Muon Working Group

Fermilab Proton Driver Workshop

2/15/2005

## 1.0 INTRODUCTION 

The proposed SCRF Linac upgrade to the present Fermilab proton source
would enable intense beams to be produced simultaneously at Main
Injector energies and at 8 GeV. This provides a unique opportunity for
the laboratory to support complementary physics programs at the two
energies. In particular, since only a small fraction of the 0.5 -- 2 MW
beam from a new 8 GeV proton driver would be needed to fill the Main
Injector, the remainder of the 8 GeV beam could be used to provide a
low-energy physics program that runs simultaneous with a long-baseline
neutrino program. An attractive option for the low energy program would
be to create a high intensity muon source with some unique
characteristics that would enable a World-class experimental program of
precision muon experiments to be conducted over the next few decades.
The main trio of experiments is the search for a permanent electric
dipole moment (EDM) of the muon, a search for lepton flavor violation in
muon decay (LFV), and a next generation muon (g-2) experiment. This
program of muon physics would need the unique characteristics available
at this new source and could not be carried out at existing facilities.
It would be complementary to the proton synchrotron being built at
J-PARC, which will be able to provide many of the necessary beam
requirements for muon physics, but will require full use of the J-PARC
accelerator to run the muon program. The ability of the low-energy muon
program to run in parallel with the high-energy neutrino program is
unique to this proposed facility, and would permit two major programs to
run in parallel, rather than serially as will be necessary at J-PARC.

In this report we give a brief description of the muon source and its
estimated performance and present the physics case for a program of low
energy muon experiments that go beyond the presently foreseen world
program.

## 2.0 THE MUON SOURCE

Low energy high precision muon experiments require both high intensity
beams and a bunch structure that is appropriate for the experiment.
Hence two ingredients needed to produce an "ideal muon source" are (i) a
high intensity proton source, and (ii) a way of producing the desired
bunch structure. Since different muon experiments require different
bunch structures the ideal source would also be sufficiently flexible to
provide different bunch structures. With an upgraded Proton Driver,
Fermilab would not only have the desired high intensity proton source,
but in the post-collider period could also utilize the Recycler Ring to
repackage the 8 GeV proton beam, yielding the desired bunch structure.

The combination of Proton Driver plus Recycler Ring would provide the
front-end for a unique muon source with intensity and flexibility that
exceed any existing facility.

The Recycler is an 8 GeV storage ring in the Main Injector tunnel that
can run at the same time as the Main Injector. The Recycler is currently
used for antiproton production for the Fermilab collider program. The
beam from the proposed Proton Driver linac that is not used to fill the
Main Injector could be used to fill the Recycler Ring approximately 10
times per second. The ring could then be emptied gradually in the 100 ms
intervals between linac pulses. The ideal muon beam structure is a
narrow pulse of muons followed by a measurement period, with the
timescale being determined by the muon lifetime. For some experiments
the extinction factor during the measurement period needs to be
extremely large, so provisions for a fast-pulsed sweeping magnet will
need to be built into the design from the beginning. The extraction
could be continuous or in bursts.

For example, the Recycler Ring could be loaded with one linac pulse of
1.5 x 10 protons every 100 ms, with one missing pulse every 1.5 seconds
for the 120 GeV Main Injector program. This provides \~1.4 x 10^22^
protons at 8 GeV per operational year (10^7^seconds). In the Recycler
each pulse of 1.5 x 10^14^ protons can be chopped into 588 bunches of
0.25 x 10^12^ protons/bunch and a pulse width of 3 ns. A fast kicker
allows for the extraction of one bunch at a time. The beam structure
made possible by the proton driver linac and the Recycler Ring is
perfect for μ-e conversion experiments, muon EDM searches and other muon
experiments where a pulsed beam is required. Slow extraction from the
Recycler Ring for μ → eγ and μ → 3e searches is also possible.

A detailed muon source design for the Proton Driver does not yet exist.
However, in the following we describe a straw-man concept and estimate
its performance. Work on a more detailed design will be necessary to
evaluate the cost of the facility, which is likely to determine its
viability. The beam requirements for the muon EDM and (g-2) experiments
are somewhat different from the LFV experiments. Muon polarization is
important for the EDM and (g-2) measurements, the figure of merit being
NA^2^, where A is the observed parity violating asymmetry which is the
product of the intrinsic asymmetry and the beam polarization. LFV decay
searches do not require polarized muons, though it has been suggested
that some mode might benefit from polarization.

![](media/image1.png){width="5.997916666666667in"
height="1.0416666666666667in"}

Figure 1. Schematic of the straw man Proton Driver muon source.

The straw-man design for the Proton Driver muon source (Figure 1) is
described in more detail in \[1\], and is based upon design concepts for
the target, pion collection, and decay channel developed for a Neutrino
Factory. The scheme consists of a carbon target within a 20 T solenoid
with a 15 cm bore. This system radially captures all particles with
transverse momentum less than 225 MeV/c. Downstream of the 20 T capture
solenoid the charged pions and their daughter muons are confined within
a lower field (1.25 T) larger bore (60 cm) solenoid decay channel. To
ensure efficient transfer between the high-field and low-field solenoids
the distance over which the field is gradually decreased must be 10 m or
more. Within this transition section the field strength B(s) is
decreased whilst the solenoid radius R(s) is increased such that B(s)
R^2^(s) is constant. The bulk of the charged pions captured within the
channel have a momentum O(100 MeV/c), and decay within a few tens of
meters (τ~0~ = 26 ns). The majority of the daughter muons survive within
the channel (τ~0~ = 2.2 μs). The muon content of the beam steadily
increases for the first 75 m, after which it gradually decreases due to
muon decay. Note that this straw man muon source captures both positive
and negative pions and muons. However, the design can be modified to
include one or more large angle-bends to enable sign selection, as
desired

The performance of the straw man muon source has been simulated using
the MARS code. The evolution of the pion and muon spectra down the decay
channel is shown in Figure 2 and the calculated fluxes are summarized in
Table 1. The scheme will yield

\~0.2 muons of each sign per incident 8 GeV proton. With 1.4 x 10^22^
protons at 8 GeV per operational year this would yield \~3 x 10^21^
muons per year. This muon flux greatly exceeds the flux required to make
progress in broad range of muon experiments. However, the muons at the
end of the decay channel have a large momentum spread and occupy a large
transverse phase space. Without further manipulation their utilization
will be very inefficient. The additional beam-manipulating interface
between the decay channel and each candidate experiment has yet to be
designed. In Japan a Phase Rotated Intense Slow Muon Source (PRISM)
based on an FFAG ring that reduces the muon energy spread (phase
rotates) is being designed. This phase rotation ring has a very large
transverse acceptance (800π mm-mrad) and a momentum acceptance of ±30%
centered at 500 MeV/c. PRISM reduces the momentum and momentum spread to
68 MeV/c and

±1-2% respectively. Hence, a PRISM-like ring downstream of the decay
channel might accept a significant fraction of the muon spectrum (Figure
2) and provide a relatively efficient way to use the available muon
flux. Explicit design work must be done to verify this, but it should be
noted that a muon selection system that utilizes only 1% of the muons
available at the end of the decay channel would still produce an
adequate muon flux for most of the cutting-edge experiments described in
the following sections.

Finally, a new 8 GeV multi-MW Proton Driver at Fermilab, together with
an appropriate target, pion capture and decay channel, would not only
provide a new World Class facility for low energy muon experiments, but
could also provide the first step towards a Neutrino Factory based on a
muon storage ring. In the initial phase the muon beam produced in the
pion decay channel would occupy a relatively large phase space. The
first generation of muon experiments would use only those muons within a
small part of this phase space. This muon source could then be upgraded
in one or more steps with the addition of phase rotation (if not already
included in the initial muon source) and cooling channels to produce a
cold muon beam occupying a phase space that fits within the acceptance
of an accelerator (perhaps the Proton Driver Linac). At this point the
number of useful muons available for low energy muon experiments will be
greatly increased. In addition, new experiments using medium energy
beams of a few GeV will also become possible. Finally, the acceleration
scheme would be upgraded to produce muon beams of 20 GeV or more and a
muon storage ring constructed with long straight sections pointing in
the desired direction. This last step would realize a Neutrino Factory,
perhaps sometime within the next two decades, or on a longer timescale
if a Linear Collider were constructed near Fermilab.

  ----------- ----------- ----------- ----------- ----------- -----------
              S=25m       50m         75m         100m        125m

  μ^+^/p      0.16        0.20        0.21        0.21        0.22

  μ^-^/p      0.16        0.20        0.21        0.21        0.21

  π^+^/p      0.095       0.051       0.030       0.020       0.014

  π^-^/p      0.087       0.044       0.025       0.016       0.011
  ----------- ----------- ----------- ----------- ----------- -----------

3.0 THE PHYSICS CASE

Atmospheric muon neutrinos have been observed to oscillate into other
neutrino flavors. This constitutes an observation of neutral Lepton
Flavor Violation (LFV). The lepton flavor violating effects in muon
decay that result from neutrino masses and mixing are suppressed by the
small neutrino masses and are not observable. However, neutrino
oscillations generically imply the existence of LFV processes at high
mass scales that can have observable effects in muon decays. These
processes can be enhanced if there is new dynamics at the TeV scale,
particularly in the presence of Supersymmetry (SUSY).

The muon anomalous dipole moment (g-2) and Electric Dipole Moment (EDM)
are sensitive to radiative corrections and a broad array of possible new
physics processes. Electric dipole moments can arise from CP violating
phases in the SUSY breaking sector, from neutrino Yukawa couplings or
from a broad range of new physics mechanisms. Although the electric
dipole moment of the electron has been measured with greater precision,
the EDM of the more massive muon is more sensitive to short-distance
effects. Additionally, because of the possibility of different flavor
mixings available from SUSY, the muon EDM could be larger than
expectations based on simple lepton mass scaling. CP violation in the
lepton sector (EDMs as well as neutrino oscillations) could have
possible cosmological implications in the form of leptogenesis. The muon
EDM experiment needs the maximum possible muon flux with polarization,
and will probably be limited by statistical sensitivity. Thus the large
muon flux available simultaneously with the high-energy neutrino beam
(and thus almost continuously) would provide a unique facility for the
EDM experiment.

The muon anomalous magnetic dipole moment a~μ ~= (g-2)/2 is sensitive to
any virtual particles that couple to muons. The Brookhaven (g-2)
collaboration has measured a value for a~μ~ that is 2.7 standard
deviations away from the expected Standard Model value. This could be
the first indication of new physics beyond the Standard Model appearing
in loop processes.

The muon working group considered two physics scenarios. In the first
scenario, Supersymmetry is discovered at the LHC. In the second
scenario, a Standard Model Higgs is discovered at a reasonable mass at
the LHC, but nothing else is found.

3.1 Scenario 1

Among the various candidates for physics beyond the Standard Model (SM),
the supersymmetric extension is considered to be one of the most
promising as it provides a solution to the hierarchy problem. In
addition, the gauge couplings measured precisely in the past decade are
consistent with the predictions of SUSY GUT models \[2\]. Supersymmetric
models, in general, contain new sources of flavor mixing in the mass
matrices of sleptons and squarks. In the event that Supersymmetry is
discovered at the LHC, the magnitude of the muon (g-2), EDM and lepton
flavor violating decays will all be enhanced. In order to understand the
detailed nature of SUSY, relevant measurements such as these will have
to be made since many of the model's parameters are only accessible from
low-energy experiments. For example, as discussed below, muon (g-2) is
sensitive to any SUSY model where tanβ is large. Once the mass spectrum
is known, (g-2) can provide a measurement of this important parameter.

![](media/image2.png){width="5.8in" height="8.3in"}

Figure 2: MARS prediction for the evolution of the pion and muon spectra
down the decay channel. The normalization corresponds to a 2 MW proton
beam at 8 GeV. Figure from Ref. \[1\].

Lepton flavor violating decays, muon (g-2) and electric dipole moments
provide the possibility to study SUSY models at low-energy. In SUSY
models, (g-2) and EDMs probe the diagonal elements of the slepton mixing
matrix, while LFV decays probe the off-diagonal terms (Figure 3). If
SUSY is observed at the LHC, LFV decays and muon moments will provide
one of the cleanest measurements of tan β and of new CP violating
phases.

![](media/image3.png){width="3.833464566929134in" height="1.0in"}

$\begin{pmatrix}
V_{\overset{\sim}{e}\overset{\sim}{e}} & V_{\overset{\sim}{e}\overset{\sim}{\mu}} \\
V_{\overset{\sim}{\mu}\overset{\sim}{e}} & V_{\overset{\sim}{\mu}\overset{\sim}{\mu}}
\end{pmatrix}$

## 3.1.1 Lepton Flavor Violating Decays

Small non-zero masses of neutrinos can be explained via the seesaw
mechanism where the left-handed neutrinos of the Standard Model acquire
a small mass from their small mixing with heavy right-handed singlet
neutrinos. However, it has yet to be demonstrated that the seesaw
mechanism can explain why mixing in the lepton sector is significantly
larger than in the quark sector. In the absence of a detailed theory of
flavor it is important to study the consequences of lepton flavor mixing
for as many systems as possible.

Current neutrino oscillation data suggests that the mass of the
right-handed neutrinos is in the range of 10^12^ to 10^15^ GeV. When the
seesaw mechanism is added into the Standard Model, the rates of lepton
flavor violating (LFV) decays is quite small:

BR(μ → eγ) α 1/M~R~^4^ \~ 10^-50^.

However, in the case of the SUSY seesaw model, which is parameterized at
the GUT scale by the right-handed Majorana mass matrix and the neutrino
Yukawa matrix, the Yukawa interactions of the neutrinos induce the
off-diagonal components at the one-loop level. The magnitude of the
effect depends on the origin of the SUSY breaking and can be sizable.
The current upper bound on the branching ratio of μ → eγ already places
severe constraints on SUSY parameters.

In the minimal supersymmetric seesaw model, LFV decays provide a tool to
indirectly study neutrino parameters and probe other aspects beyond the
large mixing angles measured in neutrino oscillations. Various seesaw
parameters can also be measured or constrained \[3\]. If SUSY breaking
originates beyond the heavy singlet neutrino mass scale, LFV slepton
masses can be induced radiatively by the Dirac Yukawa couplings of the
neutrinos. Alternatively, the light neutrino masses and mixings depend
on both the Yukawa couplings and the Majorana masses of the heavy
singlet-neutrinos. Thus, one can hope to reconstruct the physical
parameters in the heavy singlet neutrino sector entirely in terms of the
light neutrino data and low-energy observables, such as rates for LFV
processes \[4\].

The generic Feynman diagrams for the LFV decay μ → eγ are shown in
Figure 4. The branching ratio has been calculated in the context of the
Supersymmetric seesaw model and is shown in Figure 5 as a function of
right-handed neutrino mass over the relevant range of 10^12^ to 10^15^
GeV \[5\]. The branching ratio is shown for various values of tan β and
indicates that a significant fraction of parameter space will be
accessible in the near future. The MEG experiment at PSI is expected to
reach a sensitivity of 10^-13^ before the end of the decade.

![](media/image4.png){width="5.708333333333333in"
height="2.6041666666666665in"}

In many SUSY models the decay μ^+^ → e^+^e^-^e^+^ and coherent μ-e
conversion in nuclei can be related to the decay rate of μ → eγ. For
μ^+^ → e^+^e^-^e^+^ the branching ratio can be approximated as

$$\text{Br}(\mu\  \rightarrow \ 3e)\  \approx \ \frac{\alpha}{8\pi}\ \frac{8}{3}\ \left( \text{log}\ \frac{m_{\mu^{2}}}{m_{e^{2}}}\  - \ \frac{\text{11}}{4} \right)\ \text{Br}(\mu\  \rightarrow \ e\gamma)$$

$$\ 7\ x\ \text{10}^{-^{3}}\ \text{Br}(\mu\  \rightarrow \ e\gamma)$$

For μ-e conversion a similar relation holds for tan β \> 1,

$$\Gamma(\mu\  \rightarrow \ e)\  \approx \ \text{16}\alpha^{4}Z^{4_{\text{eff}}}Z|F(q^{2})|^{2}\ \text{Br}(\mu\  \rightarrow \ e\gamma)$$

where Z is the proton number in the nucleus, Z~eff~ is the effective
charge and F(q^2^) is the nuclear form factor. The μ-e conversion rate,
normalized by the muon capture rate in Ti is approximately

$$R(\mu^{-}\  \rightarrow \ e^{-};\ \text{Ti})\  \approx \ 6\ x\ \text{10}^{- 3}\ \text{Br}(\mu\  \rightarrow \ e\gamma)\text{.}$$

Different models predict different ratios between these processes so it
is useful to measure them all.

It is potentially very interesting to measure the μ-e conversion rate in
various nuclei. The conversion rate depends on the input proton and
neutron densities for each nucleus, but the dependencies can be
significantly different for various LFV couplings. Measurements of the
conversion rate in different nuclei are therefore useful for model
discrimination \[6\].

**3.1.2 Muon (g-2)**

A precision measurement of the muon anomalous magnet moment a~μ~=(g-2)/2
tests the Standard Model to the extent that both theory and experiment
are well understood. The Standard Model expectation for a~μ~ depends on
contributions from QED, the weak interaction and hadronic loops. a~μ~ is
dominated by the QED contribution, but the weak and hadronic
contributions are still significant. All three contributions must be
computed at the sub-ppm level in order to be meaningfully compared to
experiment. The QED and weak contributions are currently known with
sufficient precision. The hadronic contributions involve the strong
interaction at low energies and cannot be calculated from first
principles. The hadronic contribution can be determined from measured
e^+^e^-^ scattering cross section measurements to hadrons.

New physics will be reflected in a difference between the measured value
of a~μ~ and the theoretically calculated Standard Model value. Examples
of new physics that could impact the measured value of a~μ~ include muon
substructure, anomalous gauge couplings, leptoquarks or supersymmetry.
In particular, in a minimal supersymmetric model with degenerate
sparticle masses the contribution to a~μ~ would be substantial,
particularly for large tan β.

The current experimental value from the BNL (g-2) collaboration and the
Standard Model theoretical calculation are

a~μ~^exp^ = 116592080(58) x 10^-11^ , a~μ~^SM^ = 116591829(73) x 10^-11^
, and

a~μ~^exp^ - a~μ~^SM^ = 251(93) x 10^-11^ .

This is a big effect with a statistical significance of 2.7σ and could
be a harbinger of new physics. At the 90% C.L. one finds

215 x 10^-11^ \< a~μ ~^NP^ \< 637 x 10^-11^.

If this is indeed a New Physics effect it is larger than the electroweak
contribution and it should manifest itself in other areas as well,
including the muon EDM and muon LFV decays.

![](media/image5.png){width="4.369291338582677in"
height="4.840157480314961in"}

Supersymmetric loop effects are the favored explanation for the large
value of

a~μ~^exp^ - a~μ~^SM^. Depending on the details of supersymmetry
breaking, the supersymmetric electro-weak correction to a~μ~ can be as
large or larger than the Standard Model electro-weak correction,
consistent with (8). The supersymmetric contributions to a~μ~ include
loops with a chargino and a muon sneutrino and loops with a neutralino
and a smuon (Figure 6). They include 2 chargino and 4 neutralino states
and could, in principle, entail slepton mixing and phases. Depending on
SUSY masses, mixing and other parameters, the contribution of a~μ~^SUSY^
can span a broad range of possibilities. Supersymmetric effects on
a~μ  ~have been computed for a variety of models. Rather than focus on
specific models, we simply consider the large tan β approximation with
degenerate SUSY masses. In specific models with non-degenerate gauginos
and sleptons a more detailed analysis is required, but here we aim
simply to illustrate the scale of supersymmetry being probed. The
leading effect from Figure 6 given in the large tan β limit is \[7\]

$$|a_{\mu^{\text{SUSY}}}|\  \approx \ \frac{\alpha(M_{Z})}{8\pi\ \text{sin}^{2}\vartheta_{W}}\ \frac{m^{2_{\mu}}}{m^{2_{\text{SUSY}}}}\ \text{tan}\ \beta\ \left( 1\  - \ \frac{4\alpha}{\pi}\ \text{ln}\ \frac{m_{\text{SUSY}}}{m_{\mu}} \right)$$

$\ (\text{sgn}\ \mu)\ \text{13}\ x\ \text{10}^{- \text{10}}\ \text{tan}\ \beta\ \left( \frac{\text{100}\ \text{GeV}}{m_{\text{SUSY}}} \right)^{2}$.

From Equation (7) we then get

$$m_{\text{SUSY}}\  \approx \ (\text{74}\ \text{GeV})\ \sqrt{\text{tan}\ \beta}\text{.}$$

For large tan β in the range of 4 to 40, where the approximate results
given above should be valid, one finds

$m_{\text{SUSY}}\  \approx \ \text{100}\  - \ \text{450}\ \text{GeV}$,

precisely the range where SUSY particles are generally expected.

Deviations of a~μ~ from the value expected from the Standard Model can
be used to place upper limits on sparticle masses and can be compared to
observations made at the LHC. If the sparticle masses are measured at
the LHC, a~μ ~can be used to constrain tan β and SUSY mixing.

## 3.1.3 Electric Dipole Moments

CP violation remains an essential ingredient of almost all attempts to
explain the matter-antimatter asymmetry of the universe in spite of the
well-recognized fact that the CP violation known to be present in the
CKM matrix is insufficient to explain the puzzle. Searches for CP
violation beyond the CKM matrix are necessary to understand the
asymmetry and are also sensitive probes of new physics. Electric dipole
moments violate both parity (P) and time reversal (T) invariance. If CPT
is conserved, a permanent EDM is a signature of CP violation. In the
Standard Model, EDMs are generated only at the multi-loop level and are
many orders of magnitude below the sensitivity of foreseeable
experiments. *Observation of a non-zero EDM would therefore constitute
unambiguous evidence for new physics.* The EDM of the muon arises from
operators that are similar to (g-2), so the possible observation of a
non-zero value for a~μ ~^NP^ also motivates the search for a muon EDM.

A number of extensions to the Standard Model predict additional sources
of CP violation, a notable example being supersymmetry. In fact, current
limits on EDMs of the electron, neutron and ^199^Hg have placed some of
the most stringent constraints on various models and are highly
complementary to many other low energy constraints since they require CP
violation but no flavor violation. It is easy to find combinations of
parameters that predict large EDMs and many of the most stringent limits
on supersymmetry come from EDM data.

Using conventional scaling arguments, d~μ~ = (m~μ~/m~e~)d~e~ \< 3.3 x
10^-25^ e-cm (90% CL). However, many models of physics beyond the
Standard Model predict deviations from conventional scaling with d~μ~ \>
10^-24^ e-cm. Non-conventional scaling happens quite naturally in
supersymmetric models. In supersymmetric models with non-zero neutrino
masses and large neutrino mixings arising from the seesaw mechanism,
d~μ~ can be significantly enhanced. If the seesaw mechanism is embedded
in the framework of a left-right symmetric gauge structure, the same
interactions responsible for the right-handed Majorana neutrino masses
lead to an enhancement in d~μ~ that can be as large as

5 x 10^-23^ e-cm, even for small values of tan β \[8\]. It has also been
shown that the magnitude of d~μ~ can be greatly enhanced when the heavy
neutrino masses are non-degenerate \[9\] and the magnitude increases
with the magnitude of the neutrino Yukawa couplings and tan β (Figure
7). When the heavy neutrino masses are degenerate, only one of the CP
phases in the minimal supersymmetric seesaw model contributes to EDMs
and the CP violation in the lepton sector is entirely connected to
flavor violation, just as in the quark sector in the Standard Model.
There is no reason to believe that the heavy neutrino masses are exactly
degenerate. In addition, leptogenesis requires non-degenerate singlet
heavy neutrino masses. Leptogenesis occurs when a non-zero lepton number
is generated from the out of equilibrium decays of heavy right-handed
Majorana neutrinos with CP violation coming from the Yukawa sector. The
complex phases in the lepton mixing suggested by leptogenesis can induce
electric dipole moments.

The new physics implied by a~μ~^NP^ ≠ 0 can also induce an electric
dipole moment. Following the model-independent approach in \[10\], one
finds

$$d_{\mu}^{\text{NP}}\  \approx \ 3\ x\ \text{10}^{- \text{22}}\ \left( \frac{a_{\mu}^{\text{NP}}}{3\  \times \ \text{10}^{- 9}} \right)\ \text{tan}\ \varphi_{\text{CP}}\ e\  \cdot \ \text{cm}\  \approx \ 3\  \times \ \text{10}^{- \text{22}}\ \text{tan}\ \varphi_{\text{CP}}\ e\  \cdot \ \text{cm}$$

where φ~CP~ is a new CP-violating phase. Thus, a measurement of d~μ~ ≈
10^-24^ e • cm probes

\| tan φ~CP~\| \> 3(1) × 10^-3^ .

If a non-vanishing muon EDM is discovered, it will be unambiguous
evidence for physics beyond the Standard Model. At the currently
envisioned sensitivity, it will also imply violation of conventional
mass scaling arguments, with important implications for new physics
models. In addition, measurement of non-Standard Model contributions to
both d~μ~ and a~μ ~will provide a measurement of new CP-violating phases
with little dependence on the overall scale of new physics \[10\]. Such
information is difficult to obtain in other ways and will be highly
complementary to information provided by the LHC.

![](media/image6.png){width="4.21875in" height="3.875in"}

## 3.2 Scenario 2

If supersymmetry is not observed at the LHC, precision measurements take
on added significance since they are sensitive, through virtual loops,
to much higher mass scales than direct searches. μ-e conversion and
(g-2) are particularly sensitive to new physics beyond SUSY. LFV occurs
in almost all extensions to the Standard Model. In many scenarios LFV
occurs at a level that will be observed by the next round of
experiments.

Figure 8 shows the current branching fraction limits and mass reach for
various kaon and muon decays that violate lepton flavor number. μ-e
conversion has the greatest mass reach and is more sensitive to nearly
all new physics contributions that are not mediated by a photon. μ → eγ
is approximately 300 times more sensitive than μ-e conversion to
processes mediated by a photon. The motivation is sufficiently strong
that both measurements should be made. The relative rates for μ → eγ and
μ^−^N → e^−^N could provide important information on the underlying
physics mechanism. μ → eγ searches as well as μ → 3e searches can also
be performed with polarized muons, perhaps resulting in smaller
backgrounds. The positrons from normal muon decays are emitted
preferentially in the direction of the muon spin and follow a (1 + P~μ~
cosθ) angular distribution, where P~μ~ is the muon polarization. By
cutting away positrons that are aligned with the muon spin the
background can be reduced, with rejection factors as large as a factor
of 5 \[11\]. However, depending on the mechanism that drives the μ →
eγ or 

μ → 3e decay, a significant signal loss could also occur.

![](media/image7.png){width="1.0763779527559054in"
height="0.5866141732283464in"}![](media/image8.wmf){width="1.0783464566929133in"
height="0.5133858267716536in"}![](media/image9.wmf){width="1.0783464566929133in"
height="0.5992125984251968in"}![](media/image10.wmf){width="1.0783464566929133in"
height="0.5118110236220472in"}![](media/image11.wmf){width="1.0783464566929133in"
height="0.5929133858267717in"}![](media/image12.wmf){width="1.0783464566929133in"
height="0.5673228346456692in"}![](media/image13.wmf){width="2.5956692913385826in"
height="0.5480314960629922in"}

μ-e conversion is sensitive to a broad array of new physics processes,
shown in Figure 9, including compositeness, heavy neutrinos,
lepton-violating Higgs couplings, leptoquarks, anomalous Z-μ-e couplings
and heavy Z' exchange \[12\]. As in scenario 1, it is interesting to
measure the μ-e conversion rate in various nuclei to discriminate
between various models.

The muon (g-2) is also very sensitive to a wide range of new physics
beyond SUSY, including effects due to muon substructure, extra Higgs
bosons, leptoquarks, 2-loop pseudoscalar effects, compact extra
dimensions, anomalous W couplings, extra dimensions and radiatively
generated muon mass \[7\].

![](media/image14.wmf){width="1.45in"
height="0.7590551181102362in"}![](media/image15.wmf){width="1.45in"
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## 3.3 Other Physics Possibilities

In addition to the trio of muon processes discussed thus far, there are
a number of additional interesting measurements in muon physics that
could be pursued if adequate muon beams were available. Many high
precision studies of muonium are currently statistics limited and could
benefit from a proton driver. Muonium allows for precision tests of QCD
and lepton flavor conservation as well precision measurements of the
fine structure constant, m~μ~/m~e~ and the structure of the proton
\[13\]. The results of these measurements have provided constraints on a
number of theories beyond the Standard Model. Future progress will
require new and intense sources of muons.

Precise measurement of the muon lifetime can be used to extract a
precise value for the value of G~F~. Extraction of G~F~ is no longer
limited by theory, as the theoretical error has been reduced to \< 0.3
ppm. The MuLAN experiment at PSI is currently attempting to measure G~F~
to an accuracy of 1 ppm. A second PSI experiment, MuCAP, is measuring
the muon capture rate on the proton. This will allow for the extraction
of the induced pseudoscalar form factor g~p~, another fundamental Weak
Interaction parameter. The goal of MuCAP is a 7% measurement of g~p~.
Future progress on these and other measurements of fundamental
parameters will require high intensity muon beams with small spot size
and tiny momentum dispersion.

![](media/image21.png){width="5.409842519685039in"
height="4.827165354330709in"}

**4.0 PRESENT LIMITS AND PLANNED FUTURE EXPERIMENTS**

The history of searches for various lepton flavor-violating decays is
shown in Figure 10. In Table 2, the current best limits and the expected
sensitivity of planned future experiments is shown. In the context of
the theoretical models discussed earlier, it is entirely possible that
charged lepton flavor violation will be observed in the next round of
experiments. However, in order to accumulate significant statistics to
constrain and extract various new physics parameters, more intense
proton sources will be required.

+------------------+--------------------+------------------------------+
| **Mode**         | ## Current Limit   | **Sensitivity of Planned**   |
|                  |                    |                              |
|                  |                    | **Future Experiments**       |
+------------------+--------------------+------------------------------+
|  μ → 3e          | \< 1 x 10^-12^     |                              |
+------------------+--------------------+------------------------------+
|  μ → eγ          | \< 1.2 x 10^-11^   | 10^-13^ (MEG)                |
+------------------+--------------------+------------------------------+
| $$\frac{\Gamma(  | \< 6.1 x 10^-13^   | 10^-17^ (MECO)               |
| \mu\ A \rightarr |                    |                              |
| ow \ eA)}{\Gamma |                    | 10^-19^ (PRIME)              |
| (\mu\ A \rightar |                    |                              |
| row \ \nu A')}$$ |                    |                              |
+------------------+--------------------+------------------------------+

Over the next decade it is expected that MEG will complete its search
for μ → eγ down to a sensitivity of \~ 10^-13^ and MECO will begin
searching for μ → e conversion. Experience from these programs would
obviously be needed as input for more sensitive searches. One critical
measurement that will make no significant progress until more intense
muon beams are available is the permanent electric dipole moment of the
muon.

The search for the muon EDM requires a high intensity polarized muon
source. For a value of NP^2^ = 10^16^, where N and P are the number of
muons and their polarization, respectively, a sensitivity of 10^-24^ e
**•** cm can be reached. The layout of a polarized muon beam for an EDM
experiment would include:

1.  Pion capture using a high-field solenoid.

2.  Semi-adiabatic transfer of the captured muons to a lower solenoid
    field.

3.  Selection of the pion momentum with a curved solenoid section.

4.  Decay of pions to muons.

5.  Selection of muon momentum.

6.  Compression of the muon momentum spread using a Fixed Field
    Alternating Gradient (FFAG) phase rotator.

The first three items could also be accomplished with a high-frequency
horn magnet system. Since the momentum acceptance of the EDM storage
ring would be limited to a level of about ∆p/p ≈ 2%, the phase rotation
technique with an FFAG ring would be particularly effective to reduce
the momentum spread with minimum loss of muons. This scheme can easily
be incorporated into the muon source facility described in Section 2. A
pulsed beam with a pulse separation of \>500 μs is required in order to
allow time for the muons to decay. As described earlier, such a pulsed
beam can be achieved using the existing Recycler Ring to repackage the 8
GeV protons from the Proton Driver. Modest assumptions suggest that a
value of NP^2^ = 10^16^ can easily be achieved in one year of running.
This is consistent with estimates made by the Muon EDM Collaboration
\[14, 15\].

For muons stored in a ring, the EDM interacting with the motional
electric field in the rest frame of the muon causes the spin to precess
out of the plane of the ring, whereas the (g-2) precession is in the
plane. This can be seen in equation (14) where the first bracketed term
corresponds to the precession due to the anomalous magnet moment (g-2)
and the last term corresponds to the precession due to a non-zero EDM.

$$\overrightarrow{\omega}\  = \  - \ \frac{e}{m}\ \left\lbrack a_{\mu\ }\overrightarrow{B}\  - \ \left( a_{\mu}\  - \ \frac{1}{\gamma^{2\ } - \ 1} \right)\ \frac{\overrightarrow{\beta}\  \times \ \overrightarrow{E}}{c}\  \right\rbrack\  + \ \frac{e}{m\ }\ \left\lbrack \frac{\eta}{2}\ \left( \frac{\overrightarrow{E}}{c}\  + \ \overrightarrow{\beta}\  \times \ \overrightarrow{B} \right) \right\rbrack$$

A dedicated EDM experiment would use a radial electric field to turn off
the (g-2) precession, so that an EDM signal would be a steady build-up
of a vertical spin component. Thus the signal for an EDM is a
time-dependent asymmetry between upward and downward muon-decay
electrons. The calorimeters used to detect the electrons must be stable
over a wide range of rates from muons with early and late decay times as
the early rates will be quite high. Because of the rates involved,
calorimeter signals will have to be integrated rather than working with
single pulses.

The MEG experiment will use a surface muon beam with an intensity of 5 ×
10^8^ μ^+^/s to reach a sensitivity on μ → eγ of \~ 10^-13^. As a next
step it seems reasonable to consider experiments aiming for a
sensitivity of 10^-15^ or better. The challenge here is to design an
experiment that can handle the increase in background from accidentals.

The level of prompt background from u^+^ → e^+^ννγ depends on the cut
selection windows for the energy and opening angle of the eγ pair.
Existing technology provides sufficient detector resolution to make
these cuts effectively in order to reduce prompt backgrounds to a
negligible level. The background for μ → eγ searches will be dominated
by accidental coincidences of a positron from the predominant muon decay
mode and a photon that may originate in the decay u^+^ → e^+^ννγ or may
result from the external bremsstrahlung of an electron. In a DC beam the
rate of accidental coincidences is given by$$

$R_{\mu}^{2}f_{e}\varepsilon_{e}f_{\gamma}\varepsilon_{\gamma}\left( \frac{\Omega}{4\pi} \right)^{2}\ \pi\ \frac{{\Delta\theta}_{e\gamma}^{2}}{\Omega}\ 2\Delta tT$,

where f~e~ (f~γ~) is the e^+^ (γ) yield per stopped muon within the cut
selection window and ∆t is the cut applied on the e -- γ time
difference. For a pulsed beam the rate must be multiplied by the inverse
of the duty cycle. Near the muon decay endpoint where E~e~ and E~γ~ \~
m~μ~/2,

$$f_{e}\ \alpha\ _{E_{e}}^{{\Delta E}_{e}}\ ,\ f_{\gamma}\ \alpha\ \left( {E_{\gamma}}^{{\Delta E}_{\gamma}} \right){}^{2}\ \text{.}$$

Since the accidental background increases quadratically with the muon
stop rate, higher rate experiments will be difficult. To make further
progress, improved experimental concepts must be developed and better
detector resolutions must be attained to reduce the product

$$\frac{{\Delta E}_{e}}{E_{e}}\ \left( \frac{{\Delta E}_{\gamma}}{E_{\gamma}} \right)^{2}\ {\Delta\theta}_{e\gamma}^{2}\Delta\ t\ \text{.}$$

Subdividing the target into a row of very thin foils could reduce the
accidental background. The requirement for positrons and electrons to
originate from the same foil would reduce the background in proportion
to the number of targets. An intense muon beam from a Proton Driver
would be necessary to obtain the required muon stopping rate in a
thinner target.

To take advantage of a series of thin targets the photon detector must
have sufficient directional resolution to distinguish between adjacent
foils. Silicon pixel detectors have been proposed as a means of tracking
both the positron and the electron-positron pair from the converted
electron \[16\]. The main limitation in using pixel technology would
come from scattering in the detectors. Development of thinner pixel
detectors in the coming years would alleviate this problem. A two to
three order of magnitude improvement in sensitivity over that
anticipated by MEG can be envisioned in some scenarios \[16\].

From an experimental perspective, the decay μ → 3e offers some important
advantages because the final state contains only charged particles.
Powerful constraints on vertex quality and position can be used to
reduce accidental backgrounds. The major concern, however, are the high
rates in the tracking system that has to handle the full muon decay
spectrum. A dramatic suppression of background could be achieved by
requiring a minimal opening angle between each e^+^e^-^ pair \[17\].
However, depending on the mechanism driving the μ → 3e decay, such a
tight cut might lead to a large loss of sensitivity as well.

Coherent μ → e conversion offers a number of experimental advantages
over other LFV decays. Since the observation of the process requires the
detection of only one final state particle, accidental coincidences are
not a serious background. The signal electron is emitted with a momentum
of p~e~ ≈ m~μ~c, which corresponds to the endpoint of muon decay in
orbit, the only intrinsic background. Since the momentum distribution of
muon decay in orbit falls sharply above m~μ~c/2, energy resolutions of
1% are sufficient to keep this background below 10^-16^. There are a
number of potential beam related backgrounds that can be suppressed by
using a pulsed beam with stringent requirements on the beam suppression
between pulses when the muons are decaying and being measured.

The current experimental limit on μ → e conversion is 6 × 10^-13^. The
MECO experiment at BNL expects to reach a sensitivity of 10^-17^ and a
future project at JPARC or a Fermilab Proton Driver facility could reach
a sensitivity of 10^-19^ using an FFAG ring to reduce the momentum
spread of the muons and achieve more stops in a thin target. The FFAG
ring also allows time for the pions to decay, providing an ultra-clean
beam of pure muons. This removes the requirement of a pulsed beam with a
very small extinction value. This measurement can continue to benefit
from increased intensity. Backgrounds can be controlled if the electron
momentum resolution can be pushed below ≈ 0.5 MeV/c.

The g-2 experiment at BNL currently has a 2.7σ deviation from the
Standard Model that could be a hint of exiting new physics beyond the
Standard Model. It is critically important that this measurement be
pushed to the level where a definitive statement can be made.

The current measurement of a~μ~ = (g -- 2)/2 is accurate to (0.46 (stat)
± 0.27 (sys)) ppm. A proposed upgrade to the current program could
reduce the systematic error by a factor of 2 and increase the
statistical sample by a factor of 9. This would increase the overall
precision from 0.5 ppm to 0.2 ppm. Combined with expected near-term
improvements in theory, this could lead to a significant confrontation
with the Standard Model. This proposal is currently unfunded.

A new, dedicated g-2 experiment could improve the overall precision by a
factor of 10. This could be accomplished by reducing the systematic
error by a factor of 7 and increasing the statistical sample by a factor
of 200. A long list of improvements necessary to reduce the systematic
error has been identified by the g-2 group. These include use of a
backward muon beam, a longer π → μ decay beamline and better phase space
matching of the beamline to the ring. Particle pile-up will ultimately
limit the sensitivity of future searches. To reduce pile-up, a pulsed
beam (\<20 ns) with a high repetition rate and a minimum of 1 ms between
pulses would be most advantageous.

**5.0 SUMMARY AND CONCLUSIONS**

The discovery of neutrino oscillations has been a major breakthrough in
flavor physics. If confirmed, the deviation of (g-2) from the Standard
Model prediction would be a breakthrough in new physics at the TeV
scale. The combination of these two effects suggests that conservation
of charged lepton number is not sacred and that its violation may soon
be observable. If so, this would be an invaluable window on the physics
of lepton flavor as well as physics at the TeV scale and would yield
novel information on lepton mixing.

Experiments performed in the coming decade may well observe charged LFV
for the first time. However, a muon physics program that is
complementary to LHC physics must move beyond the initial discovery
phase and amass statistics for a number of different modes and
processes. More intense muon beams with very specific properties will be
required for this task.

The combination of a new Proton Driver and the existing Recycler Ring
can provide the front-end for a unique muon source with intensity and
flexibility that exceed any existing facility. Such a muon source would
be capable of significant progress on a broad range of experiments that
probe physics beyond the Standard Model. This program could run
simultaneously with a 120 GeV neutrino program. A detailed design for
the muon source does not currently exist, though we expect to pursue
this in the coming months.

**REFERENCES**

\[1\] S. Brice, S. Geer, K. Paul, and R. Tayloe, hep-ex/0408135
(FERMILAB-CONF-04-196-E).

\[2\] U. Amaldi, W. de Boer, H. Furstenau, Phys. Lett. **B260**, 447
(1991).

\[3\] J. Ellis, J. Hisano, M. Raidal and Y. Shimizu, hep-th/0206110
(2002).

\[4\] S. Davidson and A. Ibarra, JHEP **0109** (2001) 013.

\[5\] J. Hisano and D. Nomura, hep-ph/0004061 (2000).

\[6\] R. Kitano, M. Koike and Y. Okada, hep-ph/0203110 (2004).

\[7\] Andrzej Czarnecki and William J. Marciano, Phys. Rev. **D64**,
013014 (2001).

\[8\] K.S. Babu, B. Dutta and R.N. Mohapatra, hep-ph/0006329 (2000).

\[9\] J. Ellis, J. Hisano, M. Raidal and Y. Shimizu, CERN-TH/2001-311
(2001).

\[10\] J. L. Feng, K. Matchev and Y. Shadmi, Nucl. Phys. **B613** (2001)
366.

\[11\] Y. Kuno, A. Maki and Y. Okada, MEG TN-01/1997.

\[12\] J. Bernabeu, E. Nardi and D. Tommasini, hep-ph/9306251 (1993),

David Ng and John N. Ng, hep-ph/9308352 (1993),

B.A. Campbell and K.A. Peterson, Phys.Lett.B**192** (1987) 401,

A.  Czarnecki, W. Marciano and K. Melnikov, hep-ph/9801218 (1998).

\[13\] K. Jungmann, nucl-ex/0404013 (2004).

\[14\] A. Silenko *et al*., J-Parc Letter of Intent: Search for a
Permanent Muon Electric Dipole Moment at the 10^-24^ e **•** cm level.

\[15\] F. J.M. Farley *et al*., Phys. Rev. Lett. **93**, 052001 (2004)

\[16\] F. DeJongh, FERMILAB-TM-2292-E.

\[17\] SINDRUM Collaboration, W. Bertl *et al*., Nucl. Phys. **B260**
(1985) 1.