Name: Vladyslav Shtabovenko Date: 02/04/17-01:00:50 PM Z

Hi Pedro,

Am 02.02.2017 um 20:41 schrieb Pedro Ruiz-Femenia:
> Hi Vladyslav,
>
> thanks a lot for the detailed answer.
>
>> As to the second part of your question, the issue here is, that, in >general, Larin’s scheme doesn’t allow one to anticommute gamma^5 >matrices in a trace with an even number of gamma^5. I know that this >is what people usually do in QCD calculations and in most cases this >works out as it should, but nevertheless this trick is not >guaranteed to give correct results in all cases.
>
> I am not very familiar with Larin’s scheme, but I would have naively expected that any gamma5 scheme gives the same output for traces with an even number of gamma5’s, even if the anticommutation of gamma5 is not allowed in such a scheme. But from what you are saying it seems this is not true, right?

from the purely algebraic point of view this is actually not the case.
Consider for example the following trace of 6 Dirac matrices in the
three different schemes

«FeynCalc`

exp = GSD[p].GAD[mu].GA[5].GSD[p + q].GAD[mu].GA[5]

(* NDR*)
$BreitMaison = False;
$Larin = False;
r1 = Tr[exp]

(* BMHV*)
$BreitMaison = True;
$Larin = False;
r2 = (Tr[exp] /. {Momentum[p | q, D - 4] :> 0,
      Momentum[h : (p | q), D] :> Momentum[h, 4]}) // Simplify

(* Larin *)
$BreitMaison = False;
$Larin = True;
r3 = Tr[exp]

For BMHV I’m using the conventional prescription that the external
momenta p and q are purely 4-dimensional. So we get

r1:
-4 (-2 + D) (SPD[p, p] + SPD[p, q])

r2:
4 (-6 + D) (SP[p, p] + SP[p, q])

r3:
2/3 (-6 + D) (-3 + D) (-2 + D) (-1 + D) (SP[p, p] + SP[p, q])

In the limit D=4

r1 /. D -> 4
r2 /. D -> 4
r3 /. D -> 4

all 3 expressions reduce to the same value, as they should

-8 (SP[p, p] + SP[p, q])

However, if the trace is multiplied by a 1/epsilon pole, the results
look quite different

Series[FCReplaceD[(r1/Epsilon), D -> 4 - 2 Epsilon], {Epsilon, 0,
1}]//Normal

8 (SPD[p, p] + SPD[p, q]) - (8 (SPD[p, p] + SPD[p, q]))/Epsilon

Series[FCReplaceD[(r2/Epsilon), D -> 4 - 2 Epsilon], {Epsilon, 0,
1}]//Normal

-8 (SP[p, p] + SP[p, q]) - (8 (SP[p, p] + SP[p, q]))/Epsilon

Series[FCReplaceD[(r3/Epsilon), D -> 4 - 2 Epsilon], {Epsilon, 0,
1}]//Normal

64/3 (SP[p, p] + SP[p, q]) - (8 (SP[p, p] + SP[p, q]))/Epsilon -
  8/3 Epsilon (SP[p, p] + SP[p, q])

Of course one will also have counter-terms, so this is not the end of
the story. In particular, both BMHV and Larin’s scheme usually require
extra counter terms that restore the axial Ward identity that gets
broken in these schemes. NDR respects the axial identity but is
internally inconsistent.

So at the end, depending on what you are computing, you might use
anti-commuting gamma^5 and obtain correct results, but this is not
something you can always rely on.

In fact, in arXiv:hep-ph/9302240 Larin was clearly using anticommuting
gamma^5 on a case by case basis, explaining why this is allowed for each
particular computation. In one of the cases he explicitly says that this
doesn’t work:

“We can not anymore impose for this purpose the coincidence of the axial
and vector vertices (8) because the singlet current generates closed
fermion loops and we cannot anticommute γ 5 outside the singlet axial
vertex.”

I would also like to point out that arXiv:1506.04517 includes a very
detailed and interesting discussion of Larin’s scheme.

So, in my opinion, the general rule for Larin’s scheme is that one can
anticommute g^5 only if one understands why this is allowed in the given
calculation. Otherwise one should better not, although in many cases
people do it anyway.

>
>> On the other hand, I’m always in favor of giving people the right to
>> choose how they do their calculations, as long as they understand the
>> implications. In my view the only acceptable solution here is to
>> implement a second Larin’s scheme with the property that you >request, so that traces with an even number of g^5 will give the >same result as in NDR.
>
> I think it makes sense, since for traces with even number of gamma5’s one does not usually bother about gamma5 issues (modulo some subtlety related to your comment above (?)).
>
>> In fact, this is probably also a good point to improve the way how
>> traces are calculated in Larin’s scheme, as the current performance >is apparently very bad.
>
> Indeed: I am not sure if it is a problem of Larin’s scheme only, but the time needed to compute a trace that has a factor
>
> cv GAD[mu] + ca GAD[mu].GA[5]
>
> inside, is very much reduced is one writes the trace as two separate traces, one with the vector and one with the axial-vector piece.

Could you provide some examples (just the trace) where this happens? I
think it should be possible to optimize DiracTrick for this type of
structures to make things work faster.

Cheers,
Vladyslav

>
>> The only issue is that I quite busy at the moment, so it will take me
>> some time O(2-3 weeks) to take care of that.
>
> That’s perfectly ok, there is no hurry!
>
> Thanks again,
> Pedro
>
> Cheers,
> Vladyslav
>