Formalism

A tree-level amplitude in e⁺e^- collisions can be expressed in terms of fermion strings of the form

#math164#

#tex2html_wrap_indisplay920#(p₂, σ₂)P_-τ #tex2html_wrap_indisplay921#a₁ #tex2html_wrap_indisplay922#a₂^... #tex2html_wrap_indisplay923#a_nu(p₁, σ₁) ,

(1)

where p and σ label the initial e^± four-momenta and helicities #math165#(σ = ±1), #math166##tex2html_wrap_inline929#a_i = a^μ_iγ_ν and #math167#P_τ = #tex2html_wrap_inline931#(1 + τγ₅) is a chirality projection operator #math168#(τ = ±1). The a^μ_i may be formed from particle four-momenta, gauge-boson polarization vectors or fermion strings with an uncontracted Lorentz index associated with final-state fermions.

In the chiral representation the γ matrices are expressed in terms of 2×2 Pauli matrices σ and the unit matrix 1 as #mathletters50# giving

#math169#

#tex2html_wrap_indisplay938#a = #tex2html_wrap_indisplay939##tex2html_wrap_indisplay940##tex2html_wrap_indisplay941#, ( #tex2html_wrap_indisplay942#a)_± = a_μσ^μ_± ,

(2)

The spinors are expressed in terms of two-component Weyl spinors as

#math170#

u = #tex2html_wrap_indisplay944##tex2html_wrap_indisplay945##tex2html_wrap_indisplay946#, v = #tex2html_wrap_indisplay947#(v)^†₊#tex2html_wrap_indisplay948# (v)^†_-#tex2html_wrap_indisplay949# .#tex2html_wrap_indisplay950#3A

(3)

The Weyl spinors are given in terms of helicity eigenstates #math171#χ_λ(p) with #math172#λ = ±1 by

#math173#

u(p, λ)_±	=	(E±λ\|#tex2html_wrap_indisplay957#\|)^1/2χ_λ(p) ,
			(4)
v(p, λ)_±	=	±λ(E#tex2html_wrap_indisplay961#λ\|#tex2html_wrap_indisplay962#\|)^1/2χ_-λ(p)