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_ ffi -iV(q) = - 4mamb
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(7-127)
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where _q2 can be identified with the square relativistic momentum transfer q2 = (Pa - p~)2, and the center of mass energy square s = (Pa + Pb)2 is taken equal to (ma + mb)2. From (7-123) we can compute the amplitude ffi to order g2 as represented in Fig. 7-16. This is again a one-loop diagram with a strong ultraviolet divergence. A theory based on (7-123) is not renormalizable due to the dimensionality of the coupling constants. However, we agreed to use it only as a phenomenological description and as such to any order we may perform a finite number of ultraviolet subtractions without modifying the long-range behavior of V which we intend to find. Therefore for our purpose we are safe. The elementary contraction needed to apply perturbation theory is
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Kpv,pa(k)
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(7-129)
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Before any subtraction the Feynman rules provide us with an amplitude ffi of the form
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ffi d(k, k')
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d4k d4k' o4(k + k' - q) (2n)4(k 2 + ie)(k,2 + ie) d(k, k')
= 8g~g~Kpv,pa(k)KPv,pa(k')
+ [4g1g~(p~p'p + ppp'~)Kay,pa(k)KPv,pa(k') + (a +-> b)] + g1gWp~p'p + ppp'~)Kav,a'v'(k)KPv,P'v,(k')(p~,p'~' + p~,p,~,) + (k+->k')J
(7-130)
A convenient way to perform the ultraviolet subtractions without modifying the long-range behavior follows from the observation that for fixed s = (Pa + Pb)2, ffi is analytic (and purely imaginary) along the semiaxis q2 < O. Its cut along the real axis arises from its real part and we are only interested in the vicinity of q2 = O. We therefore write
= -:-
dm 2
2 ffi (m ) R 2 . m-q-le
+ ~ffi(q2)
(7-131)
RADIATIVE CORRECTIONS
Pb --~-....- ....- - P~ b
Figure 7-16 Interaction between neutral systems arising from electric and magnetic susceptibilities.
with I:lfii regular in the vicinity of q2 = 0 and M2 positive and arbitrary. Consequently, from (7-126) and (7-128) the asymptotic part of the potential will be given by
(7-132)
We shall soon see that fiiR(q2) behaves as - A(q2)2 for q2 -+ 0 so that the dominant part of V will be obtained by taking the limit Mr -+ 00 in (7-132), that is,
Va,(r)
16n mambr
fro dx
-x =
8n mambr
(7-133)
It remains to compute A. To do so we observe that each denominator in (7-130) may be split into two terms according to I/(k 2 + ie) = PPlk 2 - inb(F) where PP means principal part. This we
abbreviate as P - inb. Up to the polynomial d the integrand in (7-130) takes the form (P - inb)(P' - inb') = PP' - n 2bb' - in(bP' + b'P) This is, of course, nothing but the Fourier transform of [GF(x - y)F in configuration space with GF the Feynman propagator. The effect of d is then to act as a polynomial of derivatives on this quantity. Now if instead of using G} we were to use the product Gadv(x - y)Gret(x - y) with at most a support at x = y we would obtain by a Fourier transformation an ill-defined polynomial of the momenta without contribution to the real part of fii. Consequently, as far as we are concerned, the replacement 1 1 k 2 + ie k'2 + ie -+ F 1
+ i(kO/lkOI)e
1 k'2 - i(k,oljk' l)e
would lead to zero. This means that we may subtract to the previous contribution the combination (P - ineb)(P' + ine'b') = PP' + n 2ee'bb' + in(Pe'b' - P'eb) with eb standing for (kO II kO i) b(k 2). The result is the quantity
(PP' - n 2bb') - (PP'
+ n 2ee'bb') =
n 2(1
+ ee')bb'
Note that 1 + ee' = 2 if kO and k'o are of the same sign and zero otherwise, and that the 15 functions project the photons on-shell. For q2 > 0 the conservation of energy momentum k + k' = q implies that we may keep only one of the two possibilities kO > 0, k'o > 0, for instance. We have now (7-134) This lengthy derivation is nothing but an application of the Cutkosky rules (Chap. 6) to this special case. Of course, fiiR as given by Eq. (7-134) is a convergent quantity and we simply want its leading
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