Order Α2 Research Articles

Two-loop radiative corrections to e+e− → γγ∗ cross section

The increasing accuracy of current and planned experiments to measure the anomalous magnetic moment of the muon requires more precision and reliability of its theoretical calculation. For this purpose, we calculate the differential cross section for the process of annihilation of an electron-positron pair into two photons, one of which is virtual, accompanied by the emission of soft photons, taking into account radiative corrections of the order α2. The results obtained can be used to improve the accuracy of calculating the contribution of the hadron vacuum polarization to the muon anomalous moment. It is shown that all logarithmically amplified two-loop corrections can be easily found using modern theorems of soft and collinear factorizations and available one-loop results.

Two-loop corrections to Lamb shift and hyperfine splitting in hydrogen via multi-loop methods

We revisit the contributions of order α2(Zα)5m and α2(Zα)EF, respectively, to the Lamb shift and to the hyperfine splitting from mixed self-energy-vacuum-polarization diagrams, involving fermionic loop. We use modern multi-loop calculation techniques based on IBP reduction and differential equations. We construct the ϵ-regular basis [1] and explicitly demonstrate that it is compatible with the renormalization. We obtain analytic results in terms of one-fold integral involving elliptic function and dilogarithm. As a by-product, we obtain the analogous contribution for the limiting cases of heavy and light fermionic loop.

Energy Interval 1S–2S in Muonic Hydrogen and Helium

The interval of a large structure of energy spectrum (1S–2S) in muonic helium and hydrogen is calculated using a quasi-potential method in quantum electrodynamics. Corrections of the order α4, α5, and α6 determined by relativistic effects, vacuum polarization effects, nuclear structure and recoil, as well as by combined corrections including those listed above, are taken into account. The nuclear structure effects are expressed in terms of the charge radius of nuclei in the case of the one-photon interaction and electromagnetic form factors of nuclei in the case of the two-photon interaction. Numerical values obtained for the (1S–2S) interval can be used for comparison with future experimental data and for more accurate determination of the charge radii of nuclei.

The Lamb shift of the 1s state in hydrogen: Two-loop and three-loop contributions

We consider the 1s Lamb shift in hydrogen and helium ions, a quantity, required for an accurate determination of the Rydberg constant and the proton charge radius by means of hydrogen spectroscopy, as well as for precision tests of the bound-state QED. The dominant QED contribution to the uncertainty originates from α8m external-field contributions (i.e., the contributions at the non-recoil limit). We discuss the two- and three-loop cases and in particular, we revisit calculations of the coefficients B61,B60,C50 in standard notation.We have found a missing logarithmic contribution of order α2(Zα)6m. We have also obtained leading pure self-energy logarithmic contributions of order α2(Zα)8m and α2(Zα)9m and estimated the subleading terms of order α2(Zα)7m, α2(Zα)8m, and α2(Zα)9m. The determination of those higher-order contributions enabled us to improve the overall accuracy of the evaluation of the two-loop self-energy of the electron.We investigated the asymptotic behavior of the integrand related to the next-to-leading three-loop term (order α3(Zα)5m, coefficient C50 in standard notation) and applied it to approximate integration over the loop momentum. Our result for contributions to the 1s Lamb shift for the total three loop next-to-leading term is (−3.3±10.5)(α3/π3)(Zα)5m.Altogether, we have completed the evaluation of the logarithmic contributions to the 1s Lamb shift of order α8m and reduced the overall α8m uncertainty by approximately a factor of three for H, D, and He+ as compared with the most recent CODATA compilation.

Energy spectra of muonic atoms in quantum electrodynamics

Vacuum polarization, nuclear structure and recoil, radiative corrections to the hyperfine structure of S-states in muonic ions of lithium, beryllium and boron are calculated on the basis of quasipotential method in quantum electrodynamics. We consider contributions in first and second orders of perturbation theory which have the order α5 and α6 in the energy spectrum. Total values of hyperfine splittings are obtained which can be used for a comparison with future experimental data.

The photon content of the proton

The photon PDF of the proton is needed for precision comparisons of LHC cross sections with theoretical predictions. In a recent paper, we showed how the photon PDF could be determined in terms of the electromagnetic proton structure functions F2 and FL measured in electron-proton scattering experiments, and gave an explicit formula for the PDF including all terms up to next-to-leading order. In this paper we give details of the derivation. We obtain the photon PDF using the factorisation theorem and applying it to suitable BSM hard scattering processes. We also obtain the same PDF in a process-independent manner using the usual definition of PDFs in terms of light-cone Fourier transforms of products of operators. We show how our method gives an exact representation for the photon PDF in terms of F2 and FL, valid to all orders in QED and QCD, and including all non-perturbative corrections. This representation is then used to give an explicit formula for the photon PDF to one order higher than our previous result. We also generalise our results to obtain formulæ for the polarised photon PDF, as well as the photon TMDPDF. Using our formula, we derive the Pγi subset of DGLAP splitting functions to order ααs and α2, which agree with known results. We give a detailed explanation of the approach that we follow to determine a photon PDF and its uncertainty within the above framework.

Theory of the Lamb shift in muonic ions μLi, μBe, μB

The calculation of the Lamb shift 2P 1/2 - 2S 1/2 in muonic ions (μ 3 6 Li)2+, (μ 3 7 Li)2+, (μ 4 9 Be)3+, (μ 4 10 Be)3+, (μ 5 10 B)4+, (μ 5 11 B)4+, is performed taking into account the corrections of order α3, α4, α5, α6 to the vacuum polarization, nuclear structure and recoil. The dependence of numerical values of the contributions from the nuclear charge is carried out. The obtained results can be used for extracting more accurate values of nuclear charge radii of nuclei of future experiments.

Deuteron properties from muonic atom spectroscopy

Leading order (α4) finite size corrections in muonic deuterium are evaluated within a few body formalism for the μ−pn system in muonic deuterium and found to be sensitive to the input of the deuteron wave function. We show that this sensitivity, taken along with the precise deuteron charge radius determined from muonic atom spectroscopy can be used to determine the elusive deuteron D-state probability, PD, for a given model of the nucleon–nucleon (NN) potential. The radius calculated with a PD of 4.3% in the chiral NN models and about 5.7% in the high precision NN potentials is favoured most by the μ−d data.

Hyperfine structure of the S- and P-wave states of muonic deuterium

Corrections of order α5 and α6 to the hyperfine structure of the S- and P-wave states of muonic deuteriumwere calculated on the basis of the quasipotential approach in quantum electrodynamics. Relativistic corrections, vacuum-polarization and deuteron-structure effects, and recoil corrections were taken into account in this calculation. The resulting hyperfine-splitting values can be used in a comparison with experimental data obtained by the CREMA Collaboration.

Possible New Positronium Bound State

The Two-Body Dirac equations of constraint dynamics applied to QED yields an exact Sommerfeld-like solution for the spectrum of 1Jj singlet positronium states which agrees with standard perturbative results through order α4. At short distance the radial part of the wave function u = rψ has two solutions with probabilities near the origin behaving like . For J ≠ 0 only the first sign is allowable but both signs for J = 0 are well behaved. The first sign in that case corresponds to ordinary positronium (with a binding energy of about 6.8 eV). The second sign corresponds to a new positronium state with a binding energy of about 300 keV and a root-mean-square radius on the order of a Compton wavelength. The ordinary 1S positronium state decays into this new 1S state by two photon emission with c.m. energy of about 300 keV. The peculiar 1S state then annihilates promptly into two photons with c.m. energy of about 700 keV. Thus the existence of this new positronium state would be a distinctive 4 photon decay signature of ordinary singlet positronium.

Change of Cr atoms distribution in Fe85Cr15 alloy caused by 250 keV He+ ion irradiation to different doses

Redistribution of Cr atoms in a Fe85Cr15 alloy caused by its irradiation with 250keV He+ ions to different doses, D=8⋅1016, 16⋅1016 and 48⋅1016ions/cm2 was investigated by means of conversion electrons Mössbauer spectroscopy. The redistribution was expressed in terms of the Warren–Cowley short-range order parameters α1, α2 and α12 pertaining to the first (1NN), second (2NN) and both i.e. 1NN+2NN shells, respectively. Clear evidence was found, both for non-irradiated and irradiated samples that the actual distribution of Cr atoms is characteristic of the shell, and for a given shell it depends on the irradiation dose. In particular, α1 is positive, hence indicates an under population of Cr atoms in 1NN with respect to the random case, α2 is negative, giving evidence thereby that 2NN is overpopulated by Cr atoms, and α12 is weakly positive. Under the applied irradiation the number of Cr atoms in both neighbor shells decreased signifying thereby a clustering of Cr atoms. The underlying decrease of Cr concentration within the 1NN–2NN volume around the probe Fe atoms was estimated at 1.5at.% ranging between 2.1 for the lowest and 0.8at.% for the highest dose.

Quantitative estimates on the binding energy for hydrogen in non-relativistic QED, II: The spin case

The hydrogen binding energy in the Pauli–Fierz model with the spin Zeeman term is determined up to the order α3, where α denotes the fine-structure constant.

On the spin separation of algebraic two-component relativistic Hamiltonians: molecular properties.

The idea for separating the algebraic exact two-component (X2C) relativistic Hamiltonians into spin-free (sf) and spin-dependent terms [Z. Li, Y. Xiao, and W. Liu, J. Chem. Phys. 137, 154114 (2012)] is extended to both electric and magnetic molecular properties. Taking the spin-free terms (which are correct to infinite order in α ≈ 1/137) as zeroth order, the spin-dependent terms can be treated to any desired order via analytic derivative technique. This is further facilitated by unified Sylvester equations for the response of the decoupling and renormalization matrices to single or multiple perturbations. For practical purposes, explicit expressions of order α(2) in spin are also given for electric and magnetic properties, as well as two-electron spin-orbit couplings. At this order, the response of the decoupling and renormalization matrices is not required, such that the expressions are very compact and completely parallel to those based on the Breit-Pauli (BP) Hamiltonian. However, the former employ sf-X2C wave functions, whereas the latter can only use nonrelativistic wave functions. As the sf-X2C terms can readily be interfaced with any nonrelativistic program, the implementation of the O(α²) spin-orbit corrections to sf-X2C properties requires only marginal revisions of the routines for evaluating the BP type of corrections.

Reassessing the importance of the colour-singlet contributions to direct [formula omitted] production at the LHC and the Tevatron

We show that the colour-singlet contributions to the hadroproduction of J/ψ in association with a W boson are sizable, if not dominant over the colour-octet contributions. They are of two kinds, sg→J/ψ+c+W at αS3α and qq¯′→γ⋆/Z⋆W→J/ψW at order α3. These have not been considered in the literature until now. Our conclusion is that the hadroproduction of a J/ψ in association with a W boson cannot be claimed as a clean probe of the colour-octet mechanism. The rate are small even at the LHC and it will be very delicate to disentangle the colour-octet contributions from the sizable colour-singlet ones and from the possibly large double-parton-scattering contributions. During this analysis, we have also noted that, for reactions such as the production of a J/ψ by light quark–antiquark fusion, the colour-singlet contribution via an off-shell photon is of the order of the expectation from the colour-octet contribution via an off-shell gluon. This is relevant for inclusive production at low energies close to the threshold. Such an observation also likely extends to other processes naturally involving light-quark annihilation.

Stabilization of the α2 Isoform of Na,K-ATPase by Mutations in a Phospholipid Binding Pocket

The α2 isoform of Na,K-ATPase plays a crucial role in Ca(2+) handling, muscle contraction, and inotropic effects of cardiac glycosides. Thus, structural, functional, and pharmacological comparisons of α1, α2, and α3 are of great interest. In Pichia pastoris membranes expressing human α1β1, α2β1, and α3β1 isoforms, or using the purified isoform proteins, α2 is most easily inactivated by heating and detergent (α2 ≫ α3 > α1). We have examined an hypothesis that instability of α2 is caused by weak interactions with phosphatidylserine, which stabilizes the protein. Three residues, unique to α2, in trans-membrane segments M8 (Ala-920), M9 (Leu-955), and M10 (Val-981) were replaced by equivalent residues in α1, singly or together. Judged by the sensitivity of the purified proteins to heat, detergent, "affinity" for phosphatidylserine, and stabilization by FXYD1, the triple mutant (A920V/L955F/V981P, called α2VFP) has stability properties close to α1, although single mutants have only modest or insignificant effects. Functional differences between α1 and α2 are unaffected in α2VFP. A compound, 6-pentyl-2-pyrone, isolated from the marine fungus Trichoderma gamsii is a novel probe of specific phospholipid-protein interactions. 6-Pentyl-2-pyrone inactivates the isoforms in the order α2 ≫ α3 > α1, and α2VFP and FXYD1 protect the isoforms. In native rat heart sarcolemma membranes, which contain α1, α2, and α3 isoforms, a component attributable to α2 is the least stable. The data provide clear evidence for a specific phosphatidylserine binding pocket between M8, M9, and M10 and confirm that the instability of α2 is due to suboptimal interactions with phosphatidylserine. In physiological conditions, the instability of α2 may be important for its cellular regulatory functions.

High-precision spectroscopy as a test of quantum electrodynamics in light atomic systems

This paper presents a review of recent progress in high-precision calculations for the ground state and low-lying excited states of helium, including the nonrelativistic energy, relativistic corrections of α2 Ry, and quantum electrodynamic (QED) corrections of lowest order α3 Ry and next-to-leading-order α4 Ry, where α is the fine-structure constant. The calculations include the terms of order α4 Ry recently obtained by Pachucki (Phys. Rev. A, 74, 062510 (2006)). Estimates of the terms of order α5 Ry, including two-loop binding corrections, are included. Comparisons with experimental ionization energies indicate reasonably good agreement for the 1s2 1S0, 1s2s 1S0, 1s2s 3S1, and 1s2p 3Pcm states, but there is a significant discrepancy for the 1s2p 1P1 state of 5.6± 3.2 MHz. An asymptotic formula for the calculation of the Bethe logarithm for Rydberg states with large angular momentum L is presented in an Appendix. PACS Nos.: 31.30.Gs, 31.30.Jv

Fine and hyperfine structure of P-wave levels in muonic hydrogen

Corrections of order α5 and α6 are calculated for muonic hydrogen in the fine-structure interval ΔEfs = E(2P3/2) − E(2P1/2) and in the hyperfine structure of the 2P1/2-and 2P3/2-wave energy levels. The resulting values of ΔEfs = 8352.08 µeV, ΔẼhfs(2P1/2) = 7819.80 µeV, and ΔẼhfs(2P3/2) = 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.

Coherent pair production by energetic gamma rays incident on quasicrystals

We develop a Born-approximation theory of coherent pair production (CPP) of electrons by energetic gamma rays incident on an icosahedral quasicrystal, described by a schematic model (K model) that includes phonon and phason disorder. Our main result is a formula for the cross-section dσcpp/dε+ for CPP, differential with respect to the positron energy ε+ and of order α2 in the fine-structure constant α≈1/137, but which is otherwise exact. We discuss results of numerical calculations of dσcpp/dε+ versus y=ε+/k for gamma rays of energies k=20MeV, 200MeV, and 3GeV, incident on icosahedral Al–Mn–Si, described as a special case of the K model (vertex model). This consists in placing an Mn atom at each vertex of the relevant Ammann tiles. Our calculations include CPP of types A and B. Both types exhibit vertical intensity drops at irregularly distributed y-values, many of these drops being so large that they should be observable experimentally. They are analogous to the large intensity drops exhibited by coherent bremsstrahlung in quasicrystals. We predict that CPP drops also occur for realistic models of i-Al–Mn–Si at the same y-values as for the vertex model, but whose magnitudes may differ from those predicted by this model.

On the 1D Coulomb Klein–Gordon equation

For a single particle of mass m experiencing the potential −α/|x|, the 1D Klein–Gordon equation is mathematically underdefined even when α ≪ 1: unique solutions require some physically motivated prescription for handling the singularity at the origin. The procedure appropriate in most cases is to soften the singularity by means of a cutoff. Here we study the bound states of spin-zero particles in the potential −α/(|x| + R), extending the nonrelativistic results of Loudon (1959 Am. J. Phys. 27 649) to allow for relativistic effects, which become appreciable and eventually dominant for small enough mR: they are totally different from conclusions based hitherto on mathematically simple-seeming matching conditions on the wavefunction at x = 0. For realizable R, all relativistic effects remain very small; but with mR decreasing to order α2 the ground-state energy E decreases through zero, and soon after that mR reaches a finite critical value below which E becomes complex, signalling a breakdown of the single-particle theory. At this critical point of the curve E(mR) the Klein–Gordon norm changes sign: the curve has a lower branch describing a bound antiparticle state, with positive energy −E, which exists for mR between the critical and some higher value where E reaches −m. Though apparently unanticipated in this context, similar scenarios are in fact familiar for strong short-range potentials (1D or 3D), and also for strong 3D Coulomb potentials with α of order unity.

Three-loop radiative corrections to Lamb shift and hyperfine splitting

We consider three-loop radiative corrections to the Lamb shift and hyperfine splitting. Corrections of order α3(Zα)5m are the largest still unknown contributions to the Lamb shift in hydrogen. We calculate radiative corrections to the Lamb shift and hyperfine splitting generated by the diagrams with insertions of one radiative photon and electron polarization loops in the graphs with two external photons. We also obtain corrections generated by the gauge-invariant sets of diagrams with two reducible radiative photon insertions in the electron line and a polarization operator insertion in one of the radiative photons, and diagrams with two reducible radiative photon insertions in the electron line and a polarization operator insertion in one of the external photons. Corrections to the Lamb shift and hyperfine splitting generated by the diagrams with insertions of the three-loop one-particle reducible diagrams with radiative photons in the electron line are calculated in the Yennie gauge.PACS Nos.: 12.20.Ds, 31.30.Jv, 32.10.Fn, 36.10.Dr