# FeynCalc: Why do LeviCivita remain after Calc/Contract?

In (perhaps even in older versions) version FeynCalc 9.0.0 sometimes, on the amplitude level, one gets expressions of the form

$\epsilon^{\alpha q\varepsilon^*(p)\varepsilon^*(q)}$ where $\alpha$ is a Lorentz index, $p,q$ are four-momenta and the $\varepsilon^*(p)$ are polarization vectors of out/incoming vector particles (some massive some massless).

I'm sure there's some logic behind this notation but my question is more why does the LeviCivita remain in some terms even after I issue Calc[] and/or Contract[]? That really does not make sense to me:

In the squared amplitude (after summing over polarizations with DoPolarizationSums[] and after FermionSpinSum[] has been issued) I can get terms like

$$\tag{1}a + b\epsilon^{kpql}$$ where $k,p$ are incoming and $q,l$ are outgoing four-momenta.

How can I make sense of eq $(1)$ or get rid of the LeviCivita?

Also I should perhaps add that there's a bar over all fourmomenta in FeynCalc 9.0.0.

## 1 Answer

OK I think I got it. The final LeviCivita[] that remains has (of course) only Momentum[] as indices. In my case (as in most other cases) four-momentum conservation implies $$\tag{2} k+p = l+q$$ and I've got something like $$\tag{3}\epsilon^{klpq}.$$ Now substituting e.g. $k = l+q-p$ and EpsEvaluate[] the LeviCivita I get zero which is what I was looking for.

In other words, unless I define Eq. $(2)$ somewhere and (perhaps?) FeynCalc recognizes this and automatically sets Eq. $(3)$ to zero, one has to set it to zero by hand.

• Thanks for the answer, in my case, the expression involves some auxiliary vectors, therefore it is not zero. How can I tell Mathematica to make the expression explicit?
– zyy
Dec 9, 2021 at 8:04