Name: V. Shtabovenko Date: 11/06/17-02:55:22 AM Z

FYI, since our collection of example calculations started to
become a big mess, I took some time to refactor the codes, introduce
some unified notation and make things more readable.

The directory structure has slightly changed. You will still find
everything in

<https://github.com/FeynCalc/feyncalc/tree/master/FeynCalc/Examples>

but the 1-loop QED and QCD calculations that use UVPart are now in

<https://github.com/FeynCalc/feyncalc/tree/master/FeynCalc/Examples/QED/OneLoop>

<https://github.com/FeynCalc/feyncalc/tree/master/FeynCalc/Examples/QCD/OneLoop>

By the way,

<https://github.com/FeynCalc/feyncalc/blob/master/FeynCalc/Examples/QCD/OneLoop/Renormalization.m>

contains the 1-loop renormalization of QCD in MS and MSbar schemes.

Actually, Rolf, Frederik and myself had the idea to add this example
since our Berlin meeting in 2016. However, only after PaVeUVPart was
added to the development version it became possible to implement this
calculation in such an easy and straightforward way.

Now that we have QED and QCD, the next interesting example would be to
add the full 1-loop renormalization of the Standard Model.

Cheers,
Vladyslav

Am 10.07.2017 um 04:50 schrieb Vladyslav Shtabovenko:
> Dear FeynCalc users,
>
> one often encounters situations, where we are interested only in the
> UV-singularities of the dimensionally regularized loop integrals. In
> such cases the finite part and the IR poles can be discarded. Instead of
> calculating the full integral and then fishing out the UV-pole (which is
> sometimes non-trivial due to the overlapping singularities), it is
> actually much easier to pick up the UV-divergent piece right away. At
> 1-loop there are even explicit formulas for doing so.
>
> FeynCalc has a (probably not so well-known) function called UVPart that
> tries to go into that direction, but doesn’t do its job in a good way.
> The original idea was to discard all the 1-loop integrals that gave no
> UV-poles. However, there are some cases where UVPart might actually drop
> a UV divergent 1-loop integral, which is why it became necessary to do
> something about that. At this point I’d like to thank Martin Beneke for
> bringing my attention to the issues with UVPart.
>
> I removed UVPart from the current development version (aka the upcoming
> FeunCalc 9.3) because now we have something much better than that. The
> replacement for UVPart is called PaVeUVPart
>
> <https://github.com/FeynCalc/feyncalc/commit/03d506077b828ebe960bcf87e259ad9331d9c2df>
>
>
> PaVeUVPart works with Passarino-Veltman coefficient functions and
> replaces each function with its UV-divergent piece. The nice thing is
> that it works with 1-loop integrals of arbitrary rank and multiplicity.
> This is achieved by using the algorithm of Georg Sulyok
>
> <https://inspirehep.net/record/727190>
>
> As an addition to his publication, Georg has also provided  a
> Mathematica version of his program that returns UV-poles of arbitrary
> PaVe functions. I’m very grateful to him for the nice collaboration on
> integrating his code into FeynCalc.
>
> For using PaVeUVPart in renormalization calculations, have a look at
>
> <https://github.com/FeynCalc/feyncalc/blob/master/FeynCalc/Examples/QCD/QCDQuarkSelfEnergyOneLoop.m>
>
>
> <https://github.com/FeynCalc/feyncalc/blob/master/FeynCalc/Examples/QCD/QCDGluonSelfEnergyOneLoop.m>
>
>
> <https://github.com/FeynCalc/feyncalc/blob/master/FeynCalc/Examples/QED/QEDElectronSelfEnergyOneLoop.m>
>
>
> <https://github.com/FeynCalc/feyncalc/blob/master/FeynCalc/Examples/QED/QEDPhotonSelfEnergyOneLoop.m>
>
>
> And to give you some idea how to use it on standalone integrals:
>
> int = SPD[k] FAD[{k + q1, m1}, {k + q2, m2}]/(I Pi^2);
> TID[int, k, UsePaVeBasis -> True, ToPaVe -> True] //PaVeUVPart
>
> -((2 (m1^2 + m2^2 + SPD[q1, q2]))/(-4 + D))
>
>
> int = SPD[k, p1]^3 FAD[k, {k + q1, m1}, {k + q2, m2}]/(I Pi^2)
>
>
> TID[int, k, UsePaVeBasis -> True, ToPaVe -> True] //PaVeUVPart
>
>
> (SPD[p1, p1] SPD[p1, q1] + SPD[p1, p1] SPD[p1, q2])/(2 (-4 + D))
>
>
> The option UsePaVeBasis is there for performance reasons. Tensor
> reduction into coefficient functions is much faster than tensor
> reduction into scalar integrals, so that it saves your time. Once you
> have your 1-loop amplitude expressed in terms of PaVe functions,
> PaVeUVPart can overtake.
>
> Cheers,
> Vladyslav
>
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