Name: V. Shtabovenko Date: 09/14/17-02:46:02 AM Z

And here is the reply from Hiren. We will whitelist his e-mail address
asap so that he will be able to write to the mailing list directly.

-——- Weitergeleitete Nachricht ——–
Betreff: Re: A possible bug: PaXSeries (FeynHelpers)
Datum: Wed, 13 Sep 2017 17:10:54 -0400
Von: Hiren Patel <[hhpatel_at_HIDDEN-E-MAIL]>
An: V. Shtabovenko <[dev.vsht_at_HIDDEN-E-MAIL]>
Kopie (CC): [feyncalc_at_HIDDEN-E-MAIL]

Hi,

Xiu-Lei is requesting an expansion of the two-point function around
large external momentum, which is a Landau singular point. At singular
points like these, and as Vladyslav correctly points out, the result
depends crucially on the order of the following limits: d -> 4 and
mN->Infinity. The phenomenologically relevant order is to first take
the limit d->4 then expand in 1/mN, in which case the function is
represented as an *asymptotic* series, involving Log[mN] in addition to
powers of mN.

Unfortunately, as of the current version of Package-X (2.1.0),
LoopRefineSeries is unable to construct series other than power series.
And, with the default setting Analytic->False, LoopRefineSeries will
generate an error message stating that a power series does not exist
does not exist around mN->Infinity.

The workaround is, as Xiu-Lei does, to use Series to construct the
desired asymptotic expansion after having converted everything to
analytic functions Mathematica knows about.

By setting the option Analytic->True, the integral is first expanded in
1/mN assuming the number of spacetime dimensions is large enough for the
necessary derivatives to exist and hence for a power series
representation to exist. Then the limit d->4 is taken at the end, and
all the Log[mN]’s which would have appeared in the asymptotic expansion
now appear is 1/epsilon poles. You can read more about it in the
documentation here:
http://packagex.hepforge.org/Documentation/HTML/X/ref/LoopRefineSeries.html#759595167

Setting the option Analytic->True is less phenomenologically relevant,
but has some theoretical interest as the result could be understood as
the contribution from a specific region of the loop integral in the
context of the method of expansion by regions.

-Hiren Patel

> On Sep 12, 2017, at 8:25 AM, V. Shtabovenko <[dev.vsht_at_HIDDEN-E-MAIL]> wrote:
>
> Hi,
>
> I think this is related to the way how Package-X expands B-functions
> (using LoopRefineSeries), when the expansion parameter multiplies the PaVe function. Consider the following code:
>
> «X`
> diff = Table[
> {(LoopRefineSeries[
> mN^pow PVB[0, 0, mN^2, mN, mPi], {mN, Infinity, 0},
> Analytic -> True] // Normal),
> Series[
> mN^pow (DiscExpand[LoopRefine[PVB[0, 0, mN^2, mN, mPi]]]), {mN,
> Infinity, 0}] // Normal}, {pow, 0, 3}]
>
> Simplify[PowerExpand[First[#] - Last[#]]] & /@ diff
> % // TableForm
>
> For mN^0 both expansions coincide exactly, but when the power is different from 0, the difference between the two expansions is finite
> and may also involve IR singularities. The case pow=2 is relevant for your expansion.
>
> In general, the expansion of an already computed integral (where you implicitly assume mN to be finite) and the expansion on the level of the Feynman parameters, where you are expanding the Feynman parameter integral (possibly multiplied by an expansion parameter) first and the compute it are not always the same.
>
> This is because the expansion may generate new singularities that can be properly regularized when evaluating Feynman parameter integrals but not afterwards.
>
> However, the current behavior of Package-X looks a bit strange to me,
> so I’m not quite sure if this is intended or a bug.
>
> I added Hiren Patel, the developer of Package-X to the CC of this mail, so that he may explain us this behavior of his tool.
>
> Cheers,
> Vladyslav
>
>
>
> Am 12.09.2017 um 08:04 schrieb X.-L.Ren:
>> Dear Vladyslav,
>> I am doing the non-relativistic expansion (1/mN, mN is nucleon mass) of one-loop integral with the help of FeynCalc 9.2 and FeynHelpers (on Mathematica 10.4).
>> I found that the expanded result from PaXSeries is not the same as the result from Series, which is right for sure. I did something wrong when using PaXSeries or a possible bug?
>> In order to see the details, the code is given below,
>> (* Diagrams * )
>> ltdia = FCI[
>> SpinorUBarD[p, mN,
>> 1].(GSD[k].(GSD[p] - GSD[k] - mN).GSD[k]).SpinorUD[p, mN,
>> 1] FAD[{k, m\[Pi]}, {p - k, mN}]]
>> (* Do Integration *)
>> ltintOS =
>> 1/(2 mN) Collect2[
>> TID[ltdia, k, UsePaVeBasis -> True, ToPaVe -> True,
>> PaVeAutoReduce -> True], Spinor] // DiracSimplify;
>> (* 1: Using the PaXSeires[mN, infinity, 0] in PaXEvaluate *)
>> IntHBPCB = (PaXEvaluate[
>> ltintOS /. {Pair[Momentum[p, D], Momentum[p, D]] -> mN^2},
>> PaXImplicitPrefactor -> 1/(2 \[Pi])^D, PaXAnalytic -> True,
>> PaXSeries -> ] // Normal //
>> FCHideEpsilon) /. {SMP[“Delta”] -> 0} // Simplify //
>> FullSimplify
>> (* 2: Using the Seires[mN, infinity, 0] after PaXEvaluate *)
>> IntHBPCB2 = (Series[
>> PaXEvaluate[
>> ltintOS /. {Pair[Momentum[p, D], Momentum[p, D]] -> mN^2},
>> PaXImplicitPrefactor -> 1/(2 \[Pi])^D, PaXAnalytic -> True] //
>> FCHideEpsilon, {mN, \[Infinity], 0}] //
>> Normal) /. {SMP[“Delta”] -> 0} // Simplify
>> (* They are different. While, IntHBPCB2 is right.*)
>> IntHBPCB2 - IntHBPCB // Expand // FullSimplify
>> I am looking forward to your reply.
>> Thank you in advance.
>> Best regards,
>> Xiu-Lei