Name: Robert Schoefbeck Date: 11/09/05-02:40:15 PM Z

Dear Tarcer programmers,

I have been playing around with this package and found some strange behaviour:

TarcerRecurse[TFI[d, M1^2, { 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)*M1^2*M2^2) -
 ((-2 + d)*TAI[d, 0, {{1, M2}}]^2)/(4*(-8 + 3*d)*M2^2) +
 ((-7*M1^2 + 2*d*M1^2 + 3*M2^2 - d*M2^2)*TJI[d, M1^2, {{2, M1}, {1, M2}, {1, M2}}])/((-4 + d)*(-8 + 3*d)) +
 ((-7*M1^2 + 2*d*M1^2 + 3*M2^2 - d*M2^2)*TJI[d, M1^2, {{2, M2}, {1, M2}, {1, M1}}])/((-4 + d)*(-8 + 3*d)) -
 (4*(M1 - M2)*M2^2*(M1 + M2)*TJI[d, M1^2, {{2, M2}, {2, M1}, {1, M2}}])/((-4 + d)*(-8 + 3*d)) +
 (2*M2^2*(-M1 + M2)*(M1 + M2)*TJI[d, M1^2, {{2, M2}, {2, M2}, {1, M1}}])/((-4 + d)*(-8 + 3*d)) +
 (4*M2^2*(-M1 + M2)*(M1 + M2)*TJI[d, M1^2, {{3, M2}, {1, M2}, {1, M1}}])/((-4 + d)*(-8 + 3*d))

This result contains among others

TJI[d, M1^2, {{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, s^2, {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*M2^2) - (3*(-3 + d)*TJI[d, M1^2, {{1, M2}, {1, M2}, {1, M1}}])/4 +
 M1^2*TJI[d, M1^2, {{2, M1}, {1, M2}, {1, M2}}] + M2^2*TJI[d, M1^2, {{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->M^2 if M is a mass of the TFI integral because the result will typically contain 1/(s-M^2) 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, M1^2, { 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->M^2 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.

kind regards
Robert Schoefbeck