Electron form factors up to fourth order. - I

J. Schwinger:Phys. Rev.,76, 790 (1949); see alsoG. Källén:Handbuch der Physik, Vol.5 (Berlin, 1958), p. 304.

A. Peterman:Helv. Phys. Acta,30, 407 (1957);C. Sommerfield:Ann. of Phys.,5, 26 (1958). Both these authors present the correct analytic result. A previous incorrect estimate was given byR. Karplus andN. Kroll:Phys. Rev.,77, 536 (1950).P. Smrz andI. Ulehla (Czech. Journ. Phys.,10, 966 (1960)) repeated part of the calculation, providing a further check. The fourth-order magnetic moment was also re-evaluated byM. V. Terent’ev (3) by a method essentially based on dispersion relations, while in all the previous calculations use was made of Feynman-parameter techniques. Actually, dispersion relations are used in the Terent’ev work only to write down suitable multiple integral representations, which are in general manipulated to get the final result, without explicitly evaluating the discontinuities. The problem of infra-red divergences has been further overlooked, and many of the intermediate results are wrong, even if somewhatad hoc compensations make the final result correct. Despite these shortcomings, this reference has been of considerable interest for the present work, as it containsin nuce many of the techniques which were developed and widely used in this paper, in particular the choice of the « natural variable »x, eq. (0.1).

M. V. Terent’ev:Sov. Phys. JETP,16, 444 (1963).

R. Barbieri, J. A. Mignaco andE. Remiddi:Nuovo Cimento,6 A, 21 (1971).R. Barbieri andE. Remiddi:On the analytic computer-assisted evaluation of the discontinuities of the electron form factors at fourth order in QED, inProceedings of the Colloquium on Advanced Computing Methods in Theoretical Physics, Marseille, 21–45 June 1971, Vol.2, p. IV-76. The result is analytic. A first (incorrect) estimate was given byJ. Wesener, R. Bersohn andN. Kroll:Phys. Rev.,91, 1257 (1953). The first (incorrect) analytic evaluation is due toM. F. Soto jr.:Phys. Rev. Lett.,17, 1153 (1966);Phys. Rev. A,2, 734 (1970); the first correct numerical calculation toT. Appelquist andS. J. Brodsky:Phys. Rev. Lett.,24, 562 (1970);Phys. Rev. A,2, 2293 (1970). Partial results are given inR. Barbieri, J. A. Mignaco andE. Remiddi:Lett. Nuovo Cimento,3, 588 (1970) (analytic);B. Lautrup, A. Peterman andE. de Rafael:Phys. Lett.,31 B, 577 (1970) (numeric);A. Peterman:Phys. Lett.,34 B, 507 (1971) (numeric);35 B, 325 (1971) (analytic);J. A. Fox:Phys. Rev. D,5, 492 (1972).

G. Källén andA. Sabry:Dan. Mat. Fys. Medd.,29, No. 17 (1955).

A. Sabry:Nucl. Phys.,33, 401 (1962).

R. E. Cutkosky:Journ. Math. Phys.,1, 429 (1960); see alsoM. Veltman:Dispersive calculation of diagrams with arbitrarily many external lines, inProceedings of the Colloquium on Advanced Computing Methods in Theoretical Physics, Marseille, June 21–25, 1971, Vol.2, p. IV-115.

N. Nielsen:Nova Acta, Vol.90 (Halle, 1909), p. 125. See Appendix A.

M. Veltman: SCHOONSCHIP, CERN preprint, 1967 (unpublished).

R. Barbieri, J. A. Mignaco andE. Remiddi:Nuovo Cimento,11 A, 865 (1972).

The value forα can be taken fromB. N. Taylor, W. H. Parker andD. N. Langerberg:Rev. Mod. Phys.,41, 375 (1969); they giveα −1 = 137.03608(26).

M. J. Levine andJ. Wright:Phys. Rev. Lett.,26, 1351 (1971) and references therein for previous works. The error due to numerical integration of the (α/π)3 coefficient is estimated to be around 0.20.

See,e.g.,G. W. Erikson andD. R. Yennie:Ann. of Phys.,35, 271, 447 (1965).

The right result to lowest order for the 2S 1/2 level shift is ΔE=(1/2)α 4 m(α/π)·(−(1/3) log (Δɛ/m)−1/8+5/18), where Δɛ is the famous Bethe energy (15). Comparing this formula with eqs. (1.11), (1.12) we see that inm 2 F1/(2)(0) the regularizing massλ is replaced by the Bethe energy and a finite term (5/18) is added.

H. A. Bethe:Phys. Rev.,72, 339 (1947);H. A. Bethe, L. M. Brown andJ. R. Stehn:Phys. Rev.,77, 370 (1950).

R. Mills andN. Kroll:Phys. Rev.,98, 1489 (1955).

For a recent tabulation of Lamb-shift data (theoretical contributions and experimental results), seeAppelquist andBrodsky (4). Some of the theoretical contributions have been recently improved, and their numerical errors reduced, byG. W. Ericson:Phys. Rev. Lett.,27, 780 (1971).

R. T. Robiscoe andT. W. Shyn:Phys. Rev. Lett.,24, 599 (1970).

D. R. Yennie, S. C. Frautschi andH. Suura:Ann. of Phys.,13, 379 (1961).

G. Källén: inBrandeis University Summer Institute Lectures in Theoretical Physics, 1962 (New York, 1963), p. 210.

R. Jackiw:Ann. of Phys.,48, 292 (1968).

T. Appelquist andJ. R. Primack:The asymptotic behaviour of form factors in field theory, Harvard University preprint (1971);

Phys. Rev. D,1, 1144 (1970);

P. M. Fishbane andJ. D. Sullivan:Phys. Rev. D,4, 458 (1971).

See ref. (8). See alsoK. S. Kölbig, J. A. Mignaco andE. Remiddi:B.I.T.,10, 38 (1970). Miscellaneous results are contained in the book byR. Lewin:Dilogarithms and Associated Functions (London, 1958).

See the Appendix of ref. (5).

The first of these equations one finds in the Appendix of ref. (5); the second one could be new in the literature.

K. S. Kölbig: GPLOG, CERN Program Library C323.

J. A. Mignaco andE. Remiddi:Nuovo Cimento,60 A, 519 (1969).