Natural Science Forum / Physics / Particle Physics / January 2007

For my diploma thesis I must provide a calculation that reproduces the
results given on page 46 of the paper hep-ph/0309342 . For those who do
not want to look it up, I briefly explain what it is about: It concerns
the two-body scattering processes

(1) N + V => L + H,
(2) N + L => V + H,
(3) N + H => V + L

where N denotes a right-handed Majorana neutrino, L and H are the SM
lepton and Higgs doublet and V represents either a SU(2) or U(1) gauge
boson. The processes are considered in the early universe, where
SU(2)xU(1) is unbroken and L and V are massless (but N is not). The
Higgs mass is also neglected since m_H << m_N, but this is of minor
importance here. The Majorana neutrino has a Yukawa coupling to the
lepton and Higgs doublet but does not couple to the gauge bosons. (See
Feynman diagrams in the paper.)
The problem is, that not only am I unable to reproduce the result given
for (2) and (3), but the result I get makes no sense. When I calculate
just the s-channel contribution |M_s|^2, then the result is negative!
With process (1), which occurs only in the t-channel and u-channel,
there is no problem, but for both (2) and (3) which have an s-channel
contribution I get a negative amplitude squared. To make it even
stranger-if I start from (1) and use crossing to obtain the result for
(3), I get the same as the author of the paper does. But still, there
must be something wrong since the direct calculation fails.

My question is now: Is there anything special about these diagrams,
some peculiarity that I most likely have not taken into account? My
supervisor told me some stories that the diagrams might violate the
Ward identity and I should try to add something which he called
"contact terms", but he was unable to explain me properly, what this
means.  In the literature, I have not found any example where anything
like that happens. I rather assume it could be connected to the
polarization sums for the gauge bosons which maybe contain something
special here. I used the same formula as for the photons (remember that
the SU(2) gauge bosons are also massless in this case!), maybe this is
wrong? But it works for process (1), this really confuses me...

I hope someone can give me a hint what could be going on here, since it
would be a pity to leave this part out. Few people have considered
these processes and I would like to include their effect in my thesis.

Thanks in advance,
Denis Besak