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for all i Recognize PDF 417 In .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET applications. Encoding PDF417 2d Barcode In .NET Using Barcode creator for Visual Studio .NET Control to generate, create PDF417 image in .NET framework applications. 1, ... , I
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l>:J
rxi(ki  mil
2 '2 JI
(6104) obtained from Eq. (6101) by means of the identity 1
AI'" AI = (I  1)1 drxl" 'drxI<5(I Irxi) ( I rxiAi
(6105) Now the singularities of h(P) arise from the zeros of the denominator [j' = I~ rxi(kr mfJ pinching the hypercontour in (rx, q) space or meeting the boundaries IJi = rxi = O. The singularity equations read PERTURBATION THEORY
(6106a) Ai()(i = 0
= 1, . . ,I
(6106b) (6106c) (6106d) The case A' = 0 leads to Ai = 0 for all i. We discard this trivial solution and assume .9" = O. Then Eq. (6106c) gives while Eq. (6106b,d) lead to
j = 1, ... ,1 We recover the previous equations (6102) with the A replaced by the ()(. The parametric representation (691), valid when the diagram is superficially ultraviolet finite [that is, 2L  4(V  1) < 0], is obtained from Eq. (6104) by integration over the loop momenta q. Disregarding possible singularities coming from the zeros of &i'G(()(), we take.9" = QG(P, ()()  L;~ 1 ()(imr, .9i = ()(" and write singularity conditions as i = 1, ... ,1 (6107) Again A' = 0 is a trivial solution. The conditions may be rewritten as ()(j 8.9" j8()(j = 0 for each j, which implies .9" = Lj ()(j 8.9"j8()(j = 0, owing to the homogeneity of .9". To show the equivalence with the Landau equations (6102) we need to reintroduce a momentum variable ki for each internal line, defined in terms of the external momenta and of the ()( parameters. They are chosen to satisfy an equation of the type (6103) where Ai is replaced by ()(i' With this definition the condition ()(j8.9"j8()(j = o boils down to Eq. (6102a). In the search for solutions to the systems (6106) or (6107), the condition L()(i = 1 may be omitted, because of the homogeneity property of the function .9". In either representation, the solution corresponding to A.i (or (Xi) # 0 for all i is referred to as the leading singularity, while a solution where A.i (or (Xi) = 0 for i E, is called a nonleading singularity. In the associated reduced diagram R = q/, , where all lines i E, have been contracted to a point, this singularity is a leading singularity. The interest of the Landau equations is to cast the problem of determining the location of singularities in an algebraic form. However, finding the general solution ofthese equations, even for simple diagrams, is an arduous task. We shall pursue this program only in the case of real singularities. 632 Real Singularities
Real singularities are those occurring for real values of the invariants Sij = Pi' P j on the physical sheet. Notice that these real values of the invariants do not QUANTUM FIELD THEORY
necessarily correspond to a physically possible kinematical configuration. For instance, the region s = (Pa + Pb < (ma + mb is a real but nonphysical interval for elastic scattering of two onshell particles of masses ma, mb' For real values of the invariants Sij and a nonvanishing imaginary part ( ie) given to the internal masses, no singularity appears along the real contour of integration in parametric space. Therefore, singularities appear only in the limit e + O. It may be shown that any real solution of the Landau equations corresponds to a pinching of the integration contour as e + 0 and thus gives rise to a singularity of the integral. / To see this we use the parametric form (691) of the integral. Consider first a leading singularity, occurring at some real value s = s(O) of the invariants. The Landau equations 8Y'/8rxj = 0 have a real solution rxj = rx)O) (j = 1, ... ,1). In the vicinity of this point, AY'

