Triple-coalescence singularity in a dynamical atomic process

Abstract

We show that the high energy limit for amplitude of the double electron capture to the bound state of the Coulomb field of a nucleus with emission of a single photon is determined by behavior of the wave function in the vicinity of the singular triple coalescence point. PCAC: 31.25.Eb; It is well known that the solution Ψ(r 1 , r 2 ) of the Schrödinger equation for a two-electron system in the Coulomb field of infinitely heavy point nucleus is singular at the triple coalescence point r 1 = r 2 = 0 [1, 2, 3, 4]. The wave function can be presented as a generalized power series containing logarithmic terms. The behavior of Ψ(r 1 , r 2 ) near this point is not very important in calculations of the binding energy since the corresponding phase volume is small. On the contrary it becomes crucial for calculation of the local energy E(r 1 , r 2 ) = HΨ(r 1 , r 2 )/Ψ(r 1 , r 2 ) [4]. However the triple coalescence point did not manifest itself in a dynamical process until now.In this letter we present an observable effect for the first time in which the interesting behavior of the two electron wave function Ψ(r 1 , r 2 ) near the point r 1 = r 2 = 0 (as well as that near the double coalescence points r 1 = 0 and r 2 = 0) plays an important role. For that we consider the double electron capture followed by the emission of a single photon in the high energy limit. Since the first attempts to detect this process in collisions of a light atom with a heavy nucleus [5] a number of experimental [6,7] and theoretical [8,9,10] papers were devoted to this reaction.Neglecting the internal motion of the electrons in the light atom we consider the capture of two continuum electrons with equal linear momenta p 1 = 1

The pion: an enigma within the Standard Model

“…An earlier lQCD result [242] produced a PDA that was in better agreement with the DSE-RL curve; but, as anticipated in [241], improvements in the lattice simulations have produced a PDA in good agreement with the DSE-DB prediction, which is the more realistic result, as explained in section 4. Contemporary theory has therefore converged on a pointwise form for the pionʼs twist-two valence-quark PDA; and that form is [237,241,[243][244][245]:…”

Analytic matrix elements for the two-electron atomic basis with logarithmic terms

“…We will consider the basis functions (6) with the maximum power of ln r equals 2, that is j = 0, 1, 2. In accordance with the latter, in order to express the entries of the K-matrix in terms of the integrals (12), it is convenient to split this entries into groups…”

“…Using Eq. ( 5) for the Coulomb potential, one can express the entries of the U-matrix in terms of the integrals (12), as follows:…”

“…Our objective is to derive analytical expressions for the basic integral (12). To this end, first of all, changing the order of integration one obtains:…”

“…Only a few of them (in fact [8] and [10]) used the basis containing ln r, what is the necessary condition (according to the Fock expansion (1)) for obtaining the two-electron wave function with correct behavior near the point r = 0. The latter presents, so called, triple coalescence point playing an important role in many physical processes treated by the atomic and related fields of physics (see, e.g., [11,12]).…”