Second-order Matrix Elements Research Articles

Theory of Majorana-Type Heavy Ion Double Charge Exchange Reactions by Pion–Nucleon Isotensor Interactions

The theory of heavy ion double charge exchange (DCE) reactions proceeding by effective rank-2 isotensor interactions is presented. Virtual pion–nucleon charge exchange interactions are investigated as the source for induced isotensor interactions, giving rise to the Majorana DCE (MDCE) reaction mechanism. MDCE is of a generic character, proceeding through pairs of complementary (π±,π∓) reactions in the projectile and target nucleus. The dynamics of the elementary processes is discussed, where the excitation of pion–nucleon resonances are of central importance. Investigations of initial and final state ion–ion interactions show that these effects are acting as vertex renormalizations. In closure approximation, well justified by the finite pion mass, the second-order transition matrix elements reduce to pion potentials and effective two-body isotensor DCE interactions, giving rise also to two-body correlations in either of the participating nuclei. Connections to neutrinoless Majorana double beta decay (MDBD) are elucidated at various levels of the dynamics, from the underlying fundamental electro-weak and QCD scales to the physical scales of nuclear MDBD and MDCE physics. It is pointed out that heavy ion MDCE reactions may also proceed by competing electro-weak charge exchange processes, leading to lepton MDCE by electrons, positrons, and neutrinos.

Generalized parton distributions from the off-forward Compton amplitude in lattice QCD

We determine the properties of generalised parton distributions (GPDs) from a lattice QCD calculation of the off-forward Compton amplitude (OFCA). By extending the Feynman-Hellmann relation to second-order matrix elements at off-forward kinematics, this amplitude can be calculated from lattice propagators computed in the presence of a background field. Using an operator product expansion, we show that the deeply-virtual part of the OFCA can be parameterised in terms of the low-order Mellin moments of the GPDs. We apply this formalism to a numerical investigation for zero-skewness kinematics at two values of the soft momentum transfer, $t = -1.1, -2.2 \;\text{GeV}^2$, and a pion mass of $m_{\pi}\approx 470\;\text{MeV}$. The form factors of the lowest two moments of the nucleon GPDs are determined, including the first lattice QCD determination of the $n=4$ moments. Hence we demonstrate the viability of this method to calculate the OFCA from first principles, and thereby provide novel constraint on the $x$- and $t$-dependence of GPDs.

Multipole transitions to determine lifetimes and polarizabilities in Mg-like ions from Si2+ to Fm88+

The relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate the multipole ($E1,M1,E2,M2$, and $E3$) matrix elements to determine the $3s3p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{2}$ lifetime and multipole polarizabilities in Mg-like ions. The electric multipole matrix elements are determined in length and velocity forms. The calculations start from a $1{s}^{2}2{s}^{2}2{p}^{6}$ Dirac-Fock potential. First-order RMBPT is used to obtain intermediate coupling coefficients, and second-order RMBPT is used to calculate transition matrix elements. Contributions from negative-energy states are included in the second-order multipole matrix elements to ensure gauge independence of transition amplitudes. The details of our calculations of the multipole polarizabilities are illustrated for Mg-like $\mathrm{Si}{}^{2+}$, $\mathrm{Fe}{}^{14+}$, $\mathrm{Kr}{}^{24+}$, $\mathrm{Mo}{}^{30+}$, $\mathrm{Xe}{}^{42+}$, and $\mathrm{W}{}^{62+}$ ions. Our RMBPT results are compared with available theoretical results and experimental measurements. Trends of the line strengths, transition rates, and contributions in the ground-state multipole polarizabilities as functions of $Z$ are illustrated graphically in Mg-like ions with $Z=14--100$.

Formal approach to deriving analytically asymptotic formulas for static polarizabilities of atoms and ions in Rydberg states

On the basis of analytical expressions in the Fues model potential approach for the second-order Stark effect on single-electron Rydberg states in atoms and ions, general equations are derived for coefficients of polynomials in powers of the principal quantum number n in asymptotic presentations of static scalar and tensor dipole polarizabilities. The power dependence for polarizabilities of isolated Rydberg states |nl〉 at n ≫ 1 scales as n7, in contrast with that of polarizabilities for hydrogenic states, degenerate in the orbital quantum number l ⩽ n − 1, which scales as n6. This difference is demonstrated analytically in the asymptotic dependencies of the second-order matrix elements, determining the Stark shifts of the isolated and degenerate states. Analytical equations for polynomial coefficients use the data on quantum defects determined from the level energies of corresponding series of states. Numerical values of coefficients are presented for S-, P- and D-series of Rydberg states in neutral atoms of alkali-metal elements and helium in comparison with existing data of the literature. The analytical equations are also used for determining numerical values of coefficients in asymptotic polynomials for polarizabilities of Rydberg states in positive ions of alkaline-earth-metal elements.

Antenna showers with one-loop matrix elements

Abstract We consider the probability for a colour-singlet $ q\overline{q} $ pair to emit a gluon, in strongly and smoothly ordered antenna showers. We expand to second order in α s and compare to the second-order QCD matrix elements for Z → 3 jets, neglecting terms suppressed by $ {1 \left/ {{N_C^2}} \right.} $ . We give a prescription that corrects the shower to the matrix-element result at this order for both soft and hard emissions, thereby explicitly reducing its dependence on evolution- and renormalization-scale choices. We confirm that the choice of p ⊥ for both of these scales absorbs all logarithms through $ \mathcal{O}\left( {\alpha_s^2} \right) $ , and contrast this with various alternatives. We include these corrections in the vincia shower generator and study the impact on LEP event-shape and fragmentation observables. An uncertainty estimate is provided for each event, in the form of a vector of alternative weights.

Observable [formula omitted] at LHC and second-order QED matrix element in [formula omitted] decays

In a recent publication by ATLAS Collaboration a new observable, the so-called ϕη⁎ angle, was used for precise measurement of transverse Z momentum. One of the dominant systematic errors for this measurement originates from the theoretical control of QED final-state bremsstrahlung. At present, it is estimated at the 0.3% level for the shape of the ϕη⁎ distribution. In this Letter we discuss the possible effects of the second-order QED matrix element for that quantity. For that purpose, results from simulations based on the Yennie–Frautchi–Suura (YFS) exponentiation and featuring the second-order matrix elements are used and compared with the case when the matrix element is restricted to the first-order. From this study we conclude that in order to reach the precision below 0.3% for the ϕη⁎ distribution at the LHC, inclusion of the second-order QED matrix element in a respective Monte Carlo event generator is necessary.

Wavelengths and transition rates for nl–n′l′ transitions in Be-, B-, Mg-, Al-, Ca-, Zn-, Ag- and Yb-like tungsten ions

We present a comprehensive theoretical study of atomic characteristics of eight isoelectronic sequences of tungsten ions in a broad range of wavelengths and transitions. In particular, excitation energies, oscillator strengths and transition probabilities are calculated for nl–n′l′ transitions in W70 +, W69 +, W62 +, W61 +, W54 +, W44 +, W27 + and W4 + ions. Atomic structure and radiative characteristics of Be-like ([He]2lnl′, n = 2, 3), B-like ([He]2l2l′2l″), Mg-like ([Ne]3l3l′), Al-like ([Ne]3l3l′3l″), Ca-like ([Ar]3d4l), Zn-like ([Ni]4l4l′), Ag-like [Kr]4d10nl) and Yb-like ([Xe]4f145l6l′) tungsten ions are computed by the relativistic many-body perturbation theory (RMBPT) method. The calculations start from a 1s2 Dirac–Fock potential for Be- and B-like W, from the 1s22s22p6 Dirac–Fock potential for Mg- and Al-like W; from the 1s22s22p63s23p6 and 1s22s22p63s23p63d10 Dirac–Fock potentials for Ca- and Zn-like W, respectively. Evaluation of properties of Ag-like and Yb-like ions starts from a Dirac–Fock potential with only partially filled n = 4 shell (1s22s22p63s23p63d104s24p64d10) and n = 5 shell (1s22s22p63s23p63d104s24p64d104f145s25p6), respectively. First-order perturbation theory is used to obtain intermediate coupling coefficients, and second-order RMBPT is used to determine the matrix elements. The contributions from negative-energy states are included in the second-order E1 matrix elements to achieve agreement between length-form and velocity-form amplitudes. Our calculations present benchmark data for many yet unmeasured properties of tungsten ions and are in particular important in diagnostics of tungsten plasma of a broad range of temperatures as well as for future ITER plasmas.

Relativistic many-body calculations of the oscillator strengths, transition rates and polarizabilities in Zn-like ions

Transition rates, oscillator strengths and line strengths are calculated for electric-dipole (E1) transitions between even-parity 4s2, 4p2, 4s4d, 4d2, 4p4f and 4f2 states and odd-parity 4s4p, 4s4f, 4p4d and 4d4f states in Zn-like ions with the nuclear charges ranging from Z = 32 to 100. Relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate retarded E1 matrix elements in length and velocity forms. The calculations start from a 1s22s22p63s23p63d10 Dirac–Fock potential. First-order RMBPT is used to obtain intermediate coupling coefficients and second-order RMBPT is used to calculate transition matrix elements. Contributions from negative-energy states are included in the second-order E1 matrix elements to ensure the gauge independence of transition amplitudes. Transition energies used in the calculation of oscillator strengths and transition rates are from second-order RMBPT. Ground state scalar α0(4s2 1S0) polarizabilities are calculated for Zn-like ions from Z = 33 to 100. To evaluate the α0(4s2 1S0) polarizabilities, we calculate RMBPT energies for the odd-parity 4l5l′ complex with J = 1 and line strengths between the even-parity 4l4l′ complex with J = 0 and the odd-parity 4l5l′, 4l6l′ complexes with J = 1.

Some remarks on dipole showers and the DGLAP equation

It has been argued recently that parton showers based on color dipoles conflict with collinear factorization and do not lead to the correct Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation. We show that this conclusion is based on an inappropriate assumption, namely, the choice of the gluon energy as evolution variable. We further show numerically that Monte Carlo programs based on dipole showers with ``infrared-sensible'' evolution variables reproduce the DGLAP equation both in asymptotic form as well as in comparison to the leading behavior of second-order QCD matrix elements.

Separation of long-range and short-range interactions in Rydberg states of diatomic molecules

Observation and analysis of the f([script-l]=3), g([script-l]=4), and h([script-l]=5) Rydberg series of CaF in the range 13<or=n<or=17 is presented. Simultaneous analysis of the f, g, and h Rydberg series of CaF, combined with significant improvements to the long-range model for nonpenetrating Rydberg states, provides a generally applicable scheme for separating the effects of the long-range (electric multipole and polarization) and short-range (core penetration) interactions between the Rydberg electron and ion core in diatomic molecules. Techniques for rigorous assignment of nonpenetrating states, as well as extensions of the long-range model, are discussed. Explicit formulas for the first- and second-order matrix elements of the first four anisotropic electric multipole interactions (electric dipole, quadrupole, octupole, and hexadecapole) are given. The discrepancies between the observed behavior and that predicted by the long-range model, which are particularly significant for the f series, are shown to be due to penetration of the Rydberg electron wavefunction within the ion core. We show that these penetration effects can be understood within the framework of ligand field theory and conclude with a discussion of the relative contributions of the long- and short-range interactions to the quantum defects of the core-penetrating "s," "p," and "d" series of CaF.

Relativistic many-body calculations of atomic properties in Pd-like ions

Wavelengths, transition rates, and line strengths are calculated for the 85 possible multipole transitions between the excited 4p6 4d9 4f, 4p6 4d9 5l, 4p5 4d10 4f, and 4p5 4d10 5l states and the ground 4p6 4d10 state in Pd-like ions with the nuclear charges ranging from Z = 47 to 100. Relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate energies and transition rates for multipole transitions in hole–particle systems. This method is based on the relativistic many-body perturbation theory, agrees with MCDF calculations in lowest order, includes all second-order correlation corrections, and includes corrections from negative energy states. The calculations start from a [Zn]4p64d10 Dirac–Fock potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and second-order RMBPT is used to determine the matrix elements. The contributions from negative-energy states are included in the second-order multipole matrix elements. The resulting transition energies and transition rates are compared with experimental values and with results from other recent calculations. Trends of the transitions rates for the selected multipole transitions as function of Z are illustrated graphically. The Z dependence of the energy splitting for all triplet terms of the 4p64d9 4f and 4p64d9 5l configurations are shown for Z = 47–100. PACS Nos.: 31.15.Ar, 31.15.Md, 32.70.Cs, 32.30.Rj, 31.25.Jf

Relativistic many-body calculations of electric-dipole lifetimes, rates and oscillator strengths of Δn = 0 transitions between 3l−14l′ states in Ni-like ions

Transition rates, oscillator strengths and line strengths are calculated for electric-dipole (E1) transitions between odd-parity 3s23p63d94l1, 3s23p53d104l2 and 3s3p63d104l1 states and even-parity 3s23p63d94l2, 3s23p53d104l1 and 3s3p63d104l2 (with 4l1 = 4p, 4f and 4l2 = 4s, 4d) in Ni-like ions with the nuclear charges ranging from Z = 34 to 100. Relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate retarded E1 matrix elements in length and velocity forms. The calculations start from a 1s22s22p63s23p63d10 Dirac–Fock potential. First-order RMBPT is used to obtain intermediate coupling coefficients and second-order RMBPT is used to calculate transition matrix elements. Contributions from negative-energy states are included in the second-order E1 matrix elements to ensure the gauge independence of transition amplitudes. Transition energies used in the calculation of oscillator strengths and transition rates are from second-order RMBPT. Lifetimes of the 3s23p63d94d levels are given for Z = 34–100. These atomic data are important in modelling of M-shell radiation spectra of heavy ions generated in electron beam ion trap experiments and in M-shell diagnostics of plasmas.

Multipole (E1, M1, E2, M2, E3, M3) transition wavelengths and rates between 3l−15l′ excited and ground states in nickel-like ions

A relativistic many-body method is used to calculate energy and transition rates for multipole transitions in many-electron ions. This method is based on relativistic many-body perturbation theory (RMBPT), agrees with MCDF calculations in lowest order, includes all second-order correlation corrections and includes corrections from negative energy states. Reduced matrix elements, oscillator strengths and transition rates are calculated for electric-multipole (dipole (E1), quadrupole (E2) and octupole (E3)) and magnetic-multipole (dipole (M1), quadrupole (M2) and octupole (M3)) transitions between 3l−15l′ excited and ground states in Ni-like ions with nuclear charges ranging from Z = 30 to 100. The calculations start from a 1s22s22p63s23p63d10 Dirac–Fock potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and second-order RMBPT is used to determine the matrix elements. A detailed discussion of the various contributions to the dipole matrix elements and energy levels is given for nickel-like tungsten (Z = 74). The contributions from negative-energy states are included in the second-order E1, M1, E2 M2, E3 and M3 matrix elements. The resulting transition energies and transition rates are compared with experimental values and with results from other recent calculations. These atomic data are important in modelling of M-shell radiation spectra of heavy ions generated in electron beam ion trap experiments and in M-shell diagnostics of plasmas.

Relativistic many-body calculations of multipole (E1, M1, E2, M2, E3, and M3) transition wavelengths and rates between 3l −14l′ excited and ground states in nickel-like ions

Wavelengths, transition rates, and line strengths are calculated for the 76 possible multipole (E1, M1, E2, M2, E3, and M3) transitions between the excited 3s 23p 63d 94l, 3s 23p 53d 104l, and 3s3p 63d 104l and the ground 3s 23p 63d 10 states in Ni-like ions with the nuclear charges ranging from Z = 30 to 100. The relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate energies and transition rates for multipole transitions in hole–particle systems. This method is based on relativistic many-body perturbation theory, agrees with MCDF calculations in lowest-order, includes all second-order correlation corrections, and includes corrections from negative energy states. The calculations start from a 1s 22s 22p 63s 23p 63d 10 Dirac–Fock potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and the second-order RMBPT is used to determine the matrix elements. The contributions from negative-energy states are included in the second-order E1, M1, E2, M2, E3, and M3 matrix elements. The resulting transition energies and transition rates are compared with experimental values and with results from other recent calculations. As a result, we present wavelengths and transition rates data for the selected transitions that include the 76 possible multipole (E1, M1, E2, M2, E3, and M3) transitions between the excited 3s 23p 63d 94l, 3s 23p 53d 104l, and 3s3p 63d 104l states and the ground 3s 23p 63d 10 state in Ni-like ions. Trends of the line strengths for the 76 multipole transitions and oscillator strengths for the 13 E1 transitions as function of Z are illustrated graphically. The Z-dependence of the energy splitting for all triplet terms of the 3s 23p 63d 94l, 3s 23p 53d 104l, and 3s3p 63d 104l configurations are shown in the range of Z = 30–100.

Relativistic many-body calculations of excitation energies, line strengths, transition rates, and oscillator strengths in Pd-like ions

Excitation energies, line strengths, oscillator strengths, and transition probabilities are calculated for 4d–14f, 4d–15p, 4d–15f, and 4d–16p hole–particle states in Pd-like ions with nuclear charges Z ranging from 49 to 100. Relativistic many-body perturbation theory (MBPT), including the Breit interaction, is used to evaluate retarded E1 matrix elements in length and velocity forms. The calculations start from a [Kr] 4d10 closed-shell Dirac–Hartree–Fock (DHF) potential and include second- and third-order Coulomb corrections and second-order Breit–Coulomb corrections. First-order perturbation theory is used to obtain intermediate-coupling coefficients and second-order MBPT is used to determine matrix elements. Contributions from negative-energy states are included in the second-order electric-dipole matrix elements. The resulting transition energies, line strengths, and transition rates are compared with experimental values and with other recent calculations. Trends of oscillator strengths as functions of nuclear charge Z are shown graphically for all transitions from the 4d–14f, 4d–15p, 4d–15f, and 4d–16p states to the ground state. PACS Nos.: 31.15.Ar, 31.15.Md, 32.70.Cs, 32.30.Rj, 31.25.Jf

Relativistic many-body calculations of electric-dipole lifetimes, transition rates and oscillator strengths for 2l−13l′ states in Ne-like ions

Transition rates, oscillator strengths and line strengths are calculated for electric-dipole (E1) transitions between odd-parity 2s22p53s, 2s22p53d and 2s2p63p states and even-parity 2s22p53p, 2s2p63s and 2s2p63d states in Ne-like ions with the nuclear charges ranging from Z = 14 to 100. Relativistic many-body perturbation theory (RMBPT), including the Breit interaction, is used to evaluate retarded E1 matrix elements in length and velocity forms. The calculations start from a 1s22s22p6 Dirac–Fock potential. First-order RMBPT is used to obtain intermediate coupling coefficients and second-order RMBPT is used to calculate transition matrix elements. Contributions from negative-energy states are included in the second-order E1 matrix elements to ensure the gauge independence of transition amplitudes. Transition energies used in the calculation of oscillator strengths and transition rates are from second-order RMBPT. Lifetimes of the 16 even-parity and 18 odd-parity levels are given for Z = 14–100. Transition rates, line strengths and oscillator strengths are compared with critically evaluated experimental values and with results from other recent calculations. These atomic data are important in the modelling of L-shell radiation spectra of heavy ions generated in electron beam ion trap experiments and in L-shell diagnostics of plasmas.

E1, E2, M1, and M2 transitions in the nickel isoelectronicsequence

A relativistic many-body method is developed to calculate energy and transition rates for multipole transitions in many-electron ions. This method is based on relativistic many-body perturbation theory (RMBPT), agrees with MCDF calculations in lowest order, includes all second-order correlation corrections, and includes corrections from negative-energy states. Reduced matrix elements, oscillator strengths, and transition rates are calculated for electric-dipole (E1) and electric-quadrupole (E2) transitions, and magnetic-dipole (M1) and magnetic-quadrupole (M2) transitions in Ni-like ions with nuclear charges ranging from Z = 30 to 100. The calculations start from a 1s22s22p63s23p63d10 Dirac–Fock potential. First-order perturbation theory is used to obtain intermediate-coupling coefficients, and second-order RMBPT is used to determine the matrix elements. The contributions from negative-energy states are included in the second-order E1, M1, E2, and M2 matrix elements. The resulting transition energies and transition rates are compared with experimental values and withresults from other recent calculations.PACS Nos.: 32.30.Rj, 32.70.Cs, 32.80.Rm, 34.70.+e

Relativistic many-body calculations of electric-dipole lifetimes, transition rates, and oscillator strengths for n=3 states in Al-like ions

Transition rates, oscillator strengths, and line strengths are calculated for electric-dipole (E1) transitions between even-parity 3 s3 p 2, 3 s 23 d, 3 p 23 d, 3 d 23 s, and 3 d 3 states and odd-parity 3 s 23 p, 3 p 3, 3 s3 p3 d, and 3 d 23 p states in Al-like ions with the nuclear charges ranging from Z=15 to 100. Relativistic many-body perturbation theory (MBPT), including the Breit interaction, is used to evaluate retarded E1 matrix elements in length and velocity forms. The calculations start from a 1 s 22 s 22 p 6 Dirac–Fock potential. First-order MBPT is used to obtain intermediate coupling coefficients and second-order MBPT is used to calculate transition matrix elements. Contributions from negative-energy states are included in the second-order E1 matrix elements to ensure gauge-independence of transition amplitudes. Transition energies used in the calculation of oscillator strengths and transition rates are from second-order MBPT. Transition rates, line strengths, and oscillator strengths are compared with critically evaluated experimental values and with results from other recent calculations. We present data for selected transitions between 10 of the possible 73 even-parity 3 s3 p 2, 3 s 23 d states and 29 of the possible 75 odd-parity 3 s 23 p, 3 p 3, and 3 s3 p3 d states in Al-like ions. Trends of the transition rates as functions of Z are illustrated graphically for 220 of the 3220 possible transitions. Lifetimes of the 10 even-parity lower levels and the 27 odd-parity upper levels are given for Z=15–100.

Vibrational Corrections to Transition Properties

The vibrational contributions to second-order transition properties are considered in the framework of the Born—Oppenheimer approximation. It is shown that the usual formula for vibrational second-order transition matrix elements is incomplete and needs to be supplemented by a term of purely vibrational origin. This pure vibrational contribution is calculated for vibrational transition polarizabilities in LiH and BeF and found to be quite significant. Its inclusion in theoretically calculated data for the Raman intensities appears to be necessary.

Excitation energies and transition rates in magnesiumlike ions

Excitation energies, transition probabilities, and lifetimes are calculated for 3 l 13l 2-3l 33l 4 electric dipole transitions in Mg-like ions with nuclear charges Z ranging from 13 to 100. Relativistic many-body perturbation theory ~MBPT!, including the Breit interaction, is used to evaluate retarded E1 matrix elements in length and velocity forms. The calculations start from a 1 s 2 2s 2 2p 6 Dirac-Fock potential. First-order perturbation theory is used to obtain intermediate coupling coefficients, and the second-order MBPT is used to determine the matrix elements. The contributions from negative-energy states are included in the second-order E1 matrix elements to achieve agreement between length-form and velocity-form amplitudes. The resulting transition energies and lifetimes are compared with experimental values and with results from other recent calculations.