Measuring the α-particle charge radius with muonic helium-4 ions

Abstract

The energy levels of hydrogen-like atomic systems can be calculated with great precision. Starting from their quantum mechanical solution, they have been refined over the years to include the electron spin, the relativistic and quantum field effects, and tiny energy shifts related to the complex structure of the nucleus. These energy shifts caused by the nuclear structure are vastly magnified 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 puzzle^2-5, in line with recent determinations of the proton charge radius^6-9, and establishes spectroscopy of light muonic atoms and ions as a precise tool for studies of nuclear properties.

Fig. 1. Energy-level scheme and experimental setup.

Left: energy levels of interest in (μ⁴He)⁺. We drive the 2S → 2P transitions ν₁ and ν₂ (at wavelengths of 813 nm and 899 nm, respectively) and measure the 8.2-keV Lyman-α X-ray from the subsequent decay to the 1S_1/2 ground state. Indicated are the Lamb shift (LS) and the shift due to the finite nuclear size (FNS), which is proportional to $r_{\mathrm{\alpha }}^2$. Right: sketch of the experimental setup (not to scale). On the way to the He target, the muon is detected, thereby triggering the laser system. After the muon is stopped in 2 mbar of He gas at room temperature, (μ⁴He)⁺ is formed. About 1 μs after the trigger, the laser pulse arrives at the target, is coupled into the multipass cavity and distributed over the entire muon stop volume (hatched area). The pulse is produced by a Ti:Sa oscillator seeded by a continuous-wave (CW) Ti:Sa laser and pumped by a frequency-doubled pulsed thin-disk laser. The continuous-wave Ti:Sa laser is stabilized to a Fabry–Pérot (FP) cavity and referenced to a wavemeter. The Lyman-α X-rays are measured via LAAPDs (not shown) mounted above and below the cavity. SHG, second harmonic generation.

Fig. 2. The measured transitions.

The 2S → P_3/2 (left) and 2S → 2P_1/2 (right) resonances in (μ⁴He)⁺ fitted with a power-broadened Lorentzian line-shape at a fixed linewidth of 319 GHz (FWHM) given by the 2P lifetime. The black data points show the laser-induced events (number of X-rays in time coincidence with the laser light), normalized to the prompt events (number of X-rays from the cascade on formation of (μ⁴He)⁺). The horizontal band shows the background levels (with 1σ uncertainty) obtained from measured data where the laser was not triggered. The tiny bars above the resonances show the 1σ uncertainty of the fitted resonance position. Indicated are the erroneous claims from refs. ^, the hatched exclusion region from ref. and the expected resonance position using r_α from e–He scattering. The latter is compared to our radius from (μ⁴He)⁺ in the inset where the inner and outer error bands represent the experimental and total uncertainty, respectively (see equation (7)). a.u., arbitrary units.