Name: Aliaksandr Dubrouski (email_not_shown)
Date: 11/26/14-11:53:18 AM Z


Hi

Lingxiao

If the Levi-Civita tensor contracted with external four-momenta only you
can simplify them to zero in case you have two equal momenta or all four
contracted are not independent due to the momentum conservation.

Say in this case the following simplification applies (pseudo code)

Eps[Momentum[p1], Momentum[p3], Momentum[p4],
      Momentum[p1 + p2 + p4]]->Eps[Momentum[p1], Momentum[p3], Momentum[p4],
      Momentum[p1]]+Eps[Momentum[p1], Momentum[p3], Momentum[p4],
      Momentum[p2]]+Eps[Momentum[p1], Momentum[p3], Momentum[p4],
      Momentum[p4]]

Eps[Momentum[p1], Momentum[p3], Momentum[p4],
      Momentum[p1]] and Eps[Momentum[p1], Momentum[p3], Momentum[p4],
      Momentum[p4]] are zero to antisymmetry.

In case of external momenta
Eps[Momentum[p1], Momentum[p3], Momentum[p4],
      Momentum[p2]] is zero due to the conservation of total momentum
p1+p2+p3+p4=0

2014-11-26 7:34 GMT+03:00 Lingxiao Xu <noreply@feyncalc.org>:

> Hi,
> Thanks for attention.
> Here is one of my results which cantains Levi-Civita tensor contracted
> with four-momentums,
>
> 1/(8 s t
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(4\)]\))
> g^4 sw^2 (s^2 t^2 - t^4 + s^2 t u + t^3 u + t^2 u^2 - t u^3 -
> 8 s^2 t
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(2\)]\) + 4 s t^2
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(2\)]\) - 4 s^2 u
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(2\)]\) - 4 s t u
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(2\)]\) - 4 t^2 u
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(2\)]\) + 4 u^3
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(2\)]\) +
> 8 I t Eps[Momentum[p1], Momentum[p2], Momentum[-p1 - p2 - p3],
> Momentum[p3]]
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(2\)]\) +
> 8 I t Eps[Momentum[p1], Momentum[p2], Momentum[p3],
> Momentum[p3 - p4]]
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(2\)]\) +
> 8 I t Eps[Momentum[p1], Momentum[p3], Momentum[p4],
> Momentum[p1 + p2 + p4]]
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(2\)]\) + 12 s^2
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(4\)]\) + 16 s t
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(4\)]\) - 12 t^2
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(4\)]\) + 16 s u
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(4\)]\) + 8 t u
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(4\)]\) - 12 u^2
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(4\)]\) -
> 16 I Eps[Momentum[p1], Momentum[p2], Momentum[-p1 - p2 - p3],
> Momentum[p3]]
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(4\)]\) +
> 8 I Eps[Momentum[p1], Momentum[p2], Momentum[p3],
> Momentum[p3 - p4]]
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(4\)]\) +
> 8 I Eps[Momentum[p1], Momentum[p2], Momentum[p3],
> Momentum[p1 + p2 + p4]]
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(4\)]\) -
> 8 I Eps[Momentum[p1], Momentum[p3], Momentum[p4],
> Momentum[p1 + p2 + p4]]
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(4\)]\) - 56 s
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(6\)]\) + 24 t
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(6\)]\) + 8 u
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(6\)]\) - 16
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(8\)]\) -
> 4 I (s - t - u) Eps[Momentum[p1], Momentum[p2], Momentum[p3],
> Momentum[p4]] (t - 2
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(2\)]\)) +
> 4 I Eps[Momentum[p1], Momentum[p2], Momentum[p3 - p4],
> Momentum[p4]] (s t - 2 (s + t + u)
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(2\)]\) + 6
> \!\(\*SubsuperscriptBox[\(m\), \(W\), \(4\)]\)))
>
> So in this kind of condition, how can I simplify the Levi-Civita tensor
> further?
>
> Best Regards!
> Lingxiao
>
>

-- 
Regards,
            Aliaksandr Dubrouski



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