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226 November 7, 2013
If we were allowed to consider only the first of the two terms of the result (9.28),
we could obtain the desired cancellation :
⎥
3∑ ⎥⎥⎦
M j = 0 ⇒ Q W = Q D − Q U : (9.29)
ɛ→k 1
j=1
but the second term in Eq.(9.28) spoils this idea by having a quite different
algebraic structure ; no tuning of coupling constants is going to ensure that a
W W γ vertex of the form (9.2.3) can do the job.
Yang-Mills coupling
Treating the W W γ vertex as a prettified sQED vertex does not work. It means
that the photon-W interactions cannot be obtained by the minimal-substitution
rule. This should not come as a surprize since the vertex (9.2.3) is only designed
for graceful behaviour towards longitudinal photons, not towards longitudinal
W ’s. We therefore propose to replace Eq.(9.2.3) by a vertex of the form
i Q W
¯h
(
(a1 p 1 + a 2 p 2 ) ρ g µν + (a 3 p 2 + a 4 p 3 ) µ g νρ + (a 5 p 3 + a 6 p 1 ) ν g ρµ) . (9.30)
Note that because of momentum conservation each of the three terms need
contain only two of the momenta; the constants a 1,...,6 are to be determined.
This we shall do by considering several situations.
First, we condier the process of decay of a photon in a W + W − pair :
γ ∗ (q) → W + (k + , ɛ + ) W − (k − , ɛ − ) .
Kinematically this is only possible if the photon is quite off-shell, and therefore
we do not give it a polarization vector but leave its Lorentz index µ free. The
matrix element is given by
M = i¯h 1/2 Q W A µ ,
A µ = (a 1 k + + a 2 k − ) µ (ɛ + · ɛ − )
+((a 3 k − − a 4 q) · ɛ + )ɛ − µ + ((−a 5 q + a 6 k + ) · ɛ − )ɛ +
µ
= (a 1 k + + a 2 k − ) µ (ɛ + · ɛ − )
+(a 3 − a 4 )(q · ɛ + )ɛ − µ + +(a 6 − a 5 )(q · ɛ − )ɛ + µ , (9.31)
where in the last line we have used q = k + + k − and (k ± · ɛ ± ) = 0. Since even
for off-shell photons the current must be strictly conserved we require that
A µ q µ = 1 2 q2 (a 1 + a 2 )(ɛ + · ɛ − ) + (a 3 − a 4 − a 5 + a 6 )(q · ɛ + )(q · ɛ − ) = 0, (9.32)
which leads to the following relations between the six constants :
a 1 + a 2 = 0 , a 3 − a 4 = a 5 − a 6 . (9.33)