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QUANTUM FIELD THEORY
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will find no difficulty in deriving the analog of Eqs. (8-13) to (8-18) or in determining in which dimension the Fermi theory or the scalar theories <p3, <p6, (OI'<P P(<p), where P is an arbitrary polynomial, are renormalizable.
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8-1-4 Convergence Theorem
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Let us analyze more closely the relation between the convergence of Feynman integrals and power counting. We first note that as long as the propagators contain an imaginary part is it is equivalent to study the convergence in minkowskian or in euclidean space; only the latter case will be considered here.
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The limit 8 -+ +0 has been studied by Bogoliubov and Parasiuk and by Hepp. They have shown the equivalence between the absolute convergence in the euclidean region defining an analytic function of the external momenta and the convergence of the corresponding Feynman integral toward a tempered (i.e., polynomially bounded) distribution in Minkowski space in the limit 8 -+ +0.
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We use here a restrictive definition of a subdiagram g of a diagram G as a subset of vertices of G and of all the internal lines joining them in G. To each connected proper (i.e., one-particle irreducible) diagram, we associate the family !F of all its connected proper subdiagrams. Of course, ff contains G itself.
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Theorem
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If w(g) < 0 for all g E ff, then the Feynman integral corresponding to G is absolutely convergent (in the euclidean region).
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To simplify the expressions, we only present the proof in the case of a scalar theory without derivative couplings, using the parametric representation. In the euclidean region, it reads
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(8-20)
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We assume that 0 for all I and recall that Q is a quadratic, positive definite form in the external momenta and a homogeneous rational fraction of degree 1 in the ex. Hence, only the convergence at ex = 0 is questionable. The exponent being bounded at the origin plays no role. The polynomial fl> is a sum of monomials of degree L [compare with (6-86)]:
fl>(ex)
(8-21)
trees!T / '!T
Following Hepp, we divide the integration domain into sectors
where 11: is a permutation of (1, 2, ... , J). We shall prove the convergence of the integral sector by sector. To each sector corresponds a family of nested subsets 1'1 of lines of G (not necessarily subdiagrams) :
1'1 C 1'2 C ... C 1'1
== G
where 1'1 contains the lines pertaining to (ex." . .. , ex.), For notational simplicity we present the reasoning for the first sector corresponding to the identical permutation. In this sector, we
RENORMALIZA nON
perform a change of variables
0:1 0:2
= PI p~
PI PI
(8-22)
O:I-l = O:I=
the jacobian of which is
In these Pvariables the integration domain
reads for 1 ~ I ~ I - 1
We may define the superficial degree of divergence of each 1'1> even though 1'1 is not necessarily connected: (8-23) where II = I is the number of internal lines of 1'1. The number of independent loops LI reads (8-24) in terms of the number of connected parts C I and vertices VI. This generalization of formula (6-69) may be easily proved by induction. Of course, we have Ll = 0, LI = L. As a function of the P, f!l' is a polynomial, the distinct monomials of which have coefficients equal to unity. Let us prove that it is of the form
f!l' = PILI p~Lz. .. PILI [1
+ O(P)]
(8-25)
First, when all the 0: are simultaneously dilated by a factor p,
f!l'(PO:l'"'' PO:I) = pLf!l'(o:(, ... , O:I)
Let us now investigate the behavior of f!l' when only the 0: belonging to 1'1 are dilated by a factor p: (8-26) Equation (8-21) expresses f!l' in terms of trees on G. Each tree tI of G projects on 1'1 along the union of Ci2: C I connected trees tI[, ... ,tlc ,. This will contribute in (8-21) a term behaving under the dilatation (8-26) as a power of p equal to the number of lines of 1'1 that do not belong to
o/.t = til utl2 u'" utlc ,
Since the set o/.t satisfies an equation of the type (8-24) with a vanishing number of loops (it is a union of trees) and joins all vertices of 1'z, its total number of lines is Iou
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