# Nuclear-level Effective Theory of $\mu\rightarrow e$ Conversion:
Formalism and Applications

W. C. Haxton  Department of Physics, University of California, Berkeley,
California 94720, USA  Lawrence Berkeley National Laboratory, Berkeley,
California 94720, USA  Evan Rule  Department of Physics, University of
California, Berkeley, California 94720, USA  Ken McElvain  Department of
Physics, University of California, Berkeley, California 94720, USA  Michael J.
Ramsey-Musolf  Tsung-Dao Lee Institute and School of Physics and Astronomy,
Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China
Amherst Center for Fundamental Interactions, Department of Physics, University
of Massachusetts, Amherst, Massachusetts 01003, USA

###### Abstract

Over the next decade new $\mu\rightarrow e$ conversion searches at Fermilab
(Mu2e) and J-PARC (COMET, DeeMe) are expected to advance limits on charged
lepton flavor violation (CLFV) by more than four orders of magnitude. By
considering the consequence of $P$ and $CP$ on elastic $\mu\rightarrow e$
conversion and the structure of possible charge and current densities, we show
that rates are governed by six nuclear responses and a single scale,
$q/m_{N}$, where $q\approx m_{\mu}$ is the momentum transferred from the
leptons to the nucleus. To relate this result to microscopic formulations of
CLFV, we construct in nonrelativistic effective theory (NRET) the CLFV
nucleon-level interaction, pointing out the relevance of the dimensionless
scales $y=({qb\over 2})^{2}>|\vec{v}_{N}|>|\vec{v}_{\mu}|>|\vec{v}_{T}|$,
where $b$ is the nuclear size, $\vec{v}_{N}$ and $\vec{v}_{\mu}$ are the
nucleon and muon intrinsic velocities, and $\vec{v}_{T}$ is the target recoil
velocity. We discuss previous work, noting the lack of a systematic treatment
of the various small parameters. Because the parameter $y$ is not small, a
proper calculation of $\mu\rightarrow e$ conversion requires a full multipole
expansion of the nuclear response functions, an apparently daunting task with
Coulomb-distorted electron partial waves. We demonstrate that the multipole
expansion can be carried out to high precision by introducing a simplifying
local momentum $q_{\mathrm{eff}}$ for the electron. Previous work has been
limited to simple charge or spin interactions, thereby treating the nucleus
effectively as a point particle. We show that such formulations are not
compatible with the general form of the $\mu\rightarrow e$ conversion rate,
failing to generate three of the six allowed nuclear response functions. The
inclusion of the nucleon velocity $\vec{v}_{N}$ yields an NRET with 16
operators and a rate of the general form. Consequently, in the current
discovery era for CLFV, it provides the most sensible starting point for
experimental analysis, defining what can and cannot be determined about CLFV
from the highly exclusive process of $\mu\rightarrow e$ conversion. Finally,
we expand the NRET operator basis to account for the effects of
$\vec{v}_{\mu}$, associated with the muon’s lower component, generating
corrections to the CLFV coefficients of the point-nucleus response functions.
Using advanced shell-model methods, we compute $\mu\rightarrow e$ conversion
rates for a series of experimental targets, deriving bounds on the
coefficients of the CLFV operators. These calculations are the first to
include a general basis of CLFV operators, full evaluation of the associated
nuclear response functions, and an accurate treatment of electron and muon
Coulomb effects. We discuss target selection as an experimental “knob” that
can be turned to probe the microscopic origins of CLFV. We describe two types
of coherence that enhance certain CLFV operators and selection rules that
blind elastic $\mu\rightarrow e$ conversion to others. We discuss the matching
of the NRET onto higher level effective field theories, such as those
constructed at the light quark level, noting opportunities to build on
existing work in direct detection of dark matter. We discuss the relation of
$\mu\rightarrow e$ conversion to $\mu\rightarrow e+\gamma$ and $\mu\rightarrow
3e$, showing how MEG II and Mu3e results will complement those of Mu2e and
COMET. Finally we describe a accompanying script – in Mathematica and Python
versions – that can be used to compute $\mu\rightarrow e$ conversion rates in
various nuclear targets for the full set of NRET operators.

##  I Introduction

Muon-to-electron conversion, in which a muon bound to a nucleus converts to a
mono-energetic outgoing electron, occurs at an observable level only if there
are new sources of flavor violation, beyond those responsible for neutrino
mixing [1, 2, 3, 4]. It has long been recognized that $\mu\rightarrow e$
conversion and other charged lepton flavor violating (CLFV) processes (e.g.,
$\mu^{+}\rightarrow e^{+}+\gamma$ and $\mu^{+}\rightarrow e^{+}e^{-}e^{+}$)
are among our most sensitive tests of new physics beyond the standard model
[5, 6, 7]. This has motivated a series of experimental advances that, in sum,
have improved limits on $\mu\rightarrow e$ conversion rates by $\approx$12
orders of magnitude over the past 75 years [8].

The experimental attributes of $\mu\rightarrow e$ conversion are quite
attractive. Intense muon beams exist, with rates on target of $\approx
10^{11}$/s expected in the experiments now under construction. Muons can be
readily stopped in thick targets, where they quickly cascade into $1s$ Coulomb
orbits around target nuclei. Conversion occurring without nuclear excitation
produces a monoenergetic electron carrying almost all of the energy released
($\approx m_{\mu}$), generating a signal that can be distinguished from the
continuous spectrum of background electrons coming from the free decay of the
muon, $\mu^{-}\rightarrow e^{-}+\nu_{\mu}+\bar{\nu}_{e}$. If the process is
mediated by photon exchange or otherwise involves an operator that couples to
nuclear charge, the rate can be enhanced by the coherent elastic response [8].

Bounds on $\mu\rightarrow e$ conversion are typically expressed as the
branching ratio with respect to muon capture in the same nucleus. The best
current limits on $\mu\rightarrow e$ conversion correspond to branching ratios
$\approx 10^{-12}-10^{-13}$. Depending on the CLFV operator (including its
isospin, which influences the coherence), existing branching ratio limits are
testing CLFV up to energy scales of $\approx 10^{3}$ TeV. Within the next few
years Fermilab’s Mu2e [9, 10] experiment and J-PARC’s COherent Muon to
Electron Transition (COMET) [11, 12] and DeeMe [13] experiments should begin
taking data. Mu2E and COMET are designed to reach branching ratios of $\approx
10^{-17}$, probing CLFV scales up to $\approx 10^{4}$ TeV.

The nuclear physics of $\mu\rightarrow e$ conversion is somewhat unusual. As
experimentalists select out the elastic channel in order to suppress
backgrounds due to free muon decay, contributing CLFV operators are
constrained by both $P$ and $CP$ selection rules. The energy transfer to the
nucleus is negligible, while the three-momentum scale $q\approx m_{\mu}$ is
comparable to the inverse nuclear size, guaranteeing sensitivity to the
internal structure of the nucleus. These same conditions arise in another
process of considerable interest, WIMP dark matter (DM) direct detection,
where the low WIMP velocity restricts the scattering to the elastic channel in
almost all nuclei, but where the momentum transfer typically peaks at
$q\approx 100-150$ MeV. These conditions lead to unusual relations between
nucleon-level effective theories of DM direct detection and the nuclear
responses that matter to experiment, enhancing some operators and suppressing
others [14, 15].

This paper is the second of two (see [16]) in which the nonrelativistic
effective theory (NRET) formulation of $\mu\rightarrow e$ conversion is
developed at the nucleon level, and then embedded in the nucleus. As was found
in the DM studies, this embedding has significant consequences for the CLFV
physics of elastic $\mu\rightarrow e$ conversion, blinding experiment to
certain interactions, while enhancing sensitivity to others. In [16] we
demonstrated that the rate for elastic $\mu\rightarrow e$ conversion depends
on six symmetry-allowed nuclear response functions $W_{i}$, evaluated at a
momentum scale $q\approx m_{\mu}$ that probes the structure of the nucleus:
this result was obtained through inclusion of the nucleon velocity operator
$\vec{v}_{N}$. In fact, this general form of the elastic $\mu\rightarrow e$
conversion rate can be deduced without specifying the microscopic forms of the
nucleon charges and currents – all that is needed are the parity and time-
reversal properties of the available densities. The derivation is presented in
Sec. III.1. The leptonic coefficients of these response functions, denoted as
$R_{i}$, represent the CLFV physics that can be “mined” in experiments that
employ appropriately selected targets. Work prior to [16] concentrated on just
two of the response functions, associated with the simplest charge and spin
interactions. As we will illustrate in this paper, such interactions may not
be present, and even if they are, do not always dominate the conversion rate.

The Galilean-invariant nucleon-level microscopic description of CLFV we
develop allows one to relate CLFV limits obtained from different targets and
also to connect limits obtained at the nuclear level to higher-energy
formulations. A microscopic operator expansion can be used with the impulse
approximation (or its generalizations) to make these comparisons and
connections. The expansion, carried out to a given order in the available
small parameters, produces a nucleon-level nonrelativistic effective theory
(NRET) with a complete set of operators in that order.

In an NRET, all possible interactions are constructed consistent with the
applicable symmetries and the available nucleon and lepton operators, which
include the nucleon and lepton charges, their spins, the direction of the
relativistic electron $\hat{q}$, the intrinsic nucleon $\vec{v}_{N}$ and muon
$\vec{v}_{\mu}$ velocities, and the target recoil velocity $\vec{v}_{T}$.
Together with the momentum transferred from the leptons $q$, these operators
generate a hierarchy of dimensionless scales, $y\equiv({qb\over
2})^{2}>|\vec{v}_{N}|>|\vec{v}_{\mu}|>|\vec{v}_{T}|$, that should be respected
in NRETs. Here $b$, the harmonic oscillator parameter employed in building
Slater determinants for the nucleus, represents the nuclear size. In this
paper we construct three levels of NRET corresponding to successively
incorporating the physics associated with $y$, $\vec{v}_{N}$, and
$\vec{v}_{\mu}$.

We discuss past work and the approximations employed, which in general has
involved a very limited set of operators and expansions that do not respect
the above hierarchy. In particular, the large value of $y$ – which ranges from
0.20 (C) to 0.37 (Cu) for the targets we explore here, and reaches 0.54 for
the heavy target tungsten – implies significant angular momentum transfer
between the leptons and nucleus. That is, several multipoles can contribute to
the nuclear response functions that govern elastic $\mu\rightarrow e$
conversion.

Previous investigators, perhaps daunted by the task of handling the required
number of electron distorted waves, have generally avoided calculating the
full nuclear response functions. This is an uncontrolled approximation,
governed by terms of higher order in $y$ and sensitive to details of the
underlying nuclear microphysics; we show, among similar targets, errors can
range from negligible ($\approx$ 5%) to $o(1)$. Yet, as summarized in Table 1
of Sec. II, only a few calculations have gone beyond the the lowest multipole,
and those employed electron plane waves and the simplest operators, spin and
charge. All work employing Dirac solutions limited the electron waves to those
with $j=\textstyle{1\over 2}$ (so Dirac $|\kappa|$ =1), so that (at most) only
the lowest multipole operator contributes. The error induced by this Dirac
truncation generally exceeds the size of the Coulomb corrections being
evaluated.

In Sec. II we introduce a simple trick that, later in the paper, enables us to
properly evaluate the response functions: the effects of the Coulomb
interaction on the electron – its shortened wavelength in the nuclear interior
and its enhanced amplitude – can be incorporated through the introduction of
an effective electron momentum $q_{\mathrm{eff}}$. We demonstrate this
approximation is very accurate for both low- and high-$Z$ targets and for the
range of relevant Dirac partial waves. The introduction of $q_{\mathrm{eff}}$
allows one to exploit the full power of standard spherical vector harmonic
Bessel expansions to generate rather elegant expressions for the needed
response functions that include the Dirac physics of the electron and bound
muon.

Section II also includes a discussion of the upper and lower $1s$ muon radial
wave functions $g(r)$ and $f(r)$ that appear in the nuclear transition
densities of the full rate formula. While we later present such a formula, we
note that in light- and medium-mass nuclei the muon is weakly bound, with a
Bohr radius significantly larger than the nuclear size. Consequently, its
Dirac wave function varies slowly over the nucleus. Thus there are
opportunities to streamline rate formulas by taking advantage of the relative
simplicity of the muon physics. By replacing the muon wave function with a
suitably determined average value, rate formulas can be obtained in which the
nuclear and particle physics are more clearly factorized and the roles of
various small parameters more apparent. Noting that past averaging has
followed procedures more appropriate to muon capture, we suggest an improved
procedure that reproduces leading multipoles exactly but remains quite
accurate for others. We quantify errors associated with our procedure for
averaging, showing that they are quite small.

Section III begins with a derivation of the general form of the elastic
$\mu\rightarrow e$ conversion rate, using only symmetry arguments and an
inventory of available charges and currents. We then develop the nucleon-level
NRET Hamiltonian, forming the most general interaction from the available
nucleon and lepton operators. The NRET describes the nonrelativistic intrinsic
Hamiltonian, the physics that remains after the phase-space integration
removes the relative motion of the relativistic electron with respect to the
daughter nucleus. In addition to the electron direction $\hat{q}$ and the
electron and nucleon spin and charge operators, the various velocity operators
come into play: here one can make use of the hierarchy,
$|\vec{v}_{N}|>|\vec{v}_{\mu}|>|\vec{v}_{T}|$ to develop various levels of
NRET. We show that this NRET operator basis needs to include at least terms
linear in $\vec{v}_{N}$ to be complete, in the sense of reproducing the six
nuclear response functions allowed by symmetry.

For completeness, we extend the NRET to include the muon velocity operator
$\vec{v}_{\mu}$ and thus the contribution from the muon’s lower component.
This addition introduces no new response function physics, but instead
generates corrections to the coefficients $R_{i}$ proportional to $\langle
f\rangle/\langle g\rangle$, the ratio of the muon’s lower and upper
components. We stress that this distinguishes the effects of $\vec{v}_{\mu}$ –
corrections to response functions that influence rates at the level of 5% for
27Al – from those of $\vec{v}_{N}$, which generates new response functions,
sensitivity to new aspects of CLFV, and contributions to rates that can be
$o(1)$, depending on the source of the CLFV. This reflects the fact that the
leptonic vertex in elastic $\mu\rightarrow e$ is inclusive, with the sum over
partial waves guaranteeing that muon lower-component contributions are always
accompanied by upper-component contributions. In contrast, the nucleon vertex
is exclusive, with $\vec{v}_{N}$ changing the parity and $CP$ of operators it
generates, thus altering the physics.

Until the present work, no formulation of $\mu\rightarrow e$ conversion has
been available that combined the desired elements: a complete basis of
operators, the full form of the nuclear response functions $W_{i}$, and an
accurate treatment of electron and muon Coulomb effects and lepton and nucleon
velocities. In fact, the effects of $\vec{v}_{N}$ appear to have been ignored
universally.

Because the NRET basis is complete, any UV theory should reduce at low energy
to a form compatible with the NRET. More importantly, as experiments are done
with largely nonrelativistic nuclei, the NRET provides one with a complete but
minimal basis for analyzing and comparing limits or results obtained from a
range of nuclear targets. Consequently, information extracted from
$\mu\rightarrow e$ conversion limits, encoded as constraints on the $R_{i}$’s,
can be “ported upward” to constrain theories formulated at higher energy
scales, which necessarily have more degrees of freedom. Thus a “top-down”
reduction of a favored UV theory to make contact with experiment would only
need to be carried out to the nucleon-level NRET: there would be no need to
repeat the nuclear physics performed here.

At the end of Sec. III we present a simple example of such a reduction: we
construct the 20 nucleon-level relativistic operators that can arise for semi-
leptonic CLFV interactions mediated by scalar or vector mediators. We show
that these interactions reduce to a subset of the NRET operators we introduced
(reflecting the specialization to scalar and vector exchanges), under a non-
relativistic reduction that treats the velocities of bound nucleons and the
muon to first order. We also discuss at the end of this section some of the
nontrivial aspects of such matching, including the interpretation of NRET
operator coefficients derived from nuclei as effective.

This strategy follows one that has been used very effectively in studies of DM
direct detection, an elastic process with very similar kinematics. The NRET DM
formulation [14] has become a very popular bridge between the nuclear scale
and UV theories or, more commonly, effective field theories (EFTs) formulated
at higher energy scales. For example, open-source tools like DirectDM [17],
based on operator formulations at the light-quark and gluon level, were
designed to work with the analogous NRET tool DMFormFactor [15]. This
formalism that has been adopted by most of the major experimental groups [18,
19, 20, 21, 22], some of whom have developed specialized cuts helpful in
isolating specific NRET interactions. We describe in Appendix C a script for
$\mu\rightarrow e$ conversion quite analogous to DMFormFactor. By making the
code publicly available, we hope to encourage similar developments for
$\mu\rightarrow e$ conversion.

In Sec. IV (and in Appendix B) we present details of the standard multipole
formalism for embedding the nucleon-level interaction in a nucleus, leading to
the elastic $\mu\rightarrow e$ conversion rate formula. We present rates for a
succession of NRETs, according to the hierarchy of small parameters: the
extreme nonrelativistic limit, the inclusion of effects linear in
$\vec{v}_{N}$, and the inclusion of additional effects linear in
$\vec{v}_{\mu}$.

Rates are expressed as sums over CLFV coefficients $R_{i}$ multiplied by
nuclear response functions $W_{i}$, with the latter expressed as a sum over
contributing multipoles. Each $R_{i}$ is a bilinear function of the NRET
operator coefficients. This factorization can be helpful in experimental
analysis, if CLFV is discovered. It allows one to view the $W_{i}$’s as
experimental “knobs” that can be dialed to isolate the CLFV coefficients
$R_{i}$: the nuclear response functions depend on properties of the ground
state, such as its angular momentum, isospin, spin-orbit properties of valence
nucleons, and the spin-orbit structure of the core. Separating the various
contributions will be challenging, of course. Yet by selecting nuclear targets
with specific properties, it should be possible to learn more about the
operator source of the CLFV found in $\mu\rightarrow e$ conversion than would
be possible from results on a single nuclear target.

In Sec. V we discuss the nuclear physics of elastic $\mu\rightarrow e$, taking
the long-wavelength limit to isolate the leading nuclear operator for each
response function. We point out two distinct forms of coherence, one of which
is quite novel and especially relevant to targets like 27Al due to the spin-
orbit structure of the ground state. We stress that the naive counting of
operators at the nucleon NRET level no longer holds at the nuclear level: some
operators are enhanced by the two coherence mechanisms, others are suppressed
by nucleus-imposed selection rules. We note that some of the nuclear response
functions $W_{i}$ are closely related to standard-model responses, and thus
potentially could be measured in iso-elastic muon capture or elastic electron
scattering, while others are not and thus necessarily require microscopic
evaluation. We present graphics showing the wide variation in target responses
to CLFV operators; target attributes experimentalists may be able to exploit.

Section V also describes the large-basis configuration-interaction shell-model
calculations we performed to compute nuclear responses, using the best
available effective interactions. The needed one-body density matrices are
obtained for a variety of nuclear targets that either have been used in past
experiments or might be suitable choices for future ones. Calculations were
performed for 11 targets ranging from C to Cu. As the included spaces we use
are separable and the calculations are done without any basis truncations, we
can project spurious center-of-mass motion, for consistency with our Galilean-
invariant NRETs.

In Sec. VI we derive constraints on the operator coefficients – the low-energy
constants or LECs – from existing limits on $\mu\rightarrow e$ conversion and
estimate how these limits will be sharpened when Mu2e and COMET results become
available. The constraints take the form of limits on bilinear combinations of
the LECs, thereby showing what can and cannot be learned about the CLFV from
elastic $\mu\rightarrow e$ conversion. We compare these constraints to those
that can be obtained from $\mu\rightarrow e+\gamma$ and $\mu\rightarrow 3e$.

In the concluding Sec. VII we summarize this paper and describe some future
goals, including extending the current formalism to allow for nuclear
excitation. Such an extension can yield additional physics: because of the
absence of $CP$ constraints on inelastic transitions, four LECs that cannot be
probed in elastic $\mu\rightarrow e$ conversion then play a role.

Figure 1: Depiction of elastic $\mu\rightarrow e$ conversion. The nuclear
Coulomb potential binds the $1s$ initial-state muon and distorts the outgoing
electron wave function. Neglecting nuclear recoil, the electron’s energy is
the muon mass minus its Coulomb binding.

##  II Leptonic Interactions

Figure 1 depicts the conversion of a muon, bound in a $1s$ atomic orbital by
the Coulomb field of the nucleus, into an outgoing electron of fixed energy
$E_{e}\approx m_{\mu}$. If this CLFV process occurs without nuclear excitation
– the nucleus remains in its ground state – one finds

$\displaystyle E_{e}=m_{\mu}-E^{\mathrm{bind}}_{\mu}-{\vec{q}^{\,2}\over
2M_{T}},$ (1)

where $\vec{q}$ is the three-momentum transferred from the nucleus to the
electron, $m_{\mu}$ and $M_{T}$ are respectively the muon and nuclear masses,
and $E^{\mathrm{bind}}_{\mu}$ is the muon’s binding energy, defined here as a
positive quantity. Working to first order in $m_{\mu}/M_{T}$ and ignoring all
smaller quantities in $1/M_{T}$ one finds

$\displaystyle\vec{q}^{\,2}={M_{T}\over
m_{\mu}+M_{T}}\left[(m_{\mu}-E^{\mathrm{bind}}_{\mu})^{2}-m_{e}^{2}\right],$
(2)

which can be substituted into Eq. (1) to determine the nuclear recoil energy.

The CLFV Hamiltonian consists of a series of terms of the form

$\displaystyle\mathcal{\hat{H}}_{i}$ $\displaystyle=$ $\displaystyle
c_{i}~{}\int
d\vec{r}~{}\bar{\Phi}_{e}(\vec{r})\mathcal{O}_{L}\Phi_{\mu}(\vec{r})~{}\Phi^{*}_{N}(\vec{r})\mathcal{O}_{N}\Phi_{N}(\vec{r}).~{}~{}~{}$
(3)

The four-fermion form is quite general: up to small nuclear recoil effects,
the momentum transfer $\vec{q}$ is fixed, and consequently, any propagator
effects can be absorbed into the coefficients $c_{i}$. In general, the
leptonic and nucleon operators $\mathcal{O}_{L}$ and $\mathcal{O}_{N}$ (and
the corresponding currents) can be scalar or vector (in which case they must
couple to an overall scalar). The choices of the $\mathcal{O}_{L}$ and
$\mathcal{O}_{N}$, and consequently the number of coefficients $c_{i}$, are
limited by the available charge and current densities that one can form.
Elastic matrix elements of the interaction can be further limited by parity
and $CP$ selection rules. In this paper, we will construct a general operator
basis for this process, derive rates, and compare those rates to existing and
anticipated limits of $\mu\rightarrow e$ conversion. A first step is the
derivation of the Coulomb-distorted leptonic wave functions that appear in the
transition density of Eq. (3).

###  II.1 Coulomb effects

Taking the matrix element of the interaction in Eq. (3) between lepton and
nuclear states yields

$\displaystyle c_{i}\langle\mathcal{O}_{L}\rangle\int d\vec{r}\langle
f|{\Psi}^{*}_{e}(\vec{r})\Psi_{\mu}(\vec{r})~{}\sum_{i=1}^{A}\hat{O}_{N}(i)\delta(\vec{r}-\vec{r}_{i})|i\rangle~{}~{}~{}~{}$
(4)

where $|i\rangle$ and $|f\rangle$ are the initial and final nuclear states and
$\langle\mathcal{O}_{L}\rangle$ is the matrix element of the leptonic operator
between spin states. We consider here a nuclear operator that is one-body,
which we write somewhat schematically in first quantization. The nuclear
transition density thus involves a convolution of the electron and muon wave
functions with the nuclear density. The outgoing highly relativistic electron
is distorted and enhanced in the region of the nucleus by the Coulomb
potential, which is sourced by the nucleus’s extended charge distribution. The
wave function and bound-state energy of a muon in the $1s$ orbital are an
eigensolution of the Dirac equation for the same Coulomb field.

Two of the terms in the integrand of Eq. (4), the nuclear density and the
electron wave function, vary over a length sale typical of the nuclear radius:
the electron carries a momentum $q\approx m_{\mu}\approx 1/1.86$ fm, while the
shell-model oscillator parameter $b$ for 27Al is 1.85 fm. The third term, the
$1s$ muon wave function, varies more slowly for the targets considered in this
paper, which range from C to Cu. The muon’s Bohr radius in 27Al is
$a_{0}^{\mu}\approx$ 20 fm.

As noted in the Introduction, the neglect of higher multipole operators in
computing elastic $\mu\rightarrow e$ rates will in general lead to a response
function that is correct only through $o(y)$. Such multipoles will contribute
to elastic $\mu\rightarrow e$ conversion if the ground-state angular momentum
$j_{N}\geq 1$. In 27Al, where $j_{N}=\textstyle{5\over 2}$, the symmetry-
allowed charge multipoles are $M_{0}$, $M_{2}$, and $M_{4}$, while the allowed
transverse (longitudinal) spin operators are $\Sigma_{1}^{\prime}$,
$\Sigma_{3}^{\prime}$, and $\Sigma_{5}^{\prime}$ ($\Sigma_{1}^{\prime\prime}$,
$\Sigma_{3}^{\prime\prime}$, and $\Sigma_{5}^{\prime\prime}$). (The specific
forms of these operators will be given later in this paper.) The error one
induces by truncating the multipole expansion is not simple to estimate: the
numerical impact of $o(y^{2})$ corrections will depend on both the NRET
operator under study and the specific nuclear physics of the target. For
example, the $M_{2}$ operator is related to the quadrupole moment, which can
be quite enhanced in mid-shell nuclei due to collective deformation.

Most past work has retained only the leading multipole contribution, while
also making other simplifications, such as neglecting the Coulomb distortion
of the outgoing electron partial waves and treating only the simplest nuclear
operators. A relatively complete compilation of past work is given in Table 1.
In the few cases in Table 1 where the full response function was computed [30,
31, 36, 37], electron plane waves were employed and only the charge and/or
spin nuclear operators were considered. Others used distorted electron waves –
solutions of the Coulomb Dirac equation – but retained only those waves with
Dirac $|\kappa|=1$ ($j=\textstyle{1\over 2})$. This guarantees that (at most)
one multipole operator is generated; an approximation that one expects to
become increasingly problematic with increasing $A$ and $y$. While the shell-
model rate calculations reported in this paper only extend up to Cu, nearly
40% of the spin response probability in 63Cu is carried by the $J=3$
multipole.

In Sec. II.2 we show that a full multipole expansion using distorted electron
waves can be carried out to high accuracy for any of our NRET operators. This
is accomplished by employing an approximation familiar from relativistic
electron scattering: the effects of the Coulomb field in shortening the
wavelength of the electron wave function and enhancing its overlap with the
nucleus can be incorporated very accurately through the introduction of an
effective momentum $q_{\mathrm{eff}}$. We demonstrate the accuracy of the
$q_{\mathrm{eff}}$ approximation as the Dirac quantum number $\kappa$ and
target charge $Z$ are varied. This substitution allows us to complete the full
multipole expansion using standard Bessel vector spherical harmonic
expansions.

In Sec. II.3 we discuss the bound muon wave function. The muon is largely
nonrelativistic: in 27Al the ratio of lower-to-upper components at the peak of
the transition density is $\approx 0.027$. Consequently, $\mu\rightarrow e$
conversion rates are dominated by the muon’s upper component. While in our
master rate formula $g(r)$ and $f(r)$ are retained in the transition
densities, we also simplify that result by replacing these functions with
average values. This is a common practice in muon capture studies [42], as
well as in $\mu\rightarrow e$ conversion, as can be seen in Table 1. Here the
purpose of the averaging is to simplify the rate formula to bring out the
underlying physics more clearly. We also do the averaging in a way that
exactly preserves the values of leading multipoles and minimizes errors for
other multipoles. We demonstrate that the errors induced by using averages
$\langle g\rangle$ and $\langle f\rangle$ are small.

In our NRET the muon’s lower component is generated from the upper component
through the action of the muon’s velocity operator. As the NRET forces one to
think about all of the theory’s velocity operators in a consistent way, it
brings out some inconsistencies in past work. Several of the papers listed in
Table 1 included the lower component, but none prior to our work in [16]
treated the nucleon velocity operator, whose magnitude for aluminum is about
four times larger. The two velocities have a common origin: the NRET is the
Galilean invariant intrinsic Hamiltonian that describes the relative motion of
the nuclear constituents. $A-1$ of the relative (Jacobi) coordinates describe
the relative positions of the $A$ nucleons, while the $A$th Jacobi coordinate
gives the muon’s position relative to the center-of-mass of the $A$ nucleons.
NRET velocity operators are conjugate to these relative coordinates, with
$|\vec{v}_{N}|>|\vec{v}_{\mu}|$ for all targets considered here. We discuss
the muon’s lower component in Sec. II.4 and address its role in the NRET later
in the paper.

Table 1: An incomplete survey of elastic $\mu\rightarrow e$ conversion studies including the nuclear targets considered, the nuclear multipole operators evaluated, and the form of the lepton wave functions employed. $\mathcal{O}_{J;\tau}$ means that both isospin structures and all allowed $J$ were included. For the Dirac electron, all of the references surveyed restrict attention to the lowest partial waves $\kappa=\pm 1$. Besides the Dirac solution, the remaining forms of the muon wave function are all constant approximations: $\braket{}{\psi_{\mu}}{{}^{2}}_{\rho}$ is obtained by averaging the probability of the Dirac solution over the nuclear density, $|\phi^{Z}_{\mathrm{1s}}(\vec{0})|^{2}$ is the probability of the point-like Schrodinger solution evaluated at the origin, $|G(R_{N})|^{2}$ is the upper component of the muon’s Dirac wave function evaluated at the nuclear radius. Superscript ${\dagger}$ indicates that the reference considers the inelastic process as well, although the information in the table reflects only the treatment of the elastic process. Author(s) | Year [Ref] | Target | Operators | $\psi_{e}$ | $\psi_{\mu}$
---|---|---|---|---|---
Weinberg and Feinberg† | 1959 [23] | multiple | $M_{0;p}$ | Plane wave | $\braket{}{\psi_{\mu}}{{}^{2}}_{\rho}$
Marciano and Sanda | 1977 [24] | multiple | $M_{0;\tau}$ | Plane wave | $\braket{}{\psi_{\mu}}{{}^{2}}_{\rho}$
Shanker | 1979 [25] | multiple | $M_{0;\tau}$ | Dirac, $|\kappa|=1~{}~{}$ | Dirac
Kosmas and Vergados† | 1990 [26] | multiple | $M_{0;\tau}$ | Plane wave | $\braket{}{\psi_{\mu}}{{}^{2}}_{\rho}$
Chiang et al.† | 1993 [27] | multiple | $M_{0;\tau}$ | Plane wave | $\braket{}{\psi_{\mu}}{{}^{2}}_{\rho}$
Kosmas et al.† | 1994 [28] |  48Ti | $M_{0;\tau}$ | Plane wave | $\braket{}{\psi_{\mu}}{{}^{2}}_{\rho}$
Czarnecki, Marciano, and Melnikov | 1998 [29] |  27Al, 48Ti, 208Pb | $M_{0;\tau}$ | Dirac, $|\kappa|=1$ | Dirac
Siiskonen, Suhonen, and Kosmas† | 2000 [30] |  27Al, 48Ti | $M_{J;\tau}\;\Sigma^{\prime}_{J;\tau}\;\Sigma^{\prime\prime}_{J;\tau}$ | Plane wave | $\braket{}{\psi_{\mu}}{{}^{2}}_{\rho}$
Kosmas† | 2001 [31] |  48Ti, 208Pb | $M_{J;\tau}\;\Sigma^{\prime}_{J;\tau}\;\Sigma^{\prime\prime}_{J;\tau}$ | Plane wave | $\braket{}{\psi_{\mu}}{{}^{2}}_{\rho}$
Kitano, Koike, and Okada | 2002 [32] | multiple | $M_{0;\tau}$ | Dirac, $|\kappa|=1$ | Dirac
Kosmas | 2003 [33] | multiple | $M_{0;\tau}$ | Plane wave | Dirac
Cirigliano et al. | 2009 [6] | multiple | $M_{0;\tau}$ | Dirac, $|\kappa|=1$ | Dirac
Crivellin et al. | 2017 [34] |  27Al, 197Au | $M_{0;\tau}$ | Dirac, $|\kappa|=1$ | Dirac
Bartolotta and Ramsey-Musolf | 2018 [35] |  27Al | $M_{0;\tau}$ | Dirac, $|\kappa|=1$ | Dirac
Cirigliano, Davidson, and Kuno | 2018 [36] |  27Al | $M_{0;\tau}\;\Sigma^{\prime}_{J;\tau}\;\Sigma^{\prime\prime}_{J;\tau}$ | Plane wave | $|\phi_{1s}^{Z}(\vec{0})|^{2}$
Davidson, Kuno, and Saporta | 2018 [37] |  27Al, Ti | $M_{0;\tau}\;\Sigma^{\prime}_{J;\tau}\;\Sigma^{\prime\prime}_{J;\tau}$ | Plane wave | $|\phi_{1s}^{Z}(\vec{0})|^{2}$
Civitarese and Tarutina† | 2019 [38] |  208Pb | $M_{0;\tau}$ | Plane wave | $|G(R_{N})|^{2}$
Rule, Haxton, and McElvain | 2021 [16] |  27Al, Ti | $M_{J;\tau}\;\Sigma^{\prime}_{J;\tau}\;\Sigma^{\prime\prime}_{J;\tau}$ | Full Dirac ($q_{\mathrm{eff}}$) | $|\phi_{1s}^{Z_{\mathrm{eff}}}(\vec{0})|^{2}$
|  |  | $\Delta_{J;\tau}\;\tilde{\Phi}^{\prime}_{J;\tau}\;\Phi^{\prime\prime}_{J;\tau}$ |  |
Heeck, Szafron, and Uesaka | 2022 [39] | multiple | $M_{0;\tau}$ | Dirac, $|\kappa|=1$ | Dirac
Cirigliano et al. | 2022 [40] |  27Al,48Ti,197Au,208Pb | $M_{0;\tau}$ | Dirac, $|\kappa|=1$ | Dirac
Hoferichter, Menéndez, and Noël | 2022 [41] |  27Al,Ti | $M_{J;\tau}\;\Sigma^{\prime}_{J;\tau}\;\Sigma^{\prime\prime}_{J;\tau}$ | Plane wave | $\braket{}{\psi_{\mu}}{{}^{2}}_{\rho}$
|  |  | $\Phi^{\prime\prime}_{J;\tau}$ |  |

###  II.2 The electron’s distorted partial waves

Somewhat schematically, in the plane-wave limit of Eq. (4) one obtains a
nuclear transition density of the form

$\sum_{i=1}^{A}e^{-i\vec{q}\cdot\vec{r}_{i}}\hat{\mathcal{O}}_{N}(i)$

where for simplicity we have omitted the s-wave contribution of the muon. Here
$\hat{\mathcal{O}}_{N}(i)$ would be one of the NRET operators that we will
construct from nucleon charges, spins, and velocities and their longitudinal
and transverse projections. As $\vec{r}_{i}$ operates on nucleon $i$, one must
expand the exponential, regrouping powers of $\vec{r}_{i}$ with
$\hat{\mathcal{O}}_{N}(i)$ to form nuclear operators. Because
$|\vec{q}\cdot\vec{r}_{i}|\approx o(1)$ for 27Al and other targets of
interest, a full expansion of the exponential should be made. The elegant way
to do this is by an expansion in spherical Bessel functions, spherical
harmonics, and vector spherical harmonics (depending on whether
$\hat{\mathcal{O}}_{N}(i)$ is a charge or current operator). From this
construction emerge nuclear multipole operators of good angular momentum, $P$,
and $CP$. Such techniques are particularly powerful for elastic processes like
$\mu\rightarrow e$ conversion because the contributing multipolarities $J$
then are restricted by the ground-state angular momentum $j_{N}$, ($J\leq
2j_{N}$), as well as by good $P$ and $CP$ of the ground state.

However, the generalization of this construction for distorted electron
partial waves is nontrivial, especially when a large NRET operator basis is
used. The paper of Kitano, Koike, and Okada [32] listed in Table 1 begins by
discussing an operator basis somewhat analogous to our NRET. The authors also
took the electron partial waves from solutions of the Coulomb Dirac equation,
but, in estimating rates, a series of simplifications were made, including the
retention of just the $|\kappa|$=1 partial waves. In the end, the
pseudoscalar, axial vector, tensor, and space-like vector operators of their
Lagrangian were eliminated from consideration. Only the lowest multipole of
the charge operator $M_{0}$ was retained, generated by the nucleon-level
scalar interaction and the charge component of the vector interaction.

This kind of procedure is not compatible with our “bottom up” NRET approach,
which requires that we construct and evaluate all possible NRET operators,
limited only by basic symmetries: in this approach one makes no apriori
assumptions about the UV source of the CLFV, or, consequently, how it will be
manifested through specific low-energy operators in elastic $\mu\rightarrow e$
conversion. Thus we need to include the full set of NRET operators and, given
our remarks about the nuclear response functions, all contributing electron
Coulomb-distorted partial waves. To accomplish this, we show that the
distorted-wave problem can be greatly simplified by recognizing that almost
all of the distortion can be absorbed through a simple scaling of the plane-
wave solution. This allows us to complete the full multipole expansion for the
entire set of NRET operators, and when combined with muon averaging,
analytically evaluate all of the one-body matrix elements of these operators.
This is an enormous simplification with very little sacrifice of accuracy, as
we will show.

The Dirac solutions for the continuum electron and bound-state muon can be
expanded in partial waves. Assuming spherical symmetry,

$\displaystyle\psi_{\kappa}(\vec{r})=\left(\begin{array}[]{c}i{g_{\ell
j}}\left[\begin{array}[]{r}\langle\textstyle{{1\over 2},{1\over
2}}|(\ell{1\over 2})jm\rangle\\\\[5.69046pt] \langle\textstyle{{1\over
2},\mbox{-}{1\over 2}}|(\ell{1\over 2})jm\rangle\end{array}\right]\\\
{}{}\hfil\\\ -{f_{\ell j}}\left[\begin{array}[]{r}\langle{1\over 2},{1\over
2}|(\ell\pm 1\,{1\over 2})jm\rangle\\\\[5.69046pt] \langle{1\over
2},\mbox{-}{1\over 2}|(\ell\pm 1\,{1\over
2})jm\rangle\end{array}\right]\end{array}\right)$ (12)

Solutions are indexed by the parameter $\kappa=\dots,-3,-2,-1,1,2,3,\dots$
where $j=|\kappa|-\textstyle{1\over 2}$ and

$\displaystyle\kappa=\left\\{\begin{array}[]{rr}-(\ell+1)&~{}~{}\kappa<0\\\
\ell&~{}~{}\kappa>0\end{array}\right.$ (15)

The angular-spin wave functions are

$\displaystyle\langle\theta\phi|(\ell{1\over
2})jm\rangle=\sum_{m_{\ell}m_{s}}\langle\ell m_{\ell}\textstyle{1\over
2}m_{s}|jm\rangle Y_{\ell m_{\ell}}(\theta,\phi)\xi_{m_{s}}~{}~{}~{}~{}$ (16)

where $\xi_{m_{s}}$ is the Pauli spinor.

In the plane-wave limit the radial solutions are spherical Bessel functions

$\displaystyle{g_{\ell j}(r)\over q}\equiv{G_{\ell j}\over qr}$
$\displaystyle=$ $\displaystyle j_{\ell}(qr)$ $\displaystyle{f_{\ell
j}(r)\over q}\equiv{F_{\ell j}\over qr}$ $\displaystyle=$
$\displaystyle\left[{E-m\over E+m}\right]^{1\over
2}\left\\{\begin{array}[]{cc}~{}~{}j_{\ell-1}(qr)&\kappa>0\\\
-j_{\ell+1}(qr)&\kappa<0\end{array}\right.~{}~{}$ (19)

In the presence of a Coulomb potential, the radial solutions satisfy

$\displaystyle(E-\mu-V_{c}(r))G_{\ell j}$ $\displaystyle=$
$\displaystyle-{d\over dr}F_{\ell j}+{\kappa\over r}F_{\ell j}$
$\displaystyle(E+\mu-V_{c}(r))F_{\ell j}$ $\displaystyle=$
$\displaystyle{d\over dr}G_{\ell j}+{\kappa\over r}G_{\ell j}$ (20)

where $\mu$ is the reduced mass of the lepton

$\mu={mM_{T}\over m+M_{T}}$

with $m$ the lepton mass and $M_{T}$ the nuclear mass. $V_{c}(r)$ is the
Coulomb potential, which we compute using an extended nuclear charge
distribution of the form

$\displaystyle\rho(r)={n_{0}\over
1+\mathrm{exp}[{r-c\over\beta}]}~{}~{}\mathrm{with}~{}~{}\int_{0}^{\infty}dr\;r^{2}\rho(r)=Z~{}~{}~{}~{}~{}$
(21)

where the values of $c$ and $\beta$, typically determined from elastic
electron scattering, are taken from Ref. [43] (see Table 11). $Z$ is the
nuclear charge. The normalization $n_{0}$ can be evaluated analytically in
terms of the polylogarithm function $\mathrm{Li}_{n}(z)$,

$n_{0}=-{Z\over 2\beta^{3}\mathrm{Li}_{3}(-\mathrm{exp}[c/\beta])}$

as can the associated potential.

A procedure familiar from electron scattering studies is the replacement of
the Coulomb solution in the vicinity of the nucleus by a plane wave solution
with a shifted momentum, $\vec{q}\rightarrow\vec{q}_{\mathrm{eff}}$:

$\displaystyle\displaystyle{e^{i\vec{q}\cdot\vec{r}}\rightarrow{q_{\mathrm{eff}}\over
q}e^{i\vec{q}_{\mathrm{eff}}\cdot\vec{r}}~{}~{}~{}~{}r\lesssim R}$ (22)

where $\vec{q}_{\mathrm{eff}}$ is the local momentum in the Coulomb well and
$R$ the nuclear radius. This is a local replacement, not affecting the flux at
infinity. For derivations and discussion see [44, 45, 46]. The use of
$\vec{q}_{\mathrm{eff}}$ in the exponential accounts for the more rapid
oscillation of the electron wave function in the attractive Coulomb well
created by the nucleus, while $q_{\mathrm{eff}}/q$ accounts for the
enhancement of the amplitude due to that attraction. Here we determine
$q_{\mathrm{eff}}$ using a constant potential whose depth is equated to the
average of the Coulomb potential over the nuclear charge distribution

$\displaystyle\bar{V}_{c}$ $\displaystyle=$
$\displaystyle{\int_{0}^{\infty}dr\;r^{2}\rho(r)V_{c}(r)\over\int_{0}^{\infty}dr\;r^{2}\rho(r)}$
$\displaystyle\vec{q}^{\,2}_{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle{M_{T}\over
m_{\mu}+M_{T}}\left[(m_{\mu}-E_{\mu}^{\mathrm{bind}}-\bar{V}_{c})^{2}-m_{e}^{2}\right]~{}~{}~{}~{}~{}~{}$
(23)

The values we obtained for a series of possible nuclear targets are given in
Table 11.

Figure 2: The Dirac Coulomb solutions $G(r)$ and $F(r)$ for the highly
relativistic outgoing electron produced in $\mu\rightarrow e$ conversion in
27Al (green line) are compared to the free solution (orange) and to the free
solution evaluated with ${q}_{\mathrm{eff}}$ (blue dashed), for low partial
waves. The nuclear charge distribution is shown by the shading (arbitrary
normalization). The agreement between the Coulomb and free solutions evaluated
with $q_{\mathrm{eff}}$ is quite good.  Figure 3: As in Fig. 2 but for 184W.

Consequently, for an ultra-relativistic electron, we identify

$\displaystyle{G^{\mathrm{Coulomb}}_{\ell j}\over qr}$
$\displaystyle\leftrightarrow$ $\displaystyle{q_{\mathrm{eff}}\over
q}j_{\ell}(q_{\mathrm{eff}}r)$ $\displaystyle{F^{\mathrm{Coulomb}}_{\ell
j}\over qr}$ $\displaystyle\leftrightarrow$
$\displaystyle{q_{\mathrm{eff}}\over
q}\left\\{\begin{array}[]{cc}~{}~{}j_{\ell-1}(q_{\mathrm{eff}}r)&\kappa>0\\\
-j_{\ell+1}(q_{\mathrm{eff}}r)&\kappa<0\end{array}\right.$ (26)

This approximate equivalence was exploited in electron scattering studies
using the targets 40Ca and 208Pb [47]. In early work, the constant potential
used in calculating $q_{\mathrm{eff}}$ was set to the Coulomb potential at the
nuclear origin, which overestimates the attraction. A physically more
reasonable choice, as calculations later demonstrated, is to fix the potential
to its nuclear-density-weighted average, as is done here. In Figs. 2 and 3 the
shifted plane waves of Eq. (26) are compared to the numerically computed Dirac
equation solutions. These solutions and $\bar{V}_{c}$ are both generated from
the nuclear charge distribution of Eq. (21). Results are shown for 27Al and
184W for the partial waves $|\kappa|\leq 2$ to show the quality of the
$q_{\mathrm{eff}}$ approximation in various partial waves and for low- and
high-$Z$ targets. The computed $\bar{V}_{c}$’s for 27Al and 184W are -5.84 and
-18.4 MeV, respectively.

The correspondence between the $q_{\mathrm{eff}}$ approximation and the exact
Dirac solutions is very good. In 27Al, deviations are only discernible on the
far tail of the nuclear density, where contributions to the convolution are
very small. The agreement is also excellent for 184W, where the Coulomb
distortion is very large. This agreement is achieved without tuning:
$q_{\mathrm{eff}}$ is derived from $\bar{V}_{c}$, which is computed from the
same nuclear density used in the Dirac equation. $q_{\mathrm{eff}}$ was not
determined by fitting the Dirac solution.

We employ in this paper the free-electron Dirac spinor [48]

$\displaystyle U(q,s)$ $\displaystyle=$ $\displaystyle\sqrt{{E_{e}+m_{e}\over
2m_{e}}}\left(\begin{array}[]{cc}\xi_{s}\\\\[5.69046pt]
\displaystyle\frac{\vec{\sigma}\cdot\vec{q}}{E_{e}+m_{e}}\xi_{s}\end{array}\right)$
(29)

with $\xi_{s}$ the Pauli spinor. For our highly relativistic electron, the net
effect of the Coulomb interaction with the nucleus is the replacement

$\displaystyle U(q,s)e^{i\vec{q}\cdot\vec{r}}\rightarrow{q_{\mathrm{eff}}\over
q}\sqrt{{E_{e}\over 2m_{e}}}\left(\begin{array}[]{c}\xi_{s}\\\
\vec{\sigma}\cdot\hat{q}\xi_{s}\end{array}\right)e^{i\vec{q}_{\mathrm{eff}}\cdot\vec{r}}~{}~{}$
(32)

where $\hat{q}$ is the unit vector in the direction of the electron.

With $q_{\mathrm{eff}}$ one can accurately account for Coulomb distortions of
the electron while not abandoning the elegant multipole formalism of plane
waves. Table 1 includes four calculations in which full response functions
have been calculated [30, 31, 36, 37], but those calculations employed
undistorted plane waves. In all other cases, only the leading multipole has
been retained. The associated error depends on the detailed nuclear physics of
the ignored multipoles and can be large. For example, among the 11 targets we
will consider in this paper, there are three – Na, Al, Cu – with only odd
isotopes, an unpaired proton, and a ground-state angular momentum of at least
$\textstyle{3\over 2}$, so that more than one multipole contributes. We picked
one familiar NRET interaction, $\vec{\sigma}_{L}\cdot\vec{\sigma}_{N}$, and
set the isospin to be $(1+\tau_{3})/2$, so that the operator couples to the
unpaired valence proton. Evaluating rates first with the leading multipole and
then with all contributing multipoles, we find rate increases of 22.4% (Na),
4.7% (Al), and 65.4% (Cu). In the case of Al, ignoring higher multipoles
produces a modest error, but in Cu the error is nearly $o(1)$. All three
isotopes have strong spin responses. One concludes that truncating the
multipole expansion to the leading operator is not in any sense a controlled
approximation. The associated error fluctuates considerably from nucleus to
nucleus. Consequently, there is no shortcut: to assess the importance of
neglected multipoles, one must calculate them. The use of $q_{\mathrm{eff}}$
makes this practical even if one’s operator basis is quite large, as it is
here.

The $q_{\mathrm{eff}}$ substitution also helps one to see connections between
$\mu\rightarrow e$ conversion and standard-model processes. Experimental
$\mu\rightarrow$ e conversion rates are expressed as a branching ratio with
respect to the muon capture rate. In treatments like that presented here, the
use of $q_{\mathrm{eff}}$ will allow one to express the numerator and
denominator in quite similar forms (polynomials in $y$), potentially making
assessments of nuclear structure uncertainties more transparent. In some cases
it may be possible to do more. For example, isovector responses contributing
to elastic $\mu\rightarrow e$ conversion in 27Al can be related directly to
the iso-elastic process of the mirror $\beta$ decay of 27Si to the ground
state of 27Al. Such relationships could be important if CLFV is discovered,
and a more quantitative assessment of nuclear structure uncertainties becomes
a priority.

Table 2:  The ratio of the lower to upper components of the muon wave function evaluated at the radius $r_{0}$ where the monopole charge transition density peaks. Target |  $\left|\textstyle{f(r_{0})\over g(r_{0})}\right|$ | Target |  $\left|\textstyle{f(r_{0})\over g(r_{0})}\right|$ | Target |  $\left|\textstyle{f(r_{0})\over g(r_{0})}\right|$
---|---|---|---|---|---
12C | 0.014 |  27Al | 0.027 |  48Ti | 0.043
16O | 0.017 |  28Si | 0.030 |  56Fe | 0.049
19F | 0.019 |  32S | 0.036 |  63Cu | 0.061
23Na | 0.024 |  40Ca | 0.040 |  184W | 0.132

###  II.3 The $1s$ muon’s upper component

We have computed bound muon Dirac wave functions for the 11 targets listed in
Table 11. These targets are both interesting experimentally and tractable for
the shell model: high quality nuclear effective interactions are available,
allowing us to complete full-space diagonalizations with exact projection of
the center-of-mass, as we describe later. We also treat 184W, as a high-$Z$
comparison target.

Solutions were obtained by numerically integrating from the origin outward.
The eigenvalue is determined by requiring the wave function to vanish at
infinity. Results for the 27Al and 48Ti $1s$ muon states ($\kappa=-1$) are
shown in Fig. 4. The corresponding Schrödinger solutions are also plotted:
they are almost indistinguishable from the Dirac radial solutions $g(r)$. The
Dirac binding energies are nearly identical to the Schrödinger values,
exceeding the latter by just 0.18% and 0.37%, respectively.

Figure 4: Lower panels: muon $\kappa=-1$ bound state Dirac solutions $G(r)$
(orange line) and $F(r)$ (green) are shown for 27Al (left) and 48Ti (right),
along with the Schrödinger solution (blue dashed). These solutions are
computed for extended nuclear charges, using the parametrizations of Table 11,
and are properly normalized. Upper panels: $g(r)$ (orange), $f(r)$ (green),
and the Schrödinger solutions for extended (blue dashed) and point (red)
charge distributions. Also shown are the volume-weighted charge distributions
$r^{2}\rho(r)j_{0}(qr)$ and $r^{2}\rho(r)j_{2}(qr)$ (shaded). The overall
normalization (but not the relative normalization) of the two densities is
arbitrary. As the muon wave function varies slowly over the nucleus, it is
appropriate to use an average value: the black dotted line is the value
obtained by averaging over $r^{2}\rho(r)j_{0}(qr)$ (see text) .

Figure 4 shows that the muon is nearly nonrelativistic, with $g(r)$ and the
Schrödinger solution effectively indistinguishable. For the first 11 entries
of Table 2, the ratio $|f(r)/g(r)|$ evaluated at the peak of charge transition
density $r^{2}j_{0}(q_{\mathrm{eff}}r)\rho(r)$ ranges from 0.014 (C) to 0.061
(Cu). The 27Al value is 0.027, while that for the high-Z target 184W is 0.132.

The Dirac bound-state normalization

$\displaystyle\int_{0}^{\infty}dr\;\left[G^{2}_{\ell j}(r)+F^{2}_{\ell
j}(r)\right]=1$ (33)

provides an alternative measure of the relativity: the lower-component
contributions to this normalization integral are 0.2% and 0.5%, respectively,
for 27Al and 48Ti. For 184W the result is 2.2%.

One concludes that, for lighter nuclear targets, retention of just the upper
component of the muon wave function – or nearly equivalently, use of the
Schrödinger solution – can be a sensible procedure. There is a long history of
muon-capture studies in light- and medium-mass nuclei that take advantage of
the muon’s low velocity in the initial state [42]. However, the three papers
listed in Table 1 that have made this approximation either did so for a high-Z
nucleus (208Pb) [38] or equated the Schrödinger density to its point-nucleus
value [36, 37], which, as we will later discuss in this section, can
substantially overestimate the muon density even for a relatively light
nucleus like 27Al.

###  II.4 The $1s$ muon’s lower component

Despite the remarks made above, in most of the past work summarized in Table 1
the muon’s lower component was included (either explicitly in the transition
density or as an average value). In those cases where the lower component was
treated explicitly, the electron partial-wave expansion was truncated to
$|\kappa|=1$, and only the charge operator was considered: the authors thus
depend on the coherent enhancement of the monopole charge operator to dilute
contributions from higher multipoles. As we have noted, this choice of
approximations would not in general be successful for operators other than the
charge operator – so clearly would not be appropriate in an NRET – and in fact
would be a poor choice even for the charge operator, were the coupling
isovector and thus not coherent. In cases where an average value was used, the
electron was treated as a plane wave. Thus it is difficult to see a consistent
pattern of approximations in past work, even if we limit our attention to
simple operators like the charge.

All papers in Table 1 that treated the muon’s lower component omitted
internucleon velocities. Not only is this parametrically difficult to justify,
but nucleon velocities generate new operators and a new source of coherent
enhancement, as we will later see. This leads to new nuclear response
contributions to the rate that in principle might be new observables, if they
can be isolated by comparing rates in a variety of targets.

In contrast, $\vec{v}_{\mu}$ plays a relatively minor role in $\mu\rightarrow
e$ conversion. Numerically, $|\vec{v}_{N}|\gtrsim|\vec{v}_{\mu}|$ for the
nuclear targets considered here; for example,
$\sqrt{\langle\vec{v}^{\,2}_{\mu}\rangle}\approx 0.05$ for 27Al while for an
unpaired $1d_{5/2}$ valence proton
$\sqrt{\langle\vec{v}_{N}^{\,2}\rangle}\approx 0.21$. But there are more
important reasons than magnitude for $\vec{v}_{\mu}$’s relative lack of
impact.

The muon’s velocity is the operator that generates the lower component Dirac
solution $f(r)$ from the upper component $g(r)$,

$\displaystyle\psi_{\kappa=-1}^{\mu}(\vec{r})$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}\xi_{s}\\\
{\vec{\sigma}\cdot\vec{p}_{\mu}\over
2\mu^{*}}\xi_{s}\end{array}\right]ig(r)Y_{00}(\Omega)$ (36)
$\displaystyle\approx$ $\displaystyle\left[\begin{array}[]{c}\xi_{s}\\\
{\vec{\sigma}\cdot\vec{v}_{\mu}\over
2}\xi_{s}\end{array}\right]{ig(r)\over\sqrt{4\pi}}.$ (39)

as

$\displaystyle f(r)\approx{1\over 2\mu^{*}}{dg(r)\over dr}\approx{1\over
2\mu}{dg(r)\over dr}.$ (40)

Here $\mu$ is the muon’s reduced mass and $\mu^{*}$ is the effective reduced
mass. If the Coulomb potential in the vicinity of the nucleus is replaced by
an average value $\bar{V}_{c}$, then
$\mu^{*}\equiv\mu-(|E_{\mu}^{\mathrm{bind}}|+\bar{V}_{c})/2$. (In 27Al the
difference between $\mu$ and $\mu^{*}$ is 2.6%.) The three functions in Eq.
(40) are compared for the high-$Z$ target 184W in Fig. 5; differences are
small.

The leptonic amplitude involves a product of this wave function with that for
the electron, given by Eq. (32), whose upper and lower components are both
$o(1)$. Consequently, whether the operator connecting the muon to the electron
is even or odd in the Foldy-Wouthuysen sense, $\vec{v}_{\mu}$ always appears
in combination with the much larger electron velocity. Furthermore, the
leptonic amplitude is inclusive, a sum over partial waves. As we will show by
explicit calculation, these properties limit the role of $\vec{v}_{\mu}$ to
that of a nuclear form factor Coulomb correction, typically at the level of
$\approx$ 5% for the light and medium mass targets considered here. It plays
no other role in the CLFV physics of $\mu\rightarrow e$ conversion.

Figure 5: The muon’s lower-component wave function $f(r)$ computed for 184W
from the Dirac equation (solid line), and approximated as a derivative of
$g(r)$ using either $\mu$ (dotted line) or $\mu^{*}$ (dashed line). Table 3:
Exact calculations of transition probabilities involving the muon’s upper
component $g(r)$ have been calculated for the charge $M$, transverse spin
$\Sigma^{\prime}$, and longitudinal spin $\Sigma^{\prime\prime}$ operators,
evaluated using both leading multipoles (e.g., $W_{M_{0}M_{0}}$) or full
response functions (e.g., $W_{MM}$), and employing isoscalar (00), isovector
(11), and proton (pp) couplings. From each calculation we extract a value of
$R$ (see text; the top line for each target), the ratio of the exact result to
that obtained when $g(r)$ is replaced by the point Schrödinger analog. If we
take $R$ from $W^{00}_{M_{0}M_{0}}$ as a baseline, percentage variations in
$R$, representing the error introduced if the baseline is used globally, are
given in the second line.

$R$ and % variation Target $W_{M_{0}M_{0}}^{00}$ $W_{MM}^{00}$
$W_{M_{0}M_{0}}^{pp}$ $W_{MM}^{pp}$ $W_{M_{0}M_{0}}^{11}$ $W_{MM}^{11}$ 27Al
0.6566 0.6565 0.6570 0.6569 0.6434 0.6417 0.00% $-0.01\%$ 0.07% 0.06%
$-2.01\%$ $-2.26\%$ 63Cu 0.3204 0.3204 0.3200 0.3199 0.3287 0.3281 0.00%
$-0.01\%$ $-0.15\%$ $-0.16\%$ $2.58\%$ $2.38\%$ Target
$W_{\Sigma^{\prime}_{1}\Sigma^{\prime}_{1}}^{00}$
$W_{\Sigma^{\prime}\Sigma^{\prime}}^{00}$
$W_{\Sigma^{\prime}_{1}\Sigma^{\prime}_{1}}^{pp}$
$W_{\Sigma^{\prime}\Sigma^{\prime}}^{pp}$
$W_{\Sigma^{\prime}_{1}\Sigma^{\prime}_{1}}^{11}$
$W_{\Sigma^{\prime}\Sigma^{\prime}}^{11}$ 27Al 0.6475 0.6463 0.6513 0.6495
0.6561 0.6533 $-1.38\%$ $-1.56\%$ $-0.81\%$ $-1.08\%$ $-0.07\%$ $-0.49\%$ 63Cu
0.3689 0.3141 0.3673 0.3085 0.3655 0.3026 $15.11\%$ $-1.99\%$ $14.62\%$
$-3.72\%$ $14.05\%$ $-5.58\%$ Target
$W_{\Sigma^{\prime\prime}_{1}\Sigma^{\prime\prime}_{1}}^{00}$
$W_{\Sigma^{\prime\prime}\Sigma^{\prime\prime}}^{00}$
$W_{\Sigma^{\prime\prime}_{1}\Sigma^{\prime\prime}_{1}}^{pp}$
$W_{\Sigma^{\prime\prime}\Sigma^{\prime\prime}}^{pp}$
$W_{\Sigma^{\prime\prime}_{1}\Sigma^{\prime\prime}_{1}}^{11}$
$W_{\Sigma^{\prime\prime}\Sigma^{\prime\prime}}^{11}$ 27Al 0.6345 0.6339
0.6338 0.6332 0.6331 0.6323 $-3.36\%$ $-3.45\%$ $-3.46\%$ $-3.56\%$ $-3.58\%$
$-3.69\%$ 63Cu 0.3275 0.2951 0.3228 0.2921 0.3182 0.2891 $2.20\%$ $-7.92\%$
$0.74\%$ $-8.85\%$ $-0.71\%$ $-9.77\%$

Table 4:  Errors incurred when $4\pi W_{O^{g}O^{(i)f}}^{\tau\tau^{\prime}}$ is replaced by $\braket{g}\braket{f}W_{OO^{(i)}}^{\tau\tau^{\prime}}$, where $\braket{g}$ is obtained by averaging over the isoscalar monopole operator $M_{00;0}(q_{\mathrm{eff}}r)$ and $\braket{f}$ is obtained by averaging over the corresponding lower-component operator $M^{(2)}_{00;0}(q_{\mathrm{eff}}r)$. |  % variation in $\langle g\rangle\langle f\rangle$ |
---|---|---
Target | $W_{MM^{(2)}}^{00}$ | $W_{MM^{(2)}}^{11}$ | $W_{MM^{(2)}}^{10}$ | $W_{MM^{(2)}}^{01}$
27Al | 0.00% | $-5.19\%$ | $0.98\%$ | $-5.91\%$
63Cu | 0.00% | $-4.05\%$ | $-1.27\%$ | $-2.83\%$
Target | $W_{\Sigma^{\prime}\Sigma^{\prime(0)}}^{00}$ | $W_{\Sigma^{\prime}\Sigma^{\prime(0)}}^{11}$ | $W_{\Sigma^{\prime}\Sigma^{\prime(0)}}^{10}$ | $W_{\Sigma^{\prime}\Sigma^{\prime(0)}}^{01}$
27Al | $-5.58\%$ | $-6.78\%$ | $-6.20\%$ | $-6.17\%$
63Cu | $-3.26\%$ | $-4.16\%$ | $-2.69\%$ | $-4.76\%$
Target | $W_{\Sigma^{\prime}\Sigma^{\prime(2)}}^{00}$ | $W_{\Sigma^{\prime}\Sigma^{\prime(2)}}^{11}$ | $W_{\Sigma^{\prime}\Sigma^{\prime(2)}}^{10}$ | $W_{\Sigma^{\prime}\Sigma^{\prime(2)}}^{01}$
27Al | $-5.61\%$ | $-6.47\%$ | $-6.38\%$ | $-5.68\%$
63Cu | $-10.29\%$ | $-13.03\%$ | $-11.04\%$ | $-12.27\%$
Target | $W_{\Sigma^{\prime\prime}\Sigma^{\prime\prime(0)}}^{00}$ | $W_{\Sigma^{\prime\prime}\Sigma^{\prime\prime(0)}}^{11}$ | $W_{\Sigma^{\prime\prime}\Sigma^{\prime\prime(0)}}^{10}$ | $W_{\Sigma^{\prime\prime}\Sigma^{\prime\prime(0)}}^{01}$
27Al | $-2.70\%$ | $-22.12\%$ | $-2.59\%$ | $-22.11\%$
63Cu | $3.57\%$ | $3.27\%$ | $4.58\%$ | $2.37\%$
Target | $W_{\Sigma^{\prime\prime}\Sigma^{\prime\prime(2)}}^{00}$ | $W_{\Sigma^{\prime\prime}\Sigma^{\prime\prime(2)}}^{11}$ | $W_{\Sigma^{\prime\prime}\Sigma^{\prime\prime(2)}}^{10}$ | $W_{\Sigma^{\prime\prime}\Sigma^{\prime\prime(2)}}^{01}$
27Al | $-5.33\%$ | $-3.79\%$ | $-5.29\%$ | $-3.82\%$
63Cu | $-5.78\%$ | $-7.27\%$ | $-4.15\%$ | $-8.86\%$

Averaging the muon wave function: In most of the papers listed in Table 1, the
gentle variation of the muon’s wave function was exploited to remove it from
the transition density convolution integral, in favor of an average value.
Rates are reasonably insensitive to how this averaging is done, though we
point out below that past averaging has employed procedures more appropriate
for the inclusive process of muon capture. In contrast, the averaging we
employ here is designed for elastic $\mu\rightarrow e$ conversion and
reproduces the exact results for the leading upper- and lower-component
multipoles.

In this paper, as previously mentioned, we will use averaging to simplify our
full rate expression. For example, the effects of $\vec{v}_{\mu}$ are thereby
reflected in one parameter $\langle f\rangle/\langle g\rangle$, which can be
factored from all response functions. There is a second advantage gained from
averaging: it significantly simplifies the nuclear physics. After averaging,
we can exploit properties of the harmonic oscillator to analytically evaluate
all charge and current one-body matrix elements and, consequently, derive
closed-form expressions for each nuclear response function. This can be very
helpful in understanding the nuclear physics and how it effects CLFV
sensitivities in targets of interest. In addition, averaging produces
$\mu\rightarrow e$ conversion multipole operators identical to those employed
in standard-model semileptonic weak interactions [49, 42]. The ability to use
electron scattering or beta decay measurements to determine CLFV transition
densities could prove helpful in reducing nuclear structure uncertainties.

The point-nucleus $1s$ muon Schrödinger solution provides a convenient
normalization for the average density

$\displaystyle\phi_{1s}^{Z}(\vec{r}=0)\equiv\phi_{1s}^{Z}(\vec{0})={1\over\sqrt{\pi}}\left[{Z\alpha\mu
c\over\hbar}\right]^{3/2},$ (41)

where $\mu$ is the reduced mass of the muon. As the point density exceeds that
obtained by averaging over the nuclear volume, a common practice is to express
the volume-averaged result in the same functional form, but with $Z$ replaced
by an effective charge $Z_{\mathrm{eff}}<Z$,

$\bar{\phi}^{\mu}_{1s}\equiv\phi_{1s}^{Z_{\mathrm{eff}}}(\vec{0}).$

The difference between the physical and effective charges $Z-Z_{\mathrm{eff}}$
will increase as Z increases. Alternatively [42] one can express the
probability density $|\bar{\phi}^{\mu}_{1s}|^{2}$ as

$\displaystyle\left|\bar{\phi}^{\mu}_{1s}\right|^{2}\equiv
R\left|\phi^{Z}_{1s}(\vec{0})\right|^{2},$ (42)

with $R<1$ decreasing as $Z$ increases. Values of $Z_{\mathrm{eff}}$ and $R$
for various nuclear targets are given in Table 11, using the averaging
procedure we describe below. Both of these ways of expressing
$\bar{\phi}^{\mu}_{1s}$ in terms of the point Schrödinger density have been
used extensively in muon capture studies.

The transition density for elastic $\mu\rightarrow e$ conversion is a
convolution of a slowly varying muon wave function
$\phi_{1s}^{\mu}(r)={1\over\sqrt{4\pi}}g(r)$ with a more rapidly varying
multipole density whose shape is frequently nearly Gaussian (see Fig. 4). The
standard procedure for handling such convolution integrals is a Taylor
expansion around the integrand’s peak at $r_{0}$, generating errors that
depends only on the second derivative of the slowly varying function.

When an isoscalar coupling to charge exists, the resulting coherence usually
leads to a rate dominated by the isoscalar monopole operator. Here we slightly
modify the Gaussian method described above, effectively choosing $r_{0}$ so
that $g(r_{0})\equiv\langle g\rangle$ yields the exact integrated monopole
strength

$\displaystyle\left|{\phi}^{Z_{\mathrm{eff}}}_{1s}(\vec{0})\right|={1\over\sqrt{\pi}}\left[{Z_{\mathrm{eff}}\alpha\mu
c\over\hbar}\right]^{3\over
2}\equiv\left|\bar{\phi}^{\mu}_{1s}\right|\equiv{1\over\sqrt{4\pi}}\langle
g\rangle$ $\displaystyle~{}~{}~{}={|\langle
g.s.|~{}\sum_{i}j_{0}(q_{\mathrm{eff}}r_{i})Y_{00}(\Omega_{i}){1\over\sqrt{4\pi}}g(r_{i})~{}|g.s.\rangle|\over|\langle
g.s.|\sum_{i}j_{0}(q_{\mathrm{eff}}r_{i})Y_{00}(\Omega_{i})~{}|g.s.\rangle|}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}={\int
dr~{}r^{2}\rho(r)j_{0}(q_{\mathrm{eff}}r){1\over\sqrt{4\pi}}\displaystyle{g(r)}\over\int
dr~{}r^{2}\rho(r)~{}j_{0}(q_{\mathrm{eff}}r)}~{}~{}~{}~{}~{}~{}$ (43)

Here $\rho(r)$ is the ground-state expectation value of the isoscalar density.
We make this choice not only because the monopole operator is often dominant,
but also because the monopole charge density is the best understood property
of nuclei, tightly constrained by elastic electron scattering. Thus a precise
treatment of the Coulomb physics for this multipole is warranted.

The approximation induced by our averaging comes in using the same $\langle
g\rangle$ in other multipoles. Defining the muon-wave-function weighted and
unweighted multipole operators by

$\displaystyle\hat{O}_{JM}^{g}(q)$ $\displaystyle\equiv$
$\displaystyle\sum_{i=1}^{A}{1\over\sqrt{4\pi}}g(x_{i})\hat{O}_{JM}(qx_{i})$
(44)

(and similarly for $f(r)$), we have for the response function generated by
$\hat{O}_{JM}^{g}(q)$

$\displaystyle W_{O^{g}O^{\prime g}}^{\tau\tau^{\prime}}\equiv{4\pi\over
2j_{N}+1}\sum_{J}\langle j_{N}||O^{g}_{J;\tau}||j_{N}\rangle\langle
j_{N}||O_{J;\tau^{\prime}}^{\prime g}||j_{N}\rangle$
$\displaystyle\rightarrow{\langle g\rangle^{2}\over 2j_{N}+1}\sum_{J}\langle
j_{N}||O_{J;\tau}||j_{N}\rangle\langle
j_{N}||O_{J;\tau^{\prime}}^{\prime}||j_{N}\rangle~{}~{}$
$\displaystyle={4\pi|\phi_{1s}^{Z_{\mathrm{eff}}}(\vec{0})|^{2}\over
2j_{N}+1}\sum_{J}\langle j_{N}||O_{J;\tau}||j_{N}\rangle\langle
j_{N}||O_{J;\tau^{\prime}}^{\prime}||j_{N}\rangle~{}~{}$

To test the validity of this replacement, we have computed the exact $R$
(equivalent to $Z_{\mathrm{eff}}^{3}$) for other multipole choices. For the
charge operator, we vary the isospin, and we repeat the averaging using the
full response function (not just the monopole), thereby including higher
multipoles. We have also repeated these calculations for the transverse
electric spin operator $\Sigma_{J}^{\prime}$ and longitudinal spin operator
$\Sigma_{J}^{\prime\prime}$, which we will introduce later in this paper. The
results for 27Al and 63Cu are presented in Table 3. Also shown in the table is
the percent error one would make in using $R_{0}$, the reduction factor
obtained for the isoscalar monopole charge operator, in all other cases. For
27Al the average value of the variation in $R$ is 0.018 and the worse case is
a deviation of 0.037. As $R$ represents a probability, this means that use of
$R_{0}$ globally would affect limits derived on CLFV LECs at the typical level
of $\lesssim 1$% and in the worse case 2%. For 63Cu, the corresponding typical
uncertainty is $\approx 2.7$% with a worse-case deviation of 7.6%.
Consequently, one should associate uncertainties of this magnitude with the
LEC bilinears we later extract from experimental bounds. In cases where the
isoscalar monopole dominates due to its coherence, the error will be much
smaller, as this multipole is exactly reproduced by our averaging procedure.

As we will show in detail later, the lower component $f(r)$ alters the
monopole transition density through an interference term that similarly can be
used to define an optimal $\langle f\rangle$:

$\displaystyle{1\over\sqrt{4\pi}}{\langle f\rangle}\equiv{\int
dr~{}r^{2}\rho(r)j_{1}(q_{\mathrm{eff}}r){1\over\sqrt{4\pi}}\displaystyle{f(r)}\over\int
dr~{}r^{2}\rho(r)j_{1}(q_{\mathrm{eff}}r)}.$ (46)

As above, we can then test the validity of using this monopole value for
$\langle f\rangle$ in other upper-lower component multipole interference terms
(defined explicitly later in this paper),

$\displaystyle W_{O^{g}O^{(i)f}}^{\tau\tau^{\prime}}\equiv{4\pi\over
2j_{N}+1}\sum_{J}\langle j_{N}||O^{g}_{J;\tau}||j_{N}\rangle\langle
j_{N}||O_{J;\tau^{\prime}}^{(i)f}||j_{N}\rangle$
$\displaystyle\rightarrow{\langle g\rangle\langle f\rangle\over
2j_{N}+1}\sum_{J}\langle j_{N}||O_{J;\tau}||j_{N}\rangle\langle
j_{N}||O_{J;\tau^{\prime}}^{(i)}||j_{N}\rangle$
$\displaystyle={4\pi|\phi_{1s}^{Z_{\mathrm{eff}}}(\vec{0})|^{2}\over
2j_{N}+1}{\langle f\rangle\over\langle g\rangle}\sum_{J}\langle
j_{N}||O_{J;\tau}||j_{N}\rangle\langle
j_{N}||O_{J;\tau^{\prime}}^{(i)}||j_{N}\rangle$ (47)

varying the operator and isospin. The small parameter that represents the
effects of $\vec{v}_{\mu}$ is thus $\langle f\rangle/\langle g\rangle$, for
which results are shown in Table 4 for 27Al and Cu. The average deviation is
6%. As the magnitude of the ratio $|\langle f\rangle/\langle g\rangle|\approx
0.027$ for 27Al this implies a typical rate error of $\approx 0.1-0.2\%$;
perhaps less, as the leading interference term is reproduced exactly.

Previous averaging procedures. Many of the references listed in Table 1
employed averaging. But typically the authors followed the very early work
[23], in which methods more appropriate to muon capture were applied to
elastic $\mu\rightarrow e$ conversion. For example, in [27]

$\displaystyle\langle\phi_{1s}\rangle^{2}\equiv{\int
d^{3}r\;\rho(r)|\phi_{1s}^{\mu}(\vec{r}\,)|^{2}\over\int d^{3}r\;\rho(r)}~{}$
$\displaystyle~{}~{}~{}~{}~{}\Rightarrow\langle\phi_{1s}\rangle^{2}\equiv{\alpha^{3}m_{\mu}^{3}\over\pi}{Z^{4}_{\mathrm{eff}}\over
Z}.~{}~{}~{}~{}~{}~{}$ (48)

A nuclear-density-weighted average of the probability arose in Primakoff’s
early work on the inclusive process of muon capture where

$\displaystyle\sum_{f}F(E_{f})|\langle
i|\Omega\,\phi_{1s}^{\mu}(\vec{r}\,)|f\rangle|^{2}$ $\displaystyle\approx
F(\bar{E}_{f})\sum_{f}|\langle
i|\Omega\,\phi_{1s}^{\mu}(\vec{r}\,)|f\rangle|^{2}$ $\displaystyle\approx
F(\bar{E}_{f})\langle
i|\Omega\Omega^{\dagger}|\phi_{1s}^{\mu}(\vec{r}\,)|^{2}|i\rangle$
$\displaystyle\approx{\int
d^{3}r\;\rho(r)|\phi_{1s}^{\mu}(\vec{r}\,)|^{2}\over\int
d^{3}r\;\rho(r)}~{}F(\bar{E}_{f})\langle i|\Omega\Omega^{\dagger}|i\rangle$
(49)

Here $\Omega$ is a nuclear operator. After replacing the final-state phase
space $F(E_{f})$ by an average value, the sum over final states was completed
by closure. The closure approximation relates a transition probability to the
ground-state expectation value of the muon probability. However, the exclusive
process of $\mu\rightarrow e$ conversion depends on a simpler ground-state
expectation value of an amplitude containing $g(r)$, and the average over this
amplitude can be computed without approximations, producing an exact result.

Similarly, the appearance of $Z^{4}_{\mathrm{eff}}$ in Eq. (48) is another
relic of early muon capture studies, derived in phenomenological fits to
rates. The muon probability contributes a factor of $Z^{3}$. Muon capture
rates are dominated by transitions to the giant resonances, which are governed
by the Thomas-Reiche-Kuhn sum rule, which predicts a scaling of $NZ/A\approx
Z/2$. The $Z^{4}_{\mathrm{eff}}$ that arises in muon capture thus reflects a
rather complicated convolution of Coulomb and nuclear structure
approximations. The $Z_{\mathrm{eff}}$ from muon capture is not simply related
to the $Z_{\mathrm{eff}}$ of elastic $\mu\rightarrow e$ conversion, which
arises solely from the impact of the finite nuclear size in softening the
Coulomb potential at small $r$.

##  III Effective Theory Formulation

###  III.1 General construction of the response functions for $\mu\rightarrow
e$ conversion

One can construct the general form of the elastic $\mu\rightarrow e$
conversion rate by considering the constraints imposed on current and charge
matrix elements by the good parity and $CP$ (or $T$) of the nuclear ground
state. We decompose the currents $\vec{\mathcal{J}}$ and charges $\rho$ into
vector and axial-vector components labeled by their transformation properties
under $T$:

$\displaystyle\vec{\mathcal{J}}(\vec{x})$ $\displaystyle\equiv$
$\displaystyle\vec{J}(\vec{x})+\vec{J}_{5}(\vec{x}),~{}~{}~{}~{}~{}\mathcal{J}_{0}(\vec{x})\equiv\rho(\vec{x})+\rho_{5}(\vec{x}),$
$\displaystyle\vec{J}(\vec{x})$ $\displaystyle=$
$\displaystyle\vec{J}^{\mathrm{T}}(\vec{x})+\vec{J}^{\not{\mathrm{T}}}(\vec{x}),~{}~{}~{}~{}\rho(\vec{x})=\rho^{\mathrm{T}}(\vec{x})+\rho^{\not}{\mathrm{T}}(\vec{x}),~{}~{}~{}$
$\displaystyle\vec{J}_{5}(\vec{x})$ $\displaystyle=$
$\displaystyle\vec{J}_{5}^{\mathrm{T}}(\vec{x})+\vec{J}_{5}^{\not{\mathrm{T}}}(\vec{x}),~{}~{}~{}\rho_{5}(\vec{x})=\rho_{5}^{\mathrm{T}}(\vec{x})+\rho_{5}^{\not}{\mathrm{T}}(\vec{x}),~{}~{}~{}$

where T indicates time-reversal even and $\not{\mathrm{T}}$ time-reversal odd.
No further assumptions are made about the currents and charges: they can be
one-body or multi-body, etc.

A current $\vec{\mathcal{J}}$ can be decomposed into its longitudinal,
transverse electric, and transverse magnetic components. Each multipole
contributing to an elastic matrix element can be evaluated according to its
transformation properties under $P$ and $T$, with those quantities odd in
either quantity eliminated due to the good $P$ and $T$ of the nuclear ground
state [50]. We can treat the charge multipoles similarly. As the
transformation properties depend on whether the multipolarity $J$ is even or
odd, we treat these separately. A straightforward exercise, summarized in
Tables 5 and 6, identifies the allowed elastic multipoles.

Table 5:  The constraints imposed by parity and time reversal on candidate multipole operators mediating elastic $\mu\rightarrow e$ conversion, assuming nuclear ground states of good parity and $CP$. Slashes and backslashes indicate symmetry-forbidden multipoles.  |  Symmetries: Parity $\cancel{\mathrm{O}}$ Time $\bcancel{\mathrm{O}}$
---|---
Current | J even | J odd | J even | J odd | J even | J odd
$\vec{J}^{\mathrm{T}}$ | $\mathcal{L}_{J}$ | $\bcancel{\cancel{\mathcal{L}_{J}}}$ | $\mathcal{T}^{el}_{J}$ | $\bcancel{\cancel{\mathcal{T}^{el}_{J}}}$ | $\cancel{\mathcal{T}^{mag}_{J}}$ | $\bcancel{\mathcal{T}^{mag}_{J}}$
$\vec{J}^{\not}{\mathrm{T}}$ | $\bcancel{\mathcal{L}_{J}}$ | $\cancel{\mathcal{L}_{J}}$ | $\bcancel{\mathcal{T}^{el}_{J}}$ | $\cancel{\mathcal{T}^{el}_{J}}$ | $\bcancel{\cancel{\mathcal{T}^{mag}_{J}}}$ | $\mathcal{T}^{mag}_{J}$
$\vec{J}_{5}^{\mathrm{T}}$ | $\cancel{\mathcal{L}^{5}_{J}}$ | $\bcancel{\mathcal{L}^{5}_{J}}$ | $\cancel{\mathcal{T}^{5\,el}_{J}}$ | $\bcancel{\mathcal{T}^{5\,el}_{J}}$ | $\mathcal{T}^{5\,mag}_{J}$ | $\bcancel{\cancel{\mathcal{T}^{5\,mag}_{J}}}$
$\vec{J}_{5}^{\not}{\mathrm{T}}$ | $\bcancel{\cancel{\mathcal{L}^{5}_{J}}}$ | $\mathcal{L}^{5}_{J}$ | $\bcancel{\cancel{\mathcal{T}^{5\,el}_{J}}}$ | $\mathcal{T}^{5\,el}_{J}$ | $\bcancel{\mathcal{T}^{5\,mag}_{J}}$ | $\cancel{\mathcal{T}^{5\,mag}_{J}}$
Table 6:  The constraints imposed by parity and time reversal on candidate multipole operators mediating elastic $\mu\rightarrow e$ conversion, assuming nuclear ground states of good parity and CP. Slashes and backslashes indicate symmetry-forbidden multipoles. |  Symmetries: Parity $\cancel{\mathrm{O}}$ Time $\bcancel{\mathrm{O}}$
---|---
Current |  J even | J odd
$\rho^{\mathrm{T}}$ | $\mathcal{M}_{J}$ | $\bcancel{\cancel{\mathcal{M}_{J}}}$
$\rho^{\not}{\mathrm{T}}$ | $\bcancel{\mathcal{M}_{J}}$ | $\cancel{\mathcal{M}_{J}}$
$\rho_{5}^{\mathrm{T}}$ | $\cancel{\mathcal{M}^{5}_{J}}$ | $\bcancel{\mathcal{M}^{5}_{J}}$
$\rho_{5}^{\not}{\mathrm{T}}$ | $\bcancel{\cancel{\mathcal{M}^{5}_{J}}}$ | $\mathcal{M}^{5}_{J}$

We now add in constraints imposed by the quantum mechanical degrees of freedom
available to us. Nucleon charges generate a scalar density
$\rho_{sc}(\vec{x})$. Velocities generate a convection current density
$\vec{J}_{c}(\vec{x})$, and spins a magnetic density $\vec{\mu}(\vec{x})$. One
can form a dot product of spin and velocity to create a pseudoscalar density
$\rho_{ps}(\vec{x})$ and a cross product to form a vector density
$\vec{\omega}(\vec{x})$. Furthermore one can form gradients of scalars and
curls of currents. Distributing these densities according to their
transformation properties yields physical representations of the currents and
charges,

$\displaystyle\vec{J}^{\mathrm{T}}(\vec{x})$ $\displaystyle=$
$\displaystyle\vec{\omega}(\vec{x})+\vec{\nabla}\rho_{sc}(\vec{x})$
$\displaystyle\vec{J}^{\not{\mathrm{T}}}(\vec{x})$ $\displaystyle=$
$\displaystyle\vec{J}_{c}(\vec{x})+\vec{\nabla}\times\vec{\mu}(\vec{x})$
$\displaystyle\vec{J}_{5}^{\mathrm{T}}(\vec{x})$ $\displaystyle=$
$\displaystyle\vec{\nabla}\times\vec{\omega}(\vec{x})+\vec{\nabla}\rho_{ps}(\vec{x})$
$\displaystyle\vec{J}_{5}^{\not{\mathrm{T}}}(\vec{x})$ $\displaystyle=$
$\displaystyle\vec{\mu}(\vec{x})+\vec{\nabla}\times\vec{J}_{c}(\vec{x})$
$\displaystyle\rho^{\mathrm{T}}(\vec{x})$ $\displaystyle=$
$\displaystyle\rho_{sc}(\vec{x})+\vec{\nabla}\cdot\vec{\omega}(\vec{x})$
$\displaystyle\rho^{\not{\mathrm{T}}}(\vec{x})$ $\displaystyle=$
$\displaystyle\vec{\nabla}\cdot\vec{J}_{c}(\vec{x})$
$\displaystyle\rho_{5}^{\mathrm{T}}(\vec{x})$ $\displaystyle=$
$\displaystyle\rho_{ps}(\vec{x})$
$\displaystyle\rho_{5}^{\not{\mathrm{T}}}(\vec{x})$ $\displaystyle=$
$\displaystyle\vec{\nabla}\cdot\vec{\mu}(\vec{x})$ (51)

First consider the charges. Table 6 allows us to eliminate
$\rho^{\not{\mathrm{T}}}(\vec{x})$ and $\rho_{5}^{\mathrm{T}}(\vec{x})$. The
divergences can be partially integrated to yield longitudinal projections of
$\vec{\omega}(\vec{x})$ and $\vec{\mu}(\vec{x})$, terms that will already be
generated by the currents above. Consequently, the only contribution from the
charges comes from the even multipoles of

$\rho^{\mathrm{T}}(\vec{x})=\rho_{sc}(\vec{x}).$

This eliminates the candidate response function associated with
$\mathcal{M}_{j}^{5}$: elastic $\mu\rightarrow e$ conversion cannot probe
axial-charge interactions.

For the currents, we note

  1. 1.

Table 5 establishes that $\vec{J}^{\not}{\mathrm{T}}$ and
$\vec{J}_{5}^{\mathrm{T}}$ only contribute through the transverse magnetic
projection. Partially integrating the curls generates transverse electric
projectors, so we see both curls are redundant, as these terms are already
included in the transverse electric contributions of
$\vec{J}^{\not}{\mathrm{T}}_{5}$ and $\vec{J}^{\mathrm{T}}$, respectively.

  2. 2.

The gradient contribution to $\vec{J}^{\mathrm{T}}_{5}$ is longitudinal, and
thus cannot contribute to a transverse magnetic multipole.

  3. 3.

The gradient contribution to $\vec{J}^{\mathrm{T}}$ contributes to the
symmetry-allowed longitudinal multipole. But if partially integrated, it
generates a charge multipole of $\rho_{sc}$, which we have already included.

  4. 4.

The curl term in $\vec{J}^{\not}{\mathrm{T}}_{5}$ can only contribute to the
allowed transverse electric multipole. But on partially integrating, the
projector becomes magnetic, and thus redundant with the convection current
contribution to $\vec{J}^{\not}{\mathrm{T}}$.

It follows that all curls and gradients in the currents are unnecessary, so
that

$\displaystyle\vec{J}^{\not{\mathrm{T}}}(\vec{x})$ $\displaystyle=$
$\displaystyle\vec{J}_{c}(\vec{x})$
$\displaystyle\vec{J}^{\mathrm{T}}(\vec{x})$ $\displaystyle=$
$\displaystyle\vec{\omega}(\vec{x})$
$\displaystyle\vec{J}_{5}^{\not{\mathrm{T}}}(\vec{x})$ $\displaystyle=$
$\displaystyle\vec{\mu}(\vec{x})$
$\displaystyle\vec{J}_{5}^{\mathrm{T}}(\vec{x})$ $\displaystyle=$
$\displaystyle 0$ (52)

This eliminates the otherwise symmetry-allowed axial transverse magnetic
multipole $\mathcal{T}_{J}^{5\,mag}$.

We conclude that the elastic $\mu\rightarrow e$ conversion rate contains six
independent response functions,

$\displaystyle\mathcal{M}_{J}$ $\displaystyle:$
$\displaystyle~{}~{}\mathrm{even~{}J}\geq
0~{}\mathrm{multipoles~{}of~{}}\rho_{sc}(\vec{x})\rightarrow M_{J},$
$\displaystyle\mathcal{T}_{J}^{mag}$ $\displaystyle:$
$\displaystyle~{}~{}\mathrm{odd~{}~{}J}\geq
1~{}\mathrm{multipoles~{}of~{}}\vec{J}_{c}(\vec{x})\rightarrow\Delta_{J},$
$\displaystyle\mathcal{L}_{J}$ $\displaystyle:$
$\displaystyle~{}~{}\mathrm{even~{}J}\geq
0~{}\mathrm{multipoles~{}of~{}}\vec{\omega}(\vec{x})\rightarrow\Phi^{\prime\prime}_{J},$
$\displaystyle\mathcal{T}_{J}^{el}$ $\displaystyle:$
$\displaystyle~{}~{}\mathrm{even~{}J}\geq
2~{}\mathrm{multipoles~{}of~{}}\vec{\omega}(\vec{x})\rightarrow\tilde{\Phi}^{\prime}_{J},$
$\displaystyle\mathcal{L}_{J}^{5}$ $\displaystyle:$
$\displaystyle~{}~{}\mathrm{odd~{}~{}J}\geq
1~{}\mathrm{multipoles~{}of~{}}\vec{\mu}(\vec{x})\rightarrow\Sigma^{\prime\prime}_{J},$
$\displaystyle\mathcal{T}_{J}^{5\,el}$ $\displaystyle:$
$\displaystyle~{}~{}\mathrm{odd~{}~{}J}\geq
1~{}\mathrm{multipoles~{}of~{}}\vec{\mu}(\vec{x})\rightarrow\Sigma^{\prime}_{J},~{}~{}~{}~{}$
(53)

where on the right we indicate the effective one-body operators that will
generate these response functions in the NRET formalism we introduce below. An
NRET with a full basis of operators should generate these six response
functions.

###  III.2 Motivations for NRETs

The procedure in EFT is to construct a complete set of operators, constrained
by the applicable symmetries, up to a given order in an expansion in one or
more small parameters, with the expectation that these operators and their
coefficients (the low-energy constants or LECs) will encompass all of the
physics relevant to the energy or momentum scale of the construction.

Measurements of elastic $\mu\rightarrow e$ conversion are performed in nuclei,
and, as detailed above, the constraints of $P$ and $CP$ allow six response
functions. The NRET’s charge and current operators should generate these
responses, providing a microscopic basis for their evaluation. Absent a
microscopic formulation, it becomes difficult to relate CLFV constraints
obtained in different nuclei.

Another purpose of the NRET is to serve as an interface with UV formulations
of CLFV. After the nuclear physics is done, $\mu\rightarrow e$ conversion
constraints on CLFV will be encoded as limits on specific bilinear
combinations of the NRET coefficients, the LECs. These limits can then be used
as constraints on higher-level EFTs or UV theories. This is generally a more
efficient way to proceed than the alternative of repeatedly selecting one UV
theory and doing a “top down” reduction to the nuclear scale, which requires
one to repeat a series of steps for each candidate UV theory being explored.

As we noted in the Introduction, this strategy has been developed for DM
direct detection over the past decade and has proved to be very popular. The
NRET formulation of the nuclear physics has been incorporated into easily used
scripts, which others have then used as the starting point for other scripts
that treat the matching to higher-level EFTs, such as those formulated at the
light quark and gluon level. In Appendix C we describe a publicly available
script for $\mu\rightarrow e$ conversion that can be utilized in a similar
fashion.

###  III.3 Constructing the NRET

In $\mu\rightarrow e$ conversion the relativistic motion of the electron can
be easily removed, effectively replacing the electron’s velocity by the
observable $\hat{q}$, the direction of three-momentum transfer. This was done
in writing Eq. (32). (See also Appendix A.) As the remaining kinematics are
nonrelativistic, the general form of the transition amplitude can be written
somewhat schematically as

$\displaystyle\langle
f|\mathcal{L}_{\textrm{int}}|i\rangle\rightarrow\sqrt{E_{e}\over
2m_{e}}{q_{\mathrm{eff}}\over
q}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\times\left\\{\begin{array}[]{l}\int
d\vec{r}~{}\xi_{m_{f}}^{\dagger}O_{L}\xi_{m_{i}}~{}\phi^{\mu}_{1s}(\vec{r}\,)e^{-i\vec{q}_{\mathrm{eff}}\cdot\vec{r}}\\\\[4.55254pt]
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\langle
f_{N}|\sum_{i}O_{N}(i)\delta(\vec{r}-\vec{r}_{i})|i_{N}\rangle\\\\[4.55254pt]
\int
d\vec{r}~{}\xi_{m_{f}}^{\dagger}\vec{O}_{L}\xi_{m_{i}}~{}\phi^{\mu}_{1s}(\vec{r}\,)~{}e^{-i\vec{q}_{\mathrm{eff}}\cdot\vec{r}}{}\\\\[4.55254pt]
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\cdot\langle
f_{N}|\sum_{i}\vec{O}_{N}(i)\delta(\vec{r}-\vec{r}_{i})|i_{N}\rangle\end{array}\right.~{}~{}~{}~{}$
(58)

where the upper expression is taken if the operators acting between the
leptons and nucleons are scalars and the bottom if they are vectors. Here
$\xi_{m}$ is the lepton Pauli spinor.

The next step is to construct all of the candidate NRET operators that might
contribute to the above expression. Hermitian operators can be constructed
from the lepton and nucleon identity operators $1_{L}$ and $1_{N}$ and from
the four dimensionless three-vectors

$\displaystyle i\hat{q}={i\vec{q}\over|\vec{q}\,|},\ \ \ \ \vec{v},\ \ \ \
\vec{\sigma}_{L},\ \ \ \ \vec{\sigma}_{N}.$ (59)

Here $\hat{q}$ is the unit vector along the three-momentum transfer to the
leptons and $\vec{v}$ represents all bound-state relative velocities.
$\hat{q}$ provides a natural quantization direction, relative to which one can
project out the longitudinal ($\hat{q}\,\cdot$) and transverse
($\hat{q}\times$) components of nuclear currents. These building blocks for
the NRET transform as

| $\dagger$ | T | P
---|---|---|---
$\vec{\sigma}_{L},\vec{\sigma}_{N}$ | +1 | $-1$ | +1
$i\hat{q}$ | +1 | +1 | $-1$
$\vec{v}$ | +1 | $-1$ | $-1$

and thus from these Hermitian operators we can construct various semi-leptonic
CLFV interactions. From the table one can also determine the transformation
properties of the interactions under $P$ and $T$.

The velocity operator. The electron’s velocity is measured with respect to
that of the recoiling final-state nucleus. The initial-state nucleus is
assumed to be at rest. Thus, after removal of the electron, only bound-state
Jacobi velocities remain.

The standard unitary transformation to the inter-particle Jacobi velocities

$\displaystyle\dot{\vec{v}}_{N}(1)$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}\left[\vec{v}_{N}(2)-\vec{v}_{N}(1)\right]$
$\displaystyle\dot{\vec{v}}_{N}(2)$ $\displaystyle=$
$\displaystyle{1\over\sqrt{6}}\left[2\vec{v}_{N}(3)-(\vec{v}_{N}(1)+\vec{v}_{N}(2))\right]~{}\dots~{}~{}~{}$
(60)

will generate $A-1$ internucleon velocities. The last intrinsic Jacobi
velocity is that of the muon, defined relative to the center-of-mass of
nucleons in the same bound state:

$\displaystyle\dot{\vec{v}}_{\mu}$ $\displaystyle=$
$\displaystyle\left[\vec{v}_{\mu}-{1\over
A}\sum_{i=1}^{A}\vec{v}_{N}(i)\right].$ (61)

Henceforth we will drop the superscript dot, with the understanding that all
velocities are Galilean invariant and intrinsic.

All of the velocities defined are small, with
$|\vec{v}_{N}(i)|>|\vec{v}_{\mu}|$. This hierarchy can be utilized in
constructing the NRET. The logical progression is

  1. 1.

Ignore all velocities, retaining only the point-nucleus operators $1_{N}$ and
$\vec{\sigma}_{N}$.

  2. 2.

Retain $\vec{v}_{N}$ to first order but neglect the smaller $\vec{v}_{\mu}$,
which generates the muon lower component.

  3. 3.

Retain both $\vec{v}_{N}$ and $\vec{v}_{\mu}$ to first order.

These NRET choices are intimately connected to the CLFV nuclear response
functions we later generate. The first NRET – which we will label the allowed
approximation – generates three response functions, corresponding to charge,
longitudinal spin, and transverse spin. In the papers of Table 1 prior to
[16], the transverse and longitudinal spin responses were lumped together,
though their form factors and dependence on nucleon-level operators are
distinct. (This is surprising as this issue in DM direct detection has been
treated properly since the early work of Engel [51].) Thus the proper form of
the lowest-level NRET for elastic $\mu\rightarrow e$ conversion has not been
used until recently [16, 41].

The second NRET, which includes $\vec{v}_{N}$, is the minimal theory capable
of generating all six of the nuclear response functions allowed by $P$ and
$CP$, and in our view consequently plays a special role, defining what can and
cannot be learned about CLFV from elastic $\mu\rightarrow e$ conversion. The
inclusion of $\vec{v}_{N}$ makes the treatment of $\mu\rightarrow e$
conversion compatible with standard formulations of semi-leptonic weak
interactions [42, 49]. It also generates some interesting nuclear physics,
including a new form of coherence that is important for certain targets,
including 27Al. This NRET was introduced in [16]; none of the earlier papers
listed in Table 1 considered the three response functions generated by
$\vec{v}_{N}$. (See also [41].)

In the third step we extend the NRET to include $\vec{v}_{\mu}$. This does not
affect selection rules or create truly distinct responses, reflecting the fact
that the leptonic part of the amplitude is not controlled by $P$ or $CP$
selection rules, as it is inelastic and as a sum over all electron distorted
waves is done. The inclusion of $\vec{v}_{\mu}$ does generate corrections to
the nuclear response functions that we are able to cast in a relatively simple
form, thanks to our use of $q_{\mathrm{eff}}$. These corrections are typically
at the few percent level for targets of current experimental interest. In all
three NRET’s we will carry out the full multipole expansion, faithfully
treating the largest of the small parameters, $y$. We limit our treatment of
velocities to linear order as ambiguities arise in treating velocities in
higher order in bound nuclear systems [52].

We begin by developing the operators for the second NRET described above. The
nucleon spin operator can be combined with $\vec{v}_{N}$ as
$\vec{v}_{N}\cdot\vec{\sigma}_{N}$ and $\vec{v}_{N}\times\vec{\sigma}_{N}$,
but not as the rank-2 tensor
$\left[\vec{v}_{N}\otimes\vec{\sigma}_{N}\right]_{2}$, which would not
triangulate between spin-$\textstyle{1\over 2}$ nucleon states. As any
propagator effects in the scattering of the leptons off the nucleon would be
evaluated at fixed $|\vec{q}\,|$, they can be absorbed into the LECs.
Consequently, the operators can be expressed as simple products of leptonic
and nucleonic operators.

The available lepton scalars are $1_{L}$ and $i\hat{q}\cdot\vec{\sigma}_{L}$,
while the vectors are $\vec{\sigma}_{L}$ and its transverse projection
$i\hat{q}\times\vec{\sigma}_{L}$; the nucleon scalars are $1_{N}$,
$\vec{v}_{N}\cdot\vec{\sigma}_{N}$, $i\hat{q}\cdot\vec{\sigma}_{N}$,
$i\hat{q}\cdot\vec{v}_{N}$, and
$i\hat{q}\cdot(\vec{v}_{N}\times\vec{\sigma}_{N})$, while the vectors are
$\vec{\sigma}_{N}$, $\vec{v}_{N}$, $\vec{v}_{N}\times\vec{\sigma}_{N}$ and
their transverse projections. One forms all scalars, noting that products of
transverse projections can be eliminated using
$\displaystyle{(i\hat{q}\times\vec{A})\cdot(i\hat{q}\times\vec{B})=-\vec{A}\cdot\vec{B}-i\hat{q}\cdot\vec{A}~{}i\hat{q}\cdot\vec{B}}$.
We identify a total of 16 operators:

$\displaystyle\mathcal{O}_{1}$ $\displaystyle=$ $\displaystyle 1_{L}~{}1_{N},$
$\displaystyle\mathcal{O}^{\prime}_{2}$ $\displaystyle=$ $\displaystyle
1_{L}~{}i\hat{q}\cdot\vec{v}_{N},$ $\displaystyle\mathcal{O}_{3}$
$\displaystyle=$ $\displaystyle
1_{L}~{}i\hat{q}\cdot\left[\vec{v}_{N}\times\vec{\sigma}_{N}\right],$
$\displaystyle\mathcal{O}_{4}$ $\displaystyle=$
$\displaystyle\vec{\sigma}_{L}\cdot\vec{\sigma}_{N},$
$\displaystyle\mathcal{O}_{5}$ $\displaystyle=$
$\displaystyle\vec{\sigma}_{L}\cdot\left(i\hat{q}\times\vec{v}_{N}\right),$
$\displaystyle\mathcal{O}_{6}$ $\displaystyle=$ $\displaystyle
i\hat{q}\cdot\vec{\sigma}_{L}~{}i\hat{q}\cdot\vec{\sigma}_{N},$
$\displaystyle\mathcal{O}_{7}$ $\displaystyle=$ $\displaystyle
1_{L}~{}\vec{v}_{N}\cdot\vec{\sigma}_{N},$ $\displaystyle\mathcal{O}_{8}$
$\displaystyle=$ $\displaystyle\vec{\sigma}_{L}\cdot\vec{v}_{N},$
$\displaystyle\mathcal{O}_{9}$ $\displaystyle=$
$\displaystyle\vec{\sigma}_{L}\cdot\left(i\hat{q}\times\vec{\sigma}_{N}\right),$
$\displaystyle\mathcal{O}_{10}$ $\displaystyle=$ $\displaystyle
1_{L}~{}i\hat{q}\cdot\vec{\sigma}_{N},$ $\displaystyle\mathcal{O}_{11}$
$\displaystyle=$ $\displaystyle i\hat{q}\cdot\vec{\sigma}_{L}~{}1_{N},$
$\displaystyle\mathcal{O}_{12}$ $\displaystyle=$
$\displaystyle\vec{\sigma}_{L}\cdot\left[\vec{v}_{N}\times\vec{\sigma}_{N}\right],$
$\displaystyle\mathcal{O}^{\prime}_{13}$ $\displaystyle=$
$\displaystyle\vec{\sigma}_{L}\cdot\left(i\hat{q}\times\left[\vec{v}_{N}\times\vec{\sigma}_{N}\right]\right),$
$\displaystyle\mathcal{O}_{14}$ $\displaystyle=$ $\displaystyle
i\hat{q}\cdot\vec{\sigma}_{L}~{}\vec{v}_{N}\cdot\vec{\sigma}_{N},$
$\displaystyle\mathcal{O}_{15}$ $\displaystyle=$ $\displaystyle
i\hat{q}\cdot\vec{\sigma}_{L}~{}i\hat{q}\cdot\left[\vec{v}_{N}\times\vec{\sigma}_{N}\right],$
$\displaystyle\mathcal{O}^{\prime}_{16}$ $\displaystyle=$ $\displaystyle
i\hat{q}\cdot\vec{\sigma}_{L}~{}i\hat{q}\cdot\vec{v}_{N}.$ (62)

Alternatively, one can count the possibilities by noting that two leptonic
scalars, $1_{L}$ and $i\hat{q}\cdot\vec{\sigma}_{L}$, can be combined with two
hadronic scalars, $1_{N}$ and $\vec{v}_{N}\cdot\vec{\sigma}_{N}$; and four
leptonic vectors, $i\hat{q}$, $\vec{\sigma}_{L}$,
$i\hat{q}\times\vec{\sigma}_{L}$, and $\left[(i\hat{q}\otimes
i\hat{q})_{2}\otimes\vec{\sigma}_{L})\right]_{1}$, can be combined with three
hadronic vectors, $\vec{v}_{N}$, $\vec{\sigma}_{N}$, and
$\vec{v}_{N}\times\vec{\sigma}_{N}$. The tensor in the electron’s velocity
that appears in $\left[(i\hat{q}\otimes
i\hat{q})_{2}\otimes\vec{\sigma}_{L})\right]_{1}$ is a component of
$i\hat{q}\cdot\vec{\sigma}_{L}i\hat{q}$.

The construction above is analogous to that performed earlier for DM direct
detection [14, 15], and in fact we have adopted a similar labeling of the
operators. However there are some significant differences that we discuss in
Appendix A, for those familiar with this earlier work.

Each operator can have distinct couplings to protons and neutrons. Thus the
NRET interaction we employ in this paper takes the form

$\displaystyle\sum_{\alpha=n,p}~{}\sum_{i=1}^{16}c_{i}^{\alpha}\mathcal{O}_{i}^{\alpha},$
(63)

where the unknown numerical coefficients $c_{i}$ would need to be determined
by experiment or matched to some predictive UV theory. One can factorize the
space/spin and proton/neutron components of Eq. (63) by introducing isospin,
which is also useful as an approximate symmetry of the nuclear wave functions.
Thus an equivalent form for our interaction is

$\displaystyle\sum_{i=1}^{16}(c_{i}^{0}1+c_{i}^{1}\tau_{3})\mathcal{O}_{i}$
$\displaystyle=$
$\displaystyle\sum_{\tau=0,1}\sum_{i=1}^{16}c_{i}^{\tau}\mathcal{O}_{i}t^{\tau},$
(64)

where $c_{i}^{0}={1\over 2}(c_{i}^{\mathrm{p}}+c_{i}^{n})$ and
$c_{i}^{1}={1\over 2}(c_{i}^{p}-c_{i}^{n})$. The isospin matrices are
$t^{0}=1$ and $t^{1}=\tau_{3}$.

The NRET has a total of 32 parameters, associated with 16 space/spin operators
each of which can have distinct couplings to protons and neutrons. If we
exclude operators that are not associated with spin-0 or spin-1 mediators, 12
space/spin operators and 24 couplings remain. A main goal of this paper is to
determine the specific constraints that a program of elastic $\mu\rightarrow
e$ conversion measurements can place on these LECs.

The coefficients $c_{i}$ in other application of NRET would be functions of
$q^{2}$ [14],

$c_{i}=c_{i}(0)+c_{i}(2)q^{2}+c_{i}(4)q^{4}+\cdots\equiv
F_{i}({q^{2}\over\Lambda^{2}})$

In DM direct detection the presence of tree-level pions in some channels would
lead to $F_{i}\approx{1\over q^{2}+m_{\pi}^{2}}$, making $c_{i}$ a function of
the observed nuclear recoil energy. If couplings were treated as constants in
fits, a bit of information would be missing: the deduced coupling $c_{i}$
would be correct only at some average momentum. But this is not a concern in
$\mu\rightarrow e$ conversion, as $q^{2}\approx m_{\mu}^{2}$ is fixed. One can
represent $F_{i}$ by a constant $c_{i}$, and know exactly how to match to an
EFT with pions. This argument extends to interactions with massless mediators,
$F_{i}\approx 1/q^{2}$.

As defined in Eq. (64), the $c_{i}$’s carry dimensions of
$1/(\mathrm{mass})^{2}$. Conventionally limits on $\mu\rightarrow e$
conversion are expressed as a branching ratio with respect to the standard-
model muon capture rate

$\displaystyle B={\Gamma[\mu^{-}+(A,Z)\rightarrow
e^{-}+(A,Z)]\over\Gamma[\mu^{-}+(A,Z)\rightarrow\nu_{\mu}+(A,Z-1)]}.$ (65)

Consequently, it makes sense to introduce a set of dimensionless LECs
$\tilde{c}_{i}$ that are normalized to the weak scale,

$\displaystyle c_{i}\equiv\tilde{c}_{i}/v^{2}=\sqrt{2}G_{F}\,\tilde{c}_{i},$
(66)

where $v=246.2$ GeV is the Higgs vacuum expectation value and
$G_{F}=1.166\times 10^{-5}$/GeV2 is the Fermi constant. Alternatively, given
an experimental measurement (or limit) on an LEC, one can define an energy
characteristic of the CLFV,

$\Lambda_{i}^{\tau}\equiv{1\over\sqrt{|c_{i}^{\tau}}|}={v\over\sqrt{|\tilde{c}_{i}^{\tau}|}}$

This is the energy scale that one would associate with the BSM source of the
CLFV. Often, experimental sensitivities are described in terms of their reach
in $\Lambda_{i}^{\tau}$.

###  III.4 Matching

The NRET developed here, which follows closely a similar construction for DM
phenomenology, is frequently used in conjunction with various “top-down” EFT
constructions. The NRET is formulated at the nuclear level, with the impulse
approximation employed to relate observables in different nuclear targets. The
operator expansion linear in $v_{N}$ (so including interactions of order
$1/m_{N}$) is consistent with their use with nuclear wave functions derived
from the many-body Schroedinger equation [52]. The coefficients of the NRET
operators are constrained by fitting experiment. In $\mu\rightarrow e$
conversion the available information is limited in principle to the
coefficients of the six nuclear response functions that we were able to
identify by symmetry arguments. It will take great effort and some good
fortune to separate the response functions: if that is not possible, the total
rate becomes the only constraint.

The NRET forms a convenient interface between the particle physics of CLFV and
nuclear calculations of the $\mu\rightarrow e$ rate. The NRET operator basis
fits naturally with many-body nuclear methods: formulating the physics in this
basis makes form factor calculations more accessible to nuclear structure
theorists. The NRET operators must be evaluated at $q\approx m_{\mu}$ and
include some interactions that cannot be probed in standard-model semi-
leptonic interactions. The quality of the form factor calculations directly
impacts our ability to extract meaningful constraints on CLFV from
$\mu\rightarrow e$ conversion.

A recent Snowmass white paper on DM effective theory [53] nicely describes the
particle physics interface, specifically, “top-down” reductions that relate
EFTs formulated at higher energy scales to our NRET. For example, in [54, 55]
an EFT formulated for operators with quarks, gluons, and photons as the
external states, is reduced to the NRET form used here. This reduction uses
techniques familiar from heavy baryon effective theory, yielding in the DM
case products of DM and nucleon charges and currents. The end result – see
Appendix A of [54] – is a dictionary of Wilson coefficients that relate the
coefficients of the quark/gluon-level operators to the LECs $c_{i}$ of the
NRET we develop here. The reduction from a covariant EFT to a Galilean-
invariant NRET leads to correlations between the NRET operators, encoded in
the dictionary. Connections between LO charge operators and $v_{N}$-dependent
currents, or the reverse, necessarily accompany the nonrelativistic reduction.
The extensive literature on such reductions is summarized in [53].

In this way, a theorist interested in a specific formulation of CLFV at some
higher energy scale, can test its consistency with experimental constraints
encoded in the NRET LECs – more precisely, the combinations of these LECs that
are associated with six nuclear responses functions. We will use the NRET to
determine precisely what CLFV information might be available from future
studies of $\mu\rightarrow e$ conversion. There are always more candidate CLFV
operators in the starting EFT than nuclear observables, some of which play no
role in elastic $\mu\rightarrow e$ conversion due to selection rules. Thus
while $\mu\rightarrow e$ studies can establish that CLFV occurs, it will not
fully determine the particle physics origin of the symmetry violation.

The work of [54, 55] was incorporated into the code DirectDM [17] that
connects to the corresponding NRET code [15]. A similar effort is described in
[56]. Much of this work can be readily adapted to connect light quark/gluon
CLFV EFTs to our NRET. Work is underway [57] to build a script like DirectDM
as an interface to the NRET script of Appendix C.

###  III.5 NRET variations

As described earlier, there are two natural variations of this NRET
construction. The first is the neglect of all Jacobi velocities, the $A-1$
relative nucleon velocities and the muon velocity relative to the nuclear
center of mass. As the available nuclear operators are then limited to just
charge and spin, this can be regarded as the point nucleus (or allowed) limit.
In this limit we retain only six of the operators enumerated above:
$\mathcal{O}_{1}$, $\mathcal{O}_{4}$, $\mathcal{O}_{6}$, $\mathcal{O}_{9}$,
$\mathcal{O}_{10}$, and $\mathcal{O}_{11}$. We will later see that the
coefficients of the charge ($M_{J}$), longitudinal spin
($\Sigma_{J}^{\prime\prime}$), and transverse electric spin
($\Sigma_{J}^{\prime}$) response functions are determined by exactly the LECs
of these operators.

The allowed limit is a fairly simple one, and consequently it is notable that
the general form of the $\mu\rightarrow e$ conversion rate in this limit has
not appeared in past work. While four papers listed in Table 1 have considered
spin-mediated interactions, all have assumed an interaction of the form of
$\mathcal{O}_{4}$ (Gamow-Teller). This interaction yields a specific linear
combination of longitudinal and transverse electric responses. In general,
however, these two responses are entirely independent. For example, as we will
see in Sec. III.6, a pseudoscalar-pseudoscalar CLFV interaction generates only
the longitudinal response.

The second variation is to include both $\vec{v}_{N}$ and $\vec{v}_{\mu}$ to
first order. The inclusion of $\vec{v}_{\mu}$ generates new NRET operators as
well as corrections to nuclear response functions that, if we employ
averaging, enter parametrically as $\langle f\rangle/\langle g\rangle$.

The new operators generated by this inclusion are

$\displaystyle{\mathcal{O}}^{f\,\prime}_{2}$ $\displaystyle=$ $\displaystyle
i\hat{q}\cdot{\vec{v}_{\mu}\over 2}~{}1_{N}$
$\displaystyle{\mathcal{O}}^{f}_{3}$ $\displaystyle=$ $\displaystyle
i\hat{q}\cdot\left[{\vec{v}_{\mu}\over
2}\times\vec{\sigma}_{L}\right]~{}1_{N}$ $\displaystyle{\mathcal{O}}^{f}_{5}$
$\displaystyle=$ $\displaystyle\left(i\hat{q}\times{\vec{v}_{\mu}\over
2}\right)\cdot\vec{\sigma}_{N}$ $\displaystyle{\mathcal{O}}^{f}_{7}$
$\displaystyle=$ $\displaystyle{\vec{v}_{\mu}\over
2}\cdot\vec{\sigma}_{L}~{}1_{N}~{}$ $\displaystyle{\mathcal{O}}^{f}_{8}$
$\displaystyle=$ $\displaystyle{\vec{v}_{\mu}\over 2}\cdot\vec{\sigma}_{N}$
$\displaystyle{\mathcal{O}}^{f}_{12}$ $\displaystyle=$
$\displaystyle\left[{\vec{v}_{\mu}\over
2}\times\vec{\sigma}_{L}\right]\cdot\vec{\sigma}_{N}$
$\displaystyle{\mathcal{O}}^{f\,\prime}_{13}$ $\displaystyle=$
$\displaystyle\left(i\hat{q}\times\left[{\vec{v}_{\mu}\over
2}\times\vec{\sigma}_{L}\right]\right)\cdot\vec{\sigma}_{N}$
$\displaystyle{\mathcal{O}}^{f}_{14}$ $\displaystyle=$
$\displaystyle~{}{\vec{v}_{\mu}\over
2}\cdot\vec{\sigma}_{L}~{}i\hat{q}\cdot\vec{\sigma}_{N}$
$\displaystyle{\mathcal{O}}^{f}_{15}$ $\displaystyle=$
$\displaystyle~{}i\hat{q}\cdot\left[{\vec{v}_{\mu}\over
2}\times\vec{\sigma}_{L}\right]~{}i\hat{q}\cdot\vec{\sigma}_{N}$
$\displaystyle{\mathcal{O}}^{f\,\prime}_{16}$ $\displaystyle=$ $\displaystyle
i\hat{q}\cdot{\vec{v}_{\mu}\over 2}~{}i\hat{q}\cdot\vec{\sigma}_{N}$ (67)

where we employ $\vec{v}_{\mu}/2$ above, the operator that generates $f(r)$
when acting on the $1s$ muon wave function $g(r)$. We will denote interactions
associated with $\vec{v}_{\mu}$ (and thus the muon’s lower component) by

$\displaystyle\sum_{i}(b_{i}^{0}1+b_{i}^{1}\tau_{3})\mathcal{O}^{f}_{i}$
$\displaystyle=$
$\displaystyle\sum_{\tau=0,1}\sum_{i}b_{i}^{\tau}\mathcal{O}_{i}^{f}t^{\tau},$
(68)

The sum over $i$ extends only over the operators of Eq. (67), with $b_{i}$
denoting the low-energy constants associated with those operators.

Table 7:  Alternative Dirac forms of our NRET CLFV amplitudes
$\mathcal{L}_{\mathrm{int}}^{j}$ are related to linear combinations of the
Pauli forms (the operators $\mathcal{O}_{i}$). Bjorken and Drell spinor and
gamma matrix conventions are used. Here
$\chi_{e}=\left(\begin{array}[]{c}\xi_{s}\\\
\vec{\sigma}_{L}\cdot\hat{q}~{}\xi_{s}\end{array}\right)$,
$\chi_{\mu}=\left(\begin{array}[]{c}\xi_{s}\\\ 0\end{array}\right)$, and
$N=\left(\begin{array}[]{c}\xi_{s}\\\ {\vec{\sigma}_{N}\cdot\vec{v}_{N}\over
2}\xi_{s}\end{array}\right)$. The Dirac forms are expanded to first order in
$\vec{v}_{N}$ to maintain consistency with their use between Schrödinger wave
functions..
$j$ | ${\mathcal{L}}^{j}_{\mathrm{int}}$ | Pauli operator reduction |  $\displaystyle\sum_{i}c_{i}\mathcal{O}_{i}$
---|---|---|---
1 | $\displaystyle{\bar{\chi}_{e}\chi_{\mu}~{}\bar{N}N}$ | $1_{L}~{}1_{N}$ | $\mathcal{O}_{1}$
2 | $\displaystyle{\bar{\chi}_{e}\chi_{\mu}~{}\bar{N}i\gamma^{5}N}$ | $1_{L}~{}\left(i\displaystyle{{\vec{q}\over 2m_{N}}\cdot\vec{\sigma}_{N}}\right)$ | $\displaystyle{{q\over 2m_{N}}}\mathcal{O}_{10}$
3 | $\displaystyle{\bar{\chi}_{e}i\gamma^{5}\chi_{\mu}~{}\bar{N}N}$ | $\displaystyle{\left(-i\hat{q}\cdot\vec{\sigma}_{L}\right)~{}1_{N}}$ | $-\displaystyle{\mathcal{O}_{11}}$
4 | $\displaystyle{\bar{\chi}_{e}i\gamma^{5}\chi_{\mu}\bar{N}i\gamma^{5}N}$ | $\displaystyle{\left(-i\hat{q}\cdot\vec{\sigma}_{L}\right)~{}\left(i{\vec{q}\over 2m_{N}}\cdot\vec{\sigma}_{N}\right)}$ | $\displaystyle{-{q\over 2m_{N}}\mathcal{O}_{6}}$
5 | $\displaystyle{\bar{\chi}_{e}\gamma^{\mu}\chi_{\mu}\bar{N}\gamma_{\mu}N}$ | $\displaystyle{1_{L}1_{N}}$ | $\displaystyle{\mathcal{O}_{1}}$
|  | $\displaystyle{-\left(\hat{q}1_{L}-i\hat{q}\times\vec{\sigma}_{L}\right)\cdot\left(\vec{v}_{N}+i{\vec{q}\over 2m_{N}}\times\vec{\sigma}_{N}\right)}$ | $\displaystyle{+i\mathcal{O}_{2}^{\prime}-\mathcal{O}_{5}-{q\over 2m_{N}}(\mathcal{O}_{4}+\mathcal{O}_{6})}$
6 | $\displaystyle{\bar{\chi}_{e}\gamma^{\mu}\chi_{\mu}\bar{N}i\sigma_{\mu\alpha}{q^{\alpha}\over m_{N}}N}$ | $\displaystyle{-\left(\hat{q}1_{L}-i\hat{q}\times\vec{\sigma}_{L}\right)\cdot\left(-i{\vec{q}\over m_{N}}\times\vec{\sigma}_{N}\right)}$ | $\displaystyle{{q\over m_{N}}\left(\mathcal{O}_{4}+\mathcal{O}_{6}\right)}$
7 | $\displaystyle{\bar{\chi}_{e}\gamma^{\mu}\chi_{\mu}\bar{N}\gamma_{\mu}\gamma^{5}N}$ | $\displaystyle{1_{L}\left(\vec{v}_{N}\cdot\vec{\sigma}_{N}\right)-\left(\hat{q}1_{L}-i\hat{q}\times\vec{\sigma}_{L}\right)\cdot\vec{\sigma}_{N}}$ | $\mathcal{O}_{7}+i\mathcal{O}_{10}-\mathcal{O}_{9}$
8 | $\displaystyle{\bar{\chi}_{e}\gamma^{\mu}\chi_{\mu}\bar{N}\sigma_{\mu\alpha}\frac{q^{\alpha}}{m_{N}}\gamma^{5}N}$ | $\displaystyle{1_{L}\left(-i{\vec{q}\over m_{N}}\cdot\vec{\sigma}_{N}\right)}$ | $\displaystyle{-{q\over m_{N}}\mathcal{O}_{10}}$
9 | $\displaystyle{\bar{\chi}_{e}i\sigma^{\mu\nu}\displaystyle{q_{\nu}\over m_{L}}\chi_{\mu}\bar{N}\gamma_{\mu}N}$ | $\displaystyle{-{q\over m_{L}}1_{L}~{}1_{N}}$ | $\displaystyle{-{q\over m_{L}}\mathcal{O}_{1}}$
|  | $-\displaystyle{\left(-i{\vec{q}\over m_{L}}\times\vec{\sigma}_{L}\right)\cdot\left(\vec{v}_{N}+i{\vec{q}\over 2m_{N}}\times\vec{\sigma}_{N}\right)}$ | $\displaystyle{-{q\over m_{L}}\left(\mathcal{O}_{5}+{q\over 2m_{N}}\left(\mathcal{O}_{4}+\mathcal{O}_{6}\right)\right)}$
10 | $\displaystyle{\bar{\chi}_{e}i\sigma^{\mu\nu}\displaystyle{q_{\nu}\over m_{L}}\chi_{\mu}\bar{N}i\sigma_{\mu\alpha}\displaystyle{q^{\alpha}\over m_{N}}N}$ | $\displaystyle{-\left(-i{\vec{q}\over m_{L}}\times\vec{\sigma}_{L}\right)\cdot\left(-i{\vec{q}\over m_{N}}\times\vec{\sigma}_{N}\right)}$ | $\displaystyle{{q\over m_{L}}{q\over m_{N}}\left(\mathcal{O}_{4}+\mathcal{O}_{6}\right)}$
11 | $\displaystyle{\bar{\chi}_{e}i\sigma^{\mu\nu}\displaystyle{q_{\nu}\over m_{L}}\chi_{\mu}\bar{N}\gamma_{\mu}\gamma^{5}N}$ | $\displaystyle{\left(-{q\over m_{L}}1_{L}\right)\vec{v}_{N}\cdot\vec{\sigma}_{N}-\left(-i{\vec{q}\over m_{L}}\times\vec{\sigma}_{L}\right)\cdot\vec{\sigma}_{N}}$ | $\displaystyle{-{q\over m_{L}}\left(\mathcal{O}_{7}+\mathcal{O}_{9}\right)}$
12 | $\displaystyle{\bar{\chi}_{e}i\sigma^{\mu\nu}\displaystyle{q_{\nu}\over m_{L}}\chi_{\mu}\bar{N}\sigma_{\mu\alpha}\frac{q^{\alpha}}{m_{N}}\gamma^{5}N}$ | $\displaystyle{\left(-{q\over m_{L}}1_{L}\right)\left(-i{\vec{q}\over m_{N}}\cdot\vec{\sigma}_{N}\right)}$ | $\displaystyle{{q\over m_{L}}{q\over m_{N}}\mathcal{O}_{10}}$
13 | $\displaystyle{\bar{\chi}_{e}\gamma^{\mu}\gamma^{5}\chi_{\mu}\bar{N}\gamma_{\mu}N}$ | $\displaystyle{\left(\hat{q}\cdot\vec{\sigma}_{L}\right)1_{N}-\vec{\sigma}_{L}\cdot\left(\vec{v}_{N}+i{\vec{q}\over 2m_{N}}\times\vec{\sigma}_{N}\right)}$ | $\displaystyle{-i\mathcal{O}_{11}-\mathcal{O}_{8}-{q\over 2m_{N}}\mathcal{O}_{9}}$
14 | $\displaystyle{\bar{\chi}_{e}\gamma^{\mu}\gamma^{5}\chi_{\mu}\bar{N}i\sigma_{\mu\alpha}\displaystyle{q^{\alpha}\over m_{N}}N}$ | $\displaystyle{-\vec{\sigma}_{L}\cdot\left(-i{\vec{q}\over m_{N}}\times\vec{\sigma}_{N}\right)}$ | $\displaystyle{q\over m_{N}}\mathcal{O}_{9}$
15 | $\displaystyle{\bar{\chi}_{e}\gamma^{\mu}\gamma^{5}\chi_{\mu}\bar{N}\gamma_{\mu}\gamma^{5}N}$ | $\displaystyle{\left(\hat{q}\cdot\vec{\sigma}_{L}\right)\left(\vec{v}_{N}\cdot\vec{\sigma}_{N}\right)}-\vec{\sigma}_{L}\cdot\vec{\sigma}_{N}$ | $-i\mathcal{O}_{14}-\mathcal{O}_{4}$
16 | $\displaystyle{\bar{\chi}_{e}\gamma^{\mu}\gamma^{5}\chi_{\mu}\bar{N}\sigma_{\mu\alpha}\frac{q^{\alpha}}{m_{N}}\gamma^{5}N}$ | $\displaystyle{\left(\hat{q}\cdot\vec{\sigma}_{L}\right)\left(-i{\vec{q}\over m_{N}}\cdot\vec{\sigma}_{N}\right)}$ | $\displaystyle{i{q\over m_{N}}\mathcal{O}_{6}}$
17 | $\displaystyle{\bar{\chi}_{e}\sigma^{\mu\nu}\frac{q_{\nu}}{m_{L}}\gamma^{5}\chi_{\mu}\bar{N}\gamma_{\mu}N}$ | $\displaystyle{\left(-i{\vec{q}\over m_{L}}\cdot\vec{\sigma}_{L}\right)1_{N}}$ | $\displaystyle{-{q\over m_{L}}\mathcal{O}_{11}}$
|  | $\displaystyle{-i{q\over m_{L}}\left(\vec{\sigma}_{L}-\hat{q}\hat{q}\cdot\vec{\sigma}_{L}\right)\cdot\left(\vec{v}_{N}+i{\vec{q}\over 2m_{N}}\times\vec{\sigma}_{N}\right)}$ | $\displaystyle{-{q\over m_{L}}\left(i\mathcal{O}_{8}+i{q\over 2m_{N}}\mathcal{O}_{9}+i\mathcal{O}_{16}^{\prime}\right)}$
18 | $\displaystyle{\bar{\chi}_{e}\sigma^{\mu\nu}\frac{q_{\nu}}{m_{L}}\gamma^{5}\chi_{\mu}\bar{N}i\sigma_{\mu\alpha}\displaystyle{{q^{\alpha}\over m_{N}}}N}$ | $\displaystyle{-i{q\over m_{L}}\left(\vec{\sigma}_{L}-\hat{q}\hat{q}\cdot\vec{\sigma}_{L}\right)\cdot\left(-i{\vec{q}\over m_{N}}\times\vec{\sigma}_{N}\right)}$ | $\displaystyle{i{q\over m_{L}}{q\over m_{N}}\mathcal{O}_{9}}$
19 | $\displaystyle{\bar{\chi}_{e}\sigma^{\mu\nu}\frac{q_{\nu}}{m_{L}}\gamma^{5}\chi_{\mu}\bar{N}\gamma_{\mu}\gamma^{5}N}$ | $\displaystyle{\left(-i{\vec{q}\over m_{L}}\cdot\vec{\sigma}_{L}\right)\left(\vec{v}_{N}\cdot\vec{\sigma}_{N}\right)}$ | $\displaystyle{-{q\over m_{L}}\mathcal{O}_{14}}$
|  | $\displaystyle{-i{q\over m_{L}}\left(\vec{\sigma}_{L}-\hat{q}\hat{q}\cdot\vec{\sigma}_{L}\right)\cdot\vec{\sigma}_{N}}$ | $\displaystyle{-{q\over m_{L}}\left(i\mathcal{O}_{4}+i\mathcal{O}_{6}\right)}$
20 | $\displaystyle{\bar{\chi}_{e}\sigma^{\mu\nu}\frac{q_{\nu}}{m_{L}}\gamma^{5}\chi_{\mu}\bar{N}\sigma_{\mu\alpha}\frac{q^{\alpha}}{m_{N}}\gamma^{5}N}$ | $\displaystyle{\left(-i{\vec{q}\over m_{L}}\cdot\vec{\sigma}_{L}\right)\left(-i{\vec{q}\over m_{N}}\cdot\vec{\sigma}_{N}\right)}$ | $\displaystyle{{q\over m_{L}}{q\over m_{N}}\mathcal{O}_{6}}$

###  III.6 Dirac spinor NRET equivalent

In some DM and $\mu\rightarrow e$ conversion work operators are expressed as
matrix elements between Dirac spinors. For example, DM experimentalists
conducting generalized analyses of spin-dependent scattering often write
magnetic, electric dipole, and anapole operators in the Dirac forms familiar
from textbooks [PandaXII]. In Table 7 we write our NRET amplitudes in the
Dirac form, connecting this representation to the Pauli spinor equivalent. The
interactions included in the table are limited to those generated through
scalar or vector mediators. We stress that this Dirac form is merely a
rewriting of our NRET: the expansion remains one linear in $v_{N}$.

The amplitudes are constructed from the available leptonic scalars

$\bar{\chi}_{e}\chi_{\mu},~{}~{}\bar{\chi}_{e}i\gamma^{5}\chi_{\mu}$

and four-vectors

$\bar{\chi}_{e}\gamma^{\mu}\chi_{\mu},~{}~{}\displaystyle{\bar{\chi}_{e}i\sigma^{\mu\nu}{q_{\nu}\over
m_{L}}\chi_{\mu}},~{}~{}\bar{\chi}_{e}\gamma^{\mu}\gamma^{5}\chi_{\mu},~{}~{}\displaystyle{\bar{\chi}_{e}\sigma^{\mu\nu}{q_{\nu}\over
m_{L}}\gamma^{5}\chi_{\mu}}$

which can be combined with their nucleon counterparts to form all possible
scalar interactions. We can assume a four-fermion interaction: any propagator
effects can be absorbed into the LECs because $q$ is fixed, as noted earlier.
Thus there are $2^{2}+4^{2}=20$ combinations.

Including $\vec{v}_{N}$ but neglecting $\vec{v}_{\mu}$ in the spinors,

$\chi_{e}=\left(\begin{array}[]{c}\xi_{s}\\\
\vec{\sigma}_{L}\cdot\hat{q}~{}\xi_{s}\end{array}\right)~{}~{}N=\left(\begin{array}[]{c}\xi_{s}\\\
{\vec{\sigma}_{N}\cdot\vec{v}_{N}\over
2}\xi_{s}\end{array}\right)~{}~{}\chi_{\mu}=\left(\begin{array}[]{c}\xi_{s}\\\
0\end{array}\right),$

a Pauli reduction of the 20 Dirac forms yields various linear combinations of
12 of the 16 NRET operators, shown in the last column of Table 7. Thus the
Dirac representation of our NRET is rather inefficient. The four spin-velocity
operators are not needed. The most general Dirac form of the interaction is

${\mathcal{L}}_{\mathrm{int}}\equiv\sum_{j=1}^{20}~{}d_{j}{\mathcal{L}}^{j}_{\mathrm{int}}$

Isospin labels can be added to this expression. Note that expressions for the
$c_{i}$ in terms of the $d_{i}$ can be read off the table, e.g.,

$c_{1}=d_{1}-{q\over
m_{L}}d_{9}~{}~{}~{}~{}~{}~{}c_{2}=id_{5}~{}~{}~{}~{}~{}~{}\mathrm{etc.}$

so that a rate given in terms of the $c_{i}$ can be easily converted.

Despite the redundancy in the $d_{i}$’s, one might wonder whether the Dirac
operator representation offers an advantage by connecting certain $o(1)$
charge operators with their associated $o(1/m_{N})$ currents, or the reverse.
This is not the case. Of the eight cases in Table 7 where this situation
arises, four involve axial couplings, but nuclear selection rules prevent the
axial charge from playing any role in elastic $\mu\rightarrow e$ conversion.
The remaining cases are vector, where coherence enhances the charge coupling,
which typically relegates the convection current rate contribution to the
level $\lesssim 10^{-4}$. Thus, although the vector charge and convection
current contributions are isolated in distinct response functions, it is
impractical to extract the convection current piece. The situation can be
summarized by stating that the embeddings of these interactions in the nucleus
destroys any operator connections through covariance.

Table 8:  As in Table 7, but listing the additional terms generated when the
linear expansion in velocities includes $\vec{v}_{\mu}$, so that
$\chi_{\mu}=\left(\begin{array}[]{c}\xi_{s}\\\
{\vec{\sigma}_{L}\cdot\vec{v}_{\mu}\over 2}\xi_{s}\end{array}\right)$.
$j$ | ${\mathcal{L}}^{j}_{\mathrm{int}}$ | Pauli operator reduction |  $\displaystyle\sum_{i}b_{i}\mathcal{O}^{f}_{i}$
---|---|---|---
1 | $\displaystyle{\bar{\chi}_{e}\chi_{\mu}~{}\bar{N}N}$ | $-\displaystyle{1\over 2}\hat{q}\cdot\vec{v}_{\mu}~{}1_{N}-\displaystyle{i\over 2}\hat{q}\cdot[\vec{v}_{\mu}\times\vec{\sigma}_{L}]~{}1_{N}$ | $\displaystyle{i}\mathcal{O}_{2}^{f\,\prime}-\mathcal{O}_{3}^{f}$
3 | $\displaystyle{\bar{\chi}_{e}i\gamma^{5}\chi_{\mu}~{}\bar{N}N}$ | $\displaystyle{i\over 2}\vec{v}_{\mu}\cdot\vec{\sigma}_{L}1_{N}$ | $\displaystyle{i}\mathcal{O}_{7}^{f}$
5 | $\displaystyle{\bar{\chi}_{e}\gamma^{\mu}\chi_{\mu}\bar{N}\gamma_{\mu}N}$ | $\displaystyle{1\over 2}\hat{q}\cdot\vec{v}_{\mu}~{}1_{N}+\displaystyle{i\over 2}\hat{q}\cdot[\vec{v}_{\mu}\times\vec{\sigma}_{L}]~{}1_{N}$ | $-\displaystyle{i}\mathcal{O}_{2}^{f\,\prime}+\mathcal{O}_{3}^{f}$
7 | $\displaystyle{\bar{\chi}_{e}\gamma^{\mu}\chi_{\mu}\bar{N}\gamma_{\mu}\gamma^{5}N}$ | $-\displaystyle{1\over 2}\vec{v}_{\mu}\cdot\vec{\sigma}_{N}-\displaystyle{i\over 2}[\vec{v}_{\mu}\times\vec{\sigma}_{L}]\cdot\vec{\sigma}_{N}$ | $-\mathcal{O}_{8}^{f}-\displaystyle{i}\mathcal{O}^{f}_{12}$
9 | $\displaystyle{\bar{\chi}_{e}i\sigma^{\mu\nu}\displaystyle{q_{\nu}\over m_{L}}\chi_{\mu}\bar{N}\gamma_{\mu}N}$ | $\displaystyle{q\over 2m_{L}}\left(\hat{q}\cdot\vec{v}_{\mu}~{}1_{N}+i\hat{q}\cdot[\vec{v}_{\mu}\times\vec{\sigma}_{L}]~{}1_{N}\right)$ | $\displaystyle{q\over m_{L}}\left(-i\mathcal{O}_{2}^{f\,\prime}+\mathcal{O}_{3}^{f}\right)$
11 | $\displaystyle{\bar{\chi}_{e}i\sigma^{\mu\nu}\displaystyle{q_{\nu}\over m_{L}}\chi_{\mu}\bar{N}\gamma_{\mu}\gamma^{5}N}$ | $\displaystyle{q\over 2m_{L}}\left(\vec{v}_{\mu}\cdot\vec{\sigma}_{N}+i[\vec{v}_{\mu}\times\vec{\sigma}_{L}]\cdot\vec{\sigma}_{N}\right.$ | $\displaystyle{{q\over m_{L}}\left(\mathcal{O}_{8}^{f}+i\mathcal{O}^{f}_{12}\right.~{}~{}~{}~{}}$
|  | $~{}\left.-i\hat{q}\cdot[\vec{v}_{\mu}\times\vec{\sigma}_{L}]\hat{q}\cdot\vec{\sigma}_{N}-\hat{q}\cdot\vec{v}_{\mu}\hat{q}\cdot\vec{\sigma}_{N}\right)$ | $~{}~{}~{}~{}~{}~{}\left.+i\mathcal{O}_{15}^{f}+\mathcal{O}_{16}^{f\,\prime}\right)$
13 | $\displaystyle{\bar{\chi}_{e}\gamma^{\mu}\gamma^{5}\chi_{\mu}\bar{N}\gamma_{\mu}N}$ | $\displaystyle{1\over 2}\vec{v}_{\mu}\cdot\vec{\sigma}_{L}~{}1_{N}$ | $\mathcal{O}_{7}^{f}$
15 | $\displaystyle{\bar{\chi}_{e}\gamma^{\mu}\gamma^{5}\chi_{\mu}\bar{N}\gamma_{\mu}\gamma^{5}N}$ | $\displaystyle{i\over 2}[\hat{q}\times\vec{v}_{\mu}]\cdot\vec{\sigma}_{N}-\displaystyle{1\over 2}(\hat{q}\times[\vec{v}_{\mu}\times\vec{\sigma}_{L}])\cdot\vec{\sigma}_{N}$ | $\mathcal{O}^{f}_{5}+\displaystyle{i}\mathcal{O}_{13}^{f\,\prime}$
|  | $-\displaystyle{1\over 2}\vec{v}_{\mu}\cdot\vec{\sigma}_{L}~{}\hat{q}\cdot\vec{\sigma}_{N}$ | $+\displaystyle{i}\mathcal{O}_{14}^{f}$
17 | $\displaystyle{\bar{\chi}_{e}\sigma^{\mu\nu}\frac{q_{\nu}}{m_{L}}\gamma^{5}\chi_{\mu}\bar{N}\gamma_{\mu}N}$ | $\displaystyle{iq\over 2m_{L}}\vec{v}_{\mu}\cdot\vec{\sigma}_{L}~{}1_{N}$ | $\displaystyle{iq\over m_{L}}\mathcal{O}^{f}_{7}$
19 | $\displaystyle{\bar{\chi}_{e}\sigma^{\mu\nu}\frac{q_{\nu}}{m_{L}}\gamma^{5}\chi_{\mu}\bar{N}\gamma_{\mu}\gamma^{5}N}$ | $\displaystyle{q\over 2m_{L}}([\hat{q}\times\vec{v}_{\mu}]\cdot\vec{\sigma}_{N}+(i\hat{q}\times[\vec{v}_{\mu}\times\vec{\sigma}_{L}])\cdot\vec{\sigma}_{N})$ | $\displaystyle{q\over m_{L}}(-i\mathcal{O}_{5}^{f}+\mathcal{O}_{13}^{f\,\prime})$

If the muon’s lower component is included in the reduction,

$\chi_{\mu}=\left(\begin{array}[]{c}\xi_{s}\\\
{\vec{\sigma}_{L}\cdot\vec{v}_{\mu}\over 2}\xi_{s}\end{array}\right),$

and the expansion repeated to first order in velocities, the additional terms
shown in Table 8 are obtained. Note that all ten NRET operators of Eq. (67)
are generated. This occurs because the leptonic spin-velocity operator
$\vec{v}_{\mu}\times\vec{\sigma}_{L}$ arises from the coupling of leptonic
lower components, as the electron is relativistic. In contrast, such coupling
on the nucleon side would be second order in the nucleon velocity.

As already noted, the spin-velocity current
$\vec{v}_{N}\times\vec{\sigma}_{N}$ does not appear because of the restriction
to scalar and vector mediators, eliminating the NRET operators
$\mathcal{O}_{3}$, $\mathcal{O}_{12}$, $\mathcal{O}_{13}^{\prime}$, and
$\mathcal{O}_{15}$. If we include tensor-mediated interactions, e.g.,

$\displaystyle\bar{\chi}_{e}i\sigma^{\mu\nu}\gamma^{5}\chi_{\mu}\bar{N}i\sigma_{\mu\nu}\gamma^{5}N,$
(69)

they do. The equivalent Pauli form is

$\displaystyle{q\over
m_{N}}1_{L}1_{N}+2i\hat{q}\cdot[\vec{v}_{N}\times\vec{\sigma}_{N}]-2\vec{\sigma}_{L}\cdot\vec{\sigma}_{N}$
$\displaystyle~{}~{}~{}-2\vec{\sigma}_{L}\cdot(\hat{q}\times[\vec{v}_{N}\times\vec{\sigma}_{N}])-i(\hat{q}\times\vec{v}_{\mu})\cdot\vec{\sigma}_{N}$
$\displaystyle~{}~{}~{}+\vec{v}_{\mu}\cdot\vec{\sigma}_{L}\hat{q}\cdot\vec{\sigma}_{N}+(\hat{q}\times[\vec{v}_{\mu}\times\vec{\sigma}_{L}])\cdot\vec{\sigma}_{N}$
$\displaystyle={q\over
m_{N}}\mathcal{O}_{1}+2\mathcal{O}_{3}-2\mathcal{O}_{4}+2i\,\mathcal{O}_{13}^{\prime}$
$\displaystyle~{}~{}~{}-2\mathcal{O}_{5}^{f}-2i\mathcal{O}_{13}^{f\prime}-2i\mathcal{O}_{14}^{f}$
(70)

generating $\mathcal{O}_{3}$ and $\mathcal{O}_{13}^{\prime}$. In contrast to
Table 7, we will later find that the nuclear embedding of the spin-velocity
current leads to a novel enhancement of this $v_{N}$-dependent operator.
Furthermore, that enhancement can be turned on and off by picking appropriate
targets. Consequently, if $\mu\rightarrow e$ conversion were dominated by
tensor-mediated interactions, covariance could be one of the sources of
measurable operator correlations.

###  III.7 Nuclear-level LECs as constraints on CLFV

Later in this paper we will describe shell-model treatments of the nuclear
many-body physics of $\mu\rightarrow e$ conversion. We know the underlying UV
theory of the strong interaction (QCD) and have available rather precise
potentials, tuned to experimental phase shifts, describing the interactions
between composite nucleons. Yet despite this knowledge, when we treat the
nuclear problem in the truncated spaces that current technology allows in the
shell model, we rely on phenomenological effective interactions. While some
approaches may begin with the $NN$ interaction and the $g$-matrix it
generates, the use of highly restricted Hilbert spaces excludes a great deal
of high-momentum physics and induces many-body forces that are difficult to
estimate theoretically. Consequently, parameters in interactions are treated
phenomenologically, adjusted to improve agreement between shell-model
predictions and measured nuclear properties, such as energy levels. The end
results are often quite impressive; a good example of the state of the art is
[58]. Yet the relationship between the effective interaction and QCD is not
simple.

The same issues will arise for the NRET interactions we have developed at the
nucleon level, but employ in the complex nuclei where $\mu\rightarrow e$
conversion experiments are done. As described below, experience gained from
the strong interaction suggests that the NRET can still be a very successful
phenomenology for analyzing CLFV, provided we consider the LECs in the
operator expansion to be effective. Because the operator basis is complete, it
has the flexibility to account for the majority of the renormalization and
operator mixing that arises from Hilbert space truncation, provided the
operator coefficients are treated as parameters. This is in part a consequence
of the averaging of many-body charges and currents when they are evaluated in
nuclei with inert cores and a few active valence nucleons near the Fermi
surface.

That is, while one could envision extending the nucleon-level NRET just
constructed to include a complete set of two- or three-nucleon Galilean-
invariant interactions, the effects of these more complicated charge and
current operators can be largely absorbed by adjusting the one-body LECs. In
fact if one were to simplify the shell-model description of 27Al, treating the
ground state as a hole in a 28Si core, two-body operators would exactly reduce
to density-dependent effective one-body operators. In more realistic
descriptions, some true two-body corrections would persist, but qualitatively
the argument would remain valid, reflecting the combinatorics of having more
core than valence nucleons. As nuclear saturation leads to little variation in
nuclear density, the density-dependent renormalization will be only weakly
dependent on $A$: nucleons in 27Al and in Cu locally see a similar nuclear
medium.

This argument is not specific to CLFV, but instead familiar in many nuclear
contexts. One standard-model example is the axial-vector coupling $g_{A}$,
which for a free nucleon has the value $\approx$ 1.27, while the empirical
value deduced in the $2s1d$ shell from $\beta$ decay is $\approx 1.0$ [59].
Limited Hilbert spaces not only lead to operator renormalization of this sort,
but also to mixing among operators of the same symmetry. For example, the
axial-vector spin can mix with the orbital angular momentum $\vec{\ell}$ and
spin-tensor $\left[Y_{2}(\Omega_{r})\otimes\vec{\sigma}\right]_{1}$ operators.
One nice aspect of an NRET is that it includes all operators up to a
designated order, so that the construction should include all of the candidate
mixing partners.

Truncated shell model spaces are not the only source of renormalization and
mixing. Even if a complete basis of Slater determinants is used in a nuclear
calculation, new charge and current operators would arise from explicit pions,
vector mesons, and other degrees of freedom not present in such a basis. For
example, in standard-model electroweak interactions, important pion exchange
currents arise for the vector three-current and the axial charge, as the one-
and two-body interactions arise at the same order in $v/c$. Again, because of
the averaging properties of nuclei, almost all of the effects of these multi-
nucleon currents can be absorbed into one-body operators, if the operator LECs
are regarded as effective. For example, the axial charge operator and its two-
nucleon pion-range correction are both of order $\vec{v}_{N}$.
Phenomenologically, the contributions of the latter can be absorbed into an
effective coupling of the axial charge operator, significantly enhancing this
coupling [60, 61].

Consequently, there is a great deal of empirical evidence that the current
NRET formulation will be a reliable phenomenology for analyzing and
correlating $\mu\rightarrow e$ conversion results obtained over a range of
nuclei. The fitted LECs will encode the available CLFV information. The
difficulty though, will come in extracting from those phenomenological LECs
constraints on one’s favorite UV theory of CLFV.

As described in the introduction, CLFV requires physics beyond the minimal
standard model, and we know such physics exists: the discovery of flavor
violation in the oscillations of light neutrinos allows $\mu\rightarrow e$
conversion to proceed through a $W$-boson neutrino loop, though at a rate that