Landau Equations in Visual Studio .NET

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6-3-1 Landau Equations
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Two types of singularities arise in a function defined by an integral
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F(z)
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dxf(x, z)
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along a contour C, for instance, an interval (a, b) of the real axis. The function f(x, z) is assumed to be analytic in the variables x, z, but for singularities located at x = Xk(Z). Clearly F(z) is analytic for any value of z such that there exists an open neighborhood of C free of singularity. If some singularity Xk(Z) approaches the contour C, and if the contour may be deformed to avoid it, then F remains analytic. Therefore, singularities of F as a function of z are expected in two cases where the contour C may no longer be deformed:
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PERTURBA TION THEORY
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Xl (Z)
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........ XZ(Z)
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Figure 6-29 Singularity ansmg from the pinching of the contour between two singularities XI(Z) and X2(Z) of the integrand.
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1. One of the singularities xr(z) approaches one of the endpoints of the contour: as Z -+ ZO, xr(z) -+ a or b; Zo is called an endpoint singularity. 2. The contour C is pinched between two singularities Xl (z) and X2(Z) (Fig. 6-29): as Z -+ Zo, Xl (z) and X2(Z) approach the contour from below and from above;
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is a pinch singularity.
Elementary examples are provided by the following integrals:
F(z) = F(z) =
dx 2 1 -2--= -arctan_IX+Z Jz Jz
(6-99a)
dx o (x - 2)(x - z) dx
_I_In z- 2
[2(Z - I)J
(6-99b)
F(z) =
o 1- x z
-~=-ln~~ 2
1 + Jz
(6-99c)
1 - Jz
In the first example, z = 0 is a pinch singularity. As z -> 0+, the contour is trapped between the two poles x = iJz and X = - iJz. In the second case, z = 0 and z = 1 are endpoint singularities. The pinch singularity at z = 2 appears in all Riemann sheets of the logarithm, except for the first one (corresponding to the principal determination). Indeed in the first sheet, the contour of integration does not cross the point X = 2. However, going to further sheets obliges us to deform the contour of integration. Pinching may now happen. The same phenomenon occurs in the third example (6-99c). Here the singularity at z = 1 is an endpoint singularity while the appearance of the singularity z = 0 in every sheet except the first one comes from a pinching at infinity. This is clear if we change the \ntegration variable x into u = Ijx.
This discussion must be extended to functions of several complex variables and Zj, F(zj) = JH(Oidxi)!(Xi, Zj). The boundaries of the integrati~n domain, the hypercontour H, are specified by a set of analytic relations Y,.(x, z) = o. The singularities of the integrand !(Xi, Zj) take place on analytic manifolds Ys(x, z) = o. Singularities occur when the hypercontour H is pinched between two or more surfaces of singularities or when a surface of singularity meets a boundary surface. More precisely, it may be shown that a necessary condition of singularity is that there exists a set of complex parameters A.s> Xr not all equal to zero such that at Xi = xp, Zi = zJ: for all s
XrYr(XO, ZO)
for all r for all i
(6-100)
~ [2: XrY,(x, z) + 2: A.sy.(x, Z)JI
UXJ.
xO,zO
QUANTUM FIELD THEORY
The last condition expresses that the hypersurfaces are tangent at the pinching point. This is only a necessary condition. Determining whether the hypercontour is really pinched requires a detailed study. We apply these general results to the case of Feynman integrals, first expressed in minkovskian momentum space. We consider
h(P) =
The notations are the same as in Eq. (6-83), but all internal momenta ki ~ve been expressed in terms of a set of loop variables q[ and of the external momenta P. The boundaries of the integration domain are at infinity. In this elementary discussion, we shall disregard the possible occurrence of endpoint singularities at infinity and introduce no term !Jr. The singularities of the integrand are given by the equations Y i == kr = O. The Landau equations are nothing but the expression of the necessary conditions (6-100) in this case:
d4 q[ I (2n)4 )] kr -
mr + if:
(6-101)
Ai(kr -
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