Decoherence effects in reactor and Gallium neutrino oscillation experiments –
a QFT approach

Raphael<EMAIL_ADDRESS>Thomas
<EMAIL_ADDRESS>
Institut für Astroteilchenphysik, Karlsruher Institut für Technologie (KIT),
76021 Karlsruhe, Germany

###### Contents

  1. 1 Introduction
  2. 2 Single-vertex calculation
    1. 2.1 Interactions with the environment and particle decay
  3. 3 Neutrino oscillation amplitude and event rate
    1. 3.1 Derivation of the decoherence terms
    2. 3.2 Discussion of the decoherence terms
    3. 3.3 Phase-space integrals over unobserved external momenta
    4. 3.4 Classical averaging
  4. 4 Numerical estimates
    1. 4.1 Particle localizations and velocities
    2. 4.2 Effective energy-momentum spreads and decoherence parameters
  5. 5 Summary
  6. A Decoherence integral
  7. B Decay rate and scattering cross section
    1. B.1 Scattering cross section
    2. B.2 Particle decay
    3. B.3 Rate for the neutrino oscillation process
  8. C Phase-space integrals

##  1 Introduction

Conceptual questions related to quantum decoherence effects in neutrino
oscillations are the topic of ongoing discussions in the literature since
several decades. For early papers on the topic see [1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11], some examples of further investigations are e.g., [12, 13, 14, 15,
16, 17]. Recently, this discussion received further attention in the context
of short-baseline reactor [18, 19, 20, 21] and radioactive source Gallium
experiments [22, 23, 24, 25] searching for sterile neutrino oscillations [26,
27, 28, 29, 30]. These papers discuss the question whether quantum decoherence
could help to reduce tension in the data [26, 30] which arises in standard
sterile neutrino explanations, see e.g., [31, 32, 33, 34]. Possible
decoherence effects in the upcoming high-precision JUNO reactor experiment
have been discussed in Refs. [35, 36, 37].

In this paper we contribute to this discussion by adopting the quantum field
theoretical (QFT) approach to neutrino oscillations [4, 38, 6, 39, 12]. In
this approach the combined process of neutrino production, propagation, and
detection is considered as a single process whose amplitude is calculated by
usual $S$-matrix methods, adapted to the situation of macroscopically
separated production and detection regions. The neutrino is treated as an
internal line and coherence properties of the flavour transition are
completely determined by the localization of the external particles at the
source and at the detector.

We will follow largely the methods discussed by Beuthe in the review article
Ref. [12], with some modifications relevant to the experimental situations of
interest to us. In particular, we generalize the formalism to take into
account decaying particles (see also [40, 14, 41, 42]) as well as coherence
loss due to interactions of external particles with the environment. We focus
on quantities which are actually observed in experiments, for instance the
energy of the positron produced by the inverse beta-decay reaction in the
detectors of reactor neutrino experiments. The formalism naturally allows a
distinction of “quantum decoherence” (at the amplitude level) compared to
“classical averaging” (at the probability level), which are, however,
indistinguishable observationally. Our work is complementary to the recent
paper by Akhmedov & Smirnov [27], who base their argumentation on the neutrino
wave packet approach, reaching very similar conclusions as we do. A somewhat
different approach has been pursued by Jones, Marzec & Spitz [43] whose
results for the decoherence parameters differ quantitatively from ours.

We provide here some guideline on how to read this paper. In section 2 we
introduce the notation by discussing external wave packets for single vertex
processes, such as scattering or decay. Our ansatz of how to include
decoherence effects due to interactions with the environment as well as
decaying particles is introduced in section 2.1. In section 3 we sketch the
QFT calculations for the oscillation amplitudes. A reader mainly interested in
the final result may, after a look at the central expression in eq. 3.14, skip
directly to section 3.2, which contains a brief discussion of the decoherence
effects, and section 3.4, where we comment on classical averaging and stress
the equivalence of quantum and classical decoherence. Section 4 contains the
numerical estimates for reactor and Gallium source experiments, where in
section 4.1 we give some details of how we estimate the localizations of all
involved particles. Readers interested only in the main numerical results may
proceed directly to section 4.2, where we discuss the effective localization
and energy spreads and show that they are many orders of magnitude below the
observable level. We summarize our findings in section 5. Supplementary
material is provided in appendices A, B and C. In particular, appendix B
contains a discussion of how to derive standard expressions for scattering
cross sections and decay rates within our formalism, and in section B.3 we
discuss how the amplitude for the neutrino oscillation process derived in the
QFT formalism can be related to differential event rates in a neutrino
oscillation experiment.

##  2 Single-vertex calculation

In the QFT approach to neutrino oscillations the combined process of neutrino
production, propagation, and detection is evaluated. There, we face the
unusual situation that two vertices of the process (production and detection)
are macroscopically separated in space-time. Before we calculate the amplitude
for the oscillation process in section 3, we discuss first the conceptually
simpler case, where there is only one interaction region, as in standard
particle physics interaction calculations. This will serve to introduce the
formalism, fix the notation, and to discuss the modifications we introduce to
model the specific physical situations of decaying particles as well as
particles confined by frequent interactions with the environment. We consider
wave packets for both, initial and final state particles. In appendix B we
show that, in the limit of plane waves and the appropriate normalization, we
can recover standard text book expressions for scattering cross sections and
decay rates from our approach.

We define states as superpositions of momentum eigenstates
$|\mathbf{k}\rangle$ as

$|\phi\rangle=\int
d\tilde{\mathbf{k}}\phi(\mathbf{k})|\mathbf{k}\rangle\,,\qquad
d\tilde{\mathbf{k}}\equiv\frac{1}{2E_{k}}\frac{d^{3}k}{(2\pi)^{3}}\,.$ (2.1)

Three-vectors are denoted by bold-face letters. Momentum states are normalized
as

$\langle\mathbf{k^{\prime}}|\mathbf{k}\rangle=2E_{k}(2\pi)^{3}\delta^{(3)}(\mathbf{k}-\mathbf{k}^{\prime})$
(2.2)

and the wave packets as

$\int\frac{d^{3}k}{(2\pi)^{3}}|\phi(\mathbf{k})|^{2}=1\,.$ (2.3)

In the following we will consider the specific case of Gaussian wave packets.
While this is not true in general, it serves as a useful approximation for the
purpose of describing the relevant physics and for order-of-magnitude
estimates. The technical advantage is that many integrals can be performed
analytically in the case of Gaussian wave packtes. Taking into account the
above normalization condition, we have

$\phi(\mathbf{k})=\left(\frac{2\pi}{\sigma^{2}}\right)^{3/4}e^{-\frac{(\mathbf{k}-\mathbf{p})^{2}}{4\sigma^{2}}}\,.$
(2.4)

Here, $\mathbf{p}$ is the mean momentum and $\sigma$ is the momentum spread.
We will use the symbol “$\sigma$” to denote uncertainties of dimension 1
(i.e., momentum or energy) and the symbol “$\delta$” to denote uncertainties
of dimension $-1$ (i.e., space or time); they are related via
$\sigma\delta=1/2$.

Let us now consider a process with a set of $N_{i}$ initial and $N_{f}$ final
state particles, $|\phi_{i}\rangle$ and $|\phi_{f}\rangle$, respectively. They
are the product of states, each having the form as defined in eq. 2.1. Then,
the total transition probability for the process $i\to f$ is given by the
$S$-matrix [44, 45]:

$P_{if}=|\langle\phi_{f}|iT|\phi_{i}\rangle|^{2}\equiv|i\mathcal{A}|^{2}\,,$
(2.5)

where $iT$ is the non-trivial part of the $S$ matrix: $S=1+iT$. We use eq. 2.1
and the standard definition of the matrix element for momentum states [44],

$\left(\prod_{f}\langle\mathbf{k}_{f}|\right)iT\left(\prod_{i}|\mathbf{k}_{i}\rangle\right)=(2\pi)^{4}\delta^{(4)}\left(\sum_{f}k_{f}-\sum_{i}k_{i}\right)i\mathcal{M}\,,$
(2.6)

where in general $\mathcal{M}$ depends on all momenta. Then, we obtain the
transition amplitude as

$\displaystyle i\mathcal{A}$ $\displaystyle=\prod_{f}\int
d\tilde{\mathbf{k}}_{f}\phi_{f}^{*}(\mathbf{k}_{f})\prod_{i}\int
d\tilde{\mathbf{k}}_{i}\phi_{i}(\mathbf{k}_{i})(2\pi)^{4}\delta^{(4)}\left(\sum_{f}k_{f}-\sum_{i}k_{i}\right)i\mathcal{M}$
(2.7) $\displaystyle=\int d^{4}x\prod_{f}\int
d\tilde{\mathbf{k}}_{f}\phi_{f}^{*}(\mathbf{k}_{f})e^{ik_{f}x}\prod_{i}\int
d\tilde{\mathbf{k}}_{i}\phi_{i}(\mathbf{k}_{i})e^{-ik_{i}x}\,i\mathcal{M}\,.$
(2.8)

In the second step we used the Fourier-transform of the $\delta$-function to
introduce an integral over configuration space $d^{4}x$. This integral will
receive contributions from the “interaction region” in space-time, determined
by the overlap of the wave packets.

We now proceed by performing the momentum integrals by adopting the following
approximations. First, we assume that the matrix element $\mathcal{M}$ varies
slowly with momenta on the scales $\sigma_{i,f}$ and we can replace
$\mathcal{M}(\mathbf{k}_{i},\mathbf{k}_{f})\approx\mathcal{M}(\mathbf{p}_{i},\mathbf{p}_{f})$.
Then it can be pulled out of the integrals. Second, consider the exponentials
$e^{ik_{i,f}x}$. For a generic $k_{i,f}$ we have $kx=E_{k}t-\mathbf{kx}$.
Again we assume that the momentum is not too far from its mean value
$\mathbf{p}$ and approximate

$\displaystyle E_{k}\approx E_{p}+\mathbf{v}(\mathbf{k}-\mathbf{p})$ (2.9)

with the group velocity

$\displaystyle\mathbf{v}\equiv\left.\frac{\partial
E_{k}}{\partial\mathbf{k}}\right|_{\mathbf{k}=\mathbf{p}}\,.$ (2.10)

The physical implication of the approximation eq. 2.9 is that the wave packets
are not spreading in time.111Wave packet dispersion would be described by the
second order term in the expansion eq. 2.9, which is of order
$\sigma^{2}/E_{p}$. This term is small compared to typical energy widths
$\sigma_{m},\sigma_{E},\Gamma_{\rm col}$ derived below, because
$\sigma/E_{p}\ll 1$. Physically this means that on time scales relevant for
the coherent interaction the spreading of wave packets can be neglected.
Stationary states correspond to $\mathbf{v}=0$, whereas for free particles we
have $\mathbf{v}=\mathbf{p}/E_{p}$. At this point we do not need to specify
the dispersion relation and can allow for arbitrary $\mathbf{k}$ dependence of
$E_{k}$.

With these two approximations all integrals become integrals over Gaussians,
which can be performed, see e.g., [12]. The momentum integrals are all of the
type

$\displaystyle\int d\tilde{\mathbf{k}}\phi(\mathbf{k})e^{\mp
ikx}\approx\mathcal{N}e^{\mp
ipx-(\mathbf{x}-t\mathbf{v})^{2}\sigma^{2}}\,,\qquad\mathcal{N}\equiv\frac{1}{\sqrt{2E_{p}}}\frac{(2\sigma)^{3/2}}{(2\pi)^{3/4}}\,.$
(2.11)

Using this in eq. 2.8, we obtain

$\displaystyle i\mathcal{A}=i\mathcal{M}\int
d^{4}x\prod_{i,f}\mathcal{N}_{i}\mathcal{N}_{f}\exp\left[i\left(\sum_{f}p_{f}-\sum_{i}p_{i}\right)x-\sum_{i,f}(\mathbf{x}-t\mathbf{v}_{i,f})^{2}\sigma^{2}_{i,f}\right]\,.$
(2.12)

In order to simplify the exponential we introduce some notation:

$\displaystyle\Delta
E\equiv\sum_{i}E_{p_{i}}-\sum_{f}E_{p_{f}}\,,\qquad\Delta\mathbf{p}\equiv\sum_{i}\mathbf{p}_{i}-\sum_{f}\mathbf{p}_{f}\,,$
(2.13)

such that $\Delta E=0$ and $\Delta\mathbf{p}=0$ correspond to exact energy and
momentum conservation, respectively. We define the total momentum spread by

$\displaystyle\sigma_{p}^{2}\equiv\sum_{i,f}\sigma^{2}_{i,f}\,,$ (2.14)

and a weighted velocity and velocity-squared:

$\displaystyle\mathbf{v}\equiv\frac{1}{\sigma_{p}^{2}}\sum_{i,f}\sigma_{i,f}^{2}\mathbf{v}_{i,f}\,,\qquad\Sigma\equiv\frac{1}{\sigma_{p}^{2}}\sum_{i,f}\sigma_{i,f}^{2}\mathbf{v}_{i,f}^{2}\,.$
(2.15)

Using these definitions to rewrite the argument of the exponential, the
$d^{4}x$ integration can be performed as well:

$\displaystyle i\mathcal{A}$
$\displaystyle=i\mathcal{M}\prod_{i,f}\mathcal{N}_{i}\mathcal{N}_{f}\int
d^{4}x\,\exp\left[-i(\Delta
Et-\Delta\mathbf{px})-\sigma_{p}^{2}(\mathbf{x}^{2}-2\mathbf{xv}t+\Sigma
t^{2})\right]$ (2.16)
$\displaystyle=i\mathcal{M}\prod_{i,f}\mathcal{N}_{i}\mathcal{N}_{f}\,\frac{\pi^{2}}{\sigma_{p}^{3}\sigma_{e}}\exp\left[-\frac{(\Delta\mathbf{p})^{2}}{4\sigma_{p}^{2}}-\frac{(\Delta
E-\Delta\mathbf{pv})^{2}}{4\sigma_{e}^{2}}\right]\,,$ (2.17)

where the effective energy spread is obtained as

$\displaystyle\sigma_{e}^{2}\equiv\sigma_{p}^{2}(\Sigma-\mathbf{v}^{2})\,.$
(2.18)

We notice that the quantities $\sigma_{e}$ and $\sigma_{p}$ correspond to the
amounts within which energy and momentum conservation can be violated. It is
easy to see that they fulfill $0\leq\sigma_{e}\leq\sigma_{p}$. The momentum
uncertainty, eq. 2.14, is just the sum of the squares of the momentum
uncertainty of all particles. Hence, it is dominated by the particle with the
largest uncertainty. For hierarchical uncertainties, $\sigma_{e}$ is generally
dominated by the particle with the second largest uncertainty, see e.g., [12]
for a discussion. Note that “energy-momentum violation” happens for the mean
quantities $p_{i},p_{f}$, whereas for the actual 4-momenta picked at the
interaction vertex, energy-momentum conservation is exact, as manifest by the
$\delta$-function in eq. 2.7. Therefore, it is not surprising that the mean
quantities need to fulfill 4-momentum conservation only approximately.

###  2.1 Interactions with the environment and particle decay

We consider now the case that the involved particles propagate through a
medium, with frequent collisions. When an interaction happens, the particle
wave function picks up a random phase, which leads to loss of coherence. We
model this effect by restricting the time integration in eq. 2.16 from $t=0$
till the time when the next interaction happens by introducing an exponential
$e^{-t\Gamma_{\rm col}/2}$, where $\Gamma_{\rm col}$ is the collision rate.
This is motivated by considering the scattering as a Poissonian process.222See
also [42] for a similar approach. Hence, the integral in eq. 2.16 becomes

$\displaystyle\int d^{4}x\,e^{-i(\Delta
Et-\Delta\mathbf{px})-\sigma_{p}^{2}(\mathbf{x}^{2}-2\mathbf{xv}t+\Sigma
t^{2})}e^{-t\Gamma_{\rm col}/2}\Theta(t)$
$\displaystyle=\frac{\pi^{3/2}}{\sigma_{p}^{3}}e^{-\frac{(\Delta\mathbf{p})^{2}}{4\sigma_{p}^{2}}}\int_{0}^{\infty}dt\,e^{-\sigma_{e}^{2}t^{2}-it\Omega-t\Gamma_{\rm
col}/2}$ (2.19)
$\displaystyle\to\frac{\pi^{3/2}}{\sigma_{p}^{3}}e^{-\frac{(\Delta\mathbf{p})^{2}}{4\sigma_{p}^{2}}}\frac{1}{i\Omega+\Gamma_{\rm
col}/2}\,,$ (2.20)

with $\Omega\equiv\Delta E-\Delta\mathbf{pv}$, and in the last step we assumed
$\sigma_{e}\ll\Gamma_{\rm col}$. Hence, after squaring the amplitude, instead
of the energy Gaussian factor in eq. 2.17 we obtain the typical Lorentzian
shape $\propto 1/(\Omega^{2}+\Gamma_{\rm col}^{2}/4)$. This modelling
describes correctly the collisional line broadening [46], see section B.2.

Actually, we will take this picture as the physical origin of wave packets for
our external particles: the interactions with the environment lead to the
localization of the particle within a length $\delta_{x}$ which in turn
determines the momentum spread $\sigma$ via $\sigma\delta_{x}=1/2$ as well as
the mean time between two scatterings as $\delta_{x}/|\mathbf{v}|$. We adopt
now the following ansatz for the effective collision rate $\Gamma_{\rm col}$

$\Gamma_{\rm
col}^{2}\equiv\sum_{i,j}\sigma_{i,f}^{2}\mathbf{v}_{i,f}^{2}=\sigma_{p}^{2}\Sigma\,,$
(2.21)

i.e., adding the interaction rates in squares.333The reason for this ansatz is
our Gaussian approximation, see below, where naturally all spreads are added
quadratically. An alternative definition could be to add them linearly,
$\tilde{\Gamma}_{\rm col}=\sum_{i,f}\sigma_{i,f}|\mathbf{v}_{i,f}|$. It is
easy to show that $\tilde{\Gamma}_{\rm col}\geq\Gamma_{\rm col}$. Numerically
we find with the numbers from section 4 that in our cases of interest we have
$\tilde{\Gamma}_{\rm col}\approx\Gamma_{\rm col}$. Hence, $\Gamma_{\rm col}$
is dominated by the particle with the fastest interaction rate: as soon as one
of the involved particles interacts with the environment, coherence is lost.
Comparing eq. 2.21 with the expression for the energy spread in eq. 2.18, we
see that $\sigma_{e}\leq\Gamma_{\rm col}$. In section 4 we will find for our
cases of interest that $\sigma_{e}\approx\Gamma_{\rm col}$ because in most
cases $\mathbf{v}^{2}\ll\Sigma$.

As we have seen, our physical picture leads to a constraint on $\Omega$ with
Lorentzian shape. In the following we will, however, replace the Lorenzian
with a corresponding Gaussian factor with width $\Gamma_{\rm col}$. This
approximation will still capture the relevant physics, but simplify the
calculations significantly, because integrals can be taken analytically.
Hence, we recover an expression for the amplitude, similar to eq. (2.17), but
with the replacement

$\sigma_{e}^{2}\to\sigma_{e}^{2}+\Gamma_{\rm
col}^{2}=\sigma_{p}^{2}(2\Sigma-\mathbf{v}^{2})\,.$ (2.22)

In neutrino oscillation experiments neutrinos are very often produced in
particle decays, e.g., beta-decay of a nucleus in reactor experiments or pion
decay in accelerator experiments. We can model the decay of a particle in
complete analogy to the interaction with the environment by introducing an
additional factor $e^{-t\Gamma_{\rm dec}/2}$ in (2.19). Hence, the width of
the Lorentzian becomes just $\Gamma_{\rm col}+\Gamma_{\rm dec}$. As we show in
section B.2, in the limit $\Gamma_{\rm col}\gg\Gamma_{\rm dec}$ (frequent
collisions) this ansatz describes the collisional line broadening of a decay
line, whereas in the limit $\Gamma_{\rm col}\ll\Gamma_{\rm dec}$ (fast decay
and negligible collisions) it reproduces the standard definition of the decay
width in terms of the matrix element. In the Gaussian approximation we simply
obtain the effective energy spread by

$\sigma_{e}^{2}\to\sigma_{E}^{2}\equiv\sigma_{e}^{2}+\Gamma_{\rm
col}^{2}+\Gamma_{\rm
dec}^{2}=\sigma_{p}^{2}(2\Sigma-\mathbf{v}^{2})+\Gamma_{\rm dec}^{2}\,.$
(2.23)

For neutrino production in a nuclear reactor as well as the source in Gallium
experiments, the lifetimes of the relevant beta decays are typically much
longer than the interaction times with the environment (see section 4), and
therefore, it is save to negect $\Gamma_{\rm dec}$ compared to $\Gamma_{\rm
col}$. This may not be the case for experiments using pion decay, where pions
decay in a decay tunnel essentially without interacting.

Although we can combine the energy spread $\sigma_{e}$, the collision rate,
and the decay rate in the same effective quantity $\sigma_{E}$ (thanks to the
Gaussian approximation for all of them) their physical origins are different.
$1/\sigma_{e}$ corresponds to the effective interaction time due to the wave
packet overlap of all involved particels, $\Gamma_{\rm col}$ describes the
coherence loss due to interactions of each of the involved particles with the
environment, and $\Gamma_{\rm dec}$ takes into account the energy spread due
to the finite lifetime of a decaying particle.

##  3 Neutrino oscillation amplitude and event rate

$p_{A}$$p_{B}$$p_{1}$$p_{2}$$p_{3}$$p_{4}$$\bar{\nu}$ Figure 1: Feynman
diagram for the total process in an oscillation experiment.

We now move to the discussion of the amplitude relevant for neutrino
oscillation experiments consisting of neutrino production, propagation and
detection. To be specific, we consider neutrino production by the decay of a
particle $A$ into two final state particles and an anti-neutrino, $A\to
1+2+\bar{\nu}$, and anti-neutrino detection via the process $B+\bar{\nu}\to
3+4$. We have in mind reactor neutrinos, where the production process
corresponds to the beta decay of a nucleus ($A$), and the detection process is
the inverse beta decay reaction on a proton ($B$), but many of our
considerations will apply also in other circumstances with minor
modifications. The total process $A+B\to 1+2+3+4$ is illustrated in fig. 1.
The neutrino is considered as an internal propagtor and does not appear as
external particle [4, 38, 6]. We now follow the common approach [12] and
calculate the amplitude for the total process, by assuming wave packets for
all external particles, both initial state ($A,B$) as well as final state
particles ($1,2,3,4$).

We proceed in complete analogy to the discussion in section 2 but generalize
it to the case of macroscopically separated production ($P$) and detection
($D$) regions. In analogy to eq. 2.17, we obtain the following expression for
the amplitude describing production of an anti-neutrino with flavour $\alpha$
and detection of an anti-neutrino with flavour $\beta$444In the case of
reactor neutrino experiments we have of course $\alpha=\beta=e$. (see e.g.,
[12, 14] for explicit derivations):

$\displaystyle i\mathcal{A}_{\alpha\beta}=$ $\displaystyle\sum_{j}U_{\alpha
j}U_{\beta
j}^{*}\left(\prod_{i=A,B,f}\mathcal{N}_{i}\right)\int\frac{d^{4}p}{(2\pi)^{4}}i\tilde{\mathcal{M}}_{P}\frac{\not{p}-m_{j}}{p^{2}-m_{j}^{2}+i\epsilon}i\tilde{\mathcal{M}}_{D}\,e^{-ip(x_{D}-x_{P})}$
$\displaystyle\times\prod_{I=P,D}\frac{\pi^{2}}{\sigma_{pI}^{3}\sigma_{EI}}\exp\left[-\frac{(\mathbf{p}-\mathbf{p}_{I})^{2}}{4\sigma_{pI}^{2}}-\frac{(p^{0}-E_{I}-\mathbf{v}_{I}(\mathbf{p}-\mathbf{p}_{I}))^{2}}{4\sigma_{EI}^{2}}\right]\,.$
(3.1)

Here, $U_{\alpha j}$ are elements of the PMNS mixing matrix, the normalization
factors $\mathcal{N}_{i}$ are defined in eq. 2.11, $\tilde{\mathcal{M}}_{P,D}$
are the reduced matrix elements of the production and detection processes, the
sum over $j$ runs over the neutrino mass states with neutrino mass $m_{j}$,
$x_{P,D}$ are space-time points located in the production and detection
region. In the second line of eq. 3.1 we obtain two Gaussian factors related
to the approximate energy-momentum conservation at production and detection
points, with the momentum spreads $\sigma_{pI}$, energy spreads $\sigma_{EI}$,
and velocities $\mathbf{v}_{I}$ (with $I=P,D$) defined as in eq. 2.14, eq.
2.23, eq. 2.15, respectively. Furthermore, we have defined the kinematic
4-momenta of the neutrino at the production and detection vertices:

$\displaystyle p_{P}$ $\displaystyle=p_{A}-p_{1}-p_{2}\,,$ (3.2)
$\displaystyle p_{D}$ $\displaystyle=-p_{B}+p_{3}+p_{4}\,,$ (3.3)

and $E_{P},E_{D}$ are the time-components of the corresponding 4-vectors.

Next we perform the integral over $d^{3}p$ by using the Grimus-Stockinger
theorem [6], which allows us to take into account the macroscopic separation
of source and detector. In the relevant limit the propagating neutrinos go on-
shell, and we obtain

$\displaystyle i\mathcal{A}_{\alpha\beta}=$
$\displaystyle\sum_{j}\frac{U_{\alpha j}U_{\beta j}^{*}}{4\pi
L}\left(\prod_{i=A,B,f}\mathcal{N}_{i}\right)i\mathcal{M}_{P}i\mathcal{M}_{D}\,\frac{\pi^{4}}{\sigma_{pP}^{3}\sigma_{EP}\sigma_{pD}^{3}\sigma_{ED}}$
$\displaystyle\times\int\frac{dp^{0}}{2\pi}\exp\left[-ip^{0}T+ip_{j}L-f_{j}(p^{0})\right]\,,$
(3.4)

where $\mathbf{L}=\mathbf{x}_{D}-\mathbf{x}_{P}$, $L=|\mathbf{L}|$,
$\hat{\mathbf{l}}=\mathbf{L}/L$, $T=t_{D}-t_{P}$. The function $f_{j}(p^{0})$
is obtained from the exponential in the second line of eq. 3.1, where
according to the Grimus-Stockinger theorem $\mathbf{p}\to\mathbf{p}_{j}$ with

$\displaystyle\mathbf{p}_{j}=p_{j}\hat{\mathbf{l}}\,,\quad
p_{j}=\sqrt{(p^{0})^{2}-m_{j}^{2}}\,.$ (3.5)

Following Ref. [12], we rewrite $f_{j}(p^{0})$ by decomposing the vectors in
components parallel and orthogonal to $\hat{\mathbf{l}}$:

$\displaystyle
f_{j}(p^{0})=\sum_{I=P,D}\left[\frac{(p_{j}-p_{I})^{2}}{4\sigma_{pI}^{2}}+\frac{(p^{0}-E_{I}-v_{I}(p_{j}-p_{I}))^{2}}{4\sigma_{EI}^{2}}+\frac{\mathbf{p}_{I\perp}^{2}}{4\sigma_{pI}^{2}}\right]\,,$
(3.6)

where $p_{I}$ and $v_{I}$ denote the components parallel to $\hat{\mathbf{l}}$
and we have redefined

$\displaystyle E_{I}-\mathbf{v}_{I\perp}\mathbf{p}_{I\perp}\to E_{I}\,.$ (3.7)

In order to keep notation concise we do not introduce a different symbol for
this new variable.

###  3.1 Derivation of the decoherence terms

Equation 3.4 is the starting point to derive the terms leading to decoherence
effects, which will be the main focus of our considerations in the following.
We are not concerned with overall factors and focus on the interference terms.
First, we have to square the amplitude and perform an integration of the
unobservable propagation time $T$, see discussion in section B.3, eq. B.17. We
obtain a $\delta$-function from the integral $\int
dTe^{-i(p^{0}-{p^{\prime}}^{0})T}$ and obtain

$\displaystyle\overline{|\mathcal{A}_{\alpha\beta}|^{2}}\propto\int
dT|\mathcal{A}_{\alpha\beta}(T)|^{2}\propto\sum_{jk}U_{\alpha j}U_{\beta
j}^{*}U_{\alpha k}^{*}U_{\beta k}\int dp^{0}\exp\left[i\frac{\Delta
m^{2}_{kj}L}{2p^{0}}-f_{j}(p^{0})-f_{k}(p^{0})\right]\,,$ (3.8)

where we used that neutrino masses are small, $m_{j}\ll p^{0}$, and expand the
square root in the neutrino momenta as $p_{j}\approx
p^{0}-m_{j}^{2}/(2p^{0})$. We see that the time integration implies that only
neutrinos with the same energy can interfere and different neutrino energies
are summed incoherently [12, 14].

The interference terms correspond to $j\neq k$ in eq. 3.8, for which we
recognize the familiar oscillation phase depending on $\Delta m^{2}_{kj}\equiv
m_{k}^{2}-m_{j}^{2}$. Let us now simplify the discussion and specialize to the
case of two neutrino states with masses $m_{1},m_{2}$, and write

$\displaystyle m_{1}^{2}=\overline{m}^{2}-\frac{1}{2}\Delta m^{2}\,,\quad
m_{2}^{2}=\overline{m}^{2}+\frac{1}{2}\Delta m^{2}\,.$ (3.9)

We are interested only in terms proportional to $\Delta m^{2}$. Terms
depending only on the absolute neutrino mass $\overline{m}^{2}$ will lead to
(tiny) irrelevant global corrections which do not affect the interference
term. Furthermore, the quantities $m_{I}^{2}=E_{I}^{2}-p_{I}^{2}$ (for
$I=P,D$) are of order of the neutrino mass-squared $\overline{m}^{2}$ and also
independent of the neutrino mass indices $j,k$. Therefore, we neglect also
terms proportional to $m_{I}^{2}$ and set $E_{I}\approx p_{I}$. Using
$p_{j}\approx p^{0}-m_{j}^{2}/(2p^{0})$ also in the function $f_{j}(p^{0})$
and dropping all terms proportional to $\overline{m}^{2}$ and independent of
$\Delta m^{2}$, we obtain at leading order in $\Delta m^{2}$:

$\displaystyle
f_{1}(p^{0})+f_{2}(p^{0})=\sum_{I=P,D}\frac{(p^{0}-E_{I})^{2}}{2\sigma_{I,\rm
eff}^{2}}+\frac{1}{2}\left(\frac{\Delta
m^{2}}{4p^{0}\sigma_{m}}\right)^{2}+\sum_{I=P,D}\frac{\mathbf{p}_{I\perp}^{2}}{2\sigma_{pI}^{2}}$
(3.10)

with

$\displaystyle\frac{1}{\sigma_{I,\rm eff}^{2}}$
$\displaystyle\equiv\frac{1}{\sigma_{pI}^{2}}+\frac{(1-v_{I})^{2}}{\sigma_{EI}^{2}}\,,$
(3.11) $\displaystyle\frac{1}{\sigma_{m}^{2}}$
$\displaystyle\equiv\sum_{I=P,D}\left(\frac{1}{\sigma_{pI}^{2}}+\frac{v_{I}^{2}}{\sigma_{EI}^{2}}\right)\,.$
(3.12)

Already at this stage we obtain the term with $\sigma_{m}$ which potentially
can lead to decoherence. We will comment on its physical interpretation in
section 3.2.

Next we perform the integral over $p^{0}$. We expand the first term in eq.
3.10 around its minimum, which is at $p^{0}=E_{0}$ with

$\displaystyle E_{0}\equiv\sigma_{\rm
eff}^{2}\sum_{I=P,D}\frac{E_{I}}{\sigma_{I,\rm
eff}^{2}}\,,\quad\frac{1}{\sigma_{\rm
eff}^{2}}\equiv\sum_{I=P,D}\frac{1}{\sigma_{I,\rm eff}^{2}}\,.$ (3.13)

The oscillatory phase $i\Delta m^{2}L/(2p^{0})$ is expanded to first order in
$(p^{0}-E_{0})$, and we set $p^{0}=E_{0}$ in the term with $\sigma_{m}$
(ignoring higher order correction to this decoherence term). Then, the
$p^{0}$-integral can be performed with the method outlined in appendix A and
we obtain

$\displaystyle\overline{|\mathcal{A}_{\alpha\beta}|^{2}}\propto$
$\displaystyle\exp\left[i\frac{\Delta
m^{2}L}{2E_{0}}\right]\times\exp\left[-\sum_{I=P,D}\frac{\mathbf{p}_{I\perp}^{2}}{2\sigma_{pI}^{2}}\right]$
$\displaystyle\times\exp\left[-\frac{1}{2}\left(\frac{\Delta
m^{2}}{4E_{0}\sigma_{m}}\right)^{2}-\frac{1}{2}\left(\frac{\Delta
m^{2}L\sigma_{\rm
eff}}{2E_{0}^{2}}\right)^{2}-\frac{1}{2}\frac{(E_{D}-E_{P})^{2}}{\sigma_{P,\rm
eff}^{2}+\sigma_{D,\rm eff}^{2}}\right]$ (3.14)

This is a central result of the QFT approach to the oscillation process; let
us briefly comment on the terms appearing here. In the first line we have the
standard oscilation phase and a term depending on the net momentum components
orthogonal to the neutrino direction. The latter term will constrain the phase
space integrals for the orthogonal components within the mometum spreads
$\sigma_{pI}$ around zero. Because of the relabeling of the energy variables
$E_{P,D}$ in eq. 3.7, also the oscillation phase depends on
$\mathbf{p}_{I\perp}$ and in principle the integral over these components can
lead to additional non-trivial effects (we comment on it in appendix C).

Note that in eq. 3.14 we have three quantities corresponding to an effective
neutrino energy: the kinematic neutrino energies $E_{P}$ and $E_{D}$ at the
production and detection vertices defined in eqs. 3.2 and 3.3, respectively,
and $E_{0}$ which is a weighted mean value of the former two. The last term in
eq. 3.14 ensures, that all “three neutrino energies” are the same within
quantum mechanical uncertainties, determined by the sum of the effective
energy-momentum uncertainties at source and detector. Equation 3.14 is
completely symmetric with respect to source and detector.

###  3.2 Discussion of the decoherence terms

The first two terms in the second line of eq. 3.14 describe the exponential
damping due to decoherence. We provide here a brief review of the two
decoherence terms, see e.g., Refs. [12, 13, 47, 14] for more discussions. The
two terms correspond to two generic types of damping. We define

$\displaystyle\xi_{\rm loc}$
$\displaystyle=\exp\left[-\frac{1}{2}\left(\frac{\Delta
m^{2}}{4E_{\nu}\sigma_{m}}\right)^{2}\right]=\exp\left[-\frac{1}{2}\left(\frac{\Delta
m^{2}\delta_{\rm loc}}{2E_{\nu}}\right)^{2}\right]\,,$ (3.15)
$\displaystyle\xi_{\rm en}$
$\displaystyle=\exp\left[-\frac{1}{2}\left(\frac{\Delta
m^{2}L}{2E_{\nu}}\frac{\sigma_{\rm
en}}{E_{\nu}}\right)^{2}\right]=\exp\left[-\frac{1}{2}\left(\frac{\Delta
m^{2}L}{4E_{\nu}^{2}\delta_{\rm en}}\right)^{2}\right]\,,$ (3.16)

with

$\displaystyle\sigma_{m}\delta_{\rm loc}=\frac{1}{2}\,,\qquad\sigma_{\rm
en}\delta_{\rm en}=\frac{1}{2}\,.$ (3.17)

We introduced the generic energy spread $\sigma_{\rm en}$, where in eq. 3.14
we have $\sigma_{\rm en}=\sigma_{\rm eff}$, and $E_{\nu}$ is a relevant
neutrino energy, $E_{0}$ in the version obtained in eq. 3.14.

Both decoherence terms have a simple physical interpretation. Starting with
$\xi_{\rm loc}$, the interpretation in the energy-representation of this term
is that the energy-momentum uncertainty encoded in $\sigma_{m}$ needs to be
large enough, such that individual mass states can neither be resolved at the
production nor at the detection process. If energy-momentum was defined with
an accuracy better than $\Delta m^{2}/E_{\nu}$, the individual neutrino mass
states would be determined and no interference of different mass states would
be possible.

Another interpretation of this term becomes apparent in the spatial
representation: defining the oscillation length by

$\displaystyle L_{\rm osc}=2\pi\frac{2E_{\nu}}{\Delta m^{2}}$ (3.18)

the decoherence terms can be written as

$\displaystyle\xi_{\rm loc}$
$\displaystyle=\exp\left[-2\pi^{2}\left(\frac{\delta_{\rm loc}}{L_{\rm
osc}}\right)^{2}\right]\,,$ (3.19) $\displaystyle\xi_{\rm en}$
$\displaystyle=\exp\left[-2\pi^{2}\left(\frac{L}{L_{\rm osc}}\frac{\sigma_{\rm
en}}{E_{\nu}}\right)^{2}\right]\,.$ (3.20)

In this form we see that $\xi_{\rm loc}$ can also be interpreted as the
condition $\delta_{\rm loc}\ll L_{\rm osc}$, i.e., that both, source and
detection points need to be localized better than the oscillation length, by
noting that $\delta_{\rm loc}^{2}=\delta_{P}^{2}+\delta_{D}^{2}$.

Moving now to $\xi_{\rm en}$, this term says that for an experiment around the
oscillation maximum ($L/L_{\rm osc}\simeq 1$), the neutrino energy needs to be
sufficiently determined, such that $\sigma_{\rm en}\ll E_{\nu}$; for
experiments beyond the oscillation maximum the condition becomes
correspondingly stronger. Using $\sigma_{\rm en}=\sigma_{\rm eff}$ with
$\sigma_{\rm eff}$ defined in eqs. 3.11 and 3.13, this is a condition on the
energy and momentum spreads of external particles of the neutrino production
and detection processes. For $L\simeq L_{\rm osc}$, decoherence will become
relevant only if $\sigma_{\rm en}\simeq E_{\nu}$, i.e., the quantum mechanical
uncertainty on the neutrino energy needs to become comparable to the neutrino
energy itself, implying that the neutrino would not have a well defined
energy. Note that we work under the assumption that momentum spreads are small
compared to relevant momenta or energies, and therefore our approximations
adopted in section 2 to perform the integrals may not apply if $\sigma_{\rm
en}\sim E_{\nu}$.

In the above derivation we have integrated first over $T$ and then over
$p^{0}$, which is more convenient to derive decoherence terms. If the order of
the $T$ and $p^{0}$ integrals are exchanged, one can see that an exponential
factor makes sure that the $T$ integral is dominated by values of $T$
constrained by $|T-L-Lm_{j}^{2}/(2E_{0}^{2})|\lesssim 1/\sigma_{\rm eff}$,
which can be interpreted as relating $L\simeq Tv_{j}$, with $v_{j}\approx
1-m_{j}^{2}/(2E_{0}^{2})$ corresponding to the “velocity” of the neutrino with
mass $m_{j}$ [12, 48]. This suggests a wave packet interpretation of the
internal neutrino in the QFT approach [14]. In that picture the damping due to
$\xi_{\rm en}$ can be interpreted as wave packet separation of the propagating
neutrinos.

We note that the two terms $\xi_{\rm loc}$ and $\xi_{\rm en}$ have the
opposite dependence on the spreads $\sigma_{m}$ and $\sigma_{\rm en}$. This
means that for oscillations to be observable, quantum uncertainties have to be
big enough that different mass states can interfere ($\sigma_{m}\gg\Delta
m^{2}/E_{\nu}$), but small enough that interference is not damped
($\sigma_{\rm en}\ll E_{\nu}L_{\rm osc}/L$). Assuming that very roughly
$\sigma_{m}\sim\sigma_{\rm en}$, we see that there are many orders of
magnitude available to fulfill both requirements, thanks to the smallness of
$\Delta m^{2}/E_{\nu}^{2}$ or, in other words, due to the macroscopically
large oscillation length, $L_{\rm osc}E_{\nu}\gg 1$.

###  3.3 Phase-space integrals over unobserved external momenta

Equation 3.14 contains the intrinsic quantum mechanical decoherence terms. So
far we have only averaged the amplitude squared over the unobservable time $T$
(which actually does not introduce a decoherence term). All other
manipulations are performed at the amplitude level; in particular, the
integral over $p^{0}$ corresponds to an internal particle, which can be
performed already at amplitude level. In real experiments there are of course
always effects leading to additional averaging at the probability level, i.e.,
of the amplitude squared (see also section 3.4). Some of these averages are
intrinsically unavoidable and related to the physical configuration of the
experiment.

Let us consider first the case of reactor neutrino oscillation experiments; a
very similar discussion applies also to accelerator or atmospheric neutrino
experiments. In these experiments usually the neutrino energy is reconstructed
in the detector, by measuring all (or some of) the outgoing particles at the
detector, i.e., particles 3 and 4 in our example, which allows to reconstruct
$E_{D}$ via eqs. 3.3 and 3.7 with some accuracy. In contrast, inital and final
state particles at the production point usually are not observed.555We do not
consider here so-called monitored neutrino beams [49], which (at least in
principle) would allow also to reconstruct $E_{P}$. Therefore, in order to
calculate event rates $R_{D}$ in the detector, we need to integrate the
squared amplitude over the phase space of final state particles in the
production reaction, see also section B.3. By a suitable variable
transformation, one of these integrals can be chosen to be over $E_{P}$. As we
sketch in appendix C, the decoherence term emerging from this integral has the
same shape as $\xi_{\rm en}$ and can be combined with the original term
present already in eq. 3.14 such that we obtain for the event rate in the
detector

$\displaystyle R_{D}(L,E_{D})\propto\int
dE_{P}\overline{|\mathcal{A}_{\alpha\beta}|^{2}}$
$\displaystyle\propto\exp\left[i\frac{\Delta
m^{2}L}{2E_{D}}-\frac{1}{2}\left(\frac{\Delta
m^{2}}{4E_{D}\sigma_{m}}\right)^{2}-\frac{1}{2}\left(\frac{\Delta
m^{2}L\sigma_{D,\rm eff}}{2E_{D}^{2}}\right)^{2}\right]\,.$ (3.21)

Hence, the energy spread in $\xi_{\rm en}$ is given by $\sigma_{\rm
en}=\sigma_{D,\rm eff}$, i.e., it depends only on the uncertainties in the
detector, while $\sigma_{m}$ in $\xi_{\rm loc}$ remains unchanged and contains
contributions from both, the production and detection process, see eq. 3.12.

The integral over $E_{P}$ corresponds to a _classical_ sum, i.e., summing the
squared amplitude. The corresponding decoherence can therefore be considered
as emerging from classical averaging. Note, however, here this averaging is in
principle unavoidable, given the physical observables in a neutrino
oscillation experiment. In this way, eq. 3.21 depends only on the “observable
neutrino energy” $E_{D}$. Also note that $\sigma_{D,\rm eff}\geq\sigma_{\rm
eff}$. Therefore, the classical averaging due to the $E_{P}$ integration
increases effectively the decoherence, see section 3.4. The fact that only the
detection process determines the decoherence in the last term of eq. 3.21 and
all production-related uncertainties drop out follows from expanding the
oscillation phase around $E_{D}$ and from the Gaussianity of all involved
uncertainties; this result may not hold in the most general case. Note that we
keep only leading terms in the expansion parameters $\sigma/E$ with
$\sigma\in(\sigma_{\rm eff},\sigma_{P,\rm eff},\sigma_{D,\rm eff})$ and
$E\in(E_{0},E_{P},E_{D})$, and differences like $\sigma_{D,\rm
eff}/E_{D}-\sigma_{P,\rm eff}/E_{P}$ are of higher order in this expansion.
However, our result that the relevant energy spread in the $\xi_{\rm en}$ term
of eq. 3.21 is larger than $\sigma_{\rm eff}$ as in eq. 3.14 is robust.

In typical detectors of modern reactor experiments only the energy of the
outgoing positron can be determined, whereas the neutron momentum is not
measured. Hence, we need an additional phase-space integral over the neutron
momentum. This will provide another contribution to $\sigma_{\rm en}$, which
however, is at most of the same order of magnitude and typically much smaller
(see the discussion of the $\mathbf{p}_{\perp}$ integral in appendix C).

Moving to Gallium radioactive source experiments, these are pure counting
experiments, i.e., the detector does not provide energy information, whereas
the neutrino source consists of a couple of quasi-monochromatic neutrino lines
from an electron-capture decay $N\to N^{\prime}+\nu$. In this case one would
naturally integrate the phase space over $E_{D}$ instead of $E_{P}$. The
calculation is completely symmetric to the one outlined above and we obtain
the same result as in eq. 3.21 with the replacements $E_{D}\to E_{P}$ and
$\sigma_{D,\rm eff}\to\sigma_{P,\rm eff}$. Hence, in this case the energy
spread in $\xi_{\rm en}$ is set by the production process and will be
determined by the localization of the nuclei $N,N^{\prime}$ as well as the
(natural and/or thermal) linewidth of the decay (see section 4). The same
comments as above apply regarding the dropping out of $\sigma_{D,\rm eff}$.

###  3.4 Classical averaging

In realistic experiments both production point and detection point are known
only within some uncertainty, related to the size of the neutrino source and
the vertex resolution of the detector. Similarly, detectors can determine
particle energy and momentum only within certain resolutions, and hence the
neutrino energy $E_{D}$ can be reconstructed only within a finite accuracy.
These effects are taken into account in predicted event rates by convoluting
the neutrino oscillation probability with the corresponding resolution
functions. Hence, a classical average is performed.

Let us approximate these spatial and energy resolutions by Gaussians with
widths $\delta_{\rm clas}$ and $\sigma_{\rm clas}$, correspondingly:

$\displaystyle\int
dL^{\prime}\,R_{D}(L^{\prime},E_{\nu})\frac{1}{\sqrt{2\pi}\delta_{\rm
clas}}\exp\left[-\frac{(L^{\prime}-L)^{2}}{2\delta_{\rm clas}^{2}}\right]\,,$
(3.22) $\displaystyle\int
dE^{\prime}_{D}\,R_{D}(L,E^{\prime}_{D})\frac{1}{\sqrt{2\pi}\sigma_{\rm
clas}}\exp\left[-\frac{(E^{\prime}_{D}-E_{D})^{2}}{2\sigma_{\rm
clas}^{2}}\right]\,,$ (3.23)

with $R_{D}(L,E_{D})$ given in eq. 3.21. Assuming that the width of the
Gaussians is small compared to the other $L$ or $E_{D}$ dependent factors in
$R_{D}$, we can expand the oscillation phase to linear order either in
$(L^{\prime}-L)$ or $(E^{\prime}_{D}-E_{D})$ and see that the integral takes
again the same form as in appendix A. Applying eq. A.4, we obtain decoherence
terms of the same form as $\xi_{\rm loc}$ for the $L$ smearing and $\xi_{\rm
en}$ for the energy resolution. Hence, they can be combined with the
corresponding terms present in eq. 3.21, which amounts to the replacement

$\displaystyle\delta_{\rm loc}^{2}$ $\displaystyle\to\delta_{\rm
loc}^{2}+\delta^{2}_{\rm clas}\,,\qquad\sigma_{D,\rm eff}^{2}\to\sigma_{D,\rm
eff}^{2}+\sigma^{2}_{\rm clas}\,.$ (3.24)

Hence, decoherence due to classical averaging has precisely the same effect as
intrinsical quantum mechanical decoherence [5, 9, 50].666This statement is
consistent with the comments related to the integration over $E_{P}$ after eq.
3.21. These results reflect the following two (rather obvious) statements:
($i$) quantum mechanical uncertainties provide a fundamental lower bound on
classical uncertainties, and ($ii$) in order to observe effects of quantum
mechanical decoherence, classical averaging effects have to be suppressed down
to the quantum level.

##  4 Numerical estimates

The QFT formalism outlined above allows to calculate the relevant
uncertainties $\delta_{\rm loc}$ (or equivalently $\sigma_{m}$) and
$\sigma_{\rm en}$ relevant for the localization and energy spread decoherence
factors $\xi_{\rm loc}$ and $\xi_{\rm en}$, respectively, from the properties
of the involved external particles. The required input for their definitions
in eqs. 3.12 and 3.11 are the effective energy and momentum uncertainties eqs.
2.14 and 2.23, which in turn are derived from the momentum spreads of all the
external particles in the production and detection processes, as well as their
velocities, as defined in eq. 2.10. We will now evaluate $\delta_{\rm loc}$
and $\sigma_{\rm en}$ for reactor neutrino and Gallium radioactive source
experiments.

###  4.1 Particle localizations and velocities

First we need to estimate the momentum spreads $\sigma$ of all involved
particles, as well as their velocities $v$. Similar estimates have been
performed recently in [27] in the context of neutrino wave packets. The
momentum spread is calculated via the spatial localization $\delta_{x}$,
assuming the uncertainty principle $\delta_{x}\sigma=1/2$. We list the
relevant quantities for all the particles involved in the production and
detection processes in reactor and Gallium experiments in table 1. They are
estimated as follows.

| Particle |  $\delta_{x}$ [nm] |  $\sigma$ [eV] | $v$
---|---|---|---|---
Reactor ($P$) | $N$ | 0.24 | 410 | $1\times 10^{-6}$
$N\to N^{\prime}+e^{-}+\overline{\nu}_{e}$ | $N^{\prime}$ | 0.24 | 410 | $4\times 10^{-5}$
| $e^{-}$ | 260 | 0.38 | 0.99
Reactor ($D$) | $p$ | 0.1 | 990 | $5\times 10^{-6}$
$p+\overline{\nu}_{e}\to n+e^{+}$ | $n$ | $5\times 10^{6}$ | $2\times 10^{-5}$ | $5\times 10^{-3}$
| $e^{+}$ | 320 | 0.3 | 0.99
Gallium ($P$) | Cr | 0.20 | 480 | $7\text{\times}{10}^{-7}$
Cr $\to$ V $+\nu_{e}$ | V | 0.20 | 480 | $2\text{\times}{10}^{-5}$
Gallium ($D$) | Ga | 0.27 | 370 | $6\text{\times}{10}^{-7}$
Ga $+\nu_{e}\to$ Ge $+e^{-}$ | Ge | 0.27 | 370 | $1\text{\times}{10}^{-5}$
| $e^{-}$ | 310 | 0.32 | 0.83
Table 1: Spatial localization $\delta_{x}$, momentum spread
$\sigma=1/(2\delta_{x})$, and velocity $v$ of the external particles involved
in the production ($P$) and detection ($D$) processes (first column) of
reactor and Gallium source experiments.

#### Reactor experiments.

For the initial and final state nuclei $N$, $N^{\prime}$ in the production
process, via beta decay within the nuclear fuel, we assume that the
localization is determined by a typical interatomic distance [47]. We estimate
this by using that the lattice parameter of uranium oxide UO2 is
$a=5.471\times 10^{-10}$ m and has 4 U and 8 O atoms in one unit cell [51],
which gives $\delta_{x}\simeq a/12^{1/3}\simeq 0.24$ nm. For the initial state
nucleus we assume a thermal velocity $v=\sqrt{k_{B}T/m}$, where the
temperature in the nuclear fuel ranges from 700 K at the outer egde to 2000 K
in the center [52]. This is justified, as the fission products termalize on
time scales much faster than their beta decay lifetimes [27]. For nuclei with
mass numbers in the range of 80 to 160, the velocity of fission products lies
in the range of $6.3\times 10^{-7}$ to $1.5\times 10^{-6}$. For our estimates
a typical value of $v_{N}\simeq 10^{-6}$ is taken. The recoiling nucleus after
beta decay, $N^{\prime}$, is not in thermal equilibrium and we estimate its
velocity using $v=|\mathbf{p}|/E$. Assuming typical neutrino energies $E\simeq
4$ MeV and mass numbers from 80 to 160 we find $v_{N^{\prime}}\simeq 4\times
10^{-5}$.

The protons in the detector are typically bound in carbon molecules. We assume
a localization of $10^{-10}$ m corresponding to the typical size of the C–H
bound length. For the velocity we take thermal velocities at room temperature.
Concerning the neutron, after being produced it undergoes scattering in the
liquid scintilator. The spatial localization is therefore estimated by the
mean collision length $l=1/n\sigma_{\rm int}$, where we use the neutron on CH2
scattering cross section from [53]. For a typical kinetic neutron energy of
$E_{\rm kin}\simeq 10$ keV a cross section of $\sigma_{\rm int}\approx 50$ b
leads to a mean collision length of 4.8 mm. Here, the number density of CH2
was estimated as $n\simeq 1/(3d^{3})$ for an interatomic distance $d\simeq 20$
nm. The velocity is calculated for a neutron with $E_{\rm kin}\simeq 10$ keV.

For the outgoing electron and positron passing through the medium, either in
the nuclear fuel or in the detector, we proceed as follows. We consider the
mean rate of energy loss $\langle-dE/dx\rangle$ using the “Bethe equation”
[54], which we numerically integrate. As localization we take the distance
which the particle travels until it deposits one mean excitation energy $I$.
This should provide us with a good estimate for the mean free path of the
particle.

For the calculation of $\langle-dE/dx\rangle$ we need to know the number
density of elementary charges, which can be estimated from the density and
composition of the stopping medium. For the reactor fuel we use the above
mentioned properties of UO2. As a typical detector material we assume linear
alkylbenzene (LAB), $\text{C}_{6}\text{H}_{5}\text{C}_{n}\text{H}_{2n+1}$,
with $n$ ranging from 10 to 13; for definiteness we will assume $n=12$. We
calculate the number density of elementary charges as
$138N_{A}\rho/m_{\text{mol}}$, where 138 is the number of elementary
charges/electrons in one molecule of LAB, and the density of LAB is taken to
be $\rho=$0.859\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$$ [55].
Furthermore, we need the mean excitation energies $I$. As we consider
materials made up of different elements, the calculated mean excitation
energies were averaged over the contributions from the different atoms. For
atoms with high elementary charge $Z$ a good approximation is
$I=$10\text{\,}\mathrm{eV}$\cdot Z$ [56]. Thus, at the source, this is used
for $\text{UO}_{2}$, where $Z=36$ on average and thus
$I=$360\text{\,}\mathrm{eV}$$. At the detector we have LAB which we simplify
to $\text{CH}_{2}$. Then, the average excitation energy of one molecule can be
approximated to be $I\approx$48\text{\,}\mathrm{eV}$$ [56]. With these numbers
we find that an electron (positron) with an initial kinetic energy of 3 MeV
deposits one $I$ energy after traveling 260 nm (320 nm), which we take as the
localization at the neutrino production (detection).

#### Gallium source experiments.

To estimate the particle localizations at Gallium experiments we take the BEST
experiment as an example [24, 25]. At the source 51Cr undergoes electron
capture to become 51V and an electron neutrino. Approximately
$90\text{\,}\mathrm{\char 37\relax}$ of the produced neutrinos have an energy
of $750\text{\,}\mathrm{keV}$ while $10\text{\,}\mathrm{\char 37\relax}$ have
energies of $430\text{\,}\mathrm{keV}$. For the sake of defineteness we
consider the $750\text{\,}\mathrm{keV}$ neutrinos. The temperature of the
source is approximately the one of the surrounding gallium, namely
$300\text{\,}\mathrm{K}$ [24, 25]. Therefore, the thermal velocity of 51Cr
(compound of nucleus + electron to be captured) is
$v_{\text{Cr}}=\sqrt{k_{B}T/m}=$7.3\text{\times}{10}^{-7}$$. The final 51V is
not thermal, instead its velocity is calculated as
$v_{\text{V}}=p_{\text{V}}/E_{\text{V}}=$1.5\text{\times}{10}^{-5}$$. The
localizations of the 51Cr and 51V are approximately the same and estimated
from the cristal lattice dimensions. They are bound in a bcc lattice with 2
atoms in one unit cell of size
$a=$2.88\text{\times}{10}^{-10}\text{\,}\mathrm{m}$$ [57]. This leads to a
spatial localization of $\delta_{\text{Cr,V}}\approx
a/\sqrt{2}=$0.20\text{\,}\mathrm{nm}$$.

At the detection we estimate the thermal velocity of Ga at
$300\text{\,}\mathrm{K}$ to
$v_{\text{Ga}}=\sqrt{k_{B}T/m_{\text{Ga}}}=$6.3\text{\times}{10}^{-7}$$. The
velocity of Ge is not thermal, instead
$v_{\text{Ge}}=p_{\text{Ge}}/E_{\text{Ge}}=$1.15\text{\times}{10}^{-5}$$. For
the electron we obtain $v_{e^{-}}=p_{e-}/E_{e-}=$0.83$$. All momenta are
estimated to be of the order of $E_{\nu}=$750\text{\,}\mathrm{keV}$$. With the
density of gallium
$\rho\approx$6\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$$, we obtain the
number density as $n=\rho N_{A}/m_{\text{mol}}$ and
$n^{-1/3}=$2.67\text{\times}{10}^{-10}\text{\,}\mathrm{m}$$, which we take as
the localization for both the Ga and Ge atoms. For the localization of the
electron we proceed as described for the reactor case and integrate $\langle-
dE/dx\rangle$ until the electron deposites one mean excitation energy, which
for Ga is $I=$11\text{\,}\mathrm{eV}$\cdot Z=$341\text{\,}\mathrm{eV}$$ [56].
This leads to $\delta_{e}\approx 310$ nm.

Our estimates for the particles involved in the production processes reported
in table 1 are in agreement with the results of [27] within about one order of
magnitude. Ref. [27] does not provide detailed estimates for the detector
particles.

###  4.2 Effective energy-momentum spreads and decoherence parameters

Now we are in the position to calculate the relevant effective uncertainies.
For convenience we summarize again the corresponding relations, as derived in
sections 2 and 3. Here, $I=P,D$ labels production and detection processes, and
the sum is over all external particles $a$ (initial and final state) of the
respective processes:

$\displaystyle\mathbf{v}_{I}\equiv\frac{1}{\sigma_{pI}^{2}}\sum_{a\in
I}\sigma_{a}^{2}\mathbf{v}_{a}\,,\qquad\Sigma_{I}\equiv\frac{1}{\sigma_{pI}^{2}}\sum_{a\in
I}\sigma_{a}^{2}\mathbf{v}_{a}^{2}\,,$ (4.1)
$\displaystyle\sigma_{pI}^{2}\equiv\sum_{a\in
I}\sigma^{2}_{a}\,,\qquad\qquad\sigma_{eI}^{2}\equiv\sigma_{pI}^{2}(\Sigma_{I}-\mathbf{v}_{I}^{2})\,,$
(4.2) $\displaystyle\Gamma_{I,\rm col}^{2}\equiv\sum_{a\in
I}\sigma_{a}^{2}\mathbf{v}_{a}^{2}=\sigma_{pI}^{2}\Sigma_{I}\,,$ (4.3)
$\displaystyle\sigma_{EI}^{2}\equiv\sigma_{eI}^{2}+\Gamma_{I,\rm
col}^{2}+\Gamma_{I,\rm
dec}^{2}=\sigma_{pI}^{2}(2\Sigma_{I}-\mathbf{v}^{2}_{I})+\Gamma_{I,\rm
dec}^{2}\,.$ (4.4)

| Reactor | Gallium
---|---|---
| $I=P$ | $I=D$ | $I=P$ | $I=D$
$\tilde{v}_{I}$ | $2.1\times 10^{-5}$ | $5.3\times 10^{-6}$ | $8.2\text{\times}{10}^{-6}$ | $6.4\text{\times}{10}^{-6}$
$\sqrt{\Sigma_{I}}$ | $6.4\times 10^{-4}$ | $3.1\times 10^{-4}$ | $1.1\text{\times}{10}^{-5}$ | $5.1\text{\times}{10}^{-4}$
$\sigma_{pI}$ | 580 eV | 990 eV | 680 eV | 520 eV
$\sigma_{eI}$ | 0.37 eV | 0.30 eV | 0.005 eV | 0.27 eV
$\Gamma_{I,\rm col}$ | 0.37 eV | 0.30 eV | 0.0076 eV | 0.27 eV
$\sigma_{EI}\approx\sigma_{I,\rm eff}$ | 0.53 eV | 0.43 eV | 0.0092 eV | 0.375 eV
$\sigma_{m}$ | 500 eV | 390 eV
$\delta_{\rm loc}$ | 0.20 nm | 0.25 nm
$\sigma_{\rm eff}$ | 0.33 eV | 0.0092 eV
$\sigma_{\rm en}$ | 0.43 eV | 0.5 eV
$\delta_{\rm en}$ | 230 nm | 200 nm
Table 2: Effective velocities and energy-momentum spreads at neutrino
production ($P$) and detection ($D$) for reactor and Gallium source
experiments. In the lower part we give the spreads relevant for the energy and
localization decoherence terms $\xi_{\rm en}$ and $\xi_{\rm loc}$. For
convenience we give values for $\sigma$’s in eV and $\delta$’s in nm; in
natural units they are related by $\sigma_{\rm en}\delta_{\rm en}=1/2$ and
$\sigma_{m}\delta_{\rm loc}=1/2$.

The upper part of table 2 shows our results for these quantities based on the
input from table 1. Momentum spreads $\sigma_{pI}$ are dominated in all cases
by the localization of hadronic particles and are of order few 100 eV to keV.
Some comments are in order concerning the velocity. The velocity vector
$\mathbf{v}_{I}$ defined in eq. 4.1 depends on the relative orientation of the
individual particle velocities $\mathbf{v}_{a}$. The quantity $\tilde{v}_{I}$
given in table 2 is calculated by

$\displaystyle\tilde{v}_{I}\equiv\frac{1}{\sigma_{pI}^{2}}\sum_{a\in
I}\sigma_{a}^{2}|\mathbf{v}_{a}|\geq|\mathbf{v}_{I}|\,.$ (4.5)

Hence, we obtain an upper bound on $|\mathbf{v}_{I}|$. From the table we see
that effective velocities and velocity spreads are all $\ll 1$, which implies
$\sigma_{eI}\ll\sigma_{pI}$, and the $\sigma_{eI}$ are approximately given by
the momentum spreads of the final state leptons $e^{\pm}$, except for the
electron capture production (which has no final state charged lepton).
Furthermore, $|\mathbf{v}_{I}|\leq\tilde{v}_{I}\ll\sqrt{\Sigma_{I}}$ implies
that $\sigma_{eI}\approx\Gamma_{I,\rm col}$. Again, an exception is the
production in Gallium experiments, where the approximation
$\tilde{v}_{I}\ll\sqrt{\Sigma_{I}}$ is not very good; nevertheless,
$\Gamma_{P,\rm col}$ is still of the same order as $\sigma_{eP}$ also in this
case. Finally, for typical lifetimes for fission products of order 1 ms or
larger, decay widths are at most of order $\Gamma_{P,\rm dec}\sim 10^{-12}$
eV, and the decay width of 51Cr is $\Gamma_{\rm Cr,dec}\approx 4\times
10^{-22}$ eV. Hence, the decay widths are always completely negligible
compared to $\sigma_{eI}$ and $\Gamma_{I,\rm col}$ and can be savely neglected
in eq. 4.4 and we have
$\sigma_{EI}\approx\sqrt{2\Sigma_{I}}\sigma_{pI}\approx\sqrt{2}\sigma_{eI}$.
Therefore, also $\sigma_{EP}$ and $\sigma_{ED}$ are determined by the momentum
spread (or by the localization) of the outgoing charged lepton (again with the
exception of the electron-capture source).

Now we can move to the calculation of the effective uncertainties summarized
in the lower part of table 2. We start with $\sigma_{m}$, defined as

$\displaystyle\frac{1}{\sigma_{m}^{2}}\equiv\sum_{I=P,D}\left(\frac{1}{\sigma_{pI}^{2}}+\frac{v_{I}^{2}}{\sigma_{EI}^{2}}\right)\,.$
(4.6)

Here, $v_{I}$ is actually the length of the component of $\mathbf{v}_{I}$
parallel to the neutrino propagation direction. The small values of
$v_{I}\leq\tilde{v}_{I}$ imply that for reactors, $\sigma_{m}$ is dominated by
$\sigma_{pI}$, whereas in Gallium the small value of $\tilde{v}_{P}$ is
partially compensated by the small value of $\sigma_{EP}$, and the energy term
gives a non-negligible contribution to $\sigma_{m}$, although the order of
magnitude remains unchanged and $\sigma_{m}\sim\sigma_{pI}$. We find for both
type of experiments values of order

$\displaystyle\sigma_{m}\simeq(400-500)\,{\rm eV}\,,\qquad\delta_{\rm
loc}=\frac{1}{2\sigma_{m}}\simeq 0.2\,{\rm nm}\,,$ (4.7)

For the localization term $\xi_{\rm loc}$ from eq. 3.15 this implies

$\displaystyle-\ln\xi_{\rm loc}=\frac{1}{2}\left(\frac{\Delta
m^{2}}{4E_{\nu}\sigma_{m}}\right)^{2}\approx 1.3\times
10^{-19}\left(\frac{\Delta m^{2}}{1\,\rm eV^{2}}\right)^{2}\left(\frac{1\,\rm
MeV}{E_{\nu}}\right)^{2}\left(\frac{500\,\rm eV}{\sigma_{m}}\right)^{2}\,.$
(4.8)

Hence, $\xi_{\rm loc}=1$ for all practical purposes and localization
decoherence is irrelevant in reactor and Gallium experiments. We emphasize,
however, that classical spatial averaging in real experiments can be
significant, with $\delta_{\rm clas}\sim 1$ m (corresponding to typical sizes
of reactor cores or Gallium detectors), which, depending on the value of
$\Delta m^{2}$, does play an important role and has to be included in the
analyis of these types of experiments.

Moving to the energy spread, we note that

$\displaystyle\frac{1}{\sigma_{I,\rm
eff}^{2}}\equiv\frac{1}{\sigma_{pI}^{2}}+\frac{(1-v_{I})^{2}}{\sigma_{EI}^{2}}\approx\frac{1}{\sigma_{EI}^{2}}\,.$
(4.9)

In the last relation in eq. 4.9 we have used that $v_{I}\leq\tilde{v}_{I}\ll
1$ and $\sigma_{EI}\ll\sigma_{pI}$ to obtain $\sigma_{I,\rm
eff}\approx\sigma_{EI}$. Hence, for the effective energy spread we obtain

$\displaystyle\sigma_{\rm eff}\equiv\left[\sum_{I=P,D}\frac{1}{\sigma_{I,\rm
eff}^{2}}\right]^{-1/2}\approx\left[\sum_{I=P,D}\frac{1}{\sigma_{EI}^{2}}\right]^{-1/2}\approx\left\\{\begin{array}[]{l@{\quad}l}0.33\,{\rm
eV}&\text{(reactor)}\,,\\\ 0.0092\,{\rm eV}&\text{(Gallium)}\,,\\\
\end{array}\right.$ (4.12)

dominated by the outgoing $e^{\pm}$ momentum spreads for reactors and by the
electron capture decay for Gallium experiments, the latter leading to a value
of $\sigma_{\rm eff}$ more than one order of magnitude smaller.

As discussed above, $\sigma_{\rm eff}$ corresponds to the “pure quantum
mechanical” energy spread, c.f., eq. 3.14. However, even in idealized
experimental configurations there is some un-avoidable averaging, e.g., due to
phase-space integrals over unobserved momenta. Therefore, as argued in section
3.3, the relevant energy spread in reactor experiments is $\sigma_{D,\rm
eff}$, due to the averaging of unobserved momenta at the production region.
This does not change the qualitative picture, as in our approximation they are
of the same order of magnitude. Numerically we find for the spread relevant
for $\xi_{\rm en}$ the values:

$\displaystyle\sigma_{\rm en}\approx\sigma_{D,\rm eff}\approx 0.43\,{\rm
eV}\,,\qquad\delta_{\rm en}\approx 230\,{\rm nm}\qquad\text{(reactor)}\,.$
(4.13)

For the Gallium source experiments, it is more natural to integrate first over
the phase-space of the detector particles, as no momenta are measured in the
detector. This would imply $\sigma_{P,\rm eff}$ as the relevant energy spread.
Here the effect of the phase-space integration is even less important, as we
anyway have $\sigma_{P,\rm eff}\approx\sigma_{\rm eff}\approx 0.0092$ eV or
$\delta_{\rm en}\approx$11\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. However,
in this case another fundamental (though classical) averaging effect needs to
be taken into account, namely the Doppler broadening due to the thermal
motions of the source particles [47, 27]. This leads to an energy smearing
with Gaussian shape, with the width set by

$\displaystyle\sigma_{\rm Doppler}\simeq v_{\rm Cr}E_{\nu}\approx 0.5\,\rm
eV\,.$ (4.14)

We can include this effect in a straight forward way, following the discussion
in section 3.4. We see that $\sigma_{\rm Doppler}\gg\sigma_{P,\rm eff}$ and
therefore it dominates the energy spread. Using eq. 3.24 with $\sigma_{\rm
clas}=\sigma_{\rm Doppler}$, we obtain

$\displaystyle\sigma_{\rm en}\approx\sigma_{\rm Doppler}\approx 0.5\,{\rm
eV}\,,\qquad\delta_{\rm en}\approx 200\,{\rm nm}\qquad\text{(Gallium)}\,,$
(4.15)

quite similar to the values obtained for reactor experiments in eq. 4.13.

Note that Doppler broadening is in principle relevant for reactor experiments
as well. In the neutrino source it would, however, not induce additional
decoherence, as we already integrate over the effective neutrino energy in the
source and the additional smearing due to the Doppler broadening would have no
effect. However, it does contribute to the energy resolution of the detector.
Here, we consider it as part of the classical energy resolution and do not
include it in the spreads given in eq. 4.13.

For the energy spread decoherence $\xi_{\rm en}$ written as in eq. 3.20, we
find

$\displaystyle-\ln\xi_{\rm en}=2\pi^{2}\left(\frac{L}{L_{\rm
osc}}\frac{\sigma_{\rm en}}{E_{\nu}}\right)^{2}\approx 4.9\times
10^{-12}\,\left(\frac{L}{L_{\rm osc}}\right)^{2}\left(\frac{1\,\rm
MeV}{E_{\nu}}\right)^{2}\left(\frac{\sigma_{\rm en}}{0.5\,\rm
eV}\right)^{2}\,,$ (4.16)

which again implies $\xi_{\rm en}=1$, both for reactor and Gallium experiments
to very good accuracy.

The energy resolution of typical detectors in reactor experiments is in the
range $(0.03-0.06)\,{\rm MeV}\sqrt{E/{\rm MeV}}$ [58, 20, 55] and hence,
$\sigma_{\rm clas}\simeq 0.1$ MeV, about 6 orders of magnitude larger as the
intrinsic energy spread eq. 4.13. Equation 4.16 shows that for these values,
$-\ln\xi_{\rm en}$ can become of order one, which just means that the
(classical) averaging due to the energy resolution of neutrino detectors is an
important effect. On the other hand consider the upcoming high precision JUNO
reactor experiment [55], which aims to observe oscillations due to the mass
squared difference $\Delta m^{2}_{31}\simeq 2.5\times 10^{-3}$ eV2 at a
distance of about 53 km. This implies $L/L_{\rm osc}\sim 13$ and $-\ln\xi_{\rm
en}\sim 5\times 10^{-11}$ for $E_{\nu}\simeq 4$ MeV. Therefore, quantum
decoherence effects will remain completely unobservable also for JUNO.

#### Comparison with previous results.

Our results of the uncertainties in eqs. 4.13 and 4.15 are in rough agreement
with [27], but differ by a factor of $10^{3}-10^{4}$ from [43] where
$\delta_{\rm en}\simeq 10-400$ pm is obtained. The authors of Ref. [43]
consider localization scales induced by nucleon-nucleon interactions within
the nucleus as well as the size of the decaying nucleon. However, we argue
that dynamics at length scales much smaller than $1/|q|$, with $q$ being the
typical momentum transfer of the reaction, are irrelevant to the problem, as
localizations smaller than $1/|q|$ cannot be resolved. For the same reason
scattering of neutrinos with MeV energies can be considered with the nucleus
as a whole, and not on individual nucleons (unless they are effectivley
“free”, as the proton in Hydrogen), or even quarks. Therefore, it is the
localization of the nuclues, which is relevant for defining the quantum
mechanical uncertainties for the process. See also Ref. [29].

The authors of Refs. [35, 59] consider a term similar to $\xi_{\rm en}$ and
perform a phenomenological analysis using reactor data to set a lower bound on
a “coherence length”, finding $\delta>2.1\times 10^{-4}$ nm, which corresponds
to $\sigma=1/(2\delta)<0.47$ MeV. This value has also been adopted by [26] to
study the impact on sterile neutrino oscillations in short-baseline
experiments. This value of the decoherence parameter is of similar order as
the energy resolution of the detectors, 5–6 orders of magnitude larger than
the intrinsic quantum uncertainty, and therefore fully dominated by the
classical averaging effect [27].

##  5 Summary

We have used the QFT approach to neutrino oscillations to estimate quantum
decoherence effects for reactor neutrino and Gallium source experiments. In
this formalism possible decoherence effects on the oscillation probability are
fully determined by specifying the momentum spreads (or equivalently the
localizations) of the external particles in neutrino production and detection
processes. The neutrino is treated as an internal particle, which is
integrated out. It is not necessary to introduce the concept of neutrino wave
packets, however, the results can also naturally be interpreted in terms of
neutrino wave packets [14]. Throughout our calculations we adopt the
approximation of a Gaussian shape for all functions describing momentum and
energy spread.

In section 2 and section 3 we have reviewed the QFT formalism and derived the
coherence factors in terms of external particle localizations and velocities.
We introduced the concept of coherence loss due to frequent interactions with
the environment, and we find that for the cases of interest this effect is of
a similar size as the finite interaction time due to wave packet overlap of
the particles involved in the production and detection processes. Furthermore,
in our approach we focus on experimentally observable quantities, such as the
positron energy after inverse beta decay in the detector, which is used to
reconstruct the neutrino energy. We show that phase-space integrals over
unobserved particle momenta lead to additional decoherence effects, which
correspond to classical averaging of the amplitude squared, but nevertheless
are intrinsic to the experimental configuration and in principle unavoidable.
Finally, we recover the well-known result that quantum and classical
decoherence effects are experimentally indistinguishable, and
phenomenologically this distinction is unphysical. Quantum decoherence can
only be observed if all classical effects leading to energy and localization
averaging can be suppressed down to the quantum level, which of course is not
possible in real-life neutrino experiments.

In section 4 we have performed numerical estimates of decoherence effects for
reactor and Gallium radioactive source experiments. First we estimated the
localizations and velocities of all involved particles in the production and
detection processes. From these we calculate the effective decoherence
parameters relevant for the two types of decoherence related to localization
and energy spread, eqs. 3.19 and 3.20, respectively. The main results are
summarized in section 4.2 and table 2. Localization decoherence is controlled
by the factor $(\delta_{\rm loc}/L_{\rm osc})^{2}$. With typical values of
$\delta_{\rm loc}\sim 0.2$ nm, this is many orders of magnitude smaller than
any relevant oscillation length $L_{\rm osc}$ and therefore, localization
decoherence is completely negligible for practical purposes. The second
decoherence effect is related to the energy spread and around the first
oscillation maximum it is controled by $(\sigma_{\rm en}/E_{\nu})^{2}$. In the
wave packet picture this term can be interpreted as decoherence due to wave
packet separation. Both, for reactor and Gallium experiments we find values of
order $\sigma_{\rm en}\sim 0.5$ eV, about 6 to 7 orders of magnitude smaller
than typical neutrino energies $E_{\nu}$, leading again to completely
negligible decoherence effects.

Hence, in both cases (localization and energy spread) quantum decoherence
effects are completely unobservable and many orders of magnitude smaller than
classical averaging effects due to finite neutrino source and detector sizes
as well as classical energy resolution effects. This remains true even for the
upcoming high precision JUNO reactor experiment. If at some point data might
provide evidence for decoherence effects beyond classical averaging, standard
quantum mechanical uncertainties as discussed here cannot be responsible and
this would definitly require exotic new physics [60, 61, 62, 63].

### Acknowledgement

We thank E. Akhmedov for discussions and W. Grimus for useful communication
about his work [42] on a related topic. This work has been supported by the
European Union’s Framework Programme for Research and Innovation Horizon 2020
under grant H2020-MSCA-ITN-2019/860881-HIDDeN.

##  Appendix A Decoherence integral

The integrals which lead to decoherence terms are all of the type

$\displaystyle I=\int dx\,h(x)e^{i\phi(x)-g(x)}\,,$ (A.1)

where for our case of interest the complex phase is given by $\phi=\Delta
m^{2}L/(2E)$ and $x$ can be either $L$ or $E$. The real function $g(x)$ has a
minimum at $x=x_{0}$ and we expand $g$ and $\phi$ up to leading non-trivial
order around that minimum:

$\displaystyle g(x)$ $\displaystyle\approx
g(x_{0})+\frac{1}{2}g^{\prime\prime}(x_{0})(x-x_{0})^{2}\,,$ (A.2)
$\displaystyle\phi(x)$
$\displaystyle\approx\phi(x_{0})+\phi^{\prime}(x_{0})(x-x_{0})\,.$ (A.3)

The function $h(x)$ is assumed to be sufficiently smooth on scales
$1/\sqrt{g^{\prime\prime}(x_{0})}$ such that we can approximate it by
$h(x_{0})$ and pull it out of the integral. Then, the integral becomes
Gaussian and we obtain

$\displaystyle I\approx
h(x_{0})e^{i\phi(x_{0})-g(x_{0})}\sqrt{\frac{2\pi}{g^{\prime\prime}(x_{0})}}\exp\left[-\frac{(\phi^{\prime})^{2}}{2g^{\prime\prime}}\right]_{x_{0}}\,.$
(A.4)

In our applications the last exponential in this expression leads to
decoherence if the argument becomes sizeable.

Keeping quadratic terms for $\phi$ in eq. A.3 would lead to a small correction
to the phase $\phi(x_{0})$ [64], which is neglected here, as we are interested
in the situation where $\phi(x_{0})\sim 1$, i.e., around the first (or maybe
second) oscillation maximum.

##  Appendix B Decay rate and scattering cross section

We depart from the amplitude eq. 2.17 derived in section 2 for a single-vertex
process, considering the effective energy spread $\sigma_{E}$ from eq. 2.23.
From this result we obtain the total transition probability as:

$\displaystyle
P_{if}=|i\mathcal{A}|^{2}=|\mathcal{M}|^{2}\prod_{i,f}\mathcal{N}_{i}^{2}\mathcal{N}_{f}^{2}\frac{\pi^{4}}{(\sigma_{p}^{3}\sigma_{E})^{2}}\exp\left[-\frac{(\Delta\mathbf{p})^{2}}{2\sigma_{p}^{2}}-\frac{(\Delta
E-\Delta\mathbf{pv})^{2}}{2\sigma_{E}^{2}}\right]\,.$ (B.1)

Now we follow Weinberg [45] and outline how to derive decay rates or cross
sections from this expression, taking into account the wave packet
localization of all external particles. Weinberg considers a cubic box around
the interaction point in which all particles are localized. This leads to a
discretization of all momenta. Here, we adapt the argumentation to the case
when particles are localized as Gaussian wave packets, allowing for different
degrees of localization for each particle, see also [65, 66].

For each particle $a$ we can consider $V_{a}\sim 1/\sigma_{a}^{3}$ as the
volume in which the particle is confined due to its wave packet. It turns out
that a suitable definition is

$\displaystyle V_{a}=\left(\frac{\sqrt{2\pi}}{2\sigma_{a}}\right)^{3}\,.$
(B.2)

The numerical factor can be motivated as follows: define the configuration
space spread $\delta_{x}$ by $\delta_{x}\sigma=1/2$. Then
$V=(\sqrt{2\pi}\delta_{x})^{3}$ and $1/V$ would be just the prefactor of a
Gaussian in configuration space with width $\delta_{x}$.

Let us now relate the transition probability in eq. B.1 to more familiar
quantities. Consider first the final state particles. $P_{if}$ is the total
transition probability for $i\to f$ where all particles are defined via the
respective wave packets. Usually we are interested in a differential quantity,
for a transition into an infinitesimal phase space element for the outgoing
particles. Such a quantity can be obtained by multiplying the probability by
the differential number $dN_{f}$ of outgoing particles within the phase-space
element $d^{3}p_{f}$ around momentum $\mathbf{p}_{f}$:

$\displaystyle dP_{if}=P_{if}\prod_{f}dN_{f}\,,$ (B.3)

with

$\displaystyle
dN_{f}=V_{f}\frac{d^{3}p_{f}}{(2\pi)^{3}}=\frac{d^{3}p_{f}}{(2\sigma_{f})^{3}(2\pi)^{3/2}}\,,$
(B.4)

where in the second step we have used eq. B.2. Hence, each final state factor
in eq. B.1 becomes just the usual phase-space factor:

$\displaystyle\mathcal{N}_{f}^{2}dN_{f}=\frac{1}{2E_{p_{f}}}\frac{d^{3}p_{f}}{(2\pi)^{3}}=d\tilde{\mathbf{p}}_{f}\,,$
(B.5)

where we have used the definition of $\mathcal{N}$ from eq. 2.11.

Next we consider the following factors from eq. B.1:

$\displaystyle\frac{\pi^{4}}{(\sigma_{p}^{3}\sigma_{E})^{2}}e^{-[\ldots]}$
$\displaystyle=V_{I}T_{I}\,(2\pi)^{4}\frac{1}{(\sqrt{2\pi})^{4}\sigma_{p}^{3}\sigma_{E}}\exp\left[-\frac{(\Delta\mathbf{p})^{2}}{2\sigma_{p}^{2}}-\frac{(\Delta
E-\Delta\mathbf{pv})^{2}}{2\sigma_{E}^{2}}\right]$ (B.6)
$\displaystyle=V_{I}T_{I}\,(2\pi)^{4}\delta^{(3)}_{G}(\Delta\mathbf{p})\delta_{G}(\Delta
E-\Delta\mathbf{pv})$ (B.7) $\displaystyle\to
V_{I}T_{I}\,(2\pi)^{4}\delta^{(4)}\left(\sum_{f}p_{f}-\sum_{i}p_{i}\right)$
(B.8)

where we have defined the effective interaction volume and interaction time in
analogy to eq. B.2 by using the effective momentum and energy spreads from
eqs. 2.14 and 2.23:

$\displaystyle V_{I}=\left(\frac{\sqrt{2\pi}}{2\sigma_{p}}\right)^{3}\,,\qquad
T_{I}=\frac{\sqrt{2\pi}}{2\sigma_{E}}\,.$ (B.9)

In eq. B.7 we have introduced the notation $\delta_{G}$ to denote a Gaussian,
which will converge to a delta function in the limit $\sigma\to 0$ and the
limit in eq. B.8 is obtained for $\sigma_{p}\to 0$.777We use here
$\delta^{(3)}(\Delta\mathbf{p})\delta(\Delta
E-\Delta\mathbf{pv})=\delta^{(3)}(\Delta\mathbf{p})\delta(\Delta
E)=\delta^{(4)}(\Delta p)$. Finally, we use eq. B.2 also to rewrite the
initial state factors $\mathcal{N}_{i}$ in terms of volumes and obtain

$\displaystyle
d\Gamma_{if}\equiv\frac{dP_{if}}{T_{I}}=V_{I}\prod_{i}\left(\frac{1}{2E_{i}}\frac{1}{V_{i}}\right)\prod_{f}d\tilde{\mathbf{p}}_{f}\,(2\pi)^{4}\delta^{(3)}_{G}(\Delta\mathbf{p})\delta_{G}(\Delta
E-\Delta\mathbf{pv})|\mathcal{M}|^{2}\,.$ (B.10)

We interpret $T_{I}$ as the effective interaction time. Therefore, we obtain
the differential interaction rate $d\Gamma_{if}$ by dividing the differential
transition probability by the interaction time. Equation B.10 is sometimes
called Fermi’s Golden Rule.

###  B.1 Scattering cross section

Consider now the case of the scattering of two initial state particles to a
set of final state particles $AB\to f$. We get the factor
$V_{I}/V_{A}V_{B}=V_{I}n_{A}n_{B}$ in eq. B.10, with $n_{A},n_{B}$ denoting
the number density of particles $A,B$. Let us consider particle $A$ as
“target”. Then $V_{I}n_{A}$ is the number of target particles inside the
interaction volume, and the flux of incoming $B$ particles is $v_{\rm
rel}n_{B}$, with $v_{\rm rel}$ being the relative velocity. The cross section
per target particle is given by (total interaction rate) / [(number of target
particles $A$) $\times$ (flux of incoming particles $B$)]. Hence,

$\displaystyle d\sigma=\frac{d\Gamma_{if}}{V_{I}n_{A}v_{\rm
rel}n_{B}}=\frac{1}{2E_{A}2E_{B}v_{\rm
rel}}\prod_{f}d\tilde{\mathbf{p}}_{f}\,(2\pi)^{4}\delta^{(3)}_{G}(\Delta\mathbf{p})\delta_{G}(\Delta
E-\Delta\mathbf{pv})|\mathcal{M}|^{2}\,,$ (B.11)

which, in the limit $\sigma_{p},\sigma_{E}\to 0$, where the Gaussian
$\delta_{G}$ functions become true $\delta$-functions, converges to the
standard expression for a cross section, e.g., [44].

###  B.2 Particle decay

Consider now the case of a single particle in the initial state, i.e., the
decay of a particle. In this case we obtain a factor $V_{I}/V_{i}$ in eq.
B.10. We can consider $n_{i}=1/V_{i}$ as the density of initial state
particles and $V_{I}$ as the effective interaction volume, with $V_{I}\leq
V_{i}$. Hence, $V_{I}n_{i}$ is the number of initial-state particles inside
the interaction volume. Usually we are interested in the decay rate of a
single particle, denoted just by $\Gamma_{\rm dec}$. Therefore, we obtain

$\displaystyle d\Gamma_{\rm
dec}\equiv\frac{d\Gamma_{if}}{V_{I}n_{i}}=\frac{1}{2E_{i}}\prod_{f}d\tilde{\mathbf{p}}_{f}\,(2\pi)^{4}\delta^{(3)}_{G}(\Delta\mathbf{p})\delta_{G}(\Delta
E-\Delta\mathbf{pv})|\mathcal{M}|^{2}\,.$ (B.12)

In the limit $\sigma_{p}\to 0$, where the Gaussian $\delta_{G}$ functions
become true delta-functions, this expression converges to the standard
expression for the decay rate of a particle, see e.g., [44].

Let us now comment on the considerations in section 2.1 related to collisions
with the environment. We restore the Lorentzian shape of the energy spread,
see eq. 2.20, and instead of eq. B.10 we obtain

$\displaystyle
d\Gamma_{if}\equiv\frac{dP_{if}}{T_{I}}=V_{I}\prod_{i}\left(\frac{1}{2E_{i}}\frac{1}{V_{i}}\right)\prod_{f}d\tilde{\mathbf{p}}_{f}\,(2\pi)^{3}\delta^{(3)}_{G}(\Delta\mathbf{p})|\mathcal{M}|^{2}\,\frac{\Gamma}{\Omega^{2}+\Gamma^{2}/4}\,,$
(B.13)

where now we identify $T_{I}=1/\Gamma$. We recognise that the energy
conservation function $\delta_{G}(\Omega)$ is now replaced by the Lorentzian,
whose width is set by the rate $\Gamma$. We consider now two physically
different situations:

  * •

First we assume that the collision rate $\Gamma_{\rm col}$ is much larger than
the decay rate $\Gamma_{\rm dec}$ and set $\Gamma=\Gamma_{\rm col}$ in eq.
B.13. Let us consider as simple example a massive scalar particle (mass
$M_{i}$) at rest decaying into two massless scalar particles. The matrix
element is a constant, and we can perform the phase space integrals, apart
from the energy of one of the final state particles and obtain

$\displaystyle\frac{d\Gamma_{\rm
dec}}{dE_{f}}=\frac{|\mathcal{M}|^{2}}{16\pi^{2}M_{i}}\frac{\Gamma_{\rm
col}}{(M_{i}-2E_{f})^{2}+\Gamma_{\rm col}^{2}/4}\,.$ (B.14)

We observe the collisional line broadening [46], i.e., the width of the decay
line is set by the collision rate. Integrating over $E_{f}$ leads to the same
total decay rate as for $\Gamma_{\rm col}\to 0$, namely $\Gamma_{\rm
dec}=|\mathcal{M}|^{2}/(16\pi M_{i})$.

  * •

Let us now assume that $\Gamma_{\rm dec}\gg\Gamma_{\rm col}$, i.e., we can
neglect collisions. Now we can set $\Gamma=\Gamma_{\rm dec}$ in eq. B.13 and
use that the probability that the particle decays into any final state is one:
$\int dP_{if}=1$. Considering again the example of a massive scalar particle
at rest decaying into two massless scalar particles, we obtain from eq. B.13
in the limit of exact momentum conservation
$\delta^{(3)}_{G}(\Delta\mathbf{p})\to\delta^{(3)}(\Delta\mathbf{p})$:

$\displaystyle
1=\frac{|\mathcal{M}|^{2}}{8\pi^{2}M_{i}}\frac{\arctan(2M_{i}/\Gamma_{\rm
dec})}{\Gamma_{\rm dec}}\,.$ (B.15)

In the limit $\Gamma_{\rm dec}\ll M_{i}$ we can solve for $\Gamma_{\rm dec}$
and recover the standard result for the decay width $\Gamma_{\rm
dec}=|\mathcal{M}|^{2}/(16\pi M_{i})$.

###  B.3 Rate for the neutrino oscillation process

In the case of neutrino oscillations, the observable of interest is the
differential event rate at the detector $dR_{D}$, depending on the momenta of
the final states in the detector, i.e., $\mathbf{p}_{3},\mathbf{p}_{4}$ in our
notation. In the case of reactor neutrino experiments these are the positron
and neutron, whose momenta are in principle observable in the detector. We can
obtain this quantity from the transition probability
$P_{if}=|i\mathcal{A}_{\alpha\beta}|^{2}$ by the following steps:

  1. 1.

As in eq. B.3, we multiply by the phase-space elements of all final state
particles in order to obtain a differential transition probability,

$dP_{if}=P_{if}\prod_{f}dN_{f}=P_{if}dN_{1}dN_{2}dN_{3}dN_{4}\,.$ (B.16)

  2. 2.

We are interested in the event rate per single detector particle, as well as
single decaying particle at the source. Hence, we have to divide by the number
of particles in the production and detection regions, $V_{P}/V_{A}$ and
$V_{D}/V_{B}$, respectively, with the volumina defined as in eqs. B.2 and B.9.

  3. 3.

The outgoing particles in the production region are not observed. Hence we
have to integrate over their phase space
$d\tilde{\mathbf{p}}_{1}d\tilde{\mathbf{p}}_{2}$.

  4. 4.

The event rate at the detector is obtained by dividing by the effective
detection time interval $T_{D}$ defined as in eq. B.9, in analogy to the first
relation in eq. B.10.

  5. 5.

The amplitude squared still depends on production and detection times via
$T=t_{D}-t_{P}$. While the detection time in principle is observable in real-
time neutrino detectors, the production time typically is not observable (see
comments below). Hence, for a given $t_{D}$, we have to sum the contributions
of all possible production times contributing to the amplitude. This is
obtained by first dividing by the effective production time interval $T_{P}$
defined as in eq. B.9, which gives the neutrino production rate, which then
has to be integrated over all possible values of $t_{P}$. By a simple
coordinate shift, this integral is transformed into an integral over $T$ and
we obtain a time averaged amplitude-squared:

$\overline{|\mathcal{A}_{\alpha\beta}|^{2}}=\int
dT\,\frac{|\mathcal{A}_{\alpha\beta}(T)|^{2}}{T_{P}}\,.$ (B.17)

Putting everything together, we obtain for the differential event rate at the
detector

$dR_{D}=\frac{V_{P}}{V_{A}}\frac{V_{D}}{V_{B}}\,dN_{3}dN_{4}\int
dN_{1}dN_{2}\frac{\overline{|\mathcal{A}_{\alpha\beta}|^{2}}}{T_{D}}\,.$
(B.18)

Following comments are in order:

  * •

In reactor experiments typically only the energy of the outgoing positron is
used to reconstruct the neutrino energy. In this case we need also to
integrate over the phase space of the outgoing neutron in the detector.

  * •

In counting experiments, such as Gallium radioactive source experiments, the
detection reaction is not observed and we have to integrate in addition over
the phase-space of outgoing detector particles $dN_{3}dN_{4}$.

  * •

Equation B.18 corresponds to the hypothetical “event rate” assuming a single
decaying particle at production and a single detector particle. For an actual
experiment we have to sum over the number of target particles in the detector,
as well as the total number of all decays contributing to the neutrino
production at the source. The latter is a complicated calculation in case of
reactor neutrinos, amounting to an ab initio calculation of the total reactor
flux.

  * •

Above we assumed that the neutrino production time is not observable. This is
true for a large number of experiments, such as reactor, solar, or atmospheric
neutrinos. At many accelerator experiments, neutrinos are produced by a pulsed
particle beam, which in principle provides some time information on the
production. Typically the duration of the beam spills are of order
$\mathrm{\SIUnitSymbolMicro s}$, which however, in practice is much longer
compared to other relevant time scales $\sim 1/\sigma_{EP},1/\sigma_{ED}$.
Therefore, for practical purposes it is a good approximation to assume the
integration in eq. B.17 over an infinite time interval [41].

  * •

In the case of a so-called monitored neutrino beam, such as proposed by ENUBET
[49], in principle time information on neutrino production can be obtained. In
such a situation the amplitude needs to be averaged over the accuracies with
which both $t_{D}$ and $t_{P}$ can be determined, instead of an infinite time
interval. In order to affect the discussion above, the accuracy of both time
measurments, $t_{D}$ and $t_{P}$, need to be comparable to the microscopic
time scales $1/\sigma_{EP}$ and $1/\sigma_{ED}$.

  * •

Using the formalism discussed in this paper, one can show [48] that in the
(realistic) case, when decoherence effects play no role, eq. B.18 factorizes
into the decay rate $\Gamma_{P}$ of the production process, the standard
oscillation probability and the detection cross section $\sigma_{D}$ as

$dR_{D}=\frac{1}{4\pi L^{2}}\int
dE_{\nu}\frac{d\Gamma_{P}}{dE_{\nu}}\,P_{\alpha\beta}(E_{\nu},L)\,d\sigma_{D}(E_{\nu})\,,$
(B.19)

see also [11, 67, 42] for similar calculations.

##  Appendix C Phase-space integrals

As discussed in section 3.3, depending on the physical configuration,
integrals over the momenta of unobserved external particles either in the
source or in the detector are unavoidable, which leads to integrals over the
effective neutrino energy at production $E_{P}$ or at detection $E_{D}$ (or
both). Here we outline the integral of eq. 3.14 over $E_{P}$; the integration
over $E_{D}$ instead of $E_{P}$ proceeds in complete analogy.

We expand the oscillation phase around $E_{D}$ and keep terms up to linear
order in $(E_{P}-E_{D})$, using $E_{0}-E_{D}=(\sigma_{\rm
eff}^{2}/\sigma_{P,\rm eff}^{2})(E_{P}-E_{D})$. But we replace $E_{0}\to
E_{D}$ in the two decoherence terms in the second line of eq. 3.14, ignoring
higher order corrections to these terms. Then the integral is of the form
discussed in appendix A and we obtain

$\displaystyle\int dE_{P}$ $\displaystyle\exp\left[i\frac{\Delta
m^{2}L}{2E_{0}}\right]\times\exp\left[-\frac{1}{2}\frac{(E_{D}-E_{P})^{2}}{\sigma_{P,\rm
eff}^{2}+\sigma_{D,\rm eff}^{2}}\right]$
$\displaystyle\propto\exp\left[i\frac{\Delta
m^{2}L}{2E_{D}}\right]\times\exp\left[-\frac{1}{2}\left(\frac{\Delta
m^{2}L}{2E_{D}^{2}}\right)^{2}\frac{\sigma_{D,\rm eff}^{4}}{\sigma_{P,\rm
eff}^{2}+\sigma_{D,\rm eff}^{2}}\right]\,.$ (C.1)

The emerging decoherence term can be combined with the $\sigma_{\rm eff}$-term
in eq. 3.14. Using the definition of $\sigma_{\rm eff}$ in eq. 3.13 we obtain:

$\displaystyle R_{D}(L,E_{D})\propto\int
dE_{P}\overline{|\mathcal{A}_{\alpha\beta}|^{2}}$
$\displaystyle\propto\exp\left[i\frac{\Delta
m^{2}L}{2E_{D}}\right]\times\exp\left[-\frac{\mathbf{p}_{D\perp}^{2}}{2\sigma_{pD}^{2}}\right]$
$\displaystyle\times\exp\left[-\frac{1}{2}\left(\frac{\Delta
m^{2}}{4E_{D}\sigma_{m}}\right)^{2}-\frac{1}{2}\left(\frac{\Delta
m^{2}L}{2E_{D}}\frac{\sigma_{D,\rm eff}}{E_{D}}\right)^{2}\right]\,.$ (C.2)

We have dropped the term depending on $\mathbf{p}_{P\perp}$ in eq. C.2, as the
oscillation phase no longer depends on it, and therefore the integration over
$\mathbf{p}_{P\perp}$ becomes trivial. In contrast, $E_{D}$ – and therefore
the oscillation phase – does depend on $\mathbf{p}_{D\perp}$. In principle,
$\mathbf{p}_{D\perp}$ is observable, by reconstructing all particles involved
in the detection process. However, this is often not the case in realistic
situations. For instance, in reactor neutrino experiments, the momentum of the
outgoing neutron cannot be observed. Therefore, we have to integrate also over
the neutron momentum phase-space, which effectively means integrating over
$\mathbf{p}_{D\perp}$. Once again we can involve appendix A to derive a
corresponding decoherence term. There is no simple closed form for this term,
however, its size can be estimated to be of order
$\mathbf{v}_{D}^{2}\sigma_{pD}^{2}/\sigma_{D,\rm eff}^{2}$ relative to the
last term in eq. C.2. One can show that
$\mathbf{v}_{D}^{2}\sigma_{pD}^{2}/\sigma_{D,\rm eff}^{2}<1$ and in typical
cases it is actually $\ll 1$. Using the numbers from section 4 we obtain for
reactor neutrinos $\mathbf{v}_{D}^{2}\sigma_{pD}^{2}/\sigma_{D,\rm
eff}^{2}\sim 10^{-4}$. Hence this term can at most induce corrections of order
one to the last term in eq. C.2.

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