Feynman Rules for Spinor Electrodynamics in Visual Studio .NET

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6-1-2 Feynman Rules for Spinor Electrodynamics
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The total lagrangian describing the interacting system of photons, electrons, and positrons reads (6-24)
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QUANTUM FIELD THEORY
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y 1 2 J12 2 A 2 !l' = --F +-A --(o A) o 4 2 2
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(6-25)
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= - elifi'ljJ AI'
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The various terms have already been encountered in Chaps. 3 and 4. In the first one, we introduce a massive photon of mass J1 much smaller than the electron mass m, and an indefinite metric is needed. The interaction term results from the minimal coupling prescription i0l' -+ iol' - eAI' in the electron lagrangian. We shall use distinct notations for the two types of propagators involved in the contractions. The electron-positron propagator will be denoted by a solid line, oriented in the direction of the charge (e) propagation:
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x,oc
y,{3
<01 TIjJa(x)lifp(y) 10) =
ljJa(x)lifp(y) i )
(6-26)
d k e-ik.(x-y) (
(2n)4
~ - m + ie "p
As is well known, this propagator is not a symmetric function of x and y but satisfies Eq. (3-176). On the other hand, the photon propagator will be marked by a wavy line:
y, v
<01 T Ap(x)Av(Y) 10) = AAx)Av(Y)
d k e-ik.(X-Y)(_i) (gvp-kvkiJ12 (2n)4 k 2 - J12 + ie
2 kv k p/J1 ) k 2 - M2 + ie
(6-27
To avoid confusion with the electron mass m, we have set M2 == J12/ A. Notice that A and M2 are not expected to appear in the physical results. Because of charge conservation, Green functions have an equal number of IjJ and lif fields:
<01 TIjJ(X1) . ljJ(xn)lif(x n+ 1) . lif(X2n)Av1 (Y1) . Avp(Yp) 10)
(6-28)
For brevity'S sake, we have omitted spinor and vector indices in the arguments of G. As in Eq. (6-14), we derive an expression for G in terms of in-fields: G(X1 ... X2n; Y1 ... yp)
<01 TljJin(X1) lifin(X2n)A~~(Y1) A~:(yp) exp i
<01 Texpi
d4 z !l'int(Z) 10)
rd z !l'int(Z) 10)
(6-29)
PERTURBA nON THEORY
where 2'int(Z) = 2'int[ljIin(Z), li/in(Z), Ain(Z)] is supposed to be normal ordered. The "in" indices will be omitted in the sequel. Expanding (6-29) in powers of e and using Wick's theorem leads to Feynman diagrams built out of the propagators (6-26) and (6-27) and a vertex
(3 r-I
-ie(y,)"
(6-30)
This vertex possesses one vector and two spinor indices which are contracted with the corresponding indices of the photons and fermion propagators. As in the scalar case, the role of the denominator in Eq. (6-27) is to eliminate the vacuum-vacuum subdiagrams. Let us collect the signs introduced by the Wick rules for fermions. From charge conservation, there are two kinds of fermion lines in the diagram, closed loops or open lines ending at a point xi(1 :::;; i :::;; n), and coming from a point xdn + 1 :::;; k i :::;; 2n). A closed loop is composed of a sequence of propagators
r---; r---;
ljI(z 1) Ii/(Zk)ljI(Zk)Ii/(Zl) .. ljI(Zq)li/(z 1)
(6-31)
between fields appearing in interaction lagrangians. Since the latter commute under the T symbol, we may reorder the T product as
<01 T [Ii/(z 1)41(Z 1)ljI(Z dJ [1i/(;k)41(Zk)ljI(Zk)] ... [1i/(Zq)41(Zq)ljI(Zq)J ... 1 > 0
without introducing any sign. The product of contractions (6-31) is obtained after the permutation of Ii/(Z1) with an odd number of fields. Therefore, a minus sign is to be associated to each fermion loop. On the other hand, the open lines define a permutation of the points Xi: X1XkIX2Xkz XnXk, where we recall that Xk is the origin of the line ending at xp. There is then an extra sign, which is the signature of the permutation
(1,2, ... , 2n) -+ (1, kb 2, k 2 , .. , n, k n )
Diagrams which differ only by the orientation of a fermion loop contribute both only when they are topologically distinct. For instance, the two diagrams of Fig. 6-8 are clearly identical; only one of them contributes to G(Yb Y2). On the
Figure 6-8 Example of identical diagrams in spinor electrodynamics; only one of them has to be retained.
QUANTUM FIELD THEORY
Figure 6-9 The four-photon amplitude to lowest order. The diagrams (a) and (b) are not identical in configuration space. After summation over the z variables, we are left with the six distinct diagrams depicted in (c).
contrary, the first two diagrams of Fig. 6-9 contribute both to the four-photon function (photon-photon scattering). They correspond to the two distinct sets of contractions:
,---,
,---,
,---,
!ii(z l)J(Z l)l/!(Z 1)!ii(Z2)J(Z2)l/!(Z2)!ii(Z3)J(Z 3)l/!(Z3)!ii(Z4)J(Z4)l/!(Z4)
where A(Zl) is contracted with A(Y1), etc. In configuration space, the four-point function G(yr, ... , Y4) receives contributions from 4! x 3! distinct diagrams, obtained from those of Fig. 6-9athrough permutations of Zr, Z2, Z3, Z4 and Y2, Y3, Y4; the diagram of Fig. 6-9b, viz, is derived from Fig. 6-9a by permuting Y2 ~ Y4, Z2 ~ Z4. When carrying out the Z integrations, the 4! permutations over the Z give a factor that compensates the 1/4! coming from the expansion of exp i Jdz 2"int(z). We are left with six distinct diagrams (and no factor) (see Fig. 6-9c). The representation in momentum space is obtained by Fourier transformation. For a connected function, we set
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