$\nu^{\prime}(X)>1/2$ and $\nu^{\prime}(Y)>1/2$. Again,
$\nu^{\prime}(X)+\nu^{\prime}(Y)>1$, violating the feasibility of
$\nu^{\prime}(.)$.

Finally, we will prove that $\\{\nu^{*}(X)\mid X\in V_{P}\\}$ achieves the
optimal value for the vertex-pack LP’s objective function. This is true
because

$\displaystyle\sum_{X\in V_{P}}\nu^{*}(X)$ $\displaystyle=$
$\displaystyle\Big{(}\sum_{X:\text{$X$ not in any one-edge
star}}\nu^{*}(X)\Big{)}+\sum_{X:\text{$X$ in a one-edge star}}\nu^{*}(X)$
$\displaystyle=$ $\displaystyle\Big{(}\sum_{X:\text{$X$ not in any one-edge
star}}\nu^{\prime}(X)\Big{)}+\text{number of one-edge stars}$
$\displaystyle(\textrm{property {\bf P4-2}})$ $\displaystyle=$
$\displaystyle\Big{(}\sum_{X:\text{$X$ not in any one-edge
star}}\nu^{\prime}(X)\Big{)}+\sum_{X:\text{$X$ in a one-edge
star}}\nu^{\prime}(X)$ $\displaystyle=$ $\displaystyle\sum_{X\in
V_{P}}\nu^{\prime}(X)$

and $\nu^{\prime}(.)$ is an optimal solution to the vertex-pack LP. ∎

We are now ready to prove Lemma D.2. Let $\\{\nu^{*}(X)\mid X\in V_{P}\\}$ be
an optimal solution to the vertex-pack LP problem promised by Lemma D.3.
Divide $V_{P}$ into three subsets

$\displaystyle U_{A}$ $\displaystyle=$ $\displaystyle\\{X\in
V_{P}\mid\nu^{*}(X)=1\\}$ $\displaystyle U_{B}$ $\displaystyle=$
$\displaystyle\\{X\in V_{P}\mid\nu^{*}(X)=0\\}$ $\displaystyle U_{C}$
$\displaystyle=$ $\displaystyle\\{X\in V_{P}\mid\nu^{*}(X)=1/2\\}.$

By the feasibility of $\nu^{*}(.)$ to the vertex-pack LP, no vertex in $U_{A}$
can be adjacent to any vertex in $U_{C}$, and no two vertices in $U_{A}$ can
be adjacent to each other. It remains to verify that the sizes of $U_{A}$,
$U_{B}$, and $U_{C}$ meet the requirement in the first bullet of Lemma D.2. To
facilitate our subsequent argument, given a one-edge star $\star_{j}$ (for
some $j\in[\beta]$), we call it a half-half one-edge star if
$\nu^{*}(X)=\nu^{*}(Y)=1/2$, where $\\{X,Y\\}$ is the (only) edge in
$\star_{j}$. Define

$\displaystyle s$ $\displaystyle=$ $\displaystyle\text{number of half-half
one-edge stars in $\\{\star_{1},...,\star_{\beta}\\}$}.$

On the other hand, if a one-edge star $\star_{j}$ is not half-half, we call it
a 0-1 one-edge star.

By Lemma D.3, $U_{C}$ includes all the vertices in
$\pentagon_{1},...,\pentagon_{\alpha}$ and all the vertices in the half-half
one-edge stars. Hence, $|U_{C}|=k_{\mathit{cycle}}+2s$. On the other hand,
$U_{B}$ includes (i) the center of every star that has at least two edges and
(ii) exactly one vertex from every 0-1 one-edge star. In other words, every
star contributes 1 to the size of $U_{B}$ except for the half-half one-edge
stars, implying $|U_{B}|=\beta-s$. Finally,
$|U_{A}|=k-|U_{B}|-|U_{C}|=k_{\mathit{star}}-\beta-s$. This completes the
proof of Lemma D.2.

##  Appendix E Analysis of the Sampling Algorithm in Section 5

The purpose of preprocessing is essentially reorganizing the data graph
$G=(V,E)$ in the adjacency-list format, which can be easily done in $O(|E|)$
expected time. The following discussion focuses on the cost of extracting a
sample. We will first consider $\lambda\leq\sqrt{|E|}$ before attending to
$\lambda>\sqrt{|E|}$.

Case ${\lambda\leq\sqrt{|E|}}$. Let
$G_{\mathit{sub}}=(V_{\mathit{sub}},E_{\mathit{sub}})$ be an occurrence of $P$
in $G$. There exist a constant number $c$ of isomorphism bijections
$g:V_{P}\rightarrow V_{\mathit{sub}}$ between $P$ and $G_{\mathit{sub}}$ (the
number $c$ is the number of automorphisms of $P$). Fix any such bijection $g$.
Each repeat of our algorithm builds a map $f:V_{p}\rightarrow V$. It is
rudimentary to verify that
$\mathbf{Pr}[f=g]=\frac{1}{2|E|}\cdot\frac{1}{\lambda^{k-2}}$. Hence, each
occurrence is returned with probability
$\frac{c}{2|E|}\cdot\frac{1}{\lambda^{k-2}}$. It is now straightforward to
prove that our algorithm has expected sample time:
$O(|E|\cdot\lambda^{k-2}/\max\\{1,\mathrm{OUT}\\})$.

Case ${\lambda>\sqrt{|E|}}$. If $\mathrm{OUT}=0$, the two-thread approach
allows our algorithm to terminate in
$O(\mathit{polymat}_{\mathit{undir}}(|E|,\lambda,P))=O(m^{\frac{k_{\mathit{cycle}}}{2}+\beta}\cdot\lambda^{k_{\mathit{star}}-2\beta})$
time. Next, we consider only $\mathrm{OUT}\geq 1$.

For each $i\in[\alpha]$, denote by $k(\pentagon_{i})$ the number of vertices
in $\pentagon_{i}$; for each $j\in[\beta]$, denote by $k(\star_{j})$ the
number of vertices in $\star_{j}$. The fractional edge cover number of
$\pentagon_{i}$ ($i\in[\alpha]$) is
$\rho^{*}(\pentagon_{i})=k(\pentagon_{i})/2$. This means
$\sum_{i=1}^{k}\rho^{*}(\pentagon_{i})=k(\pentagon_{i})/2=k_{\mathit{cycle}}/2$.

Fix an arbitrary occurrence
$G_{\mathit{sub}}=(V_{\mathit{sub}},E_{\mathit{sub}})$ of $P$ in $G$. Let $g$
be any of the $c$ isomorphism bijections between $P$ and $G_{\mathit{sub}}$.
Each repeat of our algorithm constructs a map $f:V_{P}\rightarrow V$ through
isomorphism bijections
$f_{\pentagon_{1}},...,f_{\pentagon_{\alpha}},f_{\star_{1}},...,f_{\star_{\beta}}$.
The event $g=f$ happens if and only if all of the following events occur.

  * •

Event E${}_{\pentagon_{i}}$ ($i\in[\alpha]$): $g(A)=f_{\pentagon_{i}}(A)$ for
each vertex $A$ of $\pentagon_{i}$;

  * •

Event E${}_{\star_{j}}$ ($j\in[\beta]$): $g(A)=f_{\star_{i}}(A)$ for each
vertex $A$ of $\star_{j}$.

If $c_{\pentagon_{i}}$ ($i\in[\alpha]$) is the number of automorphisms of
$\pentagon_{i}$, we have

$\displaystyle\mathbf{Pr}[\text{{\bf E${}_{\pentagon_{i}}$}}]$
$\displaystyle=$
$\displaystyle\frac{1}{c_{\pentagon_{i}}\cdot|\mathit{occ}(G,\pentagon_{i})|}.$
(45)

Likewise, if $c_{\star_{j}}$ ($j\in[\beta]$) is the number of automorphisms of
$\star_{j}$, we have

$\displaystyle\mathbf{Pr}[\text{{\bf E${}_{\star_{j}}$}}]$ $\displaystyle=$
$\displaystyle\frac{1}{c_{\star_{j}}\cdot|\mathit{occ}(G,\star_{j})|}.$ (46)

It follows from (45) and (46) that

$\displaystyle\mathbf{Pr}[g=f]$ $\displaystyle=$
$\displaystyle\Big{(}\prod_{i=1}^{\alpha}\frac{1}{c_{\pentagon_{i}}\cdot|\mathit{occ}(G,\pentagon_{i})|}\Big{)}\cdot\Big{(}\prod_{j=1}^{\beta}\frac{1}{c_{\star_{j}}\cdot|\mathit{occ}(G,\star_{j})|}\Big{)}.$

Therefore:

$\displaystyle\mathbf{Pr}[\text{$G_{\mathit{sub}}$ sampled}]$ $\displaystyle=$
$\displaystyle
c\cdot\Big{(}\prod_{i=1}^{\alpha}\frac{1}{c_{\pentagon_{i}}\cdot|\mathit{occ}(G,\pentagon_{i})|}\Big{)}\cdot\Big{(}\prod_{j=1}^{\beta}\frac{1}{c_{\star_{j}}\cdot|\mathit{occ}(G,\star_{j})|}\Big{)}.$
(47)

As this probability is identical for all $G_{\mathit{sub}}$, we know that each
occurrence of $P$ is sampled with the same probability.

The expected number of repeats to obtain a sample occurrence of $P$ is

$\displaystyle
O\Big{(}\frac{\prod_{i=1}^{\alpha}|\mathit{occ}(G,\pentagon_{i})|\cdot\prod_{j=1}^{\beta}|\mathit{occ}(G,\star_{j})|}{\mathrm{OUT}}\Big{)}$

whereas each repeat runs in time at the order of

$\displaystyle\Big{(}\prod_{i=1}^{\alpha}\frac{|E|^{\rho^{*}(\pentagon_{i})}}{|\mathit{occ}(G,\pentagon_{i})|}\Big{)}\cdot\Big{(}\prod_{j=1}^{\beta}\frac{|E|\cdot\lambda^{k(\star_{j})-2}}{|\mathit{occ}(G,\star_{j})|}\Big{)}.$
(48)

We can now conclude that the expected sample time is at the order of

$\displaystyle\frac{1}{\mathrm{OUT}}\cdot\Big{(}\prod_{i=1}^{\alpha}|E|^{\rho^{*}(\pentagon_{i})}\Big{)}\cdot\Big{(}\prod_{j=1}^{\beta}|E|\cdot\lambda^{k(\star_{j})-2}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{|E|^{k_{\mathit{cycle}}/2}\cdot\lambda^{k_{\mathit{star}}-2}}{\mathrm{OUT}}.$

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What charged cosmic rays tell us on dark matter

Pierre Salati$\,\star$

LAPTh and Université Savoie Mont-Blanc

9 chemin de Bellevue, BP110, Annecy-le-Vieux

F-74941 Annecy Cedex, France

*<EMAIL_ADDRESS>
LAPTH-Conf-065/22

Plenary talk given at the
14th International Conference on Identification of Dark Matter
Vienna, Austria, 18-22 July 2022

## Abstract

Dark matter particles could be the major component of the haloes of galaxies.
Their mutual annihilations or decays would produce an indirect signature under
the form of high-energy cosmic-rays. The focus of this presentation is on
antimatter species, a component so rare that any excess over the background
should be easily detected. After a recap on Galactic propagation, I will
discuss positrons, antiprotons and anti-nuclei. For each of these species,
anomalies have been reported. The antiproton excess, for instance, is
currently a hot topic. Alas, it does not resist a correct treatment of
theoretical and data errors.

##  1 Introduction

A significant portion of the universe is in the form of a massive and
pressureless component, dubbed dark matter (DM). Nearly 90 years after DM was
discovered by Zwicky in the Coma galactic cluster [1], its nature is still
unresolved. Among the plethora of candidates so far proposed, weakly
interacting, massive and neutral species have attracted much attention. These
particles would pervade the DM haloes of galaxies inside which they are
expected to annihilate or decay, producing gamma-rays, neutrinos and charged
particles, in particular antimatter cosmic rays (CRs) which are the focus of
this contribution.

Positrons, antiprotons and anti-nuclei are already produced by high-energy CR
protons and helium nuclei interacting on interstellar gas, hence yielding
backgrounds against which a potential DM anomaly is to be searched under the
form of a spectral excess. Most of these backgrounds have been detected.
Understanding them is the keystone to correctly decipher and validate a DM
signal. This requires to model accurately the propagation of charged CRs
inside the turbulent magnetic fields of the Milky Way. These extend over a
region, the so-called magnetic halo, which encompasses the Galactic disk and
extends above and beneath it over a distance $L$. The thicker the magnetic
halo, the more DM is trapped inside it and the stronger the CR flux at the
Earth from DM origin. Determining this height $L$ is paramount. It requires to
properly model the propagation of charged species inside the Milky Way.

##  2 Measuring the height of the magnetic halo

Charged particles follow the Galactic magnetic lines and diffuse on their
knots. CR transport is actually described as a diffusion process in space,
with coefficient $K$ grossly scaling with rigidity as ${\cal R}^{\delta}$. A
spectral break of $K$ has been observed at high rigidity, making CR fluxes
harder above ${\cal O}(250)$ GV [2]. At rigidities of order a few GV, the
situation is unclear. Either diffusion becomes more efficient at low energy,
hence another spectral break, or Galactic convection as well as diffusive
reacceleration come into play. Both possibilities are allowed in the transport
scheme BIG proposed in [3] to fit the secondary-to-primary flux ratio B/C, a
sensitive probe of CR diffusion insofar as it scales like $L/K$. The QUAINT
and SLIM schemes are two subsets of BIG. The former corresponds to the
configurations used so far in the literature while the latter is devised for
the gifted amateur.

Measuring the flux of an unstable secondary nuclear species is necessary to
break the degeneracy between the diffusion coefficient $K$ and the height $L$
of the magnetic halo. Beryllium is a secondary species produced, like boron,
by the fragmentation of carbon nuclei impinging on interstellar gas. Its
unstable isotope 10Be has a half-lifetime $t_{1/2}$ of 1.387 Myr of the same
order of magnitude as the typical resident time of CRs inside the Galactic
disk. At high energy, 10Be behaves as if it were stable. At low energy, it
propagates over a distance $\lambda\propto(Kt_{1/2})^{1/2}$ and its flux at
the Earth scales like ${1}/{\lambda}$, hence the possibility to disentangle
$K$ from $L$.

To do so requires the flux ratios 10Be/Be and 10Be/9Be. However, these have
been essentially measured below 1 GeV/n and suffer from lack of statistics. On
the other hand, some isotopic information is contained in the elemental ratio
Be/B recently measured by AMS-02 with great precision [4]. Combining all data
sets yields a halo size $L$ of order $4.5\pm 1$ kpc as showed in [5]. The
best-fit value $L$ depends actually on the CR scheme and is respectively equal
to $4.08$ (QUAINT), $4.64$ (BIG) and $4.66$ kpc (SLIM).

##  3 Positrons, dark matter and TeV haloes

In 2008, the rise of the positron fraction reported by the PAMELA
collaboration [6] triggered a hectic activity to explain the observed excess
in terms of DM annihilations or decays. The dust has settled by now. On top of
the background of secondary positrons produced in the Galactic disk by primary
CR nuclei interacting with gas, a primary component is also contributed by
nearby pulsars as already proposed in 1988 by A. Boulares [7], hence a
positron flux

$\Phi_{e^{+}}=\Phi_{e^{+}}^{\rm sec}+\Phi_{e^{+}}^{\rm prim}({\rm
pulsars})\,.$ (1)

The primary component is underdetermined since a single pulsar is enough to
fit the observations [8, 9]. Research is currently focused on the kind of
pulsar population needed to fit the data. As showed in the comprehensive
simulation carried out in [10], most of the synthetic populations of pulsar
wind nebulae (PWNe) produce several wiggles in the positron flux which are not
observed. The possibility of just a few sources, typically around 3,
generating a flux over a wide range in energy is then favored. This result is
criticized in [11] where the stochastic nature of high-energy inverse Compton
scatterings (ICS), a prominent cooling mechanism for positrons, is showed to
erase the wiggles in the flux sourced by PWNe. This reopens the possibility
that such a feature could be used as a signature for DM.

In this context, it is somewhat difficult to extract robust bounds on the
annihilation cross section or decay rate of DM species. A conservative
approach would consist in disregarding pulsars and adding to the sole
secondary positron component the flux produced by DM, requiring that none of
the data points is overshooted. This yields upper bounds on $\langle\sigma
v\rangle$ above the thermal value of $3\times 10^{-26}\,{\rm cm^{3}\,s^{-1}}$
required to get the observed DM cosmological abundance [12]. A more agressive
possibility is to fit the data and to gauge how much DM can be added without
perturbing the fit. This yields considerably more stringent bounds which, in
the case of the $e^{+}e^{-}$ channel, lie below the thermal value for DM
masses up to 200 GeV [13, 12].

The recent discovery of $\gamma$-ray TeV haloes around pulsars [14, 15]
confirms these objects as potential sources of positrons. These are
accelerated by pulsar winds and produce TeV photons through ICS on stellar
light inside a region where space diffusion is significantly inhibited [16]
compared to the bulk of the CR magnetic halo. Current studies concentrate on
how magnetic turbulence is itself generated by the accelerated particles [17].

Authors | Data analyzed | Method | ${\cal C}^{\rm data}$ | ${\cal C}^{\rm model}$ | Significance of | Conclusive
---|---|---|---|---|---|---
|  | F or B |  |  | fit w and w/o DM | hint for DM
Cuoco et al. [18] |  $\bar{p}/p$, $p$ & He + Voyager $p$ & He | F | no | no |  $4.7\,\sigma$ (L,2dof) | yes
Cui et al. [19] |  $\bar{p}$ with prior $\Pi(\lambda)$ from CR nuclei | B | no | no |  $2_{\,}\ln\\!{K}=11$ to $54$ | Decisive
Reinert & Winkler [20] | B/C, $\bar{p}$ & AMS/PAM | F | no | ${\cal C}^{\rm XS,parents}$ |  $2.2\,\sigma$ (L,1dof) $\to$ $1.1\,\sigma$ (G) | no
Cui et al. [21] |  $\bar{p}$ with prior $\Pi(\lambda)$ from CR nuclei | B | no | no | Best-fit $m_{\chi}$ and $\langle\sigma v\rangle$ | N/A
| \+ PandaX-II, LUX, XENON1T (DD) |  |  |  |  |
Cuoco et al. [22] |  $\bar{p}/p$, $p$ & He + Voyager and CREAM $p$ & He | F | no | no |  $3.1\,\sigma$ (L,2dof) | yes
| all data with ${\cal R}>5\,{\rm GV}$ | F | no | ${\cal C}^{\rm XS}$ |  $2.9\,\sigma$ (L,2dof) | yes
|  | F |  | no |  $5.5\,\sigma$ (L,2dof) | yes
Cholis et al. [23] |  $\bar{p}/p$ with best-fit of fudge factor $N_{XS}(T_{\bar{p}})$ | F | no | no |  $4.7\,\sigma$ (L,1dof) | yes
| \+ $\bar{p}$ production in SNR shocks | F | no | no |  $3.3\,\sigma$ (L,1dof) | yes
Heisig et al. [24] | B/C, $\bar{p}$ & AMS/PAM | F |  | ${\cal C}^{\rm XS}$ |  $0.5\,\sigma$ (L,1dof) | no
|  $\bar{p}/p$, $p$ & He + Voyager $p$ & He | F |  | ${\cal C}^{\rm XS}$ |  $1.8\,\sigma$ (L,1dof) $\to$ $0.5\,\sigma$ (G) | no
Di Mauro & Winkler [12] | B/C, $\bar{p}$ & AMS/PAM | F |  | ${\cal C}^{\rm XS}$ |  $2\,\sigma$ (L,1dof) $\to$ $1\,\sigma$ (G) | no
Luque [25] | B/C, B/O, Be/C, Be/O, Be/B, Li/B, Li/Be | B | no | no | DM fit is poorer | no
| \+ 10Be/Be, 10Be/9B and $\bar{p}/p$ |  |  |  | than fit w/o DM |
Kahlhoefer et al. [26] |  $p$, He & $\bar{p}/p$ \+ Voyager $p$ & He | F+B | no | no |  $4.5\,\sigma$ (L,1dof) | TBC
Calore et al. [27] |  $\bar{p}$ \+ nuisance on $L$ from LiBeB | F |  |  |  $1.8\,\sigma$ (L,1dof) | no
Table 1: Recent analyses of a potential DM signal in the CR $\bar{p}$ flux.
The statistical method is either frequentist (F) or Bayesian (B). It makes use
($\checkmark$) or not (no) of a covariance matrix of errors for data (${\cal
C}^{\rm data}$) and theory (${\cal C}^{\rm model}$). The significance of the
DM signal is either local (L) or global (G).

##  4 Antiprotons – Trimmed hints and robust bounds

A DM excess hidden in the $\bar{p}$ flux has been claimed by many groups in
the past few years. Some studies reach the opposite conclusion. It is timely
then to analyze how a DM $\bar{p}$ signal could be extracted from the data and
to understand the crucial differences between these analyses which are
summarized in Table 1. A powerful estimator for testing the null-hypothesis,
i.e. the absence of any DM signal from observations, is the likelihood ratio

$LR({\rm null})=-2\ln\left\\{{\displaystyle\frac{{\rm
sup}_{\lambda\in\Lambda}{\cal L}(\lambda)}{{\rm
sup}_{{\\{\lambda,\mu\\}}\in\Lambda\cup M}{\cal L}(\lambda,\mu)}}\right\\}\,,$
(2)

where ${\lambda\in\Lambda}$ stands for the CR parameters while
$\mu\equiv\\{\mbox{$m_{\chi}$},\langle\sigma v\rangle,{\rm channel}\\}$
characterizes the properties of the DM species. This ratio gauges how more
likely the data were to be collected when DM is added.

Defining the likelihood function ${\cal L}(\lambda,\mu)$ requires some care
though. In principle, we could define it through a global $\chi^{2}$ measuring
the distance of both CR nuclear species and antiproton fluxes to theory

$-2\ln{\cal L}(\lambda,\mu)\equiv\chi_{\rm
LiBeB}^{2}(\lambda)+\chi_{\bar{p}}^{2}(\lambda,\mu)\,.$ (3)

However this would be extremely resources and time consuming. The situation
considerably simplifies by remarking that the $\bar{p}$ flux, which may be
expressed as the sum

$\Phi_{\bar{p}}=\Phi_{\bar{p}}^{\rm sec}(\lambda)+\Phi_{\bar{p}}^{\rm
DM}(\lambda,\mu)\,,$ (4)

is by far dominated by the contribution from secondaries produced by CR
protons and He nuclei interacting with Galactic gas. This component behaves
like a stable secondary nuclear species. It is no suprise then if the CR
parameters $\hat{\lambda}$ minimizing $\chi^{2}_{\rm LiBeB}$ should also
minimize $\chi^{2}_{\bar{p}}$, as actually observed in [27].

A more tractable yet robust definition of the likelihood consists in replacing
the distance $\chi_{\rm LiBeB}^{2}$ by the posterior probability
$\Pi(\lambda)$ yielded by an independent LiBeB analysis. A Bayesian approach,
like in [21], allows to derive the probability

${\cal P}(\mu)\propto{\displaystyle\int}{\cal
L}(\bar{p}_{\,}|_{\,}\mu,\lambda)\;\Pi(\lambda)\;d{\lambda}\,,$ (5)

which $\bar{p}$ data associate to each set $\mu$ of DM parameters. From a
frequentist perspective, $\chi_{\rm LiBeB}^{2}$ could be replaced by a
nuisance term gauging the distance between the CR parameters $\lambda$ and
those $\hat{\lambda}$ best-fitting the LiBeB data. It is even possible to go a
step further by noticing that the primary DM component of the $\bar{p}$ flux
scales as ${L^{2}}/{K}$, i.e. as $L$ once the B/C ratio is taken into account.
In [27], the nuisance term that replaces $\chi_{\rm LiBeB}^{2}$ gauges the
distance of $L$ from the LiBeB best-fit value $\hat{L}$, hence the likelihood

$-2\ln{\cal L}(L,\mu)=\left\\{{\displaystyle\frac{\log
L-\log\hat{L}}{\sigma_{\log
L}}}\right\\}^{\\!2}+\chi_{\bar{p}}^{2}(L,\hat{\lambda}_{i},\mu)\,,\text{ with
$\hat{\lambda}_{i}$ best-fitting LiBeB at fixed $L$.}$ (6)

A key ingredient in the calculation of the likelihood is the definition of the
chi-square. There’s the rub! In principle, one should take into account the
correlations between different rigidity bins $i$ and $j$, and use a covariance
matrix of errors for both data (${\cal C}^{\rm data}$) and theory (${\cal
C}^{\rm model}$), hence the definition

$\chi_{\bar{p}}^{2}\equiv\sum_{i,j}\,x_{i\,}({\cal C}^{-1})_{ij\,}x_{j}\,,$
(7)

where $x_{i}=\Phi_{\bar{p},i}^{\rm exp}-\Phi_{\bar{p},i}^{\rm th}$ while
$\cal{C}=\cal{C}^{\rm data}+\cal{C}^{\rm model}$. It is unfortunate that
$\cal{C}^{\rm data}$ is not yet published. Its structure can nevertheless be
guessed as explained in [28]. The matrix $\cal{C}^{\rm model}$ results from
uncertainties in the secondary $\bar{p}$ production cross sections, in the
fluxes of the progenitors producing them and in the CR propagation parameters
$\lambda_{i}$ other than $L$ [29]. A close inspection of Table 1 indicates
that most of the analyses where a covariant matrix of errors is used do not
point toward a DM signal, whereas the opposite conclusion is reached if
statistical and systematic experimental errors are just added in quadrature.

As a final word of caution, it is tempting to gauge the statistical
significance of the null hypothesis by interpreting the likelihood ratio as a
$\Delta\chi^{2}$ associated to the two degrees of freedom $m_{\chi}$ and
$\langle\sigma v\rangle$. However, the DM mass is not defined under the null
hypothesis. In these conditions, the global significance should be determined
by generating mock $\bar{p}$ data under the null hypothesis (only secondaries)
to build the $\Delta\chi^{2}$ law and derive the actual $p$-value [24].

Figure 1:  Left panel: The upper limit on the annihilation cross section
derived in [27] for the $b\bar{b}$ channel (blue solid line), compared to
other results involving $\bar{p}$ analyses of [30]. The red solid line
corresponds to the Giesen et al. upper bound [31]. The limits set by Cuoco et
al. [18] (orange dot-dashed curve) and Cui et al. [19] (green dashed line) are
also featured. Right panel: The upper limit on the annihilation cross section
derived in [27] for the $b\bar{b}$ channel (blue solid line), compared to
other probes. The red contour is the 95% CL contour of the fit to the Galactic
Center excess reported in [32] for the same annihilation channel. Bounds from
different samples of dwarf spheroidal galaxies (dSph) derived with a new data-
driven method [33] (green solid line), with traditional template-fitting
strategies [34] (green dashed line), and by combining Fermi-LAT and ground-
based telescopes data [35] (Glory Duck project, green dot-dashed line) are
also displayed. The yellow band are radio constraints obtained from the EMU
survey [36]. The thermal relic cross section is reported in dotted black
lines.

##  5 Anti-nuclei – The new frontier

Although not seen so far, anti-nuclei are also expected in the cosmic
radiation. They should be produced as secondary species, like in terrestrial
accelerators, by the collisions of CR primaries on Galactic gas. The General
Antiparticle Spectrometer (GAPS) is about to fly and will measure the
$\bar{p}$ flux below $200$ MeV [37]. The instrument is devised to disentangle
antiprotons from antideuterons. The AMS collaboration has recently reported
[38] a few events in the mass region from 0 to 10 GeV with charge $Z=-2$ and
rigidity ${\cal R}<50$ GV. The masses of all events are in the
${}^{3}\overline{\rm He}$ and ${}^{4}\overline{\rm He}$ mass region. The event
rate is one $\overline{\rm He}$ for $10^{8}$ ${\rm He}$. This is extremely
surprising insofar as the flux of a secondary anti-nucleus decreases by 4
orders of magnitude each time its atomic number is incremented by 1. The flux
of secondary ${}^{3}\overline{\rm He}$ is expected to be well below AMS-02
sensitivity [39, 40].

These AMS events, if confirmed, would nevertheless point to unconventional
physics. In general, DM annihilations or decays yield also a vanishingly small
flux. However, a nice counterexample has been recently proposed in [41], where
the DM particles annihilate into $b{\bar{b}}$ pairs, a fraction ${\cal
F}\simeq 0.1$ of which hadronize into $\bar{\Lambda}_{b}$ baryons. This
fraction has been measured at LEP but the event generator Pythia is short by a
factor of $3$ to reproduce it. The colored string extending between the $b$
and $\bar{b}$ quarks needs to be broken into a diquark ($ud$) and an anti-
diquark ($\bar{u}\bar{d}$) pair. In Pythia, the probability for this to
happen, dubbed probQQtoQ, is too low. It has been increased from 0.09 to 0.24
in [41] to recover the value of ${\cal F}$ measured at LEP. Increasing
probQQtoQ has also the merit to enhance the decay of $\bar{\Lambda}_{b}$
baryons into ${}^{3}\overline{\rm He}$ nuclei. In the process, two protons are
created to conserve the baryon number. Given the masses of the initial ($5.6$
GeV) and final ($4.7$ GeV) states, the nucleons and anti-nucleons are produced
at rest and the coalescence of anti-nucleons into ${}^{3}\overline{\rm He}$
occurs easily, hence the possibility to explain the putative AMS events. This
proposal has been criticized insofar as modifying probQQtoQ plays havoc with
many observations, leading to an overproduction of baryons with respect to
mesons at LEP and the LHC [42].

##  6 Conclusion

The transport of charged particles inside our Galaxy is better understood,
especially in the light of recent AMS-02 measurements of secondary nuclear
species. The size of the magnetic halo $L$ is found to be $4.5\pm 1$ kpc.
Anomalies in the fluxes of antimatter charged CRs could be an indirect
signature of annihilating DM particles. But caution must prevail. Positrons,
for instance, are most probably accelerated in PWNe and are detected as
$\gamma$-ray TeV haloes. As regards antiprotons, many groups have reported an
excess. But taking properly the errors into account, i.e. including their
correlations at different energies, makes the excess recede. Finally, AMS-02
has reported a few anti-helium events. If confirmed, this would be a major
discovery, pointing to exotic physics. If DM annihilates into $b\bar{b}$
quarks, the decay of $\bar{\Lambda}_{b}$ into ${}^{3}\overline{\rm He}$ could
produce an excess over secondaries provided that diquark formation is
enhanced. The search for this reaction at the LHC and the measurement of its
branching ratio would definitely be of great interest.

## Acknowledgements

Many thanks to the organizers for their invitation and for the kind and
inspiring atmosphere of the conference. I also would like to thank my home
Institution LAPTh for its financial support.

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