on the study of BNV, probing $m_{\chi}$ parameter space not accessible to
terrestrial nucleon decay experiments, due to experimental limitations in the
detection of a final-state photon. More than this, the dense nuclear medium
admits the study of regions for which $m_{\chi}$ exceeds the vacuum mass of
the nucleon, as well as the possibility of probing strange baryon decays. Our
final limits are reported in Figs. 14 and 15. We observe that in the regions
of parameter space to which proton decay (nuclear stability) experiments are
sensitive [114, 115], they exceed the limits we set by nearly twenty orders of
magnitude. In contrast, however, our neutron star limits exceed the
sensitivity of those from terrestrial $\Lambda$ and neutron $\beta$-decay
experiments by a comparably large amount. Let us emphasize that our limits are
likely upper bounds, and hence are conservative, in that they are determined
by the electromagnetic decay ${\cal B}\to{\chi}\gamma$ alone, although the
particle physics models we study do admit the possibility of ${\cal
B}\to\chi\,+\,\rm meson(s)$ decays as well. This latter set of decays has no
reason to be negligible compared to the electromagnetic decays in rate — and
we note Ref. [16] for specific examples computed within (in-vacuum) chiral EFT
[51]. As a result, we would expect larger ${\cal B}$ decay rates for fixed
$\varepsilon_{{\cal B\chi}}$, but the challenges in realizing a suitable
theoretical assessment of the hadronic channels prompt the conservative
approach we have espoused in this paper.

We now turn to an assessment of the limitations in our approach. One key
question concerns the largest value of $\varepsilon_{{\cal B}\chi}$,
$\varepsilon_{{\cal B}\chi}^{\rm max}$, we can possibly limit with our
formalism, in which the SM drives the dynamical response of the neutron star
to BNV. (In our work, dark-sector interactions drive the removal of $\chi$, so
that the neutron star survival constraints on the mass of $m_{\chi}$ noted in
Refs. [101, 125, 126] do not operate.) We believe a realistic assessment of
$\varepsilon_{{\cal B}\chi}^{\rm max}$ requires a study of neutron star
heating from relatively fast rates of BNV, the complexities of which lie
beyond the scope of this paper. We note, however, the outcomes of terrestrial
neutron $\beta$-decay searches [45], shown in Fig. 15, as well as limits
arising from constraints due to the charged-current structure of the SM [41],
noted in Eq. (2.1). Since $n\to\chi\gamma$ does not derive from a SM weak
process in any way, a ${\rm Br}(n\to\chi\gamma)$ limit of ${\cal O}(10^{-3})$
implies a limit on $\varepsilon_{n\chi}$ of ${\cal O}(10^{-9})$! Thus we think
these limits are severe enough that determining $\varepsilon_{{\cal
B}\chi}^{\rm max}$ precisely is not an immediate concern, but, rather, an
important topic for future investigation.

Another potential limitation may be our use of a relativistic mean-field
theory framework [17, 18, 19] in which to describe the nuclear medium within a
neutron star. This approach is computationally tractable and readily allows
for the treatment of more sophisticated models of the nucleon-nucleon
interaction than those in which it was first devised. We have employed the
chiral SU(3) hadronic model of Refs. [21, 91, 9] in this paper. This is
admittedly a model that is not QCD, and our ability to assess the errors
predicated by this choice is rather limited. We have, however, studied how our
results change within a family of EoSs, namely DS(CMF) 1-8 EoSs [92, 93], to
which it can be connected. Moreover, frankly, there is no other alternative
for the treatment of dense nuclear matter, though this may ultimately change
[127]. We note that the use of chiral effective theory has been championed in
this regard [87], but its applicability does not stretch much beyond that of
nuclear saturation density. In the future, it may be advantageous to consider
EoSs that blend the chiral effective field theory and relativistic mean-field
theory approaches [94]. Nevertheless, given our interest in order-of-magnitude
estimates, we believe that our choice is also reasonably realistic.

Different paths beckon as opportunities for future work. We believe that
studies of neutron star heating from BNV is important not only to discerning
the limits of our existing formalism, but also, crucially, to interpreting
what a significant observation of anomalous binary spin down might mean. It
strikes us that theoretical heating studies and concomitant observational
studies of neutron star cooling may be the only tangible way to tell a failure
of general relativity, in some undetermined way, from BNV.

As for other possibilities, we could consider how our results could change if
the neutron star were a hybrid star, containing a quark core [9], or how
viable models with a significant $\chi$ admixture in the neutron star (albeit
constrained by Eq. (2.1) [41]), such as that of Ref. [37], could be addressed
through modifications of our formalism. As for future terrestrial experiments
that could complement the studies of this paper, it strikes us that empirical
studies of the lifetime of SM-stable composites, such as atomic hydrogen, or
of the 9Be nucleus, could yield fruitful results.

###### Acknowledgements.

We thank the Network for Neutrinos, Nuclear Astrophysics (N3AS), and
Symmetries (N3AS) for an inspiring environment. J.M.B. acknowledges support
from the National Science Foundation, Grant PHY-1630782, the Heising-Simons
Foundation, Grant 2017-228 and the U.S. Department of Energy Office of Science
under award number DE-SC00018327. J.M.B. also thanks the Institute for Nuclear
Theory at the University of Washington for its kind hospitality and
stimulating research environment. This research was supported in part by the
INT’s U.S. Department of Energy grant No. DE-FG02-00ER41132. S.G. and M.Z.
acknowledge partial support from the U.S. Department of Energy Office of
Science under contract DE-FG02-96ER40989. S.G. thanks N3AS and the Theoretical
Division at Fermilab for gracious hospitality and lively environments and
acknowledges support from N3AS and the Universities Research Association
during sabbatical visits while completing this paper.

##  Appendix A In-Medium Electromagnetic Form Factors

In this appendix we derive the general form for electromagnetic interactions
of baryons in the context of hadronic RMF models, and explicitly show how the
electric charge and magnetic moment are to be identified from the scattering
amplitudes of baryons off of electric and magnetic potentials respectively. We
start from the Dirac equation (4.6),
$\not{p}u(p^{*})=(m^{*}+\not{\Sigma})u(p^{*})$, in which we suppressed the
spin index $\lambda$, to write

$\displaystyle\overline{u}({p^{\prime}}^{*})\left(\frac{i\sigma^{\mu\nu}\left(p^{\prime}_{\nu}-p_{\nu}\right)}{2m^{*}}\right)u(p^{*})$
$\displaystyle=\overline{u}({p^{\prime}}^{*})\left(\gamma^{\mu}-\frac{{p^{\prime}}^{\mu}+p^{\mu}}{2m^{*}}+\frac{\Sigma^{\mu}}{m^{*}}\right)u(p^{*}),$
(A.1)

in which $\sigma^{\mu\nu}\equiv(i/2)[\gamma^{\mu},\gamma^{\nu}]$ and
$\gamma^{\mu}$ are the usual Dirac matrices. The in-medium Gordon
decomposition is then given by

$\displaystyle\overline{u}({p^{\prime}}^{*})\gamma^{\mu}u(p^{*})=\overline{u}({p^{\prime}}^{*})\left(\frac{{p^{\prime}}^{\mu}+p^{\mu}}{2m^{*}}+\frac{i\sigma^{\mu\nu}q_{\nu}}{2m^{*}}-\frac{\Sigma^{\mu}}{m^{*}}\right)u(p^{*})$
(A.2)

in which we defined $q_{\nu}\equiv p^{\prime}_{\nu}-p_{\nu}$. The general form
of a vector interaction vertex, $\Gamma^{\mu}$, can be written as

$\displaystyle\Gamma^{\mu}=\gamma^{\mu}A+\left({p^{\prime}}^{\mu}+p^{\mu}\right)B+q^{\mu}C+D\Sigma^{\mu},$
(A.3)

in which $A,B,C,D$ are functions of scalar quantities (e.g., $q^{2}$).
Applying the Ward identity, $q_{\mu}\Gamma^{\mu}=0$, plus
${p^{\prime}}^{*2}=p^{*2}=m^{*2}$ and ${p^{\prime}}^{2}-p^{2}=2q\cdot\Sigma$,
yields $C=0$ and $2B=D$. The electromagnetic vertex factor can then be written
as

$\displaystyle\Gamma^{\mu}=\gamma^{\mu}F_{1}^{*}(q^{2})+\frac{i\sigma^{\mu\nu}q_{\nu}}{2m^{*}}F_{2}^{*}(q^{2}),$
(A.4)

in which $F_{1,2}^{*}$ are in principle distinct from their vacuum
counterparts $F_{1,2}$.

We now show how the electric charge can be identified in the scattering
amplitude of a baryon from a Coulomb potential $A_{\mu}=(\Phi(x),\vec{0})$.
Employing equations $\overline{u}(k^{*},\lambda)u(k^{*},\lambda)=2m^{*}$ and
$\overline{u}(k^{*},\lambda)\gamma^{0}u(k^{*},\lambda)=2E^{*}(k^{*})$, this
amplitude can be written as

$i\mathcal{M}=-ieF_{1}^{*}(0)\tilde{\Phi}(q)\left(\frac{E^{*}}{m^{*}}\right)2m^{*}\chi^{\dagger}\chi,$
(A.5)

in which $E^{*}=\sqrt{{m^{*}}^{2}+(\vec{p}^{\,*})^{2}}$, and $\chi$ is the
Pauli spinor. The electric charge ($q$) can then be identified, by considering
this scattering in the c.v. frame ($\vec{p}^{\,*}=0$), as
$q=F^{*}_{1}(0)(E^{*,({\rm c.v.})}/m^{*})=F^{*}_{1}(0)$. This can also be
understood from the time component of spin-independent conserved EM current
$J^{0}=\overline{\psi}\gamma^{0}\psi=2E^{*}$, with the Lorentz invariant
electric charge defined in the c.v. frame ($E^{*,({\rm c.v.})}=m^{*}$).

Similarly, we can identify the magnetic moment from the scattering amplitude
of a baryon from a static magnetic field potential $A_{\mu}=(0,\vec{A})$ at
small momentum transfers ($q^{2}\approx 0$), which is given by

$i\mathcal{M}=+ie\overline{u}({p^{\prime}}^{*})\left[\gamma^{i}F_{1}^{*}(0)+\frac{i\sigma^{i\nu}q_{\nu}}{2m^{*}}F_{2}(0)\right]u(p^{*})\tilde{A}^{i}(0).$
(A.6)

The first term inside the bracket can be written as

$\begin{split}\overline{u}({p^{\prime}}^{*})\gamma^{i}u(p^{*})=&\left(E^{*}+m^{*}\right)\left[\chi^{\dagger},\chi^{\dagger}\frac{\vec{\sigma}\cdot\vec{p}^{\,{}^{\prime}*}}{E^{*}+m^{*}}\right]\begin{pmatrix}0&\sigma^{i}\\\
\sigma^{i}&0\end{pmatrix}\begin{pmatrix}\eta\\\
\frac{\vec{\sigma}\cdot\vec{p}^{*}}{E^{*}+m^{*}}\eta\end{pmatrix}\\\
=&\chi^{\dagger}\left[\sigma^{i}\vec{\sigma}\cdot\vec{p}^{\,*}+\vec{\sigma}\cdot\vec{p}\,^{\prime*}\sigma^{i}\right]\eta,\end{split}$
(A.7)

in which $\sigma^{i}$ are the Pauli matrices, and $\chi$, $\eta$ represent the
spin states. This expression can be further simplified using
$\sigma^{i}\sigma^{j}=\delta^{ij}+i\epsilon^{ijk}\sigma^{k}$, such that

$\overline{u}({p^{\prime}}^{*})\gamma^{i}u(p^{*})=\chi^{\dagger}\left[\left(p^{*}+p^{\prime*}\right)^{i}-i\epsilon^{ijk}\left(p^{\prime*}-p^{*}\right)^{j}\sigma^{k}\right]\eta.$
(A.8)

The $F_{2}$ term in the scattering amplitude (A.6) already contains a factor
of $q$, and so we can evaluate it using the leading order expansion of the
spinors in the non-relativistic limit ($\vec{p}^{\,*}\ll m^{*}$), which is
given by $u(\vec{p}^{\,*}=0)=\sqrt{2m^{*}}(\chi,0)^{T}$. We also note that

$\displaystyle\frac{i}{2m^{*}}\sigma^{ij}q_{j}$
$\displaystyle=\frac{i\epsilon^{ijk}}{2m^{*}}\sigma^{k}q_{j},$ (A.9)
$\displaystyle\frac{i}{2m^{*}}\sigma^{i0}q_{0}$
$\displaystyle=\frac{q_{0}}{2m^{*}}\begin{pmatrix}0&\sigma^{i}\\\
\sigma^{i}&0\end{pmatrix},$ (A.10)

such that the spin-dependent contribution from Eq. (A.10), i.e.,
$\overline{u}({p^{\prime}}^{*})(\sigma^{i0}q_{0})u(p^{*})$ is proportional to
$q_{0}q^{j}$, which is subdominant to other terms. The term from Eq. (A.9),
i.e., $\overline{u}({p^{\prime}}^{*})(\sigma^{ij}q_{j})u(p^{*})$ is given by

$\overline{u}({p^{\prime}}^{*})\left(\frac{i}{2m^{*}}\sigma^{ij}q_{j}\right)u(p^{*})=i\epsilon^{ijk}q_{j}\left(\chi^{\dagger},0\right)\begin{pmatrix}\sigma^{k}\eta\\\
0\end{pmatrix}=i\epsilon^{ijk}q_{j}\chi^{\dagger}\sigma^{k}\eta.$ (A.11)

The amplitude in Eq. (A.6) can then be written as (note $q_{j}=-q^{j}$)

$\begin{split}i\mathcal{M}=&-e\chi^{\dagger}\left\\{\epsilon^{ijk}q^{i}\tilde{A}^{j}(0)\sigma^{k}\left[F_{1}^{*}(0)+F_{2}^{*}(0)\right]\right\\}\eta\\\
=&-2ie\left[F_{1}^{*}(0)+F_{2}^{*}(0)\right]S^{k}\tilde{B}^{k},\end{split}$
(A.12)

in which we defined the magnetic field by
$\tilde{B}^{k}\equiv-i\epsilon^{ijk}q^{i}\tilde{A}^{j}$, spin by
$\vec{S}\equiv(1/2)\chi^{\dagger}\vec{\sigma}\eta$, and the baryon g-factor
can be identified as $g^{*}=2\left[F_{1}^{*}(0)+F_{2}^{*}(0)\right]$.

##  Appendix B Nonrelativistic Limit of In-Medium Scattering

In this appendix we study the non-relativistic (NR) limit of the RMF model,
and derive the elastic scattering formalism in the Born approximation. Since
the medium effects in RMF formalism resemble an electromagnetic interaction
with a constant EM background field given by $eA^{\mu}\to\Sigma^{\mu}$, it is
instructive to consider the NR limit of baryon EM interactions in medium. We
explicitly show that the NR limit of the modified Dirac (Eq. (4.6)) solutions
under the influence of EM interactions, reduces to the two-component Pauli
spin theory, with replacements $m\to m^{*}$, $e\Phi\to e\Phi+\Sigma^{0}$,
$e\vec{A}\to e\vec{A}+\vec{\Sigma}$, in which $\Sigma^{0}$ and $\vec{\Sigma}$
are the self-energies due to the medium effects, $e$ is the baryon electric
charge, with $\Phi$ and $\vec{A}$ as the scalar and vector EM potentials
respectively. We start from the Schrodinger equation, which can be written by
denoting the Dirac wave-function ($\psi$) in two-component notation [128],
$\psi=\left(\tilde{\varphi},\tilde{\chi}\right)^{T}$, such that we have

$i\frac{\partial}{\partial t}\begin{pmatrix}\tilde{\varphi}\\\
\tilde{\chi}\end{pmatrix}=\vec{\sigma}\cdot\vec{\pi}\begin{pmatrix}\tilde{\chi}\\\
\tilde{\varphi}\end{pmatrix}+\left(e\Phi+\Sigma^{0}\right)\begin{pmatrix}\tilde{\varphi}\\\
\tilde{\chi}\end{pmatrix}+m^{*}\begin{pmatrix}\tilde{\varphi}\\\
-\tilde{\chi}\end{pmatrix},$ (B.1)

in Pauli-Dirac representation, in which
$\vec{\pi}\equiv\vec{p}-\vec{\Sigma}-e\vec{A}$. Using the definition
$\left(\tilde{\varphi},\tilde{\chi}\right)=\exp(-im^{*}t)\left(\varphi,\chi\right)$,
we can rewrite Eq. (B.1) as

$i\frac{\partial}{\partial t}\begin{pmatrix}{\varphi}\\\
{\chi}\end{pmatrix}=\vec{\sigma}\cdot\vec{\pi}\begin{pmatrix}{\chi}\\\
{\varphi}\end{pmatrix}+\left(e\Phi+\Sigma^{0}\right)\begin{pmatrix}{\varphi}\\\
{\chi}\end{pmatrix}-2m^{*}\begin{pmatrix}0\\\ {\chi}\end{pmatrix}.$ (B.2)

We note that in the NR limit, in which kinetic and interaction energies are
much smaller than $m^{*}$, the second component $\chi$ is subdominant to the
first component $\varphi$ and is approximately given by

$\chi\approx\frac{\vec{\sigma}\cdot\vec{\pi}}{2m^{*}}\varphi.$ (B.3)

We also arrive at the Pauli equation governing the first component
($\varphi$):

$i\frac{\partial}{\partial
t}\varphi=\left(\frac{\left(\vec{p}-\vec{\Sigma}-e\vec{A}\right)^{2}}{2m^{*}}-\frac{e}{2m^{*}}\vec{\sigma}\cdot\vec{B}+e\Phi+\Sigma^{0}\right)\varphi,$
(B.4)

in which $\vec{B}=\nabla\times A$. This expression can be further simplified
for a weak uniform magnetic field ($\vec{A}=\vec{B}\times\vec{r}/2$) as

$i\frac{\partial}{\partial
t}\varphi=\left(\frac{|\vec{p}^{\,*}|^{2}}{2m^{*}}-\frac{e}{2m^{*}}\left(\vec{L}^{\,*}+2\vec{S}\right)\cdot\vec{B}+e\Phi+\Sigma^{0}\right)\varphi,$
(B.5)

in which $\vec{p}^{\,*}=\vec{p}-\vec{\Sigma}$ is the kinetic three momentum,
and $\vec{L}^{*}=\vec{r}\times\vec{p}^{\,*}$ and $\vec{S}=\vec{\sigma}/2$ are
the baryon’s kinetic orbital angular momentum and spin respectively. Note that
in the n.m. frame ($\vec{\Sigma}=0$) the canonical and kinetic three momenta
are equal $\vec{p}=\vec{p}^{\,*}$.

We now construct the elastic scattering formalism off of an arbitrary
potential ($V$) in this NR limit, by turning off the EM fields, i.e.,
$\vec{A}=\Phi=0$, for the rest of this discussion. From Eq. (B.4), we deduce
the energy eigenvalues $E=|\vec{p}^{\,*}|^{2}/2m^{*}+\Sigma^{0}$, which agree
with the NR expansion of Dirac energy eigenvalues given in Eq. (4.8). The
energy eigenfunctions in position space satisfy

$-\nabla^{2}\varphi+2i\vec{\Sigma}\cdot\vec{\nabla}\varphi+\left[|\vec{\Sigma}|^{2}-2m^{*}\left(E-\Sigma^{0}\right)\right]\varphi=0,$
(B.6)

with solutions of the form

$\varphi=e^{-iEt}\left[A_{1}e^{i\vec{p}.\vec{x}}+A_{2}e^{-i\left(\vec{p}-2\vec{\Sigma}\right)\cdot\vec{x}}\right],$
(B.7)

which can also be written in a more symmetric way in terms of $\vec{p}^{\,*}$.
If we orient our coordinates such that $\vec{\Sigma}.\vec{x}>0$, then for a
positive $\vec{p}$ ($\vec{p}.\vec{x}>0$) the first term is a plane wave moving
to the right and the second term is a wave moving to the left. Therefore, we
pick the first term for incident waves in the elastic scattering problem. Let
$H_{0}$ be the Hamiltonian used in Eq. (B.4) (with $\Phi=\vec{A}=0$), and
$|k^{(+)}\rangle$ be the state that satisfies the following Schrodinger
equation in the presence of a potential $V$

$\left(E-H_{0}\right)|k^{(+)}\rangle=V|k^{(+)}\rangle,$ (B.8)

then, $|k^{(+)}\rangle$ can be found from the Lippmann-Schwinger equation:

$|k^{(+)}\rangle=|k\rangle+\frac{1}{E-H_{0}+i\varepsilon}V|k^{(+)}\rangle.$
(B.9)

The momentum representation of operator $G\equiv(E-H_{0}+i\varepsilon)^{-1}$
is given by

$\langle\vec{q}\,|G|\,\vec{q}{\,{}^{\prime}}\rangle=\delta(\vec{q}-\vec{q}^{\prime})\frac{2m^{*}}{\left(\vec{k}-\vec{\Sigma}\right)^{2}-\left(\vec{q}-\vec{\Sigma}\right)^{2}+i\varepsilon},$
(B.10)

and the position space representation is given by

$\langle\vec{r}\,|G|\,\vec{r}{\,{}^{\prime}}\rangle=\int\frac{d^{3}q}{\left(2\pi\right)^{3}}\frac{2m^{*}}{\left(\vec{k}-\vec{\Sigma}\right)^{2}-\left(\vec{q}-\vec{\Sigma}\right)^{2}+i\varepsilon}e^{i\vec{q}\cdot\left(\vec{r}-\vec{r}{\,{}^{\prime}}\right)}.$
(B.11)

We define $\vec{R}\equiv\vec{r}-\vec{r}{\,{}^{\prime}}$,
$\vec{\xi}\equiv\vec{k}-\vec{\Sigma}$ and change the variable from $\vec{q}$
to $\vec{Q}\equiv\vec{q}-\vec{\Sigma}$ such that

$\begin{split}\langle\vec{r}\,|G|\,\vec{r}{\,{}^{\prime}}\rangle=&\int\frac{d^{3}Q}{\left(2\pi\right)^{3}}\frac{2m^{*}}{\vec{\xi}^{2}-\vec{Q}^{2}+i\varepsilon}e^{i\left(\vec{Q}+\vec{\Sigma}\right)\cdot\vec{R}}\\\
=&\frac{m^{*}e^{i\vec{\Sigma}\cdot\vec{R}}}{4\pi^{2}(iR)}\int_{-\infty}^{\infty}\frac{QdQ}{\vec{\xi}^{2}-\vec{Q}^{2}+i\varepsilon}\left[e^{iQR}-e^{-iQR}\right]\\\
=&\frac{-m^{*}}{2\pi R}e^{i\vec{\Sigma}\cdot\vec{R}}e^{i\xi R},\end{split}$
(B.12)

in which we performed the angular integration in the second line, and the
complex contour integration in the third line. To characterize the scattering
problem at $r\to\infty$ we approximate the above expression for
$(r^{\prime}/r)\to 0$ using $R=|\vec{r}-\vec{r}{\,{}^{\prime}}|\approx
r-\hat{r}\cdot\vec{r}{\,{}^{\prime}}$, such that

$\begin{split}\langle\vec{r}\,|G|\vec{r}{\,{}^{\prime}}\rangle=&\left(\frac{-m^{*}}{2\pi
r}\right)e^{i\left[|\vec{k}-\vec{\Sigma}\,|r+\vec{\Sigma}\cdot\vec{r}\,\right]}\,e^{-i\left[|\vec{k}-\vec{\Sigma}\,|\hat{r}+\vec{\Sigma}\right]\cdot\vec{r}{\,{}^{\prime}}}.\end{split}$
(B.13)

We now write the asymptotic form of the Lippmann-Schwinger equation in
position space as

$\psi_{k}\left(\vec{r}\,\right)\sim\varphi_{k}\left(\vec{r}\,\right)-\frac{m^{*}}{2\pi
r}e^{i\left[|\vec{k}-\vec{\Sigma}\,|r+\vec{\Sigma}\cdot\vec{r}\,\right]}\int
d^{3}r^{\prime}\,e^{-i\left[|\vec{k}-\vec{\Sigma}\,|\hat{r}+\vec{\Sigma}\right]\cdot\vec{r}{\,{}^{\prime}}}\,V(r^{\prime})\psi_{k}(\vec{r}{\,{}^{\prime}}),$
(B.14)

in which
$\psi_{k}\left(\vec{r}\,\right)\equiv\langle\vec{r}\,|\vec{k}^{(+)}\rangle$
and
$\varphi_{k}\left(\vec{r}\,\right)\equiv\left(2\pi\right)^{-3/2}\exp\left(i\vec{k}\cdot\vec{r}\,\right)$.
The exponential outside of the integral in the second term is an ellipsoidal
wave (stretched along $\vec{\Sigma}$) which becomes spherical in the n.m.
frame ($\vec{\Sigma}=0$). The exponent inside the integral is a vector
pointing in the direction of $|\vec{k}-\vec{\Sigma}\,|\hat{r}+\vec{\Sigma}$,
which reduces to the familiar $k\hat{r}$ term in the n.m. frame. We can see
that the gradient of ellipsoidal surface is equal to the vector in the
exponent inside the integral since

$\nabla\left[|\vec{k}-\vec{\Sigma}\,|r+\vec{\Sigma}\cdot\vec{r}\,\right]=|\vec{k}-\vec{\Sigma}\,|\hat{r}+\vec{\Sigma},$
(B.15)

which suggests that the exponent
$\vec{k}{\,{}^{\prime}}\equiv|\vec{k}-\vec{\Sigma}\,|\hat{r}+\vec{\Sigma}$ is
the momentum of scattered particle in the direction of an observer at $r$.
Note that the kinetic energy of the scattered particle is given by

$T(k^{\prime})=\frac{\left(\vec{k}{\,{}^{\prime}}-\vec{\Sigma}\right)^{2}}{2m^{*}}=\frac{\left(\vec{k}-\vec{\Sigma}\right)^{2}}{2m^{*}}=T(k),$
(B.16)

and the scattering is indeed elastic. We can therefore deduce the scattering
amplitude by writing

$\psi_{k}\left(\vec{r}\,\right)\sim\left(2\pi\right)^{-3/2}\left\\{e^{i\vec{k}\cdot\vec{r}}+\frac{e^{i\left[|\vec{k}-\vec{\Sigma}\,|r+\vec{\Sigma}\cdot\vec{r}\,\right]}}{r}f(k^{\prime}|k)\right\\},$
(B.17)

in which

$f(k^{\prime}|k)=-4\pi^{2}m^{*}\int
d^{3}r\,\varphi_{k^{\prime}}^{*}(\vec{r}\,)\,V(r)\psi_{k}(\vec{r}),$ (B.18)

which is the Fourier transform of the potential in the Born approximation.

##  Appendix C In-Medium Compton Scattering

In this section we evaluate the Compton scattering cross section of baryons,
${\cal B}(p_{1})+\gamma(k_{1})\to{\cal B}(p_{2})+\gamma(k_{2})$ (see Fig. 17)
in neutron star medium, denoting the photon and baryon energies by
$\omega_{1,2}$ and $E_{1,2}$ respectively.

Figure 17: Feynman diagrams for the baryon Compton scattering ${\cal
B}(p_{1})+\gamma(k_{1})\to{\cal B}(p_{2})+\gamma(k_{2})$.

We first note that the second term in the baryon propagator defined in Eq.
(4.14) vanishes since

$(p_{1}^{*}+k_{1})^{2}-(m_{\cal
B}^{*})^{2}=2\left(E_{1}^{*}|\vec{k}_{1}|-\vec{k}_{1}\cdot\vec{p}^{\,*}_{1}\right)=2|\vec{k}_{1}|\left(\sqrt{(\vec{p}^{\,*}_{1})^{2}+(m_{\cal
B}^{*})^{2}}-\hat{k}_{1}\cdot\vec{p}^{\,*}_{1}\right)>0,$ (C.1)

and similarly, it can be shown that $(p_{1}^{*}-k_{2})^{2}-(m_{\cal
B}^{*})^{2}$ is strictly negative. The amplitude for the diagrams shown in
Fig. 17 can then be written as

$i\mathcal{M}=i\mathcal{M}_{L}+i\mathcal{M}_{R},$ (C.2)

in which

$\begin{split}i\mathcal{M}_{L}=-i\overline{u}(p_{2})&\left(\gamma^{\mu}F_{1}^{*}+\frac{i\sigma^{\mu\nu}k_{2,\nu}}{2m_{\cal
B}^{*}}F_{2}^{*}\right)\epsilon^{*}_{\mu}(k_{2})\left(\frac{\not{p}^{*}_{1}+\not{k}_{1}+m}{(p_{1}^{*}+k_{1})^{2}-(m_{\cal
B}^{*})^{2}}\right)\\\
\times&\left(\gamma^{\nu}F_{1}^{*}+\frac{i\sigma^{\nu\alpha}k_{2,\alpha}}{2m_{\cal
B}^{*}}F_{2}^{*}\right)\epsilon_{\nu}(k_{1})u(p_{1}),\end{split}$ (C.3)

and

$\begin{split}i\mathcal{M}_{R}=-i\overline{u}(p_{2})&\left(\gamma^{\nu}F_{1}^{*}+\frac{i\sigma^{\nu\alpha}k_{1,\alpha}}{2m_{\cal
B}^{*}}F_{2}^{*}\right)\epsilon_{\nu}(k_{1})\left(\frac{\not{p}^{*}_{1}-\not{k}_{2}+m}{(p_{1}^{*}-k_{2})^{2}-(m_{\cal
B}^{*})^{2}}\right)\\\
\times&\left(\gamma^{\mu}F_{1}^{*}+\frac{i\sigma^{\mu\nu}k_{2,\nu}}{2m_{\cal
B}^{*}}F_{2}^{*}\right)\epsilon^{*}_{\mu}(k_{2})u(p_{1}),\end{split}$ (C.4)

in which $F_{1,2}^{*}$ are the in-medium form factors. The interaction term in
the amplitude can be simplified using

$\left(\gamma^{\mu}F_{1}^{*}+\frac{i\sigma^{\mu\nu}k_{2,\nu}}{2m_{\cal
B}^{*}}F_{2}^{*}\right)\epsilon_{\mu}^{*}(k_{2})=\not{\epsilon}^{*}(k_{2})F_{1}^{*}-\frac{F_{2}^{*}}{2m_{\cal
B}^{*}}\not{\epsilon}^{*}(k_{2})\not{k}_{2},$ (C.5)

which follows from
$\epsilon_{\mu}(k_{1})k_{1}^{\mu}=\epsilon_{\mu}(k_{2})k_{2}^{\mu}=0$. The
spin-averaged squared amplitudes simplify to

$\begin{split}\overline{\left|\mathcal{M}_{L}\right|^{2}}=\frac{1}{4\left[(p_{1}^{*}+k_{1})^{2}-(m_{\cal
B}^{*})^{2}\right]^{2}}\,{\rm Tr}\,&\left[\left(\not{p}^{*}_{2}+m_{\cal
B}^{*}\right)\left(F_{1}^{*}+\frac{F_{2}^{*}}{2m_{\cal
B}^{*}}\not{k}_{2}\right)\gamma^{\mu}\left(\not{p}_{1}^{*}+\not{k}_{1}+m_{\cal
B}^{*}\right)\gamma^{\nu}\right.\\\
&\times\left(F_{1}^{*}-\frac{F_{2}^{*}}{2m_{\cal
B}^{*}}\not{k}_{1}\right)\left(\not{p}^{*}_{1}+m_{\cal
B}^{*}\right)\left(F_{1}^{*}-\frac{F_{2}^{*}}{2m_{\cal
B}^{*}}\not{k}_{1}\right)\\\
&\times\left.\gamma_{\nu}\left(\not{p}_{1}^{*}+\not{k}_{1}+m_{\cal
B}^{*}\right)\gamma_{\mu}\left(F_{1}^{*}+\frac{F_{2}^{*}}{2m_{\cal
B}^{*}}\not{k}_{2}\right)\right],\end{split}$ (C.6)

and

$\begin{split}\overline{\left|\mathcal{M}_{R}\right|^{2}}=\frac{1}{4\left[(p_{1}^{*}-k_{2})^{2}-(m_{\cal
B}^{*})^{2}\right]^{2}}\,{\rm Tr}\,&\left[\left(\not{p}^{*}_{2}+m_{\cal
B}^{*}\right)\left(F_{1}^{*}+\frac{F_{2}^{*}}{2m_{\cal
B}^{*}}\not{k}_{1}\right)\gamma^{\nu}\left(\not{p}_{1}^{*}-\not{k}_{2}+m_{\cal
B}^{*}\right)\gamma^{\mu}\right.\\\
&\times\left(F_{1}^{*}-\frac{F_{2}^{*}}{2m_{\cal
B}^{*}}\not{k}_{2}\right)\left(\not{p}^{*}_{1}+m_{\cal
B}^{*}\right)\left(F_{1}^{*}-\frac{F_{2}^{*}}{2m_{\cal
B}^{*}}\not{k}_{2}\right)\\\
&\times\left.\gamma_{\mu}\left(\not{p}_{1}^{*}-\not{k}_{2}+m_{\cal
B}^{*}\right)\gamma_{\nu}\left(F_{1}^{*}+\frac{F_{2}^{*}}{2m_{\cal
B}^{*}}\not{k}_{1}\right)\right],\end{split}$ (C.7)

with the cross-term given by

$\begin{split}\overline{\mathcal{M}_{L}\mathcal{M}_{R}^{\dagger}}=\frac{{\rm
Tr}\,\left[T_{\rm LR}\right]}{4\left[(p_{1}^{*}-k_{2})^{2}-(m_{\cal
B}^{*})^{2}\right]\left[(p_{1}^{*}+k_{1})^{2}-(m_{\cal
B}^{*})^{2}\right]},\end{split}$ (C.8)

in which

$\begin{split}T_{\rm LR}=&\left(\not{p}^{*}_{2}+m_{\cal
B}^{*}\right)\left(F_{1}^{*}+\frac{F_{2}^{*}}{2m_{\cal
B}^{*}}\not{k}_{2}\right)\gamma^{\mu}\left(\not{p}_{1}^{*}+\not{k}_{1}+m_{\cal
B}^{*}\right)\gamma^{\nu}\left(F_{1}^{*}-\frac{F_{2}^{*}}{2m_{\cal
B}^{*}}\not{k}_{1}\right)\left(\not{p}^{*}_{1}+m_{\cal B}^{*}\right)\\\
&\times\left(F_{1}^{*}-\frac{F_{2}^{*}}{2m_{\cal
B}^{*}}\not{k}_{2}\right)\gamma_{\mu}\left(\not{p}_{1}^{*}-\not{k}_{2}+m_{\cal
B}^{*}\right)\gamma_{\nu}\left(F_{1}^{*}+\frac{F_{2}^{*}}{2m_{\cal
B}^{*}}\not{k}_{1}\right),\end{split}$ (C.9)

with
$\overline{\mathcal{M}_{L}\mathcal{M}_{R}^{\dagger}}=\overline{\mathcal{M}_{R}\mathcal{M}_{L}^{\dagger}}$.
We now define the following Mandelstam variables

$\displaystyle s^{*}\equiv$
$\displaystyle\left(p_{1}^{*}+k_{1}\right)^{2}=(m_{\cal
B}^{*})^{2}+2p_{1}^{*}\cdot k_{1}=(m_{\cal B}^{*})^{2}+2p_{2}^{*}\cdot k_{2}$
(C.10) $\displaystyle t^{*}\equiv$
$\displaystyle\left(p_{2}^{*}-p_{1}^{*}\right)^{2}=2(m_{\cal
B}^{*})^{2}-2p_{1}^{*}\cdot p^{*}_{2}=-2k_{1}\cdot k_{2}$ (C.11)
$\displaystyle u^{*}\equiv$
$\displaystyle\left(k_{2}-p_{1}^{*}\right)^{2}=(m_{\cal
B}^{*})^{2}-2p_{1}^{*}\cdot k_{2}=(m_{\cal B}^{*})^{2}-2p_{2}^{*}\cdot k_{1},$
(C.12)

such that $s^{*}+t^{*}+u^{*}=2(m_{\cal B}^{*})^{2}$. We suppress the
superscripts (“*”) of $m_{\cal B}^{*}$, $F_{1,2}^{*}$ in some of the following
equations for convenience. The averaged amplitude-squared can be written as

$\overline{\left|\mathcal{M}\right|^{2}}=\frac{1}{16}\left(\frac{\rm
I}{(p^{*}_{1}\cdot k_{1})^{2}}+\frac{\rm II}{(p^{*}_{1}\cdot
k_{1})(p^{*}_{1}\cdot k_{2})}+\frac{\rm III}{(p^{*}_{1}\cdot
k_{1})(p^{*}_{1}\cdot k_{2})}+\frac{\rm IV}{(p^{*}_{1}\cdot
k_{2})^{2}}\right),$ (C.13)

in which

$\displaystyle{\rm I}=$ $\displaystyle 8F_{1}^{4}\left(m_{\cal B}^{4}+m_{\cal
B}^{2}(3s+u)-su\right)+4F_{2}^{2}F_{1}^{2}\left(1-\frac{s}{m_{\cal
B}^{2}}\right)\left[2m_{\cal B}^{4}-m_{\cal B}^{2}(3s+u)+s(3s-u)\right]$
$\displaystyle\,-\frac{F_{2}^{4}m_{\cal B}^{2}}{2}\left(1-\frac{s}{m_{\cal
B}^{2}}\right)^{3}\left(m_{\cal B}^{2}-u\right),$ (C.14) $\displaystyle{\rm
II}=$ $\displaystyle{\rm III}=F_{2}^{4}\left(2-\frac{s+u}{m_{\cal
B}^{2}}\right)\left(su-m_{\cal B}^{4}\right)-8F_{1}^{4}m_{\cal
B}^{2}\left(2m_{\cal B}^{2}+s+u\right)$
$\displaystyle\quad\quad-2F_{2}^{2}F_{1}^{2}\left(3m_{\cal
B}^{2}(s+u)-2\left(s^{2}+su+u^{2}\right)+\frac{su(s+u)}{m_{\cal
B}^{2}}-2m_{\cal B}^{4}\right),$ (C.15) $\displaystyle{\rm IV}=$
$\displaystyle 8F_{1}^{4}\left(m_{\cal B}^{4}+m_{\cal
B}^{2}(s+3u)-su\right)+4F_{2}^{2}F_{1}^{2}\left(1-\frac{u}{m_{\cal
B}^{2}}\right)\left[2m_{\cal B}^{4}-m_{\cal B}^{2}(s+3u)+u(3u-s)\right]$
$\displaystyle\,-\frac{F_{2}^{4}m_{\cal B}^{2}}{2}\left(m_{\cal
B}^{2}-s\right)\left(1-\frac{u}{m_{\cal B}^{2}}\right)^{3},$ (C.16)

in which we note that $\rm I$ and $\rm IV$ are related via $(s\leftrightarrow
u)$ replacement. Equation (C.13) can then be written as

$\begin{split}\overline{\left|\mathcal{M}\right|^{2}}=&\frac{2F_{1}^{4}\left[6m_{\cal
B}^{8}-m_{\cal B}^{4}\left(3s^{2}+14su+3u^{2}\right)+m_{\cal
B}^{2}(s+u)\left(s^{2}+6su+u^{2}\right)-su\left(s^{2}+u^{2}\right)\right]}{\left(m_{\cal
B}^{2}-s\right)^{2}\left(m_{\cal B}^{2}-u\right)^{2}}\\\
&\,+\frac{F_{2}^{4}\left[3m_{\cal B}^{8}-m_{\cal
B}^{4}\left(s^{2}+8su+u^{2}\right)+4m_{\cal
B}^{2}su(s+u)-s^{2}u^{2}\right]}{4m_{\cal B}^{4}\left(m_{\cal
B}^{2}-s\right)\left(m_{\cal B}^{2}-u\right)}\\\
&\,+\frac{F_{2}^{2}F_{1}^{2}\left[2m_{\cal B}^{6}-3m_{\cal
B}^{4}(s+u)+2m_{\cal B}^{2}\left(s^{2}+su+u^{2}\right)-su(s+u)\right]}{m_{\cal
B}^{2}\left(m_{\cal B}^{2}-s\right)\left(m_{\cal B}^{2}-u\right)}.\end{split}$
(C.17)

Figure 18: The Compton scattering in the rest (c.v.) frame of ${\cal
B}(p_{1})$. Note that even though there is a specific direction to the
canonical momentum $\vec{p}_{1}^{\,\rm(c.v.)}\neq 0$, the amplitude depends
only on $p^{*}_{1}$ for which $\vec{p}_{1}^{\,*,\rm(c.v.)}=0$. Therefore, we
have the freedom to choose the z-axis in the direction of the incoming photon.
Evaluating the integrated cross section in Eq. (C.24) will require specifying
the Fermi ellipsoid in the c.v. frame, which depends on
$\vec{p}_{1}^{\,\rm(n.m.)}$ (see Eq. (C.23)).

We now consider the Compton scattering in the rest (c.v.) frame of ${\cal
B}(p_{1})$ (see Fig. 18), in which $\vec{p}_{1}^{\,*}=0$. We first note that
the relationship $k_{1}\cdot k_{2}=p^{*}_{1}\cdot\left(k_{1}-k_{2}\right)$,
written in the c.v. frame, yields
$\omega_{1}\omega_{2}\left(1-\cos\theta\right)=m_{\cal
B}^{*}\left(\omega_{1}-\omega_{2}\right)$. We then arrive at the following
kinematics in the c.v. frame

$\displaystyle\omega_{2}=$
$\displaystyle\frac{\omega_{1}}{1+\frac{\omega_{1}}{m_{\cal
B}^{*}}\left(1-\cos\theta\right)},$ (C.18) $\displaystyle\vec{p}^{\,*}_{2}=$
$\displaystyle\left[-\omega_{2}\sin\theta,0,\omega_{1}-\omega_{2}\cos\theta\right],$
(C.19) $\displaystyle E^{*}_{\rm 2}=$ $\displaystyle\sqrt{(m_{\cal
B}^{*})^{2}+\omega_{1}^{2}+\omega_{2}^{2}-2\omega_{1}\omega_{2}\cos\theta}.$
(C.20)

which resembles the familiar Compton’s formula. We use these kinematical
relationships to write Eq. (C.13) in terms of $\omega_{1}$ and the scattering
angle ($\theta$) in the c.v. frame as

$\begin{split}\overline{\left|\mathcal{M}\right|^{2}}=&16F_{1}^{4}m_{\cal
B}\bigg{\\{}\frac{m_{\cal
B}^{3}}{4}\left(\cos(2\theta)+3\right)+\omega_{1}\sin^{2}\left(\theta/2\right)m_{\cal
B}^{2}(\cos(2\theta)+3)\\\ &\qquad\qquad+\omega_{1}^{2}\left[m_{\cal
B}(\cos(2\theta)+5)-2\omega_{1}(\cos(\theta)-1)\right]\sin^{4}\left(\theta/2\right)\bigg{\\}}\\\
&-8F_{1}^{2}F_{2}^{2}m_{\cal
B}\bigg{\\{}\frac{\omega_{1}^{3}}{2}\left[\cos(2\theta)-8\cos(\theta)+7\right]\sin^{2}\left(\theta/2\right)-m_{\cal
B}\omega_{1}^{2}\left[5\cos(\theta)-7\right]\sin^{2}\left(\theta/2\right)\\\
&\qquad\qquad\qquad+8\omega_{1}m_{\cal
B}^{2}\sin^{2}\left(\theta/2\right)+2m_{\cal B}^{3}\bigg{\\}}\\\
&+F_{2}^{4}m_{\cal B}\omega_{1}^{2}\left[5-\cos(2\theta)\right]\left[m_{\cal
B}-\omega_{1}(\cos(\theta)-1)\right].\end{split}$ (C.21)

The phase space integrals over the final states (see Eq. (4.15)) can be
written as

$\begin{split}\int
d\Pi_{2}&=\int\frac{d^{3}k_{2}}{\left(2\pi\right)^{3}}\frac{1}{2\omega_{2}}\frac{d^{3}p^{*}_{2}}{\left(2\pi\right)^{3}}\frac{1}{2E^{*}_{2}}\left(2\pi\right)^{4}\delta^{4}\left(k_{2}+p^{*}_{2}-k_{1}-p^{*}_{1}\right)\,[1-f_{\cal
B}(\vec{p}_{2})]\\\
&=\int\frac{\omega_{2}d\omega_{2}\,d\Omega_{2}}{16\pi^{2}E^{*}_{2}}\delta\left(\omega_{2}+\sqrt{(m_{\cal
B}^{*})^{2}+\omega^{2}_{1}+\omega^{2}_{2}-2\omega_{1}\omega_{2}\cos\theta}-\omega_{1}-m_{\cal
B}^{*}\right)[1-f_{\cal B}(\vec{p}_{2})]\\\
&=\int\frac{d\Omega_{2}}{16\pi^{2}}\frac{\omega_{2}}{E^{*}_{2}+\omega_{2}-\omega_{1}\cos\theta}[1-f_{\cal
B}(\vec{p}_{2})]=\int\frac{d\Omega_{2}}{16\pi^{2}}\frac{\omega_{2}^{2}}{m_{\cal
B}^{*}\,\omega_{1}}[1-f_{\cal B}(\vec{p}_{2})],\end{split}$ (C.22)

in which $d\Omega_{2}=d\cos(\theta)\,d\phi$ is the differential solid angle of
$\vec{k}_{2}$ in the c.v. frame, and $f_{\cal B}(\vec{p}_{2})$ is the Pauli
blocking factor for the outgoing baryon. The shape of the Fermi surface in a
general frame (such as c.v.) changes from being spherical to an ellipsoid. The
general form of $f_{\cal B}(\vec{p}_{2})$ in an arbitrary frame is given by
$\theta(E_{F}^{*}-p_{2}^{*\mu}B_{\mu}/n_{\cal B})$ [89], in which $E_{F,{\cal
B}}^{*}\equiv\sqrt{p_{F,{\cal B}}^{2}+(m_{\cal B}^{*})^{2}}$, $p_{F,{\cal B}}$
is the Fermi momentum defined in the n.m. frame, $B^{\mu}$ is the baryon
current density defined below Eq. (4.14), and $n_{\cal B}$ is the baryon
number density. Evaluating the invariant argument of the step-function in the
n.m. frame yields $\theta(E_{F}^{*}-E_{2}^{*,{\rm(n.m.)}})$, in which

$E_{2}^{*,({\rm n.m.})}=\left(\frac{E_{1}^{*,({\rm n.m.})}\,E_{2}^{*,({\rm
c.v.})}}{m_{\cal
B}^{*}}\right)+\left(\frac{\vec{p}_{1}^{\,\rm(n.m.)}\cdot\vec{p}_{2}^{\,*,({\rm
c.v.})}}{m_{\cal B}^{*}}\right).$ (C.23)

We see that even though the amplitude in the c.v. frame depends only on
$\theta$, integrating over the azimuthal angle ($\phi$) requires the explicit
coordinates of the initial baryon ${\cal B}(p_{1})$ momentum in the n.m.
frame, $\vec{p}_{1}^{\,({\rm n.m.})}$, in our chosen coordinate in Fig. 18.
Using Eq. (LABEL:eq:medium:modific:incident_beam_velocity), and noting
$v_{\mathcal{B}}^{*}=0$, $v_{\mathcal{A}}^{*}=1$ in our chosen frame (c.v.),
the in-medium Compton scattering differential cross section can be written as

$\frac{d\sigma}{d\Omega_{2}}=\frac{\omega_{2}^{2}\,\overline{\left|\mathcal{M}\right|^{2}}}{64\pi^{2}\omega_{1}^{2}\,(m_{\cal
B}^{*})^{2}}[1-f_{\cal B}(\vec{p}_{2})],$ (C.24)

in which we recover the Klein-Nishina [129, 100] formula if we set $f_{\cal
B}(\vec{p}_{2})=0$, $F_{1}=e$, $F_{2}=0$ and replace $m_{\cal B}^{*}$ by
$m_{e}$.

##  Appendix D Fermion Mixing in Dense Matter

In this appendix we evaluate the eigenvalues of a system consisting of a
neutral baryon (${\cal B}$) and a dark fermion ($\chi$) with a mixing term
between them, in the context of RMF framework. We suppress the superscript
(“*”) in baryon’s effective mass ($m_{\cal B}^{*}$) for convenience. The
Lagrangian for this system is presented in Eq. (5.1), with conjugate momenta
given by

$\Pi_{{\cal B},\chi}=\frac{\partial\mathcal{L}}{\partial\dot{\psi}_{{\cal
B},\chi}}=i\psi^{\dagger}_{{\cal B},\chi},$ (D.1)

and a coupled set of equation of motion

$\displaystyle\left(i\not{\partial}-\not{\Sigma}_{\cal B}-m_{\cal
B}\right)\psi_{\cal B}=$ $\displaystyle\varepsilon\psi_{\chi},$ (D.2)
$\displaystyle\left(i\not{\partial}-m_{\chi}\right)\psi_{\chi}=$
$\displaystyle\varepsilon\psi_{{\cal B}}.$ (D.3)

Note that the baryon current $J^{\mu}_{\cal B}\equiv\overline{\psi}_{\cal
B}\gamma^{\mu}\psi_{\cal B}$ satisfies

$\begin{split}\partial_{\mu}J^{\mu}_{\cal
B}=&\left(\partial_{\mu}\overline{\psi}_{\cal B}\right)\gamma^{\mu}\psi_{\cal
B}+\overline{\psi}_{\cal B}\not{\partial}\psi_{\cal
B}=i\varepsilon\left(\overline{\psi}_{\chi}\psi_{\cal
B}-\overline{\psi}_{{\cal B}}\psi_{\chi}\right),\end{split}$ (D.4)

and as expected is not conserved anymore, instead, the combined current
$J^{\mu}\equiv J^{\mu}_{\cal B}+J^{\mu}_{\chi}$ is conserved. The conserved
energy-momentum prescribed by the Noether’s theorem [130, 100] is given by

$\displaystyle H=$ $\displaystyle\int d^{3}x\left[\overline{\psi}_{\cal
B}\left(i\vec{\gamma}\cdot\vec{\nabla}+\not{\Sigma}_{\cal B}+m_{\cal
B}\right)\psi_{\cal
B}+\overline{\psi}_{\chi}\left(i\vec{\gamma}\cdot\vec{\nabla}+m_{\chi}\right)\psi_{\chi}+\varepsilon\left(\overline{\psi}_{{\cal
B}}\psi_{\chi}+\overline{\psi}_{\chi}\psi_{{\cal B}}\right)\right],$ (D.5)
$\displaystyle\vec{P}=$ $\displaystyle\int d^{3}x\,\left[\psi_{\cal
B}^{\dagger}\left(-i\vec{\nabla}\right)\psi_{\cal
B}+\psi_{\chi}^{\dagger}\left(-i\vec{\nabla}\right)\psi_{\chi}\right].$ (D.6)

We expand each of the fields in terms of four modes
$\omega_{1,2}^{\pm}(\vec{k})$ as

$\displaystyle\psi(x,t)=\sum_{s}\int\frac{d^{3}k}{\sqrt{2(2\pi)^{3}}}\Bigg{[}$
$\displaystyle\alpha\left(a_{1}(k,s)u_{1}(k,s)e^{-i\omega_{1}^{(+)}t+ik\cdot
x}+b_{1}^{\dagger}(k,s)v_{1}(k,s)e^{-i\omega_{1}^{(-)}t-ik\cdot x}\right)$
$\displaystyle+$
$\displaystyle\beta\left(a_{2}(k,s)u_{2}(k,s)e^{-i\omega_{2}^{(+)}t+ik\cdot
x}+b_{2}^{\dagger}(k,s)v_{2}(k,s)e^{-i\omega_{2}^{(-)}t-ik\cdot
x}\right)\Bigg{]},$ (D.7)

in which $\psi(x,t)$ stands for $\psi_{{\cal B},\chi}$, $a_{1,2}$ and
$b_{1,2}$ are the annihilation operators for particles and anti-particles,
$\omega$ stands for $\omega(\vec{k})$, we note the inequality
$\omega(\vec{k})\neq\omega(-\vec{k})$ if $\vec{\Sigma}_{\cal B}\neq 0$ (see
Eq. (4.8)), and the fact that in the presence of medium ($\Sigma^{0}_{\cal
B}\neq 0$) the particle and anti-particle energies are not equal anymore
(e.g., see Eq. (2.40) of [89]). The coefficients $\alpha$ and $\beta$ can be
found by requiring that the Hamiltonian in Eq. (D.5) is diagonal. The spinors
$u(p,s)$ and $v(p,s)$ satisfy (see Eq. (2.33) of Ref. [18])

$\displaystyle u^{\dagger}(p,s)u(p,s^{\prime})=$ $\displaystyle
v^{\dagger}(p,s)v(p,s^{\prime})=\delta_{ss^{\prime}},$ (D.8)
$\displaystyle\sum_{s}u(p,s)\overline{u}(p,s)=$
$\displaystyle\frac{\not{p}+m}{2E(p)},$ (D.9)
$\displaystyle\sum_{s}v(p,s)\overline{v}(p,s)=$
$\displaystyle\frac{\not{p}-m}{2E(p)},$ (D.10)

in which $m$ stands for $m_{\cal B},m_{\chi}$, and we make the replacements
$p\to p^{*}_{\cal B}$, $E\to E^{*}_{\cal B}$ for the baryon. We can combine
Eqs. (D.2) with (D.3) and arrive at the following equation after multiplying
the left-hand-side by $\left(\not{P}+m_{\chi}\right)\left(\not{P}^{*}+m_{{\cal
B}}\right)$:

$\begin{split}\left(P^{2}-m_{\chi}^{2}\right)\left[(P-\Sigma_{\cal
B})^{2}-m_{\cal B}^{2}\right]\psi_{\cal
B}=&\varepsilon^{2}\left(2P_{\mu}\left(P^{\mu}-\Sigma^{\mu}_{\cal
B}\right)+2m_{\cal B}m_{\chi}-\varepsilon^{2}\right)\psi_{\cal B},\end{split}$
(D.11)

in which we note the definition $P_{\mu}=i\partial_{\mu}=(H,-\vec{P})$. We now
plug in the field expansion from Eq. (D.7) into Eq. (D.11) to arrive at the
equation governing the spectrum of $\omega_{1,2}^{\pm}$ modes:

$\displaystyle\left(\omega^{2}-k^{2}-m_{\chi}^{2}\right)\left[(\omega-\Sigma^{0}_{\cal
B})^{2}-(\vec{k}-\vec{\Sigma}_{\cal B})^{2}-m_{\cal
B}^{2}\right]=2\varepsilon^{2}$
$\displaystyle\left(\omega^{2}-k^{2}-\omega\Sigma^{0}_{\cal
B}+\vec{k}\cdot\vec{\Sigma}_{\cal B}+m_{\cal B}m_{\chi}\right.$
$\displaystyle\quad\left.-\frac{\varepsilon^{2}}{2}\right),$ (D.12)

We denote the solutions in the absence of mixing ($\varepsilon=0$) by
$\omega_{0}$, and solve Eq. (D) using a perturbation series in powers of
$\delta\equiv\varepsilon/\omega_{0}$:
$\omega=\sum_{i=0}^{\infty}\omega_{i}\delta^{i}$, with the zeroth order
equation yielding

$\mathcal{O}(\delta^{0}):\quad\left[\left(\omega_{0}-\Sigma^{0}_{\cal
B}\right)^{2}-(\vec{k}-\vec{\Sigma}_{\cal B})^{2}-m_{\cal
B}^{2}\right]\left(\omega_{0}^{2}-k^{2}-m_{\chi}^{2}\right)=0,$ (D.13)

such that we get the usual spectrum for ${\cal B}$ and $\chi$:

$\displaystyle\omega_{0}^{(\pm)}(\chi)=$
$\displaystyle\pm\sqrt{k^{2}+m_{\chi}^{2}},$ (D.14)
$\displaystyle\omega_{0}^{(\pm)}({\cal B})=$ $\displaystyle\Sigma^{0}_{\cal
B}\pm\sqrt{\left(\vec{k}-\vec{\Sigma}_{\cal B}\right)^{2}+m_{{\cal B}}^{2}}.$
(D.15)

For the rest of this discussion we consider the energy spectrum in the n.m.
frame ($\vec{\Sigma}_{\cal B}^{\,({\rm n.m.})}=0$). Denoting $\Sigma^{0,({\rm
n.m.})}_{\cal B}$ by $\Sigma_{0}$ for convenience, we can rewrite Eq. (D) as

$\begin{split}&\omega^{4}-2\Sigma_{0}\omega^{3}-\left[2k^{2}+m_{\cal
B}^{2}+m_{\chi}^{2}-\Sigma_{0}^{2}+2\varepsilon^{2}\right]\omega^{2}+2\Sigma_{0}\left[\left(k^{2}+m_{\chi}^{2}\right)+\varepsilon^{2}\right]\omega\\\
&+\left(k^{2}+m_{\chi}^{2}\right)\left(k^{2}+m_{\cal
B}^{2}-\Sigma_{0}^{2}\right)+\varepsilon^{2}\left(2k^{2}-2m_{\cal
B}m_{\chi}\right)+\varepsilon^{4}=0.\end{split}$ (D.16)

We note that both of the parenthesis in Eq. (D.13) can be simultaneously equal
to zero, if $k$ satisfies the following condition at zeroth order:

$|\vec{k}|=\frac{\sqrt{\left[\left(\Sigma^{0}-m_{\cal
B}\right)^{2}-m_{\chi}^{2}\right]\left[\left(\Sigma^{0}+m_{\cal
B}\right)^{2}-m_{\chi}^{2}\right]}}{2\Sigma^{0}},$ (D.17)

We divide our solutions into two sets: the normal solutions for which the
zeroth order condition in Eq. (D.17) is not satisfied, and those for which Eq.
(D.17) is satisfied, which we denote by “*” superscript. First we write down
the general equations to third order in perturbation. The first order equation
is given by

$\mathcal{O}(\delta^{1}):\quad
2\omega_{0}\omega_{1}\left[\left(\omega_{0}-\Sigma_{0}\right)^{2}-k^{2}-m_{\cal
B}^{2}\right]+2\omega_{1}\left(\omega_{0}-\Sigma_{0}\right)\left(\omega_{0}^{2}-k^{2}-m_{\chi}^{2}\right)=0.$
(D.18)

Given the $\mathcal{O}(\delta^{0})$ equation in Eq. (D.13), we conclude that
either $\omega_{1}=0$ or the condition in Eq. (D.17) is satisfied. The second
order equation yields:

$\begin{split}\mathcal{O}(\delta^{2}):\quad&\left(\omega_{1}^{2}+2\omega_{0}\omega_{2}\right)\left[\left(\omega_{0}-\Sigma_{0}\right)^{2}-k^{2}-m_{\cal
B}^{2}\right]-2\omega_{0}^{2}\left(m_{\cal
B}m_{\chi}-\Sigma_{0}\omega_{0}+\omega_{0}^{2}-k^{2}\right)\\\
&\,+\left(2\omega_{2}\left(\omega_{0}-\Sigma_{0}\right)+\omega_{1}^{2}\right)\left(\omega_{0}^{2}-k^{2}-m_{\chi}^{2}\right)+4\omega_{0}\omega_{1}^{2}\left(\omega_{0}-\Sigma_{0}\right)=0.\end{split}$
(D.19)

Finally, the third order equation is given by

$\begin{split}\mathcal{O}(\delta^{3}):\quad&\left(2\omega_{1}\omega_{2}+2\omega_{0}\omega_{3}\right)\left[\left(\omega_{0}-\Sigma_{0}\right)^{2}-k^{2}-m_{\cal
B}^{2}\right]+2\omega_{1}\left(\omega_{1}^{2}+2\omega_{0}\omega_{2}\right)\left(\omega_{0}-\Sigma_{0}\right)\\\
&+\left(2\omega_{3}\left(\omega_{0}-\Sigma_{0}\right)+2\omega_{1}\omega_{2}\right)\left(\omega_{0}^{2}-k^{2}-m_{\chi}^{2}\right)\\\
&-2\omega_{0}^{2}\omega_{1}\left(2\omega_{0}-\Sigma_{0}\right)+2\omega_{1}\omega_{0}\left(2\omega_{2}\left(\omega_{0}-\Sigma_{0}\right)+\omega_{1}^{2}\right)=0.\end{split}$
(D.20)

First, let us assume that the condition in Eq. (D.17) doesn’t hold, in which
case $\mathcal{O}(\delta^{1})$ condition yields $\omega_{1}=0$. We then solve
for $\omega_{2}$ in $\mathcal{O}(\delta^{2})$:

$\displaystyle\omega_{2}^{(+)}(\chi)=$
$\displaystyle\frac{\sqrt{k^{2}+m_{\chi}^{2}}\left(m_{\chi}\left(m_{\cal
B}+m_{\chi}\right)-\Sigma_{0}\sqrt{k^{2}+m_{\chi}^{2}}\right)}{\left(\sqrt{k^{2}+m_{\chi}^{2}}-\Sigma_{0}\right)^{2}-k^{2}-m_{\cal
B}^{2}},$ (D.21) $\displaystyle\omega_{2}^{(-)}(\chi)=$
$\displaystyle-\frac{\sqrt{k^{2}+m_{\chi}^{2}}\left(m_{\chi}\left(m_{\cal
B}+m_{\chi}\right)+\Sigma_{0}\sqrt{k^{2}+m_{\chi}^{2}}\right)}{\left(\sqrt{k^{2}+m_{\chi}^{2}}+\Sigma_{0}\right)^{2}-k^{2}-m_{\cal
B}^{2}},$ (D.22) $\displaystyle\omega_{2}^{(+)}({\cal B})=$
$\displaystyle\frac{\left(\sqrt{k^{2}+m_{\cal
B}^{2}}+\Sigma_{0}\right)^{2}\left(m_{\cal B}\left(m_{\cal
B}+m_{\chi}\right)+\Sigma_{0}\sqrt{k^{2}+m_{\cal
B}^{2}}\right)}{\sqrt{k^{2}+m_{\cal B}^{2}}\left[\left(\sqrt{k^{2}+m_{\cal
B}^{2}}+\Sigma_{0}\right)^{2}-k^{2}-m_{\chi}^{2}\right]},$ (D.23)
$\displaystyle\omega_{2}^{(-)}({\cal B})=$
$\displaystyle-\frac{\left(\sqrt{k^{2}+m_{\cal
B}^{2}}-\Sigma_{0}\right)^{2}\left(m_{\cal B}\left(m_{\cal
B}+m_{\chi}\right)-\Sigma_{0}\sqrt{k^{2}+m_{\cal
B}^{2}}\right]}{\sqrt{k^{2}+m_{\cal
B}^{2}}\left(\Sigma_{0}\left(\Sigma_{0}-2\sqrt{k^{2}+m_{\cal
B}^{2}}\right)+m_{\cal B}^{2}-m_{\chi}^{2}\right)},$ (D.24)

which after plugging into the $\mathcal{O}(\delta^{3})$ equation yields
$\omega_{3}=0$, and so the energies of $\chi$ and ${\cal B}$ to third order
are given by

$\displaystyle\omega^{(+)}(\chi)=$
$\displaystyle\sqrt{k^{2}+m_{\chi}^{2}}+\frac{\varepsilon^{2}\left(m_{\chi}\left(m_{\cal
B}+m_{\chi}\right)-\Sigma_{0}\sqrt{k^{2}+m_{\chi}^{2}}\right)}{\sqrt{k^{2}+m_{\chi}^{2}}\left[\left(\sqrt{k^{2}+m_{\chi}^{2}}-\Sigma_{0}\right)^{2}-k^{2}-m_{\cal
B}^{2}\right]}+\mathcal{O}(\delta^{4}),$ (D.25)
$\displaystyle\omega^{(-)}(\chi)=$
$\displaystyle-\sqrt{k^{2}+m_{\chi}^{2}}-\frac{\varepsilon^{2}\left(m_{\chi}\left(m_{\cal
B}+m_{\chi}\right)+\Sigma_{0}\sqrt{k^{2}+m_{\chi}^{2}}\right)}{\sqrt{k^{2}+m_{\chi}^{2}}\left[\left(\sqrt{k^{2}+m_{\chi}^{2}}+\Sigma_{0}\right)^{2}-k^{2}-m_{\cal
B}^{2}\right]}+\mathcal{O}(\delta^{4}),$ (D.26)
$\displaystyle\omega^{(+)}({\cal B})=$
$\displaystyle\Sigma_{0}+\sqrt{k^{2}+m_{\cal
B}^{2}}+\frac{\varepsilon^{2}\left(m_{\cal B}\left(m_{\cal
B}+m_{\chi}\right)+\Sigma_{0}\sqrt{k^{2}+m_{\cal
B}^{2}}\right)}{\sqrt{k^{2}+m_{\cal B}^{2}}\left[\left(\sqrt{k^{2}+m_{\cal
B}^{2}}+\Sigma_{0}\right)^{2}-k^{2}-m_{\chi}^{2}\right]}+\mathcal{O}(\delta^{4}),$
(D.27) $\displaystyle\omega^{(-)}({\cal B})=$
$\displaystyle\Sigma_{0}-\sqrt{k^{2}+m_{\cal
B}^{2}}-\frac{\varepsilon^{2}\left(m_{\cal B}\left(m_{\cal
B}+m_{\chi}\right)-\Sigma_{0}\sqrt{k^{2}+m_{\cal
B}^{2}}\right)}{\sqrt{k^{2}+m_{\cal B}^{2}}\left[m_{\cal
B}^{2}-m_{\chi}^{2}-\Sigma_{0}\left(2\sqrt{k^{2}+m_{\cal
B}^{2}}-\Sigma_{0}\right)\right]}+\mathcal{O}(\delta^{4}),$ (D.28)

in which the negative energy solutions would be interpreted as antiparticles.
We now consider the second set of solutions assuming that Eq. (D.17) holds.
The zeroth order equation yields

$\displaystyle\omega_{0}^{*}(\chi)=\omega_{0}^{*}({\cal B})=$
$\displaystyle\frac{m_{\chi}^{2}-m_{\cal B}^{2}+\Sigma_{0}^{2}}{2\Sigma^{0}}.$
(D.29)

The first order equation $\mathcal{O}(\delta^{1})$ is trivial, and the second
order equation yields:

$\omega_{1}^{*}=\pm\sqrt{\frac{\left(m_{\cal
B}+m_{\chi}\right)\left[2\Sigma_{0}^{2}m_{\cal B}+\left(m_{\chi}-m_{\cal
B}\right)\left(m_{\cal
B}+m_{\chi}\right)^{2}\right]-\Sigma_{0}^{4}}{4\left(m_{\chi}^{2}-m_{\cal
B}^{2}-\Sigma_{0}^{2}\right)}},$ (D.30)

which we plug into the third order equation to arrive at

$\omega_{2}^{*}=\frac{\left(m_{\chi}^{2}-m_{\cal
B}^{2}\right)^{3}-\Sigma_{0}^{2}\left(m_{\chi}-m_{\cal B}\right)\left(m_{\cal
B}+m_{\chi}\right)^{3}}{4\Sigma_{0}\left(m_{\cal
B}^{2}-m_{\chi}^{2}+\Sigma_{0}^{2}\right)^{2}},$ (D.31)

such that the full energy spectrum is given by

$\begin{split}\omega^{*}_{(\pm)}=&\frac{\Sigma_{0}^{2}-m_{\cal
B}^{2}+m_{\chi}^{2}}{2\Sigma_{0}}\pm\varepsilon\Sigma_{0}\sqrt{\frac{\left(m_{\cal
B}+m_{\chi}\right)^{2}-\Sigma_{0}^{2}}{\left(m_{\cal
B}^{2}-m_{\chi}^{2}\right)^{2}-\Sigma_{0}^{4}}}\\\
&+\frac{\varepsilon^{2}\Sigma_{0}\left(m_{\chi}-m_{\cal B}\right)\left(m_{\cal
B}+m_{\chi}\right)^{3}\left[\left(m_{\cal
B}-m_{\chi}\right)^{2}-\Sigma_{0}^{2}\right]}{\left[\left(m_{\cal
B}^{2}-m_{\chi}^{2}\right)^{2}-\Sigma_{0}^{4}\right]^{2}}+\mathcal{O}(\delta^{3}).\end{split}$
(D.32)

We can see that the first order term breaks the degeneracy by splitting the
energies.

##  Appendix E Baryon Decays to $\chi+\gamma$

Here we present the full calculation of the decay of a baryon to
$\chi+\gamma$.

###  E.1 Matrix Element

The matrix element for this process is

$i\mathcal{M}=\frac{i\varepsilon_{{\cal B}\chi}g_{\cal B}e}{4m_{\cal
B}^{*}}\overline{u}_{\chi}(k_{\chi})\frac{1}{\not{k}_{\chi}-\not{\Sigma}_{\cal
B}-m_{\cal B}^{*}}\not{\epsilon}^{*}\not{k}_{\gamma}u(p^{*}_{\cal B})\,,$
(E.1)

where we note

$(\not{k}_{\chi}-\not{\Sigma}_{\cal B}-m_{\cal
B}^{*})^{-1}=\frac{\not{k}_{\chi}-\not{\Sigma}_{\cal B}+m_{\cal
B}^{*}}{(k_{\chi}-\Sigma_{\cal B})^{2}-(m_{\cal
B}^{*})^{2}}\equiv\frac{\not{k}^{*}+m_{\cal
B}^{*}}{(k^{*}_{\chi})^{2}-(m_{\cal B}^{*})^{2}}$ (E.2)

and we define the quantity $k^{*}_{\chi}\equiv k_{\chi}-\Sigma_{\cal
B}=p_{\cal B}^{*}-k_{\gamma}$. Note in the neutron star medium that energy-
momentum conservation of the total canonical momentum still holds: $p_{\cal
B}^{\mu}=k_{\gamma}^{\mu}+k_{\chi}^{\mu}$. Consideration of the kinematics
show that we need only consider the first term in the full baryon propagator
given in Eq. (4.14).

We then find the spin-summed matrix element to be

$\displaystyle|\mathcal{M}|^{2}$ $\displaystyle=\frac{\varepsilon_{{\cal
B}\chi}^{2}g_{\cal B}^{2}e^{2}(p_{\cal B}^{*}\cdot k_{\gamma})}{(m_{\cal
B}^{*})^{2}[(m_{\cal B}^{*})^{2}-(k^{*}_{\chi})^{2}]^{2}}\big{\\{}[(m_{\cal
B}^{*})^{2}-(k^{*}_{\chi})^{2}](k_{\chi}\cdot k_{\gamma})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+2(k^{*}_{\chi}\cdot
k_{\gamma})(k^{*}_{\chi}\cdot k_{\chi}+m_{\cal B}^{*}m_{\chi})\big{\\}}$ (E.3)
$\displaystyle=\frac{\varepsilon_{{\cal B}\chi}^{2}g_{\cal B}^{2}e^{2}(p_{\cal
B}^{*}\cdot k_{\gamma})}{2(m_{\cal B}^{*})^{2}(p_{\cal B}^{*}\cdot
k_{\gamma})^{2}}\left[(p_{\cal B}^{*}\cdot k_{\gamma})(k_{\chi}\cdot
k_{\gamma})+(k^{*}_{\chi}\cdot k_{\gamma})(k^{*}_{\chi}\cdot k_{\chi}+m_{\cal
B}^{*}m_{\chi})\right]$ (E.4) $\displaystyle=\frac{\varepsilon_{{\cal
B}\chi}^{2}g_{\cal B}^{2}e^{2}}{2(m_{\cal B}^{*})^{2}}\left[(p_{\cal
B}^{*}\cdot k_{\chi})+m_{\cal B}^{*}m_{\chi}\right],$ (E.5)

where we note the useful relations $(k^{*}_{\chi})^{2}=(m_{\cal
B}^{*})^{2}-2(p_{\cal B}^{*}\cdot k_{\gamma})$ and $(k^{*}_{\chi}\cdot
k_{\gamma})=(p_{\cal B}^{*}\cdot k_{\gamma})$.

###  E.2 Integrated Rates

We now address full integral over phase space,

$\frac{dn_{B}}{d\tau}=-\int\frac{d^{3}\vec{p}_{\cal
B}}{(2\pi)^{3}(2E^{*}_{\cal
B})}\frac{d^{3}\vec{k}_{\chi}}{(2\pi)^{3}(2E_{\chi})}\frac{d^{3}\vec{k}_{\gamma}}{(2\pi)^{3}(2E_{\gamma})}f_{\cal
B}(\vec{p}_{\cal
B})\times|\mathcal{M}|^{2}\times(2\pi)^{4}\delta^{(4)}\left(p_{\cal
B}-k_{\chi}-k_{\gamma}\right).$ (E.6)

In the main text, we presented the rate as an integral over the baryon Fermi
sphere of the dilated widths of individual baryons. Here, we will contrast
this approach with a more straightforward evaluation of this integral and
demonstrate that these yield consistent results, as expected.

Our first step in the evaluation of the rate is to separate the integrals over
the $\chi$ and $\gamma$ phase spaces and evaluate these first:

$\displaystyle\dfrac{dn_{B}}{d\tau}$
$\displaystyle=-\int_{0}^{p_{F,n}}\dfrac{p^{2}\,dp}{4\pi^{2}E^{*}_{\cal
B}}\,{\cal G}(p);$ (E.7) $\displaystyle{\cal G}(p)$
$\displaystyle=\int\dfrac{d^{3}\vec{k}_{\chi}}{(2\pi)^{3}(2E_{\chi})}\dfrac{d^{3}\vec{k}_{\gamma}}{(2\pi)^{3}(2E_{\gamma})}|\mathcal{M}|^{2}\times(2\pi)^{4}\delta^{(4)}\left(p_{\cal
B}-k_{\chi}-k_{\gamma}\right).$ (E.8)

We have simplified the first integral by noting that it only depends on the
magnitude of the three-momentum $|\vec{p}_{\cal B}|\equiv p$, and that we only
integrate within the neutron Fermi sphere. We tackle this second integral by
computing it in the c.m. frame of the decaying neutron. We note, however, that
the matrix element depends on $p_{\cal B}^{*}$, which has a nonvanishing
spatial component, though we will find that this is not relevant for the
ultimate evaluation of the integral.

We begin by articulating the boost between the n.m. frame and the c.m. frame.
We denote the 4-momentum of a baryon in the n.m. frame by $p_{\cal
B}^{\rm(n.m.)}=\left(E_{\cal B},\vec{p}_{\cal B}\right)$, and the vector self-
energy as $\Sigma_{\cal B}^{\rm(n.m.)}=\left(\Sigma_{\cal
B}^{0},\vec{0}\right)$, and express the c.m. frame 4-momentum $p_{\cal
B}^{\rm(c.m.)}$ only in terms of the quantities in the n.m. frame. The boost
from the n.m. to the c.m. frame is parameterized by

$\gamma=\dfrac{E_{\cal B}^{*}+\Sigma^{0}_{\cal
B}}{\sqrt{s}},\quad\gamma\beta=\dfrac{|\vec{p}_{\cal B}|}{\sqrt{s}},\quad
s\equiv(m_{\cal B}^{*})^{2}+2E_{\cal B}^{*}\Sigma^{0}_{\cal
B}+(\Sigma^{0}_{\cal B})^{2},$ (E.9)

such that

$p_{\cal B}^{\rm(c.m.)}=\Lambda\cdot p_{\cal
B}^{\rm(n.m.)}=\left(\begin{array}[]{cc}\gamma&-\gamma\beta\\\
-\gamma\beta&\gamma\end{array}\right)\left(\begin{array}[]{c}E_{\cal
B}^{*}+\Sigma^{0}_{\cal B}\\\ |\vec{p}_{\cal
B}|\hat{z}\end{array}\right)=\left(\begin{array}[]{c}\sqrt{s}\\\
\vec{0}\end{array}\right).$ (E.10)

Therefore, we write $p_{\cal B}^{*}$ in the c.m. frame as

$p_{\cal B}^{*,\rm{(c.m.)}}=\frac{1}{\sqrt{s}}\left(\begin{array}[]{c}(m_{\cal
B}^{*})^{2}+E_{\cal B}^{*}\Sigma^{0}_{\cal B}\\\ |\vec{p}_{\cal
B}|\Sigma^{0}_{\cal B}\hat{z}\end{array}\right).$ (E.11)

With

$E_{\chi}^{\rm{(c.m.)}}=\dfrac{s+m_{\chi}^{2}}{2\sqrt{s}};\qquad|\vec{k}_{\chi}^{\rm{(c.m.)}}|=\dfrac{s-m_{\chi}^{2}}{2\sqrt{s}}.$
(E.12)

we may write

$p_{\cal B}^{*,\rm{(c.m.)}}\cdot k_{\chi}^{\rm{(c.m.)}}=\frac{[(m_{\cal
B}^{*})^{2}+E_{\cal B}^{*}\Sigma^{0}_{\cal
B}](s+m_{\chi}^{2})}{2s}-\frac{(|\vec{p}_{\cal B}|\Sigma^{0}_{\cal
B})(s-m_{\chi}^{2})}{2s}\cos\theta^{*},$ (E.13)

in which $\cos\theta^{*}$ is the angle between $\vec{k}_{\chi}^{\rm{(c.m.)}}$
and $(\vec{p}_{\cal B})^{*,\rm{(c.m.)}}$, and we note that the second term
vanishes once an integration over the direction of
$\vec{k}_{\chi}^{\rm{(c.m.)}}$ is performed. We consider next the form of the
energy delta function when transformed to momentum; using
$k\equiv|\vec{k}_{\gamma}^{\rm{(c.m.)}}|=|k_{\chi}^{\rm{(c.m.)}}|$, we have

$\delta\left(\sqrt{s}-k-\sqrt{k^{2}+m_{\chi}^{2}}\right)=\delta\left(k-\dfrac{s-m_{\chi}^{2}}{2\sqrt{s}}\right)\frac{E_{\chi}}{\sqrt{s}}.$
(E.14)

Putting these pieces together, we arrive at

$\displaystyle{\cal G}(p)$ $\displaystyle=\int\frac{k\,dk}{4\pi
E_{\chi}}\delta\left(k-\dfrac{s-m_{\chi}^{2}}{2\sqrt{s}}\right)\frac{E_{\chi}}{\sqrt{s}}\times\frac{\varepsilon_{n\chi}^{2}g_{\cal
B}^{2}e^{2}}{2(m_{\cal B}^{*})^{2}}\frac{[(m_{\cal B}^{*})^{2}+E_{\cal
B}^{*}\Sigma^{0}_{\cal B}](s+m_{\chi}^{2})+2sm_{\cal B}^{*}m_{\chi}}{2s}$
(E.15) $\displaystyle=\frac{\varepsilon_{{\cal B}\chi}^{2}g_{\cal
B}^{2}e^{2}}{32\pi(m_{\cal
B}^{*})^{2}}\left(\frac{s-m_{\chi}^{2}}{s^{2}}\right)\left\\{s[(m_{\cal
B}^{*})^{2}+E_{\cal B}^{*}\Sigma^{0}_{\cal B}+2m_{\cal
B}^{*}m_{\chi}]+m_{\chi}^{2}[(m_{\cal B}^{*})^{2}+E_{\cal
B}^{*}\Sigma^{0}_{\cal B}]\right\\}.$ (E.16)

We can therefore write Eq. (E.6) as

$\dfrac{dn_{B}}{d\tau}=-\int_{0}^{p_{F,{\cal
B}}}\frac{p^{2}\,dp}{4\pi^{2}E^{*}_{\cal B}}{\cal G}(p)=-\int_{m_{\cal
B}^{*}}^{E_{F,{\cal B}}}\frac{\sqrt{(E_{\cal B}^{*})^{2}-(m_{\cal
B}^{*})^{2}}dE_{\cal B}^{*}}{4\pi^{2}}\,{\cal G}(p),$ (E.17)

which after using the definitions in Eq. (5.12) turns into the expression
given in Eq. (5.11). Using Eq. (5.9), we can also write the individual baryon
decay rate in the c.m. frame $\Gamma_{\rm c.m.}(p_{\cal B})$ as

$\begin{split}\Gamma_{\rm c.m.}(p_{\cal B})=\left(\frac{g_{\cal
B}^{2}e^{2}\varepsilon_{{\cal B}\chi}^{2}}{128\pi\,m_{\cal
B}^{*}}\right)\frac{1+\sigma^{2}+2x\sigma-\mu^{2}}{\left(1+\sigma^{2}+2x\sigma\right)^{3/2}\left(1+x\sigma\right)}&\left[\left(1+\sigma^{2}+2\sigma
x\right)\left(1+\sigma x+2\mu\right)\right.\\\ &\,\,\left.+\mu^{2}(1+\sigma
x)\right]\,.\end{split}$ (E.18)

We note that if the self-energy were to vanish ($\sigma=0$) we would recover
the vacuum decay rate reported in Eq. (2.7).

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