#  FLUKA cross sections for cosmic-ray interactions with the DRAGON2 code

P. De La Torre Luque  M. N. Mazziotta  A. Ferrari  F. Loparco  P. R. Sala  D.
Serini

###### Abstract

Secondary particles produced in spallation reactions of cosmic rays with the
interstellar gas provide valuable information that allow us to investigate the
injection and transport of charged particles in the Galaxy. A good
understanding of the cross sections of production of these particles is
crucial to correctly interpret our models, although the existing experimental
data is very scarce and uncertain. We have developed a new set of cross
sections, both inelastic and inclusive, computed with the FLUKA Monte Carlo
nuclear code and tested its compatibility with CR data. Inelastic and
inclusive cross sections have been compared to the most up-to-date data and
parameterisations finding a general good agreement.

Then, these cross sections have been implemented in the DRAGON2 code to
characterize the spectra of CR nuclei up to $Z=26$ and the secondary-to-
primary ratios of B, Be and Li. Interestingly, we find that the FLUKA cross
sections allow us to predict an energy-dependence of the B, Be and Li flux
ratios which is compatible with AMS-02 data and to reproduce simultaneously
these flux ratios with a scaling lower than $20\%$. Finally, we implement the
cross sections of production of gamma rays, calculated with FLUKA, in the
Gammasky code and compute diffuse gamma-ray sky maps and the local HI
emissivity spectrum, finding a very good agreement with Fermi Large Area
Telescope data.

##  1 Introduction

Galactic cosmic rays (CRs) are known to play an important role in the Galaxy,
regulating its evolution, shape, mass-metallicity relation, magnetic fields
and other processes [1, 2, 3, 4]. Understanding their propagation may help
reach a better interpretation of the gamma-ray emissions throughout the Milky
Way, the characterisation of environmental plasmas or even in indirect
searches of dark matter.

The widely accepted standard scenario is that collision-less scatterings of
CRs with magneto-hydrodynamic (MHD) plasma waves cause them to follow an
erratic motion that can be approximated as a diffusive transport [5, 6, 7].
These interactions extend several kiloparsecs above and below the galactic
disk [8, 9, 10, 11, 12, 13], constituting the so-called magnetised halo [14].
During their journey, CRs repeatedly cross the galactic disk undergoing
hadronic interactions (spallation reactions) with the gas in the interstellar
medium (ISM) and produce secondary particles that can be secondary CRs, gamma
rays or neutrinos [15]. These secondary particles are crucial since they bring
valuable information of the CR transport and interactions. In particular, the
flux ratios of secondary-to-primary CRs (considering as primary CRs those
mainly accelerated in astrophysical sources) are commonly used to constrain
the propagation parameters defining their transport [16, 17].

Recently, many works have claimed the need of improving cross sections
measurements on spallation reactions in order to reduce the uncertainty
related to the determination of the propagation parameters from the secondary-
to-primary flux ratios [8, 18, 19, 20, 21, 22, 23]. Due to the lack of
experimental data in most of the channels and its uncertainty, there have been
several attempts meant to develop the cross sections network purely from
physical principles, usually relying on Monte Carlo simulations and event
generators. However, as they are mainly adjusted to reproduce accelerator
data, they have demonstrated to be less accurate in the region between $100$
MeV to $\sim$ hundreds GeV than the current dedicated parameterisations [24,
25, 26, 22]. However, these simulation codes (as, for example, GEANT4 [27],
PHITS [28, 29], SHIELD-HIT [30]) have experienced a positive boost in the last
years, usually driven by radiological and medical applications, which need to
accurately describe the transport of ions in different materials and their
interactions.

This paper aims at computing the full cross sections network (including
inelastic and inclusive cross sections) needed to be implemented in CR
propagation codes using one of the most recently updated Monte Carlo nuclear
codes FLUKA [31, 32, 33] and demonstrating its compatibility with data. FLUKA
is a general purpose tool that can be used to transport particles in
arbitrarily complex geometries, including magnetic fields, through the FLUKA
combinatorial geometry. Its application in astroparticle physics studies is
one of the main goals of FLUKA, as it is demonstrated by a variety of recent
works [34, 35, 36, 37, 38], raising high expectations in the CR community.
This work updates the cross sections calculated by Ref. [39], extend its
calculations to isotopes up to Z=26 and explore in detail the predicted cross
sections for B, Be and Li production.

These cross sections are implemented in the DRAGON2 code111 Available at
https://github.com/cosmicrays/DRAGON2-Beta_version [40, 41] in order to
demonstrate that they can be used to reproduce the new and accurate CR data at
a similar level as common analytical and semi-analytical cross sections
parameterisations such as the DRAGON2 [8, 42] and GALPROP [43, 44] cross
sections, focusing on the study of the secondary CRs B, Be and Li. We employ a
Markov chain Monte Carlo (MCMC) analysis, already used in earlier works [45,
46], to determine the propagation parameters from the most recent data from
the AMS-02 collaboration [47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57].
Furthermore, we derive the cross sections for production of leptons (electrons
and positrons) and gamma rays from CR collisions with the interstellar medium
and implement them in the Gammasky code [58, 59, 60] in order to compute
gamma-ray sky maps and the local HI emissivity spectrum (see Refs. [61, 62]
for a discussion on these measurements and predictions). The FLUKA cross
sections for secondary lepton production will be discussed in a separated
paper, more dedicated to the positron spectrum. Also the production of
isotopes around the so-called iron group will be discussed in a separated
work.

This paper is organized as follows: in section 2 we discuss the main aspects
of the cross sections computations performed and show them in comparison with
relevant parameterisations and data. Then, in section 3, we demonstrate that
the FLUKA cross sections can be successfully used in a CR propagation code
like the DRAGON2 code and report the results of the MCMC analysis for the flux
ratios of B, Be and Li over C and O. Section 4 is dedicated to show the
computations of the galactic diffuse gamma-ray emission with the FLUKA cross
sections. Finally, the main highlights are discussed in section 5.

##  2 FLUKA cross sections for CR spallation interactions

One of the main advantages of using nuclear codes for the computation of cross
sections of CR interactions is that it allows us to calculate them in a broad
range of energies and species. The possible energy range covered for hadron-
hadron and hadron-nucleus interactions in FLUKA extends from a threshold
($\sim 1\leavevmode\nobreak\ \hbox{$\,{\rm MeV}$}$) up to
$10^{4}\leavevmode\nobreak\ \hbox{$\,{\rm TeV}$}$, while electromagnetic and
muon interactions can be treated from $1\leavevmode\nobreak\ \hbox{$\,{\rm
keV}$}$ ($100\leavevmode\nobreak\ \hbox{$\,{\rm eV}$}$ for photons) up to
$10^{4}\leavevmode\nobreak\ \hbox{$\,{\rm TeV}$}$. Nucleus-nucleus
interactions are also supported from $1\leavevmode\nobreak\ \hbox{$\,{\rm
MeV/n}$}$ up to $10^{4}\leavevmode\nobreak\ \hbox{$\,{\rm TeV/n}$}$. We refer
the reader to Ref. [39] for a brief description on the interaction models that
FLUKA employs for these interactions in different energy ranges. Moreover,
full information on the different models used by the code and its related
publications and references can be found in the FLUKA webpage
222http://www.fluka.org/fluka.php?id=publications&mm2=3.

a) b)

c) d)

e) f)

Figure 1: Comparison of the CROSEC and FLUKA inelastic cross sections on
hydrogen target for diverse species of projectiles relevant for CR physics: a)
protons; b) helium; c) boron; d) carbon; e) other heavier primary CRs (Ne, Mg
and Si); f) other heavier secondary CRs (${}^{19}F$, ${}^{23}Na$, ${}^{26}Al$,
${}^{27}Al$). Experimental data are also shown when they are available.
Proton-induced reactions data are taken from [63] and are assembled in tables
in http://www.oecd-nea.org/dbdata/bara.html. Data for heavier nuclei come from
a compilation using the EXFOR database [64, 65]. Full references can be found
in [22].

Here, we compute the inelastic and inclusive cross sections of all stable
isotopes from protons to iron impinging in helium and hydrogen as targets,
following the strategy of Ref. [39], from $1\leavevmode\nobreak\ \hbox{$\,{\rm
MeV/n}$}$ to $35\leavevmode\nobreak\ \hbox{$\,{\rm TeV/n}$}$, using 176 bins
equally spaced in a logarithmic scale. Ghost nuclei (i.e. those unstable
nuclei, generated via spallation reactions, which have a negligible lifetime
compared to typical CR propagation times, and whose contribution is directly
added up to the cross sections of formation of their daughter nuclei) are
carefully taken into account by making them decay before calculating the yield
of secondary particles produced after the simulated interaction. The results
can be summarized in a set of spallation and inelastic cross section tables,
with the spallation ones calculated for both the direct and cumulative
channels, which will be soon publicly available.

Some of the most important inelastic cross sections, for hydrogen as target,
computed with FLUKA are shown in Figure 1. It shows the proton, helium (as
examples of light primary CR), carbon (as an example of heavy primary CR),
boron (as an example of secondary CR) and other heavy elements (as a sample of
other CR nuclei with very few measurements available) inelastic cross sections
calculated with FLUKA and with the CROSEC code333This is the default option
for inelastic cross sections in the DRAGON2 code. (also called CRN6) [66] as a
function of energy per nucleon compared to experimental data. For protons,
FLUKA adopts the parameterisation found in [67]. Notice that in the case of
boron (panel c), the experimental data correspond to the isotopic composition
(with $A\sim 10.8$). Comparisons for the primary CRs 20Ne, 24Mg and 28Si are
shown to emphasize the differences found between both computations in channels
where the amount of experimental data is very scarce. In fact, as in the case
of the spallation inclusive cross sections, parameterisations for inelastic
(total) cross sections with few or no experimental data (as in Figure 1, f)
are just extrapolations of the better measured channels, which may
significantly bias the final results. Channels above silicon turn out to be
not very important in CR studies of light secondary CRs: they have an impact
on the production of light secondary CRs $<4\%$ for Be and Li, and $<1\%$ for
B (at $\sim$ 10 GeV). In the case of heavier species (Figure 1, e, f),
discrepancies around $10$-$20\%$ are found between the two computations at
atomic numbers above that of Ne ($Z\geq 10$). For all these heavy nuclei, the
energy dependence of their cross sections is very similar. The lack of data in
these channels makes difficult to decide which computation is more suitable,
but, in general, it seems that the inelastic cross sections calculated with
FLUKA show a good agreement to one of the most updated semi-analytical
parameterisations used in the CR community, as the CROSEC ones.

Besides, the inclusion of a rise cross sections at high energies is evident in
the FLUKA cross sections, while this is not incorporated neither in the
inelastic nor in the inclusive cross sections parameterisations, although it
has been already experimentally observed [68, 69, 70]. This fact can have
important implications on the interpretation of the break at high energies in
CR spectra, as we comment in Appendix A.

In turn, the case of inclusive cross sections is more delicate, due to the
amount of isotopes involved in the CR network. Figures 2 and 3 show the total
(i.e. cross sections of interactions with a gas with ISM composition)
cumulative (i.e. including the contribution of ghost nuclei) cross sections of
production of the main secondary CR species. The cross sections calculated
from FLUKA have associated a small statistical uncertainty, since it is a
Monte Carlo-based computation, that we highlight as a grey band. We compare
the computed FLUKA cross sections with the most widespread cross sections
parameterisations used in CR codes: the GALPROP and DRAGON2 parameterisations
and the cross sections calculated with the WNEW03 [71, 72] and YIELDX [73, 74]
(labelled in the legends as TS98) codes. These four cross section models give
similar predictions even though they follow different strategies to fit data:
whereas the Webber (WNEW03) and DRAGON2 parameterisations followed different
phenomenological formulations (and even different experimental data sets to
adjust them), the GALPROP cross sections make use of a semi-analytical
approach, using Monte Carlo codes in combination with their phenomenological
parameterisations and giving special attention to the cross sections
measurements more reliable in case there are sizeable discrepancies between
experiments in a certain energy region. On the contrary the FLUKA cross
sections derived in this work completely relies in theory-based interaction
models.

Figure 2: Comparison of the spallation cross sections of CR interactions with
ISM gas computed with FLUKA and the most widespread parameterisations, for the
production of various isotopes of B, Be and Li from 12C as projectile.

Figure 3: As in Fig.2 but for 16O as projectile.

In general, they seem to be consistent with each other, sharing some common
trends that should be remarked:

  * •

We find that the FLUKA cumulative cross sections are compatible within $30\%$
with the other cross sections parameterisations. In addition, not only the
normalization of these cross sections is compatible with the most updated
parameterisations, but their energy dependence seem to be in great agreement.

  * •

The position of the predicted resonances are, in general, in good agreement
with those parameterised from data. The main discrepancy is found in the
production of 10B, which could also be related to the cross sections from
other resonances decaying to 10B, as, e.g., 10C. We highlight here that the
different parameterizations also differ in the predicted intensity of
resonances, which can be seen from the 16O $\longrightarrow$ 9Be reaction
(upper right panel of Fig. 3). This would affect our predictions in the energy
range of the Voyager-1 data. However, most of the resonances are well
reproduced, as it can be seen in the 12C $\longrightarrow$ 11B reaction (lower
right panel of Fig. 2).

  * •

One of the main features expected in the FLUKA cross sections is their smooth
rise with energy, that is reliably observed in particle physics experiments
[70, 69, 68], but which is not incorporated in any of the current relevant
parameterisations. This logarithmic rise in the cross sections spectra is
present in the FLUKA inelastic and inclusive cross sections, clearer above
$100\leavevmode\nobreak\ \hbox{$\,{\rm GeV/n}$}$, and can affect our
predictions of the spectral index of the diffusion coefficient, $\delta$.

As demonstrated, the FLUKA code is able to make inelastic and spallation cross
sections calculations which are, in general, compatible with experimental data
and also with the most used and updated parameterisations. Given the scarcity
of cross sections measurements (especially for spallation reactions), these
nuclear codes, in combination with existent data, are expected to provide a
meaningful improvement in the cross sections sets employed for CR studies. In
the next section, the capability of the FLUKA cross sections data sets to be
consistent with CR data and to provide predictions on CR propagation is
broadly studied.

##  3 Implementation of the FLUKA cross sections in CR propagation codes and
compatibility with CR data

The implementation of these new sets of cross sections into the DRAGON2 code
has been performed in order to test the reproducibility of the current CR data
for all species up to $Z=26$. The general formula describing the propagation
of CRs solved by DRAGON2 is given by the following set of equations:

$\begin{split}\vec{\nabla}\cdot(\vec{J}_{i}-\vec{v}_{\omega}N_{i})+\frac{\partial}{\partial
p}\left[p^{2}D_{pp}\frac{\partial}{\partial
p}\left(\frac{N_{i}}{p^{2}}\right)\right]=Q_{i}+\frac{\partial}{\partial
p}\left[\dot{p}N_{i}-\frac{p}{3}\left(\vec{\nabla}\cdot\vec{v}_{\omega}N_{i}\right)\right]\\\
-\frac{N_{i}}{\tau^{f}_{i}}+\sum\Gamma^{s}_{j\rightarrow
i}(N_{j})-\frac{N_{i}}{\tau^{r}_{i}}+\sum\frac{N_{j}}{\tau^{r}_{j\rightarrow
i}}\hskip 34.14322pt\end{split}$ (3.1)

Here, $\vec{J_{i}}$ indicates the CR diffusive flux of the $i$-th species and
$N_{i}$ is the density per unit momentum. The term $\vec{v}_{\omega}$
represents the advection speed, which we assume to be null. The second term in
the left-hand side accounts for the diffusion in momentum space. The first
term in the right-hand side, $Q_{i}$, represents the distribution and energy
spectra of particle sources (injection spectra); the second term describes the
momentum losses; finally, the remaining four terms describe losses and gains
due to decay and fragmentation.

Figure 4: Spectra of the CRs injected in the simulations, compared to AMS-02
and Voyager-1 data (available up to Ne). Modulated spectra (see the text for
the details in the Fisk potential used for the data collected in different
periods) are shown as a solid line and the unmodulated spectra as a dashed
line. Voyager data is shown in green, AMS-02 data corresponding to CR species
with a significant secondary contribution (namely N, Na and Al) is shown in
magenta and AMS-02 for the main primary CR nuclei is shown in red.

The same injection and diffusion set-up as presented in Refs. [8, 45] is used
for the simulations performed with the derived FLUKA cross sections to
facilitate comparisons. We have employed two diffusion coefficient
parameterisations commonly used in previous works [75, 76, 39], which differ
in the interpretation of the high energy break observed in CR data [77, 78,
79, 80]. We refer to them as “diffusion” (Eq. 3.3) and “source” (Eq. 3.2)
hypotheses, as discussed in Ref. [45].

$D=D_{0}\beta^{\eta}\left(\frac{R}{R_{0}}\right)^{\delta}\text{ \hskip
51.21504pt {Source hypothesis}}$ (3.2)

$D=D_{0}\beta^{\eta}\frac{\left(R/R_{0}\right)^{\delta}}{\left[1+\left(R/R_{b}\right)^{\Delta\delta/s}\right]^{s}}\text{
\hskip 19.91684pt {Diffusion hypothesis}}\,\,\,,$ (3.3)

where $R_{0}$, the rigidity at which the diffusion coefficient is normalized,
is set to $4\leavevmode\nobreak\ \hbox{$\,{\rm GV}$}$. For the diffusion
coefficient of Equation 3.3, we use the values $\Delta\delta=0.14$,
$R_{b}=312\leavevmode\nobreak\ \hbox{$\,{\rm GV}$}$ and $s=0.040$, as
explained in [45] and derived in Ref. [75]. The use of the diffusion
coefficient of Eq. 3.3 implies using as injection a broken power-law (with
break at $8$ GV), while the use of that defined in Eq. 3.2 implies using a
doubly broken power-law (with breaks at $8$ GV and $320$ GV) for the
injection.

We show in Figure 4 the spectra of the main primary CRs (mainly dependent on
the injection parameters employed) compared to AMS-02. The injection spectra
used are freely adjusted to reproduced AMS-02 data. All the experimental data
from CR experiments are taken from //https://lpsc.in2p3.fr/crdb/ [81, 82] and
https://tools.ssdc.asi.it/CosmicRays/ [83]. We find here that, in order to
reproduce those spectra, we need to employ different injection parameters for
1H, 4He, C-O group (i.e. 12C and 16O), Ne-Mg-Si group (20Ne, 24Mg and 28Si),
N-Na-Al (14N, 23Na and 27Al) group and Fe (56Fe, 57Fe and 58Fe). These
discrepancies may be due to the fact that sources with different properties
(as mass or metallicity) may accelerate particles with slightly different
power-laws, but the actual reason is still a mystery (see also Ref. [84] for a
discussion).

Figure 5: Left column: secondary-over-secondary flux ratios for Li, Be and B
obtained with the FLUKA computations, together with the band of statistical
uncertainties due to the spallation cross sections calculation (since this is
a Monte Carlo computation). Right column: B, Be and Li fluxes obtained from
the simulation compared with experimental data, using the diffusion parameters
obtained from the fit of the B/C flux ratio (see section 3.2).

Given their importance to study the transport of CRs, we specially focus on
the spectra of B, Be and Li and the flux ratios among them, which are depicted
in Fig. 5. These spectra are derived from a Markov chain Monte Carlo analysis
(section 3.2) of the AMS-02 boron-over-carbon flux ratio (see Fig. 7 and upper
table 1) for the diffusion coefficient of Eq. 3.2. As it is observed in the
figure, there is a $\sim 20-25$% discrepancy in the Li and Be spectra (right
panels of Fig. 5) with these diffusion parameters, which is well within the
uncertainties usually reported in cross sections data [8]. We remark that this
discrepancy can be resolved with a $\sim 12\%$ overall increase of the B
production cross sections and a $\sim 12\%$ reduction of the Be and Li
production cross sections, which predicts the same ratios as for the case with
a $25\%$ increase of the B cross sections. Notably, the energy dependence of
the derived spectra is well compatible with that of the AMS-02 data for Be,
while the Li spectrum seems to be slightly more deviated, as easily seen from
their residuals. We notice that the use of the straight-ahead approximation
(where the kinetic energy per nucleon of the primary nucleus is conserved to
the secondary nucleus) in the DRAGON2 code can affect the predictions of light
nuclei at low energy, which would, mainly, add more uncertainty to the low
energy part of the Li spectrum. The energy dependence of their respective flux
ratios (left column), whose main dependence above $10\leavevmode\nobreak\
\hbox{$\,{\rm GeV/n}$}$ is the inclusive cross sections, shows this as well.
Specially, the predicted Li/Be flux ratio is compatible with AMS-02 data
without any scaling of the predicted cross sections above a few GeV/n, while
the other ratios need a scale to be reproduced. In addition, the energy
dependence of the predicted Li/B spectrum seems to be slightly discrepant,
although still in agreement within the $1\sigma$ uncertainties. We have
renormalised the B cross sections by $25\%$ to show this more clearly (dashed
lines). In Appendix B we show the predicted flux of another important light
secondary CR, the 3He isotope, compared to data.

The high importance of having good estimations of the cross sections of
production of B, Be and Li stands for the fact that they allow us to constrain
the magnetized halo size, the diffusion parameters, the Alfvén velocity and
other parameters important for understanding the transport of CRs. These
parameters are carefully studied in sections 3.1 and 3.2.

###  3.1 Halo size predictions with the FLUKA cross sections

Figure 6: Top panel: secondary-over-secondary flux ratios of the light
secondary CRs (Li, Be, B). In these plots, the spectrum for the best-fit halo
size value found from the 10Be/9Be ratios is also shown. Bottom panels:
10Be/9Be and 10Be/Be flux ratios. The predictions are compared to experimental
data for simulations performed with various halo sizes.

The evaluation of the magnetized halo size, i.e. the region in the Galaxy
where CRs diffuse, can be constrained from several observables, as radio [13],
X-ray and gamma-ray emission [11] from lepton synchrotron and inverse-Compton
processes or from other CR observables like unstable heavy nuclei [12], whose
mean lifetime is of the order of the propagation time.

Here, we estimate the magnetized halo size, derived with the FLUKA cross
section, from a combined analysis of the ratios of 10Be (mean life time $\sim
1.4\leavevmode\nobreak\ \hbox{$\,{\rm My}$}$ [85])), namely 10Be over 9Be and
10Be over the total Be flux. To do so, the halo size value is fitted to
experimental data on the 10Be flux ratios, from the ACE [86], IMP [87, 88],
ISEE [89], ISOMAX [90], Ulysses [91] and Voyager [92] missions, using the
curve_fit package from the scipy.optimize library, as it is done in [8], where
we refer the reader for more details about the analysis. This fit yields a
value of the halo size of $\sim 7.5^{+1.13}_{-0.95}\leavevmode\nobreak\
\hbox{$\,{\rm kpc}$}$, similar to estimations with other cross sections,
usually quoted to be around $6\leavevmode\nobreak\ \hbox{$\,{\rm kpc}$}$ [8,
9, 10] and also favored by gamma-ray [93] and radio observations [94, 95, 96].

In the top panel of Figure 6, we display the Be/B flux ratio, with B cross
sections scaled by $25\%$, for various values of the halo size, to show the
variations of the predicted ratio for different values of $H$. In the bottom
panel, we show the predicted 10Be ratios for different halo size values. We
have also included the 10Be/9Be AMS-02 preliminary data in the figure (lower
left panel of Fig. 6) to show that these predictions are compatible with these
data as well, but we did not include them in the fit procedure since they are
yet preliminary. From these figures one can conclude that a halo size value
between $5$ and $10$ kpc would be well compatible with the data. However, we
employ the best-fit halo size value in the determination of the rest of
propagation parameters from the secondary-to-primary and secondary-to-primary
flux ratios, as explained in the next section.

###  3.2 Markov chain Monte Carlo analysis of the diffusion coefficients
parameters with the FLUKA cross sections

In this section, we show and discuss the results obtained from the analyses
carried out for the flux ratios of B, Be and Li to C and O (B/C, B/O, Be/C,
Be/O, Li/C and Li/O) and its respective secondary-to-secondary flux ratios
(Be/B, Li/B, Li/Be), using the Markov chain Monte Carlo algorithm (MCMC)
presented in Ref. [45]. We also performed the combined analysis of all these
spectra presented in that work, which adds a scale factor to renormalise the
cross sections of production of B, Be and Li. The best-fit propagation
parameters and their confidence intervals are depicted for both the
parameterisations of the diffusion coefficient used, in the form of a box-plot
in Figure 7 and explicitly given in tables in appendix A. On one hand, it is
convenient to remark some of the implications of the values found in these
analyses of independent ratios:

  * •

Interestingly, an average value of $\delta\sim 0.42$ is favored by the B and
Be ratios, whereas the Li ratios predict a $\delta$ value below $0.4$.
Nevertheless, these values seem to be generally compatible within the
$2\sigma$ uncertainties in their determination.

  * •

The analyses of the Li and B ratios favor an Alfvén velocity value around
$30\leavevmode\nobreak\ \hbox{$\,{\rm km/s}$}$, not too far from theoretical
estimations [97, 98, 99] and with predictions using other cross sections
parameterisations [45]. Nevertheless, these values are in tension with
energetic considerations, as discussed by Ref. [100]. The lower values
obtained for $V_{A}$ in the analyses of the Be ratios were also found for the
GALPROP cross sections by Ref. [45].

  * •

Notably, $\eta$ is negative for all the analyses, as revealed by other recent
analyses using different cross sections (see, for example, Ref. [18]), which
reflects a change in the trend of the diffusion coefficient at low energies,
towards lower confinement of low-energy CRs. This seems to be due to the
dissipation or damping of MHD waves at the scales (Larmor radius of the
particles) associated to these energies [101, 102].

  * •

Finally, it can be observed that the $D_{0}/H$ values obtained are pretty
coincident for pairs of ratios with same secondary CR, while the ratios of
different secondary CRs significantly deviate, which is due to the spallation
cross sections related to the production of each of these secondaries. This
highlights the need to add a scale factor for each of these CRs in order to
renormalise their cross sections of production.

Figure 7: Box-plot summarising the information from the Probability
Distribution Function (PDF) of every parameter included in the MCMC analyses
performed with the FLUKA cross sections. Different colors represent the
different diffusion coefficients employed (Eq. 3.2 and Eq. 3.3). This figure
is shown in the same format as in Ref [45] to facilitate the comparison with
the results obtained from the GALPROP and DRAGON2 cross sections.

Figure 8: Top panel: Secondary-over-secondary flux ratios of B, Be and Li
computed using the propagation parameters and scaling factors found in the
combined analysis, compared to the AMS-02 experimental data. Lower panels:
Secondary-over-primary spectra (B/C, B/O, Be/C, Be/O, Li/C, Li/O) predicted
with the parameters determined by the MCMC combined analysis for the DRAGON2
cross sections. The grid lines at the level of 5% residuals are highlighted in
the lower panel in a different color for clarity. In addition, the Voyager-1
data are also included for completeness.

On the other hand, the combined analyses converge to propagation parameters
closer to those predicted by Li ratios. These analyses predict a spectral
index of the diffusion coefficient $\sim 0.36$, which is significantly lower
than the typical values found from B ratios, but it is compatible with the
values found for the GALPROP cross sections in Ref.[45]. A crucial point of
this analysis is the cross sections scale factors obtained. These are the same
for both diffusion coefficient used and are $1.18$, $0.94$ and $0.93$, for B,
Be and Li, respectively, with $1\sigma$ statistical uncertainties of $\sim\pm
0.01$. However, we highlight that there are other systematic uncertainties
related to the determination of these scale factors, like those related to the
inelastic cross sections used, and could make the total systematic
uncertainties in the determination of these scale factors to larger than $5\%$
[45]. In addition, we highlight that the diffusion coefficient obtained from
these combined analyses allows us to reproduce, at the same time, the 3He/4He
flux ratio (see Fig. 11), which evidences that the diffusion coefficient
predicted by the light secondary CRs is compatible for the different nuclei
within uncertainties (mainly cross sections uncertainties).

The spectra of secondary-to-secondary and secondary-to-primary flux ratios are
shown in Figure 8 for the best-fit parameters obtained in the combined
analysis for the diffusion coefficient of Eq. 3.3 (diffusion hypothesis for
the origin of the break). These panels demonstrate that the ratios of
secondary CRs can be successfully reproduced at the same time with the FLUKA
cross sections within $1\sigma$ uncertainties, after scaling the overall cross
sections of production of B, Be and Li which is well below the typical average
uncertainties in the experimental measurements of these cross sections for the
best-known channels: $\sim 20-30$%, which are those from 12C and 16O as
projectiles. We observe small discrepancies at low energies (below $5\%$) that
could be related with these cross sections but could also be related to the
parameterisation of the diffusion coefficient used.

It must be highlighted that very few nuclear codes have been able to produce
suitable cross sections sets intended to be used in CR propagation codes and
none has demonstrated to be able to reproduce the secondary-to-primary flux
ratios just with the use of renormalisation (scale) factors. This, in fact,
means that the FLUKA cross sections are at the level of precision of the most
updated dedicated cross sections parameterisations of CR spallation reactions.

##  4 Study of the gamma-ray diffuse emission with the FLUKA cross sections

In addition to the information about the transport of CRs from secondary CRs,
other secondary particles - namely leptons, antinuclei, gamma rays and
neutrinos - are formed and provide essential insight about acceleration
mechanisms, magnetic field configurations, environment of sources, etc. An
important fraction of the particles produced from spallation reactions of CR
nuclei with the ISM gas are short-living mesons (mainly pions), which are the
responsible of the formation of leptons and gamma rays from their decays.
Secondary electrons and positrons are mainly produced via the decay of charged
pions (accompanied by neutrinos). Gamma rays are mainly produced by the decays
of neutral mesons and by the interactions of CR leptons with magnetic and
radiation fields in the ISM, carrying important information about the
environmental conditions. The study of the galactic gamma-ray diffuse emission
is important to constrain our models and opens a new window to unveil the
mechanisms behind CR interactions and acceleration [103, 104].

In this section we demonstrate that the gamma-ray sky maps and the local HI
emissivity spectrum, produced using the FLUKA cross sections for the
production of leptons and gamma rays from CR interactions with the
interstellar gas, are in good agreement with the current experimental data. We
adjust the spectra of protons and leptons with the injection parameters tuned
to reproduce AMS-02 data and using the diffusion coefficient given in Eq. 3.3,
with the parameters obtained in the fit of the B/C spectrum (see Fig 12 in
appendix C). Here, the CR e+ and e- spectra require further considerations:
while the vast majority of CR electrons are supposed to be accelerated in
supernova remnants (SNRs), a significant amount of positrons seem to be
emitted from pulsar wind nebulae (PWN), expected to be symmetric $e^{+}$ and
$e^{-}$ pairs emitters [105, 106]. Therefore, to reproduce this component of
primary positrons and electrons we inject, in the DRAGON2 simulation, an extra
source of leptons (using a single power-law) emitting them evenly. We have
adjusted this extra component by reproducing the high energy points of the
AMS-02 positrons data. In this way, we reproduce the AMS-02 $e^{+}$ spectrum
as a sum of secondary positrons + primary positrons from extra sources (likely
PWN) and the AMS-02 $e^{-}$ spectrum as primary electrons injected by SNRs +
secondary electrons from CR interactions + extra sources (right panel of
Figure 12). We leave a detailed study of secondary leptons from FLUKA for a
separated paper.

Once the CR spectra of these species match local data, we compute the gamma-
ray diffuse emission by using the Gammasky code, a dedicated code able to
calculate sky maps of various radiative processes in our Galaxy [107, 108,
39]. Our computation is performed using the gas maps developed by the GALPROP
team [109, 110]. Each of its main components and the total gamma-ray intensity
predicted at $100\leavevmode\nobreak\ \hbox{$\,{\rm GeV}$}$ with the FLUKA
cross sections are shown in Figure 9, together with the Fermi-LAT template of
the gamma-ray diffuse emission (lower ight panel). Following the calculations
performed in Ref. [39], the IC, bremsstrahlung and hadronic cross sections of
gamma-ray production are calculated with FLUKA assuming an ISM composition
with relative abundance of H : He : C : N : O : Ne : Mg : Si =
$1:0.096:4.65\times 10^{-4}:8.3\times 10^{-5}:8.3\times 10^{-4}:1.3\times
10^{-4}:3.9\times 10^{-5}:3.69\times 10^{-5}$.

Figure 9: Gamma-ray intensity maps of the Galaxy, in Mollweide projection, for
the propagation parameters inferred from the B/C analysis with the FLUKA cross
sections. The upper-left map displays the bremsstrahlung emission, the upper-
right map the gamma-ray emission from $\pi^{0}$ decays, and the lower-left map
the IC emission. The lower-right map is the Fermi-LAT template for the diffuse
emission, derived from the 8 first years of operation (PASS 8 data; $P8\\_R3$,
taken from gll_iem_v07.fits
https://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html).

As we can see from Figure 9, the hadronic emission and bremsstrahlung
intensity maps mainly track the gas composition, with the IC component
principally probing the ISRF, which is more intense at the Galactic Center.
The most recent Fermi-LAT template for the total diffuse emission
gll_iem_v07.fits
(https://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html) is
also displayed for comparison. In this template, the contributions from large
scale structures, such as the gamma-ray Loop-I [111] and the Fermi Bubbles
[112] are also added, which explains the differences observed at high
latitudes around the Galactic Center.

Then, in order to avoid the uncertainties related to the gas and radiation
fields, we turn our attention to the gamma-ray emissivity produced in the
local ISM (LISM). In this context, the local HI emissivity is defined as the
diffuse gamma-ray flux, coming from CR interactions with the local gas per
unit of gas atom (HI). This quantity is inferred from the Fermi-LAT gamma-ray
data [61] and its determination is useful to help constraining different
propagation models. The main contribution to the local HI emissivity at low
energy is originated from Coulomb scattering of CRs with gas and electron
bremsstrahlung; a further contribution, dominant above $\sim
100\leavevmode\nobreak\ \hbox{$\,{\rm MeV}$}$, comes from the gamma-ray
emissions of unstable particles formed via nuclear reactions (hadronic
emissions). Formally, emissivity can be calculated by means of the following
equation:

$\frac{Q_{S}(E_{S})}{n_{gas}}=4\pi\int J(E)\frac{d\sigma(E_{S}|E)}{dE_{S}}dE,$
(4.1)

where $n_{gas}$ is the number of target gas atoms per unit volume and
$\sigma(E_{S}|E)$ the cross section of production of any secondary particle,
S, as for example neutrinos, gamma rays, positrons, etc. The ratio
$Q_{S}(E_{S})/n_{gas}$ is expressed in units of $\leavevmode\nobreak\
\hbox{$\,{\rm GeV}$}s^{-1}$.

The computed emissivity spectra, calculated for the propagation parameters
derived from the B/C analysis of the FLUKA cross sections, are shown for the
diffusion coefficient given in Equation 3.3 (diffusion hypothesis) in Figure
10. They are compared to the emissivity observed by the Fermi-LAT, inferred
from the recent PASS8 data set444These are public data taken from the Fermi
background models website..

Figure 10: Local HI gamma-ray emissivity spectrum for the propagation
parameters derived from the B/C analysis for the FLUKA cross sections. An
uncertainty band, for the adjustment of the spectra of p, He and electrons,
related to solar modulation uncertainty (variation of $\pm 0.1$ in the Fisk
potential) is also shown. Bremsstrahlung and hadronic emissions are also
displayed to visualize the region of dominance of each component of the total
emissivity. The experimental data is taken from Ref. [61].

In order to reproduce the local emissivity spectrum at low energies, we are
forced to include a low energy break in the injection spectrum of electrons to
prevent the bremsstrahlung emission to overpredict the data (or a too small
solar modulation potential, which is not in agreement with the rest of
observations from neutron monitors and other cosmic rays). This makes us use a
doubly broken power-law injection spectrum for CR electrons, with a low energy
break at $8$ GeV (note that this is the same break position as for protons), a
high energy break at $65$ GeV and an exponential cut-off at around $1$ TeV
(although the AMS-02 spectrum can be perfectly reproduced only with the high
energy break and the cut-off). We remark that the high-energy break and cut-
off used are in agreement with what other works have recently found [113],
however the low energy break is added ad hoc, without a clear theoretical
motivation.

Finally, it is worth mentioning that, thanks to the use of the FLUKA nuclear
code to calculate the gamma-ray production cross sections, we are able to
study the gamma-ray emission lines coming from the hadronic interactions of CR
protons and helium with ISM nuclei (even being found as traces) such as
$p+p\longrightarrow\gamma+X$ or $p+^{14}$N$\longrightarrow\gamma+X$. These
lines could be detected in the MeV region by the future space experiments like
e-Astrogam [114] and have been explored in very few works [115].

##  5 Conclusions

In this work, the FLUKA calculations for the inelastic and inclusive cross
sections important for CR spallation reactions have been presented and widely
discussed. We found that the inelastic cross sections for H, He, B and C are
in good agreement with data and are quite similar to the CROSEC predictions at
the level of a few percent. Nevertheless, elements with atomic number $Z\geq
10$ (from Ne) start to differ significantly. Then, the total cumulative
inclusive cross sections involving the production of Be, Li and B isotopes
have been tested in comparison to the most recent dedicated parameterisations,
showing a good general agreement, within $30\%$ discrepancies.

Then, we evaluate the spectra of secondary CR species after implementing the
FLUKA inelastic and inclusive cross sections in the DRAGON2 propagation code,
in order to demonstrate that every CR observable can be reproduced with
accuracy and at the level of precision of current CR parameterisations.
Notably, we have studied the B, Be and Li over C and O flux ratios under two
different parameterisations of the diffusion coefficient, and the results of
these fits and their confidence intervals have been reported. We obtain a
best-fit value of the halo size, from the 10Be ratios of
$H=7.5^{+1.13}_{-0.95}\leavevmode\nobreak\ \hbox{$\,{\rm kpc}$}$. All these
ratios favor a change in the trend of the diffusion coefficient at sub-GeV
energies and the predicted slope of the diffusion coefficient, $\delta$,
varies from $\sim 0.42$ (for B and Be ratios) to $\sim 0.39$ (for Li ratios),
which is very similar to what is found from the GALPROP and DRAGON2
parameterisations. We have also performed a combined analysis of these ratios,
which includes also scale factors intended to renormalise the overall FLUKA
cross sections of production of B, Be and Li. We have demonstrated that, with
the use of these scale factors, we are able to match AMS-02 observations of
these secondary-to-primary species within $5$% errors above $5$ GeV. The
scaled factors obtained are of $\sim 1.18$, $\sim 0.94$ and $\sim 0.93$ for B,
Be and Li, respectively, which are below the typical average experimental
uncertainties of cross sections measurements for the best known channels (i.e.
those from 12C and 16O).

Finally, we have shown the derived diffuse gamma-ray sky maps and the local HI
emissivity spectrum using the FLUKA cross sections for production of gamma
rays, implemented in the Gammasky code. The derived maps are shown to be
coherent with the standard picture of the different mechanisms of gamma-ray
emission (Inverse-Compton, bremsstrahlung and hadronic decay into photons) in
the Galaxy. In addition, we demonstrate that the local HI emissivity spectrum
can be perfectly reproduced from the models derived with the FLUKA cross
sections, although with the need of inserting a low energy break in the
injection spectrum of electrons.

In conclusion, the cross sections obtained with the FLUKA code seem to be
consistent with the current CR parameterisations and with experimental data,
and are able to reproduce every CR observable with a low level of uncertainty.
They represent a promising tool for future more precise CR studies.

## Acknowledgments

P. De la Torre is supported by the Swedish National Space Agency under
contract 117/19. We acknowledge the FLUKA collaboration for providing and
supporting the code. This work has been partly carried out using the RECAS
computing infrastructure in Bari (https://www.recas-bari.it/index.php/en/). A
particular acknowledgment goes to G. Donvito and A. Italiano for their
valuable support. This project used computing resources from the Swedish
National Infrastructure for Computing (SNIC) under project Nos. 2021/3-42 and
2021/6-326 partially funded by the Swedish Research Council through grant no.
2018-05973. Some of the discussions included in the text were triggered thanks
to the Paris-Saclay Astroparticle Symposium 2021, with the support of the P2IO
Laboratory of Excellence (programme “Investissements d’avenir”
ANR-11-IDEX-0003-01 Paris-Saclay and ANR-10-LABX-0038), the P2I research
departments of the Paris-Saclay university, as well as IJCLab, CEA, IPhT,
APPEC, the IN2P3 master projet UCMN and EuCAPT.

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##  Appendix A Summary of the MCMC results

In this appendix, the results obtained from the MCMC algorithm of each
secondary-to-primary ratio (B/C, B/O, Be/C, Be/O, Li/C, Li/O), in table 1, and
their combined analysis, in table 2, for both diffusion parameterisations used
(’Source’ and ’Diffusion’ hypotheses) are summarized. They contain the median
value (coinciding with the maximum posterior probability value) $\pm$ the
$1\sigma$ uncertainty related to its determination, assuming their PDFS to be
Gaussian, and the actual range of values contained in the 95% probability of
the distribution.

Independent analyses

Source hypothesis
---
| B/C | B/O | Be/C | Be/O | Li/C | Li/O
$\frac{D_{0}\,(10^{28}\leavevmode\nobreak\ \hbox{$\,{\rm cm^{2}/s}$})}{H\,(kpc)}$ | 0.87 $\pm$ 0.03 | 0.97$\pm$ 0.04 | 1.11$\pm$ 0.02 | 1.06 $\pm$ 0.05 | 1.22 $\pm$ 0.07 | 1.31 $\pm$ 0.06
[0.82, 0.93] | [0.89, 1.05] | [1.06, 1.16] | [0.96, 1.17] | [1.1, 1.27] | [1.15, 1.41]
$v_{A}$ (km/s) | 25.86 $\pm$ 2.16 | 32.4 $\pm$ 1.86 | 5.74 $\pm$ 5.6 | 17.02 $\pm$ 6.8 | 24.3 $\pm$ 6.11 | 37.03 $\pm$ 3.73
[21.54, 30.16] | [29.01, 35.79] | [0., 16.94] | [5.52, 29.14] | [13.98, 36.52] | [29.55, 40.83]
$\eta$ | -0.53 $\pm$ 0.13 | -0.51 $\pm$ 0.11 | -1.88 $\pm$ 0.18 | -1.85 $\pm$ 0.28 | -1.91 $\pm$ 0.34 | -1.23 $\pm$ 0.24
[-1.29, -0.8] | [-0.75, -0.31] | [-2.19, -1.52] | [-2.35, -1.29] | [-2.41, -1.29] | [-1.75, -0.92]
$\delta$ | 0.43 $\pm$ 0.01 | 0.40 $\pm$ 0.01 | 0.41 $\pm$ 0.02 | 0.42 $\pm$ 0.01 | 0.39 $\pm$ 0.01 | 0.37 $\pm$ 0.02
[0.41, 0.45] | [0.38, 0.43] | [0.39, 0.44] | [0.4, 0.44] | [0.37, 0.41] | [0.35, 0.4]

Diffusion hypothesis
---
| B/C | B/O | Be/C | Be/O | Li/C | Li/O
$\frac{D_{0}\,(10^{28}\leavevmode\nobreak\ \hbox{$\,{\rm cm^{2}/s}$})}{H\,(kpc)}$ | 0.82 $\pm$ 0.03 | 0.91 $\pm$ 0.04 | 1.1 $\pm$ 0.02 | 1.03 $\pm$ 0.05 | 1.18 $\pm$ 0.04 | 1.23 $\pm$ 0.07
[0.88, 0.76] | [0.99, 0.83] | [1.14, 1.06] | [0.94, 1.13] | [1.11, 1.26] | [1.09, 1.37]
$v_{A}$ (km/s) | 23.26 $\pm$ 2.25 | 30.3 $\pm$ 2.01 | 5.06 $\pm$ 5.2 | 13.19 $\pm$ 6.76 | 20.82 $\pm$ 3.82 | 33.1 $\pm$ 4.78
[18.6, 27.71] | [26.14, 34.16] | [0., 15.29] | [0., 25.87] | [13.18, 28.44] | [23.54, 40.64]
$\eta$ | -0.67 $\pm$ 0.13 | -0.66 $\pm$ 0.11 | -1.97 $\pm$ 0.17 | -2.02 $\pm$ 0.26 | -2.13 $\pm$ 0.19 | -1.52 $\pm$ 0.32
[-0.93, -0.41] | [-0.88, -0.44] | [-2.26, -1.63] | [-2.4, -1.74] | [-2.37, -1.73] | [-2.16, -1.04]
$\delta$ | 0.45 $\pm$ 0.01 | 0.42 $\pm$ 0.01 | 0.42 $\pm$ 0.01 | 0.43 $\pm$ 0.02 | 0.40 $\pm$ 0.02 | 0.38 $\pm$ 0.02
[0.43, 0.47] | [0.40, 0.44] | [0.40, 0.44] | [0.41, 0.45] | [0.38, 0.43] | [0.37, 0.43]

Table 1: Results obtained in the independent analysis of the flux ratios B/C,
Be/C, Li/C, B/O, Be/O and Li/O for the diffusion coefficient parameterisations
of Eqs. 3.2 and 3.3 for FLUKA the cross sections derived in this work. The
halo size value, $H$, obtained from the 10Be flux ratios was $7.5$ kpc

Combined analysis

Source hypothesis
---
$\frac{D_{0}\,(10^{28}\leavevmode\nobreak\ \hbox{$\,{\rm cm^{2}/s}$})}{H\,(kpc)}$ |  $v_{A}$ $\,{\rm(km/s)}$ | $\eta$ | $\delta$
1.33 $\pm$ 0.02 | 39.8 $\pm$ 0.15 | -0.72 $\pm$ 0.05 | 0.35 $\pm$ 0.02
[1.29, 1.37] | [39.59, 40.02] | [-0.83, -0.62] | [0.33, 0.39]
Diffusion hypothesis
$\frac{D_{0}\,(10^{28}\leavevmode\nobreak\ \hbox{$\,{\rm cm^{2}/s}$})}{H\,(kpc)}$ |  $v_{A}$ $\,{\rm(km/s)}$ | $\eta$ | $\delta$
1.31 $\pm$ 0.02 | 39.7 $\pm$ 0.33 | -0.81 $\pm$ 0.07 | 0.36 $\pm$ 0.01
[1.27, 1.35] | [39.98, 40.1] | [-0.95, -0.67] | [0.34, 0.39]

Table 2: Same as in table 1 but for the combined analysis of the B, Be and Li
flux ratios.

Here, we report tables with the $\chi^{2}$ values obtained with the best-fit
parameters of the independent and combined analyses FLUKA cross sections.
Interestingly, we observe that the difference between the $\chi^{2}$ values
from the source and diffusion hypotheses in the independent analyses are of
$\Delta\chi^{2}\sim 6$, for the B and Li ratios. This yields a significance of
$\sim 3.5\sigma$ (making use of Wilk’s theorem), which is $1.2\sigma$ less
significant than for the DRAGON2 and GALPROP parameterisations, as studied in
Ref. [45]. This is due to the inclusion of the rise of cross sections at high
energies which is not taken into account in the parameterisations and implies
that the interpretation of the origin of the break in the CR spectra is
sensitive to the cross sections used.

$\chi^{2}$ values of best fit parameters
---
| B/C | B/O | Be/C | Be/O | Li/C | Li/O
Source hypothesis Indep. analysis | 35.45 | 27.49 | 59.01 | 55.24 | 57.87 | 40.45
Diffusion hypothesis Indep. analysis | 29.43 | 22.99 | 44.47 | 47.6 | 51.38 | 34.13
Source hypothesis Comb. analysis | 51.79 | 32.8 | 72.33 | 55.92 | 63.44 | 66.56
Diffusion hypothesis Comb. analysis | 48.17 | 32.53 | 62.77 | 54.3 | 60.0 | 63.52

Table 3: $\chi^{2}$ values of the best-fit diffusion parameters found for the
MCMC analyses with the FLUKA cross sections.

##  Appendix B 3He FLUKA cross sections

In this appendix we report the predicted flux of another of the main light
secondary CRs, the 3He isotope. We have computed the full 3He production cross
sections from isotopes up to Iron ($Z=26$). The contribution to the total 3He
flux from nuclei heavier than He is, in average, $\sim 44\%$. As we observe
from Figure 11, the predicted 3He flux, with the diffusion coefficient
obtained in the combined analysis, is reasonably compatible with data in both,
energy-dependence and intensity. We also include data from the PAMELA mission
[116] in this figure, taken in the period from 2006/07 to 2008/12. It is worth
mentioning that the predicted 3He flux with the diffusion coefficient obtained
in the independent B/C analysis is overpredicted in the energy region between
$1$ GeV and $10$ GeV by around $20\%$.

Figure 11: Left panel: Modulated and unmodulated predicted spectra of He and
3He compared to AMS-02 and Voyager 1 and Pamela data. Left panel: Predicted
3He/4He flux ratio compared to Pamela and AMS-02 data. The residuals,
calculated as model-data/model are also shown. These spectra are calculated
with the diffusion coefficient obtained from the combined fit.

##  Appendix C Electrons and protons spectra

In this appendix we show the evaluated spectra of proton, helium and leptons.
These are obtained by tuning the injection spectra to reproduce experimental
data. In the case of protons and helium, we also include the Voyager-01 data.
The reproduction of the CR electrons spectrum (CRE) involves also choosing a
configuration of the magnetic field and the radiation fields. The DRAGON2 code
allows the parameterisation of three components of the magnetic field (see
Ref. [117]): one for the disc, one for the halo and a turbulent one. We set
the magnetic field parameters to $B_{disc}$ = $2\mu G$, $B_{halo}$ = $4\mu G$
and $B_{turb}$ = $7.5\mu G$ as found in Ref. [113]. The energy density of the
radiation fields, important for the energy losses related to IC scattering,
were taken from Ref. [118].

The computed proton and helium spectra are shown in Figure 12, left, compared
to Voyager-1 [119] (unmodulated) and AMS-02 [48] (using a Fisk potential
$\phi=0.6\leavevmode\nobreak\ \hbox{$\,{\rm GV}$}$). The electron and total
CRE spectra are compared to data, from Fermi-LAT [120], DAMPE [121], CALET
[122] and AMS-02 [123], in the right panel of Figure 12. These spectra are
derived using the propagation parameters inferred from the B/C analysis with
the FLUKA cross sections in the diffusion approximation. The uncertainty band
related to a variation of the Fisk potential of $\pm 0.1$ GV is included for
these spectra.

Figure 12: Left panel: Modulated and unmodulated predicted spectra of H and He
compared to AMS-02 and Voyager 1 data. Right panel: Predicted total CRE
spectra compared to AMS-02, Fermi-LAT, DAMPE and CALET data. The dashed line
represents the predicted flux of electrons, fitting to the AMS-02 electron
flux. The uncertainty band related to a variation of the Fisk potential of
$\pm 0.1$ GV is shown for the spectra in both figures.