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

David d’Enterria<EMAIL_ADDRESS>CERN, EP Department, CH-1211
Geneva, Switzerland  Van Dung Le<EMAIL_ADDRESS>University of Science -
Ho Chi Minh National University, Vietnam

###### Abstract

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 $\mathrm{e}^{+}\mathrm{e}^{-}$ and p-p collisions
at the future circular collider (FCC).

###### Contents

  1. I Introduction
  2. II Results
    1. II.1 Theoretical predictions
    2. II.2 Experimental limits
    3. II.3 Input parameters of the calculations
  3. III Rare Higgs boson decays
    1. III.1 Rare two-, three-, and four-body Higgs boson decays
    2. III.2 Exclusive Higgs decays into a gauge boson plus a meson
      1. III.2.1 Higgs decays into a photon plus a vector meson
      2. III.2.2 Higgs decays into a photon plus a flavoured meson
      3. III.2.3 Higgs decays into a Z boson plus a meson
      4. III.2.4 Higgs decays into a W boson plus a meson
    3. III.3 Radiative Higgs leptonium decays
    4. III.4 Exclusive Higgs decays into two mesons
  4. IV Rare Z boson decays
    1. IV.1 Rare three-body Z boson decays
    2. IV.2 Exclusive Z boson decays into a gauge boson plus a meson
      1. IV.2.1 Z boson decays into a photon plus a meson
      2. IV.2.2 Z boson triphoton decays
      3. IV.2.3 Z boson decays into a W boson plus a meson
    3. IV.3 Radiative Z boson leptonium decays
    4. IV.4 Exclusive Z boson decays into two mesons
  5. V Rare W boson decays
    1. V.1 Exclusive W decays into a photon plus a meson
    2. V.2 Exclusive W decays into two mesons
  6. VI Rare top quark decays
    1. VI.1 Two-body top quark decays
    2. VI.2 Three-body top quark decays
    3. VI.3 Semiexclusive top quark decays into a quark plus a meson
  7. VII Summary

##  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
$\Lambda_{\mathrm{EW}}\approx\mathcal{O}(100$ 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
$\mathcal{L}_{\mathrm{\tiny{int}}}=3$ 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”
$\mathrm{e}^{+}\mathrm{e}^{-}$ 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 $\mathrm{e}^{+}\mathrm{e}^{-}$ colliders, p-p collisions at
$\sqrt{s}=100$ 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 final-state particles, with
branching fractions below $\mathcal{B}\approx 10^{-5}$—, 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.

Figure 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, $\gamma$
(or into a gauge boson plus two neutrinos $\nu$) 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 $\rm H\to qq^{\prime}$ 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 model-independent 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.,
$\mathrm{Z}\to\gamma\,a(\gamma\gamma)$ with $a$ being an axion-like particle
[20] or a graviton [21] decaying into photons; or $\rm
H\to,A^{\prime}(\ell^{+}\ell^{-})+X$ where a dark photon A’ further decays
into an $\ell^{+}\ell^{-}$ 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 $\rm t\to Zc$, $\rm t\to\gamma c$, and $\rm t\to cg$, 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 ($\alpha$ and $\alpha_{\text{s}}$,
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 well-established 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, $\mu_{\text{hard}}\approx\Lambda_{\mathrm{EW}}$
rendering the impact of nonperturbative power corrections at typical
hadronization energy scales $\Lambda_{\text{QCD}}\approx 0.2$ GeV,
$\mathcal{O}(\Lambda_{\text{QCD}}/\Lambda_{\mathrm{EW}})\lesssim 10^{-3}$,
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
$\sim$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.

###  II.1 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
($m_{\mathrm{Q}}\gg\Lambda_{\text{QCD}}$, for $\rm Q=c,b$), 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
$m_{\mathrm{q}}/E$ (with $m_{\mathrm{q}}$ and $E$ being the mass of the
produced quark and its typical energy, respectively), and the quarks form-
factors 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,
$\log(m_{\mathrm{M}}/\Lambda_{\mathrm{EW}}$), 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 form-factors 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, $m_{\mathrm{Q}}$ and $\Lambda_{\text{QCD}}$, 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 (Bc) mesons,
where the interactions are organized as an expansion in the typical relative
velocity of the heavy quarks inside the meson state ($v^{2}\approx 0.3,0.1$
for charmonium and bottomonium, respectively). The formation of heavy
quarkonium is a multi-scale problem, with $m(\mu_{\text{hard}})\gg mv\gg
mv^{2}\gg\Lambda_{\text{QCD}}$, 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 $\mathrm{Q}\overline{\mathrm{Q}}$ pairs are
produced in colour-singlet (CS) states at the hard-scattering scale
$\mu_{\text{hard}}$. The quantum numbers of the quark pair are the same as
those of the heavy quarkonium, and the $\mathrm{Q}\overline{\mathrm{Q}}$ 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.

###  II.2 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 1 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
$\mathrm{e}^{+}\mathrm{e}^{-}$ collider. We note, somehow curiously, that the
last LEP-II operation, which integrated 2.46 fb-1 over $\sqrt{s}=189$–209 GeV
[47], featured a Higgs cross section of $\sigma_{\mathrm{H}}\approx 3$ fb
(adding Higgstrahlung and weak-boson-fusion 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
$m_{\mathrm{H}}=115$ GeV scalar boson plus an onshell Z boson, for which the
H(125 GeV) signal would not have been visible). For the next
$\mathrm{e}^{+}\mathrm{e}^{-}$ 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 1 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
$\mathrm{t}\overline{\mathrm{t}}$ threshold. The largest data sample of all
will be for Z bosons, for which $6\cdot 10^{12}$ particles will be produced.
The number of H bosons at FCC-ee includes those produced in the HZ
($1.45\times 10^{6}$) and $\mathrm{t}\overline{\mathrm{t}}$ ($+330$k events)
runs, plus via $\rm WW\to H$ at all runs ($+125$k). Although, in much more
complicated background conditions than in $\mathrm{e}^{+}\mathrm{e}^{-}$
collisions, the HL-LHC (with 3 ab-1of p-p collisions at $\sqrt{s}=14$ TeV
collected per ATLAS/CMS experiment) and the FCC-hh (p-p at $\sqrt{s}=100$ 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
$\mathrm{p}\mbox{-}\overline{\mathrm{p}}$ collisions at $\sqrt{s}=1.96$ TeV,
as there exist still a couple of competitive rare decay limits from the CDF
experiment. The hadron collider numbers in Table 1 are obtained from the
corresponding production cross sections for each particle multiplied by the
integrated luminosity quoted (plus the contribution from
$\rm\mathrm{t}\overline{\mathrm{t}}\to W^{+}W^{-}+X$ 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_NNLO parton distribution
functions (PDFs) [52]. Theoretical (scale, PDF) uncertainties are at most 10%
and not quoted.

Table 1: Total number of Higgs, Z, and W bosons, and top quarks produced (or expected to be produced) in $\mathrm{e}^{+}\mathrm{e}^{-}$ collisions at LEP and FCC-ee, as well as in $\mathrm{p}\mbox{-}\overline{\mathrm{p}}$ at Tevatron, and in p-p collisions at HL-LHC, and FCC-hh. For $\mathrm{e}^{+}\mathrm{e}^{-}$ colliders, the Z-pole, Higgstrahlung and Higgs weak-fusion, and pair production (WW, $\mathrm{t}\overline{\mathrm{t}}$) cross sections (without ISR) are indicated. At LEP and FCC-ee, the numbers of W bosons and top-quarks consider two bosons and two quarks produced per collision at the $\mathrm{e}^{+}\mathrm{e}^{-}\to\rm W^{+}W^{-},\,\mathrm{t}\overline{\mathrm{t}}$ 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 $\mathrm{t}\overline{\mathrm{t}}$ decays.  Collider | W± bosons | Z bosons | H bosons | top quarks
---|---|---|---|---
|  $\sigma$(W) |  $N$(W) |  $\sigma$(Z) |  $N$(Z) |  $\sigma$(H) |  $N$(H) |  $\sigma$($\mathrm{t}\overline{\mathrm{t}}$) |  $N$(top)
LEP | 4.0 pb | $0.8\times 10^{5}$ | 59 nb | $2\times 10^{7}$ |  $\sim$2, 1 fb |  $\sim$5 | – | –
FCC-ee | 4.0 pb | $5\times 10^{8}$ | 59 nb | $6\times 10^{12}$ | 200, 30 fb | $1.9\times 10^{6}$ | 0.5 pb | $3.8\times 10^{6}$
Increase factor LEP$\,\mapsto\,$FCC-ee | 1 | 6250 | 1 | $300,000$ | 70, 30 | $400,000$ | – | –
Tevatron (1.96 TeV, 10 fb-1) | 25.3 nb | $2.5\times 10^{8}$ | 7.6 nb | $7.6\times 10^{7}$ | 1.1 pb | $1.1\times 10^{4}$ | 7.1 pb | $1.4\times 10^{5}$
HL-LHC (14 TeV, $2\times 3$ ab-1) | 200 nb | $1.2\times 10^{12}$ | 62.5 nb | $3.8\times 10^{11}$ | 58 pb | $3.5\times 10^{8}$ | 1 nb | $1.2\times 10^{10}$
FCC-hh (100 TeV, 30 ab-1) | 1300 nb | $4.1\times 10^{13}$ | 415 nb | $1.2\times 10^{13}$ | 0.93 nb | $2.8\times 10^{10}$ | 35 nb | $2.1\times 10^{12}$
Increase factor Tevatron$\,\mapsto\,$HL-LHC | 8 | 4800 | 8.2 | 5000 | 52.7 | 31 800 | 141 | 86 000
Increase factor HL-LHC$\,\mapsto\,$FCC-hh | 6.5 | 34 | 6.7 | 32 | 16 | 80 | 35 | 175

The largest yields increases across colliders are factors of $6\cdot 10^{3}$,
$3\cdot 10^{5}$, and $4\cdot 10^{5}$ for W, Z, and H bosons, respectively,
going from LEP to FCC-ee, and a factor of $9\cdot 10^{4}$ 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 $\sim$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. 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 $\sqrt{2}$ to account for the statistical combination of two
experiments, ATLAS + CMS) with their bibliographical reference.

  2. 2.

For those channels where LHC limits exist today in measurements with a given
integrated luminosity (labelled e.g., “$\mathcal{L}_{\mathrm{\tiny{int}}}$(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 $\sqrt{\smash{2\times
3~{}\mathrm{ab}^{-1}}/\smash{\mathcal{L}_{\mathrm{\tiny{int}}}(\text{13\,TeV})}}$,
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. 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,
$\sqrt{\smash{N_{\mathrm{X}}\text{(HL-
LHC)}}/\smash{N_{\mathrm{X}}\text{(Tevatron)}}}$. According to the increase
factors quoted in Table 1, 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 $\mathcal{B}\times N_{\mathrm{X}}\gtrsim 1$, where
$N_{\mathrm{X}}$ are the FCC numbers given in Table 1 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 $\mathrm{e}^{+}\mathrm{e}^{-}$ colliders, and even more so at FCC-ee
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.

###  II.3 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 2 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
$f_{\mathrm{M}}$ and HQET first inverse moments $\lambda_{\mathrm{M}}$ (right
column).

Table 2: Numerical 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, $m_{\ell\ell}$, are given as twice the single lepton masses, ignoring tiny binding energies), except for those where a specific reference is provided.  masses and widths | couplings and CKM elements | meson masses and form-factor parameters (in GeV)
---|---|---
$m_{\mathrm{ee}}$ |  $1.02991$ MeV | $\alpha(0)$ | 1/137.036 | $m_{\mathrm{\pi^{\pm}}}$ | 0.13957 | $f_{\mathrm{\pi}}$ | 0.1304 [62]
$m_{\mu\mu}$ | 0.21126 GeV | $\alpha(m_{\mathrm{Z}})$ | 1/128.943 | $m_{\mathrm{K^{\pm}}}$ | 0.493677 | $f_{\mathrm{K}}$ | 0.1562 [62]
$m_{\tau\tau}$ | 3.5537 GeV | $\alpha_{\text{s}}(m_{{}_{\mathrm{Z}}})$ | 0.1180 | $m_{\mathrm{\rho^{\pm}}}$ | 0.77526 | $f_{\mathrm{\rho}}$ | 0.212 [62]
$m_{\mathrm{c}}$ | 1.50 GeV | $\sin^{2}{\theta_{\mathrm{w}}}$ | 0.2351 | $m_{\mathrm{K^{*\pm}}}$ | 0.89166 | $f_{\mathrm{K^{*}}}$ | 0.203 [62]
$m_{\mathrm{b}}$ | 4.18 GeV | $\rm G_{F}$ |  $1.1664\cdot 10^{-5}$ GeV-2 | $m_{\mathrm{\phi}}$ | 1.0194 | $f_{\mathrm{\phi}}$ | 0.223 [14]
$m_{\mathrm{t}}$ | 172.69 GeV | $|V_{\mathrm{ud,cs}}|$ | 0.974 | $m_{\mathrm{D^{\pm}}}$ | 1.86966 | $f_{\mathrm{D}}$ | 0.212 [63]
$m_{\mathrm{W}}$ | 80.377 GeV | $|V_{\mathrm{us,cd}}|$ | 0.225 |  |  | $\lambda_{\mathrm{D}}$ | 0.354 [64]
$m_{\mathrm{Z}}$ | 91.188 GeV | $|V_{\mathrm{cb,ts}}|$ | 0.041 | $m_{\mathrm{D_{s}^{\pm}}}$ | 1.96835 | $f_{\mathrm{D_{s}}}$ | 0.2499 [63]
$m_{\mathrm{H}}$ | 125.25 GeV | $|V_{\mathrm{ub}}|$ | 0.00382 | $m_{\mathrm{D^{*0}}}$ | 2.007 | $f_{\mathrm{D^{*0}}}$ | 1.28 $f_{\mathrm{D}}$ [65]
$\Gamma_{\mathrm{W}}$ | 2.085 GeV | $|V_{\mathrm{tb}}|$ | 0.999 | $m_{\mathrm{D^{*\pm}}}$ | 2.01027 | $f_{\mathrm{D^{*}}}$ | 1.28 $f_{\mathrm{D}}$ [65]
$\Gamma_{\mathrm{Z}}$ | 2.4955 GeV | $|V_{\mathrm{td}}|$ | 0.0086 | $m_{\mathrm{D_{s}^{*\pm}}}$ | 2.1123 | $f_{\mathrm{D_{s}^{*}}}$ | 1.26 $f_{\mathrm{D_{s}}}$ [65]
$\Gamma_{\mathrm{Z\to\ell\ell}}$ | 0.08399 GeV |  |  | $m_{\mathrm{B^{\pm}}}$ | 5.27934 | $f_{\mathrm{B}}$ | 0.190 [63]
$\Gamma_{\mathrm{H}}$ | 4.1 MeV [66] |  |  |  |  | $\lambda_{\mathrm{B}}$ | 0.338 [67]
$\Gamma_{\mathrm{H\to\gamma\gamma}}$ | $2.5\cdot 10^{-3}~{}\Gamma_{\mathrm{H}}$ |  |  | $m_{\mathrm{B^{*\pm,0}}}$ | 5.32471 | $f_{\mathrm{B^{*}}}$ | 0.941 $f_{\mathrm{B}}$ [68]
$\Gamma_{\mathrm{t}}$ | 1.331 GeV [69] |  |  |  |  | $f_{\mathrm{B_{\mathrm{s}}}}$ | 0.2303 [63]
|  |  |  |  |  | $\lambda_{\mathrm{B_{s}}}$ | 0.438 [70]
|  |  |  | $m_{\mathrm{B_{s}^{*0}}}$ | 5.415 | $f_{\mathrm{B_{\mathrm{s}}^{*}}}$ | 0.953 $f_{\mathrm{B_{\mathrm{s}}}}$ [68]
|  |  |  | $m_{\mathrm{B_{\mathrm{c}}^{\pm}}}$ | 6.27447 | $f_{\mathrm{B_{\mathrm{c}}}}$ | 0.427 [68]
|  |  |  | $m_{\mathrm{B_{\mathrm{c}}^{*\pm}}}$ | 6.32877 [68] | $f_{\mathrm{B_{\mathrm{c}}^{*}}}$ | 0.988 $f_{\mathrm{B_{\mathrm{c}}}}$ [68]

##  III Rare Higgs boson decays

###  III.1 Rare two-, three-, and four-body Higgs boson decays

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 2. The corresponding results are listed in Table 3.

Figure 2: Representative diagrams of rare 2-, 3-, and 4-body decays of the H
boson into photons and/or neutrinos.

The first calculation is that of the invisible two-body Higgs boson decay into
a neutrino-antineutrino pair, which is infinitesimal in the SM
($\mathcal{B}\approx 10^{-36}$) compared to the standard invisible (four-
neutrino) $\rm H\to ZZ^{\star}\to 4\nu$ decay ($\mathcal{B}\approx 0.1\%$)
[66]. However, since the SM assumption of massless $\nu$’s is incorrect, the
$\rm H\to\nu\overline{\nu}$ 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 $\rm H\to 3\gamma$. 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, $\rm
C(H)=0\neq C(3\gamma)=(-1)^{3}=-1$, 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 $\rm H\to 3\gamma$ 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 $\mathcal{O}(10^{-40})$ probability. Thus, any potential experimental
observation of $\rm H\to 3\gamma$ 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.

Table 3: Compilation of rare Higgs decays to two neutrinos, three or four
photons, and a photon plus two neutrinos. For each decay, we provide the
theoretical branching fraction computed here with the MadGraph5_aMC@NLO MC
(called MG5_aMC, hereafter). The last two columns indicate whether the decay
can be produced at FCC-ee/FCC-hh.

|  |  | Exp. limits | Producible at
---|---|---|---|---
|  | Branching fraction | Framework | 2023 | HL-LHC | FCC-ee | FCC-hh
$\rm H\,\to$ | $\rm\nu+\overline{\nu}$ |  $7.2$ | $\times~{}10^{-36}$ |  MG5_aMC (this work) | – | – | ✘ | ✘
$\rm\gamma+\nu+\overline{\nu}$ |  $(3.74\pm 0.01)$ | $\times~{}10^{-4}$ |  MG5_aMC (this work) | – | – | ✓ | ✓
$\rm\gamma+\gamma+\gamma$ |  $1.0$ | $\times~{}10^{-40}$ |  MG5_aMC (this work) | – | – | ✘ | ✘
$\rm\gamma+\gamma+\gamma+\gamma$ |  $(4.56\pm 0.01)$ | $\times~{}10^{-12}$ |  MG5_aMC (this work) | – | – | ✘ | ✘

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
$\mathcal{B}=3.74\cdot 10^{-4}$ that is about 20% larger than the naive
estimate given by $\mathcal{B}(\rm H\to Z\gamma)=1.54\cdot 10^{-3}$ [66]
combined with $\mathcal{B}(\rm Z\to\nu\overline{\nu})=0.200$ [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, $\mathrm{H}\to a(\gamma\gamma)\,a(\gamma\gamma)$. The Higgs
$4\gamma$ decay has a branching fraction of $\mathcal{B}=4.56\cdot 10^{-12}$,
which is 28 orders-of-magnitude larger than the 3-$\gamma$ 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 $4\gamma$ final
state alone [61, 76, 77, 78, 79] (searches for $3\gamma$ decays have focused
on Z’ resonances off the Higgs peak [61]), but unfortunately the limits have
been set only on the process $\rm H\to a(\gamma\gamma)a(\gamma\gamma)$ with
two intermediate ALPs decaying into photons. It would not be difficult for
ATLAS/CMS to recast these searches into upper bounds on the $\rm H\to 4\gamma$
“continuum” decay.

###  III.2 Exclusive Higgs decays into a gauge boson plus a meson

Since it is extremely difficult to experimentally access the Yukawa couplings
to first- ($\rm q=u,d,s$) and second- (c) generation quarks, due to the
smallness of the $\rm
H\to\mathrm{q}\overline{\mathrm{q}},\mathrm{c}\overline{\mathrm{c}}$ 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 $\rm H\to V+M$ (where $\rm V=Z,W^{\pm},\gamma$, 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 $\rm
V^{*}\to\mathrm{q}\overline{\mathrm{q}}$. 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 Higgs2̆013quark coupling
comes from the (destructive) interference of the two amplitudes. Thus, for
example, the $\rm H\to\gamma+\mathrm{J}/\psi,\gamma+\phi$ modes allow direct
access to the flavour-diagonal coupling of the Higgs to the charm and strange
quarks, respectively, while the $\rm H\to\gamma+\rho,\gamma+\omega$ decays can
probe the Higgs couplings to up and down quarks. The radiative decays $\rm
H\to\gamma+M$, where $\rm M=K^{*0},D^{*0},B_{s}^{*0},B_{d}^{*0}$ – 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].

Figure 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.

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 $f_{\mathrm{M}}$ (see Table 2). This quantity can be directly
obtained from the experimental measurements of the vector-meson leptonic
partial decay width (which proceeds through the EW annihilation
$\mathrm{q}\overline{\mathrm{q}}\to\gamma^{*},\mathrm{Z}^{*}\to\ell^{+}\ell^{-}$),
given by

$\Gamma(\mathrm{M\to\ell^{+}\ell^{-}})=\frac{4\pi
Q_{\mathrm{M}}^{2}f_{\mathrm{M}}^{2}}{3m_{\mathrm{M}}}\alpha^{2}(m_{\mathrm{M}})$
(1)

where $Q_{\mathrm{M}}$ is the relevant combination of quark electric charges,
and $m_{\mathrm{M}}$ the mass of the meson. Corrections due to the
offshellness of the photon and to the contribution of the $\rm
H\to\gamma\,Z^{*}$ process are suppressed by
$m^{2}_{\mathrm{M}}/m^{2}_{\mathrm{H}}$ and
$m^{2}_{\mathrm{M}}/(m^{2}_{\mathrm{Z}}-m^{2}_{\mathrm{M}})$, respectively,
and hence very small [14]. The direct amplitudes, which are much smaller than
the indirect ones except for the heaviest $\rm H\to\gamma+\Upsilon$ decay,
have been calculated theoretically by different groups. The hierarchy
$m_{\mathrm{M}}\ll m_{\mathrm{H}}$ implies that the vector meson is emitted at
very high energy $E_{\mathrm{M}}\gg m_{\mathrm{M}}$ 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.

####  III.2.1 Higgs decays into a photon plus a vector meson

Table 4 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 $\mathcal{B}$ values are in the range of
$\mathcal{O}(10^{-5}$–$10^{-9})$, with larger rates for decays into the
lightest vector mesons due to the inverse dependence on the meson mass of the
$\rm V^{(*)}\to\mathrm{q}\overline{\mathrm{q}}$ 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 $\rm
H\to\gamma+D,\gamma+B$ 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 4: 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
---|---|---|---|---|---|---
$\rm H\to\,\gamma$ | $+$ | $\rm M$ | Branching fraction | Framework | 2023 | HL-LHC | FCC-ee | FCC-hh
$\rm H\,\to\,\gamma$ | + | $\rm\rho^{0}$ |  $(1.68\pm 0.08)$ | $\times~{}10^{-5}$ | SCET+LCDA [14] |  $<$ $8.8$$\times~{}10^{-4}$ [81] | $\lesssim 6.8\times~{}10^{-5}$ | ✓ | ✓
$\rm\omega$ |  $(1.48\pm 0.08)$ | $\times~{}10^{-6}$ | SCET+LCDA [14] |  $<$ $1.5$$\times~{}10^{-4}$ [84] | $\lesssim 2.2\times~{}10^{-5}$ | ✓ | ✓
$\rm\phi$ |  $(2.31\pm 0.11)$ | $\times~{}10^{-6}$ | SCET+LCDA [14] |  $<$ $4.8$$\times~{}10^{-4}$ [81] | $\lesssim 3.7\times~{}10^{-5}$ | ✓ | ✓
$\rm J/\psi$ |  $(2.95\pm 0.17)$ | $\times~{}10^{-6}$ | SCET+LCDA [14] | $<$ $2.6$$\times~{}10^{-4}$ [85, 83] | $\lesssim 3.9\times~{}10^{-5}$ [55] | ✓ | ✓
$(3.01\pm 0.15)$ | $\times~{}10^{-6}$ | NRQCD (NLL)+LDME [86]
$\left(2.99^{+0.16}_{-0.15}\right)$ | $\times~{}10^{-6}$ | NRQCD+LCDA [87]
$\rm\psi(2S)$ |  $(1.3\pm 0.0)$ | $\times~{}10^{-6}$ | SCET+LCDA [14] |  $<$ $9.9$$\times~{}10^{-4}$ [85, 82] | $\lesssim 7.7\times~{}10^{-5}$ | ✓ | ✓
$\rm\Upsilon(1S)$ |  $\left(4.61^{+1.76}_{-1.23}\right)$ | $\times~{}10^{-9}$ | SCET+LCDA [14] | $<$ $4.9$$\times~{}10^{-4}$ [82] | $\lesssim 3.8\times~{}10^{-5}$ | ✘ | ✓
$\left(9.97^{+4.04}_{-3.03}\right)$ | $\times~{}10^{-9}$ | NRQCD (NLL)+LDME [86]
$\left(5.22^{+2.02}_{-1.70}\right)$ | $\times~{}10^{-9}$ | NRQCD+LCDA [87]
$\rm\Upsilon(2S)$ |  $\left(2.34^{+0.76}_{-1.00}\right)$ | $\times~{}10^{-9}$ | SCET+LCDA [14] | $<$ $5.9$$\times~{}10^{-4}$ [82] | $\lesssim 4.6\times~{}10^{-5}$ | ✘ | ✓
$\left(2.62^{+1.39}_{-0.91}\right)$ | $\times~{}10^{-9}$ | NRQCD (NLL)+LDME [86]
$\left(1.42^{+0.72}_{-0.57}\right)$ | $\times~{}10^{-9}$ | NRQCD+LCDA [87]
$\rm\Upsilon(3S)$ |  $\left(2.13^{+0.76}_{-1.12}\right)$ | $\times~{}10^{-9}$ | SCET+LCDA [14] | $<$ $5.7$$\times~{}10^{-4}$ [82] | $\lesssim 4.4\times~{}10^{-5}$ | ✘ | ✓
$\left(1.87^{+1.05}_{-0.69}\right)$ | $\times~{}10^{-9}$ | NRQCD (NLL)+LDME [86]
$\left(9.1^{+4.8}_{-3.8}\right)$ | $\times~{}10^{-10}$ | NRQCD+LCDA [87]

Figure 4: Branching ratios (in negative log scale) of exclusive $\rm
H\to\gamma+vector$-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.

####  III.2.2 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 $\rm H\to\gamma+M$
with $\rm M=K^{*0},D^{*0},B_{s}^{*0},B_{d}^{*0}$, which are all excited vector
states, as explained below. On the one hand, calculations for the inclusive
flavour-changing $\rm H\to qq^{\prime}$ rates exist [88, 89, 80], but without
the photon emission and the exclusive final-state meson formation. These
predicted FCNC Higgs branching fractions, $\mathcal{B}(\rm H\to
qq^{\prime})\equiv\mathcal{B}(\rm
H\to\mathrm{q}\overline{\mathrm{q}}^{\prime}+q^{\prime}\overline{q})$, amount
to $\mathcal{B}(\rm H\to uc)=(2.7\pm 0.5)\cdot 10^{-20}$, $\mathcal{B}(\rm
H\to ds)=1.19\cdot 10^{-11}$, $\mathcal{B}(\rm H\to db)=(3.8\pm 0.6)\cdot
10^{-9}$, and $\mathcal{B}(\rm H\to sb)=(8.9\pm 1.5)\cdot 10^{-8}$ [89]. On
the other hand, the work of [13] determined the branching fractions of
radiative Higgs flavoured-meson decays assuming (arbitrary) $\mathcal{O}(1)$
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

$\mathcal{B}\left(\mathrm{H\to\gamma+M(qq^{\prime})}\right)=\frac{1}{8\pi}\left(\frac{f_{\mathrm{M}}m_{\mathrm{M}}}{2\,\lambda_{\mathrm{M}}(\mu)}\frac{m_{\mathrm{b}}}{\varv}Q_{\mathrm{q}}\right)^{2}\frac{\mathcal{B}(\mathrm{H\to
qq^{\prime}})}{2\,m_{\mathrm{H}}\,\Gamma_{\mathrm{H}}},$ (2)

where $\varv\approx 246$ 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 $\lambda_{\mathrm{M}}(\mu)$, which plays
an important role when working out rates for exclusive decays involving charm
and bottom mesons [90, 64], we have used the values quoted in Table 2 with
$\rm\lambda_{B_{s}^{*0}}=\lambda_{B_{d}^{*0}}=\lambda_{B}$, and
$\rm\lambda_{D_{s}^{*0}}=\lambda_{D}$. For the $K^{*0}$ decay, we simply took
$\mathcal{O}(10^{-8})\times\rm\mathcal{B}(\rm H\to ds)$ for the predicted
rate. The theoretical branching fractions for such exclusive FCNC radiative
decays are extremely suppressed, in the range of
$\mathcal{O}(10^{-14}$–$10^{-26})$. 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 $\rm
H\to\gamma+K^{*0}$ channel at $\mathcal{O}(10^{-5})$ [84], to be compared with
our $\mathcal{O}(10^{-19})$ SM prediction.

Table 5: 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.

|  |  |  |  | Exp. limits | Producible at
---|---|---|---|---|---|---
$\rm H\to\,\gamma$ | $+$ | $\rm M$ | Branching fraction | Framework | 2023 | HL-LHC | FCC-ee | FCC-hh
$\rm H\,\to\,\gamma$ | + | $\rm K^{*0}$ |  $1.0$ | $\times~{}10^{-19}$ | EFT+LCDA (this work) |  $<$ $8.9$$\times~{}10^{-5}$ [84] | $\lesssim 1.3\times~{}10^{-5}$ | ✘ | ✘
$\rm D^{*0}$ |  $7.0$ | $\times~{}10^{-27}$ | EFT+LCDA (this work) | – | – | ✘ | ✘
$\rm\overline{B}^{*0}$ |  $8.6$ | $\times~{}10^{-16}$ | EFT+LCDA (this work) | – | – | ✘ | ✘
$\rm\overline{B}_{s}^{*0}$ |  $1.9$ | $\times~{}10^{-14}$ | EFT+LCDA (this work) | – | – | ✘ | ✘

Figure 5: Branching fractions (in negative log scale) of exclusive $\rm
H\to\gamma+$ flavoured-meson decays. Our predictions via Eq. (2) (blue bars)
are compared to current experimental upper limits (violet) and expected HL-LHC
bounds (orange). The vertical red 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.

####  III.2.3 Higgs decays into a Z boson plus a meson

Higgs decays of the form $\rm H\to Z+M$ involve neutral currents and are very
similar to the radiative decays just discussed. However, an important
difference with respect to the $\gamma\,+\,$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-$\gamma$-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 ($\mathrm{H}\to\mathrm{Z}+a$) [91].
Table 6 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 $\rm
H\to Z+\rho$ 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 6,
and they are in the $\mathcal{O}(10^{-2}$–$10^{-3})$ range111We note that one
could a priori also derive limits for $\mathcal{B}(\rm H\to
Z+\eta_{\mathrm{c}})$ and $\mathcal{B}(\rm H\to Z+\eta_{\mathrm{b}})$ from a
recent 95% CL upper limit on the cross section for the $\rm H\to
Z(\ell\ell)a(\gamma\gamma)$ process over $m_{a}\approx 1$–30 GeV performed by
the CMS Collaboration [91]. However, the upper bound cross sections measured
at the $m_{a}=m_{\eta_{\mathrm{c}},\eta_{\mathrm{b}}}\approx 2.9,~{}9.4$ GeV
mass points ($\sigma\lesssim 7,$ 2.1 fb, corresponding to $\mathcal{B(\rm H\to
Z+a)}\lesssim 0.002,~{}0.62\cdot 10^{-3}$ after normalizing by the Higgs
production cross section and correcting for the dilepton decays of the Z
boson) are too weak to place any sensible constrain on these $\mathcal{B}$..
In general, the experimental $\rm H\to Z+M$ limits are about a factor of ten
worse than their $\rm H\to\gamma+M$ counterparts because of the extra events
lost from the requirement of Z boson identification via dilepton decays (with
$\mathcal{B}\approx 3\%$ 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 6: 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
---|---|---|---|---|---|---
$\rm H\to\,Z$ | $+$ | $\rm M$ | Branching fraction | Framework | 2023 | HL-LHC | FCC-ee | FCC-hh
$\rm H\,\to\,Z$ | + | $\rm\pi^{0}$ |  $(2.3\pm 0.1)$ | $\times~{}10^{-6}$ | EFT+NRQM [10] | – | – | ✓ | ✓
$(2.3\pm 0.1)$ | $\times~{}10^{-6}$ | EFT+LCDA [92]
$\rm\eta$ |  $(8.3\pm 0.9)$ | $\times~{}10^{-7}$ | EFT+LCDA [92] | – | – | ✓ | ✓
$\rm\rho^{0}$ |  $(1.4\pm 0.1)$ | $\times~{}10^{-5}$ | EFT+NRQM [10] | $<$ $1.2$$\times~{}10^{-2}$ [93] | $\lesssim 1.8\times~{}10^{-3}$ | ✓ | ✓
$(7.19\pm 0.29)$ | $\times~{}10^{-6}$ | EFT+LCDA [92]
$\rm\omega$ |  $(1.6\pm 0.1)$ | $\times~{}10^{-6}$ | EFT+NRQM [10] | – | – | ✓ | ✓
$(5.6\pm 0.2)$ | $\times~{}10^{-7}$ | EFT+LCDA [92]
$\rm\eta^{\prime}$ |  $(1.24\pm 0.13)$ | $\times~{}10^{-6}$ | EFT+LCDA [92] | – | – | ✓ | ✓
$\rm\phi$ |  $(4.2\pm 0.3)$ | $\times~{}10^{-6}$ | EFT+NRQM [10] | $<$ $3.6$$\times~{}10^{-3}$ [93] | $\lesssim 5.4\times~{}10^{-4}$ | ✓ | ✓
$(2.42\pm 0.10)$ | $\times~{}10^{-6}$ | EFT+LCDA [92]
$\rm\eta_{c}$ |  $(1.0\pm 0.1)$ | $\times~{}10^{-5}$ | EFT+NRQM [10] | – | – | ✓ | ✓
$(1.0\pm 0.0)$ | $\times~{}10^{-5}$ | EFT+LCDA [94]
$\rm J/\psi$ |  $3.4$ | $\times~{}10^{-6}$ | NRQCD (NLO)+LMDE [95] | $<$ $1.9$$\times~{}10^{-3}$ [96] | $\lesssim 2.1\times~{}10^{-4}$ [57] | ✓ | ✓
$3.2$ | $\times~{}10^{-6}$ | EFT+NRQM [97]
$(2.3\pm 0.1)$ | $\times~{}10^{-6}$ | EFT+LCDA [92]
$\rm\psi(2S)$ |  $1.5$ | $\times~{}10^{-6}$ | EFT+NRQM [97] |  $<$ $6.6$$\times~{}10^{-3}$ [96] | $\lesssim 1.0\times~{}10^{-3}$ | ✓ | ✓
$\rm\eta_{b}$ |  $(2.69\pm 0.05)$ | $\times~{}10^{-5}$ | EFT+LCDA [94] | – | – | ✓ | ✓
$\left(4.739^{+0.276}_{-0.244}\right)$ | $\times~{}10^{-5}$ | EFT (NLO)+LCDA [98]
$\rm\Upsilon(1S)$ |  $1.7$ | $\times~{}10^{-5}$ | NRQCD (NLO)+LMDE [95] | – | – | ✓ | ✓
$1.7$ | $\times~{}10^{-5}$ | EFT+NRQM [97]
$(1.54\pm 0.06)$ | $\times~{}10^{-5}$ | EFT+LCDA [92]
$\rm\Upsilon(2S)$ |  $8.9$ | $\times~{}10^{-6}$ | EFT+NRQM [97] | – | – | ✓ | ✓
$(7.5\pm 0.3)$ | $\times~{}10^{-6}$ | EFT+LCDA [92]
$\rm\Upsilon(3S)$ |  $6.7$ | $\times~{}10^{-6}$ | EFT+NRQM [97] | – | – | ✓ | ✓
$(5.63\pm 0.24)$ | $\times~{}10^{-6}$ | EFT+LCDA [92]

Figure 6: Branching ratios (in negative log scale) of exclusive $\rm H\to
Z+meson$ 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.

####  III.2.4 Higgs decays into a W boson plus a meson

The charged $\rm H\to W^{\pm}M^{\mp}$ 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 7
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 $\mathcal{B}$ numerical
values computed in [12] (but have likely a typo in the $\mathcal{B}(\rm H\to
B^{*\pm}W^{\mp})=10^{-5}$ 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 $\mathcal{O}(10^{-5})$ branching fraction, is $\rm H\to
W^{\pm}+\rho^{\mp}$ 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).

Table 7: 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.  |  |  |  |  | Exp. limits | Producible at
---|---|---|---|---|---|---
$\rm H\to\,W$ | $+$ | $\rm M$ | Branching fraction | Framework | 2023 | HL-LHC | FCC-ee | FCC-hh
$\rm H\,\to\,W^{\mp}$ | + | $\rm\pi^{\pm}$ |  $(4.2\pm 0.2)$ | $\times~{}10^{-6}$ | EFT+NRQM [10] | – | – | ✓ | ✓
$(4.3\pm 0.2)$ | $\times~{}10^{-6}$ | EFT+LCDA [92]
$\rm\rho^{\pm}$ |  $(1.5\pm 0.1)$ | $\times~{}10^{-5}$ | EFT+NRQM [10] | – | – | ✓ | ✓
$(1.09\pm 0.05)$ | $\times~{}10^{-5}$ | EFT+LCDA [92]
$\rm K^{\pm}$ |  $(3.3\pm 0.1)$ | $\times~{}10^{-7}$ | EFT+NRQM [10] | – | – | ✘ | ✓
$(3.3\pm 0.1)$ | $\times~{}10^{-7}$ | EFT+LCDA [92]
$\rm K^{*\pm}$ |  $(4.3\pm 0.2)$ | $\times~{}10^{-7}$ | EFT+NRQM [10] | – | – | ✘ | ✓
$(5.6\pm 0.4)$ | $\times~{}10^{-7}$ | EFT+LCDA [92]
$\rm D^{\pm}$ |  $(5.8\pm 0.6)$ | $\times~{}10^{-7}$ | EFT+NRQM [10] | – | – | ✓ | ✓
$(5.6\pm 0.5)$ | $\times~{}10^{-7}$ | EFT+LCDA [92]
$\rm D^{*\pm}$ |  $(1.3\pm 0.1)$ | $\times~{}10^{-6}$ | EFT+NRQM [10] | – | – | ✓ | ✓
$(1.04\pm 0.14)$ | $\times~{}10^{-6}$ | EFT+LCDA [92]
$\rm D_{s}^{\pm}$ |  $(1.6\pm 0.1)$ | $\times~{}10^{-5}$ | EFT+NRQM [10] | – | – | ✓ | ✓
$(1.71\pm 0.11)$ | $\times~{}10^{-5}$ | EFT+LCDA [92]
$\rm D_{s}^{*\pm}$ |  $(3.5\pm 0.2)$ | $\times~{}10^{-5}$ | EFT+NRQM [10] | – | – | ✓ | ✓
$(2.51\pm 0.19)$ | $\times~{}10^{-5}$ | EFT+LCDA [92]
$\rm B^{\pm}$ |  $(1.6\pm 0.4)$ | $\times~{}10^{-10}$ | EFT+NRQM [10] | – | – | ✘ | ✓
$(1.54\pm 0.40)$ | $\times~{}10^{-10}$ | EFT+LCDA [92]
$\rm B^{*\pm}$ |  $(1.3\pm 0.2)$ | $\times~{}10^{-5}$ | EFT+NRQM [10] | – | – | ✓ | ✓
$(1.41\pm 0.36)$ | $\times~{}10^{-10}$ | EFT+LCDA [92]
$\rm B_{c}^{\pm}$ |  $(1.6\pm 0.2)$ | $\times~{}10^{-8}$ | EFT+NRQM [10] | – | – | ✘ | ✓
$(8.21\pm 0.83)$ | $\times~{}10^{-8}$ | EFT+LCDA [92]
Figure 7: Branching fractions (in negative log scale) of exclusive $\rm H\to
W^{\pm}+meson$ 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.

###  III.3 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 $(\ell^{+}\ell^{-})$, where
$(\ell^{+}\ell^{-})=(\mathrm{e}^{+}\mathrm{e}^{-}),\,(\mu^{+}\mu^{-}),\,(\tau^{+}\tau^{-})$
represent positronium, dimuonium, and ditauonium bound states, respectively.
Depending on the accompanying gauge boson (Z or $\gamma$), 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
$\rm H\to W^{\pm}+M$ case). We provide here an estimate of the decay rates for
such processes by considering the similar $\rm H\to\gamma+M,\,Z+M$ processes,
and changing the quarkonium form-factors by leptonium 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],

$|\phi_{n,(\ell\ell)}(r=0)|^{2}=\frac{\left(m_{\ell}\alpha(0)\right)^{3}}{8\pi
n^{3}}.$ (3)

Since the QED coupling is much smaller than the QCD one,
$\alpha(m_{\ell\ell})\ll
C_{F}\alpha_{\text{s}}(m_{\mathrm{q}\overline{\mathrm{q}}})$, and since the
charged lepton masses are smaller than the quark masses of the same
generation, $m_{\ell\ell}\ll m_{\mathrm{q}\overline{\mathrm{q}}}$, the ratio
$[\alpha(0)m_{\ell\ell}/(\alpha_{s}(m_{\mathrm{q}\overline{\mathrm{q}}})m_{\mathrm{q}\overline{\mathrm{q}}})]^{3}$
is very small, and one can anticipate that those decays will be orders-of-
magnitude more suppressed than the $\rm
H\to\gamma+M(\mathrm{q}\overline{\mathrm{q}})$ ones discussed in Section
III.2.

Figure 8: Schematic diagrams of exclusive decays of the H boson into a photon
or a Z boson plus an leptonium state in the direct (left) and indirect (right)
processes. The solid fermion lines represent leptons, the gray blob represents
the leptonium bound state.

We start by calculating the branching fraction for the $\rm
H\to\gamma+(\ell^{+}\ell^{-})$ process in which, due to charge-conjugation
conservation, the leptonium can only be in the ortho $(\ell^{+}\ell^{-})_{1}$
state. The partial width of this decay can be derived from the similar
expressions for quarkonium vector mesons [11], namely

$\displaystyle\mathcal{B}(\rm
H\to\gamma+(\ell^{+}\ell^{-})_{1})=\frac{1}{8\pi}\frac{m_{\mathrm{H}}^{2}-m_{\ell\ell}^{2}}{m_{\mathrm{H}}^{2}\,\Gamma_{\mathrm{H}}}\left|\mathcal{A}_{\text{dir}}+\mathcal{A}_{\text{ind}}\right|^{2},$
(4)

with the direct and indirect amplitudes corresponding to the diagrams of Fig.
8 left and right, respectively, and where the $\rm H\to\gamma+Z^{*}$
contribution followed by the $\rm Z^{*}\to(\ell^{+}\ell^{-})_{1}$ transition
is negligible compared to the corresponding $\rm H\to\gamma+\gamma^{*}$ one.
These two amplitudes can be written as a function of the wavefunction at the
origin $\phi_{\mathrm{n,(\ell\ell)}}(0)$, as follows

$\displaystyle\mathcal{A}_{\text{dir}}$
$\displaystyle=2Q_{\ell}\sqrt{4\pi\alpha(0)}\left(\sqrt{2}\mathrm{G_{F}}\,m_{\ell\ell}\right)^{1/2}\frac{m_{\mathrm{H}}^{2}-m_{\ell\ell}^{2}}{\sqrt{m_{\mathrm{H}}}(m_{\mathrm{H}}^{2}-m_{\ell\ell}^{2}/2-2m_{\mathrm{\ell}}^{2})}\,\phi_{\mathrm{n,(\ell\ell)}}(0),$
(5) $\displaystyle\mathcal{A}_{\text{ind}}$
$\displaystyle=\frac{Q_{\ell}\sqrt{4\pi\alpha(0)}\,f_{\ell\ell}}{m_{\ell\ell}}\left(16\pi\Gamma_{\mathrm{H\to\gamma\gamma}}\right)^{1/2}\frac{m_{\mathrm{H}}^{2}-m_{\ell\ell}^{2}}{m_{\mathrm{H}}^{2}}=-\frac{2Q_{\ell}\sqrt{4\pi\alpha(0)}}{m_{\ell\ell}^{3/2}}\left(16\pi\Gamma_{\mathrm{H\to\gamma\gamma}}\right)^{1/2}\frac{m_{\mathrm{H}}^{2}-m_{\ell\ell}^{2}}{m_{\mathrm{H}}^{2}}\,\phi_{\mathrm{n,(\ell\ell)}}(0)\,,$
(6)

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,
$f_{\mathrm{X}}$, to its wavefunction at the origin:

$\displaystyle
f^{2}_{\mathrm{X}}=4N_{\mathrm{c}}\frac{|\phi_{\mathrm{X}}(0)|^{2}}{m_{\mathrm{X}}}.$
(7)

Here, the particle X can be either a pseudoscalar/vector meson or an
ortho-/para-leptonium state, and the number of colours $N_{\mathrm{c}}=1$ is
used for the latter. We choose $\phi_{X}(0)$ to be real, and then the decay
constant $f_{\mathrm{X}}$ 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 $m_{\ell\ell}\ll m_{\mathrm{H}}$, 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 2, we determine the
radiative ortholeptonium $(n=1)$ branching fractions listed in the first three
rows of Table 8. The radiative ortholeptonium decays of the Higgs boson have
all numerically similar $\mathcal{O}(10^{-12})$ 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 $(\ell^{+}\ell^{-})_{1}$ decay, which lead
to significantly displaced triphotons (for positronium and dimuonium),
$\mathrm{e}^{+}\mathrm{e}^{-}$ pairs (for dimuonium and ditauonium), or
$\mu^{+}\mu^{-}$ pairs (for ditauonium).

We determine next the rates of the $\rm H\to Z+(\ell^{+}\ell^{-})$ 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 $\rm H\to\gamma+(\ell^{+}\ell^{-})_{1}$ 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 $\rm H\to Z+VM$ decays, where VM are quarkonia
vector mesons such as $\mathrm{J}/\psi,\Upsilon$ [97]. The corresponding $\rm
Z+(\ell^{+}\ell^{-})_{1}$ decay width can then be written as

$\displaystyle\Gamma(\rm H\to
Z+(\ell^{+}\ell^{-})_{1})=\Gamma_{1}+\Gamma_{2}+\Gamma_{3},$ (8)

where $\Gamma_{1},\,\Gamma_{2},$ and $\Gamma_{12}$ are the contributions from
$\rm H\to Z+Z^{*}\to Z+(\ell^{+}\ell^{-})_{1}$, $\rm H\to Z+\gamma^{*}\to
Z+(\ell^{+}\ell^{-})_{1}$, and their interference, respectively. Defining
$r_{\mathrm{Z}}=m_{\mathrm{Z}}^{2}/m_{\mathrm{H}}^{2}$,
$r_{\ell\ell}=m_{\ell\ell}^{2}/m_{\mathrm{H}}^{2}$,
$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+ac+bc)$, and the $\rm H\gamma Z$
effective coupling $C_{Z\gamma}\approx 5.54$ [102], these individual partial
widths can be expressed as follows

$\displaystyle\Gamma_{1}$
$\displaystyle=\frac{m_{\mathrm{H}}^{3}(g_{\ell\ell}f_{\ell\ell})^{2}}{16\pi\varv^{4}}\frac{\lambda^{1/2}(1,r_{\mathrm{Z}},r_{\ell\ell})}{(1-r_{\ell\ell}/r_{\mathrm{Z}})^{2}}[(1-r_{\mathrm{Z}}-r_{\ell\ell})^{2}+8r_{\mathrm{Z}}r_{\ell\ell}],$
$\displaystyle\Gamma_{2}$
$\displaystyle=\frac{\alpha(0)^{3}f_{\ell\ell}^{2}Q_{\ell}^{2}m_{\mathrm{H}}^{3}}{32\pi^{2}\varv^{2}\sin^{2}\theta_{\mathrm{W}}}\frac{C_{Z\gamma}^{2}}{m_{\ell\ell}^{2}}\lambda^{1/2}(1,r_{\mathrm{Z}},r_{\ell\ell})[(1-r_{\mathrm{Z}}-r_{\ell\ell})^{2}+2r_{\mathrm{Z}}r_{\ell\ell}],$
$\displaystyle\Gamma_{12}$
$\displaystyle=\frac{3\alpha(0)^{2}f_{\ell\ell}^{2}g_{\ell\ell}Q_{\ell}m_{\mathrm{H}}C_{Z\gamma}}{8\pi\cos\theta_{\mathrm{W}}\sin^{2}\theta_{\mathrm{W}}\varv^{2}}\frac{\lambda^{1/2}(1,r_{\mathrm{Z}},r_{\ell\ell})}{1-r_{\ell\ell}/r_{\mathrm{Z}}}(1-r_{\mathrm{Z}}-r_{\ell\ell}).$
(9)

with $\varv\approx 246$ GeV, and
$g_{\ell\ell}=T_{3}^{\ell}-2Q_{\ell}\sin^{2}\theta_{\mathrm{W}}=1/2+2\sin^{2}\theta_{\mathrm{W}}$
(where $Q_{\ell}$ is the lepton charge, and $T_{3}^{\ell}$ its third component
of the weak isospin). Using the expressions above with the numerical values of
Table 2, we determine the branching fractions listed in the second three rows
of Table 8. The $\rm H\to Z+(\ell^{+}\ell^{-})_{1}$ decays have
$\mathcal{O}(10^{-10}$–$10^{-13})$ 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 $\rm H\to Z+(\mathrm{e}^{+}\mathrm{e}^{-})_{1}$ final state,
with a particularly unique search for a Z boson accompanied by triphotons, or
an $\mathrm{e}^{+}\mathrm{e}^{-}$ pair222The boosted positronium would have an
extremely long path-length (its natural triphoton decay, with lifetime
$c\tau=142$ ns, would take place many km(!) away from the interaction point:
$L=(c\tau)(\beta*\gamma)\approx 1450$ km, for
$\beta\gamma\approx(m_{\mathrm{H}}-m_{\mathrm{Z}})/m_{\mathrm{(ee)}}\approx
3.4\,10^{4}$), but the bound state would be first broken into its constituent
$\mathrm{e}^{+}\mathrm{e}^{-}$ pair by the detector magnetic field [103].,
issuing from a significantly displaced vertex from the secondary positronium
decay.

Lastly, we compute the rates of the $\rm H\to Z+(\ell^{+}\ell^{-})_{0}$
decays, whose width can be written using similar expressions for the $\rm H\to
Z+M$ decays (where M is a pseudoscalar meson) [92], as follows

$\displaystyle\Gamma(\rm H\to Z+({\ell^{+}\ell^{-}})_{0})$
$\displaystyle=\frac{m_{\mathrm{H}}^{3}}{4\pi\varv^{4}}\,\lambda^{3/2}(1,r_{\mathrm{Z}},r_{\ell\ell})\,\big{|}F^{\mathrm{Z+\ell\ell_{0}}}\big{|}^{2}\;\mbox{
with
}F^{\mathrm{Z+\ell\ell_{0}}}=F_{\text{dir}}^{Z+\ell\ell_{0}}+F_{\text{ind}}^{Z+\ell\ell_{0}}.$
(10)

The contribution from the direct amplitude to the partial width (10) amounts
to

$\displaystyle
F_{\mathrm{dir}}^{Z+\ell\ell_{0}}=-f_{\ell\ell}\,a_{\mathrm{\ell}}\,\frac{m_{\mathrm{\ell}}^{2}}{m_{\mathrm{H}}^{2}}\frac{1-r_{\mathrm{Z}}^{2}+2r_{\mathrm{Z}}\ln
r_{\mathrm{Z}}}{(1-r_{\mathrm{Z}})^{3}},$ (11)

with $a_{\mathrm{\ell}}=T_{3}^{\mathrm{\ell}}/2=1/4$, which is suppressed by a
factor $m_{\ell\ell}^{2}/m_{\mathrm{H}}^{2}$ or
$m_{\mathrm{\ell}}^{2}/m_{\mathrm{H}}^{2}$, that makes it completely
negligible (the contribution to the final $(\tau\tau)_{0}$ amplitude is of
about 0.09%). The contribution from the indirect amplitude to the width (10)
reads

$\displaystyle F_{\text{ind}}^{Z+\ell\ell_{0}}$
$\displaystyle=f_{\ell\ell}\,a_{\mathrm{\ell}}$

Using the expressions above with the numerical values of Table 2, we determine
Z-plus-paraleptonium decay rates, $\rm H\to Z+(\ell^{+}\ell^{-})_{0}$, that
amount to $10^{-12}$–$10^{-16}$ (three last rows of Table 8).

All branching fractions computed here for Higgs decays into Z or $\gamma$ plus
leptonium are listed in Table 8 and shown in graphical form in Fig. 9. As
aforementioned, the decay rates are minuscule, in the
$\mathcal{O}(10^{-10}$–$10^{-16})$ 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 $\rm H\to Z+(3\gamma)$ with a displaced triphoton
vertex from the late orthopositronium $(ee)_{1}\to 3\gamma$ decay, or $\rm
H\to Z+(\ell^{+}\ell^{-})$ 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
$\mathrm{e}^{+}\mathrm{e}^{-},\mu^{+}\mu^{-}$ decays.

Table 8: Compilation of exclusive Higgs decays to a photon or a Z boson plus an ortho- $(\ell^{+}\ell^{-})_{1}$ or para- $(\ell^{+}\ell^{-})_{0}$ 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.  |  |  |  |  | Exp. limits | Producible at
---|---|---|---|---|---|---
$\rm H\to\,V$ | $+$ | $\rm(\ell\ell)$ | Branching fraction | Framework | 2023 | HL-LHC | FCC-ee | FCC-hh
$\rm H\,\to\,\gamma$ | + | $\rm(ee)_{1}$ |  $3.5$ | $\times~{}10^{-12}$ | (this work) | – | – | ✘ | ✘
$\rm(\mu\mu)_{1}$ |  $3.5$ | $\times~{}10^{-12}$ | (this work) | – | – | ✘ | ✘
$\rm(\tau\tau)_{1}$ |  $2.2$ | $\times~{}10^{-12}$ | (this work) | – | – | ✘ | ✘
$\rm H\,\to\,Z$ | + | $\rm(ee)_{1}$ |  $5.1$ | $\times~{}10^{-10}$ | (this work) | – | – | ✘ | ✓
$\rm(\mu\mu)_{1}$ |  $5.5$ | $\times~{}10^{-13}$ | (this work) | – | – | ✘ | ✘
$\rm(\tau\tau)_{1}$ |  $1.2$ | $\times~{}10^{-11}$ | (this work) | – | – | ✘ | ✘
$\rm(ee)_{0}$ |  $2.7$ | $\times~{}10^{-16}$ | (this work) | – | – | ✘ | ✘
$\rm(\mu\mu)_{0}$ |  $1.1$ | $\times~{}10^{-14}$ | (this work) | – | – | ✘ | ✘
$\rm(\tau\tau)_{0}$ |  $3.2$ | $\times~{}10^{-12}$ | (this work) | – | – | ✘ | ✘
Figure 9: Branching fractions (in negative log scale) of exclusive $\rm
H\to\gamma,Z+leptonium$ 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.

###  III.4 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 quark-antiquark 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 $\phi$ 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.4 for the similar $\rm Z\to M+M$ case).

Figure 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.

Table 9 lists the corresponding theoretical predictions and experimental
limits for concrete $\rm H\to 2(\mathrm{Q}\overline{\mathrm{Q}})$ and $\rm
H\to(\mathrm{Q}\overline{\mathrm{Q}})(\mathrm{Q}\overline{\mathrm{Q}}^{\prime})$
decay modes. The calculations are carried out in multiple frameworks (LC +
LCDA, RQM, NRQCD/NRCSM, NRQCD + LDME) and predict rates in the
$\mathcal{O}(10^{-9}$–$10^{-11})$ range333Semiexclusive decays $\rm H\to
B_{\mathrm{c}}+Q,+Q^{\prime},\,B_{\mathrm{c}}^{*}+Q+Q^{\prime}$ have much
larger rates [111] and, since they are not rare, nor exclusive, not covered
here., 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 $\rm H\to\phi+\mathrm{J}/\psi$
decay is the only process computed to date that includes the $\rm H\to
W^{*}W^{*}$ 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.

Table 9: 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.

|  |  |  |  |  | Exp. limits | Producible at
---|---|---|---|---|---|---|---
$\rm H\,\to$ | M | + | M | Branching fraction | Framework | 2023 | HL-LHC | FCC-ee | FCC-hh
$\rm H\,\to$ | $\rm\phi$ | + | $\rm J/\psi$ |  $1.0$ | $\times~{}10^{-9}$ | LC+LCDA [105] | – | – | ✘ | ✓
$\rm J/\psi$ | + | $\rm J/\psi$ |  $(5.8\text{ -- }6.0)$ | $\times~{}10^{-9}$ | NRQCD+LDME [110] | $<$ $3.8$$\times~{}10^{-4}$ [96] | $\lesssim 5.8\times~{}10^{-5}$ | ✘ | ✓
$1.7$ | $\times~{}10^{-10}$ | RQM [107]
$2.1$ | $\times~{}10^{-10}$ | RQM [109]
$(5.9\pm 2.3)$ | $\times~{}10^{-10}$ | NRQCD/NRCSM [108]
$1.5$ | $\times~{}10^{-10}$ | LC+LCDA [105]
$\rm\psi(2S)$ | + | $\rm J/\psi$ |  $\mathcal{O}(5)$ | $\times~{}10^{-11}$ | – |  $<$ $2.1$$\times~{}10^{-3}$ [96] | $\lesssim 3.2\times~{}10^{-4}$ | ✘ | ✓
$\rm\psi(2S)$ |  $(5.1\pm 2.0)$ | $\times~{}10^{-11}$ | NRQCD/NRCSM [108] |  $<$ $3.0$$\times~{}10^{-3}$ [96] | $\lesssim 4.5\times~{}10^{-4}$ | ✘ | ✓
$\rm B_{c}^{*\mp}$ | + | $\rm B_{c}^{*\pm}$ |  $(1.4\text{ -- }1.7)$ | $\times~{}10^{-10}$ | RQM [106] | – | – | ✘ | ✓
$\rm B_{c}^{\mp}$ | + | $\rm B_{c}^{\pm}$ |  $(2.0\text{ -- }3.0)$ | $\times~{}10^{-10}$ | RQM [106] | – | – | ✘ | ✓
$\rm\Upsilon(1S)$ | + | $\rm J/\psi$ |  $(2.7\text{ -- }3.6)$ | $\times~{}10^{-10}$ | NRQCD+LDME [110] | – | – | ✘ | ✓
$1.6$ | $\times~{}10^{-11}$ | LC+LCDA [105]
$\rm\Upsilon(1S)$ |  $(8.5\text{ -- }9.2)$ | $\times~{}10^{-10}$ | NRQCD+LDME [110] | $<$ $1.7$$\times~{}10^{-3}$ [96] | $\lesssim 2.6\times~{}10^{-4}$ | ✘ | ✓
$1.8$ | $\times~{}10^{-10}$ | RQM [107]
$2.3$ | $\times~{}10^{-9}$ | RQM [109]
$(4.3\pm 0.9)$ | $\times~{}10^{-10}$ | NRQCD/NRCSM [108]
$2.3$ | $\times~{}10^{-9}$ | LC+LCDA [105]
$\rm\Upsilon(2S)$ | + | $\rm\Upsilon(2S)$ |  $(1.0\pm 0.2)$ | $\times~{}10^{-10}$ | NRQCD/NRCSM [108] | – | – | ✘ | ✓
$\rm\Upsilon(3S)$ | + | $\rm\Upsilon(3S)$ |  $(5.7\pm 1.2)$ | $\times~{}10^{-11}$ | NRQCD/NRCSM [108] | – | – | ✘ | ✓
$\rm\Upsilon(mS)$ | + | $\rm\Upsilon(nS)$ |  |  | – |  $<$ $3.5$$\times~{}10^{-4}$ [96] |  $\lesssim 1.1\times~{}10^{-5}$ [57] | ✘ | ✘

Experimentally, the double-$(\mathrm{Q}\overline{\mathrm{Q}})$ and
$(\mathrm{Q}\overline{\mathrm{Q}})(\mathrm{Q}\overline{\mathrm{Q}}^{\prime})$
decays have been searched for at the LHC [96], but the current
$\mathcal{O}(10^{-3}$–$10^{-4})$ 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.

Figure 11: Branching ratios (in negative log scale) of exclusive $\rm H\to
meson+meson$ 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 H bosons to be produced at
both machines.

##  IV Rare Z boson decays

###  IV.1 Rare three-body Z boson decays

The two-body decay of the Z boson into two massless vector particles ($\rm
Z\to\gamma\gamma,gg$) 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
$\alpha_{\text{s}}(m_{{}_{\mathrm{Z}}})$ coupling constant, hadronic Z boson
partial widths, and top quark mass, have (noticeably) changed. For example,
the original $\rm Z\to 3\gamma$ calculations from [115] used
$m_{\mathrm{t}}=124$ GeV, whereas the calculation of the $\rm Z\to 3g$ decay
of [116] used $m_{\mathrm{t}}\to\infty$, and the prediction for $\rm Z\to
g+g+\gamma$ from [114] used $\alpha_{\text{s}}(m_{{}_{\mathrm{Z}}})=0.17$,
$m_{\mathrm{t}}=20$ GeV, and
$\Gamma_{\mathrm{Z}}=\Gamma_{\mathrm{Z\to\mathrm{q}\overline{\mathrm{q}}}}\approx
2$ 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 2. 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
10) change with respect to the existing ones by 10–25% up or down, except for
the $\rm Z\to\gamma+\nu\overline{\nu}$ 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.

Figure 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. Table 10: Compilation of exclusive Z decays to three
gauge bosons (photons and/or gluons, with two additional channels requiring
one soft photon, $E_{\gamma}<1$ 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.

|  |  | Exp. limits | Producible at
---|---|---|---|---
| Branching fraction | Framework | 2023 | HL-LHC | FCC-ee
$\rm Z\;\to\;g+g+g$ |  $1.9$ | $\times~{}10^{-6}$ | NLO SM [116] | $<$ $1.1$$\times~{}10^{-2}$ [118] | – | ✓
$1.9$ | $\times~{}10^{-6}$ | NLO SM (this work)
$\rm Z\;\to\;g+g+\gamma$ |  $7.0$ | $\times~{}10^{-7}$ | NLO SM EW [114] | – | – | ✓
$6.8$ | $\times~{}10^{-7}$ | NLO SM (this work)
$\rm Z\;\to\;g+g+\gamma_{\mathrm{soft}}$ |  $1.2$ | $\times~{}10^{-9}$ | NLO SM (this work) | – | – | ✓
$\rm Z\;\to\;\gamma+\gamma+\gamma$ |  $5.4$ | $\times~{}10^{-10}$ | NLO SM [115] | $<$ $2.2$$\times~{}10^{-6}$ [61] | $\lesssim 1.3\times~{}10^{-7}$ | ✓
$6.4$ | $\times~{}10^{-10}$ | NLO SM (this work)
$\rm Z\;\to\;\gamma+\gamma+\gamma_{\mathrm{soft}}$ |  $2.2$ | $\times~{}10^{-12}$ | NLO SM (this work) |  $<$ $1.5$$\times~{}10^{-5}$ [119] | $\lesssim 2.1\times~{}10^{-7}$ | ✓
$\rm Z\;\to\;\gamma+\nu+\hat{\nu}$ |  $7.2$ | $\times~{}10^{-10}$ | NLO SM [117] | – | – | ✓
$1.3$ | $\times~{}10^{-10}$ | NLO SM (this work)

Among the four rare Z boson decays shown of Fig. 12, those with final-state
gluons have the largest rates, $\mathcal{O}(10^{-6})$, whereas the pure
electroweak processes have much smaller, $\mathcal{O}(10^{-10})$, branching
fractions. Experimentally, limits exist for $\rm Z\to 3g$ and $\rm Z\to
3\gamma$ of $\mathcal{O}(10^{-2})$ and $\mathcal{O}(10^{-6})$, respectively,
although no searches have been yet performed to our knowledge for $\rm
Z\to\gamma\,g\,g$ and $\rm Z\to\gamma\,\nu\overline{\nu}$ 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-ee444In this whole
section, we will be providing future Z-boson limits expectations for the FCC-
ee alone because, although the FCC-hh will produce twice more Z bosons (Table
1), the backgrounds are much larger for this collider., as indicated by the
red vertical line in Fig. 13. We also provide estimates of the branching
fraction for the $\rm Z\to gg\,\gamma_{\text{soft}}$ and $\rm
Z\to\gamma\gamma\,\gamma_{\text{soft}}$ decays (where the soft photon has
$E_{\gamma}<1$ GeV) that can mimic the forbidden $\rm Z\to gg,\gamma\gamma$
channels if the low-energy photon goes experimentally undetected. The
branching ratios for both such “fake” Landau–Yang-violating decay modes are
$\mathcal{O}(10^{-9})$ and $\mathcal{O}(10^{-12})$, respectively.

Figure 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.

###  IV.2 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 1), 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 $\to\gamma^{*}\gamma^{(*)}$
decays. In addition, exclusive Z boson decays into flavoured mesons such as
that shown in Fig. 14 (left) probe FCNC processes.

Figure 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.

####  IV.2.1 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
($\rm
Z\to\gamma+\mathrm{q}\overline{\mathrm{q}},\mathrm{c}\overline{\mathrm{c}},\mathrm{b}\overline{\mathrm{b}}$)
are listed in Table 11, 12, and 13, 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 [120, 119], ATLAS [81, 121, 122, 123, 84], CMS [124, 125,
83, 85], and LHCb [126] collaborations. In some cases, such as for the $\rm
Z\to\gamma\eta,\,\gamma\eta^{\prime}$ channels, LEP measurements at the Z pole
still provide the best upper limits [127].

The exclusive Z-boson radiative decays to light mesons ($\rm Z\to\gamma+M$),
listed in Table 11, have theoretical branching fractions in the range of
$\mathcal{O}(10^{-8}$–$10^{-12})$, whereas the current experimental limits are
in the $\mathcal{O}(10^{-5}$–$10^{-7})$ range with seven channels searched for
by ATLAS (3), LHCb (1), CDF at Tevatron (1), and ALEPH at LEP (2). The $\rm
Z\to\gamma\omega$ decay is very close to being detected at HL-LHC because its
predicted $\mathcal{B}$ is about half of our projected limit. All channels are
producible at the FCC-ee, except the FCNC $\rm Z\to\gamma+D^{0}$ one, which is
the only one studied to date that proceeds through the right diagram of Fig.
14.

Table 11: 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.

|  |  |  |  | Exp. limits | Producible at
---|---|---|---|---|---|---
$\rm Z\,\to\,\gamma$ | $+$ | $\rm M$ | Branching fraction | Framework | 2023 | HL-LHC | FCC-ee
$\rm Z\,\to\,\gamma$ | + | $\rm\pi^{0}$ |  $(9.8\pm 1.0)$ | $\times~{}10^{-12}$ | SCET+LCDA [62] |  $<$ $2.0$$\times~{}10^{-5}$ [119] | $\lesssim 2.8\times~{}10^{-7}$ | ✓
$\rm\eta$ |  $(1.0\text{ -- }17.0)$ | $\times~{}10^{-10}$ | SCET+LCDA [128] |  $<$ $5.1$$\times~{}10^{-5}$ [127] | – | ✓
$\rm\rho^{0}$ |  $(4.19\pm 0.47)$ | $\times~{}10^{-9}$ | SCET+LCDA [62] |  $<$ $2.5$$\times~{}10^{-5}$ [81] | $\lesssim 1.8\times~{}10^{-6}$ | ✓
$\rm\omega$ |  $(2.82\pm 0.41)$ | $\times~{}10^{-8}$ | SCET+LCDA [62] |  $<$ $3.8$$\times~{}10^{-7}$ [84] | $\lesssim 5.7\times~{}10^{-8}$ | ✓
$\rm\eta^{\prime}$ |  $(3.1\text{ -- }4.8)$ | $\times~{}10^{-9}$ | SCET+LCDA [128] |  $<$ $4.2$$\times~{}10^{-5}$ [127] | – | ✓
$\rm\phi$ |  $(1.17\pm 0.08)$ | $\times~{}10^{-8}$ | LC+LCDA [129] | $<$ $9.0$$\times~{}10^{-7}$ [81] | $\lesssim 6.6\times~{}10^{-8}$ | ✓
$(1.04\pm 0.12)$ | $\times~{}10^{-8}$ | SCET+LCDA [62]
$\rm D^{0}$ |  $1.0$ | $\times~{}10^{-15}$ | SCET+LCDA [62, 126] |  $<$ $2.1$$\times~{}10^{-3}$ [126] | $\lesssim 3.8\times~{}10^{-5}$ | ✘

The exclusive Z-boson radiative decays to charmonium mesons ($\rm
Z\to\gamma+\mathrm{c}\overline{\mathrm{c}}$), collected in Table 12, have
rates in the $\mathcal{O}(10^{-8}$–$10^{-10})$ 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, $\rm Z\to\gamma+\mathrm{J}/\psi$, with $\mathcal{O}(10^{-6})$
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.

Table 12: 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.

|  |  |  |  | Exp. limits | Producible at
---|---|---|---|---|---|---
$\rm Z\,\to\,\gamma$ | $+$ | $\rm M$ | Branching fraction | Framework | 2023 | HL-LHC | FCC-ee
$\rm Z\,\to\,\gamma$ | + | $\rm\eta_{c}$ |  $(9.5\pm 0.2)$ | $\times~{}10^{-9}$ | NRQCD (NNLO+NLL) [130] | – | – | ✓
$\left(1.32^{+0.56}_{-0.54}\right)$ | $\times~{}10^{-8}$ | NRQCD+LDME [131]
$(7.42\pm 0.61)$ | $\times~{}10^{-9}$ | NRQCD (NLO+NLL) [132]
$6.6$ | $\times~{}10^{-9}$ | NRQCD+LDME [133]
$(9.4\pm 1.0)$ | $\times~{}10^{-9}$ | LC+LCDA [133]
$\rm J/\psi$ |  $\left(5.75^{+0.08}_{-0.09}\right)$ | $\times~{}10^{-8}$ | NRQCD (NNLO+NLL) [130] | $<$ $6.0$$\times~{}10^{-7}$ [85, 124] | $\lesssim 3.1\times~{}10^{-7}$ [55] | ✓
$\left(1.04^{+0.18}_{-0.16}\right)$ | $\times~{}10^{-7}$ | NRQCD+LDME [131]
$\left(8.96^{+1.51}_{-1.38}\right)$ | $\times~{}10^{-8}$ | LC+LCDA [134]
$4.5$ | $\times~{}10^{-8}$ | NRQCD+LDME [133]
$(8.8\pm 0.9)$ | $\times~{}10^{-8}$ | LC+LCDA [133]
$(9.96\pm 1.86)$ | $\times~{}10^{-8}$ | NRQCD+LDME [129]
$(8.02\pm 0.45)$ | $\times~{}10^{-8}$ | SCET+LCDA [62]
$\rm\chi_{c0}$ |  $(3.74\pm 0.05)$ | $\times~{}10^{-10}$ | NRQCD+LDME [135] | – | – | ✓
$1.4$ | $\times~{}10^{-10}$ | NRQCD+LDME [133]
$(5.0\pm 2.0)$ | $\times~{}10^{-10}$ | LC+LCDA [133]
$\rm\chi_{c1}$ |  $\left(2.383^{+0.014}_{-0.017}\right)$ | $\times~{}10^{-9}$ | NRQCD+LDME [135] | – | – | ✓
$8.7$ | $\times~{}10^{-10}$ | NRQCD+LDME [133]
$(5.6\pm 2.0)$ | $\times~{}10^{-9}$ | LC+LCDA [133]
$\rm h_{c}$ |  $\left(3.487^{+0.206}_{-0.230}\right)$ | $\times~{}10^{-9}$ | NRQCD+LDME [135] | – | – | ✓
$3.0$ | $\times~{}10^{-9}$ | NRQCD+LDME [133]
$(1.0\pm 0.4)$ | $\times~{}10^{-8}$ | LC+LCDA [133]
$\rm\chi_{c2}$ |  $\left(3.38^{+0.19}_{-0.22}\right)$ | $\times~{}10^{-10}$ | NRQCD+LDME [135] | – | – | ✓
$2.9$ | $\times~{}10^{-10}$ | NRQCD+LDME [133]
$(1.0\pm 0.4)$ | $\times~{}10^{-9}$ | LC+LCDA [133]

The exclusive Z-boson radiative decays to bottomonium mesons ($\rm
Z\to\gamma+\mathrm{b}\overline{\mathrm{b}}$), listed in Table 13, have also
branching fractions in the $\mathcal{O}(10^{-8}$–$10^{-10})$ range. Three $\rm
Z\to\gamma+\Upsilon(nS)$ channels have been searched for at the LHC, setting
limits in the $\mathcal{O}(10^{-6})$ range. The $\rm Z\to\gamma+\Upsilon(1S)$
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 13: 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.

|  |  |  |  | Exp. limits | Producible at
---|---|---|---|---|---|---
$\rm Z\,\to\,\gamma$ | $+$ | $\rm M$ | Branching fraction | Framework | 2023 | HL-LHC | FCC-ee
$\rm Z\,\to\,\gamma$ | + | $\rm\eta_{b}$ |  $(2.43\pm 0.01)$ | $\times~{}10^{-8}$ | NRQCD (NNLO+NLL) [130] | – | – | ✓
$\left(2.79^{+0.20}_{-0.19}\right)$ | $\times~{}10^{-8}$ | NRQCD+LDME [131]
$(2.8\pm 0.5)$ | $\times~{}10^{-8}$ | NRQCD (NLO+NLL) [132]
$\rm\Upsilon(1S)$ |  $(4.63\pm 0.02)$ | $\times~{}10^{-8}$ | NRQCD (NNLO+NLL) [130] | $<$ $2.8$$\times~{}10^{-6}$ [82] | $\lesssim 2.2\times~{}10^{-7}$ | ✓
$(5.41\pm 0.39)$ | $\times~{}10^{-8}$ | NRQCD+LDME [131]
$(5.61\pm 0.29)$ | $\times~{}10^{-8}$ | NRQCD+LDME [136]
$\left(4.8^{+0.3}_{-0.2}\right)$ | $\times~{}10^{-8}$ | LC+LCDA [134]
$(4.93\pm 0.51)$ | $\times~{}10^{-8}$ | NRQCD+LDME [129]
$(5.39\pm 0.16)$ | $\times~{}10^{-8}$ | SCET+LCDA [62]
$\rm\Upsilon(2S)$ |  $(2.66\pm 0.31)$ | $\times~{}10^{-8}$ | NRQCD+LDME [136] | $<$ $1.7$$\times~{}10^{-6}$ [82] | $\lesssim 1.3\times~{}10^{-7}$ | ✓
$\left(2.44^{+0.14}_{-0.13}\right)$ | $\times~{}10^{-8}$ | LC+LCDA [134]
$\rm\Upsilon(3S)$ |  $(1.93\pm 0.25)$ | $\times~{}10^{-8}$ | NRQCD+LDME [136] | $<$ $4.8$$\times~{}10^{-6}$ [82] | $\lesssim 3.7\times~{}10^{-7}$ | ✓
$\left(1.88^{+0.11}_{-0.10}\right)$ | $\times~{}10^{-8}$ | LC+LCDA [134]
$\rm\Upsilon(4S)$ |  $(1.22\pm 0.13)$ | $\times~{}10^{-8}$ | SCET+LCDA [62] | – | – | ✓
$\rm\Upsilon(nS)$ |  $\left(9.96^{+0.28}_{-0.26}\right)$ | $\times~{}10^{-8}$ | SCET+LCDA [62] | – | – | ✓
$\rm\chi_{b0}$ |  $\left(2.73^{+0.05}_{-0.04}\right)$ | $\times~{}10^{-10}$ | NRQCD+LDME [135] | – | – | ✓
$\rm\chi_{b1}$ |  $\left(1.473^{+0.010}_{-0.011}\right)$ | $\times~{}10^{-9}$ | NRQCD+LDME [135] | – | – | ✓
$\rm h_{b}$ |  $\left(9.27^{+0.36}_{-0.41}\right)$ | $\times~{}10^{-10}$ | NRQCD+LDME [135] | – | – | ✓
$\rm\chi_{b2}$ |  $\left(2.92^{+0.12}_{-0.14}\right)$ | $\times~{}10^{-10}$ | NRQCD+LDME [135] | – | – | ✓

Figure 15: Branching fractions (in negative log scale) of exclusive $\rm
Z\to\gamma+M$ 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.

####  IV.2.2 Z boson triphoton decays

As seen in Table 10, the $\rm Z\to 3\gamma$ decay is extremely suppressed in
the SM ($\mathcal{B}=6.4\cdot 10^{-10}$, 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
$\mathrm{Z}\to\gamma\,a(\gamma\gamma)$, with $a$ being an ALP [137, 20] 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
$\rm Z\to\gamma+M$ results for spin-0,2 mesons of Tables 11–13 with their
corresponding two-photon branching fractions [53]. The results are listed in
Table 14. The sum of all exclusive mesonic decay channels amounts to
$\mathcal{B}=1.8\cdot 10^{-10}$, representing an increase of about 30% from
the direct $3\gamma$ decay. The yields of SM triphoton decays will therefore
amount to about 5 000 events for the $N(\rm Z)=6\cdot 10^{12}$ 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 $\mathrm{e}^{+}\mathrm{e}^{-}$
facilities [137].

Figure 16: Triphoton decays of the Z boson: SM direct decay (left), exclusive $\rm\gamma+meson(\gamma\gamma)$ decay (center), radiative decay into a BSM (ALP or graviton) diphoton system (right). Table 14: Compilation of exclusive Z decays into triphoton final states. The predictions of relevant $\rm Z\to\gamma+M$ radiative decays listed in Tables 11–13 are combined with the diphoton branching fractions of the produced mesons, $\rm M\to\gamma\gamma$ [53]. $\rm Z\,\to\,\gamma$ | + | $X(\gamma\gamma)$ | Branching fraction
---|---|---|---
$\rm Z\,\to\,\gamma$ | + |  $\rm\pi^{0}$$(\gamma\gamma)$ | $9.7$ | $\times~{}10^{-12}$
$\rm\eta$$(\gamma\gamma)$ | $6.3$ | $\times~{}10^{-11}$
$\rm\eta^{\prime}$$(\gamma\gamma)$ | $1.1$ | $\times~{}10^{-10}$
$\rm\eta_{c}$$(\gamma\gamma)$ | $2.1$ | $\times~{}10^{-12}$
$\rm\chi_{c0}$$(\gamma\gamma)$ | $7.6$ | $\times~{}10^{-14}$
$\rm\chi_{c1}$$(\gamma\gamma)$ | $1.5$ | $\times~{}10^{-14}$
$\rm\chi_{c2}$$(\gamma\gamma)$ | $9.6$ | $\times~{}10^{-14}$
$\rm\chi_{b0}$$(\gamma\gamma)$ | $1.6$ | $\times~{}10^{-14}$
$\rm\chi_{b2}$$(\gamma\gamma)$ | $1.6$ | $\times~{}10^{-14}$
Sum |  |  | $1.8$ | $\times~{}10^{-10}$
$Z$ | $\longrightarrow$ | $\gamma\gamma\gamma$ | $6.4$ | $\times~{}10^{-10}$
Total |  |  | $8.2$ | $\times~{}10^{-10}$

####  IV.2.3 Z boson decays into a W boson plus a meson

Table 15 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 15 are shown in
graphical form in Fig. 17.

Table 15: 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.

|  |  |  |  | Exp. limits | Producible at
---|---|---|---|---|---|---
$\rm Z\,\to\,W$ | $+$ | $\rm M$ | Branching fraction | Framework | 2023 | HL-LHC | FCC-ee
$\rm Z\,\to\,W^{\mp}$ | + | $\rm\pi^{\pm}$ |  $(1.51\pm 0.01)$ | $\times~{}10^{-10}$ | SCET+LCDA [62] |  $<$ $7.0$$\times~{}10^{-5}$ [127] | – | ✓
$\rm\rho^{\pm}$ |  $(4.0\pm 0.1)$ | $\times~{}10^{-10}$ | SCET+LCDA [62] |  $<$ $8.3$$\times~{}10^{-5}$ [127] | – | ✓
$\rm K^{\pm}$ |  $(1.16\pm 0.01)$ | $\times~{}10^{-11}$ | SCET+LCDA [62] | – | – | ✓
$\rm K^{*\pm}$ |  $(1.96\pm 0.12)$ | $\times~{}10^{-11}$ | SCET+LCDA [62] | – | – | ✓
$\rm D^{\pm}$ |  $(1.99\pm 0.17)$ | $\times~{}10^{-11}$ | SCET+LCDA [62] | – | – | ✓
$\rm D_{s}^{\pm}$ |  $(6.04\pm 0.30)$ | $\times~{}10^{-10}$ | SCET+LCDA [62] | – | – | ✓

Figure 17: Branching fractions (in negative log scale) of exclusive $\rm Z\to
W^{\pm}+meson$ 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.

###  IV.3 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, $(\ell\ell)_{0}$)
or triplet (ortholeptonium, $(\ell\ell)_{1}$) states. No calculation for such
processes exist to our knowledge, but the key ingredients can be obtained from
the determination of the
$\rm\mathrm{e}^{+}\mathrm{e}^{-}\to\gamma+(\tau^{+}\tau^{-})_{0}$ cross
section at the Z pole provided in the ditauonium studies of Ref. [138].

Figure 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.

The leading-order partial decay width of a Z boson into a leptonium-plus-a-
photon can be derived from the ratio of the
$\sigma(\mathrm{e}^{+}\mathrm{e}^{-}\to\gamma+(\ell^{+}\ell^{-}))$ cross
section at the Z pole over the total resonant Z boson cross section in
$\mathrm{e}^{+}\mathrm{e}^{-}$ collisions, and amounts to

$\displaystyle\mathcal{B}\left(\rm Z\to\gamma+(\ell^{+}\ell^{-})_{0}\right)$
$\displaystyle=\frac{\alpha(0)^{4}\alpha(m_{\mathrm{Z}})^{2}}{9\cdot
256\,n^{3}}\frac{m_{\ell\ell}^{2}}{m_{\mathrm{Z}}^{2}}\frac{\left(1-4s_{w}^{2}\right)^{2}\left(8s_{w}^{4}-4s_{w}^{2}+1\right)\left(m_{\mathrm{Z}}^{2}-m_{\ell\ell}^{2}\right)}{\Gamma_{\mathrm{ee}}\Gamma_{\mathrm{Z}}s_{w}^{4}c_{w}^{4}}$
(12)

for the paraleptonium case, and to

$\displaystyle\mathcal{B}\left(\rm Z\to\gamma+(\ell^{+}\ell^{-})_{1}\right)$
$\displaystyle=\frac{\alpha(0)^{4}\alpha(m_{\mathrm{Z}})^{2}}{9\cdot
256\,n^{3}}\frac{m_{\ell\ell}^{2}}{m_{\mathrm{Z}}^{4}}\frac{\left(8s_{w}^{4}-4s_{w}^{2}+1\right)\left(m_{\mathrm{Z}}^{4}-m_{\ell\ell}^{4}\right)}{\Gamma_{\mathrm{ee}}\Gamma_{\mathrm{Z}}s_{w}^{4}c_{w}^{4}}$
(13)

for the ortho-leptonium one, where $n$ is the principal quantum number of the
leptonium resonance, and $s_{w}\equiv\sin\theta_{\mathrm{W}}$, and
$c_{w}\equiv\cos\theta_{\mathrm{W}}$ are the sine and cosine of the Weinberg
angle (Table 2). In both expressions above, $\alpha(0)$ and
$\alpha(m_{\mathrm{Z}})$ are the electromagnetic coupling evaluated at
zero555The leptonium wavefunction is proportional to $\alpha(0)^{3}$, Eq. (3),
which combined with the emission of a final-state onshell photon, gives the
$\alpha(0)^{4}$ dependence shown in Eqs. (12) and (13). and at the Z pole
mass, respectively, that, together with the rest of parameters, are listed in
Table 2. The obtained branching fractions are tabulated in Table 16, and
plotted in Fig. 19. The dependence of these Z-boson radiative rates on
$\alpha^{6}\,m_{\ell\ell}^{2}$, leads to vanishingly small branching
fractions, $\mathcal{O}(10^{-13}$–$10^{-23})$, 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 $(\ell^{+}\ell^{-})$ into photons, $\rm e^{\pm}$, and/or
$\mu^{\pm}$ [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 $\rm Z\to\gamma\,(\tau^{+}\tau^{-})_{0}$ case.

Table 16: 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.  |  |  |  |  | Exp. limits | Producible at
---|---|---|---|---|---|---
$\rm Z\,\to\,\gamma$ | $+$ | $\rm(\ell\ell)$ | Branching fraction | Framework | 2023 | HL-LHC | FCC-ee
$\rm Z\,\to\,\gamma$ | + | $\rm(ee)_{1}$ |  $7.3$ | $\times~{}10^{-21}$ | (this work) | – | – | ✘
$\rm(\mu\mu)_{1}$ |  $3.1$ | $\times~{}10^{-16}$ | (this work) | – | – | ✘
$\rm(\tau\tau)_{1}$ |  $8.9$ | $\times~{}10^{-14}$ | (this work) | – | – | ✘
$\rm(ee)_{0}$ |  $4.7$ | $\times~{}10^{-23}$ | (this work) | – | – | ✘
$\rm(\mu\mu)_{0}$ |  $2.0$ | $\times~{}10^{-18}$ | (this work) | – | – | ✘
$\rm(\tau\tau)_{0}$ |  $5.7$ | $\times~{}10^{-16}$ | (this work) | – | – | ✘
Figure 19: Branching ratios (in negative log scale) of exclusive $\rm
Z\to\gamma+leptonium$ 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.

###  IV.4 Exclusive Z boson decays into two mesons

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 $\rm V^{*}\to M$ transitions (right). Due to the coincidence of
diagrams with those of the Higgs boson (Fig. 10), and since the Z boson yields
at colliders are $\sim$1000 times larger (Table 1), 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, $\rm Z\to M+M$,
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.

Figure 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.

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 $\rm
Z\to VM+VM$ 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

$\displaystyle\Gamma_{\mathrm{Z\to
VM\,VM}}=\Gamma^{1}_{\mathrm{VM\,VM}}(1+2R+R^{2}),$ (14)

where $\Gamma^{1}_{\mathrm{VM\,VM}}$ is the partial width corresponding to the
left diagram of Fig. 20, amounting to

$\displaystyle\Gamma^{1}_{\mathrm{VM\,VM}}=\frac{2048\pi^{2}\alpha(0)\alpha_{\text{s}}(m_{{}_{\mathrm{Z}}})^{2}}{27c_{w}^{2}s_{w}^{2}m_{\mathrm{Z}}^{5}}|\phi_{\mathrm{VM}}(0)|^{4}\left(1-\frac{4m_{\mathrm{VM}}^{2}}{m_{\mathrm{Z}}^{2}}\right)^{5/2},$
(15)

and where $R$ is the ratio of the direct amplitudes $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$ corresponding, respectively, to the left and center diagrams
of Fig. 20, given by

$\displaystyle
R=\frac{\mathcal{A}_{2}}{\mathcal{A}_{1}}=-\frac{9Q_{\mathrm{q}}^{2}\alpha(0)}{8\alpha_{\text{s}}(m_{{}_{\mathrm{Z}}})}\frac{m_{\mathrm{Z}}}{m_{\mathrm{VM}}}.$
(16)

For the light meson wavefunction at the origin $\phi_{\mathrm{M}}(0)$ in Eq.
(15), we derive it from its decay constant using the Van Royen–Weisskopf
expression, Eq. (7), as follows

$\displaystyle|\phi_{\mathrm{VM}}(0)|^{2}=\frac{m_{\mathrm{VM}}f^{2}_{\mathrm{VM}}}{4N_{\mathrm{c}}},$
(17)

for $N_{\mathrm{c}}=3$ colours, and without higher-order QCD corrections. We
can only estimate the rate for the $\rm Z\to\phi+\phi$ channel because it is
the only pure $|\mathrm{q}\overline{\mathrm{q}}\rangle$ state among the light
mesons. Using Eqs. (14)–(17) and the parameters of Table 2, we obtain a
$\mathcal{B}=2.1\cdot 10^{-12}$ rate (Table 17). There is no $\rm
Z\to\phi+\phi$ 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 $\rm Z\to\pi^{0}+\pi^{0}$ 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
$|\mathrm{u}\overline{\mathrm{u}}\rangle,|\mathrm{d}\overline{\mathrm{d}}\rangle$
to physical eigenstates such as $\rho,\omega,\eta...$ would need to be
performed. One can, however, anticipate branching fractions of the same order
as the double-$\phi$ channel, $\mathcal{O}(10^{-12})$, since they only differ
from a rotation and they are enhanced by the second and third diagram of Fig.
20 involving extra $\rm\gamma^{*},W^{*}$ 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 17: 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-$\phi$ 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
---|---|---|---|---|---|---|---
$Z\,\to$ | M | $+$ | M | Branching fraction | Framework | 2023 | HL-LHC | at FCC-ee
$\rm Z\,\to$ | $\rm\pi^{0}$ | + | $\rm\pi^{0}$ |  |  | – |  $<$ $1.5$$\times~{}10^{-5}$ [119] | $\lesssim 2.1\times~{}10^{-7}$ | ✘
$\rm\phi$ | + | $\rm\phi$ |  $2.1$ | $\times~{}10^{-12}$ | EFT+NRQM (this work) | – | – | ✓

Tables 18 and 19 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, $\mathcal{B}(\rm Z\to
M+M)\lesssim 10^{-10}$, 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 $\rm
Z\to\chi_{c1}+\chi_{c1},\,\chi_{c2}+\chi_{c2},\,\chi_{c0}+\chi_{c2}$,…, 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 double-quarkonia production. All that said, and
within the relatively large theoretical uncertainties for those processes, the
largest branching fractions expected are for double-charmonium with
$\mathcal{O}(10^{-10})$ 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 18: 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.

|  |  |  |  |  | Exp. limits | Producible at
---|---|---|---|---|---|---|---
$Z\,\to$ | M | $+$ | M | Branching fraction | Framework | 2023 | HL-LHC | at FCC-ee
$\rm Z\,\to$ | $\rm\eta_{c}$ | + | $\rm J/\psi$ |  $(1.5\pm 0.4)$ | $\times~{}10^{-11}$ | NRQCD/NRCSM [140] | – | – | ✓
$(1.8\text{ -- }2.7)$ | $\times~{}10^{-11}$ | NRQCD+LDME (NLO) [142]
$2.7$ | $\times~{}10^{-14}$ | NRQCD+LDME [141]
$\rm\chi_{c0}$ |  $2.3$ | $\times~{}10^{-12}$ | NRQCD+LDME [141] | – | – | ✓
$(2.3\pm 1.0)$ | $\times~{}10^{-12}$ | LC+LCDA [141]
$\rm\chi_{c1}$ |  $5.4$ | $\times~{}10^{-14}$ | NRQCD+LDME [141] | – | – | ✘
$\rm h_{c}$ |  $2.1$ | $\times~{}10^{-13}$ | NRQCD+LDME [141] | – | – | ✓
$(1.0\pm 0.5)$ | $\times~{}10^{-12}$ | LC+LCDA [141]
$\rm\chi_{c2}$ |  $9.7$ | $\times~{}10^{-13}$ | NRQCD+LDME [141] | – | – | ✓
$(4.6\pm 2.0)$ | $\times~{}10^{-12}$ | LC+LCDA [141]
$\rm J/\psi$ | + | $\rm J/\psi$ |  $(1.1\pm 0.3)$ | $\times~{}10^{-10}$ | NRQCD/NRCSM [140] | $<$ $1.1$$\times~{}10^{-6}$ [96] | $\lesssim 1.7\times~{}10^{-7}$ | ✓
$\left(1.11^{+0.34}_{-0.24}\right)$ | $\times~{}10^{-10}$ | NRQCD+LDME [143]
$(1.1\text{ -- }1.3)$ | $\times~{}10^{-10}$ | NRQCD+LDME (NLO) [142]
$2.3$ | $\times~{}10^{-14}$ | NRQCD+LDME [141]
$2.7$ | $\times~{}10^{-11}$ | NRQCD [139]
$\rm\chi_{c0}$ |  $(1.1\text{ -- }4.1)$ | $\times~{}10^{-12}$ | NRQCD+LDME (NLO) [142] | – | – | ✓
$8.3$ | $\times~{}10^{-14}$ | NRQCD+LDME [141]
$(4.7\pm 2.0)$ | $\times~{}10^{-13}$ | LC+LCDA [141]
$\rm\chi_{c1}$ |  $(3.5\text{ -- }4.4)$ | $\times~{}10^{-12}$ | NRQCD+LDME (NLO) [142] | – | – | ✓
$3.5$ | $\times~{}10^{-15}$ | NRQCD+LDME [141]
$\rm h_{c}$ |  $1.5$ | $\times~{}10^{-12}$ | NRQCD+LDME [141] | – | – | ✓
$(9.5\pm 5.0)$ | $\times~{}10^{-12}$ | LC+LCDA [141]
$\rm\chi_{c2}$ |  $(9.6\text{ -- }24.8)$ | $\times~{}10^{-13}$ | NRQCD+LDME (NLO) [142] | – | – | ✓
$1.4$ | $\times~{}10^{-13}$ | NRQCD+LDME [141]
$(9.3\pm 4.0)$ | $\times~{}10^{-13}$ | LC+LCDA [141]
$\rm\chi_{c0}$ | + | $\rm\chi_{c1}$ |  $7.6$ | $\times~{}10^{-14}$ | NRQCD+LDME [141] | – | – | ✘
$(1.4\pm 1.0)$ | $\times~{}10^{-12}$ | LC+LCDA [141]
$\rm h_{c}$ |  $3.5$ | $\times~{}10^{-16}$ | NRQCD+LDME [141] | – | – | ✘
$\rm\chi_{c2}$ |  $6.4$ | $\times~{}10^{-15}$ | NRQCD+LDME [141] | – | – | ✘
$\rm\chi_{c1}$ | + | $\rm\chi_{c1}$ |  $3.9$ | $\times~{}10^{-16}$ | NRQCD+LDME [141] | – | – | ✘
$\rm h_{c}$ |  $2.9$ | $\times~{}10^{-14}$ | NRQCD+LDME [141] | – | – | ✘
$(6.1\pm 5.0)$ | $\times~{}10^{-13}$ | LC+LCDA [141]
$\rm\chi_{c2}$ |  $1.3$ | $\times~{}10^{-13}$ | NRQCD+LDME [141] | – | – | ✘
$(2.8\pm 2.0)$ | $\times~{}10^{-12}$ | LC+LCDA [141]
$\rm h_{c}$ | + | $\rm h_{c}$ |  $9.9$ | $\times~{}10^{-17}$ | NRQCD+LDME [141] | – | – | ✘
$\rm\chi_{c2}$ |  $2.3$ | $\times~{}10^{-16}$ | NRQCD+LDME [141] | – | – | ✘
$\rm\chi_{c2}$ | + | $\rm\chi_{c2}$ |  $1.3$ | $\times~{}10^{-16}$ | NRQCD+LDME [141] | – | – | ✘

Figure 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.  Table 19: Compilation of exclusive Z
decays to one bottomonium meson plus another meson (or anything else, in a few
cases). 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.

|  |  |  |  |  | Exp. limits | Producible at
---|---|---|---|---|---|---|---
$Z\,\to$ | M | $+$ | M | Branching fraction | Framework | 2023 | HL-LHC | at FCC-ee
$\rm Z\,\to$ | $\rm J/\psi$ | + | $\rm\Upsilon(1S)$ |  $4.6$ | $\times~{}10^{-11}$ | NRQCD [139] | – | – | ✓
$\rm\eta_{b}(1S)$ | + | $\rm\Upsilon(1S)$ |  $(1.9\pm 0.2)$ | $\times~{}10^{-11}$ | NRQCD/NRCSM [140] | – | – | ✓
$\rm\Upsilon(1S)$ | + | $\rm\Upsilon(1S)$ |  $\left(4.4^{+0.6}_{-0.3}\right)$ | $\times~{}10^{-13}$ | NRQCD/NRCSM [140] |  $<$ $1.8$$\times~{}10^{-6}$ [96] | $\lesssim 2.7\times~{}10^{-7}$ | ✓
$\rm\Upsilon(mS)$ | + | $\rm\Upsilon(nS)$ |  $2.1$ | $\times~{}10^{-12}$ | NRQCD [139] |  $<$ $3.9$$\times~{}10^{-7}$ [96] | $\lesssim 5.9\times~{}10^{-8}$ | ✓
$\rm Z\,\to$ | $\rm\Upsilon(1S)$ | + | $\rm X$ |  $(2.6\text{ -- }2.9)$ | $\times~{}10^{-6}$ | NRQCD+LDME (NLO) [144] |  $<$ $4.4$$\times~{}10^{-5}$ [145] | – | ✓
$\rm\Upsilon(2S)$ | + | $\rm X$ |  $(3.7\text{ -- }4.3)$ | $\times~{}10^{-6}$ | NRQCD+LDME (NLO) [144] |  $<$ $1.4$$\times~{}10^{-4}$ [146] | – | ✓
$\rm\Upsilon(3S)$ | + | $\rm X$ |  $(7.6\text{ -- }8.3)$ | $\times~{}10^{-6}$ | NRQCD+LDME (NLO) [144] |  $<$ $9.4$$\times~{}10^{-5}$ [146] | – | ✓
$\rm\Upsilon(nS)$ | + | $\rm X$ |  $(1.4\text{ -- }1.5)$ | $\times~{}10^{-5}$ | NRQCD+LDME (NLO) [144] |  $(1.0\pm 0.5)$$\times~{}10^{-4}$ [147] | – | ✓

Figure 22: Branching fractions (in negative log scale) of exclusive $\rm Z\to
b$-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.

##  V Rare W boson decays

###  V.1 Exclusive W decays into a photon plus a meson

Figure 23 displays the two generic diagrams contributing to exclusive decays
of a W boson into a photon (the only gauge boson kinematically accessible for
the least massive weak boson) plus a meson. Table 20 compiles the
corresponding theoretical predictions and experimental limits for branching
ratios of the eight radiative exclusive decay modes considered to date. The
theoretical branching fractions are in the $\mathcal{O}(10^{-8}$–$10^{-12})$
range, whereas current experimental upper bounds are three to five orders-of-
magnitude larger, $\mathcal{O}(10^{-4}$–$10^{-6})$. 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 20 are presented in graphical form in Fig. 24.

Figure 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. Table 20: 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.

|  |  |  |  | Exp. limits | Producible at
---|---|---|---|---|---|---
$\rm W^{\pm}\,\to\,\gamma$ | + | $\rm M$ | Branching fraction | Framework | 2023 | HL-LHC | FCC-ee | FCC-hh
$\rm W^{\pm}\,\to\,\gamma$ | + | $\rm\pi^{\pm}$ |  $(4.0\pm 0.8)$ | $\times~{}10^{-9}$ | SCET+LCDA [62] |  $<$ $1.9$$\times~{}10^{-6}$ [148, 149] | $\lesssim 2.9\times~{}10^{-7}$ | ✓ | ✓
$\rm\rho^{\pm}$ |  $(8.74\pm 1.91)$ | $\times~{}10^{-9}$ | SCET+LCDA [62] |  $<$ $5.2$$\times~{}10^{-6}$ [148] | $\lesssim 7.9\times~{}10^{-7}$ | ✓ | ✓
$\rm K^{\pm}$ |  $(3.25\pm 0.69)$ | $\times~{}10^{-10}$ | SCET+LCDA [62] |  $<$ $1.7$$\times~{}10^{-6}$ [148] | $\lesssim 2.6\times~{}10^{-7}$ | ✘ | ✓
$\rm K^{*\pm}$ |  $(4.78\pm 1.15)$ | $\times~{}10^{-10}$ | SCET+LCDA [62] | – | – | ✘ | ✓
$\rm D^{\pm}$ |  $\left(1.38^{+0.51}_{-0.33}\right)$ | $\times~{}10^{-9}$ | SCET+LCDA [62] | – | – | ✘ | ✓
$\rm D_{s}^{\pm}$ |  $\left(3.66^{+1.49}_{-0.85}\right)$ | $\times~{}10^{-8}$ | SCET+LCDA [62] | $<$ $6.5$$\times~{}10^{-4}$ [126] | $\lesssim 1.2\times~{}10^{-5}$ | ✓ | ✓
$4.7$ | $\times~{}10^{-9}$ | NRQCD+LDME [150]
$3.4$ | $\times~{}10^{-9}$ | LC+LCDA [150]
$\rm D_{s}^{*\pm}$ |  $8.9$ | $\times~{}10^{-10}$ | NRQCD+LDME [150] | – | – | ✘ | ✓
$3.4$ | $\times~{}10^{-9}$ | LC+LCDA [150]
$\rm B^{\pm}$ |  $\left(1.55^{+0.79}_{-0.60}\right)$ | $\times~{}10^{-12}$ | SCET+LCDA [62] | – | – | ✘ | ✓
$\left(2.616^{+3.146}_{-1.330}\right)$ | $\times~{}10^{-12}$ | HQET+LCDA [151]
$\left(1.99^{+2.49}_{-0.82}\right)$ | $\times~{}10^{-12}$ | SCET+LCDA666updated inputs from [151] for the result from [62] [151]

Figure 24: Branching ratios (in negative log scale) of exclusive $\rm
W^{\pm}\to\gamma+meson$ 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.

###  V.2 Exclusive W decays into two mesons

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 $\rm W^{*},\,\gamma^{*}\to M$ transitions (right). Table 21 lists
the corresponding 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 literature777The exclusive three-pion decay, $\rm
W^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}$, estimated to have a $\mathcal{B}\approx
10^{-7}$ [152], has been however searched-for by the CMS experiment at the LHC
[125].. The expected rates are truly tiny, of order
$\mathcal{O}(10^{-11}\text{--}10^{-14})$, 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
$m_{\mathrm{M}}/m_{\mathrm{W}}$, 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.

Figure 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. Table 21: 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/FCC-hh.

|  |  |  |  |  | Exp. limits | Producible at
---|---|---|---|---|---|---|---
$\rm W^{\pm}\,\to$ | M | + | M | Branching fraction | Framework | 2023 | HL-LHC | FCC-ee | FCC-hh
$\rm W^{\pm}\,\to$ | $\rm\eta_{c}$ | + | $\rm D_{s}^{\pm}$ |  $2.1$ | $\times~{}10^{-12}$ | NRCSM+LCDA [153] | – | – | ✘ | ✓
$\left(1.31^{+0.34}_{-0.22}\right)$ | $\times~{}10^{-11}$ | LC+LCDA [153]
$\rm D_{s}^{*\pm}$ |  $3.0$ | $\times~{}10^{-12}$ | NRCSM+LCDA [153] | – | – | ✘ | ✓
$\left(1.48^{+0.36}_{-0.22}\right)$ | $\times~{}10^{-11}$ | LC+LCDA [153]
$\rm J/\psi$ | + | $\rm D_{s}^{\pm}$ |  $2.6$ | $\times~{}10^{-12}$ | NRQCD+LDME [150] | – | – | ✘ | ✓
$2.1$ | $\times~{}10^{-12}$ | NRCSM+LCDA [153]
$\left(1.8^{+0.4}_{-0.2}\right)$ | $\times~{}10^{-11}$ | LC+LCDA [153]
$\rm D_{s}^{*\pm}$ |  $1.7$ | $\times~{}10^{-12}$ | NRQCD+LDME [150] | – | – | ✘ | ✓
$3.0$ | $\times~{}10^{-12}$ | NRCSM+LCDA [153]
$\left(2.03^{+0.46}_{-0.24}\right)$ | $\times~{}10^{-11}$ | LC+LCDA [153]
$\rm\psi(2S)$ | + | $\rm D_{s}^{\pm}$ |  $5.1$ | $\times~{}10^{-11}$ | NRQCD+LDME [150] | – | – | ✘ | ✓
$\rm D_{s}^{*\pm}$ |  $7.4$ | $\times~{}10^{-12}$ | NRQCD+LDME [150] | – | – | ✘ | ✓
$\rm\chi_{c0}$ | + | $\rm D_{s}^{\pm}$ |  $9.4$ | $\times~{}10^{-14}$ | NRQCD+LDME [150] | – | – | ✘ | ✓
$4.7$ | $\times~{}10^{-14}$ | NRCSM+LCDA [153]
$\left(7.1^{+3.5}_{-3.1}\right)$ | $\times~{}10^{-13}$ | LC+LCDA [153]
$\rm D_{s}^{*\pm}$ |  $1.2$ | $\times~{}10^{-13}$ | NRQCD+LDME [150] | – | – | ✘ | ✓
$8.1$ | $\times~{}10^{-14}$ | NRCSM+LCDA [153]
$\left(8.0^{+3.7}_{-3.1}\right)$ | $\times~{}10^{-13}$ | LC+LCDA [153]
$\rm\chi_{c1}$ | + | $\rm D_{s}^{\pm}$ |  $2.0$ | $\times~{}10^{-13}$ | NRQCD+LDME [150] | – | – | ✘ | ✓
$2.9$ | $\times~{}10^{-13}$ | NRCSM+LCDA [153]
$\left(7.83^{+3.40}_{-3.05}\right)$ | $\times~{}10^{-12}$ | LC+LCDA [153]
$\rm D_{s}^{*\pm}$ |  $2.0$ | $\times~{}10^{-13}$ | NRQCD+LDME [150] | – | – | ✘ | ✓
$4.0$ | $\times~{}10^{-13}$ | NRCSM+LCDA [153]
$\left(8.83^{+3.50}_{-3.06}\right)$ | $\times~{}10^{-12}$ | LC+LCDA [153]
$\rm h_{c}$ | + | $\rm D_{s}^{\pm}$ |  $1.4$ | $\times~{}10^{-13}$ | NRCSM+LCDA [153] | – | – | ✘ | ✓
$\left(2.13^{+0.97}_{-0.83}\right)$ | $\times~{}10^{-12}$ | LC+LCDA [153]
$\rm D_{s}^{*\pm}$ |  $2.0$ | $\times~{}10^{-13}$ | NRCSM+LCDA [153] | – | – | ✘ | ✓
$\left(2.4^{+1.1}_{-0.9}\right)$ | $\times~{}10^{-12}$ | LC+LCDA [153]
$\rm\chi_{c2}$ | + | $\rm D_{s}^{\pm}$ |  $3.9$ | $\times~{}10^{-14}$ | NRQCD+LDME [150] | – | – | ✘ | ✓
$9.6$ | $\times~{}10^{-14}$ | NRCSM+LCDA [153]
$\left(1.42^{+0.62}_{-0.53}\right)$ | $\times~{}10^{-12}$ | LC+LCDA [153]
$\rm D_{s}^{*\pm}$ |  $3.9$ | $\times~{}10^{-14}$ | NRQCD+LDME [150] | – | – | ✘ | ✓
$1.4$ | $\times~{}10^{-13}$ | NRCSM+LCDA [153]
$\left(1.6^{+0.7}_{-0.6}\right)$ | $\times~{}10^{-12}$ | LC+LCDA [153]
$\rm B_{s}^{0}$ | + | $\rm B_{c}^{*\pm}$ |  $2.0$ | $\times~{}10^{-12}$ | NRQCD+LDME [150] | – | – | ✘ | ✓
$\rm B_{c}^{\pm}$ |  $2.5$ | $\times~{}10^{-12}$ | NRQCD+LDME [150] | – | – | ✘ | ✓
$\rm B_{s}^{*}$ | + | $\rm B_{c}^{*\pm}$ |  $2.7$ | $\times~{}10^{-12}$ | NRQCD+LDME [150] | – | – | ✘ | ✓
$\rm B_{c}^{\pm}$ |  $2.7$ | $\times~{}10^{-12}$ | NRQCD+LDME [150] | – | – | ✘ | ✓

Figure 26: Branching fractions (in negative log scale) of exclusive $\rm
W^{\pm}\to meson+meson$ 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.

##  VI Rare top quark decays

###  VI.1 Two-body top quark decays

The top quark decays to a W boson and a bottom quark, $\rm t\to W\,b$, with a
branching fraction of nearly 100%, with the other tree-level decays $\rm t\to
W\,s$ and $\rm t\to W\,d$ comparatively suppressed by factors of $10^{-3}$ and
$10^{-4}$, respectively, as per the CKM element hierarchy $V_{\mathrm{tb}}\gg
V_{\mathrm{ts}}>V_{\mathrm{td}}$ (Table 2). The FCNC top-quark decays to a
gauge boson plus a light up-type quark (q = u or c; for charge conservation),
$\rm t\to q\,Z$, $\rm t\to\gamma\,c$, and $\rm t\to g\,c$, 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 factor888Theoretically, the b-quark running mass must be evaluated
at the top pole mass scale (at which the t quark decays) in the
$\rm\overline{MS}$ scheme, otherwise rates are artificially enhanced. given by
$f(m_{\mathrm{b}}^{2}/m_{\mathrm{W}}^{2})\approx 10^{-9}$), 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].

Figure 27: Schematic diagrams of rare FCNC two-body decays of the top quark