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+ m2)2 since 1X1, 1X2 >
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O. The integral may be computed explicitly:
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A1/2 (mI 4n2[TG(s) - TG(sd] = [ - I n 2 2s ml
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+ m~ 2 + m2 -
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1/2 A
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mI - m~ - - - - I n 2 - [s = Sl] 2s m2
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(6-113)
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In this expression the last term reminds us that the amplitude is subtracted at s = for the function introduced in (5-155a):
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A = A(S, mI, m~)
and A stands
+ mt + m1 -
2(mI
+ m~)s -
2mIm~
= [s - (ml + m2)2] [s -; (ml - m2)2]
The argument of the logarithm can be rewritten as
and in this form it is clear that in the physical sheet, corresponding to the first determination of the logarithm, TG(s) is free from singularity at s = (ml - m2)2. On the other hand, singularities at s = 0 and s = (ml - m2)2 occur in the unphysical sheets. / Even though this is not the main purpose of this subsection, we may elaborate a little on these complex singularities. The singularity at s = (ml - m2)2, also called a pseudothreshold, is seen to correspond to a solution of the Landau equations in parametric space at a negative value of 1X1 or 1X2, thus outside the initial integration interval. This means that going to unphysical sheets has forced us to deform the contour of integration from 0 ,,;; IX ,,;; 1 to a complex path. This is the analog ofthe phenomenon discussed in the example (6-99b). Similarly, the singularity at s = 0 is the analog of the case (6-99c). This singularity, which did not appear as a solution of the first set of Landau equations (6-112) is an illustration of the so-called second-type singularities that we have discarded by assuming that the boundaries at infinity of the integration domain in momentum space were irrelevant. They are not, actually, because a pinching at infinity may occur which leads to the singularity at s = 0, as seen in the form (6-111) of the integral. The Landau equations express that the hyperboloids k2 = mI and (P - k = m~ are tangent, which occurs when p2 = (ml m2)2; but the contact may also take place at infinity, when the centers coincide or are separated by a zero length vector, that is, s = p 2 = O. The appearance of a singularity at s = 0 must be a warning. The analysis of complex singularities is indeed ... complex!
QUANTUM FIELD THEORY
Figure 6-32 The triangle vertex diagram, and its dual diagram.
In the physical sheet, the imaginary discontinuity of TG(s) across the cut running from (ml
+ m2
(6-114)
+ 00 is
From this absorptive part, the amplitude TG(s) - TG (Sl) may be reconstructed by a once-subtracted dispersion relation, i.e., by the Cauchy formula along a contour encircling the cut:
TG(s) - TG(stl
= -S Sl
(m,+m,)'
ds' 1m TG(s')
(s - 51)(S - s)
(6-115)
We now turn to the vertex diagram of Fig. 6-32a:
T. (2 2 2) __ G PI> P2, P3 -
if (2n)4 (ki: d q
1 m1:)(k1 - m1)(k3 - m~)
(6-116)
with kl
q - P2, k2 = q
+ PI, and k3 =
q. The Landau equations consist of the system
Ai(kr i
mr) =
i = 1,2,3
IAiki= 0
For the leading singularity, none of the A vanish; therefore the determinant of scalar products (k i ' kj ) must vanish. This is more conveniently rewritten as
Y12 Y13 Y23 Y13 Y23
(6-117)
where Yij = ki ' kj/mimj = -(prmJ)j2mimj, (i,j, k) being a permutation of (1,2,3). Similarly, for the nonleading singularity occurring at A3 = 0, say, we must have
mr -
IY:2 Y~21
that is,
PERTURBATION THEORY 311
We recover the normal threshold at s = P~ = (ml + m2)2 while, as in the case of the bubble diagram, the singularity at P~ = (ml - m2)2 does not show up in the physical sheet. This may be seen by the analysis of the Landau equations in parametric space. The discontinuity across the cut between (ml + m2f and + 00 is (with s = p~)
TG(s
+ is) S(PI
TG(s - is) =
4IT.Ie
a+b 2i 2 2 In - (m!, m2, s) a- b
(6-118)
where and
S2 -
+ p~ + mI + m~ -
2m~) - (mI - m~)(pI - p~)
b = .Ie 1/2(mr, m~, s).Ie 1/2(pr, p~, s)
Of course, similar normal thresholds exist in the channels PI and p~. To analyze the anomalous threshold, we introduce the dual diagram of Fig. 6-32b. It is a tetrahedron in three-dimensional euclidean space with edges of square lengths Pf and kr. This is possible > 0 and the ki are on-shell, = mr. The stability conditions of internal and external since particles insure that the angles 8 i (0 S; 8i S; IT) defined through
are real. Equation (6-117) implies that the dual diagram lies in a plane. Finally, the reality condition requires the central point of Fig. 6-32b to lie inside the triangle. This means 8 1 + 82 > IT or, equivalently, cos 8 1 + cos 82 < 0, that is, (6-119)
Let us derive this result using the parametric representation, in the special case where PI = p~ == p2, m~ == m 2 The function [f' reads
[f' =
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