Dear Jongping,
Am 25.11.2016 um 07:35 schrieb Jongping Hsu:
> Dear Rolf and Vladyslav:
> Thank you very much for your help.
> Rolf has done works on qcd, could you please tell me whether there is a
> program related to the calculation of beta function or asymptotic
> freedom or third order vertex corrections (of pure gluons, without
> fermions)? If you have one, could you please send it to us for reference?
I don't have such a program now, but with FeynRules and FeynHelpers
(arXiv:1611.06793) it should be quite easy to do. For my FeynHelpers
paper I included an example code for the full 1-loop renormalization of
QED in MS, MSbar and On-Shell schemes (including evaluation of the
vertices as a consistency check). For this I needed less than a week,
with most of the time spent on checking intermediate results with the
literature and chasing some bugs in the code. The QCD case should be
only minimally more involved and my QED code can be used as a template.
It is just that I currently don't have time to do it myself: I'm
planning to graduate in Spring and need to work on the remaining papers
to come out, the thesis and of course my applications for a post doc
position.
If you have a student to assign the task to, he/she should not need too
much time for that. Chapter 4.5 of Boehm, Denner, Joos "Gauge Theories
of Strong and Electroweak Interaction" (for the QCD Lagrangian with
counter-terms) as well as Table 11.1 (for the renormalization constants)
in George Sterman's "Introduction to QFT" can serve as a good reference
to check the final results.
Actually, you can even avoid dealing with the vertices if you use QCD in
the background field gauge (following Abbott, 1981). Then the
calculation becomes very much QED-like. I actually added that model and
a simple example
(FeynCalc/Examples/FeynRules/QCDBGF/PureYMSelfEnergyInQCDBGFAtOneLoop.m)
to FC 9.2. See the instruction
in https://feyncalc.org/forum/1125.html to get the code running.
>> (2)These relations are very intriguing. If one is interested in
>> asymptotic freedom, one just calculate the UV divergent terms, which
>> show up in Bo in the OneLoop calculations. One can do subtraction at
>> finite p^2 to avoid IR divergence. Since Co is not a UV divergent
>> integral(although it may have IR divergence), one does not expect it to
>> contribute to the UV divergent amplitudes. So, why p^2 Co contribute to
>> the asymptotic freedom in the OneLoopSimplify[...] and OneLoop[...]?
I don't think that there is anything special happening here. Dimensional
regularization was originally designed in such a way that one can
regulate UV and IR singularities with the same regulator.
This means that usually one doesn't really have to worry, what is UV and
what is IR and the answer still comes out right. Even though in the
example we are talking about the C0 is IR-divergent, the 1/Epsilon pole
that appears in the final result is still UV divergent. There is no
magic involved here. It is just that if you *do* want to
distinguish between IR and UV, then you need to use different regulators
for that and keep in mind that logarithmically
divergent scaleless integrals like d^D q / q^4 are not zero, but rather
proportional to 1/Epsilon_UV - 1/Epsilon_IR. I have a short summary on
how to do this in FeynCalc 9.2 in my recent paper (arXiv:1611.06793),
end of section 4.2. However, the technique by itself is of course not
my invention. It is often used in effective field theory calculations,
see e.g. the famous NRQCD paper of Manohar, hep-ph/9701294 Sec. IV.
So, if you are interested in the gory details: Actually, in the ghost
triangle calculation there is a scaleless B0(0,0,0) function that is
normally set to zero in DR. However, if you distinguish between IR and
UV, then it is of course not zero and the IR-piece of the integral
precisely cancels the IR pole of the C0 function, while the UV pole
contributes to the final results, such that at the end the tensor
structure of the 1/Epsilon_UV pole comes out as it should and no IR pole
appears. The only thing is that for such investigations you should use
TID and not OneLoop. Furthermore, you need to install my addon
FeynHelpers, which is also descried in the above mentioned paper.
> Vladyslav's answer regarding the relation between -p^2 Co and Bo for
> asymptotic property (or UV divergence) is interesting and puzzling. I
> have some related questions:
>
> (1)Are there relations between p^4 Do(0,p^2,0,p^2,p^2,p^2,0,0,0,0) and
> Bo, and between
> p^4 Do(0,0,p^2,p^2,0,p^2,0,0,0,0) and Bo for asymptotic property? If
> yes, please let me know.
>
Yes, essentially
Do(0,p^2,0,p^2,p^2,p^2,0,0,0,0) -> (D-6)(D-3) B0(p^2,0,0)/p^4
Do(0,0,p^2,p^2,0,p^2,0,0,0,0) -> (D-4)(D-3) B0(p^2,0,0)/(2 p^4)
Such relations are very easy to obtain with automatic tools like FIRE by
A. Smirnov, see the attached notebook.
Before you ask about the poles. Actually it is the same thing as with
the vertex: The pole of D0 is an IR pole, but the pole B0 is a UV pole.
However, if one does the IBP reduction by hand, one sees that there
are log divegent terms B0(0,0,0) popping up, that are put to zero by
automatic IBP tools like FIRE. If one keeps them, then
again there will be a cancellation such that the UV pole of B0(0,0,0)
cancels the UV pole of B0(p^2,0,0) while the IR pole of B0(0,0,0)
survives in the final result.
At the end, everything comes out consistently in DR, as it should. It is
nice that one can trace where IR and UV poles go
(at least at 1-loop), but this is not really necessary in real life
calculations, provided that the calculation itself is
correct. This is why in most cases people don't really care.
>
> (3)How to avoid the error warning:".....FC Split: Error! Splitting 0
> w,r,t, {PaVe, DenPaVe, Ao,........Co,Do} failed!"? I tried to introduce
> an artificial mass for massless gluon, it does not seem to work. This
> problem suddenly happened today and all calculations are aborted. JP
Looks like a bug that gets detected in the right moment. FeynCalc is
very strict on bugs, so if it sees that something is going wrong, the
rule is that it is better to stop immediately than to return a wrong result.
Could you please send me a minimal working example to reproduce the problem?
Cheers,
Vladyslav
>
> 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/)
>
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