Expansion of Compton Hamiltonian

[sjmacewan]

Hi there,
I'm working on getting a presentation together for a graduate course I'm taking and chose to give a brief introduction on spin polarizabilities.

In the case of the nucleon, these 4 intrinsic quantities manifest themselves in a 3rd-order expansion of the Compton Scattering Hamiltonian, as seen on page 12 of

arXiv:hep-ph/9910427v2 26 Oct 1999
(http://arxiv.org/pdf/hep-ph/9910427).

I've tried to recreate this equation from scratch but honestly have no clue how this expansion came to play...is it some sort of Taylor Series term? I have little experience in multivariable expansions.

If anyone could provide some guidance on this, I would be very grateful.

 

[gtoguy97]

The expansion in question is: H = 4π(1+α_E)E⃗.E′⃗ + 8πα_M(E⃗xE′⃗).(S/2) + 16πα_M^(s)(E⃗.(S/2))(E′⃗.(S/2)).The expansion you are referring to is a multipole expansion of the scattering Hamiltonian for Compton scattering. It is an expansion in terms of the electric and magnetic multipole moments of the nucleon. In this case, the electric dipole, magnetic dipole and electric quadrupole moments are being considered. The coefficients α_E, α_M and α_M^(s) are the polarizabilities for each moment.Hope this helps!