# Fireball anti-nucleosynthesis

Michael A. Fedderke<EMAIL_ADDRESS>Perimeter Institute for
Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada  The William H. Miller
III Department of Physics and Astronomy, The Johns Hopkins University,
Baltimore, Maryland, 21218, USA  David E. Kaplan<EMAIL_ADDRESS>The
William H. Miller III Department of Physics and Astronomy, The Johns Hopkins
University, Baltimore, Maryland, 21218, USA  Anubhav Mathur<EMAIL_ADDRESS>The William H. Miller III Department of Physics and Astronomy, The Johns
Hopkins University, Baltimore, Maryland, 21218, USA  Surjeet Rajendran
<EMAIL_ADDRESS>The William H. Miller III Department of Physics and
Astronomy, The Johns Hopkins University, Baltimore, Maryland, 21218, USA
Erwin H. Tanin<EMAIL_ADDRESS>Stanford Institute for Theoretical
Physics, Stanford University, Stanford, California, 94305, USA  The William H.
Miller III Department of Physics and Astronomy, The Johns Hopkins University,
Baltimore, Maryland, 21218, USA

###### Abstract

The tentative identification of approximately ten relativistic anti-helium
($\overline{\text{He}}$) cosmic-ray events at AMS-02 would, if confirmed,
challenge our understanding of the astrophysical synthesis of heavy anti-
nuclei. We propose a novel scenario for the enhanced production of such anti-
nuclei that is triggered by isolated, catastrophic injections of large
quantities of energetic Standard Model (SM) anti-quarks in our galaxy by
physics beyond the Standard Model (BSM). We demonstrate that SM anti-
nucleosynthetic processes that occur in the resulting rapidly expanding,
thermalized fireballs of SM plasma can, for a reasonable range of parameters,
produce the reported tentative $\sim 2:1$ ratio of
${}^{3}\overline{\text{He}}$ to ${}^{4}\overline{\text{He}}$ events at AMS-02,
as well as their relativistic boosts. Moreover, we show that this can be
achieved without violating anti-deuterium or anti-proton flux constraints for
the appropriate anti-helium fluxes. A plausible BSM paradigm for the
catastrophic injections is the collision of macroscopic composite dark-matter
objects carrying large net anti-baryon number. Such a scenario would require
these objects to be cosmologically stable, but to destabilize upon collision,
promptly releasing a fraction of their mass energy into SM anti-particles
within a tiny volume. We show that, in principle, the injection rate needed to
attain the necessary anti-helium fluxes and the energetic conditions required
to seed the fireballs appear possible to obtain in such a paradigm. We leave
open the question of constructing a BSM particle physics model to realize
this, but we suggest two concrete scenarios as promising targets for further
investigation.

###### Contents

  1. I Introduction
  2. II The AMS-02 candidate anti-helium events
  3. III Fireball anti-nucleosynthesis
    1. III.1 Thermalization
    2. III.2 Relativistic fireball expansion
    3. III.3 Nuclear physics
      1. III.3.1 Preliminaries
      2. III.3.2 Expansion timescales
      3. III.3.3 Anti-nucleon abundance
      4. III.3.4 Anti-deuterium production
      5. III.3.5 Anti-deuterium burning
      6. III.3.6 Anti-helium-3 burning
      7. III.3.7 Anti-tritium burning
      8. III.3.8 Final anti-nucleosynthesis products
      9. III.3.9 Non-nuclear outputs
    4. III.4 Summary
  4. IV Propagation and detection
    1. IV.1 Cosmic rays
    2. IV.2 Other signatures (indirect detection)
  5. V A dark-matter origin for the fireballs?
    1. V.1 Collision rates
    2. V.2 Mapping to fireball parameters
    3. V.3 Towards a particle physics model
      1. V.3.1 Imploding fermion+Yukawa model
      2. V.3.2 Dark-dwarf model
  6. VI Discussion and conclusion
  7. A Derivation of fireball scaling laws
  8. B Freeze-out ratio of anti-nucleons
  9. C Simplified nuclear reaction network
  10. D Nuclear cross sections
  11. E Changes in the number of degrees of freedom
  12. F Prompt (‘burst’) or slow (‘wind’) injection
  13. G AMS-02 anti-helium sensitivity

##  I Introduction

The AMS-02 Collaboration111The Alpha Magnetic Spectrometer (AMS-02) is a
particle-physics detector located on the International Space Station [1]. has
unofficially reported [2, 3, 4, 5, 6, 7] $\mathcal{O}(10)$ highly relativistic
cosmic-ray events detected in $\sim 10$ years of data that are consistent with
tentative identification as anti-helium [6, 7].222These unofficial reports [2,
3, 4, 5, 6, 7] have taken the form of public oral presentations on behalf of
the Collaboration in the context of scientific conferences or major colloquia,
as well as the associated publicly available presentation slides. We stress
however that these data have not to date been published, have never been
officially claimed by the AMS-02 Collaboration to present a formal detection
of anti-helium cosmic rays, and have always been accompanied by disclaimers
and caveats that the origin of these candidate events requires more study.
Additionally, only partial data is available for these events. Although
publicly available mass determinations are uncertain [6], the data are
reported to be consistent with tentative identification of both
${}^{3}\overline{\text{He}}$ and ${}^{4}\overline{\text{He}}$ candidate
events, with an event ratio of roughly
$N_{{}^{3}\overline{\text{He}}}:N_{{}^{4}\overline{\text{He}}}\sim 2:1$
(albeit with small statistics and large uncertainties) [6, 7]. Additionally,
tentative identification of 7 anti-deuterium candidate events has recently
been reported [7].

While the tentative identifications of these candidate events require more
work to confirm [7], the anti-helium candidates in particular have been the
subject of extensive recent interest in the literature [8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18] because, taken at face-value, they are surprising: the
formation of complex anti-nuclei in astrophysical environments is challenging
and the rates of production implied by these candidate events are hard to
reconcile with known Standard Model (SM) physics.

Within the SM, a known source of anti-nucleus cosmic rays is spallation
induced by primary cosmic rays (hydrogen or helium) in the interstellar medium
(ISM) [12]. Spallation is, however, inefficient at producing anti-nuclei; due
to kinematics, this is particularly true for those anti-nuclei with higher
atomic mass number $A$, such as ${}^{4}\overline{\text{He}}$. Coalescence of
two anti-nucleons or anti-nuclei (or of an anti-nucleon and an anti-nucleus)
produced in spallation into a higher-$A$ anti-nucleus is probable only if
their relative kinetic energy is comparable to or below the difference in the
nuclear binding energies $E_{B}$ of the initial and final states; roughly,
$E_{B}\sim A\times\mathcal{O}(\mathinner{\mathrm{MeV}})$. On the other hand,
the threshold energy for production of nuclear anti-particles via spallation
of a primary cosmic ray against the ISM is $E_{\text{th}}\sim m_{\bar{N}}\sim
A\mu_{a}\sim A\times\mathcal{O}(\mathinner{\mathrm{GeV}})$. Because primary
cosmic-ray fluxes tend to be power laws as a function of energy (see, e.g.,
Ref. [1]), it follows that the kinetic energy of nuclear anti-particles
produced by spallation of sufficiently energetic primary cosmic rays is also
of typically $\mathcal{O}(\text{GeV})$, except very near threshold. The
formation rates of heavier anti-nuclei via coalescence of such products
therefore tend to suffer significant phase-space suppressions, leading to
strong hierarchies between the expected numbers $N$ of anti-nucleus events
with subsequently higher $A$ that would be observed at AMS-02: i.e.,
$N_{\bar{p}}\gg N_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\gg
N_{{}^{3}\overline{\text{He}}}\gg N_{{}^{4}\overline{\text{He}}}$.

These conventional astrophysics predictions for anti-nucleus cosmic ray fluxes
from spallation were quantified recently in Ref. [12], where the expected
numbers of anti-nucleus events at AMS-02 were found to scale roughly as
$N_{\bar{p}}\sim 10^{4}N_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\sim
10^{8}N_{{}^{3}\overline{\text{He}}}\sim
10^{12}N_{{}^{4}\overline{\text{He}}}$, in line with the phase-space argument
above. For model parameters that reproduce the anti-proton flux observed by
AMS-02, Ref. [12] thus found that the predicted anti-helium fluxes are always
orders of magnitude below AMS-02 sensitivity. Conversely, in order to
reproduce, e.g., the tentatively identified ${}^{4}\overline{\text{He}}$ flux,
one would have to overproduce, e.g., the observed anti-proton flux by many
orders of magnitude.

Similar challenges in producing anti-nuclei also occur in decaying or
annihilating particle dark-matter models commonly studied in the context of
indirect searches [12, 10, 11, 8, 9]. The anti-nucleus production rates in
these models suffer from similar phase-space suppressions for spallation of
primary cosmic rays.333The phase-space suppression argument made above, and
the resulting hierarchy of anti-nucleus fluxes with higher $A$, should apply
for any anti-nucleus production scenario that starts with high-energy
($E\gtrsim\mathinner{\mathrm{GeV}}$) processes. One should keep in mind
however that there are considerable variations in the predicted anti-nucleus
fluxes stemming from uncertainties in the parameters of the nuclear-
coalescence model [8, 9] and different choices of the cosmic-ray propagation
model [14]. With optimistic assumptions [14, 11], and possibly with
enhancements from $\overline{\Lambda_{b}}$ decays [15], it has been found that
it might be possible for annihilating particle dark matter to be the origin of
the tentatively identified ${}^{3}\overline{\text{He}}$ flux. However, even a
single confirmed ${}^{4}\overline{\text{He}}$ event at AMS-02 would be
challenging to explain.

If the candidate events are confirmed, explaining the presence of both the
${}^{3}\overline{\text{He}}$ and ${}^{4}\overline{\text{He}}$ events at
AMS-02, with their comparable rates, seems to require the absence of the
severe phase-space suppressions that inexorably lead to a strong hierarchical
relation of the anti-nucleus fluxes. These suppressions can be ameliorated if
the anti-nucleons that combined to form the anti-nuclei have low relative
momenta, $\Delta p\ll\mathinner{\mathrm{GeV}}$. Various beyond-the-Standard-
Model (BSM) scenarios have been proposed to achieve this.

Refs. [12, 16, 18] considered anti-nucleus production occurring in anti-
matter–dominated regions of primordial origin that have cooled down
significantly by the time of the anti-nucleus production. These scenarios
require a BSM mechanism in the early Universe to cause the requisite
matter–anti-matter segregation.

Anti-nucleus production scenarios via the decay of a new particle carrying
anti-baryon number (which may or may not be the dark matter) have been also
considered. In Ref. [13], the mass of the decaying particle was tuned to be
very close to the mass of the desired anti-nucleus in order to restrict the
final-state phase space, such that the produced particles are non-
relativistic. Such a scenario however requires several new decaying particles,
each with its own mass tuning to separately enhance the production of
${}^{4}\overline{\text{He}}$, ${}^{3}\overline{\text{He}}$, and perhaps
${\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}$. In Ref. [17], a strongly coupled
dark sector was considered, where dark hadron showers triggered by the decay
of a new particle simultaneously increase the multiplicity of the decay
products and decrease their relative momenta $\Delta p$. The final decay
products (i.e., the lightest dark bound states) then decay to SM anti-quarks
which subsequently form anti-nuclei. A challenging aspect of this scenario is
the need to model strong-coupling phenomena such as dark hadron showers.

Another challenging aspect of the tentatively identified events at AMS-02 is
their relativistic nature (i.e., large Lorentz boosts). Overcoming phase-space
suppressions of nuclear coalescence rates by considering scenarios where the
colliding particles have low relative momenta usually also results in anti-
helium products that are non-relativistic. There are however ways around this:
the anti-helium products in scenarios that start with the decay of a particle,
such as those considered in Refs. [13, 17], can be made relativistic if the
decaying particle is already boosted in the galactic rest frame. This can be
achieved if, e.g., the decaying particle is produced in turn from the earlier
decay of another heavier particle. Other works [12, 16, 18] have either
suggested acceleration mechanisms based on supernova shockwaves (similar to
the Fermi mechanism), or did not address this question.

In this paper, we propose a scenario where the anti-helium nuclei observed at
AMS-02 originate from sudden and localized “injections” (of BSM origin) of
anti-baryon number in our Galaxy in the form of energetic SM anti-quarks.
These particles subsequently thermalize into relativistically expanding,
optically thick fireballs with a net anti-baryon number. Anti-nuclei are
produced in each of these fireballs thermally through a nuclear reaction chain
similar to that operating during Big Bang Nucleosynthesis (BBN), albeit with
remarkable qualitative and quantitative differences due to the very different
(anti-)baryon–to–entropy ratios, timescales, and expansion dynamics involved.
The evolution of the fireball plasma after the initial injection of SM
particles is purely dictated by SM physics and, moreover, since this process
involves thermalization, depends only on certain bulk properties being
achieved by the injection, and for the most part not on the exact details of
the latter. However, owing to the inefficiency of weak interactions, there are
regions of parameter space where the anti-neutron–to–anti-proton ratio in the
plasma at the onset of the anti-nucleosynthetic processes may depend on the
details of the initial injection of SM particles. With that one exception, the
predicted anti-particle cosmic ray fluxes produced in our proposed scenario
are therefore both predictive and largely agnostic as to the microphysical
origin of the injections that seed these fireballs.

Assuming that the requisite injections can occur, we show that this scenario
could explain not only the tentatively identified anti-nucleus fluxes at
AMS-02, but also the relativistic Lorentz boosts of the detected particles.
The seemingly paradoxical requirements of a low-energy environment to foster
production of higher-$A$ anti-nuclei and the high energies required for
relativistic anti-nucleus products are reconciled naturally in our scenario by
the expansion dynamics of the fireball plasma: as the plasma expands, its
temperature falls adiabatically while its thermal energy is converted to bulk
kinetic energy by the work of its internal pressure. This allows a low-energy
environment for anti-nucleosynthesis to proceed, while at the same time
accelerating its products to relativistic speeds.

Of course, injections of the requisite amounts of SM anti-particles with the
correct properties to generate these fireballs cannot occur spontaneously: a
BSM mechanism is required. Suggestively, we show that collisions of certain
very heavy, macroscopic, composite dark objects (possibly a sub-fraction of
all of the dark matter) that carry SM anti-baryon number could at least
achieve a high-enough rate of injections with appropriate parameters to
explain the tentatively identified anti-helium events at AMS-02, provided that
a substantial fraction of the dark objects’ mass energy can be converted into
SM anti-quarks as a result of dark-sector dynamics triggered by the collision
(in some parameter regions, there may be a requirement to inject also a
comparably sized asymmetry of charged leptons). While this is encouraging, we
have not yet developed a detailed microphysical model that achieves the
necessary dark-sector dynamics in a way that is amenable to robust and
controlled understanding; however, we offer some speculative initial thoughts
on certain model constructions that we believe are promising avenues to
explore toward that goal. For the purposes of this paper, we ultimately leave
this question open; as it is crucial to providing a concrete realization of
the scenario we advance, however, we both intend to return to it in our own
future work and we encourage other work on it.

The remainder of this paper is structured as follows. In Sec. II, we review
the candidate anti-nuclei events observed by AMS-02. In Sec. III, we discuss
synthesis of anti-nuclei in an expanding relativistic fireball, discussing
first questions of thermalization after energy injection [Sec. III.1], then
turning to the fireball expansion dynamics [Sec. III.2], and the nuclear
reactions at play [Sec. III.3], before summarizing [Sec. III.4]. In Sec. IV,
we then discuss how the anti-nuclei thus produced would propagate to AMS-02
and give our scenario’s projections for the anti-nuclei spectra and event
rates [Sec. IV.1]; we also discuss other potential observables [Sec. IV.2]. In
Sec. V, we discuss a potential origin for the fireballs in collisions of
composite dark-matter states, showing first that the rates could work [Sec.
V.1], that the fireballs could be appropriately seeded if certain benchmarks
can be met [Sec. V.2], and then offering some speculative thoughts toward
particle physics models that may be worth further investigation to see it they
are able to achieve the necessary conditions [Sec. V.3]. We conclude in Sec.
VI. A number of appendices add relevant detail. Appendix A gives derivations
of various scaling laws for fireball expansion that we rely on in the main
text. Appendix B discusses the ratio of anti-nucleons from which the anti-
nucleosynthesis is initiated. Appendices C and D, respectively, give details
of the nuclear reaction networks and cross-sections we have used. Appendix E
discusses whether dynamical changes in the number of degrees of freedom during
fireball expansion are relevant. Appendix F discusses prompt versus slow
injections. Finally, Appendix G reviews an estimate of the AMS-02 rigidity-
dependent sensitivity to anti-helium events.

##  II The AMS-02 candidate anti-helium events

In this section, we summarize the data that is publicly available [2, 3, 4, 5,
6, 7] regarding the AMS-02 candidate anti-helium and anti-deuterium events.

# |  Event Date [mm/dd/yyyy] (Day of Year) |  $p$ [GeV] |  $m$ [GeV] | $Q/e$ | $\Gamma$ |  $\mathcal{R}$ [GV] | Ref(s).
---|---|---|---|---|---|---|---
1 |  09/26/2011 (269) | $33.1(1.6)$ | $2.93(36)$ | $-1.97(5)$ | $11.3(1.5)$ | $-16.80(92)$ | [3, 4]
2 |  12/08/2016 (—)444No later than | $40.3(2.9)$ | $2.96(33)$ |  $-2$(—) | $13.6(1.3)$ | $-20.2(1.5)$ | [2, 3]
3 |  06/22/2017 (173) | $32.6(2.5)$ | $3.81(29)$ | $-2.05(5)$ | $8.61(92)$ | $-15.9(1.3)$ | [3, 5, 7]
4 |  09/20/2022 (265) | — | $3.15(53)$ | $-2$ | — | — | [7]
Table 1: Parameters for individual candidate anti-helium events displayed in
the identified references: $p$ is momentum, $m$ is mass, $Q$ is charge ($e>0$
is the elementary charge), $\Gamma$ is the Lorentz factor, and
$\mathcal{R}=p/Q$ is rigidity. For identification purposes across the
references, the reported event date and corresponding day of year are given
(the date of event 2 is not given in the references; we give the date of its
first public presentation in Ref. [2]). Unavailable data are denoted by “—”.
We have set $\hbar=c=1$. Data for $p$, $m$, and $Q/e$ are given in the
references [speed data, $v=0.9973(5)$, is additionally given for event 2]; we
derived $\Gamma=\sqrt{1+(p/m)^{2}}$ [or $\Gamma=1/\sqrt{1-v^{2}}$ for event 2]
and $\mathcal{R}=p/Q$ and propagated uncertainties naïvely. For comparison,
$m_{{}^{3}\overline{\text{He}}}=2.81\,\mathinner{\mathrm{GeV}}$ and
$m_{{}^{4}\overline{\text{He}}}=3.73\,\mathinner{\mathrm{GeV}}$. Event # 3 is
clearly tentatively identified in Refs. [3, 7] as a candidate
${}^{4}\overline{\text{He}}$ event. Event # 4, which is similarly identified
as an ${}^{4}\overline{\text{He}}$ candidate in the oral presentation of Ref.
[7], is not included in our analysis; see discussion in text.

To our knowledge, the most recent scientific presentations on these candidate
events are Refs. [6, 7]. Ref. [6] provides graphical mass and rigidity
histogram data for 9 candidate events collected in $\sim 10\ \text{years}$ of
AMS-02 observations. Ref. [7] also appears to show an additional anti-helium
candidate event dated after Ref. [6], with a mass measurement consistent with
either ${}^{3}\overline{\text{He}}$ or ${}^{4}\overline{\text{He}}$, but
closer to the former. Table 1 shows the detailed basic parameter values for 3
of the 9 candidate events that have been shown publicly [2, 3, 4, 5, 7], as
well as the data available for the additional candidate event [7]; such
detailed data has not been presented for the other candidate events.

We base our analysis on the data for the 9 candidate events presented in Ref.
[6], which are consistent with the identification of 6 candidate
${}^{3}\overline{\text{He}}$ events, and 3 candidate
${}^{4}\overline{\text{He}}$ events in $T\sim 10\,\text{years}$ of AMS-02 data
(see discussion about $T$ below).555This identification is also consistent
with earlier presentations based on a smaller dataset with a total of 8
candidate anti-helium events [3, 5], of which 2 were tentatively identified as
${}^{4}\overline{\text{He}}$ candidates [3, 4], giving a 3:1 ratio for
${}^{3}\overline{\text{He}}:{}^{4}\overline{\text{He}}$.,666We do not
explicitly consider the “additional” event shown in Ref. [7] (event # 4 in
Table 1). This event is labeled in the slide deck for Ref. [7] to have
occurred on September 20, 2022, which is after the date of presentation of the
histograms in Ref. [6] on February 28, 2022; it is thus highly likely to be a
new event not previously discussed in past references before Ref. [7].
However, at the level of uncertainty regarding these events that we work in
this paper, whether or not we include this event has almost no relevant impact
on our discussion of the overall or relative event rates for the
${}^{3}\overline{\text{He}}$ and ${}^{4}\overline{\text{He}}$ events. Assuming
either that $T\sim 8.5\,\text{years}$ or $T\sim 10\,\text{years}$ for the data
in Ref. [6] (see discussion in main text), it is also entirely consistent
within statistical errors for one additional event to occur in the additional
integration time of relevance for the $T\sim 11\,\text{years}$ of data that
appears to be discussed in Ref. [7]. The mass histogram shown in Ref. [6] is
however more than broad enough to support an inference that some of the
candidate ${}^{3}\overline{\text{He}}$ events thus classified could in fact be
${}^{4}\overline{\text{He}}$ events, and vice versa (the additional event
shown in Ref. [7] could also be identified as either isotope within the
uncertainties). Nevertheless, throughout this paper, we adopt a $\sim 2:1$
event ratio for ${}^{3}\overline{\text{He}}$ vs. ${}^{4}\overline{\text{He}}$
as fiducial.

Of the candidate events in Ref. [6], 3 have rigidities $\mathcal{R}=p/Q$
(where $p$ is the particle momentum and $Q=qe$ its charge) in the approximate
range $-40\,\text{GV}\lesssim\mathcal{R}\lesssim-35\,\text{GV}$, while the
other 6 candidate events have rigidities in the range
$-25\,\text{GV}\lesssim\mathcal{R}\lesssim-15\,\text{GV}$. The data in Table 1
however make clear that care should be taken _not_ to associate the 3
candidate events with larger $|\mathcal{R}|$ shown in the histogram in Ref.
[6] with the 3 candidate ${}^{4}\overline{\text{He}}$ events and the remainder
with the 6 candidate ${}^{3}\overline{\text{He}}$ events: indeed, event # 3 in
Table 1 has a rigidity in the low range, but a mass most consistent with
${}^{4}\overline{\text{He}}$. One robust inference however is that all of the
candidate events for which the necessary data are available to make this
determination are highly relativistic, with $\Gamma\sim 10$ being a typical
fiducial value (an extreme range of possible Lorentz factors for the other
candidate events based on the rigidity and mass data shown in Ref. [6] is
roughly $6\lesssim\Gamma\lesssim 36$, assuming $q=-2$).

Given the $\sim 1.3\times 10^{8}$ confirmed helium events in the AMS-02 data
as of Ref. [6], the ratio of the 9 candidate anti-helium events in that
reference to the confirmed helium events is approximately $7\times 10^{-9}$.
Likewise, Ref. [7] reports $\sim 1.45\times 10^{8}$ confirmed helium events;
with 10 total anti-helium events, this yields the same ratio within
uncertainties.

Finally, for the purposes of converting total candidate anti-helium event
numbers into rate estimates, we assume that the relevant AMS-02 data-taking
period over which all 9 events discussed in Ref. [6] were observed is
$T=10\,\text{years}$. There is some uncertainty on this given available
information: the data-taking period of relevance to the 9 events reported as
of Ref. [6] may actually be slightly shorter, $T\approx 8.5\,\text{years}$
(this point is not made unambiguously clear in Ref. [6]). There is thus an
uncertainty on the required rates of $\mathcal{O}(15\%)$ arising from the
incompleteness of the publicly available information on this point; this is
however significantly smaller than the uncertainty on the rates owing to the
small statistics of the relevant event samples.

Additionally, Ref. [7] makes a new report of 7 candidate anti-deuterium
(${\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}$) events in $T\sim
11\,\text{years}$ of AMS-02 data. However, only scant information is available
regarding these events: a single histogram showing that the charge-sign–mass
product for these events lies in the range
$-2.1\leq\text{sgn}[q]\times(m/\text{GeV})\leq-1.8$.

##  III Fireball anti-nucleosynthesis

In this section, we show how an abrupt localized injection in a region of
space of a large amount of energy and anti-baryon number in the form of SM
anti-quarks can lead to formation of a locally thermalized fireball comprised
of a plasma mixture of free anti-nucleons, pions, leptons, and photons. As
this fireball expands hydrodynamically [19, 20, 21, 22, 23, 24], it cools
adiabatically, eventually permitting Standard Model anti-nucleosynthetic
processes to produce bound anti-nuclei, including anti-helium, in the hot and
dense environment. Owing to the dynamics of the expansion in the interesting
region of parameter space, the radial bulk expansion velocity of the fireball
constituents also becomes relativistic by the time of anti-nucleosynthesis,
resulting in the anti-helium thus produced being released into the
interstellar medium relativistically (assuming the fireball is located within
our galaxy). We also demonstrate that the fireball expansion shuts off anti-
nucleosynthetic processes prior to the attainment of complete nuclear
statistical equilibrium, thereby allowing the amount of
${}^{3}\overline{\text{He}}$ produced, after decays of unstable products, to
be larger than the amount of ${}^{4}\overline{\text{He}}$.

The thermalized nature of the initial fireball caused by the requisite
injection of energy and anti-baryon number conveniently erases most of the
history of how such a thermal state comes to exist. As such, the conclusions
we reach in this section as to the anti-nuclear outputs of the fireball
expansion are largely independent of the model details of how such a fireball
comes to exist; instead, they depend only on bulk physical properties of the
fireball state, such as its temperature, anti-baryon–to–entropy ratio, and
radius at certain critical points in its evolution. The exception to this is
that, due to incomplete thermalization via inefficient weak interactions, the
results we obtain can depend on the net charge on the hadronic sector that is
injected primarily via the BSM process that seeds the fireball; this
essentially becomes another parameter we must consider.

Of course, this history-erasure does not alleviate the requirement for a
concrete BSM mechanism by which the requisite energy and anti-baryon number
injection could occur; we comment on this aspect of the problem in Sec. V but
ultimately defer this to future work.

In the remainder of this section, we discuss: the thermalization following
energy and anti-baryon number injection [Sec. III.1], the dynamics of the
fireball expansion [Sec. III.2], and the anti-nucleosynthetic and other
outputs of the fireball expansion [Sec. III.3]. We summarize in Sec. III.4.

###  III.1 Thermalization

We will be mainly concerned with SM fireballs whose initial size of order
$10^{-4}\,\text{m}\lesssim R_{0}\lesssim 1\,\text{m}$, and which double in
size on characteristic timescales $\tau$ of order
$10^{-13}\,\text{s}\lesssim\tau_{0}\lesssim 10^{-9}\,\text{s}$. The injected
particles will interact through various Standard Model processes which overall
tend to bring themselves toward local thermal equilibrium.

For simplicity, we limit ourselves to the parameter space where the injected
SM energy density is such that the would-be temperature of the SM plasma is
below the QCD scale,777At higher energy densities, the plasma would lie above
the QCD phase transition, a quark-gluon plasma (QGP) would form, and one would
need more careful treatment of the evolution of the fireball back through the
phase transition as it cools.  $\sim 200\,\mathinner{\mathrm{MeV}}$. The path
toward thermalization in this regime will involve a process of hadronization
where the injected anti-quarks confine and fragment into anti-nuclei and
copious pions. The rate of hadronization is set by the QCD scale $\Lambda_{\rm
QCD}\sim 200\,\mathinner{\mathrm{MeV}}\sim\left(3\times
10^{-24}\,\text{s}\right)^{-1}$ in the rest frame of the confining or
fragmenting particles and will be time-dilated at some level in the center of
mass frame of the injected gas of particles. Unless the initial Lorentz
factors of the injected anti-quarks are extremely high, $\gamma_{*}\gtrsim
10^{11}$ (corresponding to energies $E\gtrsim
10^{10}\,\mathinner{\mathrm{GeV}}$), a case we do not consider, hadronization
occurs essentially instantaneously compared to the initial expansion timescale
of the fireball, $\tau_{0}\gtrsim 10^{-13}\,\text{s}$.

The subsequent thermalization should proceed similarly to that in analogous
setups involving hadrons found in the contexts of heavy-ion colliders [25, 26,
27] and in the early Universe before [28, 29] and during [30, 31] BBN. A
proper description of the thermalization process would require solving a
complex set of Boltzmann equations dictating the time-evolution of the energy
spectra of relevant particles and resonances. We instead provide rough
estimates of the typical rates of the processes involved. The typical rates of
strong interactions involving pions (e.g.,
$\bar{n}+\pi^{-}\rightarrow\bar{p}+\pi^{0}$), $\Gamma_{\rm strong}$;
electromagnetic interactions for charged particles that are relativistic at a
given temperature (e.g., $\gamma\gamma\leftrightarrow e^{+}e^{-}$),
$\Gamma_{\rm EM}$; and weak interactions for relativistic particles (e.g.,
$e^{+}e^{-}\leftrightarrow\bar{\nu}_{e}\nu_{e}$), $\Gamma_{\rm weak}$, are
respectively given by

$\displaystyle\Gamma_{\rm strong}$
$\displaystyle\sim\left(\frac{m_{\pi}T}{2\pi}\right)^{3/2}e^{-m_{\pi}/T}\left<\sigma_{\rm
strong}v\right>$ $\displaystyle\sim(1\times
10^{-20}\,\text{s})^{-1}\left(\frac{T}{100\,\mathinner{\mathrm{MeV}}}\right)^{3/2}e^{-m_{\pi}/T}\>,$
(1) $\displaystyle\Gamma_{\rm EM}$ $\displaystyle\sim\alpha_{\rm
EM}^{2}T\sim(1\times
10^{-19}\,\text{s})^{-1}\left(\frac{T}{100\,\mathinner{\mathrm{MeV}}}\right)\>,$
(2) $\displaystyle\Gamma_{\rm weak}$ $\displaystyle\sim
G_{F}^{2}T^{5}\sim\left(5\times
10^{-10}\,\text{s}\right)^{-1}\left(\frac{T}{100\,\mathinner{\mathrm{MeV}}}\right)^{5}\>,$
(3)

with $m_{\pi}\approx 140\,\mathinner{\mathrm{MeV}}$ and
$\left<\sigma_{\text{strong}}v\right>\sim 1\,\text{mb}$ [31, 32]. These rates
suggest that within the timescale of expansion of the initial fireball
$\tau_{0}\gtrsim 10^{-13}\,\text{s}$, the SM particles can interact
efficiently through strong and electromagnetic processes, but generically not
through weak processes, unless the temperature is significantly higher than
$100\,\mathinner{\mathrm{MeV}}$.

In the cases that we consider, strong and electromagnetic processes are
sufficient for thermally populating all SM particle species with masses below
a given temperature. For instance, anti-protons can be created from charged
pions and anti-neutrons through strong interactions, photons can be produced
via bremsstrahlung from charged pions and the decay of neutral pions, leptons
pairs can be produced in photon annihilations.

However, if the weak interactions are indeed always inefficient after SM
particle injection, charge would need to be separately conserved in the
leptonic and hadronic sectors, because reactions such as
$\pi^{+}+e^{-}\leftrightarrow\pi^{0}+\nu_{e}$ (and similar crossed or charge
conjugated reactions, as well as similar reactions with the anti-baryons) that
would allow charge to be exchanged between those two sectors would not be
efficient. Therefore, were the initial injection of particles to be such that
the the net charge is zero in both sectors (e.g., only net charge- and color-
neutral combinations of anti-quarks are injected), the charged pions would
have a chemical potential that fixes their population asymmetry to the number
of anti-protons. If the plasma temperature falls, this may have what we will
see to be undesirable consequences for our scenario such as a dramatic
depletion of the anti-proton abundance when the symmetric, thermal pion
abundance becomes Boltzmann suppressed, while strong interactions between the
asymmetric $\pi^{+}$ abundance and the $\bar{p}$ population remain efficient
for some time.888One possible injection-model-dependent solution to this issue
is an initial injection of SM particles that is still net color- and charge-
neutral, but which in addition to having a large net negative baryon number,
also has a large _charged lepton asymmetry_ and therefore baked-in opposite-
sign EM charge asymmetries in the hadronic and leptonic sectors that cannot be
removed by the inefficient weak interactions. We discuss this issue in more
detail in Sec. III.3.3 and Appendix B, as well as comments in Sec. V.3.

Immediately upon the completion of (partial) thermalization, the fireball
energy is dominated by radiation comprised of a subset of photons and
relativistic $e^{\pm}$, $\mu^{\pm}$, and $\pi^{0,\pm}$ (depending on the
temperature); while its mass is dominated by anti-baryons in the form of free
$\bar{n}$ and $\bar{p}$. The thermal pressure of the trapped radiation drives
an adiabatic expansion of the fireball, which can be described
hydrodynamically owing to the short mean free path $\ell$ of the constituent
particles, $\ell\sim
v/\Gamma_{\rm\\{strong,\,EM\\}}<1/\Gamma_{\rm\\{strong,\,EM\\}}\ll R_{0}$.
This is the topic of the next subsection.

###  III.2 Relativistic fireball expansion

We now turn to considering the dynamics of the expanding thermalized plasma.

We treat the plasma as a spherically symmetric perfect fluid and consider only
its radial expansion. In what follows, quantities defined in the comoving
rest-frame of the radially moving fluid will be marked with a prime ′, while
those defined in the fireball center-of-mass (CoM) frame are unprimed.

If $t$ is the CoM time co-ordinate and $r$ is the CoM-frame radial coordinate
centered on the fireball, then the radial expansion of the fireball is
described by two relativistic fluid equations. Respectively, these are an
equation encoding energy conservation and one encoding momentum conservation
of the fireball fluid [33, 34]:

$\displaystyle\partial_{t}\left[\gamma^{2}(\rho^{\prime}+p^{\prime})\right]+\frac{1}{r^{2}}\partial_{r}\left[r^{2}\gamma^{2}v\left(\rho^{\prime}+p^{\prime}\right)\right]$
$\displaystyle=\partial_{t}p^{\prime}\>,$ (4)
$\displaystyle\partial_{t}\left[\gamma^{2}v(\rho^{\prime}+p^{\prime})\right]+\frac{1}{r^{2}}\partial_{r}\left[r^{2}\gamma^{2}v^{2}\left(\rho^{\prime}+p^{\prime}\right)\right]$
$\displaystyle=-\partial_{r}p^{\prime}\>,$ (5)

where $\gamma(t,r)\equiv 1/\sqrt{1-[v(t,r)]^{2}}$ is the Lorentz factor
associated with the CoM-frame bulk radial velocity $v(t,r)$ of the fluid at a
given radius $r$, and $\rho^{\prime}(t,r)$ and $p^{\prime}(t,r)$ are the
comoving energy density and pressure of the fluid.

While Eqs. (4) and (5) are not analytically solvable, it is understood that
the fireball will undergo an initial phase of rapid acceleration to
relativistic speeds [33, 35], whereupon analytically tractable evolution takes
over. The acceleration is initially limited to a thin shell near the surface
of the fireball where the pressure gradient is strong. This surface expansion
then generates inward-traveling rarefaction waves which accelerate the bulk of
the fireball. Numerical simulations [36, 37, 38] suggest that the radial
layers comprising an initially static thermal fireball will accelerate to
relativistic radial velocities within the time it takes for sound waves (whose
speed in a radiation-dominated fluid is $c_{s}\approx 1/\sqrt{3}$) to cover
the initial radius $R_{0}$ of the fireball. Upon arrival at the center of the
of the fireball, the waves get reflected outward, creating a strong
underdensity at that point. This underdensity creates a hollow structure at
the center of the fireball that becomes more pronounced over time, essentially
turning the thermal plasma into a radially moving shell of density
concentration with an initial thickness $\sim R_{0}$. The radial bulk
velocities $v$ of all the radial layers of the plasma shell soon approach the
speed of light $v\approx 1$ and the resulting nearly flat velocity profile
keeps the shell thickness $\sim R_{0}$ constant until much later times.

Once the radial layers of the plasma shell are moving relativistically with
Lorentz factors $\gamma\gtrsim\text{few}$, its subsequent evolution follows
simple scaling laws, which can be parameterized by the initial temperature
$T_{0}$ and the initial radius (and thickness) $R_{0}$ when
$\gamma\sim\text{few}$, as well as the (conserved) anti-baryon–to–entropy
ratio $\bar{\eta}$ of the plasma shell [33, 35].999Because the plasma
temperature $T^{\prime}$ will be related to the shell radius $R$ through
simple scaling laws, the precise definitions of $T_{0}$ and $R_{0}$ beyond
what we have described are not important. One can more generally define
$T_{0}$ as the temperature when the radius of the shell is $R_{0}$ (or vice
versa), where the reference $R_{0}$ (or $T_{0}$) is arbitrarily chosen. So
long as the reference point is sufficiently early in the shell evolution that
the energy of the plasma is still dominated by radiation, the specific choice
of reference point is unimportant. For simplicity, in this paper we choose the
reference point to be when $\gamma\sim\text{few}$. Note that $T_{0}$ and
$\bar{\eta}$ naturally have $\mathcal{O}(1)$ radial variation over the shell;
however, thanks to the smoothing effect of sound waves, this variation is
typically mild [33]. For simplicity, our parametric estimates will assume a
single (average) value of $T_{0}$ and $\bar{\eta}$ for the entire shell. We
will however take into account the $\Delta\gamma/\gamma\sim 1$ radial
variation of the Lorentz factor between the inner and outer radius of the
shell because, as we will see, it has an important consequence at the late
stages of the shell evolution; where relevant, we assume that $\gamma$ varies
monotonically across the shell, being larger on the outer edge of the shell.

The expansion in the relativistic regime proceeds in three stages (see
Appendix A for derivations):

  1. 1.

Acceleration. As long as the shell energy density is radiation-dominated, the
internal pressure of the radiation continually accelerates the radial velocity
of the shell, converting radiation energy into anti-baryon kinetic energy in
the process. In this phase, the shell thickness in the center-of-mass frame of
the fireball $\Delta R$ remains approximately constant, $\Delta R\sim R_{0}$,
and the bulk Lorentz factor scales with the radius of the shell $R$ as
$\gamma\propto R$.

  2. 2.

Coasting. Once the radiation energy drops below the anti-baryon kinetic
energy, the radiation can no longer accelerate the shell appreciably and from
then on the shell simply coasts at its terminal Lorentz factor

$\displaystyle\Gamma\sim\frac{T_{0}}{\bar{\eta}m_{p}}\>,$ (6)

where $m_{p}$ is the proton mass.

  3. 3.

Spreading. The (assumed monotonic) variation of the Lorentz factor
$\Delta\gamma/\gamma\sim 1$ between the inner and outer radii of the shell
translates to velocity variation $\Delta v\sim\Delta\gamma/\gamma^{3}\sim
1/\gamma^{2}$. This leads to increasing shell thickness which becomes
important when $R\gtrsim\Gamma^{2}R_{0}$ and is well captured by

$\displaystyle\Delta
R\sim\left(1+\frac{1}{\gamma^{2}}\frac{R}{R_{0}}\right)R_{0}\>.$ (7)

To sum up, the Lorentz factor of the shell scales as

$\displaystyle\gamma(R)\sim\begin{cases}R/R_{0},&R\lesssim\Gamma R_{0}\>;\\\
\Gamma,&R\gtrsim\Gamma R_{0}\end{cases}\>.$ (8)

Neglecting changes in the number of degrees of freedom $g_{*}$ as various
species (pions, muons, electrons, positrons) fall out of thermal equilibrium
and get Boltzmann suppressed,101010This amounts to setting $g_{*}=2$ (from
photons) throughout. We discuss in Appendix E how accounting for changes in
$g_{*}$ yields only mild, $\mathcal{O}(1)$ quantitative changes to the picture
presented here, but leaves the qualitative evolution unchanged.  the average
comoving temperature $T^{\prime}$ of the shell scales as

$\displaystyle
T^{\prime}(R)\sim\begin{cases}T_{0}\left(R_{0}/R\right),&\phantom{i\Gamma
R_{0}<}R\lesssim\Gamma R_{0}\>;\\\ (T_{0}/\Gamma)\left(\Gamma
R_{0}/R\right)^{2/3},&\Gamma R_{0}\lesssim R\lesssim\Gamma^{2}R_{0}\>;\\\
(T_{0}/\Gamma^{5/3})\left(\Gamma^{2}R_{0}/R\right),&\phantom{i\Gamma
R_{0}<}r\gtrsim\Gamma^{2}R_{0}\end{cases}\>.$ (9)

These scaling laws are in agreement with the numerical simulations in Refs.
[39, 33, 35].

Note that the co-moving dynamical expansion timescale (i.e., $e$-folding
timescale) for the fireball, as measured by the change in its comoving
temperature,111111This turns out to be a convenient measure of fireball
expansion for our later purposes; see also Appendix C at Eq. (91). can then be
approximated as

$\displaystyle\tau^{\prime}(T^{\prime})$
$\displaystyle\equiv\frac{T^{\prime}}{\gamma\left|dT^{\prime}/dt\right|}$
$\displaystyle\sim\begin{cases}\displaystyle
R_{0},&\displaystyle\phantom{i\frac{T_{0}}{\Gamma^{5/3}}\lesssim}T^{\prime}\gtrsim\frac{T_{0}}{\Gamma}\>;\\\\[12.91663pt]
\displaystyle
R_{0}\left(\frac{T_{0}/\Gamma}{T^{\prime}}\right)^{3/2},&\displaystyle\frac{T_{0}}{\Gamma^{5/3}}\lesssim
T^{\prime}\lesssim\frac{T_{0}}{\Gamma}\>;\\\\[12.91663pt] \displaystyle\Gamma
R_{0}\left(\frac{T_{0}/\Gamma^{5/3}}{T^{\prime}}\right),&\displaystyle\phantom{i\frac{T_{0}}{\Gamma^{5/3}}\lesssim}T^{\prime}\lesssim\frac{T_{0}}{\Gamma^{5/3}}\>,\end{cases}$
(10)

where the three cases here map to the three different expansion regimes for
the fireballs discussed above. We display the evolution of this timescale for
a set of benchmark parameters that will be of interest (see Sec. III.3) in
Fig. 1.

Figure 1: The comoving expansion timescale $\tau^{\prime}$ (solid black line)
as a function of the comoving temperature $T^{\prime}$ [cf. Eq. (10)] of a
fireball with $T_{0}=100\,\mathinner{\mathrm{MeV}}$, $R_{0}=1\,\text{mm}$, and
$\bar{\eta}=10^{-2}$. The three vertical dashed lines mark, from left to
right, the transition at $T^{\prime}=T_{0}/\Gamma$ from the acceleration phase
to the coasting phase, the transition at $T^{\prime}=T_{0}/\Gamma^{5/3}$ from
the coasting phase to the spreading phase, and the point at which photon
decoupling occurs [corresponding to $R=R_{\rm thin}$; cf. Eq. (14) with
$\varepsilon n_{\bar{p}}\sim n_{\bar{B}}$]. The orange shaded region is thus
the region in which the plasma expands as a single tightly coupled fluid
through its various expansion phases (as annotated), while the unshaded region
where is the region where the plasma has decoupled.

The fluid evolution described above applies as long as the photons remain
tightly coupled with the charged particles in the plasma. Eventually, this
assumption is violated and the photons decouple; we can estimate when this
occurs as follows. Photons in the plasma scatter dominantly with the electrons
and positrons, with the Thomson cross section
$\sigma_{T}=(8\pi/3)(\alpha^{2}/m_{e}^{2})$. In our parameter space of
interest, it is the case that photon decoupling always occurs at a temperature
well below the electron mass, after any symmetric thermal population of
charged leptons have annihilated away. The opacity of the plasma is then due
to the remaining charged leptons that must exist to guarantee net neutrality
of the plasma; their identity depends on some details of the BSM injection
process because (as we will see) typical expansion timescales are too short
for any muon asymmetry that is present to be guaranteed to decay by the time
of this decoupling. In what follows, we parametrize the population of residual
positrons as having a CoM-frame density $n_{e^{+}}=\varepsilon n_{\bar{p}}$,
where $\varepsilon$ can be a BSM-injection dependent parameter, but which we
typically expect to be $\varepsilon\sim\mathcal{O}(1)$; because of the
$\sigma_{T}\propto m^{-2}$ scaling, residual muons are not relevant to our
estimates unless $\varepsilon\lesssim(m_{e}/m_{\mu})^{2}\sim 2\times 10^{-5}$.

The radial optical depth $\kappa(R)$ for a photon emitted from a radius $r\sim
R$ inside the shell and escaping to infinity is then [40]121212The integral in
Eq. (11) reflects what we see in the CoM frame of the fireball. Since the
photon is moving at the speed of light, $dr/dt=1$, the differential $dr$ can
be thought of as a time increment $dt$. The combination $(1-v)dr$ is the
length of plasma the photon overtakes in a time increment $dt=dr$. As the
photon propagates to $r=\infty$, the $n_{\bar{p}}$ eventually loses support
when the photon overtakes the outer boundary of the plasma shell. Since the
velocity difference between the photon and the outer radius of the shell is
$1-v$, this occurs only after the photon has displaced by $\Delta r=\Delta
R/(1-v)\approx 2\gamma^{2}\Delta R$. Note that $\gamma^{2}\Delta R\gtrsim R$
in any expansion regime as per Eq. (7). Hence, the integral has an effective
upper bound at $\gtrsim R$ above the lower bound, not at $\mathcal{O}(\Delta
R)$ above the lower bound.

$\displaystyle\kappa(R)$ $\displaystyle\sim\int_{\sim R}^{\infty}\varepsilon
n_{\bar{p}}\sigma_{T}(1-v)dr\sim\kappa_{0}\frac{R_{0}^{2}}{\gamma^{2}R\Delta
R}\>,$ (11)

where the anti-baryon density inside the shell is taken to be roughly constant
and scales inversely with the volume of the plasma shell $n_{\bar{B}}\propto
1/(R^{2}\Delta R)$ and $1-v\approx 1/(2\gamma^{2})$ for $v\approx 1$.131313The
optical depth for a photon on a non-radial trajectory is given by Eq. (11),
but with the replacements $v\rightarrow v\cos\theta$ and $dr\rightarrow
dr/\cos\theta$, where $\theta$ is the angle between the photon velocity and
the radial direction at the location of the photon. Due to relativistic
beaming, most of the thermal photons are concentrated within
$\theta\lesssim\gamma^{-1}\ll 1$ in the CoM frame, yielding
$1-v\cos\theta\approx
1-v+v\theta^{2}/2\approx(\gamma^{-2}+\theta^{2})/2\sim\gamma^{-2}$. Thus the
radial optical depth found in Eq. (11) provides an $\mathcal{O}(1)$-accurate
estimate for the optical depths experienced by most non-radially moving
photons. We have also defined the initial optical depth

$\displaystyle\kappa_{0}$
$\displaystyle=\varepsilon\bar{\eta}T_{0}^{3}\sigma_{T}R_{0}$
$\displaystyle\sim
10^{11}\left(\frac{\bar{\eta}}{10^{-2}}\right)\left(\frac{T_{0}}{100\,\mathinner{\mathrm{MeV}}}\right)^{3}\left(\frac{R_{0}}{1\,\text{mm}}\right)\left(\frac{\varepsilon
n_{\bar{p}}}{n_{\bar{B}}}\right)\>,$ (12)

we have set a leading coefficient $\sim(\zeta(3)/\pi^{2})g_{*}\sim 1$ and
neglected other $\mathcal{O}(1)$ factors.

We define $t_{\rm thin}$ to be the rough moment in time141414At $t=t_{\rm
thin}$, the _entire_ bulk of the plasma becomes optically thin due to the
decrease in the number density of charged particles over time. Note that most
photons that decouple around $t=t_{\rm thin}$ do so from a point located
inside the bulk of the plasma region. This temporal occurrence is distinct
from, and should not be confused with, the fact that photons can also
continually decouple from the plasma throughout its evolution by escaping from
it spatially. That is, the fireball always has a photosphere near its outer
edge, where there is a steep spatial gradient in the density of particles. The
time-dependent photospheric radius $R_{\rm PS}(t)$ of the fireball is to be
distinguished with the fireball radius at $t=t_{\rm thin}$, $R(t_{\rm thin})$.
The amount of photons escaping continually through the spatial photosphere is
set by the timescale for photons to diffuse over the thickness of the
fireball, which is extremely long due to the tiny photon mean free path. We
assume that this loss is always negligible before $t=t_{\rm thin}$ in the
parameter space we consider. where the plasma optical depth for photons in the
bulk of the fireball at $r\sim R(t_{\rm thin})$ becomes unity, $\kappa\sim 1$.
It turns out that the initial optical depth $\kappa_{0}$ is many orders of
magnitude larger than unity in our parameter space of interest; see Eq. (12).
This implies that $t_{\rm thin}$ always occurs deep into the final expansion
stage (i.e., the spreading stage), and can be estimated as

$\displaystyle R_{\rm thin}$ $\displaystyle\sim\kappa_{0}^{1/2}R_{0}$ (13)
$\displaystyle\sim
300\,\text{m}\times\left(\frac{\bar{\eta}}{10^{-2}}\right)^{1/2}\left(\frac{T_{0}}{100\,\mathinner{\mathrm{MeV}}}\right)^{3/2}$
$\displaystyle\qquad\times\left(\frac{R_{0}}{1\,\text{mm}}\right)^{3/2}\left(\frac{\varepsilon
n_{\bar{p}}}{n_{\bar{B}}}\right)^{1/2}\>.$ (14)

At the fiducial parameter point, $\tau^{\prime}(t_{\text{thin}})\sim
R_{\text{thin}}/\Gamma\sim 10^{-7}\,\text{s}$, which is less than the rest-
frame muon lifetime, as noted above. When the fireball radius hits $R_{\rm
thin}$, the bulk of the plasma becomes optically thin and a burst of photons
is released. This is also the approximate moment at which the anti-baryons and
other particles that were coupled to the fireball plasma (see Sec. III.3) are
released into the interstellar medium (see Sec. IV), assuming that the
fireball was located in our Galaxy.

###  III.3 Nuclear physics

The monotonically decreasing co-moving temperature $T^{\prime}(R)$ of the
fireball plasma given at Eq. (9) implies that it eventually becomes
thermodynamically favorable for bound anti-nuclei to form.

Were thermodynamic equilibrium among the lightest few anti-nuclei species to
be achieved (analogous to the situation in Big Bang Nucleosynthesis [BBN]),
almost the entirety of the available anti-neutron abundance would be converted
into ${}^{4}\overline{\text{He}}$, leading to a final configuration dominated
by $\bar{p}$ and ${}^{4}\overline{\text{He}}$, with only trace amounts of
other complex light anti-nuclei. Given that AMS-02 has tentatively identified
similar numbers of ${}^{3}\overline{\text{He}}$ and
${}^{4}\overline{\text{He}}$ candidates (up to a factor of a few; statistics
are small), such an outcome would not be phenomenologically viable.

To understand how to avoid this outcome, consider a key feature of how the
analogous process of ordinary BBN proceeds. The most efficient nuclear-
reaction pathway to ${}^{4}\text{He}$ has as an initial step free-neutron
capture on hydrogen to form deuterium D: $n+p\rightarrow\text{D}+\gamma$ [41],
with the D then being processed by further nuclear burning to other light
elements. However, the relatively low deuterium binding energy
$B_{\text{D}}\approx 2\,\mathinner{\mathrm{MeV}}$ and the low baryon-to-
entropy ratio $\eta\approx 6\times 10^{-10}$ during BBN make deuterium prone
to photodissociation back to free neutrons and protons: this is the famous
“deuterium bottleneck” [42, 43, 44, 45]. Consequently, ${}^{4}\text{He}$
production in BBN was delayed until the temperature of the primordial plasma
cooled down significantly below $B_{\text{D}}$, whereupon the abundance of
photons capable of dissociating deuterium was considerably Boltzmann
suppressed, enabling the deuterium abundance to rise. Crucially, in the BBN
realized in our Universe, the deuterium bottleneck was overcome while
thermodynamic equilibrium was still being maintained among the light species:
nuclear reaction rates were still sufficiently fast compared to Hubble
expansion that, once deuterium was capable of being created without being
photodissociated, it was rapidly burned to tritium and ${}^{3}\text{He}$, and
then further to ${}^{4}\text{He}$.

But standard BBN successfully overcame the deuterium bottleneck while
maintaining thermodynamic equilibrium only marginally. If the expansion rate
of the Universe were to have been sufficiently larger, neutron capture on
hydrogen would have decoupled (i.e., frozen out) before the bottleneck could
have been overcome. In that case, the output of the nucleosynthesis would not
have been dictated by thermodynamic equilibrium among the light nuclei.
Instead, the immediate nucleosynthesis products in this scenario would have
been mostly free protons and neutrons, with smaller abundances of deuterium,
tritium, ${}^{3}\text{He}$, and ${}^{4}\text{He}$ being produced in amounts
controlled by the relative rates of nuclear reactions that produce them, which
can be comparable to one another. Accounting for the fact that unstable
neutrons later decay to protons and tritium later decays to ${}^{3}\text{He}$,
the final output in this counterfactual case could easily have been such that
$n_{p}\gg n_{{}^{3}\text{He}}\sim n_{{}^{4}\text{He}}$.

In what follows, we show that parameter space exists for which the analogous
anti-nucleosynthesis occurring in our expanding fireball remains in this
“stuck in the bottleneck” regime, yielding phenomenologically viable amounts
of ${}^{3}\overline{\text{He}}$ as compared to ${}^{4}\overline{\text{He}}$.

####  III.3.1 Preliminaries

In order to separate changes in the number density of an element due to
nuclear reactions from that due to the fireball expansion, in this section we
describe the evolution of a nuclear species $i$ in terms of its fractional
abundance $X_{i}$, defined as

$\displaystyle X_{i}\equiv\frac{n_{i}^{\prime}}{n_{\bar{B}}^{\prime}}\>,$ (15)

where $n_{i}^{\prime}$ is the number density of element $i$ and
$n^{\prime}_{\bar{B}}$ is the anti-baryon number density. We consider only
$i\in\\{\bar{p},\bar{n},{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}},{\bar{\text{T}}},{}^{3}\overline{\text{He}},{}^{4}\overline{\text{He}}\\}$.
We obtain the evolution of the abundances of nuclear elements $X_{i}$ by
numerically solving the Boltzmann equations detailed in Appendix C describing
the simplified network of nuclear reactions among these species,151515We solve
a simplified, partial reaction network accounting only for light species,
which is acceptably accurate for our purposes. In principle, more accurate
results could be obtained by using a modified version of BBN nucleosynthesis
codes such as PRyMordial [46], PRIMAT [47], or AlterBBN [48].  using the
nuclear cross sections shown in Appendix D. The results of these numerical
computations are summarized in Figs. 2–5.

While those numerical results are of course more accurate, we also wish to
gain an understanding of, and intuition for, the most important nuclear
processes at work, and determine the dependencies of the final anti-helium
isotope abundances on the fireball parameters $(T_{0},R_{0},\bar{\eta})$. In
what follows, we therefore develop an analytical understanding that reproduces
the gross features of the numerical results.

The main approximation we employ in our analytical arguments is as follows. In
general, nuclear species with higher mass numbers $A$ are produced from those
with lower $A$ in a sequence of successive two-body nuclear reactions. Our
numerical analysis shows that in the fireball parameter space that yields
$X_{{}^{3}\overline{\text{He}}}\gtrsim X_{{}^{4}\overline{\text{He}}}$, a
hierarchy is maintained between the nuclear abundances with successive mass
numbers, namely $X_{\bar{n}}+X_{\bar{p}}\gtrsim X_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}\gtrsim
X_{{\bar{\text{T}}}}+X_{{}^{3}\overline{\text{He}}}\gtrsim
X_{{}^{4}\overline{\text{He}}}$, as depicted in Figs. 2 and 3. Furthermore,
that analysis shows that anti-nucleosynthesis occurs mainly at temperatures
$T^{\prime}\sim 100-200\,\mathinner{\mathrm{keV}}$ low enough that all of the
endothermic reverse nuclear reactions to the exothermic ones considered in
this subsection are negligible, except for the photodissociation of anti-
deuterium responsible for the bottleneck in the first place,161616This
endothermic process is an exception because (a) it involves photons, which are
highly abundant compared to anti-baryons due to the low anti-baryon–to–entropy
ratio $\bar{\eta}\ll 1$ we consider; and (b) the binding energy of anti-
deuterium is unusually small, $B_{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}\approx 2.2\,\mathinner{\mathrm{MeV}}$. which we thus take into
account in our analysis. These observations suggest that, instead of solving
the whole nuclear reaction network at once, we can treat the nuclear reactions
sequentially; that is, we can consider elements produced earlier in the chain
of successive two-body reactions as fixed sources for reactions later in the
chain of successive reactions, neglecting back-reaction on those sources
arising from those later reactions.

In what follows, we first discuss relevant expansion timescales, and then
proceed to discuss in turn heavier and heavier anti-nuclei synthesized in this
approximate sequential paradigm. Finally, we summarize and discuss other,
non–anti-nucleosynthetic outputs.

####  III.3.2 Expansion timescales

Anti-nucleosynthesis in the fluid rest frame of the expanding fireball is
qualitatively similar to BBN in that there are various nuclear reactions
occurring in an adiabatically expanding background [42, 43, 44, 45]. However,
it differs from the BBN in important ways. The plasma in our scenario is
spatially finite and expands relativistically into vacuum.171717Our anti-
nucleosynthesis process resembles in this respect that occurring in the
context of gamma-ray burst [49, 50, 51, 52] or heavy-ion collision [53], but
is otherwise very different. This of course leads to a non-trivial dependence
of the dynamical expansion timescale $\tau^{\prime}$ on the comoving
temperature $T^{\prime}$, as shown at Eq. (10).

The parameter space that is viable for our model is roughly
$10^{-4}\,\text{m}\lesssim R_{0}\lesssim 1\,\text{m}$ (i.e.,
$10^{-13}\,\text{s}\lesssim\tau_{0}\lesssim 10^{-9}\,\text{s}$),
$10\,\mathinner{\mathrm{MeV}}\lesssim T_{0}\lesssim
200\,\mathinner{\mathrm{MeV}}$, and $\Gamma\sim T_{0}/\bar{\eta}m_{p}\sim 10$
(i.e., $10^{-3}\lesssim\bar{\eta}\lesssim 10^{-2}$). We displayed the scalings
of $\tau^{\prime}(T^{\prime})$ in Fig. 1 for benchmark parameters in these
ranges. The results are quantitatively and qualitatively very different
compared to the Hubble time as a function of temperature during BBN; as we
will see, this leads to important differences between fireball anti-
nucleosynthesis and BBN. As we will show, fireball anti-nucleosynthesis in
this parameter space commences at temperature
$100\,\mathinner{\mathrm{keV}}\lesssim T_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}^{\prime}\lesssim 200\,\mathinner{\mathrm{keV}}$, which
satisfies $T_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}^{\prime}\lesssim
T_{0}/\Gamma^{5/3}$ and thereby always falls in the “spreading” phase of the
fireball expansion.

Figure 2: The abundances of nuclear species
$X_{i}=n^{\prime}_{i}/n^{\prime}_{\bar{B}}$ (solid and dashed colored lines,
as identified in the legend) as a function of the comoving fireball
temperature $T^{\prime}$, computed by numerically solving the Boltzmann
equations for a simplified nuclear-reaction network detailed in Appendix C,
for a fireball with $T_{0}=100\,\mathinner{\mathrm{MeV}}$,
$R_{0}=1.5\,\text{mm}$, $\bar{\eta}=10^{-2}$ ($\Gamma=10$). Also shown is
$Y_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}},\gamma}=n^{\prime}_{\gamma}(E^{\prime}_{\gamma}>Q_{D})/n^{\prime}_{\bar{B}}$
(dotted black line), the abundance of photons with energies
$E^{\prime}_{\gamma}$ above the anti-deuteron photodissociation threshold
$Q_{D}$. Initially, nuclear bound states are essentially non-existent, apart
from ${\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}$ whose abundance is kept at an
exponentially small value due to the high abundance of photodissociating
photons, $Y_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}},\gamma}\gg
X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$. As the fireball cools down,
$Y_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}},\gamma}$ decreases while
$X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$ increases. Eventually at
$T=T^{\prime}_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\sim
140\,\mathinner{\mathrm{keV}}$ [cf. Eq. (26)], $Y_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}},\gamma}$ drops sufficiently low that photodissociation becomes
inefficient, thus marking the onset of anti-nucleosynthesis: the absence of
photodissociation allows $X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$ to
rise significantly, which enables nuclear reactions involving
${\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}$ to produce ${\bar{\text{T}}}$ whose
presence, in turn, enables the production of ${}^{4}\overline{\text{He}}$. The
rise in $X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$ also enables
${}^{3}\overline{\text{He}}$ production but the produced
${}^{3}\overline{\text{He}}$ quickly converts into ${\bar{\text{T}}}$. Since
the nuclear reaction rates are proportional to the antibaryon number density
$n^{\prime}_{\bar{B}}\propto T^{\prime 3}$, most of the anti-nucleus
production occurs within the first few $e$-foldings of expansion after
$Y_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}},\gamma}$ drops below
$X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$. Nuclear species depicted in
dashed lines are stable in the timescale of fireball evolution (until it
becomes optically thin) but are expected to decay to the species shown by the
solid line of the same color during their journey to the solar system.  Figure
3: The abundances of nuclear species
$X_{i}=n^{\prime}_{i}/n^{\prime}_{\bar{B}}$ (with $i$ as annotated in the
legend) released when a fireball with an initial radius $R_{0}$ becomes
optically thin, computed by numerically solving the Boltzmann equations for a
simplified nuclear-reaction network, as detailed in Appendix C. Here, we set
$T_{0}=100\,\mathinner{\mathrm{MeV}}$ and $\bar{\eta}=10^{-2}$ ($\Gamma=10$).
While the hierarchy $X_{\bar{p}}\approx X_{\bar{n}}\gtrsim
X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\gtrsim
X_{{\bar{\text{T}}}}\gtrsim X_{{}^{4}\overline{\text{He}}}$ is maintained, the
abundances of ${\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}$, ${\bar{\text{T}}}$,
${}^{4}\overline{\text{He}}$ released by the fireball scale with $R_{0}$,
respectively, as $X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\propto R_{0}$
[cf. Eq. (29)], $X_{{\bar{\text{T}}}}\propto X_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}^{2}\propto R_{0}^{2}$ [cf. Eq. (31)],
$X_{{}^{4}\overline{\text{He}}}\propto
X_{{\bar{\text{T}}}}X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}^{2}\propto
R_{0}^{4}$ [cf. Eq. (33)]. Besides that, ${}^{3}\overline{\text{He}}$ is
produced promptly in negligible amount,
$X_{{}^{3}\overline{\text{He}}}\lesssim 10^{-6}$ [cf. Eq. (32)]. Nuclear
species depicted in dashed lines are stable on the timescale of the fireball
evolution (until it becomes optically thin) but are expected to subsequently
decay to the species shown in with solid lines of the same color on timescales
short compared to their propagation time in the Milky Way (e.g.,
${\bar{\text{T}}}$ later decays to ${}^{3}\overline{\text{He}}$ with a half-
life of $\sim 12$ years in the ${\bar{\text{T}}}$ rest frame).

####  III.3.3 Anti-nucleon abundance

At temperatures $10\,\mathinner{\mathrm{MeV}}\lesssim T^{\prime}\lesssim
200\,\mathinner{\mathrm{MeV}}$, nuclear bound states have not formed and all
the anti-baryons reside in unbound anti-neutrons and anti-protons. Anti-
neutrons can, in principle, convert to and from anti-protons through both weak
(e.g., $\bar{n}+e^{-}\leftrightarrow\bar{p}+\nu_{e}$) and strong (e.g.,
$\bar{n}+\pi^{-}\leftrightarrow\bar{p}+\pi^{0}$) processes [54, 31, 32]. If at
least one of these processes is efficient, the relative abundance of these
anti-nucleons is initially kept at its chemical equilibrium value,
$n^{\prime}_{\bar{n}}/n^{\prime}_{\bar{p}}=\text{exp}\left[-(m_{n}-m_{p})/T^{\prime}\right]$.
In the early Universe, the matter analogue of both processes were efficient at
some point after hadronization. Then, strong processes decouple first as pions
rapidly decay and annihilate away, and hence the pre-BBN freeze-out abundances
of neutrons and protons were determined by the later-occurring decoupling of
weak interactions. By contrast, in the fireball anti-nucleosynthesis scenario
we consider, the typically short timescales of the fireball expansion render
weak interactions inefficient at all times. Consequently, the anti-nucleons
freeze out as soon as the pion-mediated strong interconversion processes
become inefficient.

After the fireball has thermalized, the following strong-mediated charge
exchange reactions (SMCER) are initially in equilibrium [30] (see discussion
about $T^{\prime}_{\bar{n}\bar{p}}$ below)

$\displaystyle\bar{p}+\pi^{+}\leftrightarrow\bar{n}+\pi^{0}\quad(Q=5.9\mathinner{\mathrm{MeV}})\>;$
$\displaystyle\bar{n}+\pi^{-}\leftrightarrow\bar{p}+\pi^{0}\quad(Q=3.3\mathinner{\mathrm{MeV}})\>.$

Additionally,181818The reactions
$\pi^{0}\pi^{0}\leftrightarrow\pi^{+}\pi^{-}$,
$\pi^{0}\rightarrow\gamma\gamma$, $\pi^{0}\pi^{0}\leftrightarrow\gamma\gamma$,
and $\pi^{+}\pi^{-}\leftrightarrow\gamma\gamma$ are all in equilibrium.
$\mu_{\gamma}=0$, $\mu_{\pi^{0}}=0$, and $\mu_{\pi^{-}}=-\mu_{\pi^{+}}$. These
imply

$\displaystyle\mu_{\bar{n}}-\mu_{\bar{p}}=\mu_{\pi^{+}}\>.$ (16)

The chemical equilibrium anti-neutron–to–anti-proton ratio for $T^{\prime}\ll
m_{n}-\mu_{\bar{n}},m_{p}-\mu_{\bar{p}}$ is thus given by

$\displaystyle\left(\frac{n_{\bar{n}}^{\prime}}{n_{\bar{p}}^{\prime}}\right)_{\rm
ch}\approx
e^{-\frac{m_{n}-m_{p}}{T^{\prime}}+\frac{\mu_{\pi^{+}}}{T^{\prime}}}\>.$ (17)

where $m_{n}-m_{p}\approx 1.3\mathinner{\mathrm{MeV}}$. Note that the chemical
potential of the charged pions, $\mu_{\pi^{+}}$, depends on the physics before
the fireball has thermalized via efficient strong and EM interactions.
Therefore, it is model-dependent. For instance, it depends on whether
electroweak interactions were ever efficient in this pre-thermalization stage,
and on some details of the BSM particle injection process that seeds the
fireball. For simplicity, we neglect the chemical potential of charged pion in
our analysis here by assuming191919As we discuss Appendix B, this specific
assumption is equivalent to assuming that there is a net negative charge in
the hadronic sector of the plasma that has a certain very specific value:
defining $X_{Q}\equiv-(n^{\prime}_{Q})^{\text{hadronic}}/n^{\prime}_{\bar{B}}$
as at Eq. (70), we would have $X_{Q}=0.56$; cf. Eq. (23). This charge is
compensated by opposite charge in the leptonic sector so that the plasma as a
whole is net EM-neutral as expected from fireballs seeded by EM-neutral dark
states. However, our results as stated in the main text are unchanged
qualitatively, and change quantitatively by only $\mathcal{O}(1)$ factors, so
long as it is approximately true that $X_{Q}\sim\mathcal{O}(1/2)$ by the time
that the SMCER become inefficient. We also show in Appendix B that we may even
be able to tolerate values as small as $X_{Q}\sim 10^{-2}$, although that
changes some conclusions stated in the main text in a qualitative fashion.

$\displaystyle\left|\mu_{\pi^{+}}\right|\ll m_{n}-m_{p}\>.$ (18)

We discuss the model-dependence of $\mu_{\pi^{+}}$ and the case when
$\mu_{\pi^{+}}$ is non-negligible in Appendix B.

The pion-mediated strong interactions decouple at a temperature
$T^{\prime}=T^{\prime}_{\bar{n}\bar{p}}$ where the pion abundance becomes
sufficiently Boltzmann suppressed that the $\Gamma_{\rm strong}(T^{\prime})$
found in Eq. (1) goes below the fireball expansion rate $1/\tau^{\prime}$
found in Eq. (10). We find that, numerically,
$T^{\prime}_{\bar{n}\bar{p}}\approx 6\,\mathinner{\mathrm{MeV}}$ invariably in
the whole parameter space that is viable for our scenario. The freeze-out
value of the anti-neutron–to–anti-proton ratio
$n^{\prime}_{\bar{n}}/n^{\prime}_{\bar{p}}$ can be approximated by its
chemical-equilibrium value at that time

$\displaystyle\left.\frac{n^{\prime}_{\bar{n}}}{n^{\prime}_{\bar{p}}}\right|_{{\rm
ch},\,T_{\bar{n}\bar{p}}^{\prime}}\approx
e^{-(m_{n}-m_{p})/T_{\bar{n}\bar{p}}^{\prime}}\approx 0.8\>.$ (19)

If $\left|\mu_{\pi^{+}}\right|\gtrsim m_{n}-m_{p}$, unlike what we have
assumed, then the freeze-out value of
$n^{\prime}_{\bar{n}}/n^{\prime}_{\bar{p}}$ and our subsequent results would
change; however, as long as $\left|\mu_{\pi^{+}}\right|\lesssim
T^{\prime}_{\bar{n}\bar{p}}\approx 6\mathinner{\mathrm{MeV}}$, these changes
are only $\mathcal{O}(1)$ and most of our conclusions remain valid. See
Appendix B for further discussion.

Moreover, while neutron decay is an important phenomenon in the BBN that was
realized in the early Universe, in our scenario anti-neutrons do not decay
until well after anti-nucleosynthesis finishes. Assuming that only a small
fraction of the $\bar{p}$ and $\bar{n}$ are burned to higher nuclei (true
throughout our parameter space of interest), we will thus have, for all times
relevant for the anti-nucleosynthesis in the expanding fireball, the
following:

$\displaystyle X_{\bar{n}}(T^{\prime})$ $\displaystyle\approx
X_{\bar{n}}(T^{\prime}_{\bar{n}\bar{p}})\equiv X^{\text{ch}}_{\bar{n}};$ (20)
$\displaystyle X_{\bar{p}}(T^{\prime})$ $\displaystyle\approx
X_{\bar{p}}(T^{\prime}_{\bar{n}\bar{p}})\equiv X^{\text{ch}}_{\bar{p}}\>,$
(21)

where

$\displaystyle X^{\text{ch}}_{\bar{n}}$ $\displaystyle\approx
0.8X^{\text{ch}}_{\bar{p}}\>,$ $\displaystyle
X^{\text{ch}}_{\bar{n}}+X^{\text{ch}}_{\bar{p}}$ $\displaystyle\approx 1\>,$
(22)

implying that

$\displaystyle X^{\text{ch}}_{\bar{n}}$ $\displaystyle\approx 0.44\>,$
$\displaystyle X^{\text{ch}}_{\bar{p}}$ $\displaystyle\approx 0.56\>.$ (23)

####  III.3.4 Anti-deuterium production

Anti-deuterium is produced primarily through the reaction202020The anti-
deuterium formation releases some amount of energy density to the plasma,
given by the total binding energy of the anti-deuterium formed:
$B_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}n^{\prime}_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\sim
B_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}\bar{\eta}X_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}T^{\prime 3}$, where $B_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}\approx 2.2\,\mathinner{\mathrm{MeV}}$. This amounts to a tiny
fraction of the radiation energy density $\sim T^{\prime 4}$ in the parameter
space of our interest, where $\bar{\eta}\lesssim 10^{-2}$,
$X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\lesssim 10^{-2}$, and
$T^{\prime}\gtrsim 100\,\mathinner{\mathrm{keV}}$ when the anti-deuterium
forms. $\bar{n}+\bar{p}\rightarrow{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}+\gamma$. Initially, however, the reverse reaction
(photodissociation) is in equilibrium and the high abundance of photons with
energies above the anti-deuterium binding energy $B_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}\approx 2.2\,\mathinner{\mathrm{MeV}}$ suppresses the (quasi-
equilibrium) anti-deuterium abundance, which is given by the Saha equation:

$\displaystyle X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}},\rm
ch}\approx\bar{\eta}\left(\frac{T^{\prime}}{m_{p}}\right)^{3/2}e^{B_{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}/T^{\prime}}.$ (24)

This continues until the abundance of photons with sufficient energy to
photodissociate anti-deuterium,

$\displaystyle Y_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}\gamma}=\frac{n^{\prime}_{\gamma}(E^{\prime}_{\gamma}\gtrsim
B_{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}})}{n^{\prime}_{\bar{B}}}\sim\frac{1}{\bar{\eta}}\frac{B_{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}^{2}}{T^{\prime 2}}e^{-B_{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}/T^{\prime}}\>,$ (25)

starts to fall below the anti-deuterium abundance; i.e.,
$Y_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}\gamma}\sim
X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$. The temperature at that point
can be estimated as

$\displaystyle T_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}^{\prime}$
$\displaystyle\approx\frac{B_{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}{4.6-\ln\bar{\eta}+(7/4)\ln\left(4.6-\ln\bar{\eta}\right)}$
$\displaystyle\approx 140-170\,\mathinner{\mathrm{keV}},$ (26)

where the displayed range of values $T_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}^{\prime}$ corresponds to the range of viable anti-
baryon–to–entropy ratios $10^{-3}\lesssim\bar{\eta}\lesssim 10^{-2}$. As
mentioned earlier, in our the parameter space of interest
$T_{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}^{\prime}$ falls in the spreading
phase of the fireball expansion (i.e., it satisfies
$T_{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}^{\prime}\lesssim
T_{0}/\Gamma^{5/3}$) and, neglecting the mild logarithmic dependence on
$\bar{\eta}$, the fireball expansion timescale at decoupling of the
photodissociation reactions is

$\displaystyle\tau^{\prime}_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$
$\displaystyle\equiv\tau^{\prime}(T_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}^{\prime})$ $\displaystyle\approx 5.1\times
10^{-10}\,\text{s}\left(\frac{T_{0}}{100\,\mathinner{\mathrm{MeV}}}\right)\left(\frac{R_{0}}{\text{mm}}\right)\left(\frac{\Gamma}{10}\right)^{-2/3}\>,$
(27)

where we have set $T_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}^{\prime}=140\mathinner{\mathrm{keV}}$ (corresponding to
$\bar{\eta}=10^{-3}$). After the photodissociation of
${\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}$ decouples at
$T_{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}^{\prime}$, at which point the
fireball expansion timescale is $\tau^{\prime}_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}$, anti-deuterium production through
$\bar{n}+\bar{p}\rightarrow{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}+\gamma$ is
no longer thwarted, and so the anti-deuterium abundance rises monotonically.
At around the same time, heavier elements that rely on anti-deuterium burning
as an initial step begin to be populated sequentially. Since the product of
the fireball expansion timescale and the nuclear reaction rates that form any
of the light elements scales as $\propto n^{\prime}_{\bar{B}}\left<\sigma
v\right>\tau^{\prime}\propto T^{\prime 2}$ during the spreading phase
($T^{\prime}\lesssim T_{0}/\Gamma^{5/3}$), these nuclear reactions are most
efficient in populating the light elements in the first fireball-expansion
$e$-fold or so after the decoupling of ${\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}$ photodissociation (when anti-baryon density is the highest),
before the anti-baryon density is significantly diluted by the expansion.

Moreover, as we operate in the regime where anti-deuterium is not efficiently
burned to more complex nuclei (see the next sub-subsection), a simple estimate
for the final anti-deuterium abundance can be obtained by assuming the anti-
deuterium abundance is that generated by neutron–proton fusion reactions
operating in a single dynamical expansion timescale at the point of anti-
deuterium photodissociation freeze-out. We estimate that abundance to be

$\displaystyle X^{\text{prompt}}_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$
$\displaystyle\approx k_{\bar{2}}n^{\prime}_{\bar{B}}\left<\sigma
v\right>_{\bar{n}\bar{p}}\tau^{\prime}_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}X_{\bar{n}}^{\text{ch}}X_{\bar{p}}^{\text{ch}}$ (28)
$\displaystyle\approx 4.0\times
10^{-3}\times\left(\frac{k_{\bar{2}}}{0.60}\right)$
$\displaystyle\qquad\times\left(\frac{T_{0}}{100\,\mathinner{\mathrm{MeV}}}\right)^{2}\left(\frac{R_{0}}{\text{mm}}\right)\left(\frac{\Gamma}{10}\right)^{-5/3},$
(29)

where we took the value of cross-section $\left<\sigma
v\right>_{\bar{n}\bar{p}}$ to be that at $T^{\prime}\sim
140\,\mathinner{\mathrm{keV}}$ (the lower end of the range of values for
$T^{\prime}_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$); see Appendix D. In
writing the above results, we have used $\bar{\eta}\sim T_{0}/(\Gamma m_{p})$
and we manually inserted an $\mathcal{O}(1)$ prefactor $k_{\bar{2}}\approx
0.60$ in Eq. (28), such that the final result is in better agreement with what
we obtained by numerically solving the Boltzmann equations at this benchmark
point.

We are interested in the regime where
$X^{\text{prompt}}_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\ll 1$: i.e., the
anti-deuterium production decouples before its abundance rises to
$X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\sim 1$.

####  III.3.5 Anti-deuterium burning

During and slightly after the production of anti-deuterium, a small fraction
of it also burns through the following dominant channels:

$\displaystyle{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}+{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}$
$\displaystyle\rightarrow{}^{3}\overline{\text{He}}+\bar{n}\>,$
$\displaystyle{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}+{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}$
$\displaystyle\rightarrow{\bar{\text{T}}}+\bar{p}\>,$

with essentially equal branching fractions (both are strong-mediated nuclear
reactions). Because we are analyzing production in the regime where anti-
deuterium is not efficiently burned to more complex anti-nuclei, we may treat
the abundance of anti-deuterium as a fixed source which acts to populate the
more complex nuclei over roughly a single dynamical expansion timescale after
the anti-deuterium are produced. As such, the prompt production of
${}^{3}\overline{\text{He}}$ and ${\bar{\text{T}}}$ can be estimated as

$\displaystyle X_{{\bar{\text{T}}}}^{\text{prompt}}\approx
X_{{}^{3}\overline{\text{He}}}^{\text{prompt}}\approx
k_{\bar{3}}n^{\prime}_{\bar{B}}\left<\sigma v\right>_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}\tau^{\prime}_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}(X^{\text{prompt}}_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}})^{2}\>,$ (30)

where we have evaluated all the quantities at $T^{\prime}\sim
T^{\prime}_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$, assumed
$X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\gg
X_{{}^{3}\overline{\text{He}}},X_{{\bar{\text{T}}}}$ around the time of this
production, and manually included an $\mathcal{O}(1)$ numerical prefactor
$k_{\bar{3}}\approx 0.15$ [cf. the factor $k_{\bar{2}}$ introduced in Eq.
(28)].

Alternative production channels for ${}^{3}\overline{\text{He}}$ and
${\bar{\text{T}}}$ are ${\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}+\bar{p}\rightarrow{}^{3}\overline{\text{He}}+\gamma$ and
${\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}+\bar{n}\rightarrow{\bar{\text{T}}}+\gamma$, respectively;
however, we verified numerically that the former production channel is
negligible as long as $X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\gg
10^{-5}$ and the latter is negligible as long as $X_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}\gg 10^{-4}$, which are always satisfied in the parameter
space we consider. It is understood that these process are inefficient because
they both suffer from photon-emission suppression (i.e., they are
electromagnetic-mediated, rather than strong-mediated, nuclear reactions).

The reaction ${\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}+{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}\rightarrow{}^{4}\overline{\text{He}}+\gamma$ can also proceed
with a branching ratio of $\sim 10^{-7}$ (it is electromagnetically mediated).
Because of this small branching fraction, prompt ${}^{4}\overline{\text{He}}$
production through this channel,
$X^{\text{prompt}}_{{}^{4}\overline{\text{He}}}\sim
n^{\prime}_{\bar{B}}(10^{-7}\left<\sigma v\right>_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}})\tau_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}^{\prime}(X^{\text{prompt}}_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}})^{2}$, is negligible compared to other, deuterium–tritium-
burning channels that we discuss below so long as
$X_{{\bar{\text{T}}}}/X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\gg 10^{-7}$
around the time of production.

####  III.3.6 Anti-helium-3 burning

The strong-mediated reaction
${}^{3}\overline{\text{He}}+\bar{n}\rightarrow{\bar{\text{T}}}+\bar{p}$ is
also present. It is also extremely efficient in part because it is not Coulomb
suppressed. It thus gives the one counterexample to our earlier statement that
we can ignore back-reaction on sequentially produced species: this reaction
burns essentially all the anti-helium-3 that are produced primarily (via
${\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}+{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}\rightarrow{}^{3}\overline{\text{He}}+\bar{n}$) to anti-
tritium. As such, the anti-helium-3 abundance is maintained at a very low,
quasi-equilibrium level: we numerically found that the residual
$X_{{}^{3}\overline{\text{He}}}$ never exceeds $\sim 10^{-6}$. At the same
time, the ${\bar{\text{T}}}$ abundance is roughly doubled because our estimate
at Eq. (30) indicated roughly equal production abundances for the two $A=3$
anti-nuclei before this depletion reaction was accounted for. Our estimates of
the anti-tritium and anti-helium-3 abundances after this burning should
therefore be revised to

$\displaystyle X^{\text{burn}}_{{\bar{\text{T}}}}$ $\displaystyle\approx
2X^{\text{prompt}}_{{\bar{\text{T}}}}\approx
2k_{\bar{3}}n^{\prime}_{\bar{B}}\left<\sigma
v\right>_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}\tau^{\prime}_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}(X^{\text{prompt}}_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}})^{2}\>,$ (31) $\displaystyle
X^{\text{burn}}_{{}^{3}\overline{\text{He}}}$ $\displaystyle\lesssim
10^{-6}\>.$ (32)

####  III.3.7 Anti-tritium burning

Anti-tritium burns efficiently to anti-helium-4 through the following dominant
process:

$\displaystyle{\bar{\text{T}}}+{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}$
$\displaystyle\rightarrow{}^{4}\overline{\text{He}}+\bar{n}\>;$

assuming that $X_{{\bar{\text{T}}}}\gtrsim X_{{}^{4}\overline{\text{He}}}$
throughout the burning, we find that the prompt production is

$\displaystyle X^{\text{prompt}}_{{}^{4}\overline{\text{He}}}\approx
k_{\bar{4}}n^{\prime}_{\bar{B}}\left<\sigma
v\right>_{{\bar{\text{T}}}{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}\tau_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}^{\prime}X_{{\bar{\text{T}}}}^{\text{burn}}X_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}^{\text{prompt}}\>,$ (33)

where we again manually introduced an $\mathcal{O}(1)$ prefactor
$k_{\bar{4}}\approx 0.20$ to better match our numerical results.

####  III.3.8 Final anti-nucleosynthesis products

Figure 4: The fireball parameter space. The initial temperature $T_{0}$ and
radius $R_{0}$ of the fireball are defined at the point right after the
fireball plasma has reorganized itself into a shell moving with an average
Lorentz factor $\gamma\sim\text{few}$. The various styles of blue lines show
contours of constant values of ratio of the _injected_ anti-helium-4 and anti-
helium-3 abundances,
$N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}|_{\text{inj.}}$.
These contours were obtained by numerically solving the Boltzmann equations
for a simplified nuclear-reaction network detailed in Appendix C, while fixing
the anti-baryon–to–entropy ratio $\bar{\eta}$ such that the terminal Lorentz
factor of the shell is $\Gamma\sim T_{0}/(\bar{\eta}m_{p})=10$. The orange
lines show contours of constant $T_{0}^{4}R_{0}^{3}$, and provide estimates,
up to numerical factors, of the total injection energy in the form of anti-
quarks required to create the fireball corresponding to a parameter space
point of interest. The black star indicates the benchmark parameter point
defined at Eq. (43). The purple region is the parameter space where the
fireball would thermalize at a temperature above the QCD phase transition
(making it a quark–gluon plasma [QGP]), a regime we avoid to keep our analysis
tractable. In the the gray region, the would-be temperature of the fireball
$T_{0}$ is so low that pions are too Boltzmann suppressed to facilitate the
thermalization of the injected anti-quarks; i.e., $\Gamma_{\rm
strong}(T_{0})\lesssim R_{0}^{-1}$.  Figure 5: Output space. The _injected_
anti-helium isotope ratio
$N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}|_{\text{inj.}}$
and the typical _injected_ Lorentz factors $\Gamma$ of the anti-particles are
two key observables we seek to explain with the fireball anti-nucleosynthesis
scenario we propose in this paper. Publicly available AMS-02 tentative data
currently favor _observed_ values of
$N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}|_{\text{obs.}}\approx
1/2$ and $\Gamma\sim 10$ (this figure does not include propagation effects;
see Sec. IV). Dashed and dotted lines show the injected anti-helium isotope
ratio
$N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}|_{\text{inj.}}$
obtained by numerically solving the Boltzmann equations for a simplified
nuclear-reaction network detailed in Appendix C, while dot-dashed lines show
the analytical approximation, Eq. (41). The black star indicates the benchmark
parameter point defined at Eq. (43). The analytical prediction on the anti-
helium isotope ratio has a precise quadratic dependence
$N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}|_{\text{inj.}}\propto
c^{2}$ on the combination of parameters
$c\equiv(T_{0}/100\,\mathinner{\mathrm{MeV}})^{2}(R_{0}/\text{mm})(\Gamma/10)^{-5/3}$;
although it is only strictly speaking valid when $c\lesssim 2.1$ [see
discussion below Eq. (42)], we also show it here for $c=5$, where it is still
reasonably accurate. The numerical computation of
$N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}|_{\text{inj.}}$
confirms the expected strong dependence on $c$, but it also shows mild
sensitivity to $(T_{0},R_{0},\Gamma)$ variation orthogonal to $c$. Shown in
purple is the parameter space where fireball thermalizes into a quark gluon
plasma (QGP), a regime we avoid to keep our analysis tractable. In the
parameter space colored in gray, the would-be temperature of the fireball
$T_{0}$ is so low that pions are too Boltzmann suppressed to facilitate the
thermalization of the injected anti-quarks; i.e., $\Gamma_{\rm
strong}(T_{0})\lesssim R_{0}^{-1}$.

Anti-neutrons have a mean rest-frame lifetime of $\mathcal{O}(15)$ minutes
(decaying to an anti-proton), while anti-tritium decays to anti-helium-3 via
the beta decay
${\bar{\text{T}}}\rightarrow{}^{3}\overline{\text{He}}+e^{+}+\nu_{e}$ with a
rest-frame half-life of 12.3 years. Even accounting for Lorentz factors
$\Gamma\sim 10$, the $\bar{n}$ and ${\bar{\text{T}}}$ will decay on
$\mathcal{O}(100)\,\text{yr}$ timescales (at most) as viewed in the fireball
center-of-mass frame. After these decays, the remaining light anti-nuclei
outputs of our scenario are, at late time, given by

$\displaystyle X_{\bar{p}}$ $\displaystyle\approx 1\>,$ (34) $\displaystyle
X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$ $\displaystyle\approx
X^{\text{prompt}}_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\qquad\text{[see
Eq{}.~{}(\ref{eq:XD})]}\>,$ (35) $\displaystyle
X_{{}^{3}\overline{\text{He}}}$ $\displaystyle\approx
X_{{\bar{\text{T}}}}^{\text{burn}}$
$\displaystyle\approx\frac{2k_{\bar{3}}}{k_{\bar{2}}}\left(\frac{\left<\sigma
v\right>_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}}{\left<\sigma
v\right>_{\bar{n}\bar{p}}}\right)_{T^{\prime}_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}}\frac{(X^{\text{prompt}}_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}})^{3}}{X^{\text{ch}}_{\bar{n}}X^{\text{ch}}_{\bar{p}}}$ (36)
$\displaystyle\approx 2.5\times 10^{3}X_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}^{3}\>,$ (37) $\displaystyle X_{{}^{4}\overline{\text{He}}}$
$\displaystyle\approx\frac{k_{\bar{4}}}{k_{\bar{2}}}\left(\frac{\left<\sigma
v\right>_{{\bar{\text{T}}}{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}}{\left<\sigma
v\right>_{\bar{n}\bar{p}}}\right)_{T^{\prime}_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}}\frac{X^{\text{burn}}_{{\bar{\text{T}}}}(X^{\text{prompt}}_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}})^{2}}{X^{\text{ch}}_{\bar{n}}X^{\text{ch}}_{\bar{p}}}$ (38)
$\displaystyle\approx\frac{2k_{\bar{3}}k_{\bar{4}}}{(k_{\bar{2}})^{2}}\left(\frac{\left<\sigma
v\right>_{{\bar{\text{T}}}{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}\left<\sigma v\right>_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}}{(\left<\sigma
v\right>_{\bar{n}\bar{p}})^{2}}\right)_{T^{\prime}_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}}\frac{(X^{\text{prompt}}_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}})^{5}}{(X^{\text{ch}}_{\bar{n}}X^{\text{ch}}_{\bar{p}})^{2}}$
(39) $\displaystyle\approx 3.6\times 10^{7}X_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}^{5}\>;$ (40)

while all other products are negligible. Note that, in Eqs. (36) and (38), we
made use of Eqs. (28) and (35) to rewrite the factors of
$n_{\bar{B}}\tau^{\prime}_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$ in Eqs.
(30) and (33) in terms of $X^{\text{prompt}}_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}},\ X^{\text{ch}}_{\bar{n}},\ X^{\text{ch}}_{\bar{p}}$, and
$\langle\sigma v\rangle_{\bar{n}\bar{p}}|_{T_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}^{\prime}}$. We also inserted the numerical values of the
$\mathcal{O}(1)$ coefficients here: $2k_{\bar{3}}/k_{\bar{2}}\approx 0.50$,
$k_{\bar{4}}/k_{\bar{2}}\approx 0.33$, and
$2k_{\bar{3}}k_{\bar{4}}/(k_{\bar{2}})^{2}\approx 0.17$. If we take
$T_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}^{\prime}=140\,\mathinner{\mathrm{MeV}}$, for which
$\left<\sigma v\right>_{\bar{n}\bar{p}}\approx 2.0\,\mu\text{b}$,
$\left<\sigma v\right>_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\approx
1.9\,\text{mb}$, and $\left<\sigma
v\right>_{{\bar{\text{T}}}{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\approx
16\,\text{mb}$, the anti-helium isotope ratio injected into the interstellar
medium for a given $X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$ is given by

$\displaystyle\left.\frac{N_{{}^{4}\overline{\text{He}}}}{N_{{}^{3}\overline{\text{He}}}}\right|_{\text{inj.}}$
$\displaystyle\approx 1.4\times 10^{4}X_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}^{2}$ $\displaystyle\approx
0.22\left[\left(\frac{T_{0}}{100\,\mathinner{\mathrm{MeV}}}\right)^{2}\left(\frac{R_{0}}{\text{mm}}\right)\left(\frac{\Gamma}{10}\right)^{-5/3}\right]^{2}\>,$
(41)

where in the last line we have substituted
$X^{\text{prompt}}_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$ from Eq. (29).
Hence, to obtain comparable anti-helium-3 and anti-helium-4 abundances, the
fireball parameters $(T_{0},R_{0},\Gamma)$ must be such that the combination
of parameters

$\displaystyle\left(\frac{T_{0}}{100\,\mathinner{\mathrm{MeV}}}\right)^{2}\left(\frac{R_{0}}{\text{mm}}\right)\left(\frac{\Gamma}{10}\right)^{-5/3}\equiv
c\>,$ (42)

is an $\mathcal{O}(1)$ number; see Fig. 4. According to the simple analytical
estimates of this section, the tentative AMS-02–observed anti-helium isotope
ratio
$N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}|_{\text{inj.}}\approx
1/2$ corresponds to $c\approx 1.5$. Note that this estimate is made here for
the _injected_ ratio, without regard to the impact of propagation of the anti-
nuclei in the Galaxy or the isotope-dependent AMS-02 sensitivity, which we
discuss and account for in Sec. IV. There, we will show that
$X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\approx 5\times 10^{-3}$ is
required, corresponding to $c\approx 1.25$.

Note that the sequential production approximation we used to derive the
analytical predictions for $X_{i}$ [Eqs. (34)–(40)] is justified if
$X_{{\bar{\text{T}}}}^{\rm burn}\gtrsim X_{{}^{4}\overline{\text{He}}}^{\rm
prompt}$. This translates to
$\left.N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}\right|_{\rm
inj.}\lesssim 1$, or $c\lesssim 2.1$.

We have also numerically computed the anti-helium isotope ratios
$N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}|_{\text{inj.}}$
for different fixed values of $c$ using the set of Boltzmann equations
described in Appendix C; see Fig. 5. While these numerical results show
reasonable overall agreement with the
$N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}|_{\text{inj.}}\propto
c^{2}$ analytical scaling derived at Eq. (41), we find numerically that
$N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}|_{\text{inj.}}$
still varies mildly with $(\Gamma,T_{0},R_{0})$ if $c$ is held fixed.
Nevertheless, because we have tuned the numerical constants $k_{\bar{2}}$,
$k_{\bar{3}}$, and $k_{\bar{4}}$ in the analytical results to the numerical
computations, we find that this mild violation of the
$N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}|_{\text{inj.}}\propto
c^{2}$ scaling makes the analytical results for that isotope ratio at worst an
$\mathcal{O}(2)$ factor discrepant from the numerical results throughout the
viable parameter space. We also find that there is still reasonable agreement
of the analytical and numerical results for $c$ as large as $c\sim 5$,
notwithstanding the limitation $c\lesssim 2.1$ that we noted previously.

####  III.3.9 Non-nuclear outputs

In addition to anti-nuclei, the fireball evolution described above will result
in the injection of light SM particles throughout the galaxy.

When the fireball becomes optically thin, a burst of photons is released, with
an average energy $\sim\Gamma T_{\text{thin}}^{\prime}\sim
10\,\mathinner{\mathrm{keV}}$. We assume (and this is almost certainly the
case) that this dominates the integrated emission from the photosphere
throughout the previous expansion. Since the pions present in the fireball
remain in thermal equilibrium until at least
$T^{\prime}_{\bar{n}\bar{p}}\approx 6\,\mathinner{\mathrm{MeV}}\ll m_{\pi}$,
they are Boltzmann-suppressed prior to decoupling and therefore no significant
gamma-ray signal is expected from their decay.

Anti-neutrinos are continually produced through weak interactions, via both
inelastic weak interactions and decays. The dominant scattering production
occurs when the fireball first thermalizes and its temperature is the highest.
At this point, all the produced anti-neutrinos (which possess an average
kinetic energy $\sim T_{0}$) will escape the fireball since their mean free
path is $\ell_{\nu}\sim 1/(n\sigma)\sim 1/(G_{F}^{2}T_{0}^{5})\sim
100\,\text{mm}\gg R_{0}$, where $n\sim T_{0}^{3}$ is taken to be the (thermal)
electron or positron number density since $T_{0}\gg m_{e}$. Additional
neutrinos are produced as the pions within the fireball decay. These have an
average kinetic energy $\sim\Gamma(m_{\pi}-m_{\mu})$. There may also be a
neutrino contribution immediately on injection of the SM species that seed the
fireball, but this is model dependent.

Charged leptons present in the fireball may also be injected into the
interstellar medium after the plasma becomes optically thin. The number of
such leptons remaining upon annihilation is model dependent; it is set by the
initial charge asymmetry in the lepton sector. However, if the fireball as a
whole is electrically neutral, this injection should be dominated by positrons
with an average energy $\sim\Gamma m_{e}$, whose number cannot exceed the
total number of anti-baryons injected.

The observability of X-ray and lepton bursts is discussed in Sec. IV.2. We
find that for the benchmark parameters sufficient to explain the AMS-02
candidate anti-helium events, detection of these additional particles is
infeasible due to the low count of particles arriving at the Earth and, in
some cases, their low energies.

###  III.4 Summary

We have identified a parameter space (see Fig. 4) where a sudden and spatially
concentrated injection of energetic anti-quarks in our Galaxy triggers
(subject to certain properties of the injection) a series of events, dictated
purely by Standard Model physics, that lead to relativistic anti-helium anti-
nuclei being released with number ratios and Lorentz boosts roughly consistent
with AMS-02 observations (we discuss in Sec. IV how propagation effects modify
the observed number-ratios from the injected values we have thus far
discussed). Here, we summarize this predicted series of events and provide
benchmark values for key quantities at various points of the process; we
denote these benchmark quantities with a tilde and an appropriate subscript.

Following the anti-quark injection, the anti-quarks rapidly hadronize and
thermalize mainly via strong and electromagnetic processes into an optically
thick, adiabatically expanding fireball with conserved anti-baryon–to–entropy
ratio $\tilde{\bar{\eta}}=10^{-2}$. This fireball then undergoes a period of
rapid acceleration which turns it into a plasma shell moving with relativistic
radial speed, with an average Lorentz factor $\gamma\sim\text{few}$. Right at
the onset of this phase of its evolution, the temperature, outer radius, and
thickness of the shell are in the ballpark of

$\displaystyle\tilde{T}_{0}$ $\displaystyle=100\,\mathinner{\mathrm{MeV}},$
$\displaystyle\tilde{R}_{0}$ $\displaystyle=1.5\,\text{mm}\ ,$
$\displaystyle\Delta\tilde{R}_{0}$ $\displaystyle\sim R_{0}\>.$ (43)

The subsequent evolution of the plasma shell proceeds in three sequential
stages (see Fig. 1):

  1. 1.

Acceleration: the shell continues to radially accelerate under its own thermal
pressure with its average Lorentz factor $\gamma$ increasing linearly with the
shell’s outer radius $R$, and keeping its thickness approximately constant,
$\Delta R\sim\tilde{R}_{0}$.

  2. 2.

Coasting: as $\gamma$ is approaching close to the terminal radial bulk Lorentz
boost $\tilde{\Gamma}=10$, the shell enters the second stage of expansion
where it simply coasts with an approximately constant Lorentz factor
$\gamma\approx\tilde{\Gamma}$, again keeping its thickness approximately
constant, $\Delta R\sim\tilde{R}_{0}$.

  3. 3.

Spreading: the expansion timescale becomes long enough that the radial
velocity difference between the innermost and outermost layers of the shell
causes the shell’s thickness to increase significantly over time.

We now describe how the fireball’s particle content evolves as it expands and
cools down. Initially, while anti-nuclei are still absent due to rapid
photodissociation of ${\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}$, anti-neutrons
and anti-protons are kept in detailed balance by pion-mediated interconversion
processes such as $\bar{n}+\pi^{-}\leftrightarrow\bar{p}+\pi^{0}$. This
continues until they finally decouple at a comoving temperature
$T_{\bar{n}\bar{p}}^{\prime}\approx 6\,\mathinner{\mathrm{MeV}}$, at which
temperature their relative abundance freezes out at212121 For the purposes of
this summary discussion, we are assuming the appropriate charge asymmetry
$X_{Q}$ on the hadronic sector is achieved at injection (i.e., that
$\mu_{\pi^{+}}=0$ at $T^{\prime}=T^{\prime}_{\bar{n}\bar{p}}$); qualitatively
similar results are however obtained so long as anti-neutron–to–anti-proton
ratio remains $\mathcal{O}(1)$. See discussion in Sec. III.3.3 and Appendix B.
$n_{\bar{n}}/n_{\bar{p}}\approx 0.8$.

Rapid photodissociation of any fusion-produced ${\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}$ ceases only deep in the final (spreading) expansion stage,
when the comoving temperature, outer radius, and thickness of the plasma shell
are about

$\displaystyle\tilde{T}_{\rm D}^{\prime}$ $\displaystyle\approx
140\,\mathinner{\mathrm{keV}},$ $\displaystyle\tilde{R}_{\rm D}$
$\displaystyle\approx 2\,\text{m}\ ,$ $\displaystyle\Delta\tilde{R}_{D}\sim
2\,\text{cm}\>.$ (44)

The following nuclear reactions then proceed to produce light anti-nuclei (see
Fig. 2):

  * •

${\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}$ production through
$\bar{n}+\bar{p}\rightarrow{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}+\gamma$.

  * •

${\bar{\text{T}}}$ production either (1) directly through
${\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}+{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}\rightarrow{\bar{\text{T}}}+\bar{p}$, or (2) indirectly through
${\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}+{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}\rightarrow{}^{3}\overline{\text{He}}+\bar{n}$, followed by the
highly efficient
${}^{3}\overline{\text{He}}+\bar{n}\rightarrow{\bar{\text{T}}}+\bar{p}$. The
latter process depletes ${}^{3}\overline{\text{He}}$ and keeps its abundance
low.

  * •

${}^{4}\overline{\text{He}}$ production through
${\bar{\text{T}}}+{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}\rightarrow{}^{4}\overline{\text{He}}+\bar{n}$.

In a way somewhat analogous to how dark matter is produced in freeze-in
scenarios, these processes sequentially produce nuclear anti-particles with
their final (frozen) numbers satisfying $N_{\bar{p}}\approx N_{\bar{n}}\gtrsim
N_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\gtrsim
N_{{\bar{\text{T}}}}\gtrsim N_{{}^{4}\overline{\text{He}}}$, and essentially
no other elements (see Fig. 3).

As the shell further expands and decreases in density, the bulk of the plasma
eventually becomes transparent to photons when its comoving temperature, outer
radius, and thickness are around

$\displaystyle\tilde{T}_{\rm thin}^{\prime}$ $\displaystyle\approx
0.7\,\mathinner{\mathrm{keV}},$ $\displaystyle\tilde{R}_{\rm thin}$
$\displaystyle\sim 400\,\text{m}\ ,$ $\displaystyle\Delta\tilde{R}_{\rm thin}$
$\displaystyle\sim 4\,\text{m}\>.$ (45)

At that point, the relativistic anti-nucleosynthetic products and a burst of
X-ray photons are released from the plasma shell.

While traversing the interstellar medium, the $\bar{n}$ decay to $\bar{p}$ and
the ${\bar{\text{T}}}$ decay to ${}^{3}\overline{\text{He}}$ (these decay
timescales are very short compared to the galactic dwell-time). Each fireball
seeded with a total anti-baryon number $\bar{B}$ therefore contributes to the
nuclear anti-particle population in the interstellar medium as follows:

$\displaystyle\begin{split}\tilde{N}_{\bar{p}}&\approx\bar{B}\>,\\\
\tilde{N}_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}&\approx 5.8\times
10^{-3}\bar{B}\>,\\\ \tilde{N}_{{}^{3}\overline{\text{He}}}&\approx 3.6\times
10^{-4}\bar{B}\>,\\\ \tilde{N}_{{}^{4}\overline{\text{He}}}&\approx 1.9\times
10^{-4}\bar{B}\>.\end{split}$ (46)

Note that these specific numerical results depend on the the benchmark values
of the parameters $(T_{0},R_{0},\bar{\eta})$ that were chosen at Eq. (43) such
that the resulting _injected_ ratio of anti-helium isotopes
$\tilde{N}_{{}^{4}\overline{\text{He}}}/\tilde{N}_{{}^{3}\overline{\text{He}}}|_{\text{inj.}}$
reproduces the current AMS-02 candidate-event _observed_ value of $\approx
1/2$, and the terminal Lorentz boost of the plasma shell is $\Gamma=10$ at
injection (see Fig. 5). These injected values are however somewhat modified by
Galactic propagation effects that we discuss and account for in the next
section.

##  IV Propagation and detection

In the previous section, we showed how anti-nucleosynthesis occurring in an
expanding thermal fireball state characterized by a certain temperature,
radius, anti-baryon content, and net hadronic charge asymmetry could generate,
after unstable elements have decayed, both ${}^{3}\overline{\text{He}}$ and
${}^{4}\overline{\text{He}}$ in an isotopic ratio broadly consistent with the
candidate AMS-02 events.

In this section, we discuss how the properties of the anti-helium (and other
species) injected by such fireballs at locations within the MW are processed
by propagation from the source to the AMS-02 detector, as well as the
necessary parameters to generate event rates consistent with the candidate
AMS-02 observations.

Our analysis is predicated on the following basic assumptions: (1) all seeded
fireballs have similar $T_{0}$, $R_{0}$, and $\bar{\eta}$ parameters (and
hadronic charge asymmetries); and (2) a large enough number of anti-helium
producing fireballs have been, and continue to be, seeded at random times up
to the present day that we can neglect both spatial and temporal clumpiness in
the injection and instead model it as a temporally constant and spatially
smooth source. Additionally, motivated by having dark-matter collisions seed
the fireballs (see Sec. V), we assume that (3) the spatial distribution of the
injections is $\propto[n_{\text{NFW}}(r)]^{2}$, where $n_{\textsc{NFW}}(r)$ is
a Navarro–Frenk–White (NFW) profile [55],222222Following Ref. [14], we take
the NFW profile to be normalized such that the local average DM density is
$\rho_{0}=0.4\,\text{GeV/cm}^{3}$ [56, 57] at $r\sim 8\,\text{kpc}$ [58], and
use a scale radius $R_{s}=20\,\text{kpc}$ [14]. so that the anti-helium
injection is occurring dominantly within the Milky Way itself and is peaked
toward its center.

###  IV.1 Cosmic rays

The fireball injection model discussed in Sec. III is such that all cosmic-ray
species are initially injected in the vicinity of the fireball with a narrow
range of velocities centered around the bulk Lorentz factor232323Although many
anti-helium nuclei are produced through decay processes (and never thermalize
with the fireball) [e.g., the bulk of ${}^{3}\overline{\text{He}}$ production
is from ${\bar{\text{T}}}$ decay], the associated nuclear decay $Q$ values are
sufficiently small that the product nuclei are always non-relativistic in the
decay rest frame. As a result, their Lorentz factor in the galactic rest frame
at the time of injection does not differ appreciably from $\Gamma$.
$\Gamma\sim 10$.

The subsequent motion of these injected cosmic rays to Earth (and hence the
AMS-02 detector in low-Earth orbit) is of course diffusive in both position
and momentum space [59]. In principle, we should thus pass the fireball-
injected species to galprop [60, 61] to solve the necessary transport
equations and account for various propagation effects; see also Ref. [12].

Instead of using a modified version of galprop to study the propagation of
injected anti-particles, we argue as follows. Galprop natively solves the
transport equation for positively charged nuclei. Of course, the opposite sign
of the charge for the anti-nuclei does not impact the diffusive nature of the
transport [59, 14]. To be sure, there are additional annihilation reactions
that can occur for anti-particles interacting with the (dominantly ordinary
matter) interstellar medium (ISM), and the inelastic cross-sections scattering
with the ISM also differ somewhat for particles vs. anti-particles.
Nevertheless, at the level of precision at which we work, the annihilation
cross-sections can however reasonably be ignored for our purposes, as they
constitute a negligible correction to the total inelastic cross-section of the
anti-particle species at energies $\gtrsim 10\,\mathinner{\mathrm{GeV}}$ [62,
63, 64] and can therefore be absorbed into the $\gtrsim 10\%$ uncertainties
associated with the propagation [14]. Ignoring also any other differences in
the anti-particle vs. particle inelastic cross-sections for interaction with
the ISM, we employ galprop results for the corresponding positive charged
nuclei as _an approximation to_ the desired results for the negatively charged
anti-nuclei (e.g., we inject primary ${}^{4}\text{He}$ instead of
${}^{4}\overline{\text{He}}$, and read off results accordingly, etc.). This
approximation could of course be revisited; however, as we shall see, the
results of this approximate treatment indicate a ratio of observed anti-helium
fluxes that differs by only an $\mathcal{O}(1)$ numerical factor as compared
to the ratio injected by the fireballs; it is therefore unclear whether a
modification of propagation code as in Ref. [12] to more correctly treat the
anti-particle propagation is justified given other, larger uncertainties in
our scenario.242424We note that such modification was however important for
Ref. [12] to achieve accurate results, as one of the main issues addressed in
that work was to refine predictions for the fully propagated anti-particle
secondary fluxes produced by the primary _ordinary matter_ cosmic-ray spectra.

Specifically, we model the injection of anti-cosmic rays of species $i$ by
specifying galprop source terms for the corresponding _positively-charged,
ordinary-matter species_ , which we denote here as ${i^{+}}$:

$\displaystyle q_{i^{+}}(\bm{r},\mathcal{R})\propto
X_{i}F_{i}(\mathcal{R})[n_{\text{NFW}}(r)]^{2}\>,$ (47)

where $X_{i}$ is taken to be the isotopic abundance for the anti-particle
species $i$ from Sec. III.3, $F_{i}(\mathcal{R})$ is taken to be a narrow top-
hat function centered at the rigidity $\mathcal{R}$ corresponding252525While
the fireballs inject each species $i$ at a single rigidity, the width of the
top-hat (chosen here to be $\sim 10\%$ of the central value for numerical
reasons) is inconsequential as long as it is subdominant to the momentum-space
diffusion occurring during propagation, which we verify _a posteriori_. In
order to specify a fixed $\Gamma$, and hence a different
$\mathcal{R}=\mathcal{R}(q,m,\Gamma)$ dependence for each species, galprop had
to be run multiple times: in each run, a single species $i^{+}$ was injected
at the required rigidity $\mathcal{R}_{i}$; the output spectra from each such
run were then summed with weights $X_{i}$. to Lorentz factor $\Gamma$ for
species $i$ as injected by the fireball, and $n_{\text{NFW}}(r)$ is the NFW
profile. We of course then also read off local flux results for the species
$i^{+}$, and impute those to species $i$. The source terms are normalized such
that the imputed total injection rate of anti-baryon number, summed over all
$i$ species and integrated over the whole galprop simulation volume, is
$\Gamma_{\text{inj.}}$.

Furthermore, we approximate the decay of $\bar{n}$ to $\bar{p}$ and
${\bar{\text{T}}}$ to ${}^{3}\overline{\text{He}}$ as occurring
instantaneously at the fireball location, and we thus consider only of the
anti-particle species $i\in\\{\bar{p},{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}},{}^{3}\overline{\text{He}},{}^{4}\overline{\text{He}}\\}$ when
running galprop via the above procedure, in the ratios specified at262626As
discussed in Sec. III.3, while the numerical solutions to the Boltzmann
equations are more accurate than the analytical results at Eqs. (34)–(40), the
tuning of those analytical results to the numerics via the constants
$k_{\bar{2}}$, $k_{\bar{3}}$, and $k_{\bar{4}}$, makes the analytical results
sufficiently accurate for our purposes here, notwithstanding the minor
violation of the scaling of
$N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}|_{\text{inj.}}\propto
c^{2}$ discussed in Sec. III.3.8; see also Fig. 5.  Eqs. (34)–(40) as a
function of $X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}$.

For the galprop diffusion model, we adopt the propagation parameters for “ISM
Model I” in Ref. [65]; within the range of parameters consistent with existing
cosmic-ray observations, our results are largely insensitive to the choice of
transport model. We also neglect the modulation of the cosmic-ray fluxes at
Earth due to the heliospheric magnetic field: in the force-field
approximation, solar modulation is governed by a single parameter known as the
Fisk potential $\phi_{F}\sim 1\,\text{GV}$ which, at high energies, corrects
the flux at most by a factor $\sim e\phi_{F}/E\ll 1$ [66].

Note also that the typical mean-free path for an anti-nucleus traveling with
$\Gamma\sim 10$ is $\ell_{\text{mfp}}\sim 1\,\text{pc}$, meaning that it stays
within the galaxy for a duration $t_{\text{diff}}\sim
h_{\text{MW}}^{2}/\ell_{\text{mfp}}\sim 10^{6}\,\text{yr}$ where
$h_{\text{MW}}\sim 1\,\text{kpc}$ is the thickness of the MW disk.

A set of example post-propagation spectra at the position of the Earth is
shown in Fig. 6, alongside AMS-02 sensitivity curves or observations. In each
case, the propagated flux of injected anti-particles peaks at kinetic energies
corresponding to the Lorentz parameter $\Gamma\sim 10$ [i.e.,
$\mathcal{O}(10\,\text{GeV/nucleon})$].

Figure 6: Predicted spectra of anti-nuclei upon propagation to the Earth as a
function of kinetic energy per nucleon $E_{k}$, for different choices of the
initial Lorentz factor $\Gamma$, as annotated by the different colored lines.
The best available sensitivity of AMS to anti-helium (top panel) and anti-
deuterium (middle panel) events [67, 68, 69, 70, 71] are presented for
comparison in gray (see text). In the top panel, the solid and dashed lines
correspond to the isotopes ${}^{3}\overline{\rm He}$ and ${}^{4}\overline{\rm
He}$, respectively. The anti-proton flux observed by AMS is shown (bottom
panel) by a solid black line [72], while the _additional_ flux required to
exceed the uncertainties on the measured flux is shown as the dashed black
line in that same panel; fireball production is a negligible source of
galactic $\bar{p}$ at these parameter values. Throughout this figure, we fix
$\Gamma_{\rm inj}=4\times 10^{35}\,\text{anti-nuclei s}^{-1}$, while the
isotopic abundances at the fireball are set by $X_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}=5\times 10^{-3}$; see Eqs. (36) and (38).

The expected number of anti-helium events accumulated at AMS over 10 years (at
the 95% CL) can be computed using the output spectra from galprop and the
published AMS sensitivity to the flux ratio between anti-helium and helium
[67, 68]. This is done by recasting the anti-helium acceptance of each energy
bin in terms of this $\overline{\text{He}}/\text{He}$ sensitivity in
conjunction with published helium data [69], following the procedure outlined
in Appendix B of Ref. [15]; see our Appendix G for a brief review. Assuming
the number of events follows Poissonian statistics, and allowing for the joint
probability of the predicted numbers of ${}^{3}\overline{\text{He}}$ and
${}^{4}\overline{\text{He}}$ to deviate from their AMS-02 tentative observed
values within the 68% confidence interval, the $(X_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}},\Gamma_{\text{inj.}})$ parameter space consistent with the
AMS-02 candidate anti-helium events is shown in Fig. 7. Explicitly, we require
[73]

$\displaystyle(\Delta\chi^{2})_{{}^{4}\overline{\text{He}}}+(\Delta\chi^{2})_{{}^{3}\overline{\text{He}}}<2.3\>.$
(48)

The events for each isotope are assumed to be independent, satisfying

$\displaystyle(\Delta\chi^{2})_{i}=2\left[N_{i}^{\text{th}}-N_{i}^{\text{obs}}-N_{i}^{\text{obs}}\ln\left(\frac{N_{i}^{\text{th}}}{N_{i}^{\text{obs}}}\right)\right],$
(49)

where $N_{i}^{\text{th}}$ is the number of events for species $i$ predicted by
galprop given a set of model parameters
$\\{\Gamma_{\text{inj.}},X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}}\\}$,
whereas $N_{i}^{\text{obs}}$ is the fiducial number of candidate $i$ events
reported by AMS-02. For the benchmark $X_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}=5\times 10^{-3}$ shown by the black dot in Fig. 7, the
isotope ratio $N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}$
changes from
$N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}|_{\text{inj.}}\approx
0.33$ at injection to an observed value of
$N_{{}^{4}\overline{\text{He}}}/N_{{}^{3}\overline{\text{He}}}|_{\text{obs.}}\approx
0.55$–$0.60$ (depending on the choice of $\Gamma$).

Part of this change in the isotope ratio from injection to observation has to
do with physical propagation effects such as spallation of
${}^{4}\overline{\text{He}}$ onto the interstellar medium; however, only $\sim
10$% of the ${}^{3}\overline{\text{He}}$ abundance arises from this effect,
which moves the isotopic ratio by only an $\mathcal{O}(1)$ factor. Moreover,
that effect would tend to drive the ratio in the other direction (i.e., it
reduces ${}^{4}\overline{\text{He}}$ and increases
${}^{3}\overline{\text{He}}$). The larger part of the change arises not from
propagation effects at all, but rather from the fact that the AMS-02 anti-
helium sensitivity, which we take to be given as the same function of rigidity
$\mathcal{R}$ for all anti-helium isotopes (see Appendix G), is not flat as a
function of $\mathcal{R}$; on the other hand, the post-propagation energy-per-
nucleon spectra of ${}^{3}\overline{\text{He}}$ and
${}^{4}\overline{\text{He}}$ are almost the same (since they have the same
energy-per-nucleon at injection), resulting in rigidity distributions for the
two species that peak at different values of
$\mathcal{R}=(m/Q)\sqrt{(E/m)^{2}-1}$ (i.e., when stated in terms of the
kinetic energy per nucleon, the AMS-02 sensitivity that we assume differs for
species with different charge-to-mass ratios; see the top panel of Fig. 6).
Ultimately, however, this $\mathcal{O}(2)$-factor change could easily be
absorbed into a slightly different parameter point if any of the assumptions
leading to this effect are found to be inaccurate.

For the case of anti-deuterium, a robust detection relies on the rejection of
backgrounds (particularly $\bar{p}$ and He) that are substantially more
abundant. As described in Ref. [71], the latest published AMS-02 anti-
deuterium sensitivity curve (shown in Fig. 6) is based on an earlier
superconducting-magnet configuration for AMS-02, rather than the permanent-
magnet configuration actually in use, and cuts off at $\approx
5\,\text{GeV/nucleon}$. Nevertheless, taking that sensitivity, only injections
taking place with $\Gamma\lesssim 12$ would result in $\gtrsim 0.1$ expected
${\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}$ events in the parameter space of
interest, but it would be challenging on the basis of that sensitivity to
simultaneously account for the 7 candidate anti-deuterium events reported in
Ref. [7] in addition to the candidate anti-helium events. That said, an
updated study of the AMS-02 anti-deuteron sensitivity would be required to
accurately determine whether a single choice of parameters
$(X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}},\Gamma_{\text{inj.}})$ could
achieve this.

Finally, anti-proton events arising from the fireball are also expected to be
observed at AMS-02; indeed, as seen in Fig. 6, this flux dominates those of
the other species in the parameter regime of interest. However, conventional
astrophysical sources are responsible for an anti-proton flux that is greater
than that created by the fireballs by a factor of $\sim 10^{3}$ [72].
Producing sufficiently many events to exceed the uncertainty on this
measurement (tantalizing in light of the anti-proton excess of $\sim 10\%$ at
similar energies [65]) would require a greater anti-nucleus injection rate
than is favored by the candidate anti-helium observations; see however Fig. 9
in Appendix B (and related discussion) for an alternative parameter point that
may be more interesting from this perspective.

Figure 7: The region of parameter space for which AMS-02 would be expected to
observe 3 events of ${}^{4}\overline{\rm He}$ and 6 events of
${}^{3}\overline{\rm He}$ in $T\sim 10$ years, with the injection taking place
at $\Gamma=10$. (The allowed region is approximately identical over the range
$8.5\leq\Gamma\leq 13$.) The anti-nucleus injection rate $\Gamma_{\rm inj}$
normalizes the total rate of fireball injection of anti-cosmic rays into the
MW, whereas the individual source isotopic ratios $X_{i}$ are set by the
fireball anti-deuterium abundance $X_{{\mkern-2.0mu\bar{\mkern
2.0mu\text{D}}}}$: $X_{i}=X_{i}(X_{{\mkern-2.0mu\bar{\mkern 2.0mu\text{D}}}})$
via Eqs. (34)–(40). The predicted number of anti-helium events is allowed to
independently vary from the fiducial quantity within the 68% confidence
interval of the joint probability distribution [see Eq. (48)]. The black dot
shows the fiducial parameter values referred to in Fig. 6 and Eq. (55).

###  IV.2 Other signatures (indirect detection)

In this section, we examine the detectability of X-ray, anti-neutrino, and
positron bursts which arise from the fireball model (discussed in Sec.
III.3.9). While fireball injection of anti-nuclei is envisaged as a continuous
process compared to the galactic dwell-time for these (diffusively
transported) particles, the rapid expansion timescale $\tau^{\prime}$ and low
galactically-integrated injection event rates $\Gamma_{\text{coll.}}\sim
3\,\text{s}^{-1}$ involved (see Sec. V.1 for this estimate) mean that the non-
nuclear outputs from independent fireball injections are temporally well-
separated at the Earth and so are best treated independently (with the
exception of positrons, whose transport is also diffusive).

At the fluxes required to explain the AMS-02 candidate anti-helium events,
there should be $N_{\text{inj.}}\sim\Gamma_{\text{coll.}}T\sim 10^{9}$ events
occurring within the galaxy over $T\sim 10\,\text{yr}$. Suppose that we made
an observation to look for their non-nuclear products that has a duration
$T_{\text{obs.}}$ and that observes a fraction $f_{\text{sky}}$ of the whole
sky. In this time, we estimate that the closest observable injection event
would occur at a distance $d_{\text{inj.}}\sim
16\,\text{pc}\times(10\,\text{yr}/T_{\text{obs.}})^{1/3}\times
f_{\text{sky}}^{-1/3}$ from Earth, if we assume that the injections arise from
collisions of the sub-component composite DM states whose benchmark parameters
we discuss below in Sec. V.1.

Photons are rapidly released in a burst when the fireball arising from the
injection becomes optically thin, at size $R_{\text{thin}}\sim 400\,\text{m}$;
see Eq. (45) for this benchmark. We estimate the energy released in this burst
to be $\sim
g_{*}(T_{\text{thin}}^{\prime})^{4}\Gamma^{2}R_{\text{thin}}^{2}\Delta
R_{\text{thin}}\sim g_{*}(T^{\prime}_{\text{thin}})^{4}R_{\text{thin}}^{3}\sim
6\times 10^{27}\,\text{erg}$ [using the benchmark at Eq. (45), and $g_{*}\sim
2$], which results in the emission of $N_{\gamma}\sim 6\times 10^{35}$ photons
with average energy $\Gamma T^{\prime}_{\text{thin}}\sim
7\,\mathinner{\mathrm{keV}}$. X-ray telescopes sensitive to such events have
an effective area $A_{\gamma}\sim 1000\,\text{cm}^{2}$ [74, 75], a field of
view of at most $\Omega_{\gamma}\sim 100$ square degrees (i.e.,
$f_{\text{sky}}\sim 2\times 10^{-3}$), and mission lifetimes of
$\mathcal{O}(\text{years})$. The total expected number of photons arriving at
the detector from the closest release that would occur would be $\sim
N_{\gamma}(A_{\gamma}/d_{\text{inj.}}^{2})\sim 0.2\times
f_{\text{sky}}^{2/3}\times(T_{\text{obs.}}/10\,\text{yr})^{2/3}$. Even if we
conservatively ignore the finite field of view (i.e., set $f_{\text{sky}}\sim
1$) and take $T_{\text{obs.}}\sim 10\,\text{yr}$, this is too faint to
observe. There may additionally be a diffuse background of X-ray photons due
to the cumulative injections taking both intra- and extra-galactically over
extended periods of time, but we have not estimated this here.

Anti-neutrinos may be injected promptly when the fireball thermalizes, or
indirectly through the decay of injected pions. The fraction of the total
energy $g_{*}T_{0}^{4}R_{0}^{3}\sim 4\times 10^{32}\,\text{erg}$ [see Eq. (43)
for this benchmark; we set $g_{*}\sim 5.5$] that is carried by the prompt
anti-neutrinos is $\sim G_{F}^{2}T_{0}^{5}\tau^{\prime}\sim 1\%$ (for
$\tau^{\prime}\sim R_{0}$ as at early times; see Fig. 1), corresponding to
$N_{\nu,\,\text{scatter}}\sim 3\times 10^{34}$ particles with an average
energy of $T_{0}\sim 100\,\mathinner{\mathrm{MeV}}$. As far as
(anti-)neutrinos from decay are concerned, thermal pions have a number density
$n_{\pi}\sim(m_{\pi}T/(2\pi))^{3/2}\exp(-m_{\pi}/T)$ within the fireball, so
the number of neutrinos injected due to their decay is
$N_{\nu,\,\text{decay}}\sim n_{\pi}R^{3}(\tau^{\prime}/\tau_{\pi})\lesssim
10^{31}$, where $\tau_{\pi}\sim 10^{-8}\,\text{s}$ is the charged pion
lifetime. Their average energy is also $\sim\Gamma(m_{\pi}-m_{\mu})\sim
100\text{ MeV}$. The decay contribution is thus sub-dominant. Given a neutrino
detector in this energy range, with an area $A_{\nu}\sim(100\,\text{m})^{2}$
[76], roughly $\sim N_{\nu}A_{\nu}/(d_{\text{inj.}})^{2}\sim
10^{3}\times(T_{\text{obs.}}/10\,\text{yr})^{2/3}$ neutrinos would pass
through the detector as a result of the nearest injection in the observation
time.272727We took $f_{\text{sky}}\sim 1$ here to reflect that Earth is quite
transparent to neutrinos with energies below a few TeV, so the detector has
full-sky coverage; see, e.g., Ref. [77]. The probability of a single neutrino
interacting in the detector is $p_{\nu}\sim n_{T}G_{F}^{2}E_{\nu}^{2}d_{T}$
where $n_{T}\sim 3\times 10^{22}/\text{cm}^{3}$ is the target (e.g., water
[76]) number density, and $d_{T}$ is the target thickness. Taking $E_{\nu}\sim
100\,\mathinner{\mathrm{MeV}}$ and $d_{T}\sim 100\,\text{m}$ as relevant for
our assumptions, we find $p_{\nu}\sim 10^{-13}$, so the expected number of
detectable events282828By way of comparison, the Kamionkande-II detector
detected only 11 neutrino interactions of energies of
$\mathcal{O}(10\,\text{MeV})$ from SN1987A [78], which released $\sim 10^{58}$
neutrinos of all flavors [78] at a distance of $\sim 50\,\text{kpc}$ [79],
leading to $\sim(\text{few})\times 10^{16}$ electron anti-neutrinos passing
through the $\mathcal{O}(200\,\text{m}^{2})$ detector area (varies slightly
depending on the orientation relative to the source) [78]. Taking $E_{\nu}\sim
10\,\mathinner{\mathrm{MeV}}$ and $d_{T}\sim 10\,\text{m}$ gives $p_{\nu}\sim
10^{-16}$, which is broadly consistent with these numbers, demonstrating the
consistency of our estimate in the text.  is $\sim
10^{-10}\times(T_{\text{obs.}}/10\,\text{yr})^{2/3}\ll 1$ for any conceivable
observation duration.

Positrons injected by the fireball are potentially observable in two ways:
directly as cosmic rays, and indirectly through $511\,\text{keV}$ photons
produced when they annihilate with electrons in the ISM. However, the injected
positron energies are $\sim\Gamma m_{e}\sim 10\,\text{MeV}$, which falls below
the AMS-02 sensitivity threshold (even without accounting for further energy
loss during propagation), likely making them unobservable.292929Unsurprisingly
given that solar-modulation effects severely impact sub-GeV positrons [80],
this energy also falls below the lowest-energy positron measurements reported
by PAMELA [81], HEAT [82], CAPRICE94 [83], AMS-01 [84], or FERMI [85, 86].
Relevant Voyager 1 data for the sum of electrons and positrons down to
energies of $\mathcal{O}(10\,\mathinner{\mathrm{MeV}})$ from periods after it
crossed the heliopause are available [87] (see also Ref. [88]); we do not
however pursue these constraints further in this work.  For the 511 keV
emission, the total integrated positron injection rate from fireballs cannot
exceed the total integrated anti-baryon injection rate (cf. the discussion
about $X_{Q}<0$ in Appendix B), which from the numbers shown in Fig. 7 is
maximally $\Gamma_{\text{inj.}}\sim(\text{few})\times 10^{36}\,\text{s}^{-1}$;
this is much smaller than the integrated positron injection rate of $\sim