resolution better than 100 ps, allowing for an effective rejection of
accidental backgrounds.

####  IV.4.4 $\mu^{+}\to e^{+}\gamma$

The best limit is much more recent, $BR(\mu^{+}\to e^{+}\gamma)<4.2\times
10^{-13}$, published by the MEG collaboration in 2016 [110], and the first
result of the upgraded experiment MEG II [111] is expected within 2023. Then,
the future of this quest is much more unsettled. It was investigated in Ref.
[112] and more recent advances were considered in the HIMB science case
document [105].

By contrast to $\mu^{+}\to e^{+}e^{+}e^{-}$, the photon reconstruction cannot
provide by any means the sub-millimeter resolution that is necessary for an
effective two-particle vertexing, so that the $\mu^{+}\to e^{+}\gamma$ search
is much more limited by the accidental background. Since it originates from
accidental time coincidences, it scales with the square of the muon beam
rates, and it poses a severe limitation on the maximum rate that can be
proficiently exploited. Indeed, for a given beam rate, if the other
discriminating variables ($e^{+}\gamma$ energies, relative angle and relative
time) do not provide enough separation to keep the background yield at $O(1)$
over the full lifetime of the experiment, the sensitivity, scaling with
$S/\sqrt{B}$, becomes independent of the beam rate, the increase of which
would only overcrowd the detectors without any real statistical advantage. In
the MEG II experiment, the optimal beam rate is found to be around $5\times
10^{7}\,\mu$/s, below the maximum rate already achievable at PSI.

Additionally, the detectors of the MEG II experiment are not designed to cope
with the very high occupancy produced by a muon beam rate increased by a
factor 10 to 100 at future facilities: both in the positron spectrometer
(instrumented with a high-granularity drift chamber) and in the LXe
calorimeter, pileup events are already impacting the reconstruction of the
events at $5\times 10^{7}\,\mu$/s; moreover, at $10^{9}\,\mu$/s, aging effects
in the drift chamber would become unmanageable if specific solutions are not
found to strongly suppress them.

New options need to be investigated, aiming at an improvement of the
resolutions (to suppress the accidental background) and an increase of the
rate tolerance of the detectors. A few technical solutions were already
investigated in the HIMB science case document.

On the positron side, a silicon detector _à la_ Mu3e would provide the
necessary high rate capabilities, although the momentum resolution could be
slightly worse, due to a less favorable compromise between material budget and
number of hits. In this respect, a next generation of HV-MAPS, thinned down to
25 $\mu$m, and an optimization of the detector geometry and magnetic field
could allow to fill the gap against gaseous detectors. To now, the silicon
detector solution looks as the best option for the next-generation
experiments, although some ideas still circulate in the community, to improve
the rate capabilities of gaseous detectors with new geometries (transverse
drift chambers, transverse drift tubes _à la_ Mu2e, radial time projection
chambers), new wire materials and new, hydrocarbon-free gas mixtures.

On the photon side, the conversion technique seems to be the most promising
one to fully exploit higher beam rates. The basic idea is that, with a beam
rate up to 100 times higher, but a total conversion efficiency not much larger
than a few per cent, the total photon rate (and hence the accidental yields)
would not significantly increase, while the better resolutions achievable in
the photon reconstruction would allow to more effectively reject the
backgrounds. Moreover, a larger acceptance could be achieved with lower cost,
and the directional information extracted from the $e+e-$ pair would allow to
build a $e\gamma$ vertex, which would provide a further handle to reject the
accidental coincidences.

In the last couple of years some significant advance was made in the
understanding of the performance of the conversion technique and in the
conceptualization of a photon detector for a future experiment. Figure 30
shows a possible arrangement for a photon detector based on the conversion
technique.

Instead of a thin and dense passive converter, a scintillating crystal
instrumented with SiPM is considered. It would allow to measure the energy
deposit in the converter itself, that is the main limiting factor to the
energy resolution when a passive converter is used.

Tracking the $e^{+}e^{-}$ pair with excellent resolution and good efficiency
will require at least two or three layers of position-sensitive detectors in
close proximity to the conversion layer (within about 1.5 cm for a 1 T
magnetic field). Although silicon detectors would fit this design, the
necessity of stacking multiple conversion layers to increase the efficiency
would make this option impractical from the technical and economical point of
view, considering that tens of square meters of detectors would be needed to
guarantee a good acceptance. However, it can be shown that the geometry of
drift chambers with stereo wires do not fit this design, while time projection
chambers would be too long to provide the necessary resolutions when operated
with low-mass gas mixtures. The option that is currently under consideration
is a time projection chamber with radial drift. Simulations are on going to
assess the performance of a detector with such a geometry.

Finally, the conversion technique should be adopted without detriment to the
good photon time resolution ($\sim 70$ ps) characterizing the MEG and MEG II
experiments. It could be achieved with the addition of gaseous detectors with
fast timing, like multi-gap resistive plate chambers (mRPC), just in front of
the active converter. These detectors would be light enough to not deteriorate
the performance of the active converter, while providing the necessary time
resolution on the $e^{+}$ and $e^{-}$, after they recurled through the
spectrometer.

Figure 30: The possible layout of a $\mu^{+}\to e^{+}\gamma$ experiment based
on a silicon positron tracker and a photon conversion detector (top), with a
detail of a possible photon detector concept (bottom).

Simulations and test beam measurements on scintillating crystals for the
active converter are on-going. Preliminary results have been shown, indicating
that four layers of 3 mm thick LYSO crystals could give a conversion
efficiency of 10% and an energy resolution of about
$140{\mathrm{\,ke\kern-1.00006ptV}}$, to be combined with the resolution of
the $e^{+}e^{-}$ tracker, the simulation of which is on going.

Other innovations were also briefly discussed at the workshop, namely an
active target to get a direct information on the positron vertex and multiple
thinner targets in vacuum.

All these studies give a preliminary indication that the performance envisaged
in Ref. [112] could be realistic, so that a BR sensitivity approaching
$10^{-15}$ could be reached with beam intensities exceeding $10^{9}~{}\mu$/s,
as shown in Figure 31.

Figure 31: Sensitivity of next-generation $\mu^{+}\to e^{+}\gamma$ experiments
under different assumptions for the detector design. See Ref. [112] for a
detailed description of the underlying assumptions.

####  IV.4.5 Exotic decays

More exotic channels can be considered, where the muon decays to new, non
standard particles, that can in turn decay into standard particles or remain
invisibile (either because they are stable or their product do not interact
with the detector). A review of several channels and searches can be found in
Ref. [106].

As a case study, the decays of muons with lepton-flavor violating axion-like
particles (ALPs) in the final state were discussed at the workshop. The two
channels of interest are $\mu^{+}\to e^{+}a$ and $\mu^{+}\to e^{+}a\gamma$,
where the ALP $a$ escapes the detection.

When the mass of the ALP is zero or very low compared the energy resolutions
of the detector, the search for $\mu^{+}\to e^{+}a$, which is based on the
reconstruction of a monochromatic positron, is affected by large systematic
uncertainties because a peak at the kinematical end point of the muon decay
spectrum can be easily faked by a miscalibration of the momentum scale (e.g.
in the calibration of the magnetic field). The most recent limit, from the
TWIST collaboration [81], is the result of a careful exploration of possible
miscalibrations at the ${\mathrm{\,ke\kern-1.00006ptV}}$ level, and strong
improvements are difficult to envisage.

For higher masses, similar searches in experiments tailored for $\mu^{+}\to
e^{+}\gamma$ are typically limited by the momentum acceptance of the positron
spectrometer, which is optimized for the highest momenta, but there is an
intermediate range, where the TWIST search is statistically limited, that
could be explored in more detail already in MEG II. It should be noticed that,
although the trigger of MEG II requires the coincidence of a positron and a
photon, the large majority of the acquired events is of accidental nature, and
hence the observed positron can still be used to reconstruct signatures
without photons.

Anyway, having in mind the possibility of reconstructing the photon with high
resolutions, the $\mu^{+}\to e^{+}a\gamma$ channel can be even more
interesting in MEG II and future $\mu^{+}\to e^{+}\gamma$ experiments,
considering the background discrimination power provided by the reconstruction
of a missing mass. This possibility was examined in detail in Ref. [113]. In
this case, the available statistics is limited by the trigger requirements for
the $\mu^{+}\to e^{+}\gamma$ search, which restrict the available phase space
to a small region at high energies and large $e\gamma$ angle. The outcome of
the studies done so far is that it would be advantageous to allocate for this
specific search a relatively small data taking period (a few percent of the
total beam time), during which the trigger requirements are released, and the
beam rate reduced to enhance the prompt over accidental yield ratio. This new
data taking strategy would strongly enhance the sensitivity of MEG II to
$\mu^{+}\to e^{+}a\gamma$, and with 1 month of data taking, assuming for
instance V-A currents, the current limits for the ALP couplings would be
overcame by a factor up to 5, as shown in Fig. 32. Detailed simulations and
trigger rate estimates are ongoing within the collaboration to confirm these
results.

Being the sensitivity of these searches also limited by the background from
accidental coincidences, the same considerations done for $\mu^{+}\to
e^{+}\gamma$ at future high-intensity facilities also apply in this case.

Figure 32: Sensitivity of MEG II, with a dedicated trigger strategy and a
reduced muon beam intensity, to left-handed (V-A) ALP couplings, compared to
the current limits. See Ref. [113] for details.

###  IV.5 Other experiments

####  IV.5.1 Session Summary

In this session of the workshop, we discussed the current status and future
plans of various muonium experiments and proposals. Muonium (commonly referred
to as M or Mu, by particle or nuclear physicists, respectively) is a
$\mu^{+}e^{-}$ bound state that is produced when $\mu^{+}$ stop in matter. It
is a purely leptonic, hydrogen-like atom where the nucleus consists of a
$\mu^{+}$. This makes it an excellent venue for precision tests of QED since,
e.g., nuclear size effects do not need to be considered. Muonium experiments
typically employ “surface” muon beams: beams of antimuons produced when
positive pions decay at rest inside the production target; this two-body decay
yields a forward-going, quasi-monoenergetic ($\approx\,$4 MeV, or $\approx$ 28
MeV/$c$) $\mu^{+}$ beam that is 100% backward-polarized and can be stopped to
produce muonium in thin conversion targets.

Three categories of fundamental-physics measurements were discussed at the
Workshop: the search for M–$\overline{\rm M}$ mixing, precision M spectroscopy
measurements, and the measurement of the gravitational acceleration of M in
the Earth’s gravitational field. Published results from the first two
measurement categories date from over 20 years ago [114, 115, 116, 117], so
the field is ripe for renewed efforts. New spectroscopy measurements are in
progress: the MuSEUM [118, 119] hyperfine-splitting experiment at J-PARC and
the Mu-MASS [120, 121] 1s–2s experiment at PSI. The MACE M–$\overline{\rm M}$
mixing experiment is proposed at the Chinese Spallation Neutron Source, and
the LEMING M gravitational experiment is in an R&D stage at PSI [122]. The
PIP-II linac under construction at Fermilab should enable the world’s highest-
intensity muonium beam. All three categories of experiments thus appear worthy
of pursuit at Fermilab once PIP-II is operational, assuming a suitable high-
power PIP-II target facility is built and the currently planned accelerator is
upgraded for continuous operation.

####  IV.5.2 Muonium Theory

Rare muonium decays of interest include ${\rm M}\to e^{+}e^{-}$, ${\rm
M}\to\gamma\gamma$, and ${\rm M}\to\nu_{e}\overline{\nu}_{e}$, all three of
which would be dominated by new physics if they occur. In effective field
theory, compared to e.g. $\mu\to 3e$, they probe different combinations of
Wilson coefficients, so are worth searching for whether or not $\mu\to 3e$ is
observed.

Conversion of muonium to antimuonium — the simultaneous conversion of an
antimuon to a positron and an electron to a muon — would be an example of
double charged-lepton flavor violation. While not forbidden (neutrino
oscillation being well established), the rate of such conversion via virtual
neutrino oscillation is so low as to be essentially unobservable.
Representative new-physics diagrams contributing to muonium–antimuonium
oscillations are shown in Fig. 33. Since some new-physics mechanisms favor
M–$\overline{\rm M}$ mixing over muon-to-electron conversion, the former is
not necessarily suppressed with respect to the latter. As with rare decays,
since Mu2e and M–$\overline{\rm M}$ mixing are differently sensitive to new
physics, both should be sought as sensitively as possible.

Figure 33: Examples of new-physics diagrams contributing to M–$\overline{\rm
M}$ mixing: exchange of (a) a doubly charged Higgs boson $\Delta^{++}$, (b)
heavy Majorana neutrinos, (c) a neutral scalar $\Phi_{N}$, or (d) a bileptonic
flavor diagonal gauge boson $X^{++}$ (from [114]).

####  IV.5.3 Muonium–Antimuonium Conversion Experiment (MACE)

The MACE experiment proposes to search for muonium–antimuonium oscillation at
a high-intensity muon beamline at, e.g., the High Intensity Heavy-ion
Accelerator Facility (HIAF) or the Chinese Spallation Neutron Source (CSNS).

The existing limit ($P_{\rm M\overline{M}}<8.3\times 10^{-11}$ at 90% C.L. in
an 0.1 T magnetic field) was set by the MACS experiment at PSI in 1999 [114].
The goal of MACE is to improve on MACS sensitivity by more than two orders of
magnitude, by increasing the rate of muonium formation and improving apparatus
resolutions. The design (Fig. 34) employs a $\mu^{+}\to{\rm M}$ conversion
target made of silica aerogel placed within a cylindrical drift chamber
surrounded by a solenoid magnet. The aerogel is perforated (“laser ablated”)
with an array of small blind holes to increase its surface-to-volume ratio,
enhancing the probability that M atoms produced within escape to the
surrounding vacuum [123]. ${\rm M\to\overline{M}}$ conversion leads to an atom
consisting of a muon bound to a positron, producing a fast electron and slow
positron once the muon decays — opposite in electric charge to the decay
products of muonium. The fast electron is reconstructed and its sign and
momentum measured in a cylindrical drift chamber. The slow atomic positron
left behind when the muon decays is accelerated electrostatically and sign-
and velocity-selected in a curved solenoid channel, which transports it to a
microchannel plate (MCP), in which it annihilates, sending a pair of
$511{\mathrm{\,ke\kern-1.00006ptV}}$ gamma rays into the surrounding
calorimeter.

Figure 34: 3D cutaway view of proposed MACE apparatus. Antimuons are incident
from the left on an aerogel target in which many of them stop and convert to
muonium. If a muonium atom converts to antimuonium, its decay electron is
reconstructed by the surrounding cylindrical drift chamber and its slow
positron is accelerated and sign- and velocity-selected in the S-shaped
solenoid channel, conveying it to a microchannel plate in which it
annihilates, sending gammas to be detected in the surrounding calorimeter.

Optimization of the muonium yield and detector performance is under study. The
muonium vacuum yield for various target designs is simulated using the random-
walk method. The simulation has been validated and is believed to be
reasonably accurate. This simulation provides a basis for optimization of the
perforated silica aerogel target and indicates that a surface muon beam with a
relatively small momentum spread (2.5% or less) can increase the yield of
muons to 1.5% or higher, to be compared with the 0.5% yield in MACS. Together
with the higher expected muon intensity, this yields an improvement of about
two orders of magnitude compared to MACS. On the other hand, the background
rejection capability of MACE depends on the spatial resolutions of the
detector — in particular, those of the spectrometer and MCP. The wire
structure of the drift chamber has been modeled, and an initial reconstruction
algorithm has been implemented. By examining various readout layer layouts,
the geometry of the drift chamber can be further optimized. Current results
demonstrate vertex resolution in the range 5–10 mm. By improving the
reconstruction algorithm, the resolution of the drift chamber is expected to
be further improved, with a goal below 5 mm.

The design has been simulated in Geant4 to characterize its performance [124].
The limiting backgrounds in MACS were accidental coincidences and the rare
decay $\mu^{+}\to e^{+}e^{+}e^{-}\nu_{e}\overline{\nu}_{\mu}$ (which can fake
the signal mode if one of the positrons is missed). MACS had one event passing
its elliptical time–vertex cut (cut radii of 4.5 ns and 12 mm, respectively).
For a background-free sample with two orders of magnitude more M events, the
time, vertex distance of closest approach, and energy resolutions should be
substantially tightened with respect to those in MACS. Preliminary MACE
simulations indicate resolutions of 6.5 mm in distance of closest approach
(comparable to) and 0.65 ns (better than that in MACS) in electron–positron
time difference. Further resolution improvement is desired and is the subject
of ongoing studies.

####  IV.5.4 Muonium Spectroscopy

Muonium spectroscopy experiments aim to measure one of three transitions: the
ground state hyperfine splitting, the 1s–2s interval, or the Lamb shift. The
hyperfine and 1s–2s splittings together with $g-2$ allow an independent, muon-
only determination of the fine-structure constant $\alpha$.

The hyperfine splitting is currently being measured by the MuSEUM experiment
[118, 119] at J-PARC with an uncertainty goal of 5 Hz (1.2 ppb). This would
exceed the precision of the theoretical prediction and so can be viewed as a
measurement of the ratio $m_{\mu}/m_{e}$, which dominates the systematic
uncertainty of the prediction. In MuSEUM, muonium is created by stopping
highly polarized muons in krypton gas. Positrons will be preferentially
emitted in the spin direction (upstream). A perpendicular microwave magnetic
field is applied in order to flip the spins. The hyperfine splitting is
determined by measuring the ratio between upstream and downstream positron
emission as a function of microwave frequency.

The 1s–2s splitting (and Lamb shift) are currently being measured by Mu-MASS
[120, 121] at PSI. The uncertainty goal for the 1s–2s splitting is 10 kHz (4
ppt), which will enable a 1 ppb measurement of $m_{\mu}$. In Mu-MASS, muonium
is formed in silica, then excited to the 2s state and ionized with a pair of
lasers. The muon is then transported away from the creation point and its
emitted Lyman-alpha photon is detected in an MCP detector.

####  IV.5.5 Low-Energy Muons at Fermilab

We briefly describe the MeV Test Area (MTA) beam line at FNAL, which could be
used for low-energy muon and muonium R&D (and perhaps for competitive physics
experiments as well), and expectations for the PIP-II superconducting linac,
whose construction is in progress.

The MTA is situated near the end of Fermilab’s 400 MeV H- Linac and shielded
well enough to accept a large fraction of the protons accelerated by the Linac
(although a shielding assessment demonstrating this has yet to be finalized).
A new MTA beam line was recently designed and installed to serve an approved
muon-catalyzed-fusion R&D program; it naturally produces low-energy pions and
muons of both signs. The layout is shown in Fig. 35. Geant4 simulation studies
using a 3-cm-long rectangular W target predict surface-muon production at a
rate of order $\sim$ 10${}^{-9}\mu^{+}/$proton-on-target (POT) into the
acceptance of the beam line, which is centered at a 146.4∘ production angle.
Measurements made in the 1970s [125] suggest that, compared to the thin
graphite targets used at surface-muon beam lines at spallation-neutron
facilities such as PSI, RAL, and TRIUMF, a Ta or W target should produce more
charged pions and surface muons per POT by factors of 3 ($\pi^{+}$ and
$\mu^{+}$) and 8 ($\pi^{-}$ and $\mu^{-}$). (These factors are not borne out
in Geant4 studies, but Geant4 is not expected to be reliable for such cases;
measurements are therefore needed, and planned.) Studies to optimize the
target configuration and beam-line acceptance are in progress and may yield
higher muon rates [126]. However, for purposes of surface-muon (and muon-
catalyzed-fusion) R&D, the rates so far simulated are more than sufficient.

The PIP-II linac will of course accelerate to 800 MeV, twice the energy of
Fermilab’s existing Linac. While neither 800 nor 400 MeV is optimal for
surface-muon production, the predicted rate decrease per POT with respect to
the peak at 590 MeV is only about 15% [126]. Furthermore, especially if it is
upgraded for CW operation, most of the protons that the PIP-II linac could
accelerate will not be destined for the Main Injector and could enable a low-
energy-muon physics program at Fermilab that — even after PSI’s High-Intensity
Muon Beam (HIMB) upgrade is completed — would be the world’s most sensitive by
more than an order of magnitude.

To prepare for a future, world-leading, PIP-II muon facility, it is highly
desirable to conduct R&D now at the MTA to develop and gain experience with
the techniques that will be needed at PIP-II, such as those discussed in the
next subsection. Such a program at the MTA will also allow (for the first
time) a U.S.-based program of $\mu$SR measurements, obviating the current need
for Fermilab superconducting RF-cavity developers (and other U.S. researchers)
to take their samples to Canada for $\mu$SR surface studies at TRIUMF.

Figure 35: Photo of Fermilab MeV Test Area showing 400 MeV proton beamline at
right and pion–muon quadrupole channel at left. The protons enter traveling
into the page at the right edge of the photo, and muons (produced both within
the target and via pion decay in flight) exit out of the page at the left
edge. A target holder is situated at the intersection of the two beam lines,
followed by a small solenoid to capture charged particles produced in the
target.

####  IV.5.6 Muonium Production in Superfluid Helium

Measurements [127] (Fig. 36) have shown that below 1 K, superfluid helium
(SFHe) is an efficient converter of stopped $\mu^{+}$ to muonium. The
signature of M formation is rapid precession of the muon spin; the resulting
time evolution of the decay asymmetry in a magnetic field has been used to
explore the production of muonium when $\mu^{+}$ are incident on pure
superfluid 4He or a solution containing 0.2% 3He [127, 128]. The high
conversion efficiency and the immiscibility of hydrogen isotopes in superfluid
helium form the basis of a technique proposed by PSI’s David Taqqu [129] that
can make a monochromatic, parallel beam of muonium atoms propagating in vacuum
by stopping slow $\mu^{+}$ in a thin, horizontal SFHe layer. Such a beam would
for the first time enable an interferometric muonium gravity experiment (see
next subsection) and could also be a game-changer for other high-sensitivity
muonium measurements.

Their immiscibility implies that any muonium atoms formed in the SFHe that
reach its upper surface will be expelled perpendicular to that surface at a
velocity determined by the chemical potential of the atoms in SFHe, predicted
to be 270 K for muonium [129], implying an expelled-atom velocity of 6.3
mm/$\mu$s. (The prediction depends on isotopic mass and has been verified
experimentally for the case of deuterium [130]; this immiscibility has also
been exploited to make a functioning SFHe-coated focusing mirror for hydrogen
atoms [131].)

Figure 36: Relative production vs. temperature of muonium in superfluid helium
(blue points from [127] and recent measurements in red by a PSI–ETH-Zürich
collaboration [128]).

To maximize the efficiency of beam creation, it is desirable to suppress
muonium formation in the SFHe bulk by applying an electric field to drift the
muons away from their ionization electrons; the required field can be created
by a surface-electron pool of $10^{8}$ $e$/cm2, maintained e.g. by use of a W
tip. Such an electron pool is stable below 1 K [132]. Calculations show that
for a layer thickness of $\approx$ 300 $\mu$m, up to $\approx$ 10% of muons
entering the SFHe drift to the surface before decaying and yield escaping M
atoms.

An alternative method of achieving high M escape efficiency is under
development at PSI by our PSI and ETH-Zürich colleagues (K. Kirch et al.
[128]), in which an extremely thin ($\sim$ 1 $\mu$m) SFHe layer is employed,
such that about half of the muonium atoms formed in the bulk diffuse to the
upper surface and are expelled. To achieve a substantial $\mu^{+}$ stopping
probability in such a thin layer, the $\mu^{+}$ beam must be precooled. This
“muCool” apparatus (also proposed by Taqqu [133]) has been developed and
tested successfully, compressing the $\mu^{+}$ beam phase space by a factor
$10^{10}$ at the expense of $10^{-3}$ efficiency due to muon decay [134, 135].

####  IV.5.7 Muonium Antimatter Gravity Experiment (MAGE)

The gravitational force between antimatter and matter has yet to be directly
measured. So far only a crude experimental limit has been established:
$-65\leq\overline{g}/g\leq 110$ [136], where $\overline{g}$ is the
gravitational acceleration of antimatter at the earth’s surface (indirect
limits on $\overline{g}$, such as those derived from torsion-pendulum
experiments and lunar radar ranging [137], are based on the assumed
gravitational effects of virtual antiparticles in nuclear binding energy and
are not applicable to antimuons). Any deviation of $\overline{g}$ from $g$
would be evidence for new physics — possibly a “fifth force,” as has been
invoked to explain multiple muonic experimental anomalies [138]. From another
perspective, antimatter gravity can be viewed as a test of the weak
equivalence principle, which some authors suggest may be violated in the case
of antimatter [139, 140]. Muonium gravity is thus worth measuring as
sensitively as possible.

Measurement of muonium gravity has not heretofore been feasible. The predicted
parallel beam produced by superfluid helium is a game changer that will enable
it to be measured for the first time. This has led to the approval of the
LEMING experiment at PSI, based on the muCool apparatus and use of a micron-
thick thin film of SFHe [122]; however, LEMING is still in an R&D stage, and a
parallel M beam has yet to be demonstrated. Furthermore, the inefficiency of
muCool is likely to limit LEMING in its currently designed form to measuring
the sign of $\overline{g}$ and determining its magnitude only crudely. The
thick-film approach described in the previous subsection, combined with the
proton-beam power available from PIP-II and the use of heavy targets, should
allow MAGE at Fermilab to measure $\overline{g}$ to 1% of $g$ or better.
Depending on the progress of our R&D, a $\overline{g}$ measurement at the MTA
may also be competitive with LEMING.

MAGE and LEMING are based on the same idea [141]: create a horizontal,
parallel, monoenergetic beam of muonium propagating in vacuum and impinging on
a three-grating Mach–Zehnder-type interferometer, detecting the products of
muonium decay in time–space coincidence downstream of the third grating. The
fast positron from $\mu^{+}$ decay is easily reconstructed using scintillating
fibers. The slow atomic electron left behind when the muon decays can be
accelerated electrostatically for detection by a 2D scintillating hodoscope
array. The interferometer measures the acceleration of the beam in the
direction perpendicular to the grating slits, which (to measure gravitational
deflection) should therefore be oriented horizontally — in contrast to the
more usual atom-interferometry applications, in which gravity is an unwanted
influence and vertical slits are employed. The de Broglie waves corresponding
to the atom beam diffract in each grating, and interference of the diffracted
waves creates a sinusoidal intensity distribution at the third grating. The
details of the pattern (its phase and period) exceed the position-
reconstruction capability of particle detectors but can be measured by
translating the third grating up and down periodically over time. The phase is
proportional to the gravitational deflection, and its sign gives the direction
(deflection up or down).

The beam is created by stopping surface or “subsurface” $\mu^{+}$ in
superfluid helium, deflecting the resulting vertically propagating M beam into
the horizontal by means of a SFHe-coated surface angled at 45∘. (Subsurface
muons are surface muons created slightly deeper into the target that lose a
fraction of their energy to $dE/dx$ on the way out of the target.) Given the
predicted 6.3 mm/$\mu$s M velocity discussed above, the 2.2 $\mu$s muon
lifetime corresponds to 1.4 cm of travel, and with the statistically optimal
2-lifetime grating separation (for measurement of an acceleration $\propto
t^{2}$), the apparatus easily fits on a tabletop, or within the sample volume
of a dilution refrigerator. Since the measurement resolution is inversely
proportional to the grating pitch, the interferometer should have as fine a
grating pitch as possible, but the gratings must also be fairly large and as
open as possible (i.e., with minimal supporting struts, which potentially
could block the passage of a significant fraction of M atoms). The current
state of the art is judged to be $\approx$ 1 cm2 grating area and 100 nm
grating pitch. With $\approx$ 50%-open (approximately optimal [135]) gratings,
and an approximately parallel beam, the geometric acceptance is close to 100%,
but half of the beam is absorbed at each grating, in addition to decay losses.

The measurement resolution depends on the pitch $d$, inter-grating travel time
$t$, and interferometric contrast $C$ according to [142]

$\delta g=\frac{1}{C\sqrt{N}}\frac{d}{2\pi}\frac{1}{t^{2}}\,,$ (2)

where $N$ is the number of detected events. With the (conservative) fringe
contrast $C$ = 0.1, and 100,000 M/s detected, the resolution is then $\delta
g/g\approx(0.35/\sqrt{\rm\\#~{}days})$ for gratings placed 1 muon lifetime
apart, or $\approx(0.1/\sqrt{\rm\\#~{}days})$ with 2-lifetime grating
separation. Thus the sign of $\overline{g}$ is determined to 5 standard
deviations in less than a day of running, and the magnitude determined to 1%
(if systematic effects can be sufficiently well controlled) in about a year.
To these estimates some margin must of course be added for calibration runs
and accelerator downtime. Thus precision measurements of $\overline{g}$ (to 1%
or better) will require the highest possible M intensity.

####  IV.5.8 R&D Projects / Potential Synergies / Open Questions

Here is a list of R&D projects with some comments about potential synergies
and some open questions.

High priority

  * •

Pursue SFHe production of muonium

    * –

this could greatly improve the efficiency of muonium production for all
muonium experiments, and is required for muonium antimatter-gravity
experiments

  * •

Pursue low-energy muon production at MTA

    * –

could be used for R&D of other experiments (e.g., measure muon production at
energies similar to those expected from PIP-II and study pion and muon
production-target materials and configurations)

  * •

Make first measurement of $\bar{g}$ with muonium

    * –

this would help establish what the limiting systematics of such an experiment
are

  * •

How does AMF host all of $\mu^{-}$, $\mu^{+}$, and muonium experiments?

    * –

low-energy surface $\mu^{+}$ are needed for both decay and muonium experiments
– can both be supported at the same time?

Low priority

  * •

Determine what improved muon production from SFHe means for all muonium
experiments

    * –

can we keep the same design for muonium oscillation experiments?

    * –

can muonium spectroscopy be done in a vacuum?

  * •

Determine feasibility of other muonium experiments (e.g., M
$\rightarrow\nu\bar{\nu}$)

##  V Conclusion

Charged lepton flavor violating processes open a unique window on the physics
of generation and flavor, complementing studies pursued at collider and
neutrino experiments. Muons offer a promising avenue to search for CLFV, and a
global experimental program is underway to explore these possibilities,
promising impressive sensitivity gains in the coming decade.

A staged program of next-generation experiments and facilities exploiting the
full potential of the PIP-II accelerator has been outlined in this document.
This program comprises Mu2e-II as a near term evolution of the Mu2e
experiment, with the goal of improving the sensitivity to muon-to-electron
conversion by an order of magnitude, followed by the Advanced Muon Facility.
This new facility would provide the world’s most intense positive and negative
muon beams, enabling a suite of experiments with unprecedented sensitivity
(probing mass scales of the order of
$10^{4}-10^{5}{\mathrm{\,Te\kern-1.00006ptV}}$), and offering unique
capabilities to explore in depth the underlying New Physics in case of an
observation. This program includes many synergies with the development of a
muon collider and a beam dump dark matter experiment at FNAL.

These experiments and facilities would provide the foundation for a
comprehensive muon physics program in the next decade and beyond, an essential
component of a global program to search for New Physics. The R&D roadmap
outlined in this document should begin immediately to ensure a timely
realization of these possibilities.

###### Acknowledgements.

This work was supported by the U.S. Department of Energy, Office of Science,
Office of High Energy Physics, under Award Number DE-SC0019095, DE-SC0007884,
DE-SC0011632 and DE-AC02-06CH11357 and by the Walter Burke Institute for
Theoretical Physics. This work was partly authored by Fermi Research Alliance,
LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy,
Office of Science, Office of High Energy Physics. This work was supported by
the EU Horizon 2020 Research and Innovation Program under the Marie
Sklodowska-Curie Grant Agreement No. 101006726. J. Zettlemoyer acknowledges
support in part by the DOE grant DE-SC0011784 and the NSF grant OAC-2103889.
R. Plestid is supported by the Neutrino Theory Network Program Grant Award
under Number DEAC02-07CHI11359 and the US DOE under Award Number DE-SC0020250.
J. Tang acknowledges support in part by National Natural Science Foundation of
China under Grant No. 12075326 and Fundamental Research Funds for the Central
Universities (23xkjc017) in Sun Yat-sen University.

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##  Appendix A Workshop program

Introduction (chair: Bertrand Echenard)
000Welcome and workshop goals - Bertrand Echenard
000Theory overview - Sacha Davidson
000Mu2e-II - status and perspectives - James Miller
000AMF - status and perspectives - Robert Bernstein

Mu2e-II: Magnets (chair: Karie Badgley)
000Muon collider HTS solenoid - Luca Bottura
000HTS (high temperature superconductor) magnets - Zach Hartwig
000Field calculations - Cole Kampa

Mu2e-II: Targetry
(chair: Kevin Lynch, Michael Mackenzie, Stefan Mueller, Vitaly Pronskikh)
000Introduction - Kevin Lynch
000What we know and do not know about tungsten in accelerator environments -
Yong Joong Lee
000Granular tungsten target R&D at RAL - Dan Wilcox
000HPR R&D - Frederique Pellemoine
000Muon Collider, fluidized targets - Carlo Carelli
000Mu2e - Michael Hedges
000Mu2e-II LDRD - David Neuffer

Mu2e II: Tracker (chair: Daniel Ambrose)
000Mu2e-II Tracker Introduction - Daniel Ambrose
000Mu2e-II Tracker Intrinsic Resolution, Electronics, Readout - Richard
Bonventre
000Mu2e-II Research topics and Discussion - James Popp
000Alternative Tracker ideas - David Brown
000Simulating Vacuum effects on Resolution and Beam Flash Radiation - Andrew
Edmonds
000Mu2e lesson learned - Daniel Ambrose

Mu2e II: Calorimeter (chair: David Hitlin, Luca Morescalchi, Ivano Sarra)
000Mu2e-II Calorimeter: Overview & Requirements - Luca Morescalchi
000Crystals 1 - Ren Yuan Zhu
000Crystals 2 - Yury Davidov
000Photosensors - David Hitlin
000Alternatives - Ivano

Mu2e II: Comsic Ray Veto (chair: Edmond Craig Dukes, Craig Group, Yuri
Oksuzian)
000Welcome and goals - Craig Group
000CRV at Mu2e II Overview - Simon Corrodi
000Studies of Triangular Counters for Mu2eII - Ralf Ehrlich
000Discussion and plans - Craig Group

Mu2e-II: Trigger and DAQ (chair: Antonio Gioiosa, Gianantonio Pezzullo)
000TDAQ summary - Gianantonio Pezzullo

Mu2e-II: Physics and Sensitivity (chair: Sophie Middleton)
000Introduction - Sophie Middleton
000Stopping target - Leo Borrel
000Normalization - David Hitlin
000Exotic physics signatures - Ryan Plestid
000Implication of production target design on sensitivity - Michael Mackenzie
000Implication of tracker design on sensitivity - David Brown

AMF: Proton compressor ring (chair: Jeffrey Eldred)
000PAR Proposal - Ben Simmons
000C-PAR Concept - Jeffrey Eldred
000SNS Experience - Vasiliy Morozov
000PIP2-BD Experiment - Jacob Zettlemoyer

AMF: Muon FFA (chair: Jaroslaw Pasternak, Robert Bernstein)
000FFAs for Muons - Francois Meot
000Building blocks and requirements of PRISM-like FFA system - Jaroslaw
Pasternak
000Review of PRISM FFA ring designs - Jean-Baptiste Lagrange
000Designing injection system for PRISM FFA ring - Jaroslaw Pasternak

AMF: Conversion experiments (chair: Cole Kampa, Craig Group)
000Introduction - Craig Group
000Overview of AMF Conversion Experiment - Bertrand Echenard
000Signal Resolution Requirements - Andrei Gaponenko
000Discussing an AMF Tracker - Daniel Ambrose
000Tracker Hit Resolution - Richard Bonventre
000Tracker Design; Simulations - David Brown
000Cosmic Ray Veto Considerations - Craig Group

AMF: Decay experiments (chair: Angela Papa, Francesco Renga)
000Status of the MEG II experiment - Dylan Palo
000HiMB: status and physics cases (incl. Mu3e) - Angela Papa
000Exotic muon decays - Deigo Redigolo
000Limiting factors of future mu $\rightarrow$ e gamma searches - Francesco
Renga
000Conceptual design and R&D activities for a future mu $\rightarrow$ e gamma
search - Wataru Ootani

AMF: Other experiments (chair: Andrew Edmonds, Daniel Kaplan)
000Muonium overview - Dan Kaplan
000M-Mbar mixing - Alexey Petrov
000Proposed MACE M-Mbar experiment - Shihan Zhao
000Muonium spectroscopy - David Kawall
000Low-energy muons at Fermilab - Carol Johnstone
000Making muonium with superfluid helium - Thomas Phillips
000Muonium gravity - Daniel Kaplan

Summary Mu2e-II (chair: Frank Porter)
000PIP-II - David Neuffer
000Calorimeter - Ivano Sarra
000Trigger and DAQ - Gianantonio Pezzullo
000Magnets - Karie Badgley
000Cosmic ray veto - Simon Corrodi
000Production target - Vitaly Pronskikh
000Tracking - Mete Yucel
000Sensitivity - Sophie Middleton

Summary AMF (chair: Bertrand Echenard)
000Compressor summary - Jeffrey Eldred
000FFA accelerator summary - Robert Bernstein
000Conversion experiment summary - Cole Kampa
000Other experiment summary - Daniel Kaplan
000Decay experiments summary - Francesco Renga


# A Note On The Cross-Sperner Families

Junyao Pan

School of Sciences, Wuxi University, Wu’xi, Jiangsu,
214105 People’s Republic of China

Abstract: Let $(\mathcal{F},\mathcal{G})$ be a pair of families of $[n]$,
where $[n]=\\{1,2,...,n\\}$. If $A\not\subset B$ and $B\not\subset A$ hold for
all $A\in\mathcal{F}$ and $B\in\mathcal{G}$, then $(\mathcal{F},\mathcal{G})$
is called a Cross-Sperner pair. P. Frankl and Jian Wang introduced the
extremal problem that
$m(n)={\rm{max}}\\{|\mathcal{I}(\mathcal{F},\mathcal{G})|:\mathcal{F},\mathcal{G}\subset
2^{[n]}~{}{\rm{are~{}cross}}$-${\rm{sperner}}\\}$, where
$\mathcal{I}(\mathcal{F},\mathcal{G})=\\{A\cap
B:A\in\mathcal{F},B\in\mathcal{G}\\}$. In this note, we prove that
$m(n)=2^{n}-2^{\lfloor\frac{n}{2}\rfloor}-2^{\lceil\frac{n}{2}\rceil}+1$ for
all $n>1$. This solves an open problem proposed by P. Frankl and Jian Wang.

Keywords: Cross-Sperner Family; Extremal Problem.

Mathematics Subject Classification: 05D05.

##  1 Introduction

Let $[n]=\\{1,2,...,n\\}$ denote the standard $n$-element set and $2^{[n]}$
stand for the power set consisting of the $2^{n}$ subsets of $[n]$. A subset
of $2^{[n]}$ is called a _family_. The extremal problems of families of finite
sets have always been an interesting research direction, the readers
interested in this aspect can refer to [6]. A family $\mathcal{F}\in 2^{[n]}$
is an antichain (also known as a Sperner family) if for all distinct
$F,G\in\mathcal{F}$, neither $F\subseteq G$ nor $G\subseteq F$, i.e. $F$ and
$G$ are incomparable. One of the principal results in extremal combinatorics
is Sperner’s theorem in [12], which states that the largest size of an
antichain in $2^{[n]}$ is $\dbinom{n}{\lfloor\frac{n}{2}\rfloor}$. It is
natural to consider a generalisation of Sperner’s theorem to multiple families
of sets, such as a pair of families $(\mathcal{F},\mathcal{G})$ is said to be
Cross-Sperner if $A\not\subset B$ and $B\not\subset A$ hold for all
$A\in\mathcal{F}$ and $B\in\mathcal{G}$. The research of such objects goes
back to the 1970s when Seymour [11] deduced from a result of Kleitman [10]
that a Cross-Sperner pair $(\mathcal{F},\mathcal{G})$ in $2^{[n]}$ satisfies
$|\mathcal{F}|^{\frac{1}{2}}+|\mathcal{G}|^{\frac{1}{2}}\leq 2^{\frac{1}{2}}$.
In fact, there exist various extremal problems about Cross-Sperner families
that have been studied, for example [1, 2, 3, 4, 5, 8, 9, 10, 11, 12].

Recently, P. Frankl and Jian Wang [7] by setting
$\mathcal{I}(\mathcal{F},\mathcal{G})=\\{A\cap
B:A\in\mathcal{F},B\in\mathcal{G}\\}$, they proposed the extremal problem that
$m(n)={\rm{max}}\\{|\mathcal{I}(\mathcal{F},\mathcal{G})|:\mathcal{F},\mathcal{G}\subset
2^{[n]}~{}{\rm{are~{}Cross}}$-${\rm{Sperner}}\\}$. Concurrently, they applied
a skillful way to show that $m(n)=2^{n}-2^{d+1}+1$ for $n=2d$, for details see
[7, Theorem 5.5]. However, their approach is unable to determine $m(n)$ in
case when $n$ is an odd number. Inspired by [7, Example 5.4,Theorem 5.5], they
proposed the following problem.

######  Question 1.1.

([7, Problem 5.6] ) For $n=2d+1$, does $m(n)=2^{n}-2^{d+1}-2^{d}+1$ hold?

See [7, Example 5.4], if $\mathcal{F}=\\{A\cup Y:A\subsetneq X\\}$ and
$\mathcal{G}=\\{B\cup X:B\subsetneq Y\\}$ where $X\cup Y=[n]$ and $X\cap
Y=\emptyset$, then we say that $(\mathcal{F},\mathcal{G})$ is a Cross-Sperner
pair of _type_ $(X,Y)$. In this note, we prove the following theorem and thus
we give an affirmative answer to Question 1.1.

######  Theorem 1.2.

Let $(\mathcal{F},\mathcal{G})$ be a Cross-Sperner pair of $[n]$ such that
$|\mathcal{I}(\mathcal{F},\mathcal{G})|=m(n)$. Then
$(\mathcal{F},\mathcal{G})$ is a Cross-Sperner pair of type $(X,Y)$ for some
$X$ and $Y$ with $|X|=\lfloor\frac{n}{2}\rfloor$ and
$|Y|=\lceil\frac{n}{2}\rceil$. In particular,
$m(n)=2^{n}-2^{\lfloor\frac{n}{2}\rfloor}-2^{\lceil\frac{n}{2}\rceil}+1$ for
all $n>1$.

##  2 Proof of Theorem 1.2

Let $\dbinom{[n]}{k}$ denote the set of all $k$-subsets of $[n]$ for some $k$
with $1\leq k\leq n-1$. Next we start to prove Theorem 1.2 from the following
lemma.

######  Lemma 2.1.

Let $(\mathcal{F},\mathcal{G})$ be a Cross-Sperner pair of $[n]$ such that
$|\mathcal{I}(\mathcal{F},\mathcal{G})|=m(n)$. Then, both
$\mathcal{F}\cap\dbinom{[n]}{n-1}\neq\emptyset$ and
$\mathcal{G}\cap\dbinom{[n]}{n-1}\neq\emptyset$ hold.

Proof Proof by contradiction. Suppose that
$\mathcal{F}\cap\dbinom{[n]}{n-1}=\emptyset$. Then $|A\cap B|\leq n-3$ for
each $A\cap B\in\mathcal{I}(\mathcal{F},\mathcal{G})$ and thus
$|\mathcal{I}(\mathcal{F},\mathcal{G})|\leq 2^{n-3}$. However, [7, Example
5.4] indicate $|\mathcal{I}(\mathcal{F},\mathcal{G})|>2^{n-3}$, a
contradiction. Therefore, $\mathcal{F}\cap\dbinom{[n]}{n-1}\neq\emptyset$.
Likewise, $\mathcal{G}\cap\dbinom{[n]}{n-1}\neq\emptyset$. $\Box$

######  Corollary 2.2.

Let $(\mathcal{F},\mathcal{G})$ be a Cross-Sperner pair of $[n]$ such that
$|\mathcal{I}(\mathcal{F},\mathcal{G})|=m(n)$. Then, both
$\bigcap\limits_{A\in\mathcal{F}}{A}\neq\emptyset$ and
$\bigcap\limits_{B\in\mathcal{G}}{B}\neq\emptyset$ hold.

Proof By Lemma 2.1, we may assume that there exist two $(n-1)$-subsets $A$ and
$B$ of $[n]$ such that $A\in\mathcal{F}$ and $B\in\mathcal{G}$. According to
the definition of Cross-Sperner, it follows that
$i\in\bigcap\limits_{A\in\mathcal{F}}{A}$ and
$j\in\bigcap\limits_{B\in\mathcal{G}}{B}$ where $\\{i\\}=[n]\setminus B$ and
$\\{j\\}=[n]\setminus A$, as desired. $\Box$

######  Lemma 2.3.

Let $(\mathcal{F},\mathcal{G})$ be a Cross-Sperner pair of $[n]$ such that
$X=\bigcap\limits_{A\in\mathcal{F}}{A}$ and
$Y=\bigcap\limits_{B\in\mathcal{G}}{B}$ and $X\cup Y=[n]$. Then
$\mathcal{I}(\mathcal{F},\mathcal{G})\subseteq\mathcal{I}(\mathcal{F}^{\prime},\mathcal{G}^{\prime})$
and
$|\mathcal{I}(\mathcal{F},\mathcal{G})|\leq|\mathcal{I}(\mathcal{F}^{\prime},\mathcal{G}^{\prime})|$
where $(\mathcal{F}^{\prime},\mathcal{G}^{\prime})$ is a Cross-Sperner pair of
type $(X,Y)$.

Proof Without loss of generality, we set
$X=\bigcap\limits_{A^{\prime}\in\mathcal{F}^{\prime}}{A^{\prime}}$ and
$Y=\bigcap\limits_{B^{\prime}\in\mathcal{G}^{\prime}}{B^{\prime}}$. By the
definitions of Cross-Sperner and type $(X,Y)$, it follows that
$\mathcal{F}\subseteq\mathcal{F}^{\prime}$ and
$\mathcal{G}\subseteq\mathcal{G}^{\prime}$, as desired. $\Box$

######  Lemma 2.4.

Let $(\mathcal{F},\mathcal{G})$ be a Cross-Sperner pair of $[n]$ such that
$|\mathcal{I}(\mathcal{F},\mathcal{G})|=m(n)$. Suppose that
$X=\bigcap\limits_{A\in\mathcal{F}}{A}$ and
$Y=\bigcap\limits_{B\in\mathcal{G}}{B}$. Then, both $X\cap Y=\emptyset$ and
$X\cup Y=[n]$ hold.

Proof Assume that $m\in X\cap Y$. Set
$\mathcal{F}^{\prime}=\\{A\setminus\\{m\\}:A\in\mathcal{F}\\}$ and
$\mathcal{G}^{\prime}=\\{B\setminus\\{m\\}:B\in\mathcal{G}\\}$. Clearly,
$|\mathcal{I}(\mathcal{F},\mathcal{G})|=|\mathcal{I}(\mathcal{F}^{\prime},\mathcal{G}^{\prime})|$.
On the other hand, it is straightforward to see that
$|\mathcal{I}(\mathcal{F}^{\prime},\mathcal{G}^{\prime})|\leq m(n-1)$.
However, one easily checks that $m(n)>m(n-1)$, a contradiction. Therefore,
$X\cap Y=\emptyset$ holds.

Pick a Cross-Sperner pair $(\mathcal{H},\mathcal{K})$ of $X\cup Y$ of type
$(X,Y)$. Suppose that $X\cup Y\neq[n]$. Let $\Delta\subset[n]\setminus(X\cup
Y)$. Define $\mathcal{H}_{\Delta}=\\{A\cup\Delta:A\in\mathcal{H}\\}$ and
$\mathcal{K}_{\Delta}=\\{B\cup\Delta:B\in\mathcal{K}\\}$. It is obvious that
$|\mathcal{I}(\mathcal{H}_{\Delta},\mathcal{K}_{\Delta})|=|\mathcal{I}(\mathcal{H},\mathcal{K})|$.
Consider $\mathcal{F}$ and $\mathcal{G}$. Define
$\mathcal{F}^{\Delta}=\\{A\in\mathcal{F}:\Delta\subseteq A,A\subseteq(X\cup
Y\cup\Delta)\\}$ and $\mathcal{G}^{\Delta}=\\{B\in\mathcal{G}:\Delta\subseteq
B,B\subseteq(X\cup Y\cup\Delta)\\}$. It follows from Lemma 2.3 that

$|\mathcal{I}(\mathcal{F}^{\Delta},\mathcal{G}^{\Delta})|\leq|\mathcal{I}(\mathcal{H}_{\Delta},\mathcal{K}_{\Delta})|=|\mathcal{I}(\mathcal{H},\mathcal{K})|.$

On the other hand, for any two subsets $\Delta_{1}\subset[n]\setminus(X\cup
Y)$ and $\Delta_{2}\subset[n]\setminus(X\cup Y)$, the Lemma 2.3 implies that
$\mathcal{I}(\mathcal{F}^{\Delta_{1}},\mathcal{G}^{\Delta_{2}})\subseteq\mathcal{I}(\mathcal{H}_{\Delta_{1}\cap\Delta_{2}},\mathcal{K}_{\Delta_{1}\cap\Delta_{2}})$.
Hence, we deduce that

$|\mathcal{I}(\mathcal{F},\mathcal{G})|\leq\sum\limits_{\Delta\subset[n]\setminus(X\cup
Y)}|\mathcal{I}(\mathcal{H}_{\Delta},\mathcal{K}_{\Delta})|=(2^{n-m}-1)|\mathcal{I}(\mathcal{H},\mathcal{K})|.$

Since $|\mathcal{I}(\mathcal{H},\mathcal{K})|=2^{m}-2^{|X|}-2^{|Y|}+1$, it
follows that

$|\mathcal{I}(\mathcal{F},\mathcal{G})|\leq(2^{n-m}-1)(2^{m}-2^{|X|}-2^{|Y|}+1)<2^{n}-2^{n-|X|}-2^{n-|Y|}+2^{n-m}.$

Note that
$2^{n}-2^{n-|X|}-2^{n-|Y|}+2^{n-m}=2^{n}-2^{n-|X|}-2^{|X|}+1-(2^{n-|Y|}-2^{|X|}-2^{n-m}+1)$.
Since $n-|Y|>|X|$ and $n-|Y|>n-m$, we have $2^{n-|Y|}-2^{|X|}-2^{n-m}+1>0$.
Therefore,
$|\mathcal{I}(\mathcal{F},\mathcal{G})|<|\mathcal{I}(\mathcal{D},\mathcal{E})|$
where $\mathcal{D}$ and $\mathcal{E}$ are two Cross-Sperner families of $[n]$
of type $(D,E)$ with $|D|=|X|$ and $|E|=n-|X|$, a contradiction. $\Box$

According to Lemma 2.4 and the inequality $a+b\geq 2\sqrt{ab}$ for $a\geq 0$
and $b\geq 0$, we obtain Theorem 1.2 immediately.

##  3 Acknowledgement

We are very grateful to professor Stijn Cambie for correcting some mistakes
about the literature related to corresponding problem in the initial version.

##  References

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