revtex4-2Repair the float

<EMAIL_ADDRESS>
# Search for dark matter in the framework of Einstein-Cartan gravity at the
International Linear Collider (ILC)

Hossam Taha1,2 [  El-sayed A. El-dahshan2<EMAIL_ADDRESS>S.
Elgammal3<EMAIL_ADDRESS>
1Basic Science Department, Faculty of Engineering, The British University in
Egypt 2Physics Department, Faculty of Science, Ain Shams University 3Centre
for Theoretical Physics, The British University in Egypt

###### Abstract

This paper investigates the possibility of Dark Matter (DM) fermions
production alongside a gauge boson (A′) using a model based on Einstein-Cartan
gravity in an electron-positron linear collider, such as the ILC, that
operates at a center-of-mass energy $\sqrt{s}=500$ GeV with a detector’s
integrated luminosity of 1000 fb-1. We used the WHIZARD package as the event
generator to simulate the $e^{+}e^{-}$ interactions that lead to the
production of di-muon pairs and missing transverse energy. We specifically are
performing this study on a low mass dark gauge boson, M${}_{A^{\prime}}$= 10
GeV, that can subsequently decay into a muon pair
(A${}^{\prime}\rightarrow\mu^{+}\mu^{-})$ while aiming to set upper limits on
the free parameters’ masses of the model, such as the torsion field (ST), if
evidence for physics beyond the standard model is not found.

Keywords

Dark matter, Dark gauge boson, The International Linear Collider (ILC), Beyond
the Standard Model

††preprint: APS/123-QED

##  I Introduction

While the Standard Model of Particle Physics (SM) successfully describes many
interactions, including the strong and weak nuclear forces as well as the
electromagnetic force, it falls short in accounting for the gravitational
force. Additionally, it only accounts for about 5% of the observable universe.
It does not account for dark energy, which is believed to be the cause behind
the universe’s expansion, nor Dark Matter (DM), our focus in this work, which
is suggested as a resolution to the mystery surrounding the unidentified
portion of the universe’s mass [1, 2, 3]. DM is a non-luminous form of matter
that almost does not interact with known particles in the SM while being
strongly supported by numerous cosmological observations [4, 5, 6]. Exploring
DM requires us to go beyond the SM, considering it a low-energy manifestation
of broader theories of higher energies that could unify the four fundamental
forces [7].

We propose that linear colliders operating at relatively low energies, such as
the International Linear Collider (ILC) operating initially at a center-of-
mass energy of $\sqrt{s}=500$ GeV and a detector’s integrated luminosity of
$\mathcal{L}=$ 1000 fb-1 (with a potential upgrade to $\sqrt{s}=1000$ GeV in
RUN II) [8, 9, 10], could be well-suited for exploring potential low-mass dark
neutral gauge boson candidates. These are beyond what the Large Hadron
Collider (LHC) can reach with its current TeV energy scale.

ATLAS and CMS collaborations have previously searched for a similar dark gauge
boson ($Z^{\prime}$) predicted by the Grand Unified Theory (GUT) and
Supersymmetry [11, 12, 13, 14], but found no evidence of its existence in the
LHC’s RUN II. The CMS experiment set a 95% confidence exclusion limit for
masses in the range between 600 - 5150 GeV, while the ATLAS experiment set the
limit in the range of 600 - 5100 GeV [15, 16].

The ILC is a future electron-positron linear collider initially designed to
work as a Higgs factory to study Higgs bosons. However, it can also be
utilized in searching for new particles beyond the SM [17]. The use of
electron-positron collisions instead of the usual proton-proton collisions at
the LHC provides a cleaner background largely free of complex quark
interactions, as the underlying events of p-p collisions will be eliminated.
Additionally, it offers higher precision because electrons and positrons are,
themselves, elementary particles, which would also make this kind of collision
energy efficient as they will use the full center-of-mass energy. Furthermore,
the proposed collider utilizes the use of controllable initial particle energy
and beam polarization techniques.

An alternative formulation of general relativity, known as the Einstein-Cartan
gravity theory, allows for additional degrees of freedom through the torsion
field, that characterizes the torsion aspect of the spacetime. In the
Einstein-Cartan portal, gravity encoded in the torsion field (ST) can couple
through its axial-vector mode to all of the SM fermions. It could provide a
way to search for DM by probing the dark gauge boson A′, which couples to all
SM particles through kinetic mixing [18, 19, 20, 21]. Searching for DM usually
involves the production of SM particles plus a large missing transverse energy
(MET), taken away by the dark particles, with a signature of (SM +
$E_{T}^{miss}$). The search for a DM fermion ($\chi$) can be conducted by
investigating the missing transverse energy because the A′ decays into a SM
di-lepton ($A^{\prime}\rightarrow l^{+}l^{-})$, which can be detected.

We are utilizing the torsion model to search for DM using monte-carlo event
generators to simulate the conditions of the electron-positron interactions at
the ILC at $\sqrt{s}$ = 500 GeV $\mathcal{L}=$ 1000 fb-1, aiming to set a %95
confidence limits on the free parameters of this model.

In this paper, we begin by describing the $U(1)_{D}$ simplified torsion model
based on the Einstein-Cartan gravity theory, which predicts the possible DM
fermions production, and the free parameters associated with this model, in
addition to a light-dark gauge boson (M${}_{A^{\prime}}<$ 80 GeV), in the
second section (Section II). Then we move to describe the simulation methods
and techniques used in event generation for both the SM background and the
model signal in Section III. While in Section IV, we present the cut-based
analysis in terms of the plan and the specific cuts. This is followed in
Section V by discussing the results of the analysis and, finally, the summary
in Section VI.

##  II EINSTEIN-CARTAN torsion model

The simplified torsion model discussed in [21] proposes that Einstein-Cartan
gravity might serve as a channel for investigating the dark gauge boson (A′).
This dark boson could interact with SM particles through the coupling of the
torsion field with SM fermions (such as $e^{+}e^{-}$) leading to the creation
of DM particles ($\chi$ $\bar{\chi}$). The suggested dark gauge boson
$A^{\prime}_{\mu}$ can be produced via $e^{+}e^{-}$ pair annihilation mediated
by the heavy torsion field ($S_{\mu}$), that represents the axial–vector part
of the torsion tensor ($T^{\lambda}_{\mu\nu}$ ), which then transforms into
two DM particles ($\chi$ $\bar{\chi}$). These particles are sufficiently heavy
to radiate A′ via bremsstrahlung, which can subsequently decay into a muon
pair ($\mu^{+}$ $\mu^{-}$) as shown in figure 1

There are two scenarios in the torsion model. Our focus is on the
bremsstrahlung scenario, in which only one of the dark fermions radiates the
$A^{\prime}_{\mu}$ via, of course, bremsstrahlung. The other scenario is the
cascade scenario, in which both dark fermions radiate the A′. The terms
responsible for the interaction between the Dirac fermion $\psi$ and the
torsion field in the effective Lagrangian are given by [21]:

$\bar{\psi}i\gamma^{\mu}(\partial_{\mu}+ig_{\eta}\gamma^{5}S_{\mu}+...)\psi,$
(1)

where $\texttt{g}_{\eta}$ represents the coupling of the torsion field to
Dirac fermions. The term in the effective Lagrangian describing the coupling
between the dark gauge boson $A^{\prime}_{\mu}$ and the dark fermions, and the
coupling of the torsion field to the DM, is given by [21]:

$\bar{\chi}(i\gamma^{\mu}D_{\mu}-M_{\chi})\chi,$ (2)

where
$D_{\mu}=\partial_{\mu}+ig_{\eta}\gamma^{5}S_{\mu}+ig_{D}A^{\prime}_{\mu},M_{\chi}$
represents the mass of the dark fermion and $\texttt{g}_{D}$ is the coupling
of the dark gauge boson to the dark fermion.

The neutral dark gauge boson (A′) decays to two SM fermions. In our case, we
choose the muonic decay of A′. The highest significant branching ratio of
$A^{\prime}\rightarrow\mu^{+}\mu^{-}$ could be reached if the following mass
assumption is satisfied [21]:

$M_{A^{\prime}}<2M_{\chi}.$ (3)

The model’s free parameters include the masses of the torsion field
($M_{ST}$), the dark gauge boson ($M_{A^{\prime}}$ ), the DM fermion
($M_{\chi}$) and the coupling constants ($\texttt{g}_{\eta}$ and
$\texttt{g}_{D}$). Now, because this model was theoretically investigating the
DM production through only p-p collisions, we had to tune some of the
parameters within acceptable limits to suit our desired $e^{+}e^{-}$
colliders. Thus, $\texttt{g}_{\eta}$ was set to 0.25, while the coupling
constant $\texttt{g}_{D}$ was set to 1.2, as taken from [21], to achieve the
highest possible cross-section multiplied by the branching ratio.

Figure 1:  Signal Topology - An electron interacts with a positron, forming
the torsion field (ST) that transforms into two dark fermions, while one of
them radiates the A′ via bremsstrahlung.

Since we are interested in studying the possible production of dark fermions
at linear colliders like the ILC, along with light neutral dark gauge boson,
we have fixed the mass of A′ to be M${}_{A^{\prime}}$ = 10 GeV to satisfy the
mass condition given in equation (3). The usual signature of this process is
composed of a pair of oppositely charged muons from the A′ decay and a large
missing transverse energy resulting from the non-decaying DM fermion $\chi$,
serving as a clean channel to SM backgrounds. This makes our studied events
have a topology of $\mu^{+}\mu^{-}+E_{T}^{miss}$.

$M_{\chi}$/GeV $M_{ST}$/GeV | 1000 | 1500 | 2000 | 2500 | 3000 | 3500 | 5000
---|---|---|---|---|---|---|---
10 | $1.9508\times 10^{-03}$ | $2.8114\times 10^{-04}$ | $8.1043\times 10^{-05}$ | $3.1576\times 10^{-05}$ | $1.4904\times 10^{-05}$ | $7.9536\times 10^{-06}$ | $1.8783\times 10^{-06}$
20 | $1.7421\times 10^{-03}$ | $2.5212\times 10^{-04}$ | $7.2352\times 10^{-05}$ | $2.8380\times 10^{-05}$ | $1.3349\times 10^{-05}$ | $7.1440\times 10^{-06}$ | $1.6704\times 10^{-06}$
30 | $1.5572\times 10^{-03}$ | $2.2454\times 10^{-04}$ | $6.4489\times 10^{-05}$ | $2.5259\times 10^{-05}$ | $1.1896\times 10^{-05}$ | $6.3336\times 10^{-06}$ | $1.4848\times 10^{-06}$
40 | $1.3773\times 10^{-03}$ | $1.9971\times 10^{-04}$ | $5.720\times 10^{-05}$ | $2.2333\times 10^{-05}$ | $1.0560\times 10^{-05}$ | $5.6332\times 10^{-06}$ | $1.3234\times 10^{-06}$
50 | $1.2177\times 10^{-03}$ | $1.7578\times 10^{-04}$ | $5.0338\times 10^{-05}$ | $1.9736\times 10^{-05}$ | $9.3121\times 10^{-06}$ | $4.9667\times 10^{-06}$ | $1.1679\times 10^{-06}$
60 | $1.0682\times 10^{-03}$ | $1.5396\times 10^{-04}$ | $4.4129\times 10^{-05}$ | $1.7320\times 10^{-05}$ | $8.1510\times 10^{-06}$ | $4.3449\times 10^{-06}$ | $1.0225\times 10^{-06}$
70 | $9.3211\times 10^{-04}$ | $1.3373\times 10^{-04}$ | $3.8460\times 10^{-05}$ | $1.510\times 10^{-05}$ | $7.1015\times 10^{-06}$ | $3.7954\times 10^{-06}$ | $8.8975\times 10^{-07}$
80 | $8.0437\times 10^{-04}$ | $1.1632\times 10^{-04}$ | $3.3075\times 10^{-05}$ | $1.2999\times 10^{-05}$ | $6.1110\times 10^{-06}$ | $3.2437\times 10^{-06}$ | $7.6534\times 10^{-07}$
90 | $6.8516\times 10^{-04}$ | $9.8643\times 10^{-05}$ | $2.8312\times 10^{-05}$ | $1.1063\times 10^{-05}$ | $5.2134\times 10^{-06}$ | $2.7763\times 10^{-06}$ | $6.5276\times 10^{-07}$
100 | $5.7573\times 10^{-04}$ | $8.3185\times 10^{-05}$ | $2.380\times 10^{-05}$ | $9.3143\times 10^{-06}$ | $4.3984\times 10^{-06}$ | $2.3398\times 10^{-06}$ | $5.4989\times 10^{-07}$
110 | $4.7792\times 10^{-04}$ | $6.8901\times 10^{-05}$ | $1.9733\times 10^{-05}$ | $7.7054\times 10^{-06}$ | $3.6475\times 10^{-06}$ | $1.9370\times 10^{-06}$ | $4.5581\times 10^{-07}$
120 | $3.9017\times 10^{-04}$ | $5.6286\times 10^{-05}$ | $1.6070\times 10^{-05}$ | $6.3093\times 10^{-06}$ | $2.9750\times 10^{-06}$ | $1.5810\times 10^{-06}$ | $3.7053\times 10^{-07}$
130 | $3.1297\times 10^{-04}$ | $4.5031\times 10^{-05}$ | $1.2820\times 10^{-05}$ | $5.0107\times 10^{-06}$ | $2.3662\times 10^{-06}$ | $1.2604\times 10^{-06}$ | $2.9720\times 10^{-07}$
140 | $2.4378\times 10^{-04}$ | $3.4828\times 10^{-05}$ | $1.0046\times 10^{-05}$ | $3.9209\times 10^{-06}$ | $1.8428\times 10^{-06}$ | $9.8466\times 10^{-07}$ | $2.3183\times 10^{-07}$
150 | $1.8568\times 10^{-04}$ | $2.6585\times 10^{-05}$ | $7.5985\times 10^{-06}$ | $2.973\times 10^{-06}$ | $1.4038\times 10^{-06}$ | $7.4473\times 10^{-07}$ | $1.7509\times 10^{-07}$
160 | $1.3533\times 10^{-04}$ | $1.9427\times 10^{-05}$ | $5.5409\times 10^{-06}$ | $2.1671\times 10^{-06}$ | $1.0218\times 10^{-06}$ | $5.4587\times 10^{-07}$ | $1.2827\times 10^{-07}$
Table 1: The signal’s cross-section measurements (in picobarns) calculated for
different values of the masses $M_{ST}$ and $M_{\chi}$ in GeV for the light-
dark sector mass assumption at M${}_{A^{\prime}}$ = 10 GeV at $\sqrt{s}=500$
GeV, and couplings constants of $g_{\eta}=0.25,\leavevmode\nobreak\
g_{D}=1.2$.

##  III Simulated Signal samples and SM backgrounds

The SM background processes that produce muon pairs within the signal region
are Drell-Yan ($DY\rightarrow\mu^{+}\mu^{-}$) production, top quark pairs
production ($t\bar{t}\rightarrow\mu^{+}\mu^{-}+2b+2\nu$) and production of
diboson ($W^{+}W^{-}\rightarrow\mu^{+}\mu^{-}+2\nu$,
$ZZ\rightarrow\mu^{+}\mu^{-}+2\nu$, and $ZZ\rightarrow 4\mu$). We used WHIZARD
v2.8.5 [22], a general-purpose matrix element event generator, to generate the
model signal events. The hadronization process was done with Pythia v6 [23].
As for the simulation of the ILC detector (the ILD), the reconstruction
processes, and the simulation of read-out system response (digitization), we
used DELPHES [24], for a fast detector simulation of the ILD. The events were
generated from electron-positron collisions at the ILC with 500 GeV center-of-
mass energy, which corresponds to the circumstances of RUN I, with polarized
degrees of $P_{e^{-}}=0.8$ for the electron beams, and $P_{e^{+}}=-0.3$ for
the positron beams.

Figure 2: Plot of the cross-section (in log scale) on the y-axis vs Mχ on the
x-axis at different masses of the torsion field. Figure 3: Plot of the cross-
section (in log scale) on the y-axis at different masses of the DM ($\chi$) vs
the torsion field on the x-axis.

For different values of the DM ($\chi$) and torsion field ($ST$) masses, the
measurements of cross-section times the branching ratio ($\times$ BR) are
calculated and shown in table 1. To get a sense of the relation between each
of $\chi$ and ST masses with the cross-section, we plotted each variable
against the cross-section ($\times$ BR) in log scale, as shown in figures 2,
3. The simulated signals that we used in our analysis are private production
samples, generated using the matrix element event generator the WHIZARD. The
used Monte Carlo samples (signal and background events) in our work were
calculated at leading order (LO), as well as for their corresponding cross-
sections, and are indicated in table 2. Thus, the contributions of our
processes (SM background and signal samples) have been determined using the
Monte Carlo simulations, normalized to their cross-sections, and an integrated
luminosity of $1000fb^{-1}$.

Process |  Decay channel |  Generator |  $\sigma\times\mathrm{BR}$ (fb) |  Order
---|---|---|---|---
$DY$ | $\mu^{+}\mu^{-}$ | WHIZARD | 1852 | LO
$t\bar{t}$ | $\mu^{+}\mu^{-}+2b+2\nu$ | WHIZARD | 10.36 | LO
WW | $\mu^{+}\mu^{-}+2\nu$ | WHIZARD | 232.48 | LO
ZZ | $\mu^{+}\mu^{-}+2\nu$ | WHIZARD | 3.72 | LO
ZZ | $4\mu$ | WHIZARD | 0.47 | LO
Table 2: SM background processes that mimic our signal, with their
corresponding event generator, cross-sections multiplied by the branching
ratio for each process, and the order of calculations.

In simulating the SM processes which include muons and/or missing energy in
the final state (accounting for the neutrinos that are not detected) which can
interfere with the signal events, we used the WHIZARD package (as the ILC is
not yet operating), we are interfacing it with Pythia to model showering and
hadronization.

Step | Variable | Requirements
---|---|---
Pre-selection | $p^{\mu}_{T}\ (GeV)$ | $>10$
$|\eta^{\mu}|\ (rad)$ | $<2.4$
IsolationVar | $<0.1$
Tight selection | $|E_{T}^{\mu^{+}\mu^{-}}-{E}_{T}^{\text{miss}}|/E_{T}^{\mu^{+}\mu^{-}}\ $ | $<0.4$
Mass window (GeV) | $M_{A^{\prime}}\times 0.5<M_{\mu^{+}\mu^{-}}\ <M_{A^{\prime}}\times 1.5$
$\Delta\phi_{\mu^{+}\mu^{-},\vec{E}{{}_{T}^{miss}}}\ (rad)$ | $>2.8$
$\Delta R_{\mu^{+}\mu^{-}}$ | $<0.8$
Table 3: Summary of the final event selection used in the analysis.

##  IV Events selection and analysis strategy

The event selection for this analysis is designed to reconstruct a final state
of muons with low transverse momentum ($p_{T}$) in association with missing
transverse energy, accounting for the DM candidates ($\chi\bar{\chi}$). The
selection criteria are applied in the form of cuts on different kinematic
parameters, as detailed in table 3. Where $\eta^{\mu}$ represents the pseudo-
rapidity for the di-muon. “IsolationVar” refers to an isolation cut in
DELPHES, used to reject muons produced inside jets. In this cut, the scalar
$p_{T}$ sum of all muon tracks within a cone of $\Delta R=0.5$ around the muon
candidate, excluding the muon candidate itself, must not exceed 10% of the
muon’s $p_{T}$. Thus, each event is selected to include two oppositely charged
muons. After applying the pre-selection cuts, at which each event must contain
exactly two muons with opposite charges, the tight selection is optimized for
DM signals to distinguish them from the SM background, thereby obtaining the
best-expected limit.

Figure 4 shows the missing transverse energy distributions for events passing
the pre-selection cuts listed in table 3, which are stacked on each other. The
red histogram represents the $t\bar{t}$ background, the yellow histogram
stands for the vector boson pair backgrounds (ZZ and WW), and the cyan
histogram represents the Drell-Yan background.

The non-filled histogram represents the bremsstrahlung scenario within the
framework of the Einstein-Cartan gravity, which has been generated with fixed
values for the neutral light gauge boson A′ and DM ${\chi}$ masses ($M\chi=10$
GeV, $M_{A^{\prime}}=10$ GeV) while varying the torsion field mass, which is
represented by different colored histograms, and overlaid on the stacked
histograms. As shown in figure 4, the SM backgrounds dominate the signal
samples. Thus, applying a tighter selection is required to discriminate our
signal from the SM backgrounds. Such selection is based on the following four
variables:

Figure 4: The missing transverse energy after applying the pre-selection cuts
listed in table 3 for both the SM background and the torsion model at
different ST mass points (MST= 1000, 1500, 3000, 5000 GeV) and at a DM mass of
Mχ = 10 GeV, and M${}_{A^{\prime}}$ = 10 GeV.

(1) $|E_{T}^{\mu^{+}\mu^{-}}-{E}_{T}^{\text{miss}}|/E_{T}^{\mu^{+}\mu^{-}}$,
which is the relative difference between the di-muon’s transverse energy
$E_{T}^{\mu^{+}\mu^{-}}$, and the missing transverse energy
${E}_{T}^{\text{miss}}$.
(2) The mass of the di-muon system (M${}_{\mu^{+}\mu^{-}}$) is required to be
within $0.5\times M_{A^{\prime}}<M_{\mu^{+}\mu^{-}}<1.5\times M_{A^{\prime}}$
to preserve consistency with leptons produced by the A′ boson decay. Having
M${}_{A^{\prime}}$= 10 GeV, the mass window is then set to be in the range of
5 to 15 GeV.
(3) The azimuthal angle difference between the di-muon system and the missing
transverse energy $\Delta\phi_{\mu^{+}\mu^{-},\vec{E}{{}_{T}^{miss}}}$.
(4) $\Delta R_{\mu^{+}\mu^{-}}$, the angular distance between the two muons.
These cuts (figures; 5(a), 5(b) and 5(c)) succeed in suppressing the SM
background as seen in figure 6.

(a) $\Delta\phi_{\mu^{+}\mu^{-},\vec{E}{{}_{T}^{miss}}}$.

(b) $|E_{T}^{\mu^{+}\mu^{-}}-{E}_{T}^{\text{miss}}|/E_{T}^{\mu^{+}\mu^{-}}$.

(c) $\Delta R_{\mu^{+}\mu^{-}}$.

Figure 5: The variables employed in the tight cuts. The histograms are
normalized to unity to emphasize the overall characteristics. A vertical black
dotted line indicates the cut-off value.

##  V Results

In the di-muon channel, we used a shape-based analysis. The missing transverse
energy distributions ${E^{miss}_{T}}$ are effective discriminant variables.
The signal processes lead to relatively larger ${E^{miss}_{T}}$ values than
those of the SM backgrounds. Figure 6 shows the missing transverse energy
distribution, after applying the final event selection, highlighting a
significant decrease in the SM background processes.

The number of di-muon events passing the pre-selection and the final selection
are outlined in table 4 for each SM background process and the signal samples
with torsion mass $M_{ST}$ of 1000, 1500 GeV, $M_{\chi}$ = 10 GeV, and
$M_{A^{\prime}}$ = 10 GeV, with an integrated luminosity of 1000 fb-1. Adding
the tight selection cuts (listed in table 3) to the pre-selection resulted in
almost complete suppression of the DY process, and the other SM backgrounds
were reduced by roughly three orders of magnitude. An ad-hoc flat 10%
uncertainty is applied to cover all possible systematic effects. The total
uncertainties, including both the statistical and systematic components, have
been taken into account for the simulated signal and background samples.

Figure 6: The missing transverse energy after applying the final cuts listed in table 3. The torsion field signal at M${}_{ST}=$ 1000, 1500, 3000 and 5000 GeV are superimposed for Mχ = 10 GeV, and M${}_{A^{\prime}}$ = 10 GeV. Process |  No. of events passing pre-selection |  No. of events passing final selection
---|---|---
$DY$ | 898381 $\pm$ 89843 | 0.0 $\pm$ 0.0
$t\bar{t}$ | 6289 $\pm$ 634 | 12 $\pm$ 4
WW | 135947 $\pm$ 13600 | 90 $\pm$ 13
ZZ($4\mu$) | 2517 $\pm$ 257 | 0.0 $\pm$ 0.0
ZZ($2\mu 2\nu$) | 460 $\pm$ 51 | 0.3 $\pm$ 0.6
Total background | 145213 $\pm$ 14526 | 102 $\pm$ 14
Signal: $M_{ST}$= 1000, $M_{\chi}$=10 (GeV) | 637 $\pm$ 69 | 620 $\pm$ 67
Signal: $M_{ST}$= 1500, $M_{\chi}$=10 (GeV) | 92 $\pm$ 13 | 89 $\pm$ 13
Table 4: Number of di-muon events passing pre-selection and final selection
for both the SM background and the model signals. The total uncertainties,
including both the statistical and systematic components, have been taken into
account for the simulated signal and background samples.

We used the Asimov formula to calculate the statistical significance, $S$, as
described in [25]

$S=\sqrt{2\times((N_{s}+N_{b})log(1+\frac{N_{s}}{N_{b}})-N_{s})},$ (4)

where $N_{s},N_{b}$ are the numbers of signal and total SM background events
passing certain selection criteria. We also plotted the statistical
significance against the detector’s luminosity (figure 7) to identify the
values of MST that lead to the desired 5$\sigma$ discovery at the pre-
selection and final selection stages. Figure 7 underscores the critical role
of the tight selection criteria outlined in table 3 in distinguishing between
the signal and the SM backgrounds to achieve a 5$\sigma$ discovery. Figure
7(b) indicates that for the $M_{ST}=$ 1500 GeV mass point, a 5$\sigma$
discovery could be attained with a luminosity of 415 fb-1.

(a) Significance vs luminosity plot at the pre-selection level.

(b) Significance vs luminosity plot at the final selection level.

Figure 7: The statistical significance plots versus the ILD integrated
luminosity before (7(a)) and after (7(b)) the final event selection. The red
horizontal dotted line points to the statistical $5\sigma$ value.

In analyzing our results statistically, we used the profile likelihood method
and the modified frequentist construction CLs [26, 27] used in the asymptotic
approximation [25] to set exclusion limits at 95% confidence level on the
product of signal cross-sections and branching ratio
Br($A^{\prime}\rightarrow\mu^{+}\mu^{-}$).

Figure 8 displays the 95% upper limit on the cross-section times the branching
ratio as a function of the mass of the torsion field MST within the framework
of the Einstein-Cartan gravity model. In the context of muonic decay of
$A^{\prime}$ with coupling constant $\texttt{g}_{\eta}$ = 0.25 and
$\texttt{g}_{D}$ = 1.2, and DM mass $M_{\chi}$ = 10 (8(a)), 70 (8(b)), 158
(8(c)) GeV for M${}_{A^{\prime}}$ = 10 GeV. We have observed from figure 8
that the exclusion of torsion field (ST) production applies to the mass range
below 2171 in 8(a), 1658 in 8(b), and 1008 GeV in 8(c), as the expected median
indicates.

In figure 9, the simplified model illustrates the cross-section multiplied by
the branching ratio limit. This graph pertains to the mediator’s mass,
$M_{ST}$, and the DM mass, $M_{\chi}$. The region below the 95% expected limit
(depicted by the black dotted line) is excluded. The findings from the
collective signal regions exclude anticipated values falling within
$1008<M_{ST}<2171$ GeV.

(a) For M$\chi$= 10 GeV, the intersection with the theoretical line occurred
exactly at $M_{ST}$ = 2171 GeV.

(b) For M$\chi$= 70 GeV, the intersection with the theoretical line occurred
exactly at $M_{ST}$ = 1658 GeV.

(c) For M$\chi$= 158 GeV, the last possible intersection with the theoretical
line occurred exactly at $M_{ST}$ = 1008 GeV.

Figure 8: 95% CL upper limits on the cross-section multiplied by the branching
ratio (expected), as a function of the mediator’s mass (MST), considering the
muonic decay of A′. The black line represents the torsion model at different
$\chi$ mass points $M_{\chi}=$ 10 GeV (8(a)), 70 GeV (8(b)), and 158 GeV
(8(c)), with M${}_{A^{\prime}}$ = 10 GeV. A vertical red dotted line indicates
the upper limit value. Figure 9: The 95% CL limits set on the product of
cross-section multiplied by the branching ratios for a variety of values of
the model’s parameters (MST, Mχ). The region below the black dots is expected
to be excluded.

##  VI Summary

A search for the production of DM fermions along with a light-dark gauge boson
A′ has been conducted in the framework of Einstein-Cartan gravity. We used MC
samples in this search generated using the WHIZARD package to simulate
electron-positron collisions at $\sqrt{s}=$ 500 GeV, with an integrated
luminosity of 1000 fb-1. Results from the muonic decay mode of A′ and the
statistical and systematic combination of uncertainties are presented. We used
a benchmark signal to test the bremsstrahlung scenario, trying various torsion
field (ST) masses on the propagator. We kept the coupling constants
$\texttt{g}_{\eta}=0.25$ and $\texttt{g}_{D}=1.2$ fixed to attain the highest
possible cross-section times branching ratio, with polarized degrees of
$P_{e^{-}}=0.8$ for the electron beams, and $P_{e^{+}}=-0.3$ for the positron
beams.

By implementing the specified cuts (detailed in table 3), a significant drop
in the SM backgrounds was noticed without affecting the strength of the
signal. This, in turn, enabled us to achieve a $5\sigma$ discovery for
$M_{ST}=$ 1000 and 1500 GeV. If the mass value, $M_{ST}$, is above 1500 GeV,
achieving a 5$\sigma$ discovery would be challenging due to the shallow cross-
section (multiplied by the branching ratio). This situation may necessitate
the implementation of additional new cuts to reduce the SM backgrounds such as
$t\bar{t}$ and WW.

If the ILC does not detect this signal, we can establish upper limits on the
masses of the torsion field ($M_{ST}$) and DM $(M_{\chi})$ at the 95%
confidence level for the decay of A′ in the charged muonic channel. We have
set limits that exclude the $M_{ST}$ range from 1008 to 2171 GeV and also
exclude the ranges between 10 - 158 GeV for the DM mass ($M_{\chi}$). Further
limits can be implemented on other parameters, such as the mass of A′, in
future work.

###### Acknowledgements.

We want to thank Cao H. Nam, the author of [21], for sharing the UFO file of
the torsion model with us that was used in WHIZARD for generating signal
events. This paper is based on works supported by the Science, Technology, and
Innovation Funding Authority (STDF) under grant number 48289.

Data Availability Statement: This manuscript has no associated data or the
data will not be deposited.

## References

  * [1] M. K. Gaillard, P. D. Grannis, F. J. Sciulli, ”The standard model of particle physics”, Reviews of Modern Physics, 71(2), S96–S111, 1999. https://doi.org/10.1103/revmodphys.71.s96
  * [2] A. Crivellin, B. Mellado, ”Anomalies in particle physics and their implications for physics beyond the standard model”, Nat Rev Phys 6, 294–309, 2024. https://doi.org/10.1038/s42254-024-00703-6
  * [3] J. A. Frieman, M. S. Turner, D. Huterer, ”Dark Energy and the Accelerating Universe”, Annual Review of Astronomy and Astrophysics, 46, 385-432, 2008. https://doi.org/10.1146/annurev.astro.46.060407.145243
  * [4] R. W. Tucker, C. Wang, ”Dark matter gravitational interactions”, Classical and Quantum Gravity, vol. 15, no. 4, p. 933, 1998. https://dx.doi.org/10.1088/0264-9381/15/4/015
  * [5] D. Clowe, M. Bradač, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones, D. Zaritsky, ”A Direct Empirical Proof of the Existence of Dark Matter*”, The Astrophysical Journal, 648(2), L109, 2006. https://dx.doi.org/10.1086/508162
  * [6] C. Lage, G. R. Farrar, ”The bullet cluster is not a cosmological anomaly”, Journal of Cosmology and Astroparticle Physics, 2015(02), 038–038, 2015. https://dx.doi.org/10.1088/1475-7516/2015/02/038
  * [7] P. Langacker, ”The Standard Model and Beyond (1st ed.)”, CRC Press, 2010. https://doi.org/10.1201/b12328
  * [8] J. Brau, Y. Okada, N. Walker, ”ILC Reference Design Report Volume 1 - Executive Summary”, 2007. https://doi.org/10.48550/arXiv.0712.1950
  * [9] A. Djouadi, J. Lykken, K. Mönig, Y. Okada, M. Oreglia, S. Yamashita, ”International Linear Collider Reference Design Report Volume 2: PHYSICS AT THE ILC”, 2007. https://arxiv.org/abs/0709.1893
  * [10] N. Phinney, N. Toge, N. Walker, ”ILC Reference Design Report Volume 3 - Accelerator”, 2007. https://arxiv.org/abs/0712.2361
  * [11] M. Cvetic and S. Godfrey, “Discovery and identification of extra gauge bosons”. arXiv:hep-ph/9504216
  * [12] A. Leike, “The Phenomenology of extra neutral gauge bosons”, Phy. Rep. 317, 143, 1999. arXiv:hepph/9805494
  * [13] M. Cvetic, P. Langacker, and B. Kayser, “Determination of g-R / g-L in left-right symmetric models at hadron colliders”, Phys. Rev. Lett. 68, 2871, 1992. https://doi.org/10.1103/PhysRevLett.68.2871
  * [14] S. Dimopoulos and H. Georgi, “Softly Broken Supersymmetry And SU(5)”, Nucl. Phys. B 193, 150, 1981. https://doi.org/10.1016/0550-3213(81)90522-8
  * [15] ATLAS Collaboration, ”Search for high-mass dilepton resonances using 139 fb-1 of pp collision data collected at $\sqrt{s}=13$ TeV with the ATLAS detector”, Physics Letters B, 796, 68–87, 2019. http://dx.doi.org/10.1016/j.physletb.2019.07.016
  * [16] CMS Collaboration, ”Search for resonant and nonresonant new phenomena in high-mass dilepton final states at $\sqrt{s}$ = 13 TeV”, Journal of High Energy Physics (7), 2021. http://dx.doi.org/10.1007/JHEP07(2021)208
  * [17] P. Bambade, T. Barklow, T. Behnke, et al. ”The International Linear Collider: A Global Project”, 2019. https://arxiv.org/abs/1903.01629
  * [18] T. W. B. Kibble; Lorentz ”Invariance and the Gravitational Field”, J. Math. Phys. 2, 212–221, 1961. https://doi.org/10.1063/1.1703702
  * [19] F. W. Hehl, P. von der Heyde, G. D. Kerlick, J. M. Nester, ”General relativity with spin and torsion: Foundations and prospects”, Rev. Mod. Phys. 48(3), 393-416, 1976. https://doi.org/10.1103/RevModPhys.48.393
  * [20] M. Shaposhnikov, A. Shkerin, I. Timiryasov, S. Zell, ”Erratum: Einstein-Cartan Portal to Dark Matter [Phys. Rev. Lett. 126, 161301 (2021)]”, Phys. Rev. Lett., 127(16), 169901, 2021. https://doi.org/10.1103/PhysRevLett.127.169901
  * [21] C. H. Nam, ”Probing dark gauge boson via Einstein-Cartan portal”, Phys. Rev. D 105, 075015, 2022 https://doi.org/10.1103/PhysRevD.105.075015
  * [22] W. Kilian, T. Ohl, J. Reuter, ”WHIZARD—simulating multi-particle processes at LHC and ILC”, The European Physical Journal C, 71(9), 2011. http://dx.doi.org/10.1140/epjc/s10052-011-1742-y
  * [23] T. Sjöstrand, S. Mrenna, P. Skands, ”PYTHIA 6.4 physics and manual”, Journal of High Energy Physics (05), 026–026, 2006. http://dx.doi.org/10.1088/1126-6708/2006/05/026
  * [24] J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, M. Selvaggi, ”DELPHES 3: a modular framework for fast simulation of a generic collider experiment”, Journal of High Energy Physics, (2), 57, 2014. https://doi.org/10.1007/JHEP02(2014)057
  * [25] G. Cowan, K. Cranmer, E. Gross, O. Vitells, ”Asymptotic formulae for likelihood-based tests of new physics”, Eur. Phys. J. C 71, 1554, 2011. https://doi.org/10.1140/epjc/s10052-011-1554-0
  * [26] A. L. Read, ”Presentation of search results: the CLs technique”, J. Phys. G: Nucl. Part. Phys., 28, 2693, 2002. https://doi.org/10.1088/0954-3899/28/10/313
  * [27] T. Junk, ”Confidence level computation for combining searches with small statistics”, ”Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment”, 434(2–3), 435-443, 1999. https://doi.org/10.1016/S0168-9002(99)00498-2


# Full-Resolution Encoder–Decoder Networks with Multi-Scale Feature Fusion for
Human Pose Estimation

Jie Ou  University of Electronic Science and Technology of ChinaChengduChina
<EMAIL_ADDRESS>,  Mingjian Chen  University of Electronic Science and
Technology of ChinaChengduChina<EMAIL_ADDRESS>and  Hong Wu  University
of Electronic Science and Technology of ChinaChengduChina<EMAIL_ADDRESS>
(2021)

###### Abstract.

To achieve more accurate 2D human pose estimation, we extend the successful
encoder-decoder network, simple baseline network (SBN), in three ways. To
reduce the quantization errors caused by the large output stride size, two
more decoder modules are appended to the end of the simple baseline network to
get full output resolution. Then, the global context blocks (GCBs) are added
to the encoder and decoder modules to enhance them with global context
features. Furthermore, we propose a novel spatial-attention-based multi-scale
feature collection and distribution module (SA-MFCD) to fuse and distribute
multi-scale features to boost the pose estimation. Experimental results on the
MS COCO dataset indicate that our network can remarkably improve the accuracy
of human pose estimation over SBN, our network using ResNet34 as the backbone
network can even achieve the same accuracy as SBN with ResNet152, and our
networks can achieve superior results with big backbone networks.

Encoder-Decoder, Full Output Resolution, Spatial Attention, Multi-Scale
Feature Fusion, Human Pose Estimation.

††journalyear: 2021††copyright: acmcopyright††conference: ACM Multimedia Asia;
March 7–9, 2021; Virtual Event, Singapore††booktitle: ACM Multimedia Asia
(MMAsia ’20), March 7–9, 2021, Virtual Event, Singapore††price: 15.00††doi:
10.1145/3444685.3446282††isbn: 978-1-4503-8308-0/21/03††ccs: Computing
methodologies Activity recognition and understanding

##  1\. Introduction

2D human pose estimation, which aims to predict locations of human body joints
in an image, is a very important and challenging computer vision task. It has
been studied for decades and the traditional methods mainly rely on hand-craft
features (Liu et al., 2015). With the application of convolutional neural
networks, the accuracy of human pose estimation has been greatly improved (Wei
et al., 2016; Newell et al., 2016; Chen et al., 2018; Moon et al., 2019; Xiao
et al., 2018).

Most multi-person pose estimation methods can be divided into two categories:
top-down and bottom-up methods. The top-down methods (Papandreou et al., 2017;
Fang et al., 2017; Xiao et al., 2018; Sun et al., 2019) first use a human
detector to detect all person instances in an image, then perform single-
person pose estimation for each of them respectively. The top-down methods is
inefficient in case of crowds of persons because the run-time of the second
step is proportional to the number of persons. On the contrary, the bottom-up
methods (Papandreou et al., 2018; Pishchulin et al., 2016; Insafutdinov et
al., 2017; Cao et al., 2017) detect the joints of all persons at once and then
group the joints into each individual person by a grouping algorithm. The
bottom-up methods suffer from high complexity of joint grouping step and the
multi-scale problem in joint detection. Recently, most state-of-the-art
results have been achieved by the top-down methods. In this paper, we follow
the top-down strategy and develop an effective human pose estimation method.

Among the top-down methods, the simple baseline network (Xiao et al., 2018)
has achieved state-of-the-art results with a simple encoder-decoder
architecture. Due to its simplicity and effectiveness, it can be used as a
basic network to develop more advanced pose estimation methods. To achieve
more accurate human pose estimation, we extend the simple baseline network in
three ways. To reduce the quantization error caused by large output stride
size, two more decoder modules are appended to the end of SBN to increase the
output resolution to the same as input resolution. Then, the global context
blocks (GCBs) are added to the encoder and decoder modules to enhance them
with global context features. Furthermore, we develop a novel spatial
attention based multi-scale feature collection and distribution module (SA-
MFCD) to fuse the feature maps from different layers of encoders, refine them
with spatial attention, and distribute them to different layers of decoders.
Experimental results on the MS COCO dataset indicate that our networks can
remarkably improve the accuracy of human pose estimation over SBN, our network
with ResNet34 can even achieve the same accuracy as SBN with ResNet152, and
our network can achieve superior results with big backbone networks.

##  2\. Related Work

###  2.1. Multi-Person Pose Estimation

Most multi-person pose estimation approaches can be classified into top-down
and bottom-up approaches.

Top-Down Multi-Person Pose Estimation. Top-Down approaches first apply human
detection to an image and then predicate the joint locations of each human
instance respectively. Convolutional Pose Machine (CPM) (Wei et al., 2016)
employs a multi-stage architecture to combine the prediction of previous
stages with the input to refine the pose estimation. Newell  _et al._ (Newell
et al., 2016) propose a stacked hourglass network to predict heatmap of 2D
joint location. Mask-RCNN (He et al., 2017) extends faster-RCNN (Ren et al.,
2015) to support keypoint predication and obtains very competitive results for
human pose estimation. CPN (Chen et al., 2018) uses a GlobalNet to localize
the “simple” keypoints and a RefineNet to address the “hard” keypoints with
online hard keypoint mining. Xiao  _et al._ (Xiao et al., 2018) propose the
simple baseline network for human pose estimation which has achieved state-of-
the-art results. Recently, most state-of-the-art results have been achieved by
top-down approaches.

Bottom-Up Multi-Person Pose Estimation. Bottom-Up approaches predict all
joints first and then group them into different human poses. Some works
(Pishchulin et al., 2016; Insafutdinov et al., 2016, 2017) formulate joint
grouping via a Linear Program. Cao  _et al._ (Cao et al., 2017) use part
affinity fields to encode the relation between different joints to help joint
grouping. Newell  _et al._ (Newell et al., 2017) use a detection heatmap and a
tagging heatmap for supervising the joint detection and grouping, and then
group joints with similar tags into an individual person. Papandreou  _et al._
(Papandreou et al., 2018) develop a convolutional network to predict body
joints and their relative displacements for joint grouping. Bottom-up
approaches suffer from high complexity of joint grouping step and the multi-
scale problem in joint detection. Although some efficient grouping methods
have been proposed, bottom-up pose estimation approaches still can not win the
competition against top-down approaches.

###  2.2. Attention Mechanism and Global Context Modeling

Attention mechanism is widely used in computer vision, which focuses on
important features and suppresses unnecessary ones. It can be divided into two
categories in terms of the dimension considered, channel-wise attention and
spatial-wise attention. SENet (Hu et al., 2018) uses a bottleneck transform to
generate channel attention weights for image classification. Zhu  _et al._
(Zhu et al., 2019) propose a channel attention mechanism in their residual
attention block to refine features for image super-resolution. CBAM (Woo et
al., 2018) and BAM (Park et al., 2018) apply both channel and spatial
attention modules to refine features for various visual tasks. Chu  _et al._
(Chu et al., 2017) use attention models for human pose estimation and model
the attention from different context and resolution.

Spatial attention mechanism has also been used to model global context. The
non-local network (Wang et al., 2018) uses a query-specific spatial attention
map for each query position to aggregate context features. GCNet (Cao et al.,
2019) achieves more efficient global context modeling by using a query-
independent attention map to generate a global context feature for all query
positions and further re-calibrating the global context feature with channel
attention.

##  3\. Methodology

Figure 1. (a) The full-resolution encoder-decoder network. The red arrow lines
stand for the flows of spatial attention heatmaps (SAH) generated by GCBs, and
black arrow lines are for feature maps. (b) The global context block (GCB).
$F$ and $F^{\prime}$ are the input and output feature maps of a GCB
respectively. (c) The SA-MFCD Module.

Our multi-person pose estimation approach uses faster-RCNN (Ren et al., 2015)
to detect each person in an image, and then predicates pose for each person
instance. Our network is an extension of the successful simple baseline
network (SBN), as show in Fig. 1(a). Firstly, two more decoder modules are
appended to the end of the SBN to increase the output resolution. The encoder
and decoder modules are then enhanced by the global context blocks (GCBs) (Cao
et al., 2019) which compute global context features to enhance the original
feature maps. Furthermore, a spatial attention based multi-scale feature
collection and distribution (SA-MFCD) module is proposed to fuse feature maps
from different layers of encoders, refine them with spatial attention and
distribute them to different layers of decoders. More detailed information are
given in the following subsections.

###  3.1. Extend Simple Baseline Network to Full Output Resolution

The encoder-decoder network structure is widely used in dense prediction,
including semantic segmentation (Ronneberger et al., 2015; Fu et al., 2019),
object detection (Lin et al., 2017), and human pose estimation (Newell et al.,
2016; Xiao et al., 2018). The encoder-decoder networks contain encoders that
gradually reduce the resolutions of feature maps while capturing higher
semantic information, and decoders that gradually recover the spatial
resolutions. The output resolutions of most popular encoder-decoder networks
for human pose estimation are smaller than their input resolutions. For
example, the output resolutions of the Hourglass network (Newell et al., 2016)
and the simple baseline network (Xiao et al., 2018) are only 1/4 of their
input resolution. The outputs are then resized to the input resolution by
simple transformation, which introduces quantization errors. In (Huang et al.,
2019), the common biased data processing for human pose estimation has been
quantitatively analyzed, and it is found that the error of prediction is
mainly due to the horizontal flip operation which is also related to the
output stride size.

To alleviate this problem, we append two more decoder modules to the end of
SBN to achieve the full output resolution. Following SBN, the encoder part of
our network is a modified version of the ResNet (He et al., 2016), where the
average pooling layer and last fully-connected layers are removed and only the
convolutional layers are used. It includes a convolutional layer and a pooling
layer followed by four stages of ResBlocks, both the first convolutional layer
and pooling layer halves the resolution, and at the beginning of each Res-
stage (except for the first one), the feature map resolution is also halved by
a convolutional layer with strides=2, while the number of filters is doubled.
Each decoder is composed of a Deconvolution (Transport Convolution) layer
followed by batch normalization and ReLU. The decoder part of our network
includes five decoder modules each of which double the resolution, and finally
gets output with the same resolution as that of input. Compared to SBN, the
two additional decoder modules bring more parameters and increase the model
size. Our previous study indicates that reducing the channel sizes of the two
additional decoder modules can make a good trade-off between the prediction
accuracy and the model size. In our study, we set the channel size of the 4th
decoder to 128 and that of the 5th decoder to 32 respectively. We name this
basic network as the full resolution version of SBN (FR-SBN).

###  3.2. Enhance Encoders and Decoders by Global Context Blocks

The combination of global context information has proven useful to various
visual recognition tasks. However, a convolution layer only models pixel
relationship in a local neighborhood, and the long-range dependencies are
mainly modeled by deeply stacking convolution layers. Non-local operation
(Wang et al., 2018) is proposed to model the long-range dependencies using
self-attention mechanism (Vaswani et al., 2017). The non-local network uses a
query-specific attention map for each query position to aggregate context
features and adds the context features to the feature of the corresponding
query position. GCNet (Cao et al., 2019) is a more effective and efficient
global context modeling approach, which uses a query-independent attention map
to generate a global context feature and add it to all query positions.

In this paper, we apply the global context block (GCB) (Cao et al., 2019) to
enhance the encoder and decoder modules. GCBs are introduced after the four
Res-stages in the encoder part and the first three Decoder Models, as shown in
Fig. 1(a). The structure of GCB is shown in Fig. 1(b). A $1\times 1$
convolution is first used to generate a spatial attention heatmap (SAH) and
followed by softmax to generate attention weights. Then attention pooling is
used to obtain a global context feature. After that, a bottleneck transform is
applied to it to capture its channel-wise dependencies. Finally, the global
context feature is added to features of all positions. The SAH is also used in
SA-MFCD which is introduced in the next subsection.

###  3.3. Spatial Attention Based Multi-Scale Feature Collection and
Distribution

In previous encoder-decoder networks such as U-Net (Ronneberger et al., 2015)
and Hourglass (Newell et al., 2016), skip connections are used to preserve
spatial information at each resolution. Skip connections are formed between
the encoders and the decoders having the same output resolution to transfer
the spatial information across the network for better localization. Through
the skip connections, feature maps from the encoder are concatenated (or
summed) with the corresponding feature maps of the decoder. For simplicity,
the simple baseline network (Xiao et al., 2018) does not use skip connections,
which may limited its performance.

Different from the traditional skip connection, we propose a Spatial Attention
based Multi-Scale Feature Collection and Distribution (SA-MFCD) module which
merges multi-scale feature-maps from different layers of encoders, uses
spatial attention to refine them, and finally distributes them to different
layers of decoders. This scheme can bring multi-scale spatial and semantic
information to the decoders for recovering high-resolution feature-maps. The
details of this module is show in Fig. 1(c). This module has two different
inputs: feature maps and spatial attention maps. The feature maps generated
from the GCBs after Res-stage1, Res-stage2 and Res-stage3 are sent to SA-MFCD,
the feature maps with lower resolutions are up-sampled to match the highest
one and concatenated together, and fused by a $3\times 3$ convolution. The
SAHs generated from the three GCBs are also resized and concatenated, and
fused by a $3\times 3$ convolution. The fused feature maps are then multiplied
element-wise with the fused SAH. Finally, the refined feature maps are
distributed to Decoder2, Decoder3 and Decoder4 after being resized to
corresponding resolutions. The refined feature maps from SA-MFCD are
concatenated with the output feature maps of the deconvolution layers in the
corresponding decoders and fused by a $3\times 3$ convolution to generate the
outputs of the decoders. All of the convolution operations used here are
followed by a batch normalization layer and ReLU function.

Via SA-MFCD, feature maps with different resolutions are fused together. The
higher semantic information in lower resolution feature maps is combined with
the more refined spatial information in the higher resolution feature maps.
The spatial attention heatmaps (SAHs) with different resolutions can help
extracting useful details and suppressing background information. And the
fused feature maps are refined by the spatial attention mechanism to emphasize
the positions related to pose estimation. When combining the refined feature
maps from SA-MFCD model, the capacities of the corresponding decoders and thus
the whole network are improved.

###  3.4. Ground-truth and Loss Function

Our human pose estimation network does not directly predict the coordinates,
but do a pixel-level prediction by transforming joint position to heatmap. For
each target-person patch cropped by human detection bounding-box, there might
exist non-target person in it, we only generate ground-truth for the target
one, by a 2D Gaussian function:

(1)
$\boldsymbol{H}_{k}(x,y)=\exp\left(\frac{-\left[\left(x-x_{k}\right)^{2}+\left(y-y_{k}\right)^{2}\right]}{2\sigma^{2}}\right),$

where $H_{k}$ is the heatmap for the $k$-th
joint$(k\in\left\\{1,...,K\right\\})$, $(x_{k},y_{k})$ is the coordinations of
the $k$-th joint, $(x,y)$ specifies a pixel location in the heatmap and the
hyper-parameter $\sigma$ denotes a pre-fixed spatial variance, which is set to
8 and 12 for $256\times 192$ and $388\times 288$ input image size
respectively.

As in (Xiao et al., 2018; Fang et al., 2017), we also choose Mean-Squared
Error (MSE) as the loss function. The MSE loss function is given as:

(2)
$\mathcal{L}_{\mathrm{mse}}=\frac{1}{K}\sum_{k=1}^{K}\left\|\boldsymbol{H}_{k}-\hat{\boldsymbol{H}}_{k}\right\|_{2}^{2},$

where $\hat{H}_{k}$ refers to the predicted heatmap for the $k$-th joint.

##  4\. Experiments

###  4.1. Experiment Setup

Dataset.  _MS COCO_ keypoint dataset (Lin et al., 2014) has more than 200k
images in the wild, where more than 250k person instances are labeled with 17
human joints. We train our networks on the COCO 2017 training dataset, which
has 57k images and 150k labeled person instances, no extra data involved. We
evaluate our networks on the val2017 set (5k images) and test-dev2017 set (20k
images) respectively.

Performance Metrics. The OKS-based average precision and recall
scores111https://cocodataset.org/#keypoints-eval. are used to evaluate the
accuracy of keypoint localization. The object keypoint similarity (OKS) is
defined as the distance between the predicted points and the ground-truth
normalized by scale of the person. The mean average precision (AP) over 10 OKS
thresholds is used as the main evaluation metric. In our experiments, we
report AP (the mean of AP scores at OKS = 0.50, 0.55, …, 0.90, 0.95), AP50 (AP
at OKS = 0.50), AP75, APM for medium objects, APL for large objects, and AR
(the mean of recalls).

Implementation Details. We implemented our networks via PyTorch (Paszke et
al., 2019) deep learning framework. ResNet (He et al., 2016) pre-trained on
ImageNet is used to construct the encoder part in our network. The base
learning rate is 1e-3, droped by 0.1 at step 90 and 120, with 140 epochs in
total. We utilize Adam optimizer for training. Data augmentation is a very
important training strategy, we adopt random horizontal flip, random rotation
(-40 to +40) degrees and random scale (0.7 to 1.3).

###  4.2. Ablation Study

We first study the effectiveness of each component used to extend SBN,
including the two more decoder modules appended to SBN (FR-SBN), the GCBs used
to enhance encoders and decoders, and our SA-MFCD module. All versions of
networks use ResNet34 to build their encoder part. We evaluate these versions
of networks on the COCO val2017 set, all results are obtained over the input
size of $256\times 192$ and the same data augmentation strategies.

Table 1. Comparison of networks with different additional components Method | AP
---|---
SBN-34 | 69.8
FR-SBN | 71.0
FR-SBN+GCB | 71.6
FR-SBN+GCB+skip | 71.7
FR-SBN+GCB+SA-MFCD | 72.5

Table 1 shows the comparison results in detail. Our re-trained SBN-34 achieved
an AP of 69.8. Comparing FR-SBN with SBN-34 (71.0 vs. 69.8), it can be seen
that increase of the output resolution effectively improves AP by 1.2 points,
which may due to the reduced quantization error. It can also be found that GCB
can improve 0.6 AP points (71.6 vs. 71.0) over FR-SBN. SA-MFCD can further
improve AP by 0.9 points (72.5 vs. 71.6), while the vanilla skip-connection
(FR-SBN+GCB+skip) can only improve AP by 0.1 points (71.7 vs. 71.6). In total,
our network (FR-SBN+GCB+SA-MFCD) gets 2.7 AP point improvements over the
baseline (SBN-34).

###  4.3. Comparison with the state of the art methods

Table 2. Comparisons on the COCO validation set. OHKM = online hard keypoints mining Method | Backbone | Input size | AP | #Params | GFLOPs | Time
---|---|---|---|---|---|---
Hourglass (Newell et al., 2016) | - | $256\times 192$ | 66.9 | 25M | 14.3G | -
CPN (Chen et al., 2018) | Resnet50 | $256\times 192$ | 68.7 | 27M | 6.2G | -
CPN+OHKM (Chen et al., 2018) | Resnet50 | $256\times 192$ | 69.4 | 27M | 6.2G | -
PoseFix (Moon et al., 2019) | CPN | $256\times 192$ | 72.1 | - | - | -
SBN-50 (Xiao et al., 2018) | Resnet50 | $256\times 192$ | 70.4 | 34M | 8.9G | 2.3ms
SBN-101 (Xiao et al., 2018) | Resnet101 | $256\times 192$ | 71.4 | 53M | 12.4G | 3.3ms
SBN-152 (Xiao et al., 2018) | Resnet152 | $256\times 192$ | 72.0 | 68.6M | 15.7G | 4.8ms
ours | Resnet34 | $256\times 192$ | 72.5 | 33M | 21.3G | 3.5ms
SBN-50 (Xiao et al., 2018) | Resnet50 | $384\times 288$ | 72.2 | 34M | 20G | 5.7ms
SBN-101 (Xiao et al., 2018) | Resnet101 | $384\times 288$ | 73.6 | 53M | 27.9G | 8.0ms
SBN-152 (Xiao et al., 2018) | Resnet152 | $384\times 288$ | 74.3 | 68.6M | 35.3G | 11.8ms
ours | Resnet34 | $384\times 288$ | 74.2 | 33M | 48.0G | 8.6ms
Table 3. Comparisons on the COCO test-dev set. ∗ means use extra data Method | Backbone | Input size | AP | AP50 | AP75 | APM | APL | AR
---|---|---|---|---|---|---|---|---
Mask. (He et al., 2017) | Res.50-FPN | - | 63.1 | 87.3 | 68.7 | 57.8 | 71.4 | -
Integral. (Sun et al., 2018) | Res.101 | $256\times 256$ | 67.8 | 88.2 | 74.8 | 63.9 | 74.0 | -
CSANet (Yu et al., 2019) | Res.101 | $256\times 192$ | 72.3 | 91.2 | 80.2 | 69.3 | 77.6 | 79.1
CASNet (Yu et al., 2019) | Res.152 | $256\times 192$ | 72.8 | 91.4 | 80.9 | 69.8 | 78.3 | 79.6
UDP(Huang et al., 2019) | Res.152 | $256\times 192$ | 72.9 | 91.6 | 80.9 | 70.0 | 78.5 | 78.4
HRNet(Sun et al., 2019) | HRNet-w48 | $256\times 192$ | 74.3 | 92.4 | 82.6 | 71.2 | 79.6 | 79.7
ours | Res.34 | $256\times 192$ | 72.0 | 91.1 | 79.9 | 68.9 | 77.7 | 77.5
ours | Res.50 | $256\times 192$ | 72.3 | 91.3 | 80.1 | 69.1 | 78.2 | 77.8
ours | Res.101 | $256\times 192$ | 72.6 | 91.6 | 80.5 | 69.4 | 78.4 | 78.1
ours | Res.152 | $256\times 192$ | 73.3 | 91.3 | 80.8 | 70.1 | 78.9 | 78.8
CPN (Chen et al., 2018) | Res.Inc. | $384\times 288$ | 72.1 | 91.4 | 80.0 | 68.7 | 77.2 | 78.5
CFN (Huang et al., 2017) | - | - | 72.6 | 86.1 | 69.7 | 78.3 | 64.1 | -
CPN (ens.) (Chen et al., 2018) | Res.Inc. | $384\times 288$ | 73.0 | 91.7 | 80.9 | 69.5 | 78.1 | 79.0
SBN (Xiao et al., 2018) | Res.152 | $384\times 288$ | 73.7 | 91.9 | 81.1 | 70.3 | 80.0 | 79.0
PoseFix (Moon et al., 2019) | Res.152 | $384\times 288$ | 73.6 | 90.8 | 81.0 | 70.3 | 79.8 | 79.0
CSANet (Yu et al., 2019) | Res.50 | $384\times 288$ | 73.5 | 91.4 | 80.8 | 69.9 | 79.4 | 79.7
CSANet (Yu et al., 2019) | Res.101 | $384\times 288$ | 74.1 | 91.6 | 81.6 | 70.7 | 79.8 | 80.4
CSANet (Yu et al., 2019) | Res.152 | $384\times 288$ | 74.5 | 91.7 | 82.1 | 71.2 | 80.2 | 80.7
HRNet (Sun et al., 2019) | HRNet-w48 | $384\times 288$ | 75.5 | 92.5 | 83.3 | 71.9 | 81.5 | 80.5
UDP (Huang et al., 2019) | Res.50 | $384\times 288$ | 72.5 | 91.1 | 79.7 | 68.8 | 79.1 | 77.9
UDP (Huang et al., 2019) | Res.152 | $384\times 288$ | 74.7 | 91.8 | 82.1 | 71.5 | 80.8 | 80.0
ours | Res.34 | $384\times 288$ | 73.7 | 91.7 | 81.1 | 70.4 | 79.7 | 78.9
ours | Res.50 | $384\times 288$ | 73.9 | 91.7 | 81.3 | 70.4 | 80.0 | 79.0
ours | Res.101 | $384\times 288$ | 74.3 | 92.0 | 82.0 | 71.0 | 80.4 | 79.6
ours | Res.152 | $384\times 288$ | 74.9 | 92.2 | 82.4 | 71.6 | 80.9 | 80.2

Our method is compared to the state-of-the-art methods on the validation and
test set with two different input sizes respectively. Most of the compared
methods are built on ResNet. PoseFix (Moon et al., 2019) uses CPN as the
backbone, and UDP (Huang et al., 2019) uses SBN as the backbone. Table 2 gives
the results on the validation set. It can be found that our method achieves
the best results when the input size is $256\times 192$. When the input size
rises to $384\times 288$, our network with ResNet34 can still achieve almost
the same AP of SBN-152 (74.2 vs. 74.3) with only half of parameters and less
inference time (8.6ms vs. 11.8ms), but the GFLOPs of our network is higher
than that of SBN-152. This is because that our backbone is relatively small
and the SA-MFCD module can run in parallel with the encoder-decoder network.
The experimental results on the validation dataset indicate our network with
ResNet34 can achieve the same accuracy as SBN with ResNet152, but with less
inference time.

Table 3 gives the results on test-dev2017 set. When using the input of
$256\times 192$, our network outperforms CSANet (Yu et al., 2019) and
UDP(Huang et al., 2019) with the same backbone networks. When using the input
of $384\times 288$, our network with ResNet34 outperforms CSANet and UDP with
ResNet50. When ResNet50 is used as the backbone, our network is 0.4 points
higher than CSANet and 1.4 points higher than UDP. When using ResNet101, our
network reaches higher results than CSANet (74.3 vs. 74.1). When ResNet152 is
used, our results (74.9) are also better than UDP (74.7) and CSANet (74.5).
HRNet is the method outperforming our network at both input size settings,
which is a more advanced and elaborate network recently refreshing the state-
of-the-art of many computer visual tasks.

##  5\. Conclusions and Future Works

This paper has proposed an advanced encoder-decoder networks for human pose
estimation, which improves the effectiveness of the simple baseline network.
To the best of our knowledge, we are the first to develop network with full
output resolution for human pose estimation, which can reduce the quantization
error and recover more accurate spatial information. Global context blocks are
also used to enhance the encoders and decoders. Furthermore, the proposed
spatial attention based multi-scale feature collection and distribution module
can fuse multi-scale feature information to boost the pose estimation. Our
experimental results show that our networks can remarkably improve the
prediction accuracy over SBN, and our network with big backbone network can
achieve top ranked performance among recent works. In this paper, our network
uses ResNet to build the encoder part, and it is worth trying some more
advanced backbones in the future.

##  6\. Acknowledgement

This work was supported in part by the National Natural Science Foundation of
China (U20B2063), the Sichuan Science and Technology Program, China
(2020YFS0057), and the Fundamental Research Funds for the Central Universities
(ZYGX2019Z015).

## References

  * (1)
  * Cao et al. (2019) Yue Cao, Jiarui Xu, Stephen Lin, Fangyun Wei, and Han Hu. 2019.  Gcnet: Non-local networks meet squeeze-excitation networks and beyond. In _Proceedings of the IEEE International Conference on Computer Vision Workshops_. 0–0.
  * Cao et al. (2017) Zhe Cao, Tomas Simon, Shih-En Wei, and Yaser Sheikh. 2017\.  Realtime multi-person 2d pose estimation using part affinity fields. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 7291–7299.
  * Chen et al. (2018) Yilun Chen, Zhicheng Wang, Yuxiang Peng, Zhiqiang Zhang, Gang Yu, and Jian Sun. 2018\.  Cascaded pyramid network for multi-person pose estimation. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 7103–7112.
  * Chu et al. (2017) Xiao Chu, Wei Yang, Wanli Ouyang, Cheng Ma, Alan L Yuille, and Xiaogang Wang. 2017\.  Multi-context attention for human pose estimation. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 1831–1840.
  * Fang et al. (2017) Hao-Shu Fang, Shuqin Xie, Yu-Wing Tai, and Cewu Lu. 2017\.  Rmpe: Regional multi-person pose estimation. In _Proceedings of the IEEE International Conference on Computer Vision_. 2334–2343.
  * Fu et al. (2019) Jun Fu, Jing Liu, Yuhang Wang, Jin Zhou, Changyong Wang, and Hanqing Lu. 2019\.  Stacked deconvolutional network for semantic segmentation.  _IEEE Transactions on Image Processing_ (2019).
  * He et al. (2017) Kaiming He, Georgia Gkioxari, Piotr Dollár, and Ross Girshick. 2017.  Mask r-cnn. In _Proceedings of the IEEE International Conference on Computer Vision_. 2961–2969.
  * He et al. (2016) Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. 2016\.  Deep residual learning for image recognition. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 770–778.
  * Hu et al. (2018) Jie Hu, Li Shen, and Gang Sun. 2018.  Squeeze-and-excitation networks. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 7132–7141.
  * Huang et al. (2019) Junjie Huang, Zheng Zhu, Feng Guo, and Guan Huang. 2019\.  The Devil is in the Details: Delving into Unbiased Data Processing for Human Pose Estimation.  _arXiv preprint arXiv:1911.07524_ (2019).
  * Huang et al. (2017) Shaoli Huang, Mingming Gong, and Dacheng Tao. 2017.  A coarse-fine network for keypoint localization. In _Proceedings of the IEEE International Conference on Computer Vision_. 3028–3037.
  * Insafutdinov et al. (2017) Eldar Insafutdinov, Mykhaylo Andriluka, Leonid Pishchulin, Siyu Tang, Evgeny Levinkov, Bjoern Andres, and Bernt Schiele. 2017\.  Arttrack: Articulated multi-person tracking in the wild. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 6457–6465.
  * Insafutdinov et al. (2016) Eldar Insafutdinov, Leonid Pishchulin, Bjoern Andres, Mykhaylo Andriluka, and Bernt Schiele. 2016\.  Deepercut: A deeper, stronger, and faster multi-person pose estimation model. In _European Conference on Computer Vision_. Springer, 34–50.
  * Lin et al. (2017) Tsung-Yi Lin, Piotr Dollár, Ross Girshick, Kaiming He, Bharath Hariharan, and Serge Belongie. 2017\.  Feature pyramid networks for object detection. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 2117–2125.
  * Lin et al. (2014) Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick. 2014.  Microsoft coco: Common objects in context. In _European Conference on Computer Vision_. Springer, 740–755.
  * Liu et al. (2015) Zhao Liu, Jianke Zhu, Jiajun Bu, and Chun Chen. 2015\.  A survey of human pose estimation: the body parts parsing based methods.  _Journal of Visual Communication and Image Representation_ 32 (2015), 10–19.
  * Moon et al. (2019) Gyeongsik Moon, Ju Yong Chang, and Kyoung Mu Lee. 2019\.  Posefix: Model-agnostic general human pose refinement network. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. Long Beach, 7773–7781.
  * Newell et al. (2017) Alejandro Newell, Zhiao Huang, and Jia Deng. 2017.  Associative embedding: End-to-end learning for joint detection and grouping. In _Advances in Neural Information Processing Systems_. 2277–2287.
  * Newell et al. (2016) Alejandro Newell, Kaiyu Yang, and Jia Deng. 2016.  Stacked hourglass networks for human pose estimation. In _European Conference on Computer Vision_. Springer, 483–499.
  * Papandreou et al. (2018) George Papandreou, Tyler Zhu, Liang-Chieh Chen, Spyros Gidaris, Jonathan Tompson, and Kevin Murphy. 2018\.  Personlab: Person pose estimation and instance segmentation with a bottom-up, part-based, geometric embedding model. In _Proceedings of the European Conference on Computer Vision_. 269–286.
  * Papandreou et al. (2017) George Papandreou, Tyler Zhu, Nori Kanazawa, Alexander Toshev, Jonathan Tompson, Chris Bregler, and Kevin Murphy. 2017.  Towards accurate multi-person pose estimation in the wild. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 4903–4911.
  * Park et al. (2018) Jongchan Park, Sanghyun Woo, Joon-Young Lee, and In So Kweon. 2018\.  Bam: Bottleneck attention module.  _arXiv preprint arXiv:1807.06514_ (2018).
  * Paszke et al. (2019) Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. 2019\.  PyTorch: An imperative style, high-performance deep learning library. In _Advances in Neural Information Processing Systems_. 8024–8035.
  * Pishchulin et al. (2016) Leonid Pishchulin, Eldar Insafutdinov, Siyu Tang, Bjoern Andres, Mykhaylo Andriluka, Peter V Gehler, and Bernt Schiele. 2016.  Deepcut: Joint subset partition and labeling for multi person pose estimation. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 4929–4937.
  * Ren et al. (2015) Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. 2015\.  Faster r-cnn: Towards real-time object detection with region proposal networks. In _Advances in Neural Information Processing Systems_. 91–99.
  * Ronneberger et al. (2015) Olaf Ronneberger, Philipp Fischer, and Thomas Brox. 2015\.  U-net: Convolutional networks for biomedical image segmentation. In _International Conference on Medical Image Computing and Computer-Assisted Intervention_. Springer, 234–241.
  * Sun et al. (2019) Ke Sun, Bin Xiao, Dong Liu, and Jingdong Wang. 2019\.  Deep high-resolution representation learning for human pose estimation. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 5693–5703.
  * Sun et al. (2018) Xiao Sun, Bin Xiao, Fangyin Wei, Shuang Liang, and Yichen Wei. 2018.  Integral human pose regression. In _Proceedings of the European Conference on Computer Vision_. 529–545.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need. In _Advances in Neural Information Processing Systems_. 5998–6008.
  * Wang et al. (2018) Xiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaiming He. 2018.  Non-local neural networks. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 7794–7803.
  * Wei et al. (2016) Shih-En Wei, Varun Ramakrishna, Takeo Kanade, and Yaser Sheikh. 2016.  Convolutional pose machines. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_. 4724–4732.
  * Woo et al. (2018) Sanghyun Woo, Jongchan Park, Joon-Young Lee, and In So Kweon. 2018.  Cbam: Convolutional block attention module. In _Proceedings of the European Conference on Computer Vision_. 3–19.
  * Xiao et al. (2018) Bin Xiao, Haiping Wu, and Yichen Wei. 2018.  Simple baselines for human pose estimation and tracking. In _Proceedings of the European Conference on Computer Vision_. 466–481.
  * Yu et al. (2019) Dongdong Yu, Kai Su, Xin Geng, and Changhu Wang. 2019\.  A Context-and-Spatial Aware Network for Multi-Person Pose Estimation.  _arXiv preprint arXiv:1905.05355_ (2019).
  * Zhu et al. (2019) Leilei Zhu, Shu Zhan, and Haiyan Zhang. 2019.  Stacked U-shape networks with channel-wise attention for image super-resolution.  _Neurocomputing_ 345 (2019), 58–66.