On The Origin of Neutrino Mass and Mixing in the Standard Model
Bob McElrath
One can describe cosmological relic neutrinos by adding Lagrange multipliers to the Standard Model Lagrangian for them. The two possible Lagrange multipliers are a chemical potential, which fixes the mean neutrino/anti-neutrino asymmetry, and a Majorana mass, which fixes the mean spin-entropy. Because these neutrinos originated from a thermal bath, their entropy should be maximal, implying that each state in the background is a symmetric superposition of a neutrino and anti-neutrino. Therefore the Standard Model must be augmented by a flavor-diagonal Majorana neutrino mass matrix. This impacts the propagator via tadpole diagrams due to self-interactions. In the low-energy limit, neutrino self-interactions are entirely off-diagonal because same-flavor four-fermion operators vanish by Pauli exclusion. These interactions must be diagonalized when propagating through a bath of neutrinos, using the $U(3)$ global flavor symmetry. $U(3)$ gets broken broken down to $SO(3)$ by Majorana masses, and down to $A_4$ if the three masses are different. Thus our universe today contains tri-bimaximal mixing and Majorana neutrinos. Neutrino mixing is due to the mismatch between the flavor-diagonal Majorana mass matrix arising at finite density and the self-interaction diagonal finite density propagator. The mass hierarchy is inverted and Majorana phases are absent. Lepton number is conserved and the neutrino-less double beta decay experiment absorbs a pair of neutrinos from the relic background and will prove their Majorana nature.
majorana_mass.tex
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%  On the Origin of Neutrino Mass and Mixing in the Standard Model
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\begin{document}
\title{On The Origin of Neutrino Mass and Mixing in the Standard Model}

\author{Bob McElrath}
\email[]{bob.mcelrath@cern.ch}
\affiliation{CERN theory group, Geneva 23, CH 1211, Switzerland}

\vskip 0.2in 
\begin{abstract}
    One can describe cosmological relic neutrinos by adding Lagrange
    multipliers to the Standard Model Lagrangian for them.  The two possible
    Lagrange multipliers are a chemical potential, which fixes the mean
    neutrino/anti-neutrino asymmetry, and a Majorana mass, which fixes the mean
    spin-entropy.  Because these neutrinos originated from a thermal bath,
    their entropy should be maximal, implying that each state in the background
    is a symmetric superposition of a neutrino and anti-neutrino.  Therefore
    the Standard Model must be augmented by a flavor-diagonal Majorana neutrino
    mass matrix.  This impacts the propagator via tadpole diagrams due to
    self-interactions.  In the low-energy limit, neutrino self-interactions are
    entirely off-diagonal because same-flavor four-fermion operators vanish by
    Pauli exclusion.  These interactions must be diagonalized when propagating
    through a bath of neutrinos, using the $U(3)$ global flavor symmetry.
    $U(3)$ gets broken broken down to $SO(3)$ by Majorana masses, and down to
    $A_4$ if the three masses are different.  Thus our universe today contains
    tri-bimaximal mixing and Majorana neutrinos.  Neutrino mixing is due to the
    mismatch between the flavor-diagonal Majorana mass matrix arising at finite
    density and the self-interaction diagonal finite density propagator.  The
    mass hierarchy is inverted and Majorana phases are absent.  Lepton number
    is conserved and the neutrino-less double beta decay experiment absorbs a
    pair of neutrinos from the relic background and will prove their Majorana
    nature.  
\end{abstract}
\noindent 

\def\os{\overline{\sigma}}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{footnote}{0}

\section{Introduction}

Neutrinos have been observed to change flavor.  Originally known as the ``solar
neutrino problem'' due to the apparent deficit of solar neutrinos, today this is
well-understood as the mixing of electron-type neutrinos emitted in nuclear
reactions into the $\mu$ and $\tau$ types.  The most economical way to describe
this mixing is via an off-diagonal mass matrix.  One can implement this with
either Majorana or Dirac neutrino masses.  We currently have no experimental
evidence that neutrino masses are either Dirac or Majorana, though there are
several running and planned experiments to test this using neutrino-less double
beta decay, which can only occur with a Majorana mass.\cite{Doi:1985dx} The
current experimental situation is well-explained by a three-neutrino mixing
hypothesis.\cite{Schwetz:2008er} 

The standard cosmological model predicts the existence of a sea of relic
neutrinos if the temperature was ever above a few MeV.  The leading order
dynamics of this background that is relevant for neutrino experiments is its
Pauli blocking.  This can be included by adding Lagrange multipliers to the
Lagrangian that fix the number density and entropy.

Whenever particles propagate in a medium, an index of refraction and rescaling
of the kinetic term are induced, defined by the tadpole diagram corrections to
the self-energy.  This is a non-perturbative effect that must be re-summed using
the Dyson series into the two-point function.  

\section{Mean Field Neutrinos}

Our universe contains neutrinos, therefore when doing ``vacuum'' physics we
should fix the parameters of the background relics with which we are not
scattering.  This is the approach taken whenever one adds a chemical potential
directly to the Lagrangian.  One is averaging over the degrees of freedom
represented by the chemical potential, treating them instead as a ``mean''
field.  One can also envision this as constraining the first moment of the
thermal distribution function.

A finite density system in an arbitrary configuration can be represented by a
density matrix
\begin{equation}
    \rho_{ij}(p) = |\psi_i(p)\rangle \langle \psi_j(p)|.
    \label{eq:density}
\end{equation}
The indices $i,j$ represent the distinct quantum numbers possible for a single
momentum mode $p$, such as helicity and flavor.  The average value of this
density matrix is
\begin{equation}
    \langle \rho_{ij} \rangle = \int \frac{d^3p}{(2 \pi)^3 \sqrt{2 E}}
    \rho_{ij}(p),
    \label{eq:avgdens}
\end{equation}
In the mean field approach one assumes that the background state $\rho_{ij}$ is
independent of time.  By then adding Lagrange multipliers for the components of
$\rho_{ij}(p)$ (written in terms of the field operator), one ensures that the
result Eq.\ref{eq:avgdens} is reflected in the dynamics.  This assumes that
Eq.\ref{eq:avgdens} will be unchanged by the dynamics we are interested in
computing.

Let us consider a single massless Weyl fermion.  Its $2 \times 2$ spin-density
matrix operator $\hat \rho$ can be related to the field operators using
\begin{equation}
    v = (a_+,\  a_-); \qquad
    \hat \rho = v^\dagger|0\rangle \langle 0 | v
% This expression needs |0\rangle \langle 0| in the middle.
%    = \left(
%    \begin{array}{cc}
%        a_+^\dagger a_+ & a_+^\dagger a_- \\
%        a_-^\dagger a_+ & a_-^\dagger a_-
%    \end{array}
%    \right)
    \label{eq:v}
\end{equation}
where $a_{\pm}$ are the annihilation operators for the two helicities, and the
momentum dependence has been suppressed for notational simplicity.  The field
operators for a Weyl fermion are
\begin{equation}
    \chi(x) = \int \frac{d^3p}{(2 \pi)^3 \sqrt{2 E}}\left[a_- u_- e^{-i px} +
    a_+^\dagger v_+ e^{ipx} \right]
    \label{eq:weylstate}
\end{equation}
with the commuting state vectors in the helicity basis for neutrinos and
anti-neutrinos respectively
\begin{equation}
    u_- = {0 \choose 1}, \qquad v_+ = {1 \choose 0}.
    \label{eq:uv}
\end{equation}

In terms of the density operator Eq.\ref{eq:v}, we can find the relationship
between the density matrix and Lagrangian operators by evaluating 
$\Tr[\hat \rho\, \sigma^j]$:
\begin{eqnarray}
    \nonumber
    \Tr [\hat \rho\, \sigma^0] &=& 1 \\
    \nonumber
    \Tr[\hat \rho\, \sigma^1] &=& \frac{1}{2} 
        \left[ \chi^\dagger i \sigma^2 \chi^* - \chi^T i \sigma^2 \chi \right]
        = a_-^\dagger a_+ + a_+^\dagger a_- \\
    \nonumber
    \Tr[\hat \rho\, \sigma^2] &=& \frac{1}{2}\left[ 
        \chi^\dagger \sigma^2 \chi^* + \chi^T \sigma^2 \chi \right]
        = i(a_+^\dagger a_- - a_-^\dagger a_+) \\
    \Tr [\hat \rho\, \sigma^3] &=& \chi^\dagger \sigma^0 \chi 
        = a_+^\dagger a_+ - a_-^\dagger a_- .
    \label{eq:trrho}
\end{eqnarray}
The first of these specifies the normalization of the density matrix.  The
second and third combinations are both Majorana masses, and are related by a
lepton number phase transformation.  Finally the fourth is the number operator.

In the mean field approximation described by the momentum-averaged density
matrix, Eq.\ref{eq:avgdens}, one can now add $\mu_j \Tr[\hat\rho\,\sigma_j]$ to
the Lagrangian with constant coefficients.  This is a Lagrange multiplier, and
when finding the equations of motion for the fermion $\chi$, will ensure that
these average values specified by the density matrix are preserved.  Note that
$\mu_i$ does not transform as a Lorentz vector.

Let us examine further the nature of these operators.  
%The density matrix, assuming there is one neutrino in a volume $V$, in
%terms of $\psi$ (Eq.\ref{eq:psi}) is
%\begin{equation}
%    \langle \rho \rangle
%    = \left( \begin{array}{cc} 
%        R^2 & R\sqrt{1-R^2}e^{i(\theta-\phi)} \\
%        R\sqrt{1-R^2} e^{i(\phi - \theta)} & 1-R^2
%    \end{array}
%    \right) \frac{1}{V}
%    .
%    \label{eq:densex}
%\end{equation}
With two neutrinos in a volume $V$, the averaged density matrix is
\begin{equation}
    \langle \rho \rangle_2
    = \left( \begin{array}{cc} 
        R_2^2 & R_2\sqrt{1-R_2^2}e^{i\varphi_2} \\
        R_2\sqrt{1-R_2^2} e^{-i\varphi_2} & 1-R_2^2
    \end{array}
    \right) \frac{2}{V}
    .
    \label{eq:densex}
\end{equation}
The appropriate traces analogous to Eq.\ref{eq:trrho} are
\begin{eqnarray}
    \Tr [\langle\rho\rangle_2 \sigma^1] &=& 2 R_2 \sqrt{1-R_2^2}\cos (\varphi_2) \frac{2}{V} \\
    \Tr [\langle\rho\rangle_2 \sigma^2] &=& 2 R_2 \sqrt{1-R_2^2}\sin (\varphi_2) \frac{2}{V} \\
    \Tr [\langle\rho\rangle_2 \sigma^3] &=& (2 R_2^2-1)\frac{2}{V},
    \label{eq:tralpha}
\end{eqnarray}
where we've taken a unit volume $V$.  For a pure state of two neutrinos,
$R_2=0$, the Majorana mass term $\Tr [\langle\rho\rangle_2 \sigma^1]=0$, and
the chemical potential term $\Tr [\langle\rho\rangle_2\sigma^3]=-2/V$ is
negative.  For a pure state of anti-neutrinos, $R_2=1$ and the chemical
potential term changes sign, while the Majorana mass term remains zero.  For a
mixed state with an equal probability for each to be a neutrino or
anti-neutrino, $R_2=1/\sqrt{2}$, the Majorana mass terms are maximal, and the
chemical potential term $\Tr[\langle \rho\rangle_2\sigma^3]$ is zero.

The chemical potential $\mu_3$
%\begin{equation}
%    \mu_3 \Tr[\hat \rho \sigma_3] = \mu_3 \chi^\dagger \sigma^0 \chi.
%    \label{eq:chempot}
%\end{equation}
is expected to be related to the baryon asymmetry of the universe, and very
small
\[
    \eta_\nu = \frac{n_\nu - n_{\overline\nu}}{n_\gamma} \sim
    6\times 10^{-10}.
\]
Therefore we will neglect it and assume $\mu_3=0$ so that the number of
neutrinos and anti-neutrinos are the same.  

The state with both $\mu_1$ and $\mu_2$ zero is a superposition of a pure state
of neutrinos with a pure state of anti-neutrinos.  The averaged density matrix is
proportional to the identity.  This configuration has zero spin-entropy
\begin{equation}
    S_\pm = -k \Tr [\langle \rho \rangle \ln \langle \rho \rangle].
    \label{eq:entropy}
\end{equation}
Since the early-universe plasma from which these neutrinos originated should be
extremely close to a blackbody spectrum, as is observed for the Cosmic
Microwave Background, the entropy for this system should instead be maximal.
For the spin-entropy this corresponds to maximizing the magnitude of the
off-diagonal components, e.g.  $R=1/\sqrt{2}$, and is a state that is in an
equal superposition of neutrino and anti-neutrino. 

If we examine the thermal Fermi-Dirac distribution $f(E) = (e^{E/kT}+1)^{-1}$
for a system with zero chemical potential, we see that $f(E) \le 1/2$ for all
energies.  In other words, each momentum mode there is either a neutrino or an
anti-neutrino, but not both (which would correspond to $f(E)=1$ for that
energy).  If both were present for a single momentum mode, Fermi statistics
indicates that the two density matrices are orthogonal $\rho_1 \cdot \rho_2 =
0$, which implies that their sum is the identity.  This configuration has zero
spin-entropy also.

Each pair in the symmetric bath therefore contributes $S_\pm = k\,2 \ln 2$ to
the spin-entropy.  Expanding in Eq.\ref{eq:entropy}, $\langle \rho \rangle = 1
+ \epsilon$, where $\epsilon$ is off diagonal, we can see that only the
magnitude of the off diagonal components of $\langle \rho\rangle$ contribute to
the spin-entropy.  If the phase were random, as in the Random Phase
Approximation (RPA), these off diagonal elements would cancel in the sum,
Eq.\ref{eq:avgdens} leading again to zero spin-entropy.  This is clearly
incorrect.  Instead the condition of maximal spin-entropy implies that all
states have a fixed, common phase.  

The two masses $\mu_1$ and $\mu_2$ are related by the lepton number phase
symmetry $\chi \to e^{i \eta}\chi$.  The Standard Model conserves lepton number
for each species individually, so we can use this freedom to rotate away one of
$\mu_1$ or $\mu_2$.  In other words, the Majorana phases
$\alpha=\arctan(\mu_1/\mu_2)$ if present, can be set to zero and are
unphysical.  Only $\mu_1^2+\mu_2^2$ is physical.

Thus given the expectation that the relic neutrino background has small
chemical potential and maximal entropy, we should add to the Standard Model a
Majorana mass (which we can choose to be $\mu_1$ or $\mu_2$) for each flavor,
as well as a small chemical potential.  
%\[%begin{equation}
%\nonumber
%    \mu_3 \Tr [\hat \rho \sigma^3] + \mu_1 \Tr [ \hat \rho \sigma^1]
%    = \mu_3 \chi^\dagger \sigma^0 \chi + \frac{i \mu_2}{2}[
%        \chi^\dagger \sigma^2 \chi^* - \chi^T \sigma^2 \chi].
%\nonumber
%%    \label{eq:lms}
%\]%end{equation}
We do not add flavor off-diagonal terms because the creation and annihilation
operators in Eq.\ref{eq:trrho} are related by the Lorentz representation of the
Weyl field.  In other words, a Lorentz transformation does not perform flavor
rotations.

Thus, the Majorana mass should not be considered a ``fundamental'' Lagrangian
parameter.  Instead it is an environmental parameter; a Lagrange multiplier
analogous to the chemical potential for a system with spin-entropy.  Because the
spin-entropy is maximal and simply related to the number density, the Majorana
mass is also in effect fixing the total number of neutrinos.  Quantitatively
then, it is simply the Fermi momentum
\begin{equation}
    \mu_1 = p_F = (3 \pi^2 n)^{1/3}
    \label{eq:fermimom}
\end{equation}
where $n$ is the number density.

\section{Self-Interactions}

Neutrino self-interactions occur at tree level by $Z$ boson exchange.  The
energies of experiments which have demonstrated neutrino mixing are far below
the energy where the process $\nu \overline \nu \to \nu \overline \nu$ becomes
large because the $Z$ boson is on shell.  Therefore we may integrate the $Z$ out
using its leading order equation of motion at zero momentum, which generates
four-point neutrino contact operators.

For neutrino energies $E \ll M_Z^2/T_\nu$, only these interaction terms survive
\begin{eqnarray}
    \label{eq:effint}
    J_i^\lambda (J_i)_\lambda &=& 
    (\chi^\dagger_e \os^\lambda \chi_e
    +\chi^\dagger_\mu \os^\lambda \chi_\mu
    +\chi^\dagger_\tau \os^\lambda \chi_\tau)^2 \\
    \nonumber
    %(\\chi^\dagger_e \os_\lambda \chi_e
    %+\\chi^\dagger_\mu \os_\lambda \chi_\mu
    %+\\chi^\dagger_\tau \os_\lambda \chi_\tau)
    &=& J^\lambda_{e\mu} (J_{e\mu})_\lambda + J^\lambda_{\mu\tau}
    (J_{\mu\tau})_\lambda + J^\lambda_{\tau e} (J_{\tau e})_\lambda
\end{eqnarray}
in terms of the currents $J_i^\lambda = \epsilon^{ijk} \chi^\dagger_j
\os^\lambda \chi_k$.  The same-flavor four-fermion interactions vanished in
this equation because
\begin{equation}
    \chi^\dagger \os^\lambda \chi \chi^\dagger \os_\lambda \chi = 0
%    \overline{\psi} \gamma^\lambda P_L \psi \overline{\psi} \gamma_\lambda P_L \psi = 0,
    \label{eq:fdfourfermi}
\end{equation}
which is another way of saying that same-flavor interactions only occur in the
$p$-wave due to Pauli blocking, and vanish in the low energy limit below the $Z$
mass.
%  One can see this for instance by applying a Fierz transformation to
%Eq.\ref{eq:fdfourfermi}, or just writing out the Grassmann components.

The interactions of Eq.\ref{eq:effint} clearly do not correspond to the
flavor-diagonal kinetic terms.  Therefore we should do a basis rotation to make
them flavor diagonal before computing the Green's functions, using the Standard
Model's vacuum global symmetry $U(3)$ on the three families of leptons.  Adding
Majorana masses that are the same for all families breaks all the real
generators of this symmetry.  The remaining pure imaginary generators form a
representation of $SO(3)$.

This residual $SO(3)$ of the Standard Model with three equal Majorana masses
and its Noether current are
\begin{equation}
    \chi_{i} \to e^{\alpha_a T^a_{ij}} \chi_j; \qquad J^\lambda_i = \epsilon_{ijk}
    \chi^\dagger_j
    \os^\lambda \chi_k,
    \label{eq:majoranao3}
\end{equation}
where the Latin indices $\{i,j\}$ run over the three families and the three
generators $T^a$ are
\[
    T^{e\mu} = \left(
    \tiny
    \begin{array}{rrr}
        0 & 1 & 0 \\
        -1& 0 & 0 \\
        0 & 0 & 0
    \end{array}
    \right), \ 
    T^{\mu\tau} = \left(
    \tiny
    \begin{array}{rrr}
        0 & 0 & 0 \\
        0 & 0 & 1 \\
        0 &-1 & 0
    \end{array}
    \right), \ 
    T^{\tau e} = \left(
    \tiny
    \begin{array}{rrr}
        0 & 0 &-1 \\
        0 & 0 & 0 \\
        1 & 0 & 0
    \end{array}
    \right)
\]
with $a = \{e \mu, \mu \tau, \tau e\}$, and we can see that the currents in
Eq.\ref{eq:majoranao3} are off-diagonal relative to the flavor basis,
\begin{eqnarray}
    \nonumber
    J^\lambda_{e\mu} &=& \chi^\dagger_e \os^\lambda \chi_\mu - \chi^\dagger_\mu \os^\lambda \chi_e \\
    \label{eq:currents}
    J^\lambda_{\mu\tau} &=& \chi^\dagger_\mu \os^\lambda \chi_\tau -
    \chi^\dagger_\tau \os^\lambda \chi_\mu \\
    \nonumber
    J^\lambda_{\tau e} &=& \chi^\dagger_\tau \os^\lambda \chi_e -
    \chi^\dagger_e \os^\lambda \chi_\tau.
\end{eqnarray}

Using our vacuum flavor $SO(3)$ freedom, we can find a basis in which the
interacting currents correspond to the kinetic term fields (which remain
diagonal under a $SO(3)$ flavor rotation).  Because the couplings are all the
same there is nothing to differentiate the flavors in the vacuum, so we know
that the diagonal interaction will take the form of a sum of squares of a
current, and that current must be the same for all three fields.  The rotation
that accomplishes this is the ``tri-bimaximal mixing'' matrix, and has the form
\begin{equation}
    U = \left( \begin{array}{ccc}
        \sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} & 0 \\
        -\frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} \\
        -\frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{2}}
    \end{array}\right).
    \label{eq:tbm}
\end{equation}
The three identical currents after the rotation are
\begin{eqnarray}
     J_{\rm int}^\lambda = \frac{1}{\sqrt{3}}\left( J^\lambda_{e\mu} +
     J^\lambda_{\mu\tau} +  J^\lambda_{\tau e} \right)
    \label{eq:nc}
\end{eqnarray}
leading to an effective interaction
\begin{eqnarray}
    \label{eq:fourpointj}
    \frac{g_Z^2}{2 M_Z^2}  J_{\rm int}^\lambda  J^{\rm int}_\lambda 
    &=& \frac{g_Z^2}{6 M_Z^2} \sum_{i=1}^3 \chi^\dagger_i \os^\lambda  \chi_i
    \chi^\dagger_i \os_\lambda  \chi_i. \\
    &=& \frac{g_Z^2}{6 M_Z^2} \sum_{i=1}^3 \psi^T_i \gamma^\lambda P_L \psi_i \psi^T
    \gamma_\lambda P_L \psi_i = 0.
    \nonumber
\end{eqnarray}
where we have written the expression using real Majorana spinors in the last
line.  This is the vacuum self-interaction basis of neutrinos in the Standard
Model.  However one can see by Eq.\ref{eq:fdfourfermi} that because different
flavor fermions also normally taken to anti-commute, this rotated four-Fermi
operator vanishes too!  Therefore, at $\mathcal{O}(G_F)$ neutrinos have no
self-interactions.

Eq.\ref{eq:fourpointj} has a discrete $A_4$ symmetry, in which the legs of the
four-Fermi operator are exchanged (this is easier to see using real Majorana
spinors).  The discrete symmetry of the interchange of four objects generates
the ``tri-bimaximal'' mixing matrix, Eq.\ref{eq:tbm}.

The interaction term of Eq.\ref{eq:effint} and Eq.\ref{eq:fourpointj} are
identical.  Nothing in the vacuum Lagrangian forces us to choose this, or any
other basis.  It is the basis defined by $U$
that the neutrino self-interactions are diagonal in the low-energy limit, and
the neutrino propagator is most transparently defined
when propagating through a neutrino medium.  One can think of neutrinos as
sharing one propagator, even though they interact with charged currents as three
separate fields.  

Following the arguments of the previous section, we must now add a
flavor-diagonal Majorana mass matrix
%\begin{equation}
%    \mu_1^e \Tr[\langle \rho^e \rangle\,\sigma_1]
%    +\mu_1^\mu \Tr[\langle \rho^\mu \rangle\,\sigma_1]
%    +\mu_1^\tau \Tr[\langle \rho^\tau \rangle\,\sigma_1]
%%    (\overline{\psi}_{\nu_e},
%%     \overline{\psi}_{\nu_\mu},
%%     \overline{\psi}_{\nu_\tau})
%%    \left(
%%    \begin{array}{ccc}
%%        m_{\nu_e} & & \\
%%        & m_{\nu_\mu} & \\
%%        & & m_{\nu_\tau}
%%    \end{array}\right)
%%    \left(\begin{array}{c}
%%        \psi_{\nu_e} \\
%%        \psi_{\nu_\mu} \\
%%        \psi_{\nu_\tau}
%%    \end{array}
%%    \right)
%%    %\sum_i m_i \overline{\psi_i} \psi_i
%    \label{eq:majmasses}
%\end{equation}
to the Lagrangian.  We chose the flavor basis because one should first write
the vacuum Lagrangian in a basis with diagonal 2-point correlation functions
for all fields (especially the charged leptons) and then add Lagrange
multipliers corresponding to the degrees of freedom in that vacuum Lagrangian.
Writing masses diagonally in any other basis is equivalent however. 

This mass matrix does not have the $SO(3)$ symmetry of the vacuum Lagrangian,
and breaks the $SO(3)$ down to the discrete $A_4$ in Eq.\ref{eq:fourpointj}.
If two of the masses were equal there would be a remaining $SO(2)$ symmetry,
but we already know experimentally this is not the case.  

\section{Neutrino 2-point Green's Function}

The purpose of the analysis in the two preceding sections is to enable us to
compute the full, in-medium two-point correlation function.  This means
computing the one-particle irreducible (1PI) tadpole interaction with the
background.  The effect of these tadpoles is inherently non-perturbative, and
cannot be neglected due to a small coupling with the medium.

The propagation of a single fermion is governed by the equation of motion
\begin{equation}
    \left[ \slashchar{k} - \Sigma(k) \right] \psi = 0.
    \label{eq:eqmo}
\end{equation}
The self-energy $\Sigma(k)$ has the general form
\begin{equation}
    \Sigma(k) = m - a \slashchar{k} 
    + b \slashchar{\mu} 
    + c \left[ \slashchar{k}, \slashchar{u}\right]
    \label{eq:genselfenergy}
\end{equation}
where $u^\alpha$ is the frame in which the thermal distribution function is
defined.\cite{weldon_effective_1982}  Terms proportional to $u^\alpha$ can only
show up at $\mathcal{O}(G_F^2)$ in the low energy theory, because the Majorana
mass is Lorentz invariant.  It is an expectation value for the energy $\langle
E \rangle \ne 0$ that defines the rest frame of the medium $u^\alpha$.  Here
$a$ contributes to a rescaling of the kinetic term, and $b$ and $c$ contribute
to the index of refraction.

The really dangerous term that necessitates the diagonalization of
Eq.\ref{eq:tbm} is $a$.  If we had done these computations in a basis with an
off-diagonal mass matrix (whatever its origin), and off-diagonal
self-interactions we would have also generated off-diagonal kinetic terms, via
the $a$ and $c$ of Eq.\ref{eq:genselfenergy} that must be diagonalized in order
to have a sensible set of correlation functions.  

Using the self-interaction basis in Eq.\ref{eq:tbm}, the neutral current
interactions are entirely removed at $\mathcal{O}(G_F)$.  The finite density
contributions to Eq.\ref{eq:genselfenergy} therefore arise at
$\mathcal{O}(G_F^2)$ due to charged $W^\pm$ bosons and $p$-wave $Z$ bosons.  We
have kept the kinetic term diagonal via this rotation at the expense of
generating non-diagonal matrices for neutrino mass, neutrino chemical
potential, and charged lepton mass.  All three of these matrices break the
$A_4$ symmetry that we found among the neutral interactions at
$\mathcal{O}(G_F)$.  Therefore, all three of these generate corrections to $U$
at $\mathcal{O}(G_F^2)$.  The largest of these arises from the $\tau$ lepton,
which generates corrections 
\begin{equation}
    \delta U \sim \mathcal{O}\left(\frac{m_\tau^2 G_F}{16 \pi^2}\right) \sim
    10^{-7}
\label{eq:deltau}
\end{equation}

The contributions at $\mathcal{O}(G_F)$ found in
Refs.\cite{notzold_neutrino_1988,D'Olivo:1992vm} vanish by
Eq.\ref{eq:fdfourfermi} in the case of same-flavor interactions and are only
present when considering a fictitious experiment of a neutrino propagating
through a bath of neutrinos of a different flavor.  Radiative corrections to
these terms are also computed in Ref.\cite{langacker_standard_1992}.

\section{Discussion}
The presence of the relic neutrino background in the universe necessitates
diagonalizing neutrino self-interactions in order to compute the neutrino
propagator.  The dominant observable consequence of this is an apparent flavor
mixing due to the mismatch between the propagator's self-interaction basis and
the flavor basis in which neutrinos are created today by $W^\pm$ bosons, and
their spin-entropy defined.

The mixing matrix is given uniquely by the requirement that the interactions are
diagonal, and is of ``tri-bimaximal'' form, previously guessed but not explained
by other authors.\cite{Harrison199979,Harrison2002167}  Therefore 
\begin{equation}
    \sin^2 \theta_{12} = 1/3 \quad
    \sin^2 \theta_{23} = 1/2  \quad
    \sin^2 \theta_{13} = 0
\end{equation}
in the Standard Model, in the absence of new physics, if there exists a relic
neutrino background containing all three flavors.  The leading corrections to
this are $\mathcal{O}(m_\tau^2 G_F/16 \pi^2)\sim 10^{-7}$.

In the mixing matrix $\sin^2 \theta_{13}= 0$ also implies the absence of CP
violation, which can be understood because the relic neutrino background with
no chemical potential is itself C and CP symmetric.  A tiny $\sin^2
\theta_{13}$ will be generated by loops of charged leptons, and a further
correction as well as an extremely tiny CP violating phase will be generated by
a chemical potential $\mu_3$.

Because the $\tau^\pm$ freezes out of thermal equilibrium before the $\mu^\pm$
and $\pi^\pm$, the relic density of $\nu_\tau$ is smaller than that of $\nu_e$
and $\nu_\mu$ today.  The $\pi^\pm$ and $\mu^\pm$ are very similar in mass, so
they freeze out creating similar densities of $\nu_e$ and $\nu_\mu$.  Thus due
to cosmic expansion the density of $\nu_\tau$ has had more time to dilute than
that of $\nu_e$ and $\nu_\mu$, so the hierarchy is inverted with $\delta
m^2_{\rm sol} = m^2_{\nu_e}-m^2_{\nu_\mu}$ and $\delta m^2_{\rm atm} =
m_{\nu_e}^2 - m_{\nu_\tau}^2$.

We unambiguously predict that the Standard Model's neutrino mass is of Majorana
type.  Therefore neutrino-less double beta decay experiments will observe the
neutrino mass, and can be understood as absorbing a pair of neutrinos from the
background.  New physics which generates mixing is perturbing around
tri-bimaximal mixing, rather than no mixing.

%The discovery of neutrino mixing also qualifies as a discovery of the presence
%of background neutrinos.

\section{Acknowledgements}

We thank Jose Bernabeu, Kiwoon Choi, Radovan Dermisek, Joachim Kopp, Pasquale
Serpcio, Thomas Schwetz and Jure Zupan for valuable conversations and comments
on this manuscript.

\bibliographystyle{unsrt}
\bibliography{majorana_mass}

\end{document}

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