Finding Simplicity in Summation Expressions

[Final]

Hi,
there is a good expression for $$\sum_{s}{u_s(\vec{p})\bar{v_s}(\vec{p})}$$ ?

Thank you

 

[Avodyne]

Not that I know of. But I don't know why you would need this sum; spin sums are needed when a spin is not observed, then you want to sum the absolute square of the transition amplitude over the unobserved spin; but that will always involve u and ubar or v and vbar, but never u and vbar or v and ubar.

 

[Final]

Not always...
My problem is about Majorana's fermions:

Take the scattering $$\nu_{\tau}+\bar{\nu}_{\tau}\rightarrow \nu_e+\bar{\nu}_e$$ and the interaction $${\cal{L}}=g \sum Z_{\mu}\bar{\psi}_{\nu_l}\gamma^{\mu}(1-\gamma_5)\psi_{\nu_l}$$.

The $$\nu$$ are Majorana's fermions (i.e. $$d_r=b_r$$) with mass $$m_{\nu_{\tau}}>m_{\nu_e}$$. Compute the cross section. Here the feynman rules are quite difficult and the sums over the spin of the square of the transition amplitude involve also u vbar and v ubar!

 

[Avodyne]

For Majorana fermions, there is always a way to transform things (using spinor identities) so that you get only u ubar or v vbar. This is explained in the book by Srednicki (draft copy available free online, google to find it).