above, the fringe contrast of $C\approx 0.3$ at the exact position of G3 can
be achieved, with a few $\mu$m high-contrast region along the beam axis. Such
a measurement thus requires precise G3 positioning with $\mu$m-accuracy in the
beam axis, and below-nm-accuracy in the vertical direction.

From (2.16) we see that determining the sign of $g$ (more precisely to reach
$\Delta g/g=1$) in about one day, requires the detection of 3.2 $M$/s,
assuming a contrast $C=0.3$. Following (2.17), and taking pessimistic
estimates from Monte-Carlo simulations and initial detector and grating
studies studies by using $T_{G}=0.3$, $\varepsilon_{0}=0.75$ and
$\varepsilon_{\mathrm{det}}=0.3$, at the source we need $N_{0}\approx
1.4\times 10^{4}$ $M$/s. As a comparison the $\pi$E5 beam line at PSI can
presently deliver $3.6\times 10^{6}$ $\mu^{+}$/s at a momentum of 10 MeV/$c$
within a transverse area of about 400 mm2. At this muon momentum we may expect
a muon-to-vacuum-M conversion efficiency of about 0.1-0.3 in the best case,
when fast diffusion of the atoms can be assumed. This will result in $M$ rates
of up to $\sim 1.1\times 10^{6}$ $M$/s. These high rates may allow a further
collimation of the $M$ beam to a $5\times 1$ mm area, which would put less
strain on grating production and alignment and would cut the number of useful
$M$ atoms conservatively by a factor 5 mm2/400 mm${}^{2}=0.013$. Using these
parameters where there is room for contingency, we expect to produce the
necessary rate of $\sim 5\times 10^{4}$ $M$/s in an small area of $\sim
5\times 1$ mm2, and reach the goal sensitivity of $\Delta
g=\frac{9.8\mathrm{~{}m/s^{2}}}{\sqrt{\mathrm{\\#\;days}}}$ with present
$\mu^{+}$ sources. An increase by two orders of magnitude in $\mu^{+}$ rates
expected by the proposed HIMB project at PSI will allow shorter measurement
times and further improvements on the sensitivity to $g$.

###  2.6 Further particle-physics applications

####  2.6.1 Muon decays

LFV muon decays $\mu\to eee$ and $\mu\to e\gamma$ are flagship cases for HIMB.
In this subsection we consider other SM and BSM muon decay processes that
allow to further scrutinise the operator causing the decay of the muon, but
also allows for a more general approach to look for low-mass BSM particles
coupling e.g. to electrons or neutrinos. As we will describe there, has been a
lot of activity in the past decades to use muons as a more general laboratory
for BSM physics. Many of the results described here can in principle be
improved upon with an increased rate provided by HIMB.

The Michel decay of muons $\mu^{+}\to e^{+}(\nu_{e}\bar{\nu}_{\mu})$ played an
important role in many aspects of particle physics. Its description through
the Fermi theory with a $V\\!-\\!A$ four-fermion interaction was an early
application of effective theories and the determination of the Fermi constant
via the muon lifetime [221] is a crucial ingredient for electroweak precision
tests.

A more general analysis of muon decays is typically done allowing for right-
handed neutrinos, but neglecting their mass [222, 223]. If no information on
the positron polarisation is available, the energy spectrum of the decay
positron can be expressed in terms of four decay parameters [224], usually
denoted by $\rho$, $\delta$, $\eta$ and $P_{\mu}\,\xi$, with $P_{\mu}$ the
polarisation of the muon. The presence of BSM physics results in deviations of
these parameters from their SM value. Earlier searches for right-handed
currents [225] have been generalised [226] and lead to stringent constraints
on $\rho$, $\delta$, $\eta$ and $\xi$.

The most recent and most precise measurements of the muon decay parameters
$\rho$, $\delta$, and $\mathcal{P}_{\mu}\xi$ were obtained by the TWIST
experiments stopping a highly polarised beam in a silver target located inside
a 2 T solenoid and surrounded by tracking and timing detectors [227, 228].
These measurements were not statistically limited, but relied on a precise
control of all relevant systematic effects. In that sense, such measurements
will not profit from the increase in muon rates offered by HIMB. This is
especially true when measuring $\mathcal{P}_{\mu}\xi$ where a very precise
knowledge of the muon beam polarisation is paramount and which is almost
impossible to achieve at a high-acceptance beamline such as HIMB. However, the
measurements of $\rho$ and $\delta$ could profit from a beam such as HIMB-
muCool (see Section 4.2), which would allow to stop the beam in an extremely
thin target thus greatly reducing systematic uncertainties related to the
passage of the Michel positron through the target and would allow a very
compact detector arrangement potentially reducing the systematic uncertainties
related to detector response and calibrations.

Interpreting the muon decay parameters in terms of Wilson coefficients of a
general EFT containing SM particles and some number of sterile neutrinos
provides a framework to combine muon decay constraints with more general BSM
explorations. Contrary to [222, 223] this requires to allow for non-negligible
sterile neutrino masses, as e.g. in [229, 230, 231]. In addition to the Michel
decay, also the RMD $\mu\to e(\nu\bar{\nu})\gamma$ and the rare muon decay
$\mu\to e(\nu\bar{\nu})(ee)$ can be used. The additional visible final-state
particles can help to analyse potential effects in more detail. On the other
hand it has to be kept in mind that the low energy scale of the process
severely limits the scale of BSM physics that can be tested.

While the rate of the RMD and rare decay with HIMB is no issue at all,
analysing these processes would require to collect a certain amount of data
disregarding the kinematic cuts used in cLFV searches. For the rare decay in
connection with Mu3e this is probably more realistic than for the RMD in
connection with a MEG-like experiment.

Regarding more general searches for low-mass BSM, in recent years theorists
have actively explored new physics prospects below $\sim$ GeV [232], and
various experiments have found hints for new particles below the muon mass.
For instance, there are neutrino experiments which individually find evidence
for an additional massive neutrino, and as a possible explanation for the 8Be
anomaly in nuclear decays [233] a 17 MeV boson has been suggested. The fact
that a 17 MeV boson coupled to electrons is difficult to confirm or rule out
motivates the search for deviations in SM muon decays.

There are also numerous theoretically well motivated scenarios with nearly
massless ALP. They appear as Goldstone bosons and possibly have even LFV
couplings. This can lead to muon decays where there are BSM particles among
the invisibles [234, 232, 235]. Searches for the two-body decay $\mu\to eX$
with $X$ an ALP were carried out at TRIUMF [225, 236]. For sufficiently large
$m_{X}$ such a process would lead to a bump in the positron energy spectrum.
For nearly massless $X$ the endpoint of the spectrum is modified and a careful
comparison to the SM prediction is required. In this context there is also the
possibility to use a forward detector to increase the sensitivity, since the
SM background is strongly suppressed in the forward direction [237, 238, 239].
The sensitivity of $\mu A\to eA$ experiments to emission of an ALP is explored
in [240]. Several decades ago Cristal Box looked for $\mu\to eX\gamma$ [241]
and $\mu\to e\gamma\gamma$ [242]. Regarding the latter, the two photons could
arise from an ALP decay $X\to\gamma\gamma$ or a contact interaction (of
dimension 8). More recently, MEG has looked for $\mu\to eX\to e\gamma\gamma$
[243] and improved the limits for the branching ratio $X\to\gamma\gamma$ in
the range $m_{X}=20-45$ MeV.

Another prominent example of a BSM particle that can be looked for in muon
decays is a dark photon. For Mu3e there are first studies [35] for dark photon
and LFV two-body decays. While they can be carried out in phase I of Mu3e, a
dedicated experimental search with HIMB would improve the sensitivity.

Astrophysics can constrain particles with masses
$\stackrel{{\scriptstyle<}}{{{}_{\sim}}}$ few MeV that are feebly coupled to
the first generation, and energy frontier experiments exclude new particles
with large couplings — but for new particles coupled to the first and/or
second generation, there remains an allowed triangle [232], beyond
astrophysics, the energy frontier and current intensity frontier bounds, which
precision $\mu$ decays could explore.

An $X$ boson coupled to neutrinos could alleviate [244] the current $H_{0}$
tension [245, 246] in cosmology, and be produced via $\mu\to e\nu\bar{\nu}X$.
Some current experimental constraints on this scenario have been discussed
[247], but the theoretical study [248] suggests that meson decays could be
more sensitive to such an $X$ than $\mu\to e\nu\bar{\nu}X$, because $X$
emission removes the chiral suppression of the meson decays.

To summarise, there are many low-mass BSM physics scenarios that can affect
muon decay processes. A general EFT extension as well as a collection of
simplified low-mass BSM models are being implemented for muon decay processes
in the framework McMule [249]. Together with the precise prediction of SM muon
decays this gives a theoretical basis for a detailed investigation of the
potential impact of future measurements with HIMB.

####  2.6.2 Muonium - antimuonium oscillations

Bruno Pontocorvo predicted there might be $M\overline{M}$ oscillations [250],
in analogy to $K^{0}\kern-1.60004pt\kern
1.99997pt\bar{\kern-1.99997ptK}{}^{0}$ mixing, even before $M$ was discovered
[251]. In the past, $M\overline{M}$ oscillations were searched for in Ar gas
[252], at TRIUMF [253, 254], with foils at LAMPF [255] and also with
$\text{SiO}_{2}$ powder targets at LAMPF [256, 257, 258], and most recently
with the same approach at PSI [259, 260]. The current best upper limit for the
probability of spontaneous muonium to antimuonium oscillation is
$P_{M\overline{M}}<8.3\times 10^{-11}$ (90% CL) and was obtained with the MACS
apparatus at PSI.

Muonium has a lifetime of $2.2\,\mu\text{s}$ and decays into a fast positron
($e^{+}_{\mu}$) from the muon decay and a slow electron ($e^{-}_{\text{A}}$)
from the bound electron. The decay of an antimuonium is signalled by a final
state with the opposite electrical charges, i.e., a fast electron
($e^{-}_{\mu}$) and a slow positron ($e^{+}_{\text{A}}$).

To study the different beam setups in more detail, a straw-man GEANT4 [261]
simulation setup emulating the MACS apparatus was developed as shown in Figure
26 (left). A target is embedded inside a five-layer tracker with total length
of 1m. In a region extending over 2 cm along the $\mu^{+}$ beam direction, an
electric field of 1.5 kV/cm provides acceleration for $e^{+}_{\text{A}}$, and
a solenoidal magnetic volume bends the accelerated $e^{+}_{\text{A}}$ onto a
sensitive volume. This simulation allows the modelling and estimation of (1)
the decay characteristics of the $M$, (2) the acceptance (defined as the
fraction of $M$ decays inside the active detector volume), and (3) the
detection efficiency as a function of the $M$ kinetic energy
($E^{\text{kin}}_{M}$) and production divergence. For a real detector, a
configuration allowing the simultaneous measurement of $M$ and $\overline{M}$
(e.g. with the accelerating electric field perpendicular to the $\mu^{+}$
direction) would be chosen.

Figure 26: (left) Straw-man GEANT4 setup emulating the MACS apparatus. The
$\mu^{+}$ beam enters from the left and hits a target (hidden in the beam
pipe) inside the 5-layer tracker. (right) Development of the upper limit of
the $M\overline{M}$ oscillation probability vs. the number of produced
$N_{M}(=N_{\text{Mu}})$ in the acceptance of the experiments. The results are
taken verbatim from the publications and consist of upper limits at 90%CL and
95%CL (no re-scaling has been performed as the difference is not visible on
this plot). The lines are to guide the eye.

In all setups, the simulated detector setup was kept constant. Therefore, the
fraction of $M$ decays inside the active volume ($f_{D}$) decreases strongly
with increasing $E^{\text{kin}}_{M}$. Table 5 illustrates this behaviour.

Table 5: Expected $M$ decays per second ($N_{M}^{\text{decay}}$) in vacuo for
different beam intensities $N_{\mu^{+}}$ and $\mu^{+}\to M$ (in vacuo)
conversion efficiencies $\eta_{M}$. The factor $f_{D}$ specifies the fraction
of $M$ decaying inside the detector volume corresponding to the $M$ kinetic
energy $E^{\text{kin}}_{M}$.

Target | $N_{\mu^{+}}\,[\text{s}^{-1}]$ | $\eta_{M}$ | $E^{\text{kin}}_{M}[\,\text{eV}]$ | $f_{D}$ |  $N_{M}^{\text{decay}}$$\,[\text{s}^{-1}]$
---|---|---|---|---|---
HIMB-3/Al(a) | $1.0\times 10^{10}$ | 0.001 | 20000 | 0.07 | $7.0\times 10^{5}$
HIMB-3/$\text{SiO}_{2}$(b) | $1.0\times 10^{10}$ | 0.028 | 0.026 | 1.00 | $2.8\times 10^{8}$
HIMB-3/aerogel | $1.0\times 10^{10}$ | 0.015 | 0.026 | 1.00 | $1.5\times 10^{8}$

(a)Assuming $\eta_{M}$ as in [255] (b)Target not stable over time.

To significantly improve on the current best upper limit
$P_{M\overline{M}}<8.3\times 10^{-11}$ (90% CL) of [260], a minimum of
$N_{M}^{\text{decay}}=10^{12}$ is required, cf. Figure 26 (right). This is
reachable with aerogel targets with relatively short measurements, and may
also be achieved with foils.

####  2.6.3 Atomic parity violation in muonic atoms

A muonic atom is formed when a negative muon comes to rest in a material and
subsequently gets captured by a nearby atom. After this atomic capture, the
muon quickly cascades down to the $1s$ atomic orbital, emitting Auger
electrons and X-rays. Due to its relatively large mass, there is significant
overlap between the muon wave function and the atomic nucleus. This makes this
system particularly attractive to study short range interactions between the
muon and a nucleus/nucleon.

Muonic atoms are being used to determine absolute nuclear charge radii, most
recently by the CREMA collaboration for low Z nuclei [262, 263, 264] and by
the muX collaboration for high-Z radioactive elements [265].

Another compelling physics case for muonic X-rays is APV which arises from the
mixing of the opposite parity $2s_{1/2}$ and $2p_{1/2}$ atomic levels,
resulting in an E1 admixture in the otherwise pure M1 $2s_{1/2}-1s_{1/2}$
transition [266]. The most straightforward PV observable is the angular
correlation between the $2s-1s$ X-ray and the electron from the $\mu^{-}$
decay in orbit.

Observing the single photon $2s-1s$ transition is a challenge however. For
Z$\backsimeq$30 nuclei, this transition has a BR of $\mathcal{O}(10^{-4})$. At
lower Z two-photon and Auger transitions completely dominate the depopulation
of the 2s level. A clean detection of this X-ray is hampered by a dominant
background from Compton scattered $(n>2)p-1s$ X-rays. In addition, the SM
amplitude of a PV observable is expected to be smaller than $10^{-3}$. For
boron (Z=5) or neon (Z=10) the PV $e^{-}\gamma$ correlation is expected to be
at the 10-2 or 10-3 level, respectively. Additional challenges arise however
due to the Auger depopulation of the $2s$ level, necessitating a low pressure
gas target to avoid repopulating the electric orbitals [267]. Furthermore, the
small energy difference between the $2s$ and $2p$ levels, responsible for the
large APV, makes it hard to resolve the $2p-1s$ and $2s-1s$ transitions with
traditional X-ray detectors.

Figure 27: Left: Sketch of the muonic X-ray setup currently used at the
$\pi$E1 beam area, with a thin muon counter, a small target surrounded by
Michel electron detectors, and a HPGe detector array. Right: Accepting higher
incoming muon rate requires the development of larger targets and/or a
confining magnetic field, and the use of novel high-resolution photon
detectors such as LaBr3 detectors and microcalorimeters.

The muX collaboration has been investigating the reach of a $2s-1s$ APV
experiment for Z$\backsimeq$30 nuclei. Deploying a HPGe detector array at the
$\pi$E1 beamline, the transition was observed in Kr and Zn for the first time
[265]. Despite being the photon detector of choice for many applications in
the $0.110$ $\text{\,}\mathrm{MeV}\text{/}$ energy range, HPGe detectors have
long charge collection times and are susceptible to neutron damage, limiting
the instantaneous and integrated rate, which in turns limits a muX-style APV
experiment (Figure 27) to a sensitivity of $\mathcal{O}(10^{-3})$ on a PV
observable such as the $e^{-}\gamma$ correlation.

To fully profit from the high $\mu^{-}$ rates at HIMB, or even from the
current intensities available at the $\pi$E5 beam area, alternative high-
resolution photon detector technologies have to be considered. One promising
technology are high-resolution inorganic scintillators such as doped LaBr3
detectors, where resolutions of better than 2 % at
$1\text{\,}\mathrm{MeV}\text{/}$ are on the horizon. A rough estimate with 50
$1\times 1$ inch detectors placed $25\text{\,}\mathrm{cm}\text{/}$ from a muon
target shows that a statistical precision on the $e^{-}\gamma$ correlation of
better than $10^{-3}$ is possible.

For Z$\backsimeq$30 nuclei this would mean an $\mathcal{O}(1)$ SM test.
Searching for muon-specific dark forces, as suggested in [268], this level of
sensitivity can be competitive when only taking the low-energy $\mu$ two-loop
constraints from normal APV. Taking into account LHC bounds from non-resonant
di-lepton data [269], the sensitivity of a muon APV experiment to $Q_{W}$
needs to be similar as the 133Cs APV measurement, which is at the 1% level. As
a consequence, from this perspective only a measurement with low-Z nuclei is
worth pursuing as the APV amplitude is experimentally accessible due to the
near degeneracy of the $2s_{1/2}$ and $2p_{1/2}$ states. This energy split
quickly increases with a Z4 dependence, yielding APV amplitudes $<10^{-3}$ for
Z$\geq$30 nuclei. To realize such APV measurement with sufficient sensitivity,
R&D efforts need to be put into novel detector technologies, as well as in
efficient low-pressure gas-targets.

The HIMB $\mu^{-}$ intensities also open up the opportunity to apply ultra-
high resolution $\gamma$ detectors such as crystal spectrometers (see e.g.
[270]) and state of the art cryogenic microcalorimeters (see e.g. [271, 272])
to muonic X-rays spectroscopy. A first possible physics case are bound-state
QED tests using exotic atoms as suggested in [273]. Such a measurement can
probably be done with a fairly simple target station.

####  2.6.4 Muon conversion

Searches for the conversion of the muon into an electron in the field of the
nucleus have been performed at PSI in the past and indeed the current best
limit on this so far unobserved process is given by the SINDRUM II
collaboration in [24]. Large efforts are currently being undertaken by the
Mu2e and COMET collaborations [28, 27] to perform these measurements at
Fermilab and J-PARC, respectively, aiming at improving the sensitivity of the
SINDRUM II experiment by a factor $10^{4}$. The search for muon conversion
profits from the pulsed structure of the Fermilab and J-PARC proton
accelerators as after the formation of the muonic atom a certain amount of
time can be waited until the background events stemming from the proton pulse
on the muon production target has significantly dropped.

The experiments at PSI needed to adopt a different method to reduce such
backgrounds, as the quasi-continuous $50\text{\,}\mathrm{MHz}\text{/}$
structure of the proton accelerator at PSI does not allow to implement this
strategy. Instead the experiment relied on making the muon beam as pure as
possible by strongly reducing the amount of pions transported by the beamline
responsible for the largest backgrounds. Figure 28 shows the setup employed by
the SINDRUM II experiment. It consisted of the actual spectrometer shown at
the end of the beamline used to accurately track and measure the particles
emitted from the muon conversion gold target placed at its centre. A transport
solenoid connected the spectrometer to the $\pi$E5 beamline. At the beginning
of the transport solenoid an $8\text{\,}\mathrm{mm}\text{/}$ thick CH2
degrader was placed. The degrader was used to separate the surviving pions in
the beam at the momentum of $52\text{\,}\mathrm{MeV}\text{/}$/$c$ through
their much shorter range in material compared to muons. The thickness was
chosen to stop all pions while most of the muons continued further downstream
– guided by the transport solenoid – to the spectrometer. Figure 1 in [24]
shows the difference in range for muons and pions in the degrader (however
keep in mind that this simulation was performed for monoenergetic beams thus
exaggerating the separation power).

Figure 28: Experimental setup of the SINDRUM II experiment. Muons and pions
are produced by the proton beam at Target E and are transported by the $\pi$E5
beamline. A degrader located at the beginning of the transport solenoid
separates the pions and muons and guides the muons to the SINDRUM II
spectrometer located at its end. Picture reproduced from [24], where also more
details can be found.

In order to see how such an experiment could be improved by HIMB at PSI, let
us examine what the improvement in muon rate could be and what strategies
could be employed to combat backgrounds at the same time:

  * •

Muon rate:
SINDRUM II was running at a beam momentum of
$52\text{\,}\mathrm{MeV}\text{/}$/$c$ in the $\pi$E5 area of PSI with an
estimated rate of around $2\times 10^{6}$ $\mu^{-}$/s available at that time.
Due to the limited focusing power of the solenoids used in the HIMB beamline
compared to quadrupoles, the momentum reach of the HIMB muon beamlines are
limited. Good transmission should be achieved up to a momentum of about
$40\text{\,}\mathrm{MeV}\text{/}$/$c$, at which point it is expected to reach
about $10^{8}$ $\mu^{-}$/s. At this momentum the muon/pion separation
separation through a degrader is obviously not as good as in the case of
SINDRUM II – see Figure 29. However, due to the lower momentum the pion
content of the beamline is approximately an order of magnitude lower to start
with.

  * •

Backgrounds:
BG in the SINDRUM II experiment originated mostly from pions decaying in front
of or stopping in the degrader. High-energy electrons generated in these
decays could reach the spectrometer and mimic the electron produced in the
muon conversion process. Different strategies can be employed to reduce the BG
to levels much below the ones achieved by SINDRUM II to fully exploit the
increased muon rate. i) A spectrometer with modern detector technology as,
e.g., employed by the Mu3e experiment (see Section 2.2) will much improve the
identification of the electron emitted in the muon conversion process and thus
allow for tighter cuts and correspondingly reduced backgrounds. ii)
Developments on active targets for such kind of experiments will also provide
an additional handle to reduce BG by uniquely identifying events originating
from the conversion target. iii) Using a curved transport solenoid with the
degrader at its entrance would eliminate a direct line-of-sight between the
degrader and the spectrometer thus stopping any high-energy electrons
originating at the degrader from reaching the spectrometer. Similar concepts
are also employed by the Mu2e and COMET experiments [28, 27].

Figure 29: Range of negative pions and muons in a CH2 degrader at
$40\text{\,}\mathrm{MeV}\text{/}$/$c$ and with a momentum resolution of 3%. A
degrader with a thickness a bit above $3\text{\,}\mathrm{mm}\text{/}$ would be
appropriate to nicely separate pions and muons.

Based on the above considerations an improvement of around two orders of
magnitude over the sensitivity of the SINDRUM II experiment seems feasible by
performing a dedicated and new experiment at HIMB. Given the fact that both
Mu2e and COMET aim at reaching four orders of magnitude improved sensitivity,
the reach of such an experiment at PSI is certainly very limited. The
situation could of course change if either these experiments are not able to
reach their ultimate sensitivity or detect the muon conversion process with a
relatively high branching ratio. At this point, an experiment at HIMB could
become interesting – especially as it is hard for the experiments at pulsed
beams to work with high-Z conversion targets, which would reveal some insights
into the underlying physics of the conversion process [274].

##  3 Muon spin spectroscopy and material science with HIMB

Muon spin spectroscopy typically uses positive muons as highly sensitive local
magnetic probes to study a broad range of research topics in solid-state
physics, chemistry and materials science [275]. The experimental techniques
referred to as muSR (for Muon Spin Rotation, Relaxation, Resonance or
Research) are universally applicable since muons can be implanted in any
material. The spin-$\frac{1}{2}$ muons are produced with 100% spin
polarisation, providing muSR with a great advantage compared to other local
probe methods such as NMR or ESR which typically rely upon a tiny thermal
equilibrium spin polarisation. The muon is a very sensitive probe of local
static and dynamic magnetic properties of materials. Its short mean lifetime
of 2.2 $\mu$s and relatively large gyromagnetic ratio of $2\pi\times 135.5$
MHz/T makes it suitable for measurements of magnetic fields ranging from
$\sim\mu$T to several Tesla, and fluctuating on a time scales from pico- to
microseconds. The muon decays into two neutrinos and an energetic positron
which is emitted preferentially along the direction of the muon spin, thus
providing information on the time evolution of the muon spin polarisation in
the investigated material.

Applications of muSR include a large variety of topics in condensed matter
research with a focus on the investigation of novel and unconventional
magnetic and superconducting materials. As a local probe, muSR is a powerful
tool in this context since it allows studying the detailed nature of magnetic
and superconducting phases on a microscopic scale and as a function of control
parameters such as doping, temperature, applied magnetic field or pressure.
One of the particular advantages of muSR in such measurements is its
sensitivity to both the superconducting and magnetic volume fractions, and to
their respective order parameters. In the case of superconductors, fundamental
microscopic parameters such as the magnetic penetration depth can be
determined absolutely. In magnetic materials, muSR is extremely sensitive to
small magnetic moments and short range magnetic order. In addition, it is
possible to discriminate between static and dynamic magnetism and to
characterise magnetic fluctuations within a unique time window, which falls
between that of neutron scattering and NMR. Thanks to the volume fraction
sensitivity of muSR, the technique is often used to complement scattering
probes like X-ray diffraction and neutron scattering where phase volume
information is difficult to obtain.

In addition to the common applications of muSR in magnetism and related
subjects, it is used to investigate molecular dynamics, charge transport
phenomena such as polaron motion in conducting polymers, electron and spin
transport as well as ion conduction in technologically relevant battery
materials. As a ”light isotope” of the proton the $\mu^{+}$ can form the
hydrogen-like exotic atom muonium [$M=(\mu^{+}\,e^{-})$] which may substitute
for hydrogen in insulators, semiconductors and organic materials and provide
accurate spectroscopic information on hydrogen levels in these materials and
their potential for practical applications. As a ”light proton isotope”, the
muon can also serve as a test model for light particle diffusion, radical
formation, and chemical reactions with the largest known kinetic isotope
effect [276].

The majority of the muSR experiments use one of the following experimental
setups, which vary in the orientation of the initial muon spin direction with
respect to the external field and the physical information that can be
obtained. Very roughly speaking, ZF muSR allows studying the static magnetic
order parameter, the volume fraction and magnetic correlation times.
Longitudinal-field (LF, a magnetic field is applied along the initial muon
spin direction) muSR is used to discriminate between static and dynamic ground
states and to investigate magnetic fluctuations, as well as spin or ion
diffusion rates. Transverse-field (TF, a magnetic field is applied
perpendicular to the initial muon spin direction) muSR is often used to
determine fundamental length scales in superconductors and to measure the
q-integrated local susceptibility (Knight shift).

In time-integrated measurements the time evolution of the polarisation is not
determined, and the measurements can be carried out at very high muon rates
(also at continuous mean beams, because one does not need to have only one
muon at a given time in the sample as in time-differential measurements, see
Section 3.1). Time-integrated experiments provide information about the
energy-level scheme of the muon spin coupled to other spins species (electrons
and/or nuclei). Examples are avoided-level-crossing spectroscopy (ALC) or
radio-frequency $\mu$SR (RF-$\mu$SR).

###  3.1 Different muon sources around the world

Condensed matter research using muSR requires intense muon beams which at
present are available at four large scale facilities: the ISIS facility at the
Rutherford Appleton Laboratory (UK), the MUSE facility at J-PARC (Japan), the
CMMS facility at TRIUMF (Canada) and the SmuS facility at PSI. The first two
are pulsed muon sources with 50 Hz and 25 Hz repetition rates, respectively,
while those at TRIUMF and at PSI deliver continuous muon beams. Pulsed and
continuous muon sources complement each other allowing for different kinds of
experiments. At pulsed sources it is possible to determine very small muon
spin relaxation rates due to the virtually zero background between the pulses,
but these sources are strongly limited in time resolution due to the 50 – 80
ns width of the muon pulses. In contrast, at continuous sources the time
resolution can be up to a factor of thousand better since it is solely
determined by the detector construction and the read-out electronics. However,
the incoming muon rate has to be reduced compared to a pulsed beam facility to
minimise pile-up events. In a traditional muSR setup at a continuous beam, an
incoming muon is registered in a muon counter, which opens a data gate of
typically 10 $\mu$s length. A valid muSR event is given after the detection of
the decay positron in one of the scintillators of the positron spectrometer.
In case of a pileup event, where a second muon or a second positron is
observed within the data gate, it is not possible to determine which muon
belongs to which positron, and the event is discarded. At a beam rate of
$4\times 10^{4}$/s, the accepted rate of $1.8\times 10^{4}$/s is at maximum
for a data gate of 10 $\mu$s. At beam rates $>4\times 10^{4}$/s, the
increasing pile-up probability causes a reduction of the accepted rate to
$<1.8\times 10^{4}$/s. In contrast, at pulsed beams, the accepted rate of
incoming muons can easily exceed $10^{5}$/s.

The rate limitation at a continuous muon beam could be overcome, if the
stopping position of the incoming muon in the studied sample could be detected
and the corresponding emitted positron tracked. In this case, each decay
positron could be assigned unambiguously to its parent muon by vertex
reconstruction, thus allowing for multiple muons to stop in the sample at any
given moment. This makes muSR measurements with at least ten times higher
incoming muon rates feasible. This paradigm shift for muSR measurements at a
continuous source will be discussed in detail in Section 3.3.

###  3.2 Current status of muSR at PSI

The Laboratory for Muon Spin Spectroscopy (LMU) develops and operates six
state-of-the-art instruments of the SmuS facility, the worldwide most powerful
muon source for condensed-matter research. The six instruments are permanently
installed at five separate secondary muon beamlines of the HIPA cyclotron
complex. The different instruments provide the users with a variety of
experimental capabilities with respect to temperature, magnetic field,
pressure, time resolution, measurement geometry, probing depth and minimal
sample size to fulfil the various requirements for the broad scientific
spectrum addressed by the Swiss and international user community (see Table
6). To maintain the leading international research position, the muSR
instruments need to be kept at the state-of-the-art. This requires a
continuous development of new experimental capabilities to be able to address
topical and novel research questions.

Instrument |  $\mu^{+}$ energy | T-range | B-range |  Comments
---|---|---|---|---
DOLLY | 4 MeV | 0.25 – 300 K | 0 – 0.6 T |  Unique device for uniaxial strain up to 1 GPa.
FLAME | 4 MeV | 0.02 – 300 K | 0 – 3.5 T |  Flexible and Advanced muSR Environment. Begin operation in 2022.
GPD | 15 – 60 MeV | 0.30 – 500 K | 0 – 0.6 T |  Unique hydrostatic pressure up to 2.8 GPa.
GPS | 4 MeV | 1.60 – 1000 K | 0 – 0.78 T |  General Purpose Instr. Unique MORE capability (Muons On REquest).
HAL-9500 | 4 MeV | 0.01 – 300 K | 0 – 9.5 T |  Unique combination of low T and very high B. Record time resolution in muSR of 60 ps.
LEM | 1 – 30 keV | 2.30 – 600 K | 0 – 0.3 T |  Unique LEmu beam and LEmuSR spectrometer. Thin films & heterostructures. $\mu^{+}$ range up to 200 nm.
s-Ne | 1 – 20 keV |  |  |  23 kHz moderated $\mu^{+}$.
s-Ar |  |  |  |  14 kHz moderated $\mu^{+}$.
Table 6: The instruments of SmuS with several unique features. T-range and
B-range denote the available range of temperatures and magnetic fields,
respectively. At all instruments, the muon spin can be rotated between 10∘ and
90∘ by so-called spin-rotators ($\vec{E}\times\vec{B}$ fields), or special
asymmetric quadrupole settings of the muon beam line (GPD instrument). This
allows zero-field-, LF-, and TF-muSR measurements. LEM, s-Ne: solid neon
moderator; s-Ar: solid argon moderator. The LEmu rates are for a proton
current of 2 mA, ”slanted” target E [277].

The research programme at LMU/SmuS includes the “classical” applications of
muSR in magnetic materials, superconductors, and semiconductors, as described
at the beginning of Section 3 but also goes well beyond this scope. Frontier
research topics of recent years include high-spin molecules [278, 279, 280],
low-dimensional magnets [281, 282], quantum spin liquids [283, 284], organic
superconductors [285], conducting polymers [286, 287], liquid crystals [288],
topological materials [289, 290, 291, 292], defect regions in novel solar-cell
materials [293, 294], and defects in semiconductor device heterostructures and
near-surface regions [295, 296, 297].

Despite the very successful research program at SmuS, there are limitations of
muSR for the further evolution of the technique towards novel applications and
research directions. These limitations, summarised below, are determined by
the muon beam and detection characteristics of the muSR spectrometers.

  * •

Large sample area: The sample area must be $>4\times 4$ mm2 due to the large
phase space of typical muon beams, the limited incoming muon rate and the
difficulty in determining whether the muon stopped in the sample or not. For
example, this makes it difficult to contribute to the field of some novel
quantum materials, which often can be grown with high quality or single
crystalline form only in small amounts and small cross sections ($\leq 1\times
1$ mm2).

  * •

Limited muon rate: As discussed in Section 3.1, the maximum incoming muon rate
is limited to about $4\times 10^{4}$/s due to the pileup problem at continuous
muon beams, and the inability of detecting the muon/positron vertex. The
accepted muon rate, i.e. those events without pileup, is then limited to only
$1.8\times 10^{4}$/s for a 10-$\mu$s data gate of a typical muSR experiment at
a continuous beam. This limits the statistical precision of an experiment,
compared to pulsed muon sources with $>10^{5}$/s accepted muon rate, and it
makes it often impossible to investigate very subtle effects like very slow
magnetic dynamics or time reversal symmetry breaking in unconventional
superconductors.

  * •

Limited data gate: Due to the continuous beam, uncorrelated muon-positron
events produce a small time-independent background in the muSR spectra.
Therefore, the signal to noise ratio gets progressively worse for longer
measurement times limiting the available data gate length. This obstructs the
measurement of long relaxation times and reduces the frequency resolution for
ultra precise Knight shift measurements. These problems can partially be
avoided by the use of the so-called Muons On REquest system (MORE) [298] which
however reduces the count rate.

  * •

Limited pressure: S$\mu$S has the world leading muSR high-pressure facility.
Due to the limitations in beam spot size and sample size – causing problems
with background of muons stopping in the pressure cells, or problems with the
mechanical stability of large samples in the device for applying uniaxial
pressure – the maximum pressures are 1 GPa and 2.8 GPa for uniaxial and
hydrostatic pressures, respectively.

  * •

Inaccessible depth range: Due to available beam energies at reasonable rates,
the depth range of muons is limited to either $<200$ nm [LEmu], or $\geq 100$
$\mu$m by $\mu^{+}$ with energies $>4$ MeV, provided by the surface- or decay-
muon beams. The currently accessible depth ranges are shown in Figure 30.
There is a big inaccessible gap in the so-called sub-surface muon beam region,
which is at energies $E_{\mu}$ between $30$ keV and $\sim 3.0$ MeV,
corresponding to muon momenta $p_{\mu}$ of $2.5$ MeV/$c$ and $25$ MeV/$c$,
respectively.

The reason for the range gap is the strong dependence on momentum of sub-
surface muon beam intensity $I(p)$ [299]

$I(p)\propto p^{3.5}.$ (3.1)

Figure 30: Left: calculated muon stopping ranges in Cu as a function of muon
energy. Right: example for the range gap in the determination of the
activation energy $E_{i}$ of shallow muonium as a function of mean depth
$\langle z\rangle$ in CdS and ZnO (Reprinted figure with permission from
[300]. Copyright 2014 by the American Physical Society.).

Already at 25 MeV/$c$, the beam intensity drops by a factor of two, making it
unfeasible to go to even lower beam momenta/energies for muSR applications.
Additional complications arise due to the fact that in a standard muSR
spectrometer, the muons have to cross a 200-$\mu$m-thin scintillator to detect
the incoming muon, plus additional thin metal windows in the radiation shield
of the cryostat where the sample is located. In these materials, the muons
lose tens to hundreds of keV, which requires an incoming beam energy in the
MeV range. Note, that this momentum dependence of beam intensity does not
apply to the low-energy muon beam at $E_{\mu}<30$ keV: these muons are
generated by the moderation of a highest-intensity surface muon beam with a
$I>2\times 10^{8}$/s in a cryogenic solid moderator (solid argon or solid
neon) to generate LEmu with a rate up to $2.3\times 10^{4}$/s at the moderator
target, see Table 6, and [208, 301, 302].

###  3.3 New opportunities for muSR at HIMB: key science drivers

HIMB in combination with the novel vertex reconstruction (see Figure 31) –
based on the development of a new generation of Si-Pixel detectors (see
Section 4.3) – will lead to a ”quantum leap” of the muSR technique, enabling
new research directions/science for muSR by much faster, higher statistics and
more efficient measurements, extending the depth range of muSR, introducing
lateral spatial resolution, and by being able to apply external stimuli with
unprecedented levels. The envisaged determination of the muon decay vertex
with a precision of $\lesssim 1$ mm will

  * •

enable to measure small samples of cross section $\leq 1$ mm2, at least twenty
times smaller than currently possible. It will also allow measuring multiple
samples in parallel, thus, for example, determining the phase diagram as a
function of doping in one measurement. This is extremely important for the
efficient and fast characterisation of novel quantum materials, which can be
often grown in small amounts only.

  * •

enable to avoid the pile-up problem by tracking each muon and its
corresponding decay positron. The incoming muon beam rate can be significantly
increased by at least one order of magnitude compared to the current limit of
$\sim 4\times 10^{4}$/s. Experiments can be carried out much faster, therefore
more efficiently using precious beam time at the two to three times
oversubscribed instruments. Due to the significantly enhanced statistics,
subtle effects can be detected, for example to measure very slow magnetic
fluctuations in quantum materials or detect ultra weak magnetic fields in time
reversal symmetry breaking superconductors.

  * •

enable to avoid uncorrelated background in the muSR spectra which allows
extending the accessible data gate by at least a factor two. This opens the
possibility to measure significantly longer relaxation times and to increase
the frequency resolution for high field measurements (Knight shift).

  * •

enable the mapping of imposed lateral variation with $\sim$ mm resolution of
sample parameters under chemical, thermal or pressure gradients. This allows
measuring extended portions of phase diagrams in one sample or stimulating
gradient induced properties like time reversal symmetry breaking in
superconductors by thermal gradients.

  * •

enable new opportunities for muSR under extreme conditions. With smaller
samples of $\sim$ mm size, the application of about ten times larger uniaxial
strain (10 GPa) and hydrostatic pressures (30 GPa) will become possible,
allowing the exploration of so far inaccessible regions in the phase diagrams
of novel materials.

Other external stimuli such as illumination or the study under DC or RF
electromagnetic fields will become possible with at least one order of
magnitude higher intensities/fields. The use of small samples will ease the
mitigation of sample heating by more efficient cooling, and it will allow
building very compact and mechanically stable and reliable setups. Possible
applications are, e.g., novel investigations of charge carrier lifetimes and
dynamics in semiconductors by photo-generated carriers, vortex dynamics in
superconductors under DC currents, skyrmion dynamics under DC or RF fields, or
quantum information processing using molecular spins or skyrmions in pump-
probe experiments (see Section 3.3.3).

Figure 31: Sketch of the vertex detection scheme for future muSR instruments.

Besides the exciting new opportunities offered by the combination of HIMB and
vertex reconstruction, HIMB will enable i) novel applications in energy
materials by closing the range gap between 200 nm and 200 $\mu$m (Section
3.3.2 and Table 7), ii) new pump-probe experiments at continuous muon beams
(Section 3.3.3), iii) low-energy muon applications with an order of magnitude
intensity gain (Section 3.3.4), and iv) significantly improved applications of
negative muons in non-destructive, depth-selective elemental analysis studies
(Section 3.3.5).

####  3.3.1 Novel quantum materials

Core activities in condensed matter physics and materials science are the
search for, understanding and tuning of phases of quantum materials. muSR as a
powerful local probe technique can provide unique insights in the properties
of unconventional magnetic and superconducting materials. A large variety of
new materials, including those with topologically non-trivial band structures,
offer the possibility of realising interesting phenomena, where muSR is a very
sensitive experimental technique for their characterisation.

The possibility of measuring ten to hundred times smaller samples, at ten to
hundred times higher rates, will have a huge impact on the use of muSR for the
investigation of novel quantum materials. In the following, we give a short
overview of possible muSR applications with HIMB, which currently can only be
carried out to a limited extent or are even unfeasible.

  * •

Higher uniaxial pressure up to 10 GPa due to smaller sample size, which means
higher mechanical stability of sample and setup and better homogeneity of
samples as well as higher hydrostatic pressure of up to 30 GPa by using high-
pressure anvil cells with small sample volumes.

    * –

Realisation and detection of quantum spin liquids by systematic and symmetry-
lowering control of the magnetic interactions by uniaxial strain in kagome,
triangular and Kitaev systems.

    * –

Tuning of flat-band superconductivity in kagome-lattice systems, for example
in LaRu3Si2, XV3Sb5 (X=K, Rb, Cs) and investigation of the interplay of TRSB,
CDW order and superconductivity in these systems.

    * –

Investigation of 3D CDW order beyond 1 GPa uniaxial strain in YBa2Cu3O7, where
only small high quality crystals are available.

    * –

Hydrostatic and uniaxial pressure tuning of the competition of CDW and
superconductivity, e.g. in NbSe2.

    * –

Uniaxial pressure tuning of stripe order and pair density wave
superconductivity in La2-xBaxCuO4.

    * –

Tuning of semiconductor to metal transitions in magnetic layered transition
metal dichalgoneides (TMC), e.g. 2H-MoTe2.

    * –

Pressure tuning of superconducting Weyl semi-metal 1T-MoTe2.

    * –

Stabilisation and control of topological fermions by tuning of magnetic
competition, e.g. in the ferromagnetic Weyl semi-metal Co3Sn2S2.

    * –

Novel low-temperature topological magnetic states in TbMn6Sn6: pushing
topological properties towards room temperature.

    * –

Strain-induced topological magnon transition in ferromagnetic kagome systems.

    * –

Tuning of mechanically soft quantum magnets, e.g. CsFeCl3 and TlCuCl3 which is
only possible with small specimens.

    * –

Lifting degeneracy of multi-component unconventional superconductivity, e.g.
study of TRSB in Sr2RuO4 under strain.

    * –

Elucidating the source of unconventional high temperature magnetism in Sr2RuO4
using high pressure.

    * –

Local probe investigation of the magnetic and superconducting phase diagram of
FeSe under hydrostatic pressures larger than 2 GPa.

    * –

Strain-tuning of nematicity in Fe-based superconductors.

    * –

Investigation of two-dome superconductivity in the temperature-pressure phase
diagram of K0.8Fe1.7Se2.

    * –

Black phosphorus as an emerging 2D material: Superconducting phases at
pressures up to 20 GPa.

    * –

Pressure induced superconductivity in elemental Fe.

  * •

Small samples allow application of stronger external stimuli (higher current
densities, electric fields or luminosity).

    * –

Current induced tuning of magnetic properties of iridates.

    * –

Current induced modification of charge order in the quantum chain systems,
e.g. in (La,Sr,Ca)14Cu24O41.

    * –

Microwave / RF induced Skyrmion motion.

    * –

Current induced vortex lattice dynamics.

    * –

Manipulation of spin states in molecular magnets using light and microwave
irradiation.

    * –

Spectroscopic investigation of muonium states in semiconductors and
insulators.

  * •

Simultaneous measurements of multiple and high quality samples.

    * –

Much more efficient exploration of phase diagrams (transition temperatures and
order parameters) of unconventional superconductors and novel magnets, while
thermal and magnetic history are identical for all samples.

    * –

Precise measurements of Knight shifts and line shapes in superconductors, spin
liquids and topological materials, using a reference sample (e.g. Ag) for in-
situ calibration of magnetic field and detection of field drifts.

  * •

High statistics measurements and low positron background (allows long data
gates of $20$ $\mu$s).

    * –

Small changes in relaxation rates and internal magnetic fields to investigate
TRSB in superconductors.

    * –

Subtle features in muSR depolarisation function at long times: quantum
entanglement and quantum decoherence in F-Mu-F systems.

    * –

Measurement of decoherence times of spin states in single molecule/ion
magnets.

    * –

Volume fraction of spontaneous magnetic fields in TRSB in unconventional
superconductors.

    * –

Thermal gradients as a source of TRSB fields in unconventional
superconductors.

  * •

Laterally resolved muSR measurements.

    * –

Thermal gradient effects on materials, e.g. spin Seeback effect.

    * –

Homogeneity of phases in magnetic and superconducting sample.

    * –

Measurement of the effect of light irradiation intensity on properties of
materials in a single shot, e.g. by irradiating with a gradient of light
intensity.

This list – although by far not complete – demonstrates the versatile future
opportunities for muSR with HIMB combined with vertex reconstruction, allowing
a significantly increased access to sample phase space of quantum materials by
muSR and thus enabling the exploration of new research avenues by muSR.

####  3.3.2 Energy materials

In recent years, LEmuSR has been successfully extended to the study of
semiconductor interfaces to provide valuable information – with unprecedented
depth resolution of a few nanometres – about charge carrier profiles/gradients
and their manipulation by illumination, as well as defect regions in
technologically relevant Si and SiC and its oxide interfaces, and defect
regions in CIGS based solar cell heterostructures [293, 294, 295, 296, 297,
303]. The goal of these studies is to obtain a better microscopic
characterisation of device interfaces with a depth resolution superior to
standard characterisation techniques, in order to be able to adapt the growth
conditions and achieve optimum device efficiency.

Another important extension of muSR demonstrated its capability to measure the
Li+ ion diffusion coefficient, $D$, in battery materials [304, 305]. In
contrast, conventional NMR is unable to provide useful information due to the
dominant effect of localised magnetic moments on the NMR spin-lattice
relaxation. With the knowledge of $D$, other crucial parameters relevant for
the performance of battery materials can be derived, such as the reactive
surface area, diffusion pathway, and density of mobile ions.

So far, these muSR studies are either limited to thin layers/heterostructures
with a thickness $<200$ nm, or to bulk material. In many cases, semiconductor
and solar cell devices as well as real life battery devices have thicknesses
in the range of hundreds of nanometres to tens of $\mu$m. Therefore, it would
be highly appealing to use such a sensitive local probe technique as muSR to
study interface and bulk regions in real devices in-operando, to gain
unparalleled insights into their function and to enable the development of
better devices in future. With the expected intensities of low momentum sub-
surface muons at HIMB, as shown in Table 7, the study of buried layers in so
far inaccessible depth ranges will become possible, allowing the extension of
muSR to become a new in-operando technique for contributing to the solution of
important societal problems in energy research.

Table 7: Expected sub-surface muon rates at HIMB for muSR applications, where we assume a surface muon rate of $10^{8}$/s being available in the new $\mu$H3 beamline, where the GPS and FLAME instruments are located. This rate is expected downstream of the $\mu$H3 spin-rotator. Beam rates calculated using (3.1). Such low beam energies will require the development of new muon entrance detectors. $\mu^{+}$ momentum |  $\mu^{+}$ energy | beam rate | mean implantation depth
---|---|---|---
2.5 MeV/$c$ | 30 keV |  $1.7\times 10^{4}$/s |  $\sim 180$ nm
3.2 MeV/$c$ | 50 keV |  $4.0\times 10^{4}$/s |  $\sim 350$ nm
3.8 MeV/$c$ | 70 keV |  $7.4\times 10^{4}$/s |  $\sim 500$ nm
4.6 MeV/$c$ | 100 keV |  $1.5\times 10^{5}$/s |  $\sim 700$ nm

####  3.3.3 Pulsed beam and pump-probe

The new $\mu$H3 beamline at HIMB is expected to deliver a surface muon rate of
$>10^{8}$/s downstream of the $\mu$H3 spin-rotator (see also Table 7). The
spin-rotator will significantly cut the initially large phase space, allowing
for pulsing/chopping of the continuous surface muon beam. A possible pulse
scheme is sketched in Figure 32.

Figure 32: Left: pulsing scheme for chopping the surface muon beam. A possible
timing scheme, adjusted to the muon life time, is T = 1 $\mu$s, and T′ = 19
$\mu$s, resulting in a 50 kHz repetition rate. During the pulse of length T,
on average about 100 $\mu^{+}$ will stop in the muSR spectrometer. Right:
expected polarisation functions $p_{z}(t)$ [306].

Figure 33: Left: Example for the measurement of spin-lattice relaxation in
pulsed $\beta$-NMR using radioactive 8Li+ implanted in SrTiO3 (Reprinted
figure with permission from [306]. Copyright 2006 by the American Physical
Society.). Right: Scheme of multiple instruments on the same muon beamline.

During the pulse of a length of T = 1 $\mu$s, about 100 $\mu^{+}$ will enter
the muSR spectrometer on average at an incoming rate of $10^{8}$/s. Between
the pulses, the measurement is free of accidental background, similar to a
pulsed muon beam facility. This technique is already in use at the $\beta$-NMR
facility at TRIUMF, where a beam of radioactive 8Li+ ions with tunable
energies between 1 and 30 keV is used for depth-resolved condensed matter
physics applications in thin films or near-surface regions, see Figure 33,
similar to LEmuSR. Compared to muSR, 8Li+ $\beta$-NMR is sensitive to much
longer times scales due to its orders of magnitude longer lifetime of 1.2 s.
For a much more efficient use of beamtime in this scheme, the beam could be
split to serve different spectrometers at the same time, as shown on the right
side of Figure 33.

Such a scheme of pulsed muon beams, with an average muon rate of $5\times
10^{6}$/s, will enable new research directions at a continuous muon beam
facility:

  * •

The possibility of measuring the effect of external stimuli synchronised with
the muon pulse will allow novel muSR applications in quantum information
processing with molecular spin systems, magnetic skyrmions etc. For example,
quantum states can be prepared in molecular magnet systems and the decoherence
time of these states can be measured by muSR. Magnetic skyrmions have the
potential to be used as information carriers in future spintronics devices,
susceptible to manipulation by very small electric currents or
magnetic/electric field pulses. In a magnetic skyrmion system, collective
magnon modes can be resonantly stimulated by GHz microwave excitation – FMR –
and probed by its effect on the muSR polarisation and the local internal field
at the muon site. This combination of FMR-muSR has the potential to examine
magnon modes while accessing the internal fields and fluctuations by muSR.

  * •

In general, such a pump-probe scheme can be applied to many systems to gain
information about the dynamics in the MHz to GHz range by using light or
microwave stimuli, and probe with the muon as a very sensitive local magnetic
field sensor. For example, excess carrier lifetimes in semiconductors can be
measured by photoexcited muSR [307].

  * •

With this kind of a pulsed beam – similar to a pulsed muon beam facility but
with much better time resolution – a long data gate of 20 $\mu$s width can be
employed due to the very low positron background to measure very small
relaxation rates, low muon spin precession frequencies and better resolved
split frequencies (e.g. in semiconductors with shallow muonium states). High-
statistics measurements would become possible – without vertex reconstruction
– to measure very small changes of muSR parameters (see also Section 3.3.1).

  * •

In this pulsing scheme with an effective muon rate in the MHz range, a muSR
spectrum can be recorded with sufficient statistics within seconds. This will
allow to extend muSR to a new field of applications: the study of transient
states with seconds lifetimes. Examples are slow magnetic dynamics, slow
chemical reactions and diffusion processes, and vortex creep in
superconductors.

####  3.3.4 Low-energy muons

The current techniques for generating low-energy $\mu^{+}$ with energies in
the eV- and keV- range at continuous muon beam facilities are based on the
moderation of a 4-MeV surface muon beam in either a thin film of a solid rare
gas (argon or neon) [208, 301] or in low-pressure He gas placed in
superimposed electric and magnetic fields (muCool, see Section 4.2 and [308]).
The conversion efficiencies to the eV range are typically small, between
$10^{-5}$ and $10^{-4}$ per incoming $\mu^{+}$. This requires very high
surface muon beam intensities of $>10^{8}$/s to be able to generate $10^{3}$/s
– $10^{4}$/s low-energy $\mu^{+}$. This order of magnitude of LEmu rates is at
the lower limit for practical LEmuSR applications. Obviously, an increase of
the 4-MeV surface muon beam rate on the moderator target by one or even two
orders of magnitude would have an enormous impact on the entire research
program with low-energy muons.

The PSI LEM facility is the leading facility for low-energy $\mu^{+}$
applications, delivering up to 23 kHz moderated muons. However, taking into
account beam transport and detection efficiencies of the LEM facility, the
maximum recorded LEmuSR event rate is less than 2.7 kHz at the moment, far
below the pile-up limit at a continuous muon beam. This reduces the available
statistics of a typical LEmuSR measurement to $10^{6}$ to $10^{7}$ recorded
muon decays in the muSR histograms, and thus the investigation of small muSR
signals or subtle changes in depolarisation rates and local magnetic fields is
not feasible. An additional limitation is the large LEmu beam spot with a RMS
width of 6 – 7 mm. This requires a minimum sample cross section of $10\times
10$ mm2. For optimum usage of the beam, samples with sizes of $25\times 25$
mm2 are required. Using a collimator in front of the LEmuSR start detector
with 10 mm or 15 mm diameter reduces the beam spot size to RMS values of 3 – 4
mm, while losing 45 – 60% of beam rate.

  * •

An increase of LEmu rate by one order of magnitude obviously leads to
significantly improved LEmuSR experiments with much better statistics and
faster measurements. Small samples with $5\times 5$ mm2 cross section or even
smaller will become possible using beam collimation, while compensating the
beam loss in the collimator by the increased initial rate after moderation.
Such small samples will enable new LEmuSR applications on novel, small size
quantum materials and device structures. In addition, experiments under
external stimuli such as illumination, electric fields or electric currents
can be carried out with much higher light intensities and current densities
without excessive heating, therefore expanding the parameter range of LEmuSR
to new regions. Current experiments to directly measure the local magnetic
field generated by current-induced spin accumulation at the surface/interface
(spin Hall effect) are limited by sample heating in the large cross section
samples [309]. In samples with 5 times smaller area, the current densities can
be increased by a factor 5 (in-plane current) or even a factor 25 (out-of-
plane current), therefore bringing the local magnetic field generated by spin
accumulation closer to the detection limit of LEmuSR. Furthermore, the quality
and homogeneity of electrical contacts can be much better controlled with
small sample areas $\ll 25\times 25$ mm2, which is the ideal sample size at
LEM at the moment. This is important for experiments on new device structures,
such as reversible spin storage in metal/metal-oxide/fullerene heterojunctions
[310], but also for future applications to study charge carrier dynamics and
accumulation/depletion in the presence of defects at technologically important
semiconductor interfaces [303] and solar cell devices [293, 294]. For the
latter, LEmuSR is used as a local probe technique to characterise these
interfaces with a resolution of a few nanometre with the overall goal to
obtain important insights in optimising the growth conditions of these
devices.

  * •

With a significant increase of LEmu rate, about twenty times lower
temperatures of $\sim 100$ mK will become feasible at the LEM facility. At the
moment, the lowest temperature of $\sim 2.3$ K is achieved with a home-made
4He cryostat, where the 6-cm-diameter opening in the radiation shield of the
cryostat888LEmu would stop in any of the typically used $\mu$m-thick radiation
shield windows. leads to a thermal load of $\sim 200$ mW on the sample. A DR,
needed to reach $\leq 100$ mK, has a cooling power of only $\sim\mu$W. This
requires a cooling of all surfaces “seen” by the sample to 4 K. Since one
needs an opening of several cm diameter in the radiation shield, all the beam
pipes with direct view on the sample have to be cooled with liquid 4He. This
means, that the existing LEmu beam at LEM needs to be extended by a section
with another bend, where from the bend towards the DR the entire beam section
will be cooled. This extension of the beamline will cause a significant loss
of at least 50% in transmission, which can be compensated by an increase of
the initial LEmu rate. The possibility of achieving twenty times lower
temperatures down to 100 mK or less will open new fascinating applications of
LEmuSR in various fields of condensed matter physics, for example:

    * –

Direct determination of the magnetic penetration depth and coherence length in
low temperature superconductors, e.g. the type-I superconductor Al with large
$\kappa$ and a Tc of 1.6 K.

    * –

Magnetic and superconducting properties of 2D electron gases at interfaces,
where coexistence of magnetic and superconducting order at LaAlO3/SrTiO3
interfaces (T${}_{c}\sim 200$ mK) was observed [311, 312]. LEmuSR can measure
the vortex broadening of the magnetic field distribution in the
superconducting state to determine the magnetic penetration depth as a
function of temperature. This allows determining the symmetry of the
superconducting order parameter.

    * –

Direct probing of TRSB magnetic fields at the surface of unconventional
superconductors with low Tc, e.g. SrRuO4 with T${}_{c}\sim 1.5$ K [313].

    * –

Extension of LEmuSR to determine the magnetic penetration depth in
unconventional heavy-fermion superconductors. For example, UTe2 with
T${}_{c}\sim 1.6$ K is showing signs of spin-triplet pairing, TRSB, multiple
order parameters and chiral surface states [314].

    * –

Proximity effects with low-temperature superconductors.

    * –

Low-temperature magnetism: frustrated magnetic systems, spin-ice and quantum
spin liquid systems, and its properties near surfaces or interfaces.

  * •

A beam of LEmu can be used to efficiently generate muonium in vacuum, which is
essential for performing high-precision muonium spectroscopy to provide
stringent tests of bound state QED and to determine fundamental parameters of
the Standard Model (such as muon mass, muon magnetic moment, etc.), as
described in Section 2.5. At the LEM facility, the efficient generation of
thermal muonium in vacuum has been demonstrated [195, 196], which will enable
a new muonium 1s-2s experiment with up to hundred times improved precision
[209]. Recently, the efficient generation of a muonium beam in the metastable
2s state [315] has been achieved in the LEM facility, which enabled a ten
times more precise measurement of the 2s-Lamb shift in muonium [210]. It is
obvious, that the precision of muonium spectroscopy would tremendously benefit
from an order of magnitude increase in LEmu rate.

  * •

A very exciting future opportunity – also pursued at the MUSE facility at
J-PARC [316] – is the generation of muon microbeams by re-acceleration of the
small phase space of the moderated muon beams to hundreds of keV or even MeV.
The moderated muons from muCool might be better suited for this purpose
compared to the moderation in a solid argon or neon layer due to the smaller
phase space of muCool. However, the extraction of muons from the muCool
apparatus and routine operation at a large phase space surface muon beam still
need to be demonstrated. A re-accelerated beam of moderated muons would have a
much smaller phase space compared to the existing sub-surface and surface muon
beams. This would allow focusing the beam on spot sizes of $<1$ mm2, tunable
implantation depths from hundreds of nanometre to the $\mu$m range, and the
realisation of a transmission muon microscope [317] – a novel extension of
muon applications to material science. Obviously, this is a further route to
completely new muon applications offered by HIMB.

An upgrade of the $\mu$E4 beamline is currently being studied, where the use
of solenoids instead of quadrupole magnets can yield an improved transmission
and focusing on the moderator target. With this upgrade, an increase of about
a factor of two of muons on moderator [318] can be expected. However, for full
exploitation of the capabilities of LEmu experiments, LEmu rates of
$>10^{5}$/s are highly desirable.

####  3.3.5 Elemental analysis and muSR

The use of negative muons for non-destructive, elemental analysis by the
detection of element-specific muonic X-rays has been tested some decades ago
[319, 320], while its routine use has been initiated since 2010 at the pulsed
muon sources at J-PARC [321] and ISIS [322]. A main drawback of these studies
at pulsed muon sources to date is the use of HPGe detectors, which have long
dead times of 5 - 10 $\mu$s. This means, that per muon pulse of width 50 - 100
ns, on average only one muonic X-ray can be detected, resulting in an event
rate determined by the repetition rate of the muon facility (50 Hz at ISIS,
and 25 Hz at J-PARC). This pile-up problem does not exist at a continuous muon
beam with a rate of even 100 kHz, where the average distance in time between
two muons is 10 $\mu$s. A feasibility study at PSI in 2018 has demonstrated
(for the setup, see Figure 27, that at the existing $\pi$E1 beamline with
$\mu^{-}$ rates of about 30 kHz and beam momentum of $\sim 30$ MeV/$c$, the
statistics can be increased by more than three orders of magnitude compared to
the pulsed muon beam facilities. In addition, active collimation is possible
at a continuous muon beam leading to a significant reduction of background.
This enables faster measurements with much better signal-to-background ratio,
higher sensitivity to low-concentration elements, and the resolution of fine
structure splittings and a much better separation of adjacent muonic X-ray
lines from different elements.

HIMB has the potential to lead to a significant boost of $\mu^{-}$
applications in elemental analysis:

  * •

With the higher rates, even faster measurements will be possible, and in-
operando measurements of devices such as Li-batteries will become feasible,
where the diffusion of Li ions through charging/discharging cycles can be
monitored as a function of depth.

  * •

Lower beam momenta $<10$ MeV/$c$ will become feasible, extending the range of
muon stopping depths from $\sim 1~{}\mu$m to $>1$ mm.

  * •

Due to the high rates, it will become possible to perform tomography, a very
exciting novel extension to image the distribution of elements inside bulky
samples and artefacts.

  * •

With HIMB, $\mu^{-}$ could potentially be used in “parasitic mode”: the
extraction of muons from the new production target H is made by solenoids,
transporting $\mu^{+}$ and $\mu^{-}$ in the same way. At the first bending
magnet, a ”parasitic beamline” can be installed to transport $\mu^{-}$ –
deflected by the opposite angle with respect to the $\mu^{+}$ – into a
separate beamline, where the momentum can be adjusted by a second bending
magnet. In this way, a dedicated experimental area for $\mu^{-}$ research
would become available without the need to share the beamtime with one of the
overbooked $\mu^{+}$SR instruments.

With higher intensity $\mu^{-}$ beams, a new kind of $\mu^{-}$SR experiments
will become possible: ”X-ray triggered $\mu^{-}$SR” with routing of histograms
for individual elements, i.e. ”element-specific” $\mu^{-}$SR. This will open
new possibilities to measure element specific diffusion rates and local
magnetic fields at element specific lattice sites.

##  4 The HIMB project & related beam and detector developments

###  4.1 The HIMB project

Figure 34: Layout of the new solenoid beamlines in the experimental hall of
PSI. The new 20-mm long target TgH is located at the same place as the
previous 5-mm long target TgM. The two new beamlines MUH2 and MUH3 replace the
two existing beamlines $\pi$M1 and $\pi$M3.

The HIMB project aims at delivering $\mathcal{O}(10^{10})$ surface muons per
second (positive muons stemming from pion decays at rest close to the surface
of the production target [299]) to experiments. To do so, the existing TgM
station located in the experimental hall at PSI on the high-energy proton
beamline of the HIPA accelerator and its connected beamlines $\pi$M1 and
$\pi$M3 will be completely rebuilt. Figure 34 shows how the layout of the new
target and beamlines will look like. While the existing graphite target
features a length of $5\text{\,}\mathrm{m}\mathrm{m}$, this will be increased
to $20\text{\,}\mathrm{m}\mathrm{m}$ in order to achieve the required rate.
Additionally, the target is slanted at 10 degrees with respect to the proton
beam such that the overall length of the graphite slab can be increased to
$100\text{\,}\mathrm{m}\mathrm{m}$, while keeping the effective length as seen
by the proton beam at $20\text{\,}\mathrm{m}\mathrm{m}$. This optimisation of
the target geometry increases the rate of emitted surface muons by about a
factor of two.

Figure 35: Cross-sectional view of the two capture solenoids together with the
new target TgH in the foreground. The black lines show trajectories of surface
muons emitted from the target.

The key to reach the intensity goal is to use capture solenoids with a 500-mm
aperture located only $250\text{\,}\mathrm{m}\mathrm{m}$ away from the target.
Figure 35 shows a cross-section through the two capture solenoids overlaid by
the trajectories of surface muons emitted from the target. Already from this
picture alone it is clear that a large fraction of the emitted surface muons
can be captured by this arrangement and transported further. The capture
solenoids will be radiation-hard and are based on the design of two existing
solenoids built for the $\mu$E4 beamline at PSI [208]. In order to transport
the large phase space accepted by the capture solenoids the rest of the
beamline also relies on solenoids for focusing combined with large-aperture
dipoles. Overall a combined capture and transport efficiency of around 10% is
achieved compared to the typical efficiencies of much less than 1% found in
the other secondary beamlines of PSI.

HIMB is embedded in the larger IMPACT project that combines two major upgrades
of the PSI facilities: HIMB and TATTOOS [323]. The IMPACT project team
consisting of people from the PSI divisions NUM (Research with Neutrons and
Muons), NES (Nuclear Energy and Safety), BIO (Biology and Chemistry), GFA
(Large Research Facilities), and LOG (Logistics) is currently preparing its
Conceptual Design Report to be completed by the end of 2021 with the aim of
having the project accepted to the 2023 Swiss Roadmap for Research
Infrastructures. If accepted, its implementation would occur during the period
2025-2028 with regular user operations starting in 2029.

Based on the transported phase space, two scenarios for the beam spot at the
final focus were formulated for the HIMB science case workshop depending on
the final divergence achievable at the end of the beamline.

HIMB-5:

  * •

$10^{10}$ $\mu^{+}$/s at 28 MeV/$c$ and $\sim$10% momentum-bite (FWHM)

  * •

beam spot $\sigma_{x,y}\sim 50$ mm

HIMB-3:

  * •

$10^{10}$ $\mu^{+}$/s at 28 MeV/$c$ and $\sim$10% momentum-bite (FWHM)

  * •

beam spot $\sigma_{x,y}\sim 30$ mm

These scenarios and especially the achievable rate of $10^{10}$ $\mu^{+}$/s
formed the basis for the different science cases developed for the workshop
and described in this document.

Figure 36: Momentum spectra of positive and negative particles registered at
the entrance of the capture solenoid.

Of course the HIMB beamlines do not only transport surface muons, but also
other particles such as negative muons, electrons, positrons, and pions.
Figure 36 shows the momentum spectrum of all these particles at the entrance
to the capture solenoid. The beamlines are designed for a good transport
efficiency up to about 40 MeV/$c$ with the dipoles capable of reaching 80
MeV/$c$. This allows, e.g., the use of pion beams for calibration purposes
albeit not at the highest intensities.

At 28 MeV/$c$ the rates at the end of the beamline are expected to be $2\times
10^{8}$/s for negative muons, $2\times 10^{10}$/s for positrons, and $7\times
10^{9}$/s for electrons.

One interesting suggestion raised during the workshop was the notion to study
the option of creating a parasitic $\mu^{-}$ beam by splitting at the first
dipole of the channel the positive and negative muons and thus creating the
possibility to simultaneously run experiments with positive and negative muons
in two legs of the beamline (see also Section 3.3.5). The feasibility of such
an approach is currently under investigation.

###  4.2 muCool beam development

Experiments with muons at the high-intensity frontier, as experiments
searching for lepton flavour violation (see Section 2.1), typically make use
of secondary beam lines with large acceptance tuned to transport surface muons
with momentum $p=28$ MeV/$c$ (equivalent to 4 MeV kinetic energy). These muons
are copiously created by $\pi^{+}$-decay from pions stopping close to the
surface of the pion production target (proton target). Muons of lower momenta,
from $\pi^{+}$ decaying below the target surface, can also be extracted and
transported by tuning the secondary beam lines for the corresponding momentum.
However, because of the momentum straggling in the target, the intensity of
these sub-surface muon beams decreases rapidly with momentum
($p^{3.5}$-dependence [299]). The muon scattering in the production target,
combined with the large area of the production target and the large acceptance
of the secondary beamline, results in muon beams with poor phase space
quality. For example the $\mu$E4 beam line which has presently the largest
muon flux of about $5\times 10^{8}\;\mu^{+}$/s at the moderator is having
$\sigma_{x,y}\approx 20$ mm, $\theta_{x,y}\approx 150$ mrad at a momentum of
28 MeV/$c$ and a momentum spread $\Delta p/p\approx 7$% (at FWHM) [208]. Other
beamlines have slightly smaller phase space at the cost of muon flux [324,
277].

The muCool project aims to transform these beams of 28 MeV/$c$ momentum to a
beam with momentum of about 1.5 MeV/$c$ (10 keV kinetic energy) with 0.1 keV
energy spread and a phase space of $\sigma_{x}\sigma_{\theta_{x}}=20$ mm mrad
with an efficiency ranging from $2\cdot 10^{-5}$ to $2\cdot 10^{-4}$. Because
such a beam can be easily focused to a sub-mm size it is well suited for
precision experiments requiring a target with small transverse area and small
stopping power or requiring muons to be stopped close to the surface.
Moreover, such a beam is well suited for re-acceleration: beams from few keV
to 100 MeV energy can be thus obtained by combining the muCool device with a
suited acceleration stage opening various possibilities including storage
experiments such as the muon EDM and the muon $g-2$ where a small phase space
is required.

#### The proposed muCool compression scheme

In the proposed muCool scheme [325], a surface muon beam propagating in the
$-z$-direction is slowed down in a He gas target featuring a strong electric
($E$) field inside a strong magnetic ($B$) field as shown in Figure 37. In the
slowing-down process, the muon energy is rapidly reduced to the eV range where
the E-field becomes important. The E-field, in conjunction with the B-field
and gas density gradients, leads to drifting of the slowed-down muons
drastically reducing their initially large spatial extent. In this drift
process in the gas, the muons are guided to a mm-sized spot.

Figure 37:  Schematic diagram of the muCool device. A surface muon beam is
stopped in a cryogenic He gas target with a vertical temperature gradient
inside a 5-T field. The extent of the stopped muons is reduced first in the
transverse ($y$), then in the longitudinal ($z$) direction using a complex
arrangement of E-field and gas density gradient. The compressed muon beam is
then extracted through an orifice into vacuum and re-accelerated along the
$-z$-direction.

The drift velocity of the $\mu^{+}$ in a gas with E- and B-fields is given by
[326]

$\displaystyle\vec{v}_{D}=\frac{\mu|\vec{E}|}{1+\omega^{2}/\nu^{2}}\left[\hat{E}+\frac{\omega}{\nu}\hat{E}\times\hat{B}+\frac{\omega^{2}}{\nu^{2}}\left(\hat{E}\cdot\hat{B}\right)\hat{B}\right].$
(4.1)

In (4.1) $\mu$ is the muon mobility, $\omega=eB/m$ the cyclotron frequency of
the muon, $\nu$ the average $\mu^{+}-$He collision rate, and $\hat{E}$ and
$\hat{B}$ the unit vectors of the electric and magnetic fields, respectively.

The spatial extent of the muon stop distribution decreases by making
$\vec{v}_{D}$ position-dependent, so that $\mu^{+}$ stopped at different
locations in the target drift in different directions, and converge to a small
spot. This can be achieved by applying a complex E-field pointing in different
directions at different positions, and by making the collision frequency $\nu$
position-dependent through gas density gradients.

The muCool setup has been conceived as a sequence of stages having various
density and electric field conditions. In the first stage, which is at
cryogenic temperatures, the muon beam is stopped and compressed in
$y$-direction (transverse compression). In the second stage, which is at room
temperature, the muon beam is compressed in $z$-direction (longitudinal
compression). In the third stage, the muons are extracted from the gas target
into vacuum. This extraction is followed by re-acceleration in $-z$-direction
to keV energy, and extraction from the B-field.

#### Demonstration of transverse, longitudinal and mixed compressions

The technology and principle at the core of the muCool cooling scheme, i.e.,
the ability to move the muon in the He gas with B-field and complex
arrangements of E-field and gas density, has been validated in various
experiments [327, 308, 328]. The time evolution of the number of decay-
positrons hitting the plastic scintillators placed around the targets were
used in these experiments to reveal the average motion of the muon ensemble
with increasing (decreasing) number of counts indicating muons approaching
(leaving) the acceptance region of the considered scintillator.

Figure 38:  (Left) Sketch of the target used to test the transverse
compression at cryogenic temperature with temperature gradient. (Middle)
Geant4 simulation of muon trajectories starting at $x\approx-15$ mm and
drifting with time in $+x$-direction while compressing in the $y$-direction.
The approximate positions of two plastic scintillators (red, blue) used to
measure decay positrons are indicated. (Right) Measured and simulated time
spectra for the two plastic scintillators of middle panel. The time zero is
given by a counter detecting the muon entering the target. The counts are
lifetime compensated.

Figure 39:  (Left) Sketch of the setup used to test the longitudinal
compression at room temperature. The scintillators T1 and T2 in coincidence
detect the $\mu^{+}$ accumulating around $z\approx 0$. The blue curve
indicates the region of acceptance for coincident events. (Middle) Simulated
$\mu^{+}$ trajectories. (Right) Measured and simulated time spectra for
various HV at the target mid plane: negative HV (red) attracting the muons to
the central plane, positive HV (green) repelling them and no HV (black).

Figure 40:  (Left) Simulated muon trajectories in the mixed transverse-
longitudinal compression target with a vertical density gradient at cryogenic
temperatures, $E_{x}$ and $E_{y}$ components as in the transverse compression
target, and an $E_{z}$ component pointing to the target mid-plane at $z=0$.
The positions of the plastic scintillators pairs (A,B,C) used to expose the
compression are also shown. (Right) Measured time spectra in the scintillator
pairs in coincidence.

Figure 38 shows a schematic of the setup, a simulation of muon trajectories in
the target, and two measurements of the transverse compression [328]. Good
agreement between simulated and measured time spectra was observed confirming
the validity of our simulation tools and target technology. Similarly Figure
40 demonstrates longitudinal compression.

To avoid the challenging connection between transverse (at cryogenic T) and
longitudinal (at room T) stages, a target has been developed and commissioned
in which both transverse and longitudinal compressions occur simultaneously
(see Figure 40) [329]. Data analysis is still ongoing, but the measured time
spectra indicate that a muon stop distribution with volume $\Delta
x\times\Delta y\times\Delta z=10\times 10\times 60$ mm3 can be transformed in
5 $\mu$s into a beam drifting in $x$-direction with 10 eV kinetic energy
capable of passing an aperture of $\Delta y\times\Delta z=1\times 1$ mm2 size
with an efficiency of about 50% (excluding muon decay losses).

#### Vacuum extraction and re-acceleration

A possible extraction scheme that we are presently simulating is shown in
Figure 41. It is based on a mixed-compression target modified to allow
$\mu^{+}$ extraction from the gas target into “vacuum” (low density gas
region) through an orifice of $1\times 1$ mm2 aperture. To compensate for the
He atoms leaving the target through the same orifice, new He gas is
continuously injected at the orifice perpendicular to the $\mu^{+}$ motion
which acts as a gas barrier for the target gas. The injected gas needs to be
efficiently evacuated through a system of differentially pumped regions to
maintain a good beam quality while the muons are drifted in $+x$-direction
using suited electrodes. These muons are eventually coupled into the re-
acceleration stage where acceleration in $-z$-direction using ring-shaped
electrodes biased at decreasing potentials is occurring.

Figure 41:  3D-rendering of the mixed-compression target modified to include
an orifice, a He gas inlet, strips of electrodes for muon extraction and drift
in $+x$-direction and ring-electrodes for re-acceleration in $-z$-direction.
In blue the sapphire plates are shown defining the temperature of the top and
bottom target walls, in orange the electrodes defining the electric fields,
and in grey the plastic frame of the target.

#### Projected performance of the complete muCool beam

This section briefly discusses the estimated performance of the HIMB-muCool
scheme assuming a muCool target with a single active region for mixed-
compression with a performance based on the commissioned mixed-compression
target. The efficiencies of the various stages are also summarised in Table 8.

  * •

Coupling the beam into the solenoid
The HIMB beam has an average momentum of 27.7 MeV/$c$, a large transverse
phase space $\sigma_{x}\sigma_{\theta_{x}}=1910$ mm mrad and a large momentum
spread of 3.5 MeV/$c$ (FWHM) corresponding to 12.6% momentum bite. Geant4
simulations show that 36% of the muons are back-reflected when injecting on-
axis the HIMB beam into a 5 T solenoid of 60 mm inner coil diameter. This
fraction becomes 44% for $14^{\circ}$ tilt between solenoid and beamline axis
to inject the muons 50 mm off the solenoid axis.

  * •

Coupling the beam into the target
Geant4 simulations show that the fringe field of a 5 T solenoid with 60 cm
inner-coil diameter focuses the HIMB beam to
$\sigma_{x}\approx\sigma_{y}\approx 11$ mm. Hence, only a fraction of 48% of
the muons entering the solenoid would impinge on the entrance face (end-cap)
of the already commissioned mixed-compression target. For a mixed target with
a three times larger side wall (but same target extent in $x$-direction) this
fraction increases to 83%. The feasibility to realise this larger target with
the adequate electric field has still to be demonstrated by R&D.

  * •

Stopping in the active region of the gas target
The large momentum spread and transverse momentum distribution of the HIMB
beam, in conjunction with the short length of the active region and low
density of the target leads to an overall small stopping probability in the
region of interest. Geant4 simulations predict that with an optimised
moderation at the target entrance, 0.41% of the muons impinging on the
entrance face (end-cap) of the target eventually stop in the target’s active
region. Here we assumed a 50 mm long active region as realised in the
commissioned mixed-compression target with a pressure of 10 mbar and a
temperature gradient from 6 K to 22 K. We should be able to improve this
stopping probability by at least a factor of 1.7 by operating the target at
colder temperatures (4 to 16 K gradient) and by elongating the extent of the
active region from 50 mm to 60 mm while keeping the same compression
efficiency. Moreover, we are presently evaluating a possible reduction of the
momentum spread by introducing some dispersion in the HIMB beam in conjunction
with a position-dependent moderation.

  * •

Compression and drift to the orifice region
Preliminary analysis of data taken with the commissioned mixed-compression
target demonstrates that muons drift and compress with an average kinetic
energy smaller than 10 eV from the stopping region of 50 mm length and 10 mm
height to the orifice region with an efficiency of 80% in about 5 $\mu$s. When
including lifetime losses this efficiency is reduced to 8%.

  * •

Extraction from the gas target through the orifice
From preliminary electric fields layouts, Geant4 simulations predict that the
compressed muon beam can be extracted from the target with an efficiency of
about 50% (neglecting lifetime losses) in about 0.7 $\mu$s though an orifice
with $1$ mm2 aperture. Including lifetime-losses, this probability becomes
thus 40%.

  * •

Coupling to the re-acceleration region
Preliminary simulations of the muon motion from the orifice to the first
accelerating ring (where the electrostatic re-acceleration is starting) show
that a transport with 90% efficiency (without decay-losses) in 0.5 $\mu$s can
be obtained. This efficiency is thus about 70%, when accounting for the muon
lifetime. The performance of this transport strongly depends on the electric
field that can be applied, and the distance muons have to travel before a gas
density suited for re-acceleration is reached. Note that muons are injected in
the first ring with transverse energies of $\mathcal{O}$(20 eV) and sub-mm
size.

  * •

Re-acceleration to 10 keV energy
In the static re-acceleration process guided by a series of ring electrodes at
decreasing (positive) HV, the only losses are the ones related to the decay
during the time of flight. They have been estimated to be about 20%. Some
degradation of the beam quality might occur in this stage due to collisions
with the rest gas (to be investigated).

  * •

Extraction from the B-field
The extraction of the 10 keV muons from the solenoid is obtained by
transporting the muons from the central region of the solenoid at 5 T to a
region of 0.01 T field where a grid with 70% transmission terminates abruptly
the magnetic field lines. In this transport from high-field to low-field, the
transverse beam energy is decreased by a factor of 500 while the beam radius
increases by $\sqrt{500}$. At the grid the abrupt termination of the B-field
produces a radial field component that impart an azimuthal momentum to the
muons of about 3 keV/c (assuming a grid with 2 mm wide apertures) resulting
effectively in a beam of 2 mrad divergence. Hence the beam is extracted into a
field-free region with a phase space of $\sigma_{x}\sigma_{\theta_{x}}\approx
20$ mm mrad (unnormalised) at 10 keV.

Table 8: The first column shows estimated baseline efficiencies of the various muCool stages using the commissioned mixed-compression target as a reference point. We assume here a target with only a single active region (10 mbar, 6-22 K, 50 mm long). The second column shows the room for possible improvements after dedicated R&D, still assuming a single active compression region. All numbers include muon-decay losses.  Baseline | Possible | Description
---|---|---
Efficiency | Improvements |
$5.6\cdot 10^{-1}$ |  | Coupling to the 5 T solenoid with 60 mm coil diameter
$4.8\cdot 10^{-1}$ | $\times 2$ | Impinging on the target entrance-face
$4.1\cdot 10^{-3}$ | $\times 1.6$ | Stopping probability in active region of the target
$8\cdot 10^{-2}$ | $\times 1.5$ |  Compression towards the orifice (within 5 $\mu$s)
$4\cdot 10^{-1}$ | $\times 1.3$ | Extraction from the orifice
$7\cdot 10^{-1}$ |  |  Drift from orifice to re-acceleration region (in $\sim 0.5$ $\mu$s)
$8\cdot 10^{-1}$ |  | Re-acceleration and transport to the iron grid
$7\cdot 10^{-1}$ |  | Transmission through the iron grid terminating the B-field
$1.4\cdot 10^{-5}$ | $\times 6$ | Total baseline compression efficiency (and possible improvement)

As can be seen from Table 8, which summarises the baseline efficiencies of the
various muCool stages, a total baseline compression efficiency of $1.4\cdot
10^{-5}$ is expected from the muCool setup applied to the HIMB. Still assuming
a target with a single active region, we expect some room for improving this
efficiency with dedicated R&D up to a value of $1\cdot 10^{-4}$. Moreover,
because the stopping probability in the active region of the commissioned
mixed-compression target is smaller than 1%, a long muCool target could be
realised having multiple active regions in series along the $z$-direction,
each of them with its own extraction orifice. With some degradation of the 6D
phase space, the various outputs can be combined to produce a single more
intense muon beam. Alternatively, each of the outputs can be extracted
individually from the magnet and sent to various experiments operating
simultaneously. The number of active regions in a muCool target in principle
can be as high as 100, but practically the required vacuum quality in the
acceleration stage will probably limit their number to a few.

In summary, the muCool target transforms the HIMB input beam of 4 MeV energy,
with 1 MeV energy spread (at FWHM), and $\sigma_{x}\sigma_{\theta_{x}}=1910$
mm mrad (unnormalised) into a beam of 10 keV with $<0.1$ keV energy spread (at
FWHM) and $\sigma_{x}\sigma_{\theta_{x}}\approx 20$ mm mrad (unnormalised)
with an efficiency ranging from $2\cdot 10^{-5}$ to $1\cdot 10^{-4}$ assuming
a target with a single active region. Such a beam can be focused to sub-mm
sized targets and after re-acceleration is well suited for storage-ring
experiments. Its efficiency can be further improved by implementing multiple
active regions in the same elongated target.

#### Post-acceleration from 1 to 60 MeV energy

The muon beam delivered by the muCool setup can be further accelerated to the
energy appropriate for experiments such as the muon EDM and the muon $g-2$
measurements, where the required muon beam energy ranges from around 1 MeV to
60 MeV or higher. One of the most important parameters of the muon accelerator
is the transmission rate: the path length should be minimised because muons
are quickly decaying. Circular accelerators therefore may not be optimal, at
least in our energy range, due to the fact that the length allocated for
acceleration is quite short in comparison to the total path length.
Conversely, the linear accelerator (linac) is efficient in this aspect and
thus is our baseline choice. As described above, the initial acceleration up
to several tens of keV energy will be accomplished simply by setting the
ground of the muCool target at positive HV so that the muons leaving the
muCool setup are at a kinetic energy basically given by this HV. This
electrostatic pre-acceleration provides muons to the linac at a proper energy
in a continuous way. For some experiments it may be more convenient to have an
extraction in bunches containing several muons. Accumulating the muons for
several microseconds before letting them enter the re-acceleration stage is
feasible, but at the cost of a largely reduced muon flux due to the 2 $\mu$s
muon lifetime.

In any case, we design the downstream linacs to be compatible with CW
operation as well as a pulsed operation. Among various types of linacs, an RFQ
would be well suited to take over the muon beam after the initial acceleration
because an RFQ acts like an “all-in-one” cavity, which focuses, bunches and
accelerates the beam. An acceleration of negative muonium atoms using an RFQ
has been demonstrated in [330]. Our preliminary investigation arrived at a
parameter set shown in Table 9. It is proposed to divide the RFQ into two
parts, one for capturing and bunching, and the other for the acceleration.
This increases flexibility in parameter optimisation drastically.

Two scenarios with a 352-MHz RF system and a 750-MHz RF system are currently
considered. The higher frequency system offers a shorter accelerator at the
expense of a smaller acceptance and a slightly higher RF power consumption.
The transmission efficiency can be sufficiently high for the estimated
emittance of 0.55 mm mrad rms (normalised). The muon flux degradation is about
10% for an accelerator length of 5 m.

Table 9: RFQ main parameters for two RF frequencies. The normalised emittance corresponds to a geometrical emittance of 40 mm mrad at 10 keV. The intermediate energy is the kinetic energy of the muon beam at the end of the capture section.  Parameters | Values for | Values for
---|---|---
| RF=352 MHz | RF=750 MHz
Input emittance (rms, normalised) | 0.55 mm mrad | 0.55 mm mrad
Input energy, kinetic | 20 keV | 20 keV
Intermediate energy, kinetic | 170 keV | 190 keV
Output energy, kinetic | 2 MeV | 2 MeV
Output energy spread | 0.75% | 0.55%
Capture section voltage, vane-to-vane | 21 kV | 28 kV
Acceleration section voltage, vane-to-vane | 65 kV | 50 kV
Total length (Capturing/bunching + Accel.) | 2.4 + 3.0 m | 2.0 + 2.4 m
Beam power | Negligible | Negligible
Power consumption | 300 kW (CW) | 340 kW (CW)
Transmission efficiency $\mu$-decay not included | 96% | 94%

A rough estimation of the costs to build such an accelerator is 250k CHF/m for
the accelerating structure and 2.3 million CHF for the 352-MHz RF system. The
cost of the RF system includes estimates for a 300-kW solid state power
amplifier, waveguides, cooling infrastructure, a local signal source with
LLRF-and interlock-system, support structures and cabling. Not included are
shielding, manpower, maintenance and running cost. The 750-MHz RF system would
be as expensive as the 352-MHz RF system.

In case where the above estimated cost is prohibitive, we may consider to
reuse the 500-MHz system of the SLS which is currently in operation but will
be available after the ongoing SLS upgrade provided that the performance of
the RFQ is similar at 500 MHz. We need to clarify if this option is cost-
effective: how long it can keep up and how much it costs to be refurbished if
necessary. Another option is to reduce the duty factor from CW (100%) to a
lower value that can be accepted by the experiments. Obviously, this will save
running costs, and we also expect a lower cost for the accelerating structure
since the cooling part can be simplified.

The power dissipation is estimated for CW operation and fully dominated by the
RF wall loss since the muon beam power is negligible. A consumption of several
hundred kW is feasible as already demonstrated in high power proton linacs
[331]. It is also possible to utilise a superconducting cavity, which offers a
highly energy-efficient operation at the expense of higher initial
construction cost and increased engineering complexity.

The obtained parameters are for the final energy of 2 MeV. Although it is hard
to change the output energy once we built the RFQ, it is straightforward at
the design stage. We simply decrease(increase) the length of the second part
of the RFQ cavity if the desired beam energy is lower(higher) than 2 MeV. The
final energy is a key parameter driving the overall costs.

The beam is further accelerated for the higher energy applications. The
Lorentz beta is still $\ll 1$ at the exit of the RFQ, hence a drift tube linac
may be employed. The accelerator chain proposed here is indeed similar to an
established layout of proton linac facilities, see e.g. [332]. Other options
are, however, to be considered before finalising the accelerator design.

#### Selected applications for muCool beams (with or without re-acceleration):

This new beam opens the way for next generation experiments with muons and
muonium where the reduced phase-space is of great advantage spanning
applications in fundamental particle physics, atomic physics and materials
science through the $\mu$SR technique.

##### Materials science.

The muCool beam can greatly benefit $\mu$SR investigations by delivering muons
simultaneously to several $\mu$SR instruments at the maximum rate allowed by
pile-up effects with energy tunability and sub-mm size. Because the pile-up
effects in the typically 10 $\mu$s-long observation time window become
increasingly unsustainable for rates exceeding $4\cdot 10^{4}$ $\mu/$s
(without vertexing, see Section 3.3), the full HIMB-muCool potential could be
exploited by distributing the keV-energy sub-mm beam between several $\mu$SR
instruments operating simultaneously.

Even assuming the baseline efficiency of $1.4\cdot 10^{-5}$, the muCool setup
injected with $10^{10}\mu$/s from HIMB and provided with an appropriate muon
distribution system would be able to deliver muons simultaneously to several
$\mu$SR instruments with keV energies similar to the present LEM beamline
[208] (best performing low-energy beam-line world-wide) but with a sub-mm
transverse size instead of the presently cm-size. The number of beams that
could be extracted from the muCool setup can be further increased by improving
the overall efficiency of the muCool setup and by realising a long muCool
target having multiple active regions in series along the $z$-direction, each
of them with its own extraction orifice. Owing to the limited rate acceptance
of the $\mu$SR instruments (for cw beams), there is no need to merge the beams
exiting the various orifices. Each of these beams could be extracted from the
solenoid at slightly different off-axis positions with negligible distortion
of their beam quality (relative to on-axis extraction).

Hence, the HIMB-muCool scheme would be able to deliver numerous beams, with
their number limited only by the technical capability of realising the needed
vacuum requirements at the re-acceleration stages. Each of these beam could be
easily transported to a compact $\mu$SR instrument in a background-free
environment capable of analysing sub-mm samples at 50 kHz rate and keV
energies. Pile-up effects could be further reduced because the sub-mm low-
energy beams can be easily “kicked away” to prevent “second muons” entering
the setup in the measurement time window of the “first muon”. Combined with
high-rates capability, this feature allows the implementation of a muon-on-
demand scheme that minimises the dead time of the $\mu$SR spectrometer: as
soon as a decay-positron is detected in the spectrometer the next muon is
allowed to enter.

The small phase space of the muCool beam is also well suited for efficient re-
acceleration to higher energies as discussed previously. Among others, the
energy range between 30 keV to a few hundred keV presently not accessible at
PSI could be covered.

To reduce the number of $\mu$SR instruments, it would also be possible to
focus several of the extracted beams from the same muCool target into various
samples (or at various locations of a larger sample) mounted in the same
$\mu$SR spectrometer. Using a positron tracking system (as described in
Section 3) it would then be possible to distinguish uniquely the muon decays
originating from the various samples in order to disentangle the corresponding
time spectra.

Summarising, the HIMB-muCool scheme would improve by an order of magnitude the
sample throughput of the PSI $\mu$SR facility especially in the keV-energy
regime because the muCool device could deliver multiple low-energy beams with
50 kHz rates clean of “second muons”. At the same time the transverse beam
size would be reduced by more than two orders of magnitude, tremendously
benefiting investigations of small samples (see Section 3). Moreover, it also
allows muon re-acceleration to energies that are presently not accessible.

##### High quality muonium sources.

Muon to vacuum-muonium conversion is very efficient for keV-energy muons
[195]. Hence, the sub-mm muCool beam at keV-energy could be converted into a
high-brightness muonium source. This novel muonium source could be exploited
to vastly improve on the precision of muonium spectroscopy [209], and for the
investigation of the gravitational interaction of muonium [333].

As detailed in Section 2.5, the HIMB-muCool beam opens the way for improving
by an order of magnitude the ongoing spectroscopy measurements in muonium:
below 1 kHz for the 1S-2S transition and down to few Hz for the HFS. The high
muonium rates from the HIMB-muCool will allow accurate studies of systematic
uncertainties, the implementation of novel experimental schemes less prone to
it, and laser spectroscopy of other transitions.

The combination of the various transition measurements in muonium will lead
to: stringent tests of bound-state QED (purely leptonic systems), the best
determination of fundamental constants such as the muon mass and the muon
magnetic moment, an independent determination of the muon $g-2$ with
uncertainty comparable with the present discrepancy, and searches for possible
new (beyond the standard model) muon-electron couplings.

As detailed in Section 2.5, the HIMB-muCool beam will also greatly benefit the
measurement of the muonium acceleration in the Earth’s gravitational field, as
a test of the equivalence principle for anti-matter and second-generation
particles.

Note in conclusion that several output beams delivered by the muCool setup
could be merged on the same spot of a muon-to-muonium converter yielding a
muonium source with a flux of several $10^{5}\,$muonium/s emitted from a 1 mm
spot size.

##### Storage ring experiments.

As detailed in Section 2.4, the search for a muon EDM and the precise
measurement of the muon $g-2$ represent well motivated channels for physics
beyond the Standard Model. The currently ongoing experiment at Fermilab aims
at a measurement of $g-2$ with statistical uncertainty of 0.1 ppm and a
similar uncertainty from systematics [189]. With the same apparatus, searches
of the muon EDM with a final sensitivity of $1\cdot 10^{-21}$ e cm will be
accomplished. A second collaboration at J-PARC also aims at the same
quantities with a combined (statistics + systematic) precision of 0.45 ppm for
the $g-2$, and a sensitivity of $1\cdot 10^{-21}$ e cm for the muon EDM.

The muonEDM collaboration at PSI proposes (see Section 2.4) a search for the
muon EDM based on the frozen-spin technique applied to a compact muon storage
ring [193]. Preliminary studies show that a sensitivity of $6\cdot 10^{-23}$ e
cm could be reached in the PSI experiment using the $\mu$E1 beam at 125
MeV/$c$ delivering $2\cdot 10^{8}$ $\mu^{+}$/s with an average polarisation of
better than 93% and assuming a B-field of 3 T. Because of the small phase
space acceptance of the storage ring, the coupling efficiency for the $\mu$E1
beam is only $2.5\cdot 10^{-4}$ so that only $5\cdot 10^{4}$ $\mu^{+}$/s are
eventually stored in orbit.

The sensitivity to the muon EDM is given by (see Section 2.4)

$\sigma(d_{\mu})=\frac{\hbar}{2\beta\gamma cBP\sqrt{N}\alpha\tau}$ (4.2)

where $B$ is the field strength, $P$ the muon polarisation, $N$ the number of
stored muons, $\alpha$ the average decay-asymmetry, $\tau$ the muon lifetime,
and can be improved with the muCool beam for two reasons: Firstly, owing to
its small phase-space the muCool beam can be efficiently coupled into the
storage ring resulting in a larger rate of stored muons $N$. Secondly, owing
again to its small phase-space the muCool beam can be efficiently accelerated
to larger momenta to increase $\gamma$ and $\beta$. With larger $\gamma$ and
$\beta$ also the B-field can be increased to keep the optimal radius of
curvature. For a storage ring with 5 T field the optimal momentum is around
190 MeV/$c$.

A muCool target performing approximately at the baseline efficiency followed
by a post-acceleration stage would deliver muons with a rate of about
$1.4\cdot 10^{5}$ s-1 so that the rate of muons injected in orbit is increased
by a factor of 3 compared to the experiment at the $\mu$E1 beamline. R&D on
the muCool target has the potential to significantly improve the muon rate
delivered to the storage ring. Yet, using a cw beam, i.e., in the scenario
where a single muon at a time is analysed in the storage ring, it is
essentially impossible to improve the sensitivity beyond $1\cdot 10^{-23}$ e
cm per year of data taking (here we assumed 5 T field and 190 MeV/$c$
momentum). This limitation is originating from pile-up effects.

A small improvement can be obtained by bunching the muCool beam: 5 muons per
bunch with 100 kHz repetition rate would result for example in a sensitivity
of $8\cdot 10^{-24}$ e cm per year of measurement. While formation of these
bunches is technically possible, the muon-decay losses over the needed
accumulation period are severe due to the 2 $\mu$s lifetime of the muon. These
losses can only be compensated by realising a muCool target with several
active regions and by recombining the various outputs at the cost of some
energy spread of $\mathcal{O}$(1 keV).

The same setup used to search for the muon EDM can be used, with minor
modifications, to measure the muon $g-2$. As detailed in Section 2.4 with a
field strength of 6 T and muon on request at 125 MeV/$c$ (that requires in
concrete terms a cw beam delivering about $5\cdot 10^{5}\mu$/s) a statistical
sensitivity of 0.1 ppm can be reached similar to the ongoing Fermilab
experiment. Also in this case, bunching of the muCool beam could be used to
improve the measurement down to 0.06 ppm.

An alternative layout is being investigated in which the $g-2$ measurement is
performed by injecting muons of only 1 MeV energy into a 17 T solenoid that is
acting as mini storage ring. One of the advantages of this scheme is that it
requires only the RFQ-acceleration stage but not the linac, reducing size and
cost.

Hence, HIMB-muCool has the capability to contribute to two flagship quantities
of the present particle physics landscape with high potential for new physics.

In conclusion, the HIMB-muCool complex provides a beam (or multiple beams)
that push forward our capabilities where we are already world-leading such as
$\mu$SR, muonium spectroscopy and fundamental particle physics with muons,
while expanding our activities to include storage-ring and gravity
experiments.

###  4.3 Detector developments

To take full advantage of the increased rate of muons provided by HIMB, a new
generation of silicon pixel detectors is required. Silicon pixel detectors are
used in particle physics for the reconstruction of trajectories of charged
particles (tracking) and the identification of vertices from the decay of non-
stable particles. Tracking is based on pattern recognition and track fitting
and thus requires detectors with high granularity and excellent spatial
resolution.

PSI has a long-standing tradition in the design, development and construction
of silicon pixel detectors. These detectors have proven their functionality
for example in the CMS experiment at the CERN LHC [334, 335, 336] and in
applications in x-ray photon science at SLS and SwissFEL [337, 338, 339]. The
pixel detectors used in this context are built using the so-called hybrid
technology, in which the particle sensing element and the necessary readout
electronics are separate entities, connected using the bump-bonding technique
[340].

An alternative approach for pixel detectors is the monolithic technology,
where the sensor and readout electronics are parts of the same entity. While
CMOS based MAPS [341] are well known in particle physics, only recently
commercially available technologies enable depleted substrates and thus DMAPS
[31, 342]. The depleted substrate is mandatory for a radiation resistant and
fast signal generation and thanks to the full amplification stage and readout
architecture being implemented in the silicon sensor chip, much smaller active
thickness can be used as the smaller charge signal can be processed without
performance degradation. The material budget per layer is significantly
reduced as only a single silicon layer is needed and the reduced leakage
current due to the lower silicon volume results in relaxed cooling
requirements. This makes DMAPS the most attractive pixel detector concept for
future experiments.

In fact, the pixel detector for the Mu3e experiment is being constructed in an
HVMAPS technology [32], which allows a high voltage being applied to the
silicon substrate to achieve the depletion zone. The MuPix family of chips
that has been developed over the past few years [61] allows to build a
tracking detector with an unprecedented low-material design [26]. Low-material
detectors are key for the reconstruction of the trajectory of the decay
positrons to minimise the effect of multiple scattering on the spatial
resolution.

The MuPix chips have an active area of $20\times 20$ mm2 and are thinned down
to a thickness of 50$\,\mu$m. Using thin high-density interconnect circuits,
made of two layers of a substrate of polyimide (10$\,\mu$m) and aluminium
(12$\,\mu$m), a detector layer ends up with a radiation length of $\approx
1.15\times 10^{-3}X_{0}$. The pixel readout is triggerless and always-on,
i.e., every hit in a chip is read out with a time-stamp having a resolution of
15 ns or better. With a pixel pitch of 80$\,\mu$m, a position resolution of
23$\,\mu$m is achieved.

The next generation of pixel detectors will provide increased granularity in
space and time to distinguish particle locations and their interaction times,
which is necessary to cope with the high muon flux delivered by HIMB. The
evolution of these detector concepts in view of particle physics experiments
at HIMB are discussed in Section 2.2.4, Section 2.3.5, and Section 2.4.2.

Pixel detectors also enable new scientific approaches in muon experiments in
solid-state physics and materials science, in particular muSR. The spatial and
temporal resolution provided by the pixel detector allows for example the
measurement of multiple samples in parallel in the same exposure, the
separation of domains with different magnetic properties in custom samples as
well as to overcome the current limitation in muon rate (as discussed in
Section 3).

A concept for a prototype pixel detector for the general purpose surface muon
instrument GPS is shown in Figure 42. The pixel detector is built from two
concentric cylinders surrounding the cryostat in which the sample is placed.
The cylinder axis is placed perpendicular to the muon beam. The pixel detector
is used to measure the trajectory of the incoming muon and the outgoing
positron, and to reconstruct the muon decay vertex in the sample.

Figure 42: (Left) Drawing of a prototype pixel detector for the general
purpose surface muon instrument. The pixel detector is built from two
cylindrical layers surrounding the cryostat in which the samples is placed.
(Right) Drawing of a pixel detector module. The sensitive element is a DMAPS
chip of the MuPix family developed in the context of the Mu3e experiment.

Due to the low momentum of the muons ($<30$ MeV), the spatial resolution of
the detector will be dominated by multiple scattering effects and building the
detector from minimum material will be crucial. The prototype detector uses a
DMAPS chip of the MuPix family as a sensitive element and all components
needed for readout, power and cooling of the detector are placed outside the
active area. The prototype detector is built from 34 modules with a total
active area of 136 cm2. The radius of the inner and outer pixel layers are 21
mm and 31 mm, respectively. The detector acceptance determined for a sample of
100’000 simulated muons with an average momentum of 27 MeV is 56%.

The effect of multiple scattering on the resolution has been studied assuming
a sensor thickness of 50$\,\mu$m and was found to be about 0.65 mm at the muon
decay point. The possibility of thinning the sensors to thicknesses of less
than 50$\,\mu$m is currently being studied. With thinner sensors with a
thickness of 30$\,\mu$m the resolution can be improved to less than 0.50 mm.

##  5 Conclusions

Two new high-intensity muon beams HIMB at PSI with up to two orders of
magnitude higher and completely unprecedented intensities of surface muons
will boost muon science in many ways. Flavour physics is a hot topic in
particle physics, with charged lepton flavour searches and muon
electromagnetic moment measurements in particular. Such experiments at HIMB
will remain at the forefront of research for the next two decades or longer.
Other fundamental atomic, nuclear and particle physics experiments will
complement these flagship experiments, testing essentially all known
interactions with considerable discovery potential for physics beyond the
current Standard Model of particle physics. Likewise, in condensed matter
physics HIMB will open up completely new regions of experimental phase space,
for smaller sample sizes, for faster spatially and time-resolved measurements,
and for depth profiling in so far inaccessible regions. All experiments at
HIMB will considerably benefit from technological advances in detector
technology, target technology and sample environment. Application of novel
technologies, in particular of thin and fast tracking detectors and of new
muon beam cooling schemes will further push the boundaries of what is
presently possible in muon physics. The physics case for HIMB will continue to
develop over time and lead to a definition of an initial physics program at
this eagerly awaited new facility. The cases presented here already motivate a
very strong and unique program for the next two decades.

## Acknowledgements

The science case presented here is built on the work of numerous people over
many years and is a true community effort. It is a pleasure to thank all our
colleagues at PSI and the external users for their contributions and support.
A special thank you goes to all speakers and other participants of the HIMB
Science Case workshop of April 2021.

We gratefully acknowledge support by the Swiss National Science Foundation
(SNF) through the grants PCEFP2_181198, PCEFP2_181117, PP00P2_176884, 200441,
197052, 197346; by the European Research Council (ERC) through CoG. 725039,
818053; by the European Union’s Horizon 2020 research and innovation programme
through grant 858199; by the German Research Foundation (DFG) under Project WA
4157/1; and by the JSPS Core-to-Core Program, A. Advanced Research Networks
JPJSCCA20180004 and JSPS KAKENHI Grant Number JP21H04991 (Japan).

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