# Consistent explanation for the cosmic-ray positron excess in $p$-wave Breit-
Wigner enhanced dark matter annihilation

Yu-Chen Dinga,b Yu-Lin Kua,b Chun-Cheng Weia,b Yu-Feng Zhoua,b,c,d aCAS key
laboratory of theoretical Physics, Institute of Theoretical Physics, Chinese
Academy of Sciences, Beijing 100190, China; bSchool of Physics, University of
Chinese Academy of Sciences, Beijing 100049, China; cSchool of Fundamental
Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study,
UCAS, Hangzhou 310024, China; dInternational Center for Theoretical Physics
Asia-Pacific, Beijing/Hangzhou, China.

###### Abstract

Dark matter (DM) annihilation in the galactic halo can be enhanced relative to
that in the early Universe due to the Breit-Wigner enhancement, if the DM
particles annihilate through a narrow resonance. Although the $s$-wave Breit-
Wigner enhancement can provide a consistent explanation for both the observed
cosmic-ray (CR) positron excess and the DM thermal relic density, it is
severely constrained by the observations of gamma rays from dwarf spheroidal
satellite galaxies (dSphs) and the cosmic microwave background (CMB), which
have relatively lower allowed DM annihilation cross section and typical DM
velocities than that in the galactic halo. Furthermore, in the $s$-wave Breit-
Wigner enhancement, the case where the resonance mass is below a threshold
(twice the DM mass) is ruled out due to the monotonically increasing
annihilation cross section with decreasing DM velocity. In this work, we
consider Breit-Wigner enhanced $p$-wave DM annihilation. We explore the
parameter regions which can simultaneously account for the CR positron excess
and DM thermal relic density without violating the constraints from dSphs
gamma rays and CMB. We show that the velocity-dependent cross section in this
scenario can peak around the typical DM velocity in the galactic halo for the
resonance mass both above and below the threshold. Moreover, the highly
suppressed annihilation cross section at extremely low DM velocity can evade
the constraints from dSphs gamma rays and CMB easily, which results in larger
allowed parameter regions than that in the $s$-wave case.

##  I Introduction

The existence of non-baryonic cold dark matter (DM) as the dominant component
of matter in the Universe has been established by numerous astrophysical and
cosmological observations, yet its particle nature remains a mystery, which
constitutes one of the most profound questions in fundamental physics. Weakly
interacting massive particles (WIMPs), as a popular DM candidate, can
naturally obtain the observed DM abundance through self-annihilation into the
standard model (SM) final states during the process of thermal freeze-out. The
annihilation of DM into the SM particles in the local Universe, and
specifically inside our Galaxy, can result in its indirect detection via
energetic annihilation products such as high energy gamma rays, charged
leptons, protons, and anti-protons, etc.

Recently, the AMS-02 Aguilar:2019owu  collaboration has updated their result
of the cosmic-ray (CR) positron flux, which indicates a hardening at $\sim 10\
\rm{GeV}$ up to $\sim 300\ \rm{GeV}$. The result of AMS-02 is consistent with
the previous experiments of PAMELA Adriani:2013uda  and Fermi-LAT
FermiLAT:2011ab  but with much higher accuracy. The observed spectral feature
is unexpected in conventional astrophysical theories where the majority of
positrons are believed to originate from the collisions between primary CR
nuclei and interstellar gas, hence strongly suggests the existence of
additional positron sources. While it is possible that some unaccounted
astrophysical sources are behind this positron excess, such as nearby pulsar
wind nebulae Hooper:2008kg ; Yuksel:2008rf ; Profumo:2008ms ; Hooper:2017gtd
and supernovae remnants Blasi:2009hv ; Hu:2009bc ; Fujita:2009wk , the DM
explanation to this excess has also been widely explored Bergstrom:2008gr ;
Cirelli:2008pk ; Bergstrom:2009fa ; Lin:2014vja ; Jin:2013nta ; Jin:2014ica .
In the scenario of DM annihilation, the absence of an associated antiproton
excess indicates that DM should dominantly annihilate into leptonic final
states Aguilar:2016kjl . The $e^{+}e^{-}$ channel leads to a very sharp
spectral structure that cannot fit the observed positron flux. For the
$\mu^{+}\mu^{-}$ and $\tau^{+}\tau^{-}$ channels, a typical DM mass of
$m_{\chi}\sim\mathcal{O}(1)\ \rm{TeV}$ and velocity-weighted annihilation
cross section of
$\langle\sigma_{\text{ann}}v_{\text{rel}}\rangle\sim\mathcal{O}(10^{-24})\
\rm{cm^{3}\ s^{-1}}$ are favored by the AMS-02 positron data Cirelli:2008pk ;
Bergstrom:2009fa ; Cholis:2008hb ; Jin:2014ica .

The minimal $s$-wave DM annihilation model with constant (velocity-
independent) cross section has been stringently constrained by many
observations. For instance, a typical annihilation cross section of
$\langle\sigma_{\text{ann}}v_{\text{rel}}\rangle\sim 3\times 10^{-26}\
\rm{cm^{3}\ s^{-1}}$ for WIMP DM candidate is required by the precisely
measured DM relic density from Planck Aghanim:2018eyx , which is about two
orders of magnitude lower than that required by the AMS-02 data. This is
usually referred to as the “boost factor” problem for DM annihilation
explanation of CR positron excess Cirelli:2008pk ; Cholis:2008hb , which is
unlikely to be explained by the local DM density clumps Diemand:2008in ;
Springel:2008by . Gamma rays produced from DM annihilation in regions with
high DM density, especially in dwarf spheroidal satellite galaxies (dSphs),
can be used to constrain DM properties Ackermann:2015zua . Relevant
observation from Fermi-LAT has set severe upper limits on DM annihilation
cross section, which exclude the large cross section favored by the AMS-02
positron data for some DM annihilation channels Lin:2014vja ; Xiang:2017jou .
During the epoch of recombination, the charged particles generated from DM
annihilation could affect the temperature and polarization anisotropy of the
cosmic microwave background (CMB) Chen:2003gz ; Padmanabhan:2005es ;
Slatyer:2015kla ; Slatyer:2015jla ; Galli:2009zc . The precise measurement
performed by Planck has placed severe constraints on DM annihilation cross
section Aghanim:2018eyx . The constraints from CMB are more stringent than
that from dSphs gamma rays for some DM annihilation channels, and are free of
some astrophysical uncertainties compared to the results from CR observations,
which arise from the large-scale structure formation, DM density profile, etc.
Slatyer:2015jla . To reconcile the contradictions mentioned above, a more
detailed DM annihilation model is required.

The typical velocity of DM particles in our galactic halo ($\sim 10^{-3}c$,
with $c$ the speed of light) is much higher than that in nearby dSphs ($\sim
10^{-4}c$) and during recombination ($\sim 10^{-8}c$), but two orders of
magnitude lower than that during thermal freeze-out of WIMPS. Therefore, if
the DM annihilation cross section is velocity-dependent in a non-monotonic way
and peaks around the typical DM velocity in the galactic halo, it can survive
the constraints from the CMB and dSphs gamma rays, and still generate the
desired boost factor. In our recent work Ding:2021zzg , we considered the
Sommerfeld mechanism to reconcile the contradictions mentioned above, which
occurs when the annihilating particles at low velocity self-interact via a
long-range attractive potential https://doi.org/10.1002/andp.19314030302 ;
Hisano:2003ec ; Hisano:2002fk ; Cirelli:2007xd ; ArkaniHamed:2008qn ;
Pospelov:2008jd ; March-Russell:2008klu ; Iengo:2009ni ; Cassel:2009wt .
Although the $s$-wave Sommerfeld-enhanced DM annihilation is highly
constrained due to the monotonically increasing cross section with decreasing
DM velocity, the $p$-wave Sommerfeld-enhanced DM annihilation is still a
viable possibility. Due to the additional velocity-square factor in $p$-wave
annihilation cross section and the saturation of Sommerfeld enhancement under
a certain DM velocity, the underlying DM annihilation cross section can
naturally have the desired peak structure.

Breit-Wigner enhancement is another extensively studied mechanism for
velocity-dependent cross section, which occurs when particles annihilate via a
narrow resonance Ibe:2008ye ; Guo:2009aj ; Bi:2009uj ; Bi:2011qm ; Bai:2017fav
; Li:2015tka ; Xiang:2017jou . In Ref. Xiang:2017jou , Breit-Wigner enhanced
$s$-wave DM annihilation was considered to reconcile the contradictions
mentioned above. The case where the resonance mass is below a threshold (twice
the DM mass) is ruled out due to the monotonically increasing cross section
with decreasing DM velocity. When the resonance mass is above the threshold,
parameter regions exist to account for these observations consistently but
only the parameters with extremely tiny values of mass deviation from the
threshold and decay width of the resonance can survive.

In this work, we consider DM particles annihilating through a $p$-wave process
with Breit-Wigner enhancement. We take into account the impact of the kinetic
decoupling effect on DM thermal relic density. We systematically calculate the
velocity-dependent astrophysics factor ($J$-factor) for the $p$-wave Breit-
Wigner enhanced DM annihilation for 15 nearby dSphs and derive the
corresponding upper limits of the DM annihilation cross section with the dSphs
gamma-ray data from Fermi-LAT. Through a combined analysis of the observation
data of CR positron excess, DM thermal relic density, dSphs gamma rays, and
CMB, we obtain the parameter regions that can explain all these observables
consistently in the $p$-wave Breit-Wigner enhanced DM annihilation model. We
find that, unlike the $s$-wave scenario where the parameter regions that can
explain the large boost factor are severely limited by the constraints from
dSphs gamma rays and CMB, in the $p$-wave scenario, due to the extra velocity-
square factor, the cross section is highly suppressed at extremely low DM
velocity hence the constraints from dSphs gamma rays and CMB are alleviated.
In addition, whether the resonance mass is above or below the threshold, there
exist parameter regions that can account for all these observations
simultaneously. For the same reason as in our recent work Ding:2021zzg , we do
not consider the constraints from other observables, such as the gamma rays
from the galactic center (GC) observed by H.E.S.S. Abdallah:2016ygi  and the
isotropic gamma ray background (IGRB) measured by Fermi-LAT Ackermann:2015tah
due to the additional uncertainties they involve.

This paper is organized as follows. In Sec. II, we derive the AMS-02 favored
DM mass and annihilation cross section through fitting to the updated AMS-02
CR positron data in the DM annihilation scenario. In Sec. III, we briefly
describe the Breit-Wigner enhancement mechanism. In Sec. IV, we discuss the
constraints from the observed DM thermal relic density. In Sec. V, we
calculate the upper limits of the DM annihilation cross section from gamma-ray
data of 15 nearby dSphs measured by Fermi-LAT. In Sec. VI, we discuss the CMB
bound on the DM annihilation cross section given by Planck. In Sec. VII, we
identify the parameter regions that can simultaneously explain all these
relevant observations. We summarize our conclusions in Sec. VIII.

##  II DM annihilation explanation of the AMS-02 positron excess

The propagation of CR positrons in the galactic halo can be described by
diffusion models Ginzburg:1990sk ; Strong:2007nh . For the CR source and
propagation, we adopt the same model from our previous work in Ref.
Ding:2021zzg . We use the public code GALPROP v54 Strong:1998pw ;
Moskalenko:2001ya ; Strong:2001fu ; Moskalenko:2002yx ; Ptuskin:2005ax  to
solve the propagation equation numerically. We focus on the annihilation
processes of DM particles $\chi$ to final states of two leptons
$\mu^{+}\mu^{-}$ and $\tau^{+}\tau^{-}$. The Monte-Carlo event generator
PYTHIA 8.2 Sjostrand:2014zea  is used to simulate the positron spectrum
produced by DM annihilation. The DM particle mass $m_{\chi}$, velocity-
weighted annihilation cross section
$\langle\sigma_{\text{ann}}v_{\text{rel}}\rangle$, and an extra normalization
factor of the background flux of CR positrons $C_{e^{+}}$ are free parameters
to be determined. We use the Bayesian analysis to perform a global fit to the
latest AMS-02 positron data Aguilar:2019owu . The prior distributions of the
free parameters are uniform distributions whose prior ranges are listed in
Tab. 1. The likelihood function is assumed to be Gaussian

$\mathcal{L}=\prod_{i}\rm{exp}[-\frac{(\Phi_{i}-\Phi_{\rm{exp,i}})^{2}}{2\sigma^{2}_{\rm{exp,i}}}]\
,$ (1)

where $\Phi_{i}$ is the predicted value, $\Phi_{\rm{exp,i}}$ and
$\sigma_{\rm{exp,i}}$ are the measured value and uncertainty of the CR
positron flux in energy bin $i$. To efficiently explore the parameter space,
we adopt the MultiNest sampling algorithm Feroz:2007kg ; Feroz:2008xx ;
Feroz:2013hea . We use $\chi^{2}=-2\rm{ln}\mathcal{L}$ to characterize the
goodness-of-fit.

channel | $\rm{log_{10}}(m_{\chi}/\rm{GeV})$ | $\rm{log_{10}}(\langle\sigma_{\text{ann}}v_{\text{rel}}\rangle/\rm{cm^{3}\ s^{-1}})$ | $C_{e^{+}}$ | $\chi^{2}/\text{d.o.f}$
---|---|---|---|---
Prior range: [1, 4] | Prior range: [-26, -21] | Prior range: [0.1, 10]
(Mean, $\sigma$) | Best-fit | (Mean, $\sigma$) | Best-fit | (Mean, $\sigma$) | Best-fit
$\mu^{+}\mu^{-}$ | $2.85\pm 0.02$ | 2.85 | $-23.65\pm 0.03$ | -23.66 | $1.69\pm 0.02$ | 1.69 | 52.22/32
$\tau^{+}\tau^{-}$ | $3.24\pm 0.04$ | 3.24 | $-22.81\pm 0.05$ | -22.8 | $1.53\pm 0.04$ | 1.53 | 21.46/32
Table 1: Prior ranges, posterior means, standard deviations, and best-fit
values of $m_{\chi}$, $\langle\sigma_{\text{ann}}v_{\text{rel}}\rangle$, and
$C_{e^{+}}$ for $\mu^{+}\mu^{-}$ and $\tau^{+}\tau^{-}$ channels. The values
of $\chi^{2}/\text{d.o.f}$ are also listed as an estimation of the goodness-
of-fit.

Figure 1:  Left) Regions of DM mass and annihilation cross section favored by
the AMS-02 positron data Aguilar:2019owu  at 95% C.L. The blue (green) region
corresponds to the annihilation channel of $\mu^{+}\mu^{-}$
($\tau^{+}\tau^{-}$). In the assumption of velocity-independent DM
annihilation cross section, the solid blue (green) curve represents the upper
limits at $95\%$ C.L. derived from the Fermi-LAT data on dSphs gamma rays for
the $\mu^{+}\mu^{-}$ ($\tau^{+}\tau^{-}$) channel, which are taken from Ref.
Ackermann:2015zua , and the dashed blue (green) curve stands for the upper
limit at $95\%$ C.L. derived from the Planck measurements of CMB
Aghanim:2018eyx  for the $\mu^{+}\mu^{-}$ ($\tau^{+}\tau^{-}$) channel. Right)
Best-fit CR positron fluxes (solid lines) for two different annihilation
channels. The dashed and dotted blue curves stand for the contributions from
DM annihilation and secondary positron backgrounds. The blue and green curves
correspond to the $\mu^{+}\mu^{-}$ and $\tau^{+}\tau^{-}$ channels,
respectively.

The best-fit values, posterior means, and standard deviations of the fitting
parameters and the goodness-of-fit for $\mu^{+}\mu^{-}$ and $\tau^{+}\tau^{-}$
channels are summarized in Tab. 1. In the left panel of Fig. 1, we show the
regions favored by the AMS-02 positron data in the
$(m_{\chi},\langle\sigma_{\text{ann}}v_{\text{rel}}\rangle)$ plane at $95\%$
C.L., which suggest that $m_{\chi}\approx 1-1.4\ \rm{TeV}$ with
$\langle\sigma_{\text{ann}}v_{\text{rel}}\rangle\approx 3.2-5.2\times
10^{-24}\ \rm{cm^{3}\ s^{-1}}$ is favored for the $\mu^{+}\mu^{-}$ channel and
$m_{\chi}\approx 1.4-2.2\ \rm{TeV}$ with
$\langle\sigma_{\text{ann}}v_{\text{rel}}\rangle\approx 1.1-2.1\times
10^{-23}\ \rm{cm^{3}\ s^{-1}}$ for the $\tau^{+}\tau^{-}$ channel. The upper
limits at $95\%$ C.L. derived from the Fermi-LAT data on dSphs gamma rays
Ackermann:2015zua  and the Planck measurement of CMB Aghanim:2018eyx  (see
Sec. VI for details) in the assumption of constant DM annihilation cross
section are also shown for comparison. The AMS-02 favored regions are
compatible with the constraints from dSphs gamma rays in $\mu^{+}\mu^{-}$
channel, but in strong tension with that in $\tau^{+}\tau^{-}$ channel due to
tremendous photons from the hadronic decay of tauons. In both channels, the
AMS-02 favored regions are totally excluded by the constraints from CMB. In
the right panel of Fig. 1, we show the calculated CR positron flux with the
best-fit DM parameters, as well as the secondary positron backgrounds. The
positron spectra from DM annihilation through the $\mu^{+}\mu^{-}$ channel are
relatively narrow compared with the observations. Nevertheless, DM
annihilation through the $\tau^{+}\tau^{-}$ channel predicts broader spectra,
which is in good agreement with the AMS-02 positron data. This can also be
seen from the values of $\chi^{2}/\text{d.o.f}$ in Tab. 1. Hence, we mainly
focus on the $\tau^{+}\tau^{-}$ channel in the following analysis. The
resonance coupling dominantly to tauons can be achieved via the flavor-
specific scenarios which allow the mediator to couple dominantly to one flavor
in a technically natural way Batell:2017kty ; Marsicano:2018vin , or the
lepton-specific scenarios in which the coupling of the mediator to one flavor
is proportional to the corresponding lepton mass Batell:2016ove ;
DAmbrosio:2002vsn .

##  III Breit-Wigner enhancement

We consider a simplified model where two fermionic DM particles directly
annihilate into a pair of charged leptons via a narrow resonance $R$. We adopt
two auxiliary parameters $\delta$ and $\gamma$ to express the mass $M$ and
decay width $\Gamma$ of the resonance

$\delta=1-M^{2}/4m^{2}_{\chi}\ \rm{and}\ \gamma=\Gamma/M\ .$ (2)

When the resonance mass is close to twice the DM mass and the decay width is
narrow, the annihilation cross section is sensitive to the relative velocity
of incident DM particles $v_{\rm{rel}}$. For a vector resonance, the DM
annihilation is an $s$-wave process and the corresponding cross section has a
general formula of Ibe:2008ye ; Bi:2009uj ; Bi:2011qm ; Xiang:2017jou

$\sigma_{\rm{ann}}v_{\rm{rel}}=\frac{a}{(\delta+v_{\rm{rel}}^{2}/4c^{2})^{2}+\gamma^{2}}\
,$ (3)

where $a$ is a global factor. For a scalar resonance, the DM annihilation is a
$p$-wave process and the corresponding cross section has an extra velocity-
square factor. For simplicity, we parameterize the cross section with a global
factor $b$ as

$\sigma_{\rm{ann}}v_{\rm{rel}}=\frac{bv_{\rm{rel}}^{2}/c^{2}}{(\delta+v_{\rm{rel}}^{2}/4c^{2})^{2}+\gamma^{2}}\
.$ (4)

In both cases, for a given relative velocity of incident DM particles, when
$\delta>0$, namely, $M<2m_{\chi}$, the DM annihilation cross section increases
monotonically with decreasing $\delta$ and $\gamma$. However, for the case of
$\delta<0$, namely, $M>2m_{\chi}$, the DM annihilation cross section remains
increasing monotonically with decreasing $\gamma$ while reaching a maximum at
$\delta=-v_{\rm{rel}}^{2}/4c^{2}$. In the case of vector resonance, the
annihilation cross section scales as $c^{4}/v^{4}_{\rm{rel}}$ when
$v_{\rm{rel}}^{2}/c^{2}\gg|\delta|,\gamma$, and remains almost unchanged with
$v_{\rm{rel}}$ due to the saturation of the Breit-Wigner enhancement when
$v_{\rm{rel}}^{2}/c^{2}\ll|\delta|,\gamma$. In the case of scalar resonance,
when $v_{\rm{rel}}^{2}/c^{2}\gg|\delta|,\gamma$, the annihilation cross
section scales as $c^{2}/v^{2}_{\rm{rel}}$. When
$v_{\rm{rel}}^{2}/c^{2}\ll|\delta|,\gamma$, the Breit-Wigner enhancement
saturates, and the annihilation cross section scales as
$v^{2}_{\rm{rel}}/c^{2}$.

##  IV Constraints from DM thermal relic density

Figure 2: The evolution of DM density $Y$ as a function of $x$. The blue,
green, cyan, and red lines correspond to $x_{\rm{kd}}=50,\ 5\times 10^{2},\
5\times 10^{3},\ \infty$, respectively. Parameters are set at
$\delta=10^{-7},\ \gamma=10^{-7}$, $m_{\chi}=1.7\ \rm{TeV}$, and $b$ is fixed
to reproduce the correct DM thermal relic density in the case without kinetic
decoupling. The orange line is the evolution of $Y_{\rm{eq}}$, and the black
line is the observed DM relic abundance from Planck Aghanim:2018eyx .  Figure
3:  Value of $\left<\sigma_{\rm{ann}}v_{\rm{rel}}\right>$ calculated from Eq.
(6) with $v_{0}=220\ \rm{km\ s^{-1}}$ as a function of $\delta$ and $\gamma$,
where DM mass is fixed at the best-fit value of AMS-02 positron data in
$\tau^{+}\tau^{-}$ channel $m_{\chi}=1.7\ \rm{TeV}$ and $b$ is derived from
the correct DM thermal relic density. $x_{\rm{kd}}$ in the lower, middle, and
upper two panels are $50,\ 500,\ \infty$, respectively. The left column is for
the case of $\delta>0$, and the right column is for the case of $\delta<0$.

The thermal evolution of DM density can be described by the Boltzmann equation
Kolb:1988aj

$\frac{dY}{dx}=-\sqrt{\frac{\pi}{45}}m_{\rm{pl}}m_{\chi}g_{\ast}^{-1/2}g_{\ast
s}x^{-2}\left<\sigma_{\rm{ann}}v_{\rm{rel}}\right>\left(Y^{2}-Y_{eq}^{2}\right)\
,$ (5)

where $Y=n_{\chi}/s$ is the DM number density $n_{\chi}$ normalized by the
entropy density $s$, $Y_{\rm{eq}}=0.145(g/g_{\ast s})x^{3/2}e^{-x}$ is the DM
density in equilibrium Jungman:1995df , $g$ is the number of degrees of
freedom of the DM particle, $x=m_{\chi}/T$, $T$ is the temperature of the
thermal bath, $m_{\rm{pl}}\approx 1.22\times 10^{19}\ \rm{GeV}$ is the Planck
mass, $g_{\ast}$ and $g_{\ast s}$ are the effective relativistic degrees of
freedom for energy and entropy density, respectively Steigman:2012nb . If the
DM particles are in thermal equilibrium with temperature $T_{\chi}$, the
velocity distribution of DM particles is the Maxwell-Boltzmann distribution,
characterized by the most probable velocity $v_{0}$, which is related to the
DM temperature as $v_{0}/c=\sqrt{2T_{\chi}/m_{\chi}}$. Then, the velocity-
averaged cross section of the $p$-wave Breit-Wigner enhanced DM annihilation
can be expressed as

$\langle\sigma_{\rm{ann}}v_{\rm{rel}}\rangle=\frac{b}{v_{0}^{3}}\sqrt{\frac{2}{\pi}}\int_{0}^{+\infty}\mathrm{d}v_{\rm{rel}}\
v_{\rm{rel}}^{2}e^{-\frac{v_{\rm{rel}}^{2}}{2v_{0}^{2}}}\frac{v_{\rm{rel}}^{2}/c^{2}}{(\delta+v_{\rm{rel}}^{2}/4c^{2})^{2}+\gamma^{2}}\
.$ (6)

The DM relic abundance has a relation with the value of $Y$ in the present day
($x=x_{\infty}$) as Chen:2013bi

$\Omega_{\chi}h^{2}\approx 2.76\times
10^{8}Y(x_{\infty})\left(\frac{m_{\chi}}{\rm{GeV}}\right)\ ,$ (7)

and the corresponding observed value from Planck is $\Omega_{\chi}h^{2}\simeq
0.1193$ Aghanim:2018eyx .

After freeze-out (when $Y-Y_{\rm{eq}}=\mathcal{O}(Y_{\rm{eq}})$, typically,
$x_{f}\equiv m_{\chi}/T_{f}\sim\mathcal{O}(10)$), DM particles remain
kinematically coupled to the thermal bath through elastic scattering. Kinetic
decoupling occurs later, when the momentum transfer rate drops below the
Hubble expansion rate. The temperature of DM kinetic decoupling $T_{\rm{kd}}$
depends on the model details of DM annihilation Bi:2011qm ; Bai:2017fav . To
perform a model-independent analysis, we take $T_{\rm{kd}}$ as a free
parameter in this work. Before the kinetic decoupling of DM, the DM particles
have the same temperature with the thermal bath $T_{\chi}=T$. After that,
$T_{\chi}$ drops as $a^{-2}$, while $T$ drops as $a^{-1}$, where $a$ is the
scale factor of the Universe, and so $T_{\chi}=T^{2}/T_{\rm{kd}}$ Chen:2001jz
; Hofmann:2001bi ; Bringmann:2006mu ; Bi:2011qm .

In Fig. 2, we show the evolution of DM density with different values of DM
kinetic decoupling temperature $x_{\rm{kd}}\equiv m_{\chi}/T_{\rm{kd}}=50,\
5\times 10^{2}$ and $5\times 10^{3}$. We also give the results in the limit of
$x_{\rm{kd}}=\infty$, corresponding to the case without kinetic decoupling.
After kinetic decoupling, the DM annihilation becomes more significant due to
the more rapid decrease in the DM temperature and more significant increase in
Breit-Wigner enhancement, which reduces the DM abundance more efficiently. If
the kinetic decoupling occurs much later than the freeze-out, namely,
$x_{\rm{kd}}\gg x_{f}$, such effect becomes not important compared with the
case without kinetic decoupling. However, if the kinetic decoupling occurs at
nearly the same epoch as that for the freeze-out, the efficient DM
annihilation would reduce DM relic abundance by about one order of magnitude.

Under the requirement of reproducing the observed DM relic density, the value
of $b$ in the expression of the DM annihilation cross section can be
determined for given parameters $m_{\chi}$, $\delta$ and $\gamma$. For a given
velocity distribution of DM particles characterized by the most probable
velocity $v_{0}$, the $p$-wave Breit-Wigner enhanced DM annihilation cross
section $\left<\sigma_{\rm{ann}}v_{\rm{rel}}\right>$ can be calculated from
the determined value of $b$ through Eq. (6). In Fig. 3, we show the predicted
value of $\left<\sigma_{\rm{ann}}v_{\rm{rel}}\right>_{\rm{halo}}$ with
$v_{0}=v_{\rm{halo}}\approx 220\ \rm{km\ s^{-1}}$ in the galactic halo as a
function of $\delta$ and $\gamma$ after the constraints from the DM thermal
relic density, for three typical values of DM kinetic decoupling temperature
$x_{\rm{kd}}=50,\ 500,\ \infty$ in both cases of $\delta>0$ and $\delta<0$.
The DM mass is fixed at the best-fit value $m_{\chi}=1.7\ \rm{TeV}$ for AMS-02
positron data in the $\tau^{+}\tau^{-}$ channel. In the case of $\delta>0$,
although smaller values of $\delta$ and $\gamma$ lead to stronger Breit-Wigner
enhancement at freeze-out and hence a smaller $b$, corresponding predicted
value of $\left<\sigma_{\rm{ann}}v_{\rm{rel}}\right>_{\rm{halo}}$ is larger
due to the rapid increase of the rescale factor in Eq. (6) with decreasing
$\delta$ and $\gamma$. In the case of $\delta<0$,
$\left<\sigma_{\rm{ann}}v_{\rm{rel}}\right>_{\rm{halo}}$ reaches a maximum at
$\delta\sim-v_{\rm{halo}}^{2}/c^{2}$ and $\gamma\lesssim
v_{\rm{halo}}^{2}/c^{2}$, which is also derived from the behavior of the
rescale factor in Eq. (6). The figure shows that the case of $\delta<0$ allows
larger boost factor than the case of $\delta>0$. As mentioned earlier, since a
smaller $x_{\rm{kd}}$ (or higher kinetic decoupling temperature) reduces the
DM relic abundance more efficiently, corresponding value of $b$ and hence the
value of $\left<\sigma_{\rm{ann}}v_{\rm{rel}}\right>_{\rm{halo}}$ are required
to be smaller in order to reproduce the observed DM thermal relic density.

##  V Constraints from dSphs gamma-ray observations from Fermi-LAT

###  V.1 Gamma-ray flux from DM annihilation in dSphs and $p$-wave Breit-
Wigner enhanced $J$-factor

The dSphs of the Milky Way are promising targets for the indirect detection of
DM annihilation due to their large DM concentration, low diffuse galactic
gamma-ray foreground, and lack of conventional astrophysical gamma-ray
production mechanisms Mateo:1998wg ; Grcevich:2009gt . The gamma-ray flux from
DM self-annihilation within an energy range ($E_{\min}$, $E_{\max}$) from a
solid angle $\Delta\Omega$ can be expressed as Ding:2021zzg

$\Phi_{\gamma}(\Delta\Omega,E_{\min},E_{\max})=\frac{C}{8\pi
m_{\chi}^{2}}\int_{E_{\min}}^{E_{\max}}\frac{dN_{\gamma}}{dE_{\gamma}}{dE_{\gamma}}\times
J\ ,$ (8)

where $C$ is the velocity-independent part of the DM annihilation cross
section, and the $J$-factor contains the information of the DM distribution
and the velocity-dependent part of the cross section. In the simplest case
where the DM annihilation cross section is velocity independent,
$C=\sigma_{\rm{ann}}v_{\rm{rel}}$, and the corresponding $J$-factor is given
by

$J_{0}=\int_{\Delta\Omega}d\Omega\int_{\text{l.o.s}}\rho^{2}(\mathbf{r})dl\ ,$
(9)

where $\rho(\mathbf{r})$ is the DM density distribution in dSphs. The $J_{0}$
factor contains only the astrophysical information. In the case of $s$-wave
Breit-Wigner enhanced DM annihilation, $C=a$, and the corresponding $s$-wave
$J$-factor can be written as

$J_{s}=\int_{\Delta\Omega}d\Omega\int_{l.o.s}dl\int d^{3}\bm{v}_{1}\int
d^{3}\bm{v}_{2}f\left(\bm{r},\bm{v}_{1}\right)f\left(\bm{r},\bm{v}_{2}\right)\frac{1}{(\delta+v_{\text{rel}}^{2}/4c^{2})^{2}+\gamma^{2}}\
,$ (10)

where $f\left(\bm{r},\bm{v}\right)$ is the phase-space distribution of DM
particles within dSphs. Similarly, in the case of $p$-wave Breit-Wigner
enhanced DM annihilation, $C=b$, and the corresponding $p$-wave $J$-factor is

$J_{p}=\int_{\Delta\Omega}d\Omega\int_{l.o.s}dl\int d^{3}\bm{v}_{1}\int
d^{3}\bm{v}_{2}f\left(\bm{r},\bm{v}_{1}\right)f\left(\bm{r},\bm{v}_{2}\right)\frac{v_{\text{rel}}^{2}/c^{2}}{(\delta+v_{\text{rel}}^{2}/4c^{2})^{2}+\gamma^{2}}\
.$ (11)

Name | $J_{p}\ [\times 10^{25}]$ | $J_{p}\ [\times 10^{25}]$ | $J_{p}\ [\times 10^{21}]$ | $J_{p}\ [\times 10^{17}]$ | $J_{0}\ [\times 10^{18}]$
---|---|---|---|---|---
$(\delta,\gamma)=(-10^{-8},10^{-8})$ | $(10^{-8},10^{-8})$ | $(10^{-6},10^{-6})$ | $(10^{-4},10^{-4})$
Bootes I | $19.0$ | $11.9$ | $14.9$ | $15.0$ | $6.3$
Canes Venatici II | $1.3$ | $1.0$ | $1.1$ | $1.1$ | $0.8$
Carina | $2.7$ | $2.0$ | $2.3$ | $2.3$ | $1.3$
Coma Berenices | $17.0$ | $12.3$ | $14.5$ | $14.6$ | $10.0$
Draco | $36.6$ | $19.1$ | $26.1$ | $26.3$ | $6.3$
Fornax | $8.7$ | $4.3$ | $6.1$ | $6.3$ | $1.6$
Hercules | $2.5$ | $1.7$ | $2.1$ | $2.1$ | $1.3$
Leo II | $0.9$ | $0.6$ | $0.8$ | $0.8$ | $0.4$
Leo IV | $1.9$ | $1.3$ | $1.5$ | $1.5$ | $0.8$
Sculptor | $18.8$ | $10.5$ | $13.9$ | $13.7$ | $4.0$
Segue 1 | $54.7$ | $41.6$ | $46.9$ | $47.0$ | $31.6$
Sextans | $7.7$ | $5.0$ | $6.3$ | $6.3$ | $2.5$
Ursa Major II | $46.8$ | $32.3$ | $39.8$ | $39.8$ | $20.0$
Ursa Minor | $39.9$ | $19.5$ | $28.2$ | $28.3$ | $6.3$
Willman 1 | $21.8$ | $16.4$ | $18.9$ | $18.7$ | $12.6$
Table 2: The values of $p$-wave Breit-Wigner enhanced $J$-factor $J_{p}$ (in
units of $\mathrm{GeV^{2}\ cm^{-5}}$) of 15 dSphs at four typical values of
$(\delta,\gamma)=(-10^{-8},10^{-8}),(10^{-8},10^{-8}),(10^{-6},10^{-6})$, and
$(10^{-4},10^{-4})$. The last column is the values of the normal $J$-factor
$J_{0}$ taken from Ackermann:2015zua .  Figure 4: The ratio of the $p$-wave
Breit-Wigner enhanced $J$-factor to the normal one $J_{p}/J_{0}$ of dSph Segue
1 as a function of $\delta$ and $\gamma$. The left panel is for the case of
$\delta>0$, and the right panel is for the case of $\delta<0$.

We calculate the spherically-symmetric and isotropic phase-space distribution
$f_{\chi}\left(r,v\right)$ of DM particles in 15 nearby dSphs (listed in Tab.
2). The typical velocity of DM particles in these 15 dSphs is
$v_{\rm{dSph}}\sim 10-20\ \rm{km\ s^{-1}}$. The method is based on the
Eddington’s formula 2008gady.book…..B  and the assumption of Navarro-Frenk-
White (NFW) profile Navarro:1996gj  of DM density in dSphs. The details of the
method can be found in our previous work Ding:2021zzg .

Instead of an integral over the line of sight, the expression of $J_{p}$ can
be rewritten in an integral over the radial distance from the center of the
target dSph as

$\displaystyle J_{p}$
$\displaystyle=\frac{32\pi^{3}}{D^{2}}\int_{0}^{r_{\max}}r^{2}dr\int_{0}^{v_{\rm{esc}}(r)}v_{1}^{2}f_{\chi}\left(r,v_{1}\right)dv_{1}\int_{0}^{v_{\rm{esc}}(r)}v_{2}^{2}f_{\chi}\left(r,v_{2}\right)dv_{2}$
(12)
$\displaystyle\times\int_{0}^{\pi}\sin\theta\frac{(v_{1}^{2}+v_{2}^{2}-2v_{1}v_{2}\cos\theta)/c^{2}}{(\delta+(v_{1}^{2}+v_{2}^{2}-2v_{1}v_{2}\cos\theta)/4c^{2})^{2}+\gamma^{2}}d\theta\
,$

where $D$ is the distance from the solar system to the center of the dSph
under consideration, whose value can be taken from Ref. Ackermann:2015zua .
The integral over solid angle is performed over a circular region with
$\Delta\Omega\sim 2.4\times 10^{-4}\ \rm{sr}$, namely, angular radius of
$\beta=0.5^{\circ}$, which corresponds to the maximal radius
$r_{\rm{max}}=D\cdot\rm{sin}\beta$. This integral range is large enough to
envelop the region where there is significant DM annihilation for the NFW
profile Ackermann:2013yva . The integration over the DM velocity is cut off at
the escape velocity $v_{\mathrm{esc}}(r)$ at radius $r$.

In Tab. 2, we show the values of $J_{p}$ at four typical values of
$(\delta,\gamma)=(-10^{-8},10^{-8}),(10^{-8},10^{-8}),(10^{-6},10^{-6}),(10^{-4},10^{-4})$
and $J_{0}$ for 15 nearby dSphs. The dSphs that have similar $J_{0}$ factors
may have different DM velocity distributions and hence quite different values
of $J_{p}$ factor. For instance, the value of $J_{p}$ factor at
$(\delta,\gamma)=(10^{-4},10^{-4})$ of Ursa Minor ($28.3\times
10^{17}~{}\mathrm{GeV^{2}\ cm^{-5}}$) is almost twice the value of Bootes I
($15\times 10^{17}~{}\mathrm{GeV^{2}\ cm^{-5}}$), although their $J_{0}$
factors are nearly the same ($6.3\times 10^{18}~{}\mathrm{GeV^{2}\ cm^{-5}}$).
Consequently, the relative importance of $J_{p}$ factors among the dSphs in
their contribution to the gamma-ray flux has changed. For instance, Ursa Minor
(Bootes I) becomes more (less) important in $J_{p}$ than in $J_{0}$. However,
Segue 1 (Leo II) remains the brightest (faintest) dSph among all the 15 dSphs.
In Fig. 4, we show the ratio of the $p$-wave Breit-Wigner enhanced $J$-factor
$J_{p}$ to the normal one $J_{0}$ of dSph Segue 1 as a function of $\delta$
and $\gamma$ in both cases of $\delta>0$ and $\delta<0$. Approximately,
$J_{p}$ scales as
$J_{0}\cdot(v_{\rm{dSph}}^{2}/c^{2})/((\delta+v_{\rm{dSph}}^{2}/c^{2})^{2}+\gamma^{2})$.
Hence, $J_{p}$ increases monotonically with decreasing $\delta$ and $\gamma$
in the case of $\delta>0$. When $\delta<0$, $J_{p}$ remains increasing
monotonically as $\gamma$ decreases while reaching a maximum at $\delta\sim-
v_{\rm{dSph}}^{2}/c^{2}$. $J_{p}$ is smaller than $J_{0}$ when
$|\delta|,\gamma\gtrsim v_{\rm{dSph}}/c$, but is greater than $J_{0}$ when
$|\delta|,\gamma\lesssim v_{\rm{dSph}}/c$, which can also be seen from Tab. 2.
When $|\delta|,\gamma\gg v_{\rm{dSph}}^{2}/c^{2}$, $J_{p}$ has almost the same
values in both cases of $\delta>0$ and $\delta<0$.

###  V.2 Upper limits on the DM annihilation cross section

Figure 5:  Upper limits on DM annihilation cross section at $95\%$ C.L. as a
function of $\delta$ and $\gamma$ derived from a combined analysis on the
Fermi-LAT gamma-ray data of the 15 nearby dSphs Ackermann:2015zua . The DM
annihilation channel is $\tau^{+}\tau^{-}$, and $m_{\chi}$ is fixed at $1.7\
\rm{TeV}$. The left panel is the for case of $\delta>0$, and the right panel
is for the case of $\delta<0$. Figure 6:  The same upper limits with Fig. 5 as
a function of $m_{\chi}$. The blue, cyan and purple curves correspond to the
parameters of $(|\delta|,\gamma)=(10^{-8},10^{-8}),(10^{-7},10^{-7})$ and
$(10^{-6},10^{-6})$, respectively. The red line is the upper limits on
constant DM annihilation cross section, which are taken from Ref.
Ackermann:2015zua . The left panel is for the case of $\delta>0$, and the
right panel is $\delta<0$. In both panels, the green regions are favored by
the AMS-02 CR positron data Aguilar:2019owu  at $95\%$ C.L.

We use the six years of Fermi-LAT PASS8 data on gamma rays from the 15 nearby
dSphs listed in Tab. 2, and adopt the same approach of standard delta-log-
likelihood as in our recent work Ding:2021zzg  to place constraints on the
velocity-independent part of the DM annihilation cross section $C$. With the
$p$-wave Breit-Wigner enhanced $J$-factor $J_{p}$ calculated in the previous
subsection, it is straightforward to calculate the gamma-ray flux produced
from the DM annihilation in dSphs through Eq. (8). We derive the upper limits
on $b$ at $95\%$ C.L. from the dSphs gamma-ray data from Fermi-LAT
Ackermann:2015zua  as a function of $m_{\chi}$ for specific values of $\delta$
and $\gamma$. To facilitate the comparison with the AMS-02 data, we convert
the obtained upper limits on the value of $b$ to that on
$\langle\sigma_{\rm{ann}}v_{\rm{rel}}\rangle$ through Eq. (6) with
$v_{0}=v_{\rm{halo}}$.

In Fig. 5, we show the upper limits as a function of $\delta$ and $\gamma$
with DM mass fixed at the best-fit value $m_{\chi}=1.7\ \rm{TeV}$ for AMS-02
positron data in the $\tau^{+}\tau^{-}$ channel in both cases of $\delta>0$
and $\delta<0$. When $\delta>0$, the upper limits are more stringent at
smaller values of $\delta$ and $\gamma$, which can be expected from the
adverse behavior of $J_{p}$ factor. In the case of $\delta<0$, the most
stringent upper limits are located at $\delta\sim-v_{\rm{dSph}}^{2}/c^{2}$ and
$\gamma\lesssim v_{\rm{dSph}}^{2}/c^{2}$, where the values of $J_{p}$ factor
are the largest. The upper limits reach a maximum at $\delta\sim-
v_{\rm{halo}}^{2}/c^{2}$ and $\gamma\lesssim v_{\rm{halo}}^{2}/c^{2}$ due to
the large rescale factor in Eq. (6) in this region. Fig. 5 shows that, only
within $|\delta|,\gamma\lesssim 10^{-7}$, the constraints from dSphs gamma
rays would exclude the AMS-02 favored cross section. In Fig. 6, we show the
upper limits as a function of DM mass at three typical values of
$(|\delta|,\gamma)=(10^{-8},10^{-8}),(10^{-7},10^{-7})$ and
$(10^{-6},10^{-6})$ in both cases of $\delta>0$ and $\delta<0$. The upper
limits on constant DM annihilation cross section are also shown for
comparison. The figure shows that the upper limits on $p$-wave Breit-Wigner
enhanced DM annihilation cross section are much weaker than that on constant
cross section at $(|\delta|,\gamma)=(10^{-7},10^{-7})$ and
$(10^{-6},10^{-6})$, and can be compatible with the constraints from AMS-02
positron data. While for lower values of
$(|\delta|,\gamma)=(10^{-8},10^{-8})$, the contradiction still exists. This is
in agreement with Fig. 5.

##  VI Constraints from the CMB

Figure 7:  Left) Parameter regions which can simultaneously account for the
AMS-02 CR positron excess Aguilar:2019owu  and DM thermal relic density
Aghanim:2018eyx  in the $(\delta,\gamma)$ plane within the scenario of
$p$-wave Breit-Wigner enhanced DM annihilation for $\delta<0$. The blue,
green, and red regions correspond to $x_{\rm{kd}}=50,\ 500$ and $\infty$,
respectively. The gray region is excluded by the combined analysis of the
Fermi-LAT gamma-ray data of 15 nearby dSphs Ackermann:2015zua  at $95\%$ C.L.
The DM annihilation channel is $\tau^{+}\tau^{-}$. Right) The same as the left
panel but within the scenario of $s$-wave Breit-Wigner enhanced DM
annihilation. Parameter region above the orange line is excluded by the CMB
data from Planck Aghanim:2018eyx  at $95\%$ C.L.  Figure 8:  Parameter regions
which can simultaneously account for the AMS-02 CR positron excess
Aguilar:2019owu  and DM thermal relic density Aghanim:2018eyx  in the
$(\delta,\gamma)$ plane within the scenario of $p$-wave Breit-Wigner enhanced
DM annihilation for $\delta>0$. The green and red regions correspond to
$x_{\rm{kd}}=500$ and $\infty$, respectively. The gray region is excluded by
the combined analysis of the Fermi-LAT gamma-ray data of 15 nearby dSphs
Ackermann:2015zua  at $95\%$ C.L. The DM annihilation channel is
$\tau^{+}\tau^{-}$.

The recombination history of the Universe can be potentially modified by
energy injection into gas, photon-baryon plasma and background radiation from
DM annihilation, which can lead to modifications in the temperature and
polarization power spectra of CMB Chen:2003gz ; Padmanabhan:2005es . Thus, the
measurement on the anisotropy of CMB can be used to constrain the particle
nature of DM Galli:2009zc ; Slatyer:2009yq ; Finkbeiner:2011dx . An advantage
of this indirect detection method over the CR observations is that it is free
of some astrophysical uncertainties such as the DM density profile, large-
scale structure formation, etc. Slatyer:2015jla . Using a parameter
$f_{\rm{eff}}$ to describe the fraction of the energy released by the DM
annihilation process that is transferred to the intergalactic medium around
the redshift to which the CMB anisotropy data are most sensitive, namely,
$z\sim 600$, the upper limits on the DM annihilation cross section at $95\%$
C.L. recently reported by the Planck collaboration is Aghanim:2018eyx

$f_{\rm{eff}}\langle\sigma_{\rm{ann}}v_{\rm{rel}}\rangle/m_{\chi}\leq
3.2\times 10^{-28}\ \mathrm{cm^{3}\ s^{-1}\ GeV^{-1}}\ .$ (13)

We take the value of $f_{\rm{eff}}\approx 0.16$ for $\mu^{+}\mu^{-}$ and
$\approx 0.15$ for $\tau^{+}\tau^{-}$ channel at $m_{\chi}\sim\mathcal{O}(1)\
\rm{TeV}$ Madhavacheril:2013cna . Assuming a velocity-independent DM
annihilation cross section, we show in Fig. 1 the upper limits at $95\%$ C.L.
on DM annihilation cross section as a function of DM mass from the CMB data,
which indicate strong contradiction with the AMS-02 favored parameter regions.
For the $p$-wave Breit-Wigner enhanced DM annihilation, the cross section is
expected to be strongly suppressed by the velocity-square factor due to the
extremely low DM velocity of $\sim 1\ \mathrm{m\cdot s^{-1}}$ in the epoch of
recombination, which results in rather weak constraints from CMB.

##  VII Combined analysis

Figure 9:  $p$-wave Breit-Wigner enhanced DM annihilation cross section as a
function of the DM most probable velocity $v_{0}$ with the global factor $b$
fixed by the DM thermal relic density (blue solid). The DM annihilation
channel is $\tau^{+}\tau^{-}$, and the parameters are set at $\delta=2\times
10^{-7},\ \gamma=2\times 10^{-7},\ m_{\chi}=1.7\ \rm{TeV}$, and
$x_{\rm{kd}}=500$. For comparison, the case of $s$-wave Breit-Wigner enhanced
DM annihilation is also shown (cyan dashed). The green region represents the
cross section favored by AMS-02 positron data Aguilar:2019owu  at $v_{0}\sim
220\ \mathrm{km\ s^{-1}}$. The red region corresponds to the upper limits at
$95\%$ C.L. from a combined analysis of the Fermi-LAT gamma-ray data from the
15 nearby dSphs Ackermann:2015zua  with $v_{0}\sim 10-20\ \mathrm{km\
s^{-1}}$. The purple vertical line indicates the cross section excluded by
Planck CMB observations Aghanim:2018eyx  at $95\%$ C.L. with $v_{0}\sim 1\
\mathrm{m\ s^{-1}}$.

To identify the parameter space that can simultaneously explain the AMS-02
positron excess and reproduce the correct DM thermal relic density while
evading the constraints from dSphs gamma rays and CMB in the $p$-wave Breit-
Wigner enhanced DM annihilation, we perform a scan in the $(\delta,\gamma)$
plane. We consider three typical DM kinetic decoupling temperatures of
$x_{\rm{kd}}=50,500,\infty$ as benchmark values. In the left panel of Fig. 7,
we show the constraints at $95\%$ C.L. from these observations on parameter
regions for the case of $\delta<0$. To give the required boost factor which
can account for the CR positron excess, the allowed parameters are limited in
a narrow range within $|\delta|,\gamma\lesssim 10^{-6}$, which depends on
$x_{\rm{kd}}$. A lower kinetic decoupling temperature allows larger values of
$|\delta|$ and $\gamma$. The parameter regions within $|\delta|,\gamma\lesssim
10^{-7}$ are excluded by the constraints from dSphs gamma rays. Due to the
strong velocity suppression of the annihilation cross section during
recombination, the CMB data imposes almost no additional constraints to the
parameter regions. As a comparison, we also show the results of $s$-wave
Breit-Wigner enhanced DM annihilation in the right panel of Fig. 7, which
perform quite differently. It shows that most of the parameter regions that
can provide the required boost factor are excluded by the CMB data, and dSphs
gamma-ray data places stricter constraints than CMB. Due to the monotonic
increase in the DM annihilation cross section with decreasing DM velocity in
the case of $\delta>0$, the $s$-wave Breit-Wigner enhancement cannot
accommodate the observations of CR positron excess, dSphs gamma rays, and CMB
simultaneously. While in $p$-wave Breit-Wigner enhanced DM annihilation,
$\delta>0$ is still allowable. The corresponding allowed parameter regions are
shown in Fig. 8, which are similar with the case of $\delta<0$ in the left
panel of Fig. 7 except that the Breit-Wigner enhancement is not large enough
to explain the AMS-02 positron excess when the DM kinetic decoupling
temperature has the same scale as the freeze-out temperature
($x_{\rm{kd}}=50$).

According to the results above, only tiny values of $|\delta|$ and $\gamma$
($\lesssim 10^{-6}$) can survive all the constraints mentioned above. The
narrow decay width of the resonance can be realized through weak coupling of
the resonance with other particles. As for how to naturally realize the tiny
deviation of the resonance mass from the threshold, one possible way proposed
in Ref. Bai:2017fav  is through the nontrivial flavour symmetry-breaking in
the dark sector, where a resonance with a mass of almost $2m_{\chi}$ is
realized by assigning a particular symmetry-breaking mode and the tiny mass
deviation is induced by loop effect.

To illustrate how the DM annihilation cross section varies in different epochs
and regions of the Universe, in Fig. 9, we show
$\langle\sigma_{\rm{ann}}v_{\rm{rel}}\rangle$ as a function of the velocity
$v_{0}$ through Eq. (6) for a typical parameter set of $m_{\chi}=1.7\
\mathrm{TeV}$, $\delta=2\times 10^{-7}$, $\gamma=2\times 10^{-7}$, and
$x_{\rm{kd}}=500$. To reproduce the correct DM thermal relic density, the
cross section at freeze-out ($v_{0}\sim c/3$) is fixed at $\sim 6.3\times
10^{-28}~{}\mathrm{cm^{3}\ s^{-1}}$. When the velocity decreases, the cross
section increases as $c^{2}/v_{0}^{2}$ due to the Breit-Wigner enhancement,
and reaches a maximum of $\sim 1.5\times 10^{-23}~{}\mathrm{cm^{3}\ s^{-1}}$
at $v_{0}\sim 220~{}\mathrm{km\cdot s^{-1}}$, which is large enough to provide
the AMS-02 positron excess. As the velocity decreases further, the Breit-
Wigner enhancement saturates and the cross section decreases as
$v_{0}^{2}/c^{2}$. The cross section at $v_{0}\sim 10~{}\mathrm{km\cdot
s^{-1}}$ is almost one order of magnitude smaller than the upper limits
derived from a combined analysis of Fermi-LAT gamma-ray data on 15 nearby
dSphs. Finally, the cross section at $v_{0}\sim 1~{}\mathrm{m\cdot s^{-1}}$
becomes so low that it no longer suffers from the constraints from CMB
observations. We also show the velocity dependence of the $s$-wave Breit-
Wigner enhanced DM annihilation cross section with the same parameter set,
whose value at freeze-out is also fixed by the observed DM thermal relic
density. The cross section increases as $c^{4}/v_{0}^{4}$ and then remains
almost constant as the velocity decreases. The result shows that although the
$s$-wave Breit-Wigner enhancement can account for the large boost factor in
the case of $\delta>0$, it is incompatible with the observations of dSphs
gamma rays and CMB due to the monotonic increase in annihilation cross section
with decreasing DM velocity.

##  VIII Conclusions

In summary, $p$-wave Breit-Wigner enhanced DM annihilation can have
complicated velocity dependence compared with that for the $s$-wave case. We
have taken into account the impact of the kinetic decoupling effect on DM
thermal relic density. We have systematically calculated the velocity-
dependent $J$-factor for the $p$-wave Breit-Wigner enhanced DM annihilation
for 15 nearby dSphs and derived the corresponding upper limits of the DM
annihilation cross section through a combined analysis of the dSphs gamma-ray
data. We have identified the parameter regions that can simultaneously explain
the AMS-02 positron excess and the DM thermal relic density while evading the
constraints from dSphs gamma rays and CMB. Our results show that the allowed
parameter regions in $p$-wave Breit-Wigner enhanced DM annihilation are larger
than that in the case of $s$-wave. In addition, whether the resonance mass is
above or below the threshold, there exist parameter regions that can explain
these observations consistently.

###### Acknowledgements.

This work is supported in part by The National Key R&D Program of China No.
2017YFA0402204, the Key Research Program of the Chinese Academy of Sciences
(CAS), Grant NO. XDPB15, the CAS Project for Young Scientists in Basic
Research YSBR-006, and the National Natural Science Foundation of China (NSFC)
No. 11825506, No. 11821505, and No. 12047503.

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