Pictures Paths Particles Processes

304 November 7, 2013
× ∆ µα∆ νβ − ∆ µβ ∆ να
2(s − M 2 )
× u(p 3 ) ɛ αβκλ p κ (A ′ + B ′ γ 5 ) γ λ v(p 4 ) ,
s = p · p , p = p 1 + p 2 = p 3 + p 4 . (10.231)
Because of the current conservation and the antisymmetry of the vertices, we
may replace ∆ µα ∆ νβ − ∆ µβ ∆ να by 2g µα g νβ . Furthermore, since
we have
ɛ µνρσ p ρ ɛ µν κλ p κ = 2 ( p σ p λ − s g σλ) (10.232)
1
M = −2i¯h
s − M
( (
2
v(p 1 ) ( A(m 2 − m 1 ) + B(m 1 + m 2 )γ 5) u(p 2 )
−s
× u(p 3 ) ( A ′ (m 3 − m 4 ) − B ′ (m 3 + m 4 )γ 5) )
v(p 4 )
(
v(p 1 ) ( A + Bγ 5) γ µ u(p 2 )
× u(p 3 ) ( A ′ + B ′ γ 5) ))
γ µ v(p 4 )
. (10.233)
Here m j is the mass of momentum p j . Note that, in contrast to e.g. the case of
QED, m 1 = m 2 or m 3 = m 4 is not necessary for current conservation. We can
now investigate several situations. In the first place, if M ≠ 0 the amplitude has
a pole for some nonzero s value, which we may take as the signal of a particle.
The second term in brackets in Eq.(10.233) then tells us that, indeed, a spin-1
particle has been exchanged 32 . The occurrence of the first term is, then, not
surprising : a similar contribution is found in e.g. the W exchange in muon
decay. Secondly, we may take M = 0. In that case, the second term no longer
has a pole. It can therefore not survive a truncation argument, and must not be
counted as coming from any particle propagation. The first term does survive ;
if we also assume flavour conservation so that m 1 = m 2 and m 3 = m 4 , the only
degree of freedom that propagates is, indeed, that of a pseudoscalar.
32 We can measure this, for instance by looking at the angular distribution of the produced
fermion-antifermion pair ; see also Appendix 10.12.