Name: Vladyslav Shtabovenko Date: 12/26/16-09:10:58 PM Z

Hi Jongping,

Am 26.12.2016 um 03:35 schrieb Jongping Hsu:
> Hi,Vladyslav,
> Happy Holidays!

happy holidays to you too.

> I need some more relations to check the consistency of the OneLoop result:
> What is the relation between B0(p^2,0,0) and
> F=E0(0,0,p^2,0,p^2,0,p^2,p^2,p^2,p^2,0,0,0,0,0) ?

Just use the code that I sent you previously:

1156.html

with

FAD[{q, 0, 3}, {q + p, 0, 2}]/(I Pi^2)

All your scalar integrals follow the same pattern, so
if you play a bit with the integers m and n in

FAD[{q, 0, m}, {q + p, 0, n}]

you can obtain all those relations for scalar integrals that you are
looking for.

>
> When I used TID, e.g., “ampssPT = (TID[#, q, ToPaVe -> True,
> UsePaVeBasis -> True] & /@
> JPamp) “, I got almost 20 new and unfamiliar functions, E3,
> E4,…E444,..D002, D333, etc.
> Is there a source or reference that one can get the relation between
> them and, say, B(p^2,0,0)?

The simplest way would be not to use the “ToPaVe” and “UsePaVeBasis”
options, so that you end up with scalar integrals that can be IBP
reduced. Otherwise I’m not aware of an existing symbolic codes to
evaluate higher order Passarino-Veltman functions, although this
probably should be using possible formulas from the work of Denner:

https://arxiv.org/abs/0709.1075
https://arxiv.org/abs/hep-ph/0212259
https://arxiv.org/abs/hep-ph/0509141

All those A,B,C,D,E,… PaVe functions are defined according to Denner.
The scalar integrals (those with 0 subscript) are quite simple, c.f. for
example the LoopTools manual
<<http://www.feynarts.de/looptools/LT28Guide.pdf>>

and the appendix of the

https://arxiv.org/pdf/1604.06792.pdf

for the general formula.

The coefficient functions (those with subscripts different from 0) are
more complicated. Depending on the kinematics it might be possible to
write them down only as Feynman parameter integrals, although others
might be reducible into scalar integrals.

Cheers,
Vladyslav

> Thanks. JP
>
> HSU Jongping,
> Chancellor Professor
> Department of Physics
> Univ. of Massachusetts Dartmouth,
> North Dartmouth, MA 02747. FAX (508)999-9115
> http://www.umassd.edu/engineering/phy/people/facultyandstaff/jong-pinghsu/
> recent monograph: Space-Time Symmetry and Quantum Yang–Mills Gravity
> (https://sites.google.com/site/yangmillsgravity123/)
>
> ————————————————————————
> *From: *“Vladyslav Shtabovenko” <[noreply_at_HIDDEN-E-MAIL]>
> *To: *[feyncalc_at_HIDDEN-E-MAIL]
> *Sent: *Sunday, December 25, 2016 2:57:28 PM
> *Subject: *Re: Question about TID in FC9.2.0
>
>
>>> Unfortunately, at the moment one cannot evaluate it numerically directly
>>> from FeynHelpers (while developing the add-on my main focus were
>>> symbolic evaluations). However, you can easily do something like
>>>
>>> exp = PaXDiLog[2.7, 1] /. PaXDiLog -> X`DiLog
>>> Export[“exp.m”, exp]
>>>
>>> Quit[]
>>> «X`
>>> Import[“exp.m”]
>>>
>>> to obtain the numerical value from Package-X. I’ll contact the developer
>>> of Package-X to see if we can find a better solution…
>
> Sorry, ignore this part. Somehow I completely forgot that the 1-loop
> library of Package-X is loaded not immediately but during the first call
> of PaXEvaluate.
>
> So if you have already done some calculations with PaXEvaluate on a
> running kernel, then
>
> exp = PaXDiLog[2.7, 1] /. PaXDiLog -> X`DiLog
>
> is sufficient. On a fresh kernel just call PaXEvaluate once (with any
> input) and then it will work as well:
>
> $LoadAddOns = {“FeynHelpers”};
> «FeynCalc`
> PaXEvaluate[1]
> exp = PaXDiLog[2.7, 1] /. PaXDiLog -> X`DiLog
>
> Cheers,
> Vladyslav
>
> Am 25.12.2016 um 20:33 schrieb Vladyslav Shtabovenko:
>> Dear Xiu-Lei,
>>
>> actually, when doing the expansion by the means of Package-X, the LO
>> coefficient of the 1/mN expansion turns out to be zero (provided that I
>> got your example right):
>>
>> SPD[p4, p4] = mN^2;
>> XC0 = C0[SPD[p4], SPD[q], SPD[p4 + q], mN^2, mpi^2, mpi^2] //
>> ExpandScalarProduct;
>> XC0Re = PaXEvaluate[XC0, PaXC0Expand -> True,
>> PaXSeries -> , PaXAnalytic -> True] // Normal
>>
>> Doing the expansion with Series afterwards
>>
>> SPD[p4, p4] = mN^2;
>> XC0 = C0[SPD[p4], SPD[q], SPD[p4 + q], mN^2, mpi^2, mpi^2] //
>> ExpandScalarProduct;
>> XC0Re = PaXEvaluate[XC0, PaXC0Expand -> True] // Normal;
>> Series[XC0Re, {mN, Infinity, 0}] // Normal // Simplify
>>
>> produces several suspicious terms, like Sqrt[-SPD[q,q]]. As you probably
>> know, Mathematica is not always careful when choosing the branch cuts of
>> logs and square roots and does not really provide options to control
>> that consistently, so I would rather trust the output of Package-X
>> (which takes care of those things in a special way internally) than the
>> output of Series.
>>
>> By the way, the author of Package-X has released several fixes
>> in the meantime (ver 2.0.3 being the most current). One can update
>> Package-X manually, by downloading the tarball from
>> packagex.hepforge.org or via FeynHelpers’ installer
>>
>> Import[“https://raw.githubusercontent.com/FeynCalc/feynhelpers/master/\
>> install.m”]
>> InstallPackageX[]
>>
>> As to the second part of your question:
>>
>> PaXDiLog is just a placeholder that for the DiLog of Package-X. Its
>> relation to PolyLog is described in 1503.01469, Sec VI.
>>
>> Unfortunately, at the moment one cannot evaluate it numerically directly
>> from FeynHelpers (while developing the add-on my main focus were
>> symbolic evaluations). However, you can easily do something like
>>
>> exp = PaXDiLog[2.7, 1] /. PaXDiLog -> X`DiLog
>> Export[“exp.m”, exp]
>>
>> Quit[]
>> «X`
>> Import[“exp.m”]
>>
>> to obtain the numerical value from Package-X. I’ll contact the developer
>> of Package-X to see if we can find a better solution…
>>
>> I agree that PaXDiLog[Complex[-1,-6],-0.2] does not look correct at all.
>> Could you provide a minimal working code example that generates this
>> weird expression?
>>
>> I also wish you happy holidays.
>>
>> Cheers,
>> Vladyslav
>>
>> Am 25.12.2016 um 10:51 schrieb Xiu-Lei Ren:
>>> Dear Vladyslav,
>>>
>>> Thank you very much for your quick reply. It helps a lot.
>>>
>>> However, when i try to obtain the analytic expressions of triangle
>>> diagram mentioned in the previous email, I also encountered two
>>> questions about PaXDiLog.
>>>
>>> In order to avoid unexpected results when performing Dimension -> 4, I
>>> use the recommended FeynHelper–Package-X.
>>>
>>> When I do this, the treatment of pave coefficient C0 is necessary.
>>> In my case, (I am handling the two-nucleon scattering with two-pion
>>> exchange.
>>> mN, mpi deonte as nucleon and pion masses, p4 is the momentum of
>>> outgoing nucleon, q is the transfer momentum between two nucleons.)
>>>
>>> XC0 = C0[p4^2, q^2, (p4+q)^2, mN^2, mpi^2, mpi^2]
>>>
>>> should be replaced by using
>>>
>>> XC0Re = PaXEvaluate[XC0, PaXC0Expand -> True]//Normal
>>>
>>> Apparently, the output is lengthy with conditions.
>>>
>>> Then, perform the 1/mN expansion,
>>>
>>> Series[XC0Re, {mN, infty, 0}]//Normal
>>>
>>> The result always contains Li2 functions (PaXDiLog).
>>>
>>> 1) How one can transfer PaXDiLog to PolyLog?
>>>
>>> Furthermore, when I do the numerical evaluation for checking,
>>> I also find another problem about PaXDiLog.
>>>
>>> 2) e.g. PaXDiLog[Complex[-1,-6],-0.2], it cannot give a numerical value.
>>>
>>> Could you kindly let me know how to handle these problem?
>>>
>>> Merry Christmas and happy new year.
>>>
>>> Cheers,
>>> Xiu-Lei
>>>
>>
>
>