A question on the limitations of findDivergences

[Vladyslav Shtabovenko]

Dear Erik,

in Lemma 5.1 of 1401.4361 it is written that all nonzero coefficients of the second Symanzik
polynomial should be positive to guarantee absolute convergence of the Feynman parametric
integrand when checking the behavior of different sets of Feynman parameters.

I’m wondering if my understanding of this statement is correct, since in practice one certainly
encounters parametric integrals where some of the coefficients are negative, but the analytic
regularization still works. So I’d like to better understand the existing limitations of the analytic
regularization. More specifically, are there examples of loop integrals, where findDivergences
would explicitly fail to recognize a divergence so that it would remain unregularized?
So far I’ve encountered none, even when calculating some masters obtained from asymptotic
expansions of more complicated integrals.

The background to my question is that I would like to add some functionality similar to
findDivergences and dimregPartial to FeynCalc (with all due acknowledgements of your
work and that of F. Brown both in the code and in the accompanying publication).
However, here I’m not so much interested in calculating parametric integrals (which I
anyhow prefer to do using HyperInt) but rather analyzing the UV and IR-behavior of
master integrals and checking how auxiliary masses added to some of the propagators
(as a tool for regulating IR-divergences) improve the IR-behavior of the integral. Here
it would be nice to know where such an analysis could potentially fail due to undetected
singularities inside the integration domain.

Cheers,
Vladyslav

[Vladyslav Shtabovenko]

I realized that my question might be somewhat misleading, so let me clarify that I explicitly exclude
all divergences that arise through particular choices of kinematic invariants (like p^2=4m^2, p^2 = 0 etc.).
So it is understood that if masses and momenta squared are evaluated at special kinematic points,
their values should be explicitly plugged in before calculating the integral. It seems to me that under
such conditions findDivergences should be able to reveal all potential ep-poles, but of course I might be wrong.

‌

[Erik Panzer]

Dear Vladyslav,

it is true that the positivity of the coefficients is not necessary. You can prove the result more generally under the assumption that the polynomials have no zeroes inside the integration domain (i.e. positive values of the Schwinger parameters). See Theorem 2.3 in Euler-Mellin integrals and A-hypergeometric functions for the precise statement (it is a bit tricky, see the ‘totally non-vanishing’ condition: also the polynomials obtained by only keeping monomials on some fixed face of the Newton polytope, have to be free of zeros).

I am not aware of any more general results. In particular, there are indeed examples of Feynman integrals where the power counting on the facets of the Newton polytope is not sufficient.

For explicit examples, see section 3 in Expansion by regions: revealing potential and Glauber regions automatically. In these cases, you e.g. get terms of the form (x-y)^2+… or (x-y)*(z-w)+… in the 2nd Symanzik polynomial of the parametric representation, and the poles at x=y, or (x,z)=(y,w), inside the integration domain, lead to additional contributions to poles in epsilon.

Now in those specific examples, there are obvious changes of variables to translate those poles back to the familiar cases (e.g. by splicing up the integration domain into x<y and x>y, so that those new poles lie on the boundaries of the subdomains), hence you can obtain the power counting condition from analyzing these reparametrized pieces.

In general, however, I am not aware of any results for arbitrary polynomials (or Feynman integrals without any restriction on the kinematics, for that matter).

By the way, if you plan to implement some UV/IR power counting, please do it not the way it is done in HyperInt. That is very slow, and only works in simple situations (integrals with a Euclidean region). The best and completely general way to obtain the constraints (and hence the necessary vectors for dimregPartial) is to obtain the faces of the Newton Polytope of the Lee-Pomeransky polytope, like explained in the paper I linked above.

With best wishes,

Erik

[Vladyslav Shtabovenko]

Dear Erik,

many thanks for your detailed reply. I agree with you that once one starts expanding Feynman parametric integrals in some special ways
and/or doing variable transformations as well as making use of the Cheng-Wu theorem, the resulting integrand can become very different
from what it originally was. In this case it is more or less clear that one can run into divergences that findDivergences would not be able to
identify.

As far as the expansion by regions is concerned, I would actually rather avoid performing it on the level of Feynman parametric integrals.
The strategy I’ve been successfully employing up to now is to use Smirnov’s asy.m to reveal the contributing regions and then find a
sequence of loop momentum shifts (for each region!) that would allow me to expand the propagator representation in internal and
external 4-momenta and masses. Upon the expansion I’d just run several IBP reductions to arrive at a final set of master integrals and
then calculate those directly. I would not claim that this method works for every integral and every region, but so far I didn’t encounter
any cases where it would fail. This is also what I meant with using findDivergences when doing asymptotic expansion: When I have my
final set of master integrals (all in propagator representation!) needed for the asymptotic expansion of the given integral, I would calculate
these masters using HyperInt and there findDivergences works as intended.

This is why I’m curious if you are aware of any integrals in the propagator representation, whose Feynman parametric representation would
contain divergences that cannot be detected with findDivergences? Of course under the assumptions that there are no singularities related
to the kinematics and that one doesn’t perform any variable transformations or expansions but simply takes the “vanilla“ Feynman parametric
representation.

By the way, if you plan to implement some UV/IR power counting, please do it not the way it is done in HyperInt. That is very slow, and only works in simple situations (integrals with a Euclidean region). The best and completely general way to obtain the constraints (and hence the necessary vectors for dimregPartial) is to obtain the faces of the Newton Polytope of the Lee-Pomeransky polytope, like explained in the paper I linked above.

Thanks for the nice suggestion! Actually, I find that your implementation is a very useful and powerful tool. The first run of findDivergences is usually fast enough, but it indeed becomes slower when one analyzes a polynomial
that already has been regularized. However, the performance still seems to be fine enough for me. My Mathematica port of your code can be found here:

https://feyncalc.github.io/FeynCalcBookDev/FCFeynmanFindDivergences.html
https://feyncalc.github.io/FeynCalcBookDev/FCFeynmanRegularizeDivergence.html
https://github.com/FeynCalc/feyncalc/blob/master/FeynCalc/LoopIntegrals/FCFeynmanFindDivergences.m
https://github.com/FeynCalc/feyncalc/blob/master/FeynCalc/LoopIntegrals/FCFeynmanRegularizeDivergence.m

Again, do you perhaps have some examples of integrals lacking a Euclidean region, where findDivergences would not work properly?
I found this example from pySecDec
https://github.com/gudrunhe/secdec/blob/master/examples/triangle2L_split/generate_triangle2L_split.py

which is described as “a 2-loop, 3-point, 6-propagator integral without a Euclidean region due to special kinematics“
but here there seem to be no additional divergences apart from what your algorithm can identify (I yet have to do a numerical
check of the HyperInt result, but it reports no singularities during the integration).

Cheers,
Vladyslav