“Indeed, self-interaction is the most basic of all processes allowed by quantum field theory, but spin and charge forbid point-like self-couplings for all particles but the Higgs.”
There is a shorter version of this statement in an IAS web page
“…the Higgs particle does not have any charge or spin. It is also the only known particle with the ability to interact with itself.” – Nima Arkani-Hamed
I was confused because we know that a charged particle (electron, for example) creates an electromagnetic field and in turn gets influenced by its own electromagnetic field. This is self-interaction through a field. The problem of self-interaction has a long history in physics. There is so much literature on the problem of electron self-interaction. Nima Arkani-Hamed is one of the most knowledgeable physicists in the world, he surely knows that particles other than the Higgs can self-interact through a field too.
Then, I encountered Tomasso Dorigo’s explanation of “self-coupling” and realized that “self-coupling” is not the same as self-interaction through a field. Self-coupling refers to self-cloning of a particle and interaction with its clone. Nima Arkani-Hamed was careful enough to say “point-like self-coupling” to differentiate this type of interaction from the self-interaction through a field.
” ‘Self coupling’ means that a particle can split into two copies of themselves (in the simplest realization): definitely not something trivial to do for matter particles, and indeed, fermions cannot do it, for a number of reasons (charge conservation, or angular momentum conservation, come to mind). Bosons instead are not prevented by the conservation of angular momentum rules, so the question is whether they possess or not other quantum numbers that would change in a 1->2 or 1->3 process. Photons cannot split into two photons, for example, because of the conservation of charge conjugation.”
“A boson which is perfectly happy to split into copies of itself is the gluon. The presence of a self-coupling of gluons is in fact built into the theory of gluons – Quantum Chromodynamics, QCD – because QCD is based on a non-abelian gauge group, SU(3). For the Higgs, the group-theory reason of the existence of three- and four-particle vertices is to be found in the very electroweak symmetry breaking mechanism, which in the unitary gauge does the miracle of creating mass degrees of freedom to W and Z bosons. I will not bother you with mathematical expressions, but the two things really have the same source in the standard model: the Higgs has self couplings if EWSB works the way it is supposed to. So comparing measured and predicted Higgs boson self couplings is the ultimate dream of the Higgs phenomenologist !”
from Tomasso Dorigo’s blog post http://www.science20.com/quantum_diaries_survivor/ideas_higgs_couplings-91586
The broad generalization “spin and charge forbid point-like self-couplings for all particles but the Higgs” come with caveats.
I am still confused but feeling a little better!