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(10-124)
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It was already noted in the previous subsection that only space-like values of virtual annihilation photon polarization (index fl) contribute to the splitting.
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Moreover, the charge conjugation matrix C = iy2 yO = ( .0
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is odd in the representation we are using, meaning that tr [Cy( is to be interpreted as
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Using this remark, a straightforward (but long and painful) calculation yields
(1) !l.Ea (2) nIX + !l.Eab + ba = -y;:;z ( 1 -
4IX) t --;: tr (<fJo yC) tr (C Y<fJo)
(1:~ (1- :IX)i
:IX)[XiO"(-i0"2)x!] o [XH-i0"2)O"Xl]l<fJoI <fJo 12 <S2)
(10-125)
The above expression has still to be multiplied by [1 - w(4m 2 )] to include the radiative correction due to the vacuum polarization to order IX. In this formalism this effect arises from the second-order v"D- 1 v" term
(10-126)
leading to an extra contribution
!l.E~~) :~
l<fJol <S2)
(10-127)
INTEGRAL EQUATIONS AND BOUND-STATE PROBLEMS
-P/2
-P/2
-P/2
-P/2
Figure 10-9 Two-photon annihilation diagrams.
The last piece of the puzzle is the two-photon annihilation term with (minus) its imaginary part equal to half the singlet level width (Fig. 10-9). Its expression is found by adding a new term in the Bethe-Salpeter kernel. It is best to return to Eq. (10-84) and to approximate the wave function by its nonrelativistic value at the origin, thereby obtaining the leading as effect in the even-charge conjugation channel. To keep track of various coefficients it is simpler to use the relativistic notation with u(1), v(2) being spinors at rest: 2 4 IlES2) . r 2y _ ia [ [2 d k v 1 2 ({Jo 2 + ieJ [(P _ k)2 + ieJ v(2)y f12 _ _ m yl'u(1) y - IT - n [k
x u(l) (Yl' f12
_1 _ m Yv + Yv
-f12
~ _ m YI')V(2)
+ in)
(10-128)
Evaluation of the integral and of the spin matrix element leaves us with a coefficient proportional to the projector on the singlet state:
IlE~2) y
ir 2y 2
~ [({JO[2 <2 m2
S2)(2 - 2ln 2
(10-129)
with the imaginary part giving r 2y = mas /2 as found in (5-128). We have now the complete value of the hyperfine splitting in positronium to order as. Taking Eqs. (10-119), (10-121), and (10-122) for the exchange channel and (10-125), (10-127), and (10-129) for the annihilation one, and subtracting the values in the S = 1 and S = 0 states, we find the Karplus and Klein result
IlE ts
"2 a2
[7"3 - (32 + )a] 9
2 In 2 ~
(10-130)
This value is not yet sufficient for a comparison with the experimental result. The reader will appreciate the effort needed to extract the (1.6 terms. The difficulty lies partly in a good control of the recoil effects, i.e., of a correct treatment of the relativistic Coulomb wave functions, which leads in fact to terms of order (1.6 In (1/(1.). Other effects such as vacuum polarization in the exchange channel also contribute to this term, so that the correction is of the type ((1.2/2) Ryd [B(l.2 In (1/(1.) + C]. The latest value, reported by Lepage in 1977, is B = -i. When added to (10-130) this would yield the theoretical prediction 2.033774 x 105 MHz, to be compared with (10-7.9). The Bethe-Salpeter formalism may also be applied to obtain radiative corrections to the decay widths, as well as to other systems such as the hydrogen atom. As an exercise the reader may
508 QUANTUM FIELD THEORY
derive, within the present formalism, the excited spectrum ofpositronium to order
(;(4:
GLJ= -(jLJ~~
1 , 2L+ 1
GLJ= -~~+-(jLO+
1 2L+ 1
1 - 2(jLO 2(2L+ 1)
3L+4 (L+ 1)(2L+ 3) 1 --L(L+ 1) (3L- 1) L(2L-1)
J=L+1 J=L J=L-1
(10-131)
How does this compare to the triplet splitting ofEq. (10-80)
NOTES
The field equations appear in the work of F. J, Dyson, Phys. Rev., voL 75, p. 1736, 1949, and J, Schwinger, Proc. Nat. Acad. Sc., voL 37, pp. 452 and 455,1951. Bound-state equations have a long history which can be traced from the book of H, A. Bethe and E. E. Sal peter, "Quantum Mechanics of One and Two Electron Atoms," Springer-Verlag, Berlin, 1957, Modern developments were prompted by the work of these authors, Phys. Rev., voL 82, p. 309, 1951, voL 84, p. 1232, 1951, by J. Schwinger in the article just quoted, and by M. Gell-Mann and F. Low, Phys. Rev., voL 84, p. 350, 1951. The ladder exchange was treated by Y. Nambu, Progr. Theor. Phys., voL 5, p. 614, 1950, by G. C. Wick, Phys. Rev., voL 96, p. 1124, 1954, and by R. E. Cutkosky, Phys. Rev., voL 96, p. 1135, 1954. The review of N. Nakanishi, Suppl. Prog. Theor. Phys., voL 43, p. 1, 1969, covers his own work together with the many contributions up to the late 1960s. The perturbation theory of the hyperfine splitting was developed by E. E. Salpeter, Phys. Rev., voL 87, p. 328, 1952, and by R. Karplus and A. Klein, Phys. Rev., vol. 87, p. 848, 1952, in the positronium case. The recent accurate measurements are due to A. P. Mills and G. H. Bearman, Phys. Rev. Lett., voL 34, p. 246, 1975, and P. O. Egan, W. E. Frieze, V. W. Hughes, and M. H. Yam, Phys. Lett., ser. A, voL 54, p. 412, 1975. The excited triplet splitting reported in the text is from A. P. Mills, S. Berko, and K. F. Canter, Phys. Rev. Lett., voL 34, p. 1541, 1975. A. Stroscio has reviewed the subject of positronium in Phys. Rep., ser. C, voL 22, p. 215, 1975. Recent contributions are those by G. P. Lepage in Phys. Rev., ser. A, voL 16, p. 863, 1977, and by G. T. Bodwin and D. R. Yennie in Phys. Rep., ser. C, voL 43, p. 267, 1978, including reference to earlier work. The positronium spectrum is discussed in J. Schwinger, "Particles, Sources and Fields," voL 2, Addison-Wesley, Reading, 1973.
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