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In the square modulus of the amplitude, an average and a sum over initial and final polarizations are understood. After integration of the energy momentum conservation (j function
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PERTURBATION THEORY
The traces are readily evaluated
+ m)yAp'l + m)] = 4(PlvPl. p - gvpPl P'l + Pl pP'lv + m2gvp) tr [YvU l + m)yp(pl. + m)] tr [yV(P2 + m)yPU z. + m)] = 32[(Pl P2 + (Pl pzY + 2m2(pl pz. - Pl P2)]
tr [YvU l and
+ m)yApz. + m)yV = -2pZ.YpPl + 4m(pz.p + Pip) - 2m 2yp tr [Yv(h + m)yp(pz. + m)yV(p2 + m)yP(P'l + m)] = - 32 [(Pi P2)2 - 2m2pl P2]
Yv(Pl
Therefore,
= _1_ {(Pl. P2 + (Pl pz. + 2m2(pl pz. - Pl P2)
2m 4 [(P'l - Pl ]2
+ (Pi pd + (Pl Pl. + 2m 2(Pi P'l - Pi P2)
[(pz. - Pl ]2 2 2 (Pl P2 - 2m Pl P2} + (' - Pi )2(p'2 - Pi )2 Pi
(6-41)
We may express all invariants in terms of the energy E and scattering angle 8: Pl P2 = 2E2 - m 2
Pi P'l = E2(1 - cos 8) Pi pz. = E2(1
+ m2 cos 8
+ cos 8) - m2 cos 8
We finally obtain the M~ller formula (1932) 2 da ex 2(2E2 - m2 (E2 - m )2 ( 2)2 sin 4 8 - sin 2 8 + (2E2 _ m 2 1 + sin 2 8 dO. = 4E2(E2 - m
(6-42)
In the ultrarelativistic limit, where m/E -+ 0,
da dO.
E2 sin 4 8
(_4___2_ + !) = ~ ( 1 + 1 + 1)
sin 2 8
4E2 sin 4 8/2
cos 4 8/2
(6-43)
and in the nonrelativistic limit, E2 ~ m2, v2 = (E2 - m 2)/E2,
da dO.
I = (m 4v (sin48 ex)2 1
sin 2 8
1 1) sin 2 8/2 cos 2 8/2 (6-44)
ex (m
)2 16v 1
(1 sin 4 8/2
+ cos4 8/2 -
a result obtained first by Mott (1930). We let the reader reexpress these cross sections in the laboratory frame. It is also instructive to compare (6-44) with the nonrelativistic limit of the Rutherford formula for Coulomb scattering, where we set Z = 1 with a reduced mass equal to m/2.
QUANTUM FIELD THEORY
Electron
Electron PI
Positron
Positron
Figure 6-15 General form of the diagrams contributing to electron-positron scattering.
Let us now consider electron-positron scattering. The kinematics and lowestorder diagrams are depicted in Figs. 6-15 and 6-16. Polarization indices are omitted and in Fig. 6-16 four-momenta are oriented according to the charge flow. The scattering amplitude may then be obtained from (6-37) by substituting
u(pd -+ U(Pl) U(PJ') -+ U(PJ') U(P2) -+ v(q'r) u(pz) -+ V(ql) PI -+ PI Pl-+ P'l P2 -+ -ql pz -+ -ql
(6-45)
and by changing the sign of the amplitude. The center of mass cross section is readily computed:
with
.'T12 = 1
_1_ {(PI q'l 2m 4
+ (PI ql)2 -
2m2(pl ql - PI .q'r) [(P'l - Pl)2]2
+ (PI ql)2 + (PI p'r)2 + 2m2(pl . Pl + PI . q'r) [(PI + ql)2]2
(6-46)
Figure 6-16 Lowest-order contributions to electron-positron scattering.
PERTURBATION THEORY
It is then straightforward to write the expression of the cross section:
da e - e +
-----;m-
8E 4 - m4 (2E2 - m 2 = 2E2 "4 - E2(E2 - m 2)(1 - cos 8) + 2(E2 - m2 (1 - cos 8 2E4( -1 + 2 cos 8 + cos 2 8) + 4E 2m 2(1 - cos 8)(2 + cos 8) + 2m4 cos 2 8J
16E 4 (6-47)
The ultrarelativistic and nonrelativistic limits are respectively
da e - e +
dQur
IX2 - -- [1 + cos 2 4
8/2 sin 8/2
1 +- (1 + cos 2 8) 2
2 cos 8/2J sin 2 8/2
(6-48) (6-49)
(mIX
)2 -.,--1---.------16v sin 8/2
Notice that the annihilation diagram does not contribute to the latter case. These expressions are due to Bhabha (1936).
The results of Eqs. (6-42) and (6-47) may be compared with experimental data. At low energies we show in Fig. 6-17 some data of Ashkin, Page, and Woodward (1954) for electron-electron scattering at 90 degrees. M ller's formula (6-42) gives a good agreement, in contrast with the corresponding expression [to be derived below, Eq. (6-65)] which would apply to spinless particles. Results for
OL-_ _--L_ _ _---L_ _--I
0.5 1.0
E Lab (MeV)
Figure 6-17 Experimental data for electron-electron and electron-positron scattering at e = 90 as a function of the incident electron energy in the laboratory frame. (a) Electron-electron scattering. The solid line represents the M ller formula, the broken one the M ller formula when the spin terms are omitted. (b) Electron-positron scattering. The solid line is the Bhabha formula, the broken one the prediction when annihilation terms are deleted. (From A. Ashkin, L. A. Page, and W. M. Woodward, Phys. Rev., vol. 94, p. 357, 1974.)
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