[a,b]Jan Heisig

# Revisiting the AMS-02 antiproton excess: The role of correlated errors

Michael Korsmeier  Martin Wolfgang Winkler

###### Abstract

Cosmic-ray antiprotons are a remarkable diagnostic tool for the study of
astroparticle physics’ processes in our Galaxy. While the bulk of measured
antiprotons is consistent with a secondary origin, several studies have found
evidence for a subdominant primary component in the AMS-02 data. In this
proceedings article, we revisit the excess considering systematic errors that
could affect the signal. Of particular importance are unknown correlations in
the AMS-02 systematic errors, the dominant of which are associated with the
cross sections for cosmic-ray absorption in the detector. We compute their
correlations in a careful reevaluation of nuclear scattering data, utilizing
the Glauber-Gribov theory to introduce a welcomed redundancy that we explore
in a global fit. The inclusion of correlated errors has a dramatic effect on
the significance of the signal. In particular, the analysis becomes more
sensitive to the diffusion model at low rigidities. For a minimal extension
beyond single-power-law diffusion, the global significance drops below
$1\sigma$ severely questioning the robustness of the finding.

##  1 Introduction

Cosmic-ray antiprotons are promising messengers for the study of astroparticle
physics’ processes in our Galaxy. The bulk of measured antiprotons is
consistent with a secondary origin, coming from scatterings of primary cosmic
rays off the interstellar gas in the stellar disk. However, the precise data
from the AMS-02 experiment [1] allows us to search for a subdominant primary
component of antiprotons, _e.g._ from dark matter annihilation.

Several groups have reported a statistical preference for such a component in
the rigidity range $10\\!-\\!20$ GV which is compatible with a dark-matter
signal [2, 3, 4, 5, 6, 7, 8, 9]. It hints at a dark-matter mass
$50\>\text{GeV}\lesssim m_{\text{DM}}\lesssim 100\>\text{GeV}$ and an
annihilation cross section consistent with thermal freeze-out, $\langle\sigma
v\rangle\\!\sim\\!10^{-26}\>\text{cm}^{3}/\text{s}$. Intriguingly, dark matter
with almost identical properties has also been considered as an explanation of
the gamma-ray Galactic center excess [10]. However, while the various
independent analyses agree on the preferred dark-matter properties the
significance of the excess is highly controversial ranging from around
$1\sigma$ to above $5\sigma$, see [11] for a recent review on the subject.

The discrepancy could point to unaccounted systematic uncertainties that
affect the different analyses to a different extent or mimic the excess
altogether. One of the leading uncertainties comes from antiproton production
cross sections that enter the background prediction. Their inclusion has,
indeed, reduced the antiproton excess in [5] while its effect has been found
to be marginal in [7].

Another important source of correlated uncertainties comes from the AMS-02
measurement itself. In the rigidity range of interest, systematics dominate
which could exhibit correlations. While no covariance matrix has been provided
by the collaboration, their inclusion can have a dramatic effect on the
significance of the signal as shown in [7]. The most relevant source of
correlated errors in the rigidity range of interest is the uncertainties in
the cross sections for cosmic-ray absorption in the detector the reported
fluxes are corrected for. Measurements of the involved inelastic nucleon-
nucleus cross sections are sparse and often subject to considerable
systematics. Furthermore, error correlations were unknown.

Here we summarize recent results [12] addressing this lack of knowledge. We
employ the Glauber-Gribov theory of inelastic scattering [13, 14] to provide a
prediction for the absorption cross section based on the input quantities of
the model, namely the nucleon-nucleon scattering cross sections and nuclear
density functions. These quantities are subject to independent measurements
and, hence, introduce a welcomed redundancy that allows us to reduce the
uncertainties and compute the error correlations via a global fit of a large
experimental dataset. Including the systematic error correlation, we perform a
spectral search for dark matter in the AMS-02 antiproton data.111For brevity,
we concentrate on the cosmic-ray propagation setup corresponding to [7].
However, reference [12] also includes results for the complementary
propagation setup used in [5]. This allows us to draw more robust conclusions
on the existence of the antiproton excess and point to the decisive quantities
its statistical significance depends on.

The remainder of this paper is organized as follows. In Sec. 2 we summarize
the various sources of systematic errors subject to correlations. In Sec. 3 we
present our results for the nucleon-nucleus absorption cross sections and the
corresponding correlation matrix. Finally, in Sec. 4 we discuss the
implications for the tentative signal before drawing our conclusions in Sec.
5.

##  2 Systematic uncertainties in the AMS-02 data

In the rigidity range ${\cal R}\sim 2\\!-\\!40$ GV, systematics dominate the
uncertainties in the AMS-02 antiproton data. Figure 1 shows all sub-
contributions of the systematic error in the $\bar{p}$ flux (left) and
$\bar{p}/p$ flux ratio (right). The most relevant contributions come from
cross sections for cosmic-ray absorption in the detector. The reported fluxes
are corrected by the corresponding loss through interactions with the detector
material in the upper layers. The computation of the associated error
correlations is one of the main subjects of the work presented here and is
discussed in Sec. 3.

Figure 1:  Relative systematic errors of the AMS-02 $\bar{p}$ (left) and
$\bar{p}/p$ data (right). The contributions in the legend are ordered
according to their size at ${R\cal}=10$ GV as indicated by the arrow. Taken
from [12]

For the $\bar{p}$ flux, the effective acceptance error is of similar
relevance. It is a residual systematic error in the effective folded
acceptance and is estimated by a comparison of Monte Carlo simulation and
direct measurements in several parts of the detector. As it affects
antiprotons and protons in the same way, it cancels out in the $\bar{p}/p$
ratio. A data-driven analysis performed in [12] implies that the effective
acceptance error is subject to short-range correlations (in rigidity).

Further systematics play a subleading role being small compared to either the
previous systematics or the statistical error. They comprise the unfolding
error, associated with the migration of events into neighboring rigidity bins,
the template shape and selection error, arising from the choice of the
template shape and cuts on the track shape, the rigidity scale error
addressing misalignments of the tracker planes and uncertainties in the
magnetic field map, and the error associated to the geomagnetic cut-off. For
these contributions, we adopt the correlation lengths estimated in [15].

##  3 Global fit of nucleon-nucleus absorption cross sections

Within the Glauber-Gribov model of inelastic scattering [13, 14] the nucleon-
nucleus absorption cross sections can be expressed as

$\sigma_{\text{abs}}^{NA}(p_{\text{lab}})=\int\\!\mathrm{d}^{2}b\left[1-\left(1-\frac{2\,\text{Im}\chi_{N}(\boldsymbol{b},p_{\text{lab}})}{A}\right)^{\\!A\,}\right]+\Delta\sigma_{\text{inel.\,screen.}}(p_{\text{lab}})$
(1)

where $\boldsymbol{b}$ is the impact parameter, $N$ and $A$ denote the
incident nucleon and the atomic mass number of the nucleus, respectively, and

$\text{Im}\chi_{N}(b,p_{\text{lab}})=\frac{1}{2}\sum_{j=1}^{A}\sigma_{NN_{j}}(p_{\text{lab}})\int_{0}^{\infty}\\!\\!\\!\mathrm{d}q\,q\,J_{0}(bq)\,\mathrm{e}^{-\beta_{NN_{j}}(p_{\text{lab}})\,q^{2}/2}\int_{0}^{\infty}\\!\\!\\!\mathrm{d}r\,r^{2}J_{0}(rq)\,\rho_{j}(r)\,,$
(2)

is the (imaginary part of the) phase-shift function assuming a spherical
symmetric nuclear density function, $\rho(r)$. The input quantities of this
model are the total nucleon-nucleon cross sections,
$\sigma_{NN_{j}}(p_{\text{lab}})$, the slope of the respective differential
inelastic cross sections in the forward direction,
$\beta_{NN_{j}}(p_{\text{lab}})$, and the parameters of the nuclear density
function chosen to be the harmonic oscillator shell model. The sum in eq. (2)
runs over all constituents, $N_{j}$, of the nucleus and $J_{n}$ denotes the
Bessel function of the first kind. The term
$\Delta\sigma_{\text{inel.\,screen.}}(p_{\text{lab}})$ accounts for inelastic
screening corrections [16], see [12] for more details. We use analytical
parametrizations for $\sigma_{NN_{j}}(p_{\text{lab}})$,
$\beta_{NN_{j}}(p_{\text{lab}})$ [12] fitted to the cross-section data
collected in Ref. [17, 18].

The most relevant process is absorption on carbon nuclei as the AMS-02
detector material mainly contains carbon ($\sim 80\%$). To compute the
correlations in the cross-section uncertainties we perform a global fit to the
$\bar{p}$C absorption cross section data [19, 20, 21, 22, 23] within the above
model varying all input parameters according to their nominal values and
correlated uncertainties derived in a separate fit. The result is shown in
Fig. 2.222We obtain analogous results for the $p$C absorption cross section
entering the covariance matrices of the $\bar{p}/p$ flux ratio and $p$ flux,
see [12] for details. We obtain the correlation matrix for
$\sigma_{\text{abs}}$ (in terms of the $p_{\text{lab}}$-values that correspond
to the rigidity bins of the AMS-02 data) directly from the statistical sample
of the global fit.

Figure 2:  $\bar{p}$C absorption cross section as a function of the projectile
momentum $p_{\text{lab}}$. The solid dark green curve and green shaded band
denote our best-fit cross section and $1\sigma$ uncertainty, respectively.
Data points (containing $1\sigma$ error bars) of different experiments are
denoted by individual symbols. For comparison, we show the cross section used
in the AMS-02 analyses stemming from an implementation in Geant 4 (gray dashed
curve and gray shaded band).

##  4 Implications for dark matter

To investigate the significance of a possible dark-matter signal in the AMS-02
antiproton data we perform global cosmic-ray fits with and without a
contribution from dark-matter annihilation. We model the latter by assuming
annihilation into $\bar{b}b$ and considering the dark-matter mass and
annihilation cross section as free parameters. Following the approach of [7],
we use a minimal set of fluxes that allows us to simultaneously constrain the
propagation and dark-matter model parameters, namely the $\bar{p}/p$ flux
ratio [1] and the proton and helium fluxes [24, 25]. We model the primary
spectra by broken power laws and employ the standard diffusion coefficient
with a slope $\delta$ but with additional freedom at low rigidities
parametrized by $\eta$:

$D_{xx}\propto\beta^{\eta}\mathcal{R}^{\delta}\,.$ (3)

(Here, $\beta$ is the particle’s velocity.) Negative values of $\eta$ resemble
a low-rigidity break in diffusion that has recently been considered to improve
the fit to secondary nuclear cosmic rays [26, 27].

We take into account convection and reacceleration and use GALPROP [28] for
the numerical solution of the diffusion equation. Solar modulation is
implemented via the standard force-field approximation but with individual
Fisk potentials for both charge signs. We exclude AMS-02 data for
$\mathcal{R}<5\>\text{GV}$ in the fit where the approximation becomes
questionable. To constrain solar modulation, we include Voyager data on
protons and helium [29]. Uncertainties in the production cross sections of
antiprotons are taken into account through the covariance matrix derived in
[5].

The correlations in the AMS-02 data are taken into account by summing up the
individual covariance matrices for all contributions listed in Sec. 2. Of
particular relevance are the correlations in the cross sections error
discussed in Sec. 3. As the absorption-corrected fluxes scale inversely
proportional to the cosmic-ray survival probability
($P=\mathrm{e}^{-n\sigma_{\text{abs}}}$, where $n$ accounts for the amount of
detector material traversed) at linear order in the cross-section error, the
correlation matrices for the measured fluxes are identical to the ones for the
cross sections. Note that the correlation matrix for the flux ratio contains
the contributions from both the $\bar{p}$C and $p$C cross section weighted by
the relative magnitude of the individual uncertainties.

Figure 3:  Antiproton-to-proton flux ratio of the fit without (left) and with
dark matter (right). The solid red and blue curves denote the best-fit spectra
at the top of the atmosphere with and without correlations in the AMS-02
errors, respectively. The dashed curves denote the corresponding interstellar
fluxes. The error bars of the AMS-02 data points denote the statistical and
systematic uncertainties according to the diagonal entries of the total
experimental covariance matrix.

The best-fit results for the $\bar{p}/p$ ratio without (left) and with (right)
dark matter are shown in Fig. 3. To highlight their impact, we display the
results without (blue) and with (red) the derived AMS-02 correlation matrices.
The corresponding global significance of the excess drops from $1.8\,\sigma$
to $0.5\,\sigma$. Accordingly, we do not find any significant preference for
dark matter in the data.

However, the situation is not fully conclusive. Considering the involved
(correlated) uncertainties and the effect of the diffusion parameter $\eta$
one by one, we can make an interesting observation. Taking into account the
AMS-02 correlation matrices only (no antiproton production cross section
uncertainties and setting $\eta=1$) increases the significance to above
$5\sigma$. This is because errors are no longer treated uncorrelated and hence
the fit provides less freedom to accommodate spectral features that are not in
line with those potentially caused by the correlations. Taking into account
the uncertainties from antiproton production cross section in addition yields
a significance slightly above $3\sigma$ while only the inclusion of these
uncertainties plus the extra freedom in the diffusion coefficient at low
rigidities allows us to reconcile the tension in the data without introducing
dark matter. Hence, the presence of the excess decisively depends on the
diffusion model at low rigidities. Interestingly, uncertainties in nuclear
cross sections are, again, a limit factor in establishing the diffusion model
[30].

Finally, we note that a more recent measurement of the 7-year $\bar{p}$ flux
by AMS-02 [31] shows systematic deviations from the previous measurement at
around 20 GV which potentially decrease the significance of the excess even
further [32].

##  5 Conclusions

The AMS-02 antiproton data provides us with a powerful tool to search for
dark-matter annihilation in our Galaxy. Interestingly, data exhibits a
residual component at a rigidity $\mathcal{R}\sim 10\\!-\\!20\>\text{GV}$ that
has been interpreted as a signal from dark-matter annihilation with a mass
around 100 GeV. However, systematic uncertainties at few percent level become
important such as the antiproton production cross section uncertainties.

Here, we have considered another source of uncertainties, namely correlations
in the AMS-02 systematic errors that potentially have a large effect on the
significance of the signal. The most relevant of these stem from cross
sections for cosmic-ray absorption in the detector for which we computed the
full covariance for the first time. These correlations are vital to fully
exploit the precision of the experiment. Their inclusion reveals that the
excess is not robust. However, the significance of the signal decisively
depends on the diffusion model at low rigidities – a subject currently under
active investigations. Again, uncertainties in nuclear cross sections that
enter its inference are the limiting factor and motivate dedicated
experimental efforts in the future.

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