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veff = 0 ) + hy(1) + h2 0eff) + ... eff eff in .NET framework
2 veff = 0 ) + hy(1) + h2 0eff) + ... eff eff PDF417 2d Barcode Scanner In .NET Framework Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in VS .NET applications. Draw PDF417 2d Barcode In Visual Studio .NET Using Barcode encoder for VS .NET Control to generate, create PDF417 image in VS .NET applications. (9117) Recognize PDF417 2d Barcode In VS .NET Using Barcode reader for .NET framework Control to read, scan read, scan image in VS .NET applications. Barcode Creation In Visual Studio .NET Using Barcode printer for .NET Control to generate, create barcode image in .NET framework applications. Using (9114) we have first
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To comply with the prescription of normal ordering we should also have added a term in q>2. Otherwise we have to include tadpole diagrams, corresponding to a contraction of two fields at the same vertex. This is precisely the only divergent oneloop diagram of the twopoint function r(2) (Fig. 95a), and it contributes a quadratically divergent term proportional to q>2 in the expansion of V;;~: Aq>2
4 d p i (2n)4 p2  m 2 + is
To this order we also have the logarithmic divergence of the fourpoint function r(4) associated with the diagram of Fig. 95b: 4 d p 1 i (Aq>2)2 4 2 (2n)4 (p2  m 2 + is)2 FUNCTIONAL METHODS
Figure 95 Divergent diagrams to the oneloop order.
Higher powers of q> lead to convergent integrals. Renormalization introduces counterterms needed to insure that d[,(2) dp2
(m~hYs) = 1 (9120) and a constraint on the fourpoint function giving the meaning of the renormalized coupling constant, for instance, its value at the onshell symmetric point S: pr = m~hYS
+ Pj =
%m~hYs
i=lj
(9121) P4)(Pi) Is =
AphYs
These conditions are useful when dealing with some actual application to a scattering problem. They are, however, cumbersome to compute quantities in the effective action expanded at zero external momenta. Up to a finite renormalization, we can replace (9120) and (9121) by [,(2)(0) = _m 2
dp 2 ) ~(O) ['(4)(0) = A
(9122) To emphasize the distinction we have introduced the label "physical" to the mass and coupling occurring in the previous normalizations. The virtue of Eqs. (9122) is that they can be translated as conditions on the effective renormalized potential and function Zeff(q as dq>2
Ye~O) = m
dq>4 Yeff(O) = A
(9123) Zeff(O) These requirements are obviously fulfilled to order 11 as shown by (9118). At this level we also have =1 (9124) eff The counterterms in q>2 and q>4 cancel the corresponding terms in the higher orders of Yeff and Zeff' Consequently, the correct oneloop contribution to the Z(O) QUANTUM FIELD THEORY
(9125) It is useful at this stage to perform the Wick rotation Po + ipo. We denote by k the corresponding euclidean fourmomentum so that 4 1 d k [( Acp2/2 ) Acp2/2 1 ( Acp2/2 ~p =:2 (2n)4 In 1 + k2 + m2  k 2 + m2 +:2 k2 + m 2 (9126) Performing the integral and adding this expression to the zerothorder term given by (9118) yields 2 m cp2 Acp4 [(Acp2 Ac(2) Acp2 Acp2 Yeff=2+::t:!+(8n T+m2 In 1+2m2  T :2T+m2 + ... )2 ( (9127) We observe that the largecp behavior of V(cp) is modified by the quantum corrections. To exhibit it more transparently, it is useful to introduce yet another set of normalization conventions to be able to set m = o. We may define a new coupling constant AM such that (9128) From (9127) it follows that
AM = A +
(8~2 [6 In ~~: + 16 + o (;:)J
We use this definition in (9127) and consider the limiting massless theory for which cp4 ALcp4 ( cp2 25) (9129) Yeff(CP) = AM 4T + (16n In M2 ""6 + ... a result due to Coleman and Weinberg. It is not possible to directly set m = 0 in Eq. (9127). This arises from the structure of the ultraviolet subtractions in Eq. (9126), the second of which, designed to enforce the condition (d 4Yeff/dcp4)(0) = 0, introduces an infrared divergence in the limit m = O. We have to choose an arbitrary but nonvanishing subtraction point cp = M to define the massless theory. Stated differently, the r functions at zero momentum generated by the expansion of (9127) are singular as 111 goes to zero. The behavior when cp + CIJ is clear from the fact that the dimension of V is four so that dominant terms must be proportional to cp4 up to logarithms in cp/M. Note that the arbitrariness in the point M implies that a related change in M and AM must leave Yeff invariant. We encounter here a manifestation of the renormalization group to be discussed at length later.

