors, while the remaining four detectors are in the third experimental hall, which is further away.
For the results presented here, we use the most up-to-date data from the two experiments, corresponding to 2200 days of data from RENO [20] and 1958 days of data from Daya Bay [21]. The necessary information on all technical details, including the baselines, thermal power, fission fractions, and efficiencies, is obtained from Refs. [20,22,23] for RENO and Refs. [21,24,25] for Daya Bay. In our statistical analyses, we account for several sources of systematic uncertainties. We include uncertainties related to the thermal power for each core and to the detection efficiencies, uncertainties on the fission fractions, a shape uncertainty for each energy bin in our analyses, and an uncertainty on the energy scale.
We define the χ 2 function for RENO as Here, R where F i and N i are the event numbers in the ith energy bin at the far and near detector, respectively. R dat,i are the background-subtracted observed event ratios, while R exp,i ( p, α) are the expected event ratios for a given set of oscillation parameters p. The uncertainty for each bin is given by σ RENO i . The last term contains penalty factors for all of the systematic uncertainties α k with expectation value µ k and standard deviation σ k . Finally the number of bins is given by N RENO .
Similarly, for Daya Bay, we define Here, we take the ratios between the far and the first near detector and between the two near detectors, as was done in Ref. [26]. To calculate the expected number of events and the χ 2 functions for each experiment, we use GLoBES [27,28]. We use reactor fluxes as parameterized in Ref. [29] and the inverse beta-decay cross section from Ref. [30]. We analyze the data from each experiment independently and also perform combined analyses, where we use In order to validate our treatment of the two data sets, we first assume a perfectly coherent source and compare our results to those published by RENO and Daya Bay. Hence, we first consider the case p = (∆m 2 31 , θ 13 ). The solar parameters are fixed to sin 2 θ 12 = 0.32 and ∆m 2 21 = 7.55 × 10 −5 eV 2 [10]. As already mentioned, this choice is inconsequential for the results presented here. The grey and blue ellipses in Fig. 2 correspond to the region of the oscillation parameter space consistent, at the 90 and 99% CL respectively (for two degrees of freedom), with data from RENO (left) and Daya Bay (center), along with the combined result (right). The combined analysis is clearly dominated by the Daya Bay data. These results agree quantitatively very well with those presented in Refs. [20] and [21].
Next, we allow for the possibility that the wave-packet width σ is not infinite, and extend the set of model parameters: p = (∆m 2 31 , θ 13 , σ). Marginalizing over σ, the regions of the sin 2 θ 13 -∆m 2 31 parameter space consistent with the different data sets are depicted in Fig. 2 as closed, empty contours (solid at the 99% CL, dashed at the 90% CL). Not surprisingly, the allowed regions on the ∆m 2 31 -sin 2 θ 13 plane are larger once one allows for finite σ values. The region of parameter space in the "combined" case is noticeably smaller than that allowed by Daya Bay data. This is a consequence of the fact that the L/E values probed by RENO and Daya Bay are slightly different and the shapes * In our numerical calculations, we use the full three-neutrino description, Eqs. (4), (6) and (7). of the allowed regions are slightly different. In particular, the best-fit point in the case of Daya Bay shifts more than that of RENO once finite values of σ are allowed. The result depicted in Fig. 2 (center) is in qualitative agreement with the results obtained by the Daya Bay collaboration in Ref. [31] using a smaller data set [32]. The definition of σ in Ref. [31], however, is different from ours so direct comparisons are less straight forward. Fig. 3 (left) and Fig. 3 (right) depict the allowed regions of the σ-sin 2 θ 13 and σ-∆m 2 31 parameter spaces, respectively, marginalizing over the absent parameter. For small enough values of σ, there is a clear (anti)correlation between σ and ∆m 2 31 (sin 2 θ 13 ). These correlations are also manifest in the anticorrelation between ∆m 2 31 and sin 2 θ 13 observed in Fig. 2.
It is straight forward to understand qualitatively the allowed regions in Fig. 3 (left) and Fig. 3 (right). Both RENO and Daya Bay probe L/E values that include the first oscillation maximum associated to ∆m 2 31 while all other maxima are outside the reach of the two experiments. Decoherence effects "flatten" the oscillation maximum, an effect that can be partially compensated by increasing sin 2 θ 13 . Hence, for smaller values of σ (stronger decoherence), one can obtain a decent fit to the data by increasing sin 2 θ 13 relative to the value obtained in the perfectly-coherent hypothesis. Decoherence effects also shift the position of the first oscillation maximum to smaller L/E values. This is simple to understand and is well illustrated in the red, dashed curve in Fig. 1. This can be compensated by lowering the size of ∆m 2 31 (longer wave-length). Hence, for smaller values of σ (stronger decoherence), one can obtain a decent fit to the data by decreasing ∆m 2 31 relative to the value obtained in the perfectly-coherent hypothesis. When σ is large enough, decoherence effects are outside the reach of Daya Bay and RENO and hence the horizontal allowed regions in Fig. 3 (left) and Fig. 3 (right) extend to arbitrarily large σ. Marginalizing over ∆m 2 31 and sin 2 θ 13 , we extract the reduced χ 2 (σ), depicted relative to its minimum value in Fig. 4; the minimum corresponds to σ = 2.01 × 10 −4 nm. Arbitrarily large values of σ are allowed at better than the 90% CL and we translate the information in Fig. 4 into the lower bound σ > 0.89 × 10 −4 nm at 90% CL, combining data from RENO and Daya Bay. For E = 3 MeV and ∆m 2 31 = 2.5 × 10 −3 eV 2 , this translates into L coh 13 > 1.8 km. This is consistent with the naive expectation that RENO and Daya Bay should be sensitive to L coh 13 O(1 km).

IV. SENSITIVITY OF THE JUNO EXPERIMENTAL SETUP
In this section we study the sensitivity of the future JUNO experiment [33] to constrain or measure the neutrino wave-packet width σ. We first estimate the sensitivity of JUNO to σ assuming future JUNO data are consistent with no decoherence effects, σ → ∞. Next, we check the potential of JUNO to establish and measure the presence of decoherence assuming the future JUNO data are consistent with σ = 2.01 × 10 −4 nm, the best-fit value of σ from current reactor data, discussed in the previous section.
In order to simulate JUNO data, we make use of information from Ref. [34]. In particular, we assume the 10reactor configuration. Thermal powers and baselines can be found in Ref. [33] while fluxes, cross sections, and fission fractions are fixed to the ones we used in our analyses of Daya Bay data. When computing oscillation probabilities, we ignore matter effects, which are subdominant. For more details, we refer readers to Refs. [35,36]. There, it was demonstrated that, when pursuing oscillation analyses, matter effects primarily impact, very slightly, the extraction of best-fit values but are negligible when it comes to uncertainties and the sensitivity to other effects, including the mass ordering [35,36]. Our statistical analyses are performed with We assume JUNO will run for 6 years, corresponding to 1800 days of data taking [34] and we do not assume the existence of a near detector. The systematic uncertainties are virtually the same as the ones discussed in the last section but, in order to account for the absence of a near detector, we include an overall flux-normalization uncertainty due to unknowns in the reactor flux spectrum. When simulating data, unless otherwise noted, we assume the true values of the oscillation parameters to be those spelled out in Eq. (5) and, as discussed earlier, assume the mass-ordering is known to be normal. We expect very similar results if it turns out that the mass-ordering is known to be inverted when JUNO takes data. Furthermore, since the impact of the mass-ordering on JUNO data is very different from the effects of non-trivial decoherence, we also expect similar results if one were to assume, in the data analysis, that the mass-ordering is not known. We do not pursue this line of investigation further as it combines different goals of JUNO in a complicated, and not especially illuminating, way. One of the main goals of JUNO is to determine the neutrino mass-ordering by performing an exquisite measurement of the oscillation probability as a function of energy with a baseline that is long enough so both ∆m 2 21 and ∆m 2 31 effects can be observed. Allowing for the hypothesis that σ is finite will, of course, render such an analysis more challenging. Determining how much more challenging is outside the aspirations of this manuscript.

A. Ruling Out Decoherence
Here, we simulate data consistent with no decoherence (σ → ∞) and analyze them as discussed above. Fig. 5 depicts the allowed regions of the σ-mixing angles (left) and σ-∆m 2 's (right) parameter spaces. When generating these twodimensional regions, we marginalize over all absent parameters. Fig. 5 reveals that the precision with which JUNO can measure the different oscillation parameters is not significantly impacted by allowing for the possibility that σ is finite. The reason for this is that JUNO is sensitive to several oscillation maxima and minima associated to the short oscillation lengths and the degeneracies observed in Daya Bay and RENO are completely lifted. Furthermore, the absence of decoherence effects associated with L coh 13 and L coh 23 preclude observable L coh 12 effects since L coh 12 /L coh 13 ∼ 300 and sin 2 θ 13 effects are clearly visible. Fig. 6 depicts the reduced χ 2 (σ), obtained upon marginalizing over all four oscillation parameters. These data would translate into σ > 2.33 × 10 −3 nm at the 90% CL. This is more than a factor of 20 stronger than the current bound from RENO and Daya Bay, obtained in the last section.

B. Observing and Measuring Decoherence
Here, we simulate data consistent with the solar parameters from Eq. (5) and the best-fit value obtained from the analysis of Daya Bay and RENO data performed in the last section. For the decoherence parameter we set σ = 2.01 × 10 −4 nm, while for the standard neutrino oscillation parameters we have ∆m 2 31 = 2.63 × 10 −3 eV 2 and sin 2 θ 13 = 0.0231. Fig. 1 reveals that the impact of decoherence is very strong in JUNO, and we expect the nodecoherence hypothesis to be completely ruled out. Furthermore, the short-wavelength oscillations are completely erased, rendering the measurements of ∆m 2 31 and ∆m 2 32 impossible. It is very clear that, under these circumstances, JUNO is completely insensitive to the mass ordering. Fig. 7 depicts the allowed regions of the σ-mixing angles (sin 2 θ 12 on the top, left and sin 2 θ 13 on the top, right) and σ-∆m 2 's (∆m 2 21 on the bottom, left and ∆m 2 31 on the bottom, right) parameter spaces. When generating all two-dimensional regions, we marginalize over all absent parameters. As advertised, there is no sensitivity to ∆m 2 31 (Fig. 7 [bottom, right]). Nonetheless, averaged-out effects of the short-wavelength oscillations remain and one can measure sin 2 θ 13 with finite, albeit poorer, precision (cf. Fig. 5 [left]). Long-wavelength effects are still present and hence both ∆m 2 21 and sin 2 θ 12 can be measured, see Fig. 7 (left). Similar to what we observe for RENO and Daya Bay, measurements of the oscillation frequency and amplitude are strongly correlated with those of σ. Smaller sigma translate into larger sin 2 θ 12 in order to compensate for the flattened-out oscillation probability while smaller sigma translate into smaller ∆m 2 21 in order to compensate for the shift of the oscillation maximum to larger energies (smaller L/E). These degeneracies lead to a less precise determination of the solar parameters (cf. Fig. 5). Fig. 8 depicts the reduced χ 2 (σ), relative to the minimum value. A clear measurement of the neutrino-wave-packet width can be extracted: σ = 2.01 +0.14 −0.13 × 10 −4 nm. The no-decoherence hypothesis is ruled out at more than ten σ.

V. CONCLUSIONS
Neutrinos observed in all neutrino oscillation experiments, to date, can either be treated as perfectly incoherente.g., solar neutrino experiments modulo Earth matter-effects -or perfectly coherent -e.g., Daya Bay and RENOsuperpositions of the mass eigenstates. The position-dependent loss of coherence expected, in principle, of neutrinos produced and detected under any circumstances, has never been observed.
Here, we explore how well reactor antineutrino experiments can constrain or measure the loss of coherence of reactor antineutrinos. For concreteness, we assume that decoherence effects are captured by the size of the neutrino wave-packet, σ. A perfectly coherent neutrino beam corresponds to σ → ∞ while an incoherent superposition of mass eigenstates is associated to σ = 0. We expect reactor neutrino experiments to be excellent laboratories to study decoherence given the high statistics, the compactness of sources and detectors, including good position resolution, excellent event-by-event energy reconstruction, and very long baselines.
We find that current reactor data from Daya Bay and RENO constrain σ > 8.9 × 10 −5 nm while future data from JUNO should be sensitive to σ < 2.3 × 10 −3 nm, a factor of 20 more sensitive than the current data. If σ ∼ few × 10 −4 nm, in perfect agreement with current reactor neutrino data, we expect decoherence effects to be clearly visible in JUNO, as illustrated in Fig. 1. In this case, σ should be measured in JUNO with good precision.
One can naively estimate that, for neutrinos produced in nuclear reactors and detected via inverse beta-decay, σ should be, at least, of order of the typical interatomic spacing that characterizes the fuel inside the nuclear reactor, which we anticipate is safely outside the sensitivity of JUNO. For the sake of reference, for pure, solid uranium, lattice parameters are of order 0.1-1 nm. JUNO is, however, sensitive to other distance scales associated with electron antineutrinos from beta-decay, including the typical size of the beta-decaying nuclei -around 10 −5 nm -or the inverse of the neutrino energy, 1/E ∼ 10 −4 nm. The discovery of nontrivial decoherence effects in JUNO would indicate that our understanding of the coherence of neutrino sources (or quantum mechanics?) is, at least, incomplete.
Outside of the decoherence effects discussed here, other new phenomena can impact the survival probability of reactor antineutrinos, including very fast neutrino decay into lighter neutrinos or new, very light particles [37,38] and a variety of new-physics effects [39][40][41][42][43][44]. These new-physics effects modify the survival probability in a way that is qualitatively different from the decoherence effects discussed here so we do not expect, assuming the data are not consistent with the standard three-neutrino paradigm, that it would be difficult to distinguish strong decoherence in neutrino propagation from other new physics. A quantitative study of how well one can distinguish different, new phenomena with JUNO is outside the scope of this manuscript.