1 Physics discovery potential

1.1 Overview

“There is a strong scientific case for an electron-positron collider, complemen-tary to the LHC, that can study the properties of the Higgs boson and other particles with unprecedented precision and whose energy can be upgraded.” [25]

This strategic guideline from the 2013 update of the European Strategy for Particle Physics (ESPP 2013) unambiguously defines the high standards to be met by the future e⁺e⁻collider, quite possibly the next high-energy collider to be built.

Since its inception, the FCC-ee study has aimed at delivering the e⁺e⁻ collider conceptual design that best complies with this guideline, and consequently offers, in a cost-effective fashion, the broadest physics discovery potential and the most ambitious perspectives for future developments.

As a result of the renewed worldwide interest for e⁺e⁻physics and the pertaining discovery potential since the observation of the Higgs boson at the LHC, the FCC is not alone in this quest. In the absence of convincing hints for physics beyond the standard model (BSM) in the LHC data so far, the situation has significantly evolved since 2013, so that today no fewer than four e⁺e⁻ collider designs are contemplated to study the properties of the Higgs boson and other standard model (SM) particles with an unprecedented precision:

– the International Linear Collider (ILC [13]) project, for which the above guideline was originally tailored, now focusses on studying the Higgs boson with a centre-of-mass energy of 250 GeV [14,15];

– the Compact Linear Collider (CLIC [16]), whose lowest centre-of-mass energy point was reduced from 500 to 380 GeV [17], in order to better study the Higgs boson and the top quark;

– the Circular Electron Positron Collider (CEPC [18–20]), in a 100 km tunnel in China, able to study the Z, the W, and the Higgs boson, with centre-of-mass energies from 90 to 250 GeV;

– the Future e⁺e⁻ Circular Collider in a new ∼100 km tunnel at CERN (FCC-ee, formerly called TLEP [8,21]), which can study the entire electroweak (EW) sector (Z and W bosons, Higgs boson, top quark) with centre-of-mass energies between 88 and 365 GeV.

The baseline luminosities expected to be delivered at the ILC, CLIC, CEPC, and FCC-ee centre-of-mass energies are illustrated in Figure1.1.

Fig. 1.1. Baseline luminosities expected to be delivered (summed overall interaction points) as a function of the centre-of-mass energy√

s, at each of the four worldwide e⁺e⁻collider projects: ILC (blue square), CLIC (green upward triangles), CEPC (black downward trian-gles), and FCC-ee (red dots), drawn with a 10% safety margin. The FCC-ee performance data are taken from Section 2, the latest CEPC parameters are taken from [20], and the linear collider luminosities are taken from [15,17].

The FCC-ee delivers the highest rates in a clean, well-defined, and precisely pre-dictable environment, at the Z pole (91 GeV), at the WW threshold (161 GeV), as a Higgs factory (240 GeV), and around the t¯t threshold (340–365 GeV), to several interaction points. It also provides high precision centre-of-mass energy calibration at the 100 keV level at the Z and WW energies, a feature unique to circular col-liders¹. The FCC-ee is therefore genuinely best suited to offer extreme statistical precision and experimental accuracy for the measurements of the standard model particle properties, it opens windows to detect new rare processes, and it furnishes opportunities to observe tiny violations of established symmetries.

Historically, such precise measurements or subtle observations have been pre-cursors for the discovery of new phenomena and new particles, and for a deeper understanding of fundamental physics. These historical precedents have also shown the important role played by lower-energy precision measurements when establish-ing road-maps for higher-energy machines. In the second half of the 1970s, precision measurements of neutral currents led scientists to infer the existence of the W and Z bosons, as well as the values of their masses, from which the dimensions of the LEP tunnel were determined. The W and Z were then observed in the early 1980’s at the

1 A circular e⁺e⁻Higgs factory, LEP3, had also been proposed in the LHC tunnel back in 2011 [21,26]. With respect to the FCC-ee, the LEP3 facility would have had the advantage of reusing existing infrastructure, at the severe expense of a much reduced sensitivity to new phenomena, with (i) a luminosity smaller by a factor 4–5 at the Z, WW, and Higgs operation points; (ii) the impossibility of a precise energy calibration at the WW threshold; (iii) the inability to measure the top-quark properties; and (iv) the lack of a vibrant perspective for subsequent energy-frontier exploration in the same tunnel.

Years

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

]

Luminosity [ab

0 50 100 150 200

Z pole WW

10

HZ

10

Top

10

Fig. 1.2. Operation model for the FCC-ee, as a result of the five-year conceptual design study, showing the integrated luminosity at the Z pole (black), the WW threshold (blue), the Higgs factory (red), and the top-pair threshold (green) as a function of time. The hatched area indicates the shutdown time needed to prepare the collider for the highest energy runs.

CERN Sp¯pS collider with masses in the predicted range. Subsequently, as described in more detail in Section1.2, the CERN LEP e⁺e⁻collider measured the properties of the Z and W bosons with high precision in the 1990’s [27,28]. These precise mea-surements could determine in a definitive way the number of light, active neutrinos, as well as allow inferring the mass of the so far unseen top quark, which was soon observed at the Tevatron within the predicted mass range. With mtop fixed by the Tevatron measurement, the ensemble of precision measurements at LEP/SLC, at the Tevatron, and from low energy inputs led in turn to a ±30% accurate prediction for the mass of the Higgs boson, which was observed in 2012 at the LHC within the predicted mass range. It is important to note that these predictions were based on the Standard Model with no additional particle content with respect to that known today.

With the Higgs boson discovery, the standard model seems complete and its predictions have no more flexibility beyond the uncertainties in the theoretical cal-culations and in the input parameters. Several experimental facts, however, reveal without any doubt that new phenomena must exist: non-baryonic dark matter; the cosmological baryon-antibaryon asymmetry; the finite albeit extremely small neu-trino masses, etc., are all evidence for physics beyond the standard model. The agreement between the predicted and observed W, top and Higgs masses, and the null result of experiments at colliders so far, are an indication that either the new physics scale is too high and/or the pertaining couplings are too small. Any new hint would be a major discovery, whether it is the observation of a new particle, a new so-far unobserved phenomenon, or a non-trivial deviation from the standard model predictions.

As a result, the next accelerator project must allow the broadest possible field of research. This is the case for the FCC. To begin with, the FCC-ee would measure the Z, W, Higgs, and top properties in e⁺e⁻ collisions, either for the first time

or with orders of magnitude increases in precision, thereby giving access to either much higher scales or much smaller couplings. The FCC-ee is the most powerful of all proposed e⁺e⁻ colliders at the electroweak (EW) scale – all things being equal, in particular the duration of operation (Fig. 1.2). The FCC-ee proposes a broad, multifaceted exploration to

1. measure a comprehensive set of electroweak and Higgs observables with high precision,

2. tightly constrain a large number of the parameters of the standard model, 3. unveil small but significant deviations with respect to the standard model

predic-tions,

4. observe rare new processes or particles, beyond the standard model expectations, and, therefore, maximise opportunities for major fundamental discoveries. The FCC-ee also mFCC-eets the last part of the ESPP 2013 guideline (“[. . . ] and whose energy can be upgraded”) in the most ambitious manner, as the FCC-ee tunnel is designed to subsequently host the FCC-hh, a hadron collider with a centre-of-mass energy of 100 TeV. Combined with the FCC-ee measurements, the FCC-hh physics reach at the energy and precision frontiers is likely to be unbeatable. The multiple synergies between the FCC-ee and FCC-hh physics opportunities are discussed in Vol. 1 of this Conceptual Design Report.

The primary goal of the FCC-ee design study was to demonstrate the feasibility of the accelerator. This goal has been successfully met, confirming and even exceeding the original luminosity expectations (Figs. 1.1 and 1.2). Great confidence can be given in the integration of the detectors at the collision points, and in the ability to reach the beam energy calibration targets. The exploration of the physics capabilities is still at a preliminary stage. Nevertheless the studies presented in the next sections provide a flavour of the extraordinary physics potential of the FCC-ee.

1.2 Precision electroweak measurements

Since the early work by Veltman [29], it has been known that the electroweak quan-tum corrections are sensitive to particles with electroweak couplings and with masses much higher than accessible directly with the centre-of-mass energies available. The case of the top quark and Higgs boson were particularly interesting: despite their high masses, their effect would indeed not decouple. Further studies in the late 1980s led to the realisation that these quantum corrections could be separated into blocks with different sensitivities. Accurate measurements of these observables thus become sensitive to the possible presence of further particles coupled to the SM interactions in a broader sense. The FCC-ee enables precision measurements of the Z, the W, the Higgs boson and the top quark properties, together with those of input parameters to the standard model, such as the electromagnetic and strong coupling constants at the Z mass scale, thereby providing sensitivity to new particles with masses of up to 10–70 TeV.

1.2.1 Current situation

As briefly mentioned above, the Z lineshape parameters (the Z mass mZ, the Z width Γ_Z, and the peak cross section σ⁰) fitted to the per-mil precision measurements of fermion pair production cross sections at and around the Z pole [27] performed at LEP, were sensitive to the yet unobserved top quark and to a lesser extent to the putative Higgs boson, as illustrated in the Feynman diagrams of Figure 1.3.

Fig. 1.3. Schematic representation of the perturbative expansion for calculating the cross section for e⁺e⁻annihilation into a pair of leptons or quarks (denoted f, for fermions); the representative higher order diagrams involving quantum loops with a top quark or a Higgs boson are indicated.

Similarly, the measurements of fermion pair asymmetries allow the determination of the effective weak mixing angle sin²θ_W^eff, the value of which is predicted in the SM from the relation:

sin²θ^eff_Wcos²θ^eff_W =παQED(m²_Z)

√2GFm²_Z × (1 + ∆κ), (1.1)

where α_QED(m²_Z) is the electromagnetic coupling constant evaluated at the Z pole, GF is the Fermi constant, and ∆κ is a small correction factor that depends on the top quark and Higgs boson masses via the graphs displayed in Figure 1.3.

The magnitude of the second graph of Figure 1.3 is proportional to the square of the top quark mass. It is much larger than that of the third one, proportional to log(m_H/m_Z), and amounts to about ten times the LEP measurement accuracy. As a consequence, LEP was able to predict the mass of the top quark within the SM (assuming that no other particle but the Higgs boson would impact the radiative corrections) [27]:

m^SM_top= 173⁺¹³₋₁₀GeV. (1.2)

The W boson mass is in turn predicted within the SM from the relation:

m^SM_W =

"

παQED(m²_Z)

√

2GFsin²θ^eff_W × 1 1 − ∆r

#¹₂

, (1.3)

where ∆r is yet another small correction factor that depends on m_top and m_H. Numerically, the W mass was predicted from the LEP measurements at the Z pole with a remarkable precision (including the above uncertainty on the top quark mass and the absence of knowledge of the Higgs boson at the time) [27]:

m^SM_W = 80.362^+0.032_−0.031GeV. (1.4) By increasing its centre-of-mass energy to above the W⁺W⁻ production thresh-old, LEP did measure the W mass directly, in agreement with equation (1.4) and with a similar precision [28]. The Tevatron later improved this precision by about a factor two [30], and observed for the first time the top quark [31,32], at the mass predicted by LEP (Eq. (1.2)) in the context of the standard model and nothing else.

Today, the W boson and top quark masses are directly measured with the following accuracies [33]:

m^direct_W = 80.379 ± 0.012 GeV, (1.5)

m^direct_top = 173.3 ± 0.4(exp) ± 0.5(theory) GeV. (1.6) The direct measurements of mW and mtop were then used to determine the magnitude of the second graph of Figure1.3, and made the third graph become the

dominant unknown term of the perturbative expansion. As a consequence, the LEP and Tevatron measurements were able to infer the existence of a Higgs boson and to predict its mass within the SM:

m^SM_H = 98⁺²⁵₋₂₁ GeV. (1.7)

The LHC observed the production of the Higgs boson in 2012 for the first time, at a mass well compatible with this prediction in the context of the standard model and nothing else. The current overall situation of the standard model fit to the precision measurements available to date is summarised in Figure 1.4. The fit prediction for the W mass and the weak mixing angle [34] within the SM, namely

mW= 80.3584 ± 0.0055m_top± 0.0025m_Z± 0.0018α_QED

± 0.0020α_S± 0.0001m_H± 0.0040theory GeV

= 80.358 ± 0.008_total GeV,

sin²θ_W^eff = 0.231488 ± 0.000029m_top± 0.000015m_Z± 0.000035α_QED

± 0.000010α_S± 0.000001m_H± 0.000047theory

= 0.23149 ± 0.00007total, (1.8)

are also very compatible with the world average of their direct measurements within current uncertainties:

mW= 80.379 ± 0.012 GeV, and sin²θ_W^eff = 0.23153 ± 0.00016. (1.9)

1.2.2 Opportunities at the Z pole

Electroweakly-coupled new physics would appear either as additional/different con-tributions to the perturbative expansion of the electroweak observable predictions, similar to those shown in Figure 1.3, or as modifications of the tree-level couplings to leptons and quarks. From the agreement between the predictions and the direct measurements, it follows that the effect of new physics, if any, must be smaller than the current uncertainties. The next significant step in this quest is therefore to drastically reduce these uncertainties, typically by one order of magnitude or more.

In this section, it is assumed that theoretical uncertainties can be brought, by the calculation of missing QED, EW and QCD higher orders within the standard model and nothing else, to a level similar to, or smaller than, that of the experimental uncertainties. This issue is addressed briefly in Section 1.5. Numerically, the FCC-ee is able to deliver about 10⁵ times the luminosity that was produced by LEP at the Z pole, i.e. typically 1.5 × 10¹¹Z → µ⁺µ⁻ or τ⁺τ⁻decays and 3 × 10¹²hadronic Z decays. Measurements with a statistical uncertainty up to 300 times smaller than at LEP (from a few per mil to 10⁻⁵) are therefore at hand.

Forward-backward and polarisation asymmetries at the Z pole are a powerful experimental tool to measure sin²θ^eff_W, which regulates the difference between the right-handed and left-handed fermion couplings to the Z. With unpolarised incoming beams, the amount of Z polarisation at production is

A_e=gL,e2− gR,e2

g_L,e²+ g_R,e²

2ve/ae

1 + (v_e/a_e)², with v_e/a_e≡ 1 − 4 sin²θ_W^eff, (1.10) by definition of the effective weak mixing angle sin²θ^eff_W. The resulting forward-backward asymmetry at the Z pole amounts to A^ff_FB = ³₄AeAf. The experimental

Fig. 1.4. From reference [35]: Contours of 68% and 95% confidence level obtained from fits of the standard model to the precision measurements available to date, in the (mtop, mW) plane. The grey area is the result of the fit without the direct measurements of the W, top, and Higgs masses, while the narrower blue area includes the Higgs boson mass measurement at the LHC. The horizontal and vertical green bands and the combined green area indicate the 1σ regions of the mWand mtop measurements (world averages).

control of the longitudinal polarisation of each of the beams can be made with the foreseen polarimeter (Sect.2.7) with great accuracy.

From the experimental point of view, the e⁺e⁻ → Z → µ⁺µ⁻process is a golden channel for an accurate measurement of A^ff_FB. The dominant source of experimental uncertainty arises from the knowledge of the centre-of-mass energy. Indeed, in the vicinity of the Z pole, A^µµ_FB exhibits a strong√

s dependence

A^µµ_FB(s) ' 3

4A_eA_µ×

"

1 + 8π√

2αQED(s) m²_ZGF 1 − 4 sin²θ^eff_W
²

s − m²_Z 2s

#

, (1.11)

caused by the off-peak interference between the Z and the photon exchange in the process e⁺e⁻ → µ⁺µ⁻. As suggested in Section2.7, a continuous measurement with resonant depolarisation of non-colliding bunches should allow a reduction of this uncertainty to below 0.1 MeV. The resulting uncertainty on A^µµ_FB amounts to 9 × 10⁻⁶ (a factor three larger than the statistical uncertainty), which propagates to an uncertainty on sin²θ^eff_W of 6 × 10⁻⁶. Among the other asymmetries to be mea-sured at the FCC-ee, the τ polarisation asymmetry in the τ → πντ decay mode provides a similarly accurate determination of sin²θ^eff_W, with a considerably reduced

√s dependence. In addition, the scattering angle dependence of the τ polarisation asymmetry provides an individual determination of both Ae and Aτ , which allows, in combination with the A^µµ_FBand the three leptonic partial width measurements, the vector and axial couplings of each lepton species to be determined. Similarly, heavy-quark forward-backward asymmetries (for b heavy-quarks, c heavy-quarks and, possibly s heavy-quarks) together with the corresponding Z decay partial widths and the precise knowledge of

Aefrom the τ polarisation, provide individual measurements of heavy-quark vector and axial couplings.

An experimental precision better than 5 × 10⁻⁶ is therefore a robust target for the measurement of sin²θ^eff_W at the FCC-ee, corresponding to more than a thirty-fold improvement with respect to the current precision of 1.6 × 10⁻⁴ (Eq. (1.9)).

Individual measurements of leptonic and heavy quark couplings are achievable, with a factor of several hundred improvement on statistical errors and, with the help of detectors providing better particle identification and vertexing, by up to two orders of magnitude on systematic uncertainties.

For this accuracy to be instrumental in constraining new physics, the parametric uncertainties of the sin²θ_W^effSM prediction (Eq. (1.8)) need to be brought to a similar level. The largest parametric uncertainty on the prediction, 3.5 × 10⁻⁵, arises from the limited knowledge of the electromagnetic coupling constant evaluated at the Z mass scale. It is hoped that this figure can be reduced by a factor of two to three with a better determination of the hadronic vacuum polarisation, in part with future low-energy e⁺e⁻ data and in part with the use of perturbative QCD [36]. The large luminosity offered by the FCC-ee allows a direct determination of α_QED(m²_Z) to be contemplated [37], from the slope of the muon forward-backward asymmetry as a function of the centre-of-mass energy in the vicinity of the Z pole (Eq. (1.11)). As displayed in Figure1.5, the statistical uncertainty of this measurement is minimum just below (√

s = 87.9 GeV) and just above (√

s = 94.3 GeV) the Z pole. It is shown in reference [37] that the experimental precision on α_QEDcan be improved by a factor 3–4 with 40 ab⁻¹at each of these two points. Because most systematic uncertainties are common to both points and almost perfectly cancel in the slope determination, the experimental uncertainty is statistics dominated as long as the centre-of-mass energy spread (90 MeV at the Z pole) can be determined to a relative accuracy better than 1%, which is deemed achievable at the FCC-ee every few minutes [38].

More studies are needed to understand if the α_QED(m²_Z) determination can profit from the centre-of-mass energy dependence of other asymmetries.

An experimental relative accuracy of 3 × 10⁻⁵ on αQED(m²_Z) can be achieved at the FCC-ee, from the measurement of the muon forward-backward asymmetry with 40 ab⁻¹ of centre-of-mass energies ∼3 GeV below and ∼3 GeV above the Z pole.

The corresponding parametric uncertainties on the sin²θ^eff_W and m_WSM predictions are accordingly reduced from 3.5 × 10⁻⁵ and 1.8 MeV to 9 × 10⁻⁶ and 0.5 MeV, respectively.

The next parametric uncertainty to address at the Z pole is that arising from the Z mass. The Z mass and width were determined at LEP from the line shape scan to be m_Z= 91187.5 ± 2.1 MeV and Γ_Z = 2495.2 ± 2.3 MeV, with data taken mostly at √

s = 89.4, 91.2, and 93 GeV. The statistical errors of 1.2 MeV and 2 MeV would be reduced below 5 keV and 8 keV at the FCC-ee, with data taken

s = 89.4, 91.2, and 93 GeV. The statistical errors of 1.2 MeV and 2 MeV would be reduced below 5 keV and 8 keV at the FCC-ee, with data taken