Two-photon Decay Rates Research Articles

Two-photon decay rates in heliumlike ions: Finite-nuclear-mass effects

The spontaneous two-photon decay rates from the $1s2s\phantom{\rule{0.16em}{0ex}}^{1}S_{0}$ level to the ground state $1{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S_{0}$ in helium and its isoelectronic ions through neon ($Z=10$) are calculated, including the effects of finite nuclear mass. In all cases the length and velocity results agree to eight or more figures, demonstrating that the theoretical formulation correctly takes into account the effects of mass scaling, mass polarization, and motion of the nucleus in the center-of-mass frame. Algebraic relationships are derived and tested relating the expansion coefficients in powers of $\ensuremath{\mu}/M$ for mass polarization, where $\ensuremath{\mu}$ is the electron reduced mass and $M$ is the nuclear mass. Muonic, pionic, and antiprotonic helium are included as extreme test cases where mass polarization is large. The results are compared with experiment and previous calculations.

Vacuum polarization and finite-nuclear-size effects in the two-photon decay of hydrogenlike ions

The total two-photon decay rate of hydrogen-like ions is studied using relativistic quantum electrodynamics. In particular, we analyse how finite nuclear size and QED vacuum polarization corrections affect the decay rate. To calculate these corrections, a finite basis set method based on $B$-splines is used for the generation of quasi-complete atomic spectra and, hence, of the relativistic Green's function. By making use of this $B$-spline approach, high precision calculations have been performed for the $2s_{1/2} \to 1s_{1/2} + 2\gamma$ and $2p_{1/2} \to 1s_{1/2} + 2\gamma$ decay of hydrogen-like ions along the entire isoelectronic sequence. The results of these calculations show that both, QED and finite nuclear size effects, are comparatively weak for the $2s_{1/2} \to 1s_{1/2} + 2\gamma$ transition. In contrast, they are much more pronounced for the $2p_{1/2}\to 1s_{1/2} + 2\gamma$ decay, where, for hydrogen-like Uranium, the decay rate is reduced by 0.484% due to the finite nuclear size and enhanced by 0.239% if the vacuum polarization is taken into account.

Ward identity of the vector current and the decay rate of ηc→γγ in lattice QCD

Using a recently proposed method arXiv:1910.11597 (Yu Meng et al.), we study the two-photon decay rate of $\eta_c$ using two $N_f=2$ twisted mass gauge ensembles with lattice spacings $0.067$fm and $0.085$fm. The results obtained from these two ensembles can be extrapolated in a naive fashion to the continuum limit, yielding a result that is consistent with the experimental one within two standard deviations. To be specific, we obtain the results for two-photon decay of $\eta_c$ as $\mathcal{B}(\eta_c\rightarrow 2\gamma)= 1.29(3)(18)\times 10^{-4}$ where the first error is statistical and the second is our estimate for the systematic error caused by the finite lattice spacing. It turns out that Ward identity for the vector current is of vital importance within this new method. We find that the Ward identity is violated for local current with a finite lattice spacing, however it will be restored after the continuum limit is taken.

Improved model-independent constraints on the recombination era and development of a direct projection method

ABSTRACT The unparalleled precision of recent experiments such as Planck have allowed us to constrain standard and non-standard physics (e.g. due to dark matter annihilation or varying fundamental constants) during the recombination epoch. However, we can also probe this era of cosmic history using model-independent variations of the free electron fraction, Xe, which, in turn, affects the temperature and polarization anisotropies of the cosmic microwave background. In this paper, we improve on the previous efforts to construct and constrain these generalized perturbations in the ionization history, deriving new optimized eigenmodes based on the full Planck 2015 likelihood data, introducing the new module Fearec++. We develop a direct likelihood sampling method for attaining the numerical derivatives of the standard and non-standard parameters, and discuss complications arising from the stability of the likelihood code. We improve the amplitude constraints of the Planck 2015 principal components constructed here, μ1 = −0.09 ± 0.12, μ2 = −0.17 ± 0.20, and μ3 = −0.30 ± 0.35, finding no indication for departures from the standard recombination scenario. The error constraint on the third mode has been improved by a factor of 2.5. We utilize an efficient eigenanalyser that keeps the cross-correlations of the first three eigenmodes to ${\rm Corr\left(\mu \, \mu ^{\prime }\right)}\lt 0.1$ per cent after marginalization for all the considered data combinations. We also propose a new projection method for estimating constraints on the parameters of non-standard recombination scenarios. As an example, using our eigenmode measurements, this allows us to recreate the Planck constraint on the two-photon decay rate, A2s1s = 7.60 ± 0.64, giving an error estimate to within ≃ 0.05σ of the full MCMC result. The improvements on the eigenmode analysis using the Planck data will allow us to implement this new method for analysis with fundamental constant variations in the future.

Can non-standard recombination resolve the Hubble tension?

The inconsistent Hubble constant values derived from cosmic microwave background (CMB) observations and from local distance-ladder measurements may suggest new physics beyond the standard $\Lambda$CDM paradigm. It has been found in earlier works that, at least phenomenologically, non-standard recombination histories can reduce the $\gtrsim 4\sigma$ Hubble tension to $\sim 2\sigma$. Following this path, we vary physical and phenomenological parameters in RECFAST, the standard code to compute ionization history of the universe, to explore possible physics beyond standard recombination. We find that the CMB constraint on the Hubble constant is sensitive to the Hydrogen ionization energy and $2s \rightarrow 1s$ two-photon decay rate, both of which are atomic constants, and is insensitive to other details of recombination. Thus, the Hubble tension is very robust against perturbations of recombination history, unless exotic physics modifies the atomic constants during the recombination epoch.

Quasirelativistic Coulomb Sturmian basis sets for atomic physics problems: illustration of their applicability to two-photon decay rates in hydrogen-like atoms

The continuum and discrete quasirelativistic one-electron Sturmian basis sets associated with the second-order Dirac–Coulomb equation are constructed by solving appropriate Sturm–Liouville problems. The potential-weighted orthogonality and closure relations obeyed by the members of such sets are discussed. The special case where both quasirelativistic Coulomb Sturmian basis sets are used to obtain solutions to the Gell-Mann–Feynman equation is presented. It is proved that for the continuum quasirelativistic Sturm–Liouville problem, the spectrum of potential strengths is continuous and covers the whole real axis. We have also introduced the complete outgoing continuum quasirelativistic Coulomb wavefunctions and presented the discretization of its radial components. The compact representation of the fully relativistic and quasirelativistic Coulomb Green’s functions is outlined. By making use of the Sturmian expansion of the Coulomb Green’s functions, closed-form expressions of two-photon spontaneous decay rates are derived for arbitrary multipole channels. For the sake of assessing the domain of applicability of quasirelativistic basis sets, an illustrative application is then carried out for two-photon decay rates for hydrogen-like ions. A comparison of present results with reference values previously obtained within a fully relativistic scheme (Bona et al 2014) is undertaken.

Properties of Charmonium States in a Phenomenological Approach

We investigate the spectrum and decay rates of charmonium states within the framework of the non relativistic quark model by employing a Coulomb like potential from the perturbative one gluon exchange and the linear confining potential along with the potential derived from instanton vacuum to account for the hyperfine mass splitting of charmonium states in variational approach. We predict radiative E1, M1, two-photon, lepton and two-gluon decay rates of low lying charmonium states. An overall agreement is obtained with the experimental masses and decay widths. We have estimated the branching ratio of two gluons decaying into light hadrons.

Relativistic semiempirical-core-potential calculations in Ca+,Sr+ , and Ba+ ions on Lagrange meshes

Relativistic atomic structure calculations are carried out in alkaline-earth-metal ions using a semiempirical-core-potential approach. The systems are partitioned into frozen-core electrons and an active valence electron. The core orbitals are defined by a Dirac-Hartree-Fock calculation using the grasp2k package. The valence electron is described by a Dirac-like Hamiltonian involving a core-polarization potential to simulate the core-valence electron correlation. The associated equation is solved with the Lagrange-mesh method, which is an approximate variational approach having the form of a mesh calculation because of the use of a Gauss quadrature to calculate matrix elements. Properties involving the low-lying metastable $^2D_{3/2,5/2}$ states of Ca$^{+}$, Sr$^{+}$, and Ba$^{+}$ are studied, such as polarizabilities, one- and two-photon decay rates, and lifetimes. Good agreement is found with other theory and observation, which is promising for further applications in alkali-like systems.

Relativistic two-photon decay rates with the Lagrange-mesh method

Relativistic two-photon decay rates of the $2s_{1/2}$ and $2p_{1/2}$ states towards the $1s_{1/2}$ ground state of hydrogenic atoms are calculated by using numerically exact energies and wave functions obtained from the Dirac equation with the Lagrange-mesh method. This approach is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. Highly accurate values are obtained by a simple calculation involving different meshes for the initial, final and intermediate wave functions and for the calculation of matrix elements. The accuracy of the results with a Coulomb potential is improved by several orders of magnitude in comparison with benchmark values of the literature. The general requirement of gauge invariance is also successfully tested, down to rounding errors. The method provides high accuracies for two-photon decay rates of a particle in other potentials and is applied to a hydrogen atom embedded in a Debye plasma simulated by a Yukawa potential.

Angular distribution and forward–backward asymmetry of the Higgs-boson decay to photon and lepton pair

The Higgs-boson decay h -> gamma l+ l- for various lepton states l = (e, mu, tau) is analyzed. The differential decay width and forward-backward asymmetry are calculated as functions of the dilepton invariant mass in a model where the Higgs boson interacts with leptons and quarks via a mixture of scalar and pseudoscalar couplings. These couplings are partly constrained from data on the decays to leptons, h -> l+ l-, and quarks h -> q \bar{q} (where q = (c, b)), while the Higgs couplings to the top quark are chosen from the two-photon and two-gluon decay rates. Nonzero values of the forward-backward asymmetry will manifest effects of new physics in the Higgs sector. The decay width and asymmetry integrated over the dilepton invariant mass are also presented.

Two-photon transitions with cascades: Two-photon transition rates and two-photon level widths

An ambiguity of the separation of cascades from ``pure'' two-photon decay is confirmed with accurate numerical calculations in a gauge-invariant way. A direct evaluation of the two-photon decay width of excited states in H-like ions via the imaginary part of two-loop self-energy is presented. We demonstrate that there is a fundamental difference between the level width and the transition probability in the presence of cascades. The two-photon widths are shown to be different from the two-photon decay rates for the transitions including cascades.

Magnetic field induced transition rates in Ne- and Be-like ions for plasma diagnostics and E1M1 two-photon decay rate determination

We report on theoretical results of magnetic field induced transitions (MITs) in Ne- and Be-like ions without nuclear spin for two applications. Firstly, MITs are promising candidates in the determination of magnetic fields in plasmas. In our work on Ne-like ions we present accurate theoretical MIT rates for 2p6 1So – 2p53s 3Po,2 [1]. Furthermore, for Be-like ions, it has been proposed to extract the rate of the E1M1 two-photon transition 2s2 1S0 – 2s2p 3P0 by measuring the lifetime of the 3P0 state using a storage ring, which involves an external magnetic field. The MIT rates are carefully evaluated and shown to be of the same order as the E1M1 rates [2].

Two-photon decay rates of hydrogenlike ions revisited by using Dirac-Coulomb Sturmian expansions of the first order

A fully relativistic multipole scheme is formulated to study two-photon emission processes in hydrogenlike ions with an infinitely heavy, pointlike, and spinless nucleus of charge up to 100. By making use of the Sturmian expansion of the Dirac-Coulomb Green function of the first order constructed by Szmytkowski, closed-form expressions are derived for arbitrary multipole channels. In the nonrelativistic limit, well-known formulas established previously are retrieved. For the sake of assessing the effectiveness of our approach, numerical applications are then carried out for two-photon decay rates of the selected $2{s}_{1/2}$ and $2{p}_{1/2}$ atomic states. To this end, radial integrals, the most crucial quantities involved in the matrix elements, are treated with great care by means of two suitable techniques that agree with each other quite closely so that very accurate values are obtained regardless of the choice of parameters, such as radial quantum numbers and orders of spherical Bessel functions of the first kind. In addition, the convergence and stability of computations are checked in connection with the intermediate-state summation, which appears within the second-order perturbation theory. As expected, the gauge invariance of our fully relativistic multipole numbers is confirmed. Relativistic effects, and the influence of the negative spectrum of the complete set of Dirac-Coulomb Sturmians of first order and retardation truncations in the transition operator are examined. Finally, a comparison is undertaken of our two-photon relativistic calculations with refined predictions of other authors based on finite basis-set methods widely employed over the past decades.

Top anomalous magnetic moment and the two-photon decay of the Higgs boson

We compute the dependence of the Higgs to two-photon decay rate ${\ensuremath{\Gamma}}_{h\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}}$ on the top quark gyromagnetic factor ${g}_{t}$ in the heavy top limit and evaluate the expected change for one-loop SM correction to ${g}_{t}$. Our results are general and allow consideration of further modifications of ${g}_{t}$, and we predict the resultant ${\ensuremath{\Gamma}}_{h\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}}$.

Effect of an external magnetic field on the determination of E1M1 two-photon decay rates in Be-like ions

In this work we report on ab initio theoretical results for the magnetic-field-induced $2s\phantom{\rule{0.16em}{0ex}}2p\phantom{\rule{0.16em}{0ex}}{}^{3}{P}_{0}\ensuremath{\rightarrow}2{s}^{2\phantom{\rule{0.16em}{0ex}}1}{S}_{0}$ E1 transition for ions in the beryllium isoelectronic sequence between $Z=5$ and 92. It has been proposed that the rate of the E1M1 two-photon transition $2s\phantom{\rule{0.16em}{0ex}}2p\phantom{\rule{0.16em}{0ex}}{}^{3}{P}_{0}\ensuremath{\rightarrow}2{s}^{2\phantom{\rule{0.16em}{0ex}}1}{S}_{0}$ can be extracted from the lifetime of the ${}^{3}{P}_{0}$ state in Be-like ions with zero nuclear spin by employing resonant recombination in a storage ring. This experimental approach involves a perturbing external magnetic field. The effect of this field needs to be evaluated in order to properly extract the two-photon rate from the measured decay curves. The magnetic-field-induced transition rates are carefully evaluated, and it is shown that, with a typical storage-ring field strength, it is dominant or of the same order as the E1M1 rate for low- and mid-$Z$ ions. Results for several field strengths and ions are presented, and we also give a simple $Z$-dependent formula for the rate. We estimate the uncertainties of our model to be within $5%$ for low- and mid-$Z$ ions and slightly larger for more highly charged ions. Furthermore we evaluate the importance of including both perturber states, ${}^{3}{P}_{1}$ and ${}^{1}{P}_{1}$, and it is shown that excluding the influence of the ${}^{1}{P}_{1}$ perturber overestimates the rate by up to $26%$ for the mid-$Z$ ions.

Interelectronic-interaction effects on the two-photon decay rates of heavy He-like ions

Based on a rigorous quantum electrodynamics (QED) approach, a theoretical analysis is performed for the two-photon transitions in heavy He-like ions. Special attention is paid to the interelectronic-interaction corrections to the decay rates that are taken into account within the two-time Green-function method. Detailed calculations are carried out for the two-photon transitions $2 {}^{1}{S}_{0}\ensuremath{\rightarrow}1 {}^{1}{S}_{0}$ and $2 {}^{3}{S}_{1}\ensuremath{\rightarrow}1 {}^{1}{S}_{0}$ in He-like ions within the range of nuclear numbers $Z=28$--92. The total decay rates together with the spectral distributions are given. The obtained results are compared with experimental values and previous calculations.

Calculation of two-photon decay rates of hydrogen-like ions by using B-polynomials

A new approach is laid out to investigate two-photon atomic transitions. It is based on the application of the finite-basis solutions constructed from the Bernstein polynomial (B-polynomial) sets. We show that such an approach provides a very promising route for the relativistic second-order (and even higher-order) calculations since it allows for analytical evaluation of the involved matrices elements. In order to illustrate possible applications of the method and to verify its accuracy, detailed calculations are performed for the 2s1/2 → 1s1/2 transition in neutral hydrogen and hydrogen-like ions, which are compared with the theoretical predictions based on the well-established B-spline basis-set approach.

NRQCD matrix elements for S-wave bottomonia and [formula omitted] with relativistic corrections

We determine the leading-order nonrelativistic quantum chromodynamics (NRQCD) matrix element 〈O1〉ϒ and the ratio 〈q2〉ϒ, for ϒ=ϒ(nS) with n=1, 2, and 3 by comparing the measured values for Γ[ϒ→e+e−] with the NRQCD factorization formula in which relativistic corrections are resummed to all orders in the heavy-quark velocity v. The values for 〈q2〉ϒ, which is the ratio of order-v2 matrix element to 〈O1〉ϒ, are new. They can be used for NRQCD predictions involving ϒ(nS) and ηb(nS) with relativistic corrections. As an application, we predict the two-photon decay rates for the spin-singlet states: Γ[ηb(1S)→γγ]=0.512−0.094+0.096 keV, Γ[ηb(2S)→γγ]=0.235−0.043+0.043 keV, and Γ[ηb(3S)→γγ]=0.170−0.031+0.031 keV.

Two-photon decays of highly excited states in hydrogen

The relativistic and nonrelativistic approaches for the calculations of the two-photon decay rates of highly excited states in hydrogen are compared. The dependence on the principal quantum number (n) of the ns, nd and np initial states is investigated up to n = 100 for the nonresonant emissions. For the ns states together with the main E1E1 channel the contributions of higher multipoles (M1M1, E2E2 and E1M2) are considered. For the np states the E1M1 and E1E2 channels are evaluated. Moreover, the simple analytical formula for the E1M1 decay is derived in the nonrelativistic limit.

Two-photon transitions in Ca+, Sr+ and Ba+ ions

Two-photon (2E1) decay rates are calculated for metastable 3dj states in Ca+, 4dj states in Sr+ and 5dj states in Ba+ to evaluate contributions of these transitions to the corresponding lifetimes. The calculations are carried out using the relativistic single-double method, where single and double excitations of Dirac–Fock wavefunctions are included to all orders of perturbation theory. We find that lowest-order calculations of the two-photon rates are strongly modified when correlation corrections are included.