SMP-23-005

SMP-23-005

# Observation of ${\PGg\PGg\to\PGt\PGt}$ in proton-proton collisions and
limits on the anomalous electromagnetic moments of the lepton

###### Abstract

The production of a pair of $\PGt$ leptons via photon-photon fusion,
${\PGg\PGg\to\PGt\PGt}$, is observed for the first time in proton-proton
collisions, with a significance of 5.3 standard deviations. This observation
is based on a data set recorded with the CMS detector at the LHC at a center-
of-mass energy of 13and corresponding to an integrated luminosity of 138.
Events with a pair of $\PGt$ leptons produced via photon-photon fusion are
selected by requiring them to be back-to-back in the azimuthal direction and
to have a minimum number of charged hadrons associated with their production
vertex. The $\PGt$ leptons are reconstructed in their leptonic and hadronic
decay modes. The measured fiducial cross section of ${\PGg\PGg\to\PGt\PGt}$ is
$\sigma^{\text{fid}}_{\text{obs}}=12.4^{+3.8}_{-3.1}\unit{fb}$. Constraints
are set on the contributions to the anomalous magnetic moment ( and ) and
electric dipole moments ($d_{\PGt}$) of the $\PGt$ lepton originating from
potential effects of new physics on the $\PGg\PGt\PGt$ vertex:
$\and=0.0009_{-0.0031}^{+0.0032}$ and $\abs{d_{\PGt}}<2.9\times
10^{-17}\,e\cm$ (95% confidence level), consistent with the standard model.

##  0.1 Introduction

The production of lepton pairs via photon-photon fusion in proton-proton
($\Pp\Pp$) collisions draws significant interest since it is a pure quantum
electrodynamics (QED) process at leading order, for which the theoretical
cross section can be calculated with uncertainties below 1% [1]. If both
protons remain intact after their electromagnetic interaction, the final state
has a particularly clean topology characterized by two leptons emitted back-
to-back in azimuth and without any hadronic activity surrounding it. Although
the photon-fusion production of dileptons in hadronic collisions has been
measured precisely for electrons and muons [2, 3, 4, 5, 6, 7, 8], a similar
experimental precision is lacking for $\PGt$ leptons [9, 10]. The photon-
fusion channel has long been proposed to probe the $\PGg-\PGt$ coupling at
hadron colliders [11, 12, 13, 14], because the short $\PGt$ lepton lifetime
makes other measurements difficult.

The most general form of photon-lepton coupling ($ie\Gamma^{\mu}$) satisfying
Lorentz invariance is:

$\Gamma^{\mu}=\gamma^{\mu}F_{1}\left(q^{2}\right)+\frac{\sigma^{\mu\nu}q_{\nu}}{2m}\left[iF_{2}\left(q^{2}\right)+F_{3}\left(q^{2}\right)\gamma_{5}\right],$
(1)

where $\sigma^{\mu\nu}=i[\gamma_{\mu},\gamma_{\nu}]/2$ is the spin tensor
proportional to the commutator of the gamma matrices, $q$ is the momentum
transfer between incoming and outgoing leptons of mass $m$, and
$F_{i}\left(q^{2}\right)$ ($i=1,2,3$) are form factors that contain the
complete information about the electric and magnetic couplings of the lepton.
Beyond serving as a sensitive test of QED interactions, the photon-photon
production process also offers insights into the anomalous electromagnetic
moments of the lepton ($\Pell=\Pe,\Pgm,\PGt$),
$F_{2}\left(0\right)=a_{\Pell}\equiv(g_{\Pell}-2)/2$ and
$F_{3}\left(0\right)=-2md_{\Pell}/e$, and is a powerful probe of physics
beyond the standard model (BSM). In this expression, the gyromagnetic ratio
$g_{\Pell}$ is a constant term that relates the magnetic moment of the lepton
to its spin, and $d_{\Pell}$ is the lepton anomalous electric dipole moment.
The one-loop contribution to $a_{\Pell}$ is equal to the so-called ”Schwinger
term”, which is common to all leptons [15]:

$a_{\Pell}=\frac{\alpha}{2\pi}\simeq 0.00116,$ (2)

where $\alpha$ is the fine-structure constant.

The anomalous magnetic moment of the electron has been measured to agree with
the standard model (SM) with an impressive precision of twelve significant
digits [16, 17]. Measurements of $a_{\mu}$ by the Muon g-2 Collaboration reach
a precision of ten significant digits and appear in conflict with certain
theoretical predictions [18, 19, 20]. If BSM effects scale with the squared
lepton mass, deviations of $\and$ from the SM predictions would be more than
two orders of magnitude larger than they are for $a_{\mu}$. The photon-fusion
production of a pair of $\PGt$ leptons probes the $\PGt\PGt\PGg$ vertex and
can be used to measure $\and$, which is predicted to be $\and=1.17721\pm
0.00005\times 10^{-3}$ in the SM [21]. This same process can also be studied
to constrain $d_{\PGt}$, which, since there is no appreciable violation of
charge-conjugation and parity symmetries in the charged lepton sector of the
SM, is predicted to have a nonzero but tiny value [22].

The signal in this analysis is
$\Pp\Pp\xrightarrow{\PGg\PGg}\Pp^{(*)}\PGt^{+}\PGt^{-}\Pp^{(*)}$, where
$\Pp^{(*)}$ denotes a final-state proton that stays intact or fragments after
the photon emission. When both protons radiate coherently a photon and remain
intact after their interaction, the process is called exclusive or elastic.
Since the protons do not dissociate, these events are characterized by the
absence of any hadron produced, other than those from the $\PGt$ lepton
decays. The photons emitted in the elastic processes are almost on-shell,
limiting their virtuality to small values ($q^{2}\to 0$) [1], and the produced
$\PGt$ leptons are back-to-back in the azimuthal direction. When one or both
incoming protons fragment as a result of photon emission, the process is
called either single- or double-dissociative. Single- and double-dissociative
events can also give rise to an elastic-like signature when the proton
remnants are not reconstructed because they fall outside of the detector
acceptance. Figure 1 shows the three diagrams considered here, contributing to
the ${\PGg\PGg\to\PGt\PGt}$ process in $\Pp\Pp$ collisions.

Figure 1: Feynman diagrams for the production of $\PGt$ lepton pairs by
photon-photon fusion. The exclusive (left), single proton dissociation
(middle), and double proton dissociation (right) topologies are shown.

The ${\PGg\PGg\to\PGt\PGt}$ process has been observed by the ATLAS and CMS
Collaborations in ultraperipheral collisions of lead ions [9, 10], and used to
set constraints on $\and$. The observation was driven by signal events with a
di-$\PGt$ invariant mass, $m_{\PGt\PGt}$, typically below 20. This paper
describes the first measurement of the photon-fusion production of a pair of
$\PGt$ leptons in $\Pp\Pp$ collisions. Because of the trigger requirements,
experimentally accessible signal events have $m_{\PGt\PGt}>50\GeV$. The
$\PGt\PGt$ production in this phase space has a much lower cross section, but
the integrated luminosity is much higher than the one achievable with heavy
ion collisions at the LHC. Additionally, BSM effects from new heavy particles
[23] are enhanced at high mass, such that constraints on BSM scenarios with
non-SM $\and$ and $d_{\PGt}$ values can be set using both the rate of the
signal and its $m_{\PGt\PGt}$ distribution. Outside of LHC experiments,
constraints on $\and$ were previously set by the DELPHI, OPAL, and L3
experiments [24, 25, 26]. Constraints on $d_{\PGt}$ were determined by the
Belle, L3, OPAL, and ARGUS Collaborations [27, 28, 25, 26].

This paper is based on $\Pp\Pp$ collision data at $\sqrt{s}=13\TeV$ collected
with the CMS detector in 2016–2018, corresponding to an integrated luminosity
of 138. To select the signal, we apply so-called exclusivity criteria, which
rely on the kinematics of the di-$\PGt$ system and on the activity in the
tracker part of the detector around the di-$\tau$ vertex. In particular, by
selecting events with no track within 0.1of the di-vertex, we can reduce the
background contributions by about three orders of magnitude for a signal
efficiency of around 50%. Such a strategy has been used previously to identify
similar photon-photon processes in $\Pp\Pp$ collisions [8, 29, 30, 31, 32,
33]. Four different final states, corresponding to different decay modes of
the $\PGt$ leptons, are studied to measure the signal strength: $\Pe\PGm$,
$\Pe\tauh$, $\PGm\tauh$, and $\tauh\tauh$, where $\tauh$ denotes a $\PGt$
lepton decaying hadronically. Events with two reconstructed muons are used to
derive corrections to the simulated samples. The visible mass of the $\PGt$
candidates, $m_{\text{vis}}$, computed as the invariant mass of the visible
decay products of the two $\PGt$ leptons, is used as a biased estimator of
$m_{\PGt\PGt}$ to extract the significance of the signal and set constraints
on $\and$ and $d_{\PGt}$. Tabulated results are provided in the HEPData record
for this analysis [34].

##  0.2 The CMS detector

The central feature of the CMS apparatus is a superconducting solenoid of 6m
internal diameter, providing a magnetic field of 3.8T. Within the solenoid
volume are a silicon pixel and strip tracker, a lead tungstate crystal
electromagnetic calorimeter (ECAL), and a brass and scintillator hadron
calorimeter (HCAL), each composed of a barrel and two endcap sections. Forward
calorimeters extend the pseudorapidity coverage provided by the barrel and
endcap detectors. Muons are measured in gas-ionization detectors embedded in
the steel flux-return yoke outside the solenoid. A more detailed description
of the CMS detector, together with a definition of the coordinate system used
and the relevant kinematic variables, is presented in Ref. [35].

Events of interest are selected using a two-tiered trigger system. The first
level, composed of custom hardware processors, uses information from the
calorimeters and muon detectors to select events at a rate of around 100kHz
within a fixed latency of 4 [36]. The second level, known as the high-level
trigger, consists of a farm of processors running a version of the full event
reconstruction software optimized for fast processing, and reduces the event
rate to around 1kHz before data storage [37].

##  0.3 Signal and background modeling

Signal samples corresponding to the elastic photon-fusion production of a
$\PGt$ lepton pair are generated using the gamma-UPC generator [38], which is
implemented in the v3.5 code [39, 40, 41]. The gamma-UPC generator derives
photon fluxes from electric dipole and charge form factors for protons and
ions, and includes realistic hadronic survival probabilities for them.

The photon fluxes used here are those derived from the elastic charge form
factor of the proton. Samples generated with the SuperChic 4 generator [42]
are used as a cross-check, as described in Section 0.7.

In this analysis, $\and$ and $d_{\PGt}$ are parameterized in using the
SMEFTsim package [43, 44], as suggested in Ref. [14]. The corresponding BSM
Lagrangian in the SM effective field theory (SMEFT) approach consists of two
dimension-6 operators modifying $\and$ and $d_{\PGt}$ at tree level:

$\mathcal{L}_{\text{BSM}}=\frac{C_{\PGt
B}}{\Lambda^{2}}\overline{L}_{L}\sigma^{\mu\nu}\tau_{R}HB_{\mu\nu}+\frac{C_{\PGt
W}}{\Lambda^{2}}\overline{L}_{L}\sigma^{\mu\nu}\tau_{R}\sigma^{i}HW^{i}_{\mu\nu}+\text{h.c.},$
(3)

where $\Lambda$ is the scale of BSM physics, $L_{L}$ represents the left-
handed $\PGt$ lepton doublet, $\tau_{R}$ corresponds to the right-handed
$\PGt$ lepton, $H$ is an SU(2) scalar doublet, $B_{\mu\nu}$ is the weak
hypercharge field, $W^{i}_{\mu\nu}$ represents the weak isospin field, and
$C_{\PGt B}$ and $C_{\PGt W}$ are the Wilson coefficients. Using the
SMEFTsim_general_alphaScheme_UFO model [43, 44], after electroweak symmetry
breaking, the $\PGt\PGt\PGg$ vertex is parameterized as:

$V_{\tau\tau\gamma}=ie\gamma^{\mu}-\frac{v\sqrt{2}}{\Lambda^{2}}\left[\operatorname{Re}\big{[}C_{\PGt\PGg}\big{]}+\operatorname{Im}\big{[}C_{\PGt\PGg}\big{]}i\gamma_{5}\right]\sigma^{\mu\nu}q_{\nu},$
(4)

where $C_{\PGt\PGg}=\cos\theta_{\mathrm{W}}C_{\PGt
B}-\sin\theta_{\mathrm{W}}C_{\PGt W}$, $v\approx 246\GeV$ is the Higgs vacuum
expectation value, and $\theta_{\mathrm{W}}$ is the weak mixing angle. Using
Eqs. (1) and (4), deviations from the SM predictions of $\and$ and $d_{\PGt}$
can be expressed as:

$\delta\and=\frac{2m_{\PGt}}{e}\frac{\sqrt{2}v}{\Lambda^{2}}\operatorname{Re}\big{[}C_{\PGt\PGg}\big{]}$
(5)

and

$\delta
d_{\PGt}=\frac{\sqrt{2}v}{\Lambda^{2}}\operatorname{Im}\big{[}C_{\PGt\PGg}\big{]}.$
(6)

For simplicity and without loss of generality, since $\delta\and$ and $\delta
d_{\PGt}$ only depend on the linear combination of $C_{\PGt W}$ and $C_{\PGt
B}$, we set $C_{\PGt W}$ to 0 in the signal simulation. The coefficient
$C_{\PGt B}$ alone is modified to simulate different $C_{\PGt\PGg}$ values,
which can be reinterpreted as linear combinations of $C_{\PGt W}$ and $C_{\PGt
B}$. The BSM contributions to the $\PZ\to\PGt\PGt$ vertex cancel for $C_{\PGt
W}=-\tan\theta_{\mathrm{W}}C_{\PGt B}$, but other values of $C_{\PGt W}$ would
result in variations in the Drell–Yan (DY) process prediction that are
negligible with respect to the systematic uncertainties of this background,
which are described in Section 0.9. The simulation of different anomalous
electromagnetic moments is performed using matrix element reweighting [45],
scanning over a range of $C_{\PGt B}$ values. In the simulation, we set
$\Lambda$ to 2, but the results were verified to be independent of the scale
of new physics. Since the anomalous electromagnetic moments of the $\PGt$
lepton depend on a linear combination of $C_{\PGt W}/\Lambda^{2}$ and $C_{\PGt
B}/\Lambda^{2}$, the analysis results can also be interpreted in this two-
dimensional plane. The form factor formalism of Eq. (1) and the SMEFT approach
are equivalent assuming $q^{2}=0$.

The gamma-UPC generator is also used to simulate the elastic photon-fusion
production of pairs of electrons, muons, and $\PW$ bosons, which are
backgrounds in the signal region (SR). Single- and double-dissociative
processes are not simulated but estimated from data as described later.

The 2.6.5 event generator is used to generate events originating from the
$\PZ/\PGg^{*}+\text{jets}$ and $\PW+\text{jets}$ processes, as well as from
diboson production with two leptons and two neutrinos in the final state.
These processes are simulated at next-to-leading order (NLO) with the FxFx jet
matching and merging [41]. The generator is also used for the simulation of
the minor contribution from other diboson production events, whereas 2.0 [46,
47, 48, 49, 50] is used for and single top quark production. The generators
are interfaced with 8.240 [51] to model parton showering and hadronization,
the underlying event activity, as well as the decay of the $\PGt$ leptons. The
parameters affecting the description of the underlying event are set to the
CP5 tune [52]. The NNPDF3.1 parton distribution function (PDF) set [53, 54,
55] at next-to-NLO accuracy is used for background simulations. Additional
$\Pp\Pp$ interactions per bunch crossing (pileup) are added to the simulated
samples with a distribution that is chosen to match that observed in the data.
Generated events are processed through a [56] simulation of the CMS detector.

##  0.4 Object reconstruction

The global event reconstruction (also called particle-flow event
reconstruction [57]) aims to reconstruct and identify each particle in an
event, with an optimized combination of all subdetector information. In this
process, the identification of the particle type (photon, electron, muon,
charged hadron, or neutral hadron) plays an important role in the
determination of the particle direction and energy. Photons are identified as
ECAL energy clusters not linked to the extrapolation of any charged particle
trajectory to the ECAL. Electrons are identified as a primary charged-particle
track and potentially many ECAL energy clusters corresponding to this track
extrapolation to the ECAL and to possible bremsstrahlung photons emitted along
the way through the tracker material. Muons are identified as tracks in the
central tracker consistent with either a track or several hits in the muon
system and associated with calorimeter deposits compatible with the muon
hypothesis. Charged hadrons are identified from the charged-particle tracks
that are not associated with electrons or muons. Finally, neutral hadrons are
identified as HCAL energy clusters not linked to any charged-hadron
trajectory, or as a combined ECAL and HCAL energy excess with respect to the
expected charged-hadron energy deposit.

The electron momentum is estimated by combining the energy measurement in the
ECAL with the momentum measurement in the tracker. The momentum resolution for
electrons with transverse momentum $\pt\approx 45\GeV$ from $\PZ\to\Pe\Pe$
decays ranges 1.6–5.0%. It is generally better in the barrel region than in
the endcaps, and also depends on the bremsstrahlung energy emitted by the
electron as it traverses the material in front of the ECAL [58, 59].

Muons are measured in the pseudorapidity range $\abs{\eta}<2.4$, with
detection planes made using three technologies: drift tubes, cathode strip
chambers, and resistive-plate chambers. The efficiency to reconstruct and
identify muons is greater than 96%. Matching muons to tracks measured in the
silicon tracker results in a relative resolution of 1% in the barrel and 3% in
the endcaps for muons with up to 100 [60].

Hadronic lepton decays () are reconstructed from jets, using the hadrons-plus-
strips algorithm [61], which combines 1 or 3 tracks with energy deposits in
the calorimeters clustered as strips. Neutral pions are reconstructed as
strips with dynamic size in $\eta$-$\phi$ (where $\phi$ is the azimuthal angle
in radians) from reconstructed electrons and photons, where the strip size
varies as a function of the $\pt$ of the electron or photon candidate. The
candidates are reconstructed in four possible decay modes (DMs): a single
charged hadron ($\mathrm{h}^{\pm}$), a charged hadron with at least one
neutral pion ($\mathrm{h}^{\pm}+\PGpz$(s)), three charged hadrons with no
neutral pion ($\mathrm{h}^{\pm}\mathrm{h}^{\mp}\mathrm{h}^{\pm}$), or 3
charged hadrons with at least one neutral pion
($\mathrm{h}^{\pm}\mathrm{h}^{\mp}\mathrm{h}^{\pm}+\PGpz$(s)). To distinguish
genuine decays from jets originating from the hadronization of quarks or
gluons, and from electrons or muons, the DeepTau algorithm [62] is used.
Information from all individual reconstructed particles near the axis is
combined with properties of the $\tauh$ candidate in the event. The tight
working point is used to separate $\tauh$ candidates from jets; its efficiency
is about 65% for $\tauh$ with $20<\pt<100\GeV$ [62]. The loosest working
point, used in the background estimation procedure, has an efficiency above
98%. The rate of a jet to be misidentified as by the DeepTau algorithm depends
on the and quark flavor of the jet. In simulated events from boson production
in association with jets, it has been estimated to be below the percent level
for the tight working point. The misidentification rate for electrons (muons)
is 2.60 (0.03)% for a genuine identification efficiency of 80 ($>$99)%.

The missing transverse momentum vector is computed as the negative vector sum
of all the particle-flow candidates in an event, and its magnitude is denoted
as [63]. The is modified to correct the energy scale of the reconstructed jets
in the event. Anomalous high-events can be due to a variety of reconstruction
failures, detector malfunctions or noncollision backgrounds. Such events are
rejected by event filters that are designed to identify more than 85–90% of
the spurious high-events with a mistagging rate less than 0.1% [63].

Tracks with $\pt>0.5\GeV$ and $\abs{\eta}<2.5$ are used in this analysis to
measure the hadronic activity in the event. The silicon tracker used in 2016
measured charged particles within the range $\abs{\eta}<2.5$. For nonisolated
particles of $1<\pt<10\GeV$ and $\abs{\eta}<1.4$, the track resolutions were
typically 1.5% in and 25–90 (45–150)in the transverse (longitudinal) impact
parameter [64]. At the start of 2017, a new pixel detector was installed [65];
the upgraded tracker measured particles up to $\abs{\eta}<3.0$ with typical
resolutions of 1.5% in and 20–75in the transverse impact parameter [66] for
nonisolated particles of $1<\pt<10\GeV$.

During the 2016–2017 data taking, a gradual shift in the timing of the inputs
of the ECAL first-level trigger in the region at $\abs{\eta}>2.0$ caused a
specific trigger inefficiency [67]. For events containing an electron (a jet)
with larger than $\approx$50($\approx$100), the efficiency loss is
$\approx$10–20% in the region $2.5<\abs{\eta}<3.0$, depending on , $\eta$, and
time. Correction factors were computed from data and applied to the acceptance
evaluated by simulation.

##  0.5 Event selection

Events in the SR are selected in the $\Pe\PGm$, $\Pe\tauh$, $\PGm\tauh$, and
$\tauh\tauh$ final states, which account for 94% of the possible final states
with a pair of $\PGt$ leptons. The $\PGm\PGm$ final state is used to derive
corrections to:

  * •

the pileup track density in all simulations, described in Section 0.6.1;

  * •

the hard scattering track multiplicity in the DY and diboson simulations,
described in Section 0.6.2;

  * •

the elastic photon-fusion simulations to include dissociative contributions,
described in Section 0.7;

  * •

the acoplanarity distribution in the DY simulation, described in Section
0.8.1.

In the $\Pe\PGm$ final state, events are recorded with a combination of
triggers requiring an electron and a muon, with a threshold of 23for the
leading lepton, and 8 (12)for the subleading muon (electron). Offline, the
leading (subleading) lepton is required to have $\pt>24$ (15). The subleading
lepton threshold is chosen to reduce background processes with misidentified
jets. In the $\Pe\tauh$ ($\PGm\tauh$) final states, events are recorded with
single-electron (single-muon) triggers with $\pt$ thresholds ranging between
25 and 32 (24 and 27)depending on the data-taking year. Additional events with
or as low as 24 (19)and $\abs{\eta}<2.1$ can be selected using triggers
requiring also the presence of a $\tauh$ candidate with $\pt>20$ (27)and
$\abs{\eta}<2.1$. In these two final states, the $\tauh$ is required to be
above 30offline and the background from $\PW+\text{jets}$ is reduced by
requiring the transverse mass $\mT$ between the or and $\ptvecmiss$ to be less
than 75. The events in the $\tauh\tauh$ final state are selected with triggers
requiring the presence of two loosely isolated $\tauh$ candidates with
$\pt>35$–40, depending on the data-taking year, and $\abs{\eta}<2.1$. The
offline thresholds are $\pt>40\GeV$ and $\abs{\eta}<2.1$ for all the data-
taking periods. A trigger requiring a single muon with $\pt>24$–27is used to
select events in the $\PGm\PGm$ final state. Offline, the leading (subleading)
muon is required to have $\pt>26$–29 (10)and $\abs{\eta}<2.4$. The invariant
mass of the two muons, $m_{\PGm\PGm}$, must be greater than 50.

In all final states, the two objects are required to be well-identified and
isolated, to have opposite sign (OS) charge, to be separated from each other
by $\Delta R=\sqrt{\smash[b]{(\Delta\eta)^{2}+(\Delta\phi)^{2}}}>0.5$, and to
satisfy $\abs{d_{z}(\Pell,\Pell^{\prime})}<0.1\cm$, where
$\abs{d_{z}(\Pell,\Pell^{\prime})}$ is the difference in the longitudinal
impact parameters of the two objects. In each final state, we veto events that
have additional identified and isolated electrons or muons, such that one
event can enter at most one final state.

The criteria listed above and summarized in Table 0.5 constitute the baseline
selection. Additional requirements are applied to define the SR or the control
regions (CRs) used to derive corrections. Additional requirements increasing
the signal-to-background ratio in the SR using the characteristics of the
exclusive process signature are described in the following paragraphs.

Baseline selection criteria used in the different final states. The electrons,
muons, and $\tauh$ are required to be well identified and isolated. The and
pseudorapidity ranges correspond to different sets of triggers, and different
data-taking periods.  $\Pe\PGm$ $\Pe\tauh$ $\PGm\tauh$ $\tauh\tauh$ $\PGm\PGm$
$\pt^{\Pe}$ () $>15/24$ $>25$–33 $\abs{\eta^{\Pe}}$ $<2.5$ $<2.1$–2.5
$\pt^{\PGm}$ () $>24/15$ $>21$–29 $>26$–29/10 $\abs{\eta^{\PGm}}$ $<2.4$
$<2.1$–2.4 $<2.4$ $\pt^{\tauh}$ () $>30$–35 $>30$–32 $>40$
$\abs{\eta^{\tauh}}$ $<2.1$–2.3 $<2.1$–2.3 $<2.1$ $m_{\PGm\PGm}$ () $>50$ OS
yes yes yes yes yes $\abs{d_{z}(\Pell,\Pell^{\prime})}$ (cm) $<0.1$ $<0.1$
$<0.1$ $<0.1$ $<0.1$ $\Delta R(\Pell,\Pell^{\prime})$ $>0.5$ $>0.5$ $>0.5$
$>0.5$ $>0.5$ $\mT(\Pe/\PGm\pt,\ptvecmiss)$ () $<75$ $<75$

The acoplanarity, defined as:

$A=1-\abs{\Delta\phi(\Pell,\Pell^{\prime})}/\pi,$ (7)

is typically small for events from the elastic photon-fusion production, for
which the leptons are produced back-to-back. In the case of $\PGt$ leptons,
$A$ is computed using the visible decay products, which are closely aligned
with the original $\PGt$ lepton before decay, especially at high . In the SR,
the events are required to have $A<0.015$. This requirement has a signal
efficiency ranging from 95% in the $\Pe\PGm$ final state to $>99\%$ in the
$\tauh\tauh$ final state.

Elastic signal events are also characterized by a low track multiplicity
($N_{\text{tracks}}$) at the dilepton vertex. The dilepton vertex position
along the $z$ axis is calculated as the average $z$ of the two objects. The
variable $N_{\text{tracks}}$ is estimated by counting the number of tracks
that have a $z$ coordinate within a 0.1wide window centered around the
dilepton vertex, removing the tracks that were used to build the lepton
candidates. The window size is chosen such that about 50% of the signal events
have $N_{\text{tracks}}=0$. This fraction is larger far from the beamspot
position ($z_{\text{BS}}$), where the contribution from pileup tracks is less
important. The SRs are defined with $N_{\text{tracks}}=0$ or 1. About 25% of
the signal events have $N_{\text{tracks}}=1$.

A low pileup activity in the tracker, corresponding to low $N_{\text{tracks}}$
values at the interaction vertex, impacts the identification of electrons,
muons, and $\tauh$ candidates, typically leading to a higher identification
and isolation efficiency. Scale factors are computed to match the simulated
efficiencies to the observed efficiencies for objects surrounded by a low
track activity at their vertex.

##  0.6 Corrections to the simulated track multiplicity

The simulations are corrected to accurately describe $N_{\text{tracks}}$. We
derive corrections for two sources of tracks: tracks originating from the
pileup (Section 0.6.1) and tracks originating from the hard scattering and
underlying event activity (Section 0.6.2). They are derived from the dimuon CR
and applied to simulations in the SRs.

###  0.6.1 Tracks originating from pileup

While the pileup multiplicity in simulation is reweighted to match that
observed in the data, the pileup track density along the $z$ axis in
simulation needs to be corrected to better describe $N_{\text{tracks}}$ at the
dilepton vertex. First, the simulations are corrected, independently for each
data-taking period, such that the $z_{\text{BS}}$ and beamspot width
($\sigma_{\text{BS}}$) distributions are similar to the observed ones. In data
collected between 2016–2018, $z_{\text{BS}}$ ($\sigma_{\text{BS}}$) is
approximately in the range between -0.5 and +1.2(3.1 and 4.2). The pileup
tracks are redistributed accordingly along the $z$ axis, as explained below.
Simulated events originally all have $\sigma_{\text{BS}}=3.5\cm$, whereas the
observed $\sigma_{\text{BS}}$ distribution is spread between about 3.0 and
3.8. Simulated events are assigned randomly corrected beamspot properties,
$z_{\text{BS}}^{\text{corr}}$ and $\sigma_{\text{BS}}^{\text{corr}}$,
following the corresponding profiles in data. The $z$ positions of pileup
tracks, identified as such using generator-level information, are corrected
for beamspot effects:

$z^{\text{corr}}=z_{\text{BS}}^{\text{corr}}+\frac{\sigma_{\text{BS}}^{\text{corr}}}{\sigma_{\text{BS}}^{\text{sim}}}\left(z-z_{\text{BS}}^{\text{sim}}\right).$
(8)

A residual correction to the pileup track density is derived in a
$\PZ\to\PGm\PGm$ CR, defined with the selection criteria for the $\PGm\PGm$
final state detailed in Section 0.5, with the additional requirement that
$m_{\PGm\PGm}$ is within 15of the $\PZ$ boson mass, $m_{\PZ}$. The track
multiplicity is counted in data and in simulation in windows of 0.1along the
$z$ axis, for windows centered at least 1away from the reconstructed dimuon
vertex, as illustrated in Fig. 2. This last requirement ensures that all
tracks in the windows come from pileup and not from the hard-scattering
interaction. Weights are derived as functions of the pileup track multiplicity
and window $z$ position so as to match the $N_{\text{tracks}}$ distribution of
the observed events. Distributions of the number of pileup tracks,
$N_{\text{tracks}}^{\text{PU}}$, are shown for the measured events, and for
the uncorrected and beamspot-corrected simulations in Fig. 3, for windows with
different $z$ positions. About one third of the windows situated at the
beamspot center do not have any pileup track. This fraction reaches about 50
(80)% one (two) beamspot width(s) away from the beamspot center. The
correction to the simulated event weight is in the range 0.89–1.00 (0.93–1.16)
for $N_{\text{tracks}}^{\text{PU}}=0$ ($N_{\text{tracks}}^{\text{PU}}=1$) over
the whole $z$ axis and data-taking periods. It is evaluated at the z position
of the selected dilepton candidate.

Figure 2: Schematic view of the 0.1wide windows probed along the $z$ axis to
derive corrections to the pileup track density in simulation. Windows within
1from the dimuon vertex, illustrated with the red box, are discarded so as not
to count tracks from the hard-scattering interaction. The green curve
indicates the probability distribution of z-coordinates for PU vertices in the
beamspot.

Figure 3: Distribution of $N_{\text{tracks}}^{\text{PU}}$ in windows of
0.1width along the $z$ axis for the observed events (black), uncorrected
simulation (red), and beamspot-corrected simulation (blue) for data collected
in 2017. The windows shown here are located at the beamspot center (upper
left), and one (upper right) or two (lower) beamspot widths away from the
center. The ratio of beamspot-corrected simulation to observation (lower
plots) is taken as a residual correction to the simulations. The last bin
includes the overflow. Similar distributions and corrections are derived
independently for the other data-taking periods.

###  0.6.2 Tracks originating from the hard scattering and underlying event
activity

The only major background in the SR estimated using simulation is the DY
production of a lepton pair. The number of tracks coming from the hard
scattering interaction is not well described in the simulation, and
corrections are derived from the same $\PZ\to\PGm\PGm$ CR, in which
$m_{\PGm\PGm}$ is required to be within 15of $m_{\PZ}$, by comparing the
number of tracks around the reconstructed dimuon vertex between data and
simulation.

As shown in Fig. 4, the simulation is split into several components depending
on the number of tracks with $\pt>0.5\GeV$ and $\abs{\eta}<2.5$, identified as
originating from the hard scattering interaction using generator-level
information, and not matched to the selected muons. The total number of
reconstructed tracks, excluding the two muon tracks, is the sum of these
tracks and those associated with pileup interactions:
$N_{\text{tracks}}=N_{\text{tracks}}^{\text{HS}}+N_{\text{tracks}}^{\text{PU}}$.
Events from the exclusive photon-fusion production of a pair of muons or $\PW$
bosons, which contribute significantly only at low $N_{\text{tracks}}$, are
estimated from simulation and subtracted.

The pileup track density is first corrected with the weights described in
Section 0.6.1, whereas the reweighting factors for each DY component with a
given number of $N_{\text{tracks}}^{\text{HS}}$ is determined iteratively by
matching the simulation to the observed data, starting from events with
$N_{\text{tracks}}=0$, to which only the simulated component with
$N_{\text{tracks}}^{\text{HS}}=0$ contributes. The corrections are determined
separately for the different data-taking years. We checked that the
requirement for $m_{\PGm\PGm}$ be compatible with $m_{\PZ}$ introduces a bias
for events with large dilepton mass, which is negligible with respect to other
systematic uncertainties described in Section 0.9. The event weight correction
is as low as $0.61\pm 0.04$ ($0.76\pm 0.04$) for
$N_{\text{tracks}}^{\text{HS}}=0$ ($N_{\text{tracks}}^{\text{HS}}=1$). This
correction is also applied to the minor diboson background with 2 leptons and
2 neutrinos in the final state, which has the same generator settings as the
DY simulation.

Figure 4:  Distribution of the number of reconstructed tracks in a 0.1wide
window in the $z$ direction, centered on the dimuon reconstructed vertex, for
acoplanarity $A<0.015$, in data collected in 2017. The DY simulation is split
into several components based on the number of reconstructed tracks
originating from the hard interaction. The red line shows the simulation
before the correction. The black points show the observed data after
subtracting the expected background contribution from the
${\PGg\PGg\to\PGm\PGm}$ and ${\PGg\PGg\to\PW\PW}$ processes (dashed orange
line). The last bin includes the overflow. Similar distributions and
corrections are derived independently for the other data-taking periods. The
ratios between the observed data, from which the exclusive background
contributions have been subtracted, and the DY prediction before (red) and
after the corrections (black), are shown in the lower panel. The region with
the selection requirement $N_{\text{tracks}}=0$ or 1 used in the SR is
highlighted with the orange shaded area in the lower panel.

##  0.7 Signal estimation

The signal is modeled using events of elastic $\Pp\Pp$ scattering. The
semidissociative and fully dissociative production modes are much less likely
to satisfy the acoplanarity and $N_{\text{tracks}}$ requirement, but they
contribute a nonnegligible number of events to the SRs because of their larger
cross sections.

The prediction from the elastic photon-fusion production is rescaled to
account for the semidissociative and fully dissociative contributions. The
scaling factor comes from data observed in a $\PGm\PGm$ CR, defined with the
baseline selection summarized in Table 0.5 and $N_{\text{tracks}}=0$ or 1 and
$A<0.015$ as additional requirements. To perform this measurement, the
inclusive backgrounds, dominated by the DY production of a muon pair, are
estimated together from observed events. Their $m_{\PGm\PGm}$ distribution is
taken from observed events with $3<N_{\text{tracks}}<7$. These thresholds
ensure that the photon-fusion contributions are negligible, while preserving
the $m_{\PGm\PGm}$ distribution. The compatibility of the $m_{\PGm\PGm}$
distributions in inclusive events with $N_{\text{tracks}}<2$ and
$3<N_{\text{tracks}}<7$ was verified in simulation. The $m_{\PGm\PGm}$
distribution is scaled so that the inclusive background normalization in the
range $\abs{m_{\PGm\PGm}-m_{\PZ}}<15\GeV$ matches the event count observed in
the same range. The ${\PGg\PGg\to\PGm\PGm}$ prediction for the elastic
production is then rescaled to match the difference between the experimental
data and the inclusive backgrounds away from the $\PZ$ peak region, as shown
in Fig. 5.

Assuming no dependence on the dilepton mass, $m_{\Pell\Pell}$, the scaling
factor to include the dissociative contributions is $2.70\pm 0.04$ ($2.71\pm
0.10$) for $N_{\text{tracks}}=0$ (1), where the uncertainty is statistical
only. This is compatible with the value predicted by the SuperChic generator,
using the same acoplanarity requirement $A<0.015$. The scale factor is also
fitted as a linear function of $m_{\PGm\PGm}$, which demonstrates a better
$\chi^{2}$, and, in that case, is equal to $2.36+0.0034~{}m_{\PGm\PGm}/\GeVns$
($2.38+0.0032~{}m_{\PGm\PGm}/\GeVns$) for $N_{\text{tracks}}=0$ (1). The fit
probability is also better with a linear function for the predictions of the
SuperChic generator, and a comparable parameterization is obtained. A linear
dependency of the scaling factor with $m_{\Pell\Pell}$ is assumed as a nominal
correction in the SRs, whereas the flat value is considered as a systematic
uncertainty as detailed in Section 0.9. Elastic events in the SR, coming from
the ${\PGg\PGg\to\PGt\PGt}/\Pe\Pe/\PGm\PGm/\PW\PW$ simulations, are all
rescaled with this correction as a function of the dilepton or diboson mass,
to include semidissociative and fully dissociative contributions.

Figure 5:  Measurement of the scale factor for the elastic exclusive signal in
$\PGm\PGm$ events for $N_{\text{tracks}}=0$ (left) or 1 (right), and
$A<0.015$. The shape of the inclusive background (blue) is estimated from the
observed data in the $3\leq N_{\text{tracks}}\leq 7$ sideband, and rescaled to
fit the observed data in $75<m_{\PGm\PGm}<105\GeV$. The scale factor is fitted
in the lower ratio panel with constant (red) and linear (blue) functions. The
vertical error bars indicate the statistical uncertainty in the number of
observed events.

##  0.8 Background estimation

In the SR, the dominant backgrounds are the DY production of a lepton pair and
processes with a jet misidentified as an electron, a muon, or a $\tauh$
candidate, collectively denoted as “jet mis-ID background”. This background is
dominantly composed of $\PW+\text{jets}$ and quantum chromodynamics (QCD)
multijet events. The photon-fusion production of a pair of $\PW$ bosons is
reduced with the acoplanarity requirement, and is significant only in the
$\Pe\PGm$ final state, where its expected contribution is about 4% of the
exclusive di-production. The photon-fusion production of an electron (muon)
pair is about 50 (4)% of the expected signal contribution in the $\Pe\tauh$
($\PGm\tauh$) final state. When there is no exclusivity requirement relying on
$A$ or $N_{\text{tracks}}$, the $\ttbar$, inclusive diboson, and single top
quark processes also contribute.

###  0.8.1 Drell–Yan background

The DY background is estimated using simulation. The corrections described in
Section 0.6 are applied to improve the description of
$N_{\text{tracks}}^{\text{PU}}$ and $N_{\text{tracks}}^{\text{HS}}$. The
acoplanarity is observed to be mismodeled by simulation in a $\PGm\PGm$ CR,
defined with the baseline selection criteria from Section 0.5 and further
enriched in DY events by requiring $\abs{m_{\PGm\PGm}-m_{\PZ}}<15\GeV$. An
acoplanarity correction, measured in two-dimensional bins of the of both
muons, is derived in this CR by matching the simulation to the observed data.
The correction is obtained by fitting the ratio of data-to-simulation with a
polynomial for $A<0.35$. For illustrative purposes, the distributions and
correction obtained for all muon bins merged together are shown in Fig. 6,
using data collected in 2018. This acoplanarity correction is applied to
simulated DY events in the SR on the basis of the generated $\PGt$ lepton
before their decays.

Figure 6:  Acoplanarity distribution for the observed events and in DY
simulation before correction, in the 2018 data-taking period. The background
prediction is normalized to match the observed yield and only the statistical
uncertainty is shown. The data-to-simulation ratio is fitted with a polynomial
to obtain the correction. The selection criterion $A<0.015$ used in the SR is
highlighted with the orange shaded area in the lower panel.

The size of the DY simulation in the final state of a pair of electrons,
muons, or $\PGt$ leptons is limited, and large statistical uncertainties arise
when requiring $N_{\text{tracks}}=0$ or 1. To reduce this effect, the DY
contribution in the SR is estimated from simulated events with
$N_{\text{tracks}}<10$, which constitute about 10% of the total number of DY
events. The upper bound ensures a similar event topology as in the SR,
ensuring that no bias is introduced to the $m_{\PGt\PGt}$ distributions, as
verified in simulation. The distribution derived with $N_{\text{tracks}}<10$
is scaled to predict the DY events with $N_{\text{tracks}}=0$ or 1. The scale
factors are derived in the $\Pe\PGm$ final state with a simulation with a
large number of $\PZ/\PGg*\to\PGt\PGt$ events with only leptonic $\PGt$
decays. All the corrections described in Section 0.6 are applied to predict
the DY yield in the $\Pe\PGm$ final state with $N_{\text{tracks}}=0$, 1, or
$<10$. The scale factors are equal to $(2.48\pm 0.05)\%$ and $(5.10\pm
0.07)\%$, for $N_{\text{tracks}}=0$ and 1, respectively. In the $\Pe\PGm$
final state, the large simulation with leptonic $\PGt$ decays is used instead
of reweighting events with $N_{\text{tracks}}<10$.

###  0.8.2 Backgrounds with misidentified jets in the $e\tau_{h}$ and
$\mu\tau_{h}$ final states

The mis-ID background with a jet misidentified as a $\tauh$ candidate is
estimated from the experiment. In the $\Pe\tauh$ and $\PGm\tauh$ final states,
the dominant processes contributing to the mis-ID background are
$\PW+\text{jets}$ and QCD multijet events. The jet misidentification factors
(MFs) depend on the process because of different topologies and jet flavors,
and they are therefore measured in two different CRs. A CR enriched in
$\PW+\text{jets}$ events is built by requiring $\mT>75\GeV$, whereas a QCD-
enriched CR is obtained by requiring the two objects to have same-sign (SS)
charge. There is no requirement on $N_{\text{tracks}}$ at this stage. In these
CRs, the MFs are measured for each data-taking period, separately for each
$\tauh$ DM ($\text{DM}^{\tauh}$), by taking the ratio of events in which the
$\tauh$ candidate passes the SR $\tauh$ identification criteria, to events in
which the $\tauh$ candidate fails these criteria but passes looser ones.
Events where the $\tauh$ candidate is genuine or corresponds to a
misidentified electron or muon are evaluated using simulation and subtracted
from the CR. The MFs are fitted as functions of the $\tauh$ and range from
about 0.03 for the $\mathrm{h}^{\pm}\mathrm{h}^{\mp}\mathrm{h}^{\pm}+\PGpz$(s)
DM, to about 0.40 for the $\mathrm{h}^{\pm}$ DM. The MFs are measured in
events recorded with single-electron or -muon triggers. The triggers that
select $\tauh$ candidates in addition to electrons or muons, called “cross-
triggers”, apply loose isolation requirements to the $\tauh$ candidates at
trigger level, such that the MFs are modified. The MF measured in events
selected with cross-triggers is measured to be 1.7 times higher than the
average MF in events selected with single-lepton triggers, and this factor is
applied as a multiplicative correction to the MFs for events recorded with a
cross-trigger.

The track activity around the di-$\PGt$ system impacts the calculation of the
$\tauh$ isolation, which enters the $\tauh$ identification discriminator, such
that the MFs are expected to depend on $N_{\text{tracks}}$. For each $\tauh$
DM, the MFs determined for given ranges of $N_{\text{tracks}}$ are divided by
the average MF measured inclusively in $N_{\text{tracks}}$ to obtain a
multiplicative $N_{\text{tracks}}$-dependent correction,
$\omega(N_{\text{tracks}},\text{DM}^{\tauh})$, to the - and decay-mode-
dependent MFs determined previously. The corrected MFs can then be written,
for each CR, as:

$\text{MF}^{\text{corr}}(\pt^{\tauh},\text{DM}^{\tauh},N_{\text{tracks}})=\text{MF}(\pt^{\tauh},\text{DM}^{\tauh})\omega(N_{\text{tracks}},\text{DM}^{\tauh}).$
(9)

The $\omega(N_{\text{tracks}},\text{DM}^{\tauh})$ corrections are measured for
the combination of the different data-taking years. Because statistical
uncertainties are large at low $N_{\text{tracks}}$,
$\omega(N_{\text{tracks}},\text{DM}^{\tauh})$ is fitted with an exponential
function for $N_{\text{tracks}}$ less than 10 or 15, depending on the $\tauh$
DM, as shown in Fig. 7 for the $\mathrm{h}^{\pm}+\PGpz$(s) DM in the
$\Pe\tauh$ final state.

Figure 7: Multiplicative $N_{\text{tracks}}$-dependent corrections to the
$\tauh$ MFs, $\omega(N_{\text{tracks}},\text{DM}^{\tauh})$, in the $\Pe\tauh$
final state, in the high-$\mT$ (left) and SS (right) CRs, for the
$\mathrm{h}^{\pm}+\PGpz$(s) DM. The purple shaded area corresponds to the fit
uncertainty. The vertical error bars indicate the statistical uncertainty in
the MF correction factors measured in individual $N_{\text{tracks}}$ ranges.

The total MF, $\text{MF}^{\text{tot}}$, is computed as a weighted average of
the MFs determined in the high-$\mT$ and SS CRs, according to the expected
relative fraction of $\PW+\text{jets}$ and QCD multijet events:

$\text{MF}^{\text{tot}}=x^{\text{QCD}}\text{MF}^{\text{corr}}_{\text{SS}}+(1-x^{\text{QCD}})\text{MF}^{\text{corr}}_{\text{high-\mT}},$
(10)

where $x^{\text{QCD}}=N^{\text{QCD}}/(N^{\text{QCD}}+N^{\PW})$. The number of
$\PW+\text{jets}$ events passing the baseline selection, $N^{\PW}$, is
determined from simulations, whereas the same quantity for QCD multijet
events, $N^{\text{QCD}}$, is taken as the difference between data and
simulated predictions for other processes in the SS CR.

To estimate the mis-ID background in the SR in the $\Pe\tauh$ and $\PGm\tauh$
final states, events passing the SR selection with the exception that the
$\tauh$ candidate fails the $\tauh$ nominal identification but passes looser
criteria, are reweighted with $\text{MF}^{\text{tot}}$. Contributions from
events with real $\tauh$, electrons, or muons are estimated from simulation
and subtracted from the anti-isolated region.

###  0.8.3 Backgrounds with misidentified jets in the $\tau_{h}\tau_{h}$ final
state

In the $\tauh\tauh$ final state, the mis-ID background is mostly composed of
QCD multijet events and the MFs are determined in a single CR where the
$\tauh$ candidates have SS charge. The MFs for the leading (subleading)
$\tauh$ are measured in events in which the subleading (leading) $\tauh$
passes or fails the tight identification criteria. They are in the range
0.05–0.38 depending on the $\tauh$ DM.

The $N_{\text{tracks}}$-dependent correction is derived in a similar way as in
the $\Pe\tauh$ and $\PGm\tauh$ final states, and is between 1.8 and 2.3 for
events with $N_{\text{tracks}}=0$. In the $\tauh\tauh$ state, the number of
events selected at low $N_{\text{tracks}}$ is larger because of the inversion
of the isolation of both $\tauh$ candidates, and contributions from exclusive
dijet production become significant in the lowest $N_{\text{tracks}}$ bins,
such that no fit is performed to the
$\omega(N_{\text{tracks}},\text{DM}^{\tauh})$ distributions. The values for
$N_{\text{tracks}}=0,1$ are used without constraints events with
$N_{\text{tracks}}>1$.

In the SR, events in which the leading (subleading) $\tauh$ is a misidentified
jet are estimated by scaling by the MFs events where the leading (subleading)
$\tauh$ is anti-isolated and the subleading (leading) $\tauh$ is isolated.
These contributions include events where both the leading and subleading
$\tauh$ candidates are misidentified jets. To remove the double counting,
events with two misidentified jets are estimated by scaling events where the
two $\tauh$ candidates are anti-isolated with the product of the MFs for each
object, and are subtracted.

###  0.8.4 Backgrounds with misidentified jets in the $e\mu$ final state

In the $\Pe\PGm$ final state, the mis-ID background with a jet misidentified
as an electron or a muon, dominantly composed of $\PW+\text{jets}$ and QCD
multijet events, is estimated from events where the electron and muon have SS
charge. Events passing the SR selection with the exception that the electron
and muon have SS charge are reweighted with a two-dimensional scale factor
dependent on the electron and muon . This factor is measured by taking the
ratio of OS-to-SS events where the muon isolation is inverted, subtracting
nonjet contributions estimated from simulation. It is in the range 1.4–2.1,
depending on the lepton .

A bias is introduced by the inversion of the muon isolation, which modifies
the background composition, and a correction to the scale factor is measured
by taking the ratio of the scale factors measured in events where the electron
identification is inverted, and the muon isolation is inverted or not. The
correction ranges between 1.0 and 2.7 depending on the lepton .

A multiplicative correction for the dependency of the scale factor on
$N_{\text{tracks}}$ is measured by fitting the ratio of the scale factors in
exclusive ranges of $N_{\text{tracks}}$ over the average inclusive scale
factor. At $N_{\text{tracks}}=0$, the correction is about $0.89\pm 0.07$,
where the uncertainty is mostly from statistical origin.

Although the normalization of the jet mis-ID background is estimated with the
procedure described above, its $m_{\text{vis}}$ distribution is taken from SS
data events with $N_{\text{tracks}}<10$, from which backgrounds other than the
jet mis-ID component are subtracted using their simulated expectations. This
method reduces statistical fluctuations in the $m_{\text{vis}}$ templates.

###  0.8.5 Other backgrounds

Minor contributions from diboson events to the SR, mostly $\PW\PW$ events, are
estimated from simulation. The exclusive ${\PGg\PGg\to\PW\PW}$,
${\PGg\PGg\to\Pe\Pe}$, and ${\PGg\PGg\to\PGm\PGm}$ events, which contribute
mostly to the $\Pe\PGm$, $\Pe\tauh$, and $\PGm\tauh$ final states,
respectively, are estimated from the elastic production simulation and
rescaled with the same correction as the signal samples to include the
dissociative components.

The agreement between data and predictions is checked in events passing the
selection without the $N_{\text{tracks}}$ and acoplanarity requirements. The
single top quark, $\ttbar$, and Higgs boson processes, which do not contribute
to the SR with $N_{\text{tracks}}<2$, are included using their prediction from
simulation.

##  0.9 Systematic uncertainties

Systematic uncertainties are considered as nuisance parameters in the
statistical procedure to extract the signal significance, the signal strength
$\mu$, and constraints on $\and$ and $d_{\PGt}$. They are treated with either
Gaussian (shape uncertainties) or log-normal (normalization uncertainties)
function priors included in the likelihood function. The signal strength $\mu$
is defined as the ratio between the observed cross section of the
${\PGg\PGg\to\PGt\PGt}$ process to the theoretical cross section predicted by
the gamma-UPC generator for the elastic production component corrected by the
multiplicative experimental scaling to include dissociative contributions.

The integrated luminosities for the 2016, 2017, and 2018 data-taking years
have 1.2–2.5% individual uncertainties [68, 69, 70], whereas the overall
uncertainty for the 2016–2018 period is 1.6%. The uncertainty in the DY
(diboson production) theoretical cross section is 2 (5)% [71].

The uncertainties in the electron and muon identification, isolation, and
triggering efficiencies are up to 2% per object. Uncertainties related to the
$\tauh$ trigger efficiency depend on $\text{DM}^{\tauh}$ and are up to 5%.
Several uncertainties, stemming from statistical and systematic sources in the
measurement, affect the $\tauh$ identification, with different correlations
across data-taking years and various dependencies on the decay mode and of the
$\tauh$ candidate. This uncertainty is up to 5% in the range considered in
this analysis, and propagates directly to the measured signal strength.
Electrons and muons misidentified as $\tauh$ candidates, coming mostly from
the $\PZ/\PGg*\to\Pe\Pe$ and $\PZ/\PGg*\to\PGm\PGm$ processes, respectively,
have normalization uncertainties on the order of 10%. The uncertainty in the
$\tauh$ energy scale is up to 1.2%, whereas electrons (muons) misidentified as
$\tauh$ have energy scale uncertainties up to 5 (1)%. Multiplicative
corrections to the identification and isolation scale factors to account for
the low-$N_{\text{tracks}}$ environment are derived for $\tauh$ candidates and
electrons, as well as electrons and muons misidentified as $\tauh$ candidates.
Scale factors for muons do not need to be adjusted for the
low-$N_{\text{tracks}}$ environment. The related uncertainties are 2.1 and
2.0% for $\tauh$ candidates and electrons, and 22 and 15% for electrons and
muons misidentified as $\tauh$ candidates, respectively.

The uncertainty in the $N_{\text{tracks}}^{\text{PU}}$ correction amounts to
2%, which covers the effects of the beamspot width and position correction.
The uncertainty in the $N_{\text{tracks}}^{\text{HS}}$ correction is taken to
have the same magnitude as the relative fraction of
${\PGg\PGg\to\PGm\PGm}/\PW\PW$ events in the CR where the correction is
derived, and it amounts to 6.5 (1.5)% in the $N_{\text{tracks}}=0$ (1)
category. The statistical uncertainty in the correction is negligible with
respect to this systematic component. This uncertainty contributes a 7% effect
in the measured signal strength as shown in Fig. 8 (second line) and is among
the leading systematic uncertainties in this analysis.

The correction to the acoplanarity distribution in simulated DY events leads
to a 5% normalization uncertainty, arising from the finite granularity used to
bin the correction as a function of the of the two leptons. An uncertainty of
2.0 (1.4)% is associated with the estimation of the DY events in the SR with
$N_{\text{tracks}}=0$ (1) from simulated events with $N_{\text{tracks}}<10$.
Uncertainties arising from the renormalization and factorization scales, as
well as from the PDF set, are also taken into account for the DY simulation,
after propagation to the acoplanarity correction.

The uncertainty in the predictions of the jet mis-ID background in final
states with $\tauh$ candidates comes from several sources:

  * •

statistical uncertainty in the inclusive MFs, growing linearly to 50% for
$\pt^{\tauh}=300\GeV$, independently for each DM and data-taking year;

  * •

statistical uncertainty in the $N_{\text{tracks}}$ correction, amounting to 18
(16, 6)% in the $\Pe\tauh$ ($\PGm\tauh$, $\tauh\tauh$) SRs;

  * •

systematic uncertainty in the inclusive MFs, coming from the inversion of
selection requirements to define the CRs, reaching up to 10%;

  * •

systematic uncertainty in the calculation of the relative fraction of
$\PW+\text{jets}$ and QCD multijet events in the $\Pe\tauh$ and $\PGm\tauh$
SRs, leading to a 9% normalization effect;

  * •

systematic uncertainty in the $N_{\text{tracks}}$ correction, arising from the
choice of the fit function and fit range, with an effect up to 10% for
$N_{\text{tracks}}=0$.

Statistical uncertainties in the OS-to-SS scaling factor used to evaluate the
mis-ID background in the $\Pe\PGm$ final state are included, independently for
each bin in the electron and muon where the measurement is performed. A
systematic uncertainty of 10% is added to cover for a potential difference in
the background composition in the SR and CR. The extrapolation of the scaling
factor to $N_{\text{tracks}}=0$ has a total uncertainty of 8%, arising from
the choice of the fit function and from the limited number of events in the
measurement.

Most theoretical and experimental uncertainties cancel when renormalizing the
signal prediction using the scaling factor derived from the $\PGm\PGm$ CR for
the ${\PGg\PGg\to\PGm\PGm}$ process. The uncertainty in the scaling factor is
composed of a normalization component of statistical origin, amounting to 1.3
(3.7)% for $N_{\text{tracks}}=0$ (1), and a component with a shape dependence
on $m_{\PGt\PGt}$. For the latter component, we consider as a 1 standard
deviation (s.d.) effect the hypothesis that the scale factor does not depend
on $m_{\Pell\Pell}$, and is symmetrized with respect to the nominal hypothesis
of a linear dependence with $m_{\Pell\Pell}$. This $m_{\PGt\PGt}$-dependent
uncertainty is the leading uncertainty in the measurement of the
${\PGg\PGg\to\PGt\PGt}$ process, as shown in Fig. 8 (first line), and
contributes a 16% uncertainty to the measured signal strength. Modifying the
$N_{\text{tracks}}$ range used to extract the inclusive processes in the scale
factor measurement has a much smaller effect and the corresponding uncertainty
is neglected. These uncertainties are also considered for the photon-fusion
$\PW\PW$, $\Pe\Pe$, and $\PGm\PGm$ production.

Statistical uncertainties in the number of simulated background events in the
SR or observed event yields in the CRs used to derive the jet mis-ID
background are considered in all bins of the distributions using the approach
from Ref. [72]. The systematic uncertainty in the pileup modeling is included
by varying by $\pm$4.6% the total inelastic cross section used to calculate
simulated pileup distributions [73].

The impacts of the leading systematic uncertainties in the measurement of the
signal strength, which is described in Section 0.10.1, are shown in Fig. 8.
All the systematic uncertainties are summarized in Table 0.9.

Figure 8: Postfit values of the nuisance parameters (black markers), shown as
the difference of their best-fit values, $\hat{\theta}$, and prefit values,
$\theta_{0}$, relative to the prefit uncertainties $\Delta\theta$. The
horizontal error bars indicate the uncertainties in these measured postfit
values. The impact $\Delta\hat{\mu}$ of the nuisance parameter on the signal
strength is computed as the difference of the nominal best fit value of $\mu$
and the best fit value obtained when fixing the nuisance parameter under
scrutiny to its best fit value $\hat{\theta}$ plus/minus its postfit
uncertainty (blue shaded area). The nuisance parameters are ordered by their
impact, and only the 20 highest ranked parameters are shown.

Summary of the systematic uncertainties considered in the analysis. The
sources of the uncertainties, the processes they affect, and their magnitudes
are indicated.  Uncertainty Process Magnitude Luminosity All simulations 1.6%
DY cross section DY 2% Inclusive diboson cross section $\PW\PW$, $\PW\PZ$,
$\PZ\PZ$ 5% $\Pe$ ID, iso, trigger All simulations up to 2% $\Pe$ ID
low-$N_{\text{tracks}}$ correction All simulations 1% $\Pgm$ ID, iso, trigger
All simulations ${<}2\%$ $\tauh$ ID All simulations 1–5% $\tauh$ trigger All
simulations up to 5% $\Pe\to\tauh$ mis-ID $\PZ/\PGg*\to\Pe\Pe$ and
${\PGg\PGg\to\Pe\Pe}$ ${<}10\%$ $\Pgm\to\tauh$ ID $\PZ/\PGg*\to\Pgm\Pgm$ and
${\PGg\PGg\to\PGm\PGm}$ ${<}10\%$ $\tauh$ energy scale All simulations
${<}1.2\%$ $\Pe\to\tauh$ energy scale $\PZ/\PGg*\to\Pe\Pe$ and
${\PGg\PGg\to\Pe\Pe}$ ${<}5\%$ $\Pgm\to\tauh$ energy scale
$\PZ/\PGg*\to\Pgm\Pgm$ and ${\PGg\PGg\to\PGm\PGm}$ ${<}1\%$ $\tauh$ ID
low-$N_{\text{tracks}}$ correction All simulations 2.1% $\Pe$ ID
low-$N_{\text{tracks}}$ correction All simulations 2.0% $\Pe\to\tauh$ ID
low-$N_{\text{tracks}}$ correction $\PZ/\PGg*\to\Pe\Pe$ and
${\PGg\PGg\to\Pe\Pe}$ 22% $\Pgm\to\tauh$ ID low-$N_{\text{tracks}}$ correction
$\PZ/\PGg*\to\Pgm\Pgm$ and ${\PGg\PGg\to\PGm\PGm}$ 15%
$N_{\text{tracks}}^{\text{PU}}$ reweighting All simulations 2%
$N_{\text{tracks}}^{\text{HS}}$ reweighting DY and inclusive VV 1.5–6.5%
Acoplanarity correction DY 5% DY extrapolation from $N_{\text{tracks}}<10$ DY
simulation 1.4–2.0% $\mu_{R}$, $\mu_{f}$ DY simulation Shape PDF DY simulation
Shape jet$\to\tauh$ MF, extrapolation with $\pt^{\tauh}$ jet$\to\tauh$ mis-ID
bkg. ${<}50\%$ jet$\to\tauh$ MF, $N_{\text{tracks}}$ extrapolation (stat.)
jet$\to\tauh$ mis-ID bkg. 6–18% jet$\to\tauh$ MF, inversion of CR selection
jet$\to\tauh$ mis-ID bkg. ${<}10\%$ jet$\to\tauh$ MF, $x^{\text{QCD}}$
fraction jet$\to\tauh$ mis-ID bkg. 9% jet$\to\tauh$ MF, $N_{\text{tracks}}$
extrapolation (syst.) jet$\to\tauh$ mis-ID bkg. ${<}10\%$ jet$\to\Pe/\Pgm$ OS-
to-SS (stat.) jet$\to\Pe/\Pgm$ mis-ID bkg. ${<}20\%$ jet$\to\Pe/\Pgm$ OS-to-SS
(syst.) jet$\to\Pe/\Pgm$ mis-ID bkg. 10% jet$\to\Pe/\Pgm$ OS-to-SS
$N_{\text{tracks}}$ extrapolation jet$\to\Pe/\Pgm$ mis-ID bkg. 8% Elastic
rescaling (stat.) $\PGg\PGg\to\PGt\PGt/\Pgm\Pgm/\Pe\Pe,\PW\PW$ 1.3–3.7%
Elastic rescaling (syst., shape) $\PGg\PGg\to\PGt\PGt/\Pgm\Pgm/\Pe\Pe,\PW\PW$
Mass-dependent Limited statistics All processes Bin-dependent Pileup
reweighting All simulations Event-dependent

##  0.10 Results

The significance of the ${\PGg\PGg\to\PGt\PGt}$ process, as well as the
constraints on $\and$ and $d_{\PGt}$, are extracted with a binned maximum
likelihood fit from the $m_{\text{vis}}$ distributions in the four final
states and two $N_{\text{tracks}}$ bins, shown in Figs. 9–10. The CMS
statistical analysis and combination tool COMBINE is used [74]. The last bins
include events up to $m_{\text{vis}}=500\GeV$, and events at higher
$m_{\text{vis}}$ are discarded to ensure a sufficient difference with the
$\Lambda$ scale used in the BSM interpretation. This upper threshold removes
SM signal events at the percent level. The lower $m_{\text{vis}}$ thresholds
depend on the final state because of different thresholds arising from trigger
requirements. The sources of systematic uncertainties described in Section 0.9
are included in the statistical procedure as nuisance parameters. The number
of data events in the $N_{\text{tracks}}=0$ SRs, with the additional
requirement that $m_{\text{vis}}$ is greater than 100to reduce the DY
background contribution, is compared in Table 0.10 to the background and
signal yields in the different final states.

Figure 9: Observed and predicted $m_{\text{vis}}$ distributions in the
$\Pe\PGm$ (upper left), $\Pe\tauh$ (upper right), $\PGm\tauh$ (lower left),
and $\tauh\tauh$ (lower right) final states for events with
$N_{\text{tracks}}=0$, the lower panels showing the observed/expected ratio.
The observed data and their associated Poissonian statistical uncertainty are
shown with black markers with vertical error bars. The minor inclusive diboson
background contribution is drawn together with the DY background in the
$\Pe\PGm$, $\Pe\tauh$, and $\PGm\tauh$ final states. The predicted background
distributions correspond to the result of the global fit. The signal
distribution is normalized to its best fit signal strength. The uncertainty
band accounts for all sources of background and signal uncertainty, systematic
as well as statistical, after the global fit. In the fit, $\and$ and
$d_{\PGt}$ are fixed to their SM values. The ratio of the total predictions
for an illustrative value of $\and=0.008$ to those with SM electromagnetic
couplings is shown with a blue line in the lower panel of each plot.

Figure 10: Observed and predicted $m_{\text{vis}}$ distributions in the
$\Pe\PGm$ (upper left), $\Pe\tauh$ (upper right), $\PGm\tauh$ (lower left),
and $\tauh\tauh$ (lower right) final states for events with
$N_{\text{tracks}}=1$. The description of the histograms is the same as in
Fig. 9.

Observed and predicted event yields per final state in the signal-enriched
phase space with $m_{\text{vis}}>100\GeV$ and $N_{\text{tracks}}=0$. The
signal and background yields are the result of the global fit including all
sources of uncertainties.  Process $\Pe\PGm$ $\Pe\tauh$ $\PGm\tauh$
$\tauh\tauh$ $\PZ/\PGg^{*}\to\PGt\PGt$ $3.2\pm 0.5$ $8.2\pm 1.2$ $16.6\pm 2.9$
$18.4\pm 3.1$ $\PZ/\PGg^{*}\to\Pe\Pe/\Pgm\Pgm$ $4.0\pm 1.2$ $1.2\pm 0.6$ Jet
mis-ID $5.2\pm 0.8$ $12.3\pm 2.9$ $15.9\pm 3.6$ $17.9\pm 2.8$ Inclusive VV
$2.8\pm 0.3$ $0.23\pm 0.02$ $0.33\pm 0.3$ ${\PGg\PGg\to\Pe\Pe}/\Pgm\Pgm$
$9.2\pm 2.3$ $1.3\pm 0.2$ ${\PGg\PGg\to\PW\PW}$ $2.7\pm 0.6$ $0.16\pm 0.04$
$0.4\pm 0.1$ Total bkg. $13.9\pm 1.3$ $34.1\pm 4.8$ $35.7\pm 4.4$ $36.3\pm
4.2$ Signal $9.5\pm 2.0$ $12.4\pm 2.6$ $31.6\pm 6.7$ $26.1\pm 5.8$ Total
$23.4\pm 1.7$ $46.5\pm 5.4$ $67.3\pm 6.8$ $62.4\pm 6.2$ Observed 24 54 57 70

###  0.10.1 Observation of ${\PGg\PGg\to\PGt\PGt}$

The observed (expected) significance, calculated with the asymptotic
approximation [75] using COMBINE, is 5.3 (6.5) s.d. for the exclusive
${\PGg\PGg\to\PGt\PGt}$ process. This constitutes the first observation of
this process in $\Pp\Pp$ collisions. The corresponding observed (expected)
significances per final state are 2.3, 3.0, 2.1, and 3.4 (3.2, 2.1, 3.9, and
3.9) s.d. in the $\Pe\PGm$, $\Pe\tauh$, $\PGm\tauh$, and $\tauh\tauh$ final
states, respectively. We measure a best fit signal strength of
$\hat{\mu}=0.75^{+0.21}_{-0.18}$, where the systematic uncertainty dominates
over the statistical uncertainty ($\hat{\mu}=0.75^{+0.17}_{-0.14}\syst\pm
0.11\stat$).

The $N_{\text{tracks}}$ distribution for events with $N_{\text{tracks}}<10$ is
shown for the combination of final states in Fig. 11 for events with
$A<0.015$, as in the SR, and $m_{\text{vis}}>100\GeV$, so as to reduce the DY
background contribution. The signal contribution is visible as an excess of
events over the inclusive background in the first bins, whereas the agreement
between prediction and observation in the other bins demonstrates a good
background modeling, more specifically in terms of $N_{\text{tracks}}$
corrections to the simulations and MFs used to predict the background with
misidentified jets.

Figure 11: Observed and predicted $N_{\text{tracks}}$ distributions for events
passing the SR selection but with the relaxed requirement
$N_{\text{tracks}}<10$ and the additional requirement
$m_{\text{vis}}>100\GeV$, combining the $\Pe\PGm$, $\Pe\tauh$, $\PGm\tauh$,
and $\tauh\tauh$ final states together. The acoplanarity requirement $A<0.015$
is applied. The observed data and their associated Poissonian statistical
uncertainty are shown with black markers with vertical error bars. The
inclusive diboson background contribution is drawn together with that of the
$\ttbar$ process. The predicted distributions are adjusted to the result of
the global fit performed with the $m_{\text{vis}}$ distributions in the SRs,
and the signal distribution is normalized to its best fit signal strength. The
lower panel shows the difference between the observed events and the
backgrounds, as well as the signal contribution. Systematic uncertainties are
assumed to be uncorrelated between final states to draw the uncertainty band.

The fiducial cross section of the ${\PGg\PGg\to\PGt\PGt}$ process is measured
using only the SR with $N_{\text{tracks}}=0$, because it is not possible to
extract the origin of the track in the SR with $N_{\text{tracks}}=1$ given the
procedure used to estimate dissociative contributions. In this measurement,
signal events not belonging to the fiducial region but entering the
reconstruction-level categories are constrained to their expected
normalizations and distributions, within uncertainties. The cross section of
the ${\PGg\PGg\to\PGt\PGt}$ process is measured in a fiducial phase space
defined to be as close as possible to the reconstructed event selection. All
variables used in the definition of the fiducial region are calculated at the
generator level after parton showering and hadronization, and the lepton
momentum includes the momenta of photons radiated within a cone of $\Delta
R<0.1$ centered on the lepton. The visible $\PGt$ lepton decay products at the
generator level are required to have an acoplanarity $A<0.015$ and an
invariant mass less than 500. The di-$\PGt$ invariant mass, including the
neutrinos, is required to be greater than 50. There must not be any stable
charged particle with $\pt>0.5\GeV$ and $\abs{\eta}<2.5$ outside of the $\PGt$
lepton decay products. The other requirements depend on the final state. In
the $\Pe\PGm$ final state, the leading lepton must satisfy $\pt>24\GeV$
whereas the requirement for the subleading lepton is $\pt>15\GeV$. The
electron (muon) must have $\abs{\eta}<2.5$ (2.4). In the $\Pe\tauh$
($\PGm\tauh$) final state, the electron (muon) is required to have
$\pt>25\GeV$ and $\abs{\eta}<2.5$ ($\pt>21\GeV$ and $\abs{\eta}<2.4$), whereas
the visible $\tauh$ must have $\pt>30\GeV$ and $\abs{\eta}<2.3$. In addition,
the transverse mass $\mT(\Pe/\PGm,\ptvecmiss)$ must be less than 75. In the
$\tauh\tauh$ final state, the sum of the visible $\tauh$ decay products at the
generator level must be greater than 40for both $\tauh$ candidates, and the
visible momentum vectors must lie within $\abs{\eta}<2.3$. Other di-$\PGt$
final states are considered to be outside of the fiducial region. The
definition of the fiducial region is summarized in Table 0.10.1. Using the SRs
with $N_{\text{tracks}}=0$, the fiducial cross section is measured to be
$\sigma^{\text{fid}}_{\text{obs}}=12.4^{+3.8}_{-3.1}\unit{fb}$. This can be
compared with the prediction from the gamma-UPC generator for the elastic
production component of the signal, rescaled by the correction based on
control samples in the measured events to include dissociative contributions:
$\sigma^{\text{fid}}_{\text{pred}}=16.5\pm 1.5\unit{fb}$, where the
uncertainty is dominated by the data-driven correction.

Selection criteria to define the fiducial cross section. Events where the two
$\PGt$ leptons decay both to electrons or to muons, with neutrinos, are
considered to be outside the fiducial region. All requirements are applied
using generator-level quantities, as detailed in the text.  $\Pe\PGm$
$\Pe\tauh$ $\PGm\tauh$ $\tauh\tauh$ $\pt^{\Pe}$ () $>15/24$ $>25$
$\abs{\eta^{\Pe}}$ $<2.5$ $<2.5$ $\pt^{\PGm}$ () $>24/15$ $>21$
$\abs{\eta^{\PGm}}$ $<2.4$ $<2.4$ $\pt^{\tauh}$ () $>30$ $>30$ $>40$
$\abs{\eta^{\tauh}}$ $<2.3$ $<2.3$ $<2.3$ $\Delta R(\Pell,\Pell^{\prime})$
$>0.5$ $>0.5$ $>0.5$ $>0.5$ $\mT(\Pe/\PGm,\ptvecmiss)$ [] $<75$ $<75$ $A$
$<0.015$ $<0.015$ $<0.015$ $<0.015$ $m_{\text{vis}}$ () $<500$ $<500$ $<500$
$<500$ $N_{\text{tracks}}$ 0 0 0 0

###  0.10.2 Constraints on the anomalous electromagnetic moments of the $\PGt$
lepton

Constraints on $\and$ and $d_{\PGt}$ are set independently by performing a
binned likelihood scan with $\and$ and $d_{\PGt}$ as single parameters of
interest. Unlike in Section 0.10.1, $\and$ and $d_{\PGt}$ are allowed to
float, one at a time, whereas the normalization and $m_{\text{vis}}$
distribution of the ${\PGg\PGg\to\PGt\PGt}$ process under the SM hypothesis is
constrained to its predicted value, estimated from the elastic production
simulation [38] and weighted with the scale factor from the $\PGm\PGm$ CR to
include dissociative contributions. Varying $\and$ and $d_{\PGt}$ from their
SM values modifies both the normalization of the signal process and its
$m_{\text{vis}}$ distribution. In particular, the number of predicted signal
events increases with $m_{\text{vis}}$ for large $\abs{\and}$ values, as shown
in the ratio panels of Figs. 9–10 for an illustrative value of $\and=0.008$.

The combination of all final states and years, using SRs with
$N_{\text{tracks}}=0$ or 1, gives an observed best fit value of
$\and=0.0009^{+0.0016}_{-0.0015}\syst^{+0.0028}_{-0.0027}\stat$, equivalent to
$\and=0.0009_{-0.0031}^{+0.0032}$, at 68% confidence level (). The
corresponding expected best fit value is $\and=0.0012_{-0.0043}^{+0.0041}$, as
shown in Fig. 12 (left). Contrary to the signal strength measurement assuming
SM values for $\and$ and $d_{\PGt}$, the statistical uncertainty dominates in
the measurement of $\and$ because the sensitivity is driven by the
high-$m_{\text{vis}}$ bins where BSM effects could be enhanced. The
corresponding observed (expected) constraint at 95% is: $-0.0042<\and<0.0062$
($-0.0051<\and<0.0072$). The 68% constraint on $d_{\PGt}$ is
$\abs{d_{\PGt}}<1.7\times 10^{-17}\,e\cm$ ($\abs{d_{\PGt}}<2.3\times
10^{-17}\,e\cm$), with a best fit value of $d_{\PGt}=0.0\times
10^{-17}\,e\cm$, whereas the 95% interval is $\abs{d_{\PGt}}<2.9\times
10^{-17}\,e\cm$ ($\abs{d_{\PGt}}<3.4\times 10^{-17}\,e\cm$). These results are
compared with constraints from other experiments in Fig. 13. If the
measurement were performed using information about the $m_{\text{vis}}$
distribution and with a floating normalization for the ${\PGg\PGg\to\PGt\PGt}$
process, the expected precision in the measurement of $\and$ would decrease by
about 50%.

Figure 12: Expected and observed negative log-likelihood as a function of
$\and$ (left) and $d_{\PGt}$ (right), for the combination of all SRs in all
data-taking periods.

Figure 13: Measurements of $\and$ (left) and $d_{\PGt}$ (right) performed in
this analysis, compared with previous results from the OPAL, L3, DELPHI,
ARGUS, Belle, ATLAS, and CMS experiments [25, 26, 24, 28, 27, 9, 10].
Confidence intervals at 68 and 95% are shown with thick black and thin green
lines, respectively. The SM values of the $\PGt$ anomalous electromagnetic
moments, $\and=1.2\times 10^{-3}$ and $d_{\PGt}=-7.3\times 10^{-38}\,e\cm$,
are indicated with the dashed blue lines.

The constraints on $\and$ and $d_{\PGt}$ can be converted to two-dimensional
constraints on the real and imaginary parts of the Wilson coefficients
$C_{\PGt B}$ and $C_{\PGt W}$ divided by $\Lambda^{2}$, using Eq. (5). The 95%
intervals in the plane of the normalized Wilson coefficients are shown in Fig.
14.

Figure 14: Expected and observed 95% constraints on the real (left) and
imaginary (right) parts of the Wilson coefficients $C_{\PGt B}$ and $C_{\PGt
W}$ divided by $\Lambda^{2}$. The SM value is indicated with a cross. The blue
shaded areas indicate excluded regions.

##  0.11 Summary

The photon-fusion production of a pair of $\PGt$ leptons,
${\PGg\PGg\to\PGt\PGt}$, has been observed for the first time in proton-proton
collisions, with a significance of 5.3 standard deviations. The $\PGt$ leptons
are reconstructed in their leptonic and hadronic decay modes. The signal has
been identified by requiring low track activity around the di-$\PGt$ vertex
and low azimuthal acoplanarity between the $\PGt$ candidates. Data in a
control region with two muons were used to determine corrections for the
simulations to accurately model the track multiplicity and to predict the
signal contribution in the final state of two $\PGt$ leptons. The signal
strength, fiducial cross section, and constraints on the anomalous
electromagnetic moments of the $\PGt$ lepton have been extracted using the
di-$\PGt$ invariant mass distributions in four di-$\PGt$ final states. The
measured fiducial cross section of ${\PGg\PGg\to\PGt\PGt}$ is
$\sigma^{\text{fid}}_{\text{obs}}=12.4^{+3.8}_{-3.1}\unit{fb}$. The anomalous
$\PGt$ magnetic moment is determined to be $\and=0.0009_{-0.0031}^{+0.0032}$,
whereas the electric dipole moment of the $\PGt$ lepton is constrained to
$\abs{d_{\PGt}}<2.9\times 10^{-17}\,e\cm$ at 95% confidence level. They are
both in good agreement with the predictions of the standard model of particle
physics, and the measurements do not show any evidence for the presence of new
physics that would modify the electromagnetic moments of the $\PGt$ lepton.
This is the most stringent limit on the $\PGt$ lepton magnetic moment to date,
improving on the previous best constraints by nearly an order of magnitude.

###### Acknowledgements.

We congratulate our colleagues in the CERN accelerator departments for the
excellent performance of the LHC and thank the technical and administrative
staffs at CERN and at other CMS institutes for their contributions to the
success of the CMS effort. In addition, we gratefully acknowledge the
computing centers and personnel of the Worldwide LHC Computing Grid and other
centers for delivering so effectively the computing infrastructure essential
to our analyses. Finally, we acknowledge the enduring support for the
construction and operation of the LHC, the CMS detector, and the supporting
computing infrastructure provided by the following funding agencies: SC
(Armenia), BMBWF and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES,
FAPERJ, FAPERGS, and FAPESP (Brazil); MES and BNSF (Bulgaria); CERN; CAS,
MoST, and NSFC (China); MINCIENCIAS (Colombia); MSES and CSF (Croatia); RIF
(Cyprus); SENESCYT (Ecuador); ERC PRG, RVTT3 and MoER TK202 (Estonia); Academy
of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); SRNSF
(Georgia); BMBF, DFG, and HGF (Germany); GSRI (Greece); NKFIH (Hungary); DAE
and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF
(Republic of Korea); MES (Latvia); LMTLT (Lithuania); MOE and UM (Malaysia);
BUAP, CINVESTAV, CONACYT, LNS, SEP, and UASLP-FAI (Mexico); MOS (Montenegro);
MBIE (New Zealand); PAEC (Pakistan); MES and NSC (Poland); FCT (Portugal);
MESTD (Serbia); MCIN/AEI and PCTI (Spain); MOSTR (Sri Lanka); Swiss Funding
Agencies (Switzerland); MST (Taipei); MHESI and NSTDA (Thailand); TUBITAK and
TENMAK (Turkey); NASU (Ukraine); STFC (United Kingdom); DOE and NSF (USA).
Individuals have received support from the Marie-Curie program and the
European Research Council and Horizon 2020 Grant, contract Nos. 675440,
724704, 752730, 758316, 765710, 824093, 101115353, 101002207, and COST Action
CA16108 (European Union); the Leventis Foundation; the Alfred P. Sloan
Foundation; the Alexander von Humboldt Foundation; the Science Committee,
project no. 22rl-037 (Armenia); the Belgian Federal Science Policy Office; the
Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture
(FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie
(IWT-Belgium); the F.R.S.-FNRS and FWO (Belgium) under the “Excellence of
Science – EOS” – be.h project n. 30820817; the Beijing Municipal Science &
Technology Commission, No. Z191100007219010 and Fundamental Research Funds for
the Central Universities (China); the Ministry of Education, Youth and Sports
(MEYS) of the Czech Republic; the Shota Rustaveli National Science Foundation,
grant FR-22-985 (Georgia); the Deutsche Forschungsgemeinschaft (DFG), under
Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306, and
under project number 400140256 - GRK2497; the Hellenic Foundation for Research
and Innovation (HFRI), Project Number 2288 (Greece); the Hungarian Academy of
Sciences, the New National Excellence Program - ÚNKP, the NKFIH research
grants K 131991, K 133046, K 138136, K 143460, K 143477, K 146913, K 146914, K
147048, 2020-2.2.1-ED-2021-00181, and TKP2021-NKTA-64 (Hungary); the Council
of Science and Industrial Research, India; ICSC – National Research Center for
High Performance Computing, Big Data and Quantum Computing and FAIR – Future
Artificial Intelligence Research, funded by the NextGenerationEU program
(Italy); the Latvian Council of Science; the Ministry of Education and
Science, project no. 2022/WK/14, and the National Science Center, contracts
Opus 2021/41/B/ST2/01369 and 2021/43/B/ST2/01552 (Poland); the Fundação para a
Ciência e a Tecnologia, grant CEECIND/01334/2018 (Portugal); the National
Priorities Research Program by Qatar National Research Fund;
MCIN/AEI/10.13039/501100011033, ERDF “a way of making Europe”, and the
Programa Estatal de Fomento de la Investigación Científica y Técnica de
Excelencia María de Maeztu, grant MDM-2017-0765 and Programa Severo Ochoa del
Principado de Asturias (Spain); the Chulalongkorn Academic into Its 2nd
Century Project Advancement Project, and the National Science, Research and
Innovation Fund via the Program Management Unit for Human Resources &
Institutional Development, Research and Innovation, grant B37G660013
(Thailand); the Kavli Foundation; the Nvidia Corporation; the SuperMicro
Corporation; the Welch Foundation, contract C-1845; and the Weston Havens
Foundation (USA).

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