# Study of the wave packet treatment of neutrino oscillation at Daya Bay

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## Abstract

The disappearance of reactor \(\bar{\nu }_e\) observed by the Daya Bay experiment is examined in the framework of a model in which the neutrino is described by a wave packet with a relative intrinsic momentum dispersion \(\sigma _\mathrm{{rel}}\). Three pairs of nuclear reactors and eight antineutrino detectors, each with good energy resolution, distributed among three experimental halls, supply a high-statistics sample of \(\bar{\nu }_e\) acquired at nine different baselines. This provides a unique platform to test the effects which arise from the wave packet treatment of neutrino oscillation. The modified survival probability formula was used to fit Daya Bay data, providing the first experimental limits: \(2.38 \times 10^{-17}< \sigma _\mathrm{{rel}} < 0.23\). Treating the dimensions of the reactor cores and detectors as constraints, the limits are improved: \(10^{-14} \lesssim \sigma _\text {rel} < 0.23\), and an upper limit of \(\sigma _\text {rel}<0.20\) (which corresponds to \(\sigma _x \gtrsim 10^{-11}\,\mathrm{{cm }}\)) is obtained. All limits correspond to a 95% C.L. Furthermore, the effect due to the wave packet nature of neutrino oscillation is found to be insignificant for reactor antineutrinos detected by the Daya Bay experiment thus ensuring an unbiased measurement of the oscillation parameters \(\sin ^22\theta _{13}\) and \(\varDelta m^2_{32}\) within the plane wave model.

## 1 Introduction

### 1.1 Neutrino oscillation in the plane wave approximation

The neutrino, a light electrically neutral fermion participating in weak interactions, was suggested by Pauli to save the conservation of energy and momentum in nuclear \(\beta \)-decays. Since then, three flavors of neutrinos \(\nu _\alpha = (\nu _e, \nu _\mu , \nu _\tau )\) were discovered, each produced or detected in association with a corresponding lepton \(\ell _\alpha =(e,\mu ,\tau )\). The neutrinos, which are completely parity-violating in their weak interactions, suggested that the gauge group of the electro-weak sector of the remarkably successful Standard Model (SM) should be built using fermions with left-handed chirality. Given the unique properties of neutrinos, studies of them may reveal a path to physics beyond the SM. In the past, experiments observing solar and atmospheric neutrinos brought increased attention to neutrino physics due to long-standing discrepancies between detection rates and no-oscillation models. Despite an impressive number of proposed solutions to these problems, all were successfully resolved by the hypothesis of neutrino oscillation, first proposed by Pontecorvo [1, 2] in the late 1950’s. Neutrino oscillation is a phenomenon firmly established in experiment, which has been observed with solar [3, 4, 5], atmospheric [6, 7], particle accelerator [7, 8] and reactor [9, 10, 11, 12] neutrinos.

*p*is the momentum of the neutrino. The time evolution of the state in Eq. (1) is expressed as

*t*is approximated by the traveled distance

*L*.

The underlying theory, assuming a plane wave approximation, was developed in the middle of the 1970s [13, 14, 15]. Although successful in explaining a wide range of neutrino experiments, it is well known that this approximation is not self-consistent, and leads to a number of paradoxes [16, 17]. The applicability of the plane wave approximation is discussed in detail in Refs. [16, 18, 19, 20]. After the first theory was developed, Refs. [21, 22, 23, 24] pointed out the necessity of a wave packet treatment of neutrino oscillation.

### 1.2 Wave packet treatment of neutrino oscillation

The wave packet is a coherent superposition of different waves whose momenta are distributed around the most probable value, with a certain “width” or dispersion. Therefore, a wave packet is localized in space-time as well as in energy-momentum space. The wave packet formalism facilitates the resolution of the paradoxes of the plane wave theory, and predicts the existence of a coherence length. The latter arises due to the different group velocities of a pair \(\nu _k\) and \(\nu _j\), which causes a separation in space over time.

The propagation distance over which a wave (classical or quantum) preserves a certain degree of coherence is known as a *coherence length*. It is important in many branches of physics. Some examples of classical physics include optics, radio-band systems, holography and telecommunications engineering. Superconductivity, superfluidity and lasers are known as examples of highly coherent quantum systems. Coherence is important in the already available technology of quantum cryptography and in the future technologies of quantum computing. Coherence in neutrino oscillation, being quantum by nature, also exhibits some features of classical systems: two waves \(\nu _k\) and \(\nu _j\) propagating with different group velocities break the coherence in the quantum state, like in Eq. (1), at distances exceeding the coherence length, similarly to what happens in optics when a wave packet propagates far enough in a medium such that the speed of a wave component with certain frequency depends on the refraction index. The smallness of the difference of neutrino masses suggests that the coherence length of neutrino oscillation is the largest available among all known phenomena.

After the pioneering studies [21, 22, 23], the wave packet models of neutrino oscillation were developed in roughly two varieties. The first one relies on a relativistic quantum mechanical (QM) formalism that does not predict the dispersion of the neutrino wave packet in momentum space, such as in Refs. [18, 19, 25]. The second one is based on calculations within quantum field theory (QFT), describing all external particles involved in neutrino production and detection as wave packets while treating neutrinos as virtual particles. The neutrino wave-function is then calculated rather than postulated. The effective momentum dispersion of the neutrino wave function depends on the kinematics of neutrino production and detection and on the momentum dispersions of the external particles, as in Refs. [26, 27, 28, 29, 30, 31, 32]. Both approaches predict a number of observable effects, like a quantitative condition on the coherence of mass eigenstates in the production–detection processes, as well as a loss of coherence.

In wave packet models, the intrinsic momentum dispersion \(\sigma _p\) of the neutrino wave packet is an effective quantity comprising the microscopic momenta dispersions of all particles involved in the production and detection of the neutrino. A non-zero value of \(\sigma _p\) leads with time to the *decoherence* in the quantum superposition of massive neutrinos which results in a vanishing oscillation pattern of \(\nu _\alpha \rightarrow \nu _\beta \) transitions. In addition, the oscillation pattern is smeared further in the reconstructed energy spectrum due to a non-zero experimental resolution \(\delta _E\) of the neutrino energy.

Despite considerable progress in building wave packet models, none of these approaches provides a solid quantitative theoretical estimate of \(\sigma _p\) or of the spatial width \(\sigma _x=1/2\sigma _p\). Theoretical estimates vary by orders of magnitude, associating the dispersion of the neutrino wave packet with various scales; for example, uranium nucleus diameter (\(\sigma _x \simeq 10^{-12}\) cm, \(\sigma _p\simeq 10\) MeV), atomic or inter-atomic distances (\(\sigma _x \simeq (10^{-8}-10^{-7})\) cm, \(\sigma _p\simeq (10^3-10^2)\) eV), pressure broadening (\(\sigma _x \simeq 10^{-4}\) cm, \(\sigma _p\simeq 0.1\) eV), etc. While most of the discussions in the current literature does not include calculations of the neutrino wave function from first principles for any type of neutrino experiment,^{1} it also lacks quantitative experimental investigations of decoherence effects in neutrino oscillation inferred from the finite size of the neutrino wave function.^{2}

It has been pointed out that a loss of coherence of neutrino mass eigenstates would lead to an event rate smaller than that expected for coherent neutrino states [16]. However, a quantitative study of decoherence effect from the absolute event rate measurements of past reactor experiments [38, 39, 40, 41, 42, 43] is subject to the significant uncertainties in the model predictions of the reactor antineutrino flux.

The day–night asymmetry of solar neutrinos provides an evidence that solar neutrinos come to the Earth in an incoherent mixture [44]. However these data do not provide any quantitative information about the size of a neutrino wave packet because of an averaging over the large volume of the Sun.

One of the motivations of this paper is to provide the first quantitative study of a possible loss of coherence in the quantum state of neutrinos following the wave packet treatment of neutrino oscillations, using data from the Daya Bay Reactor Neutrino Experiment. The second motivation is to demonstrate that the oscillation parameters estimated with the plane wave approximation are unbiased. The oscillation probability formula modified by the wave packet contribution, which is discussed further, has two distinctive features: it depends on \(\varDelta m^2_{kj}/p^2\sigma _\mathrm{{rel}}\) via the so-called localization term and on \(L\varDelta m^2_{kj}\sigma _\mathrm{{rel}}/p\) via the term responsible for the loss of coherence with distance, where \(\sigma _\mathrm{{rel}}=\sigma _p/p\). The large statistics, good energy resolution, and multiple baselines of the Daya Bay experiment make its data valuable in the study of these quantum decoherence effects in neutrino oscillation.

## 2 Analysis

### 2.1 Neutrino oscillation in a wave packet model

^{3}The plane wave state in (1) is replaced by a wave packet describing a neutrino produced as flavor \(\alpha \):

*P*in \(f_P(p)\), \(p_P\) and \(\sigma _{pP}\) indicates the quantities at production. In configuration space the state in Eq. (4) describes a wave packet with mean coordinate \(x_P\) at time \(t_P\). The state in Eq. (4) is normalized to unity. Similarly, a wave packet state at detection \(|\widetilde{\nu }_\beta (p_D; t_D,x_D)\rangle \) is defined as the state given by Eq. (4).

*p*and momentum dispersion \(\sigma _p\) comprising the details of production and detection

^{4}

*L*:

^{5}The probability in Eq. (8) contains three quantities with dimensions of length:

It is always possible for the given values of *p* and *L* to identify the domain of \(\sigma _p\) where Eqs. (3) and (8b) are numerically almost identical to each other (see Sect. 2.2).

### 2.2 Sensitivity of Daya Bay experiment to neutrino wave packet

The Daya Bay experiment is composed of two near underground experimental halls (EH1 and EH2) and one far underground hall (EH3). Each of the experimental halls hosts identically designed antineutrino detectors (ADs). EH1 and EH2 contain two ADs each, while EH3 contains four ADs. Electron antineutrinos are produced in three pairs of nuclear reactors via \(\beta \) decays of neutron-rich daughters of the fission isotopes \({}^{235}\text {U}\), \({}^{238}\text {U}\), \({}^{239}\text {Pu}\) and \({}^{241}\text {Pu}\), and detected via the inverse \(\beta \) decay (IBD). The coincidence of the prompt (\(e^+\) ionization and annihilation) and delayed (*n* capture on Gd) signals efficiently suppresses the backgrounds, which amounted to less than 2% (5%) of the IBD candidates in the near (far) halls [49]. The Gd-doped liquid scintillator target is a cylinder of three meters in both height and diameter. The detectors have a light yield of about 165 photoelectrons/MeV and a reconstructed energy resolution \(\delta _E/E\approx 8\%\) at 1 MeV of deposited energy in the scintillator. More details on the experimental setup are contained in Refs. [49, 50, 51, 52].

The number of IBD candidates and mean distances of the three experimental halls to the pairs of reactor cores

Halls | IBD candidates | Mean distance, m | ||
---|---|---|---|---|

Daya Bay | Ling Ao | Ling Ao II | ||

EH1 | 613,813 | 365 | 860 | 1310 |

EH2 | 477,144 | 1348 | 481 | 529 |

EH3 | 150,255 | 1909 | 1537 | 1542 |

One can meet claims in literature that the smallest among \(\sigma _p\) and \(\delta _E\) determines the decoherence effects in neutrino oscillations. In what follows, we provide some qualitative and analytical arguments showing the actual interplay of intrinsic momentum dispersion \(\sigma _p\) of neutrino wave packet and \(\delta _E\). The latter is sometimes erroneously considered as an upper extreme value of \(\sigma _p\). The width (\(\varGamma \simeq \sigma _p\)) of a hadronic resonance which is typically much larger than an experimental energy resolution \(\delta _E\) provides a well-known counter-example, illustrating that \(\sigma _p\) could be much larger than \(\delta _E\).

For relatively large values of \(\sigma _p\simeq \delta _E\), the effects of these two parameters on the observed energy spectra might appear similar, however they are distinct. First, they have different physical origins: while \(\sigma _p\) is governed by the most localized particle in the production and detection of the neutrino, \(\delta _E\) is determined by the energy depositions of the final state particles in the detector, the amount and efficiency of detection devices used to observe such depositions. In particular, considering a liquid scintillator detector surrounded by a number of PMTs as an example, one could hypothesize modifications in the number of PMTs, their efficiencies or even in the light yield. Such variations would modify the energy resolution \(\delta _E\) correspondingly, leaving intact the microscopic processes determining \(\sigma _p\) and, respectively, the number of neutrino interactions in the detector. Second, these effects can also be distinguished from their order of occurrence since the microscopic processes used in the energy estimation occur later in time with respect to the neutrino interaction in the detector. Third, their effects are not identical. In particular, as described in Sec. 2.1, the limit \(\sigma _p\rightarrow 0\) leads to the decoherence of neutrino oscillation in contrast to the impact of energy resolution which does not lead to any smearing in the reconstructed energy spectrum in the limit \(\delta _E\rightarrow 0\).

^{6}:

*p*with \(E_\mathrm{vis}\), and \(L^\text {coh}_\text {det}\) is given by \(L^\text {coh}_\text {rec}\), replacing \(\sigma _p\) with \(\delta _E\). The interplay of \(\sigma _p\) and \(\delta _E\) is illustrated by the effective coherence length \(L^\text {coh}_\text {eff}\), which is dominantly determined by the smallest among \(L^\text {coh}_\text {rec}\) and \(L^\text {coh}_\text {det}\), or by the largest among \(\sigma _p\) and \(\delta _E\). Therefore, the effective energy dispersion \(\sigma _p^\text {eff}\) is determined by \((\sigma _p^\text {eff})^2=\sigma _p^2+\delta _E^2\).

The following provides simple numerical estimates of Daya Bay sensitivity to wave packet effects on neutrino oscillations.

For a typical momentum of \(p=4\) MeV of detected reactor \(\bar{\nu }_e\), the oscillation would be suppressed for two distinctive domains of \(\sigma _\text {rel}\). The domain \(\sigma _\text {rel}\gtrsim O(0.1)\) corresponds to significant contributions from *L*-dependent interference-suppressing terms and corrections to the oscillation phase \(\varphi ^d_{32}\) in Eq. (8), while the \(D^2_{kj}\) term is negligibly small. For example, at \(L=L_{32}^\text {osc}/2\) the exponential suppression reaches its maximum \(\text {e}^{-\pi /8}\) at \(\sigma _\text {rel}=1/\sqrt{2\pi }\simeq 0.4\). Correspondingly, the coherence and dispersion lengths read \(L_{32}^\text {coh}\simeq 2.2\) km and \(L_{32}^\text {d}\simeq 2\) km. At larger values of \(\sigma _\text {rel}\) and at a fixed distance the spatial dispersion of neutrino wave packets partially compensates the loss of coherence due to the spatial separation of \(\nu _k\) and \(\nu _j\).

*L*–independent term, while the

*L*-dependent terms are negligibly small. Thus, the region of \(O(2.8\times 10^{-17}) \ll \sigma _\text {rel} \ll O(0.1)\) is where the wave packet impact on neutrino oscillation is negligible for the Daya Bay experiment.

For illustrative purposes Fig. 1 shows the ratio of the observed to expected numbers of IBD events assuming no oscillation using the data collected at the near and far experimental halls as a function of reconstructed visible energy \(E_\text {vis}\). Figure 1 also shows the expected ratio for neutrino oscillation with the plane wave and wave packet models with \(\sigma _\text {rel}\) of 0.33 and \(8\times 10^{-17}\) as examples.

Both model expectations are shown with the oscillation parameters fixed to their best-fit values within the plane wave model.^{7} For this set of parameters, the wave packet models with \(\sigma _\text {rel}=0.33\) and with \(\sigma _\text {rel}=8\times 10^{-17}\) are inconsistent with the data by about five standard deviations, thus motivating the chosen values of \(\sigma _\text {rel}\). The two panels illustrate how the visible energy spectra are modified in the near and far halls depending on the intrinsic dispersion of the neutrino wave packet. Remarkably, most changes in the energy spectra due to \(\sigma _\text {rel}\) are in opposite directions for near and far halls, which can be explained qualitatively as follows. As mentioned above, the extremes \(\sigma _p\rightarrow 0\) and \(\sigma _p\rightarrow \infty \) would yield fully decoherent neutrinos with the oscillation probability given by Eq. (12). Antineutrinos detected at the near halls experience a relatively small oscillation in the plane wave approach. The values of \(\sigma _\text {rel}\) selected for Fig. 1 make the \(\bar{\nu }_e\) partially decoherent and \(P_\text {ee}\) tend towards Eq. (12), predicting a *smaller* number of surviving \(\bar{\nu }_e\) as compared to the plane wave formula. The distance at which the far detectors of the Daya Bay experiment are placed is tuned to observe the maximal oscillation effect due to \(\varDelta m^2_{32}\). Partial decoherence of the \(\bar{\nu }_e\) tends to reduce the oscillation, thus predicting a *larger* number of survived \(\bar{\nu }_e\) with respect to the plane wave formula. This feature of Daya Bay provides additional sensitivity to the decoherence effects and makes such a study less sensitive to the predicted reactor \(\bar{\nu }_e\) spectrum.

^{8}:

### 2.3 Statistical framework

As the goodness-of-fit measure we use \(\chi ^2(\varvec{\eta }) = (\mathbf {d}-\mathbf {t}(\varvec{\eta }))^TV^{-1}(\mathbf {d}-\mathbf {t}(\varvec{\eta }))\), where \(\mathbf {d}\) is a data vector containing detected numbers of IBD candidates in energy bins and in different detectors, while \(\mathbf {t}(\varvec{\eta })\) is the corresponding theoretical model vector which depends on constrained and unconstrained parameters \(\varvec{\eta }\). All constraints of the model as well as expected fluctuations in the number of IBD events are encompassed in the covariance matrix *V*. The model vector \(\mathbf {t}(\varvec{\eta })\) comprises expected numbers of IBD and background events. All constrained parameters (or systematic uncertainties) relevant for the Daya Bay oscillation analyses were taken into account in this analysis. These are mainly associated with the reactor antineutrino flux, background predictions and the detector response modeling. The uncertainty of the detector response is dominant. Details can be found in Refs. [49, 52].

*N*. The confidence regions are produced by means of two statistical methods: the conventional fixed-level \(\varDelta \chi ^2\) analysis and the Feldman–Cousins method [54]. The marginalized \(\varDelta \chi ^2\) statistic is

*p*-value of the observed dataset and the model.

*n*degrees of freedom (\(n=1,2\) for one and two dimensional confidence regions) were used for the fixed-level \(\varDelta \chi ^2\) analysis. Toy Monte Carlo sampling was used to determine \(\varDelta \chi ^2_{1-\alpha }\) of the statistic in Eq. (19) with the Feldman–Cousins method.

## 3 Results and discussion

Figure 2 displays the allowed regions in \((\varDelta m^2_{32},\sigma _{\text {rel}})\) and \((\sin ^2 2\theta _{13},\sigma _{\text {rel}})\) obtained with both the fixed-level \(\varDelta \chi ^2\) and the Feldman–Cousins methods, which are found to be consistent. For the values of \(\sigma _\text {rel}\lesssim 10^{-16}\) the decoherence effects lead to strong correlations between \(\varDelta m^2_{32}, \sin ^22\theta _{13}\) and \(\sigma _\text {rel}\), yielding smaller values of \(\varDelta m^2_{32}\) and larger values of \(\sin ^22\theta _{13}\). These correlations are expected taking into account the explicit form of \(1-P_\text {ee}(L)\) in Eq. (13). The coefficients of \(\sigma _\text {rel}\) correlation with \(\sin ^22\theta _{13}\) and \(\varDelta m^2_{32}\) are found to be \(-0.98\) and 0.96 respectively. For \(\sigma _\text {rel}\gtrsim O(0.1)\), these correlations are found to be significantly weaker. The absolute values of the corresponding correlation coefficients are smaller than \(10^{-5}\).

## 4 Summary

We performed a search for the footprint of the neutrino wave packet which should show itself through specific modifications of the neutrino oscillation probability. The reported analysis of the Daya Bay data provides, for the first time, an allowed interval of the intrinsic relative dispersion of neutrino momentum \(2.38 \times 10^{-17}< \sigma _\mathrm{{rel}} < 0.23\). Taking into account the actual dimensions of the reactor cores and detectors, we find that the lower limit \(\sigma _\mathrm{rel} > 10^{-14}\) corresponds to the regime when the localization term is vanishing, thus allowing us to put an upper limit: \(\sigma _\text {rel}<0.20\) at a 95% C.L. This upper limit of \(\sigma _\text {rel}\) implies that \(\sigma _x \gtrsim 10^{-11}\,\text { cm }\) exceeds size of any nucleus thus excluding a theoretical possibility of neutrino wave function to be formed at nuclear scales.

The current limits are dominated by statistics. With three years of additional data the upper limit on \(\sigma _\text {rel}\) is expected to be improved by about 30%. The allowed decoherence effect due to the wave packet nature of neutrino oscillation is found to be insignificant for reactor antineutrinos detected by the Daya Bay experiment thus ensuring an unbiased measurement of the oscillation parameters \(\sin ^22\theta _{13}\) and \(\varDelta m^2_{32}\) within the plane wave model.

## Footnotes

- 1.
Recently, a first calculation which consistently treats the full pion-neutrino-environment quantum system and calculates the decoherence effects for neutrinos produced in two-body decays was published in Ref. [33].

- 2.
Attention to the decoherence phenomena in neutrino oscillation is increasing and the literature discusses possible decoherence effects due to physics beyond the SM like quantum gravity [34, 35, 36, 37], differing from the considerations of this paper, which studies the consequences of a self-consistent way to describe neutrino oscillation within the minimally extended Standard Model hosting non-zero mass neutrinos.

- 3.
While a neutrino travels in the three-dimensional space, the transverse part of its wave function essentially leads to the \(1/L^2\) dependence of the flux [45] and does not affect significantly the oscillation pattern.

- 4.
- 5.
Since the QFT approach considers both neutrino production and detection one finds that \(\sigma _\mathrm{{rel}}\), being a relativistic invariant, is actually a function of kinematic variables involved in the production and detection processes as well as of momentum dispersions of wave packets describing all involved particles [48]. Therefore, in comparing the QM and QFT approaches, we may treat the QM \(\sigma _\mathrm{{rel}}\) as that of the QFT approach averaged over the kinematic variables of all external wave packets involved in neutrino production and detection.

- 6.
The actual implementation of the detector effects in this analysis was performed numerically without approximations.

- 7.
The following values of the oscillation parameters were used in Fig. 1: \(\varDelta m^2_{21} = 7.53 \times 10^{-5}\,\text { eV}^2\), \(\varDelta m^2_{32}=2.45\times 10^{-3}\,\text { eV}^2\), \(\sin ^22\theta _{12}=0.846\), \(\sin ^22\theta _{13}=0.0852\).

- 8.
The best-fit values of the oscillation parameters \(\sin ^22\theta _{13}\) and \(\varDelta m^2_{32}\) are different from our previous publication [49] because of a different implementation of systematic uncertainties and another choice of \(E_\text {vis}\) binning.

## Notes

### Acknowledgements

Daya Bay is supported in part by the Ministry of Science and Technology of China, the U.S. Department of Energy, the Chinese Academy of Sciences, the CAS Center for Excellence in Particle Physics, the National Natural Science Foundation of China, the Guangdong provincial government, the Shenzhen municipal government, the China General Nuclear Power Group, Key Laboratory of Particle and Radiation Imaging (Tsinghua University), the Ministry of Education, Key Laboratory of Particle Physics and Particle Irradiation (Shandong University), the Ministry of Education, Shanghai Laboratory for Particle Physics and Cosmology, the Research Grants Council of the Hong Kong Special Administrative Region of China, the University Development Fund of The University of Hong Kong, the MOE program for Research of Excellence at National Taiwan University, National Chiao-Tung University, and NSC fund support from Taiwan, the U.S. National Science Foundation, the Alfred P. Sloan Foundation, the Ministry of Education, Youth, and Sports of the Czech Republic, the Joint Institute of Nuclear Research in Dubna, Russia, the RFBR research program, the National Commission of Scientific and Technological Research of Chile, and the Tsinghua University Initiative Scientific Research Program. We acknowledge Yellow River Engineering Consulting Co., Ltd., and China Railway 15th Bureau Group Co., Ltd., for building the underground laboratory. We are grateful for the ongoing cooperation from the China General Nuclear Power Group and China Light and Power Company.

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