By Damiano Anselmi
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21). 21), which absorbs also a combinatorial factor 1/2. It is easy to check that the counting of combinatorial factors works: ~ (~) -1~+~ (~) -1~+~ (~) -1~+~ (~) -]~=~+~. The sum 81 8 = (e) + (1) + (g) + (h) + (i) can be arranged as 81 + 82 + 83 , wher = (e) 1 1 + 3(h) + 3(i), 82 = (1) 2 + 3(h), 83 = (g) 2 + 3(i). 33) are free of subdivergences. Indeed, the subdiagrams of (e) are obtained freezing the loop to the left or the one to the right. In the former case the sub divergence is subtracted by (i)/3, in the latter case by (h)/3.
Uklt[ . okf,Ln om r Okf,Ll .. okf,Ln am I ,. is sufficiently large [Ie] - n - r becomes negative. Thus, if we oI dl"II' llIiat e t he integral a sufficient number of times with respect to the 1"1 11<11 momenta and/ or the masses it becomes convergent. Said in equivaIII \\'ords, the differentiation kills the divergent pa rt. hat the divergent part must be polynomial in the masses and the I 11 '111 ;11 momenta. This is a crucial property of renormalization. It is called 1,1/ "/' //1 of the counterterms, b ecause the Fourier transform of a polynomial d 11, /' momenta is a finite sum of delta functions and derivatives of delta 111111' 1iOlls , which are distributions localized in a single point.
I'heorem 1 Once subdivergences are subtracted, the overall divergences of dlllgrams are always local and polynomial in the masses. Call Va any vertex of the initia l lagr angian and VL, L ~ 1, any L-loop 1'1II Ll1terterms. Denote the set of VL'S, L ~ 0 , with VL . Call GL a diagram ('II. I/ ' fl 'l m J. "'h'N()"'~/II , I i:. 11 v('l'l,i('('~ of Vo ami 9L 1,11(' ~<'I of S il l diagrams. Call CL a diagram of order L construcLed wilh at least one ('0 111 terterm VN, N > 0, but no VL. Call CL the set of such diagrams.