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1 2 (0 /l<P) 2 - m <p 2 2:
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1:1 2 m5 2 ..1,0 4 (u CPo) - - CPo - - CPo
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where This is sometimes referred to as the renormalized lagrangian, an unfortunate denomination, since its coefficients are infinite. The lagrangian seR has the same expression as the initial one, up to the replacements cP -+ CPo, ..1,-+..1,0, m -+ mo. We recall that in perturbation theory
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..1,0(..1" Aim) = ..1,[1
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mo(A, Aim) = 1 + O(Ah) m Z(A, Aim) = 1 + O(A2h2)
The last relation is particular to the cp4 interaction. A priori we would expect Z = 1 + O(h). The fact that renormalization boils down to a redefinition of the parameters implies that unrenormalized (or bare) and renormalized Green functions are related through 1) 1 G(n)( Plo , Pm m, IL -- z-n/2G(n) (Plo , Pn, mo, 1L0, A) R reg (8-40) r~)(plo ... ' Pn, m, A) = zn/2 rr~~(Plo ... ' Pn, mo, ..1,0, A) This relation holds for connected and proper functions respectively. Indeed, the connected renormalized functions are computed from the lagrangian seR = se + /).se by adding a source term jcP, while the unrenormalized ones may be computed from seR (in the presence of a regularization) by coupling a source to CPo, that is, jcpo. Taking n derivatives with respect to j leads to the announced relation. The limit A -+ 00 is understood on the right-hand side of Eqs. (8-40). Finally, proper functions are obtained after elimination of one-particle reducible diagrams, an operation which commutes with renormalization, and amputation by the renormalized or unrenormalized propagator respectively-whence the change in the power of Z.
8-2-2 Bogoliubov's Recursion Formula In the preceding subsection, counterterms have been associated to proper functions, i.e., to a sum of Feynman diagrams computed to a given order. The existence of normalization conditions enables us to associate a counterterm to each superficially divergent diagram, computed using the requirement that it satisfies the normalization condition. A subtlety is involved when owing to cancellations caused by symmetljies (Bose or Fermi symmetry, internal symmetry, etc.) individual diagrams may be more divergent than their sum to a given order. In that case, it is safer to consider only sets of diagrams-gauge-invariant sets in quantum electrodynamics-that exhibit these cancellations. We shall avoid
390 QUANTUM FIELD THEORY
this unessential complication in the sequel by restricting ourselves to a scalar nonderivative theory. Let us now investigate the effect of lower-order counterterms on a given proper Feynman diagram G. Indeed, G may contain superficially divergent proper sub diagrams y: w(y) ~ O. To each of these subdiagrams we may associate a counterterm of order h L , (L y being the number of loops of y). Let G denote the integrand of the Feynman diagram G in momentum space, PiG the one when all lowerorder counterterms are taken into account, and f!llG the renormalized integrand that leads to a finite integral. If G is superficially convergent, for w(G) < 0 (8-41)
However, if w(G) ~ 0, f!llG differs from PiG by the contribution of the counterterm attached to G itself. Alternatively, PiG has to be subtracted to yield f!llG that leads to a finite integral satisfying the normalization condition. After integration over the internal momenta S(f!llG - PiG) must be a polynomial of degree less or equal to w( G) in the independent external momenta of G. In the framework of the intermediate renormalization (subtraction at zero momentum), PiG - f!llG will be nothing but the Taylor expansion TGPiG of PiG in the external momenta at the origin, up to the order w(G) included. We shall write for w(G)
(8-42)
We are left with the problem of relating PiG to the initial integrand G. The difference between PiG and G comes from the counterterms associated with the renormalization parts y of G. Following Zimmermann, this refers to proper superficially divergent subdiagrams, hence containing all the lines of G that join two of their vertices, according to our convention of Sec. 8-1-4. The contribution of the counterterm associated with y is - TyPiy, and when inserted in the diagram G in place of y this gives (8-43) Here TyPiy denotes the Taylor expansion of the modified integrand Piy, in the independent external momenta of y, up to the order w(y) (included).
This Taylor expansion is not as well defined as in the case of G itself, since the distinction between internal and external independent momenta requires more care. We shall admit here that a definition of these external variables is always possible and refer the scrupulous reader to the references quoted in the notes.
On the other hand, Gjy stands for the contribution to the integrand of the lines and vertices external to y in the initial diagram. It contains in particular the propagators pertaining to the lines that join y to the rest of G. (See Fig. 8-3 for an illustration.) Equation (8-43) contains the contribution of the counterterm relative to y (and possibly of those contributing to Piy). It is now easy to write the contribution of two counterterms relative to two disjoint renormalization
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