The operator $\sigma_{\alpha\beta}$ is the $4\times 4$ spin tensor defined in
Eq. (1.26). We would like to point out some interesting features of the Pauli
term most notably that the spin tensor itself is not Hermitian with

$\displaystyle\sigma_{\alpha\beta}^{{\dagger}}=\gamma_{0}\sigma_{\alpha\beta}\gamma_{0}\,.$
(3.10)

However, the conjugate of the Lagrangian term in Eq. (3.9)

$\displaystyle\left(\nu^{{\dagger}}\gamma_{0}\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}\nu\right)^{{\dagger}}=\nu^{{\dagger}}\sigma_{\alpha\beta}^{{\dagger}}F^{\alpha\beta}_{\mathrm{ext}}\gamma_{0}\nu=\nu^{{\dagger}}\gamma_{0}\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}\nu\,,$
(3.11)

is Hermitian. More about the spin tensor’s properties will be elaborated on in
Section 3.1.

The Majorana magnetic moment matrix acts in flavor space. It satisfies the
following constraints (Giunti and Studenikin, 2015) for CPT symmetry reasons
and the anticommuting nature of fermions

$\displaystyle\mu_{\ell\ell^{\prime}}^{{\dagger}}=\mu_{\ell\ell^{\prime}}\,,\qquad\mu_{\ell\ell^{\prime}}^{\mathrm{T}}=-\mu_{\ell\ell^{\prime}}\,,$
(3.12)

i.e. the AMM matrix $\mu_{\ell\ell^{\prime}}$ is Hermitian and fully anti-
symmetric. This requires that the transition magnetic moment elements are
purely imaginary while all diagonal AMM matrix elements vanish

$\displaystyle\mu_{\ell\ell^{\prime}}=\begin{pmatrix}\mu_{ee}&\mu_{e\mu}&\mu_{e\tau}\\\
\mu_{\mu e}&\mu_{\mu\mu}&\mu_{\mu\tau}\\\ \mu_{\tau
e}&\mu_{\tau\mu}&\mu_{\tau\tau}\end{pmatrix}\xrightarrow{\mathrm{Majorana}}\mu_{\ell\ell^{\prime}}=\begin{pmatrix}0&i\mu_{e\mu}&-i\mu_{e\tau}\\\
-i\mu_{e\mu}&0&i\mu_{\mu\tau}\\\
i\mu_{e\tau}&-i\mu_{\mu\tau}&0\end{pmatrix}\,.$ (3.13)

We can combine the mass term in Eq. (3.8) and AMM contribution in Eq. (3.9)
into a single effective Lagrangian

$\displaystyle\mathcal{L}_{\mathrm{eff}}^{\mathrm{Maj.}}$
$\displaystyle=\mathcal{L}_{\mathrm{kinetic}}^{\mathrm{Maj.}}+\mathcal{L}_{\mathrm{mass}}^{\mathrm{Maj.}}+\mathcal{L}_{\mathrm{AMM}}^{\mathrm{Maj.}}\,,$
(3.14) $\displaystyle\mathcal{L}_{\mathrm{eff}}^{\mathrm{Maj.}}$
$\displaystyle=\mathcal{L}_{\mathrm{kinetic}}^{\mathrm{Maj.}}-\frac{1}{2}\bar{\nu}_{\ell}\left(M_{\ell\ell^{\prime}}+\mu_{\ell\ell^{\prime}}\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}\right)\nu_{\ell^{\prime}}\;.$
(3.15)

Eq. (3.15) is our working Lagrangian. For later convenience we define the
generalized mass-dipole matrix $\mathcal{M}_{\ell\ell^{\prime}}$ present in
Eq. (3.15) as

$\displaystyle\mathcal{M}_{\ell\ell^{\prime}}(\bm{E},\bm{B})\equiv
M_{\ell\ell^{\prime}}+\mu_{\ell\ell^{\prime}}\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}\,,\qquad\mathcal{M}_{\ell\ell^{\prime}}^{{\dagger}}=\gamma_{0}\mathcal{M}_{\ell\ell^{\prime}}\gamma_{0}\,.$
(3.16)

As neutrinos must propagate as energy eigenstates, our objective is to find
the eigenvalues of Eq. (3.15) rather than Eq. (3.7). As the mass eigenvalues
are modified by the presence the EM interactions
$m\rightarrow\widetilde{m}(\bm{E},\bm{B})$ so will the mixing matrix, leading
to modifications of Eq. (3.4). These electromagnetic components then
facilitate time-dependant oscillation among the free-particle mass eigenstates
(Giunti and Studenikin, 2015).

Additionally we may consider matter effects via the weak interaction. Electron
(anti)neutrinos passing through matter preferentially interact via weak
charge-current (via the $W^{\pm}$ boson) with electrons which make up the bulk
of charged leptons in most matter. The neutral-current (via the $Z_{0}$ boson)
however affects all flavors and couples to the neutrons within the medium as
the electron and proton contributions cancel in charge neutral matter. This
can be represented, MSW effect aside, as the weak charge-current $V_{CC}$ and
neutral-current $V_{NC}$ effective potentials (Pal, 1992; Greiner and Müller,
2009) which contribute to the action as

$\displaystyle\mathcal{L}_{\mathrm{matter}}^{\mathrm{Maj.}}$
$\displaystyle=\bar{\nu}_{\ell}(\gamma_{0}V_{\ell\ell^{\prime}})\nu_{\ell^{\prime}}\,,\qquad
V_{\ell\ell^{\prime}}=\begin{pmatrix}V_{CC}+V_{NC}&0&0\\\ 0&V_{NC}&0\\\
0&0&V_{NC}\end{pmatrix}\,,$ (3.17) $\displaystyle V_{CC}$
$\displaystyle=\sqrt{2}G_{F}\hbar^{2}c^{2}n_{e}\,,\qquad
V_{NC}=-\frac{1}{2}\sqrt{2}G_{F}\hbar^{2}c^{2}n_{n}\,.$ (3.18)

The coefficient $G_{F}$ is the Fermi constant, $n_{e}$ is the number density
of electron matter and $n_{n}$ is the number density of neutrons within the
medium. We note that $V_{\ell\ell^{\prime}}\gamma_{0}$ behaves like the zeroth
component of a vector-potential. As written, Eq. (3.17) is approximately true
for non-relativistic matter.

####  3.2.2 Chiral properties of the relativistic Pauli dipole

While the Pauli dipole was introduced and discussed in Section 1.1.2, we will
further elaborate on details directly relevant to neutrinos. The
electromagnetic dipole behavior of the neutrino depends on mathematical
properties of the tensor product
$\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}$. We prefer to work in the
Weyl (chiral) spinor representation where the EM contribution is diagonal in
spin space. Therefore we evaluate the product
$\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}$ in the Weyl
representation following Feynman and Gell-Mann (1958) yielding

$\displaystyle-\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}=\begin{pmatrix}\bm{\sigma}\cdot(\bm{B}+i\bm{E}/c)&0\\\
0&\bm{\sigma}\cdot(\bm{B}-i\bm{E}/c)\end{pmatrix}\equiv\begin{pmatrix}\bm{\sigma}\cdot\bm{f}_{+}&0\\\
0&\bm{\sigma}\cdot\bm{f}_{-}\end{pmatrix}\,,$ (3.19)

where we introduced the complex electromagnetic field form
$\bm{f}_{\pm}=\bm{B}\pm i\bm{E}/c$ showing sensitivity to both magnetic and
electric fields. The eigenvalues of Eq. (3.19) were also discussed in Section
2.4. As this expression is diagonal in the Weyl representation, it does not
exchange handedness when acting upon a state. This is explicitly understood by
the fact that Eq. (3.19) commutes with $\gamma^{5}$. Since left and right-
handed neutrinos are not remixed by magnetic moments, sterile right-handed
neutrinos do not need to be introduced. We can also see explicitly in Eq.
(3.19) its non-Hermitian character, see Eq. (3.15), of the EM spin-field
coupling. Specifically this is mirrored in the complex field’s $\bm{f}_{\pm}$
relation to its complex conjugate $(\bm{f}_{\pm})^{*}=\bm{f}_{\mp}$. The
complex EM fields have a Hermitian $(\bm{B})$ and anti-Hermitian $(i\bm{E})$
part.

Taking the product of $\bm{f}_{\pm}$ with its complex conjugate we find

$\displaystyle\frac{1}{2}\left(\bm{\sigma}\cdot\bm{f}_{\pm}\right)\left(\bm{\sigma}\cdot\bm{f}_{\mp}\right)=T_{\mathrm{ext}}^{00}\mp\sigma_{i}T_{\mathrm{ext}}^{0i}\,,$
(3.20)

where we recognize the stress-energy tensor $T_{\mathrm{ext}}^{\alpha\beta}$
component $T_{\mathrm{ext}}^{00}$ for field energy density and
$T_{\mathrm{ext}}^{0i}$ momentum density respectively

$\displaystyle
T_{\mathrm{ext}}^{00}=\frac{1}{2}\left(B^{2}+E^{2}/c^{2}\right)\,,\qquad
T_{\mathrm{ext}}^{0i}=\frac{1}{c}\varepsilon_{ijk}E_{j}B_{k}\,.$ (3.21)

As we will see in Section 3.3, Eq. (3.20) will appear in the EM-mass
eigenvalues of our effective Lagrangian Eq. (3.14). Using the identity in Eq.
(3.19) and Eq. (3.20) we also find the interesting relationship

$\displaystyle\frac{1}{2}\left(\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}\right)\left(\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}\right)^{{\dagger}}=\gamma_{0}\left(T_{\mathrm{ext}}^{00}\gamma_{0}+T_{\mathrm{ext}}^{0i}\gamma_{i}\right)\,.$
(3.22)

Now that we have elaborated on the relevant EM field identities, we turn back
to the magnetic dipole and flavor rotation problem.

###  3.3 Electromagnetic-flavor mixing for two generations

Considering experimental data on neutrino oscillations, it is understood that
either the two lighter (normal hierarchy) or the two heavier (inverted
hierarchy) neutrino states are close together in mass. If the electromagnetic
properties of the neutrino do indeed lead to flavor mixing effects, then it is
likely the closer pair of neutrino mass states that are most sensitive to the
phenomenon we explore. In the spirit of Bethe (1986), we therefore explore the
$N=2$ two generation $(\nu_{e},\nu_{\mu})$ toy model.

Following the properties established in Eq. (3.12) and Eq. (3.16) we write
down the two generation mass and dipole matrices as

$\displaystyle M_{\ell\ell^{\prime}}=\begin{pmatrix}m_{e}^{\nu}&{\delta m}\\\
{\delta
m}&m_{\mu}^{\nu}\end{pmatrix}\,,\qquad\mu_{\ell\ell^{\prime}}=\begin{pmatrix}0&i\delta\mu\\\
-i\delta\mu&0\end{pmatrix}\,.$ (3.23)

The AMM coupling $\delta\mu$ is taken to be real with a pure imaginary
coefficient. While the mass elements $(m_{e}^{\nu},m_{\mu}^{\nu},{\delta m})$
are generally complex, we choose in our toy model for them to be fully real

$\displaystyle m_{e}^{\nu}=(m_{e}^{\nu})^{*}\,,\qquad
m_{\mu}^{\nu}=(m_{\mu}^{\nu})^{*}\,,\qquad\delta m=\delta m^{*}\,,$ (3.24)

making the mass matrix $M_{\ell\ell^{\prime}}$ Hermitian. This allows us to
more easily evaluate and emphasize the EM contributions to mixing rather than
complications arising from the mass matrix.

Using Eq. (3.23) and Eq. (3.24), we write the mass-dipole matrix in Eq. (3.16)
in terms of $2\times 2$ flavor components as

$\displaystyle\mathcal{M}_{\ell\ell^{\prime}}=\begin{pmatrix}m_{e}^{\nu}&{\delta
m}+i\delta\mu\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}/2\\\ {\delta
m}-i\delta\mu\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}/2&m_{\mu}^{\nu}\end{pmatrix}\,,\qquad\mathcal{M}_{\ell\ell^{\prime}}^{{\dagger}}=\gamma_{0}\mathcal{M}_{\ell\ell^{\prime}}\gamma_{0}\,.$
(3.25)

As noted before, this matrix is not Hermitian due to the inclusion of the spin
tensor, therefore it is not guaranteed to satisfy an algebraic eigenvalue
equation in its present form which is a requirement for well behaved masses.

This can be remedied by recalling that any arbitrary complex matrix can be
diagonalized into its real eigenvalues $\lambda_{j}$ by the biunitary
transform

$\displaystyle W_{\ell
j}^{{\dagger}}\mathcal{M}_{\ell\ell^{\prime}}Y_{\ell^{\prime}j^{\prime}}=\lambda_{j}\delta_{jj^{\prime}}\,,$
(3.26)

where $Y_{\ell j}$ and $W_{\ell j}$ are both unitary matrices. Taking the
complex conjugate of Eq. (3.26), we arrive at

$\displaystyle(W_{\ell
j}^{{\dagger}}\mathcal{M}_{\ell\ell^{\prime}}Y_{\ell^{\prime}j^{\prime}})^{{\dagger}}=Y_{\ell
j^{\prime}}^{{\dagger}}\gamma_{0}\mathcal{M}_{\ell\ell^{\prime}}\gamma_{0}W_{\ell^{\prime}j}=\lambda_{j}\delta_{jj^{\prime}}\,,$
(3.27) $\displaystyle Y_{\ell j}=\gamma_{0}W_{\ell j}\rightarrow W_{\ell
j}^{{\dagger}}\mathcal{M}_{\ell\ell^{\prime}}\gamma_{0}W_{\ell^{\prime}j^{\prime}}=\lambda_{j}\delta_{jj^{\prime}}\,.$
(3.28)

As $Y_{\ell j}$ and $W_{\ell j}$ are related by a factor of $\gamma_{0}$ based
on the conjugation properties of Eq. (3.25), this lets us eliminate $Y_{\ell
j}$ and diagonalize using a single unitary matrix $W_{\ell j}$. The related
matrix $\mathcal{M}_{\ell\ell^{\prime}}\gamma_{0}$ is Hermitian

$\displaystyle(\mathcal{M}_{\ell\ell^{\prime}}\gamma_{0})^{{\dagger}}=\mathcal{M}_{\ell\ell^{\prime}}\gamma_{0}\,,$
(3.29)

and also equivalent to the root of the Hermitian product of Eq. (3.25)

$\displaystyle(\mathcal{M}\mathcal{M}^{{\dagger}})_{\ell\ell^{\prime}}=\left((\mathcal{M}\gamma_{0})(\mathcal{M}\gamma_{0})\right)_{\ell\ell^{\prime}}\,.$
(3.30)

Therefore a suitable unitary transformation $W_{\ell j}$ rotates flavor
$\ell$-states into magnetized mass $j$-states. The eigenvalues
$\lambda_{j}^{2}$ of
$(\mathcal{M}\mathcal{M}^{{\dagger}})_{\ell\ell^{\prime}}$ are the squares of
both signs of the eigenvalues of $\mathcal{M}_{\ell\ell^{\prime}}\gamma_{0}$.
We write this property (with flavor indices suppressed) as

$\displaystyle W^{{\dagger}}(\mathcal{M}\mathcal{M}^{{\dagger}})W$
$\displaystyle=W^{{\dagger}}(\mathcal{M}\gamma_{0})WW^{{\dagger}}(\mathcal{M}\gamma_{0})W=\mathrm{diag}(\lambda_{1}^{2},\lambda_{2}^{2})\,.$
(3.31)

We associate $\lambda_{j}=\widetilde{m}_{j}(\bm{E},\bm{B})$ with $j\in 1,2$ as
the effective EM-mass states which are field dependant in this basis.

####  3.3.1 Separating electromagnetic-mass mixing into two rotations

The matrix $W_{\ell j}$ mixes flavor states into a new basis distinct from the
free-particle case however this rotation must smoothly connect with the free-
particle case in the limit that the electromagnetic fields go to zero. We
proceed to evaluate $W_{\ell j}$ breaking the rotation into two separate
unitary transformations:

  * (a)

Rotation matrix $V_{\ell k}^{{\dagger}}(\ell\rightarrow k)$ converting from
flavor to free-particle mass

  * (b)

Rotation matrix $Z_{kj}^{\mathrm{ext}{\dagger}}(k\rightarrow j)$ converting
from free-particle mass to EM-mass

Guided by Eq. (3.2) we write

$\displaystyle\nu_{j}=W^{{\dagger}}_{\ell
j}\nu_{\ell}=Z_{kj}^{\mathrm{ext}{\dagger}}V_{\ell
k}^{{\dagger}}\nu_{\ell}\,.$ (3.32)

In the limit that the EM fields go to zero, the electromagnetic rotation
becomes unity $Z_{kj}^{\mathrm{ext}}\rightarrow\delta_{kj}$ thereby ensuring
the EM-mass basis and free-particle mass basis become equivalent. The rotation
$Z_{kj}^{\mathrm{ext}}$ can then be interpreted as the external field forced
rotation. While our argument above is done explicitly for the two generation
case, it can be generalized to accommodate three generations of neutrinos as
well.

According to Eq. (3.7), the mass matrix in Eq. (3.23) can be diagonalized in
the two generation case by a one parameter unitary mixing matrix $V_{\ell k}$
given by

$\displaystyle V_{\ell k}(\theta)=\begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{pmatrix}\,.$ (3.33)

For a real Hermitian $2\times 2$ mass matrix, the rotation matrix $V_{\ell k}$
is real and only depends on the angle $\theta$. The explicit form of the EM-
field related rotation $Z_{kj}^{\mathrm{ext}}$ introduced in Eq. (3.32) is

$\displaystyle
Z_{kj}^{\mathrm{ext}}(\omega,\phi)=\begin{pmatrix}\cos\omega&e^{i\phi}\sin\omega\\\
-e^{-i\phi}\sin\omega&\cos\omega\end{pmatrix}\,,\qquad W_{\ell
j}(\theta,\omega,\phi)=V_{\ell
k}(\theta)Z_{kj}^{\mathrm{ext}}(\omega,\phi)\,,$ (3.34)

where $Z_{kj}^{\mathrm{ext}}$ depends on the real angle $\omega$ and complex
phase $\phi$. The full rotation $W_{\ell j}$ therefore depends on three
parameters when broken into free-particle rotation and EM rotation.

The eigenvalues of the original Hermitian mass matrix in Eq. (3.23) are given
by

$\displaystyle
m_{1,2}=\frac{1}{2}\left(m_{e}^{\nu}+m_{\mu}^{\nu}\mp\sqrt{|\Delta
m_{0}|^{2}+4\delta m^{2}}\right)\,,\qquad|\Delta
m_{0}|=|m_{\mu}^{\nu}-m_{\mu}^{e}|\,.$ (3.35)

We assign $m_{1}$ to the lower mass $(-)$ root and $m_{2}$ with the larger
mass $(+)$ additive root. The rotation angle $\theta$ in Eq. (3.33) is then
given by

$\displaystyle\sin 2\theta=\sqrt{\frac{4\delta m^{2}}{|\Delta
m_{0}|^{2}+4\delta m^{2}}}\,,\qquad\cos 2\theta=\sqrt{\frac{|\Delta
m_{0}|^{2}}{|\Delta m_{0}|^{2}+4\delta m^{2}}}\,.$ (3.36)

In our toy model, the off-diagonal imaginary transition magnetic moment
$\mu_{\ell\ell^{\prime}}$ commutes with the real valued mixing matrix $V_{\ell
k}$ and the following relations hold

$\displaystyle V_{\ell
k}^{{\dagger}}\mu_{\ell\ell^{\prime}}V_{\ell^{\prime}k^{\prime}}=(V^{{\dagger}}V)_{k\ell^{\prime}}\mu_{\ell^{\prime}k^{\prime}}=\mu_{kk^{\prime}}=\begin{pmatrix}0&i\delta\mu\\\
-i\delta\mu&0\end{pmatrix}\,.$ (3.37)

We see that the Majorana transition dipoles in our model are off-diagonal in
both flavor and mass basis. Therefore the real parameter unitary matrix in Eq.
(3.37) cannot rotate a pure imaginary matrix at least in the two generation
case. We apply the rotation in Eq. (3.33) to Eq. (3.29) yielding

$\displaystyle V_{\ell
k}^{{\dagger}}(\mathcal{M}_{\ell\ell^{\prime}}\gamma_{0})V_{\ell^{\prime}k^{\prime}}$
$\displaystyle=V_{\ell
k}^{{\dagger}}M_{\ell\ell^{\prime}}\gamma_{0}V_{\ell^{\prime}k^{\prime}}+V_{\ell
k}^{{\dagger}}(\mu_{\ell\ell^{\prime}}\sigma_{\alpha\beta}\gamma_{0}F^{\alpha\beta}_{\mathrm{ext}}/2)V_{\ell^{\prime}k^{\prime}}\,,$
(3.38) $\displaystyle V_{\ell
k}^{{\dagger}}(\mathcal{M}_{\ell\ell^{\prime}}\gamma_{0})V_{\ell^{\prime}k^{\prime}}$
$\displaystyle=\begin{pmatrix}m_{1}\gamma_{0}&i\delta\mu\sigma_{\alpha\beta}\gamma_{0}F^{\alpha\beta}_{\mathrm{ext}}/2\\\
-i\delta\mu\sigma_{\alpha\beta}\gamma_{0}F^{\alpha\beta}_{\mathrm{ext}}/2&m_{2}\gamma_{0}\end{pmatrix}\equiv\begin{pmatrix}\mathcal{A}&i\mathcal{C}\\\
-i\mathcal{C}&\mathcal{B}\end{pmatrix}\,,$ (3.39)

where we have defined implicitly the Hermitian elements
$(\mathcal{A},\mathcal{B},\mathcal{C})$. Applying now both rotations to Eq.
(3.29) yields

$\displaystyle W_{\ell
j}^{{\dagger}}(\mathcal{M}_{\ell\ell^{\prime}}\gamma_{0})W_{\ell^{\prime}j^{\prime}}$
$\displaystyle=Z^{\mathrm{ext}{\dagger}}\begin{pmatrix}\mathcal{A}&i\mathcal{C}\\\
-i\mathcal{C}&\mathcal{B}\end{pmatrix}Z^{\mathrm{ext}}=\lambda_{j}\delta_{jj^{\prime}}\,.$
(3.40)

Eq. (3.40) is therefore the working matrix equation which needs to be solved
to identify the EM rotation parameters. As discussed before, this means that
the rotation angle $\omega$ and phase $\phi$ are in general functions of
electromagnetic fields.

####  3.3.2 Effective electromagnetic-mass eigenvalues

We will now solve for the rotation parameters necessary to define the EM-mass
basis which acts as a distinct propagating basis for neutrinos in external
fields. Considering that the $j$-columns vectors $\bm{v}^{(j)}$ of
$Z_{kj}^{\mathrm{ext}}$ as eigenvectors for each eigenvalue $\lambda_{j}$

$\displaystyle
Z_{kj}^{\mathrm{ext}}=v_{k}^{(j)}=\begin{pmatrix}\bm{v}^{1}&\bm{v}^{2}\end{pmatrix}\,,$
(3.41)

Eq. (3.40) has the meaning of an eigenvalue equation

$\displaystyle\begin{pmatrix}\mathcal{A}&i\mathcal{C}\\\
-i\mathcal{C}&\mathcal{B}\end{pmatrix}Z^{\mathrm{ext}}=Z^{\mathrm{ext}}\begin{pmatrix}\lambda_{1}&0\\\
0&\lambda_{2}\end{pmatrix}\rightarrow\begin{pmatrix}\mathcal{A}&i\mathcal{C}\\\
-i\mathcal{C}&\mathcal{B}\end{pmatrix}\bm{v}^{(j)}=\lambda_{j}\bm{v}^{(j)}\,.$
(3.42)

Given the eigenvalue equation defined in Eq. (3.42), the effective EM-masses
are then solutions to the characteristic polynomial

$\displaystyle(\mathcal{A}-\lambda_{j}\gamma_{0})(\mathcal{B}-\lambda_{j}\gamma_{0})-\mathcal{C}^{2}=0\,,$
(3.43)

which we obtained by taking the determinant of Eq. (3.42) over flavor but not
spin space. It is useful to define the following identities for the off-
diagonal element

$\displaystyle\mathcal{C}^{2}=\delta\mu^{2}\left(\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}\right)\left(\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}\right)^{{\dagger}}=2\delta\mu^{2}\gamma_{0}\left(T_{\mathrm{ext}}^{00}\gamma_{0}+T_{\mathrm{ext}}^{0i}\gamma_{i}\right)\,,$
(3.44)

and for the diagonal elements

$\displaystyle(\mathcal{B}-\mathcal{A})^{2}=|m_{2}-m_{1}|^{2}=|\Delta
m|^{2}\,,\qquad(\mathcal{A}+\mathcal{B})\gamma_{0}=m_{1}+m_{2}\,.$ (3.45)

Eq. (3.44) was obtained using the expression in Eq. (3.22). Because of the
spinor behavior of each element, the eigenvalues are obtained with
$\gamma_{0}$ coefficients. Eq. (3.43) therefore has the roots
$\lambda_{1,2}=\widetilde{m}_{1,2}(\bm{E},\bm{B})$

$\displaystyle\widetilde{m}_{1,2}(\bm{E},\bm{B})\\!=\\!\frac{1}{2}\left(m_{1}\\!+\\!m_{2}\\!\mp\\!\sqrt{|\Delta
m|^{2}\\!+\\!8\delta\mu^{2}\gamma_{0}\left(T_{\mathrm{ext}}^{00}\gamma_{0}+T_{\mathrm{ext}}^{0i}\gamma_{i}\right)}\right)\\!,$
(3.46)
$\displaystyle\boxed{\widetilde{m}_{1,2}(\bm{E},\bm{B})\\!=\\!\frac{m_{1}\\!+\\!m_{2}}{2}\\!\mp\\!\frac{1}{2}\sqrt{|\Delta
m|^{2}\\!+\\!8\delta\mu^{2}\gamma_{0}\left(\gamma_{0}\frac{1}{2}\left(B^{2}\\!+\\!\frac{E^{2}}{c^{2}}\right)\\!+\\!\bm{\gamma}\\!\cdot\\!(\frac{\bm{E}}{c}\times\bm{B})\right)}}$
(3.47)

The EM-mass eigenstates $\widetilde{m}(\bm{E},\bm{B})$ depends on the energy
density $T_{\mathrm{ext}}^{00}$ of the EM field and the spin projection along
the EM momentum density $T_{\mathrm{ext}}^{0i}$. However the coefficient
$\delta\mu^{2}$ is presumed to be very small, therefore the EM contribution
only manifests in strong EM fields or where the free-particle case has very
nearly or exactly degenerate masses, $\Delta m\to 0$. When the the
electromagnetic fields go to zero, the EM-masses in Eq. (3.47) reduce as
expected to the free-particle result.

The complex phase in Eq. (3.34) has the value $\phi=\pi(n-1/2)$ with $n\in
0,\pm 1,\pm 2...$ making the complex exponential in Eq. (3.34) pure imaginary.
Curiously, the phase is not field dependant, but tied to the fact that the
Majorana moments are pure imaginary quantities. Complex phases in mixing
matrices are generally associated with CP violation such as the Dirac phase
$\delta$ in Eq. (3.3) which suggests that CP violation in the neutrino sector
can be induced in the presence of external EM fields. Some implications of CP
violation from transition moments are discussed in Nieves (1982). Analysis of
the three generation case is required to show this explicitly, but we
postulate that the constant valued complex phases would be replaced with field
dependant quantities $\delta\to\delta(\bm{E},\bm{B})$.

We note that the solution in Eq. (3.47) actually contain four distinct EM-mass
eigenstates $\widetilde{m}_{j}^{s}(\bm{E},\bm{B})$ with the lower $(j=1)$ and
upper $(j=2)$ masses and the additional spin splitting from the alignment
$(s=+1)$ or anti-alignment $(s=-1)$ of the neutrino spin with the momentum
density of the external EM field. Spin splitting vanishes for the pure
electric or magnetic field cases. For good spin eigenstates $s\in\pm 1$, we
can rewrite Eq. (3.44) with EM fields explicitly as

$\displaystyle\mathcal{C}^{2}_{s}(\bm{E},\bm{B})=2\delta\mu^{2}\left(\frac{1}{2}(B^{2}+E^{2}/c^{2})+s|\bm{E}/c\times\bm{B}|\right)\,.$
(3.48)

The above expression within the square is positive definite, therefore Eq.
(3.48) is always real. Spin splitting requires that we consider separate
rotations for each spin state as the rotation angle $\omega_{s}$ depends on
the spin quantum number

$\displaystyle\sin 2\omega_{s}=\sqrt{\frac{4\mathcal{C}_{s}^{2}}{|\Delta
m|^{2}+4\mathcal{C}_{s}^{2}}}\,,\qquad\cos 2\omega_{s}=\sqrt{\frac{|\Delta
m|^{2}}{|\Delta m|^{2}+4\mathcal{C}_{s}^{2}}}\,.$ (3.49)

The expressions in Eq. (3.49) are mathematically similar to that of the free-
particle case written in Eq. (3.36) in the two flavor generation model with
the off-diagonal mass being replaced with the EM dependant quantity
$\mathcal{C}_{s}$.

###  3.4 Strong field (degenerate mass) and weak field limits

The rotation angles in Eq. (3.49) reveal two distinct limits where EM-masses
are dominated by either:

  * (a)

Intrinsic mass splitting $\mathcal{C}_{s}\ll|\Delta m|^{2}$ with
$\omega_{s}\rightarrow 0$

  * (b)

EM mass splitting $\mathcal{C}_{s}\gg|\Delta m|^{2}$ with
$\omega_{s}\rightarrow\pi/4$

For the first case where the masses are not degenerate or the fields are weak,
we obtain the expansion

$\displaystyle\lim_{\mathcal{C}_{s}\ll|\Delta
m|^{2}}\widetilde{m}_{1,2}^{s}(E,B)=\frac{1}{2}\left(m_{1}+m_{2}\mp|\Delta
m|\left(1+\frac{2\mathcal{C}_{s}^{2}}{|\Delta m|^{2}}+\ldots\right)\right)\,,$
(3.50)

which as stated before reduces to the free-particle case at lowest order.

In the opposite limit, where the masses are very nearly degenerate or fields
are strong, the EM-mass eigenvalues in Eq. (3.47) can be approximated by the
series

$\displaystyle\lim_{\mathcal{C}_{s}\gg|\Delta
m|^{2}}\widetilde{m}_{1,2}^{s}(E,B)=\frac{1}{2}\left(m_{1}+m_{2}\mp
2\mathcal{C}_{s}\left(1+\frac{|\Delta
m|^{2}}{8\mathcal{C}_{s}^{2}}+\ldots\right)\right)$ (3.51)

For fully degenerate free-particle masses $m_{1}=m_{2}$, this reduces to

$\displaystyle\lim_{|\Delta m|^{2}\to
0}\widetilde{m}_{1,2}^{s}(E,B)=m_{1}\mp\mathcal{C}_{s}\,.$ (3.52)

Eq. (3.52) indicates that for degenerate free-particle masses, the effective
splitting $|\Delta m_{\mathrm{EM}}|\equiv\mathcal{C}_{s}$ between masses
arises purely from the electromagnetic interaction of the neutrinos. We return
to this interesting insight in our final comments.

Because of the bounds in Eq. (3.1) on the effective neutrino magnetic moment,
we can estimate the field strength required for an external magnetic field to
generate an electromagnetic mass splitting of $|\Delta
m_{\mathrm{EM}}|=10^{-3}$ eV which is a reasonable comparison to intrinsic
splitting based on the experimental limits on neutrino masses. Using the upper
limit for the neutrino effective moment of $\mu_{\nu}^{\mathrm{eff}}\sim
10^{-10}\mu_{B}$ we obtain

$\displaystyle\left.\frac{\mathcal{C}_{s}}{\mu_{\nu}^{\mathrm{eff}}}\right\rvert_{\vec{E}=0}=\frac{10^{-3}\,\mathrm{eV}}{10^{-10}\mu_{B}}\approx
1.7\times 10^{11}\,\mathrm{T}\,.$ (3.53)

This is near the upper bound of the magnetic field strength of magnetars
(Kaspi and Beloborodov, 2017) which are of the order $10^{11}$ Tesla. In this
situation, the EM contribution to the mass splitting rivals the estimated
inherent splitting (Workman et al., 2022) of the two closer in mass neutrinos.
Primordial magnetic fields (Grasso and Rubinstein, 2001) in the Early Universe
may also present an environment for significant EM neutrino flavor mixing as
both the external field strength and density of neutrinos would be very large
(Rafelski et al., 2023a). The magnetic properties of neutrinos may also have
contributed alongside the charged leptons in magnetization in the Early
Universe (Steinmetz et al., 2023) prior to recombination.

While the above estimate was done with astrophysical systems in mind, we note
that strong electrical fields should also produce EM-mass splitting. Therefore
environments near to high $Z$-nuclei is also of interest (Bouchiat and
Bouchiat, 1974, 1997; Safronova et al., 2018) as weak interactions violate
parity. Should neutrinos have abnormally large transition magnetic dipole
moments, then they should exhibit mass splitting from the neutrino’s
electromagnetic dipole interaction which may compete with the intrinsic mass
differences of the free-particles.

##  Chapter 4 Matter-antimatter origin of cosmic magnetism

We investigate the hypothesis that the observed intergalactic magnetic fields
(IGMF) are primordial in nature, predating the recombination epoch.
Specifically, we explore the role of the extremely large electron-positron
$(e^{+}e^{-})$ pair abundance in the temperature range of $2000\text{
keV}>T>20\text{ keV}$ which only disappeared after Big Bang nucleosynthesis
(BBN). We review the status of cosmic magnetism in Section 4.1 which motivates
our study. Section 4.2 discusses the extreme electron-positron abundance
during this epoch. The statistical and thermodynamic theory of the electron-
positron gas is described in Section 4.3. Section 4.4 describes the
relativistic paramagnetism of the electron-positron gas. We propose in Section
4.5 a model of self-magnetization caused by spin polarization within the
individual species in the gas.

This chapter serves primarily as a review of our work in Steinmetz et al.
(2023) and portions of Rafelski et al. (2023a) where we propose that the early
universe electron-positron plasma was a highly magnetized environment. We will
use natural units $(c=\hbar=k_{B}=1)$ unless otherwise noted.

NOTE: The letter $\mu$ within this chapter will refer exclusively to charged
chemical potential and _not_ magnetic moment as before.

###  4.1 Short survey of magnetism in the universe

Macroscopic domains of magnetic fields have been found in all astrophysical
environments from compact objects (stars, planets, etc.); interstellar and
intergalactic space; and surprisingly in deep extra-galactic void spaces.
Considering the ubiquity of magnetic fields in the universe (Giovannini, 2018,
2004; Kronberg, 1994), we search for a common primordial mechanism initiate
the diversity of magnetism observed today. In this chapter, IGMF will refer to
experimentally observed intergalactic fields of any origin while primordial
magnetic fields (PMF) refers to fields generated via early universe processes
possibly as far back as inflation. The conventional elaboration of the origins
for cosmic PMFs are detailed in (Gaensler et al., 2004; Durrer and Neronov,
2013; Batista and Saveliev, 2021).

IGMF are notably difficult to measure and difficult to explain. The bounds for
IGMF at a length scale of $1{\rm\ Mpc}$ are today (Neronov and Vovk, 2010;
Taylor et al., 2011; Pshirkov et al., 2016; Jedamzik and Saveliev, 2019;
Vernstrom et al., 2021)

$\displaystyle 10^{-8}{\rm\ G}>B_{\mathrm{IGMF}}>10^{-16}{\rm\ G}\,.$ (4.1)

We note that generating PMFs with such large coherent length scales is
nontrivial (Giovannini, 2023) though currently the length scale for PMFs are
not well constrained (Batista and Saveliev, 2021). Faraday rotation from
distant radio active galaxy nuclei (AGN) (Pomakov et al., 2022) suggest that
neither dynamo nor astrophysical processes would sufficiently account for the
presence of magnetic fields in the universe today if the IGMF strength was
around the upper bound of $B_{\mathrm{IGMF}}\simeq 30-60{\rm\ nG}$ as found in
Vernstrom et al. (2021). Such strong magnetic fields would then require that
at least some portion of the IGMF arise from primordial sources that predate
the formation of stars.

Magnetized baryon inhomogeneities which in turn would produce anisotropies in
the cosmic microwave background (CMB) (Jedamzik and Abel, 2013; Abdalla et
al., 2022). Jedamzik and Pogosian (2020) propose further that the presence of
a magnetic field of $B_{\mathrm{PMF}}\simeq 0.1{\rm\ nG}$ could be sufficient
to explain the Hubble tension.

Figure 4.1: Qualitative plot of the primordial magnetic field strength over
cosmic time. All figures are printed in temporal sequence in the expanding
universe beginning with high temperatures (and early times) on the left and
lower temperatures (and later times) on the right.

Our motivating hypothesis is outlined qualitatively in Figure 4.1 where PMF
evolution is plotted over the temperature history of the universe. The
descending blue band indicates the range of possible PMF strengths. The
different epochs of the universe according to $\Lambda\mathrm{CDM}$ are
delineated by temperature. The horizontal lines mark two important scales: (a)
the Schwinger critical field strength given by

$\displaystyle B_{\mathrm{C}}=\frac{m_{e}^{2}}{e}\simeq 4.41\times
10^{13}\,\mathrm{G}\,.$ (4.2)

where electrodynamics is expected to display nonlinear characteristics and (b)
the upper field strength seen in magnetars of $\sim 10^{15}\,\mathrm{G}$. A
schematic of magnetogenesis is drawn with the dashed red lines indicating
spontaneous formation of the PMF within the early universe plasma itself. The
$e^{+}e^{-}$ era is notably the final epoch where antimatter exists in large
quantities in the cosmos (Rafelski et al., 2023a).

###  4.2 Electron-positron abundance

As the universe cooled below temperature $T\\!=\\!m_{e}$ (the electron mass),
the thermal electron and positron comoving density depleted by over eight
orders of magnitude. At $T_{\mathrm{split}}=20.3\text{ keV}$, the charged
lepton asymmetry (mirrored by baryon asymmetry and enforced by charge
neutrality) became evident as the surviving excess electrons persisted while
positrons vanished entirely from the particle inventory of the universe due to
annihilation.

Figure 4.2: Number density of electron $e^{-}$ and positron $e^{+}$ to baryon
ratio $n_{e^{\pm}}/n_{B}$ as a function of photon temperature in the universe.
See text for further details. In this work we measure temperature in units of
energy (keV) thus we set the Boltzmann constant to $k_{B}=1$. Figure courtesy
of Cheng Tao Yang.

The electron-to-baryon density ratio $n_{e^{-}}/n_{B}$ is shown in Figure 4.2
as the solid blue line while the positron-to-baryon ratio $n_{e^{+}}/n_{B}$ is
represented by the dashed red line. These two lines overlap until the
temperature drops below $T_{\mathrm{split}}=20.3\text{ keV}$ as positrons
vanish from the universe marking the end of the $e^{+}e^{-}$ plasma and the
dominance of the electron-proton $(e^{-}p)$ plasma. The two vertical dashed
green lines denote temperatures $T\\!=\\!m_{e}\simeq 511\text{ keV}$ and
$T_{\mathrm{split}}=20.3\text{ keV}$. These results were obtained using charge
neutrality and the baryon-to-photon content (entropy) of the universe; see
details in Rafelski et al. (2023a). The two horizontal black dashed lines
denote the relativistic $T\gg m_{e}$ abundance of
$n_{e^{\pm}}/n_{B}=4.47\times 10^{8}$ and post-annihilation abundance of
$n_{e^{-}}/n_{B}=0.87$. Above temperature $T\simeq 85\text{ keV}$, the
$e^{+}e^{-}$ primordial plasma density exceeded that of the Sun’s core density
$n_{e}\simeq 6\times 10^{26}{\rm\ cm}^{-3}$ (Bahcall et al., 2001).

Conversion of the dense $e^{+}e^{-}$ pair plasma into photons reheated the
photon background (Birrell et al., 2014) separating the photon and neutrino
temperatures. The $e^{+}e^{-}$ annihilation and photon reheating period lasted
no longer than an afternoon lunch break. Because of charge neutrality, the
post-annihilation comoving ratio $n_{e^{-}}/n_{B}=0.87$ (Rafelski et al.,
2023a) is slightly offset from unity in Figure 4.2 by the presence of bound
neutrons in $\alpha$ particles and other neutron containing light elements
produced during BBN epoch.

The abundance of baryons is itself fixed by the known abundance relative to
photons (Workman et al., 2022) and we employed the contemporary recommended
value $n_{B}/n_{\gamma}=6.09\times 10^{-10}$. The resulting chemical potential
needs to be evaluated carefully to obtain the behavior near to
$T_{\mathrm{split}}=20.3\text{ keV}$ where the relatively small value of
chemical potential $\mu$ rises rapidly so that positrons vanish from the
particle inventory of the universe while nearly one electron per baryon
remains. The detailed solution of this problem is found in Fromerth et al.
(2012); Rafelski et al. (2023a) leading to the results shown in Figure 4.2.

###  4.3 Theory of thermal matter-antimatter plasmas

To evaluate magnetic properties of the thermal $e^{+}e^{-}$ pair plasma we
take inspiration from Ch. 9 of Melrose’s treatise on magnetized plasmas
(Melrose, 2013). We focus on the bulk properties of thermalized plasmas in
(near) equilibrium.

We consider a homogeneous magnetic field domain defined along the $z$-axis as

$\displaystyle\bm{B}=(0,\,0,\,B)\,,$ (4.3)

with magnetic field magnitude $|\bm{B}|=B$. Following Chapter 2, we reprint
the microscopic energy (Eq. (2.16) in different notation) of the charged
relativistic fermion within a homogeneous magnetic field given by

$\displaystyle
E^{n}_{\sigma,s}(p_{z},{B})=\sqrt{m_{e}^{2}+p_{z}^{2}+e{B}\left(2n+1+\frac{g}{2}\sigma
s\right)}\,,$ (4.4)

where $n\in 0,1,2,\ldots$ is the Landau orbital quantum number, $p_{z}$ is the
momentum parallel to the field axis and the electric charge is $e\equiv
q_{e^{+}}=-q_{e^{-}}$. The index $\sigma$ in Eq. (4.4) differentiates electron
$(e^{-};\ \sigma=+1)$ and positron $(e^{+};\ \sigma=-1)$ states. The index $s$
refers to the spin along the field axis: parallel $(\uparrow;\ s=+1)$ or anti-
parallel $(\downarrow;\ s=-1)$ for both particle and antiparticle species.

aligned: $s=+1$ anti-aligned: $s=-1$ electron: $\sigma=+1$ $U_{\rm Mag}>0$
$U_{\rm Mag}<0$ positron: $\sigma=-1$ $U_{\rm Mag}<0$ $U_{\rm Mag}>0$

Figure 4.3: Organizational schematic of matter-antimatter $(\sigma)$ and
polarization $(s)$ states with respect to the sign of the non-relativistic
magnetic dipole energy $U_{\rm Mag}$ obtainable from Eq. (4.4).

The reason Eq. (4.4) distinguishes between electrons and positrons is to
ensure the correct non-relativistic limit for the magnetic dipole energy is
reached. Following the conventions found in Tiesinga et al. (2021), we set the
gyro-magnetic factor $g\equiv g_{e^{+}}=-g_{e^{-}}>0$ such that electrons and
positrons have opposite $g$-factors and opposite magnetic moments relative to
their spin; see Figure 4.3.

We recall the conventions established in Section 1.3. Conservation of magnetic
flux requires that the magnetic field through a comoving surface $L_{0}^{2}$
remain unchanged. The magnetic field strength under expansion (Durrer and
Neronov, 2013) starting at some initial time $t_{0}$ is then given by

$\displaystyle B(t)=B_{0}\frac{a^{2}_{0}}{a^{2}(t)}\rightarrow
B(z)=B_{0}\left(1+z\right)^{2}\,,$ (4.5)

where $B_{0}$ is the comoving value obtained from the contemporary value of
the magnetic field today. Magnetic fields in the cosmos generated through
mechanisms such as dynamo or astrophysical sources do not follow this scaling
(Pomakov et al., 2022). It is only in deep intergalactic space where matter
density is low are magnetic fields preserved (and thus uncontaminated) over
cosmic time.

From Eq. (1.61) and Eq. (4.5) there emerges a natural ratio of interest which
is conserved over cosmic expansion

$\displaystyle\boxed{b\equiv\frac{e{B}(t)}{T^{2}(t)}=\frac{e{B}_{0}}{T_{0}^{2}}\equiv
b_{0}={\rm\ const.}}$ (4.6) $\displaystyle 10^{-3}>b_{0}>10^{-11}\,,$ (4.7)

given in natural units ($c=\hbar=k_{B}=1$). We computed the bounds for this
cosmic magnetic scale ratio by using the present day IGMF observations given
by Eq. (4.1) and the present CMB temperature $T_{0}=2.7{\rm\ K}\simeq
2.3\times 10^{-4}\text{ eV}$ (Aghanim et al., 2020).

####  4.3.1 Eigenstatess of magnetic moment in cosmology

As statistical properties depend on the characteristic Boltzmann factor $E/T$,
another interpretation of Eq. (4.6) in the context of energy eigenvalues (such
as those given in Eq. (4.4)) is the preservation of magnetic moment energy
relative to momentum under adiabatic cosmic expansion. The Boltzmann
statistical factor is given by

$\displaystyle x\equiv\frac{E}{T}\,.$ (4.8)

We can explore this relationship for the magnetized system explicitly by
writing out Eq. (4.8) using the KGP energy eigenvalues written in Eq. (4.4) as

$\displaystyle
x_{\sigma,s}^{n}=\frac{E_{\sigma,s}^{n}}{T}=\sqrt{\frac{m_{e}^{2}}{T^{2}}+\frac{p_{z}^{2}}{T^{2}}+\frac{eB}{T^{2}}\left(2n+1+\frac{g}{2}\sigma
s\right)}\,.$ (4.9)

Introducing the expansion scale factor $a(t)$ via Eq. (1.61), Eq. (4.5) and
Eq. (4.6). The Boltzmann factor can then be written as

$\displaystyle
x_{\sigma,s}^{n}(a(t))=\sqrt{\frac{m_{e}^{2}}{T^{2}(t_{0})}\frac{a(t)^{2}}{a_{0}^{2}}+\frac{p_{z,0}^{2}}{T_{0}^{2}}+\frac{eB_{0}}{T_{0}^{2}}\left(2n+1+\frac{g}{2}\sigma
s\right)}\,.$ (4.10)

This reveals that only the mass contribution is dynamic over cosmological
time. The constant of motion $b_{0}$ defined in Eq. (4.6) is seen as the
coefficient to the Landau and spin portion of the energy. For any given
eigenstate, the mass term drives the state into the non-relativistic limit
while the momenta and magnetic contributions are frozen by initial conditions.

In comparison, the Boltzmann factor for the DP energy eigenvalues are given by

$\displaystyle
x_{\sigma,s}^{n}|_{\mathrm{DP}}=\sqrt{\left(\sqrt{\frac{m_{e}^{2}}{T^{2}}+\frac{eB}{T^{2}}\left(2n+1+\sigma
s\right)}+\frac{eB}{2m_{e}T}\left(\frac{g}{2}-1\right)\sigma
s\right)^{2}+\frac{p_{z}^{2}}{T^{2}}}\,,$ (4.11)

which scales during FLRW expansion as

$x_{\sigma,s}^{n}(a(t))|_{\mathrm{DP}}=\\\
\sqrt{\left(\sqrt{\frac{m_{e}^{2}}{T_{0}^{2}}\frac{a(t)^{2}}{a_{0}^{2}}+\frac{eB_{0}}{T_{0}^{2}}\left(2n+1+\sigma
s\right)}+\frac{eB_{0}}{2m_{e}T_{0}}\frac{a_{0}}{a(t)}\left(\frac{g}{2}-1\right)\sigma
s\right)^{2}+\frac{p_{z,0}^{2}}{T_{0}^{2}}}\,.$ (4.12)

While the above expression is rather complicated, we note that the KGP Eq.
(4.10) and DP Eq. (4.11) Boltzmann factors both reduce to the Schödinger-Pauli
limit as $a(t)\rightarrow\infty$ thereby demonstrating that the total magnetic
moment is protected under the adiabatic expansion of the universe.

As noted in in Section 2.4 and Section 3.3.2, higher order non-minimal
magnetic contributions can be introduced to the Boltzmann factor such as
$\sim(e/m)^{2}B^{2}/T^{2}$. The reasoning above suggests that these terms are
suppressed over cosmological time driving the system into minimal
electromagnetic coupling with the exception of the anomalous magnetic moment.
It is interesting to note that cosmological expansion then serves to ‘smooth
out’ the characteristics of more complex electrodynamics erasing them from a
statistical perspective in favor of minimal-like dynamics.

####  4.3.2 Magnetized fermion partition function

To obtain a quantitative description of the above evolution, we study the bulk
properties of the relativistic charged/magnetic gasses in a nearly homogeneous
and isotropic primordial universe via the thermal Fermi-Dirac or Bose
distributions.

The grand partition function for the relativistic Fermi-Dirac ensemble is
given by the standard definition

$\displaystyle\ln\mathcal{Z}_{\mathrm{total}}=\sum_{\alpha}\ln\left(1+\Upsilon_{\alpha_{1}\ldots\alpha_{m}}\exp\left(-\frac{E_{\alpha}}{T}\right)\right)\,,\qquad\Upsilon_{\alpha_{1}\ldots\alpha_{m}}=\lambda_{\alpha_{1}}\lambda_{\alpha_{2}}\ldots\lambda_{\alpha_{m}}$
(4.13)

where we are summing over the set all relevant quantum numbers
$\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{m})$. We note here the
generalized the fugacity $\Upsilon_{\alpha_{1}\ldots\alpha_{m}}$ allowing for
any possible deformation caused by pressures effecting the distribution of any
quantum numbers.

In the case of the Landau problem, there is an additional summation over
$\widetilde{G}$ which represents the occupancy of Landau states (Greiner et
al., 2012b) which are matched to the available phase space within $\Delta
p_{x}\Delta p_{y}$. If we consider the orbital Landau quantum number $n$ to
represent the transverse momentum $p_{T}^{2}=p_{x}^{2}+p_{y}^{2}$ of the
system, then the relationship that defines $\widetilde{G}$ is given by

$\displaystyle\frac{L^{2}}{(2\pi)^{2}}\Delta p_{x}\Delta
p_{y}=\frac{eBL^{2}}{2\pi}\Delta
n\,,\qquad\widetilde{G}=\frac{eBL^{2}}{2\pi}\,.$ (4.14)

The summation over the continuous $p_{z}$ is replaced with an integration and
the double summation over $p_{x}$ and $p_{y}$ is replaced by a single sum over
Landau orbits

$\displaystyle\sum_{p_{z}}\rightarrow\frac{L}{2\pi}\int^{+\infty}_{-\infty}dp_{z}\,,\qquad\sum_{p_{x}}\sum_{p_{y}}\rightarrow\frac{eBL^{2}}{2\pi}\sum_{n}\,,$
(4.15)

where $L$ defines the boundary length of our considered volume $V=L^{3}$.

The partition function of the $e^{+}e^{-}$ plasma can be understood as the sum
of four gaseous species

$\displaystyle\ln\mathcal{Z}_{e^{+}e^{-}}=\ln\mathcal{Z}_{e^{+}}^{\uparrow}+\ln\mathcal{Z}_{e^{+}}^{\downarrow}+\ln\mathcal{Z}_{e^{-}}^{\uparrow}+\ln\mathcal{Z}_{e^{-}}^{\downarrow}\,,$
(4.16)

of electrons and positrons of both polarizations $(\uparrow\downarrow)$. The
change in phase space written in Eq. (4.15) modify the magnetized $e^{+}e^{-}$
plasma partition function from Eq. (4.13) into

$\displaystyle\ln\mathcal{Z}_{e^{+}e^{-}}=\frac{e{B}V}{(2\pi)^{2}}\sum_{\sigma}^{\pm
1}\sum_{s}^{\pm
1}\sum_{n=0}^{\infty}\int_{-\infty}^{\infty}\mathrm{d}p_{z}\left[\ln\left(1+\lambda_{\sigma}\xi_{\sigma,s}\exp\left(-\frac{E_{\sigma,s}^{n}}{T}\right)\right)\right]\,$
(4.17)
$\displaystyle\Upsilon_{\sigma,s}=\lambda_{\sigma}\xi_{\sigma,s}=\exp{\frac{\mu_{\sigma}+\eta_{\sigma,s}}{T}}\,,$
(4.18)

where the energy eigenvalues $E_{\sigma,s}^{n}$ are given in Eq. (4.4). The
index $\sigma$ in Eq. (4.17) is a sum over electron and positron states while
$s$ is a sum over polarizations. The index $s$ refers to the spin along the
field axis: parallel $(\uparrow;\ s=+1)$ or anti-parallel $(\downarrow;\
s=-1)$ for both particle and antiparticle species.

We are explicitly interested in small asymmetries such as baryon excess over
antibaryons, or one polarization over another. These are described by Eq.
(4.18) as the following two fugacities:

  * (a)

Chemical fugacity $\lambda_{\sigma}$

  * (b)

Polarization fugacity $\xi_{\sigma,s}$

For matter $(e^{-};\ \sigma=+1)$ and antimatter $(e^{+};\ \sigma=-1)$
particles, a nonzero relativistic chemical potential $\mu_{\sigma}=\sigma\mu$
is caused by an imbalance of matter and antimatter. While the primordial
electron-positron plasma era was overall charge neutral, there was a small
asymmetry in the charged leptons (namely electrons) from baryon asymmetry
(Fromerth et al., 2012; Canetti et al., 2012) in the universe. Reactions such
as $e^{+}e^{-}\leftrightarrow\gamma\gamma$ constrains the chemical potential
of electrons and positrons (Elze et al., 1980) as

$\displaystyle\mu\equiv\mu_{e^{-}}=-\mu_{e^{+}}\,,\qquad\lambda\equiv\lambda_{e^{-}}=\lambda_{e^{+}}^{-1}=\exp\frac{\mu}{T}\,,$
(4.19)

where $\lambda$ is the chemical fugacity of the system.

We can then parameterize the chemical potential of the $e^{+}e^{-}$ plasma as
a function of temperature $\mu\rightarrow\mu(T)$ via the charge neutrality of
the universe which implies

$\displaystyle
n_{p}=n_{e^{-}}-n_{e^{+}}=\frac{1}{V}\lambda\frac{\partial}{\partial\lambda}\ln\mathcal{Z}_{e^{+}e^{-}}\,.$
(4.20)

In Eq. (4.20), $n_{p}$ is the observed total number density of protons in all
baryon species. The chemical potential defined in Eq. (4.19) is obtained from
the requirement that the positive charge of baryons (protons, $\alpha$
particles, light nuclei produced after BBN) is exactly and locally compensated
by a tiny net excess of electrons over positrons.

We then introduce a novel polarization fugacity $\xi_{\sigma,s}$ and
polarization potential $\eta_{\sigma,s}=\sigma s\eta$. We propose the
polarization potential follows analogous expressions as seen in Eq. (4.19)
obeying

$\displaystyle\eta\equiv\eta_{+,+}=\eta_{-,-}\,,\quad\eta=-\eta_{\pm,\mp}\,,\quad\xi_{\sigma,s}\equiv\exp{\frac{\eta_{\sigma,s}}{T}}\,.$
(4.21)

An imbalance in polarization within a region of volume $V$ results in a
nonzero polarization potential $\eta\neq 0$. Conveniently since antiparticles
have opposite signs of charge and magnetic moment, the same magnetic moment is
associated with opposite spin orientations. A completely particle-antiparticle
symmetric magnetized plasma will have therefore zero total angular momentum.

##### Euler-Maclaurin integration

Before we proceed with the Boltzmann distribution approximation which makes up
the bulk of our analysis, we will comment on the full Fermi-Dirac distribution
analysis. The Euler-Maclaurin formula (Abramowitz et al., 1988) is used to
convert the summation over Landau levels $n$ into an integration given by

$\sum^{b}_{n=a}f(n)-\int^{b}_{a}f(x)dx=\frac{1}{2}\left(f(b)+f(a)\right)\\\
+\sum_{i=1}^{j}\frac{b_{2i}}{(2i)!}\left(f^{(2i-1)}(b)-f^{(2i-1)}(a)\right)+R(j)\,,$
(4.22)

where $b_{2i}$ are the Bernoulli numbers and $R(j)$ is the error remainder
defined by integrals over Bernoulli polynomials. The integer $j$ is chosen for
the level of approximation that is desired. Euler-Maclaurin integration is
rarely convergent, and in this case serves only as an approximation within the
domain where the error remainder is small and bounded; see Greiner et al.
(2012b) for the non-relativistic case. In this analysis, we keep the zeroth
and first order terms in the Euler-Maclaurin formula. We note that
regularization of the excess terms in Eq. (4.22) is done in the context of
strong field QED (Greiner and Reinhardt, 2008) though that is outside our
scope.

Using Eq. (4.22) allows us to convert the sum over $n$ quantum numbers in Eq.
(4.17) into an integral. Defining

$\displaystyle
f_{\sigma,s}^{n}=\ln\left(1+\Upsilon_{\sigma,s}\exp\left(-\frac{E_{\sigma,s}^{n}}{T}\right)\right)\,,$
(4.23)

Eq. (4.17) for $j=1$ becomes

$\ln\mathcal{Z}_{e^{+}e^{-}}=\frac{e{B}V}{(2\pi)^{2}}\sum_{\sigma,s}^{\pm
1}\int_{-\infty}^{+\infty}dp_{z}\\\
\left(\int_{0}^{+\infty}dnf_{\sigma,s}^{n}+\frac{1}{2}f_{\sigma,s}^{0}+\frac{1}{12}\frac{\partial
f_{\sigma,s}^{n}}{\partial n}\bigg{\rvert}_{n=0}+R(1)\right)$ (4.24)

It will be useful to rearrange Eq. (4.4) by pulling the spin dependency and
the ground state Landau orbital into the mass writing

$\displaystyle
E^{n}_{\sigma,s}={\tilde{m}}_{\sigma,s}\sqrt{1+\frac{p_{z}^{2}}{{\tilde{m}}_{\sigma,s}^{2}}+\frac{2e{B}n}{{\tilde{m}}_{\sigma,s}^{2}}}\,,$
(4.25)
$\displaystyle\varepsilon_{\sigma,s}^{n}(p_{z},{B})=\frac{E_{\sigma,s}^{n}}{{\tilde{m}}_{\sigma,s}}\,,\qquad{\tilde{m}}_{\sigma,s}^{2}=m_{e}^{2}+e{B}\left(1+\frac{g}{2}\sigma
s\right)\,,$ (4.26)

where we introduced the dimensionless energy $\varepsilon^{n}_{\sigma,s}$ and
effective polarized mass ${\tilde{m}}_{\sigma,s}$ which is distinct for each
spin alignment and is a function of magnetic field strength ${B}$. The
effective polarized mass ${\tilde{m}}_{\sigma,s}$ allows us to describe the
$e^{+}e^{-}$ plasma with the spin effects almost wholly separated from the
Landau characteristics of the gas when considering the plasma’s thermodynamic
properties.

With the energies written in this fashion, we recognize the first term in Eq.
(4.24) as mathematically equivalent to the free particle fermion partition
function with a re-scaled mass $m_{\sigma,s}$. The phase-space relationship
between transverse momentum and Landau orbits in Eq. (4.14) and Eq. (4.15) can
be succinctly described by

$\displaystyle p_{T}^{2}\sim 2eBn\,,\qquad 2p_{T}dp_{T}\sim 2eBdn\,,\qquad
d\bm{p}^{3}=2\pi p_{T}dp_{T}dp_{z}$ (4.27)
$\displaystyle\frac{eBV}{(2\pi)^{2}}\int_{-\infty}^{+\infty}dp_{z}\int_{0}^{+\infty}dn\rightarrow\frac{V}{(2\pi)^{3}}\int
d\bm{p}^{3}$ (4.28)

which recasts the first term in Eq. (4.24) as

$\displaystyle\ln\mathcal{Z}_{e^{+}e^{-}}=\frac{V}{(2\pi)^{3}}\sum_{\sigma,s}^{\pm
1}\int
d\bm{p}^{3}\ln\left(1+\Upsilon_{\sigma,s}\exp{\left(-\frac{m_{\sigma,s}\sqrt{1+p^{2}/m_{\sigma,s}^{2}}}{T}\right)}\right)+\ldots$
(4.29)

As we will see in the proceeding section, this separation of the ‘free-like’
partition function can be reproduced in the Boltzmann distribution limit as
well. This marks the end of the analytic analysis without approximations.

####  4.3.3 Boltzmann approach to electron-positron plasma

Since we address the temperature interval $200\text{ keV}>T>20\text{ keV}$
where the effects of quantum Fermi statistics on the $e^{+}e^{-}$ pair plasma
are relatively small, but the gas is still considered relativistic, we will
employ the Boltzmann approximation to the partition function in Eq. (4.17).
However, we extrapolate our results for presentation completeness up to
$T\simeq 4m_{e}$.

| aligned: $s=+1$ | anti-aligned: $s=-1$
---|---|---
electron: $\sigma=+1$ | $+\mu+\eta$ |  $+\mu-\eta$
positron: $\sigma=-1$ | $-\mu-\eta$ |  $-\mu+\eta$

Table 4.1: Organizational schematic of matter-antimatter $(\sigma)$ and
polarization $(s)$ states with respect to the chemical $\mu$ and polarization
$\eta$ potentials as seen in Eq. (4.32). Companion to Table 4.3.

The partition function shown in equation Eq. (4.17) can be rewritten removing
the logarithm as

$\displaystyle\ln{\mathcal{Z}_{e^{+}e^{-}}}=\frac{e{B}V}{(2\pi)^{2}}\sum_{\sigma,s}^{\pm
1}\sum_{n=0}^{\infty}\sum_{k=1}^{\infty}\int_{-\infty}^{+\infty}\mathrm{d}p_{z}\frac{(-1)^{k+1}}{k}\exp\left({k\frac{\sigma\mu+\sigma
s\eta-{\tilde{m}}_{\sigma,s}\varepsilon^{n}_{\sigma,s}}{T}}\right)\,,$ (4.30)
$\displaystyle\sigma\mu+\sigma
s\eta-{\tilde{m}}_{\sigma,s}\varepsilon_{\sigma,s}^{n}<0\,,$ (4.31)

which is well behaved as long as the factor in Eq. (4.31) remains negative. We
evaluate the sums over $\sigma$ and $s$ as

$\ln{\mathcal{Z}_{e^{+}e^{-}}}=\frac{e{B}V}{(2\pi)^{2}}\sum_{n=0}^{\infty}\sum_{k=1}^{\infty}\int_{-\infty}^{+\infty}\mathrm{d}p_{z}\frac{(-1)^{k+1}}{k}\times\\\
\left(\
\exp\left(k\frac{+\mu+\eta}{T}\right)\exp\left(-k\frac{{\tilde{m}}_{+,+}\varepsilon_{+,+}^{n}}{T}\right)\right.+\exp\left(k\frac{+\mu-\eta}{T}\right)\exp\left(-k\frac{{\tilde{m}}_{+,-}\varepsilon_{+,-}^{n}}{T}\right)\qquad\\\
+\exp\left(k\frac{-\mu-\eta}{T}\right)\exp\left(-k\frac{{\tilde{m}}_{-,+}\varepsilon_{-,+}^{n}}{T}\right)+\left.\exp\left(k\frac{-\mu+\eta}{T}\right)\exp\left(-k\frac{{\tilde{m}}_{-,-}\varepsilon_{-,-}^{n}}{T}\right)\right)$
(4.32)

We note from Figure 4.3 that the first and forth terms and the second and
third terms share the same energies via

$\displaystyle\varepsilon_{+,+}^{n}=\varepsilon_{-,-}^{n}\,,\qquad\varepsilon_{+,-}^{n}=\varepsilon_{-,+}^{n}\,.\qquad\varepsilon_{+,-}^{n}<\varepsilon_{+,+}^{n}\,,$
(4.33)

Eq. (4.33) allows us to reorganize the partition function with a new
magnetization quantum number $s^{\prime}$ which characterizes paramagnetic
flux increasing states $(s^{\prime}=+1)$ and diamagnetic flux decreasing
states $(s^{\prime}=-1)$. This recasts Eq. (4.32) as

$\ln{\mathcal{Z}_{e^{+}e^{-}}}=\frac{e{B}V}{(2\pi)^{2}}\sum_{s^{\prime}}^{\pm
1}\sum_{n=0}^{\infty}\sum_{k=1}^{\infty}\int_{-\infty}^{+\infty}\mathrm{d}p_{z}\frac{(-1)^{k+1}}{k}\\\
\left[2\xi_{s^{\prime}}\cosh\frac{k\mu}{T}\right]\exp\left(-k\frac{{\tilde{m}}_{s^{\prime}}\varepsilon_{s^{\prime}}^{n}}{T}\right)$
(4.34)

with dimensionless energy $\varepsilon_{s^{\prime}}^{n}$, polarization mass
$\tilde{m}_{s^{\prime}}$, and polarization $\eta_{s^{\prime}}$ redefined in
terms of the moment orientation quantum number $s^{\prime}$

$\displaystyle{\tilde{m}}_{s^{\prime}}^{2}=m_{e}^{2}+e{B}\left(1-\frac{g}{2}s^{\prime}\right)\,,$
(4.35)
$\displaystyle\eta\equiv\eta_{+}=-\eta_{-}\qquad\xi\equiv\xi_{+}=\xi_{-}^{-1}\,,\qquad\xi_{s^{\prime}}=\xi^{\pm
1}=\exp\left(\pm\frac{\eta}{T}\right)\,.$ (4.36)

We introduce the modified Bessel function $K_{\nu}$ (see Ch. 10 of Letessier
and Rafelski (2023)) of the second kind

$\displaystyle
K_{\nu}\left(\frac{m}{T}\right)=\frac{\sqrt{\pi}}{\Gamma(\nu-1/2)}\frac{1}{m}\left(\frac{1}{2mT}\right)^{\nu-1}\int_{0}^{\infty}\mathrm{d}p\,p^{2\nu-2}\exp\left({-\frac{m\varepsilon}{T}}\right)\,,$
(4.37) $\displaystyle\nu>1/2\,,\qquad\varepsilon=\sqrt{1+p^{2}/m^{2}}\,,$
(4.38)

allowing us to rewrite the integral over momentum in Eq. (4.34) as

$\displaystyle\frac{1}{T}\int_{0}^{\infty}\\!\\!\mathrm{d}p_{z}\exp\\!\left(\\!{-\frac{k{\tilde{m}}_{s^{\prime}}\varepsilon_{s^{\prime}}^{n}}{T}}\\!\right)\\!=\\!W_{1}\\!\\!\left(\frac{k{\tilde{m}}_{s^{\prime}}\varepsilon_{s^{\prime}}^{n}(0,{B})}{T}\right)\,.$
(4.39)

The function $W_{\nu}$ serves as an auxiliary function of the form
$W_{\nu}(x)=xK_{\nu}(x)$. The notation $\varepsilon(0,{B})$ in Eq. (4.39)
refers to the definition of dimensionless energy found in Eq. (4.26) with
$p_{z}=0$. The standard Boltzmann distribution is obtained by summing only
$k=1$ and neglecting the higher order terms.

We take advantage again of Euler-Maclaurin integration Eq. (4.22) and
integrate the partition function. After truncation of the series and error
remainder, the partition function Eq. (4.30) can then be written in terms of
modified Bessel $K_{\nu}$ functions of the second kind and cosmic magnetic
scale $b_{0}$, yielding

$\displaystyle\boxed{\ln\mathcal{Z}_{e^{+}e^{-}}\simeq\frac{T^{3}V}{\pi^{2}}\sum_{s^{\prime}}^{\pm
1}\left[\xi_{s^{\prime}}\cosh{\frac{\mu}{T}}\right]\left(x_{s^{\prime}}^{2}K_{2}(x_{s^{\prime}})+\frac{b_{0}}{2}x_{s^{\prime}}K_{1}(x_{s^{\prime}})+\frac{b_{0}^{2}}{12}K_{0}(x_{s^{\prime}})\right)}\,,$
(4.40) $\displaystyle
x_{s^{\prime}}=\frac{{\tilde{m}}_{s^{\prime}}}{T}=\sqrt{\frac{m_{e}^{2}}{T^{2}}+b_{0}\left(1-\frac{g}{2}s^{\prime}\right)}\,.$
(4.41)

The latter two terms in Eq. (4.40) proportional to $b_{0}K_{1}$ and
$b_{0}^{2}K_{0}$ are the uniquely magnetic terms present in powers of magnetic
scale Eq. (4.6) containing both spin and Landau orbital influences in the
partition function. The $K_{2}$ term is analogous to the free Fermi gas
(Greiner et al., 2012b) being modified only by spin effects.

This ‘separation of concerns’ can be rewritten as

$\displaystyle\ln\mathcal{Z}_{\mathrm{S}}=\frac{T^{3}V}{\pi^{2}}\sum_{s^{\prime}}^{\pm
1}\left[\xi_{s^{\prime}}\cosh{\frac{\mu}{T}}\right]\left(x_{s^{\prime}}^{2}K_{2}(x_{s^{\prime}})\right)\,,$
(4.42)
$\displaystyle\ln\mathcal{Z}_{\mathrm{SO}}=\frac{T^{3}V}{\pi^{2}}\sum_{s^{\prime}}^{\pm}\left[\xi_{s^{\prime}}\cosh{\frac{\mu}{T}}\right]\left(\frac{b_{0}}{2}x_{s^{\prime}}K_{1}(x_{s^{\prime}})+\frac{b_{0}^{2}}{12}K_{0}(x_{s^{\prime}})\right)\,,$
(4.43)

where the spin (S) and spin-orbit (SO) partition functions can be considered
independently. When the magnetic scale $b_{0}$ is small, the spin-orbit term
Eq. (4.43) becomes negligible leaving only paramagnetic effects in Eq. (4.42)
due to spin. In the non-relativistic limit, Eq. (4.42) reproduces a quantum
gas whose Hamiltonian is defined as the free particle (FP) Hamiltonian plus
the magnetic dipole (MD) Hamiltonian which span two independent Hilbert spaces
$\mathcal{H}_{\mathrm{FP}}\otimes\mathcal{H}_{\mathrm{MD}}$. The non-
relativistic limit is further discussed in Section 4.3.4.

Writing the partition function as Eq. (4.40) instead of Eq. (4.30) has the
additional benefit that the partition function remains finite in the free gas
$({B}\rightarrow 0)$ limit. This is because the free Fermi gas and Eq. (4.42)
are mathematically analogous to one another. As the Bessel $K_{\nu}$ functions
are evaluated as functions of $x_{\pm}$ in Eq. (4.41), the ‘free’ part of the
partition $K_{2}$ is still subject to spin magnetization effects. In the limit
where ${B}\rightarrow 0$, the free Fermi gas is recovered in both the
Boltzmann approximation $k=1$ and the general case $\sum_{k=1}^{\infty}$.

####  4.3.4 Non-relativistic limit of the magnetized partition function

While we label the first term in Eq. (4.29) as the ‘free’ partition function,
this is not strictly true as the partition function dependant on the magnetic-
mass we defined in Eq. (4.26). When determining the magnetization of the
quantum Fermi gas, derivatives of the magnetic field $B$ will not fully vanish
on this first term which will resulting in an intrinsic magnetization which is
distinct from the Landau levels.

This represents magnetization that arises from the spin magnetic energy rather
than orbital contributions. To demonstrate this, we will briefly consider the
weak field limit for $g=2$. The effective polarized mass for electrons is then

$\displaystyle\tilde{m}_{+}^{2}$ $\displaystyle=m_{e}^{2}\,,$ (4.44)
$\displaystyle\tilde{m}_{-}^{2}$ $\displaystyle=m_{e}^{2}+2eB\,,$ (4.45)

with energy eigenvalues

$\displaystyle E_{n}^{+}$ $\displaystyle=\sqrt{p_{z}^{2}+m_{e}^{2}+2eBn}\,,$
(4.46) $\displaystyle E_{n}^{-}$
$\displaystyle=\sqrt{\left(E_{n}^{+}\right)^{2}+2eB}\,.$ (4.47)

The spin anti-aligned states in the non-relativistic (NR) limit reduce to

$\displaystyle E_{n}^{-}|_{\mathrm{NR}}\approx
E_{n}^{+}|_{\mathrm{NR}}+\frac{eB}{m_{e}}\,.$ (4.48)

This shift in energies is otherwise not influenced by summation over Landau
quantum number $n$, therefore we can interpret this energy shift as a shift in
the polarization potential from Eq. (4.21). The polarization potential is then

$\displaystyle\eta_{e}^{\pm}=\eta_{e}\pm\frac{eB}{2m_{e}}\,,$ (4.49)

allowing us to rewrite the partition function in Eq. (4.30) as

$\displaystyle\ln\mathcal{Z}_{e^{-}}|_{NR}=\frac{eBV}{(2\pi)^{2}}\sum_{s^{\prime}}^{\pm}\sum_{n=0}^{\infty}\sum_{k=1}^{\infty}\int_{-\infty}^{+\infty}dp_{z}\frac{(-1)^{k+1}}{k}2\cosh(k\beta\eta_{e}^{s^{\prime}})\lambda^{k}\exp(-k\epsilon_{n}/T)\,,$
(4.50)
$\displaystyle\epsilon_{n}=m_{e}+\frac{p_{z}^{2}}{2m_{e}}+\frac{eB}{2m_{e}}\left(n+1\right)\,.$
(4.51)

Eq. (4.50) is then the traditional NR quantum harmonic oscillator partition
function with a spin dependant potential shift differentiating the aligned and
anti-aligned states. We note that in this formulation, the spin contribution
is entirely excised from the orbital contribution. Under Euler-Maclaurin
integration, the now spin-independant Boltzmann factor can be further
separated into ‘free’ and Landau quantum parts as was done in Eq. (4.29) for
the relativistic case. We note however that the inclusion of anomalous
magnetic moment spoils this clean separation.

####  4.3.5 Electron-positron chemical potential

In presence of a magnetic field in the Boltzmann approximation, the charge
neutrality condition Eq. (4.20) becomes

$\displaystyle\sinh\frac{\mu}{T}=n_{p}\frac{\pi^{2}}{T^{3}}\left[\sum_{s^{\prime}}^{\pm
1}\xi_{s^{\prime}}\\!\left(\\!x_{s^{\prime}}^{2}K_{2}(x_{s^{\prime}})\\!+\\!\frac{b_{0}}{2}x_{s^{\prime}}K_{1}(x_{s^{\prime}})\\!+\\!\frac{b_{0}^{2}}{12}K_{0}(x_{s^{\prime}}\\!)\\!\right)\\!\right]^{-1}\\!.$
(4.52)

Eq. (4.52) is fully determined by the right-hand-side expression if the spin
fugacity is set to unity $\eta=0$ implying no external bias to the number of
polarizations except as a consequence of the difference in energy eigenvalues.
In practice, the latter two terms in Eq. (4.52) are negligible to chemical
potential in the bounds of the primordial $e^{+}e^{-}$ plasma considered and
only becomes relevant for extreme (see Figure 4.4) magnetic field strengths
well outside our scope.

Figure 4.4: The chemical potential over temperature $\mu/T$ is plotted as a
function of temperature with differing values of spin potential $\eta$ and
magnetic scale $b_{0}$.

Eq. (4.52) simplifies if there is no external magnetic field $b_{0}=0$ into

$\displaystyle\sinh\frac{\mu}{T}=n_{p}\frac{\pi^{2}}{T^{3}}\left[2\cosh\frac{\eta}{T}\left(\frac{m_{e}}{T}\right)^{2}K_{2}\left(\frac{m_{e}}{T}\right)\right]^{-1}\,.$
(4.53)

In Figure 4.4 we plot the chemical potential $\mu/T$ in Eq. (4.52) and Eq.
(4.53) which characterizes the importance of the charged lepton asymmetry as a
function of temperature. Since the baryon (and thus charged lepton) asymmetry
remains fixed, the suppression of $\mu/T$ at high temperatures indicates a
large pair density which is seen explicitly in Figure 4.2. The black line
corresponds to the $b_{0}=0$ and $\eta=0$ case.

The para-diamagnetic contribution from Eq. (4.43) does not appreciably
influence $\mu/T$ until the magnetic scales involved become incredibly large
well outside the observational bounds defined in Eq. (4.1) and Eq. (4.6) as
seen by the dotted blue curves of various large values $b_{0}=\\{25,\ 50,\
100,\ 300\\}$. The chemical potential is also insensitive to forcing by the
spin potential until $\eta$ reaches a significant fraction of the electron
mass $m_{e}$ in size. The chemical potential for large values of spin
potential $\eta=\\{100,\ 200,\ 300,\ 400,\ 500\\}\,\text{ keV}$ are also
plotted as dashed black lines with $b_{0}=0$.

It is interesting to note that there are crossing points where a given
chemical potential can be described as either an imbalance in spin-
polarization or presence of external magnetic field. While spin potential
suppresses the chemical potential at low temperatures, external magnetic
fields only suppress the chemical potential at high temperatures.

The profound insensitivity of the chemical potential to these parameters
justifies the use of the free particle chemical potential (black line) in the
ranges of magnetic field strength considered for cosmology. Mathematically
this can be understood as $\xi$ and $b_{0}$ act as small corrections in the
denominator of Eq. (4.52) if expanded in powers of these two parameters.

###  4.4 Relativistic paramagnetism of electron-positron gas

The total magnetic flux within a region of space can be written as the sum of
external fields and the magnetization of the medium via

$\displaystyle{B}_{\mathrm{total}}={B}+\mathcal{M}\,.$ (4.54)

For the simplest mediums without ferromagnetic or hysteresis considerations,
the relationship can be parameterized by the susceptibility $\chi$ of the
medium as

$\displaystyle{B}_{\mathrm{total}}=(1+\chi){B}\,,\qquad\mathcal{M}=\chi{B}\,,\qquad\chi\equiv\frac{\partial\mathcal{M}}{\partial{B}}\,,$
(4.55)

with the possibility of both paramagnetic materials $(\chi>1)$ and diamagnetic
materials $(\chi<1)$. The $e^{+}e^{-}$ plasma however does not so neatly fit
in either category as given by Eq. (4.42) and Eq. (4.43). In general, the
susceptibility of the gas will itself be a field dependant quantity.

In our analysis, the external magnetic field always appears within the context
of the magnetic scale $b_{0}$, therefore we can introduce the change of
variables

$\displaystyle\frac{\partial b_{0}}{\partial{B}}=\frac{e}{T^{2}}\,.$ (4.56)

The magnetization of the $e^{+}e^{-}$ plasma described by the partition
function in Eq. (4.40) can then be written as

$\displaystyle\mathcal{M}\equiv\frac{T}{V}\frac{\partial}{\partial{B}}\ln{\mathcal{Z}_{e^{+}e^{-}}}=\frac{T}{V}\left(\frac{\partial
b_{0}}{\partial{B}}\right)\frac{\partial}{\partial
b_{0}}\ln{\mathcal{Z}_{e^{+}e^{-}}}\,,$ (4.57)

Magnetization arising from other components in the cosmic gas (protons,
neutrinos, etc.) could in principle also be included. Localized
inhomogeneities of matter evolution are often non-trivial and generally be
solved numerically using magneto-hydrodynamics (MHD) (Melrose, 2013; Vazza et
al., 2017; Vachaspati, 2021) or with a suitable Boltzmann-Vlasov transport
equation. An extension of our work would be to embed magnetization into
transport theory (Formanek et al., 2021a). In the context of MHD, primordial
magnetogenesis from fluid flows in the electron-positron epoch was considered
in Gopal and Sethi (2005); Perrone et al. (2021).

We introduce dimensionless units for magnetization ${\mathfrak{M}}$ by
defining the critical field strength

$\displaystyle{B}_{C}\equiv\frac{m_{e}^{2}}{e}\,,\qquad{\mathfrak{M}}\equiv\frac{\mathcal{M}}{{B}_{C}}\,.$
(4.58)

The scale ${B}_{C}$ is where electromagnetism is expected to become subject to
non-linear effects, though luckily in our regime of interest, electrodynamics
should be linear. We note however that the upper bounds of IGMFs in Eq. (4.1)
(with $b_{0}=10^{-3}$; see Eq. (4.6)) brings us to within $1\%$ of that limit
for the external field strength in the temperature range considered.

The total magnetization ${\mathfrak{M}}$ can be broken into the sum of
magnetic moment parallel ${\mathfrak{M}}_{+}$ and magnetic moment anti-
parallel ${\mathfrak{M}}_{-}$ contributions

$\displaystyle{\mathfrak{M}}={\mathfrak{M}}_{+}+{\mathfrak{M}}_{-}\,.$ (4.59)

We note that the expression for the magnetization simplifies significantly for
$g\\!=\\!2$ which is the ‘natural’ gyro-magnetic factor (Evans and Rafelski,
2022; Rafelski et al., 2023b) for Dirac particles. For illustration, the
$g\\!=\\!2$ magnetization from Eq. (4.57) is then

$\displaystyle{\mathfrak{M}}_{+}$
$\displaystyle=\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\xi\cosh{\frac{\mu}{T}}\left[\frac{1}{2}x_{+}K_{1}(x_{+})+\frac{b_{0}}{6}K_{0}(x_{+})\right]\,,$
(4.60) $\displaystyle-{\mathfrak{M}}_{-}$
$\displaystyle=\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\xi^{-1}\cosh{\frac{\mu}{T}}\left[\left(\frac{1}{2}+\frac{b_{0}^{2}}{12x_{-}^{2}}\right)x_{-}K_{1}(x_{-})+\frac{b_{0}}{3}K_{0}(x_{-})\right]\,,$
(4.61) $\displaystyle x_{+}$ $\displaystyle=\frac{m_{e}}{T}\,,\qquad
x_{-}=\sqrt{\frac{m_{e}^{2}}{T^{2}}+2b_{0}}\,.$ (4.62)

As the $g$-factor of the electron is only slightly above two at $g\simeq
2.00232$ (Tiesinga et al., 2021), the above two expressions for
${\mathfrak{M}}_{+}$ and ${\mathfrak{M}}_{-}$ are only modified by a small
amount because of anomalous magnetic moment (AMM) and would be otherwise
invisible on our figures.

####  4.4.1 Evolution of electron-positron magnetization

In Figure 4.5, we plot the magnetization as given by Eq. (4.60) and Eq. (4.61)
with the spin potential set to unity $\xi=1$. The lower (solid red) and upper
(solid blue) bounds for cosmic magnetic scale $b_{0}$ are included. The
external magnetic field strength ${B}/{B}_{C}$ is also plotted for lower
(dotted red) and upper (dotted blue) bounds. Since the derivative of the
partition function governing magnetization may manifest differences between
Fermi-Dirac and the here used Boltzmann limit more acutely, out of abundance
of caution, we indicate extrapolation outside the domain of validity of the
Boltzmann limit with dashes.

Figure 4.5: The magnetization ${\mathfrak{M}}$, with $g\\!=\\!2$, of the
primordial $e^{+}e^{-}$ plasma is plotted as a function of temperature. Figure
made in collaboration with Cheng Tao Yang.

We see in Figure 4.5 that the $e^{+}e^{-}$ plasma is overall paramagnetic and
yields a positive overall magnetization which is contrary to the traditional
assumption that matter-antimatter plasma lack significant magnetic responses
of their own in the bulk. With that said, the magnetization never exceeds the
external field under the parameters considered which shows a lack of
ferromagnetic behavior.

The large abundance of pairs causes the smallness of the chemical potential
seen in Figure 4.4 at high temperatures. As the universe expands and
temperature decreases, there is a rapid decrease of the density $n_{e^{\pm}}$
of $e^{+}e^{-}$ pairs. This is the primary the cause of the rapid paramagnetic
decrease seen in Figure 4.5 above $T\\!=\\!21\text{ keV}$. At lower
temperatures $T<21\text{ keV}$ there remains a small electron excess (see
Figure 4.2) needed to neutralize proton charge. These excess electrons then
govern the residual magnetization and dilutes with cosmic expansion.

An interesting feature of Figure 4.5 is that the magnetization in the full
temperature range increases as a function of temperature. This is contrary to
Curie’s law (Greiner et al., 2012b) which stipulates that paramagnetic
susceptibility of a laboratory material is inversely proportional to
temperature. However, Curie’s law applies to systems with fixed number of
particles which is not true in our situation; see Section 4.4.3.

A further consideration is possible hysteresis as the $e^{+}e^{-}$ density
drops with temperature. It is not immediately obvious the gas’s magnetization
should simply ‘degauss’ so rapidly without further consequence. If the very
large paramagnetic susceptibility present for $T\simeq m_{e}$ is the origin of
an overall magnetization of the plasma, the conservation of magnetic flux
through the comoving surface ensures that the initial residual magnetization
is preserved at a lower temperature by Faraday induced kinetic flow processes
however our model presented here cannot account for such effects.

Early universe conditions may also apply to some extreme stellar objects with
rapid change in $n_{e^{\pm}}$ with temperatures above $T\\!=\\!21\text{ keV}$.
Production and annihilation of $e^{+}e^{-}$ plasmas is also predicted around
compact stellar objects (Ruffini et al., 2010; Ruffini and Vereshchagin, 2013)
potentially as a source of gamma-ray bursts.

####  4.4.2 Dependency on g-factor

As discussed at the end of Section 4.4, the AMM of $e^{+}e^{-}$ is not
relevant in the present model. However out of academic interest, it is
valuable to consider how magnetization is effected by changing the $g$-factor
significantly.

Figure 4.6: The magnetization $\mathfrak{M}$ as a function of $g$-factor
plotted for several temperatures with magnetic scale $b_{0}=10^{-3}$ and
polarization fugacity $\xi=1$.

The influence of AMM would be more relevant for the magnetization of baryon
gasses since the $g$-factor for protons $(g\approx 5.6)$ and neutrons
$(g\approx 3.8)$ are substantially different from $g\\!=\\!2$. The influence
of AMM on the magnetization of thermal systems with large baryon content
(neutron stars, magnetars, hypothetical bose stars, etc.) is therefore also of
interest (Ferrer and Hackebill, 2019, 2023).

Eq. (4.60) and Eq. (4.61) with arbitrary $g$ reintroduced is given by

$\displaystyle{\mathfrak{M}}=\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\sum_{s^{\prime}}^{\pm
1}\xi_{s^{\prime}}\cosh{\frac{\mu}{T}}\left[C^{1}_{s^{\prime}}(x_{s^{\prime}})K_{1}(x_{s^{\prime}})+C^{0}_{s^{\prime}}K_{0}(x_{s^{\prime}})\right]\,,$
(4.63) $\displaystyle
C^{1}_{s^{\prime}}(x_{\pm})=\left[\frac{1}{2}-\left(\frac{1}{2}-\frac{g}{4}s^{\prime}\right)\left(1+\frac{b^{2}_{0}}{12x^{2}_{s^{\prime}}}\right)\right]x_{s^{\prime}}\,,\qquad
C^{0}_{s^{\prime}}=\left[\frac{1}{6}-\left(\frac{1}{4}-\frac{g}{8}s^{\prime}\right)\right]b_{0}\,,$
(4.64)

where $x_{s^{\prime}}$ was previously defined in Eq. (4.41).

In Figure 4.6, we plot the magnetization as a function of $g$-factor between
$4>g>-4$ for temperatures $T\\!=\\!\\{511,\ 300,\ 150,\ 70\\}\text{ keV}$. We
find that the magnetization is sensitive to the value of AMM revealing a
transition point between paramagnetic $({\mathfrak{M}}>0)$ and diamagnetic
gasses $({\mathfrak{M}}<0)$. Curiously, the transition point was numerically
determined to be around $g\simeq 1.1547$ in the limit $b_{0}\rightarrow 0$.
The exact position of this transition point however was found to be both
temperature and $b_{0}$ sensitive, though it moved little in the ranges
considered.

It is not surprising for there to be a transition between diamagnetism and
paramagnetism given that the partition function (see Eq. (4.42) and Eq.
(4.43)) contained elements of both. With that said, the transition point
presented at $g\approx 1.15$ should not be taken as exact because of the
approximations used to obtain the above results.

It is likely that the exact transition point has been altered by our taking of
the Boltzmann approximation and Euler-Maclaurin integration steps. It is known
that the Klein-Gordon-Pauli solutions to the Landau problem in Eq. (4.4) have
periodic behavior (Steinmetz et al., 2019; Evans and Rafelski, 2022; Rafelski
et al., 2023b) for $|g|=k/2$ (where $k\in 1,2,3\ldots$).

These integer and half-integer points represent when the two Landau towers of
orbital levels match up exactly. Therefore, we propose a more natural
transition between the spinless diamagnetic gas of $g=0$ and a paramagnetic
gas is $g=1$. A more careful analysis is required to confirm this, but that
our numerical value is close to unity is suggestive.

####  4.4.3 Magnetization per lepton

Despite the relatively large magnetization seen in Figure 4.5, the average
contribution per lepton is only a small fraction of its overall magnetic
moment indicating the magnetization is only loosely organized. Specifically,
the magnetization regime we are in is described by

$\displaystyle\mathcal{M}\ll\mu_{B}\frac{N_{e^{+}}+N_{e^{-}}}{V}\,,\qquad\mu_{B}\equiv\frac{e}{2m_{e}}\,,$
(4.65)

where $\mu_{B}$ is the Bohr magneton and $N=nV$ is the total particle number
in the proper volume V. To better demonstrate that the plasma is only weakly
magnetized, we define the average magnetic moment per lepton given by along
the field ($z$-direction) axis as

$\displaystyle|\vec{m}|_{z}\equiv\frac{\mathcal{M}}{n_{e^{-}}+n_{e^{+}}}\,,\qquad|\vec{m}|_{x}=|\vec{m}|_{y}=0\,.$
(4.66)

Statistically, we expect the transverse expectation values to be zero. We
emphasize here that despite $|\vec{m}|_{z}$ being nonzero, this doesn’t
indicate a nonzero spin angular momentum as our plasma is nearly matter-
antimatter symmetric. The quantity defined in Eq. (4.66) gives us an insight
into the microscopic response of the plasma.

Figure 4.7: The magnetic moment per lepton $|\vec{m}|_{z}$ along the field
axis as a function of temperature. Figure made in collaboration with Cheng Tao
Yang.

The average magnetic moment $|\vec{m}|_{z}$ defined in Eq. (4.66) is plotted
in Figure 4.7 which displays how essential the external field is on the ‘per
lepton’ magnetization. The $b_{0}=10^{-3}$ case (blue curve) is plotted in the
Boltzmann approximation. The dashed lines indicate where this approximation is
only qualitatively correct. For illustration, a constant magnetic field case
(solid green line) with a comoving reference value chosen at temperature
$T_{0}=10\text{ keV}$ is also plotted.

If the field strength is held constant, then the average magnetic moment per
lepton is suppressed at higher temperatures as expected for magnetization
satisfying Curie’s law. The difference in Figure 4.7 between the non-constant
(blue solid curve) case and the constant field (solid green curve) case
demonstrates the importance of the conservation of primordial magnetic flux in
the plasma, required by Eq. (4.5). While not shown, if Figure 4.7 was extended
to lower temperatures, the magnetization per lepton of the constant field case
would be greater than the non-constant case which agrees with our intuition
that magnetization is easier to achieve at lower temperatures. This feature
again highlights the importance of flux conservation in the system and the
uniqueness of the primordial cosmic environment.

###  4.5 Polarization potential and ferromagnetism

Up to this point, we have neglected the impact that a nonzero spin potential
$\eta\neq 0$ (and thus $\xi\neq 1$) would have on the primordial $e^{+}e^{-}$
plasma magnetization. In the limit that $(m_{e}/T)^{2}\gg b_{0}$ the
magnetization given in Eq. (4.63) and Eq. (4.64) is entirely controlled by the
spin fugacity $\xi$ asymmetry generated by the spin potential $\eta$ yielding
up to first order $\mathcal{O}(b_{0})$ in magnetic scale

$\lim_{m_{e}^{2}/T^{2}\gg
b_{0}}{\mathfrak{M}}=\frac{g}{2}\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\sinh{\frac{\eta}{T}}\cosh{\frac{\mu}{T}}\left[\frac{m_{e}}{T}K_{1}\left(\frac{m_{e}}{T}\right)\right]\\\
+b_{0}\left(g^{2}-\frac{4}{3}\right)\frac{e^{2}}{8\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\cosh{\frac{\eta}{T}}\cosh{\frac{\mu}{T}}K_{0}\left(\frac{m_{e}}{T}\right)+\mathcal{O}\left(b_{0}^{2}\right)$
(4.67)

Given Eq. (4.67), we can understand the spin potential as a kind of
‘ferromagnetic’ influence on the primordial gas which allows for magnetization
even in the absence of external magnetic fields. This interpretation is
reinforced by the fact the leading coefficient is $g/2$.

We suggest that a variety of physics could produce a small nonzero $\eta$
within a domain of the gas. Such asymmetries could also originate
statistically as while the expectation value of free gas polarization is zero,
the variance is likely not.

As $\sinh{\eta/T}$ is an odd function, the sign of $\eta$ also controls the
alignment of the magnetization. In the high temperature limit Eq. (4.67) with
strictly $b_{0}=0$ assumes a form of to lowest order for brevity

$\displaystyle\lim_{m_{e}/T\rightarrow
0}{\mathfrak{M}}|_{b_{0}=0}=\frac{g}{2}\frac{e^{2}}{\pi^{2}}\frac{T^{2}}{m_{e}^{2}}\frac{\eta}{T}\,,$
(4.68)

While the limit in Eq. (4.68) was calculated in only the Boltzmann limit, it
is noteworthy that the high temperature (and $m\rightarrow 0$) limit of Fermi-
Dirac distributions only differs from the Boltzmann result by a
proportionality factor.

The natural scale of the $e^{+}e^{-}$ magnetization with only a small spin
fugacity ($\eta<1\text{ eV}$) fits easily within the bounds of the predicted
magnetization during this era if the IGMF measured today was of primordial
origin. The reason for this is that the magnetization seen in Eq. (4.60), Eq.
(4.61) and Eq. (4.67) are scaled by $\alpha{B}_{C}$ where $\alpha$ is the fine
structure constant.

####  4.5.1 Hypothesis of ferromagnetic self-magnetization

One exploratory model we propose is to fix the spin polarization asymmetry,
described in Eq. (4.21), to generate a homogeneous magnetic field which
dissipates as the universe cools down. In this model, there is no external
primordial magnetic field $({B}_{\mathrm{PMF}}=0)$ generated by some unrelated
physics, but rather the $e^{+}e^{-}$ plasma itself is responsible for the
field by virtue of spin polarization.

Figure 4.8: The spin potential $\eta$ and chemical potential $\mu$ are plotted
under the assumption of self-magnetization through a nonzero spin polarization
in bulk of the plasma. Figure made in collaboration with Cheng Tao Yang.

This would obey the following assumption of

$\displaystyle{\mathfrak{M}}(b_{0})=\frac{\mathcal{M}(b_{0})}{{B}_{C}}\longleftrightarrow\frac{B}{{B}_{C}}=b_{0}\frac{T^{2}}{m_{e}^{2}}\,,$
(4.69)

which sets the total magnetization as a function of itself. The spin
polarization described by $\eta\rightarrow\eta(b_{0},T)$ then becomes a fixed
function of the temperature and magnetic scale. The underlying assumption
would be the preservation of the homogeneous field would be maintained by
scattering within the gas (as it is still in thermal equilibrium) modulating
the polarization to conserve total magnetic flux.

The result of the self-magnetization assumption in Eq. (4.69) for the
potentials is plotted in Figure 4.8. The solid lines indicate the curves for
$\eta/T$ for differing values of $b_{0}=\\{10^{-11},\ 10^{-7},\ 10^{-5},\
10^{-3}\\}$ which become dashed above $T\\!=\\!300\text{ keV}$ to indicate
that the Boltzmann approximation is no longer appropriate though the general
trend should remain unchanged.

Figure 4.9: The number density $n_{e^{\pm}}$ of polarized electrons and
positrons under the self-magnetization model for differing values of $b_{0}$.
Figure courtesy of Cheng Tao Yang.

The dotted lines are the curves for the chemical potential $\mu/T$. At high
temperatures we see that a relatively small $\eta/T$ is needed to produce
magnetization owing to the large densities present. Figure 4.8 also shows that
the chemical potential does not deviate from the free particle case until the
spin polarization becomes sufficiently high which indicates that this form of
self-magnetization would require the annihilation of positrons to be
incomplete even at lower temperatures.

This is seen explicitly in Figure 4.9 where we plot the numerical density of
particles as a function of temperature for spin aligned $(+\eta)$ and spin
anti-aligned $(-\eta)$ species for both positrons $(-\mu)$ and electrons
$(+\mu)$. Various self-magnetization strengths are also plotted to match those
seen in Figure 4.8. The nature of $T_{\rm split}$ changes under this model
since antimatter and polarization states can be extinguished separately.
Positrons persist where there is insufficient electron density to maintain the
magnetic flux. Polarization asymmetry therefore appears physical only in the
domain where there is a large number of matter-antimatter pairs.

####  4.5.2 Matter inhomogeneities in the cosmic plasma

In general, an additional physical constraint is required to fully determine
$\mu$ and $\eta$ simultaneously as both potentials have mutual dependency (see
Section 4.5). We note that spin polarizations are not required to be in
balanced within a single species to preserve angular momentum.

The CMB (Aghanim et al., 2020) indicates that the early universe was home to
domains of slightly higher and lower baryon densities which resulted in the
presence of galactic super-clusters, cosmic filaments, and great voids seen
today. However, the CMB, as measured today, is blind to the localized
inhomogeneities required for gravity to begin galaxy and supermassive black
hole formation.

Such acute inhomogeneities distributed like a dust (Grayson et al., 2023) in
the plasma would make the proton density sharply and spatially dependant
$n_{p}\rightarrow n_{p}(x)$ which would directly affect the potentials
$\mu(x)$ and $\eta(x)$ and thus the density of electrons and positrons
locally. This suggests that $e^{+}e^{-}$ may play a role in the initial
seeding of gravitational collapse. If the plasma were home to such localized
magnetic domains, the nonzero local angular momentum within these domains
would provide a natural mechanism for the formation of rotating galaxies
today.

Recent measurements by the James Webb Space Telescope (JWST) (Yan et al.,
2023; Adams et al., 2023; Haro et al., 2023) indicate that galaxy formation
began surprisingly early at large redshift values of $z\gtrsim 10$ within the
first 500 million years of the universe requiring gravitational collapse to
begin in a hotter environment than expected. The observation of supermassive
black holes already present (Larson et al., 2023) in this same high redshift
period (with millions of solar masses) indicates the need for local high
density regions in the early universe whose generation is not yet explained
and likely need to exist long before the recombination epoch.

##  Chapter 5 Outlook and key results

#### Chapter 2: Dynamics of charged particles with arbitrary magnetic moment

We highlighted the comparison of magnetic moment dynamics of DP and KGP
formulation of relativistic quantum mechanics. The DP equation breaks up the
magnetic moment into an underlying spinor structure part inherent to the Dirac
equation, and a dedicated anomalous part. In contrast, for the KGP, the entire
effect of magnetic moment is contained in a single Pauli term irrespective of
the magnetic moment’s size. We find that the two models disagree in their
predicted energy levels for the homogeneous magnetic field and the Coulomb
field.

For the KGP-Landau levels, Eq. (2.16), we have a simple dependence on the full
magnetic moment $g$-factor and have the correct non-relativistic reduction at
lowest order. This simplicity allows, for the KGP equation, the
straightforward analysis of physical systems and elegant expressions for their
solutions. Thus Dirac’s beauty principle favors heavily the KGP considering
the Landau levels.

In the case of the Coulomb field there are weak fields differences in both the
Lamb shift and fine structure; the contribution to the Lamb shift and fine
structure splitting are proportional in KGP to: $g^{2}/8-1/2$, Eq. (2.41) and
$g^{2}/8$, Eq. (2.49) respectively, rather than: DP $g/2-1$ Eq. (2.43)) and
$g/2-1/2$, Eq. (2.51)) respectively.

For strong fields, both DP and KGP share the behavior of a shrinking
particle/antiparticle gap for the ground state when $|g|>2$, though the
expressions differ from each other. The models are extremely different in both
their predicted energy values and the ultimate fate of the states as $B$
increases. For the KGP equation, the gap vanishes in very strong magnetic
fields Eq. (2.56a). KGP also shows state-merging behavior for states of the
same total angular momentum.

#### Chapter 3: Dynamic neutrino flavor mixing through transition moments

We have incorporated electromagnetic effects in the Majorana neutrino mixing
matrix by introducing an anomalous transition magnetic dipole moment. We have
described the formalism for three generations of neutrinos and explicitly
explored the two generation case as a toy model.

In the two generation case, we determined the effect of electric and magnetic
fields on flavor rotation in Eq. (3.34) by introducing an electromagnetic
flavor unitary rotation $Z_{kj}^{\mathrm{ext}}$. We presented remixed mass
eigenstates $\widetilde{m}(E,B)$ in Eq. (3.47) which are the propagating mass-
states in a background electromagnetic field
$F^{\alpha\beta}_{\mathrm{ext}}(x^{\mu})$. These EM-mass eigenstates were also
further split by spin aligned and anti-aligned states relative to the external
field momentum density. There is much left to do to explore further the
nascent connection between the spin and the flavor via transition magnetic
moments.

The transition dipole moments are the origin of dynamical flavor mixing. While
our focus was on Majorana neutrinos, Dirac-type fermions (neutrinos included)
may also have non-zero transition dipole moments. These could remix flavor in
the presence of strong external background fields. Here quarks are of special
interest because they are not only electrically charged, but have color charge
as well. This means quarks could in principle possess one, or both, EM and
color-charge transition dipole moments, leading to dynamical effects in the
CKM mixing matrix within hadrons as well as in quark-gluon plasma.

More speculatively, as transition dipoles act as a mechanism to generate mass
by virtue of EM energy density $T_{\mathrm{ext}}^{00}$ as seen in Eq. (3.47),
an analogous consequence of our work could arise in the presence of a dark
vector field in the Universe coupled to neutrinos, resulting in off-diagonal
masses in flavor. Massless neutrinos could then obtain dynamical non-zero
masses in the Universe by virtue of their interactions originating in dark
transition moments.

#### Chapter 4: Matter-antimatter origin of cosmic magnetism

We characterized the primordial magnetic properties of the early universe
before recombination. We studied the temperature range of $2000\text{ keV}$ to
$20\text{ keV}$ where all of space was filled with a hot dense electron-
positron plasma (to the tune of 450 million pairs per baryon) which occurred
within the first few minutes after the Big Bang. We note that our chosen
period also includes the era of Big Bang Nucleosynthesis.

We found that subject to a primordial magnetic field, the early universe
electron-positron plasma has a significant paramagnetic response due to
magnetic moment polarization. We considered the interplay of charge chemical
potential, baryon asymmetry, anomalous magnetic moment, and magnetic dipole
polarization on the nearly homogeneous medium.

This novel approach to high temperature magnetization shows that the
$e^{+}e^{-}$-plasma paramagnetic response (see Eq. (4.60) and Eq. (4.61)) is
dominated by the varying abundance of electron-positron pairs, decreasing with
decreasing $T$ for $T\\!<\\!m_{e}c^{2}$. This is unlike conventional
laboratory cases where the number of magnetic particles is constant.

We find that electron-positron magnetization rapidly vanishes as the number of
pairs depletes as the universe cools. This therefore presents an opportunity
for induced currents to facilitate inhomogeneities in the early universe. We
also presented a simple model of self-magnetization of the primordial
electron-positron plasma which indicates that only a small polarization
asymmetry is required to generate significant magnetic flux when the universe
was very hot and dense.

##  Chapter 6 Future research efforts

This chapter contains unpublished work and is exploratory; though we hope to
complete this research in the near future. Section 6.1 extends the KGP wave
equation to also include chromomagnetic dipole moments necessary for quarks in
quantum chromodynamics (QCD). We show explicitly that the dipoles are linear
in field tensors and do not mix spin degrees of freedom.

In Section 6.2, we explore CP violation in the neutrino sector in terms of the
Jarlskog invariant and modifications to the mass matrix. The CP symmetry of
the electric dipole is discussed. This section follows the conventions found
in Chapter 3.

The final Section 6.3 is a short description of a personal passion project to
integrate spin dynamics into the five dimensional theory of Kaluza-Klein which
unifies electromagnetism and gravitation classically. While I have written
much on the topic, it remains unfinished and will only occupy a small section
of this dissertation.

###  6.1 Klein-Gordon-Pauli extensions to non-Abelian fields

The KGP approach to wave equations is robust and is useful not only for
electromagnetic interactions: This is a first look at adding the strong
interaction into KGP inspired in part by Labun and Rafelski (2012). Quarks
participate in the strong color interaction of quantum chromodynamics (QCD);
therefore they should present a $g$-factor for both their electromagnetic
dipole $g_{\mathrm{EM}}$ as well as their QCD chromomagnetic dipole
$g_{\mathrm{QCD}}$.

Since color charges follows a more complex $SU(3)$ group structure unlike the
more straight forward $U(1)$ of electromagnetism, the “color magnetism” of QCD
requires more than just the analogous Pauli term to describe color dipole
moments owing due to the fact that QCD has non-Abelian (non-commuting) gluon
gauge fields $\mathcal{A}^{\alpha}$.

The quarks (like all particles) should obey the quantum mechanical analogue of
the energy-momentum relation seen in Eq. (1.51) with the only theoretical
difference being a modified covariant derivative. The covariant derivative,
written in terms of kinetic momentum, should appear as

$\displaystyle\mathrm{EM+QCD}:\qquad
i\hbar\widetilde{\nabla}=\pi^{\alpha}=p^{\alpha}-q_{\mathrm{EM}}A^{\alpha}-q_{\mathrm{QCD}}\mathcal{A}^{\alpha}\,,$
(6.1)

where $q_{\mathrm{EM}}$ is the electric charge of the quarks
$q_{\mathrm{EM}}/e\in\pm 1/3,\pm 2/3$ and $q_{\mathrm{QCD}}$ is the color
charge coupling strength. In many texts the symbol $g_{s}$ is used for the
color coupling strength, but we circumvent that notation using
$q_{\mathrm{QCD}}$ to avoid confusion with $g$-factor.

We follow the conventions in Greiner et al. (2006). The eight $3\times 3$
Gell-Mann matrices $\lambda^{a}$ are embedded into each independent field as

$\displaystyle\mathcal{A}^{\alpha}\equiv\frac{1}{2}\lambda^{a}\mathcal{A}^{\alpha}_{a}\,,\qquad
a\in 1\ldots
8\,,\qquad[\lambda^{a},\lambda^{b}]=\frac{i}{2}f^{abc}\lambda^{c}\,,$ (6.2)

where $\mathcal{A}_{a}^{\alpha}$ are the individual fields for each gluon
species in a given representation and $f^{abc}$ is the $SU(3)$ antisymmetric
structure function. The non-commuting behavior of the Gell-Mann matrices
captures the non-Abelian structure of the gauge fields. The gluon field
strength tensor $\mathcal{G}^{\alpha\beta}$ is then

$\displaystyle\mathcal{G}^{\alpha\beta}=\partial^{\alpha}\mathcal{A}^{\beta}-\partial^{\beta}\mathcal{A}^{\alpha}+\frac{i}{\hbar}q_{\mathrm{QCD}}\left[\mathcal{A}^{\alpha},\mathcal{A}^{\beta}\right]\,,$
(6.3)
$\displaystyle\left[\mathcal{A}^{\alpha},\mathcal{A}^{\beta}\right]=\frac{1}{4}\mathcal{A}^{\alpha}_{a}\mathcal{A}^{\beta}_{b}\left[\lambda^{a},\lambda^{b}\right]=\frac{i}{8}\mathcal{A}^{\alpha}_{a}\mathcal{A}^{\beta}_{b}f^{abc}\lambda_{c}\,.$
(6.4)

Following the same procedure in Section 1.2.2, we can generalize the energy-
momentum relation and obtain the EM+QCD variant of the KGP equation for
quarks. We find that the resulting quark-KGP equation is

$\displaystyle\gamma_{\alpha}\gamma_{\beta}\pi^{\alpha}\pi^{\beta}=\eta_{\alpha\beta}\pi^{\alpha}\pi^{\beta}-\frac{q_{\mathrm{EM}}\hbar}{2}\sigma_{\alpha\beta}F^{\alpha\beta}-\frac{q_{\mathrm{QCD}}\hbar}{2}\sigma_{\alpha\beta}\mathcal{G}^{\alpha\beta}\,,$
(6.5)
$\displaystyle\left(\eta_{\alpha\beta}\pi^{\alpha}\pi^{\beta}-\frac{q_{\mathrm{EM}}\hbar}{2}\sigma_{\alpha\beta}F^{\alpha\beta}-\frac{q_{\mathrm{QCD}}\hbar}{2}\sigma_{\alpha\beta}\mathcal{G}^{\alpha\beta}\right)\Psi=m_{q}^{2}c^{2}\Psi_{q}\,,$
(6.6)

which mirrors the electromagnetic case except for the extension of a
chromomagnetic Pauli term (Morgan, 1995). We note that $m_{q}$ is the quark
mass and that the field $\Psi_{q}$ is a color triplet of spinors for: red,
green, blue:

$\displaystyle\Psi_{q}=\begin{pmatrix}\Psi_{r}\\\ \Psi_{g}\\\
\Psi_{b}\end{pmatrix}\,.$ (6.7)

As only the non-commuting portion of Eq. (6.6) (written explicitly in Eq.
(6.4)) is off-diagonal in color, this means that the additional non-commuting
chromomagnetic term acts as a transition matrix between different quark
colors.

This method (based on the commutator of the kinetic momentum) suggests that
the color and electromagnetic $g$-factors both have a natural value of
$g_{\mathrm{EM}}\\!=\\!g_{\mathrm{QCD}}\\!=\\!2$. We can generalize Eq. (6.5)
to allow for arbitrary EM and QCD dipole moments $g_{\mathrm{EM}}$ and
$g_{\mathrm{QCD}}$ respectively as

$\displaystyle\boxed{\left(\eta_{\alpha\beta}\pi^{\alpha}\pi^{\beta}-\frac{g_{\mathrm{EM}}}{2}\frac{q_{\mathrm{EM}}\hbar}{2}\sigma_{\alpha\beta}F^{\alpha\beta}-\frac{g_{\mathrm{QCD}}}{2}\frac{q_{\mathrm{QCD}}\hbar}{2}\sigma_{\alpha\beta}\mathcal{G}^{\alpha\beta}\right)\Psi=m_{q}^{2}c^{2}\Psi}\,.$
(6.8)

While in electromagnetism, DP and KGP approaches only differ in the presence
of strong EM fields and are otherwise identical in the weak field limit, this
cannot be equally said in QCD. The perturbative limit which justifies the DP
approach for leptons in QED is possible due to the small value of the fine
structure constant which is not true in QCD. Only for high momentum
interactions (such as those present in quark-gluon-plasma (QGP) or in
energetic collisions) is the perturbative approach applicable (Choudhury and
Lahiri, 2015). We emphasize however that the DP approach is only valid where
the dipole moment is obtainable via perturbative expansion which may not hold
if the $g$-factor results from non-perturbative physics.

The dipole characteristics (both electromagnetic and chromomagnetic) of the
top-quark is of particular interest (Labun and Rafelski, 2013; Vryonidou and
Zhang, 2018) because of top-quark’s strong coupling to the Higgs and potential
BSM physics. We note that current studies focus on a DP approach to
chromomagnetism (Zhang and Willenbrock, 2011; Zhang, 2012; Buarque Franzosi
and Zhang, 2015). There is also the added complexity of both the
chromomagnetic and magnetic $g$-factors differing from the natural value
independently of one another $g_{\mathrm{EM}}\neq g_{\mathrm{QCD}}\neq 2$.
Further study of the KGP approach to chromomagnetism should be pursued. To our
knowledge, there is no formulation of Eq. (6.8) as an effective field theory
in the manner of Fleming et al. (2001); Bauer et al. (2001) for quarks or for
electromagnetically charged fermions as was discussed in Section 1.2.1.

###  6.2 Electromagnetic forced neutrino CP violation

Electromagnetic processes in the neutrino sector may yield measurable effects
in two aspects of neutrino physics:

  * (a)

Neutrino oscillation which is evidence of the non-zero mass eigenstates

  * (b)

Charge-parity (CP) violation in the neutrino sector which occurs due to the
presence of at least three generations of neutrinos or additional CP violating
interactions

Our motivation is to explore the effect of strong electromagnetic fields on
neutrino CP violation by analysis of the electromagnetic dipole interaction
and determining its influence on the Jarlskog invariant $J$ (Jarlskog, 1985b,
a, 2005) which controls the size of CP violation.

One important aspect of neutrino physics is the size of the CP violation
(Xing, 2001; Giunti and Kim, 2007; Huber et al., 2022) which can yield
insights not only in fundamental physics but cosmology as well. One of the
goals of modern neutrino experiments such as DUNE Abi et al. (2020) is to
better characterize such effects in astrophysical contexts such as supernova
(Abi et al., 2021; Sajjad Athar et al., 2022), solar neutrinos (Akhmedov and
Martínez-Miravé, 2022) and magnetars (Lichkunov et al., 2020). Additionally,
there may be a connection (Pehlivan et al., 2014; Balaji et al., 2020a, b)
between CP violation in the neutrino sector and the anomalous magnetic moment
(AMM) of the neutrino.

Amplified CP violation is already expected in matter (Harrison and Scott,
2000) such as when travelling through the Earth’s crust due to the abundance
of electrons and the lack of muons and taus which preferentially affect the
electron neutrinos via the weak interaction. This effect may also play a role
in the primordial universe where neutrinos would propagate through incredibly
matter dense gasses such as during the electron-positron epoch (Rafelski et
al., 2023a). Matter modifies CP violation because it changes the effective
Hamiltonian of the propagating neutrinos, therefore we should look to other
possible sources for modification of the Hamiltonian for CP violation
amplification or suppression.

The source of direct CP violation in the neutrinos is ultimately attributable
to the mismatch between the mass matrices of the charged flavors
$(e,\mu,\tau)$ and the neutral flavors $(\nu_{e},\nu_{\mu},\nu_{\tau})$. The
situation is analogous to the quark sector, where instead the relation is
between the upper $(u,c,t)$ and lower quark $(d,s,b)$ flavors.

Therefore mass matrix for charged leptons does not commute with the mass
matrix of the neutral leptons and cannot be simultaneously diagonalized except
for special cases or degeneracy among the mass eigenstates. We can
characterize CP violation by introducing the mixing matrices $V_{\ell k}$
(following the notation in Chapter 3) which diagonalize the individual mass-
matrices $M_{\ell\ell^{\prime}}$ as follows

$\displaystyle V_{\ell
l}^{{\dagger}}(M^{\nu}M^{\nu{\dagger}})_{\ell\ell^{\prime}}V_{\ell^{\prime}k^{\prime}}=D_{\ell\ell^{\prime}}^{\nu}=\mathrm{diag}(m_{1}^{2},m_{2}^{2},m_{3}^{2})\,,$
(6.9) $\displaystyle
D_{\ell\ell^{\prime}}=\mathrm{diag}(m_{e},m_{\mu},m_{\tau})\,,$ (6.10)

We have specifically defined the charged leptons flavor states as being
simultaneously mass eigenstates without a loss of generality. We will not
consider oscillation among the charged leptons, though that may be an avenue
of further study Akhmedov (2007).

####  6.2.1 Jarlskog invariant

The intrinsic CP violation inherent to these mass matrices can be described
using the Jarlskog invariant $J$. We first define the commutator of the
charged and neutral lepton mass matrices as

$\displaystyle[M^{\nu}M^{\nu{\dagger}},D^{2}]_{\ell\ell^{\prime}}=C_{\ell\ell^{\prime}}\,.$
(6.11)

As the elements of the neutrino mixing matrix $V_{\ell k}$ can be
experimentally determined, the size of the commutator can be expressed using
Eq. (6.9) and Eq. (6.10) in terms of the mass eigenstates of the neutrino mass
matrix

$\displaystyle[V(D^{\nu})^{2}V^{{\dagger}},D^{2}]_{\ell\ell^{\prime}}=C_{\ell\ell^{\prime}}\,.$
(6.12)

The matrix $C_{\ell\ell^{\prime}}$ can be unwieldy, so following the procedure
by (Jarlskog, 1985b, a, 2005), we take the determinant of
$C_{\ell\ell^{\prime}}$ in Eq. (6.12) which extracts the invariant quantity
associated with the size of the CP violation present. Specifically we are
interested in the imaginary portion given by

$\displaystyle\mathrm{Im}\left[\mathrm{det}(C_{\ell\ell^{\prime}})\right]=2\left(\Delta_{12}\Delta_{23}\Delta_{13}\right)\left(\Delta_{e\mu}\Delta_{\mu\tau}\Delta_{e\tau}\right)J\,,$
(6.13)

where $J$ is the invariant quantity of interest. We define $\Delta_{ij}$ via
the eigenstates of the mass matrices as

$\displaystyle\Delta_{ij}\equiv m^{2}_{i}-m^{2}_{j}\,.$ (6.14)

We can also define the real portion of the determinant and define two
quantities $R$ and $J$ together. These two scalars are written in terms of the
components of the mixing matrix $V_{\ell k}$ as

$\displaystyle{\cal
J}_{ikjl}=\mathrm{Im}\left[V_{ij}V_{kl}V^{*}_{il}V^{*}_{kj}\right]=J\sum_{m,n}\epsilon_{ikm}\epsilon_{jln}\,,$
(6.15) $\displaystyle{\cal
R}_{ikjl}=\mathrm{Re}\left[V_{ij}V_{kl}V^{*}_{il}V^{*}_{kj}\right]=R\sum_{m,n}\epsilon_{ikm}\epsilon_{jln}\,.$
(6.16)

The benefit of $J$ is it captures the ‘size’ of CP violation in a single value
and is identically zero for systems which preserve charge-parity symmetry.
From Eq. (6.13) we can see that CP violation vanishes if the commutator of the
mass matrices is purely real, and thus the mixing matrix is also purely real,
or if there is degeneracy among the mass eigenstates which absorbs a degree of
freedom.

While generally CP violation comes exclusively from the presence of three
flavor generations in the Standard Model, we would like to expand the usage of
the J-invariant to encompass a family of effects via modification of the mass
matrix as was done in Chapter 3. This would encompass CP violation from the
number of flavor generations, physical systems (matter, strong fields, etc…)
which effectively break CP, or in CP violating terms in the Lagrangian.

####  6.2.2 Toy model: CP violation amplification

While the specific change to neutrino mixing and CP violation depends on the
model of the neutrino dipole moment, a demonstrative model would be to assume
that the magnetic moment matrix was simply proportional to the natural mass
matrix of the neutrino at low order. Therefore we substitute

$\displaystyle
M_{\ell\ell^{\prime}}^{\nu}\rightarrow(M_{\ell\ell^{\prime}}^{\nu})^{\prime}=M_{\ell\ell^{\prime}}^{\nu}(1+\kappa\sigma_{\alpha\beta}F^{\alpha\beta})\,.$
(6.17)

If the same mechanism which produced neutrino masses also produced their
dipoles through some BSM physics, this would not be an unreasonable
assumption.

As the modified mass matrix commutes entirely with the original mass matrix,
the mathematical structure of the commutator in Eq. (6.11) is unchanged

$\displaystyle\left[(M^{\nu})^{\prime},M^{\nu}\right]=0\,\rightarrow\mathrm{Im}[\mathrm{det}(C_{\ell\ell^{\prime}})]=\mathrm{Im}[\mathrm{det}(C_{\ell\ell^{\prime}}^{\prime})]\,.$
(6.18)

The determinant calculation is then identical between the two mass matrices.
While the overall determinant is fixed, the individual elements which make up
the determinant are modified. This yields

$\displaystyle\mathrm{Im}\left[\mathrm{det}(C_{\ell\ell^{\prime}})\right]$
$\displaystyle=\left(\Delta_{12}\Delta_{23}\Delta_{13}\right)\left(\Delta_{e\mu}\Delta_{\mu\tau}\Delta_{e\tau}\right)J\,,$
(6.19)
$\displaystyle\mathrm{Im}\left[\mathrm{det}(C_{\ell\ell^{\prime}}^{\prime})\right]$
$\displaystyle=\left(\Delta_{12}^{\prime}\Delta_{23}^{\prime}\Delta_{13}^{\prime}\right)\left(\Delta_{e\mu}\Delta_{\mu\tau}\Delta_{e\tau}\right)J^{\prime}\,,$
(6.20)

where we have added primes to denote new values due to remixing. A similar
result occurs in matter in Harrison and Scott (2000). The ratio of modified
$J^{\prime}$ to $J$ is the amplification (or suppression) of CP violation
given by

$\displaystyle\mathcal{R}=\frac{J^{\prime}}{J}=\frac{\left(\Delta_{12}\Delta_{23}\Delta_{13}\right)}{\left(\Delta_{12}^{\prime}\Delta_{23}^{\prime}\Delta_{13}^{\prime}\right)}\,.$
(6.21)

In our simple proportionality model, this simplifies to

$\displaystyle\mathcal{R}=\frac{1}{\left(1+\kappa\sigma_{\alpha\beta}F^{\alpha\beta}\right)^{6}}\,.$
(6.22)

To first order, and evaluating $\sigma_{\alpha\beta}F^{\alpha\beta}$ for
homogeneous magnetic fields, Eq. (6.22) can be expressed as

$\displaystyle\mathcal{R}=1\pm 12\kappa B+\mathcal{O}(B^{2})\,.$ (6.23)

where we allow for aligned and anti-aligned spin states. This suggests the
possibility that CP violation having spin dependence in strong EM fields.

####  6.2.3 Electric dipole moments and CP symmetry

Here we state the standard picture of CP violation through an electric dipole.
The structure of the Pauli term in Eq. (1.31) informs us how to construct the
relativistic electric dipole moment (EDM); see Knecht (2004); Jegerlehner
(2017). The generalization to include the electric dipole is

$\displaystyle\delta\mu\rightarrow\delta\tilde{\mu}\equiv\delta\mu+i\epsilon\gamma^{5}\,,$
(6.24)

where $\epsilon$ is the EDM of the particle. As the natural electric dipole
within the Dirac equation is zero, the presence of $\epsilon$ is always
considered anomalous. The EDM Pauli Lagrangian term is

$\displaystyle\mathcal{L}_{\mathrm{EDM}}=-{\bar{\psi}}\left(i\epsilon\gamma^{5}\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}\right)\psi\,,$
(6.25)

which is of interest because of the inclusion of $\gamma^{5}$. Taking
advantage of the properties of $\gamma^{5}$, we can write the EDM in Eq.
(6.25) as

$\displaystyle\gamma^{5}\sigma_{\mu\nu}=\frac{i}{2}\varepsilon_{\mu\nu\alpha\beta}\sigma^{\alpha\beta}\,\rightarrow\mathcal{L}_{\mathrm{EDM}}=+{\bar{\psi}}\left(\epsilon\frac{1}{2}\sigma^{\alpha\beta}F_{\alpha\beta}^{*}\right)\psi\,.$
(6.26)

which is more closely analogous to the structure of the AMM in Eq. (1.31)
making use of the dual form of the electromagnetic field tensor shown in Eq.
(1.29).

Following a procedure similar to the one found in Section 1.1, Eq. (6.26)
reduces in the non-relativistic limit to the Hamiltonian density EDM
interaction

$\displaystyle\mathcal{H}_{\mathrm{EDM}}\approx\epsilon\chi^{{\dagger}}\bm{\sigma}\cdot\bm{E}\chi\,.$
(6.27)

The electric dipole is important because the presence of one would signify
charge-parity (CP) violation in the theory which provides a method to
distinguish between matter and antimatter. We discuss briefly using the parity
(P) and time (T) symmetries of the relevant spin $\bm{s}$ and field vectors
$(\bm{E},\bm{B})$ and their inner products. The even and odd symmetries of
each are printed in Table 6.1.

| $\bm{E}$ | $\bm{B}$ | $\bm{s}$ | $\bm{s}\\!\cdot\\!\bm{E}$ | $\bm{s}\\!\cdot\\!\bm{B}$
---|---|---|---|---|---
| T symmetry
---
$(t\rightarrow-t)$
even | odd | odd | odd |  even
| P symmetry
---
$(\bm{x}\rightarrow-\bm{x})$
odd | even | even | odd |  even
| PT symmetry
---
$(x^{\alpha}\rightarrow-x^{\alpha})$
odd | odd | odd | even |  even

Table 6.1: Time (T), parity (P) and PT symmetries of electric $\bm{E}$,
magnetic $\bm{B}$, spin $\bm{s}$ three-vectors and the inner products which
describe the dipole Hamiltonian terms.

The EDM term Eq. (6.27) is overall T-odd and P-odd while the magnetic dipole
is T-even and P-even. While EDM dipoles are common in molecular systems, no
electric dipole has ever been measured for an elementary particle nor
composite particles like the proton or neutron despite extensive searching. As
a point of comparison, the EDM of the electron is excluded (Andreev et al.,
2018; Roussy et al., 2023) by a bound of $|\epsilon_{e}/c|<4.1\times
10^{-30}\,e\,\mathrm{cm}$.

For CPT symmetry to hold, Table 6.1 implies that both the electric and
magnetic dipoles must be C-even. C-symmetry is a more complicated concept to
discuss as it is only well-defined relativistically where particle and
antiparticle states are simultaneously described by the theory. The Dirac
spinor charge conjugates as

$\displaystyle
C:\psi\rightarrow\psi_{c}=\eta_{c}C(\bar{\psi})^{\mathrm{T}}=\eta_{c}C\gamma_{0}^{\mathrm{T}}\psi^{*}\,,$
(6.28)

where $C$ is the charge conjugation matrix satisfying the conjugation relation

$\displaystyle-C\gamma_{\alpha}^{\mathrm{T}}C^{-1}=\gamma_{\alpha}\,,$ (6.29)

and $\eta_{c}$ is an arbitrary complex phase. The exact matrix expression of
$C$ depends on the representational basis used (Dirac, Weyl, Majorana, etc…).
We’re specifically interesting in the following conjugations

$\displaystyle
C:\bar{\psi}\gamma_{\alpha}\psi\rightarrow-\bar{\psi}\gamma_{\alpha}\psi\,,\qquad
C:\bar{\psi}\sigma_{\alpha\beta}\psi\rightarrow-\bar{\psi}\sigma_{\alpha\beta}\psi\,,\qquad
C:A^{\alpha}\rightarrow-A^{\alpha}\,.$ (6.30)

As the spin density $\bar{\psi}\sigma_{\alpha\beta}\psi$ and vector potential
$A^{\alpha}$ (and thus $F^{\alpha\beta}$) are both odd under charge
conjugation, the combination present in the AMM Lagrangian Eq. (1.31) is
C-even under charge conjugation. The same is true of the EDM Lagrangian Eq.
(6.26) which is more easily seen when cast in terms of the dual tensor.

We note that the fields $\psi$, $\bar{\psi}$ and $A^{\alpha}$ are considered
dynamical _operators_ within a quantum field theory. C-symmetry is therefore
broken when considering an externally fixed background field such as
$A_{\mathrm{ext}}^{\alpha}(x)$ as $C:A_{\mathrm{ext}}^{\alpha}(x)\rightarrow
A_{\mathrm{ext}}^{\alpha}(x)$. This distinction bears some importance when
discussing the neutrino flavor rotation in Chapter 3. There is also interest
in EDMs in the background of curved spacetime (Filho et al., 2023).

###  6.3 Spin in 5D Kaluza-Klein theory

The ‘miracle’ of Kaluza-Klein (Kaluza, 1921; Klein, 1926) is that both 4D
gravitation and electromagnetism emerge from a higher dimension 5D
gravitational theory. While the specific importance of Kaluza’s result to the
physical world (if there is one) has yet to be revealed, the ideas of Kaluza-
Klein have been extensively used to showcase the emergence of unified physics
from higher dimensional geometries (Ortin, 2015).

In the modern context, we understand that Kaluza-Klein in its original form
cannot strictly be correct as we are operating purely in the classical regime
and we are missing the incorporation of the weak and strong interactions as
well. With all that said, there is still value in exploring the implications
of Kaluza-Klein in that Kaluza’s miracle might very well not be an accident,
but a natural result from a fuller more complete implementation of these ideas
(Overduin and Wesson, 1997).

The main goal of this research effort is to obtain the spin dynamics of a test
particle in the context of a 5D Kaluza-Klein style theory in the same spirit
as was accomplished for the general relativistic torque equations

$\displaystyle\frac{Dp^{\mu}}{D\tau}+\frac{1}{2}u^{\nu}s^{\rho\sigma}R^{\mu}_{\
\nu\rho\sigma}=0\,,\qquad\frac{Ds^{\mu\nu}}{D\tau}+2u^{[\mu}p^{\nu]}=0\,,$
(6.31)

which couples spin and precession to a curved spacetime. An additional term is
also required to accommodate spin in the kinematic equation. Eq. (6.31) is
known as the Mathisson-Papapetrou-Dixon (MPD) equations (Mathisson, 1937;
Papapetrou, 1951; Dixon, 1970; Mathisson, 2010). The rank-four tensor
$R^{\mu}_{\ \nu\rho\sigma}$ is the Rienmann curvature defined by

$\displaystyle R^{\mu}_{\
\nu\rho\sigma}V^{\nu}=2\nabla_{[\rho}\nabla_{\sigma]}V^{\mu}\,,$ (6.32)

where $V^{\mu}$ is any arbitrary vector and the notation $[\rho,\sigma]$
indicates a commutator of indices. The significance of Eq. (6.31) is that
particle motion will deviate away from traditional geodesic motion due to a
coupling of the curvature to the spin.

To promote the above equations into the Kaluza-Klein framework, we consider
Lorentz five-vectors $\hat{P}^{A}$ which are dressed with a hat and capital
Latin letters indices $A\in(0-3,5)$. The position five-vector is then denoted
by $\hat{x}^{A}=(x^{\mu},x_{5})$ where $x_{5}$ is the fifth coordinate
position. The 5D Einstein-Hilbert action is

$\displaystyle\mathcal{S}[\hat{g}_{AB}]=\frac{c^{4}}{16\pi\hat{G}}\int\mathrm{d}\hat{x}^{5}\sqrt{-\hat{g}}\hat{R}\,.$
(6.33)

The variable $\hat{G}$ is the 5D gravitational constant and $\hat{R}$ is the
Ricci scalar in the 5D space-time. Kaluza-Klein theories are usually expressed
as Ricci-flat theories with the scalar curvature $\hat{R}=0$, but in general
nothing prevents us from including matter or other fields in the five-
dimensional bulk by adding terms to the above action.

In a 5D spacetime under Klein’s cylindrical condition, a particle under free-
fall motion manifests as accelerated motion analogous to Eq. (6.31) caused by
the electromagnetic force and an additional scalar force in the four-
dimensional sector. Because particles with spin deviate from geodesics in
free-fall, there should ultimately be spin precession generated by
electromagnetism, the scalar field, and gravitation.

Therefore we propose a classical spin five-vector

$\displaystyle\hat{s}^{A}=(s^{\alpha},s_{5})\,,$ (6.34)

which relates the classical four-spin Eq. (2.76) discussed in Section 2.5 to a
new fifth component of spin $s_{5}$. We note that analogously the fifth
component of five-momentum is related to the mass of the particle; it is our
suggestion that the fifth component of spin is related to the invariant spin
magnitude. It is the study of this object that will be left to future
publications.

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