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PERTURBA nON THEORY
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(6-32)
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where all momenta are incoming. Let us now summarize the Feynman rules for the computation of G Draw all possible topologically distinct diagrams.and c associate the factors enumerated in Table 6-1. Then contract all spin or indices along the fermion lines (and thus compute a trace for each closed loop) and all vector indices along the photon lines. Finally, carry out all the integrations over internal momenta.
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Table 6-1 Feynman rules for spinor electrodynamics
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1. External lines Incoming fermion
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Outgoing fermion
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Incident photon
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2. Vertices
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3. Propagators
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4 d p ( i (211:)4 p - m
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- - -I
+ is
4 d k ( (211:)4
(gV2p -
kvk p/JJ k - Jl2 + is
2 kvkp/Jl ) k 2 - M2 + is
4. A minus sign for every closed fermion loop and a global sign, depending on the configuration of external lines, computed as indicated above
QUANTUM FIELD THEORY
Figure 6-10 A diagram of spinor electrodynamics without any symmetry factor.
Figure 6-11 Two diagrams with opposite orientations of the spinor loop.
For a diagram of order r (diagram with r vertices), the integrations over the r variables z and their permutations cancel the factor l/r! coming from the expansion of exp i Jd4 z 2'int(Z). Therefore, with this set of rules, when only topologically distinct diagrams are taken into account, no symmetry factor occurs in fermion electrodynamics. The reader is invited to compare the diagram of Figs. 6-8a or 6-10 with the analogous scalar diagrams of Figs. 6-6c or 6-3a. The former have no symmetry factor when a single orientation is selected for each fermion loop; the latter have weights respectively of 1/2! and (1/2 !)2.
Furry's theorem All the diagrams containing a fermion loop with an odd number of vertices may be omitted in the computation of a Green function. Indeed, the two orientations of the loop (Fig. 6-11) lead to contributions of opposite sign. To show that, we write the contribution corresponding to the first orientation as
G 1 = tr [1'",SF(Zl, zS)1'",SF(Z" zS-tl1''''_1 "'1'",SF(Z2, Zl)J
and we appeal to the existence of a matrix C satisfying [compare with (3-176)J
CS F(X,y)C 1 = S~(y, x) C1'"C1
(6-33)
-1'/
in order to insert CC- 1 between propagators and rewrite
G1 = (-1)' tr [1'!,S~(z" Zl)yJ,S~(Zs- t, ZS)1'~_I" '1'!,SJ;(zt, Z2)J
(-1)' tr [1'",SF(Zt, Z2)1'", "'1'",SF(Z" Zl)J
(6-34)
Up to the sign ( - I)S, this is exactly the contribution of the second orientation. For odd s, the two contributions cancel, which proves the statement.
6-1-3 Electron-Electron and Electron-Positron Scattering
As an illustration of the diagrammatic machinery, let us compute the cross section for electron-electron scattering to lowest order. We start from the reduction formula derived in the previous chapter:
X U(P'l, 8'l)(i x'l - m)u(p'z, 8'z)(i X2 - m)
x (- i ~XI - m)u(pl' 8d( - i iiX2 - m)u(p2' 82)
....,.
....,.
<01 Tl/I(X'l)l/I(x'z)l/i(x 1) l/i(X2) 10)C
(6-35)
PERTURBATION THEORY
PI, I
P2' 2
Figure 6-12 General form of the diagrams contributing to electronelectron scattering.
The notations are summarized on Fig. 6-12; notice that here P'I, P2 are the outgoing electron momenta. To lowest order, the two diagrams of Fig. 6-13 contribute to the Green function, and Z2 = 1. The Dirac operators and the integrations in (6-35) compensate the factors attached to the external lines and put the external momenta on the mass shell. Use of momentum conservation at every vertex implies in this case that no integration is left. The two diagrams have a relative minus sign and Sfi = (2n)4 c5 4(p'1 + P2 - PI - P2)( - ie)2
_ , , v
- U(Pl' el)Y U(Pl, 1'1)( -z)
(gv k(I)2 _ fl2 + k(I)2 _ ) U(P2' e2)Y U(P2' 1'2) k~l)k~I)/fl2 W)k~)/fl2 _ , ,
_(' ' k~2)k~) k~2)k~2) fl2) + U P2, 1'2)Yv U(p 1,1'1 )( -z.) (gv P k(2)2 _ fl2 / fl2 + k(2)2 _ /M2
_(' ') p ( )] U Pb 1'1 Y U P2, 1'2 (6-36)
where
k(l)
= PI k(2) = PI
- P'1
P2 - P2
- P2 =
pi - P2
Therefore, W)ii(pi, e'dyVU(Pb 1'1) = ii(P'I, ei)(PI - P'1)U(Pb I'd = 0 and
k~2)ii(P2' e2)yVU(Pb
1'1)
= ii(P2,e2)(PI - P2)U(Pb I'd = 0
The terms in kvkp in the propagators give no contribution, and in the computation of the elastic cross section we may take the fl = 0 limit without encountering any infrared divergence: Sfi = - ie 2(2n)4c5 4(pi + P2 - PI - P2).07
=:E;,-p',
Figure 6-13 Lowest-order contributions to electron-electron scattering.
QUANTUM FIELD THEORY
p; (E, p')
PI (E, p)
p~(E,
_p')
Figure 6-14 Kinematics of electron-electron scattering in the center of mass frame.
with
or _
::1 -
ii(p't, 81)yVU(PI, 81)ii(p'z, 8'z)yvU(P2, 82) ( '- PI PI ) 2 ii(p'z., 8'z)YvU(Pt, 81)ii(p't, 8't)YvU(P2' 82) (PI - p'z.)2
(6-37)
The antisymmetry of the initial or final states is now manifest. There is no factor due to the identity of the particles, but of course the total cross section will be obtained by integrating over only half of the final phase space. Let us compute the differential cross section for unpolarized initial beams, when the final polarizations are not observed. The kinematics of the reactiop. in the center of mass frame is represented in Fig. 6-14, where () is the scattering angle in this frame, the energy E is conserved, and we denote 1pi = 1p' 1= P = JE2 - m 2. Using the general formula (5-13), with due care paid to the fermion normalization, we get
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