caption captionlabel 1em single PSI short = PSI, long = Paul Scherrer
Institut, long-indefinite = the MAPS short = MAPS, long = monolithic active
pixel sensor DMAPS short = D-MAPS, long = depleted monolithic active pixel
sensor HVMAPS short = HVMAPS, long = high-voltage monolithic active pixel
sensor HVCMOS short = HV-CMOS, long = high-voltage complementary metal-oxide-
semiconductor HDI short = HDI, long = high-density interconnect, long-
indefinite = a, short-indefinite = an EDM short = EDM, long = electric dipole
moment MDM short = MDM, long = magnetic dipole moment AMM short = AMM, long =
anomalous magnetic moment HVP short = HVP, long = hadronic vacuum polarisation
HLbL short = HLbL, long = hadronic light-by-light scattering BR short = BR,
long = branching ratio EFT short = EFT, long = effective field theory MC short
= MC, long = Monte-Carlo LFV short = LFV, long = lepton flavour violation MFV
short = MFV, long = minimal flavour violation cLFV short = cLFV, long =
charged lepton flavour violation HEP short = HEP, long = high energy physics
PCB short = PCB, long = printed circuit board SM short = SM, long = Standard
Model of particle physics BSM short = BSM, long = beyond the Standard Model
MSSM short = MSSM, long = minimal supersymmetric Standard Model LHC short =
LHC, long = Large Hadron Collider DAQ short = DAQ, long = data acquisition
muSR short = $\mu$SR, long = muon spin rotation LEmuSR short = LE-$\mu$SR,
long = low-energy muon spin rotation LEmu short = LE-$\mu^{+}$, long = low-
energy muon SmuS short = S$\mu$S, long = Swiss Muon Source LMU short = LMU,
long = Laboratory for Muon Spin Spectroscopy HIMB short = HIMB, long = High-
Intensity Muon Beams LEM short = LEM, long = Low Energy Muons HPGe short =
HPGe, long = high-purity germanium APV short = APV, long = atomic parity
violation PV short = PV, long = parity violation SFHe short = SFHe, long =
superfluid helium NMR short = NMR, long = nuclear magnetic resonance MRI short
= MRI, long = magnetic resonance imaging ESR short = ESR, long = electron spin
resonance FMR short = FMR, long = ferromagnetic resonance ZF short = ZF, long
= zero-field HIPA short = HIPA, long = high-intensity proton accelerator DC
short = DC, long = direct current RF short = RF, long = radio frequency RFQ
short = RFQ, long = radio frequency quadrupole CDW short = CDW, long = charge
density wave TRSB short = TRSB, long = time-reversal symmetry breaking CIGS
short = CIGS, long = copper indium gallium selenide RMS short = RMS, long =
root mean square QED short = QED, long = quantum electrodynamics HFS short =
HFS, long = hyperfine splitting FNAL short = FNAL, long = Fermi National
Accelerator Laboratory SES short = SES, long = single event sensititivity GPU
short = GPU, long = graphics processing unit RMD short = RMD, long = radiative
muon decay GasPM short = Gas PM, long = gaseous photo multiplier tube SiPM
short = SiPM, long = silicon photomultiplier MPPC short = MPPC, long = multi-
pixel photon counter LYSO short = LYSO, long = Lutetium–Yttrium
oxyorthosilicate TPC short = TPC, long = time projection chamber RPC short =
RPC, long = resistive plate chamber mRPC short = mRPC, long = multi-layer
resistive plate chamber DR short = DR, long = dilution refrigerator BG short =
BG, long = background TDC short = TDC, long = time-to-digital converter LXe
short = LXe, long = liquid xenon pTC short = LXe, long = pixelated timing
counter CDCH short = CDCH, long = cylindrical drift chamber SLS short = SLS,
long = Swiss Light Source ALP short = ALP, long = axion-like particle MPGD
short = MPGD, long = micro-pattern gaseous detector VUV short = VUV, long =
vacuum ultraviolet

# Science Case for the new High-Intensity Muon Beams HIMB at PSI

Edited by A. Knecht, F. Meier Aeschbacher, T. Prokscha, S. Ritt, A. Signer

M. Aiba  Paul Scherrer Institut, 5232 Villigen PSI, Switzerland  A. Amato
Paul Scherrer Institut, 5232 Villigen PSI, Switzerland  A. Antognini  Paul
Scherrer Institut, 5232 Villigen PSI, Switzerland  Institute for Particle
Physics and Astrophysics, ETH Zurich, 8093 Zürich, Switzerland  S. Ban  ICEPP,
The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan  N.
Berger  Institute for Nuclear Physics and PRISMA+ Cluster of Excellence,
Johannes Gutenberg University Mainz, Germany  L. Caminada  Paul Scherrer
Institut, 5232 Villigen PSI, Switzerland  Physik-Institut, Universität Zürich,
8057 Zürich, Switzerland  R. Chislett  Department of Physics and Astronomy,
University College London, WC1E 6BT, United Kingdom  P. Crivelli  Institute
for Particle Physics and Astrophysics, ETH Zurich, 8093 Zürich, Switzerland
A. Crivellin  Paul Scherrer Institut, 5232 Villigen PSI, Switzerland  Physik-
Institut, Universität Zürich, 8057 Zürich, Switzerland  G. Dal Maso  Paul
Scherrer Institut, 5232 Villigen PSI, Switzerland  Institute for Particle
Physics and Astrophysics, ETH Zurich, 8093 Zürich, Switzerland  S. Davidson
LUPM, Université de Montpellier, Place Eugène Bataillon, 34095 Montpellier,
France  M. Hoferichter  Albert Einstein Center for Fundamental Physics,
Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, 3012
Bern, Switzerland  R. Iwai  Institute for Particle Physics and Astrophysics,
ETH Zurich, 8093 Zürich, Switzerland  T. Iwamoto  ICEPP, The University of
Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan  K. Kirch  Paul Scherrer
Institut, 5232 Villigen PSI, Switzerland  Institute for Particle Physics and
Astrophysics, ETH Zurich, 8093 Zürich, Switzerland  A. Knecht  Paul Scherrer
Institut, 5232 Villigen PSI, Switzerland  U. Langenegger  Paul Scherrer
Institut, 5232 Villigen PSI, Switzerland  A. M. Lombardi  CERN, 1211 Geneva
23, Switzerland  H. Luetkens  Paul Scherrer Institut, 5232 Villigen PSI,
Switzerland  F. Meier Aeschbacher  Paul Scherrer Institut, 5232 Villigen PSI,
Switzerland  T. Mori  ICEPP, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,
Tokyo, 113-0033, Japan  J. Nuber  Paul Scherrer Institut, 5232 Villigen PSI,
Switzerland  Institute for Particle Physics and Astrophysics, ETH Zurich, 8093
Zürich, Switzerland  W. Ootani  ICEPP, The University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo, 113-0033, Japan  A. Papa  Paul Scherrer Institut, 5232
Villigen PSI, Switzerland  Dipartimento di Fisica E. Fermi & INFN Sezione di
Pisa, Largo Bruno Pontecorvo, Edificio C, 208, 56127 Pisa, Italy  T. Prokscha
Paul Scherrer Institut, 5232 Villigen PSI, Switzerland  F. Renga  INFN Sezione
di Roma, Piazzale A. Moro 2, 00185 Roma, Italy  S. Ritt  Paul Scherrer
Institut, 5232 Villigen PSI, Switzerland  M. Sakurai  Institute for Particle
Physics and Astrophysics, ETH Zurich, 8093 Zürich, Switzerland  Z. Salman
Paul Scherrer Institut, 5232 Villigen PSI, Switzerland  P. Schmidt-Wellenburg
Paul Scherrer Institut, 5232 Villigen PSI, Switzerland  A. Schöning
Physikalisches Institut, Universität Heidelberg, Im Neuenheimer Feld 226,
69120 Heidelberg, Germany  A. Signer  Paul Scherrer Institut, 5232 Villigen
PSI, Switzerland  Physik-Institut, Universität Zürich, 8057 Zürich,
Switzerland  Corresponding author<EMAIL_ADDRESS>A. Soter  Institute
for Particle Physics and Astrophysics, ETH Zurich, 8093 Zürich, Switzerland
L. Stingelin  Paul Scherrer Institut, 5232 Villigen PSI, Switzerland  Y.
Uchiyama  ICEPP, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo,
113-0033, Japan  F. Wauters  Institute for Nuclear Physics and PRISMA+ Cluster
of Excellence, Johannes Gutenberg University Mainz, Germany

###### Abstract

In April 2021, scientists active in muon physics met to discuss and work out
the physics case for the new High-Intensity Muon Beams (HIMB) project at PSI
that could deliver of order $10^{10}$ s-1 surface muons to experiments. Ideas
and concrete proposals were further substantiated over the following months
and assembled in the present document. The high intensities will allow for
completely new experiments with considerable discovery potential and unique
sensitivities. The physics case is outstanding and extremely rich, ranging
from fundamental particle physics via chemistry to condensed matter research
and applications in energy research and elemental analysis. In all these
fields, HIMB will ensure that the facilities S$\mu$S and CHRISP on PSI’s High
Intensity Proton Accelerator complex HIPA remain world-leading, despite the
competition of muon facilities elsewhere.

###### Contents

  1. 1 Introduction
  2. 2 Particle physics with HIMB
    1. 2.1 Muon flavour physics
    2. 2.2 The Mu3e experiment
      1. 2.2.1 Mu3e final focus
      2. 2.2.2 Muon stopping target
      3. 2.2.3 Mu3e signal and background
      4. 2.2.4 Mu3e phase II detector design
      5. 2.2.5 Online data processing at high rates
      6. 2.2.6 Mu3e-Gamma
    3. 2.3 $\mu\rightarrow e\gamma$
      1. 2.3.1 General introduction
      2. 2.3.2 Gaseous positron tracker
      3. 2.3.3 Photon calorimeter
      4. 2.3.4 Photon conversion spectrometer
      5. 2.3.5 Silicon pixel sensor for $\mu\rightarrow e\gamma$
    4. 2.4 Muon moments
      1. 2.4.1 ${g-2}$ and muon electric dipole moment
      2. 2.4.2 A ${g-2}$ / EDM measurement at PSI
      3. 2.4.3 Prospects for a search of the muon electric dipole moment
      4. 2.4.4 Prospects for a high-precision measurement of the muon ${g-2}$
      5. 2.4.5 Conclusion
    5. 2.5 Muonium
      1. 2.5.1 Development of novel muonium sources at PSI
      2. 2.5.2 Spectroscopy
      3. 2.5.3 Gravitational free fall experiment of muonium
    6. 2.6 Further particle-physics applications
      1. 2.6.1 Muon decays
      2. 2.6.2 Muonium - antimuonium oscillations
      3. 2.6.3 Atomic parity violation in muonic atoms
      4. 2.6.4 Muon conversion
  3. 3 Muon spin spectroscopy and material science with HIMB
    1. 3.1 Different muon sources around the world
    2. 3.2 Current status of muSR at PSI
    3. 3.3 New opportunities for muSR at HIMB: key science drivers
      1. 3.3.1 Novel quantum materials
      2. 3.3.2 Energy materials
      3. 3.3.3 Pulsed beam and pump-probe
      4. 3.3.4 Low-energy muons
      5. 3.3.5 Elemental analysis and muSR
  4. 4 The HIMB project & related beam and detector developments
    1. 4.1 The HIMB project
    2. 4.2 muCool beam development
    3. 4.3 Detector developments
  5. 5 Conclusions
  6. Acknowledgements

##  1 Introduction

Muon physics covers research fields from fundamental particle physics to
materials science. Muons are leptons with either negative or positive electric
charge, such as electrons and positrons, but 207 times more massive and
unstable. Muons are efficiently produced in weak decays of pions which are
usually produced by proton beams hitting nuclei in some target material. Muons
themselves decay again weakly, with a lifetime of about 2.2 $\mu$s. They have
spin 1/2 and a magnetic moment.

In particle physics, since their first discovery, muons played an important
role to help develop the theory and establish the present SM. Muons are today
used to perform some of the most sensitive tests to probe the limits of our
theoretical understanding. In almost all presently existing tensions of
experimental measurements with precision predictions of the SM muons are
involved, suggesting they might be playing a key role in finding and
establishing the breaking of this best theory to date. With a similarly long
history, a broad range of research topics in solid-state physics, chemistry
and materials science is being addressed by muon spin spectroscopy, usually
using positive muons as highly sensitive local magnetic probes.

In all these fields, the availability of low-momentum, high-intensity muon
beams is a prerequisite, and technological progress from muon production to
detector development and sample environments boosted reach and capabilities of
the research over decades.

World-wide, several large-scale facilities provide muons to experiments and
user instrumentation. They are located at RAL (UK), at J-PARC (Japan), at
TRIUMF (Canada) and at PSI in Switzerland. They are all active in condensed
matter research, while broader programs in particle physics are pursued mostly
at J-PARC and at PSI. A dedicated program in particle physics also exists at
the muon campus of FNAL (USA). Several other accelerator facilities are
developing or considering a future muon physics program, among them the CSNS
(China), RAON (Korea), SNS (USA), and the ESS (Sweden). Some facilities
provide pulsed, others continuous muon beams, complementing each other and
allowing for different kinds of experiments.

The High Intensity Proton Accelerator facility HIPA at PSI [1] provides one of
the most powerful proton beams to target stations, with 1.4 MW average beam
power presently only matched by the pulsed SNS. HIPA has a 50 MHz time
structure leading to quasi-continuous beams of slow muons. Many experiments
today use so-called ‘surface muons’, positive muons generated in the decay of
positive pions stopped close to the surface of a production target. These
muons have well-defined momenta, are fully polarised and can be efficiently
transported into secondary beam areas. The muon beams at PSI are presently
leading the high-intensity frontier, with surface muon rates of order $10^{8}$
s-1, and PSI is home to world-leading research in particle physics (see [2]
for a recent review) and condensed matter research with muons (see [3, 4, 5,
6, 7, 8] for reviews).

Installation of two new HIMB, proposed within the IMPACT project (Isotope and
Muon Production using Advanced Cyclotron and Target technology) at PSI, will
constitute a leap forward for muon physics. Surface muon intensities will be
boosted to $10^{10}$ s-1, serving particle physics in one and condensed matter
research in a second beamline. These unprecedented muon intensities will allow
for completely new experiments.

The interested national and international muon science community met for a
HIMB physics case workshop, April 6-9, 2021 at PSI, as a kick-off event for
the work presented in this paper. While we report on ongoing work and many
aspects will be worked out over the coming years and many new, additional
applications and ideas for HIMB will continue to appear, we have assembled
here a compelling physics case documenting unprecedented opportunities for
muon science. HIMB will secure a world-wide leading position of fundamental
and applied muon science at PSI and attract many national and international
user groups. Specific findings of future research cannot be predicted,
however, it is clear that HIMB will have great impact. Projects to be
conducted on this new facility will push the limit of the known far into
presently unknown territory with plenty of opportunities for ground-breaking
discovery, furthering basic knowledge and understanding of nature, development
of novel technologies and fertilising spin-offs to other fields.

This paper is structured according to the two main use cases of the two
beamline branches of HIMB as follows: Opportunities for particle physics are
discussed in Section 2. Muon spin spectroscopy and materials science follow in
Section 3. In Section 4 we deal with facility aspects, a short description of
the lay-out and the properties of HIMB as well as further add-ons and
technological aspects.

##  2 Particle physics with HIMB

###  2.1 Muon flavour physics

Flavour is arguably the least understood sector of the SM. There is still no
compelling explanation as to why three copies (or flavours) of matter field
families exist. Put specifically for charged leptons, why is there a muon and
a tau in addition to the electron? According to the SM, the only difference
between the various flavours is the coupling to the Higgs field, resulting in
(widely) different masses.

Given the need to go beyond the SM it is natural to explore the sector that is
least understood. Does the muon indeed behave precisely as the electron? There
have been several recent measurements that cast doubt on this statement. The
so-called $B$-anomalies seem to indicate that the decay of some $B$ mesons
involving a $\mu^{+}\,\mu^{-}$ pair in the final state do not precisely follow
the pattern expected from corresponding decays with an electron-positron or
tau pair in the final state [9, 10, 11]. Also, measurements of the anomalous
magnetic moment (AMM) of the muon [12, 13] are in tension with the prediction
of the SM [14]. Thus, it is not inconceivable that the role of the muon in
particle physics will evolve from “who ordered that” to being the key to
unlock the door to physics beyond the SM. Ever more precise investigations of
processes involving muons are required to address these questions.

In the SM without right-handed (and therefore with massless) neutrinos, lepton
flavour is conserved. Hence, a muon cannot decay into an electron without a
muon neutrino and an electron antineutrino in the final state. The neutrinos
are required to balance muon and electron flavour. However, this symmetry is
accidental. Writing down all operators with the fields of the SM, compatible
with gauge invariance, Lorentz invariance, and renormalisability (i.e. of
dimension 4 or less) it just so happens that there is no operator that
violates lepton flavour symmetry. It is by no means a fundamental ingredient
in the construction of the SM. And, more importantly, we know that this
symmetry is broken in nature. Indeed, neutrinos have tiny but non-vanishing
masses and as a result they oscillate, i.e. they change their flavour. This
can be seen for example by letting muons decay and observe at a distant
detector the emergence of electrons. This is precisely the observation of
lepton flavour violation. Accordingly, the SM has to be modified by the
introduction of right-handed neutrinos. This also leads cLFV in muon decays
investigated at PSI such as $\mu\to e\gamma$ or $\mu\to eee$. In the SM with
massive neutrinos, these decays can happen, albeit with a BR smaller than
$10^{-54}$. Thus, while there is nothing sacred about lepton flavour, for
practical purposes cLFV processes are forbidden in the SM and any measurement
of such a process is a clear signal of BSM physics. This offers a unique
opportunity to search for BSM physics and has triggered a wealth of
experimental and theoretical activities [15].

Similar to the introduction of right-handed neutrinos, a generic BSM model
induces cLFV. As one typical example we mention supersymmetry. Even in the
MSSM the soft breaking terms include numerous cLFV operators and currently
there is no satisfactory explanation as to why they are absent or suppressed.
The situation is similar for virtually all extensions of the SM. Thus, from a
BSM point of view there is absolutely nothing exotic about cLFV. To the
contrary, the absence of cLFV is exotic.

This can also be understood from an EFT point of view. If we assume BSM
physics is at a high scale well beyond the energy scale of the process, we can
parameterise BSM effects through operators of dimension larger than 4. Using
the SM fields, there is a single dimension 5 operator [16], closely linked to
neutrino masses. At dimension 6, there are numerous operators including
contact interactions (four-fermion operators) and dipole interactions among
others [17, 18]. The flavour-diagonal dipole operators contribute to the EDM
and AMM of the corresponding particles. Their off-diagonal variants directly
induce for example the cLFV decay $\mu\to e\gamma$. Similarly, off-diagonal
four-fermion operators contribute to decays like $\mu\to eee$. Including a
flavour-diagonal dimension 6 operator to parameterise BSM effects, but
avoiding the corresponding cLFV operators requires either the introduction of
an additional symmetry as explanation, or a fine-tuning for which there is
currently no theoretical justification at all. In this connection it should
also be mentioned that the conventional split into contributions from dipole
and contact interactions is applicable at the high scale only. Through
renormalisation-group evolution these operators mix and contact interactions
directly impact, e.g., the $\mu\to e\gamma$ process [19, 20].

The mixing of different effects also implies that a diverse program is
required to investigate cLFV. In case a signal is measured in a single
process, it is impossible to pinpoint what precisely causes this cLFV decay.
As for the current tension for the AMM of the muon, there will be many
possible explanations. In order to narrow down the nature of BSM that causes a
potential cLFV decay it is imperative to measure or constrain as many such
processes as possible. While this also concerns cLFV tau decays, the most
stringent limits come from cLFV muon decays.

Figure 1: The history of limits for branching ratios of the three golden cLFV
channels. Symbols that are not filled in indicate projected measurements. Plot
modified from [21].

There are three golden cLFV muon-decay channels that have dramatic impact. In
addition to the already mentioned $\mu\to e\,\gamma$, these are the decay
$\mu\to e\,(e^{+}e^{-})$ (or $\mu\to 3e$ for short) as well as the conversion
of a muon to an electron in the field of a nucleus, $\mu\,N\to e\,N$. As
illustrated in Figure 1, there is a long history of searches for such decays.
The current best limits on the BR for all three decays have been obtained at
PSI with $\mathrm{BR}(\mu\to 3e)<1\cdot 10^{-12}$ [22], $\mathrm{BR}(\mu\to
e\gamma)<4.2\cdot 10^{-13}$ [23] and $\mathrm{BR}(\mu\,N\to e\,N)<7\cdot
10^{-13}$ [24].

The sensitivity to $\mu\to e\,\gamma$ and $\mu\to 3e$ will be improved further
in the coming years. Using the currently existing beamline $\pi$E5 at PSI with
$10^{8}\,\mu\,\mathrm{s}^{-1}$, MEG II [25] and phase I of Mu3e [26] aim to
reach a sensitivity of $\mathrm{BR}(\mu\to e\gamma)=6\cdot 10^{-14}$ and
$\mathrm{BR}(\mu\to 3e)\sim 2\cdot 10^{-15}$, respectively.

To improve the sensitivity for the third golden channel, $\mu\,N\to e\,N$, it
is advantageous to use a pulsed beam. At J-PARC and Fermilab a dedicated
effort is ongoing by COMET [27] and Mu2e [28] with the long-term goal to
increase the sensitivity for muon conversion by several orders of magnitude.
For the other two channels a continuous muon beam is better suited. As
illustrated in Figure 1, improving upon MEG II and phase I of Mu3e requires an
increased muon intensity, as provided by the HIMB project. In fact, from the
beginning it was envisaged to have a phase II for Mu3e and in Section 2.2 the
decisive impact of HIMB for Mu3e will be described in detail. Prospects for
measurements of $\mathrm{BR}(\mu\to e\gamma)$ beyond MEG II will be discussed
in Section 2.3.

There are several observables linked to cLFV that can also be investigated
more precisely with HIMB. As already mentioned, the AMM and the EDM of the
muon are related to the same dipole operator, albeit with diagonal flavour
indices. Plans for future PSI activities in this direction are discussed in
Section 2.4. Muonium, a bound state of a positive muon with an electron
$M=(\mu^{+}\,e^{-})$ also serves as a clean probe for testing cLFV through
oscillation into antimuonium $\overline{M}=(\mu^{-}\,e^{+})$. More generally,
muonium serves as a laboratory for QED and gravity tests. This will be
elaborated upon in Section 2.5. In Section 2.6 we will touch upon further
possible applications of HIMB for particle physics with muons. Taken together,
HIMB is an essential facility to fully explore the opportunities provided by
high-intensity, low-energy experiments. It will ensure PSI remains at the
forefront of this branch of particle physics, lead to further insights into
its theory landscape [29] and trigger progress in experimental techniques that
often has applications well beyond particle physics as well.

###  2.2 The Mu3e experiment

The Mu3e experiment at PSI is aiming to search for the LFV decay of the muon
in the reaction $\mu^{+}\rightarrow e^{+}e^{+}e^{-}$ with unprecedented
sensitivity.

The ultimate experimental sensitivity depends mainly on the achievable energy
(momentum) resolution and has been estimated [30] to be
$\mathcal{O}(${10}^{-16}$$) SES for the new and innovative conceptual design
of the Mu3e experiment, and assuming state-of-the-art detector technologies.
Another limitation comes from the available muon rate which, for a given
number of muon stops, not only determines the total running time of the
experiment but also the accidental BG rate, and thus the maximum achievable
sensitivity.

The validity of the experimental concept and the performance of the applied
high-rate detector technologies will be tested during Mu3e phase I at the
$\pi$E5 compact muon beam line, with an anticipated start of data taking in
the year 2023. Experience collected during phase I will provide important
input for the design of the phase II detector.

Figure 2: Schematic view of the Mu3e phase I experiment. Taken from [26].

The design of the Mu3e phase I experiment is sketched in Figure 2 and shortly
described here. A full description of the experimental setup is given in the
Mu3e phase I technical design report [26]. Positively charged muons with a
momentum of about $p=$28\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/c$ (“surface
muons”) pass a moderator and are then stopped on a hollow double cone shaped
target. The $10\text{\,}\mathrm{c}\mathrm{m}$ long muon stopping target is
surrounded by two layers of silicon pixel sensors which have a pixel size of
$$80\text{\,}\mu\mathrm{m}$\times$80\text{\,}\mu\mathrm{m}$$. Tracking is
completed by the two outer pixel layers in the central region, and the up- and
downstream recurl stations which register hits from back-curling tracks in the
strong solenoidal magnetic fields of $B=$1\text{\,}\mathrm{T}$$. All silicon
pixel layers use the monolithic MuPix chip produced in HVMAPS technology [31,
32]. The ultra-thin pixel modules have a radiation length of only about
$1\text{\times}{10}^{-3}\text{\,}\mathrm{X}_{\mathrm{0}}$ and are cooled by an
innovative gaseous helium system. The timing system consists of scintillating
fibres in the central region and scintillating tiles placed up- and downstream
from the central region. Both timing detector systems provide sub-nanosecond
resolution to measure time coincidences. Vertex and timing information
combined are crucial to suppress accidental BG, mainly originating from Bhabha
scattered positrons in combination with ordinary Michel decays, see Section
2.2.3. The data are continuously readout and processed by a GPU based filter
farm which searches for three tracks pointing to a common vertex (3-prong) and
reconstructs the charge and momentum of the particles.

In the following the proposed detector design for Mu3e phase II is presented,
taking into account the HIMB constraints as well as the expected BG.

####  2.2.1 Mu3e final focus

The transverse beam emittance
$\epsilon_{\textrm{trans}}=\epsilon_{x}\cdot\epsilon_{y}$ of the HIMB will be
about one order of magnitude larger compared to the compact muon beam line
used for Mu3e phase I. The so-called matched beam size depends on the magnetic
field $B$ of the experimental solenoid

$\displaystyle\sigma_{x,y}^{\textrm{matched}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2\epsilon_{x,y}\,p}{e\,B}}$ (2.1)

with $p$ being the muon momentum at the stopping target. To achieve an overall
muon stopping efficiency similar to Mu3e phase I the diameter of the stopping
target ($38\text{\,}\mathrm{m}\mathrm{m}$ in phase I) would have to be
significantly increased.

However, a wider muon stopping target is detrimental to the requirement of an
excellent vertex resolution which is mainly given by the track extrapolation
uncertainties due to multiple scattering. As a consequence the accidental BG
rejection scales with the muon stopping target diameter as $\approx 1/D^{2}$.
Thus a small target diameter, and correspondingly small beam size, is favoured
to achieve highest sensitivity in an accidental BG limited environment, see
also discussion in Section 2.2.3

Figure 3: $1\text{\,}\sigma$ beam envelopes for the HIMB-5 scenario (see
Section 4.1) and a solenoid field of $B=$1\text{\,}\mathrm{T}\text{/}$$ for an
unmatched beam optics (left) and a matched beam optics (right) with a
$600\text{\,}\mu\mathrm{m}$ mylar moderator placed inside the solenoid
entrance region. The $1\text{\,}\sigma$ beam envelope is determined by
transport matrices (left) and using particle tracking including energy loss
and multiple scattering in the moderator (right). The right plot is taken from
[33] and has been modified.

Consequently, one of the design challenges for Mu3e at HIMB is the
minimisation of the matched beam size which can be achieved, according to
(2.1), by three measures:

  * •

reduction of the muon beam momentum $p$,

  * •

increase of the magnetic field strength $B$ of the experiment,

  * •

special optics with oscillating beam envelope.

Simulation studies show that the initial muon beam momentum of about
$28\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/$c$ can be decreased using a
moderator to about $16\text{\,}\text{/}$ $\pm$
$4\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/$c$ without significantly losing
muons. In order to profit from the lower beam momentum, the moderator must be
placed in a region of small beam waist where the beam dispersion is large and
the additional contribution from multiple scattering in the moderator is
negligible. Several designs with different placements of the moderator,
different type of moderators (solid and gaseous), and with and without pre-
solenoids have been simulated by the method of single particle tracking. The
simulation also considers the fringe field of the solenoid which significantly
influences the trajectories of the very low momentum muons. For the example of
a design with a solid moderator in the entrance region of the Mu3e solenoid,
the $1\text{\,}\sigma$ envelope is shown in Figure 3 (right). The most
critical parameter for the muon transmission efficiency is the size of the
beam pipe inside the experimental magnet. Simulation studies show that the
transmission efficiency scales approximately linear with the beam-pipe radius
in the region $R=30$ \- $50\text{\,}\mathrm{m}\mathrm{m}$. Whether the beam-
pipe radius can be increased with respect to the phase I design
($30\text{\,}\mathrm{m}\mathrm{m}$) needs to be answered by detailed technical
design studies.

The second option is to increase the experimental solenoid field strength.
According to (2.1) the beam size would shrink $\propto 1/\sqrt{B}$. At the
same time, however, particles from the muon decay would be bent more strongly,
leading to a larger acceptance loss of low-momentum tracks, if the detector
system is not resized accordingly. Considering the already very tight space
constraints of the Mu3e phase I design, a simple miniaturisation with two
inner and two outer pixel layers would be very challenging. A better option is
to add a third inner pixel layer. With three closely stacked pixel layers low-
momentum tracks can be reconstructed which do not reach the outer pixel layers
due to the strong magnetic field. The higher redundancy also helps to reduce
the hit combinatorics in the track reconstruction. Ideally, one of the three
pixel layers would also provide very precise time information (timing layer).
Designs with three inner pixel layers are discussed in the context of a
combined Mu3e and MEG search in Section 2.2.6.

The third option is to use a special beam optics where the final focus of the
muon beam is intentionally not matched with the experimental solenoid field.
In that case the beam envelope inside the experiment oscillates with a wave
length of $\approx$60\text{\,}\mathrm{c}\mathrm{m}$$ for
$B=$1\text{\,}\mathrm{T}\text{/}$$ and
$p=$28\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/c$, see Figure 3 (left). This
beam setup provides periodic minima which are significantly smaller than the
equilibrium beam size. Such a design, however, is difficult to realise for
several reasons: A) the beam pipe must be large to include all periodic beam
waists, B) the minima are smeared out due to dispersion effects especially
after placing a moderator, and C) the region around the minimum is relatively
narrow, $\approx$10\text{\,}\mathrm{c}\mathrm{m}$$, thus excluding a long muon
stopping target which is favoured to reduce accidental BG.

####  2.2.2 Muon stopping target

Accidental BG can be best suppressed by using a long muon stopping target and
by ensuring excellent vertex resolution, so that two muons decaying at the
same time can be distinguished by tracking (vertexing). For a given beam
momentum, the longitudinal radiation length $X_{L}$ is fixed by the required
stopping power and given by $X_{L}\propto l\rho$, with $\rho$ being the
average density of the muon stopping target. The transverse direction is the
preferred detection plane. The average transverse radiation length is given by
the product of target density and radius which should ideally match the beam
size, $X_{T}\propto r\rho=X_{L}\,r/l$. Since $X_{T}$ affects the signal
particles (e.g. multiple scattering) and the generation of accidental BG
(Bhabha), the primary design goal111The relation between longitudinal and
transverse radiation length holds, irrespective of the detailed stopping
target geometry and material (gas, solid, etc.). is a long muon stopping
target and a small beam size, see Figure 4.

Figure 4: Longitudinal view of the Mu3e muon stopping target region for a) a
short and wide, and b) a long and narrow geometry. The different brightnesses
represent different average densities of the stopping material. The inner
pixel tracking layers are shown in grey. Figure 5: Transverse view of the Mu3e
target region. Multiple scattering in the first detector layer leads to
extrapolation uncertainties, defining the region susceptible to accidental BG
(a). This region can be made smaller by reducing material in the first layer
(b) or by reducing the transverse extent of the target region (c).

The impact of the radial target size on the directional momentum resolution is
illustrated in Figure 5. The accidental BG rate depends on the vertex
resolution which is mainly determined by multiple scattering in the first
tracking layer. The vertex resolution can be improved by reducing the material
in the first tracking layer or by using a smaller target radius, thus reducing
the extrapolation uncertainties.

For Mu3e phase II, a hollow double cone structure similar to phase I is
considered as well as new ideas like silicon aerogel or gaseous targets. Gas
is favourable for constructing a long muon stopping target and has also the
advantage of a very small transverse radiation length. Active (instrumented)
targets, which provide direct information about the muons stop/decay position,
have the potential to improve the tracking, and therefore momentum resolution.
However, the potential improvement is very small and probably not worth the
effort222Active targets could also be used to reject accidental BG by
identifying and rejecting close muon stops in space and time, which however
also reduces the effective muon stopping rate. .

Several target designs and materials have been simulated in a
$B=$1\text{\,}\mathrm{T}\text{/}$$ field and optimised with the goal to
equally distribute the muon stops along the beam direction [33]. For a muon
stopping target radius of $19\text{\,}\mathrm{m}\mathrm{m}$, stopping rates of
about $2\text{\times}{10}^{9}\text{\,}\mu\mathrm{/}\mathrm{s}$ were calculated
for solid state targets with different geometries (double cone, tilted plane,
helicoid). The flattest stopping distribution on the longitudinal axis was
achieved with a $30\text{\,}\mathrm{c}\mathrm{m}$ long gaseous ethane target
at atmospheric pressure. The corresponding total stopping rate is about
$2.5\text{\times}{10}^{9}\text{\,}\mu\mathrm{/}\mathrm{s}$ and slightly higher
as for the solid targets. These estimates were obtained for a rather large
beam-pipe radius of $50\text{\,}\mathrm{m}\mathrm{m}$. Which target design
eventually is chosen will also depend on the technical feasibility to build a
long ($\gg$$10\text{\,}\mathrm{c}\mathrm{m}$) stopping target with low
density.

####  2.2.3 Mu3e signal and background

Detailed studies of signal efficiencies and BG suppression have been performed
in the context of Mu3e phase I and are documented in the technical design
report [26]. Assuming a muon stopping rate of
$1\text{\times}{10}^{8}\text{\,}\mu\mathrm{/}\mathrm{s}$ a single event
sensitivity of $2\text{\times}{10}^{-15}$ is expected.

In order to reach the ultimate sensitivity goal of ${10}^{-16}$ at HIMB,
higher muon stopping rates and an improved detector are required. In the
following, some of the challenges for almost completely suppressing BG whilst
keeping a reasonable signal efficiency in a much harsher environment are
discussed.

For HIMB, we assume that a surface muon beam with
$2\text{\times}{10}^{9}\text{\,}\mu\mathrm{/}\mathrm{s}$ is stopped in the
experiment and the muons decay at rest. The signal signature is then two
positrons and one electron originating from a common vertex, coincident in
time and with a four-momentum sum corresponding to a stopped muon. All BG
processes that can mimic this signal topology need to be suppressed to an
expectation of less than one BG event in the signal region over the lifetime
of the experiment. The two main BG classes, namely the rare muon decay
$\mu^{+}\rightarrow e^{+}e^{-}e^{+}\nu\bar{\nu}$ and accidental combinations
of two positrons and an electron are discussed below.

Besides an excellent BG suppression, a decent efficiency for the signal
process is required in order to achieve the sensitivity goal in a reasonable
amount of time. The efficiency is on the one hand limited by the detector
geometry, which does not cover the area around the beam and defines a lower
limit on reconstructable transverse momenta (roughly
$10\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/$c$ transverse momentum for the
phase I Mu3e design). On the other hand, any gaps or deficient parts in the
detector will affect the efficiency. As all three tracks have to be fully
reconstructed, detector inefficiencies enter the final efficiency with a high
power. The exact signal efficiency depends on the signal kinematics, which in
turn depends on the type of new physics. For the phase I design it ranges from
about 10% for pure dipole operators to about 19% for four-fermion operators
[34, 35, 26]. With the planned improvements for the phase II detector design,
see Section 2.2.4, the signal efficiency is expected to increase by at least
50% compared to phase I.

The process $\mu^{+}\rightarrow e^{+}e^{-}e^{+}\nu\bar{\nu}$ is allowed in the
SM and occurs with a BR of $3.4\text{\times}{10}^{-5}$ [36]. It is
indistinguishable from the signal process except for the energy and momentum
carried away by the neutrinos. The invariant mass of the $e^{+}e^{-}e^{+}$
system has been calculated to next-to-leading order [37]. Close to the
endpoint, the spectrum falls roughly with the sixth power of the _visible
mass_ and the branching fraction falls below ${10}^{-16}$ about
$$1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/c^{2}$ below the endpoint, driving
the requirements for the invariant mass resolution.

Combinations of different processes that in total have two positrons and one
electron in the final state can mimic the signal signature if their origins in
space and time are not resolvable by the detector and their combined
kinematics match the signal. These accidental BGs scale with powers of the
muon decay rate $R$. The ordinary muon decay is a very rich source of
positrons. So when studying accidental BGs, sources of electrons have to be
investigated, where those that appear in coincidence with a positron are
particularly dangerous. Since most electron production processes scale with
the amount of material seen by positrons leaving the target region transverse
to the beam $X_{T}$, see Figure 4, the best BG mitigation is to reduce all
material to the minimum. Furthermore, accidental BGs can be reduced by good
timing ($\sigma_{t}$) and vertex ($\sigma_{v}$) resolution as well as good
resolution for the kinematics, e.g. the three-particle invariant mass
resolution $\sigma_{M_{3}}$, and the centre-of-mass-system momentum resolution
$\sigma_{p}$.

Detailed simulation studies for the phase I Mu3e experiment have shown that
the most frequent accidental BG is from the combination of a Michel positron
which undergoes Bhabha scattering in the detector material (leading to an
$e^{+}e^{-}$ pair) with another Michel positron. In order to match the signal
kinematics, both Michel positrons must have energies close to the maximum
allowed energy333Note that this BG has its kinematic endpoint $1$ electron
mass _above_ the muon mass, as the electron originates from the detector
material and is not created in the muon decay.. The scattering of a positron
at $\approx$53\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/c$ with an electron at
rest creates an $e^{+}e^{-}$ pair with an invariant mass of
$\approx$7\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/c^{2}$ which can be used to
suppress this BG with little signal loss if the two particle invariant mass
resolution $\sigma_{M_{2}}$ is sufficiently small.

A similar BG topology is produced by a photon conversion combined with a
Michel positron. The rarity of
$\approx$53\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ photons combined with the
small amount of material in the muon stopping target region strongly
suppresses this BG and obviates the need for a two-particle invariant mass cut
(which would remove the expected signal kinematics in case of dominating
dipole operators).

The third variety of accidental BG with two coincident particles combining
with a Michel positron takes the $e^{+}e^{-}$ pair from a rare muon decay
$\mu^{+}\rightarrow e^{+}e^{-}e^{+}\nu\bar{\nu}$. A detailed study of this BG
was performed for the phase I Mu3e experiment [38], a simple rate
extrapolation produces 17 events in the signal region for running the phase I
apparatus under HIMB conditions, giving a target for the necessary
improvements in kinematic, vertex and timing resolution.

Backgrounds with three unrelated particles can arise from combinations of any
of the above processes which include one electron, or from Compton scattering
with two Michel positrons. The BG rates scale with the muon rate to the third
power, but are also suppressed by the vertex and timing resolution squared.
Extrapolations from the phase I simulation show that at muon rates of a few
${10}^{9}$ per second this kind of BG is completely negligible if the detector
performance is sufficient to deal with the “2+1” accidental BG.

####  2.2.4 Mu3e phase II detector design

Figure 6: Sketch of an elongated detector design for Mu3e phase II at HIMB.

To further optimise the signal sensitivity it is planned to expand the
geometrical acceptance for tracks with respect to Mu3e phase I design. This
can be achieved by extending the pixel sensor instrumentation in up- and
downstream directions. With the experience from Mu3e phase I, we believe that
significantly larger tracking modules can be produced. For Mu3e phase II, we
consider a vertex detector length of about $30\text{\,}\mathrm{c}\mathrm{m}$,
see Figure 6, or even longer. Currently, the longest pixel modules for Mu3e
phase I have a length of $36\text{\,}\mathrm{c}\mathrm{m}$. Twice as long
pixel modules could be produced with a new pixel sensor generation with
improved readout capabilities, like daisy chaining of the data. With such a
design the requirements will be less stringent for the flexprint, where the
individual pixel sensors are glued onto.

The detector technologies of Mu3e phase I were specifically chosen to prove
the high-rate capability of the experiment for phase II and to pave the road
for extremely high beam stopping rates in excess of $10^{9}\,\mu$/s.
Nonetheless, for Mu3e phase II several improvements with respect to the phase
I detector are necessary to further suppress BG and to exploit the full HIMB
potential. In addition to the extended vertex region and the increase of the
overall acceptance, the scintillating fibre detector, see Figure 2, needs to
be exchanged due to occupancy limitations. The alternative technology has to
provide a significantly higher granularity and a time resolution of about
$100\text{\,}\mathrm{p}\mathrm{s}$, or better. Therefore, the development of a
monolithic silicon pad detector (PicoPix) based on the
$130\text{\,}\mathrm{nm}\text{/}$ SiGe BiCMOS process from IHP444Institute for
High Performance Microelectronics in Frankfurt/Oder, Germany was started. This
technology was proven in test beams to provide time resolutions as good as
$\mathcal{O}($100\text{\,}\mathrm{p}\mathrm{s}$)$ for a design with sub-
millimeter pixel size [39]. Replacing the scintillating fibre detector by a
layer of PicoPix sensors would provide 4-dimensional information (time and
spatial coordinates). The high granularity of these sensors will allow to
place them at small radii close to the muon stopping target, see Figure 6. The
PicoPix detector can be regarded as a fifth tracking layer, in addition to the
four standard pixel layers, for which a new generation of MuPix sensors with
improved time resolution and readout capabilities will be used. Running Mu3e
at HIMB with about 20 times higher beam rates compared to phase I will
substantially increase the hit combinatorial problem in the online track
finding on the filter farm, see Section 2.2.5. The generally much improved
time resolution of the tracking detectors will help to mitigate the
combinatorial problem.

The final layout of the phase II design will be subject of further simulation
studies. They will address the possibility to increase the magnetic field and
to change the radius of the outer pixel layer to further improve the momentum
resolution, see also Section 2.2.6. During the workshop also an idea was
discussed to use the most inner pixel layer to stop the muons. With such a
concept the rate of muon stops could be further increased. The feasibility of
this idea needs to be experimentally studied.

####  2.2.5 Online data processing at high rates

The data rate produced by the Mu3e detector at HIMB is three to four orders of
magnitude larger than what can technologically and economically be saved to
mass storage. The experiment thus relies on a fast and efficient on-line
reconstruction, which identifies interesting events in real time. For the
phase I experiment it was shown [40, 41, 42] that a compact and affordable
farm of a dozen PCs with GPU is capable of finding and fitting all relevant
particle tracks and identifying three-track vertices, all using algorithms
developed specifically for the multiple scattering dominated regime in which
Mu3e operates [43, 44, 45, 46]. Assuming a four-layer tracking on the filter
farm, the computing power needed scales roughly with the third power of the
rate; even with optimistic assumptions about the evolution of GPU until the
start of data taking at HIMB, this factor 8000 cannot be compensated simply by
more and newer hardware. One key to tackling the online reconstruction
challenge is improved time resolution of the pixel sensors, in particular in
the vertex region. If a resolution of the order of
$1\text{\,}\mathrm{n}\mathrm{s}$ can be reached, the GPU only based approach
is likely viable even at HIMB rates. Alternatively, large associative memories
can be used for matching with pre-computed patterns; this option introduces
additional complexity into the system, but we can profit from experience
gained in the ATLAS hardware track trigger projects [47, 48].

####  2.2.6 Mu3e-Gamma

Figure 7: Transverse cross sections of conceptual Mu3e-$\gamma$ designs (top)
and calculated momentum resolution of charged particles from the muon decays
as function of the momentum for three example elevation angles $\lambda$
(bottom). The designs are optimised for the detection of $\mathrm{\mu^{+}\to
e^{+}\gamma}$ for a solenoid field of $B=$1.3\text{\,}\mathrm{T}$$ (left) and
$B=$2.6\text{\,}\mathrm{T}$$ (right). Photons are detected in a dedicated
three-layer active converter pixel spectrometer which measures the ionisation
loss of the $e^{+}e^{-}$ pair in silicon. See text for more details.

Considering the very high rates of the HIMB with up to
$1\text{\times}{10}^{10}\text{\,}\mu\mathrm{/}\mathrm{s}$, a
$\gamma\rightarrow e^{+}e^{-}$ converter can be considered for photon
detection, despite the in general low conversion (detection) efficiency. By
tracking the $e^{+}e^{-}$ pair, a photon converter allows for a precise
measurement of the photon position, a good measurement of the photon direction
and a very precise measurement of the photon energy. This makes the converter
option very interesting for the search $\mu^{+}\to e^{+}\gamma$, see also
section Section 2.3.4. In particular an active converter is of high interest
as it further improves the excellent energy resolution of a photon converter
tracker by directly measuring the energy loss of the $e^{+}e^{-}$ pair in the
converter, which is the main limitation for the energy resolution.

The feasibility of a passive photon converter in the Mu3e phase I experiment
has been studied in [49] using a GEANT4 simulation. In the studied design the
converter consists of $0.1\text{\,}\mathrm{m}\mathrm{m}$ gold and the
$e^{+}e^{-}$ pair is reconstructed in a two-layer HVMAPS spectrometer with a
pixel size of $80\text{\,}\mu\mathrm{m}$. For photons with energy
$E_{\gamma}=$53\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$, a resolution of
about $250\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ (peak) was found, to be
compared with the in average $220\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$
energy loss of a single MIP in $0.1\text{\,}\mathrm{m}\mathrm{m}$ gold. A
significant contribution to the uncertainty comes from the unknown position of
the conversion point and the therefore unknown path length of the $e^{+}e^{-}$
pair in the converter material. Also energy straggling significantly
contributes to the energy resolution which is more severe for large Z
materials due to the larger possible energy transfer in collisions.

Both contributions to the energy resolution can be significantly reduced if
thick silicon sensors are used as active converter which measures directly the
energy loss in the depleted region. To make this concept work the sensor
should be thick and the depleted region which is relevant for charge
collection should be large. A charge collection region of about
$600\text{\,}\mu\mathrm{m}$ can be achieved if a high ohmic substrate (e.g.
$\gtrsim$15\text{\,}\mathrm{k}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$$)
is depleted at about $200\text{\,}\mathrm{V}$. Using three such active
converter layers results in a total radiation length of about 2%. The
resulting photon conversion (detection) efficiency is on one hand much smaller
than for photon calorimeters, on the other hand such a detector provides and
excellent photon energy resolution of about
$100\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$, more than an order of magnitude
better than the upgraded MEG II xenon calorimeter. For not too small pixel
sizes the active converter serves as spectrometer for the reconstruction of
the $e^{+}e^{-}$ without the need of any additional tracking detectors, thus
largely simplifying the design.

Two possible designs of a Mu3e detector combined with a three-layer active
converter pixel spectrometer are shown in Figure 7 (Mu3e-$\gamma$). The
designs are optimised for different solenoid fields and provide an excellent
momentum resolution of about
$\sigma_{p}\lesssim$250\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$/c$ over a
large momentum range. This is achieved by choosing a spacing of the tracking
layers that ensures that for all particle momenta at least one section of the
track (a section is defined between two detection layers) approximately
fulfils the condition of a half turn where the multiple scattering uncertainty
on the track momentum vanishes, see discussion in [44]. Because of the
excellent momentum resolution of the Mu3e-$\gamma$ designs, this concept is
very well suited for a combined search of $\mu^{+}\rightarrow e^{+}e^{+}e^{-}$
and $\mu^{+}\rightarrow e^{+}\gamma$.

The $B=$1.3\text{\,}\mathrm{T}$$ design in Figure 7 is optimised to provide a
large kinematic acceptance555Assuming standalone tracking in the three
innermost pixel layers.
($p_{T}\gtrsim$8\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/c$). With such a
design even very exotic processes like $\mu\rightarrow eeeee\nu\nu$ can be
detected with good efficiency. This design requires the instrumentation of
large detector areas with pixel sensors which is costly and creates a high
power consumption. The instrumented areas are almost a factor two smaller for
the $B=$2.6\text{\,}\mathrm{T}$$ design, however for the price of a smaller
kinematic acceptance
($p_{T}\gtrsim$15\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/c$). The larger
solenoid field has also the advantage of better muon beam focusing which leads
to an increase of the muon stopping rate of almost a factor $2$ compared to
the $B=$1.3\text{\,}\mathrm{T}$$ design, see the discussion in Section 2.2.1.

In all designs, charged particles from the muon decays remain inside a radius
of $r\leq$26\text{\,}\mathrm{c}\mathrm{m}$$ and cannot reach the active
converter pixel spectrometer. The BG rate for photon detection is therefore
very small, thus making it together with the excellent photon energy
resolution a promising instrument to search for $\mu\rightarrow e^{+}\gamma$
or any other decay involving photons in the final state. Examples are the
search for exotic LFV decays such as $\mu^{+}\rightarrow e^{+}X\gamma$ with
$X$ being a pseudo-scalar axion, and the search for dark photons
$\mu^{+}\rightarrow e^{+}\nu_{e}\bar{\nu_{\mu}}A^{\prime}$ with $A^{\prime}$
being a long living dark photon which weakly mixes with SM particles and
decays in flight $A^{\prime}\rightarrow e^{+}e^{-}$ after a few picoseconds.

For the $\mu\rightarrow e^{+}\gamma$ search an excellent timing resolution is
crucial. This could be achieved by using a dedicated timing layer in one of
the inner three pixel layers for measuring the $e^{+}$ timing and by
integrating a $\sigma_{t}\lesssim$100\text{\,}\mathrm{p}\mathrm{s}$$ TDC in
the active converter pixel layers. For monolithic silicon pixel detectors time
resolutions of $\mathcal{O}($100\text{\,}\mathrm{p}\mathrm{s}$)$ have only
been achieved with small prototypes, so far [39]. If such a time resolution
can be achieved on a large scale chip in a large scale tracker system will be
an important research topic for the coming years. Alternatively, a dedicated
timing detector (e.g. scintillator) could be added in front to the converter
station666The timing layer would measure the time of the $e^{+}e^{-}$ pair
after a half turn. to measure the photon timing.

###  2.3 $\mu\rightarrow e\gamma$

####  2.3.1 General introduction

The state-of-the-art of the $\mu^{+}\to e^{+}\gamma$ search is represented by
the MEG II experiment [25] at PSI.

The best current upper limit on the BR of the $\mu^{+}\to e^{+}\gamma$ decay
has been set by the MEG experiment at PSI as $\mathrm{BR}(\mu\to
e\gamma)<4.2\cdot 10^{-13}$ [23]. An upgrade of the experiment to MEG II has
been carried out. MEG II preserves the concept of the previous experiment
while improving the detector performances roughly by a factor 2 for all the
kinematic variables and aiming at running up to $7\cdot 10^{7}\mu^{+}$/s. The
expected final sensitivity is $\mathrm{BR}(\mu\to e\gamma)=6\cdot 10^{-14}$
for a data-taking period of 3 years. The MEG II experiment has successfully
completed its engineering phase and it has just entered into the physics run
mode (run2021). For all technical aspects we refer to the most recent document
[50].

The signature of a $\mu^{+}\to e^{+}\gamma$ decay at rest is a back-to-back,
mono-energetic, time coincident $\gamma$ and $e^{+}$. There are two main BG
sources, the dominant being the accidental coincidences between a high-energy
positron from the principal decay $\mu^{+}\to e^{+}\nu\overline{\nu}$ (Michel
decay) and a high-energy photon from positron annihilation-in-flight or
bremsstrahlung or from the RMD $\mu^{+}\to e^{+}\nu\overline{\nu}\gamma$. The
other source comes from the RMD itself, when neutrinos take off only a small
amount of energy.

Figure 8: A sketch of the MEG II experiment with all the key elements. Table 1: Comparison of the $e^{+}$ and $\gamma$ kinematic variable resolutions (in $\sigma$) with the MEG (measured) and MEG II (expected) apparatus. | MEG | MEGII
---|---|---
$E_{e}$(core) [keV] | 306 | 130
$\theta_{e}$(core) [mrad] | 9.4 | 5.3
$\phi_{e}$ (core) [mrad] | 8.7 | 3.7
$t_{e}$ [ps] | 70 | 35
$u_{\gamma}$ [mm] | 5 | 2.4
$v_{\gamma}$[mm] | 5 | 2.2
$w_{\gamma}$ [mm] | 6 | 3.1
$E_{\gamma}$ (w $<$ 2 cm) [$\%$] | 2.4 | 1.1
$E_{\gamma}$ (w $>$ 2 cm) [$\%$] | 1.7 | 1.0
$t_{\gamma}$ [ps] | 67 | 60
$t_{e\gamma}$ [ps] | 122 | 84
Tracking efficiency [$\%$] | 65 | 78
CDCH-pTC matching efficiency [$\%$] | 45 | 90
Gamma efficiency [$\%$] | 63 | 69
Trigger efficiency [$\%$] | 99 | 99

In MEG II, positive surface muons with a momentum of 28 $\mathrm{MeV}/c$ are
stopped in a thin slanted polyethylene target (thickness 140 $\mathrm{\mu m}$;
angle $15$ deg), located at the centre of the apparatus.

All kinematic variables of the $\gamma$ (energy $E_{\gamma}$, time
$t_{\gamma}$ and interaction point $X_{\gamma}$) are measured using a LXe
calorimeter. All kinematic variables of the $e^{+}$ are measured with a
spectrometer made of a CDCH combined with plastic scintillators coupled to
SiPM - the so-called pTC mounted inside a gradient magnetic field. The
performance of the experiment is continuously monitored by a variety of
calibration methods. All signals are recorded with custom designed waveform
digitisers up to 5 Gsample/s with the DRS4 chip. A flexible trigger system
allows to select $\mu^{+}\to e^{+}\gamma$ candidate events together with pre-
scaled calibration data.

A sketch of the MEG II experiment with all the key elements is shown in Figure
8. Table 1 summarises the performances of MEG (measured) and MEG II
(expected).

Pushing down the sensitivity of the $\mu^{+}\to e^{+}\gamma$ search requires
both beam rate increase and improved background rejection capability. The SES
scales as

$\mathrm{SES}=\frac{1}{R\cdot T\cdot A\cdot\epsilon}\,,$ (2.2)

where $R$ is the beam rate, $T$ is the acquisition time, $A$ is the
geometrical acceptance and $\epsilon$ the product of all the efficiencies
(detection efficiency, selection efficiency, etc.).

The BR of the accidental BG $B_{acc}$ scales as

$B_{acc}=R\cdot\Delta E_{e}\cdot(\Delta E_{\gamma})^{2}\cdot\Delta
T_{e\gamma}\cdot(\Delta\Theta_{e\gamma})^{2}\,,$ (2.3)

where $\Delta E_{e}$, $\Delta E_{\gamma}$, $\Delta T_{e\gamma}$ and
$\Delta\Theta_{e\gamma}$ are the positron energy, gamma energy, relative
positron and gamma timing and relative positron and gamma angle resolutions.

These formulas suggest that, given a very good detector geometrical acceptance
and detector efficiency, the most effective parameter to improve sensitivity
is the beam rate. On the other hand the BR of the accidental BG – the dominant
BG – increases linearly with the beam rate. In order to really benefit from a
beam-rate increase therefore relies on detectors able to simultaneously
sustain higher beam intensity and provide better kinematic variable
resolutions.

While the measurement of the positron kinematic variables is conceptually
delineated and the spectrometer option is the most competitive one, leaving
just room for the specific technology to be selected, the measurement of the
gamma kinematic variables is in principle open to two different approaches:
either detecting the gamma directly via a calorimeter or converting it into an
electron-positron pair and then measuring it via a spectrometer.

The calorimeter option offers a higher detection efficiency compared to the
conversion one. Nonetheless, the latter could be the favourite choice at a
very high beam rate in order to keep the BG under control, due to the better
kinematic resolutions. The capability of sustaining a higher beam rate
compensates for the drop in terms of efficiency. The best option as a function
of the beam rate is defined by the BG regime. Figure 10 shows it in a
schematic way.

Figure 9: A sketch of the sensitivity as a function of the beam rate for three
different scenarios: a) calorimeter (black line) - used as a reference; b)
Improved performing calorimeter (red line); c) photon conversion (blue line).

Figure 10: A sketch of the different experimental approaches as a function of
the beam rate.

In the next sections the different experimental approaches summarised in
Figure 10 will be presented. While a calorimeter with improved performance
could still be appealing, based on the current and foreseen technologies
external photon conversion seems to be more promising at very high beam
intensities, as expected with HIMB. Photon conversion can be further improved
if complemented with an active target.

####  2.3.2 Gaseous positron tracker

A precise reconstruction of the kinematics of 52.8 MeV/$c$ positrons can only
be achieved with a very light tracker in a magnetic field. Indeed, at such low
momentum, the contribution of the multiple Coulomb scattering to the tracking
resolutions tends to become dominant, so that an extremely good position
resolution is not necessary if it is compensated by a reduced material budget.
Despite the rapid evolution of ultra-thin silicon detectors, this aspect makes
gaseous detectors still competitive for this application.

The state of the art is represented by the CDCH of the MEG II experiment [51].
With a gas mixture of helium and isobutane in 90:10 volume concentrations and
cells made of 20 $\mu$m gold-plated tungsten sense wires and 50 $\mu$m silver-
plated aluminium field wires, with a minimum cell size of $3.5\times 3.5$ mm2
and a full-stereo geometry, this detector is designed to provide 100 $\mu$m
single-hit resolution in the plane transverse to the beam axis, with an
average radiation length of about 350 m. The resulting momentum and angular
resolutions are expected to be around 100 keV and 5 mrad, respectively. In
consideration of the technical difficulties for building a chamber with
thinner wires, and the intrinsic limitations to the achievable single-hit
resolution in a drift chamber, we cannot expect any dramatic improvement of
these performances, although there are proposals to refine the track
reconstruction exploiting the detection of single ionisation clusters [52] and
to use carbon monofilaments to build wires with reduced density [53].

For these reasons, in terms of performances, the MEG II CDCH can be considered
the benchmark for the next generation of gaseous trackers for $\mu\to
e\gamma$. There is anyway a strong limitation to the use of such a kind of
detector at beam intensities much higher than $10^{8}~{}\mu$/s: the MEG II
CDCH is expected to undergo a 25% gain loss per year in the inner and most
illuminated region at $7\times 10^{7}~{}\mu$/s, due to wire ageing effects
[25]. While in MEG II an increase of the sense wire voltage can be applied to
compensate for this loss, with a much higher beam intensity the ageing rate
would clearly become unmanageable.

As a possible solution, a radical change in the geometry of the drift chamber,
with wires lying in planes transverse to the beam axis, would reduce the
positron rate per unit length in the inner wires, so suppressing the ageing
rate by a factor of about 10. This design requires the wire support structures
and the readout electronics to be distributed in between the tracker and the
outer detectors (positron timing and photon detectors), whose performances
could deteriorate by the consequently higher material budget.

As an alternative approach, the replacement of the drift chamber with a TPC
can be considered, so to strongly reduce the material budget (thanks to the
absence of wires in the tracking volume) and exploit the high particle flow
capabilities of modern TPC readout by MPGD. The typical geometry of a $\mu\to
e\gamma$ experiment requires anyway a relatively long detector along the beam
axis (2 m for the MEG II CDCH). If a conventional TPC is used, with the
electrons drifting along the beam axis, the deterioration of the longitudinal
position resolution due to the electron diffusion would spoil the detector
performances. Moreover, simulations show that the current density in the
readout detector and the space charge density in the tracking volume would be
prohibitive also with the most advanced MPGD. We considered, as a possible
alternative, a radial TPC, i.e. with electrons drifting radially, toward the
external surface of the cylindrical detector. This surface could be
instrumented with a cylindrical MPGD (e.g. a cylindrical GEM [54] or
$\mu$RWell [55]). The shorter drift distance would allow to reduce the readout
current and space charge densities, down to values that are comparable with
the ones expected in the GEM-TPC of the upgraded ALICE experiment [56]. It
supports the feasibility of such a solution, although big challenges need to
be faced. Beside the ones already experienced in ALICE, there is the specific
need of keeping the material budget on the external surface of the detector
very low, which would require a dedicated effort to integrate the front-end
electronics with its cooling in the structure of the readout MPGD.

####  2.3.3 Photon calorimeter

The photon measurement is the most difficult and crucial part of the $\mu\to
e\gamma$ experiment. A significant improvement of the photon detector
performance would be a key to higher sensitivities of future $\mu\to e\gamma$
experiments. Photon measurements with a LXe scintillation detector have been
carried out in the MEG/MEG II experiment. This calorimetric approach has been
nicely working with the muon beam intensity up to $7\times
10^{7}\,\mu/\mathrm{s}$ in the MEG II experiment. The performance of the
calorimeter would, however, be limited at higher beam intensities above
$10^{8}\,\mu/\mathrm{s}$ foreseen at future $\mu\to e\gamma$ experiments
aiming at $\mathcal{O}(10^{-15})$ sensitivity.

The challenges for new calorimetry for incoming experiments at the intensity
frontier is to provide detectors with ultra-precise time resolution and
supreme energy resolution. Two very promising materials on the market are
BrilLanCe (cerium doped lanthanum bromide, $\mathrm{LaBr_{3}(Ce)}$) and LYSO
$\mathrm{Lu_{2(1-x)}}$ $\mathrm{Y_{2x}SiO_{5}(Ce)}$, supported by recent
developments aiming at providing relatively large crystals.

Cerium doped lanthanum bromide stands out due to its ultra-high light yield
(1.65 $\times$ NaI(Tl)) and by a more than an order of magnitude faster decay
time compared to NaI(Tl). With these properties together with its high
density, $\mathrm{LaBr_{3}(Ce)}$ is the ideal medium for calorimetry limited
only by the currently available crystal sizes on the market.

Due to recent developments, larger crystals up to a radius $R$ =
$4.45\text{\,}\mathrm{c}\mathrm{m}$ and a length $L$ =
$20.3\text{\,}\mathrm{c}\mathrm{m}$ can be produced commercially. A
calorimeter built from such a large crystal is a potential candidate for the
detection of photons at energies from few tens up one hundred MeV. This
corresponds to the interesting energy range of current cLFV experiments. Thus
$\mathrm{LaBr_{3}(Ce)}$ may be a suitable candidate for future experiments in
this sector.

LYSO on the other hand exhibits a very high density, comparable to BGO and
thus features short radiation length ${X_{0}}$ and Molière radius ${R_{M}}$.
Despite the fact that the light yield is only roughly
$70\text{\,}\mathrm{\char 37\relax}\text{/}$ of NaI and the decay time roughly
three times longer compared to $\mathrm{LaBr_{3}(Ce)}$, its density makes LYSO
an attractive candidate as well, especially considering that the available
crystal size is one of the limiting factors.

Table 2 summarises the main scintillation properties compared to the widely
used scintillation media. For a quick comparison a figure of merit (F.o.M.) is
defined as the square root of the ratio of the scintillation decay time
$\mathrm{\tau}$ and the product of the light yield (LY) and the density
$\mathrm{\rho}$.

Table 2: Main scintillation properties for widely used scintillating media. A
F.o.M. is given as defined in the text.

Scintillator | Density | Light yield | Decay time | F.o.M.
---|---|---|---|---
|  $\mathrm{\rho}$ ($\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{3}$) | LY (ph/keV) |  $\mathrm{\tau}$ (ns) | $\mathrm{\sqrt{\tau/(\rho\cdot LY)}}$
$\mathrm{LaBr_{3}(Ce)}$ | 5.08 | 63 | 16 | 0.22
LYSO | 7.1 | 27 | 41 | 0.46
YAP | 5.35 | 22 | 26 | 0.47
LXe | 2.89 | 40 | 45 | 0.62
NaI(Tl) | 3.67 | 38 | 250 | 1.34
BGO | 7.13 | 9 | 300 | 2.16

The result is a detector with a high photosensor granularity, high rate
sustainability, maximal photosensor coverage area, optimal geometrical
acceptance and insensitivity to magnetic fields. Due to the small thickness of
modern photosensors of a few mm, even the radiation impinging area can be
covered with minimal impact on photons passing through. This feature allows
for a double readout scheme, where the MPPC are mounted also on the
front/entrance face. The smaller the MPPC pixel size the smaller the
saturation effect. In addition, the granularity due to the MPPC allows some
geometrical reconstruction of the event. A sketch of a detector assembly is
shown in Figure 12.

Figure 11: A sketch of the scintillation light distribution associated to a 55
MeV gamma event impinging on ($R$=3.5 cm, $L$=16 cm) LYSO crystal coupled to
Hamamatsu S13360-6025PE MPPC (top) and the typical collected charge
distribution on the front and back face (bottom).

Figure 12: Energy resolution for different crystal sizes, where $\sigma$ and
$\mu$ are the standard deviation and the mean . ”Available” crystals are shown
in black, ”Large” sized crystals in red and the ”Ultimate” crystals in blue.

Figure 12 shows the results obtained with different crystal sizes of both
$\mathrm{LaBr_{3}(Ce)}$ and LYSO up to the so-called ultimate dimension, not
yet available on the market. The detectors are exposed to 55 MeV gammas
impinging on the entrance face. Two types of MPPC were selected to be used as
a model in the simulation: Hamamatsu S13360-6025PE and sensL MicroFj-60035TSV.
For crystals of 15 cm diameter and 20 cm and 16 cm of length for
$\mathrm{LaBr_{3}(Ce)}$ and LYSO respectively, an energy resolution up to
around $1\%$ for $\mathrm{LaBr_{3}(Ce)}$ and $0.4\%$ for LYSO is obtained. In
order for $\mathrm{LaBr_{3}(Ce)}$ to fully benefit from its higher light
yield, one has to go for even larger crystals of about 40 cm diameter. In such
a configuration, resolutions around $0.3\%$ and better has been quoted also
for the $\mathrm{LaBr_{3}(Ce)}$.

Figure 13: Timing resolutions for the available ($R$=4.45 cm, $L$=20.32 cm)
$\mathrm{LaBr_{3}(Ce)}$ and ($R$=3.5 cm, $L$=16 cm) LYSO crystals. Here the
$\mathrm{LaBr_{3}(Ce)}$ is coupled to the Hamamatsu S13360-6025PE. For the
LYSO both MPPC options (Hamamatsu S13360-6025PE and sensL MicroFJ-60035TSV)
are displayed.

Examples of time distributions are shown in Figure 13. Timing resolutions
below 50 ps are in the reach. The Monte-Carlo simulations are based on the
Geant4 libraries with dedicated code that includes the MPPC response and the
whole electronic chain up to the DAQ, based on waveform digitisers with a
sampling frequency up to 5 GSample/s. The reconstructed algorithms are based
on waveform analysis. The simulations are supported by measurements done with
available LaBr3(Ce) crystals with sizes of ($R$=3.81 cm, $L$=7.62 cm) and
($R$=1.27 cm, $L$=10.16 cm) coupled to either photomultiplier tubes or MPPC
and the characterisation of the MPPC response.

In conclusion, new detectors have been considered here based on either large
LaBr3(Ce) or LYSO crystals coupled to MPPC showing very promising results for
high-energy $\mathcal{O}(50)$ MeV photon calorimetry. Independent of the
specific detector assembly, simultaneous energy, timing and position
resolutions below 0.5 MeV, 50 ps and a few mm appear to be feasible. Such
results put this new calorimetry at the detector forefront for particle
physics research at beam intensity frontiers.

####  2.3.4 Photon conversion spectrometer

Photon measurements based on a pair conversion spectrometer would be an
alternative and viable option with several advantages over the calorimetric
approach with higher beam intensity. In the pair conversion spectrometer, the
incident photon is converted to an electron-positron pair in a thin converter
and the conversion pair is then measured with a tracker. More precise
measurements of the energy and position as well as higher rate capabilities
are expected. In addition, the photon direction can be measured with the
conversion spectrometer since the emission of the conversion pair is highly
boosted along the incident photon direction. This would provide a possibility
to further reduce the accidental background together with a widely spread
distribution of the stopping muon. It must be noted that the precise
measurement of the photon direction is not possible with the calorimetric
approach. On the other hand, a major challenge for the conversion spectrometer
is the low detection efficiency due to the low conversion efficiency with a
thin conversion layer. It can be mitigated to some extent with multiple
conversion layers. The pair spectrometer was employed in the previous $\mu\to
e\gamma$ experiment, MEGA (1985–1999)[57]. It consisted of three photon
conversion layers, each of which was based on two 250 $\mu$m thick lead
converters, with the overall detection efficiency of 5%. The signal statistics
can be improved by increasing the beam intensity, limited by the rate
capability of the detectors.

Another crucial limiting factor of the conversion spectrometer is the energy
loss of the conversion pair inside the converter. Figure 15 shows the
simulated result for the energy sum of the conversion pair emitted from a 560
$\mu$m thick lead converter with signal photon injection, where a significant
tail due to the energy loss in the converter can be seen. This leads to an
inefficiency for the signal photon and/or deterioration of the energy
resolution.

Figure 14: Energy sum distribution of the conversion pair emitted from 560
$\mu$m thick lead converter with signal photon injection.

Figure 15: Possible structure of the active conversion spectrometer.

It is proposed here to use a so-called “active converter” where the energy
loss of the conversion pair in the converter is measured by the converter
itself. Figure 15 shows a possible structure of the conversion spectrometer
composed of an active converter, a tracking layer, and a timing layer. The
incident photon is converted in the active converter layer which measures the
energy loss of the conversion pair in the converter. The possible technology
for the active converter is a scintillator coupled with a photo-detector or
silicon detector. The tracks of the conversion pair are measured in the
tracking layer. It is not a harsh environment since the Michel positrons do
not reach the conversion spectrometer placed outside the positron spectrometer
in the proposed design as shown later. A gaseous detector such as a drift
chamber or a TPC can therefore be used as a tracker, although thin silicon
detectors can be an attractive candidate technology as well. The timing layer
placed in front of the active converter measures the timing of the returning
conversion pair, with which the photon timing can be extracted after
correcting for the time-of-flight of the conversion pair. The timing layer is
required to be low-mass to avoid the photon conversion before the active
converter, although it can basically be vetoed by placing a thin scintillator
layer between the active converter and the timing layer. The use of mRPC
detectors is considered as a candidate technology. The timing can also be
measured by the active converter if its timing resolution is good enough.

Figure 16 shows a possible layout of the $\mu\to e\gamma$ experiment with the
photon conversion spectrometer. The photon spectrometer with four layers of
the active converters is placed outside the positron spectrometer. Silicon
detectors are envisaged as a technology option for the positron spectrometer
because of their high rate capability. The design of the positron spectrometer
is inspired by the Mu3e detector, which would provide a possibility of a
concurrent search for $\mu\to eee$ in this setup. One of the most important
improvements in the experimental design compared to the MEG II experiment is
the enhanced signal acceptance especially for the zenith angle, which is $\pm
60^{\circ}$, while it is $\pm 20^{\circ}$ for the MEG II experiment. The
enhanced zenith-angle acceptance would enable us to measure the angular
distribution of $\mu\to e\gamma$ with polarised muon beam after a possible
discovery of $\mu\to e\gamma$ decay in the MEG II experiment. Together with
the measurement of the BR, we could pin-down the underlying new physics behind
$\mu\to e\gamma$.

Figure 16: Possible layout of $\mu\to e\gamma$ experiment with photon
conversion spectrometer.

The materials for the active converter are being investigated from the
viewpoint of the expected performance of the active converter such as
conversion efficiency and the detector resolutions. One of the crucial
parameters is the critical energy, above which the energy loss by
bremsstrahlung dominates over ionisation energy loss. Since the active
converter can basically measure the ionisation energy loss only, the critical
energy should ideally be higher than the energies of the conversion pair,
which range from 0 to 52.8 MeV. The critical energy is known to have an
empirical dependence on the atomic number as $610\,\mathrm{MeV}/(Z+1.24)$
[58]. The critical energies of LYSO and YAP as good candidates for
scintillator materials from the viewpoint of the detection efficiency are 12
MeV and 23 MeV, respectively, which are not sufficiently high compared to the
energies of the conversion pair. On the other hand, the critical energy for
plastic scintillator, which is not ideal from the viewpoint of the detection
efficiency, is 93 MeV and thus sufficiently high. The comparison among the
candidate materials is not straightforward since some part of the energy loss
by bremsstrahlung can be absorbed and measured by the converter. Simulation
studies are in progress to compare the performance of different materials as
the active converter. The energy loss measurement can be ruined when the
returning conversion pair hits the converter. A segmentation of the
scintillator into cells is therefore required for the active converter.

The GasPM and SiPM are under consideration as the photodetector for the
scintillator converter. The GasPM consists of a photocathode and RPC as
electron multiplier. The technology of the ultra-low mass RPC with diamond-
like carbon electrodes developed for the MEG II radiative decay counter [59]
can be used for the RPC of the GasPM. The development of the large-area
photocathode sensitive to visible scintillation light is, however, extremely
challenging. The combination of the VUV scintillator and the CsI photocathode
sensitive to VUV light is also under investigation because CsI photocathodes
are much easier to handle. The scintillation readout by SiPM is also viable
option. A good light collection efficiency and its small position dependence
have been already demonstrated by the CALICE scintillator calorimeters [60].

Further studies on the design and the performance of the photon conversion
spectrometer are in progress, including detailed simulation studies and
prototype tests for the candidate technologies.

####  2.3.5 Silicon pixel sensor for $\mu\rightarrow e\gamma$

With the advent of monolithic silicon pixel sensors, which can be thinned to a
few tens of micrometers, silicon detectors became an attractive alternative
for tracking low-energy particles, especially for high-rate applications,
where gaseous detectors suffer from sparks and ageing effects. Ultra-light
mechanical designs combined with gaseous helium cooling of the sensors allow
for the construction of silicon tracking layers with a thickness of about 1
per mill of radiation length, as demonstrated by the Mu3e Phase I experiment.
A similar design could also be used for precisely measuring the trajectory of
the positron in the $\mu\rightarrow e\gamma$ search. Since positrons from this
decay are mono-energetic a further optimisation of the Mu3e spectrometer can
be done and applied to MEG to achieve a momentum resolution of about 100
keV/$c$. This optimisation aims at detecting the decay positrons after curling
half a turn ($180\text{\,}\mathrm{\SIUnitSymbolDegree}\text{/}$ in the
solenoid field). This can be achieved by operating the Mu3e pixel detector in
a magnetic field of about $2.5\text{\,}\mathrm{T}\text{/}$ (instead of
$1.0\text{\,}\mathrm{T}\text{/}$) or by increasing the radius of the outer
pixel layer with respect to the Mu3e phase I design. In order to fully profit
from the high rate capability of the silicon detector technology (in terms of
radiation hardness) the sensor is required to provide a very good time
resolution as well. The HVMAPS sensor used for the Mu3e phase I experiment has
a time resolution of about $5\text{\,}\mathrm{n}\mathrm{s}$ and would
perfectly fulfil the requirements of a $\mu\rightarrow e\gamma$ search at
HIMB. For the $\mu\rightarrow e\gamma$ search the first tracking layer must be
as thin as possible in terms of radiation length. The thinner the sensor the
less multiple scattering affects the measurement of the initial positron
direction, which is derived from the hit positions in the first two tracking
layers and is crucial for the topological BG rejection (back-to-back signature
of signal).

Figure 17: Sketch of the active muon stopping target concept for the
reconstruction of $\mu\to e\gamma$ events. Extrapolated positron tracks are
assigned to hits (muon decay) in the active target to improve the vertex
information (direction of the positron and photon).

The experimentally proven radiation hardness of the HVCMOS process [61] also
allows to use HVMAPS for beam monitoring applications and the usage as active
muon stopping target such that the muon decay position is directly measured.
The achievable resolution is a few tens of microns and much smaller than the
uncertainty of a few hundreds of micrometer by extrapolating the reconstructed
positron trajectory back to the muon stopping target, see Figure 17. By
directly measuring the muon decay vertex, the direction of the signal
positrons and photons can be more precisely determined and BG not exhibiting
the back-to-back topology can be more efficiently suppressed. This concept,
however, only works if the active muon stopping target is not too thick since
multiple scattering inside the material would otherwise compromise the
measurement of the positron direction. Studies are ongoing where the signal
response of thinned HVMAPS is measured and a small reduction of the hit
detection efficiency has been observed with the MuPix10 sensor for substrate
thickness below 50 $\mu$m. This efficiency loss, however, might be recovered
with an improved amplifier design in future sensor versions. Furthermore, in
order to make this concept viable, the density of muon stops must not be too
high as otherwise ambiguities in the matching of positron tracks with the muon
decay positrons will reduce the track linking efficiency. This requirement
defines a limit on the maximum muon stop rate, depending on the maximum
distance (radius) of the first tracking layer and the thickness of the first
tracking layer in case of a silicon tracker.

Figure 18: Sketch of an active converter silicon sensor implemented as HVMAPS.
The ionisation loss is proportional to the measured signal amplitude. The
active zone where the charge (electrons) is collected is about 40 times larger
than the inactive zone where the CMOS circuits are placed for readout.

The possibility to implement an active conversion target (see Section 2.3.4)
in a depleted silicon detector as photon detector was already discussed in the
context of a combined Mu3e-MEG experiment in Section 2.2.6. A sketch of such a
detector is shown in Figure 18. Photons with an energy
$\gtrsim$10\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ convert in the sensor
material and create an $e^{+}e^{-}$ pair which undergoes ionisation loss.
Electron-hole pairs are produced in the silicon. The electrons drift to the
anode (n-well) and are registered by the readout circuitry which is
implemented in CMOS. An interesting feature of this concept is that the active
converter can also be used as high precision $e^{+}e^{-}$ spectrometer if the
pixel size is not too large. The goal is to reach an energy resolution of
about $\sigma_{E}\approx$100\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$$ for
$53\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ photons.

For the development of the active converter pixel sensor the following
challenges need to be tackled. Assuming a substrate resistivity of about
$\gtrsim$15\text{\,}\mathrm{k}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$$
a depletion voltage of about $200\text{\,}\mathrm{V}$ is required to fully
deplete $600\text{\,}\mu\mathrm{m}$ of silicon. The pixel and guard-ring
design must therefore allow for a breakdown voltage
$\geq$200\text{\,}\mathrm{V}$$. The pixel size should not be larger than
$150\text{\,}\mu\mathrm{m}$ for the $B=$2\text{\,}\mathrm{T}\text{/}$$ setup
so that spatial hit uncertainties do not deteriorate the momentum resolution.
The pixel sensor circuitry has to provide a time resolution of
$\sigma_{t}\lesssim$100\text{\,}\mathrm{p}\mathrm{s}$$. The TDC and the
readout electronics has to be capable to provide a time resolution of a
$\sigma_{t}\lesssim$100\text{\,}\mathrm{p}\mathrm{s}$$. The amplitude of the
signal should be measured with a relative accuracy of 10% or better.
Furthermore all active converter layers must not have any significant amount
of support material, making the detector design very challenging. Significant
R&D effort will be required to realise such an active converter pixel sensor.

In summary, monolithic silicon detectors are a very promising detector
technology for search for $\mu\to e\gamma$. They provide excellent tracking in
a high-rate experiment at HIMB and can also be used as active muon stopping
target and active photon converter.

###  2.4 Muon moments

Measurements of magnetic and electric dipole moments have historically
provided unique insights into new physics as well as stringent tests of the
SM. There is huge potential for a future muon ($g-2$) experiment at PSI, using
re-accelerated muons from a new low-energy, high-brightness source (muCool)
coupled to the future HIMB. This complements ongoing efforts at Fermilab and
J-PARC, potentially further increasing the precision. Similarly, there is
potential to improve the sensitivity of a search for a muon EDM to better than
$1\text{\times}{10}^{-23}\text{\,}\mathrm{e}\text{/}$. A null result sets a
stringent limit on a currently poorly constrained Wilson coefficient and a
discovery of a muon EDM would prove the existence of physics beyond the SM.

####  2.4.1 ${g-2}$ and muon electric dipole moment

The AMM of the electron and muon have a long tradition as precision tests of
the SM, going back to Schwinger’s famous prediction
$a_{\ell}=(g-2)_{\ell}/2=\alpha/(2\pi)$ [62, 63]. For the electron, the
precision at which SM prediction and direct measurement can be confronted is
approaching $10^{-13}$, with [64]

$a_{e}^{\text{exp}}=1\,159\,652\,180.73(28)\times 10^{-12}.$ (2.4)

Regarding the SM prediction, the mass-independent $4$-loop QED coefficient is
known semi-analytically [65], so that the dominant uncertainties, both at the
level of $10^{-14}$, arise from the numerical evaluation of the $5$-loop
coefficient [66] and hadronic contributions [67].777 For the $5$-loop QED
coefficient there is a $4.8\sigma$ tension between [66] and [68] in the
evaluation of diagrams without closed lepton loops. The present limiting
factor thus concerns independent input for the fine-structure constant
$\alpha$, with the current most precise measurements in atom interferometry
[69, 70] differing by $5.4\sigma$,

$\displaystyle a_{e}^{\text{SM}}[\text{Cs}]$
$\displaystyle=1\,159\,652\,181.61(23)\times 10^{-12},$ $\displaystyle
a_{e}^{\text{SM}}[\text{Rb}]$ $\displaystyle=1\,159\,652\,180.25(10)\times
10^{-12},$ (2.5)

i.e., $2.5\sigma$ above and $1.6\sigma$ below (2.4), respectively.

The value for the muon provides an intriguing hint of new physics given the
$4.2\sigma$ discrepancy between the theoretical predication and current
world’s best measurement. The experimental value is

$a_{\mu}^{\text{exp}}=116\,592\,061(41)\times 10^{-11},$ (2.6)

with a combined precision of $0.35\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$
after averaging the Run-1 result from the Fermilab experiment [12, 71, 72, 73]
and the Brookhaven measurement [13]. The resulting $4.2\sigma$ tension with
the SM prediction [14]

$\displaystyle a_{\mu}^{\text{SM}}=116\,591\,810(43)\times 10^{-11}$ (2.7)

will be scrutinised at subsequent runs at Fermilab [74] and at J-PARC [75],
with a precision goal of $0.14\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ and
$0.45\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$, respectively. In particular,
the J-PARC experiment will pioneer a new experimental technique that does not
rely on the so called “magic momentum” in a storage ring, see also [76], with
a set-up similar to the experiment proposed here, see Section 2.4.4.

The uncertainty in the SM prediction (2.7) is completely dominated by hadronic
contributions, with HVP and HLbL at
$0.34\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ and
$0.15\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ producing a total precision of
$0.37\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$. This value reflects the current
recommendation from the muon $g-2$ Theory Initiative [14], which is based on
major input from [77, 66, 78, 79, 80, 81, 82, 83, 84, 67, 85, 86, 87, 88, 89,
90, 91, 92, 93, 94]. While the current precision suffices for the comparison
to the precision of the current measured value, further improvements both on
HVP and HLbL are required to match the experimental projections, including new
$e^{+}e^{-}\to\text{hadrons}$ data, consolidated lattice-QCD calculations of
HVP, and direct input on space-like HVP from the proposed MUonE experiment
[95, 96]. A first lattice calculation of HVP at sub-percent precision [97] has
been reported recently, indicating a tension with the data-driven
determination [98, 99, 100, 101, 102]. Other developments include new data for
$e^{+}e^{-}\to\pi^{+}\pi^{-}$ from SND [103] and improved radiative
corrections [104]. For HLbL, a recent evaluation in lattice QCD agrees with
the phenomenological result at a similar level of precision [105], but the
sub-leading contributions to HLbL [106, 107, 108, 109, 110, 111] need to be
better understood to meet the target precision set by experiment.

The current discrepancy between the measured and theoretical value is larger
than the electroweak contribution and so requires some form of enhancement
mechanism, but there are several well-motivated examples that provide this.
One example concerns new light, weakly coupled models that provide an
explanation via the small mass of the new particle [112, 113, 114, 115, 116,
117, 118, 119]. Alternatively, solutions with new heavy particles above the
electroweak scale are possible if the chirality flip originates from a large
coupling to the SM Higgs, instead of the muon Yukawa coupling in the SM,
leading to a chiral enhancement that allows for viable solutions for particle
masses up to tens of TeV [120, 121, 122, 123].

The most-studied theoretical framework of BSM physics above the electroweak
scale that can explain $g-2$ of the muon is the MSSM. Here, the necessary
chiral enhancement is provided by $\tan\beta\equiv v_{u}/v_{d}$, where $v_{u}$
and $v_{d}$ are the vacuum expectation values of the two Higgs doublets
$H_{u}$ and $H_{d}$, which give mass to up-type and down-type fermions,
respectively. A large value of $\tan\beta\approx 50$ is suggested by
top–bottom Yukawa coupling unification [124, 125], which can thus provide a
natural explanation [126, 127, 128].

Another possible explanation, also motivated by the hints for lepton flavour
universality violation in semi-leptonic $B$ decays, are leptoquarks. In fact,
two scalar leptoquark representations can provide a chiral enhancement factor
of $m_{t}/m_{\mu}\approx 1600$ [129, 130, 131, 132, 133, 134]. This allows for
a BSM explanation with perturbative couplings that is not in conflict with
direct LHC searches. It is furthermore very predictive as it involves only two
free couplings whose product is determined by $g-2$. Thus, correlated effects
in $h\to\mu^{+}\mu^{-}$ [134], and, to a lesser extent, in
$Z\to\mu^{+}\mu^{-}$ [133, 135] arise.

Alternatively, there exist many heavy BSM explanations such as composite or
extra-dimensional models [136, 137, 138] or models with vector-like leptons
[121, 139, 140, 141, 142, 143, 122, 144, 145], including in addition a second
Higgs doublet [146, 147, 148](2HDM). Note that also a pure 2HDM can provide a
solution, either via Barr–Zee diagrams in the 2HDM-X [149, 150, 151, 152, 153,
154] or through a lepton-flavour-violating $\tau\mu$ coupling [155, 156, 157,
158], which is, however, strongly constrained from $h\to\tau\mu$ searches.

Unlike the magnetic dipole moment of the muon, the EDM is predicted to be well
below the reach of current experiments and so any measurement would be an
indication of new physics. In comparison to other EDM bounds [159], the limit
on the muon EDM is particularly weak: the current direct limit
$d_{\mu}<$1.5\text{\times}{10}^{-19}\text{\,}\mathrm{e}\text{/}$$ [160] is ten
orders of magnitude higher than the one for
$d_{e}<$1.1\text{\times}{10}^{-29}\text{\,}\mathrm{e}\text{/}$$ [161]. One
reason for this is that in the past, the muon EDM was often discarded as a
valuable BSM probe by rescaling the limits on $d_{e}$, assuming MFV [162, 163,
164, 165] (by the ratio $m_{\mu}/m_{e}$), with a resulting limit of
$d_{\mu}<$2.3\text{\times}{10}^{-27}\text{\,}\mathrm{e}\text{/}$$, which is
orders of magnitude below the direct limit. However, MFV is, at least to some
extent, an ad-hoc symmetry imposed to allow for light particle spectra, in
particular within the MSSM, where this reduces the degree of fine-tuning in
the Higgs sector while respecting at the same time flavour constraints.
However, since the LHC did not discover any new particles directly [166, 167],
the concept of naturalness is challenged. Furthermore, LHCb, Belle, and BaBar
discovered significant tensions in semi-leptonic $B$ decays [168, 169, 170,
171, 172, 173, 174, 175, 176, 177] implying a discrepancy significantly above
the $5\sigma$-level within a global analysis [178, 179, 180, 181, 182]. These
hints for BSM physics point towards a significant violation of lepton flavour
universality and are therefore not compatible with MFV in the lepton sector
[183].

Further, the $4.2\sigma$ tension in the muon $g-2$ sets the expected size for
the muon EDM if the $CP$-violating phase in the respective Wilson coefficient
is sizeable. That is, even though the value of $g-2$ is not directly related
to the EDM, any BSM contribution would result from the real part of the Wilson
coefficient whose imaginary part determines the EDM, with an
$\mathcal{O}(1)$-phase (in case of a chirally enhanced explanation) leading to
a muon EDM of the order of ${10}^{-22}\text{\,}\mathrm{e}\text{/}$. Moreover,
while $g-2$ by itself does not conflict the MFV paradigm, solutions with
chiral enhancement mentioned above can violate the MFV scaling [139, 122], and
particularly in the case of leptoquarks they even must do so, in order to
respect the bounds from $\mu\to e\gamma$ [133]. At the same time, such
scenarios automatically provide an a priori free phase, leading to a large EDM
unless the phase happens to be small.

Therefore, it is well-motivated that the BSM flavour structure goes beyond
MFV, a notion sometimes contested on grounds of naturalness arguments.
However, this does not mean that a conflict with $\mu\to e\gamma$ arises, as
in the limit of vanishing neutrino masses lepton flavour is conserved, and
thus it is possible to completely disentangle the muon from the electron EDM.
This can even be achieved via a symmetry, such as $L_{\mu}-L_{\tau}$ symmetry
[184, 185, 186], which, even after its breaking, protects the electron EDM and
$g-2$ from BSM contributions [142]. Also from an EFT point of view [187, 122,
188], it is clear that the muon EDM can be large and that its measurement is
the only way of determining the imaginary part of the associated Wilson
coefficient.

For these reasons, both a high-precision measurement of the muon $g-2$ and a
dedicated muon EDM experiment are highly motivated, and would be valuable
contributions in the search for BSM physics in low-energy precision
observables.

####  2.4.2 A ${g-2}$ / EDM measurement at PSI

Currently the most precise measurement of the muon $g-2$ is being performed at
Fermilab, with an expected relative precision of
$0.14\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$. The second experiment proposed
at J-PARC [75] will have a precision of about
$0.45\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$. Both experiments also intend to
use the same data to search for the muon EDM with a precision of about
$1\text{\times}{10}^{-21}\text{\,}\mathrm{e}\text{/}$ [189, 75].

The possible experiment at PSI would provide an independent measurement of the
muon $g-2$ with a different experimental technique and systematic
uncertainties, as well as providing insight into uncharted terrain by
improving on the sensitivity to a muon EDM.

Both measurements observe the spin precession $\vec{\omega}$ of a muon in a
storage ring with an electric field $\vec{E}$ and magnetic field $\vec{B}$
given by

$\vec{\omega}=\frac{q}{m}\left[a\vec{B}-\left(a+\frac{1}{1-\gamma^{2}}\right)\frac{\vec{\beta}\times\vec{E}}{c}\right]+\frac{q}{m}\frac{\eta}{2}\left(\vec{\beta}\times\vec{B}+\frac{\vec{E}}{c}\right).$
(2.8)

Certain choices of the muon momentum and the combination of electric and
magnetic fields permit measurements with strongly reduced systematic effects.

The muEDM collaboration at PSI proposes a search for the muon EDM with a
sensitivity of about $6\text{\times}{10}^{-23}\text{\,}\mathrm{e}\text{/}$ for
a year of data taking using muons with a momentum of
$|\vec{p}|=$125\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}$c$,
$|\vec{\beta}|=|\vec{v}|/c=0.77$, and an average polarisation of better than
$P=93\%$ from the $\mu$E1 beam line at PSI with a particle flux of up to
$2\times 10^{8}$ $\mu^{+}$/s. The concept is based on the frozen-spin
technique [190, 191] combined with a spiral injection into a magnetic field of
$B=$3\text{\,}\mathrm{T}$$, similar as in the J-PARC $(g-2)$/muEDM experiment
[192, 75]. A sketch of the experiment is shown in Figure 19.

Figure 19: Sketch of the helix muEDM instrument, not to scale.

The search profits from the large electric field in the rest frame of the muon
$|E^{\ast}|=|\gamma
c\vec{\beta}\times\vec{B}|=$1.1\text{\,}\mathrm{G}\mathrm{V}\mathrm{/}\mathrm{m}$$.
A radial electrode system provides an electric field $\vec{E}_{\mathrm{f}}$
perpendicular to the motion of the muon and the magnetic field $\vec{B}$,
hence $\vec{\beta}\cdot\vec{B}=\vec{\beta}\cdot\vec{E}_{\mathrm{f}}=0$, and
$\vec{B}\cdot\vec{E}_{\mathrm{f}}=0$. By adjusting the strength of the
electric field $\vec{E}_{\mathrm{f}}$ such that

$a\vec{B}=\left(a-\frac{1}{\gamma^{2}-1}\right)\frac{\vec{\beta}\times\vec{E_{\mathrm{f}}}}{c},$
(2.9)

it is possible to cancel the anomalous precession term in (2.8), which
simplifies the spin precession to

$\vec{\omega}=\frac{q}{m}\frac{\eta}{2}\left(\vec{\beta}\times\vec{B}+\frac{\vec{E_{\mathrm{f}}}}{c}\right),$
(2.10)

known as the frozen-spin technique.

The statistical sensitivity of this proposal is limited by the maximum muon
momentum and the large lateral phase at $\mu$E1. As we outline in the
following, by using a dedicated beam line with sub millirad divergence and a
momentum of $P=$210\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/c$, this concept
has the potential to improve the sensitivity of the muEDM search to better
than $1\text{\times}{10}^{-23}\text{\,}\mathrm{e}\text{/}$. Without the
electric field, i.e., with both electrodes grounded, this same setup measures
the anomalous magnetic moment and could reach a relative statistical precision
of the muon $g-2$ of about $0.1\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$.

####  2.4.3 Prospects for a search of the muon electric dipole moment

The sensitivity for a muon EDM experiment deploying the frozen-spin technique
[190] with $E\approx aBc\beta\gamma^{2}$ is given by

$\sigma(d_{\mu})=\frac{\hbar}{2\beta cP\gamma B\sqrt{N}\alpha\tau}.$ (2.11)

Figure 20 illustrates expected sensitivities to a muon EDM in the case that a
new dedicated beam could deliver muons with a phase space optimally adapted
for injection. We assume an overall positron detection efficiency of 70%. As a
benchmark case we consider the solenoid based layout described in the letter
of intent [193] with a muon orbit of $r=$0.14\text{\,}\mathrm{m}$$. Note that
the sensitivity at $B=$3\text{\,}\mathrm{T}$$, the nominal field value for the
experiment proposed for the $\mu$E1 beamline, will be improved by a factor two
if the next muon can be injected on request, whenever the previous positron
decay was confirmed or a time-out of five times the laboratory lifetime is
reached. In this single-muon-at-a-time scenario it is possible to improve the
sensitivity to about $1\text{\times}{10}^{-23}\text{\,}\mathrm{e}\text{/}$.
Figure 20b shows that with a mean muon multiplicity above three, i.e. in
average three muons are injected at a time, and a magnetic field above
$B=$5\text{\,}\mathrm{T}$$ a sensitivity of better than
$8\text{\times}{10}^{-24}\text{\,}\mathrm{e}\text{/}$, c.f. current limit
$d_{\mu}<$1.5\text{\times}{10}^{-19}\text{\,}\mathrm{e}\text{/}$$ [160], in a
year of measurement can be reached. This could be accomplished with a muon
rate from muCool of $5\text{\times}{10}^{5}\text{\,}\text{/}\mathrm{s}$ and a
repetition rate of $100\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ (see Table 3
for a comparison of potential sensitivities).

(a)

(b)

Figure 20: Sensitivity landscape for muon EDM searches. (a) Single muon-on-demand: Sensitivity increases with magnetic field strength keeping the radius constant. Electric fields above $8\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$ are very difficult to obtain for large electrodes with extremely thin material thickness (positron transmission). The region within the circle indicates the possible parameter space for a future muEDM search using re-accelerated muons from muCool. (b) Increase in sensitivity with multiplicity of muons per injection for the case $r=$0.14\text{\,}\mathrm{m}$$, as function of the magnetic field. Table 3: Comparison of two future scenarios on a re-accelerated high-brightness muon beam with the concept proposed for $\mu$E1. A measurement with a sensitivity better than $1\text{\times}{10}^{-23}\text{\,}\mathrm{e}\text{/}$ can only be realised with a larger magnetic field and higher momentum, which in turn also requires a higher electric field. Note that injection of more than one muon per measurement also requires a longer time-out, which partially compensates for the increase in injected muons. |  $\mu$E1 | HIMB muCool
---|---|---
|  $125\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/$c$ |  $125\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/$c$ |  $210\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/$c$
$E$-Field (MV/m) | 2 | 2 | 8
$B$-Field (T) | 3 | 3 | 5
radius (m) | 0.14 | 0.14 | 0.14
${\mathrm{e}^{+}}$/year (1 muon) | $7.3\text{\times}{10}^{11}$ | $4\text{\times}{10}^{12}$ | $3\text{\times}{10}^{12}$
Sensitivity/ year (1 muon) | 6 | 2.3 | 1
($1\text{\times}{10}^{-23}\text{\,}\mathrm{e}\text{/}$) |  |  |
Sensitivity/ year (3 muons) | - | 1.9 | 0.8
($1\text{\times}{10}^{-23}\text{\,}\mathrm{e}\text{/}$) |  |  |

####  2.4.4 Prospects for a high-precision measurement of the muon ${g-2}$

The HIMB in combination with a muon cooling stage and a re-acceleration
beamline means that the same experimental setup, without the electric field
applied, can be used to make a dedicated measurement of the magnetic moment of
the muon. In this AMM measurement in the muEDM setup (AMMiED) no electric
field will be applied and (2.8) will reduce to

$\vec{\omega}=\frac{q}{m}\left(a\vec{B}\right)$ (2.12)

in the absence of an EDM. In the case of using the same apparatus as for the
search for an EDM the central electrode should be removed, to reduce multiple
scattering, while the outer electrode is simply grounded.

The relative sensitivity of the anomalous frequency measurement is given by

$\sigma(a_{\mu})_{\mathrm{rel}}=\frac{\sqrt{2}\quad
m_{\mu}}{P\sqrt{N}\gamma\tau\alpha~{}ea_{\mu}B}.$ (2.13)

#### Measurement of $\boldsymbol{g-2}$ in muEDM setup

For the sensitivity calculation, shown in Figure 21, we assume the same values
for $P$ and $\alpha$ as in the muEDM scenario and a muon on request operation
scheme with an integral positron detection efficiency of 70%. It can be seen
that for a field strength of $B=$6\text{\,}\mathrm{T}$$ a statistical
sensitivity of $0.1\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ can be reached,
matching the expected final precision of the experiment at Fermilab. An
improvement can be seen when the muon multiplicity is increased with the
potential to achieve about $0.06\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$, or
to match the Fermilab sensitivity with a lower field strength of
$B=$4\text{\,}\mathrm{T}$$.

In terms of the systematic uncertainties, the experimental design benefits
from the more compact magnet and hence a high field homogeneity is expected,
as demonstrated in MRI technologies. Further investigations need to be
undertaken in order to fully understand the exact levels of these
uncertainties.

Figure 21: Statistical sensitivity in ppb for a measurement of the muon $g-2$.
For this calculation the radius was fixed to $r=$0.14\text{\,}\mathrm{m}$$ and
$P=qBr$.

Both previously discussed scenarios for a muon EDM and AMM measurement are
based on the assumption of a highly efficient re-acceleration of muons to at
least $125\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/$c$ from the future low-
energy, high-brightness source muCool. The caveat of such a scenario is the
difficulty of a low loss extraction of muons from the high magnetic field
region in muCool to the first acceleration stage and subsequent further
acceleration. Although the initial phase space density and lateral extension
within the muCool source are excellent, significant losses will appear due to
muon decay and a mismatch of final beam and the experimental acceptance phase
space, resulting in the likely situation of one single-muon-on-request-
scenario.

#### Tiny all magnetic $\boldsymbol{g-2}$ storage ring

An alternative concept, similar to first ideas presented during the PSI
workshop 2010 by D. Taqqu, is to use non-relativistic muons, with an energy of
$1\,\mathrm{MeV}/c$, in a tiny storage ring as depicted in Figure 22. This
scenario circumvents the challenges of a long re-acceleration beam line by
proposing to use only one RFQ acceleration stage within the same high magnetic
field of muCool and transporting the muons into a high magnetic solenoid field
of $17\text{\,}\mathrm{T}$. The muons longitudinal motion along the field
lines is stopped by a short quadrupolar pulse similar as in the PSI muEDM
concept. The muon is then confined to an orbit of about
$6\text{\,}\mathrm{m}\mathrm{m}$ diameter by using a weakly focusing magnetic
field configuration. A combination of scintillating fibres and a segmented
calorimeter detects the decay positron. Although, sub nanosecond timing
resolution for scintillating fibres has been reported, a direct measurement of
the muon spin precession frequency to extract $g$ and not $g-2$ seems still
too ambitious as it would require a resolution of better than
$40\text{\,}\mathrm{p}\mathrm{s}$. Instead we propose to use a central
scintillating fibre bundle within the tiny storage orbit to distinguish
between in and outwards going positrons for a measurement of the anomalous
precession frequency.

Figure 22: Sketch of a hyper compact $g-2$ storage ring experiment, TiAMMo, using muons with $E=$1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$ from a moderately re-accelerated low-energy, high-brightness muon source in a $17\text{\,}\mathrm{T}$ magnetic field. Table 4: Comparison of principle parameters of Fermilab E989 [74], J-PARC [75] and our proposals (AMMiED and TiAMMo). | Fermilab E989 | J-PARC | AMMiED | TiAMMo
---|---|---|---|---
Muon momentum | $$30.9\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$/c$ | $$300\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/c$ | $$125\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/c$ | $$14.6\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/c$
Lorentz $\gamma$ | 29.3 | 3 | 1.56 | 1
Polarisation | 100% | 50% | 95% | 95%
Magnetic field $B$ | $1.45\text{\,}\mathrm{T}$ | $3\text{\,}\mathrm{T}$ | $6\text{\,}\mathrm{T}$ | $17\text{\,}\mathrm{T}$
Focusing field | E-quadrupoles | weakly focusing (wf) | wf | wf
Cyclotron period | $149\text{\,}\mathrm{n}\mathrm{s}$ | $7.4\text{\,}\mathrm{n}\mathrm{s}$ | $3.8\text{\,}\mathrm{n}\mathrm{s}$ | $0.4\text{\,}\mathrm{n}\mathrm{s}$
AMM precession period | $4.37\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ | $2.11\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ | $1.05\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ | $0.37\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$
Number of detected $e^{+}$ | $1.6\text{\times}{10}^{11}$ | $5.7\text{\times}{10}^{11}$ | $7\text{\times}{10}^{12}$ | $1.7\text{\times}{10}^{12}$
AMM precision (stat.) | $0.1\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ | $0.45\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ | $0.1\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ | $0.08\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$
AMM precision (sys.) | $0.1\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ | $0.07\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ | – | –

####  2.4.5 Conclusion

There is potential to investigate the tantalising hints of new physics by
measuring the magnetic dipole moment of the muon using a different method and
to reach four orders of magnitude below the current muon EDM limit using a
novel low-energy, high-brightness muon source coupled to the proposed HIMB. In
this section we discussed possible prospects for a muon EDM measurement, which
will ideally require a highly efficient re-acceleration to muon momenta of
about $210\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$/$c$. For a measurement
scheme based on a single-muon-on-request we show that a sensitivity of
$\sigma(d_{\mu})=$1\text{\times}{10}^{-23}\text{\,}\mathrm{e}\text{/}$$ may be
expected within one year of data taking, with a potential improvement if
multiple muons are stored at the same time. Similarly, we presented two
possible scenarios for a high-precision measurement of the muon $g-2$ with a
statistical sensitivity better than 0.1 ppm in one year of data taking,
matching or even surpassing the prospected sensitivity of the currently
ongoing effort at Fermilab (see Table 4). Again higher sensitivity could be
achieved if the source provided sufficient muons to permit a storage of more
than one muon at a time within the storage ring. The tiny non-relativistic
storage ring proposal has the advantage to avoid a long re-acceleration beam
line. The price for this is a re-acceleration scheme within a solenoid guiding
field and a high magnetic field for storage of $B=$17\text{\,}\mathrm{T}$$
resulting in an orbit of less than $6\text{\,}\mathrm{m}\mathrm{m}$ diameter.

###  2.5 Muonium

####  2.5.1 Development of novel muonium sources at PSI

A high-intensity, low-emittance atomic muonium $M=(\mu^{+}e^{-})$ beam is
being developed at PSI, which would enable improving the precision of $M$
spectroscopy measurements, and may allow a direct observation of the $M$
gravitational interaction.

Improvements in measuring the 1S-2S transition frequency using state-of-the
art laser spectroscopy techniques is strongly motivated by recent experiments
measuring the muon AMM [12], and other searches for BSM physics. Measurement
of the free fall of $M$ atoms would be the first test of the weak equivalence
principle using elementary antimatter ($\mu^{+}$) and a purely leptonic
system.

Both experiments rely on the high-intensity, continuous muon beams at PSI with
significant benefits expected from the proposed HIMB upgrade. The especially
challenging gravity experiment additionally relies on a proposed novel atomic
$M$ source, with cold atoms converted in the surface of SFHe.

Present state-of-the-art vacuum muonium sources are room temperature, porous
materials that allow combination of the muon with an electron from the bulk,
and a following quick diffusion inside the nanoscopic pores, with laser
ablated silica aerogel being one of the best room temperature converters. Such
sources provide $\sim 3\%$ muon-to-vacuum $M$ conversion using surface
$\mu^{+}$ beams of 28 MeV/$c$ momentum [194].

Mesoporous materials have been shown to convert $\mu^{+}$ to vacuum $M$ with
efficiencies of 40% (20%) at room temperature (down to 100 K) when using
highly moderated, keV energy muons from the LEM beamline at PSI [195, 196];
this has an intensity four orders-of-magnitude lower than a surface muon beam.
Future developments of moderated low-energy muons (the muCool experiment
described in Section 4.2) may reach one to two orders of magnitude higher
rates. However, such converters produce a $M$ beam with broad (thermal) energy
and angular ($\sim\cos\theta$) distributions. Improving the source quality by
cooling these samples results in lower emission rates, with no observable
emission below $\sim 50$ K due to the decreased diffusion constant, and to the
sticking of $M$ to the pore walls that occurs unavoidably with any
conventional $M$ converter [195, 197].

A newly proposed experiment at PSI plans to tackle the above difficulties by
using SFHe as a vacuum $M$ converter, due to its inert nature that rejects
impurities from its bulk even at the lowest temperatures [198, 199, 200].
Based on the calculated and measured chemical potentials of hydrogen isotopes
in SFHe [201, 202], $M$ atoms are expected to have a high chemical potential
in the liquid compared to the thermal energies, implying that they would be
ejected from the SFHe surface with a well defined longitudinal velocity of
$v_{M}\sim 6300$ m/s only broadened by the thermal spread which is expected to
fall below the Landau velocity ($v_{t}\approx 50$ m/s).

Assuming high muon-to-vacuum-muonium conversion rates on the order of 10-20%
in SFHe, the present $\pi$E5 beamline at PSI could provide a flux of
$1-5\times 10^{5}$ $M$/s from the cold atomic beam, while the HIMB beamline
could offer up to an order of magnitude higher rates.

####  2.5.2 Spectroscopy

Pure leptonic systems, such as positronium and muonium are free of finite-size
effects and therefore are ideal systems to test bound state QED [203]. They
are essential to extract fundamental constants such as the muon mass and
magnetic moment. Those systems can also be used to search for new physics,
including dark-sector particles and new muonic forces [204], as well as
testing Lorentz/CPT symmetries [205] or measure the effect of gravity on
antimatter via the gravitational redshift [206].

Past $M$ spectroscopy experiments were conducted between $1980-2000$ at
TRIUMF, RAL and LAMPF (see [207] for a recent review). As a result of the
difficulty in obtaining a high flux of $\mu^{+}$ and the necessity to slow
down the muons so that $M$ into vacuum can be formed efficiently, all those
experiments were essentially limited by statistics, or statistics-related
systematic effects [207]. With its intense $\mu^{+}$ beam, PSI harbours
tremendous opportunities for improving $M$ spectroscopy experiments. Higher
statistics makes it possible to implement experimental techniques that are
systematically more robust. In this respect the LEM beamline at PSI [208]
plays a crucial role as recently demonstrated by the Mu-MASS collaboration
[209] which improved the determination of the $M$ Lamb shift (LS) by one order
of magnitude [210].

A recent review of the ongoing measurements of the 1S-2S transition and the LS
of muonium in the context of the Mu-MASS experiment at PSI is given in [211].
A description of the progress of the Muonium Spectroscopy Experiment Using
Microwave (MuSEUM) that is ongoing at J-PARC aiming to improve the muonium HFS
can be found in [212]. Here we point out the impact that HIMB would have on
muonium spectroscopy. HIMB would increase by orders of magnitude the available
statistics of these experiments. This will also allow for accurate studies of
the systematic effects and for the implementation of new measurement schemes,
such as the employment of an enhancement cavity with a larger laser beam to
reduce AC-stark shift in the 1S-2S measurement or the use of separated
oscillatory fields spectroscopy to measure Lamb shift, fine structure and HFS.

HIMB would allow to push the experimental projected accuracy of the ongoing
MuMASS and MUSEUM experiment by an order of magnitude, below 1 kHz for the
1S-2S transition and to the few Hz level for the HFS. Assuming the theoretical
accuracy could also be improved to match the experimental one, combining those
measurement will result in a very stringent test of bound state QED [213]. It
will also allow to improve the determination of fundamental constants such as
the muon mass, the muon magnetic dipole moment, and a determination of the
Rydberg constant independent of finite size effects as well as the
determination of the fine structure constant at a level comparable with the
current best determinations [70, 69] .

Moreover, as pointed out recently [214], $M$ spectroscopy at this level of
precision would provide an independent determination of the muon AMM with an
uncertainty at the level of the current tension. This is illustrated in Figure
23.

Figure 23: Muon AMM determined from the combined measurements at BNL and FNAL
(red) [12], SM calculations with leading-order HVP evaluated from
$e^{+}e^{-}\to$ hadrons data (green) [14] or using a recent lattice QCD result
(gray) [97]. Also shown are the projected sensitivities in muonium (dashed)
with the currently planned spectroscopy improvements (blue) and its ultimate
improvement which could be reached at HIMB (magenta), centred around the
current experimental average. The orange band shows the four-fold improved
FNAL standard deviation expected in the near future. Adapted from [214].

$M$ spectroscopy offers also the possibility to search for new light bosons
coupled to electrons and muons [204]. This is very interesting since those
could provide an explanation in terms of new physics of the muon AMM. In this
case, a dark force between the electron and the antimuon could be mediated for
example by a new scalar or a new vector gauge boson giving rise to Yukawa-like
attractive potentials [215]

$\displaystyle V_{ss}(\vec{r})=-$ $\displaystyle
g_{e}^{s}g^{s}_{\mu}\,\frac{e^{-m_{s}r}}{4\pi r}$ (2.14) $\displaystyle
V_{vv}(\vec{r})=-$ $\displaystyle g^{v}_{e}g^{v}_{\mu}\frac{e^{m_{v}r}}{4\pi
r}$ $\displaystyle+$
$\displaystyle\frac{g^{v}_{e}g^{v}_{\mu}}{4}\bigg{(}\vec{\sigma}_{e}\cdot\vec{\sigma}_{\mu}\Big{(}\frac{1}{r^{3}}+\frac{m_{v}}{r^{2}}+\frac{m_{v}^{2}}{r}-\frac{8\pi}{3}\delta(\vec{r})\Big{)}$
$\displaystyle\qquad-(\vec{\sigma}_{e}\cdot\hat{r})(\vec{\sigma}_{\mu}\cdot\hat{r})\Big{(}\frac{3}{r^{3}}+\frac{3m_{v}}{r^{2}}+\frac{m_{v}^{2}}{r}\Big{)}\bigg{)}\,\frac{e^{-m_{v}r}}{4\pi
m_{e}m_{\mu}}$ (2.15)

where $V_{ss}$ and $V_{vv}$ are for scalar and vector potentials,
respectively, $m_{s}$ is the scalar boson mass, $m_{v}$ is the vector boson
mass, and $g^{s,v}_{e},g^{s,v}_{\mu}$ are the coupling strengths to electrons
and antimuons. The effect of such new forces on the measured transitions can
be estimated by applying first order perturbation theory.

In Figure 24, we present the current and projected sensitivity with HIMB of
muonium spectroscopy to new physics. The constraints on
$g^{s}_{e},\,g^{s}_{\mu}$ as a function of the scalar/vector mass, which are
nearly identical in the mass range considered here, are compared to the region
favoured by the $g-2$ muon anomaly, considering the bounds from the electron
gyromagnetic factor [64]. However, as pointed out in Section 2.4.1, the
situation for the electron AMM should be clarified since the two latest
determinations of the fine structure constant using atom interferometry differ
from each other by 5 $\sigma$ [69, 70]. We do not present results from
experiments at the intensity frontier since those typically depend on
assumptions on the decay channels. An example is the recent results of the
NA64 experiment placing stringent bounds on new bosons with the assumption
that those would decay invisibly [216]. The combination of the Lamb shift and
1S$-$2S $M$ measurements provides the most stringent laboratory constraint
excluding that a new scalar/vector boson with a mass $<10\,$keV could
contribute to the muon $g-2$ anomaly.

Figure 24: Constraints from $M$ spectroscopy on $g^{s}_{e},g^{s}_{\mu}$ as a
function of the scalar/vector mass. The solid black line is the constraint
from the $M$ 1S$-$2S measurement [217] while the blue line is from the recent
Lamb shift measurement at PSI [210]. The dashed lines are the projections of
the uncertainty that would be enabled by HIMB. The orange band represents the
region suggested by the $g-2$ muon anomaly considering the lower bound from
the measurement of the electron gyromagnetic factor for the scalar case, while
the hatched region is for the vector one.

Another interesting application is that $M$ spectroscopy is sensitive to
Lorentz and CPT violating effects. In the context of the Standard Model
Extension (SME) this would improve the current limits [205]. Moreover, a test
of the gravitational behaviour of second generation leptons via the
gravitational redshift [206] will become possible.

The realisation of HIMB would thus greatly expand the physics reach of $M$
spectroscopy.

####  2.5.3 Gravitational free fall experiment of muonium

The novel cold atomic beam under development would enable the measurement of
the gravitational fall of $M$ atoms by means of atom interferometry. However,
such an experiment is inherently challenging due to the short lifetime or the
muon ($\tau\approx 2.2~{}\mu$s) which allows a mere $\Delta
x=\frac{1}{2}gt^{2}=600$ pm gravitational fall in a time of $t=5\tau$.
Measuring small deflections like this needs a precise knowledge of the initial
momentum of the atoms, and requires strict momentum selection. Two periodic
gratings (G1 and G2) with horizontal slits of pitch $d$ and spaced by a
distance $L$ could be used to achieve this momentum selection as shown in
Figure 25.

Figure 25: A three-grating interferometer used to measure the gravitational
interaction of $M$ atoms. The quantum diffraction pattern caused by the
gratings G1 and G2 with a fully coherent beam is given in grey. Classical
trajectories (red and dashed lines) are shown to illustrate the effect of
gravity on the measured interference pattern appearing at G3. The vertical
shift of the interference pattern caused by the gravitational acceleration $g$
is detected by measuring the transmitted $M$ rate while scanning G3 in
vertical direction. See details in text.

The classical and quantum regime of this device is characterised by the de
Broglie wavelength of the atoms, $\lambda=h/p$, and grating pitch $d$ in terms
of the Talbot length, $L_{T}=d^{2}/\lambda$, which is approximately 18 microns
for thermal $M$ atoms with $\lambda_{M}\approx 0.56$ nm. Trajectory selection
for both classical and quantum regimes of this device by G1 and G2 will result
in an intensity pattern with the same periodicity $d$ at a distance $L$ after
G2. Gravitational acceleration and deflection of the atoms causes a phase
shift $\delta\phi$ of this pattern in the vertical direction as
$\delta\phi=2\pi gT^{2}/d$, where $T=L/v_{M}$ is the $M$ time of flight
between each pair of gratings.

Direct observation of this sub-micron pattern and sub-nanometer shifts needed
for measuring $M$ gravity would be extremely hard. It is possible however to
carry out an indirect measurement using a third grating (G3) of the same pitch
$d$, placed at distance $L$ from G2. By counting the total rate of $M$ atoms
transmitted through G3 as a function of the G3 vertical position $\Delta x$
the phase shift can be measured.

The contrast of the intensity pattern $C$ is defined by the ratio of the
amplitude and the average yield $C=A/A_{0}$ as shown in Figure 25. This
contrast strongly depends on the transverse coherence length of the beam,
$\ell_{0}$, that determines how many slits of G1 are illuminated with a
coherent wavefront. This coherence length in relation to the beam width
$w_{0}$ and the interferometer parameters (the grating periodicity $d$ and
distances $L$) together with the de Broglie wavelength ($\lambda$) of the
atoms is sufficient to estimate the interferometer performance in the first
approximation. In analogy to statistical optics (Van Cittert-Zernike theorem
[218]), we can relate the transverse coherence length of the $M$ beam to the
transverse momentum distribution of the atoms as
$\ell_{0}=\frac{1}{2}\frac{\lambda}{\alpha}\approx 16$ nm, where $\alpha$ is
the angular spread of the cold $M$ source.

The sensitivity in measuring the gravitational acceleration $g$ is given by
[219]

$\Delta g=\frac{1}{2\pi T^{2}}\frac{d}{C\sqrt{N}}\;,$ (2.16)

where $N$ is the number of $M$ atoms transmitted through G3 and measured by
the detector given by

$N=N_{0}\;\varepsilon_{0}\,e^{-(t_{0}+2T)/\tau}\;(T_{G})^{3}\;\varepsilon_{\mathrm{det}}\,,$
(2.17)

with $N_{0}$ being the number of $M$ atoms produced at the $M$ source, and
$\varepsilon_{0}$ the $M$ transport efficiency from the source to G1. The $M$
decay is accounted for by the third term $e^{-(t_{0}+2T)/\tau}$, where $t_{0}$
is the time of flight from the source to G1. The number of detected $M$ atoms
is further reduced by the $M$ detection efficiency
$\varepsilon_{\mathrm{det}}$, and by the limited transmission $T_{G}$ of a
single grating. The short lifetime of the muon necessitates a gain in
sensitivity by using a small grating pitch $d$. Maximal sensitivity, as a
tradeoff between phase shift $\delta\phi$ and statistics $N$, is obtained for
$T\approx 6-8~{}\mu$s corresponding to an interferometer length of 40-50 mm.

A calculation of the interferometer parameters to extract the contrast $C$,
uses an approximation of the $M$ source with a Gaussian Schell-model beam
[220], and adapted mutual intensity functions that are widely used to describe
the propagation of partially coherent light [218]. Using realistic parameters
on the initial beam size and quality expected from the superfluid source