In the Standard Model, using quantum field theory it is conventional to use the helicity basis to simplify calculations (of cross sections, for example). In this basis, the spin is quantized along the axis in the direction of motion of the particle.
Spinors The two-component helicity eigenstates ξ λ {\displaystyle \xi _{\lambda }} satisfy
σ ⋅ p ^ ξ λ ( p ^ ) = λ ξ λ ( p ^ ) {\displaystyle \sigma \cdot {\hat {p}}\xi _{\lambda }\left({\hat {p}}\right)=\lambda \xi _{\lambda }\left({\hat {p}}\right)\,}
where
σ {\displaystyle \sigma \,} are the Pauli matrices,
p ^ {\displaystyle {\hat {p}}\,} is the direction of the fermion momentum,
λ = ± 1 {\displaystyle \lambda =\pm 1\,} depending on whether spin is pointing in the same direction as p ^ {\displaystyle {\hat {p}}\,} or opposite. To say more about the state, ξ λ {\displaystyle \xi _{\lambda }\,} we will use the generic form of fermion four-momentum:
p μ = ( E , | p → | sin θ cos ϕ , | p → | sin θ sin ϕ , | p → | cos θ ) {\displaystyle p^{\mu }=\left(E,\left|{\vec {p}}\right|\sin {\theta }\cos {\phi },\left|{\vec {p}}\right|\sin {\theta }\sin {\phi },\left|{\vec {p}}\right|\cos {\theta }\right)\,}
Then one can say the two helicity eigenstates are
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