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0= (1  .'Y,2IG )(  .'Y~/I")(l  .'Y; 21,)(  .'Y,;;21,,) [.9'2(1X) e Q(P.")] Ip,= p,= p='= I in .NET framework
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(.icpd(PIP)mfk.m(.icpd
k::::;;2IY2 m> 21y n+m':::;; 2/y\ Each individual term in the sum is homogeneous of degree k
+ n in .ic. But
+ n:S: 2(1" + I,,)  m:S: 2(1" + I"  == 2Ia
Therefore the action of the last subtraction (1  .'Y,21G ) which retains only the terms of degree
RENORMALIZATION
larger than  21 G gives zero. It is a mere exercise of bookkeeping to generalize this proof to arbitrary situations. Finally, it is easy to show that if we perform subtractions on a homogeneity scale of subsets of parameters that do not correspond to superficially divergent proper subdiagrams, their effect drops out in the complete expression. Therefore, the final result of this analysis is (857) where the product runs over the (2 1G  1) nonempty sets of (X parameters. Again, the result does not depend on the ordering. The merit of this last expression is twofold. First, it is independent of the topology of the diagram. Second, it enables us to understand, at least qualitatively, the arguments for the proof of convergence. Indeed, for any family g of parameters (x, we may bring the operator (1  p~21,) to the left of the product in (857), and it follows from (854) that the integrand behaves as p;21,+ 1 and hence is integrable in pg since the measure stilI contains a factor p~I,l dpg. That the possible singularity of an arbitrary subset of the (X is integrable is unfortunately insufficient to insure the convergence of the integral. For instance, in the integral the singularities corresponding to (Xl .... 0, (X2 # 0, or to (X2 but integration over (Xl leads to the divergent integral .... 0, (Xl # 0, or to
(X2 ....
0 are integrable, The proof that such phenomena do not occur for Feynman integrals is tedious. We will not reproduce it here. As in Sec. 814, we have to divide the integration domain into sectors and appeal to the homogeneity properties of the parametric functions. We conclude this long and technical analysis by stating once again the important result of the BogoliubovParasiukHepp theorem. The subtraction operation described in Eqs. (845), (849), or (857) yields an absolutely convergent integral, and defines an analytic function of the momenta in the euclidean region and a tempered distribution in the Minkowski domain. 825 Finite Renormalizations
We have considered so far the subtractions of infinities. But the previous developments on the construction of counterterms, on the multiplicative character of the renormalization, or on the algebra of subtractions apply as well to finite renormalizations. This term refers to the operations required to modify the normalization conditions, thereby changing by a finite amount the (renormalized) parameters of the theory. Such is the case for instance, if we want to modify the normalization conditions (830) and (835) into (836). More generally, sticking to our cp4 theory, let us consider the following set of renormalization conditions, depending on an arbitrary mass scale J1. : (2)( r R P2) Ip2=It' = J1. 2  m 2
o r (2)(p) Ip2=1t2 = 2 R
rk4 ) Is" = (858) where Sit is defined as in Eq. (835) but with m replaced by J1..
QUANTUM FIELD THEORY
This is a perfectly sensible choice since it is satisfied to lowest order, and it interpolates between conditions (830) and (835), and (836). It is, however, safer to choose /1 such that the renormalization points p2 = /1 2 and S" lie inside the analyticity regions of the two and fourpoint functions respectively. Otherwise, the above condition should be understood to hold for the real part of the amplitude only. The theory depends now on two mass scales: m is the mass entering the propagator in Feynman diagrams and J1. specifies the renormalization point. As for the physical mass, defined as the pole of the complete propagator, it is some function of m, J1., and A, and may be computed order by order in perturbation theory. Also, the residue at the pole is no longer one, and the computation of S matrix elements should take it into account. How are two renormalized theories corresponding to two different choices of J1. related Clearly each one may be reconstructed from the other through the introduction of finite counterterms, determined order by order to implement the new conditions. As in the case of the infinite renormalization, this is in turn equivalent to a redefinition of the parameters m and A of the theory, provided we also allow for a finite wavefunction renormalization of the field. These parameters being equal to the value of the Green functions at a given point J1., changing J1. into J1.' amounts to changing the parameters m, A (and 1) into m', A', and z. Hence (859) where m', A.', and z are functions of m, A, J1., and J1.', computable order by order in perturbation theory. This will be illustrated on the twopoint functions of the cp3 theory in a sixdimensional space. This renormalizable theory has the merit of having a nontrivial wavefunction renormalization to the oneloop approximation. If .Pint =  A13! : cp3:, the selfenergy of Fig. 87 reads

