'2 _ _fl_o_ _ Z 2 fl - 1 + 6)(0) 3flo in .NET

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'2 _ _fl_o_ _ Z 2 fl - 1 + 6)(0) 3flo
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(7-66)
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Here flo stands for the (infinite) bare mass. Therefore, fJ,2 A2 = Z3fl5A2. In a massive vector boson theory the real mass fl would be related to fJ, by a finite renormalization factor.
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It is now necessary to convince ourselves that higher Green functions are finite, since we have now exhausted our freedom of choice by selecting the measurable mass, coupling constant, and pole residues. This analysis is of interest because there still exists a potentially dangerous amplitude for photon-photon scattering. If we were dealing with spin-zero charged bosons, a new physical constant, apparently absent at the level of Born diagrams, would emerge from the needs of renormalization. In this case we have to give the normalization of the charged particle elastic scattering amplitude. The mechanism responsible for ultraviolet divergences in one-loop integrals may be characterized as follows. If k denotes the momentum flowing through the loop we are interested in the large-k behavior of the integrand in the Feynman k integral. Each fermion propagator contributes a factor 1 1- 1 and each boson propagator a factor 1 1- 2. Vertices with a YJ1. coupling are k independent. Proper k functions are classified according to the number of fermion external lines zero, two, four, etc. Typical diagrams are shown in Fig. 7-13.
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QUANTUM FIELD THEORY
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Figure 7-13 Examples of one-loop diagrams with zero, two, four, ... , external fermion lines.
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If A stands for some ultraviolet cutoff, the behavior of the integral will be
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4 - IF - 2IB > 0
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4 - IF - 2In= 0
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with IF(I B denoting the number of fermion (boson) lines in the loop. The integral ) will be superficially divergent if W(G)
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4 - IF - 2I B ;:::: 0
(7-67)
where w(G) is called the superficial degree of divergence. In electrodynamics the kinematics somewhat complicates the matter. We have already encountered examples such as in the case of vacuum polarization (IB = 0, IF = 2) where the above counting seemed to indicate a quadratic divergence w = 2, but current conservation allowed the extraction of a factor (k 2 g Pa - k Pk1 leaving only a logarithmic divergence. Similarly for the electron self-energy w = 1, but again the fact that L was found proportional to the dimensional quantities m and p reduced the degree of divergence by one unit. For the vertex function the counting is correct and w = O. But even in this case only one form factor was found divergent. We may express w(G) in terms of the (even) number of external fermionic lines EF and of the number EB of bosonic external lines by using the fact that two fermion lines and one boson line meet at each vertex. Along the loop the number of vertices V is equal to the number of internal propagators V= IF and according to the previous argument 2V= 2IF
+ IB
V= 2In+ EB
+ EF
since each propagator meets two vertices. Eliminating V, IF, and IB we find
(7-68)
Therefore we derive the following table:
RADIATIVE CORRECTIONS
Superficial degree of divergence
Effective degree of divergence
o o o
Excluded by charge conjugation invariance
(7-69)
As the number of external lines is increased the integral becomes more convergent and table (7-69) shows that we have exhausted all possibilities except for photonphoton scattering. However, the latter is in fact convergent, again due to gauge invariance, which dictates some kinematical structure leaving a well-behaved integral. This follows from our earlier discussion of the Euler-Heisenberg effective lagrangian (Sec. 3-3-4) and wiII be discussed in detail below (Sec. 7-3-1). To order h we now have a finite predictive theory satisfying perturbatively the general principles of unitarity and causality, since the counterterms amount to subtractions in dispersion relations, i.e., they do not modify the analyticity properties nor the discontinuity rules. Another confirmation stems from the fact that we are using a local hermitian lagrangian. Let us now return to Eq. (7-65) and reorganize the various terms as
1 2 4Z3F
1 A + 2" Jl5 Z 3A2 - 2" (0- Af + Z2
[i2"ljJ $ljJ -~
(m - (jm)ljJljJ
(7-70)
- ZleljiJljJ
This expression leads to a natural interpretation. The fields ljJ, ljJ, and A occurring here are renormalized fields corresponding to propagators with residue equal to
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