Radiative Corrections to Coulomb Scattering in Visual Studio .NET

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7-2-2 Radiative Corrections to Coulomb Scattering
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As an example let us consider Coulomb scattering on a nucleus of small Z so that Zrx may still be considered as a small parameter. Atomic corrections (a screening effect in a first approximation) will be neglected. Using the effective interaction (7-75) let us compute all corrections up to order rx 3 to the Mott cross section [Eq. (2-130)] for unpolarized electrons. If E and p stand for the energy and magnitude of the electron three-momentum and 13 denotes its velocity 13 = piE, we recall that the scattering cross section at angle 0 was given by
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= 4p2f3 2 sin4 (012) 1 - 13
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(7-79)
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Since the first corrections will be of order Z2 rx 3 it will be mandatory to include the second Born approximation whose interference with the leading term will
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QUANTUM FIELD THEORY
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produce a contribution of order Z3 0: 3 . This is comparable with the effect of vertex and vacuum polarization renormalization provided Z is small enough. The interest of pushing calculations that far is that it also enables us to discuss another aspect of infrared singularities related to the long-range character of the Coulomb forces. The latter induces an infinite phase shift on the scattered plane waves. To prevent it we may introduce a screening factor which in a consistent theory would be related to the fictitious photon mass }1. In order to show that this type of infrared divergence is cancelled in the cross section we shall distinguish it by introducing an independent screening length (J and by studying the limit (J -+ 0 separately. The diagrams contributing to the scattering amplitude are shown in Fig. 7-14 with the external potential Ac equal to the screened Coulomb interaction ar 4 Ac = 0 A~(x) = - Ze e- = Ze d q 8(qo) eiq x 4nr (2n)3 q2 - (J2
The first three diagrams (1), (2), and (3) of Fig. 7-14 yield
S123 = - iZe 2(2n)8(po - Po)
2 1 2 ii(p')YO {I q -(J
- 2 coth 2<p
+ ~ [(1 + In .E..)(2<P coth 2<p n m
d<p' <p' tanh <p' -
~ <p tanh <p
(7-80)
(1 - coth3 9<p2+!] _ -.!L ~ slllh<p2<p } u(p) <p)(<p coth - 1) . m n
with p. pi = m2 cosh 2<p and q = pi - p. The fourth diagram adds the quantity
S4 = (-zZe)
4 d k [2n8(Po - ko) i 2n8(kO - pO) ] (2n)4 u(p) (P' _ k)2 _ (J2 ey ~ _ m + is ey (k _ p _ (J2 u(p)
From the kinematical constraints, pO = p'O = E and 1p 1= 1pi I. Set
II =
"2 (p + P )12 =
[(pi _ k)2
+ (J2] [(p _ + (J2] [(p _
+ (J2] (p2 _ + (J2] (p2 _
. k 2 + ze) k 2 + is)
d k [(pi _ k
k k
Then S4 takes the form
S4 = - 2 i - - 2n8(po - po)ii(p')[m(l1 - 12) + yOE(l1
Z 2 0:2
+ Jz)]u(p)
(7-81)
We are, of course, only interested in the small (J limits of II and 12 Feynman's identity
+ )')(b + ).)
df3 [f3a
+ (1
1 - f3)b
+ ).]
RADIATIVE CORRECTIONS
enables us to write
where M2 = (J2
+ 4f3(1 -
"2 (p + p )a 12 =
f II
O df3 2M OM
d k (p2 _ k 2 + is)[;k _
p + M2]
(3)p'. Similarly,
(3)p2 sin 2 (8/2) and P = f3p
df3 20Pa - 2M oM
P 0) fd k (p2 _ k 2 + is)[(k _ P + M2] 1
+ (1 -
The intermediate integral is readily performed with the result that 2 d3k 1 = in In p - P + iM 2 + is) [(k - p + M2] (p2 _ k P p + P + iM
with an adequate choice of branch cuts to define the logarithm. The corresponding expressions for II and 12 in the limit (J -+ 0 are
II =
I 2p sin (8/2) n -"-------'-----'-2ip sin (8/2) (J
(7-82)
2p3 cos 2 (8/2) 2
{!: [1 _ sin 1
(8/2)
1 In 2p sin (8/2) sin 2 (8/2) (J
+ In ~J}
2p (7-83)
Inserting these expressions in Eq. (7-81) we obtain the last matrix element. The unpolarized cross section is therefore given by d(J 4Z 20(2m 2 1 (7-84) dO = Iq 14 2 Lpoll u(p') Tu(P) 12 with T standing for the coefficient of (iZe 2/1 q 12) 2nc5(pO - p,O) in the sum S 123 S4, that is,
T= l(1
+ A) + yO
2~ B + C
[(1 + In ~}2<P coth 2<p - 1) - 2 coth 2<p f: d<p' <p' tanh <p' n sinh 2<p
tanh <p
(7-85)
1 IJ ZO( + ( 1 - 3" coth 2 ) (<p coth <p - 1) + 9 - 2n 2 Iq 12 E(I1 + 12 ) <p
B = - ~ ----:---c-:-<p_::__
ZO( -2n- m Iq 12 (II - 12 ) 2
To be consistent, in Eq. (7-84) we keep only terms up to order 0(3. Note that the only compJex quantities are II and 12 . The sum over polarizations yields
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