# Neutrino-(anti)neutrino forward scattering potential for massive neutrinos
at low energies

P. Dedin Neto  Gleb Wataghin Institute of Physics, University of Campinas,
R. Sérgio Buarque de Holanda, 777 ,Campinas, 13083-859, Brazil
<EMAIL_ADDRESS>E. Kemp  Gleb Wataghin Institute of Physics, University
of Campinas,
R. Sérgio Buarque de Holanda, 777 ,Campinas, 13083-859, Brazil
<EMAIL_ADDRESS>
###### Abstract

In this work, we calculate expression for the potential due to
neutrino-(anti)neutrino forward scattering at low energies ($E<<m_{Z^{0}}$)
for ultra-relativistic massive neutrinos ($E>>m_{\nu}$), a representative
regime within astrophysical scenarios. There is a broadly used expression for
this potential in the literature, which, however, lacks an explicit derivation
from basic principles of quantum field theory. Therefore, this paper has the
intention to guide the reader through the steps and concepts to derive this
potential, trying to be clear and pedagogical. Moreover, we used a rigorous
approach concerning the massive nature of the neutrinos, using massive
quantized neutrino fields throughout the entire process, while the usual
approach is to consider massless neutrino fields at the interaction. In this
context, we explicitly show the validity of the massless neutrino fields
approximation at the ultra-relativistic regime, as expected. As the last step,
we connect the potential expression to the density matrix formalism, which is
a usual framework for works considering neutrino-neutrino interactions. We
also discuss some theoretical details through the paper, such as the normal
ordering of quantum operators and the implications of massive fields in the
neutrino state at its production.

###### keywords:

neutrinos; neutrino-neutrino potential; collective effects.

Received (Day Month Year)Revised (Day Month Year)

PACS Nos.: include PACS Nos.

##  1 Introduction

When neutrinos cross a matter profile, they are subject to a non-zero forward
scattering potential due to charged leptons in the medium. That was first
recognized by Wolfenstein [1], who showed that this potential induces an
effective mass in the neutrinos, affecting their oscillation. After that,
Mikheyev and Smirnov [2] showed that the matter profile along the neutrino
trajectory could lead to resonant regions, where drastic modifications in the
flavor transition probability occur. These works resulted in the well-
established MSW effect. Some authors noted that a similar effect could occur
for neutrino-neutrino interactions while propagating in a medium with a high
density of neutrinos, which can be the case for a supernova environment and
the early universe as well. However, it was recognized later by Pantaleone [3,
4] that the neutrino-neutrino interactions have off-diagonal elements in
flavor basis (Figure 1), allowing the neutrino to exchange flavor in the
interactions. Since then, a large collection of works have been published
analyzing the effects of such neutrino-neutrino interactions. These effects
are often called ”collective oscillations ” [5], where the neutrinos evolve in
a coupled manner.

[horizontal=i1 to i2] i1 [particle=$\nu_{e}(p_{1})$] – [fermion] a [dot] –
[fermion] i2 [particle=$\nu_{\mu}(p_{2})$], f1 [particle=$\nu_{\mu}(p_{2})$] –
[fermion] a – [fermion] f2 [particle=$\nu_{e}(p_{1})$], ;

(a)

[horizontal=i1 to i2] i1 [particle=$\nu_{e}(p_{1})$] – [fermion] a [dot] –
[fermion] i2 [particle=$\nu_{e}(p_{2})$], f1 [particle=$\nu_{\mu}(p_{2})$] –
[fermion] a – [fermion] f2 [particle=$\nu_{\mu}(p_{1})$], ;

(b)

Figure 1: Diagram of a neutrino-neutrino interaction ($\nu_{e}-\nu_{\mu}$) at
low energies ($E_{\nu}<<m_{Z^{0}}$). At the left, we have a forward scattering
without flavor exchange, and at the right, we have the equivalent with the
flavor exchange.

Although most of the papers in the field use an expression for the neutrino-
neutrino effective potential, most of them do not show how this expression is
derived, relying on the results of a limited number of sources. Among these
sources, the main one is [6], where the authors have an extensive discussion
about the EoM (Equation of Motion) in a Heisenberg picture approach. However,
the paper does not show explicitly how we can arrive at the potential matrix
that rules the neutrino evolution (Equation 5.2 in [6]) starting from the
interaction Hamiltonian (Equation 5.1 in [6]). Other works [7, 8] use an
approach that averages the interaction Lagrangian or Hamiltonian over the
background of neutrinos, similar to what is done for an electron background.
However, it is not clear how this can be done, given that the neutrino fields
in the Hamiltonian operator can act both in the state of the propagating
neutrino or the state of background neutrinos. This average is allowed in the
neutrino-electron interaction because the electron fields can act just on
electron states. Other works [9, 10] derive the potential using Finite
Temperature QFT (Quantum Field Theory), based on propagators in a thermal
medium instead of the vacuum ones.

Given this picture, our intention in this paper is to derive the neutrino-
neutrino potential at low energies (less than $Z^{0}$ mass) in the clearest
way possible by showing the calculation steps and discussing the relevant
concepts. We will work only on the ground of the standard QFT, trying to keep
the document as accessible as possible. In this context, we will derive the
potential using second quantized fields, in a similar way to what is done
elsewhere [11, 12, 13] for the electron-neutrino forward scattering potential.
Moreover, we decided to be rigorous about the massive nature of the neutrino.
To the best of the authors’ knowledge, this is the first time that these
calculations are done explicitly using massive neutrino fields at the
interaction. Previous works consider the neutrino field as massless at the
interaction, which is a valid approximation in the ultra-relativistic regime,
as we will show. In this way, we develop our calculations on a consistent
ground regarding the massive nature of the neutrinos with no further
complications.

We organize this paper as follows. In section 2, we calculate the forward
scattering potential due to neutrino-neutrino interactions. In the first part
of this section, we discuss the neutrino state at the production, its
propagation, and the Hamiltonian governing its evolution. After this
preliminary discussion, we calculate the neutrino-neutrino potential itself,
followed by section 3, where we do the same calculations considering neutrino-
antineutrino interactions. In the next section 4, we connect the potential
expressions to the density matrix formalism, which is extensively used in the
case of neutrino-neutrino phenomenology due to its connection to the
polarization vector formalism. Finally, in section 5, we draw our conclusions
about this work.

##  2 Neutrino-Neutrino Potential

In this section, we derive the neutrino-neutrino forward scattering potential.
Before starting the calculations, we make a preliminary discussion about the
neutrino state at the production, its evolution, and possible interactions.
Here we were strongly influenced by [11], borrowing most of its notation.

###  2.1 Preliminaries

#### Neutrino state at the production

The first problem that arises when working with massive neutrino fields is the
definition of the neutrino state at the production. The reason is that the
flavor fields that diagonalize the Standard Model interactions do not have
well-defined masses. If we look at the production process of a neutrino
$\nu_{\alpha}$, like in (1), its flavor is defined by the charged lepton
partner $l^{+}_{\alpha}$ in the reaction111Neutrinos may also be produced by
neutral current interactions. However, the flavor definition comes
experimentally from charged current interactions, like the one in equation 1.

$P_{I}\rightarrow P_{F}+l^{+}_{\alpha}+\nu_{\alpha}$ (1)

where $P_{I}$ and $P_{F}$ are, respectively, the initial and final particles
involved in the process. It is usual to consider the produced neutrino as a
flavor eigenstate defined as follows.

$\ket{\nu_{\alpha}}\equiv\sum_{k}U^{*}_{\alpha k}\ket{\nu_{k}}$ (2)

However, as mentioned before, since flavor fields do not have well-defined
masses, the quantized flavor fields do not construct a well-behaved Fock space
with physical masses [14], i.e., they cannot create or annihilate physical
neutrino flavor states. The correct way is to consider the created neutrino
$\nu_{\alpha}$ as a superposition of mass eigenstates $\ket{\nu_{k}}$, which
come from quantized massive fields $\nu_{k}$ with well-defined masses. Each
mass eigenstate have its own amplitude of production $\mathcal{A}^{P}_{\alpha
k}=\bra{P_{F}+l^{+}_{\alpha}+\nu_{k}}\mathcal{S}\ket{P_{I}}$ given by the
allowed interactions. Therefore, we can write the produced neutrino state as
follows

$\ket{\nu^{P}_{\alpha}}=N\sum_{k}\mathcal{A}^{P}_{\alpha
k}\ket{\nu_{k}}=N\sum_{k}U^{*}_{\alpha k}{M}^{P}_{\alpha k}\ket{\nu_{k}}$ (3)

where $N$ is just a normalization constant and we have made the decomposition
${A}^{P}_{\alpha k}=U^{*}_{\alpha k}{M}^{P}_{\alpha k}$. Note that this state
is generally not equal to the usual one defined in equation 2. However, if the
experiment measuring $\ket{\nu^{P}_{\alpha}(t)}$ is not sensible to the mass
at the time of production, we can consider the production amplitude as the
same for all mass eigenstates (${M}^{P}_{\alpha k}\approx{M}^{P}_{\alpha}$).
Since this experimental limitation is always verified, we can safely use

$\ket{\nu^{P}_{\alpha}}\approx N{M}^{P}_{\alpha}\sum_{k}U^{*}_{\alpha
k}\ket{\nu_{k}}=\sum_{k}U^{*}_{\alpha k}\ket{\nu_{k}}$ (4)

Therefore, the usual flavor eigenstate as in equation 2 consists in a good
approximation 222A deeper discussion in this topic can be found in [11]
chapter 8.. Being a bit more rigorous, as the neutrino will be a particle
localized in space, a better description is given by considering the neutrino
as a wave packet. This consideration reflects in an amplitude of production
$\mathcal{A}^{P}_{\alpha k}(\vec{p},h)$ that depends on the uncertainties
involved in the process.

$\ket{\nu^{P}_{\alpha}}=N\sum_{k,h}\int\frac{dp^{3}}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{k}}}\mathcal{A}^{P}_{\alpha
k}(\vec{p},h)\ket{\nu_{k}(E_{k},\vec{p},h)}$ (5)

where $\vec{p}$ is the momentum, $E_{k}$ the energy, and $h$ the helicity.
Again, if ${M}^{P}_{\alpha k}\approx{M}^{P}_{\alpha}$ we have the following
common expression

$\begin{split}\ket{\nu^{P}_{\alpha}}&\approx
N\sum_{h}\int\frac{dp^{3}}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{k}}}\mathcal{M}^{P}_{\alpha}(\vec{p},h)\sum_{k}U^{*}_{\alpha
k}\ket{\nu_{k}(E_{k},\vec{p},h)}\\\
&=\sum_{h}\int\frac{dp^{3}}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{k}}}\varphi_{\alpha}(\vec{p},h)\ket{\nu_{\alpha}(\vec{p},h)}\end{split}$
(6)

where $\varphi_{\alpha}(\vec{p},h)$ is the helicity dependent momentum
distribution of the neutrino created with a charged lepton of flavor $\alpha$.
As the reader can see, we arrived at the same expressions usually used for the
produced neutrino state, considering it a plane wave or a wave packet. We just
followed a more rigorous approach once we are interested in treating carefully
the neutrino massive nature.

#### Neutrino Evolution

In the Schrodinger picture, the neutrino evolution is described by the
Hamiltonian

$i\frac{d}{dt}\ket{\nu_{\alpha}^{P}(t)}=H\ket{\nu_{\alpha}^{P}(t)}$ (7)

The Hamiltonian can be divided into a free and an interacting part. We can
further split the last into a contribution due to interaction with ordinary
matter (electrons, protons, and neutrons) and another due to interaction with
the neutrinos in the environment.

$H=H_{0}+H_{int}=H_{vac}+V_{matter}+V_{\nu\nu}$ (8)

In this paper, we will focus on the neutrino-neutrino $V_{\nu\nu}$
contribution and derive its expression. Moreover, we will only consider the
neutrino-neutrino forward scattering, which is equivalent to considering a
free streaming neutrino propagating through a neutrino dense environment.

At this point, one may be concerned about the spatial distribution of the
neutrino state being evolved in the equation 7 and how this distribution may
affect with which particles it is interacting. As we will see, the forward
scattering potential is uniform over all the space. However, the neutrino will
be limited to a finite space, given by the size of its wave packet, and will
interact only with the particles in this range, i.e., when we have a wave
packet overlap. As the neutrino propagates, the number of particles (neutrino
or electrons) interacting with it may change, given an interacting Hamiltonian
that depends on time over the neutrino trajectory.333For the reader interested
in the microscopic behavior of the neutrino potential, in the ordinary matter
case, we recommend the reference [15]..

#### Neutrino Fields

As we will discuss below, the Hamiltonians can be written in terms of the
second quantized neutrino fields. Considering the neutrinos as Dirac particles
with well-defined masses, the Fourier expansion is given by the following
expressions 444As we are in the Schrodinger picture, the operators are time-
independent. Therefore, we can use the fields that are solutions of the free
Dirac equation, given that we can make this picture equivalent to the
Interaction picture at a given time $t_{0}$, which is described by free
fields..

$\begin{split}\nu_{k}(\vec{x})&=\sum_{\vec{p},s}\frac{1}{\sqrt{2VE_{\vec{p}}}}\left[a^{s}_{k}(\vec{p})u^{s}_{k}(\vec{p})e^{i\vec{p}\cdot\vec{x}}+b^{s\dagger}_{k}({\vec{p}})v^{s}_{k}(\vec{p})e^{-i\vec{p}\cdot\vec{x}}\right]\end{split}$
(9a)

$\begin{split}\overline{\nu}_{k}(\vec{x})&=\sum_{\vec{p},s}\frac{1}{\sqrt{2VE_{\vec{p}}}}\left[a^{s\dagger}_{k}(\vec{p})\overline{u}^{s}_{k}(\vec{p})e^{-i\vec{p}\cdot\vec{x}}+b^{s}_{k}({\vec{p}})\overline{v}^{s}_{k}(\vec{p})e^{i\vec{p}\cdot\vec{x}}\right]\end{split}$
(9b)

In this paper, we adopt a normalization in a finite volume $V$ with periodic
conditions in the Fourier expansion of the fields. Although it is a less
realistic approach, it will give us properly normalized states which are
better to work with. Moreover, our final results will be volume-independent,
allowing us to take the infinite volume limit at the end. See appendix A for
more information about our conventions.

The flavor fields are related to the massive fields in equation 9a and 9b by
the PMNS (Pontecorvo-Maki-Nakagawa-Sakata) matrix which diagonalizes the
standard model weak interactions.

$\nu_{\alpha}(x)=\sum_{k}U_{\alpha k}^{*}\nu_{k}(x)$ (10a)

$\overline{\nu}_{\alpha}(x)=\sum_{k}U_{\alpha k}\overline{\nu}_{k}(x)$ (10b)

As already mentioned, these flavor fields do not have well-defined mass and do
not generate a physical Fock space of flavor states. One usual way to work
around this issue is to consider massless neutrino fields at the interaction,
so that the flavor fields have a well-defined mass ($m=0$). That is a good
approximation, given that the mass contribution is suppressed by the neutrino
energy in the ultra-relativistic limit, as we will see. However, we will use
massive fields through the entire calculation process, taking the ultra-
relativistic limit in the final expression.

#### Standard Model Interaction Hamiltonian

To calculate the potential $V_{\nu\nu}$, we will use the effective Hamiltonian
density $\mathcal{H}_{\nu\nu}(x)$ at low energies. This energy regime is
representative of astrophysical scenarios, such as in a supernova. In the
standard model, it is given by the following neutral current Hamiltonian
density

$\begin{split}\mathcal{H}_{eff}^{(NC)}(x)&=\sum_{\alpha,\beta}\frac{G_{F}}{\sqrt{2}}[\overline{\nu}_{\alpha}(x)\gamma^{\rho}(1-\gamma^{5})\nu_{\alpha}(x)][\overline{\nu}_{\beta}(x)\gamma_{\rho}(1-\gamma^{5})\nu_{\beta}(x)]\\\
&=\frac{G_{F}}{\sqrt{2}}j^{\rho}_{Z,\nu}(x)j^{Z,\nu}_{\rho}(x)\end{split}$
(11)

The neutrino neutral currents $j^{\rho}_{Z,\nu}(x)$ can be written in terms of
the flavor fields $\nu_{\alpha}(x)$ or massive fields $\nu_{k}(x)$, using the
PMNS matrix to rotate one into the other.

$\begin{split}j^{\rho}_{Z,\nu}(x)&=\sum_{\alpha}\overline{\nu}_{\alpha}(x)\gamma^{\rho}(1-\gamma^{5})\mathbf{\nu}_{\alpha}(x)\\\
&=\sum_{\alpha,k}U_{\alpha k}U^{*}_{\alpha
j}[\overline{\nu}_{\alpha}(x)\gamma^{\rho}(1-\gamma^{5})\mathbf{\nu}_{j}(x)]\\\
&=\sum_{k}\overline{\nu}_{k}(x)\gamma^{\rho}(1-\gamma^{5})\mathbf{\nu}_{k}(x)\\\
\end{split}$ (12)

That shows us that the neutral current is invariant under the rotation of the
fields, as stated by the GIM mechanism [16]. Therefore, we can write the
interaction Hamiltonian in terms of the massive fields without changing its
format.

$\begin{split}\mathcal{H}_{eff}^{(NC)}(x)&=\sum_{i,j}\frac{G_{F}}{\sqrt{2}}[\overline{\nu}_{i}(x)\gamma^{\rho}(1-\gamma^{5})\nu_{i}(x)][\overline{\nu}_{j}(x)\gamma_{\rho}(1-\gamma^{5})\nu_{j}(x)]\\\
&=\sum_{i,j}\frac{G_{F}}{\sqrt{2}}[\overline{\nu}_{i}(x)\Gamma^{\rho}\nu_{i}(x)][\overline{\nu}_{j}(x)\Gamma_{\rho}\nu_{j}(x)]\end{split}$
(13)

Usually, the notation $\Gamma^{\rho}$ represents any Dirac matrix
multiplication. Here we use it just as a short notation, replacing it with the
one corresponding to the SM weak neutral currents when necessary, as in the
first line of equation 13.

#### Normal Ordering

When using the second quantized Hamiltonians described above, we may find
divergent expectation values. That is a common problem in quantum field
theory, and it comes from the ambiguity in the ordering of the operators when
we are quantizing a classical theory555For example, the harmonic oscillator
Hamiltonian, which inspires the quantization of fields, can be written as
$H=(1/2)(\omega q-ip)(\omega q+ip)$ or $H=(1/2)(\omega q+ip)(\omega q-ip)$
because the variables commute in the classical theory. In quantum mechanics,
one corresponds to $H=\hslash\omega a^{\dagger}a$ whereas the other to
$H=\hslash\omega aa^{\dagger}$, which are incompatible by a factor of
$\hslash\omega[a,a^{\dagger}]$. However, both are equal in the classical limit
($\hslash\rightarrow 0$) as expected. A good discussion on this can be found
in section 2.3 of Professor David Tong QFT notes
http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf. An example of this
divergence occurs when one calculates the matter potential for antineutrinos
as shown in equation 14.

$\bra{f}[\overline{\nu}(x)\Gamma^{\rho}\nu(x)]\ket{i}\propto\sum_{k,k^{\prime}}\bra{f}b(k)b^{\dagger}(k^{\prime})\ket{i}=\sum_{k,k^{\prime}}\left(-\bra{f}b^{\dagger}(k^{\prime})b(k)\ket{i}+\bra{f}\delta_{kk^{\prime}}\ket{i}\right)$
(14)

Due to the anti-commutation rule for the creation and annihilation operators,
a term proportional to $\delta_{kk^{\prime}}$ appears, which will be divergent
and independent of the initial and final particles, i.e., it is no zero even
for the vacuum. In the neutrino literature, it is usual to purely omit this
divergent delta term without making any comment about it. However, there is a
simple argument to ignore this infinity. What is physically meaningful is the
energy above the vacuum, being irrelevant if we add the same constant, finite
or not, to the energy of all the states. A usual prescription to get well-
behaved operators in QFT, with zero vacuum expectation value, is to normal
order them ($:A:$)666Operation where all the annihilation operators go to the
right.. In our case, the operator is the interaction Hamiltonian. Therefore,
to avoid this problem with divergent vacuum expectation values, we will take
the normal ordering of the Hamiltonian from the start of our calculations.

$\mathcal{H}_{\nu\nu}(x)\rightarrow:\mathcal{H}_{\nu\nu}(x):$ (15)

###  2.2 The Neutrino-Neutrino potential

Completed our preliminary discussion aimed to establish a common ground in the
topic, in this section we will calculate the potential due to neutrino-
neutrino forward scattering. Although the neutrino is in general described by
a wave packet due to uncertainties, as in equation 6, the forward scattering
implies no momentum exchange. Therefore, we can calculate the scattering
potential for each momentum eigenstate individually. Considering a forward
scattering
$\nu_{\alpha}(p_{1},s_{1}),\nu_{\alpha^{\prime}}(p_{2},s_{2})\rightarrow\nu_{\beta}(p_{1},s_{1}),\nu_{\beta^{\prime}}(p_{2},s_{2})$
of neutrinos with well-defined momenta $p_{i}$, helicity $h_{i}$ and flavor,
we calculate the potential by the matrix element of the Hamiltonian given the
initial and final states.

$\begin{split}V_{\alpha\beta}&=\frac{1}{4}\sum_{s_{1},s_{2}}\int
dx\frac{\bra{\nu_{\beta}(p_{1},s_{1}),\nu_{\beta^{\prime}}(p_{2},s_{2})}:\mathcal{H}_{\nu\nu}(x):\ket{\nu_{\alpha}(p_{1},s_{1}),\nu_{\alpha^{\prime}}(p_{2},s_{2})}}{\sqrt{\bra{i}\ket{i}\bra{f}\ket{f}}}\\\
&=\frac{1}{4}\sum_{s_{1},s_{2}}\int
dx\frac{G_{F}}{\sqrt{2}}\frac{\mathcal{V}_{\alpha\beta}(x)}{\sqrt{\bra{i}\ket{i}\bra{f}\ket{f}}}\end{split}$
(16)

The non-primed indices $\alpha$ and $\beta$ are respectively the initial and
final flavors of the neutrino with momentum $p_{1}$, while the primed ones
($\alpha^{\prime}$ and $\beta^{\prime}$) are related to the neutrino with
momentum $p_{2}$. As usual, we have taken the average over the initial
helicity state, given our ignorance about it. The inner product in the
denominator is necessary for right normalization (more information in appendix
C). However, as we will be using a finite volume normalization with periodic
conditions, the normalization of the states will give a Kronecker delta and
this denominator will simply be equal to one.

Transforming the flavor states into mass eigenstates by using equation 2, the
potential density can be written as

$\begin{split}&\mathcal{V}_{\alpha\beta}(x)=\\\
&=\sum_{i^{\prime\prime},j^{\prime\prime}}\bra{\nu_{\beta}(p_{1},s_{1}),\nu_{\beta^{\prime}}(p_{2},s_{2})}:[\overline{\nu}_{i^{\prime\prime}}(x)\Gamma^{\rho}\nu_{i^{\prime\prime}}(x)][\overline{\nu}_{j^{\prime\prime}}(x)\Gamma_{\rho}\nu_{j^{\prime\prime}}(x)]:\ket{\nu_{\alpha}(p_{1},s_{1}),\nu_{\alpha^{\prime}}(p_{2},s_{2})}\\\
&=\sum_{i,j}\sum_{i^{\prime},j^{\prime}}U^{*}_{\alpha
i}U^{*}_{\alpha^{\prime}i^{\prime}}U_{\beta j}U_{\beta^{\prime}j^{\prime}}\\\
&\times\sum_{i^{\prime\prime},j^{\prime\prime}}\bra{\nu_{j}(p_{1},s_{1}),\nu_{j^{\prime}}(p_{2},s_{2})}:[\overline{\nu}_{i^{\prime\prime}}(x)\Gamma^{\rho}\nu_{i^{\prime\prime}}(x)][\overline{\nu}_{j^{\prime\prime}}(x)\Gamma_{\rho}\nu_{j^{\prime\prime}}(x)]:\ket{\nu_{i}(p_{1},s_{1}),\nu_{i^{\prime}}(p_{2},s_{2})}\\\
&=\sum_{i,j}\sum_{i^{\prime},j^{\prime}}U^{*}_{\alpha
i}U^{*}_{\alpha^{\prime}i^{\prime}}U_{\beta
j}U_{\beta^{\prime}j^{\prime}}\mathcal{V}^{i^{\prime},j^{\prime}}_{ij}(x)\\\
\end{split}$ (17)

If we focus on the term $\mathcal{V}^{i^{\prime},j^{\prime}}_{ij}(x)$ and make
the Fourier expansion of the fields, only the terms that annihilate and create
two neutrinos will survive ($\propto a^{\dagger}aa^{\dagger}a$).

$\begin{split}&\mathcal{V}^{i^{\prime},j^{\prime}}_{ij}(x)=\\\
&=\sum_{i^{\prime\prime},j^{\prime\prime}}\bra{\nu_{j}(p_{1},s_{1}),\nu_{j^{\prime}}(p_{2},s_{2})}:[\overline{\nu}_{i^{\prime\prime}}(x)\Gamma^{\rho}\nu_{i^{\prime\prime}}(x)][\overline{\nu}_{j^{\prime\prime}}(x)\Gamma_{\rho}\nu_{j^{\prime\prime}}(x)]:\ket{\nu_{i}(p_{1},s_{1}),\nu_{i^{\prime}}(p_{2},s_{2})}\\\
&=\sum_{i^{\prime\prime},j^{\prime\prime}}\sum_{\begin{subarray}{c}\vec{q}\vec{q^{\prime}}\vec{k}\vec{k^{\prime}}\\\
ss^{\prime}hh^{\prime}\end{subarray}}\frac{e^{ix(q-q^{\prime}+k-k^{\prime})}}{\sqrt{2VE_{q}}\sqrt{2VE_{q^{\prime}}}\sqrt{2VE_{k}}\sqrt{2VE_{k^{\prime}}}}[\overline{u}_{i^{\prime\prime}}^{s}(q)\Gamma^{\rho}u_{i^{\prime\prime}}^{s^{\prime}}(q^{\prime})][\overline{u}_{j^{\prime\prime}}^{h}(k)\Gamma_{\rho}u_{j^{\prime\prime}}^{h^{\prime}}(k^{\prime})]\\\
&\times\bra{\nu_{j}(p_{1},s_{1}),\nu_{j^{\prime}}(p_{2},s_{2})}:a^{s\dagger}_{i^{\prime\prime}}(q)a^{s^{\prime}}_{i^{\prime\prime}}(q)a^{h\dagger}_{j^{\prime\prime}}(k)a^{h^{\prime}}_{j^{\prime\prime}}(k^{\prime}):\ket{\nu_{i}(p_{1},s_{1}),\nu_{i^{\prime}}(p_{2},s_{2})}\\\
\end{split}$ (18)

Normal ordering the operators inside $::$, opening the definition of the
states, and using anticommutation rules, we can write the matrix element as
follows.

$\begin{split}&\bra{\nu_{j}(p_{1},s_{1}),\nu_{j^{\prime}}(p_{2},s_{2})}:a^{s\dagger}_{i^{\prime\prime}}(q)a^{s^{\prime}}_{i^{\prime\prime}}(q^{\prime})a^{h\dagger}_{j^{\prime\prime}}(k)a^{h^{\prime}}_{j^{\prime\prime}}(k^{\prime}):\ket{\nu_{i}(p_{1},s_{1}),\nu_{i^{\prime}}(p_{2},s_{2})}\\\
=&-\bra{\nu_{j}(p_{1},s_{1}),\nu_{j^{\prime}}(p_{2},s_{2})}a^{s\dagger}_{i^{\prime\prime}}(q)a^{h\dagger}_{j^{\prime\prime}}(k)a^{s^{\prime}}_{i^{\prime\prime}}(q^{\prime})a^{h}_{j^{\prime\prime}}(k^{\prime})\ket{\nu_{i^{\prime}}(p_{2},s_{2}),\nu_{i}(p_{1},s_{1})}\\\
=&-\bra{0}\uline{a^{s_{1}}_{j}(p_{1})a^{s_{2}}_{j^{\prime}}(p_{2})a^{s\dagger}_{i^{\prime\prime}}(q)a^{h\dagger}_{j^{\prime\prime}}(k)}\dotuline{a^{s^{\prime}}_{i^{\prime\prime}}(q^{\prime})a^{h}_{j^{\prime\prime}}(k^{\prime})a^{s_{2}\dagger}_{i^{\prime}}(p_{2})a^{s_{1}\dagger}_{i}(p_{1})}\ket{0}\\\
=&-[(\uline{\delta_{i^{\prime\prime}j^{\prime}}\delta_{qp_{2}}\delta_{j^{\prime\prime}j}\delta_{kp_{1}}-\delta_{i^{\prime\prime}j}\delta_{qp_{1}}\delta_{j^{\prime\prime}j^{\prime}}\delta_{kp_{2}}})(\dotuline{\delta_{j^{\prime\prime}i^{\prime}}\delta_{k^{\prime}p_{2}}\delta_{i^{\prime\prime}i}\delta_{q^{\prime}p_{1}}-\delta_{j^{\prime\prime}i}\delta_{k^{\prime}p_{1}}\delta_{i^{\prime\prime}i^{\prime}}\delta_{q^{\prime}p_{2}}})]\end{split}$
(19)

Where we have omitted the deltas in helicity for simplicity, given that it has
the same behavior as the momentum index. We use the underline and dots just to
help the reader identify from which commutation each delta came. Note that
summing over the momentum (and helicity) of the fields, as requested in
equation 18, there will be only the four terms listed in table 1. Two of them
(1,4) correspond to an exchange of mass eigenstates between the interacting
neutrinos
$\ket{\nu_{i}(p_{1}),\nu_{i^{\prime}}(p_{2})}\rightarrow\ket{\nu_{i^{\prime}}(p_{1}),\nu_{i}(p_{2})}$,
where the neutrino with momentum $p_{1}$ change from $\ket{\nu_{i}}$ to
$\ket{\nu_{i}^{\prime}}$ 777This can alternatively be seen as a momentum
exchange, although we still in a forward scattering with a neutrino with
momentum $p_{1}$ and another with momentum $p_{2}$. The other two (2,3)
correspond to no exchange of eigenstates
$\ket{\nu_{i}(p1),\nu_{i^{\prime}}(p_{2})}\rightarrow\ket{\nu_{i}(p_{1}),\nu_{i^{\prime}}(p_{2})}$.
The presence of two terms for each case came from the freedom of the neutrino
field to act in $\ket{\nu_{i}(p_{1})}$ or $\ket{\nu_{i^{\prime}}(p_{2})}$.

Table 1: Possibilities of annihilation and creation of the neutrinos in terms of the initial and final states. Numb. | $(\vec{q},\vec{k},\vec{q^{\prime}},\vec{k^{\prime}})$ | $\mathcal{V}^{i^{\prime},j^{\prime}}_{ij}(x)$ |  Eigenstate exchange
---|---|---|---
1 | $(\vec{p}_{2},\vec{p}_{1},\vec{p}_{1},\vec{p}_{2})$ | $-[\overline{u}^{s_{2}}_{j^{\prime}}(p_{2})\Gamma^{\rho}u^{s_{1}}_{i}(p_{1})][\overline{u}^{s_{1}}_{j}(p_{1})\Gamma_{\rho}u^{s_{2}}_{i^{\prime}}(p_{2})]\delta_{ji^{\prime}}\delta_{j^{\prime}i}$ | yes
2 | $(\vec{p}_{2},\vec{p}_{1},\vec{p}_{2},\vec{p}_{1})$ | $+[\overline{u}^{s_{2}}_{j^{\prime}}(p_{2})\Gamma^{\rho}u^{s_{2}}_{i^{\prime}}(p_{2})][\overline{u}^{s_{1}}_{j}(p_{1})\Gamma_{\rho}u^{s_{1}}_{i}(p_{1})]\delta_{ji}\delta_{j^{\prime}i^{\prime}}$ | no
3 | $(\vec{p}_{1},\vec{p}_{2},\vec{p}_{1},\vec{p}_{2})$ | $+[\overline{u}^{s_{1}}_{j}(p_{1})\Gamma^{\rho}u^{s_{1}}_{i}(p_{1})][\overline{u}^{s_{2}}_{j^{\prime}}(p_{2})\Gamma_{\rho}u^{s_{2}}_{i^{\prime}}(p_{2})]\delta_{ji}\delta_{j^{\prime}i^{\prime}}$ | no
4 | $(\vec{p}_{1},\vec{p}_{2},\vec{p}_{2},\vec{p}_{1})$ | $-[\overline{u}^{s_{1}}_{j}(p_{1})\Gamma^{\rho}u^{s_{2}}_{i^{\prime}}(p_{2})][\overline{u}^{s_{2}}_{j}(p_{2})\Gamma_{\rho}u^{s_{1}}_{i}(p_{1})]\delta_{ji^{\prime}}\delta_{j^{\prime}i}$ | yes

As the reader can see, the eigenstate exchange terms (1,4) have different
momenta spinors inside the same square bracket. However, to use the trace
techniques (Appendix B), we have to have spinors with the same momenta in
these square brackets. Therefore, we need to exchange spinors in the different
square brackets. This operation will give us only a minus sign. The reader can
see this by two reasoning: (i)When doing a Fierz transformation, we already
account for a minus sign due to fermionic anti-commutation. Therefore, an
equivalent transformation with spinors will have to have a minus sign ; (ii)
If we do the Fierz transformation for these terms before expanding the fields,
the order of the ladder operators would be three commutations from our
ordering in equation 18, which differs from our results by a minus sign.

Going back to the potential density in equation 17 and using the results above
we have

$\begin{split}\mathcal{V}_{\alpha\beta}(x)&=\sum_{i,i^{\prime}}\frac{U^{*}_{\alpha
i}U_{\beta
i}U^{*}_{\alpha^{\prime}i^{\prime}}U_{\beta^{\prime}i^{\prime}}}{2VE_{p_{1}}2VE_{p2}}2[\overline{u}^{s_{1}}_{i}(p_{1})\Gamma^{\rho}u^{s_{1}}_{i}(p_{1})][\overline{u}^{s_{2}}_{i^{\prime}}(p_{2})\Gamma_{\rho}u^{s_{2}}_{i^{\prime}}(p_{2})]\\\
&+\sum_{i,i^{\prime}}\frac{U^{*}_{\alpha
i}U_{\beta^{\prime}i}U^{*}_{\alpha^{\prime}i^{\prime}}U_{\beta
i^{\prime}}}{2VE_{p_{1}}2VE_{p2}}2[\overline{u}^{s_{1}}_{i}(p_{1})\Gamma^{\rho}u^{s_{1}}_{i}(p_{1})][\overline{u}^{s_{2}}_{i^{\prime}}(p_{2})\Gamma_{\rho}u^{s_{2}}_{i^{\prime}}(p_{2})]\\\
\end{split}$ (20)

Note that the space dependence despairs in the potential. That happens because
there is no momentum exchange in the forward scattering, resulting in a
potential that is uniform over space. For plane waves, the state of the
particles spreads over all the space, and the interaction happens everywhere.
However, for real particles, the interaction happens only when there is an
overlap of their wave packets. When the overlap ceases, so does the
interaction. Going back to equation 20, we can use the trace techniques
developed in the appendix B to simplify the spinor and Dirac matrices
multiplications.

$\overline{u}^{(h)}(p)\gamma^{\mu}(1-\gamma^{5})u^{(h)}(p)=2(p^{\mu}-ms^{\mu}_{h})$
(21)

If we consider the ultra-relativistic limit $E>>m$, we may write

$2(p^{\mu}-ms^{\mu}_{h})=2\left(E-h|\vec{p}|,\vec{p}-h\frac{E\vec{p}}{|\vec{p}|}\right)\overset{E\approx|\vec{p}|}{\approx}\left\\{\begin{matrix}(4E,4\vec{p}),\;\;\;\;\;\text{
for }h=-1\\\ (2\frac{m^{2}}{E},0),\;\;\;\;\;\;\text{ for
}h=+1\end{matrix}\right.$ (22)

As the neutrino masses are below the $eV$ scale [17, 18, 19], we have $E>>m$
for the most relevant cases. Therefore, the positive helicity component is
suppressed by a factor of $m^{2}/E$, as already expected for ultra-
relativistic particles. Here we can see that the consideration of a massless
neutrino field in the interaction used in some works [6, 20] is a good
approximation in this energy regime. Therefore, we can write the potential in
equation 16 as follows.

$\begin{split}V_{\alpha\beta}&=\sqrt{2}G_{F}\int
dx\sum_{i,i^{\prime}}[U^{*}_{\alpha i}U_{\beta
i}U^{*}_{\alpha^{\prime}i^{\prime}}U_{\beta^{\prime}i^{\prime}}+U^{*}_{\alpha
i}U_{\beta^{\prime}i}U^{*}_{\alpha^{\prime}i^{\prime}}U_{\beta
i^{\prime}}]\frac{(p_{1}^{\rho}-m_{i}s^{\rho}_{s_{1}})(p_{2\rho}-m_{j}s_{\rho}^{s_{2}})}{2VE_{p_{1}}2VE_{p2}}\\\
&\overset{E\approx|\vec{p}|}{\approx}\sqrt{2}G_{F}\int
dx\sum_{i,i^{\prime}}[U^{*}_{\alpha i}U_{\beta
i}U^{*}_{\alpha^{\prime}i^{\prime}}U_{\beta^{\prime}i^{\prime}}+U^{*}_{\alpha
i}U_{\beta^{\prime}i}U^{*}_{\alpha^{\prime}i^{\prime}}U_{\beta
i^{\prime}}]\frac{4(E_{p_{1}}E_{p_{2}}-\vec{p}_{1}\cdot\vec{p}_{2})}{2VE_{p_{1}}2VE_{p2}}\\\
&=\frac{\sqrt{2}G_{F}}{V}(\delta_{\beta\alpha}\delta_{\beta^{\prime}\alpha^{\prime}}+\delta_{\beta\alpha^{\prime}}\delta_{\beta^{\prime}\alpha})(1-\hat{p}_{1}\cdot\hat{p}_{2})\end{split}$
(23)

Where we have used the unitarity of the PMNS matrix $UU^{\dagger}=\mathbb{1}$.
We can see, as pointed by J. Pantaleone [3, 4], that in the flavor basis the
potential have diagonal $\delta_{\beta\alpha}$ and off-diagonal
$\delta_{\beta\alpha^{\prime}}$ elements (Figures 2 and 3 respectively). As
already found by [9], the potential felt by a neutrino interacting with a
neutrino with the same flavor ($\alpha=\alpha^{\prime}$) is twice the one felt
in the interaction with another flavor ($\alpha\neq\alpha^{\prime}$). That
happens because the process that exchanges the flavor is indistinguishable
from the one where there is no exchange, given that the interacting neutrinos
have the same flavor.

[vertical=a to b] i1 [particle=$\nu_{\alpha}(p_{1})$] – [fermion] a –
[fermion] f1 [particle=$\nu_{\alpha}(p_{1})$], a – [boson, edge label=$Z^{0}$]
b, b – [fermion] i2 [particle=$\nu_{\alpha^{\prime}}(p_{2})$], f2
[particle=$\nu_{\alpha^{\prime}}(p_{2})$] – [fermion] b, ;

(a)

[horizontal=i1 to i2] i1 [particle=$\nu_{\alpha}(p_{1})$] – [fermion] a [dot]
– [fermion] i2 [particle=$\nu_{\alpha^{\prime}}(p_{2})$], f1
[particle=$\nu_{\alpha^{\prime}}(p_{2})$] – [fermion] a – [fermion] f2
[particle=$\nu_{\alpha}(p_{1})$], ;

(b)

Figure 2: Diagrams that represent the neutrino-neutrino forward scattering
where there is no exchange of flavor between $\nu_{\alpha}(p_{1})$ and
$\nu_{\alpha^{\prime}}(p_{2})$.

(a)

[horizontal=i1 to i2] i1 [particle=$\nu_{\alpha}(p_{1})$] – [fermion] a [dot]
– [fermion] i2 [particle=$\nu_{\alpha}(p_{2})$], f1
[particle=$\nu_{\alpha^{\prime}}(p_{2})$] – [fermion] a – [fermion] f2
[particle=$\nu_{\alpha^{\prime}}(p_{1})$], ;

(b)

Figure 3: Diagrams that represent the neutrino-neutrino forward scattering
where there is an exchange of flavor between $\nu_{\alpha}(p_{1})$ and
$\nu_{\alpha^{\prime}}(p_{2})$.

Considering a propagating neutrino $\nu_{\alpha}(p_{1},s_{1})$, we can write
the potential as a matrix in the flavor space of this neutrino, which will
depend on the other neutrino flavor $\alpha^{\prime}$.

$V_{\nu\nu}=\frac{\sqrt{2}G_{F}}{V}(1-\hat{p}_{1}\cdot\hat{p}_{2})\begin{pmatrix}2\delta_{\alpha^{\prime},e}+\delta_{\alpha^{\prime},\mu}+\delta_{\alpha^{\prime},\tau}&\delta_{\alpha^{\prime},e}&\delta_{\alpha^{\prime},e}\\\
\delta_{\alpha^{\prime},\mu}&2\delta_{\alpha^{\prime},\mu}+\delta_{\alpha^{\prime},e}+\delta_{\alpha^{\prime},\tau}&\delta_{\alpha^{\prime},\mu}\\\
\delta_{\alpha^{\prime},\tau}&\delta_{\alpha^{\prime},\tau}&2\delta_{\alpha^{\prime},\tau}+\delta_{\alpha^{\prime},e}+\delta_{\alpha^{\prime},\mu}\end{pmatrix}$
(24)

Note that we may remove a diagonal term $\mathbb{1}_{3\times
3}(\delta_{\alpha^{\prime},e}+\delta_{\alpha^{\prime},\mu}+\delta_{\alpha^{\prime},\tau})$
that does not contribute to the phase difference in the evolution, but only to
an overall phase.

$V^{\prime}_{\nu\nu}=\frac{\sqrt{2}G_{F}}{V}(1-\hat{p}_{1}\cdot\hat{p}_{2})\begin{pmatrix}\delta_{\alpha^{\prime},e}&\delta_{\alpha^{\prime},e}&\delta_{\alpha^{\prime},e}\\\
\delta_{\alpha^{\prime},\mu}&\delta_{\alpha^{\prime},\mu}&\delta_{\alpha^{\prime},\mu}\\\
\delta_{\alpha^{\prime},\tau}&\delta_{\alpha^{\prime},\tau}&\delta_{\alpha^{\prime},\tau}\end{pmatrix}$
(25)

Although this expression for the potential is similar to the one shown by J.
Pantaleone [3, 4], the recent literature usually uses the one in terms of the
density matrix. To find this common expression, we have to connect the
potential to the density matrix formalism, what we do in section 4. Before
doing this, we will do the same calculations for the neutrino-antineutrino
interaction, which has its peculiarities.

##  3 Neutrino-Antineutrino Potential

In the case of a neutrino $\nu_{\alpha}(p_{1})$ interacting with an
antineutrinos $\overline{\nu}_{\alpha^{\prime}}(p_{2})$, we only need to
change the initial and final states in our potential density expression.

$\begin{split}&\mathcal{V}_{\alpha\beta}(x)=\\\
&=\sum_{i^{\prime\prime},j^{\prime\prime}}\bra{\nu_{\beta}(p_{1},s_{1}),\overline{\nu}_{\beta^{\prime}}(p_{2},s_{2})}:[\overline{\nu}_{i^{\prime\prime}}(x)\Gamma^{\rho}\nu_{i^{\prime\prime}}(x)][\overline{\nu}_{j^{\prime\prime}}(x)\Gamma_{\rho}\nu_{j^{\prime\prime}}(x)]:\ket{\overline{\nu}_{\alpha^{\prime}}(p_{2},s_{2}),\nu_{\alpha}(p_{1},s_{1})}\\\
&=\sum_{i,j}\sum_{i^{\prime},j^{\prime}}U^{*}_{\alpha
i}U_{\alpha^{\prime}i^{\prime}}U_{\beta j}U^{*}_{\beta^{\prime}j^{\prime}}\\\
&\times\sum_{i^{\prime\prime},j^{\prime\prime}}\bra{\nu_{j}(p_{1},s_{1}),\overline{\nu}_{j^{\prime}}(p_{2},s_{2})}:[\overline{\nu}_{i^{\prime\prime}}(x)\Gamma^{\rho}\nu_{i^{\prime\prime}}(x)][\overline{\nu}_{j^{\prime\prime}}(x)\Gamma_{\rho}\nu_{j^{\prime\prime}}(x)]:\ket{\overline{\nu}_{i^{\prime}}(p_{2},s_{2}),\nu_{i}(p_{1},s_{1})}\\\
&=\sum_{i,j}U^{*}_{\alpha i}U_{\alpha^{\prime}i^{\prime}}U_{\beta
j}U^{*}_{\beta^{\prime}j^{\prime}}\mathcal{V}^{i^{\prime},j^{\prime}}_{ij}(x)\\\
\end{split}$ (26)

Focusing on the term $\mathcal{V}^{i^{\prime},j^{\prime}}_{ij}(x)$ and making
the Fourier expansion of the fields, only the terms that annihilate/create a
neutrino and an antineutrino will survive ($\propto
a^{\dagger}ab^{\dagger}b$). That will make the calculations a bit different
from the neutrino-neutrino case because the neutrino field can either
annihilate a neutrino or create an antineutrino. The opposite occurs for its
adjoint. With that in mind, the calculations go as follows.

$\begin{split}&\mathcal{V}^{i^{\prime},j^{\prime}}_{ij}(x)=\\\
&=\sum_{i^{\prime\prime},j^{\prime\prime}}\bra{\nu_{j}(p_{1},s_{1}),\overline{\nu}_{j^{\prime}}(p_{2},s_{2})}:[\overline{\nu}_{i^{\prime\prime}}(x)\Gamma^{\rho}\nu_{i^{\prime\prime}}(x)][\overline{\nu}_{j^{\prime\prime}}(x)\Gamma_{\rho}\nu_{j^{\prime\prime}}(x)]:\ket{\overline{\nu}_{i^{\prime}}(p_{2},s_{2}),\nu_{i}(p_{1},s_{1})}\\\
&=\sum_{i^{\prime\prime},j^{\prime\prime}}\sum_{\begin{subarray}{c}\vec{q}\vec{q^{\prime}}\vec{k}\vec{k^{\prime}}\\\
ss^{\prime}hh^{\prime}\end{subarray}}\frac{e^{ix(q-q^{\prime}+k-k^{\prime})}}{\sqrt{2VE_{q}}\sqrt{2VE_{q^{\prime}}}\sqrt{2VE_{k}}\sqrt{2VE_{k^{\prime}}}}\left\\{[\overline{u}_{i^{\prime\prime}}^{s}(q)\Gamma^{\rho}u_{i^{\prime\prime}}^{s^{\prime}}(q^{\prime})][\overline{v}_{j^{\prime\prime}}^{h}(k)\Gamma_{\rho}v_{j^{\prime\prime}}^{h^{\prime}}(k^{\prime})]\mathcal{A}_{qq^{\prime}kk^{\prime}}^{i^{\prime\prime}j^{\prime\prime}}\right.\\\
&+\left.[\overline{u}_{i^{\prime\prime}}^{s}(q)\Gamma^{\rho}u_{j^{\prime\prime}}^{s^{\prime}}(k^{\prime})][\overline{v}_{j^{\prime\prime}}^{h}(k)\Gamma_{\rho}v_{i^{\prime\prime}}^{h^{\prime}}(q^{\prime})]\mathcal{A}_{qq^{\prime}kk^{\prime}}^{i^{\prime\prime}j^{\prime\prime}}\right\\}\end{split}$
(27)

Where we already consider a Fierz transformation
($\nu_{i^{\prime\prime}}(q^{\prime})\leftrightharpoons\nu_{j^{\prime\prime}}(k^{\prime})$)
when necessary to separate the particles and antiparticles spinors inside
different square brackets. That will be useful to apply the trace techniques
of appendix B. Note that in the last line we separate the terms with (right)
and without (left) a Fierz transformation. As in the neutrino-neutrino case,
there are four possible terms, according to the choice of which state a field
will act on. We list them in the table 2, where the possibilities (3,4) are
the ones with Fierz transformation.

Table 2: Possibilities of annihilation and creation of the neutrino and antineutrino in terms of the initial and final states. Numb. | $(\vec{q},\vec{q^{\prime}},\vec{k},\vec{k^{\prime}})$ | $\mathcal{A}_{qq^{\prime}kk^{\prime}}^{i^{\prime\prime}j^{\prime\prime}}$
---|---|---
1 | $(\vec{p}_{1},\vec{p}_{1},\vec{p}_{2},\vec{p}_{2})$ |  $\bra{f}:a^{s\dagger}_{i^{\prime\prime}}(q)a^{s^{\prime}}_{i^{\prime\prime}}(q^{\prime})b^{h}_{j^{\prime\prime}}(k)b^{h^{\prime}\dagger}_{j^{\prime\prime}}(k^{\prime}):\ket{i}$= $-(\delta_{ji}\delta_{j^{\prime}i^{\prime}})(\delta_{qp_{1}}\delta_{q^{\prime}p_{1}}\delta_{kp_{2}}\delta_{k^{\prime}p_{2}})$
2 | $(\vec{p}_{2},\vec{p}_{2},\vec{p}_{1},\vec{p}_{1})$ |  $\bra{f}:b^{s}_{i^{\prime\prime}}(q)b^{s^{\prime}\dagger}_{j^{\prime\prime}}(q^{\prime})a^{h\dagger}_{j^{\prime\prime}}(k)a^{h^{\prime}}_{i^{\prime\prime}}(k^{\prime}):\ket{i}$= $-(\delta_{ji}\delta_{ji^{\prime}})(\delta_{qp_{2}}\delta_{q^{\prime}p_{2}}\delta_{kp_{1}}\delta_{k^{\prime}p_{1}})$
3 | $(\vec{p}_{1},\vec{p}_{2},\vec{p}_{2},\vec{p}_{1})$ |  $\bra{f}:a^{s\dagger}_{i^{\prime\prime}}(q)a^{h^{\prime}}_{j^{\prime\prime}}(k^{\prime})b^{h}_{j^{\prime\prime}}(k)b^{h^{\prime}\dagger}_{i^{\prime\prime}}(q^{\prime}):\ket{i}$= $-(\delta_{ii^{\prime}}\delta_{jj^{\prime}})(\delta_{qp_{1}}\delta_{q^{\prime}p_{2}}\delta_{kp_{2}}\delta_{k^{\prime}p_{1}})$
4 | $(\vec{p}_{2},\vec{p}_{1},\vec{p}_{1},\vec{p}_{2})$ |  $\bra{f}:b^{s}_{i^{\prime\prime}}(q)b^{h^{\prime}\dagger}_{j^{\prime\prime}}(k^{\prime})a^{h\dagger}_{j^{\prime\prime}}(k)a^{s^{\prime}}_{i^{\prime\prime}}(q^{\prime}):\ket{i}$= $-(\delta_{ii^{\prime}}\delta_{jj^{\prime}})(\delta_{qp_{2}}\delta_{q^{\prime}p_{1}}\delta_{kp_{1}}\delta_{k^{\prime}p_{2}})$

Going back to the potential density in equation 26 and using the results
above, we have the following expression.

$\begin{split}\mathcal{V}_{\alpha\beta}(x)=&-\sum_{i,i^{\prime}}\frac{U^{*}_{\alpha
i}U_{\alpha^{\prime}i^{\prime}}U_{\beta
i}U^{*}_{\beta^{\prime}i^{\prime}}}{2VE_{p_{1}}2VE_{p2}}2[\overline{u}^{s_{1}}_{i}(p_{1})\Gamma^{\rho}u^{s_{1}}_{i}(p_{1})][\overline{v}^{s_{2}}_{i^{\prime}}(p_{2})\Gamma_{\rho}v^{s_{2}}_{i^{\prime}}(p_{2})]\\\
&-\sum_{i,j}\frac{U^{*}_{\alpha i}U_{\alpha^{\prime}i}U_{\beta
j}U^{*}_{\beta^{\prime}j}}{2VE_{p_{1}}2VE_{p2}}2[\overline{u}^{s_{1}}_{i}(p_{1})\Gamma^{\rho}u^{s_{1}}_{j}(p_{1})][\overline{v}^{s_{2}}_{j}(p_{2})\Gamma_{\rho}v^{s_{2}}_{i}(p_{2})]\\\
\end{split}$ (28)

One difficulty that arises in the equation 28 is the presence of
(anti)particle spinors with different masses in the last line, forbidding us
to use the properties developed in appendix B. These terms come from the
process where we annihilate a particle-antiparticle pair of equal mass
$i=i^{\prime}$ and create another pair of any mass $j=j^{\prime}$, as
indicated in figure 4.

[horizontal=a to b] i1 [particle=$\overline{\nu}_{i}(p_{2})$] – [anti fermion]
a – [anti fermion] f1 [particle=$\nu_{i}(p_{1})$],

a – [boson, edge label=$Z^{0}$] b, b – [fermion] i2
[particle=$\nu_{j}(p_{1})$],

f2 [particle=$\overline{\nu}_{j}(p_{2})$] – [fermion] b, ;

(a)

[horizontal=i1 to i2] i1 [particle=$\nu_{i}(p_{1})$] – [fermion] a [dot] –
[fermion] i2 [particle=$\overline{\nu}_{j}(p_{2})$], f1
[particle=$\overline{\nu}_{i}(p_{2})$] – [fermion] a – [fermion] f2
[particle=$\nu_{j}(p_{1})$], ;

(b)

Figure 4: Annihalation of a neutrino and an antineutrino with the same mass
$m_{i}$ which allows the creation of a pair $\nu-\overline{\nu}$ of any mass
$m_{j}$.

One way to get around this problem is to consider that the masses are about
the same $m_{i}\approx m_{j}\approx m_{\overline{\nu}}$. In any case, its
contribution will be suppressed by the energy as already found in the
neutrino-neutrino potential. Making this consideration, we can use the
following relations from appendix B.

$\overline{v}^{(h)}(p)\gamma^{\mu}(1-\gamma^{5})v^{(h)}(p)=2(p^{\mu}+m_{\overline{\nu}}s^{\mu}_{h})$
(29)
$2(p^{\mu}+m_{\overline{\nu}}s^{\mu}_{h})=2\left(E+h|\vec{p}|,\vec{p}+h\frac{E\vec{p}}{|\vec{p}|}\right)\overset{E\approx|\vec{p}|}{\approx}\left\\{\begin{matrix}(4E,4\vec{p}),\;\;\;\;\;\text{
for }h=+1\\\ (2\frac{m_{\overline{\nu}}^{2}}{E},0),\;\;\;\;\;\;\text{ for
}h=-1\end{matrix}\right.$ (30)

Considering the neutrinos as ultra-relativistic, the potential in equation 16,
for the case of neutrino-antineutrino interactions, can be written as

$\begin{split}V_{\alpha\beta}&=-\sqrt{2}G_{F}\int
dx\sum_{i,i^{\prime}}[U^{*}_{\alpha i}U_{\alpha^{\prime}i^{\prime}}U_{\beta
i}U^{*}_{\beta^{\prime}i^{\prime}}+U^{*}_{\alpha
i}U_{\alpha^{\prime}i}U_{\beta
i^{\prime}}U^{*}_{\beta^{\prime}j}]\frac{(p_{1}^{\rho}-m_{\nu}s^{\rho}_{s_{1}})(p_{2\rho}+m_{\overline{\nu}}s_{\rho}^{s_{2}})}{2VE_{p_{1}}2VE_{p2}}\\\
&\overset{E\approx|\vec{p}|}{\approx}-\sqrt{2}G_{F}\int
dx\sum_{i,i^{\prime}}[U^{*}_{\alpha i}U_{\alpha^{\prime}i^{\prime}}U_{\beta
i}U^{*}_{\beta^{\prime}i^{\prime}}+U^{*}_{\alpha
i}U_{\alpha^{\prime}i}U_{\beta
i^{\prime}}U^{*}_{\beta^{\prime}j}]\frac{4(E_{p_{1}}E_{p_{2}}-\vec{p}_{1}\cdot\vec{p}_{2})}{2VE_{p_{1}}2VE_{p2}}\\\
&=-\frac{\sqrt{2}G_{F}}{V}(\delta_{\beta\alpha}\delta_{\beta^{\prime}\alpha^{\prime}}+\delta_{\alpha\alpha^{\prime}}\delta_{\beta\beta^{\prime}})(1-\hat{p}_{1}\cdot\hat{p}_{2})\end{split}$
(31)

Therefore, there are a flavor diagonal contribution $\delta_{\beta\alpha}$
(Figure 5) and a $\delta_{\beta\beta^{\prime}}$ contribution (Figure 6) ,
which is diagonal if $\beta=\alpha$ or off-diagonal if $\beta\neq\alpha$.

[vertical=a to b] i1 [particle=$\nu_{\alpha}(p_{1})$] – [fermion] a –
[fermion] f1 [particle=$\nu_{\alpha}(p_{1})$], a – [boson, edge label=$Z^{0}$]
b, b – [anti fermion] i2 [particle=$\overline{\nu}_{\alpha^{\prime}}(p_{2})$],
f2 [particle=$\overline{\nu}_{\alpha^{\prime}}(p_{2})$] – [anti fermion] b, ;

(a)

[horizontal=i1 to i2] i1 [particle=$\nu_{\alpha}(p_{1})$] – [fermion] a [dot]
– [fermion] i2 [particle=$\nu_{\alpha}(p_{1})$], f1
[particle=$\overline{\nu}_{\alpha^{\prime}}(p_{2})$] – [fermion] a – [fermion]
f2 [particle=$\overline{\nu}_{\alpha^{\prime}}(p_{2})$], ;

(b)

Figure 5: Diagrams that represent the neutrino-antineutrino interaction where
there is no exchange of flavor between $\nu_{\alpha}(p_{1})$ and
$\overline{\nu}_{\alpha^{\prime}}(p_{2})$

[horizontal=a to b] i1 [particle=$\overline{\nu}_{\alpha}(p_{2})$] – [anti
fermion] a – [anti fermion] f1 [particle=$\nu_{\alpha}(p_{1})$],

a – [boson, edge label=$Z^{0}$] b, b – [fermion] i2
[particle=$\nu_{\beta}(p_{1})$],

f2 [particle=$\overline{\nu}_{\beta}(p_{2})$] – [fermion] b, ;

(a)

[horizontal=i1 to i2] i1 [particle=$\nu_{\alpha}(p_{1})$] – [fermion] a [dot]
– [fermion] i2 [particle=$\overline{\nu}_{\beta}(p_{2})$], f1
[particle=$\overline{\nu}_{\alpha}(p_{2})$] – [fermion] a – [fermion] f2
[particle=$\nu_{\beta}(p_{1})$], ;

(b)

Figure 6: Neutrino-antineutrino annihilation which allows the creation of a
new flavor $\beta$, which may be equal to the initial one $\beta=\alpha$ or
different $\beta\neq\alpha$.

Again, considering a propagating neutrino $\nu_{\alpha}(p_{1},s_{1})$, we can
write the potential as a matrix in the flavor space of this neutrino.

$V_{\nu\overline{\nu}}=-\frac{\sqrt{2}G_{F}}{V}(1-\hat{p}_{1}\cdot\hat{p}_{2})\begin{pmatrix}2\delta_{\alpha^{\prime},e}+\delta_{\alpha^{\prime},\mu}+\delta_{\alpha^{\prime},\tau}&\delta_{\alpha^{\prime},\mu}&\delta_{\alpha^{\prime},\tau}\\\
\delta_{\alpha^{\prime},e}&\delta_{\alpha^{\prime},e}+2\delta_{\alpha^{\prime},\mu}+\delta_{\alpha^{\prime},\tau}&\delta_{\alpha^{\prime},\tau}\\\
\delta_{\alpha^{\prime},e}&\delta_{\alpha^{\prime},\mu}&\delta_{\alpha^{\prime},e}+\delta_{\alpha^{\prime},\mu}+2\delta_{\alpha^{\prime},\tau}\end{pmatrix}$
(32)

As we did in equation 24, we can remove the contribution proportional to the
diagonal, finally getting the expression for the neutrino-antineutrino
potential.

$V^{\prime}_{\nu\overline{\nu}}=-\frac{\sqrt{2}G_{F}}{V}(1-\hat{p}_{1}\cdot\hat{p}_{2})\begin{pmatrix}\delta_{\alpha^{\prime},e}&\delta_{\alpha^{\prime},\mu}&\delta_{\alpha^{\prime},\tau}\\\
\delta_{\alpha^{\prime},e}&\delta_{\alpha^{\prime},\mu}&\delta_{\alpha^{\prime},\tau}\\\
\delta_{\alpha^{\prime},e}&\delta_{\alpha^{\prime},\mu}&\delta_{\alpha^{\prime},\tau}\end{pmatrix}$
(33)

Note that this potential matrix in the flavor basis is, apart from a minus
sign, the transpose from the neutrino-neutrino one in equation 25 when looking
at the flavor content of the interacting neutrino $\nu_{\alpha^{\prime}}$.

##  4 Connection to Density Matrix Formalism

Finally, let us connect the potentials derived above to the density matrix
formalism. We will consider a neutrino quantum state $\ket{\psi_{\nu}(t)}$
going into a forward scattering with another neutrino in a state
$\ket{\phi_{\nu}(t)}$ or antineutrino $\ket{\phi_{\overline{\nu}}(t)}$.
Therefore, we can define the density matrix which represents the background of
neutrinos as an incoherent mixture of each neutrino quantum state.

$\rho_{p_{2}}(t)=\sum_{\nu^{\prime}}n_{\nu^{\prime},p_{2}}\ket{\phi_{\nu^{\prime},p_{2}}(t)}\bra{\phi_{\nu^{\prime},p_{2}}(t)}\rightarrow[\rho_{p_{2}}(t)]_{\alpha\beta}\equiv\bra{\nu_{\alpha}}\rho_{p_{2}}(t)\ket{\nu_{\beta}}$
(34)

$\overline{\rho}_{p_{2}}(t)=\sum_{\overline{\nu}^{\prime}}n_{\overline{\nu}^{\prime},p_{2}}\ket{\phi_{\overline{\nu}^{\prime},p_{2}}(t)}\bra{\phi_{\overline{\nu}^{\prime},p_{2}}(t)}\rightarrow[\overline{\rho}_{p_{2}}(t)]_{\beta\alpha}\equiv\bra{\overline{\nu}_{\alpha}}\overline{\rho}_{p_{2}}(t)\ket{\overline{\nu}_{\beta}}$
(35)

Where $\ket{\phi_{\nu^{\prime},p_{2}}(t)}$ is the state of the neutrino with
momentum $\vec{p}_{2}$ at time $t$ elapsed after its creation, and
$n_{\nu^{\prime},p_{2}}$ is density occupation number of this state. The index
$\nu^{\prime}$ represents any other quantum number for the neutrinos with
momentum $\vec{p}_{2}$, such as flavor or mass eigenvalues. Note that we
define the matrix elements $[\rho_{p_{2}}(t)]_{\alpha\beta}$ and
$[\overline{\rho}_{p_{2}}(t)]_{\beta\alpha}$ with inverted indices, as adopted
in [6, 5] . This allows $\rho$ and $\overline{\rho}$ to have the same
transformation when changing from one basis to another. Note also that we
could consider the density matrix of the entire neutrino ensemble by summing
over the momentum $\vec{p}_{2}$. However, we will keep the momentum
dependence, as it will be useful in the calculations below.

Considering the evolution equation 7 for the neutrino, we can work with each
component of the momentum basis separately instead of a full wave packet as in
equation 6. We can do this simplification because we are considering only
forward scattering. Therefore, the evolution equation for each eigenstate with
momentum $\vec{p}_{1}$ is the following in the flavor basis

$\begin{split}i\frac{d}{dt}\sum_{\alpha}\psi_{\alpha}(t)\ket{\nu_{\alpha}(p_{1})}=&\sum_{\alpha}\psi_{\alpha}(t)H\ket{\nu_{\alpha}(p_{1})}=\sum_{\alpha,\beta}\psi_{\alpha}(t)\ket{\nu_{\beta}(p_{1})}\bra{\nu_{\beta}(p_{1})}H\ket{\nu_{\alpha}(p_{1})}\\\
\end{split}$ (36)

To calculate the matrix element we have to consider the background final and
initial states $\ket{\phi_{\nu^{\prime},p_{2}}(t)}$ and then sum over the
momentum $\vec{p}_{2}$ considering an occupation number distribution
$N_{\nu^{\prime},p_{2}}$. The same has to be done for the antineutrinos states
$\ket{\phi_{\overline{\nu}^{\prime},p_{2}}(t)}$. As we are interested in the
neutrino-neutrino potential, we will ignore here the vacuum and matter
potential components of the Hamiltonian.

$\begin{split}&i\frac{d}{dt}\sum_{\alpha}\psi_{\alpha}(t)\ket{\nu_{\alpha}(p_{1})}=\\\
&=\sum_{\nu^{\prime}p_{2}}\sum_{\alpha,\beta}N_{\nu^{\prime},p_{2}}\psi_{\beta}(t)\bra{\nu_{\beta}(p_{1}),\phi_{\nu^{\prime},p_{2}}(t)}V_{\nu\nu}\ket{\phi_{\nu^{\prime},p_{2}}(t),\nu_{\alpha}(p_{1})}\ket{\nu_{\beta}(p_{1})}\\\
\\\
&+\sum_{\overline{\nu}^{\prime}p_{2}}\sum_{\alpha,\beta}N_{\overline{\nu}^{\prime},p_{2}}\psi_{\beta}(t)\bra{\nu_{\beta}(p_{1}),\phi_{\overline{\nu}_{,}p_{2}}(t)}V_{\nu\overline{\nu}}\ket{\phi_{\overline{\nu}^{\prime},p_{2}}(t),\nu_{\alpha}(p_{1})}\ket{\nu_{\beta}(p_{1})}\\\
\end{split}$ (37)

Decomposing the background neutrino and antineutrino states into the flavor
basis, we have

$\begin{split}&i\frac{d}{dt}\sum_{\alpha}\psi_{\alpha}(t)\ket{\nu_{\alpha}(p_{1})}=\\\
&=\sum_{\nu^{\prime}p_{2}}\sum_{\begin{subarray}{c}\alpha,\beta\\\
\alpha^{\prime},\beta^{\prime}\end{subarray}}N_{\nu^{\prime},p_{2}}\psi_{\alpha}(t)\phi^{*}_{\nu^{\prime}\beta^{\prime}}(t)\phi_{\nu^{\prime}\alpha^{\prime}}(t)\ket{\nu_{\beta}(p_{1})}\bra{\nu_{\beta}(p_{1}),\nu_{\beta^{\prime}}(p_{2})}V_{\nu\nu}\ket{\nu_{\alpha}(p_{1}),\nu_{\alpha^{\prime}}(p_{2})}\\\
&+\sum_{\overline{\nu}^{\prime}p_{2}}\sum_{\begin{subarray}{c}\alpha,\beta\\\
\alpha^{\prime},\beta^{\prime}\end{subarray}}N_{\overline{\nu}^{\prime},p_{2}}\psi_{\alpha}(t)\phi^{*}_{\overline{\nu}^{\prime}\beta^{\prime}}(t)\phi_{\overline{\nu}^{\prime}\alpha^{\prime}}(t)\ket{\nu_{\beta}(p_{1})}\bra{\nu_{\beta}(p_{1}),\overline{\nu}_{\beta^{\prime}}(p_{2})}V_{\nu\overline{\nu}}\ket{\nu_{\alpha}(p_{1}),\overline{\nu}_{\alpha^{\prime}}(p_{2})}\\\
\end{split}$ (38)

Using the results from sections 2 and 3 for the Hamiltonian matrix element, we
can write

$\begin{split}&i\frac{d}{dt}\sum_{\alpha}\psi_{\alpha}(t)\ket{\nu_{\alpha}(p_{1})}=\\\
&+\sum_{\nu^{\prime}p_{2}}\sum_{\alpha,\alpha^{\prime}}\sqrt{2}G_{F}\frac{N_{\nu^{\prime},\vec{p}_{2}}}{V}\psi_{\alpha}(t)\phi^{*}_{\nu^{\prime}\alpha^{\prime}}(t)\phi_{\nu^{\prime}\alpha^{\prime}}(t)(1-\hat{p}_{1}\cdot\hat{p}_{2})\ket{\nu_{\alpha}(p_{1})}\\\
&+\sum_{\nu^{\prime}p_{2}}\sum_{\alpha,\alpha^{\prime}}\sqrt{2}G_{F}\frac{N_{\nu^{\prime},\vec{p}_{2}}}{V}\psi_{\alpha}(t)\phi^{*}_{\nu^{\prime}\alpha}(t)\phi_{\nu^{\prime}\alpha^{\prime}}(t)(1-\hat{p}_{1}\cdot\hat{p}_{2})\ket{\nu_{\alpha^{\prime}}(p_{1})}\\\
&-\sum_{\overline{\nu}^{\prime}p_{2}}\sum_{\alpha,\alpha^{\prime}}\sqrt{2}G_{F}\frac{N_{\overline{\nu}^{\prime},\vec{p}_{2}}}{V}\psi_{\alpha}(t)\phi^{*}_{\overline{\nu}^{\prime}\alpha^{\prime}}(t)\phi_{\overline{\nu}^{\prime}\alpha^{\prime}}(t)(1-\hat{p}_{1}\cdot\hat{p}_{2})\ket{\nu_{\alpha}(p_{1})}\\\
&-\sum_{\overline{\nu}^{\prime}p_{2}}\sum_{\alpha,\beta}\sqrt{2}G_{F}\frac{N_{\overline{\nu}^{\prime},\vec{p}_{2}}}{V}\psi_{\alpha}(t)\phi^{*}_{\overline{\nu}^{\prime}\beta}(t)\phi_{\overline{\nu}^{\prime}\alpha}(t)(1-\hat{p}_{1}\cdot\hat{p}_{2})\ket{\nu_{\beta}(p_{1})}\\\
\end{split}$ (39)

Note that the terms proportional to $\ket{\nu_{\alpha}(p_{1})}$ only
contribute to the diagonal and with the same amplitude independent of the
flavor $\alpha$. Therefore, these terms only contribute to an overall phase
and can be ignored. Writing the result in the matrix format

$\begin{split}i\frac{d}{dt}\begin{bmatrix}\psi_{e}(t)\\\ \psi_{\mu}(t)\\\
\psi_{\tau}(t)\end{bmatrix}&=\sum_{\nu^{\prime},\vec{p}_{2}}\sqrt{2}G_{F}(1-\hat{p}_{1}\cdot\hat{p}_{2})n_{\nu^{\prime},\vec{p}_{2}}\begin{bmatrix}|\phi_{p_{2}}^{e}|^{2}&\phi_{p_{2}}^{\mu*}\phi_{p_{2}}^{e}&\phi_{p_{2}}^{\tau*}\phi_{p_{2}}^{e}\\\
\phi_{p_{2}}^{e*}\phi_{p_{2}}^{\mu}&|\phi_{p_{2}}^{\mu}|^{2}&\phi_{p_{2}}^{\tau*}\phi_{p_{2}}^{\mu}\\\
\phi_{p_{2}}^{e*}\phi_{p_{2}}^{\tau}&\phi_{p_{2}}^{\mu*}\phi_{p_{2}}^{\tau}&|\phi_{p_{2}}^{\tau}|^{2}\\\
\end{bmatrix}\begin{bmatrix}\psi_{e}(t)\\\ \psi_{\mu}(t)\\\
\psi_{\tau}(t)\end{bmatrix}\\\
&-\sum_{\overline{\nu}^{\prime},\vec{p}_{2}}\sqrt{2}G_{F}(1-\hat{p}_{1}\cdot\hat{p}_{2})n_{\overline{\nu}^{\prime},\vec{p}_{2}}\begin{bmatrix}|\phi_{p_{2}}^{e}|^{2}&\phi_{p_{2}}^{e*}\phi_{p_{2}}^{\mu}&\phi_{p_{2}}^{e*}\phi_{p_{2}}^{\tau}\\\
\phi_{p_{2}}^{\mu*}\phi_{p_{2}}^{e}&|\phi_{p_{2}}^{\mu}|^{2}&\phi_{p_{2}}^{\mu*}\phi_{p_{2}}^{\tau}\\\
\phi_{p_{2}}^{\tau*}\phi_{p_{2}}^{e}&\phi_{p_{2}}^{\tau*}\phi_{p_{2}}^{e}&|\phi_{p_{2}}^{\tau}|^{2}\\\
\end{bmatrix}\begin{bmatrix}\psi_{e}(t)\\\ \psi_{\mu}(t)\\\
\psi_{\tau}(t)\end{bmatrix}\end{split}$ (40)

Where the occupation number $N_{\nu^{\prime},\vec{p}_{2}}$ divided by the
volume turns into a density occupation number
$n_{\overline{\nu}^{\prime},\vec{p}_{2}}$. Using the matrix density definition
in equations 34 and 35 , we can rewrite this as

$i\frac{d}{dt}\ket{\psi_{\nu}(t)}=\sum_{\vec{p}_{2}}\sqrt{2}G_{F}(1-\hat{p}_{1}\cdot\hat{p}_{2})[\rho_{\vec{p}_{2}}(t)-\overline{\rho}_{\vec{p}_{2}}(t)]\ket{\psi_{\nu}(t)}$
(41)

This is finally the neutrino-neutrino potential format usually used in other
works. There are alternative definitions of the density matrix that may lead
to a slightly different expression, depending on other variables rather than
the momentum. However, this momentum-dependent density matrix can easily be
translated into the neutrino spectrum and vice-versa, which is helpful for
many applications.

###  4.1 Polarization Vector Formalism

One useful application for the neutrino-neutrino potential in the format of
equation 41 is the so-called Polarization Vector formalism. If we consider
only two families of neutrinos (e.g. $\nu_{e}$ and $\nu_{x}$ 888This can be
applied to supernovae, given that $\nu_{\mu}$ and $\nu_{\tau}$ are produced in
the same quantity and have the same interactions in the propagation and
detection.) the matrices in the evolution equation will be $2\times 2$ complex
matrices, which can be decomposed in Pauli matrices
$(\sigma_{1},\sigma_{2},\sigma_{3})$. The coefficients of this expansion work
as components of a vector in a 3-dimensional space, which are called
Polarization Vectors. To see this, let us rewrite the evolution of the
neutrino state $\ket{\psi_{\nu}(t)}$ in terms of its density matrix.

$i\frac{d\rho(t)}{dt}=[H,\rho(t)],\;\;\;\rho(t)\equiv\ket{\psi_{\nu}(t)}\bra{\psi_{\nu}(t)}$
(42)

We can therefore decompose the system in Pauli matrices.

$H=-\frac{1}{2}\vec{\sigma}\cdot\vec{B}\text{,}\;\;\;\;\;\rho=\frac{1}{2}\mathbf{1}+\frac{1}{2}\vec{\sigma}\cdot\vec{P}\text{,}\;\;\;\;\;\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$
(43)

Note that we ignore the Hamiltonian component proportional to the identity,
which is irrelevant for the evolution. With this decomposition, the evolution
equation for the neutrino became a precession equation due to the anti-
commutation rules of the Pauli matrices. In this picture, the polarization
vector of the neutrino $\vec{P}$ precesses around the Hamiltonian vector
$\vec{B}$.

$\frac{d}{dt}\vec{P}(t)=\vec{P}(t)\times\vec{B},\;\;\;\;\;\left(i\frac{d}{dt}P_{i}(t)=\frac{i}{2}P_{i}(t)B_{j}[\sigma_{i},\sigma_{j}]\right)$
(44)

In the case of neutrino-neutrino interactions, considering
$H=V_{\nu\nu}+V_{\nu\overline{\nu}}$, we have the following evolution
equation.

$\frac{d}{dt}\vec{P}_{\nu,\vec{p_{1}}}=\vec{P}_{\nu,\vec{p_{1}}}\times\sum_{\vec{p}_{2}}\sqrt{2}G_{F}(1-\hat{p}_{1}\cdot\hat{p}_{2})(\vec{P}_{\nu,\vec{p}_{2}}-\vec{P}_{\overline{\nu},\vec{p}_{2}})$
(45)

Therefore, a given neutrino polarization vector precesses around all the
others. This gives us a good pictorial view of the neutrino evolution inside a
high-density neutrino environment. Besides, this formalism is highly used in
works trying to solve the neutrino evolution in a high-density neutrino
environment, such as in supernovae.

##  5 Conclusions

In this paper, we have derived the potential due to neutrino-(anti)neutrino
forward scattering at the low energy regime. In our approach, we took the
Standard Model Hamiltonian for neutral current interactions in terms of the
neutrinos second quantized field and found the matrix element of it given the
initial and final states. The final expressions can be found in equations 25
for neutrino-neutrino interactions and 33 for neutrino-antineutrino, or in
equation 41 using the density matrix formalism.

To the best of the authors’ knowledge, this is the first time that these
calculations were explicitly done using massive neutrino fields at the
interaction. This approach shows explicitly how the mass of the neutrino
appears in the potential expression. As expected, the neutrino mass
contribution is suppressed by the energy in the ultra-relativistic limit,
given the well-known result of the massless approximation.

The expression for the potential found in this paper can be applied for
neutrinos in the energy regime of $E>>m_{\nu}$ (ultra-relativistic) and
$E<m_{Z_{0}}$ (low energy). That is the case for supernova neutrinos, which
was our motivation to derive this potential. The authors intend to use these
results in future works searching for solutions to collective oscillations in
supernova neutrinos. With the potential in terms of the density matrix, we can
easily translate it to the neutrino polarization vectors formalism, in which
we intend to explore the neutrino evolution in a supernova environment.

##  Appendix A Conventions

Here we describe our conventions regarding the neutrinos quantized fields and
states. We take the following Fourier expansion for the quantized neutrino
fields, using a finite normalization volume $V$ with periodic conditions,

$\begin{split}\nu_{\alpha}(x)&=\sum_{\vec{p},s}\frac{1}{\sqrt{2VE_{\vec{p}}}}\left[a^{s}_{\vec{p}}u^{s}(\vec{p})e^{-ipx}+b^{s\dagger}_{\vec{p}}v^{s}(\vec{p})e^{ipx}\right]\end{split}$
(46)

$\begin{split}\overline{\nu}_{\alpha}(x)&=\sum_{\vec{p},s}\frac{1}{\sqrt{2VE_{\vec{p}}}}\left[a^{s\dagger}_{\vec{p}}\overline{u}^{s}(\vec{p})e^{ipx}+b^{s}_{\vec{p}}\overline{v}^{s}(\vec{p})e^{-ipx}\right]\end{split}$
(47)

where the anti-commutation relations are

$\\{a^{s\dagger}_{p},a^{s^{\prime}\dagger}_{p^{\prime}}\\}=\\{b^{s\dagger}_{p},b^{s^{\prime}\dagger}_{p^{\prime}}\\}=\delta_{\vec{p}\vec{p}^{\prime}}\delta_{ss^{\prime}}$
(48)

Therefore, the fields obey the canonical equal-time anticommutation rules

$\\{\nu_{\alpha}(t,\vec{x}),\nu^{\dagger}_{\beta}(t,\vec{y})\\}=\delta^{(3)}(\vec{x}-\vec{y})\delta_{\alpha\beta}$
(49)

The one particle states are defined as

$\ket{\nu(p,s)}=a^{s\dagger}_{\vec{p}}\ket{0}\text{,}\;\;\;\;\;\ket{\overline{\nu}(p,s)}=b^{s\dagger}_{\vec{p}}\ket{0}$
(50)

where their normalization is given by

$\bra{\nu(p,s)}\ket{\nu(p^{\prime},s^{\prime})}=\bra{\overline{\nu}(p,s)}\ket{\overline{\nu}(p^{\prime},s^{\prime})}=\delta_{\vec{p}\vec{p}^{\prime}}\delta_{ss^{\prime}}$
(51)

Note that for Majorana neutrinos $a^{s\dagger}_{p}=b^{s\dagger}_{p}$.

##  Appendix B Trace Techniques

Here we derived some properties of spinors and Dirac matrices that are useful
to our work. The resulting properties may be found by the name of ”trace
techniques” in the literature, as they translate the product of spinors and
Dirac matrices into trace expressions.

###  B.1 Spinor to Trace

Lets consider a spinor that obeys the Dirac equation

$(\not{p}-m)u^{(h)}(p)=0$ (52) $\overline{u}^{(h)}(p)(\not{p}-m)=0$ (53)

The helicity projector can be written as

$P_{h}=\frac{1+\gamma^{5}\not{s}_{h}}{2}$ (54)

where

$s^{\mu}_{h}=h\left(\frac{|\vec{p}|}{m},\frac{E}{m}\frac{\vec{p}}{|\vec{p}|}\right)$
(55)

Also, lets consider the completeness relation

$\sum_{h=1}^{2}u^{h}(p)\overline{u}^{h}(p)=P_{h=1}u^{h}(p)\overline{u}^{h}(p)+P_{h=2}u^{h}(p)\overline{u}^{h}(p)=\not{p}+m$
(56)

Note that this expression is a $4\times 4$ matrix, which can be seen due to
the Dirac matrix in $\not{p}=\gamma^{\mu}p_{\mu}$. In general, all the
vertices factors in SM interactions can be written as
$\overline{u}^{h}(p)\Gamma u^{h}(p)$ where $\Gamma$ is a combination of Dirac
matrices. We can use the relations above to turn this interaction factor into
a trace expression.

$\begin{split}\overline{u}^{(h)}(p)\Gamma
u^{(h)}(p)=&\sum_{m,n}\overline{u}^{(h)}_{n}(p)\Gamma_{nm}u^{(h)}_{m}(p)=\sum_{m,n}u^{(h)}_{m}(p)\overline{u}^{(h)}_{n}(p)\Gamma_{nm}\\\
=&\sum_{m,n}\left[P_{h}(\not{p}+m)\right]_{mn}\Gamma_{nm}=\sum_{m}\left[P_{h}(\not{p}+m)\Gamma\right]_{mm}\\\
=&Tr\left[P_{h}(\not{p}+m)\Gamma\right]=Tr\left[\frac{1+\gamma^{5}\not{s}_{h}}{2}(\not{p}+m)\Gamma\right]\\\
=&Tr\left[(\not{p}+m)\left(\frac{1+\gamma^{5}\not{s}_{h}}{2}\right)\Gamma\right]\end{split}$
(57)

Note that going from the first line to the second, we have used only one
helicity component of equation 56. Note also that
$[\gamma^{5}\not{s}_{h},\not{p}]=0$. In the same way, we can derive the
following expression for the sum over the helicity states.

$\sum_{h=1}^{2}\overline{u}^{(h)}(p)\Gamma
u^{(h)}(p)=Tr\left[(\not{p}+m)\Gamma\right]$ (58)

###  B.2 Weak SM interactions

In the specific case of weak interactions of the Standard Model, we have the
following V-A structure $\Gamma=\gamma^{\mu}(1-\gamma^{5})$. Therefore,
equation 57 became

$\begin{split}\overline{u}^{(h)}(p)\gamma^{\mu}(1-\gamma^{5})u^{(h)}(p)=&\frac{1}{2}Tr\left[(\not{p}+m)(1+\gamma^{5}\not{s}_{h})\gamma^{\mu}(1-\gamma^{5})\right]=\\\
=&\frac{1}{2}Tr[(\not{p}+m)(\gamma^{\mu}-\gamma^{\mu}\gamma^{5}+\gamma^{5}\not{s}_{h}\gamma^{\mu}-\gamma^{5}\not{s}_{h}\gamma^{\mu}\gamma^{5})]=\\\
=&\frac{p_{\nu}}{2}Tr[\gamma^{\nu}\gamma^{\mu}-\gamma^{\nu}\gamma^{\mu}\gamma^{5}+\gamma^{\nu}\gamma^{5}\gamma^{\sigma}s^{h}_{\sigma}\gamma^{\mu}-\gamma^{\nu}\gamma^{5}\gamma^{\sigma}s^{h}_{\sigma}\gamma^{\mu}\gamma^{5}]\\\
+&\frac{m}{2}Tr[\gamma^{\mu}-\gamma^{\mu}\gamma^{5}+\gamma^{5}\gamma^{\sigma}s^{h}_{\sigma}\gamma^{\mu}-\gamma^{5}\gamma^{\sigma}s^{h}_{\sigma}\gamma^{\mu}\gamma^{5}]\end{split}$
(59)

Using the following proprieties999And the trace property
$Tr[A+B]=Tr[A]+Tr[B]$.

$\begin{split}&Tr[\gamma^{\nu}\gamma^{\mu}]=4g^{\mu\nu}\\\
&Tr[\gamma^{\nu}\gamma^{\mu}\gamma^{5}]=0\\\
&Tr[\gamma^{\nu}\gamma^{5}\gamma^{\sigma}\gamma^{\mu}]=0\\\
&Tr[\gamma^{\nu}\gamma^{5}\gamma^{\sigma}\gamma^{\mu}\gamma^{5}]=Tr[\gamma^{\nu}\gamma^{\sigma}\gamma^{\mu}]=0\\\
\end{split}$ (60)

$\begin{split}&Tr[\gamma^{\mu}]=0\\\ &Tr[\gamma^{\mu}\gamma^{5}]=0\\\
&Tr[\gamma^{5}\gamma^{\sigma}\gamma^{5}]=0\\\
&Tr[\gamma^{5}\gamma^{\sigma}\gamma^{\mu}\gamma^{5}]=Tr[\gamma^{\sigma}\gamma^{\mu}]=4g^{\sigma\mu}\\\
\end{split}$ (61)

we finally have our desired relation.

$\overline{u}^{(h)}(p)\gamma^{\mu}(1-\gamma^{5})u^{(h)}(p)=2(p^{\mu}-ms^{\mu}_{h})$
(62)

We can use the same reasoning for the equation 58 and obtain the relation
bellow.

$\sum_{h=1}^{2}\overline{u}^{(h)}(p)\gamma^{\mu}(1-\gamma^{5})u^{(h)}(p)=4p^{\mu}$
(63)

##  Appendix C About the Potential Normalization

The one-particle states of momentum $\vec{p}$ are defined in a way that their
normalization is Lorentz invariant.

$\bra{\vec{q}}\ket{\vec{p}}=(2\pi)^{3}2E_{\vec{p}}\delta^{(3)}(\vec{p}-\vec{q})$
(64)

However, as in quantum mechanics, these states are not properly normalized to
one in the continuum regime, and therefore can not describe physical
particles. One way to get rid of this is to use wave packets properly
normalized to represent particles. Another way is to consider finite volume
normalization with periodic boundary conditions. That shows us that amplitudes
calculated using states such as in 64 will not be properly normalized.
However, the right normalization using these states can be achieved by the
following process.

$\mathcal{A}_{norm}=\frac{\mathcal{A}_{fi}}{\sqrt{\bra{f}\ket{f}\bra{i}\ket{i}}}=\frac{\mathcal{A}_{fi}}{\prod_{i}\sqrt{E_{i}}\prod_{f}\sqrt{E_{f}}}=\frac{\bra{f}\mathcal{S}\ket{i}}{\sqrt{\bra{f}\ket{f}\bra{i}\ket{i}}}$
(65)

## Acknowledgments

This work was supported by São Paulo Research Foundation (FAPESP) grants no.
2019/08956-2 and no. 14/19164-6.

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