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Figure 10-7 Second-order exchange: (a) the retarded interaction Yb to second order and (b) crossed photon exchange. Broken lines represent the instantaneous exchange, notched lines the retarded one, and wavy lines the covariant photon propagator.
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QUANTUM FIELD THEORY
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for K (and K) its zeroth-order approximation, i.e., the wave function at the origin in configuration space K(p) ---+ f d4 p K(p) In short, 4IIX 2 2) _ 2 L\Ebb+x - (2n)2 [IPo [
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Yl Y2 f (k2 d+kiB)2 ({(~ -)11 )12 k
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(f/2 + + mh (Jtl2 - + mh (Y Y~k6-[(k o + m 2) - vi + iB] [(k o - m)2 - w 2 - iB] k2 yi(f/2 +
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)11")12)}
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(10-120)
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+ mh YlvY2 V(f/2 + + mh Y21')
[(k o +m -w 2 +iB]2
The second term is, of course, the crossed photon exchange. The notation w is for ~k2 + m2 and P has been taken here equal to (2m, 0). We proceed to the matrix algebra by taking into account the fact that a spherical average over k may be performed. Thus the part contributing to the splitting is
((y y~ ~~ -)11" )l2)(~ + + m} (~ ---+ 'IV
+ m)2 (y y~ ~~ -)11" )12))
j <01 "02> k6
(Yi(~ + +m}YIVY2(~ + +m)2YzI')
---+
j <01 "02> (3k6 - 2k2)
and we are left with 2 L\Eb~)+x= 8iIX [IPO[2<01"02> 3n
roo Jo k2dkfOO
dk o (k6 - k 2 + iB
x { [(k o + m)2 - w 2 + iB] [(k o - m)2 - w 2 + iB]
+ [(k o - m - w 2 + iB]2
3k6 - 2k2
We reinstate a photon mass J1 and, using similar techniques as above, we find
L\EW+x
= :::
[IPO[2 <01 "02>
(~+ In ~)
(10-121)
The origin of the infrared divergence lies here in the use of a free two-particle propagator, but as expected this approximation is justified since the In (m/J1) terms cancel between (10-119) and (10-121). Second-order radiative corrections L\Eb1J modify the vertices and the Vb potential through vacuum polarization. The latter correction does not affect the singlettriplet splitting to order IX s since it is a short-range effect while the former may
INTEGRAL EQUATIONS AND BOUND-STATE PROBLEMS
be taken into account by including the anomalous magnetic moments of the electron and positron, i.e., by mUltiplying the dominant term in (10-119) by (1 + 1Y./2n)2. Thus to the required order
~Eb}t =
::: 1CPo 12 <til' ti2>
(10-122)
We turn now to the annihilation part ~E~l) in Eq. (10-109), where the replacement of p2 by 4m 2 in the denominator was justified for the present calculation. A new difficulty arises here since in this contribution we implicitly encounter part of the vertex charge renormalization by including the Coulomb wave function. This is clearly indicated in Fig. 10-8. Care must be paid to the way in which the subtraction is carried out, because of the noncovariant splitting of the one-photon exchange potential. To restore the covariance of the procedure it is necessary to include the second-order terms VbD- l Va + VaD- l lib. This has the effect of completing the photon propagator to its covariant form. As a matter of fact, it follows from the approximation of a free two-particle propagator in the secondorder energy displacement that the Vb potential acts immediately before or after the annihilation vertex. Of course, the leading 1Y.4 contribution is insensitive to this effect. Thus we shall directly combine ~E~l) + ~E~1,~ba into
~E~l) + ~E~~)+ba =
4n(2:)4 2 zi m
f d4p' tr {[K(p') + ~K'(p')]YIlC}
+ !X;~p)
f d4p tr {Cyll[K(p) + ~K(p)]}
4 2 ptr[CyIlK(p)] =4nlY.m
fW(p2 +d:
21Y.2/2)2 tr{CYIl[(l
(1 - !X~~p) + 4:2JCPO}
(10-123)
f d4p tr [Cyll~K(p)] = 2iIY. f d4p {tr [CyIlD-l(p/)(p~/p2._ !Xl . !X2)CPO] P -/1 +ze
_ tr [CyIlD-l(p, 0)Y1VY2VCPO]} p2 _ A2 + ie
Figure 10-8 Annihilation diagrams: (a) lowest-order term and (b) second-order contribution of the crossed term VaD- 1 Vt, + Vt,D- 1 Va.
506 QUANTUM FIELD THEORY
Note that this time in K(p) we have kept the complete two-particle free propagator, while <fJo is still to be understood to involve the product Xl X2 of two-component spinors describing the polarization of the state and hence appears as a matrix under the trace sign. In !l.K(p) we have approximated the Coulomb wave function by its value at the origin in momentum space and used an infrared cutoff fl. The vertex renormalization is obtained as in Chap. 7 by subtracting the similar expression at a large photon mass A and using the renormalization constant Zl = Z2 in the Feynman gauge computed by the same method as in Eq. (7-34):
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