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After this rotation, we get in the euclidean region
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exp - [Q (P a) + " a m2 J (4n;2L[.9'G(a)p I I
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The factor qG) in front of Io(P) [compare with (6-83)] contains i V coming from the expansion of exp [i Jd 4 z 2';nt(z)] to the Vth order. We set qG) = iV C(G) and rewrite (6-93) as
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qG) . C(G) _ _ - - Io(P) - 1 - - Io(P)
S(G)
S(G)
For instance, in the Atp4 theory, C(G) = ( - At. We recall from the analysis of Sec. 6-2-2 that the proper functions have been identified with the il(n) defined through the Legendre transformation (6-73) and (6-75). Therefore, thanks to this last factor i, these functions r(n) coincide with the real, euclidean Green functions r(n):
r(n)(p 1, . . . , Pn) = r(n)(h , Pn)
defined as sums of the contribution [C(G)!S(G)J IG(P) of each diagram. It is easy to see, by repeating backward the same computation as in Sec. 6-2-3, that
PERTURBATION THEORY
(6-94) where now k/ are euclidean fuur-momenta, kr = I~~o (ki Here, as in Eq. (6-93), the integrand enjoys positivity properties. Of course, Eq. (6-94) means that f may be computed by euclidean Feynman rules: Vertex Propagator
instead of - iA.
instead of
(6-95)
k -m
From these rules, we learn that connected Green functions (with external propagators) are related by (6-96) in the coincidence region (6-92). In configuration space, these functions read
Gc X!, ... , Xn -
-( )_ f
TIn d
4 qk
(2n)
- , ' e - iq,' x, (2n )4 i5 4" q..) Gc(ql, ... , qn) (L.,
and a similar expression for f(fPc)' Collecting all factors i in Eqs. (6-71b), (6-72), (6-73), and (6-96), it is then straightforward to check that f( fPc) is the Legendre transform of G : c (6-97) where fPc and j are related by (6-98) The lowest-order euclidean functional in the fP4 theory is
rtree(fPc)= -
d4X 'E[fPc(X)] - =
~ d4X 2k~O iAfPc8k fPc
m A. + TfPc2 + 4!
fPc4)
Notice the change in sign of the (8 ofP)2 term as compared to (6-82). The euclidean theory may be considered as a statistical mechanical theory, the weight of each field configuration being exp [ - d4z 2'E( fP)]. This will appear more transparent after the introduction of functional integrals (Chap. 9). As an exercise, the reader may discuss the meaning of the Wick rotation when spin i fields are present, namely, what happens to the l' matrices and what is the relation between t/I and 1[/. He or she may then write the Feynman rules for euclidean quantum electrodynamics.
6-3 ANALYTICITY PROPERTIES
In the last section of Chap. 5, analyticity properties of amplitudes were derived from the general principle of local causality. As a typical example, the two-body forward elastic scattering amplitude has been proved to be analytic in the energy variable, under suitable conditions on the values of the masses of the particles. The
QUANTUM FIELD THEORY
analyticity domain is a cut plane with two branch points at s = M~ and u = + m~) - s = M~ (at t = 0), corresponding to the lowest intermediate states in the direct (s) and crossed (u) channels. The discontinuities across the cuts were related through the optical theorem to the total cross section of the channel. Therefore, we expected that the amplitude will have singularities along the real axis at all values of the energy corresponding to the opening of a new threshold, i.e., the onset of a new possible final state. For instance, the nN -+ nN forward scattering amplitude is expected to have singularities at s (or u) = (mN + m,,)2, (mN + 2m,,)2, etc. In the present section, we shall present a sketchy discussion of analyticity properties of Feynman integrals. The interest of such a study is threefold. First, it is important to test in the framework of perturbation theory the general analyticity properties derived rigorously. Whenever we are able to prove from general axioms the existence of an analyticity domain, the contribution of an arbitrary diagram must go through the successive steps of the proof and must therefore enjoy these analyticity properties, or rather their equivalent perturbative expression. However, when the masses are such that the general proof fails, the study of individual diagrams may prove useful. Second, such a study allows to search for complex singularities or to study analyticity in several variables. Even though we suspect that the perturbation series does not converge and therefore that the exact amplitudes may have different properties from those of individual diagrams, this investigation can be suggestive of fruitful conjectures. Finally, dispersion relations in one or several variables may be a useful device for the computation of Feynman amplitudes. We shall frequently refer to the physical sheet of a scattering amplitude. We have in mind the region reached by analytic continuation, starting from threshold and using the Feynman iB prescription. In this process we do not consider the possibility of crossing the various cuts arising from the singularity structure of the amplitude.
2(m;
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