# TASI Lectures on Future Colliders

TH Department, CERN, 1211 Geneva, Switzerland
E-mail

###### Abstract:

These lectures review the main motivations for future high-energy colliders,
focusing on the understanding of electroweak symmetry breaking and on the
search for physics beyond the Standard Model. The open questions and the
challenges are common to all future projects; for concreteness, I will use
studies of the potential of $\rm e^{+}e^{-}$ and pp circular colliders to
provide examples of the anticipated physics reach.

CERN-TH-2019-073

##  1 Introduction

Several projects for future high-energy particle colliders are under
consideration in various regions worldwide, to complement and extend the
physics reach of CERN’s Large Hadron Collider (LHC). These include:

  * •

the $\rm e^{+}e^{-}$ International Linear Collider (ILC [1, 2, 3, 4]),
exploiting an established acceleration technology based on superconducting
(SC) radio frequency (RF) cavities, operating initially at $\sqrt{s}=250$ GeV,
with an upgrade path in principle up to 1000 GeV.

  * •

the $\rm e^{+}e^{-}$ Compact Linear Collider (CLIC [5, 6, 7, 8]), relying on
the RF field produced by a drive beam, to induce the accelerating gradients
required to push the center of mass energy from an initial value of
$\sqrt{s}=380$ GeV, up to 1.5 and eventually 3 TeV.

  * •

the Future Circular Collider (FCC [9, 10, 11, 12]) and the CEPC/SppC [13, 14],
two similar projects promoted by CERN and China respectively, envisioning a
staged facility, enabled by a 100 km circular ring, designed to deliver $\rm
e^{+}e^{-}$ collisions at energies in the range $M_{Z}<\sqrt{s}<365$ GeV, pp
collisions up to $\sqrt{S}\sim 100$ TeV, ep collisions at $\sqrt{S}=3.5$ TeV,
as well as heavy ion collisions.

This variety of layouts (circular or linear), beam types (electrons or
protons) and energies, reflects slightly different priorities for the physics
targets and observables, as well as a different judgement on the overall
balance between physics returns, technological challenges and feasibility,
time scales for completion and exploitation, and financial/political
realities.

If approved today, the $\rm e^{+}e^{-}$ projects in this list could in
principle begin delivering physics results at some point during the decade
2030-40, and operate for the following 15-25 years, depending on the
technology and upgrade path. Beyond this, but with an unspecified time scale,
ideas are on the table for a possible subsequent generation of even more
ambitious lepton-collider projects, which I only mention here: linear electron
accelerators based on multi-GeV/m gradient technologies like plasma wake
fields or lasers [15], and muon circular colliders [16, 17].

The FCC and SppC proton colliders would face a preparatory phase longer than
the $\rm e^{+}e^{-}$ colliders, mostly because of the R&D period required to
produce reliable and affordable SC bending magnets with the 16 T magnetic
field needed to keep 50 TeV protons in orbit in the 100 km ring (A 12 T
option, based on high-Tc SCs, is considered for SppC, but is itself far from
being established).

No matter what the energy or the technology, all these projects share common
goals, driven by the need to clarify several outstanding open issues in
particle physics. This need singles out the next generation of colliders
beyond the LHC as unique and indispensable exploratory tools to continue
driving the progress in our understanding of nature.

In these lectures, I will review the main motivations for future high-energy
colliders and discuss their physics potential. I will mostly cover topics such
as Higgs and physics beyond the Standard Model (BSM), and, while I will recall
some basic theoretical background, I will give for granted that students know
those topics from their studies, or from other lectures in the TASI program.
In particular, in 2018 students have been exposed, among others, to excellent
lectures on the Higgs boson [18]), supersymmetry and dark matter [19], QCD at
colliders [20], effective field theory [21] and flavour [22]. I refer to
these, and to the great lectures on LHC physics by our host Tilman Plehn [23],
for the necessary background material.

I will illustrate the value of the physics reach through concrete examples of
the FCC physics potential. No attempt is made here to compare the FCC against
the other projects, as the point of these lectures is not to promote one
project over another: I choose here the FCC since it is the project that I
know better, and the one that, in terms of breadth, variety and physics
performance, best illustrates how ambitious the targets of a future collider
can be.

I will include also a few exercises here and there. They are meant to
stimulate your thinking, they are simple, do not necessarily require
calculations and are mostly for a qualitative discussion. But if you take them
seriously, some of them could be the seed for interesting work!

##  2 Where we stand

In almost ten years of studies at the LHC, the picture of the particle physics
landscape has greatly evolved. The legacy of this first phase of the LHC
physics programme can be briefly summarised in three points: a) the discovery
of the Higgs boson, and the start of a new phase of detailed studies of its
properties, aimed at revealing the deep origin of electroweak (EW) symmetry
breaking; b) the indication that signals of new physics around the TeV scale
are, at best, elusive; c) the rapid advance of theoretical calculations, whose
constant progress and reliability underline the key role of ever improving
precision measurements, from the Higgs to the flavour sectors. Last but not
least, the LHC success has been made possible by the extraordinary
achievements of the accelerator and of the detectors, whose performance is
exceeding all expectations, supporting the confidence in the ability of the
next generation of colliders to achieve what they promise.

###  2.1 The puzzling origin of the Higgs field

The years that preceded the discovery of the Higgs boson have been
characterized by a general strong belief in its existence, justified by the
success of the Standard Model (SM), and by the confidence that EW symmetry
breaking (EWSB) is indeed driven by the basic dynamics of the Higgs mechanism,
as described in the SM. Starting from this assumption, the theoretical
speculations focused on identifying possible solutions to the hierarchy
problem, namely the extreme fine tuning required to achieve the decoupling of
the Higgs and EW mass scale from the phenomena expected to emerge at much
higher energy scales, up to the Planck scale. These speculations led to the
consideration of several possible scenarios of new physics, from supersymmetry
to large extra dimensions, which would provide natural solutions to the
hierarchy problem by introducing new degrees of freedom, new symmetries, or
new dynamics at the TeV scale. The opportunity to combine the solution of the
hierarchy problem with the understanding of experimental facts such as the
existence of dark matter, or of the features of flavour phenomena, gave
further impetus to these theoretical efforts, and to the many experimental
studies dedicated to the search for BSM manifestations.

The conceptual simplicity and appeal of several of these scenarios, justified
optimism that their concrete manifestations would appear “behind the corner”.
After all, the SM itself was born as the simplest possible model in which to
embed an elegant explanation (the Higgs mechanism) to the problem of
justifying the mass of the weak force carrier and parity non-conservation, and
this simple framework DID work! Why shouldn’t the next step beyond the SM be
accomplished by similarly elegant and “simple” proposals?

Lack of evidence of new physics at the TeV scale, made even more compelling by
the hundreds of inconclusive searches scrupolously carried out by the LHC
experiments, has not removed, however, the need to continue addressing the
original motivations for a BSM extension of the SM. If anything, this has made
the open issues even more intriguing, and challenging. But, while the efforts
to review the underlying perspective on the hierarchy problem and naturalness
continue, we should focus on a perhaps even more basic question: who ordered
the Higgs? Where does the famous “mexican hat” Higgs potential come from? This
appears like a pointless trivial question. A sort of mexican-hat potential
must be there, it’s a necessary ingredient in the realization of EWSB, without
it we would not be here discussing it. But what is its true origin?

To understand the value of this question, it is useful to compare the dynamics
of the Higgs field with that of electromagnetism (EM), or of any of the other
known fundamental forces in nature. All properties of EM arise from a simple
principle, the gauge principle. Coulomb’s law has no free parameter, except
the overall scale of the electric charge, absorbed in the definition of the
charge unit. The quantization of the charge may have a deep origin in quantum
mechanical properties such as anomaly cancellation, or in the algebraic
structure of the representations of larger gauge groups in which EM is
embedded. The sign of the electric force, positive or negative depending on
the relative sign of the interacting charges, follows from the spin-1 nature
of the photon. The $1/r^{2}$ behaviour follows from the Gauss theorem, or
charge conservation, or gauge invariance, depending on how we want to phrase
it. We do not know why nature has chosen gauge symmetry as a guiding
principle, although this appears as an unavoidable consequence of the
existence of interactions mediated by massless particles, which are the basis
of the long-range forces needed to sustain our existence. But gauge symmetry
appears everywhere, e.g. in the zero modes of a string theory, or as a result
of compactification in Kaluza-Klein gravitational theories. Gauge symmetry is
therefore intimately related to possible deeper properties of nature.

On the contrary, nothing is fundamental in the Higgs potential, there doesn’t
seem to be any fundamental symmetry or underlying principle that controls its
structure. To fix the notation for further use, we shall write this potential
as

$V(\Phi)=-\frac{\mu^{2}}{2}|\Phi|^{2}+\frac{\lambda}{4}|\Phi|^{4}\;,$ (1)

where $\Phi$ is the SU(2) doublet scalar field and $\phi=v+H$ is the real part
of the neutral component ($v$ being its vacuum expectation value). The
condition of minimum of the potential ($\partial V/\partial\phi|_{\phi=v}=0$)
and the Higgs mass definition ($m_{H}^{2}=\partial^{2}V/\partial\phi^{2}$)
lead to the relations $\mu^{2}=m_{H}^{2}/2$ and $\lambda=m_{H}^{2}/2v^{2}$,
expressed in terms of the measured Higgs mass and of Fermi’s coupling
$v=(\sqrt{2}G_{F})^{-1/2}\sim 246$ GeV. The sign and value of the parameters
$\mu^{2}$ and $\lambda$ are a priori arbitrary. A negative sign in front of
the quadratic term is required to achieve symmetry breaking, but is not
required by any symmetry. The positive sign of $\lambda$ is necessary for the
stability of the potential at large $\phi$ but, again, is not dictated by
anything: it could be negative, and the potential could be stabilized at
larger $\phi$ values by higher-order terms. Even the functional form is not
fundamental: the underlying gauge symmetry only requires the potential to
depend on $|\Phi|^{2}$, and the quartic form could simply represent the
leading terms in the power expansion of a more complex functional dependence
of the potential.

The SM Higgs potential has therefore the features of an effective potential,
as in other natural phenomena. Spontaneous symmetry breaking, in fact, is not
a process unique to the EW theory. There are many other examples in nature
where the potential energy of a system is described by a mexican-hat
functional form, leading to some order parameter acquiring a non-zero
expectation value. A well know case is that of superconductivity. The Landau-
Ginzburg theory (L-G) [24] is a phenomenological model that describes the
macroscopic behaviour of type-1 superconductors. This model contains a scalar
field $\phi$, with free energy given by

$F=\frac{1}{2m}|(-i\hbar\nabla-2eA)\phi|^{2}+\alpha|\phi|^{2}+\frac{\beta}{2}|\phi|^{4}+...$
(2)

where the ellipsis denote additional terms not relevant to this discussion.
This equation describes a scalar field of charge $Q=2$ with a mass and a
quartic interaction. These parameters are temperature dependent. At high
temperature the mass-squared is positive and the scalar field has a vanishing
expectation value throughout the superconductor. However, below the critical
temperature $T_{c}$ the mass-squared is negative, leading to a non-vanishing
expectation value of $\phi$ throughout the superconductor. This expectation
value essentially generates a mass for the photon within the superconductor,
leading to the basic phenomenology of superconductivity.

Similarly to the Higgs mechanism, the L-G theory is a phenomenological model,
which offers no explanation as to the fundamental origin of the parameters of
the model. It also does not explain the fundamental origin of the scalar field
itself. Ultimately, these questions were answered by Bardeen, Cooper, and
Schrieffer, in the celebrated BCS theory of superconductivity [25]. The scalar
field is a composite of electrons, the Cooper pairs, and its mass relates to
the fundamental microscopic parameters describing the material. The $\alpha$
and $\beta$ parameters can therefore be calculated from first principles,
starting from the underlying dynamics, namely the electromagnetic interactions
inside the metal, subject to the rules of quantum mechanics and to the phonon
interactions within the solid.

The analogy of the Higgs with the L-G model is striking, with the exception
that the model is relativistically invariant and the gauge forces non-Abelian.
Unlike with superconductivity, currently neither the fundamental origin of the
SM scalar field nor the origin of the mass and self-interaction parameters in
the Higgs scalar potential are known. The SM itself does not provide a
dynamical framework that allows us to predict the shape of the Higgs
potential. This must follow from a theory beyond the SM. What are possible
scenarios? An obvious option is a mechanism analogous to BCS: the Higgs could
be the bound state of a pair of fermions, strongly coupled by a new
fundamental (possibly gauge-) interaction, whose dynamics determines the
properties of the Higgs field. Another well known framework is supersymmetry:
elementary scalar fields appear as a result of the symmetry itself, and the
Higgs potential is likewise determined by the symmetry. For example, in the
minimal supersymmetric SM (MSSM) the Higgs self-coupling is not a free
parameter, but is related to the weak gauge coupling. The parameters that
characterize supersymmetry breaking modify the supersymmetric predictions for
the Higgs interactions, and ultimately the dynamics of EW symmetry breaking
would be calculable from the fundamental properties of supersymmetry breaking.

Now that the Higgs boson has been discovered, and the basic phenomenology of
EWSB established, the next stage of exploration for any future high energy
physics programme is to determine their microscopic origins. And the obvious
place where one should look for hints is the Higgs boson itself, exploring in
detail all of its properties. As of today, we are not aware of any other
experimental context except particle colliders, where the question of the
origin of the Higgs potential can be studied.

Another aspect of the Higgs dynamics makes it appear very different than EM or
other gauge forces. This is the (lack of) decoupling between short- and long-
distance interactions, or between low- and high-energy modes. The Gauss
theorem, or gauge invariance, teaches us that the charge of an electron can be
obtained by measuring the integral of the flux of its field through any closed
surface surrounding the electron. Using the surface of a sphere of small or
large radius will give the same value. Possible additional unknown
interactions of the electron at very short distances do not modify the charge
we measure at large distances. In the case of the Higgs potential, its
parameters $\mu^{2}$ and $\lambda$ receive instead dominant contributions from
any short-distance Higgs interaction. Self-energy loop diagrams for the Higgs
boson shift the Higgs mass squared by amounts proportional to the mass scale
of the particles in the loop. Given that we anticipate the existence in nature
of other fundamental mass scales much larger than the weak scale, notably the
Planck scale, the mass of the Higgs boson is intrinsically unpredictable, and
its small value rather unnatural: this is the so called hierarchy problem.
This puzzle could be resolved if there were an additional new microscopic
scale near the weak scale, involving new particles and interactions governed
by symmetries that decouple the Higgs mass from short-distance contributions.

Figure 1: Renormalization group running of the Higgs self-coupling $\lambda$,
and its dependence on the top and Higgs masses, and on the strong coupling
constant $\alpha_{s}(M_{Z})$. From Ref. [26].

The Higgs quartic coupling $\lambda$ is modified only logarithmically by loop
corrections, but the effect of running to high energy can be dramatic. At
leading order, and neglecting small contribtions from the gauge couplings, the
renormalization group running of $\lambda$ is given by the following
expression:

$\beta(\lambda)\,=\,\frac{d\lambda}{d\log
Q^{2}}\,=\,\frac{3}{4\pi^{2}}\left(\lambda^{2}+\lambda
y_{t}^{2}-y_{t}^{4}\right)\;,$ (3)

where $y_{t}=m_{t}/v\sim 0.7$ is the top Yukawa coupling, and
$\lambda=\lambda(Q^{2})$ is the running Higgs selfcoupling. It is
straightforward to verify that, for the actual values of the top and Higgs
masses, $\beta(\lambda)<0$, and $\lambda(Q^{2})$ is driven to smaller values
at large $Q^{2}$. A complete analysis, including higher order terms (see e.g.
[26]), indicates that $\lambda$ turns negative at energies in the range of
$10^{10}$ GeV, as shown in Fig. 1. A negative $\lambda$ would make the
potential unstable, and short-distance quantum fluctuations could therefore
potentially destabilize the SM Higgs vacuum. The timescale for “our” vacuum to
run away, calculated with the given values of top and Higgs mass, is much
longer than the age of the universe, making the vacuum metastable and
consistent with observation [27]. But it is disturbing, once more, that the
dynamics of the Higgs field be influenced so much by physics taking place at
scales much higher than the weak scale!

Both the puzzle of the hierarchy problem and the issue of the metastability of
the Higgs vacuum point to the existence of a more fundamental layer behind EW
symmetry breaking, and galvanise the need to understand the deeper origin of
the Higgs potential.

###  2.2 More exploration targets for future colliders

Even setting to the side the key issue of the origin of the Higgs, there are
other very concrete reasons why the Higgs deserves further study, and may
provide a window to undiscovered phenomena. As it carries no spin and is
electrically neutral, the Higgs may have so-called ‘relevant’ (i.e.
dimension-4) interactions (e.g. $|\Phi|^{2}|S|^{2}$) with a scalar particle,
S, living in sectors of particle physics that are otherwise totally decoupled
from the SM interactions. These interactions, even if they only take place at
very high energies, remain relevant at low energies – contrary to interactions
between new neutral scalars and the other SM particles. The possibility of new
hidden sectors already has strong experimental support: there is overwhelming
evidence from astrophysical observations that a large fraction of the observed
matter density in the universe is invisible. This so-called dark matter (DM)
makes up 26% of the total energy density in the universe and more than 80% of
the total matter [28]. Despite numerous observations of the astrophysical
properties of DM, not much is known about its particle nature. This makes the
discovery and identification of DM one of the most pressing questions in
science, a question whose answer may hinge on the role of the Higgs boson.

The current main constraints on a particle DM candidate $\chi$ are that it: a)
should gravitate like ordinary matter, b) should not carry colour or
electromagnetic charge, c) is massive and non-relativistic at the time the CMB
forms, d) is long lived enough to be present in the universe today
($\tau\gg\tau_{\rm universe}$), and e) does not have too strong self-
interactions ($\sigma/M_{\rm DM}\lesssim 100~{}{\rm GeV}^{-3}$). While no SM
particles satisfy these criteria, they do not pose very strong constraints on
the properties of new particles to play the role of DM. In particular the
allowed range of masses spans almost 80 orders of magnitude. Particles with
mass below $10^{-22}$ eV would have a wave length so large that they wipe out
structures on the kPc (kilo-Parsec) scale and larger [29], disagreeing with
observations, while on the other end of the scale micro-lensing and MACHO
(Massive Astrophysical Compact Halo Objects) searches put an upper bound of
$2\times 10^{-9}$ solar masses or $10^{48}$ GeV on the mass of the dominant DM
component [30, 31, 32]. We shall discuss later on how future colliders can
attack this pressing question, providing comprehensive exploration of the
class of ‘thermal freeze-out’ DM, which picks out a particular broad mass
range as a well-motivated experimental target, as well as unique probes of
weakly coupled dark sectors.

Returning to the matter which is observable in the Universe, the SM alone
cannot explain baryogenesis, namely the origin of the dominance of matter over
antimatter that we observe today. Since the matter-antimatter asymmetry was
created in the early universe when temperatures and energies were high, higher
energies must be explored to uncover the new particles responsible for it, and
the LHC can only start this search. In particular, a well-motivated class of
scenarios, known as EW baryogenesis theories, can explain the matter-
antimatter asymmetry by modifying how the transition from the high-temperature
EW-symmetric phase to the low-temperature symmetry-broken phase occurred.
Independently of the problem of the matter-antimatter asymmetry, there is the
question of the nature of the EW phase transition (EWPT): was it a smooth
cross-over, as predicted by the SM, or a first-order one, as possible in BSM
scenarios (and as necessary to enable EW baryogenesis)? Since this phase
transition occurred at temperatures near the weak scale, the new states
required to modify the transition would likely have mass not too far above the
weak scale, singling out future 100 TeV colliders as the leading experimental
facility to explore the nature of this foundational epoch of the early
Universe.

Another outstanding question lies in the origin of the neutrino masses, which
the SM alone cannot account for. As with DM, there are numerous models for
neutrino masses that are within the discovery reach of future lepton and
hadron colliders, as discussed in Ref. [9].

These and other outstanding questions might also imply the existence of
further spatial dimensions, or larger symmetries that unify leptons and quarks
or the known forces. The LHC’s findings notwithstanding, higher energy and
larger statistics will be needed to explore these fundamental mysteries more
deeply and possibly reveal new paradigm shifts.

##  3 The way forward with future colliders

Since the mid 70’s, the path to establish experimentally the SM was clear:
discover the gauge bosons and complete the fermion sector (e.g. determine the
number of SM-like neutrino species and eventually discover the top quark),
test strong and EW interactions at the level of quantum corrections (comparing
precise measurements and accurate theoretical predictions), test the CKM
framework of flavor phenomena, and discover the Higgs boson. Having
accomplished all this, the situation today is less well defined. In spite of
the fact that the formulation of the open problems, as reviewed in the
previous Section, is rather clear, there is however no experimental approach
known today that can guarantee conclusive answers. This is underscored by the
fact that our prejudices on where to look have not given results. One of the
main questions we face in planning our future is therefore “why don’t we see
as yet any sign of the new physics that we confidently expected to be present
around the TeV scale?”. The question admits two possible answers: (i) the mass
scale of the new physics lies beyond the LHC reach, or, (ii) while being
within LHC’s reach, its manifestations are elusive and escaped so far the
direct search. These two scenarios are a priori equally likely, but they
clearly impact in different ways the future of our field, and thus the
assessment of the physics potential of possible future facilities. Our safest
hedge is therefore the readiness to cover both scenarios, via an experimental
programme relying on higher precision and sensitivity (to address possible
elusive signatures), and on an extended energy and mass reach relative to the
LHC.

A possible way to assess the value of a future collider facility is to
consider the following three criteria:

  * •

The guaranteed deliverables. This criterion is what I refer to as the “value
of measurements”: the new information that we can collect to probe the SM to a
deeper level, pushing further the exploration of particles and processes that
are still poorly known. The main targets of this component of the programme
include of course the Higgs boson, the gauge bosons and EW interactions at
energies above the EWSB scale, the flavor phenomena, in particular those
related to the least known fermions, such as the top quark or the tau lepton.

  * •

The discovery potential. While the emergence of phenomena beyond the SM cannot
be guaranteed, a future facility must promise a significant extension of
today’s sensitivity to new physics, addressing the most relevant and
compelling BSM scenarios under consideration, and with sufficient flexibility
to accommodate new ideas. The increase in the reach for direct discovery at
the highest masses should be accompanied by the increased sensitivity
throughout the whole mass range, thanks to higher precision and statistics.
The mass reach for direct discovery should ideally match the sensitivity reach
obtained indirectly via precision measurements.

  * •

Conclusive answers. Unless an actual discovery is made, no experiment can
provide conclusive answers to general questions such as “what is DM?”, “do
supersymmetry or new Z’ bosons exist?”. Lack of evidence can be evaded by
pushing the relevant mass spectrum beyond reach. But there exist important,
less generic, questions, for which it is reasonable to expect that a
conclusive answer can be found below well-defined mass scales. Some examples
were given before: did EWSB induce a first order phase transition? Is DM made
of particles coupled to the SM via the weak interaction? Do neutrino masses
arise from the weak scale? Even negative answers, if firm, would be of great
value, since they would force us to focus the searches elsewhere. While
current experiments (LHC and others) could find partial answers, conclusive
statements are expected to require higher energy and sensitivity, setting
performance targets for the evaluation of future experiments.

These lectures will present an overview of the physics potential of the
various elements of the FCC programme, in the light of those three criteria.

###  3.1 The Role of FCC-ee

The capabilities of circular $\rm e^{+}e^{-}$ colliders are well illustrated
by LEP, which occupied the LHC tunnel from 1989 to 2000\. Its point-like
collisions between electrons and positrons and precisely known beam energy
allowed the four LEP experiments to test the SM to new levels of precision,
particularly regarding the properties of the W and Z bosons. Putting such a
machine in a 100 km tunnel and taking advantage of advances in accelerator
technology such as superconducting radio-frequency cavities would offer even
greater levels of precision on a greater number of processes. For example, it
would be possible to adapt the collision energy during about 15 years of
operation, to examine physics at the Z pole, at the WW production threshold,
at the peak of ZH production, and above the $\rm t\overline{t}$ threshold.
Controlling the beam energy at the 100 keV level would allow exquisite
measurements of the Z and W boson masses, whilst collecting samples of up to
$10^{13}$ Z and $10^{8}$ W bosons, not to mention several million Higgs bosons
and top quark pairs. The experimental precision would surpass any previous
experiment and challenge cutting edge theory calculations.

FCC-ee would quite literally provide a quantum leap in our understanding of
the Higgs. Like the W and Z gauge bosons, the Higgs receives quantum EW
corrections typically measuring a few per cent in magnitude due to
fluctuations of massive particles such as the top quark. This aspect of the
gauge bosons was successfully explored at LEP, but now it is the turn of the
Higgs – the keystone in the EW sector of the SM. The millions of Higgs bosons
produced by FCC-ee, with its clinically precise environment, would push the
accuracy of the measurements to the per mille level, accessing the quantum
underpinnings of the Higgs and probing deep into this hitherto unexplored
frontier. In the process $\rm e^{+}e^{-}\to HZ$, the mass recoiling against
the Z has a sharp peak that allows a unique and absolute determination of the
Higgs decay-width and production cross section. This will provide an absolute
normalisation for all Higgs measurements performed at the FCC, enabling exotic
Higgs decays to be measured in a model independent manner.

The high statistics promised by the FCC-ee programme goes far beyond precision
Higgs measurements. Other signals of new physics could arise from the
observation of flavour changing neutral currents or lepton-flavour-violating
decays, by the precise measurements of the Z and H invisible decay widths, or
by direct observation of particles with extremely weak couplings, such as
right-handed neutrinos and other exotic particles. The precision of the FCC-ee
programme on EW measurements would allow new physics effects to be probed at
scales as high as 100 TeV, anticipating what the FCC-hh must focus on.

###  3.2 The Role of FCC-hh and FCC-eh

The FCC-hh would operate at seven times the LHC energy, and collect about 10
times more luminosity. The discovery reach for high-mass particles – such as
Z′ or W′ gauge bosons corresponding to new fundamental forces, or gluinos and
squarks in supersymmetric theories – will increase by a factor five or more,
depending on the final statistics. The production rate of particles already
within the LHC reach, such as top quarks or Higgs bosons, will increase by
even larger factors. During the planned 25 years of data taking, a total of
more than $10^{10}$ Higgs bosons will be created, several thousand times more
than collected by the LHC through Run 2 and 200 times more than will be
available by the end of its operation. These additional statistics will enable
the FCC-hh experiments to improve the separation of Higgs signals from the
huge backgrounds that afflict most LHC studies, overcoming some of the
dominant systematics that limit the precision attainable at the LHC. While the
ultimate precision of most Higgs properties can only be achieved with FCC-ee,
several demand complementary information from FCC-hh. For example, the direct
measurement of the coupling between the Higgs and the top quark requires that
they be produced together, requiring an energy beyond the reach of the FCC-ee.
At 100 TeV, almost $10^{9}$ out of the $10^{12}$ top quarks produced will
radiate a Higgs boson, allowing the top-Higgs interaction to be measured at
the 1% level – several times better than at the HL-LHC and probing deep into
the quantum structure of this interaction. Similar precision can be reached
for Higgs decays that are too rare to be studied in detail at FCC-ee, such as
those to muon pairs or to a Z and a photon. All of these measurements will be
complementary to those obtained with FCC-ee and will use them as reference
inputs to precisely correlate the strength of the signals obtained through
various production and decay modes.

One respect in which a 100 TeV proton-proton collider would really come to the
fore is in revealing how the Higgs behaves in private. The rate of Higgs pair
production events, which in some part occur through Higgs self-interactions,
would grow by a factor of 40 at FCC-hh, with respect to 14 TeV, and enable
this unique property of the Higgs to be measured with an accuracy reaching 5%.
Among many other uses, such a measurement would comprehensively explore
classes of models that rely on modifying the Higgs potential to drive a strong
first order phase transition at the time of EW symmetry breaking, a necessary
condition to induce baryogenesis.

FCC-hh would also allow an exhaustive exploration of new TeV-scale phenomena.
Indirect evidence for new physics can emerge from the scattering of W bosons
at high energy – where the Higgs boson plays a key role in controlling the
rate growth – from the production of Higgs bosons at very large transverse
momentum, or by testing the far ‘off-shell’ nature of the Z boson via the
measurement of lepton pairs with invariant masses in the multi-TeV region. The
plethora of new particles predicted by most models of symmetry-breaking
alternatives to the SM can be searched for directly, thanks to the immense
mass reach of 100 TeV collisions. The search for DM, for example, will cover
the possible space of parameters of many theories relying on weakly
interacting massive particles, guaranteeing a discovery or ruling them out.
Several theories that address the hierarchy problem will also be conclusively
tested. For supersymmetry, the mass reach of FCC-hh pushes beyond the regions
motivated by the hierarchy problem alone. For composite Higgs theories, the
precision Higgs coupling measurements and searches for new heavy resonances
will fully cover the motivated territory. A 100 TeV proton collider will even
confront exotic scenarios such as the twin Higgs, which are extremely
difficult to test. These theories predict very rare or exotic Higgs decays,
possibly visible at FCC-hh thanks to its enormous Higgs production rates.

The FCC-eh collider could operate in synchronous, symbiotic operation
alongside the pp collider. The facility would serve as the most powerful,
high-resolution microscope onto the substructure of matter ever built. High-
energy ep collisions would provide precise information on the quark and gluon
structure of the proton, and how they interact. FCC-eh would complement and
enhance the study of the Higgs, and broaden the new physics searches also
performed at FCC-hh and FCC-ee, with a specific focus on phenomena such as
quark substructure, leptoquarks, heavy sterile neutrinos and long-lived
particles.

While not discussed at all in these lectures, FCC-hh would also enable the
continuation of the LHC successful programme of heavy ion collisions,
extending studies of the thermodynamic behaviour of QCD of crucial relevance
to multiple topics, ranging from the fundamental properties of quantum field
theory, to cosmology and astrophysics.

##  4 Higgs boson properties

Indirect information about the Higgs boson is accessible through precision EW
measurements, as proven by the global fits to the LEP and SLC data, which set
very tight constraints on the Higgs mass well before its discovery. But,
following the Higgs discovery, the most direct way to test the Higgs
properties is to produce it and observe its decay features. With the knowledge
of the Higgs mass, the SM predicts uniquely its couplings to each SM particle,
and therefore all production and decay rates are fixed. Since our target is to
explore the origin of EWSB, and possibly identify the underlying BSM phenomena
that trigger it, we must be open however to all sorts of deviations from the
SM. For example, while the couplings of the Higgs to the gauge bosons are
determined by the Higgs quantum numbers (an SU(2) doublet), the existence of
an additional Higgs scalar, acquiring its own expectation value, could lead to
a mixing in the Higgs sector, and the mass eigenstate at 125 GeV could couple
to the W and the Z with a slightly reduced strength. The existence of
additional Higgses opens the door to the possibility that different fermions
couple to different Higgses, modifying the direct relation between fermion
mass and Yukawa coupling to the 125 GeV state (as in the case in
supersymmetry). The study of Higgs couplings, threfore, requires as much as
possible a model independent approach.

Establishing the gauge couplings of the known fermions was relatively
straightforward, since they are quantized and the fermion assignment to a
gauge group representation ranges over a discrete set of possibilities.
Deviations are possible of course, but only in presence of additional BSM
interactions, that appear at low energy as operators of dimension higher than
4. The basic, leading-order and renormalizable interactions of SM fermions are
therefore easily established experimentally. That the top quark is an $SU(3)$
color triplet, for example, can follow from the analysis of its production
rate and decay patterns111This is so straightforward in principle, that I am
not even sure there has ever been an explicit experimental analysis to confirm
that the top quark is a triplet. I leave it to you as an exercise to list the
data and signatures that could be used to confirm it.. On the contrary, the
leading-order Higgs couplings are a priori a generic real (or complex) number,
and the confidence on whether they agree or not with the SM will always only
be conditional to the precision of the available data.

Figure 2: Basic Higgs production processes in hadronic collisions. From the
top left, clockwise: gluon-gluon fusion (ggH), vector boson fusion (VBF),
associate production with a gauge boson (VH), and with a $\rm t\bar{t}$ pair
(ttH). V=$\rm W^{\pm}$ or Z0, throughout.

The dominant Higgs production channels, in hadronic collisions like at the
LHC, are shown in Fig. 2. In these examples, the production rates are
proportional to the coulings to the gauge bosons, or to the top quark. In the
ideal world in which the strong coupling $\alpha_{s}$, the partonic densities
(PDFs) and the QCD matrix elements were perfectly known, counting events in a
given $\rm H\to Y$ decay mode would provide a measurement of
$g_{X}^{2}\,\Gamma_{Y}/\Gamma_{H}$, where $g_{X,Y}$ are the Higgs couplings to
initial and final state state, and $\Gamma_{Y}\propto g_{Y}^{2}$
($\Gamma_{H}$) is the partial (total) decay width. If we could observe every
possible Higgs decay, summing over all Y states for a given production channel
X would allow the measurement of $g_{X}$, since
$\sum_{Y}\Gamma_{Y}=\Gamma_{H}$. At the LHC and in general in hadronic
collisions, this is hardly possible: several SM decay modes with a substantial
branching ratio (BR), like $\rm H\to c\bar{c}$, are very difficult to measure,
and possible exotic Higgs decays are also likely to escape detection. A
completely model-independent extraction of Higgs couplings in hadronic
collisions can therefore only reach a limited precision, independently of the
theoretical challenge of properly calculating the QCD part of the reactions.

As we show in the next sections, the measurements at an electron collider can
provide the needed input of $\Gamma_{H}$, and open the way for a powerful
synergetic programme of precision measurements with the next generation of
hadron colliders.

A complete compilation and critical review of the Higgs coupling measurement
prospects covering all proposed future colliders can be found in Ref. [33].

###  4.1 Higgs coupling measurements at FCC-ee

Figure 3: Left: the $\rm e^{+}e^{-}\to HZ$ production process. Right: Higgs
signal and background contributions to the recoil mass in $\rm
e^{+}e^{-}\to\mu^{+}\mu^{-}X$ events.

The determination of $\Gamma_{H}$ is however possible at future $\rm
e^{+}e^{-}$ colliders, operating above the ZH threshold. Here, the production
of a Higgs boson can be reconstructed, in a model-independent way, with the
so-called recoil-mass technique. One considers $\rm
e^{+}e^{-}\to\mu^{+}\mu^{-}+X$ final states, and for each event defines the
recoil mass as
$m^{2}_{Recoil}=P_{X}^{2}=(p_{e^{+}}+p_{e^{-}}-p_{\mu^{+}}-p_{\mu^{-}})^{2}$.
Most $\rm\mu^{+}\mu^{-}+X$ final states arise from ZZ production ($\rm
Z[\to\mu^{+}\mu^{-}]\,Z[\to X]$), in which case $P_{X}$ is the momentum of the
second Z, and the recoil mass equals (up to finite-width and experimental
resolution effects) the Z mass. A further contribution comes from WW
production ($\rm W^{+}[\to\mu^{+}\nu]\,W^{-}[\to\mu^{-}\bar{\nu}]$), in which
case $P_{X}$ represents the missing momentum, and the recoil mass is a broad
continuum. In the case of $\rm Z[\to\mu^{+}\mu^{-}]\,H[\to X]$ production (see
the left image of Fig. 3), the recoil mass coincides with the Higgs mass,
regardless of the H decay mode. These three contributions are shown, for the
simulation of an FCC-ee experiment, in Fig. 3. A global fit of the recoil mass
spectrum returns the total number of Higgs produced in $\rm e^{+}e^{-}\to
HZ[\to\mu^{+}\mu^{-}]$, and a direct measurement of the HZZ coupling,
$g_{ZZ}$. If we now focus on events with the $\rm H\to ZZ^{*}\to 4\ell$ decay,
and consider that their rate is proportional to
$g_{ZZ}^{2}\times\Gamma_{ZZ}/\Gamma_{H}\propto g_{ZZ}^{4}/\Gamma_{H}$, the
knowledge of $g_{ZZ}$ allows to extract $\Gamma_{H}$ in a model-independent
way.

Exercise: discuss, in a qualitative way, to which extent EW radiative
corrections or BSM effects influence this line of reasoning, and whether they
affect the “model-independent” argument.
Exercise: discuss, in a qualitative way, the backgrounds under the H peak in
the recoil mass spectrum, and how they can be estimated, and subtracted,
precisely.
Exercise: discuss how the recoil mass observable can be used to determine the
presence of exotic (in particular, invisible) H decays.

Having established the value of $\Gamma_{H}$, further dedicated measurements
allow to determine the absolute value of the Higgs couplings to all particles
accessible via decay modes or production channels. Assuming SM couplings, the
statistical precision that can be achieved for several BRs measurable at FCC-
ee is summarized in Table 1 (for the details of the reconstruction of
individual final states, see e.g. [10]). These include the results obtained
from the run just above the Higgs threhsold, at 240 GeV, and the runs above
the $t\bar{t}$ thresholds, where the VBF process $e^{+}e^{-}\to\nu\bar{\nu}H$,
shown in Fig. 4, becomes relevant.

Figure 4: Left: the $\rm e^{+}e^{-}\to H\nu\bar{\nu}(e^{+}e^{-})$ production process. Right: Higgs production rates for the leading $\rm e^{+}e^{-}$ processes in the FCC-ee energy range. Table 1: Relative statistical uncertainty on the measurements of event rates, providing $\sigma_{\rm HZ}\times{\rm BR}({\rm H\to XX})$ and $\sigma_{\rm\nu\bar{\nu}H}\times{\rm BR}({\rm H\to XX})$, as expected from the FCC-ee data [10]. This is obtained from a fast simulation of the reference FCC-ee detector and consolidated with extrapolations from full simulations of similar linear-collider detectors (SiD and CLIC). All numbers indicate 68% C.L. intervals, except for the 95% C.L. sensitivity in the last line. The accuracies expected with $5~{}{\rm ab}^{-1}$ at 240 GeV are given in the middle columns, and those expected with $1.5~{}{\rm ab}^{-1}$ at $\sqrt{s}=365$ GeV are displayed in the last columns. $\sqrt{s}$ (GeV) | $240$ | $365$
---|---|---
Luminosity (${\rm ab}^{-1}$) | 5 | $1.5$
$\delta(\sigma{\rm BR})/\sigma{\rm BR}$ (%) | HZ | ν$\overline{\mbox{n}}$ H | HZ | ν$\overline{\mbox{n}}$ H
${\rm H\to any}$ | $\pm 0.5$ |  | $\pm 0.9$ |
${\rm H\to b\bar{b}}$ | $\pm 0.3$ | $\pm 3.1$ | $\pm 0.5$ | $\pm 0.9$
${\rm H\to c\bar{c}}$ | $\pm 2.2$ |  | $\pm 6.5$ | $\pm 10$
${\rm H\to gg}$ | $\pm 1.9$ |  | $\pm 3.5$ | $\pm 4.5$
${\rm H\to W^{+}W^{-}}$ | $\pm 1.2$ |  | $\pm 2.6$ | $\pm 3.0$
${\rm H\to ZZ}$ | $\pm 4.4$ |  | $\pm 12$ | $\pm 10$
${\rm H\to}$ ττ | $\pm 0.9$ |  | $\pm 1.8$ | $\pm 8$
${\rm H\to}$ γγ | $\pm 9.0$ |  | $\pm 18$ | $\pm 22$
${\rm H\to}$ µ+µ- | $\pm 19$ |  | $\pm 40$ |
${\rm H\to invis.}$ | $<0.3$ |  | $<0.6$ |
Table 2: Precision determined in the $\kappa$ framework of the Higgs boson couplings and total decay width, as expected from the FCC-ee data, and compared to the projections for HL-LHC [34]. All numbers indicate 68% CL sensitivities, except for the last line which gives the 95% CL sensitivity on the ”exotic” branching fraction, accounting for final states that cannot be tagged as SM decays. The FCC-ee accuracies are subdivided in three categories: the first sub-column give the results of the model-independent fit expected with $5~{}{\rm ab}^{-1}$ at 240 GeV, the second sub-column in bold – directly comparable to the other collider fits – includes the additional $1.5~{}{\rm ab}^{-1}$ at $\sqrt{s}=365$ GeV, and the last sub-column shows the result of the combined fit with HL-LHC. The fit to the HL-LHC projections alone (first column) requires two additional assumptions to be made: here, the branching ratios into ${\rm c\bar{c}}$ and into exotic particles are set to their SM values. From [9].  Collider | HL-LHC | FCC-ee240+365
---|---|---
Lumi (${\rm ab}^{-1}$) | 3 | $5_{240}$ | $+1.5_{365}$ | $+$ HL-LHC
Years | 25 | 3 |  $+$4 |
${}\delta\Gamma_{\rm H}/\Gamma_{\rm H}$ (%) | SM | 2.7 | 1.3 | 1.1
${}\delta g_{\rm HZZ}/g_{\rm HZZ}$ (%) |  1.5 | 0.2 | 0.17 | 0.16
${}\delta g_{\rm HWW}/g_{\rm HWW}$ (%) |  1.7 | 1.3 | 0.43 | 0.40
${}\delta g_{\rm Hbb}/g_{\rm Hbb}$ (%) |  3.7 | 1.3 | 0.61 | 0.56
${}\delta g_{\rm Hcc}/g_{\rm Hcc}$ (%) |  SM | 1.7 | 1.21 | 1.18
${}\delta g_{\rm Hgg}/g_{\rm Hgg}$ (%) |  2.5 | 1.6 | 1.01 | 0.90
${}\delta g_{\rm H\mbox{tt}}/g_{\rm H\mbox{tt}}$ (%) | 1.9 | 1.4 | 0.74 | 0.67
${}\delta g_{\rm H\mbox{\textmu\textmu}}/g_{\rm H\mbox{\textmu\textmu}}$ (%) | 4.3 | 10.1 | 9.0 | 3.8
${}\delta g_{\rm H\mbox{gg}}/g_{\rm H\mbox{gg}}$ (%) | 1.8 | 4.8 | 3.9 | 1.3
${}\delta g_{\rm Htt}/g_{\rm Htt}$ (%) |  3.4 | – | – | 3.1
BREXO (%) | SM | $<1.2$ | ${\bf<1.0}$ | ${\bf<1.0}$

In practice, the width and the couplings are determined with a global fit,
which closely follows the logic of Ref. [35]. The results of this fit are
summarised in Table 2 and are compared to the same fit applied to HL-LHC
projections [34]. Table 2 also shows that the extractions of $\Gamma_{\rm H}$
and of $g_{\rm HWW}$ from the global fit are significantly improved by the
addition of the WW-fusion process at $\sqrt{s}=365$ GeV, as a result of the
correlation between the HZ and ν$\overline{\mbox{n}}$ H processes. In
particular the Higgs EW couplings have a permille-level precision, and the
couplings to the tau, the bottom and charm quarks and the effective couplng to
the gluon reach the percent level or better.

Several SM couplings are left out of these projections: to the lightest quarks
(u, d, s), to the electron, to the top quark, to the $\rm Z\gamma$ pair, and
the Higgs self-coupling. To access the light quarks, several ideas have been
proposed: exclusive decays to hadronic resonances, such as $\rm H\to Vh$ ($\rm
h=\phi,\;\rho)$, V=W/Z/γ) [36, 37, 38, 39], light-jet tagging techniques [40],
or kinematical distributions of the Higgs boson in hadronic collisions [41,
42]. Experimental searches for exclusive radiative hadronic decays have
started already at the LHC [43], to at least establish upper limits, even
though well beyond the SM expectations. Given the small BRs, an electron
collider will barely have sufficient statistics to gain the required SM
sensitivity. At the FCC-ee, the most promising channel is H$\to$γρ[$\to$ππ],
with about 40 events expected [44]. A future hadron collider will have much
more events to play with, but backgrounds and experimental conditions will be
extremely challenging, and only detailed simulations will be able to establish
their true potential.

To probe the Hee coupling, the best hope appears to be the direct resonant
production in $\rm e^{+}e^{-}\to H$. The low rate demands high luminosity, and
a tuning of the beam energy to exactly match $m_{H}/2$. Preliminary studies
[44] indicate that a 3$\sigma$ observation requires an integrated luminosity
of 90ab-1, namely several years of dedicated running at 125 GeV.

While the direct access to the Htt coupling in an $\rm e^{+}e^{-}$ collider
requires a center-of-mass energy of 500 GeV and more, FCC-ee will expose an
indirect sensitivity to it, through its effect at quantum level on the ${\rm
t\bar{t}}$ cross section just above production threshold, $\sqrt{s}=350$ GeV.
The precise measurement of $\alpha_{s}$ from the runs at the Z pole will allow
the QCD effects to be disentangled from those of the top Yukawa coupling at
the ${\rm t\bar{t}}$ vertex, to achieve a precision of $\pm 10\%$ [10].

To access in a direct way the top Yukawa coupling, and to improve to the
percent level the measurement of small BR decays such as H$\to$γγ, Zγ and
µ+µ-, we can then appeal to the huge statistics available to a hadron
collider.

###  4.2 Higgs couplings measurements at FCC-hh

Two elements characterise Higgs production at the FCC-hh: the large statistics
(see Table 3), and the large kinematic range, which, for several production
channels, probes $p_{T}$ in the multi-TeV region (see Fig. 5).

Table 3:  Higgs production event rates for selected processes at 100 TeV ($N_{100}$) and 27 TeV ($N_{27}$), and statistical increase with respect to the statistics of the HL-LHC ($N_{100/27}=\sigma_{100/27~{}\mathrm{TeV}}\times 30/15$ ab-1, $N_{14}=\sigma_{14~{}\mathrm{TeV}}\times 3$ ab-1). | gg$\to$H | VBF | WH | ZH | tt̄H | HH
---|---|---|---|---|---|---
$N_{100}$ | $24\times 10^{9}$ | $2.1\times 10^{9}$ | $4.6\times 10^{8}$ | $3.3\times 10^{8}$ | $9.6\times 10^{8}$ | $3.6\times 10^{7}$
$N_{100}/N_{14}$ | 180 | 170 | 100 | 110 | 530 | 390
$N_{27}$ | $2.2\times 10^{9}$ | $1.8\times 10^{8}$ | $5.1\times 10^{7}$ | $3.7\times 10^{7}$ | $4.4\times 10^{7}$ | $2.1\times 10^{6}$
$N_{27}/N_{14}$ | 16 | 15 | 11 | 12 | 24 | 19

These factors lead to an extended and diverse sensitivity to possible
deviations of the Higgs properties from their SM predictions: the large rates
enable precise measurements of branching ratios for rare decay channels such
as γγ or µµ, and push the sensitivity to otherwise forbidden channels such as
τµ. The large kinematic range can be used to define cuts improving the signal-
to-background ratios and the modelling or experimental systematics, but it can
also amplify the presence of modified Higgs couplings, described by higher-
dimension operators, whose impact grows with $Q^{2}$. Overall, the Higgs
physics programme of FCC-hh is a fundamental complement to what can be
measured at FCC-ee, and the two Higgs programmes greatly enrich each other.
This section contains some examples of these facts, and documents the current
status of the precision projections for Higgs measurements. A more extensive
discussion of Higgs production properties at 100 TeV and of possible
measurements is given in Ref. [45].

Figure 5: Production rates of Higgs bosons at high $p_{T}$, for various
production channels at 100 TeV and 30 ab-1.

Figure 5 shows the Higgs rates above a given $p_{T}$ threshold, for various
production channels. It should be noted that these rates remain above the
level of one million up to $p_{T}\sim 1$ TeV, and there is statistics for
final states like H$\to$bb̄ or H$\to$ττ extending up to several TeV.
Furthermore, for
$p_{T}(\mathrm{H})\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}$1
TeV, the leading production channel becomes tt̄H, followed by vector boson
fusion when
$p_{T}(\mathrm{H})\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}$2
TeV. The analysis strategies to separate various production and decay modes in
these regimes will therefore be different to what is used at the LHC. Higgs
measurements at 100 TeV will offer many new options and precision
opportunities with respect to the LHC, as it happened with the top quark
moving from the statistics-hungry Tevatron to the rich LHC.

Exercise: discuss possible strategies to separate the different Higgs
production processes in the various ranges of $p_{T}(H)$ shown in Fig. 5.

Figure 6: Left: Integrated transverse momentum rates (20 ab-1) for a photon
pair with mass close to the Higgs mass: signal and QCD background. Right:
$S/B$, significance of the signal, and potential statistical accuracy of the
sample. From Ref. [45]

For example, as shown in Ref. [45], $S/B$ improves for several final states at
large $p_{T}$. In the case of the important γγ final state, Fig. 6 shows that
$S/B$ increases from $\sim 3\%$ at low $p_{T}$ (a value similar to what
observed at the LHC), to
$\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}1$ at
$p_{T}\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}300$ GeV. In
this range of few hundred GeV, some experimental systematics will also
improve, from the determination of the energies (relevant e.g. for the mass
resolution of H$\to$γγ or bb̄) to the mitigation of pile-up effects.

Exercise: why do you think S/B improves at large $p_{T}(\rm H)$ for a process
like $\rm gg\to H[\to$γγ]+jet?

Figure 7: The uncertainty on the reconstruction efficiency of electrons,
photons and muons as a function of transverse momentum. An optimistic (solid)
and a conservative (dashed) scenario are considered.

Figure 8: Projected precision for the rate measurement of various Higgs final
states, in the gg$\to$H production channel. The label “lumi” indicates the
inclusion of a 1% overall uncertainty. The systematic uncertainty “syst” is
defined in the text.

The analyses carried out so far for FCC-hh are still rather crude when
compared to the LHC standards, but help to define useful targets for the
ultimate attainable precision and the overall detector performance. The
details of the present detector simulations for Higgs physics at FCC-hh are
contained in Ref. [46].

The target uncertainties considered include statistics (taking into account
analysis cuts, expected efficiencies, and the possible irreducible
backgrounds) and systematics (limited here to the identification efficiencies
for the relevant final states, and an overall 1% to account for luminosity and
modelling uncertainties). While these estimates do not reflect the full
complexity of the experimental analyses in the huge pile-up environment of
FCC-hh, the systematics assumptions that were used are rather conservative.
Significant improvements in the precision of reconstruction efficiencies would
arise, for example, by applying tag-and-probe methods to large-statistics
control samples. Modelling uncertainties will likewise improve through better
calculations, and broad campaigns of validation against data. By choosing here
to work with Higgs bosons produced at large $p_{T}$, the challenges met by
triggers and reconstruction in the high pile-up environment are eased. The
projections given here are therefore considered to be reasonable targets for
the ultimate precision, and useful benchmarks to define the goals of the
detector performance.

The consideration of the reconstruction efficiency of leptons and photons is
relevant in this context since, to obtain the highest precision by removing
global uncertainties such as luminosity and production modelling, ratios of
different decay channels can be exploited. The reconstruction efficiencies are
shown in Fig. 7 as a function of $p_{T}$. The uncertainties on the electron
and photon efficiencies are assumed to be fully correlated, but totally
uncorrelated from the muon one. The curves in Fig. 7 reflect what is
achievable today at the LHC, and it is reasonable to expect that smaller
uncertainties will be available at the FCC-hh, due to the higher statistics
that will allow statistically more powerful data-driven fine tuning. For
example, imposing the identity of the Z boson rate in the ee and µµ decay
channels will strongly correlate the e and µ efficiencies.

The absolute uncertainty expected in the measurement of the production and
decay rates for several final states is shown in Fig. 8, as a function of the
minimum $p_{T}(\rm H)$. The curves labeled by “stat+syst” include the optimal
reconstruction efficiency uncertainties shown in Fig. 7. The curves labeled by
“stat+syst+lumi” include a further 1%, to account for the overall uncertainty
related to luminosity and production systematics. The luminosity itself could
be known even better than that by using a standard candle process such as Z
production, where both the partonic cross section and the PDF luminosity will
be pinned down by future theoretical calculations, and by the FCC-eh,
respectively. Notice that the gg luminosity in the mass range between $m_{H}$
and several TeV will be measured by FCC-eh at the few per mille level.

Several comments on these figures are in order. First of all, it should be
noted that the inclusion of the systematic uncertainty leads to a minimum in
the overall uncertainty for $p_{T}$ values in the range of few hundred GeV.
The very large FCC-hh statistics make it possible to fully benefit from this
region, where experimental systematics are getting smaller. The second remark
is that the measurements of the Higgs $p_{T}$ spectrum can be performed with a
precision better than 10%, using very clean final states such as γγ and
4$\ell$, up to $p_{T}$ values well in excess of 1 TeV, allowing the possible
existence of higher-dimension operators affecting Higgs dynamics to be probed
up to scales of several TeV.

Independently of future progress, the systematics related to production
modelling and to luminosity cancel entirely by taking the ratio of different
decay modes, provided selection cuts corresponding to identical fiducial
kinematic domains for the Higgs boson are used. This can be done for the final
states considered in Fig. 8. Ratios of production rates for these channels
provide absolute determinations of ratios of branching ratios, with
uncertainties dominated by the statistics, and by the uncorrelated systematics
such as reconstruction efficiencies for the different final state particles.
These ratios are shown in Fig. 9. The curves with the systematics labeled as
“cons” use the conservative reconstruction uncertainties plotted in Fig. 7.

Figure 9: Projected precision for the measurement of ratios of rates of different Higgs final states, in the gg$\to$H production channel. The systematic uncertainty labels are defined in the text. Table 4:  Target precision for the parameters relative to the measurement of various Higgs decays, ratios thereof, and of the Higgs self-coupling $\lambda$. Notice that Lagrangian couplings have a precision that is typically half that of what is shown here, since all rates and branching ratios depend quadratically on the couplings. Observable | Parameter | Precision | Precision
---|---|---|---
|  | (stat) | (stat+syst+lumi)
$\mu=\sigma$(H)$\times$B(H$\to$ γγ) | $\delta\mu/\mu$ | 0.1% | 1.5%
$\mu=\sigma$(H)$\times$ B(H$\to$µµ) | $\delta\mu/\mu$ | 0.28% | 1.2%
$\mu=\sigma$(H)$\times$B(H$\to$ 4µ) | $\delta\mu/\mu$ | 0.18% | 1.9%
$\mu=\sigma$(H)$\times$B(H$\to$ γµµ) | $\delta\mu/\mu$ | 0.55% | 1.6%
$\mu=\sigma$(HH)$\times$B(H$\to$γγ)B(H$\to$bb̄) | $\delta\lambda/\lambda$ | 5% | 7.0%
$R=$ B(H$\to$µµ)/B(H$\to$4µ) | $\delta R/R$ | 0.33% | 1.3%
$R=$ B(H$\to$γγ)/B(H$\to$ 2e2µ) | $\delta R/R$ | 0.17% | 0.8%
$R=$ B(H$\to$γγ)/B(H$\to$ 2µ) | $\delta R/R$ | 0.29% | 1.4%
$R=$ B(H$\to$µµγ)/B(H$\to$µµ) | $\delta R/R$ | 0.58% | 1.8%
$R=\sigma$(tt̄H)$\times$ B(H$\to$ bb̄)/$\sigma$(tt̄Z)$\times$B(Z$\to$ bb̄) | $\delta R/R$ | 1.05% | 1.9%
$B$(H$\to$ invisible) |  $B@$95%CL | $1\times 10^{-4}$ | $2.5\times 10^{-4}$
Figure 10: Integrated luminosity evolution of the H$\to$invisible branching
ratio, under various systematics assumptions.

These results are summarised in Table 4, separately showing the statistical
and systematic uncertainties obtained in our studies. As remarked above, there
is in principle room for further progress, by fully exploiting data-driven
techniques to reduce the experimental systematics. At the least, one can
expect that these potential improvements will compensate for the current
neglect of other experimental complexity, such as pile-up. The most robust
measurements will involve the ratios of branching ratios. Taking as a given
the value of the HZZ coupling (and therefore $B$(H$\to 4\ell$)), which will be
measured to the few per-mille level by FCC-ee, from the FCC-hh ratios it could
be possible to extract the absolute couplings of the Higgs to γγ (0.4%), µµ
(0.7%), and Zγ(0.9%).

Exercise: discuss the possible role of precise measurememts of ratios of BRs
in exploring the microscopic origin of potential deviations from the SM
expectations. Which type of models can give rise to deviations in the ratios
considered here? Which models would leave no signatures in these ratios?

The ratio with the tt̄Z process is considered for the tt̄H process, as
proposed in Ref. [47]. This allows the removal of the luminosity uncertainty,
and reducing the theoretical systematics on the production modelling below 1%.
An updated study of this process, including the FCC-hh detector simulation, is
presented in Ref. [46]. Assuming FCC-ee will deliver the expected precise
knowledge of $B$(H$\to$bb̄), and the confirmation of the SM predictions for
the Ztt̄ vertex, the tt̄H/tt̄Z ratio should therefore allow a determination of
the top Yukawa coupling to 1%.

The limit quoted in Table 4 on the decay rate of the Higgs boson to new
invisible particles is obtained from a study of large missing-$E_{T}$
signatures. The analysis, discussed in detail in Ref. [46], relies on the
data-driven determination of the leading SM backgrounds from W/Z+jets. The
integrated luminosity evolution of the sensitivity to invisible H decays is
shown in Fig. 10. The SM decay H$\to$4ν, with branching ratio of about
$1.1\times 10^{-3}$, will be seen after $\sim$1 ab-1, and the full FCC-hh
statistics will push the sensitivity to $2\times 10^{-4}$.

Exercise: if the Higgs admits an important decay rate to non-SM particles,
$\Gamma_{H}$ will increase. A larger width will reduce all BRs by a common
factor. Assuming that these decay signatures are elusive in a hadron collider,
discuss how ratios of BRs could still be used to learn more about their
origin.

Last but not least, Table 4 reports a 7% expected precision in the extraction
of the Higgs self-coupling $\lambda$. This result is discussed in more detail
in a later Section, with other probes of the Higgs self-interaction.

Figure 11: Left: precision in the determination of the scattering of same-sign longitudinal W bosons, as function of luminosity, for various kinematic cuts. Right: sensitivity of the longitudinal boson scattering cross section w.r.t. deviations of the WWH coupling from its SM value ($\kappa_{W}=1$), for various selection cuts on the final-state dilepton invariant mass. The vertical bars represent the precision of the measurement, for 30 ab-1. Table 5: Constraints on the HWW coupling modifier $\kappa_{W}$ at 68% CL, obtained for various cuts on the di-lepton pair invariant mass in the $\rm W_{L}W_{L}\to HH$ process. $m_{l^{+}l^{+}}$ cut |  $>50$ GeV |  $>200$ GeV |  $>500$ GeV |  $>1000$ GeV
---|---|---|---|---
$\kappa_{W}\in$ | [0.98,1.05] | [0.99,1.04] | [0.99,1.03] | [0.98,1.02]

###  4.3 Longitudinal Vector Boson Scattering

The scattering of the longitudinal components of vector bosons is particularly
sensitive to the relation between gauge couplings and the VVH coupling. A
thorough analysis of same-sign $\rm W_{L}W_{L}$ scattering, in the context of
the FCC-hh detector performance studies, is documented in Ref. [46]. The
extraction of the $\rm W^{\pm}_{L}W^{\pm}_{L}$ signal requires the removal of
large QCD backgrounds ($\rm W^{\pm}W^{\pm}$+jets, WZ+jets) and the separation
of large EW background of transverse-boson scattering. The former is
suppressed by requiring a large dilepton invariant mass and the presence of
two jets at large forward and backward rapidities. The longitudinal component
is then extracted from the scattering of transverse states by exploiting the
different azimuthal correlations between the two leptons. The precision
obtained for the measurement of the $\rm W_{L}W_{L}$ cross section as a
function of integrated luminosity, is shown in Fig. 11 (left). The three
curves correspond to different assumptions about the rapidity acceptance of
the detector and drive the choice of the detector design, setting a lepton
(jet) acceptance out to $|\eta|=4(6)$. The small change in precision when
increasing the jet cut from $p_{T}>30$ to $p_{T}>50$ GeV indicates a strong
resilience of the results against the presence of large pile-up. The quoted
precision, reaching the value of 3% at 30 ab-1, accounts for the systematic
uncertainties of luminosity (1%), lepton efficiency (0.5%), PDF (1%) and the
shape of the distributions used in the fit (10%). The right plot in Fig. 11
shows the impact of rescaling the WWH coupling by a factor $\kappa_{W}$. The
effect is largest at the highest dilepton invariant masses, as expected. The
measurement precision, represented by the small vertical bars, indicates a
sensitivity to $\delta\kappa_{W}$ at the percent level, as shown also in Table
5.

For a review of the Higgs measurements at HE-LHC, see Refs. [9, 34].

##  5 Precision EW measurements

The Higgs boson forms an integral part of the EW sector, and its properties
are deeply intertwined with those of EW phenomena. A thorough program of EW
measurements goes hand in hand with the study of Higgs properties, and is an
essential complement to it. At the FCC, EW interactions can be studied from
multiple perspectives, extending by large factors all previous targets of
precision and energy reach. We summarize here the main results from the
existing studies, documented in more detail in Ref. [9, 10].

The FCC-ee run at the $\rm Z^{0}$ peak will deliver about $2\times 10^{5}$
times the LEP statistics, with about $10^{11}$ $\rm\mu^{+}\mu^{-}$ or
$\rm\tau^{+}\tau^{-}$ final states, and $3\times 10^{12}$ hadronic decays. The
larger statistics w.r.t. LEP will be accompanied by significant efforts to
minimize the systematic uncertainties. For example, the beam energies will be
measured more precisely, and better detectors will improve the efficiency of
b-tagging or the precision in the absolute luminosity determination.
Significant theoretical improvements in the calculation of higher-loop EW and
QCD corrections are also foreseen, and necessary to fully exploit the
potential improvement by over two orders of magnitude in the statistical
precision.

$\alpha_{\rm EM}(M_{Z})$ is a crucial input parameter to interpret SM
precision observables. The EM coupling at the scale of the electron mass is
the best known fundamental constant of nature, but its renormalization group
evolution to the scale of weak interactions is subject to important
uncertainties, due to non-perturbative hadronic physics, which enters the
photon self-energy corrections as $Q^{2}$ evolves through the region of the
hadronic resonances. The systematic uncertainties of the experimental data on
$\sigma(\rm e^{+}e^{-}\to$hadrons$)$ end up dominating the precision of the
extrapolation to $Q^{2}=M_{Z}^{2}$, which is limited to the level of
$10^{-4}$. Dedicated runs at $\sqrt{s}$=87.7 and 93.9 GeV will extract
$\alpha_{\rm EM}(M_{Z})$ directly (namely without an extrapolation in $Q$)
from the energy-dependence of the $\rm\mu^{+}\mu^{-}$ forward-backward
asymmetry, improving the current uncertainty by a factor of 4.

Forward-backward and polarization asymmetries will also allow to reduce by a
factor of 30-50 the uncertainty in $\sin^{2}\theta_{W}$. I recall that today’s
determination of the weak mixing angle, $\sin^{2}\theta^{\rm lept}_{\rm
eff}=0.23153\pm 0.00015$ [48], is dominated by the combination of two precise
measurements (the b-quark forward-backward asymmetry from LEP and the left-
right polarization asymmetry from SLD), which differ among themselves by 3.2
standard deviations. While future HL-LHC data [49] will provide an independent
determination of the weak mixing angle with a precision approaching the
current LEP/SLD one, using the lepton charge asymmetry in Z boson decays, it
is only with the future FCC-ee data that this puzzling result will be
clarified.

The Z-decay asymmetries will also help improving the measurement of vector and
axial couplings of the leptons and of the charm and bottom quarks. In absence
of a reliable technique to distinguish Z decays to the different lighter
quarks (u, d and s), the most precise determination of their couplings to the
neutral current will come from FCC-eh. There, a simultaneous fit to the light
quarks EW couplings and to the PDFs, using both charged and neutral current
data, will disentangle the individual quarks and allow the measurement of
their respective vector and axial couplings. The projection in in Fig. 12 for
the precision of all fermionic couplings, from a global fit [50, 51] to both
FCC-ee and FCC-eh data treating each lepton and quark flavour as independent,
shows the improvement expected with respect to today’s knowledge.

Figure 12: Sensitivity, at the 1-$\sigma$ level, to deviations of the neutral
current couplings resulting from a global EFT fit at the dimension-6 level to
EW precision measurements at FCC-ee and FCC-eh. From [9].

The measurement of the total Z width $\Gamma_{Z}$, and of its visible
fraction, will allow to extract the invisible component of $\Gamma_{Z}$.
Today, the number of neutrino species obtained from the LEP data is
$N_{\nu}=2.984\pm 0.008$, which is low by two standard deviations. A deficit
in the neutrino counting from Z decays could be attributed to a violation of
unitarity in the neutrino mixing matrix, or to the presence of right-handed
neutrinos [52]. FCC-ee will improve the precision on $N_{\nu}$ by almost a
factor of 10, down to 0.001.

The $10^{8}$ pairs of W bosons produced at the two energies of
$\sqrt{s}=$157.5 and 162.5 GeV will reduce the uncertainty on the W mass,
$m_{W}$, to 0.5 MeV, and of its width to 1.2 MeV. The limited statistics of W
bosons from LEP2 left us with a puzzling $\sim 3\sigma$ discrepancy between
the decay branching ratio of the W to the tau lepton, $(11.38\pm 0.21)\%$, and
to the e and µ($(10.71\pm 0.16)\%$ and $(10.63\pm 0.15)\%$, resp.). FCC-ee can
reduce these uncertainties by almost two orders of magnitude, greatly
increasing the sensitivity to possible violations of lepton flavour
universality, a topic that is receiving great attention nowadays. For
comparison, the $10^{11}$ Z decays to individual leptons will allow to test
neutral-current lepton universality at the level of of $10^{-5}$. τ
semileptonic decays, and the τ lifetime, could achieve a sensitivity to
deviations from lepton flavour universality in weak charged currents at a
similar level.

Figure 13:  Contours of 68% confidence level in the $(m_{\rm top},m_{\rm W})$
plane obtained from fits of the SM to the EW precision measurements offered by
the FCC-ee, under the assumption that all relevant theory uncertainties can be
reduced to match the experimental uncertainties: the red ellipse is obtained
from the FCC-ee measurements at the Z pole, while the blue ellipses arise from
the FCC-ee direct measurements of the W and top masses. The two dotted lines
around the SM prediction illustrate the uncertainty from the Z mass
measurement if it were not improved at the FCC-ee. The green ellipse
corresponds to the current W and top mass uncertainties from the Tevatron and
the LHC. The potential future improvements from the LHC are illustrated by the
black dashed ellipse. Plot from Ref. [10], obtained using the GFitter code
introduced in [53].

A collection of the various EW precision measurements possible at the FCC-ee,
including those relative to the top quark properties, is shown in Table 6.
Their overall impact in testing the SM relation between the top and W masses
is shown in Fig. 13.

Table 6: Measurement of selected EW quantities at FCC-ee, compared with the present precisions [10] Observable | present |  |  | FCC-ee | FCC-ee | Comment and
---|---|---|---|---|---|---
| value | $\pm$ | error | Stat. | Syst. | dominant exp. error
$\mathrm{m_{Z}~{}(keV)}$ | 91186700 | $\pm$ | 2200 | 5 | 100 | From Z line shape scan
|  |  |  |  |  | Beam energy calibration
$\mathrm{\Gamma_{Z}~{}(keV)}$ | 2495200 | $\pm$ | 2300 | 8 | 100 | From Z line shape scan
|  |  |  |  |  | Beam energy calibration
$\mathrm{R_{\ell}^{Z}}~{}(\times 10^{3})$ | 20767 | $\pm$ | 25 | 0.06 | 0.2-1.0 | ratio of hadrons to leptons
|  |  |  |  |  | acceptance for leptons
$\mathrm{\alpha_{s}(m_{Z})}~{}(\times 10^{4})$ | 1196 | $\pm$ | 30 | 0.1 | 0.4-1.6 | from $\mathrm{R_{\ell}^{Z}}$ above [54]
$\mathrm{R_{b}}~{}(\times 10^{6})$ | 216290 | $\pm$ | 660 | 0.3 | ¡60 | ratio of $\rm{b\bar{b}}$ to hadrons
|  |  |  |  |  | stat. extrapol. from SLD [55]
$\mathrm{\sigma_{had}^{0}}~{}(\times 10^{3})$ (nb) | 41541 | $\pm$ | 37 | 0.1 | 4 | peak hadronic cross-section
|  |  |  |  |  | luminosity measurement
$\mathrm{N_{\nu}}(\times 10^{3})$ | 2991 | $\pm$ | 7 | 0.005 | 1 | Z peak cross sections
|  |  |  |  |  | Luminosity measurement
$\mathrm{sin^{2}{\theta_{W}^{\rm eff}}}(\times 10^{6})$ | 231480 | $\pm$ | 160 | 3 | 2 - 5 | from $\mathrm{A_{FB}^{{\mu}{\mu}}}$ at Z peak
|  |  |  |  |  | Beam energy calibration
$\mathrm{1/\alpha_{QED}(m_{Z})}(\times 10^{3})$ | 128952 | $\pm$ | 14 | 4 | small | from $\mathrm{A_{FB}^{{\mu}{\mu}}}$ off peak [56]
$\mathrm{A_{FB}^{b},0}~{}(\times 10^{4})$ | 992 | $\pm$ | 16 | 0.02 | 1-3 | b-quark asymmetry at Z pole
|  |  |  |  |  | from jet charge
$\mathrm{A_{FB}^{pol,\tau}~{}(\times 10^{4})}$ | 1498 | $\pm$ | 49 | 0.15 | ¡2 | τ polarisation and charge asymmetry
|  |  |  |  |  | τ decay physics
$\mathrm{m_{W}~{}(MeV)}$ | 80350 | $\pm$ | 15 | 0.5 | 0.3 | From WW threshold scan
|  |  |  |  |  | Beam energy calibration
$\mathrm{\Gamma_{W}~{}(MeV)}$ | 2085 | $\pm$ | 42 | 1.2 | 0.3 | From WW threshold scan
|  |  |  |  |  | Beam energy calibration
$\mathrm{\alpha_{s}(m_{W})}(\times 10^{4})$ | 1170 | $\pm$ | 420 | 3 | small | from $\mathrm{R_{\ell}^{W}}$ [57]
$\mathrm{N_{\nu}}(\times 10^{3})$ | 2920 | $\pm$ | 50 | 0.8 | small | ratio of invis. to leptonic
|  |  |  |  |  | in radiative Z returns
$\mathrm{m_{top}~{}(MeV)}$ | 172740 | $\pm$ | 500 | 17 | small | From $\mathrm{t\bar{t}}$ threshold scan
|  |  |  |  |  | QCD errors dominate
$\mathrm{\Gamma_{top}~{}(MeV)}$ | 1410 | $\pm$ | 190 | 45 | small | From $\mathrm{t\bar{t}}$ threshold scan
|  |  |  |  |  | QCD errors dominate
$\mathrm{\lambda_{top}/\lambda_{top}^{SM}}$ | 1.2 | $\pm$ | 0.3 | 0.1 | small | From $\mathrm{t\bar{t}}$ threshold scan
|  |  |  |  |  | QCD errors dominate
$\mathrm{ttZ~{}couplings}$ |  | $\pm$ | 30% |  $0.5-1.5$% | small | From $\mathrm{E_{CM}=365GeV}$ run

###  5.1 Complementarity of EW and Higgs measurements

The best framework to expose the complementarity between Higgs and EW probes
of new physics at large scales, is that of the SM effective field theory
(SMEFT [58, 59, 60]). Here one assumes that, as in the SM, the Higgs boson
transforms as doublet of $SU(2)_{L}$, and considers all $SU(2)_{L}\times
U(1)_{Y}$ invariant operators, classified according to their dimension:

${\cal L}_{\mathrm{Eff}}=\sum_{d=4}^{\infty}\frac{1}{\Lambda^{d-4}}{\cal
L}_{d}={\cal L}_{\mathrm{SM}}+\frac{1}{\Lambda}{\cal
L}_{5}+\frac{1}{\Lambda^{2}}{\cal L}_{6}+\ldots,~{}~{}~{}~{}~{}~{}{\cal
L}_{d}=\sum_{i}C_{i}{\cal O}_{i}.$ (4)

At dimension 4, one finds the SM itself. At dimension 5 appear the operators
that generate Majorana neutrino masses. At dimension 6 one first finds
operators that, parameterizing the low-energy behaviour of new interactions
beyond the EW scale, lead to modifications of EW and Higgs observables.
Focusing on lepton and baryon-number conserving operators, and assuming
flavour universality, leaves a basis of 59 dim-6 operators [59].

Figure 14 shows the constraints that future FCC-ee data can impose on the
coefficients of the subset of operators that play a role in the EW and Higgs
observables discussed so far. On the EW side, one has the following 10
operators:

$\displaystyle{\cal O}_{\phi
D}=\left|\phi^{\dagger}D^{\mu}\phi\right|^{2},\quad{\cal O}_{\phi
WB}=\left(\phi^{\dagger}\sigma_{a}\phi\right)W_{\mu\nu}^{a}B^{\mu\nu},$
$\displaystyle{\cal
O}_{\phi\psi}^{(1)}=(\phi^{\dagger}~{}\\!\stackrel{{\scriptstyle\leftrightarrow}}{{D}}\\!_{\mu}\phi)(\overline{\psi}^{i}\gamma^{\mu}\psi^{i}),\quad{\cal
O}_{\phi
F}^{(3)}=(\phi^{\dagger}~{}\\!\stackrel{{\scriptstyle\leftrightarrow}}{{D}}\\!^{\\!~{}a}_{\mu}\phi)(\overline{F}^{i}\gamma^{\mu}\sigma_{a}F^{i}),\quad{\cal
O}_{ll}=\left(\overline{l}\gamma_{\mu}l\right)\left(\overline{l}\gamma^{\mu}l\right),$
(5)

where $\phi$ is the Higgs scalar doublet, $\psi$ runs over all the 5 types of
SM fermion multiplets, while $F$ only refers to the 2 types of SM left-handed
fermion doublets. The $\phi$ field can induce effects in processes where an
explicit Higgs particle is present, or can influence EW observables
indirectly, when $\phi$ is set to its vacuum expectation value. The other 8
operators shown in Fig 14 are mostly affecting Higgs observables:

$\displaystyle{\cal O}_{\phi
G}=\phi^{\dagger}\phi~{}\\!G_{\mu\nu}^{A}G^{A~{}\\!\mu\nu},\quad{\cal O}_{\phi
W}=\phi^{\dagger}\phi W_{\mu\nu}^{a}W^{a~{}\\!\mu\nu},\quad{\cal O}_{\phi
B}=\phi^{\dagger}\phi B_{\mu\nu}B^{\mu\nu},\quad{\cal
O}_{\phi\Box}=(\phi^{\dagger}\phi)\Box(\phi^{\dagger}\phi),$
$\displaystyle{\cal
O}_{\mu\phi}=(\phi^{\dagger}\phi)(\bar{l}^{2}\phi~{}\\!\mu),\quad{\cal
O}_{\tau\phi}=(\phi^{\dagger}\phi)(\bar{l}^{3}\phi~{}\\!\tau),\quad{\cal
O}_{b\phi}=(\phi^{\dagger}\phi)(\bar{q}^{3}\phi\\!b),\quad{\cal
O}_{c\phi}=(\phi^{\dagger}\phi)(\bar{q}^{2}\tilde{\phi}~{}\\!c).$ (6)

The sensitivities to the ratios $C_{i}/\Lambda^{2}$ are reported in Fig. 14 as
95$\%$ probability bounds on the interaction scale, $\Lambda/\sqrt{C_{i}}$,
associated to each operator. Notice that, in the same way that the energy
scale associated to the Fermi constant $G_{\rm F}^{-{1}/{2}}$ differs from the
W boson mass by a factor related to the weak coupling, the interaction scale
defined by $\Lambda/\sqrt{C_{i}}$ does not correspond exactly to the mass of
new particles.

Few remarks are in order. First of all, comparison [10] with the current
constraints shows that the FCC-ee data will increase by a factor of 4-5 the
sensitivity to new energy scales. This matches well the factor of 7 increase
in energy of the FCC-hh w.r.t. the LHC, which will lead to a comparable
incease in the direct sensitivity to new physics. In other words, a 100 TeV pp
collider is typically properly scaled to search for the microscopic orgin of
possible deviations observed by the FCC-ee precision measurements. In absolute
terms, the scales that can be probed in the case of weakly-interacting new
physics ($C_{I}\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}1$)
range between a few and few tens of TeV, while they can go up to O(100) TeV in
the case of strongly interacting forces. The second remark is that Fig. 14
underscores the complementarity between EW and Higgs observables: even though
operators considered here (except ${\cal O}_{ll}$) include both Higgs and
gauge bosons or fermions, their constraints come primarily from either EW or
Higgs measurements. Both types of measurements are therefore necessary for a
systematic exploration of BSM contributions induced via SMEFT operators. The
respective scale sensitivity observed for both types of operators is rather
consistent, with most limits on interaction scales in the range of 20-30 TeV,
showing that the precision targets set by the Higgs and EW programs are
coherent.

Figure 14:  FCC-ee constraints on various EFT operators, using EW and Higgs
observables, separately and combined. Darker shades of each color indicate the
results neglecting all SM theory uncertainties. Plot from Ref. [10], obtained
using the HEPfit framework [51].

##  6 Precision versus sensitivity

In the previous sections, we focused on the very precise measurements that the
FCC enables, both in the Higgs and in the EW sector of the SM. These
measurements have a value per se, independently of whether we eventually find
discrepancies w.r.t. the SM expectations. We need these precise measurements
to test up to which point our understanding of fundamental EW phenomena is
controlled by the SM. But we also need precise measurements because, the day
new physics were found some place, we shall need as much data as possible to
evaluate and constrain the many models that will be proposed to explain it. In
this context, data that agree with the SM can be as useful as data that may
not agree with it. Precise data in agreement with the SM, for example, can
help us rule out claims of observations of new physics that were to openly
clash with established measurements.

So, the success of a precision physics program is not necessarily tied to the
discovery of discrepancies with the SM, but builds on reliable, unbiased and
ever more accurate measurements of how nature behaves. Treating the achievable
precision as a gauge of the sensitivity to new physics phenomena, however, is
critical to examine and characterize the potential relevance of a given
measurement in a specific BSM context. It is also a useful way to evaluate the
reach of different facilities, which might not measure exactly the same
observable: the interpretation of their measurements in terms of constraints
on some common BSM phenomenon can be seen as a useful standard candle for
their comparison, to assess synergies and complementarities.

Figure 15:  Example of high-$Q$ Higgs production processes.

In the context of the FCC, there is great richness of synergies and
complementarities. The same new physics model can manifest itself via
departures in indirect precision measurements at FCC-ee, and directly via the
production of new particles at FCC-hh. But low- and high-energy observables
can also both participate in building indirect evidence for new phenomena
living at scales that are beyond the direct discovery reach. This interplay
can be shown with a general example, using the EFT language. Let us consider a
dimension-6 operator $O$, parameterizing at energies $E<\Lambda$ the effects
of some new physics present at a scale $\Lambda$. This induces a contribution
to the Lagrangian given by $O/\Lambda^{2}$, and leads to corrections
$\delta_{if}=\langle f|O|i\rangle/\Lambda^{2}$ to the transition between the
initial and final states $|i\rangle$ and $|f\rangle$. By dimensional analysis,
$\delta_{if}$ will scale, relative to a dimension-4 SM contribution, like
$\delta_{if}\propto(\mu/\Lambda)^{2}$, where $\mu$ is a mass scale
characteristic of the $i\to f$ transition. In the case of a Higgs decay, or of
Higgs production at threshold as in $gg\to H$, the only possible scales are
$m_{H}$ or $v$. Taking e.g. $\mu=m_{H}$, gives $\delta_{if}\propto
m_{H}^{2}/\Lambda^{2}\sim 1.5\%\times({\rm 1~{}TeV}/\Lambda)^{2}$. If the
transition involves another large kinematical scale $Q$, with
$m_{H}<Q<\Lambda$, as in the case of Higgs production at large $p_{T}$ (see
Fig. 15), the deviation could scale like $Q$, $\delta_{if}\propto(Q/{\rm
1\;TeV})^{2}\times({\rm 1\;TeV}/\Lambda)^{2}$. The impact of the EFT operator
would therefore be greatly enhanced at large $Q$. Assuming $\Lambda=1$ TeV,
for example, a measurements at a scale $Q\sim 400$ GeV would lead to an effect
of approximatey 15%, ten times larger than for the decay or production at
threshold. This means that, at large $Q$, one can detect the presence of new
physics effects with lesser precision than is required at smaller
$Q$222Needless to say the EFT formalism looses validity once the hard scale
becomes large enough to induce direct manifestation of new physics. . A hadron
collider cannot provide precision comparable to an electron collider, but its
higher kinematical reach can compensate for this!

Several studies of this interplay between precision and sensitivity from high-
energy indirect measurements have appeared in the literature (see e.g. [61,
62, 63, 64, 65, 66, 67, 68]) covering both Higgs and EW observables. Several
of these papers test these ideas in the context of the LHC, and the
sensitivity comparison is therefore drawn w.r.t. the LEP precision
measurements. We shall give here a couple of examples specific to the 100 TeV
collider, where the kinematic reach and the large Higgs production rates make
this approach even more powerful.

###  6.1 Example: the VVHH coupling

Figure 16 shows the diagrams for Higgs-pair production via vector boson
fusion, studied in Ref. [69]. At large invariant mass of the Higgs pair,
$m_{HH}$, the triple-Higgs coupling contribution is suppressed by the $1/s$
off-shell Higgs propagator,

Figure 16:  Leading order Feynman diagrams for Higgs-pair production via
vector boson fusion.

and the amplitude is controlled by the behaviour of the longitudinal-
longitudinal component of the amplitude, characterised by the destructive
interference between the first two diagrams:

$A({\rm V_{L}V_{L}\to HH})\sim\frac{\hat{s}}{v^{2}}(c_{2V}-c_{V}^{2})+{\cal
O}(m_{W}^{2}/\hat{s})\;.$ (7)

Here, $c_{2V}$ and $c_{V}$ represent, respectively, the coefficients of the
VVHH and VVH couplings, normalised to their SM values.
$\delta_{c}=c_{2V}^{2}-c_{V}$ vanishes in the SM and in models where the
$SU(2)$ symmetry is linearly realized, and the growth of the amplitude with
energy is suppressed. In composite Higgs models based e.g. on an $SO(5)/SO(4)$
symmetry [70], on the other hand, $c_{V}=\sqrt{1-\xi}$ and $c_{2V}=1-2\xi$
($\xi\sim v^{2}/\Lambda^{2}$), so the scattering amplitude grows with energy
as $A\sim\hat{s}/\Lambda^{2}$.

Figure 17:  Left: HH invariant mass spectrum in vector boson scattering.
Right: probability density distribution for $\delta c_{2V}$, assuming
$c_{V}=1$ in Eq. 7. From Ref. [69].

The study of Ref. [69] considered the HH$\to$4b final state, applying boosted-
jet tagging techniques, given the high $p_{T}$ of the Higgs bosons at high
$m_{HH}$. The impact of $\delta_{c}\neq 0$ is visible in Fig. 17, which shows
the $m_{HH}$ distribution in the SM in a $c_{V}=1$, $c_{2V}=0.8$ scenario, and
the expected backgrounds. $c_{V}$ will be measured with a few per-mille
precision at FCC-ee (independently of whether it agrees or not with the SM),
and the constraints on $\delta_{c}$ at FCC-hh will translate directly into a
constraint on $c_{2V}$. The detailed study of Ref. [69] projects a sensitivity
to deviations of $c_{2V}$ from its SM value of better than $\pm 1\%$, in spite
of the much coarser precision in the measurement of the HH rates! See Ref.
[69] for the discussion of the validity range of the EFT approximation.

###  6.2 Example: Drell-Yan at large mass

Drell-Yan (DY) production is another example where energy can complement
precision, achieving sensitivity to new high-scale phenomena [71]. The total
production rates of W± and Z0 bosons at 100 TeV are about 1.3 and 0.4 µb,
respectively, i.e. samples of ${\cal O}(10^{10-11})$ leptonic decays per ab-1!
A large fraction of these events probe very high energies. Figure 18, left
panel, shows the integrated spectra of the W boson transverse mass
($M^{2}_{T}=2p_{T,\ell}p_{T,\nu}(1-\cos\theta_{\ell\nu})$) and of the γ/Z
dilepton mass.

Exercise: The right panel of Fig. 18 shows the invariant mass spectrum of
gauge boson pairs. Compare these rates to those for lepton-pair production
(left panel), and discuss the reasons for the large differences.

Figure 18: Left: integrated lepton transverse (dilepton) mass distribution in
$\rm pp\to W^{*}\to\ell$ν ($\rm pp\to Z^{*}/$γ${}^{*}\to\ell^{+}\ell^{-}$).
One lepton family is included, with $|\eta_{\ell}|<2.5$. Right: Integrated
invariant mass spectrum for the production of gauge boson pairs in the central
kinematic range $|y|<1.5$. No branching ratios included.

In presence of new physics, large corrections to the SM prediction can arise
from the $W$ and $Y$ oblique parameters defined by the following dim-6 EFT
operators [72]:

$\hat{W}=-\frac{W}{4m_{W}^{2}}(D_{\rho}W^{a}_{\mu\nu})^{2}\quad,\quad\hat{Y}=-\frac{Y}{4m_{W}^{2}}(\partial_{\rho}B_{\mu\nu})^{2}\;.$
(8)

These parameters capture the universal modifications of the EW gauge boson
propagators, and are already constrained, at the per mille level, from LEP-2
precision measurements and from the W mass measurements at Tevatron and LHC
(as well as other precision measurements at the Z pole at LEP/SLD). The FCC-hh
DY statistics at very high mass will contribute, with the precision
measurements at lower energy by FCC-ee, to improve the current constraints by
two orders of magnitude, as shown in Table 7. In terms of a new physics scale
$\Lambda$ defined by $g_{*}^{2}/\Lambda^{2}=(W,Y)/4m_{W}^{2}$, the FCC-hh
reach corresponds to
$\Lambda\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}g_{*}\times
80$ TeV, a sensitivity that could be matched by a multi-TeV lepton collider
such as CLIC. We should remark that the low- and high-energy approaches should
not be seen as alternative, but as synergetic: should deviations be observed
in either of them, the other measurement would serve as an independent probe
to confirm the SM departure, and to help pinning down its origin.

Table 7:  Reach at 95% CL on W and Y from different experiments. The bounds from neutral DY are obtained by setting the unconstrained parameter to zero. From Ref. [71]. | LEP | ATLAS 8 | CMS 8 | LHC 13 | FCC-hh | FCC-ee
---|---|---|---|---|---|---
luminosity |  $2\times 10^{7}\,Z$ |  19.7 fb-1 |  20.3 fb-1 |  0.3 ab-1 |  3 ab-1 |  10 ab-1 |  $10^{12}\,Z$
NC |  W $\times 10^{4}$ | $[-19,3]$ | $[-3,15]$ | $[-5,22]$ | $\pm 1.5$ | $\pm 0.8$ | $\pm 0.04$ | $\pm 1.2$
|  Y $\times 10^{4}$ | $[-17,4]$ | $[-4,24]$ | $[-7,41]$ | $\pm 2.3$ | $\pm 1.2$ | $\pm 0.06$ | $\pm 1.5$
CC | W $\times 10^{4}$ | — |  $\pm 3.9$ | $\pm 0.7$ | $\pm 0.45$ | $\pm 0.02$ | —

Further applications of the huge DY lever arm in $Q^{2}$ are (i) the
determination of the running of EW couplings, by measuring the transverse
(invariant) mass spectrum of (di)leptons produced by far off-shell W (Z)
bosons [73], and (ii) the indirect search for new heavy EW particles (like
gauginos in supersymmetry) using the distortion of the DY shape near their
production threshold [74].

##  7 The Higgs potential

As we discussed at the beginning of these lectures, understanding the origin
of the Higgs potential is among the most, if not the most, outstanding target
of future colliders. As an essential part of this understanding, we must start
measuring it. Let us briefly rediscuss the relations between the parameters of
the Higgs potential and the physical observables, in a context slightly more
general than the SM. To simplify the notation, let us consider a single real
field, and consider the following simple generalization of the SM potential
(you could consider repeating the exercise with a more general functional
form):

$V(\phi)=-\frac{\mu^{2}}{2}\phi^{2}+\frac{\lambda}{n}\phi^{n}\;,$ (9)

$n$ must be even, and of course $n=4$ in the SM. The two key relations
obtained by setting $\langle\phi\rangle=v$ at the minimum of the potential,
and defining $m_{\phi}^{2}=\partial^{2}V(\phi)/\partial\phi^{2}|_{\phi=v}$
give:

$v^{n-2}=\frac{\mu^{2}}{\lambda}\quad,\quad m_{\phi}^{2}=(n-2)\mu^{2}\;.$ (10)

We stress that, while dimensional analysis makes the Higgs mass proportional
to $\mu^{2}$, its precise value is not defined by the dynamics near the origin
$\phi\sim 0$, where the quadratic term dominates, but it is defined by the
dynamics around the minimum $\phi=v$, which could be far away from the origin.
This is reflected by the coefficient $(n-2)$ in Eq. 10, whose specific value
depends on the shape of the potential at large $\phi$. In other words, the
quadratic term of the potential only provides the overall scale of the Higgs
mass, but the specific value is controlled by the Higgs dynamics at the
minimum, where the higher-order terms of the potential are important: in the
same way that the masses of SM particles are related to the strength of their
interaction with the Higgs, it is reasonable that the mass of the Higgs be
related to its own self-interaction. Therefore studying the structure of the
Higgs potential is also a way to address the question of “what gives mass to
the Higgs”.

Now, the potential above has three parameters, $\mu^{2}$, $\lambda$ and the
power $n$. However we only have two measurements, the Higgs mass and its
expectation value $v$. The two are absolute numbers, independent of the shape
of the potential: the mass is what we measure in the experiments, while $v$ is
given by the relation $M_{W}=gv$, where $g$ is the weak gauge coupling (recall
also that $v=(\sqrt{2}G_{F})^{-1/2}$). For each $n$, we can extract a value
for $\lambda$ and $\mu^{2}$, but we won’t be able to determine what the shape
of the potential is. In other words, the only thing we know experimentally
about the Higgs potential is that its second derivative at the origin is
negative, to drive $v$ away from 0. We have no experimental evidence, as of
today, that $n=4$ rather than 6. To make progress in learning about the
structure of $V(\phi)$, we need a further measurement, sensitive to the power
$n$ in Eq. 9. Expanding the potential around its minimum, the cubic term
controls the cubic self-interaction, and its strength is given by:

$\lambda_{\phi\phi\phi}=\frac{\partial^{3}V}{\partial\phi^{3}}|_{\phi=v}=(n-1)\frac{m_{\phi}^{2}}{v}\;.$
(11)

Assuming a Higgs potential given by Eq. 9, and given that $m_{\phi}$ and $v$
are known, the measurement of the Higgs cubic self-interaction would be
directly a measurement of $n$. More in general, it is clear that the Higgs
self-coupling is a mandatory measurement to start learning about the Higgs
potential. Notice one point: the change from $n=4$ (as in the SM), to
something like $n=6$, has a big impact on the self-coupling, namely it
increases it by a factor of 5/3. With this modification of the Higgs
potential, there is no continuous knob that allows to smoothly change
$\lambda_{\phi\phi\phi}$ from its SM value. However large, this change would
have still failed to be detected experimentally, so for all we know this is
still an open option. So we should be open to the possibility that
$\lambda_{\phi\phi\phi}$ differs from its SM value by ${\cal O}(1)$, and a 20%
uncertainty in its determination might already explore possible deviations at
the 5$\sigma$ level.

Needless to say, the most likely scenario assumes the presence of the
$\lambda|H|^{4}$ SM quartic coupling333The presence of a quartic coupling is
unavoidable, even if, for some odd reason, the underlying fundamental theory
did not have such a coupling: the quartic would in fact be generated via
radiative corrections at the one-loop order, starting from the $|H|^{6}$
term., supplemented by other higher-order terms, like $1/\Lambda^{2}|H|^{6}$.
In this case, $v^{2}/\Lambda^{2}$ becomes a continuous tunable parameter,
which can alter the SM Higgs self-coupling $\lambda$ by arbitrarily small
values:

$\delta\kappa_{\lambda}\equiv\frac{\delta\lambda}{\lambda}\sim\frac{2v^{4}}{m_{H}^{2}\Lambda^{2}}\;.$
(12)

One more small remark444Notice that obviously $\lambda_{\phi\phi\phi}$ has
mass dimension 1, independently of the form of the potential, since it’s the
parameter of a 3-boson coupling.: we can rewrite Eq. 11 as
$m_{\phi}^{2}=v\lambda_{\phi\phi\phi}/(n-1)$. This highlights the fact that,
contrary to the case of all other SM particles, where the strength $y$ of
their interaction with the Higgs uniquely defines their mass as $m=yv$, the
mass vs $v$-times-coupling relation for the Higgs has an additional dependence
on $n$. So, until we measure the Higgs self-coupling, we cannot claim to know
“how” the Higgs gets its mass ….

###  7.1 Higgs Self-Coupling Probes at FCC

At the FCC two approaches to measure the Higgs self-coupling, known as
‘direct’ and ‘indirect’, can be followed. In the former the Higgs self-
coupling enters at tree-level and in the latter it enters via processes at the
loop level. In general, to unambiguously probe the coefficient of any
dimension-6 operator one should perform a global fit of all available
observations, including the effects of all operators in all physical
observables. In the case that the heavy particles contribute dominantly to
some subset of dimension-6 operators, it is well-motivated to include only
those operators in the analysis, even though the result now becomes model-
dependent, owing to the assumption made that other operator coefficients are
small.

In the case of Higgs pair production at FCC-hh, for example via the gluon-
fusion process gg$\to$HH, the Higgs self-coupling enters at tree level. The
‘direct’ constraint on $\delta\kappa_{\lambda}$ from a global analysis [75]
would require the inclusion of other dimension-6 operators that may
contribute, for example those involving gluons or top quarks that give rise to
the diagrams in the second row of Fig. 19. But assuming that the deviations
induced by these additional operators are already constrained to be small, as
justified by the high precision achievable at the FCC in the measurement of
ggh and ggt couplings, one may directly extract a constraint on
$\delta\kappa_{\lambda}$ from the study of Higgs-pair production.

Figure 19:  Feynman diagrams contributing to double Higgs production via gluon
fusion: in the SM (upper set) and with higher-dimension operators affecting
the tthh, ggh and gghh couplings, respectively (lower set).

For the case of indirect constraints the situation is somewhat different. For
a given single-Higgs production process one has the tree-level process but
also higher loop processes in which the Higgs self-coupling may enter. The
first discussion of such effects was in [76], however, it was more recently
emphasised for current and future experiments in [77]. Using these effects one
may search for the influence of a modified Higgs self-coupling on experimental
observables involving a single Higgs boson. Such probes are known as
‘indirect’. For indirect probes, if additional dimension-6 operators
contribute to this process at tree-level then, for a comparable magnitude of
coefficient, they would lead to deviations that are a loop factor greater than
from the self-coupling effects. Thus to assume that the self-coupling
modification is the leading source of deviations in an observable would
necessarily imply the assumption that the other operators that contribute at
tree-level have coefficients that are smaller than the self-coupling by more
than a loop factor. This assumption is too strong to cover a wide range of
scenarios for new physics beyond the SM that might modify the self-coupling,
thus for an indirect probe it is only realistic to perform a global analysis,
allowing all dimension-6 operators to enter the scattering process whilst
considering all available measurements to over constrain the system of unknown
coefficients.

###  7.2 FCC-ee: Indirect Probe

With the large luminosity delivered at 240 and 365 GeV, the FCC-ee has
privileged sensitivity to the Higgs self-coupling by measuring its centre-of-
mass-energy-dependent effects at the quantum level on single Higgs observables
[77], such as the HZ and the ννH production cross sections, representative
diagrams of which are displayed in Fig. 20.

Figure 20:  From Ref. [78], sample Feynman diagrams illustrating the effects
of the Higgs trilinear self-coupling on single Higgs process at next-to-
leading order. Figure 21:  Indirect measurements of the Higgs self-coupling at
FCC-ee combining runs at different energies.

In Fig. 21 the results of a global analysis for FCC-ee are shown. This figure
is taken from [78], which the reader is referred to for details. Notably,
through a global analysis which allows for the presence of all dimension-6
operators with arbitrary coefficients, at FCC-ee a robust and model-
independent measurement of the Higgs self-coupling can be made with a
precision of $44\%$, to be compared with the HL-LHC projection, in which the
precision is around $50\%$. This would establish, at better than $95\%$
confidence level, a non-zero value for the Higgs self-coupling and would probe
a scale of $\Lambda\sim 1$ TeV in Eq. 12. In this analysis the FCC-ee
precision electroweak measurements at lower energies are equally important to
fix extra parameters that would otherwise enter the global Higgs fit and open
flat directions that cannot be resolved.

###  7.3 FCC-hh: Direct Probes

At FCC-hh, the Higgs self-coupling can be probed directly via Higgs-pair
production. The cross sections for several production channels are given [79]
in Table 8, where the quoted systematics reflect today’s state of the art, and
are therefore bound to be significantly improved by the time of FCC-hh
operations.

Table 8:  Higgs-pair cross sections rates for various production processes [79]. The first uncertainty corresponds to the scale choice, the second combines $\alpha_{S}$ and PDF systematics (PDF4LHC15NNLO), the third estimates finite-$m_{top}$ effects in the NNLO contribution to the gg channel. |  $\sigma$[100 TeV](fb) |  $\sigma$[27 TeV](fb)
---|---|---
gg$\to$HH | $1.22\times 10^{3}{+0.9\%\atop-3.2\%}\pm 2.4\%\pm 4.5\%_{m_{t}}$ | $140{+1.3\%\atop-3.9\%}\pm 2.5\%\pm 3.4\%_{m_{t}}$
HHjj | $80.5\pm 0.5\%\pm 1.8\%$ | $1.95\pm 2\%\pm 2.4\%$
W+HH | $4.7\pm 1\%\pm 1.8\%$ | $0.37\pm 0.4\%\pm 2.1\%$
W-HH | $3.3\pm 4\%\pm 1.9\%$ | $0.20\pm 1.3\%\pm 2.7\%$
ZHH | $8.2\pm 5\%\pm 1.7\%$ | $0.41\pm 3\%\pm 1.8\%$
tt̄HH | $82.1\pm 8\%\pm 1.6\%$ | $0.95{+1.7\%\atop-4.5\%}\pm 3.1\%$

The most studied channel, in view of its large rate, is gluon fusion (see Fig.
19). In the SM there is a large destructive interference between the diagram
with the top-quark loop and that with the self-coupling. While this
interference suppresses the SM rate, it makes the rate more sensitive to
possible deviations from the SM couplings, the sensitivity being enhanced
after NLO corrections are included, as shown in the case of gg$\to$HH in Ref.
[80], where the first NLO calculation of $\sigma$(gg$\to$HH) inclusive of top-
mass effects was performed. For values of $\kappa_{\lambda}$ close to 1,
$1/\sigma_{\rm HH}d\sigma_{\rm HH}/d\kappa_{\lambda}\sim-1$, and a measurement
of $\kappa_{\lambda}$ at the few percent level requires the measurement and
theoretical interpretation of the Higgs-pair rate at a similar level of
precision. Table 8 shows that the current theoretical systematics on the
signal is at the 5% level (for a complete discussion see [81]), which is
already competitive with the statistical and experimental systematics, to be
presented shortly. It is reasonable to predict a further reduction to the
percent level.

The full analysis of the Higgs pair observation potential and its
interpretation in terms of the Higgs self-coupling is rather complex, due to
the presence of significant backgrounds and experimental challenges, ranging
from the tagging of b quarks to the optimization of the mass resolution in the
reconstruction of the Higgs boson peaks. All the details can be found in Refs.
[9, 46]. I limit myself here to showing the bottom line, Fig. 22, with the
results relative to the most significant channel, $\rm HH\to b\bar{b}$γγ.

Figure 22: Expected precision on the Higgs self-coupling modifier
$\kappa_{\lambda}$ with no systematic uncertainties (only statistical), 1%
signal uncertainty, 1% signal uncertainty together with 1% uncertainty on the
Higgs backgrounds (left) and assuming respectively $\times 1$, $\times 2$,
$\times 0.5$ background yields (right).)

Decay modes other than HH$\to$bb̄γγ have also been considered in the detector
performance studies. These include bb̄ZZ*[$\to$4$\ell$] ($\ell=$e,µ),
bb̄WW*[$\to$2j$\ell$ν], and 4b+jet. A summary of the target precision in the
measurement of $\kappa_{\lambda}$ is given in Table 9, where the results were
obtained with the baseline detector performance parameters, and a 1%
systematics on the rates of the signals and of the leading backgrounds.

Table 9:  Precision of the direct Higgs self-coupling measurement in gg$\to$HH production, for various decay modes, from the FCC-hh detector performance studies. | bb̄γγ | bb̄ZZ*[$\to$4$\ell$] | bb̄WW*[$\to$2j$\ell$ν] | 4b+jet
---|---|---|---|---
$\delta\kappa_{\lambda}$ | 6.5% | 14% | 40% | 30%

Additional studies, of a more phenomenological nature, have appeared in the
literature, see e.g. Refs. [82, 83]. Their results are consistent with those
presented above. To summarise, within the stated assumptions on the expected
performance of the FCC-hh detector, a precision target on the Higgs self-
coupling of $\delta\kappa_{\lambda}=$ 5% in the gg$\to$HH channel appears
achievable, by exploiting several techniques and decay modes, and assuming the
future theoretical progress in modelling signals and backgrounds.

###  7.4 The EW phase transition

Aside from the need to clarify the deep origin of the Higgs potential, there
is a second very interesting issue raised by EWSB: what was the nature of the
transition that, during the big bang, led to the broken symmetry phase in
which we live today? This is more than a matter of curiosity: the EW phase
transition (EWPT) could have been the seed for the out-of-equilibrium state
needed to freeze into a matter-antimatter asymmetric world some primordial
baryon number and CP violating interactions [84]. This process of EW
baryogenesis (EWBG [85, 86]), relying on the SM baryon number violation
induced at high temperatures by sphalerons [87], requires a strong first-order
EWPT, and a sufficent source of CP asymmetry.

These conditions, unfortunately, are not met by the SM with a 125 GeV Higgs.
On one side the CKM source of CP violation is too weak [88]. On the other
[89], the SM Higgs potential for
$m_{H}\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}70$ GeV causes
the transition from the high-temperature vacuum at $H=0$ to smoothly cross
over to the $H=v$ vacuum, without creating the potential barrier needed for a
first-order transition. The required conditions, however, can be met in a
variety of BSM scenarios. CP violation relevant to the matter-antimatter
asymmetry can arise from new interactions over a broad range of mass scales,
possibly well above 100 TeV (for a recent review, see e.g. Ref. [90]).
Exhaustively testing these scenarios may therefore go beyond the scope of the
FCC. For the phase transition to be sufficiently strong, on the other hand,
one expects new particles to exist with masses typically below one TeV, whose
interactions with the Higgs boson modify the Higgs potential. Should they
exist, these particles and interactions could appear at FCC, setting a key
scientific opportunity and priority for the FCC.

An important task to be addressed by a future collider, therefore, is to
establish in a conclusive way whether or not the EWPT has been of strong first
order. This was first introduced as a possible no-lose target for future
colliders in Ref. [91], which considered in detail “nightmare scenarios”,
namely models particularly challenging to assess at colliders. The challenge
can be met by probing the existence of suitable particles and interactions,
either through a direct search, or indirectly through the modifications
induced on the Higgs interactions. A more complete overview of the studies
carried out so far in the context of future colliders in given in Ref. [9].
Here we present just one example, given by a theory with an extra real scalar
S, coupled as follows:

$\displaystyle V(H,S)$ $\displaystyle=$
$\displaystyle-\mu^{2}(H^{\dagger}H)+\lambda(H^{\dagger}H)^{2}+\frac{a_{1}}{2}S(H^{\dagger}H)$
(13)
$\displaystyle+\frac{a_{2}}{2}S^{2}(H^{\dagger}H)+\frac{b_{2}}{2}S^{2}+\frac{b_{3}}{3}S^{3}+\frac{b_{4}}{4}S^{4}\;.$

As a singlet under the SM, S can only be produced at colliders in association
with, or in the decay of, Higgs bosons, which makes its rate highly
suppressed. This type of model is therefore among those experimentally most
challenging. Depending on the values of the $a_{i}$ and $b_{i}$ parameters,
several scenarios are possible, where S itself may or may not acquire an
expectation value, at temperatures higher or lower than H. Some of these
scenarios will lead to acceptable phenomenology and to a strong first-order
EWPT, others will not. In general, the 125 GeV Higgs boson will be a mixture
of H and S, which we call here h1, and h2 will be the heavier state. The
mixing between h1 and h2 will make it possible for h2 to be produced in gluon
fusion, and, when sufficiently heavy, to decay to h1 pairs. Figure 23, from
Ref. [92], shows the significance of the searches for $\rm h_{2}\to
h_{1}h_{1}\to$ bb̄γγ, at the HL-LHC and FCC-hh, for various luminosities. The
shaded areas correspond to the envelope of results obtained for combination of
parameters in Eq. 13 satisfying the strong first-order EWPT conditions. The
energy and luminosity of FCC-hh are necessary, but sufficient, to meet the
5$\sigma$ goal for the full spectrum of viable models.

Figure 23: Discovery potential for the singlet-induced strong first order EWPT
using resonant di-Higgs production combining 4τ and bb̄γγ final states [92].
Vertical axis gives significance as a function of the singlet-like scalar mass
$m_{2}$ for the HL-LHC (blue band) and the FCC-hh with 3 ab-1 and 30 ab-1
(purple and red bands, respectively).

The observation of these signatures could then be correlated to precise
measurements of the Higgs couplings, in order to learn more about the
underlying origin of these signals, and break possible degeneracies among
various interpretations. Figure 24, from Ref. [93], shows the size of
deviations, from their SM value, of the Higgs triple self-coupling and its
coupling to a pair of Z0 bosons. The points in the scatter plot correspond to
parameters of the S scalar model fulfilling the condition of a strong first-
order EWPT, and detectable directly as in Fig. 23. The projected precision at
FCC-ee (for the ZHH coupling) and at FCC-hh (using a conservative $\pm 10\%$
uncertainty), can cover most of the parameter range.

Figure 24: Correlation between changes in the HZZ coupling (vertical axis) and
the HHH coupling scaled to its SM value (horizontal axis), for a space scan of
the S singlet model parameters leading to a first-order phase transition.
Adapted from Refs. [93], where similar plots, for a broad set of BSM models,
are also shown.

Independently of the issue of EWBG, collisions among bubbles created by a
strong phase transition during the early universe, can lead [94, 95] to the
generation of a stochastic background of primordial gravitational waves (GWs),
adding a further possible remarkable signature for a strong first-order EWPT.
Reference [93] studied the models discussed above, in the context of the
observation potential of future satellite GW interferometers, like eLISA [96].

Establishing the nature of the EWPT is therefore a target with a broad range
of phenomenological ramifications, which makes the FCC a facility of unique
value!

##  8 Direct searches for new physics

The LHC experiments have already published many more than 1000 papers
reporting the results of searches for new physics. While there is certainly
some duplication (the same search was repeated at different energies, and by
different experiments), this number gives a sense of the immense variety of
models that can be probed by a collider. The FCC facility will allow to extend
all of these searches, and to probe new parameter ranges, both at the high-
and at the lower-end of the mass spectrum. Many more models will emerge in the
future, adding to the immense discovery potential enabled by the energies and
rates available at the FCC. In this lecture we give a brief overview of some
of the new features that characterize this potential. In the case of FCC-hh, a
more extensive review can be found in Ref. [97].

###  8.1 The reach at high mass

To first approximation, if the beam energy of a pp collider goes up by a
factor $z$, we expect the mass reach to go up by a similar factor, provided
the integrated luminosity is properly increased by a factor $z^{2}$. This is
because the partonic cross section to produce an object of mass M scales
typically like $\sigma(M)\sim 1/M^{2}$, and therefore
$\sigma(zM)\sim\sigma(M)/z^{2}$. This is confirmed by Fig. 25, which shows the
projections for the discovery or exclusion reach at the FCC-hh (or at the HE-
LHC), as a function of the equivalent HL-LHC targets, and assuming backgrounds
scale with energy like the signals.

Figure 25:  Left (right): the discovery/exclusion mass reach for resonance
searches at FCC-hh (HE-LHC), as a function of the equivalent target for the
HL-LHC [98].

However things can be more subtle. For example, when potential signals are
subject to large backgrounds, it is important to consider whether the signal
and the backgrounds evolve with energy in the same way or not. Or detector
properties can force changes of analysis strategies (for better or for worse)
when the mass scale of final states is increased, and this can modify the
discovery opportunities at larger energies and mass. Finally, in specific
cases one may not be interested in extending the discovery reach in mass, but
in sensitivity. This is the case of the search for rare or forbidden decay BRs
of a given particle, say of the Higgs, or the search for superweakly coupled
resonances at low masses. In all of these cases, the energy scaling of the
discovery reach does not admit a simple universal rule, and should be
considered on a case by case basis. We shall return to these issues in Sect.
8.4, and we start now with an overview of the general search potential at the
highest masses.

###  8.2 $s$-channel resonances

Searching for heavy resonances produced in the $s$ channel probes a large
variety of BSM scenarios, from the existence of new gauge interactions, to the
existence of excited quarks, a signal of quark’s compositness. The signatures
are typically straightforward, with mass peaks in various two-body final
states. The measurement of the heaviest objects potentially within the reach
of a 100 TeV collider, however, poses new challenges relative to the LHC. For
example, in this multi-TeV domain, hadronic decays of final-state top quarks
or W/Z gauge bosons appear as collimated jets, and require enhanced
calorimeter granularity to apply substructure discriminators tagging the
object and reducing the potentially overwhelming QCD dijet backgrounds.
Reference [99] describes in more detail the detector features that are
necessary to fully exploit the discovery potential at the highest masses, and
provides concrete examples of analyses for many BSM models. We summarize here
some of the main results.

The sensitivity to new Z′ gauge bosons decaying to leptons is shown in Fig.
26, for a set of extra-U(1) models considered in the literature [100, 101].
The right plot of this figure shows the luminosity required for the 5$\sigma$
discovery of a sequential SM (SSM) Z′ (namely a gauge boson with couplings to
the SM particles identical to those of the Z), as a function of its mass.

Figure 26: Left (centre): Cross section times branching ratio 95%CL limits
versus mass, for the ee+µµ(ττ) final states, compared to the expectations of
several Z′ models. Right: luminosity required for the $5\sigma$ discovery of a
SSM Z′ in the dilepton channel, versus the hypothetical resonance mass. From
Ref. [99].

To explore the sensitivity to hadronic final states, three scenarios were
considered in [99]: a Z′ in $\rm q\bar{q}\to$ Z${}^{\prime}\to\rm t\bar{t}$, a
Randall-Sundrum graviton [102] in $\rm gg/q\bar{q}\to G_{RS}\to W^{+}W^{-}\to$
jets, and an excited quark resonance [103, 104] in $\rm qg\to Q^{*}\to$ dijet.
The 5$\sigma$ discovery range reaches 18 TeV (24) for $\rm
Z_{SSM}^{\prime}\to$tt̄ ($\rm Z_{TC2}^{\prime}$), 22 TeV for $\rm G_{RS}\to
WW$ and 40 TeV for the excited quark. The corresponding exclusion limits are
shown in Fig. 27.

Figure 27: Sensitivity to various resonance models using fully hadronic final
states. For left to right: excited quarks Q∗ in the dijet channel; Z′ in the
$\rm t\bar{t}$ hadronic channel (SSM and top-assisted technicolour [105], TC2,
models); $\rm G_{R}S$ in the WW fully hadronic channel. From Ref. [99].

A summary of the discovery reach for various models, for various integrated
luminosities at 27 and 100 TeV, is shown in Fig. 28. You can verify that the
gain in reach, from 27 to 100 TeV, is indeed the expected approximate factor
of 4\. We also notice that, with the assumed target integrated luminosities
for HE-LHC and FCC-hh, 15 and 30ab-1 respectively, the discovery reach is
close to saturation, as the gain obtained by further increasing the luminosity
to 100ab-1 is rather marginal.

Figure 28: Summary of a $5\sigma$ discovery reach as a function of the
resonance mass for different luminosity scenario of FCC-hh and HE-LHC.

###  8.3 Pair production of new particles

Looking beyond resonances, heavy particles with SM gauge charges feature in
many scenarios beyond the SM. Typically, the production cross section may be
calculated for a given particle mass and gauge representation. As a hadron
collider, the FCC-hh discovery reach for new coloured particles is extensive.
As a simple example for illustration, the pair production cross section for
colour octet particles of various spins at the LHC versus FCC-hh is shown in
Fig. 29. At a given mass value the cross section at 100 TeV is orders of
magnitude greater than at 14 TeV. Not only does this demonstrate a significant
increase in the discovery potential at FCC-hh, but also implies that if a new
particle were discovered in the HL-LHC runs, it would be possible to study
this particle in significantly greater qualitative detail at FCC-hh.

Figure 29:  The pair production cross section for colour octets of various
spins.

Dedicated studies have been carried out in the context of specific models,
taking into proper account backgrounds and analysis opportunities. As an
example, Fig. 30 shows the reach for pair production of supersymmetric
particles. The reach for the degenerate light-generation squarks extends up to
15 TeV, for pair-produced gluinos up to 10-15 TeV, depending on the decay
mode, and goes up to 20 TeV for the associated production of a squark and a
gluino, assuming equal mass. More details on the discovery reach for stop
squarks, decaying to a top quark and a neutralino, are shown in the right plot
of Fig. 30, for different values of the neutralino mass (see [106] for the
discussion of the analysis).

Figure 30:  Left: Projected direct FCC-hh $2\sigma$ and $5\sigma$ discovery
reach for supersymmetric Higgsinos, Winos, sleptons, stops, squarks, and
gluinos (see Ref. [97] for details). Right: 5$\sigma$ discovery reach for
pair-produced stops at FCC-hh. The area below the solid red (black) curve
represents the expected exclusion and the $\pm 1\sigma$ contours for the
nominal (conservative) scenario of associated systematic uncertainties [106].

###  8.4 Considerations on the energy scaling of production rates at hadron
colliders

There are circumstances in which it might make sense to optimize the beam
energy choice, as a function of a specific physics goal, or when
considerations of cost and technology may limit our ambition to reach the
highest possible energy. For example, if a particle were discovered at the
LHC, we might want to optimize the design of the next collider in order to
collect as soon as possible a statistics large enough to study its properties.
Appendix A presents, through simple analytic approximations, some general
qualitative features of production rates, which help understanding the main
properties of their dependence on the pp center-of-mass energy555The results
shown in this Section use parton luminosities calculated from real proton
PDFs, not from the approximated forms used in Section A.. We learn there that
what drives the energy dependence of cross sections is just the partonic
luminosity: once the initial state is assigned, there is no need to specify
the details of the produced final state, and it is possible therefore to make
general statements valid for a large class of models. We apply this concept to
the case of particles that could be discovered at the HL-LHC, produced by gg
or $\rm q\bar{q}$ initial states. Figure 31 shows the growth of the production
rates as a function of beam energy, relative to the rates at 14 TeV, for
systems of mass $m_{X}$ between 0.5 and 6 TeV. These could be $s$-channel
resonances (e.g. $\rm q\bar{q}\to Z^{\prime}$, with $m_{Z^{\prime}}=m_{X}$) or
new particles pair-produced (e.g. gluinos $\rm\tilde{g}$ in $\rm
gg\to\tilde{g}\tilde{g}$, with $m_{\tilde{g}}\sim m_{X}/2$). Few comments are
in order.

Figure 31: Scaling with beam energy of the production cross section for final
states of various masses, in the gg and $\rm q\bar{q}$ channels. The final
state X is characterized here by its mass: it can be a single resonance, as in
$\rm q\bar{q}toZ^{\prime}$, with $m_{X}=m_{Z^{\prime}}$, or it could be a pair
production, as in $\rm gg\to Q\bar{Q}$, in which case
$m_{X}\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}2m_{Q}$.

To start with, the figure shows clearly that the gain with energy is most
significant for the heaviest final states. Furthermore, objects produced in
the $\rm gg$ channel receive a much larger boost in rate than objects produced
in the $\rm q\bar{q}$ channel.

Exercise: discuss and explain the origin of these behaviors.

In practice, a 6 TeV object produced in the $\rm gg$ channel can see a factor
of 40,000 increase in going from 14 to 100 TeV, while for a 0.5 TeV object
produced in the $\rm q\bar{q}$ channel (this could be, for example, a very
weakly coupled 500 GeV $\rm Z^{\prime}$, or a pair of 250 GeV electroweakinos
from supersymmetry), the rate goes up only by a factor of 10! In the former
case, it is clear that one would like to go to 100 TeV to study in real detail
the new discovery. In the latter case, one might argue that it is more
economical to just sit at the HL-LHC for a bit longer, or maybe to move to a
TeV-scale $\rm e^{+}e^{-}$ collider, where the study of a weakly interacting
sub-TeV particle is likely cleaner than in a hadron collider. You can play
with Fig. 31 and with your favourite new physics scenarios accessible to the
LHC, and make up your mind on what is the best way to go after discovery!

In my mind, when dealing with the further exploration of possible LHC
discoveries, nothing beats the additional power provided by the higher energy
of a 100 TeV collider. The higher energy can turn out to be crucial not just
in studying this object with greater statistics (even if only by a factor of
10 as in the 500 GeV case discussed above), but it allows us to explore the
other new particles that will most likely accompany any new BSM scenario, of
which the LHC discovery can just be a first signal of. For example, the new
resonance could be the first state in a tower of Kaluza-Klein (KK) states, and
we may want to search for further KK recurrences. If these states were EW
gauginos, we would want to search for the heavier supersymmetric particles.
And so on…

Another important consideration is that, as the LHC has shown, a greater
kinematical reach can lead to greater benefits even when dealing with lighter
objects, which are produced well below the kinematic limit of the collider.
Let me discuss a couple of examples: $\rm H\to b\bar{b}$ and the search for
resonances in the 100 GeV mass range.

It is well known that the huge QCD background makes it almost impossible to
pull out the $\rm H\to b\bar{b}$ signal from the inclusive $\rm gg\to H$
sample, in spite of the fact that we are well below the kinematic threshold
and that the inclusive Higgs production rate is huge. At high energy, however,
one can consider the $\rm VH$ ($\rm V=W,Z$) associated production, in a
configuration of large invariant mass of the $\rm VH$ pair, where the Higgs is
highly boosted. The large momentum of the b-pair gives a higher b-tagging
efficiency, allows the use of jet-substructure techniques, where the b-pair is
confined within a broad jet with a two-prong substructure, and gives a more
precise measurement of the invariant mass of $\rm b\bar{b}$ jet. This makes
the peak structure of the signal at $m_{bb}\sim m_{H}$ more pronounced, and
better visible over the continuum background from $\rm q\bar{q}\to Vb\bar{b}$
and $\rm q\bar{q}\to VZ[\to b\bar{b}]$. This technique has allowed ATLAS [107]
and CMS [108] to discover the $\rm H\to b\bar{b}$ decay, with a signal
enhanced by the requirement of
$p_{T}(H)\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}150$ GeV. For
the use of these signals at high transverse momentum, in the study of SM EFT
operators, see e.g. [65].

Figure 32: Signal and background diagrams for the production of a hadronic
resonance. Top row: inclusive production. Bottom row: production at large
$p_{T}$ in association with a jet. Similar processes, with a $\rm q\bar{q}$
initial state, are possible with the emission of photons or W/Z bosons.

A second important example is the search for a weakly interacting hadronic
resonance $V\to jj$ with mass of (few) hundred GeV. This is relevant in the
search of potential candidates for mediators between the SM and a DM sector
[109]. The search for low-mass hadronic resonances is difficult, because of
the huge QCD dijet background. On one side this background forces high
thresholds on the minimum jet $E_{T}$ at the trigger level, strongly
suppressing the signal. On the other, what’s left of the resonant signal ends
up being swamped by the large QCD continuum. Requiring the resonance to be
highly boosted (see Fig. 32), will significantly improve the situation. The
large $E_{T}$ of the jets from the V decay, or of the recoil jet, will
maximize the trigger efficiency. The S/B ratio will also be strongly enhanced,
since the signal initial state will be $\rm qg$, rather than $\rm q\bar{q}$,
gaining a factor of $\rm g(x)/\bar{q}(x)$. Jet substructure techniques will
also help, since the structure of a high-$E_{T}$ fat jet coming from the decay
of a color-singlet resonance has distinctive features that can be used to
separate it from a generic QCD high-$E_{T}$ jet. Selecting events with a jet
of $E_{T}>500$ GeV, CMS [110] could probe resonance masses down to 100 GeV,
with a sensitivity orders of magnitude better than any previous measurement at
previous hadron colliders, in spite of the fact that this mass region was well
within their kinematic reach. In this powerful analysis, even the hadronic