Rare and exclusive few-body decays of the Higgs, Z, W bosons, and the top quark

We perform an extensive survey of rare and exclusive few-body decays -- defined as those with branching fractions $\mathcal{B} \lesssim 10^{-5}$ and two or three final particles -- of the Higgs, Z, W bosons, and the top quark. Such rare decays can probe physics beyond the Standard Model (BSM), constitute a background for exotic decays into new BSM particles, and provide precise information on quantum chromodynamics factorization with small nonperturbative corrections. We tabulate the theoretical $\mathcal{B}$ values for almost 200 rare decay channels of the four heaviest elementary particles, indicating the current experimental limits in their observation. Among those, we have computed for the first time ultrarare Higgs boson decays into photons and/or neutrinos, H and Z radiative decays into leptonium states, radiative H and Z quark-flavour-changing decays, and semiexclusive top-quark decays into a quark plus a meson, while updating predictions for a few other rare H, Z, and top quark partial widths. The feasibility of measuring each of these unobserved decays is estimated for p-p collisions at the high-luminosity Large Hadron Collider (HL-LHC), and for $e^+e^-$ and p-p collisions at the future circular collider (FCC).


I. INTRODUCTION
With the discovery of the Higgs boson at the CERN Large Hadron Collider (LHC) about ten years ago [1,2], the full particle content of the Standard Model (SM) of particle physics has become fully fixed.Among the 17 existing elementary particles (6 quarks, 6 leptons, 4 gauge bosons, and the scalar boson), the top quark, the Higgs and the electroweak (W, Z) bosons are the most massive ones.Studying in detail the properties of the four heaviest elementary particles, with masses around the electroweak scale Λ EW ≈ O(100 GeV), is an important priority in precision SM studies and in searches for new physics beyond it (BSM).At the LHC, the large center-of-mass energies and integrated luminosities (up to L int = 3 ab −1 at the end of the high-luminosity, HL-LHC, phase) [3] available in proton-proton (p-p) collisions, as well as the many ab −1 to be integrated in the very clean "background-free" e + e − collision environment of the next planned lepton collider facilities, such as the Future Circular collider (FCC-ee) [4] or the CEPC [5], will allow the collection of very large H, W, Z, and top-quark data samples.In addition, and despite much more complicated background conditions than at e + e − colliders, p-p collisions at √ s = 100 TeV planned at FCC-hh [6] will produce unprecedented numbers of W, Z, Higgs, and top particles.The very large data samples expected at these machines will make it possible to measure many of their rare few-body decays -understood here as decays into two or three finalstate particles, with branching fractions below B ≈ 10 −5 -, which remain unobserved to date.Broadly speaking, in this work we consider three types of rare few-body decays shown in Fig. 1: (i) decays into three lighter gauge bosons (or one gauge boson plus two neutrinos), (ii) decays into a lighter gauge boson plus a single hadronic (or leptonic) bound system in the form of a quarkonium (or leptonium) state, and (iii) exclusive decays into two quarkonium bound states.The reason for the rarity of the first type of decays is the fact that they proceed through suppressed (heavy particle) loops, whereas the second and third decay modes occur very scarcely because the probability to form a single (let alone double) -onium bound state, out of the outgoing quarks or leptons of the primary decay, is very small.FIG. 1. Examples of schematic diagrams of rare and exclusive two-and three-body decays of the Higgs, Z, and W bosons, and of the top quark.The leftmost diagram shows a rare decay into two or three gauge bosons V = Z, W, γ (or into a gauge boson plus two neutrinos ν) through virtual loops (grey circle).The center and rightmost diagrams show, respectively, typical exclusive decays into a gauge boson (mostly a photon) plus an onium bound state, and into two onium states (dashed blobs).
The incentives for the study of such decays are multiple, with some specificities depending on the decaying "mother" particle.A first general motivation for their study is the possibility that new physics phenomena alter some very rare partial decay widths.Precision tests of suppressed (or forbidden) processes in the SM -such as flavour changing neutral currents (FCNC), lepton flavour violating, or lepton flavour universality violating processes -are powerful probes of BSM physics that have been mostly studied so far in b-quark decays [7,8].Unlike in the latter case where, due to the relatively low masses of the B hadrons involved, large power corrections to the decay rates lead to sizable theoretical uncertainties, power corrections in decays of the electroweak and Higgs bosons and top quark are under better theoretical control thanks to the large boost of the final-state hadrons.More concrete motivations are succinctly mentioned next for each particle.In the case of the Higgs boson, very rare decays with photons and/or neutrinos (Fig. 1, left) lead to experimental signatures that are identical to potential exotic BSM Higgs decays [9,10], and therefore need to be well estimated as background(s) for the latter.Exclusive Higgs decays such as those shown in Fig. 1 (center) are sensitive to the Yukawa couplings of the charm and lighter quarks [11][12][13][14][15], as well as to FCNC H → qq ′ decays, which are otherwise very difficult to access experimentally due to the smallness of the quark masses involved, and/or their heavily suppressed loop-induced rates [10].In the Z boson case, specific decay modes allow probing FCNC couplings (which are loop-and Glashow-Iliopoulos-Maiani (GIM) [16] suppressed processes in the SM) in a modelindependent way [17,18].In addition, rare decays of the H and Z bosons shown in Fig. 1 where the onium state decays into diphotons or dileptons, constitute a background for different searches for exotic BSM decays [19], such as e.g., Z → γ a(γγ) with a being an axion-like particle [20] or a graviton [21] decaying into photons; or H →, A ′ (ℓ + ℓ − ) + X where a dark photon A' further decays into an ℓ + ℓ − lepton pair [22].The measurements of exclusive decays of the W boson [23][24][25] and of the top quark [26] have also been suggested e.g., as an alternative means to determine the W boson and top quark masses via two-or three-body invariant mass analysis, free of invisible neutrinos or (messy) jets involved in the inclusive decay modes.In the top quark case, the study of its suppressed radiative decay rates, induced by the off-diagonal parts of the quark dipole moments, has been of interest for many decades because they also provide an experimentally clean probe for new physics [27].Of particular importance are precision studies of rare FCNC decays such as t → Zc, t → γc, and t → cg, for which many BSM extensions can enhance their branching ratios by orders of magnitude, thereby yielding compelling phenomenology [27,28].
Theoretically, the calculation of the partial widths of the few-body decays schematically shown in Fig. 1 (left) are carried out through an expansion of the underlying virtual loops in the electroweak (EW) and/or quantum chromodynamics (QCD) couplings (α and α s , respectively), at a given order of perturbative accuracy.First calculations were computed at leading order (LO), but more recent results exist at next-to-leading-order (NLO) accuracy.With regards to the center and right diagrams of Fig. 1, the formalism of perturbative QCD factorization [29][30][31][32][33] is a wellestablished approach to study and compute rates for hard exclusive processes with individual hadrons in the final state.Within this framework, short-distance physics at the energy scale of the initial heavy particle is appropriately separated from the long-distance dynamics governing the formation of the final hadron(s), and the decay amplitudes can be therefore obtained from the convolution of perturbatively calculable hard-scattering functions and nonlocal hadronic matrix elements.These latter objects, called light-cone distribution amplitudes (LCDAs), are nonperturbative and scale-dependent functions that encode the infrared physics of the final hadron formation.The decay amplitudes are formally given as expansions in the ratio of the two (hard and soft) scales in the problem, given by the large energy released E in the process and the final hadron mass, respectively.In the cases of interest in this work, the hard scale is set by the heavy mass of the decaying bosons or top quark, µ hard ≈ Λ EW rendering the impact of nonperturbative power corrections at typical hadronization energy scales Λ QCD ≈ 0.2 GeV, O(Λ QCD /Λ EW ) ≲ 10 −3 , under control.The studies of exclusive decays of the H, Z, W bosons, and top quark, therefore not only provide a sensitive test of the SM but, in particular, also stringent tests of the QCD factorization formalism, including constraints on poorly known aspects of the nonperturbative formation of hadronic bound states.In the case of final states with charm and bottom quarks, they can bring forward valuable insights into partially conflicting mechanisms of heavy quarkonium production [34,35].
The main purpose of this work is to present a comprehensive summary of the current theoretical and experimental status of rare and exclusive few-body decays of the four heaviest SM particles.We have first collected all calculations and experimental upper limits for rates of rare and exclusive decays existing in the literature, revised them, and complemented them with ∼40 additional channels estimated here for the first time.In total, we provide a list of about 200 predicted rare decays branching ratios, and identify those that are potentially observable at the HL-LHC, or at FCC, and those with negligible rates (unless some BSM physics enhances them).This document should therefore help guide and prioritize future experimental and theoretical studies of the different channels.The paper is organized as follows.Section II provides an overall description of the theoretical and experimental details of the branching ratios compiled for each decay channel, as well as an explanation of how estimates of future experimental upper limits are derived.Sections III, IV, V, and VI present, respectively, a detailed list of all rare decays of the Higgs, Z, and W bosons, and top quark, including revised branching fractions for a few channels, as well as ultrarare H decays, H and Z decays into leptonium states, radiative H and Z quark-flavour-changing decays, and semiexclusive top-quark decays into a quark plus a meson, computed here for the first time.The paper is closed with a summary of the main findings.

II. RESULTS
The rare decays results covered in this work are organized in tables per decaying particle listing their theoretical branching fractions, their current experimental upper limits, and future bounds expected at HL-LHC, FCC-ee, and FCC-hh colliders.Here, we discuss first the theoretical models collected, followed by an explanation of the method used to estimate future experimental limits.The numerical values of the relevant SM parameters used in the (re)calculations of a few branching fractions are also provided.

A. Theoretical predictions
For all rare decays collected here, we indicate the theoretical framework used to calculate their corresponding branching fractions.All channels with hadronic final states (Fig. 1, center and right) have been obtained through various implementations of the QCD factorization formalism, using different prescriptions to describe the meson form-factors and to evolve them using the renormalization group equations.We provide here a succinct description of each model so that their acronyms listed below are understandable.The concrete models employed mostly depend on the identity (mass) of the underlying quarks and associated final hadron(s), which defines their kinematics and relevant energy scales.Thus, for example, the form-factors of heavy (charm and/or bottom) mesons also include short-distance physics at energy scales of the heavy-quark mass (m Q ≫ Λ QCD , for Q = c, b), which are often described in terms of long-distance matrix elements (LDMEs).
A common method for computing exclusive light-hadron production in high-energy decays is given by the so-called amplitude expansion in the light cone (LC), where the perturbatively small expansion parameter is the chirality factor m q /E (with m q and E being the mass of the produced quark and its typical energy, respectively), and the quarks formfactors are nonperturbative objects described by LCDAs [33] that encode the internal motion of the quark-antiquark pair inside their bound state, constrained by QCD sum rules [36,37].Alternatively, many decay modes involving light mesons have been calculated using Soft-Collinear Effective Theory (SCET) [38], where QCD factorization is rephrased in the language of effective field theory (EFT) [39] to properly address the problem of the multiple scales appearing in the calculations.The SCET framework allows proper resummation of the large logarithms in the ratios of scales given by the mass of the mother particle and that of the mesonic (M) final state, log(m M /Λ EW ), which spoil the convergence of the perturbative expansion (likewise, such large logarithms can arise in situations where the hadrons are light and highly energetic).In a few other calculations, the EFT approach is combined with formfactors based on the nonrelativistic quark model (NRQM) [40].Final-state mesons containing one light and one heavy quark are instead often described in Heavy-Quark Effective Theory (HQET) where the LCDA describes the hadronic physics at two distinct scales, m Q and Λ QCD , of which the former should be tractable by perturbative methods [41].In HQET, the hadronic matrix elements are expressed as a combination of perturbatively computable coefficients and new, suitably defined, hadronic matrix elements that exploit the constraints provided by heavy quark symmetries on the nonperturbative matrix elements at low scales [42].
A third class of models is used for systems with a heavy quark-antiquark bound system, such as charmonium and bottomonium, and also bottom-charm (B c ) mesons, where the interactions are organized as an expansion in the typical relative velocity of the heavy quarks inside the meson state (v 2 ≈ 0.3, 0.1 for charmonium and bottomonium, respectively).The formation of heavy quarkonium is a multi-scale problem, with m(µ hard ) ≫ mv ≫ mv 2 ≫ Λ QCD , and the simple light-quark static limit is not sufficient for these systems.A nonrelativistic description is often used instead, called Non-Relativistic QCD (NRQCD) [43].Calculations are done at LO, NLO, or next-to-NLO (NNLO) accuracy, resumming in some cases also leading (LL) or next-to-leading (NLL) logarithms.In the NRQCD factorization, the long-distance nature of heavy quarkonium is factorized into the NRQCD LDMEs.Several approaches can be combined with NRQCD to describe the evolution of the heavy-flavour quark pair into a heavy quarkonium meson, such as the colour-singlet model (CSM), where the QQ pairs are produced in colour-singlet (CS) states at the hard-scattering scale µ hard .The quantum numbers of the quark pair are the same as those of the heavy quarkonium, and the QQ LDME in the CSM can be estimated from decay rates measurements, or in a potential model.The colour-octet model (COM), which requires the extra radiation of a gluon, and is not fully adequate for exclusive heavy quarkonia final states, has also been considered in a few cases.An alternative relativistic quark model (RQM) [44], which shares several basic features with NRQCD, has been also employed in exclusive processes with heavy quarkonia.

B. Experimental limits
As we will see below, predictions for rare decays of the electroweak and Higgs bosons, and the top quark, have branching fractions in the 10 −5 to 10 −15 range (or even down to 10 −22 for positronium + photon final states).Such tiny branching fractions are very challenging experimentally, and no such decays have yet been observed for any of the particles.To provide an idea of the size of the data samples of W ± , Z, H, and t particles discussed in this work, Table I collects their total number produced in past, current, and future colliders.The LEP numbers of W and Z bosons are obtained from the LEP-I and LEP-II electroweak summaries [45,46], and indicate that no rare decay mode with rates below 10 −7 and 10 −5 for the Z and W bosons, respectively, has ever been probed in the clean experimental conditions of an e + e − collider.We note, somehow curiously, that the last LEP-II operation, which integrated 2.46 fb −1 over √ s = 189-209 GeV [47], featured a Higgs cross section of σ H ≈ 3 fb (adding Higgstrahlung and weak-bosonfusion production processes) [48], and therefore LEP-II did produce a few Higgs(125 GeV) boson counts (however, the Higgs searches at the time were optimized for the associated production of a m H = 115 GeV scalar boson plus an onshell Z boson, for which the H(125 GeV) signal would not have been visible).For the next e + e − machine, we focus on FCC-ee because it is the planned facility with the largest data samples expected to be collected [49].The FCC-ee numbers in Table I cover the time span of the baseline 15-years program with four interaction points and four dedicated runs under consideration: Z-pole, WW threshold, HZ Higgstrahlung, and tt threshold.The largest data sample of all will be for Z bosons, for which 6 • 10 12 particles will be produced.The number of H bosons at FCC-ee includes those produced in the HZ (1.45 × 10 6 ) and tt (+330k events) runs, plus via WW → H at all runs (+125k).Although, in much more complicated background conditions than in e + e − collisions, the HL-LHC (with 3 ab −1 of p-p collisions at √ s = 14 TeV collected per ATLAS/CMS experiment) and the FCC-hh (p-p at √ s = 100 TeV with 30 ab −1 ) will truly serve as W, Z, Higgs, and top factories.We also list the number of the four massive particles produced at the Tevatron in p-p collisions at √ s = 1.96TeV, as there exist still a couple of competitive rare decay limits from the CDF experiment.The hadron collider numbers in Table I are obtained from the corresponding production cross sections for each particle multiplied by the integrated luminosity quoted (plus the contribution from tt → W + W − + X decays for the W ± counting).The cross sections at hadron colliders have been either obtained from the existing literature [50] or, when not readily available or not fully up-to-date, have been recomputed at next-to-next-to-leading-order (NNLO) accuracy with mcfm v.8.0 [51] with the NNPDF3.1_NNLOparton distribution functions (PDFs) [52].Theoretical (scale, PDF) uncertainties are at most 10% and not quoted.TABLE I. Total number of Higgs, Z, and W bosons, and top quarks produced (or expected to be produced) in e + e − collisions at LEP and FCC-ee, as well as in p-p at Tevatron, and in p-p collisions at HL-LHC, and FCC-hh.For e + e − colliders, the Z-pole, Higgstrahlung and Higgs weak-fusion, and pair production (WW, tt) cross sections (without ISR) are indicated.At LEP and FCCee, the numbers of W bosons and top-quarks consider two bosons and two quarks produced per collision at the e + e − → W + W − , tt thresholds.For hadron machines, the integrated luminosities and the NNLO production cross section for each particle are indicated.All top quark numbers are for pair production (i.e., the number of events are multiplied by two), and the number of W bosons is the (W + + W − ) sum including also the contributions from tt decays.

Collider
W ± bosons Z bosons H bosons top quarks The largest yields increases across colliders are factors of 6 • 10 3 , 3 • 10 5 , and 4 • 10 5 for W, Z, and H bosons, respectively, going from LEP to FCC-ee, and a factor of 9 • 10 4 for top quarks from Tevatron to HL-LHC.As of today (end of 2023), measurements at LEP, Tevatron, and LHC have been able to set upper limits at 95% confidence level (CL) in about one-fourth of the ∼200 rare channels considered here.The most stringent experimental rare decays limits are listed in the tables below, including in some cases new results which are not yet available in the 2023 PDG decays listings [53].One key goal of our work is to provide reasonable expectations of the achievable upper bounds at the HL-LHC, FCC-ee, and FCC-hh.The corresponding extrapolations are obtained through three different means: 1.For a few decay channels, there exist dedicated ATLAS and/or CMS studies that have determined the expected limits at the end of the HL-LHC phase [54][55][56][57][58][59].For such cases, we provide directly the expected limits (multiplied by √ 2 to account for the statistical combination of two experiments, ATLAS + CMS) with their bibliographical reference.
2. For those channels where LHC limits exist today in measurements with a given integrated luminosity (labelled e.g., "L int (13 TeV)" next), but for which no dedicated studies exist for HL-LHC, we estimate the latter by assuming that they will be statistically improved by the size of the final data sample, namely by the ratio of squared-root integrated luminosities √ 2 × 3 ab −1 /L int (13 TeV), where the factor of two assumes an ATLAS + CMS combination.For individual LHC measurements carried out with the full Run-2 integrated luminosity of around 140 fb −1 , this translates into an expected HL-LHC improvement of more than a factor of six.Bounds estimated that way are conservative, as they ignore improvements in the data analyses (e.g., optimization of the event selection, enlarged categorization depending on the production mode, adoption of more advanced statistical profiling methods, etc.), increased particle production cross sections (from collision energies rising from 13 to 14 TeV), and possible multi-experiment combinations (e.g., adding results from LHCb) for some channels.We have checked that our simple statistical approximation here gives limits comparable to those obtained with the method 1) above whenever dedicated studies are available.
3. For those channels where the best limits today are from Tevatron, the corresponding HL-LHC extrapolation is obtained also statistically, by multiplying the current bound by the square-root of the ratio of the number of decaying particles X between Tevatron and HL-LHC, √ N X (HL-LHC)/N X (Tevatron).According to the increase factors quoted in Table I, this implies an improvement of about a factor of 70 in the expected W and Z bosons upper bounds.Such an estimate is conservative as, in general, the overall acceptance/efficiency of the ATLAS/CMS detectors is larger than that of CDF.
Finally, for the FCC-ee and FCC-hh cases, whenever a given decay channel has a branching fraction commensurate with the expected number of particles produced (i.e., whenever B × N X ≳ 1, where N X are the FCC numbers given in Table I for any given particle), it will appear listed as "producible at FCC-ee/FCC-hh".Assuming that potential backgrounds are small and/or well under control, as is often the case at e + e − colliders, and even more so at FCCee where very precise detector performances are being required [49], observation can be achieved with a handful of events.Such an assumption is unwarranted at hadron colliders, such as FCC-hh, where at least 100 signal events are typically required to suppress backgrounds in searches for rare prompt decays of Higgs and Z bosons [60,61].Future detector and experimental analysis progresses at the FCC-hh could make it less critical than for the LHC.It is conceivable that much advanced data analysis techniques will be available by the time this new energy-frontier collider starts to operate, which can render observable some producible decays.

C. Input parameters of the calculations
For the numerical evaluation of various theoretical expressions for branching fractions computed in this work, the input parameters listed in Table II are used.This include leptonium and quark and boson masses and widths (left column), couplings and Cabibbo-Kobayashi-Maskawa (CKM) matrix elements (center column), and hadron masses and form-factor parameters: decay constants f M and HQET first inverse moments λ M (right column).Figure 2 shows rare decays of the H boson into two neutrinos, three photons, a photon plus two neutrinos, and four photons, respectively.Calculations of rates for these processes, which proceed via virtual loops in the SM, do not exist to our knowledge.Since they are very suppressed and have no obvious phenomenological impact, they are not included in the standard Higgs decay codes [66], although they are intriguing for diverse reasons exposed below.We have therefore computed them with the MadGraph5_aMC@NLO program [71] with QCD and EW loop corrections up to NLO accuracy [72,73] using the input parameters of Table II.The corresponding results are listed in Table III.The first calculation is that of the invisible two-body Higgs boson decay into a neutrino-antineutrino pair, which is infinitesimal in the SM (B ≈ 10 −36 ) compared to the standard invisible (four-neutrino) H → ZZ ⋆ → 4ν decay (B ≈ 0.1%) [66].However, since the SM assumption of massless ν's is incorrect, the H → νν decay can receive extra contributions depending on the mechanism of neutrino mass generation actually realized in nature.Therefore, it is a process worth keeping an eye on, when considering invisible Higgs decays and accounting for massive neutrinos.The second ultrarare Higgs decay of interest here is H → 3γ.At variance with the naive assumption that it is forbidden for a scalar particle to decay into three photons, because such a process violates charge-conjugation (C) symmetry, C(H) = 0 C(3γ) = (−1) 3 = −1, such a decay is possible, albeit extremely suppressed, for EW-induced decays.In a situation akin to the case of the triphoton decay of the scalar positronium state (para-positronium), which is forbidden in QED but not in the EW theory [74,75], such a Higgs decay can proceed through W loops. Since H → 3γ violates C-symmetry, it must also violate parity in order to conserve CP and, therefore, the final state must be composed of three spatially symmetric photons with vanishing total angular momentum.As a consequence, this partial width features a utterly small O(10 −40 ) probability.Thus, any potential experimental observation of H → 3γ would require an enhancement of the SM rate by up to 40 orders-of-magnitude, and it would be a strong signal of BSM physics.The third diagram of Fig. 2 shows the photon-plus-neutrinos decay, which proceeds through Z and W loops and, since the neutrino pair goes undetected, it appears experimentally as an unbalanced monophoton decay of the Higgs boson.Such a final state is shared by many exotic BSM Higgs decays [9], and it is worth to compute its SM rate.This decay is actually the least rare considered in this whole survey, and has a branching fraction of B = 3.74 • 10 −4 that is about 20% larger than the naive estimate given by B(H → Zγ) = 1.54 • 10 −3 [66] combined with B(Z → νν) = 0.200 [53].This is so because extra W-induced channels (not shown in Fig. 2) contribute to the amplitude.Last but not least, it is interesting to compute the 4-photon decay of the Higgs boson as it may constitute a background for exotic decays into a pair of light axion-like or scalar particles, each of which further decays into two photons, H → a(γγ) a(γγ).
The Higgs 4γ decay has a branching fraction of B = 4.56 • 10 −12 , which is 28 orders-of-magnitude larger than the 3-γ one, as C and P conservation are not an issue here, and the rate is only suppressed due to the presence of heavy charged-particle loops.Among all rare channels discussed in this section, LHC searches have been performed to date for the 4γ final state alone [61,[76][77][78][79] (searches for 3γ decays have focused on Z' resonances off the Higgs peak [61]), but unfortunately the limits have been set only on the process H → a(γγ)a(γγ) with two intermediate ALPs decaying into photons.It would not be difficult for ATLAS/CMS to recast these searches into upper bounds on the H → 4γ "continuum" decay.

B. Exclusive Higgs decays into a gauge boson plus a meson
Since it is extremely difficult to experimentally access the Yukawa couplings to first-(q = u, d, s) and second-(c) generation quarks, due to the smallness of the H → qq, cc decay widths, and the very large QCD jet backgrounds at the LHC, it has been proposed to constrain those couplings via rare exclusive decays into a photon (or a gauge boson) plus a vector meson [11][12][13][14][15].The relevant Feynman diagrams for the H → V + M (where V = Z, W ± , γ, and M = meson) decays in the SM are shown in Fig. 3.The first diagram represents LO amplitudes where the Higgs boson directly couples to a quark-antiquark pair that radiates a gauge boson and forms the mesonic bound state.The second diagram depicts indirect contributions to the decay amplitude, where the scalar boson decays first into two gauge bosons, one of which transforms (offshell) into a hadron state via V * → qq.The third diagram shows the radiative FCNC decay into a photon plus a flavoured meson, through a double W loop.The indirect diagram provides the dominant contribution to the decay rates due to the smallness of the Yukawa couplings to the first-and second-generation quarks, and the largest sensitivity to the Higgs 2013quark coupling comes from the (destructive) interference of the two amplitudes.Thus, for example, the H → γ + J/ψ, γ + ϕ modes allow direct access to the flavour-diagonal coupling of the Higgs to the charm and strange quarks, respectively, while the H → γ + ρ, γ + ω decays can probe the Higgs couplings to up and down quarks.The radiative decays H → γ + M, where M = K * 0 , D * 0 , B * 0 s , B * 0 d -which in the SM can only proceed through a virtual W boson because the photon (and Z) splitting preserves flavour, or through the direct process in BSM scenarios with flavour-violating Higgs decays -provide possibilities to probe BSM flavour-violating q-q' Higgs couplings [13,80].The indirect amplitude, in which the virtual photon or Z boson couples to the vector meson through the matrix element of a local current, can be parameterized in terms of a single hadronic parameter: the vector-meson decay constant f M (see Table II).This quantity can be directly obtained from the experimental measurements of the vectormeson leptonic partial decay width (which proceeds through the EW annihilation qq → γ * , Z * → ℓ + ℓ − ), given by where Q M is the relevant combination of quark electric charges, and m M the mass of the meson.Corrections due to the offshellness of the photon and to the contribution of the H → γ Z * process are suppressed by , respectively, and hence very small [14].The direct amplitudes, which are much smaller than the indirect ones except for the heaviest H → γ + Υ decay, have been calculated theoretically by different groups.The hierarchy m M ≪ m H implies that the vector meson is emitted at very high energy E M ≫ m M in the Higgs boson rest frame.The constituent partons of the vector meson can thus be described by energetic particles moving collinear to the direction of M, and perturbative QCD factorization, either in the SCET or NRQCD incarnation, can be employed to compute them.

Higgs decays into a photon plus a vector meson
Table IV lists the corresponding theoretical predictions and experimental limits for Higgs decays into a photon plus a vector meson, and Fig. 4 displays them in graphical form.Theoretical B values are in the range of O(10 −5 -10 −9 ), with larger rates for decays into the lightest vector mesons due to the inverse dependence on the meson mass of the V ( * ) → qq transitions of the dominant indirect process, as per Eq.(1).Experimental upper bounds have been set for all decays at the LHC [81][82][83][84][85] except for the H → γ + D, γ + B ones, but it seems that evidence for a few of those SM decays will only possible at FCC-ee.The heaviest bottomonium radiative decays will require the number of Higgs bosons produced in a machine like FCC-hh.TABLE IV.Compilation of exclusive Higgs decays to a photon plus a meson.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it, as well as the current experimental upper limit and that estimated for HL-LHC.The last two columns indicate whether the decay can be produced at FCC-ee/FCC-hh.

Exp. limits
Producible at FIG. 4. Branching ratios (in negative log scale) of exclusive H → γ + vector-meson decays.Most recent theoretical predictions (blue bars) compared to current experimental upper limits (violet) and expected HL-LHC bounds (orange).The red vertical lines indicate the FCC-ee (solid) and FCC-hh (dashed) reach based only on the total number of H bosons to be produced at both facilities.

Higgs decays into a photon plus a flavoured meson
The third diagram of Fig. 3 shows the radiative decay into a photon plus a flavoured meson.In the SM, this FCNC process can only proceed through a double W loop, and we are not aware of any theoretical calculation of these rates to date.We estimate here the branching fractions for H → γ + M with M = K * 0 , D * 0 , B * 0 s , B * 0 d , which are all excited vector states, as explained below.On the one hand, calculations for the inclusive flavour-changing H → qq ′ rates exist [80,88,89], but without the photon emission and the exclusive final-state meson formation.These predicted FCNC Higgs branching fractions, , and B(H → sb) = (8.9± 1.5) • 10 −8 [89].On the other hand, the work of [13] determined the branching fractions of radiative Higgs flavoured-meson decays assuming (arbitrary) O(1) flavour-changing Yukawa couplings.Combining the results of [13,89], the branching ratios of Higgs radiatively decaying into flavored neutral mesons can be determined through the following EFT + LCDA-based expression where ≈ 246 GeV is the Higgs vacuum expectation value, and all other quantities have been already defined.For the numerical evaluation of the meson HQET first inverse moment λ M (µ), which plays an important role when working out rates for exclusive decays involving charm and bottom mesons [64,90], we have used the values quoted in Table II with λ B * 0 For the K * 0 decay, we simply took O(10 −8 ) × B(H → ds) for the predicted rate.The theoretical branching fractions for such exclusive FCNC radiative decays are extremely suppressed, in the range of O(10 −14 -10 −26 ).Therefore, flavoured meson decays of the Higgs boson appear utterly rare to be visible at any current or future collider, and therefore a very clean probe of BSM physics that may enhance Higgs FCNC decays.One experimental limit has been recently set for the H → γ + K * 0 channel at O(10 −5 ) [84], to be compared with our O(10 −19 ) SM prediction.TABLE V. Compilation of exclusive Higgs decays to a photon plus a flavoured meson.For each decay, we provide the branching fraction predicted using Eq. ( 2), as well as the current experimental upper limit and that estimated for HL-LHC.The last two columns indicate whether the decay can be produced at FCC-ee/FCC-hh.

Higgs decays into a Z boson plus a meson
Higgs decays of the form H → Z + M involve neutral currents and are very similar to the radiative decays just discussed.However, an important difference with respect to the γ + meson decays, is that the interference terms between the direct and indirect amplitudes are smaller.Although these decays are less useful for measuring the Higgs couplings to light quarks, they probe the important effective H-γ-Z coupling, and provide independent constraints on the meson distribution amplitudes.In addition, they constitute a background for exotic BSM decays of the Higgs boson into, e.g., a Z boson plus an ALP (H → Z + a) [91].Table VI compiles the theoretical predictions and experimental limits for Higgs decay branching ratios into a Z boson plus a vector meson, and Fig. 6 presents them in graphical form.All decays have rates in the 10 −5 -10 −6 range with small differences across models for the same channel, except for the H → Z + ρ case where the calculations of [12] and [92] differ by a factor of six, seemingly because the former had neglected the indirect contributions.Experimental upper bounds have been set for a few channels listed in Table VI, and they are in the O(10 −2 -10 −3 ) range 1 .In general, the experimental H → Z+M limits are about a factor of ten worse than their H → γ + M counterparts because of the extra events lost from the requirement of Z boson identification via dilepton decays (with B ≈ 3% for each lepton pair).The HL-LHC will be able to set upper bounds about 100 times below the expected SM branching fractions, and it appears that experimental evidence will only be possible at FCC-ee for all of them.Bottomonia-plus-Z decays have the largest rates, but no bound has been set to date for them.TABLE VI.Compilation of exclusive Higgs decays to Z boson plus a meson.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it, as well as the current experimental upper limit and that estimated for HL-LHC.The last two columns indicate whether the decay can be produced at FCC-ee/FCC-hh.

Exp. limits
Producible at Theoretical BR Current limits HL-LHC 14 TeV, 2x3 ab −1 FCC-ee, N(H)=1.9 × 10 6 FCC-hh, N(H)=2.8 × 10 10 FIG. 6. Branching ratios (in negative log scale) of exclusive H → Z + meson decays.The most recent theoretical predictions (blue bars) are compared to current experimental upper limits (violet) and expected HL-LHC bounds (orange).The red vertical lines indicate the FCC-ee (solid) and FCC-hh (dashed) reach based only on the total number of H bosons to be produced at both machines.

Higgs decays into a W boson plus a meson
The charged H → W ± M ∓ decays (corresponding to the diagrams of Fig. 3 with W bosons in the final state) differ qualitatively from the neutral radiative decays discussed above, because the W attaches itself to a charged current, and one can probe flavour-violating couplings of the Higgs boson.The theoretical complication is that the W has both transverse and longitudinal polarizations, yielding lengthier analytical expressions [12,92].Table VII lists the theoretical predictions for Higgs decays into a W ± boson plus a charged meson, and Fig. 7 presents them in graphical form.The rates for these processes range roughly between 10 −5 and 10 −10 with small differences between models for the same final state.The EFT + NRQM theoretical predictions of Ref. [10], which update the B numerical values computed in [12] (but have likely a typo in the B(H → B * ± W ∓ ) = 10 −5 rate, which should be 10 −10 ), agree with the alternative EFT + LCDA rates of Ref. [92].There has been no experimental search to date at the LHC for any of these 11 decays.The most promising channel, with a O(10 −5 ) branching fraction, is H → W ± + ρ ∓ with the rho meson decaying into two pions with almost 100% probability, but seemingly experimental evidence for 7 (all) of them will only be possible at FCC-ee (FCC-hh).

C. Radiative Higgs leptonium decays
Figure 8 shows the diagrams of the decay of the Higgs boson into a photon or a Z boson plus a leptonium state (ℓ + ℓ − ), where (ℓ + ℓ − ) = (e + e − ), (µ + µ − ), (τ + τ − ) represent positronium, dimuonium, and ditauonium bound states, respectively.Depending on the accompanying gauge boson (Z or γ), leptonium can be produced in spin triplet (ortho) and/or spin singlet (para) states.Such decays are similar to the exclusive radiative decays into mesons shown in Fig. 3, but changing the quark lines for lepton lines (of course, in this case, there is no unlike-flavour leptonic bound state possible, at variance with the H → W ± + M case).We provide here an estimate of the decay rates for such processes by considering the similar H → γ + M, Z + M processes, and changing the quarkonium form-factors by leptonium EFT+LCDA [92] ones.The overall probability for forming an onium bound state is determined by one single parameter: its radial wavefunction at the origin of the space coordinate, which for leptonium bound states (of principal quantum number n) reads [99], Since the QED coupling is much smaller than the QCD one, α(m ℓℓ ) ≪ C F α s (m qq ), and since the charged lepton masses are smaller than the quark masses of the same generation, m ℓℓ ≪ m qq , the ratio [α(0)m ℓℓ /(α s (m qq )m qq )] 3 is very small, and one can anticipate that those decays will be orders-of-magnitude more suppressed than the H → γ + M(qq) ones discussed in Section III B. We start by calculating the branching fraction for the H → γ + (ℓ + ℓ − ) process in which, due to charge-conjugation conservation, the leptonium can only be in the ortho (ℓ + ℓ − ) 1 state.The partial width of this decay can be derived from the similar expressions for quarkonium vector mesons [11], namely -log(BR) with the direct and indirect amplitudes corresponding to the diagrams of Fig. 8 left and right, respectively, and where the H → γ + Z * contribution followed by the Z * → (ℓ + ℓ − ) 1 transition is negligible compared to the corresponding H → γ + γ * one.These two amplitudes can be written as a function of the wavefunction at the origin ϕ n,(ℓℓ) (0), as follows where for the second equality of the indirect component, we have adopted the Van Royen-Weisskopf formula [100,101] to relate the leptonium decay constant, f X , to its wavefunction at the origin: Here, the particle X can be either a pseudoscalar/vector meson or an ortho-/para-leptonium state, and the number of colours N c = 1 is used for the latter.We choose ϕ X (0) to be real, and then the decay constant f X comes with a phase that decides the interference between indirect and direct amplitudes.In this work, we choose the (positive) phase that yields a destructive interference similar to the quarkonium case [11].Since m ℓℓ ≪ m H , plugging Eq. ( 3) into ( 6), one can see that the indirect amplitudes are independent of the leptonium masses.Using Eqs. ( 4)-( 6) with the numerical values of Table II, we determine the radiative ortholeptonium (n = 1) branching fractions listed in the first three rows of Table VIII.The radiative ortholeptonium decays of the Higgs boson have all numerically similar O(10 −12 ) rates for the three leptons, they have not been searched-for to date at the LHC, and only a future machine such as the FCC-hh can try to set upper limits on them at about 10 times their SM values.The experimental search has a very clean signature characterized by a secondary vertex from the boosted (ℓ + ℓ − ) 1 decay, which lead to significantly displaced triphotons (for positronium and dimuonium), e + e − pairs (for dimuonium and ditauonium), or µ + µ − pairs (for ditauonium).
We determine next the rates of the H → Z + (ℓ + ℓ − ) decays.At variance with the photon case, these decays have a massive final-state gauge boson that can be in a longitudinal polarization state.As a consequence, both scalar (para-) and vector (ortho-) leptonium states can be produced, whereas in the H → γ + (ℓ + ℓ − ) 1 case, the leptonium could only be a (transversely polarized) ortho state.We consider first the Higgs decays into Z-boson-plus-ortholeptonium by properly adapting the theoretical expressions for the similar H → Z + VM decays, where VM are quarkonia vector mesons such as J/ψ, Υ [97].The corresponding Z + (ℓ + ℓ − ) 1 decay width can then be written as where Γ 1 , Γ 2 , and Γ 12 are the contributions from and their interference, respectively.Defining , and the HγZ effective coupling C Zγ ≈ 5.54 [102], these individual partial widths can be expressed as follows with ≈ 246 GeV, and g ℓℓ = T ℓ 3 − 2Q ℓ sin2 θ W = 1/2 + 2 sin 2 θ W (where Q ℓ is the lepton charge, and T ℓ 3 its third component of the weak isospin).Using the expressions above with the numerical values of Table II, we determine the branching fractions listed in the second three rows of Table VIII.The H → Z + (ℓ + ℓ − ) 1 decays have O(10 −10 -10 −13 ) rates, and they have not been searched-for to date at the LHC.Given the negligible rates, only a high-luminosity machine such as FCC-hh would be able to provide limits approaching the SM values for the most probable H → Z + (e + e − ) 1 final state, with a particularly unique search for a Z boson accompanied by triphotons, or an e + e − pair 2 , issuing from a significantly displaced vertex from the secondary positronium decay.
Lastly, we compute the rates of the H → Z + (ℓ + ℓ − ) 0 decays, whose width can be written using similar expressions for the H → Z + M decays (where M is a pseudoscalar meson) [92], as follows The contribution from the direct amplitude to the partial width (10) amounts to with a ℓ = T ℓ 3 /2 = 1/4, which is suppressed by a factor m 2 ℓℓ /m 2 H or m 2 ℓ /m 2 H , that makes it completely negligible (the contribution to the final (ττ) 0 amplitude is of about 0.09%).The contribution from the indirect amplitude to the width (10) reads Using the expressions above with the numerical values of Table II, we determine Z-plus-paraleptonium decay rates, H → Z + (ℓ + ℓ − ) 0 , that amount to 10 −12 -10 −16 (three last rows of Table VIII).
All branching fractions computed here for Higgs decays into Z or γ plus leptonium are listed in Table VIII and shown in graphical form in Fig. 9.As aforementioned, the decay rates are minuscule, in the O(10 −10 -10 −16 ) range, and in the absence of BSM effects enhancing them, only a machine producing as many Higgs bosons as the FCC-hh can attempt to observe the H → Z + (3γ) with a displaced triphoton vertex from the late orthopositronium (ee) 1 → 3γ decay, or H → Z + (ℓ + ℓ − ) with a displaced dielectron from the magnetic-field-induced breaking of orthopositronium into its constituents.Limits on final states containing the dimuonium and ditauonium states can be set by searching for very displaced secondary vertices from their e + e − , µ + µ − decays.

D. Exclusive Higgs decays into two mesons
Figure 10 shows representative diagrams of the exclusive decay of the Higgs boson into two mesons, which can proceed through a multitude of intermediate states coupling to the scalar particle: quarks, gluons, virtual EW gauge bosons.First estimates of these processes were performed ignoring the internal motion of the produced quarkantiquark pairs [104], and then further improved within different approaches for the meson-pair formation [105][106][107][108][109][110].Except for a channel involving the ϕ meson, only exclusive decays involving charmonium and/or bottomonium final states have been computed to date, i.e., no calculations of decays to a pair of light mesons exist to our knowledge (due to the difficulties explained in Section IV D for the similar Z → M + M case).) decay modes.The calculations are carried out in multiple frameworks (LC + LCDA, RQM, NRQCD/NRCSM, NRQCD + LDME) and predict rates in the O(10 −9 -10 −11 ) range 3 , with some differences in the results for the same final state driven by the partial inclusion of the diagrams shown in Fig. 10.As a matter of fact, the existing predictions have not fully included all processes shown in the figure with, e.g., the W-induced and quark "crossed" decays often considered subleading and not added to the rates.The H → ϕ + J/ψ decay is the only process computed to date that includes the H → W * W * intermediate diagram.The direct (quark-induced) contributions are only relevant for the heavier double bottomonia with larger Yukawa couplings, and final decays involving charmonium ignore them.

′
) decays have been searched for at the LHC [96], but the current O(10 −3 -10 −4 ) limits are 4-5 orders-of-magnitude larger than the predictions.Their production at the HL-LHC, as well as at any future lepton collider, appears unfeasible, and only a machine like FCC-hh will have the Higgs production rates required to reach those decay modes, as indicated in Fig. 11.TABLE IX.Compilation of exclusive Higgs decays to two mesons.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it, as well as the current experimental upper limit and that estimated for HL-LHC.The last two columns indicate whether the decay can be produced at FCC-ee/FCC-hh.
-log(BR) The two-body decay of the Z boson into two massless vector particles (Z → γγ, gg) is forbidden by the Landau-Yang theorem, which states that a spin-1 boson cannot decay into two massless bosons [112,113].On the other hand, three-body decays into gauge bosons, or into a photon plus two neutrinos, are possible but can only be induced at the loop level in the SM (Fig. 12).Calculations for these virtual processes exist since many years [114][115][116][117] and, to our knowledge, have not been revised since, although key underlying parameters, such as the α s (m Z ) coupling constant, hadronic Z boson partial widths, and top quark mass, have (noticeably) changed.For example, the original Z → 3γ calculations from [115] used m t = 124 GeV, whereas the calculation of the Z → 3g decay of [116] used m t → ∞, and the prediction for Z → g + g + γ from [114] used α s (m Z ) = 0.17, m t = 20 GeV, and Γ Z = Γ Z→qq ≈ 2 GeV.We have therefore recalculated those Z boson rare decay branching fractions with MG5_aMC in the SM framework with QCD and EW loop corrections computed up to NLO accuracy using the input parameters of Table II.To validate our setup, we compare first the results that we obtain with MG5_aMC to the old ones with the same input parameters used in the past, finding good agreement in all cases (except one, see next).The updated branching fractions (Table X) change with respect to the existing ones by 10-25% up or down, except for the Z → γ + νν case, where our result is six times smaller than that estimated in [117].This older work quoted a partial width for the triangle diagram which is smaller than our results for this contribution, whereas the relative impact of box diagrams was not clearly specified.So we suspect that a larger cancellation of contributions is present in our setup compared to that in this previous study.TABLE X. Compilation of exclusive Z decays to three gauge bosons (photons and/or gluons, with two additional channels requiring one soft photon, E γ < 1 GeV, as explained in the text) and a photon plus two neutrinos.For each decay, we provide the predicted branching fraction and the theoretical approach used to compute it, as well as the current experimental limit and that estimated for HL-LHC.The last column indicates whether the decay can be produced at FCC-ee.Among the four rare Z boson decays shown of Fig. 12, those with final-state gluons have the largest rates, O(10 −6 ), whereas the pure electroweak processes have much smaller, O(10 −10 ), branching fractions.Experimentally, limits exist for Z → 3g and Z → 3γ of O(10 −2 ) and O(10 −6 ), respectively, although no searches have been yet performed to our knowledge for Z → γ g g and Z → γ νν decays.The study of the 3-gluon decay appears hopeless at hadronic machines given the huge QCD trijet backgrounds, although the triphoton decay can be constrained to 100 times the expected SM value at the HL-LHC.Observation of all such decays appears only feasible at a lepton collider such as FCC-ee 4 , as indicated by the red vertical line in Fig. 13.We also provide estimates of the branching fraction for the Z → gg γ soft and Z → γγ γ soft decays (where the soft photon has E γ < 1 GeV) that can mimic the forbidden Z → gg, γγ channels if the low-energy photon goes experimentally undetected.The branching ratios for both such "fake" Landau-Yang-violating decay modes are O(10 −9 ) and O(10 −12 ), respectively.-log(BR)

B. Exclusive Z boson decays into a gauge boson plus a meson
Figure 14 shows the schematic diagrams of the exclusive decay of the Z boson into a photon (or a W boson) plus a meson.Due to the similarity of these diagrams to those of the Higgs boson (Fig. 3), and since the Z boson yields at colliders are about three orders-of-magnitude larger than for the scalar boson (Table I), the study of such processes provides valuable information on, both, theoretical elements (SCET and NRQCD validation, LCDAs/LDMEs constraints, etc.) and experimental aspects (optimization of search techniques) for the corresponding studies of exclusive Higgs boson decays.The observation of any such exclusive decays would provide unique opportunities to reconstruct the Z boson from isolated hadrons, and would improve our knowledge of meson transition form-factors, which describe the M → γ * γ ( * ) decays.In addition, exclusive Z boson decays into flavoured mesons such as that shown in Fig. 14 (left) probe FCNC processes.

Z boson decays into a photon plus a meson
The theoretical predictions and experimental limits for the branching fractions of the radiative Z boson decay into light, charm, and bottom mesons (Z → γ + qq, cc, bb) are listed in Table XI, XII, and XIII, respectively, and shown in graphical form in Fig. 15.After forty years from their discovery, no hadronic radiative decay of the electroweak bosons has yet been observed, despite searches performed by the CDF [119,120], ATLAS [81,84,[121][122][123], CMS [83,85,124,125], and LHCb [126] collaborations.In some cases, such as for the Z → γη, γη ′ channels, LEP measurements at the Z pole still provide the best upper limits [127].
The exclusive Z-boson radiative decays to light mesons (Z → γ + M), listed in Table XI, have theoretical branching fractions in the range of O(10 −8 -10 −12 ), whereas the current experimental limits are in the O(10 −5 -10 −7 ) range with seven channels searched for by ATLAS (3), LHCb (1), CDF at Tevatron (1), and ALEPH at LEP (2).The Z → γω decay is very close to being detected at HL-LHC because its predicted B is about half of our projected limit.All channels are producible at the FCC-ee, except the FCNC Z → γ + D 0 one, which is the only one studied to date that proceeds through the right diagram of Fig. 14.TABLE XI.Compilation of exclusive Z decays to a photon plus a light meson.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it, as well as the current experimental upper limit and that estimated for HL-LHC.The last column indicates whether the decay can be produced at FCC-ee.The exclusive Z-boson radiative decays to charmonium mesons (Z → γ + cc), collected in Table XII, have rates in the O(10 −8 -10 −10 ) range as computed within multiple approaches (LC, SCET, NRQCD), which are in general consistent among themselves for the same process.Only one single channel has been searched for, Z → γ + J/ψ, with O(10 −6 ) experimental upper bounds [85,124].This decay may be visible at HL-LHC, because its predicted rate is about 1/5 of our projected limit.All channels are producible at FCC-ee.

Exp. limits
The exclusive Z-boson radiative decays to bottomonium mesons (Z → γ + bb), listed in Table XIII, have also branching fractions in the O(10 −8 -10 −10 ) range.Three Z → γ + Υ(nS) channels have been searched for at the LHC, setting limits in the O(10 −6 ) range.The Z → γ+Υ(1S) channel might be visible at HL-LHC, as the predicted branching fraction is about 1/4 of our projected limit.As for the photon-plus-charmonium case, all photon-plus-bottomonium channels are producible at FCC-ee.TABLE XII.Compilation of exclusive Z decays to a photon plus a charmonium state.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it, as well as the current experimental upper limit and that estimated for HL-LHC.The last column indicates whether the decay can be produced at FCC-ee.(1.0 ± 0.4) × 10 −9 LC+LCDA [133] TABLE XIII.Compilation of exclusive Z decays to a photon plus a bottomonium state.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it, as well as the current experimental upper limit and that estimated for HL-LHC.The last column indicates whether the decay can be produced at FCC-ee.-log(BR) -log(BR)

Z boson triphoton decays
As seen in Table X, the Z → 3γ decay is extremely suppressed in the SM (B = 6.4•10 −10 , as computed here) making of such a final state a particularly clean means to search for exotic Z decays into BSM particles with EW couplings.Many new physics scenarios contain new light (pseudo)scalar or tensor particles that couple to the electroweak (or Higgs) bosons and decay primarily to photons.Searches for Z → γ a(γγ), with a being an ALP [20,137] or a graviton [21] decaying into photons, have attracted an increasing interest in the last years [19].Figure 16 shows possible diagrams leading to the triphoton decay of the Z boson.The direct decay (left diagram) and the exclusive radiative decays into C-even mesons followed by their diphoton decays (center) can mimic the BSM processes signals (right), and constitute a background to the latter.The direct and mesonic backgrounds have been neglected so far in all ALP/graviton upper limits set in studies based on current and future triphoton Z decay data.Such an assumption is justified so far given the null sensitivity to such rare SM processes, but it will not be the case at the FCC-ee facility where, as we see below, a few thousands such events are expected during the whole Z-pole run.Here, we quantify the triphoton branching fractions from meson decays by combining the exclusive Z → γ + M results for spin-0,2 mesons of Tables XI-XIII with their corresponding two-photon branching fractions [53].The results are listed in Table XIV.The sum of all exclusive mesonic decay channels amounts to B = 1.8 • 10 −10 , representing an increase of about 30% from the direct 3γ decay.The yields of SM triphoton decays will therefore amount to about 5 000 events for the N(Z) = 6 • 10 12 Z bosons expected to be collected during the full Z-pole operation at the FCC-ee.Searches for ALPs and gravitons will have to be carried out on top of such a relatively large number of background events that have been neglected so far in the derivation of ALP limits at future e + e − facilities [137].

Z boson decays into a W boson plus a meson
Table XV lists the theoretical predictions for Z boson decays into a W ± boson plus a charged vector meson (corresponding to the diagrams of Fig. 14 with W bosons in the final state).All branching fractions are in the 10 −10 -10 −11 range, and only two channels have been searched for in the Z-pole run at LEP. Experimental observation will only be possible at FCC-ee for all of them.The same results compiled in Table XV are shown in graphical form in Fig. 17.TABLE XV.Compilation of exclusive Z decays to W plus a meson.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it, as well as the current experimental upper limit and that estimated for HL-LHC.The last column indicates whether the decay can be produced at FCC-ee.

C. Radiative Z boson leptonium decays
Figure 14 shows the diagrams for the decay of the Z boson into a photon plus a leptonium state.Depending on the relative helicities of the outgoing leptons, their bound state can be in the spin singlet (paraleptonium, (ℓℓ) 0 ) or triplet (ortholeptonium, (ℓℓ) 1 ) states.No calculation for such processes exist to our knowledge, but the key ingredients can be obtained from the determination of the e + e − → γ + (τ + τ − ) 0 cross section at the Z pole provided in the ditauonium studies of Ref. [138].
The leading-order partial decay width of a Z boson into a leptonium-plus-a-photon can be derived from the ratio of the σ(e + e − → γ + (ℓ + ℓ − )) cross section at the Z pole over the total resonant Z boson cross section in e + e − collisions, and amounts to for the paraleptonium case, and to for the ortho-leptonium one, where n is the principal quantum number of the leptonium resonance, and s w ≡ sin θ W , and c w ≡ cos θ W are the sine and cosine of the Weinberg angle (Table II).In both expressions above, α(0) and α(m Z ) are the electromagnetic coupling evaluated at zero5 and at the Z pole mass, respectively, that, together with the rest of parameters, are listed in Table II.The obtained branching fractions are tabulated in Table XVI, and plotted in Fig. 19.The dependence of these Z-boson radiative rates on α 6 m 2 ℓℓ , leads to vanishingly small branching fractions, O(10 −13 -10 −23 ), with ditauonium (positronium) featuring the largest (smallest) values.No experimental search has been conducted to date, although the relatively long lifetime of the leptonium objects (significantly boosted in the decay of the much heavier Z boson) would lead to a clean signature characterized by a displaced vertex from secondary decays of the (ℓ + ℓ − ) into photons, e ± , and/or µ ± [99], akin to many BSM long-lived particles (but with an invariant mass at the corresponding leptonium mass).Given the negligible rates, only an FCC-ee run at the Z pole would be able to provide limits approaching the SM values for the Z → γ (τ + τ − ) 0 case.TABLE XVI.Compilation of exclusive Z decays into a photon plus a ground state (n = 1) of ortho-and para-leptonium.For each decay, we provide the prediction of its branching fraction computed via Eq.( 12) or (13).The last column indicates whether the decay can be produced at FCC-ee.with those of the Higgs boson (Fig. 10), and since the Z boson yields at colliders are ∼1000 times larger (Table I), the study of such processes provides valuable information for the corresponding Higgs rates.The two-body decays into the same pair of pseudo-scalar mesons, Z → M + M, are quantum-mechanically forbidden due to the presence of two identical final-state particles and conservation of angular momentum (thus violating the spin-statistics theorem), but the mesons M can be in a vector state.The decays of Z bosons into double quarkonia, first studied in 1990 [139], have been investigated as a means to provide clean information on the quarkonium bound-state dynamics at large momentum transfers.As explained below, there is still a relatively large uncertainty in the theoretical predictions, but this is still the most promising place to search for exclusive double-charmonia/bottomonia decays.To our knowledge, there are no calculations of exclusive decays of the Z boson involving light mesons, but only into charmonium and/or bottomonium states.We have evaluated here for the fist time the double light vector meson decay Z → VM + VM via the two direct (left and center) diagrams of Fig. 20, using the EFT + NRQM formalism of [140] in which the width can be written as

Exp. limits
where Γ 1 VM VM is the partial width corresponding to the left diagram of Fig. 20, amounting to and where R is the ratio of the direct amplitudes A 1 and A 2 corresponding, respectively, to the left and center diagrams of Fig. 20, given by For the light meson wavefunction at the origin ϕ M (0) in Eq. ( 15), we derive it from its decay constant using the Van Royen-Weisskopf expression, Eq. ( 7), as follows for N c = 3 colours, and without higher-order QCD corrections.We can only estimate the rate for the Z → ϕ+ϕ channel because it is the only pure |qq⟩ state among the light mesons.Using Eqs. ( 14)-( 17) and the parameters of Table II, we obtain a B = 2.1 • 10 −12 rate (Table XVII).There is no Z → ϕ + ϕ experimental search performed to date, and only a machine like FCC-ee can provide enough Z bosons to start producing the decay.The table also quotes the forbidden Z → π 0 + π 0 channel, for which the CDF upper bound could be improved by a factor 100 at the HL-LHC.To provide a reasonable evaluation for exclusive double decays into other light mesons, rotation from flavour eigenstates |uu⟩, |dd⟩ to physical eigenstates such as ρ, ω, η... would need to be performed.One can, however, anticipate branching fractions of the same order as the double-ϕ channel, O(10 −12 ), since they only differ from a rotation and they are enhanced by the second and third diagram of Fig. 20 involving extra γ * , W * contributions.The scenario where Z decays into light pseudoscalar plus vector mesons suffers from the same complications of mixed states, but their expected rates are expected to be of the same order-of-magnitude as the double vector meson case because they have the same enhancing effect from the photon propagator contribution (Fig. 20, center) [140].
TABLE XVII.Compilation of exclusive Z decays to a pair of light mesons.For the forbidden two-pion decay, we provide the current experimental upper limit.For the double-ϕ decay, we provide the branching fraction predicted using Eqs.( 14)-( 17).The last column indicates whether the decay can be produced at FCC-ee.
Exp. limits Producible at Tables XVIII and XIX list the theoretical predictions and experimental limits for concrete two-meson decay modes with pairs of charmonium mesons and bottomonium plus other mesons, respectively.The same results are shown in graphical form in Figs.21 and 22.The decay rates are very small, B(Z → M + M) ≲ 10 −10 , and quite sensitive to the model details and ingredients.All heavy-quarkonium channels have rates obtained within the NRQCD + LDMEs formalism.Alternative LC calculations show a good agreement with the latter [141] but, unfortunately, they can only provide predictions for final states with different mesons or with mesons in different excited states.The NRQCD predictions from the work [141] show discrepancies with other later works, because they did not consider diagrams involving a photon propagator (Fig. 20, center) that can enhance the rates by more than two orders-of-magnitude.In contrast, this latter study provides results for some excited states that have not been calculated elsewhere (although their results for channels such as Z → χ c1 + χ c1 , χ c2 + χ c2 , χ c0 + χ c2 ,..., which cannot be predicted using LC formalism, may need to be revisited for the same reasons stated above).In addition, recent calculations including higher-order QED [142] and QCD + QED [143] corrections yield significant enhancements in the branching ratios for doublequarkonia production.All that said, and within the relatively large theoretical uncertainties for those processes, the largest branching fractions expected are for double-charmonium with O(10 −10 ) rates.Despite a few searches carried out at LEP (and one at the LHC), experimental observation will only be possible at FCC-ee for about half of them.TABLE XVIII.Compilation of exclusive Z decays to two charmonium mesons.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it, as well as the current experimental upper limit and that estimated for HL-LHC.The last column indicates whether the decay can be produced at FCC-ee.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it, as well as the current experimental upper limit and that estimated for HL-LHC.The last column indicates whether the decay can be produced at FCC-ee.  ).Four channels have been searched for to date at the LHC (3 by CMS, 1 by ATLAS, and 1 by LHCb), and the HL-LHC will improve the existing limits by about one order-of-magnitude, but still far from any possible observation.The FCC-ee will be able to produce events for three decay modes, and the FCC-hh will produce all of them.The results listed in Table XX are presented in graphical form in Fig. 24.TABLE XX.Compilation of exclusive W decays to a photon plus a meson.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it, as well as the current experimental upper limit and that estimated for HL-LHC.The last two columns indicate whether the decay can be produced at FCC-ee/FCC-hh. a updated inputs from [151] for the result from [62] B. Exclusive W decays into two mesons   responding theoretical predictions and experimental limits for concrete decay modes, all of them involving charm and/or bottom quarks mesons.We are not aware of any study of exclusive W-boson decays involving one or two light mesons in the literature 6 .The expected rates are truly tiny, of order O(10 −11 -10 −14 ), with relatively large uncertainties.The work of [153] compared the branching fractions predicted within two approaches (LC and NRCSM), which differ by factors of 5-10, whereas Ref. [150] computed those within the NRQCD framework.For decays with a small ratio m M /m W , the results from the LC approach should be preferable.Differences among predictions for the same process appear in some cases as due to not consistently considering all diagrams of Fig. 25.The effect of the center diagram in Fig. 25 was partially discussed in [153], but none of the two theoretical works that have so far studied these processes, has considered the rightmost diagram of Fig. 25.Experimentally, none of the listed decays has been searched for at LEP, Tevatron, or the LHC to date.Only FCC-hh has possibilities to produce about eight decay channels (Fig. 26), but the expected tiny rates seemingly preclude their observation in the complex hadron-collider environment.The top quark decays to a W boson and a bottom quark, t → W b, with a branching fraction of nearly 100%, with the other tree-level decays t → W s and t → W d comparatively suppressed by factors of 10 −3 and 10 −4 , respectively, as per the CKM element hierarchy V tb ≫ V ts > V td (Table II).The FCNC top-quark decays to a gauge boson plus a light up-type quark (q = u or c; for charge conservation), t → q Z, t → γ c, and t → g c, occur only at the level of quantum loop corrections (Fig. 27), and are extremely suppressed in the SM.More precisely, they are 1-loop-, CKM-, and GIM-suppressed (with a GIM suppression factor 7 given by f (m 2 b /m 2 W ) ≈ 10 −9 ), leading to tiny amplitudes.In many BSM models, however, the GIM suppression can be relaxed, and one-loop diagrams mediated by new bosons may also contribute, yielding effective couplings orders of magnitude larger, and correspondingly enhancing such ultrarare branching fractions [27,28].TABLE XXII.Compilation of rare two-body top quark decays.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it, as well as the current experimental upper limit and that estimated for HL-LHC.The last two columns indicate whether the decay can be produced at FCC-ee/FCC-hh.NLO SM [28] Table XXII lists the branching fractions of FCNC top decays, and Fig. 28 displays them graphically.The predicted SM rates are tiny, O(10 −12 -10 −17 ), and very sensitive to the value of the b-quark mass used to calculate them 7 .The decay branching fractions t → Z q, t → γ q, t → g q, t → H q, are further suppressed for q = u compared to the q = c case, by a CKM factor (|V ub |/|V cb |) 2 ≈ 0.008.Experimentally, FCNC decays of the top quark have attracted lots of interest in the quest for BSM phenomena at the LHC, and all eight channels have been searched for.The current experimental upper limits are set in the O(10 −4 -10 −5 ) range, and will be improved by about a factor of ten at the end of the HL-LHC.In the absence of BSM effects enhancing such FCNC top decays, only the FCC-hh will reach top production rates capable of probing the t → g c decay.-log(BR) Theoretical BR Current limits HL-LHC 14 TeV, 2x3 ab −1 FCC-ee, N(t)=3.8 × 10 6 FCC-hh, N(t)=2.1 × 10 12 FIG.28.Branching fractions (in negative log scale) of rare two-body t → V + q, H + q decays (with V = gauge boson, and q = u, c).The most recent theoretical predictions (blue bars) are compared to current experimental upper limits (violet) and expected HL-LHC bounds (orange).The red vertical lines indicate the FCC-ee (solid) and FCC-hh (dashed) reach based only on the total number of top quarks expected at both facilities.

B. Three-body top quark decays
Thanks to its large mass, the top quark has different rare decays kinematically accessible involving the presence of multiple heavy bosons in the final state [27,160].The possibility of 3-body radiative decays of the top quark t → W b X, where X can be a Z or a Higgs boson has been considered, e.g., in Refs.[161,162] (the cases X = γ, g are simply NLO real QED or QCD corrections to the dominant tW b decay, for which one needs also an energy threshold to avoid infrared/collinear divergences in the decay rates, and not considered here as they are not rare).The corresponding diagrams are shown in Fig. 29.Properly taking into account the finite width effects is key to compute any close-to-threshold 3-body decays with (partially) offshell particles [157,162].
Table XXIII collects the branching fractions for rare three-body decays of the top quark, and Fig. 30 presents them in graphical form.No experimental search has been performed to date.By an amusing coincidence, the t → Z W b decay is kinematically allowed because m t ≈ (m W +m b )+m Z to within a few GeV, which can be satisfied within the O(2 GeV) widths of the onshell electroweak bosons (Table II).The corresponding branching fraction is B(t → Z W b) ≈ 2 • 10 −6 and, although no experimental limits exist yet from the LHC data, it could be potentially discovered at the HL-LHC, and for sure observed at FCC-ee and FCC-hh.The branching fraction of t → H W b has been estimated here for the first time using MG5_aMC with the parameters listed in Table II, and found to be B ≈ 2 • 10 −10 .Other decays are highly GIM-suppressed (Fig. 29, bottom) and have much smaller rates.The t → ZZq is below O(10 −13 ), and likely impossible to observe anywhere unless some BSM mechanism enhances it.The top quark can also decay into three up-type quarks, either through the same diagrams shown in Fig. 27 where the emitted boson further decays/splits into uu or cc, or through a virtual W boson exchange.This process is dominated by the gluon splitting into two up-type quarks shown in the Fig. 29 (lower right panel), and has a decay rate of O(10 −12 ) [163] commensurate with that of the "parent" t → u g two-body decay (Table XXII).

C. Semiexclusive top quark decays into a quark plus a meson
There are also theoretical studies of semiexclusive top quark decays in which interactions among the decay quarks (either from the W decay, or combining the primary bottom quark with the W-decay quarks) lead to the formation of final states with one meson recoiling against a jet [26,[166][167][168].Such decays can provide a new method to measure the top-quark mass via a two-body (jet-meson) invariant mass analysis, with different systematic uncertainties of those of the currently existing approaches [26].Typical diagrams for the process are shown in Fig. 31 with the meson recoiling against a q = u, c quark (right), or against a b quark (left).[26].Four channels are producible at FCC-ee, and all of them can be produced at FCC-hh.TABLE XXIV.Compilation of exclusive decays of the top quark into a c or u quark plus a meson.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it, as well as the current experimental upper limit and that estimated for HL-LHC.The last two columns indicate whether the decay can be produced at FCC-ee/FCC-hh. -log(BR) W * decays into M = π ± , D s ± have been given in Refs.[27,166] in the EFT + LCDA approach, based on the expression where f 2 M is the decay constant of the meson, and |V qq ′ | 2 the relevant CKM matrix element, and where the last numerical equality is obtained with the parameters of Table II.We extend here the study of semiexclusive t → meson + b-jet decays to include also the M = ρ ± , K ± , D ± , D ± s , B ± , B ± c cases (and their corresponding excited states) by employing the decay constants from the recent compilation [62] for the light mesons, and from lattice calculations [65,68] for the heavy ones (Table II).The resulting t → b + M branching fractions are tabulated in Table XXV, and also shown in graphical form in Fig. 33.Estimated theoretical branching fractions are in the O(10 −7 -10 −13 ) range.No experimental search has been performed to date at the LHC.Although no channel is producible in the clean FCC-ee environment, most channels will be accessible at FCC-hh.TABLE XXV.Compilation of exclusive decays of the top quark into a b quark plus a meson.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it, or derived here using Eq.(18).The last two columns indicate whether the decay can be produced at FCC-ee/FCC-hh.

VII. SUMMARY
We have presented a comprehensive survey of the theoretical and experimental status of about 200 rare and exclusive few-body decays of the four heaviest Standard Model (SM) particles: the Higgs, the electroweak (W, Z) bosons, and the top quark.Rare decays are defined here as those having branching fractions below B ≈ 10 −5 , and we focus on those with two-or three-particles in the final state.Such decay processes remain experimentally unobserved and only upper limits have been set for about 50 of them.The study of these decay processes provides a powerful window into physics beyond the Standard Model (BSM) either directly, by probing SM suppressed or forbidden processes (such as flavour-changing neutral currents, FCNC, or spin-selection-rules violating decays), or indirectly as SM backgrounds to multiple exotic BSM decays (e.g., into axion-like particles ALPs, gravitons, or dark photons).Additionally, such decays offer a unique opportunity to constrain the light-quark Yukawa couplings, can provide alternative means to measure the W and/or top-quark masses, and help improve our understanding of quantum chromodynamics (QCD) factorization with small nonperturbative corrections.
First, we have systematically collected and organized in tabular form the theoretical branching fractions of almost 200 rare decay channels, indicating the model(s) used for the calculations, while providing any existing experimental upper limits on them.Among those, we have estimated for the first time the rates of about 40 new processes including ultrarare Higgs boson decays into photons and/or neutrinos (with rates 10 −12 -10 −40 ), radiative H and Z boson decays into leptonium states (with rates 10 −10 -10 −23 ), exclusive radiative H quark-flavour-changing decays (with rates 10 −14 -10 −27 ), exclusive Z decays into a pair of ϕ mesons (with an O(10 −12 ) rate), three-body H W b top-quark decay (with an O(10 −10 ) rate), and semiexclusive top-quark decays into a quark plus a meson (with rates 10 −7 -10 −13 ).In addition, we have revised and updated predictions for a few other rare Z-boson and top-quark partial widths.We have also studied in detail all Z decay channels leading to a triphoton final state, and found that the sum of all relevant exclusive photon-plus-meson decays amounts to B(Z → γ + M → 3γ) = 1.8 • 10 −10 , which is about one-third of the direct rate estimated here to be B(Z → 3γ) = 6.4 • 10 −10 .All such decays will need to be taken into account as backgrounds for future searches of Z → γa(γγ) processes, where a can be a BSM particle such as spin-0 (axion-like) or spin-2 (graviton-like) object.
Secondly, the feasibility of measuring each of these unobserved decays has been estimated for p-p collisions at the high-luminosity Large Hadron Collider (HL-LHC), as well as for e + e − and p-p collisions at the future circular  collider (FCC-ee and FCC-hh).From the number of H, Z, W bosons, and top quarks expected to be produced at the HL-LHC, FCC-ee, and FCC-hh colliders, and by statistically extrapolating the current 95% confidence-level limits set, we provide estimates of the ultimately achievable experimental upper bounds (or observations) in those machines.Among those, in Table XXVI we have selected a few interesting channels that can be observed and/or deserve further study in p-p collisions at the HL-LHC.The last column indicates the approximate ratio of theoretical to expected experimental rates, B(th)/B(exp).Two reasons motivate this selection of channels: either (i) they have relatively large rates, O(10 −5 ) (with a measurement having been attempted or not, to date), or (ii) they have lower rates, B ≲ 10 −8 , but measurements of upper bounds have been performed, and our projections indicate B(th)/B(exp) not far from unity.
For the Higgs boson, we list first the rare H → 4γ decay that has not been directly searched for at the LHC to date, but limits exist on the process H → a(γγ)a(γγ) with two intermediate ALPs decaying into photons [61,[76][77][78][79].Although its rate makes its observation impossible at the LHC in the absence of enhancing BSM effects, it would not be difficult for ATLAS/CMS to recast any current and future similar ALP searches into upper bounds on the H → 4γ "continuum" decay.Next, we find that the HL-LHC can set upper bounds on the exclusive H → γ ρ and H → γ J/ψ decays at about four and ten times their expected SM values, respectively.Similarly, we find that HL-LHC can potentially observe Z → γ ω, Z → γ J/ψ, and Z → γ Υ(1S ) with ratios between the theoretical rates and projected experimental limits as large as 1/2, 1/3, and 1/4, respectively.In the W boson case, the least suppressed rare decays are W → γ π, γ D s but with experimental bounds expected at about two orders-of-magnitude their SM values.Finally, the semiexclusive decay of the top quark into a charm jet and B meson (identified experimentally as a b-jet) has O(10 −5 ) rates that make it observable at the HL-LHC [26], and the three-body t → W Z b decay can occur in about one out of one million top-quark decays, and a search should also be attempted.Of course, the HL-LHC channel selection of Table XXVI is driven by the SM rates and their potential visibility, but as explained in this work there are many other suppressed decays of the four heaviest particles that can be enhanced in many BSM scenarios, and should be active part of target searches in the next years.
Finally, we have shown that a future high-luminosity e + e − electroweak factory, such as the FCC-ee with 6 • 10 12 Z bosons, 5 • 10 8 W bosons, 1.9 • 10 6 Higgs bosons, and 3.8 • 10 6 top quarks produced in very clean experimental conditions, can discover about half of the 200 rare decays discussed here.Eventually, the FCC-hh with huge data samples expected, can produce most of the decays channels listed in this work, although their experimental observation will be difficult in a challenging hadronic collider environment.We hope that this document can help guide and prioritize upcoming experimental and theoretical studies of rare and exclusive few-body decays of the most massive SM particles, as well as further motivate BSM searches, at the LHC and future colliders.

FIG. 3 .
FIG. 3. Schematic diagrams of exclusive decays of the H boson into a meson plus a gauge boson: direct (left), indirect (center), and W-loop (right) processes.The solid fermion lines represent quarks, and the gray blob represents the mesonic bound state.
FIG. 7. Branching fractions (in negative log scale) of exclusive H → W ± + meson decays.The theoretical predictions are shown as blue bars.The red vertical lines indicate the FCC-ee (solid) and FCC-hh (dashed) reach based only on the total number of H bosons to be produced at both facilities.

− 1 FCC 10 FIG. 9 .
FIG.9.Branching fractions (in negative log scale) of exclusive H → γ, Z + leptonium decays: The theoretical predictions computed here are shown as blue bars.The red vertical lines indicate the FCC-ee (solid) and FCC-hh (dashed) reach based only on the total number of H bosons to be produced at both facilities.

FIG. 10 .
FIG. 10.Schematic diagrams of exclusive decays of the H boson into two mesons.The wavy lines indicate gauge bosons, the solid fermion lines represent quarks and the gray blobs the meson bound states.

FIG. 12 .
FIG. 12. Representative diagrams of rare 3-body decays of the Z boson into gluons and/or photons and into a photon plus neutrinos.The solid fermion lines represent quarks in the first and second diagrams, and quarks and leptons in the third one.

− 1 FCC 12 FIG. 13 .
FIG.13.Branching fractions (in negative log scale) of rare three-body Z boson decays.The most recent theoretical predictions (blue bars) are compared to current experimental upper limits (violet) and expected HL-LHC bounds (orange).The red vertical line indicates the expected FCC-ee reach based only on the total number of Z bosons to be produced.

FIG. 14 .
FIG. 14.Schematic diagrams of exclusive decays of the Z boson into a photon or W boson plus a meson.The solid fermion lines represent quarks, and the gray blobs the mesonic bound state.

− 1 FCC 12 FIG. 15 .
FIG.15.Branching fractions (in negative log scale) of exclusive Z → γ + M decays, with M being light (upper), charm (middle), and bottom (lower) mesons.The most recent theoretical predictions (blue bars) are compared to current experimental upper limits (violet) and expected HL-LHC bounds (orange).The red vertical line indicates the expected FCC-ee reach based only on the total number of Z bosons to be produced.

− 1 FCC 12 FIG. 17 .
FIG. 17. Branching fractions (in negative log scale) of exclusive Z → W ± + meson decays.The most recent theoretical predictions (blue bars) are compared to current experimental upper limits (violet) and expected HL-LHC bounds (orange).The red vertical line indicates the expected FCC-ee reach based only on the total number of Z bosons to be produced.

FIG. 18 .
FIG. 18. Schematic diagrams of exclusive decays of the Z boson into para-(left) and ortho-(right) leptonium plus a photon.The solid fermion lines represent leptons, the gray blobs represent the leptonium bound state.

Figure 20 displays
Figure 20 displays the diagrams of exclusive Z boson decays into two mesons, with contributions from direct quark decays (left and center), as well as from indirect V * → M transitions (right).Due to the coincidence of diagrams

− 1 FCC 12 FIG. 19 .
FIG.19.Branching ratios (in negative log scale) of exclusive Z → γ + leptonium decays.Theoretical predictions computed here are shown as blue bars.The red vertical line indicates the expected FCC-ee reach based on the total number of Z bosons to be produced.

FIG. 20 .
FIG. 20.Schematic diagrams of exclusive decays of the Z boson into two mesons.The solid fermion lines represent quarks, and the gray blobs the mesonic bound state.

12 FIG. 21 .
FIG.21.Branching fractions (in negative log scale) of exclusive Z boson decays into two light mesons (upper) or two charmonium mesons (lower).The most recent theoretical predictions (blue bars) are compared to current experimental upper limits (violet) and expected HL-LHC bounds (orange).The red vertical line indicates the expected FCC-ee reach based only on the total number of Z bosons to be produced.

12 FIG. 22 .
FIG.22.Branching fractions (in negative log scale) of exclusive Z → b-meson + X decays.The most recent theoretical predictions (blue bars) are compared to current experimental upper limits (violet) and expected HL-LHC bounds (orange).The red vertical line indicates the expected FCC-ee reach based only on the total number of Z bosons to be produced.

FIG. 23 .
FIG. 23.Schematic diagrams of exclusive decays of the W bosons into a photon plus a meson.The solid fermion lines represent quarks, and the gray blobs represent the mesonic bound state.

Figure 25
Figure 25 shows the schematic diagrams of the exclusive decay of the W boson into two mesons proceeding through quark decays (left and center), or through intermediate W * , γ * → M transitions (right).Table XXI lists the cor-

− 1 FCC 13 FIG. 24 .
FIG.24.Branching ratios (in negative log scale) of exclusive W ± → γ + meson decays.Most recent theoretical predictions (blue bars) are compared to current experimental upper limits (violet) and expected HL-LHC bounds (orange).The red vertical lines indicate the FCC-ee (solid) and FCC-hh (dashed) reach based only on the total number of W bosons to be produced at both machines.

FIG. 25 .
FIG. 25.Schematic diagrams of exclusive decays of the W boson into two mesons.The solid fermion lines represent quarks, and the gray blobs the mesonic bound state.

− 1 FCC 13 FIG. 26 .
FIG. 26.Branching fractions (in negative log scale) of exclusive W ± → meson + meson decays: Most recent theoretical predictions are shown as blue bars.The red vertical lines indicate the FCC-ee (solid) and FCC-hh (dashed) reach based only on the total number of W bosons expected at both facilities.

FIG. 27 .
FIG. 27.Schematic diagrams of rare FCNC two-body decays of the top quark into a gauge (g, Z, or γ) boson or a Higgs boson plus an up-type quark (q = c, u).

FIG. 29 .
FIG. 29.Representative diagrams of rare three-body decays of the top quark into a pair of bosons plus a quark at tree-level (upper), and via loops into ZZq (lower left) and into three quarks (lower right).The outgoing quarks are either b, or up-type quarks (q = c, u), by charge conservation.

FIG. 30 .
FIG. 30.Branching fractions (in negative log scale) of rare three-body top quark decays: Most recent theoretical predictions are shown as blue bars.The red vertical lines indicate the FCC-ee (solid) and FCC-hh (dashed) reach based only on the total number of top quarks expected at both facilities.

FIG. 31 .
FIG. 31.Schematic diagrams of semiexclusive two-body decays of the top quark into a meson plus a q = c, u quark (left) or a b quark (right).

9
+0.2 −0.1 × 10 −11 NRQCD+LDME [26] --✘ The right diagram of Fig. 31 corresponds to the process of a top quark decaying through an offshell W boson with virtual mass m W * ≈ m M close to a mesonic resonance M that recoils against a b-jet.Estimates for the partial width of

− 1 FCC 12 FIG. 32 .
FIG. 32.Branching fractions (in negative log scale) of semiexclusive t → c/u quark + meson decays.Most recent theoretical predictions are shown as blue bars.The red vertical lines indicate the FCC-ee (solid) and FCC-hh (dashed) reach based only on the total number of top quarks expected at both facilities.

t 12 FIG. 33 .
FIG.33.Branching fractions (in negative log scale) of semiexclusive t → b quark + meson decays.The theoretical rates obtained with Eq. (18) are shown as blue bars.The red vertical lines indicate the FCC-ee (solid) and FCC-hh (dashed) reach based only on the total number of top quarks expected at both facilities.

TABLE II .
[53]rical values of SM parameters used here in theoretical calculations of various branching fractions.Most of the values are from the PDG[53](the leptonium masses, m ℓℓ , are given as twice the single lepton masses, ignoring tiny binding energies), except for those where a specific reference is provided.

TABLE VII .
Compilation of exclusive Higgs decays to W boson plus a meson.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it.The last two columns indicate whether the decay can be produced at FCC-ee/FCC-hh.

TABLE VIII .
Compilation of exclusive Higgs decays to a photon or a Z boson plus an ortho-(ℓ + ℓ − ) 1 or para-(ℓ + ℓ − ) 0 leptonium state (only the ground states, n = 1, are considered).For each decay, we provide the prediction of its branching fraction computed here.The last two columns indicate whether the decay can be produced at FCC-ee/FCC-hh.

Table
IX lists the corresponding theoretical predictions and experimental limits for concrete H → 2(QQ) and H → (QQ)(QQ

TABLE XIX .
Compilation of exclusive Z decays to one bottomonium meson plus another meson (or anything else, in a few cases).
Table XXI lists the cor-

TABLE XXI .
Compilation of exclusive W decays into two mesons.For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it.The last two columns indicate whether the decay can be produced at FCC-ee/FCChh.

TABLE XXIII .
Compilation of rare three-body top quark decays (u 1,2 = u, c quarks).For each decay, we provide the predicted branching fraction(s) and the theoretical approach used to compute it.The last two columns indicate whether the decay can be produced at FCC-ee/FCC-hh.

TABLE XXVI .
Selection of rare and exclusive decays of the H, Z, W bosons and top quark potentially observable in pp(14 TeV) collisions at the HL-LHC.The last column indicates the approximate ratio of theoretical to our expected projected rates, B(th)/B(exp).