# Measuring CP violation in rare $W$ decays at the LHC

Peng-Cheng Lu<EMAIL_ADDRESS>School of Physics, Shandong University, Jinan,
Shandong 250100, China  Zong-Guo Si<EMAIL_ADDRESS>School of Physics,
Shandong University, Jinan, Shandong 250100, China  Zhe Wang
<EMAIL_ADDRESS>School of Physics, Shandong University, Jinan, Shandong
250100, China  Xing-Hua Yang<EMAIL_ADDRESS>School of Physics and
Optoelectronic Engineering, Shandong University of Technology, Zibo, Shandong
255000, China

###### Abstract

Heavy Majorana neutrinos beyond the standard model can simultaneously explain
the origin of tiny neutrino masses and matter-antimatter asymmetry in our
Universe. The existence of heavy Majorana neutrinos will also lead to lepton
number violation and the rare lepton-number-violating $W$ decays are possible.
With contributions from two different Majorana neutrinos, a nonzero CP
asymmetry may be generated from the rate difference between $W$ decay and its
CP-conjugate process. The aim of this paper is to investigate the prospects
for measuring CP violation in rare $W$ decays via Majorana neutrinos at the
LHC. Our calculations show that the induced CP asymmetry is independent of the
Majorana neutrino mass for $10~{}\rm GeV<m_{N}<70~{}\rm GeV$. Such a CP
asymmetry if observed, would in turn provide unambiguous evidence of new
physics beyond the standard model.

###### pacs:

14.60.Pq, 14.60.St

##  I INTRODUCTION

The standard model (SM) of particle physics has proved to be extremely
successful in describing all the known fundamental particles and interactions.
However, there are still several questions that remain unexplained. For
example, one of the main mysteries is the origin of tiny neutrino masses.
Within the framework of SM, neutrinos are predicted to be massless. However,
the discovery of neutrino oscillations has firmly indicated that neutrinos are
massive particles and lepton flavors are mixed ParticleDataGroup:2020ssz .
Another important mystery is how to explain the observed matter-antimatter
asymmetry of the Universe. It is argued that the baryon asymmetry in our
Universe can be characterized by the baryon-to-photon density ratio
$\eta=n_{B}/n_{\gamma}$. From a careful analysis of recent Planck measurements
of the cosmic microwave background, the value of $\eta$ has been determined to
a good degree of accuracy: $\eta\simeq(6.12\pm 0.03)\times 10^{-10}$
Planck:2018vyg , which is just too large compared to the SM expectation. To go
beyond the SM, the simplest way to accommodate neutrino masses is to introduce
$n$ right-handed Majorana neutrino fields $N_{\rm R}$, which can couple to the
left-handed neutrino fields through Yukawa interactions to form Dirac mass
terms $\overline{\nu_{\rm L}}M_{\rm D}N_{\rm R}$. As being SM gauge singlets,
the introduced right-handed neutrino fields $N_{\rm R}$ are also allowed to
couple to their charge conjugate fields to constitute Majorana mass terms
$\overline{N^{c}_{\rm L}}M_{\rm R}N_{\rm R}$. This is known as the famous
type-I seesaw mechanism and the tiny neutrino masses can be given by
$M_{\nu}\approx-M_{\rm D}M^{-1}_{\rm R}M^{\rm T}_{\rm D}$ Minkowski:1977sc ;
Mohapatra:1979ia . In this canonical seesaw mechanism, the smallness of the
left-handed neutrinos can be attributed to the heaviness of the right-handed
Majorana neutrinos. As for the origin of the baryon asymmetry in our Universe,
one popular explanation is leptogenesis mechanism Fukugita:1986hr . In this
case, the CP-violating and out-of-equilibrium decays of the heavy Majorana
neutrinos thermally produced in the early Universe may first generate lepton
number asymmetry, and the latter may be subsequently converted into baryon
number asymmetry through sphaleron processes Manton:1983nd ; Klinkhamer:1984di
.

The key point behind the seesaw mechanism and the associated leptogenesis
mechanism is the existence of the heavy Majorana neutrinos, which will cause
the violation of lepton number by two units ($\Delta L=2$) simultaneously and
the lepton-number-violating processes such as neutrinoless double-beta decay
($0\nu\beta\beta$) Furry:1939qr ; Elliott:2004hr  may be possible.The
underlying process with $\Delta L=2$ can be generically expressed as a $W$
decay via Majorana neutrino exchange, which can be identified by the signature
of same-sign dilepton in the final state. This rare $W$ decay, to be specific,
$W^{-}\rightarrow\ell_{1}^{-}N\rightarrow\ell_{1}^{-}\ell_{2}^{-}(q\bar{q}^{\prime})^{+}$,
has been well studied in the literature (for a review, see e.g. Refs.
Deppisch:2015qwa ; Antusch:2016ejd ; Cai:2017mow ). For example, in the mass
range of $M_{N}<M_{W}$, the $\Delta L=2$ same-sign dilepton production signal
has been explored in rare meson decays Atre:2005eb ; Cvetic:2010rw ;
Wang:2014lda ; Dong:2013raa ; Cvetic:2016fbv , tau lepton decays
Gribanov:2001vv ; Kobach:2014hea ; Yuan:2017xdp  and even top quark decays
Bar-Shalom:2006osy ; Si:2008jd ; Delepine:2012nea ; Liu:2019qfa . For heavy
Majorana neutrinos with masses above $M_{W}$, the same signal has been
extensively investigated at various collider experiments, such as the
electron-positron colliders Buchmuller:1991tu ; Gluza:1993gf ; Cvetic:1998vg ;
delAguila:2005ssc , electron-proton colliders Buchmuller:1990vh ;
Ingelman:1993ve ; Liang:2010gm ; Lindner:2016lxq ; Li:2018wut ; Gu:2022muc
and proton-proton collidersKeung:1983uu ; Datta:1993nm ; Han:2006ip ;
Atre:2009rg ; Chao:2009ef ; Fuks:2020att . The difference between the rates of
$W^{-}\rightarrow\ell_{1}^{-}\ell_{2}^{-}(q\bar{q}^{\prime})^{+}$ and its CP-
conjugate process
$W^{+}\rightarrow\ell_{1}^{+}\ell_{2}^{+}(\bar{q}q^{\prime})^{-}$ may induce a
nonzero CP asymmetry, which arises from the significant interference of
different Majorana neutrinos. The generated CP violation effects can serve as
smoking gun for new physics beyond the SM. A great deal of work has been done
for trying to measure the CP violation effects in the decays of meson
Cvetic:2015naa ; Cvetic:2020lyh ; Godbole:2020jqw ; Zhang:2020hwj  and tau
lepton Zamora-Saa:2016ito ; Tapia:2019coy . Recently, Ref. Najafi:2020dkp
explored the CP violation in rare $W$ decays at the LHC, but the CP violation
effect produced in this case is influenced by the initial parton distribution
functions in proton. To evade this conflict, in a recent work Lu:2021vzj , we
investigated the possibility for measuring CP violation in $t\bar{t}$ pair
production and rare decay at the LHC. As the $W^{-}$ and $W^{+}$ bosons
respectively arise from the decays of $\bar{t}$ and $t$ quarks, they have the
same quantity. Different from the previous studies, in this paper, we explore
the discovery prospects for measuring CP violation in $W^{+}W^{-}$ pair
production and rare decay at the LHC, where the $W^{-}$ and $W^{+}$ bosons are
produced directly from $pp$ collisions. In principle, the number $n$ of
introduced right-handed Majorana neutrinos in seesaw mechanism is a free
parameter. Since two neutrino mass-squared differences between light neutrinos
have been observed, $n\geq 2$ is needed. A nonzero direct CP asymmetry also
requires the existence of at least two different Majorana neutrinos. For
illustration, we take two heavy Majorana neutrinos in consideration, and the
general case with more Majorana neutrinos can be analyzed in a similar way.

This paper is organized as follows. A theoretical framework for heavy Majorana
neutrinos is briefly introduced in Section II. The rare $W$ decays via
Majorana neutrinos are discussed in Section III. In Section IV, we explore the
experimental prospects for measuring CP vioaltion at the LHC. Finally, we
conclude in Section V.

##  II Heavy Majorana neutrinos beyond SM

Throughout this paper, we only consider a minimal extension of the SM by
introducing two right-handed Majorana neutrinos. In the notation of Ref.
Atre:2009rg , the flavor eigenstates $\nu_{\ell}$ (with $\ell=e,\mu,\tau$) of
three active neutrinos can be expressed by the mass eigenstates of light and
heavy Majorana neutrinos:

$\nu_{\ell L}=\sum_{m=1}^{3}V_{\ell m}\nu_{mL}+\sum_{m^{\prime}=1}^{2}R_{\ell
m^{\prime}}N^{c}_{m^{\prime}L}\;.$ (1)

Therefore, the weak charged-current interaction Lagrangian can now be written
in terms of the mass eigenstates as follows:

$\displaystyle-\mathcal{L}_{\rm
cc}=\frac{g}{\sqrt{2}}W^{+}_{\mu}\sum_{\ell=e}^{\tau}\sum_{m=1}^{3}V_{\ell
m}^{\ast}\overline{\nu_{m}}\gamma^{\mu}P_{L}\ell+\frac{g}{\sqrt{2}}W^{+}_{\mu}\sum_{\ell=e}^{\tau}\sum_{m^{\prime}=1}^{2}R_{\ell
m^{\prime}}^{\ast}\overline{N^{c}_{m^{\prime}}}\gamma^{\mu}P_{L}\ell+\text{h.c.}\;.$
(2)

Here, $V_{\ell m}$ is just the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix
Pontecorvo:1957cp ; Maki:1962mu  responsible for neutrino oscillations. Note
that $R_{\ell m^{\prime}}$ describes the light-heavy neutrino mixing and can
be generally parameterized as

$R_{\ell m^{\prime}}=\left|R_{\ell m^{\prime}}\right|e^{i\phi_{\ell
m^{\prime}}},~{}~{}~{}\ell=e,\mu,\tau,~{}~{}~{}m^{\prime}=1,2\;.$ (3)

It is worth pointing out that the heavy Majorana neutrinos in the conventional
type-I seesaw mechanism are usually too heavy (e.g. at order of GUT-scale) and
their mixings with light neutrinos are severely suppressed. However, there
also exists some low-scale seesaw scenarios with heavy Majorana neutrino
masses being much lower and the strength of light-heavy neutrino mixings being
large enough (for a review, see e.g. Refs. Xing:2009in ; Chen:2011de ). In
this paper, we will adopt a model-independent phenomenological approach by
taking the heavy neutrino masses $m_{N}$ and light-heavy neutrino mixings
$R_{\ell m^{\prime}}$ as free parameters.

Constraints on the free parameters $m_{N}$ and $R_{\ell m^{\prime}}$ mentioned
above can be derived from experimental observations, and a detailed summary
can be found in Ref. Deppisch:2015qwa . In our calculations, to be
conservative, we take

$\displaystyle\left|R_{ei}\right|^{2}=1.0\times 10^{-7}\;,~{}~{}\left|R_{\mu
i}\right|^{2}=\left|R_{\tau i}\right|^{2}=1.0\times
10^{-5}\;,~{}~{}~{}i=1,2\;,$ (4)

for $10~{}\rm GeV<m_{N}<70~{}\rm GeV$, which are consistent with
$0\nu\beta\beta$-decay searches GERDA:2020xhi , a global fit to lepton flavor
and electro-weak precision data Fernandez-Martinez:2016lgt , a reanalysis of
the Large Electron Positron (LEP) collider data DELPHI:1996qcc  as well as
direct searches by the Large Hadron Collider (LHC) experiments CMS:2018jxx .

##  III CP violation in rare $W$ decays

The rare $W$ decays can be induced by heavy Majorana neutrinos. Here, we
consider both the rare lepton-number-violating decay of $W^{-}$ and $W^{+}$
bosons via two intermediate on-shell Majorana neutrinos $N_{i}$ (with $i=1,2$)
(depicted in Fig. 1):

$\displaystyle W^{-}(p_{1})$
$\displaystyle\to\ell_{\alpha}^{-}(p_{2})+N_{i}(p_{N})\to\ell_{\alpha}^{-}(p_{2})+\ell_{\beta}^{-}(p_{3})+q(p_{4})+\bar{q}^{\prime}(p_{5})\;,$
$\displaystyle W^{+}(p_{1})$
$\displaystyle\to\ell_{\alpha}^{+}(p_{2})+N_{i}(p_{N})\to\ell_{\alpha}^{+}(p_{2})+\ell_{\beta}^{+}(p_{3})+\bar{q}(p_{4})+q^{\prime}(p_{5})\;,$
(5)

where $\alpha,\beta=e,\mu,\tau$. Note that $p_{1}$, $p_{2}$, etc. denote the
four-momentum of the corresponding particles, respectively. The squared matrix
elements averaged (summed) over the initial (final) particles for the process
in Eq. (III) can be obtained as follows:

$\displaystyle\overline{\left|{\cal
M}_{\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}}\right|^{2}}=$
$\displaystyle\frac{g^{6}}{m_{W}^{2}}\left|V_{qq^{\prime}}\right|^{2}\left(1-\frac{1}{2}\delta_{\alpha\beta}\right)\left|D_{W}\left(p_{w}^{2}\right)\right|^{2}$
$\displaystyle\times\left\\{m_{N_{1}}^{2}\left|R_{\alpha 1}R_{\beta
1}\right|^{2}{\cal T}_{1}+m_{N_{2}}^{2}\left|R_{\alpha 2}R_{\beta
2}\right|^{2}{\cal T}_{2}\right.$
$\displaystyle\left.+m_{N_{1}}m_{N_{2}}\left|R_{\alpha 1}R_{\alpha 2}R_{\beta
1}R_{\beta 2}\right|{\rm Re}\left[e^{\pm i\Delta\phi}{\cal
T}_{12}\right]\right\\}\;,$ (6)

where $p_{w}=p_{4}+p_{5}$.
$D_{W}\left(p^{2}\right)=1/(p^{2}-m_{W}^{2}+im_{W}\Gamma_{W})$ where $m_{W}$,
$\Gamma_{W}$ are the mass and total decay width of $W$ boson. Note that
$V_{qq^{\prime}}$ is the Cabibbo-Kobayashi-Maskawa (CKM) matrix element
Cabibbo:1963yz ; Kobayashi:1973fv  and set to be diagonal with unit entries
for simplicity. Also note that $R_{\alpha i}$ (with $\alpha=e,\mu,\tau$ and
$i=1,2$) is the light-heavy neutrino mixing matrix element defined in Eq. (3)
and the complex phase $\Delta\phi=\phi_{\alpha 2}-\phi_{\alpha 1}+\phi_{\beta
2}-\phi_{\beta 1}$ coming from the significant interference between $N_{1}$
and $N_{2}$ can serve as a new source of CP violation. The explicit
expressions of ${\cal T}_{i}$ ($i=1,2$) and ${\cal T}_{12}$ are shown in
Appendix A.

Figure 1: Feynman diagrams for rare $W^{-}$ decay via heavy Majorana neutrino
exchange.

Furthermore, the corresponding differential decay width can then be expressed
as

$\displaystyle
d\Gamma_{W^{\pm}\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}(q\bar{q}^{\prime})^{\mp}}=\frac{1}{2m_{W}}\overline{\left|{\cal
M_{\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}}}\right|^{2}}d{\cal L}ips_{4}\;,$ (7)

with $d{\cal L}ips_{4}$ being the Lorentz invariant phase space of the four
final particles. The decay modes of heavy Majorana neutrino have been well
studied in Ref. Atre:2009rg . As the decay width of heavy Majorana neutrino is
much smaller than its mass in the mass range of our interest, the narrow-width
approximation (NWA) Uhlemann:2008pm  can then be applied. Therefore, the total
decay width for the process in Eq. (III) can be factorized as follows:

$\displaystyle\Gamma_{W^{\pm}\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}(q\bar{q}^{\prime})^{\mp}}\approx\Gamma_{W^{\pm}\to\ell_{\alpha}^{\pm}N_{i}}\cdot{\rm
Br}\left(N_{i}\to\ell_{\beta}^{\pm}(q\bar{q}^{\prime})^{\mp}\right)=\left(2-\delta_{\alpha\beta}\right)\cdot
S_{\alpha\beta}\cdot\Gamma_{0}\;,$ (8)

where $\Gamma_{0}$ is a function of Majorana neutrino mass and the “effective
mixing parameter” $S_{\alpha\beta}$ is defined as

$\displaystyle S_{\alpha\beta}=\frac{\left|R_{\alpha i}R_{\beta
i}\right|^{2}}{\sum_{\ell=e}^{\tau}\left|R_{\ell i}\right|^{2}}\;.$ (9)

In our numerical calculations, we employ the following approximations:

$\displaystyle
m_{N_{2}}=m_{N_{1}}+\Gamma_{N_{1}}/2\;,~{}~{}\Gamma_{N_{2}}\approx\Gamma_{N_{1}}\;.$
(10)

The normalized branching ratio of
$W^{-}\to\ell_{\alpha}^{-}\ell_{\beta}^{-}(q\bar{q}^{\prime})^{+}$ and
$W^{+}\to\ell_{\alpha}^{+}\ell_{\beta}^{+}(\bar{q}q^{\prime})^{-}$ as a
function of $m_{N_{1}}$ are shown in Fig. 2. For illustration, the CP phase
difference is set to $\Delta\phi=0,+\pi/2,-\pi/2,\pi$. It is found that, the
normalized branching ratio decreases slowly as the Majorana neutrino mass
increases for $10~{}\rm GeV<m_{N_{1}}<70~{}\rm GeV$. When the Majorana
neutrino mass is close to the $W$ boson mass, the normalized branching ratio
falls off sharply. The difference between the rates of
$W^{-}\rightarrow\ell_{1}^{-}\ell_{2}^{-}(q\bar{q}^{\prime})^{+}$ and
$W^{+}\rightarrow\ell_{1}^{+}\ell_{2}^{+}(\bar{q}q^{\prime})^{-}$ may induce
the CP asymmetry, which can be defined as

$\displaystyle{\cal A}_{\rm
CP}=\frac{\Gamma_{W^{-}\to\ell_{\alpha}^{-}\ell_{\beta}^{-}(q\bar{q}^{\prime})^{+}}-\Gamma_{W^{+}\to\ell_{\alpha}^{+}\ell_{\beta}^{+}(\bar{q}q^{\prime})^{-}}}{\Gamma_{W^{-}\to\ell_{\alpha}^{-}\ell_{\beta}^{-}(q\bar{q}^{\prime})^{+}}+\Gamma_{W^{+}\to\ell_{\alpha}^{+}\ell_{\beta}^{+}(\bar{q}q^{\prime})^{-}}}\;.$
(11)

As also can be seen in Eq. (III), this CP asymmetry arises from the
interference of contributions from two different heavy Majorana neutrinos.
Therefore, in order to generate such a CP asymmetry, the following two
necessary conditions must be satisfied: (i) the existence of at least two
heavy Majorana neutrinos and (ii) the non-zero CP phase difference
$\Delta\phi$. The numerical results of ${\cal A}_{\rm CP}$ versus the Majorana
neutrino mass $m_{N_{1}}$ for various values of $\Delta\phi$ are shown in Fig.
3. Taking $m_{N_{1}}=10~{}\rm GeV$ for example, the value of ${\cal A}_{\rm
CP}$ as a function of $\Delta\phi$ is also displayed in Fig. 3. It is obvious
that, for fixed $\Delta\phi$, ${\cal A}_{\rm CP}$ is independent of the
Majorana neutrino mass in the mass range of $10~{}\rm GeV<m_{N}<70~{}\rm GeV$.
Furthermore, it can also be found that the CP asymmetry vanishes for
$\Delta\phi=0,\pi$, and ${\cal A}_{\rm CP}\to-{\cal A}_{\rm CP}$ for
$\Delta\phi\to-\Delta\phi$. When $\Delta\phi\approx\pm 3\pi/5$, the maximal
value of $|{\cal A}_{\rm CP}|_{\rm max}\approx 0.22$ can be reached.

Figure 2: The normalized branching ratio of (a)
$W^{-}\to\ell_{\alpha}^{-}\ell_{\beta}^{-}(q\bar{q}^{\prime})^{+}$ and (b)
$W^{+}\to\ell_{\alpha}^{+}\ell_{\beta}^{+}(\bar{q}q^{\prime})^{-}$ versus
Majorana neutrino mass $m_{N_{1}}$ for $\Delta\phi=0,+\pi/2,-\pi/2,\pi$.

Figure 3: (a) The value of ${\cal A}_{\rm CP}$ as a function of $m_{N_{1}}$
for $\Delta\phi=0,+\pi/2,-\pi/2,\pi$. (b) The value of ${\cal A}_{\rm CP}$ as
a function of $\Delta\phi$ for $m_{N_{1}}=10~{}\rm GeV$.

##  IV CP violation in $W^{+}W^{-}$ pair production and rare decay at the LHC

With unprecedented high energy and high luminosity, the LHC will offer a great
opportunity to probe new physics beyond the SM. In this paper, we explore the
prospects for measuring CP violation in rare $W$ decays at the LHC and
consider the following process

$\displaystyle pp\to W^{\pm}W^{\mp}\to\ell^{\pm}\ell^{\pm}+4j\;,$ (12)

where the $W^{\pm}W^{\mp}$ pairs are produced directly from $pp$ collisions.
In the following numerical calculations, we restrict our study to the same-
sign dimuon production channel and employ CTEQ6L1 Pumplin:2002vw  for the
parton distribution functions in proton. In Fig. 4, the total cross sections
for the process in Eq. (12) are shown as a function of $m_{N_{1}}$ for
$\Delta\phi=0,+\pi/2,-\pi/2,\pi$ at 14 TeV and 100 TeV LHC, respectively. With
the integrated luminosity of $\cal{L}$=300 ${\rm fb}^{-1}$, there only a few
events produced for $m_{N_{1}}<70~{}\rm GeV$ at 14 TeV LHC, and more than ten
events can be produced at 100 TeV LHC.

Figure 4: The total cross sections for (a) $pp\to
W^{+}W^{-}\to\mu^{+}\mu^{+}4j$ and (b) $pp\to W^{+}W^{-}\to\mu^{-}\mu^{-}4j$
at 14 TeV LHC, (c) $pp\to W^{+}W^{-}\to\mu^{+}\mu^{+}4j$ and (d) $pp\to
W^{+}W^{-}\to\mu^{-}\mu^{-}4j$ at 100 TeV LHC versus Majorana neutrino mass
$m_{N_{1}}$ with $\Delta\phi=0,+\pi/2,-\pi/2,\pi$.

Analogously, the difference between $\sigma(pp\to\mu^{+}\mu^{+}4j)$ and
$\sigma(pp\to\mu^{-}\mu^{-}4j)$ can also lead to a non-zero CP asymmetry,
which can be expressed as

$\displaystyle\overline{{\cal A}_{\rm
CP}}=\frac{\sigma(pp\to\mu^{+}\mu^{+}4j)-\sigma(pp\to\mu^{-}\mu^{-}4j)}{\sigma(pp\to\mu^{+}\mu^{+}4j)+\sigma(pp\to\mu^{-}\mu^{-}4j)}.$
(13)

In this case, the underlying CP violation effect is similarly caused by the
rate difference between rare $W^{-}$ decay and its CP-conjugate process.
Therefore, the CP asymmetry defined in Eq. (13) is equivalent to that given in
Eq. (11). In Fig. 5, we display the value of $\overline{{\cal A}_{\rm CP}}$
with respect to $m_{N_{1}}$ and $\Delta\phi$, respectively. As expected, the
CP asymmetry in Fig. 5 behaves almost the same as that in Fig. 3.

Figure 5: (a) The value of $\overline{{\cal A}_{\rm CP}}$ as a function of
$m_{N_{1}}$ for $\Delta\phi=0,+\pi/2,-\pi/2,\pi$. (b) The value of
$\overline{{\cal A}_{\rm CP}}$ as a function of $\Delta\phi$ for
$m_{N_{1}}=10~{}\rm GeV$.

In order to simulate detector response, the lepton and jet energies are
smeared according to the assumption of Gaussian resolution parametrization

$\displaystyle\frac{\delta(E)}{E}=\frac{a}{\sqrt{E}}\oplus b,$ (14)

where $\delta(E)/E$ represents the energy resolution and $\oplus$ denotes a
sum in quadrature. In our calculations, we take $a=5\%$, $b=0.55\%$ for
leptons and $a=100\%$, $b=5\%$ for jets, respectively CMS:2007sch ;
ATLAS:2009zsq . The isolated leptons and jets are identified by angular
separation, which can be defined as

$\displaystyle\Delta R_{ij}=\sqrt{\Delta\phi^{2}_{ij}+\Delta\eta^{2}_{ij}}\;,$
(15)

with $\Delta\phi_{ij}$ ($\Delta\eta_{ij}$) being the azimuthal angle
(rapidity) difference of the corresponding particles.

As the jets from rare $W^{\pm}$ decay $W^{\pm}\to\ell^{\pm}\ell^{\pm}jj$ are
much softer than those in the hadronic decay of $W^{\mp}\to jj$, the former
two jets can be merged into one fat jet. Therefore, we only require three jets
($n_{j}=3$) in the final state of our signal process. To quantify the signal
observability, we impose the following basic acceptance cuts on leptons and
jets (referred as cut-I)

$\displaystyle p_{T}^{\ell}>10~{}{\rm
GeV}\;,\;|\eta^{\ell}|<2.8\;,\;p_{T}^{j}>15~{}{\rm
GeV}\;,\;|\eta^{j}|<3.0\;,\;0.4<\Delta R_{\ell j}<3.5\;,\;n_{j}=3\;.$ (16)

Our signal process in Eq. (12) consists of two same-sign dileptons and three
jets. To purify the signal, the missing transverse energy are required to
satisfy (referred as cut-II)

$\displaystyle{E\\!\\!\\!/}_{T}<20~{}{\rm GeV}\;.$ (17)

For our signal process, the main backgrounds in SM come from $pp\to
W^{\pm}W^{\pm}W^{\mp}W^{\mp}$, $pp\to W^{\pm}W^{\pm}W^{\mp}j$ and $pp\to
W^{\pm}W^{\pm}W^{\mp}Z$. To be specific, the SM backgrounds are simulated by
MadGraph5_aMC@NLO Alwall:2014hca . The parton shower is performed with
Pythia-8.2 Sjostrand:2006za . Jets-clustering is done with the anti-$k_{t}$
algorithm Cacciari:2008gp .

Comparing our signal process with the backgrounds, we further fully
reconstruct the two Ws. One W boson which decay hadronically can be
reconstructed from the two jets ($j_{1},j_{2}$). The invariant mass of this
two jets is closest to $m_{W}$. After reconstructing one W boson, the
remaining ingredients are grouped to reconstruct the other W boson. We adopt
the following cut (referred as cut-III)

$\displaystyle|M_{j_{1}j_{2}}-m_{W}|<20~{}{\rm GeV}\;,\;|M_{\ell\ell
j_{3}}-m_{W}|<20~{}{\rm GeV}\;,$ (18)

where $j_{3}$ refers to the left jet and $\ell\ell$ are the two same-sign
dileptons.

After implementing all the above cuts, we obtain the total cross sections for
the signal and background processes at 14 TeV and 100 TeV, respectively. For
illustration, we have used $m_{N_{1}}=10~{}\rm GeV$ and $\Delta\phi=\pi/2$. At
14 TeV, the signal cross section after all cuts is only $2.32\times 10^{-3}$
$\mathrm{fb}$, which is too small to be detected experimentally. We list the
total cross sections for the signal and background processes at 100 TeV LHC in
Table 1. The statistical significance $S/\sqrt{B}$ with integrated luminosity
of $\cal{L}$=300 $\mathrm{fb}^{-1}$ and $\cal{L}$=3000 $\mathrm{fb}^{-1}$ are
also given, where $S$ and $B$ denote respectively the signal and background
event numbers after all cuts. It is shown that the signal cross section after
all cuts is remains $1.40\times 10^{-2}$ $\mathrm{fb}$ at 100 TeV LHC and the
corresponding statistical significance can reach 4.34 (13.74) with
$\cal{L}$=300 $\mathrm{fb}^{-1}$ ($\cal{L}$=3000 $\mathrm{fb}^{-1}$), which
will offer us a great opportunity to explore CP violation effects in rare $W$
decays at the LHC. After adopting all the kinematic cuts, we display
respectively $\overline{{\cal A}_{\rm CP}}$ versus $m_{N_{1}}$ and
$\Delta\phi$ at 100 TeV LHC in Fig. 6. It can be found that the CP asymmetry
is almost unchanged after all the selection cuts.

Table 1: The cross sections for the signal and background processes at 100 TeV LHC after all cuts. Also shown is the statistical significance $S/\sqrt{B}$ with integrated luminosity of $\cal{L}$=300 $\mathrm{fb}^{-1}$ and $\cal{L}$=3000 $\mathrm{fb}^{-1}$. For illustration, we have used $m_{N_{1}}=10~{}\rm GeV$ and $\Delta\phi=\pi/2$.  | 100 TeV
---|---
|  $\sigma_{S}$ [$\mathrm{fb}$] |  $\sigma_{pp\to W^{\pm}W^{\pm}W^{\mp}W^{\mp}}$ [$\mathrm{fb}$] |  $\sigma_{pp\to W^{\pm}W^{\pm}W^{\mp}j}$ [$\mathrm{fb}$] |  $\sigma_{pp\to W^{\pm}W^{\pm}W^{\mp}Z}$ [$\mathrm{fb}$]
Cut-I | $2.29\times 10^{-2}$ | $2.9\times 10^{-2}$ | $5.56$ | $1.53\times 10^{-2}$
Cut-II | $1.98\times 10^{-2}$ | $5.45\times 10^{-4}$ | $1.59\times 10^{-1}$ | $2.88\times 10^{-4}$
Cut-III | $1.40\times 10^{-2}$ | $1.21\times 10^{-5}$ | $3.10\times 10^{-3}$ | $3.02\times 10^{-6}$
$S/\sqrt{B}$ with $\cal{L}$=300 $\mathrm{fb}^{-1}$ | 4.34
$S/\sqrt{B}$ with $\cal{L}$=3000 $\mathrm{fb}^{-1}$ | 13.74

Figure 6: (a) The value of $\overline{{\cal A}_{\rm CP}}$ after all cuts as a
function of $m_{N_{1}}$ for $\Delta\phi=0,+\pi/2,-\pi/2,\pi$. (b) The value of
$\overline{{\cal A}_{\rm CP}}$ after all cuts as a function of $\Delta\phi$
for $m_{N_{1}}=10~{}\rm GeV$.

##  V Summary

The simplest way to extend the SM is to introduce the heavy Majorana neutrinos
and allow for lepton number violation. The introduced heavy Majorana neutrinos
can simultaneously explain the tiny neutrino masses via seesaw mechanism and
the baryon asymmetry of the Universe via leptogenesis. In this paper, we
explore the prospects for measuring CP violation in rare $W$ decays at the
LHC, where the CP asymmetry between
$W^{-}\to\ell_{\alpha}^{-}\ell_{\beta}^{-}(q\bar{q}^{\prime})^{+}$ and
$W^{+}\to\ell_{\alpha}^{+}\ell_{\beta}^{+}(\bar{q}q^{\prime})^{-}$ arises from
the significant interference of contributions from two different Majorana
neutrinos. We find that, the CP asymmetry for fixed $\Delta\phi$ is basically
independent of the Majorana neutrino mass in the mass range of our interest
$10~{}\rm GeV<m_{N}<70~{}\rm GeV$. Taking $m_{N_{1}}=10~{}\rm GeV$ and
$\Delta\phi=\pi/2$ as an example, we investigate the possibility of measuring
such CP violation at 14 TeV and 100 TeV LHC. Although the signal cross section
at 14 TeV LHC is too small to be detected experimentally, the high energy 100
TeV LHC will offer us a great opportunity to explore CP violation effects. The
measurement of such CP violation would provide us much important information
about the underlying new physics.

## Acknowledgements

P. C. Lu, Z. G. Si, Z. Wang and X. H. Yang thank the members of the Institute
of theoretical physics of Shandong University for their helpful discussions.
This work is supported in part by National Natural Science Foundation of China
(Grants No. 11875179, 11775130) and Natural Science Foundation of Shandong
Province (Grant No. ZR2021QA040).

##  Appendix A Formalism for rare $W$ decay

The functions ${\cal T}_{i}$ ($i=1,2$) and ${\cal T}_{12}$ in Eq. III can be
respectively expressed as

$\displaystyle{\cal T}_{i}=$
$\displaystyle\left|D_{N_{i}}\left(p_{N}^{2}\right)\right|^{2}\cdot{\cal
F}-{\rm
Re}\left[D_{N_{i}}\left(p_{N}^{2}\right)D_{N_{i}}^{\ast}\left({p_{N}^{\prime}}^{2}\right)\right]\cdot{\cal
I}$ $\displaystyle+{\rm
Im}\left[D_{N_{i}}\left(p_{N}^{2}\right)D_{N_{i}}^{\ast}\left({p_{N}^{\prime}}^{2}\right)\right]\cdot{\cal
J}\;,$ (19) $\displaystyle{\cal T}_{12}=$
$\displaystyle\left[D_{N_{1}}\left(p_{N}^{2}\right)D_{N_{2}}^{\ast}\left({p_{N}}^{2}\right)+D_{N_{1}}\left({p_{N}^{\prime}}^{2}\right)D_{N_{2}}^{\ast}\left({p_{N}^{\prime}}^{2}\right)\right]\cdot{\cal
F}$
$\displaystyle-\left[D_{N_{1}}\left(p_{N}^{2}\right)D_{N_{2}}^{\ast}\left({p_{N}^{\prime}}^{2}\right)+D_{N_{1}}\left({p_{N}^{\prime}}^{2}\right)D_{N_{2}}^{\ast}\left(p_{N}^{2}\right)\right]\cdot{\cal
I}$
$\displaystyle+i\left[D_{N_{1}}\left(p_{N}^{2}\right)D_{N_{2}}^{\ast}\left({p_{N}^{\prime}}^{2}\right)-D_{N_{1}}\left({p_{N}^{\prime}}^{2}\right)D_{N_{2}}^{\ast}\left(p_{N}^{2}\right)\right]\cdot{\cal
J}\;,$ (20)

where $p_{N}=p_{2}-p_{1}$ and $p_{N}^{\prime}=p_{1}-p_{3}$.
$D_{N_{i}}\left(p^{2}\right)$ $(i=1,2)$ is the Breit-Wigner propagator and can
be defined as

$\displaystyle
D_{N_{i}}\left(p^{2}\right)=\frac{1}{p^{2}-m_{N_{i}}^{2}+im_{N_{i}}\Gamma_{N_{i}}}\;,$
(21)

with $m_{N_{i}}$ and $\Gamma_{N_{i}}$ being the mass and total decay width of
the two Majorana neutrinos $N_{1}$ and $N_{2}$.

The explicit expressions of ${\cal F}$, ${\cal I}$ and ${\cal J}$ introduced
in Eq. A and Eq. A can be given by

$\displaystyle{\cal F}=$ $\displaystyle 16(p_{4}\cdot
p_{3})\left[m_{W}^{2}(p_{5}\cdot p_{2})+2(p_{5}\cdot p_{1})(p_{1}\cdot
p_{2})\right]\;,$ (22) $\displaystyle{\cal I}=$ $\displaystyle
8\biggl{\\{}-(p_{4}\cdot p_{3})\left[m_{W}^{2}(p_{5}\cdot p_{2})+2(p_{5}\cdot
p_{1})(p_{1}\cdot p_{2})\right]-(p_{4}\cdot p_{2})\left[m_{W}^{2}(p_{5}\cdot
p_{3})+2(p_{5}\cdot p_{1})(p_{1}\cdot p_{3})\right]$
$\displaystyle+(p_{2}\cdot p_{3})\left[m_{W}^{2}(p_{4}\cdot
p_{5})+2(p_{4}\cdot p_{1})(p_{5}\cdot p_{1})\right]\biggr{\\}}\;,$ (23)
$\displaystyle{\cal J}=$ $\displaystyle 8\left[m_{W}^{2}+2(p_{5}\cdot
p_{1})\right]\epsilon_{p_{1}p_{2}p_{3}p_{5}}\;,$ (24)

where
$\epsilon_{p_{1}p_{2}p_{3}p_{5}}=\epsilon_{\mu\nu\rho\sigma}p_{1}^{\mu}p_{2}^{\nu}p_{3}^{\rho}p_{5}^{\sigma}$.

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Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

CLEF 2023: Conference and Labs of the Evaluation Forum, September 18–21, 2023,
Thessaloniki, Greece

<EMAIL_ADDRESS>] [1]

<EMAIL_ADDRESS>]

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[1]Corresponding author.

# Gpachov at CheckThat! 2023: A Diverse Multi-Approach Ensemble for
Subjectivity Detection in News Articles

Georgi Pachov  Sofia University “St. Kliment Ohridski”, Bulgaria  Mohamed bin
Zayed University of Artificial Intelligence, UAE  Dimitar Dimitrov  Ivan
Koychev  Preslav Nakov

(2023)

# Notebook for the CheckThat Lab at CLEF 2023

Georgi Pachov  Sofia University “St. Kliment Ohridski”, Bulgaria  Mohamed bin
Zayed University of Artificial Intelligence, UAE  Dimitar Dimitrov  Ivan
Koychev  Preslav Nakov

(2023)

###### Abstract

The wide-spread use of social networks has given rise to subjective,
misleading, and even false information on the Internet. Thus, subjectivity
detection can play an important role in ensuring the objectiveness and the
quality of a piece of information. This paper presents the solution built by
the Gpachov team for the CLEF-2023 CheckThat! lab Task 2 on subjectivity
detection. Three different research directions are explored. The first one is
based on fine-tuning a sentence embeddings encoder model and dimensionality
reduction. The second one explores a sample-efficient few-shot learning model.
The third one evaluates fine-tuning a multilingual transformer on an altered
dataset, using data from multiple languages. Finally, the three approaches are
combined in a simple majority voting ensemble, resulting in 0.77 macro F1 on
the test set and achieving 2nd place on the English subtask.

###### keywords:

Subjectivity detection Sentence Embeddings Few-shot learning Transformer
Ensemble Natural Language Processing Deep Learning

##  1 Introduction

Subjectivity is a feature of language and a form of bias, in which whenever a
person is sharing information, it comes out skewed by the speaker’s own
personal preferences, beliefs and views. In today’s interconnected world,
where opinions and biases travel fast and far, subjectivity detection can be a
very important piece in order to ensure information reporting is done in a
clear, objective and unbiased fashion.

Specifically, subjectivity in news and media articles can be nuanced, subtle
and difficult to identify. Detection of subjectivity in such texts can play an
important role in identifying potentially misleading or malicious texts and in
detecting fake news online.

In Task 2 of CheckThat! Lab at CLEF 2023 [1, 2], systems are required to
distinguish whether a sentence from a news article expresses the subjective
view of the author or presents an objective view on the covered topic instead.
This is a binary classification task in which systems have to identify whether
a text (a sentence or a paragraph) is subjective or objective.

This paper explores the effects of fine-tuning a large pre-trained language
model on the subjectivity task. Additionally, we have examined the angles of
few-shot-learning and fine-tuning a sentence embedding model. Finally, an
ensemble method is proposed to unify all three into one solution.

##  2 Related Work

While sentiment analysis can be regarded as a classic NLP task with lots of
research already available on the subject, subjectivity classification is
deemed to be a slightly less popular research topic. [3] looks at the
subjectivity detection task as a way to improve sentiment analysis classifiers
by excluding neutral (objective) sentences. They offer a broad survey on
published subjectivity detection methods, categorizing them into syntactic
(keyword-spotting, lexical affinity, statistical methods), semantic (parse
trees, convolutional neural networks, extreme learning machines) and multi-
modal (BiLSTM, multiple-kernel learning).

In [4], authors explore multi-task learning with hard parameter sharing via
Neural Tensor Network. They demonstrate that using a single network with
shared layers while learning on two semantically related datasets can improve
performance on both datasets.

In [5], authors compare Word2Vec and BERT embedding models in the context of
subjectivity detection. With additional classification models to process the
embedding outputs, authors demonstrate the superiority of BERT embeddings in
high-resource settings, while showing Word2Vec embeddings can be more
efficient in low-resource settings. In the current challenge, pairing an
embedding encoder with various classification models is also explored.

In [6], authors compare the performance of pure transformer models against a
variety of more specialized methods for short text classification. Their
results show superior performance of the transformer models and part of the
research performed in this paper is influenced by their findings.

##  3 Data and Baseline Solution

For the subjectivity detection task, datasets in 6 different languages were
provided - Arabic, Dutch, English, German, Italian and Turkish. A total of 7
datasets were available - one for each language, and one for the multilingual
version of the task. The English dataset contained a total of 1019 examples.
800 of the provided examples were labeled as training, the other 219 as
validation. A baseline
solution111https://gitlab.com/checkthat_lab/clef2023-checkthat-
lab/-/tree/main/task2/baseline is provided by competition organizers, which
consists of a sentence encoder model, producing sentence embeddings, which are
then classified with Logistic Regression.

An interesting insight was that most of the sequences were relatively short.
For the English dataset, the average number of words in a sequence was 23,
while 90% of sequences consisted of 40 words or less.

The English training set is imbalanced, with 64% of samples labeled objective
and 36% - subjective. This imbalance is not present in the validation set.

##  4 Experiments and Evaluation

Three research directions were explored, each of them resulting in a separate
solution. The final program used for submission is a simple majority voting
ensemble of the three solutions. The first research direction explores what is
achievable using sentence embeddings. The second one looks at a few-shot-
learning model and dual-stage fine-tuning. The third is based on fine-tuning a
pre-trained transformer model, also utilizing training data from the other
languages available for the task.

All evaluations of experiments are done on the English validation set,
provided by the organizers. All research directions will be described in more
detail in the next subsections.

###  4.1 Sentence Embeddings

Multiple experiments with sentence embeddings were conducted. All of them were
based on using a pre-trained sentence embedding encoder model [7] . The
following ideas were explored:

  * •

Using more powerful classifiers on top of sentence embeddings output

  * •

Using dimensionality reduction

  * •

Fine-tuning the sentence embeddings encoder in the context of subjectivity
detection

Initially, the baseline solution provided by organizers used sentence
embeddings and a simple Logistic Regression on top to produce classification
outputs. More complex classifiers were tested. Multiple different classifiers
yielded improvements, measured on the validation set. The most performant was
LogisticRegression (from sklearn), using ElasticNet penalty, balanced class
weights, ‘saga’ solver and 0.5 as regularization constant.

A potential place for improvement was related to sentence embeddings
dimensionality. Due to only having 800 training examples, embeddings of
dimensionality 384 could prove challenging for a classifier. Using
dimensionality reduction, information from embedding vectors can be further
compressed in a way that could make it easier for classifiers to find a proper
decision boundary.

Best performance was achieved using PCA with 110 remaining components (out of
384), which explained 92.5% of total variance. Experiments with different
classifiers and dimensionality reduction are summarized in the first column of
Table 1.

While dimensionality reduction can help classifiers, the sentence embeddings
themselves were generally created for a very broad category of NLP tasks.
Fine-tuning the embeddings encoder for subjectivity detection proved to be
helpful for all of the tested classifiers.

Embeddings were fine-tuned using cosine similarity loss in a contrastive
learning manner. The new similarity label of a pair of sentences consisted of
two equally weighted components - their original similarity and their label-
based similarity. Label-based similarity is defined as 1 if the two sentences
are from the same class and 0 otherwise. This can be summarized with the
following equation:

$New\\_Similarity\\_Label(A,B)=0.5*Similarity(A,B)+0.5*(class(A)==class(B))$
(1)

To generate training samples, N objective and N subjective sentences were
randomly selected. Training pairs were generated - each sentence was paired
with every other sentence and a similarity label was generated using Formula
1. In total, 2N*(2N-1) training pairs were generated.

All of the tested classifiers performed better using the fine-tuned embeddings
(with N=100) instead of original embeddings. All experiments with classifiers,
dimensionality reduction and fine-tuned sentence embeddings are summarized in
Table 1. For both the original and the fine-tuned embeddings, best performance
was achieved through PCA and Linear Regression with elastic net penalty. All
macro F1 scores are measured on the (English) validation set.

Table 1: Classifiers and dimensionality reduction with original vs fine-tuned embeddings encoder (Macro F1) Classifier$\backslash$Embedding | Original Sentence Embeddings | Fine-Tuned Embeddings (N=100)
---|---|---
Baseline (SBERT + LR) | 0.74 | 0.76
SVM | 0.76 | 0.78
ElasticNet | 0.77 | 0.8
PCA + ElasticNet | 0.78 | 0.81

###  4.2 Few-Shot Learning

The second research direction explored what can be achieved with few-shot
learning. Experiments are based on the SetFit model from HuggingFace [8].
While conceptually similar to the idea of fine-tuning sentence embeddings, the
SetFit model has numerous advantages, including faster training, better sample
efficiency and a dual-stage fine-tuning mechanism. In the first stage, the
classification head is frozen and embeddings are fine-tuned. Vice versa in the
second stage.

Experiments and results for this approach are outlined in Table 2. Results are
similar to fine-tuning sentence embeddings encoder, but achieved while using a
lot less information (samples) from the task-specific dataset, which showed
promise of low variance and good generalization capabilities.

Table 2: Few-shot learning experiments with SetFit Fine-tuning regime | Number of samples | Macro F1
---|---|---
Single stage | 10 | 0.79
Single stage | 20 | 0.8
Dual stage | 20 | 0.81
Dual stage | 64 | 0.8
Dual stage | 100 | 0.79

###  4.3 Transformer Fine-Tuning

The third research direction was based on the idea of fine-tuning a
transformer model to the dataset provided. Lots of transformer models are
available in the HuggingFace hub [9]. BERT [10], RoBERTa [11] and DeBERTa [12]
were used throughout the experiments. Limitations in hardware capabilities
restricted the experiments to only ’base’ and ’large’ variants of the models.
Xlarge and xxlarge models were not tested. Initially, experiments were
performed using the English dataset only. Later on, additional experiments
were performed including data from other languages.

Table 3 summarizes the experiments and results when using the English dataset.
Fine-tuning a DeBERTa-v2-large achieved the best performance on the validation
set.

Table 3: Transformer experiments with English dataset Transformer Model | Architecture | Macro F1
---|---|---
BERT | base | 0.77
BERT | large | 0.78
RoBERTa | base | 0.81
DeBERTa-v2 | large | 0.82

Additional experiments were performed using data from other languages.
Multiple training datasets were created. The first one consisted of all the
available data for all languages. The second one consisted of English, Arabic
and Turkish - the languages who had published baseline solutions with higher
F1 than English. The third one contained training samples from English and
German translated to English using a neural model [13]. While creating the
datasets, English validation samples were always manually excluded as a final
step to prevent data leakage.

For all three datasets, multilingual transformer models were used - bert-base-
multilingual [10], mDeBERTa [14] and xlm-roberta-base [15].

Table 4 summarizes the results. An xlm-roberta-base model was fine-tuned using
all available data, achieving a macro F1 score of 0.83, which showed slight
improvement over using English-only dataset. The best results were obtained by
using English and German translated to English, fitted to xlm-roberta-base
model.

Table 4: Transformer experiments with multilingual datasets Model | Architecture | Training Set | F1 macro
---|---|---|---
bert-multilingual | base | All data | 0.81
xlm-roberta | large | All data | 0.81
mdeberta-v3 | base | All data | 0.82
xlm-roberta | base | All data | 0.83
xlm-roberta | base | English, Arabic and Turkish | 0.82
xlm-roberta | base | English, German translated to English | 0.84

###  4.4 Ensemble

Best performing solutions from all three research directions were chosen and
formed into an ensemble (Figure 1). The final solution was a simple majority
voting ensemble from the following solutions:

  * •

A fine-tuned sentence embeddings encoder, producing improved sentence
embeddings (in the context of this task). The encoder output passes through
dimensionality reduction (384 dimensions reduced to 110). The reduced
embeddings are then classified through LogisticRegression with equally
weighted L1 and L2 penalties, ‘saga’ solver, balanced class weights and 0.5
regularization constant.

  * •

A few-shot learning SetFit model trained using dual-stage fine-tuning
procedure with N=20 sentence pairs.

  * •

An xlm-roberta-base model, fine-tuned on English and German translated to
English.

The final submitted ensemble was able to achieve 0.85 macro F1 on English
validation set.

Figure 1: Schema of the ensemble

##  5 Conclusions

The final solution achieved macro F1 of 0.85 on validation set and 0.77 on
test set. This indicates generalization issues, likely stemming from
overfitting the solution to the validation set.

Each of the three separate methods was analyzed on the test set, to identify
possible causes for the significantly lower performance on the test set. All
components of the solution perform worse on the test set than on the
validation set, indicating a more difficult test set. The first method however
incurs a bigger loss of performance than the others. This is likely caused by
an issue in the fine-tuning procedure, resulting in the encoder model
producing highly specialized embeddings, which lose a bigger part of their
pretrained semantics than optimal and thus generalize poorly. Less examples
used for contrastive learning, smaller learning rate, or using a holdout set
to check for generalization issues could have mitigated this problem.

Results also indicate that the best performer is an xlm-roberta-base model
trained on English and translated German. All test and validation F1 scores
are summarized in Table 5.

Table 5: Test scores of individual solutions and ensemble Solution | Val F1 score | Test F1 score
---|---|---
Fine-tuned S-BERT, PCA, ElasticNet | 0.81 | 0.7
SetFit model | 0.81 | 0.76
xlm-roberta-base | 0.84 | 0.79
Majority voting ensemble | 0.85 | 0.77
Table 6: Per-class precision, recall and f1-scores of the final solution Class | Precision | Recall | F1
---|---|---|---
subjective | 0.83 | 0.71 | 0.77
objective | 0.73 | 0.84 | 0.78

Table 6 indicates that the solution is biased to expect more frequent
‘objective’ examples, hence the higher recall but lowered precision. This is
likely stemming from the initial class imbalance in the English training set,
where 64% of examples are labeled objective. While this imbalance doesn’t seem
to have a major impact on final F1 scores, a more balanced solution could have
been achieved by using any of the well-known class-imbalance techniques, e.g.
sample weighing or choosing a different threshold.

Overall, xlm-roberta model trained on English and translated German seem to be
the best performer. The few-shot-learning approach yielded decent test
results. The fine-tuned SBERT encoder method seems to not be general enough
and is reducing the performance of the full solution. Submitting predictions
from only the transformer model would have resulted in 0.79 macro F1, which
could have won the English subtask challenge.

##  6 Future Work

As indicated by Table 5, the transformer-based solution proved to be the most
effective and robust out of the ones used in the ensemble. Further experiments
can be conducted with newer and more promising versions of existing
transformer models. For example, DeBERTa-V3 [14] is reported to improve
performance of the original DeBERTa model with 1.37% on the GLUE benchmark.
This can prove relevant to the task of subjectivity classification as well.

Additionally, due to resource constraints, transformer models with ’xlarge’
and ’xxlarge’ architectures were not used for this research. For the same
reason, very little hyperparameter tuning was performed. Using a larger model
and exploring bigger hyperparameter space can potentially improve the results.

Finally, the transformed-based approach used for subtask 2A (English) can also
be attempted and used for the other 5 languages in the task - Arabic, Dutch,
German, Italian and Turkish.

###### Acknowledgements.

This research is partially funded by Project UNITe BG05M2OP001-1.001-0004
funded by the OP "Science and Education for Smart Growth", co-funded by the EU
through the ESI Funds, and partially financed by the European Union-
NextGenerationEU, through the National Recovery and Resilience Plan of the
Republic of Bulgaria, project SUMMIT, No BG-RRP-2.004-0008.

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