QUANTUM FlliLD THEORY in .NET

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QUANTUM FlliLD THEORY
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it would be sufficient to keep the term multiplying the projector over positive energies. However, to reach the next correction we are forced to use the complete expression. Contributions of order p2 may fortunately be dropped. This is so because they will generate a correction of relative order IY provided p2 is associated to a factor insuring a sufficient ultraviolet convergence. Replacing Eo whenever possible by 2m a satisfactory approximation for D- 1 (t) is
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im + a:1 + m21J.2/4) 2m [(w + m) ri!t!(w-m) - (w -
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p) (1 _ 2mp)
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(10-117)
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Using the same principle as before and keeping in the matrix element only the term responsible for the hyper fine splitting we find
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The remaining part of the integral being rotationally invariant the right-hand side may be further replaced by -(k2/4m 2H<0"1 0"2)' Inserting those expressions into (10-115) and performing the t integral yields 3 3 3 ~E~l) = _ 21J. [<Po [2 <0"1 0"2) d p d p' dk o 3n (2n)3 (p2 + 1J.2 m 2/4)2(p'2 + 1J. 2m 2/4)2(kil - k 2 + it;)
x (k o - w -
~, + 2m + it; -
+w +
~, - 2m -
it;)
The integral over ko is easily obtained by closing the contour in the upper half plane leaving 3 3 41J. d p dV ~E~l) = 3(2n [<Po [2 <0"1' 0"2) (p2 + 1J.2 m2/4)2(p,2 + 1J. 2m2/4
x --
[k[ [ (w + m)(w' + m) 4ww' w + w' + [k [ - 2m
(ww' - m 2) (w - m)(w' - m) + ----'----,---'---';--::-0----::'--- ] w + w' + [k [ w + w' + [k [ + 2m (10-118)
The last term can be omitted since the integral converges when we set a 2 equal to zero in both denominators. We use the identity (w + m)(w' - m) + (w - m)(w' + m) = 2(ww' - m2) and the symmetry of the integral to drop in the middle term one of the a 2 in the denominators, therefore obtaining d3p d 3p' (w' + m) [k [ -2 2 2
p2(p'2
a j4)2 4(w
+ m)ww'(w + w' + [k [)
If we rescale p' ---> (maj2)p' and p ---> mp and keep the dominant a- 1 contribution this yields a simple integral
INTEGRAL EQUATIONS AND BOUND-STATE PROBLEMS
2 dp P dp' p'Z --(41ll mZa 0 ~1 + pZ (1 + Jl + pZ)(1 + P + Jl + pZ) 0 (1 + p'Z)Z
(21l/ --(I-ln2) mZa
The first integral in (10-118) contains the dominant eJect. We write it as d3p d 3p' (w + m)(w' + m) [k [ Za Zj4)Z(p'Z + mZa Z (pZ + m j4JZ 4ww' w + w' + [k [ - 2m
n (2n)3 ="2 mZa z
d 3p d 3p' Za Zj4)Z(p'Z + mZ Zj4)Z (pZ + m a
[(W + m)(w' + m)
4ww' w
[k [ ] + w' + [k[ _ 2m - 1
We cannot apply here the same method as above in a straightforward way. If we were to do so we would generate an extra infrared divergence from the energy denominator (w + w' + [k [ - 2m). To overcome this difficulty we introduce artificially a photon mass /1, thereby replacing [k [ by z + /1z 'in the denominator. If /1 is such that maj2 /1 m we can proceed as before. The remaining integral is then easily evaluated as
(2n)3 [ (m)] mZa 1 + In
Collecting our results we have
L\E~l) =
3m l
2nrx 1 11 <til' til) <Po
[1 - (1 + ~
(10-119)
The rx 4 contribution agrees, of course, with the original estimate. The next correction contains a spurious infrared divergence due to our way of estimating integrals (it is, of course, correct with f1 chosen as explained above). When all terms of the same magnitude will be collected this In (m/f1) should (and will) disappear. Instead of proceeding directly to the evaluation of the annihilation contribution we shall turn to the second-order effect to exhibit the announced cancellation. It would, however, be foolish to compute the second-order effect of the Vb potential without introducing the correction to the Bethe-Salpeter kernel due to the crossed exchange of two photons (Fig. 1O-7b) to maintain gauge covariance. We denote the sum of these contributions by L\EW+x' In applying Eq. (10-108) we are going to make a further simplification. We observe that in this relation we have replaced the intermediate propagation of the electron-positron pair by the free propagation. This amounts in Fig. 10-7a to neglect the instantaneous interaction between the two Vb interactions. In the same spirit we may substitute
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