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1 n - + -;=c----,-------,.-:-:-o=
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[In(1- p2/4)]2
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K:2: 1
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INTEGRAL EQUATIONS AND BOUND-STATE PROBLEMS
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This is not the only disease of this model. It is possible to study corrections to the sensible set of solutions (K = 0) which reproduces to lowest order the nonrelativistic result, Eqs. (10-75) and (10-76). The result of this analysis shows that 2n8/A, with 8 = 1 - P2/4, has an expansion of the form
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2n8 T = 1 + 8(all In 8 + ad + 82[a21(1n 8)2 + a22ln 8 + a23] + ...
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with 8 of order A. These logarithmic terms are not present in a more physical approximation than the ladder series. If this were not enough, Nakanishi has shown that certain solutions have a negative norm! They are called "ghosts," and it is unclear whether they result from the inadequacy of the approximation or from a deeper inconsistency of the theory. The previous study may be generalized to the inhomogeneous equation for the amplitude, in particular to high-energy behavior in the crossed channel (corresponding to the exchanged particles). It is possible to find a Regge behavior sa(t) with a computable trajectory function a(t).
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Various attempts have been made to introduce an effective potential (or quasipotential) approximation to reduce the number of degrees of freedom of the relativistic bound-state problem, in particular the troublesome relative time. Even though they lead to interesting practical results, they mutilate one way or another the exact theory and generally introduce spurious singularities.
10-3 HYPERFINE SPLITTING IN POSITRONIUM
It should not be concluded that relativistic weak binding corrections cannot be obtained for two-body systems that agree with experiment. On the contrary, the positronium states give an example of a successful agreement. This will serve to illustrate the theory. To be completely fair, we should admit that accurate predictions require some artistic gifts from the practitioner. As yet no systematic method has been devised to obtain the corrections in a completely satisfactory way. We quote here some of the significant results and refer to Secs. 2-3 and 5-2 for preliminary investigations. Even though in the study of positronium we restrict ourselves to an almost pure electromagnetic system, some of the methods are useful in other instances such as models of quark bound states of hadrons. The energy difference between the higher triplet (ortho) and lower singlet (para) ground states of positronium, denoted respectively 13 S1 and 11 So (in the spectroscopic notation n 2S + 1 L J ), has now been measured with great accuracy. The values quoted for this hyperfine splitting fiE ts are
fiE ts = 2.033870 (16) x 105 MHz
(Mills and Bearman)
fiE ts
= 2.033849 (12) x 105 MHz
(Egan, Frieze, Hughes, and Yam)
(10-79)
QUANTUM FIELD THEORY
The interval is sometimes also called the fine structure of positronium. Recently Mills, Berko, and Canter have also measured the spacing between n = 2 triplet excited levels
(10-80)
We recall that all these states are unstable. The ground-state radiative width has already been discussed to lowest order (Sec. 5-2).
We may understand the magnitude and sign of the singlet-triplet splitting in positronium by noticing that it corresponds to the sum of two effects. The magnetic interaction is given by the Fermi estimate discussed in the context of the hydrogen atom (Sec. 2-3-2). In terms of the electron-positron parameters with gyro magnetic factors equal to 2, it reads
where [((!o [2 is the square of the nonrelativistic wave function at the origin for a system of two equalmass particles [((!O[2 = (mrx)3/8n. We have also to deal with a new effect corresponding to the annihilation channel, as was mentioned in the early discussion of electron-positron scattering (Sec. 6-1-3). If we restrict ourselves to the lowest-order effect it corresponds to the one-photon channel contributing an s-wave interaction energy of order rx to the triplet state only because of its odd charge conjugation. The desired energy shift may be computed from an effective potential identified with the corresponding T-matrix element at the threshold (S = I + iT) multiplied by [((!O[2 From Chap. 6, the scattering amplitude at the threshold is
u(2)yV u(1)u(1)Y v v(2)
(2m)2
We have been careful about the signs and u(v) denotes the electron (positron) spinor at the threshold
u(l) =
(~,)
u(l) = (xI. 0)
v(2) = CuT(2) = (
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