VI - Ci in .NET framework

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VI - Ci
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Hence the power of p in the corresponding monomial of f!l' is II - Iou = I - VI + Ci 2: I - VI + C I = L I The lowest power of p is reached by the trees tI of G such that Ci = Cz, that is, by these trees that project according to
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All these trees tI may be generated by constructing independent connected trees in each connected part of 1'1 and completing their union into a tree of G. Therefore, their total contribution
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QUANTUM FIELD THEORY
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to f1J' factorizes into f1J'y" that is, the polynomial f1J' attached to YI by the rule (8-21), times f1J'G/y" where the reduced diagram Girl is obtained by contracting all the lines and vertices of each connected part of YI into a unique vertex. We thus conclude that
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We now return to the variables /3. Obviously f31 is the homogeneity factor of the parameters pertaining to YI = G, f31- 1 that of Y1- 1, ... , etc. Since the yare nested, Eq. (8-27) may be used repeatedly:
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f1J'(/3r /3~ ... f31, /3~ ... /31, ... , f31- 1f31, f31) = f31 L1 f1J'(/3r /37-1, /3~ ... /31- h , /37- h 1) = f31LI/3~~ll' [f1J'YI_l(fJI f31-2, /3~ ... /37-2, ... , /31-2, 1)f1J'G/YI_J1) + 0(/37-1)]
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(8-28) with the result (8-25). The coefficient in front of the leading term is 1, since a unique monomial of the initial polynomial f1J' may have this combination of powers of the /3. We return to the integral (8-20) in the sector ~. Under the assumption of the theorem, namely, that w(g) < 0 for any proper diagram g, it is easy to show that w(y) < 0 for any subset Y, proper or not. Therefore the integrand is majorized up to a factor by
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and since WI < 0, the integral 0 II d/31 /31- w, - 1 is absolutely convergent at the origin. We conQED clude that the integral (8-20) is absolutely convergent in every sector.
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For a scalar theory without derivative coupling, the foregoing proof has also shown that as soon as a proper subdiagram has a nonnegative superficial degree of divergence the Feynman integral is divergent. Indeed, there is a divergence in at least one sector and, the integral being positive definite, no cancellation may occur. On the other hand, we have encountered examples in electrodynamics of cancellations between different terms appearing in the numerator of the Feynman integrand in momentum space. For instance, we have shown that the vacuum polarization is only logarithmically rather than quadratically divergent and that light-by-light scattering is convergent. Later, we shall have to show that after subtraction of all subdiagrams such that w(g) ~ 0 the integral is absolutely convergent. Notice that the previous considerations on absolute convergence justify a posteriori the manipulationsinterchange of orders of integrations or changes of integration variables-performed in the derivation of (8-20). As a bonus we also find how many terms have to be subtracted to the conventional propagator (8-2) in order to regularize the theory. For instance, in the <p4 theory, the replacement (k 2 - m 2 )-1 -+ (k 2 - m 2 ) -1 - (k 2 - A2) - 1 makes every diagram finite but the one-loop tadpole, since the superficial degree of divergence now reads
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w(G) = 4L- 41 = 4(1- V)
which is negative for V> 1. The one-loop tadpole may be either regularized separately or discarded by the use of a Wick ordering prescription. In this way,
RENORMALIZA nON
we end up with a finite regularized theory. Such an analysis may be carried out in the case of quantum electrodynamics (see Sec. 8-4-2). A useful corollary of the previous theorem is the following. If a diagram G has no superficially divergent subdiagram, w(g) < 0 for all subdiagrams g =I- G, but is itself superficially divergent, w( G) 2:: 0, then the divergent part of its amplitude is a polynomial of degree less or equal to w(G) in the external momenta P and in the internal masses. Indeed, since w(G) measures the degree of homogeneity of Ia(P) in the momenta and masses, the [w(G) + l]-th derivatives with respect to the Pi and the ml have a degree minus one, and hence are superficially convergent. By virtue of the theorem, the derivatives [OW+l/(OP)W+l]Ia(P) [or (OW+ 1/omw+ 1 )Ia, or mixed derivatives of this order] are finite.
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