Theory of the muonic hydrogen-muonic deuterium isotope shift

Corrections of the α³, α⁴, and α⁵ orders are calculated for the Lamb shift of the 1S and 2S energy levels of muonic hydrogen μp and muonic deuterium μd. The nuclear structure effects are taken into account in terms of the charge radii of the proton r _p and deuteron r _d for one-photon interaction, as well as in terms of the electromagnetic form factors of the proton and deuteron for the case of one-loop amplitudes. The μd-μp isotope shift for the 1S-2S splitting is found to be equal to 101003.3495 meV, which can be treated as a reliable estimate when conducting the corresponding experiment with an accuracy of 10^?6. The fine-structure intervals E(1S)-8E(2S) in muonic hydrogen and muonic deuteron are calculated.     

Emergence and space–time structure of lump solution to the (2$$$$1)-dimensional generalized KP equation

We derive the relativistic Hamiltonian of hydrogen atom in dynamical non-commutative spaces (DNCS or τ-space). Using this Hamiltonian we calculate the energy shift of the ground state as well the 2P _1/2, 2S _1/2 levels. In all the cases, the energy shift depends on the dynamical non-commutative parameter τ. Using the accuracy of the energy measurement, we obtain an upper bound for τ. We also study the Lamb shift in DNCS. Both 2P _1/2 and 2S _1/2 levels receive corrections due to dynamical non-commutativity of space which is in contrast with the non-dynamical non-commutative spaces (NDNCS or ??-space) in which the 2S _1/2 level receives no correction.     

Fine and hyperfine structure of <Emphasis Type="Italic">P</Emphasis>-wave levels in muonic hydrogen

Corrections of order α ⁵ and α ⁶ are calculated for muonic hydrogen in the fine-structure interval ΔE ^fs = E(2P _3/2) − E(2P _1/2) and in the hyperfine structure of the 2P _1/2-and 2P _3/2-wave energy levels. The resulting values of ΔE ^fs = 8352.08 μeV, ΔẼ ^hfs(2P _1/2) = 7819.80 μeV, and ΔẼ ^hfs(2P _3/2) = 3248.03 μeV provide reliable guidelines in performing a comparison with relevant experimental data and in more precisely extracting the experimental value of the (2P–2S) Lamb shift in the muonic-hydrogen atom. Original Russian Text ? A.P. Martynenko, 2008, published in Yadernaya Fizika, 2008, Vol. 71, No. 1, pp. 126–136.     

Hyperfine structure of the <Emphasis Type="Italic">S</Emphasis> levels of the muonic helium ion

Corrections of the α⁵ and α⁶ orders to the energy spectrum of the hyperfine splitting of the 1S and 2S levels of the muonic helium ion are calculated with the inclusion of the electron vacuum polarization effects, nuclear-structure corrections, and recoil effects. The values ΔE ^hfs(1S) = ?1334.56 meV and ΔE ^hfs(2S) = ?166.62 meV obtained for hyperfine splitting values can be considered as reliable estimates for comparison with experimental data. The hyperfine structure interval Δ₁₂ = 8ΔE ^hfs(2S) ? ΔE ^hfs(1S) = 1.64 meV can be used to verify QED predictions.     

Fine Structure of Beta Decay Strength Function and Anisotropy of Isovector Nuclear Dencity Component Oscillations in Deformed Nuclei

The experimental measurement data on the fine structure of beta-decay strength function S_β(E) in spherical, transitional, and deformed nuclei are analyzed. Modern high-resolution nuclear spectroscopy methods made it possible to identify the splitting of peaks in S_β(E) for deformed nuclei. By analogy with splitting of the peak of E1 giant dipole resonance (GDR) in deformed nuclei, the peaks in S_β(E) are split into two components from the axial nuclear deformation. In this report, the fine structure of S_β(E) is discussed. Splitting of the peaks connected with the oscillations of neutrons against protons (E1GDR), of proton holes against neutrons (peaks in S_β(E) of β⁺/EC-decay), and of protons against neutron holes (peaks in S_β(E) of β^–-decay) is discussed.     

Isotopic shift of the ground state of neon from the experimental results concerning the absorption of 0.63-μm laser radiation

Isotopic shifts of the 3s[3/2] ₁ ⁰ –2р ⁶(¹ S ₀) and 3s'–2р ⁶(¹ S ₀) transitions, equal, respectively, to 417 ± 20 and–98 ± 20 MHz, have been measured using the 0.63-μm radiation of a helium–neon laser and opto-magnetic resonances induced by the interference of the reactive components of fields in overlapping areas of the emissions of isotopic atoms. Combining these results with the absolute specific mass shift of the 3p[5/2]₂ level (–647 MHz), the isotopic mass shift of the ground state of neon equal to 3223 ± 30 MHz, and its specific mass shift equal to–9782 ± 30 MHz have been determined.     

Method for taking into account hard-photon emission in four-fermion processes

A method for taking into account hard-photon emission in four-fermion processes proceeding in the s channel is described. The application of this method is exemplified by numerically estimating one-loop electroweak corrections to observables (cross sections and asymmetries) of the reaction e^?e⁺ → μ^?μ⁺(γ) involving longitudinally polarized electrons and proceeding at energies below the Z-resonance energy.     

Estimating of CP-violation in <Emphasis Type="Italic">B</Emphasis><Superscript>0</Superscript>→ψ(2<Emphasis Type="Italic">S</Emphasis>)π<Superscript>0</Superscript> decay

I present estimates of CP-violating asymmetries in the non-leptonic charmonium two-body B⁰→ψ(2S)π⁰ decay and the same decays of B⁺→ψ(2S)π⁺ and B⁺→ψ(2S)K⁺ These estimates are based on QCD and improved QCD factorization approach making use of next-to-leading order (NLO) contributions. The CP-violating asymmetry for B⁰→ψ(2S)π⁰ decay is not available, according to the same calculations, it is expected if it can be measured in the future its value will be S_ψ(2S)π0(B⁰ → ψ(2S)π⁰)= 0.662 ± 0.197 and C_ψ(2S)π0(B⁰ → ψ(2S)π⁰)= 0.024 ± 0.007.     

Die Hyperfeinstrukturaufspaltung des Grundzustandes2 S 1/2 und das magnetische Kerndipolmoment von Au197

The hyperfine structure of the groundstate 6s ² S _1/2 and the nuclear magnetic dipole moment of gold 197 have been studied by the atomic beam magnetic resonance technique. A special high frequency arrangement is described. The hyperfine structure separationΔ v was determined fromΔF=1 transitions. The magnetic dipole momentμ _I was measured by a direct method. The experiments yield the following results:Δv (²S_1/2)=(6099,309±0,010) Mc/secμ _I (Au¹⁹⁷)=+(0,1445±0,0014)μ _K.     

Precision measurement of the rate of nuclear muon capture in the muonic hydrogen atom and the determination of the pseudoscalar form factor of the nucleon

The rate Λ _S of nuclear muon capture by a proton from the hyperfine singlet ground state of the µp atom has been measured using a new experimental method based on a time-projection chamber operating in an ultrapure hydrogen gas at a pressure of 10 atm. The capture rate has been determined from the difference between the measured lifetime of the negative muon in hydrogen and the world average lifetime of the positive muon. The analysis of 10% of the collected statistics (2 × 10¹⁰) of µe decays yields the muon capture rate Λ _S = 725.0 ± 17.4 s^?1, from which the pseudoscalar form factor of the nucleon, g _P (q _c ² = ?0.88m _µ ² ) = 7.3 ± 1.1, is determined. The further analysis of the collected experimental data should improve the precision of this measurement by a factor of 3.     

Collision frequency shift of hyperfine transitions in atomic hydrogen at a low temperature

The experimental results on the collision frequency shift of ESR in diluted two-and three-dimensional atomic hydrogen at ultralow temperature have been analyzed. The apparent contradiction between experiment and theory is resolved by taking into account the relation between the symmetry of the state of two atoms and their total electron spin S. For example, transitions between symmetric and antisymmetric states of a pair of atoms induced by a symmetric perturbation are forbidden. If symmetric and antisymmetric states are, respectively, pure electronic triplets (S= 1) and singlets (S = 0), this results in the cancellation of the singlet-triplet transitions. Thus, the collision frequency shift of b → c and b → a transitions vanishes in a completely electron and nuclear spin-polarized gas (hyperfine state b). The comparison is performed with experiments in ultracold alkali vapors.     

Nuclear magnetic relaxation of spin<Emphasis Type="Italic">I</Emphasis>=1/2 scalar coupled to a quadrupolar nucleus in the presence of cross-correlation effects

The rigorous treatment of relaxation for the dipolar-multipolarAX spin system (I=1/2,S>1/2) in the presence of the dipolarI-S coupling, anisotropy chemical shift and quadrupolar interaction ofS spin is proposed. The calculations of the spin evolution under the relaxation Hamiltonian are based on the second-order time-dependent perturbation theory and are carried out in the operator representation. For this task the double commutator identities of the type [[I _±S _z ⁿ ,A _q ^μp ]A _?q ^μp ] and [[I _zS _z ⁿ ,A _q ^μp ]A _?q ^μp. ] are derived. The fist-order differential equations for the evolution of longitudinal two-spin orderI _zS _z ⁿ , z=magnetization ofS spinS _z ⁿ and coherences <I _±S _z ⁿ > in the spin systemIS with scalar coupling between spin 1/2 and quadrupolar spinS>1/2 were obtained. These equations are used to get equations for the evolutions of each component of the multiplet structure of spinI. The imaginary part of the cross-correlation spectral density function and indirect spin-spin coupling Hamiltonian are taken into account. Equations for the longitudinal components of theI spin spectrum in the presence of cross-correlation effects were obtained also. Longitudinal and transverse relaxation times and cross-relaxation times in the presence of cross-correlation D-CSA, Q-CSA, Q-D were analyzed.     

<Emphasis Type="Italic">g</Emphasis> factor of Li-like ions with a nonzero nuclear spin

A relativistic theory of the g factor of Li-like ions with a nonzero nuclear spin is considered for the 1s ² 2s state. A correction to the atomic g factor for the magnetic-dipole hyperfine interaction is calculated including the one-electron contribution, as well as the contribution of interelectronic-interaction effects of the order of 1/Z. Along with corrections for the interelectronic interaction, quantum electrodynamic effects, nuclear recoil, and finite nuclear size, this correction allows high-precision theoretical values for the g factor of Li-like ions with a nonzero nuclear spin to be obtained. The results can be used for refining the nuclear magnetic moments from comparison with experimentally determined values of the g factor.     

Coulomb deexcitation of muonic hydrogen within the quantum close-coupling method

The Coulomb deexcitation of muonic hydrogen in collisions with the hydrogen atom has been studied in the framework of the fully quantum-mechanical close-coupling method for the first time. The calculations of the l-averaged cross sections of the Coulomb deexcitation are performed for (μp)_n and (μd)_n atoms in the initial states with the principal quantum number n = 3–9 and at relative energies E = 0.1–100 eV. The obtained results for the n and E dependences of the Coulomb deexcitation cross sections drastically differ from the semiclassical results. An important contribution of the transitions with Δn > 1 to the total Coulomb deexcitation cross sections (up to ～37%) is predicted.     

Nonplanar contribution to the four-loop universal anomalous dimension of the twist-2 Wilson operators in the $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills theory

The result of the direct component calculation of the nonplanar contribution to the four-loop anomalous dimension of the Konishi operator in the \(\mathcal{N}\) = 4 supersymmetric Yang-Mills theory is reported. The result contains only the ζ(5) term proportional to the ζ(5) contribution in the planar case, which comes only from wrapping corrections. The previous calculations of the leading transcendent contribution to the anomalous dimension of the twist-2 operators for first three even moments are also expanded to the nonplanar case and the same results as in the planar case are obtained up to a general factor. These two results imply that the nonplanar contribution of the four-loop universal anomalous dimension of the twist-2 Wilson operators with an arbitrary Lorentz spin j is proportional to S ₁ ² (j)ζ(5). This result provides a nonstandard square logarithmic asymptotic behavior ln² j for large Lorentz spins j of the operators.     

Study of background processes with the formation of neutrons in nuclear reactions in the energy range of 26–32 kev

The nature of background processes accompanying astrophysical nuclear reactions induced by hydrogen, helium, and neon ions in deuterated targets with small cross sections has been studied in calculations and experiments. The experiments have been performed at a Hall pulsed plasma accelerator in the ion energy range of 26–32 keV. The yield of background neutrons and γ-quanta with energies below 4 MeV in the proton-induced D(p, γ)³He reaction is primarily due to the presence of a natural impurity of gaseous deuterium in gaseous hydrogen and the chain of D(D, ³He)n → (n, γ) or (n, n'γ) reactions. A small contribution comes from the chain of D(¹H, ¹H)D → D(D, ³He)n → (n, γ) or (n, n'γ) reactions. It has been shown that background neutrons and γ-quanta from the D(4He, γ)6Li reaction are entirely due to the chain of D(⁴He, ⁴He)D → D(D, ³He)n → (n, γ) or (n, n'γ) reactions. It has been shown that the yield of neutrons and γ-ray photons detected at the interaction of neon ions with deuterated targets is also entirely due to the chain of elastic- scattering reactions of neon ions on deuterons in the target and to subsequent inelastic processes of interaction of deuterons accelerated at elastic scattering with other deuterons of the target. The main contribution to the yields of background neutrons and γ-quanta comes from doubly charged neon ions. The main conclusion is that the explanation of the yield of neutrons and γ-quanta at the interaction of hydrogen, helium, and neon ions with deuterated targets does not require “exotic” theoretical models.     

Uncertainty budget for determinations of mean isomer shift from Mössbauer spectra

The magnetite/maghemite content within iron oxide nanoparticles can be determined using the mean isomer shift (\(\overline {\delta }\)). However, accurate characterisation of the composition is limited by the uncertainty associated with \(\overline {\delta }\). We have identified four independent sources of uncertainty and developed a quantitative expression for the uncertainty budget. Sources of uncertainty are categorised as follows: that from the fitting of the Mössbauer spectrum (σ _fit), that of the calibration of the α-Fe reference spectrum (σ _cal), thermal corrections to the spectrum due to second order Doppler shift (SODS) (σ _Δδ ) and other experimental errors (σ _err). Each contribution is discussed in detail using ⁵⁷Fe Mössbauer spectra obtained from an iron oxide nanoparticle system at temperatures between 16 K and 295 K on different spectrometers in two different laboratories.     

On Probability Domains

Motivated by IF-probability theory (intuitionistic fuzzy), we study n-component probability domains in which each event represents a body of competing components and the range of a state represents a simplex S _n of n-tuples of possible rewards–the sum of the rewards is a number from [0,1]. For n=1 we get fuzzy events, for example a bold algebra, and the corresponding fuzzy probability theory can be developed within the category ID of D-posets (equivalently effect algebras) of fuzzy sets and sequentially continuous D-homomorphisms. For n=2 we get IF-events, i.e., pairs (μ,ν) of fuzzy sets μ,ν∈[0,1] ^X such that μ(x)+ν(x)≤1 for all x∈X, but we order our pairs (events) coordinatewise. Hence the structure of IF-events (where (μ ₁,ν ₁)≤(μ ₂,ν ₂) whenever μ ₁≤μ ₂ and ν ₂≤ν ₁) is different and, consequently, the resulting IF-probability theory models a different principle. The category ID is cogenerated by I=[0,1] (objects of ID are subobjects of powers I ^X ), has nice properties and basic probabilistic notions and constructions are categorical. For example, states are morphisms. We introduce the category S _n D cogenerated by \(S_{n}=\{(x_{1},x_{2},\ldots ,x_{n})\in I^{n};\:\sum_{i=1}^{n}x_{i}\leq 1\}\) carrying the coordinatewise partial order, difference, and sequential convergence and we show how basic probability notions can be defined within S _n D.     

Gleichzeitige Messung von Hyperfeinstruktur,Starkeffekt und Zeemaneffekt des85Rb19F mit einer Molekülstrahl-Resonanzapparatur

A molecular beam resonance apparatus with electric quadrupole lenses asA- andB-fields and with superimposed parallel electric and magnetic transition-fields was used. Molecules in different rotational statesJ, m _J are separated by theA-field. Spectra of molecules in different vibrational states are resolved by their different Starkeffect energies. By this means the following electric and magnetic properties of the molecule could be measured in the rotational stateJ=1 and vibrational statesv=0 and 1: The magnetic and electric dipole moment of the molecule, the scalar and the tensor nuclear dipole — dipole interactiond _s andd _T, the nuclear spinrotational interactionc _F andc _Rb, the nuclear quadrupole interactioneqQ, the nuclear magnetic moment μ_Rb, the anisotropy of the diamagnetic susceptibility ξ_⊥?ξ_∥, the anisotropy of the diamagnetic shielding of the external field by the electrons at the position of the nuclei σ_⊥?σ_∥. Using these quantities it was possible to calculate the quadrupole moment and a weighted quadrupole moment of the electronic charge distribution. The results are: (J=1,v=0) μ_el=8,5464 (17) debμ _J/J=?29,79(2)x10^?6 μ _B d _s/h=0,36(23) kHzd _T/h=0,69(22)kHzc _F/h=10,42(70) kHzc _Rb/h=0,479 (48) kHz.eqQ _Rb/h=?70,3410(26) MHzμ(1?σ_S)Rb=1,3474(5) μ_k (ξ⊥-ξ _∥)=12(6)×10^?30 erg/Gauß² (σ_⊥-σ∥)_Rb=?3,8(2,1)×10^?4 (σ⊥-σ _∥)_F=?2,6(3)×10^?4     

Probabilities of radiative dipole transitions of parahelium in an electric field

Quadratic Stark corrections to the wave functions, matrix elements, and probabilities of transitions between the singlet states ¹ S ₀ and ¹ P ₁ of helium atoms are calculated. The coefficients of the polynomials that depend on the effective principal quantum number of the upper level v _f and that approximate the numerical values of the polarizabilities, the quadratic corrections to the wave functions, and the probabilities of transitions to highly excited Rydberg states with large v _f are determined. The results of calculations testify that the probabilities of all σ transitions n _i ¹ S ₀ → n _f ¹ P ₁ and π transitions to the states with n _f > n _i /2 are decreased with increasing electric field strength, except for the transition 2¹ S ₀ → 2¹ P ₁, whose probability increases both for σ and for π transitions.