# Complementary Signals of Lepton Flavor Violation at a High-Energy Muon
Collider

Samuel<EMAIL_ADDRESS>Qianshu
<EMAIL_ADDRESS>and Matthew<EMAIL_ADDRESS>
Department of Physics, Harvard University, Cambridge, MA 02138

(August 27, 2024)

###### Abstract

A muon collider would be a powerful probe of flavor violation in new physics.
There is a strong complementary case for collider measurements and precision
low-energy probes of lepton flavor violation (as well as CP violation). We
illustrate this by studying the collider reach in a supersymmetric scenario
with flavor-violating slepton mixing. We find that the collider could discover
sleptons and measure the slepton and neutralino masses with high precision,
enabling event reconstruction that could cleanly separate flavor-violating new
physics signals from Standard Model backgrounds. The discovery reach of a
high-energy muon collider would cover a comparably large, and overlapping,
range of parameter space to future $\mu\to e$ conversion and electron EDM
experiments, and unlike precision experiments could immediately shed light on
the nature of new physics responsible for flavor violation. This
complementarity strengthens the case that a muon collider could be an ideal
energy-frontier laboratory in the search for physics beyond the Standard
Model.

###### Contents

  1. 1 Introduction
  2. 2 Charged Lepton Flavor Violation in the MSSM
    1. 2.1 Low-Energy Signatures of Lepton Flavor Violation
  3. 3 The MSSM at a High-Energy Muon Collider
    1. 3.1 Measurement of the Slepton and Neutralino Masses
  4. 4 Direct Reach for Flavor-Violating Signatures
  5. 5 Beyond the Minimal Spectrum: LFV, EDM and Collider Complementarity
    1. 5.1 Complementarity with the electron EDM: minimal flavor violation
    2. 5.2 Complementarity with eEDM and $\mu\to e\gamma$ via stau mixing
  6. 6 Conclusions
    1. Acknowledgments
  7. A Flavor and Supersymmetry Breaking
    1. A.1 Gauge mediation with subdominant gravity mediation
    2. A.2 A variant: GMSB with gauged $L_{\mu}-L_{\tau}$
    3. A.3 Should gravity mediation violate flavor maximally?
  8. B Details of the Mass Measurements
    1. B.1 Background Mitigation
    2. B.2 Unbinned Likelihood Procedure
    3. B.3 Additional Results and Discussion
  9. C Slepton Pair Reconstruction Analysis

##  1 Introduction

Since discovering the Higgs boson, the Large Hadron Collider has continued to
push the energy frontier, and in the absence of any clear signs of new
physics, has set ever-stronger constraints on potential extensions to the
Standard Model (SM). At the same time, a great deal of what we know about the
SM and what lies beyond it comes from precision tests of Standard Model
particles. In fact, depending on the assumptions about physics beyond the
Standard Model (BSM), our strongest constraints on new states come from
searches for the electron and neutron electric dipole moments, constraints on
flavor-violating muon decays, or measurements of Kaon mixing. If one assumes
that new physics responsible for flavor-violating neutral currents arises with
anarchic, $\mathcal{O}(1)$ coefficients at a common scale, the scale of new
physics must be $\gtrsim 10^{6}\,\textrm{TeV}$ [1]: far beyond the reach of
colliders.

However, given that the SM itself has a very non-generic flavor structure—the
fermion masses span five orders of magnitude, and the quark mixing matrix has
a clear, hierarchical structure—it is natural to think that the structure of
physics beyond the Standard Model will be non-generic as well. Coupling this
reasoning with the Higgs hierarchy problem, the sensitivity of the electroweak
scale to quantum corrections, there is good reason to expect that we might
find flavorful new physics near the TeV scale. This accentuates the need for
new colliders to probe yet shorter distances and higher energies.

In this light, a high-energy muon collider is a uniquely well-suited option. A
muon collider would have immense capability to extend our reach for new
physics to much higher masses by colliding fundamental objects at multi-TeV
energies [2, 3, 4, 5]. In doing so, such a collider can probe the same physics
that we hope to see signals of in low-energy experiments. This raises the
tantalizing prospect of not only discovering new physics at the TeV scale, but
learning qualitatively and quantitatively about its structure, so that we can
make headway on the fundamental puzzles of the Standard Model.

Among the numerous processes that can be tested reciprocally at both low- and
high-energies, we will focus in this paper on those with charged lepton flavor
violation (LFV). As many of the most sensitive searches for LFV involve muons,
these are the most naturally linked with a muon collider. Indeed, we will see
in an explicit example that searches for mixed-flavor final states at a
collider and processes such as $\mu\to e\gamma$, $\mu\to 3e$ and $\mu$-to-$e$
transitions are probing the same UV physics. An investigation of
complementarity with LFV probes is also timely, as a number of experiments
searching for the aforementioned processes are expected to come online with
orders of magnitude greater sensitivity than their predecessors in the coming
decade [6, 7].

Throughout this work, we will focus on the minimal supersymmetric standard
model (MSSM) as a canonical example of how lepton flavor violation can arise
in models of beyond the Standard Model (BSM) physics. Absent some underlying
symmetry structure, the spectrum of the MSSM naturally yields an assortment of
new contributions to flavor-changing processes and sources of CP violation.
However, while the MSSM contains a large number of new parameters, we will see
that only a few of them are necessary to describe the lepton-flavor violating
signatures at a high-energy muon collider and the complementary signals in
low-energy probes.

While our main goal is to understand the complementarity of collider and low-
energy probes, we will also revisit the expectations for lepton flavor
violation rates in the MSSM. We will mostly work within a simple, minimal
framework that leads to both collider and low-energy signals. Within this
framework, we will highlight the regions of parameter space that are
particularly well motivated in concrete models, with an emphasis on how flavor
violation in the MSSM is intimately connected to how supersymmetry is broken
at higher scales. This framework serves as a useful benchmark for not only the
collider reach, but also for prospective measurements of $\mu\to e\gamma$,
$\mu\to 3e$ and $\mu$-to-$e$ conversion in nuclei.

Realistically, of course, nature is unlikely to subscribe to a simple
construction that is most convenient for studying particular processes. With
this in mind, we will also explore how moving beyond this minimal framework
changes the expected rates for LFV processes at low energies. As we will see,
the next-to-minimal ingredients also give rise to a nonzero electric dipole
moment (EDM) for the electron. This further reinforces the potential of a
high-energy muon collider in being particularly well-suited to explore the
same fundamental physics being tested at low energies.

The rest of this paper is structured as follows: in Section 2, we review the
minimal scenario in which charged lepton flavor violation arises in the MSSM
and discuss the contributions to LFV processes at low energies. Section 3 is
devoted to a discussion of signatures of the MSSM at a muon collider,
including an estimate of how precisely the slepton and neutralino masses can
be measured. In Section 4, we present the muon collider reach for the flavor-
violating processes at high-energies and compare to future constraints Mu3e,
Mu2e, and PRISM/PRIME. Finally, in Section 5 we discuss how the LFV signatures
can change when we include states beyond the minimal ingredients, and the
complementarity with searches for electric dipole moments. We conclude in
Section 6. The appendices contain more discussion on how flavor-violating
terms arise in realistic models of supersymmetry breaking and provide more
details on the slepton and neutralino mass measurement and the reconstruction
procedure used to compute the collider reach.

##  2 Charged Lepton Flavor Violation in the MSSM

Charged lepton flavor violation arises in the MSSM when there are
contributions to the slepton masses that are not diagonal in the same basis as
the SM leptons. When this occurs, the physical sleptons are mixtures of
different flavors, and their interactions with the SM leptons and
neutralinos/charginos will be flavor-violating.

As the Yukawa interactions dictated by supersymmetry are necessarily aligned
with the lepton interactions, the flavor-violating terms must arise from soft
supersymmetry-breaking terms, which depend on the underlying model of
supersymmetry breaking. In this sense, the flavor structure of supersymmetry
is inextricably tied with understanding the dynamics of supersymmetry
breaking. We will say more about this in Appendix A.

For our purposes, it will suffice to consider only the right-handed sleptons
and the lightest neutralino of the full MSSM spectrum. The left-handed and
color-charged scalars, as well as the gluino, are assumed to acquire soft
masses with parametrically larger values that lie outside the reach of the
high-energy collider we consider. We will further assume that the lightest
neutralino is pure Bino, $\tilde{B}$, with mass $M_{1}$. Aside from being
phenomenologically convenient, this assumption is also quite natural, as the
left-handed and color-charged particles typically receive additional
contributions to their masses from $SU(2)_{L}$ and $SU(3)_{c}$ gauge
interactions that the right-handed sleptons and Bino do not. This spectrum
bears some resemblance to those studied from a UV perspective in ref. [8] in
the context of the hierarchy problem and the muon $(g-2)$. The LHC bounds were
compared to the flavor constraints for a similar spectrum in ref. [9] as well.

For our collider studies, we will be interested in the case that $M_{1}$ is
less than the slepton masses, so that the sleptons decay via
$\tilde{\ell}_{i}\to\ell_{j}\tilde{B}$, with the neutralino appearing as
missing momentum. Finally, for simplicity, we will largely ignore the effects
of the stau, and assume that its mass is reasonably well separated from the
selectron and smuon. We will revisit the effects of stau mixing with the other
sleptons again at the end of the section and in much more detail in Section 5.

The result of this suite of motivated assumptions is that the flavor-violating
slepton spectrum is reduced to a $2\times 2$ mixing problem. We write the mass
matrix of the right-handed selectron and smuon as

$\begin{pmatrix}\tilde{e}_{R}^{\dagger}&\tilde{\mu}_{R}^{\dagger}\end{pmatrix}\begin{pmatrix}\mathcal{M}_{R}^{2}\end{pmatrix}\begin{pmatrix}\tilde{e}_{R}\\\
\tilde{\mu}_{R}\end{pmatrix}\equiv\begin{pmatrix}\tilde{e}_{R}^{\dagger}&\tilde{\mu}_{R}^{\dagger}\end{pmatrix}\begin{pmatrix}m_{R}^{2}+\Delta^{RR}_{ee}&\Delta^{RR}_{e\mu}\\\
\big{(}\Delta^{RR}_{e\mu}\big{)}^{*}&m_{R}^{2}+\Delta^{RR}_{\mu\mu}\end{pmatrix}\begin{pmatrix}\tilde{e}_{R}\\\
\tilde{\mu}_{R}\end{pmatrix}$ (1)

where $m_{R}^{2}$ is the flavor universal part of the mass matrix, and the
$\Delta^{RR}_{ij}$ represent flavor non-universal entries. The matrix
$\mathcal{M}_{R}^{2}$ in Eq. (1) is diagonalized by a unitary matrix $U_{R}$
parametrized by a mixing angle $\theta_{R}$ and a phase $\varphi_{R}$,

$U_{R}^{\dagger}\mathcal{M}_{R}^{2}U_{R}=\textrm{diag}(m_{\tilde{e}_{1}}^{2},m_{\tilde{e}_{2}}^{2}),\quad\text{where}\quad
U_{R}=\begin{pmatrix}\cos\theta_{R}&-\mathrm{e}^{i\varphi_{R}}\sin\theta_{R}\\\
\mathrm{e}^{-i\varphi_{R}}\sin\theta_{R}&\cos\theta_{R}\end{pmatrix}.$ (2)

From Eq. (1), we can compute the average and difference of the mass-squared
eigenvalues, the mixing angle $\theta_{R}$ and the phase $\varphi_{R}$:

$\displaystyle\overline{m^{2}}$
$\displaystyle\equiv\frac{1}{2}\big{(}m_{\tilde{\ell}_{1}}^{2}+m_{\tilde{\ell}_{2}}^{2}\big{)}=m_{R}^{2}+\frac{1}{2}\big{(}\Delta^{RR}_{ee}+\Delta^{RR}_{\mu\mu}\big{)}$
$\displaystyle\Delta m^{2}$ $\displaystyle\equiv
m_{\tilde{\ell}_{2}}^{2}-m_{\tilde{\ell}_{1}}^{2}=\sqrt{\big{|}\Delta^{RR}_{ee}-\Delta^{RR}_{\mu\mu}\big{|}^{2}+4\big{|}\Delta^{RR}_{e\mu}\big{|}^{2}}$
$\displaystyle\sin 2\theta_{R}$
$\displaystyle=\frac{2|\Delta^{RR}_{e\mu}|}{\sqrt{|\Delta^{RR}_{ee}-\Delta^{RR}_{\mu\mu}|^{2}+4|\Delta_{e\mu}|^{2}}}$
$\displaystyle\varphi_{R}$ $\displaystyle=\arg(\Delta^{RR}_{e\mu})$ (3)

where we’ve defined $\overline{m^{2}}$ and $\Delta m^{2}$ for later
convenience.

The phase $\varphi_{R}$ can be removed from the slepton mass matrix by phase
rotations on $\tilde{e}_{R}$, $e_{R}$ and $e_{L}$ such that all (s)electron
interaction vertices and the electron mass stay real. Without the
$\varphi_{R}$ phase, the slepton mixing problem can thus be completely
described in our scenario by the mean slepton mass squared,
$\overline{m^{2}}$, the mass splitting, $\Delta m^{2}/\overline{m^{2}}$, and
the mixing angle, $\sin(2\theta_{R})$. To make the parameters responsible for
flavor violation more manifest, we will rewrite the slepton mass matrix as

$\mathcal{M}_{R}^{2}=\overline{m^{2}}\mathds{1}+\frac{\Delta
m^{2}}{2}U_{R}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}U^{\dagger}_{R}.$ (4)

The off-diagonal element of the second term breaks the $U(2)$ rotation that
mixes the sleptons, and is proportional to $\Delta m^{2}\sin(2\theta_{R})$.
This is why the usual normalized parameter describing the amount of flavor
violation, $\delta^{RR}_{\mu e}$, is [10]

$\delta^{RR}_{\mu e}\equiv\frac{\big{(}\mathcal{M}_{R}^{2}\big{)}_{\mu
e}}{\sqrt{\big{(}\mathcal{M}_{R}^{2}\big{)}_{\mu\mu}\big{(}\mathcal{M}_{R}^{2}\big{)}_{ee}}}=\frac{\Delta
m^{2}}{2\overline{m^{2}}}\sin(2\theta_{R}).$ (5)

As we will discuss more in Section 4, a high-energy muon collider can study
the flavor violation in the MSSM via slepton pair production with searches for
the flavor-violating decays [11, 12]. As we will see, the three parameters
above—along with the Bino mass, $M_{1}$—suffice to completely characterize the
reach. Before turning to the collider reach, however, we first review the low-
energy signatures of LFV. We will see that these same four parameters largely
govern the lepton flavor violating signatures in muon decays and $\mu$-to-$e$
transitions as well.

###  2.1 Low-Energy Signatures of Lepton Flavor Violation

In this section, we discuss the constraints from low-energy precision
measurements of muons. In particular, we’ll focus on the constraints from
$\mu\to e\gamma$, $\mu\to 3e$ and $\mu$-to-$e$ conversion in atomic nuclei,
since these arise without any field content beyond the mixed right-handed
sleptons and the Bino that we will also consider for the collider reach.
Beyond this minimal scenario, additional constraints from flavor-violating
mixings with the stau and searches for the electron electric dipole moment can
arise, but these depend on additional parameters, so we delay a discussion of
them until Section 5.

There is a large literature on the rate of charged lepton flavor violation
processes in supersymmetric theories, too large to review here; key papers and
useful entry points to this literature include [13, 14, 15, 16, 17, 18, 19,
20, 21, 22]. Here, we will only extract some results in particular limits and
regions of parameter space that are of interest for our work.

In the situations of interest to us, the dominant contributions to CLFV
processes are from contributions to dipole operators. Given a general dipole
operator below the electroweak scale,

${\cal
L}\supset\mathcal{A}^{R}_{ij}\,{\bar{\ell}}_{i}\sigma^{\mu\nu}P_{R}\ell_{j}F_{\mu\nu}+\mathrm{h.c.},$
(6)

the $\mu\to e\gamma$ decay rate is given by

$\mathrm{Br}(\mu\to
e\gamma)=\frac{48\pi^{3}\alpha}{m_{\mu}^{2}G_{F}^{2}}\left(|\mathcal{A}^{R}_{e\mu}|^{2}+|\mathcal{A}^{R}_{\mu
e}|^{2}\right).$ (7)

The minimal set of ingredients we consider already leads to a contribution to
the dipole operator above, as illustrated in Fig. 1. This diagram has a mass
insertion on the external muon line. Because the diagram is not 1LPI, one way
to interpret this is that integrating out the sleptons and bino generates an
operator of the form
${\bar{e}}_{j}^{\dagger}{\overline{\sigma}}^{\mu}\overset{\leftrightarrow}{D^{\nu}}{\bar{e}}_{i}B_{\mu\nu}$,
which becomes the more familiar dipole operator
$L_{j}{\overline{\sigma}}^{\mu\nu}{\bar{e}}_{i}B_{\mu\nu}$ only after applying
the equation of motion for the lepton fields.

Figure 1: The minimal one-loop contribution to the $\mu\to e\gamma$ dipole
associated with the flavor-violating slepton collider signal of interest. The
right-handed selectron and smuon are both relatively light, and contribute to
$\mu\to e\gamma$ through a loop involving a bino and the off-diagonal slepton
mass matrix contribution $\delta_{e\mu}^{RR}$.

The mixing of ${\tilde{e}}_{R}$ and ${\tilde{\mu}}_{R}$ leads to two slepton
mass eigenstates ${\tilde{\ell}}_{1,2}$. Then the dipole amplitude in this
case is

$\mathcal{A}^{R}_{\mu
e}=m_{\mu}\frac{\alpha_{Y}}{8\pi}\sin(2\theta_{R})\left[\frac{1}{m^{2}_{{\tilde{\ell}}_{1}}}{\bar{A}}\left(\frac{|M_{1}|^{2}}{m^{2}_{{\tilde{\ell}}_{1}}}\right)-\frac{1}{m^{2}_{{\tilde{\ell}}_{2}}}{\bar{A}}\left(\frac{|M_{1}|^{2}}{m^{2}_{{\tilde{\ell}}_{2}}}\right)\right],$
(8)

where the loop function is defined as (in notation following [23]; see also
[13, 24])

${\bar{A}}(r)=\frac{2r^{3}+3r^{2}-6r+1-6r^{2}\log(r)}{6(1-r)^{4}}.$ (9)

In the case that the sleptons are very nearly degenerate, the mixing angle
$\theta_{R}$ can be large. Nonetheless, the $\mu\to e\gamma$ rate is
suppressed by the approximate degeneracy, because the two terms inside the
brackets nearly cancel. Indeed, at the beginning of the section we discussed
that flavor violation is controlled by the product $\Delta
m^{2}\sin(2\theta_{R})$. Expanding in small $\Delta m^{2}$, we find

$\mathcal{A}^{R}_{\mu e}\approx
m_{\mu}\frac{\alpha_{Y}}{\pi}\frac{\delta^{RR}_{\mu
e}}{\overline{m^{2}}}f_{1n}\left(\frac{|M_{1}|^{2}}{\overline{m^{2}}}\right),$
(10)

where

$f_{1n}(r)=\frac{-1+9r+9r^{2}-17r^{3}+6r^{2}(r+3)\log(r)}{24(r-1)^{5}}.$ (11)

This latter result appears in the literature via the mass insertion
approximation; see, e.g., [21]. These results are related via
$(r{\bar{A}}(r))^{\prime}=4f_{1n}(r)$.

Thus, even when the right-handed selectron and smuon are heavily mixed, so
that each slepton mass eigenstate has an $O(1)$ probability of decaying to
either an electron or a muon, the $\mu\to e\gamma$ rate is still suppressed
due to the mass degeneracy between the states. The mass insertion
approximation continues to be valid, as reflected in the smallness of
$\delta^{RR}_{\mu e}$, despite the large mixing angle. This is a consequence
of a super-GIM mechanism: in the limit of degenerate masses, the total rate is
proportional to $\sum_{i}U_{1i}U_{2i}^{*}$, where $U_{ji}$ denotes the matrix
element between flavor eigenstate $j$ and mass eigenstate $i$. By unitarity,
this sum is zero. As a result, when flavor mixing is very large due to a near-
degeneracy in masses, constraints from dedicated CLFV experiments become
weaker, and relatively light new states are allowed.

The current limit $\textrm{BR}(\mu\to e\gamma)<4.2\times 10^{-13}$ ($90\%$
C.L.) was set by the MEG experiment [25]. The upgrade MEG-II will improve this
bound to $<6\times 10^{-14}$ [26]. The constraints on slepton mixing in our
scenario from this constraint will be shown in Section 4, along with
constraints from other lepton flavor violating processes to which we now turn.

The minimal dipole contribution in Eq. (10) also contributes to the $\mu\to
3e$ decay rate. In the situations we are interested here, the dipole
contribution dominates over box and penguin diagrams, and the branching ratio
can be reliably estimated as [13, 16]

$\textrm{BR}(\mu\to
3e)=\frac{\alpha}{3\pi}\Big{(}\log\frac{m_{\mu}^{2}}{m_{e}^{2}}-\frac{11}{4}\Big{)}\,\textrm{BR}(\mu\to
e\gamma)\simeq\frac{1}{162}\,\textrm{BR}(\mu\to e\gamma).$ (12)

The current bound on $\mu\to 3e$ decays comes from the SINDRUM experiment
[27], which set a limit $\textrm{BR}(\mu\to 3e)<1.09\times 10^{-12}$. This
constraint will significantly improve, to $5\times 10^{-15}$ with the first
phase of the Mu3e experiment [28], with further upgrades expected to push the
sensitivity to $<10^{-16}$ in the future.

The dipole operator can also dominate the contributions to the $\mu$-to-$e$
conversion in nuclei, particularly if the squarks are much heavier than the
sleptons. Assuming dipole dominance, the conversion rate in a nucleus $N$
takes the simple form [14, 20]

$\displaystyle\textrm{CR}(\mu\to e)_{N}=\textrm{BR}(\mu\to
e\gamma)\times\begin{cases}\frac{\alpha}{3}&N=\textrm{Al},\\\
\frac{\alpha}{2}&N=\textrm{Au}.\end{cases}$ (13)

The best constraints available now are from SINDRUM II, which set a limit on
the conversion rate in Au at $<7.0\times 10^{-13}$ [29]. There are a number of
upcoming experiments that are expected to improve these bounds significantly
using aluminum nuclei: COMET is projected to set a limit of $7\times 10^{-15}$
($2.6\times 10^{-17}$) in Phase-I (Phase-II) [30, 31], while Mu2e projects an
eventual Phase-II sensitivity of $2.5\times 10^{-18}$ [32, 33]. A future
experiment, PRISM/PRIME, aims to eventually push this sensitivity to the $\sim
10^{-19}$ level [34, 35, 6].

Thus far, we have completely ignored the effects of mixing with the stau. As
emphasized in [36] and further in [19, 37, 18], the SUSY contributions to
dipole operators in theories with flavor violation can be dominated by mixing
with the third generation, due to the much larger Yukawa couplings that are
then accessible. We will discuss these effects, along with analogous
contributions to the electron electric dipole moment, in Section 5.

##  3 The MSSM at a High-Energy Muon Collider

In Section 4, we will discuss how the physics leading to the low-energy LFV
signals described in the previous section can be studied directly at a high-
energy muon collider. First, it is worth reviewing the prospects of a high-
energy muon collider in more generality—in particular the capabilities in
searching for superpartners.

A high-energy muon collider combines the advantages of proton and electron-
positron machines. Because of their larger mass, muons produce less
synchrotron radiation than electrons and positrons, so they can be readily
accelerated to TeV-scale energies in relatively compact circular accelerators.
At the same time, the muons are not composite objects like protons, so they
can utilize the full center of mass energy in their collisions to produce
heavy new states. Because the colliding objects are second-generation fermions
themselves, muon colliders have a particular aptitude for testing physics with
generation-specific couplings [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50]. Finally, high-energy muons have a large probability to radiate collinear
gauge bosons in their collisions [51, 52, 53, 3, 54], and can thus act
efficiently as “vector boson colliders”, with large rates for VBF processes
that can be used both to search for new physics [55, 56, 57, 58, 59, 60, 61,
62, 63] and to make precision measurements of the electroweak sector [64, 65,
66, 67, 68, 69, 70, 71, 72, 73]. That the muons are color-neutral, and produce
far fewer hadronic backgrounds, makes them particularly well-suited for this
purpose.

These advantages come at the cost of working with unstable particles, which
are more challenging to produce with high-intensities while maintaining a
high-quality, low-emittance beam [74]. The decays of the muons also lead to a
large “beam-induced background” (BIB), which must be effectively mitigated by
a combination of shielding and precision timing in the detectors [75]. Recent
advances have begun demonstrating that these challenges can be overcome [76,
77, 78, 79], however, coinciding with a surge of interest in the physics
potential of a high-energy muon collider facility [80, 81, 4, 5, 82].

In what follows, we will consider muon colliders with center of mass energies
of $3$, $10$ and $30\,\textrm{TeV}$, with corresponding integrated
luminosities of $1$, $10$, and $10\,\textrm{ab}^{-1}$. As for the detectors,
we will only assume coverage up to $|\eta|<2.5$, and require that charged
particles have $p_{T}\geq 25\,\textrm{GeV}$, and otherwise ignore any
reconstruction efficiencies or secondary effects of the BIB. As we are
primarily interested in signals involving high-energy charged leptons, we do
not expect details of the detector design to significantly affect any of our
results.

The direct reach for superpartners at high-energy muon colliders was explored
in detail in ref. [3]. There, the rates for a number of simplified model
production processes were estimated and some of the leading candidates for
discovery signatures were discussed, particularly in relation to the hierarchy
problem. The prospects for discovering superpartners were also studied in the
context of WIMP candidates in refs. [83, 84, 85]. In particular, [84] provides
a proof-of-concept that a disappearing track signature can be searched for
even in the presence of heavy contamination due to BIB.

In general, any superpartner with electroweak quantum numbers, such as the
sleptons of interest to us here, will be pair-produced in the $s$-channel with
a rate that falls like $1/s$, but is still at the $O(\text{few
tens})\leavevmode\nobreak\ \textrm{ab}$ level, even at a $30\,\textrm{TeV}$
collider. As the superpartner mass approaches $\sqrt{s}/2$, the $s$-channel
cross section scales like the velocity $\beta$ for fermionic superpartners and
like $\beta^{3}$ for scalar ones. This leads to roughly an order of magnitude
drop in the cross section for scalars with masses $\sim 0.9\times\sqrt{s}/2$,
but with sufficient luminosity the reach for electroweak particles can extend
all the way up to half the center of mass energy.

In summary, a high-energy muon collider would have the reach to discover any
new charged superpartners if they have masses less than approximately half the
center of mass energy. In the scenarios we have discussed in Section 2, this
means that the first signals of supersymmetry could be seen in the pair
production of sleptons, perhaps in tandem with signatures from low-energy
experiments. The results in the next subsection, and in Section 4, will
demonstrate that beyond discovering the new states, a muon collider can
measure a number of properties of the spectrum and deliver interesting lessons
about the nature of the underlying supersymmetric theory.

###  3.1 Measurement of the Slepton and Neutralino Masses

As we will discuss in detail in Section 4, knowledge of the slepton and
neutralino masses can be used to suppress the backgrounds to flavor-violating
processes, allowing for more precise studies of the SUSY spectrum. It is thus
worthwhile to understand how well these masses can be measured at a muon
collider, assuming that sleptons and neutralinos are within the collider
reach, and will be discovered via flavor-conserving channels. The measurement
of superpartner masses at a lepton collider is a well-studied problem, and a
number of detailed studies have been performed for $e^{+}e^{-}$ colliders with
energies in the $250\,\textrm{GeV}$ – $3\,\textrm{TeV}$ range [86, 87, 88, 89,
90].

However, there are several important differences in performing this
measurement at a high-energy muon collider. Synchrotron radiation is highly
suppressed relative to $e^{+}e^{-}$ beams of comparable energies, and as a
result, beamsstrahlung effects can essentially be ignored. While initial state
radiation exists, the collision energy is expected to be much more sharply
defined than at, e.g., CLIC, and designs typically project a spread on the
beam energy of $\delta E/E\sim 0.1\%$ [2, 91, 92]. On the other hand, SM
backgrounds at higher energy colliders are expected to be quite large, and the
rates for vector boson fusion (VBF) processes especially are significantly
larger than at lower energies. This is compounded by the fact that collisions
at a muon collider suffer from a large beam-induced background. Methods of
effectively mitigating the BIB are the subject of ongoing research, but a
common proposal is to make use of “shielding nozzles” that cover the far-
forward regions of the detector, suppressing the BIB at the cost of detector
coverage [75]. Such a scheme makes it difficult to veto events with forward-
going leptons, which could otherwise greatly reduce the VBF background. A
threshold scan to determine the masses may be possible, but likely only for a
small range of energies, making a shape-based determination more important.

In light of these differences, an updated investigation of the attainable
precision is warranted. A comprehensive analysis requires not only a precise
understanding of the detector performance, but also a careful treatment of the
systematic uncertainties, both of which are beyond the scope of this work.
However, the problem at hand is primarily kinematic, and depends only on
measuring the energies of charged leptons, so a reasonable estimate of the
precision can be obtained with simple parton-level simulations of the relevant
processes. These estimates will still be enlightening in understanding how the
unique muon collider environment can be dealt with, and suffice for mitigating
the backgrounds to the flavor-violating processes we are interested in.444A
similar study has been performed in ref. [93]. Here, we consider a different
set of cuts to mitigate the $\gamma\gamma$ induced background, and a different
set of muon collider energies.

As in the $e^{+}e^{-}$ collider case, the slepton and neutralino masses can be
measured at a muon collider via the flavor-conserving pair production
processes:

$\displaystyle\mu^{+}\,\mu^{-}$
$\displaystyle\to\tilde{\mu}_{R}^{+}\,\tilde{\mu}_{R}^{-}\to\mu^{+}\,\mu^{-}\,\tilde{B}\,\tilde{B},$
$\displaystyle\mu^{+}\,\mu^{-}$
$\displaystyle\to\tilde{e}_{R}^{+}\,\tilde{e}_{R}^{-}\to
e^{+}\,e^{-}\,\tilde{B}\,\tilde{B},$ (14)

by measurements of the final state leptons.555For the purposes of estimating
the precision attainable on the masses, we will assume that the sleptons are
flavor eigenstates, i.e., that it is the selectron mass we are measuring in
the final states with $e^{+}e^{-}$ and missing momentum, and the smuon mass in
final states with $\mu^{+}\mu^{-}$. This is only strictly true in the limit
$\sin 2\theta_{R}\to 0$ (or $\Delta m^{2}/\overline{m^{2}}\to 0$, when the
masses are identical), but we evaluate the precision for all
$\sin(2\theta_{R})$, taking the effect on the signal cross section into
account. Energy and momentum conservation dictates that the slepton energy
equal the beam energy, and the lab-frame energy of the charged lepton decay
product is therefore fully determined by the (rest frame) decay angle,
$\theta_{0}$. The lepton energy is then found to be,

$E_{\ell}=\frac{\sqrt{s}}{4}\bigg{(}1-\frac{M_{1}^{2}}{m_{\tilde{\ell}}^{2}}\bigg{)}\big{(}1+\beta\cos\theta_{0}\big{)}.$
(15)

In the above, $\beta=\sqrt{1-4m_{\tilde{\ell}^{2}}/s}$ is the velocity of the
slepton in the lab frame. The lepton energies are distributed uniformly
between the two limits of Eq. (15) obtained when $\cos\theta_{0}=1$
($E_{\textrm{max}}$) and $\cos\theta_{0}=-1$ ($E_{\textrm{min}}$). These
endpoints, $E_{\textrm{min}}$, $E_{\textrm{max}}$, are therefore related to
the slepton and neutralino masses via:

$m_{\tilde{\ell}}^{2}=s\,\frac{E_{\textrm{min}}E_{\textrm{max}}}{(E_{\textrm{min}}+E_{\textrm{max}})^{2}},\quad
M_{1}^{2}=m_{\tilde{\ell}}^{2}\bigg{(}1-\frac{2(E_{\textrm{min}}+E_{\textrm{max}})}{\sqrt{s}}\bigg{)}.$
(16)

The problem of measuring the slepton and neutralino masses then becomes simply
the problem of measuring the endpoints of the slepton energy distribution in
the presence of backgrounds and detector effects.

In practice, the endpoints of the spectrum (and the corresponding uncertainty)
can be determined via an unbinned likelihood fit to the data. This requires a
functional description of the expected shapes of the distribution, both for
signal and background events. To determine the expected precision on the
masses, we simulate a large sample of the signal and background events. Toy
datasets with the expected number of events (after applying a simple set of
cuts to mitigate the background) can then be generated, and the likelihood fit
performed. Repeating this procedure for a large number of toys gives an
estimate of the expected precision on the slepton and neutralino masses. More
details of this procedure are left to Appendix B, where we describe the
simulation procedure, the selection cuts used to mitigate the backgrounds, the
fitted shapes of the signal and background distributions, and the details of
the likelihood analysis.

Figure 2:  Contours of the expected precision on the selectron, smuon and
neutralino masses as a function of $\sin 2\theta_{R}$ and $\Delta
m^{2}/\overline{m^{2}}$ for fixed $m_{\tilde{\ell}}$ and $M_{1}$ (left) and as
a function of $M_{1}$ and $\sin 2\theta_{R}$ for fixed $m_{\tilde{\ell}}$ and
$\Delta m^{2}/\overline{m^{2}}$ (right), all at $\sqrt{s}=10\,\textrm{TeV}$,
assuming $10\,\textrm{ab}^{-1}$ of data.

The results of the likelihood analysis for $\sqrt{s}=10\,\textrm{TeV}$ are
summarized in Fig. 2. We show contours of the expected precision (at 95% C.L.)
on the selectron and smuon masses ($m_{\tilde{\mu}}$ and $m_{\tilde{e}}$) and
on the neutralino mass, $M_{1}$, in orange, blue, and green, respectively. In
the left panel, the precision is shown as a function of the slepton mixing
angle, $\sin 2\theta_{R}$ and mass splitting $\Delta m^{2}/\overline{m^{2}}$,
as defined in Eq. (2), with the average slepton mass fixed to
$3\,\textrm{TeV}$ and $M_{1}=1.5\,\textrm{TeV}$. The uncertainties on the
smuon and neutralino masses vary from $\sim 45$ to $60\,\textrm{GeV}$ and
$\sim 20$ to $25\,\textrm{GeV}$, increasing slightly towards larger mixing
angles where the $\mu^{+}\mu^{-}\tilde{B}\tilde{B}$ cross section decreases.
Correspondingly, the rate for the $e^{+}e^{-}$ final state process increases,
and the precision on the selectron varies from $\sim 350$ to
$300\,\textrm{GeV}$.

The right panel shows the same contours but as a function of $\sin
2\theta_{R}$ and $M_{1}$, with the mean slepton mass
$m_{\tilde{\ell}}=3\,\textrm{TeV}$ and $\Delta m^{2}/\overline{m^{2}}=0.1$. In
this plane, the same behavior for large mixing angles is observed, while the
precision on all the masses improves for small $M_{1}$ but gets rapidly worse
as $M_{1}$ approaches $m_{\tilde{\ell}}$, where the cross section drops
significantly due to the small phase space available for the leptons and the
endpoints approach the kinematic limits of the detector, where they are harder
to measure accurately.

The results for $\sqrt{s}=3$ and $30\,\textrm{TeV}$ show very similar
behavior, with the precision on the selectron, smuon, and neutralino varying
from 25 to 50, 10 to 50 and 20 to $60\,\textrm{GeV}$, respectively at
$3\,\textrm{TeV}$ and from 600 to 750, 200 to 350 and 100 to
$200\,\textrm{GeV}$ at $30\,\textrm{TeV}$. We show the results here for the
benchmark luminosities assumed above, but the precision on each of the masses
scales like the square root of the integrated luminosity, absent any
systematic effects that we have neglected.

##  4 Direct Reach for Flavor-Violating Signatures

Charged lepton flavor violation can be searched for in a high-energy muon
collider through the direct production process $\mu^{+}\mu^{-}\rightarrow\mu
e+X$. In the simplified MSSM model of interest described in Section 2, the
signal process is

$\mu^{+}\mu^{-}\rightarrow\mu e\tilde{B}\tilde{B},$ (17)

where $\tilde{B}$ is the lightest neutralino that we have assumed to be a pure
Bino which would appear as missing momentum. The dominant contribution to the
signal process is pair production,

$\mu^{+}\mu^{-}\rightarrow\tilde{\ell}_{i}\tilde{\ell}_{j}\rightarrow\mu
e\tilde{B}\tilde{B},$ (18)

as shown in Fig. 3, where $\tilde{\ell}_{i,j}$ are any of $\tilde{\ell}_{1}$
and $\tilde{\ell}_{2}$, the slepton mass eigenstates that are mixtures of the
flavor eigenstates $\tilde{\mu}_{R}$ and $\tilde{e}_{R}$. The signal cross
section is completely determined by the four parameters in the model,

$\overline{m^{2}},\;\;\frac{\Delta
m^{2}}{\overline{m^{2}}},\;\;\sin(2\theta_{R}),\;\;M_{1},$ (19)

as are the low-energy observables discussed in Section 2.1. This allows a
direct comparison of the muon collider constraint with bounds coming from low-
energy observables. The strong complementarity between the high-energy and
low-energy experiments will be demonstrated in detail in the rest of this
section.

Figure 3: Main production mechanism for the flavor-violating signal at the
muon collider. For the $t$-channel process, $\tilde{\ell}_{i}$ and
$\tilde{\ell}_{j}$ can be any of the two slepton mass eigenstates,
$\tilde{\ell}_{1}$ and $\tilde{\ell}_{2}$.

There are two backgrounds to the signal process. The first is
$\tilde{\tau}^{+}\tilde{\tau}^{-}$ pair production decaying into
$\tau^{+}\tau^{-}\tilde{B}\tilde{B}$, with the pair of taus decaying
leptonically to $\mu$-$e$ final states (the staus cannot directly decay into
$\mu$ or $e$ because our minimal MSSM model outlined in Section 2 assumes that
flavor violation is confined in the $\mu$-$e$ sector).666Of course, in general
all mixing parameters will be present, and one would need a more complicated
global fit. Nonetheless, we believe our results are a useful starting point
for calibrating expectations about the muon collider’s capabilities. But this
background has a negligible cross section compared to the more dominant
Standard Model background discussed in the next paragraph, even when the
$\tilde{\tau}$ is light, because of the branching ratio suppression coming
from demanding specific leptonic decay of the $\tau$s.

The second, larger background for the signal process is Standard Model pair
production of $W^{+}W^{-}$ that decay leptonically to $\mu$-$e$ final states.
Even though the background process has a significantly larger cross section in
the range of collider energies and model parameters we consider, the
background can be rejected efficiently by reconstructing the neutralino
kinematics, using the measurement of the slepton and neutralino masses as
discussed in Section 3.1. The kinematic reconstruction procedure is discussed
in detail in Appendix C. The key quantities from the reconstruction analysis
are the “reconstructed energies” for the two neutralinos, which are the
neutralino energies calculated without the $y$-component of their three-
momenta (see Appendix C for the definition of the frame of reference and the
formula in terms of visible particle momenta),

$\mathcal{E}_{\tilde{B}_{a}}\equiv\sqrt{p^{2}_{\tilde{B}_{a,x}}+p^{2}_{\tilde{B}_{a,z}}+m^{2}_{\tilde{B}}},\quad\mathcal{E}_{\tilde{B}_{b}}\equiv\sqrt{p^{2}_{\tilde{B}_{a,x}}+(p_{{\rm
inv}}-p_{\tilde{B}_{a,z}})^{2}+m^{2}_{\tilde{B}}}.$ (20)

Here $\\{a,b\\}$ labels the two neutralinos and their corresponding sleptons
and leptons, i.e. $\tilde{B}_{a}$ and $\ell_{a}$ are produced from
$\tilde{\ell}_{a}$ decay, and $\tilde{B}_{b}$ and $\ell_{b}$ are produced from
$\tilde{\ell}_{b}$ decay. The reconstructed energies put a lower bound on the
true total energy carried by the neutralinos

$E_{\tilde{B}_{a}}+E_{\tilde{B}_{b}}\geq\mathcal{E}_{\tilde{B}_{a}}+\mathcal{E}_{\tilde{B}_{b}}.$
(21)

By energy conservation, $\mathcal{R}\equiv E_{\rm
CM}-E_{\ell_{a}}-E_{\ell_{b}}-\mathcal{E}_{\tilde{B}_{a}}-\mathcal{E}_{\tilde{B}_{b}}$
must be positive. Since we do not know which of the two slepton mass
eigenstates is involved in the particular signal event that we are
reconstructing the kinematics for, we calculate $\mathcal{R}$ for all four
possibilities of $\tilde{\ell}_{a}=\tilde{\ell}_{1,2}$ and
$\tilde{\ell}_{b}=\tilde{\ell}_{1,2}$, take the largest value of $\mathcal{R}$
among the four cases as our final result, and demand that $\max\mathcal{R}>0$.
While for signal events $\max\mathcal{R}>0$ is almost always true, background
events rarely satisfy this criterion since the events are being reconstructed
using the wrong particle masses. The precise efficiency varies with collider
energy and particle masses, but in general signal events pass $\sim 98\%$ of
the time, while background event efficiencies are around $\sim 0.2\%$.

The reconstruction efficiency for the signal events becomes worse once we take
into account the uncertainty in the measurements of slepton and neutralino
masses found in Section 3.1. We will use the mass measurement uncertainty
$\Delta m$ as an estimate of how far the measured mass is from the true mass,

$m_{{\rm measured}}=m_{{\rm true}}\pm\Delta m.$ (22)

Specifically for the purpose of kinematic reconstruction, the sign of $\Delta
m$ matters. For example, if we do kinematic reconstruction analysis with a
slepton mass that is lower than the true value, what the analysis deems the
“correct” phase space would be smaller than the true phase space spanned by
the signal events, thus rejecting more signal events. Among the eight
different combinations of $\Delta m$ signs for the three particle masses,
$m_{\tilde{\ell}_{1}}$, $m_{\tilde{\ell}_{2}}$, and $M_{1}$, the following
“low-low-high” combination lowers the signal reconstruction efficiency the
most,

$m_{\tilde{\ell}_{1}{\rm measured}}=m_{\tilde{\ell}_{1}{\rm true}}-\Delta
m_{\tilde{\ell}_{1}},\;\;m_{\tilde{\ell}_{2}{\rm
measured}}=m_{\tilde{\ell}_{2}{\rm true}}-\Delta
m_{\tilde{\ell}_{2}},\;\;m_{\tilde{B}{\rm measured}}=m_{\tilde{B}{\rm
true}}+\Delta m_{\tilde{B}},$ (23)

because the phase space that the reconstruction analysis considers correct is
the smallest. The signal efficiency of the $\mathcal{R}>0$ cut in the “low-
low-high” configuration strongly depends on the value of the true masses and
the mass errors. For example, in the extreme case where the $m_{\tilde{B}{\rm
measured}}$ becomes higher than $m_{\tilde{\ell}_{1,2}{\rm measured}}$, the
signal efficiency would approach zero. In our study, the muon collider reach
for lepton flavor-violating signal will be computed in two ways, first with
the true particle masses in the reconstruction analysis, second with the “low-
low-high” particle masses which will yield a weaker collider reach, capturing
the uncertainty in collider reach due to the error in the mass measurements.
Now, because the background distribution of $\mathcal{R}$ is much wider than
the signal distribution, it is in principle possible to place the
reconstruction cut at a negative value, $\max\mathcal{R}>-C$, to improve
signal efficiency for the “low-low-high” configuration without passing many
background events. We do not consider this optimization of the kinematic
reconstruction cut in our study, and simply cut at $\max\mathcal{R}>0$ for all
events and mass configurations.

In the rest of the section, we will discuss in detail the direct reach of
flavor-violating processes at muon colliders with $\sqrt{s}=3$, 10, and 30
TeV. All signal and background events are simulated using MadGraph5 [94]. For
the signal events we use an extended MSSM UFO model that allows general flavor
violation in the lepton sector, created using the FeynRules packages from
[95].

Figure 4: 5$\sigma$ discovery reach of the lepton-flavor violating MSSM by a 3
TeV muon collider with 1 ${\rm ab}^{-1}$ of data in the $\Delta
m^{2}/\overline{m^{2}}$ vs $\sin 2\theta_{R}$ plane and the $\sin 2\theta_{R}$
vs $M_{1}$ plane (red band). The width of the collider reach band shows the
uncertainty due to inexact slepton and neutralino mass measurements. The gray
contours are cross sections of the signal process. Both plots assume a mean
slepton mass of 1 TeV. In the left plot we fix the neutralino mass $M_{1}=500$
GeV, while in the right figure $\Delta m^{2}/\overline{m^{2}}$ is fixed to
0.1. The current (solid) and expected (dashed) limits from low-energy lepton
flavor violation experiments are indicated by the blue, purple and green
lines. The purple and blue shaded lightly shaded regions indicate parameters
preferred in Gauge-Mediated Supersymmetry Breaking scenarios and flavor-
dependent mediator scenarios, respectively (as described in Appendix A).

Fig. 4 shows the $5\sigma$ reach of a 3 TeV muon collider with 1 ${\rm
ab}^{-1}$ of data, assuming a mean slepton mass of $\overline{m^{2}}$ = (1
TeV$)^{2}$. The remaining three-dimensional parameter space of the model is
sliced in two different ways: in the $\Delta m^{2}/\overline{m^{2}}$ vs $\sin
2\theta_{R}$ plane where we assume $M_{1}$ = 500 GeV and in the $\sin
2\theta_{R}$ vs $M_{1}$ plane where we assume $\Delta m^{2}/\overline{m^{2}}$
= 0.1. In both planes, the reach in $\sin 2\theta_{R}$ with fixed $\Delta
m^{2}/\overline{m^{2}}$ or fixed $M_{1}$ will improve by a facor of $10^{1/4}$
if collider luminosity increases by a factor of 10, since the signal cross
section is proportional to $(\sin 2\theta_{R})^{2}$. In the $\Delta
m^{2}/\overline{m^{2}}$ vs $\sin 2\theta_{R}$ plane, we see that the collider
reach deteriorates quickly in $\sin 2\theta_{R}$ as the mass splitting becomes
small. This suppression was pointed out in ref. [11] as a consequence of a
quantum interference effect between the two mass eigenstates contained in the
flavor eigenstates. In particular, the probability of the gauge eigenstate
selectron, $\tilde{e}$, decaying into a final state containing a muon,
$f_{\mu}$, is calculated as the product of the time-integrated probability of
the selectron oscillating into a smuon gauge eigenstate, $\tilde{\mu}$, and
the probability of a smuon decaying into a final state containing a muon. The
probability $P(\tilde{e}\rightarrow f_{\mu})$ is found to be777The
$\bar{m}^{2}$ in this formula is the square of the average,
$\bar{m}^{2}=(\frac{1}{2}(m_{\tilde{\ell}_{1}}+m_{\tilde{\ell}_{2}}))^{2}$,
which naturally appears since the probability is due to interference between
two mass eigenstates that evolve with different energies,
$E=m_{\tilde{\ell}_{1,2}}$. Elsewhere in the paper we refer to the average of
the square,
$\overline{m^{2}}=\frac{1}{2}(m_{\tilde{\ell}_{1}}^{2}+m_{\tilde{\ell}_{2}}^{2})$.
In the limit of small mass splitting, the two values approach each other,
$\overline{m}^{2}-\bar{m}^{2}=\frac{1}{4}\Delta m^{2}$.

$\begin{split}P(\tilde{e}\rightarrow
f_{\mu})&=P(\tilde{e}\rightarrow\tilde{\mu}){\rm Br}(\tilde{\mu}\rightarrow
f_{\mu})\\\ &=\frac{\int_{0}^{\infty}\differential
t|\innerproduct{\tilde{\mu}}{\tilde{e}(t)}|^{2}}{\int_{0}^{\infty}\differential
t\innerproduct{\tilde{e}(t)}{\tilde{e}(t)}}{\rm Br}(\tilde{\mu}\rightarrow
f_{\mu})\\\ &=\frac{1}{2}\sin 2\theta_{R}\frac{\left(\Delta
m^{2}\right)^{2}}{4\bar{m}^{2}\Gamma^{2}+\left(\Delta m^{2}\right)^{2}}{\rm
Br}(\tilde{\mu}\rightarrow f_{\mu}),\end{split}$ (24)

where $\Gamma$ is the widths of the two mass eigenstates which are assumed to
be equal, and $\tilde{e}(t)$ is the time-evolved wavefunction of a selectron
eigenstate prepared at $t=0$. $\tilde{e}(t)$ is a mixture of selectron and
smuon eigenstates due to the oscillation effect, and the oscillation effect
vanishes as the mass splitting between the two eigenstates approaches zero.
This interference effect can also be understood from a Feynman diagram
perspective without using the language of flavor oscillations. When the two
mass eigenstates have similar masses, the two pair production diagrams shown
in Fig. 5, which are the dominant contributions to the flavor-violating
process $\mu^{+}\mu^{-}\rightarrow\mu e\tilde{B}\tilde{B}$, cancel due to the
opposite sign of the flavor-violating coupling, $\pm\sin\theta_{R}$. The
narrow-width approximation is breaking down here because the mass splitting
between the two intermediate states becomes smaller than their individual
widths, and the interference effect cannot be neglected.

Figure 5: When mass splitting of the two mass eigenstates are small,
interference between the two Feynman diagrams suppresses signal cross section
due to the opposite coupling.

The width of the collider reach contour indicates the uncertainty coming from
error in slepton and neutralino mass measurements. In the $\Delta
m^{2}/\overline{m^{2}}$ vs $\sin 2\theta_{R}$ plane, the uncertainty varies
mildly since the mass measurement errors are relatively small compared to the
available phase space. In the $\sin 2\theta_{R}$ vs $M_{1}$ plane, as $M_{1}$
becomes closer to $m_{\tilde{\ell}_{1}}$ and $m_{\tilde{\ell}_{2}}$, the
available phase space for $e$/$\mu$ final states becomes smaller. When the
measured masses are the same as the true masses, the reconstruction efficiency
of the signal events is roughly constant as $M_{1}$ increases, but background
events have smaller reconstruction efficiency since it is more difficult for
the event kinematics to land within the small phase space. Thus the best
possible collider reach (lower edge of the red band) is stronger as $M_{1}$
increases. On the other hand, the reach in the “low-low-high” configuration
(upper edge of the red band) defined earlier in the section deteriorates
quickly as $M_{1}$ increases, because the smuon and bino mass measurement
uncertainties increase quickly with $M_{1}$, as shown in Fig. 2. At around
$M_{1}\approx 750$ GeV, the mass measurement errors are so large such that

$m_{\tilde{\ell}_{1,2}{\rm measured}}\approx M_{1{\rm measured}},$ (25)

causing the kinematic reconstruction analysis efficiency to approach zero for
the signal events. Therefore the width of the collider reach diverges at
$M_{1}\gtrsim 750$ GeV, as shown in the right panel of Fig. 4.

Figure 6: 5$\sigma$ discovery reach of lepton-flavor violating MSSM from a 10
TeV (top) or 30 TeV (bottom) muon collider, both with 1 ${\rm ab}^{-1}$ of
data, in $\Delta m^{2}/\overline{m^{2}}$ vs $\sin 2\theta_{R}$ plane and $\sin
2\theta_{R}$ vs $M_{1}$ plane (red band). The width of the collider reach band
shows the uncertainty due to inexact slepton and neutralino mass measurements.
The gray contours are cross sections of the signal process. The expected
limits from low-energy lepton flavor violation experiments are indicated by
the blue, purple and green dashed lines. The purple and blue shaded lightly
shaded regions indicate parameters preferred in Gauge-Mediated Supersymmetry
Breaking scenarios and flavor-dependent mediator scenarios, respectively.

The straight lines in blue, purple, and green in Fig. 4 shows current (solid)
and projected (dashed) constraints coming from low-energy experiments. The
contours show a great complementarity between the high energy muon collider
and low-energy experiments. A 3 TeV muon collider would produce competitive
bounds with the projected Mu2e Stage II experiment. Furthermore, the
complementary slicing of the parameter space by the muon collider and the low-
energy experiments helps elucidate the underlying mechanism of lepton flavor
violation when a flavor-violating signal is discovered at one of the
experiments.

Lastly, the lightly shaded purple and blue regions in Fig. 4 indicate
motivated parameter regions from different UV origins of the flavor mixing.
Large mixing angles are motivated in models involving gauge-mediated
supersymmetry breaking (GMSB), indicated by the purple region, while larger
mass splittings are motivated in scenarios where the messengers carry flavor-
dependent charges, such as models where $L_{\mu}-L_{\tau}$ is gauged,
indicated by the blue regions. We see that a 3 TeV muon collider will
constrain a significant fraction of the motivated parameter ranges.

Fig. 6 shows the $5\sigma$ reach of a 10 TeV and 30 TeV muon collider with 10
${\rm ab}^{-1}$ of data, assuming a mean slepton mass of $\overline{m^{2}}$ =
(3 TeV$)^{2}$ and (10 TeV$)^{2}$, respectively. Many qualitative features of
the collider reach contours are similar to the 3 TeV case. We again observe
the interference effect which deteriorates the collider reach at small mass
splitting in the $\Delta m^{2}/\overline{m^{2}}$ vs $\sin 2\theta_{R}$ plane,
and the divergence of the uncertainty width with $M_{1}$ in the $\sin
2\theta_{R}$ vs $M_{1}$ plane. The strength of the high energy muon collider
compared to low-energy experiments is more evident at higher energies. We see
that a 10 TeV muon collider will probe parameter space unreached by the final
stage PRISM/PRIME experiment.

##  5 Beyond the Minimal Spectrum: LFV, EDM and Collider Complementarity

###  5.1 Complementarity with the electron EDM: minimal flavor violation

One of the most stringent precision constraints on sleptons and neutralinos
arises from electron EDM experiments, among which the best current limit is
from the ACME II experiment [96]. The electron EDM can arise at one loop in
diagrams involving a CP-violating phase (see, e.g., [97]). There is no direct
analogue of the minimal $\mu\to e\gamma$ diagram in Fig. 1, because only the
off-diagonal $\delta^{RR}$ entries admit a complex phase. As a result, any
comparison of collider reach with the electron EDM constraint necessarily
involves additional assumptions about the SUSY spectrum and interactions. For
realizations of SUSY with large CP-violating phases, the EDM constraint can
already rule out slepton masses of order 10 TeV [98]. However, many well-
motivated scenarios for SUSY breaking predict smaller phases. In this section,
we will discuss some plausible assumptions about the dominant sources of the
EDM, and compare the reach of precision experiments with collider experiments.
Further discussion of the underlying assumptions about SUSY breaking may be
found in Appendix A.

Figure 7:  One-loop diagrams contributing to the electron EDM involving
higgsinos or right-handed selectrons, but without flavor violation. Both
diagrams may be sensitive to a subleading, CP-violating phase $\arg(\mu
M_{1})$ arising from gravity-mediated contributions. At left: a diagram that
is sensitive to $\mu$ through the mixing of ${\tilde{e}}_{L}$ and
${\tilde{e}}_{R}$. At right: a diagram involving only ${\tilde{e}}_{R}$
together with bino–higgsino mixing.

In models with minimal flavor violation, the electron EDM will be proportional
to $m_{e}$ itself. In this case, the leading contribution to the electron EDM
could arise from the diagrams in Fig. 7. Both diagrams are sensitive to the
phase $\arg(\mu M_{1})$, where the higgsino mass parameter $\mu$ enters in one
diagram through the mixing of left- and right-handed selectrons, and in the
other diagram through bino-higgsino mixing. In the limit where the left-handed
selectron and higgsino are much heavier than the right-handed selectron and
bino, the sum of these contributions can be estimated as

$d_{e}\approx-e\frac{\alpha_{Y}}{4\pi}\frac{m_{e}\,\tan\beta}{m_{{\tilde{\ell}}_{R}}^{2}}\mathrm{Im}\left[\frac{\mu
M_{1}}{m_{{\tilde{\ell}}_{L}}^{2}}+\frac{M_{1}}{\mu}\right]B\left(\frac{|M_{1}|^{2}}{m_{{\tilde{\ell}}_{R}}^{2}}\right),$
(26)

where the loop function $B(r)$ (following the notation of [99]) is defined as

$B(r)=\frac{1-r^{2}+2r\log r}{2(1-r)^{3}}.$ (27)

This formula captures the full diagram up to precision
$O(m_{{\tilde{\ell}}_{R}}^{2}/m_{{\tilde{\ell}}_{L}}^{2})$ and
$O(|M_{1}/\mu|^{2})$. In our numerical results below, we will include the full
one-loop contribution of the mixed left- and right-handed selectrons and the
mixed bino and higgsinos.

Figure 8: Complementarity of a 10 TeV muon collider with EDM and $\mu\to
e\gamma$ experiments. This figure is similar to the upper row of Fig. 6, but
also shows the electron EDM constraint from the contributions shown in Fig. 7.
The phase in the $\mu$ term is assumed to originate from a subdominant gravity
mediated contribution of order $m_{3/2}\sim\sqrt{\delta}\mu$. The solid blue
curve displays the constraint from the current ACME II experiment [96]. The
dashed blue curve shows a hypothetical (near-)future constraint.

To compare the reach of electron EDM experiments to LFV and muon collider
experiments, we must make additional assumptions about the SUSY spectrum. If
the CP-violating phase $\arg(M_{1}\mu)$ is large, then the EDM experiment is
much more stringent. However, it is plausible that this phase vanishes at
leading order and obtains only subdominant contributions. If the off-diagonal
mass matrix terms are of order $m_{3/2}^{2}$ and the component of $\mu$ with a
CP-violating phase is of order $m_{3/2}$, then we have
$\arg(\mu)\sim\sqrt{\delta}$. We follow this assumption in plotting
constraints in Fig. 8, taking $\delta$ to be related to the parameters
$\sin(2\theta_{R})$ and $\Delta m^{2}/\overline{m^{2}}$ on the two plot axes
via Eq. (5). We further assume that the left-handed sleptons and higgsinos are
sufficiently massive that they will not be directly produced at a 10 TeV muon
collider (6 TeV and 5 TeV, respectively) and will have a subdominant effect on
the $\mu\to e\gamma$ rate, so that the other curves in the figure are
unchanged from Fig. 6. The result, illustrated in Fig. 8, is that ACME II is
already a stringent constraint on the parameter space, but the muon collider
can probe regions that are not yet excluded. Furthermore, models in which
$\arg(M_{1}\mu)$ is smaller than $\sqrt{\delta}$ by an order of magnitude
would continue to evade EDM experiments, while remaining accessible to LFV and
muon collider experiments.

###  5.2 Complementarity with eEDM and $\mu\to e\gamma$ via stau mixing

Figure 9:  One-loop diagrams contributing to the electron EDM in the presence
of flavor violation, proportional to the flavor-violating parameters
$\delta^{LL}_{e\tau}$ and $\delta^{RR}_{\tau e}$, which will in general carry
$O(1)$ CP-violating phases when arising from gravity mediation.

Flavor violation opens up the possibility that the dominant source of the
electron EDM will be sensitive to larger couplings, such as the tau Yukawa
[36, 19, 37, 18]. These contributions are enhanced by $y_{\tau}/y_{e}\approx
3.5\times 10^{3}$, but also suppressed by mixing effects, as illustrated in
Fig. 9. In the limit in which the right-handed sleptons are all nearly
degenerate with mass $m_{{\tilde{\ell}}_{R}}$, and the left-handed sleptons
are all nearly degenerate with a much larger mass $m_{{\tilde{\ell}}_{L}}$, we
can estimate the electron EDM induced by loops of the right-handed selectron
and stau states and a bino:

$d_{e}\approx
e\frac{\alpha_{Y}}{2\pi}\frac{m_{\tau}\,\tan\beta}{m_{{\tilde{\ell}}_{L}}^{2}m_{{\tilde{\ell}}_{R}}^{2}}\mathrm{Im}(\mu
M_{1}\delta^{LL}_{e\tau}\delta^{RR}_{\tau
e})f_{2n}\left(\frac{|M_{1}|^{2}}{m_{{\tilde{\ell}}_{R}}^{2}}\right).$ (28)

Here (in notation following [36]) the loop function is

$f_{2n}(r)=\frac{1+(4-5r)r+2r(2+r)\log r}{4(1-r)^{4}}.$ (29)

As in the $\mu\to e\gamma$ context discussed in Section 2.1, this loop
function is related to the function $B(r)$ appearing in (27) by a derivative:
$(rB(r))^{\prime}=2f_{2n}(r)$. Also as in that discussion, the mass insertion
approximation here is valid even when the states
${\tilde{e}}_{R},{\tilde{\tau}}_{R}$ are maximally mixed with each other (and
similary ${\tilde{e}}_{L},{\tilde{\tau}}_{L}$), provided the $\delta$’s are
small. The equation (28) gets corrections of relative size
$m_{{\tilde{\ell}}_{R}}^{2}/m_{{\tilde{\ell}}_{L}}^{2}$. Another approximate
expression has been derived in the limit where all of the sleptons are
degenerate at a common scale $m_{\tilde{\ell}}$ [36],

$d_{e}\approx
e\frac{\alpha_{Y}}{4\pi}\frac{m_{\tau}\,\tan\beta}{m_{\tilde{\ell}}^{4}}\mathrm{Im}(\mu
M_{1}\delta^{LL}_{e\tau}\delta^{RR}_{\tau
e})f_{4n}\left(\frac{|M_{1}|^{2}}{m_{\tilde{\ell}}^{2}}\right)$ (30)

Here

$f_{4n}(r)=\frac{-3-44r+36r^{2}+12r^{3}-r^{4}-12r(3r+2)\log r}{6(1-r)^{6}}.$
(31)

In our numerical results, we use the complete one-loop correction from the set
of mixed ${\tilde{e}}_{L,R},{\tilde{\tau}}_{L,R}$ states, rather than either
of these approximate formulas.

An analogous contribution to the $\mu\to e\gamma$ decay rate can enhance it
beyond the minimal contribution that we discussed in Sec. 2.1. The
corresponding Feynman diagram is precisely Fig. 9, but with either
$e_{L},{\tilde{e}}_{L}$ replaced by $\mu_{L},{\tilde{\mu}}_{L}$ or the
analogous change to the right-handed (s)electron. The corresponding amplitude
can be estimated in exactly the same way, but without taking the imaginary
part:

${\cal A}^{R}_{\mu
e}\approx\frac{\alpha_{Y}}{2\pi}\frac{m_{\tau}\,\tan\beta}{m_{{\tilde{\ell}}_{L}}^{2}m_{{\tilde{\ell}}_{R}}^{2}}\mu
M_{1}\delta^{LL}_{\mu\tau}\delta^{RR}_{\tau
e}f_{2n}\left(\frac{|M_{1}|^{2}}{m_{{\tilde{\ell}}_{R}}^{2}}\right).$ (32)

When the $\delta$ values are relatively large, this effect dominates over the
minimal contribution due to the large ratio $m_{\tau}/m_{\mu}$. However,
because it is quadratic in the $\delta$’s while the minimal contribution is
linear in $\delta^{RR}_{e\mu}$, we expect that in most of the parameter space
it is subleading.

Figure 10: Complementarity of a 10 TeV muon collider with EDM and $\mu\to
e\gamma$ experiments. This figure is similar to Fig. 6, but also shows the
electron EDM and $\mu\to e\gamma$ effects from the stau mixing contributions
shown in Fig. 9. The solid blue curve displays the constraint from the current
ACME II experiment [96]. The dashed blue curves show hypothetical future EDM
constraints. The dotted green and purple curves are the low-energy LFV
constraints from Fig. 6, while the dashed green curves add the effects from
stau mixing. We see that the stau mixing effect, being quadratic in $\delta$,
is only important at larger $\delta$ values.

We illustrate the size of these stau-mixing-induced EDM and LFV signals, in
comparison to collider reach, in Fig. 10. In this case, we do not include the
contributions discussed in the preceding subsection (assuming $\arg(M_{1}\mu)$
is negligible), but instead assuming a maximal phase in the products of
$\delta$ parameters appearing in Eq. (28) and Eq. (32). For plotting purposes,
we take both $\delta$ parameters to be determined by $\frac{\Delta
m^{2}}{\overline{m^{2}}}$ and $\sin(2\theta_{R})$ as in Eq. (5), even though
strictly speaking that was an equation only for $\delta^{RR}_{e\mu}$. At least
in certain classes of models, all of the $\delta^{LL}$ and $\delta^{RR}$
values could be expected to be parametrically of the same size. For the
$\mu\to e\gamma$ contribution, we add the minimal contribution from Eq. (10)
to that of Eq. (32), taking each of the $\delta$ parameters to have phase
$\pi/4$. (Choosing different phases would have only a mild effect on the low-
energy LFV curves in these log-log plots.) The figure shows that ACME II is
already a significant constraint on the parameter space. EDM experiments will
need to improve by multiple orders of magnitude to fully cover the muon
collider’s range in this scenario, because the phase scales as $\delta^{2}$
(though dramatic experimental improvements are expected in the next decade, so
this is possible). Furthermore, we see that at larger values of $\delta$, the
new stau-mediated contribution to $\mu\to e\gamma$ has a significant effect on
the bounds (dashed green and purple curves) compared to the minimal
contribution (dotted green and purple curves). As expected, this effect is
negligible for smaller $\delta$, as in the PRISM/PRIME curves.

In this section we have focused on some particular contributions to the
electron EDM and $\mu\to e$ processes, with a small number of additional
particles involved beyond those contributing to our collider observables. In
general, the full spectrum of superpartners could participate in loops
contributing to these processes. For example, $\mu\to e$ conversion in nuclei
can proceed via box diagrams involving squarks, sleptons, and neutralinos or
charginos [13, 17]. However, provided that we remain in the limit where the
SUSY spectrum is not split, and we consider the squarks to be modestly heavier
than the sleptons, it is reasonable to expect that the contributions we have
considered here provide the correct order of magnitude for the observables.
More complete discussions of the relative size of various contributions,
especially in the split SUSY limit where the variation is more dramatic, may
be found in, e.g., [20].

##  6 Conclusions

A high-energy muon collider is an exciting possibility for advancing beyond
the Standard Model, offering a unique combination of a clean lepton-collider
environment for precision measurements, the high energy reach conventionally
thought of as the domain of hadron colliders, and both muon fusion processes
taking advantage of the full beam energy and vector boson fusion processes
sampling a range of lower energies. By colliding muons, new opportunities open
up for flavor physics previously inaccessible at colliders based on first-
generation fermions. At the same time, new precision experiments are improving
constraints on CP-violating observables like the electron EDM and lepton
flavor violating observables like $\mu\to e\gamma$ by orders of magnitude. By
the time any future collider is built, we may already have dramatic
discoveries or stringent constraints from such experiments. Thus, it is
important to explore the interplay between them.

In this paper, we have shown that there is a powerful complementarity between
these different experiments. A muon collider offers the opportunity to
discover new electromagnetically-interacting particles, like sleptons, and to
measure their mass and the mass of their decay products with high precision.
Such precision measurements further facilitate clean measurements of the rate
of flavor-violating processes. We have illustrated this with the case of
mixing between right-handed selectrons and smuons. In regions of parameter
space with sizable mixing, as motivated by gauge-mediated supersymmetry
breaking with subleading (e.g., gravity-mediated) flavor violation, flavor-
violating signals could be measured even for fractional mass splittings at the
sub-percent level.

The minimal model responsible for the collider signal that we considered also
gives rise to $\mu\to e$ decay and conversion processes in the right range to
be probed by next-generation experiments. These are complementary, in the
sense that each of these experiments can probe parameter space that is
inaccessible to the other. Furthermore, they would complement each other well
in their region of overlap. A discovery at an LFV experiment could occur
first, but this would only tell us a single branching ratio. We would know
that physics beyond the Standard Model exists, but we would lack the detailed
information needed to build a new, more complete Standard Model. The only way
to unravel the physics ultimately responsible for the flavor-violating decay
would be to reach high energies and directly produce the particles
responsible. We have shown, in an example, that a muon collider could do this.

Models responsible for these signals could also, generically, produce a
detectable electron EDM. However, the EDM signal requires CP violation,
whereas the LFV and collider signals do not. EDM experiments, then, could
measure the complex phase associated with the couplings that might be
discovered by the other experiments. A discovery of new physics with an
associated small CP phase would (much like the Strong CP problem of the
Standard Model) be an important hint about underlying dynamics. It could
point, for instance, to the possibility that CP is a spontaneously broken
gauge symmetry in our universe, broken dynamically in a way that generates a
large CKM phase but suppresses other phases.

In the end, we will need a large suite of experiments to uniquely fingerprint
the source of physics beyond the Standard Model. LFV and EDM experiments are
making major strides, and a decisive discovery of new physics could come in
the near future. In such a case, the crucial next step is to turn to an
experiment that can fully characterize the new physics. Only a collider has
the power to produce multiple new particles and measure their masses and
couplings, unveiling the origin of higher-dimension operators. A muon collider
is an ideal choice for such a task.

### Acknowledgments

The work of SH, QL and MR is supported in part by the DOE grant DE-SC0013607
and the by the Alfred P. Sloan Foundation Grant No. G-2019-12504. QL and MR
are also supported by the NASA Grant 80NSSC20K0506.

##  Appendix A Flavor and Supersymmetry Breaking

###  A.1 Gauge mediation with subdominant gravity mediation

In our phenomenological studies, we have simply assumed some structure of
flavor-conserving and flavor-violating soft SUSY-breaking terms. In this
appendix, we provide some brief comments on possible underlying models of SUSY
breaking responsible for such structures. Model building is not our primary
goal in this paper, so we will keep our comments brief, but having an
approximate underlying model in mind will allow us to provide estimates not
only of the relative reach of collider observables and precision CLFV
observables, but also of electric dipole moment (EDM) observables.

Gauge-mediated supersymmetry breaking (GMSB, reviewed in [100, 101]) naturally
provides flavor-universal supersymmetric soft masses, because the different
generations of matter fields have identical gauge interactions. GMSB also
produces CP-preserving supersymmetric soft masses. A weakness of GMSB models
is that they do not automatically explain the $\mu$ and $b_{\mu}$ terms, which
are needed to give mass to higgsinos and to explain electroweak symmetry
breaking. This requires extensions of the underlying model; examples include
[102, 103, 104]. Some of these extensions predict no nonzero physical CP phase
in the gaugino and higgsino sector. GMSB models predict small $A$-terms,
generated radiatively from the gaugino masses and $\mu$. Thus, flavor- and CP-
violating observables can be highly suppressed in GMSB models.

The soft terms generated by GMSB from messenger fields (which carry SM gauge
charges) at a mass scale $M_{\mathrm{mess}}$ with SUSY-breaking F-term VEV $F$
are of order

$m_{\mathrm{soft}}\sim\frac{c\alpha_{r}}{4\pi}\frac{F}{M_{\mathrm{mess}}},$
(33)

where $\alpha_{r}$ is determined by the most significant gauge coupling (e.g.,
$\alpha_{s}$ for squarks and gluinos, or $\alpha_{W}$ for left-handed sleptons
or winos) and $c$ is a factor that we will take to be order-one.888In many
models, the gauginos are in fact parametrically lighter than the scalars due
to a “gaugino screening” phenomenon [105, 106, 107, 108], leading to split
SUSY phenomenology. We have chosen not to focus on split spectra with very
light gauginos in this work. Some conditions under which this assumption holds
are explained in [107]. We will assume that gaugino screening is not
significant, so that slepton and neutralino masses are the same order of
magnitude. One expects in the most minimal variants of GMSB that there are
light right-handed sleptons and binos, with somewhat heavier left-handed
sleptons and winos, and squarks and gluinos an order of magnitude heavier.
However, variants of GMSB can allow a much wider range of spectra [109, 110].
The gravitino mass is

$m_{3/2}=\frac{F_{0}}{\sqrt{3}M_{\mathrm{Pl}}},$ (34)

where $F_{0}\equiv\mathrm{e}^{\langle K\rangle/2}\langle W\rangle/M_{\rm Pl}$
is expected to be of the same order as the F-term VEV $F$ appearing in the
MSSM soft terms. In our studies, we have focused on decays
${\tilde{\ell}}\to\ell{\tilde{\chi}}^{0}$, and omitted subsequent decays to
the gravitino, like ${\tilde{\chi}}^{0}\to\gamma{\tilde{G}}$. The lifetime for
such decays is

$c\tau\sim\frac{16\pi F_{0}^{2}}{m_{{\tilde{\chi}}^{0}}^{5}}=\frac{48\pi
m_{3/2}^{2}M_{\mathrm{Pl}}^{2}}{m_{{\tilde{\chi}}^{0}}^{5}}=1.7\,\mathrm{km}\left(\frac{m_{3/2}}{1\,\mathrm{GeV}}\right)^{2}\left(\frac{10\,\mathrm{TeV}}{m_{{\tilde{\chi}}^{0}}}\right)^{5}.$
(35)

Thus, the decays happen well outside the collider and our approach is
consistent, provided that $m_{3/2}\gtrsim 100\,\mathrm{MeV}$. This threshold
is several orders of magnitude below $m_{{\tilde{\chi}}^{0}}$, meaning that it
is consistent to assume that GMSB dominates over gravity mediation.

Gravity mediation is the effect of generic Planck-suppressed operators, which
contribute to soft SUSY-breaking terms in addition to GMSB. Such contributions
are generally assumed to be flavor-violating, that is, we expect to have terms
like

$\int\mathrm{d}^{4}\theta\,\frac{X^{\dagger}X}{M_{\rm
Pl}^{2}}c_{ij}L_{i}^{\dagger}L_{j}\sim
c_{ij}m_{3/2}^{2}{\tilde{\ell}}_{i}^{\dagger}{\tilde{\ell}}_{j},$ (36)

where $c_{ij}$ is a matrix in flavor space with order-one entries. (This
assumption is not so easily justified, despite being a common one; we discuss
this further below in §A.3.) In this way, GMSB with subdominant gravity
mediation can explain a nearly flavor-symmetric SUSY spectrum with subleading
flavor-violating effects, as we have assumed. We then expect the off-diagonal
entries in $\delta^{LL}$ and $\delta^{RR}$ to be of order

$\delta^{LL}_{\text{off-diag}}\sim\delta^{RR}_{\text{off-
diag}}\sim\left(\frac{m_{3/2}}{m_{{\tilde{\ell}}_{L,R}}}\right)^{2}.$ (37)

One can also consider gravity-mediated contributions to the left-right slepton
mixing parameters $\delta^{LR}$. These could arise from $A$-terms of order
$m_{3/2}$. However, one could consider a scenario in which the fields $X$
acquiring F-term VEVs also carry charges under other symmetries, which would
suppress the gravity-mediated contributions to $A$-terms (as well as to
gaugino masses). Thus, we will assume that the flavor-violating slepton
mixings are localized in $\delta^{LL}$ and $\delta^{RR}$.

Focusing on the right-handed selectron and smuon studied in the main text, in
terms of the parametrization Eq. (1), gauge mediation motivates the choice

$\Delta^{RR}_{ij}=c_{ij}m_{3/2}^{2},$ (38)

with the $c_{ij}$ order-one numbers. The universal contribution $m_{R}^{2}$ is
a sum of a gauge-mediated contribution and a D-term contribution:

$m_{R}^{2}=\left({\tilde{m}}_{\ell_{R}}^{2}\right)_{\mathrm{GMSB}}+m_{Z}^{2}\cos(2\beta)\sin^{2}\theta_{W}+\cdots,$
(39)

where $\cdots$ stands in for the universal part of loop corrections. We can
absorb various other non-universal, subleading contributions into the
definition of the $\Delta_{ij}$ terms, including the Yukawa contributions of
$m_{e}^{2}$ and $m_{\mu}^{2}$ on the diagonal, and shifts due to mixing with
the left-handed sleptons, which for the smuon is of order
$m_{\mu}^{2}\mu^{2}\tan^{2}\beta/{\tilde{m}}_{\ell_{L}}^{2}$ and is subleading
provided that $m_{3/2}\gg m_{\mu}\tan\beta$. From Eq. (2), we see that for
generic, order-one choices of the $c_{ij}$ coefficients, the mixing
$\sin(2\theta_{R})$ is $O(1)$.

As with the slepton soft masses, we could also imagine that the higgsino mass
parameter $\mu$ is a sum of a dominant term of order $m_{\mathrm{soft}}$ and a
subleading, gravity-mediated term of order $m_{3/2}$. To avoid stringent EDM
constraints, the leading contribution should not carry a CP-violating phase,
but the subleading contribution could. This may be rather model-dependent, as
any structure or symmetries invoked to solve the $\mu$ problem in GMSB could
also affect whether the Giudice-Masiero contribution to the $\mu$ and
$b_{\mu}$ terms [111] is allowed at order $m_{3/2}$ or is further suppressed.

###  A.2 A variant: GMSB with gauged $L_{\mu}-L_{\tau}$

The Standard Model (without right-handed neutrinos) can be extended by gauging
an anomaly-free $U(1)$ symmetry which is the difference of lepton numbers in
two generations [112, 113]. The most interesting case is $L_{\mu}-L_{\tau}$,
as its gauging can explain why the muon neutrino and tau neutrino are nearly
maximally mixed [114]. This gauge symmetry must be higgsed, in order for
neutrino masses to be generated.

The GMSB formalism can be extended to include contributions from messengers
charged under higgsed gauge fields [115, 116]. If messengers of SUSY breaking
carry charge under $L_{\mu}-L_{\tau}$, then the soft terms for the muon and
tau lepton fields will have additional contributions. Unlike conventional
GMSB, then, this theory would predict a spectrum with nearly-degenerate smuons
and staus (and the corresponding sneutrinos) but with selectrons having a
different mass. For our purposes, a detailed discussion of the resulting
spectrum is not necessary; we simply take this as one possible justification
for a spectrum in which the right-handed selectron is lighter, by an $O(1)$
factor, than the right-handed smuon and stau, leading to a larger value of
$\Delta\overline{m^{2}}$. Again, subdominant effects of gravity mediation can
contribute flavor-violating soft terms.

###  A.3 Should gravity mediation violate flavor maximally?

The claim that Planck-suppressed operators should have completely generic
flavor structure arises from the expectation that theories of quantum gravity
have no approximate global symmetries at the Planck scale. However, there is
some tension between this point of view and the expectation that the flavor
structure of the Standard Model—the pattern of hierarchical masses and
mixings—is explained in a natural way by some physics beyond the Standard
Model at high energies. This point seems to be under-appreciated, so we will
make a few remarks on how the tension might be resolved. One way to explain
small Yukawa couplings and mixings is by invoking horizontal symmetries, such
as a $U(1)$ symmetry under which different generations carry different charges
[117, 118, 119]. In a gravitational theory, one should demand that such a
symmetry is either gauged or is an approximate (accidental) symmetry enforced
by a gauge symmetry (perhaps a discrete one). In the case that it is a gauge
symmetry, it may be anomalous, and hence the corresponding gauge boson could
have a large Stueckelberg (4d Green-Schwarz) mass. In any event, such a gauge
symmetry would forbid a term of the form
$\int\mathrm{d}^{4}\theta\,X^{\dagger}XL_{i}^{\dagger}L_{j}$ where $L_{i}$ and
$L_{j}$ carry different flavor gauge charges. Left-handed lepton superfields
could have equal horizontal charges and completely anarchic couplings, as
suggested by neutrino mixing [120], but this would not be expected for right-
handed leptons or quarks. In such models a phenomenon of alignment, in which
the quark mass matrices and squark mass-squared matrices are approximately
diagonal in the same basis, can suppress flavor-changing neutral currents
[119]. It is possible that CP is a fundamental symmetry spontaneously broken
only by fields with horizontal charge, which can suppress EDMs [121]; this was
recently discussed in the lepton sector in [23, 122].

Although models with such flavor textures are interesting in their own right,
an extension of the models can allow the generic Planck-suppressed flavor
violation that is conventionally assumed. If there are flavor-charged fields
that acquire F-term VEVs at a common scale, then the operators take the form
$\int\mathrm{d}^{4}\theta\,X_{k}^{\dagger}X_{l}L_{i}^{\dagger}L_{j}$ where the
horizontal charge of $X_{k}^{\dagger}X_{l}$ is nonzero and compensates that of
$L_{i}^{\dagger}L_{j}$. Of course, this constitutes an additional source of
flavor-symmetry breaking. Nonetheless, it could be extremely subdominant to
the leading flavor breaking, which is often assumed to arise from SUSY-
preserving VEVs, $\langle Z\rangle/\Lambda\sim\lambda\sim 0.2$, generated at a
much higher scale. In other words, flavor symmetries are broken twice: once at
a high scale, generating the correct textures of Yukawa terms, but without
breaking supersymmetry; and a second time at low scale, breaking supersymmetry
in a substantially flavor-violating manner, but producing only small
corrections to the Yukawa textures.

One could consider other explanations of flavor structure and the associated
gravity-mediated soft terms. For example, Yukawa couplings are sometimes
suppressed not by a symmetry but by physical separation (small wavefunction
overlap) of fields in extra dimensions. The three-field overlap integral
involved in generating a coupling like $H_{d}L_{i}{\bar{E}}_{j}$ is quite
different from the four-field overlap integral generating a coupling to
$X^{\dagger}XL_{i}^{\dagger}L_{j}$, and (if the wavefunction of $X$ is
relatively flat) the latter may be much less suppressed than the former.

Producing complete models along these lines is beyond the scope of this work,
where our primary aim is collider phenomenology and an assessment of the
complementarity between different experiments. Nonetheless, we believe that if
new sources of flavor violation are observed in experiment it will be
important to think clearly about how they relate to the flavor problem of
explaining the SM Yukawa textures, and considerations along the lines that we
have sketched here may be relevant.

##  Appendix B Details of the Mass Measurements

In this Appendix we provide more details on the measurement of the slepton and
neutralino masses obtained by fitting to the charged lepton energy spectrum,
as described in Eqs. (3.1) and (16). In Appendix B.1, we discuss the
backgrounds, details of their generation, and the selection cuts used to
mitigate them. In Appendix B.2, we provide more details on the likelihood
procedure used to fit the slepton and neutralino masses, and how the resulting
precision was determined. More discussion of the results, including results
for other center of mass energies not presented in Fig. 2 is provided in
Appendix B.3.

###  B.1 Background Mitigation

The leading backgrounds to the processes in Eq. (3.1) are
$\ell^{+}\ell^{-}\nu\bar{\nu}$ production (via intermediate $W$ bosons) and
lepton pair production via neutral vector boson fusion. The importance of the
latter is particular to high-energy muon colliders, where the forward remnants
may not be tagged as a result of the shielding for the beam-induced
background, thus mimicking the missing momentum of the neutralinos in the
signal process.999The referee brought to our attention the possibility that a
forward muon can potentially be detected even in the presence of a BIB-
shielding nozzle, since a muon is sufficiently heavy to penetrate the nozzle
and reach a forward detector. If the muon detection coverage is increased from
$\eta=2.5$ to $\eta=4$, then the neutral-VBF background is completely
eliminated after applying all the kinematic cuts described in the text. There
is potentially room for a better cut strategy. For example, a less aggressive
cut on the transverse component of $p_{\ell^{+}}+p_{\ell^{-}}$ might be
sufficient to eliminate all of the neutral-VBF background, but we leave such
detailed cut optimization for future work. There are also backgrounds from
$\tau$ pair production and charged vector boson fusion (which is kinematically
similar to the neutral VBF background, but suppressed by the $W$ parton
luminosity), as well as other reducible backgrounds from detector effects, but
these are subdominant after imposing simple cuts and we therefore neglect them
in this estimate.

To isolate the signal process we impose the following cuts, based in part on
those of ref. [93]. In all cases, we require exactly one $\ell^{+}$ and one
$\ell^{-}$, each with $|\eta_{\ell}|<2.5$ and $p_{T,\ell}>25\,\textrm{GeV}$.
We demand that $|p_{\ell^{+}}+p_{\ell^{-}}|>250,750$, or $2500\,\textrm{GeV}$,
and that $p_{\textrm{miss}}>0.5,1.5$ or $5\,\textrm{TeV}$ for $\sqrt{s}=3,10$
and $30\,\textrm{TeV}$ respectively. We further demand that
$|\cos(\phi_{\ell^{+}}-\phi_{\ell^{-}})|<0.95$ to reduce back-to-back
topologies. Finally, we require that the transverse component of
$p_{\ell^{+}}+p_{\ell^{-}}$ be greater than $250,750$ and $2500\,\textrm{GeV}$
at $3,10$, and $30\,\textrm{TeV}$. This final cut is added to substantially
limit the VBF background: in the limit where the remnant muons continue
exactly in the forward direction (i.e., in the limit used for the PDF
approximation to the background), the underlying
$\gamma\gamma\to\ell^{+}\ell^{-}$ process has exactly zero transverse
momentum. In the full $2\to 4$ process, a small amount of transverse momentum
is imparted to the remnants, and the rejection is no longer perfect, but still
removes a great deal of the events.

The backgrounds are generated with MadGraph5 [94] and analyzed at parton
level. To generate VBF background events, in order to avoid collinear
singularities, we impose a generator-level cut on the maximum allowed
pseudorapidity of the muons, $|\eta|<\eta_{\textrm{max}}$. Since in the
$|\eta|\to\infty$ limit the cut on the transverse momentum of the central di-
lepton system removes all the events, we need only take $\eta_{\textrm{max}}$
large enough such that the cross section after applying the selection cuts is
constant. We generated events with $\eta_{\textrm{max}}$ ranging from $4.0$ to
$6.0$, finding stable results, and took $|\eta|<5.0$ as the generator-level
cut for the final analysis. The final, post-cut cross sections of the
different background processes at the various center of mass energies are
shown in Table 1.

$\sqrt{s}$ |  $3\,\textrm{TeV}$ | $10\,\textrm{TeV}$ | $30\,\textrm{TeV}$
---|---|---|---
$W^{+}W^{-}$ | $0.616$ | $0.609$ | $3.03\times 10^{-4}$
VBF ($e^{+}e^{-}$)9 | $0.0256$ | $7.41\times 10^{-3}$ | $2.68\times 10^{-4}$
VBF ($\mu^{+}\mu^{-}$)9 | $0.0314$ | $0.0969$ | $4.16\times 10^{-4}$
Table 1: The cross sections, in fb, for the background processes at various
center of mass energies after all the selection cuts described in the text.

To mimic the imperfect resolution of the detectors, the energies of the events
were manually smeared by a Gaussian, with the width taken as a function of the
energy and pseudorapidity of the particle that mimics the E-Cal resolution of
the Muon Collider Collaboration Delphes Card [123].

###  B.2 Unbinned Likelihood Procedure

Given a finite set of data, $\\{x_{i}\\}$ distributed according to a known
distribution $f(x,\theta_{a})$ containing unknown parameters, $\theta_{a}$,
one can estimate the parameters of the underlying distribution by maximizing
the (extended) likelihood function of the data, $L(\\{x_{i}\\};\theta_{a})$.
In the present situation, the expected functional form of the signal
distribution whose parameters we wish to endpoint is simple, so an unbinned
likelihood fit to the data is an ideal way to estimate their values. For more
details on maximum likelihood methods, see refs. [124, 125].

As discussed below Eq. (15), the distribution of the lepton energies in
slepton pair production is expected to be uniform between $E_{\textrm{min}}$
and $E_{\textrm{max}}$. We can therefore take the (normalized) signal
probability density function (p.d.f.) for a lepton of energy $E$ as

$f_{\textrm{sig}}(E;\hat{E}_{\textrm{min}},\hat{E}_{\textrm{max}},\sigma_{1},\sigma_{2})=\frac{1}{\hat{E}_{\textrm{max}}-\hat{E}_{\textrm{min}}}\begin{cases}\frac{1}{2}\bigg{[}1+\textrm{erf}\Big{(}\frac{E-\hat{E}_{\textrm{min}}}{\sqrt{2}\sigma_{1}}\Big{)}\bigg{]},&E<\frac{1}{2}(\hat{E}_{\textrm{max}}-\hat{E}_{\textrm{min}}),\\\\[12.0pt]
\frac{1}{2}\,\textrm{erfc}\Big{(}\frac{E-\hat{E}_{\textrm{max}}}{\sqrt{2}\sigma_{2}}\Big{)}\phantom{\bigg{|}},&E>\frac{1}{2}(\hat{E}_{\textrm{max}}-\hat{E}_{\textrm{min}}),\\\\[12.0pt]
\end{cases}$ (40)

where $\hat{E}_{\textrm{min}}$ and $\hat{E}_{\textrm{max}}$ are the estimators
for the lower- and upper-endpoints, respectively. The parameters
$\sigma_{1,2}$ describe the width of the error functions at the endpoints,
included to account for the smearing of the edges of the distribution due to
the detector energy resolution.

We must also introduce template p.d.f.s for the backgrounds. These can be
chosen to follow simple functional forms based on the energy distributions
obtained from simulation. The $WW$ background leads to events with lepton
energies ranging up to $\sqrt{s}/2$, while the leptons from the VBF background
have energies peaked at low values. After the selection cuts, we find
reasonable descriptions of the background distributions using a uniform
distribution for the $WW$ background and a shape similar to a Gamma
distribution for the VBF events:

$\displaystyle
f_{\textrm{bkg.,VBF}}(E;a_{1},a_{2})\propto\bigg{(}\frac{E}{a_{1}}\bigg{)}^{a_{2}}\exp(-E/a_{1})$
(41) $\displaystyle f_{\textrm{bkg.,}WW}(E)\propto\frac{1}{E_{\textrm{beam}}}$
(42)

Here, the $a_{i}$ are nuisance parameters that are fit to the background
shapes based on simulation. The proportionality factors are taken (for given
values of the nuisance parameters) such that the integral over energies up to
$E_{\textrm{beam}}=\sqrt{s}/2$ is unity.

Figure 11: Normalized histograms of the slepton-pair signal and the $WW$ and
VBF backgrounds, together with plots of the normalized p.d.f.s, with best fit
values of the nuisance parameters.

With the normalized signal and background p.d.f.s in hand, we can compute the
negative extended log-likelihood, $L$, of a set of $n$ observed events with
energies $\\{E_{i}\\}$:

$-\log
L(n,\\{E_{i}\\};\mu_{\textrm{sig.}},\hat{E}_{\textrm{min}},\hat{E}_{\textrm{max}},\theta_{m})=\mu_{\textrm{tot.}}-\sum_{i=1}^{n}\,\log\Big(\mu_{\textrm{tot.}}\,p(E_{i};\mu_{\textrm{sig.}},\hat{E}_{\textrm{min}},\hat{E}_{\textrm{max}},\theta_{m})\Big{missing}).$
(43)

Here, the $\mu_{a}$ indicate the mean number of expected events for the signal
and background processes after the selection cuts detailed in the previous
section, assuming a given luminosity, and
$\mu_{\textrm{tot.}}=\mu_{\textrm{sig.}}+\mu_{\textrm{bkg.,VBF}}+\mu_{\textrm{bkg.,}WW}$.
The $\theta_{m}$ is short for the set of nuisance parameters for the signal
and background p.d.f.s, $\theta_{m}=\\{\sigma_{1},\sigma_{2},a_{1},a_{2}\\}$
described in Eqs. (40) and (41). The likelihood for an individual event,
$p(E_{i};\dots)$ is given by

$p(E;\mu_{\textrm{sig.}},\hat{E}_{\textrm{min}},\hat{E}_{\textrm{max}},\theta_{m})=\\\\[6.0pt]
\frac{\mu_{\textrm{sig.}}}{\mu_{\textrm{tot.}}}f_{\textrm{sig}}(E;\hat{E}_{\textrm{min}},\hat{E}_{\textrm{max}},\sigma_{1},\sigma_{2})+\frac{\mu_{\textrm{bkg.,VBF}}}{\mu_{\textrm{tot.}}}f_{\textrm{bkg.,VBF}}(E;a_{1},a_{2})+\frac{\mu_{\textrm{bkg.,}WW}}{\mu_{\textrm{tot.}}}f_{\textrm{bkg.,}WW}(E)$
(44)

We take $\mu_{\textrm{bkg.,VBF}}$ and $\mu_{\textrm{bkg.,}WW}$ as known from
simulation and evaluate $\mu_{\textrm{sig.}}$, $\hat{E}_{\textrm{min}}$, and
$\hat{E}_{\textrm{max}}$ as functions of the estimated slepton and neutralino
masses, $\hat{m}_{\tilde{\ell}_{R}}$, $\hat{m}_{\tilde{\chi}_{1}^{0}}$. The
signal rate, $\mu_{\textrm{sig}}$ is also a function of the mixing angle and
mass splitting, $\sin 2\theta_{R}$ and $\Delta m^{2}/\overline{m^{2}}$.

Given the expected number of events for the signal and background, we create
toy datasets by randomly selecting $n_{\textrm{sig.}}$,
$n_{\textrm{bkg.,VBF}}$, and $n_{\textrm{bkg.,}WW}$ events from the
corresponding generated samples, with $n_{a}$ drawn from a Poisson
distribution with the corresponding mean, $\mu_{a}$. We can then evaluate the
extended log-likelihood (or more practically, the negative log of the
likelihood or NLL) on the toy datasets. For large enough sample sizes, Wilks’
theorem [126, 127] implies that $2\Delta\textrm{NLL}$ follows a $\chi^{2}$
distribution, which we can use to construct confidence intervals for the
slepton and neutralino masses. This process can be repeated for a number of
toy datasets with the mean values of the NLL allowing us to estimate the
expected precision on the masses.

Because the cross sections for the
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\tilde{B}\tilde{B}$ process are roughly an
order of magnitude larger than the same process with $e^{+}e^{-}$ in the final
state, the measurement of the smuon mass is generally more precise than the
measurement of the selectron mass. This can be significantly ameliorated,
however, by using the fact that the neutralino mass measured in both processes
is the same, and is driven mostly by the $\mu^{+}\mu^{-}$ measurement. To
improve the precision on the selectron mass, we therefore add the likelihood
as a function of the neutralino mass (marginalized over the smuon mass) to the
likelihood obtained in the $e^{+}e^{-}$ measurement.

The resulting $1$ and $2\sigma$ contours for several toy datasets, as well as
the mean, are illustrated for $\sqrt{s}=10\,\textrm{TeV}$ with both
$\mu^{+}\mu^{-}$ (blue) and $e^{+}e^{-}$ (green) final states in Fig. 12. In
practice, since the number of signal events tends to be relatively large at
all the energies considered, we find a stable mean with 10 toy datasets. This
process can be repeated for different values of the true masses and mixing
angles, and the resulting likelihoods are marginalized over to obtain one-
dimensional likelihoods for each mass that can be converted to an expected
precision.

Figure 12:  Contours of $\Delta\textrm{NLL}$ evaluated for the various toy
datasets, as well as the average $\Delta\textrm{NLL}$ for ten such toys, both
for the $\mu^{+}\mu^{-}$ (blue) and $e^{+}e^{-}$ (green) channels, at 68%
(solid) and 95% (dashed) C.L.

###  B.3 Additional Results and Discussion

The results summarized here, and in Section 3.1, indicate that, despite the
additional challenges present at a muon collider, slepton and neutralino
masses can be measured with $\sim$ few $\%$ precision. Our analysis neglects
several important components that would be part of a more rigorous study: we
have made no effort to include systematics associated with our
parameterization of the background shapes, and taken all the nuisance
parameters as fixed. These require a more detailed handling of the detector
effects, but we expect they can be handled in a way that does not
significantly affect the achievable precision.

Furthermore, as noted below Eq. (3.1), we have assumed that the eigenstates we
are measuring are flavor eigenstates. Explicitly, while the cross sections for
$\mu^{+}\mu^{-}\to\mu^{+}\mu^{-}\tilde{B}\tilde{B}$ and $\mu^{+}\mu^{-}\to
e^{+}e^{-}\tilde{B}\tilde{B}$ are taken to vary as a function of the mixing
angle, we are assuming throughout that the masses measured in $\mu^{+}\mu^{-}$
and $e^{+}e^{-}$ final states are respectively the “smuon” and “selectron”
masses, $m_{\tilde{\mu}}$ and $m_{\tilde{e}}$. This is certainly the case in
the limit $\sin 2\theta_{R}\to 0$, and is also a safe assumption when $\Delta
m^{2}\to 0$, since then the two mass eigenstates are degenerate. The only
potential difficulty arises for large $\sin 2\theta_{R}$ and large $\Delta
m^{2}/\overline{m^{2}}$, in which case there are two distinct masses (and
therefore two distinct endpoints) that would be measured in both flavor final
states. This is a problem, for instance, where $\Delta
m^{2}/\overline{m^{2}}=0.1$, in which case the slepton mass splitting for
$m_{\tilde{\ell}}=3\,\textrm{TeV}$ is $\Delta m=150\,\textrm{GeV}$, which is
greater than the expected precision shown Fig. 2. However, the large $\sin
2\theta_{R}$ and large $\Delta m^{2}/\overline{m^{2}}$ regions are also where
the projected low-energy constraints are the strongest, and in principle it is
straightforward to generalize our analysis to measure two endpoints in the
same signal spectrum. Given that the muon collider reach extends to relatively
small mixing angles, we do not expect these issues to significantly affect our
results, and we leave a detailed study of this scenario to future work.

##  Appendix C Slepton Pair Reconstruction Analysis

In this section we describe in detail how the two neutralino momenta in the
process

$\mu^{+}\mu^{-}\rightarrow\tilde{\ell}^{\pm}_{1,2}\tilde{\ell}^{\mp}_{1,2}\rightarrow
e^{\pm}\mu^{\mp}\tilde{B}\,\tilde{B}$ (45)

can be partially reconstructed from the single missing momentum measured in
the detector, given knowledge of the slepton and neutralino masses. To avoid
confusion with the $\\{1,2\\}$ label for the slepton mass eigenstates, we
label the two lepton and neutralinos in the event final state by $\\{a,b\\}$.

Assume that we have measured two lepton three-momenta ${\vec{p}}_{\ell a}$,
${\vec{p}}_{\ell b}$, with magnitude $E_{\ell i}=|{\vec{p}}_{\ell i}|$. The
invisible momentum is taken to be ${\vec{p}}_{\text{inv}}=-{\vec{p}}_{\ell
a}-{\vec{p}}_{\ell b}$. The total center-of-mass energy of the collision is
$E_{\rm CM}$. We assume that $\ell_{a}$ and (invisible) $\tilde{B}_{a}$
reconstruct ${\tilde{\ell}}_{a}$ and that $\ell_{b}$ and (invisible)
$\tilde{B}_{b}$ reconstruct ${\tilde{\ell}}_{b}$, where $\tilde{\ell}_{a,b}$
can be any of $\tilde{\ell}_{1,2}$.

We label the neutralino momenta as ${\vec{p}}_{\tilde{B}a}$ and
${\vec{p}}_{\tilde{B}b}$. These momenta can be parameterized in terms of one
unknown 3-momentum $\vec{k}$, where

${\vec{p}}_{\tilde{B}a}=\frac{1}{2}{\vec{p}}_{\text{inv}}+{\vec{k}},\qquad{\vec{p}}_{\tilde{B}b}=\frac{1}{2}{\vec{p}}_{\text{inv}}-{\vec{k}}.$
(46)

We can rewrite the neutralino momenta in terms of the unknown 3-momentum
$\vec{k}$. Writing this in terms of the lepton energies, we find:

$E_{\tilde{B}a}^{2}-E_{\tilde{B}b}^{2}=(E_{\tilde{B}a}+E_{\tilde{B}b})(E_{\tilde{B}a}-E_{\tilde{B}b})=2{\vec{p}}_{\text{inv}}\cdot{\vec{k}}.$
(47)

Using the total energy conservation we have

$E_{\tilde{B}a}-E_{\tilde{B}b}=\frac{2{\vec{p}}_{\text{inv}}\cdot{\vec{k}}}{E_{\rm
CM}-E_{\ell a}-E_{\ell b}}.$ (48)

The sleptons also need to satisfy the mass-shell conditions,
$(E_{\tilde{B}i}+E_{\ell i})^{2}-({\vec{p}}_{\tilde{B}i}+{\vec{p}}_{\ell
i})^{2}=m_{{\tilde{\ell}}i}^{2}$. Using that the lepton and neutralino are on-
shell, we find that

$E_{\tilde{B}i}=\frac{{\vec{p}}_{\tilde{B}i}\cdot{\vec{p}}_{\ell i}}{E_{\ell
i}}+\frac{1}{2E_{\ell i}}\left(m_{{\tilde{\ell}}_{i}}^{2}-M_{1}^{2}\right).$
(49)

Plugging these formulas into (48), using the parametrization (46), and writing
$\vec{p}_{\rm inv}=-\vec{p}_{\ell a}-\vec{p}_{\ell b}$, we get two linear
equations involving the two unknown quantities ${\vec{k}}\cdot{\vec{p}}_{\ell
a}$, ${\vec{k}}\cdot{\vec{p}}_{\ell b}$, in terms of particle masses and
visible momenta and energies only. The solution to these two linear equations
is

$\displaystyle{\vec{k}}\cdot{\vec{p}}_{\ell a}$
$\displaystyle=+\left[\frac{E_{\rm CM}E_{\ell a}}{2}+\frac{E_{\ell a}}{2E_{\rm
CM}}\left(m_{{\tilde{\ell}}a}^{2}-m_{{\tilde{\ell}}b}^{2}\right)-\frac{1}{2}\left(m_{{\tilde{\ell}}a}^{2}-M_{1}^{2}+E_{\ell
a}^{2}-{\vec{p}}_{\ell a}\cdot{\vec{p}}_{\ell b}\right)\right],$
$\displaystyle{\vec{k}}\cdot{\vec{p}}_{\ell b}$
$\displaystyle=-\left[\frac{E_{\rm CM}E_{\ell b}}{2}+\frac{E_{\ell b}}{2E_{\rm
CM}}\left(m_{{\tilde{\ell}}b}^{2}-m_{{\tilde{\ell}}a}^{2}\right)-\frac{1}{2}\left(m_{{\tilde{\ell}}b}^{2}-M_{1}^{2}+E_{\ell
b}^{2}-{\vec{p}}_{\ell a}\cdot{\vec{p}}_{\ell b}\right)\right].$ (50)

Thus, we have determined two of the three components of the unknown vector.
From this, we can calculate

$\displaystyle{\vec{p}}_{\tilde{B}a}\cdot{\vec{p}}_{\text{inv}}$
$\displaystyle=\frac{1}{2}E_{\rm CM}(E_{\ell b}-E_{\ell a})-\frac{E_{\ell
a}+E_{\ell b}}{2E_{\rm
CM}}\left(m_{{\tilde{\ell}}a}^{2}-m_{{\tilde{\ell}}b}^{2}\right)+\frac{1}{2}\left(2E_{\ell
a}^{2}+m_{{\tilde{\ell}}a}^{2}-m_{{\tilde{\ell}}b}^{2}+2{\vec{p}}_{\ell
a}\cdot{\vec{p}}_{\ell b}\right),$
$\displaystyle{\vec{p}}_{\tilde{B}b}\cdot{\vec{p}}_{\text{inv}}$
$\displaystyle=\frac{1}{2}E_{\rm CM}(E_{\ell a}-E_{\ell b})-\frac{E_{\ell
1}+E_{\ell b}}{2E_{\rm
CM}}\left(m_{{\tilde{\ell}}b}^{2}-m_{{\tilde{\ell}}a}^{2}\right)+\frac{1}{2}\left(2E_{\ell
b}^{2}+m_{{\tilde{\ell}}b}^{2}-m_{{\tilde{\ell}}a}^{2}+2{\vec{p}}_{\ell
a}\cdot{\vec{p}}_{\ell b}\right).$ (51)

Note that these equations are exchanged by $a\leftrightarrow b$ and sum to
$|{\vec{p}}_{\text{inv}}|^{2}=E_{\ell a}^{2}+E_{\ell b}^{2}+2{\vec{p}}_{\ell
a}\cdot{\vec{p}}_{\ell b}$, as they should.

The third component of the unknown momentum should be fixed by conservation of
energy. Therefore, if we find that the components we already found lead to
more energy than $E_{\rm CM}$, we conclude the event is not reconstructible.
At this point, it is useful to introduce a coordinate system. We take the
invisible 3-momentum to point along the $z$-axis, and the lepton 3-momenta to
lie in the $(x,z)$ plane. We can translate our constraints on
${\vec{k}}\cdot{\vec{p}}_{\ell a}$ and ${\vec{k}}\cdot{\vec{p}}_{\ell b}$ into
the $x$ and $z$ components of $k$, with the $y$-component unknown (and only
increasing the energy).

We take the invisible 3-momentum and the lepton 3-momenta to be:

${\vec{p}}_{\text{inv}}=(0,0,p_{\text{inv}}),\qquad{\vec{p}}_{\ell a}=(p_{\ell
x},0,p_{\ell z}),\qquad{\vec{p}}_{\ell b}=(-p_{\ell
x},0,-p_{\text{inv}}-p_{\ell z}).$ (52)

Note that these quantities can be easily calculated starting from a general
frame: $p_{\text{inv}}=|{\vec{p}}_{\ell a}+{\vec{p}}_{\ell b}|$, $p_{\ell
z}=({\vec{p}}_{\ell a}\cdot{\vec{p}}_{\text{inv}})/p_{\text{inv}}$, and
$p_{\ell x}=\sqrt{|{\vec{p}}_{\ell a}|^{2}-p_{\ell z}^{2}}$. In this frame, we
have

${\vec{p}}_{\tilde{B}a}=(p_{\tilde{B}a,x},p_{\tilde{B}a,y},p_{\tilde{B}a,z}),\qquad{\vec{p}}_{\tilde{B}b}=(-p_{\tilde{B}a,x},-p_{\tilde{B}a,y},p_{\text{inv}}-p_{\tilde{B}a,z}).$
(53)

Using Eq. (50), we can solve for the unknown components $p_{\tilde{B}a,x}$ and
$p_{\tilde{B}a,z}$:

$\displaystyle p_{\tilde{B}a,x}$ $\displaystyle=\frac{1}{2E_{\rm
CM}p_{\text{inv}}p_{\ell x}}\Big{[}E_{\rm CM}^{2}(-E_{\ell b}p_{\ell
z}+E_{\ell a}(p_{\text{inv}}+p_{\ell
z}))+(m_{{\tilde{\ell}}a}^{2}-m_{{\tilde{\ell}}b}^{2})(E_{\ell b}p_{\ell
z}+E_{\ell a}(p_{\text{inv}}+p_{\ell z}))\leavevmode\nobreak\ +$
$\displaystyle\qquad\leavevmode\nobreak\ E_{\rm
CM}\big{(}p_{\text{inv}}(M_{1}^{2}-m_{{\tilde{\ell}}a}^{2}+{\vec{p}}_{\ell
a}\cdot{\vec{p}}_{\ell b}-E_{\ell a}^{2})+p_{\ell z}(E_{\ell
b}^{2}-m_{{\tilde{\ell}}a}^{2}+m_{{\tilde{\ell}}2}^{2}-E_{\ell
a}^{2})\big{)}\Big{]},$ $\displaystyle p_{\tilde{B}a,z}$
$\displaystyle=\frac{1}{2p_{\text{inv}}}\Big{[}E_{\rm CM}(E_{\ell b}-E_{\ell
a})-\frac{E_{\ell a}+E_{\ell b}}{E_{\rm
CM}}\big{(}m_{{\tilde{\ell}}a}^{2}-m_{{\tilde{\ell}}b}^{2}\big{)}+\big{(}E_{\ell
a}^{2}-E_{\ell
b}^{2}+m_{{\tilde{\ell}}a}^{2}-m_{{\tilde{\ell}}b}^{2}+2{\vec{p}}_{\ell
a}\cdot{\vec{p}}_{\ell b}\big{)}\Big{]}.$ (54)

Now, the total energy of the two neutralinos is bounded below:

$E_{\tilde{B}a}+E_{\tilde{B}b}\geq\mathcal{E}_{\tilde{B}a}+\mathcal{E}_{\tilde{B}b}\equiv\sqrt{p_{\tilde{B}a,x}^{2}+p_{\tilde{B}a,z}^{2}+M_{1}^{2}}+\sqrt{p_{\tilde{B}a,x}^{2}+(p_{\text{inv}}-p_{\tilde{B}a,z})^{2}+M_{1}^{2}}.$
(55)

If $\mathcal{E}_{\tilde{B}a}+\mathcal{E}_{\tilde{B}b}$ is larger than $E_{\rm
CM}-E_{\ell a}-E_{\ell b}$, then no real value of $p_{\tilde{B}a,y}$ will lead
to a valid solution of the system.

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