# Research on electron and positron spectrum in the high-energy region based
on the gluon condensation model

Jin-tao Wu  Ming-jun Feng  Jian-hong Ruan11footnotetext: Corresponding author.

###### Abstract

Electron(positron), proton and nuclei can be accelerated to very high energy
by local supernova remnants (SNR). The famous excesses of electron and proton
(nuclei) potentially come from such kind of local sources. Recently, the DAMPE
experiment measured the electron spectrum (including both electrons and
positrons) of cosmic rays with high-accuracy. It provides an opportunity to
further explore the excess of electrons. According to the gluon condensation
(GC) theory, once GC occurs, huge number of gluons condense at a critical
momentum, and the production spectrum of electron and proton showing typical
GC characteristics. There are exact correlations between the electron and
proton spectrum from a same GC process. It is possible to interpret the power-
law break of cosmic rays in view of GC phenomenon, and predict one from
another based on the relations between electron and proton spectrum. In this
work, we point out the potential existence of a second excess in the electron
spectrum, the characteristics of this excess is derived from experimental data
of proton. We hope that the future DAMPE experiments will confirm the
existence of this second excess and support the result of GC model.

##  1 Introduction

In recent decades, much important developments have been made in space-based
and ground-based cosmic rays (CRs) experiments. With the advent of these next-
generation experiments, measurements of CRs are entering a high-precision era
and revealing a range of new phenomena. In particular, in recent years,
collaborations such as Alpha Magnetic Spectrometer (AMS-02) [1, 2], Fermi
Large Area Telescope (Fermi-LAT) [3], DArk Matter Particle Explorer (DAMPE)
[4, 5], and so on have provided more accurate experimental data of electron
and positron (or $e^{-}+e^{+}$) and proton spectrum. Depending on the particle
species, these data cover an energy range from about 1 GeV to several TeV. The
CRs with energies above 1 TeV usually come from distances within 1 kpc from
the solar system, highlighting the importance of nearby sources as detection
targets. The energy spectrum of CRs may be influenced by nearby sources such
as pulsar wind nebulae (PWNe) [6, 7, 8, 9, 10, 11, 12], supernova remnants
(SNRs) [13, 14, 15, 16, 17, 18, 19], and dark matter (DM) particle
annihilation or decay [20, 21, 22, 23, 24, 25, 26]. A commonly accepted idea
is that the detected spectrum should be a combination of contributions
originating from background and local SNR sources. The energy spectra of both
($e^{-}+e^{+}$) and protons are precisely interpreted using this approach with
appropriate nearby sources. The Payload for Antimatter Matter Exploration and
Light-nuclei Astrophysics (PAMELA) [27] discovered an excess of positron
spectra above 20 GeV, a remarkable finding confirmed by the sharp drop in the
positron spectrum observed by AMS-02 at an energy of 284 GeV [28]. The DAMPE
experiment found that the electron spectrum follows a power-law shape at
energies up to approximately 0.9 TeV, followed by a spectral break-off at the
same energy [4]. Moreover, the electron spectrum exhibits a single power law
below 0.9 TeV without any noticeable exotic features.

The AMS-02 experiment has also shown the significant excess in the nuclei
(protons, He, C, O) spectrum at energies from $\sim$200 GeV [29]. The
hardening of the proton spectrum has been confirmed by the DAMPE experiment,
which identified a significant excess of proton at 13.6 TeV [5]. This spectrum
excess was previously established observationally by the ATIC [30] and NUCLEON
[31]. The nucleon excess may also come from the same SNR [32, 33, 34, 16, 35,
36].

The gluon condensation (GC) model, proposed by Zhu Wei et al., is obtained
from the QCD kinetic evolution equation in condition of very high interaction
energies [37, 38, 39, 40, 41]. GC is the state in which the gluons in proton
(or nucleus) condense at a critical momentum, and the distribution of gluons
appearing as delta-function like. When high-energy cosmic rays entering the
atmosphere, the secondary particles such as electron-positron, pion, proton-
antiproton, etc., are produced efficiently by proton-proton or proton-nucleus
($p-p(A)$) collisions. If GC happens, the producing electron and proton will
be affected significantly and their energy spectra will show typical GC
features. Specifically, since $p+p(A)\rightarrow
p+\bar{p}+\pi^{\pm}+\pi^{0}+\text{others}$, the excess of leptons observed in
CRs may come from the secondary process $\pi^{0}\rightarrow\gamma\rightarrow
e^{+}+e^{-}$ in the GC state. The proton-antiproton excess may also be derived
from the same $p-p(A)$ collision. Therefore, an existing association between
the spectrum of electron and proton excesses and a correlation of GC
thresholds between them is given in the literature [42]. Based on the
measurement of electron of DAMPE, we assume a secondary additional excess of
electron with the position and magnitude estimated by the data of proton.

This paper is organized as follows. In section 2, both the spatially-dependent
propagation (SDP) model of CRs, the cosmic ray background SNR and its
injection spectrum, local SNR and the GC model are introduced briefly. Section
3 presents computational results obtained by using the DRAGON2 [43, 44]
numerical package to solve the SDP function of the background CRs while
considering synchrotron effects, inverse Compton scattering effects, and
inverse Compton loss effects [45, 46, 47], and also solves the function of the
local source and briefly analyzes them with experimental data. Finally,
section 4 presents the conclusions.

##  2 Model description

###  2.1 Spatially-dependent propagation

The CRs diffuses in the galactic halo via scattered magnetic waves and
magnetohydrodynamic (MHD) turbulence after injection into interstellar space.
The shape of the diffusive halo is usually approximated to be a cylinder, with
its radial boundary equal to the Galactic radius $R=20~{}\text{kpc}$, while
its half thickness $z_{h}$ needs to be fixed by the CRs data. Both CRs sources
and interstellar medium (ISM) are mainly diffused in the Galactic disk, the
width $z_{s}$ is roughly regarded as $200~{}\text{pc}$, which is much less
than the halo’s thickness. In addition to the diffusion effect, the
convection, reacceleration, and fragmentation processes exist in the transport
of CRs particles by colliding with the ISM. Because of ionization and Coulomb
scattering, CRs particles with low energies will further loss their energies.
The transport equation is generally written as follows [48, 49]:

$\displaystyle\frac{\partial\psi}{\partial t}$
$\displaystyle=Q(\boldsymbol{r},p)+\nabla\cdot\left(D_{xx}\nabla\psi-\boldsymbol{V}_{c}\psi\right)+\frac{\partial}{\partial
p}\left[p^{2}D_{pp}\frac{\partial}{\partial p}\frac{\psi}{p^{2}}\right]$ (2.1)
$\displaystyle-\frac{\partial}{\partial
p}\left[\dot{p}\psi-\frac{p}{3}\left(\nabla\cdot\boldsymbol{V}_{c}\right)\psi\right]-\frac{\psi}{\tau_{f}}-\frac{\psi}{\tau_{r}}.$

Here $\psi=\mathrm{d}n/\mathrm{d}p$ is the CRs density per particle momentum
$p$ at position $\boldsymbol{r}$ and is associated with the flux as
$\boldsymbol{J}=\psi c/4\pi$, $Q(\boldsymbol{r},p)$ is the source function,
$D_{xx}$ and $D_{pp}$ are the diffusion coefficients in the coordinate and
momentum space, $\boldsymbol{V}_{c}$ is the convection velocity, $\dot{p}$ is
the energy loss rate, $\tau_{f}$ and $\tau_{r}$ are the fragmentation and
radioactive decaying time scales. The momentum diffusion coefficient $D_{pp}$
is associated to $D_{xx}$ as
$D_{pp}D_{xx}=\frac{4p^{2}v_{A}^{2}}{3\delta(4-\delta^{2})(4-\delta)}$, in
which $v_{A}$ is the Alfvén velocity [50].

The SDP model divides the entire diffusive region into two pieces. The
Galactic disk and its surrounding areas are regarded as the inner halo (IH),
and the vast regions outside IH are known as the outer halo (OH) [51, 52]. The
spatially-dependent diffusion coefficient can be parameterized as follows
[53]:

$D_{xx}(r,z,{R})=D_{0}F(r,z)\beta^{\eta}\left(\frac{{R}}{{R}}_{0}\right)^{\delta(r,z)}.$
(2.2)

Here $\beta=v/c$ is the velocity of CR particles, ${R}=pc/Ze$ is rigidity
function, and the relationship between $F(r,z)$ and $\delta(r,z)$ and the
source distribution function $F(r,z)$ can be described as [54, 55]:

$\displaystyle F(r,z)=\left\\{\begin{array}[]{ll}\displaystyle
g(r,z)+[1-g(r,z)]\left(\frac{z}{\xi{z_{h}}}\right)^{n},&|z|\leq\xi{z_{h}}\\\
1,&|z|>\xi{z_{h}},\end{array}\right.$ (2.3)

and

$\displaystyle\delta(r,z)=\left\\{\begin{array}[]{ll}\displaystyle
g(r,z)+[\delta_{0}-g(r,z)]\left(\frac{z}{\xi{z_{h}}}\right)^{n},&|z|\leq\xi{z_{h}}\\\
\delta_{0},&|z|>\xi{z_{h}}.\end{array}\right.$ (2.4)

Here $g(r,z)=N_{m}/[1+f(r,z)]$, the IH region indicated by the half-thickness
of $\xi z_{h}$, which $\xi$ is determined by fitting the hardening of the
proton spectrum, and the size of the OH region is $(1-\xi)z_{h}$.

###  2.2 Background supernova remnants

The SNRs are generally considered as the most probable source for the
acceleration of CRs. The charged particles are accelerated to a power-law
distribution through diffusive shock acceleration. The distribution of the
SNRs is approximated as axisymmetric, which is generally parameterized as:

$f(r,z)=\left(\frac{r}{r_{\odot}}\right)^{\alpha}\exp\left[-\frac{\beta\left(r-r_{\odot}\right)}{r_{\odot}}\right]\exp\left(-\frac{|z|}{z_{s}}\right),$
(2.5)

where $r_{\odot}\equiv 8.5~{}\text{kpc}$ represents the distance from the
solar system to the Galactic center. In this work, the parameters $\alpha$ and
$\beta$ are taken as 1.09 and 3.87 respectively [56]. The density of CRs
sources descends exponentially along the vertical height from the Galactic
plane, with $z_{s}=100~{}\text{pc}$.

For the injection spectra of lepton and nuclei, we take a broken power-law
form as given in paper [53],

${Q}={Q}_{0}\left\\{\begin{array}[]{ll}\displaystyle\left(\frac{{R}_{\mathrm{br}1}}{{R}_{0}}\right)^{\nu_{2}}\left(\frac{{R}}{{R}_{\mathrm{br}1}}\right)^{\nu_{1}},&{R}\leq{R}_{\mathrm{br}1}\\\
\displaystyle\left(\frac{{R}}{{R}_{0}}\right)^{\nu_{2}},&{R}_{\mathrm{br}1}<{R}\leq{R}_{\mathrm{br}2}\\\
\displaystyle\left(\frac{{R}_{\mathrm{br}2}}{{R}_{0}}\right)^{\nu_{2}}\left(\frac{{R}}{{R}_{\mathrm{br}2}}\right)^{\nu_{3}},&{R}>{R}_{\mathrm{br}2},\end{array}\right.$
(2.6)

where ${Q}_{0}$, $\nu_{1,2,3}$, ${R}_{0}$ and ${R}_{\mathrm{br}1,2}$ are the
normalization, power index, reference rigidity, and broken rigidity
respectively.

###  2.3 Local source

The local sources of CRs are found to be within $\sim$1 kpc around the solar
system [57, 58]. For the high-energy particles, the synchrotron in an
interstellar magnetic field and inverse Compton scattering is far more
important than fragmentation. So the eq. (2.1) of leptons and nucleus can be
simplified to [14, 59, 60]:

$\frac{\partial\varphi}{\partial
t}-D_{xx}\Delta\varphi+\frac{\partial}{\partial
E}(\dot{E}\varphi)=Q_{\text{int}}(E,t)\delta(\boldsymbol{r}-\boldsymbol{r}^{\prime}).$
(2.7)

The energy loss rate $\dot{E}$ for leptons is approximated to
$\dot{E}=-bE^{2}$ in the Thomson limit, and for the nuclei, this can be
omitted reliably. The analytical solutions for leptons and protons are as
follows [61]:

$\varphi(r,t,E)=\frac{Q(E)}{[4\pi D(E)t]^{3/2}}\cdot\exp[-r^{2}/4D(E)t].$
(2.8)

Here, the diffusion coefficient $D(E)$ only relate to energy, so eq. (2.2)
becomes

$D(E)=D_{0}\beta^{\eta}\left(\frac{E}{E_{0}}\right)^{\delta_{0}}.$ (2.9)

###  2.4 The GC Model

The electrons and protons can be injected into the interstellar space from the
local SNR, in which GC may happen. If so, the following processes
$p+p(A)\rightarrow\pi^{\pm}+\pi^{0}+p+\bar{p}+(\text{others})$ and $\pi^{0}\to
e^{+}+e^{-}$ will contribute the corresponding GC-characteristic energy
spectrum. According to the theory of Zhu et al. [42], the positron and
electron spectra take the following form

$\displaystyle Q_{e}(E)=\left\\{\begin{array}[]{ll}\displaystyle
C_{e}(E_{\mathrm{GC}})\left(\frac{E}{E_{\mathrm{GC}}}\right)^{-\beta_{1}}\left[\frac{1}{\beta_{2}}\left(\frac{E}{E_{\mathrm{GC}}}\right)^{-\beta_{2}}+\left(\frac{1}{\beta_{3}}-\frac{1}{\beta_{2}}\right)\right],&\text{
if }E\leq E_{\mathrm{GC}}\\\
\displaystyle\frac{C_{e}}{\beta_{3}}(E_{\mathrm{GC}})\left(\frac{E}{E_{\mathrm{GC}}}\right)^{-\beta_{1}-\beta_{3}},&\text{
if }E>E_{\mathrm{GC}},\end{array}\right.$ (2.10)

and for protons

$\displaystyle Q_{p}(E)=\left\\{\begin{array}[]{ll}\displaystyle
C_{p}\left(E_{\mathrm{GC}}\right)\left(\frac{E}{E_{\mathrm{GC}}}\right)^{-\beta_{2}},&\text{
if }E\leq E_{\mathrm{GC}}\\\ \displaystyle
C_{p}\left(E_{\mathrm{GC}}\right)\left(\frac{E}{E_{\mathrm{GC}}}\right)^{-\beta_{3}},&\text{
if }E>E_{\mathrm{GC}}.\end{array}\right.$ (2.11)

Here the parameters $\beta_{1}$ and $\beta_{2}$ represent the propagating loss
of electrons and gamma-rays or protons, respectively.
$\beta_{3}=\beta_{2}+2\beta_{p}-1$, where $\beta_{p}$ denote the acceleration
mechanism of scattering protons. The coefficient $C_{e}(C_{p})$ incorporates
the kinematic factor into the flux dimension and the percentage of $\pi^{0}\to
2\gamma$ and $\gamma+\gamma\to e^{-}+e^{+}$, $E_{\mathrm{GC}}$ is the GC-
threshold energy for electron and proton, respectively.

If there are two different GC sources, each one come up with electrons and
protons. We define the coefficient $C$ in eqs. (2.10) and (2.11) as
$C_{e}^{1}$, $C_{p}^{1}$ and $C_{e}^{2}$, $C_{p}^{2}$ for each source
respectively. According to article [42], there are relations,

$\frac{E_{\text{GC}(e)}}{E_{\text{GC}(p)}}\simeq\frac{m_{\pi}}{2m_{p}},$
(2.12)

and

$\frac{C_{e}^{1}}{C_{p}^{1}}=\frac{C_{e}^{2}}{C_{p}^{2}}.$ (2.13)

##  3 Prediction of the second excess of electron spectrum

The cosmic ray spectra of lepton and proton are typically considered to
incorporate the contributions of nearby sources to the background. In general,
the propagation parameters for the background are determined by fitting the
observations of protons, helium, and B/C ratios. Figure 1(a) and figure 1(c)
depict the spectrum of electron, figure 1(b) and figure 1(d) illustrate the
spectrum of proton.

We assume that both the electron and proton spectra include contributions from
two local GC sources, represented by the yellow and green solid lines,
respectively. The first source (yellow, source 1) is probably from the Geminga
SNR, which has an age of $3.24\times 10^{5}\text{~{}years}$ and is located at
a distance of $0.33\text{~{}kpc}$ from the solar system [62]. The parameters
for the second source (green) of proton, including its distance ($r$) and age
($t$), are inferred from the proton spectrum, considering the larger
uncertainty of proton data, we give two extreme possibilities of the second
source, and distinguish them as source 2(I) and source 2(II).

The spectrum of proton is depicted in figures 1(b) and 1(d). Lots of models
have been proposed to explain its characteristics. One type of models consider
the spectrum as one local source adding the background spectrum, which is
represented by the solid pink line in figures 1(b) and 1(d) [63, 64]. This
background line of proton (Background(ref.)) exhibits an abrupt drop around
$2\times 10^{6}\text{~{}GeV}$, which is not a naturally expected pow-law
decay. So, we prefer the red dashed line (Background) which is derived from
fitting the AMS-02 experimental data and extending it. The observed spectrum
of proton can be fitted by using the Background and taking into account the
contribution of two GC sources. However, the local source in high-energy
region cannot be precisely determined due to large errors, and so, two extreme
possibilities are considered and displayed as the green solid lines in figures
1(b) (source 2(I)) and (1(d)) (source 2(II)). The local sources of the GC
effect contribute to the excess spectrum of electron and proton. The electron
spectra (solid green line) in figures 1(a) (source 2(I)) and 1(c) (source
2(II)) are obtained by using equations eqs. (2.12) and (2.13) with the
corresponding data of proton. To summarize, the analysis process is as
follows:

(1). Determine the background spectrum of electron (the red dashed line in
figure 1(a) and 1(c)). We take the background propagation parameters of
electron as in work [53, 65] , $D_{0}=9.65\times
10^{28}\mathrm{~{}cm}^{2}\cdot\mathrm{s}^{-1}$ at the reference rigidity
${R}_{0}=4\mathrm{~{}GeV}$, $\delta_{0}=0.65$, $N_{m}=0.24$, $\xi=0.12$,
$z_{h}=5\mathrm{~{}kpc}$, and the Alfvén velocity is
$v_{A}=6\mathrm{~{}km}\cdot\mathrm{s}^{-1}$, and other parameters as shown in
table 1. With these parameters and using the DRAGON2 [43, 44] package to solve
the transport equations, we get the background spectrum of the electron.

(2). Determine the contribution of electron from GC source 1 (the solid yellow
line in figure 1(a) and 1(c)). The location and age of source 1 are inferred
from the experimental data. We assume that GC source 1 occurring in Geminga
SNR, the corresponding parameters are in table 2. By using the DRAGON2
package, we get the contribution from GC source 1. For this source 1,
$E_{\mathrm{GC}(e)}^{1}\simeq 880\mathrm{~{}GeV}$.

(3). Determine the background spectrum of proton (the red dashed line in
figures 1(b) and 1(d)). The purple solid line in the two figures are one kind
of background in some works [63, 64], it’s not naturally pow-law like and we
consider it should have complex combination. In this work we take the red
dashed line as background which is fitting the AMS02-data and extending.

background | $Q_{0}[\text{m}^{-2}\text{sr}^{-1}\text{s}^{-1}\text{GeV}^{-1}]$ | $\nu_{1}$ | ${R}_{br1}[\text{GeV}]$ | $\nu_{2}$ | ${R}_{br2}[\text{GeV}]$ | $\nu_{3}$
---|---|---|---|---|---|---
electron | $1.18\times 10^{-4}$ | 1.44 | 5.0 | 2.79 | 1500 | 3.8
Table 1: The injected spectrum parameters of background for electron. GC source | source | $C[\mathrm{GeV}^{-1}]$ | $E_{\mathrm{GC}}[\mathrm{TeV}]$ | $\beta_{1}$ | $\beta_{2}$ | $\beta_{3}$
---|---|---|---|---|---|---
electron | source 1 | $4.11\times 10^{45}$ | 0.88 | 0.2 | 0.3 | 2.9
source 2(I) | $1.33\times 10^{46}$ | 24.0 | 0.2 | 0.3 | 2.9
source 2(II) | $2.11\times 10^{39}$ | 110 | 0.2 | 0.3 | 2.9
proton | source 1 | $1.12\times 10^{52}$ | 11.9 | - | 1.0 | 3.6
source 2(I) | $3.71\times 10^{47}$ | 324 | - | 1.0 | 3.6
source 2(II) | $5.75\times 10^{45}$ | 1500 | - | 1.0 | 3.6
Table 2: The injected spectrum parameters of GC for electron and proton.

(4). Determine the contribution of proton from source 1 (the solid yellow line
in figure 1(b) and 1(d)). Since we consider the first excess of electron
spectrum is from GC source 1, the source will contribute an excess of proton
at the same time with electron. With the eq. (2.12) and (2.13) in GC model, we
get the GC threshold of $E_{\mathrm{GC}(p)}^{1}\simeq
11.9\mathrm{~{}TeV},C_{p}^{1}\simeq 1.12\times 10^{52}$. By using the DRAGON2
package and the other parameters in table 2, the contribution from source 1 is
obtained.

(5). Determine the excess of electron and proton from source 2 (the solid
green line in figure 1). For the data of electron, the upper energy limit at
present is about several TeV. So, in the higher energy region, we use the data
of proton to speculate contribution from the source 2. For the spectrum of
proton, adding the contributions from two sources to the background should be
consistent with the data. While considering the larger uncertainties, we take
two extremes as in 1(b) (source 2(I)) and 1(d) (source 2(II)).

For source 2(I), The GC threshold of electron
$E^{\mathrm{2(I)}}_{\mathrm{GC}(e)}\simeq 24.0\mathrm{~{}TeV}$, is obtained by
using the GC threshold of proton $E^{\mathrm{2(I)}}_{\mathrm{GC}(p)}\simeq
324.0\mathrm{~{}TeV}$ with eq. (2.12) [40]. Equation (2.13) is used to
determine the relationship between $C_{e}$ and $C_{p}$ for sources 1 and 2.
Experimental data provides an unambiguous value of $C_{e}^{1}\simeq 4.11\times
10^{45}$ for electron (source 1) and $C_{p}^{1}\simeq 1.12\times 10^{52}$ for
proton (source 1). Thus, only $C_{e}$ or $C_{p}$ of source 2 needs to be
confirmed. Comparing with data of proton, we found
$C_{p}^{\mathrm{2(I)}}\simeq 3.71\times 10^{47}$ at a GC threshold of
$324.0\mathrm{~{}TeV}$. Substituting this value along with $C_{e}^{1}$ and
$C_{p}^{1}$ into eq. (2.13) yields $C_{e}^{\mathrm{2(I)}}\simeq 1.33\times
10^{46}$, at a GC threshold of $24.0\mathrm{~{}TeV}$. With the other
parameters in table 2, the electron and proton spectra of source 2(I) in
figures 1(a) and 1(b) (solid green line) are obtained.

For source 2(II), considering the maximum energy of RUNJOB, the maximum GC
threshold energy of proton source 2 is assumed to be
$E^{\mathrm{2(II)}}_{\mathrm{GC}(p)}\simeq 1500\mathrm{~{}TeV}$, accordingly,
for electron source 2, $E^{\mathrm{2(II)}}_{\mathrm{GC}(e)}\simeq
110\mathrm{~{}TeV}$. The coefficient for proton source 2 at a GC threshold of
$1500\mathrm{~{}TeV}$ is $C_{p}^{\mathrm{2(II)}}\simeq 5.75\times 10^{45}$,
which in turn yields the $C_{e}$ for electron source 2 at a threshold of
$110\mathrm{~{}TeV}$ as $C_{e}^{\mathrm{2(II)}}\simeq 2.11\times 10^{39}$.
Along with other parameters in table 2, the electron and proton spectra of
source 2 in figures 1(c) and 1(d) (solid green line) are obtained.

(a) (b) (c) (d)

Figure 1: The right panel displays the calculated proton spectra, while the
left panel shows the total spectrum of positron and electron. Both panels
feature red dashed lines representing the background spectrum, which was
calculated by Dragon 2. The yellow and green lines represent the local source
spectrum, while the black line represents the total spectrum obtained by
pulsing the background and local source spectra. The electron data points
released by the AMS-02 (blue) [1], DMAPE (red) [4], and Fermi-LAT (purple) [3]
experiments are shown in blue, red, and purple, respectively . Meanwhile, the
proton data from the AMS-02 (blue) [2], DMAPE (red) [5], ATIC (green) [30],
CREAM (azure) [66], RUNJOB (purple) [67], and JACEE (pink) [68] experiments
were included.

##  4 Conclusion

In this work, we propose a second excess of electron spectra according to GC
model. GC happens in very high energy collision, which the energy scale
exceeds the maximum energy in Large Hadron Collider (LHC). In a GC process,
$p+p(A)\rightarrow\pi^{\pm}+\pi^{0}+p+\bar{p}+\text{(others)}$ and
$\pi^{0}\rightarrow e^{+}+e^{-}$, the electron and proton injection spectra
are shown as eqs. (2.10) and (2.11) respectively. The excess of electron and
proton originate from a same GC source shows definite relationship between
their GC thresholds as eq. (2.12). And for the excesses from two different GC
sources, eq. (2.13) establishes the relationship between their $C_{e}$ and
$C_{p}$ for electron and proton of sources 1 and 2. So we can estimate one
from another. With the electron and proton data from AMS-02 and DAMPE
measurements, we assume that the first excesses are from GC source 1. In
higher energy region, the data of proton shown another excess, if it is from a
GC source 2, there should be a second excess in the spectrum of electron.
Because of the large uncertainties of the data (proton) in high energy region,
the GC thresholds of the second excess of electron and proton cannot be
accurately determined, so two extreme possibilities are considered. To obtain
the $C_{p}$ for proton and $C_{e}$ for electron of source 2, the assumption is
made that the proton Background(ref.) conforms well to reality at high energy
about $10^{6}~{}\text{GeV}$.

In summary, according to the measurements for proton, we assume that electron
spectrum exists second excess. The thresholds for electron and proton excess
are determined through the GC model and measurements. We find that the
observed first excess of electron spectra is near
$E_{\mathrm{GC}(e)}^{1}\simeq 880\mathrm{~{}GeV}$ and that of proton
$E_{\mathrm{GC}(p)}^{1}\simeq 11.9\mathrm{~{}TeV}$ which may originate from
the Geminga SNR. We further hypothesize that the minimum possible second
excess in electron spectra is near $E_{\mathrm{GC}(e)}^{\mathrm{2(I)}}\simeq
24.0\mathrm{~{}TeV}$ and in proton spectra is about
$E_{\mathrm{GC}(p)}^{\mathrm{2(I)}}\simeq 324\mathrm{~{}TeV}$, while the
maximum possible second excess in electron spectra is near
$E_{\mathrm{GC}(e)}^{\mathrm{2(II)}}\simeq 110\mathrm{~{}TeV}$ and in proton
spectra is about $E_{\mathrm{GC}(p)}^{\mathrm{2(II)}}\simeq
1500\mathrm{~{}TeV}$. Presently, measurements of electrons and protons above
$40\mathrm{~{}TeV}$ are still inaccurate with significant errors. We
anticipate that forthcoming precise measurements will validate our
conjectures.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China
(No.11851303).

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