DESY 21-146

ICAS 066/21

IFIC/21-33

$Z^{\prime}$-explorer 2.0: reconnoitering the dark matter landscape

Víctor Martín Lozano(a)†, Rosa María Sandá Seoane(b)⋆, Jose Zurita(c),⋄

(a) Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg,
Germany

(b) International Center for Advanced Studies (ICAS)
UNSAM, Campus Miguelete, 25 de Mayo y Francia, (1650) Buenos Aires, Argentina

(c) Instituto de Física Corpuscular, CSIC-Universitat de València, E-46980
Paterna, Valencia, Spain

###### Abstract

We introduce version 2.0 of $Z^{\prime}$-explorer, a software tool which
provides a simple, fast and user-friendly test of models with an extra $U(1)$
gauge boson ($Z^{\prime}$) against experimental LHC results. The main novelty
of the second version is the inclusion of missing energy searches, as the
first version only included final states into SM particles. Hence
$Z^{\prime}$-explorer 2.0 is able to test dark matter models where the
$Z^{\prime}$ acts as an s-channel mediator between the Standard Model and the
dark sector, a widespread benchmark employed by the ATLAS and CMS experimental
collaborations. To this end, we perform here the first public reinterpretation
of the most recent ATLAS mono-jet search with 139 fb-1. In addition, the
corresponding searches in the visible final states have also been updated. We
illustrate the power of our code by re-obtaining public plots, and also
showing novel results. In particular, we study the cases where the
$Z^{\prime}$ couples strongly to top quarks (top-philic), where dark matter
couples with a mixture of vector and axial-vector couplings, and also perform
a scan in the parameter space of a string inspired Stückelberg model.
$Z^{\prime}$-explorer 2.0 is publicly available on GitHub.

E-mail: $\dagger$<EMAIL_ADDRESS>$\star$<EMAIL_ADDRESS>$\diamond$<EMAIL_ADDRESS>
##  1 Introduction

Modern model-building requires confronting the parameter space (masses,
couplings) of a New Physics model against a large and ever-growing variety of
experimental constraints. The reinterpretation of LHC searches can then range
from trivial to highly-involved to (almost) impossible. The difficulty of the
task can depend on several factors, including the implicit and explicit
assumptions of a given study, the degree of model-independence and the public
availability of the necessary information to reproduce the reported bounds
outside of the specific collaboration (e.g. experimental efficiencies). A
fruitful dialogue between the phenomenological and experimental communities is
already ongoing [1] which has considerable helped the public reinterpretation
effort.

We can classify the re-interpretation codes in two broad categories. First and
foremost, the _general_ codes provide a framework where any LHC study can be
implemented, and those results applied to arbitrary models, e.g. CheckMATE [2,
3], MadAnalysis5 [4, 5] and GAMBIT [6]. This flexibility on the model-
independence comes with the price of requiring the generation of a large
number MonteCarlo events in order to confront a single point in parameter
space with the experimental data. The second category, _dedicated_ codes,
narrows the applicability by focusing on a specific model framework (e.g.
HiggsBounds [7] and HiggsSignals [8] for extended scalar sectors, or SModelS
[9] for simplified models), which allows to bypass the CPU-time consuming
event generation and thus allow a fast exploration of the parameter space. In
the latter category belongs our tool, $Z^{\prime}$-explorer [10] which focuses
on models where the SM is augmented with a new $U(1)$ gauge boson, dubbed
$Z^{\prime}$.

$Z^{\prime}$-models are widely used in a variety of contexts, see e.g. [11,
12] for a review. Of our particular interest is a $Z^{\prime}$ boson acting as
a s-channel mediator to the dark sector [13, 14]. The existence of the mono-X
process $pp\to Z^{\prime}X\to\chi\chi X$, where $X=j,\gamma,Z,W,\dots$ is a
Standard Model (SM) particle that must be present for the event to be recorded
on tape (“triggered”), requires that the $Z^{\prime}$ couples to both the
Standard Model and the dark matter candidate $\chi$. This provides a robust
foundation to the existing mediator search programme at the LHC, encompassing
both _visible_ (i.e. Standard Model) and _invisible_ (dark sector) final
states. Having a fast and flexible tool for the simultaneous reinterpretation
of the broad palette of experimental searches is a desirable addition to the
model-builder’s toolkit. A first step in that direction was done in
$Z^{\prime}$-explorer 1.0, where the whole suite of visible final states was
implemented [10], requiring as user input only the $Z^{\prime}$ mass and its
coupling to SM particles, which avoids the need for event generation.

In this article we extend the capabilities of $Z^{\prime}$ explorer to further
include searches with missing energy. To that effect, we perform (to the best
of our knowledge) the first public reinterpretation and validation of the
ATLAS mono-jet study with 139 fb-1 of total integrated luminosity [15], which
provides the most stringent constraints from the whole set of mono-X searches
when $X$ comes from initial state radiation [16]. When compared to ZPEED [17]
(which included all $Z^{\prime}$ decays into di-jet and di-lepton channels),
$Z^{\prime}$-explorer additionally includes the $WW$ and $Zh$ channels (from
version 1.0) and the missing energy studies described in this article (version
2.0).

Our code goes beyond the public results presented by ATLAS on two directions.
First and foremost, we allow for arbitrary couplings to every SM fermion. The
oversimplifying assumption of having one single common quark coupling $g_{q}$
and/or a single lepton coupling $g_{l}$ can be dropped, revealing an
interesting interplay between the visible and invisible sectors. We note that
in this case, since $Z^{\prime}$-explorer does yet not include constraints
from flavor physics (which can nonetheless be easily obtained from e.g. flavio
[18]), the user must taken them separately into account. Second, we consider a
general coupling structure for the $Z^{\prime}-\chi-\chi$ vertex, going beyond
the two benchmark setups commonly used by the experimental collaboration: the
vector and axial-vector scenarios. Our results are implemented in
$Z^{\prime}$-explorer 2.0, which is publicly available on GitHub [19]. Since
$Z^{\prime}$-explorer does not require simulating events, it is appropriate
for a thorough scanning the parameter space when compared to general re-
interpretation codes.

The current article is structured as follows. In section 2 we review the
fundamentals of $Z^{\prime}$ models, presenting the parametrization used
within $Z^{\prime}$ explorer, which coincides with the one adopted by the Dark
Matter Working Group (DMWG) [14].111Except that the DMWG employs the vector
(V) - axial-vector (A) basis for the couplings, instead of the left-right used
in $Z^{\prime}$-explorer. In section 3 we present the validation of the ATLAS
mono-jet study [15]. We exemplify the impact of $Z^{\prime}$-explorer by
applying it to a series of examples in section 4, and we reserve section 5 for
our conclusions. Technical details regarding the software implementation are
left for Appendix A.

##  2 $Z^{\prime}$ models: theoretical framework

In this section we present a brief summary of the theoretical framework (for
details we refer the reader to [13]) and describe in a nutshell the main
features of the dark sector implementation in $Z^{\prime}$-explorer 2.0, while
the technical details are presented in Appendix A.

We augment the SM with a new $U(1)$ gauge boson $Z^{\prime}$ and a SM singlet
Dirac fermion $\chi$. The latter is our dark matter candidate, rendered stable
by a discrete $\mathbf{Z}_{2}$ symmetry, as customary in dark matter models.
The SM fermions $f$ and the dark matter $\chi$ have arbitrary left- and right-
handed couplings, $g_{f,\chi}^{L,R}$ to the $Z^{\prime}$ mediator. The
relevant Lagrangian then reads

$\mathcal{L}\supset
Z^{\prime}_{\mu}\left[\sum_{f}\left(g_{f_{L}}\bar{f}_{L}\gamma^{\mu}f_{L}+g_{f_{R}}\bar{f}_{R}\gamma^{\mu}f_{R}\right)+g_{\chi_{L}}\bar{\chi}_{L}\gamma^{\mu}\chi_{L}+g_{\chi_{R}}\bar{\chi}_{R}\gamma^{\mu}\chi_{R}\right]\,,$
(1)

where the index $f$ runs over all the SM fermions (for the treatment of
neutrinos see Appendix A). This Lagrangian coincides with that adopted by the
ATLAS and CMS collaborations [13] when all the leptonic couplings are set to
zero, and when the $Z^{\prime}$ has either vector ($g_{f_{L}}=g_{f_{R}}$) or
axial-vector ($g_{f_{L}}=-g_{f_{R}}$) couplings to the SM quarks and to the
dark matter $\chi$. While it has been shown that the different couplings in
equation 1 must obey non-trivial constraints among them if unitarity and gauge
invariance are imposed [20, 21], we consider this framework a minimal
parametrization of richer models, that could feature more than one mediator
and/or new dark states beyond $Z^{\prime}$ and $\chi$. Hence each left- and
right-handed (or axial and vector) coupling is treated as a free parameter by
$Z^{\prime}$-explorer. We also contemplate the possibility of $Z^{\prime}$
decays into new, non-SM states (other than $\chi\chi$) parametrized by an
unknown decay width $\Gamma_{xx}$. In comparison with $Z^{\prime}$-explorer
1.0, the new parameters the user must input are the dark matter mass
$m_{\chi}$ and the dark matter couplings to the $Z^{\prime}$,
$g_{\chi_{L,R}}$.

A foundational ingredient of $Z^{\prime}$-explorer’s philosophy is to employ
the _expected_ limits instead of the observed ones. This is an approximation
that holds as long as no large fluctuations in the data are present. The
rationale behind it is to be able to disentangle the experimental sensitivity
of a given channel from these fluctuations. In its current form,
$Z^{\prime}$-explorer would not be appropriate in a discovery (or large
significant fluctuation) scenario. We emphasize, again, that the goal of
$Z^{\prime}$-explorer is to set exclusion bounds on the parameter space given
by equation 1. Hence throught this work all limits and background events
reported correspond to the _expected_ case, unless noted otherwise.

A key constituent of $Z^{\prime}$-explorer is the narrow-width approximation
(NWA), which allows to factorize the LHC processes involving $Z^{\prime}$ as a
_production cross section_ times a corresponding branching ratio in a given
channel. It is then important to check if the approximation is fulfilled. The
calculation of the $Z^{\prime}$ partial width at leading order (LO) is
straightforward and full expressions are presented in Ref. [13]. In the limit
where $m_{f,\chi}\lesssim M_{Z^{\prime}}$ the $Z^{\prime}$ width is
independent of the left- and right-handed coupling structure, reading

$\frac{\Gamma_{Z^{\prime}}}{M_{Z^{\prime}}}=\frac{1}{12\pi}\left(g_{\chi}^{2}+\sum_{f}N_{C}g_{f}^{2}\right)\,,$
(2)

where $N_{C}$ is the number of color of each SM fermion $f$: 3 for quarks and
1 for charged leptons. $Z^{\prime}$-explorer does not stop the execution of
the code, but prints a warning in the output if the width-to-mass ratio of
equation 2 is larger than 5 %. For a leptophobic $Z^{\prime}$ with a single
$g_{q}$ coupling being equal to (4 times smaller than) $g_{\chi}$, this 5 %
threshold is reached for $g_{q}=0.35(0.25)$, which sets a ballpark value for
these couplings. We note that even if the width is small, interference effects
can be important if the lepton decay rates are appreciable, as discussed in
reference [17]. This work introduced the public code ZPEED. In comparison with
$Z^{\prime}$-explorer, ZPEED outputs the likelihoods for a given point in
parameter space while including the relevant interference effects, but it does
not include SM final states with gauge bosons and scalars ($W^{+}W^{-},Zh$)
and neither does include invisible decays.

The model described by equation 1 has been implemented in the Universal
Feynman Output [22] (UFO) format in reference [23]. Along this work, we will
employ MadGraph5_aMC@NLO [24] for event generation at the parton level, Pythia
8 [25, 26] for showering and hadronization, and Delphes 3.4.2 [27] for
detector simulation, using the default ATLAS card. We will work at leading-
order (LO) accuracy in the strong coupling, which is justified since next-to-
leading order QCD effects have been found to be quite mild [28].

A series of sanity checks has been performed with the UFO model. On one hand,
we have verified the validity of the partial widths expressions by numerically
comparing them to those obtained with MadWidth [29]. On the other hand, we
have studied several kinematical distributions (in particular missing
transverse energy, and the leading jet momenta and the jet pseudorapidities)
to verify that, as expected, the shape of these distributions (which are those
employed by ATLAS and CMS) depends only upon $M_{Z^{\prime}}$ and $m_{\chi}$,
while being independent of all couplings. This is a crucial assumption upon
which $Z^{\prime}$-explorer relies.

The quark couplings obviously enter in the production cross section for the
$pp\to Z^{\prime}j$ process. The two main processes for $pp\to Z^{\prime}j$
are shown in Fig. 1, in the first one a pair of quarks annihilate to produce a
$Z^{\prime}$ while a gluon is radiated from one of them (left diagram from
Fig. 1). The other process is initiated by a gluon and a quark. The gluon
splits into two quarks, one of them annihilates with the other inital quark to
give a $Z^{\prime}$ while the other is radiated. We have also verified that
the different quark contributions from the initial state do not interfere with
each other, namely that

$\sigma(pp\to
Z^{\prime}j)=\sum_{q_{i}}g_{q_{i}}^{2}\left[\sigma(q_{i}\bar{q_{i}}\to
Z^{\prime}g)+\sigma(q_{i}g\to Z^{\prime}q_{i})+\sigma(\bar{q}_{i}g\to
Z^{\prime}\bar{q}_{i})\right]\,.$ (3)

holds. We note that the different cross sections $\sigma$ in the square
brackets are _not_ parton level matrix-elements, but they also include the
corresponding convolution with the parton distribution functions (PDF). Here
we assume that CP is conserved, hence the different between $q_{i}g$ and
$\bar{q}g$ initial states is only due to the PDFs.

$Z^{\prime}$$g$$q$$q$$Z^{\prime}$$g$$q$$q$

Figure 1: Feynman diagrams of the $Z^{\prime}$ production with a radiated jet
from the inital state. This process could be initated either by a pair of
quarks radiating a gluon (left) or a pair gluon quark where a quark is
radiated (right).

For the fast evaluation of these production cross sections, we follow a
procedure analogous to the one used in $Z^{\prime}$-explorer 1.0. For the
visible channels, the relevant process under consideration is
$q_{i}\bar{q_{i}}\to Z^{\prime}$. Due to the trivial scaling with the $g_{q}$
couplings and the lack of interference among the different channels, it is
enough to make a fine scan of $\sigma(q_{i}\bar{q_{i}}\to Z^{\prime})$ in the
1-dimensional parameter space given by $M_{Z^{\prime}}$. Here the task is
slightly more complicated since now the different cross sections from equation
3 must be scanned in the two dimensional mass plane $M_{Z^{\prime}}-m_{\chi}$.
Nonetheless, this is not an obstacle, and with a fine scan of this mass plane
the cross section of any given point can be numerically estimated with great
accuracy, as described in Appendix A.

##  3 ATLAS search for $Z^{\prime}$ mediators to the dark sector

In this section we perform the validation of the ATLAS mono-jet search with
139 fb-1 and explain the implementation in $Z^{\prime}$-explorer 2.0. We
comment on the shortcomings of our validation while making suggestions for the
improvement of the reinterpretation material.

There is a large number of mono-X searches conducted by the ATLAS [15, 30, 31,
32, 33] and CMS collaborations [34, 35, 36, 37]. In our $Z^{\prime}$ setup,
with only one mediator and one dark matter particle, the $X$ particle comes
exclusively from initial state radiation (ISR) and it has been shown that in
that case the mono-jet search trumps over all the others [16]. We will then
focus on the latest ATLAS mono-jet analysis [15].222While this work was in the
final stages of completion, updated monojet results by the CMS collaboration
were made public [34]. Their sensitivity to the axial-vector scenario is
comparable with the ATLAS result.. Nonetheless, for a successful validation of
this channel we will also consider an older version of the study, using 36.1
fb-1 [38] of total integrated luminosity, since each search present different
benchmark points. The latter study had been implemented in the Physics
Analysis Database (PAD) of MadAnalysis5 [39], which provided a useful cross-
check of our signal cutflows 333We are indebted to Benjamin Fuks for providing
us with MadAnalysis5 v.1.9.35.. Nonetheless, due to the modular nature of
$Z^{\prime}$-explorer, and the goal to keep it as flexible as possible for
future updates, we plan to implement the neglected mono-X studies in a future
release including more general dark matter sectors. In that case, final state
radiation (FSR) also plays an important role and hence there is no “a-priori”
predominance of the mono-jet channel.

The ATLAS study with 139 fb-1 considers events that passes the
$E_{T}^{miss}>200$ GeV trigger, and requires the presence of at least one
energetic jet $j_{1}$ fulfiling $p_{T,j_{1}}>250$ GeV and $|\eta|<2.4$, while
up to a maximum of 4 jets with $p_{T}>30$ GeV and $|\eta|<2.8$ are allowed. In
all cases, there is a minimum azimuthal distance requirement between each jet
and the direction of the transverse missing momentum,
$\Delta\phi(jet,\vec{p_{T}^{miss}})>0.4(0.6)$ for $E_{T}^{miss}$ > 250 GeV
($E_{T}^{miss}\in[200-250]$ GeV). To complete the event selection, events with
leptons ($e,\mu,\tau$) or photons are vetoed.

After this event selection, the analysis is perfomed using two sets of signal
regions: an _inclusive_ selection (IM1, IM2, …) and an _exclusive_ one (EM1,
EM2, …). As their names suggest, the inclusive analysis requires
$E_{T}^{miss}$ above a certain threshold value, while the exclusive analysis
requires an specific interval. The exclusive signal regions (which are those
employed by $Z^{\prime}$-explorer), with their $E_{T}^{miss}$ thresholds, are
detailed in Table 1. The previous version of this study uses a similar
cutflow, albeit with fewer signal regions, tighter cuts on $E_{T}^{miss}$ and
looser requirements on $e,\mu$ and no $\tau$ neither photon vetoes. Since the
exclusion curves are only derived for the exclusive bins, we will restrict
ourselves to the EMi regions. For completeness, we also present in Table 1 the
expected (predicted) number of background events in each of them.

Exclusive | EM0 | EM1 | EM2 | EM3 | EM4 | EM5 | EM6
---|---|---|---|---|---|---|---
$E_{T}^{miss}$ [GeV] | 200-250 | 250-300 | 300-350 | 350-400 | 400-500 | 500-600 | 600-700
Predicted | 1783000 | 753000 | 314000 | 140100 | 101600 | 29200 | 10000
Exclusive | EM7 | EM8 | EM9 | EM10 | EM11 | EM12 |
$E_{T}^{miss}$ [GeV] | 700-800 | 800-900 | 900-1000 | 1000-1100 | 1100-1200 | >1200 |
Predicted | 3870 | 1640 | 754 | 359 | 182 |  |
Table 1: Exclusive (EM0-EM12) signal regions defined in ATLAS monojet search
at $13$ TeV and $139$ fb-1, together with the expected number of background
events.

For our validation, we turned to the public available material. The latest
study uses as a benchmark point $M_{Z^{\prime}}=2000$ GeV and $m_{\chi}=1$
GeV, while the previous version employs $M_{Z^{\prime}}=1000$ GeV and
$m_{\chi}=400$ GeV, and a universal coupling to quarks $g_{q}$ is set to 0.25,
while $g_{\chi}$ is fixed to 1. These two benchmark points represent different
kinematic regimes (in the former we can neglect the effect of $m_{\chi}$ while
in the other the dark decay is close to the threshold), and hence the
combination of both provide crucial information for the proper
reinterpretation.

The ATLAS study only present results for the vector mediator and axial-vector
mediator cases. ATLAS provides an official cutflow for one benchmark point (in
the 139 fb-1 version), and the $E_{T}^{miss}$ distribution for the signal
benchmark and the background. The previous version of this study presented in
addition (for the other benchmark point) the leading jet transverse momentum
$p_{T}(j_{1})$ distribution, which provides a welcome additional cross-check
of the validation procedure.

Regarding the 95 % confidence level (CL) exclusions, ATLAS presents the
results for the observed upper limits (UL) on the product of the signal cross-
section, acceptance and efficiency using the inclusive selection, and the 95 %
CL exclusion curves (“Brazilian flags”) using the exclusive selection. This
creates an additional difficulty for the validation, as i) only the _observed_
UL are reported, and not the expected ones, ii) the use of different
selections between the UL and the exclusion curves prevents the derivation of
the Brazilian flags from the UL and iii) correlations among the different bins
are not reported.

We present in figure 2 the missing energy distributions obtained for the
axial-vector case by ATLAS, together with the result of our simulations. In
the left panel we present the 36 fb-1 study, while in the right panel we
present the results for 139 fb-1. In order to match the published
distributions, we need to include a global k-factor of $0.80\pm 0.02$, whereas
the older version (for a different benchmark point and with less background
events, hence a larger statistical error) required $0.82\pm 0.02$. We can then
conclude that our event generation pipeline is validated, although as
mentioned additional distributions, and potentially additional benchmark
points would be a desirable addition, following also the recommendations in
Ref. [1]. In passing, we note that we have also investigated here the impact
of jet-matching / merging together with the different options for the
dynamical scale choices offered by MadGraph5_aMC@NLO. The effect is at the
percent level, and as such can be ignored if we take into account that ATLAS
reports 10% of uncertainty in the signal modeling, and 10 % of uncertainty
stemming from the parton distribution functions (PDF).

(a)  (b)

Figure 2: Missing transverse energy distributions for a signal point with
$g_{q}=0.25$, $g_{\chi}=1$, and (a) $M_{Z^{\prime}}=1000$ GeV, $m_{\chi}=400$
GeV; (b) $M_{Z^{\prime}}=2000$ GeV, $m_{\chi}=1$ GeV in the axial-vector
mediator scenario. A flat k-factor of 0.82 (0.80) has been applied in the left
(right) panel.

We then proceed to derive the 95 % C.L exclusion contours for the axial
mediator model, which we present in the left panel of figure 3. We have
considered here 366 points in the 2D mass plane. We display the contours where
the significance $z=s_{i}/\sqrt{b_{i}}$ is 2,4,6 and 8, together with the
ATLAS 95 % C.L expected exclusion (dashed black line). Considering that the
latter is derived from a different event selection than the former, the
agreement is striking. We also provide, in the right panel of figure 3 the
most sensitive EM signal region for all of our scanned points. This is an
interesting plot for two reasons. First and foremost, it provides important
insight on the impact of each signal region, and can help to better understand
the limits and suggest improvements to the search. Second, this is a
relatively simple piece of information to provide, and that would be helpful
as now the published table with the 95 % UL on the signal rates could be
directly employed. For completeness we present the analogous of figure 3 for
the vector mediator case, in figure 4. We see that the situation is completely
analogous, except that the excluded regions do not fully due to the difference
in cross sections for each case.

(a)  (b)

Figure 3: (left) 95 % C.L exclusion contours for the significance
$z={2,4,6,8}$ and the exclusive selection in Table 1, for an axial-vector
mediator, with $g_{\chi}=1$ and $g_{q}=0.25$. (right) Most sensitive exclusive
bin for all the reference points considered in the $(M_{Z^{\prime}},m_{\chi})$
plane, for an axial-vector ($b$) mediator with $g_{\chi}=1$ and $g_{q}=0.25$,
where the sensitivity is estimated in the Gaussian limit.

(a)  (b)

Figure 4: Same as figure 3, for the vector mediator case.

From these two figures we then conclude that the ATLAS mono-jet search, in
spite of lacking some detailed validation material, has been reproduced with
an exceptionally good agreement, and hence we will consider the implementation
of this study in $Z^{\prime}$-explorer 2.0 as validated, and we make the code
publicly available on GitHub [19].

##  4 Numerical results

In this section we present several plots obtained with $Z^{\prime}$-explorer
2.0, which serve as an illustration of its capabilities. For each example, we
will present the exclusions in the two-dimensional $M_{Z^{\prime}}-m_{\chi}$
plane, and we will adopt the following convention to display our results. The
most sensitive channel in a given point in the mass plane is indicated by its
_shape_ (stars, circles, diamonds, etc). If a given point is allowed (i.e. not
excluded) then the shape is shown in a color other than black, while excluded
points have a shape in black.

We will start first working directly on the parametrization of equation 1,
where each individual coupling is considered a free parameter. Within this
framework, we will restrict ourselves to a few slices of the parameter space,
stressing the non-trivial interplay among the different search channels. In
second place, we will consider a complete model, a string inspired Stückelberg
portal, where there are non trivial correlations among the different
couplings.

###  4.1 Simplified Model

We start first by examining the benchmark couplings of
$g_{q}=0.25,g_{\chi}=1,g_{l}=0$ used by the ATLAS monojet-study, and for
comparison we also include a slightly altered version with $g_{\chi}=1.5$.
These two _leptophobic_ cases are shown in the left and right panel of figure
5.

(a)  (b)

Figure 5: Excluded parameter space and most sensitive channel for the case
where the dark matter couples with vector couplings, for two leptophobic
$g_{l}=0$ scenarios, fixing $g_{q}=0.25$ and $g_{\chi}=1$ (left) and
$g_{\chi}=1.5$ (right). The shape of each point shows the most sensitive
channel, while a black (colored) point indicates if it is excluded (allowed).

From the figure one clearly sees the impact of the mono-jet search. Clearly
when the dark matter channel is kinematically closed, the situation is
relatively simple: depending on the specific mass, a hadronic channel (either
$jj$, $b\bar{b}$ or $t\bar{t}$) provide the strongest constraints. The
excluded masses range then from 0.5 to 4.5 TeV approximately, with the upper
value being due to a reduction of the production cross section, while the
lower value is due to the fact that $Z^{\prime}$-explorer only included
information on the visible channels from 400 GeV. This limitation is due to
the fact that for lower masses, the inclusive $H_{T}$ trigger is less
efficient due to the overwhelming QCD multi-jet background 444Nonetheless,
there are efforts to reduce this threshold and extend the analysis into lower
masses, see e.g. [40, 41, 42].. The mono-jet channel is only relevant for the
on-shell region, but as soon as it is open it dominates the exclusion, up to a
value of $M_{Z^{\prime}}$ of about 1 (1.5) TeV in the left (right) panel.
Comparing the shape of the mono-jet exclusion among both panels, we see that
it is relevant for the case where $g_{\chi}$ dominates over $g_{q}$.

In second place we consider the case where the $Z^{\prime}$ decays almost
exclusively to the top quark (top-philic $Z^{\prime}$) while still being
leptophobic $g_{l}=0$. Note that a non zero $g_{q}$ for $u,c,d,s$ needs to be
added in order to produce the $Z^{\prime}$ at the LHC. For simplicity and to
avoid having to consider FCNC bounds, we set consider all fermions coupling
vectorially, and set $g_{u}=0.1$, and set $g_{\chi}=1$. We present in the left
(right) panel of figure 6 the different exclusions for a medium (small)
$g_{t}$ couplings: 0.25 (0.1).

(a)  (b)

Figure 6: Excluded parameter space and most sensitive channel for the case
where the dark matter couples with vector couplings, for two top-philic
scenarios (and also leptophobic), fixing $g_{u}=0.25$ and $g_{\chi}=1$, with
$g_{t}=0.25(0.1)$ in the left (right) panel and $g_{\chi}=1.5$ (right).

As expected, this figure illustrates that when the $Z^{\prime}$ couples more
strongly to tops, the sensitive channels in light di-jet resonance become
ineffective, and the mono-jet channel can be the most sensitive one in a large
fraction of parameter space. We note that in particular, the mono-jet channel
can be the most sensitive up to masses of about 2.5 TeV (obviously having the
$Z^{\prime}\to\chi\chi$ channel open).

For the next example, we drop the assumption of the $Z^{\prime}$ being
leptophobic, and hence consider both vector and axial vector scenarios, with
$g_{q}=0.1$, $g_{\chi}=1$ and $g_{l}=0.01(0.1)$ for the vector (axial-vector)
case. The results plots are shown in the left and right panel of figure 7

(a)  (b)

Figure 7: Same as figure 5 and 6, yet allowing for a non-zero coupling to
leptons. We set $g_{q}=0.1$ and $g_{\chi}=1$, and show the results for
$g_{l}=0.01(0.1)$ in the left (right) panel, for the vector (axial-vector)
case.

From the figure we clearly see that the lepton channels are very sensitivie,
and if they are open they can dominate the exclusions. For large enough
couplings the mono-jet search is much less sensitive than the di-lepton
channels, except for the case where $g_{l}=0.01$, where more points have mono-
jet as they most sensitive probe.

Next, we examine dropping the assumption that the $Z^{\prime}$ must be either
a vector or an axial-vector. We then consider a generic mixing angle among
both possibilities, first for the quarks, and then for the dark matter $\chi$,
both in figure 8. In each figure we fix the absolute value of $g_{\chi}=1$ and
$g_{q}=0.1$. We fix also one of the couplings in vector type, and for the
other coupling, we consider three phases: 30∘, 45∘ and 60∘ ($\theta=0^{\circ}$
corresponds to a pure axial-vector case, and $\theta=90^{\circ}$ to a pure
vector case). What see in the figures is that the exclusions are rather
similar, only presenting small variations. We thus conclude that while one can
in principle be sensitive to the phase, in practical terms the variation with
the angle is fairly small and can be hard to extract, from a potential signal,
the $\theta$ phase.

(a)  (b)  (c)  (d)  (e)  (f)

Figure 8: In the left (right), excluded parameter space and most sensitive
channel for the case where the SM quarks (dark matter) couple (couples) with
both vector and axial-vector couplings, fixing $g_{q_{V}}=0.1$ and
$|g_{\chi}|=1$. The upper, middle and lower panels show the case where the
phase $\theta$ between the vector and axial-vector case are 30∘, 45∘ and 60∘.

###  4.2 Stückelberg portal from intersecting D6 branes

Matter field | $Q_{A}$ | $Q_{B}$ | $Q_{C}$ | $Q_{D}$ | $Y$
---|---|---|---|---|---
$Q_{L}$ | 1 | -1 | 0 | 0 | 1/6
$q_{L}$ | 1 | 1 | 0 | 0 | 1/6
$U_{R}$ | -1 | 0 | 1 | 0 | -2/3
$D_{R}$ | -1 | 0 | -1 | 0 | 1/3
$L$ | 0 | -1 | 0 | -1 | -1/2
$E_{R}$ | 0 | 0 | -1 | 1 | 1
$N_{R}$ | 0 | 0 | 1 | 1 | 0
Table 2: Charge assignment of the SM matter particles under the four $U(1)$
factors of Ref. [43].

Next we will test the power of the tool $Z^{\prime}$-explorer with a complete
model that involves more complicated relations among the couplings of the
$Z^{\prime}$. This particular model consists of a Stückelberg portal that
could arise from intersecting D6 branes that was studied in Refs. [44, 45,
46]. We will focus on a particular gauge sector given by

$\displaystyle SU(3)_{c}\times SU(2)_{L}\times U(1)_{V}^{A}\times
U(1)_{V}^{B}\times U(1)_{V}^{C}\times U(1)_{V}^{D}\times U(1)_{h}^{m}\times
G_{h},$ (4)

where the subscript $V$ and $h$ corresponds to visible and hidden sector
respectively, $U(1)_{h}^{m}$ are the $m$ abelian gauge factors that only
couple to the hidden sector while $G_{h}$ is the semi-simple part of the
hidden gauge group. Within the visible sector there are four abelian gauge
factor that couple to visible matter fields. Each of these factors555These
factors arise from the different overlapping branes that intersect while
obtaining the SM from type IIA string theory, this realization is the so-
called Madrid quivers that is one of the simplest realistic models of
intersecting D6 branes [43]. A stack of $N$ branes usually hosts a gauge group
$U(N)\cong SU(N)\times U(1)$. has a charge $Q_{\alpha}$ under which the SM
matter particles are charged. The charges of the SM matter particles in terms
of the four visible $U(1)$ are listed in Table 2. In order to have an anomaly-
free model the quark sector is split in two assignments, being
$Q_{L}=t_{L},b_{L}$ and $q_{L}=u_{L},c_{L},d_{L},s_{L}$. If we have a look at
Table 2 we can see that the different charges are related with different
physical properties. $Q_{A}$ is related with baryon number, $Q_{D}$ with
lepton number while $Q_{B}$ and $Q_{C}$ are related with the left and right
nature of the matter particle.

To obtain the SM a combination of the four charges must give rise to the
hypercharge, in this case and according to Table 2 we have,

$\displaystyle Q^{Y}=\frac{1}{6}(Q_{A}-3Q_{C}+3Q_{D}).$ (5)

This combination remains massless before electroweak symmetry breaking. The
other three $U(1)$ gauge bosons left acquire masses by the Stückelberg
mechanism [47, 48]. Here, we will assume that only one of these bosons is
light enough to produce interesting phenomenology at collider energies while
the other two are heavy enough to be decoupled from any effect. So any matter
field $\psi_{\alpha}$ will couple to the lightest $Z^{\prime}$ boson as a
combination of the different charges to the $U(1)$ factors,

$\displaystyle g_{\alpha}^{Z^{\prime}}=aQ_{\alpha A}+bQ_{\alpha B}+cQ_{\alpha
C}+dQ_{\alpha D}+\sum_{i=1}^{m}h_{i}Q_{\alpha i}^{h},$ (6)

where $h_{i}$ are the couplings to the hidden sector. Once we know how the
$Z^{\prime}$ couples to matter and also the information of the charge
assignment given in Table 2 we can derive the actual couplings of the SM
particles to the $Z^{\prime}$. As we mentioned before the quark sector is
divided between the first and second generation and the third one, as they are
charged differently they will consequently have different couplings. The
couplings for the first and second generation of quarks are,

$\displaystyle g_{L}^{u,c,d,s}=(a+b),\quad g_{R}^{u,c}=(-a+c),\quad
g_{R}^{d,s}=(-a-c).$ (7)

The couplings for the top and bottom quarks read,

$\displaystyle g_{L}^{t,b}=(a-b),\quad g_{R}^{t}=(-a+c),\quad
g_{R}^{b}=(-a-c).$ (8)

And finally the couplings to the leptons are,

$\displaystyle g_{L}^{\ell,\nu}=(-b-d),\quad g_{R}^{\ell}=(-c+d).$ (9)

As we can see in the lepton sector there is no distinctions among families.
For the sake of simplicity we will consider that the dark matter couples to
the $Z^{\prime}$ boson with two different couplings representing the left and
right ones, $g_{\chi}^{L}$ and $g_{\chi}^{R}$.

One of the aspects that makes this kind of portals attractive to explain the
dark matter sector is the fact that the SM couplings to the $Z^{\prime}$ can
be quite general. In that way the dark matter particle will couple differently
through the $Z^{\prime}$ to protons and neutrons, since the up and down
couplings are different. Thus, this will be translated to non-zero isospin
violation, $f_{n}/f_{p}$, as the general case. This affects directly to the
phenomenology of dark matter since the dark matter direct detection
experiments rely on the particle dark matter scattering off nuclei [49].

Once we have set the model and obtained the couplings we can now test it
against $Z^{\prime}$-explorer. In order to scrutinise the parameter space of
the model we have chosen different scenarios that are phenomenologically
interesting, they are shown in Table 3.

Scenario | $a$ | $b$ | $c$ | $d$ | $g_{\chi}^{L}$ | $g_{\chi}^{R}$
---|---|---|---|---|---|---
1 | 0.07 | 0.00058 | 0.01 | 0.006 | 1.0 | -1.0
2 | -0.025 | -0.005 | 0.005 | 0.025 | 1.0 | 1.0
3 | 0.1 | -0.01 | 0.01 | 0.01 | 1.0 | 1.0
Table 3: Different values of the couplings for the three phenomenological
scenarios to be tested against $Z^{\prime}$-explorer 2.0.

First of all we examine Scenario 1 where we have an amount of isospin
violation that makes Xenon based direct detection of dark matter experiments
less sensitive to the $Z^{\prime}$. This kind of scenarios are usually called
Xe-phobic. The amount of isospin violation that one needs to obtain this is
$f_{n}/f_{p}=-0.7$, so in our construction this is achieved by a combination
of the left and right parameters such as $b/c\simeq 0.0588$.666Details on how
the amount of isospin violation is related with the parameters of the model
can be found in Ref. [46] The parameters that define this scenario are listed
in Table 3 while we scan the mass of the dark matter particle and the
$Z^{\prime}$ in the ranges $m_{\chi}\in(0-1.5)$ TeV and $m_{\chi}\in(0-8)$
TeV. The results are depicted in the left plot of figure. 9.

(a)  (b)  (c)

Figure 9: Excluded parameter space and most sensitive channel for the case of
the scenarios listed in Table 3.

The results for Scenario 1 shows the versatility in searches performed by
$Z^{\prime}$-explorer. First of all we can see that for low dark matter masses
and when this decay channel is kinematically allowed the monojet search can
exclude up to dark matter masses of $m_{\chi}\sim 400$ GeV for masses of the
$Z^{\prime}$ up to 1.1 TeV. In the same mass region of the $Z^{\prime}$ but
for larger masses of the dark matter that closes the $Z^{\prime}\to\chi\chi$
channel, the leptonic searches are the most sensitive ones. The electron and
muon searches can exclude all the region between $M_{Z^{\prime}}=(0.4-1.1)$
TeV. While the exclusion for larger masses of $Z^{\prime}$ is due to the
$b$-quark searches.

In the second scenario we have chosen the parameters in such a way that the
b-quark couplings gets shrunk. This makes the coupling to leptons be
predominant and that is the reason why we call this scenario leptophilic even
if there are some parts of the parameter region where the decay to quarks
(including the $b$-quark) can be significant. The monojet channel is the most
sensitive one for $Z^{\prime}$ masses up to 1.6 TeV when the dark matter decay
channel is open. However, the power of exclusion only reaches until
$M_{Z^{\prime}}$=1 TeV. When the dark matter channel is not kinematically
allowed the electron and muon searches can cover the mass range of
$M_{Z^{\prime}}=(400-2000)$ GeV. One then can find different most sensitive
channels for the rest of the parameter space. However, the tiny values of the
couplings and the large mass of the $Z^{\prime}$ make the total cross section
small enough so it lies under the reach of the experimental searches.

The third and last scenario corresponds to the opposite to Scenario 2. In this
case the values of the parameters are chosen in such a way that the vector and
axial couplings of the leptons to the $Z^{\prime}$ vanish. This can be seen in
the results of the right plot in figure. 9. The most sensitive channels in all
the parameter space are the monojet, $b\bar{b}$, $t\bar{t}$ and dijet. The
monojet region here is very clear. When the dark matter decay channel is
kinematically open the monojet search excludes up to masses of
$M_{Z^{\prime}}$. In this mass range but for a non-kinematically allowed dark
matter channel the two most sensitive channels are $t\bar{t}$ and dijet, but
they are not powerful enough to exclude. In the range of masses between
$M_{Z^{\prime}}=(1-4.5)$ TeV the $b\bar{b}$ search is the most sensitive
allowing us to exclude masses that are lower than $M_{Z^{\prime}}\lesssim 3.2$
TeV.

##  5 Conclusions

In this work we have presented the expansion of the code $Z^{\prime}$ explorer
to further include missing energy searches, where $Z^{\prime}$ plays the role
of a mediator between the dark and visible sectors. We have implemented within
the existing $Z^{\prime}$-explorer framework, which already included all LHC
searches for $Z^{\prime}$ bosons with masses above the weak scale, the ATLAS
mono-jet study with 139 fb-1 of data. We have found a good agreement between
our simulations and the ATLAS results, which allowed us to make an exploratory
journey of several $Z^{\prime}$ models.

In passing, we also commented on the difficulties encountered in the
reinterpretation of the mono-jet study in view of the available
reinterpretation material. While we had indeed found a good agreement, the
material at our disposal was not always sufficient. For instance, we have
benefited from the fact that two different versions of this study used each a
different single benchmark point. Only having had one benchmark, the
validation of the study would have lacked robustness. We would have also
benefited from having access to the 95 % C.L. upper limits instead of only
having the 95 % C.L. exclusion contours, and it would have been desirable that
both exclusive and inclusive selections results were presented for the upper
limits and exclusion contours.

We have studied the non-trivial interplay between the visible and invisible
channels for dark matter, and showed how they depend on the different coupling
choices. In a first step, we have take a bottom-up approach, considering the
different $Z^{\prime}$ couplings as free, independent parameters. We have
departed from the simplifying assumptions of universal couplings for quarks
and leptons, showing the rich palette of phenomenological possibilities. We
have also explored the impact of the dark matter coupling neither vectorially
nor axially to the dark matter particle, showing that even for a non-trivial
configuration that involves more channels $Z^{\prime}$ arises as a powerful
tool to test the different regions of the parameter space. These excursions in
the simplified model space highlight the importance of having a broad program
of experimental searches, as no single channel dominates over the whole
parameter space.

However, in explicit $Z^{\prime}$ models which derive from top-down
considerations, there often exists correlations among the different couplings.
We have then considered a Stückelberg portal that are obtained as low energy
effective actions in some string compactifications with intersecting branes in
type IIa string theory. In this construction the $Z^{\prime}$ couples to the
matter fields via a combination of four different charges. Consequently, the
matter fields the coupling structure is more complicated involving more
parameters and correlations between couplings. With the aid of
$Z^{\prime}$-explorer 2.0, we have derived the exclusion limits in the
$M_{Z^{\prime}}-m_{\chi}$ plane for different benchmark scenarios. Even for a
more complex coupling construction $Z^{\prime}$-explorer is to be a powerful
tool. While different parameter choices in the Stückelberg portal seem to
prefer different visible channels, the mono-jet channeld added in this version
appears ubiquitiously in the three scenarios explored. This specific search
then has a crucial role in testing models that have $Z^{\prime}$ when the dark
matter channel is open.

Along the way, we have also highlighted the potential improvements that could
be added to the code, for example the addition of flavor bounds, considering
$Z^{\prime}$ in the GeV range (known as _dark photons_ in the jargon, they are
well covered by DarkCast [50]), extrapolating the results to the High-
Luminosity LHC (HL-LHC and future colliders), or enlarging the particle
content of the dark sector. We leave this interesting options, among others,
as possibles avenues for improvement for a new version.

## Acknowledgements:

We would like to thank Ezequiel Alvarez for his collaboration in early stages
of this work. We are indebted to Benjamin Fuks and Dipan Sengupta for
correspondence regarding the MadAnalysis5 PAD implementation of reference [51]
and to Boyu Gao for useful cross-checks of the event generation setup. We
would also like to thank Giuliano Gustavino, Valerio Ippolito and Steven Worm
for useful communication regarding technical details of the ATLAS analysis,
and Marie-Hélène Genest and Nishita Desai for useful discussions. We also
thank Diego Mouriño for assistance with the code. JZ is supported by the
Generalitat Valenciana (Spain) through the plan GenT program
(CIDEGENT/2019/068). VML is funded by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC
2121 “Quantum Universe” - 39083330. We also want to thank the organizers of
the “(Re)interpreting the results of new physics searches at the LHC” and “Río
de la Plata Ph-Exp Institute” workshops and the IFLP where the starting point
of this project took off.

##  Appendix A $Z^{\prime}$ explorer 2.0: implementation details

###  A.1 Backend

As previously described, Z’-explorer 2.0 extend the scope of the software’s
first version to include final states with missing transverse energy. The
implementation for the SM (_visible_) channels remains unchanged, see [10] for
further details. In this version we have modifieded some of the visible
channels to include the most sensitive searches up to date. The updated
references for each visible channel are: $jj$ [52], $b\bar{b}$ [52],
$t\bar{t}$ [53], $e^{+}e^{-}$ [54], $\mu^{+}\mu^{-}$ [54], $\tau^{+}\tau^{-}$
[55], $W^{+}W^{-}$ [56], $Zh$ [57]. Therefore, in this appendix we focus on
the implementation of the new DM final state.

To compute the production cross section in the new decay channel, the software
uses previously generated and recorded the leading order (LO) $pp\to
Z^{\prime}j$ cross section (instead of just $q\bar{q}\to Z^{\prime}$, used for
visible channels) at $\sqrt{s}=13$ TeV with MadGraph5_aMC@NLO, customizing the
UFO-model described in section 2, to set $g_{q_{A}}=g_{q_{V}}=1$ for only one
quark in the proton each time, for $M_{Z^{\prime}}\in[0.01,2,5]$ TeV, with a
step of 0.01 TeV. We have explicitly verified that the initial states with $b$
quarks have a negligible contribution to this process. We employ default
settings of MadGraph5_aMC@NLO, and in particular the fast detector simulation
uses the ATLAS default card from Delphes 3.4.2.

These cross sections grids, stored in the repository as /cards/AXIAL(VEC)_Zpj,
are invoked during the program execution: the predicted production cross
section for a benchmark point is simply the sum of the four contributions of
quarks ($u$, $d$, $c$ and $s$), each of them adjusted by the sum of the
corresponding squared couplings. That is

$\sigma(pp\to
Z^{\prime}j)^{g_{q_{A}},\,g_{q_{V}}}=\sum\limits_{q}\left[\sigma(pp\to
Z^{\prime}j)^{g_{q_{A}}=1}(g_{q_{A}})^{2}+\sigma(pp\to
Z^{\prime}j)^{g_{q_{V}}=1}(g_{q_{V}})^{2}\right]$ (10)

The software selects inside the simulations the record with the mass
$M_{Z^{\prime}}$ that is closest to the one in the input card at the
corresponding benchmark point. The axial and vector couplings are directly
estimated from the chiral coupling in the input card (“incard”)

$g_{f_{V}}=\frac{1}{2}(g_{f_{R}}+g_{f_{L}})\,,\qquad
g_{f_{A}}=\frac{1}{2}(g_{f_{R}}-g_{f_{L}})\,.$ (11)

The production cross-section times branching ratio
($\mathcal{BR}(Z^{\prime}\to\chi\bar{\chi})$) estimated for the given
benchmark point are used to re-scale the experimental limits in [15] for the
axial-vector mediator scenario, within the NWA.777The case for the vector
mediator can be trivially derived from the axial one, so we will describe the
latter only, for simplicity. The number of events in each exclusive signal
region (coming from our recast previously described) are stored for the set of
points in the ($M_{Z^{\prime}},\,m_{\chi}$) plane mentioned in section 3, in
different text files in /cards/DM/AXIAL folder. The software selects the file
for the ($M_{Z^{\prime}},\,m_{\chi}$) pair that is closest to the one in the
input card at the corresponding benchmark point, through the determination of
the euclidean norm. As mentioned in the main text, this is the main difference
with respect to the first version: having upper limits and exclusion countours
in 2 dimensions instead of only one. With the corresponding events selected
and properly re-scaled, the software selects the most sensitive EM signal
region comparing to background (also stored in /cards/DM/), and estimates the
strength for the $\chi\bar{\chi}$ channel.

###  A.2 Frontend

The input card (/incard/card_1.dat) required by in Z’-explorer 2.0 is quite
similar to the one needed in its first version. The user must provide the old
requirements for the visible channels: $M_{Z^{\prime}}$ (in TeV), the
$Z^{\prime}$ couplings to all SM-fermions, except for neutrinos, which
information is required through the partial width $\Gamma_{\nu\nu}$, as for
the $W^{+}W^{-}$ ($\Gamma_{WW}$) and the $Zh$ ($\Gamma_{Zh}$) decay channels.
These entries are required as partial width since there is not a unique
Lorentz structure in their couplings. Additionally, for the new channel, we
require the fermionic DM mass $m_{\chi}$ (in TeV), and its couplings to
$Z^{\prime}$. The total width to other non-SM particles can be added in the
computation as $\Gamma_{xx}$. Each entry is one row (with twenty-six columns),
and corresponds to one benchmark point within the model to be tested by
$Z^{\prime}$-explorer 2.0. The NP parameters should be specified in the
following order (with each column separated by spaces)

$M_{Z^{\prime}}\hskip 2.84544ptg_{u_{L}}\hskip 2.84544ptg_{u_{R}}\hskip
2.84544ptg_{d_{L}}\hskip 2.84544ptg_{d_{R}}\hskip 2.84544ptg_{c_{L}}\hskip
2.84544ptg_{c_{R}}\hskip 2.84544ptg_{s_{L}}\hskip 2.84544ptg_{s_{R}}\hskip
2.84544ptg_{b_{L}}\hskip 2.84544ptg_{b_{R}}\hskip 2.84544ptg_{t_{L}}\hskip
2.84544ptg_{t_{R}}\hskip 2.84544ptg_{e_{L}}\hskip 2.84544ptg_{e_{R}}\hskip
2.84544ptg_{\mu_{L}}\hskip 2.84544ptg_{\mu_{R}}\hskip
2.84544ptg_{\tau_{L}}\hskip 2.84544ptg_{\tau_{R}}\hskip
2.84544pt\Gamma_{\nu\nu}\hskip 2.84544pt\Gamma_{WW}\hskip
2.84544pt\Gamma_{Zh}\hskip 2.84544ptm_{\chi}\hskip 2.84544ptg_{\chi_{L}}\hskip
2.84544ptg_{\chi_{R}}\hskip 2.84544pt\Gamma_{xx}$

where $g_{f_{L}}$ ($g_{f_{R}}$) is the coupling of $Z^{\prime}$ to the
corresponding Left (Right) fermion. To obtain the right form of the
$g_{f_{L/R}}$ couplings from a given NP model, interactions between
$Z^{\prime}$ and SM fermions must be written in the parametrization of
equation (1).

The instruction for running the software remains unchanged. In the command
line type

    >./program.out

The software displays on command shell a summary of the tasks performed during
execution. The addition of the new invisible channel does not significantly
change the speed of the performance from the first version
($\mathcal{O}(10^{3})$ points can be processed in no more than 2 seconds in a
regular CPU). Z’-explorer 2.0 prints in the output file (/output/1.dat) the
incard information for each benchmark point (to ease the after-processing),
followed by

$\mathcal{S}_{jj}\hskip 5.69046pt\mathcal{S}_{bb}\hskip
5.69046pt\mathcal{S}_{tt}\hskip 5.69046pt\mathcal{S}_{ee}\hskip
5.69046pt\mathcal{S}_{\mu\mu}\hskip 5.69046pt\mathcal{S}_{\tau\tau}\hskip
5.69046pt\mathcal{S}_{\nu\nu}\hskip 5.69046pt\mathcal{S}_{WW}\hskip
5.69046pt\mathcal{S}_{Zh}\hskip 5.69046pt\mathcal{S}_{\chi\chi}\hskip
5.69046pt\mathcal{S}_{xx}\hskip 5.69046pt\Gamma_{Z^{\prime}}\hskip
5.69046ptWARNING:\Gamma_{Z^{\prime}}>5$

where $\mathcal{S}$ is the calculated strength in each channel (already
defined in [10]), $\Gamma_{Z^{\prime}}$ is the $Z^{\prime}$ total width, and
the last column is a warning for when the input parameters invalidate the
narrow width approximation (displays $1$ if $\Gamma_{Z^{\prime}}>5\%$ and $0$
otherwise). Nonetheless, this warning does not halt the execution, so the
resulting rate is computed using NWA even if it is not a good approximation.
The parameter $\mathcal{S}_{xx}$ can be regarded as a dummy variable to
eventually add experimental information of other possible non-SM $Z^{\prime}$
channels. For further details about running and the additional information
that can be extracted from Z’-explorer 2.0, visit the GitHub repository [19].

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# One-Shot Video Inpainting

Sangjin Lee Suhwan Cho Sangyoun Lee

###### Abstract

Recently, removing objects from videos and filling in the erased regions using
deep video inpainting (VI) algorithms has attracted considerable attention.
Usually, a video sequence and object segmentation masks for all frames are
required as the input for this task. However, in real-world applications,
providing segmentation masks for all frames is quite difficult and
inefficient. Therefore, we deal with VI in a one-shot manner, which only takes
the initial frame’s object mask as its input. Although we can achieve that
using naive combinations of video object segmentation (VOS) and VI methods,
they are sub-optimal and generally cause critical errors. To address that, we
propose a unified pipeline for one-shot video inpainting (OSVI). By jointly
learning mask prediction and video completion in an end-to-end manner, the
results can be optimal for the entire task instead of each separate module.
Additionally, unlike the two-stage methods that use the predicted masks as
ground truth cues, our method is more reliable because the predicted masks can
be used as the network’s internal guidance. On the synthesized datasets for
OSVI, our proposed method outperforms all others both quantitatively and
qualitatively.

##  1 Introduction

Video inpainting (VI) is a task that aims to remove a designated object in a
given video sequence and fill in that area with plausible content. General
approaches for VI take a video sequence and object segmentation masks for all
frames as their input. However, in real-world applications, obtaining object
segmentation masks for every frame is usually difficult and labor-intensive.
To address this issue, we expand on an approach that is better suited to
practical scenarios, namely one-shot video inpainting (OSVI). Compared to a
conventional VI that requires full-frame segmentation masks as shown in Figure
1 (a), OSVI requires only the initial frames’s object mask and internally
predicts those of subsequent frames while filling the missing content for all
frames; this is depicted in Figure 1 (b).

Figure 1: Visualized comparison between the definitions of (a) conventional
video inpainting and (b) one-shot video inpainting.

OSVI can be achieved by sequentially connecting a video object segmentation
(VOS) network and a VI network as a two-stage pipeline. Using the initial
frame object annotation, the VOS network first generates that object’s entire
frame masks. Then, using full-frame object masks as its input, the VI network
fills in the object regions with plausible content that befits the background.
However, this two-stage approach suffers from critical issues when directly
applied to OSVI. Here, we will discuss this approach’s two main problems.
First, since the entire network is not trained in an end-to-end manner, the
results are optimal for each separate module but not as a whole. This property
that two different networks are trained based on respective objectives leads
to sub-optimal results for OSVI. Second, existing VI algorithms assume that
the object masks received as input are always accurate. Therefore, if the
predicted masks generated from a VOS model, which could be inaccurate, are
provided as their input, the system will severely break down. This phenomenon
will be particularly more drastic for flow-based VI methods where reference
pixels are directly warped to the target region.

To address these concerns, we propose an end-to-end learnable baseline network
that consists of mask prediction and video completion modules sharing a single
encoder’s embedded features. Unlike existing VI algorithms, we erase the
object region at the feature level by adopting a novel transformer
architecture. Our proposed method demonstrates its effectiveness for OSVI on
synthetic datasets both quantitatively and qualitatively. Compared with two-
stage methods, which are mainly naive combinations of existing VOS and VI
methods, our method surpasses all of them by a large margin.

Our main contributions can be summarized as follows:

  * •

We expand VI to OSVI, which refers to erasing a designated object in a video
sequence only using a single frame annotation.

  * •

We propose a novel end-to-end learnable algorithm that can handle OSVI more
effectively when compared to existing two-stage approaches.

  * •

Our proposed method outperforms all existing methods on synthetic datasets.

##  2 Related Work

Object removal. Due to its usefulness in diverse vision applications, such as
video editing, object removal has been of interest recently. Shetty et al.
(Shetty, Fritz, and Schiele 2018) propose a GAN-based two-stage network
composed of a mask generator and an image inpainter for object removal in an
image. DynaFill (Bešić and Valada 2020) presents a new type of network using
depth completion on the inpainted image. The depth map predicted from the
inpainted image of a previous frame is used as guidance for current frame
image inpainting. AutoRemover (Zhang et al. 2020) takes masks as its input and
detects shadows using them. By using object masks and extracted shadow masks,
the network can erase their shadows as well as objects in a video. VORNet
(Chang, Yu Liu, and Hsu 2019) fills the missing object regions only with box
annotations as weak supervision to reduce annotation efforts.

Video object segmentation. Semi-supervised VOS is a pixel-level classification
task that tracks and segments an arbitrary target object in a video. To
efficiently handle an arbitrary target object, early works, such as OSVOS
(Caelles et al. 2017), OnAVOS (Voigtlaender and Leibe 2017) and OSVOS-S
(Maninis et al. 2018), are based on online learning that trains a network
during test time. However, since online learning is impractical for real-world
applications because of its huge computational cost at test time, recent
methods are based on feature matching that compares query frame features to
reference frame features without online learning. VideoMatch (Hu, Huang, and
Schwing 2018) extracts features from an initial frame and a query frame and
then matches them at pixel-level using a soft matching layer. FEELVOS
(Voigtlaender et al. 2019) extends the use of pixel-level matching by
employing initial and previous frames as reference ones; it also uses the
matching output as the network’s internal guidance. CFBI (Yang, Wei, and Yang
2020) improves FEELVOS by employing background matching as well as foreground
matching. To fully utilize the information from all past frames as well as
initial and previous frames, STM (Oh et al. 2019a) proposes the space–time
memory network, in which query frame features are densely matched to the
memory features built from all past frames, covering all space–time locations.
AFB-URR (Liang et al. 2020) and XMem (Cheng and Schwing 2022) improves the
memory construction scheme to design a model robust to long videos.

Figure 2: Overall architecture of our proposed algorithm. Solid lines indicate
an operation is performed every frame, and dotted lines indicate an operation
is performed after all solid line operations are done. Dotted lined operation
takes a full-frame information as its input and generates full-frame
predictions as its output at once. Red and purple letters refer to the
information flows of the mask prediction and video completion, respectively.

Video inpainting. Early works (Huang et al. 2016; Wexler, Shechtman, and Irani
2004) use patch-based optimization to restore missing regions with valid
regions for VI. However, these methods have some limitations. For example,
they face difficulty in representing dynamic motions while maintaining
temporal consistency, in addition to requiring high computational costs. To
address these issues, various deep learning-based methods have been proposed
within the past few years. These studies can be divided into two categories:
patch-based approaches (Lee et al. 2019; Oh et al. 2019b; Zeng, Fu, and Chao
2020; Li et al. 2020; Liu et al. 2021) and flow-based approaches (Kim et al.
2019; Xu et al. 2019; Gao et al. 2020; Zou et al. 2021; Li et al. 2022).
Patch-based approaches aim to fill missing regions using direct pixel-level
feature matching. CPNet (Lee et al. 2019) uses an alignment network to fill
missing regions by using the different object positions with background
matching. OPN (Oh et al. 2019b) proposes a similarity matching network that
extracts and compares key and value features for each frame. It also suggests
a post-processing network called the TCN, which uses a recurrent structure to
increase temporal consistency. STTN (Zeng, Fu, and Chao 2020) is composed of
an encoder–decoder architecture and multi-layer multi-head spatio-temporal
transformers trained using a 3D discriminator for temporal consistency. BSCA
(Li et al. 2020) proposes a boundary-aware short-term module for adjacent
frame matching and a dynamic long-term context aggregation module for temporal
consistency. FuseFormer (Liu et al. 2021) develops the transformer-based
algorithms by proposing a method of efficiency patch splitting and composing.
By contrast, flow-based approaches exploit the optical flow map for VI. VINet
(Kim et al. 2019) proposes a flow warping-based recurrent network that refers
to adjacent frames. DFVI (Xu et al. 2019) restores flow maps extracted using a
pre-trained flow estimation network by performing forward–backward pixel
propagation. FGVC (Gao et al. 2020) presents a flow–edge completion technique
for the sharpness of flow map edges, a color propagation in the gradient
domain for reducing warping error, and a non-local flow to resolve the chronic
problem of flow maps; the latter namely being that they cannot obtain
clearness when adjacent frames lack in information. TSAM (Zou et al. 2021)
proposes a method that applies flow warping to the feature level based on the
DNN architecture. E2FGVI (Li et al. 2022) shows the problems of previous flow-
based pipelines and solves them by presenting the unified pipeline in an end-
to-end manner.

##  3 Proposed Approach

###  3.1 Problem Formulation

Let us denote a video as $I:=\\{I^{i}\in[0,255]^{3\times H0\times
W0}~{}|~{}i=0,1,...T-1\\}$, ground truth object masks as
$M:=\\{M^{i}\in\\{0,1\\}^{H0\times W0}~{}|~{}i=0,1,...T-1\\}$, and a ground
truth clean video as $O:=\\{O^{i}\in[0,255]^{3\times H0\times
W0}~{}|~{}i=0,1,...T-1\\}$. Our goal is to generate the predicted object masks
$\hat{M}$ and clean video $\hat{O}$, where $I$ and $M^{0}$ are provided as
input.

###  3.2 Network Overview

An overview of our proposed network is shown in Figure 2. It consists of two
modules, a mask prediction module and a video completion module. First, the
mask prediction module predicts the segmentation masks for all frames using
the segmentation mask from the initial frame and the features extracted from
each frame. The predicted masks and the extracted features are stored in a
feature bank for reuse. As the semantic cues obtained from the mask prediction
module are transferred to the video completion module without gradient
disconnection, the entire network can be trained in an end-to-end manner.

###  3.3 Mask Prediction Module

Memory construction. To propagate the object information of the initial frame,
i.e., ground truth segmentation mask, we employ a memory-based architecture as
in STM (Oh et al. 2019a). From $I^{0}$, we first extract base features
$X^{0}$. Then, key features $K^{0}\in\mathbb{R}^{C_{K}\times HW}$ and value
features $V^{0}\in\mathbb{R}^{C_{V}\times HW}$ can be obtained as

$\displaystyle K^{0}=Conv(X^{0})~{},$ (1) $\displaystyle
V^{0}=Conv(X^{0}\oplus down(M^{0}))~{},$ (2)

where $\oplus$ and $down$ indicate channel concatenation and adaptive spatial
pooling, respectively. The extracted key and value features are then stored in
memory to be used for future frame prediction.

One of the memory architecture’s main advantages is its ability to
simultaneously store an arbitrary number of frames in memory. To fully exploit
dense cues of a video, we store the key and value features of every five
frames in the memory as well as the initial frame during the inference.
Assuming $N$ frames are stored in the memory, the memory key and value
features can be defined as $K_{M}\in\mathbb{R}^{C_{K}\times NHW}$ and
$V_{M}\in\mathbb{R}^{C_{V}\times NHW}$, respectively.

Memory read. Based on the accumulated information (memory key and value
features) obtained from past frames, the goal is to predict the query frame
mask. To this end, we first extract query key features
$K_{Q}\in\mathbb{R}^{C_{K}\times HW}$ from the query frame image and compare
them to memory key features $K_{M}$ to calculate visual similarity. Then, we
transfer the object cues, i.e., the memory value features $V_{M}$, to the
query frame and the query frame value features
$V_{Q}\in\mathbb{R}^{C_{V}\times HW}$ can be obtained. This process is
identical to popular self-attention mechanism (Vaswani et al. 2017), and can
be represented as

$\displaystyle sim=Softmax(K_{M}^{T}\otimes K_{Q})~{},$ (3) $\displaystyle
V_{Q}=V_{M}\otimes sim~{},$ (4)

where $Softmax$ and $\otimes$ indicate a query-wise softmax operation and
matrix multiplication.

Mask prediction. As $V_{Q}$ contains object cues of the query frame, object
segmentation masks $\hat{M}$ can be obtained by decoding it. To do that, we
gradually increase the spatial size using the low-level features extracted
from the encoder via skip connections. To obtain better feature
representations, we add a CBAM (Woo et al. 2018) layer after every skip
connection.

Figure 3: Visualized pipelines of (a) masked multi-head attention and (b)
spatial multi-head attention.

###  3.4 Video Completion Module

Masked multi-head attention. In our framework, visual features are extracted
from a video sequence and stored in the feature bank to be used for both mask
prediction and video completion modules. Unlike conventional VI methods where
object features are excluded, the features in our model contain object
information since they should also be used to track a designated object.
Considering object features may be referenced when filling the missing region,
this will severely degrade the performance of the system. To address this
issue, we propose a novel masked multi-head attention (MMHA) for transformer
architecture.

Figure 4: Architecture of our transformer block.

The MMHA’s visualized workflow is shown in Figure 3 (a). Given that $F$
represents input features of a transformer block and $\hat{m}$ is a
downsampled version of $\hat{M}$, we can obtain a mask guidance matrix
$G\in[0,1]^{TH^{\prime}W^{\prime}\times TH^{\prime}W^{\prime}}$ as
$1-(1-\hat{m})\otimes(1-\hat{m})^{T}$, after reshaping $\hat{m}$ to
$TH^{\prime}W^{\prime}\times 1$. Then, the mask guidance matrix is masked to a
key–query similarity map to remove the object region from consideration.
However, as a softmax operation comes after this process, we adopt a value
substitution operation instead of a simple multiplication. This process can be
represented as

$masking(s,G)=\begin{cases}-\infty&G<0.5\\\ s&otherwise~{},\end{cases}$ (5)

where $s\in\mathbb{R}^{TH^{\prime}W^{\prime}\times TH^{\prime}W^{\prime}}$
indicates the key–query matching scores. This allows us to diminish the
influence of the object regions even after the softmax operation. In summary,
the MMHA plays two roles: 1) it prevents object regions from bringing semantic
cues from other object regions; 2) it forces object regions not to affect
other valid regions by limiting content propagation.

Spatio-temporal transformer. As shown in Figure 4, we divide the general
transformer block into a temporal transformer block (TTB) and a spatial
transformer block (STB). The architectures of both are identical to that of
the general transformer architecture, i.e., the sequential connection of
multi-head attention and multi-layer perceptron. However, the TTB exchanges
semantic cues between multiple frames, while the STB internally exchanges
semantic cues in a single frame. Furthermore, due to the difference in the
attention layer, they have different objectives. The TTB is used to erase
object cues and fill object regions by leveraging the cues from other frames,
while the STB is used to internally fill those regions within a single frame.

Given that $F_{l}$ is $l$-th features where $l\in\\{0,1,...,L-1\\}$, and $L$
is the number of total transformer blocks, the transformer block formula can
be defined as

$\displaystyle
F^{\prime}_{l}=\mathbf{MMHA}(\mathbf{LN}(F_{l}),\hat{M})+F_{l}~{},$ (6)
$\displaystyle\hat{F}_{l}=\mathbf{MLP}(\mathbf{LN}(F^{\prime}_{l}))+F^{\prime}_{l}~{},$
(7) $\displaystyle
F^{\prime\prime}_{l}=\mathbf{SMHA}(\mathbf{LN}(\hat{F}_{l}))+\hat{F}_{l}~{},$
(8) $\displaystyle
F_{l+1}=\mathbf{MLP}(\mathbf{LN}(F^{\prime\prime}_{l}))+F^{\prime\prime}_{l}~{},$
(9)

where $\mathbf{MLP}$ and $\mathbf{LN}$ indicate multi-layer perceptron and
layer normalization layer (Ba, Kiros, and Hinton 2016), respectively.

Table 1: Quantitative evaluation on the synthesized datasets. VOS |  VI | DAYT | BLDA
---|---|---|---
|  |  PSNR $\uparrow$ |  SSIM $\uparrow$ |  LPIPS $\downarrow$ |  PSNR $\uparrow$ |  SSIM $\uparrow$ |  LPIPS $\downarrow$
FRTM |  CPNet |  27.038 |  0.9324 |  0.0946 |  29.998 |  0.9644 |  0.0451
|  STTN |  25.357 |  0.9118 |  0.1032 |  28.167 |  0.9388 |  0.0552
|  FGVC |  27.082 |  0.9300 |  0.0922 |  30.114 |  0.9650 |  0.0409
|  FuseFormer |  28.495 |  0.9343 |  0.0901 |  34.115 |  0.9661 |  0.0422
CFBI |  CPNet |  31.985 |  0.9591 |  0.0616 |  33.716 |  0.9728 |  0.0336
|  STTN |  30.388 |  0.9400 |  0.0618 |  31.139 |  0.9477 |  0.0414
|  FGVC |  32.124 |  0.9541 |  0.0484 |  34.491 |  0.9749 |  0.0240
|  FuseFormer |  34.309 |  0.9623 |  0.0472 |  38.427 |  0.9751 |  0.0282
BMVOS |  CPNet |  31.910 |  0.9587 |  0.0596 |  33.405 |  0.9723 |  0.0325
|  STTN |  30.173 |  0.9389 |  0.0615 |  30.799 |  0.9464 |  0.0420
|  FGVC |  31.941 |  0.9523 |  0.0487 |  33.719 |  0.9737 |  0.0248
|  FuseFormer |  34.055 |  0.9607 |  0.0480 |  36.679 |  0.9742 |  0.0289
TBD |  CPNet |  32.479 |  0.9597 |  0.0585 |  35.202 |  0.9755 |  0.0274
|  STTN |  30.469 |  0.9405 |  0.0600 |  32.021 |  0.9502 |  0.0362
|  FGVC |  32.105 |  0.9533 |  0.0490 |  35.247 |  0.9766 |  0.0202
|  FuseFormer |  34.296 |  0.9618 |  0.0479 |  38.785 |  0.9771 |  0.0246
Ours |  35.518 |  0.9650 |  0.0515 |  38.959 |  0.9783 |  0.0268

###  3.5 Loss Function

To learn the mask prediction module, we apply mask loss for network training.
It is a standard cross-entropy loss, as described in Eqn. 10, where $p$
indicates a single pixel location in $M$.

$\displaystyle L_{mask}=-\sum_{p}\log P(\hat{M}=M)$ (10)

For the video completion module, we use $L_{1}$ loss by separately applying it
to the object and valid regions. We weight the loss based on the number of
pixels in the area as each video has different ratios of object and valid
regions. The losses are defined as

$L_{object}=\frac{\|M\odot(\hat{O}-O)\|_{1}}{\|M\|_{1}}~{},$ (11)
$L_{valid}=\frac{\|(1-M)\odot(\hat{O}-O)\|_{1}}{\|1-M\|_{1}}~{},$ (12)

where $\odot$ indicates Hadamard product. To ensure high perceptual quality,
we also employ adversarial loss. The discriminator $D$ takes $O$ and $\hat{O}$
as its input, and outputs 1 if input seems real and 0 if input seems fake. By
contrary, the network is trained make the discriminator make a wrong decision
by generating a real-like fake image. The loss function for the discriminator
is formulated as

$\displaystyle L_{dis}=\mathbb{E}_{O}[\log
D(O)]+\mathbb{E}_{\hat{O}}[\log(1-D(\hat{O}))]~{},$ (13)

where the loss function for the network is formulated as

$\displaystyle L_{adv}=\mathbb{E}_{\hat{O}}[\log D(\hat{O})].$ (14)

In conclusion, total loss for our network can be defined as follows.

$\displaystyle L_{total}=L_{mask}+L_{object}+L_{valid}+L_{adv}$ (15)

###  3.6 Network Training

We adopt two datasets for network training, COCO (Lin et al. 2014) and
YouTube-VOS 2018 (Xu et al. 2018). As COCO is an image dataset, we randomly
augment each image to generate videos, following the protocol in STM (Oh et
al. 2019a) and TBD (Cho et al. 2022b). We resize all training videos to a
$240\times 432$ resolution and use them as clean videos. To generate input
videos, we attach objects scrapped from other training snippets. The batch
size is set to four and each snippet contains seven frames. We use an Adam
optimizer (Kingma and Ba 2014) for network optimization.

##  4 Experiments

###  4.1 Experimental Setup

Datasets. To validate the effectiveness of our method, we synthesize the
testing videos using DAVIS (Perazzi et al. 2016; Pont-Tuset et al. 2017),
YouTube-VOS 2018 (Xu et al. 2018), and BL30K (Cheng, Tai, and Tang 2021)
datasets. We design two synthesized datasets, DAYT and BLDA, which mean DAVIS
objects on YouTube-VOS 2018 videos and BL30K objects on DAVIS videos. DAYT and
BLDA consist of 88 and 120 video sequences, each consisting of input images,
object masks, and output clean images.

Detailed settings. To compare our method with existing two-stage methods, we
use state-of-the-art VI and VOS methods with publicly available pre-trained
models. As DAVIS and BL30K are used to construct objects in our testing
datasets, we do not adopt VOS models trained on those datasets. For a fair
comparison, we use the official code and pre-trained models for all methods.
All experiments are implemented on a single RTX A6000 GPU.

Figure 5: Qualitative comparison between state-of-the-art two-stage methods and our proposed method. Green regions indicate the object regions to be removed. Table 2: Ablation study on our proposed algorithm on the synthesized datasets. Version | DAYT | BLDA
---|---|---
|  PSNR $\uparrow$ |  SSIM $\uparrow$ |  LPIPS $\downarrow$ |  PSNR $\uparrow$ |  SSIM $\uparrow$ |  LPIPS $\downarrow$
w/o mask loss |  24.624 |  0.9247 |  0.1075 |  28.218 |  0.9561 |  0.0589
w/o mask guidance |  33.298 |  0.9558 |  0.0638 |  36.713 |  0.9719 |  0.0353
SMHA $\rightarrow$ MMHA |  26.970 |  0.9287 |  0.0955 |  29.647 |  0.9580 |  0.0529
no end-to-end training |  34.871 |  0.9633 |  0.0539 |  37.292 |  0.9762 |  0.0304
w/o encoder sharing |  32.324 |  0.9526 |  0.0694 |  35.790 |  0.9703 |  0.0375
Ours |  35.518 |  0.9650 |  0.0515 |  38.959 |  0.9783 |  0.0268

###  4.2 Quantitative Results

We quantitatively compare our method to existing two-stage methods in Table 1.
For the VOS models, we adopt FRTM (Robinson et al. 2020), CFBI (Yang, Wei, and
Yang 2020), BMVOS (Cho et al. 2022a), and TBD (Cho et al. 2022b). For the VI
models, CPNet (Lee et al. 2019), STTN (Zeng, Fu, and Chao 2020), FGVC (Gao et
al. 2020), and FuseFormer (Liu et al. 2021) are used. Among the VI models,
FuseFormer shows the highest performance. When compared to other two-stage
methods, it obtains the best accuracy on DAYT with CFBI, and on BLDA with TBD.
We can also observe that if segmentation masks are not accurate enough,
satisfactory OSVI performance cannot be achieved even if the state-of-the-art
VI method is adopted. This supports the claim that a unified OSVI pipeline
capable of effectively handling such noises or errors is needed. On both
datasets, our method significantly outperforms all previous methods in PSNR
and SSIM metrics. Quantitative results demonstrate the superiority of our
method compared to state-of-the-art two-stage methods.

###  4.3 Qualitative Results

In Figure 5 and Figure 6, we qualitatively compare our method to state-of-the-
art two-stage methods. As a VOS model, TBD (Cho et al. 2022b) is adopted for
all VI models because it quantitatively outperforms other methods in Table 1.
From Figure 5, we can conclude that optical flow-based methods, such as FGVC
(Gao et al. 2020), are inadequate for OSVI. This is because un-erased objects
may be copied and pasted to fill the object regions, and make the flow-based
method produce the artifacts or some afterimages. Our method shows the
clearest and most accurate object removal quality. Qualitative results are
also compared in video form in Figure 6.

###  4.4 Ablation Study

In Table 2, we conduct an ablation study on various components of our method.
Each different version is quantitatively compared on two synthesized datasets,
DAYT and BLDA.

Mask loss. We conduct a study on the use of mask loss to figure out how mask
supervision affects object removal quality. Removing mask loss leads to a
drastic performance degradation as small errors in object regions lead to
critical errors in the entire system. This demonstrates the need for mask
supervision via mask loss is an essential component.

Mask guidance. To filter out object cues extracted from the feature encoder,
we apply the mask guidance matrix in MMHA. The quantitative results
demonstrate the effectiveness of this filtering process. Compared to the model
version without filtering, the model version with filtering shows better
performance on all metrics.

Multi-head attention. As TTB focuses on filling the object regions missing the
object cues, we utilize STB to focus on filling the object regions that cannot
be filled in TTB. Therefore, we apply the MMHA method only to TTB and not to
STB. To prove its functionality, we evaluate the model version designed with
MMHA instead of SMHA. It turns out that the employed masked attention was
ineffective, in which it could not fill the object regions. Thus, the
collaborative employment of masked attention for temporal information and
naive attention for spatial information is required for OSVI.

Figure 6: Example input and output videos of different one-shot video
inpainting algorithms.

End-to-end training. To demonstrate the effectiveness of end-to-end network
training, we compare the models with and without the end-to-end training
protocol. For the model without end-to-end training, the video completion
module takes ground truth masks and extracted features without gradients to be
separately trained from the mask prediction module. As observed from the
table, end-to-end network training enables the network to be more effective.
Meaningful improvements are observable for all metrics.

Encoder sharing. We also conduct an experiment to see the difference between
models with single and multiple encoders. It is interesting to observe that
using a single encoder shows much better performance than using separate
encoders. This demonstrates that mask prediction and video completion share
common complementary properties. Moreover, jointly learning them ensures a
more appropriate network for OSVI.

##  5 Conclusion

Inspired by inefficiency of conventional VI methods for real-world
applications, we deal with the task of OSVI only requiring a single-frame
manual annotation. To better deal with this without merely sequentially
connecting existing VOS and VI algorithms, we propose an end-to-end learnable
baseline model. By a significant margin, our method outperforms all existing
two-stage methods both quantitatively and qualitatively on two synthesized
datasets. We believe our research makes a step towards efficient and
applicable VI.

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