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I'm evaluating the one-gluon-exchange tensor part of a phenomenological potential between two particles, and it involves a term like this:

[tex]S_{ij}=3(\vec{\sigma_i}\hat{r_{ij}})(\vec{\sigma_j}\hat{r_{ij}})-\vec{\sigma_i}\vec{\sigma_j}[/tex]

With [tex]r_{ij}[/tex] the unit vector in the direction along the axis from the first to the second particle

The second term [tex]\vec{\sigma_i}\vec{\sigma_j}[/tex] is very easy to evaluate, it yields -3 for S=0 and 1 for S=1

But I can't solve the first term, whatever I try I always end up with terms including [tex]r^{z}_{ij}[/tex] the proyection of the unit vector along de Z-axis. I suppose that implies that it doesn't only depend on S but also on Sz, which doesn't sound right to me.

Can anyone help me? or even better guide me in the right direction?

Thanks