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Gregory S. Adkins

Publications and source records attributed to Gregory S. Adkins.

18 recordsLinked to original sources
arXiv2024

Relativistic and Recoil Corrections to Vacuum polarization in muonic systems: Three--photon exchange, gauge invariance and numerical values

For an accurate theoretical description of muonic bound systems, it is crucial to consistently treat relativistic and recoil corrections to vacuum polarization. The one-loop vacuum-polarization effect is by far the dominant quantum electrodynamic (QED) energy correction for bound muons, being of order $α(Zα)^2 m_r$, where $α$ is the fine-structure constant, $Z$ is the nuclear charge number, and $m_r$ is the reduced mass. Gauge invariance of the relativistic and recoil corrections to vacuum polarization of order $α(Zα)^4 m_r$ is investigated with respect to nonretarded and standard, renormalized variants of Coulomb gauge. The invariance is shown after including three-photon exchange diagrams. Our derivation is based on an adapted form of Nonrelativistic Quantum Electrodynamics for bound muon systems (NRQED$_μ$), which is a version of NRQED where the hard scale is set at the muon mass instead of the electron mass. Updated values for the gauge-independent corrections for one-muon ions with nuclear charge numbers $Z = 1,2,6$ are presented.

hep-ph↗
arXiv2022

Recoil corrections to the energy levels of hydrogenic atoms

We have completed the calculation of pure-recoil corrections of order $(Z α)^6$ to Coulombic bound states of two spin-1/2 fermions without approximation in the particle masses. Our result applies to systems of arbitrary mass ratio such as muonium and positronium, and also hydrogen and muonic hydrogen (with the neglect of proton structure effects). We have shown how the two-loop master integrals that occur in the relativistic region can be computed in analytic form, and suggest that the same method can be applied to the three-loop integrals that would be present in a calculation of order $(Z α)^7$ corrections.

hep-ph↗
arXiv2019

Coulomb expectation values in $D=3$ and $D=3-2ε$ dimensions

We explore the quantum Coulomb problem for two-body bound states, in $D=3$ and $D=3-2ε$ dimensions, in detail, and give an extensive list of expectation values that arise in the evaluation of QED corrections to bound state energies. We describe the techniques used to obtain these expectation values and give general formulas for the evaluation of integrals involving associated Laguerre polynomials. In addition, we give formulas for the evaluation of integrals involving subtracted associated Laguerre polynomials--those with low powers of the variable subtracted off--that arise when evaluating divergent expectation values. We present perturbative results (in the parameter $ε$) that show how bound state energies and wave functions in $D=3-2ε$ dimensions differ from their $D=3$ dimensional counterparts and use these formulas to find regularized expressions for divergent expectation values such as $\big \langle \bar V^3 \big \rangle$ and $\big \langle (\bar V')^2 \big \rangle$ where $\bar V$ is the $D$-dimensional Coulomb potential. We evaluate a number of finite $D$-dimensional expectation values such as $\big \langle r^{-2+4ε} \partial_r^2 \big \rangle$ and $\big \langle r^{4ε} p^4 \big \rangle$ that have $ε\rightarrow 0$ limits that differ from their three-dimensional counterparts $\big \langle r^{-2} \partial_r^2 \big \rangle$ and $\big \langle p^4 \big \rangle$. We explore the use of recursion relations, the Feynman-Hellmann theorem, and momentum space brackets combined with $D$-dimensional Fourier transformation for the evaluation of $D$-dimensional expectation values. The results of this paper are useful when using dimensional regularization in the calculation of properties of Coulomb bound systems.

quant-ph↗
arXiv2018

The hydrogen atom in $D=3-2ε$ dimensions

The nonrelativistic hydrogen atom in $D=3-2ε$ dimensions is the reference system for perturbative schemes used in dimensionally regularized nonrelativistic effective field theories to describe hydrogen-like atoms. Solutions to the $D$-dimensional Schrödinger-Coulomb equation are given in the form of a double power series. Energies and normalization integrals are obtained numerically and also perturbatively in terms of $ε$. The utility of the series expansion is demonstrated by the calculation of the divergent expectation value $\langle (V')^2 \rangle$.

quant-ph↗
arXiv2016

Angular decomposition of tensor products of a vector

The tensor product of $L$ copies of a single vector, such as $p_{i_1} ... p_{i_L}$, can be analyzed in terms of angular momentum. When $p_{i_1} ... p_{i_L}$ is decomposed into a sum of components $( p_{i_1} ... p_{i_L} )^L_\ell$, each characterized by angular momentum $\ell$, the components are in general complicated functions of the $p_i$ vectors, especially so for large $\ell$. We obtain a compact expression for $( p_{i_1} ... p_{i_L} )^L_\ell$ explicitly in terms of the $p_i$ valid for all $L$ and $\ell$. We use this decomposition to perform three-dimensional Fourier transforms of functions like $p^n \hat p_{i_1} ... \hat p_{i_L}$ that are useful in describing particle interactions.

math-ph↗
arXiv2016

Positronium energy levels at order $m α^7$: Product contributions in the two-photon-annihilation channel

Ongoing improvements in the measurement of positronium transition intervals motivate the calculation of the $O(m α^7)$ corrections to these intervals. In this work we focus on corrections to the spin-singlet parapositronium energies involving virtual annihilation to two photons in an intermediate state. We have evaluated all contributions to the positronium S-state energy levels that can be written as the product of a one-loop correction on one side of the annihilation event and another one-loop correction on the other side. These effects contribute $ΔE = -0.561971(25) m α^7/π^3$ to the parapositronium ground state energy.

hep-ph↗
arXiv2015

Three-photon-annihilation contributions to positronium energies at order $m α^7$

Positronium spectroscopy (n=1 hyperfine splitting, n=2 fine structure, and the 1S-2S interval) has reached a precision of order 1 MHz. Vigorous ongoing efforts to improve the experimental results motivate the calculation of the positronium energy levels at order $m α^7$. In this work we present the result for a complete class of such contributions--those involving virtual annihilation of positronium to three photons in an intermediate state. We find an energy shift of $2.6216(11) m α^7/(n π)^3$ for the spin-triplet S state with principal quantum number n.

hep-ph↗
arXiv2014

Positronium energy levels at order $m α^7$: light-by-light scattering in the two-photon-annihilation channel

Recent and ongoing experimental work on the positronium spectrum motivates new efforts to calculate positronium energy levels at the level of three loop corrections. We have obtained results for one set of such corrections involving light-by-light scattering of the photons produced in a two-photon virtual annihilation process. Our result is an energy shift $1.58377(8) m α^7/π^3$ for the n=1 singlet state, correcting the ground state hyperfine splitting by -6.95 kHz. We also obtained a new and more precise result for the light-by-light scattering correction to the real decay of parapositronium into two photons.

hep-ph↗
arXiv2014

Positronium hyperfine splitting at order $m α^7$: light-by-light scattering in the two-photon-exchange channel

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.

hep-ph↗
arXiv2013

Lorentz and CPT violation in the hydrogen spectrum

We have studied the effect of hypothetical violations of Lorentz and CPT symmetry by calculating the corrections to the energy levels of hydrogen induced by the Standard-Model Extension (SME). Hydrogen studies are interesting because the energy levels of hydrogen can be measured with great precision and the theory for hydrogen based on the Standard Model (SM) is well understood. We obtained corrections through order α^2 times the SME parameters for all levels of hydrogen and applied them to determine the SME corrections to the transition frequency for the 2S-1S transition.

hep-ph↗
arXiv2013

Three-dimensional Fourier transforms, integrals of spherical Bessel functions, and novel delta function identities

We present a general approach for evaluating a large variety of three-dimensional Fourier transforms. The transforms considered include the useful cases of the Coulomb and dipole potentials, and include situations where the transforms are singular and involve terms proportional to the Dirac delta function. Our approach makes use of the Rayleigh expansion of exp(i p.r) in terms of spherical Bessel functions, and we study a number of integrals, including singular integrals, involving a power of the independent variable times a spherical Bessel function. We work through several examples of three-dimensional Fourier transforms using our approach and show how to derive a number of identities involving multiple derivatives of 1/r, 1/r^2, and delta(\vec r).

math-ph↗
arXiv2012

Higher order corrections to the hydrogen spectrum from the Standard-Model Extension

We have studied the effects of the Standard-Model Extension (SME) on hydrogen as a realization of new physics effects that incorporate Lorentz and CPT violation. Specifically, we calculated the SME-induced energy level shifts at order $α^2$ times the SME parameters. We obtained contributions at this order both from the non-relativistic effective Hamiltonian for motion of a spin-1/2 particle in the presence of SME interactions and also from SME corrections to the propagator for exchange photons. We applied our result to the $2S-1S$ transition in hydrogen, which has been measured with extremely high precision. The results obtained in this work give the leading SME corrections for this transition.

hep-ph↗
arXiv2010

Search for CP and CPT violation in positronium decay

Positronium, the electron-positron bound state, is described to a good approximation by pure QED. The states of positronium have definite values of C and P. Consequently, positronium is an attractive system for the investigation of possible violations of the discrete symmetries in the leptonic sector. We discuss signals for CP and CPT violation in the decay of spin-polarized orthopositronium and show where such correlations might arise in the context of the Standard-Model Extension.

hep-ph↗
arXiv2007

Orbital precession due to central-force perturbations

We calculate the precession of Keplerian orbits under the influence of arbitrary central-force perturbations. Our result is in the form of a one-dimensional integral that is straightforward to evaluate numerically. We demonstrate the effectiveness of our formula for the case of the Yukawa potential. We obtain analytic results for potentials of the form V(r) = αr^n and V(r) = α\ln(r/λ) in terms of the hypergeometric function {_2F_1} (1/2-n/2,1-n/2; 2; e^2), where e is the eccentricity. Our results reproduce the known general relativistic (n=-3), constant force (n=1), and cosmological constant (n=2) precession formulas. Planetary precessions are often used to constrain the sizes of hypothetical new weak forces--our results allow for more precise, and often stronger, constraints on such proposed new forces.

gr-qc↗
arXiv2006

Cosmological perturbations on local systems

We study the effect of cosmological expansion on orbits--galactic, planetary, or atomic--subject to an inverse-square force law. We obtain the laws of motion for gravitational or electrical interactions from general relativity--in particular, we find the gravitational field of a mass distribution in an expanding universe by applying perturbation theory to the Robertson-Walker metric. Cosmological expansion induces an ($\ddot a/a) \vec r$ force where $a(t)$ is the cosmological scale factor. In a locally Newtonian framework, we show that the $(\ddot a/a) \vec r$ term represents the effect of a continuous distribution of cosmological material in Hubble flow, and that the total force on an object, due to the cosmological material plus the matter perturbation, can be represented as the negative gradient of a gravitational potential whose source is the material actually present. We also consider the effect on local dynamics of the cosmological constant. We calculate the perihelion precession of elliptical orbits due to the cosmological constant induced force, and work out a generalized virial relation applicable to gravitationally bound clusters.

gr-qc↗
arXiv2005

Analytic evaluation of the amplitudes for orthopositronium decay to three photons to one-loop order

The matrix element for the decay of orthopositronium to three photons can be expressed in terms of three independent amplitudes. We describe the analytic evaluation of these amplitudes, both to lowest order and with the inclusion of all one-loop corrections. We use these amplitudes to find precise values for the one-loop correction to the orthopositronium decay rate Gamma_1=-10.286606(10) (alpha/pi) Gamma_{LO}, and for the order-alpha^2 "square" correction to the decay rate Gamma_2(square) = 28.860(2) (alpha/pi)^2 Gamma_{LO}, where Gamma_{LO} is the lowest order rate. We give in explicit form the function describing the one-loop correction to the distribution in phase space of the final state photons.

hep-ph↗
arXiv2003

Two-loop corrections to the decay rate of parapositronium

Order $α^2$ corrections to the decay rate of parapositronium are calculated. A QED scattering calculation of the amplitude for electron-positron annihilation into two photons at threshold is combined with the technique of effective field theory to determine an NRQED Hamiltonian, which is then used in a bound state calculation to determine the decay rate. Our result for the two-loop correction is $5.1243(33)$ in units of $(α/π)^2$ times the lowest order rate. This is consistent with but more precise than the result $5.1(3)$ of a previous calculation.

hep-ph↗