Pictures Paths Particles Processes

238 November 7, 2013
9.4 The Higgs sector
9.4.1 The Higgs hypothesis
Fully longitudinal scattering
Having pursued the consequences of unitarity in processes where a single external
spin-1 particle is longitudinally polarized, we must of course also face the
more taxing case in which, perhaps, all external spin-1 particles are longitudinally
polarized : surely this is the most dangerous case from the point of view
of unitarity. In doing so, we must however take into account the fact that the
notion of longitudinal polarization is not strictly a Lorentz-invariant one since a
generic Lorentz boost will mix longitudinal and transverse degrees of freedom.
It therefore behooves us to specify in which particular Lorentz frame the particles
are assumed to be longitudinally polarized. To this end we introduce a
vector c µ with
c · c = 1 ;
the frame in which ⃗c = 0 is defines the appropriate Lorentz frame. In these
notes we shall take c µ to be proportional to the total momentum involved in the
scattering process, that is, the external vector particles are assumed to be purely
longitudinal in the centre-of-mass frame of the scattering 18 . The longitudinal
polarization of an on-shell vector particle with momentum p µ and mass m is
then given by
ɛ L µ = N L
m
(p µ − m2
c · p cµ )
, N L −2 = 1 − m2
(c · p) 2 , (9.80)
which expression is well-defined as long as ⃗p ≠ 0. We see that, as before,
ɛ L = p/m + O ( m/p 0) . In the cases studied so far, the subleading terms in ɛ L
have only led to safe terms so that they could be neglected 19 ; now, this is no
longer automatically the case.
W W → ZZ
The first Gedanken process 20 is
W + (p 1 , ɛ 1 ) W − (p 2 , ɛ 2 ) → Z 0 (p 3 , ɛ 3 )Z 0 (p 4 , ɛ 4 )
18 That this is not a trivial point becomes clear when we realize that in ‘W W scattering’ at
the LHC, say, the centre-of-mass frame of the scattering does not coincide with the laboratory
frame, in which the detector is at rest, and in which the polarization analysis of the produced
bosons is presumably performed.
19 From the point of view of restoring unitarity, not that of actually getting the cross section
right!
20 As I write these notes, this is still a true Gedanken process. As usual, with improving technology
and the commissioning of higher-energy machines, Gedanken processes are gradually
turned into actual ones...