aainstitutetext: CERN, CH-1211 Geneva 23, Switzerlandbbinstitutetext: INFN
Sezione di Torino, via P. Giuria 1, 10125, Torino, Italy

# Measuring the Higgs self-coupling via Higgs-pair production at a 100 TeV p-p
collider

Michelangelo L. Mangano b ,Giacomo Ortona a and Michele Selvaggi
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Higgs pair production provides a unique handle for measuring the strength of
the Higgs self interaction and constraining the shape of the Higgs potential.
Among the proposed future facilities, a circular 100 TeV proton-proton
collider would provide the most precise measurement of this crucial quantity.
In this work, we perform a detailed analysis of the most promising decay
channels and derive the expected sensitivity of their combination, assuming an
integrated luminosity of 30 ab-1. Depending on the assumed detector
performance and systematic uncertainties, we observe that the Higgs self-
coupling will be measured with a precision in the range 3.4 - 7.8% at 68%
confidence level.

##  1 Introduction

The steady progress of the LHC experiments keeps improving our knowledge of
the Higgs properties Aad:2019mbh ; Sirunyan:2018koj . The long-term prospects
for the high-luminosity phase of the LHC (HL-LHC) set important precision
goals Cepeda:2019klc , reaching the level of few percent for several of the
Higgs couplings to gauge bosons and fermions. Beyond this, the per-mille level
frontier is opened by a future generation of Higgs factories deBlas:2019rxi .
The measurement of the Higgs self-coupling, the key parameter controlling the
shape of the Higgs potential, will remain however elusive for a long time.
Aside from providing clues to the deep origin of electroweak (EW) symmetry
breaking (EWSB), the determination of the Higgs potential has implications for
a multitude of fundamental phenomena, ranging from the nature of the EW phase
transition (EWPT) in the early universe Kajantie:1996qd , to the (meta)
stability of the EW vacuum Cabibbo:1979ay ; Hung:1979dn ; Lindner:1985uk ;
Sher:1988mj ; Degrassi:2012ry . This measurement sets therefore a primary
target among the promised guaranteed deliverables of any future collider
programme. Comparative assessments of the potential of different collider
options, relying on studies carried out through the years in preparation for
their design studies, have recently appeared in two reports deBlas:2019rxi ;
DiMicco:2019ngk . The $\pm 50\%$ precision projected for the HL-LHC
Cepeda:2019klc  can be improved by a factor up to 2 at future $e^{+}e^{-}$
colliders deBlas:2019rxi ; Blondel:2018aan , exploiting the impact of
radiative corrections induced by the Higgs self-coupling on single-H
production at several energies below the onset of on-shell Higgs-pair (HH)
production McCullough:2013rea . The direct measurement of HH production at
$\sqrt{s}\geq 1$ TeV will provide stronger, and independent, measurements,
reaching 10% and 9% for the ILC at $\sqrt{s}=1$ TeV Fujii:2019zll  and CLIC at
$\sqrt{s}=3$ TeV Charles:2018vfv , respectively. These measurements will
require a longer time scale, as they will be possible only at the last stage
of the proposed ILC and CLIC programmes. On these timescales, comparable or
even better precision could be possible via the study of HH production at a
future high-energy proton-proton (pp) collider, like the 100 TeV Future
Circular Collider 111For the sake of simplicity, we shall just refer in the
following to FCC-hh. (FCC-hh Abada:2019lih  or the SPPC CEPCStudyGroup:2018ghi
).

HH production in hadronic collisions has long been considered as an ideal
probe of the Higgs self-coupling Baur:2002rb ; Blondel:2002nta ;
Gianotti:2002xx , and much work along these lines has been done since the
Higgs discovery. Some of the most recent work, in the context of future
colliders, is documented in Refs. Yao:2013ika ; Barr:2014sga ; Liu:2014rva ;
He:2015spf ; Kumar:2015kca ; Fuks:2015hna ; Papaefstathiou:2015iba ;
Cao:2016zob ; Bishara:2016kjn ; Contino:2016spe ; Banerjee:2018yxy ;
Goncalves:2018yva ; Homiller:2018dgu ; Chang:2018uwu ; Biekotter:2018jzu ;
Borowka:2018pxx ; L.Borgonovi:2642471 ; Li:2019uyy ; Agrawal:2019bpm ;
Banerjee:2019jys ; Park:2020yps . The best estimates, obtained in these
studies, of the sensitivity to the Higgs self-coupling at the FCC-hh have used
the $\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ decay channel, leading to an
achievable precision between 5-10%, using this channel alone. A study focusing
on the $\mathrm{\mathrm{b\bar{b}}}\tau\tau$ and
$\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ final states Banerjee:2018yxy
in the boosted regime achieved a sensitivity of 8% and 20%, respectively. The
most up-to-date result, performed by the FCC-hh collaboration Abada:2019lih ;
L.Borgonovi:2642471  quotes a precision of 5-7%, driven by the
$\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ channel.

The goal of the present study is to extend the scope of previous projections
summarized in Ref. Abada:2019lih  and to provide a refined and comprehensive
reference for the combined prospect for the Higgs self-coupling measurement at
the FCC-hh. We improve on previous studies and show that further optimization
of the most sensitive Higgs decay channels using multi-variate techniques is
possible. When interpreted in the framework of the Standard Model (SM), the
combination of these measurements of HH production allows to reach a precision
on the tri-linear Higgs self-coupling in the range
$\delta_{\kappa_{\lambda}}=3.4-7.8\%$, significantly improving previous
estimates.

This article is organized as follows. We introduce the theoretical framework,
discussing the relation between Higgs self-coupling and HH production, in
Section 2, and we present in Section 3 the event generation tools used for
this study. The detector modeling, event simulation and analysis frameworks
are discussed in Section 4. In Section 5 we introduce the general measurement
strategy and the procedure that we use for the signal extraction and to derive
the expected precision on the self-coupling. The analyses of the three most
sensitive decay channels $\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$,
$\mathrm{\mathrm{b\bar{b}}}\tau\tau$ and
$\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ final states and their
combination are presented in Section 6. Section 7 summarizes our results and
our conclusions.

##  2 The theoretical framework

Perturbing the Higgs potential around its minimum, leads to the general
expression:

$\mathcal{L}_{h}=\frac{1}{2}m_{\mathrm{H}}^{2}H^{2}+\lambda_{3}H^{3}+\lambda_{4}H^{4},$
(1)

where $m_{\mathrm{H}}$ is Higgs boson mass and $\lambda_{3}$ and $\lambda_{4}$
are respectively the trilinear and quartic Higgs self-couplings. In the SM the
self-couplings are predicted to be
$\lambda_{3}^{SM}={m_{\mathrm{H}}^{2}}/{2v}$,
$\lambda_{4}^{SM}={m_{\mathrm{H}}^{2}}/{8v^{2}}$, where $v$ is the vacuum
expectation value (vev) of the Higgs field. The Higgs vev is known from its
relation to Fermi constant, $v=(\sqrt{2}G_{F})^{-1/2}=246$ GeV, and the
discovery of the Higgs particle at the LHC Aad:2012tfa ; Chatrchyan:2012ufa
has fixed the last remaining free parameter of the SM, the Higgs mass
$m_{\mathrm{H}}$ Aad:2015zhl . Beyond the SM, corrections to $\lambda_{3}$ and
$\lambda_{4}$, as well as higher-order terms, are possible.

To this day, large departures from the SM potential are perfectly compatible
with current observations Aad:2019uzh ; Sirunyan:2018two . This makes it
possible, for example, to contemplate BSM models where the modified Higgs
potential allows for a strong first order EW phase transition (SFOPT) in the
early universe, instead of the smooth cross-over predicted in the SM (for a
recent discussion of the interplay between collider observables and models
with a SFOPT, see e.g. Ref. Ramsey-Musolf:2019lsf ). In the context of SM
modifications of the Higgs properties deFlorian:2016spz  parameterized by
effective-field-theories (EFTs), it is well known that changes of the Higgs
potential are often correlated with changes of other couplings, such as those
of the Higgs to the EW gauge bosons. In many instances, a very precise
measurement of the latter can be as powerful in constraining new physics as
the self-coupling measurement Katz:2014bha . For example, Ref. Huang:2016cjm
considered models for SFOPT with an extra real scalar singlet, and showed that
a measurement of the HZZ coupling $g_{HZZ}$ with a precision of $\sim 1\%$ can
rule out most of the parameter space that could be probed by a measurement of
the self-coupling with a $\sim 50\%$ precision (see Fig. 1 of that paper).
Should a deviation from the SM be observed in $g_{HZZ}$, however, a large
degeneracy would be present in the set of allowed parameters. For example,
Fig. 1 of Ref. Huang:2016cjm  shows that a $\sim 2\%$ deviation in $g_{HZZ}$
would be compatible, in this class of models, with any value of
$1\lesssim\lambda_{3}/\lambda_{3}^{SM}\lesssim 2$. A precise direct
measurement of $\lambda_{3}$ is therefore necessary, independently of what
other observables could possibly probe, and is an indispensable component of
the Higgs measurement programme.

Another remark is in order: the relation between the Higgs self-coupling and
HH production properties is unambiguous only in the SM. Beyond the SM, the HH
production rate could be modified not only by a change in the Higgs self-
coupling, but also by the presence of BSM interactions affecting the HH
production diagrams. These could range from a modified top Yukawa coupling, to
higher-order EFT operators leading to local vertices such as ggHH
Azatov:2015oxa , WWHH Bishara:2016kjn  or $\mathrm{\mathrm{t\bar{t}}HH}$
Grober:2010yv ; Contino:2012xk . The measurement of an anomalous HH production
rate, therefore, could not be turned immediately into a shift of
$\lambda_{3}$; rather, its interpretation should be made in the context of a
complete set of measurements of both Higgs and EW observables, required to pin
down and isolate the coefficients of the several operators that could
contribute. In view of this, it is not possible to predict an absolute degree
of precision that can be achieved on the measurement of $\lambda_{3}$, since
this will depend on the ultimate $\lambda_{3}$ value, on the specific BSM
framework leading to that value, and on the ancillary measurements that will
be available as additional inputs. As is customary in the literature
222However, see for example Ref. DiVita:2017vrr  and Refs. Azatov:2015oxa ;
Capozi:2019xsi ; Biekotter:2018jzu , for global studies of the Higgs self-
coupling in presence of multiple anomalous couplings, at $e^{+}e^{-}$ and pp
colliders, respectively., we shall therefore focus on the context of the SM,
neglecting the existence of interactions influencing the HH production, except
for the presence of a pure shift in $\lambda_{3}$. The precision with which
$\lambda_{3}$ can be measured under these conditions has been for a long time
the common standard by which the performance of future experiments is gauged,
and we adopt here this perspective. Our results remain therefore indicative of
the great potential of a hadron collider in the exploration of the Higgs
potential.

Figure 1: Diagrams contributing to Higgs pair production: (a) gluon fusion,
(b) vector-boson fusion, (c) double Higgs-strahlung and (d) double Higgs
bremsstrahlung off top quarks. The trilinear Higgs self-coupling is marked in
red.

##  3 The theoretical modeling of signals and backgrounds

The signal and background processes are modeled with the MadGraph5_aMC@NLO
Alwall:2014hca  and Powheg Frixione:2007vw ; Alioli:2010xd  Monte Carlo (MC)
generators, using the parton distribution functions (PDF) set NNPDF3.0
Ball:2014uwa  from the Lhapdf Buckley:2014ana  repository. The evolution of
the parton-level events is performed with Pythia8 Sjostrand:2014zea ,
including initial and final-state radiation (ISR, FSR), hadronization and
underlying event (UE). The generated MC events are then interfaced with the
Delphes deFavereau:2013fsa  software to model the response of the FCC-hh
detector, as described in Section 4.2. The full event generation chain is
handled within the integrated FCC collaboration software (Fccsw) fccsw_web .
The event yields for the background and signal samples are normalized to the
integrated luminosity of $\mathcal{L}_{int}=30\text{ ab}^{-1}$.

###  3.1 The HH production processes

At $\sqrt{s}=\text{100 TeV}$, the dominant HH production modes are, in order
of decreasing cross section, gluon fusion ($\mathrm{ggHH}$), vector boson
fusion ($\mathrm{VBF\,HH}$), associated production with top pairs
($\mathrm{\mathrm{t\bar{t}}HH}$) and double Higgs-strahlung ($\mathrm{VHH}$).
A subset of diagrams for these processes is given in Fig. 1. Single top
associated production is also a possible production mode but it is neglected
in this study. The cross-section calculations Baglio:2012np ;
deFlorian:2013jea ; Grigo:2014jma ; deFlorian:2016uhr ; Borowka:2016ypz ;
Dreyer:2018qbw ; Grazzini:2018bsd ; Baglio:2018lrj ; Davies:2019dfy ;
Baglio:2020ini  for these main production mechanisms, reported also in Refs.
hhxswg ; deFlorian:2016spz ; DiMicco:2019ngk , are given in Table 1. We note
that the relative rate of the sub-dominant modes ($\mathrm{VBF\,HH}$,
$\mathrm{\mathrm{t\bar{t}}HH}$ and $\mathrm{VHH}$) increases significantly
from $\sqrt{s}=\text{14 TeV}$ to $\sqrt{s}=\text{100 TeV}$. In particular,
associated top pair production becomes as important as vector boson fusion,
and together they contribute to nearly 15% of the total HH cross section.

Process |  $\sigma$(14 TeV) |  $\sigma$(100 TeV) | accuracy | K-factor
---|---|---|---|---
$\mathrm{ggHH}$ | $36.69\pm 5.3\%$ | $1224\pm 5.6\%$ | $\mathrm{\mathrm{NNLO}_{FTapprox}}$ | 1.08
$\mathrm{VBF\,HH}$ | $2.05\pm 2.1\%$ | $82.8\pm 2.1\%$ | N3LO | 1.15
$\mathrm{\mathrm{t\bar{t}}HH}$ | $0.949\pm 2.9\%$ | $82.1\pm 7.8\%$ | $\mathrm{NLO}$ | 1.38
$\mathrm{VHH}$ | $0.982\pm 1.8\%$ | $16.23\pm 2.9\%$ | $\mathrm{NNLO}$ | 1.40
Table 1: Signal cross sections ($\sigma$, in fb) for HH production, including
the QCD corrections recommended by the LHC Higgs Cross Section Working Group
hhxswg ; deFlorian:2016spz . For each process, scale variations have been
symmetrized and added in quadrature to PDF+$\alpha_{\text{S}}$ uncertainties.
For the $\mathrm{ggHH}$ process, we added in quadrature also the dominant
uncertainty induced by the finite $m_{top}$ corrections. The cross sections of
$\mathrm{W^{-}HH}$, $\mathrm{W^{+}HH}$ and $\mathrm{ZHH}$ processes have been
summed together in a single $\mathrm{VHH}$ line and their uncertainties have
been summed in quadrature.

The $\mathrm{ggHH}$ MC events have been generated at next-to-leading order
(NLO) with the full top mass dependence using Powheg Alioli:2010xd ;
Heinrich:2019bkc . The $\mathrm{VBF\,HH}$, $\mathrm{\mathrm{t\bar{t}}HH}$ and
$\mathrm{VHH}$ events were instead generated at leading order (LO) with
MadGraph5_aMC@NLO. All the HH production mechanisms feature the interference
between diagrams that depend on the self-coupling with diagrams that do not.
This leads to a non-trivial total cross section dependence on $\lambda_{3}$,
as shown in Fig. 2, and has crucial implications for the self-coupling
measurement strategy, as discussed in Section 5. In order to account for this
non-trivial dependence of the cross section on the self-coupling, the MC
samples for the signal processes have been generated for several possible
values of $\kappa_{\lambda}=\lambda_{3}/\lambda_{3}^{\mathrm{SM}}$ within the
interval $\kappa_{\lambda}\in$ [0.0,3.0]. In order to match the MC inclusive
cross section prediction with the cross sections of Table 1, we correct the
event normalisation by means of a constant K-factor (shown in the last column
of Table 1). We note that in principle the K-factors are
$\kappa_{\lambda}$-dependent. In this work, the (process dependent) K-factor
is derived for $\kappa_{\lambda}=1$ and applied to correct the cross section
at values of $\kappa_{\lambda}\neq 1$. This is justified by the explicit
calculation of the N3LO corrections at $\kappa_{\lambda}\neq 1$ for the VBF
production channel Dreyer:2018qbw , and by the study of the $\kappa_{\lambda}$
dependence of the NNLO/NLO ratio for $\mathrm{ggHH}$ in Ref.Amoroso:2020lgh .
In the latter case, the shape variation of kinematical distributions for
$\kappa_{\lambda}\neq 1$ from NLO to NNLO is small compared to the overall
size of the difference between $\kappa_{\lambda}\neq 1$ and
$\kappa_{\lambda}=1$. The total cross section obtained with this procedure as
a function of $\kappa_{\lambda}$ is shown in Fig. 2. The merging of the NLO
parton-level configurations with the parton-shower evolution is realized in
the Powheg samples with Pythia8. In Fig. 2 the transverse momentum of the HH
system $p_{\text{T}}^{\mathrm{HH}}$ is shown as a validation of the NLO
merging procedure. For the $\mathrm{VBF\,HH}$, $\mathrm{\mathrm{t\bar{t}}HH}$
and $\mathrm{VHH}$ samples, Pythia8 simply adds the regular parton shower to
the LO partonic final states.

Figure 2: (a) Cross section of the $\mathrm{ggHH}$, $\mathrm{VBF\,HH}$,
$\mathrm{\mathrm{t\bar{t}}HH}$, and $\mathrm{VHH}$ processes as a function of
$\kappa_{\lambda}=\lambda_{3}/\lambda_{3}^{\mathrm{SM}}$. (b) Transverse
momentum spectrum of the HH system in $\mathrm{ggHH}$ NLO events after parton-
shower merging for $\kappa_{\lambda}=0$, $\kappa_{\lambda}=1$,
$\kappa_{\lambda}=2$ and $\kappa_{\lambda}=3$.

The Higgs self-coupling can be probed via a number of different Higgs boson
decay channels. Given the small cross section, at least one of the Higgs
bosons is required to decay to a pair of b-quarks. Here, we consider the three
most promising channels:
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$,
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\tau\tau$ and
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$. The di-
Higgs system decay in the various modes is performed by the Pythia8 program
and the respective branching fractions
$\text{BR({$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$})}=0.00262$,
$\text{BR({$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\tau\tau$})}=0.072$
and
$\text{BR({$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$})}=0.33$
are taken from Ref. deFlorian:2016spz , assuming
$m_{\mathrm{H}}=125.10\,{\mathrm{\ Ge\kern-1.00006ptV}}$.

###  3.2 The background processes

The background processes for the channels under study can be classified in
irreducible, reducible and instrumental backgrounds. Irreducible backgrounds
feature the presence in the matrix element of the exact same final state as
the $\mathrm{ggHH}$ signal process. These include for example prompt
$\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ (QCD) production, or
$\mathrm{Zb\bar{b}}$ with $\mathrm{Z\rightarrow b\bar{b}(\tau\tau)}$. We
define as reducible background the processes that contain the same final state
particles as the signal, but also additional particles that can be used as
handles for discrimination. This is the case for instance of
$\mathrm{\mathrm{t\bar{t}}H}$, $\mathrm{H}\rightarrow\gamma\gamma$ as a
background for the
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ channel or the
$\mathrm{t\bar{t}}$ background for the
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\tau\tau$ channel. Finally,
we call as instrumental the background processes that mimic the signal final
state due to a mis-reconstruction of the event in the detector. An
instrumental background for the
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ channel is the
$\gamma+\mathrm{jets}$ process where one the jets gets accidentally
reconstructed as an isolated photon. Special care has to be given to such
backgrounds as they strongly depend on the details of the detector
performance.

Single-Higgs production constitutes a background for all di-Higgs final
states. The four main production modes, gluon fusion ($\mathrm{ggH}$), vector
boson fusion ($\mathrm{VBF\,H}$), top pair associated production
($\mathrm{\mathrm{t\bar{t}}H}$) and Higgs-strahlung ($\mathrm{VH}$), have been
simulated at LO, including up to two extra MLM-matched jets Mangano:2006rw ;
Alwall:2007fs , using MadGraph5_aMC@NLO. The $\mathrm{ggH}$ matrix element was
generated using the full top mass dependence. The rates of single-Higgs
processes have been normalised to the most accurate cross-section calculations
at $\sqrt{s}=\text{100 TeV}$ Contino:2016spe . The normalisation K-factor for
the $\mathrm{ggH}$ process includes corrections up to $\mathrm{N^{3}LO}$,
while the $\mathrm{VBF\,H}$, $\mathrm{\mathrm{t\bar{t}}H}$ and $\mathrm{VH}$
modes include corrections up to $\mathrm{NNLO}$.

Top-induced backgrounds, in particular top-pair production
($\mathrm{t\bar{t}}$), constitute a large background for the
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\tau\tau$ final state, and to
a minor degree for the
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ final
state. This process was generated at LO using MadGraph5_aMC@NLO with up to two
extra MLM-matched jets. The total cross section is normalised to match the
$\mathrm{NNLO}$ prediction at $\sqrt{s}=\text{100 TeV}$. The Drell-Yan
($\text{Z}/\gamma^{*}\text{+jets}$) and di-boson backgrounds are also mainly
relevant for the $\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\tau\tau$
and $\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ final
states. These are generated at LO with MadGraph5_aMC@NLO by directly requiring
the presence of $\mathrm{\mathrm{b\bar{b}}}\tau\tau$ (or
$\mathrm{jj\mathrm{b\bar{b}}}$ for the
$\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ channel) final state at the
matrix element level. The pure QCD contribution,
$\mathrm{jj\mathrm{b\bar{b}}}$ has also been generated with MadGraph5_aMC@NLO
at order $\mathcal{O}(\alpha_{\text{S}}^{3})$. The next contribution,
$\text{Z}/\gamma^{*}\text{+jets}$, corresponding to
$\mathrm{jj\mathrm{b\bar{b}}}$ and $\mathrm{\mathrm{b\bar{b}}}\tau\tau$ was
generated at order $\mathcal{O}(\alpha_{\text{S}}^{2}\alpha_{\text{EW}})$. The
latter includes for example the $\mathrm{Z\rightarrow b\bar{b}(\tau\tau)}$
process. The final contribution, generated at order
$\mathcal{O}(\alpha_{\text{S}}\alpha_{\text{EW}}^{2})$, includes the pure EW
processes such as ZZ and ZH. When this background is included, the single-
Higgs ZH mode discussed earlier is indeed omitted. For the pure QCD
contribution we simply assume a conservative $K=2$ correction to the MC LO
cross section. For the processes at orders $\mathcal{O}(\alpha_{\text{EW}})$
and $\mathcal{O}(\alpha_{\text{EW}}^{2})$ we employ K-factors that match to
the $\mathrm{NNLO}$ Drell-Yan and di-boson $\sqrt{s}=\text{100 TeV}$
predictions. The last class of relevant background processes for the the
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ and the
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\tau\tau$ final states are
the $\mathrm{ttZ}$ and $\mathrm{ttW}$ processes. These were also generated at
LO using MadGraph5_aMC@NLO and normalized to the highest accuracy NLO cross-
section calculations.

The largest background contribution for the
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ final state are
QCD multijet production with one or more prompt photons in the final states,
$\gamma\gamma+\mathrm{jets}$ and $\gamma+\mathrm{jets}$ respectively. For the
$\gamma\gamma+\mathrm{jets}$ process we generated the matrix element of
$\gamma\gamma$ plus two partons with MadGraph5_aMC@NLO, where partons are
generated in the 5-flavour (5F) scheme to allow for mis-reconstructed light
and c-quark jets. In order to maximise the MC event efficiency in the signal
region, the $\gamma\gamma+\mathrm{jets}$ process was generated with the
$|m_{\gamma\gamma}-125|<10$ GeV requirement at parton level. The
$\gamma+\mathrm{jets}$ process was instead generated as $\gamma$ plus three
partons in the final state, again in the 5F scheme. Both these processes were
generated at LO and a conservative K=2 correction factor on the LO prediction
to account for higher order predictions was applied. The
$\mathrm{t\bar{t}}\gamma\gamma$ process was also considered for this channel
and its contribution was found to be negligible.

##  4 The experimental and analysis framework

The FCC project is described in detail in its Conceptual Design Reports
Abada:2019zxq ; Benedikt:2018csr . We focus here on the 100 TeV pp collider,
FCC-hh, designed to operate at instantaneous luminosities up to
$\mathcal{L}=3\times 10^{35}~{}\mbox{cm${}^{-2}$~{}s${}^{-1}$}$. For our study
we adopt the reference total integrated luminosity of
$\mathcal{L}_{int}=30\text{ ab}^{-1}$, to be achieved after 20 years of
operations, possibly combining the statistics of two general purpose
detectors. The analysis of these data will set challenging requirements to the
detector design and performance, which will reflect on the physics potential
in general, and in particular on the measurement of the HH cross sections. We
summarize here the main features of the current detector design, as
implemented in the Delphes deFavereau:2013fsa  simulation tool used for our
study.

###  4.1 Detector requirements

A detector operating within the FCC-hh environment will have to isolate the
hard-scattering event from up to 1000 pile-up (PU) simultaneous collisions per
bunch-crossing. Extreme detector granularity together with high spatial and
timing resolution are therefore needed. In addition, to meet the high
precision goal in key physics channels such as
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$, an excellent
photon energy resolution is needed. This requires a small calorimeter
stochastic term 333The resolution in a calorimeter can be expressed as
$\frac{\sigma_{E}}{E}=\frac{N}{E}\oplus\frac{S}{\sqrt{E}}\oplus C$, where $N$,
$S$, and $C$ are usually referred respectively as the _noise_ , _stochastic_
and _constant_ terms. in an environment of large PU noise, which in turn can
be achieved via a large sampling fraction and a fine transverse and
longitudinal segmentation. Finally, physics processes occurring at moderate
energy scales ($Q=100{\mathrm{\ Ge\kern-1.00006ptV}}-1{\mathrm{\
Te\kern-1.00006ptV}}$) will be produced at larger rapidities compared to the
LHC. Therefore high precision calorimetry and tracking need to be extended up
to $|\eta|<6$.

A prototype of a baseline FCC-hh detector that could fulfill the above
requirements has been designed for the FCC CDR Benedikt:2018csr ;
Aleksa:2019pvl ; Selvaggi:2019xyd . The detector has a diameter of 20 m and a
length of 50 m, with dimensions comparable to the ATLAS detector. A central
detector (covering a region up to $|\eta|<2.5$) contains a silicon-based
tracker, a Liquid Argon (LAr) electromagnetic calorimeter (ECAL) and a
Scintillating Tile Hadron calorimeter (HCAL) inside a 4 T solenoid with a free
bore diameter of 10 m. The muon chambers are based on small Monitored Drift
Tube technology (sMDTs). The tracking volume has a radius of 1.7 m with the
outermost layer lying at 1.6 m from the interaction point (IP) in the central
and the forward regions, providing the full lever arm up to $|\eta|=3$. The
ECAL has a thickness of 30 radiation lengths and provides, together with the
HCAL, an overall calorimeter thickness of than 10.5 nuclear interaction
lengths. The transverse segmentation of both the electromagnetic and hadronic
calorimeters is $\sim 4$ times finer than the present ATLAS Aad:2008zzm  and
CMS calorimeters Bayatian:2006nff . A high longitudinal segmentation in the
ECAL is needed to ensure a high sampling fraction, hence a small stochastic
term and in turn the good photon energy resolution required in order to
maximise the efficiency of the $\mathrm{H}\rightarrow\gamma\gamma$
reconstruction. In order to reach good performances at large rapidites
($2.5<|\eta|<6$), the forward parts of the detector are placed at 10 m from
the interaction point along the beam axis. Two forward solenoids with an inner
bore of 5 m provide the required bending power for forward tracking. The
integrated forward calorimeter system (ECAL and HCAL) is fully based on LAr
due to its instrinsic radiation hardness. Coverage up to $|\eta|=6$ is
feasible by placing the forward system at a distance z$=$16.6 m from the IP in
the beam direction and at r$=$8 cm in the transverse direction. The FCC-hh
baseline detector performance has been studied in full Geant4
Agostinelli:2002hh  simulations and parameterised within the fast simulation
framework Delphes deFavereau:2013fsa ; delphes_card_fcc .

###  4.2 Detector simulation and object reconstruction

The reconstruction of the MC-generated events in the FCC-hh detector is
simulated with the Delphes framework. Delphes makes use of a parameterised
detector response in the form of resolution functions and efficiencies. The
Delphes simulation includes a track propagation system embedded in a magnetic
field, electromagnetic and hadron calorimeters, and a muon identification
system. Delphes produces physics objects such as tracks, calorimeter deposits
and high level objects such as isolated leptons, jets, and missing energy.
Delphes also includes a particle-flow reconstruction that combines tracking
and calorimeter information to form particle-flow candidates, i.e charged
hadrons, neutral hadrons and photons. Such particles are then used as input
for jet clustering, missing energy, and isolation variables. In the following
we will focus on describing the key parameters of the FCC-hh detector
implementation in Delphes that are relevant for the self-coupling analysis
presented here.

Jets are clustered by the $\text{anti-}k_{\text{T}}$ algorithm Cacciari:2008gp
with a parameter R$=$0.4. For leptons ($\ell=e,\mu$) and photons ($\gamma$),
the relative isolation $I_{\text{rel}}$ is computed by summing the
$p_{\text{T}}$ of all particle-flow candidates in a cone around the particle
of interest an dividing by the particle’s $p_{\text{T}}(e,\mu,\gamma)$.
Isolated objects, such as photons originating from a
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ decay,
typically feature a small value of $I_{\text{rel}}$. The reconstruction and
identification (ID) efficiencies for leptons and photons are parameterised as
function of $p_{\text{T}}$ and pseudo-rapidity $\eta$.

We note that the effect of pile-up is not simulated directly by overlaying
minimum bias events to the hard scattering. Although Delphes allows for such
possibility, including in the simulation up to 1000 pile-up interactions would
result in an overly conservative object reconstruction performance for the
simple reason that the current Delphes FCC-hh setup does not possess the well-
calibrated pile-up rejection tools that will necessarily be employed for a
detector operating in such conditions, and so far in the future. These
techniques will include the use of picosecond (ps) timing detectors as well as
advanced machine-learning-based techniques for pile-up mitigation. For the
present LHC detectors, as well as for presently approved future detectors (the
ATLAS and CMS Phase II detectors) it is already the case that such techniques
allow to recover the nominal detector performance in the absence of pile-up
Contardo:2020886 ; CERN-LHCC-2015-020 . The level of degradation of the
$\lambda_{3}$ measurement precision caused by the deterioration of the
performance of specific physics objects (for example the photon energy
resolution and reconstruction efficiency or the b-tagging efficiency) has been
quantified in previous studies L.Borgonovi:2642471 . The impact of degrading
the photon energy resolution due to pile-up contamination was studied in full
simulation with up 1000 pile-up interactions in Ref Aleksa:2019pvl . The
degraded resolution was then propagated in Delphes and the effect on the
$\lambda_{3}$ precision was found to be approximately 1% (or 20% in relative
terms). We stress however that this level of degradation should be considered
as a worse case scenario given that a simple sliding window algorithm was
used, and timing information was not exploited. In Ref. L.Borgonovi:2642471
we have also studied the impact of degrading the photon reconstruction or,
equivalently, of increasing the jet-to-photon probability, which also showed
an effect of 1% on the $\lambda_{3}$ precision. Since a full-fledged event
simulation and object reconstruction does not exist at this stage for the FCC-
hh detector, the assumed object efficiencies result from extrapolations from
the LHC detectors. In order to account for a possible degradation of the
detector performance in the presence of pile-up, we define 3 baseline
scenarios:

  * •

scenario I: optimistic – target detector performance, similar to Run 2 LHC
conditions

  * •

scenario II: realistic – intermediate detector performance

  * •

scenario III: conservative – pessimistic detector performance, assuming
extrapolated HL-LHC performance using present-day algorithms

The assumptions on the performance of various physics objects for each
baseline scenario, are summarized in Table 2. As mentioned previously, a
dominant background for
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ analysis is the
$\gamma+\mathrm{jets}$ process. The probability for a hard scattering jet to
be mis-reconstructed as an isolated photon is small, $\mathcal{O}(10^{-3})$,
in current LHC detectors, thanks to the excellent angular resolution of
present calorimeters. As we noted in Section 4.1, the assumed granularity for
the FCC-hh detector is a factor 2-4 better than present LHC detectors. We make
however the conservative choice of assuming a $\text{j}\rightarrow\gamma$
fake-rate $\epsilon_{j\rightarrow\gamma}=0.0007\cdot
e^{-p_{\text{T}}[GeV]/187}$, which is of the same magnitude as in the HL-LHC
detectors ATLAS:2016ukn . In addition, we account for the probability for a
pile-up jet to be reconstructed as photon by further multiplying the fake rate
by a factor 2. This factor has been derived by simulating 1000 PU collision
with the CMS Phase-II detector in Delphes, applying a pile-up ID mistag rate
of 10 % (from Ref. CERN-LHCC-2015-020 ) and applying the fake-rate probability
for calibrated pile-up jets given in Figure 5(b) in Ref. ATLAS:2016ukn . This
procedure is used for Scenario I. For Scenario II and III we multiply the
above fake-rate by factors of 2 and 4 respectively. For leptons we neglect
possible fake jets contributions since these are negligible at the momemtum
scale relevant for the
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\tau\tau$ final state.
Delphes also provides heavy flavour tagging, in particular $\tau$ (hadronic)
and $b$-jet identification. Both hadronic $\tau$’s and $b$-jets are
reconstructed using the total visible 4-momentum of the jet. The tagging
efficiencies rely on a parameterisation of the (mis-)identification
probability as a function of ($p_{\text{T}}$, $\eta$). Again, since we cannot
yet derive such performance from full-simulation, we assume efficiencies and
mistag rates of the same order as for the (HL-)LHC detectors. For Scenario I,
the efficiencies for $\tau$ and $b$-jets are modelled after the CMS
performance given in Refs. CMS-DP-2019-033 ; CMS-DP-2017-013 . For scenarios
II and III a degradation of the efficiencies for a constant mistag-rate
probability is assumed. It should be noted that this is a conservative
assumption, since present estimates for heavy-flavour tagging in LHC Phase II
conditions project a similar performance as in present conditions, due to
superior tracking and high-precision timing detectors CMS:2667167 ; CERN-
LHCC-2017-009 . For $\tau$ and $b$-jets we also consider two definitions, a
"Medium" (M) and a "Tight" (T) working point, in order to operate at an
optimal signal-to-background rejection in each decay channel. As shown in
Table 2, we also consider the impact of photon and b-jet energy resolutions.
Both are relevant since all di-Higgs processes are resonant and the
reconstructed Higgs mass directly affects the final sensitivity. The di-photon
resolution for scenarios I, II and III was directly determined from full-
simulation respectively with 0, 200, and 1000 pile-up interactions (see Ref.
Aleksa:2019pvl ). For the di-jet invariant mass, for scenario I we assume the
invariant mass resolution as obtained with a multi-variate regression
technique in CMS Run 2 (in Ref. Sirunyan:2018kst ), whereas for scenario II
and III respectively we assume a factor 1.5 and 2 degradation compared to
scenario I. The $\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ channel is a
special case, since two di-jet invariant masses are reconstructed (see Section
6.3). In that case, as a conservative assumption, we assume that only the
Higgs candidate with the largest $p_{\text{T}}$ is affected by the
$m_{\mathrm{{bb}}}$ resolution assumed for the scenarii I to III, while the
sub-leading Higgs candidate mass has the resolution of Scenario III.

parameterisation | scenario I | scenario II | scenario III
---|---|---|---
b-jet ID eff. | 82-65% | 80-63% | 78-60%
b-jet c mistag | 15-3% | 15-3% | 15-3%
b-jet l mistag | 1-0.1% | 1-0.1% | 1-0.1%
$\tau$-jet ID eff | 80-70% | 78-67% | 75-65%
$\tau$-jet mistag (jet) | 2-1% | 2-1% | 2-1%
$\tau$-jet mistag (ele) | 0.1-0.04% | 0.1-0.04% | 0.1-0.04%
$\gamma$ ID eff. | 90 | 90 | 90
jet $\rightarrow$ $\gamma$ eff. | 0.1 | 0.2 | 0.4
$m_{\gamma\gamma}$ resolution [GeV] | 1.2 | 1.8 | 2.9
$m_{bb}$ resolution [GeV] | 10 | 15 | 20
Table 2: Performance of physics objects for the various scenarios. Objects
efficiencies and mistag rates are given for a representative
$p_{\text{T}}\approx 50{\mathrm{\ Ge\kern-1.00006ptV}}$. For b and
$\tau$-tagging (and their respective mistag rates) numbers for two different
working points are given (Medium and Tight).

###  4.3 Systematic uncertainties

Systematic uncertainties can play a major role on the expected sensitivity of
the self-coupling measurement. Several assumptions have been made on the
possible evolution of theoretical and experimental sources of uncertainties in
order to present a realistic estimate of the physics potential of FCC-hh for
the channels considered here. In particular, for each uncertainty source, we
defined three possible scenarios, following the general principle introduced
in Section 4.2. We note that the intermediate assumptions are almost
equivalent to those made for HL-LHC projections Cepeda:2019klc ;
deBlas:2019rxi .

Uncertainty source | scenario I | scenario II | scenario III | Processes
---|---|---|---|---
b-jet ID eff. /b-jet | 0.5% | 1% | 2% | single H, HH, ZZ
$\tau$-jet ID eff. /$\tau$ | 1% | 2.5% | 5% | single H, HH, ZZ
$\gamma$ ID eff. /$\gamma$ | 0.5% | 1% | 2% | single H, HH
$\ell$ = $e$-$\mu$ ID efficiency | 0.5% | 1% | 2% | single H, HH, ZZ
luminosity | 0.5% | 1% | 2% | single H, HH, ZZ
theoretical cross section | 0.5% | 1% | 1.5% | single H, HH, ZZ
Table 3: Summary of the sources of systematic uncertainties in the 3
scenarios. The last column indicates the processes that are affected by the
corresponding source of uncertainty. For each given object (b-jet, $\tau$-jet,
$\gamma$, lepton), the quoted uncertainty on reconstruction and identification
efficiency is applied as many times as the object appears in the final-state.

A detailed list of the systematic uncertainties considered is presented in
Table 3 for all the channels, together with the processes affected by each
uncertainty. The numbers in the table refer to the individual contributions to
the overall yield uncertainty. In particular, we consider uncertainties on:

  * •

theoretical cross-section, affecting the single-Higgs and ZZ backgrounds. Due
to their moderate yields we assume these backgrounds to be estimated from
Monte Carlo at the FCC-hh. We also assume these two processes to be well known
and well reproduced by Monte Carlo simulations at the FCC-hh, with an overall
uncertainty varied between 0.5% and 1.5% depending on the scenario.
Furthermore, we include a similar theoretical uncertainty on the HH cross
section, affecting the interpretation of the HH rate measurement in terms of
$\mu$ and $\kappa_{\lambda}$.

  * •

luminosity. We assume that the integrated luminosity will be known at FCC-hh
at least as well as at the LHC. For this reason, we assume a conservative
estimate of 2% and an optimistic (intermediate) estimate of 0.5% (1%),
reflecting future opportunities to extract the luminosity from hard processes
like Z production. As for the theoretical uncertainties, the luminosity
affects both the signal and the single-Higgs and ZZ backgrounds.

  * •

experimental uncertainties on objects reconstruction and identification
efficiencies:

    * –

b-jets: for each b-jet, we assume a 0.5%, 1.0%, and 2.0% uncertainty for the
optimistic, intermediate and conservative scenarios, respectively. Since we
expect this to be one of the dominant uncertainties, it is applied during
event simulation accounting for the $p_{\text{T}}$ dependence of the b-jet
efficiency uncertainty (taken from Ref. Sirunyan:2017ezt ). This procedure
allows to take into account the effect of the uncertainty on the shape of the
resulting BDT distribution.

    * –

$\tau$-jets: for each jet originated from the hadronic decay of a $\tau$-jet
we assume an uncertainty of 1.0%, 2.0% and 5.0% for the optimistic,
intermediate and conservative scenarios, respectively.

    * –

leptons: we assume the same uncertainty on the lepton identification and
reconstruction efficiency for electrons and muons: a 0.5%, 1.0%, and 1.5%
uncertainty for the optimistic, intermediate and conservative scenarios,
respectively.

    * –

photons: we assume that the photon related uncertainties will be comparable to
electrons. For this reason we assign a systematic uncertainty to photon
reconstruction of 0.5%, 1.0%, and 2.0% for the optimistic, intermediate and
conservative scenarios, respectively.

The uncertainty on the energy resolution of photons has been studied in Ref.
L.Borgonovi:2642471 . Degrading the photon resolution by a relative 100% has
an effect of order 1% on the self-coupling precision. According to Ref.
Aad:2014nim , the uncertainty on the resolution, parametrised as an additional
constant term, amounts to 0.4% in the barrel. Assuming a photon energy E$=$60
GeV, a nominal constant term of 0.8% and a stochastic term of 10%, this
results in a relative difference of less than 5% (in relative terms 444More
precisely the relative energy resolution is 1.57% with the additional 0.4%
constant term and 1.51% without the additional 0.4% constant term ) on the
photon energy resolution. Such a degradation has an effect, at first order, of
less than 0.1% of the self-coupling measurement (assuming that a degradation
of more than 100% has an effect of 1% on the self-coupling precision). A
similar argument applies to the effect of the jet energy resolution, which is
measured by ATLAS with a relative precision of 7% according to Ref.
Aaboud:2019ibw . We therefore neglect such source of systematic uncertainties.
As far as the scale uncertainty goes, for both photons and jets, the above
references show that they amount respectively to 0.1% and 1%. We have then
explicitly verified that the effect of shifting the scale of both photon and
jets by 1% results in a relative change in the significance of order 0.5%,
translating at worst into a relative change in the self-coupling precision
555We assume a conservative factor 2 between the precision on the self-
coupling and the precision on the self-coupling. See Section 5 for a
discussion on how the two are related of 1%. We conclude therefore that also
the photon and jet (and necessarily hadronic $\tau$’s) energy scale have a
negligible impact on the self-coupling precision. Moreover, we stress that the
effect of a degradation of the detector performance, especially in terms of
photons and b-jet energy resolution, is probed by studying the various
scenarios given in Table 2. The absolute performance degradation considered in
Scenario II and III largely overcomes the corresponding systematic
uncertainties on the object resolution that were neglected.

We assume that several backgrounds will be measured with high statistical
accuracy from “side bands” or “control regions”. This is the case for example
for the $\mathrm{t\bar{t}}$, QCD, and non-single-Higgs backgrounds (with the
exception of ZZ, that we assume to be predicted by the Monte Carlo) that
dominate the $\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ and
$\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ channels background contributions. In
these cases, while there is no uncertainty associated to the normalisation of
the backgrounds, the statistical uncertainty due to the possible fluctuations
of the number of events in the side bands is considered in the fit.

When performing the fit for the combination across different channels,
systematic uncertainties with the same physical origin are considered fully
correlated across processes and final states. Otherwise they are considered as
completely uncorrelated, with the notable exception of the b-tagging
efficiency (and mistag rate) uncertainty, which is correlated across channels
but not across processes. We therefore consider separate shape b-tagging and
mistag uncertainties for each process (single H, ZZ, and HH), correlated
across the various channels. For example, the b-tagging uncertainty affecting
the single-Higgs production is correlated across all channels, but is
uncorrelated from the the b-tagging uncertainty affecting double Higgs, ZZ,
and so on. This reflects possible differences in the properties of b-tagged
jets created in different processes.

##  5 Signal extraction methodology

As mentioned in Section 3.1, the cross section for HH production has a non-
trivial dependence on the self-coupling modifier
$\kappa_{\lambda}=\lambda_{3}/\lambda_{3}^{\mathrm{SM}}$ due to the presence,
at LO, of diagrams that contain the trilinear interaction vertex ($(S)$) as
well as diagrams that do not ($(T)$), as shown in Fig. 1. In Fig. 1,
$(T)$-diagrams appear on the left column while $(S)$-diagrams are shown on the
right. The $(S)$ and $(T)$ contributions are present in all HH-production
mechanisms. Moreover, the contribution of the interference term between $(S)$
and $(T)$ is highly non trivial. For the $\mathrm{ggHH}$ and
$\mathrm{VBF\,HH}$ modes, the total cross section reaches a minimum
respectively at $\kappa_{\lambda}\approx$ 2.5 and $\kappa_{\lambda}\approx$
1.8, while the slopes of the $\mathrm{\mathrm{t\bar{t}}HH}$ and $\mathrm{VHH}$
cross sections carry little dependence on $\kappa_{\lambda}$ (see Fig. 2).

At first order one can write:

$\mu(\kappa_{\lambda})=1+(\kappa_{\lambda}-1)\frac{d\mu}{d\kappa_{\lambda}}\Bigr{|}_{\text{SM}},$
(2)

where we define $\mu=\sigma/\sigma_{\mathrm{SM}}$ as the signal strength. One
can measure $\lambda_{3}$ (or alternatively $\kappa_{\lambda}$) by measuring
the total HH production cross section. It follows that:

$\delta_{\kappa_{\lambda}}=\frac{\delta_{\mu}}{\frac{d\mu}{d\kappa_{\lambda}}\Bigr{|}_{\text{SM}}},$
(3)

where $\delta_{\kappa_{\lambda}}$ and $\delta_{\mu}$ are respectively the
uncertainty on the self-coupling modifier and on the signal strength. It can
be noted that at first order the precision of the self-coupling measurement is
determined by the slope of the cross section (or $\mu$) at
$\kappa_{\lambda}=1$ and by the uncertainty on the measurement of the total
cross section. Since $\frac{d\mu}{d\kappa_{\lambda}}\Bigr{|}_{\text{SM}}$ is a
given parameter, in order to maximise the precision on the self-coupling, we
have to maximise the precision on the cross section, or equivalently on $\mu$.
Assuming all other standard model parameters are known with better precision
than the expected precision on $\kappa_{\lambda}$ 666This assumes that for
instance the top Yukawa parameter will be known with
$\delta_{y_{t}}/y_{t}$$\approx$ 1%. The studies of Refs. Plehn:2015cta ;
L.Borgonovi:2642471  show that such precision is achievable at the FCC-hh,
using the ttZ coupling measured at FCC-ee Janot:2015yza , the relative weight
of the $(S)$ and $(T)$ amplitudes (and their interference) is determined by
the magnitude of $\kappa_{\lambda}$.

The magnitude of $\kappa_{\lambda}$ impacts not only the total HH rate, as
discussed above, but also the HH production kinematic observables. Notably,
the invariant mass of the HH pair $m_{\mathrm{hh}}$ is highly sensitive to the
value of the self-coupling. This can be easily understood by noting that
configurations with large $m_{\mathrm{hh}}$ are mostly suppressed in the $(S)$
amplitude (not in $(T)$ diagrams). Vice-versa, the phase-space region near
threshold, at $m_{\mathrm{hh}}$ $\gtrsim$ 2$m_{\mathrm{H}}$, maximises the
$(S)$ contribution. The $m_{\mathrm{hh}}$ distribution is shown for the
$\mathrm{ggHH}$ and $\mathrm{\mathrm{t\bar{t}}HH}$ processes respectively in
Figs. 3 and 3. For $\mathrm{ggHH}$ the dependence is distorted at values of
$\kappa_{\lambda}$ $\approx$ 2 due to the large destructive interference
between $(S)$ and $(T)$. The transverse momenta of the two Higgs bosons
($p_{\text{T}}(h_{1})$, and $p_{\text{T}}(h_{2})$) also display a large
dependence on $\kappa_{\lambda}$, as shown in Figs. 4 and 4.

The general strategy for providing the best possible accuracy on the self-
coupling will therefore rely on maximizing the cross-section precision by
using obervables that are able to discriminate between signal and backgrounds
as well as exploiting the shapes of observables that are highly sensitive to
the value of $\kappa_{\lambda}$. The signal over background optimisation is
largely dependent on the class of background and will be addressed in the
discussion specific to each channel below. However a common theme is that
typically the strategy to obtain a high $\mathrm{S/B}$ ratio relies heavily on
the reconstruction of the mass peak of the two Higgs bosons. In addition we
will make use of the $m_{\mathrm{hh}}$ observable, and the Higgs particles
transverse momentum ($p_{\text{T}}(h_{1})$, and $p_{\text{T}}(h_{2})$)
differential distributions to further improve the sensitivity on
$\kappa_{\lambda}$.

Figure 3: Higgs pair invariant-mass distribution in $\mathrm{ggHH}$ (a) and
$\mathrm{\mathrm{t\bar{t}}HH}$ (b) events for $\kappa_{\lambda}=0$,
$\kappa_{\lambda}=1$, $\kappa_{\lambda}=2$ and $\kappa_{\lambda}=3$.

Figure 4: Transverse momentum spectra of the leading (a) and sub-leading (b)
Higgs boson in $\mathrm{ggHH}$ events for $\kappa_{\lambda}=0$,
$\kappa_{\lambda}=1$, $\kappa_{\lambda}=2$ and $\kappa_{\lambda}=3$.

##  6 Determination of the Higgs self-coupling

While the Higgs pair can be reconstructed in a large variety of final states,
only the most promising ones are considered here:
$\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$, $\mathrm{\mathrm{b\bar{b}}}\tau\tau$
and $\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$. For each of these final
states, the event kinematical properties are combined within boosted decision
trees (BDTs) to form a powerful single observable that optimally discriminates
between signal and backgrounds. The BDT discriminant is built using the ROOT-
TMVA package Brun:1997pa ; Hocker:2007ht . The statistical procedure and the
evaluation of the systematic uncertainties are summarized in Appendix A and
4.3, respectively.

For a similar analysis in the case of HL-LHC, see Ref. Adhikary:2017jtu  and
the studies by the ATLAS and CMS collaborations ATL-PHYS-PUB-2018-053 ; CMS-
PAS-FTR-18-019 , contributed to Ref. Cepeda:2019klc .

###  6.1 The $\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ channel

Despite its small branching fraction, the
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ channel is by
far the most sensitive decay mode for measuring the Higgs self-coupling. The
presence of two high $p_{\text{T}}$ photons in the final state, together with
the possibility of reconstructing the decay products of both Higgses without
ambiguities and with high resolution, provide a clean signature with a large
$\mathrm{S/B}$. The largest background processes are single-Higgs production
and the QCD continuum $\gamma\gamma+\mathrm{jets}$ and $\gamma+\mathrm{jets}$.
A discussion of the simulation of these processes was given in Section 3.2.

####  6.1.1 Event selection

In the $\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ channel, events are required
to contain at least two isolated photons and two b-tagged jets with the
requirement $p_{\text{T}}(\gamma,\mathrm{b})$ > 30 GeV and
$|\eta(\gamma,\mathrm{b})$| < 4.0. The leading photon and b-jet are further
required to have $p_{\text{T}}(\gamma,\mathrm{b})$ > 35 GeV. The Higgs
candidates 4-momenta are formed respectively from the two reconstructed b-jets
and photons with the largest $p_{\text{T}}(\gamma,\mathrm{b})$. The b-jets are
identified with the "Medium" working point criterion, defined in Table 2.
Since the $\gamma\gamma+\mathrm{jets}$ process was generated with a parton-
level requirement (see Section 3.2) on $m_{\gamma\gamma}$, we further require
the events to pass the loose selection $|m_{\gamma\gamma}-125|<7$ GeV. The
efficiency of the full event selection for the SM signal sample is
approximately 26%. For an integrated luminosity $\mathcal{L}_{int}=30\text{
ab}^{-1}$ this event selection yields approximately 10k Higgs pair events,
125k single Higgs, 2.6M $\mathrm{jj}\gamma\gamma$ and 7M
$\gamma+\mathrm{jets}$ events for Scenario I. The trigger efficiency for the
above selection is assumed to be 100% efficient.

In order to maximally exploit the kinematic differences between signal and
background, a boosted decision tree (BDT) is trained using most of the
available kinematic information in the event:

  * •

The 3-vector components of the leading ($\gamma_{1}$) and subleading photon
($\gamma_{2}$): transverse momentum ($p_{\text{T}}^{\mathrm{\gamma_{1}}}$,
$p_{\text{T}}^{\mathrm{\gamma_{2}}}$), pseudo-rapidity ($\eta_{\gamma_{1}}$,
$\eta_{\gamma_{2}}$), and azimutal angle ($\phi_{\gamma_{1}}$,
$\phi_{\gamma_{2}}$).

  * •

The 3-vector components of the leading ($\mathrm{b}_{1}$) and subleading b-jet
($\mathrm{b}_{2}$): transverse momentum
($p_{\text{T}}^{\mathrm{\mathrm{b}_{1}}}$,
$p_{\text{T}}^{\mathrm{\mathrm{b}_{2}}}$), pseudo-rapidity
($\eta_{\mathrm{b}_{1}}$, $\eta_{\mathrm{b}_{2}}$), and azimutal angle
($\phi_{\mathrm{b}_{1}}$, $\phi_{\mathrm{b}_{2}}$).

  * •

The 3-vector components of the leading ($\mathrm{j}_{1}$) and subleading
additional reconstructed jets in the event ($\mathrm{j}_{2}$): transverse
momentum ($p_{\text{T}}^{\mathrm{\mathrm{j}_{1}}}$,
$p_{\text{T}}^{\mathrm{\mathrm{j}_{2}}}$), pseudo-rapidity
($\eta_{\mathrm{j}_{1}}$, $\eta_{\mathrm{j}_{2}}$), and azimutal angle
($\phi_{\mathrm{j}_{1}}$, $\phi_{\mathrm{j}_{2}}$). If no additional jets are
found, dummy values are given to these variables.

  * •

The 4-vector components of the $\mathrm{H}\rightarrow\gamma\gamma$ candidate:
transverse momentum ($p_{\text{T}}^{\mathrm{\gamma\gamma}}$), pseudo-rapidity
($\eta_{\gamma\gamma}$), azimutal angle ($\phi_{\gamma\gamma}$) and invariant
mass ($m_{\gamma\gamma}$).

  * •

The 4-vector components of the $\mathrm{H\rightarrow b\bar{b}}$ candidate:
transverse momentum ($p_{\text{T}}^{\mathrm{\mathrm{bb}}}$), pseudo-rapidity
($\eta_{\mathrm{bb}}$), azimutal angle ($\phi_{\mathrm{bb}}$) and invariant
mass ($m_{\mathrm{{bb}}}$).

  * •

The 4-vector components of the Higgs pair candidate: transverse momentum
($p_{\text{T}}^{\mathrm{\mathrm{hh}}}$), pseudo-rapidity
($\eta_{\mathrm{hh}}$), azimutal angle ($\phi_{\mathrm{hh}}$) and invariant
mass ($m_{\mathrm{hh}}$).

  * •

The number of reconstructed b-jets $N_{b}$, the number of light jets $N_{l}$
and the total number of jets $N_{j}=N_{b}+N_{l}$.

In a future FCC-hh experiment, identification algorithms for photon and heavy-
flavour will make use of the information of the invariant mass of the photon
or jet candidate. Therefore we have to assume that the parameterised
performance of the identification efficiency of such objects (in Delphes)
already accounts for these variables. As a result, the photon and jet mass are
not used as input variables in the BDT discriminant. The $m_{\gamma\gamma}$,
$m_{\mathrm{{bb}}}$, and $m_{\mathrm{hh}}$ observables, shown respectively in
Figs. 5, 5 and 5 provide most of the discrimination against the background.

Figure 5: Invariant mass spectra of the $\mathrm{H}\rightarrow\gamma\gamma$
(a), $\mathrm{H\rightarrow b\bar{b}}$ (b), HH (c) candidates after applying
the event pre-selection. The SM Higgs pair process is normalized to 50 times
the expected yield with $\mathcal{L}_{int}=30\text{ ab}^{-1}$.

The QCD ($\gamma+\mathrm{jets}$ and $\gamma\gamma+\mathrm{jets}$) and single-
Higgs background processes possess different kinematic properties, and are
therefore treated in separate classes. In QCD backgrounds, the final photons
and jets tend to be softer and at higher rapidity. Conversely, the photon-pair
candidates in single-Higgs processes often originate from a Higgs decay. As a
result, while the $m_{\gamma\gamma}$ observable is highly discriminating
against QCD, it is not against single-Higgs processes. In order to maximally
exploit these kinematic differences we perform a separate training for each
class of backgrounds, producing two multivariate discriminants:
$\mathrm{BDT_{H}}$ and $\mathrm{BDT_{QCD}}$. During the training, each
background within each class is weighted according to the relative cross
section.

Figure 6: Spectrum of SM signal (a), the QCD (b) and single Higgs (c)
backgrounds in the ($\mathrm{BDT_{H}}$,$\mathrm{BDT_{QCD}}$) plane.

The discrimination against single-Higgs backgrounds ($\mathrm{BDT_{H}}$) is
largely driven by the $m_{\mathrm{{bb}}}$ variable, followed by
$m_{\mathrm{hh}}$ and $p_{\text{T}\mathrm{bb}}$. Additional separation power,
in particular against the $\mathrm{\mathrm{t\bar{t}}H}$ background, is
provided by the number of reconstructed jets $N_{j}$ as well as
$p_{\text{T}}^{\mathrm{\mathrm{j}_{1}}}$ and
$p_{\text{T}}^{\mathrm{\mathrm{j}_{2}}}$. On the other hand, the
discrimination against QCD is driven by the $m_{\gamma\gamma}$ and
$m_{\mathrm{{bb}}}$ variables, followed by $p_{\text{T}\gamma\gamma}$,
$p_{\text{T}\mathrm{bb}}$, $p_{\text{T}}^{\mathrm{\gamma_{1}}}$,
$p_{\text{T}}^{\mathrm{\mathrm{b}_{1}}}$, $m_{\mathrm{hh}}$,
$p_{\text{T}}^{\mathrm{\gamma_{2}}}$ and
$p_{\text{T}}^{\mathrm{\mathrm{b}_{2}}}$.

The output of the BDT discriminant is shown in the
$(\mathrm{BDT_{H}},\,\mathrm{BDT_{QCD}})$ plane for the signal and the two
background components in Figs. 6, 6 and 6, respectively. As expected, the
signal (background) enriched region clearly corresponds to large (small)
$\mathrm{BDT_{H}}$ and $\mathrm{BDT_{QCD}}$ values. We note that the
multivariate discriminant correctly identifies the two main components
($\mathrm{ggH}$ and $\mathrm{\mathrm{t\bar{t}}H}$) within the single-Higgs
background. The $\mathrm{ggH}$ background, as opposed to
$\mathrm{\mathrm{t\bar{t}}H}$, is more “signal-like” and populates a region of
high $\mathrm{BDT_{H}}$ and $\mathrm{BDT_{QCD}}$.

####  6.1.2 Signal Extraction and results

The expected precision on the signal strength
$\mu=\sigma/\sigma_{\mathrm{SM}}$ and on the self-coupling modifier
$\kappa_{\lambda}=\lambda_{3}/\lambda_{3}^{\mathrm{SM}}$ are obtained from a
2-dimensional fit of the $(\mathrm{BDT_{H}},\,\mathrm{BDT_{QCD}})$ output,
following the procedure described in Appendix A. The results are shown in
Figs. 7 and 7. The various lines correspond to the different scenarios
described in Sections 4.2 and 4.3. From Figure 7 one can extract the
symmetrized 68% and 95% confidence intervals for the various scenarios. The
expected precision for $\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ is summarized
in Table 4 for each of these assumptions. Depending on the assumed scenario,
the Higgs self-coupling can be measured with a precision of 3.8-10% at 68% C.L
using the $\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ channel alone. We note that
the achievable precision is largely dependent on the assumptions on the
detector configuration and the systematic uncertainties. Such result needs to
be compared against the precision of $\delta_{\kappa_{\lambda}}$$=$3.4-7.4%,
obtained using statistical uncertainties alone. It is clear that the
performance of the detector. In particular degrading the photon and jet energy
resolution can have a substantial impact on the achievable precision.

Figure 7: Expected negative log-Likelihood scan as a function the signal strenth $\mu=\sigma/\sigma_{\mathrm{SM}}$ (a) and trilinear self-coupling modifier $\kappa_{\lambda}=\lambda_{3}/\lambda_{3}^{\mathrm{SM}}$ (b) in the $\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ channel. The various lines correspond to the different systematic uncertainty assumptions summarized in Table 3. The black dashed line shows the likelihood profile when only the statistical uncertainty is included under scenario I. @68% CL | scenario I | scenario II | scenario III
---|---|---|---
$\delta_{\mu}$ | stat only
---
stat + syst
| 2.5
---
2.8
| 3.6
---
4.4
| 5.6
---
7.5
$\delta_{\kappa_{\lambda}}$ | stat only
---
stat + syst
| 3.4
---
3.8
| 4.8
---
5.9
| 7.4
---
10.0
Table 4:  Expected precision at 68% CL on the di-Higgs production signal
strength and Higgs self coupling using the
$\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ channel at the FCC-hh with
$\mathcal{L}_{int}=30\text{ ab}^{-1}$. The symmetrized value
$\delta=(\delta^{+}+\delta^{-})/2$ is given in %.

###  6.2 The $\mathrm{\mathrm{b\bar{b}}}\tau\tau$ channel

The $\mathrm{\mathrm{b\bar{b}}}\tau\tau$ channel is very attractive thanks to
the large branching fraction (7.3%) and the relatively clean final state. As
opposed to the $\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ channel, the
$\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}}\tau\tau$ decay cannot be
fully reconstructed due to the presence of $\tau$ neutrinos in the final
state. We consider mainly two channels here: the fully hadronic final state
$\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{h}$, and the semi-leptonic one,
$\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{\ell}$($\ell=e,\mu$).

As spelled out in Section 3.2, several processes act as background for the
$\mathrm{\mathrm{b\bar{b}}}\tau\tau$ final state. The largest background
contributions are QCD and $\mathrm{t\bar{t}}$. QCD is a background mainly for
the $\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{h}$ decay channel. However, the
absence of prompt missing energy in QCD events makes this background
reducible. We have verified that it can be suppressed entirely and therefore
has been safely neglected here. Moreover, analyses using CMS data
Sirunyan:2017djm  show that QCD is overall a subdominant background at the
LHC, and negligible in the signal region. As a result recent CMS Phase II
projections neglect this background altogether CMS-PAS-FTR-18-019 . In order
of decreasing magnitude, the largest backgrounds are
$\text{Z}/\gamma^{*}\text{+jets}$ single Higgs, ttV and ttVV, where V=W,Z.

####  6.2.1 Event selection

Events are required to contain at least two b-jets with
$p_{\text{T}}(\mathrm{b})$ > 30 GeV and $|\eta(\mathrm{b})$| < 3.0. We require
at least, and not exactly, two bjets in order not to suppress the
$\mathrm{\mathrm{t\bar{t}}HH}$ signal contribution. For the
$\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{\ell}$ final state the presence is
required of exactly one isolated ($I_{\text{rel}}~{}<~{}0.1$) lepton
$\ell=e,\mu$ with $p_{\text{T}}(\ell)$ > 25 GeV and $|\eta(\ell)$| < 3.0 and
at least one hadronically tagged $\tau$-jet with $p_{\text{T}}(\tau_{h})$ > 45
GeV and $|\eta(\tau_{h})$| < 3.0. For the
$\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{h}$ final state, we require at least
two hadronically tagged $\tau$-jet with $p_{\text{T}}(\tau_{h})$ > 45 GeV and
$|\eta(\tau_{h})$| < 3.0. The hadronic $\tau$ is identified according to the
"Tight" criterion defined in Table 2. This ensures a highly efficient
rejection of the QCD background (at the cost of a smaller $\tau_{h}$
efficiency) which strengthens the solidity of our assumption of neglecting the
QCD background altogether. In what follows we refer to a $\tau$-candidate as
the lepton $\ell=e,\mu$ or the $\tau$-jet. In particular the $\tau$ 4-momentum
is defined as the sum of the 4-momenta of the visible $\tau$ decay products.
In order to maximally exploit the kinematic differences between the signal and
the dominant $\mathrm{t\bar{t}}$ background, we build a multivariate BDT
discriminant using as an input the following kinematic properties:

  * •

The 3-vector components of the leading ($\tau_{1}$) and subleading
$\tau$-candidate ($\tau_{2}$): transverse momentum
($p_{\text{T}}^{\mathrm{\tau_{1}}}$, $p_{\text{T}}^{\mathrm{\tau_{2}}}$),
pseudo-rapidity ($\eta_{\tau_{1}}$, $\eta_{\tau_{2}}$), and azimutal angle
($\phi_{\tau_{1}}$, $\phi_{\tau_{2}}$).

  * •

The 3-vector components of the leading ($\mathrm{b}_{1}$) and subleading b-jet
($\mathrm{b}_{2}$): transverse momentum
($p_{\text{T}}^{\mathrm{\mathrm{b}_{1}}}$,
$p_{\text{T}}^{\mathrm{\mathrm{b}_{2}}}$), pseudo-rapidity
($\eta_{\mathrm{b}_{1}}$, $\eta_{\mathrm{b}_{2}}$), and azimutal angle
($\phi_{\mathrm{b}_{1}}$, $\phi_{\mathrm{b}_{2}}$).

  * •

The 3-vector components of the leading ($\mathrm{j}_{1}$) and subleading
additional reconstructed jets in the event ($\mathrm{j}_{2}$): transverse
momentum ($p_{\text{T}}^{\mathrm{\mathrm{j}_{1}}}$,
$p_{\text{T}}^{\mathrm{\mathrm{j}_{2}}}$), pseudo-rapidity
($\eta_{\mathrm{j}_{1}}$, $\eta_{\mathrm{j}_{2}}$), and azimutal angle
($\phi_{\mathrm{j}_{1}}$, $\phi_{\mathrm{j}_{2}}$). If no additional jets are
found, dummy values are given to these variables.

  * •

The 4-vector components of the $\mathrm{H}\rightarrow\tau\tau$ candidate:
transverse momentum ($p_{\text{T}}^{\mathrm{\tau\tau}}$), pseudo-rapidity
($\eta_{\tau\tau}$), azimutal angle ($\phi_{\tau\tau}$) and invariant mass
($m_{\tau\tau}$).

  * •

The 4-vector components of the $\mathrm{H\rightarrow b\bar{b}}$ candidate:
transverse momentum ($p_{\text{T}}^{\mathrm{\mathrm{bb}}}$), pseudo-rapidity
($\eta_{\mathrm{bb}}$), azimutal angle ($\phi_{\mathrm{bb}}$) and invariant
mass ($m_{\mathrm{{bb}}}$).

  * •

The 4-vector components of the Higgs pair candidate: transverse momentum
($p_{\text{T}}^{\mathrm{\mathrm{hh}}}$), pseudo-rapidity
($\eta_{\mathrm{hh}}$), azimutal angle ($\phi_{\mathrm{hh}}$) and invariant
mass ($m_{\mathrm{hh}}$).

  * •

The transverse missing energy $p_{\text{T}}^{\textrm{miss}}$.

  * •

The transverse mass of each $\tau$-candidate, computed as
$m_{\textrm{T}}=\sqrt{2p_{\text{T}}^{\mathrm{\tau}}p_{\text{T}}^{\textrm{miss}}-\vec{p_{\text{T}}}^{\tau}\cdot\vec{p_{\text{T}}}^{\textrm{miss}}}$.

  * •

The event s-transverse mass $m_{\textrm{T2}}$ as defined in Refs Lester:1999tx
; Barr:2003rg .

  * •

The number of reconstructed b-jets $N_{b}$, the number of light jets $N_{l}$
and the total number of jets $N_{j}=N_{b}+N_{l}$.

The $m_{\tau\tau}$ and $m_{\textrm{T2}}$ observables are shown in Figs. 8,9
and 8,9 for the $\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{h}$ and
$\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{\ell}$ final states respectively.
These provide, together with the $m_{\mathrm{{bb}}}$ observable, the largest
discrimination against the $\mathrm{t\bar{t}}$ background. Additional
discrimination against the $\mathrm{t\bar{t}}$ background is provided by
$p_{\text{T}}^{\mathrm{\tau\tau}}$ and $p_{\text{T}}^{\mathrm{\mathrm{bb}}}$
and the $p_{\text{T}}^{\textrm{miss}}$ followed by $m_{\mathrm{hh}}$,
$\eta_{\tau\tau}$, $\eta_{\mathrm{bb}}$ and the $p_{\text{T}}$ and $\eta$ of
$\tau_{2}$, $\tau_{1}$, $\mathrm{b}_{2}$, $\mathrm{b}_{1}$, $\mathrm{j}_{2}$
and $\mathrm{j}_{1}$, and finally $N_{j}$. The output of the BDT discriminant
is shown in Figs. 8 and 9 for the $\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{h}$
and $\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{\ell}$ final states.

Figure 8: Distributions in the $\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{h}$
final state of the invariant mass of the $\tau\tau$ pair (left),
$m_{\textrm{T2}}$ (center), and output of the BDT multi-variate discriminant
(right).

Figure 9: Distributions in the $\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{\ell}$
final state of the invariant mass of the $\tau\tau$ pair (left),
$m_{\textrm{T2}}$ (center), and output of the BDT multi-variate discriminant
(right).

####  6.2.2 Signal Extraction and results

The expected precision on the signal strength and the Higgs self-coupling are
derived from a maximum likelihood fit on the BDT observable, according to the
prescription described in Appendix A. The
$\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{h}$ and
$\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{\ell}$ channels are considered
separately with their relative set of systematic uncertainties and then
combined assuming a 100% correlation on equal sources of uncertainties among
the two channels. The combined expected precision on the
$\mathrm{\mathrm{b\bar{b}}}\tau\tau$ channel is shown in Figs. 10 and 10. The
various lines correspond to the different scenarios described in Sections 4.2
and 4.3. From Fig. 10 one can extract the 68% and 95% confidence intervals for
the various systematics assumptions. Depending on the assumed scenario, using
the $\mathrm{\mathrm{b\bar{b}}}\tau\tau$ channel, the Higgs pair signal
strength and Higgs self-coupling can be measured respectively with a precision
of $\delta_{\mu}=5.8-7.9\%$ and $\delta_{\kappa_{\lambda}}=9.8-13.8\%$ at 68%
C.L 777We note that the precision on the signal strength is higher for
$\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{\ell}$ compared to
$\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{h}$. The opposite is true of the
self-coupling precision. This inversion can be explained by a smaller slope
$\frac{d\mu}{d\kappa_{\lambda}}\Bigr{|}_{\text{SM}}$ (see Section 5) for the
$\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{\ell}$ channel, caused by different
kinematic and acceptance requirements.. Despite the large signal event rate in
the $\mathrm{\mathrm{b\bar{b}}}\tau\tau$ channel, the sensitivity is limited
by the large background. Therefore, the $\mathrm{\mathrm{b\bar{b}}}\tau\tau$
channel is statistically dominated at the FCC-hh with
$\mathcal{L}_{int}=30\text{ ab}^{-1}$, and the achievable precision is only
moderately dependent on the assumptions on the systematic uncertainties. We
note that Ref. Banerjee:2018yxy  quotes a precision of
$\delta_{\kappa_{\lambda}}=13\%$ using the resolved semi-leptonic final state,
which is consistent with the result presented here of
$\delta_{\kappa_{\lambda}}=14-18\%$. The same study shows that further
precision can be obtained using the boosted topology, which has not been
considered here.

Figure 10: Expected negative log-Likelihood scan as a function the signal strength $\mu=\sigma/\sigma_{\mathrm{SM}}$ (a) and trilinear self-coupling modifier $\kappa_{\lambda}=\lambda_{3}/\lambda_{3}^{\mathrm{SM}}$ (b) in the $\mathrm{\mathrm{b\bar{b}}}\tau\tau$ channel (combination of the $\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{h}$ and $\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{\ell}$ channels). The various lines correspond to the different systematic uncertainties assumptions summarized in Table 3. The black dashed line shows the likelihood profile when only the statistical uncertainty is included under scenario I. @68% CL | scenario I | scenario II | scenario III
---|---|---|---
$\delta_{\mu}$($\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{h}$)  | stat only
---
stat + syst
| 8.7
---
9.0
| 10.7
---
11.1
| 10.9
---
11.6
$\delta_{\kappa_{\lambda}}$($\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{h}$)  | stat only
---
stat + syst
| 12.6
---
13.2
| 16.0
---
16.7
| 16.5
---
17.7
$\delta_{\mu}$($\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{\ell}$)  | stat only
---
stat + syst
| 7.4
---
7.6
| 8.2
---
8.8
| 9.2
---
10.4
$\delta_{\kappa_{\lambda}}$($\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{\ell}$)  | stat only
---
stat + syst
| 14.3
---
14.7
| 16.3
---
17.5
| 18.6
---
21.0
$\delta_{\mu}$(comb.)  | stat only
---
stat + syst
| 5.6
---
5.8
| 6.5
---
7.0
| 7.0
---
7.9
$\delta_{\kappa_{\lambda}}$(comb.)  | stat only
---
stat + syst
| 9.4
---
9.8
| 11.4
---
12.2
| 12.3
---
13.8
Table 5:  Expected precision at 68% CL on the di-Higgs production cross-
section and Higgs self coupling using the $\mathrm{\mathrm{b\bar{b}}}\tau\tau$
channel at the FCC-hh with $\mathcal{L}_{int}=30\text{ ab}^{-1}$. The
symmetrized value $\delta=(\delta^{+}+\delta^{-})/2$ is given in %.

###  6.3 The $\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ channel

The $\mathrm{HH}\rightarrow\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ decay
mode has the largest branching fraction among all possible Higgs-pair decays.
Despite the presence of soft neutrinos from semi-leptonic b decays (that may
impact negatively the reconstructed hadronic Higgs-mass resolution), the Higgs
decays into b-jets can be fully reconstructed. However, due do the fully
hadronic nature of this decay mode, this channel suffers from the presence of
very large QCD backgrounds and hence features a relatively small
$\mathrm{S/B}$. Moreover, a combinatorial ambiguity affects the possibility to
correctly associate the four b-jets to the two parent Higgs candidates.

We consider mainly the case where the Higgs candidates are only moderately
boosted, leading to four fully resolved b-jets. The boosted analysis, where
the Higgs candidates are sufficiently boosted to decay into a single large
radius jet Behr:2015oqq ; deLima:2014dta , provides less sensitivity to the
self-coupling measurement and was discussed in previous
studiesBanerjee:2018yxy ; L.Borgonovi:2642471 . The main backgrounds to this
final state are QCD and $\mathrm{t\bar{t}}$, followed by $\mathrm{Zb\bar{b}}$,
single-Higgs production and ZZ.

####  6.3.1 Event selection

In order to fulfill our initial assumption of fully efficient online triggers,
the event selection starts by requiring the presence of at least four b-jets
with $p_{\text{T}}(\mathrm{b})$ > 30 GeV and $|\eta(\mathrm{b})$| < 4.0. The
b-jets are identified with the "Medium" working point defined in Table 2. The
Higgs candidates are reconstructed as the pairing of b-jet pairs that
minimizes the difference between the invariant masses of the two b-jet pairs.
The Higgs candidate with the largest (smallest) $p_{\text{T}}$ is named
$h_{1}$ ($h_{2}$).

The following variables are then used as input to a multivariate BDT
discriminant to ensure an optimal discrimination versus the dominant QCD
background:

  * •

The 3-vector components of the four leading b-jets in the event ($b_{1}$,
$b_{2}$, $b_{3}$, $b_{4}$): transverse momenta
($p_{\text{T}}^{\mathrm{b_{i}}}$), pseudo-rapidities ($\eta_{b_{i}}$), and
azimutal angles ($\phi_{b_{i}}$, $\text{i}=1..4$).

  * •

The 3-vector components of the leading ($\mathrm{j}_{1}$) and subleading
additional reconstructed jets in the event ($\mathrm{j}_{2}$): transverse
momentum ($p_{\text{T}}^{\mathrm{\mathrm{j}_{1}}}$,
$p_{\text{T}}^{\mathrm{\mathrm{j}_{2}}}$), pseudo-rapidity
($\eta_{\mathrm{j}_{1}}$, $\eta_{\mathrm{j}_{2}}$), and azimutal angle
($\phi_{\mathrm{j}_{1}}$, $\phi_{\mathrm{j}_{2}}$). If no additional jets are
found, dummy values are given to these variables.

  * •

The 4-vector components of the leading ($h_{1}$) and subleading ($h_{2}$)
Higgs candidates: transverse momentum ($p_{\text{T}}^{\mathrm{h_{1}}}$,
$p_{\text{T}}^{\mathrm{h_{2}}}$), pseudo-rapidity ($\eta_{h_{1}}$,
$\eta_{h_{2}}$), azimutal angle ($\phi_{h_{1}}$, $\phi_{h_{2}}$) and invariant
mass ($m_{h_{1}}$, $m_{h_{2}}$).

  * •

The 4-vector components of the Higgs-pair candidate: transverse momentum
($p_{\text{T}}^{\mathrm{\mathrm{hh}}}$), pseudo-rapidity
($\eta_{\mathrm{hh}}$), azimutal angle ($\phi_{\mathrm{hh}}$) and invariant
mass ($m_{\mathrm{hh}}$).

The $m_{h_{1}}$ and $m_{\mathrm{hh}}$ observables are shown respectively in
Figs. 11 and 11. The $m_{h_{1}}$ distribution shows that the procedure
described above correctly associates the b-jet pairs to the parent Higgs
particle for the signal, and to the parent Z particle for the
$\mathrm{Zb\bar{b}}$ and ZZ backgrounds. Thanks to their resonant nature, the
$m_{h_{1}}$ and $m_{h_{2}}$ distributions provide the largest discrimination
against the QCD background. A substantial discrimination against the QCD
background is provided, in decreasing order of importance, by
$p_{\text{T}}^{\mathrm{b_{2}}}$, $p_{\text{T}}^{\mathrm{b_{4}}}$,
$p_{\text{T}}^{\mathrm{b_{3}}}$ and $p_{\text{T}}^{\mathrm{b_{1}}}$, followed
by $m_{\mathrm{hh}}$, $p_{\text{T}}^{\mathrm{\mathrm{hh}}}$,
$p_{\text{T}}^{\mathrm{\mathrm{j}_{1}}}$,
$p_{\text{T}}^{\mathrm{\mathrm{j}_{2}}}$, and the $\eta_{b_{i}}$. The output
of the BDT discriminant is shown in Fig. 11 for the signal and various
background contributions.

Figure 11: Distributions in the $\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$
final state of the highest $p_{\text{T}}$ reconstructed Higgs invariant mass
(left), the Higgs pair invariant mass (center), and the output of the BDT
multi-variate discriminant (right).

####  6.3.2 Results

The expected precision on the signal strength and the Higgs self-coupling are
derived from a 1D maximum likelihood fit on the BDT discriminant, according to
the prescription described in Section A. The combined expected precision on
the $\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ channel is shown in Figs. 12
and 12. The coloured lines correspond to the different systematic
uncertainties assumptions summarized in Table 3. The 68% and 95% confidence
intervals on $\delta_{\mu}$ and $\delta_{\kappa_{\lambda}}$ for the various
systematics assumptions can be extracted from Figs. 12 and 12 and the results
are summarized in Table 6. Depending on the assumed scenario, using the
$\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ channel, the Higgs pair signal
strenth and Higgs self-coupling can be measured respectively with a precision
of $\delta_{\mu}=8-18\%$ and $\delta_{\kappa_{\lambda}}=18-32\%$ at 68% C.L.
As for the $\mathrm{\mathrm{b\bar{b}}}\tau\tau$ case, due to the huge QCD
background this channel is statistically limited. We note that, despite the
large statistical uncertainty, the systematic uncertainties play a large role
in this channel. This is the result of the significant contamination in the
signal region from single-Higgs background events. Since this background is
assumed to be estimated from Monte Carlo, its uncertainty, even though at the
percent level, is reflected in a larger signal systematics.

Figure 12: Expected Negative log-Likelihood scan as a function the signal strength $\mu=\sigma/\sigma_{\mathrm{SM}}$ (a) and trilear self-coupling modifier $\kappa_{\lambda}=\lambda_{3}/\lambda_{3}^{\mathrm{SM}}$ (b) in the $\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ channel. The various lines correspond to the different systematic uncertainties assumptions summarized in Table 3. The black dashed line shows the likelihood profile when onlythe statistical uncertainty is included under scenario I. @68% CL | scenario I | scenario II | scenario III
---|---|---|---
$\delta_{\mu}$ | stat only
---
stat + syst
| 8.4
---
10.4
| 9.1
---
12.2
| 10.8
---
17.9
$\delta_{\kappa_{\lambda}}$ | stat only
---
stat + syst
| 18.0
---
22.3
| 20.0
---
27.1
| 24.2
---
32.0
Table 6:  Expected precision at 68% CL on the di-Higgs production cross-
section and Higgs self coupling using the
$\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ channel at the FCC-hh with
$\mathcal{L}_{int}=30\text{ ab}^{-1}$. The symmetrized value
$\delta=(\delta^{+}+\delta^{-})/2$ is given in %.

###  6.4 Combined precision

Figure 13: Expected negative log-Likelihood scan as a function of the
trilinear self-coupling modifier
$\kappa_{\lambda}=\lambda_{3}/\lambda_{3}^{\mathrm{SM}}$ in all channels, and
their combination. The solid line corresponds to the scenario II for
systematic uncertainties. The band boundaries represent respectively scenario
I and III. The dashed line represents the sensitivity obtained including
statistical uncertainties only, under the assumptions of scenario I. Figure
14: Expected precision on the Higgs self-coupling as a function of the
integrated luminosity.

When combining results from the various channels, the systematic uncertainties
from the various sources affecting those processes that we assume to be
estimated from Monte Carlo simulations (HH, single Higgs, and ZZ) are
accounted for as follows.
Lepton (e/$\mu$, $\tau$) uncertainties are correlated across all process and
across the $\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{h}$ and
$\mathrm{\mathrm{b\bar{b}}}\tau_{h}\tau_{\ell}$ channels. In the
$\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ channel, the photon uncertainty for
the single and double Higgs processes are correlated. The luminosity
uncertainty is correlated across all these processes and all channels. The
same applies, for each process independently, to the overall normalisation
uncertainty. The uncertainties on lepton ID, luminosity and normalisation are
assumed to affect only the overall normalization of signal and background
shapes and to not introduce a significant deformation of their shapes. For the
b-jets ID, we take into account both the shape and normalisation uncertainty
due to the b-jets systematic uncertainties. This is achieved by shifting the
efficiency for each jet by the ($p_{\text{T}}$-dependent) values reported in
Table 2, and re-computing the resulting BDT distributions. As discussed in
section 4.3, this uncertainty is correlated across different channels when it
affects the same process. Overall normalisation uncertainties are cancelled
out when a background is estimated from a control region. Moreover, we expect
all control regions to be well-populated at the FCC-hh. For these reasons, we
assume the systematic uncertainties affecting those backgrounds that are most
likely to be estimated from well populated control regions or side bands, such
as QCD, Zbb, photon(s)+jets, and $\mathrm{t\bar{t}}$, to be negligible and we
do not include them in the fit procedure.

The combined expected negative log-Likelihood scan is shown in Fig. 13. The
expected precision for the single channels is also shown. For completeness, we
introduced in the combination also the
$\mathrm{\mathrm{b\bar{b}}ZZ~{}(4}\ell\mathrm{)}$ channel, which provides a
sensitivity similar to the 4b channel. This decay channel was not re-optimized
in this study and the result of the analysis is documented in Ref
L.Borgonovi:2642471 . The expected combined precision on the Higgs self-
coupling obtained after combining the channels
$\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$,
$\mathrm{\mathrm{b\bar{b}}}\tau\tau$,
$\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ and
$\mathrm{\mathrm{b\bar{b}}ZZ~{}(4}\ell\mathrm{)}$ can be inferred from the
intersection of black curves with the horizontal 68% and 95% CL lines. The
expected statistical precision for Scenario I, neglecting systematic
uncertainties, can be read from the dashed black line in Fig. 13, and gives
$\delta_{\kappa_{\lambda}}=3.0$% at 68% CL. The solid line corresponds to
scenario II, while the boundaries of the shaded area represent respectively
the alternative scenarios I and III. From the shaded black curve one can infer
the final precision when including systematic uncertainties. Depending on the
assumptions, the expected precision for the Higgs self-coupling is
$\delta_{\kappa_{\lambda}}=3.4-7.8$% at 68% CL. The signal strength and self-
coupling precision for the combination are summarized in Table 7.

The expected precision on the Higgs self-coupling as a function of the
integrated luminosity is shown in Fig. 14, for the three scenarios, with and
without systematic uncertainties. With the most aggressive scenario I, a
precision of $\delta_{\kappa_{\lambda}}=10\%$ can be reached with only 3 ab-1
of integrated luminosity, whereas approximately 20 ab-1 are required for the
most conservative scenario III. Therefore, assuming scenario I, the 10% target
should therefore be achievable during the first 5 years of FCC-hh operations,
combining the datasets of two experiments. Even including the duration of the
FCC-ee phase of the project, and the transition period from FCC-ee to FCC-hh,
this timescale is competitive with the time required by the proposed future
linear colliders, which to achieve this precision need to complete their full
programme at the highest beam energies.

As already discussed, the value of the self-coupling coupling can
significantly alter both the Higgs pair production cross section and the event
kinematic properties. In order to explore the sensitivity to possible BSM
effects in Higgs pair production, a multivariate BDT discriminant was
optimised against the backgrounds for several values of $\kappa_{\lambda}$ in
the range $0<\kappa_{\lambda}<3$, in order to maximise the achievable
precision for values of $\kappa_{\lambda}\neq 1$. The BDT training has been
performed only for the $\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ channel, which
dominates the overall sensitivity, whereas for the other channels we
conservatively employ the BDT trained at $\kappa_{\lambda}=1$. The obtained
precision as a function of $\kappa_{\lambda}$ is shown in Fig. 15 888We stress
once more that, as discussed in Section 2, precision projections for
$\kappa_{\lambda}\neq 1$ are tied to a scenario in which only $\lambda_{3}$ is
modified, and other BSM effects on the HH cross section are assumed to be
negligible. For a recent study of the BSM modifications to kinematical
distributions in presence of multiple anomalous couplings, see Ref.
Capozi:2019xsi ..

It can be seen that the overall precision follows the behaviour of the HH
production cross section as function of $\kappa_{\lambda}$ given in Figure 2.
The best precision, $\delta_{\kappa_{\lambda}}$$\approx$ 2%, is reached at
$\kappa_{\lambda}=0$ where the value $\frac{d\mu}{d\kappa_{\lambda}}$ is
large. Conversely, the maximum uncertainty
$\delta_{\kappa_{\lambda}}$$\approx$ 60% is obtained at
$\kappa_{\lambda}\approx 2.5$, and corresponds to the minimum of the total HH
cross section, where $\frac{d\mu}{d\kappa_{\lambda}}$ $\to 0$ . As can be
expected, the likelihood function presents a broad second minimum999The first
minimum being at the probed value of $\kappa_{\lambda}$ in correspondence of
the minimum of the HH cross section at $\kappa_{\lambda}=2.5$. The presence of
this minimum is the reason behind the asymmetric behaviour of the
uncertainties for the points near $\kappa_{\lambda}=2.5$. If the measurement
is performed close enough to $\kappa_{\lambda}=2.5$ the likelihood falls in
the second minimum before reaching the 68% C.L. threshold, thus enlarging the
measurement uncertainty in one direction. It should be noted that, while the
HH cross section is roughly symmetric around $\kappa_{\lambda}=2.5$, we do not
expected the uncertainties to be symmetric as well, as the kinematic behaviour
of the HH system are quite different between $\kappa_{\lambda}<2.5$ and
$\kappa_{\lambda}>2.5$. It can also be noticed that when switching on the
systematic uncertainties the precision at small $\kappa_{\lambda}$ degrades
compared to the SM case. This reflects the fact that the HH kinematics at
$\kappa_{\lambda}\approx 0$ are similar to the single-Higgs background.

@68% CL | scenario I | scenario II | scenario III
---|---|---|---
$\delta_{\mu}$ | stat only
---
stat + syst
| 2.2
---
2.4
| 2.8
---
3.5
| 3.7
---
5.1
$\delta_{\kappa_{\lambda}}$ | stat only
---
stat + syst
| 3.0
---
3.4
| 4.1
---
5.1
| 5.6
---
7.8
Table 7:  Combined expected precision at 68% CL on the di-Higgs production
cross- and Higgs self coupling using all channels at the FCC-hh with
$\mathcal{L}_{int}=30\text{ ab}^{-1}$. The symmetrized value
$\delta=(\delta^{+}+\delta^{-})/2$ is given in %. Figure 15: Expected
precision on the Higgs self-coupling as a function of the $\kappa_{\lambda}$
value for each scenario. To improve readability, the positions of scenario I
(III) bands are slightly offset in the negative (positive) direction along the
$\kappa_{\lambda}$ axis.

##  7 Conclusions and perspectives

The precise measurement of the Higgs self-coupling must be a top priority of
future high-energy collider experiments. Previous studies on the potential of
a 100 TeV pp collider have discussed the sensitivity of various decay
channels, often based on simple rectangular cut-based analyses 101010Just
before the public release of this work, we learned of a similar study
presented in Ref. Park:2020yps , using a multivariate analysis of the
$\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ final state. While many aspects of
the two studies are different, in particular for what concerns the
consideration of systematic uncertainties, there is quantitative agreement on
the improvements induced by the use of multivariate analysis.. In the present
study the measurement strategy has been optimized in the
$\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$, $\mathrm{\mathrm{b\bar{b}}}\tau\tau$
and $\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ channels using machine
learning techniques. For the first time, a precise set of assumptions of
detector performances and possible sources of systematic uncertainties has
been defined and used to derive the achievable precision. Consistently with
our previous findings, the $\mathrm{\mathrm{b\bar{b}}}\gamma\gamma$ channel
drives the final sensitivity, with an expected precision of
$\delta_{\kappa_{\lambda}}=3.8-10.0\%$ depending on the detector and
systematic assumptions. The $\mathrm{\mathrm{b\bar{b}}}\tau\tau$ and
$\mathrm{\mathrm{b\bar{b}}\mathrm{b\bar{b}}}$ channels provide instead a less
precise single channel measurement, respectively of
$\delta_{\kappa_{\lambda}}=10-14\%$ and $\delta_{\kappa_{\lambda}}=22-32\%$.

The final combined sensitivity across all considered channels leads to an
expected precision at the FCC-hh on the Higgs self-coupling
$\delta_{\kappa_{\lambda}}=3.4-7.8\%$ with an integrated luminosity of
$\mathcal{L}_{int}=30\text{ ab}^{-1}$. By considering the most agressive
detector and systematics assumptions, the 10% threshold can be achieved with
$\sim 3$ ab-1, corresponding to $\sim 3-5$ years of early running at the
start-up luminosity of $5\times 10^{34}$ cm-2 s-1.

This work shifts the perspective on the Higgs self-coupling ultimate precision
at FCC from being statistics dominated to systematics dominated. This is a
crucial new development: on one side it gives us confidence that the design
parameters of FCC-hh are well tailored to reach, unique among all proposed
future collider facilities, the few-percent level of statistical precision. On
the other hand, it calls for a more thorough assessment of all systematic
uncertainties. For example, we should validate the estimates presented in this
work through full simulations of more realistic detector designs in the
presence of pile-up, and explore all other possible handles to further reduce
them. The huge FCC statistics will provide multiple control samples, well
beyond what discussed in our paper, that could be used for these purposes, and
in particular to pin down the background rates with limited reliance on
theoretical calculations. At this level of precision, however, the theoretical
uncertainties on the HH signal will play an important role in the extraction
of the self-coupling from the measured production rate. As indicated in
Section 4.3, we assumed in our study an uncertainty on the HH cross sections
ranging from 0.5 to 1.5%. This would need the theoretical predictions to
improve relative to today’s knowledge, requiring the extension of the
perturbative order by at least one order beyond today’s known NLO with full
top mass dependence Borowka:2016ypz , and possibly beyond the N3LO in the
$m_{top}\to\infty$ limit, recently achieved Chen:2019lzz ; Chen:2019fhs . This
will be very challenging, and it is impossible today to estimate the
asymptotic reach in theoretical precision. Nevertheless, the innovative
technical progress we have witnessed in the recent years encourages us to
assume that the necessary improvements are possible within the several decades
that separate us from the first FCC-hh run.

In conclusion, this study strengthens the evidence that a 100 TeV pp collider,
with integrated luminosity above 3 ab-1, can measure the Higgs self-coupling
more precisely than any other proposed project, on a competitive time scale.

###### Acknowledgements.

We would like to thank the FCC group at CERN. In particular we thank Alain
Blondel and Patrick Janot for helpful suggestions, Clement Helsens, Valentin
Volkl and Gerardo Ganis for the valuable help and support on the FCC software.
We also thank Gudrun Heinrich and Stephen Jones for helpful discussions on NLO
Higgs pair event generation. Finally, we would like to acknowledge the
$\mathrm{\mathrm{b\bar{b}}ZZ~{}(4}\ell\mathrm{)}$ channel study from Lisa
Borgonovi, Elisa Fontanesi and Sylvie Brabant, which we have used as one of
the inputs for our determination of the self-coupling combined sensitivity.

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##  Appendix A Statistical procedure

The statistical methodology used in this paper relies on the strategy adopted
by the ATLAS and CMS Collaborations, and described in Ref. LHC-HCG . A
detailed description of the procedures used in this paper are described in
more detail in Refs. Chatrchyan:2012tx ; Chatrchyan:2012ufa . The Combine
software package combine  has been used as statistical and fitting tool to
produce the final results. Combine is based on the standard LHC data modeling
and handling toolkits RooFit Verkerke:2003ir  and RooStats Moneta:2010pm  and
it is developed and maintained by the CMS collaboration.

The parameter of interest (POI) tested in these results are either the
trilinear coupling modifier
$\kappa_{\lambda}=\lambda_{3}/\lambda_{3}^{\mathrm{SM}}$ or the double Higgs
signal strength $\mu=\sigma/\sigma_{\mathrm{SM}}$, defined as the ratio
between the (expected) measured double Higgs yield and its SM expectation.

In the model, the POI $\alpha=k_{\lambda}$ or $\alpha=\mu$ is estimated with
its corresponding confidence intervals using a profile likelihood ratio test
statistic $q(\alpha)$ Cowan:2010st ; LHC-HCG , in which experimental or
theoretical uncertainties are incorporated via nuisance parameters (NP). Given
a of POI $\alpha$ that depends on the set of NP $\vec{\theta}$, $q(\alpha)$ is
defined as:

$q(\alpha)=-2\ln\left(\frac{L\big{(}\alpha\,,\,{\hat{\vec{\theta}}}_{\alpha}\big{)}}{L(\hat{\alpha},\hat{\vec{\theta}})}\right).$
(4)

An individual NP represents a single source of systematic uncertainty. Its
effect is therefore considered fully correlated between all of the the final
states included in the fit that share a dependency on it, as will be discussed
later in this section.

The quantities $\hat{\alpha}$ and $\hat{\vec{\theta}}$ denote the
unconditional maximum likelihood estimates of the parameter value, while
$\hat{\vec{\theta}}_{\alpha}$ denotes the conditional maximum likelihood
estimate for fixed values of the POI $\alpha$.

The likelihood functions in the numerator and denominator of Eq. (4) are
constructed using products of probability density functions (PDFs) of signal
and background for the various discriminating variables used in the input
analyses, as well as constraint terms for the NPs. The PDFs are built from the
BDT distributions described in Section 6. It should be noted that while the
signal shape depends on the value on $\kappa_{\lambda}$, that dependence is
relatively mild. Given the expected precision of $\mathcal{O}(10\%)$ on the
measurement of $k_{\lambda}$ at the FCC-hh, the effect of its variation on the
signal lineshape is minimal when performing the measurement at a given value
of $\kappa_{\lambda}$ and can be safely neglected. The effects on acceptance
and selection efficiencies of varying $k_{\lambda}$ or $\mu$ are instead
considered in the fit. The expected precision on $\kappa_{\lambda}$ and $\mu$
is assessed by performing a likelihood fit on a a pseudo-data set that has
been constructed assuming $\mu=1$ and $k_{\lambda}=1$, using the asymptotic
approximation as described in Cowan:2010st ; Junk:2016jag .