# Towards an independent determination of muon g-2 from muonium spectroscopy

Cédric Delaunay<EMAIL_ADDRESS>Laboratoire d’Annecy-le-Vieux
de Physique Théorique LAPTh, CNRS – USMB, BP 110 Annecy-le-Vieux, F-74941
Annecy, France  Ben Ohayon<EMAIL_ADDRESS>Institute for Particle Physics and
Astrophysics, ETH Zürich, CH-8093 Zürich, Switzerland  Yotam Soreq
<EMAIL_ADDRESS>Physics Department, Technion – Israel Institute
of Technology, Haifa 3200003, Israel

###### Abstract

We show that muonium spectroscopy in the coming years can reach a precision
high enough to determine the anomalous magnetic moment of the muon below one
part per million (ppm). Such an independent determination of muon $g-2$, which
is not limited by hadronic uncertainties, would certainly shed light on the
$\sim 2\,$ppm difference currently observed between spin-precession
measurements and ($R$-ratio based) Standard Model predictions. The magnetic
dipole interaction between electrons and (anti)muons bound in muonium gives
rise to a hyperfine splitting (HFS) of the ground state which is sensitive to
the muon anomalous magnetic moment. A direct comparison of the muonium
frequency measurements of the HFS at J-PARC and the 1S-2S transition at PSI
with theory predictions will allow to extract muon $g-2$ with high precision.
Improving the accuracy of QED calculations of these transitions by about one
order of magnitude is also required. Moreover, the good agreement between
theory and experiment for the electron $g-2$ indicates that new physics
interactions are unlikely to affect muonium spectroscopy down to the envisaged
precision.

††preprint: LAPTH-021/21

Introduction. The long and winding road that leads to the discovery of physics
beyond the Standard Model (SM) may well go through loops. The magnetic moments
of the electron and the muon

$\displaystyle\vec{\mu}_{\ell}=g_{\ell}\left(\frac{q_{\ell}}{2m_{\ell}}\right)\vec{S}_{\ell}\,,$
(1)

($\ell=e,\mu$) are shining examples. Indeed, quantum fluctuations in the
vacuum of all known and unknown fields inevitably causes their
$g_{\ell}$-factors to deviate from the prediction of the Dirac equation by an
“anomalous” part $a_{\ell}\equiv(g_{\ell}-2)/2$ [1]. Therefore, any accurate
measurement of the latter, together with equally accurate theoretical
predictions, open the door to new physics (NP) phenomena.

The Muon g-2 collaboration at the Fermi National Accelerator Laboratory (FNAL)
recently determined [2, 3, 4] the muon anomalous magnetic moment to 460 parts-
per-billion (ppb) from spin-precession measurements. The central value is in
good agreement with an earlier measurement [5] at the Brookhaven National
Laboratory (BNL), leading to a new experimental world average [2]

$\displaystyle a_{\mu}^{\rm exp}=116\,592\,061(41)\times 10^{-11}\,,$ (2)

to $0.35$ parts-per-million (ppm).

The SM prediction for $a_{\mu}$ includes contributions from electromagnetic
[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25],
strong and weak interactions involving virtual exchange of photons, leptons,
hadrons and $W$, $Z$ and Higgs bosons. Its evaluation was recently reviewed by
the international theory community, finding [26]

$\displaystyle a_{\mu}^{\rm SM}=116\,591\,810(43)\times 10^{-11}\,,$ (3)

where the theory uncertainty is dominated by uncertainties in the
(nonperturbative) hadronic vacuum polarisation (HVP) and hadronic light-by-
light (HLbL) contributions. This result is 4.2 standard deviation smaller than
the experimental average, suggesting a possible contribution from NP with a
magnitude of

$\displaystyle\Delta a_{\mu}\equiv a_{\mu}^{\rm exp}-a_{\mu}^{\rm SM}=(251\pm
59)\times 10^{-11}\,,$ (4)

which is about $2\,$ppm. The SM prediction in Eq. (3) relies on a leading-
order (LO) HVP contribution of $a_{\mu}^{\rm HVP-
LO}(e^{+}e^{-})=6\,931(40)\times 10^{-11}$ [26, 14, 10, 11, 12, 13, 15]
derived from experimental $e^{+}e^{-}\to$ hadrons data (the so-called
$R$-ratio method) at low-energy using dispersion relations [27, 28, 29].

An alternative determination of the LO-HVP contribution stems from an ab-
initio calculation in lattice QCD [30]. The latest result from the Budapest-
Marseille-Wuppertal collaboration (BMWc) $a_{\mu}^{\rm HVP-LO}({\rm
lattice})=7\,075(55)\times 10^{-11}$ [31], consistent with previous
calculations [32, 33, 34, 35, 36] but with a three-fold reduced uncertainty,
is about $1\,$ppm larger than the $R$-ratio value and agrees with the
experimental average in Eq. (2) within two standard deviations.

The present status of muon $g-2$ is therefore puzzling. Imagining that the
discrepancy holds either between experiment and a converged theory (currently
$2\,$ppm), or between the two theoretical calculations (currently $1\,$ppm),
then an independent determination of $a_{\mu}$ to this level, demanding as it
would be, is of immense importance.

An alternative strategy for determining $a_{\mu}$ would be probing magnetic
properties of muons bound in atoms instead of free particles. Muonium
($M\equiv\mu^{+}e^{-}$) is the bound state of an antimuon and an electron. It
is a purely leptonic bound state, free of nuclear structure effects usually
hampering the theoretical accuracy in ordinary atoms, for which high-order QED
calculations are available. In contrast, muonium spectroscopy is currently
limited by experimental uncertainties, being mostly of statistical origin
[37]. In this letter, we show that by pushing muonium spectroscopy to its
limits, both theoretical and experimental, a determination of $a_{\mu}$ is
possible with ${\cal O}$($1\,$ppm) precision. This completely different
approach would shed a new light on the current puzzle.

Muon $g-2$ from muonium spectroscopy. One of the most precisely measured
spectral line in muonium is the hyperfine splitting (HFS) of its 1S ground
state $\nu_{\rm HFS}$ at $12\,$ppb [38]. The HFS originates from the
interaction of the electron spin with the magnetic field sourced by the
magnetic dipole moment of the (static) antimuon. For electronic states without
orbital momentum, the LO Hamiltonian is a Fermi contact interaction between
the electron and muon magnetic moments [39]

$\displaystyle H_{\rm
HFS}=-\frac{2\mu_{0}}{3}\vec{\mu}_{e}\cdot\vec{\mu}_{\mu}\,\delta^{3}(r)\,,$
(5)

where $\mu_{0}=2\alpha h/(e^{2}c)$ is the vacuum permeability. As it involves
explicitly the magnetic moment of the muon, it is in principle sensitive to
its anomalous part.

$\nu_{i}$ | quantity |  | $u_{r}$ |  | Refs.
---|---|---|---|---|---
(unit) |  | current | ongoing | ultimate |
| QED | 8.1 | 5.7 | 0.7 | [40, 41]
| HVP | $\mathcal{O}(10^{-2})$ |  |  | [42]
1S-2S | $R_{\infty}$ | 1.9 | 0.65 |  | [43, 41]
(ppt) | $\alpha$ | $\mathcal{O}(10^{-3})$ |  |  | [43]
| exp | $3.99\times 10^{3}$ | 4.1 | 1.6 | [40, 44]
| QED | 16 | 2.2 | 0.2 | [45, 46]
| HVP | 0.33 | 0.18 |  | [43, 15]
HFS | $\alpha$ | 0.30 | 0.16 |  | [43, 47]
(ppb) | $R_{\infty}$ | $\mathcal{O}(10^{-3})$ |  |  | [43]
| exp | 12 | 2.2 | 0.90 | [38, 48]
Table 1:  Uncertainty budget for the 1S-2S transition (in ppt) and 1S HFS (in ppb) in muonium. The first column summarizes the current status of the relative uncertainties $u_{r}(x)=u(x)/x$. The second one (ongoing) indicates the milestone set by the Mu-MASS and MuSEUM experiments. The last one presents the foreseeable improvements necessary for a muonium determination of $a_{\mu}$ at sub-ppm level. Blank entries correspond to quantities that do not need further improvement.  parameter | quantity |  | $u_{r}$ |
---|---|---|---|---
(unit) |  | current | ongoing | ultimate
|  $\nu_{\rm 1S-2S}$(exp) | 825 | 0.84 | 0.34
$m_{e}/m_{\mu}$ | QED(1S-2S) | 1.7 | 1.2 | 0.1
(ppb) | $R_{\infty}$ | 0.40 | 0.13 |
| total | 825 | 1.5 | 0.37
|  $\nu_{\rm 1S-2S}$(exp) | 708 | 0.73 | 0.29
|  $\nu_{\rm HFS}$(exp) | 10 | 1.9 | 0.77
| QED(1S-2S) | 1.4 | 1.0 | 0.07
$a_{\mu}$ | QED(HFS) | 14 | 1.9 | 0.2
(ppm) | HVP(HFS) | 0.29 | 0.16 |
| $R_{\infty}$ | 0.35 | 0.13 |
| $\alpha$ | 0.26 | 0.14 |
| total | 708 | 3.0 | 0.88
Table 2: Uncertainty budget for the determination of $m_{e}/m_{\mu}$ and
$a_{\mu}$ from precision muonium spectroscopy. Same as Table 1.

Besides the fine-structure constant, $\alpha$, and the Rydberg constant
$R_{\infty}=\alpha^{2}m_{e}c/(2h)$, two parameters are endemic to muonic
physics at low energy: the electron-muon mass ratio, $m_{e}/m_{\mu}$, and the
muon magnetic moment. Hydrogen spectroscopy determines $R_{\infty}$ at $1.9\,$
parts-per-trillion (ppt) [43], making it the most accurately known constant in
physics.111A more recent 1S-3S frequency measurement [49] provides another
determination of $R_{\infty}$ at $3.5\,$ppt consistent with the CODATA value.
Combining $R_{\infty}$ with accurate measurements of the electron and rubidium
atomic masses and the $h/m_{\rm Rb}$ ratio gives the currently best
determination of $\alpha$ at $81\,$ppt [47].222This value disagrees with the
previous best determination of $\alpha$ [50] by $5.4\sigma$ [47], possibly due
to systematic errors [51]. While puzzling, this discrepancy is too small
(relative to uncertainties in muonium) to affect significantly present and
future determinations of $a_{\mu}$. The muonic constants must be extracted
from (at least) two other independent observables. $m_{e}/m_{\mu}$ is known at
$19\,$ppb from comparing the measured ground-state HFS of muonium [38] with
the SM prediction [52]. Since we envisage to use this observable to extract
$a_{\mu}$, $m_{e}/m_{\mu}$ must be obtained by other means. The current second
best determination of the electron-muon mass ratio is at $120\,$ppb, coming
from a measurement of the (total) muon magnetic moment [38]. However, this
cannot be used either as it clearly depends on $a_{\mu}$. Another way to
extract $m_{e}/m_{\mu}$ is to measure a muonium line that is (mostly)
independent of the magnetic moment. To date, the only possibility is the
Lyman-$\alpha$ line between the 1S and 2S states.

The theory prediction for the 1S-2S transition frequency in muonium is

$\displaystyle\nu_{\rm
1S-2S}=\frac{3}{4}\frac{R_{\infty}c}{(1+m_{e}/m_{\mu})}\left[1+\delta_{\rm
1S-2S}\right]\,,$ (6)

where the muon mass enters as a recoil contribution of $\sim 0.5\%$ and
subleading corrections in $\delta_{\rm 1S-2S}\sim{\cal O}(\alpha^{2})$ are
known up to three-loop QED with 20 kHz uncertainty [40].

For the ground-state HFS the theory prediction reads

$\displaystyle\nu_{\rm
HFS}=\frac{16}{3}(1+a_{\mu})\frac{m_{e}}{m_{\mu}}\frac{R_{\infty}c\alpha^{2}}{(1+m_{e}/m_{\mu})^{3}}\left[1+\delta_{\rm
HFS}\right]\,,$ (7)

where $\delta_{\rm HFS}\sim{\cal O}(\alpha/\pi)$ gathers corrections [53]
beyond the Hamiltonian in Eq. (5) from relativistic, radiative (including the
anomalous magnetic moment $a_{e}$ of the electron), recoil, radiative-recoil,
weak and hadronic contributions. The theory uncertainty $\delta_{\rm HFS}$ is
about 70 Hz [45, 52], dominated by unknown three-loop QED contributions to the
radiative-recoil term in $\delta_{\rm HFS}$ [52].

Since the $\delta_{\rm HFS,1S-2S}$ corrections above only weakly depend on
fundamental constants, the leading contributions to $\nu_{\rm HFS}$ and
$\nu_{\rm 1S-2S}$ are sufficient to estimate the sensitivity of muonium
spectroscopy to muon $g-2$. Combining Eqs. (6) and (7) allows to separately
determine $m_{e}/m_{\mu}$ and $a_{\mu}$, providing a simple estimate of their
uncertainties

$\displaystyle u^{2}(m_{e}/m_{\mu})$ $\displaystyle\simeq$ $\displaystyle
u_{r}^{2}(R_{\infty})+u_{r}^{2}(\nu_{\rm 1S-2S})$ (8)
$\displaystyle+\delta_{\rm 1S-2S}^{2}u_{r}^{2}(\delta_{\text{1S-2S}})\,,$
$\displaystyle u^{2}(a_{\mu})$ $\displaystyle\simeq$ $\displaystyle
u_{r}^{2}(m_{e}/m_{\mu})+4u_{r}^{2}(\alpha)+u_{r}^{2}(\nu_{\rm HFS})$ (9)
$\displaystyle+\delta_{\rm HFS}^{2}u_{r}^{2}(\delta_{\text{HFS}})\,,$

where $u_{r}(x)\equiv u(x)/x$ and $u(x)$ is one standard deviation of the
observable $x$, and $u_{r}(\delta_{i})$ denotes the relative theory
uncertainty of $\delta_{i}$ that is not associated with the $R_{\infty}$,
$\alpha$ and $m_{e}/m_{\mu}$ parameters.

A least-square adjustment (see Supplemental Material, which includes Refs.
[54, 55, 56, 57]) using 2018 CODATA recommended values for $R_{\infty}$ and
$\alpha$, and including the state-of-the-art calculation (see Ref. [43] and
references therein) of $\delta_{\rm HFS,1S-2S}$ yields

$\displaystyle a_{\mu}^{M}=116\,637(82)\times 10^{-8}\,,$ (10)

and $m_{e}/m_{\mu}=4\,836\,329(4)\times 10^{-9}$, which is larger
($a_{\mu}^{M}-a_{\mu}^{\rm exp}\simeq 4.5\times 10^{-7}$) but consistent with
both the experimental value in Eq. (2) and the theoretical result in Eq.
(3).The large uncertainty $u(a_{\mu})=8.2\times 10^{-7}$ dominated by the
experimental $u_{r}(\nu_{\rm 1S-2S})=4.0\times 10^{-9}$ [40]. Note that this
determination of $a_{\mu}$ assumes that muonium theory follows SM predictions.
As shown below, contributions beyond the SM ones related to an hypothetical NP
coupling to electrons are sufficiently constrained not to affect the least-
square adjustment.

Expected improvements in muonium physics. The present data is lacking
precision to provide a competitive determination of muon $g-2$. However the
situation is expected to dramatically improve in the near future thanks to new
experimental techniques and more accurate QED calculations. We summarize the
developments planned at the next round of experiments at PSI and J-PARC and
show that, together with the ongoing theory improvement, they will allow for
an extraction of $a_{\mu}$ at few ppm. Moreover, we outline and argue on the
feasibility of the refinements necessary to bring $u_{r}(a_{\mu})$ below the
ppm-level.

1S-2S transition The first necessary ingredient is to improve the electron-
muon mass ratio from muonium 1S-2S spectroscopy. Spectroscopy of the 1S-2S
transition in hydrogenic atoms relies on two-photon excitation with a UV
laser, operating at a $244\,$nm wavelength for muonium. The very high
transition frequency makes possible measurements at ppt precision, with a
$145$ kHz natural linewidth due to the muon lifetime.

One of the main challenges is the low excitation efficiency, as the 1S-2S
transition is a two-photon transition. In order to increase the transition
probability, previous 1S-2S measurements utilized a high-power pulsed laser.
This however came at the cost of broadening the linewidth to $20\,$MHz
associated with the laser pulse-width. Another main systematic uncertainty of
$10$ MHz also originated from the pulsed interaction. A high-power pulsed
laser changes its frequency during every pulse, an effect known as chirping
which is notoriously difficult to compensate for.

To circumvent the limitations of pulsed laser excitation, the MuoniuM lAser
SpectroScopy (Mu-MASS) experiment [44] at PSI utilizes a cavity-enhanced
continuous wave (CW) excitation [58]. The reduced excitation efficiency in CW
operation is compensated by the use of the low-energy-muon (LEM) beamline [59]
paired with new methods to obtain slow muonium atoms emitted into vacuum after
production in mesoporous thin SiO2 films [60]. With such techniques the Mu-
MASS uncertainty goal was set to $10\,$kHz ($4\,$ppt) [44].

On the theory side the $\nu_{\rm 1S-2S}$ uncertainty was quoted at $20\,$kHz
in Ref. [40]. To the best of our knowledge this figure has not been updated,
despite the recent improvement in QED calculations for hydrogen-like atoms
[61, 41]. By rescaling the theory uncertainties for hydrogen [43] to the muon
mass (discarding nuclear finite-size and polarizability contributions) we
assess the current theory uncertainty of $\nu_{\rm 1S-2S}$ to be at $14\,$kHz
($5.7\,$ppt).

The Rydberg constant is also expected to improve in the next few years,
anticipating a full resolution of the so-called proton radius puzzle [62]. The
very precise determination of the proton radius from muonic hydrogen [63, 64]
improves the theoretical precision in hydrogen by about one order-of-
magnitude, which then becomes limited by bound-state QED calculations. The
current QED uncertainty in hydrogen is roughly $1\,$kHz [41], allowing in
principle a three-fold [65] more precise determination of $R_{\infty}$
relative to the latest CODATA [43].

Using Eq. (8) the above values (referred to as “ongoing” in Tables 1 and 2)
yield an expected precision on $m_{e}/m_{\mu}$ at $1.5\,$ppb.

Ground state HFS The major improvement to the electron-muon mass ratio
considered above opens up the possibility to obtain a value for $a_{\mu}$ with
few ppm uncertainty, comparable to the current difference in Eq. (4), granted
that the ground state HFS is improved as well. The current best measurement of
$\nu_{\rm HFS}$ was done with a chopped beam at LAMPF and limited by
statistics [38]. The Muonium Spectroscopy Experiment Using Microwave (MuSEUM)
experiment will improve the statistical uncertainty by using the high-
intensity pulsed muon beam at J-PARC [66, 67] as well as Rabi-oscillation
spectroscopy [68]. Moreover, recently a thorough optimization of the microwave
cavity has been done to drive down the systematic uncertainties to the ppb
level [48]. This will allow for more precise HFS measurements by about one
order-of-magnitude [48] compared to previous LAMPF measurements. We take
$10\,$Hz ($2.2\,$ppb) as an estimate for the uncertainty goal of the MuSEUM
experiment, which is compatible with the ‘several ppb’ in. [48]. (See also
Refs. [69, 70] and references therein.) Another important systematic
uncertainty is due to the pressure shift from the finite gas density
environment [71] in which HFS measurement are performed. This can be overcome
by measuring the HFS in vacuum using a low-energy muon beam.

Excluding the muon mass uncertainty, the theoretical ground state HFS
calculation is currently limited by bound-state QED to around $70\,$Hz
accuracy [52] coming from uncalculated sets of three-loop diagrams. Efforts to
improve this calculation are ongoing, quoting a goal of $10\,$Hz [45].

Collecting the theoretical and experimental values discussed above we estimate
with Eq. (9) the expected uncertainty for extracting $a_{\mu}$ from the
ongoing effort in muonium spectroscopy as about $3\,$ppm (shown in blue on
Fig. 1). Such rousing prospects, comparable to the difference in Eq. (4),
would already contribute to the muon $g-2$ puzzle.

Further improvements Next, we explore how much the above uncertainties can be
further reduced. We based our estimations on known experimental techniques,
arguing that the necessary theoretical improvements can be reasonably achieved
along the way.

An improvement on the 1S-2S frequency determination from the Mu-MASS goal of
$10\,$kHz to few kHz is expected [44]. Since final systematic uncertainties
are estimated at the kHz-level [44], this precision could be accommodated by
an order-of-magnitude increase in statistics. Such an increase could either
come from an improved high-energy muon beam rate, as considered in the High-
intensity Muon Beam (HIMB) upgrade [72, 73, 74] at PSI, and/or a higher
efficiency in muon moderation, as is pursued by the MuCool collaboration [75,
76, 77]. Moreover, another effort in improving the 1S-2S measurement is under
consideration at J-PARC [78]. A 1S-2S measurement at $4\,$kHz would be
sufficient to make the $m_{e}/m_{\mu}$ uncertainty on $\nu_{\rm HFS}$
subdominant.

Such an improved experimental precision must be supplemented by higher
accuracy calculations. Following recent progress in such calculations [61, 41,
79], the main limitation for the theoretical uncertainty comes from unknown
radiative-recoil corrections of $\mathcal{O}[\alpha(Z\alpha)^{6}(m/M)]$ [41].
This correction is likely to be calculated in the near future since the recent
convergence on the proton radius [80, 62, 81, 82, 83, 84] makes it the
limiting factor to reducing the Rydberg constant uncertainty, as well as the
Deuterium radius deduced from the H-D isotope shift [85]. The pure recoil
correction at $\mathcal{O}[(Z\alpha)^{6}(m/M)^{2}]$ is also required for kHz-
accuracy in M. It is partially calculated [86] and there is no known obstacle
towards a complete result.

Regarding the ground state HFS, a further improvement upon the ongoing
efforts, both in experiment and theory, is more demanding. Experimentally the
muon-lifetime constraint on the linewidth already poses a major challenge. A
measurement at $10\,$Hz precision already requires to resolve the line center
to $~{}10^{-4}$ of the linewidth, an achievement similar to the recent 2S-4P
frequency measurement in hydrogen [87]. Further improving the precision with
better line-shape modeling would constitute a premiere in spectroscopy. A
complementary approach consists in converting extremely high statistics into
narrower linewidths by post-selecting so-called ‘old muonium’ atoms [88, 38],
namely muons which have not decayed after several lifetimes. Therefore,
despite the challenges, we imagine that HFS measurements could be done at the
ppb-level ($\simeq 4\,$Hz).

As regards systematic uncertainty, down to the ppb-level the main one is still
considered to be the quadratic pressure shift [38], which could be reduced
with lower pressures, relying on higher statistics. Another promising option
would be to mix gases such as He and Kr which have opposite pressure shift
contributions [89]. The needed high rates could be accommodated by the HIMB
upgrade at PSI that will deliver $\sim 10^{10}$ antimuons per second, which is
three orders of magnitude higher than the CW muon beam used for the HFS
measurement at LAMPF. Leveraging the technical advancements by the MuSEUM
collaboration [68, 48], together with a $\sim 100$-days beamtime, we conclude
that a sub-ppb experimental uncertainty in $\nu_{\rm HFS}$ could be
envisioned.

The corresponding theory prediction to the level of $4$ Hz accuracy is
demanding as well. However, as long as the uncertainty from HVP contributions
(presently about $0.8\,$Hz [15]) is subdominant, completing the required
bound-state QED calculations in the same timescale as experimental milestones
could be reasonably envisaged. Indeed, ongoing work already set a goal of a
few Hz [46], which would suffice to make $\nu_{\rm HFS}$ limited by
experiment.

Figure 1: Anomalous magnetic moment of the muon determined from spin-
precession measurements at BNL and FNAL (red) [2], Standard Model calculations
with LO HVP evaluated from $e^{+}e^{-}\to$ hadrons data (green) [26] or
lattice QCD (gray) [31] and the projected sensitivities in muonium (dashed)
with the currently planned spectroscopy improvements (blue) and its ultimate
improvement (magenta), centered around the current experimental average. The
orange band shows the four-fold improved FNAL standard deviation expected in
the near future.

Assuming that the 1S-2S transition and ground state HFS in muonium are pushed
to the aforementioned limits would bring down the $a_{\mu}$ uncertainty to
$u(a_{\mu})\simeq 103\times 10^{-11}$ ($0.88$ ppm) which is about half of the
current difference between theory and experiment.

Summary and Outlook. We proposed an alternative strategy to extract the
anomalous magnetic moment of the muon based on highly-precise measurements and
theoretical QED calculations in muonium. Current data on the 1S-2S transition
and ground state HFS frequencies allows to determine $a_{\mu}$ at $\sim
700\,$ppm, largely limited by statistics in the 1S-2S measurement. A new
program, partly motivated by the present muon $g-2$ puzzle, of highly-precise
muonium spectroscopy and theory will significantly improve the situation in
the coming decade, reducing the $a_{\mu}$ uncertainty to about $3\,$ppm, and
even below the ppm-level at a future stage. As shown in the Supplemental
Material, which includes Refs. [90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100,
101, 102, 103], this indirect determination which relies on bound-state QED
calculation is unlikely to be affected by NP, given present constraints on
exotic forces coupled to electrons.

Such an independent value of $a_{\mu}$ cannot be competitive with spin-
precession measurements that are expected to reach $\sim 0.1\,$ppm accuracy in
a few years [2, 104]. However it would most certainly help interpreting the
current difference between these measurements and the ($R$-ratio based) SM
prediction, were it to persist over the same timescale. A muonium value of
$a_{\mu}$ favoring today’s experimental average, along with $\delta_{\rm
1S-2S}$ and $\delta_{\rm HFS}$ consistent with the SM, would strengthen the
evidence of NP dominantly coupled to muons (assuming the $R$-ratio
determination of the HVP contribution is a faithful figure of the SM
prediction). Conversely, a value consistent with the current $R$-ratio
prediction would indicate a potential unknown systematic with the BNL/FNAL
measurements, or point to a non-trivial NP dynamics. In the latter case,
precision measurements of additional muonium lines, like the 2S-2P Lamb shift
[105, 106] ongoing at PSI, would help discriminating among different
possibilities. Either way, this highly motivates to push the frontier of
precision muonium physics as much as possible.

Acknowledgements. We thank Michael Eides and Yevgeny Stadnik for useful
discussions and Paolo Crivelli, Michael Eides, Gilad Perez and in particular
Klaus Jungmann for valuable comments on the manuscript. BO acknowledges
support from the European Union’s Horizon 2020 research and innovation program
under the Marie Skłodowska-Curie grant agreement No. 101019414. YS is
supported by the United States-Israel Binational Science Foundation (BSF)
(NSF-BSF program grant No. 2018683), by the Israel Science Foundation (grant
No. 482/20) and by the Azrieli foundation. YS is Taub fellow (supported by the
Taub Family Foundation).

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Towards an independent determination of muon g-2 from muonium spectroscopy

Supplemental Material
Cédric Delaunay, Ben Ohayon and Yotam Soreq

##  I Least-square adjustment

The electron-to-muon mass ratio $m_{e}/m_{\mu}$ and the muon anomalous
magnetic moment $a_{\mu}$ are extracted from muonium data through a least-
square adjustment. We follow the method and notation used by the CODATA
collaboration, as described in their 1998 review [54]. Our adjustment is based
on the 1999 [40] measurement of the 1S-2S transition frequency at the
Rutherford Appleton Laboratory (RAL), the 1982 [55] and 1999 [38] measurements
of the ground state HFS at the Clinton P. Anderson Meson Physics Facility
(LAMPF). It includes also inputs $\delta E({\rm 1S})$, $\delta E({\rm 2S})$
and $\delta E({\rm HFS})$ representing additive corrections to the 1S, 2S
energies $E_{M}$ and the 1S HFS, respectively, associated to unknown high-
order QED contributions. They are assigned zero values and an uncertainty
given by the theory uncertainty of the energy level calculations. Input data
are summarized in Table S1.

input datum | value | relative uncertainty | identification | reference
---|---|---|---|---
$\nu_{\rm 1S-2S}$ | $2\,455\,528\,941.0(9.8)\,{\rm MHz}$ | $4.0\times 10^{-9}$ | RAL-99 | [40]
$\nu_{\rm HFS}$ | $4\,463\,302\,776(51)\,{\rm Hz}$ | $1.2\times 10^{-8}$ | LAMPF-99 | [38]
$\nu_{\rm HFS}$ | $4\,463\,302.88(16){\rm kHz}$ | $3.6\times 10^{-8}$ | LAMPF-82 | [55]
$\delta E({\rm 1S})/h$ | $0.000(14)\,{\rm MHz}$ | $4.3\times 10^{-12}$ | theory | [43]
$\delta E({\rm 2S})/h$ | $0.0(1.8)\,{\rm kHz}$ | $2.2\times 10^{-12}$ | theory | [43]
$\delta E({\rm HFS})/h$ | $0.000(70)\,{\rm kHz}$ | $1.6\times 10^{-8}$ | theory | [52]
Table S1: Input data for the determination of $m_{e}/m_{\mu}$ and $a_{\mu}$
from muonium spectroscopy. For the additive corrections $\delta E({\rm 1S})$
and $\delta E({\rm 2S})$ the quoted $u_{r}$ are relative to the corresponding
state energies, and to the 1S HFS frequency for $\delta E({\rm HFS})/h$.

The data is compared to the best available theory predictions for $\nu_{\rm
1S-2S}$ and $\nu_{\rm HFS}$. The full expressions in terms of $R_{\infty}$,
$\alpha$, $m_{e}/m_{\mu}$ and $a_{\mu}$ (including latest developments in the
$n$S levels energies [61] and the 1S HFS [56, 57]) are nicely summarized in
the 2018 CODATA review [43]. (The muonium energies are obtained from hydrogen
ones upon changing the reduced mass and discarding the nuclear finite-size and
polarizability contributions.) Given their very small uncertainty compared to
those in muonium, we assume $R_{\infty}$ and $\alpha$ are exactly known
constants fixed to their 2018 CODATA recommended values [43]:
$R_{\infty}c=3\,289\,841\,960\,250.8\,$kHz and
$\alpha^{-1}=137.035\,999\,084$. The observational equations relating the
input data in Table S1 and the remaining adjusted constants including
$m_{e}/m_{\mu}$, $a_{\mu}$ and the additive corrections $\delta^{\rm th}_{\rm
1S}$, $\delta^{\rm th}_{\rm 2S}$, $\delta^{\rm th}_{\rm HFS}$ are given in
Table S2.

input datum | observational equation
---|---
$\nu_{\rm 1S-2S}$ | $\nu_{\rm 1S-2S}\doteq[E_{M}({\rm 2S};m_{e}/m_{\mu})+\delta_{\rm 2S}^{\rm th}-E_{M}({\rm 1S};m_{e}/m_{\mu})-\delta_{\rm 1S}^{\rm th}]/h$
$\nu_{\rm HFS}$ | $\nu_{\rm HFS}\doteq\nu_{\rm HFS}^{\rm th}(m_{e}/m_{\mu},a_{\mu})+\delta_{\rm HFS}^{\rm th}/h$
$\delta E({\rm 1S})/h$ | $\delta E({\rm 1S})\doteq\delta_{\rm 1S}^{\rm th}$
$\delta E({\rm 2S})/h$ | $\delta E({\rm 2S})\doteq\delta_{\rm 2S}^{\rm th}$
$\delta E({\rm HFS})/h$ | $\delta E({\rm HFS})\doteq\delta_{\rm HFS}^{\rm th}$
Table S2: Observational equations used for the determination of
$m_{e}/m_{\mu}$ and $a_{\mu}$ from muonium spectroscopy.

The theory uncertainty of the $n$S energy calculation is dominated by the
unknown higher-order terms in the relativistic-recoil QED contribution.
Reference [40] quotes a QED uncertainty of $20\,$kHz for the overall 1S-2S
transition frequency. To our knowledge there is no updated value published in
the literature. Following Ref. [43], we reevaluated the muonium 1S and 2S
energy uncertainties using the latest developments in hydrogen calculations,
yielding a total uncertainty of $u[\delta E({\rm 1S})/h]\simeq 14\,$kHz and
$u[\delta E({\rm 2S})/h]\simeq 1.8\,$kHz, with a $99\%$ correlation between
them.

After two iterations, the least-square adjustment described above converged to
the values reported in the main text: $m_{e}/m_{\mu}=4\,836\,329(4)\times
10^{-9}$ and $a_{\mu}=116\,637(82)\times 10^{-8}$, the former reproducing the
value derived in Ref. [40]. The Birge ratio $R_{b}=0.68$ and all normalized
residuals are found to be below unity, showing self-consistency of the input
data.

The least-square adjustment presented above can be straightforwardly improved
to include future more precise measurements and theory calculations (including
other muonium transitions). Possible new physics contributions (as discussed
below) can be accounted for by additional adjusted constants for the
associated mass scale and couplings to electrons and muons.

##  II New physics contributions to muonium energy levels

The new physics (NP) contributions from the exchange of a scalar or vector of
mass $m$ is given in first-order perturbation theory by the overlap of the
non-relativistic potential

$V_{\rm NP}(r)=\alpha_{\rm
NP}e^{-mr}\left[\frac{1}{r}+\frac{8\pi(\vec{S}_{e}\cdot\vec{S}_{\mu})}{3m_{e}m_{\mu}}\left[\frac{m^{2}}{4\pi
r}-\delta^{3}(r)\right]\right]\,,$ (S1)

with the modulus square of the Schrödinger wavefunction for hydrogen-like
atoms [95]

$\psi_{nlm}(r,\theta,\phi)=R_{nl}(r)Y_{l}^{m}(\theta,\phi)\,,$ (S2)

where $Y_{l}^{m}(\theta,\phi)$ are spherical harmonics (normalized as
$\int_{-1}^{1}d\cos\theta\int_{0}^{2\pi}d\phi\
Y_{l}^{m}(\theta,\phi)Y_{l^{\prime}}^{m^{\prime}}(\theta,\phi)=\delta_{ll^{\prime}}\delta_{mm^{\prime}}$)
and

$R_{nl}(r)=\sqrt{\left(\frac{2}{na_{M}}\right)^{3}\frac{(n-l-1)!}{2n(n+l)!}}e^{-\rho/2}\rho^{l}L_{n-l+1}^{2l+1}(\rho)\,,\quad{\rm
with}\quad\rho\equiv\frac{2r}{na_{M}}\,,$ (S3)

is the radial wavefunction expressed in terms of the reduced Bohr radius
$a_{M}=(1+m_{e}/m_{\mu})/(\alpha m_{e})$ and the generalized Laguerre
polynomials of degree $n-l+1$, $L_{n-l+1}^{2l+1}(x)$. The NP contribution to
the 1S-2S frequency shift (arising from the spin-independent part of $V_{\rm
NP}$) is

$\delta_{\rm 1S-2S}^{\rm NP}=\frac{8\alpha_{\rm
NP}}{3\alpha}I_{12}(ma_{M})\,,$ (S4)

where

$\displaystyle I_{12}(ma_{M})$ $\displaystyle\equiv$ $\displaystyle a_{M}\int
d^{3}r\,\frac{e^{-mr}}{r}\left[|\psi_{200}^{2}(r,\theta,\phi)|^{2}-|\psi_{100}^{2}(r,\theta,\phi)|^{2}\right]=a_{M}\int_{0}^{\infty}drre^{-mr}\left[R_{20}(r)^{2}-R_{10}(r)^{2}\right]$
(S5) $\displaystyle=$
$\displaystyle\frac{1+2(ma_{M})^{2}}{4(1+ma_{M})^{4}}-\frac{4}{(2+ma_{M})^{2}}\,,$

with the asymptotic behaviors $I_{12}(x\to 0)=-3/4$ and
$I_{12}(x\to\infty)\simeq-7/(2x^{2})$. Similarly, its counterpart to the 1S
HFS shift (arising from the spin-spin interaction part of the potential) is

$\delta_{\rm HFS}^{\rm NP}=\frac{\alpha_{\rm NP}}{4\alpha}J_{1}(ma_{M})\,,$
(S6)

where

$\displaystyle J_{1}(ma_{M})$ $\displaystyle\equiv$ $\displaystyle
a_{M}^{3}\int
d^{3}r\,e^{-mr}\left[\frac{m^{2}}{r}-4\pi\delta^{3}(r)\right]|\psi_{100}(r,\theta,\phi)|^{2}=a_{M}^{3}\left[-R_{10}(0)^{2}+m^{2}\int_{0}^{\infty}drre^{-mr}R_{10}(r)^{2}\right]$
(S7) $\displaystyle=$ $\displaystyle-16\frac{1+ma_{M}}{(2+ma_{M})^{2}}\,,$

with $J_{1}(x\to 0)=-4$ and $J_{1}(x\to\infty)\simeq-16/x$.

Note that the large mass ($m\gg a_{M}^{-1}$) behavior of $J_{1}$ is at odds
with naive expectations from effective field theory which predicts a
decoupling of the HFS contribution like $1/m^{2}$ (or faster). The $1/m$
scaling is a spurious artefact of the non-relativistic approximation used to
obtain the potential in Eq. (S1). While this approximation is certainly valid
for long-range interactions, it breaks down for new spin-dependent forces
whose range is shorter than the Compton wavelength of the electron. This is
manifest from the appearance of a $\delta$-function term in Eq. (S1)
characterized by the scale $\sqrt{m_{e}m_{\mu}}$. This $\delta$-function stems
from the assumption that the electron couples to the magnetic field created by
a static muon located at $r=0$. For scalar/vector masses above that scale, the
short-distance dynamics is not properly resolved, yielding a spurious $1/m$
scaling. The expected $1/m^{2}$-asymptotic of $J_{1}$ should be recovered
within a bound-state theory, like the Bethe-Salpeter formalism [96], where
both the electron and the antimuon are relativistic.

To what extent NP could contaminate muonium observables given the discrepancy
currently observed in muon $g-2$? In order to address this question, we fix
the muon coupling $y_{\mu}$, assuming the NP contribution at one-loop to the
anomalous magnetic moment of leptons [92]

$\displaystyle\Delta a_{\ell}^{\rm
NP}=\frac{y_{\ell}^{2}}{8\pi^{2}}\int_{0}^{1}dz\frac{(1-z)^{2}\zeta(z)}{(1-z)^{2}+z(m/m_{\ell})^{2}}\,,$
(S8)

[$\zeta(z)\equiv(1+z)$ for scalars and $2z$ for vectors] saturates Eq. (4).
The electron counterpart is constrained by the good agreement between the
measured electron $g-2$ [97] and the QED prediction [7] based on $\alpha$
determined from the recent $h/m_{\rm Rb}$ measurement at LKB [47], yielding
[47]

$\displaystyle\Delta a_{e}\equiv a_{e}^{\rm exp}-a_{e}^{\rm SM}(\alpha_{\rm
LKB})=(4.8\pm 3.0)\times 10^{-13}\,,$ (S9)

where the uncertainty is dominated by $a_{e}^{\rm exp}$. The maximal possible
NP contributions (as function of $m$) to $\delta_{\rm 1S-2S}$ and $\delta_{\rm
HFS}$, assuming $\Delta a_{e}^{\rm NP}\leq 10.8\times 10^{-13}$ and $\Delta
a_{\mu}^{\rm NP}=259\times 10^{-11}$ are shown as blue lines in Fig. S1. The
NP contribution to the 1S-2S transition is suppressed above $a_{M}^{-1}$, due
to the $1/m^{2}$ scaling of $\delta_{\rm 1S-2S}^{\rm NP}$, and goes below
$10\,$kHz at about $m\sim 1\,$MeV. At lower masses, below $m\sim 300\,$keV,
the electron coupling is strongly constrained (gray lines) by stellar cooling
considerations [98]. The remaining gap can be covered by the order-of-
magnitude improvement on $a_{e}^{\rm exp}$ [99, 100]. This shows that the
forthcoming 1S-2S frequency measurement at Mu-Mass will be typically safe from
NP contamination, thus allowing a determination $m_{e}/m_{\mu}$ at the ppb
level. Similar conclusion holds for the $10\,$Hz accuracy goal on measuring
the ground state HFS at MuSEUM. Note that, with additional assumptions on the
scalar/vector decays, fixed-target experiments constrain the electron coupling
more stringently up to one order of magnitude below the GeV-scale e.g. [101,
102]. Taking these bounds at face value further excludes NP contributions to
$\delta_{\rm 1S-2S}$ and $\delta_{\rm HFS}$ below the ultimate precision goal
envisaged in the previous section.

NP could also enter indirectly through a modification of the Rydberg constant
(and in turn also $\alpha$). Since $R_{\infty}$ is determined from hydrogen
spectroscopy, this is only possible assuming an additional coupling to
protons. However, the overall consistency among several highly-precise
hydrogen measurements strongly restricts NP interactions between electrons and
protons, still allowing to extract $R_{\infty}$ with an uncertainty comparable
to that of assuming the SM [103]. Hence we conclude that a reliable
determination of $a_{\mu}$ from a comparison of future muonium spetroscopy
measurements with their SM predictions is expected.

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Figure S1:  NP contributions to $\delta_{\rm 1S-2S}$ (left) and $\delta_{\rm
HFS}$ (right). Solid red lines indicate the expected accuracy of the ongoing
Mu-Mass ($10\,$kHz) and MuSEUM ($10\,$Hz) experiments. The blue (gray) shaded
area is excluded by the electron $g-2$ measurement [97] (stellar cooling
constraints [98]). The dotted green line denotes the NP contribution from the
relation $y_{e}=(\alpha/\pi)\times y_{\mu}$ naively expected from kinetic
mixing at one-loop in the vector case. The right-hand side plot is cut at
$m=\sqrt{m_{e}m_{\mu}}\simeq 7\,$MeV, which is the validity range of the spin
dependent effetive potentail in Eq. (S1).