The CREMA Collaboration

# The helion charge radius from laser spectroscopy of muonic helium-3 ions

Karsten Schuhmann  Institute for Particle Physics and Astrophysics, ETH
Zurich, 8093 Zurich, Switzerland  Luis M. P. Fernandes  LIBPhys, Physics
Department, Universidade de Coimbra, P-3004-516 Coimbra, Portugal François Nez
Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS, ENS-PSL Research
University, Collège de France, 4 place Jussieu, case 74, 75005 Paris, France
Marwan Abdou Ahmed  Institut für Strahlwerkzeuge, Universität Stuttgart, 70569
Stuttgart, Germany  Fernando D. Amaro  LIBPhys, Physics Department,
Universidade de Coimbra, P-3004-516 Coimbra, Portugal Pedro Amaro  Laboratório
de Instrumentação, Engenharia Biomédica e Física da Radiação (LIBPhys-UNL) e
Departamento de Física da Faculdade de Ciências e Tecnologia da Universidade
Nova de Lisboa, Monte da Caparica, 2892-516 Caparica, Portugal  François
Biraben  Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS, ENS-PSL
Research University, Collège de France, 4 place Jussieu, case 74, 75005 Paris,
France  Tzu-Ling Chen  Physics Department, National Tsing Hua University,
Hsincho 300, Taiwan  Daniel S. Covita  i3N, Universidade de Aveiro, Campus de
Santiago, 3810-193 Aveiro, Portugal  Andreas J. Dax  Paul Scherrer Institute,
5232 Villigen, Switzerland  Marc Diepold  Max-Planck-Institut für
Quantenoptik, 85748 Garching, Germany  Beatrice Franke  Max-Planck-Institut
für Quantenoptik, 85748 Garching, Germany  Sandrine Galtier  Laboratoire
Kastler Brossel, UPMC-Sorbonne Universités, CNRS, ENS-PSL Research University,
Collège de France, 4 place Jussieu, case 74, 75005 Paris, France  Andrea L.
Gouvea  LIBPhys, Physics Department, Universidade de Coimbra, P-3004-516
Coimbra, Portugal Johannes Götzfried  Max-Planck-Institut für Quantenoptik,
85748 Garching, Germany  Thomas Graf  Institut für Strahlwerkzeuge,
Universität Stuttgart, 70569 Stuttgart, Germany  Theodor W. Hänsch  Max-
Planck-Institut für Quantenoptik, 85748 Garching, Germany  Malte Hildebrandt
Paul Scherrer Institute, 5232 Villigen, Switzerland  Paul Indelicato
Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS, ENS-PSL Research
University, Collège de France, 4 place Jussieu, case 74, 75005 Paris, France
Lucile Julien  Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS,
ENS-PSL Research University, Collège de France, 4 place Jussieu, case 74,
75005 Paris, France  Klaus Kirch  Institute for Particle Physics and
Astrophysics, ETH Zurich, 8093 Zurich, Switzerland  Paul Scherrer Institute,
5232 Villigen, Switzerland  Andreas Knecht  Paul Scherrer Institute, 5232
Villigen, Switzerland  Franz Kottmann  Institute for Particle Physics and
Astrophysics, ETH Zurich, 8093 Zurich, Switzerland  Paul Scherrer Institute,
5232 Villigen, Switzerland  Julian J. Krauth  Max-Planck-Institut für
Quantenoptik, 85748 Garching, Germany  Johannes Gutenberg Universität Mainz,
Institut für Physik, QUANTUM, & Exzellenzcluster PRISMA+, 55099 Mainz, Germany
Yi-Wei Liu  Physics Department, National Tsing Hua University, Hsincho 300,
Taiwan  Jorge Machado  Laboratório de Instrumentação, Engenharia Biomédica e
Física da Radiação (LIBPhys-UNL) e Departamento de Física da Faculdade de
Ciências e Tecnologia da Universidade Nova de Lisboa, Monte da Caparica,
2892-516 Caparica, Portugal  Cristina M. B. Monteiro  LIBPhys, Physics
Department, Universidade de Coimbra, P-3004-516 Coimbra, Portugal Françoise
Mulhauser  Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
Boris Naar  Institute for Particle Physics and Astrophysics, ETH Zurich, 8093
Zurich, Switzerland  Tobias Nebel  Max-Planck-Institut für Quantenoptik, 85748
Garching, Germany  Joaquim M. F. dos Santos  LIBPhys, Physics Department,
Universidade de Coimbra, P-3004-516 Coimbra, Portugal José Paulo Santos
Laboratório de Instrumentação, Engenharia Biomédica e Física da Radiação
(LIBPhys-UNL) e Departamento de Física da Faculdade de Ciências e Tecnologia
da Universidade Nova de Lisboa, Monte da Caparica, 2892-516 Caparica, Portugal
Csilla I. Szabo  Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS,
ENS-PSL Research University, Collège de France, 4 place Jussieu, case 74,
75005 Paris, France  David Taqqu  Institute for Particle Physics and
Astrophysics, ETH Zurich, 8093 Zurich, Switzerland  Paul Scherrer Institute,
5232 Villigen, Switzerland  João F. C. A. Veloso  i3N, Universidade de Aveiro,
Campus de Santiago, 3810-193 Aveiro, Portugal  Andreas Voss  Institut für
Strahlwerkzeuge, Universität Stuttgart, 70569 Stuttgart, Germany  Birgit
Weichelt  Institut für Strahlwerkzeuge, Universität Stuttgart, 70569
Stuttgart, Germany  Aldo Antognini<EMAIL_ADDRESS>Institute for
Particle Physics and Astrophysics, ETH Zurich, 8093 Zurich, Switzerland  Paul
Scherrer Institute, 5232 Villigen, Switzerland  Randolf Pohl  pohl@uni-
mainz.de Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
Johannes Gutenberg Universität Mainz, Institut für Physik, QUANTUM, &
Exzellenzcluster PRISMA+, 55099 Mainz, Germany

###### Abstract

Hydrogen-like light muonic ions, in which one negative muon replaces all the
electrons, are extremely sensitive probes of nuclear structure, because the
large muon mass increases tremendously the wave function overlap with the
nucleus. Using pulsed laser spectroscopy we have measured three 2S-2P
transitions in the muonic helium-3 ion ($\mu^{3}$He+), an ion formed by a
negative muon and bare helium-3 nucleus. This allowed us to extract the Lamb
shift $E(2P_{1/2}-2S_{1/2})=1258.598(48)^{\rm exp}(3)^{\rm theo}$
$\mathrm{meV}$, the 2P fine structure splitting $E_{\rm FS}^{\rm
exp}=144.958(114)$ $\mathrm{meV}$, and the 2S-hyperfine splitting (HFS)
$E_{\rm HFS}^{\rm exp}=-166.495(104)^{\rm exp}(3)^{\rm theo}$ $\mathrm{meV}$
in $\mu^{3}$He+. Comparing these measurements to theory we determine the rms
charge radius of the helion (3He nucleus) to be $r_{h}$= 1.97007(94)
$\mathrm{fm}$. This radius represents a benchmark for few nucleon theories and
opens the way for precision tests in 3He atoms and 3He-ions. This radius is in
good agreement with the value from elastic electron scattering, but a factor
15 more accurate. Combining our Lamb shift measurement with our earlier one in
$\mu^{4}$He+ we obtain $r_{h}^{2}-r_{\alpha}^{2}=1.0636(6)^{\rm exp}(30)^{\rm
theo}\,\mathrm{fm}^{2}$ to be compared to results from the isotope shift
measurements in regular He atoms, which are however affected by long-standing
tensions. By comparing $E_{\rm HFS}^{\rm exp}$ with theory we also obtain the
two-photon-exchange contribution (including higher orders) which is another
important benchmark for ab-initio few-nucleon theories aiming at understanding
the magnetic and current structure of light nuclei.

muonic atoms, Lamb shift, laser spectroscopy, helion, nuclear charge radius

##  I Introduction

Precise and accurate values of nuclear radii are necessary for advancing the
theory prediction in simple atomic and molecular systems, that could be
exploited to test bound-state Quantum Electro Dynamics (QED) predictions, to
determine fundamental constants such as the Rydberg constant and the electron
mass and to search for new physics Antognini:2022xoo . Moreover these radii
represent rigorous benchmarks for nuclear structure theory with the peculiar
enhanced sensitivity to the long range behaviour of the nuclear wave
functions. Spectacular advances have been obtained recently for nucleon-
nucleon currents and form factors of light nuclei using nuclear interactions
derived from chiral effective field theory which can be systematically
improved Ji:2018:nuclPOL ; Epelbaum:2019jbv ; Piarulli:2012bn . On the
experimental side, laser spectroscopy of muonic atoms with their exquisite
sensitivity to nuclear properties has been recently established
Antognini:2013:Science_mup2 ; Pohl:2016:mud ; Krauth:2021foz . Owing to the
200 times larger muon mass compared to the electron mass, these muonic atoms
(ions) where a single negative muon is orbiting a bare nucleus, have a
dramatically enhanced sensitivity to the nuclear charge radii compared to
regular atoms. This sensitivity arises from the enhanced overlap of the atomic
wavefunction with the nucleus which scales with the third power of the
orbiting particle mass (reduced mass). Indeed the leading-order finite-
nuclear-size effect takes the form

$\displaystyle\Delta E_{\mathrm{FNS}}(n,l)$
$\displaystyle=\frac{2}{3n^{3}}(Z\alpha)^{4}m_{r}^{3}r^{2}\delta_{l0}\,,\ $
(1)

where $n$ is the principal quantum number, $l$ the angular momentum, $\alpha$
the fine structure constant, $Z$ the atomic number, $m_{r}$ the reduced mass,
and $r$ the nuclear charge radius. The Kronecker $\delta_{l0}$ indicates that
only S-states energy levels are affected in leading approximation by the
finite size due to their overlap with the nucleus.

In this work, we present the measurement of three $\mathrm{2S\rightarrow 2P}$
transitions of the muonic helium-3 ion $\mu^{3}$He+ (a two-body ion formed by
a negative muon and a bare 3He nucleus) as shown in Fig. 1 from which we
extracted the 2S-2P Lamb shift, the 2S hyperfine splitting and the 2P fine
splitting. From the Lamb shift we have then extracted the rms charge radius of
the helion $r_{h}$ with an unprecedented relative precision of $7\times
10^{-4}$, improving the previous best value Sick:2014:HeZemach  from elastic
electron scattering by a factor of 15. From the comparison between 2S-HFS and
theory, we extracted the two-photon-exchange contribution which is the
leading-order nuclear structure dependent contribution for the HFS. Hence,
these measurements provide important benchmarks for nuclear theories, and pave
the way for advancing precision tests of two- and three-body QED when combined
with ongoing efforts in regular He atoms Shiner:1995:heliumSpec ;
Rooij:2011:HeSpectroscopy ; CancioPastor:2012:PRL108 ; Patkos:2016:HeIso ;
Zheng:2017:He_Iso ; Rengelink:2018:4He ; PhysRevA.101.062507 ;
vanderWerf:2023cuv  and He+ ions Moreno:2023amv ; Krauth:2019:HePlus .

The 2S-2P transition frequencies were measured by pulsed laser spectroscopy at
wavelengths around 850-940 nm (frequencies of 310-350 THz) to an accuracy of
about 20 GHz corresponding to relative accuracies of about 50 ppm. The
resonances were exposed by detecting the $K_{\alpha}$ X-ray of 8 keV energy
emitted from $\mathrm{2P\rightarrow 1S}$ de-excitation following a successful
laser transition from the 2S to the 2P state. The 50 ppm measurement precision
has to be compared to the energy shift caused by the finite-size effect (see
Eq. (1)) that contributes as much as 25% to the 2S-2P energy splitting owing
to its $m_{r}^{3}Z^{4}$ dependence. The binding energy of this hydrogen-like
system, scaling as $Z^{2}m_{r}$, is strongly enhanced compared to hydrogen,
while the atomic size (Bohr radius) scaling as $1/Zm_{r}$ is strongly reduced
making this atom immune to external perturbation. Because of its $m_{r}Z^{4}$
dependence, the decay rate from the 2P state is also vastly increased
resulting in a 2P-linewidth of 319 GHz. This broad linewidth represents by far
the main limitation to our experimental precision. Having in mind these energy
scales, sensitivity to nuclear properties and immunity to external
perturbations helps framing the requirements for the spectroscopy experiment
and understanding of the experimental setup.

Figure 1: Scheme (not to scale) of the $n=2$ energy levels in $\mu^{3}$He+ and
the measured transitions. Due to the negative magnetic moment of the helion,
the ordering of the hyperfine levels is reversed.

##  II Principle and experimental setup

$\mu^{3}$He+ ions are formed in highly excited states by stopping a keV-energy
muon in a low-pressure (2 mbar) 3He gas target at room temperature and placed
in a 5 T solenoid. The newly formed $\mu^{3}$He+ ions deexcite to the 1S state
in a fast (ns-scale) and complex cascade process with a fraction of about 1%
reaching the metastable 2S state whose lifetime is approximately 1.7 $\mu$s at
this target pressure. This lifetime is sufficiently long to enable pulsed
laser spectroscopy of the 2S-2P splitting. For this purpose the muon entering
the target is being detected to trigger a laser system that delivers a pulse
to excite the 2S into the 2P state after a delay of about 1 $\mathrm{\upmu
s}$. The laser pulse is injected into a multipass cell formed by two elongated
mirrors that fold the light back and forth to illuminate the elongated muon
stopping volume. A successful laser excitation is established by detecting the
8 keV energy $K_{\alpha}$ X-ray from the 2P de-excitation into the ground
state. This is accomplished using two rows of large area avalanche photodiodes
(LAAPDs) placed above and below the muon stopping volume, respectively and
covering 30% solid angle. The 2S-2P resonances are eventually obtained by
plotting the number of $K_{\alpha}$ X-rays detected in time coincidence with
the laser light as a function of the laser frequency.

Figure 2: Sketch of the experimental setup. CT cyclotron trap, MEC muon
extraction channel, Ti:Sa Titanium-sapphire laser, Yb:YAG thin-disk laser, SHG
second harmonic generation. The arrows indicate the electric field of the
$E\times B$ Wien-filter.

A sketch of the setup is shown in Fig. 2. The experiment has been performed at
the $\pi$E5 beamline of the CHRISP facility at the Paul Scherrer Institute
(PSI), Switzerland. Here, $102\,\mathrm{MeV}/c$ negative pions are injected
tangentially into a cyclotron trap (CT) formed by two coils generating a
B-field with a magnetic-bottle geometry Gotta:2021vke . A fraction of the
muons from the pion decays are trapped in the B-field of the CT. Passing
multiple times a thin foil placed in the trap mid-plane, these muons are
decelerated down to 20-40 keV energy. At this low energy, the electric field
which is applied along the trap axis imparts sufficient longitudinal momentum
to the muons so that they can escape the magnetic confinement of the CT in
axial direction. The escaping muons are coupled into a toroidal magnetic
field, the muon extraction channel (MEC), that acts as a momentum filter. The
MEC transports the muons into a region of lower background where the
spectroscopy experiment can be performed shielded from the large background of
neutrons, gammas and other charged particles present in the CT. From the MEC
toroidal field of 0.1 T, the muon beam is focused into the 5 T solenoid.
Before entering the He target with an average rate of about 500 s-1, the muons
are passing two stacks of few $\mu$g/cm2 thin carbon foils placed at high
voltages that serve two purposes: firstly they reduce the muon energy to few
keV while obtaining some frictional cooling Muehlbauer:1999:FrictCool , and
secondly they generate a muon entrance signal to trigger the laser system. The
muon entrance signal is obtained by detecting the secondary electrons ejected
from the carbon foils by the passing muon. The foils are biased to accelerate
the electrons towards two plastic scintillators coupled to photomultipliers.
An $E\times B$ Wien filter separates the muon from the electrons generated in
the first stack of foils. To increase the laser trigger quality, i.e., to fire
the laser only for muons which have maximal probability to stop in the He gas
target, a fine selection of the muon momentum is performed. This is achieved
requiring a coincidence with the optimal time-of-flight between the two
electron signals from the two carbon stacks. The muons then pass a thin (2
$\mu$g/cm2) entrance window of Formvar before slowing down in the low-density
He gas giving rise to a 20 cm long stopping distribution. The gas in the
target is kept clean by a circulation system with a cold trap.

The detection of a randomly arriving muon triggers the pulsed laser system.
With a delay of about 1 $\mathrm{\upmu s}$ the laser pulses of 5 mJ energy and
70 ns pulse length are coupled through a 0.6 mm diameter hole into a multi-
pass cell formed by two elongated laser mirrors Vogelsang:2014:Cavity  which
are shaped to match the muon stopping distribution. The laser system consists
of a thin-disk Yb:YAG oscillator-amplifier system Antognini:2009:Disklaser ;
Schuhmann:2015:TDL , a frequency doubling stage (SHG), and a Ti:Sapphire
oscillator. The Ti:Sapphire oscillator is injection seeded by a Ti:Sapphire
single-frequency cw laser which is stabilized to a calibrated Fabry-Perot and
referenced to a wavemeter with a 60 MHz absolute accuracy. The wavemeter was
calibrated using transition lines in 133Cs and 83Kr. From the laser laboratory
the pulses are sent to the target region over a distance of $\mathrm{\sim
20\,m}$. An active pointing stabilization ensures a stable coupling of the
laser pulse into the optical cell. A $\lambda$/2 retardation plate combined
with a polarizer is used to adjust from time to time the pulse energy and
maintain it roughly at 5 mJ compensating for possible drifts. The time and
spatial distribution of the laser light in the cell is monitored using several
photodiodes embedded into the mirror substrates which detect the small amount
of light leaking through the mirror coatings.

A waveform analysis was applied to the LAAPDs signals to improve energy and
time resolutions and to disentangle electrons from X-ray events yielding a
FWHM energy resolution of 16% at 8.2 keV Diepold:2015:xray . As the experiment
is taking place with only one muon at a time in the setup, the detection of
the muon-decay electron after the detection of a 8 keV X-ray is used to
sharpen the event selection: a reduction of the background by more than an
order of magnitude was observed when implementing this muon-decay electron
cut. To increase the detection efficiency for these MeV electrons which are
curling in the magnetic field, four plastic scintillators were placed radially
around the He target as shown in Fig. 2.

##  III Measurement and Data Analysis

Figure 3: Measured $\mathrm{2S}\rightarrow\mathrm{2P}$ transitions in
$\mu^{3}$He+. The black data points show the number of laser-induced
$K_{\alpha}$ X-ray events normalized to prompt $K_{\alpha}$ events. The data
are fitted with a line shape model detailed in the main text. This model
applies only at the measured points, and the colored lines only connect the
fit points. The fitted center frequencies including their uncertainties are
indicated by the colored points with error bar above the resonances. The
yellow bands indicate the average $\pm 1\sigma$ backgrounds obtained from
events where the laser was not fired.

The events used to expose the three 2S-2P resonances shown in Fig. 1 had to
fulfill the following sequence: a muon is detected in the entrance counter, a
$K_{\alpha}$ X-ray is detected in the LAAPDs in time coincidence with the
laser light in the cavity, and a decay electron is detected afterwards in the
electron counters (or LAAPDs). To obtain the resonances shown in Fig. 3, the
number of these events is plotted versus the laser frequency and normalized to
the number of prompt $K_{\alpha}$ X-rays so that variations of the number of
muons and X-ray detection efficiency do not affect the results. The laser
frequency was alternated on the two sides of the resonance every 1-2 hours to
reduce a possible distortion of the measured resonance due to variations of
the performance of the experimental setup. The injected pulse energy was
limited to 5 mJ to avoid significant saturation effects and optically-induced
damage of the cavity coating. Because a minimum pulse-to-pulse separation of
about 2 ms was imposed on the laser system for increased stability, about half
of the entering muons do not trigger the laser system. Yet these ”laser-off
events” are used to precisely measure the average background yielding the
yellow bands in Fig. 3.

The transition (1) took about two weeks of continuous measurement, the
transitions (2) and (3) have been measured at once, within about three weeks
of continuous data taking. For the first transition, on resonance we observed
a rate of 7-8 events/h including a background rate of $\sim$1.5 events/h.

The center frequency of the measured lines is obtained by fitting the data
with a line shape model. The model is a Lorentzian corrected for saturation
effects and variations of the laser pulse energy measured for every shot. It
can therefore only be evaluated at the position of each data point. This leads
to the distortion in the line shape seen in Fig. 3. From the line shape fit we
obtain following transition frequencies:

$\displaystyle\nu_{\rm exp}^{(1)}\equiv\nu(2P_{3/2}^{F=2}-2S_{1/2}^{F=1})$
$\displaystyle=347.212(20)^{\rm stat}(1)^{\rm sys}\,\mathrm{THz}$ (2)
$\displaystyle\nu_{\rm exp}^{(2)}\equiv\nu(2P_{3/2}^{F=1}-2S_{1/2}^{F=0})$
$\displaystyle=312.830(21)^{\rm stat}(1)^{\rm sys}\,\mathrm{THz}$ (3)
$\displaystyle\nu_{\rm exp}^{(3)}\equiv\nu(2P_{1/2}^{F=1}-2S_{1/2}^{F=1})$
$\displaystyle=310.814(20)^{\rm stat}(1)^{\rm sys}\,\mathrm{THz}.$ (4)

The fit was done with a fixed linewidth of $\Gamma=318.7$ GHz at FWHM. A
separate fit with a free width resulted in widths that agreed with the
theoretical one. Simply fitting Lorentzians yields line centers in agreement
with the ones from the line shape model. The measured transition frequencies
yield the energy splittings

$\displaystyle\Delta E_{\rm exp}^{(1)}$ $\displaystyle=1435.951(81)\,{\rm
meV}$ (5) $\displaystyle\Delta E_{\rm exp}^{(2)}$
$\displaystyle=1293.759(86)\,{\rm meV}$ (6) $\displaystyle\Delta E_{\rm
exp}^{(3)}$ $\displaystyle=1285.425(81)\,{\rm meV}$ (7)

via the relation $1\,\mathrm{meV}\,\widehat{=}\,241.798935\,\,\mathrm{GHz}$.

The uncertainty of about 20 GHz, corresponding to about $6\times
10^{-2}\,\Gamma$, is dominated by far by statistics. The largest systematic
uncertainty stems from an upper limit to quantum interference effects
Brown:2013:QI  ($<5\times 10^{-4}\,\Gamma=0.16$ GHz) Amaro:2015:muonicQI . The
uncertainty on the laser frequency is dominated by the chirp and the
calibration of the wavemeter and conservatively given by 0.1 GHz. We corrected
for the Zeeman shift caused by the 5T field by maximally 0.3 GHz (depending on
the transition) so that the uncertainty of this correction is negligibly small
in our context. Other systematic uncertainties (light shift, collisional
effects, Stark shift etc.) are negligible compared with the precision of the
measurement due to the strong binding, the small atomic size and the large
separation between energy levels.

##  IV Experimental results

From the three transition measurements between 2S and 2P states with different
fine and hyperfine sublevels, it is possible to determine three quantities: we
choose the Lamb shift $E_{\rm LS}=\Delta E(2P_{1/2}-2S_{1/2})$, the 2S
hyperfine splitting $E_{\rm HFS}$, and the 2P fine splitting $E_{\rm
FS}=\Delta E(2P_{3/2}-2P_{1/2})$. The relations between the measured
transition energies and these quantities are given by (see also Methods):

$\displaystyle\Delta E_{\rm exp}^{(1)}=E_{\rm LS}~{}-~{}\frac{1}{4}E_{\rm
HFS}+E_{\rm FS}-~{}\,9.23945(26)\;\mathrm{meV}\mbox{~{} ~{}}$ (8)
$\displaystyle\Delta E_{\rm exp}^{(2)}=E_{\rm LS}~{}+~{}\frac{3}{4}E_{\rm
HFS}+E_{\rm FS}+15.05305(44)\;\mathrm{meV}\mbox{~{} ~{}}$ (9)
$\displaystyle\Delta E_{\rm exp}^{(3)}=E_{\rm LS}~{}-~{}\frac{1}{4}E_{\rm
HFS}\hskip 28.45274pt-14.80851(18)\;\mathrm{meV}\mbox{~{} ~{}}$ (10)

where $E_{\rm HFS}<0$ and the numerical values of the last terms in Eqs.
(8-10) arise from the 2P fine and hyperfine splittings, and include the
contribution due to the mixing of the F=1 levels. These contributions can be
calculated with great precision, because the 2P wave function of the hydrogen-
like muonic He ion has negligible overlap with the nucleus, resulting in
negligible contributions from nuclear size and structure corrections.

Note that the two most recent theory papers by Karshenboim et al.,
Karshenboim:2017jtz  and Pachucki et al. Pachucki:2023:CODATA  use different
conventions for the definition of the $2P_{1/2}$ and $2P_{3/2}$ centroids,
which results in differing definitions of the Lamb shift. To obtain the
constant terms in Eqs. (8-10) we have used the 2P levels calculated by
Karshenboim et al., and modified them to account for the different
definitions, such that our final result for the Lamb shift follows the
convention of Pachucki et al. (see Methods).

We can solve the system of equations to obtain the experimental values of the
Lamb shift, the 2S HFS und the 2P fine splitting:

$\displaystyle E_{\rm LS}^{\rm exp}$
$\displaystyle=~{}1258.612(~{}86)\,\mathrm{meV}$ (11) $\displaystyle E_{\rm
HFS}^{\rm exp}$ $\displaystyle=-166.485(118)\,\mathrm{meV}$ (12)
$\displaystyle E_{\rm FS}^{\rm exp}$
$\displaystyle=~{}~{}144.958(114)\,\mathrm{meV}.$ (13)

The experimental value of the fine splitting $E_{\rm FS}^{\rm exp}$ is in
excellent agreement with predictions $E_{\rm FS}^{\rm
th}=144.979(5)\,\mathrm{meV}$ Karshenboim:2017jtz , demonstrating consistency
between our three muonic transitions measurements on the one hand, and the
correctness of the theory of the 2P levels on the other. Owing to its much
smaller uncertainty and consistency with measurements, we can use the theory
value of the fine splitting to solve the system of equations Eqs. (8)-(10) to
obtain improved values of the Lamb shift and 2S-HFS:

$\displaystyle E_{\rm LS}^{\rm exp}$
$\displaystyle=~{}1258.598(~{}48)^{\mathrm{exp}}(3)^{\mathrm{theo}}\,\mathrm{meV}$
(14) $\displaystyle E_{\rm HFS}^{\rm exp}$
$\displaystyle=-166.496(104)^{\mathrm{exp}}(3)^{\mathrm{theo}}\,\mathrm{meV}\,.$
(15)

The theoretical uncertainties are from the $\pm 0.005$ meV estimated higher-
order corrections to the fine structure in Ref. Karshenboim:2017jtz .

##  V The helion charge radius and the isotopic shift

The theory prediction of the Lamb shift has been recently updated accounting
for the contributions of various groups. It reads Pachucki:2023:CODATA

$\displaystyle E_{\rm LS}^{\mathrm{th}}(r_{h}^{2})=$ $\displaystyle
1644.348(8)\,\mathrm{meV}-103.383\,r_{h}^{2}\,\mathrm{meV}/\,\mathrm{fm}^{2}$
(16) $\displaystyle+15.499(378)\,\mathrm{meV}~{}\mbox{~{} ~{} ~{}}$

where the first term accounts for all QED corrections independent of the
nuclear structure, the term proportional to $r_{h}$2 accounts for the finite-
size correction including radiative corrections to it, and the last term is a
sum of all higher-order nuclear structure dependent contributions
Carlson:2016cii ; Ji:2018:nuclPOL ; Pachucki:2018yxe  which are dominated by
the nuclear two- and three-photon exchange contributions (2PE and 3PE,
respectively). Comparing this theory prediction with the measured Lamb shift
$E_{\rm LS}^{\rm exp}$ we obtain the rms charge radius of the helion

$\begin{split}r_{h}&=1.97007(12)^{\rm exp}(93)^{\rm
theo}\,\mathrm{fm}=1.97007(94)\,\mathrm{fm}.\end{split}$ (17)

This value is 15 times more precise than the previous best value from elastic
electron-3He scattering of 1.973(14) $\mathrm{fm}$ Sick:2014:HeZemach , and in
perfect agreement with it (see Fig. 2).

Figure 4: Recent determinations of the 3He nucleus (helion) charge radius. The
dark and light bands indicate the experimental and total uncertainty in our
measurement. The value of Sick of 1.973(14) fm Sick:2014:HeZemach  is from the
world data on elastic electron scattering. The other values are recent
predictions from nuclear few-body theory: Piarulli 1.962(4) fm Piarulli:2012bn
; Marcucci:2015rca , Nevo Dinur 1.979(10) fm NevoDinur:2019:zemach , and
LENPIC collaboration 1.955(34) fm PhysRevC.103.054001 ; LENPIC:2022cyu  222We
obtained this value using the point-proton structure radius and procedure
explained in Ref. PhysRevC.103.054001 .

Our value could be further improved by almost an order of magnitude by
advancing the predictions for the two-photon-exchange and three-photon-
exchange contributions both for the nucleus and the nucleons
Pachucki:2023:CODATA ; Antognini:2022xoo ; Ji:2018:nuclPOL .

It is interesting to compare this value with the helion charge radius as
obtained from most recent nuclear theories which uses chiral effective field
theory to describe the nuclear interaction and ab-initio methods to solve the
quantum-mechanical few-body problem. Figure. 2 shows some of the most recent
results taken from Ref. Marcucci:2015rca ; Piarulli:2012bn ; NevoDinur:2018hdo
; PhysRevC.103.054001 ; LENPIC:2022cyu  depicting an overall satisfactory
agreement between the measured value and the various predictions, and
highlighting the role of the helion charge radius as benchmark for precision
nuclear theory.

Figure 5: Squared charge radius difference $r_{h}^{2}-r_{\alpha}^{2}$ from
measurements of the isotope shift in ordinary He atoms Shiner:1995:heliumSpec
; Rooij:2011:HeSpectroscopy ; CancioPastor:2012:PRL108 ; Patkos:2016:HeIso ;
Zheng:2017:He_Iso ; Rengelink:2018:4He ; PhysRevA.101.062507 ;
vanderWerf:2023cuv , compared to our value from muonic ions. The dark and
light bands indicate the experimental and total uncertainty of our
determination. The values of Shiner Shiner:1995:heliumSpec  and Cancio Pastor
CancioPastor:2012:PRL108  have been corrected for improved theory calculations
PhysRevA.95.062510 , but may lack a systematic correction due to quantum
interference effects Brown:2013:QI , as suggested in Ref.
Marsman:2014:QIshifts . The value of Zheng Zheng:2017:He_Iso  may have to be
corrected for a systematic Doppler shift Wen:2023:PRA107 . The most recent
work from Amsterdam by van der Werf vanderWerf:2023cuv  supersedes the results
from van Rooij Rooij:2011:HeSpectroscopy  and Rengelink Rengelink:2018:4He  by
the same group.

Spectroscopy of ”normal” helium atoms can not yet provide precise values of
the helion and alpha-particle charge radii, given the present uncertainty of
the three-body atomic theory. Yet, in the isotopic shift, several
cancellations take place in the theory PhysRevA.95.062510  of the energy
levels, so that values of $r_{h}^{2}-r_{\alpha}^{2}$ Shiner:1995:heliumSpec ;
Rooij:2011:HeSpectroscopy ; CancioPastor:2012:PRL108 ; Patkos:2016:HeIso ;
Zheng:2017:He_Iso ; Rengelink:2018:4He  can be obtained where $r_{\alpha}$ is
the alpha particle (4He) charge radius. The scattering of the values obtained
so far shown in Fig. 5 however reveals some tensions that highlight the
challenges faced by both theory and experiments.

It is thus interesting to address this quantity by considering the isotopic
shift in muonic helium ions and presenting it in the form:
Pachucki:2023:CODATA

$r_{h}^{2}-r_{\alpha}^{2}=-\frac{E_{\mathrm{LS}}^{\mathrm{exp}}(\mu^{3}\mathrm{He}^{+})}{103.383\mathrm{\,\frac{meV}{fm^{2}}}}+\frac{E_{\mathrm{LS}}^{\mathrm{exp}}(\mu^{4}\mathrm{He}^{+})}{106.209\mathrm{\,\frac{meV}{fm^{2}}}}+0.2585(30)\,\mathrm{fm^{2}}$
(18)

to take advantage of some cancellations in the nuclear structure
contributions. Inserting the measured Lamb shift in $\mu^{4}$He+ of
$E_{\mathrm{LS}}^{\mathrm{exp}}(\mu^{4}\mathrm{He}^{+})=1378.521(48)$ meV
Krauth:2021foz  and $E_{\mathrm{LS}}^{\mathrm{exp}}(\mu^{3}\mathrm{He}^{+})$
from Eq. (11) into this expression Diepold:2016:mu4HeTheo ;
Pachucki:2023:CODATA  we obtain

$r_{h}^{2}-r_{\alpha}^{2}=1.0636(6)^{\rm exp}(30)^{\rm
theo}\,\mathrm{fm}^{2}.$ (19)

This value can be compared with various isotopic shift measurements obtained
in regular He atoms with two electrons Shiner:1995:heliumSpec ;
Rooij:2011:HeSpectroscopy ; CancioPastor:2012:PRL108 ; Patkos:2016:HeIso ;
Zheng:2017:He_Iso ; Rengelink:2018:4He ; PhysRevA.101.062507 , where long-
standing discrepancies exist, see Fig. 5. Several novel systematic shifts were
recently identified in these measurements Marsman:2014:QIshifts ;
Wen:2023:PRA107 ; vanderWerf:2023cuv , but nevertheless our result deviates by
3.6$\sigma$ from the most recent and most precise value by the Amsterdam group
vanderWerf:2023cuv . Our measurements in muonic helium ions do not require the
exquisite experimental accuracy of $10^{-12}$ reached in normal helium atoms,
and are not sensitive to systematic effects. On the other hand, our results
are limited by the nuclear structure effects which are much larger in muonic
systems Ji:2018:nuclPOL ; NevoDinur:2018hdo ; Hernandez:2019zcm ;
PhysRevC.103.024313 ; Acharya:2022drl ; Muli:2022jma . Clearly, more work is
required to understand this $3.6\sigma$ deviation.

For completeness we quote here the updated Lamb shift theory in $\mu^{4}$He+
from Ref. Pachucki:2023:CODATA

$\begin{split}E_{\rm
LS}^{\mathrm{th}}(r_{\alpha}^{2})=&1668.491(7)\,\mathrm{meV}-106.209r_{\alpha}^{2}\;\,\mathrm{meV}/\,\mathrm{fm}^{2}\\\
&+9.276(433)\,\mathrm{meV}\;.\end{split}$ (20)

Combined with the Lamb shift in $\mu^{4}$He+ we have measured in Ref.
Krauth:2021foz  this yields an updated 4He (alpha particle) charge radius of
$r_{\alpha}=1.6786(12)$ fm with a 45% larger uncertainty compared to our
previous determination which used the theory summarized in
Diepold:2016:mu4HeTheo . The new $r_{\alpha}$ is obtained using the two-
photon-exchange (TPE) contribution calculated solely from ab-initio theory
Ji:2018:nuclPOL  while previously we split it into a third-Zemach moment
(Friar radius) contribution obtained from electron elastic scattering and a
polarizability contribution from few-nucleon theories Ji:2018:nuclPOL .
Because this splitting could lead to some inconsistencies Pachucki:2023:CODATA
, we now opt for the solution fully based on few-body theories Ji:2018:nuclPOL
.

##  VI Nuclear-structure contribution for the 2S HFS

By comparing the measured 2S-HFS $E_{\rm HFS}^{\rm exp}$ in $\mu^{3}$He+ with
the corresponding theory prediction Franke:2016:mu3HeTheo

$E^{\rm th}_{\rm HFS}=-172.7457(89)\,\mathrm{meV}+E_{\rm HFS}^{\rm
nucl.struct.}$ (21)

we extract the nuclear structure-dependent contributions (2PE and higher
orders) to the 2S hyperfine splitting

$E^{\rm nucl.struct.}_{\rm HFS}=6.25(10)\,\mathrm{meV},$ (22)

with an uncertainty arising basically only from the statistical uncertainties
of the measurements. Subtracting the elastic part of the two-photon-exchange
contribution $\Delta E_{\rm 2PE}^{\rm Zemach}=2.5836\,{\rm
meV/fm}~{}r_{Z}=6.53(4)$ meV, where $r_{Z}=2.528(16)$ fm is the Zemach radius
of the 3He nucleus Sick:2014:HeZemach ; NevoDinur:2018hdo  from $E^{\rm
nucl.struct.}_{\rm HFS}$, we can obtain a value for the hitherto unknown
polarizability contribution to the $\mathrm{2S}$ hyperfine splitting of
$-0.28(10)\,\mathrm{meV}$ (that includes also higher-order contributions).
This represents an important benchmark to refine our understanding of the
magnetic structure of the 3He nucleus. It also allows, using appriopiate
scaling, to predict the nuclear structure contribution of the ground-state
hyperfine splitting in $\mu^{3}\mathrm{He}^{+}$ restricting considerably the
range where this reasonance has to be searched for.

##  VII Conclusion and Outlook

Laser spectroscopy of $\mu^{3}$He+ and $\mu^{4}$He+ Krauth:2021foz  ions
provides precision values of 3He and 4He charge radii and two-photon-exchange
contributions, that serve as benchmarks Ekstrom:2015:PRC91 ; Lynn:2017:PRC ;
Vanasse:2018:PRC  for few-body ab-initio nuclear theories Hammer:2020:RMP ,
driving the theory to new levels of precision. The nuclear theory
Hammer:2020:RMP  is challenged to scrutinize its approximations
Marcucci:2015rca , to systematically improve the nuclear interaction
Epelbaum:2019jbv ; Krebs:2019aka ; PhysRevLett.126.092501 ; Ekstrom:2015:PRC91
; Lynn:2017:PRC ; Vanasse:2018:PRC  and the formalism Ji:2018:nuclPOL ;
NevoDinur:2018hdo ; Hernandez:2019zcm ; PhysRevC.103.024313  while finding
novel methods to evaluate the uncertainties Acharya:2022drl ; Muli:2022jma .

The precise knowledge of $r_{h}$ and $r_{\alpha}$ reduces the uncertainty of
the nuclear-structure-dependent contributions in He and He+ Moreno:2023amv .
This paves the way for bound-state QED tests for two-body (He+) and three-body
(He) systems to an unprecedented level of accuracy Karshenboim:2019iuq ;
Yerokhin:2019:LS ; Antognini:2022xoo . The Lamb shift theory in He has
undergone a spectacular advance in recent years, completing the calculation up
to terms of order $\alpha^{7}m$ Patkos:2021wam . Once the remaining
discrepancies are solved there, electronic helium atoms and ions will provide
more accurate charge radii Krauth:2019:HePlus ; Moreno:2023amv , because of
the smaller higher-order nuclear structure effects in these electronic
systems. Using Eqs. (16),(20), our measurements in muonic He ions will then
yield the most precise experimental values for the nuclear structure
contributions (beyond the leading finite-size effect). Finally, of course,
comparison of results from electronic vanderWerf:2023cuv  and muonic systems
can be used to search for physics beyond the Standard Model Antognini:2022xoo
.

##  VIII Methods

To arrive at Eqs. (8-10) we used the most recent calculations for the 2P level
structure by Karshenboim et al. Karshenboim:2017jtz , and of the 2S Lamb shift
by Pachucki et al. Pachucki:2023:CODATA .

However, these papers use a different convention for the definition of the
$2P_{1/2}$ and $2P_{3/2}$ level centroids, and of the Lamb shift. These
differences come from the treatment of the Barker-Glover (BG) and Brodsky-
Parsons (BP) terms.

Both terms are not part of what Karshenboim et al. Karshenboim:2017jtz  denote
as ”Lamb shift”, but it is named ”Unperturbed quantum mechanics” therein, and
their sum is -0.0032 meV (Tab. X No. 0), which is the sum of the ”BP*” and
”BG*” terms listed in Tab.II, therein.

The ”BG*” term of 0.1265 meV in Ref. Karshenboim:2017jtz  is also already
included in the Lamb shift in Pachucki et al. Pachucki:2023:CODATA , term as
”$(Z\alpha)^{4}$ recoil”, term III.C, hence this term contributes equally to
the $2P_{1/2}$-$2S_{1/2}$ energy difference in both conventions.

The ”BP*” term of -0.1298 meV, however, is not considered in Pachucki et al.
Pachucki:2023:CODATA , because it only arises because of the hyperfine level
mixing of the F=1 levels.

Thus, to make the Lamb shift of Pachucki et al. Pachucki:2023:CODATA
compatible with the definitions of the 2P levels in Karshenboim et al.
Karshenboim:2017jtz , we define the position of our $2P_{1/2}$ centroid as

$\displaystyle E(2P_{1/2})=E(2S_{1/2})+E_{\mathrm{LS}}-0.1298\,{\rm meV}\,,$
(23)

where 0.1298 meV is the ”BP*” term in Ref. Karshenboim:2017jtz , Tab.II, No.
0.4. This ensures that our measured Lamb shift $E_{\mathrm{LS}}$ agrees with
the convention of Pachucki, while the FS and 2P-HFS splittings follow the
convention of Karshenboim, where the $2P_{1/2}$ and $2P_{3/2}$ levels are the
weighted average of the physical hyperfine levels.
This gives for the transition energies we measured

$\displaystyle\Delta E_{\rm exp}^{(1)}$ $\displaystyle=E_{\rm LS}+E_{\rm
BP*}-\frac{1}{4}E_{\rm HFS}+E_{\rm FS}+\frac{3}{8}E_{\rm HFS}^{(P3/2)}$ (24)
$\displaystyle\Delta E_{\rm exp}^{(2)}$ $\displaystyle=E_{\rm LS}+E_{\rm
BP*}+\frac{3}{4}E_{\rm HFS}+E_{\rm FS}-\frac{5}{8}E_{\rm HFS}^{(P3/2)}$ (25)
$\displaystyle\Delta E_{\rm exp}^{(3)}$ $\displaystyle=E_{\rm LS}+E_{\rm
BP*}-\frac{1}{4}E_{\rm HFS}\hskip 28.45274pt+\frac{1}{4}E_{\rm
HFS}^{(P1/2)}\,,$ (26)

where $E_{\rm HFS}^{(P1/2)}=-58.7150(7)$ meV and $E_{\rm
HFS}^{(P3/2)}=-24.2925(7)$ meV are the 2P hyperfine level splittings
Karshenboim:2017jtz  and $E_{\mathrm{BP*}}=-3/4*0.17302(2)$ meV (see Tab. XIV
in Ref. Karshenboim:2017jtz ) is the above-mentioned ”BP*” term of $-0.1298$
meV Karshenboim:2017jtz .

##  IX Acknowledgments

This paper is dedicated to Ingo Sick and Wim Vassen. We thank Sonia Bacca,
Kjeld Eikema, Evgeny Epelbaum, Franziska Hagelstein, Savely Karshenboim,
Simone Li Muli, Vadim Lensky, Laura Marcucci, Ulf Meißner, Krzysztof Pachucki,
Leo Simons, and Kees Steinebach for fruitful discussions. We acknowledge
support from the European Research Council (ERC) through StG. #279765, and
CoG. #725039; from the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy EXC PRISMA+ (390831469),
European Union’s Horizon 2020 research and innovation programme under grant
agreement STRONG – 2020 - No 824093; from FEDER and FCT via project PTDC/FIS-
NUC/0843/2012, from FCT Contract No. SFRH/BPD/76842/2011, and
SFRH/BPD/92329/2013, SFRH/BD/52332/2013; from DFG_GR_3172/9-1, from SNF
projects 2200021L_138175, 200020_159755, 200021_165854, 200020_197052 and by
the ETH Femtosecond and Attosecond Science and Technology (ETH-FAST)
initiatives as part of the NCCR MUST program.

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