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J<l) + J<2) + f(3) + J<4) and divide the coefficient of the multilinear combination
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To recover the scattering amplitude we have simply to replace
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by the sum
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TIi J<i) by 4! With the substitution fpa =
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As a shorthand notation we identify f/ with a matrix, and the trace symbol runs over the four values of the Lorentz index. It is then easy to see that (f .J
4 tr f4 - 2(tr P
From the above instructions the amplitude M is found to be
= - ~ 1.- [5(tr J<1)f(2)tr J<3)J<4) + tr J<llJ(3) tr J<2)J<4) 4
+ tr f(1)J<4)tr J<2lJ(3 ) - 7 tr(f(1lJ(2)f(3)f(4) + J<2)f(1)f(3)J<4) + f(3)J<1)J<2lJ(4) + J<2)J<3lJ(1lJ(4) + J<3)J<2)J<1)J<4) + J<1)f(3)J<2lJ(4 )J
The unpolarized cross section involves the average of the absolute square of M:
IMI2 = i L IMI2
In the center of mass frame it reads 3 3 1 d k3 d k4 4 4 d(J = 4(k1 k ) 2W3(2n)3 2W4(2n)3 (2n) b (k3 2
+ k4 -
k1 - k 2) M
1 -64w2(2n)2 IMI2 dO
To perform the sum over polarizations we note that " Jr*arp' a' - - (g pp' kak a' + k p k p,g aa' - g p a'kak p' L. p J e
g p'
ak p k a')
A tedious calculation yields
4 d(J 1 1 0: 2 (90)2m 8 dO = (2n 2w
x 139 [(k1 k2 (k3 k4
+ (k1 k3)2(k2 k4)2 + (k1 k 4)2(k 2 k3 J
The second invariant combination which could have entered this expression, that is, Ldistinct permutations (k1 k2)(k2 k3)(k3 k4)(k4 k 1) is equal to half the quantity under brackets in Eq. (7-99). If 0 is the center of mass scattering angle we have k1 k2 = k3 k4 = 2w 2 k1 k 3 = k2 k4 = w 2(1 - cos 0) k1 k4 = k 2 .k3 = w 2(1 + cos 0) so that the unpolarized differential cross section is equal to
d(J dO
= (2n)2 (90)2
4(w )6
1 m2 (3
+ cos
20) 2
w/m 1
From Bose statistics this expression is symmetrical in the exchange e--+ n - e. To obtain the total elastic cross section it is therefore required to integrate only over half the unit sphere in solid angle, with the result
= 2n (90 11
1 139 (56) (W)6 1
w/m 1
This result agrees with our previous estimate therefore producing the required coefficient a.
As an exercise the reader may compute the coefficient b entering the high-energy behavior (7-95) of the cross section. The photon-photon induced interaction implies the possibility of a coherent scattering on charged targets when two of the lines in the diagram of Fig. 7-15 refer to Coulomb interactions with a nucleus. This can be related to pair creation in the nuclear field and may be compared with the Compton process. An alternative process worth investigating is the photon "splitting" amplitude when a photon hitting a target (-Ze) yields two photons.
7-3-2 Lamb Shift
The 1947 measurement by Lamb and Retherford of the shift between the hydrogen states 2S 1/2 and 2Pl/2 predicted to be degenerate according to the Dirac theory has been a memorable event stimulating the development of quantum field theory. Here we shall content ourselves with a simple account of the effect, leaving aside the more basic questions related to the formulations of a relativistic boundstate theory. In fact what Lamb and Retherford measured was the 2P3 / 2 2S 1/2 transitions as a function of an applied magnetic field. In the limit of zero field the observed value was approximately 1000 MHz lower than what could be expected from the fine structure interval mrx4 /32 '" 10,960 MHz. To analyze this discrepancy we shall use, following the original work of Bethe, a combination of Dirac's theory and radiative corrections, as discussed earlier in this chapter. We have seen that the vacuum polarization, by modifying the effective potential, had the effect of increasing the binding of the 2S 1/2 level by 27 MHz relative to the 2P1/ 2 state. This cannot therefore be the main effect since the experimental observation requires the 2S 1/2 level to lie well above the 2P1/ 2 one. Of course, a full theory requires to take into account all effects of the same order and to include nuclear corrections (magnetic moment, recoil, form factors, polarizability, etc.). Moreover, all excited levels are metastable, meaning that each one has a natural line width. In Chap. 2 we recalled Welton's argument implying the interaction of the bound electron with the fluctuating vacuum electric field and leading to an estimated shift with the correct sign and order of magnitude:
4 Z4 rx S 1 bEn I ~ - m - - In ~ 1:51 0 , 3n n3 Zrx'
We now want to give a more quantitative evaluation. Our starting point will be the effective interaction (7-77) for electrons inter-
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