**Next message:**Maksym: "Error! DiracTrick has encountered a fatal problem and must abort the computation. The problem reads: Incorrect combination of dimensions and g^5 scheme!"**Previous message:**Vladyslav Shtabovenko: "Re: Tarcer Problem with RankLimit"**Next in thread:**V. Shtabovenko: "Re: UV-Divergences of 1-loop integrals"**Reply:**V. Shtabovenko: "Re: UV-Divergences of 1-loop integrals"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Mail actions:**[ respond to this message ] [ mail a new topic ]

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

**Next message:**Maksym: "Error! DiracTrick has encountered a fatal problem and must abort the computation. The problem reads: Incorrect combination of dimensions and g^5 scheme!"**Previous message:**Vladyslav Shtabovenko: "Re: Tarcer Problem with RankLimit"**Next in thread:**V. Shtabovenko: "Re: UV-Divergences of 1-loop integrals"**Reply:**V. Shtabovenko: "Re: UV-Divergences of 1-loop integrals"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Mail actions:**[ respond to this message ] [ mail a new topic ]

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