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The matrix element reads explicitly
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where we have used some Fierz reshuffling. Therefore we find
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The term + 0', 0'2)/2) is nothing but S2 where S = (0', + 0'2)/2 is the total spin; thus it contributes a positive amount to the triplet energy only. To lowest order, the total ground-state spin dependence to the energy displacement is thus expected to be
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[((!O[2 /0',. 0'2 + S2)
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To obtain the hyperfine splitting we subtract the values corresponding to S t'1E ts
1 and 0. Thus
(10-81)
rx Ryd
INTEGRAL I1QUATIONS AND BOUND-STATE PROBLEMS
where Ryd is the Rydberg constant Ryd = - - = 3.28984 2
mIX z
10 15 Hz
(10-82)
Eq. (10-81) predicts!1E = 2.04 x 10 5 MHz, which compares favorably with the experimental results up to order IX. This is fortunately two orders of magnitude greater than the two-photon decay rate of the singlet state 1Z, = IX3 Ryd. Hence the computation of the hyperfine splitting will be an accurate test both of quantum electrodynamics and of the application of the Bethe-Salpeter equation.
10-3-1 General Setting
Our presentation will be modeled after the original work of Karplus and Klein. The starting point is the analog of Eq. (10-26) written in the electron-positron channel. The bound-state amplitude X = <0 [I/I(x)li/(y) [P) stands for a 4 x 4 matrix in the space of Dirac indices. It should be emphasized that X is a gauge-covariant quantity. Consequently, it is not indifferent to choose a particular gauge if X satisfies an approximately soluble equation. In practice, since relative velocities are small, we expect the nonrelativistic approximation to the binding energy to be a reasonable starting point. This is
Eo=2m--4n 2
mlY
(10-83)
and X is given by a Pauli-Schrodinger wave function. A noncovariant radiation gauge would seem the most appropriate to derive this result. This is, however, a dangerous choice since higher corrections will require renormalization. On the other hand, we know that to implement gauge covariance in a relativistic manner it is necessary to include in the kernel V an infinite series of crossed diagrams. We are facing a dilemma which will be resolved in practice in the following way. We will use a covariant gauge, to be specific the Feynman one, and will have to separate an instantaneous interaction from a retarded one. This is because we are unable to solve exactly the equivalent of the Wick-Cutkosky relativistic equation. The retarded interaction will then be treated as a perturbation on the same footing as higher-order terms in V. If we are led to smaller and smaller corrections which do agree with experiment, this procedure may at least be considered as a useful method in spite of its poor theoretical foundations. This will be true of the corrections of order 0: 3 Ryd. The development of the subject shows, however, that even the experts have some trouble with higher orders required for a comparison with the very accurate experimental resolution. Several improvements have been suggested such as expanding X in terms of Lorentz invariant scalar amplitudes multiplying covariant quantities, carrying out a complete angular analysis on the Wick rotated amplitude or attempting to find an equivalent but soluble form of the relativistic ladder approximation. Ignoring at first radiative corrections, we write the equation satisfied by X
QUANTUM FIELD TIIEORY
in momentum space as
(~ +p-m)x(~ -p+m)=(VB+Va)x
VBX _. IX
I 41[3
d p' , Il (p _ p,)2 YIlX(P )y
(10-84)
VaX = - i 4: 3
~2 yll
d p' tr [y IlX(P')]
where VB represents one-photon exchange in the crossed t channel and v" is the one-photon annihilation interaction. Even at this early stage the complexity is somehow frightening. It will be convenient to work with the quantity K obtained by acting with the antisymmetric charge conjugation matrix C on the second index of X: (10-85) This amplitude has the same transformation properties as one corresponding to the particle-particle channel except that the latter has no electromagnetic bound state. It satisfies
(10-86)
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