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.

Neutrino oscillation is a quantum phenomenon of quasi-periodic change of neutrino flavor \(\nu _\alpha \rightarrow \nu _\beta \) with time. This phenomenon originates in the non-equivalence of neutrino flavor \(\nu _\alpha \) and mass \(\nu _k=(\nu _1,\nu _2,\nu _3)\) eigenstates, differences in their masses, and an assumption that the produced and detected neutrino states are coherent superpositions of neutrino mass eigenstates:

$$\begin{aligned} |\nu _\alpha (p)\rangle =\sum _{k=1}^3 V_{\alpha k}^*|\nu _k(p)\rangle , \end{aligned}$$

where \(V_{\alpha k}\) is an element of the unitary PMNS-matrix, named after Pontecorvo, Maki, Nakagawa, Sakata, and p is the momentum of the neutrino. The time evolution of the state in Eq. (1) is expressed as

$$\begin{aligned} |\nu _\alpha (t;p)\rangle =\sum _{k=1}^3 V_{\alpha k}^*\text {e}^{-iE_kt}|\nu _k(p)\rangle , \end{aligned}$$

where \(E_k=\sqrt{p^2+m_k^2}\). This leads to the oscillatory behavior of the probability to detect a neutrino originally of flavor \(\alpha \) as having flavor \(\beta \):

$$\begin{aligned} P_{\alpha \beta }(L)= & {} |\langle \nu _\beta (p)|\nu _\alpha (t;p)\rangle |^2\nonumber \\= & {} \sum _{k,j=1}^{3}V_{\alpha k}^* V_{\beta j}^* V_{\beta k}^{\phantom {*}} V_{\alpha j}^{\phantom {*}} \text {e}^{-i2\pi L/L^\text {osc}_{kj}}, \end{aligned}$$

where \(L^\text {osc}_{kj}= 4\pi p/\varDelta m_{kj}^2\) is the oscillation length due to the non-zero differences \(\varDelta m^2_{kj}=m^2_k-m^2_j\), and time 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,Footnote 1 it also lacks quantitative experimental investigations of decoherence effects in neutrino oscillation inferred from the finite size of the neutrino wave function.Footnote 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

Measured energy spectra of \(\bar{\nu }_e\) interactions are compared to a prediction using a QM wave packet model of neutrino oscillation which is briefly outlined in what follows. We simplify the consideration by examining a one-dimensional wave packet of the neutrino.Footnote 3 The plane wave state in (1) is replaced by a wave packet describing a neutrino produced as flavor \(\alpha \):

$$\begin{aligned} |\widetilde{\nu }_\alpha (p_P;t_P,x_P)\rangle = \sum _{k=1}^3 V_{\alpha k}^*\int \frac{dp}{2\pi } f_P(p) \text {e}^{-i\phi _{P}(p)}| \nu _k(p) \rangle , \end{aligned}$$

with \(\phi _{P}(p) = E_k t_P - p x_P\). \(f_P(p)\) is the wave function of the neutrino in momentum space and is assumed to be Gaussian:

$$\begin{aligned} f_P(p) = \left( \frac{2\pi }{\sigma _{pP}^2}\right) ^{\frac{1}{4}} \text {e}^{-\frac{(p-p_P)^2}{4\sigma ^2_{pP}}}, \end{aligned}$$

where the subscript 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).

A projection of \(|\widetilde{\nu }_\alpha (p_P;t_P,x_P)\rangle \) onto \(\langle \widetilde{\nu }_\beta (p_D;t_D,x_D)|\) produces the flavor-changing amplitude

$$\begin{aligned} \mathscr {A}_{\alpha \beta }(p;t_D-t_P, L,\sigma _p)\equiv \langle \widetilde{\nu }_\beta (p_D;t_D,x_D)|\widetilde{\nu }_\alpha (p_P;t_P,x_P)\rangle , \end{aligned}$$

which depends on \(L\equiv x_D-x_P\), time difference \(t_D-t_P\) and on the effective mean neutrino momentum p and momentum dispersion \(\sigma _p\) comprising the details of production and detectionFootnote 4

$$\begin{aligned} p = \frac{p_P \sigma _{pD}^2 + p_D \sigma _{pP}^2}{\sigma _{pP}^2+\sigma _{pD}^2},\quad \frac{1}{\sigma _p^2}= \frac{1}{\sigma _{pP}^2}+\frac{1}{\sigma _{pD}^2}. \end{aligned}$$

The probability \(|\mathscr {A}_{\alpha \beta }(p;t_D-t_P, L,\sigma _p)|^2\) should be integrated over usually unobservable variables – production time \(t_P\) (or, equivalently, over \(t_D-t_P\)) and most probable momentum \(p_P\) to get an experimentally observable oscillation probability, which does not depend anymore on time \((t_D-t_P)\) but does depend on L:

$$\begin{aligned} P_{\alpha \beta }(L)&=\int \frac{dt_P dp_P}{2\pi } |\mathscr {A}_{\alpha \beta }(p;t_D-t_P, L,\sigma _p)|^2 \end{aligned}$$
$$\begin{aligned}&=\sum _{k,\,j=1}^3\frac{ V^*_{\alpha k} V^{\phantom {*}}_{\beta k}V^{\phantom {\dagger }}_{ \alpha j} V^*_{\beta j} }{\root 4 \of {1 +\big (L/L^\mathrm{{d}}_{kj}\big )^2}} \mathrm{{e}}^{- \frac{\big (L/L^\mathrm{{coh}}_{kj}\big )^2}{1+\big (L/L^\mathrm{{d}}_{kj}\big )^2} -\mathrm {D}^2_{kj}} \mathrm{{e}}^{-i\widetilde{\varphi }_{kj}}, \end{aligned}$$

where the phase \(\widetilde{\varphi }_{kj}\) is the sum of the plane wave phase \(\varphi _{kj} = 2\pi L/L^\mathrm{{osc}}_{kj}\) and correction \(\varphi ^d_{kj}\) due to the dispersion of the wave packet: \(\widetilde{\varphi }_{kj} = \varphi _{kj} + \varphi ^d_{kj}\), with

$$\begin{aligned} \varphi ^\mathrm{{d}}_{kj} = - \frac{L/L^\mathrm{{d}}_{kj}}{1+\big (L/{L^\mathrm{{d}}_{kj}}\big )^2} \left( \frac{L}{L^\text {coh}_{kj}}\right) ^2 + \frac{1}{2} \arctan { \frac{L}{L^\mathrm{{d}}_{kj}}}. \end{aligned}$$

Oscillation probability formulas similar to Eq. (8) but neglecting wave packet dispersion were obtained in several studies (see, for example, Refs. [18, 29, 31, 46]). Equation (8) has appeared as a particular case of a more general consideration within QFT with relativistic wave packets [32]. Relativistic invariance suggests that \(\sigma _x E_k\) (and thus \(\sigma _p/E_k\)) should be a Lorentz invariant [16, 47]. Up to a typically tiny correction of the order of \(m_k^2/p^2\), \(\sigma _\mathrm{{rel}}\) should also be a relativistic invariant, at least when neutrinos remain relativistic. In the QM approach adopted in Eqs. (4)–(8) the only possibility to preserve Lorentz invariance is for \(\sigma _\mathrm{{rel}}\) to be a constant.Footnote 5 The probability in Eq. (8) contains three quantities with dimensions of length:

$$\begin{aligned} L^\mathrm{{osc}}_{kj}= & {} \frac{4\pi p}{\varDelta m^2_{kj}}, \qquad L^\mathrm{{coh}}_{kj} =\frac{L^\mathrm{{osc}}_{kj}}{\sqrt{2} \pi \sigma _\mathrm{{rel}}},\nonumber \\ L^\mathrm{{d}}_{kj}= & {} \frac{L^\mathrm{{coh}}_{kj}}{2\sqrt{2}\sigma _\mathrm{{rel}}}, \end{aligned}$$

where \(L^\mathrm{{osc}}_{kj}\) is the usual oscillation length of a pair of neutrino states \(|\nu _k\rangle \) and \(|\nu _j\rangle \), \(L^\mathrm{{coh}}_{kj}\) is interpreted as the neutrino coherence length, i.e. the distance at which the interference of neutrino mass eigenstates vanishes, and finally \(L^\mathrm{{d}}_{kj}\) is the dispersion length, i.e. a distance at which the wave packet is doubled in its spatial dimension due to the dispersion of waves moving with different velocities. The term

$$\begin{aligned} \mathrm {D}^2_{kj} = \frac{1}{2} \left( \frac{\varDelta m^2_{kj}}{4 p^2\sigma _\text {rel}} \right) ^2= \frac{1}{4} \left( \frac{\varDelta m^2_{kj}}{\sigma _{m^2}} \right) ^2 =\left( \frac{\sqrt{2}\pi \sigma _x}{L_{kj}^\text {osc}} \right) ^2 \end{aligned}$$

suppresses the coherence of massive neutrino states \(|\nu _k\rangle \) and \(|\nu _j\rangle \) if \(\varDelta m^2_{kj}\gg \sigma _{m^2}\), where \(\sigma _{m^2}= 2\sqrt{2}p\sigma _p\) could be interpreted as an uncertainty in the neutrino mass squared [22]. \(\mathrm {D}^2_{kj}\) can be seen from another perspective as the localization term suppressing the oscillation if \(\sqrt{2}\pi \sigma _x\gg L_{kj}^\text {osc}\), where \(\sigma _x=(2\sigma _p)^{-1}\) is the width of neutrino wave packet in the configuration space.

It is worth mentioning that terms in Eq. (8) which correspond to the interference of \(\nu _k\) and \(\nu _j\) states also get suppressed by the denominator \(\root 4 \of {1 +(L/L^{\text {d}}_{kj})^2}\) and vanish for both limits \(\sigma _p\rightarrow 0\) and \(\sigma _p\rightarrow \infty \), reducing the oscillation probability in Eq. (8) to the non-coherent sum

$$\begin{aligned} P_{\alpha \beta }=\sum _k|V_{\alpha k}|^2|V_{\beta k}|^2, \end{aligned}$$

which does not depend on energy and distance. The oscillation probability in Eq. (8b) is not reduced to the plane wave formula in Eq. (3) in the limit \(\sigma _p\rightarrow 0\) because of the integration over an unobservable production time \(t_P\) in Eq. (8a) which is necessary in a self-consistent consideration. Let us observe, that a time average of Eq. (3) also leads to non-coherent formula in Eq. (12).

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).

For the \(\bar{\nu }_e\) at Daya Bay, \(1-P_\mathrm{ee}\) is expressed as

$$\begin{aligned}&\frac{1}{2}\sin ^22\theta _{12}\cos ^4\theta _{13}\nonumber \\&\quad \times \,\left( 1-\frac{\exp {\left[ -\frac{\left( L /L^\text {coh}_{21}\right) ^2}{1+\left( L/{L^\text {d}_{21}} \right) ^2}-D^2_{21}\right] }}{\root 4 \of {1 + \left( L/ {L^\text {d}_{21}}\right) ^2}}\cos {(\varphi _{21}} +\varphi _{21}^\text {d})\right) \nonumber \\&\quad +\,\frac{1}{2}\cos ^2\theta _{12}\sin ^22\theta _{13}\nonumber \\&\quad \times \, \left( 1-\frac{\exp {\left[ -\frac{\left( L /L^\text {coh}_{31}\right) ^2}{1+\left( L/{L^\text {d}_{31}} \right) ^2}-D^2_{31}\right] }}{\root 4 \of {1 + \left( L/ {L^\text {d}_{31}}\right) ^2}}\cos {(\varphi _{31}} +\varphi _{31}^\text {d})\right) \nonumber \\&\quad +\,\frac{1}{2}\sin ^2\theta _{12}\sin ^22\theta _{13}\nonumber \\&\quad \times \,\left( 1-\frac{\exp {\left[ -\frac{\left( L /L^\text {coh}_{32}\right) ^2}{1+\left( L/{L^\text {d}_{32}} \right) ^2}-D^2_{32}\right] }}{\root 4 \of {1 + \left( L/ {L^\text {d}_{32}}\right) ^2}}\cos {(\varphi _{32}} +\varphi _{32}^\text {d})\right) .\nonumber \\ \end{aligned}$$

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 studies in this paper are based on data acquired in the 6-AD period when there were two ADs in EH1, one AD in EH2 and 3 ADs in EH3, with the addition of the 8-AD period from October 2012 to November 2013, a total of 621 days. The number of IBD candidates used in this analysis, and the mean baselines of the three experimental halls to each pair of reactor cores, are summarized in Table 1. The expected numbers of IBD events are convolutions of the reactor-to-target expectation with the detector-response function. The reactor-to-target expectation takes into account the antineutrino fluxes from each reactor core including non-equilibrium and spent nuclear fuel corrections, first order in \(1/m_p\) (\(m_p=\) proton mass) IBD cross-section accounting for the positron emission angle [53], and the oscillation survival probability \(P_\text {ee}\) given by Eq. (3) for the plane wave model and by Eq. (8) for the wave packet model. The detector response-function accounts for energy loss in the inner acrylic vessel, liquid scintillator and electronics non-linearity and energy resolution \(\delta _E\).

Table 1 The number of IBD candidates and mean distances of the three experimental halls to the pairs of reactor cores

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 \(\delta _E\rightarrow 0\).

In order to illustrate analytically an interplay of \(\sigma _p\) and \(\delta _E\), let us consider the exponential in the oscillation probability in Eq. (8) convolved with a Gaussian energy resolution, as a function of the reconstructed energy \(E_\mathrm{vis}\), assuming \(\delta _E\ll p\), infinite dispersion length \(L^\text {d}\), neglecting the \(D^2\) term, and suppressing mass eigenstate indices for the sake of compactnessFootnote 6:

$$\begin{aligned}&\frac{1}{\sqrt{2\pi }\delta _E}\int dp \; \exp (-i \; 2\pi L/L^\text {osc} - (L/L^\text {coh})^2 \nonumber \\&\quad \quad - (p-E_\mathrm{vis})^2/2\delta _E^2) \nonumber \\&\quad \simeq \exp {(-i \; 2\pi L/L^\text {osc}_\text {rec} - (L/L^\text {coh}_\text {eff})^2)}, \end{aligned}$$

where \(L^\text {osc}\) and \(L^\text {coh}\) are given by Eq. (10) and the effective coherence length comprises both the intrinsic \(\sigma _p\) and detector resolution \(\delta _E\):

$$\begin{aligned} \left( \frac{1}{L^\text {coh}_\text {eff}}\right) ^2 = \left( \frac{1}{L^\text {coh}_\text {rec}}\right) ^2 + \left( \frac{1}{L^\text {coh}_\text {det}}\right) ^2, \end{aligned}$$

where \(L^\text {osc}_\text {rec}\) and \(L^\text {coh}_\text {rec}\) are given by \(L^\text {osc}\) and \(L^\text {coh}\) replacing 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 \(\nu _j\).

The domain \(\sigma _\text {rel}\lesssim O(2.8\times 10^{-17})\) corresponds to \(D_{32}^2 \gtrsim 1\), which is significant in suppressing the interference in Eq. (8) through the 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.

Fig. 1
figure 1

Ratios of the observed to expected numbers of IBD events in the absence of oscillation as a function of reconstructed visible energy \(E_\mathrm{{vis}}\). The data are grouped by near (EH1 \(+\) EH2) and far (EH3) halls, displayed in the upper and in the bottom panels respectively, with the error bars representing the statistical uncertainties. Superimposed solid lines are ratios assuming neutrino oscillations within the plane wave model (PW) with the best-fit values of \(\sin ^22\theta _{13}\) and \(\varDelta m^2_{32}\) obtained with the plane wave model. The ratios using the wave-packet model (WP) assume \(\sigma _\mathrm{{rel}} = 0.33\) (dashed line) and \(\sigma _\mathrm{{rel}}= 8\times 10^{-17}\) (dotdashed line), as two examples. The green lines correspond to the wave packet model ratios assuming the best-fit values of \(\sin ^22\theta _{13}\) and \(\varDelta m^2_{32}\) obtained with the plane wave model and thus, inconsistent with the data by about five standard deviations. The red lines correspond to the wave packet model ratios assuming the best-fit values of \(\sin ^22\theta _{13}\) and \(\varDelta m^2_{32}\) obtained within the wave packet model, yielding a much better agreement with the data. All ratios enter the region below \(2m_e\), which corresponds to the IBD threshold, because of detector response effects like energy reconstruction and absorption in the inner acrylic vessel (see details in Refs. [49, 52])

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.Footnote 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.

The data can be reasonably well described by

$$\begin{aligned}&\varDelta m^2_{32}=2.17\times 10^{-3}\,\text { eV}^2, \quad \sin ^22\theta _{13}=0.102,\nonumber \\&\sigma _\text {rel}=8\times 10^{-17}, \quad \chi ^2/\text {ndf} = 246.8/(256-4), \end{aligned}$$

and by

$$\begin{aligned}&\varDelta m^2_{32}=2.16\times 10^{-3}\text { eV}^2, \quad \sin ^22\theta _{13}=0.097,\nonumber \\&\sigma _\text {rel}=0.33, \quad \chi ^2/\text {ndf} = 253.8/(256-4). \end{aligned}$$

These results demonstrate that one could obtain reasonable fits of the data within the wave packet model with certain values of \(\sigma _\text {rel}\) and yield best-fit values of the oscillation parameters which differ from the corresponding best-fit values with the plane wave model, assuming normal mass hierarchyFootnote 8:

$$\begin{aligned}&\varDelta m^2_{32}=2.45\times 10^{-3}\,\text { eV}^2,\quad \sin ^22\theta _{13}=0.0852, \nonumber \\&\quad \chi ^2/\text {ndf} = 245.9/(256-3). \end{aligned}$$

However, Eqs. (1617) do not correspond to the global minimum of the \(\chi ^2\) discussed below because \(\sigma _\text {rel}\) was fixed to two arbitrary values for illustrative purposes. In order to find the global minimum we performed a detailed statistical analysis of the allowed region of \(\sigma _\text {rel}\).

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].

The analysis was done with four unconstrained parameters \(\sigma _{\text {rel}}\), \(\varDelta m^2_{32}\), \(\sin ^22\theta _{13}\) and reactor flux normalization 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

$$\begin{aligned} \varDelta \chi ^2(\varvec{\eta }') = \min \limits _{\varvec{\eta }{\setminus }\varvec{\eta }'} \chi ^2(\varvec{\eta }) -\min \limits _{\varvec{\eta }}\chi ^2(\varvec{\eta }), \end{aligned}$$

where \(\varvec{\eta }=(\sigma _{\text {rel}}, \varDelta m^2_{32}, \sin ^22\theta _{13}, N)\) and \(\varvec{\eta }'\) is its subspace with parameters of interest (\(\varvec{\eta }'=\sigma _\text {rel}\) for one dimensional interval, and \(\varvec{\eta }'=(\sigma _\text {rel},\varDelta m^2_{32}\)) or \(\varvec{\eta }'=(\sigma _\text {rel},\sin ^22\theta _{13}\)) for two dimensional regions), and both are used to determine the p-value of the observed dataset and the model.

The closed interval corresponding to the \(100\times (1-\alpha )\%\) confidence level (C.L.) is constructed for both the fixed-level \(\varDelta \chi ^2\) analysis and the Feldman–Cousins method as the region of \(\varvec{\eta }'\) which satisfies:

$$\begin{aligned} \varDelta \chi ^2(\varvec{\eta }') < \varDelta \chi ^2_{1-\alpha }, \end{aligned}$$

where \(\varDelta \chi ^2_{1-\alpha }\) is the \((1-\alpha )\)-th quantile of the statistic in Eq. (19). The tabulated values of the quantile \(\chi ^2_{n;1-\alpha }\) of the \(\chi ^2_n\) distribution with 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.

An open confidence interval can be constructed if neutrinos are assumed to be produced and detected coherently, which is equivalent to assuming \(\sigma _\text{ rel } \gg 10^{-16}\). In this case, instead of using Eq. (19), an upper bound on \(\sigma _\text {rel}\) can be computed using the modified statistic [55]

$$\begin{aligned} \varDelta \chi ^2_\text {up}(\sigma _{\text {rel}}) = {\left\{ \begin{array}{ll} \varDelta \chi ^2(\sigma _{\text {rel}}) &{}\text{ if } \hat{\sigma }_{\text {rel}} < \sigma _{\text {rel}} \\ 0 &{}\text{ if } \hat{\sigma }_{\text {rel}} > \sigma _{\text {rel}}, \end{array}\right. } \end{aligned}$$

with \(\hat{\sigma }_{\text {rel}}\) representing the best-fit value. In the fixed-level \(\varDelta \chi ^2\) analysis the \(100\times (1-\alpha )\%\) C.L. upper limit is given by:

$$\begin{aligned} \varDelta \chi ^2(\sigma _{\text {rel}}) \le \chi ^2_{1;1-2\alpha }. \end{aligned}$$

For example, in order to set a 95% C.L. upper limit, the quantile \(\chi ^2_{1;0.9}=2.71\) was used. The Feldman–Cousins method automatically produces the proper interval using the interval construction in Eq. (20).

Fig. 2
figure 2

Allowed regions of \((\varDelta m^2_{32},\sigma _\mathrm{{rel}})\) (top) and of \((\sin ^2 2\theta _{13},\sigma _\mathrm{{rel}})\) (middle) parameters obtained with fixed-level \(\varDelta \chi ^2\) (contours corresponding to \(1\sigma \), \(2\sigma \), \(3\sigma \) C.L., dashed lines) and within the Feldman–Cousins (contours corresponding to \(1\sigma \), \(2\sigma \) C.L., solid lines) methods. Bottom panel shows the marginalized \(\varDelta \chi ^2(\sigma _\mathrm{{rel}})\) statistic given by (19) vs \(\sigma _\mathrm{{rel}}\). Note the break in the abscissa and the change from a logarithmic to linear scale

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 \(10^{-5}\).

The best-fit point corresponds to

$$\begin{aligned}&\varDelta m^2_{32}=1.59\times 10^{-3}\,\text { eV}^2, \quad \sin ^22\theta _{13}=0.160,\nonumber \\&\sigma _\text {rel}=4.0\times 10^{-17}, \quad \chi ^2/\text {ndf} = 245.9/(256-4), \end{aligned}$$

with the p-value 0.596 which is smaller than the p-value 0.614 with the plane wave model given by Eq. (18). The allowed region for \(\sigma _\text {rel}\) at a 95% C.L. reads:

$$\begin{aligned} 2.38 \times 10^{-17}< \sigma _\mathrm{{rel}} <0.23. \end{aligned}$$

Taking the average momentum \(p=4\) MeV of detected reactor \(\bar{\nu }_e\), the interval in Eq. (24) can be translated to \(10^{-11}\, \text { cm } \lesssim \sigma _x\lesssim 1\,\text { km}\). The upper bound of Eq. (24) ensures that the coherence is preserved during at least almost two oscillation half-cycles: \(L^\text{ coh }_{32}>1.94\;L^\text{ osc }_{32}/2\) while the dispersion length is larger than almost three oscillation half-cycles: \(L^\text{ d }_{32}>2.96\;L^\text{ osc }_{32}/2\).

The lower limit of Eq. (24) (\(\sigma _x\lesssim 1\,\text { km}\)) obtained by constraining the \(D^2_{kj}\) term is much weaker than an obvious constraint of \(\sigma _x\lesssim 2\) m which follows from the consideration that the \(\sigma _x\) (which equals 1/2\(\sigma _ p\)) of \(\bar{\nu }_e\) wave packets detected by the Daya Bay Experiment does not exceed the dimensions of the reactor cores and detectors. Taking this constraint into account, \(\sigma _p\gtrsim 5\times 10^{-8}\) eV, which for the average momentum \(p=4\) MeV, translates into \(\sigma _\text {rel}\gtrsim 10^{-14}\). Such a \(\sigma _\text {rel}\) corresponds to the regime where \(D^2_{kj}\ll 1\) and the localization term can be safely neglected, which allows us, using the modified statistic for an open interval in Eq. (21), to put an upper limit of:

$$\begin{aligned} \sigma _\text {rel}<0.20, \text { at a } 95\%\text { C.L.} \end{aligned}$$

Future reactor experiments at baselines of approximately 50 km such as JUNO [56] and RENO-50 [57] would be able to improve the upper limit on \(\sigma _\text {rel}\) by more than an order of magnitude due to about 20 oscillation cycles to be detected and unprecedented resolution of visible energy of \(\delta _{E}/E\simeq 3\%/\sqrt{E}\). We estimate the following sensitivity of JUNO: \(3.8\times 10^{-17}<\sigma _\text {rel}<0.01\) at a 95% C.L. The lower limit cannot be improved by JUNO because of smaller statistics of expected \(\bar{\nu }_e\) interactions with respect to Daya Bay and independence of \(D^2_{kj}\) on the baseline.

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.