Hard three-loop corrections to hyperfine splitting in positronium and muonium

Abstract

We consider hard three-loop corrections to hyperfine splitting in muonium and positronium generated by the diagrams with closed electron loops. There are six gauge-invariant sets of such diagrams that generate corrections of order mα 7 . The contributions of these diagrams are calculated for an arbitrary electron-muon mass ratio without expansion in the small mass ratio. We obtain the formulae for contributions to hyperfine splitting that in the case of small mass ratio describe corrections for muonium and in the case of equal masses describe corrections for positronium. First few terms of the expansion of hard corrections in the small mass ratio were earlier calculated for muonium analytically. We check numerically that the new results coincide with the sum of the known terms of the expansion in the case of small mass ratio. In the case of equal masses we obtain hard nonlogarithmic corrections of order mα 7 to hyperfine splitting in positronium.

One more hard three-loop correction to parapositronium energy levels

“…Analogous sum of hard contributions to HFS of the six gauge invariant sets of scattering diagrams was calculated earlier [7,8,11] ∆E = −1.291 7 (1) mα 7 π 3 = −5.6720 (4) kHz.…”

“…Hard corrections due to the diagrams with two annihilation photons are generated by seven gauge invariant sets of diagrams that are similar to the respective seven gauge invariant sets of non-annihilation diagrams in muonium and positronium [8,11]. All these diagrams can be obtained by two-loop radiative insertions in the skeleton diagrams with two annihilation photons in Fig.…”

“…Many hard nonlogarithmic corrections of order mα 7 were calculated recently in a rapid succession [7][8][9][10][11][12][13]. These corrections are generated both by the annihilation and nonannihilation diagrams.…”

Calculation of higher order corrections to positronium energy levels

“…The ones that have been done involve "ultrasoft" corrections [18], "hard" (i.e. high-energy) corrections in the two-photon-exchange channel [19][20][21][22][23], corrections in the three-photon-annihilation channel [24], corrections in the two-photon-annihilation channel [25][26][27][28], and corrections in the one-photon-annihilation channel [29].…”

History and Some Aspects of the Lamb Shift

“…A period of intense theoretical development followed Bethe's calculation, characterized by calculations of the energy levels of the H atom, and QED in general, done with greater and greater precision and complexity. Some of the key developments from 1950 to about 1970 are in the papers [12,[51][52][53][54][55][56]; from 1980 to 2000 are in [57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74]; and from 2000 to current are in [75][76][77][78][79][80][81][82][83][84][85][86][87][88][89]. Theorists applied themselves to compute the numerous other effects leading to the total shift between the 2s 1/2 and 2p 1/2 levels, as well as for other levels, including relativistic corrections, center of mass effects, recoil corrections, radiative recoil corrections, nuclear size and spin effects, and more rigorous, more precise and higher order calculations of the radiative shifts (for reviews, see [1][2][3][4]…”