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

DOI 10.1140/epjc/s10052-017-4970-y

Regular Article - Experimental Physics

Study of the wave packet treatment of neutrino oscillation at Daya

Bay

Daya Bay Collaboration

a,∗ dayabay.ihep.ac.cn

Received: 27 January 2017 / Accepted: 2 June 2017

© The Author(s) 2017. This article is an open access publication

Abstract

The disappearance of reactor

¯ν

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

σ

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

¯ν

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

prob-ability formula was used to fit Daya Bay data, providing

the first experimental limits: 2.38 × 10

−17

_{< σ}

rel

< 0.23.

Treating the dimensions of the reactor cores and detectors as

constraints, the limits are improved: 10

−14

 σ

rel

< 0.23,

and an upper limit of

σ

rel

< 0.20 (which corresponds to

σx

 10

−11

_{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

reac-tor antineutrinos detected by the Daya Bay experiment thus

ensuring an unbiased measurement of the oscillation

param-eters sin

2

2θ

13

and

Δm

232

within the plane wave model.

Contents

1 Introduction

. . . .

2 Analysis

. . . .

3 Results and discussion

. . . .

Summary

. . . .

References

. . . .

∗_{The full list of contributors is displayed at the end of the article.}

a_{e-mail:[email protected]}

1 Introduction

1.1 Neutrino oscillation in the plane wave approximation

The neutrino, a light electrically neutral fermion

participat-ing in weak interactions, was suggested by Pauli to save the

conservation of energy and momentum in nuclear

β-decays.

Since then, three flavors of neutrinos

να

= (ν

e, νμ, ντ

) were

discovered, each produced or detected in association with a

corresponding lepton

_α

= (e, μ, τ). 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

discrep-ancies 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

–

5

], atmospheric [

6

,

7

], particle accelerator [

7

,

8

] and

reactor [

9

–

12

] neutrinos.

Neutrino oscillation is a quantum phenomenon of

quasi-periodic change of neutrino flavor

ν

_α

→ ν

_β

with time. This

phenomenon originates in the non-equivalence of neutrino

flavor

ν

_α

and mass

νk

= (ν

1

, ν

2

, ν

3

) eigenstates, differences

in their masses, and an assumption that the produced and

detected neutrino states are coherent superpositions of

neu-trino mass eigenstates:

|ν

α(p) = 3



k₌₁

where V

_αk

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

|ν

α

(t; p) =

3



k=1

V

_αk∗

e

−i Ekt

|ν

k(p),

₍₂₎

where Ek

=



p

2

_{+ m}

2

k

. This leads to the oscillatory

behav-ior of the probability to detect a neutrino originally of flavor

α as having flavor β:

P

_αβ

(L) = |ν

_β

(p)|ν

_α

(t; p)|

2

=

3



k, j=1

V

_αk∗

V

_{β j}∗

V

_βk

V

_αj

e

−i2π L/Losck j

,

₍₃₎

where L

osc_{k j}

= 4πp/Δm

2_{k j}

is the oscillation length due to

the non-zero differences

Δm

2_{k j}

= m

2_k

− m

2_j

, and time t is

approximated by the traveled distance L.

The underlying theory, assuming a plane wave

approxi-mation, was developed in the middle of the 1970s [

13

–

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

dis-cussed in detail in Refs. [

16

,

18

–

20

]. After the first theory

was developed, Refs. [

21

–

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

prob-able value, with a certain “width” or dispersion.

There-fore, 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

νk

and

ν

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

νk

and

ν

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

neu-trino masses suggests that the coherence length of neuneu-trino

oscillation is the largest available among all known

phenom-ena.

After the pioneering studies [

21

–

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

dis-persion of the neutrino wave packet in momentum space,

such as in Refs. [

18

,

19

,

25

]. The second one is based on

cal-culations within quantum field theory (QFT), describing all

external particles involved in neutrino production and

detec-tion as wave packets while treating neutrinos as virtual

par-ticles. The neutrino wave-function is then calculated rather

than postulated. The effective momentum dispersion of the

neutrino wave function depends on the kinematics of

neu-trino production and detection and on the momentum

dis-persions of the external particles, as in Refs. [

26

–

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

disper-sion

σ

p

of the neutrino wave packet is an effective quantity

comprising the microscopic momenta dispersions of all

parti-cles involved in the production and detection of the neutrino.

A non-zero value of

σp

leads with time to the decoherence

in the quantum superposition of massive neutrinos which

results in a vanishing oscillation pattern of

ν

_α

→ ν

_β

transi-tions. In addition, the oscillation pattern is smeared further in

the reconstructed energy spectrum due to a non-zero

experi-mental resolution

δE

of the neutrino energy.

Despite considerable progress in building wave packet

models, none of these approaches provides a solid

quan-titative theoretical estimate of

σp

or of the spatial width

σx

= 1/2σ

p. Theoretical estimates vary by orders of

magni-tude, associating the dispersion of the neutrino wave packet

with various scales; for example, uranium nucleus diameter

(σx

 10

−12

_cm,

_σ

_p

_{ 10 MeV), atomic or inter-atomic}

distances (σx

 (10

−8

_{− 10}

−7

_{) cm, σ}

p

 (10

3

− 10

2

)

eV), pressure broadening (σx

 10

−4

_cm,

_σp

_{ 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

1 _{Recently, a first calculation which consistently treats the full}

also lacks quantitative experimental investigations of

deco-herence 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

–

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

inco-herent 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

quan-tum 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

demon-strate that the oscillation parameters estimated with the plane

wave approximation are unbiased. The oscillation

probabil-ity formula modified by the wave packet contribution, which

is discussed further, has two distinctive features: it depends

on

Δm

2_{k j}

/p

2

σ

rel

via the so-called localization term and on

L

Δm

2_{k j}

σ

rel

/p via the term responsible for the loss of

coher-ence with distance, where

σ

rel

= σ

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

¯ν

e

interactions are compared to a

prediction using a QM wave packet model of neutrino

oscilla-tion which is briefly outlined in what follows. We simplify the

consideration by examining a one-dimensional wave packet

of the neutrino.

3

The plane wave state in (

1

) is replaced by a

Footnote 1 continued

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 [34–37], 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/L2dependence of the flux [45] and does not affect significantly the oscillation pattern.

wave packet describing a neutrino produced as flavor

α:

|ν

α

(pP

; t

P, xP

) =

3



k=1

V

_αk∗



d p

2π

fP

(p)e

−iφP(p)

|ν

k(p),

(4)

with

φP(p) = Ekt

P

− px

P

. fP

(p) is the wave function of the

neutrino in momentum space and is assumed to be Gaussian:

fP

(p) =



2π

σ

2 p P



1 4

e

− (p−pP )2 4σ2_{p P}

,

(5)

where the subscript P in fP

(p), pP

and

σp P

indicates the

quantities at production. In configuration space the state in

Eq. (

4

) describes a wave packet with mean coordinate xP

at

time tP

. The state in Eq. (

4

) is normalized to unity. Similarly,

a wave packet state at detection

|ν

_β

(pD

; t

D, xD

) is defined

as the state given by Eq. (

4

).

A projection of

|ν

_α

(pP

; t

P

, x

P) onto νβ

(pD

; t

D, xD

)|

produces the flavor-changing amplitude

A

αβ

(p; tD

−t

P

, L, σp)≡ν

β

(pD

; t

D, xD)|να

(pP

; t

P

, x

P),

(6)

which depends on L

≡ x

D

− x

P

, time difference tD

−t

P

and

on the effective mean neutrino momentum p and

momen-tum dispersion

σp

comprising the details of production and

detection

4

p

=

p

Pσ 2 p D

+ p

D

σ

2 p P

σ

2 p P

+ σ

2 p D

,

1

σ

2 p

=

1

σ

2 p P

+

1

σ

2 p D

.

(7)

The probability

|A

_αβ

(p; tD

− t

P

, L, σp)|

2

should be

inte-grated over usually unobservable variables – production time

tP

(or, equivalently, over tD

−t

P) and most probable

momen-tum p

P

to get an experimentally observable oscillation

prob-ability, which does not depend anymore on time

(tD

− t

P

)

but does depend on L:

P

_αβ

(L) =



dtP

d pP

2

π

|A

αβ

(p; tD

− t

P, L, σp)| 2

(8a)

=

3



k, j=1

V

_αk∗

V

_βk

V

_αj

V

_{β j}∗ 4



1

+



L

/L

d_{k j}

2

e

−



L/Lcohk j

2

1+



L/Ldk j

2

−D2 k j

e

−iϕk j

,

(8b)

where the phase



ϕk j

is the sum of the plane wave phase

ϕk j

= 2π L/L

osc

k j

and correction

ϕ

k jd

due to the dispersion of

4 _{The momentum integral in Eq. (6) is calculated by expanding E}

k=



p2_{+ m}2

k in a Taylor series up to second order around the effective

the wave packet:



ϕk j

= ϕ

k j

+ ϕ

_{k j}d

, with

ϕ

d k j

= −

L/L

d k j

1

+



L

/L

d_{k j}

2



L

L

coh_{k j}



2

+

1

2

arctan

L

L

d_{k j}

.

(9)

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

con-sideration within QFT with relativistic wave packets [

32

].

Relativistic invariance suggests that

σx

Ek

(and thus

σ

p/Ek

)

should be a Lorentz invariant [

16

,

47

]. Up to a typically tiny

correction of the order of m

2_k

/p

2

_,

_σ

rel

should also be a

rela-tivistic invariant, at least when neutrinos remain relarela-tivistic.

In the QM approach adopted in Eqs. (

4

)–(

8

) the only

possi-bility to preserve Lorentz invariance is for

σ

rel

to be a

con-stant.

5

The probability in Eq. (

8

) contains three quantities

with dimensions of length:

L

osc_{k j}

=

4πp

Δm

2 k j

,

L

coh_{k j}

=

L

osc k j

√

2πσ

rel

,

L

dk j

=

L

coh_{k j}

2

√

2σ

rel

,

(10)

where L

osc_{k j}

is the usual oscillation length of a pair of

neu-trino states

|ν

k

 and |ν

j

, L

cohk j

is interpreted as the neutrino

coherence length, i.e. the distance at which the interference

of neutrino mass eigenstates vanishes, and finally L

d_{k j}

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

D

2k j

=

1

2



Δm

2 k j

4 p

2

_σ

rel



2

=

1

4



Δm

2 k j

σm

2



2

=

√

2πσx

L

osc_{k j}



2

(11)

suppresses the coherence of massive neutrino states

|ν

k

 and

|ν

j

 if Δm

2k j

σ

m2

, where

σm

2

= 2

√

2 pσp

could be

inter-preted as an uncertainty in the neutrino mass squared [

22

].

D

2_{k j}

can be seen from another perspective as the localization

term suppressing the oscillation if

√

2πσx

L

osc_{k j}

, where

σx

= (2σ

p)−1

_{is the width of neutrino wave packet in the}

configuration space.

5_{Since the QFT approach considers both neutrino production and}

detection one finds thatσrel, being a relativistic invariant, is actually

a function of kinematic variables involved in the production and detec-tion 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 QMσrel as that of the QFT

approach averaged over the kinematic variables of all external wave packets involved in neutrino production and detection.

It is worth mentioning that terms in Eq. (

8

) which

corre-spond to the interference of

νk

and

ν

j

states also get

sup-pressed by the denominator



4

₁

+ (L/L

d

k j

)

2

and vanish for

both limits

σ

p

→ 0 and σ

p

→ ∞, reducing the oscillation

probability in Eq. (

8

) to the non-coherent sum

P

_αβ

=



k

|V

αk

|

2

|V

βk

|

2

,

(12)

which does not depend on energy and distance. The

oscilla-tion probability in Eq. (

8b

) is not reduced to the plane wave

formula in Eq. (

3

) in the limit

σp

→ 0 because of the

inte-gration over an unobservable production time tP

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

σp

where Eqs. (

3

) and (

8b

) are

numer-ically almost identical to each other (see Sect.

2.2

).

For the

¯ν

e

at Daya Bay, 1

− P

ee

is expressed as

1

2

sin

2

_2θ

12

cos

4

θ

13

×

⎛

⎜

⎜

⎝1 −

exp

−

L/Lcoh₂₁ 2 1+L/Ld₂₁ 2

− D

2 21



4



1

+



L

/L

d₂₁

2

cos

(ϕ

21

+ ϕ

21d

)

⎞

⎟

⎟

⎠

+

1

2

cos

2

_θ

12

sin

2

2θ

13

×

⎛

⎜

⎜

⎝1 −

exp

−

L/Lcoh31 2 1+L/Ld 31 2

− D

312



4



1

+



L/L

d₃₁

2

cos

(ϕ

31

+ ϕ

31d

)

⎞

⎟

⎟

⎠

+

1

2

sin

2

_θ

12

sin

2

2θ

13

×

⎛

⎜

⎜

⎝1 −

exp

−

L/Lcoh32 2 1+L/Ld₃₂ 2

− D

2 32



4



1

+



L/L

d₃₂

2

cos

(ϕ

32

+ ϕ

32d

)

⎞

⎟

⎟

⎠ .

(13)

2.2 Sensitivity of Daya Bay experiment to neutrino wave

packet

The Daya Bay experiment is composed of two near

ground experimental halls (EH1 and EH2) and one far

under-ground 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

β decays of neutron-rich daughters of the

fis-Table 1 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

sion isotopes

235

U,

238

U,

239

Pu and

241

Pu, and detected via

the inverse

β 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

scintilla-tor 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

δE/E ≈ 8% at 1 MeV of deposited energy in the

scintilla-tor. More details on the experimental setup are contained in

Refs. [

49

–

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

experimen-tal halls to each pair of reactor cores, are summarized in

Table

1

. The expected numbers of IBD events are

convolu-tions 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

correc-tions, 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

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

scintil-lator and electronics non-linearity and energy resolution

δE.

One can meet claims in literature that the smallest among

σp

and

δE

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

σp

of neutrino wave packet

and

δE

. The latter is sometimes erroneously considered as

an upper extreme value of

σp. The width (Γ  σp) of a

hadronic resonance which is typically much larger than an

experimental energy resolution

δE

provides a well-known

counter-example, illustrating that

σ

p

could be much larger

than

δE

.

For relatively large values of

σp

 δ

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

σ

p

is governed by the most localized

particle in the production and detection of the neutrino,

δ

E

is

determined by the energy depositions of the final state

par-ticles in the detector, the amount and efficiency of detection

devices used to observe such depositions. In particular,

con-sidering a liquid scintillator detector surrounded by a number

of PMTs as an example, one could hypothesize

modifica-tions in the number of PMTs, their efficiencies or even in

the light yield. Such variations would modify the energy

res-olution

δE

correspondingly, leaving intact the microscopic

processes determining

σp

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

σp

→ 0 leads to the

deco-herence 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

δ

E

→ 0.

In order to illustrate analytically an interplay of

σp

and

δE

,

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

vis

, assuming

δE

 p,

infinite dispersion length L

d

, neglecting the D

2

term, and

suppressing mass eigenstate indices for the sake of

compact-ness

6

:

1

√

2πδE



d p exp

(−i 2π L/L

osc

− (L/L

coh

)

2

−(p − E

vis

)

2

/2δ

2E

)

 exp (−i 2π L/L

osc

rec

− (L/L

coheff

)

2

_),

(14)

where L

osc

and L

coh

are given by Eq. (

10

) and the effective

coherence length comprises both the intrinsic

σp

and detector

resolution

δE

:



1

L

coh_eff



2

=



1

L

coh rec



2

+



1

L

coh_det



2

,

(15)

where L

oscrec

and L

cohrec

are given by L

osc

and L

coh

replacing

p with E

vis

, and L

coh_det

is given by L

cohrec

, replacing

σp

with

δE

. The interplay of

σp

and

δE

is illustrated by the effective

coherence length L

coh_eff

, which is dominantly determined by

the smallest among L

cohrec

and L

cohdet

, or by the largest among

σ

p

and

δE

. Therefore, the effective energy dispersion

σ

peff

is

determined by

(σ

eff_p

)

2

= σ

2_p

+ δ

2_E

.

6 _{The actual implementation of the detector effects in this analysis was}

The following provides simple numerical estimates of

Daya Bay sensitivity to wave packet effects on neutrino

oscil-lations.

For a typical momentum of p

= 4 MeV of detected

reac-tor

¯ν

e, the oscillation would be suppressed for two

distinc-tive domains of

σ

rel

. The domain

σ

rel

 O(0.1) corresponds

to significant contributions from L-dependent

interference-suppressing terms and corrections to the oscillation phase

ϕ

d

32

in Eq. (

8

), while the D

k j2

term is negligibly small. For

example, at L

= L

osc₃₂

/2 the exponential suppression reaches

its maximum e

−π/8

at

σ

rel

= 1/

√

2

π  0.4.

Correspond-ingly, the coherence and dispersion lengths read L

coh₃₂

 2.2

km and L

d₃₂

 2 km. At larger values of σ

rel

and at a fixed

distance the spatial dispersion of neutrino wave packets

par-tially compensates the loss of coherence due to the spatial

separation of

νk

and

ν

j

.

The domain

σ

rel

 O(2.8 × 10

−17

) corresponds to

D

2₃₂

 1, which is significant in suppressing the

interfer-ence in Eq. (

8

) through the L–independent term, while the

L-dependent terms are negligibly small. Thus, the region of

O(2.8 × 10

−17

_{)  σ}

rel

 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

exper-imental halls as a function of reconstructed visible energy

E

vis

. Figure

1

also shows the expected ratio for neutrino

oscil-lation with the plane wave and wave packet models with

σ

rel

of 0.33 and 8

× 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

σ

rel

= 0.33 and with σ

rel

= 8 × 10

−17

are inconsistent

with the data by about five standard deviations, thus

moti-vating the chosen values of

σ

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

neu-trino wave packet. Remarkably, most changes in the energy

spectra due to

σ

rel

are in opposite directions for near and

far halls, which can be explained qualitatively as follows. As

mentioned above, the extremes

σp

→ 0 and σ

p

→ ∞ would

yield fully decoherent neutrinos with the oscillation

proba-bility given by Eq. (

12

). Antineutrinos detected at the near

halls experience a relatively small oscillation in the plane

wave approach. The values of

σ

rel

selected for Fig.

1

make

the

¯ν

e

partially decoherent and P

ee

tend towards Eq. (

12

),

pre-dicting a smaller number of surviving

¯ν

e

as compared to the

plane wave formula. The distance at which the far detectors

7_{The following values of the oscillation parameters were used in Fig.1:} Δm2 21= 7.53 × 10−5 eV2,Δm232 = 2.45 × 10−3 eV2, sin22θ12 = 0.846, sin22θ13= 0.0852. 0.96 0.98 1.00

R

obs

/R

pred ,no osc

EH1 + EH2

θ

13

, Δm

232

fixed

θ

13

, Δm

232

free

1 2 3 4 5 6 7 8

E

vis

, MeV

0.90 0.94 0.98

EH3

data PW WP (σrel= 8 · 10−17) WP (σrel= 0.33)

Fig. 1 Ratios of the observed to expected numbers of IBD events in the absence of oscillation as a function of reconstructed visible energy

E_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 sin2_2θ

13andΔm232obtained

with the plane wave model. The ratios using the wave-packet model (WP) assumeσrel = 0.33 (dashed line) and σrel = 8 × 10−17(dot– dashed line), as two examples. The green lines correspond to the wave

packet model ratios assuming the best-fit values of sin22θ13andΔm232

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 sin22θ13and Δm2

32obtained within the wave packet model, yielding a much better

agreement with the data. All ratios enter the region below 2me, 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])

of the Daya Bay experiment are placed is tuned to observe

the maximal oscillation effect due to

Δm

2₃₂

. Partial

decoher-ence of the

¯ν

e

tends to reduce the oscillation, thus predicting

a larger number of survived

¯ν

e

with respect to the plane wave

formula. This feature of Daya Bay provides additional

sensi-tivity to the decoherence effects and makes such a study less

sensitive to the predicted reactor

¯ν

e

spectrum.

The data can be reasonably well described by

Δm

2

32

= 2.17 × 10

−3

eV

2

_{, sin}

2

_2θ

13

= 0.102,

and by

Δm

2 32

= 2.16 × 10

−3

eV

2

_{, sin}

2

_2θ

13

= 0.097,

σ

rel

= 0.33, χ

2

/ndf = 253.8/(256 − 4).

(17)

These results demonstrate that one could obtain reasonable

fits of the data within the wave packet model with certain

val-ues of

σ

rel

and yield best-fit values of the oscillation

parame-ters which differ from the corresponding best-fit values with

the plane wave model, assuming normal mass hierarchy

8

:

Δm

2

32

= 2.45 × 10

−3

eV

2

, sin

2

2

θ

13

= 0.0852,

χ

2

_{/ndf = 245.9/(256 − 3).}

(18)

However, Eqs. (

16

,

17

) do not correspond to the global

mini-mum of the

χ

2

discussed below because

σ

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

σ

rel

.

2.3 Statistical framework

As the goodness-of-fit measure we use

χ

2

(η) = (d −

t

(η))

T

V

−1

(d − t(η)), where d is a data vector containing

detected numbers of IBD candidates in energy bins and in

different detectors, while t(η) is the corresponding

theoreti-cal model vector which depends on constrained and

uncon-strained parameters

η. All constraints of the model as well as

expected fluctuations in the number of IBD events are

encom-passed in the covariance matrix V . The model vector t

(η)

comprises expected numbers of IBD and background events.

All constrained parameters (or systematic uncertainties)

rel-evant 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

parame-ters

σ

rel

,

Δm

2₃₂

, sin

2

2θ

13

and reactor flux normalization N .

The confidence regions are produced by means of two

statis-tical methods: the conventional fixed-level

Δχ

2

analysis and

the Feldman–Cousins method [

54

]. The marginalized

Δχ

2

statistic is

Δχ

2

_(η

_{) = min}

η\η

χ

2

_{(η) − min}

η

χ

2

_(η),

₍₁₉₎

where

η = (σ

rel

, Δm

2₃₂

, sin

2

2θ

13

, N) and η

is its subspace

with parameters of interest (

η

= σ

rel

for one dimensional

interval, and

η

= (σ

rel

, Δm

232

) or

η

= (σ

rel

, sin

2

2θ

13

) for

8_{The best-fit values of the oscillation parameters sin}2_2θ

13andΔm232

are different from our previous publication [49] because of a different implementation of systematic uncertainties and another choice of Evis

binning.

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

× (1 − α)%

confidence level (C.L.) is constructed for both the fixed-level

Δχ

2

_{analysis and the Feldman–Cousins method as the region}

of

η

which satisfies:

Δχ

2

_(η

_{) < Δχ}

2

1−α

,

(20)

where

Δχ

₁2_−α

is the

(1 − α)-th quantile of the statistic in

Eq. (

19

). The tabulated values of the quantile

χ

_n2_;1−α

of the

χ

2

n

distribution with n degrees of freedom (n

= 1, 2 for one

and two dimensional confidence regions) were used for the

fixed-level

Δχ

2

analysis. Toy Monte Carlo sampling was

used to determine

Δχ

2

1−α

of the statistic in Eq. (

19

) with the

Feldman–Cousins method.

An open confidence interval can be constructed if

neu-trinos are assumed to be produced and detected coherently,

which is equivalent to assuming

σ

rel

10

−16

. In this case,

instead of using Eq. (

19

), an upper bound on

σ

rel

can be

computed using the modified statistic [

55

]

Δχ

2

up

(σ

rel

) =



Δχ

2

_(σ

rel

) if ˆσ

rel

< σ

rel

0

if

ˆσ

rel

> σ

rel

,

(21)

with

ˆσ

rel

representing the best-fit value. In the fixed-level

Δχ

2

_{analysis the 100}

_{× (1 − α)% C.L. upper limit is given}

by:

Δχ

2

_(σ

rel

) ≤ χ

12;1−2α

.

(22)

For example, in order to set a 95% C.L. upper limit, the

quan-tile

χ

₁2_;0.9

= 2.71 was used. The Feldman–Cousins method

automatically produces the proper interval using the interval

construction in Eq. (

20

).

3 Results and discussion

Figure

2

displays the allowed regions in

(Δm

2₃₂

, σ

rel

) and

(sin

2

_2θ

13

, σ

rel

) obtained with both the fixed-level Δχ

2

and

the Feldman–Cousins methods, which are found to be

con-sistent. For the values of

σ

rel

 10

−16

the decoherence

effects lead to strong correlations between

Δm

2₃₂

, sin

2

2θ

13

and

σ

rel

, yielding smaller values of

Δm

2₃₂

and larger

val-ues of sin

2

2θ

13

. These correlations are expected taking into

account the explicit form of 1

− P

ee

(L) in Eq. (

13

). The

coefficients of

σ

rel

correlation with sin

2

2θ

13

and

Δm

2₃₂

are

found to be

−0.98 and 0.96 respectively. For σ

rel

 O(0.1),

these correlations are found to be significantly weaker. The

absolute values of the corresponding correlation coefficients

are smaller than 10

−5

.

1.0 1.5 2.0 2.5 3.0

Δm

2 32

,

10

− 3

eV

2 10−17 ₁₀−16

σ

rel 1 4 9

Δ

χ

2 0.2 0.4 0.1 0.2 0.3 0.4

sin

2

2

θ

13

Δχ

2

_{Feldman-Cousins}

1 σ

2 σ

3 σ

1 σ

2 σ

Fig. 2 Allowed regions of(Δm2₃₂, σrel) (top) and of (sin22θ13, σrel)

(middle) parameters obtained with fixed-levelΔχ2_(contours

corre-sponding to 1σ, 2σ, 3σ C.L., dashed lines) and within the Feldman– Cousins (contours corresponding to 1σ, 2σ C.L., solid lines) methods.

Bottom panel shows the marginalizedΔχ2(σrel) statistic given by (19)

vsσrel. Note the break in the abscissa and the change from a logarithmic

to linear scale

The best-fit point corresponds to

Δm

2

32

= 1.59 × 10

−3

eV

2

, sin

2

2

θ

13

= 0.160,

σ

rel

= 4.0 × 10

−17

, χ

2

/ndf = 245.9/(256 − 4),

(23)

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

σ

rel

at a 95% C.L. reads:

2.38 × 10

−17

< σ

rel

< 0.23.

(24)

Taking the average momentum p

= 4 MeV of detected

reactor

¯ν

e

, the interval in Eq. (

24

) can be translated to

10

−11

cm

 σ

x

 1 km. The upper bound of Eq. (

24

)

ensures that the coherence is preserved during at least almost

two oscillation half-cycles: L

coh₃₂

> 1.94 L

osc₃₂

/2 while the

dispersion length is larger than almost three oscillation

half-cycles: L

d

> 2.96 L

osc

/2.

The lower limit of Eq. (

24

) (σx

 1 km) obtained by

constraining the D

_{k j}2

term is much weaker than an obvious

constraint of

σx

 2 m which follows from the consideration

that the

σx

(which equals 1/2σp) of

¯ν

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,

σ

p

 5 × 10

−8

eV, which for the average

momen-tum p

= 4 MeV, translates into σ

rel

 10

−14

. Such a

σ

rel

corresponds to the regime where D

2_{k j}

 1 and the

localiza-tion 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:

σ

rel

< 0.20, at a 95% C.L.

(25)

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

σ

rel

by more than an order of

magnitude due to about 20 oscillation cycles to be detected

and unprecedented resolution of visible energy of

δE

/E 

3%/

√

E. We estimate the following sensitivity of JUNO:

3

.8 × 10

−17

< σ

rel

< 0.01 at a 95% C.L. The lower limit

cannot be improved by JUNO because of smaller statistics

of expected

¯ν

e

interactions with respect to Daya Bay and

independence of D

_{k j}2

on the baseline.

Summary

We performed a search for the footprint of the neutrino wave

packet which should show itself through specific

modifica-tions 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

neu-trino momentum 2

.38 × 10

−17

< σ

rel

< 0.23. Taking into

account the actual dimensions of the reactor cores and

detec-tors, we find that the lower limit

σ

rel

> 10

−14

corresponds

to the regime when the localization term is vanishing, thus

allowing us to put an upper limit:

σ

rel

< 0.20 at a 95% C.L.

This upper limit of

σ

rel

implies that

σx

 10

−11

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

σ

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

measure-ment of the oscillation parameters sin

2

2θ

13

and

Δm

232

within

the plane wave model.

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.

Author list F. P. An: Institute of Modern Physics, East China University of Science and Technology, Shanghai. A. B. Balantekin: University of Wisconsin, Madison, WI 53706, USA.

H. R. Band: Department of Physics, Yale University, New Haven, CT 06520, USA. M. Bishai: Brookhaven National Laboratory, Upton, NY 11973, USA.

S. Blyth: Department of Physics, National Taiwan University, Taipei; National United University, Miao-Li. D. Cao: Nanjing University, Nanjing.

G. F. Cao: Institute of High Energy Physics, Beijing. J. Cao: Institute of High Energy Physics, Beijing. W. R. Cen: Institute of High Energy Physics, Beijing. Y. L. Chan: Chinese University of Hong Kong, Hong Kong. J. F. Chang: Institute of High Energy Physics, Beijing.

L. C. Chang: Institute of Physics, National Chiao-Tung University, Hsinchu. Y. Chang: National United University, Miao-Li.

H. S. Chen: Institute of High Energy Physics, Beijing. Q. Y. Chen: Shandong University, Jinan.

S. M. Chen: Department of Engineering Physics, Tsinghua University, Beijing. Y. X. Chen: North China Electric Power University, Beijing.

Y. Chen: Shenzhen University, Shenzhen.

J.-H. Cheng: Institute of Physics, National Chiao-Tung University, Hsinchu. J. Cheng: Shandong University, Jinan.

Y. P. Cheng: Institute of High Energy Physics, Beijing. Z. K. Cheng: Sun Yat-Sen (Zhongshan) University, Guangzhou. J. J. Cherwinka: University of Wisconsin, Madison, WI 53706, USA. M. C. Chu: Chinese University of Hong Kong, Hong Kong.

A. Chukanov: Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia. J. P. Cummings: Siena College, Loudonville, NY 12211, USA.

J. de Arcos: Department of Physics, Illinois Institute of Technology, Chicago, IL 60616, USA. Z. Y. Deng: Institute of High Energy Physics, Beijing.

X. F. Ding: Institute of High Energy Physics, Beijing. Y. Y. Ding: Institute of High Energy Physics, Beijing.

M. V. Diwan: Brookhaven National Laboratory, Upton, NY 11973, USA.

M. Dolgareva: Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia.

J. Dove: Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. D. A. Dwyer: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.

W. R. Edwards: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. R. Gill: Brookhaven National Laboratory, Upton, NY 11973, USA.

M. Gonchar: Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia. G. H. Gong: Department of Engineering Physics, Tsinghua University, Beijing. H. Gong: Department of Engineering Physics, Tsinghua University, Beijing. M. Grassi: Institute of High Energy Physics, Beijing.

W. Q. Gu: Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai Laboratory for Particle Physics and Cosmology, Shanghai.

M. Y. Guan: Institute of High Energy Physics, Beijing.

L. Guo: Department of Engineering Physics, Tsinghua University, Beijing. X. H. Guo: Beijing Normal University, Beijing.

Z. Guo: Department of Engineering Physics, Tsinghua University, Beijing. R. W. Hackenburg: Brookhaven National Laboratory, Upton, NY 11973, USA. R. Han: North China Electric Power University, Beijing.

S. Hans: Brookhaven National Laboratory, Upton, NY 11973, USA; Department of Chemistry and Chemical Technology, Bronx Community College, Bronx, NY 10453, USA.

M. He: Institute of High Energy Physics, Beijing.

K. M. Heeger: Department of Physics, Yale University, New Haven, CT 06520, USA. Y. K. Heng: Institute of High Energy Physics, Beijing.

A. Higuera: Department of Physics, University of Houston, Houston, TX 77204, USA. Y. K. Hor: Center for Neutrino Physics, Virginia Tech, Blacksburg, VA 24061, USA. Y. B. Hsiung: Department of Physics, National Taiwan University, Taipei.

B. Z. Hu: Department of Physics, National Taiwan University, Taipei. T. Hu: Institute of High Energy Physics, Beijing.

W. Hu: Institute of High Energy Physics, Beijing.

E. C. Huang: Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. H. X. Huang: China Institute of Atomic Energy, Beijing.

X. T. Huang: Shandong University, Jinan.

P. Huber: Center for Neutrino Physics, Virginia Tech, Blacksburg, VA 24061, USA. W. Huo: University of Science and Technology of China, Hefei.

G. Hussain: Department of Engineering Physics, Tsinghua University, Beijing. D. E. Jaffe: Brookhaven National Laboratory, Upton, NY 11973, USA.

P. Jaffke: Center for Neutrino Physics, Virginia Tech, Blacksburg, VA 24061, USA. K. L. Jen: Institute of Physics, National Chiao-Tung University, Hsinchu.

S. Jetter: Institute of High Energy Physics, Beijing.

X. P. Ji: School of Physics, Nankai University, Tianjin; Department of Engineering Physics, Tsinghua University, Beijing. X. L. Ji: Institute of High Energy Physics, Beijing.

J. B. Jiao: Shandong University, Jinan.

R. A. Johnson: Department of Physics, University of Cincinnati, Cincinnati, OH 45221, USA.

D. Jones: Department of Physics, College of Science and Technology, Temple University, Philadelphia, PA 19122, USA. J. Joshi: Brookhaven National Laboratory, Upton, NY 11973, USA.

L. Kang: Dongguan University of Technology, Dongguan.

S. H. Kettell: Brookhaven National Laboratory, Upton, NY 11973, USA.

S. Kohn: Department of Physics, University of California, Berkeley, CA 94720, USA.

M. Kramer: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA; Department of Physics, University of California, Berkeley, CA 94720, USA.

K. K. Kwan: Chinese University of Hong Kong, Hong Kong. M. W. Kwok: Chinese University of Hong Kong, Hong Kong.

T. Kwok: Department of Physics, The University of Hong Kong, Pokfulam, Hong Kong. T. J. Langford: Department of Physics, Yale University, New Haven, CT 06520, USA. K. Lau: Department of Physics, University of Houston, Houston, TX 77204, USA. L. Lebanowski: Department of Engineering Physics, Tsinghua University, Beijing. J. Lee: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.

J. H. C. Lee: Department of Physics, The University of Hong Kong, Pokfulam, Hong Kong. R. T. Lei: Dongguan University of Technology, Dongguan.

R. Leitner: Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic. C. Li: Shandong University, Jinan.

D. J. Li: University of Science and Technology of China, Hefei. F. Li: Institute of High Energy Physics, Beijing.

G. S. Li: Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai Laboratory for Particle Physics and Cosmology, Shanghai. Q. J. Li: Institute of High Energy Physics, Beijing.

S. Li: Dongguan University of Technology, Dongguan.

S. C. Li: Department of Physics, The University of Hong Kong, Pokfulam, Hong Kong; Center for Neutrino Physics, Virginia Tech, Blacksburg, VA 24061, USA.

W. D. Li: Institute of High Energy Physics, Beijing. X. N. Li: Institute of High Energy Physics, Beijing. Y. F. Li: Institute of High Energy Physics, Beijing. Z. B. Li: Sun Yat-Sen (Zhongshan) University, Guangzhou. H. Liang: University of Science and Technology of China, Hefei.

C. J. Lin: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. G. L. Lin: Institute of Physics, National Chiao-Tung University, Hsinchu. S. Lin: Dongguan University of Technology, Dongguan.

S. K. Lin: Department of Physics, University of Houston, Houston, TX 77204, USA. Y.-C. Lin: Department of Physics, National Taiwan University, Taipei.

J. J. Ling: Sun Yat-Sen (Zhongshan) University, Guangzhou.

J. M. Link: Center for Neutrino Physics, Virginia Tech, Blacksburg, VA 24061, USA. L. Littenberg: Brookhaven National Laboratory, Upton, NY 11973, USA.

B. R. Littlejohn: Department of Physics, Illinois Institute of Technology, Chicago, IL 60616, USA. D. W. Liu: Department of Physics, University of Houston, Houston, TX 77204, USA.

J. L. Liu: Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai Laboratory for Particle Physics and Cosmology, Shanghai. J. C. Liu: Institute of High Energy Physics, Beijing.

C. W. Loh: Nanjing University, Nanjing.

C. Lu: Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA. H. Q. Lu: Institute of High Energy Physics, Beijing.

K. B. Luk: Department of Physics, University of California, Berkeley, CA 94720, USA; Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.

Z. Lv: Department of Nuclear Science and Technology, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an. Q. M. Ma: Institute of High Energy Physics, Beijing.

X. Y. Ma: Institute of High Energy Physics, Beijing. X. B. Ma: North China Electric Power University, Beijing. Y. Q. Ma: Institute of High Energy Physics, Beijing.

Y. Malyshkin: Instituto de Física, Pontificia Universidad Católica de Chile, Santiago, Chile.

D. A. Martinez Caicedo: Department of Physics, Illinois Institute of Technology, Chicago, IL 60616, USA.

R. D. McKeown: California Institute of Technology, Pasadena, CA 91125, USA; College of William and Mary, Williamsburg, VA 23187, USA. I. Mitchell: Department of Physics, University of Houston, Houston, TX 77204, USA.

M. Mooney: Brookhaven National Laboratory, Upton, NY 11973, USA.

Y. Nakajima: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.

J. Napolitano: Department of Physics, College of Science and Technology, Temple University, Philadelphia, PA 19122, USA. D. Naumov: Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia.

E. Naumova: Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia. H. Y. Ngai: Department of Physics, The University of Hong Kong, Pokfulam, Hong Kong. Z. Ning: Institute of High Energy Physics, Beijing.

J. P. Ochoa-Ricoux: Instituto de Física, Pontificia Universidad Católica de Chile, Santiago, Chile. A. Olshevskiy: Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia.

H.-R. Pan: Department of Physics, National Taiwan University, Taipei.

J. Park: Center for Neutrino Physics, Virginia Tech, Blacksburg, VA 24061, USA. S. Patton: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.

V. Pec: Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic.

J. C. Peng: Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. L. Pinsky: Department of Physics, University of Houston, Houston, TX 77204, USA.

C. S. J. Pun: Department of Physics, The University of Hong Kong, Pokfulam, Hong Kong. F. Z. Qi: Institute of High Energy Physics, Beijing.

M. Qi: Nanjing University, Nanjing.

X. Qian: Brookhaven National Laboratory, Upton, NY 11973, USA.

N. Raper: Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, NY 12180, USA. J. Ren: China Institute of Atomic Energy, Beijing.

R. Rosero: Brookhaven National Laboratory, Upton, NY 11973, USA.

B. Roskovec: Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic; Instituto de Física, Pontificia Universidad Católica de Chile, Santiago, Chile.

X. C. Ruan: China Institute of Atomic Energy, Beijing.

H. Steiner: Department of Physics, University of California, Berkeley, CA 94720, USA; Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.

G. X. Sun: Institute of High Energy Physics, Beijing. J. L. Sun: China General Nuclear Power Group, Shenzhen. W. Tang: Brookhaven National Laboratory, Upton, NY 11973, USA.

D. Taychenachev: Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia. K. Treskov: Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia. K. V. Tsang: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. C. E. Tull: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. N. Viaux: Instituto de Física, Pontificia Universidad Católica de Chile, Santiago, Chile. B. Viren: Brookhaven National Laboratory, Upton, NY 11973, USA.

V. Vorobel: Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic. C. H. Wang: National United University, Miao-Li.

M. Wang: Shandong University, Jinan.

N. Y. Wang: Beijing Normal University, Beijing. R. G. Wang: Institute of High Energy Physics, Beijing.

W. Wang: Sun Yat-Sen (Zhongshan) University, Guangzhou; College of William and Mary, Williamsburg, VA 23187, USA. X. Wang: College of Electronic Science and Engineering, National University of Defense Technology, Changsha.

Y. F. Wang: Institute of High Energy Physics, Beijing.

Z. Wang: Department of Engineering Physics, Tsinghua University, Beijing. Z. Wang: Institute of High Energy Physics, Beijing.

Z. M. Wang: Institute of High Energy Physics, Beijing.

H. Y. Wei: Department of Engineering Physics, Tsinghua University, Beijing. L. J. Wen: Institute of High Energy Physics, Beijing.

K. Whisnant: Iowa State University, Ames, IA 50011, USA.

C. G. White: Department of Physics, Illinois Institute of Technology, Chicago, IL 60616, USA. L. Whitehead: Department of Physics, University of Houston, Houston, TX 77204, USA. T. Wise: Department of Physics, Yale University, New Haven, CT 06520, USA.

H. L. H. Wong: Department of Physics, University of California, Berkeley, CA 94720, USA; Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.