Hi,
the behaviour is reliable. It is just an implementation of Tarasov's algorithm with some minor
extensions of Rainer Scharf and myself.
If you can take the on-shell limit afterwards: fine.
If not, you have to evaluate the basis integrals or find relations between them.
Unfortunately I am not currently in the position to further develop Tarcer, however
you may want to contact Georg Weiglein from Durham who should be able to help you
or point you to further resources:
http://www.cpt.dur.ac.uk/Members/Georg.Weiglein/
Regards,
Rolf Mertig
GluonVision GmbH
Robert Schoefbeck wrote:
> Dear Tarcer programmers,
>
> I have been playing around with this package and found some strange
> behaviour:
>
>
> TarcerRecurse[TFI[d, M12, { 0, 0, 1, 0, 0}, {{1, M2}, {1, 0}, {0, 0}, {1,
> M1}, {1, M2}}]]
>
>
> gives
>
> -((-2 + d)2*(M1 - M2)*(M1 + M2)*TAI[d, 0, {{1, M1}}]*TAI[d, 0, {{1,
> M2}}])/(4*(-4 + d)*(-8 + 3*d)*M12*M22) -
> ((-2 + d)*TAI[d, 0, {{1, M2}}]^2)/(4*(-8 + 3*d)*M22) +
> ((-7*M12 + 2*d*M12 + 3*M22 - d*M22)*TJI[d, M12, {{2, M1}, {1, M2},
> {1, M2}}])/((-4 + d)*(-8 + 3*d)) +
> ((-7*M12 + 2*d*M12 + 3*M22 - d*M22)*TJI[d, M12, {{2, M2}, {1, M2},
> {1, M1}}])/((-4 + d)*(-8 + 3*d)) -
> (4*(M1 - M2)*M22*(M1 + M2)*TJI[d, M12, {{2, M2}, {2, M1}, {1,
> M2}}])/((-4 + d)*(-8 + 3*d)) +
> (2*M22*(-M1 + M2)*(M1 + M2)*TJI[d, M12, {{2, M2}, {2, M2}, {1,
> M1}}])/((-4 + d)*(-8 + 3*d)) +
> (4*M22*(-M1 + M2)*(M1 + M2)*TJI[d, M12, {{3, M2}, {1, M2}, {1,
> M1}}])/((-4 + d)*(-8 + 3*d))
>
> This result contains among others
>
> TJI[d, M12, {{3, M2}, {1, M2}, {1, M1}}
>
> and therefore could be reduced further in principle. Since the momentum
> also is a mass squared argument I was able to compute it by hand using
> differential equations and get a result up to order (d-4)0. However, the
> integral and some others too, appear with a 1/(d-4) in front, so the
> series of the result is not a simple sum of the serieses of these
> remaining J-integrals, but the (d-4)0 term of the result will get a (d-4)
> contribution from the J's. I can't compute the finite part of the J's with
> differential equations but instead I calculated
>
> TarcerRecurse[TFI[d, s2, {0, 0, 1, 0, 0}, {{1, M2}, {1, 0}, {0, 0}, {1,
> M1}, {1, M2}}]] /. s -> M1
>
> which simply gives
>
> ((-2 + d)*TAI[d, 0, {{1, M2}}]^2)/(8*M22) - (3*(-3 + d)*TJI[d, M12, {{1,
> M2}, {1, M2}, {1, M1}}])/4 +
> M12*TJI[d, M12, {{2, M1}, {1, M2}, {1, M2}}] + M22*TJI[d, M12, {{2,
> M2}, {1, M2}, {1, M1}}]
>
> In cases where my initial problem of the unrecursed J's does NOT appear I
> am usually NOT able to take the limit s->M2 if M is a mass of the TFI
> integral because the result will typically contain 1/(s-M2) terms.
> del'Hospital suggests that this factor differentiates the integral it goes
> with and i think this causes the J's to be unreduced in the case above.
>
> My question is simply if this behaviour is reliable. To sum up: I compute
> TFI[d, M12, { 0, 0, 1, 0, 0}, {{1, M2}, {1, 0}, {0, 0}, {1, M1}, {1,
> M2}}]
> in two ways:
> 1) directly. Tarcerrecurse gives nonstandard J-integrals and appaerantly
> even needs their (d-4)-term
> 2) putting the external momentum to s, and taking the limit afterwards ->
> nice result; but: I cant bring back s->M2 in many other cases and I cant
> know in advance when to apply this trick.
>
> I would like to hear your thoughts on this thing, if possible.
>
> PS: I have posted this message also on the forum but got some strange
> message doing so.
>
>
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