Finding Simplicity in Summation Expressions

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SUMMARY

The discussion centers on the complexities of summation expressions involving Majorana fermions, specifically in the context of the scattering process \(\nu_{\tau}+\bar{\nu}_{\tau}\rightarrow \nu_e+\bar{\nu}_e\). Participants highlight that while spin sums typically involve combinations of \(u\) and \(\bar{u}\) or \(v\) and \(\bar{v}\), the presence of Majorana fermions introduces scenarios where \(u\) and \(\bar{v}\) or \(v\) and \(\bar{u}\) may appear. The Feynman rules for these interactions are noted as challenging, but it is established that spinor identities can simplify the expressions to only include \(u\bar{u}\) or \(v\bar{v}\). Reference is made to Srednicki's book for further clarification on these transformations.

PREREQUISITES
  • Understanding of Majorana fermions and their properties
  • Familiarity with Feynman rules in quantum field theory
  • Knowledge of spinor algebra and identities
  • Basic concepts of scattering processes in particle physics
NEXT STEPS
  • Study the Feynman rules for Majorana fermions in detail
  • Learn about spinor identities and their applications in quantum field theory
  • Review the scattering processes involving neutrinos and their interactions
  • Read Srednicki's book on quantum field theory for comprehensive insights
USEFUL FOR

Particle physicists, quantum field theorists, and students studying neutrino interactions and Majorana fermions will benefit from this discussion.

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

Thank you
 
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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.
 
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!
:rolleyes:
 
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).
 

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