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Figure 6-31 A diagram with an intermediate state of masses
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ml, ... , ml and the corresponding reduced diagram.
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(6-110)
is a normal threshold of the amplitude. Physically, s(O) is the smallest value of the invariant s such that the physical state of masses mI, ... , ml can be created.
Let us show that such an intermediate state does correspond to a solution of Landau's equations. We equate to zero all a pertaining to lines that do not belong to the intermediate state, say al + 10 . , a]. Then for the reduced diagram
9'(a1o ... ,aZ,O, ... ,O) The equations o9'/oai = 0 have a solution at to be a zero and a maximum of 9'.
aimr
L.,i=1
Js =
I m" ai
mi- /(Im!-I), and this is easily seen
Contrary to naive expectations, there exist other real singularities, called anomalous thresholds. These singularities are somehow troublesome since their contribution to the absorptive part of the amplitude cannot be directly related through unitarity to physical processes. The condition for the existence of anomalous thresholds in a scattering amplitude corresponds to a situation where the axiomatic derivation of dispersion relations is no longer valid. It is therefore important to control the possible occurrence of these new singularities. This is a difficult problem and we shall only illustrate this phenomenon on simple examples in the next subsection.
In momentum space, the distinction between normal and anomalous thresholds may be interpreted in terms of the dimension of the space spanned by the internal momenta ki of the reduced diagram, at the solution of the Landau equations. For a normal threshold, these momenta form a onedimensional space. This is obvious from the set of equations (6-102) for the reduced diagram depicted in Fig. 6-30. These equations express that all the momenta ki (i = 1, ... , l) are collinear, and hence collinear with the external momentum P = I~EG Pv ' On the contrary, the dimension is larger than 1 one for an anomalous threshold.
6-3-3 Real Singularities of Simple Diagrams
The general considerations will be illustrated on one-loop diagrams. First we consider the bubble diagram of Fig. 6-30. We are interested in analyticity properties in the variable s = p2, where P is the total energy momentum entering one or the other vertex. The diagram may represent a selfenergy contribution in a cp3 theory, or a scattering amplitude in a cp4 theory, etc. At any rate, the solution of the Landau equations is trivial. We write the integral in momentum space as (6-111) Here instead of h(P) defined in (6-83) and (6-101) we consider the quantity To(P) = ( - i)v+ I Ic(P) that contributes additively to the scattering amplitude of Eq. (5-171). The Landau equations express the existence of two real numbers Al and A2 such that
PERTURBATION THEORY 309
Al(e-mI)=O
A2[(P - k)2 - mn = 0 Alk. - A2(P - k).
(6-112)
This means that P and k are collinear, and it follows that mIAI = m~A~, S = p 2 = (ml m2)2. The value (ml + m2)2 is the expected normal threshold, while the other one, (ml - m2)2, will soon be shown not to occur on the physical sheet. It seems that we are cheating, since the integral (6-111) is logarithmically ultraviolet divergent and requires a renormalization. Only the value of TG(p2) subtracted at some point is meaningful, but this subtraction does not modify either the Landau equations or the singularity structure. In parametrIc space, the subtracted integral reads
TG(s) - TG(sd = -
i l dlXl dIX2 15(1 0
(4n)
(1X1
+ 1X2)
1X1 2
1X2)
[ 1X11X2S/(1X1 1X11X2St/(1X1
+ 1X2) + 1X2) -
1X1mI - 1X2m~ ] 1X1mI - 1X2m~
as is readily seen from Eq. (6-90) if the integration over A is carried out after subtraction of the integrand at an arbitrary value Sl. The singularities come from possible zeros of [j' = 1X11X2S/(1X1 + 1X2) - 1X1mI - 1X2m~ and the equations O[j'/OlXj = 0 (i = 1,2) lead to
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