We consider an impact of hadronic light-by-light scattering on the muonium hyperfine structure. A shift of the hyperfine interval $Δν({\rm Mu}) _{\rm\tiny HLBL}$ is calculated with the light-by-light scattering approximated by exchange of pseudoscalar and pseudovector mesons. Constraints from the operator product expansion in QCD are used to fix parameters of the model similar to the one used earlier for the hadronic light-by-light scattering in calculations of the muon anomalous magnetic moment. The pseudovector exchange is dominant in the resulting shift, $Δν({\rm Mu})_{\rm\tiny HLBL}= -0.0065(10) {Hz}$. Although the effect is tiny it is useful in understanding the level of hadronic uncertainties.
We compute O(alpha^3 ln alpha) relative corrections to the ground state hyperfine splitting of a QED two body bound state with different masses of constituents. The general result is then applied to muonium and positronium. In particular, a new value of the muon to electron mass ratio is derived from the muonium ground state hyperfine splitting.
Contributions of hadronic effects to the muonium physics and anomalous magnetic moment of muon are considered. Special attention is paid to higher-order effects and the uncertainty related to the hadronic contribution to the hyperfine structure interval in the ground state of muonium.
We discuss hadronic effects in the muonium hyperfine structure and derive an expression for the hadronic contribution to the hfs interval in form of the one-dimensional integral of the cross section of e+e- annihilation into hadrons. Higher-order hadronic contributions are also considered.
The contribution of hadronic vacuum polarization (HVP) to the hyperfine splitting of the muonium ground state is evaluated with the account of modern experimental data on the cross section of $e^+e^- \to$ annihilation into hadrons.
A new result for the hadronic vacuum polarization correction to the muonium hyperfine splitting (HFS) is presented: Delta nu(had - vp) = (0.233 +- 0.003) kHz. Compared with previous calculations, the accuracy is improved by using the latest data on e+e- -> hadrons. The status of the QED prediction for HFS is discussed.
We make use of recent accurate results obtained for the pion and kaon vector form factors within a chiral unitary approach in order to calculate the decay widths of the $τ$ lepton to these mesons and also to evaluate the contribution of this two mesons to the anomalous magnetic moment of the muon, the running of the fine structure constant and the muonium hyperfine splitting.
We present results for the corrections of order alpha^2 (Z 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.
We examine the muonium ($μ^+e^-$)-antimuonium ($μ^-e^+$) system in the models which accomodate the dilepton gauge bosons, and study their contributions to the ground state hyperfine splitting in ``muonium''. We also consider the exotic muon decay $μ^+\to e^+ +\overline ν_e +ν_μ$ mediated by the dilepton gauge boson, and obtain a lower bound $(M_{X^{\pm }}/g_{3l})>550 \rm GeV$ at 90\% confidence level for the singly-charged dilepton mass using the unitarity relation of the Kobayashi-Maskawa matrix for the 3-family case.
The contribution of pseudoscalar pole diagrams to the hadronic light-by-light corrections of order alpha^3 E_F to the ground state hyperfine splitting in hydrogenic atom is calculated in the pseudoscalar pole terms approximation. The vector dominance model for the form factor of the transition of a pseudoscalar meson into two photons is used. With the account of new experimental data on the cross sections sigma (e^+e^-\to ρ, ω\to π^0(η)γ) in the SND and CMD-2 experiments some other hadronic corrections to the muonium hyperfine splitting are calculated.
The hyperfine structure of the ground states in a number of positronium hydrides (TPs, DPs, ${}^{1}$HPs) and MuPs ($μ^{+} e^{-}_2 e^{+}$) is determined with the use of highly accurate variational wave functions. We also evaluate the probabilities of various processes in the MuPs system, including the $(e^{-}, e^{+})$-pair annihilation and its conversion into the charge conjugate system $μ^{+} e^{-}_2 e^{+} \to μ^{-} e^{-} e^{+}_2$.
Terms contributing to the hyperfine structure of the muonium ground state at the level of few tenths of kHz have been evaluated. The $α^2 (Zα)$ radiative correction has been calculated numerically to the precision of 0.02 kHz. Leading $\ln (Zα)$ terms of order $α^{4-n} (Zα)^n , n=1,2,3,$ and some relativistic corrections have been evaluated analytically. The theoretical uncertainty is now reduced to 0.17 kHz. At present, however, it is not possible to test QED to this precision because of the 1.34 kHz uncertainty due to the muon mass.
The complete contribution to the muonium hyperfine splitting of relative order alpha^3(m_e/m_mu)ln(alpha) is calculated. The result amounts to 0.013 kHZ, much smaller than suggested by a previous estimate, and leads to a 2-sigma shift of the most precise value for the muon-electron mass ratio, with the error reduced by approximately 30%. Analogous contributions are calculated for the positronium hyperfine splitting: (217/90-17*ln{2}/3)m_e(alpha^7/pi)ln(1/alpha) \approx -0.32 MHz; the remaining theoretical uncertainty should be well below experimental error, leaving discrepancies of 2.5-sigma a…
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.
This is the second of a series of papers on the radiative corrections of order $α^2 (Zα)$, $α(Zα)^2$, and various logarithmic terms of order $α^4$, to the hyperfine structure of the muonium ground state. This paper deals with the $α(Zα)^2$ correction. Based on the NRQED bound state theory, we isolated the term of order $α(Zα)^2$ exactly. Our result $+16.904 2 (11) α(Zα)^2 E_F / π$ for the non-logarithmic part is consistent with the $α(Zα)^2$ part of Sapirstein's calculation and the recent result of Pachucki, and reduces the numerical uncertainty in the $α(Zα)^2$ term by two orders of magnitude…
This is the first of a series of papers on a systematic application of the NRQED bound state theory of Caswell and Lepage to higher-order radiative corrections to the hyperfine structure of the muonium ground state. This paper describes the calculation of the $α^2 (Zα)$ radiative correction. Our result for the complete $α^2 (Zα)$ correction is 0.424(4) kHz, which reduces the theoretical uncertainty significantly. The remaining uncertainty is dominated by that of the numerical evaluation of the nonlogarithmic part of the $α(Zα)^2$ term and logarithmic terms of order $α^4$.
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 $α^2(Zα)(m/M)E_F$ to hyperfine splitting in muonium generated by these diagrams.
In contrast to other atomic systems, in true muonium ($μ^+μ^-$) 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 $\mathcal{O}(mα^7)$ 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.
We calculate radiative-recoil corrections of order $α^2(Zα)(m/M)E_F$ to hyperfine splitting in muonium generated by the diagrams with electron and muon polarization loops. These corrections are enhanced by the large logarithm of the electron-muon mass ratio. The leading logarithm cubed and logarithm squared contributions were obtained a long time ago. The single-logarithmic and nonlogarithmic contributions calculated here improve the theory of hyperfine splitting, and affect the value of the electron-muon mass ratio extracted from the experimental data on the muonium hyperfine splitting.
The two-photon-annihilation contribution to the true muonium hyperfine splitting arising from $e$ and $τ$ loops is obtained analytically at order $m_μα^6$. The contribution to the hyperfine splitting is $-2.031092873 m_μα^6/n^3π^2=-793.926988/n^3$ MHz. The contribution to the triplet true muonium decay rate has also been obtained and was found to be $9.825708266 m_μα^6/n^3π^2=3840.737698/n^3$ MHz. Additional results have been computed for other purely leptonic bound states.
We calculate corrections of order $α^3(Zα)E_F$ 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 -2.63 Hz. The total contribution of all known corrections of order $α^3(Zα)E_F$ is equal to -4.28 Hz.
We calculate three-loop radiative-recoil corrections to hyperfine splitting in muonium generated by the diagrams with the first order electron and muon polarization loop insertions in graphs with two exchanged photons. These corrections are enhanced by the large logarithm of the electron-muon mass ratio. The leading logarithm squared contribution was obtained a long time ago. Here we calculate the single-logarithmic and nonlogarithmic contributions. We previously calculated the three-loop radiative-recoil corrections generated by two-loop polarization insertions in the exchanged photons. The c…
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.
Motivated by recent progress of theory and experiment on the anomalous magnetic moment of the electron, a_e, we update the hadronic contributions to a_e. Using our up-to-date compilation of e^+ e^- \to hadrons data, we find the leading order hadronic contribution a_e^{had,LO,VP} = (1.866 \pm 0.010_{exp} \pm 0.005_{rad}) \cdot 10^{-12} and the next-to-leading order hadronic contribution a_e^{had,NLO,VP} = (- 0.2234 \pm 0.0012_{exp} \pm 0.0007_{rad}) \cdot 10^{-12}, where the first and second errors are from the error of the experimental data and the uncertainty in the treatment of radiative cor…
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 $mα^7$. 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 th…
We consider three-loop radiative-recoil corrections to hyperfine splitting in muonium. These corrections are enhanced by the large logarithm of the electron-muon mass ratio. The leading logarithm cubed and logarithm squared contributions were obtained a long time ago. We calculate the single-logarithmic and nonlogarithmic contributions of order $α^3(m/M)E_F$ generated by gauge invariant sets of diagrams with one- and two-loop polarization insertions in diagrams with two exchanged photons and radiative photons, and by diagrams with one-loop radiative photon insertions both in the electron and m…
The method of NRQED is an adaptation of QED to bound systems. While it is fully equivalent to QED, it enables us to explore the QED bound states systematically as an expansion in the fine structure constant $α$ and velocity ($\sim Zα$) of the bound electron. I describe how to construct the NRQED Hamiltonian choosing recent works on the $α(Zα)$, $α^2 (Zα)$,$α(Zα)^2$, $α(Zα)^3$, and $α^2 (Zα)^2$ radiative corrections to the hyperfine structure of the muonium ground state as examples.
Radiative-recoil corrections to hyperfine splitting in muonium of orders $α(Zα)(m/M)^2E_F$ and $(Z^2α)(Zα)(m/M)^2E_F$ are calculated. These corrections are of the second order in the small electron-muon mass ratio. An analytic expression $[(-6 \ln2- \frac{3}{4})α(Zα) - \frac{17}{12} (Z^2 α) (Zα)](\frac{m}{M})^2 E_F$ is obtained. Numerically the correction is equal to $-0.0351\:\mbox{kHz}$ and is of the same order of magnitude as the expected accuracy of the current Los Alamos experiment to measure the hyperfine splitting.
We consider three-loop radiative-recoil corrections to hyperfine splitting in muonium generated by the diagrams with electron and muon vacuum polarizations. We calculate single-logarithmic and nonlogarithmic contributions of order $α^3(m/M)E_F$ generated by gauge invariant sets of diagrams with electron and muon polarization insertions in the electron and muon factors. Combining the new contributions with our older results we present complete result for all three-loop radiative-recoil corrections generated by the diagrams with electron and muon polarization loops.
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 curren…
We consider a set of radiative contributions from one-loop lepton vacuum polarization to the hyperfine splitting of true muonium. Improving previous results, we obtain values for the electron-loop coefficients and extract their leading dependence upon $β=m_μ/m_e$. The coefficients are $C^μ_{20,1{\rm VPCT}}=0.734931603β$, $C^μ_{20,1{\rm VPCA}}=0.551198702β$, and $C^μ_{20,1{\rm VPT}}=0.419149715β$. The mass-independent contribution from three one-loop vacuum polarization is $C_{40,{\rm VPX}}=\frac{1}{127575}\left(-5255-2016π^2+51840ζ(3)\right)$. Contributions from $τ$ up to $\mathcal{O}(m_μα^8)$…
Electroweak second order shifts of muonium ($μ^+e^-$ bound state) energy levels are calculated for the first time. Calculation starts from on-shell one-loop elastic $μ^+ e^-$ scattering amplitudes in the center of mass frame, proceed to renormalization and to derivation of muonium matrix elements by using the momentum space wave functions. This is a reliable method unlike the unjustified four-Fermi approximation in the literature. Corrections of order $αG_F$ (with $α\sim 1/137$ the fine structure constant and $G_F$ the Fermi constant) and of order $αG_F /(m_Z a_B)$ (with $m_Z$ the Z boson mass…
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 ($a_μ$) and tau lepton ($a_τ$), 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, $α(M_Z^2)$. Combining the results for the hadronic vacuum polarisation contributions with recent updates for the hadronic light-by-light corrections, the electromagnetic and th…
The leading-order hadronic vacuum polarization contribution to the hyperfine splitting of true muonium is reevaluated in two ways. The first considers a more complex pionic form factor and better estimates of the perturbative QCD contributions. The second, more accurate method directly integrates the Drell ratio $R(s)$ to obtain $C_{1,\rm hvp}=-0.0489(3)$. This corresponds to an energy shift in the hyperfine splitting of $ΔE^μ_{hfs,\rm hvp}=276196(51)$ MHz.
We present a new independent evaluation of the hadronic and QCD contributions to the QED running coupling α(M_Z) and to the muonium hyperfine splitting ν. We obtain: Δα_{had}=2770(17)10^{-5} and Δν_{had}=232.5(2.5) Hz. Combined with the QED and Electroweak Standard Model contributions, they lead to: α^{-1}(M_Z)=128.926(25) and to the Fermi energy splitting ν_F=4~459~031~783(229) Hz, where for the latter, we have used, in addition, the precise measurement of the muonium hyperfine splitting ν_{exp}. We use ν_F in order to predict the ratios of masses m_μ/m_e=206.768~276(11) and of the magnetic m…
For the S-states of muonium and positronium, the hyperfine shifts to the order $α^6$ of a recently derived two-fermion equation with explicit $\cal CPT$-invariance are checked against the results of a nonrelativistic reduction, and the leading $α^8$ shifts are calculated. An additional hyperfine operator is discovered which can milden the singularity for $r\to 0$ of the Dirac hyperfine operator, such that the resulting extended operator can be used nonperturbatively. The binding correction to magnetic moments is mentioned.
We have calculated the contribution to the positronium hyperfine splitting at order $m α^7$ of the light-by-light scattering process in the exchange of two photons between the electron and positron. Our result is $ΔE = -0.235355(8) m α^4 \left ( \fracαπ \right )^3 = -1.034 kHz$. As a check of our approach we confirm earlier evaluations of the analogous correction for a bound system (such as muonium) with unequal masses.
A relativistic wave equation for bound states of two fermions with arbitrary masses which are exposed to a magnetic field is derived from quantum electrodynamics. The interaction kernels are based upon the generalized invariant M-matrices for inter-fermion and fermion-field interactions. As an application we calculate the energy corrections in a weak homogeneous B field to obtain the Zeeman splitting of the hyperfine structure and g-factors in the lowest order (i.e. to alpha^4). Lande g-factors are presented for several of the first excited states of hydrogen, muonium, and muonic-hydrogen.
The contribution of hadronic vacuum polarization to the hyperfine splitting of the muonic hydrogen ground state is calculated with the account of experimental data on the cross section of e^+e^-\to annihilation into hadrons and the dipole parameterization of electromagnetic proton form factors.
We calculate radiative corrections to the Lamb shift of order $α^3(Zα)^5m$ and radiative corrections to hyperfine splitting of order $α^3(Zα)E_F$ generated by the diagrams with insertions of radiative photons and electron polarization loops in the graphs with two external photons. We also obtain the radiative-recoil correction to hyperfine splitting in muonium generated by the diagrams with the $τ$ polarization loop.
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.
We consider three-loop radiative-recoil corrections to hyperfine splitting in muonium generated by diagrams with one-loop radiative photon insertions both in the electron and muon lines. An analytic result for these nonlogarithmic corrections of order $α(Z^2α)(Zα)(m/M)\widetilde E_F$ is obtained. This result constitutes a next step in the implementation of the program of reduction of the theoretical uncertainty of hyperfine splitting below 10 Hz.
We consider three-loop radiative-recoil corrections to hyperfine splitting in muonium due to insertions of one-loop polarization operator in the muon factor. The contribution produced by electron polarization insertions are enhanced by the large logarithm of the electron-muon mass ratio. We obtained all single-logarithmic and nonlogarithmic radiative-recoil corrections of order $α^3(m/M)E_F$ generated by the diagrams with electron and muon polarization insertions.
Muonium is a bound state composed of an antimuon and an electron, and it constitutes a hydrogen-like atom. Because of the absence of the hadronic matter in the bound state, the muonium is a useful probe to explore new physics being free from the hadronic uncertainties. The process of the muonium-to-antimuonium transition is considered to be effective to identify fundamental interactions which relate to the lepton flavor and lepton number violation. New experiments are being planned at J-PARC in Japan and CSNS in China, and it is expected to attract more attention in the near future. In this pa…
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 $α^3(m/M)E_F$ generated by the diagrams with electron and muon polarization insertions.
Ground-state hyperfine splittings in hydrogen and muonium are very well measured. Their difference, after correcting for magnetic moment and reduced mass effects, is due solely to proton structure--the large QED contributions for a pointlike nucleus essentially cancel. The rescaled hyperfine difference depends on the Zemach radius, a fundamental measure of the proton, computed as an integral over a product of electric and magnetic proton form factors. The determination of the Zemach radius, (1.043 +/- 0.016) fm, from atomic physics tightly constrains fits to accelerator measurements of proton …
Two recent experimental observations at the ATOMKI Institute of the Hungarian Academy of Sciences (regarding the angular emission pattern of electron-positron pairs from nuclear transitions from excited states in 8Be and 4He) indicate the possible existence of a particle of a rest mass energy of roughly 17 MeV. The so-called X17 particle constitutes a virtual state in the process, preceding the emission of the electron-positron pair. Based on the symmetry of the nuclear transitions ($1^+$ to $0^+$ and $0^-$ to $0^+$), the X17 could either be a vector, or a pseudoscalar particle. Here, we calcu…
We consider hyperfine splitting of 1s and, in part, of 2s levels in light hydrogen-like atoms: hydrogen, deuterium, tritium, helium-3 ion, muonium and positronium. We discuss present status of precision theory and experiment for the hfs intervals. We pay a special attention to a specific difference, D_{21} = 8 E_{hfs}(2s) - E_{hfs}(1s), which is known experimentally for hydrogen, deuterium and ^3He^+ ion. The difference is weakly affected by the effects of the nuclear structure and thus may be calculated with a high accuracy. We complete a calculation of the fourth order QED contributions to t…
The calculation of hadronic polarizability contribution of the nucleus to hyperfine structure of muonic hydrogen and helium is carried out within the unitary isobar model and experimental data on the polarized structure functions of deep inelastic lepton-proton and lepton-deuteron scattering. The calculation of virtual absorption cross sections of transversely and longitudinally polarized photons by nucleons in the resonance region is performed in the framework of the program MAID.
In this paper we discuss a quantity that can be derived from the hadronic spectrum - the ratio between the color hyperfine splitting of a K meson and that of a Sigma baryon. It will be shown that within the constituent quark model this ratio depends only on the ratio of contact probabilities within the hadrons. We compute this ratio assuming several phenomenological potential models, and show that the best agreement with data is obtained from the Cornell potential - Coulomb + linear. Thus the analysis of color hyperfine interactions corroborat es the preference for the Cornell potential based …
Two-photon exchange contributions to elastic electron-scattering are reviewed. The apparent discrepancy in the extraction of elastic nucleon form factors between unpolarized Rosenbluth and polarization transfer experiments is discussed, as well as the understanding of this puzzle in terms of two-photon exchange corrections. Calculations of such corrections both within partonic and hadronic frameworks are reviewed. In view of recent spin-dependent electron scattering data, the relation of the two-photon exchange process to the hyperfine splitting in hydrogen is critically examined. The imaginar…
The ultrafine splitting of heavy quarkonium with next-to-next-to-next-to-next-to-leading-order accuracy
Universidad Complutense de Madrid（马德里康普顿斯大学）；Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology（巴塞罗那高等物理研究所（IFAE），巴塞罗那科学技术研究院）；Universitat Autònoma de Barcelona（巴塞罗那自治大学）
I summarize our self-contained determinations of the lowest order hadronic contributions to the anomalous magnetic moments a_{μ,τ} of the muon and tau leptons, the running QED coupling α(M_Z) and the muonium hyperfine splitting ν. Using as a revised estimate of the light-by light scattering: a_μ(LL)=85(18) 10^{-11}, we deduce: a_μ^{SM} = 116 591 861(78) 10^{-11}, a_τ^{SM} = 117 759(7) 10^{-8}, giving: a_μ^{SM}-a_μ^{exp}=162(170) 10^{-11}. We also obtain: α^{-1}(M_Z)=128.926(25) and the Fermi energy splitting: ν_F^{SM}=4 459 031 783(229) {Hz}. Lower bounds on some eventual new physics are given…
Hadronic vacuum polarization in hydrogen-like atoms and ions amid the interplay of recoil and finite-size effects
Institut für Kernphysik and PRISMA ++ Cluster of Excellence, Johannes Gutenberg-Universität Mainz（美因茨约翰内斯·古腾堡大学核物理研究所与PRISMA++卓越集群）；LPNHE, Sorbonne Université, Université Paris Cité, CNRS/IN2P3（巴黎萨克雷大学、巴黎西岱大学及法国国家科学研究中心/欧洲核子研究组织高能物理实验室）
