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III II I I I I in .NET framework
III II I I I I Reading PDF 417 In Visual Studio .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in .NET framework applications. PDF 417 Creation In VS .NET Using Barcode creator for VS .NET Control to generate, create PDF 417 image in Visual Studio .NET applications. I I I I 1I
Scanning PDF417 In .NET Framework Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications. Bar Code Printer In .NET Framework Using Barcode printer for .NET framework Control to generate, create bar code image in Visual Studio .NET applications. Figure 107 Secondorder 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. Read Bar Code In VS .NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in VS .NET applications. PDF417 Generator In Visual C#.NET Using Barcode drawer for VS .NET Control to generate, create PDF417 image in Visual Studio .NET applications. QUANTUM FIELD THEORY
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Reading USS Code 39 In VB.NET Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET applications. Printing Barcode In ObjectiveC Using Barcode encoder for iPhone Control to generate, create bar code image in iPhone applications. )11")12)} Printing ANSI/AIM Code 39 In VB.NET Using Barcode creator for Visual Studio .NET Control to generate, create Code 39 image in VS .NET applications. Barcode Creation In VS .NET Using Barcode printer for ASP.NET Control to generate, create barcode image in ASP.NET applications. (10120) 1D Generator In Visual C# Using Barcode creation for .NET framework Control to generate, create Linear 1D Barcode image in VS .NET applications. Create UPCA In Java Using Barcode creation for Java Control to generate, create GTIN  12 image in Java applications. + 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 ~) (10121) The origin of the infrared divergence lies here in the use of a free twoparticle propagator, but as expected this approximation is justified since the In (m/J1) terms cancel between (10119) and (10121). Secondorder 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 shortrange effect while the former may INTEGRAL EQUATIONS AND BOUNDSTATE PROBLEMS
be taken into account by including the anomalous magnetic moments of the electron and positron, i.e., by mUltiplying the dominant term in (10119) by (1 + 1Y./2n)2. Thus to the required order ~Eb}t =
::: 1CPo 12 <til' ti2>
(10122) We turn now to the annihilation part ~E~l) in Eq. (10109), 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. 108. Care must be paid to the way in which the subtraction is carried out, because of the noncovariant splitting of the onephoton exchange potential. To restore the covariance of the procedure it is necessary to include the secondorder 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 twoparticle 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} (10123) f d4p tr [Cyll~K(p)] = 2iIY. f d4p {tr [CyIlDl(p/)(p~/p2._ !Xl . !X2)CPO] P /1 +ze
_ tr [CyIlDl(p, 0)Y1VY2VCPO]} p2 _ A2 + ie
Figure 108 Annihilation diagrams: (a) lowestorder term and (b) secondorder 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 twoparticle free propagator, while <fJo is still to be understood to involve the product Xl X2 of twocomponent 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. (734):

