Muonium Hyperfine Splitting Research Articles

Update of Muonium 1S–2S transition frequency

We present an updated value of the Muonium 1S–2S transition frequency, highlighting contributions from different QED corrections as well as the large uncertainty in the Dirac contribution, stemming from the uncertainty of the electron to muon mass ratio. Improving the measurement of this spectral line would allow to extract a more accurate determination of fundamental constants, such as the electron to muon mass ratio or, combined with the Muonium hyperfine splitting, an independent value of the Rydberg constant. Furthermore, we report on the current status of the Mu-MASS experiment, which aims at measuring the Muonium 1S–2S transition frequency at a 10hbox {kHz} uncertainty level.Graphic abstract

Precise measurement of the hyperfine splitting in muonium with a high intensity pulsed muon beam at J-PARC

At J-PARC, the MuSEUM (Muonium Spectroscopy Experiment Using Microwave) collaboration aims to precisely measure the ground-state hyperfine splitting of muonium atoms arising from the muon and electron spins. The pulsed muon beam is stopped in a krypton gas cell to form muonium atoms. The transitions of spin states are induced with a microwave cavity, which are then measured by positron counters. After the previously performed successful measurements with a nearly-zero magnetic field, we are currently planning a measurement with the 2.9T magnetic field by measuring two Zeeman-split sub-levels, so that increased statistics will allow us to more precisely determine the transition frequency down to ∼1ppb. Moreover, a new microwave cavity with a unique geometry is being designed to perform the measurement at an even stronger field of 2.9T in the future.

Measurement of the transition frequency from 2S1/2, F = 0 to 2P1/2, F = 1 states in Muonium

Muons are puzzling physicists since their discovery when they were first thought to be the meson predicted by Yukawa to mediate the strong force. The recent result at Fermilab on the muon g-2 anomaly puts the muonic sector once more under the spotlight and calls for further measurements with this particle. Here, we present the results of the measurement of the 2S1/2, F = 0 → 2P1/2, F = 1 transition in Muonium. The measured value of 580.6(6.8) MHz is in agreement with the theoretical calculations. A value of the Lamb shift of 1045.5(6.8) MHz is extracted, compatible with previous experiments. We also determine the 2S hyperfine splitting in Muonium to be 559.6(7.2) MHz. The measured transition being isolated from the other hyperfine levels holds the promise to provide an improved determination of the Muonium Lamb shift at a level where bound state QED recoil corrections not accessible in hydrogen could be tested. This result would be sensitive to new physics in the muonic sector, e.g., to new bosons which might provide an explanation of the g-2 muon anomaly and allow to test Lorentz and CPT violation. We also present the observation of Muonium in the n = 3 excited state opening up the possibility of additional precise microwave measurements.

G−2 of charged leptons, α(MZ2) , and the hyperfine splitting of muonium

Following updates in the compilation of $e^+e^-\rightarrow{\rm hadrons}$ data, this work presents re-evaluations of the hadronic vacuum polarisation contributions to the anomalous magnetic moment of the electron ($a_e$), muon ($a_\mu$) and tau lepton ($a_\tau$), to the ground-state hyperfine splitting of muonium and also updates the hadronic contributions to the running of the QED coupling at the mass scale of the $Z$ boson, $\alpha(M_Z^2)$. Combining the results for the hadronic vacuum polarisation contributions with recent updates for the hadronic light-by-light corrections, the electromagnetic and the weak contributions, the deviation between the measured value of $a_\mu$ and its Standard Model prediction amounts to $\Delta a_{\mu} = (28.02 \pm 7.37) \times 10^{-10}$, corresponding to a muon $g-2$ discrepancy of $3.8\sigma$.

Hyperfine splitting in muonium: Accuracy of the theoretical prediction

In the last twenty years, the theory of hyperfine splitting in muonium developed without any experimental input. Finally, this situation is changing and a new experiment on measuring hyperfine splitting in muonium is now in progress at J-PARC. The goal of the MuSEUM experiment is to improve by an order of magnitude experimental accuracy of the hyperfine splitting and muon-electron mass ratio. Uncertainty of the theoretical prediction for hyperfine splitting will be crucial for comparison between the forthcoming experimental data and the theory in search of a possible new physics. In the current literature estimates of the error bars of the theoretical prediction differ roughly by a factor of two. We explain the origin of this discrepancy and obtain the theoretical prediction for the muonium hyperfine splitting ΔνHFSth(Mu)=4463302872(515)Hz,δ=1.2×10−7.

Hadronic vacuum polarization in true muonium

In order to reduce the theoretical uncertainty in the prediction, the leading-order hadronic vacuum polarization contribution to the hyperfine splitting of true muonium is reevaluated in two ways. A more complex pionic form factor and better estimates of the perturbative QCD contributions are used to study the model dependence of the previous calculation. The second, more accurate method directly integrates the Drell ratio $R(s)$ to obtain ${C}_{1,\mathrm{HVP}}=\ensuremath{-}0.04874(9)$. This corresponds to an energy shift in the hyperfine splitting (HFS) of $\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{HFS},\mathrm{HVP}}^{\ensuremath{\mu}}=\ensuremath{-}8202(16)$ MHz and represents a factor-of-50 reduction in the theoretical uncertainty from hadronic sources. We also compute the contribution in positronium, which is too small at present to detect.

Hyperfine splitting in muonium and positronium

Calculation of hard three-loop corrections of order [Formula: see text] to hyperfine splitting in muonium and positronium is reviewed. All these contributions are generated by the graphs with photon, electron and/or muon loop radiative insertions in the two-photon exchange diagrams. We calculate contributions of six gauge invariant sets of diagrams.

Hyperfine splitting in a true muonium toO(mμα6): Two-photon-annihilation contribution from other flavors

The two-photon-annihilation contribution to the true muonium hyperfine splitting arising from $e$ and $\tau$ loops is obtained analytically at order $m_\mu\alpha^6$. The contribution to the hyperfine splitting is $-2.031092873 m_\mu\alpha^6/n^3\pi^2=-793.926988/n^3$ MHz. The contribution to the triplet true muonium decay rate has also been obtained and was found to be $9.825708266 m_\mu\alpha^6/n^3\pi^2=3840.737698/n^3$ MHz. Additional results have been computed for other purely leptonic bound states.

Hard three-loop corrections to hyperfine splitting

We consider hard three-loop nonlogarithmic corrections of order [Formula: see text] to hyperfine splitting in muonium and positronium. All these contributions are generated by the graphs with photon, electron and/or muon loop radiative insertions in the two-photon exchange diagrams. We calculate contributions of six gauge invariant sets of diagrams.

Hard three-loop corrections to hyperfine splitting in positronium and muonium

We consider hard three-loop corrections to hyperfine splitting in muonium and positronium generated by the diagrams with closed electron loops. There are six gauge-invariant sets of such diagrams that generate corrections of order $m\alpha^7$. The contributions of these diagrams are calculated for an arbitrary electron-muon mass ratio without expansion in the small mass ratio. We obtain the formulae for contributions to hyperfine splitting that in the case of small mass ratio describe corrections for muonium and in the case of equal masses describe corrections for positronium. First few terms of the expansion of hard corrections in the small mass ratio were earlier calculated for muonium analytically. We check numerically that the new results coincide with the sum of the known terms of the expansion in the case of small mass ratio. In the case of equal masses we obtain hard nonlogarithmic corrections of order $m\alpha^7$ to hyperfine splitting in positronium.

Electroweak corrections to the true muonium hyperfine splitting

In contrast to other atomic systems, in true muonium ($\mu^+\mu^-$) the leading-order $Z$ boson corrections to the hyperfine splitting are shown to be experimentally accessible in the near future. This contribution ($-109$ MHz) constitutes a necessary contribution to a full $\mathcal{O}(m\alpha^7)$ calculation of the true muonium hyperfine splitting. This would enable a number of solutions to the muon problem to be constrained. Additionally, we compute the general expression for a pseudovector coupling to particle-antiparticle bound states at leading order, including the annihilation channel.

Three-loop contributions to hyperfine splitting: Muon loop light-by-light insertion and other closed lepton loops

The muon and tauon light-by-light scattering contributions to hyperfine splitting in muonium are calculated. These results conclude calculation of all hard three-loop contributions to hyperfine splitting containing graphs with closed fermion loops. We discuss the special role that the lepton anomalous magnetic moments play in these calculations. The full result for all three-loop radiative-recoil corrections to hyperfine splitting generated by the graphs with closed lepton loops is presented.

Muon loop light-by-light contribution to hyperfine splitting in muonium.

Three-loop corrections to hyperfine splitting in muonium, generated by the gauge-invariant sets of diagrams with muon and tauon loop light-by-light scattering blocks, are calculated. These results complete calculations of all light-by-light scattering contributions to hyperfine splitting in muonium.

Loopy constraints on leptophilic dark matter and internal bremsstrahlung

A sharp and spatially extended peak in an astrophysical gamma ray spectrum would provide very strong evidence for the existence of dark matter (DM), given that there are no known astrophysical processes that could mimic such a signal. From the particle physics perspective, perhaps the simplest explanation for a gamma ray peak is internal bremsstrahlung in DM annihilation through a charged t-channel mediator η close in mass to the DM particle χ. Since DM annihilation to quarks is already tightly constrained in this scenario, we focus here on the leptophilic case. We compute the electromagnetic anapole and dipole moments that DM acquires at 1-loop, and we find an interesting enhancement of these moments if the DM particle and the mediator are close in mass. We constrain the DM anapole and dipole moments using direct detection data, and then translate these limits into bounds on the DM annihilation cross section. Our bounds are highly competitive with those from astrophysical gamma ray searches. In the second part of the paper, we derive complementary constraints on internal bremsstrahlung in DM annihilation using LEP mono-photon data, measurements of the anomalous magnetic moments of the electron and the muon, and searches for lepton flavor violation. We also comment on the impact of the internal bremsstrahlung scenario on the hyperfine splitting of true muonium.

Light-by-light scattering nonlogarithmic corrections to hyperfine splitting in muonium

We consider three-loop corrections to hyperfine splitting in muonium generated by the gauge invariant set of diagrams with virtual light-by-light scattering block. These diagrams produce both recoil and nonrecoil contributions to hyperfine splitting. Recoil corrections are enhanced by large logarithms of the muon-electron mass ratio. Both nonrecoil and logarithmically enhanced radiative-recoil corrections where calculated some time ago. Here we calculate nonlogarithmic radiative-recoil corrections generated by the insertions of the light-by-light scattering block.

Light-by-light scattering single-logarithmic corrections to hyperfine splitting in muonium

We consider three-loop radiative-recoil corrections to hyperfine splitting in muonium generated by the gauge invariant set of diagrams with virtual light-by-light scattering block. These corrections are enhanced by the large logarithms of the electron-muon mass ratio. We present the results of an analytic calculation of the single-logarithmic radiative-recoil corrections of order ${\ensuremath{\alpha}}^{2}(Z\ensuremath{\alpha})(m/M){E}_{F}$ to hyperfine splitting in muonium generated by these diagrams.

Hadronic contributions to the anomalous magnetic moment of the electron and the hyperfine splitting of muonium

Motivated by recent progress of theory and experiment on the anomalous magnetic moment of the electron, ae, we update the hadronic contributions to ae. Using our up-to-date compilation of e+e−→hadrons data, we find the leading order hadronic contribution aehad,LO,VP=(1.866±0.010exp±0.005rad)⋅10−12 and the next-to-leading order hadronic contribution aehad,NLO,VP=(−0.2234±0.0012exp±0.0007rad)⋅10−12, where the first and second errors are from the error of the experimental data and the uncertainty in the treatment of radiative corrections, respectively. These values are compatible with earlier evaluations by other groups, but have significantly improved uncertainties due to the more precise input data used. We also update the leading order hadronic contribution to the ground state hyperfine splitting of muonium, obtaining ΔνMuhad,VP=(232.68±1.25exp±0.72rad) Hz. This value is consistent with the most precise evaluation in the literature and reduces its error by a factor of two.

Three-loop contribution to hyperfine splitting in muonium: Polarization corrections to light-by-light scattering contributions

We calculate corrections of order ${\ensuremath{\alpha}}^{3}(Z\ensuremath{\alpha}){E}_{F}$ to hyperfine splitting in muonium generated by the gauge invariant set of diagrams with polarization insertions in the light-by-light scattering diagrams. This nonrecoil contribution turns out to be $\ensuremath{-}2.63$ Hz. The total contribution of all known corrections of order ${\ensuremath{\alpha}}^{3}(Z\ensuremath{\alpha}){E}_{F}$ is equal to $\ensuremath{-}4.28$ Hz.

Radiative-nonrecoil corrections of orderα2(Zα)EFto the hyperfine splitting of muonium

We present results for the corrections of order ${\ensuremath{\alpha}}^{2}(Z\ensuremath{\alpha}){E}_{F}$ to the hyperfine splitting of muonium. We compute all the contributing Feynman diagrams in dimensional regularization and a general covariant gauge using a mixture of analytical and numerical methods. We improve the precision of previous results.

Radiative recoil corrections to hyperfine splitting: Polarization insertions in the electron factor

We consider three-loop radiative-recoil corrections to hyperfine splitting in muonium due to insertions of one-loop polarization operator in the electron factor. The contribution generated by electron polarization insertions is a cubic polynomial in the large logarithm of the electron-muon mass ratio. The leading logarithm cubed and logarithm squared terms are well known for some time. We calculated all single-logarithmic and nonlogarithmic radiative-recoil corrections of order $\alpha^3(m/M)E_F$ generated by the diagrams with electron and muon polarization insertions.