This allows us to compute SJ<-i as in VS .NET

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This allows us to compute SJ<-i as
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+ ei(Pj-q)'y-i(p,-q)'x ii(p I, !X)t. q _ m + if; t I u(p '" !X,)] 1 I
QUANTUM FIELD THEORY
We perform the integrals over x, y, and q, so that
SJ<-i = -ie 2 (2n)4(j4(kf
+ Pf -
k i - Pi) m tiU(Pi, IXi)
x [U(Pf' IXf)fiJ Pi
+;i _
ii(Pf, IXfMi Pi
-,L _
m fiJU(Pi, IXi)]
(5-104)
Integrals over configuration space variables have generated the energy momentum conservation (j function. The is prescription in fermion propagators can be dropped since by adding or subtracting a light-like vector (k) to an on-shell momentum we necessarily leave the mass shell. From the discussion given in Sec. 5-1, the coefficient in brackets is the reduced transition matrix element
(5-105)
The two terms occurring in!!7 can be represented by Feynman diagrams (Fig. 5-3). External lines carry the polarization functions s, U, or U and the corresponding momenta. Internal lines correspond to the propagators. Vertices which represent the interaction ey are such that energy momentum is conserved at each of them. This explains why the first propagator corresponds to k i + Pi = kf + Pf and the second to Pi - k f = Pf - k i Total energy momentum is therefore conserved throughout the diagram. The limit f.1 -+ 0 for photons is trivial and it is easily checked that if one of the polarizations is longitudinal (index 3) the contribution vanishes. We can therefore limit ourselves to the two transverse polarizations. Using (5-105) let us compute the scattering of photons on unpolarized electrons when the final electron polarization is not detected. This amounts to averaging the probabilities on initial (electron) polarizations and summing over the final ones. Up to kinematical factors we therefore have to evaluate 1-La,.aI I!!7f <-i 12. The scattering cross section is given by Eq. (5-13) with the appropriate modifications due to the presence of an initial electron [flux factor (2/m)Pi k i ] and of an electron in the final state [phase space factor m d 3 pf/(2n pJ] , leading to the expression
_!'" d(J - 2 L... (2n)
a,.a
4(4)
+ kf
3 .Iorf<-i 12~ 2kd3(2 f )3 (2 )3pf k md O k,)::; 2 .. k.
n Pf
Figure 5-3 Lowest order Feynman diagrams for the Compton effect.
ELEMENTARY PROCESSES
Let us use laboratory variables with
Hence
dkJ d pfd 3 4 kfb (pf+kf-Pi-ki)=(kf )2/ d(pJ+kJ) / dO.
where 0. is the emitted photon solid angle measured with respect to the incident momentum. We have
dpJ + dkJ _ 1 _1_ d(pJ)2 _ 1 _1_ ~ [2 dk O + 2pfO dkO - + 2pfOdkO m f f f d(J = ct 2
(k _ k)2] _ Pf' kf f , 0kO Pf f
~ L 1u" . U12
2 "!.",
(kJmf dO. Pi'kiPf'kf
where the kinematical factor is derived from momentum conservation
+ k i)2 = (Pf + kf )2
Pi ki
= pf'kf ~b mki =
For photons we use the notation k = kO =
Ikl, and we are left with
(5-106)
Rationalizing the denominators and using Dirac's equation the matrix element can be rewritten
U(pf> ctf)0f Pi
+ ~i _
m ti
+ ti Pi - : f _
m h )U(Pi' cti)
_ (Pi + {{i + m Pi - ({f + m ) 2 k ti - ti 2, k h U(Pi' cti) = U(Pf' ctf) h
_ -( )[J 2(Pi+ki)'Si- i~i_J.2(Pi-kf) Sf+ f~fJ ( . .) - U Pf' ctf f'f 2 k f" 2 k Up" ct, Pi' i Pi' f
The transverse photon polarizations Si, sf can be chosen orthogonal to the "time" axis Pi and to k i (for Si) or kf (for sf). Hence the previous matrix element takes the form
We should have noted that the total contribution of the two Feynman diagrams is invariant under the exchange
(5-107)
permuting initial and final photon variables with a sign reversal of the momentum.
QUANTUM FIELD THEORY
This is an example of a new symmetry of transition amplitudes, called crossing symmetry, which will be elaborated later (Sec. 5-3-2). Using
p+m) u~(p, r:x)udp, r:x) = ( - 2 -
(5-108)
we can now sum over electron polarizations
(5-109)
with (5 = yOotyO and the trace is to be taken over Dirac indices. In the present , case we find the cumbersome expression
(5-110)
The trace can be divided into four terms: two diagonal ones, each of which is obtained from the other by the substitution (5-107), and two crossed terms. The latter are equal since the trace of an odd number of y matrices vanishes and tr (YlY2 ... Y2n)
tr (Y2n'" Y2Yl)
as follows by transposition and relating Y to yT through the charge conjugation matrix C. Two nontrivial traces remain to be computed. We try to make the best possible use of the identities kr = k} = 0, sf = s} = - 1 by applying repeatedly the identity r/l~ = - ~r/l + 2a' b to move the corresponding momenta adjacent to each other
Tl = tr [htil'i{Pi
+ m)I'iih(Pf + m)]
= tr [htil'iPil'ihhpf]
= 2Pi' ki tr ( f ilti tdfP f) = 2Pi' ki tr ( f~dfPf) = 8Pi' ki (2sf ' k i Sf' Pf + ki ' Pf) = 8Pi' k i [2(Sf' ki + kf ' Pi]
where at the last stage we have used momentum conservation which implies ki'Pf=kf'Pi and sf Pf=sf (Pi+ki-kf)=sf ki. (We have chosen Sf'Pi= Si . Pi = 0). Similarly,
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