<table><caption>**Table 5** Constraints from flavour, high $p_T$, and EWPOs on the mass of the gauge bosons predicted by flavour deconstruction</caption><thead><tr><th></th><th>Deconstructed SU(2)<sub>L</sub></th><th>Deconstructed U(1)<sub>Y</sub></th></tr></thead><tbody><tr><td>Electroweak: Z-pole &amp; WW-threshold</td><td>9 TeV (5 TeV if exc. m<sub>W</sub>)</td><td>2 TeV</td></tr><tr><td>Flavour: B<sub>s</sub> → μμ (up-alignment)</td><td>7.5 TeV</td><td>2 TeV</td></tr><tr><td>High p<sub>T</sub>: Drell–Yan pp → ee, μμ, ττ</td><td>4.5 TeV</td><td>3.5 TeV</td></tr><tr><td>EW projection FCC-ee<br>(on and above Z-pole &amp; W-pole)</td><td>30 TeV</td><td>7 TeV</td></tr></tbody></table>

### 2.3.3 Flavour deconstruction at FCC-ee

To give an example of concrete scenarios explored through a combination of the unprecedented precision on Z-pole and WW-threshold observables with excellent capabilities as a flavour factory, a special class of theories based on ‘flavour deconstruction’ [117] is considered, whereby part of the electroweak symmetry is resolved into three generation-specific copies $G_i$ (with the Higgs boson coupled to $G_3$). This symmetry, which arises, e.g., from gauge-flavour unification [118] or from extra-dimensions [119], is broken to the SM in two steps that generate hierarchical fermion masses and mixings. The last breaking step, $G_{12} \times G_3 \to G$, should occur close to the TeV scale to avoid destabilising the Higgs potential, delivering TeV-mass gauge bosons in the adjoint of $G$ coupled mostly to the third generation.

These electroweakly-charged and flavour non-universal gauge bosons give rise to a rich phenomenology across EWPOs, flavour observables, and high-energy LHC measurements, which has been quantified by considering deconstructed hypercharge [120] and deconstructed $SU(2)_L$ [121] (Table 5). Firstly, EWPOs are shifted at tree-level because the heavy gauge bosons directly couple to the Higgs boson. Constraints from LEP are already strong and the expected precision brought by FCC-ee would all but cover the ‘natural’ regime in which these flavour models can be reconciled with the hierarchy problem. Secondly, being intrinsically non-universal theories of flavour, many rare B- and τ-decays receive tree-level shifts that will also be measured with excellent precision at Tera-Z. While more detailed studies are needed to fully assess these prospects, key processes will be $B \to K^{(*)}\sqrt{\bar{v}}$, $b \to s\tau\tau$ transitions that are a unique opportunity at FCC-ee, and LFUV measurements in $\tau$ decays. In these models, the shift in $B \to K^{(*)}\sqrt{\bar{v}}$ is directly correlated to that in $B_s \to \mu\mu$, which should be measured to few percent precision at HL-LHC [122]. Lastly, these models are constrained by high-energy Drell–Yan measurements of $pp \to \ell\ell$, that should improve by nearly a factor of two after HL-LHC.

Other flavour deconstruction models have been proposed where, for instance, a new force-carrying gauge boson field is associated with a TeV-scale $U(1)_{Y_3} Z'$, coupling primarily to the third family of fermionic fields [123]. These models, currently favoured by measurements of B decays in tension with the SM predictions, can be thoroughly tested by FCC-ee.

To summarise, FCC-ee is an optimal machine for probing theories of flavour deconstruction because it achieves comparable sensitivity across diverse measurements traditionally separated into ‘electroweak’ and ‘flavour’ categories, but whose effects in such flavour models are inextricably tied. Viewing the electroweak and flavour programmes of FCC-ee together, and in combination with high-$p_T$ and low-$p_T$ HL-LHC data, is therefore crucial to understanding its full power.

### 2.3.4 Heavy neutral leptons

In its minimal version, the SM Lagrangian does not accommodate neutrino masses, but this historical accident can easily be overcome by minimally extending the SM. Nonzero neutrino masses offer many theoretical and phenomenological opportunities to address pending questions in the understanding of Nature.

Massive neutrinos can oscillate in flavour space, as experimentally observed since 1998. With three families, this phenomenon naturally leads to the possibility of breaking CP symmetry in the leptonic sector. Furthermore, fermion (or lepton) number violation might be observed. In the SM, fermion number conservation (FNC) stems from charge conservation for charged fermions and from angular momentum conservation for massless neutrinos [124]. Thus, FNC is considered to be an accidental conservation law, which has no reason to hold if neutrinos are massive. The combination of the two possibilities opens the door to shedding light on the question of the existence of matter via leptogenesis.

When the FNC rule is relaxed, active neutrinos can mix with right-handed neutrinos, via Yukawa interactions mediated by the Higgs field. This minimal scenario is a concrete realisation of the (type-I) see-saw mechanism (Fig. 28). The right-handed neutrinos, which do not carry electric charge, weak isospin, or colour, have no interaction that would distinguish the
<figure><img src="image_34.png" /><figcaption>**Fig. 28** Schematic illustration of the type-I see-saw mechanism. Left: the left-handed lepton EW doublets $L_L$ have Yukawa-like interactions $Y_N$, with the Higgs EW doublet $\phi$, mediated by the heavy right-handed neutrinos $N_R$. Right: at energies below the mass $M_R$ of these heavy neutrinos, an effective fermion-number-violating dimension-5 operator, involving two left-handed leptons and two Higgs fields, describes the low-energy degrees of freedom. After electroweak symmetry breaking, a Majorana mass is generated for the active left-handed neutrinos. This mass is proportional to the square of the original Yukawa coupling, to the squared Higgs vacuum expectation value $v$, and to $1/M_R$. Heavy Neutral Leptons (HNLs), aligned with the right-handed neutrinos in the limit $M_R \gg v$, complete the spectrum</figcaption></figure>

particle from the antiparticle: they are naturally Majorana particles. At energies much below the Majorana mass, the mixing is described by an effective dimension-5 Weinberg operator that, after electroweak symmetry breaking, generates an effective mass for the active neutrinos. Having two mass terms for each family, the neutrinos undergo level splitting thus the ‘see-saw’, with typically light active neutrinos and heavy sterile neutrinos. Unlike for other SM fermions, the Yukawa coupling is not proportional to the mass of the light neutrinos, but could either be made proportional to the geometric mean of the masses of the light and heavy particles, or be enhanced with a symmetry that protects the masses of the light neutrinos [125]. The resulting heavy mass eigenstates are called Heavy Neutral Leptons (HNLs) and are nearly sterile particles. The ensuing rich phenomenology includes the possible observation of LLPs, for which, in the mass range 20–80 GeV, a thorough and conclusive search can be performed at FCC-ee, during the Tera-Z run.

Generalising beyond this minimal scenario, FCC-ee can search for HNLs in the large sample of Z bosons produced resonantly. The Z decay to $\nu_L + \text{HNL}$ followed by the decays of the heavy neutral lepton HNL $\to \ell + \text{W}^*$, HNL $\to \nu + \text{Z}^*$, leads to a wealth of final state signatures that can be exploited at FCC-ee [126]. The Z branching ratio to HNLs is proportional to the squares of the mixing angles $U_{ij}^2$ between the SM neutrinos and the HNL, where i, j run over the three neutrino flavours.

The very large number of Z boson decays allows the exploration of very low $U_{ij}^2$ values, for HNL masses below the Z mass. The explored values may result in HNLs with a measurable path in the detectors, yielding spectacular signatures with jets or leptons produced far from the interaction vertex, with no SM backgrounds. In particular, for decay lengths in the detector between 1 mm and 2 m, a discovery would be possible for HNL masses up to 60–70 GeV down to $U^2$ values of $\mathcal{O}(10^{-11})$. For larger HNL masses, up to the kinematic limit of the Z mass, a prompt analysis would be needed, which, because of significant SM backgrounds, would allow discovery for $U^2$ values of $\mathcal{O}(10^{-9})$.

Detailed simulation studies were performed to verify the experimental feasibility of these analyses and to determine the corresponding requirements on detector design. The reach in parameter space was first studied in a toy model commonly used to compare different experimental approaches, rather than a complete model for neutrino masses. The model features a single Majorana HNL mixing with a single flavour of active neutrinos. The signal samples and all of the SM decays of the Z boson, as well as four-fermion processes yielding the same final state as the HNL decays, were passed through a parametrised simulation of the IDEA detector (Sect. 6) and then analysed. Both semi-leptonic and fully leptonic decays of the HNL were studied, for the cases of mixing with an electron neutrino and with a muon neutrino. The fully leptonic decay mode provides a clean experimental final state, with a good reach for long-lived decays, but with a significant price in branching fraction. The semileptonic decay HNL $\to \ell\nu jj$ has a branching fraction $\sim 50\%$, allows full kinematic reconstruction of the neutrino decay, and was studied both for long-lived and prompt decays. The results are shown in Fig. 29 and documented in Refs. [127–132].

The conclusion of these studies is that searches at FCC-ee enable the HNL discovery over a mass range beyond the reach of specialised detectors for LLP searches being developed for HL-LHC and for mixing values much smaller than those covered by future searches at HL-LHC, for both prompt and long-lived signatures. Besides the work based on parametrised detector simulations, the model is also being studied [136] through a detailed GEANT4 simulation of the ILD detector (Sect. 6).

The models featuring a single HNL are useful for assessing, in a simplified way, the parameter space coverage of the experiments. In order to explain the observed neutrino oscillations, however, at least two HNLs are needed. A realistic model [137] featuring two Majorana neutrinos with coupling to all three flavours of active neutrinos was studied in the fully leptonic final state featuring two electrons or muons in the final state. The patterns of couplings were chosen such as to be compatible with neutrino oscillation data based on the benchmarks proposed in Ref. [138]. As for the single-neutrino analyses, the study was performed with the parametrised simulation of the IDEA detector and featured a full background analysis. An estimate of the reach was obtained for different coupling scenarios, corresponding to both normal and inverted neutrino mass hierarchy.
<figure><img src="image_35.png" /><figcaption>**Fig. 29** Discovery potential in the $m_N - |U_{\mu N}|^2$ plane. The FCC-ee potential is based on the decay channel HNL $\to \ell\nu jj$ and is shown as a red (green) line for the prompt (long-lived) analyses described in the text. The blue line shows the reach of a search for long-lived particles in the decay channel HNL $\to \mu^+\mu^-\nu$. The dashed green line bounds the area where, out of $6 \times 10^{12}$ Z bosons, three events are produced with visible HNL decays inside an FCC-ee detector, i.e., with a displacement smaller than 5 m and larger than 0.5 mm (based on the analytical formulas in Ref. [133]). The requirement to explain the light neutrino masses imposes a lower bound, indicated as a pink band, on the total HNL mixing (summed over flavours). The width of this band indicates the uncertainty in this lower bound due to the current lack of knowledge about the absolute scale and the ordering of the light neutrino masses. Light neutrino oscillation data can be explained anywhere above this band, in particular in models where the neutrino masses are protected by a symmetry related to approximate lepton-number conservation. Furthermore, this region could also accommodate the observed matter-antimatter asymmetry via a leptogenesis mechanism [134]. The existing limits from LHC searches are shown as turquoise areas. The expected discovery potential of projected experimental searches based on long baseline experiments are shown as green areas and are taken from the website accompanying Ref. [135], where all the original works are cited</figcaption></figure>

The phenomenology of the 'Symmetry Protected Seesaw Scenario' [139,140] has been presented in Refs. [141,142]. This model adds two right-chiral neutrinos, required to explain the observed light neutrino mass squared differences, provided with a protective lepton number-like symmetry (LNLS) to ensure that the two heavy neutral leptons form an almost degenerate pair of pseudo-Dirac neutrinos. The main phenomenological signature of this model is the production of lepton flavour violating final states, with a probability that oscillates with the length of the flight path of the HNL in the detector [143]. The region in parameter space where this oscillation would be detectable at FCC-ee is mapped in detail in the phenomenological study documented in Ref. [144]. An experimental study was performed based on this work, with the same parametrised detector simulation and software as the studies described above [145]. For HNL masses below 35 GeV, for favourable ratios of the HNL width and of the mass difference between the two pseudo-Dirac states, a striking oscillation signal would be observed, allowing the measurement of the mass difference.

A large fraction of the model parameter space consistent with the see-saw mechanism, including those with symmetry-enhanced neutrino masses, such as the inverse [146] or linear [147,148] models, would be probed. For masses around 40 GeV, the FCC-ee sensitivity would reach the smallest value of the mixing angle theoretically compatible with the mass range experimentally still allowed for the active neutrinos.

With HNLs (as well as with other light and feebly-coupled particles, such as ALPs), FCC-ee is simultaneously a discovery and a precision tool, in a single collider. In view of the cost and time scales associated with the construction of each new collider, such an advantage cannot be overstated. Indeed, the number of HNLs that can be detected during the Z-pole run grows as $|U_{\ell N}|^4$ if their decay length in the laboratory exceeds the detector size, or as $|U_{\ell N}|^2$ if it is smaller than the effective detector dimensions. As a direct consequence, up to a million HNL decays can be observed during the Z-pole run if $|U_{\ell N}|$ is near the current upper experimental limits [133]. Such a large sample opens the way for the measurement of several quantities, from which important information about the underlying particle physics model can be extracted, as the following: (i) The branching ratios of HNL decays into different SM generations can be measured with per-cent accuracy [149], which allows the consistency with neutrino oscillation data to be checked; many parameters of the see-saw model to be constrained [150] (including CP-violating phases and the absolute neutrino mass scale); the leptogenesis hypothesis [149] to be tested; and underlying flavour and CP symmetries [151] to be probed. (ii) The number of HNL events and the measurement of the HNL lifetime can reveal information about the violation of lepton number [130,133] as well as underlying flavour and CP symmetries [151]. (iii) Finally, the angular distribution and the energy spectrum of the HNL decay products can provide information regarding the lepton number violation [152] and the HNL mass splitting [143–145].
The combination of these observables may provide important information about the HNL role in particle physics and cosmology, in particular in the context of neutrino masses and leptogenesis. It can also shed light on the properties of the underlying particle-physics theory within which the see-saw model is embedded (such as flavour and CP symmetries).

### 2.3.5 Dark matter and dark sectors

Understanding the origin/nature of dark matter (DM) is a central question in contemporary physics, connecting particle physics and astrophysics. Despite decades of searches across experiments, the favoured DM candidates, the WIMPs, have not been found, and the lower bounds on their masses are progressively pushed beyond the reach of TeV-class $e^{+}e^{-}$ colliders.

Mono-photon searches at FCC-ee can play a significant role in probing the so far unexplored parameter range allowed by the WIMP relic density constraints [153]. In particular, predictive models of leptophilic candidates with a thermal origin can be tested. Missing energy signatures at FCC-ee can probe much of the parameter space for which DM direct annihilation into a dilepton yields the observed relic density in Higgs-like models with mass-proportional couplings to charged leptons [154]. Models of asymmetric dark matter can also be probed at FCC-ee, with a sensitivity to DM-lepton interactions improved by almost an order of magnitude [155]. Additionally, it would be possible to detect spin 0, 1, and 1/2 DM at FCC-ee in the context of simple, consistent, and renormalisable field theories that provide the correct DM abundance and satisfy direct detection, indirect detection, and collider constraints [156].

Hidden sectors, consisting of new, invisible particles that interact almost imperceptibly with the SM, are rapidly gaining attention as they could hold the answer, not only to the dark matter problem but also to a variety of other open questions in the field. The dark sector could contain a multitude of hidden particles. After all, the visible sector is non-minimal, so there is no good reason why the dark sector should contain only one type of dark matter particle and nothing else. Benefiting from a clean environment, high luminosity, and large acceptance, FCC-ee can directly scrutinise the $\mathcal{O}(100)$ GeV mass range for so-far unknown particles, with interactions that would otherwise have been too feeble to detect.

Dark sectors typically exhibit a stable particle, fundamental or composite, that could be a DM candidate, and one or more mediator particles coupled to the SM via a neutral portal. The spin of the mediator particle defines the portal: vector, e.g., dark photon; scalar or pseudoscalar, e.g., Higgs portal; or fermion, e.g., sterile neutrino. These well-motivated dark sectors can be thoroughly explored at FCC-ee in a mass-coupling range that is inaccessible by any other means with the same sensitivity. Furthermore, small couplings of dark sector particles to SM particles could give rise to LLPs and other unconventional collider signatures, such as dark showers.

Dark photons may mix kinetically with the SM hypercharge gauge boson. This possibility can be studied in $e^{+}e^{-} \to A'\gamma$ production, where the dark photon $A'$ decays to $\mu^{+}\mu^{-}$. The sensitivity to small coupling values could be significant, reaching around $\sim 2 \times 10^{-4}$ for $m_{A'}$ ranging from 10 to 80 GeV at FCC-ee [157]. Alternatively, dark photons or dark $Z'$ bosons can also be investigated via $e^{+}e^{-} \to A'H$ or $Z'H$ production at the WW threshold or above [158].

Finally, the associated production of a neutrino and a dark sector fermion can also be searched for at FCC-ee in mono-photon signatures [159]. This kind of search has sensitivity up to mass values in excess of 1 TeV. Other searches, such as invisible decays of a dark Higgs boson [160] or Z boson decays to an invisible dark photon [161] can also be exploited.

### 2.3.6 Axion-like particles

Axion-like particles (ALPs) are generically expected in many BSM extensions, most famously in QCD axion models originally introduced to tackle the strong CP problem [162]. In these models, ALPs can be heavier than the QCD axion and serve as mediators to the dark sector. Astrophysical and beam-dump experiments constrain the ALP-photon coupling at low masses, but are less stringent in the 0.1–100 GeV range [125,163], potentially accessible at FCC-ee via $e^{+}e^{-} \to a\gamma$ [162] and $e^{+}e^{-} \xrightarrow{\gamma\gamma} (e^{+}e^{-})a$ [164], with the ALP $a$ decaying as $a \to \gamma\gamma$. Various aspects of the ALP phenomenology at FCC-ee are addressed in Refs. [165–168]

The FCC-ee sensitivity to the $e^{+}e^{-} \to a\gamma$ channel was studied in Ref. [169] with the parametrised simulation of the IDEA detector. Two final states were addressed: the ‘prompt’ case, where the ALP decays near the interaction vertex, and the ‘monojet’ case, where the ALP decays outside the detector, yielding the signature of a monochromatic photon recoiling against missing energy and mass. For both channels, a full kinematic analysis was developed to separate the signal from irreducible SM backgrounds.

Sensitivity to photon couplings down to $< 10^{-3} \text{ TeV}^{-1}$ can be obtained at FCC-ee in the $m_a$ range from 0.1 to 100 GeV, extending current limits by more than two orders of magnitude in the Tera-Z run. In particular, the ‘monophoton’ signature
<figure><img src="image_36.png" /><figcaption>**Fig. 30** Projected sensitivity for ALPs in the photon coupling vs. ALP mass plane from the three following processes at FCC-ee: $e^{+}e^{-} \rightarrow \gamma a \rightarrow 3\gamma$ (yellow area) [169], photon-fusion $\gamma\gamma \rightarrow a \rightarrow 2\gamma$ (salmon area) [164], and $e^{+}e^{-} \rightarrow \gamma a \rightarrow \gamma + \text{INV}$ (orange area). Existing limits (in grey) are adapted from Refs. [125,135]. Also shown are projected exclusion limits at 95% C.L. on the ALP-photon coupling as a function of the ALP mass expected from searches for $\gamma\gamma \rightarrow a \rightarrow \gamma\gamma$ in pp (violet), pPb (dark pink), and PbPb (orange) collisions at FCC-hh [171]</figcaption></figure>

would cover a difficult region in the vicinity of 1 GeV, which is not accessible to beam dump experiments. In an intermediate region in mass and at small coupling values, the ALP lifetime is large enough to give rise to measurable paths inside the detector, providing LLP signatures that could be accessible by exploiting the pointing capabilities of the calorimeters.

The ALP coupling to Z and Higgs bosons can also be probed via $e^{+}e^{-} \rightarrow Za$ or $Ha$, with visible Z boson decays or $H \rightarrow b\bar{b}$, and either $a \rightarrow \gamma\gamma$ or $a \rightarrow \ell^{+}\ell^{-}$. Decays to SM particles other than photons are less constrained and provide an additional opportunity for ALP discovery at FCC-ee. A preliminary study of the sensitivity of FCC-ee to ALPs decaying into two gluons is presented in Ref. [170].

The projected sensitivity of the ALP search is illustrated in Fig. 30, where the importance of FCC-ee is conspicuous in probing ALPs coupling to photons with a lifetime below cosmological/astrophysical scales, i.e., with a mass above 1 MeV.

### 2.3.7 Exotic decays of the Higgs and Z bosons

The large Higgs boson event sample at FCC-ee allows a direct search for exotic decays of the Higgs boson. Exotic decays are predicted by a variety of BSM theories [172–175], including extended scalar sectors, SUSY, and dark sector models with Higgs or vector portals. Higgs branching ratios up to four orders of magnitude lower than the expected HL-LHC reach will be probed at FCC-ee, giving access to entirely new channels [176].

Substantial work in preparing searches for Higgs boson decays into long-lived particles (LLPs) is ongoing, focusing on signatures with displaced vertices [177]. The results suggest that FCC-ee is sensitive to long-lived scalars with decay lengths of order 1 mm to 10 m, with the peak sensitivity for decay lengths around 0.3 m, as shown in Fig. 31. Additional studies, comparing different detector setups for exotic Higgs decays, have also been initiated [178].

Searches for exotic Z decays to LLPs can also be carried out at FCC-ee during the Z pole run. In models of R-parity-violating supersymmetry, studied in Ref. [179], the FCC-ee sensitivity to long-lived light neutralinos in Z decays exceeds that of hadron colliders or of future dedicated LLP experiments by three orders of magnitude. Comparable gains in sensitivity can be obtained in other searches, such as Z decays into hidden mesons [180].

The available parameter space of new SM-neutral vector bosons (Z') that couple exclusively to leptons could be significantly extended, offering the best performance of all future collider projects in the kinematically allowed mass range (from 5 to 360 GeV) [181].

In brief, by taking full advantage of the large Tera-Z event samples, the dark sector programme at FCC-ee extends the intensity-frontier discovery potential significantly. It provides access to very-weakly-interacting dark-sector particles with couplings so small that they are usually considered typical of beam-dump experiments, but in a mass range wholly inaccessible to any other intensity-frontier experiment. Given that there is no preferred mass range for such states, this mass-range extension by more than two orders of magnitude provides unique scientific opportunities in particle physics.
<figure><img src="image_37.png" /><figcaption>**Fig. 31** Estimate of the sensitivity to LLPs in exotic Higgs portal decays; adapted from Ref. [175]. Solid (dashed) lines correspond to dedicated light (heavy) LLP search strategies</figcaption></figure>

<figure><img src="image_38.png" /><figcaption>**Fig. 32** Summary of the 5σ discovery reach, as a function of the resonance mass, for different FCC-hh luminosity scenarios. From Ref. [190]</figcaption></figure>

### 2.3.8 Other new physics searches

Other possible studies at FCC-ee include, for example, searches for lepton-flavour violation (LFV) in different manners [182–187], compositeness [107], leptoquarks [188], new scalars [189], and more. In this instance, the production of hidden-valley light quarks would perturb the QCD cascade, generating correlations among the final state particles that are absent in the SM. A detailed experimental study is in progress.

### 2.3.9 Complementarity and synergy between FCC-ee and FCC-hh

The FCC-hh will complement and substantially extend the FCC-ee physics reach in nearly all possible directions. The seven-fold centre-of-mass energy increase with respect to LHC enhances the potential for observing new particles at mass scales up to 40 TeV, as shown in Fig. 32. Indirectly, it will be sensitive to energies well above its kinematic reach of 100 TeV, for example in the tails of Drell–Yan distributions. Should any deviations from SM expectations be observed at FCC-ee, FCC-hh has the potential to pinpoint its microscopic origin. Some specific synergies between FCC-ee and FCC-hh in this regard are highlighted in the next paragraphs.

As already alluded to in Sect. 2.2, there are many synergies between FCC-ee and FCC-hh that come together to make the integrated FCC physics programme more than the sum of its parts. For example, the absolute determinations of Higgs
<figure><img src="image_39.png" /><figcaption>**Fig. 33** The projected sensitivity of searches for a WIMP triplet DM candidate in final states with disappearing tracks at FCC-hh [10]. Adapted from Ref. [193]</figcaption></figure>

couplings and its total width at FCC-ee is necessary to fully exploit the potential of FCC-hh that is only able to measure relative coupling ratios. The clean environment of FCC-ee enables the precise determination of SM parameters necessary to reduce uncertainties for FCC-hh. Moreover, certain types of BSM physics can only be accessed by FCC-ee and not FCC-hh, and vice versa. For instance, the nature of the electroweak phase transition will be probed by the precise measurement of the Higgs self-coupling and the search for additional Higgs partners at FCC-hh, together with potential deviations in the Higgs couplings at FCC-ee.

More concretely, if a deviation in $\kappa_Z$ were observed at the level of 0.5%, this hint would be less than half of one sigma at HL-LHC (with some unavoidable assumptions), but would correspond to an unambiguous $5\sigma$ discovery at FCC-ee. In Ref. [191], it was shown that for a BSM modification of this coupling new physics would have to show up at an energy scale of

$$E_{\text{Max}} \approx \frac{1.1\text{TeV}}{\sqrt{|\kappa_Z - 1|}}. \qquad (1)$$

Thus, a deviation of $5\sigma$ at FCC-ee would correspond to an energy scale of 15 TeV, essentially guaranteeing that FCC-hh would be able to discover the heavy states ultimately responsible for the low energy Higgs coupling deviation, illustrating a compelling, dynamic, interplay between FCC-ee and FCC-hh.

This synergy also extends to astrophysics and cosmology. Beyond-the-Standard-Model physics leading to a phase transition of some new sector, anywhere from the electroweak to the TeV scale, could yield gravitational wave signatures accessible to the next-generation gravitational wave observatories, such as LISA, the origin of which could then potentially be directly accessed at FCC-hh [92]. The detection of high-energy gamma rays in astrophysical observatories such as CTA [192] may be due to a TeV-scale WIMP, but with a large degree of uncertainty inherent to such indirect observation methods. The FCC-hh will be directly sensitive to the entire upper mass range, of around 1 and 3 TeV for triplet and doublet thermal WIMP dark matter, respectively [10], far beyond the reach of any conventional dark matter direct detection experiment, as illustrated in Fig. 33. Cosmic-ray physics will furthermore benefit from hadron collision data at unprecedented energy and luminosity.

The multiplicity of partons in the proton makes FCC-hh the most versatile high-energy parton collider for the broadest possible exploration of elementary particle processes. Similarly to LHC, its capabilities may be augmented by hosting complementary detectors for neutrino physics, dark sectors, and long-lived particles that may otherwise escape undetected. Together, FCC-ee and FCC-hh can truly enter the tens of TeV scale, indirectly and directly, to fully explore the open questions that LHC has only begun to touch upon.

## 2.4 Selected topics in flavour physics

The peculiar structure of quark and lepton masses, as well as the quark mixing angles, is ad hoc within the SM and is likely the low-energy imprint of some new dynamics. This structure implies approximate flavour symmetries that give rise to the
<table><caption>**Table 6** Yields of heavy-flavoured particles produced at FCC-ee for $6 \times 10^{12}$ Z decays [194]</caption><thead><tr><th>Particle species</th><th>B<sup>0</sup></th><th>B<sup>+</sup></th><th>B<sup>0</sup><sub>s</sub></th><th>Λ<sub>b</sub></th><th>B<sup>+</sup><sub>c</sub></th><th>c<span style="text-decoration: overline">c</span></th><th>τ<sup>-</sup>τ<sup>+</sup></th></tr></thead><tbody><tr><td>Yield (×10<sup>9</sup>)</td><td>370</td><td>370</td><td>90</td><td>80</td><td>2</td><td>720</td><td>200</td></tr></tbody></table>

strong suppression of a series of flavour-changing processes within the SM, whose precise experimental study allows probing dynamics at scales well above the electroweak scale. It is necessary to push the precision flavour frontier further and the FCC-ee programme offers unique opportunities in this respect. Despite the fact that many tests in this sector have been performed in the recent past, without reporting significant deviations from the SM, the discovery potential remains high.

One class of unique opportunities concerns b-hadron and $\tau$ decays, and, in particular, $b \to \tau$ transitions. These processes directly test models where new physics is coupled mainly to the third generation – a general feature of many explicit BSM scenarios. Moreover, the class of models that can be tested indirectly via such studies is also the one less constrained by direct searches (given the suppressed couplings to light quarks). Here, the FCC-ee programme offers a particularly large margin of improvement over the accuracy expected from currently approved flavour physics experiments, both at HL-LHC and at lower-energy $e^{+}e^{-}$ colliders (in particular Belle II at Super KEKB). The key features of the flavour-physics programme during the Z-pole run at FCC-ee can be summarised as follows.

- Clean environment (as in B-factories), with momentum and tagging efficiency of the pair-produced b's, c's, and $\tau$'s from Z decays, with ~10 times more $b\bar{b}$ and $c\bar{c}$ pairs than the total collected by Belle II (Table 6).

- Boosted b's and $\tau$'s, leading to a significantly higher efficiency (compared to B-factories) for modes with missing energy (especially multiple-$\nu$ modes) and inclusive modes, as well as smaller uncertainties in lepton ID efficiencies.

These features lead to the identification of the following class of observables as particularly promising: (i) neutral-current rare b- and c-hadron decays with $\tau^{+}\tau^{-}$ and $\nu\bar{\nu}$ pairs in the final state; (ii) charged-current b-hadron decays with a $\tau\nu$ pair in the final state; (iii) CP-violating observables in b- and c-hadron decays involving neutral particles in the final states ($\pi^{0}$, $K_{S}$, $\gamma$, ...); (iv) lepton flavour violating $\tau$ decays; (v) precision tests of lepton universality in $\tau$ decays.

In addition to these classical flavour studies via low-energy probes, a truly unique opportunity offered by FCC-ee is the possibility of determining flavour parameters from W decays, as well as searching for flavour-violating decays of Z and Higgs bosons. Within the SM, these processes are possibly too rare to be detected, but their search is an effective way to constrain or discover BSM dynamics. Some further ideas uniquely relevant to the FCC-ee flavour programme are discussed in Refs. [194–196]

The wide-ranging FCC-ee flavour physics programme cannot be easily summarised in a few pages. The most interesting classes of observables are listed below, together with a discussion of a few examples that illustrate the physics reach and the diversity of this programme.

### 2.4.1 Lepton universality tests in $\tau$ decays

The major improvement expected at FCC-ee in the precision of measurements of the leptonic $\tau$ decay branching fraction, the $\tau$ mass, $m_{\tau}$, and the $\tau$ lifetime, $\tau_{\tau}$, corresponds to a jump of more than one order of magnitude (from $10^{-3}$ to $10^{-4}$) in tests of universality between third-generation and light leptons [197, 198]. An illustration of the FCC-ee reach on these measurements is shown in the left panel of Fig. 34 [199]. With this level of precision, models addressing the hint of non-universality observed in $b \to c\tau\nu$ decays [200] could either be unambiguously confirmed, leading to a major discovery, or ruled out. Theoretical work (on electroweak and radiative corrections), achievable with present knowledge but not yet available, is needed to reduce the theoretical systematic uncertainties on these measurements well below the projected experimental systematic uncertainties.

### 2.4.2 Lepton flavour violating $\tau$ decays

Searches for LFV $\tau$ decays are null tests of the SM and, thus, are pristine targets for indirect NP searches. A variety of explicit models, predicting rates not far from current exclusion bounds, exist in the literature [201–203], and FCC-ee will offer the ultimate experimental sensitivity on virtually all $\tau$ LFV decay modes [198, 204, 205]. For example, the 90% CL projected limit on $B(\tau \to 3\mu)$ would be $2 \times 10^{-11}$ in absence of signal [199], which is three orders of magnitude below the current
<figure><img src="image_40.png" /><figcaption>**Fig. 34** Left: direct measurements of $B(\tau \to \ell\nu\bar{\nu})$ branching fraction and $\tau$ lifetime (ellipse) compared to the SM prediction (band); the width of the SM band is determined by the $\tau$ mass uncertainty. From Ref. [199]. Right: FCC-ee impact in constraining, at 95% CL, the coefficients of representative dimension-six LFV operators normalised to the 1 TeV scale. Adapted from Ref. [201], scaled to the baseline FCC-ee luminosities</figcaption></figure>

<figure><img src="image_41.png" /><figcaption></figcaption></figure>

limit ($2.1 \times 10^{-8}$ [35]) and more than an order of magnitude better than the final Belle II projection (with $50 \text{ ab}^{-1}$). The impact of such bounds is illustrated in the right panel of Fig. 34.

### 2.4.3 Rare b-hadron decays with $\tau^+\tau^-$ pairs in the final state

In all these processes, there is an unexplored gap of about three orders of magnitude between SM predictions [206–208] and experimental data [209,210] – a gap that represents a wide unexplored parameter range of motivated new-physics models [211–213]. At the SM predicted rates, FCC-ee is expected to be able to measure these decays, in particular $B^0 \to K^{*0}\tau^+\tau^-$ [214,215].

### 2.4.4 Charged-current b-hadron decays with a $\tau\nu$ pair in the final state

The persisting anomalies in exclusive $b \to c\tau\nu$ decays [216] give a strong phenomenological motivation for deeper studies of these processes [185,217–219]. A unique opportunity offered by the FCC-ee programme would be the experimental access to the clean inclusive [220,221] and leptonic [222,223] modes, as well as the suppressed $b \to u$ transitions [224,225]. For example, the measurement of the $B \to \tau\nu$ branching ratio at FCC-ee [226] is expected to yield ultimate precision on $|V_{ub}|$, with an estimated relative uncertainty of only 1%, which is a factor of three below current best estimates [216] and more than a factor of two below the projected precision from the $B \to \tau\nu$ measurement at Belle II [227].

### 2.4.5 Rare b- and c-hadron decays to di-neutrino final states

A recent sensitivity study [228] has shown that the branching ratios of the rare $B^0 \to K^{*0}\nu\bar{\nu}$, $B^0 \to K_S^0\nu\bar{\nu}$, $B_s \to K_S^0\nu\bar{\nu}$, and $\Lambda_b^0 \to \Lambda\nu\bar{\nu}$ decays could be determined at FCC-ee with precisions of 0.53%, 1.20%, 3.37%, and 9.86%, respectively, relative to the current SM estimates [229]. In particular, the precision on the $B$ decay modes would exceed that expected from Belle II by an order of magnitude, while the $B_s$ and $\Lambda_b^0$ modes, accessible at FCC-ee, would uniquely constrain possible new physics in $b \to \bar{s}\nu\bar{\nu}$ transitions, as illustrated in the left panel of Fig. 35.

Furthermore, the large number of expected events could uniquely allow more detailed differential studies of decay kinematics beyond simple branching ratios. Such measurements can also efficiently probe possible BSM effects in terms of the
<figure><img src="image_42.png" /><figcaption>**Fig. 35** Left: comparison between the current constraint on the relevant effective b → sνν̄ transition operator coefficients ($C_{L,R}$) due to existing measurements [230] (cyan band) and the (systematic-error dominated) sensitivities predicted at FCC-ee (blue, orange, green, and red bands) [228]. The projected regions denote the 68% probability of the marginal posterior density, assuming that all observables are SM-</figcaption></figure>

like. The Wilson coefficients are normalised to an effective scale of 6.5 TeV, up to the CKM factors $V_{ts} \times V_{tb}$ [228]. Right: current (from D-¯D mixing) and projected (from FCC-ee searches for $h \to cu$) limits on effective charm-up flavour-changing Yukawa couplings of the Higgs boson [187]. The 68%, 95%, and 99.7% CL regions allowed by D-¯D mixing data are depicted from darker to lighter red

production of light invisible particles in the final state, mimicking the missing energy signature of the neutrino pair in the SM [231].

In the charm sector, processes mediated by the $c \to u\bar{v}$ transition are unique probes of up-quark FCNCs because they do not receive long-distance QCD contributions plaguing related non-leptonic or rare semileptonic decays [232]. In addition, SM gauge invariance and approximate $U(2)$ flavour symmetry relate these quark-flavour-changing processes to rare (semi)leptonic kaon decays [233–235], thus making them crucial in (over)constraining BSM effects in FCNCs involving the first two quark generations. Finally, the light dark sector coupling to $c \to u$ quark currents [236,237] can be probed with the missing energy signature, similar to the $b \to s$ transition in the b-quark sector. Currently, the only experiment probing these processes is BES III, which set the first upper limit, $B(D^0 \to \pi^0 \nu\bar{\nu}) < 2.1 \times 10^{-4}$ at 90% CL [238]. Based on naïve luminosity scaling, this bound could be improved by more than two orders of magnitude at FCC-ee and would thus make it relevant to probe flavour $U(2)$ invariant BSM scenarios [235].

### 2.4.6 Rare decays and CPV studies with neutrals

The discovery of CP violation in singly Cabibbo-suppressed D decays by LHCb [239] opened a new and largely unexplored venue for CPV studies. One of the outstanding puzzles is to understand whether the observed size of CPV is due to SM effects enhanced by long distance QCD dynamics, or a possible signal of NP [240,241]. Recent phenomenological studies using isospin [242,243] and $U$-spin [244,245] expansions have shown that the measurements of decay rates and CP asymmetries in related D and $D_s$ decays to final states involving $\pi^0$ and $K_S^0$, which are not possible at LHCb but would be accessible at FCC-ee, could yield a definite answer. Important representative studies also include CP violation in radiative singly Cabibbo suppressed modes [246–248] and the doubly-radiative decay $D^0 \to \gamma\gamma$ required for the SM prediction of $D^0 \to \mu^+\mu^-$ [249].

In the B sector, rates and CP asymmetries of rare radiative decays, mediated by the $b \to s\gamma$ transition, such as $B \to K^*\gamma$ and $B \to X_s\gamma$, are most-sensitive probes of quark flavour violating dipole transitions [250] in both minimally violating and $U(2)$ flavour models; they are also crucial probes of flavour dynamics in composite Higgs models [251]. Improvements in the understanding of these decays [252,253] are needed to fully leverage the prospective sizes of the FCC-ee event samples [254]. Recent steps in this direction are documented in Refs. [255,256]. Finally, time-dependent studies of rare decays, such as $B_s \to \mu^+\mu^-$ [257,258], $B \to K_S\ell^+\ell^-$ [259,260], $B_s \to \phi\mu^+\mu^-$ [261], and $B \to K_S\nu\bar{\nu}$ [262], offer novel and complementary probes of CP violation beyond the SM, some only accessible at FCC-ee.
<figure><img src="image_43.png" /><figcaption>**Fig. 36** Sensitivities to relative BSM amplitudes ($h_d$ and $h_s$) in $B_d$ and $B_s$ mixings: current values (left), combined LHCb (50 fb$^{-1}$) and Belle II (50 ab$^{-1}$) projections (centre), and FCC-ee projections (right). From</figcaption></figure>

Ref. [273]. A limiting factor of the FCC-ee sensitivity is the uncertainty on $V_{cb}$, expected to be reduced with the WW threshold run and from improved flavour-tagging algorithms

### 2.4.7 On-shell Z, W, and Higgs flavour-changing decays

Flavour-changing decays offer unique novel probes of SM and BSM flavour dynamics and can be searched for in inclusive final states (profiting from the unparalleled jet-flavour tagging capabilities of the detectors) or in exclusive final states [263]. In particular, $W \to bc$ decays could yield the ultimate precision on $|V_{cb}|$, with a relative uncertainty well below 1% [264,265]. Similarly, rare four-body Z decays could uniquely determine individual flavour neutrino couplings [266]. Furthermore, lepton-flavour-violating Higgs and Z decays involving $\tau$ leptons in the final state would improve existing LEP and LHC constraints by orders of magnitude [51,267,268]. Finally, FCC-ee could also directly probe FCNC decays of Z and Higgs bosons [187], possibly even reaching SM expectations for the former (in the case of $Z \to \text{bs}$) and surpassing indirect constraints from neutral meson oscillation measurements for the latter, as shown in the right panel of Fig. 35. Such direct measurements are also particularly important to lift degeneracies in BSM fits and disentangle possible UV sources of low energy FCNC effects [269].

### 2.4.8 Combined studies and synergy between the flavour and EW programmes

Besides the interest of the specific observables illustrated above, probably the most interesting aspect of the FCC flavour programme lies in the possibility of combining several precision flavour-violating and flavour-conserving measurements. This combination offers a unique opportunity to characterise a possible NP signal, when it emerges, as illustrated in the next two paragraphs.

#### New-physics in $B-\bar{B}$ mixing

Measurements of $\phi_s$ from $B_s \to J/\psi \phi$ and $B_s \to \phi\phi$ at FCC-ee could challenge present theory uncertainties [270,271]. Similarly, the full exploitation of experimental precision on $\Delta m_s$ will hinge upon the determination of $|V_{cb}|$ at or below the sub-per-cent level [272,273] from either (inclusive) semileptonic B decays, leptonic $B_c$ decays [222,223], or on-shell $W^+ \to \bar{b}c$ decays [264,265]. As an illustration of the increased sensitivity achieved by combining all these measurements, the NP reach in the $B_{s,d}$ mixing amplitudes is shown in Fig. 36. Mainly due to the increased precision on $|V_{cb}|$, the effective NP scale sensitivity would increase by a factor of three compared to current measurements and by a factor of 1.5 compared to the ultimate precision expected at Belle II. It is worth stressing that FCC-ee will also deliver the ultimate determination of the CKM-matrix $\gamma$ angle from $B \to \text{DK}$ decays [274], allowing a per-mil test of CKM unitarity in the $b \to s$ triangle [275–277] (one order of magnitude better than the current sensitivity), as well as the possibility to test the SM predictions for mixing-induced semileptonic CP asymmetries in both $B_d$ and $B_s$ decays [278–280].

#### Synergy with the EW programme

A particularly interesting synergy emerges when combining flavour-violating and EW observables. The discovery potential offered by this combination in a variety of NP models has only recently begun to be explored [106,281,282]. An illustration of this potential is presented in Fig. 37 in the context of models describing generic NP coupled mainly to the third generation, as defined in Ref. [106].
<figure><img src="image_44.png" /><figcaption>**Fig. 37** Constraints on semileptonic effective operators describing new physics coupled mainly to the third generation with minimal $U(2)$ breaking [106]. The grey regions show the result of the current fit (68%, 95%, and 98% CL), including flavour, electroweak, and Drell-Yan data (the slight tension with the SM is driven by $b \to c\tau\bar{\nu}$ data). The small ellipses are the result of an hypothetical fit with FCC-ee projected uncer-</figcaption></figure>

<figure><img src="image_45.png" /></figure>

tainties on flavour and electroweak observables, assuming a signal compatible with current data. The coloured bands are the individual constraints of the observables in this scenario. The FCC constraints follow from the projections of a precision of 3%, 1.6%, 0.01%, and 20% on the $B(B \to K\nu\bar{\nu})$, $B(B_c \to \tau\bar{\nu})$, $B(\tau \to \mu\nu\bar{\nu})$, and $B(B \to K\tau^+\tau^-)$ values, respectively. From Ref. [283]

As can be seen in this case, different observables probe the same parameter space in different directions, and their combination is essential to fully characterise the hypothetical NP framework, with, in particular, the synergy between rare B decays, rare $\tau$ decays, and the flavour-conserving effective couplings of the Z boson.

## 2.5 FCC-hh specificities compared to high-energy lepton colliders

With operation expected to start in the mid-2040s, the guaranteed deliverables and primary targets of FCC-ee are the precise and model-independent measurement of a broad range of key couplings of the Higgs boson, a fifty-to thousand-fold improvement in the precision of EW parameters, and a precise determination of the top quark mass and EW couplings. A 3 TeV muon collider has also been proposed, for a timescale similar to that of FCC-ee [284]. The best-measured Higgs couplings, $\kappa_Z$ and $\kappa_W$, will be measured at FCC-ee with expected uncertainties of 0.11% and 0.29%, respectively (Table 3), much better than the 0.9% and 0.4% uncertainties foreseen at a 3 TeV muon collider [285]. This observation, combined with the Tera-Z programme and the $10^6$ top pairs at FCC-ee, as compared to orders of magnitude fewer at a 3 TeV muon collider, renders FCC-ee the obvious priority for Higgs, electroweak, top, and flavour physics to follow HL-LHC [286]. As the longer-term ambition of the high-energy-physics community is to explore the highest achievable energies, this section considers the question of what should follow FCC-ee as a high-energy exploration facility, comparing and contrasting frontier ($\gtrsim 10^7$ TeV) lepton collider options with FCC-hh.

### 2.5.1 Generalities

Exploration in particle physics cannot be quantified by the volume of parameter space explored, nor by the scale of the energy reached. Indeed, LHC has shown that the microscopic world that can be explored by hadron colliders is so multidimensional and rich that no metric can fairly convey the depth with which the laws of nature are explored by colliding protons. Yet, looking to future high-energy facilities, this is precisely what must be attempted.

The core attribute of a proton collider is that it does not simply collide a single particle and an antiparticle but, instead, collides anything found in the proton; any of the lightest five quarks and antiquarks, the gluons, the photon, and the EW gauge
bosons. Thus, a proton collider has many particle colliders in one, operating at the highest plausibly achievable energies, with an array of production channels for discovery and exploration that is factorially greater than what can be achieved by colliding single fundamental particles – besides the additional possibility of colliding heavy ion in the same machine. The price to be paid for this great leap in exploratory power is the inherent messiness of the final state from colliding composite objects. However, this challenge has been amply met at LHC and there is no reason to expect something else at a future proton collider.

Were some hints for new physics at high energies to appear in the Run 3 of LHC or at HL-LHC, it is highly unlikely that identification of the microscopic production channel from which it emerged would be possible at HL-LHC. Depending on the new physics at stake, the FCC-ee precision data may exhibit interesting patterns of deviations that would refine the interpretation of these hints and give precious information about their quantum origin. At high energy, however, the only way to guarantee further exploration and characterisation of whatever may arise at LHC would be to collide protons once again, with greater energy and larger collected event samples, possibly activating additional production channels and dynamical regimes: this is the only way to cover all possible production channels.

Also, it is worth noting that the discovery of the Higgs boson opened windows beyond the gauge paradigm to a new class of fundamental forces and to the flavour puzzle. Modern particle physics exploration should be equipped to navigate this new world. The case, for example, of a new neutral scalar coupled to fermions with a strength proportional to their mass is enlightening. Clearly, the third generation quarks provide the most immediate access to such a sector, whether through production by $b\bar{b}$ annihilation or through gluon fusion via a top quark loop. This is, after all, how the Higgs boson was discovered. Such scenarios should not be blind spots for any future facility and, indeed, a future proton collider would provide excellent discovery opportunities for states that do not carry electroweak charges. Some more concrete, yet still general, future discovery possibilities are considered below.

### 2.5.2 Resonance searches

A key exploratory task for any high-energy exploration facility is to search thoroughly for new high-mass resonances. There are two complementary approaches to probe the existence of high-mass states: the direct search for a mass peak and the indirect evidence emerging from deviations in precise measurements of processes accurately predicted within the SM. The indirect approach extends the sensitivity of a collider well beyond its kinematic reach. As clearly shown by LEP2, the superior precision of a lepton collider can therefore match the sensitivity of direct searches at hadron colliders of much higher centre-of-mass energies. The limitation of the indirect approach, however, is that the interpretation of such deviations is very much dependent on model assumptions. For example, the new state masses can vary significantly as a function of the assumptions on their couplings. Outside a well-defined theoretical BSM framework, for which there is no compelling prejudice today, this ambiguity significantly limits the ability to point to new measurements that would clarify the source of new physics. Direct search, when possible, therefore provides a preferable (yet synergistic) approach, when the indirect and direct sensitivity of lepton and hadron colliders are numerically comparable.

For this reason, the focus here is on the comparison of FCC-hh and high-energy lepton colliders to the case of direct detection. Present theory priors do not favour any particular possibility, thus the breadth of couplings that can be explored is a paramount consideration in planning for the future. For this purpose, a proton collider has a further unique virtue, sampling a wide array of initial states, from gluons to photons or EW gauge bosons and quarks of a variety of flavours, and from neutral initial states to charged or even coloured, any of which could lead to the formation of a new resonance. For example, 15 $q\bar{q}^{(l)}$ initial states can create exotic bosonic resonances, both in flavour-conserving and flavour-violating channels. Besides, 20 V + q initial states could create excited quark states (with V = γ, g), or new heavy quarks through charged (V = W) or neutral (V = Z) current transitions. Furthermore, 8 VV' initial states can give rise to EW resonances or axion-like particles (V, V' = γ, W, Z), or to gg resonances.

Another key component of exploration combines breadth with energy. As a figure of merit, proton colliders can be compared with lepton colliders to assess their relative reach in energy for resonances. Following Ref. [287], an optimistic scenario is considered, where a new high-energy resonance (serendipitously) resides at the kinematic limit of a lepton collider and can be produced by that initial state.<fnref n="1" /> The energy of proton collisions that would be necessary to produce the same number of resonant events, given the same integrated luminosity, is a natural figure of merit here.

The resulting equivalent CM energy is shown in Fig. 38, under the assumption that the parton-level production cross sections are a factor $\beta = 1, 10, 100$ larger than the lepton-collider production cross section on resonance for $q\bar{q}$ and gg initial

<footnote n="1">¹ For lower masses, radiative-return production would be required to enable discovery.</footnote>
<figure><img src="image_46.png" /><figcaption>Fig. 38 Energies of equivalent event numbers, assuming equal integrated luminosities, between a high-energy muon collider and a proton collider; $\beta = 1$ corresponds to the same partonic cross section between muons and proton partons while larger values of $\beta$ correspond to enhanced partonic cross sections for quarks and gluons, as would be the case for resonances coupled primarily through QCD. Note that $\beta < 1$ would require muons to be more strongly coupled to new resonances than quarks, requiring muon flavour to play a role</figcaption></figure>

states. It is noteworthy that, even under the significant assumption that the production cross section via leptons matches that of quarks or gluons ($\beta = 1$), a 100 TeV proton collider significantly exceeds, for such a resonance, the reach of a 10 TeV lepton collider. For larger parton production cross sections relative to leptons, the gap in coverage between the two classes of facilities further increases. Such a comparison, however, also conceals the richness of the physics programme for such > 10 TeV resonances at a proton collider.

To go further, the direct discovery prospects for narrow resonances of mass $M_R > 10$ TeV, which would be inaccessible to a 10 TeV muon collider, is investigated. In the narrow width approximation, at a proton collider operating at proton CM energy $\sqrt{s}$, the production cross section for a resonance R of spin S coupled to the initial state partons ‘yy’ and decaying into the final state ‘xx’ is

$$ \sigma = r \frac{C_{yy}}{s}, \qquad (2) $$

where, for gluons and quarks, the factor $C_{yy}$ is related to the parton luminosity as

$$ \begin{aligned} C_{gg} &= \frac{\pi^2}{8} \int_{\tau}^{1} \frac{dx}{x} f_g(x) f_g(\tau x), \\ C_{q\bar{q}} &= \frac{4\pi^2}{9} \int_{\tau}^{1} \frac{dx}{x} [f_q(x) f_{\bar{q}}(\tau x) + f_{\bar{q}}(x) f_q(\tau x)], \end{aligned} \qquad (3) $$

where $\tau = M_R^2/s$ and the parton distribution functions $f_{g,q,\bar{q}}$ are evaluated at $Q^2 = M_R^2$. The parameter $r$ encodes all the model-dependent factors,

$$ r = (2S + 1)B_{yy}B_{xx} \frac{\Gamma_R}{M_R}. \qquad (4) $$

The width-to-mass ratio essentially encodes the microscopic coupling and hence parametrises the nature of the underlying theory. For these purposes, whenever $\Gamma_R/M_R \lesssim 0.1$, the theory is approximately perturbative and the resonance effectively narrow. In Eq. (4), $B_{yy}$ is the branching ratio into the partons whose collision produced the resonance and $B_{xx}$ is the branching ratio into whichever final state is under consideration. The latter could more prosaically be a pair of SM particles or something much more exotic, such as pairs of long-lived particles.

Figure 39 shows the number of $pp \to R \to xx$ events for a variety of production channels at masses above 10 TeV. This number of events is weighted relative to the model-dependent factor $r$. Simply as an illustrative example, a scalar resonance could plausibly have $r = 10^{-2}$. Thus, in this case, the total number of events in Fig. 39 should be multiplied by $10^{-2}$. The number of events plausibly available, hence the breadth of the exploration, to a 100 TeV proton collider for resonances above the 10 TeV scale is vast. Smaller values of $r$ are also possible, with events occurring above energies of 10 TeV even for extremely narrow resonances, down to $r \sim 10^{-8}$. For example, the SM Higgs boson has $r \approx 3 \times 10^{-5}$.
<figure><img src="image_47.png" /><figcaption>Fig. 39 The number of resonance-production events at a 100 TeV proton collider, relative to the model-dependent factor r defined in Eq. (4). A non-exhaustive variety of initial states yy is considered and the final-state decay product xx is unspecified</figcaption></figure>

If the final states were rare collider objects, such as long-lived particles, then these many-orders of magnitude of events would be highly visible, offering a vast programme of not only discovery but also characterisation. Presumably, for such exotic decays, the discovery reach would essentially saturate kinematic and PDF limits, up to ~50 TeV. On the other hand, if the final states were not rare but visible, such as to photons or leptons, then such a resonance would also stand out above the background. This is confirmed, for example, for the Sequential Standard Model scenario reach for leptonic decays shown in Fig. 32, where the reach extends beyond 40 TeV. These are the easiest discovery opportunities that would maximise the opportunities of the statistical reach of the large event samples.

Finally, even in scenarios where the final state has a significant background, the number of events is large enough that a discovery would still be expected. This point is confirmed by detailed studies for specific scenarios. For instance, Fig. 32 also shows the discovery reach for a variety of s-channel resonances. Even for a dijet final state, for which there is overwhelming background, the discovery reach extends to the 40 TeV scale.

### 2.5.3 Pair production of new particles

Figure 40 shows the result of a similar exercise as for the resonances, again following Ref. [287] and assuming the same integrated luminosity for each collider, and wondering what energy of muon collider would produce the same number of pair-production events as a proton collider operating at a CM energy $\sqrt{s_p}$. To do this, it is assumed that the muon collider can produce states close to threshold and, considering partonic cross sections

$$ \hat{\sigma}_{ij \to xx}(s_\mu) = \beta \hat{\sigma}_{\mu^+\mu^- \to xx}(s_\mu) \qquad (5) $$

(where ij = gg, q\bar{q} are the initial state partons and xx are the pair-produced new states), it is also assumed that the energy dependence of the cross section is inversely proportional to the partonic energy-squared. As before, $\beta \approx 1$ qualitatively corresponds to new states that couple equally strongly to the muons and the partons. Such scenarios would involve the production of purely electroweak states, or states with flavour-independent couplings. Instead, $\beta \approx 10-100$ corresponds to states that may also carry colour. Going beyond this, states that only couple to gluons or couple proportionally to fermion masses would have $\beta \gg 100$.

It can be seen from Fig. 40 that only for the case of $\beta \approx 1$, hence essentially only purely EW states, does a 10 TeV muon collider have an estimated reach extending beyond that of a proton collider. This is confirmed in the simple example of supersymmetry, where the estimated discovery reach for coloured particles shown in Fig. 41 for a 100 TeV proton collider extends beyond 10 TeV, hence beyond the kinematic limit of a 10 TeV muon collider. For purely electroweak particles, the proton collider reach saturates at around the 5 TeV scale, close to the kinematic limit of a 10 TeV muon collider.

This treatment has, however, been relatively naïve in the sense that it is essentially rooted in the gauge paradigm. It can instead be assumed that, at these energy scales, the new physics couples only through dimension-6 contact interaction. In this case, the production cross section at the partonic level scales proportional to the partonic energy-squared. Such a possibility is plausible if the states that connect the two sectors are heavy.
<figure><img src="image_48.png" /><figcaption>**Fig. 40** Same as Fig. 38, but for pair-produced resonances</figcaption></figure>

<figure><img src="image_49.png" /><figcaption>**Fig. 41** Summary of the 5σ discovery reach for different SUSY particles. From Ref. [288]</figcaption></figure>

Figure 42 shows the approximately equivalent colliders under this assumption, where the outcome is very different. In this case, a 100 TeV proton collider exceeds the reach of a 10 TeV muon collider even assuming comparable partonic cross sections at $s_{\mu}$, which may not be the case.

## 2.6 Physics reach of alternative FCC-hh $\sqrt{s}$ options

The reduction of the FCC tunnel length, with respect to the CDR, from 100 to 90.7 km, and the range of values for the magnetic field strength of dipoles that might be ready for mass production on the timescale of the final definition of the FCC-hh accelerator, motivate a study of beam-energy scenarios alternative to the canonical $\sqrt{s} = 100$ TeV used for the CDR physics studies.

The first results of this study are presented here, considering the cases of $\sqrt{s} = 80$ and 120 TeV. The former is motivated by the combined effect of a reduction in ring size and a less aggressive assumption, with respect to the CDR, for the strength of Nb$_3$Sn dipoles finally available (14 T instead of 16 T). The higher energy value is relevant in scenarios where high-temperature superconducting (HTS) dipoles, in the range of 20 T, become available. A more precise and detailed assessment of the full operational scenarios, including therefore the estimates of the collider luminosity, is underway. The definition of these scenarios will be modulated by assumptions about, and interplay between, key elements such as overall power consumption and experimental pileup. The results of these more complete studies, and their impact on the physics programme, will be documented at a later stage.

For this report, the same luminosity setting is assumed for all three energies, 80, 100, and 120 TeV, namely a total of 30 ab$^{-1}$, obtained by combining the event samples collected by two general-purpose detectors. The goal of the study is not to provide
<figure><img src="image_50.png" /><figcaption>**Fig. 42** Same as Fig. 40, but for production through a higher-dimension operator</figcaption></figure>

<table><caption>**Table 7** Projected uncertainties of Higgs coupling measurements at FCC-hh, for different $\sqrt{s}$ scenarios, with an integrated luminosity of 30 ab$^{-1}$</caption><thead><tr><th rowspan="2">Coupling</th><th>100 TeV</th><th>80 TeV</th><th>120 TeV</th></tr><tr><th>CDR baseline</th><th></th><th></th></tr></thead><tbody><tr><td>δg<sub>Hγγ</sub>/g<sub>Hγγ</sub> (%)</td><td>0.4</td><td>0.4</td><td>0.4</td></tr><tr><td>δg<sub>Hμμ</sub>/g<sub>Hμμ</sub> (%)</td><td>0.65</td><td>0.7</td><td>0.6</td></tr><tr><td>δg<sub>HZγ</sub>/g<sub>HZγ</sub> (%)</td><td>0.9</td><td>1.0</td><td>0.8</td></tr><tr><td>δg<sub>Hμμ</sub>/g<sub>Hμμ</sub> (%)</td><td>0.65</td><td>0.7</td><td>0.6</td></tr></tbody></table>

an accurate update of the 100 TeV results but rather to focus on the loss or gain in performance for some of the key deliverables highlighted in the FCC-hh CDR, namely the precision study of Higgs properties, the search for high-mass resonances, and the potential to discover or exclude scenarios of WIMP dark matter candidates. The chosen energies (80 and 120 TeV) may not match precisely the values that will emerge from the more detailed accelerator studies. Preliminary indications suggest, however, that they provide a good reference for the physics performance with dipole field strengths ranging between 12 and 20 T.

## 2.6.1 Higgs properties

The CDR has shown that the baseline FCC-hh programme can improve to sub-percent level the precision of Higgs decays such as $\gamma\gamma$, $Z\gamma$, and $\mu\mu$, which are too rare to be detected with that kind of precision at FCC-ee or any other proposed $e^{+}e^{-}$ Higgs factory. This precision is achieved by considering the ratio of Higgs bosons produced above some $p_{T}$ threshold and decaying to a given channel, relative to those decaying to four leptons. For the latter, the branching ratio is expected to be known from FCC-ee with a per-mil precision. The ideal value of the $p_{T}$ cut is defined by the optimum balance between statistical and systematic uncertainties; typical values are in the 100–200 GeV range. In this $p_{T}$ range, the Higgs production rate is reduced (increased) by about 30% (35%), when changing $\sqrt{s}$ from 100 to 80 (120) TeV. Folding this modified statistical uncertainty with the systematic uncertainties leads to the projections for the measurement precision of Higgs couplings shown in Table 7. The 1% statistical precision expected at 100 TeV for the determination of the $t\bar{t}H / t\bar{t}Z$ production cross-section ratios is modified to 1.2% and 0.85%, at 80 and 120 TeV, respectively.

The evolution of the Higgs self-coupling measurement is shown in Table 8. Three different scenarios for the systematic uncertainties are considered [85]: (I) corresponds to systematic uncertainties comparable to those of the ATLAS and CMS experiments during the Run 2 of LHC; (II) is a conservative estimate based on 2019 projections for the HL-LHC performance; and (III) is an intermediate scenario. Significant progress has been made since 2019 and the results, even for the baseline 100 TeV operations, will be updated by the time of the ESPP discussions.
<table><caption>**Table 8** Projected uncertainties of SM Higgs self-coupling measurements at FCC-hh, for different $\sqrt{s}$ scenarios, under different assumptions regarding the experimental systematic uncertainties, as defined in Ref. [85]</caption><thead><tr><th>Systematic uncertainties scenario</th><th>100 TeV<br>CDR baseline</th><th>80 TeV</th><th>120 TeV</th></tr></thead><tbody><tr><td>I (%)</td><td>3.4</td><td>3.8</td><td>3.1</td></tr><tr><td>II (%)</td><td>5.1</td><td>5.6</td><td>4.7</td></tr><tr><td>III (%)</td><td>7.8</td><td>8.4</td><td>7.3</td></tr></tbody></table>

### 2.6.2 WIMP DM search

A key result obtained for the CDR is the proof that an FCC-hh experiment could conclusively discover, or exclude, broad classes of dark matter candidates. For example, weakly interacting neutral components of doublet or triplet weak isospin fermions, such as the higgsino and wino gauge fermions in supersymmetric theories, are required by cosmology to have masses smaller than about 1 and 3.5 TeV, respectively. It was shown in Ref. [289] that a search for disappearing charged tracks in events with large missing transverse energy could fully cover these mass ranges (as shown by the red bands in the left-side plots of Fig. 43). The analysis has been repeated at 80 TeV, rescaling the signal strength while keeping the same background level, leading therefore to a conservative extrapolation to lower energy. The results are shown in the right-side plots of Fig. 43. The $\mathcal{O}(10\%)$ loss in reach still enables the $5\sigma$ discovery up to the cosmology limit, even for the more critical case of the higgsino. Is is expected that a re-optimisation of the analysis should cover the relevant mass range down to slightly smaller collision energies, possibly above $\sqrt{s} = 70$ TeV.

### 2.6.3 High-mass reach

The general potential of FCC-hh to directly explore the existence of new large-mass particles is discussed above. The beam energy is clearly the main driver for the mass reach. The detector performance and the pileup environment have a minor impact on the energy-dependence of this reach, a dependence that can be studied with a simple extrapolation based on the evolution with energy of the partonic luminosities, using, e.g., the Collider Reach tool [290]. Table 9 shows the extrapolation to 80 and 120 TeV of the $5\sigma$ reach for several possible s-channel resonances. Details of the 100 TeV analysis and the definition of the various models listed here can be found in Ref. [190]. A reduction of the collision energy to 80 TeV leads to a 15–20% loss in mass reach, while running at 120 TeV would increase the reach by 10–15%. It goes without saying that, to this date, there is no specific model that precisely requires the existence of such new resonances in some of the mass ranges that would be (or not be) covered at the different energies. The impact of the beam-energy change is reduced for particles well below the kinematic endpoint, whose discovery reach is mostly limited by weak couplings, backgrounds or systematic uncertainties (as, for example, in the case of DM WIMPS discussed above).

In conclusion, the beam-energy dependence has been assessed for a few reference physics results used in the CDR as key performance benchmarks for FCC-hh, assuming equal systematic uncertainties and integrated luminosities at the energies $\sqrt{s} = 80, 100$, and 120 TeV. Under these assumptions, key measurements such as the per-cent level precision of Higgs boson couplings, or the discovery of a class of electroweak DM candidates within their allowed mass range, remain feasible, with target results changing by $\mathcal{O}(10-15\%)$ with respect to 100 TeV. The performance reduction at 80 TeV does not compromise the ambitions of the FCC-hh programme and, in principle, could be compensated by improved systematic uncertainties and higher luminosity. More complete studies, reflecting the most recent projections for the FCC-hh accelerator performance at the different plausible energies, are underway.

## 2.7 A forward-physics facility at FCC-hh

The immense breadth of the physics programme available to a hadron-collider facility is well proven by LHC, with its programme of pp, pN, and NN collisions (where N stands for heavy ion). The four large LHC experiments are accompanied by a multitude of smaller, dedicated experiments, which further push the exploitation of the scientific opportunities. A first exploration of the opportunities offered by the programme of heavy ion collisions at FCC-hh was presented in Ref. [57] and in the CDR [10]. The additional unique opportunities that could arise from the exploitation of the high intensity beam of energetic particles, including neutrinos, produced by pp collisions in the forward region, is briefly discussed here. At
<figure><img src="image_51.png" /><figcaption>**Fig. 43** The 5σ discovery reach for wino and higgsino dark matter candidates, via disappearing-track signals for $\sqrt{s} = 100$ (left) and 80 TeV (right). The red band corresponds to the alternative pixel-detector layout proposed in Ref. [289], where the details of the analysis are discussed</figcaption></figure>

<table><caption>**Table 9** Beam-energy dependence of the projected 5σ discovery reach, with the new particle mass in TeV, for s-channel BSM resonances, in various decay modes. The definition of the various models can be found in Ref. [190]</caption><thead><tr><th>Resonance</th><th>100 TeV</th><th>80 TeV</th><th>120 TeV</th></tr></thead><tbody><tr><td>Q<sup>*</sup></td><td>40</td><td>33</td><td>46</td></tr><tr><td>Z'<sub>TC2</sub> → t<span style="text-decoration: overline">t</span></td><td>23</td><td>20</td><td>26</td></tr><tr><td>Z'<sub>SSM</sub> → t<span style="text-decoration: overline">t</span></td><td>18</td><td>15</td><td>20</td></tr><tr><td>G<sub>RS</sub> → WW</td><td>22</td><td>19</td><td>25</td></tr><tr><td>Z'<sub>SSM</sub> → ℓℓ</td><td>43</td><td>36</td><td>50</td></tr><tr><td>Z'<sub>SSM</sub> → ττ</td><td>18</td><td>15</td><td>20</td></tr></tbody></table>

LHC, these particles are studied with the far-forward experiments FASER($\nu$) [291–294] and SND@LHC [295,296], and new dedicated experiments have been proposed in the context of a forward physics facility (FPF) [297,298] operating concurrently with HL-LHC. A FPF-like suite of far-forward experiments integrated within FCC-hh would provide unique opportunities for neutrino physics, QCD studies, and BSM searches. Such FPF@FCC could be located around 1.5 km away from the IP (Fig. 44), where a dedicated cavern would be excavated aligned with the line-of-sight (LoS). It would have a length between 100 and 500 m, depending on the detector setup. Representative applications of the FPF@FCC experiments are summarised here. A more detailed discussion can be found in Ref. [299].
<figure><img src="image_52.png" /><figcaption>**Fig. 44** High-energy light particles (neutrinos, LLPs/FIPs) produced in FCC-hh collisions could be detected at the FPF@FCC, located at around 1.5 km away from the IP, enabling sensitivity to a variety of SM and BSM signatures</figcaption></figure>

2.7.1 Neutrino physics

For neutrino detection, two options are considered: a FASERν2-like detector [297], dubbed FCCν, with a length of 6.6 m, and a deeper variant, FCCν(d), with a length of 66 m. These detectors could collect [299] up to $10^9$ electron/muon neutrinos and $10^7$ tau neutrinos for $\mathcal{L}_{pp} = 30 \text{ ab}^{-1}$, an increase of several orders of magnitude with respect to the (HL-)LHC yields. These neutrinos would exhibit the highest energies ever achieved in a laboratory (up to 40 TeV), overlapping with those from astrophysical sources. The event yields forecast at the FPF@FCC would outperform all previous, ongoing, and future (proposed) neutrino experiments for all three generations, as indicated in Fig. 45. These unprecedented event rates allow the precise fingerprinting of neutrino properties, such as their flavour universality and non-standard interactions.

As a representative application, the electromagnetic properties of neutrinos are considered, long recognised as a window to new physics [300], in particular on the measurement of the neutrino charge radius $\langle r_ν^2 \rangle$, which modifies the neutral-current DIS cross section for neutrinos. It can be extracted [301,302] by searching for deviations in the ratio between NC and CC DIS events as compared to the $\langle r_ν^2 \rangle = 0$ baseline. The upper left panel of Fig. 46 shows the projected sensitivity to $\langle r_ν^2 \rangle$ at FPF@FCC considering only statistical uncertainties. World-leading bounds would be achieved, measuring the (SM) neutrino charge radius for $ν_e$ and $ν_μ$, and reaching down to five times the SM value for $ν_τ$. Realising this potential for precision neutrino physics demands an excellent modelling of CC and NC neutrino DIS interactions within the detector, combined with high-precision event generators for neutrino DIS [303–305].
<figure><img src="image_53.png" /><figcaption>**Fig. 45** The muon and tau neutrino scattering yields as a function of $E_ν$ at FCC-ν and other experiments</figcaption></figure>

## 2.7.2 BSM sensitivity

The FPF@FCC is sensitive to a variety of BSM models [299], including dark Higgs bosons [306], relaxion-type scenarios, quirks [307–310], and millicharged particles (mCPs). These BSM signatures fall, in most cases, outside the coverage of the main FCC-hh detectors, and hence such a far-forward facility would markedly increase the discovery capabilities of FCC-hh. The sensitivity to dark Higgs bosons, *D*-type quirks, and mCPs of the FPF@FCC is summarised in Fig. 46. The enormous rates of forward Higgs production at 100 TeV enable the discovery of LLPs from Higgs decays with masses as large as 62 GeV (≈ $m_Η/2$) and couplings as small as $\sin\theta \sim 10^{-8}$ (beyond the reach of FCC-ee); of quirks with masses up to 14 TeV for a broad range of confinement scales $\Lambda$; and of mCPs with masses up to several hundreds of GeV and charge $q \sim 10^{-3}e$, closing the gap between (future) accelerator-based bounds and dark matter direct detection searches.

## 2.7.3 QCD and hadronic structure

The enormous samples of multi-TeV neutrinos available at the FPF@FCC (Fig. 45) enable high-resolution probes of the unpolarised and polarised structure of protons and of heavy nuclei, as summarised by the representative applications of Fig. 47.

First, high-energy neutrino DIS structure functions [311,312] provide a precise quark/antiquark flavour separation at large-$x$, which in turn reduces the theoretical uncertainties entering high-mass searches at FCC-hh. Second, neutrino DIS on a polarised target [313,314] resolves the spin structure of the proton [315,316] with complementary information to related experiments, such as those at the Electron-Ion Collider [317]. Third, detecting neutrinos originating from proton-lead collisions provides information on nuclear structure [318–320] down to $x \sim 10^{-9}$, where nuclear PDFs are unconstrained and novel dynamical regimes of QCD, such as the Colour Glass Condensate, are expected to dominate. This kinematic region is also of prime importance for astroparticle physics experiments.

## 2.7.4 Summary

Integrating far-forward experiments into FCC-hh would extend its scientific potential in several synergetic directions from QCD and neutrino physics to BSM searches in a cost-efficient manner. While at this early stage no attempt has been made to optimise the accelerator infrastructure or define the FPF@FCC detector technology more precisely, the results shown here motivate further studies of how to fully realise this unique physics potential.
<figure><img src="image_54.png" /><figcaption>**Fig. 46** Top left: sensitivity of the FCCν detectors to the neutrino charge radius. Top right: reach for a dark Higgs boson $\phi$ at the FPF@FCC. Bottom left: discovery potential for D-type quirks with mass $m_Q$ and confinement scale $\Lambda$. Bottom right: Sensitivity to mCPs with mixing parameter $\epsilon$.</figcaption></figure>

<figure><img src="image_55.png" /><figcaption>**Fig. 47** Projected constraints from the FPF@FCC experiments on the unpolarised (left) and polarised (middle) PDFs, and on the nuclear PDFs of lead nuclei (right)</figcaption></figure>

### 3 Theoretical calculations

To fully leverage the significantly improved experimental precision in Z-pole observables, W boson and top quark masses, $b$ and $\tau$ decays, as well as a broad array of Higgs observables, it is essential to obtain Standard Model predictions with an accuracy that matches the anticipated statistical uncertainties of FCC-ee. The expectation that most systematic experimental uncertainties can be reduced to the level of the statistical precision further underscores this requirement. Such theoretical predictions are necessary for both inclusive and exclusive processes, with the latter requiring integration with Monte Carlo generators. They encompass a wide range of technical aspects, including fixed-order perturbative corrections, resummation calculations, improved parton showers, non-perturbative hadronization effects, and lattice QCD. Any discrepancies between
experimental data and Standard Model predictions (anomalies) could provide crucial insights, potentially guiding the way towards new discoveries. In addition, detailed precision analyses of BSM effects within concrete models and effective theories will open up a broad spectrum of new prospects.

On top of that, FCC-hh has unique capabilities for testing SM phenomena at ultra-high energies, particularly the mechanism of electroweak symmetry breaking and broad coverage for direct particle discovery. Theory calculations are needed to evaluate (often large) backgrounds, expected signal rates, and to optimise the experimental search strategies. The high precision reached by the LHC, due to further improve in the HL-LHC era, has been and remains a powerful stimulus for the theory community to enhance the calculation reliability. This progress is fully aligned with the future needs of FCC-hh.

A very active effort (in lattice and in perturbative calculations) is ongoing worldwide to improve the theoretical control of flavour observables, in view of the current and forthcoming experimental progress made by LHCb and Belle II. This effort will continue alongside the experimental advances made by these experiments in the next decades, undertaking the required steps towards the ultimate precision goals required by FCC. Suitably refined precision calculations for flavour observables should also be identified and relevant actions must be planned accordingly. Present and future work focuses on (a) calculating higher-order perturbative QCD and EW corrections (including, for the latter, their matching at the weak scale), and (b) understanding an order-of-magnitude better than before the separation and interplay between perturbative and non-perturbative contributions with the goal of constraining the latter from data as much as possible. Improved lattice calculations, from algorithmic progress and hardware developments, are a further essential ingredient.

For all the measurement areas where theoretical improvements are needed to match the FCC precision goals – EW, QCD, flavour, BSM – the large interplay between the development of EW and QCD calculations, in terms of impact on the total theoretical systematic uncertainties and of computational challenges and tools, is remarkable. Two-loop QCD corrections are often of comparable size to one-loop EW corrections, and they both play a critical role at the required level of precision. Their interplay is thus essential, and often it is the source of added complexity when both EW and QCD corrections are mixed (e.g., in the presence of off-shell decays of weak bosons to quarks). On the formal side, progress towards the understanding of multi-scale multi-loop diagrams is of common value to both EW and QCD, as well as to BSM studies. Techniques inspired by effective field theory (EFT) are becoming ubiquitous, and tools initially developed for flavour physics (soft-collinear effective theory, or SCET) are now relevant to describe QCD event shapes, resummations, jet-vetoed final states, and much more. This global synergy of developments in different specific areas opens a whole new dimension for future coordinated theoretical efforts.

If any deviation from the SM expectations were observed at FCC-ee/hh, the interpretation would require calculations to the requisite precision in BSM models or models with higher-dimensional EFT operators to the requisite precision. These higher-order corrections can be straightforwardly achieved by adapting the techniques developed for standard model (QED, EW, and QCD) calculations. Therefore, improving the accuracy of SM predictions at large currently remains the main priority of the community of experts on radiative corrections. This section focuses on a brief review of the status, future needs and prospects for theoretical improvements of relevance to EW, Higgs, jets, and top quark measurements, including the MC event-generator perspective, and the actions under way to support and coordinate such efforts.

## 3.1 Electroweak corrections

To meet the precision goals of FCC-ee, significant advances in calculations of higher-order radiative corrections and in MC generators will be needed [38, 40, 321]. For instance, electroweak NNLO corrections for various pair production processes ($e^+e^- \to f\bar{f}$, $e^+e^- \to \gamma\gamma$, $e^+e^- \to W^+W^-$, $e^+e^- \to ZH$) are needed, as well as MC tools for the simulation of multiple photon radiation beyond leading-logarithmic (LL) approximation. Even higher perturbative orders, including three-loop corrections in the full SM and leading four-loop corrections, are required to interpret precision measurements at the Z pole. Table 10 provides a few illustrative examples of precision quantities and the theory calculations required to extract them from data.

Additional theory input is necessary for the interpretation of the experimentally determined values of these quantities, i.e., to test the validity of the SM and probe possible physics beyond the SM, as illustrated in Table 11 for some of the same examples as above. In this context, the three-loop $\alpha_S^3$ corrections to the semileptonic $b \to c$ decay [322] and the three-loop QED corrections to the muon decay [322, 323] were recently calculated in the Fermi approximation. These results are a milestone in perturbative calculations and an important step towards precision calculations for FCC-ee. In particular, the $\alpha_S^3$ QED contribution translates to a shift of the muon lifetime of ($-9 \pm 1$) $\times 10^{-8}$ µs. With the current measurement of $\tau_\mu = 2.1969811 \pm 0.0000022$ µs, the new correction terms are almost two orders of magnitude smaller than the experimental uncertainty. Thus, an updated value of $G_F$ can only be extracted once the latter has been improved.
<table><caption>**Table 10** A few sample precision quantities of interest for the FCC-ee programme, their current and projected experimental uncertainties, and the required theory input for their *extraction* from the data. The last</caption><thead><tr><th>Quantity</th><th>Current precision</th><th>FCC-ee stat. (syst.) precision</th><th>Required theory input</th><th>Theory status as of today</th><th>Needed theory improvement<sup>a</sup></th></tr></thead><tbody><tr><td>m<sub>Z</sub> (MeV)</td><td>2.0</td><td>0.004 (0.1)</td><td>Non-resonant e<sup>+</sup>e<sup>-</sup> → f<span style="text-decoration: overline">f</span>, initial-state radiation (ISR)</td><td>NLO, ISR logarithms up to 6th order</td><td>NNLO for e<sup>+</sup>e<sup>-</sup> → f<span style="text-decoration: overline">f</span></td></tr><tr><td>Γ<sub>Z</sub> (MeV)</td><td>2.3</td><td>0.004 (0.012)</td><td></td><td></td><td></td></tr><tr><td>sin<sup>2</sup> θ<sub>eff</sub><sup>l</sup></td><td>1.6 × 10<sup>-4</sup></td><td>1.2 (1.2) × 10<sup>-6</sup></td><td></td><td></td><td></td></tr><tr><td>m<sub>W</sub> (MeV)</td><td>9.9</td><td>0.18 (0.16)</td><td>Lineshape of e<sup>+</sup>e<sup>-</sup> → NLO (e<sup>+</sup>e<sup>-</sup> → 4f or WW near threshold → EFT framework)</td><td>NNLO for e<sup>+</sup>e<sup>-</sup> → WW, W → f<span style="text-decoration: overline">f</span>' in EFT setup</td><td></td></tr><tr><td>HZZ coupling<sup>b</sup></td><td></td><td>0.1%</td><td>Cross section for e<sup>+</sup>e<sup>-</sup> → ZH</td><td>NLO EW plus partial NNLO QCD/EW</td><td>Full NNLO EW</td></tr><tr><td>m<sub>top</sub> (MeV)</td><td>290</td><td>4.2 (4.9)</td><td>Threshold scan e<sup>+</sup>e<sup>-</sup> → N<sup>3</sup>LO QCD, NNLO EW, resummations up to NNLL, O(30 MeV) scale uncert.</td><td>Matching fixed orders with resummations, merging with MC, α<sub>S</sub> (input)</td><td></td></tr></tbody></table>

two columns show the current state of the art for calculations of this theory input and higher-order calculations needed to reach the FCC-ee precision target. More details can be found in Ref. [40]

<sup>a</sup>The necessary theory calculations mentioned are a minimum baseline; additional partial higher-order contributions may also be required

<sup>b</sup>No absolute value for the HZZ coupling can be extracted from the LHC data without additional assumptions

<table><caption>**Table 11** Required theory calculations for the *prediction* of the listed precision quantities within the SM. These predictions are needed for comparison with the quantities extracted from data (Table 10), for the</caption><thead><tr><th>Quantity</th><th>Required theory input</th><th>Theory status as of today</th><th>Needed theory improvement<sup>a</sup></th></tr></thead><tbody><tr><td>Γ<sub>Z</sub></td><td>vertex corrections for Z → f<span style="text-decoration: overline">f</span></td><td>NNLO + partial higher orders</td><td>N<sup>3</sup>LO EW + partial higher orders</td></tr><tr><td>sin<sup>2</sup> θ<sub>eff</sub><sup>l</sup></td><td></td><td></td><td></td></tr><tr><td>m<sub>W</sub></td><td>SM corrections to the muon decay rate</td><td>NNLO + partial higher orders</td><td>N<sup>3</sup>LO EW + partial higher orders</td></tr></tbody></table>

purpose of testing the validity of the SM and probing BSM physics. More details in Ref. [40]

<sup>a</sup>The mentioned needed theory calculations are a minimum baseline; additional partial higher-order contributions may also be required

The Fermi constant $G_F$ is one of several fundamental parameters that need to be precisely determined in order to make quantitative predictions for electroweak precision observables (EWPOs) within the SM or concrete BSM realisations through data-theory comparisons. Other examples of such quantities include the strong coupling $\alpha_S$ and the top-quark mass $m_{top}$. Many of these parameters can be obtained from FCC-ee data with much reduced uncertainty compared to today (Table 2), but their extraction from the experimental measurements requires significant theory input (as shown in Sect. 3.2 for $\alpha_S$ and in the last row of Table 10 for $m_{top}$).

Similarly, the interpretation of cross section measurements requires the precise determination of the luminosity through measurements of small-angle Bhabha scattering and/or large-angle photon-pair production (Sect. 6.13). For this purpose, the rates for these processes need to be computed at least with NNLO QED corrections, logarithmically enhanced higher-order effects, and a careful treatment of virtual hadronic corrections (which appear at NLO for Bhabha scattering and NNLO for two-photon production) [324,325].

To achieve the precision goals outlined in Tables 10 and 11, significant improvements in calculation techniques for higher-order radiative corrections are required, including full two-loop corrections for $2 \to 2$ scattering processes, as well as decay processes with full three-loop corrections and approximate four-loop contributions in a large-mass expansion.

Recently, significant progress has been achieved in semi-numerical multi-loop calculation techniques based on dispersion relations [326,327] and on series solutions of differential equations [328–333]. These techniques have enabled the first calculations of electroweak two-loop corrections for the main Higgs boson production process at FCC-ee, $e^+e^- \to ZH$ [327, 334], and they are expected to be useful for calculations of NNLO corrections for other pair production processes as well.

The approach of Refs. [326,327] exploits dispersion relations and Feynman parameters to express one subloop of a two-loop diagram in terms of integrals with up to three variables. Then the second subloop becomes a simple one-loop integral, with well-known analytical results. For divergent diagrams, the singularities can be removed with systematically
constructed subtraction terms, which can also be evaluated analytically. These procedures lead to two- or three-dimensional finite numerical integrals that can be evaluated with good precision within minutes on a single CPU core for a single diagram class. The method does not use any reduction to master integrals, which saves computing resources.

Another powerful method for multi-loop integrals is the method of differential equations [335–339]. While it can be difficult to find analytical solutions to these equations, it was recently realised that solutions can be efficiently obtained to arbitrarily high precision, using deep series expansions [328,329,331–333]. By matching series about different expansion points, solutions can be constructed that consistently extend across thresholds and other singular points. To fully determine the differential equation solutions, boundary values are also needed, and can be evaluated analytically or numerically for special kinematic points (such as zero momentum or unphysical Euclidean momentum). This step is made particularly simple with the auxiliary flow method, which uses boundary conditions at infinity of an unphysical (auxiliary) variable, where they can be evaluated in terms of algebraic recurrence relations [330].

The series expansion approach is not only restricted to two-loop integrals, but is also a promising tool for three-loop SM corrections. It requires a reduction to master integrals, which is the computational bottleneck for this method. Recent developments in integral reduction methods [340–342] can help to push the envelope on this front. In addition, the integral reduction can be performed numerically with much lower computing resources, and the functional dependence on kinematic variables can be reconstructed with suitable expansions or interpolations. Progress was also made towards analytical solutions of differential equations for Feynman integrals, using functions beyond multiple polylogarithms [343,344].

For a reliable phenomenological description of $e^{+}e^{-} \rightarrow ZH$, one of the next steps is to combine the NNLO result with a calculation of the process $e^{+}e^{-} \rightarrow \text{H}\bar{\text{f}}\text{f}$, where the fermion pair may be off the Z resonance. These off-shell contributions are important because of the relatively large decay width of the Z boson. For the projected FCC-ee precision, it is sufficient to compute $e^{+}e^{-} \rightarrow \text{H}\bar{\text{f}}\text{f}$ at NLO, which can be straightforwardly accomplished with existing automated tools.

## 3.2 QCD precision calculations

This section addresses some of the main theoretical challenges regarding quantum chromodynamics (QCD) calculations on the path towards FCC-ee. The considerations below are inspired by previous reviews [345] and largely by discussions that took place at topical FCC-ee workshops [321,346].

### 3.2.1 QCD studies in $Z/\gamma^{*} \rightarrow \text{jets}$

The very large FCC-ee integrated luminosities at the Z boson pole and at higher energies provide an excellent opportunity to expand the knowledge of strong interactions in QCD final states, both in the area of jet physics and for the extraction of the strong coupling constant, $\alpha_S(m_Z^2)$, from hadronic observables in $Z/\gamma^* \rightarrow \text{jets}$ events.

#### Jet physics and shape observables

Fine details of QCD final states can be investigated through a range of jet observables such as event shapes (designed to describe the geometric properties of hadronic events), jet rates, and jet substructure observables.

Owing to their sensitivity to QCD radiation, these observables are widely used to extract $\alpha_S$ [100], calibrate non-perturbative hadronisation models [347], or study QCD dynamics within jets [348]. Fully differential calculations for the process $e^{+}e^{-} \rightarrow Z/\gamma^{*} \rightarrow q\bar{q} + X$ at N$^3$LO in QCD ($\alpha_S^3$) for massless partons in the final state can be derived starting from the results of Refs. [349–353] with the inclusive cross section at N$^3$LO [354]. Similarly, the production of heavy (notably bottom) quarks, $e^{+}e^{-} \rightarrow Z/\gamma^{*} \rightarrow Q\bar{Q} + X$, can be described at NNLO in QCD using the predictions of Refs. [355–358]. For higher jet multiplicities, the computation of QCD radiative corrections with massless final-state partons has been extended to NLO for $e^{+}e^{-} \rightarrow Z/\gamma^{*} \rightarrow n$ jets with $n = 5$ [359] and $n = 6, 7$ [360].

The essential NNLO QCD calculations for final states with four or five jets are beyond the current state of the art. With the recent progress in the calculation of the necessary two-loop scattering amplitudes (see, e.g., Refs. [361–363] for the five-point case), these NNLO predictions will arguably become available in the coming years. Future developments in the computation of such multi-scale amplitudes may benefit from novel computational techniques discussed in Sect. 3.1 and in Refs. [329,364–374]. A recent review of modern computational methods can be found in Refs. [321,375].

The perturbative description of kinematic regimes that require the all-order resummation of radiative corrections has also improved substantially in the past decade, and the state-of-the-art calculations for standard global event shapes and jet rates in $e^{+}e^{-} \rightarrow Z/\gamma^{*} \rightarrow q\bar{q} + X$ have reached NNLL order and beyond [376–394]. Resummations for multijet final states are desirable for QCD phenomenology at FCC-ee, while currently only a limited number of predictions is available
beyond the NLL order for $e^{+}e^{-} \to Z/\gamma^{*} \to q\bar{q}g + X$ observables [395–398]. Further progress is therefore necessary in this area, requiring dedicated analytical and numerical resummation techniques. Furthermore, the computation of non-global observables [399,400], sensitive to the geometric pattern of soft QCD radiation, has recently been pushed to the NNLL order [401–404], and a number of dedicated phenomenological applications to $e^{+}e^{-}$ collisions are becoming available [400, 402, 404–411]. Computational techniques for jet substructure observables have also witnessed outstanding progress in the last decade, resulting in an array of applications at lepton colliders to the study of observables measured on groomed jets [412–415] or the study of fragmentation and spin correlations in jet physics [416–422].

## Measurements of the strong coupling constant

Among the main challenges to match FCC-ee experimental accuracy is the reduction of the uncertainties of the QCD parameters, and notably the strong coupling constant, $\alpha_S(m_Z^2)$ [100]. The current world average [35], which reaches a ~0.8% uncertainty ($\alpha_S(m_Z^2) = 0.1180 \pm 0.0009$), is mainly constrained by the precision of lattice calculations [423], expected to be improved by a factor of two in the next decade [424,425].

At FCC-ee, a precision of 0.1% on $\alpha_S(m_Z^2)$ can be contemplated (Table 2). The most precise determination comes from a combined fit of three EW pseudo-observables at the Z pole: the Z boson hadronic partial width, the total hadronic cross section at the resonance peak, and the ratio of hadronic to leptonic branching fractions. These inclusive quantities are particularly suitable to extract $\alpha_S$ given the small non-perturbative hadronisation corrections, which scale with the centre-of-mass energy $Q$ as $(\Lambda_{QCD}/Q)^6$ [426]. With the $6 \times 10^{12}$ Z bosons produced at FCC-ee, the total experimental uncertainty in $\alpha_S$ extractions from fits of the above quantities is of the order of 0.1% [427], hence requiring a substantial reduction of the corresponding theoretical uncertainties. The status of theory computations for these observables is well-advanced: QCD corrections are known up to $N^4$LO [428–430] and $N^3$LO corrections for massive bottom quarks are also known in a power series in $m_b^2/Q^2$ [358]. On the other hand, EW and mixed QCD-EW corrections are available at least up to two loops (see, e.g., Refs. [431–433] and references therein), as discussed in Sect. 3.1 of this report.

Other very accurate extractions of $\alpha_S$ at FCC-ee can be derived from $\tau$ [434,435] and W [427,436] hadronic and leptonic decays, using high-order perturbative QCD computations. Existing studies on the $\alpha_S$ extraction from the decay of EW bosons indicate that the calculation of higher-order corrections<fnref n="2" /> up to $O(\alpha_S^5)$, $O(\alpha_S^3)$, and $O(\alpha_S, \alpha^3)$ or $O(\alpha_S^2, \alpha^2)$ for QCD, EW, and QCD $\oplus$ EW, respectively, are needed to match the expected statistical experimental precision on inclusive W and Z hadronic observables [427]. In the case of $\tau$ decays, open theoretical questions are related to the treatment of non-perturbative effects [440–442], as well as to the difference between extractions relying on contour-improved (CIPT) [443,444] and fixed-order (FOPT) perturbative calculations adopted in the fits [445,446]. Recent investigations indicate that CIPT calculations might require a more robust estimate of non-perturbative corrections [445,447,448], with potential ways forward discussed in Refs. [449,450]. Given these open questions, $\alpha_S$ fits based on CIPT have been excluded from the latest world average [35]. A deeper understanding of these aspects is necessary for robust determinations of $\alpha_S$ from $\tau$ decays at FCC-ee.

The sensitivity to $\alpha_S$ of differential observables, e.g., in the final states discussed in the previous section, makes them also suitable for precise extractions of the strong coupling. Currently, different $\alpha_S$ determinations from these observables are included in the world average [376–378,451–465], and can differ from each other by up to a few standard deviations. Besides the high-accuracy perturbative ingredients described above, such fits require input on hadronisation effects. Non-perturbative radiation induces $O(\Lambda_{QCD}^p/Q^p)$ changes in the observables (with typically $p=1$) that must be estimated to achieve the desired precision. Aside from the different observables considered in the fit, a central difference between these different $\alpha_S$ determinations lies in how hadronisation corrections are evaluated (with either Monte Carlo models or analytic techniques) [426,466–477]. The yet suboptimal internal consistency of $\alpha_S$ determinations in $e^{+}e^{-}$ jet observables hints at unsatisfactory modelling of non-perturbative QCD dynamics, which poses a major bottleneck for precision phenomenology at FCC-ee and requires substantial improvements in our understanding of this kinematic regime. First innovative steps towards this ambitious goal are being taken within a dispersive model of the QCD coupling, which has recently been used to extract the leading non-perturbative scaling in three-jet final states [478–481].

Data from FCC-ee will be very beneficial for deepening the understanding of hadronisation corrections. On the one hand, energies higher than those of previous lepton colliders would arguably justify the further development of analytic models based on a power expansion in $\Lambda_{QCD}/Q$. On the other hand, the energy span and experimental accuracy of FCC-ee are instrumental in gaining better control of non-perturbative dynamics in MC generators, which will be beneficial in all measurements expected at FCC-ee, such as $e^{+}e^{-} \to t\bar{t}$, $e^{+}e^{-} \to W^{+}W^{-}$, and $e^{+}e^{-} \to ZH$. A complementary approach is given by the design of observables with reduced sensitivity to hadronisation, e.g., through jet-substructure techniques [412–415], which open

<footnote n="2">² State-of-the-art calculations can be found in, e.g., Refs. [428,429,432,437–439].</footnote>
promising avenues for complementary extractions of $\alpha_S$ [482]. An in-depth study of the effectiveness of these techniques at FCC-ee energies, as well as the estimate of the remaining hadronisation corrections, are highly desirable in the coming years [482,483].

### 3.2.2 QCD aspects of Higgs physics

Reaching theoretical uncertainties aligned with the projections of Table 3 for Higgs precision studies requires dedicated developments in different areas of both QCD and EW calculations. While EW corrections are discussed in Sect. 3.1, this section focuses on QCD aspects. The clean experimental conditions at FCC-ee allow a detailed and exclusive study of the hadronic decays of the Higgs boson. Partial widths are currently theoretically known at the per-cent level, with a parametric uncertainty on $\alpha_S(m_Z^2)$ that will be significantly reduced at FCC-ee with the expected 0.1% precision achieved in this parameter. More specifically, in the case of $H \to b\bar{b}$, N$^4$LO QCD corrections are known in the limit of massless bottom quarks [430,484,485], and N$^4$LO QCD corrections to $H \to gg$ have been computed in the heavy-top-mass limit [430]. The simulation of Higgs decays is paramount both for correcting the fiducial acceptance of the experiments and for studying kinematic distributions of the decay products and jet observables. These distributions are sensitive to quark Yukawa couplings [486,487] or to new physics states [172,176,488–490]. The sensitivity to light-quark Yukawa couplings can be enhanced by means of modern quark and gluon tagging techniques [491].

A key application is the extraction of the strange-quark Yukawa coupling, for which preliminary studies have shown promising results (Sect. 4.5.1). Among the challenges in the realisation of this measurement, a relevant theory bottleneck is the separation of the $H \to s\bar{s}$ decay from the Dalitz decay of a Higgs to a pair of either gluons (QCD mediated) or photons (EW mediated) followed by a gluon/photon splitting into strange quarks. Initial investigations [492] indicate that this background can be drastically suppressed by a cut in the invariant mass of the pair of jets originating from the fragmentation of the two strange quarks, $m_{j_1j_2} \gtrsim 100$ GeV. A robust theoretical control of the $H \to s\bar{s}$ signal in this region requires accurate perturbative calculations as well as the resummation of logarithmic corrections stemming from soft-gluon radiation off the final-state strange quarks near the Higgs mass threshold, $m_{j_1j_2} \sim m_H$. Together with advances in perturbative calculations, a second essential element in this endeavour is the development of improved hadronisation models to distinguish the non-perturbative fragmentation of (strange) quarks from that of gluons. Such models are instrumental for the reliable training of jet taggers, and their calibration within future Monte Carlo generators will highly benefit from the precise QCD data collected at FCC-ee.

Recently, considerable steps have been taken in the calculation of theoretical predictions for Higgs decays, which are essential for the implementation of the above ideas. The predicted kinematic distributions of the $H \to b\bar{b}$ decay products are known up to N$^3$LO in the limit of massless b quarks [493–498] and NNLO (and partially beyond) mass corrections are available [499–506]. Similarly, differential QCD predictions for $H \to gg$ are now available up to N$^3$LO in the large-top-mass limit [507]. Finite quark mass corrections to $H \to gg$ are relevant at the level of precision foreseen at FCC-ee and could be included up to NNLO in QCD in the near future with state-of-the-art calculations [500,508–514]. Calculations of hadronic event shapes and jet resolutions at NLO [515–517] and NNLO [518] in QCD have also been performed in recent years, including the treatment of mass logarithms in the relevant virtual amplitudes [519–527] and of Sudakov logarithms in event-shape distributions [528]. Similar to Z boson decays, the relatively low-energy scale involved in the Higgs boson decays ($\sim m_H$) implies that non-perturbative QCD effects have a sizeable impact on most differential distributions of hadronic Higgs decay products. Therefore, the considerations made in Sect. 3.2.1 apply here as well. The FCC-ee Higgs programme will benefit enormously from future developments in the modelling of hadronisation effects.

### 3.2.3 QCD modelling of the top-quark threshold

The properties of top quark, such as its mass, width, and EW couplings (Table 2) are scheduled to be measured at FCC-ee with runs at centre-of-mass energies between 340 and 365 GeV. The top mass can be extracted with high precision at FCC-ee through a threshold scan, where the top-quark pair is non-relativistic. Suitable definitions of short-distance top-mass schemes unaffected by ambiguities related to infrared physics are available [529–533], and can be exploited for precise predictions of the $\sigma_{t\bar{t}}$ vs. $\sqrt{s}$ lineshape. Although NNLO and N$^3$LO fixed-order QCD calculations are available [534,535], $\sigma_{t\bar{t}}$ receives a substantial contribution from Coulomb-type interactions in this non-relativistic kinematic regime. These effects can be accurately described in the context of effective field theories derived from non-relativistic QCD [536–538], valid when the top quark velocity is of order $\alpha_S$. A lot of effort has been devoted to computing predictions in this framework, including
QCD effects up to N³LO [539] (see also Refs. [540,541] and references therein), approximate NNLL renormalisation-group improved corrections [542], and the inclusion of EW effects within an analogous EFT framework [543].

The description of the final state $W^+W^-b\bar{b} + X$ requires the inclusion of non-resonant channels that do not involve the creation of a pair of on-shell top quarks. These non-resonant channels critically demand embedding the aforementioned non-relativistic EFT into the unstable particle EFT [544,545], where current predictions reach NNLO accuracy for the non-resonant part [543]. Projections for FCC-ee quote an expected theoretical accuracy for the top mass of $\mathcal{O}(50)$ MeV [543] in the potential-subtracted top-mass scheme [532] (see also Refs. [546,547]), with $\mathcal{O}(30)$ MeV scale uncertainties [77,548]. Improving further on these results represents a formidable challenge for the field of precision calculations, far beyond the current state of the art. The main steps include the computation of N⁴LO corrections in the non-relativistic EFT framework, as well as the description of QED effects at NNLL both in the collinear limit (e.g., ISR [549,550]) and in the soft limit (as discussed in Ref. [551]). Moreover, the optimal exploitation of the FCC-ee measurements might also require the N³LO calculation for the non-resonant channels.

The theoretical description of differential distributions is less accurate than that of the inclusive quantities just discussed, and reaches either NLO or NNLO accuracy only for specific observables [552,553]. Further progress is needed in these computations, which are central to controlling precisely the effect of experimental cuts. Some aspects of these calculations pose considerable theoretical challenges, for instance, concerning the differential calculations in the non-relativistic limit, or the assessment of non-factorisable radiative corrections to the decays of the two top quarks [554].

### 3.3 Monte Carlo event generators

Among the theoretical developments necessary for the FCC-ee physics programme, the area of Monte Carlo (MC) event generators (see, e.g., Ref. [555] for a review of the current state of the art) plays a pivotal role. These tools are instrumental not only for the accurate simulation of QCD and QED effects, but also for the calibration of the detectors as well as the training of analysis tools. The precision expected at FCC-ee requires a significant improvement in MC generators, with respect to their previous generation, far exceeding the current state of the art. The following paragraphs summarise some of the corresponding challenges, regarding QCD and EW (notably QED) aspects.

#### 3.3.1 QCD aspects

Monte Carlo generators simulate QCD corrections in three stages: the hard scattering at high momentum transfer; the parton shower stage, in which the system evolves towards low momentum scales; and the non-perturbative stage, which implements the transition of final state partons into hadrons. A first area where substantial improvement is necessary concerns the formulation of parton shower algorithms, given that current public tools are generally limited to leading logarithmic (LL) accuracy. The precision goals of FCC-ee arguably demand NNLL accuracy or beyond, which is currently the subject of widespread investigations within the community. Specifically, state-of-the-art algorithms achieving NLL accuracy for broad ranges of observables have recently been formulated [419,556-568] with novel techniques that help bridge the field of parton showers with QCD resummations [556-559,561,562,564]. Significant progress has also been made in the formulation of amplitude-level evolution [569,570], which would ultimately allow a systematic treatment of soft quantum interference effects in parton showers. Beyond NLL, several conceptual and technical problems become relevant, spanning from the inclusion of higher-order matrix elements [571-576], to the formulation of consistent schemes for the treatment of virtual corrections in the soft and collinear limits [388,422,576-579], and to the realisation of matching schemes to NLO matrix element corrections that preserve the shower accuracy [580]. The significant progress in the understanding of these aspects has recently resulted in the first class of NNLL parton-shower algorithms [581] capable of achieving this perturbative accuracy for event-shape observables at lepton colliders. This progress suggests that fully general NNLL parton showers may become available for phenomenology applications in the coming years.

A second aspect with much scope for improvement is the matching of parton showers to higher-order calculations for the hard scattering. Presently, an array of techniques allows NLO [582-587] or NNLO [588-592] QCD calculations (and possibly beyond [593]) to be matched to LL parton showers. These techniques have been applied to the differential simulation of Higgs boson decays [594,595], but will arguably need to be revisited in light of the recent progress in advancing the accuracy of parton showers [580].

Further necessary developments concern the accurate production of particle pairs at threshold (e.g., $t\bar{t}$ or $W^+W^-$), for which significant challenges arise from the inclusion of effects related to non-relativistic dynamics and unstable particle decay (Sect. 3.2.3) within event generators. Notable progress in the development of resonance-aware matching has been made
for top-pair production in hadronic collisions [553,596,597]. However, a full description of the hierarchy of scales involved in the process (top-quark velocity $v$, $m_{\text{top}}$, and width $\Gamma_{\text{top}}$) is not available in existing generators and requires significant conceptual advances.

A final area that requires significant improvement in view of the precision required at FCC-ee is the modelling of non-perturbative corrections, especially in the fragmentation of light partons and heavy quarks, central in the simulated performance of flavour tagging algorithms. Possible advances may come from a combination of generators with higher perturbative accuracy, as discussed above, and more versatile parametrisations and tuning of non-perturbative effects, for which the use of machine learning technology offers a promising alternative to current models [598–600]. The tuning of these models benefit from the ability to select large high-purity data samples enriched with specific flavours (such as gluons and b quarks), essential for the direct training of jet taggers on experimental data. Valuable experimental data for the calibration and testing of non-perturbative corrections could be collected from measurements of observables performed with hadronic invariant masses below the Z-boson resonance. Such measurements are accessible either via dedicated runs at lower $\sqrt{s}$ or from hard initial-state radiation away from the Z-pole run. Preliminary feasibility studies [601,602] indicate that $\mathcal{O}(10^9)$ events can be collected at various energy points over the 20–80 GeV hadronic mass range, enabling a variety of measurements useful for the development and testing of non-perturbative models.

Finally, the experimental accuracy expected in resonance production, such as $e^{+}e^{-} \rightarrow W^{+}W^{-} \rightarrow q\bar{q}'q''\bar{q}'''$ or $e^{+}e^{-} \rightarrow t\bar{t}$, will likely demand improved models of colour reconnection [603], which will be calibrated with the accurate data available for these processes.

### 3.3.2 QED aspects

The tools based on Yennie, Frautschi, and Suura (YFS) exponentiation [604] provide a promising framework for systematically improving the precision of QED (and QCD) radiative processes in MC generators. While the soft logarithms, arising from the real and virtual enhanced regions of phase space, are resummed to infinite order in the YFS formalism, the remaining collinear logarithms are not. These algorithms can be incorporated order by order, and the associated divergences can be regularised by including masses for all leptons. The YFS treatment is completely exclusive for multiple photon emission.

The KKMC event generator uses a variant of YFS exponentiation called Coherent Exclusive Exponentiation (CEEX). The most recent version, 5.00.2, of the KKMC package [605] includes improvements for the simulation of fermion pair production. Work is ongoing to include additional collinear contributions in the YFS framework [606], beam spread parametrisations, and better descriptions of tau decays [607].

YFS resummation has been implemented in the SHERPA-3 series in an automatic framework [608] for both the initial (ISR) and final (FSR) state QED radiation. Initial-final state interference (IFI) QED effects within the YFS framework are currently implemented in KKMC but not in SHERPA, for which they remain an essential target for future improvements, especially for the prediction of forward-backward asymmetries at FCC-ee. Using SHERPA's automated matrix element generators, the ISR resummation can be applied to any $e^{+}e^{-}$ process, while final state radiation is currently restricted to leptonic states only. The matching of this resummation to higher-order perturbative corrections is also automated within SHERPA, including the effects of collinear logarithms. The procedure renders finite the higher-order corrections, which by themselves can be infrared divergent. One-loop electroweak corrections are provided by automated tools such as RECOLA [609], while SHERPA automatically creates the YFS subtraction term such that the end result is infrared finite. The real corrections can also be included in a similar fashion using internal automated tools. Since the subtraction has been automated within the SHERPA framework, the current limitation on the perturbative accuracy is missing higher-order corrections, in particular multiloop calculations. However, as these calculations become available, it is relatively simple to include them within the YFS resummation framework.

These higher-order corrections can be provided by external electroweak libraries, which contain process-dependent multi-loop matrix elements and that can be interfaced with MC tools. The DIZET 6.45 [610] package contains most of the currently known higher-order corrections for Z-pole physics. For systematic extensions to higher levels of precision, however, a flexible object-oriented code library would be more suitable, which is the goal of the new GRIFFIN project [611]. Its object-oriented structure also simplifies the interfacing with MC generators, fitting tools, and other external programs.

The study of aspects related to ISR has been the subject of recent work [612]. An alternative approach to YFS factorisation for the description of ISR is based on collinear factorisation, as in Refs. [613–615]. Unlike what happens in the YFS framework, a systematic resummation of collinear logarithms is implemented, while soft corrections are only approximately described. Recent work has expanded this formulation to the NLL order [549,550], which is relevant for the precision targets of FCC-ee [615]. Such a formalism also offers a promising avenue for an alternative and independent simulation of QED radiation in parton showers.
## 3.4 Organisation and support of future activities to improve theoretical precision

As discussed in Sect. 3.1, the important targets for FCC precision physics have been defined. Theoretical progress is limited by several factors, ranging from the analytic control and classification of the integral structures due to appear at higher-loop order to the numerical challenges of integration or event generation. A rather diverse expertise is needed to address all the different aspects of the problem. Most calculations to improve the precision of EW and QCD observables at FCC-ee are therefore currently ongoing as part of efforts by individual groups, historically engaged in this activity. The physics groups of the FCC Feasibility Study are overseeing the coordination of the different efforts, via their regular meetings and dedicated workshops. The workshops are an opportunity to discuss and share information on the progress of the various groups, as well as to refine the definition of the precision targets and milestones. Recent examples of these workshops include ‘Precision EW and QCD Calculations for the FCC Studies: methods and tools’ [38], ‘Precision calculations for future $e^+e^-$ colliders: targets and tools’ [321], ‘Parton Showers for future $e^+e^-$ colliders’ [346], and, in 2024, ‘Frontiers in precision phenomenology: resummation, amplitudes, and subtraction’ [616].

The Future Colliders Unit at CERN has made resources available to host researchers engaged in FCC studies and, in particular, precision studies, either as long-term Scientific Associates (SASS) or as short-term visitors (STV). These visitors interact with the current CERN Theory Group staff and fellows, providing a constant presence of world experts on EW and QCD precision calculations.

Activities related to HL-LHC and its precision goals are intimately connected with FCC studies: the LHC precision needs are pushing both the fields of QCD, EW, and mixed QCD $\oplus$ EW calculations. On the QCD side, higher-order perturbative calculations develop technology that can be adapted to EW higher-loop cases, in addition to pushing the precision of QCD predictions for FCC-ee at the Z peak. A more profound understanding of non-perturbative effects of interest to the LHC (e.g., the study of leading and subleading power corrections) directly impacts the leading theoretical uncertainties ultimately affecting the precision of, for example, the extraction of $\alpha_S$ at FCC-ee. The sensitivity of LHC measurements to EW effects, furthermore, is pushing the development of higher-order EW calculations, with direct benefit for the pure EW programme of FCC-ee. As a consequence, all the efforts of CERN and of the wider community, dedicated to precision physics at the LHC must be seen as milestones towards the achievement of the FCC-ee precision goals.

In this context, a new initiative is being undertaken by the LHC Physics Centre at CERN (LPCC), with the creation of a global effort dedicated to coordinate and support studies in the domain of event generators and multi-loop calculations, also building on the success of the former MCnet network [617]. This activity, in addition to continuing the coordinated Monte Carlo development work established by MCnet – opening it to MC codes not covered by MCnet so far and preserving the student mentoring and educational activities – will support research to improve the projected computing performance of event generators and of multi-loop codes, exploring the opportunities offered by new hardware high-performance computing architectures (e.g., GPUs). The numerical complexity and CPU cost are recognised as a limiting factor in implementing the highest-available theoretical precision in event generators that can be used for realistic studies of the experimental data. These improvements are, therefore, a critical step towards the realistic study of FCC-ee detector performance in the context of precision observables.

To summarise: the theoretical field of precision studies for FCC-ee is multifaceted, touching on diverse challenges in both the formal and numerical fields and thus calling on diverse expertise. The structures that have been put in place during the FCC feasibility study allow these developments to be addressed and coordinated, engaging the most qualified and experienced researchers in the community. These steps towards the goal of matching theoretical and experimental uncertainties are important and promising, in order to optimally exploit the data from FCC-ee and, later, from FCC-hh. While it will be years before the necessary precision goals are attained, the efforts initiated during the period of the FCC feasibility study have began defining a clear roadmap. The approval of the FCC project will catalyse the engagement of the theory community. The past and ongoing successes in enhancing the precision of LHC predictions, beyond any previous expectation, give confidence in the feasibility of attaining the FCC theory precision targets.

# 4 Detector requirements

## 4.1 Introduction

The detector performance specifications required by the physics programme of a future electron-positron Higgs factory operating at the ZH production threshold and above, have been studied in the past for linear colliders. The different FCC-ee
experimental environment, on the one hand, and the rich FCC-ee electroweak, QCD, and flavour physics programme offered by the very large event samples anticipated at the Z resonance (the so-called ‘Tera-Z’ run), on the other, come with specific and entirely new challenges. The statistical uncertainties expected on key electroweak measurements, both at the Z peak and at the WW threshold, call for a superb control of the systematic uncertainties, and put commensurate demands on the acceptance, construction quality and stability of the detectors. The specific discovery potential of feebly coupled particles in the huge FCC-ee event samples should also be kept in mind when designing the detectors.

General considerations on the requirements for an FCC-ee detector were outlined in Refs. [36,54]. In this section, a few benchmark analyses [53] are selected and their sensitivity dependence on various aspects of the detector performance are studied. In some cases, these studies result in a quantified performance requirement. It is understood that the extremely broad range of FCC-ee measurements has not yet been completely covered, and that some important requirements may be missing. The analyses performed for this report make use of the simulation of the performance of the detector concepts considered so far and described in Sects. 6.2 (CLD), 6.3 (IDEA), and 6.4 (ALLEGRO). For the reader’s convenience, a brief account of these three concepts is given below, in Sect. 4.2; they are an important input to further optimisations and to the development of new concepts that might be better adapted to parts of the FCC-ee physics programme. As mentioned in Sect. 1, FCC-ee will accommodate four interaction points. A single experiment may not be perfect in all aspects; a configuration with 4 IPs allows a range of detector solutions that will cover all physics opportunities.

Most of the studies reported here have been performed using computer-generated events processed through a fast detector simulation performed with the DELPHES package. The chosen baseline is the IDEA detector [10] described in Sect. 6.3. More details about the DELPHES simulations used for the studies reported here can be found in Sect. 8.5. To investigate the impact of the detector performance, variations around this detector baseline have been studied, either by smearing the reconstructed objects at the analysis level or by producing dedicated event samples. For some tracker-related studies, the performance expected with the CLD detector [618] has also been looked at. The CLD main tracker consists of a set of layers of silicon sensors, in sharp contrast with the gaseous drift chamber of the IDEA detector. One study using a GEANT simulation of the ALLEGRO noble-liquid calorimeter has also been made. A few preliminary performance studies or physics analyses have been performed using a full simulation of the CLD detector, as shown in Sect. 4.10. After a brief overview of the detector benchmarks implemented in the simulations used for the studies reported here, the following sections address, in turn, the requirements on the various subdetectors.

## 4.2 Brief overview of the current detector concepts

Two detector concepts have been studied for FCC-ee at the time of the CDR: CLD, a consolidated option based on the detector design developed for CLIC, with a silicon tracker and a 3D-imaging highly-granular calorimeter; and IDEA, an innovative design developed specifically for FCC-ee, with a short-drift wire chamber and a dual-readout calorimeter. Since then, a third concept, ALLEGRO, has been proposed and is being developed, with an electromagnetic calorimeter based on a noble liquid. The three concepts considered so far have similar overall dimensions, with a length of 11–13 m and a height of 10–12 m. More details on these detector concepts are given in Sect. 6.

### 4.2.1 *The IDEA detector concept*

The tracking system of IDEA<fnref n="3" /> includes a silicon vertex detector (VXD) surrounded by a low mass drift chamber and an external layer of silicon micro-strip detectors. In the version used for the studies reported here, the VXD consists of a cylindrical ‘barrel’ section made of five single layers of sensors, at radii between $R = 1.2$ and $31.5$ cm, and of two ‘endcap’ sections (one on each side of the interaction point), each made of three disks of single layers of sensors. The drift chamber (DC) has a total length of 4 m and consists of 112 layers of wires, placed at radii between 35 and 200 cm. It allows ‘continuous tracking’ and particle identification, as will be shown below. The total amount of DC material crossed by a particle emitted at $90^\circ$ from the detector axis (defined by the bisector of the two beam axes) is 1.6% of a radiation length. The DC is surrounded by a silicon wrapper that provides one precise spatial point at the end of the particle trajectory and measures the time-of-arrival of charged particles. A finely-segmented, 20 cm deep, crystal electromagnetic calorimeter [619], surrounds the silicon wrapper and is placed inside a thin superconducting solenoid that provides a 2 T magnetic field parallel to the detector axis. Outside the coil, a dual readout (DR) fibre calorimeter completes the calorimeter system, resulting in an excellent energy resolution

<footnote n="3">³ Innovative detector for an electron-positron accelerator.</footnote>
for both electromagnetic and hadron showers. Finally, the muon system consists of layers of chambers based on μRWELL detectors embedded in the magnet return yoke.

### 4.2.2 The CLD detector concept

The CLD<fnref n="4" /> detector has been adapted to the FCC-ee specificities from the CLIC detector model; it features a silicon pixel vertex detector and a silicon tracker, followed by a highly granular calorimeter. For the simulations used here, the barrel section of the VXD consists of three double-layers of sensors, at radii ranging between 1.2 and 5.8cm, and two endcap sections, consisting of three disks each, built as double-layer devices. The silicon tracker is made of six barrel layers, at radii from 12.7cm to 2.1m, and of two sets of eleven endcap disks. The material budget for the tracker modules is estimated to be 1.1–2.1% of a radiation length per layer. The tracker is surrounded by a silicon-tungsten electromagnetic calorimeter (ECAL), which is 30cm or 22 X₀ deep, and by a scintillator-steel hadron calorimeter (HCAL), that extends up to R = 3.6m; the combined ECAL plus HCAL thickness corresponds to 6.5 interaction lengths. The very high granularity of the calorimeter makes it particularly well suited for ‘particle flow’ reconstruction techniques that, in CLD, are the key for reaching very good resolutions. A superconducting solenoid, delivering a 2T magnetic field, surrounds the calorimeter, at a radius of about 3.7m. The iron return yoke is instrumented with resistive plate chambers (RPC) that form the muon detection system. A detailed description of the CLD detector can be found in Ref. [618].

### 4.2.3 The ALLEGRO detector concept

The design of ALLEGRO<fnref n="5" /> is more recent than the IDEA and CLD designs. Its tracking system consists of a silicon vertex detector and a main tracker, which could either also be made of silicon sensors or be a gaseous detector. Surrounding the tracker is a high granularity noble-liquid ECAL, which could be made of lead (or tungsten) and liquid argon or, alternatively, of tungsten and liquid krypton. The magnet coil separates the ECAL from the high-granularity HCAL, which could be made of steel absorber plates interleaved with scintillator tiles, as envisaged for the CLD calorimeter. For the muon system, several options are under consideration.

## 4.3 Measurement of the tracks of charged particles

The detection of charged particles and the measurement of their momenta before they traverse a large amount of high density material is crucial for all physics analyses and for a successful particle-flow reconstruction [620]. This section deals with properties that are mostly related to the main tracker: the precise measurement of the track momentum and angles, and the reconstruction of highly-displaced vertices. The determination of the track impact parameters, which is largely provided by the vertex detector, is treated in Sect. 4.4. The measurement of quantities related to the ionisation energy per unit length, useful to identify the nature of a charged particle, is covered in Sect. 4.5.

### 4.3.1 Track momentum resolution: the measurement of the Higgs boson mass

The track momentum resolution is a key handle in many analyses. In particular, it directly impacts the reconstructed masses, which are often used to suppress backgrounds. Both IDEA and CLD tracker designs would offer an excellent track momentum resolution. For example, for 10 GeV (50 GeV) muons emitted at an angle of 90° with respect to the detector axis, the momentum resolution is about 0.5‰ (1.5‰) with the very light drift chamber of IDEA and about 2.5‰ (3‰) with the heavier full silicon tracker of CLD, the latter being dominated by the effect of multiple scattering. Some examples from B physics, showing how the exceptional momentum resolution of the IDEA tracker separates the signals from the backgrounds, are presented in Sects. 4.5.3 and 4.8.

In this section, the needs on the track momentum resolution are illustrated with the measurement of the Higgs boson mass, $m_H$, which needs to be known to better than 4 MeV (its intrinsic width), in view of a potential run at the Higgs resonance (Sect. 9.4).

This measurement, described in detail in Ref. [65], exploits the Higgs-strahlung process, i.e., the associated production of a Z and a Higgs boson. Since, at a lepton collider, the total energy and momentum of the final state are well known, the mass

<footnote n="4">⁴ CLIC-like detector.</footnote>

<footnote n="5">⁵ A lepton-lepton collider experiment with granular read-out.</footnote>
<figure><img src="image_56.png" /><figcaption>**Fig. 48** Left: fits to the distribution of the Higgs boson recoil mass in ZH events, where the Z decays into muons, assuming an ideal momentum resolution (red), such that the resolution on the recoil mass is determined by the beam energy spread, or the momentum resolution of the</figcaption></figure>

IDEA (black and blue) or CLD (green) detectors. The blue curve corresponds to a 3 T magnetic field, instead of the 2 T field used for the other distributions. Right: corresponding Higgs boson mass measurements

of the system that recoils against the Z, called recoil mass and denoted $m_{\text{recoil}}$, can be reconstructed exclusively considering the Z decay products, irrespective of the Higgs boson decay. The $m_{\text{recoil}}$ distribution exhibits a sharp peak at $m_H$, so that a fit to this distribution provides a precise measurement of the Higgs boson mass. Experimentally, the channel where the Z decays into a pair of muons offers the best resolution. A fit to the $m_{\text{recoil}}$ distribution for $Z(\mu\mu)H$ events at $\sqrt{s} = 240$ GeV is displayed in the left panel of Fig. 48, for various assumptions on the muon momentum resolution, while the right panel shows the result of a likelihood fit of the distribution of signal events in the presence of background, with the same assumptions. Even with a perfect measurement of the momentum of the muons, the width of the $m_{\text{recoil}}$ peak would be limited by the beam energy spread (BES), which amounts$^6$ to 0.185% of the beam energy at $\sqrt{s} = 240$ GeV, so that $m_H$ would be determined with an uncertainty of 3.95 MeV (including the contribution of other sub-dominant systematic uncertainties [65]), for an integrated luminosity of 10.8 ab$^{-1}$.

The decay muons, with typical momentum of $\mathcal{O}(50)$ GeV, should be measured with a momentum resolution better than the BES, so that the mass resolution be not limited by the momentum measurement.$^7$ This goal is achieved with the IDEA detector (black curves in Fig. 48), only considering the muon decay channel. The full silicon tracker of CLD, in its current implementation, performs less well (green curves) because of the larger amount of material, which leads to increased multiple scattering, the factor that dominates the muon momentum resolution, even for the relatively high momentum range of interest. However, the goal of 4 MeV on the measured Higgs boson mass would most likely be achieved by combining the muon and electron decay channels (Sect. 4.6). The blue curves in Fig. 48 show what would be expected, with the IDEA tracker, if the detector magnetic field were increased from 2 to 3T;$^8$ the result is a 33% improvement of the momentum resolution and a 14% improvement on the total mass uncertainty.

<footnote>$^6$ The beam energy spread values used for the studies reported here are taken from Ref. [621].</footnote>

<footnote>$^7$ Having a momentum resolution for $\mathcal{O}(50)$ GeV muons better than or comparable to the BES is also an important requirement in the search for $Z \to \tau\mu$ lepton flavour violating decays, at $\sqrt{s} = 91$ GeV. The analysis strategy requires a clear tau decay in one hemisphere and a beam-energy muon in the other, in order to suppress the $Z \to \tau\tau$ background [198].</footnote>

<footnote>$^8$ For the highest luminosity runs at the Z peak, the magnetic field of the detector is constrained to not exceed about 2 T, as explained in Sect. 5. At $\sqrt{s} = 240$ GeV, the blow-up of the beam emittance that results from a higher field is negligible and running, for example, with a 3 T field, which would provide a better track momentum resolution, is not ruled out.</footnote>
<figure><img src="image_57.png" /><figcaption>**Fig. 49** Statistical uncertainty in the peak position of the measured dimuon invariant mass distribution, expected with the full FCC-ee event sample of the Z lineshape scan (125 ab⁻¹ at the Z peak and 40 ab⁻¹ at each off-peak energy). This uncertainty is shown assuming an ideal detector resolution (leftmost bars), the resolution of the IDEA tracker (middle bars), and that of the CLD tracker (rightmost bars)</figcaption></figure>

### 4.3.2 Track momentum resolution: the Z width and the stability of the momentum scale

The point-to-point uncertainty on the centre-of-mass energy in the scan of the Z lineshape is, together with the knowledge of the BES and the point-to-point uncertainty on the integrated luminosity, a dominant source of systematic uncertainty in the Z width measurement. The reconstructed peak position of the dimuon invariant mass distribution in $e^{+}e^{-} \rightarrow \mu^{+}\mu^{-}$ events provides a measurement of the centre-of-mass energy. As proposed in Ref. [22] and developed in Ref. [622], the aforementioned uncertainty can be assessed from the difference in this reconstructed peak position between the energy points used in the lineshape scan. More details about the method are given in Sect. 9.2. Figure 49 shows the statistical uncertainty with which the peak position is expected to be determined at $\sqrt{s} = 87.9, 91.2$, and $94.3$ GeV, with the full FCC-ee event sample. With the track momentum resolution provided by the IDEA tracker, this precision reaches 20 keV for the two off-peak energies, such that the difference in the collision energy at these two points can be measured with an uncertainty of about 28 keV, leading to an 11 keV uncertainty on the extracted Z width. With the resolution offered by the CLD tracker, this difference is measured with two times larger uncertainty, resulting in an uncertainty of 22 keV in the Z width.

Such a precision, however, requires that the scale of the momentum measurements (and, in particular, of the magnetic field) be stable at the level of a few $10^{-7}$ (20 keV over $\sqrt{s}$, to ensure a 20 keV uncertainty on the peak position) or, at least, that its variations be monitored at that level. A monitoring at this level of precision may be difficult to achieve with magnetic NMR probes. However, the large samples of well-known resonances, in particular of $K_S$ decaying into $\pi^{+}\pi^{-}$, provide an in-situ monitoring at this exceptional precision. An efficient and high purity algorithm that reconstructs $K_S \rightarrow \pi^{+}\pi^{-}$ decays is presented below. It allows a $K_S$ to be reconstructed in about every second Z event, when the Z decays hadronically. With a mass resolution better than 400 keV with the IDEA detector, the position of the $K_S$ mass peak can be determined with a relative uncertainty of $2.3 \times 10^{-9}$ with an integrated luminosity of $40 \text{ ab}^{-1}$ at $\sqrt{s} = 87.9$ GeV (and even better at $\sqrt{s} = 94.3$ GeV, given the larger cross section). This dataset could, hence, be split into, e.g., 100 subsamples in time, and the $K_S$ resonance be reconstructed in 100 bins, to ensure monitoring of the scale stability at the required level of $2.3 \times 10^{-7}$.

### 4.3.3 Angular resolutions

The polar and azimuthal angles of the momentum of a prompt charged particle, at its production vertex, are given by the angles of the corresponding track at its distance of closest approach to the detector axis. The angular resolutions in the two detector concepts considered so far vary between about 20 µrad for high momentum particles produced in the central region of the detector and a few mrad for soft forward particles [618,623]. The contribution of these angular resolutions to the uncertainty in the recoil mass shown above is negligible compared to that of the momentum resolution. For the reconstruction of the mass of heavy-flavoured hadrons, the momenta of the daughter particles must be taken at the decay vertex of the hadron and the resolution of the azimuthal angle crucially relies on the reconstruction of this decay vertex (see Sect. 4.4). Nevertheless, for all examples considered involving B mesons (see, e.g., Sects. 4.5.3 and 4.8), the mass resolution remains completely dominated by the momentum resolution.
<figure><img src="image_58.png" /><figcaption>**Fig. 50** The $K_S$ reconstruction efficiency for kaons emitted in $B^+ \to (K_S\pi^0)_D K^+$ decays, as a function of the distance $L_{xyz}$ between the $K_S$ decay vertex and the interaction point, for several thresholds on the $K_S$ mass candidate, for the IDEA (left) and CLD (right) detector concepts</figcaption></figure>

A precise determination of the polar and azimuthal angles is crucial for exploiting the over-constrained kinematics of many processes in $e^+e^-$ collisions. That is the case, in particular, of muon pair production. As shown in Ref. [22], for $e^+e^- \to \mu^+\mu^-(\gamma)$ dimuon events, the crossing angle and the longitudinal momentum imbalance can be reconstructed event-by-event from the sole measurement of the polar and azimuthal angles of both muons. The BES, which is a crucial ingredient in the extraction of the Z width at Tera-Z, can then be derived from the width of the longitudinal momentum imbalance distribution. To ensure that the BES uncertainty has a negligible effect on the extracted Z width, muon tracks from Z decays must be measured with an angular resolution of 0.1 mrad or better, a requirement fulfilled [618,623] by the detector concepts presented in the CDR. Moreover, the mean of the longitudinal momentum imbalance distribution provides the longitudinal boost. Its determination constrains the energy losses of the beams in their separate rings [47] and, consequently, has a crucial influence on the precision of the Z and W mass measurements.

Another example that exploits the angles of dimuon events is the determination of the centre-of-mass energy well above the WW threshold, where the resonant depolarisation method cannot be applied. In particular, at $\sqrt{s} = 240$ GeV, the centre-of-mass energy needs to be determined to $O(1)$ MeV in order not to spoil the Higgs boson mass determination from the recoil mass. The over-constrained kinematics of radiative return events $e^+e^- \to Z(\gamma)$ with $Z \to \mu^+\mu^-$ (accompanied by an initial-state radiated photon along the direction of one of the beams) allows the determination of $\sqrt{s}$ from the precise knowledge of the Z mass and the muon angles alone. The target precision for the $\sqrt{s}$ measurement may set tighter requirements on the angular resolutions than those given above.

### 4.3.4 Number of tracker layers and highly-displaced vertices

The reconstruction of long-lived particles (LLPs) that decay into charged particles within the tracker volume after having crossed the innermost layers of the vertex detector requires that the main tracker be able to efficiently identify such highly-displaced vertices. Typical use cases are the searches for LLPs predicted in many models that extend the Standard Model (Sect. 2.3 and Refs. [129,177]) or the reconstruction of decays that involve $K_S$ or $\Lambda$ hadrons.

As an example, the case of $K_S$ mesons produced in $B^+ \to K^+D^0$ decays, followed by the decay of the $D^0$ meson into a $K_S\pi^0$ pair, is considered here. The reconstruction of the $K_S \to \pi^+\pi^-$ decay is detailed in Ref. [275]. The $K_S$ candidates are built from pairs of opposite-charge particle tracks that do not come from the primary vertex and that can be fitted to a common vertex, with a reconstructed mass close to the nominal $K_S$ mass.

The $K_S$ reconstruction efficiency obtained with the IDEA detector is shown in Fig. 50 (left), as a function of the distance between the $K_S$ decay vertex and the interaction point, $L_{xyz}$. The efficiency is defined for the $K_S$ mesons that come from $B^+ \to (K_S\pi^0)_D K^+$ decays for which the daughter pions satisfy the acceptance requirement $|\cos\theta| < 0.95$. Within a mass window of $\pm 5$ MeV, the efficiency remains higher than 80%, as long as the $K_S$ meson decays within 1.5 m from the interaction point.<fnref n="9" /> At larger distances, the efficiency drops because the pion tracks only traverse a small fraction of the tracker. The right panel of Fig. 50 shows the corresponding efficiency for events processed through a DELPHES simulation of the CLD detector. As expected, the performance is much worse than that of IDEA. Steps corresponding to the positions of the tracker layers are

<footnote n="9">⁹ The drop in efficiency observed at small flight distances is due to the correlation of the flight distance with the $K_S$ momentum: the pion tracks from $K_S$ mesons that decay close to the IP are softer and more affected by multiple scattering or may lead to curling tracks ('loopers').</footnote>
<figure><img src="image_59.png" /><figcaption>**Fig. 51** Tracking efficiency evaluated with full simulation of the CLD and IDEA detectors using the ML-based approaches, as a function of $p_T$ (left) and of the angle $\Delta_{MC}$ to the closest generated charged particle (right)</figcaption></figure>

clearly visible. For example, since there are only four barrel layers at a radial distance larger than 40 cm, the efficiency for selecting 5-hit tracks displaced by more than 40 cm vanishes in the central region, explaining the first step seen in the figure. Also, the $K_S$ mesons that decay within a few tens of centimetres from the IP have a lower detection efficiency than that of the IDEA drift chamber, given the larger amount of material of the full silicon tracker and the correspondingly larger multiple scattering and worse resolutions on the $K_S$ reconstructed vertex and mass.

While some optimisations could be made to the full silicon tracker considered here, it is clear that an efficient reconstruction of $K_S \to \pi^+\pi^-$ decays and, more generally, of long-lived particles that lead to late appearing tracks calls for a highly transparent tracker, a large tracking volume, and a considerable number of measurement layers.

### 4.3.5 Work ahead

#### Tracker acceptance and efficiencies

The search for the $B \to K^*\tau\tau$ rare decay, described in more detail in Sect. 4.4.2, sets requirements on the track efficiencies. Demanding that both taus decay into three prongs is particularly useful for this analysis and leads to a final state with six soft pion tracks. A momentum acceptance down to 100–150 MeV is necessary to maintain a good signal efficiency. With a magnetic field of 2 T, the requirement of a minimum number of, e.g., 5 hits to reconstruct a track, implies that there be at least five measurement layers at a radius smaller than 33 cm (50 cm) to reconstruct a track with a transverse momentum of 100 MeV (150 MeV). This requirement on the position of the layers of the vertex detector and of the innermost layers of the main tracker is fulfilled by both the IDEA and the CLD tracker designs [618,623]. As an illustration, the tracking efficiency as a function of transverse momentum is shown in Fig. 51 (left), for tracks reconstructed with a novel machine-learning algorithm, discussed in Sect. 4.10. In addition, the track reconstruction efficiency as a function of the distance to the closest charged hadron in a jet is shown in Fig. 51 (right). It indicates that CLD features a slightly larger efficiency in dense hadronic environments, possibly due to a better separation capability of two close-by tracks, driven by the superior single-point spatial resolution provided by silicon sensors.

The measurement of the luminosity from $e^+e^- \to \gamma\gamma$ events (Sect. 4.6), to a precision of about $10^{-5}$, requires a very precise knowledge of the large background from $e^+e^- \to e^+e^-$ events. The targeted precision will set a requirement on the $e/\gamma$ separation that, in turn, will constrain the tracker inefficiency and the precision with which this inefficiency is known. In addition, the ratio of the partial width of the Z boson decay into hadrons to that into muons, $R_\mu$, will be measured with a statistical precision of about $5 \times 10^{-6}$ at Tera-Z. Counting the number of $\mu^+\mu^-$ events with a similar level of systematic uncertainty sets a very challenging requirement on the knowledge of the tracking (and muon chamber) efficiency.

#### Track angles

The angular resolutions provided by the current tracker designs comply, with a good safety margin, with the requirements set by the studies done so far, as shown above. Further studies are needed to check if other measurements set tighter requirements. Beside the measurement of $\sqrt{s}$ from the radiative return events previously mentioned, methods that are being developed to precisely determine dilepton acceptance in-situ (Sect. 4.6.3) may call for better angular resolutions.
### 4.3.6 Preliminary conclusions

The performance of a gaseous tracker and of a full silicon tracker have been shown and quantified with several examples.

- Having a large number of measurement points along the tracks, as offered by a gaseous tracker, is crucial for an efficient reconstruction of $K_S$ and $\Lambda$ hadrons or other long-lived particles that decay into charged particles, and is a clear bonus for an experiment with a strong focus on flavour or BSM physics.

- The momentum resolution offered by both designs looks adequate for Higgs boson measurements. This statement probably holds as well for most electroweak measurements, with the notable exception of the Z width measurement. For flavour physics at the Z peak, where low momentum tracks are involved, a low mass gaseous tracker is advantageous since the momentum resolution is minimally affected by multiple scattering.

- The tracker volume, extending to a radius of about 2 m, may have to be reduced a little in order to free some space to accommodate a detector dedicated to charged-hadron particle identification (which may be needed in particular for the CLD option, see Sect. 4.5). A reduction by $\mathcal{O}(20)$ cm would degrade the momentum resolution by about 15%, as shown in Sect. 6.6. This effect might be partly compensated by reducing the amount of material in the CLD tracker layers.

## 4.4 Requirements for the vertex detector

The measurement of the track impact parameters is driven by the performance of the vertex detector, as it provides very precise spatial points in the tracker layers closest to the beams. A precise measurement of these parameters is crucial for reconstructing vertices, for efficient identification of heavy quarks and of taus, and for accurate lifetime measurements. The resolution on these parameters typically scales as

$$ \sigma(d_0) = a \oplus \frac{b}{p \sin^{3/2} \theta}, \qquad (6) $$

where the asymptotic term, $a$, is driven by the single hit resolution, while the second term represents the contribution of multiple scattering and depends on the material of the vertex detector layers and of the beam pipe. The radial distance of the first layer of the vertex detector is also crucial [491]. In the simulations used for the studies reported here, the radius of the beam pipe is 1 cm, that of the innermost layer of the vertex detector is 1.2 cm, and the material crossed by a particle emitted at a polar angle $\theta$ before the first VXD measurement corresponds to 0.67% / $\sin\theta$ of a radiation length.

### 4.4.1 Heavy-flavour tagging and the Higgs boson coupling to charm quarks

Previous flavour tagging studies made for linear colliders [624,625] concluded that $a \approx 5$ µm and $b \approx 15$ µm/GeV (in Eq. 6) are adequate for measuring the Higgs boson couplings to bottom and charm quarks. These requirements have been revisited in the context of the present study. In particular, major progress has been made in recent years in the field of flavour tagging algorithms. Machine-learning approaches, on one hand, and the ever increasing computing power, on the other, have significantly improved the performance of flavour tagging with respect to that of the algorithms that were the state-of-the-art a decade ago, so that more ambitious goals can be contemplated today. The algorithm developed for the studies reported here is based on PARTICLENET [626] and uses state-of-the-art jet representations and an advanced graph neural network architecture. This algorithm is referred to as PARTICLENETIDEA and is described in more detail in Ref. [491]. With this new tool, an efficient and pure identification of jets induced by gluons or even strange quarks (see Sect. 4.5.1) is available and the tagging efficiency of bottom and charm quarks is much larger than what was achieved with the previous generation of algorithms, for the same purity.

This algorithm is a key ingredient in the measurement of the Higgs boson couplings to bottom, charm, and strange quarks, as well as to gluons, as described in detail in Ref. [66]. The analysis exploits the ZH events at $\sqrt{s} = 240$ GeV and considers the channels where the Z boson decays to pairs of muons or electrons, to jets, or to neutrinos, the latter channel currently providing the best sensitivity. The probabilities that each jet originates from a bottom, charm, or strange quark, or from a gluon, are determined by the PARTICLENETIDEA algorithm and are used to categorise events into orthogonal categories, each enriched in one of the different Higgs boson decay modes and depleted in the others.

The branching ratios of the Higgs boson decays to $b\bar{b}$, $c\bar{c}$, $s\bar{s}$, and $gg$ pairs are measured in all categories with a simultaneous fit to the recoil mass distributions (and, for the $Z \to \nu\bar{\nu}$ channel, to the visible mass distributions).
<figure><img src="image_60.png" /><figcaption>**Fig. 52** Decay chain under study for a measurement of the $B \to K^*\tau\tau$ branching fraction</figcaption></figure>

The anticipated precision of the coupling measurements is summarised in Sect. 2.2. With an integrated luminosity of 10.8 ab⁻¹, the $H \to b\bar{b}$ and $H \to c\bar{c}$ event yields (cross section times branching ratio) can be measured with a precision of 0.21% and 1.66%, respectively.

The dependence of this precision on alternative beam pipe and vertex detector configurations has been studied and compared to the sensitivity obtained with the baseline IDEA detector design. Beam pipes with twice or half the nominal material budget have been studied. A vertex detector with the barrel layers shifted by 0.5 cm towards larger radii and an option where the heavier beam pipe is complemented by a single-point resolution degraded by a factor of 2 have also been considered. The measurement of the $H \to c\bar{c}$ and $H \to b\bar{b}$ branching fractions is marginally improved or degraded for the assumed better or worse scenarios. A transverse impact parameter resolution of 2–3 µm is achieved with the baseline IDEA vertex detector. A resolution worsened by a factor of 2 remains largely sufficient to identify displaced tracks originating from boosted D and B meson decays produced in $H \to b\bar{b}$ and $c\bar{c}$ decays and has, therefore, a small impact on the expected precision. A comprehensive discussion on the impact of vertex detector assumptions on the tagging performance can be found in Ref. [627].

### 4.4.2 Reconstruction of vertices and the measurement of the $B \to K^*\tau\tau$ branching fraction

With the current version of the IDEA detector, the primary vertex, for example in events where a Z decays to hadrons, is typically reconstructed with a resolution of 2–3 µm in the x and z coordinates, and of a few tens of nm in the y direction (driven by the r.m.s. of the beam position in the vertical direction, as shown in Table 14, Sect. 5), using a beamspot constraint in the vertex fit. For displaced vertices, the resolution depends very much on the track multiplicity of the vertex, on the track momenta and angles, and on the angular separation of the tracks. In the various B-physics processes that have been looked into so far, the resolution on the 3D distance between the interaction point and the displaced vertex typically varies between ~10 and ~80 µm. For example, in the $B_s \to D_s K$ decay, where the $D_s$ decays to $K K\pi$, the resolution of 15 µm obtained on the $B_s$ decay vertex is sufficient for time-dependent CP violation measurements, since it is $O(40)$ times smaller than the distance travelled by the $B_s$ during one oscillation period.

However, some processes of interest for flavour physics impose very strong demands on the reconstruction of vertices. So far, the most stringent requirements on the performance of the vertex detector come from the measurement of the branching ratio of the, yet unobserved, $B \to K^*\tau\tau$ decay. This branching fraction is predicted to be very small in the Standard Model, at the level of $O(10^{-7})$, and the current experimental upper limit is larger than this prediction by three orders of magnitude. New physics contributions can potentially result in a significant enhancement of this branching fraction. Consequently, its measurement is an important goal of the FCC-ee physics programme. A detailed analysis of the feasibility of this measurement at the Z pole is reported in Ref. [215], and is summarised below.

When both $\tau$ leptons decay into three charged pions and a neutrino, there are enough kinematic constraints to completely reconstruct the decay, as illustrated in Fig. 52. The $\tau$ decay vertices (TV) are obtained from their daughter pion tracks. The secondary vertex (SV) is reconstructed from the two tracks (the kaon and the pion) produced by the $K^*$ decay. The flight direction of each $\tau$ is then determined from the line that joins the SV to each corresponding TV. The component of the neutrino momentum that is transverse to this flight direction is obtained as the opposite of the transverse component of the momentum of the three charged pions. The $\tau$ mass constraint finally provides the remaining longitudinal component of the
**Fig. 53** Distribution of the mass of the B → K*ττ candidates after the full selection, assuming that the secondary and tertiary vertices can be reconstructed with a resolution of 5 µm (20 µm) in the transverse (longitudinal) direction. The normalisation corresponds to a total of 6 × 10¹² produced Z bosons

<figure><img src="image_61.png" /></figure>

**Fig. 54** Statistical uncertainty on the B → K*ττ branching fraction expected from an event sample corresponding to 6 × 10¹² Z bosons, as a function of the resolution of the position of the secondary and tertiary vertices, in the transverse direction. The blue star shows the precision expected with the version of the IDEA detector used in the nominal simulations. The additional star symbols show how the sensitivity improves in alternative simulations of the IDEA detector

<figure><img src="image_62.png" /><figcaption>Precision of BF measurement as function of the resolution</figcaption></figure>

neutrino momentum. Clearly, this reconstruction relies critically on the precise reconstruction of the secondary and tertiary vertices.

Several b → c̄cs and b → cτν transitions lead to final states that are similar to that of the signal, apart from the presence of additional neutrinos and neutral pions; a powerful π⁰ identification is needed to suppress these backgrounds. A π⁰ identification efficiency of 80% is assumed here. A multi-variate analysis allows the background to be reduced to a manageable, albeit still large, level. As an example, Fig. 53 shows the distribution of the mass of the B candidates for a case where the resolution of the position of the secondary and tertiary vertices is 20 µm in the longitudinal direction and 5 µm in the transverse direction.<fnref n="10" /> The remaining background under the B mass peak is dominated by irreducible background processes.

The numbers of signal (N) and background events are determined in the 5.0–6.0 GeV range of invariant mass from a maximum likelihood fit. The significance of the signal in this range is used as a conservative figure of merit of the reconstruction performance. This significance has been determined for several assumptions on the vertex resolution. It shows a very strong

<footnote n="10">¹⁰ For each tertiary or secondary vertex, the longitudinal direction is defined by the flight direction of the decaying particle.</footnote>
dependence on the resolution of the position of the secondary and tertiary vertices in the transverse direction,<fnref n="11" /> as can be seen from the black dots in Fig. 54. With an event sample corresponding to 6 × 10¹² produced Z bosons, a resolution better than ~4.3 µm is needed to see evidence (3σ) of the decay mode under study and it should be better than ~2.2 µm for the observation (5σ) of this decay. With the version of the IDEA detector that is used as a baseline for this report this resolution is only 5 µm, as shown by the blue star symbol in Fig. 54. To reach the evidence and observation thresholds, the resolution on the track impact parameters needs to be improved by about 10% and 40%, respectively, with respect to the nominal IDEA resolutions.<fnref n="12" />

In order to investigate if and how these improvements can be achieved, additional DELPHES signal samples have been produced, with an alternative tracker description, with less tracker material<fnref n="13" /> and an improved single hit resolution. The results are shown in Fig. 54 by the additional star symbols. An improvement of the single hit resolution in the VXD layers by 30% (i.e., from 3 to 2 µm in the barrel layers) brings the sensitivity beyond the 3 σ evidence threshold, as shown by the yellow symbol. This threshold is also reached with a 50% reduction of the VXD material, as shown by the magenta symbol. However, the impact of this sole improvement is limited by the material of the beam pipe, which causes charged particles to experience multiple scattering before they enter the detector. Indeed, in the nominal IDEA simulations used here, the material of the beam pipe is twice that of a VXD layer, so that to profit from the reduction in the material of the VXD layers the beam pipe transparency also needs to be increased. The green symbol in Fig. 54 shows that, by reducing by 50% the material of both the beam pipe and the VXD layers, the sensitivity increases to about 3.5 σ, under the SM hypothesis for the considered branching fraction. Additional handles will be exploited to increase the sensitivity beyond the 5 σ discovery level.

### 4.4.3 Vertex resolutions and heavy-flavour electroweak precision observables

The FCC-ee operation at the Z-pole offers an immense potential for the measurement of the heavy-quark electroweak precision observables, the partial decay-width ratios $R_{b(c)} = \Gamma(Z \to b\bar{b}(c\bar{c})) / \Gamma(Z \to \text{hadrons})$, and the heavy-quark forward-backward production asymmetries $A_{\text{FB}}^{b(c)}$. A huge improvement of the statistical uncertainty is expected, with respect to the LEP measurements (by up to a factor of 2000), thanks not only to the large luminosity increase but also to the improvements in vertex detector technologies and tagging algorithms. Experimental strategies that go beyond the current state-of-the-art methods also have to be developed, in order to reduce the systematic uncertainties in these measurements, down to a level commensurate with the expected statistical precision. Earlier measurements have shown that the contamination from light quarks is a large source of systematic uncertainty. Taking advantage of the very large number of Z bosons that will be produced at FCC-ee, a novel tagging technique has been pioneered in Ref. [628] for measuring $R_b$ and $A_{\text{FB}}^b$, which relies on the exclusive reconstruction of a selected list of b-hadron decay modes. It allows hemispheres containing a b-hadron to be tagged with a purity larger than 99.8% and, using selected decay modes of charged B mesons, the charge of the hemisphere is determined unambiguously. After removing the contamination from light quark processes, the remaining source of systematic uncertainty for the $R_b$ measurement is the correlation of tagging efficiencies between the two hemispheres of an event. This uncertainty has been studied in detail in Ref. [628], using generated events passed through a full simulation (including the reconstruction steps) of the CLD detector. In particular, an improved selection of the b-hadron tracks based on their displacement with respect to the luminous region has been shown to minimise the impact on the uncertainty of the exact knowledge of the correlation. The hemisphere correlation would need to be known with a relative precision of only 10% to ensure that the corresponding systematic uncertainty on $R_b$ is similar to the statistical uncertainty, at the level of 0.015%. This results in a $R_b$ determination that is 20 times more precise than the current value, with a lot of potential to do even better. For the measurement of $A_{\text{FB}}^b$, the remaining source of systematic uncertainty is the size of the QCD corrections. By exploiting acolinearity [629] or b-hadron energy selection requirements, these corrections would need to be known with a relative uncertainty of 1% for the statistical and systematic uncertainties to contribute equally to the total uncertainty. As a result, an improvement by a factor of 50 compared to the precision of the average of the LEP measurements would be obtained.

<footnote n="11">¹¹ The dependence on the resolution in the longitudinal direction is much milder.</footnote>

<footnote n="12">¹² Restricting the invariant mass to the 5.2–5.6 GeV range would yield a local significance of the signal of about 5 σ, but this value is obtained assuming an accurate knowledge of the functional shapes of signal and background candidates, which requires a study that goes beyond the scope of this work.</footnote>

<footnote n="13">¹³ The decay of interest comprises eight charged particles plus two neutrinos and, hence, features low momentum charged particles to be reconstructed. The reduction of the material is therefore one natural improvement to test, in view of minimising multiple Coulomb scattering in the resolution sources.</footnote>
<figure><img src="image_63.png" /><figcaption>Fig. 55 Distribution of the pointing variable log(1 − Ω) for signal and background events, where ¯ D 0 → K $^{+}$π − decays (and their charge-conjugates) are used to tag charmed hemispheres. The dashed and dash-dotted histograms show how the discrimination improves when the resolution of track impact parameters is improved by 25% and 50%, respectively, with respect to the nominal IDEA performance</figcaption></figure>

These very small uncertainties motivate the use of exclusively reconstructed hadrons to measure R$_{c}$ as well. A first study, described in Ref. [630], exploits ¯ D 0 → K $^{+}$π − decays. The main contamination arises from Z → b ¯ b events and needs to be reduced as much as possible to reach the expected statistical precision of 5 . 3 × 10 − 5 (corresponding to an improvement by a factor of about 60 compared to the statistically most precise measurement from the SLD Collaboration [631]). This contamination can be reduced by demanding that the ‘pointing angle’, i.e., the angle between the ¯ D 0 momentum and the line that joins the primary vertex and the decay vertex of the ¯ D $^{0}$, be close to zero. Figure 55 illustrates how a cut on the cosine of this angle, Ω, separates the signal from the background. The measurement of Ω clearly depends on the reconstruction of the vertices, and the figure also shows how the discrimination power increases when the resolution on the longitudinal and transverse track impact parameters improves.

4.4.4 Alignment, overall scale of the detector, and the measurement of the tau lifetime

Precise measurements of the mass, the lifetime, and the leptonic branching fraction of the tau offer a crucial test of lepton flavour universality (LFU) since, up to small radiative and electroweak corrections, the following relation [440,632] holds between the muon and tau masses, their lifetimes τ$_{µ}$ and τ$_{τ}$ , their charged-current coupling constants g$_{µ}$ and g$_{τ}$ , and their branching fractions into electrons and neutrinos:

$$
\left( \frac{g_{\tau}}{g_{\mu}} \right)^{2} \approx \frac{\tau_{\mu} \mathcal{B}(\tau \rightarrow e\nu_{\tau}\bar{\nu}_{e})}{\tau_{\tau} \mathcal{B}(\mu \rightarrow e\nu_{\mu}\bar{\nu}_{e})} \left( \frac{m_{\mu}}{m_{\tau}} \right)^{5}. \qquad (7)
$$

This test, which can be seen as a determination of the Fermi constant from the tau, is currently limited by the precision of the τ lifetime, 290 . 3 ± 0 . 5 fs, and by that of the B ( τ → eν$_{τ}$ ¯ ν$_{e}$ ) branching fraction, 17 . 82 ± 0 . 04%. The measurement of B ( τ → µν$_{τ}$ ¯ ν$_{µ}$ ) allows additional similar tests of ( g$_{τ}$/g$_{e}$ ) and ( g$_{µ}$/g$_{e}$ ). With the ∼ 2 × 10 11 tau pair events expected at Tera-Z, and a much improved determination of the three key ingredients 14 of Eq. (7), FCC-ee has the potential to test LFU at an unprecedented level. In particular, the statistical uncertainty in the tau lifetime is expected to be about 15 ppm [199]. The current measurements, from the Belle and LEP experiments, are statistically limited and, for the two single most precise measurements (Belle [633] and DELPHI [634]), the alignment of the vertex detector is the dominant source of systematic uncertainty. While this alignment could, at first sight, appear to be a concern in view of a lifetime measurement at the level of 15 ppm (which corresponds to a few tens of nm on the flight distance of τ leptons produced at 91 GeV), a more careful investigation shows that this is not the case [199]. Indeed, the offset on the decay length caused by misalignment effects averages to zero, to first order, when integrating over the azimuthal angle, as was first noted in Ref. [635]. The expected

<footnote n="14">14 The knowledge of the τ lepton mass is also expected to be much improved from pair production threshold measurements at a next-generation tau-factory.</footnote>
systematic uncertainty due to misalignment, scaled from the value quoted by DELPHI according to the luminosity increase, is below 4 ppm. The stability of the detector alignment will be tracked over time with continuous updates, as presently already done by LHCb [636-638], so that no significant systematic uncertainty from this source is expected. Other sources of systematic uncertainty have been considered in Ref. [199]. One of them reflects the knowledge of the overall length scale of the vertex detector. For this uncertainty to be smaller than one half of the statistical uncertainty on the τ lifetime, that scale should be known to better than 5 ppm. At LEP and at the B-factories this scale was only known to 100 ppm. However, recent developments made for the MUonE experiment indicate that a precision of 5 ppm is within reach, thanks to the use of optical techniques [639]. The leading systematic uncertainties are expected to be due to the modelling of initial state radiation and to the knowledge of the tau mass, and a total systematic uncertainty of 16 ppm on the tau lifetime is deemed within reach [199].

### 4.4.5 Preliminary conclusions

- One example has been shown where the physics outcome of FCC-ee would gain from having better vertex detector performance than the baseline detectors considered so far.

- Minimising the amount of material in front of the vertex detector is important. Indeed, the material of the beam pipe is a limiting factor in some cases, in particular for observing the rare $B \to K^*\tau\tau$ decay.

- It should be noted that these requirements will have to be reached despite the tight constraints set by the FCC-ee environment on the readout electronics of the detector: (i) its power budget is constrained (power-pulsing the electronics is not possible with collisions occurring every ~20 ns) and (ii) it should handle a hit rate of about 200 MHz/cm² in the innermost vertex detector layer during the Z run (Sect. 5.4).

## 4.5 Requirements for charged hadron particle identification

Charged hadron particle identification (PID) is essential for the FCC-ee flavour physics programme and brings significant benefits for other areas. The momentum range over which good PID capabilities are necessary is broad, in sharp contrast with the PID needs of a B-factory. For flavour physics, for example, it extends from the lowest momentum of the reconstructed charged-particle tracks up to about 40 GeV. In a gaseous tracker like the IDEA drift chamber, the determination of the number of ionisation clusters per unit length ($dN/dx$) is a promising approach that should allow charged kaons to be efficiently separated from charged pions in a large momentum range, from about 1.5 GeV to several tens of GeV. The expected resolution on $dN/dx$ is about 2%,<fnref n="15" /> significantly better than that of the well-known measurement of ionisation energy per unit length ($dE/dx$).

Time-of-flight (TOF) measurements, in a single layer at a distance of about 2 m from the interaction point, fill the gap around 1 GeV,<fnref n="16" /> where the average energy loss per unit length of kaons and pions is similar. However, they can only provide a $3\sigma \pi$/K separation at low momenta, up to about 3 GeV with the 30 ps resolution assumed for the studies reported here.<fnref n="17" /> Innovative solutions are being studied that would provide good PID capabilities in the absence of a specific energy loss measurement, over the large momentum range of interest and despite the tight spatial constraints. A conceptual design exists for a compact RICH detector, which looks promising and could comfortably cover the required momentum range [640-642]. An overview of the motivations for PID and of the possible detector solutions can be found in Ref. [45]. This section illustrates the demands on PID performance with the measurement of the Higgs boson coupling to strange quarks and with two B-physics processes.

### 4.5.1 Strange tagging and the Higgs boson coupling to strange (and charmed) quarks

The progress in the development of flavour-tagging algorithms in recent years allows, for the first time, a relatively efficient and pure tagging of jets induced by strange quarks. Various analysis efforts are ongoing within the circular and linear colliders communities. They all rely heavily on the identification power of state-of-the-art machine learning approaches.

Figure 56 illustrates the performance of strange-quark tagging with the baseline IDEA detector used for this report, using the PARTICLENETIDEA algorithm [491]. As shown in the left panel, when no PID information is provided to the algorithm, the separation power between jets induced by strange quarks and those induced by u or d quarks is limited, as it mostly

<footnote n="15">¹⁵ The promising performance of the $dN/dx$ approach as determined from calculations, in particular the upper momentum bound, is being checked with test beam data.</footnote>

<footnote n="16">¹⁶ This statement holds for a magnetic field of 2 T; the case of a higher field value is briefly considered in Sect. 4.9.1.</footnote>

<footnote n="17">¹⁷ Even with a 10 ps resolution, the momentum range would only extend up to about 5 GeV.</footnote>
<figure><img src="image_64.png" /><figcaption>**Fig. 56** Performance of strange-quark tagging in jet samples from $Z(\nu\bar{\nu})H$ events at $\sqrt{s} = 240$ GeV, where the Higgs boson decays into quarks of a well-defined flavour or into gluons. The left panel shows the impact of PID on the separation power between jets induced by strange</figcaption></figure>

<figure><img src="image_65.png" /><figcaption>quarks and those induced by u or d quarks. The right panel shows how the algorithm separates strange jets from all other flavours. The 's vs. ud' (black) curve in the right panel is identical to the red curve in the left panel</figcaption></figure>

comes from the slightly harder energy spectrum of the constituents of a strange jet.<fnref n="18" /> When the specific energy loss measured along the tracks with the cluster counting technique is also used, the misidentification probability of jets induced by u or d quarks is reduced by about one order of magnitude, for the same s-quark tagging efficiency. In the kinematic range considered here, spanned by quarks produced in $Z(\nu\bar{\nu})H(q\bar{q})$ events at $\sqrt{s} = 240$ GeV, the time-of-flight information only brings a mild performance improvement. The right panel of Fig. 56 shows how the algorithm separates strange jets from all other flavours, the separation from ud jets being, as expected, the most challenging. For a strange-quark tagging efficiency of 80% (90%), the mis-tag efficiency of ud jets reaches 20% (40%).

This tagging algorithm has been used to categorise $Z(\ell\ell, jj)H(jj)$ events (where $\ell$ stands for e, μ or ν) into mutually orthogonal categories enriched in one of the different Higgs boson decays (Sect. 4.4.1) and to extract the Higgs boson branching fractions to $b\bar{b}$, $c\bar{c}$, $s\bar{s}$, and gg. With an integrated luminosity of $10.8 \text{ ab}^{-1}$, the Higgs boson branching fraction to strange quarks would be measured with a 105% uncertainty. More details can be found in Ref. [66].

The anticipated precision on the Higgs boson branching fractions to $c\bar{c}$ and $s\bar{s}$ has been explored for several particle identification capabilities and the result is shown in Fig. 57. Given that charged kaons from strange quark hadronisation feature a hard momentum spectrum, it is crucial to have an efficient $K/\pi$ separation at large momenta, as provided by the $dN/dx$ cluster counting technique. It is remarkable that the expected precision on the measurement of the $H \to s\bar{s}$ branching fraction, obtained by combining $dN/dx$ with the time-of-flight technique, is almost identical to the asymptotic performance obtained with perfect PID capabilities. Conversely, if no PID is assumed, or only assuming ToF, the $H \to s\bar{s}$ branching fraction measurement cannot be performed, given an expected uncertainty in excess of 300%. The PID also plays a non-negligible role in the $H \to c\bar{c}$ branching fraction measurement, by enabling the identification of secondary charged kaons produced in D meson decays, as shown in the right panel of Fig. 57. A comprehensive discussion on the impact of PID on flavour tagging performance can be found in Ref. [627].

### 4.5.2 Separation of $K^{\pm}$ from $\pi^{\pm}$ and measurement of $b \to s\bar{v}$

The trillions of $b\bar{b}$ events that will be collected, in a clean environment, during the Tera-Z run at FCC-ee makes it the ideal facility to study rare decays of b hadrons. Among those, the modes corresponding to the $b \to s\bar{v}$ transition are of considerable interest within the flavour physics community.<fnref n="19" /> A first sensitivity study for FCC-ee, using the decays $B \to K^*\nu\bar{\nu}$

<footnote n="18">¹⁸ The number of reconstructed $K_S \to \pi^+\pi^-$ decays from displaced tracks associated with the jets (see Sect. 4.3.4) is not yet explicitly included in the set of variables used by the PARTICLENETIDEA algorithm; adding this variable may improve the performance shown here.</footnote>

<footnote n="19">¹⁹ The first evidence for this transition has been recently obtained by the Belle II Collaboration in the $B^+ \to K^+\nu\bar{\nu}$ decay channel [643]. In the future, with an integrated luminosity of $50 \text{ ab}^{-1}$, Belle II is expected to measure the corresponding branching fraction with $\mathcal{O}(10\%)$ experimental precision [227].</footnote>
<figure><img src="image_66.png" /><figcaption>**Fig. 57** Relative uncertainty in the $H \to s\bar{s}$ (left) and $H \to c\bar{c}$ (right) branching fractions, as expected for several assumptions on particle identification performance setups</figcaption></figure>

<figure><img src="image_67.png" /><figcaption>**Fig. 58** Sensitivity to the $B \to K^*\nu\bar{\nu}$ (left) and $B_s \to \phi\nu\bar{\nu}$ (right) branching fractions, as a function of the $\pi/K$ separation power, with respect to the sensitivity assuming perfect PID</figcaption></figure>

and $B_s \to \phi\nu\bar{\nu}$, is detailed in Ref. [228]. According to the Standard Model, both decays should have branching fractions around $10^{-5}$. The current experimental upper limit on the branching fraction of $B \to K^*\nu\bar{\nu}$ is larger than this prediction by a factor of two, while for $B_s \to \phi\nu\bar{\nu}$, the upper limit is two orders of magnitude larger. The study looks for a $K^*$ or a $\phi$ resonance in a hemisphere with a large amount of missing energy and runs a multi-variate analysis, following the strategy described in Ref. [226]. Assuming a perfect PID, the $B \to K^*\nu\bar{\nu}$ ($B_s \to \phi\nu\bar{\nu}$) signal can be selected with an efficiency of 3.7% (7.4%), for a signal-to-background ratio of 0.17 (0.13). With $6 \times 10^{12}$ Z bosons produced, this corresponds to a sensitivity of 0.5% for $B \to K^*\nu\bar{\nu}$ and of 1.2% for $B_s \to \phi\nu\bar{\nu}$. Here, sensitivity is defined as the relative precision expected on the branching fraction, computed as $\sqrt{S+B}/S$, where S and B are the expected numbers of signal and background events, respectively. The dependence of this sensitivity on the performance of the $\pi/K$ separation that the detector would achieve is shown in Fig. 58. With a separation power of $2\sigma$, the loss in sensitivity, compared to what is expected with perfect PID, is marginal, but it degrades quickly as the performance worsens. The momentum of the kaons involved in these decays peaks at about 3 GeV, but extends up to 15–20 GeV, hence the PID performance offered by time-of-flight measurements alone is unlikely to be sufficient for this analysis.
**Fig. 59** Mass distributions of the $B_s$ candidates in the $B_s \to D_s K \to \phi(KK)\pi K$ decay channel, prior to any PID cut (left) and after requiring that the ‘bachelor’ (the particle accompanying the reconstructed $D_s$) must be identified as a kaon (right). The momentum of the ‘bachelor’ is required to be larger than 1.5 GeV

<figure><img src="image_68.png" /></figure>

### 4.5.3 Separation of $K^{\pm}$ from $\pi^{\pm}$ and measurement of $B_s \to D_s K$

The measurement of time-dependent CP asymmetries in the decay $B_s \to D_s K$ is a well-known method to extract the $\gamma$ angle of the CKM matrix, which is currently only known with a precision of 4 degrees. The prospects for this measurement at FCC-ee have been first studied in Ref. [276], using the $D_s \to \phi(KK)\pi$ decay channel. The presence of the $\phi$ resonance and the absence of neutral particles among the final decay products makes this signal easy to reconstruct. A precision on the $\gamma$ angle of a few tenths of a degree was shown to be within reach at FCC-ee, using only this decay mode.<fnref n="20" />

The study reported in Ref. [276], which considered a few exclusive background processes and used a simple parametrisation of the detector response, has been redone for this report with DELPHES to simulate the response of the IDEA detector and accounting for the inclusive $Z \to b\bar{b}$ and $Z \to c\bar{c}$ backgrounds. The analysis starts by exploiting the mass resolution to reconstruct the $\phi$ candidates and, afterwards, the $D_s$ candidates, without any PID requirement. In a second step, $B_s$ candidates are built by fitting the $D_s$ and another track (the ‘bachelor’) to a common vertex [644]. The invariant mass of the $B_s$ candidates satisfying minimal kinematic and vertex $\chi^2$ cuts is shown in the left panel of Fig. 59. The signal, shown in dark green, is very well separated from the ten times larger $B_s \to D_s \pi$ background, shown in red. However, running the same selection over the inclusive $Z \to b\bar{b}$ sample shows that there are other sources of backgrounds that contaminate the signal mass peak, in addition to the exclusive backgrounds that were considered in Ref. [276] and that are also shown in Fig. 59. The right panel of this figure shows that this background is largely suppressed by requiring that the ‘bachelor’ particle must be identified as a kaon, which is done by using a PID cut that is 95% efficient on the signal. The PID cut uses the ratio of the likelihoods for the ‘bachelor’ to be a kaon or a pion and combines the $dN/dx$ measurement of the track, determined with a resolution of about 2%, and the measurement of its TOF at 2 m from the interaction point, with a 30 ps resolution. Even for this mode with charged particles only, PID capabilities appear to be mandatory to extract the signal with a decent signal-to-background ratio. Since the momentum distribution of the kaons from $B_s \to D_s K$ decays is hard, time-of-flight measurements alone would not lead to an acceptable background suppression. For example, in a mass window that contains 95% of the reconstructed $B_s$ candidates (the signal), the background contamination decreases from 48%, without any PID, to 33% if only the TOF measurement is used, and to 19% if both the $dN/dx$ and TOF measurements are used (assuming the PID capabilities of IDEA). This last value would degrade to 24% with a twice worse $dN/dx$ resolution.

### 4.5.4 Work ahead

- The measurement of time-dependent CP asymmetries in the $B_s$ system requires the flavour of the meson ($B_s$ or $\bar{B}_s$) to be tagged at the time of production. Various algorithms are typically combined to achieve this goal. A powerful method uses the charge of the kaons in the event that are not associated with the signal decay, which typically have momenta well below 10 GeV. The performance of flavour tagging is studied in order to provide requirements on PID at low momentum. For example, the determination of the angle $\gamma$ of the CKM matrix, from the $B_s \to D_s K$ process, or of the phase $\beta_s$, from the $B_s \to J/\psi \phi$ decay, could serve as benchmark measurements. The identification of very soft fragmentation

<footnote n="20">²⁰ A similar sensitivity is expected from the Phase-2 LHCb upgrade, with an integrated luminosity of 300 fb⁻¹ [273]. The FCC-ee sensitivity can be improved by including other $D_s$ decay modes, in particular those that include neutral particles (see Sect. 4.6.2).</footnote>
<table><caption>**Table 12** Expected energy resolution of the different electromagnetic calorimeter options considered in the studies of Ref. [41]</caption><thead><tr><th rowspan="2">Technology</th><th colspan="2">EM energy resolution</th></tr><tr><th>Stochastic term</th><th>Constant term</th></tr></thead><tbody><tr><td>Highly granular Si/W based</td><td>15–17%</td><td>1%</td></tr><tr><td>Dual readout fibre (ECAL+HCAL)</td><td>11%</td><td>&lt; 1%</td></tr><tr><td>Hybrid crystal (dual readout)</td><td>3%</td><td>&lt; 1%</td></tr><tr><td>Highly granular noble liquid based ECAL</td><td>8–10%</td><td>&lt; 1%</td></tr></tbody></table>

kaons (produced when a b (anti-)quark hadronises into a $B_s$ meson), for which the PID performance brought by TOF measurements is relevant, should also play a key role in the measurement of the branching fraction of the $B_s \to \nu\bar{\nu}$ decay.

- The determination of $V_{cs}$ from W decays may benchmark the performance of strange-quark tagging for working points with a much higher purity than that needed for the determination of the coupling of the Higgs boson to the strange quark. Alternatively, $V_{cs}$ may be measured in events where exclusive decays of charmed hadrons into kaons are selected, calling for a good $\pi$/K separation. More generally, electroweak measurements involving strange quarks, such as $R_s$ or $A_{FB}^s$, are studied for the requirements on the PID performance.

- A very precise test of lepton flavour universality in the tau sector is provided by the simultaneous measurements, at the $10^{-5}$ level, of the tau lifetime and of the branching fraction of its decay to electrons (Sect. 4.4.4). The latter requires a $e/\pi$ separation controlled at the $10^{-5}$ level, which might be difficult to achieve from calorimetry measurements alone. The requirements on $dE/dx$ or $dN/dx$ measurements should also be studied in the context of $e/\pi$ separation.

### 4.5.5 Preliminary conclusions

- It is not only for flavour physics that charged-hadron PID is needed. In particular, the potential for constraining the coupling of the Higgs boson to the strange quark provides a strong motivation for having PID, up to high momenta. Similarly, powerful strange tagging offers new prospects for electroweak and top measurements in the strange sector, such as measurements of the $Z \to s\bar{s}$ branching fraction [645] and forward-backward asymmetry, or of the $V_{ts}$ CKM element via rare $t \to Ws$ decays [646].

- The momentum range to be covered, extending from $O(150)$ MeV to 40 GeV, is challenging. Current studies show that the $\pi$/K separation offered by the cluster counting approach, in the IDEA drift chamber, is probably adequate. A degradation of the resolution of the energy loss measurement by a factor of 2, with respect to what is expected from the baseline IDEA $dN/dx$ performance, would have a non-negligible impact on the determination of the Higgs coupling to strange quarks. Such a degradation would correspond to the expected $dE/dx$ performance.

- In the absence of specific energy loss measurements in the tracker, there is a conceptual design for a compact RICH detector that looks promising and could comfortably cover the required momentum range. Detailed simulation studies are needed to evaluate the impact of such a detector on particle-flow reconstruction performance.

## 4.6 Requirements for electromagnetic calorimetry

This section summarises the needs for the measurement of electromagnetic (EM) showers, which may, but does not have to, be performed in a dedicated electromagnetic calorimeter (ECAL). In the baseline IDEA concept, the EM showers are measured using a finely segmented crystal ECAL with dual readout capabilities [619]. In a variant of this concept, considered in Ref. [10], the electromagnetic and hadron showers are both measured in a dual readout (DR) calorimeter that uses scintillation and Cerenkov fibres. The CLD concept uses a high-granularity ECAL built as a sandwich of tungsten plates and silicon sensors. A high granularity noble liquid ECAL is also under consideration in ALLEGRO. These concepts have complementary strengths, ranging from an energy resolution of $\sigma_E/E \simeq 3\%/\sqrt{E}$ for the crystal-based ECAL to an extreme transverse resolution of $\sim 2 \times 2 \text{ mm}^2$ for the fibre DR calorimeter, if each fibre is equipped with a SiPM readout device, or to a high longitudinal segmentation for the Si/W or noble liquid solution. More details are given in Table 12. The next paragraphs illustrate the requirements on electromagnetic calorimetry from representative analyses. Benchmark processes for which the performance is solely driven by the energy resolution are described first, followed by one case for which the transverse granularity is crucial, and by examples that place stringent requirements on both the energy resolution and the granularity.