FeynCalc manual (development version)

 

FVLND

FVLND[p,mu,n,nb] denotes the positive component in the lightcone decomposition of the Lorentz vector pμp^{\mu } along the vectors n and nb in DD dimensions. It corresponds to 12nμ(pnˉ)\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.

See also

Overview, Pair, FVLPD, FVLRD, SPLPD, SPLND, SPLRD, MTLPD, MTLND, MTLRD.

Examples

FVLND[p, \[Mu], n, nb]

12nμ(nbp)\frac{1}{2} n^{\mu } (\text{nb}\cdot p)

StandardForm[FVLND[p, \[Mu], n, nb] // FCI]

12  Pair[LorentzIndex[μ,D],Momentum[n,D]]  Pair[Momentum[nb,D],Momentum[p,D]]\frac{1}{2} \;\text{Pair}[\text{LorentzIndex}[\mu ,D],\text{Momentum}[n,D]] \;\text{Pair}[\text{Momentum}[\text{nb},D],\text{Momentum}[p,D]]

Notice that the properties of n and nb vectors have to be set by hand before doing the actual computation

FVLPD[p, \[Mu], n, nb] FVLND[q, \[Mu], n, nb] // Contract

14(n  nb)(np)(nbq)\frac{1}{4} (n\cdot \;\text{nb}) (n\cdot p) (\text{nb}\cdot q)

FVLPD[p, \[Mu], n, nb] FVLPD[q, \[Mu], n, nb] // Contract

14  nb2(np)(nq)\frac{1}{4} \;\text{nb}^2 (n\cdot p) (n\cdot q)

FCClearScalarProducts[]
SPD[n] = 0;
SPD[nb] = 0;
SPD[n, nb] = 2;
FVLPD[p, \[Mu], n, nb] FVLND[q, \[Mu], n, nb] // Contract

12(np)(nbq)\frac{1}{2} (n\cdot p) (\text{nb}\cdot q)

FVLPD[p, \[Mu], n, nb] FVLPD[q, \[Mu], n, nb] // Contract

00

FCClearScalarProducts[]