Name: Vladyslav Shtabovenko (email_not_shown)
Date: 06/24/17-09:31:29 PM Z


This issue is not so easy to fix automatically. What happens here is
that after the tensor reduction your integral contains several pure
1-loop integrals.

ToTFI handles those my multiplying them with a dummy 1-loop integral
1/qQ^2-mM^2. Which gives a 2-loop integral that one can process with
TARCER. Ideally, the dummy 1/qQ^2-mM^2 remains factorized after the
reduction so that one can simply replace it by unity.

However, if the 1-loop integral contains certain loop momentum-dependent
scalar products in the numerator, then the dummy mass mM may enter the
IBP reduced integrals in a non-trivial way, which is what you observe.

You can avoid that by performing tensor reduction of the 1-loop
integrals with TID beforehand, e.g.

exp = FAD[{p2, I mG}, {p1, I mG}, {p1, I mG}, {p1, I mG}, {p1,
      I mG}, {p1, I mG}] FVD[p2, Lor1] FVD[p2, Lor2] SPD[p1,
     p1] SPD[p1, q]^2 // FCMultiLoopTID[#, {p1, p2}] &

oneLoopSelect[expr_] :=
   FCLoopIsolate[expr, {p1, p2},
     Head -> oneLoop] //. {oneLoop[
        ex_] /; (FreeQ[ex, p1] && ! FreeQ[ex, p2]) :> tid[ex, p2],
     oneLoop[ex_] /; (FreeQ[ex, p2] && ! FreeQ[ex, p1]) :>
      tid[ex, p1]} /. oneLoop -> Identity

tmp = oneLoopSelect[exp] /. tid -> TID
ToTFI[tmp, p1, p2, q] // TarcerRecurse

I will add a check in ToTFI to detect cases where the dummy integral
does not factorize, but even then ToTFI will have no other choice than
to abort the evaluation. It would also not be clever to let ToTFI always
call TID on 1-loop integrals, as this might lead to some unwanted
effects (e.g. extreme proliferation of terms on higher rank 2-point
functions).

Cheers,
Vladyslav

Am 23.06.2017 um 11:12 schrieb Philipp Schicho:
> To put it correctly this seems to be a but in ToTFI which occurs when converting the expression
>
> FAD[{p2, I mG}, {p1, I mG}, {p1, I mG}, {p1, I mG}, {p1, I mG}, {p1,
> I mG}] FVD[p2, Lor1] FVD[p2, Lor2] SPD[p1, p1] SPD[p1, q]^2 //FCMultiLoopTID[#,{p1,p2}]&//ToTFI[#,p1,p2]&
>
> =
>
> -(1/(2 (-1 + D)))(FVD[q, Lor1] FVD[q, Lor2] -
> MTD[Lor1, Lor2] SPD[q, q]) Tarcer`TFI[2 + D,
> FeynCalc`SPD[
> q, q], {{4, Complex[0, 1] mG}, {2, FeynCalc`ToTFI`Private`mM}, {0,
> 0}, {0, 0}, {0, 0}}] + (1/(2 (-1 + D)))
> mG^2 (FVD[q, Lor1] FVD[q, Lor2] -
> MTD[Lor1, Lor2] SPD[q, q]) Tarcer`TFI[2 + D,
> FeynCalc`SPD[
> q, q], {{5, Complex[0, 1] mG}, {2, FeynCalc`ToTFI`Private`mM}, {0,
> 0}, {0, 0}, {0, 0}}] - (1/((-1 + D) SPD[q, q]))
> mG^2 (-FVD[q, Lor1] FVD[q, Lor2] +
> MTD[Lor1, Lor2] SPD[q, q]) Tarcer`TFI[D,
> FeynCalc`SPD[q, q], {0, 0, 2, 0,
> 0}, {{4, Complex[0, 1] mG}, {1, Complex[0, 1] mG}, {0, 0}, {0,
> 0}, {0, 0}}] + (1/((-1 + D) SPD[q, q]))
> mG^4 (-FVD[q, Lor1] FVD[q, Lor2] +
> MTD[Lor1, Lor2] SPD[q, q]) Tarcer`TFI[D,
> FeynCalc`SPD[q, q], {0, 0, 2, 0,
> 0}, {{5, Complex[0, 1] mG}, {1, Complex[0, 1] mG}, {0, 0}, {0,
> 0}, {0, 0}}] + ((D FVD[q, Lor1] FVD[q, Lor2] -
> MTD[Lor1, Lor2] SPD[q, q]) Tarcer`TFI[D,
> FeynCalc`SPD[q, q], {0, 0, 2, 2,
> 0}, {{4, Complex[0, 1] mG}, {1, Complex[0, 1] mG}, {0, 0}, {0,
> 0}, {0, 0}}])/((-1 + D) SPD[q, q]^2) - (1/((-1 + D) SPD[q, q]^2))
> mG^2 (D FVD[q, Lor1] FVD[q, Lor2] -
> MTD[Lor1, Lor2] SPD[q, q]) Tarcer`TFI[D,
> FeynCalc`SPD[q, q], {0, 0, 2, 2,
> 0}, {{5, Complex[0, 1] mG}, {1, Complex[0, 1] mG}, {0, 0}, {0,
> 0}, {0, 0}}]
>
> The RankLimit {2,10} however seems to work so far.
>
> Cheers,
> Philipp
>



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