FVLN[p,mu,n,nb] denotes the positive component in the
lightcone decomposition of the Lorentz vector p^{\mu } along the vectors n and
nb. It corresponds to \frac{1}{2}
n^{\mu} (p \cdot \bar{n}).
If one omits n and nb, the program will use
default vectors specified via $FCDefaultLightconeVectorN
and $FCDefaultLightconeVectorNB.
Overview, Pair, FVLP, FVLR, SPLP, SPLN, SPLR, MTLP, MTLN, MTLR.
FVLN[p, \[Mu], n, nb]\frac{1}{2} \overline{n}^{\mu } \left(\overline{\text{nb}}\cdot \overline{p}\right)
StandardForm[FVLN[p, \[Mu], n, nb] // FCI]\frac{1}{2} \;\text{Pair}[\text{LorentzIndex}[\mu ],\text{Momentum}[n]] \;\text{Pair}[\text{Momentum}[\text{nb}],\text{Momentum}[p]]
Notice that the properties of n and nb
vectors have to be set by hand before doing the actual computation
FVLP[p, \[Mu], n, nb] FVLN[q, \[Mu], n, nb] // Contract\frac{1}{4} \left(\overline{n}\cdot \overline{\text{nb}}\right) \left(\overline{n}\cdot \overline{p}\right) \left(\overline{\text{nb}}\cdot \overline{q}\right)
FVLP[p, \[Mu], n, nb] FVLP[q, \[Mu], n, nb] // Contract\frac{1}{4} \overline{\text{nb}}^2 \left(\overline{n}\cdot \overline{p}\right) \left(\overline{n}\cdot \overline{q}\right)
FCClearScalarProducts[]
SP[n] = 0;
SP[nb] = 0;
SP[n, nb] = 2;FVLP[p, \[Mu], n, nb] FVLN[q, \[Mu], n, nb] // Contract\frac{1}{2} \left(\overline{n}\cdot \overline{p}\right) \left(\overline{\text{nb}}\cdot \overline{q}\right)
FVLP[p, \[Mu], n, nb] FVLP[q, \[Mu], n, nb] // Contract0
FCClearScalarProducts[]