Hyperfine Splitting In Muonic Hydrogen Research Articles

The muon beam monitor for the FAMU experiment: design, simulation, test, and operation

FAMU is an INFN-led muonic atom physics experiment based at the RIKEN-RAL muon facility at the ISIS Neutron and Muon Source (United Kingdom). The aim of FAMU is to measure the hyperfine splitting in muonic hydrogen to determine the value of the proton Zemach radius with an accuracy better than 1%. The experiment has a scintillating-fibre hodoscope for beam monitoring and data normalisation. In order to carry out muon flux estimation, low-rate measurements were performed to extract the single-muon average deposited charge. Then, detector simulation in Geant4 and FLUKA allowed a thorough understanding of the single-muon response function, which is crucial for determining the muon flux. This work presents the design features of the FAMU beam monitor, along with the simulation and absolute calibration measurements in order to enable flux determination and enable data normalisation.

Status of the detector setup for the FAMU experiment at RIKEN-RAL for a precision measurement of the Zemach radius of the proton in muonic hydrogen

The FAMU experiment at RIKEN-RAL is a muonic atom experiment with the aim to determine the Zemach radius of the proton by measuring the 1s hyperfine splitting in muonic hydrogen. The activity of the FAMU Collaboration in the years 2015–2023 enabled the final optimisation of the detector-target setup as well as the gas working condition in terms of temperature, pressure and gas mixture composition. The experiment has started its data taking in July 2023. The status of the detector setup for the 2023 experimental runs, for the beam characterisation and muonic X-ray detection in the 100–200 keV energy range, is presented and discussed.

Diffusion of muonic hydrogen in hydrogen gas and the measurement of the 1$s$ hyperfine splitting of muonic hydrogen

The CREMA collaboration is pursuing a measurement of the ground-state hyperfine splitting (HFS) in muonic hydrogen (\muμp) with 1 ppm accuracy by means of pulsed laser spectroscopy. In the proposed experiment, the \muμp atom is excited by a laser pulse from the singlet to the triplet hyperfine sub-levels, and is quenched back to the singlet state by an inelastic collision with a H_22 molecule. The resulting increase of kinetic energy after this cycle modifies the \muμp atom diffusion in the hydrogen gas and the arrival time of the \muμp atoms at the target walls. This laser-induced modification of the arrival times is used to expose the atomic transition. In this paper we present the simulation of the \muμp diffusion in the H_22 gas which is at the core of the experimental scheme. These simulations have been implemented with the Geant4 framework by introducing various low-energy processes including the motion of the H_22 molecules, i.e. the effects related with the hydrogen target temperature. The simulations have been used to optimize the hydrogen target parameters (pressure, temperatures and thickness) and to estimate signal and background rates. These rates allow to estimate the maximum time needed to find the resonance and the statistical accuracy of the spectroscopy experiment.

Experimental determination of the energy dependence of the rate of the muon transfer reaction from muonic hydrogen to oxygen for collision energies up to 0.1 eV

We report the experimental determination of the collision-energy dependence of the muon transfer rate from the ground state of muonic hydrogen to oxygen at near-thermal energies. A sharp increase by nearly an order of magnitude in the energy range 0--70 meV was found that is not observed in other gases. The results set a reliable reference for quantum-mechanical calculations of low-energy processes with exotic atoms and provide firm ground for the measurement of the hyperfine splitting in muonic hydrogen and the determination of the Zemach radius of the proton by the FAMU collaboration.

Axial-vector exchange contribution to the hyperfine splitting

We revisit the contribution of axial-vector mesons to the hyperfine splitting of muonic hydrogen. We focus our attention on the doubly-virtual asymptotic behavior of the relevant form factors of axial-vector mesons, together with their coupling to nucleons based on resonance saturation and short-distance constraints. Among others, we find significant differences with respect to previous studies, including an opposite sign and a $\sim50\%$ effect of the doubly-virtual high-energy behavior.

Tensor meson contribution to the Lamb shift and hyperfine splitting in muonic hydrogen

We calculate the tensor meson exchange contribution to the interaction operator of muon and proton, which is determined by the tensor meson coupling with two photon state. For the construction of transition form factor T → γγ we use the monopole parametrization over photon four-momenta and experimental data on the decay width ΓTγγ. It is shown that tensor meson f2(1270) exchange gives essential contribution to the Lamb shift in muonic hydrogen ΔΕLs (2P – 2S) which should be taken into account for a comparison with precise experimental data.

Excitation probability of the hyperfine ground state of Doppler-broadened muonic hydrogen through dipole-forbidden transition

We study the magnetic dipole induced excitation probability of the hyperfine ground state of Doppler-broadened muonic hydrogen (pμ−) by a single nanosecond laser pulse with pulse durations of 20, 2, and 0.2 ns pulses with a Gaussian temporal shape such that the pulse bandwidth is, respectively, narrower than, comparable to, and broader than the Doppler width at 300 K. We numerically solve a set of density matrix equations to obtain the excitation probability at the various laser intensities and detunings. For the resonant 20 ns pulse, the excitation probability increases very slowly as the laser intensity increases. The width of the excitation probability lineshape mainly comes from the Doppler broadening and hardly increases with intensity. For the resonant 0.2 ns pulse a complete population inversion occurs at a certain laser intensity, and the excitation probability oscillates between 0 and unity as the intensity further increases. Owing to the inherent pulse bandwidth the width of the excitation probability lineshape is very broad even at the low intensity, and it increases further as the intensity increases. For the resonant 2 ns pulse at a moderate intensity (∼1011 W/cm2) the excitation probability is higher than those by the 20 and 0.2 ns pulses with a relatively narrow width. Our study can serve as a guideline for the development of the relevant laser source, depending on the purpose to excite muonic hydrogen, i.e., precise measurement of the ground state hyperfine splitting of muonic hydrogen or production of spin-polarized muonic hydrogen.

Measurement of the proton Zemach radius from the hyperfine splitting in muonic hydrogen atom

Muonic hydrogen is a bound state of a proton and a negative muon. Its Bohr radius is 200 times smaller than that of an electronic hydrogen atom. Therefore, a spectroscopy of the muonic hydrogen is highly sensitive to the finite size effect of proton. Recent years, the proton charge radius was determined by the laser spectroscopy of the Lamb shifts in muonic hydrogen atom. The experiment determined the proton charge radius significantly smaller than the results of past measurements. This anomaly is called “proton radius puzzle” and it has been an important unsolved problem in subatomic physics. Towards solving the puzzle, a new measurement of the ground-state hyperfine splitting in muonic hydrogen was proposed. The hyperfine splitting of muonic hydrogen derives the proton Zemach radius, which is defined as a convolution of the charge distribution with the magnetic moment distribution. This experiment aims to determine the proton Zemach radius with 1% precision by a measurement of the decay electron angular asymmetry. In order to test the feasibility of the laser spectroscopy, a preliminary experiment to measure the hyperfine quenching rate was proposed.

The proton size puzzle: experiment vs theory.

Current status of the proton size puzzle from experimental and theoretical points of view is briefly discussed. The interest to these studies is primarily related to experiments conducted by the CREMA collaboration (Charge Radius Experi- ments with Muonic Atoms) with muonic hydrogen and deuterium by methods of laser spectroscopy. As a result a more accurate value of the proton charge radius was found to be rp = 0:84184(67) fm, which is different from the value recommended by CODATA for 7σ. In the second part we discuss recent calculations of the contribution of light pseudoscalar (PS) and axial-vector (AV) mesons to the interaction operator of a muon and a proton in muonic hydrogen atom, with the coupling of mesons to the muon being via two-photon intermediate state. Numerical estimates of the contributions to the hyperfine structure of the spectrum of the S and P levels are presented. It is shown that such contribution to the hyperfine splitting in muonic hydrogen is rather important for a comparison with precise experimental data.

Two-photon exchange correction to the hyperfine splitting in muonic hydrogen

We reevaluate the Zemach, recoil and polarizability corrections to the hyperfine splitting in muonic hydrogen expressing them through the low-energy proton structure constants and obtain the precise values of the Zemach radius and two-photon exchange (TPE) contribution. The uncertainty of TPE correction to S energy levels in muonic hydrogen of 105 ppm exceeds the ppm accuracy level of the forthcoming 1S hyperfine splitting measurements at PSI, J-PARC and RIKEN-RAL.

The contribution of pseudoscalar and axial-vector mesons to hyperfine structure of muonic hydrogen

In the framework of the quasipotential method in quantum electrodynamics we calculate the contribution of light pseudoscalar (PS) and axial-vector (AV) mesons to the interaction operator of a muon and a proton in muonic hydrogen atom. The coupling of mesons with the muon is via two-photon intermediate state. The parametrization of the transition form factor of two photons into PS and AV mesons, based on the experimental data on the transition form factors and QCD asymptotics, is used. Numerical estimates of the contributions to the hyperfine structure of the spectrum of the S and P levels are presented. It is shown that such contribution to the hyperfine splitting in muonic hydrogen is rather important for a comparison with precise experimental data.

The contribution of axial-vector mesons to hyperfine structure of muonic hydrogen

The contribution from the axial-vector meson exchange to the potential of the muon–proton interaction in muonic hydrogen induced by anomalous axial-vector meson coupling to two photon state is calculated. It is shown that such contribution to the hyperfine splitting in muonic hydrogen is large and important for a comparison with precise experimental data. In the light of our result, the proton radius “puzzle” is discussed.

Model-independent determination of the two-photon exchange contribution to hyperfine splitting in muonic hydrogen

We obtain a model-independent prediction for the two-photon exchange contribution to the hyperfine splitting in muonic hydrogen. We use the relation of the Wilson coefficients of the spin-dependent dimension-six four-fermion operator of NRQED applied to the electron-proton and to the muon-proton sectors. Their difference can be reliably computed using chiral perturbation theory, whereas the Wilson coefficient of the electron-proton sector can be determined from the hyperfine splitting in hydrogen. This allows us to give a precise model-independent determination of the Wilson coefficient for the muon-proton sector, and consequently of the two-photon exchange contribution to the hyperfine splitting in muonic hydrogen, which reads delta {overline{E}}_{pmu, mathrm{H}mathrm{F}}^{mathrm{TPE}}(nS)=-frac{1}{n^3}1.161(20) meV. Together with the associated QED analysis, we obtain a prediction for the hyperfine splitting in muonic hydrogen that reads Epμ,HFth(1S) = 182.623(27) meV and Epμ,HFth(2S) = 22.8123(33) meV. The error is dominated by the two-photon exchange contribution.

Single pion contribution to the hyperfine splitting in muonic hydrogen

A detailed discussion of the long-range one-pion exchange (Yukawa potential) contribution to the 2S hyperfine splitting in muonic hydrogen which had, until recently, been disregarded is presented. We evaluate the relevant vertex amplitudes, in particular $\pi^0\mu^+\mu^-$, combining low energy chiral expansions together with experimental data on $\pi^0$ and $\eta$ decays into two leptons. A value of $\Delta{E}^\pi_{HFS}= -(0.09 \pm 0.06)\ \mu\rm{eV}$ is obtained for this contribution.

Hyperfine splitting in muonic hydrogen constrains new pseudoscalar interactions

We constrain the possibility of a new pseudoscalar coupling between the muon and proton using a recent measurement of the 2S hyperfine splitting in muonic hydrogen.

Toward the measurement of the hyperfine splitting in the ground state of muonic hydrogen

The recent Lamb shift experiment at PSI and the controversy about proton size revived the interest in measuring the hyperfine splitting in muonic hydrogen and extracting the proton Zemach radius. The efficiency of the experimental method depends on the energy dependence of the muon transfer rate to higher-Z gases in the near epithermal energy range. As long as the available experimental data only give the average transfer rate in the whole epithermal range, and the detailed theoretical calculations have not yet been verified, an experiment has been started for the measurement of the transfer rate in thermalized gas target at different temperatures and extracting from the data an estimate of the transfer rate for arbitrary energies. We outline the underlying mathematical method and estimate its accuracy.

Theoretical and computational study of the energy dependence of the muon transfer rate from hydrogen to higher-Z gases

The recent PSI Lamb shift experiment and the controversy about proton size revived the interest in measuring the hyperfine splitting in muonic hydrogen as an alternative possibility for comparing ordinary and muonic hydrogen spectroscopy data on proton electromagnetic structure. This measurement critically depends on the energy dependence of the muon transfer rate to heavier gases in the epithermal range. The available data provide only qualitative information, and the theoretical predictions have not been verified. We propose a new method by measurements of the transfer rate in thermalized target at different temperatures, estimate its accuracy and investigate the optimal experimental conditions.

Theory of the 2S–2P Lamb shift and 2S hyperfine splitting in muonic hydrogen

The 7σ discrepancy between the proton rms charge radius from muonic hydrogen and the CODATA-2010 value from hydrogen spectroscopy and electron-scattering has caused considerable discussions. Here, we review the theory of the 2S–2P Lamb shift and 2S hyperfine splitting in muonic hydrogen combining the published contributions and theoretical approaches. The prediction of these quantities is necessary for the determination of both proton charge and Zemach radii from the two 2S–2P transition frequencies measured in muonic hydrogen; see Pohl et al. (2010) [9] and Antognini et al. (2013) [71].

Proton-structure corrections to hyperfine splitting in muonic hydrogen

We present the derivation of the formulas for the proton structure-dependent terms in the hyperfine splitting of muonic hydrogen. We use compatible conventions throughout the calculations to derive a consistent set of formulas that reconcile differences between our results and some specific terms in earlier work. Convention conversion corrections are explicitly presented, which reduce the calculated hyperfine splitting by about $46$ ppm. We also note that using only modern fits to the proton elastic form factors gives a smaller than historical spread of Zemach radii and leads to a reduced uncertainty in the hyperfine splitting. Additionally, hyperfine splittings have an impact on the muonic hydrogen Lamb shift and proton radius measurement, however the correction we advocate has a small effect there.

“Hyperfine splitting in muonic hydrogen: QED corrections of the α2 order,”

“Hyperfine splitting in muonic hydrogen: QED corrections of the α2 order,”