ELEMENTARY PROCESSES in .NET framework

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ELEMENTARY PROCESSES
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5-1-6 Fermions
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We generalize the preceding formalism to the case where spin i fermions, described by the interpolating spinor field I/I(x), are present in the initial or final states. The normalization constant affecting the field is traditionally called Z Z in electromagnetism so that in the weak sense
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I/I(x)----'> Z~/zI/Iout(x)
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t-+ oo in
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(5-53)
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The vacuum expectation value of the anticommutator has the form
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i<ol {I/Ia(X), i/Jp(y)}
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0> 1 = i L [<01 I/Ia(O) In><nl i/Jp(O) 10> e- ip".
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(x- y)
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The matrix spectral density
Pap(q)
= (2n)3 L i5 4 (Pn - q)<OI I/Ia(O) In><nl i/Jp(O) 10>
can be expanded as a combination of the sixteen independent 4 x 4 matrices. From relativistic invariance it follows that
Ifwe deal with a theory invariant under parity, as is the case for electromagnetism, a unitary operator f l> exists such that
f l>
10> = 10 >
This leads to the condition
and requires the terms involving y5 to vanish, that is, P3 = P4 = O. Consequently,
p(q) = Pl(qZ)4
+ Pz(qZ)
Pi(qZ) = 0
if qZ < 0
From PCT in variance the antiunitary operator and acts on the fi~lds as
leaves the vacuum invariant
This enables us to relate the second term in the anticommutator to the first according to
Hence
(5-54)
QUANTUM FIELD THEORY
Using the support properties of Pi(q2) on the positive real axis we find a superposition of free-field anticommutators:
i <01 {l/Jin(x), ~in(y)}
10) =
- (i~x
+ m)Ll(x 3
y, m) = S(x - y, m)
(5-55)
Ll(x - y m) ,
d k 2kO(2n)3
(e-ik.(x- y) _ eik.(x- y )
In the interacting case we find
i<OI {l/Ja(X),
~p(y)} 10) = =
too dJ12 [Pl(J12)i x + P2(J1 2)] ap Ll(x y,J1) P2(J12)]<5ap Ll(x - y, J1)}
y, J1)
too dJ12{Pl(J12)Sap(X + [J1Pl(J12) -
(5-56)
It is understood that J1 is the positive square root of J12. An analogous formula holds for the propagator with the same spectral densities PI and P2:
Tl/J(x)~( )10) =
i f d k e-ik.(X-y)f d 2 (2n)4 J1
~Pl(J12) + P2(J12)
P - J12 + ie
(5-57)
From the relation pt(q) = yOp(q)yO it follows that PI and P2 are real. Moreover, since the quantities tr [yOp(q)] and tr [yO(q - J1)p(q)(q - J1)] (with q2 = J12) are proportional respectively to the positive quantities La I<OIl/Ja(O) In) 12 and 2 La I<01 [(i - J1)l/J] a In) 1 we have the following positivity conditions: , (5-58) Finally, it is possible to extract the one-particle contribution. Using invariance under parity, we come to the conclusion that J1Pl(J12) - P2(J12) has a support only on the continuum states, so that in (5-56) the first term only receives a contribution from the discrete one-particle state for J1 = m. This assumes, of course, that we can isolate a one-particle state from the continuum, and is not valid in the strict sense with massless particles such as photons present. The small fictitious mass of the photons will therefore be removed at the final stage of the calculations after coping with possible infrared divergences. Otherwise the single (charged)-particle pole becomes a gauge-dependent branch-point singularity, and the treatment gets slightly cumbersome. Assuming, therefore, that PI contains a term of the form Z2<5(J12 - m 2) and that ml stands for the continuum threshold,
i<OI {l/J(x),
~(y)} 10 ) =
Z 2S(x - y,m) - f: dJ12[Pl(J12)i x
+ P2(J12)]Ll(x -
y,J1)
(5-59)
If we let XO = yO the left-hand side reduces to the canonical anticommutator i<5 3 (x - y)y~p. Using the identities
ELEMENTARY PROCESSES
i1(X - y, ,u)!XO=yO = 0
it therefore follows that
1 = Z2
+ L~ d,u2 Pl(,u2)
which, combined with positivity (5-58), leads to the same conclusion as in the scalar case, namely,
0:::; Z2 < 1
(5-60)
From this point, we can proceed to the reduction formulas as in Sec. 5-1-3. Dirac operators replace Klein-Gordon ones. The free field has the Fourier decomposition
ljIin(X) =
f(~:;3
=~ 1 [bin(k, 8)u(k, 8) e- ik x + dln(k, 8)v(k, 8) eik . X]
with fixed helicity spinors satisfying v(k, 8) = CuT(k, 8). The creation and annihilation operators are expressed as
. bin(k, 8) = dtn(k,8) = bln(k,8) = din(k, 8) =
d 3x u(k, 8) e ik . x yOljlin(X) d 3x V(k,8) e-ik.xyOljlin(X)
(5-61)
d3xiJ/in(k,8)yOe-ik.xu(k,8) d 3x iJ/in(k, 8)YO e ik . x v(k, 8)
A sign ambiguity affects one-particle states of given helicity and momentum. Ignoring finite size wave packets we can write for an electron, say, in the initial state [compare with (5-26)] <out! bln(k, 8) lin) =
t~i~oo Z2 1/ 2
d 3x <out! iJ/(x)yO lin) e- ik . x u(k, 8)
= <out! bZut(k, 8) lin) - iZ- 1 / 2
d 3x [<out! iJ/(x) lin)
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