# Markov chain Monte Carlo analyses of the flux ratios of B, Be and Li with
the DRAGON2 code

P. De La Torre Luque  M. N. Mazziotta  F. Loparco  F. Gargano  D. Serini

###### Abstract

Recent cosmic-ray measurements are challenging our models of propagation in
the Galaxy. A good characterization of the secondary cosmic rays (B, Be, Li
and sub-iron species) is crucial to constrain these models and exploit the
precision of modern CR experiments. In this work, a Markov chain Monte Carlo
analysis has been implemented to fit the experimental flux ratios between B,
Be and Li and their flux ratios to the primary cosmic-ray nuclei C and O. We
have fitted the data using two different parametrizations for the spallation
cross sections. The uncertainties in the evaluation of the spectra of these
secondary cosmic rays, due to spallation cross sections, have been taken into
account by introducing scale factors as nuisance parameters in the fits,
assuming that this uncertainty is mostly due to the normalization of the cross
sections parametrizations. We have also tested two different formulations for
the diffusion coefficient, which differ in the origin of the high energy
hardening ($\sim 200\leavevmode\nobreak\ \hbox{$\,{\rm GeV/n}$}$) of cosmic
rays. Additionally, two different approaches are used to scale the cross
sections, one based on a combined analysis of all the species (“combined”
analysis) and the other reproducing the high energy spectra of the secondary-
to-secondary flux ratios of Be/B, Li/B, Li/Be (“scaled” analysis). This allows
us to make a better comparison between the propagation parameters inferred
from the cross sections parametrizations tested in this work. This novel
analysis has been successfully implemented using the numerical code DRAGON2
dedicated to cosmic-ray propagation to reproduce the cosmic-ray nuclei data up
to $Z=14$ from the AMS-02 experiment. In general, it is found that the ratios
of Li favor a harder spectral index of the diffusion coefficient, but
compatible with the other ratios inside the observed $2\sigma$ uncertainties.
In addition, it is shown that, including these scale factors, the secondary-
to-primary flux ratios can be simultaneously reproduced, obtaining that the
scale factor associated to the cross sections of boron production is the
lowest one, whereas that associated to Li production is the largest one.

##  1 Introduction

The high precision measurements provided by recent cosmic-ray missions such
AMS-02 [1, 2, 3] and DAMPE [4, 5] together with the increasing quality and
quantity of gamma-ray data, thanks to the Fermi-LAT [6, 7], HAWC [8, 9, 10]
and H.E.S.S. [11, 12, 13], are requiring improved models on particle
acceleration and transport for their interpretation [14]. The transport of
Galactic cosmic rays (CRs) is governed by their interactions with the magneto-
hydrodynamical turbulence generated in the plasma of the interstellar medium
(ISM). This mechanism is responsible for the observed isotropy in their
arrival directions at Earth [15, 16] and is usually described with a diffusive
transport equation.

Injection and acceleration of primary CRs is believed to take place mainly in
supernova remnants (SNRs) [17] (although also pulsars [18, 19] and OB stars
[20] are likely CR sources) and it is explained, at least in general
principles, by the diffusive shock acceleration (DSA) model [21, 22, 23, 24].
In turn, the study of secondary CRs (i.e. those generated from the spallation
reactions of primary CRs with gas nuclei in the ISM) provides crucial
information about the propagation of CRs; the amount of matter traversed
(grammage) or path length [25, 26], the size of the magnetised halo [27, 28,
29] or the spectrum of self-generated turbulence [30, 31] can, in fact, be
inferred from the measured fluxes of secondary CR nuclei.

In particular, the flux ratio of secondary to primary CRs is sensitive to the
propagation parameters [32, 33], whereas it is mostly insensitive to the
source spectrum of primary species. As a first order approximation, the flux
ratio between the $k$-th species of secondary CRs and the $i$-th species of
primaries can be written as:

$\frac{J_{k}}{J_{i}}\left(R\right)\approx\frac{\tau_{diff}(E)Q_{k}(E)}{J_{i}(R)}\sim\sigma_{i\rightarrow
k}(R)\tau_{diff}(R)\propto\sigma_{i\rightarrow k}(R)/D(R)$ (1.1)

where $R$ is the rigidity of CR particles. In the previous equation we
indicate with $Q_{k}=n_{ISM}\sigma_{i\rightarrow k}J_{i}$ the source term for
secondary CRs, where $\sigma_{i\rightarrow k}$ is the production cross section
of the secondary CR species $k$ from the interaction of the primary CR species
$i$ with the gas in the ISM (spallation reactions) and $n_{ISM}$ is the
density of gas in the ISM (consisting of H:He = 1:0.1 and traces of heavier
elements [34]). Finally, $\tau_{diff}\propto 1/D(R)$ is the mean diffusion
time of CRs in the Galaxy, where $D(R)$ is the diffusion coefficient used to
describe CR propagation in the Galaxy.

The spallation cross sections are one of the main ingredients in CR
propagation computations, since they are crucial for evaluating the fluxes of
different secondary CR species. However, due to the lack of experimental data
and to the large number of interaction channels involved in the production of
each isotope, the uncertainties on the cross sections are still quite large.
These uncertainties propagate to the predicted fluxes of secondary CRs, which
are therefore also uncertain. In order to reduce the impact of the lack of
data, we make use of parametrizations and extrapolations of the spallation
cross sections involved in the CR network [35]. This can be done due to the
fact that the energy dependence of the cross sections is generally better
constrained and most of the uncertainties in the parametrizations are mainly
dominated by their normalization, as recently quantified in ref. [36].

Recently, some authors have proposed to use other secondary-to-primary flux
ratios to cross check the validity of the cross sections parametrizations
[37], concluding that this cross check will help reducing the systematic
uncertainties related to the determination of the propagation parameters. In
this work, we have implemented two different and well known cross sections
parametrizations in a customised version111This version is public at
https://github.com/tospines/Customised-DRAGON2_beta/ [38] of the DRAGON2 CR
propagation code [39, 40]. We have then optimized the propagation parameters
to reproduce the AMS-02 experimental data on secondary-to-primary flux ratios
of B, Be and Li to C and O as well as the corresponding secondary-to-secondary
flux ratios. In this way we study how spallation cross sections
parametrizations affect the determination of the propagation parameters for
the different secondary CRs. This work is the follow-up of our recent paper
[41], where we investigated how the choice of cross sections affects the
predictions on the fluxes of the light secondary CRs B, Be and Li, and
demonstrated that a simultaneous fit of the fluxes of these secondary CRs can
be achieved above a few GeV/n taking into account the uncertainties in the
normalization of the cross sections used.

There are very few works involving the ratios of secondary Li and Be nuclei
(see [42, 43]). This is mainly due to the fact that experimental fluxes of Li
and Be were poorly measured before AMS-02 (for a comparison, see ref. [44]).
In addition, the uncertainties on the predicted fluxes of Li and Be [43] are
larger than those on the flux of B. In this work, two different strategies
have been adopted to reduce the systematic uncertainties related to the the
spallation cross sections parametrizations using combined analyses of the
secondary-to-primary flux ratios with a nuisance parameter to account for the
cross sections normalization uncertainties and the secondary-to-secondary flux
ratios between B, Be and Li.

This paper is organised as follows: in section 2 we illustrate the details of
the Markov chain Monte Carlo (MCMC) algorithm and the analyses performed 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). Then, in section 3, we report and discuss the results on the
determination of the diffusion parameters using what we call “standard
analyses” for the different diffusion coefficient and cross sections
parametrizations adopted, taking into account the uncertainties related to the
normalization of spallation cross sections for Be, B and Li production by
means of the use of nuisance parameters. In section 3.2, we employ another
approach that allows us to compare the diffusion parameters found for
different secondary CRs after tuning the normalization of the cross sections
parametrizations in a similar way as explored in [41]. Finally, in sect. 4 we
discuss the main conclusions of this work.

##  2 Determination of the propagation parameters

A spectral break (the rigidity where the CR spectrum changes its slope) at
around $300\leavevmode\nobreak\ \hbox{$\,{\rm GV}$}$, has been observed by
many experiments in primary and secondary species [45, 1, 46, 47, 48].
Although a quantitative analysis of B/C data favours a scenario with a break
due to a diffusion mechanism (“diffusion hypothesis”) [49, 42], the scenario
of a break in the injection spectra (“source hypothesis”) or a combination of
both mechanisms (supported by the fact that different primary species point to
different break positions [50]) can not be discarded. In this work we have
therefore tested two well known parametrizations of the diffusion coefficient,
that differ in the physical interpretation of the high energy part of the CR
spectra, as explained above, and are given by the following equations:

$D=D_{0}\beta^{\eta}\left(\frac{R}{R_{0}}\right)^{\delta}\text{ \hskip
51.21504pt {Source hypothesis}}$ (2.1)
$D=D_{0}\beta^{\eta}\frac{\left(R/R_{0}\right)^{\delta}}{\left[1+\left(R/R_{b}\right)^{\Delta\delta/s}\right]^{s}}\text{
\hskip 19.91684pt {Diffusion hypothesis}}$ (2.2)

where the reference rigidity is set to $R_{0}=4\leavevmode\nobreak\
\hbox{$\,{\rm GV}$}$.

As in [41], where we refer to for full details about the set-up of the CR
propagation simulations, we solved the complete propagation equation using the
DRAGON2 code for a diffusion-reacceleration model, adopting a 2D ISM gas
distribution. In both diffusion models we use the values of the Alfvén
velocity (that accounts for CR reacceleration), $V_{A}$, the normalization of
the diffusion coefficient $D_{0}$, the spectral index, $\delta$, and the
$\beta$ (particle’s speed, in speed of light units) exponent, $\eta$, as the
parameters that we will include in our analyses, and we refer to them as
diffusion parameters. In equation 2.2 there are three additional parameters:
the rigidity break ($R_{b}$), the change in spectral index
($\Delta\delta=\delta-\delta_{h}$, where $\delta_{h}$ is the spectral index
for rigidities above the break), and a smoothing parameter $s$, used to allow
a soft transition around the hardening position. In our analyses, we fix these
parameters to the values found in ref. [49], which were determined from a
detailed analysis of the AMS-02 fluxes of protons and helium (with less
experimental uncertainties than heavier nuclei at high energies). These values
are: $\Delta\delta=0.14\pm 0.03$, $R_{b}=(312\pm 31)\leavevmode\nobreak\
\hbox{$\,{\rm GV}$}$ and $s=0.040\pm 0.0015$. The fact that these parameters
are fixed from the primary fluxes allows us to avoid degeneracies with the
evaluation of the main propagation parameters, leading to a more rigid
determination. We have observed that, varying these parameters within their
$1\sigma$ uncertainty values only affects the determination of the spectral
index, $\delta$, by a maximum variation of $\pm 1\%$, which should be taken as
an additional systematic uncertainty in the determination of $\delta$.

The mathematical model used to parametrize the injection spectra of primary
nuclei is a doubly broken power law, with a high-energy break at
$335\leavevmode\nobreak\ \hbox{$\,{\rm GeV}$}$ for protons and at
$200\leavevmode\nobreak\ \hbox{$\,{\rm GeV/n}$}$ for heavier elements, and a
low-energy break at $7\leavevmode\nobreak\ \hbox{$\,{\rm GeV/n}$}$ for all
primary CRs, when we treat the diffusion parameters in the source hypothesis,
whereas we use a simple broken power law when we treat the diffusion
hypothesis, with low-energy break in the injection spectra at
$7\leavevmode\nobreak\ \hbox{$\,{\rm GeV/n}$}$. Then, to account the solar
modulation effect, we make use of the Force-field approximation [51], which is
characterized by the value of the Fisk potential. We set the Fisk potential to
the value found in [41], corresponding to $\phi=0.61\leavevmode\nobreak\
\hbox{$\,{\rm GV}$}$ (obtained from the NEWK neutron monitor experiment
222http://www01.nmdb.eu/station/newk/ neutron monitor experiment - see [52,
53] \- in combination with Voyager-1 data [54]) for the period between May,
2011 to May, 2016, that corresponds to the period of collected B, Be and Li
data by the AMS-02 collaboration. The source distribution used in these
computations is that from [55] and the gas distribution function is taken from
[56]. We argue here that, while the source distribution does not seem to have
any impact on our results, the gas distribution used can have a sizeable
impact on the determination of the scale factors. In particular, we observed
changes in the secondary-to-secondary fluxes below 5% at energies above
$10\leavevmode\nobreak\ \hbox{$\,{\rm GeV/n}$}$ when using different gas
distributions, and these changes can be even larger at low energies .
Furthermore, the authors of ref. [57] report differences in the propagation
parameters derived in the case of 2D and 3D spatial distributions of the gas.

Basically, our strategy consists of an iterative procedure, which starts
taking a set of propagation parameters ($D_{0}$, $V_{A}$, $\eta$ and $\delta$)
as first guess; then, a fit of the ratios of Li, Be and B to C and O to the
AMS-02 data is performed to determine these parameters and their associated
uncertainties. Before the fit, the injection spectra are adjusted by fitting
the AMS-02 data for C, N, O, Ne, Mg and Si. If the guessed diffusion
parameters are very different (out of 1 $\sigma$ uncertainty) from those
predicted by the fit, the injection spectra are adjusted again with the new
predicted set of diffusion parameters and the fit is performed again to obtain
a more refined prediction on the diffusion parameters. Convergence is reached
when the output diffusion parameters after the last iteration are consistent
(within 1 $\sigma$ uncertainty) with those used in the previous iteration to
adjust the injection spectra. As expected, one-two iterations are needed,
since the secondary-to-primary flux ratios are highly independent of the
injection parameters. In this way, the injection spectra of primary CRs are
always adjusted to reproduce AMS-02 measurements . We remark that in this
analyses it is crucial the use of the current AMS-02 data, including the
recently published fluxes of Mg, Si and Ne [58], thus providing accurate
measurements of the fluxes of the main primary CR species producing B, Be and
Li [1, 46, 59, 60, 61, 47].

In order to perform the fit of the ratios, a Markov chain Monte Carlo (MCMC)
procedure, relying on Bayesian inference, has been implemented to get the
probability distribution functions for a set of diffusion parameters ($D_{0}$,
$V_{A}$, $\eta$ and $\delta$) to describe the data (in this case the AMS-02
data) and their confidence intervals. The technical details are presented in
the next section.

###  2.1 MCMC and minimization algorithm

Bayesian inference is used to get the posterior probability distribution
function (PDFs), $\mathscr{P}$, for a set of diffusion parameters to explain
the AMS-02 data. To evaluate it, the prior PDF, $\Pi$, and the likelihood,
$\mathscr{L}$, must be defined. These three terms are related by:

$\mathscr{P}(\vec{\theta}|\vec{D})\propto\mathscr{L}(\vec{D}|\vec{\theta})\Pi(\vec{\theta})$
(2.3)

where $\vec{\theta}=\\{\theta_{1},\theta_{2},...,\theta_{m}\\}$ is the set of
parameters, $\vec{D}$ is the data set used (CR flux ratios measured by AMS-02)
and $\mathscr{P}(\vec{\theta}|\vec{D})$ is the posterior PDF for the
parameters.

The posterior probability is calculated from equation 2.3 by a MCMC algorithm
which uses, instead of the classical Metropolis-Hastings algorithm, a modified
version of the Goodman & Weare algorithm [62]. This method is directly
implemented in the emcee module of Python (see ref. [63] for full technical
information).

| Parameters range
---|---
$D_{0}$ ($10^{28}cm^{2}/s$) | [5.5 , 9.5]
$v_{A}$ ($km/s$) | [0 , 40]
$\eta$ | [-3. , 0.]
$\delta$ | [0.33, 0.53]

Table 1: Parameter space considered in the analyses.

In this procedure a “grid” of simulations is built for $\sim 10000$ regularly
spaced combinations of parameters in order to interpolate them. The range of
parameters used to build this grid is shown in table 1. For each of these
combinations, we compute the flux ratios among different CR species and carry
out interpolations for other combinations of propagation parameters by using
the python tool RegularGridInterpolator from the Scipy module. In other words,
we create a sort of four-dimensional “matrix” (a dimension for each diffusion
parameter) and, to each of the points of the matrix, we associate a vector
containing the simulated flux ratios at different energies. The diffusion
equation is integrated in 171 points with equal spacing in a logarithmic
scale, from $10\leavevmode\nobreak\ \hbox{$\,{\rm MeV/n}$}$ to $\sim
100\leavevmode\nobreak\ \hbox{$\,{\rm TeV/n}$}$. The errors coming from this
interpolation were estimated comparing different simulated fluxes of different
nuclei with the interpolated fluxes, obtaining errors always much smaller than
1%. Errors in the interpolation of fluxes in different energies are completely
negligible.

The likelihood $\mathscr{L}$ is set to be a Gaussian function (as in many
other studies, e.g. [64] or [65]):

$\mathscr{L}(\vec{D}|\vec{\theta})=\prod_{i,j}\frac{1}{\sqrt{2\pi\sigma_{i,j}^{2}}}\exp\left[-\frac{(\Phi_{i,j}(\vec{\theta})-\Phi_{i,j,data})^{2}}{2\sigma^{2}_{i,j}}\right]$
(2.4)

where $\Phi_{i,j}(\vec{\theta})$ is the flux ratio computed in the simulation,
$\Phi_{i,j,data}$ is the corresponding flux ratio measured by AMS-02 and
$\sigma_{i,j}$ is its associated error for the $i$-th flux ratio and for the
$j$-th energy bin. In this approach we do not take into account possible
correlations among data, as the public data from the AMS-02 Collaboration do
not include the full covariance matrix [42, 66]. Therefore, the experimental
uncertainties are taken to be the sum in quadrature of statistical and
systematic errors reported by the AMS-02 Collaboration. The prior PDF is
defined as a uniform distribution for all the parameters:

$\Pi(\vec{\theta})=\left\\{\begin{array}[]{ll}\prod_{i}\cfrac{1}{\theta_{i,max}-\theta_{i,min}}&\text{for
$\theta_{i,min}<\theta_{i}<\theta_{i,max}$}\\\ 0&\text{elsewhere}\\\
\end{array}\right.$ (2.5)

Then, in order to take into account cross sections uncertainties, we make use
of a nuisance parameter for each of the secondary CRs analysed (i.e. B, Be and
Li), which enables a renormalization (or scaling) of the parametrization used
for their production cross sections. We argue that applying evenly this scale
factor to every isotopic species is a simplification, but given that there is
no isotopic data on CR spectra above $10\leavevmode\nobreak\ \hbox{$\,{\rm
GeV/n}$}$ this scaling can not be precisely applied for individual isotopes.
The application of this scaling on the cross sections parametrizations can
lead to a subtle change in the energy dependence of the low energy part
($\sim$ below $1\leavevmode\nobreak\ \hbox{$\,{\rm GeV/n}$}$) of the spectra
of B, Be and Li. We highlight that, even though our determination of the
propagation parameters focuses on AMS-02 data, our predictions on isotopic
fluxes seem to be also compatible with the isotopic data from the ACE/CRIS
experiment [67] (although at the level of $2\sigma$ for Li isotopes) and from
a balloon borne experiment [68], at energies around $100\leavevmode\nobreak\
\hbox{$\,{\rm MeV/n}$}$. The associated nuisance term is usually defined as
(eq. 4 of ref. [42]):

$\mathscr{N}_{X}=\sum_{i}\frac{\left(y_{i,X}-\hat{y}_{i,X}\right)^{2}}{\sigma_{i,X}^{2}}$
(2.6)

where $\hat{y}_{i,X}$ is the experimental cross section value at the $i$-th
point of energy and $\sigma_{i,X}$ its associated experimental uncertainty
referred to the secondary CR species $X$, that can be B, Be or Li. The term
$y_{i,X}$ is defined as $y_{i,X}\equiv\mathcal{S}_{X}\cdot\hat{y}_{i,X}$,
where $\mathcal{S}_{X}$ is the nuisance parameter to be adjusted, referred to
the species X, and independent of energy.

As the nuisance parameters are independent of energy, eq. 2.6 can be rewritten
as:

$\mathscr{N}_{X}=(\mathcal{S}_{X}-1)^{2}\sum_{i}\left(\frac{\hat{y}_{i,X}}{\sigma_{i,X}}\right)^{2}=(\mathcal{S}_{X}-1)^{2}N\left\langle(\hat{y}/\sigma)^{2}\right\rangle_{X}$
(2.7)

Here $N$ stands for the number of experimental data points and
$\langle\sigma/\hat{y}\rangle_{X}$ is the average relative error for a given
channel of production of $X$. Given the lack of cross sections experimental
data, we use the reaction channels with 12C and 16O as projectiles. Then, the
final expression for the nuisance term associated to a secondary CR species
$X$ is:

$\mathscr{N}_{X}=(\mathcal{S}_{X}-1)^{2}\sum_{X^{\prime}}(N_{C\rightarrow
X^{\prime}}\langle(\hat{y}/\sigma)^{-2}\rangle_{C\rightarrow
X^{\prime}}+N_{O\rightarrow
X^{\prime}}\langle(\hat{y}/\sigma)^{-2}\rangle_{O\rightarrow X^{\prime}})$
(2.8)

where $X^{\prime}$ stands for all possible isotopes of the CR species $X$ (for
example, for B it means the sum for the isotopes 10B and 11B). The nuisance
terms are therefore originated from Gaussian distributions with mean 1 and
sigma given by $\langle\sigma/\hat{y}\rangle$ (these relative uncertainties
are $>20\%$, as shown in ref. [41]), and can be interpreted as penalty terms
that prevent the scaling factors to be far from 1.

With this strategy we are including the normalization uncertainty of the cross
sections in the analysis, thus enabling the cross sections to be scaled up or
down, with each secondary being more penalized as the scaling associated to
its production cross section deviates from 1. The effect of scaling all the
production cross sections causes the fluxes to be scaled by the same factor.
Therefore the fluxes of the CR species $X$ in the likelihood will be scaled of
a factor $\mathcal{S}_{X}$. These changes in the inclusive cross sections of
B, Be and Li production result in small changes of the inelastic (often called
fragmentation) cross sections. We have checked that the impact of these
changes in the energy dependence of the flux ratios among secondary CRs is
negligible and we estimate that the uncertainty associated to this effect
should be of, at most, $1\%$. Inelastic cross sections are a subdominant term
in the propagation equation and the overall variation of the inelastic cross
sections obtained in this work is very small given that most of the inelastic
reactions produce He and protons.

Nevertheless, when studying each ratio independently, the cross sections
uncertainties are mostly absorbed by the normalization of the diffusion
coefficient, as also reported in ref. [42], which means that $\mathcal{S}_{X}$
will remain equal to 1 and no conclusions on these uncertainties can be
obtained. Combining the secondary-to-primary ratios of different species
allows accessing this information. In order to include the full correlation
between cross sections and CR fluxes, the secondary-over-secondary ratios are
included in this combined analysis. As we have seen, they are a perfect tool
to study cross sections parametrizations with very small uncertainty at high
energies. Therefore including these ratios in the analysis can make the
determination of the diffusion coefficient more robust, providing information
about the correlations between the secondary species. In this way, we use the
likelihood function defined using the secondary-over-secondary ratios above
$30\leavevmode\nobreak\ \hbox{$\,{\rm GeV}$}$, since, at these energies, their
spectra is mostly dependent on the ratio of the cross sections of production
of the secondary CRs involved.

In conclusion, the logarithm of the likelihood function used to carry out the
combined analysis of Li, B and Be is given by:

$\ln\mathscr{L}^{Total}=\sum^{Li,Be,B/(C,O,Li,Be,B)}_{F}\ln(\mathscr{L}(F))+\frac{1}{2}\sum^{B,Be,Li}_{X}\mathscr{N}_{X}$
(2.9)

where $F$ indicates the flux ratios (six secondary-over-primary ratios and
three secondary-over-secondary ratios) and $X$ indicates the cross sections
channels included (those coming from 12C and 16O projectiles only).

##  3 Results

The algorithm explained above has been applied to find the optimal diffusion
parameters that provide the best fit of the spectra of each of the ratios of
B, Be and Li to C and O (B/C, B/O, Be/C, Be/O, Li/C, Li/O) independently, for
both diffusion models (eqs. 2.1 and 2.2). In section 3.1, we report the
results of this analysis for each of flux ratios independently. Additionally,
in this section we report the results for a combined analysis of the spectra
of the secondary CRs B, Be and Li. In this analysis we combine the studied
secondary-to-primary flux ratios and the secondary-over-secondary ratios Be/B,
Li/B and Li/Be (at $E>30\leavevmode\nobreak\ \hbox{$\,{\rm GeV}$}$, in order
to avoid the uncertainties at low energy, largely studied in [41]) as well as
the nuisance parameters accounting for uncertainties related to the
normalization of the cross sections parametrizations.

Then, in section 3.2, we employ another analysis in order to consider, in a
different way, the uncertainties in the normalization of the cross sections
parametrizations and obtain the best-fit diffusion parameters from the
secondary-to-primary flux ratios. This strategy consist of re-scaling the
cross sections of production of B, Be and Li by reproducing the high energy
part of the spectra of the secondary-to-secondary flux ratios Be/B, Li/B and
Li/Be, provided that this part of their spectra is intimately related to their
cross sections and rather independent of the diffusion parameters used, as
discussed in [41]. After re-scaling the spallation cross sections used, we
apply the MCMC analysis to each of the secondary-to-primary ratios. In this
way, we can compare individually the parameters inferred from each secondary
CR after “correcting” their cross sections. Although similar analysis have
been presented in the past based on refitting the cross sections
parametrizations using updated cross sections measurements (see, e.g., ref.
[69]), the novelty here consists of renormalizing the cross sections
parametrizations from a combination of the high energy part of the secondary-
to-secondary spectra, taking the minimum possible rescaling which is able to
simultaneously reproduce these ratios. In fact, this strategy has been
successfully implemented for the study of antiprotons in ref. [70], where the
author shows that, considering the scale factors calculated in this way, the
antiproton-to-proton spectrum from the AMS-02 experiment can be reproduced
with a high precision.

These analyses are carried out using two different spallation cross sections
parametrizations: the DRAGON2 cross sections333Publicly available as ASCII
files at
https://raw.githubusercontent.com/cosmicrays/DRAGON2-Beta_version/master/data/crxsecs_fragmentation_Evoli2019_cumulative_modified.dat
[41, 40, 71] and the GALPROP 444Available as ASCII files at
https://dmaurin.gitlab.io/USINE/input_xs_data.html#nuclei-xs-nuclei [72, 73]
cross sections parametrizations. It must be stressed here that the halo size
values were determined in an independent way, from the ${}^{10}Be/^{9}Be$ flux
ratios (which are roughly independent of the propagation parameters), as it is
shown in section 5 of [41]. Interestingly, the halo size values obtained from
both parametrizations are very similar and compatible within $1\sigma$
statistical uncertainty: $6.76\leavevmode\nobreak\ \hbox{$\,{\rm kpc}$}$ and
$6.93\leavevmode\nobreak\ \hbox{$\,{\rm kpc}$}$ for the DRAGON2 and GALPROP
parametrizations, respectively.

###  3.1 Standard analysis

The results of the standard analyses are summarized in Figure 1. The posterior
PDF of each parameter is usually very close to a Gaussian distribution
function, although some of the PDFs obtained from the combined analyses of the
GALPROP cross sections with the source hypothesis of the diffusion coefficient
exhibit asymmetric tails.

Figure 1: Box-plots representing the PDFs resulting from the MCMC algorithm
used to find the best propagation parameters to describe AMS-02 results with
diffusion coefficient parametrizations of eq. 2.1, and with the DRAGON2 and
GALPROP cross sections. The orange solid lines within each box indicate the
medians of the corresponding PDFs, while the dashed cyan lines indicate the
mean values; the colored boxes correspond to the interquartile ranges (IQR)
from the $25$th (Q1) to the $75$th (Q3) percentile; finally, the black lines
correspond to the range $[Q1-1.5IQR,Q3+1.5IQR]$ which, in case of gaussian
PDFs contains the $99.3\%$ of the probability.

An explicit summary of the median values and of the ranges of propagation
parameters within the 68 and 95% of their PDFs is given in form of tables
(tables 3, 4 and 5) in the appendix A for each of the cross sections and
diffusion coefficient parametrizations. In addition, the corner plots with the
PDFs for the combined analyses are also shown in the appendix, for both the
source and diffusion hypotheses.

Very similar results (always within 2$\sigma$) are found for the propagation
parameters analyzed in both hypotheses for the CR hardening. In general, the
diffusion hypothesis favors smaller values of $D_{0}$, $V_{A}$ and $\eta$ and
larger values of $\delta$. This is something expected, since the slope
(controlled by the $\delta$ parameter) in the source hypothesis tends to be
smaller, to compensate the hardening observed in CR spectra at the highest
energies. Therefore, if $\delta$ is larger, $D_{0}$ must be smaller to balance
this change (see the contour plots in appendix A where the correlations
between parameters are explicitly shown). In the following, some important
remarks are discussed for these results.

####  3.1.1 Comparison among the trend of different ratios

In general, the relative uncertainties in the determination of propagation
parameters are larger in the $\eta$ and $V_{A}$ parameters, for the
independent analyses, mainly due to their degeneracy and the need of more data
points at the low energy region.

The propagation parameters obtained for each pair of ratios involving the same
secondary (e.g. B/C and B/O) are compatible with each other (although in the
case of Be, for the GALPROP parametrizations, we observe discrepancies up to
the level of $2\sigma$). Small discrepancies can arise, mainly due to the
production of secondary carbon, which is important at low energies.

Each of the ratios (also in the combined analysis and with both cross sections
sets used) favors a negative value of $\eta$, theoretically motivated by
dissipation of MHD waves [74] or appearance of non-resonant interactions at
low energies [75]. The Be ratios show clearly smaller values of this parameter
for every cross sections set used, but this seems to be due to the halo size
used in the simulations. The degeneracy between these two parameters makes one
compensate the defects of the other, thus requiring much lower $\eta$ values.

Interestingly, the value of $\delta$ is usually around 0.42-0.45 for the B and
Be ratios (although the prediction of the Be/C ratio from the GALPROP
parametrization is lower by at least $0.02$ units) while a lower value is
favored for the Li ratios (between 0.36 and 0.40). The fact that the Li
spectrum is harder at high energies (i.e. smaller $\delta$ value) than those
of other species was pointed out in ref. [76] and lead the authors to suggest
the presence of primary Li. Nevertheless, provided that the uncertainties
related to the spallation cross sections for Li production are specially
large, this is not clear at all (and a proof is the large difference in the
inferred $D_{0}$ value between both cross sections parametrizations for the Li
flux ratios).

The values of $V_{A}$ obtained are smaller than $30\leavevmode\nobreak\
\hbox{$\,{\rm km/s}$}$ for the B ratios ($\sim 26\leavevmode\nobreak\
\hbox{$\,{\rm km/s}$}$), while the Li ratios yields larger values (around
$35\leavevmode\nobreak\ \hbox{$\,{\rm km/s}$}$). On the other hand, the
$V_{A}$ values for Be ratios are much smaller, usually compatible with
$V_{A}=0$. This is due to the degeneracy with the halo size employed in the
simulations. Therefore, excluding the Be ratios, an average value of
$V_{A}\sim 30\leavevmode\nobreak\ \hbox{$\,{\rm km/s}$}$ is found, which seems
to be in agreement with the expected values (see refs. [77, 78, 79]).

Finally, as we see from Figure 2, the secondary-over-primary flux ratios are
perfectly reproduced when making each fit individually. This means that the
degrees of freedom available in the parametrizations of the diffusion
coefficient are enough to suitably match observations, although there is a
clear improvement on the predictions when using the break in the diffusion (eq
2.2). On the other hand, it should be mentioned that the level of uncertainty
appreciably changes when taking into account correlated errors from the AMS-02
experiment.

Figure 2: Predicted secondary-over-primary ratios with the propagation
parameters determined independently in the MCMC analysis for the DRAGON2 cross
sections parametrizations and for both hypothesis of the CR fluxes hardening.
The parametrizations of the diffusion coefficients allow perfect reproduction
of the shapes of the ratios. Fits with same quality are achieved for the
GALPROP cross sections too.

####  3.1.2 Comparison among different cross sections sets

Comparing both cross sections parametrizations by implementing a change in
their normalization is, in fact, a way to see if the energy dependence of
these parametrizations is compatible with the flux ratios of secondary CRs
simultaneously. In general, the different cross sections give very similar
predictions. One of the important differences are the $\eta$ and $V_{A}$
values obtained for the Be ratios with the DRAGON2 cross sections, which are
larger than those obtained with GALPROP cross sections. Remarkably, we also
observe an important difference between both parametrizations in the predicted
$\delta$ value obtained from the Be ratios. However, this might be also
related to the degeneracy with the halo size value used.

The most meaningful difference is that, as discussed above, for both cross
sections sets, the value of $\delta$ determined by the Li ratios is
significantly smaller (often more than $2\sigma$) than the one determined by
the other ratios. This points to the fact that, in both, parametrizations, the
overall slope of the cross sections for Li production at high energies tends
to be softer than it should be to be consistent with the $\delta$ values
obtained from the B and Be analyses. Nonetheless, the best-fit $\delta$ value
for the Li ratios with the DRAGON2 cross sections is closer to the value
inferred from the B ratios, which might indicate that updating cross sections
measurements can help to solve this issue.

Finally, it is worth noticing that the $D_{0}$ values obtained are slightly
different in the independent analyses of different secondary CRs, which mostly
depends on the normalization of the spallation cross sections parametrizations
in each channel. This is prevented by adding an overall scaling factor
($\mathcal{S}$) on the cross sections in the combined analysis. The relative
differences of $D_{0}$ obtained and the secondary-over-secondary ratios for
each of the cross sections sets reveal the relative degree of scaling on the
production cross sections needed for each of the secondary CRs. These scaling
factors, determined as nuisance parameters, are shown in Figure 3 and
discussed in detail down below and in section 3.2.

Cross sections scale factors

Figure 3: Corner plot of the scaling (nuisance) parameters determined by the
MCMC combined analysis. A dashed line indicating scaling = 1 is added as
reference for each of the panels.

Figure 4: 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.

Figure 5: Same as in Fig. 4 but with the GALPROP cross sections.

####  3.1.3 Comparing combined vs independent analyses of the ratios

The most remarkable fact seems to be that the values inferred from the
combined analysis tend to be closer to those inferred from the Li ratios,
leading to smaller $\delta$ values in both parametrizations and larger $V_{A}$
values for the GALPROP cross sections.This could imply that this analysis
compensates possible deviations in the adjustment of the energy dependence of
the cross sections of Li production with modifications of the diffusion
parameters. Only the DRAGON2 cross sections show a median value of $\delta$
higher than 0.40, but also in this case it is below the median for B and Be
ratios. This is likely related to the lack of experimental data on cross
sections of Li production (implying a poorer parametrization of the Li
production cross sections) and may cause an important bias in our
determination of the propagation parameters (and also in the determination of
the nuisance parameters) in the combined analysis. In general, it seems that
the parameters found in the combined and independent analyses are compatible
for both parametrizations. While the best-fit parameters found in the
independent analyses are similar for both cross sections parametrizations, the
predicted best-fit value for $\delta$ significantly differ in the combined
analyses of both parametrizations.

The results of the combined analysis are shown in Figs. 4 and 5 for the
DRAGON2 and GALPROP cross sections, respectively. As we see, the results of
the combined analysis with the DRAGON2 cross sections show an almost perfect
fit of all the ratios, with the largest discrepancies for the Be ratios, which
could be due to the halo size used. In turn, with the GALPROP cross sections
the predictions are noticeably discrepant for Be and B ratios at low energies,
while only the Li ratios are well reproduced within experimental errors. This
could be due to possible deviations in the adjustment of the shape describing
the cross sections, as commented above, but other authors have solved this
discrepancy by adding an extra source of lithium (related to Li production
from novae) [76].

From Figure 3 we see for the GALPROP cross sections that, while the scaling
factors for Li and Be are the largest ($\sim$26 and $10\%$ respectively), the
scaling for B is the smallest ($1-2\%$ up) and it is even compatible with no
scaling. However, the DRAGON2 cross sections parametrizations determine a
scaling for B, Be and Li of $5\%$, $1\%$ and $3\%$ ($\pm$1%), respectively.

To make a rough estimation of the total uncertainty in the determination of
these scaling factors we must consider that they are mostly fixed by the high
energy part of the secondary-over-secondary ratios. Thus, a $\sim 3$%
uncertainty, due to the inelastic (fragmentation) cross sections, is
reasonable for the flux of secondary CRs at high energies (see [80]). Moreover
the uncertainties in the high energy part of these ratios, associated to the
determination of the diffusion parameters, are $\sim(3-4)\%$, as shown in
[41], due to the effect of multi-step spallation reactions. This makes the
total uncertainties in the determination of the scaling factors to be at least
of $\pm 5\%$ (assuming to be uncorrelated, i.e. summed in quadrature). On top
of this, there is some additional uncertainty associated to the energy
dependence of the cross sections parametrizations, which can affect the
determination of the scaling factors by an additional $\sim\pm(3-5)$%.
Therefore, we estimate that the uncertainty in the determination of the
scaling factors can be as high as $\sim\pm 7-8$%, which is well below half of
the average experimental uncertainty associated to the cross sections
measurements [41]. Furthermore, we remark that using a different gas
distribution can affect the determination of these scaling factors, since up
to a $5\%$ change has been observed in the secondary-to-secondary flux ratios
above a few GeV/n.

Figure 6: Secondary-over-secondary flux ratios of Li, Be and B computed using
the propagation parameters and scaling factors found in the combined analysis
for each of the cross sections sets compared to the AMS-02 experimental data.
The grid lines at the level of $5\%$ residuals are highlighted in the lower
panel in a different color in order to guide the eye.

In Figure 6 we show a comparison of the predicted secondary-to-secondary
spectra of Li, Be and B with the AMS-02 data using the propagation parameters
and scaling factors obtained in the combined analysis for each of the cross
sections sets. As we see, the secondary-over-secondary ratios predicted with
the DRAGON2 cross sections are consistent with data within the level of $5\%$
(highlighted residual grid lines). The Li/Be and Be/B ratios are the ones with
slightly larger discrepancies with respect to experimental data. In the case
of the GALPROP parametrizations, the discrepancies at low energy are larger
(often above $5\%$ at low energy) and the energy dependence of the predicted
Be/B ratio seems to be slightly different to that shown by the experimental
AMS-02 Be/B ratio. We remind that these discrepancies could change when
evaluating the secondary-to-secondary flux ratios using a different gas
distribution.

In order to investigate how important this bias is in the determination of the
propagation parameters, we explore, in the next section, a different procedure
that could allow us to obtain a more reliable set of propagation parameters
after correcting their production cross sections.

###  3.2 “Scaled” analysis

To prevent the propagation parameters from being biased, we have implemented
an alternative strategy. This procedure allows us to use the propagation
parameters extracted from the B ratios, since the amount of data for the cross
sections of B production allows a better parametrization of these channels.

| $\mathcal{S}_{B}$ | $\mathcal{S}_{Be}$ | $\mathcal{S}_{Li}$
---|---|---|---
| Fit | Comb | Fit | Comb | Fit | Comb
DRAGON2 | 1.04 | 1.05 | 0.975 | 0.985 | 0.945 | 0.97
GALPROP | 0.94 | 1.01 | 0.865 | 0.895 | 1.18 | 1.26

Table 2: Comparison between the scaling factors inferred in the combined
analyses (Comb) and in the fit of the secondary-over-secondary ratios measured
by AMS-02 (Fit) for the cross sections sets used. These scaling factors are
the same for both diffusion hypotheses

used. The $1\sigma$ statistical errors on this determination are $\sim 1\%$.

First, we fit the AMS-02 secondary-over-secondary ratios by adjusting the
scaling factors of the B, Be and Li production cross sections, including also
the nuisance terms (eq. 2.8), and using the diffusion parameters obtained in
the B/O analysis of each cross sections set (see Figure 1). We use the
diffusion parameters inferred from this ratio as reference since we expect
that the prediction of the flux ratios of B to be more accurate than for the
other secondary CRs. In fact B is mainly produced from interactions of C and O
projectiles, and the interactions channels of B production are those with more
cross sections measurements available. In addition, the fraction of secondary
oxygen is negligible, so that the oxygen nuclei entering in this ratio are
pure primaries in the full energy range (see ref. [36]). Then, we apply the
same independent MCMC analysis to each of the ratios and compare the
propagation parameters.The idea behind the first step is that all possible
scalings that fit the AMS-02 secondary-over-secondary ratios and lay inside
cross sections data uncertainties are allowed candidates, but the preferred
one is that combination of scaling factors which involves the minimum change
from the original parametrization. In order to carry out this analysis we use
again a MCMC procedure, in which the free parameters are the scaling factors,
$\mathcal{S}_{B}$, $\mathcal{S}_{Be}$ and $\mathcal{S}_{Li}$, and the data to
be fitted are just the secondary-over-secondary ratios above
$20\leavevmode\nobreak\ \hbox{$\,{\rm GeV/n}$}$, with a penalty factor
proportional to the nuisance term of eq. 2.8. This will avoid biasing the
scaling factors obtained by the cross sections parametrizations when combining
the different species.

In Table 2, a comparison among the scaling factors obtained with both analysis
approaches is shown. As we see, while the scaling factors from both procedures
are quite similar for the DRAGON2 cross sections, they are considerably
different for the GALPROP cross sections. This indicates that the bias due to
the shape of the cross sections can be relevant. In fact, these differences
are higher than the $\pm$4% uncertainties related to variations in the
diffusion coefficient that we assume, although they are lower than $\pm$8%. We
stress the fact that scaling factors for the DRAGON2 cross sections are
<$5.5\%$ and that, as expected, the B scaling is the closest to 1 in the
GALPROP parametrization.

Figure 7: Secondary-over-secondary flux ratios predicted by the scaled DRAGON2
and GALPROP cross sections parametrizations compared to the AMS-02 data. They
are fitted to reproduce the high energy part of the spectra, while the low
energy part of the Li/Be and Be/B ratios mainly depends on the halo size value
used.

These scaled secondary-over-secondary flux ratios (figure 7) are expected to
show larger discrepancies at low energies, while the high energy part should
be well reproduced. Then, with these scaling factors, the independent analysis
of each of the ratios is performed for the DRAGON2 and GALPROP cross sections
and the results are shown in Figure 8. Again, the results are summarized in
the tables 6 and 7 in the appendix A.

Figure 8: Box-plots representation of the diffusion parameters obtained in the
independent analyses of the Li,Be,B/C,O AMS-02 ratios for the scaled GALPROP
and DRAGON2 cross sections for both hypothesis on the diffusion coefficient.
The meaning of the lines and of the boxes is the same as in Fig. 1.

It is worth remarking here that analyzing the ratios after scaling the cross
sections does not lead to just a change in the predicted normalization of the
diffusion coefficient ($D_{0}$), but also to slight changes in the slope
($\delta$) and in the low-energy parameters (although never by more than
2$\sigma$). As we can see, the GALPROP cross sections predict different
propagation parameters for each secondary CR species, meaning that the
analysis must be performed including more degrees of freedom for the shape of
their cross sections energy spectrum. This is probably related to the fact
that the Be and Li scaling factors deviate more than $10\%$, since the cross
sections at low energy are always better constrained than at high energy. As
expected, the diffusion parameters inferred from the ratios of B are the most
reliable. On the other hand, we see that for the DRAGON2 parametrizations,
both the $D_{0}$ and $\delta$ values are nearly identical for the B and Be
ratios, while they are significantly different for the Li ratios. From these
results, we can see that a $\delta$ value of $0.43-0.45$ is favored when
excluding the Li predictions, and this value goes down to an average of
$\sim$0.41 when including it.

##  4 Conclusions

In this work, we have studied the two parametrizations of the diffusion
coefficient that best-fit with the theoretical expectations and are suitable
to reproduce the experimental data for the two most accurate cross sections
parametrizations (namely DRAGON2 and GALPROP cross sections). We have
presented the MCMC analyses of the secondary-over-primary and secondary-over-
secondary flux ratios, for the Li, Be and B, in order to determine the
propagation parameters that best reproduce AMS-02 data for the two cross
sections sets and both diffusion hypotheses (with and without high energy
break). This combination of different secondary CRs provides a way to mitigate
the effect of cross sections uncertainties in the determination of the
propagation parameters, which are mostly dominated by their normalization in
the cross sections parametrizations.

In general, we see that the propagation parameters obtained in the independent
analysis are very similar for the cross sections studied, favoring a
$\delta\sim 0.37-0.45$, specially hard for Li ratios. The combined analysis
tends to $\delta\sim 0.41-0.44$ for the DRAGON2 cross sections, while
$\delta=0.37-0.40$ for the GALPROP cross sections. In addition, we have
demonstrated that the secondary-over-primary ratios of B, Be and Li to C and O
can be simultaneously reproduced with low discrepancies including a
renormalization of the production cross sections. We underline that the
DRAGON2 cross sections yield an almost perfect fit of all these ratios with
respect to the AMS-02 and Voyager-1 data at the same time with very low
corrections on the normalization of the B, Be and Li production cross
sections. On the other hand, the GALPROP cross sections lead to large
discrepancies that are due to the shape of the cross sections energy spectra.
Nevertheless, the propagation parameters determined in this combined analysis
could be biased by deviations in the adjustment of the energy dependence of
the cross sections parametrizations from the different secondary CRs, as it
may be the case of the combined analysis with the GALPROP parametrizations,
specially favoring the values found from the Li ratios.

In order to investigate and prevent this bias, which might be particularly
important for parametrizations based on experimental cross sections data,
because of the uncertainty on the cross sections of Li production, an
alternative procedure has been explored to determine the diffusion parameters
differently. After applying the analysis presented in section 3.2, for the
GALPROP and DRAGON2 cross sections, we have determined that the shape of the
energy spectrum for the cross sections of Li production needs to be slightly
adjusted in order to get the same prediction for the diffusion parameters in
all the ratios. For the DRAGON2 spallation cross sections, the inferred
diffusion parameters are compatible within $1\sigma$ for the diffusion
hypothesis of the diffusion coefficient for the Be and B with $\delta\sim
0.43-0.45$. In turn, when using the GALPROP cross sections, we infer more
disperse values of $\delta\sim 0.37-0.46$. It seems that a change on the shape
of the GALPROP cross sections of Be and Li production is needed, which is
probably related to the fact that both the Li and Be cross sections energy
spectra need scaling factors deviating of more than $10\%$ to reproduce the
secondary-over-secondary ratios. When excluding the predictions from the Li
ratios, both the DRAGON2 and GALPROP cross sections favor a larger $\delta$
value, of around $0.44$ for the DRAGON2 parametrizations and $\sim 0.42$ for
the GALPROP ones. Isotopic cosmic-ray data will be a valuable tool to adjust
the different spallation cross section parametrizations in the various
isotopic channels.

## Acknowledgments

We remark the crucial help of Daniele Gaggero in the development of the paper
and his examinations during the evolution of the ideas commented here.

Many thanks to the instituto de física teórica (IFT) in Madrid for hosting
Pedro De la Torre for a long stay there and specially to the DAMASCO (DArk
MAtter AStroparticles and COsmology) group for their support and valuable
conversations related to this work.

This work has been 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.

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##  Appendix A Summary of the MCMC results

In this appendix, the corner plots obtained from the MCMC algorithm of the
combined analysis for each cross sections set and both diffusion
parametrizations used (’Source’ and ’Diffusion’ hypotheses) are shown (figs. 9
and 10) to visualize the probability distribution functions (PDFs) of each
propagation parameter and the correlations between them. These corner plots
contain the diffusion parameters ($D_{0}$, $\delta$, $\eta$ and $V_{A}$) as
well as the scaling factors included as nuisance parameters. Furthermore, we
are summarizing the results for the PDFs of each of the propagation parameters
found in form of tables. First, we show the results for the combined analysis,
then, the parameters found in the independent analyses of each secondary-to-
primary ratio (B/C, B/O, Be/C, Be/O, Li/C, Li/O) and, finally, the analysis of
each of the ratios after scaling the original cross sections according to the
secondary-over-secondary ratios. The tables 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 region.

Propagation parameters PDF - Source hypothesis

Figure 9: Corner plot representing the PDF of every parameter included in the
combined MCMC analysis with the diffusion coefficient parametrization of eq.
2.1 for the DRAGON2 and GALPROP cross sections parametrizations. The inner
panels represent the correlations between parameters with the 1 and 2 $\sigma$
surface probabilities highlighted.

Propagation parameters PDF - Diffusion hypothesis

Figure 10: Corner plot as in figure 9 but for the diffusion coefficient
parametrization of eq. 2.2.

Combined analysis

| Diffusion hypothesis | Source hypothesis
---|---|---
| DRAGON2 | GALPROP | DRAGON2 | GALPROP
$\frac{D_{0}\,(10^{28}\leavevmode\nobreak\ \hbox{$\,{\rm cm^{2}/s}$})}{H\,(kpc)}$ | 1.11 $\pm$ 0.02 | 1.14 $\pm$ 0.03 | 1.16 $\pm$ 0.02 | 1.26 $\pm$ 0.03
[1.06, 1.30] | [1.08, 1.30] | [1.11, 1.15] | [1.21, 1.32]
$V_{A}$ (km/s) | 26.92 $\pm$ 1.28 | 33.32 $\pm$ 1.23 | 30.20 $\pm$ 1.28 | 38.21 $\pm$ 1.11
[24.52, 29.48] | [31.03, 35.78] | [27.80, 32.17] | [35.99, 40.01]
$\eta$ | -0.75 $\pm$ 0.06 | -0.80 $\pm$ 0.07 | -0.58 $\pm$ 0.06 | -0.44 $\pm$ 0.7
[-0.87, -0.65] | [-0.92, -0.66] | [-0.68, -0.46] | [-0.58, -0.36]
$\delta$ | 0.42 $\pm$ 0.01 | 0.40 $\pm$ 0.01 | 0.41 $\pm$ 0.01 | 0.37 $\pm$ 0.01
[0.39, 0.44] | [0.38, 0.42] | [0.38, 0.43] | [0.35, 0.39]

Table 3: Results obtained for the combined ratios analysis for each cross
sections sets and diffusion coefficient parametrizations. Maximum posterior
probability value with its standard deviation (assuming the PDF to be a
Gaussian) and the range of parameters within $95\%$ of confidence level are
shown for each parameter.

Independent analyses - Source hypothesis

DRAGON2 parameters - 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}$})}{H\,(kpc)}$ | 1.02 $\pm$ 0.05 | 1.04$\pm$ 0.05 | 1.09 $\pm$ 0.05 | 1.06 $\pm$ 0.07 | 1.37 $\pm$ 0.05 | 1.33 $\pm$ 0.04
[0.94., 1.12] | [0.95, 1.14] | [1.03, 1.19] | [0.96, 1.19] | [1.27, 1.45] | [1.25, 1.39]
$V_{A}$ (km/s) | 25.72 $\pm$ 3.22 | 20.65 $\pm$ 2.65 | 11.74 $\pm$ 7.24 | 29.65 $\pm$ 7.19 | 36.49 $\pm$ 2.68 | 38.12 $\pm$ 1.66
[19.72, 32.16.] | [24.85, 34.95] | [0., 26.26] | [15.27, 40.75] | [31.07, 40.31] | [34.9, 40.66]
$\eta$ | -0.66 $\pm$ 0.12 | -0.51 $\pm$ 0.11 | -1.15 $\pm$ 0.16 | -0.94 $\pm$ 0.19 | -0.37 $\pm$ 0.11 | -0.29 $\pm$ 0.08
[-0.80, -0.42] | [-0.72, -0.33] | [-1.47, -0.83] | [-1.32, -0.6] | [-0.59, -0.15] | [-0.45, -0.15]
$\delta$ | 0.43 $\pm$ 0.01 | 0.41 $\pm$ 0.01 | 0.44 $\pm$ 0.01 | 0.43 $\pm$ 0.02 | 0.39 $\pm$ 0.01 | 0.38 $\pm$ 0.01
[0.41, 0.45] | [0.39, 0.43] | [0.41, 0.46] | [0.39, 0.47] | [0.37, 0.41] | [0.36, 0.40]

GALPROP parameters - 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}$})}{H\,(kpc)}$ | 0.99 $\pm$ 0.04 | 0.98 $\pm$ 0.04 | 1.26 $\pm$ 0.03 | 1.10 $\pm$ 0.05 | 0.98 $\pm$ 0.05 | 1.00 $\pm$ 0.05
[0.92, 1.07] | [0.92, 1.07] | [1.22, 1.32] | [1.05, 1.15] | [0.88, 1.08] | [0.91, 1.07]
$V_{A}$ (km/s) | 27.17 $\pm$ 2.61 | 29.11 $\pm$ 2.14 | 6.31 $\pm$ 7.20 | 10.50 $\pm$ 7.48 | 32.37 $\pm$ 2.04 | 32.40 $\pm$ 1.66
[21.95, 32.25] | [25.07, 33.39] | [0., 15.05] | [0., 25.46] | [28.65, 36.45] | [29.28, 35.04]
$\eta$ | -0.54 $\pm$ 0.12 | -0.29 $\pm$ 0.09 | -1.56 $\pm$ 0.19 | -1.15 $\pm$ 0.19 | -0.6 $\pm$ 0.20 | -0.32 $\pm$ 0.14
[-0.78, -0.30] | [-0.45, -0.11] | [-1.94, -1.18] | [-1.53, -0.78] | [-1.0, -0.44] | [-0.60, -0.10]
$\delta$ | 0.43 $\pm$ 0.01 | 0.42 $\pm$ 0.01 | 0.39 $\pm$ 0.01 | 0.42 $\pm$ 0.01 | 0.38 $\pm$ 0.01 | 0.37 $\pm$ 0.01
[0.41, 0.45] | [0.40, 0.45] | [0.37, 0.41] | [0.40, 0.44] | [0.36, 0.40] | [0.36, 0.39]

Table 4: 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 parametrization
of eq. 2.1 for each of the cross sections sets analysed.

Independent analyses - Diffusion hypothesis

DRAGON2 parameters - 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}$})}{H\,(kpc)}$ | 0.97 $\pm$ 0.04 | 0.98$\pm$ 0.04 | 1.07 $\pm$ 0.03 | 1.00 $\pm$ 0.05 | 1.30 $\pm$ 0.04 | 1.28 $\pm$ 0.05
[0.90., 1.05] | [0.91, 1.06] | [1.03, 1.14] | [0.92, 1.11] | [1.21, 1.37] | [1.18, 1.36]
$V_{A}$ (km/s) | 22.02 $\pm$ 3.54 | 26.53 $\pm$ 2.54 | 7.60 $\pm$ 6.70 | 14.19 $\pm$ 8.76 | 33.31 $\pm$ 2.72 | 36.60 $\pm$ 2.47
[14.94, 28.86.] | [21.45, 31.43] | [0., 20.00] | [0, 29.27] | [27.87, 37.49] | [31.64, 40.00]
$\eta$ | -0.78 $\pm$ 0.13 | -0.65 $\pm$ 0.10 | -1.30 $\pm$ 0.16 | -1.16 $\pm$ 0.21 | -0.54 $\pm$ 0.12 | -0.38 $\pm$ 0.11
[-1.04, -0.54] | [-0.75, -0.55] | [-1.60, -0.98] | [-1.58, -0.74] | [-0.78, -0.36] | [-0.60, -0.22]
$\delta$ | 0.44 $\pm$ 0.01 | 0.43 $\pm$ 0.01 | 0.45 $\pm$ 0.01 | 0.45 $\pm$ 0.02 | 0.41 $\pm$ 0.01 | 0.39 $\pm$ 0.01
[0.42, 0.46] | [0.41, 0.45] | [0.43, 0.47] | [0.41, 0.47] | [0.39, 0.43] | [0.37, 0.41]

GALPROP parameters - 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}$})}{H\,(kpc)}$ | 0.94 $\pm$ 0.04 | 0.94 $\pm$ 0.03 | 1.24 $\pm$ 0.02 | 1.08 $\pm$ 0.03 | 0.89 $\pm$ 0.04 | 0.91 $\pm$ 0.03
[0.87, 1.04] | [0.88, 1.00] | [1.21, 1.28] | [1.04, 1.15] | [0.81, 0.97] | [0.85, 0.96]
$V_{A}$ (km/s) | 24.44 $\pm$ 2.97 | 27.08 $\pm$ 1.91 | 4.74 $\pm$ 4.99 | 7.64 $\pm$ 7.23 | 29.11 $\pm$ 2.20 | 29.32 $\pm$ 1.52
[18.90, 29.54] | [23.26, 30.86] | [0., 14.72] | [0., 22.10] | [24.71, 33.21] | [26.28, 31.76]
$\eta$ | -0.66 $\pm$ 0.12 | -0.39 $\pm$ 0.09 | -1.68 $\pm$ 0.16 | -1.27 $\pm$ 0.19 | -0.94 $\pm$ 0.20 | -0.61 $\pm$ 0.11
[-0.90, -0.18] | [-0.57, -0.23] | [-1.98, -1.36] | [-1.65, -0.91] | [-1.34, -0.62] | [-0.83, -0.41]
$\delta$ | 0.45 $\pm$ 0.01 | 0.44 $\pm$ 0.01 | 0.40 $\pm$ 0.01 | 0.42 $\pm$ 0.01 | 0.40 $\pm$ 0.01 | 0.39 $\pm$ 0.01
[0.43, 0.47] | [0.42, 0.46] | [0.38, 0.43] | [0.40, 0.44] | [0.37, 0.42] | [0.37, 0.41]

Table 5: Same as in 4 but for the diffusion coefficient parametrization of eq.
2.2.

Independent scaled analyses - Source hypothesis

DRAGON2 parameters - 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}$})}{H\,(kpc)}$ | 1.09 $\pm$ 0.05 | 1.09$\pm$ 0.05 | 1.06 $\pm$ 0.02 | 1.04 $\pm$ 0.06 | 1.30 $\pm$ 0.05 | 1.28 $\pm$ 0.05
[1.0, 1.19] | [1.01, 1.19] | [1.01, 1.16] | [0.94, 1.16] | [1.20, 1.40] | [1.18, 1.35]
$V_{A}$ (km/s) | 26.58 $\pm$ 3.54 | 30.72 $\pm$ 2.78 | 11.80 $\pm$ 7.05 | 21.06 $\pm$ 6.06 | 35.81 $\pm$ 2.55 | 37.63 $\pm$ 1.82
[18.5, 33.16] | [25.16, 35.58] | [0., 25.48] | [8.96, 30.15] | [30.71, 40.01] | [32.99, 40.50]
$\eta$ | -0.71 $\pm$ 0.12 | -0.55 $\pm$ 0.11 | -1.17 $\pm$ 0.18 | -0.94 $\pm$ 0.18 | -0.32 $\pm$ 0.11 | -0.21 $\pm$ 0.09
[-0.95, -0.57] | [-0.77, -0.35] | [-1.24, -0.81] | [-1.28, -0.58] | [-0.54, -0.10] | [-0.41, -0.03]
$\delta$ | 0.42 $\pm$ 0.01 | 0.41 $\pm$ 0.01 | 0.44 $\pm$ 0.01 | 0.43 $\pm$ 0.02 | 0.39 $\pm$ 0.01 | 0.38 $\pm$ 0.01
[0.40, 0.44] | [0.39, 0.43] | [0.41, 0.45] | [0.39, 0.46] | [0.37, 0.41] | [0.36, 0.41]

GALPROP parameters - 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}$})}{H\,(kpc)}$ | 0.89 $\pm$ 0.04 | 0.93$\pm$ 0.04 | 0.90 $\pm$ 0.02 | 0.93 $\pm$ 0.03 | 1.14 $\pm$ 0.04 | 1.11 $\pm$ 0.03
[0.81, 0.98] | [0.86, 0.94] | [1.01, 1.08] | [0.87, 1.01] | [1.06, 1.22] | [1.05, 1.17]
$V_{A}$ (km/s) | 25.56 $\pm$ 2.80 | 27.09 $\pm$ 2.16 | 5.50 $\pm$ 5.33 | 6.06 $\pm$ 6.28 | 35.61 $\pm$ 1.71 | 34.94 $\pm$ 1.30
[19.96, 30.70.] | [22.93, 31.41] | [0., 16.16] | [0, 18.72] | [32.19, 37.17] | [32.34, 37.0]
$\eta$ | -0.90 $\pm$ 0.14 | -0.60 $\pm$ 0.11 | -2.69 $\pm$ 0.20 | -2.17 $\pm$ 0.25 | -0.17 $\pm$ 0.11 | -0.10 $\pm$ 0.08
[-1.17, -0.62] | [-0.82, -0.38] | [-2.99, -2.29] | [-2.58, -1.67] | [-0.39, +0.05] | [-0.26, +0.04]
$\delta$ | 0.44 $\pm$ 0.01 | 0.43 $\pm$ 0.01 | 0.39 $\pm$ 0.01 | 0.42 $\pm$ 0.02 | 0.38 $\pm$ 0.01 | 0.38 $\pm$ 0.01
[0.41, 0.46] | [0.41, 0.45] | [0.37, 0.41] | [0.40, 0.44] | [0.36, 0.41] | [0.36, 0.41]

Table 6: 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 parametrization
of eq. 2.1 with the scaled cross sections sets analysed.

Independent scaled analyses - Diffusion hypothesis

DRAGON2 parameters - 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}$})}{H\,(kpc)}$ | 1.03 $\pm$ 0.04 | 1.03$\pm$ 0.04 | 1.04 $\pm$ 0.03 | 0.98 $\pm$ 0.05 | 1.22 $\pm$ 0.04 | 1.22 $\pm$ 0.04
[0.95, 1.11] | [0.95, 1.11] | [1.0, 1.11] | [0.92, 1.09] | [1.14, 1.30] | [1.14, 1.30]
$V_{A}$ (km/s) | 22.20 $\pm$ 4.29 | 27.25 $\pm$ 2.76 | 8.60 $\pm$ 6.77 | 13.84 $\pm$ 8.22 | 32.61 $\pm$ 2.66 | 35.36 $\pm$ 2.12
[13.62, 27.32] | [21.73, 32.47] | [0., 22.14] | [0, 29.27] | [27.29, 37.83] | [31.08, 38.56]
$\eta$ | -0.84 $\pm$ 0.13 | -0.69 $\pm$ 0.10 | -1.29 $\pm$ 0.16 | -1.16 $\pm$ 0.20 | -0.48 $\pm$ 0.13 | -0.37 $\pm$ 0.09
[-1.10, -0.6] | [-0.89, -0.49] | [-1.61, -0.97] | [-1.36, -0.78] | [-0.74, -0.26] | [-0.75, -0.19]
$\delta$ | 0.44 $\pm$ 0.01 | 0.43 $\pm$ 0.01 | 0.45 $\pm$ 0.01 | 0.45 $\pm$ 0.02 | 0.40 $\pm$ 0.01 | 0.39 $\pm$ 0.01
[0.42, 0.46] | [0.41, 0.45] | [0.43, 0.47] | [0.42, 0.47] | [0.39, 0.42] | [0.36, 0.41]

GALPROP parameters - 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}$})}{H\,(kpc)}$ | 0.83 $\pm$ 0.04 | 0.85 $\pm$ 0.04 | 1.02 $\pm$ 0.03 | 0.92 $\pm$ 0.03 | 1.08 $\pm$ 0.03 | 1.07 $\pm$ 0.03
[0.78, 0.92] | [0.79, 0.92] | [0.98, 1.07] | [0.88, 0.96] | [1.02, 1.14] | [1.01, 1.12]
$V_{A}$ (km/s) | 21.95 $\pm$ 3.14 | 24.82 $\pm$ 2.31 | 5.14 $\pm$ 4.83 | 4.91 $\pm$ 5.46 | 33.77 $\pm$ 1.85 | 33.67 $\pm$ 1.39
[15.75, 28.23] | [19.20, 29.04] | [0., 14.80] | [0., 15.83] | [30.07, 36.73] | [30.89, 36.15]
$\eta$ | -1.09 $\pm$ 0.16 | -0.73 $\pm$ 0.11 | -2.70 $\pm$ 0.17 | -2.31 $\pm$ 0.21 | -0.32 $\pm$ 0.10 | -0.21 $\pm$ 0.07
[-1.44, -0.79] | [-0.57, -0.23] | [-2.96, -2.36] | [-2.69, -1.90] | [-0.52, -0.12] | [-0.35, -0.07]
$\delta$ | 0.46 $\pm$ 0.01 | 0.44 $\pm$ 0.01 | 0.40 $\pm$ 0.01 | 0.42 $\pm$ 0.01 | 0.40 $\pm$ 0.01 | 0.39 $\pm$ 0.01
[0.44, 0.48] | [0.42, 0.47] | [0.38, 0.41] | [0.40, 0.43] | [0.38, 0.42] | [0.37, 0.40]

Table 7: Same as in 6 but for the diffusion coefficient parametrization of eq.
2.2.

Finally, here we report tables with the $\chi^{2}$ values obtained with the
best fits parameters of the independent and combined analyses for the GALPROP
and DRAGON2 cross sections. As previously reported, the inclusion of a break
in the injection improves the quality for all the simulated ratios. By
comparing the $\chi^{2}$ values for both hypotheses in the independent
analysis, we observe that both cross sections yield similarly good fits.
Interestingly, the maximum $\Delta\chi^{2}$ between both cross sections is
observed for the Be ratios. This could be due to the halo size value used for
each parametrization but it could also be related to the cross sections of Be
production by themselves.

Besides, we observe that the difference of $\chi^{2}$ values between the
source and diffusion hypotheses in the independent analyses is of
$\Delta\chi^{2}\sim 10$ for the ratios of Li and B (roughly unaffected by the
halo size value), both for DRAGON2 and GALPROP. Applying Wilk’s theorem we can
translate this difference in $\chi^{2}$ values to significance, finding it to
be of around $4.7\sigma$.

Finally, from these tables we show that the combined analyses yield better
$\chi^{2}$ values for the DRAGON2 cross sections than for the GALPROP ones as
displayed in 4 and 5.

$\chi^{2}$ values of best-fit parameters $-$ GALPROP
---
| B/C | B/O | Be/C | Be/O | Li/C | Li/O
Source hypothesis Indep. analysis | 41.79 | 37.22 | 44.94 | 43.6 | 40.27 | 36.82
Diffusion hypothesis Indep. analysis | 31.87 | 26.42 | 47.08 | 43.03 | 32.42 | 27.64
Source hypothesis Comb. analysis | 92.79 | 152.35 | 176.64 | 141.12 | 69.78 | 42.58
Diffusion hypothesis Comb. analysis | 86.87 | 154.06 | 169.79 | 138.94 | 64.56 | 34.86

Table 8: $\chi^{2}$ values of the best-fit diffusion parameters found for the
MCMC analyses with the GALPROP cross sections.

$\chi^{2}$ values of best-fit parameters $-$ DRAGON2
---
| B/C | B/O | Be/C | Be/O | Li/C | Li/O
Source hypothesis Indep. analysis | 40.8 | 37.48 | 35.61 | 33.25 | 39.76 | 39.96
Diffusion hypothesis Indep. analysis | 31.55 | 27.3 | 33.94 | 31.98 | 30.37 | 29.04
Source hypothesis Comb. analysis | 41.69 | 38.47 | 55.42 | 53.68 | 44.14 | 52.46
Diffusion hypothesis Comb. analysis | 32.43 | 29.04 | 54.01 | 51.51 | 34.24 | 41.69

Table 9: Same as in table 8, but for the DRAGON2.