QUANTUM FIELD THEORY in VS .NET

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QUANTUM FIELD THEORY
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LpoII u(p')Tu(p)12 =
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p + m f P' + m)
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2m 2m - f32 sin 2 (0/2)]
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+ 2 Re A)(2E2)[1
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+ 2B(2E 2)f32 sin 2 (0/2) + 2 Re C (2mE) + 0(0: 2)
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All the dependence on the screening factor (J which was occurring in the imaginary parts of A and C has disappeared as expected, and the elastic cross section including the first nontrivial radiative corrections reads
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(:~) Mot! {I + 2: [(1 + In ~)(2<P coth 2<p - 2 coth 2<p
1 <p f32 sin 2 (0/2) ] coth2 <p) (13 (<p coth <p - 1) + 9 - sinh 2<p 1 - f32 sin 2 (0/2)
d<p' <p' tanh <p' -
tanh <p
+ Zo:n
f3 sin (0/2) [1 - sin (O/2)]} 1 _ f32 sin 2 (0/2)
(7-86)
We recall that E and p are the electron energy and momentum, f3 = piE its velocity, 0 the scattering angle, and cosh 2<p = p' p'/m2, sinh <p = p/m sin (0/2), and Iq 12 = 4m 2 sinh 2 <p = 4p2 sin 2 (0/2). In Eq. (7-86) we are left with the genuine infrared divergence. To obtain a sensible result we will have to compute and add the cross section for emitting soft photons.
7-2-3 Soft Bremsstrahlung
In Sec. 5-2-4 we have already encountered electron bremsstrahlung in a Coulomb field. We obtained the Bethe-Heitler formula exhibiting a typical dw/w spectrum, where w is the photon energy or, what amounts to the same, the electron energy loss. If we consider a typical energy resolution I1E we would expect by integration a probability of radiation of the order of Z 20:3 In (I1E/J1), which might well compensate the analogous term in the elastic cross section. Our present objective is hence to integrate the Bethe-Heitler expression on photon variables in a range which we define as w :s;; 11E. We shall assume that J1 I1E E. An objection may be raised to the effect that in Chap. 5 we practically assumed that J1 ~ o. In fact, the only dangerous term in the limit J1/E I1E/E -+ 0 arises from the propagators. We may therefore use the formula (5-151) valid for vanishing photon momentum k. If this contribution is denoted [(d(J/dQ)(I1E)lnelastic' we have
d(J ] [ -(I1E)
(d(J) dQ Mott
inelastic
{J):5,AE
2 2 3 d k e2 [ 2p' pi - - - - - -] m m 2w(2n k pk' pi (k' p (k' p')2
(7-87)
RADIATIVE CORRECTIONS
The resolution could vary with the kinematical conditions, which would require a more careful evaluation using the complete Bethe-Heitler expression. Here we only want to demonstrate the basic compensation mechanism. In formula (7-87) we have in principle E = E' + w, but we are only looking for the dominant contributions in 11E. Therefore, whenever allowed we shall take the limit E' ~ E, Ip I~ Ipi I. Of course, W = (k 2 + /1 2 )1/2. Let B denote the coefficient of (do/dQ)Mott. Using Feynman's technique we write
B = 4n2
w:s;tlE
2 2 3 d k [ 2p pi m m ] 2w (k p)2 - (k p)2 - (k' p'f
0 0 0
where P = i [(p + pi) + Z(p' - p)J. Taking into account that Ik I < [(I1E)2 - /1 2 ] '" I1E we find
2 4n 2 (In 211E _ Po-P /1
~ln Po + IPI) 21 P I po-ipi
Inserting this in the expression of B with the notation p = Ipl, pi = Ip'l, and P = IPI, which we hope will not confuse the reader, we are left with onedimensional integrals over z:
B=- 21nIX {
211E [
p plf + 1 1 ] E E + P E' E' + pi -1+dz +-In--+-In-2 -1 po(zf - p(zf 2p E - P 2p' E' - pi
1 Po(z) I Po(z) + P(Z)} PO(Z)2 - p(zf P(z) n Po(z) - P(z)
p' plf+l d
-1 Z
We observe that po(zf - p(zf notations. Consequently, pOplf+l 1 -2 - 1 dz P0 ()2 - p()2 Z Z
m2 (cosh 2 <P -
sinh 2 <p), using our previous
cosh 2<p
dz cos h 2 <p -
. h2 SIll
2<p coth 2<p
For the last integral we introduce the variable ~ = P(z)I[3Po(z) and take the limit E' ~ E, pi ~ P up to corrections of order I1EIE. We see that ~ is also equal to [cos 2 (fJI2) + Z2 sin 2 (fJI2)] 1/2 with
p' pi f + 1 dz 1 Po(z) In Po(z) + P(z) 2 -1 P5(Z) - p 2(z) P(z) Po(z) - P(z)
1 - [32 cosh2<p [3 sin (fJI2)
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