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+ .::. [In
(mr)2
_1 - 2y e-2m,
~ + ...J 3
Q(r) =
{ 1+
4n 1/2(mr)3/2
+ ...
mr 1
with y equal to Euler's constant - $:;, du In u e- = 0.5772 ... According to the definition of charge Q( co) = 1. We see again that as r decreases Q(r) increases, even becoming infinite as r tends to zero. The approximation used in Eq. (7-23) is only valid in the mean.
-----":---------......
......
...., - - - - - 2S 1/ 2 Figure 7-6 The vacuum polarization contribution to the 2S 1/2
2P 1/2 hydrogen splitting.
RADIATIVE CORRECTIONS
Finally, we remark that the above conventions amount to the use of a photon propagator with residue equal to unity at the photon pole (when J12 ---> 0). The factor Z3"I/2 attached to photon lines in scattering amplitudes has therefore been replaced by one.
7-1-2 Electron Propagator
The first nontrivial contribution to the one-particle irreducible Green function with two external electron lines (also called electron self-energy) is shown diagrammatically on Fig. 7-7 and reads
." . 2
-zL....(P) = (-ze)
d k 1[ (2n)4 i k 2
g pa
_ fl2
+ ie + (k 2 -
(1 - A)kpk a
+ ie)(Ak 2 -
+ ie)
(7-25) This expression, a 4 x 4 matrix function of p, unfortunately suffers from all possible diseases. It has an apparent linear ultraviolet divergence (meaning that the integrand looks like Jd 4 k/1 k 1 3 for large k). We shall see that this can be reduced to a logarithmic divergence. Infrared divergences might also creep in for small k and special values of the external momentum; we therefore keep fl2 -# 0 to be on the safe side. Also gauge dependence is expected, as would be the case in nonrelativistic physics for a wave function of a charged particle. To deal with meaningful quantities it is necessary, as before, to use a regularization of the free propagators. To avoid cumbersome notation we shall assume that a cutoff A2 has been introduced in an appropriate manner, without being more explicit. We write (7-26) in such a way that Ia(p) corresponds to the choice of Feynman gauge (A = 1), and use the parametric representation of propagators to express Ia,b after integration over the loop momentum k as
(7-27a)
Figure 7-7 Electron self-energy to lowest order.
QUANTUM FIELD THEORY
exp { I. [
+ 1X3) 1X1 + 1X2 + 1X3
1X2(1X1
p2 -
1X1/1
2 /1 1X3 , IL
1X2m2J}
(7-27b)
The masses are understood as having a vanishing negative imaginary part, ensuring convergence for the large IX. As in the case of photons, the self-energy parts are classified according to the number of loops, and the propagator itself is obtained by summing the series (Fig. 7-8):
i i i m + p - m [ - i~:)p)J p - m
+ p m [~
i i i il)p)]~,/,~ [ - i~)p)]~,/,~ p-m p-m
+ ... = P-m- L( p)
(7-28)
We proceed by computing La(p, A), where the cutoff A will soon make its appearance when we try to integrate over a common dilatation factor of the IX parameters:
La(p, A) = 2:
tXl tXl
dlXl dIX2
<5(1 -
1X1 -
1X2)
dp (2m o P
.( 2 2 2 1X1P) e'P Cl, Cl 2P -Cl,/1 -Cl2 m )
(7-29)
The integral over p is only logarithmically divergent due to the extraction of the kinematical factors m and p, in contradistinction to the classical linear divergence of self-mass (Chap. 1). Were m so large that Ip I A m, corresponding to a static classical limit, then we would recover this linear divergence in A. In order to have a relativistic invariant regularization we may subtract from La the contribution of a fictitious massive photon with /1 replaced by A. The present value of A is therefore unrelated to the one used in the vacuum polarization. Should it be necessary, for large but finite cutoff, to study the A-dependent parts, it would be wise to introduce a uniform regularization procedure. We therefore use the substitution
OO dp e'P (Cl,Cl2P 2 -Cl,/1 2 -Cl2m 2) -+ fOO dp (e'P (Cl,Cl2P 2 -Cl,/1 2 -Cl2m 2) _ e-1PCl1 A2 ) . . . _ _ f
-+--0-+-
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