Figure 7-8 The electron propagator expressed as a power series in the self-energy ~. in .NET framework

Draw PDF-417 2d barcode in .NET framework Figure 7-8 The electron propagator expressed as a power series in the self-energy ~.

Figure 7-8 The electron propagator expressed as a power series in the self-energy ~.
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where all other massive parameters have been neglected as compared to A2 when the latter tends to infinity. The identity
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IX L (p, A) = 2n f1 d[3 (2m
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[ [3A2 ] [3p) In m2(1 _ [3) + [3Ji - [3(1 - [3)p2 - if:
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(7-30)
The A dependence affects the coefficients of a linear term in p, the remaining expression being finite. The situation is slightly complicated by an infrared problem. As long as /1 2 remains finite the quadratic form appearing in the logarithm is positive for p2 < (m + /1 Indeed, for 0 :::;; [3 :::;; 1 and p2 below threshold we have 0:::;; [3(1 - [3) :::;;{-and (1 - [3)m 2 + [3/12 - [3(1 - [3)p2 ;:::: (1 - [3)m 2 + [3/12 - [3(1 - [3)(m + /1
In this region will be "real," that is, == = Beyond this point will acquire an "imaginary" part corresponding to the process
La yOLat yO La.
[(1 - [3)m - [3/1]2
Virtual electron
real electron + photon
This is analogous to the photon case beyond the threshold for pair creation. The bothersome point here is that for /1 2 -+ 0 the threshold coincides with the physical mass shell p2 = m2. In the real world it will not be possible to isolate in the propagator the electron pole from a cut starting at the same location. This explains our insistence on keeping /1 2 > O. When /1 2 = 0, one of the zeros of the quadratic form reaches the limit [3 = 1 of the integration interval, and expansions around the singular point p2 = m 2 are no longer possible.
A similar but distinct phenomenon occurs also in the photon case at much higher orders (four or more loops) corresponding to intermediate states with three, five, ... , photons. It leads to powertype corrections to the large-distance behavior of the photon propagator with extremely small coefficients. Due to its smallness this effect has not yet been measured.
We may, however, consider the limit /1 2 -+ 0 for values p2 < m 2 without encountering singularities. This has the virtue of greatly simplifying the integral in Eq. (7-30), leading to the result
IX A2 La(p, A, /1 = 0) = 2n {In m2 ( 2m 2P + 2m [m2 p2 p2 1+
p2 In ( 1 - m 2)]
1[3 - 2P 2 +
4 m - (P2)2 ( p2) m2]} (p2 In 1 - m2 + 7
(7-31)
As p2 tends to m 2 this expression remains finite but its derivatives become infinite.
QUANTUM FIELD THEORY
Insertion of such a self-energy in formula (7-28) obviously shifts the position of the propagator pole and modifies its residue. Two equivalent attitudes are again possible. Either we call mo (and eo) the parameters occurring in the original lagrangian, define m to be the modified position of the pole, and reexpress the Green functions in terms of physical mass and coupling constant. Or we avoid mentioning bare parameters and introduce counterterms in the lagrangian in such a way as to keep the position and residue of this pole unchanged. We may do that by expanding L considered as a function of HP = p2) around p = m:
L(p, A) =
c5m =
c5m(A) - [Z21(A) - 1](P - m)
+ Z21(A) LR(P)
(7-32)
L (p, A) Jp=m
-1 Z2 (A) - 1] = 8p (p, A) J p=m
To order h the factor Z21(A) in front of LR(P) can be ignored and the first equation appears as a Taylor expansion since and However, it is more transparent to interpret this relation as
In the Feynman gauge we use (7-30) to extract c5m and Z21. In fact, for c5m we can ignore the infrared infinities and apply Eq. (7-31): c5m a (A)
m - In A2 4n m2
30( (
1) +2
(7-33)
To extract Z21(A) - 1 we have to return to (7-30) and keep the contributions singular or finite when J1.2 -+ O. This yields
J1.2 9 = -0( [1 In - 2 + In - 2 + - + 0 (J1.)] - A2 2n 2 m m
(7-34)
The renormalized value of L is obtained by subtracting c5m - (Z21 - 1)(p - m). In the range p2 - m 2 J1.2 we use (7-31) with the result
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