Renormalization in Parametric Space in Visual Studio .NET

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8-2-4 Renormalization in Parametric Space
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It is possible to reexpress the subtraction prescriptions in the parametric representation. This formulation allows a simpler proof of the convergence theorem. We shall outline the major steps. As a preliminary example, consider again the one-loop self-energy diagram (Fig. 8-7) of the sixdimensional "cp3 theory. Its value in the euclidean region is
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+ rxz)]p l . We
find a quadratic divergence due to the non-
Figure 8-7 The bubble diagram.
RENORMALIZA nON
integrability at the origin of the parametric space. If we use the intermediate renormalization, the renormalized form will be obtained after subtraction of the truncated Taylor expansion in p 2 at the origin
dl IR(P 2 ) = I(P 2 ) - 1(0) - P 2 ~2 dP
(8-51) Under a simultaneous dilatation of the parameters lXI, 1X2 by a factor A, the subtracted integrand will behave as r 1. Owing to the extra factor A arising from the measure dlXl d1X2, the integral will be convergent at the origin. Now the subtraction in p 2 may be transmuted into a subtraction over the parameter A, if we recall the homogeneity properties of Q and fY' and observe that
Q(pP, IX)
Q(P, p21X)
(8-52) This explicit example leads us to the following definition. Letf(p) be a function of p such that pPf(p) is differentiable at the origin for some integer p. We define the operator :!/k as
k+Pl
:!/kf(p) = p-PI
s. dp
[pPi(P)]p=o
(8-53)
with k integer. It is straightforward to check that this definition does not depend on PI 2: P and that it could be generalized to p noninteger. The expression (8-53) makes sense provided k + P 2: 0, which allows k to be negative. By convention, we set ifk+p<O The essential property of this generalized Taylor expansion is that
(1 - :!/k) f(p)
O(pk+ 1)
(8-54)
In the previous example, the renormalized integrand reads
(1- :!/-4){[fY'(p21X)]-3 e-Q(P,P'.)} [p=1
where the notation implies that the subtractions are performed at p = 0 and the function is then evaluated at p = 1. These operations are generalized to arbitrary situations. Returning to a four-dimensional theory, the translation of formulas (8-49) yields the following renormalized amplitude:
(8-55)
~(P, IX) =
I T1
U yEU
(_:!/;,21,) {[fY'(IXW2 e-Q(P,.)} [p,= 1
The operator :!/p, acts on the parameters IXI pertaining to the renormalization part 1', after a rescaling IXI ..... P;IXI; Iy denotes the number of internal lines of 1'. The subtractions are performed at p, = 0 and
QUANTUM FIELD THEORY
: : , . , , : )([[IX L_____ J
: : KTIIX .
r--------------,
, , ,
'- _____________ .J
XID<
;-------------~
; ______________.J1 L
Figure 8-8 Overlapping divergences to be studied in parametric space.
the result evaluated at p, = 1. Finally, these operations are performed on all the y belonging to a forest U, and a summation over the U is carried out (U = 0 corresponds to the identity with no subtraction). In Eq. (8-55), we have taken for simplicity a scalar theory without derivative couplings, considered in the euclidean region and renormalized at the origin in momentum space. The generalization to nonzero spin, derivative couplings, higher dimensions, etc., offers no difficulty-only for notational ones. This formulation of subtractions in parametric space enjoys remarkable algebraic properties. It may be shown that the complete expression (8-55) is independent of the ordering of the Taylor subtractions, although two particular .'Y pertaining to overlapping subdiagrams do not commute in general. Moreover, for a given diagram, or a finite set of diagrams, there exists an upper bound on the powers p. Consequently, the operators .'Yp depend only on the number of internal lines I" and no longer on the particular topology of the diagram. Even the last reference to the topology in (8-55), namely, the enumeration of forests and renormalization parts, may be disposed of. First, we prove that (8-56) where the product runs over all the renormalization parts of the diagram. In other words, the subtractions pertaining to overlapping subdiagrams drop out. As an illustration let us verify this property on the example of Fig. 8-8. In four-dimensional space, the diagram G together with the subdiagrams yl, Y2, and yare renormalization parts. Showing the equivalence between Eqs. (8-55) and (8-56) amounts to proving the identity
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