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4.8 Fermionic Matrix Elements
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F1 -> DiracChain[Spinor[k[2], ME, -1], 6, Lor[1], Spinor[k[1], ME, 1]] *
DiracChain[Spinor[k[3], MT, 1], 6, Lor[1], Spinor[k[4], MT, -1]]
In physical observables such as the cross-section, where only the square of the amplitude
or interference terms can enter, these spinor chains can be evaluated without reference to a
concrete representation for the spinors. The point is that in terms like |M|2 or 2 Re(M1 M∗2 )
only products (Fi Fj) of spinor chains appear and these can be calculated using the density
matrix for spinors

 1 (1 ± λγ ) p/
for massless fermions†
5
{uλ ( p)ūλ ( p), vλ ( p)v̄λ ( p)} = 2
 1 (1 + λγ /
s)( p/ ± m) for massive fermions
2
5
where λ = ± and s is the helicity reference vector corresponding to the momentum p. s
is the unit vector in the direction of the spin axis in the particle’s rest frame, boosted into
the CMS. It is identical to the longitudinal polarization vector of a vector boson, and fulfills
s · p = 0 and s2 = −1.
In the unpolarized case the λ-dependent part adds up to zero, so the projectors become
∑
λ =±
uλ ( p)ūλ ( p) = p/ + m ,
∑
λ =±
vλ ( p)v̄λ ( p) = p/ − m .
Technically, one can use the same formula as in the polarized case by putting λ = 0 and
multiplying the result by 2 for each external fermion.
FormCalc supplies the function HelicityME to calculate these so-called helicity matrix elements (Fi Fj).
† In
the limit E ≫ m the vector s becomes increasingly parallel to p, i.e. s ∼ p/m, hence
)
p/( p/ + m) = p2 + m/
p = m( p/ + m)
p/
E≫m
⇒ (1 + λγ5/s)( p/ ± m) −→ 1 + λγ5
( p/ ± m) = (1 ± λγ5 ) p/ .
m
p/( p/ − m) = p2 − m/
p = −m( p/ − m)