Quantum field-theoretical description of neutrino oscillations in magnetic
field
Vadim Egorov, Igor Volobuev

Skobeltsyn Institute of Nuclear Physics, Moscow State University

119991 Moscow, Russia

###### Abstract

It is shown that the processes of neutrino oscillations in a magnetic field
can be consistently described in the framework of a new quantum field-
theoretical approach without use of the neutrino flavor states. It is based on
the Feynman diagram technique with a modified distance-dependent propagator,
which takes into account the geometry of neutrino oscillation experiments.
Processes of neutrino oscillations in a magnetic field, where the neutrinos
are detected through the weak charged- and neutral-current interaction, have
been studied and numerical calculations have been carried out for some
specific examples. Implications for the solar neutrinos are briefly discussed,
and formulas for the asymptotic values of the normalized probability of solar
neutrino oscillation processes are derived, which coincide with the observable
ratio of the measurable neutrino flux to that predicted by the standard solar
model.

##  1 Introduction

The Standard Model (SM) allows one to describe, with high accuracy, a great
amount of various elementary particle interaction processes in the framework
of the S-matrix formalism and Feynman diagram technique. However, there is a
belief that it cannot describe the phenomena of neutral kaon and neutrino
oscillations, the latter being under intense theoretical and experimental
investigation nowadays. The standard way to describe the phenomenon is the
quantum-mechanical approach in terms of plane waves [1, 2, 3, 4]. Although
being straightforward, it is believed to be inconsistent, because the
production of the neutrino flavor states is described within the SM (which is
a gauge field theory), whereas their evolution is described within quantum
mechanics. Such a description seems to be eclectic, since quantum field theory
includes quantum mechanics as an indispensable part and must be able to
describe all quantum phenomena. Moreover, the production of states without
definite mass leads to violation of energy-momentum conservation [5, 6, 7, 8].
This problem is supposedly solved in the framework of the quantum-mechanical
approach it terms of wave packets [3, 9], although this description turns out
to be very bulky. Thus, the construction of a consistent and convenient
description of neutrino oscillations within quantum field theory is of current
interest.

The first attempt to describe neutrino oscillations in the framework of
quantum field theory was made back in 1982 in paper [10]. In this paper,
within the standard perturbative S-matrix formalism, it was assumed that
virtual neutrino mass eigenstates were produced and detected in the charged-
current interactions with nuclei. The matrix elements of the charged weak
hadron current between the initial and final states of the nuclei were
approximated by delta functions of their positions separated by a fixed
distance, whereas all the incoming and outgoing leptons were described by
plane waves. This approximation fixed the distance between the production and
detection points of the neutrinos and, in the momentum representation,
resulted in the appearance of distance-dependent neutrino propagators. Hence,
neutrino oscillations were regarded as interference of the amplitudes of
processes mediated by different neutrino mass eigenstates. In the subsequent
studies the delta functions, which seemed to be a rather too rough
approximation and formally contradicted the S-matrix formalism, were replaced
by localized wave packets for describing the states of the nuclei [5, 6, 7,
11]. However, the calculations in the framework of this approach turned out to
be very bulky and complicated. The reason is that the standard perturbative
formalism of S-matrix is not suitable for describing processes taking place at
finite space and time intervals.

It is a common knowledge that the presence of external fields and matter
affects neutrino oscillations [3]. The influence of magnetic fields on
neutrino oscillations, as well as its possible implications for the solar
neutrino problem were repeatedly considered in the framework of the standard
quantum-mechanical description based on the use of the neutrino flavor states
in papers [12, 13, 14, 15, 16, 17, 18]. However, up to now, there were no
papers dealing with the description of neutrino oscillations in external
fields and matter within quantum field theory without use of the flavor
states. In the present paper, we develop such a quantum field-theoretical
description of neutrino oscillations in a magnetic field. Our approach makes
use of a modified perturbative formalism adapted for describing processes
passing at finite space and time intervals [19, 20, 21, 22, 23]. The formalism
is based on the Feynman diagram technique in the coordinate representation
supplemented with modified rules of passing to the momentum representation.
The latter reflect the geometry of neutrino oscillation experiments and lead
to the Feynman propagator of virtual neutrino mass eigenstates in the momentum
representation being modified. Namely, a distance-dependent propagator of
neutrino mass eigenstates in the momentum representation arises, while the
rest of the Feynman rules in this representation are kept intact. The
description in terms of plane waves allows one to avoid cumbersome
calculations, while catching the essence of the phenomenon.

In describing neutrino oscillations in a magnetic field, we assume that
neutrinos are produced and detected through the charged- and neutral-current
interaction with nuclei and electrons in the absence of field, but the
propagation of the particles occurs in a region of magnetic field. In Section
2 we give a brief review of the main ideas of the approach. In Section 3 we
apply the formalism to the case of neutrino oscillation in a magnetic field,
deriving the oscillation probability. In Section 4 we consider specific
examples with solar neutrinos, namely neutrino production in ${}^{15}\rm O$
decay or electron capture by ${{}^{7}{\rm Be}}$ and detection by Ga-Ge or
Cherenkov detectors, and derive useful formulas for the asymptotic values of
the normalized probability of solar neutrino oscillation processes, which
coincide with the observable ratio of the measurable neutrino flux to that
predicted by the standard solar model.

##  2 Basics of the approach

First, we note that neutrino oscillation experiments are characterized by a
specific geometry, where the distance between a neutrino source and a detector
is much larger than their sizes. For this reason one can consider the source
and detector to be pointlike and describe the oscillation process by just one
parameter of geometric origin: the distance between the centers of the source
and detector. Moreover, one can use the one-dimensional approximation, where
the neutrino momenta are directed along the line connecting the centers, which
is usually called the plane wave approximation in the standard quantum-
mechanical approach [3].

In what follows we use the one-dimensional approximation. Also, we work in the
framework of the minimal extension of the Standard Model by the right neutrino
singlets. The interaction Lagrangian of the leptons takes the form

$\begin{split}L_{\rm int}^{\rm
lep}=&-\frac{g}{2\sqrt{2}}\left(\sum_{i,k=1}^{3}\bar{l}_{i}\,\gamma^{\mu}\left(1-\gamma^{5}\right)U_{ik}\,\nu_{k}\,W^{-}_{\mu}+{\rm
h.c.}\right)+\frac{g\sin^{2}\theta_{\rm w}}{\cos\theta_{\rm
w}}\sum_{i=1}^{3}\bar{l}_{i}\,\gamma^{\mu}\,l_{i}\,Z_{\mu}-\\\
&-\frac{g}{4\cos\theta_{\rm
w}}\sum_{i=1}^{3}\bar{l}_{i}\,\gamma^{\mu}\left(1-\gamma^{5}\right)l_{i}\,Z_{\mu}+\frac{g}{4\cos\theta_{\rm
w}}\sum_{k=1}^{3}\bar{\nu}_{k}\,\gamma^{\mu}\left(1-\gamma^{5}\right)\nu_{k}\,Z_{\mu}\,,\end{split}$
(1)

where $l_{i}$ denotes the field of the charged lepton of the $i$th generation,
$U_{ik}$ is the PMNS-matrix, $\nu_{k}$ stands for the field of the neutrino
state with definite mass $m_{k}$, and $\theta_{\rm w}$ is the Weinberg angle.

Let us first recall the basics of the approach and consider the case of
neutrino oscillations in vacuum. To this end, we consider a process, where a
neutrino is produced and detected through the charged-current interaction with
nuclei. In the lowest order of perturbation theory the process is described by
the following diagram:

$e^{+}(q)$$x$$W^{+}$$\nu_{i}(p_{\rm n})$$e^{-}(k)$$y$$W^{+}$(2)

The points of production $x$ and detection $y$ are supposed to be separated by
a fixed macroscopic distance $L$ along a unit vector
$\vec{n},\,\vec{y}-\vec{x}=L\vec{n}$. The intermediate virtual neutrino mass
eigenstate is described by the Feynman propagator in the coordinate
representation. The amplitude corresponding to the diagram must be summed over
all the three neutrino mass eigenstates, $i=1,2,3$.

The initial and final nuclei and particles are assumed to be described by
plane waves. Their 4-momenta are denoted as it is shown in the diagram: $q$,
$k$, $p_{\rm n}$ correspond to the positron, electron and intermediate virtual
neutrino, respectively. The filled circles in the diagram represent the matrix
elements of the weak charged hadron current between the states of initial
nuclei 1 $\left({}^{A_{1}}_{Z_{1}}{\rm X}\right)$ and 2
$\left({}^{A_{2}}_{Z_{2}}{\rm X}\right)$ and final nuclei $1^{\prime}$
$\left({}^{A_{1}}_{Z_{1}-1}{\rm X}\right)$ and $2^{\prime}$
$\left({}^{A_{2}}_{Z_{2}+1}{\rm X}\right)$:

$\begin{split}j_{\mu}^{(1)}\left({\vec{P}^{(1)},\vec{P}^{(1^{\prime})}}\right)&=\left<{}^{A_{1}}_{Z_{1}-1}{\rm
X}\left({\vec{P}^{(1^{\prime})}}\right)\right|j_{\mu}^{({\rm
h})}\left|{}^{A_{1}}_{Z_{1}}{\rm X}\left({\vec{P}^{(1)}}\right)\right>,\\\
j_{\rho}^{(2)}\left({\vec{P}^{(2)},\vec{P}^{(2^{\prime})}}\right)&=\left<{}^{A_{2}}_{Z_{2}+1}{\rm
X}\left({\vec{P}^{(2^{\prime})}}\right)\right|j_{\rho}^{({\rm
h})}\left|{}^{A_{2}}_{Z_{2}}{\rm
X}\left({\vec{P}^{(2)}}\right)\right>,\end{split}$ (3)

the nuclei 4-momenta being denoted by
$P^{(l)}=\left({E^{(l)},\vec{P}^{(l)}}\right)$, $l=1,1^{\prime},2,2^{\prime}$.

The amplitude of the process in the coordinate representation can be
constructed using the Feynman rules formulated, for example, in textbook [24].
The passing to the momentum representation is performed by integrating the
amplitude with respect to $x$ and $y$ over Minkowski space. However, such a
straightforward integration would result in losing the information about the
space-time interval between the production and detection events. Thus, in
order to be able to describe processes passing at finite space and time
interval one has to fix somehow the distance between the neutrino production
and detection points. To this end, in paper [10] the matrix elements of hadron
currents (3) in the coordinate representation were assumed to be proportional
to the delta functions
$\delta(\vec{x}-\vec{x}_{1}),\,\,\delta(\vec{y}-\vec{x}_{2})$ with
$\vec{x}_{2}-\vec{x}_{1}$ fixed, $\vec{x}_{2}-\vec{x}_{1}=L\vec{n},$ which, in
the approximation of Fermi’s interaction, gave the desired result.

In our approach, instead of fixing the positions of the initial and final
nuclei, we just fix the distance $L$ between the interaction points along the
unit vector $\vec{n}$ directed from the source to the detector by introducing
the delta function $\delta(\vec{n}(\vec{y}-\vec{x})-L)$ into the integrand,
which gives a generalization of the standard perturbative formalism to the
case of processes passing at finite distances.

The introduction of the delta function is formally equivalent to replacing the
standard Feynman propagator $S^{\rm c}_{i}(y-x)$ of the neutrino mass
eigenstate $\nu_{i}$ in the coordinate representation by $S^{\rm
c}_{i}(y-x)\,\delta(\vec{n}(\vec{y}-\vec{x})-L)$. The Fourier transform of
this expression was called in paper [19] the distance-dependent propagator of
the neutrino mass eigenstate $\nu_{i}$ in the momentum representation:

$S^{\rm c}_{i}\left({p,\vec{n},L}\right)\equiv\int{{\rm d}^{4}z\,{\rm e}^{{\rm
i}pz}\,S^{\rm c}_{i}\left(z\right)\delta\left({\vec{n}\vec{z}-L}\right)}\,.$
(4)

This integral can be evaluated exactly:

$S^{\rm c}_{i}(p,\vec{n},L)={\rm
i}\,\frac{\hat{p}+\vec{\gamma}\vec{n}\left(\vec{p}\vec{n}-\sqrt{(\vec{p}\vec{n})^{2}+p^{2}-m^{2}_{i}}\right)+m_{i}}{2\sqrt{(\vec{p}\vec{n})^{\,2}+p^{2}-m^{2}_{i}+{\rm
i}\varepsilon}}\,{\rm e}^{-{\rm
i}\left(\vec{p}\vec{n}-\sqrt{(\vec{p}\vec{n})^{\,2}+p^{2}-m^{2}_{i}}\,\right)L}\,,$
(5)

where $\hat{p}=\gamma^{\mu}p_{\mu}$.

According to the Grimus-Stockinger theorem [6] the virtual particles
propagating over macroscopic distances are almost on the mass shell, and for
the momenta $\vec{p}$ satisfying
$\left|{p^{2}-m_{i}^{2}}\right|/(\vec{p}\vec{n})^{\,2}\ll 1$ the distance-
dependent propagator can be brought to the simple form

$S_{i}^{\rm c}\left({p,\vec{n},L}\right)={\rm
i}\,\frac{{\hat{p}+m_{i}}}{{2{\vec{p}\vec{n}}}}\,{\rm e}^{{\rm
i}\frac{{p^{2}-m_{i}^{2}}}{{2{\vec{p}\vec{n}}}}L}\,.$ (6)

In particular, this approximation is always valid for the neutrino momenta
$\vec{p}$ directed along the vector $\vec{n}$, $\vec{p}\vec{n}=|\vec{p}|,$
which are the only momenta needed for calculating the amplitudes.

Distance-dependent propagator (6) differs from the one found in paper [10]:
there the propagator is the projection operator ${\hat{p}+m_{i}}$ multiplied
by a spherical wave, whereas in our case this operator is multiplied by the
exponential similar to that appearing in the standard plane wave
approximation. It is also necessary to note that a distance-dependent
propagator very similar to ours has been obtained in recent paper [25] within
a path integral approach.

Finally, using expression (6) instead of the usual Feynman propagator for
constructing the amplitude in the momentum representation allows one to
consistently describe neutrino oscillations in vacuum [20, 21, 22, 23]. It is
worth noting that in this case one can also use the time-dependent propagator,
which is obtained by fixing the time interval $T$ between the neutrino
production and detection events. This is due to the fact that, for the
neutrinos almost on the mass shell, the space and time intervals are related
by the standard formula

$L=\frac{|\vec{p}|}{p^{0}}T.$ (7)

However, this relation is no longer true for neutrinos in an external field,
and one has to use distance-dependent propagator (6) in this case.

##  3 Neutrino oscillations in a magnetic field

Now let us turn a background electromagnetic field on. Neutrinos are able to
interact with it through quantum loops, which provides the neutrino with,
among others, anomalous dipole magnetic moment. We will take it into account
but neglect the transition moments, which are usually assumed to be much
smaller. This means that the equations of motion for different neutrino mass
eigenstates are not coupled. Thus, the equation of motion of a neutrino mass
eigenstate in an external electromagnetic field takes the form

$\left({{\rm i}\gamma^{\mu}\partial_{\mu}-m_{i}-\frac{{\rm
i}}{2}\mu_{0}m_{i}F_{\mu\nu}\gamma^{\mu\nu}}\right)\nu_{i}\left(x\right)=0\,,$
(8)

where the magnetic moment of the $i$th neutrino mass eigenstate, proportional
to its mass, is $\mu_{i}=\mu_{0}m_{i}$ and
$\gamma^{\mu\nu}=\frac{1}{2}\left[{\gamma^{\mu},\gamma^{\nu}}\right]$. In the
Standard Model, the parameter $\mu_{0}={{3eG_{\rm
F}}\mathord{\left/{\vphantom{{3eG_{\rm
F}}{8\sqrt{2}\pi^{2}}}}\right.\kern-1.2pt}{8\sqrt{2}\pi^{2}}}$, which gives,
for the experimentally allowed neutrino masses, the neutrino magnetic moments
at least 10 orders of magnitude smaller than the Bohr magneton, but they may
be much larger in SM extensions.

For a homogeneous electromagnetic field, Green’s function of equation (8) in
the momentum representation reads

$\begin{split}S_{i}^{\rm c}\left(p\right)=&\ {\rm
i}\left\\{{\left({p^{2}-m_{i}^{2}}\right)\left({p^{2}-m_{i}^{2}+{\rm
i}\varepsilon}\right)-\mu_{0}^{2}m_{i}^{2}\left[{\left({p^{2}+m_{i}^{2}}\right)F_{\mu\nu}F^{\mu\nu}-4F_{\mu\nu}p^{\nu}F^{\mu\sigma}p_{\sigma}}\right]+\vphantom{\frac{1}{4}}}\right.\\\
&+\frac{1}{4}\mu_{0}^{4}m_{i}^{4}\left[{\left({F_{\mu\nu}F^{\mu\nu}}\right)^{2}+\left({F_{\mu\nu}\tilde{F}^{\mu\nu}}\right)^{2}}\right]\bigg{\\}}^{-1}\left\\{{\left({p^{2}-m_{i}^{2}}\right)\left({\hat{p}+m_{i}}\right)-\vphantom{\frac{1}{4}}}\right.\\\
&-\frac{1}{2}\mu_{0}^{2}m_{i}^{2}F_{\mu\nu}F^{\mu\nu}\left({\hat{p}-m_{i}}\right)-2\mu_{0}^{2}m_{i}^{2}F_{\mu\nu}F^{\nu\sigma}p_{\sigma}\gamma^{\mu}+2\mu_{0}m_{i}^{2}\tilde{F}_{\mu\nu}p^{\nu}\gamma^{\mu}\gamma^{5}+\\\
&+{\rm
i}\mu_{0}m_{i}\left[{\frac{1}{2}\left({p^{2}+m_{i}^{2}}\right)F_{\mu\nu}-\frac{1}{4}\mu_{0}^{2}m_{i}^{2}F^{\rho\sigma}\left({F_{\rho\sigma}F_{\mu\nu}+\tilde{F}_{\rho\sigma}\tilde{F}_{\mu\nu}}\right)-2F_{\mu\rho}p^{\rho}p_{\nu}}\right]\gamma^{\mu\nu}-\\\
&-\frac{{\rm
i}}{2}\mu_{0}^{2}m_{i}^{3}F_{\mu\nu}\tilde{F}^{\mu\nu}\gamma^{5}\bigg{\\}}\,.\end{split}$
(9)

Here
$\tilde{F}^{\mu\nu}=-\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$,
$\varepsilon^{0123}=-1$.

First, we consider a homogeneous magnetic field $\vec{H}$:
$F_{\mu\nu}=\varepsilon_{\mu\nu k0}H^{k}$, $k=1,2,3$. Then the neutrino
dispersion relation following from the denominator of Green’s function (9),

$\left({p^{0}}\right)^{2}=\vec{p}^{\,2}+m_{i}^{2}+\mu_{0}^{2}m_{i}^{2}\vec{H}^{2}\pm
2\mu_{0}m_{i}\sqrt{\vec{p}^{\,2}\vec{H}_{\bot}^{2}+m_{i}^{2}\vec{H}^{2}}\,,$
(10)

where $\vec{H}_{\bot}$ denotes the component of the magnetic field $\vec{H}$
transverse to the direction of neutrino propagation
$\vec{n}={{\vec{p}}\mathord{\left/{\vphantom{{\vec{p}}{\left|{\vec{p}}\right|}}}\right.\kern-1.2pt}{\left|{\vec{p}}\right|}}$,
coincides, mutatis mutandis, with the dispersion relation for the neutron in a
magnetic field, which was first derived in paper [26] and recently reproduced
within the standard approach in paper [17] for the neutrinos.

Since the neutrino magnetic moment is extremely small,
$\mu_{0}^{2}m_{i}^{2}\vec{H}^{2}\ll\vec{p}^{\,2}$, we neglect the terms of
order 2 and higher in $\mu_{0}$. Substituting the Green’s function in the
coordinate representation in definition (4) of the distance-dependent
propagator, taking the neutrino momentum to be parallel to $\vec{n}$ and
neglecting also the neutrino masses everywhere except in the exponential, we
arrive at the distance-dependent propagator of the neutrino mass eigenstate in
a homogeneous magnetic field in the momentum representation:

$S_{i}^{\rm c}\left({p,L,\vec{H}}\right)={\rm i}\,\frac{{\hat{p}\left({1-{\rm
i}\vec{\gamma}\vec{j}}\right)}}{{4\left|{\vec{p}}\right|}}\,{\rm e}^{{\rm
i}\frac{{p^{2}-m_{i}^{2}+2\mu_{0}m_{i}\left|{\vec{p}}\right|H_{\bot}}}{{2\left|{\vec{p}}\right|}}L}+{\rm
i}\,\frac{{\hat{p}\left({1+{\rm
i}\vec{\gamma}\vec{j}}\right)}}{{4\left|{\vec{p}}\right|}}\,{\rm e}^{{\rm
i}\frac{{p^{2}-m_{i}^{2}-2\mu_{0}m_{i}\left|{\vec{p}}\right|H_{\bot}}}{{2\left|{\vec{p}}\right|}}L}.$
(11)

Here $H_{\bot}=|{\vec{H}_{\bot}}|$, we suppose
$m_{i}^{2}\vec{H}^{2}\ll\vec{p}^{\,2}\vec{H}_{\bot}^{2}$, and

$\vec{j}\equiv\frac{{\left[{\vec{n}\times\vec{h}}\right]}}{{\sqrt{1-\left({\vec{n}\vec{h}}\right)^{2}}}},\qquad\vec{h}\equiv\frac{{\vec{H}}}{{\left|{\vec{H}}\right|}},\qquad\vec{j}^{2}=1,$
(12)

the square brackets denoting the vector product. Comparing expressions (11)
and (6) one can conclude that, in the magnetic field, each neutrino mass
eigenstate splits into two states corresponding to two possible spin
orientations and energies. This effect is in full agreement with the quantum-
mechanical expectations. The numerators in the exponents, $p^{2}-m_{i}^{2}\mp
2\mu_{0}m_{i}\left|{\vec{p}}\right|H_{\bot}$, measure the deviation of the
virtual neutrinos from the mass shell, which is consistent with dispersion
relation (10) in our approximation.

We see that the distance-dependent propagator essentially depends only on the
transverse component of a magnetic field. In what follows, we will neglect the
longitudinal component of the field and assume the magnetic field to be
transverse. Further, propagator (11) has been derived for the case of a
constant homogeneous magnetic field. Nevertheless, it can be used for the
transverse magnetic field, the magnitude of which varies along the neutrino
path adiabatically, i.e., if the condition

$\left|{\mu_{0}m_{\max}(\vec{n}\vec{\nabla})\left|{\vec{H}}\right|}\right|\ll\frac{|\vec{p}|}{d}$
(13)

is fulfilled, where $d$ is the characteristic size of the field region and
$m_{\max}$ is the largest neutrino mass. This condition refines the
adiabaticity condition for the magnetic field introduced earlier in paper
[15]. It takes into account the neutrino magnetic moments, the size of the
field region and guarantees that the term $({p^{2}-m_{i}^{2}})/2|{\vec{p}}|$
in the exponential can be considered to be constant along the path. However,
since now the magnetic field varies along the path, the field $H_{\bot}$ in
formula (11) should be replaced by the mean field

$\overline{H}=\frac{1}{L}\int\limits_{0}^{L}H\left({l}\right){\rm d}l.$ (14)

Now we will use distance-dependent propagator (11) with these amendments to
calculate the probability of an actual process. Let us first consider a
process analogous to the one described by diagram (2), where the neutrinos are
produced and detected in the absence of field, but propagate through a region
of magnetic field (for which the assumptions above are true). In what follows
we will work in the approximation of Fermi’s interaction. Then the amplitude
of the process in the momentum representation, where the distance is fixed and
equals $L$, looks as follows:

$\begin{split}M=\ &{-{\rm i}}\frac{{G_{\rm F}^{2}}}{{8\left|{\vec{p}_{\rm
n}}\right|}}\,j_{\rho}^{(2)}\left({\vec{P}^{(2)},\vec{P}^{(2^{\prime})}}\right)\bar{u}\left(\vec{k}\right)\gamma^{\rho}\left({1-\gamma^{5}}\right)\hat{p}_{\rm
n}\times\\\
&\times\sum\limits_{i=1}^{3}{\left|{U_{1i}}\right|^{2}\left[{\left({1-{\rm
i}\vec{\gamma}\vec{j}}\right){\rm e}^{{\rm i}\frac{{p_{\rm
n}^{2}-m_{i}^{2}+2\mu_{0}m_{i}\left|{\vec{p}_{\rm
n}}\right|\overline{H}}}{{2\left|{\vec{p}_{\rm n}}\right|}}L}+\left({1+{\rm
i}\vec{\gamma}\vec{j}}\right){\rm e}^{{\rm i}\frac{{p_{\rm
n}^{2}-m_{i}^{2}-2\mu_{0}m_{i}\left|{\vec{p}_{\rm
n}}\right|\overline{H}}}{{2\left|{\vec{p}_{\rm
n}}\right|}}L}}\right]}\times\\\
&\times\gamma^{\mu}\left({1-\gamma^{5}}\right)v\left(\vec{q}\right)j_{\mu}^{(1)}\left({\vec{P}^{(1)},\vec{P}^{(1^{\prime})}}\right).\end{split}$
(15)

Here we omit fermion polarization indices for brevity; the notations for the
momenta are the same as in diagram (2) explained in the paragraph above
formula (3). Again, neutrino masses are neglected everywhere, except in the
exponential.

The squared modulus of the amplitude (15), averaged over the polarizations of
the incoming nuclei and summed over the polarizations of the outgoing
particles and nuclei (the operation of averaging and summation is denoted by
the angle brackets), factorizes in the approximation of zero neutrino masses.
The latter means that we also neglect the terms proportional to $p_{\rm
n}^{2}$, which, due to the Grimus-Stockinger theorem [6], is of the order of
neutrino masses squared. In (15), the terms containing the vector $\vec{j}$
vanish, and we arrive at the result

$\left\langle{\left|M\right|^{2}}\right\rangle=\left\langle{\left|{M_{\rm
P}}\right|^{2}}\right\rangle\left\langle{\left|{M_{\rm
D}}\right|^{2}}\right\rangle\frac{1}{{4\vec{p}_{\rm
n}^{\,2}}}\,P_{ee}\left({\left|{\vec{p}_{\rm
n}}\right|,L,\overline{H}}\right),$ (16)

where

$\left\langle{\left|M_{\rm P}\right|^{2}}\right\rangle=4G_{\rm
F}^{2}\left[-{g^{\mu\nu}\left({qp_{\rm n}}\right)+q^{\mu}p_{\rm
n}^{\nu}+p_{\rm n}^{\mu}q^{\nu}-{\rm
i}\varepsilon^{\mu\nu\rho\sigma}q_{\rho}\left({p_{\rm
n}}\right)_{\sigma}}\vphantom{\left({p_{\rm
n}}\right)_{\rho}}\right]W_{\mu\nu}^{(1)}$ (17)

is the squared modulus of the amplitude of the production process;

$\left\langle{\left|M_{\rm D}\right|^{2}}\right\rangle=4G_{\rm
F}^{2}\left[-{g^{\mu\nu}\left({p_{\rm n}k}\right)+p_{\rm
n}^{\mu}k^{\nu}+k^{\mu}p_{\rm n}^{\nu}-{\rm
i}\varepsilon^{\mu\nu\rho\sigma}\left({p_{\rm
n}}\right)_{\rho}k_{\sigma}}\right]W_{\mu\nu}^{(2)}$ (18)

is the squared modulus of the amplitude of the detection process;

$W_{\mu\nu}^{(l)}=W_{\mu\nu}^{(l,{\rm S})}+{\rm i}W_{\mu\nu}^{(l,{\rm
A})}=\left\langle{j_{\mu}^{(l)}\left({j_{\nu}^{(l)}}\right)^{+}}\right\rangle,\quad
l=1,2,$ (19)

are the nuclear tensors characterizing the interaction of nuclei 1,
$1^{\prime}$ and 2, $2^{\prime}$ with the leptons, their symmetric parts
$W_{\mu\nu}^{(l,{\rm S})}$ being real and the anti-symmetric ones ${\rm
i}W_{\mu\nu}^{(l,{\rm A})}$ being imaginary;

$\begin{split}P_{ee}\left({\left|{\vec{p}_{\rm
n}}\right|,L,\overline{H}}\right)&=\ 1-\sum\limits_{\scriptstyle
i,k=1\hfill\atop\scriptstyle
k<i\hfill}^{3}{\left|{U_{1i}}\right|^{2}\left|{U_{1k}}\right|^{2}\left\\{{\sin^{2}\left[{\left({\frac{{\Delta
m_{ik}^{2}}}{{4\left|{\vec{p}_{\rm n}}\right|}}-\frac{{\mu_{0}\Delta
m_{ik}\overline{H}}}{2}}\right)L}\right]+}\right.}\\\
&+\sin^{2}\left[{\left({\frac{{\Delta m_{ik}^{2}}}{{4\left|{\vec{p}_{\rm
n}}\right|}}-\frac{{\mu_{0}\Sigma
m_{ik}\overline{H}}}{2}}\right)L}\right]+\sin^{2}\left[{\left({\frac{{\Delta
m_{ik}^{2}}}{{4\left|{\vec{p}_{\rm n}}\right|}}+\frac{{\mu_{0}\Sigma
m_{ik}\overline{H}}}{2}}\right)L}\right]+\\\
&\left.{+\sin^{2}\left[{\left({\frac{{\Delta
m_{ik}^{2}}}{{4\left|{\vec{p}_{\rm n}}\right|}}+\frac{{\mu_{0}\Delta
m_{ik}\overline{H}}}{2}}\right)L}\right]}\right\\}-\sum\limits_{i=1}^{3}{\left|{U_{1i}}\right|^{4}\sin^{2}\left({\mu_{0}m_{i}\overline{H}L}\right)}\end{split}$
(20)

is the neutrino oscillation probability depending on the neutrino momentum
${\vec{p}_{\rm n}}$, $L$ is the distance between the source and detector and
$\overline{H}$ denotes the mean value of the transverse magnetic field; we
also introduce the notations similar to the usual ones $\Delta
m_{ik}^{2}\equiv m_{i}^{2}-m_{k}^{2}$:

$\Delta m_{ik}\equiv m_{i}-m_{k},\qquad\Sigma m_{ik}\equiv m_{i}+m_{k}.$ (21)

In the case of two neutrino flavors formula (20) is consistent with the ones
derived in papers [17, 18] for a homogeneous magnetic field, although it looks
different.

We note that, since the neutrinos are produced and detected in the absence of
magnetic field and the field is transverse to the neutrino path, the energy
and momentum are conserved in the process under consideration. Following the
prescription formulated in papers [20, 21, 22], in order to find the
probability of the process, before integrating over the phase volume of the
final particles, one must multiply expression (16) not only by the delta
function of energy-momentum conservation, but also by a delta function, which
would guarantee that the momentum $\vec{p}_{\rm n}$ of the intermediate
neutrinos is directed along the vector $\vec{n}$. Since we have calculated the
squared modulus of the amplitude in the approximation of massless neutrinos,
it is natural to calculate the probability in the same approximation, i.e., to
choose $p_{\rm n}^{2}=0$. A special value of $p_{\rm n}$ satisfying these
conditions will be denoted by $p$, where $\vec{p}$ is directed from the source
to the detector and $p^{2}=0$.

Thus, we multiply the squared modulus of the amplitude by the delta function
of energy-momentum conservation
$\left({2\pi}\right)^{4}\delta\big{(}P^{(1)}+P^{(2)}-P^{(1^{\prime})}-q-P^{(2^{\prime})}-k\big{)}$
and also by the delta function
$2\pi\,\delta\left({P^{(1)}-P^{(1^{\prime})}-q-p}\right)$, as well as
substitute $p$ instead of $p_{\rm n}$ everywhere in (16) [20]. In so doing, we
fix the momentum of the intermediate neutrinos, whose direction is determined
by the source-detector relative position, and, after the integration over the
phase volume, find the differential probability ${{{{\rm
d}^{3}W}\mathord{\left/{\vphantom{{{\rm d}^{3}W}{{\rm
d}^{3}p}}}\right.\kern-1.2pt}{{\rm d}^{3}p}}}$ of the process with a definite
neutrino momentum. Since the experimental situation determines only the
direction of neutrino momentum, but not its magnitude, we must also integrate
${{{{\rm d}^{3}W}\mathord{\left/{\vphantom{{{\rm d}^{3}W}{{\rm
d}^{3}p}}}\right.\kern-1.2pt}{{\rm d}^{3}p}}}$ with respect to
$\left|{\vec{p}}\right|$ over the admissible values. The final probability of
the process under consideration reads:

$\frac{{{\rm d}W}}{{{\rm
d}\Omega}}=\int\limits_{\left|{\vec{p}}\right|_{\min}}^{\left|{\vec{p}}\right|_{\max}}{\frac{{{\rm
d}^{3}W}}{{{\rm d}^{3}p}}\left|{\vec{p}}\right|^{2}{\rm
d}|{\vec{p}}|}=\int\limits_{\left|{\vec{p}}\right|_{\min}}^{\left|{\vec{p}}\right|_{\max}}{\frac{{{\rm
d}^{\rm 3}W_{\rm P}}}{{{\rm d}^{3}p}}\,W_{\rm
D}\,P_{ee}\left({\left|{\vec{p}}\right|,L,\overline{H}}\right)\left|{\vec{p}}\right|^{2}{\rm
d}|{\vec{p}}|}.$ (22)

Here

$\frac{{{\rm d}^{3}W_{\rm P}}}{{{\rm
d}^{3}p}}=\frac{1}{{2E^{(1)}}}\frac{1}{{\left({2\pi}\right)^{3}2p^{0}}}\int{\frac{{{\rm
d}^{3}q}}{{\left({2\pi}\right)^{3}2q^{0}}}\frac{{{\rm
d}^{3}P^{(1^{\prime})}}}{{\left({2\pi}\right)^{3}2E^{(1^{\prime})}}}\left.{\left\langle{\left|{M_{\rm
P}}\right|^{2}}\right\rangle}\right|_{p_{\rm
n}=p}\left({2\pi}\right)^{4}\delta\left({P^{(1)}-P^{(1^{\prime})}-q-p}\right)}$
(23)

is the differential probability of the decay of nucleus 1 into nucleus
$1^{\prime}$, positron and a massless fermion with momentum $\vec{p}$, which
coincides with the sum of differential probabilities of the decay of nucleus 1
into nucleus $1^{\prime}$, positron and all three neutrino mass eigenstates;

$W_{\rm D}=\frac{1}{{2E^{(2)}2p^{0}}}\int{\frac{{{\rm
d}^{3}k}}{{\left({2\pi}\right)^{3}2k^{0}}}\frac{{{\rm
d}^{3}P^{(2^{\prime})}}}{{\left({2\pi}\right)^{3}2E^{(2^{\prime})}}}\left.{\left\langle{\left|{M_{\rm
D}}\right|^{2}}\right\rangle}\right|_{p_{\rm
n}=p}\left({2\pi}\right)^{4}\delta\left({P^{(2)}+p-P^{(2^{\prime})}-k}\right)}$
(24)

is the probability of the scattering process of the massless fermion and
nucleus 2 with the production of nucleus $2^{\prime}$ and an electron, which
coincides with the sum of the probabilities of the scattering processes of all
three neutrino mass eigenstates and nucleus 2. The lower limit of integration
$\left|{\vec{p}}\right|_{\min}$ is determined by the threshold of the
registration process and the upper one $\left|{\vec{p}}\right|_{\max}$ is
determined by the energy-momentum conservation in the production vertex. In
what follows, we assume the initial nuclei 1 and 2 to be at rest and put their
momenta $\vec{P}^{(1)}$, $\vec{P}^{(2)}$ equal to zero, then the integration
limits are given by [27]

$\left|{\vec{p}}\right|_{\min}=\frac{{\left({M_{2^{\prime}}+m_{e}}\right)^{2}-M_{2}^{2}}}{{2M_{2}}},\qquad\left|{\vec{p}}\right|_{\max}=\frac{{M_{1}^{2}-\left({M_{1^{\prime}}+m_{e}}\right)^{2}}}{{2M_{1}}},$
(25)

where $M_{l}$, $l=1,1^{\prime},2,2^{\prime}$, are the nuclear masses and
$m_{e}$ is the electron mass. In the next section we will apply formula (22)
to specific neutrino oscillation processes.

It is easy to verify that one arrives at an expression of the same form as
(22), if the production process is not a nuclear decay but electron capture:

$e^{-}+{{}^{A_{1}}_{Z_{1}}{\rm X}}\to{{}^{A_{1}}_{Z_{1}-1}{\rm X}}+\nu_{i}.$
(26)

The only difference is that the differential decay probability (23) must be
replaced by the differential scattering probability. Oscillating factor (20)
does not change. However, since the reaction has only two particles in the
final state, the magnitudes of the momenta are already fixed by energy-
momentum conservation. Hence, the differential probability ${{{{\rm
d}^{3}W_{\rm P}}\mathord{\left/{\vphantom{{{\rm d}^{3}W_{\rm P}}{{\rm
d}^{3}p}}}\right.\kern-1.2pt}{{\rm d}^{3}p}}}$ for reaction (26) is singular,
and the integration with respect to neutrino momentum magnitude leads to

$\frac{{{\rm d}W}}{{{\rm
d}\Omega}}=\int\limits_{\left|{\vec{p}}\right|_{\min}}^{\left|{\vec{p}}\right|_{\max}}{\frac{{{\rm
d}^{\rm 3}W_{\rm P}}}{{{\rm d}^{3}p}}\,W_{\rm
D}\,P_{ee}\left({\left|{\vec{p}}\right|,L,\overline{H}}\right)\left|{\vec{p}}\right|^{2}{\rm
d}|{\vec{p}}|}=\frac{{{\rm d}W_{\rm P}}}{{{\rm d}\Omega}}\left.{W_{\rm
D}}\right|_{\left|{\vec{p}}\right|=\left|{\vec{p}}\right|^{*}}P_{ee}\left({\left|{\vec{p}}\right|^{*},L,\overline{H}}\right),$
(27)

where $\left|{\vec{p}}\right|^{*}$ is the neutrino momentum magnitude selected
by the energy-momentum conservation in the production process and ${{{{\rm
d}W_{\rm P}}\mathord{\left/{\vphantom{{{\rm d}W_{\rm P}}{{\rm
d}\Omega}}}\right.\kern-1.2pt}{{\rm d}\Omega}}}$ is the differential
probability of production of the neutrino moving in the direction of the
detector. Since the initial particles always have a momentum distribution, one
must average probability (27) over the momenta of these particles.

In the same way we can consider a process, where the neutrino is detected
through both the charged- and neutral-current interactions with an electron.
The process is described by the diagrams

$e^{+}(q)$$x$$W^{+}$$\nu_{i}(p_{\rm
n})$$\nu_{i}(k_{2})$$y$$Z$$e^{-}(k_{1})$$e^{-}(k)$(28)

.

$e^{+}(q)$$x$$W^{+}$$\nu_{k}(p_{\rm
n})$$e^{-}(k)$$y$$W^{+}$$e^{-}(k_{1})$$\nu_{i}(k_{2})$(29)

The amplitude corresponding to diagram (29) should be summed over the type $k$
of the intermediate neutrino mass eigenstate, because all of them contribute.
Since only the final electron is detected in the experiment, the probability
of the process with $i$th neutrino mass eigenstate in the final state should
be summed over $i$ to give the probability of registering an electron.

In order to use the foregoing formulas without redefinitions, we retain the
previous notations for particles’ 4-momenta and nuclear values. Besides, we
introduce the missing notations $k_{1}$ and $k_{2}$ for the 4-momenta of the
incoming electron and outgoing neutrino $\nu_{i}$, respectively.

Using the approximation of Fermi’s interaction and distance-dependent
propagator (11), keeping the neutrino masses only in the exponential, one can
write down the amplitude corresponding to neutral-current diagram (28) in the
momentum representation as follows:

$\begin{split}&M_{i}^{{\rm nc}}={\rm i}\frac{{G_{\rm
F}^{2}}}{{8\left|{\vec{p}_{\rm
n}}\right|}}U_{1i}^{*}\,j_{\mu}^{(1)}\left({\vec{P}^{(1)},\vec{P}^{(1^{\prime})}}\right)\bar{\nu}_{i}\left({\vec{k}_{2}}\right)\left({1+\gamma^{5}}\right)\gamma^{\rho}\,\hat{p}_{\rm
n}\times\\\ &\hphantom{M_{i}^{{\rm nc}}=}\times\left[{\left({1-{\rm
i}\vec{j}\vec{\gamma}}\right){\rm e}^{{\rm i}\frac{{p_{\rm
n}^{2}-m_{i}^{2}+2\mu_{0}m_{i}\left|{\vec{p}_{\rm
n}}\right|\overline{H}}}{{2\left|{\vec{p}_{\rm n}}\right|}}L}+\left({1+{\rm
i}\vec{j}\vec{\gamma}}\right){\rm e}^{{\rm i}\frac{{p_{\rm
n}^{2}-m_{i}^{2}-2\mu_{0}m_{i}\left|{\vec{p}_{\rm
n}}\right|\overline{H}}}{{2\left|{\vec{p}_{\rm
n}}\right|}}L}}\right]\gamma^{\mu}\left({1-\gamma^{5}}\right)v\left({\vec{q}}\right)\times\\\
&\hphantom{M_{i}^{{\rm
nc}}=}\times\left[{\left({-\frac{1}{2}+\sin^{2}\theta_{\rm
w}}\right)\bar{u}\left({\vec{k}}\right)\gamma_{\rho}\left({1-\gamma^{5}}\right)u\left({\vec{k}_{1}}\right)+\sin^{2}\theta_{\rm
w}\,\bar{u}\left({\vec{k}}\right)\gamma_{\rho}\left({1+\gamma^{5}}\right)u\left({\vec{k}_{1}}\right)}\right].\end{split}$
(30)

The amplitude corresponding to charged-current diagram (29) summed over the
index $k$ reads

$\begin{split}&M_{i}^{{\rm cc}}=-{\rm i}\frac{{G_{\rm
F}^{2}}}{{8\left|{\vec{p}_{\rm
n}}\right|}}U_{1i}^{*}\,j_{\mu}^{(1)}\left({\vec{P}^{(1)},\vec{P}^{(1^{\prime})}}\right)\bar{u}\left({\vec{k}}\right)\left({1+\gamma^{5}}\right)\gamma^{\rho}\,\hat{p}_{\rm
n}\times\\\ &\hphantom{M_{i}^{{\rm
cc}}=}\times\sum\limits_{k=1}^{3}{\left|{U_{1k}}\right|^{2}\left[{\left({1-{\rm
i}\vec{j}\vec{\gamma}}\right){\rm e}^{{\rm i}\frac{{p_{\rm n}^{\rm
2}-m_{k}^{2}+2\mu_{0}m_{k}\left|{\vec{p}_{\rm
n}}\right|\overline{H}}}{{2\left|{\vec{p}_{\rm n}}\right|}}L}+\left({1+{\rm
i}\vec{j}\vec{\gamma}}\right){\rm e}^{{\rm i}\frac{{p_{\rm n}^{\rm
2}-m_{k}^{2}-2\mu_{0}m_{k}\left|{\vec{p}_{\rm
n}}\right|\overline{H}}}{{2\left|{\vec{p}_{\rm
n}}\right|}}L}}\right]}\times\\\ &\hphantom{M_{i}^{{\rm
cc}}=}\times\gamma^{\mu}\left({1-\gamma^{5}}\right)\upsilon\left({\vec{q}}\right)\cdot\bar{\nu}_{i}\left({\vec{k}_{2}}\right)\gamma_{\rho}\left({1-\gamma^{5}}\right)u\left({\vec{k}_{1}}\right).\end{split}$
(31)

The squared modulus of the total amplitude $M_{i}=M^{\rm nc}_{i}+M^{\rm
cc}_{i}$, averaged and summed over particles’ polarizations, factorizes in the
approximation $p_{\rm n}^{2}=0$. Following the procedure described above and
summing the differential probability over the final neutrino type $i$, we
arrive at the probability of detecting an electron in the process at hand:

$\frac{{{\rm d}W}}{{{\rm
d}\Omega}}=\int\limits_{\left|{\vec{p}}\right|_{\min}}^{\left|{\vec{p}}\right|_{\max}}{\frac{{{\rm
d}^{\rm 3}W_{\rm P}}}{{{\rm d}^{3}p}}\,W_{\rm
D}\left({L,\overline{H}}\right)\left|{\vec{p}}\right|^{2}{\rm d}|{\vec{p}}|}.$
(32)

Here the differential production probability ${{{{\rm d}^{3}W_{\rm
P}}\mathord{\left/{\vphantom{{{\rm d}^{3}W_{\rm P}}{{\rm
d}^{3}p}}}\right.\kern-1.2pt}{{\rm d}^{3}p}}}$ is given by (23) and the total
detection probability $W_{\rm D}\left({L,\overline{H}}\right)$, which is the
sum of the detection probabilities for each neutrino mass eigenstate in the
final state, reads

$\begin{split}&W_{\rm D}\left({L,\overline{H}}\right)=\frac{{G_{\rm
F}^{2}m_{e}}}{{2\pi}}\left\\{{\left[{\left({1-4\sin^{2}\theta_{\rm
w}+8\sin^{4}\theta_{\rm w}}\right)\Delta T-4\sin^{2}\theta_{\rm
w}\left({\frac{{\sin^{2}\theta_{\rm
w}}}{{T_{\max}}}-\frac{{m_{e}}}{{4\left|{\vec{p}}\right|^{2}}}}\right)\Delta
T^{2}\,+}\right.}\right.\\\ &\hphantom{W_{\rm
D}\left({L,\overline{H}}\right)=\frac{{G_{\rm
F}^{2}m_{e}}}{{2\pi}}\bigg{\\{}\bigg{[}}\left.{+\,\frac{4}{3}\frac{\sin^{4}\theta_{\rm
w}}{{\left|{\vec{p}}\right|^{2}}}\Delta
T^{3}}\right]\sum\limits_{i=1}^{3}{\left|{U_{1i}}\right|^{2}\cos^{2}\left({\mu_{0}m_{i}\overline{H}L}\right)+}\\\
&\hphantom{W_{\rm D}\left({L,\overline{H}}\right)=\frac{{G_{\rm
F}^{2}m_{e}}}{{2\pi}}\bigg{\\{}}\left.{+\,8\sin^{2}\theta_{\rm w}\left[{\Delta
T-\frac{{m_{e}}}{{4\left|{\vec{p}}\right|^{2}}}\Delta
T^{2}}\right]P_{ee}\left({\left|{\vec{p}}\right|,L,\overline{H}}\right)}\right\\},\end{split}$
(33)

where

$\Delta T\equiv T_{\max}-T_{\min}\,,\qquad\Delta T^{2}\equiv
T_{\max}^{2}-T_{\min}^{2}\,,\qquad\Delta T^{3}\equiv
T_{\max}^{3}-T_{\min}^{3}\,,$ (34)
$T_{\max}=\frac{{2\vec{p}^{\,2}}}{{2\left|{\vec{p}}\right|+m_{e}}}.$ (35)

Here $T_{\max}$ is the maximum kinetic energy of the final electron in the
detection process, determined by energy-momentum conservation, and $T_{\min}$
is the minimum kinetic energy of the electron accessible for registrations by
a specific detector. The lower integration limit
$\left|{\vec{p}}\right|_{\min}$ in (32) is connected with $T_{\min}$ by a
relation similar to (35), resolving which we get

$\left|{\vec{p}}\right|_{\min}=\frac{1}{2}\left({T_{\min}+\sqrt{T_{\min}\left({T_{\min}+2m_{e}}\right)}}\right).$
(36)

It is easy to verify that the detection probability (33) is expressed through
the oscillating factor (20) and the Standard Model neutrino scattering
probabilities $W_{\nu_{e}e}$ and $W_{\nu_{\mu}e}$, calculated for the same
values of $T_{\min}$ and $T_{\max}$, by the relation

$W_{\rm
D}=P_{ee}\left({\left|{\vec{p}}\right|,L,\overline{H}}\right){W_{\nu_{e}e}\left({\left|{\vec{p}}\right|}\right)}+\left(\sum\limits_{i=1}^{3}{\left|{U_{1i}}\right|^{2}\cos^{2}\left({\mu_{0}m_{i}\overline{H}L}\right)}-P_{ee}\left({\left|{\vec{p}}\right|,L,\overline{H}}\right)\right)W_{\nu_{\mu}e}\left({\left|{\vec{p}}\right|}\right),$
(37)

which differs from the standard one in vacuum by an extra field-dependent
oscillating factor taking into account the neutrino spin rotation.

##  4 Specific examples

Let us discuss some examples, where neutrinos are produced in the decay of
${{}^{15}{\rm O}}$

${{}^{15}{\rm O}}\to{{}^{15}{\rm N}}+e^{+}+\nu_{i}$ (38)

or in the electron capture reaction

${{}^{7}{\rm Be}}+e^{-}\to{{}^{7}{\rm Li}}+\nu_{i}$ (39)

and detected by a gallium-germanium detector in the reaction

$\nu_{i}+{{}^{71}{\rm Ga}}\\\ \to{{}^{71}{\rm Ge}}+e^{-}.$ (40)

Both reactions contribute to the solar neutrino flux, the first reaction
having a wide energy spectrum and the second one producing almost
monoenergetic neutrinos.

We start with the ${{}^{15}{\rm O}}$-source. Nuclear reactions (38), (40) are
allowed transitions [28], which means that we can neglect the nucleon
positions and momenta and consider the interaction or decay of a nucleon as if
it were at rest. Thus, neglecting the dependence of the nuclear form-factors
on the momentum transfer [28] and also neglecting the possible contribution of
the excited states of the final nuclei, one can approximate the product of the
differential probability of neutrino production and the probability of
neutrino detection by the function

$\displaystyle\begin{split}\frac{{{\rm d}^{3}W_{\rm P}}}{{{\rm
d}^{3}p}}\,W_{\rm D}=&\
C\sqrt{\left({\left|{\vec{p}}\right|_{\max}-\left|{\vec{p}}\right|}\right)\left({\left|{\vec{p}}\right|_{\max}-\left|{\vec{p}}\right|+2m_{e}}\right)}\left({\left|{\vec{p}}\right|_{\max}-\left|{\vec{p}}\right|+m_{e}}\right)\times\\\
&\times\sqrt{\left({\left|{\vec{p}}\right|-\left|{\vec{p}}\right|_{\min}}\right)\left({\left|{\vec{p}}\right|-\left|{\vec{p}}\right|_{\min}+2m_{e}}\right)}\left({\left|{\vec{p}}\right|-\left|{\vec{p}}\right|_{\min}+m_{e}}\right).\end{split}$
(41)

Here $C$ is a constant, the explicit form of which is unimportant for us,
because we will normalize the probability (22) so that it equals unity at the
point $L=0$; $\left|{\vec{p}}\right|_{\max}$ and
$\left|{\vec{p}}\right|_{\min}$ are the same as the integration limits in
(22), determined by (25). For the production and detection processes under
consideration we have:

$\left|{\vec{p}}\right|_{\max}=1732{\ \rm
keV,}\quad\left|{\vec{p}}\right|_{\min}=232{\ \rm keV}.$ (42)

Below the following values of the mixing angles are used [29]:

$\sin^{2}{\theta_{12}}=0.307,\qquad\sin^{2}{\theta_{23}}=0.545,\qquad\sin^{2}{\theta_{13}}=2.18\cdot
10^{-2}.$ (43)

Let the constant $\mu_{0}$ take the value predicted by the Standard Model,
$\mu_{0}={{3eG_{\rm F}}\mathord{\left/{\vphantom{{3eG_{\rm
F}}{8\sqrt{2}\pi^{2}}}}\right.\kern-1.2pt}{8\sqrt{2}\pi^{2}}}=9.488\cdot
10^{-26}{\ \rm eV^{-2}}$. Taking into account the experimental restrictions
for the masses of the normally ordered neutrinos [29], which is the most
likely scenario,

$\Delta m_{21}^{2}=7.53\cdot 10^{-5}{\ \rm eV}^{2},\qquad\Delta
m_{32}^{2}=2.45\cdot 10^{-3}{\ \rm eV}^{2},$ (44)

and the cosmological limit [30]

$m_{1}+m_{2}+m_{3}<0.120{\ \rm eV},$ (45)

we first take one set of masses,

$m_{1}=0.0114{\ \rm eV},\qquad m_{2}=0.0143{\ \rm eV},\qquad m_{3}=0.0515{\
\rm eV},$ (46)

and then the second one,

$m_{1}=6\cdot 10^{-4}{\ \rm eV},\qquad m_{2}=8.7\cdot 10^{-3}{\ \rm eV},\qquad
m_{3}=0.0503{\ \rm eV}.$ (47)

The results of numerical integration of formula (22) in the case of
homogeneous magnetic field with these parameters for the two sets of masses,
in the absence of external magnetic field and for its transverse values
$10^{15}$, $10^{16}$, $10^{17}$ G are presented in Fig. 1.

a) The first set of masses (46).

b) The second set of masses (47).

Figure 1: Normalized probabilities (22) of the neutrino oscillation processes
in constant homogeneous magnetic field of various magnitude; neutrino
production in the ${{}^{15}{\rm O}}$ decay and detection by a Ga-Ge detector.

Such huge magnitudes of the magnetic field, close to the magnetism limit of
neutron stars $10^{18}$ G, are taken just for an illustration, in order to
show both the momentum-dependent and field-dependent oscillations in one
picture. As it was discussed in [23], the momentum-dependent oscillations fade
out at a rather short distance from the source due to the presence of the
neutrino momentum spread, and at larger distances only the field-dependent
oscillations are basically left.

Now let us consider a source, where reaction (39), ${{}^{7}{\rm
Be}}+e^{-}\to{{}^{7}{\rm Li}}+\nu_{i}$, takes place. This reaction also
belongs to the allowed transitions, and one can neglect nuclear form-factors
in calculating the matrix elements. In the non-relativistic approximation the
squared modulus of the production amplitude, averaged and summed over the
polarizations of the nuclei and particles, can be written as [31]

$\left\langle{\left|{M_{\rm P}}\right|^{2}}\right\rangle=q^{0}p_{\rm
n}^{0}\left({C_{0}+C_{1}\vec{v}_{e}\vec{v}_{\nu}}\right),$ (48)

where $C_{0,1}$ are constants, $q^{0}$ is the initial electron energy, $p_{\rm
n}^{0}$ is the final neutrino energy and $\vec{v}_{e}$, $\vec{v}_{\nu}$ are
the electron and neutrino velocities, respectively. Reaction (39), having a
two-particle final state, produces neutrinos with a fixed energy, when the
initial particles have definite energies. The neutrino energy has only a
thermal broadening due to the spread in the energies of the initial particles.
Since the electron mass is 4 orders of magnitude smaller then the ${{}^{7}{\rm
Be}}$ nucleus mass, it is the electron energy spread that mainly contributes
to the neutrino energy broadening. Taking the plasma temperature to be
$T=1.5\cdot 10^{7}$ K, as in the center of the Sun, which is consistent with
the non-relativistic approximation, we average probability (27) with the
Maxwell-Boltzmann distribution for the initial electron momentum. As a result,
the term with the velocities vanishes, and, neglecting the exited states of
${{}^{7}{\rm Li}}$ nucleus, we arrive at the expression

$\begin{split}&\overline{\frac{{{\rm d}W}}{{{\rm
d}\Omega}}}=C\int\limits_{\Delta}^{\infty}{\sqrt{\left({\left|{\vec{p}}\right|-\left|{\vec{p}}\right|_{\min}}\right)\left({\left|{\vec{p}}\right|-\left|{\vec{p}}\right|_{\min}+2m_{e}}\right)}\left({\left|{\vec{p}}\right|-\left|{\vec{p}}\right|_{\min}+m_{e}}\right)\times}\\\
&\hskip 170.71652pt\times{\rm
e}^{-\frac{\left|{\vec{p}}\right|-\Delta}{{kT}}}\sqrt{\left|{\vec{p}}\right|-\Delta}\,P_{ee}\left({\left|{\vec{p}}\right|,L,\overline{H}}\right)\left|{\vec{p}}\right|^{2}{\rm
d}|{\vec{p}}|,\end{split}$ (49)

where $\Delta=M_{{\rm Be}}+m_{e}-M_{{\rm Li}}=862$ keV is the reaction energy
release (the neutrino mass is neglected), $\left|{\vec{p}}\right|_{\min}$ for
a Ga-Ge detector is presented in (42), $\left|{\vec{p}}\right|$ is the
neutrino momentum magnitude and $C$ is a constant. Performing the numerical
integration in the latter formula for two sets of masses (46)–(47) and a Ga-Ge
detector, one arrives at the results depicted in Fig. 2.

a) The first set of masses (46).

b) The second set of masses (47).

Figure 2: Averaged normalized probabilities (49) of the neutrino oscillation
processes in constant homogeneous magnetic field of various magnitude;
neutrino production in the ${{}^{7}{\rm Be}}$-electron interaction and
detection by a Ga-Ge detector.

We see, once again, that the narrower neutrino momentum distribution leads to
larger coherence lengths of the momentum-dependent oscillations. Namely, in
the case of ${{}^{7}{\rm Be}}\,+\,e^{-}$ source of temperature $T=1.5\cdot
10^{7}$ K the coherence length turns out to be about 10 000 km. We do not
present the corresponding figure, because the oscillations actually merge into
a band on this scale. The curves in Fig. 2 b) resemble those obtained in [18]
for the spin-flavor oscillations of monoenergetic neutrinos in homogeneous
magnetic field.

Borexino [32] and GEMMA [33] experiments restrict the neutrino magnetic moment
from above by the value about $2.8\cdot 10^{-11}\,\mu_{\rm B}=8.3\cdot
10^{-18}{\ \rm eV^{-1}}$, where $\mu_{\rm
B}=e\hbar\mathord{\left/{\vphantom{e\hbar{2m_{e}c}}}\right.\kern-1.2pt}{2m_{e}c}$
is the Bohr magneton. This bound is 10 orders of magnitude larger than the
value, predicted with the Standard Model $\mu_{0}$, which allows 10 orders of
magnitude weaker magnetic field for the same effect.

The results depicted in Figs. 1, 2 refer to the case of a homogeneous magnetic
field present along the entire neutrino path. If the neutrinos travel in
magnetic field only a part of the path, for fairly large values of $L$, where
the momentum-dependent oscillations fade out, the normalized probabilities of
all processes are defined only by the magnetic field and go to the asymptotic
value

$W_{\rm
asym}=\sum\limits_{i=1}^{3}\left|{U_{1i}}\right|^{4}-\sum\limits_{i=1}^{3}{\left|{U_{1i}}\right|^{4}\sin^{2}\delta_{i}}\,,\quad\delta_{i}=\mu_{0}m_{i}\int\limits_{D}H\left({l}\right){\rm
d}l\,,$ (50)

where $\sum\nolimits_{i=1}^{3}\left|{U_{1i}}\right|^{4}\simeq 0.550$ is the
asymptotic value in vacuum, $\delta_{i}$ is the phase accumulated by the
neutrino mass eigenstate $\nu_{i}$ on its path and $D$ is the field region
with respect to coordinate $l$ along the neutrino trajectory.

Let us consider, for example, the solar neutrinos, which are produced in the
solar core. As we have shown, even for monoenergetic neutrino sources the
coherence length is of the order of 10 000 km. Therefore, when the neutrinos
come to the convective zone, all the momentum-dependent oscillations fade out,
and only the oscillations due to the solar magnetic field present in this zone
remain. These oscillations for $\mu_{1}=2.8\cdot 10^{-11}\,\mu_{\rm B}$, i.e.,
near the upper limit set by the Borexino and GEMMA experiments [32, 33], the
solar magnetic field $10^{4}$ G used in paper [15], and the first mass set
(46) are shown in Fig. 4, where the asymptotic value is given by

$W_{\rm asym}^{\rm
sol}=\sum\limits_{i=1}^{3}\left|{U_{1i}}\right|^{4}-\sum\limits_{i=1}^{3}{\left|{U_{1i}}\right|^{4}\sin^{2}\left(\mu_{0}m_{i}\overline{H}L_{\rm
conv}\right)},\quad\overline{H}L_{\rm conv}=\int\limits_{\scriptstyle{\rm
convective}\atop\scriptstyle{\rm zone}}H\left({l}\right){\rm d}l\,,$ (51)

with $\mu_{0}\overline{H}\simeq 5\cdot 10^{-13}$ and $L_{\rm conv}\simeq
200\,000$ km. We note that, for the neutrino energy above 200 keV, the size of
the field region $L_{\rm conv}$ and the chosen value of the neutrino magnetic
moment, adiabaticity condition (13) is fulfilled with a great accuracy. Since
the thickness of the solar convective zone $L_{\rm conv}$ is fixed and the
product $\mu_{0}\overline{H}$ varies depending on the model, it is useful to
plot the asymptotic value of the normalized probability as a function of this
product, which is shown in Fig. 4 (again, the first neutrino mass set is
chosen).

Figure 3: Normalized probability of the neutrino oscillation process in the
convective zone of the Sun.

Figure 4: Asymptotic value (51) as a function of $\mu_{0}\overline{H}$. Text
for easy picture alignment.

The normalized probability discussed in this paper can be considered as the
ratio of the number of neutrinos detected in the presence of oscillations to
the number of neutrinos that would have been detected in the absence of
oscillations. For the processes with only charged-current interaction its
asymptotic value does not depend on the production and detection processes,
and for this reason it gives a theoretical prediction for the ratio of the
flux of solar neutrinos measured in an experiment with a Ga-Ge (or Cl-Ar)
detector to that predicted by the standard solar model. This ratio for the
GALLEX + GNO experiments is $0.58\pm 0.07$ and for the SAGE experiment
$0.59\pm 0.07$ [34], which is well consistent with the curve in Fig. 4. We see
that there are many values of $\mu_{0}\overline{H}$ giving the asymptotic
value above the lower experimental limit (dotted line). This means that, if we
obtain an experimental value of the neutrino magnetic moments, we will be able
to obtain restrictions on the solar magnetic field from the solar neutrino
experiments, and vice versa.

In the case of neutrino detection through both charged- and neutral-current
interaction the situation is different, because, as it is clearly seen in
formula (37), the oscillating factor
$P_{ee}\left({\left|{\vec{p}}\right|,L,\overline{H}}\right)$ does not
factorize. For this reason the asymptotic value of the normalized probability
depends not only on the production process, but also on the energy range, in
which the neutrinos are detected. Here we consider the same production
processes, the ${}^{15}\rm O$ decay and the electron capture by ${{}^{7}{\rm
Be}}$, and the detection by a Cherenkov detector, which is capable of
measuring the neutrino energy.

For ${}^{15}\rm O$ decay (38), neglecting the dependence of the nuclear form-
factors on the momentum transfer, the differential probability of neutrino
production can again be approximated by the function

$\frac{{{\rm d}^{3}W_{\rm P}}}{{{\rm
d}^{3}p}}=C\sqrt{\left({\left|{\vec{p}}\right|_{\max}-\left|{\vec{p}}\right|}\right)\left({\left|{\vec{p}}\right|_{\max}-\left|{\vec{p}}\right|+2m_{e}}\right)}\left({\left|{\vec{p}}\right|_{\max}-\left|{\vec{p}}\right|+m_{e}}\right),$
(52)

where the maximum neutrino momentum $\left|{\vec{p}}\right|_{\max}=1732$ keV.
We take $\left|{\vec{p}}\right|_{\min}=421$ keV, which corresponds to a water-
based Cherenkov detector. The results of numerical integration for the two
neutrino mass sets are depicted in Fig. 5.

a) The first set of masses (46).

b) The second set of masses (47).

Figure 5: Normalized probabilities (32) of the neutrino oscillation processes
in constant homogeneous magnetic field of various magnitude; neutrino
production in the ${}^{15}\rm O$ decay and detection by a water-based
Cherenkov detector.

The asymptotic values for the oscillation processes with neutral current can
be found as follows. Substituting the asymptotic value of
$P_{ee}\left({\left|{\vec{p}_{\rm n}}\right|,L,\overline{H}}\right)$ (20) into
formula (32) for the probability and using the explicit form of $W_{\rm
D}\left({L,\overline{H}}\right)$ (33), we get the asymptotic value of the
normalized probability for such processes in the form

$\begin{split}W_{{\rm
asym}}=&\sum\limits_{i=1}^{3}{\left|{U_{1i}}\right|^{4}}-\sum\limits_{i=1}^{3}{\left|{U_{1i}}\right|^{4}\sin^{2}\delta_{i}}\
+\\\ &\,+C_{\rm
nc}\left({1-\sum\limits_{i=1}^{3}{\left|{U_{1i}}\right|^{2}\sin^{2}\delta_{i}}-\sum\limits_{i=1}^{3}{\left|{U_{1i}}\right|^{4}}+\sum\limits_{i=1}^{3}{\left|{U_{1i}}\right|^{4}\sin^{2}\delta_{i}}}\right),\end{split}$
(53)

where $\delta_{i}$ are the phases defined in Eq. (50) and the coefficient
$C_{\rm nc}$ is given by

$C_{\rm
nc}=\int\limits_{\left|{\vec{p}}\right|_{\min}}^{\left|{\vec{p}}\right|_{\max}}{\frac{{{\rm
d}^{3}W_{\rm P}}}{{{\rm
d}^{3}p}}\,W_{\nu_{\mu}e}\left|{\vec{p}}\right|^{2}{\rm
d}|{\vec{p}}|}\,\left(\int\limits_{\left|{\vec{p}}\right|_{\min}}^{\left|{\vec{p}}\right|_{\max}}{\frac{{{\rm
d}^{3}W_{\rm P}}}{{{\rm d}^{3}p}}\,W_{\nu_{e}e}\left|{\vec{p}}\right|^{2}{\rm
d}|{\vec{p}}|}\right)^{-1},$ (54)

$W_{\nu_{\alpha}e}$ being the scattering probability of a massless neutrino
flavor state $\alpha$ at an electron, calculated within the framework of the
Standard Model. The term with $C_{\rm nc}$ takes into account the contribution
of the neutral current. Numerical evaluation of $C_{\rm nc}$ for the neutrino
production in the ${}^{15}\rm O$ decay in our approximation gives $C_{\rm
nc}=0.210$. Using the explicit form of $W_{\nu_{\alpha}e}$, one can estimate
$C_{\rm nc}$ in the interval $420\,{\rm
keV}\leq\left|{\vec{p}}\right|_{\min}<\left|{\vec{p}}\right|_{\max}\leq
14\,{\rm MeV}$ as $0.177<C_{\rm nc}<0.321$.

Finally, let us consider production reaction (39) of electron capture by
${{}^{7}{\rm Be}}$. Similar to the case of only the charged-current
interaction, averaging over the electron momentum distribution in the source,
we get the process probability in the form

$\overline{\frac{{{\rm d}W}}{{{\rm
d}\Omega}}}=C\int\limits_{\Delta}^{\infty}{{\rm
e}^{-\frac{\left|{\vec{p}}\right|-\Delta}{{kT}}}\sqrt{\left|{\vec{p}}\right|-\Delta}\,W_{\rm
D}\left({L,\overline{H}}\right)\left|{\vec{p}}\right|^{2}{\rm d}|{\vec{p}}|}.$
(55)

Assuming the plasma temperature to be $T=1.5\cdot 10^{7}$ K and performing the
numerical integration, we obtain the plots presented in Fig. 6.

a) The first set of masses (46).

b) The second set of masses (47).

Figure 6: Averaged normalized probabilities (55) of the neutrino oscillation
processes in constant homogeneous magnetic field of various magnitude;
neutrino production in the electron capture by ${}^{7}{\rm Be}$ and detection
by a Cherenkov detector.

For this process with the solar neutrinos the coherence length is also of the
order of tens of thousand kilometers, and the asymptotic value of the
normalized probability is given by the same expression (53), but the
coefficient $C_{\rm nc}$ is calculated with the neutrino production
probability in the electron capture by ${{}^{7}{\rm Be}}$. The numerical
integration gives $C_{\rm nc}=0.224$, which corresponds to the asymptotic
value in vacuum $W_{{\rm asym}}=0.652$. The ratio of the neutrino flux
measured by the Borexino collaboration to that predicted for ${{}^{7}{\rm
Be}}$ by the standard solar model is $0.62\pm 0.05$ [35]. This is again in a
good agreement with our theoretical value and leaves room for a contribution
from the solar magnetic field, which is always negative.

##  5 Conclusion

In the present paper we have shown that it is possible to give a consistent
quantum field-theoretical description of neutrino oscillations in a magnetic
field in the Standard Model minimally extended by the right neutrino singlets
without use of the neutrino flavor states. The description is performed in
terms of plane waves and is based on the Feynman diagram technique in the
coordinate representation supplemented with modified rules of passing to the
momentum representation. These rules reflect the experimental setting and give
rise to the distance-dependent propagators of neutrino mass eigenstates. The
distance-dependent propagators in a magnetic field are explicitly calculated
and found to split into the sum of two terms corresponding to two possible
neutrino spin orientations and energies.

Processes of neutrino oscillation have been considered, where the neutrinos
are produced and detected through the charged- and neutral-current weak
interactions with nuclei and electrons in the absence of magnetic field, but
the propagation of the neutrinos takes place in a region of magnetic field.
Formulas for the probabilities of the oscillation processes have been derived
and an agreement with the results obtained in the standard quantum-mechanical
description is shown.

An important new result derived within the framework of the approach is
formulas (50) and (53) for the asymptotic values of the normalized probability
of processes, where neutrinos are detected through the interaction with nuclei
or electrons. For a particular process, this value is the observable ratio of
the measurable neutrino flux to that predicted by the standard solar model.
These formulas can be immediately compared with the experimental data. For the
neutrino detection through the interaction with electrons, also limits on the
neutral current contribution to the asymptotic value are set.

The neutrino production in the solar core in ${}^{15}\rm O$ decay and electron
capture by ${{}^{7}{\rm Be}}$ and detection by Ga-Ge and Cherenkov detectors
has been studied. Numerical computations of the normalized probabilities of
these processes have been performed and the results have been found to be in a
good agreement with the experimental data.

The advantages of the approach are the technical simplicity and physical
transparency. It makes use of only the standard tools of perturbative quantum
field theory and does not need wave packets and the neutrino flavor states. In
fact, we have shown that the Standard Model extended by the right neutrinos is
capable of describing not only scattering processes, but also quantum
processes passing at finite space and time intervals, like particle
oscillations, in the framework of the standard Feynman diagram technique and
the modified perturbative formalism.

Finally, we have to note that we have not touched upon the problem of neutrino
oscillations in matter and have chosen the examples of the production and
detection processes in the energy range, where the influence of oscillations
in matter is expected to be weak. A quantum field-theoretical description of
oscillations in matter can be developed along the lines set forth in the
present paper, although the problem is rather complicated, because one has to
invert analytically a 12$\times$12 matrix to find Green’s function of neutrino
mass eigenstates in matter. Such a description is rigorous and very different
from the standard one. Hence the results may also differ. This is a matter of
a special investigation.

## Acknowledgments

The authors are grateful to E. Boos, A. Lobanov, A. Pukhov and Yu. Tchuvilsky
for interesting and useful discussions. Special thanks are due to M. Smolyakov
for reading the manuscript. Analytical calculations of the amplitudes have
been carried out with the help of the CompHEP and Reduce packages. The work of
V. Egorov was supported by the Foundation for the Advancement of Theoretical
Physics and Mathematics “BASIS”.

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# Inverse mean curvature flow and Ricci-pinched three-manifolds

Gerhard Huisken  Gerhard Huisken
University of Tübingen
Department of Mathematics
Auf der Morgenstelle 10
72076 Tübingen, Germany
<EMAIL_ADDRESS>and  Thomas Koerber  Thomas Koerber
University of Vienna
Faculty of Mathematics
Oskar-Morgenstern-Platz 1
1090 Vienna, Austria
https://orcid.org/0000-0003-1676-0824
<EMAIL_ADDRESS>
###### Abstract.

Let $(M,g)$ be a complete, connected, non-compact Riemannian three-manifold
with non-negative Ricci curvature satisfying
$Ric\geq\varepsilon\,\operatorname{tr}(Ric)\,g$ for some $\varepsilon>0$. In
this note, we give a new proof based on inverse mean curvature flow that
$(M,g)$ is either flat or has non-Euclidean volume growth. In conjunction with
the work of J. Lott [lott] and of M.-C. Lee and P. Topping [leetopping], this
gives an alternative proof of a conjecture of R. Hamilton recently proven by
A. Deruelle, F. Schulze, and M. Simon [deruelleschulzesimon] using Ricci flow.

August 28, 2024

##  1\. Introduction

Let $(M,g)$ be a complete, connected, non-compact Riemannian three-manifold.
Recall that $(M,g)$ is called Ricci-pinched if there is $\varepsilon>0$ such
that

(1) $\displaystyle Ric\geq\varepsilon R\,g.$

Here, $Ric$ and $R$ denote the Ricci curvature and the scalar curvature of
$(M,g)$, respectively. The following theorem has been conjectured by R.
Hamilton [chowluni, Conjecture 3.39] and by J. Lott [lott, Conjecture 1.1]. It
has been proven by A. Deruelle, F. Schulze, and M. Simon
[deruelleschulzesimon, Theorem 1.3] under the additional assumption that
$(M,g)$ has bounded curvature. M.-C. Lee and P. Topping [leetopping, Theorem
1.2] have subsequently shown that this additional assumption can be dispensed
with.

######  Theorem 1 ([leetopping, Theorem 1.2]).

Let $(M,g)$ be a complete, connected, non-compact Riemannian three-manifold
that is Ricci-pinched. Then $(M,g)$ is flat.

######  Remark 2.

Previous results in the direction of Theorem 1 have been obtained by B.-L.
Chen and X.-P. Zhu [chenzhu, Main Theorem II] and by J. Lott [lott, Theorem
1.4].

######  Remark 3.

R. Hamilton has shown that every complete, connected, compact Riemannian
three-manifolds that is Ricci-pinched is either flat or smoothly isotopic to a
spherical space form; see [hamilton, Main Theorem] and the discussion on
[deruelleschulzesimon, p. 4].

To describe the contributions in [lott, deruelleschulzesimon, leetopping], let
$p\in M$ and

$\operatorname{AVR}=\frac{3}{4\,\pi}\,\lim_{r\to\infty}\frac{|B_{r}(p)|}{r^{3}}$

be the asymptotic volume ratio of $(M,g)$. Note that, by the Bishop-Gromov
theorem, $\operatorname{AVR}$ is well-defined, independent of $p$, and
satisfies $\operatorname{AVR}\in[0,1]$. J. Lott [lott] has shown that, if
$(M,g)$ is Ricci-pinched and has bounded curvature, then there exists a
smooth, Ricci-pinched solution of Ricci flow coming out of $(M,g)$. By
performing a detailed asymptotic analysis of this flow, J. Lott has proven
that, if $(M,g)$ is not flat, then $(M,g)$ has positive asymptotic volume
ratio; see [lott, Corollary 1.7]. Subsequently, M.-C. Lee and P. Topping
[leetopping] have shown that the assumption that $(M,g)$ has bounded curvature
can be dispensed with. By contrast, if $(M,g)$ has positive asymptotic volume
ratio, A. Deruelle, F. Schulze, and M. Simon [deruelleschulzesimon] have
observed that the asymptotic cone of $(M,g)$ is a three-dimensional Alexandrov
space with non-negative curvature. Moreover, the previous work of A. Deruelle
[Deruelle] respectively of F. Schulze and M. Simon [SchulzeSimon] implies the
existence of an expanding soliton solution with non-negative curvature coming
out of the asymptotic cone of $(M,g)$. Using their stability result
[deruelleschulzesimon, Theroem 1.2] to compare this solution with the Ricci-
pinched solution constructed by J. Lott [lott], they have concluded that
$(M,g)$ is in fact flat.
The goal of this paper is to provide a new proof based on inverse mean
curvature flow of Theorem 1 under the additional assumption that $(M,g)$ has
positive asymptotic volume ratio.

######  Theorem 4.

Let $(M,g)$ be a complete, connected, non-compact Riemannian three-manifold
that is Ricci-pinched. If $(M,g)$ has positive asymptotic volume ratio, then
$(M,g)$ is isometric to flat $\mathbb{R}^{3}$.

######  Remark 5.

In conjunction with the results of J. Lott [lott] and of M.-C. Lee and P.
Topping [leetopping], Theorem 4 provides a new proof of Theorem 1.

######  Remark 6.

Our technique extends to the case where $(M,g)$ has superquadratic volume
growth; see Theorem 11.

We now outline the proof of Theorem 4. Suppose, for a contradiction, that
$(M,g)$ is not flat. Since $(M,g)$ has non-negative Ricci curvature and is
Ricci-pinched, the scalar curvature of $(M,g)$ must be strictly positive at
one point. It follows that there is a closed outward-minimizing hypersurface
$\Sigma\subset M$ such that

(2) $\displaystyle\int_{\Sigma}H^{2}\,\mathrm{d}\mu<16\,\pi.$

Here, $\mathrm{d}\mu$ and $H$ denote the area element and the mean curvature
of $\Sigma$, respectively. Recall that a nested family
$\\{E_{t}\\}_{t=0}^{\infty}$ with strictly mean-convex boundary $\partial
E_{t}\subset M$ flows by inverse mean curvature flow if

$\frac{dx}{dt}=\frac{1}{H}\,\nu.$

Here, $x$ and $\nu$ are the position and the outward normal of $\partial
E_{t}$, respectively. Using that $(M,g)$ has positive asymptotic volume ratio,
the results of L. Mari, M. Rigoli and A. Setti [maririgolisetti], which, in
turn, build on previous work of R. Moser [Moser], show that there exists a
weak solution $\\{E_{t}\\}_{t=0}^{\infty}$ of inverse mean curvature flow with
$\partial^{*}E_{0}=\Sigma$ in the sense of the work of T. Ilmanen and the
first-named author [huiskenilmanen]. Here, $\partial^{*}$ denotes the reduced
boundary. Using (2) and that $(M,g)$ is Ricci-pinched, we show that

$\lim_{t\to\infty}\int_{\partial^{*}E_{t}}H^{2}\,\mathrm{d}\mu=0.$

By contrast, the work of V. Agostiniani, M. Fogagnolo, and L. Mazzieri
[agostinianifogagnolomazzierei] implies that, for every $t\geq 0$,

$\int_{\partial^{*}E_{t}}H^{2}\,\mathrm{d}\mu\geq
16\,\pi\,\operatorname{ARV},$

a contradiction.

### Acknowledgments

The second-named author acknowledges the support of the Lise-Meitner-Project
M3184 of the Austrian Science Fund. This work originated during the authors’
visit to the Hebrew University during the first-named author’s Mark Gordon
Distinguished Visiting Professorship. The authors thank the Hebrew University
for its hospitality. The authors thank Or Hershkovits and Miles Simon for
helpful discussions.

##  2\. Proof of Theorem 4

In this section, we assume that $(M,g)$ is a complete, connected, non-compact
Riemannian three-manifold with non-negative Ricci curvature satisfying (1) for
some $\varepsilon>0$.
The goal of this section is to prove Theorem 4.
We recall the following result of S.-H. Zhu [zhu] which extends previous work
of R. Schoen and S.-T. Yau [SchoenYau, Theorem 3].

######  Proposition 7.

If $(M,g)$ is not flat, then $(M,g)$ is diffeomorphic to $\mathbb{R}^{3}$.

###### Proof.

Using (1), we see that there is $p\in M$ with $Ric(p)>0$. The assertion
follows from [zhu]. ∎

Let $\Sigma\subset M$ be a closed surface with area measure $\mathrm{d}\mu,$
designated normal $\nu$, and mean curvature $H$ with respect to $\nu$. In
Lemma 8 below, $\accentset{\circ}{h}$ denotes the traceless second fundamental
form of $\Sigma$.

######  Lemma 8.

If $\operatorname{genus}(\Sigma)\geq 1$, there holds

(3) $\displaystyle
2\,\int_{\Sigma}Ric(\nu,\nu)+|\accentset{\circ}{h}|^{2}\,\mathrm{d}\mu\geq\int_{\Sigma}H^{2}\,\mathrm{d}\mu$

and, if $\operatorname{genus}(\Sigma)=0$, there holds

(4) $\displaystyle
2\,\int_{\Sigma}Ric(\nu,\nu)\,\mathrm{d}\mu\geq\varepsilon\,\bigg{(}16\,\pi-\int_{\Sigma}H^{2}\,\mathrm{d}\mu\bigg{)}.$

###### Proof.

Integrating the contracted Gauss equation and using the Gauss-Bonnet theorem,
we have

$\int_{\Sigma}H^{2}\,\mathrm{d}\mu=16\,\pi\,(1-\operatorname{genus}(\Sigma))+2\,\int_{\Sigma}|\accentset{\circ}{h}|^{2}\,\mathrm{d}\mu+\int_{\Sigma}4\,Ric(\nu,\nu)-2\,R\,\mathrm{d}\mu.$

Using that $Ric\geq 0$, we have $R\geq Ric(\nu,\nu)$. This implies (3).
In the case where $\operatorname{genus}(\Sigma)=0$, we have, using that
$Ric\geq 0$,

$2\,\int_{\Sigma}R\,\mathrm{d}\mu\geq 16\,\pi-\int_{\Sigma}H^{2}.$

In conjunction with (1), we obtain (4). ∎

######  Lemma 9.

Suppose that $(M,g)$ is not flat. There exists a sequence
$\\{\Sigma_{i}\\}_{i=1}^{\infty}$ of closed surfaces $\Sigma_{i}\subset M$
with

$\lim_{i\to\infty}\int_{\Sigma_{i}}H^{2}\,\mathrm{d}\mu=0.$

###### Proof.

In the case where $\operatorname{AVR}=0$, the assertion follows from
[agostinianifogagnolomazzierei, Theorem 1.1].
In the case where $\operatorname{AVR}>0$, using that $(M,g)$ is not flat and
that $Ric\geq 0$, we see that there is $p\in M$ with $R(p)>0$. Consequently,

(5) $\displaystyle\int_{B_{r}(p)}H^{2}\,\mathrm{d}\mu<16\,\pi$

provided that $r>0$ is sufficiently small; see, e.g., [mondino, Proposition
3.1]. Let $\Sigma^{\prime}\subset M$ be the minimizing hull of $B_{r}(p)$; see
[huiskenilmanen, p. 371]. Using [huiskenilmanen, (1.15)], we see that

$\int_{\Sigma^{\prime}}H^{2}\,\mathrm{d}\mu\leq\int_{B_{r}(p)}H^{2}\,\mathrm{d}\mu.$

By [maririgolisetti, Remark 1.6 and Theorem 1.7], there exists a proper weak
solution $\\{E_{t}\\}_{t=0}^{\infty}$ of inverse mean curvature flow in the
sense of [huiskenilmanen, p. 368] such that
$\Sigma^{\prime}=\partial^{*}E_{0}$. Let $\Sigma_{t}=\partial^{*}E_{t}$. By
Proposition 7 and [huiskenilmanen, Lemma 4.2], $\Sigma_{t}$ is connected for
every $t\geq 0$. According to the results in [huiskenilmanen, §5] and
[Heidusch, Korollar 5.6], $\Sigma_{t}$ is of class $W^{2,2}\cap C^{1,1}$ and
there holds

(6)
$\displaystyle\int_{\Sigma_{0}}H^{2}\,\mathrm{d}\mu\geq\int_{\Sigma_{t}}H^{2}\,\mathrm{d}\mu+2\,\int_{0}^{t}\int_{\Sigma_{s}}Ric(\nu,\nu)+|\accentset{\circ}{h}|^{2}\,\mathrm{d}\mu\,\mathrm{d}s$

for every $t\geq 0$. Clearly, the function

(7) $\displaystyle[0,\infty)\to\mathbb{R},\qquad
t\mapsto\int_{\Sigma_{t}}H^{2}\,\mathrm{d}\mu$

is non-increasing. We claim that

$\lim_{t\to\infty}\int_{\Sigma_{t}}H^{2}\,\mathrm{d}\mu=0.$

Indeed, suppose, for a contradiction, that there is $\delta>0$ such that

$\int_{\Sigma_{t}}H^{2}\,\mathrm{d}\mu\geq\delta$

for every $t\geq 0$. Shrinking $\delta>0$, if necessary, and using (5), we may
assume that

$\int_{\Sigma_{t}}H^{2}\,\mathrm{d}\mu\leq 16\,\pi-\delta$

for every $t\geq 0$. Using Lemma 8, we have

$2\,\int_{\Sigma_{s}}Ric(\nu,\nu)+|\accentset{\circ}{h}|^{2}\,\mathrm{d}\mu\geq\min\\{1,\varepsilon\\}\,\delta$

for every $t\geq 0$. In conjunction with (6), we see that

$\int_{\Sigma_{t}}H^{2}\,\mathrm{d}\mu<0$

for every $t\geq 16\,\pi\,\min\\{1,\varepsilon\\}^{-1}\,\delta^{-1}$. The
assertion follows from this contradiction. ∎

###### Proof of Theorem 4.

Let $(M,g)$ be a connected, complete, non-compact Riemannian three-manifold
that is Ricci-pinched and has positive asymptotic volume ratio
$\operatorname{AVR}$. Suppose, for a contradiction, that $(M,g)$ is not flat
$\mathbb{R}^{3}$. By Lemma 9, there exists a closed surface $\Sigma\subset M$
with

$\int_{\Sigma}H^{2}\,\mathrm{d}\mu<16\,\pi\,\operatorname{AVR}.$

As this is incompatible with [agostinianifogagnolomazzierei, Theorem 1.1], the
assertion follows. ∎

##  Appendix A Superquadratic volume growth

In this section, we assume that $(M,g)$ is a complete, connected, non-compact
Riemannian three-manifold with non-negative Ricci curvature satisfying (1) for
some $\varepsilon>0$. Moreover, we assume that there is $p\in M$ and
$\alpha>0$ with

(8) $\displaystyle 0<\lim_{r\to\infty}\frac{|B_{r}(p)|}{r^{1+\alpha}}<\infty.$

Note that, by the Bishop-Gromov theorem, $\alpha\leq 2$.
The goal of this section is to give an alternative proof, based on inverse
mean curvature flow, of the fact that $(M,g)$ is either flat or has
subquadratic volume growth, that is, $\alpha\leq 1$.

######  Lemma 10.

Suppose that $(M,g)$ is not flat. Then there exists a proper weak solution
$\\{E_{t}\\}_{t=0}^{\infty}$ of inverse mean curvature flow such that, for
every $t\geq 0$,

$\int_{\partial^{*}E_{t}}H^{2}\,\mathrm{d}\mu<16\,\pi\,e^{-t}.$

###### Proof.

By Lemma 9, there is a closed, outward-minimizing surface $\Sigma\subset M$
satisfying

$\varepsilon\,(16\,\pi-\int_{\Sigma}H^{2}\,\mathrm{d}\mu)\geq\int_{\Sigma}H^{2}\,\mathrm{d}\mu.$

By [maririgolisetti, Remark 1.6 and Theorem 1.7], there is a proper weak
solution $\\{E_{t}\\}_{t=0}^{\infty}$ of inverse mean curvature flow with
${\partial^{*}}E_{0}=\Sigma$. Let $\Sigma_{t}=\partial^{*}E_{t}$. As in the
proof of Lemma 9, we see that the function (7) is non-increasing so that, for
every $t\geq 0$,

(9)
$\displaystyle\varepsilon\,(16\,\pi-\int_{\Sigma_{t}}H^{2}\,\mathrm{d}\mu)\geq\int_{\Sigma_{t}}H^{2}\,\mathrm{d}\mu.$

Again, as in the proof of Lemma 9, using (9) and Lemma 8, we see that

$\int_{\Sigma}H^{2}\,\mathrm{d}\mu\geq\int_{\Sigma_{t}}H^{2}\,\mathrm{d}\mu+\int_{0}^{t}\int_{\Sigma_{s}}H^{2}\,\mathrm{d}\mu\,\mathrm{d}s.$

Using that

$\int_{\Sigma}H^{2}\,\mathrm{d}\mu<16\,\pi,$

the assertion follows. ∎

######  Theorem 11.

Let $(M,g)$ be a complete, connected, non-compact Riemannian three-manifold
that is Ricci-pinched and satisfies (10) for some $\alpha>0$. If $(M,g)$ is
not flat, then $\alpha\leq 1$.

###### Proof.

Let $\\{E_{t}\\}_{t=0}^{\infty}$ be a proper weak solution of inverse mean
curvature flow as in Lemma 10. Let $\Sigma_{t}=\partial^{*}E_{t}$. By
[huiskenilmanen, Lemma 5.1] there holds $H>0$ $\mathrm{d}\mu$-almost
everywhere on $\Sigma_{t}$ for almost every $t\geq 0$. Using Hölder’s
inequality, we have, for almost every $t\geq 0$,

(10)
$\displaystyle|\Sigma_{t}|^{2}\leq\int_{\Sigma_{t}}H^{-1}\,\mathrm{d}\mu\,\left(\int_{\Sigma_{t}}H^{2}\,\mathrm{d}\mu\right)^{1/2}\,|\Sigma_{t}|^{1/2}.$

Recall from [huiskenilmanen, Exponential Growth Lemma 1.6], that

(11) $\displaystyle|\Sigma_{t}|=|\Sigma_{0}|\,e^{t}.$

In conjunction with Lemma 10 and (10), we obtain that, for almost every $t\geq
0$,

(12)
$\displaystyle\int_{\Sigma_{t}}H^{-1}\,\mathrm{d}\mu\geq\sqrt{\frac{|\Sigma_{0}|^{3}}{16\,\pi\,}}e^{2\,t}.$

Arguing as in [huiskenilmanen, §5], we see that

$|E_{t}|\geq|E_{0}|+\int_{0}^{t}\,\int_{\Sigma_{s}}H^{-1}\,\mathrm{d}\mu\,\mathrm{d}s$

for every $t\geq 0$. In conjunction with (12) and (11), it follows that

(13) $\displaystyle\liminf_{t\to\infty}|\Sigma_{t}|^{-2}\,|E_{t}|>0.$

Using the results on fake distances in the context of $p$-harmonic functions
in [maririgolisetti, Theorem 1.4 and Theorem 5.4] and on inverse mean
curvature flow in [huiskenilmanen, Minimizing Hull Property 1.4 and Theorem
2.2 ii)], we see that there is a function $\rho\in C^{0,1}(M)$ called fake
distance in [maririgolisetti] and a number $\delta>0$ with the following
properties:

  * $\circ$

There holds, for every $t\geq 0$,

$\\{x\in M:\rho(x)<\delta\,e^{t/2}\\}\subset E_{t}\subset\\{x\in
M:\rho(x)\leq\delta^{-1}\,e^{t/2}\\}.$

  * $\circ$

There holds, for every $r\geq 1$,

$B_{r}(p)\subset\\{x\in M:\rho(x)\leq r\\}\subset
B_{\delta^{-1}\,r^{2/\alpha}}(p).$

  * $\circ$

$\partial\\{x\in M:\rho(x)\leq r\\}$ is outward-minimizing for ever $r>0$.

  * $\circ$

There holds, for every $r\geq 0$,

$|\partial\\{x\in M:\rho(x)\leq r\\}|=4\,\pi\,r^{2}.$

Clearly,

$|E_{t}|\leq|\\{x\in
M:\rho(x)\leq\delta^{-1}\,e^{t/2}\\}|\leq|B_{\delta^{-1-2/\alpha}\,e^{t/\alpha}}(p)|.$

Moreover,

$|\Sigma_{t}|\geq|\partial\\{x\in
M:\rho(x)\leq\delta\,e^{t/2}\\}|=4\,\pi\,\delta^{2}\,e^{t}.$

In conjunction with (13), we conclude that

$\liminf_{t\to\infty}\frac{|B_{\delta^{-1-2/\alpha}\,e^{t/\alpha}}(p)|}{16\,\pi\,\delta^{4}\,e^{2\,t}}>0.$

Consequently,

(14) $\displaystyle\liminf_{r\to\infty}\frac{|B_{r}(p)|}{r^{2\,\alpha}}>0.$

This is incompatible with (8) unless $\alpha\leq 1$. The assertion follows. ∎

######  Remark 12.

Let $M=\mathbb{R}\times S^{1}\times S^{1}$ and $g$ be the flat metric. Note
that $(M,g)$ is Ricci-pinched and satisfies $\eqref{volume growth}$ with
$\alpha=0$. Moreover, $(M,g)$ is foliated by tori with area equal to
$4\,\pi^{2}$. In light of (11), there can be no proper weak solution of
inverse mean curvature flow in $(M,g)$ regardless of the initial data. This
suggests that inverse mean curvature flow is not an adequate technique to
study the case where $\alpha\leq 1$.

## References