IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT. VOL. 44, NO. 2. APRIL 1995 48 1

Corrections to Hyperfine Splitting and Lamb Shift Induced by Diagrams with Second-Order Radiative lnsertions in the Electron Line

Michael I. Eides, Savely G. Karshenboim, and Valery A. Shelyuto

Abstract-Contributions to HFS and to the Lamb shift intervals of order (r2(Zrr)5 induced by a gauge invariant set of 19 topolog- ically different graphs with two radiative photons inserted in the electron line are considered. Corrections both to HFS and Lamb shift induced by nine of these 19 diagrams have been calculated in the Fried-Yennie gauge.

I. INTRODUCTION ECENT theoretical work on high-order corrections to R hypefine splitting and Lamb shift concentrated on cal-

culation of recoil corrections of relative order ~ v ( Z a ) ~ ( m / M ) (see [ll-[3] for the Lamb shift and 141 and 151 for HFS) and on calculation of nonrecoil contributions of order a2 (201)~. These corrections are of immediate phenomenological interest for HFS measurements in the ground state of muonium and for n = 2 Lamb shift measurements in hydrogen.' Their magnitude may run up to several kilohertz for both intervals. This is to be compared with current experimental uncertainty of 0.16 kHz for HFS in muonium [7] and 1.9 kHz for the Lamb shift in hydrogen [8]. Further experimental progress is envisaged at least in the case of muonium hyperfine splitting measurements 191.

We have shown recently that there exist six gauge invariant sets of diagrams, which produce corrections of order ~ t ~ ( Z a ) ~ to HFS [ 111 (see Fig. 1) . All these diagrams may be obtained with the help of different radiative insertions from the skeleton diagram (see Fig. 2 ) , which contains two external photons attached to the electron line.

Contribution to hyperfine splitting, produced by diagrams in Fig. 1, is given by the expression (see [4])

where I C , = (0, k) is the momentum of the extemal photons measured in the electron mass units and

where vi, = m/(l + m / M ) is the reduced mass of the electron-muon system and a, is the anomaly of the muon

Manuscript received July 1, 1994; revised October IS, 1994. M. I. Eides is with the Petersburg Nuclear Physics Institute, Gatchina, St.

S. G . Karshenboim and V. A. Shelyuto are with the D. 1. Mendeleev Institute

IEEE Log Number 9409135. ' Several leading logarithmic corrections of higher order that nevertheless

Petersburg 188350, Russia.

of Metrology, S t . Petersburg 198005. Russia.

turn out to be numerically significant were recently obtained in [6) .

TABLE I

CLOSED ELECTRON LOOPS CORRECTIONS TO HFS INDUCED BY THE DIAGRAMS WITH

magnetic moment. The function F(k) is connected with the numerator structure of each particular graph and describes radiative corrections to the skeleton diagram. It is normalized on the skeleton numerator contribution.

Five of six sets of diagrams relevant for the corrections of order a2(Zn)'m to HFS contain graphs with closed elec- tron loops. Contributions induced by polarization operator insertions in external photons [see Fig. I(a) and (b)] and by simultaneous insertion of a radiative photon in the electron line and a one-loop polarization operator in the extemal photon [see Fig. l(c)] have been calculated in analytic form [l 11. The correction induced by polarization operator insertions in radiative photons [see Fig. l(d)J was obtained in semianalytic form as a one-dimensional integral, where the integrand is itself a complete elliptic integral [ 121. Contribution induced by the fifth gauge invariant set of graphs containing light by light scattering insertions [see Fig. l(e)] was reduced to a three-dimensional integral over Feynman parameters. This was calculated numerically 1131. All respective results are presented in Table I.

Contributions to the Lamb shift induced by the same sets of gauge invariant diagrams also were obtained recently [14].

We discuss below all possible contributions to HFS and to the Lamb shift of order ~t'(Zrr)~ that are induced by the last and most complicated set of diagrams, with two radiative photons inserted in the electron line (see Fig. 3). Results of the calculations of all contributions both to HFS and to the Lamb shift induced by the diagrams containing one-loop electron self-energy as a subgraph, by the diagram containing two one- loop vertices, and by the diagram with overlapping two-loop self-energy insertion are presented below 1151, [ 161. Other groups announced recently their results for the contribution of the diagrams in Fig. 3 to HFS 1171 and to the Lamb shift [181.

0018-9456/94$05.00 0 1994 IEEE

482 IEEE TRANSACTlONS ON INSTRUMENTATION AND MEASIJREMENT. VOL. 44. NO. 2. APRIL 1995

b C

d e f Fig. I Six gauge invariant sets of graphs producing corrections of order 0 2 ( Z c u ) E ~ .

Fig. 2. Skeleton integral

11. CONTRIBUTIONS TO THE ENERGY SPLITTINGS

There are 19 topologically different diagrams with insertions of two radiative photons in the electron line and two extemal photons that are relevant for calculation of corrections of order tr*(Ztr)’ to HFS and to the Lamb shift. The simplest way to describe these graphs is to realize that they may be obtained from three graphs for the two-loop electron self-energy by insertion of two extemal photons in all possible ways.

As was shown in [4] for HFS and in (101 for the Lamb shift, to obtain contribution to the energy splitting one has to calculate matrix elements of the diagrams described above between free electron spinors with all extemal electron lines on the mass shell, project these matrix elements on the respective spin state, and multiply the result by the square at the origin of the Schrodinger-Coulomb wave function.

It should be mentioned that some of the diagrams under consideration also contain contributions of previous order in Z a . Lower order terms are proportional to the exchanged momentum squared, and to get rid of them one has to subtract all low-frequency terms proportional to the exchanged momentum squared in the low-frequency asymptotes of the matrix elements when they exist.

Actual calculation of the matrix elements is impeded by the ultraviolet and infrared divergences. To get rid of ultra- violet problems we work only with renormalized (subtracted) graphs and subgraphs. Infrared problems are as usual more difficult to deal with than the ultraviolet ones. We choose the Fried-Yennie (FY) gauge for the radiative photons. The advan- tage of this choice of gauge is connected with mild behavior of all matrix elements in the IR region of integration momenta in

a C

e d f a

h d K

I m n C

P c1 Y S

Fig. 3. insertions in the electron linc.

Nineteen topologically different diagrams, with two radiative photocl

the FY gauge and with the possibility to perform on-mass-shell renormalization without introduction of the infrared photon mass [4]. This last feature of the FY gauge is especially important for us as we want to push analytical calculations as far forward as possible, and working without the photon mass gives us the chance to obtain much simpler analytical formulas. On the other hand, working in the FY gauge without the photon mass makes it absolutely necessary to pay special attention to the infrared behavior of the integrand functions

EIDES PI a/,: CORRECTIONS TO HYPERFINE SPLITTING AND LAMB SHIFT

a b C

a

483

.*EF x w m 9 4 0

-6.660... 2.955 ... 3.932 ... -2.223...

-3.903... -5.235...

TABLE 11 CORRECTIONS TO HFs AND LAMB SHIFT

I DEI HFS 1 Lamb 1

e f R

4.566 ... 5.056 ... -3.401... -1.017...

2.682 ... -0.146... Y

h 1

33/16 153180 -1.984.. . 1.749 ...

and to perform cancellation of spurious IR divergences prior to integration.

Our general approach to actual calculation consists in con- sidering universal diverging skeleton integrals corresponding to the electron line with two external photons. Radiative corrections induce additional factors in the integrands, which convert respective integrals into convergent ones. After tedious calculations we obtained the results presented in Table 11.

Calculations of the contributions induced by the remaining ten graphs in Fig. 3 (M. Eides and V. Shelyuto) are in the final stage now. Calculations of the contributions induced by all diagrams in Fig. 3 has been recently completed 1191, [20).

ACKNOWLEDGMENT

This work was completed during the visit by one of the authors (M. Eides) to Penn State University. He is deeply grateful to his colleagues at Penn State, and especially to H. Grotch, for their kind hospitality.

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