Name: Vladyslav Shtabovenko Date: 11/26/14-12:36:42 PM Z

Hi Aliaksandr,

good point! For the code Lingxiao has posted, one replacement rule is
indeed enough to simplify the whole thing

exp // ReplaceAll[#, {p1 -> p3 + p4 - p2}] &

returns

1/(8*s*t*Subscript[m, W]^4)

Cheers,
Vladyslav

On 26/11/14 11:53, Aliaksandr Dubrouski wrote:
> 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_at_HIDDEN-E-MAIL]
> <mailto:[noreply_at_HIDDEN-E-MAIL]>>:
>
> 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