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§ CONDITIONS FOR FLOATABILITY BY CAPILLARITY

[]Sketch of a spherical solid particle, of radius $R$ and density $\rho_{\rm sol}$, floating at the surface
of a liquid with density $\rho_{\rm liq}$. The meniscus (blue line) is represented by the equation $z=f(x)$, the surface
tension of the interface is $\sigma$, while the contact angle between the
solid and the liquid is denoted $\theta_c$; $\theta_{\rm f}$ is the filling angle. The difference of height between
the plane of flotation and the surface of the unperturbed liquid is $h_0$. The local tangent to the meniscus has the angle
$\psi$ with the horizontal direction. The gravity is represented by $\overrightarrow{g}$.
This figure is our reconstruction of Fig. 1 published by lee_2018, it is also very similar to Fig. 2 in

In this frame, let us define a spherical solid particles; of radius $R$
and density $\rho_{\rm sol}$. This object is assumed to be deposited at the surface of a liquid characterized by a density $\rho_{\rm liq}$
and a surface tension $\sigma$ (see Fig. <ref>). According to the literature <cit.>, this particle undergoes three forces.
The most obvious of them, is the weight

\begin{equation}
F_M = \frac{4\pi}{3} \rho_{\rm sol} R^3 g
\end{equation}

The second one is the capillary force, whose vertical component $F_C$ has the expression

\begin{equation}
F_C = 2\pi \, R \, \sin \theta_f \, (\sigma \sin \beta)
\end{equation}

where $\theta_f$ is the filling angle and $\beta$ includes both the contact angle $\theta_c$
and $\theta_f$ (see Fig. <ref>: $\beta=\theta_c-\theta_f$). Taking into account forces caused by air and liquid, coming
from the classical Archimede's principle, and also generated by liquid above the line of flotation (i.e. corresponding to the
“layer” of thickness $h_0$ in Fig. <ref>), the resulting buoyancy force is

\begin{equation}
\begin{split}
F_B = &    \frac{4\pi}{3} R^3 \rho_{\rm air} g
+ \frac{\pi}{3} R^3 (2 + 3 \cos \theta_f - \cos^3 \theta_f) \times \\
& (\rho_{\rm liq} - \rho_{\rm air}) g
- \pi R^2 \sin^2\theta_f \, h_0 \, (\rho_{\rm liq} - \rho_{\rm air}) \, g
\end{split}
\end{equation}

The flotation of the particle is reached at equilibrium when we have the force balance

\begin{equation}\label{balance}
F_M= F_C + F_B
\end{equation}

Unfortunately, this equation cannot be solved alone since it contains two unknowns: the filling angle $\theta_f$ and $h_0$; other parameters,
particularly the contact angle $\theta_c$, specify the nature of the system, and are assumed to be given. As a requirement, we have to include
the Young-Laplace equation

\begin{equation}
\kappa_{m} = \frac{\rho_{\rm liq} - \rho_{\rm air}}{2 \sigma} g z
\end{equation}

which governs the profile of the meniscus (blue line in Fig. <ref>).  In this equation, $\kappa_{m}$ is the local mean curvature given
by the non-linear differential equation

\begin{equation}\label{mean_curvature}
\kappa_{m} = -\frac{1}{2} \, \left( \frac{f''}{(1+f'^2)^{3/2}} + \frac{f'}{x (1 + f'^2)^{1/2}} \right)
\end{equation}

where $z= f(x)$ is the equation of the meniscus profile. According to the dedicated literature <cit.>, the system of equations
(<ref>)-(<ref>) can be reformulated under the form of a pair of differential equations, solved numerically using
a shooting method <cit.>. In Fig. <ref> (a) we show an example of computed menisci profiles, illustrating the influence
of the density ratio $D = (\rho_{\rm sol} - \rho_{\rm air}) / (\rho_{\rm liq} - \rho_{\rm air})$.

By expanding and reformulating Eq. <ref>, we can write

\begin{equation}\label{reformulated}
\frac{g}{6\sigma}( \rho_{\rm liq} -\rho_{\rm air}) R^2
= \frac{\sin \theta_f \, \sin (\theta_c - \theta_f) }{4D - (2 + 3 \cos\theta_f - \cos^3 \theta_f)
+ 3\frac{h_0}{R} \sin^2\theta_f}
\end{equation}

Since, in the present context, the density of the solid is larger than the liquid one, the ratio $D$ remains larger than 1
(a typical value is around $\sim 2$); in addition, we have $ -2 \le 3\cos\theta_f - \cos^3 \theta_f \le +2$ for all the permitted
$\theta_f$ values, which are in the range $0^{\rm o}-180^{\rm o}$. As a consequence, the denominator of the right-hand side of
Eq. <ref> is positive. Then, the liquid being denser than the air, $\sin(\theta_c - \theta_f)$ has to be positive and,
importantly, the filling angle $\theta_f$ has to remain smaller than the contact angle $\theta_c$.

For $\theta_f \lesssim 45^{\rm o}$, the term $3\cos\theta_f - \cos^3 \theta_f$ remains reasonably close to $2$, and
$3 h_0/R \sin^2 \theta_f$ is negligible, in such a case

\begin{equation}
\frac{g}{6\sigma}(\rho_{\rm liq} -\rho_{\rm air}) R^2 \simeq
\frac{\sin \theta_f \, \sin (\theta_c - \theta_f) }{4D - 4}
\end{equation}

It can be easily shown <cit.> that the radius has a maximum value
$R_{\rm max}$ for $\theta_f = \theta_c/2$ and

\begin{equation}\label{Rmax}
R_{\rm max} \simeq \sqrt{\frac{3\sigma}{2 (\rho_{\rm sol} - \rho_{\rm liq}) g}} \, \sin \frac{\theta_c}{2}
\end{equation}

More Information and Advice:


ATL-PHYS-PROC-2022-114

Direct top quark mass measurements with the ATLAS and CMS detectors

Marco Vanadia, on behalf of the ATLAS and CMS collaborations00footnotetext:
Copyright 2022 CERN for the benefit of the ATLAS and CMS collaborations.
CC-BY-4.0 license.

INFN Sezione di Roma Tor Vergata

> This article presents a review of recent results on direct top quark mass
> measurements performed by the ATLAS and CMS collaborations on $pp$
> collisions collected during Run 2 of the LHC at $\sqrt{s}=$13 TeV.

> PRESENTED AT
>
>
>
>
> $15^{\mathrm{th}}$ International Workshop on Top Quark Physics
> Durham, UK, 4–9 September, 2022

##  1 Introduction

The top quark is the heaviest elementary particle in the Standard Model (SM)
of particle physics. A precise measurement of its mass $m_{t}$ is crucial for
Electroweak precision tests and for predictions for physics phenomena within
the SM and beyond. A review of recent direct measurements of $m_{t}$ at the
LHC [1] is presented here. These measurements reconstruct $m_{t}$ from the
kinematics of the decay products of the top quark. Typically they fit an
observable measured in data with templates generated as a function of the top
quark mass parameter of the Monte Carlo (MC) generators. The ATLAS [2] and CMS
[3] collaborations published the most precise direct $m_{t}$ measurements of
172.69$\pm$0.48 and 172.44$\pm$0.49 GeV [4, 5], obtained by combining several
measurements in different signatures performed on $pp$ collisions collected
during Run 1 of the LHC. A combined value of 174.30$\pm$0.65 GeV has been
obtained by Tevatron experiments [6]. The results reviewed in this article are
obtained on $\sqrt{s}=$13 TeV $pp$ collisions from Run 2 of the LHC.

##  2 Measurements

A $m_{t}$ measurement by the CMS Collaboration on a data set of 36
$\mathrm{fb^{-1}}$ in $t\overline{t}$ $\ell$+jets events is reported in Ref.
[7]. Events with exactly 1 isolated electron or muon and at least 4 hadronic
jets are selected. Exactly 2 jets are required to be identified as $b-$jets by
a tagging algorithm. A $\chi^{2}$ quantity is defined using the measured
kinematic properties of the reconstructed objects and their resolution,
imposing the $W$ boson mass $m_{W}$ as a constraint; $m_{t}$ is a free
parameter of the fit. The fitted top mass $m_{t}^{\mathrm{fit}}$ for events
with goodness-of-fit GOF$>$0.2 is the main observable used in the analysis.
The measurement is improved by using additional observables for those events:
the reconstructed $m_{W}^{\mathrm{reco}}$, the ratio between the invariant
mass of the $\ell+b-$jet system $m_{\ell b}$ and $m_{t}^{\mathrm{fit}}$, and
the ratio $R^{\mathrm{reco}}_{bq}$ between the scalar sum of the
$p_{\mathrm{T}}$ of the $b-$jets and the scalar sum of the $p_{\mathrm{T}}$ of
the light-jets. These provide sensitivity to the jet energy scale. A fifth
observable used in the analysis is $m_{\ell b}$ for events with GOF$<$0.2.
Backgrounds are obtained from simulations. The $m_{t}$ is measured with a
Maximum Likelihood (ML) profiled fit. Systematic uncertainties are treated as
nuisance parameters (NPs); with this technique the data is used to obtain more
information on the uncertainties and possibly to constrain them, even if
additional studies are typically needed to support eventual strong
constraints. The result of the measurement, shown in Fig. 1, is
$m_{t}$$=171.77\pm 0.38$ GeV, the measurement with the smallest uncertainty to
date. The uncertainty is dominated by systematic effects; the most relevant
ones are associated with the $b-$jet energy measurement and the modelling of
the final state radiation (FSR) and color reconnection (CR) in simulation. The
measurement is lower than the world average based on previous measurements,
but this is mostly associated with the correlation model chosen for the FSR
uncertainty, where variations for different branching types ($g\rightarrow
gg$, $g\rightarrow qg$, $q\rightarrow qg$ for $q=u,d,s,c$ and $q\rightarrow
qg$ for $q=b,t$) are kept uncorrelated. It is shown that by correlating those
variations, as done in previous measurements, the difference with those
results can be significantly reduced.

(a)  Figure 1: Post-fit distribution of $m_{t}^{\mathrm{fit}}$ in the (left)
muon and (right) electron channels for the measurement in Ref. [7].

A measurement performed by the CMS collaboration on a data set of 36
$\mathrm{fb^{-1}}$ in the single-top $t-$channel is published in Ref. [8].
This channel has lower statistics and purity than $t\overline{t}$ ones, but it
explores a different kinematic region and is subject to partially independent
systematic uncertainties. Events with exactly 1 isolated electron or muon and
exactly 2 jets are selected. Exactly 1 jet is required to be identified as a
$b-$jet by a dedicated algorithm; events with no $b-$jets are used to define a
validation region for the analysis. The transverse mass of the charged lepton
plus neutrino system $m_{\mathrm{T}}$ is required to be greater than 50 GeV,
to reduce background contributions. The QCD multijet background is controlled
using a data-driven technique, exploiting events with non-isolated leptons and
events with low $m_{\mathrm{T}}$. Other backgrounds are obtained from
simulations. $m_{t}$ is directly reconstructed using the kinematic properties
of the selected objects, with the momentum of the neutrino inferred from
$\vec{p}_{\mathrm{T}}^{~{}\mathrm{miss}}$ and imposing $m_{W}$ as a
constraint. In order to improve the signal-over-background ratio, a Boosted
Decision Tree (BDT) method is defined, using several kinematic quantities as
input, of which the most powerful one is the angular separation between the
two jets in the event. For the final selection events are required to be
associated with a BDT score greater than 0.8, a value optimised to reduce the
total uncertainty of the measurement. The final observable is defined as
$\zeta=ln(m_{t}^{\mathrm{reco}}/\mathrm{1~{}GeV})$. After the QCD multijet
background is subtracted, the other contributions, modelled with analytical
functions, are fitted to the data, as shown in Fig. 2. The peak position of
the distribution is sensitive to the $m_{t}$ value, which is measured to be
$m_{t}=172.13^{+0.76}_{-0.77}$ GeV. The uncertainty is dominated by systematic
effects associated with the jet energy scale (JES), $b-$jet identification,
and modelling of FSR, CR and $b-$hadronization effects.

(a)

(b)

Figure 2: (a) Distribution of the BDT score in the muon channel and (b) fit to
the final observable for the measurement in Ref. [8]

Ref. [9] presents a measurement performed by the ATLAS collaboration on a data
set of 36 $\mathrm{fb^{-1}}$ in $t\overline{t}$ $\ell$+jets events with a
$b-$hadron decaying semileptonically to a muon, $B\rightarrow\mu+X$. The
invariant mass of the isolated lepton and of the muon produced by the
$b-$hadron decay, $m_{\ell\mu}$, is a purely leptonic observable sensitive to
the $m_{t}$ value. It is largely insensitive to jet-related uncertainties,
providing a complementary sensitivity with respect to jet-based measurements.
Events with exactly 1 isolated $e$ or $\mu$ and at least 4 jets are selected,
one of which must be identified as a $b-$jet by a dedicated algorithm.
$B\rightarrow\mu+X$ decays are identified using a soft muon tagging (SMT)
algorithm, requiring that a muon (SMT muon) is identified within $\Delta
R<0.4$ from the axis of one the selected jets. $V+$jets and QCD multijet
backgrounds are determined using data-driven techniques, while other
backgrounds are modelled using MC samples. A detailed study is performed on
$e^{+}e^{-}\rightarrow Z\rightarrow b\overline{b}$ data to tune the parameters
controlling the $b-$quark fragmentation modelling in MC. This procedure
assumes the universality of the $b-$fragmentation for lepton and hadron
colliders, as supported by existing data in the phase space relevant for the
analysis, within the quoted uncertainties. A ML profiled fit is performed on
$m_{\ell\mu}$ using systematic uncertainties as NPs, as shown in Fig. 3. The
final result is $m_{t}$$=$174.41$\pm$0.39 (stat.)$\pm$0.66 (syst.)$\pm$0.25
(recoil) GeV. Dominant systematic uncertainties are associated with the
modelling of the $b-$hadron decay. The recoil uncertainty is associated with
an intrinsic ambiguity in the definition of the recoil for gluon radiation
from the $b-$quark in $t\rightarrow Wb$ decays for the signal MC simulation.
This uncertainty was developed after input from the MC authors and will need
to be carefully addressed by future measurements.

(a)

(b)

Figure 3: Post-fit distribution of $m_{\ell\mu}$ for events with an isolated
lepton and a SMT muon with charges with (a) opposite sign (OS) and (b) same
sign (SS) for the measurement in Ref. [9]

Finally, Ref. [10] presents a measurement by the ATLAS collaboration on the
full Run 2 data set (139 $\mathrm{fb^{-1}}$) in the $t\overline{t}$ $\ell\ell$
channel. Events with two oppositely charged $e/\mu$, and at least two jets,
are selected. Exactly two jets are required to be identified as $b-$jets by a
tagging algorithm. A veto is applied in the $ee$ and $\mu\mu$ channels on the
invariant mass of the two leptons $m_{\ell\ell}$, to suppress contributions
from decays of known resonances. A new method, based on a Deep Neural Network
(DNN), is used to reconstruct the event topology by finding the appropriate
$b-$jet/$\ell$ pairing, using several kinematic quantities as an input. Events
with a high probability of being correctly reconstructed by the DNN are
selected. Then the $b-$jet/$\ell$ pair with the highest $p_{\mathrm{T}}$ is
used in the rest of the analysis, and only events where the $b-$jet in the
pair is the leading one in the event are retained. A final selection requiring
the $p_{\mathrm{T}}$ of the pair to be greater than 160 GeV is applied. The
modelling of this variable yields a significant difference between MC and
data, which is at least partly expected due to missing higher order
corrections to the top quark $p_{\mathrm{T}}$; a careful study showed that the
impact of such mismodelling is well controlled within the uncertainties quoted
for the measurement. The invariant mass $m_{\ell b}^{\mathrm{High}}$ of that
pair is used as the final observable, and an unbinned ML fit is performed to
extract the $m_{t}$ as shown in Fig. 4. This results in
$m_{t}$$=$172.63$\pm$0.20 (stat.)$\pm$0.67 (syst.)$\pm$0.37 (recoil) GeV, with
dominant uncertainties associated with modelling effects (matrix element
generator, ISR/FSR, CR) and JES. The recoil uncertainty, discussed above, is
taken into account for the first time for a jet-based measurement; without it,
the result improves by 17% with respect to the precision of the one based on
Run 1 data. An additional check performed on non-resonant and off-shell
effects showed that these are well within the quoted modelling uncertainty.

(a)

(b)

Figure 4: (a) DNN score for the selected $b-$jet/$\ell$ pair and (b) $m_{\ell
b}^{High}$ distribution for the measurement in Ref. [10]

##  3 Conclusion

A review of recent direct top quark measurements by the ATLAS and CMS
collaborations on LHC Run 2 data was presented. The measurements reach a level
of precision comparable with that of the theoretical uncertainties on the
definition of $m_{t}$ itself, posing a significant challenge on their
interpretation [11]. Furthermore, they are now limited by systematic
uncertainties associated with the modelling in MC simulations, which
inevitably introduces a dependency of the experimental results on the specific
simulation. With the LHC providing already more data with the ongoing Run 3,
and the potential of the Run 2 data set not yet fully explored, the next years
will pose a great challenge for experimental and theoretical physicists, which
are called on to build a solid framework for improving and interpreting the
$m_{t}$ measurements.

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