A lower rate at small charge
The calculation reports a lower decay rate for a charged scalar particle bound to a light nucleus than for its free rate. The leading difference is a quadratic suppression in the product of alpha, the fine-structure constant, and Z, the nuclear charge. The result comes from a scalar-QED model that combines numerical and analytical calculations for the decay rate.
In plain language, the reported bound-to-free rate ratio starts at 1, falls by one-half of the square of alpha Z, then receives a positive term equal to 17/24 of the fourth power of alpha Z. Terms beginning at fifth order are not shown. It is a small-alpha-Z expansion, so it is a low-charge description rather than an exact expression at arbitrary charge.
Where the correction enters
At fourth order, the calculation separates the contribution from free pions from the effect of the Coulomb field on the outgoing charged pion. The free-pion part has a coefficient of -5/8. Coulomb distortion does not change the leading quadratic term and first appears at fourth order, with a leading correction equal to 4/3 of the fourth power of alpha Z.
That split is central to the numerical method. The free-pion contribution is evaluated separately through a one-dimensional representation, while the partial-wave sum, which adds angular-momentum components, is applied only to the Coulomb-distortion correction. The continuum integrals are handled with adaptive eighth-order DOP853 Runge-Kutta integration and composite Gauss-Legendre quadrature.
A numerical check across the range
Across the displayed range, the model evaluates nuclear charges from 1 through 10 inclusive. At Z=1, the reported free-pion suppression is 2.6627 and the Coulomb correction is 3.817; the interacting suppression is 2.6624 and the normalized correction is 1.346. At Z=10, the same four figures are 268.55, 42,720, 264.28 and 1.507, respectively. The displayed suppression and correction measures are larger at the upper endpoint than at the lower one.
An internal numerical check supports the reorganization. For five nuclei, direct and subtracted calculations give Coulomb-correction values agreeing in all four quoted significant digits. The subtracted implementation needs a partial-wave cutoff about 25 to 30 percent lower, and the cutoff is chosen so the estimated tail is smaller than half of the last quoted digit of the correction.
A useful benchmark with clear limits
For light nuclei, the all-orders numerical calculation agrees with the analytical expansion. As Z decreases, the Coulomb correction divided by the fourth power of alpha Z approaches 4/3, the coefficient of the leading analytical correction. The expansion remains a perturbative description, with higher-order terms beyond those shown.
The model is deliberately spare. It treats the nucleus as pointlike, infinitely heavy and spinless, takes both final pions to be massless, and neglects QCD effects. It also leaves out recoil, nuclear polarization and finite nuclear size. Those assumptions keep the result within a simplified scalar-QED setup and mean that the calculation does not establish the behavior of a more complete bound-state system.
It is an arXiv version 1 preprint dated 29 September 2026, supported by the Natural Sciences and Engineering Research Council of Canada.
Paper data and sources
Original title: Bound-state decay of a charged scalar particle in an external Coulomb field
Authors: Andrzej Czarnecki, Artem O. Davydov
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-09-29
DOI: Not available
Original paper · Full text