Momentum shifts in FCLoopBasisSplit ?

[MJKirk]

Hi Vlad,

Is there any chance of FCLoopBasisSplit being extended to also use shifts in momentum to see if a 2-loop integral is factorisable (or is there some other function that does this)?

E.g. taking the example from the documentation:

FCI@FAD[{q1, m}, {q2, m}, {p1 - p2, 0}] 
FCLoopBasisSplit[%, {q1, q2}]

(which works nicely and already finds some of my factorisable diagrams) but then doing a simple change

FCI@FAD[{q1 + q2, m}, {q2, m}, {p1 - p2, 0}] 
FCLoopBasisSplit[%, {q1, q2}]

it no longer sees that this is separable (by q1 -> q1 - q2).

[vsht]

Hi Matthew,

FCLoopBasisSplit is a really old routine function with very limited functionality...

If you know into which integrals your 2-loop integrals can factorize (I suppose there will be some bubbles and tadpoles only), it is easier to use FCLoopFindIntegralMappings with the PreferredIntegrals option. The last section of the manual describes exactly this case

https://feyncalc.github.io/FeynCalcBookDev/FCLoopFindIntegralMappings.html

[vsht]

Actually, since I had it on my todo list for some time, I added several new routines to the dev version that can do the splitting
of factorizing integrals automatically. Please have a look at

https://feyncalc.github.io/FeynCalcBookDev/FCLoopFactorizingQ.html

https://feyncalc.github.io/FeynCalcBookDev/FCLoopFactorizingSplit.html

https://feyncalc.github.io/FeynCalcBookDev/FCLoopCreateFactorizingRules.html

[MJKirk]

From a quick test it seems like these do exactly what I want - thanks!

[vsht]

It won't work that well for integrals with numerators, though.

If there are only denominators, then checking for factorization is a purely algorithmic step: you only need to check if the Symanzik U-polynomial factorizes. From there one can split the integral into propagators according to the Feynman parameters in each factor and do some heuristics to work out the real and spurious loop momenta in each subintegral.

Irreducible numerators ruin this logic, because they do not appear in the U-polynomial. So at the moment I count as integrals as nonfactorizable. I think that one could do a bit better than that, but with lots of extra heuristics and still having factorizable cases not being reliably recognized.

[MJKirk]

Good to know (luckily all my master integrals have no numerators)