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RENORMALIZATION
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Figure 8-3 Subtracting an internal renormalization part y in the integrand f
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parts Yl and Y2, where disjoint means that they have no common line or vertex: Yl n Y2 = 0. This contribution reads (8-44) where G/{ Y"Y2 } refers to the contribution to the initial integrand of all lines and vertices of G except for those belonging to Y1 or Y2. The reason why we only consider disjoint renormalization parts is that otherwise both could not be replaced by their corresponding counterterm. This enumeration of the possible contributions to fiG may be pursued. The resulting recursion formula, due to Bogoliubov, reads
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{Yl, ... ,y,} Ya n Yb=0
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G/{y" ... ,y,}
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(8-45)
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The first term on the right-hand side is the initial integrand, and the sum runs over all families of disjoint renormalization parts. Equation (8-45) after iteration gives, together with Eqs. (8-41) and (8-42), the expression of the renormalized integrand. If G contains no superficially divergent proper sub diagram, then fiG = G. This was the case of all diagrams studied in Chap. 7. For one-loop diagrams, either w(G) < 0 and ~G = G, or w(G) ;:::: 0 and ~G = (1 - TG) G.
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Let us now present simple examples involving more loops. For the sake of simplicity, we consider the theory of a scalar field cp endowed with the interaction .Pint = - Acp3/3! in a six-dimensional space-time. Though such a theory suffers from serious pathologies-its hamiltonian is not bounded from below-it does make sense to develop its perturbative expansion. It is a renormalizable theory (in six dimensions). The power counting ofEq. (8-18) is replaced by w(G) = 6L - 21 = 6 - 2E. Consider the diagram of Fig. 8-4. The subdiagram y inside the dotted box is the only renormalization part,
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Figure 8-4
Nested divergences.
QUANTUM FIELD THEORY
r-----~-=1_---1
:. . _____~_=J--- _.J
Gh 2
Figure 8-5 Overlapping divergences.
besides G itself. According to (8-45),
ilkG
= fG
+ f G/y( -- Tyilky)
~y=fy
whence we deduce
~G = fG
/ Gy( -- Tyfy)
== (1 -- Ty)fG
with a slight abuse of notations, and, following (8-42),
~G =
(1 --
TG)~G =
(1 -- TG)(l -- Ty)fG
(8-46)
We see that to the nested diagrams l' c G have been associated two factors (1 -- T). This would still be true if we were to add more (parallel) rungs to this diagram. It is a good exercise to verify that the renormalized integrand leads to a finite integral. We turn to the diagram of Fig. 8-5. It is superficially divergent [w(G) = 2], as well as its two renormalization parts 1'1 and 1'2 [W(1'd = W(1'2) = 0]. Here a new phenomenon arises; 1'1 and 1'2 are neither nested nor disjoint but overlapping. They share one line and two vertices, but none is included in the other. Consequently, Eq. (8-45) leads to
~G =
/y,( G -- Ty,ilk,,)
/y,( G -- T"ilk,,)
ilk" = f" ilk" = f" If we write TyfG
== f G/yTyfy we finally have
~G =
(1 -- Ty, -- Ty,)fG
~G = (1 -- TG)ilk G = (1 -- TG)(l -- T" -- T,,)fG
(8-47)
We observe that ~G is not equal to (1 -- TG)(l -- 0,)(1 -- T,,)fG. The supplementary terms, namely, (1 -- TG)(T" T,,), just correspond to overlapping sub diagrams. This diagram of Fig. 8-5 will be analyzed further in the case of four-dimensional quantum electrodynamics in Sec. 8-4-4.
8-2-3 Zimmermann's Explicit Solution
The preceding examples suggest the general solution of the recursion equation (8-45). Following Zimmermann, a forest of renormalization parts will be defined as a family U of proper superficially divergent subdiagrams ')I such that either Yr if ')II and
')12 E
')12
')12 C ')II ')II n')l2 =
(8-48)
RENORMALlZA nON
We recall that Yl n Y2 = 0 means that these subdiagrams have no common vertex nor line. A forest may be empty. Moreover, we denote by V the forests of G such that, if G is itself superficially divergent, G does not belong to V. Of course, if G is not superficially divergent, the two sets of forests {U} and {V} are identical. If a consistent set of internal momenta has been chosen to make the operations Ty meaningful, two such operations pertaining to disjoint Yl, Y2 commute, whereas if Yl C Y2, we understand that Ty I will stand to the right of Tn [compare the examples of Eqs. (8-46) and (8-47)]. Under these circumstances, let us show that
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