# Cosmic-ray antiproton excess from annihilating tensor dark matter.

H<EMAIL_ADDRESS>corresponding
author Departamento de Física, Universidad de Guanajuato, Lomas del Bosque
103, Fraccionamiento Lomas del Campestre, 37150, León, Guanajuato, México.  M.
<EMAIL_ADDRESS>Departamento de Física, Universidad de
Guanajuato, Lomas del Bosque 103, Fraccionamiento Lomas del Campestre, 37150,
León, Guanajuato, México.  S. Rodríguez333email<EMAIL_ADDRESS>Facultad de Ciencias Físico-Matemáticas, Universidad Autónoma de Coahuila,
Edificio A, Unidad Camporredondo, 25000, Saltillo, Coahuila, México.

###### Abstract

In this work we show that the excess of antiprotons in the range
$E_{K}=10-20~{}GeV$ reported by several groups in the analysis of the AMS-02
Collaboration data, can be explained by the production of antiprotons in the
annihilation of dark matter with a $(1,0)\oplus(0,1)$ space-time structure
(tensor dark matter). First, we calculate the proton and antiproton flux from
conventional mechanisms and fit our results to the AMS-02 data, confirming the
antiproton excess. Then we calculate the antiproton production in the
annihilation of tensor dark matter. For the window $M\in[62.470,62.505]~{}GeV$
to which the measured relic density, XENO1T results and the gamma ray excess
from the galactic center constrain the values of the tensor dark matter mass,
we find sizable contributions of antiprotons in the excess region from the
annihilation into $\bar{b}b$ and smaller contributions from the $\bar{c}c$
channel. We fit our results to the AMS-02 data, finding an improvement of the
fit for these values of $M$.

##  I Introduction

One of the major challenges in particle physics, astrophysics and cosmology is
the elucidation of the nature of dark matter. There is compelling evidence for
the existence of dark matter from galaxy rotation curves and observables
required to fit the precision measurements of the cosmic microwave background,
among them, the dark matter relic density (a recent review on the subject can
be found in Lin:2019uvt ). For decades, a great effort has been put in the
measurement of observables that help us to constrain the values of the mass
and possible couplings to the standard model fields of this type of matter
that conforms around $26\%$ of the energy budget in the universe. Several
experiments have been designed aiming to detect signals of non-gravitational
dark matter interactions in direct, indirect and collider searches. In
particular, instruments such as PAMELA and AMS-02 have measured with good
precision the antimatter cosmic-ray spectrum Adriani:2011cu ;
PhysRevLett.114.171103 ; PhysRevLett.115.211101 ; PhysRevLett.117.231102 ,
which may include antimatter produced in the annihilation of dark matter.

In the past decade, several reports indicate the existence of an excess of
antiprotons in the $\sim 10-20~{}GeV$ region of the AMS-02 Collaboration data,
and many studies have identified a consistency with an additional contribution
from annihilating dark matter Bringmann:2014lpa ; Cirelli:2014lwa ;
Hooper:2014ysa ; Cui:2016ppb ; Cuoco:2017rxb ; PhysRevLett.118.191102 ;
Cholis:2019ejx ; PhysRevD.96.123010 ; PhysRevD.97.023003  (see however
alternative explanations to the excess regarding secondary cosmic-rays and
systematic errors in Heisig:2020nse ; Boudaud:2019efq ). It is interesting to
note that most of these studies scan independently the values of
$\langle\sigma v_{r}\rangle$ for the annihilation of dark matter into standard
model channels producing antiprotons in subsequent processes, and the dark
matter mass, $M$, to conclude that it is possible to explain this excess if
dark matter has a mass $M\sim 60$ GeV and an annihilation cross-section into
standard model particles of the order of the thermal relic cross section
($\langle\sigma v_{r}\rangle\sim 10^{-26}cm^{3}/seg$).

On the other side, an excess in the gamma ray signal from the galactic center
has been claimed by several groups Hooper:2010mq ; Boyarsky:2010dr ;
Hooper:2011ti ; Abazajian:2012pn ; Macias:2013vya ; TheFermi-LAT:2017vmf ;
Gordon:2013vta  and the possibility that this excess be explained by the
annihilation of dark matter has been explored scanning independently the
values of the annihilation cross section $\langle\sigma v_{r}\rangle$ for
different standard model channels producing gamma rays in the final state, and
the dark matter mass, concluding that similar values can explain the gamma ray
excess.

Recently, we proposed an unconventional $(1,0)\oplus(0,1)$ space-time
structure for dark matter Hernandez-Arellano:2018sen . In this formalism, dark
matter fields are described by a six-component Dirac-like spinor transforming
in the $(1,0)\oplus(0,1)$ representation of the Homogenous Lorentz Group
(HLG). This theory uses the covariant basis of $6\times 6$ matrices for this
space constructed in Gomez-Avila:2013qaa  and it is based on simultaneous
parity and Poincarè projectors, which yields an equation of motion which
differs from the one proposed by Weinberg for these fields long ago
Weinberg:1964cn . The new equation of motion can be derived from a suitable
Lagrangian and the theory turns out to have second class constraints, which
can be solved using conventional Dirac’s method which replaces Poisson
brackets by Dirac brackets. The details of the classical theory and its
canonical quantization where given in Napsuciale:2015kua , where it is shown
that Fock states can be constructed and the formalism do not suffer from the
well known tachyonic solutions of the Weinberg equation (for a comprehensive
list of references on this point see e.g. Eeg:1972us  and for a recent
proposal to use high spin fields in the Weinberg’s theory for dark matter
applications in an effective field theory approach in spite of these
complications we refer to Criado:2020jkp ).

The field transforming in the $(1,0)\oplus(0,1)$ representation space was
named spin-one matter field in Napsuciale:2015kua  because it is the direct
generalization of the $(1/2,0)\oplus(0,1/2)$ (Dirac) representation under
which transform matter fields in the standard model. Conventionally, fields
transforming in the $(1,0)\oplus(0,1)$ are described by a tensor with two
antisymmetric Lorentz indices and are named tensor fields. We stick to the
historical name here and call the proposed dark matter field transforming in
this representation tensor dark matter field (TDM).

There is a chirality operator ($\chi$) in the theory and TDM fields can be
splitted into left and right fields but, unlike the Dirac theory, the kinetic
term of TDM fields couple them, thus charged tensor dark matter cannot have
chiral gauge interactions, although vector interactions are allowed
Napsuciale:2015kua . The effective theory for interactions with standard model
fields in a hidden scenario yields three dimension-four operators. Two of them
couple TDM with the Higgs boson, among which one of them respects parity and
the other one violates this symmetry. The corresponding dimensionless
couplings are denoted as $g_{s}$ and $g_{p}$ respectively. The third operator
couples the $Z^{0}$ and $\gamma$ to the higher multipoles of TDM, and the
corresponding coupling is denoted as $g_{t}$.

In a first panoramic study of the constraints from available data, it was
shown that the measured relic density and invisible $Z$ width constrain the
TDM mass to $M>43~{}GeV$. Also, XENON1T upper bounds for the spin independent
dark matter-nucleon cross section Aprile:2017iyp , constrain the spin portal
coupling to $g_{t}<10^{-4}$ for TDM masses of the order of hundreds of GeV and
sensible values, $g_{s}\leq 10^{-1}$, are compatible with these upper bounds
Hernandez-Arellano:2018sen . Similarly, indirect detection upper bounds to the
annihilation of dark matter into $\bar{b}b$, $\tau^{+}\tau^{-}$ and
$\mu^{+}\mu^{-}$ are satisfied for a wide range of values of $M$.

Recently, we studied the possibility that the excess in the gamma-ray spectrum
from our galactic center be explained by the annihilation of tensor dark
matter Hernandez-Arellano:2019qgd . In a comprehensive study of the possible
mechanisms, we found that the gamma ray excess can be accounted for by the
annihilation of TDM only if $M\in[62.470,62.505]~{}GeV$. For a given $M$ in
this window, the measured relic density, $\Omega^{exp}_{DM}h^{2}=0.1186\pm
0.0020$ Ade:2015xua ; Patrignani:2016xqp , yields the corresponding value
$g_{s}(M)$ (see Fig.10 of Hernandez-Arellano:2019qgd ), which takes values in
the range $g_{s}(M)\in[0.98,1.01]\times 10^{-3}$. These values of the TDM mass
are at the Higgs resonance for non-relativistic dark matter, thus we refined
our previous calculations to focus on the resonance region and to account for
the resonant effects in the calculation of the relic density, finding these
values compatible with its experimental value, the XENON1T upper bounds on the
dark matter-nucleon spin independent cross section, the upper bounds to the
annihilation of dark matter into $\bar{b}b$, $\tau^{+}\tau^{-}$ and
$\mu^{+}\mu^{-}$ from indirect detection experiments Bergstrom:2013jra ;
Aguilar:2013qda ; Drlica-Wagner:2015xua ; Fermi-LAT:2016uux  and also with the
upper bounds for the annihilation into $\gamma\gamma$ from the FermiLAT
Ackermann:2015lka  and HESS Abdallah:2018qtu  collaborations.

For the values of $M\in[62.470,62.505]\text{ GeV}$ and the correlated values
$g_{s}(M)\in[0.98,1.01]\times 10^{-3}$, we get $\langle\sigma v_{r}\rangle$ of
the order of the thermal cross section for the annihilation into a fermion
pair. Fermion pairs (quark and leptons) produced in the annihilation of TDM
hadronize (in the case of quark jets) or decay into hadronic states (in the
case of leptons) producing other particles, including antiprotons. This signal
could be sizable enough to be detected on Earth by any of the current anti-
matter measuring experiments, including PAMELA and AMS-02. Annihilation into
$\bar{b}b$ is expected to play a major role in the antiproton production, but
there could also be important contributions from the $\bar{c}c$ and
$\tau^{+}\tau^{-}$ channels.

In this paper we explore the possibility that the antiproton excess be
explained by antiproton production in the annihilation of TDM. Our work is
organized as follows. In the next section we calculate the antiproton and
proton cosmic-ray spectrum in the galaxy using standard methods, reproduce
results in the literature for the production of antiprotons from standard
sources and confirm an excess of antiprotons with $E_{K}$ around
$10-20~{}GeV$. In section III, we include the contribution from TDM
annihilation into fermions. We fit the obtained flux to AMS-02 data and
compare the corresponding $\chi^{2}$ to the analogous results without TDM
contributions to check if there is a clear preference for TDM annihilation
contributions. Our conclusions are given in Section IV.

##  II Antiproton excess in the cosmic-ray spectrum

The need to collect data on the existing antimatter in the universe that help
us in the uncovering of the mystery of the observed baryon asymmetry, prompted
the search and eventual detection of cosmic ray (CR) antiprotons during the
1970’s Golden:1979bw ; bogomolov1979stratospheric . Following several
measurements and model proposals for the generation of antiparticles in the
Galaxy, it was concluded that CR antiprotons are produced after interactions
between high-energy nuclei (cosmic ray primaries, i.e. those accelerated by
remnants of supernovae) and matter (mostly hydrogen and helium Ferriere:2001rg
) within the interestellar medium (ISM), thus they are called secondary
antiprotons. Antiprotons could be produced directly (primary antiprotons),
either in regions consisting of antimatter (not yet observed) PhysRev.171.1344
, or it could be a product of evaporation of primordial black holes
Barrau:2001ev . However, the spectrum of antiprotons observed at Earth is in
agreement with mostly that of secondary origin PhysRevLett.87.271101 . The
particular kinematics of their production makes their energy spectrum to have
a maximum near $2~{}GeV$ and then decrease at energies around tenths of $GeV$
in a steeper form than that of protons Moskalenko:2001ya . This results in a
steep decrease in the antiproton-to-proton ratio at these energies. The
antiproton flux is thus determined by CR propagation and interaction of nuclei
with interestelar gas, processes which suffer from large uncertainties that
must be treated carefully in order to predict the flux with high precision.

The first source of systematic uncertainty comes from the CR propagation which
involves several complicated processes such as diffusion, convection, re-
acceleration and loss of energy. The parameters of choice to model such
effects must account for secondary-to-primary nuclei ratios, among which are
Boron-to-Carbon (B/C) and other nuclei Strong_1998 . A second source of
uncertainty is the effect of solar modulation on the CR spectra osti_4797303 .
When CR enter the Solar System, the solar magnetic field modifies their
spectra. This mainly acts on the low energy part of the spectrum, but it is
difficult to estimate it with precision since it requires the modeling of the
solar wind and its effects, which change through time. For an overview of the
effects of solar modulation of the cosmic rays entering the heliosphere, see
Ref. Potgieter_2013 . In addition to these, another important source of
uncertainties in CR antiproton production is the limited characterization of
the cross sections for the production, annihilation and scattering of these
particles. In this section we will take this into account and will make use of
the parameter choice in the first model in Ref. Cholis:2019ejx  which yields
the best fit to data, to evaluate the proton and antiproton flux from cosmic
rays.

###  II.1 Modeling the antiproton and proton cosmic-ray spectrum in the galaxy

We make use of the tool GALPROP WEBRUNGALPROP ; VLADIMIROV20111156 ;
MOSKALENKO2005156 , which takes care of solving the transport equation to
yield the local flux of the primary and secondary cosmic ray species. Adopting
the first model in Table I of Cholis:2019ejx , we use the input parameters
listed in Table 1.

Parameter | Value
---|---
$\delta$ | 0.40
$z_{L}$ (kpc) | 5.6
$D_{0}$ ($cm^{2}s^{-1}$) | $4.85\times 10^{28}$
$v_{A}$ (km/s) | 24.0
${\alpha}_{1}$ | 1.88
$dv_{c}/d|z|$ (km/s/kpc) | 1.0
${\alpha}_{2}$ | 2.38
$R_{br}$ (GV) | 11.7
Table 1: Cosmic-ray injection and propagation model parameters used, adopted
from Ref. Cholis:2019ejx .

The diffusion coefficient is defined by

$D_{xx}(R)=\beta D_{0}(R/4\text{ GV})^{\delta},$ (1)

where $\delta$ is the diffusion index and $\beta\equiv v/c$. The details of
this choice of parameters and the process of CR injection, diffusion,
convection and reacceleration can be found in Ref. Cholis:2019ejx .

A study of the antiproton production cross section was done in Ref.
diMauro:2014zea , where it was found that different procedures led to
equivalent results, resulting in an uncertainty of 10-20%. In order to account
for this uncertainty, we employ the same approach as formulated in
PhysRevD.95.123007 , where we multiply the antiproton flux prior solar
modulation by an scaling factor in the form of an energy-dependent function,
defined by

$N_{CS}(k_{ISM})=a+b\Big{[}ln\Big{(}\frac{k_{ISM}}{GeV}\Big{)}\Big{]}+c\Big{[}ln\Big{(}\frac{k_{ISM}}{GeV}\Big{)}\Big{]}^{2}+d\Big{[}ln\Big{(}\frac{k_{ISM}}{GeV}\Big{)}\Big{]}^{3},$
(2)

where $k_{ISM}$ is the kinetic energy of the cosmic ray in the interstellar
medium, before entering the Solar System. The fourth term of this scaling
factor, proportional to
$\Big{[}ln\Big{(}\frac{k_{ISM}}{GeV}\Big{)}\Big{]}^{3}$, can be omitted and
still reach a adequate fit, as we will show later.

We must also consider the effects of solar modulation. The differential flux
at Earth, $dN^{\oplus}/dE_{kin}$ in terms of $k_{ISM}$, is obtained as
osti_4797303

$\frac{dN^{\oplus}}{dE_{kin}}(k_{ISM})=\frac{(k_{ISM}-|Z|e\Phi(R)+m)^{2}-m^{2}}{(k_{ISM}+m)^{2}-m^{2}}\frac{dN^{ISM}}{dk_{ISM}}(k_{ISM}),$
(3)

where ${dN^{ISM}}/{dk_{ISM}}$ is the differential flux prior to the effects of
solar modulation, $E_{kin}$, $|Z|e$ and $m$ are the kinetic energy, charge and
mass of the cosmic ray. To obtain the flux in terms of $E_{kin}$, we use the
equivalence $E_{kin}=k_{ISM}-|Z|e\Phi(R)$. The modulation potential, $\Phi$,
can be described as a function that depends on time, the rigidity
($R=\sqrt{k_{ISM}(k_{ISM}+2m_{p})}$) and charge of the cosmic ray prior
entering the Solar System. We follow the analytic expression constructed in
Ref. PhysRevD.93.043016 :

$\Phi(R,t,q)=\phi_{0}\Big{(}\frac{|B_{tot}(t)|}{4\text{
nT}}\Big{)}+\phi_{1}N^{\prime}(q)H(-qA(t))\Big{(}\frac{|B_{tot}(t)|}{4\text{
nT}}\Big{)}\Big{(}\frac{1+(R/R_{0})^{2}}{\beta(R/R_{0})^{3}}\Big{)}\Big{(}\frac{\alpha(t)}{\pi/2}\Big{)}^{4},$
(4)

where $\beta$ is the velocity, $R_{0}\equiv 0.5\text{ GV}$ and $B_{tot}$ is
the strength of the heliospheric magnetic field (HMF) at Earth, which has a
polarity $A(t)$, $H$ denotes the Heaviside function and $\alpha$ is the tilt
angle of the heliospheric current sheet. $N^{\prime}(q)\neq 1$ when the HMF
does not have a well-defined polarity.

We allow $\phi_{0}\in[0.32,0.38]$ GV and $\phi_{1}\in[0,16]$ GV in order to
stand within the uncertainties for the modulation potential described in Ref.
PhysRevD.93.043016 . The values of $B_{tot}$, $\alpha$ and
$N^{\prime}(q)H(-qA(t))$ averaged over six-month intervals during the
observation period by AMS-02 can be found in Table II of PhysRevD.95.123007 .
We take these values to evaluate the potential and the flux for each period
and then take the average of the obtained values for the final result.

There is an additional parameter, the local ISM gas density normalization,
taken as an energy-independent factor, $g_{ISM}$. In total, there are seven
free parameters that will be used for the fit: $\phi_{0}$, $\phi_{1}$, $a$,
$b$, $c$, $d$ and $g_{ISM}$. The flux ratio is defined as follows.

$R_{\bar{p}/p}=\frac{\Phi_{\bar{p}}}{\Phi_{p}}=g_{ISM}\frac{\frac{dN^{\oplus}_{\bar{p}}}{dE_{kin}}}{\frac{dN^{\oplus}_{p}}{dE_{kin}}}.$
(5)

###  II.2 Results of the fit to the antiproton-proton ratio without dark
matter

Figure 1: Best fit to the AMS-02 data on the a) antiproton flux and b)
antiproton-to-proton ratio for $d=0$ (solid line) and $d\neq 0$ (dashed line).

The results for the best fit to the AMS-02 antiproton flux and antiproton-to-
proton ratio data PhysRevLett.117.091103  are shown in Fig. 1. We performed
the fit for the two cases, $d=0$ (fit 1) and $d\neq 0$ (fit 2). A summary of
the fitting parameters and the corresponding value of $\chi^{2}$ for each case
is given in Table 2. The difference (residual) between the antiproton-to-
proton ratio for each fit and the AMS-02 data is shown in Fig. 2. We can see
that there is an excess in the residual around $\sim$10-20 GeV and in the
higher end of the spectrum.

Fit | $\phi_{0}/GV$ | $\phi_{1}/GV$ | a | b | c | d | $g_{ISM}$ | $\chi^{2}$
---|---|---|---|---|---|---|---|---
1 | 0.3257 | 16 | 1.1579 | -0.1632 | 0.0216 | 0 | 1.1844 | 1.0561
2 | 0.32 | 16 | 1.1549 | -0.1301 | 0.0038 | 0.0023 | 1.1842 | 0.9243
Table 2: Best-fit parameters to the AMS-02 antiproton-to-proton ratio data
considering secondary antiprotons produced in the ISM, for $d=0$ (fit 1) and
$d\neq 0$ (fit 2). Figure 2: Antiproton-to-proton ratio residual from the
AMS-02 data from Ref. PhysRevLett.117.091103 .

##  III Antiproton production from annihilating tensor dark matter

In the effective field theory for hidden tensor dark matter, the leading
interacting terms for the simplest case of a $U(1)_{D}$ dark gauge group, turn
out to be dimension-four and are given by Hernandez-Arellano:2018sen

$\mathcal{L}_{int}=\bar{\psi}(g_{s}\mathbf{1}+ig_{p}\chi)\psi\phi^{\dagger}\phi+g_{t}\bar{\psi}M_{\mu\nu}\psi
B^{\mu\nu}+\mathcal{L}_{selfint},$ (6)

where $\psi$ is a six-component ”spinor” describing TDM, $\phi$ is the
standard model Higgs doublet and $g_{s}$, $g_{p}$ and $g_{t}$ are
dimensionless constants. The first two operators yield parity-respecting and
parity-violating Higgs portals and the third couple the photon and $Z^{0}$ to
higher multipoles of tensor dark matter and is therefore dubbed spin portal.
After spontaneous symmetry breaking, this Lagrangian yields

$\mathcal{L}_{int}=\frac{1}{2}\bar{\psi}(g_{s}\mathbf{1}+ig_{p}\chi)\psi\left(H+v\right)^{2}+g_{t}C_{W}\bar{\psi}M_{\mu\nu}\psi
F^{\mu\nu}-g_{t}S_{W}\bar{\psi}M_{\mu\nu}\psi Z^{\mu\nu},$ (7)

where $C_{W}$, $S_{W}$, $H$, $v$, are the cosine and sine of the Weinberg
angle, the Higgs field and the Higgs vacuum expectation value, and
$F^{\mu\nu},Z^{\mu\nu}$ stand for the stress tensors of the electromagnetic
and $Z^{0}$ fields.

Tensor dark matter can annihilate into fermion-anti-fermion through the Higgs
portals and the spin portal in Eq. (7). We refer the reader to Ref. Hernandez-
Arellano:2018sen  for technical details of the calculation of the cross
sections for the annihilation of TDM into standard model particles. The cross
section for the annihilation into a fermion-anti-fermion pair is Hernandez-
Arellano:2018sen

$\displaystyle(\sigma v_{r})_{\bar{f}f}(s)$ $\displaystyle=\frac{N_{c}}{144\pi
M^{4}\sqrt{s}}\frac{\sqrt{s-4m_{f}^{2}}}{(s-M^{2})}\left[\frac{m_{f}^{2}\left(s-4m_{f}^{2}\right)\left(g_{p}^{2}s\left(s-4M^{2}\right)+g_{s}^{2}\left(6M^{4}-4M^{2}s+s^{2}\right)\right)}{\left(\left(s-M_{H}^{2}\right)^{2}+\Gamma_{H}^{2}M_{H}^{2}\right)}\right.$
$\displaystyle\left.+\frac{2g_{t}^{2}M_{Z}^{2}S_{W}^{2}s\left(s-4M^{2}\right)\left(2M^{2}+s\right)\left(2\left(A_{f}^{2}-2B_{f}^{2}\right)m_{f}^{2}+s\left(A_{f}^{2}+B_{f}^{2}\right)\right)}{3v^{2}\left(\left(s-M_{Z}^{2}\right)^{2}+\Gamma_{Z}^{2}M_{Z}^{2}\right){}^{2}}\right.$
$\displaystyle\left.+\frac{32C_{W}^{2}Q_{f}^{2}g_{t}^{2}M_{W}^{2}S_{W}^{2}\left(s-4M^{2}\right)\left(2M^{2}+s\right)\left(2m_{f}^{2}+s\right)}{3v^{2}s}\right.$
$\displaystyle\left.-\frac{16A_{f}C_{W}Q_{f}g_{t}^{2}M_{W}M_{Z}S_{W}^{2}\left(s-4M^{2}\right)\left(2M^{2}+s\right)\left(2m_{f}^{2}+s\right)}{3v^{2}\left(\left(s-M_{Z}^{2}\right)^{2}+\Gamma_{Z}^{2}M_{Z}^{2}\right)}\right],$
(8)

where $N_{c}=3$ for quarks and $N_{c}=1$ for leptons, $m_{f}$, $Q_{f}$
correspond to the mass of the fermion and its charge in units of $e>0$,
respectively, and

$A_{f}=2T^{(3)}_{f}-4Q_{f}\sin^{2}\theta_{W},\qquad B_{f}=-2T^{(3)}_{f}.$ (9)

In the non-relativistic limit, only the parity-conserving Higgs portal
contributions survive and the velocity averaged cross section reads

$\langle\sigma
v_{r}\rangle_{\bar{f}f}=\frac{N_{c}g^{2}_{s}m^{2}_{f}(M^{2}-m^{2}_{f})^{\frac{3}{2}}}{12\pi
M^{3}[(4M^{2}-M^{2}_{H})^{2}+M^{2}_{H}\Gamma^{2}_{H}]}+{\cal{O}}(v^{2}_{r}).$
(10)

Notice that $\langle\sigma v_{r}\rangle_{\bar{f}f}$ and the TDM mass $M$ are
not independent parameters, thus the predictions are specific to the nature
(space-time structure) of TDM. In order to keep consistency with the measured
dark matter relic density, upper limits by XENON1T for the spin-independent
dark matter-nucleon cross-section, indirect detection limits and the gamma-ray
excess at the Milky Way galactic center, we consider in our calculations
$M\in[62.470,62.505]\text{ GeV}$. For a given value of $M$ the corresponding
value of $g_{s}(M)$ is fixed by the measured relic density given in Fig. 10 of
Ref. Hernandez-Arellano:2019qgd .

We use the PPC4DMID code Cirelli_2011  to obtain the antiproton flux produced
in the hadronization of quarks or the hadronic decay of the $\tau$ lepton
produced in the annihilation of TDM, and its propagation in the ISM. The code
uses Monte Carlo simulations in order to find the differential spectra of
antiprotons, including electroweak corrections. Although protons are also
produced by the same mechanism, its contribution to the overall proton flux is
very small and can be neglected. Only the antiproton production is large
enough to be relevant in the calculation of the antiproton to proton flux
ratio. The contributions of the relevant channels to the antiproton-proton
ratio are shown in Fig. 3, where we can see that the $\bar{b}b$ and $\bar{c}c$
channels yield an antiproton flux with a maximum in the energy range of the
excess in the AMS-02 data and it is dominated by the $\bar{b}b$ channel.

Figure 3: Antiproton-to-proton flux ratio for antiprotons produced in the
annihilation of TDM for $M=62.488$ GeV and the correlated coupling
$g_{s}(M)=1.001\times 10^{-3}$, for the $\bar{b}b$, $\bar{c}c$,
$\tau^{+}\tau^{-}$ and light quark channels.

###  III.1 Results of the fit to the antiproton-proton ratio including tensor
dark matter contributions

In order to asses the statistical relevance of this contribution we fitted the
total flux to the AMS-02 data for both cases, $d=0$ and $d\neq 0$, for fixed
value of the mass M. Scanning the whole range $M\in[62.470,62.505]~{}GeV$ we
obtain results shown in Fig. 4 for the $\chi^{2}$ of the fit including TDM
contributions and compare it to the value for the best fit without TDM
contributions obtained in the previous section for $d=0$ and $d\neq 0$. We can
see that the fit improves for all values of $M\in[62.470,62.505]~{}GeV$ in
both cases. In these plots we also mark with a point the minimal value of
$\chi^{2}$ for the considered values of $M$. These points correspond to the
best fit including TDM contributions in each case. The parameters
corresponding to these points are given in Table 3. In the case $d=0$, the
best fit is obtained for $M=62.4839~{}GeV$ and the corresponding value of the
spin portal coupling compatible with the measured relic density is
$g_{s}(M)=1.0053\times 10^{-3}$. As to the $d\neq 0$ case the best fit yields
$M=62.4877~{}GeV$ and the corresponding coupling is $g_{s}(M)=1.0009\times
10^{-3}$.

Figure 4: $\chi^{2}$ value as a function of $M$, for $d=0$ (left) and $d\neq 0$ (right). The dashed lines are the $\chi^{2}$ values without the TDM annihilation contributions in Section II. The points correspond to the minimal value in each case. Fit | $M/GeV$ | $\phi_{0}/GV$ | $\phi_{1}/GV$ | a | b | c | d | $g_{ISM}$ | $\chi^{2}$
---|---|---|---|---|---|---|---|---|---
3 | 62.4839 | 0.32 | 16 | 1.1216 | -0.1599 | 0.0247 | 0 | 1.1371 | 0.8783
4 | 62.4877 | 0.32 | 16 | 1.1225 | -0.1691 | 0.0290 | -0.00043 | 1.1310 | 0.8772
Table 3: Best-fit parameters to the AMS-02 antiproton-to-proton ratio for
$d=0$ (Fit 3) and $d\neq 0$ (Fit 4), including antiprotons from TDM
annihilation.

The separate signals for the parameters of the best-fit results in Table 3 are
shown in Fig. 5, where data points correspond to the antiproton-proton
residual ratio from the AMS-02 data for antiprotons produced in the ISM and
the continuous lines correspond to antiprotons produced in the annihilation of
TDM. We can see in these plots that TDM contributions account for the
antiprotons excess extracted from the AMS-02 data and this result is not
sensitive to the value of $d$.

Figure 5: Antiproton-to-proton ratio pure residual (annihilation contributions
not included) from the AMS-02 data PhysRevLett.117.091103  for the parameters
in the best fit for $d=0$ (top) and $d\neq 0$ (bottom) in Table 3. In both
cases we also show the contributions from TDM annihilation into $\bar{b}b$ and
$\bar{c}c$ for the corresponding value of $M$ in Table 3.

##  IV Conclusions

In this work we study the production of antiprotons in the annihilation of
tensor dark matter and the possibility that it explains the cosmic-ray
antiprotons excess with $E_{K}$ around $10-20~{}GeV$ in the AMS-02 data
claimed by several groups. First we use the GALPROP tool to solve the
transport equation in order to evaluate the proton and antiproton flux
produced in the interstellar medium. The largest uncertainty ($10-20~{}\%$) in
this calculation comes from the estimate of the cross-section for the
antiproton production in the ISM. Following PhysRevD.95.123007 , we introduce
a scaling factor in the antiproton to proton flux ratio to study effects of
this uncertainty. The fits are improved by this scaling factor and our results
confirm an excess of antiprotons in the $\sim 10-20$ GeV region of the
spectrum.

Then, we use the PPC4DMID code to calculate the flux of antiprotons produced
in the hadronization and propagation of quarks or in the hadronic decays of
leptons produced in the annihilation of tensor dark matter. The corresponding
velocity averaged cross sections depend only on the TDM mass $M$ and the
coupling $g_{s}$ of the parity-conserving Higgs portal in the effective
theory. The value of the TDM mass is constrained to the narrow windows
$M\in[62.470,62.505]~{}GeV$ by the measured relic density, the gamma ray
excess from the galactic center and the upper bounds for the dark matter-
nucleon spin independent cross section measured by XENON and, for a given
value of $M$, the measured dark matter relic density fixes the value
$g_{s}(M)$ which takes values in the range $g_{s}(M)\in[0.98,1.01]\times
10^{-3}$, such that we have actually only one independent parameter. These
range of values are also consistent with the upper bounds for the annihilation
of dark matter into $\bar{b}b$, $\tau^{+}\tau{-}$, $\mu^{+}\mu^{-}$ and
$\gamma\gamma$ from indirect detection searches in gamma ray physics
Hernandez-Arellano:2019qgd . We find a sizable antiproton flux from the
annihilation of tensor dark matter for these values of $M$. The main
contribution comes from TDM annihilation into $\bar{b}b$, the $\bar{c}c$
channel contribution being roughly two orders of magnitude smaller and other
contributions negligible. This antiproton flux has a maximum for $E_{K}$ in
the region of the antiproton excess.

In order to asses the statistical significance of antiprotons produced in the
annihilation of tensor dark matter we perform fits to the AMS-02 data for the
antiproton to proton ration including these contributions for fixed $M$,
finding that it improves the fits for the whole range of mass
$M\in[62.470,62.505]\text{ GeV}$. The best fit is obtained for
$M=62.4877~{}GeV$ and the corresponding coupling for the parity-conserving
Higgs portal in the affective theory is fixed by the measured relic density to
$g_{s}(M)=1.0009\times 10^{-3}$. We conclude that the antiproton excess in the
AMS-02 data can be explained by antiprotons produced in the annihilation of
tensor dark matter.

##  V Acknowledgements

We thank Ilias Cholis for his help in the use of GALPROP to model proton and
antiproton flux. H.H.A acknowledges financial support from CONACyT, México to
pursue her PhD program.

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