Name: Vladyslav Shtabovenko (email_not_shown)
Date: 05/31/16-10:01:53 PM Z


Hi Maxim,

let me contribute to the resolution of the discrepancy.

The point is that at the end of the computation OneLoop converts
all the D-dimensional 4-vectors and metric tensors to 4-dimensions.

This is also what one normally does when doing computation by hand,
so there is no issue here. However, if the output of OneLoop is not the
full expression that you want to evaluate but just a part of it, one has to
be careful.

In particular, if the output of OneLoop contains metric tensors and you
want to contract it with D-dimensional metric tensors, then you have to
convert your expression back to D-dimensions first. Otherwise, from
contracting g^{mu nu} g_{mu nu} you obtain 4 instead of D, which gives
the rise to the discrepancy that you have been observing.

The correct way of writing the code is

ScalarProduct[q1, q1] = 0;
ScalarProduct[q2, q2] = 0;
ScalarProduct[q3, q3] = 0;
den = FAD[{q, 0}, {q - q1, 0}, {q - q1 - q2, 0}, {q - q1 - q2 - q3,
    0}];

ex1 = Contract[(OneLoop[q, den*SP[q, q]*FV[q, mu]*FV[q, nu]] //
      ChangeDimension[#, D] &)*MTD[mu, nu]] // PaVeReduce

ex2 = OneLoop[q, den*SP[q, q]^2] // PaVeReduce //
  ChangeDimension[#, D] &

and the difference is zero, as it should be.

Cheers,
Vladyslav

> -----
----- ðÅÒÅÎÁÐÒÁ×ÌÅÎÎÏÅ ÓÏÏÂÝÅÎÉÅ ----------
> ïÔ: *Nefedov Maxim*
> äÁÔÁ: ÓÕÂÂÏÔÁ, 11 ÍÁÑ 2013 Ç.
> ôÅÍÁ: Box-diagrams, rational parts and OneLoop
> ëÏÍÕ: feyncalc@feyncalc.org
>
> Hi!
> Trying to calculate box diagrams with OneLoop (Mathematica 7 + FC 8.2.0)
> I obtained the different results in the seemingly equivalent calculations:
> ------------------------------------------------------------------
> ScalarProduct[q1, q1] = 0;
> ScalarProduct[q2, q2] = 0;
> ScalarProduct[q3, q3] = 0;
> den = FAD[{q, 0}, {q - q1, 0}, {q - q1 - q2, 0}, {q - q1 - q2 - q3,
> 0}];
>
> (*Doing OneLoop with q^2*q^mu*q^nu in numerator and then contracting with
> g_munu*)
> ex1 = PaVeReduce[
> Contract[OneLoop[q, den*SP[q, q]*FV[q,mu]*FV[q, nu]]*
> MT[mu, nu]]]
>
> (*Doing OneLoop with q^4 in numerator*)
> ex2 = PaVeReduce[OneLoop[q, den*SP[q, q]^2]]
> ------------------------------------------------------------------
> The results are different:
>
> ex1-ex2=I*Pi^2/2
>
> It looks like that in this two cases, the rational part of the answer (the
> part which is finite in the limit D->4, but not proportional to the basis
> scalar integrals) is treated in a different way, and I can not guess how to
> use OneLoop to obtain always the correct answers.
> Thanks in advance for any help.
> Maxim Nefedov



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