Majorana fermions do not have vector interaction and hence there is no spin-independent direct search cross section for majorana like WIMPs. Can anyone please show me the proof ?
Essentially because for Majorana spinors ψ̅ γμ ψ = - ψ̅c γμ ψc (sorry for not being able to type the bar over c, you should read psi-c-bar) where ψ is a real four component spinor and c denotes charge conjugation. You can see the book: Particle and Astroparticle Physics by Utpal Sarkar, pages 142-143.
PS: It is best to prove this just by brute force.
The book by Utpal Sarkar is excellent. In addition you may find information in the article "All electromagneticc form factors by N. Nowakowski et al. [Eur. J. Phys. 26(2005) 545-560]. Study equations(2.21) to (2.25) and discussion just after equation (2.29). When you reduce the form factors of that article to the non-relativistic limit the vector terms connect upper spin to low spin components and are very small. You need and axial-vector term to obtain a lerge component. this form factor is called the anapole.
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