Article Measuring the α-particle charge radius with muonic helium-4 ions https://doi.org/10.1038/s41586-021-03183-1 Julian J. Krauth1,2,12 ✉, Karsten Schuhmann3,4, Marwan Abdou Ahmed5, Fernando D. Amaro6, Pedro Amaro7, François Biraben8, Tzu-Ling Chen9, Daniel S. Covita10, Andreas J. Dax4, Received: 3 July 2020 Marc Diepold1, Luis M. P. Fernandes6, Beatrice Franke1,13, Sandrine Galtier8,14, Accepted: 24 November 2020 Andrea L. Gouvea6, Johannes Götzfried1, Thomas Graf5, Theodor W. Hänsch1,11, Jens Hartmann11, Malte Hildebrandt4, Paul Indelicato8, Lucile Julien8, Klaus Kirch3,4, Published online: 27 January 2021 Andreas Knecht4, Yi-Wei Liu9, Jorge Machado7, Cristina M. B. Monteiro6, Françoise Mulhauser1, Open access Boris Naar4, Tobias Nebel1, François Nez8, Joaquim M. F. dos Santos6, José Paulo Santos7, Csilla I. Szabo8,15, David Taqqu3,4, João F. C. A. Veloso10, Jan Vogelsang1,16, Andreas Voss5, Check for updates Birgit Weichelt5, Randolf Pohl1,2 ✉, Aldo Antognini3,4 ✉ & Franz Kottmann3,4 The energy levels of hydrogen-like atomic systems can be calculated with great precision. Starting from their quantum mechanical solution, they have been refned over the years to include the electron spin, the relativistic and quantum feld efects, and tiny energy shifts related to the complex structure of the nucleus. These energy shifts caused by the nuclear structure are vastly magnifed in hydrogen-like systems formed by a negative muon and a nucleus, so spectroscopy of these muonic ions can be used to investigate the nuclear structure with high precision. Here we present the measurement of two 2S–2P transitions in the muonic helium-4 ion that yields a precise determination of the root-mean-square charge radius of the α particle of 1.67824(83) femtometres. This determination from atomic spectroscopy is in excellent agreement with the value from electron scattering1, but a factor of 4.8 more precise, providing a benchmark for few-nucleon theories, lattice quantum chromodynamics and electron scattering. This agreement also constrains several beyond-standard-model theories proposed to explain the proton-radius puzzle2–5, in line with recent determinations of the proton charge radius6–9, and establishes spectroscopy of light muonic atoms and ions as a precise tool for studies of nuclear properties. The α particle is the nucleus of the helium-4 (4He) atom and it con- In atomic spectroscopy, I = 0 results in the absence of any hyperfine sists of two protons and two neutrons, tightly bound by the strong structure, which simplifies the interpretation of atomic spectra sub- nuclear force. It is one of the most-studied atomic nuclei and its prop- stantially. However, so far, no determination of rα exists from atomic erties are of great importance for understanding the nuclear forces spectroscopy. In fact, so far, the only absolute radii determined by and the development of modern nuclear physics10,11. Its simplicity laser spectroscopy are for the proton and the deuteron7,8,12, because the makes the α particle a favourable target for a variety of precision required combination of sufficiently precise measurements and theory studies: its zero nuclear spin (I = 0) means that it can be described calculations exists only for atomic hydrogen (H) and deuterium (D) by a simple charge distribution, without magnetization distribu- (atomic number Z = 1). For elements with Z > 1, laser spectroscopy has tion, quadrupole or higher moments. In elastic electron scattering, yielded only differences of charge radii within an isotopic chain13–18 by the α particle can thus be described by a single charge-monopole measuring the same atomic transition in various isotopes to eliminate form factor. From the analysis of electron scattering world data, a the common energy shifts related to the interaction among electrons. root-mean-square (r.m.s.) charge radius of the α particle rα = 1.681(4) For the determination of absolute radii from He atoms (three-body fm was extracted1, where the value in brackets indicates the 1σ uncer- system with two electrons), theory is not yet advanced enough19. Suffi- tainty in the last digit(s). ciently precise experiments with the H-like He+ ion, where the two-body 1Max Planck Institute of Quantum Optics, Garching, Germany. 2QUANTUM, Institut für Physik & Exzellenzcluster PRISMA, Johannes Gutenberg-Universität Mainz, Mainz, Germany. 3Institute for Particle Physics and Astrophysics, ETH Zurich, Zurich, Switzerland. 4Paul Scherrer Institute, Villigen, Switzerland. 5Institut für Strahlwerkzeuge, Universität Stuttgart, Stuttgart, Germany. 6LIBPhys-UC, Department of Physics, University of Coimbra, Coimbra, Portugal. 7Laboratory for Instrumentation, Biomedical Engineering and Radiation Physics (LIBPhys-UNL), Department of Physics, NOVA School of Science and Technology, NOVA University Lisbon, Caparica, Portugal. 8Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Research University, Collège de France, Paris, France. 9Physics Department, National Tsing Hua University, Hsincho, Taiwan. 10i3N, Universidade de Aveiro, Aveiro, Portugal. 11Ludwig-Maximilians-Universität, Fakultät für Physik, Munich, Germany. 12Present address: LaserLaB, Department of Physics and Astronomy, Vrije Universiteit, Amsterdam, The Netherlands. 13Present address: TRIUMF, Vancouver, British Columbia, Canada. 14Present address: Institut Lumière Matière, University of Lyon, Université Claude Bernard Lyon 1, CNRS, Villeurbanne, France. 15Present address: Theiss Research, La Jolla, CA, USA. 16Present address: Department of Physics, Lund University, Lund, Sweden. ✉e-mail: [email protected]; [email protected]; [email protected] Nature | Vol 589 | 28 January 2021 | 527 Article 20 cm 2P (P4He)+ 3/2 Setup 2P 1/2 Cavity – Muon detection Q1 P Laser LS Q2 20 m 70 ns, 3.9 mJ Target: Trigger He at 2 mbar 2S1/2 FNS O/2 813 nm, 899 nm Disk laser Disk laser Ti:Sa CW Ti:Sa oscillator amplier oscillator CW seed 8.2 keV 1,030 nm 515 nm FP X-ray SHG O meter 1S1/2 Pump laser Spectroscopy laser Fig. 1 | Energy-level scheme and experimental setup. Left: energy levels of About 1 μs after the trigger, the laser pulse arrives at the target, is coupled into 4 + interest in (μ He) . We drive the 2S → 2P transitions ν1 and ν2 (at wavelengths of the multipass cavity and distributed over the entire muon stop volume 813 nm and 899 nm, respectively) and measure the 8.2-keV Lyman-α X-ray from (hatched area). The pulse is produced by a Ti:Sa oscillator seeded by a the subsequent decay to the 1S1/2 ground state. Indicated are the Lamb shift (LS) continuous-wave (CW) Ti:Sa laser and pumped by a frequency-doubled pulsed 2 and the shift due to the finite nuclear size (FNS), which is proportional to rα. thin-disk laser. The continuous-wave Ti:Sa laser is stabilized to a Fabry–Pérot Right: sketch of the experimental setup (not to scale). On the way to the He (FP) cavity and referenced to a wavemeter. The Lyman-α X-rays are measured target, the muon is detected, thereby triggering the laser system. After the via LAAPDs (not shown) mounted above and below the cavity. SHG, second muon is stopped in 2 mbar of He gas at room temperature, (μ4He)+ is formed. harmonic generation. theory of H is applicable, will soon be available20,21. Thus, the present contribution computed using a state-of-the-art ab initio few-nucleon work, with a muonic ion, provides the first, to our knowledge, deter- approach26. mination of a nuclear charge radius from laser spectroscopy beyond The last term in equation (1) is the sum of the calculated elastic and the proton and the deuteron. the unknown inelastic three-photon exchange (3PE) contributions. The In light muonic atoms and ions, a single muon orbits a bare nucleus. former was used to estimate the latter, considering the cancellation 27 Owing to the large muon mass mμ ≈ 200me, where me denotes the elec- of the two terms observed in muonic deuterium , as detailed in the tron mass, the muon’s Bohr radius is smaller than the electron’s Bohr Methods. We assign to the sum a conservative 100% uncertainty (1σ). orbit in the corresponding H-like ion by a factor of about 200. This The 2P fine structure23 results in a roughly 2003 ≈ 8 million times increased overlap of the Δ=Etheo 146.1828(3)meV muon’s wave function with the nucleus and a correspondingly increased 2P3/21−2P /2 (2) sensitivity to nuclear properties, such as the nuclear charge radius. This finite extension of the nucleus modifies the so-called Lamb shift22, is about half as large as the finite-size effect and can be calculated with which is the energy difference between the 2S1/2 and 2P1/2 states. Here great precision due to the absence of both the hyperfine structure and we present the first measurement of the 2P–2S energy splittings in the leading-order nuclear finite-size effects. the H-like muonic He ion (μ4He)+. Combined with the corresponding The experimental determination of the Lamb shift and the fine theoretical prediction, our measurement yields a precise determina- structure of (μ4He)+ reported here follows the technique of our pre- 2,3,6 tion of the α-particle charge radius rα. vious muonic H and muonic D measurements . About 500 muons The lowest atomic levels in (μ4He)+ are sketched in Fig. 1 (left). The per second from the world’s most intense beam of negative muons at Lamb shift is dominated by pure quantum electrodynamics (QED) ultralow energy (a few kiloelectronvolts) at the πE5 beam line of the effects, in particular, vacuum polarization, which is vastly enhanced Paul Scherrer Institute (Switzerland) are stopped in 2 mbar of He gas in muonic atoms23 (see Methods), but the effect of the finite nuclear at room temperature. The slowing down in the He gas occurs through size amounts to as much as 20% of the total energy splitting.