# Aligning a Majorana fermion’s anapole moment with an external current
through photon emission mediated by the fermion’s generalized polarizabilities

Kiana Walter  Kobi Hall  David C. Latimer  Department of Physics, University
of Puget Sound, Tacoma, WA 98416-1031

###### Abstract

The sole static electromagnetic property of a spin-$\frac{1}{2}$ Majorana
fermion is its anapole moment. Though they cannot couple to single real
photons, these particles can interact with electric currents through virtual
photons. If a Majorana fermion is immersed in a background current, there is
an energy difference between the spin states of the fermion; the higher energy
state has its anapole moment antialigned with the current. In this paper, we
address the ability of a system of initially unpolarized Majorana fermions to
achieve some degree of polarization relative to a static background current.
In considering processes that allow the Majorana fermion’s spin to flip to the
lower-energy state, we focus upon two irreversible processes: the spontaneous
emission of two real photons and the emission of a single real photon emitted
in virtual Compton scattering. Both of these processes involve coupling to
photons via the fermion’s polarizaibilities. We compute the spin-flip
transition rates for these processes using a low-energy expansion of the
Hamiltonian and construct a toy model to showcase how these rates depend upon
the underlying parameters within a model. Applying these ideas to a thermal
dark matter (DM) model, we find that when the DM thermally decouples from the
Standard Model plasma in the early universe, two-photon emission is negligible
but partial polarization for the DM medium can proceed via virtual Compton
scattering if sufficient currents exist.

##  I Introduction

The self-conjugate nature of Majorana fermions severely restricts their
coupling to the electromagnetic field. For a spin-$\frac{1}{2}$ Majorana
fermion, its charge, electric dipole moment, and magnetic dipole moment must
vanish identically. Its sole static electromagnetic property is the anapole
moment Kayser (1982); Nieves (1982). A classical model of the anapole exists:
a toroidal current distribution; see, for example, Ref. Gray et al. (2010).
This system has a nonzero anapole moment along its symmetry axis, and
additionally, its charge and electric and magnetic dipole moments vanish. As a
result, the classical torus does not interact with static electric or magnetic
fields, but it does experience a torque in the presence of an external current
density, which acts to align the torus’s anapole moment with the current. This
notion carries over faithfully to the realm of particle physics. Majorana
fermions do not couple to single real photons for the same reasons that the
classical torus does not respond to electric or magnetic fields. But, as with
the torus, the interaction Hamiltonian between a fermion’s anapole moment,
$\mathbf{a}$, aligned with the particle’s spin and an external current
density, $\mathbf{J}$, is given by
$H_{\text{anapole}}=-\mathbf{a}\cdot\mathbf{J}$ Zeldovich (1957). From this
Hamiltonian, we see that particles with anapole moments aligned with a
background external current are in a lower energy state than those with
moments antialigned with the current. The background current results in an
energy difference between these two states by an amount $\Delta E=2aJ$.

Suppose a system of initially unpolarized identical spin-$\frac{1}{2}$
Majorana fermions were subjected to a uniform constant current density.
Assuming sufficient interaction, the system will come into thermal equilibrium
after some time and exhibit a degree of polarization according to the
Boltzmann distribution, $n(\uparrow)/n(\downarrow)=\exp(\Delta E/T)$ (assuming
spin-up fermions are aligned with the current). The time scale over which this
occurs is dependent upon the interaction rate for processes that admit a spin
flip of the Majorana fermion. The dominant mode of interaction is through the
exchange of a single photon with the current, which, at the individual
particle level, is a reversible process. The ability to polarize the system
via such reversible processes crucially relies upon sufficient interactions
with the environment to achieve thermalization. But, in addition to reversible
interactions, there are also effectively irreversible processes that allow the
Majorana fermion to flip to the lower-energy state with the anapole moment
aligned with the external current. We will study these subdominant
irreversible processes in this paper and take up the reversible anapole
interactions in subsequent work.

For a Majorana fermion with anapole moment antialigned with an external
current, photon emission is one irreversible mechanism in which the fermion
can flip its spin to the lower energy state; however, spontaneous emission of
a single photon is not possible because anapoles cannot couple to a single
real photon. But a Majarona fermion can couple to two real photons via its
model-dependent electric and magnetic polarizabilities Radescu (1985). Through
these polarizabilities, spontaneous emission of two photons is possible.

Generically, a spin-$\frac{1}{2}$ fermion has sixteen independent
polarizabilties, which can be classified by how the two-photon interaction
transforms under parity ($\mathcal{P}$) and time reversal ($\mathcal{T}$)
transformations. There are six polarizabilities that are separately invariant
under $\mathcal{P}$\- and $\mathcal{T}$-transformations Prange (1958) and an
additional four polarizabilities that are $\mathcal{P}$-odd and
$\mathcal{T}$-even Chen et al. (2001); the remaining six polarizabilities are
$\mathcal{P}$-odd and $\mathcal{T}$-odd Gorchtein (2008). The Majorana
fermion’s self-conjugate nature constrains this list somewhat, requiring the
four $\mathcal{P}$-odd and $\mathcal{T}$-even polarizabilities to vanish
Latimer (2016). This leaves a dozen modes through which Majorana fermions can
interact with two real photons. In this work, we will focus upon only those
which are $\mathcal{P}$\- and $\mathcal{T}$-even because the other couplings
are likely to be suppressed (given that they require $\mathcal{P}$\- and
$\mathcal{T}$\- violation). Of these six $\mathcal{P}$\- and
$\mathcal{T}$-even polarizabilities, two are spin independent – the familiar
electric and magnetic polarizabilites, while four are spin dependent Ragusa
(1993, 1994). It is these spin-dependent polarizabilities, that will admit a
spin flip of the Majorana fermion via the spontaneous emission of two real
photons.

The study of particle polarizabilities has a long history in nuclear physics
in the context of real Compton scattering (RCS) Prange (1958); Jacob and
Mathews (1960); Hearn and Leader (1962); Bardeen and Tung (1968); Bernard et
al. (1991); Hemmert et al. (1998); Babusci et al. (1998); Holstein et al.
(2000); Bedaque and Savage (2000); Chen et al. (2001); Bernard et al. (2003);
Hildebrandt et al. (2004); Holstein et al. (2005); Gorchtein (2008) because
photons can be used as a clean probe of nucleon structure. More recently,
virtual Compton scattering (VCS) has emerged as an additional probe Guichon et
al. (1995); Vanderhaeghen (1996); Scherer et al. (1996); Metz and Drechsel
(1996); Pasquini and Boffi (1996); Drechsel et al. (1997, 1998); Hemmert et
al. (1997, 2000); L’vov et al. (2001); Gorchtein (2010). In VCS, the incoming
photon that interacts with the target nucleon is virtual with four momentum
satisfying $q^{2}=-Q^{2}<0$ rather than $q^{2}=0$ as in RCS. In the low energy
limit, one may describe VCS via an effective interaction using generalized
polarizabilities which are now functions of the initial photon’s four
momentum, $Q^{2}$. Focusing upon only the $\mathcal{P}$\- and
$\mathcal{T}$-even interactions, the ostensibly ten generalized
polarizabilities Guichon et al. (1995) can be reduced to six after applying
charge conjugation and crossing symmetries Drechsel et al. (1998). These six
terms should match up with the RCS polarizabilities in the limit in which the
initial photon becomes real, but care must be taken in how this limit is
realized. The low energy expansion of the VCS amplitude developed by Gorchtein
Gorchtein (2010) provides a scheme that continuously connects the VCS and RCS
polarizabilities.

Given the presence of a background current, a Majorana fermion can undergo
single photon emission by coupling to charged particles through its
generalized polarizabilities in VCS. The charged particles can interact with
the Majorana fermion via a virtual photon, and the Majorana fermion can emit a
real final-state photon in the process. As with RCS, some of the generalized
polarizabilites are spin dependent and, thus, able to effect a spin flip. We
will compute the transition rate for this process as well as spontaneous two-
photon emission.

In what follows, we first introduce the effective anapole interaction between
the Majorana fermion and background current and construct a toy model to show
how this interaction emerges from a more complete theory. We then consider the
effective interaction between two real photons and the Majorana fermion.
Focusing upon RCS in the low-energy limit, we compute the Majorana fermion’s
polarizabilities. With these, we determine the spontaneous two-photon emission
rate for a fermion undergoing a spin flip to the lower energy state. Finally
we move on to a discussion of the low-energy expansion of the VCS amplitude.
We follow Gorchtein’s scheme in Ref. Gorchtein (2010) to determine the
generalized polarizabilities for our toy model. Gorchtein uses the Breit
frame, so we shift the results over to the lab frame in order to allow us to
compute the interaction rate for spin-flip under single photon emission. This
process is not spontaneous, so we also determine the rate at which the
Majorana fermion’s anapole moment transitions from aligned to antialigned with
the current through VCS.

The ability of a current to polarize a collection of Majorana fermions is
intrinsically interesting, but it could have consequential relevance to
neutrino physics or dark matter (DM) models Ho and Scherrer (2013); Gao et al.
(2014); Anchordoqui et al. (2015); Chua and Wong (2016); Kumar et al. (2016);
Ahmed et al. (2018); Neves et al. (2021). Because we only consider Majorana
fermions that are non-relativistic, our results are not directly applicable to
neutrinos, though generalizing our results to relativistic particles could be
an interesting avenue to pursue. But, for DM thermal relics, once the universe
cools to the point at which the DM decouples from the plasma of Standard Model
particles, local electric currents could result in regions of polarized DM
that would persist to present day. Using reasonable parameters for the toy
model, we estimate the rate at which two-photon emission and VCS occur around
the time that DM decouples from the thermal bath. We find that two-photon
emission is essentially forbidden in this era, but the VCS process could be
appreciable in the presence of a sufficiently large, yet still non-
relativistic, current.

This possibility has implications for indirect DM detection experiments. In
such experiments, the detection of high-energy Standard Model particles could
be ascribed to DM annihilation, if the detected particles have no other
astrophysical origin. Majorana fermions can rather generically annihilate into
photons if they are in an $s$-wave state Latimer (2017); Chua and Wong (2016),
though sometimes this mode is naturally suppressed Kumar et al. (2016); Ho and
Scherrer (2013); Gao et al. (2014). Regardless, non-observation of high-energy
photons from DM annihilation results in an upper bound on the annihilation
cross section. However, if a region of DM were partially polarized in the
early universe by a background current, this polarization would persist to the
present, and the polarization would suppress $s$-wave annihilation because
this mode requires the two annihilating DM particles to have opposite spins.
In such a region, an indirect detection experiment would result in an overly
stringent bound on the annihilation cross section if the analysis assumed an
unpolarized DM medium.

##  II Anapole moment and toy model

When considering the electromagnetic interactions of a Majorana fermion,
$\chi$, we take an effective field theory approach, expressing the Lagrangian
density in terms of the electromagnetic field strength tensor,
$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$. For the single-
photon interaction represented in Fig. 1, the effective Lagrangian is

$\mathcal{L}_{\text{anapole}}=f_{\text{a}}(q^{2})\bar{\chi}\gamma^{\mu}\gamma^{5}\chi\partial^{\nu}F_{\mu\nu}.$
(1)

The static anapole moment is determined in the limit of vanishing photon
momentum, $q\to 0$; that is, we set $f_{\text{a}}:=f_{\text{a}}(q^{2}=0)$. A
few remarks are in order. First, reflecting the fact that Eq. 1 is an
effective Lagrangian, the anapole moment has mass dimension $[M]^{-2}$.
Second, it is clear from the structure of the Lagrangian term that the
interaction vanishes for real, transverse photons given that $q^{2}=0$ and
$\epsilon\cdot q$, where $\epsilon^{\mu}$ is the photon’s polarization vector,
viz. $A^{\mu}=\epsilon^{\mu}e^{-iq\cdot x}$. Finally, the derivative of the
field tensor can be replaced with an external density
$J^{\mu}=\partial_{\nu}F^{\mu\nu}$, highlighting the coupling between the
Majorana fermion’s anapole moment and an external current.

Figure 1:  Anapole vertex for a Majorana fermion.

For our calculations, we will be working in the low-energy limit. The Majorana
fermion will be non-relativistic, $|\mathbf{p}|,|\mathbf{p}^{\prime}|\ll
m_{\chi}$, and momentum transfer will be small $|\mathbf{q}|\ll m_{\chi}$. In
this regime, the full Dirac four-spinor is not needed to describe the particle
dynamics. In the low energy limit, the spatial parts of the Lagrangian, Eq.
(1), dominate resulting in the anapole interaction Hamiltonian

$H_{\text{anapole}}=-f_{a}\bm{\sigma}\cdot\mathbf{J},$ (2)

which acts on Pauli spinors, $\xi$. In this two state system, the energy
difference between spinors antialigned and aligned with the current is
$\omega_{0}=2f_{a}J$.

A particle’s anapole moment $f_{a}$ is a model-dependent quantity whose
parameter dependences can only be determined from a more fundamental theory.
It is useful to consider a toy model to see how model parameters enter into
the effective coupling for the anapole moment and for the polarizabilities
that follow. For our toy model, we suppose the Majorana fermion couples to a
charged scalar, $\phi$, and a Dirac fermion, $\psi$. In order for the anapole
moment to be nonzero, the coupling must be parity violating. For our toy
model, we assume maximal parity violation. Explicitly, we take the interaction
term in the Lagrangian density to be

$\mathcal{L}_{\text{int}}=g\bar{\psi}\frac{1}{2}(1-\gamma^{5})\chi\phi^{*}+\mathrm{h.c.}.$
(3)

To simplify our expressions, we will also assume that the mass of the scalar
particle dominates the fermion masses.

Figure 2:  One-loop diagrams that determine the Majorana fermion’s anapole
moment.

The anapole moment can be determined from the one-loop Feynman diagrams in
Fig. 2. For the computation, we use the Feynman rules for Majorana fermions
developed in Refs. Denner et al. (1992a, b). In particular, we must consider
both orientations of the Dirac fermion flow in loop diagrams. Per the
effective Lagrangian in Eq. (1), the anapole vertex has the form
$if_{a}(q^{2}\gamma^{\mu}-\not{\hbox{\kern-3.0pt$q$}}q^{\mu})\gamma^{5}$;
therefore, to determine the anapole moment, it is sufficient to compute the
diagrams in Fig. 2 out to order $\mathcal{O}(q^{2})$. We execute the
calculations in Mathematica using Package-X 2.0 developed in Ref. Patel
(2017). The resulting moment is then given by

$f_{a}\approx\frac{eg^{2}}{(4\pi)^{2}M_{\phi}^{2}}\left[\frac{1}{3}\log\left(\frac{M_{\phi}^{2}}{m_{\psi}^{2}}\right)-\frac{1}{2}\right],$
(4)

where we keep only the leading terms in $M_{\phi}\gg m_{\chi},m_{\psi}$. We
note that the structure of this anapole moment is similar to the charge radius
of the neutrino in the analogous limits Bernabéu et al. (2000).

##  III Spontaneous two-photon emission

###  III.1 Effective Hamiltonian

To compute the interaction of the Majorana fermion with real photons, we
introduce an effective Hamiltonian that couples two photons via the fermion’s
polarizabilities. At low energies, the interactions can be expressed rather
simply in terms of the photons’ electric and magnetic field components. For
the $\mathcal{P}$\- and $\mathcal{T}$-invariant interactions, two of these
interaction terms are spin independent, while the remaining four are spin
dependent and involve derivatives of the fields. Following the parametrization
for the spin-dependent terms laid out in Ragusa Ragusa (1993, 1994), we write
the low-energy expansion (LEX) of the Hamiltonian as

$\displaystyle H_{\text{pol}}=$
$\displaystyle-\frac{1}{2}4\pi(\alpha_{E}\mathbf{E}\cdot\mathbf{E}+\beta_{M}\mathbf{B}\cdot\mathbf{B})$
$\displaystyle+4\pi\tfrac{1}{2}\gamma_{1}\mathbf{E}\cdot[\bm{\sigma}\times(\bm{\nabla}\times\mathbf{B})]-4\pi(\gamma_{2}+\gamma_{4})\mathbf{B}\cdot[\bm{\nabla}(\bm{\sigma}\cdot\mathbf{E})]$
$\displaystyle+4\pi\gamma_{3}\mathbf{E}\cdot[\bm{\nabla}(\bm{\sigma}\cdot\mathbf{B})]+4\pi\left(\tfrac{1}{2}\gamma_{2}+\gamma_{4}\
\right)\mathbf{B}\cdot[\bm{\sigma}\times(\bm{\nabla}\times\mathbf{E})],$ (5)

where the coefficients represent the various polarizabilities. This
Hamiltonian is appropriate both for real Compton scattering, as in Fig. 3(a),
and the emission of two photons, Fig. 3(b).

Figure 3:  Two-photon interactions with a Majorana fermion by virtue of the
particle’s polarizability. (a) Real Compton scattering. (b) Two-photon
emission.

Particle polarizabilities can be easily understood within the context of
classical electromagnetism. In a two-photon interaction with the Majorana
fermion, one can heuristically think of one photon’s electric or magnetic
field inducing an electric or magnetic dipole with which the second photon can
interact. The simplest examples are the spin-independent polarizabilities. For
instance, in the presence of an electric field, an electromagnetic object will
typically deform; the positively charged constituents feel a tug in the
direction of the field while the negatively charged bits feel a tug in the
opposite direction. To leading order, the field induces an electric dipole in
the same direction of the field, $\mathbf{p}=\alpha_{E}\mathbf{E}$. This
dipole will lower the energy of the system via its interaction with the field
$\sim-\mathbf{p}\cdot\mathbf{E}$, resulting in a potential term in the
Hamiltonian $-\frac{1}{2}\alpha_{E}E^{2}$. The contribution from the spin-
independent magnetic polarizability, $\beta_{M}$, is similar, contributing to
the system energy an amount $-\frac{1}{2}\beta_{M}B^{2}$. Higher order
interactions between the particle and (derivatives of) the electromagnetic
field can result in spin-dependent polarizabilities Ragusa (1993, 1994).
Because these moments involve derivatives of the fields, their contribution to
the Hamiltonian will be $\mathcal{O}(\omega^{3})$, with $\omega$ the photon
energy. For a spin-$\frac{1}{2}$ particle, the two leading order spin-
dependent electric dipole moments are
$-\frac{1}{2}\gamma_{1}\bm{\sigma}\times(\bm{\nabla}\times\mathbf{B})$ and
$-\gamma_{3}\bm{\nabla}(\bm{\sigma}\cdot\mathbf{B})$, and the corresponding
spin-dependent magnetic dipole moments are
$(\gamma_{2}+\gamma_{4})\bm{\sigma}\times(\bm{\nabla}\times\mathbf{E})$ and
$-(\tfrac{1}{2}\gamma_{2}+\gamma_{4})\bm{\nabla}(\bm{\sigma}\cdot\mathbf{E})$.
The coefficients $\gamma_{j}$ are known as the spin-dependent
polarizabilities. We note that our seemingly peculiar expressions for these
coefficients are consistent with a prevailing definition of the spin-dependent
polarizabilities in the literature, e.g., in Ref. Hemmert et al. (1998).

Before moving on to discuss two-photon emission, we will make a few comments
on the RCS process. Our reasons are twofold. First, we will explicitly show
that our expression is consistent with the leading order LEX of the RCS
amplitude found in the literature. Second, we would like to have an expression
for the RCS Hamiltonian matrix element that can be more easily compared to
that for the VCS Hamiltonian. One challenge in the literature has been to show
that the VCS result limits to the RCS case whenever the incident photon
becomes real. In Ref. Gorchtein (2010), Gorchtein finds an appropriate linear
combination of fields such that the matrix element for the RCS Hamiltonian is
recovered from his expression for the VCS Hamiltonian in the appropriate
limits. We will show this for the non-Born terms in our work. To facilitate
this comparison, we will write a matrix element for the RCS Hamiltonian in
terms of the photon momentum and polarization three-vectors, enforcing the
transverse condition
$\mathbf{q}\cdot\bm{\epsilon}=0=\mathbf{q}^{\prime}\cdot\bm{\epsilon}^{\prime}$
and the on-shell condition $q^{2}=0=q^{\prime 2}$; that is, we will write the
Hamiltonian in terms of the Fourier modes of the fields. Our expression will
be valid in both the center-of-mass frame and the Breit frame. The Breit frame
requires the incoming and outgoing fermion three-momenta satisfy
$\mathbf{p}^{\prime}=-\mathbf{p}$, and this fact forces the incident and
outgoing photon energies to be the same $q^{0}=q^{\prime 0}=:\omega$.

To set our notation, we work with the quantized vector field

$A^{\mu}(x)=\sum_{\lambda}\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\frac{1}{\sqrt{2\omega}}\left[\epsilon^{\mu}(k,\lambda)a_{\mathbf{k},\lambda}e^{-ik\cdot
x}+\epsilon^{*\mu}(k,\lambda)a^{\dagger}_{\mathbf{k},\lambda}e^{ik\cdot
x}\right],$ (6)

where we sum over polarization states $\lambda$. We work in a gauge with
$A^{0}=0$ so that the polarization vectors are purely spatial. The electric
and magnetic fields are found through the usual relations:
$\mathbf{E}=-\frac{\partial}{\partial t}\mathbf{A}$ and
$\mathbf{B}=\bm{\nabla}\times\mathbf{A}$. For the RCS process, the initial and
final real photons have momentum and polarization $(q,\epsilon)$ and
$(q^{\prime},\epsilon^{\prime})$, respectively. Omitting some normalization
factors, we express the RCS Hamiltonian matrix elements in terms of the
electromagnetic field Fourier components

$\displaystyle\langle
q^{\prime},\bm{\epsilon}^{\prime}|H_{\text{pol}}|q,\bm{\epsilon}\rangle\sim$
$\displaystyle[\omega^{2}\alpha_{E}+(\mathbf{q}\cdot\mathbf{q}^{\prime})\beta_{M}]\bm{\epsilon}\cdot\bm{\epsilon}^{\prime*}-\beta_{M}({\mathbf{q}}\cdot{\bm{\epsilon}}^{\prime*})(\mathbf{q}^{\prime}\cdot\bm{\epsilon})$
$\displaystyle+i\omega[\omega^{2}\gamma_{1}-(\mathbf{q}\cdot\mathbf{q}^{\prime})(\gamma_{2}+2\gamma_{4})][\bm{\sigma}\cdot({\bm{\epsilon}}^{\prime*}\times\bm{\epsilon})]$
$\displaystyle+i\omega\gamma_{2}(\bm{\epsilon}^{\prime*}\cdot\bm{\epsilon})[\bm{\sigma}\cdot(\mathbf{q}^{\prime}\times\mathbf{q})]$
$\displaystyle+i\omega\gamma_{4}\\{(\mathbf{q}^{\prime}\cdot\bm{\epsilon})[\bm{\sigma}\cdot(\bm{\epsilon}^{\prime*}\times\mathbf{q})]-(\mathbf{q}\cdot\bm{\epsilon}^{\prime*})[\bm{\sigma}\cdot(\bm{\epsilon}\times\mathbf{q}^{\prime})]\\}$
$\displaystyle+i\omega\gamma_{3}\\{({\bm{\epsilon}}\cdot\mathbf{q}^{\prime})[\bm{\sigma}\cdot(\bm{\epsilon}^{\prime*}\times\mathbf{q}^{\prime})]-(\bm{\epsilon}^{\prime*}\cdot\mathbf{q})[\bm{\sigma}\cdot(\bm{\epsilon}\times\mathbf{q})]\\}.$
(7)

This expression is consistent with the low-energy behavior of the RCS matrix
element in Ref. Hemmert et al. (1998), as applied to a Majorana fermion with
vanishing charge and static dipole moments.

###  III.2 Polarizabilities in the toy model

Figure 4:  One-loop diagrams that contribute to the real Compton scattering
amplitude. Both orientations of fermion flow within the loop are considered,
and the diagrams with the “crossed” photons lines are not shown.

Polarizabilities are model-dependent quantities, so we turn again to the model
introduced in Sec. II to examine how model parameters can enter into these
coefficients. A sample of the one-loop Feynman diagrams that contribute to the
RCS process is shown in Fig. 4. Again, we execute the calculations in
Mathematica using the package developed in Ref. Patel (2017). We compute the
amplitude to $\mathcal{O}(\omega^{3})$ in the Breit frame and then extract the
polarizability coefficients through comparison with the expression in Eq. (7).
Keeping only the leading order terms in this limit, we find the Majorana
fermion’s spin-independent polarizabilities, $\alpha_{E}$ and $\beta_{M}$, and
spin-dependent polarizabilities, $\gamma_{j}$, in the Breit (or CM) frame to
be

$\displaystyle\alpha_{E}\approx$
$\displaystyle\frac{e^{2}g^{2}}{(4\pi)^{3}}\frac{m_{\chi}}{M_{\phi}^{4}}\Bigg{[}\frac{2}{3}\log\left(\frac{M_{\phi}^{2}}{m_{\psi}^{2}}\right)-\frac{5}{6}\Bigg{]},$
(8) $\displaystyle\beta_{M}\approx$
$\displaystyle\frac{e^{2}g^{2}}{(4\pi)^{3}}\frac{m_{\chi}}{M_{\phi}^{4}}\Bigg{[}\frac{2}{3}\log\left(\frac{M_{\phi}^{2}}{m_{\psi}^{2}}\right)-\frac{13}{6}\Bigg{]},$
(9) $\displaystyle\gamma_{1}\approx$
$\displaystyle-\frac{e^{2}g^{2}}{(4\pi)^{3}}\frac{1}{3}\frac{1}{M_{\phi}^{2}m_{\psi}^{2}},$
(10) $\displaystyle\gamma_{2}\approx$
$\displaystyle\frac{e^{2}g^{2}}{(4\pi)^{3}}\frac{1}{M_{\phi}^{4}}\Bigg{[}\frac{2}{3}\log\left(\frac{M_{\phi}^{2}}{m_{\psi}^{2}}\right)-\frac{3}{2}\Bigg{]},$
(11) $\displaystyle\gamma_{3}\approx$
$\displaystyle\frac{e^{2}g^{2}}{(4\pi)^{3}}\frac{1}{6}\frac{1}{M_{\phi}^{2}m_{\psi}^{2}},$
(12) $\displaystyle\gamma_{4}\approx$
$\displaystyle-\frac{e^{2}g^{2}}{(4\pi)^{3}}\frac{1}{6}\frac{1}{M_{\phi}^{2}m_{\psi}^{2}},$
(13)

where we consider the approximation in which the scalar mass dominates
$m_{\psi},m_{\chi}\ll M_{\phi}$. These values are consistent with the
independent calculation in Ref. Latimer (2016).

###  III.3 Rate of spontaneous two-photon emission

We now turn to the spontaneous emission of two photons, Fig. 3(b), from a
Majorana fermion immersed in a steady background current $\mathbf{J}$. We take
the the current to be in the $z$-direction. Spin-down Majorana fermions,
$\xi_{-}$, will have an energy of $\omega_{0}=2f_{a}J$ greater than a spin-up,
$\xi_{+}$, counterparts. We will assume that the Majorana fermion is at rest
relative to the current both before and after photon emission; that is, we
neglect recoil effects which are small because $m_{\chi}\gg\omega_{0}$.
Shifting the calculation to the particle rest frame (from the previously
discussed Breit or CM frame) has one additional implication. The
polarizability coefficients that appear in the low-energy Hamiltonian, Eq.
(5), for two-photon interactions are frame dependent; however, the corrections
to the polarizabilities incurred in boosting from the the Breit frame to the
particle’s rest frame are sub-dominant and thus negligible when considering
only leading order terms. Given this, we employ the polarizabilities
calculated in Eqs. (8-13).

To compute the transition rate for spontaneous emission of two photons, we use
Fermi’s golden rule

$\mathrm{d}^{6}\Gamma_{\text{spont}}=2\pi|\langle
f|H_{\text{pol}}|i\rangle|^{2}\rho_{\text{f}}.$ (14)

Here the initial state is a spin down Majorana fermion with $\mathbf{p}=0$,
and the final state consists of a spin-up fermion with $\mathbf{p}^{\prime}=0$
along with two photons with momenta and polarizations $q_{1},q_{2}$ and
$\bm{\epsilon}_{1}^{(\alpha)},\bm{\epsilon}_{2}^{(\beta)}$, respectively. For
our calculations, we will sum over final photon polarizations, $\alpha$ and
$\beta$, and integrate over the density of photon final states

$\rho_{f}=\prod_{j=1}^{2}\frac{\mathrm{d}^{3}q_{j}}{(2\pi)^{3}}.$ (15)

Because the transition from initial to final state involves a spin flip for
the Majorana fermion, the spin-independent terms in the Hamiltonian, Eq. (5),
have vanishing matrix element. As such, there are four contributions to the
amplitude coming from the spin-dependent terms . As an example, let us
consider in some detail just the first spin-dependent electric polarizability:
$H_{\gamma_{1}}=4\pi\tfrac{1}{2}\gamma_{1}\mathbf{E}\cdot[\bm{\sigma}\times(\bm{\nabla}\times\mathbf{B})]$.
The matrix element for this process is

$\displaystyle\langle\text{f}|H_{{\gamma}_{1}}|\text{i}\rangle=$
$\displaystyle
i\gamma_{1}\frac{\pi}{\sqrt{\omega_{1}\omega_{2}}}(\omega_{1}\omega_{2}^{2}-\omega_{2}\omega_{1}^{2})\bm{\sigma}_{\text{fi}}\cdot\left({\bm{\epsilon}_{2}^{(\beta)*}}\times{\bm{\epsilon}_{1}^{(\alpha)*}}\right),$
(16)

where we set
$\bm{\sigma}_{\text{fi}}=\langle\xi_{+}|\bm{\sigma}|\xi_{-}\rangle$. To
compute the total transition rate for this, we integrate over the final phase
space of the photons and sum over photon polarizations. For a transition
mediated solely by the $\gamma_{1}$ term in Eq. (5), we compute a spin-flip
rate

${\Gamma_{\text{spont}}}_{\gamma_{1}}=\frac{2\gamma_{1}^{2}{\omega_{0}}^{9}}{2835\pi}.$
(17)

The other contributions to two-photon spontaneous emission are computed in a
similar fashion. Generally, we must sum over these so that possible
interfering terms can be given proper account. After doing so (and summing
over photon polarization states), we find the spontaneous two-photon emission
transition rate to be

$\Gamma_{\text{spont}}=(2\gamma_{1}^{2}+10\gamma_{2}^{2}-3\gamma_{2}\gamma_{3}+12\gamma_{3}^{2}+20\gamma_{2}\gamma_{4}-2\gamma_{3}\gamma_{4}+12\gamma_{4}^{2}+\gamma_{1}\gamma_{2}+4\gamma_{1}\gamma_{3}+3\gamma_{1}\gamma_{4})\frac{\omega_{0}^{9}}{2835\pi}.$
(18)

From the model-depend polarizabilities, Eq. (13), we see that at leading order
$\gamma_{2}$ is smaller than the remaining spin-dependent coefficients by a
factor of $m_{\psi}^{2}/M_{\phi}^{2}$. For what remains, we define
$\gamma:=\gamma_{3}$ and find $\gamma_{1}=-2\gamma$ along with
$\gamma_{4}=-\gamma$. Neglecting $\gamma_{2}$, we can approximate the
transition rate for this model as

$\Gamma_{\text{spont}}=\frac{32\gamma^{2}\omega_{0}^{9}}{2835\pi}.$ (19)

##  IV Single-photon emission via VCS

###  IV.1 Effective Hamiltonian

We now consider single-photon emission mediated by the interaction with a
charged particle, coupled to the Majorana fermion via generalized
polarizabilties. Though spontaneous emission of a single photon is not
possible, a Majorana fermion can emit a single real photon in the virtual
Compton scattering process depicted in Fig. 5. Here, the Majorana fermion
interacts with a charged fermion, represented by $\Psi$ with charge $e$. The
amplitude for this process is

$\mathcal{M}_{\text{VCS}}=ie\bar{u}(k^{\prime})\gamma_{\mu}u(k)\frac{1}{q^{2}}\mathcal{T}^{\mu\nu}_{\text{VCS}}\epsilon^{\prime*}_{\nu}(q^{\prime},\lambda^{\prime}),$
(20)

where $\mathcal{T}^{\mu\nu}_{\text{VCS}}$ is the VCS tensor that encodes the
physics within the shaded vertex in Fig. 5 and $q=k-k^{\prime}$.

Figure 5:  Single-photon emission from a Majorana fermion via virtual Compton
scattering.

We can actually cast this amplitude in a more suggestive form by projecting
the Dirac bilinear from the charged fermion, $\Psi$, onto a set of three
orthogonal normalized vectors, $\epsilon^{\mu}(q,\lambda)$,

$\bar{u}(k^{\prime})\gamma^{\mu}u(k)=\sum_{\lambda}\Omega_{\lambda}\epsilon^{\mu}(q,\lambda).$
(21)

To do so, we first note that the off-shell photon momentum, $q$, is orthogonal
to Dirac bilinear, $\bar{u}(k^{\prime})\not{\hbox{\kern-3.0pt$q$}}u(k)=0$,
because on-shell spinors satisfy
$\not{\hbox{\kern-3.0pt$k$}}u(k)=m_{\Psi}u(k)$. Given this, two of the of the
vectors, $\epsilon^{\mu}(q,\lambda=\pm)$, can be chosen to be wholly spatial.
Setting $q^{\mu}=(\omega,\mathbf{q})$, these two basis vectors are
$\epsilon^{\mu}(q,\lambda=\pm)=(0,\bm{\epsilon}_{\pm})$ with the spatial part
orthogonal to the photon’s three-momentum,
$\bm{\epsilon}_{\pm}\cdot\mathbf{q}=0$. For normalized $\bm{\epsilon}_{\pm}$,
then these two four-vectors have norm
$\epsilon^{\mu}(q,\lambda=\pm)\epsilon_{\mu}(q,\lambda=\pm)=-1$. The third
vector, $\epsilon^{\mu}(q,\lambda=0)$, must be time-like because the off-shell
photon’s momentum is spatial, $q^{2}:=-Q^{2}<0$, viz. $|\mathbf{q}|>\omega$.
Then, the longitudinal polarization vector is given by
$\epsilon^{\mu}(q,\lambda=0)=\frac{1}{\sqrt{Q^{2}}}(|\mathbf{q}|,\omega\hat{\mathbf{q}})$.
This has norm $\epsilon^{\mu}(q,\lambda=0)\epsilon_{\mu}(q,\lambda=0)=+1$. The
Dirac blinear can then be decomposed as in Eq. (21) where the coefficients are
$\Omega_{\lambda}=(\pm)\bar{u}(k^{\prime})\gamma^{\mu}u(k)\epsilon_{\mu}(q,\lambda)$
with the $+$ sign taken for $\lambda=0$ and $-$ for $\lambda=\pm$. With this
change, we can write the VCS amplitude as

$\mathcal{M}_{\text{VCS}}=\frac{ie}{q^{2}}\sum_{\lambda}\Omega_{\lambda}\epsilon_{\mu}(q,\lambda)\mathcal{T}^{\mu\nu}_{\text{VCS}}\epsilon^{\prime*}_{\nu}(q^{\prime},\lambda^{\prime}).$
(22)

In Fig. 5, the sub-diagram involving the two photons and Majorana fermion is
identical to that of RCS, in Fig. 3(a), except for the fact that the incident
photon is virtual. Because the incident photon can have a longitudinal mode,
the contraction between $\epsilon^{\mu}(q,\lambda)$ and $q^{\mu}$ is not
necessarily zero; viz., $\epsilon\cdot q\neq 0$. This can result in additional
structures present in the low-energy expansion of the VCS Hamiltonian, but in
the end, one should be able to recover the RCS Hamiltonian in the limit in
which the incident photon becomes real. With an appropriate linear combination
of terms, this limiting process has been achieved by Gorchtein Gorchtein
(2010).

To develop the LEX for the VCS Hamiltonian, Gorchtein finds it convenient to
work in the Breit frame, which requires that the Majorana fermion’s momentum
pre- and post-interaction satisfy $\mathbf{p}^{\prime}=-\mathbf{p}$. This
choice renders the same value for initial and final photon energies, $\omega$,
which could not be achieved in, say, the center-of-mass frame because
$q^{2}\neq 0$. Also, Gorchtein opts to work in the Coulomb gauge with spatial
polarization vectors. Recall, above in the Lorenz gauge, the longitudinal mode
$\epsilon^{\mu}(q,\lambda=0)$ was time-like with $\epsilon\cdot q=0$. Gauge
invariance allows us to redefine this vector so that it only has spatial
components. This does not impact the overall VCS amplitude by virtue of the
Ward identity, $q_{\mu}\mathcal{T}^{\mu\nu}=0$. From
$\epsilon^{\mu}(q,\lambda=0)$, we subtract
$\frac{|\mathbf{q}|}{\omega\sqrt{Q^{2}}}q^{\mu}$ defining the longitudinal
polarization vector as
$\tilde{\epsilon}^{\mu}(q,\lambda=0)=-\frac{\sqrt{Q^{2}}}{\omega}(0,\hat{\mathbf{q}})$.
We note that this new vector, $\tilde{\epsilon}^{\mu}(q,\lambda=0)$, is no
longer normalized nor orthogonal to $q^{\mu}$. Going forward, we will use the
three spatial polarization vectors
$\\{\bm{\epsilon}_{\pm},\bm{\epsilon}_{0}\\}$, dropping the tilde notation for
the longitudinal mode.

With these conventions, Gorchtein determines the leading order contribution to
the VCS Hamiltonian matrix element in a LEX is

$\displaystyle H_{\text{VCS}}\sim$ $\displaystyle
4\pi\alpha_{E}(Q^{2})\mathbf{E}\cdot\mathbf{E}^{\prime}+4\pi\beta_{M}(Q^{2})\mathbf{B}\cdot\mathbf{B}^{\prime}$
$\displaystyle+4\pi
i[\gamma_{1}(Q^{2})-\gamma_{2}(Q^{2})-2\gamma_{4}(Q^{2})]\frac{1}{2}\\{\bm{\sigma}\cdot[\mathbf{E}\times(\mathbf{q}^{\prime}\times\mathbf{B}^{\prime})]-\bm{\sigma}\cdot[\mathbf{E}^{\prime}\times(\mathbf{q}\times\mathbf{B})]\\}$
$\displaystyle+4\pi
i\gamma_{2}(Q^{2})\frac{1}{2}\\{\bm{\sigma}\cdot[\mathbf{q}^{\prime}\times(\mathbf{E}^{\prime}\times\mathbf{B})]-\bm{\sigma}\cdot[\mathbf{q}\times(\mathbf{E}\times\mathbf{B}^{\prime})]\\}$
$\displaystyle+4\pi
i\left[\frac{1}{2}\gamma_{2}(Q^{2})+\gamma_{3}(Q^{2})+\gamma_{4}(Q^{2})\right]\\{(\bm{\sigma}\cdot\mathbf{B})(\mathbf{q}\cdot\mathbf{E}^{\prime})-(\bm{\sigma}\cdot\mathbf{B}^{\prime})(\mathbf{q}^{\prime}\cdot\mathbf{E})\\}$
$\displaystyle-4\pi
i\left[\frac{1}{2}\gamma_{2}(Q^{2})+\gamma_{4}(Q^{2})\right](\bm{\sigma}\cdot\bm{\Delta})\\{\mathbf{B}\cdot\mathbf{E}^{\prime}+\mathbf{E}\cdot\mathbf{B}^{\prime}\\},$
(23)

where the spin-independent, $\alpha_{E}(Q^{2}),\beta_{M}(Q^{2})$, and spin-
dependent, $\gamma_{j}(Q^{2})$, generalized polarizabilities are functions of
the initial photon’s invariant “mass”, $Q^{2}$, and
$\Delta^{\mu}:=q-q^{\prime}=p^{\prime}-p$. Gorchtein’s definitions of
$\gamma_{j}(Q^{2})$ mirror their analogues in common RCS parameterizations,
e.g., see Ref. Hemmert et al. (1998). To aid comparison with RCS, we will
write this expression wholly in terms of the electromagnetic Fourier modes
(still within the Breit frame)

$\displaystyle H_{\text{VCS}}\sim$
$\displaystyle(\omega^{2}\alpha_{E}+(\mathbf{q}\cdot\mathbf{q}^{\prime})\beta_{M})\bm{\epsilon}\cdot\bm{\epsilon}^{\prime*}-\beta_{M}({\bm{\epsilon}}^{\prime*}\cdot{\mathbf{q}})(\bm{\epsilon}\cdot\mathbf{q}^{\prime})$
$\displaystyle+i\omega[\tfrac{1}{2}(|\mathbf{q}|^{2}+|\mathbf{q}^{\prime}|^{2})\gamma_{1}-(\mathbf{q}\cdot\mathbf{q}^{\prime})(\gamma_{2}+2\gamma_{4})][\bm{\sigma}\cdot({\bm{\epsilon}}^{\prime*}\times\bm{\epsilon})]$
$\displaystyle+i\omega\gamma_{2}(\bm{\epsilon}^{\prime*}\cdot\bm{\epsilon})[\bm{\sigma}\cdot(\mathbf{q}^{\prime}\times\mathbf{q})]$
$\displaystyle+i\omega\gamma_{4}\\{(\bm{\epsilon}\cdot\mathbf{q}^{\prime})[\bm{\sigma}\cdot(\bm{\epsilon}^{\prime*}\times\mathbf{q})]-(\bm{\epsilon}^{\prime*}\cdot\mathbf{q})[\bm{\sigma}\cdot(\bm{\epsilon}\times\mathbf{q}^{\prime})]\\}$
$\displaystyle+i\omega\gamma_{3}\\{({\bm{\epsilon}}\cdot\mathbf{q}^{\prime})[\bm{\sigma}\cdot(\bm{\epsilon}^{\prime*}\times\mathbf{q}^{\prime})]-(\bm{\epsilon}^{\prime*}\cdot\mathbf{q})[\bm{\sigma}\cdot(\bm{\epsilon}\times\mathbf{q})]\\}$
$\displaystyle-i\omega\tfrac{1}{2}\gamma_{1}(\bm{\epsilon}\cdot\mathbf{q})[\bm{\sigma}\cdot(\bm{\epsilon}^{\prime*}\times\mathbf{q})]+i\omega(\tfrac{1}{2}\gamma_{2}+\gamma_{4})(\bm{\epsilon}\cdot\mathbf{q})[\bm{\sigma}\cdot(\bm{\epsilon}^{\prime*}\times\mathbf{q}^{\prime})],$
(24)

where we suppress the $Q^{2}$ dependence of the generalized polariziabilities.
Note that the last two terms in Eq. (24) vanish for a transverse photon.
Additionally, if the incident photon were real, then in the Breit (and COM)
frame, we would have $|\mathbf{q}|^{2}=|\mathbf{q}^{\prime}|^{2}=\omega^{2}$.

###  IV.2 Generalized polarizabilities in the toy model

Before computing the spin-flip transition rate mediated by the VCS vertex, we
first compute the generalized polarizabilities for the toy model introduced in
Sec. II. The Feynman diagrams that contribute to the generalized
polarizabilities are the same as those in RCS; see Fig. 4. The only
modifications are that we set $q^{2}=-Q^{2}$ and $q\cdot\epsilon\neq 0$. We
work in the Breit frame so that we may easily compare the resulting
polarizabilities with those from RCS. As before, we use Package-X 2.0 in
Mathematica to compute the diagrams Patel (2017) and carry out the low energy
expansion. We treat both $q$ and $q^{\prime}$ as small variables and carry all
computations out to fifth order in these small quantities before truncating.
To simplify expressions, we also assume $M_{\phi}\gg m_{\psi},m_{\chi}$,
keeping at most terms that are $\mathcal{O}(M_{\phi}^{-4})$.

For our toy model, we find the LEX of the VCS tensor in the Breit frame to be

$\displaystyle\mathcal{T}^{\mu\nu}_{\text{VCS}}\epsilon_{\mu}\epsilon^{\prime*}_{\nu}=$
$\displaystyle\frac{e^{2}g^{2}}{(4\pi)^{2}}\Bigg{\\{}\frac{m_{\chi}^{2}}{M_{\phi}^{4}}\left[\left(\frac{4}{3}\log\left(\frac{M_{\phi}^{2}}{m_{\psi}^{2}}\right)-\frac{5}{3}\right)\omega^{2}+\left(\frac{4}{3}\log\left(\frac{M_{\phi}^{2}}{m_{\psi}^{2}}\right)-\frac{13}{3}\right)\mathbf{q}\cdot\mathbf{q}^{\prime}\right](\bm{\epsilon}\cdot\bm{\epsilon}^{\prime})$
$\displaystyle-\frac{m_{\chi}^{2}}{M_{\phi}^{4}}\left[\frac{4}{3}\log\left(\frac{M_{\phi}^{2}}{m_{\psi}^{2}}\right)-\frac{13}{3}\right]({\bm{\epsilon}}^{\prime*}\cdot{\mathbf{q}})(\bm{\epsilon}\cdot\mathbf{q}^{\prime})$
$\displaystyle+i\omega\left[-\frac{2}{3}\frac{m_{\chi}}{m_{\psi}^{2}M_{\phi}^{2}}\left(\tfrac{1}{2}|\mathbf{q}|^{2}+\tfrac{1}{2}|\mathbf{q}^{\prime}|^{2}-\mathbf{q}\cdot\mathbf{q}^{\prime}\right)\right][\bm{\sigma}\cdot({\bm{\epsilon}}^{\prime*}\times\bm{\epsilon})]$
$\displaystyle+i\omega\frac{m_{\chi}}{M_{\phi}^{4}}\left[\frac{4}{3}\log\left(\frac{M_{\phi}^{2}}{m_{\psi}^{2}}\right)-3\right](\bm{\epsilon}^{\prime*}\cdot\bm{\epsilon})[\bm{\sigma}\cdot(\mathbf{q}^{\prime}\times\mathbf{q})]$
$\displaystyle-i\omega\frac{1}{3}\frac{m_{\chi}}{m_{\psi}^{2}M_{\phi}^{2}}\\{(\bm{\epsilon}\cdot\mathbf{q}^{\prime})[\bm{\sigma}\cdot(\bm{\epsilon}^{\prime*}\times\mathbf{q})]-(\bm{\epsilon}^{\prime*}\cdot\mathbf{q})[\bm{\sigma}\cdot(\bm{\epsilon}\times\mathbf{q}^{\prime})]\\}$
$\displaystyle+i\omega\frac{1}{3}\frac{m_{\chi}}{m_{\psi}^{2}M_{\phi}^{2}}\\{({\bm{\epsilon}}\cdot\mathbf{q}^{\prime})[\bm{\sigma}\cdot(\bm{\epsilon}^{\prime*}\times\mathbf{q}^{\prime})]-(\bm{\epsilon}^{\prime*}\cdot\mathbf{q})[\bm{\sigma}\cdot(\bm{\epsilon}\times\mathbf{q})]\\}$
$\displaystyle-i\omega\frac{2}{3}\frac{m_{\chi}}{m_{\psi}^{2}M_{\phi}^{2}}(\bm{\epsilon}\cdot\mathbf{q})[\bm{\sigma}\cdot(\bm{\epsilon}^{\prime*}\times\mathbf{q})]+i\omega\frac{2}{3}\frac{m_{\chi}}{m_{\psi}^{2}M_{\phi}^{2}}(\bm{\epsilon}\cdot\mathbf{q})[\bm{\sigma}\cdot(\bm{\epsilon}^{\prime*}\times\mathbf{q}^{\prime})].\Bigg{\\}}$
(25)

In the limit in which the incident photon becomes real, viz. $q^{2}=0$ and
$\epsilon\cdot q=0$, we find the VCS amplitude is equivalent to the RCS
amplitude, with the VCS generalized polarizabilities collapsing to the RCS
ones calculated above; that is, $\alpha_{E}(Q^{2}=0)$, $\beta_{M}(Q^{2}=0)$,
and $\gamma_{j}(Q^{2}=0)$ correspond to their RCS counterparts in Eqs. (8-13).
However, the additional structures that are present for the longitudinal modes
of the incident photon, those proportional to $\bm{\epsilon}\cdot\mathbf{q}$,
do not bear out the expectations from Gorchtein’s LEX in Eq. (24). From Eq.
(24), we expect the new VCS terms, relative to the RCS amplitude, to be
$i\omega(-\tfrac{1}{2}\gamma_{1})(\bm{\epsilon}\cdot\mathbf{q})[\bm{\sigma}\cdot(\bm{\epsilon}^{\prime*}\times\mathbf{q})]+i\omega(\tfrac{1}{2}\gamma_{2}+\gamma_{4})(\bm{\epsilon}\cdot\mathbf{q})[\bm{\sigma}\cdot(\bm{\epsilon}^{\prime*}\times\mathbf{q}^{\prime})]$.
Referring to the polarizabilities in our toy model, Eqs (10-13), it is clear
that we need to add two additional terms to Eq. (23): $4\pi
i\gamma_{5}\omega(\bm{\epsilon}\cdot\mathbf{q})[\bm{\sigma}\cdot(\bm{\epsilon}^{\prime}\times\mathbf{q})]$
and $4\pi
i\gamma_{6}\omega(\bm{\epsilon}\cdot\mathbf{q})[\bm{\sigma}\cdot(\bm{\epsilon}^{\prime}\times\mathbf{q}^{\prime})]$.
In our toy model, these new coefficients are found to be
$\gamma_{5}\approx-\frac{e^{2}g^{2}}{(4\pi)^{3}}\frac{1}{2}\frac{1}{M_{\phi}^{2}m_{\psi}^{2}}$
and
$\gamma_{6}\approx\frac{e^{2}g^{2}}{(4\pi)^{3}}\frac{1}{2}\frac{1}{M_{\phi}^{2}m_{\psi}^{2}}$.

###  IV.3 Rate of single-photon emission via VCS

With these generalized polarizabilities in hand, we can now compute the spin
flip transition rate for the Majorana fermion in a background current
$\mathbf{J}=J\hat{\mathbf{z}}$ that proceeds via the VCS Hamiltonian, Fig. 5.
As before, we note that spin-down Majorana fermions, $\xi_{-}$, will have an
energy of $\omega_{0}=2f_{a}J$ greater than spin-up, $\xi_{+}$, counterparts.
We, again, work in the rest frame of the Majorana fermion. Because the
generalized polarizabilities are frame dependent, they will pick up
corrections when boosting from the Breit frame to the particle’s rest frame;
in particular, there are terms that are
$\mathcal{O}\left(\frac{Q^{2}}{m_{\chi}^{2}}\right)$. These corrections are
subdominant, so we neglect them here. Additionally, the VCS amplitude in the
rest frame is modified because $\omega\neq\omega^{\prime}$, which we
accommodate.

The transition rate for the spin-flip process is give by

$\Gamma_{\text{VCS}}=n_{\Psi}v_{\Psi}\sigma_{\text{VCS}},$ (26)

where $n_{\Psi}$ is the number density of charged fermions incident upon the
Majorana fermion, $v_{\Psi}$ their average speed, and $\sigma_{\text{VCS}}$
the total cross section for the VCS process. To compute the total cross
section, we sum over the final photon polarization states $\lambda^{\prime}$
and integrate over the final available phase space for the charged fermion
$\Psi$, real photon, and Majorana fermion.

In our calculations, we will assume that the charged fermion is relativistic
so that the particle’s energy can be equated with the magnitude of its
momentum, $E_{\Psi}\approx|\mathbf{k}|$. Likewise, for the charged fermion
post-interaction, we will neglect the particle’s mass. We will also assume the
charged fermion to be unpolarized and sum over its final spin states. After
integrating over the Majorana fermion’s final momentum, $\bm{\Delta}$, the
cross section becomes

$\mathrm{d}^{6}\sigma_{\text{VCS}}=\frac{\mathrm{d}^{3}k^{\prime}\,\mathrm{d}^{3}q^{\prime}}{(4\pi)^{5}v_{\Psi}m_{\chi}^{2}|\mathbf{k}||\mathbf{k}^{\prime}|\omega^{\prime}}\delta({\textstyle\sum}E_{i}-E_{f})\big{|}\textstyle\sum_{\lambda^{\prime}}\mathcal{M}_{\text{VCS}}\big{|}^{2},$
(27)

with the VCS amplitude expressed previously in Eq. (22), assuming now that we
average over (sum over) the initial (final) charged fermion’s spin states. We
have made some additional kinematical simplifications. The Majorana fermion is
taken to be at rest initially, and post-interaction we assume the impact on
recoil to be minimal so that $p^{\prime\mu}\approx(m_{\chi},\bm{\Delta})$.
Overall, momentum conservation in the interaction yields
$\mathbf{k}-\mathbf{k}^{\prime}=\mathbf{q}=\mathbf{q}^{\prime}+\bm{\Delta}$
while energy conservation simplifies to
$|\mathbf{k}|=|\mathbf{k}^{\prime}|+\omega^{\prime}-\omega_{0}$.

The portion of the full VCS amplitude coming from the VCS tensor can be cast
in terms of the generalized polarizabilities using our results from Eq. (25).
Recall, the polarization vectors, $\bm{\epsilon}$, in Eq. (25) refer to the
virtual photon and, in fact, we sum over the three-polarization states for the
full VCS amplitude. Working in the Coulomb gauge, this sum can be written as
$\sum_{\lambda}\Omega_{\lambda}\epsilon^{\mu}(q,\lambda)=\tilde{K}^{\mu}$
where
$\tilde{K}^{j}=\bar{u}(k^{\prime})\gamma^{j}u(k)-\bar{u}(k^{\prime})\gamma^{0}u(k)\omega^{-1}q^{j}$
and the temporal component vanishes by construction, $\tilde{K}^{0}=0$. The
leading order contribution to a spatial component of $\tilde{K}$ is given by
the second term,
$\tilde{K}^{j}\approx-\bar{u}(k^{\prime})\gamma^{0}u(k)\omega^{-1}q^{j}$; that
is, the longitudinal polarization of the virtual photon is dominant in this
process. Focusing only upon the spin-dependent processes for the Majorana
fermion, the square of the amplitude becomes

$\displaystyle|\mathcal{M}_{\text{VCS}}|^{2}=$
$\displaystyle\bigg{|}\frac{1}{2}\frac{e}{q^{2}}\sum_{\lambda,s,s^{\prime}}\Omega_{\lambda}\epsilon_{\mu}(q,\lambda){\mathcal{T}^{\mu\nu}_{\text{VCS}}}_{\text{spin}}\epsilon^{\prime*}_{\nu}\bigg{|}^{2}$
(28) $\displaystyle\approx$
$\displaystyle(|\mathbf{k}||\mathbf{k}^{\prime}|+\mathbf{k}^{\prime}\cdot\mathbf{k})\left(8\pi
m_{\chi}\frac{e}{q^{2}}\right)^{2}\bigg{|}\sum_{\lambda^{\prime}}\Big{\\{}[\gamma_{5}|\mathbf{q}|^{2}+\tfrac{1}{2}\gamma_{1}|\mathbf{q}^{\prime}|^{2}-(\gamma_{2}+\gamma_{4})(\mathbf{q}\cdot\mathbf{q}^{\prime})][\bm{\sigma}\cdot({\bm{\epsilon}}^{\prime*}\times\mathbf{q})]$
$\displaystyle+(\gamma_{2}+\gamma_{4})(\bm{\epsilon}^{\prime*}\cdot\mathbf{q})[\bm{\sigma}\cdot(\mathbf{q}^{\prime}\times\mathbf{q})]$
$\displaystyle+[(\tfrac{1}{2}\gamma_{2}+\gamma_{4}+\gamma_{6})|\mathbf{q}|^{2}+\gamma_{3}({\mathbf{q}}\cdot\mathbf{q}^{\prime})][\bm{\sigma}\cdot(\bm{\epsilon}^{\prime*}\times\mathbf{q}^{\prime})]\Big{\\}}\bigg{|}^{2},$
(29)

where $s$ $(s^{\prime})$ indicate the initial (final) spin states of the
charged fermion, and we neglect the charged fermion’s mass.

The integration over the phase space of the of the final photon is routine,
shifting over to spherical coordinates for the phase space variables. In terms
of integration limits, the minimum momentum transfer to the Majorana fermion
must be sufficient to produce a real photon of energy $\omega_{0}$ so we
require $|\mathbf{q}|\geq\omega_{0}$; this provides an upper limit for the
final-state photon’s energy,
$|\mathbf{k}^{\prime}|\leq|\mathbf{k}|-\omega_{0}$. Finally, we assume that
the momentum, $\mathbf{k}$, of the initial charged fermion is randomly
oriented as it would be in a plasma. In our final expression for the cross
section, we average over all possible directions $\mathbf{k}$. In the end, we
find the leading order contribution to the cross section is

$\displaystyle\sigma=$
$\displaystyle\frac{e^{2}}{\pi}\bigg{\\{}|\mathbf{k}|^{6}\bigg{(}\frac{1}{528}\gamma_{1}^{2}+\frac{1}{180}\gamma_{2}^{2}+\frac{1}{280}\gamma_{3}^{2}+\frac{1}{166}\gamma_{4}^{2}+\frac{43}{540}\gamma_{5}^{2}+\frac{1}{45}\gamma_{6}^{2}$
(30)
$\displaystyle+\frac{1}{264}\gamma_{1}\gamma_{3}+\frac{1}{62}\gamma_{1}\gamma_{5}+\frac{1}{166}\gamma_{2}\gamma_{4}-\frac{1}{97}\gamma_{2}\gamma_{6}+\frac{1}{62}\gamma_{3}\gamma_{5}+\frac{1}{83}\gamma_{4}\gamma_{6}\bigg{)}$
(31)
$\displaystyle+|\mathbf{k}|^{5}\omega_{0}\bigg{(}\frac{1}{64}\gamma_{1}^{2}+\frac{1}{30}\gamma_{2}^{2}+\frac{3}{101}\gamma_{3}^{2}+\frac{6}{163}\gamma_{4}^{2}+\frac{31}{90}\gamma_{5}^{2}+\frac{2}{15}\gamma_{6}^{2}$
(32)
$\displaystyle+\frac{5}{159}\gamma_{1}\gamma_{3}+\frac{8}{83}\gamma_{1}\gamma_{5}+\frac{6}{163}\gamma_{2}\gamma_{4}-\frac{5}{84}\gamma_{2}\gamma_{6}+\frac{8}{83}\gamma_{3}\gamma_{5}+\frac{9}{122}\gamma_{4}\gamma_{6}\bigg{)}\Bigg{\\}}$
(33)

neglecting terms that are higher order in $\omega_{0}$111Note, in Eq. 33 the
coefficients to each term were evaluated via numerical integration and then
rationalized to a precision of one part in ten thousand.. For the toy model,
we again define
$\gamma:=\gamma_{3}=\frac{e^{2}g^{2}}{(4\pi)^{3}}\frac{1}{6}\frac{1}{M_{\phi}^{2}m_{\psi}^{2}}$,
and then we can write the cross section in terms of this parameter. The
resulting rate of spin-transition, from antialigned to aligned, through the
VCS channel is

$\Gamma_{\text{VCS}}=\alpha\gamma^{2}n_{\Psi}|\mathbf{k}|^{5}\left(3.8|\mathbf{k}|+17.7\omega_{0}\right),$
(34)

where $\alpha$ is the fine structure constant.

One important distinction, relative to the two-photon spontaneous emission
process considered previously, is that in the VCS process the spin of the
Majorana fermion can flip from aligned with the external current to
antialigned. That is, if the virtual photon from the charged fermion, $\Psi$,
is sufficiently energetic, namely $\omega>\omega_{0}$, then the Majorana
fermion can be converted into the more energetic antialigned state through
single photon emission. This process competes with the one in which the
Majorana fermion’s spin flips to the lower energy state. For a collection of
unpolarized Majorana fermions, it is the difference between these two rates
that determines the rate at which the system can be polarized through the VCS
process. After computing the aligned-to-antialigned transition rate, we find
the difference in spin-flip rates to be

$\Delta\Gamma_{\text{VCS}}=35.5\alpha\gamma^{2}n_{\Psi}|\mathbf{k}|^{5}\omega_{0},$
(35)

to leading order for the toy model.

##  V  Discussion

Because dark matter interactions are highly suppressed, we anticipated at the
outset of this calculation that the ability to align with a background current
the spins of a collection of DM Majorana fermions, via either two-photon
spontaneous emission or single-photon emission through VCS, is nearly
impossible in all but the most extreme environments. Present day, reasonable
parameter estimates for the size of the anapole moment and polarizabilities
show that the irreversible spin-flip transitions discussed above are
essentially forbidden in terrestrial experiments and stellar environments.
But, in the early universe, the plasma density is so large that the
differential spin-flip rates for the VCS process could be significant if
sufficiently large currents exist.

In the very early universe when temperatures are much greater than the DM
mass, $T\gg m_{\chi}$, reversible spin-flip interactions can assist in
polarizing a DM medium in the presence of a background current. Because DM is
in thermal equilibrium with the thermal bath, one would expect that the
distribution of DM states would follow the Boltzmann distribution with the
ratio of states aligned with a background current to those antialigned given
by $n(\uparrow)/n(\downarrow)=\exp(\omega_{0}/T)$. As the universe cools,
$T\lesssim m_{\chi}$, DM becomes non-relativistic, and because it interacts so
weakly, DM eventually decouples from the thermal bath. At freeze out, the DM
system is fully decoupled, and the DM’s comoving number density essentially
becomes constant. After freeze out, irreversible processes like two-photon
emission and VCS are the only available mechanisms for polarization. We
examine these here.

The freeze out temperature, $T_{f}$, and relic DM density are largely
determined by the DM annihilation cross section. For a large annihilation
cross section, the DM stays in thermal equilibrium until later times (that is,
lower temperatures) resulting in greater DM annihilation, reducing the
ultimate relic density of the particle. Conversely, for smaller annihilation
cross sections, freeze occurs at a higher temperature, and the relic DM
density is greater. To precisely determine the temperature at which thermal DM
begins to drop out of equilibrium one must solve the Boltzmann equation, but
reliable estimates of the freeze out temperature and relic density, accurate
to a few percent, have been developed.

Before estimating the freeze out temperature, it is first useful to expand the
annihilation cross section as a power series in velocity. Because DM is non-
relativistic at freeze out, the lower velocity modes will be the most
significant annihilation channels. This expansion essentially encodes the
temperature dependence of the annihilation mode given that
$\langle|v|\rangle\sim T^{\frac{1}{2}}$ for non-relativistic particles. In
many models, a single annihilation mode dominates. In this case, we can
parametrize the thermally averaged annihilation cross section as
$\langle\sigma_{\text{ann}}|v|\rangle=\sigma_{0}x^{-n}$ where we employ the
dimensional parameter $x:=\frac{m_{\chi}}{T}$ and $\sigma_{0}$ has no explicit
temperature dependence Kolb and Turner (1990). The $s$-wave annihilation mode
corresponds to $n=0$, $p$-wave to $n=1$, and so on. Assuming a dominant
annihilation mode, one may estimate that freeze out occurs when

$x_{f}\simeq\log\left[0.076\,\,g_{*}^{-\frac{1}{2}}(n+1)M_{\text{Pl}}m_{\chi}\sigma_{0}\right]-\left(n+\frac{1}{2}\right)\log[\log[0.076\,\,g_{*}^{-\frac{1}{2}}(n+1)M_{\text{Pl}}m_{\chi}\sigma_{0}]],$
(36)

where $g_{*}$ represents the relativistic degrees of freedom at freeze out and
$M_{\text{Pl}}$ is the Planck mass Scherrer and Turner (1986); Kolb and Turner
(1990). Once the freeze out temperature is determined, one may estimate the
relative DM relic density present today

$\displaystyle\Omega_{\text{DM}}=$
$\displaystyle\frac{3.79\,s_{0}}{\rho_{\text{crit}}g_{*}^{\frac{1}{2}}M_{\text{Pl}}}\frac{(n+1)x_{f}^{n+1}}{\sigma_{0}},$
(37)

where $s_{0}$ represents the universe’s present entropy density and
$\rho_{\text{crit}}$ is the critical energy density Scherrer and Turner
(1986); Kolb and Turner (1990). We use the values for the necessary
astrophysical data found in Ref. P. A. Zyla, et al. (2020) (Particle Data
Group).

To determine the freeze out temperature for the toy model, we must first
examine its annihilation modes. The $s$-wave annihilation of DM into two
photons proceeds via the polarizability terms in the LEX of the interaction
Hamiltonian, Eq. (5). In terms of kinematics, the initial momenta of the DM
particles are $\mathbf{p}\sim\mathbf{p}^{\prime}\sim 0$. Momentum conservation
results in two back-to-back final state real photons with momenta
$q=m_{\chi}(1,\hat{\mathbf{q}})$ and
$q^{\prime}=m_{\chi}(1,-\hat{\mathbf{q}})$; additionally, the photons are
transverse,
$\mathbf{q}\cdot\bm{\epsilon}=\mathbf{q}^{\prime}\cdot\bm{\epsilon}^{\prime}=0$.
Given these kinematic constraints, an examination of the RCS Hamiltonian
matrix elements in Eq. (7) suggests that the only surviving term in the
$s$-wave annihilation amplitude has the form $\mathcal{M}_{s}\sim
m_{\chi}^{4}[\gamma_{1}+\gamma_{2}+2\gamma_{4}][\bm{\sigma}\cdot({\bm{\epsilon}}^{\prime*}\times\bm{\epsilon})]\sim
g^{2}e^{2}\frac{m_{\chi}^{4}}{M_{\phi}^{4}}[\bm{\sigma}\cdot({\bm{\epsilon}}^{\prime*}\times\bm{\epsilon})]$,
using the polarizabilities in Eqs. (10-13). A more refined treatment of the
mass scales involved in the process result in amplitudes with different mass
dependencies Latimer (2017), but if the loop masses dominate the DM mass,
$M_{\phi},m_{\psi}\gg m_{\chi}$, then indeed the $s$-wave annihilation
amplitude scales as $\mathcal{M}_{s}\sim\frac{m_{\chi}^{4}}{M^{4}}$ where
$M\sim M_{\phi},m_{\psi}$. With this amplitude, the annihilation cross section
for the $s$-wave process scales as $\sigma_{s}\sim\frac{m_{\chi}^{6}}{M^{8}}$.
We will see that this mass dependence suppresses the $s$-wave annihilation
mode relative to the $p$-wave mode. We also note that even if we have the
alternate mass hierarchy $M_{\phi}\gg m_{\chi}\gg m_{\psi}$, $p$-wave
annihilation still dominates the $s$-process for large ($\mathcal{O}$ (100
GeVs)) DM masses because the $p$-wave channel has a multitude of kinematically
available final states Latimer (2017).

In $p$-wave annihilation, the non-relativistic DM particles couple via the
anapole moment to a virtual photon which pair produces charged particles like
an electron and positron. This annihilation cross section in the center of
mass frame has been worked out previously Ho and Scherrer (2013)

$\sigma_{p}v_{\text{rel}}=\frac{2}{3}\alpha
f_{a}^{2}m_{\chi}^{2}v_{\text{rel}}^{2}.$ (38)

The mass dependence of the anapole moment in our toy model is
$f_{a}\sim\frac{1}{M_{\phi}^{2}}$, so the $p$-wave annihilation cross section
scales as $\sigma_{p}\sim\frac{m_{\chi}^{2}}{M_{\phi}^{4}}$. This mode
dominates the $s$-wave cross section, so we assume annihilation proceeds
predominantly through the $p$-wave mode. Expressing this is terms of the
dimensionless factor $x$, the thermally averaged cross section is

$\langle\sigma_{p}|v_{\text{rel}}|\rangle=\frac{2}{3}\alpha
N_{f}f_{a}^{2}m_{\chi}^{2}\langle|v_{\text{rel}}|^{2}\rangle=\sigma_{0}x^{-1}$
(39)

with $\sigma_{0}=4\alpha N_{f}f_{a}^{2}m_{\chi}^{2}$. As in Ref. Ho and
Scherrer (2013), the factor $N_{f}$ accounts for the sum over all the
kinematically available final states weighted by the square of the particles’
charges. For instance, an electron-positron pair contribute a term of 1 to the
sum, whereas an up-antiup pair contribute a term of
$\left(\frac{2}{3}\right)^{2}\times 3$ to the sum, where the factor of 3
accounts for the different color states. Because the $p$-wave annihilation
mode is much larger than the $s$-wave mode for reasonable model parameters, we
set $n=1$ in Eqs. (36, 37).

After freeze out, the only mechanism by which polarization of a DM medium can
be achieved is via irreversible processes like those discussed above. But this
can only happen if the spin-flip rate aligning the anapole moment with the
background current is much greater than the universe’s expansion rate. When DM
decouples, the universe is in a radiation dominated era, and the Hubble
parameter is given by Kolb and Turner (1990)

$H=1.66g_{\star}^{\frac{1}{2}}\frac{T^{2}}{M_{\text{Pl}}}.$ (40)

In order for the DM to achieve some degree of polarization, we require that
the irreversible spin-flip rate is much greater than the Hubble parameter
around freeze out; $\Gamma\gg H$.

Let us consider specific model parameters to see if this inequality can be
achieved. Collider experiments place meaningful constraints on the size of
possible anapole moments for particles with masses on the order of hundreds of
GeVs. Using an effective-field-theory approach, a recent study, Ref. Alves et
al. (2018), finds that the high-luminosity LHC runs constrain the anapole
moment for a 500 GeV particle to be smaller than $f_{a}=2\times
10^{-6}\mu_{N}$ fm. Given this value for the anapole moment, we can use Eq.
(37) to directly relate the DM’s anapole moment to the freeze-out temperature,
if we assume the entirety of the relic DM is due to this Majorana fermion.
Taking $f_{a}=2\times 10^{-6}\mu_{N}$ fm, we find that freeze out occurs at
$T_{f}=17$ GeV. Then, using Eq. (36), this freeze out temperature and anapole
moment correspond to a DM mass of $m_{\chi}=419$ GeV. This is reasonably
consistent with the constraints from Ref. Alves et al. (2018). We will also
need to estimate the size of the polarizabilties. To do so requires an
assumption about the mass of the charged fermion $\psi$ to which the Majorana
fermion couples. We set $m_{\psi}=100$ GeV, which yields
$\gamma:=\gamma_{3}\sim 6.6\times 10^{-13}$ GeV-4, neglecting an
$\mathcal{O}(1)$ logarithm, $\log(M_{\phi}/m_{\psi})$.

Turning our attention to spin-flip processes, we now examine whether
spontaneous two-photon emission is viable at freeze out for DM with this
particular anapole moment. The rate depends heavily on the energy difference
between the two spin states, $\omega_{0}=2f_{a}J$, dictated by the background
current density. We will assume the current is due to some net drift velocity
of charged particles in the relativistic thermal plasma,
$J=en_{q}v_{\text{drift}}$. The factor $n_{q}$ is the number density of charge
carriers summed over all the charged relativistic species weighted by the
absolute value of their charge; e.g., electrons are weighted by $1$ while up
quarks have a weight of $\frac{2}{3}$.

Below the freeze out temperature of $17$ GeV, the charged relativistic species
are exclusively fermionic, and the number density for a single relativistic
species of fermions is

$n=\frac{3}{4}\frac{\zeta(3)}{\pi^{2}}gT^{3}$ (41)

where $\zeta$ is the Riemann zeta function and $g$ is the degrees of freedom
for the particular species Kolb and Turner (1990). Around the freeze out
temperature, we have $J=1.1\,v_{\text{drift}}\,T^{3}$ which yields an energy
difference of $\omega_{0}=18$ MeV, taking $v_{\text{drift}}=1$ to represent an
upper limit on the drift speed. From Eq. (19), we find that the two-photon
spontaneous emission rate at freeze out is $\Gamma_{\text{spont}}\leq
4.6\times 10^{-19}$ s-1, which corresponds to a transition time that is on the
order of the Hubble time. Comparing this to the universe’s expansion rate at
$T_{f}=17$ GeV, we find $\frac{\Gamma_{\text{spont}}}{H}\sim 8.3\times
10^{-28}$. Thus, the rate of spontaneous two-photon emission is negligible
around DM freeze out for these model parameters.

We now consider the differential spin-flip rate for the VCS process. In Sec.
IV.3, we assumed that VCS was effectuated by a single charged fermion of
momentum $\mathbf{k}$, though in the early universe VCS would be due to an
incoherent sum of interactions with various fermion species in the
relativistic plasma. To incorporate in the differential spin-flip rate, Eq.
(35), several flavors of fermions with differing charges, we substitute for
$n_{\Psi}$ a sum over the relativistic flavors of fermions, at the given
temperature, with each fermion’s contribution weighted by the square of its
charge. Additionally, the fermion momentum is given by the thermal average
Kolb and Turner (1990)

$\langle|\mathbf{k}|\rangle=\frac{7\pi^{4}}{180\,\zeta(3)}T.$ (42)

Evaluating the degrees of freedom at $17$ GeV, we compare the differential
spin-flip rate to the Hubble parameter

$\frac{\Delta\Gamma_{\text{VCS}}}{H}=28M_{\text{Pl}}f_{a}\gamma^{2}T^{9}v_{\text{drift}}.$
(43)

At freeze out for the model parameters under consideration, we find that the
differential spin-flip rate is sufficiently large if $v_{\text{drift}}\gg
0.03$.

The existence of such currents in the early universe would break the
homogeneity assumed in the cosmological principle, but inhomogeneities must
occur to seed the formation of galaxies. In the literature, we find that the
production of electric currents in the early universe has been discussed in
the inflationary period Turner and Widrow (1988), around the electroweak phase
transition Sigl et al. (1997); Baym et al. (1996), and around the QCD phase
transition Quashnock et al. (1989); Cheng and Olinto (1994); Sigl et al.
(1997); de Souza and Opher (2008). These currents are expected to dampen
rather quickly, compared to the Hubble time Siegel and Fry (2006), but their
ability to locally polarize DM does not rely on long time scales.

It is beyond the scope of this present paper to determine if sufficient
currents exist to polarize a DM medium at freeze out, but if they do exist,
there are implications for DM indirect detection experiments. Supposing a DM
medium were to be partially polarized at freeze out by an electric current,
the DM polarization in the local medium will be locked in place because
subsequent interactions with the thermal bath will be too feeble to disrupt
it. The consequence of such polarized regions, present day, is the suppression
of the rate of $s$-wave annihilation which, by definition, requires vanishing
total angular momentum. For a Majorana fermion, $s$-wave annihilation results
in two final state photons, coupled via the fermion’s polarizabilities, and an
indirect DM detection experiment could look for high-energy photons as the
annihilation products of DM. If no photons were observed, the experiment
places an upper bound on the annihilation cross section as a function of DM
mass. However, if the detector were within a region of partially polarized DM,
the established bound on the $s$-wave annihilation cross section would be an
overestimate if the DM medium were assumed to be unpolarized.

##  VI conclusion

In this paper, we studied the ability of background currents to polarize
Majorana fermions through irreversible processes involving photon emission. If
a Majorana fermion has a non-vanishing anapole moment, then a background
current results in an energy difference between states with spins aligned and
antialigned with the current. Because the other static electromagnetic
properties of a Majarona fermion necessarily vanish, the ability of a Majorana
fermion to flip its spin in the presence of such a background current requires
higher order interactions. We focused upon two possibilities. The first
involved spontaneous two-photon emission which proceeds via the particle’s
polarizability. The second involved emission of a single photon as would occur
in virtual Compoton scattering; this proceeds by virtue of the particle’s
generalized polarizabilities.

Both processes involve spin-dependent effective couplings that carry mass
dimension [M]-4 with a GeV-mass scale. As such, they are highly suppressed
making the transition rate from the state in which the anapole moment is
antialigned with the current to the one in which it is aligned vanishingly
small in terrestrial experiments or stellar environments. However, if DM is
constituted by massive Majorana fermions, the early universe allows for the
possibility that these spin-flip interactions can occur at an appreciable
rate. We focused on the time at which DM decouples from the thermal bath. By
considering reasonable parameter values for a thermal DM model, we find that
spontaneous two-photon emission is negligible at this time, but single-photon
emission via VCS can be appreciable. This has consequences for present-day
indirect DM searches. If a region of DM were polarized by a background current
at freeze out, then the persistent polarization would suppress $s$-wave
annihilation, potentially confounding limits on the annihilation cross section
set by the indirect detection experiment.

Our focus in this paper has been on irreversible interactions that flip the
spin of a Majorana fermion in a current. If the Majorana fermions are not in
thermal equilibrium, then there are no other mechanisms by which a collection
of these particles can be polarized. However, if we consider an earlier time
in the universe before freeze out, Majorana fermions interact sufficiently to
thermalize, and reversible interactions can facilitate the spin-flip
transition. Because these interactions occur via the anapole coupling and not
a polarizability, it is likely that the rate of interaction is dramatically
higher. The situation is akin to what occurs in NMR physics. The transition
rate for the proton’s magnetic dipole moment to align with a $\sim 10$ T
magnetic field via spontaneous emission of a single photon, an irreversible
process, was found to be $10^{-22}$ s-1 Purcell (1946). But NMR relaxation
times, due to reversible spin-spin interactions or spin-lattice interactions,
are on the order of seconds. If our Majorana fermions can interact with the
thermal bath, then polarization might be more easily achieved. We will pursue
this line of thought in future work.

##  VII ACKNOWLEDGMENTS

DCL thanks the Kavli Institute for Theoretical Physics for its hospitality
during the completion of this work. This research was supported in part by the
National Science Foundation under Grant No. NSF PHY-1748958.

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