FeynCalc manual (development version)

Light-cone formalism

See also

Overview.

Notation for light-cone components

FeynCalc is equipped with special symbols that facilitate calculations involving light-cone vectors. The default nn and nˉ\bar{n} vectors are defined via the global variables $FCDefaultLightconeVectorN and $FCDefaultLightconeVectorNB. By default those are set to FCGV["n"] and FCGV["nb"] to avoid possible conflicts with user-defined variables

{$FCDefaultLightconeVectorN, $FCDefaultLightconeVectorNB}

{FCGV(n),FCGV(nb)}\{\text{FCGV}(\text{n}),\text{FCGV}(\text{nb})\}

These names can be of course changed. A particularly convenient choice is to use n and nb. Notice that these commands must be evaluated at the beginning of every FeynCalc session

$FCDefaultLightconeVectorN = n;
$FCDefaultLightconeVectorNB = nb;

Apart from this you must also explicitly define the values of the scalar products n2n^2, nˉ2\bar{n}^2 and n⋅nˉn \cdot \bar{n}

FCClearScalarProducts[]
ScalarProduct[n] = 0;
ScalarProduct[nb] = 0;
ScalarProduct[n, nb] = 2;

The Plus, Minus and peRpendicular components of 4-vectors are called FVLP, FVLN and FVLR respectively. The plus and minus components are immediately rewritten into forms involving nn and nˉ\bar{n}. The perpendicular component is a separate entity that cannot be simplified further.

{FVLP[p, \[Mu]], FVLN[p, \[Mu]], FVLR[p, \[Mu]]}

{12nb‾μ(n‾⋅p‾),12n‾μ(nb‾⋅p‾),p‾μ⊥}\left\{\frac{1}{2} \overline{\text{nb}}^{\mu } \left(\overline{n}\cdot \overline{p}\right),\frac{1}{2} \overline{n}^{\mu } \left(\overline{\text{nb}}\cdot \overline{p}\right),\overline{p}^{\mu }{}_{\perp }\right\}

{FVLPD[p, \[Mu]], FVLND[p, \[Mu]], FVLRD[p, \[Mu]]}

{12  nbμ(n⋅p),12nμ(nb⋅p),pμ⊥}\left\{\frac{1}{2} \;\text{nb}^{\mu } (n\cdot p),\frac{1}{2} n^{\mu } (\text{nb}\cdot p),p^{\mu }{}_{\perp }\right\}

It is also possible to specify your own symbols for the light-cone vectors thus overriding what is set via the global variables

FVLR[p, mu, myN, myNB]
% // FCI // StandardForm

p‾mu⊥\overline{p}^{\text{mu}}{}_{\perp }

(*Pair[LightConePerpendicularComponent[LorentzIndex[mu], Momentum[myN], Momentum[myNB]], LightConePerpendicularComponent[Momentum[p], Momentum[myN], Momentum[myNB]]]*)

Internally, the perpendicular component is implemented as an extra head wrapped around such internal symbols as LorentzIndex or Momentum. This head is called LightConePerpendicularComponent and has 3 arguments. The last two arguments specify the light-cone vectors.

?LightConePerpendicularComponent

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The pattern introduced for 4-vectors can be also found when working scalar products, metric tensors or Dirac matrices

{SPLP[p, q], SPLN[p, q], SPLR[p, q]}

{12(n‾⋅p‾)(nb‾⋅q‾),12(n‾⋅q‾)(nb‾⋅p‾),p‾⋅q‾⊥}\left\{\frac{1}{2} \left(\overline{n}\cdot \overline{p}\right) \left(\overline{\text{nb}}\cdot \overline{q}\right),\frac{1}{2} \left(\overline{n}\cdot \overline{q}\right) \left(\overline{\text{nb}}\cdot \overline{p}\right),\overline{p}\cdot \overline{q}_{\perp }\right\}

{SPLPD[p, q], SPLND[p, q], SPLRD[p, q]}

{12(n⋅p)(nb⋅q),12(n⋅q)(nb⋅p),p⋅q⊥}\left\{\frac{1}{2} (n\cdot p) (\text{nb}\cdot q),\frac{1}{2} (n\cdot q) (\text{nb}\cdot p),p\cdot q_{\perp }\right\}

{MTLP[\[Mu], \[Nu]], MTLN[\[Mu], \[Nu]], MTLR[\[Mu], \[Nu]]}

{12n‾νnb‾μ,12n‾μnb‾ν,gˉμν⊥}\left\{\frac{1}{2} \overline{n}^{\nu } \overline{\text{nb}}^{\mu },\frac{1}{2} \overline{n}^{\mu } \overline{\text{nb}}^{\nu },\bar{g}^{\mu \nu }{}_{\perp }\right\}

{GALP[\[Mu]], GALN[\[Mu]], GALR[\[Mu]]}

{12nb‾μγˉ⋅n‾,12n‾μγˉ⋅nb‾,γˉμ⊥}\left\{\frac{1}{2} \overline{\text{nb}}^{\mu } \bar{\gamma }\cdot \overline{n},\frac{1}{2} \overline{n}^{\mu } \bar{\gamma }\cdot \overline{\text{nb}},\bar{\gamma }^{\mu }{}_{\perp }\right\}

{GSLP[\[Mu]], GSLN[\[Mu]], GSLR[\[Mu]]}

{12γˉ⋅n‾(nb‾⋅μ‾),12(n‾⋅μ‾)γˉ⋅nb‾,γˉ⋅μ‾⊥}\left\{\frac{1}{2} \bar{\gamma }\cdot \overline{n} \left(\overline{\text{nb}}\cdot \overline{\mu }\right),\frac{1}{2} \left(\overline{n}\cdot \overline{\mu }\right) \bar{\gamma }\cdot \overline{\text{nb}},\bar{\gamma }\cdot \overline{\mu }_{\perp }\right\}

Contracting the full metric tensor with the perpendicular component returns the latter

MT[\[Mu], \[Nu]] MTLR[\[Mu], \[Rho]]
% // Contract

gˉμνgˉμρ⊥\bar{g}^{\mu \nu } \bar{g}^{\mu \rho }{}_{\perp }

gˉνρ⊥\bar{g}^{\nu \rho }{}_{\perp }

The dimensionality of the perpendicular component is 22 in 44-dimensions and D−2D-2 in DD-dimensions

MT[\[Mu], \[Nu]] MTLR[\[Mu], \[Nu]]
% // Contract

gˉμνgˉμν⊥\bar{g}^{\mu \nu } \bar{g}^{\mu \nu }{}_{\perp }

22

MTD[\[Mu], \[Nu]] MTLRD[\[Mu], \[Nu]]
% // Contract

gμνgμν⊥g^{\mu \nu } g^{\mu \nu }{}_{\perp }

D−2D-2

Dirac matrices with light-cone components

Dirac algebra involving matrices contracted to light-cone momenta or having particular light-cone components is fully supported. The general strategy followed by DiracSimplify is to move all perpendicular components to the very right of the chain.

ex1 = GALR[p] . GA[\[Mu], \[Nu]]

γˉp⊥.γˉμ.γˉν\bar{\gamma }^p{}_{\perp }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }

ex1 // DiracSimplify

−12n‾μ(γˉ⋅nb‾).γˉp⊥.γˉν⊥−12nb‾μ(γˉ⋅n‾).γˉp⊥.γˉν⊥+12n‾ν(γˉ⋅nb‾).γˉp⊥.γˉμ⊥+14n‾νnb‾μ(γˉ⋅n‾).(γˉ⋅nb‾).γˉp⊥+12nb‾ν(γˉ⋅n‾).γˉp⊥.γˉμ⊥−14n‾μnb‾ν(γˉ⋅n‾).(γˉ⋅nb‾).γˉp⊥+n‾μnb‾νγˉp⊥+γˉp⊥.γˉμ⊥.γˉν⊥-\frac{1}{2} \overline{n}^{\mu } \left(\bar{\gamma }\cdot \overline{\text{nb}}\right).\bar{\gamma }^p{}_{\perp }.\bar{\gamma }^{\nu }{}_{\perp }-\frac{1}{2} \overline{\text{nb}}^{\mu } \left(\bar{\gamma }\cdot \overline{n}\right).\bar{\gamma }^p{}_{\perp }.\bar{\gamma }^{\nu }{}_{\perp }+\frac{1}{2} \overline{n}^{\nu } \left(\bar{\gamma }\cdot \overline{\text{nb}}\right).\bar{\gamma }^p{}_{\perp }.\bar{\gamma }^{\mu }{}_{\perp }+\frac{1}{4} \overline{n}^{\nu } \overline{\text{nb}}^{\mu } \left(\bar{\gamma }\cdot \overline{n}\right).\left(\bar{\gamma }\cdot \overline{\text{nb}}\right).\bar{\gamma }^p{}_{\perp }+\frac{1}{2} \overline{\text{nb}}^{\nu } \left(\bar{\gamma }\cdot \overline{n}\right).\bar{\gamma }^p{}_{\perp }.\bar{\gamma }^{\mu }{}_{\perp }-\frac{1}{4} \overline{n}^{\mu } \overline{\text{nb}}^{\nu } \left(\bar{\gamma }\cdot \overline{n}\right).\left(\bar{\gamma }\cdot \overline{\text{nb}}\right).\bar{\gamma }^p{}_{\perp }+\overline{n}^{\mu } \overline{\text{nb}}^{\nu } \bar{\gamma }^p{}_{\perp }+\bar{\gamma }^p{}_{\perp }.\bar{\gamma }^{\mu }{}_{\perp }.\bar{\gamma }^{\nu }{}_{\perp }

ex2 = GALR[p] . GA[\[Mu], \[Nu]] . GALR[p]

γˉp⊥.γˉμ.γˉν.γˉp⊥\bar{\gamma }^p{}_{\perp }.\bar{\gamma }^{\mu }.\bar{\gamma }^{\nu }.\bar{\gamma }^p{}_{\perp }

ex2 // DiracSimplify

−2γˉμ⊥.γˉν⊥+4gˉμν⊥+12n‾νnb‾μ(γˉ⋅n‾).(γˉ⋅nb‾)−12n‾μnb‾ν(γˉ⋅n‾).(γˉ⋅nb‾)+2n‾μnb‾ν-2 \bar{\gamma }^{\mu }{}_{\perp }.\bar{\gamma }^{\nu }{}_{\perp }+4 \bar{g}^{\mu \nu }{}_{\perp }+\frac{1}{2} \overline{n}^{\nu } \overline{\text{nb}}^{\mu } \left(\bar{\gamma }\cdot \overline{n}\right).\left(\bar{\gamma }\cdot \overline{\text{nb}}\right)-\frac{1}{2} \overline{n}^{\mu } \overline{\text{nb}}^{\nu } \left(\bar{\gamma }\cdot \overline{n}\right).\left(\bar{\gamma }\cdot \overline{\text{nb}}\right)+2 \overline{n}^{\mu } \overline{\text{nb}}^{\nu }

Notice that when entering particular light-cone components of Dirac matrices, the standard trick for entering multiple indices does not work. This is because the 2nd and 3rd arguments are reserved for user-specified light-cone vectors

GALR[mu1, myN, myNB]
% // FCI // StandardForm

γˉmu1⊥\bar{\gamma }^{\text{mu1}}{}_{\perp }

(*DiracGamma[LightConePerpendicularComponent[LorentzIndex[mu1], Momentum[myN], Momentum[myNB]]]*)

Instead, you should put your list of indices into curly brackets

GALR[{\[Mu], \[Nu], \[Rho]}]

γˉμ⊥.γˉν⊥.γˉρ⊥\bar{\gamma }^{\mu }{}_{\perp }.\bar{\gamma }^{\nu }{}_{\perp }.\bar{\gamma }^{\rho }{}_{\perp }

ex3 = GALR[p] . GALR[{\[Mu], \[Nu]}] . GALR[p]

γˉp⊥.γˉμ⊥.γˉν⊥.γˉp⊥\bar{\gamma }^p{}_{\perp }.\bar{\gamma }^{\mu }{}_{\perp }.\bar{\gamma }^{\nu }{}_{\perp }.\bar{\gamma }^p{}_{\perp }

ex3 // DiracSimplify

4gˉμν⊥−2γˉμ⊥.γˉν⊥4 \bar{g}^{\mu \nu }{}_{\perp }-2 \bar{\gamma }^{\mu }{}_{\perp }.\bar{\gamma }^{\nu }{}_{\perp }

ex4 = DiracTrace[GA[\[Rho], \[Sigma]] . GALR[{\[Mu], \[Nu]}]]

tr(γˉρ.γˉσ.γˉμ⊥.γˉν⊥)\text{tr}\left(\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma }.\bar{\gamma }^{\mu }{}_{\perp }.\bar{\gamma }^{\nu }{}_{\perp }\right)

ex4 // DiracSimplify

4gˉμσ⊥gˉνρ⊥−4gˉμρ⊥gˉνσ⊥+4gˉρσgˉμν⊥4 \bar{g}^{\mu \sigma }{}_{\perp } \bar{g}^{\nu \rho }{}_{\perp }-4 \bar{g}^{\mu \rho }{}_{\perp } \bar{g}^{\nu \sigma }{}_{\perp }+4 \bar{g}^{\rho \sigma } \bar{g}^{\mu \nu }{}_{\perp }

ex5 = DiracTrace[GA[\[Rho], \[Sigma]] . GA[5] . GALR[{\[Mu], \[Nu]}]]

tr(γˉρ.γˉσ.γˉ5.γˉμ⊥.γˉν⊥)\text{tr}\left(\bar{\gamma }^{\rho }.\bar{\gamma }^{\sigma }.\bar{\gamma }^5.\bar{\gamma }^{\mu }{}_{\perp }.\bar{\gamma }^{\nu }{}_{\perp }\right)

ex5 // DiracSimplify

2in‾ρϵˉμ⊥ν⊥σ⊥  nb‾+2inb‾ρϵˉμ⊥ν⊥σ⊥n‾−2in‾σϵˉμ⊥ν⊥ρ⊥  nb‾−in‾σnb‾ρϵˉμ⊥ν⊥n‾  nb‾−2inb‾σϵˉμ⊥ν⊥ρ⊥n‾+in‾ρnb‾σϵˉμ⊥ν⊥n‾  nb‾−4iϵˉμ⊥ν⊥ρ⊥σ⊥2 i \overline{n}^{\rho } \bar{\epsilon }^{\mu _{\perp }\nu _{\perp }\sigma _{\perp }\;\overline{\text{nb}}}+2 i \overline{\text{nb}}^{\rho } \bar{\epsilon }^{\mu _{\perp }\nu _{\perp }\sigma _{\perp }\overline{n}}-2 i \overline{n}^{\sigma } \bar{\epsilon }^{\mu _{\perp }\nu _{\perp }\rho _{\perp }\;\overline{\text{nb}}}-i \overline{n}^{\sigma } \overline{\text{nb}}^{\rho } \bar{\epsilon }^{\mu _{\perp }\nu _{\perp }\overline{n}\;\overline{\text{nb}}}-2 i \overline{\text{nb}}^{\sigma } \bar{\epsilon }^{\mu _{\perp }\nu _{\perp }\rho _{\perp }\overline{n}}+i \overline{n}^{\rho } \overline{\text{nb}}^{\sigma } \bar{\epsilon }^{\mu _{\perp }\nu _{\perp }\overline{n}\;\overline{\text{nb}}}-4 i \bar{\epsilon }^{\mu _{\perp }\nu _{\perp }\rho _{\perp }\sigma _{\perp }}

In order to handle Dirac matrices involving light-cone components effectively, one needs to define some ordering. The current (hard-coded) choice is that the perpendicular component always get anticommuted to the very right of each Dirac chain, while every remaining occurence of (γ⋅nˉ)(γ⋅n)(\gamma \cdot \bar{n}) (\gamma \cdot n) is changed to (γ⋅n)(γ⋅nˉ)(\gamma \cdot n) (\gamma \cdot \bar{n}).

Introducing light-cone components by hand

In some calculations one might end up with a mixture of explicit light-cone components and generic Lorentz tensors. If those tensors admit a particularly simple representation in terms of light-cone components, it can be enforced using the function ToLightConeComponents

For example, the following expression cannot be simplified any further

ex6 = GS[nb, vp]

(γˉ⋅nb‾).(γˉ⋅vp‾)\left(\bar{\gamma }\cdot \overline{\text{nb}}\right).\left(\bar{\gamma }\cdot \overline{\text{vp}}\right)

Now let us suppose that (v′)μ(v')^{\mu} can be actually written as αnμ+nˉμ/(4α)\alpha n^\mu + \bar{n}^{\mu}/(4 \alpha). We can implement this as follows

SP[vp, nb] = 2*alpha;
SP[vp, n] = 2*1/(4 alpha);
LightConePerpendicularComponent[Momentum[vp], Momentum[n], Momentum[nb]] = 0;
FV[vp, mu]
% // ToLightConeComponents

vp‾mu\overline{\text{vp}}^{\text{mu}}

alphan‾mu+nb‾mu4  alpha\text{alpha} \overline{n}^{\text{mu}}+\frac{\overline{\text{nb}}^{\text{mu}}}{4 \;\text{alpha}}

However, this will not make FeynCalc automatically simplify the Dirac chain

ex6 // DiracSimplify

(γˉ⋅nb‾).(γˉ⋅vp‾)\left(\bar{\gamma }\cdot \overline{\text{nb}}\right).\left(\bar{\gamma }\cdot \overline{\text{vp}}\right)

Using ToLightConeComponents we can explicitly rewrite vp in the chain in terms of the light-cone components and hence enforce the desired simplification. In fact, the function will also automatically simplify some common expressions such γ⋅nˉγ⋅nˉ=γ⋅nγ⋅n=0\gamma \cdot \bar{n} \gamma \cdot \bar{n} = \gamma \cdot n \gamma \cdot n = 0

ex6 // ToLightConeComponents
% // DiracSimplify

alpha(γˉ⋅nb‾).(γˉ⋅n‾)\text{alpha} \left(\bar{\gamma }\cdot \overline{\text{nb}}\right).\left(\bar{\gamma }\cdot \overline{n}\right)

4  alpha−alpha(γˉ⋅n‾).(γˉ⋅nb‾)4 \;\text{alpha}-\text{alpha} \left(\bar{\gamma }\cdot \overline{n}\right).\left(\bar{\gamma }\cdot \overline{\text{nb}}\right)

Such simplifications inside ToLightConeComponents can be disabled using the option DotSimplify

ex6 // ToLightConeComponents[#, DotSimplify -> False] &
% // DiracSimplify

(γˉ⋅nb‾).(alphaγˉ⋅n‾+γˉ⋅nb‾4  alpha)\left(\bar{\gamma }\cdot \overline{\text{nb}}\right).\left(\text{alpha} \bar{\gamma }\cdot \overline{n}+\frac{\bar{\gamma }\cdot \overline{\text{nb}}}{4 \;\text{alpha}}\right)

4  alpha−alpha(γˉ⋅n‾).(γˉ⋅nb‾)4 \;\text{alpha}-\text{alpha} \left(\bar{\gamma }\cdot \overline{n}\right).\left(\bar{\gamma }\cdot \overline{\text{nb}}\right)

Reductions of loop integrals with numerators involving light-cone components

int = FVLRD[p, \[Mu]] SFAD[p, p - q]

pμ⊥(p2+iη).((p−q)2+iη)\frac{p^{\mu }{}_{\perp }}{(p^2+i \eta ).((p-q)^2+i \eta )}

TID[int, p]

qμ⊥2(p2+iη).((p−q)2+iη)\frac{q^{\mu }{}_{\perp }}{2 (p^2+i \eta ).((p-q)^2+i \eta )}

Differentiations

FourDivergence cannot yet differentiate w.r.t light-cone components directly. However, the same effect can be easily achieved by first differentiating w.r.t the usual 4-momentum and then contracting the free index with the corresponding metric tensor

ex = FV[p1, \[Mu]]/SP[p1]

p1‾μp1‾2\frac{\overline{\text{p1}}^{\mu }}{\overline{\text{p1}}^2}

Differentiating w.r.t p1,+p_{1,+}, p1,−p_{1,-} or p1,⊥p_{1,\perp}

MTLN[\[Nu], \[Rho]] FourDivergence[ex, FV[p1, \[Rho]]] // Contract

n‾νnb‾μ2p1‾2−n‾νp1‾μ(nb‾⋅p1‾)p1‾4\frac{\overline{n}^{\nu } \overline{\text{nb}}^{\mu }}{2 \overline{\text{p1}}^2}-\frac{\overline{n}^{\nu } \overline{\text{p1}}^{\mu } \left(\overline{\text{nb}}\cdot \overline{\text{p1}}\right)}{\overline{\text{p1}}^4}

MTLP[\[Nu], \[Rho]] FourDivergence[ex, FV[p1, \[Rho]]] // Contract

n‾μnb‾ν2p1‾2−nb‾νp1‾μ(n‾⋅p1‾)p1‾4\frac{\overline{n}^{\mu } \overline{\text{nb}}^{\nu }}{2 \overline{\text{p1}}^2}-\frac{\overline{\text{nb}}^{\nu } \overline{\text{p1}}^{\mu } \left(\overline{n}\cdot \overline{\text{p1}}\right)}{\overline{\text{p1}}^4}

MTLR[\[Nu], \[Rho]] FourDivergence[ex, FV[p1, \[Rho]]] // Contract

gˉμν⊥p1‾2−2p1‾μp1‾ν⊥p1‾4\frac{\bar{g}^{\mu \nu }{}_{\perp }}{\overline{\text{p1}}^2}-\frac{2 \overline{\text{p1}}^{\mu } \overline{\text{p1}}^{\nu }{}_{\perp }}{\overline{\text{p1}}^4}