# A Simple and Natural Interpretations of the DAMPE Cosmic Ray
Electron/Positron Spectrum within Two Sigma Deviations

Jia-Shu Niu<EMAIL_ADDRESS>Institute of Theoretical Physics, State Key
Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University,
Taiyuan, 030006, China  CAS Key Laboratory of Theoretical Physics, Institute
of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, No.
19A Yuquan Road, Beijing 100049, China  Tianjun Li<EMAIL_ADDRESS>CAS Key
Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese
Academy of Sciences, Beijing, 100190, China  School of Physical Sciences,
University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing
100049, China  Fang-Zhou Xu  Institute of Modern Physics and Center for High
Energy Physics, Tsinghua University, Beijing 100084, China

###### Abstract

The DArk Matter Particle Explorer (DAMPE) experiment has recently announced
the first results for the measurement of total electron plus positron fluxes
between 25 GeV and 4.6 TeV. A spectral break at about 0.9 TeV and a tentative
peak excess around 1.4 TeV have been found. However, it is very difficult to
reproduce both the peak signal and the smooth background including spectral
break simultaneously. We point out that the numbers of events in the two
energy ranges (bins) close to the 1.4 TeV excess have $1\sigma$ deficits. With
the basic physics principles such as simplicity and naturalness, we consider
the $-2\sigma$, $+2\sigma$, and $-1\sigma$ deviations due to statistical
fluctuations for the 1229.3 GeV bin, 1411.4 GeV bin, and 1620.5 GeV bin.
Interestingly, we show that all the DAMPE data can be explained consistently
via both the continuous distributed pulsar and dark matter interpretations,
which have $\chi^{2}\simeq 17.2$ and $\chi^{2}\simeq 13.9$ (for all the 38
points in DAMPE electron/positron spectrum with 3 of them revised),
respectively. These results are different from the previous analyses by
neglecting the 1.4 TeV excess. At the same time, we do a similar global
fitting on the newly released CALET lepton data, which could also be
interpreted by such configurations. Moreover, we present a $U(1)_{D}$ dark
matter model with Breit-Wigner mechanism, which can provide the proper dark
matter annihilation cross section and escape the CMB constraint. Furthermore,
we suggest a few ways to test our proposal.

##  I Introduction

Because of the strong radiative cooling via synchrotron and inverse Compton
scattering (ICS) processes, the TeV electrons can only travel a short distance
of about a few kpc in the Milky Way. Therefore, the nearby Cosmic Ray (CR)
sources such as pulsars (Shen, 1970; Harding and Ramaty, 1987; Aharonian  _et
al._ , 1995; Chi  _et al._ , 1996; Zhang and Cheng, 2001) and dark matter (DM)
(Bergström, 2000; Bertone  _et al._ , 2005; Bergström  _et al._ , 2009) can be
probed via the high energy electrons and positrons. The spectra of the cosmic
ray electrons and positrons (CREs) have been measured up to TeV energy scales
by the ground-based and space-borne experiments, for example, HESS (Aharonian
_et al._ , 2008; HESS collaboration, 2009), VERITAS (Staszak and for the
VERITAS Collaboration, 2015; Holder, 2017), FermiLAT (Abdollahi  _et al._ ,
2017; Meehan  _et al._ , 2017), AMS-02 (AMS collaboration, 2014a, b, 2013a,
c), and CALET (collaboration, 2017a; Adriani  _et al._ , 2018). In particular,
the excesses of the electrons (Chang  _et al._ , 2008; Aharonian  _et al._ ,
2008; Fermi-LAT collaboration, 2009; Aguilar  _et al._ , 2014, 2014) and
positrons (Adriani  _et al._ , 2009; Fermi-LAT collaboration, 2012; Aguilar
_et al._ , 2013; Accardo  _et al._ , 2014) have been discovered as well.

Recently, the DArk Matter Particle Explorer (DAMPE), which is a new generation
space-borne experiment to measure CRs and was launched in December 2015, has
announced the first results of high energy CR electron plus positron
($e^{-}+e^{+}$) flux from 25 GeV to 4.6 TeV with unprecedentedly high quality
(Ambrosi  _et al._ , 2017). The energy resolution of the DAMPE is better than
1.5% at TeV energies, and the hadron rejection power is about $10^{5}$. Thus,
DAMPE is able to reveal (fine) structures of the electron and positron fluxes.
The main DAMPE spectrum can be fitted by a smoothly broken power-law model
with a spectral break around 0.9 TeV, which confirms the previous results by
HESS experiment (Aharonian  _et al._ , 2008; HESS collaboration, 2009). And
there exists a tentative peak-like flux excess around 1.4 TeV. Thus, the DAMPE
results have stimulated the extensive studies (Gu and He, 2017; Fang  _et al._
, 2017; Fan  _et al._ , 2017; Yuan  _et al._ , 2017a; Duan  _et al._ , 2017;
Gu, 2017a; Chao and Yuan, 2017; Tang  _et al._ , 2017; Zu  _et al._ , 2017;
Liu and Liu, 2017; Cao  _et al._ , 2017; Athron  _et al._ , 2017; Chao  _et
al._ , 2017; Gao and Ma, 2017; Niu  _et al._ , 2018a; Jin  _et al._ , 2017;
Huang  _et al._ , 2017; Duan  _et al._ , 2017; Cao  _et al._ , 2017a; Ghorbani
and Ghorbani, 2017; Nomura and Okada, 2017; Gu, 2017b; Li  _et al._ , 2017;
Chen  _et al._ , 2017; Yang  _et al._ , 2017; Ding  _et al._ , 2017; Ge and
He, 2017; Liu and Liu, 2017; Okada and Seto, 2017; Sui and Zhang, 2017; Cao
_et al._ , 2017b; Han  _et al._ , 2017; Niu  _et al._ , 2018a; Cholis  _et
al._ , 2017; Fowlie, 2017). The spectral break can be explained by the broad
distributed pulsars, pulsar wind nebulae (PWNe), supernova remnants (SNRs)
(Fang  _et al._ , 2017; Yuan  _et al._ , 2017a), and by the dark matter
annihilation and decay in the galaxy halo (Yuan  _et al._ , 2017a; Niu  _et
al._ , 2018a; Athron  _et al._ , 2017; Jin  _et al._ , 2017). Also, the
tentative peak is always interpreted by local pulsars, PWNe, and SNRs (Fang
_et al._ , 2017; Yuan  _et al._ , 2017a; Cholis  _et al._ , 2017)), and by the
DM sub-halos, clumps, and mini-spikes (Fan  _et al._ , 2017; Duan  _et al._ ,
2017; Gu and He, 2017; Athron  _et al._ , 2017; Liu and Liu, 2017; Cao  _et
al._ , 2017; Jin  _et al._ , 2017; Huang  _et al._ , 2017; Yang  _et al._ ,
2017; Ge and He, 2017). Another important interpretation ascribes observed CR
spectrum puzzling features to a nearly 2-3 Myr Super Nova (Kachelrieß  _et
al._ , 2015, 2018), which could naturally explain not only proton to helium
ratio, positron and anti-proton fluxes, but also plateau in the cosmic ray
dipole anisotropy.

However, one can easily show that it is impossible to explain both the
spectral break and the tentative peak simultaneously (Fang  _et al._ , 2017;
Yuan  _et al._ , 2017a; Cholis  _et al._ , 2017; Jin  _et al._ , 2017; Huang
_et al._ , 2017)). In addition, we have 74, 93, and 33 events for three
continuous bins or energy ranges [1148.2, 1318.3] GeV, [1318.3, 1513.6] GeV,
and [1513.6, 1737.8] GeV, respectively, which for simplicity we shall call
1229.3 GeV bin, 1411.4 GeV bin, and 1620.5 GeV bin (Ambrosi  _et al._ , 2017).
The number of events and fluxes for these bins are given in Table 1. From
Figure 2 of the DAMPE’s paper (Ambrosi  _et al._ , 2017), it is obvious that
the 1411.4 GeV bin has a little bit more than $3\sigma$ excess, while the
1229.3 GeV bin and 1620.5 GeV bin have about $1\sigma$ deficits. Therefore, it
is very difficult to explain the events in these three bins, especially the
first two, no matter by the pulsar or dark matter interpretations.

Energy Bins (GeV) |  $N$ (original) |  $\Phi(e^{-}+e^{+})\pm\sigma_{\mathrm{stat}}\pm\sigma_{\mathrm{sys}}$ (original) |  $N$ (revised) |  $\Phi(e^{-}+e^{+})\pm\sigma_{\mathrm{stat}}\pm\sigma_{\mathrm{sys}}$ (revised) | $\Delta N$ | $\Delta N_{2\sigma}$
---|---|---|---|---|---|---
$[1148.2,~{}1318.3]$ | 74 | $(4.38\pm 0.53\pm 0.14)\times 10^{-8}$ | 92 | $(5.44\pm 0.48\pm 0.14)\times 10^{-8}$ | +18 |  $\pm$18
$[1318.3,~{}1513.6]$ | 93 | $(4.99\pm 0.53\pm 0.17)\times 10^{-8}$ | 73 | $(3.92\pm 0.60\pm 0.17)\times 10^{-8}$ | -20 |  $\pm$20
$[1513.6,~{}1737.8]$ | 33 | $(1.52\pm 0.28\pm 0.06)\times 10^{-8}$ | 39 | $(1.80\pm 0.26\pm 0.06)\times 10^{-8}$ | +6 |  $\pm$12
Table 1: The original and revised numbers of events and fluxes, $\Delta N$,
and $\Delta N_{2\sigma}$ for the 1229.3 GeV bin, 1411.4 GeV bin, and 1620.5
GeV bin. Here, $\Delta N$ and $\Delta N_{2\sigma}$ are the adjusted numbers of
events and the numbers of events for $2\sigma$ deviations from statistical
fluctuations. Thus, we should require $|\Delta N|\leq|\Delta N_{2\sigma}|$.

From the theoretical physics point of view, we would like to explain nature
with basic principles such as simplicity and naturalness, or say truth and
beauty! In the words of Sir Isaac Newton, “Truth is ever to be found in the
simplicity, and not in the multiplicity and confusion of things.” Therefore,
to explain all the DAMPE data via a simple and natural way, we propose that
the excess in the 1411.4 GeV bin and the deficits in the 1229.3 GeV bin and
1620.5 GeV bin arise from the $+2\sigma$, $-2\sigma$, and $-1\sigma$
deviations due to statistical fluctuations, which happened frequently in
collider experiments. Remarkably, we can indeed explain all the DAMPE data
consistently via the pulsar and dark matter interpretations, which have
$\chi^{2}\simeq 17.2$ and $\chi^{2}\simeq 13.9$ (for all the 38 points in
DAMPE electron/positron spectrum with 3 of them revised), respectively. Our
results are different from the previous analyses by neglecting the 1.4 TeV
excess (Niu  _et al._ , 2018a). As a comparison, the newly released CALET
lepton data is used to do a similar global fitting, which could give us some
more supports on the origin of the lepton excess. In addition, we present a
$U(1)_{D}$ dark matter model with Breit-Wigner mechanism, which can provide
the proper dark matter annihilation cross section and escape the CMB
constraint. Furthermore, we suggest a few ways to test our proposal as well as
the 1.4 TeV excess.

##  II Statistical Fluctuations

In the DAMPE’s paper (Ambrosi  _et al._ , 2017), the numbers of events and the
CRE fluxes with $1\sigma$ statistical and systematic errors have been given in
its Table 1. To evaluate the uncertainties for numbers of the events, we need
to understand their relations. The relation between the number of events and
fluxes in each energy bin is (Ambrosi  _et al._ , 2017; AMS collaboration,
2014a)

$\Phi(e^{-}+e^{+})=\frac{N(E)\cdot(1-\varepsilon_{\mathrm{bg}}(E))}{A_{\mathrm{eff}}(E)\cdot
T\cdot\Delta E}\cdot\varepsilon_{\mathrm{other}}(E),$ (1)

where $N$ is the number of ($e^{-}+e^{+}$) events, $A_{\mathrm{eff}}$ is the
effective detector acceptance, $T$ is the operating time, $\Delta E$ is the
energy range of the bin, $\varepsilon_{\mathrm{bg}}$ is the background
fraction of the events, and $\varepsilon_{\mathrm{other}}$ represents the
effects caused by other mechanisms which were not given in the Table 1 of Ref.
(Ambrosi  _et al._ , 2017).

Taking $T=530$ days and $\varepsilon_{\mathrm{other}}=1.3$, we can reproduce
the corresponding results in the 1229.3 GeV bin, 1411.4 GeV bin, and 1620.5
GeV bin within the uncertainty $<0.1\%$. Consequently, we use the formula

$\Phi(e^{-}+e^{+})=\frac{N(E)\cdot(1-\varepsilon_{\mathrm{bg}}(E))}{A_{\mathrm{eff}}(E)\cdot
T\cdot\Delta E}\cdot 1.3$ (2)

in this letter to calculate the fluxes in these bins.

We calculate the $2\sigma$ deviations for the number of events ($\Delta
N_{2\sigma}$) from the flux statistical fluctuations as follows

$\Delta
N_{2\sigma}=\frac{\Delta\Phi(e^{-}+e^{+})_{2\sigma_{\mathrm{stat}}}}{\Phi(e^{-}+e^{+})}\cdot
N~{}.$ (3)

Thus, for the 1229.3 GeV bin, 1411.4 GeV bin, and 1620.5 GeV bin, we obtain
$\Delta N_{2\sigma}=\pm 18,~{}\pm 20,~{}\pm 12$, respectively. Assume
$-2\sigma$, $+2\sigma$, and $-1\sigma$ deviations for these bins from
statistical fluctuations, we have $\Delta N=+18,~{}-20,~{}+6$, respectively.
Therefore, the revised numbers of events for the 1229.3 GeV bin, 1411.4 GeV
bin, and 1620.5 GeV bin, are 92, 73, and 39, respectively.

Furthermore, we reestimate the statistical uncertainties in these bins based
on the revised numbers of events via the formula

$\Delta N_{1\sigma}\simeq\frac{1}{\sqrt{N}}~{},$ (4)

and then calculate the corresponding fluxes and their statistical
uncertainties. The systematical uncertainties are assumed to be invariant. All
the detailed information for these three bins are given in Table 1. By the
way, as a cross check, with Eq. (4), we have reproduced similar $1\sigma$
statistical uncertainties of the original fluxes in the DAMPE’s paper (Ambrosi
_et al._ , 2017).

##  III Fitting Procedure

In CR theory, the CR electrons are considered to be accelerated during the
acceleration of CR nuclei at the sources, e.g. SNRs. On the other hand, the CR
positrons are produced as secondary particles from CR nuclei interaction with
the interstellar medium (ISM) (Adriani  _et al._ , 2009; AMS collaboration,
2013b; HEAT collaboration, 1997; AMS-01 collaboration, 2007). From the
observed spectra of positrons and electrons (AMS collaboration, 2013a, 2014c,
2014b, 2014a), we can conclude that there should exist some extra sources
producing electron-positron pairs. As we stated in the first section, these
extra sources could be astrophysical sources or DM annihilation or decay. As a
result, the CREs data contains (i) the primary electrons; (ii) the secondary
electrons; (iii) the secondary positrons; (iv) the extra source of electron-
positron pairs. If we want to study the properties of the extra source, we
should deduct the primary electrons and secondary electrons/positrons first.

The primary electrons are always assumed to have a power-law injection and the
secondary electrons/positrons are determined mainly by the CR proton and
helium nucleus interact with ISM. Consequently, we should do global fitting to
all these related data simultaneously which can avoid the bias of choosing the
lepton background parameters.

The public code dragon 111https://github.com/cosmicrays/DRAGON (Evoli  _et
al._ , 2008) was used to do numerical calculations. Some custom modifications
are performed in the original code, such as the possibility to use specie-
dependent injection spectra, which is not allowed by default in dragon.

In view of some discrepancies when fitting the new data (Jóhannesson  _et al._
, 2016), we use a factor $c_{\,\mathrm{He}}$ to re-scale the helium-4
abundance (which has a default value of $7.199\times 10^{4}$), which helps us
to get a better global fitting.

###  III.1 Background

In this work, we use the widely used diffusion-reacceleration model which can
give a consistent fitting results to the AMS-02 nuclei data (see for e.g.,
(Niu and Li, 2018; Yuan  _et al._ , 2017b; Yuan, 2018; Zhu  _et al._ , 2018)).
In the whole propagation region, a uniform diffusion coefficient
($D_{xx}=D_{yy}=D_{zz}=D_{0}\beta\left(R/R_{0}\right)^{\delta}$) is employed
to describe the propagation .

The hardening of the nuclei spectra at $\sim 300\,\mathrm{GeV}$ (which has
been observed by ATIC-2 (Panov  _et al._ , 2006), CREAM (Ahn  _et al._ ,
2010), PAMELA (PAMELA collaboration, 2011a), and AMS-02 (AMS collaboration,
2015a, b)) is considered by adding breaks in the primary source injections. At
the same time, considering the observed significant difference in the slopes
of proton and helium (of about $\sim 0.1$ (PAMELA collaboration, 2011b; AMS
collaboration, 2015a, b)), we use separate primary source spectra settings for
proton and helium. In summary, for nuclei primary source injections, each of
them has 2 breaks at rigidity $R_{\,\mathrm{A}1}$ and $R_{\,\mathrm{A}2}$. The
corresponding slopes are $\nu_{\,\mathrm{A}1}$ ($R\leq R_{\,\mathrm{A}1}$),
$\nu_{\,\mathrm{A}2}$ ($R_{\,\mathrm{A}1}<R\leq R_{\,\mathrm{A}2}$) and
$\nu_{\,\mathrm{A}3}$ ($R>R_{\,\mathrm{A}3}$).

For CR electrons primary source, we use one break $R_{e}$ for electron primary
source, and the corresponding slopes are $\nu_{e1}$ ($R\leq R_{e}$) and
$\nu_{e2}$ (($R>R_{e}$)).

In order to take into account the uncertainties when calculating the secondary
CR particles’ fluxes, we employ parameters $c_{\,\bar{\text{p}}}$ and
$c_{e^{+}}$ to re-scale the calculated secondary flux to fit the data (Tan and
Ng, 1983; Duperray  _et al._ , 2003; Kappl and Winkler, 2014; di Mauro  _et
al._ , 2014; Lin  _et al._ , 2015). Note that the above mentioned
uncertainties may not be simply represented by a constant factor, but most
probably are energy dependent (Delahaye  _et al._ , 2009; Mori, 2009). Here we
expect that a constant factor could be a simple assumption.

The force-field approximation (Gleeson and Axford, 1968) is used to describe
the effects of solar modulation effects. $\phi_{\mathrm{nuc}}$,
$\phi_{\,\bar{\text{p}}}$ and $\phi_{e^{+}}$ are used to modulate the local
interstellar spectra of nuclei (proton and helium), anti-proton and positrons
respectively, which based on the charge-sign dependence of solar modulation.
On the other hand, Because the DAMPE lepton data $\gtrsim 20\,\mathrm{GeV}$,
we did not consider the modulation effects on electrons (or leptons).

###  III.2 Extra Sources

We consider both pulsar and DM scenarios to generate the CRE excesses in the
observed spectrum by the DAMPE experiment. For the pulsar scenario, a
continuous distributed pulsar background was used (Lin  _et al._ , 2015; Niu
_et al._ , 2018a). The injection spectrum of such sources is assumed to be a
power law with an exponential cutoff

$q_{e}^{\,\mathrm{psr}}(p)=N_{\,\mathrm{psr}}(R/\mathrm{10\,\mathrm{GeV}})^{-\nu_{\,\mathrm{psr}}}\exp{(-R/R_{\mathrm{c}})},$
(5)

where $N_{\,\mathrm{psr}}$ is the normalization factor, $\nu_{\,\mathrm{psr}}$
is the spectral index, and $R_{\mathrm{c}}$ is the cutoff rigidity. For the DM
scenario, we employ the Einasto profile (Navarro  _et al._ , 2004; Merritt
_et al._ , 2006; Einasto, 2009; Navarro  _et al._ , 2010)

$\rho(r)=\rho_{\odot}\exp\left[-\left(\frac{2}{\alpha}\right)\left(\frac{r^{\alpha}-r_{\odot}^{\alpha}}{r_{s}^{\alpha}}\right)\right],$
(6)

with $\alpha\approx 0.17$, $r_{s}\approx 20\,\mathrm{kpc}$, and
$\rho_{\odot}\approx 0.39\,\mathrm{GeV}\,\mathrm{cm}^{-3}$ is the local DM
relic density (Catena and Ullio, 2010; Weber and de Boer, 2010; Salucci  _et
al._ , 2010; Pato  _et al._ , 2010; Iocco  _et al._ , 2011). And the source
term, which we use to add the CRE particles from the annihilations of the
Majorana DM particles, is

$Q(\boldsymbol{r},p)=\frac{\rho(\boldsymbol{r})^{2}}{2m_{\chi}^{2}}\langle\sigma
v\rangle\sum_{f}\eta_{f}\frac{dN^{(f)}}{dp},$ (7)

where $\langle\sigma v\rangle$ is the velocity-averaged DM annihilation cross
section multiplied by DM relative velocity (referred as cross section),
$\rho(\boldsymbol{r})$ is the DM density distribution, and $dN^{(f)}/dp$ is
the injection energy spectrum of CREs from DM annihilating into the Standard
Model (SM) final states via leptonic channels $f{\bar{f}}$ ($e^{-}e^{+}$,
$\mu\bar{\mu}$, and $\tau\bar{\tau}$) with $\eta_{f}$ ($\eta_{e}$,
$\eta_{\mu}$, and $\eta_{\tau}$) the corresponding branching fractions. Here,
we normalized $\eta_{f}$ as $\eta_{e}+\eta_{\mu}+\eta_{\tau}=1$.

The parameters related to the extra source of the leptons for pulsar scenario
is $(N_{\,\mathrm{psr}},\nu_{\,\mathrm{psr}},R_{c})$, and for DM scenario is
$(m_{\chi},\langle\sigma v\rangle,\eta_{e},\eta_{\mu},\eta_{\tau})$.

###  III.3 Data Sets and Parameters

As in Ref. (Niu  _et al._ , 2018a), we perform a global fitting on the data
set including the proton fluxes from AMS-02 and CREAM (AMS collaboration,
2015a; Ahn  _et al._ , 2010) helium flux from AMS-02 and CREAM222The CREAM
data was used as the supplement of the AMS-02 data because it is more
compatible with the AMS-02 data when $R\gtrsim 1\,\mathrm{TeV}$. (AMS
collaboration, 2015b; Ahn  _et al._ , 2010), $\,\bar{\text{p}}/\text{p}$ ratio
from AMS-02 (AMS collaboration, 2016), positrons flux from AMS-02 (AMS
collaboration, 2014b), and CRE flux from DAMPE (Ambrosi  _et al._ , 2017),
which could account for the primary electrons, the secondary leptons, and the
extra leptons in a self-consistent way 333The errors used in our global
fitting are the quadratic summation over statistical and systematic errors..
Moreover, the employed AMS-02 positron flux is used to calibrate the positron
contribution in the DAMPE CRE flux in energy region $\lesssim
300\,\mathrm{GeV}$444The systematics between AMS-02 and DAMPE are dealt with
by employing a re-scale factor $c_{e^{+}}$ on positron flux.. The framework of
the fitting procedure is the same as our previous work (Niu  _et al._ , 2018a;
Niu and Li, 2018), where the details can be found.

Considering the systematics between different CREs spectra observed by
different experiments (see in Fig. 1 and 2), we take the newly released CREs
spectrum from CALET (Adriani  _et al._ , 2018) as a comparison to do global
fitting as that on DAMPE CREs spectrum. Because both of these experiments are
implemented in space and have a similar ability to obtain CREs data, a
reasonable explanation on lepton excess should explain both of them
simultaneously. In this case, DAMPE and CALET could be considered as the
maximum and minimum CREs flux examples.

Altogether, the data set in our global fitting is

$\displaystyle D=$
$\displaystyle\\{D^{\text{AMS-02}}_{\,\mathrm{p}},D^{\text{AMS-02}}_{\,\mathrm{He}},D^{\text{AMS-02}}_{\,\bar{\text{p}}/\text{p}},D^{\text{CREAM}}_{\,\mathrm{p}},$
$\displaystyle
D^{\text{CREAM}}_{\,\mathrm{He}},D^{\text{AMS-02}}_{e^{+}},D^{\text{DAMPE}}_{e^{-}+e^{+}}/D^{\text{CALET}}_{e^{-}+e^{+}}\\}~{}.$

The parameter sets for pulsar scenario is

$\displaystyle\boldsymbol{\theta}_{\,\mathrm{psr}}=$
$\displaystyle\\{D_{0},\delta,z_{h},v_{A},|N_{\,\mathrm{p}},R_{\,\mathrm{p}1},R_{\,\mathrm{p}2},\nu_{\,\mathrm{p}1},\nu_{\,\mathrm{p}2},\nu_{\,\mathrm{p}3},$
$\displaystyle
R_{\,\mathrm{He}1},R_{\,\mathrm{He}2},\nu_{\,\mathrm{He}1},\nu_{\,\mathrm{He}2},\nu_{\,\mathrm{He}3},|\phi_{\mathrm{nuc}},\phi_{\,\bar{\text{p}}},c_{\,\mathrm{He}},c_{\,\bar{\text{p}}},|$
$\displaystyle
N_{\,\mathrm{e}},R_{\,\mathrm{e}1},\nu_{\,\mathrm{e}1},\nu_{\,\mathrm{e}2},|$
$\displaystyle N_{\,\mathrm{psr}},\nu_{\,\mathrm{psr}},R_{c},|$
$\displaystyle\phi_{e^{+}},c_{e^{+}}\\}~{},$

for DM scenario is

$\displaystyle\boldsymbol{\theta}_{\,\mathrm{DM}}=$
$\displaystyle\\{D_{0},\delta,z_{h},v_{A},|N_{\,\mathrm{p}},R_{\,\mathrm{p}1},R_{\,\mathrm{p}2},\nu_{\,\mathrm{p}1},\nu_{\,\mathrm{p}2},\nu_{\,\mathrm{p}3},$
$\displaystyle
R_{\,\mathrm{He}1},R_{\,\mathrm{He}2},\nu_{\,\mathrm{He}1},\nu_{\,\mathrm{He}2},\nu_{\,\mathrm{He}3},|\phi_{\mathrm{nuc}},\phi_{\,\bar{\text{p}}},c_{\,\mathrm{He}},c_{\,\bar{\text{p}}},|$
$\displaystyle
N_{\,\mathrm{e}},R_{\,\mathrm{e}1},\nu_{\,\mathrm{e}1},\nu_{\,\mathrm{e}2},|$
$\displaystyle m_{\chi},\langle\sigma
v\rangle,\eta_{e},\eta_{\mu},\eta_{\tau},|$
$\displaystyle\phi_{e^{+}},c_{e^{+}}\\}~{}.$

Note that, most of these 2 scenarios’ parameters in the set
$\boldsymbol{\theta}_{\,\mathrm{psr}}$ and
$\boldsymbol{\theta}_{\,\mathrm{DM}}$ is the same with each other except those
which account the extra sources of lepton.

##  IV Results

When the Markov Chains have reached their equilibrium state, we take the
samples of the parameters as their posterior probability distribution
functions. The best-fit values, statistical mean values, standard deviations
and allowed intervals at $95\%$ CL for parameters in set
$\boldsymbol{\theta}_{\,\mathrm{psr}}$ and
$\boldsymbol{\theta}_{\,\mathrm{DM}}$ for DAMPE and CALET are shown in
Appendix, Table 2 (DAMPE, pulsar scenario), Table 3 (CALET, pulsar scenario),
Table 4 (DAMPE, DM scenario), and Table 5 (CALET, DM scenario), respectively.
For best fit results of the global fitting, we got $\chi^{2}/d.o.f=243.13/299$
(DAMPE, pulsar scenario), $\chi^{2}/d.o.f=229.98/301$ (CALET, pulsar
scenario), $\chi^{2}/d.o.f=262.94/297$ (DAMPE, DM scenario), and
$\chi^{2}/d.o.f=265.03/299$ (CALET, DM scenario).

###  IV.1 Background

The best-fitting results and the corresponding residuals of the proton flux,
helium flux and $\,\bar{\text{p}}/\text{p}$ ratio for pulsar scenario is
showed in Appendix, Fig. 5, for DM scenario is showed in Appendix, Fig. 6. In
these figures, we can see that the nuclei data is perfectly reproduced, which
would provide a good precondition for the fitting on the lepton data.

Although the main purpose of this work does not focus on the CR propagation
models, we would like to emphasize some points here: (i) As shown in Appendix,
Table 2, Table 3, Table 4, and Table 5, we got larger best-fit values of
$D_{0}$ and $z_{h}$ than previous works (Niu and Li, 2018). This mainly
because the newly released AMS-02 nuclei spectra favor large values of $D_{0}$
and $z_{h}$.555A similar global fitting result can be found in Ref. (Niu  _et
al._ , 2018b), which use a different numerical tool galprop to do calculation.
Moreover, the employed 2 breaks in nuclei primary source injection strengthen
the classical degeneracy between $D_{0}$ and $z_{h}$ based on the data set we
used in this work (without B/C), which both got larger best-fit values in this
work. (ii) The employed 2 breaks in the nuclei primary source injection
accounted for the observed hardening in the observed spectra, other than use
only one break and let $\delta$ compromise the different slopes in high energy
regions, which lead to a smaller value of $\delta$ and fitting uncertainties
on $\delta$ ($\lesssim 0.01$) than previous works.

###  IV.2 Extra Sources

The fitting results of the pulsar and DM scenario on DAMPE and CALET CREs
spectrum are given in Figs. 1 and 2 respectively, which also shows some CREs
spectrum from other experiments. Cleaner fitting results are shown in Figs. 3
and 4.666Here, one should note that both proton and positron flux from AMS-02
are well fitted in such configurations, but the coincidence of the proton and
positron flux in this work is caused by chance. From these figures, we can
conclude that both scenarios could provide a excellent fittings to the DAMPE
CREs spectrum within $3\sigma$ fitting deviation, which do not need to employ
extra local sources. At the same time, the CALET CREs spectrum could also be
fitted by both scenarios, although the fitting result is not that good because
of a suspected bump at about 0.9 - 1 $\,\mathrm{TeV}$.

For the best fit result on the DAMPE and CALET CREs spectrum, we got
$\chi^{2}\simeq 17.2$ (DAMPE, pulsar scenario), $\chi^{2}\simeq 16.1$ (CALET,
pulsar scenario), $\chi^{2}\simeq 13.9$ (DAMPE, DM scenario) and
$\chi^{2}\simeq 25.7$ (CALET, DM scenario). This indicate that CALET CREs data
disfavors the DM scenario because of the defective fit on the suspected bump
at about 0.9 - 1 $\,\mathrm{TeV}$. This need more events accumulated in the
future.

The detailed results of the constraints on the parameters could be found in
Appendix, Table 2, Table 3, Table 4, and Table 5. Most of the parameters are
slightly different between DAMPE and CALET fitting because of the systematics
between them. One point we want to mention is that in the fitting results of
CALET CREs data, DM scenario, $\eta_{e}\simeq 0.930$, $\eta_{\mu}\simeq
0.042$, and $\eta_{\tau}\simeq 0.028$, which is largely different from the
DAMPE results.

In the following of this work, we will focus on the analysis of the DAMPE
results, whose method could be extended to deal with the CALET results without
difficulties.

Figure 1: The global fitting results of the DAMPE and CALET lepton flux for
pulsar scenario. The $2\sigma$ (deep color) and $3\sigma$ (light color) bounds
of total fitted results (red), contribution from background (blue) and pulsar
(green) are also shown in the figure. And we have $\chi^{2}\simeq 17.2$
(DAMPE) and $\chi^{2}\simeq 16.1$ (CALET). (Data sources: DAMPE (Ambrosi  _et
al._ , 2017), CALET (Adriani  _et al._ , 2018), Fermi (collaboration, 2017b),
HESS (Aharonian  _et al._ , 2008; HESS collaboration, 2009) and AMS-02 (AMS
collaboration, 2014a).)

Figure 2: The global fitting results of the DAMPE and CALET lepton flux for DM
scenario. The $2\sigma$ (deep color) and $3\sigma$ (light color) bounds of
total fitted results (red), contribution from background (blue) and DM (green)
are also shown in the figure. And we have $\chi^{2}\simeq 13.9$ (DAMPE) and
$\chi^{2}\simeq 25.7$ (CALET).

Figure 3: The global fitting results and the corresponding residuals to the
lepton (DAMPE and CALET) and positron (AMS-02) flux for pulsar scenario. The
$2\sigma$ (deep color) and $3\sigma$ (light color) bounds of total fitted
results (red), contribution from background (blue) and pulsar (green) are also
shown in the figure. Different from Fig. 1 and Fig. 2, all the experiment data
which did not participate in the global fitting has not been not represented.

Figure 4: The same as Fig. 3, but for DM scenario.

For the pulsar scenario, the fitting results give $\nu_{\,\mathrm{psr}}\simeq
0.62$, which is obviously different from the fitting results in previous works
(see for e.g., (Profumo, 2012)). In standard pulsar models, the injection
spectrum indices of CREs from pulsars are always in the range
$\nu_{\,\mathrm{psr}}\in[1.0,2.4]$ (Reynolds, 1988; Thompson  _et al._ , 1994;
Fierro  _et al._ , 1995). As a result, more attention should be paid in future
researches. This may indicate: (i) there is something wrong or inaccuracy with
the classical pulsar CRE injection model; (ii) the CRE excess is not
contributed primarily by pulsars. Moreover, the cut-off is $R_{c}\simeq 692$
GV. In the previous work (Niu  _et al._ , 2018a) where the 1.4 TeV peak excess
was neglected, we obtained that the spectral index of the injection is
$\nu_{\,\mathrm{psr}}\simeq 0.65$ and the cut-off is $R_{c}\simeq 650$ GV.
Thus, there exist about $+5\%$ and $-5\%$ deviations for
$\nu_{\,\mathrm{psr}}$ and $R_{c}$, respectively.

For the DM scenario, we obtain $\langle\sigma v\rangle\simeq 4.07\times
10^{-23}\,\mathrm{cm}^{2}\,\mathrm{s}^{-1}$ and $m_{\chi}\simeq
1884\,\mathrm{GeV}$. The value of $\langle\sigma v\rangle$ is about 3 orders
larger than that of thermal DM (Jungman  _et al._ , 1996). To explain this
discrepancy, we will present a concrete model in the next section. Moreover,
we have $\eta_{e}\simeq 0.465$, $\eta_{\mu}\simeq 0.510$, and
$\eta_{\tau}\simeq 0.025$. So the DM annihilation into $\tau\bar{\tau}$ is
highly suppressed, which provides some hints to construct an appropriate DM
model. In our previous work (Niu  _et al._ , 2018a) where the 1.4 TeV peak
excess was neglected, we have $\langle\sigma v\rangle\simeq 1.48\times
10^{-23}\,\mathrm{cm}^{2}\,\mathrm{s}^{-1}$, $m_{\chi}\simeq
1208\,\mathrm{GeV}$, $\eta_{e}\simeq\eta_{\mu}\simeq 0.5$, while $\eta_{\tau}$
is highly suppressed. Thus, we have similar results on branching fractions,
but different DM masses and annihilation cross sections.

##  V Model Building

Because we have $\eta_{e}\sim 0.465$, $\eta_{\mu}\sim 0.510$, and
$\eta_{\tau}\sim 0.025$, the constraints from the Fermi-LAT observations of
dwarf spheroidal galaxies Fermi-LAT collaboration (2011); Geringer-Sameth and
Koushiappas (2011); Sming Tsai  _et al._ (2012); Fermi-LAT collaboration
(2015); Li  _et al._ (2015); Profumo  _et al._ (2017) can be avoided Yuan  _et
al._ (2017a). To escape the constraints from the Planck observations of CMB
anisotropies Planck collaboration (2015), we employ the Breit-Wigner mechanism
Feldman  _et al._ (2008); Ibe  _et al._ (2008); Guo and Wu (2009); Bi  _et
al._ (2009, 2011); Hisano  _et al._ (2011); Bai  _et al._ (2017); Xiang  _et
al._ (2017). We consider the dark $U(1)_{D}$ model where the SM fermions and
Higgs fields are neutral under it. We introduce one SM singlet Higgs field
$S$, one chiral fermionic dark matter particle $\chi$, and three pairs of the
vector-like particles (${\widehat{XE}}_{i},{\widehat{XE}}_{i}^{c})$, whose
quantum numbers under the $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}\times
U(1)_{D}$ are

$\displaystyle
S:(\mathbf{1},\mathbf{1},\mathbf{0},\mathbf{2})~{},~{}~{}~{}\chi:(\mathbf{1},\mathbf{1},\mathbf{0},\mathbf{-1})$
$\displaystyle{\widehat{XE}}_{i}:(\mathbf{1},\mathbf{1},\mathbf{-1},\mathbf{-2})~{},~{}~{}~{}{\widehat{XE}}^{c}_{i}:(\mathbf{1},\mathbf{1},\mathbf{1},\mathbf{2})~{}.~{}\,$
(8)

The relevant Lagrangian is

$\displaystyle-{\cal L}$ $\displaystyle=$ $\displaystyle-
m_{S}^{2}|S|^{2}+\frac{\lambda}{2}|S|^{4}+\left(M^{V}_{ij}{\widehat{XE}}^{c}_{i}{\widehat{XE}}_{j}\right.$
(9)
$\displaystyle\left.+y_{ij}S{\widehat{E}}_{i}^{c}{\widehat{XE}}_{j}+yS\chi\chi+{\rm
H.C.}\right)~{},~{}\,$

where ${\widehat{E}}_{i}^{c}$ are the right-handed charged leptons. For
simplicity, we choose $M^{V}_{ij}=M^{V}_{i}\delta_{ij}$ and
$y_{ij}=y_{i}\delta_{ij}$. After $S$ acquires a Vacuum Expectation Value
(VEV), the $U(1)_{D}$ gauge symmetry is broken down to a $Z_{2}$ symmetry
under which $\chi$ is odd. Thus, $\chi$ is a DM matter candidate. For
simplicity, we assume that the mass of $U(1)_{D}$ gauge boson is about twice
of $\chi$ mass, i.e., $M_{Z^{\prime}}\simeq 2m_{\chi}$, while the Higgs field
$S$ and vector-like particles are heavier than $M_{Z^{\prime}}$. Moreover,
${\widehat{E}}_{i}^{c}$ and ${\widehat{XE}}_{i}^{c}$ will be mixed due to the
$M^{V}_{i}{\widehat{XE}}^{c}_{i}{\widehat{XE}}_{i}$ and
$y_{i}S{\widehat{E}}_{i}^{c}{\widehat{XE}}_{i}$ terms, and we obtain the mass
eigenstates $E_{i}^{c}$ and $XE_{i}^{c}$ by neglecting the tiny charged lepton
masses

$\displaystyle\left(\begin{array}[]{c}E_{i}^{c}\\\
XE_{i}^{c}\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta_{i}&\sin\theta_{i}\\\
-\sin\theta_{i}&\cos\theta_{i}\end{array}\right)\left(\begin{array}[]{c}{\widehat{E}}_{i}^{c}\\\
{\widehat{XE}}_{i}^{c\prime}\end{array}\right)~{},~{}\,$ (16)

where $\tan\theta_{i}=-y\langle S\rangle/M^{V}_{i}$.

Neglecting the charged lepton masses again, we obtain

$\displaystyle\sigma v=\sum_{i=1}^{3}\frac{g^{\prime
4}\sin^{2}\theta_{i}}{6\pi}\frac{s-m_{\chi}^{2}}{(s-m_{Z^{\prime}}^{2})^{2}+(m_{Z^{\prime}}\Gamma_{Z^{\prime}})^{2}}~{},~{}\,$
(17)

where $m_{\chi}=y\langle S\rangle$, and $g^{\prime}$ and $M_{Z^{\prime}}$ are
the gauge coupling and gauge boson mass for $U(1)_{D}$ gauge symmetry.

For $m_{Z^{\prime}}\simeq 2m_{\chi}$, $Z^{\prime}$ decays dominantly into
leptons, and the decay width is

$\displaystyle\Gamma_{Z^{\prime}}=\sum_{i=1}^{3}\frac{g^{\prime
2}\sin^{2}\theta_{i}}{6\pi}m_{Z^{\prime}}~{}.~{}\,$ (18)

To explain the DM best fit results, we choose

$\displaystyle g^{\prime}\simeq 0.028,~{}m_{\chi}\simeq
1884~{}\mbox{GeV},~{}\frac{m_{Z^{\prime}}-2m_{\chi}}{m_{Z^{\prime}}}\simeq
3.0\times 10^{-6},~{}$ $\displaystyle\sin\theta_{e}\simeq
0.21~{},~{}~{}~{}\sin\theta_{\mu}\simeq
0.22~{},~{}~{}~{}\sin\theta_{\tau}\simeq 0.05~{}.~{}\,$ (19)

And then we obtain $\langle\sigma v\rangle\simeq 4.07\times
10^{-23}\mbox{cm}^{3}\mbox{s}^{-1}$, and
$\eta_{e}:\eta_{\mu}:\eta_{\tau}\simeq 0.465:0.510:0.025$. Of course, there
exists fine-tuning between $m_{Z^{\prime}}$ and $m_{\chi}$ , which deserves
further study. For some solutions, see Ref. Bai  _et al._ (2017).

##  VI Discussions and Conclusion

First, we would like to point out that if the numbers of events in the 1229.3
GeV bin and 1411.4 GeV bin are exchanged, we can also explain the DAMPE’s data
similarly. Of course, the most important question is how to test our proposal
that there exists statistical fluctuations in the 1229.3 GeV bin, 1411.4 GeV
bin, and 1620.5 GeV bin. For the data analyses, we suggest that one chooses
different energy ranges to study the data again. For example, we can shift the
energy ranges by $\pm 50$ GeV and $\pm 100$ GeV for the high energy bins, and
then study the corrsponding events and fluxes. In the future, DAMPE will
provide us more accurate spectrum data reaching up to $\sim 10\,\mathrm{TeV}$,
which can give us a unprecedented opportunity to study the origin and
propagation of CREs. We predict that the CRE spectrum would be more
continuous. In particular, the peak excess in the 1411.4 GeV bin as well as
the deficits in the 1229.3 GeV bin and 1620.5 GeV bin will all decrease!
Moreover, if the 1.4 TeV peak signal was proved to be correct, we do need a
local source of high energy CREs. Other experiment is needed as a cross check
if such signal arises from DM annihilation, for example, our recent work (Niu
_et al._ , 2018c) proposed a novel scenario to probe the interaction between
DM particles and electrons for the DM mass range $5\,\mathrm{GeV}\lesssim
m_{\chi}\lesssim 10\,\mathrm{TeV}$.

In summary, with the simplicity and naturalness physics principle, we proposed
that there exists the $-2\sigma$, $+2\sigma$, and $-1\sigma$ deviations due to
statistical fluctuations for the 1229.3 GeV bin, 1411.4 GeV bin, and 1620.5
GeV bin of the DAMPE data. Interestingly, we showed that all the DAMPE data
can be explained consistently via both the pulsar and dark matter
interpretations, which have $\chi^{2}\simeq 17.2$ and $\chi^{2}\simeq 13.9$
(for all the 38 points in DAMPE electron/positron spectrum with 3 of them
revised), respectively. These results are different from the previous analyses
by neglecting the 1.4 TeV excess. At the same time, we employed the newly
released CALET CREs spectrum to do a similar global fitting, which cold also
be fitted by continuous distributed pulsar and DM scenarios. Moreover, we
presented a $U(1)_{D}$ dark matter model with Breit-Wigner mechanism, which
can provide the proper dark matter annihilation cross section and escape the
CMB constraint. Furthermore, we suggested a few ways to test our proposal.

###### Acknowledgements.

We would like to thank Xiao-Jun Bi and Yi-Zhong Fan for helpful discussions,
and thank D. Maurin, et al. Maurin  _et al._ (2014) for collecting the
database and associated online tools for charged cosmic-ray measurements. This
research was supported in part by the Projects 11475238 and 11875062 (from
National Science Foundation of China), and the Projects 11747601 (from Key
Research Program of Frontier Sciences, Chinese Academy of Sciences). The
calculation in this paper are supported by HPC Cluster of SKLTP/ITP-CAS.

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## Appendix

ID | Prior | Best-fit | Posterior mean and | Posterior 95%
---|---|---|---|---
| range | value | Standard deviation | range
$D_{0}\ (10^{28}\,\mathrm{cm}^{2}\,\mathrm{s}^{-1})\ ^{a}$ | [1, 20] | 14.32 | 14.24$\pm$0.21 | [13.55, 14.83]
$\delta$ | [0.1, 1.0] | 0.322 | 0.318$\pm$0.006 | [0.305, 0.336]
$z_{h}\ (\,\mathrm{kpc})$ | [0.5, 30.0] | 25.88 | 25.61$\pm$0.94 | [23.86, 27.35]
$v_{A}\ (\,\mathrm{km}/\,\mathrm{s})$ | [0, 80] | 41.33 | 40.82$\pm$0.64 | [39.04, 42.67]
$N_{\,\mathrm{p}}\ ^{b}$ | [1, 8] | 4.47 | 4.47$\pm$0.01 | [4.44, 4.49]
$R_{\,\mathrm{p}1}\ (\,\mathrm{GV})$ | [1, 30] | 25.90 | 25.60$\pm$0.45 | [24.62, 27.14]
$R_{\,\mathrm{p}2}\ (\,\mathrm{GV})$ | [60, 1000] | 454.80 | 466.64$\pm$14.92 | [416.01, 502.10]
$\nu_{\,\mathrm{p}1}$ | [1.0, 4.0] | 2.190 | 2.197$\pm$0.013 | [2.159, 2.227]
$\nu_{\,\mathrm{p}2}$ | [1.0, 4.0] | 2.461 | 2.456$\pm$0.007 | [2.445, 2.479]
$\nu_{\,\mathrm{p}3}$ | [1.0, 4.0] | 2.336 | 2.351$\pm$0.012 | [2.318, 2.365]
$R_{\,\mathrm{He}1}\ (\,\mathrm{GV})$ | [1, 30] | 11.89 | 12.09$\pm$0.19 | [11.58, 12.72]
$R_{\,\mathrm{He}2}\ (\,\mathrm{GV})$ | [60, 1000] | 247.27 | 246.63$\pm$10.39 | [220.06, 279.23]
$\nu_{\,\mathrm{He}1}$ | [1.0, 4.0] | 2.184 | 2.191$\pm$0.013 | [2.155, 2.220]
$\nu_{\,\mathrm{He}2}$ | [1.0, 4.0] | 2.417 | 2.422$\pm$0.007 | [2.404, 2.434]
$\nu_{\,\mathrm{He}3}$ | [1.0, 4.0] | 2.205 | 2.215$\pm$0.014 | [2.185, 2.245]
$\phi_{\mathrm{nuc}}\ (\,\mathrm{GV})$ | [0, 1.5] | 0.73 | 0.74$\pm$0.02 | [0.70, 0.77]
$\phi_{\,\bar{\text{p}}}\ (\,\mathrm{GV})$ | [0, 1.5] | 0.31 | 0.27$\pm$0.03 | [0.20, 0.36]
$c_{\,\mathrm{He}}$ | [0.1, 10.0] | 4.056 | 3.81$\pm$0.11 | [3.61, 4.18]
$c_{\,\bar{\text{p}}}$ | [0.1, 10.0] | 1.382 | 1.36$\pm$0.03 | [1.28, 1.44]
$\log(N_{\,\mathrm{e}})\ ^{c}$ | [-4, 0] | -1.9479 | -1.938$\pm$0.008 | [-1.957, -1.923]
$\log(R_{\,\mathrm{e}}/\,\mathrm{GV})$ | [0, 3] | 1.636 | 1.65$\pm$0.02 | [1.59, 1.71]
$\nu_{\,\mathrm{e}1}$ | [1.0, 4.0] | 2.433 | 2.57$\pm$0.06 | [2.37, 2.69]
$\nu_{\,\mathrm{e}2}$ | [1.0, 4.0] | 2.392 | 2.39$\pm$0.01 | [2.36, 2.42]
$\log(N_{\,\mathrm{psr}})\ ^{d}$ | [-8, -4] | -6.14 | -6.14$\pm$0.02 | [-6.17, -6.11]
$\nu_{\,\mathrm{psr}}$ | [0, 3.0] | 0.63 | 0.67$\pm$0.03 | [0.55, 0.73]
$\log(R_{c}/\,\mathrm{GV}))$ | [2, 5] | 2.84 | 2.83$\pm$0.03 | [2.76, 2.91]
$\phi_{e^{+}}\ (\,\mathrm{GV})$ | [0, 1.5] | 1.395 | 1.37$\pm$0.02 | [1.34, 1.42]
$c_{e^{+}}$ | [0.1, 10.0] | 5.22 | 5.10$\pm$0.11 | [4.99, 5.28]

a Here $D_{0}$ is defined at the reference rigidity $R=4\,\mathrm{GV}$.
b $N_{\,\mathrm{p}}$ is the post-propagated normalization flux of protons at
100 GeV in unit
$10^{-2}\,\mathrm{m}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}\,\mathrm{GeV}^{-1}$.
c $N_{\,\mathrm{e}}$ is the post-propagated normalization flux of electrons at
25 GeV in unit
$\,\mathrm{m}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}\,\mathrm{GeV}^{-1}$.
d $N_{\,\mathrm{psr}}$ is the post-propagated normalization flux of electrons
at 300 GeV in unit
$\,\mathrm{m}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}\,\mathrm{GeV}^{-1}$.

Table 2:  Constraints on the parameters of DAMPE CREs spectrum, pulsar
scenario. The prior interval, best-fit value, statistic mean, standard
deviation and the allowed range at $95\%$ CL are listed for parameters. For
best fit values, we have $\chi^{2}/d.o.f.=243.13/299$.

ID | Prior | Best-fit | Posterior mean and | Posterior 95%
---|---|---|---|---
| range | value | Standard deviation | range
$D_{0}\ (10^{28}\,\mathrm{cm}^{2}\,\mathrm{s}^{-1})\ ^{a}$ | [1, 20] | 16.74 | 14.40$\pm$1.13 | [12.50, 17.36]
$\delta$ | [0.1, 1.0] | 0.289 | 0.314$\pm$0.014 | [0.279, 0.342]
$z_{h}\ (\,\mathrm{kpc})$ | [0.5, 30.0] | 24.14 | 24.73$\pm$2.07 | [19.58, 28.86]
$v_{A}\ (\,\mathrm{km}/\,\mathrm{s})$ | [0, 80] | 46.73 | 41.76$\pm$2.84 | [38.01, 48.69]
$N_{\,\mathrm{p}}\ ^{b}$ | [1, 8] | 4.46 | 4.45$\pm$0.02 | [4.42, 4.49]
$R_{\,\mathrm{p}1}\ (\,\mathrm{GV})$ | [1, 30] | 28.34 | 26.80$\pm$1.61 | [25.12, 29.90]
$R_{\,\mathrm{p}2}\ (\,\mathrm{GV})$ | [60, 1000] | 596.36 | 486.39$\pm$78.40 | [428.26, 712.11]
$\nu_{\,\mathrm{p}1}$ | [1.0, 4.0] | 2.223 | 2.181$\pm$0.026 | [2.125, 2.245]
$\nu_{\,\mathrm{p}2}$ | [1.0, 4.0] | 2.483 | 2.463$\pm$0.013 | [2.435, 2.495]
$\nu_{\,\mathrm{p}3}$ | [1.0, 4.0] | 2.352 | 2.342$\pm$0.016 | [2.307, 2.371]
$R_{\,\mathrm{He}1}\ (\,\mathrm{GV})$ | [1, 30] | 13.00 | 12.30$\pm$0.60 | [11.41, 13.51]
$R_{\,\mathrm{He}2}\ (\,\mathrm{GV})$ | [60, 1000] | 255.16 | 239.77$\pm$32.42 | [190.45, 297.17]
$\nu_{\,\mathrm{He}1}$ | [1.0, 4.0] | 2.204 | 2.172$\pm$0.022 | [2.127, 2.225]
$\nu_{\,\mathrm{He}2}$ | [1.0, 4.0] | 2.436 | 2.418$\pm$0.013 | [2.391, 2.449]
$\nu_{\,\mathrm{He}3}$ | [1.0, 4.0] | 2.244 | 2.232$\pm$0.024 | [2.197, 2.276]
$\phi_{\mathrm{nuc}}\ (\,\mathrm{GV})$ | [0, 1.5] | 0.72 | 0.71$\pm$0.03 | [0.65, 0.75]
$\phi_{\,\bar{\text{p}}}\ (\,\mathrm{GV})$ | [0, 1.5] | 0.06 | 0.21$\pm$0.09 | [0.01, 0.33]
$c_{\,\mathrm{He}}$ | [0.1, 10.0] | 4.10 | 4.09$\pm$0.31 | [3.72, 4.79]
$c_{\,\bar{\text{p}}}$ | [0.1, 10.0] | 1.438 | 1.35$\pm$0.07 | [1.20, 1.49]
$\log(N_{\,\mathrm{e}})\ ^{c}$ | [-4, 0] | -1.993 | -1.988$\pm$0.012 | [-2.004, -1.978]
$\log(R_{\,\mathrm{e}}/\,\mathrm{GV})$ | [0, 3] | 2.207 | 1.74$\pm$0.28 | [1.12, 2.47]
$\nu_{\,\mathrm{e}1}$ | [1.0, 4.0] | 2.504 | 2.55$\pm$0.05 | [2.47, 2.65]
$\nu_{\,\mathrm{e}2}$ | [1.0, 4.0] | 2.475 | 2.45$\pm$0.03 | [2.38, 2.52]
$\log(N_{\,\mathrm{psr}})\ ^{d}$ | [-8, -4] | -6.17 | -6.18$\pm$0.03 | [-6.25, -6.14]
$\nu_{\,\mathrm{psr}}$ | [0, 3.0] | 0.69 | 0.67$\pm$0.14 | [0.30, 0.97]
$\log(R_{c}/\,\mathrm{GV}))$ | [2, 5] | 2.84 | 2.80$\pm$0.11 | [2.54, 3.05]
$\phi_{e^{+}}\ (\,\mathrm{GV})$ | [0, 1.5] | 1.44 | 1.41$\pm$0.04 | [1.37, 1.49]
$c_{e^{+}}$ | [0.1, 10.0] | 5.45 | 5.24$\pm$0.25 | [5.00, 5.48]

a Here $D_{0}$ is defined at the reference rigidity $R=4\,\mathrm{GV}$.
b $N_{\,\mathrm{p}}$ is the post-propagated normalization flux of protons at
100 GeV in unit
$10^{-2}\,\mathrm{m}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}\,\mathrm{GeV}^{-1}$.
c $N_{\,\mathrm{e}}$ is the post-propagated normalization flux of electrons at
25 GeV in unit
$\,\mathrm{m}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}\,\mathrm{GeV}^{-1}$.
d $N_{\,\mathrm{psr}}$ is the post-propagated normalization flux of electrons
at 300 GeV in unit
$\,\mathrm{m}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}\,\mathrm{GeV}^{-1}$.

Table 3:  Constraints on the parameters of CALET CREs spectrum, pulsar
scenario. The prior interval, best-fit value, statistic mean, standard
deviation and the allowed range at $95\%$ CL are listed for parameters. For
best fit values, we have $\chi^{2}/d.o.f.=229.98/301$.

ID | Prior | Best-fit | Posterior mean and | Posterior 95%
---|---|---|---|---
| range | value | Standard deviation | range
$D_{0}\ (10^{28}\,\mathrm{cm}^{2}\,\mathrm{s}^{-1})\ ^{1}$ | [1, 20] | 15.12 | 15.16$\pm$0.13 | [14.87, 15.62]
$\delta$ | [0.1, 1.0] | 0.334 | 0.334$\pm$0.003 | [0.323, 0.342]
$z_{h}\ (\,\mathrm{kpc})$ | [0.5, 30.0] | 28.57 | 28.59$\pm$0.20 | [28.13, 29.15]
$v_{A}\ (\,\mathrm{km}/\,\mathrm{s})$ | [0, 80] | 42.81 | 42.92$\pm$0.64 | [41.72, 45.02]
$N_{\,\mathrm{p}}\ ^{b}$ | [1, 8] | 4.49 | 4.49$\pm$0.01 | [4.46, 4.52]
$R_{\,\mathrm{p}1}\ (\,\mathrm{GV})$ | [1, 30] | 25.02 | 25.03$\pm$0.22 | [24.59, 25.68]
$R_{\,\mathrm{p}2}\ (\,\mathrm{GV})$ | [60, 1000] | 461.99 | 463.00$\pm$5.36 | [449.86, 478.37]
$\nu_{\,\mathrm{p}1}$ | [1.0, 4.0] | 2.186 | 2.187$\pm$0.009 | [2.166, 2.209]
$\nu_{\,\mathrm{p}2}$ | [1.0, 4.0] | 2.463 | 2.462$\pm$0.006 | [2.447, 2.474]
$\nu_{\,\mathrm{p}3}$ | [1.0, 4.0] | 2.324 | 2.326$\pm$0.010 | [2.308, 2.345]
$R_{\,\mathrm{He}1}\ (\,\mathrm{GV})$ | [1, 30] | 10.91 | 10.90$\pm$0.13 | [10.52, 11.22]
$R_{\,\mathrm{He}2}\ (\,\mathrm{GV})$ | [60, 1000] | 239.35 | 238.80$\pm$8.65 | [215.60, 262.77]
$\nu_{\,\mathrm{He}1}$ | [1.0, 4.0] | 2.172 | 2.170$\pm$0.010 | [2.146, 2.192]
$\nu_{\,\mathrm{He}2}$ | [1.0, 4.0] | 2.410 | 2.409$\pm$0.006 | [2.395, 2.421]
$\nu_{\,\mathrm{He}3}$ | [1.0, 4.0] | 2.205 | 2.205$\pm$0.010 | [2.180, 2.231]
$\phi_{\mathrm{nuc}}\ (\,\mathrm{GV})$ | [0, 1.5] | 0.77 | 0.77$\pm$0.01 | [0.73, 0.79]
$\phi_{\,\bar{\text{p}}}\ (\,\mathrm{GV})$ | [0, 1.5] | 0.43 | 0.42$\pm$0.02 | [0.36, 0.46]
$c_{\,\mathrm{He}}$ | [0.1, 10.0] | 4.50 | 4.53$\pm$0.14 | [3.66, 4.66]
$c_{\,\bar{\text{p}}}$ | [0.1, 10.0] | 1.56 | 1.56$\pm$0.02 | [1.51, 1.61]
$\log(N_{\,\mathrm{e}})\ ^{c}$ | [-4, 0] | -1.933 | -1.934$\pm$0.006 | [-1.945, -1.923]
$\log(R_{\,\mathrm{e}}/\,\mathrm{GV})$ | [0, 3] | 1.676 | 1.680$\pm$0.025 | [1.63, 1.76]
$\nu_{\,\mathrm{e}1}$ | [1.0, 4.0] | 2.55 | 2.55$\pm$0.02 | [2.51, 2.60]
$\nu_{\,\mathrm{e}2}$ | [1.0, 4.0] | 2.37 | 2.37$\pm$0.01 | [2.35, 2.39]
$\log(m_{\chi}/\,\mathrm{GeV})$ | [1, 6] | 3.275 | 3.270$\pm$0.005 | [3.258, 3.275]
$\log(\langle\sigma v\rangle)\ ^{d}$ | [-28, -18] | -22.39 | -22.40$\pm$0.03 | [-22.48, -22.36]
$\eta_{e}$ | [0, 1] | 0.465 | 0.460$\pm$0.011 | [0.438, 0.476]
$\eta_{\mu}$ | [0, 1] | 0.510 | 0.513$\pm$0.009 | [0.499, 0.530]
$\eta_{\tau}$ | [0, 1] | 0.025 | 0.027$\pm$0.007 | [0.016, 0.037]
$\phi_{e^{+}}\ (\,\mathrm{GV})$ | [0, 1.5] | 1.36 | 1.36$\pm$0.01 | [1.33, 1.41]
$c_{e^{+}}$ | [0.1, 10.0] | 5.19 | 5.19$\pm$0.09 | [5.09, 5.33]

a Here $D_{0}$ is defined at the reference rigidity $R=4\,\mathrm{GV}$.
b $N_{\,\mathrm{p}}$ is the post-propagated normalization flux of protons at
100 GeV in unit
$10^{-2}\,\mathrm{m}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}\,\mathrm{GeV}^{-1}$.
c $N_{\,\mathrm{e}}$ is the post-propagated normalization flux of electrons at
25 GeV in unit
$\,\mathrm{m}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}\,\mathrm{GeV}^{-1}$.
d $(\langle\sigma v\rangle)$ is in unit $\,\mathrm{cm}^{3}\,\mathrm{s}^{-1}$

Table 4:  Constraints on the parameters of DAMPE CREs spectrum, DM scenario.
The prior interval, best-fit value, statistic mean, standard deviation and the
allowed range at $95\%$ CL are listed for parameters. For best fit values, we
have $\chi^{2}/d.o.f.=262.94/297$.

ID | Prior | Best-fit | Posterior mean and | Posterior 95%
---|---|---|---|---
| range | value | Standard deviation | range
$D_{0}\ (10^{28}\,\mathrm{cm}^{2}\,\mathrm{s}^{-1})\ ^{a}$ | [1, 20] | 16.68 | 16.00$\pm$0.75 | [14.72, 18.22]
$\delta$ | [0.1, 1.0] | 0.304 | 0.307$\pm$0.009 | [0.285, 0.327]
$z_{h}\ (\,\mathrm{kpc})$ | [0.5, 30.0] | 25.93 | 27.64$\pm$1.72 | [22.52, 28.79]
$v_{A}\ (\,\mathrm{km}/\,\mathrm{s})$ | [0, 80] | 47.07 | 45.92$\pm$1.93 | [43.02, 51.38]
$N_{\,\mathrm{p}}\ ^{b}$ | [1, 8] | 4.47 | 4.45$\pm$0.02 | [4.44, 4.49]
$R_{\,\mathrm{p}1}\ (\,\mathrm{GV})$ | [1, 30] | 27.80 | 27.09$\pm$0.93 | [26.12, 29.56]
$R_{\,\mathrm{p}2}\ (\,\mathrm{GV})$ | [60, 1000] | 569.43 | 592.75$\pm$40.33 | [505.98, 713.39]
$\nu_{\,\mathrm{p}1}$ | [1.0, 4.0] | 2.201 | 2.186$\pm$0.019 | [2.152, 2.230]
$\nu_{\,\mathrm{p}2}$ | [1.0, 4.0] | 2.471 | 2.458$\pm$0.010 | [2.4457, 2.485]
$\nu_{\,\mathrm{p}3}$ | [1.0, 4.0] | 2.349 | 2.363$\pm$0.029 | [2.312, 2.376]
$R_{\,\mathrm{He}1}\ (\,\mathrm{GV})$ | [1, 30] | 11.77 | 11.46$\pm$0.32 | [11.19, 12.18]
$R_{\,\mathrm{He}2}\ (\,\mathrm{GV})$ | [60, 1000] | 225.19 | 268.93$\pm$29.10 | [220.29, 343.72]
$\nu_{\,\mathrm{He}1}$ | [1.0, 4.0] | 2.178 | 2.160$\pm$0.015 | [2.134, 2.198]
$\nu_{\,\mathrm{He}2}$ | [1.0, 4.0] | 2.421 | 2.414$\pm$0.010 | [2.398, 2.430]
$\nu_{\,\mathrm{He}3}$ | [1.0, 4.0] | 2.234 | 2.221$\pm$0.026 | [2.185, 2.256]
$\phi_{\mathrm{nuc}}\ (\,\mathrm{GV})$ | [0, 1.5] | 0.72 | 0.69$\pm$0.02 | [0.66, 0.73]
$\phi_{\,\bar{\text{p}}}\ (\,\mathrm{GV})$ | [0, 1.5] | 0.15 | 0.17$\pm$0.05 | [0.03, 0.25]
$c_{\,\mathrm{He}}$ | [0.1, 10.0] | 4.89 | 4.96$\pm$0.21 | [4.56, 5.58]
$c_{\,\bar{\text{p}}}$ | [0.1, 10.0] | 1.48 | 1.44$\pm$0.05 | [1.40, 1.56]
$\log(N_{\,\mathrm{e}})\ ^{c}$ | [-4, 0] | -1.990 | -1.994$\pm$0.011 | [-2.006, -1.980]
$\log(R_{\,\mathrm{e}}/\,\mathrm{GV})$ | [0, 3] | 1.789 | 1.764$\pm$0.036 | [1.71, 1.87]
$\nu_{\,\mathrm{e}1}$ | [1.0, 4.0] | 2.54 | 2.57$\pm$0.05 | [2.47, 2.61]
$\nu_{\,\mathrm{e}2}$ | [1.0, 4.0] | 2.43 | 2.39$\pm$0.03 | [2.39, 2.47]
$\log(m_{\chi}/\,\mathrm{GeV})$ | [1, 6] | 3.109 | 3.109$\pm$0.003 | [3.103, 3.116]
$\log(\langle\sigma v\rangle)\ ^{d}$ | [-28, -18] | -22.25 | -22.21$\pm$0.11 | [-22.41, -22.21]
$\eta_{e}$ | [0, 1] | 0.930 | 0.918$\pm$0.032 | [0.861, 0.967]
$\eta_{\mu}$ | [0, 1] | 0.042 | 0.039$\pm$0.022 | [0.008, 0.078]
$\eta_{\tau}$ | [0, 1] | 0.028 | 0.043$\pm$0.027 | [0.006, 0.090]
$\phi_{e^{+}}\ (\,\mathrm{GV})$ | [0, 1.5] | 1.42 | 1.44$\pm$0.02 | [1.40, 1.49]
$c_{e^{+}}$ | [0.1, 10.0] | 5.22 | 5.30$\pm$0.16 | [5.08, 5.46]

a Here $D_{0}$ is defined at the reference rigidity $R=4\,\mathrm{GV}$.
b $N_{\,\mathrm{p}}$ is the post-propagated normalization flux of protons at
100 GeV in unit
$10^{-2}\,\mathrm{m}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}\,\mathrm{GeV}^{-1}$.
c $N_{\,\mathrm{e}}$ is the post-propagated normalization flux of electrons at
25 GeV in unit
$\,\mathrm{m}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}\,\mathrm{GeV}^{-1}$.
d $(\langle\sigma v\rangle)$ is in unit $\,\mathrm{cm}^{3}\,\mathrm{s}^{-1}$

Table 5:  Constraints on the parameters of CALET CREs spectrum, pulsar
scenario. The prior interval, best-fit value, statistic mean, standard
deviation and the allowed range at $95\%$ CL are listed for parameters. For
best fit values, we have $\chi^{2}/d.o.f.=265.03/299$.

Figure 5: The global fitting results and the corresponding residuals to the
proton flux, helium flux, $\,\bar{\text{p}}/\text{p}$ ratio, and positron flux
for 2 scenarios. The $2\sigma$ (deep red) and $3\sigma$ (light red) bound are
also showed in the figures.

Figure 6: The global fitting results and the corresponding residuals to the
proton flux, helium flux, $\,\bar{\text{p}}/\text{p}$ ratio, and positron flux
for 2 scenarios. The $2\sigma$ (deep red) and $3\sigma$ (light red) bound are
also showed in the figures.


# Enchancing Semi-Supervised Learning for Extractive Summarization with an
LLM-based pseudolabeler

Gaurav Sahu
University of Waterloo
<EMAIL_ADDRESS>
Olga Vechtomova
University of Waterloo
Issam H. Laradji
ServiceNow Research

###### Abstract

This work tackles the task of extractive text summarization in a limited
labeled data scenario using a semi-supervised approach. Specifically, we
propose a prompt-based pseudolabel selection strategy using GPT-4. We evaluate
our method on three text summarization datasets: TweetSumm, WikiHow, and
ArXiv/PubMed. Our experiments show that by using an LLM to evaluate and
generate pseudolabels, we can improve the ROUGE-1 by 10-20% on the different
datasets, which is akin to enhancing pretrained models. We also show that such
a method needs a smaller pool of unlabeled examples to perform better.

Enchancing Semi-Supervised Learning for Extractive Summarization with an LLM-
based pseudolabeler

Gaurav Sahu University of Waterloo<EMAIL_ADDRESS>Olga Vechtomova
University of Waterloo Issam H. Laradji ServiceNow Research

##  1 Introduction

Text summarization is a crucial problem in today’s data-driven era. For
instance, more and more enterprises now have customer service chatlogs and
wish to create an automated text summarization system to enhance the user
experience. The internet, too, is flooded with an overwhelming number of
articles to read. With the exponential growth of digital content, the need for
automated summarization techniques that can swiftly process and distill
information becomes paramount (Koupaee and Wang, 2018; Feigenblat et al.,
2021; Cohan et al., 2018). However, in real-life scenarios, we often do not
have enough labeled data to employ a fully supervised summarization model.
Therefore, we explore the landscape of semi-supervised methods to solve the
task. While supervised methods heavily rely on annotated data, incorporating
semi-supervised learning mitigates the burdens of obtaining large-scale
labeled datasets, thereby extending the applicability of text summarization
models to various domains and languages. More specifically, we tackle the task
of extractive summarization, where the summary is constructed by copying lines
of the input document (Radev et al., 2004; Li et al., 2006).

Semi-supervised approaches for text summarization are heavily under-explored.
Earliest works on extractive summarization through semi-supervision include
(Amini and Gallinari, 2001; Wong et al., 2008), where both works reformulate
the task of extractive text summarization as text classification. First, they
train an unsupervised language model to learn sentence representations. Next,
they train that model on a limited labeled dataset on the binary
classification task of predicting whether or not a sentence will be present in
the summary. More recently, Liu and Lapata (2019) and Zhuang et al. (2023)
employed a similar strategy using large language models (LLMs) like BERT as
their backbones. While these works have shown promising results, they all
primarily focus on the development of a better pretrained model in the first
step. While pretraining is crucial, we also need a better strategy to use the
unlabeled samples during the teacher-student learning phase. Specifically, we
need a selection mechanism to choose the most impactful pseudolabels first
during the iterative teacher-student learning algorithm.

Inspired by the recent success of using LLMs for generating and improving
pseudolabels for various tasks Sahu et al. (2022, 2023); Dziri et al. (2023),
in this work, we focus on improving the process of selecting good-quality
pseudolabels during the training of the summarization model. Specifically, we
propose a novel prompt-based strategy for selecting higher-quality
pseudolabels using a large language model like GPT-4 OpenAI (2023). We
evaluate our approach on three text summarization datasets: TweetSumm
(Feigenblat et al., 2021), WikiHow (Koupaee and Wang, 2018), and ArXiv/PubMed
Cohan et al. (2018). We show that our method can enhance the performance of a
standard teacher-student learning algorithm, which primarily relies on the
teacher’s confidence to determine a pseudolabel’s quality. To the best of our
knowledge, we are the first to show that a good selection and pseudolabeling
strategy can achieve improvements akin to developing a better pretrained
model.

##  2 Methodology

This section describes our confidence-guided pseudolabel selection with GPT-4
score approach for training self-supervised summarization models. We employ a
standard teacher-student training framework during training. 1) First, we
train a fully-supervised model (teacher) on the limited labeled examples in
the dataset and then use it to generate pseudolabels for the unlabeled
dataset. Next, we select a subset of 50 pseudolabels with the highest teacher
confidence. We use PreSumm with BERTbase backbone as our teacher. Therefore,
we reformulate the task of generative extractive summaries to that of
classification, where the model predicts whether or not the given sentence in
the input text will be present in the output summary. We then construct the
extractive summary from the top-$n$ sentences according to the model’s
predicted probability. To define model confidence for a generated summary
(pseudolabel), let $p_{ij}$ denote the probability of sentence $i$ in
unlabeled datapoint $u_{j}$ being selected in the summary. We then compute
model confidence ($C_{j}$) on an input text $u_{j}$ as follows:

$C_{j}=\frac{\sum_{i=1}^{n}(p_{ij})}{n},$ (1)

where $n$ is the number of sentences in the generated summary (pseudolabel).
After selecting top 50 examples using model confidence, we prompt GPT-4 to
output a rating between 0-100 based on the following criteria:

  * •

The summary should be concise, ideally with less than $n$ sentences.

  * •

The summary should capture the key points from the input text. Do not penalize
if it doesn’t include any minor additional details.

  * •

The summary should be extractive, i.e., directly taken from the conversation
without any modifications or removal of symbols.

  * •

The response should contain only the numerical rating.

Figure 1 shows our prompt template for scoring a pseudolabel. We choose the
top 5 unlabeled examples with the highest GPT-4 score for the pseudolabels.
Finally, we prompt GPT-4 to pseudolabel each unlabeled example and append them
to the training set for the next cycle. Figure 2 shows our prompt template for
generating summaries using GPT-4. Note that we use the same mechanism when
prompting as PreSumm since we want to generate extractive summaries.

Figure 1: Prompt for scoring pseudolabels. The demonstration highlights the
two main parts of the prompt: Part 1 contains the text and summary pair, and
Part 2 lists the evaluation criteria. Note: for brevity, we do not include the
full text from the evaluation criteria. Please refer to Section 2 for details
about the evaluation criteria. Figure 2: Prompt for generating pseudolabels.
We attach a line ID for each sentence of the input text and instruct GPT-4 to
use those line IDs in its response.

##  3 Experimental Setup

This section describes our experimental setup, including the datasets used,
training and evaluation details, and the baselines run.

###  3.1 Datasets

We use three datasets for extractive summarization.

1) TweetSumm Feigenblat et al. (2021) is a real-world customer service chat
dataset. The dataset has 1100 conversations between a customer and an agent,
and each conversation has three human-annotated extractive summaries. The
training set has 858 dialogs, out of which we randomly sample a subset of 50
examples as the initial training set and treat the remaining as the unlabeled
set. We use the official validation and test sets for evaluation.

2) WikiHow (Koupaee and Wang, 2018) contains WikiHow articles with their
headlines as abstrative summaries. The dataset has over 180k articles, with
around 168k training articles, and 6000 test and validation articles.

3) ArXiv/PubMed (Cohan et al., 2018) is a collection of scientific articles
from PubMed and ArXiv with their abstracts as their summaries. The dataset has
close to 325k articles, with nearly 300k training articles and 12.5k test and
validation articles.

Notably, the WikiHow and ArXiv/PubMed datasets do not have extractive labels.
Therefore, we follow the same preprocessing steps as the original PreSumm
paper (Liu and Lapata, 2019) to convert the abstraction datasets to extractive
datasets. Since the training corpus is humongous for WikiHow and ArXiv/PubMed,
we limit the pool of unlabeled examples by randomly sampling a subset of 1,000
articles in each training cycle.

We choose the above three datasets to cover a diverse set of scenarios:
TweetSumm contains real-world conversations and human-annotated extractive
summaries, WikiHow contains medium-sized documents, and ArXiv/PubMed contains
long documents and summaries.

###  3.2 Training and Evaluation Details.

We use the TransformerSum
repository111https://transformersum.readthedocs.io/en/latest/ to implement our
training pipeline using bert-base-uncased as the backbone for PreSumm
architecture. We select 50 labeled examples as the initial training set and
add 5 new pseudolabels at the end of each cycle. We terminate the experiment
after 50 cycles, giving a final dataset size of 300 (50 labeled and 250
pseudolabeled examples). We set the summary size ($n$) to 4 for TweetSumm and
8 for WikiHow and ArXiv/PubMed We base these summary sizes on the average
summary size of the labeled training set.

We report the test ROUGE-1 (R-1), ROUGE-2 (R-2), and ROUGE-L (R-L) F-1 scores
(Lin, 2004) for the three datasets. ROUGE-1 and ROUGE-2 scores measure the
$n$-gram overlap between the predicted and the ground truth summaries, whereas
ROUGE-L also takes into consideration the order of $n$-grams. We run each
experiment for three random seeds and report the mean and standard deviation
in Table 1.

| TweetSumm | WikiHow | ArXiv/Pubmed
---|---|---|---
Method | R-1 (%) | R-2 (%) | R-L (%) | R-1 (%) | R-2 (%) | R-L (%) | R-1 (%) | R-2 (%) | R-L (%)
Fully Supervised Models
PreSumm (50 labels) | 48.7 (1.5) | 37.1 (1.1) | 47.9 (2.4) | 23.4 (1.9) | 3.2 (0.8) | 12.1 (1.1) | 26.4 (1.3) | 7.3 (0.9) | 13.5 (1.1)
PreSumm (300 labels) | 61.2 (1.5) | 51.1 (2.1) | 60.3 (1.7) | 26.6 (1.2) | 7.6 (0.6) | 16.7 (1.3) | 29.7 (0.8) | 10.8 (0.9) | 18.5 (1.2)
PreSumm (all labels) | 67.0 (0.7) | 58.1 (0.4) | 65.6 (0.5) | 31.1 (0.4) | 8.9 (0.9) | 19.1 (0.3) | 34.8 (0.6) | 12.3 (0.4) | 19.5 (0.5)
SSPT (50 labels) | 59.6 | 50.2 | 58.9 | - | - | - | - | - | -
Semi-Supervised Models with Different Pseudolabel Selection Strategies
TSL (50:500) | 49.0 | 37.7 | 48.2 | - | - | - | - | - | -
TSL (500:500) | 58.96 | 48.32 | 58.19 | - | - | - | - | - | -
Random | 45.4 (1.4) | 32.4 (1.9) | 42.5 (1.8) | 22.1 (1.7) | 2.4 (1.5) | 9.6 (1.5) | 23.3 (1.4) | 6.1 (1.2) | 12.4 (1.3)
Confidence | 49.7 (1.6) | 39.5 (1.4) | 49.4 (1.3) | 24.5 (0.6) | 4.8 (1.1) | 12.8 (1.0) | 27.6 (1.1) | 7.7 (0.7) | 14.2 (1.2)
\+ GPT4 relabeling | 57.8 (1.2) | 50.3 (0.5) | 58.9 (1.2) | 26.4 (0.3) | 7.3 (0.5) | 16.4 (0.8) | 28.7 (0.9) | 9.5 (1.1) | 17.1 (0.8)
Confidence w/ GPT4 score | 52.3 (1.6) | 42.8 (0.7) | 51.0 (1.4) | 25.2 (0.7) | 5.6 (0.5) | 13.1 (1.0) | 27.7 (0.9) | 7.9 (0.2) | 15.5 (1.3)
\+ GPT4 relabeling (Ours) | 58.9 (1.4) | 50.4 (0.8) | 59.4 (1.5) | 26.1 (0.4) | 7.2 (0.6) | 15.9 (0.9) | 29.1 (0.7) | 9.7 (1.2) | 17.7 (0.6)

Table 1: Mean ROUGE F-1 Scores (with standard deviation) on the three
datasets. SSPT (Mixed strategy) and TSL (50:500) and TSL (500:500) results
taken from Zhuang et al. (2023).

###  3.3 Baselines

We compare our method with the following baselines:

1) PreSumm (Liu and Lapata, 2019). The original PreSumm model that pretrains a
BERT model for summarization. The BERT encoder creates sentence
representations, which are then used to predict whether or not a sentence will
be present in the summary. We train three PreSumm models – one on a limited
training set with 50 labeled examples, another with 300 labeled examples, and
one on the complete training set.

2) Self-supervised Pre-training (SSPT) (Zhuang et al., 2023). Similar to
PreSumm, where, instead of using vanilla BERT as the sentence encoder, it uses
an enhanced BERT model obtained by pertaining on self-supervised tasks
designed specifically for dialog summarization. Since the work focuses solely
on dialogs, it only evaluates on the TweetSumm dataset. We report results for
the “Mixed strategy" as it is shown to be the best model in the paper.

3) Teacher-Student Learning (TSL) (Zhuang et al., 2023). The teacher-student
learning framework uses a slightly different formulation for model confidence
than ours, as given below:

$C_{j}=\sum_{i=1}^{n}(C_{ij})/n_{j}$ (2)

In Equation 2, $C_{ij}=p_{ij}q_{ij}+(1-p_{ij})(1-q_{ij})$, where $p_{ij}$ is
the probability of sentence $i$ being selected for summary for dialog $j$
estimated by the teacher model, and $q_{ij}=1\ \textrm{if}\ {p_{ij}}\
\textrm{in top 4, else}\ 0$. Notably, this formulation also considers the
teacher’s confidence in the sentences it chooses to discard from the summary.
We report the performance of TSL (50:500) as it was the closest to our setup
(50 labeled examples and 500 unlabeled examples).

4) Confidence-guided pseudolabel selection with GPT4 (Ours). Our proposed
method for selecting pseudolabels using PreSumm and GPT-4 confidence, followed
by pseudolabeling with GPT-4. We denote this model by “Confidence w/ GPT4
score + GPT4 relabeling" in Table 1.

5) Random. We randomly add five pseudolabels from the unlabeled set in each
cycle instead of using model confidence and GPT4 relabeling as in 4).

##  4 Results

We now discuss the results in Table 1. The first block in Table 1 shows the
performance of fully supervised PreSumm models trained on three scenarios. The
second block displays the performance of semi-supervised approaches using
different strategies for selecting pseudolabels.

###  4.1 Comparison of Fully supervised models

Intuitively, we note that including more labels leads to better rouge scores
on all the datasets; however, improvements on WikiHow and ArXiv/PubMed are
less pronounced than on the TweetSumm dataset. This might be because the
extractive labels for WikiHow and Arxiv/PubMed were created using a greedy
heuristic that matches sentences from an abstractive summary with the input
article. We must also note that the PreSumm (300 labels) model is a proxy for
an oracle model, as it simulates the best-case scenario where all the
pseudolabels are high-quality. Finally, we observe that SSPT (50 labels) is
comparable to PreSumm (300 labels), suggesting an efficient knowledge transfer
for the enhanced encoder in SSPT.

###  4.2 Effect of Selection Strategy

First, we note that all pseudolabel selection strategies for semi-supervised
models outperform the random baseline. This affirms the importance of a good
pseudolabel selection strategy, especially in a low-resource setup. The random
baseline performs worse than the PreSumm (50 labels), suggesting that the
majority of the pseudolabels generated by the model are low-quality. Using
model confidence leads to slight performance gains on all the datasets, and
adding GPT4-score further improves the results (Confidence v/s Confidence w/
GPT4 score). This suggests that the summarization model also generates some
high-quality pseudolabels, and using GPT4 to rate those pseudolabels is
beneficial. PreSumm’s confidence is insufficient, as it seems misaligned with
the actual ground truth distribution (incorrectly has higher confidence for
bad-quality summaries). This misalignment might also be due to an extremely
data-scarce experiment setting (only 50 labeled examples).

###  4.3 Effect of Relabeling

Relabeling with GPT-4 leads to significant boosts in ROUGE scores across the
board (Confidence v/s Confidence + GPT4 relabeling and Confidence w/ GPT4
score + GPT relabeling). We note that the ROUGE-1 improves from 52.3 to 58.7
on the TweetSumm dataset, 25.2 to 26.1 on the WikiHow dataset, and 27.7 to
29.1 on the ArXiv/PubMed dataset. GPT4 relabeling leads to good results even
when using just PreSumm’s confidence to scoring pseudolabels, suggesting that
most GPT4 pseudolabels are high-quality. The performance after GPT4 relabeling
is very close to the fully supervised PreSumm with 300 labels, even though we
do not include any examples in the relabeling prompt (making this GPT4’s zero-
shot performance). We may further improve the model performance by including
some examples in the prompt.

Our proposed method outperforms TSL (50:500) and TSL (500:500) despite working
in a smaller labeled:unlabeled dataset ratio of 50:250.

##  5 Conclusion and Future Work

This work proposes a novel strategy for selecting pseudolabels to enhance
semi-supervised approaches for low-resource text summarization. We show that
adding a GPT4 critic on top of a teacher’s confidence leads to the selection
of better quality pseudolabels. Critically, pseudolabeling with GPT4 leads to
significant performance gains, making the use of LLMs like GPT4 for semi-
supervised approaches highly promising.

We also identify further improvements that can be made to our approach. For
instance, we relabel all the top 50 examples selected by PreSumm. This is
cheaper than pseudolabeling every unlabeled datapoint, but there’s a
possibility that GPT-4 relabels an already good-quality summary generated by
PreSumm. Therefore, we need better evaluation strategies to decide whether we
should relabel an unlabeled datapoint in the first place. Such a method would
further reduce costs for using closed-source models like GPT-4. Alternatively,
we may also explore the feasibility of open-source models for this task. We
hope this work furthers research toward incorporating LLMs to address such
problems for low-resource semi-supervised tasks.

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