11institutetext:  LPSC, Université Grenoble Alpes, CNRS/IN2P3, 53 avenue des
Martyrs, 38026 Grenoble, France 22institutetext: Kapteyn Astronomical
Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The
Netherlands 33institutetext: LAPTh, Université Savoie Mont Blanc & CNRS,
Chemin de Bellevue, 74941 Annecy Cedex, France 44institutetext: Niels Bohr
International Academy & Discovery Center, Niels Bohr Institute, University of
Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark

# The importance of Fe fragmentation for LiBeB analyses

Is a Li primary source needed to explain AMS-02 data?

D. Maurin<EMAIL_ADDRESS>E. Ferronato Bueno
<EMAIL_ADDRESS>Y. Génolini<EMAIL_ADDRESS>L.
Derome 11 M. Vecchi 22

(Received / Accepted)

###### Abstract

Context. High-precision data from AMS-02 on Li, Be, and B provide the best
constraints on Galactic cosmic-ray transport parameters.

Aims. We re-evaluate the impact of Fe fragmentation on the Li, Be, and B
modelling. We discuss the consequences on the transport parameter
determination and reassess whether a primary source of Li is needed to match
AMS-02 data.

Methods. We renormalised several cross-section parametrisations to existing
data for the most important reactions producing Li, Be, and B. We used the
usine code with these new cross-section sets to re-analyse Li/C, Be/C, and B/C
AMS-02 data.

Results. We built three equally plausible cross-section sets. Compared to the
initial cross-section sets, they lead to an average enhanced production of Li
($\sim 20-50\%$) and Be ($\sim 5-15\%$), while leaving the B flux mostly
unchanged. In particular, Fe fragmentation is found to contribute to up to 10%
of the Li and Be fluxes. Used in the combined analysis of AMS-02 Li/C, Be/C,
and B/C data, the fit is significantly improved, with an enhanced diffusion
coefficient ($\sim 20\%)$. The three updated cross-section sets are found to
either slightly undershoot or overshoot the Li/C and B/C ratios: this strongly
disfavours evidence for a primary source of Li in cosmic rays. We stress that
isotopic cosmic-ray ratios of Li (and to a lesser extent Be), soon to be
released by AMS-02, are also impacted by the use of these updated sets.

Conclusions. Almost no nuclear data exist for the production of Li and B
isotopes from Ne, Mg, Si, and Fe, whereas these reactions are estimated to
account for $\sim 20\%$ of the total production. New nuclear measurements
would be appreciated and help to better exploit the high-precision AMS-02
cosmic-ray data.

###### Key Words.:

Astroparticle physics – Cosmic rays – Diffusion – Nuclear reactions

##  1 Introduction

Galactic cosmic-ray (GCR) Li, Be, and B (hereafter LiBeB for short) isotopes
are present in minute amount in the Solar System, but are seen in excess in
cosmic-ray (CR) data (e.g. Tatischeff & Gabici 2018). These CR species are
denoted secondary species, as they are generally assumed to be produced only
by a nuclear interaction of heavier CR species on the interstellar medium
(ISM). The dominant channels for this production are the direct production of
LiBeB from C and O CR fluxes (e.g. Génolini et al. 2018). The latter fluxes
are among the most abundant CR species, and are of primary origin, that is
they result almost solely from the diffusive shock wave acceleration of the
ISM material. The study of secondary species, or secondary-to-primary ratios,
plays a central role in CR physics because they calibrate the transport
parameters in the Galaxy. The latter can give insight into the micro-physics
of transport in the turbulent medium (e.g. Génolini et al. 2019), but it is
also a central ingredient for many related studies (e.g. electron and positron
spectra, $\gamma$-ray diffuse emissions, indirect searches for dark matter in
CRs).

The AMS-02 experiment on-board the International Space Station has collected
an unprecedented number of CRs (Aguilar et al. 2021a); the data published by
the collaboration reach the few percent level of precision. The LiBeB data,
which display a spectral break at $\sim 200$ GV (Aguilar et al. 2018a), have
been used by several authors to study the transport of GCRs (Génolini et al.
2017, 2019; Evoli et al. 2019; Weinrich et al. 2020b; Boschini et al. 2020a,
b; Yuan et al. 2020; De La Torre Luque et al. 2021a). The above studies assume
no extra source of LiBeB, but it remains possible to have a small amount of
secondary production inside the acceleration site (Mertsch et al. 2021;
Kawanaka & Lee 2021), or even to have a primary source of Li (Kawanaka &
Yanagita 2018) from nova explosions (Hernanz 2015).

All model calculations rely on a network of CR fragmentation reactions.
Uncertainties on these reactions range from $10\%$ to $20\%$, and are a
limiting factor to take full advantage of the high-precision CR data (Génolini
et al. 2018). For this reason, possible excesses or mismatches (between the
model and the data) must be robustly checked against nuclear uncertainties
(among others). A new methodology to account for and propagate these
uncertainties was proposed in Derome et al. (2019). The main idea was to start
from a given production nuclear dataset and, while fitting LiBeB data, to
allow the most relevant reactions to vary around their central values;
penalties prevent cross sections from wandering far away from their expected
values and uncertainties. We successfully used this approach in Weinrich et
al. (2020b) to show that models with a pure secondary LiBeB component give an
excellent fit to the AMS-02 data, provided that a few cross-section values
were varied by $\sim 15\%$; similar but independent studies reached the same
conclusions (De La Torre Luque et al. 2021a, b; Korsmeier & Cuoco 2021).
Conversely, Boschini et al. (2020b) recently argued for a hint of primary Li
in CRs, though they do not exclude an overall bias in the Li production cross
sections.

At a first glance, there is not much to do to improve the main LiBeB
production channels. Indeed, comprehensive inspections of the nuclear cross-
section parametrisations were carried out by several independent teams twenty
years ago (Silberberg et al. 1998; Webber et al. 2003; Moskalenko et al. 2003;
Moskalenko & Mashnik 2003), and were also re-inspected in more recent studies
(Reinert & Winkler 2018; Evoli et al. 2018). However, a closer look shows that
Evoli et al. (2018) only focussed on progenitors up to O, while Reinert &
Winkler (2018) did so on those up to Si. This means that ‘recent’ nuclear
data, for reactions involving nuclei heavier than Si (up to Ni), may have been
skipped in this re-evaluation. The most comprehensive set of parametrisations,
taking advantage of all nuclear data up to Ni, are those of Webber (Webber et
al. 2003) and of the galprop team111https://galprop.stanford.edu/. These
parametrisations were, for instance, extensively used in Génolini et al.
(2018) to rank the most important production channels for LiBeB; in these
ranking, Fe (as a progenitor) only appears at the percent level. A closer
inspection of the nuclear physics literature indeed indicates that important
data, related to the fragmentation of Fe into LiBeB, only became available
after these parametrisations were established; this means that Fe
fragmentation into LiBeB must be re-evaluated.

This paper is organised as follows: in Sect. 2, we recall the setup used for
the propagation and the motivation for our renewed interest in Fe
fragmentation for LiBeB calculations. In Sect. 3, we gather and use available
nuclear data to update cross-section parametrisations of the most important
progenitors, highlighting the enhanced production of Li (with regard to Be and
B) from Fe. In Sect. 4, we show a ranking of the most important progenitors of
LiBeB, providing a complementary view of the rankings proposed in Génolini et
al. (2018). In Sect. 5, we discuss the impact updated cross sections have on
the derived transport parameters. In particular, we repeat and update our
recent analysis of AMS-02 LiBeB data (Weinrich et al. 2020b), in order to
address the interpretation of a primary source of Li in CRs. We also
illustrate how isotopic ratios of Li, Be, and B fluxes (e.g. 6Li/Li and
7Li/Li) are sensitive to the uncertainties on the production cross sections.
We then conclude in Sect. 6.

Additional comparisons and checks can be found in the appendices. App. A
identifies the progenitors whose updated cross sections make up most of the
observed changes in the Li/C, Be/C, and B/C ratios (hereafter LiBeB/C for
short). App. B shows that the use of the new AMS-02 F, Ne, Na, Mg, Al, Si, and
Fe data only marginally impact the LiBeB/C calculations, compared to our
previous analysis (Weinrich et al. 2020b) in which only AMS-02 C, N, and O
data were available. App. C shows the constraints set by the updated cross
sections on various combinations of the transport parameters (only the
simplest transport configuration is discussed in the main text).

##  2 Propagation setup and ingredients

###  2.1 CR flux calculation

The quasi-isotropic flux $\psi^{k}(E)$ of a CR ion is related to its
differential density $N^{k}(E)$ by $\psi^{k}=vN^{k}/(4\pi)$, with $N^{k}$
calculated from the transport equation (Berezinskii et al. 1990). In this
study, we consider a 1D geometry (e.g. Maurin et al. 2010), with a thin plan
(half-thickness $h=100$ pc) in which the sources and the gas lie, hence where
energy losses and nuclear reactions occur. We assume that CRs diffuse
isotropically in a thick halo of half-size $L$, with a diffusion coefficient
$K(R)$; without loss of generality for our analysis, we set $L=5$ kpc
(Weinrich et al. 2020a, b). Further assuming steady-state, the transport
equation in this geometry becomes a second order differential equation along
the vertical spatial coordinate and on energy. To highlight the role of
nuclear interactions, which are the prime focus of this study, we write below
the (simplified) equation for our model (energy loss and gain terms, decay
terms, and convective transport omitted):

$\displaystyle-K\frac{d^{2}\\!N^{k}}{dz^{2}}\\!=\\!2h\,\delta(z)\left[Q^{k}\\!+\\!\\!\\!\sum_{i\,\in\rm
ISM}\\!\\!n_{i}\left(\sum_{s}v^{s}\,\\!\sigma^{s+i\to k}_{\rm
prod}N^{s}\\!-v^{k}\,\\!\sigma^{k+i}_{\rm inel}N^{k}\right)\right].\\!\\!$ (1)

The left-hand side term corresponds to diffusion, while the right-hand side
has a primary source term, $Q^{k}(E)$, and two terms associated with nuclear
interactions, written generically as $\sum_{i}n_{i}v\,\sigma(E)$. We use
$n_{\rm ISM}=1\leavevmode\nobreak\ {\rm cm}^{3}$ ($90\%$ H and $10\%$ He in
number) and $\sigma$ is a nuclear interaction, either corresponding to a net
loss (destruction cross section $\sigma_{\rm inel}$) or to a secondary source
term (production cross section $\sigma_{\rm prod}$, straight-ahead
approximation assumed); the latter must sum over all possible CR progenitors
$s$ (heavier than the produced fragments of interest $k$). For CR elements
below Ni, the above set of equations couples $\sim 100$ species, which
involves a nuclear network of $\gtrsim 1000$ reactions. The above coupled set
of equations can be solved starting from the heaviest species—which is assumed
to be a primary—and solving down to the lightest species. This ensures that
all secondary contributions (from heavier species) are accounted for at each
step $k$.

All the analyses presented below rely on the usine
package222https://lpsc.in2p3.fr/usine (Maurin 2020), and we stress that the
analyses below solve the full transport equation, that is with energy gains
and losses, decay terms, and possibly a constant convection term in the 1D
geometry (Génolini et al. 2019).333The same model and approach was used in
several of our previous analyses of AMS-02 B/C and LiBeB/C nuclei (Génolini et
al. 2017, 2019; Weinrich et al. 2020b), and we refer readers to these
publications for more details. Inelastic cross-sections are an important
ingredient of the calculation, but their uncertainties are subdominant in the
total error budget (e.g. Génolini et al. 2019), so we do not discuss them in
this paper; the parametrisation of Tripathi et al. (1997, 1999) are used.

###  2.2 Transport parameters

Even in the simple propagation framework used, transport parameters come in
different flavours. For instance, Génolini et al. (2019) explored several
configurations, with or without convection and re-acceleration, and different
parametrisations of the diffusion coefficient $K(R)$. It was found that
several of them were able to give a very good fit to LiBeB data (Weinrich et
al. 2020b). To ease the discussion and without loss of generality, most of our
calculations will be carried out in the so-called SLIM configuration. The
latter consists of pure diffusion (no convection, no reacceleration) with a
break at both low- (Génolini et al. 2019; Vittino et al. 2019; Weinrich et al.
2020b) and high-rigidity (Génolini et al. 2017, 2019; Reinert & Winkler 2018;
Niu & Xue 2020) in the diffusion coefficient:

$K(R)={\beta}K_{0}\;{\left\\{1+\left(\frac{R_{l}}{R}\right)^{\frac{\delta-\delta_{l}}{s_{l}}}\right\\}^{s_{l}}}{\left\\{\frac{R}{1\,{\rm
GV}}\right\\}^{\delta}}\,{\left\\{1+\left(\frac{R}{R_{h}}\right)^{\frac{\delta-\delta_{h}}{s_{h}}}\right\\}^{-s_{h}}}.$
(2)

The high-rigidity break parameters $R_{h}$, $\delta_{h}$, and $s_{h}$ are
fixed to the values found in Génolini et al. (2019); we also enforce
$s_{l}=0.04$. We stress that for the rest of this section, the other
parameters are fixed to $\delta_{l}=-0.74$, $R_{l}$=4.53 GV, and
$\delta=0.51$, $K_{0}=0.0389$ kpc2 Myr-1 (Weinrich et al. 2020b). We later see
in Sect. 5 how these parameters are impacted when re-analysing the LiBeB data
with updated cross-section values. We also discuss how these results change
with alternative convection and reacceleration models in App. C.

###  2.3 Nuclear cross sections: relevance of Fe for LiBeB?

Production cross sections are the backbone of CR studies, and several
parametrisations are publicly available from the galprop or usine codes: (i)
W03 (Webber et al. 2003) relies on an unreleased version of the wnew code
(Webber et al. 1998a, b, c), which is a semi-empirical parametrisation fitted
on existing data; (ii) S01 (A. Soutoul, private communication) is very similar
to W03 but with different parameters; (iii) Galp (Moskalenko et al. 2001;
Moskalenko & Mashnik 2003) is based on systematic fits on existing nuclear
data, where Galp--opt12 and Galp--opt22 pre-fit parametrisations come
respectively from the 1998 version of the wnew code or from the 2000 version
of the yieldx code (semi-analytical formulae fitted on data, Silberberg et al.
1998; Tsao et al. 1998). More details on all these parametrisations can be
found in Génolini et al. (2018).

Many LiBeB/C recent studies considered CR progenitors up to Si at most
(Reinert & Winkler 2018; Evoli et al. 2018; Génolini et al. 2019). Actually,
most studies addressing a possible mismatch in the Li production considered
parents up to Si (Weinrich et al. 2020b; Boschini et al. 2020b; De La Torre
Luque et al. 2021b; Korsmeier & Cuoco 2021), and those considering progenitors
up to Fe (Boschini et al. 2020a, b) relied on the Galp--opt12 production cross
sections.

Figure 1: Relative difference on Li/C (top), Be/C (middle), and B/C (bottom)
between $R_{\rm Fe}$ (progenitors up to Fe, $Z=26$) and $R_{\rm Si}$
(progenitors up to Si, $Z=14$) as a function of rigidity. The comparisons are
carried out for four cross-section parametrisations available in the
literature: Galp--opt22 (dotted lines), S01 (dash-dotted lines), W03 (dotted
lines), and Galp--opt12 (solid lines). The enhanced production observed with
Galp--opt22 motivates the re-analysis of the LiBeB production cross sections.

To highlight the importance of these different settings (cross-section set and
heaviest CR parent considered), we show in Fig. 1 the relative difference
between calculations of Li/C, Be/C, and B/C ratios (from top to bottom)
performed with progenitors up to Fe, compared to a reference calculation with
progenitors only up to Si. The various line styles and colours encode the use
of the four cross-sections parametrisations discussed above, that is W03, S01,
Galp--opt12, and Galp--opt22. The solid black lines (i.e. Galp--opt12) show
that the choice of the heaviest progenitor (Fe instead of Si) has a marginal
impact ($\lesssim 3\%$) on all ratios, in line with checks that calculations
up to Si are accurate enough to interpret LiBeB/C data (Maurin et al. 2001;
Génolini et al. 2019). The use of S01 (dash-dotted blue) and W03 (dashed
green) parametrisations lead to similar conclusions, though with a $\sim 5\%$
increase of Be/B (middle panel); W03 does not provide production cross
sections for Li. Only with Galp--opt22 (grey-dashed lines) do we see a
significant and differential increase for Li/C ($\sim 15\%$), Be/C ($\sim
10\%$), and B/C ($\sim 5\%$). A close inspection shows that large differences
exist between specific reactions for these different parametrisations, in
particular for the fragmentation of 56Fe into LiBeB isotopes.

This discrepancy, corresponding to an increased production of light nuclei in
Fe fragmentation, might explain the mismatch seen in several analyses of Li CR
data (Weinrich et al. 2020b; Boschini et al. 2020b; De La Torre Luque et al.
2021b; Korsmeier & Cuoco 2021). This clearly motivates a re-investigation of
the nuclear data to see which of the above parametrisations is favoured by
recent nuclear data.

##  3 Updating the LiBeB production cross sections

In this section, we review the wealth of nuclear data relevant for our study,
briefly mentioning the different measurement techniques (Sect. 3.1). We then
detail a procedure to update the cross-section parametrisations, highlighting
the most impacted reactions (Sect. 3.2) and bracketing by how much Fe
fragmentation impacts the LiBeB/C calculations (Sect. 3.3). In the rest of the
paper, the original cross-section sets considered are the Galp--opt12 and Galp
--opt22 parametrisations, and to avoid confusion, the new (updated) cross-
section sets are denoted OPT12 and OPT22 respectively (a hybrid third set,
OPT12up22, is also considered).

###  3.1 Nuclear data

Most of the nuclear cross sections relevant to CR studies are from the 1980s
and 1990s, when it was realised that these inputs were essential to interpret
elemental CR measurements. Back then, CR data were gaining in precision over a
wider energy range, a situation quite similar to the one observed nowadays
(with the advent of AMS-02, DAMPE, CALET, and PAMELA instruments). On the one
hand, a wealth of nuclear data was provided by the Transport Collaboration et
al. (1990), who studied isotopic production cross sections through direct
reactions, that is by measuring secondaries produced by ion beams (mostly
$Z<26$) colliding liquid hydrogen or carbon and methylene CH2 targets (and
using a ${\rm CH}_{2}-{\rm C}$ subtraction technique). This technique was
notably used by Bill Webber and his colleagues, in the energy range
$\sim$400–800 MeV/n (Webber & Brautigam 1982; Ferrando et al. 1988; Webber et
al. 1990d, b, c, a, 1998c, 1998a, 1998b; Knott et al. 1996; Chen et al. 1997a,
b; Knott et al. 1997). On the other hand, Michel and Leya’s group (Michel et
al. 1989, 1995, 1997; Dittrich et al. 1990a; Bodemann et al. 1993; Sudbrock et
al. 1996; Leya et al. 1998, 2005, 2006; Ammon et al. 2008) as well as
Sisterson’s group (Sisterson et al. 1992, 1994, 1997, 2000; Sisterson & Caffee
1998; Kim et al. 2002; Sisterson & Vincent 2006) relied on indirect reactions,
where a target of heavy elements is irradiated by a high-energy proton beam.
With this technique, the cross sections are determined by
$\gamma$-spectrometry whenever radioactive elements are produced, and after
chemical processing by mass spectroscopy for the long-lived and stable
isotopes. Although the target materials are generally not composed of pure
isotopes, this technique gives a good approximation of the isotopic production
cross sections from the dominant isotope in nature. We stress that this second
technique does not directly provide the required cross sections, but must
reconstruct the desired ones via measurements at different times (to account
for the decay chain of short-lived nuclei)—see, for instance, Titarenko et al.
(2008) and references therein for the steps to derive individual cross
sections from cumulative ones.

The data collected for this work come from these two different techniques. Our
starting point is the galprop cross-section database, assembled in the file
isotope_cs.dat of the corresponding code444https://galprop.stanford.edu/). We
then performed extensive checks of the references therein—and also a few
others pointed out in Reinert & Winkler (2018)—along with systematic
extractions from the exfor database555https://www.nndc.bnl.gov/exfor/exfor.htm
(Otuka et al. 2014). The reactions under scrutiny are those involving the most
important progenitors of Li, Be, and B isotopes and ghosts (see below), that
is (i) B, C, N, O, Ne, Mg, and Si, as highlighted in Génolini et al. (2018);
(ii) specific production of some light isotopes from Li and Be progenitors
(for completeness of the $Z\leq 6$ progenitors), and (iii) Fe which is the
main novelty of this study. Indeed, new relevant fragmentation data from 56Fe
(Napolitani et al. 2004; Herbach et al. 2006; Villagrasa-Canton et al. 2007;
Titarenko et al. 2008, 2011) were published seemingly after the galprop
database was last updated. Finally, our cross-checks lead us to replace some
of the Read & Viola (1984) compiled data (used in the galprop file) by the
original ones, and a few were even removed (because they corresponded to
cumulative cross sections, i.e. with the contributions of short lives nuclei).

The updated dataset, for the specific channels leading to LiBeB, is more
complete than the one used in Génolini et al. (2018). The data gathered are
shown in Figs. 2, 3, and 4. Despite these additional datasets, the reader can
witness the scarcity of the data at energies relevant for GCRs studies (above
GeV/n) and even the absence of data for some important channels. Recent
reconsideration of this issue (Génolini et al. 2018) has already triggered a
promising new experimental programme (Unger & NA61 Collaboration 2019) to
improve the precision of these nuclear data and fill some of the gaps in
existing measurements.

###  3.2 From data to updated parametrisations

Providing updated cross sections and the procedure to do so depends on three
factors: the energy range we are interested in, the energy coverage of nuclear
cross-section data, and the underlying model assumed for the production cross
sections. The relevant energy range for AMS-02 data is a few GV to a few
TeV/n, so that roughly, production cross sections of interest are in the GeV/n
to TeV/n range. However, beside a few data points at 300 GeV/n (Raisbeck et
al. 1975), and awaiting the promising new data at 13.5 GeV/n from the
NA61/SHINE collaboration (Unger & NA61 Collaboration 2019; Amin & NA61/SHINE
2022), the bulk of nuclear data are below a few GeV/n. Furthermore, many
reactions of interest for CRs are still lacking measurements or only have a
few data points. In this context, very few reactions have a full coverage in
energy, that is from tens of MeV/n (reaction threshold) up to multi-GeV/n,
where it is generally assumed that the cross-sections become constant with
energy. For this reason, it is necessary to have an underlying model to
provide the cross sections for the missing reactions, or at least to provide
an energy dependence to be normalised on the few existing data. This is in
spirit close to the approach followed by the galprop team.

Of course, other approaches exist to improve, extend, and validate isotopic
production cross sections (e.g. Boudard et al. 2013; Cheng & Feng 2021; Tao et
al. 2021; Werneth et al. 2021; Dan et al. 2021). These works are more focussed
on the microphysics. They also have a broader scope, in order to tackle a much
larger variety of reactions and situations. However, they are not necessarily
optimised for CR physics. For the time being, there is not a key advantage to
using these other approaches, thus we rely here on updates and rescaling of
the Galp--opt12 and Galp--opt22 parametrisations.

####  3.2.1 Procedure

After several trials and errors, we came up with a quite flexible procedure to
update the production cross sections on hydrogen. This procedure proves to be
quite successful in matching the data, in order to handle (i) the variety of
situations for any given reaction (completeness or scarcity of data,
discrepant data), (ii) the several energy regimes of the production process
(steep rise at threshold, possible bumps and deeps, and asymptotically
constant behaviour at high energy), and (iii) the fact that neither a simple
rescaling of the Galp--opt12 and Galp--opt22 cross sections, nor a generic
(and at the same time specific) enough formula can be satisfactorily used to
fit all reactions. For any given reaction, this procedure goes in four steps
detailed below.

##### Step 1: rescaling over the whole energy range.

To do so, (i) we calculate the ratio $r=$ data/model for the associated
nuclear data; (ii) from these $r$, we form a spline ${\cal S}_{r}$ that is
used as an interpolation function (constant if used for extrapolation); (iii)
we multiply the original model by ${\cal S}_{r}$. A logarithmic smoothing
kernel (0.15 decade in energy) is then applied to ensure that the cross
sections cannot have large variations on small energy scales666This smoothing
cures situations of nearby discrepant data points, while its main drawback is
to slightly shift the threshold energy of the reaction to higher energy if
there are no data; otherwise see Step 2..

##### Step 2: dedicated fit for the low-energy rise.

Near the reaction threshold, in order to reproduce the quick rise and possible
bump, if enough data are available (typically $>6$ data points), we directly
fit a second order polynomial in $\log-\log$ scale enforcing convexity. This
fit, that accounts for data uncertainties, replaces Step 1’s result in the
low-energy regime.

##### Step 3: high-energy behaviour.

Above 1.5 GeV/n, we consider two alternatives. We either keep the possible
energy-dependant behaviour obtained at Step 1 (referred to as no-fit HE later
on), or enforce a constant value based on a fit (accounting for data
uncertainties) on existing data at these energies777We chose 1.5 GeV/n in
order to have one or multiple points to fit a constant, but this is far from
being an ideal choice, in a region where some energy dependence might
remain—see the prediction of the wnew code in Fig. 5 of Webber et al. (2003)
as an illustration. In any case, the residual dependence of the models is
generally within the nuclear data uncertainties (see the same figure in Webber
et al. 2003). Waiting for more data in this energy range, using 1.5 GeV/n or a
higher value (e.g. 2 GeV/n) would change a few cross-section values here and
there, but we checked that it has a minor impact on the global calculation and
all our conclusions.. The latter configuration is taken as default, used
throughout our paper, unless explicitly stated otherwise.

Figure 2: Comparison of models and nuclear data for the fragmentation cross
sections of 56Fe, 28Si, 24Mg, 20Ne, and 16O into Li (left), Be (centre), and B
(right) isotopes and associated ghosts. The references for the data points are
gathered in Fig. 5.

Figure 3: Same as 2, but for 15,14N, 12C, and 10,11B progenitors. The
references for the data points are gathered in Fig. 5.

Figure 4: Same as 2, but for and 10,9,7Be and 7Li progenitors. The references
for the data points are gathered in Fig. 5.  Figure 5: References for all data
points shown in Figs. 2, 3, and 4, sorted from the most recent to the older
ones. Specific labels from the original galprop files are [imos], [We96], and
[NU98], which correspond to Igor Moskalenko’s additional data, Bill Webber
private communication on Ramaty et al. (1997), and data retrieved from Nuclex
(Ivanov et al. 1998) respectively.

##### Step 4: scaling for He targets.

Finally, not only cross sections on H matter, but cross sections on He ($\sim
10\%$ in number in the ISM) are also necessary. As a last step, we rescale the
re-evaluated cross sections on H using the empirical formulae of Ferrando et
al. (1988)888See also Reinert & Winkler (2018) for a discussion of possible
shortcomings of this scaling relation.. These formulae, however, if naively
applied, lead to arbitrarily large enhancements for the extrapolation to small
fragments (in particular Li) and low energies, so that we enforce an ad hoc
maximal enhancement $(\sigma^{\rm He}/\sigma^{\rm H})_{\rm max}=6$. This can
be compared to the few data points of Yashima et al. (2004), for the
fragmentation of $Z=24$ and $Z=26$ into 7Be: at 100 (resp. 230 MeV/n),
$\sigma^{\rm He}/\sigma^{\rm H}=1.8$ (resp. $2.0$) for Cr, whereas
$\sigma^{\rm He}/\sigma^{\rm H}=17$ for Fe999At 100 MeV/n, in order to
calculate this ratio for Fe, we used for $\sigma^{\rm H}$ the data point from
Michel et al. (1997) because no corresponding data point was available from
Yashima et al. (2004). (resp. $11$). These measurements show irregularities
that could probably not be captured by the scaling proposed by Ferrando et al.
(1988). However, as will be seen later, the contribution of Fe fragmentation
for the production of Li and Be reaches at most $10\%$: even a factor 3 error
on the production of He (which is only $10\%$ in number) would amount to a
mere $\lesssim 3\%$ bias on the total production; moreover, these enhanced
production of light fragments only applies below a few hundreds of MeV/n, that
is much below the range where AMS-02 collects data. Nonetheless, they might be
of interest to interpret very low-energy Voyager data (Cummings et al. 2016).

####  3.2.2 Updated cross sections and most impacted reactions

As already said, we consider the most important CR progenitors for LiBeB
production. We show the data and our new cross sections for the following CR
isotopes: 56Fe, 28Si, 24Mg, 20Ne, and 16O progenitors in Fig. 2; then 14,15N,
12C, and 10,11B progenitors in Fig. 3; and finally 7,9,10Be and 7Li
progenitors in Figs. 4. The legend and associated references for all the data
presented are listed in Fig. 5.

In Figs. 2, 3, and 4, the three columns (left, centre, and right) show the
reactions related to the production of Li, Be, and B respectively. They
include the direct production of stable (or CR-relevant) isotopes, but also
the contributions from short-lived nuclei (dubbed ghosts), whose final decay
state is one of these CR isotopes (Letaw et al. 1984; Maurin 2001): we recall
that the cumulative cross-sections required for CR propagation corresponds to

$\displaystyle\sigma^{\rm c}(X+H\to Y)$ $\displaystyle=$
$\displaystyle\sigma(X+H\to Y)$ $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\\!\sum_{G\,\in\,{\rm ghosts}}\\!\\!\\!\sigma(X+H\to
G)\cdot{\cal B}r\,(G\\!\to\\!Y)\,,$

where ${\cal B}r\,(G\\!\to\\!Y)$ is the branching ratio for the ghost $G$ to
end up as $Y$ (see the third row of Table 1 for their values). In each panel
of Figs. 2–4, the data (symbols) and updated models (OPT12 in thin lines and
OPT22 in thick lines) are colour-coded according to the fragments considered.
It is worth highlighting a few significant features: (i) when plenty of data
exist, our procedure is very successful in matching the data at all energies;
(ii) where only very few data exist, our procedure roughly amounts to a global
rescaling of the initial cross sections; (iii) Galp--opt12 and Galp--opt22
parametrisations can have very different energy dependences, and these
differences remain in OPT12 and OPT22 when only a few data are available (e.g.
compare the thin and thick lines in the right-centre panel of Fig. 2, i.e.
24Mg in B through 11C); (iv) in several very relevant cases for which Galp--
opt12 was null, the updated cross section OPT12 was replaced by a constant
value passing through the few data points (e.g. top-left panel of Fig. 2, that
is 56Fe in Li); (v) as already said, many reactions still lack any data (e.g.
bottom-left panel of Fig. 3, that is 10Be in Li). We finally stress that for
many reactions, Galp--opt12 and Galp--opt22 initial parametrisations were
observed to be one and the same, so that their updated counterparts OPT12 and
OPT22 are also one and the same; this peculiarity comes from the fact that
these initial parametrisations were probably already rescaled to pre-2004
nuclear data for their use in galprop.

Table 1: High-energy value of the rescaling factors ${\cal R}_{XS}^{method}=\sigma^{updated}_{XS}/\sigma^{init}_{XS}$ for the production of Li, Be, and B using a production cross-section set $XS$ built from $method$; we take $method$ here to be the rescaling with constant cross section above 1.5 GeV/n (dubbed ‘fit $>\\!1.5$ GeV/n’, see Sect. 3.2.1 and text for details). The first column shows the list of progenitors (or projectiles) and the first three lines show (i) the considered CR element, (ii) the associated list of fragments (isotopes and ghosts ${\cal B}{\rm r}>5\%$), and (iii) the branching ratios of the ghosts into CRs (in parenthesis). Entries in the table sometimes show two factors for the two updated $XS$ sets considered (${\cal R}_{XS=\,\rm{\tt OPT12}{}}\,|\,{\cal R}_{XS=\,\rm{\tt OPT22}{}}$); however, most of the time a single number is shown, the original Galp--opt12 and Galp--opt22 sets being similar for most reactions. The following special keys are also used: $\times$ means no-data available (hence no possible re-evaluation of the cross section), and $\infty$ refers to an ill-defined ${\cal R}$ (null value for $\sigma^{init}_{XS}$), though the updated cross-section value $\sigma^{updated}_{XS}$ is well defined. | Li |  | Be |  | B
---|---|---|---|---|---
| ${}^{6}{\rm Li}$ | ${}^{7}{\rm Li}$ | ${}^{6}{\rm He}$ |  ${}^{8}{\rm He}\\!\\!$ |  | ${}^{7}{\rm Be}$ | ${}^{9}{\rm Be}$ | ${}^{10}{\rm Be}$ |  ${}^{11}{\rm Li}$ | ${}^{9}{\rm Li}$ |  | ${}^{10}{\rm B}$ | ${}^{11}{\rm B}$ | ${}^{11}{\rm C}$ | ${}^{10}{\rm C}$ | ${}^{11}{\rm Be}$ |  ${}^{11}{\rm Li}$
|  |  | (100%) | (16%) |  |  |  |  | (85%) | (49%) |  |  |  | (100%) | (100%) | (97%) | (7.8%)
${}^{56}{\rm Fe}$ |  $\infty$—0.8 |  $\infty$—1 |  $\infty$—0.6 | $\times$ |  | 15—0.8 | 21—1.4 | 19—0.7 | $\times$ |  $\infty$—0.3 |  | 20—0.8 | 15—1 | 2.0—1.7 | $\times$ | $\times$ | $\times$
${}^{28}{\rm Si}$ | $\times$ | $\times$ | $\times$ | $\times$ |  | 1 | 1.05 | 1.02 | $\times$ | 0.4 |  | $\times$ | $\times$ | 0.5—1.2 | $\times$ | $\times$ | $\times$
${}^{24}{\rm Mg}$ | $\times$ | $\times$ | $\times$ | $\times$ |  | 1.04 | 2.04 | 0.95 | $\times$ | 0.6 |  | $\times$ | $\times$ | 0.5—1.1 | $\times$ | $\times$ | $\times$
${}^{20}{\rm Ne}$ | $\times$ | $\times$ | $\times$ | $\times$ |  | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  | $\times$ | $\times$ | 0.95 | 1 | $\times$ | $\times$
${}^{16}{\rm O}$ | 1.4 | 0.96 | $\infty$ | $\times$ |  | 0.98 | 1.41 | 1.18 | $\times$ | 0.7 |  | 0.96 | 0.4 | $\infty$ | $\infty$ | $\infty$ | $\times$
${}^{15}{\rm N}$ | 1 | 1 | $\times$ | $\times$ |  | 1 | 1 | 1 | $\times$ | 0.5 |  | 1.34 | 1.17 | 1 | $\times$ | $\times$ | $\times$
${}^{14}{\rm N}$ | 1 | 1 | $\infty$ | $\times$ |  | 1.18 | 0.94 | 1.02 | $\times$ | 0.8 |  | 0.91 | 0.6 | $\infty$ | $\infty$ | $\times$ | $\times$
${}^{12}{\rm C}$ | 1.1 | 0.94 | $\infty$ | $\times$ |  | 0.94 | 0.91 | 1.04 | $\times$ | $\infty$ |  | 1.08 | 0.92 | 1.04 | 0.7 | $\times$ | $\times$
${}^{11}{\rm B}$ | 1 | 1 | $\times$ | $\times$ |  | 1.06—1.16 | 1.04 | 0.4 | $\times$ | 0.97 |  | 0.93 | … |  $\infty$—0.7 |  $\infty$—1.16 | $\times$ | $\times$
${}^{10}{\rm B}$ | $\times$ | $\times$ | $\times$ | $\times$ |  | 0.94 | $\times$ | $\times$ | … | $\times$ |  | … | … | … |  $\infty$—1.75 | … | …
${}^{10}{\rm Be}$ | $\times$ | $\times$ | $\times$ | $\times$ |  | $\times$ | $\times$ | … | … | $\times$ |  | $\times$ | … | … | $\times$ | … | …
${}^{9}{\rm Be}$ | 1 | 1 | $\times$ | $\times$ |  | 1 | … | … | … | $\times$ |  | … | … | … | … | … | …
${}^{7}{\rm Be}$ | $\times$ | $\times$ | $\times$ | … |  | … | … | … | … | … |  | … | … | … | … | … | …
${}^{7}{\rm Li}$ | 1 | … | $\times$ | … |  | $\infty$ | … | … | … | … |  | … | … | … | … | … | …

In order to have a more concise view of the main changes in these derived
cross sections, we gather in Table 1 the ratio of the updated to initial cross
sections above 1.5 GeV/n, ${\cal R}=\sigma^{\rm updated}/\sigma^{\rm
initial}$, that is in a regime where we enforce the cross sections to be
constant. The first column and first row respectively show the progenitors and
fragments that appear in Figs. 2–4. The main comments we can make on this
table are: (i) while data exist for the most important LiBeB contributors (C,
N, and O, see Sect. 4), the less important Si, Mg, and Ne channels have no
data at all for the production of Li and 10,11B; (ii) contributions from the
secondary species 10B and 7Be into Li—that peak at GeV/n energies, see
Génolini et al. (2018)—also lack data; (iii) there is also no data for the
production of 8He and 11Li ghost nuclei, however, owing to their small
branching ratios and small cross-section predictions (compared to the other
isotopes), these reactions are not expected to even reach the percent level of
contribution overall; (iv) surprisingly, the initial cross sections Galp--
opt12 and Galp--opt22 for the fragmentation of 12C and 16O into the short-
lived 10C and 11C nuclei were null (a closer inspection indicates that these
contributions were probably added to the direct production of 10Be and 11B;
(v) the most striking numbers are those for the fragmentation of 56Fe,
indicating that the initial Galp--opt12 (left-hand side numbers in the cells)
significantly undershot the data, whereas the initial Galp--opt22 (right-hand
side numbers in the cells) predictions were more or less in line with the data
(with an overall small overshoot though). Looking back at Fig. 1, this latter
behaviour seems to confirm and favour the case for an enhanced production of
Li with regard to both Be and B (dotted lines).

Table 2: Ratio of high-energy rescaling factors between two methods tested, $\tilde{\cal R}={\cal R}^{{\rm fit}\leavevmode\nobreak\ >1.5\leavevmode\nobreak\ {\rm GeV/n}}/{\cal R}^{\rm no-fit}$, with ‘fit $>\\!1.5$ GeV/n’ for which the cross-section is constant above 1.5 GeV/n (fit on existing data and their errors) and ‘no-fit’ for which the model goes through all available points. First column (projectiles) and line (fragments) are as in Table 1. Otherwise, we only show here reactions for which the two re-scaling methods used in $\tilde{\cal R}$ differ by more than $5\%$. | Be |  | B
---|---|---|---
| ${}^{7}{\rm Be}$ | ${}^{9}{\rm Be}$ | ${}^{10}{\rm Be}$ | ${}^{9}{\rm Li}$ |  | ${}^{10}{\rm B}$ | ${}^{10}{\rm C}$
|  |  |  | (49%) |  |  | (100%)
${}^{56}{\rm Fe}$ | 0.9 | 0.9 | 1.2 | … |  | … | …
${}^{16}{\rm O}$ | 1.4 | 2.4 | 3.9 | 1.75 |  | 0.67 | …
${}^{14}{\rm N}$ | 0.8 | … | 0.7 | … |  | … | …
${}^{12}{\rm C}$ | … | … | 1.1 | … |  | 1.11 | 0.6

Table 2 shows the ratio $\tilde{\cal R}={\cal R}^{{\rm
fit}\leavevmode\nobreak\ >1.5\leavevmode\nobreak\ {\rm GeV/n}}/{\cal R}^{\rm
no-fit}$, that is the comparison of two different choices for the high-energy
update of the cross-section parametrisations: the numerator, ${\cal R}^{{\rm
fit}\leavevmode\nobreak\ >1.5\leavevmode\nobreak\ {\rm GeV/n}}$, corresponds
to the rescaling factors shown in Table 1 where cross sections are enforced to
be constant above $1.5$ GeV/n; the denominator, ${\cal R}^{\rm no-fit}$,
corresponds to the case in which the cross sections follow closely all
structures seen in the high-energy data (see Sect. 3.2.1). This $\tilde{\cal
R}$ factors gives a hint of the variability of the data at high energy, and
how sensitive the cross-section modelling is to these data. For readability,
Table 2 only shows values significantly different from 1, that is reactions
directly sensitive to this rescaling choice. We see that the fragmentation of
56Fe into Be isotopes and ghosts, and also that of 16O into Be and 10B, are
significantly dependent on the two renormalisation procedure used. For a
clearer illustration of this dependence, we show in Fig. 6 the production of
Be from 56Fe (left panel) and 16O (right panel). The updated cross sections
(thick coloured lines) clearly split above 1.5 GeV/n, and the case of
$\sigma^{\mathrm{{}^{16}O+H\to^{10}Be}}$ illustrates the difficulty to favour
one or the other approach, as the high-energy points are clearly below those
at $\sim 1$ GeV/n, though with much larger error bars. We also show on these
plots the initial cross sections (thin black lines), to highlight the fact
that moving from the old to the new cross sections is often more than a mere
rescaling.

Figure 6: Comparison of original (Galp--opt22, thin black lines labelled ‘OLD
XS’) and updated (OPT22, thick coloured lines labelled ‘NEW XS’) production
cross sections, for 56Fe (left) and 16O (right) fragmenting into Be isotopes.
The two branches seen in some of the updated cross sections above 1.5 GeV/n
illustrate two different procedures to update them (fit of a constant or
spline passing through the data, see Sect. 3.2.1). The ratio of the values in
these two branches at high energy corresponds to the values $\tilde{\cal
R}={\cal R}^{{\rm fit}\leavevmode\nobreak\ >1.5\leavevmode\nobreak\ {\rm
GeV/n}}/{\cal R}^{\rm no-fit}$ shown in Table 2.

###  3.3 Impact on Li/C, Be/C, and B/C ratios

To start this subsection, we recall that we considered two different initial
cross-section parametrisations, namely Galp--opt12 and Galp--opt22, rescaled
to nuclear data and updated into OPT12 and OPT22. We also build a third cross-
section set, where cross sections without data are taken from Galp--opt12,
while those with data and updated are taken from OPT22; this hybrid
configuration is denoted OPT12up22101010The motivation behind this set is that
Galp--opt22 provides a more realistic energy dependence for Fe fragmentation
(Galp--opt12 has null values), though it is not clear, for reactions without
data, whether Galp--opt22 or Galp--opt12 is more appropriate..

Figure 7: Enhancement of Li/C (orange thick lines), Be/C (green lines), and
B/C (blue thin lines) flux ratios with regard to calculations performed using
Galp--opt12 (used in all recent studies). The three sets of curves correspond
to the updated production cross-section sets OPT12 (dashed lines), OPT12up22
(solid lines), and OPT22 (dash-dotted lines).

We can now provide and bracket the impact these updated production cross
sections have for the calculation of CR LiBeB/C ratios, as shown in Fig. 7.
These curves tell us that heavy species up to Fe matter for the LiBeB
production (see Fig. 1). As can be seen, whereas B is not impacted ($\lesssim
4\%$) by the use of the updated cross-section sets, both Be and Li are
significantly increased with regard to calculations performed with the
original set Galp--opt12. These enhancements are also more important for Li
than for Be, and this ordering holds true for the three updated cross-section
sets. To better understand the origin of the increased production of Li and
Be, we show in App. A the impact of individual progenitors. It confirms that
Fe updated cross sections are the main driver of the Li and Be enhancements;
the update of O and Si cross-sections is also contributing to this
enhancement. More precisely, we can identify three different origins for the
changes in the predicted (LiBeB)/C ratios, with the combined impact of (e.g.
at 10 GV): (i) accounting for 56Fe fragmentation when moving from Galp--opt12
to Galp--opt22 that can be read off Fig. 1 ($+15\%$ for Li, $+10\%$ for Be,
and $+4\%$ for B); (ii) renormalising Galp--opt22 into OPT22 for 56Fe
reactions and other progenitors as read off the right panel of Fig 14 ($+6\%$
for Li, $+2\%$ for Be, and $-4\%$ for B); (iii) accounting for the differing
cross sections values for unmeasured reactions in Galp--opt12 and Galp--opt22.
The latter is mostly significant for Li, as read off Fig. 7, indicating that
the global change moving from Galp--opt12 to OPT12up22 can be even larger than
the sum of the two first effects ($+50\%$ for Li, $+10\%$ for Be, and $+2\%$
for B).

Concerning the spread between the OPT12, OPT12up22, and OPT22 sets in Fig. 7,
we observe that the enhancement for Be is relatively well predicted ($6\%$
spread between the updated cross-section sets), while the spread on Li remains
quite large ($\sim 10-20\%$). This spread originates from the different values
of the cross sections in the three updated sets. Indeed, for Fe, OPT22 (dash-
dotted lines) provides a growing production cross sections with energy,
whereas it is constant in OPT12 (see top left panel of Fig. 2). The difference
between OPT12up22 (solid lines) and OPT22 (dash-dotted lines) is more subtle
and related to cross sections not rescaled: in the former set based on Galp--
opt22 parametrisation, these cross sections are globally larger than in the
latter set based on Galp--opt12. It is not clear whether the observed spread
could be reduced by also applying our procedure to less-important cross-
section reactions, or whether this difference is related to still unmeasured
reactions; this certainly deserves further inspection. Finally, it is also
difficult to decide which of the three updated cross-section sets proposed is
the most motivated, all the more because, for the few reactions studied here
and that were not renormalised to the data before, the cross sections from
Galp--opt12 were found on average to undershoot the data, while those from
Galp--opt22 to overshoot them (see Table 1).

##  4 Updated ranking for Li production

As underlined several times already, a thorough study and rankings of the most
important reactions (or projectiles) involved in the production of Li to N
fluxes were provided in Génolini et al. (2018). Owing to the important role of
Fe fragmentation—that was not appreciated at that time—these ranking need to
be updated. Instead of directly updating Table V-VII and X-XII of Génolini et
al. (2018), we provide a complementary view here, grouping contributions of
specific CR elements. In words, for any given charge of a CR element $Z_{i}$,
we can track all the contributions leading directly or indirectly to Li, Be,
or B. Mathematically-speaking, we can define the $f_{i\to j}$ coefficients,
which correspond to the fractional contribution of a CR isotope $i$ ending up
in a CR isotope $j$, via direct (1-step) or multi-step production. These
coefficients read (see Eq. A1 in Génolini et al. 2018)

$f_{i\to j}=f^{\rm 1-step}_{ij}+\sum_{i>k>j}f^{\rm
2-step}_{ikj}+\sum_{i>k>l>j}f^{\rm 3-step}_{iklj}+\dots,$ (4)

where $i>k$ means that the CR isotope $i$ is heavier than $k$. Denoting
$\sigma_{ij}$ the cross section of $i$ fragmenting in $j$ (for any ISM
target), the $f_{i\to j}$ coefficients can be calculated from

$f_{i\to j}=\frac{\psi_{j}^{\rm sec}\big{(}\sigma_{kk^{\prime}}\\!=\\!0\;{\rm
for}\;k>i\,\&\,k^{\prime}\leq k\big{)}-\psi_{j}^{\rm
sec}\big{(}\sigma_{ik}\\!=\\!0\;{\rm for}\;k\leq i\big{)}}{\psi_{j}^{\rm
sec}}\;,$ (5)

which by construction, ensures $\sum_{i}f_{i\to j}=1$. We can then define the
fractional contribution $f_{Z_{i}\to Z_{j}}$ of a CR element of charge $Z_{i}$
into a CR element of charge $Z_{j}$ (with $n_{Z_{j}}$ isotopes) to be

$f_{Z_{i}\to Z_{j}}=\frac{1}{n_{Z_{j}}}\times\displaystyle\sum_{\forall
Z(i)=Z_{i},\,\forall Z(j)=Z_{j}}f_{i\to j}\,,$ (6)

which also satisfies $\sum_{Z_{i}}f_{Z_{i}\to Z_{j}}=1$.

Figure 8: Colour-coded ranked progenitors (elements) of the Li, Be, and B
(from top to bottom) fluxes as a function of rigidity. Calculations are based
on OPT12up22 production cross-section set with fluxes modulated at 700 MV. For
readability, only contributing fractions larger than $1\%$ are shown.

Figure 8 shows from top to bottom, using OPT12up22, the quantities
$f_{Z_{i}\to{\rm Li}}$, $f_{Z_{i}\to{\rm Be}}$, and$f_{Z_{i}\to{\rm B}}$ as a
function of rigidity. Only progenitor elements having contributing fractions
$>1\%$ are shown, in order not to overcrowd the plot. As in Génolini et al.
(2018), we recover that C and O are the main progenitors of Li, Be, and B, but
at variance with these results, with the use of the updated cross-section
sets, Fe can provide up to $\sim 10\%$ of Li and Be (and $\sim 5\%$ of B) at
very high rigidity111111These numbers remain consistent with the values
reported for Fe in the ‘Max’ column of Tables V-VII of Génolini et al. (2018)
at 10 GeV/n, which were calculated with Galp--opt22. We stress that the column
‘Mean’ resulted from an average of S01, W03, Galp--opt12, and Galp--opt22 (see
their Sect. 2.3), where 3 out of these 4 parametrisations give a negligible
contribution of Fe into LiBeB (see Fig. 7 in this paper).. The remaining
contributors are Mg, Ni, Ne, and N, that is the progenitors whose cross
sections into LiBeB were updated in this study. We do not detail the multi-
step production accounted for in the above plot. For instance, for Li, it
amounts to $\lesssim 15\%$ of the total production and peaks at at few GV
(Génolini et al. 2018). Among the important two-step reactions for the
production of Li, one finds 16O$\to^{11}$B$\to^{7}$Li ($\sim 1.5\%$), for
which the two associated production cross sections are already updated in our
procedure. However, for some others, like the first ranking two-step reaction
16O$\to^{15}$N$\to^{7}$Li ($\sim 1.9\%$), only one cross section is updated.
It is not straightforward to quantify the benefit one would get by
renormalising also the relevant cross sections for the most important two-step
production channels. It should be a subdominant correction, but updating these
cross sections should nevertheless be part of the global strategy to improve
the CR flux predictions.

We finally comment on the rigidity dependence of the contributing fractions
seen in Fig. 8. We recall that the heavier the species, the larger the
inelastic cross section, implying a more damped flux at low rigidities (when
the interaction rate is larger than the diffusion rate). As a result, ratios
of heavier-to-lighter primary CRs, while asymptotically constant at high
energy (we assume the same injection spectrum for all species heavier than C
here), decrease with decreasing energy below a few tens of GeV/n; this
behaviour was highlighted in Fig. 14 of Putze et al. (2011) and is also seen
in the recent AMS-02 Fe/O data (see Fig. 3 of Aguilar et al. 2021b). Coming
back to Fig. 8, heavy elements represent a growing fraction of the total CR
species with growing energy, and this explains why the contributing fraction
of Fe in LiBeB (red dash-dotted curves) grows with energy121212Another subtler
energy dependence comes from the fact that $f_{Z_{i}\to Z_{j}}$ includes
direct and multi-step contributions (from $f_{i\to j}$ coefficients) in
variable amounts, with the multi-step production peaking at a few GeV/n
(Génolini et al. 2018)., with respect to C and O contributions.

##  5 Consequences of the updated cross sections: LiBeB/C re-analysis

To assess the impact of using the updated production cross-section sets, we
re-analyse the AMS-02 LiBeB/C data. We stress that throughout this section,
the cross sections used for the propagation runs are the cumulative ones.
These cumulative production cross-section sets are generated from the
individual reactions, following Eq. (3.2.2). We thus have three sets of
cumulative cross sections that we keep denoting OPT12, OPT12up22, and OPT22
for simplicity.

##### Methodology

We repeat below the analysis of Weinrich et al. (2020b), which is based on
AMS-02 Li/C, Be/C, and B/C data from 2011-2016 (Aguilar et al.
2018a)131313Data for 7 years were recently presented in Aguilar et al.
(2021a), but for the sake of comparison with our earlier analyses, we prefer
to stick to the same AMS-02 dataset as used in Weinrich et al. (2020b, a).. In
detail, we perform a $\chi^{2}$ analysis with the minuit minimiser (James &
Roos 1975), where $\chi^{2}$ values are obtained from the sum over (i) the
quadratic distance between the model and data, accounting for a covariance
matrix ${\cal C}$ of data uncertainties (Derome et al. 2019; Génolini et al.
2019; Weinrich et al. 2020b); (ii) a Gaussian-distributed nuisance parameter
${\cal N}_{\phi}$ for the Solar modulation Fisk potential $\phi$ (Gleeson &
Axford 1967, 1968)) with mean $\mu_{\phi}=676$ MV (appropriate for the LiBeB
data taking period) and $\sigma_{\phi}=100$ MV (Weinrich et al. 2020b); (iii)
a series of Gaussian-distributed nuisance parameters ${\cal N}^{r}_{\rm XS}$
associated to a selection of LiBeB production reactions (Derome et al. 2019;
De La Torre Luque et al. 2021b; Korsmeier & Cuoco 2021); the latter parameters
combine a shift in the normalisation and a tilt on the slope of the cross
sections, as introduced in Derome et al. (2019). Formally, this leads to

$\chi^{2}=(y_{\rm data}-y_{\rm model})\,{\cal C}^{-1}\,(y_{\rm data}-y_{\rm
model})+{\cal N}_{\phi}+\sum_{r=0}^{n_{r}}{\cal N}^{r}_{\rm XS},$ (7)

where $r$ runs over the $n_{r}$ cross-section reactions used as nuisance (see
below). It is also useful to isolate the specific contribution of the nuisance
parameters to the total $\chi^{2}$, that is

$\chi^{2}_{\rm nui}/{n_{\rm nui}}{}\equiv\left({\cal
N}_{\phi}+\sum_{r=0}^{n_{r}}{\cal N}^{r}_{\rm XS}\right)/(1+n_{r}),$ (8)

which tells us how far away from our initial guess the post-fit values are. We
recall that a good fit of the model to the data typically correspond to
$\chi^{2}_{\rm min}/{\rm dof}\sim 1$ and should have $\chi^{2}_{\rm
nui}/{n_{\rm nui}}{}\lesssim 1$ (i.e. nuisance parameters within $1\sigma$ of
their central values on average).

##### Post-fit values $\mu_{p}$ for cross-section nuisance parameters

The selection of a few most impacting cross sections to use as nuisance are
the same as in Weinrich et al. (2020b)141414Specific numbers for the
normalisation factors $\mu_{\rm XS}$ and their uncertainties $\sigma_{\rm XS}$
can be found in Table B.1 of Weinrich et al. (2020b). and they only involve
reactions on H targets. For the inelastic interactions the most impacting
reactions for LiBeB are 12C+H, 7Be+H, and 11B+H. The production reactions are
however the most impacting ones: for each element considered in the analysis,
we associate a single proxy reaction $p$, related to the production of an
isotope (of this element) from the most impacting progenitor. For Li, Be, and
B studied here, the associated proxies are $p\mathrm{(Li)=^{16}O+H\to^{6}Li}$,
$p\mathrm{(Be)=^{16}O+H\to^{7}Be}$, and $p\mathrm{(C)=^{12}C+H\to^{11}B}$.
This means that in order to obtain a good fit to LiBeB/C, the associated proxy
cross-section values $\sigma_{p}$ were renormalised into
$\mu_{p}\times\sigma_{p}$, where $\mu_{p}$ are the post-fit values.

We stress that using a single isotopic cross section as a proxy to account for
the uncertainties in an element production is mandatory to avoid degeneracies
that would appear, were several isotopic cross sections used for the same
element (Derome et al. 2019). However, an undesired side-effect of using a
single proxy is that it biases the prediction of isotopic fluxes and ratios.
As we want to use the post-fit values of these proxies to draw conclusions
about the need for primary source terms for LiBeB, and also to make
predictions for future isotopic ratios measured by AMS-02, these proxies must
be handled with care.

##### Contributing fractions $f_{r}$ for relevant reactions

We recall that a given reaction $r$ (from a CR progenitor into a CR fragment)
only contributes to a fraction $f_{r}$ of the overall element production (from
all possible CR progenitors). For the Li, Be, and B elements, these fractions
have been calculated in Génolini et al. (2018). We report below, for each
proxy $p$ considered, the $f_{r}$ associated to that proxy (in boldface) and
also to the other isotopes (for the element involved)151515We assume here that
these contributing fractions are energy-independent, though it is not exactly
the case., that is

$\displaystyle f_{r=\mathrm{{}^{16}O+H\to(^{6}Li,\,^{7}Li)}}$ $\displaystyle=$
$\displaystyle({\bf 15\%},\,12\%),$ $\displaystyle
f_{r=\mathrm{{}^{16}O+H\to(^{7}Be,\,^{9}Be,\,^{10}Be)}}$ $\displaystyle=$
$\displaystyle({\bf 19\%},\,6\%,\,1.3\%),$ (9) $\displaystyle
f_{r=\mathrm{{}^{12}C+H\to(^{10}B,\,^{11}B)}}$ $\displaystyle=$
$\displaystyle(10\%,\,{\bf 33\%}).$

##### From $\mu_{p}$ to global production correction $\mu_{Z}^{(p)}$

When considering a secondary species $Z$, we can calculate a global correction
factor to apply for the overall production of the element $Z$ (from any
progenitor), as opposed to the single correction $\mu_{p}$ applied on the
proxy $p$ (i.e. single progenitor into a single isotope). We would obtain
exactly the same secondary-to-primary elemental ratio (as with the single
proxy reaction correction) when applying the correction factor $\mu_{Z}^{(p)}$
to all individual production cross sections into isotopes of the element $Z$.
This global correction factor $\mu_{Z}^{(p)}$ is obtained from

$(\mu_{Z}^{(p)}-1)=(\mu_{p}-1)\times f_{p}\,.$ (10)

The benefit of this procedure is that the global factor $\mu_{Z}^{(p)}$ now
directly tells us by how much the overall production of an element must be
modified to match the data: if $\mu_{Z}^{(p)}>1$ (resp. $<1$), the overall
production must be larger (reps. smaller) that what we assumed in the
production cross-section set. Considering, for instance, B with its proxy
$p=\mathrm{{}^{12}C+H\to^{11}B}$, assuming $\mu_{p}=1.2$ and applying the
above equation would give $\mu_{Z}^{(p)}=(1.2-1)\times 33\%+1=1.067$. This
would mean that overall, the production reactions of $B$ are off and must be
increased by $6.7\%$ to match AMS-02 LiBeB/C data. The $\mu_{Z}^{(p)}$ values
are used for the discussion about evidences for LiBeB primary source terms in
Sect. 5.2 (see Fig. 12).

##### Unbiased isotopic production cross sections via $\mu_{\rm iso}$

Applying a correction on the proxy reaction only (e.g.
$\mu_{p=\mathrm{{}^{12}C+H\to^{11}B}}=1.2$ in the above example) changes the
predictions for CR isotopic ratios without real justification. To avoid this,
one possibility is to redistribute this modification democratically between
the isotopes (e.g. 10B and 11B here). In practice, we build new production
cross-section sets (from our updated sets), applying the correction factor
$\mu_{Z}^{{\rm iso}}$ for all reactions from the progenitor (used in the
proxy) into the isotopes of the element (to which the proxy is associated),
with

$(\mu_{Z}^{{\rm iso}}-1)\times\sum_{i\in Z}f_{i}=(\mu_{p}-1)\times f_{p}\,.$
(11)

For instance, the proxy $p=\mathrm{{}^{12}C+H\to^{11}B}$ involves the
progenitor 12C, so that using the above equation, we apply the correction
$\mu_{Z}^{{\rm iso}}=[(1.2-1)\times 33\%]/43\%+1=1.155$161616We calculated
$\sum_{i\in{\rm
B}}f_{i}=f_{\mathrm{{}^{12}C+H\to^{10}B}}+f_{\mathrm{{}^{12}C+H\to^{11}B}}=43\%$.
to both the cross sections $\sigma_{{}^{12}{\rm C+H}\to^{10}{\rm B}}$ and
$\sigma_{{}^{12}{\rm C+H}\to^{11}{\rm B}}$ (instead of using the single proxy
correction $\mu_{p}\times\sigma_{\mathrm{{}^{12}C+H\to^{11}B}}$). These new
sets are identified by the suffix ‘-U’ (for unbiased)—that is we build
OPT12-U, OPT12up22-U, and OPT22-U—and are used for isotopic ratio calculations
in Sect. 5.3 (and Fig. 13); and we explicitly checked that these sets leave
unchanged the predictions for the elemental fluxes (as expected).

Now that we are done with the definitions, we can move on to the
interpretation of our LiBeB/C re-analysis results: first, we discuss the
impact of the new cross-section sets OPT12, OPT12up22, and OPT22 on the
transport parameters (Sect. 5.1); second, we address the need for a primary
lithium source (Sect. 5.2); finally, we study the impact of the updated cross-
section sets on CR isotopic ratios (Sect. 5.3).

###  5.1 Impact on transport parameters

In this subsection, we show the constraints set on the SLIM parameters at
$L=5$ kpc (see App. C for the constraints on BIG and QUAINT). The
minimisations involve 7 nuisance parameters (one for Solar modulation and 6
for cross sections) and 201 LiBeB/C data points. We show in Fig. 9 the best-
fit transport parameters of SLIM, that is, from top to bottom, the
normalisation and slope of the diffusion coefficient ($K_{0}$ and $\delta$),
and the low-rigidity break parameters ($\delta_{l}$ and $R_{l}$, see Eq. 2)
for different cross-section sets; the next-to last and last panels show
$\chi^{2}_{\rm min}/{\rm dof}$ and $\chi^{2}_{\rm nui}/{n_{\rm nui}}{}$
respectively, see Eqs. (7) and (8). We show on the right-hand side of the plot
the values obtained in this analysis with the three updated cross-section sets
OPT12, OPT12up22, and OPT22. For comparison purpose, we also report, on the
left-hand side of the plot, the parameters obtained in Weinrich et al. (2020b)
from the same LiBeB analysis but with the original set Galp--opt12 (violet
stars), and those obtained even earlier from the same cross-section set in
Génolini et al. (2019) where only a fit to AMS-02 B/C data was
performed171717The original analysis was based on $L=10$ kpc and the values
for $L=5$ kpc are obtained using the scaling relations of Weinrich et al.
(2020a). (cyan crosses). The latter result is shown to recall that, with
respect to a combined analysis of several secondary-to-primary species, the
analysis of a single ratio is both less constraining (larger error bars) and
possibly biased by systematics in the production cross sections (Derome et al.
2019; Weinrich et al. 2020b).

Figure 9: Best-fit transport parameters (and uncertainties) from the fit to
AMS-02 LiBeB/C data using different cross-section sets: from left to right,
the original Galp--opt12 set used in Génolini et al. (2019) and Weinrich et
al. (2020b), and the three updated sets obtained in this paper (OPT12,
OPT12up22, and OPT22). The next-to-last and last panels show the
$\chi^{2}_{\rm min}$/dof and $\chi^{2}_{\rm nui}/{n_{\rm nui}}$ values for the
associated runs—see Eqs (7) and (8) respectively.

The striking features seen in Fig. 9 for the combined LiBeB/C fit are the
following. Firstly, when going from OPT12 to OPT22, the normalisation $K_{0}$
(second panel) grows, whereas the low-rigidity break $\delta_{l}$ becomes
consistent with $0$ (i.e. no break). The change in $K_{0}$ is directly linked
to the updated cross sections, as more production means less grammage to
produce the same amount of secondary species, which in turns means a larger
diffusion coefficient (at fixed halo size $L$). The overall impact on the
diffusion coefficient $K(R)$ is also illustrated in Fig. 10 where $1\sigma$
envelopes are shown.

Figure 10: Best-fit and 1$\sigma$ envelopes for the diffusion coefficient, Eq.
(2), associated to the parameters shown in Fig. 9 and for the same cross-
section sets ($L=5$ kpc). For readability, we do not show the envelopes for
OPT12 and OPT22 as they are similar to those for OPT12up22.

Secondly, not only do the updated cross-section sets improve the global fit
substantially, but they also reduce the deviations of the nuisance parameters
from their priors. In particular, the best $\chi^{2}/$dof is obtained for
OPT12up22181818We recall that the fact that $\chi^{2}/$dof is significantly
larger than 1 is caused by the two low-rigidity Be/B points (upturn) that
cannot be well-fitted by the models, see Weinrich et al. (2020b). (black
circle), with a very significant improvement ($\chi^{2}_{\rm min}$ goes from
$258.3$ to $237.5$) compared to the use of the original Galp--opt12 set in
Weinrich et al. (2020b) analysis (violet star). This OPT12up22 set also gives
the smallest $\chi^{2}_{\rm nui}/{n_{\rm nui}}$ value, making it the most
favoured by the LiBeB/C data.

Figure 11: Flux ratios (top), residuals (centre) and $\tilde{z}$-scores
(bottom) for B/C (blue circles), Be/C (green triangles), and Li/C (orange
squares). The models (top panel) are calculated from the best-fit transport
parameters (combined LiBeB/C analysis) using the original Galp--opt12 set
(thick violet dotted line) or our updated sets: OPT12 (dashed grey line),
OPT12up22 (solid black line), and OPT22 (dashed-dotted grey line). The
residuals and $\tilde{z}$-score are shown for SLIM only—the $\tilde{z}$-score
is related to the usual $z$-score by a rotation in a base where the covariance
matrix of data systematics is diagonal (see text for details).

For illustration purpose, the best-fit models for Li/C, Be/B, and B/C ratios
are reported (SLIM configuration for the various cross-section sets) and
compared to the data in the top panel of Fig. 11. For the specific OPT12up22
cross-section set, we further show in the middle and bottom panels
respectively, the residuals with regard to the total uncertainties
(systematics and statistic combined) and the rotated score $\tilde{z}$; the
latter quantity accounts for the role of correlations in the data
systematics191919As introduced in Boudaud et al. (2020), a graphical
representation of the ‘rotated’ score (denoted $\tilde{z}$-score) is better
suited to provide an unbiased graphical view of the difference between the
model and data, when correlations exists between the data bins. The rotated
base is defined so that the covariance matrix of uncertainties is diagonal,
$\tilde{\cal C}=U{\cal C}U^{\rm T}$, with $U$ an orthogonal rotation matrix.
In the rotated base, rotated rigidities are
$\tilde{R}_{i}=\sum_{j}U_{ij}^{2}\,R_{j}$ (that remain close to the original
$R_{i}$ values for AMS-02 data), and the rotated residuals become
$\tilde{z}_{i}=\tilde{x}_{i}/\sqrt{{\cal C}_{ii}}$ with
$\tilde{x}_{i}\equiv\sum_{j}U_{ij}({\rm model}_{j}-{\rm data}_{j})$. By
construction, $\chi^{2}=\sum_{i}\tilde{z}_{i}^{2}$ (distance between the model
and data, forgetting about the nuisance parameters), and we further stress
that for a $\chi^{2}/n_{\rm data}$ close to 1, the histogram of
$\tilde{z}_{i}$ values should follow a centred Gaussian of width unity (see
right-hand side of the bottom panel in Fig. 11).. Overall, the above plot is
very similar to the corresponding one shown in Fig. 2 of Weinrich et al.
(2020b), that is the model matches the data quite well. Moreover, very similar
$\chi^{2}/n_{\rm data}$ values (as reported in the legend of the second panel)
and goodness-of-fit are obtained for the BIG and QUAINT configurations (not
shown).

###  5.2 Is a primary source of Li needed

Figure 12: Correlation between log${}_{10}(K_{0})$ and the normalisation
factor $\mu_{Z}^{(p)}$ (see Eq. 10), that is the correction factor applied (on
the total production cross section of element $Z$) in order to get the best-
fit on LiBeB/C data. The three elements considered are colour-coded: $Z=3$
(Li) in orange, $Z=4$ (Be) in green, and $Z=5$ (B) in blue. The $1\sigma$
correlation ellipses are shown for analyses with different cross section sets
(in model SLIM): from left to right, stars are for the original OPT12,
downward triangles for the update OPT12, circles for OPT12up22, and upward
triangles for OPT22. The horizontal grey dashed line highlights $\mu_{Z}=1$
(i.e. no modification needed for the production cross sections).

We recall that in Boschini et al. (2020b), using the propagation code galprop
and no nuisance parameters on the cross sections, the authors proposed the
presence of a primary Li contribution to correct for a 20-25% deficit between
the model and the data. Conversely, using nuisance parameters and relying on
the same cross-section set Galp--opt12, Weinrich et al. (2020b) found no
mismatch between the model and the data, but rather the need for a 13%
increase of the production cross section of Li (see also De La Torre Luque et
al. 2021a, b for similar conclusions, though with slightly different numbers).
The latter conclusion was reached by inspecting $\mu_{Z}^{(p)}$ values
calculated from Eq. (10), reproduced here with star symbols (and their
associated ellipses) in the left-hand side of Fig. 12: in Weinrich et al.
(2020b)’s analysis, to match the LiBeB/C data with the Galp--opt12 cross-
section set, the overall production of Li had to be increased by $13\%$, and
that for B (resp. Be) had to be decreased by 3% (resp. 6%).

This picture changes when considering our updated cross-section sets: moving
from the original Galp--opt12 to the updated sets (from left to right in the
plot), the total Li production goes from a significant increase ($+13\%$) to a
mild decrease ($-5\%$), whereas the total B production evolves in the opposite
direction (from $-3\%$ to $+10\%$); the overall production of Be remains
stable at $\sim-5\%$. This shows that there is no need for a primary source of
Li, and that we cannot trust the fact that, taken at face value, we would need
now a primary source of B. Our analysis merely reinforces the conclusions of
Weinrich et al. (2020b) and De La Torre Luque et al. (2021a): AMS-02 LiBeB/C
data can be reproduced without invoking the presence of a new primary source
because mismatches are likely to be associated to cross-section uncertainties.
We also note in passing that OPT12up22, which was our (slightly) favoured
configuration from the nuclear data perspective (see Sect. 3.2), is the one
that best-fit AMS-02 LiBeB/C data (see Fig. 9). It is also the one for which
$\mu_{Z}^{(p)}$ values are the closest to 1 (within $\pm 5\%$, see empty
circles in Fig. 12), that is minimal modifications needed for the production
cross sections.

Figure 12 and the discussion are based on the SLIM configuration, but we
checked that the BIG and QUAINT configurations lead to the same
decrease/increase of the Li, Be, and B production cross sections (and their
uncertainties), that is same values as those shown in Fig. 12 along the $y$
axis. Our conclusion against the evidence of a Li primary source in CRs is
thus independent of the propagation configuration and robust.

###  5.3 Impact on isotopic ratios

It is also interesting to see whether the updated cross-section sets predict
different isotopic contents for the LiBeB elements. The HELIX project (Park et
al. 2019) and the AMS-02 (Aguilar et al. 2019) experiment both have the
capability to separate light isotopes from a few hundreds of MeV/n up to $\sim
10$ GeV/n , whereas past experiments only provide data on a much limited
range; the most recent data on $Z=3-5$ isotopic fluxes are from ACE (de Nolfo
et al. 2006), AMS-01 (Aguilar et al. 2011), ISOMAX (Hams et al. 2004), and
PAMELA (Menn et al. 2018; Nozzoli & Cernetti 2021).

Figure 13: Isotopic ratios of Li, Be, and B (from top to bottom panel)
calculated from the best-fit models to AMS-02 LiBeB/C (at fixed $L=5$ kpc).
The production cross-section sets considered are the original Galp--opt12
(dotted lines) and the updated sets OPT12 (dashed lines), OPT12up22 (solid
lines), and OPT22 (dash-dotted lines); the suffix ‘-U’ in the labels recalls
that we use here unbiased cross-section sets, see Eq. (11). The data are from
IMP (Garcia-Munoz et al. 1977b, a, 1981), Balloon ’73/08 (Hagen et al. 1977),
Balloon ’77/05 (Buffington et al. 1978), Balloon ’77/09 (Webber & Kish 1979),
Balloon ’70/06 (Bjarle et al. 1979), ISEE3 (Wiedenbeck & Greiner 1980; Mewaldt
et al. 1981; Krombel & Wiedenbeck 1988), HEIST (Gibner et al. 1992), Ulysses
(Connell 1998), Voyager1&2 (Lukasiak 1999), SMILI-II (Ahlen et al. 2000),
ISOMAX (Hams et al. 2004), ACE (Yanasak et al. 2001; de Nolfo et al. 2006),
AMS-01 (Aguilar et al. 2011), PAMELA (Menn et al. 2018; Nozzoli & Cernetti
2021).

We show in Fig. 13 various isotopic ratios of Li, Be, and B, calculated from
the best-fit models (parameters shown in Sect. 5.1) for various ‘unbiased’
production cross-section sets. We see that the updated sets predict very
similar isotopic ratios for all species, except for Li isotopes, for which the
largest difference is seen for 7Li/6Li; this is not so surprising since all
sets were renormalised to the same nuclear data (though the underlying
original cross-section models have different energy dependences, and possibly
different prediction for unmeasured reactions). The largest difference for
7Li/6Li is at high energy and disappears at low energy. It is likely to be
associated with Fe fragmentation: as seen in Fig. 8, the contribution from Fe
to the total production vanishes at low energy, and as seen on the top left
panel of Fig. 3, this is where the updated cross-section values differ most
for both 6Li and 7 Li. The difference between Galp--opt12 (dashed lines) and
the updated cross-section sets (all other lines) is also maximal for Li
isotopes: the origin of the difference can be read off Table 1, where one can
see that the production cross sections from 16O and 12C have been increased
for 6Li and decreased for 7Li. The differences seen for the other isotopes are
smaller, but also result from the updated cross sections listed in Table 1.
This behaviour is qualitatively (but not quantitatively) the same if we
consider the BIG and QUAINT transport configurations instead of SLIM.

We finally want to stress that the suffix ‘-U’ in the labels of Fig. 13)
indicate that our predictions were made with the ‘unbiased’ cross-section
sets. In these sets, the corrections applied to cross-sections for a given
element are distributed equally between the various isotopes of this
element—parameter $\mu_{\rm iso}$ in Eq. (11). We recognise that this
procedure is debatable. Indeed, once the fitting procedure to secondary
elements decides that the overall production of the said element must be
slightly increased (or decreased), there is no real clear path to choose how
to distribute this modification over the various isotopes (i.e. modification
of a specific isotope or equally distributed between isotopes). Using more
proxy reactions (e.g. one proxy per isotope) must be backed-up by isotopic
ratio data in the fit, otherwise the various proxy parameters are degenerated;
in that latter case, post-fit value would have no predictive power and would
lead to overestimated variations in the predicted isotopic ratios. As
illustrated in Fig. 13, current data (symbols) are clearly too scattered and
imprecise to get a stronger handle on the isotopic production cross sections.
Moreover, at this stage, it is difficult to quantify whether our updated (and
‘unbiased’) cross-section sets provide an improvement over the use of the
original Galp--opt12-U set. The forthcoming AMS-02 and HELIX data are awaited
to make some progress on these issues.

##  6 Conclusions

We have revisited and updated the production cross sections of Li, Be, and B
species. We showed that contributions from Fe favour an enhanced production of
Li—and to a lesser extent Be—with respect to B. We emphasised that most (and
possibly all) previous interpretations of the AMS-02 Li, Be, and B data rely
on a cross-section set that does not account for these enhancements. The
consequences of using updated cross sections are the following202020These
cross-section sets will appear at some point in the next release of the usine
code. They can be obtained on demand in the meantime.: (i) in the ranking of
the most important progenitors for LiBeB, Fe becomes the third most important
element, responsible for $\sim 10\%$ of Li and Be212121In practice, this
slightly modifies the priority order (for new measurement campaigns in
beam+target experiments) discussed in Fig. 5 of Génolini et al. (2018)., but
only $\sim 4\%$ for B; (ii) the combined analysis of the AMS-02 Li/C, Be/C,
and B/C ratios lead to a 20% larger value of the diffusion coefficient
(compared to those obtained in our previous investigation), which is not
without consequences for the predictions made, for instance, on secondary
antiprotons (Boudaud et al. 2020)222222It means that this is another worrying
systematics to consider when trying to uncover or set limits on dark matter in
this channel (e.g. Calore et al. 2022).; (iii) the updated cross-section sets
also impact the predictions for the isotopic ratios of Li, Be, and B—in
particular, uncertainties on the production of Be isotopes may be an issue for
the determination of the halo size of the Galaxy (see the companion paper,
Maurin et al. 2022); (iv) via the inspection of the post-fit values of the
nuisance parameters (on the production cross section), we find that the
updated cross-section sets, expected to be closer to the truth, indeed provide
a more consistent description of the Li/C, Be/C, and B/C data—this is
reassuring and validates to some extent our approach.

We recall that the motivation for this re-analysis was the interpretation of a
mismatch between models and AMS-02 Li data, interpreted by some authors as a
possible hint for a primary source of Li. We unambiguously showed here that
the need for an extra Li component completely disappears, comforting the
studies arguing for systematics in the production cross sections instead; at
variance with these studies, however, we showed that these systematics were
mostly attributed to an underestimated production of Li from Fe. Actually, the
different plausible cross-section sets we built either lead to the need for a
$5-10\%$ increase or decrease of Li and B production cross sections; our
favoured cross-section set (that also happens to provide the best-fit to Li/C,
Be/B, and B/C data) even leads to a very small excess of Li with respect to
the data. For this reason, we caution against any hasty interpretation of
these overshoots or undershoots: they merely reflect the presence of
systematics or still missing important reactions in the production cross-
section sets.

The next step of this work would be to generalise our normalisation procedure
(on nuclear data) to sub-dominant production channels. It could also be useful
to make systematic comparisons to state-of-the-art cross-section codes like
SPACS (Schmitt et al. 2014), FRACS (Mei 2017), or others, in order to better
quantify systematics related to unmeasured cross sections. The rescaling
procedure should then be extended to the production of secondary species with
$Z>5$, as all the elements up to Fe are within reach of the AMS-02 experiment.
On the nuclear data side, new measurements for the production of Li and B
isotopes from C and O, but also from Ne, Mg, Si, and Fe progenitors would be
of great value: these progenitors amount to $\sim 20\%$ of the total
production of Li and B, while many of these reactions remain unmeasured or
with only a couple of data points.

###### Acknowledgements.

D.M. thanks his AMS-02 colleagues for interesting discussions that lead to the
implementation of a ranking per progenitor shown in Fig. 8. We thank the
Center for Information Technology of the University of Groningen for their
support and for providing access to the Peregrine high-performance computing
cluster. This work was supported by the Programme National des Hautes Energies
of CNRS/INSU with INP and IN2P3, co-funded by CEA and CNES. Y.G. acknowledges
support from Villum Fonden under project no. 18994.

##  Appendix A Impact of updated reactions

We showed in the main text the global impact of the updated cross sections on
the calculated Li/C, Be/C, and B/C ratios (Fig. 6). We show here a more
detailed view, identifying the progenitors whose updated cross sections make
up most of the observed changes in these ratios. However, we do not comment on
the energy dependence seen in the figures, as they are already discussed in
the main text (see Sect. 3.3).

Figure 14: Global impact (thick black lines) of updated cross-sections in Li/C
(top panels), Be/C (centre panels), and B/C (bottom panels) against broken-
down impact per progenitor (colour-coded), as a function of rigidity. Rows
show only progenitors whose relative impact is larger than 3% (resp. $2\%$ and
$1\%)$ for Li/C (resp. Be/C and B/C); all calculations are at a fiducial Solar
modulation level of 700 MV. The columns show different ways to update the
cross-section sets: updating the original Galp--opt12 into OPT12 (left
panels); updating the original Galp--opt22 into OPT22 (right panels), or the
hybrid approach OPT12up22 (centre panels). See text for details.

In the left panels of Fig. 14, the reference calculation relies on the
original Galp--opt12 cross-section set, whereas the new calculation relies on
the associated updated set OPT12. The maximal impact from updating the cross
sections (thick solid lines) typically reaches $+20\%$ on Li/C (top), $+10\%$
on Be/C (centre), and $-5\%$ on B/C (bottom). In all three ratios, the
culprits for these changes are the fragmentation of 56Fe (green double-dashed
line) but also of 16O. This is in line with the values observed for the
rescaling factors ${\cal R}$ shown in Table 1, where ${\cal R}$ varies the
most for the 56Fe and 16O progenitors.

In the right panels, the reference calculation is now the original Galp--opt22
set, whereas the new calculation is based on the associated updated set OPT22.
In that case, updating 56Fe reactions has a more limited impact on the flux
ratios. This is consistent with the observation that the initial cross-section
set Galp--opt22 was slightly overshooting the nuclear data: in Table 1, the
rescaling for 56Fe (second number in the entry associated to OPT22) is, on
average, close to 1 (i.e. the model prediction was already close to the data).
In addition, with Fe fragmentation ranking at the $\lesssim 10\%$ level (see
Fig. 8), the combination of these last two numbers falls in the percent range
at most. In this context, the most significant impact in LiBeB/C is from
updating 16O fragmentation reactions.

Finally, the middle panels shows the hybrid case, where the initial
parametrisation is Galp--opt12 (for all measured and unmeasured cross
sections), but where we substitute for measured cross section an update of the
original Galp--opt22 (so called OPT12up22 set); we recall that this was
motivated by the more realistic energy dependence provided in the Galp--opt22.
This configuration is expected to be the most realistic, and we see that the
ratios are maximally impacted, with Li/C maximally enhanced (up to $\sim
40\%$). Moreover, these changes are now dominated by the updated Fe
fragmentation.

##  Appendix B Impact of recent F, Ne, Na, Mg, Al, Si, and Fe AMS-02 data

In a propagation run, while we do not fit the primary species, we rescale
primary source abundances on a specific high-energy point. As illustrated in
Génolini et al. (2019), this procedure is sufficient to match primary CR
progenitors of LiBeB (provided we fix correctly the universal source slope).
Moreover, the secondary-to-primary ratios are insensitive to this source slope
(Maurin et al. 2002; Génolini et al. 2015).

Figure 15: Impact of using the recently published AMS-02 data in the LiBeB/C
calculation. The calculation relies on the use of C, N, O, F, Ne, Na, Mg, Al,
Si, and Fe data from AMS-02, and HEAO-3 data for all other elements, whereas
our previous analysis (Weinrich et al. 2020b), relied on C, N, O AMS-02 data
(only available LiBeB progenitors from this experiment at the time) and HEAO-3
data for all other progenitors.

In practice, to ensure the most accurate calculation, it is best to use the
most-recent data available to normalise these source abundances. We use AMS-02
elemental data at 50 GV when available and HEAO3 data (Engelmann et al. 1990)
at 10.6 GeV/n otherwise.

In our previous LiBeB/C analysis (Weinrich et al. 2020b), only AMS-02 data for
the progenitors C, N, and O were available (Aguilar et al. 2017, 2018b). For
this publication, we can rely on AMS-02 data for more progenitors, namely F,
Ne, Na, Mg, Al, Si, and Fe (Aguilar et al. 2021, 2020, 2021b), but also C, N,
and O fluxes for 7 years of data (Aguilar et al. 2021a). Figure 15 illustrates
by how much the Li/C (thick yellow dashed line), Be/C (green dashed line), and
B/C (thin blue dashed line) ratios are impacted when these new data are taken
into account in the calculation. We see that Li/C and Be/C are the most
impacted ones, but only at a few percent level. This is understood as follows:
first, N (resp. Na and Ne) contributes to the percent level (resp. negligibly)
to LiBeB production (Génolini et al. 2018), so that a small discrepancy
between AMS-02 and HEAO3 data would have no impact on the calculated LiBeB;
second, Fe is responsible for up to tens of percent of Li, a few percent on
Be, and almost no impact on B. This ordering is consistent with the one seen
in Fig. 15, and with the fact that AMS-02 data for Fe are slightly above HEAO3
ones (Aguilar et al. 2021b).

CALET has recently published new data for the C, O and Fe fluxes (Adriani et
al. 2020, 2021), which were found to be respectively $\sim 30\%$ and $\sim
27\%$ lower than the AMS-02 ones. This difference for C and O would certainly
impact the production of lighter elements, but taking ratios of elements, as
we do here, cancels out these yet to be explained differences.

##  Appendix C Further checks on transport parameters

In the main text, we only discuss results for the SLIM propagation
configuration (see Sect. 5.1). For completeness, we show in Fig. 16 the
constraints set on the transport parameters in the two other configurations
QUAINT and BIG (Génolini et al. 2019); these configurations were considered in
several of our previous studies (Génolini et al. 2019; Boudaud et al. 2020;
Weinrich et al. 2020a, b), in particular in the context of dark matter
analyses (Génolini et al. 2021; Calore et al. 2022). The respective free
parameters for the three configurations are: (i) for SLIM, $K_{0}$, $\delta$,
$R_{l}$, and $\delta_{l}$, that is diffusion normalisation and slope with a
low-rigidity break; (ii) for QUAINT, $K_{0}$, $\delta$, $\eta$, $V_{a}$,
$V_{c}$, that is diffusion without a low-rigidity break but a possible upturn
in the non-relativistic regime ($\beta$ replaced by $\beta^{\eta}$ in Eq. 2),
re-acceleration, and convection; (iii) for BIG, $K_{0}$, $\delta$, $R_{l}$,
and $\delta_{l}$, $V_{a}$, and $V_{v}$, that is diffusion with a low-rigidity
break, reacceleration, and convection.

Figure 16: Same as Fig. 9, but also showing the parameter constraints for the
QUAINT and BIG propagation configurations.

As in SLIM, BIG and QUAINT show a better $\chi^{2}_{\rm min}$/dof and
$\chi^{2}_{\rm nui}/{n_{\rm nui}}$ for all the updated cross-section sets,
with OPT12up22 providing the best-fit for all transport configurations. Also,
moving from OPT12 to OPT12up22, and OPT22, the significance of the low-
rigidity break decreases ($\delta_{l}$ consistent with zero for the latter
set). However, at variance with SLIM, the parameter $K_{0}$ is no longer
sensitive to the production cross-section set considered, that is whether we
use the original Galp--opt12 set (violet stars) or the updated ones (grey and
black symbols). Actually, although $V_{a}$ and $V_{c}$ are consistent for all
the different cross-section sets, their best-fit values and $1\sigma$ upper
limits are larger for the updated sets: convection and reacceleration dominate
over diffusion below tens of GV (e.g. Derome et al. 2019; Vecchi et al. 2022),
and the fit prefers to modify these parameters instead of adjusting $K_{0}$.

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