Gluonic NNLO polarized matching in M scheme

[giacomomagni]

NNLO polarised matching conditions have been introduced in eko in #221, with main reference being https://arxiv.org/pdf/2211.15337.pdf. However according to the authors is still unclear how to convert the gluonic operators:
A_gg, A_gq to the M scheme used for all the other polarised ingredients.
No numerical test or benchmark was possible since NNLO matching is not currently implemented anywhere else.

[felixhekhorn]

• my current understanding is that the M scheme is defined by [1, Eq. (2.12)]
• whether that equation also hold for OME I don't know, but given that this relation is introduced at such a "high" level I expect it to be universal
• we know from [1, below Eq. (2.17)] $z_{gg} = z_{qg} = 0$ so I wonder if there is any additional transformation needed. To be specific: if OMEs follow a similar transformation as splitting functions (which might be reasonable) we can conclude from [1, Eq. (2.19)] that $A_{gg,M}^2= A_{gg,L}^2$

[1] https://arxiv.org/pdf/1409.5131.pdf

[giacomomagni]

• we know from [1, below Eq. (2.17)] zgg=zqg=0 so I wonder if there is any additional transformation needed. To be specific: if OMEs follow a similar transformation as splitting functions (which might be reasonable) we can conclude from [1, Eq. (2.19)] that Agg,M2=Agg,L2

Actually we might have some more hints. By looking at eq 85 and 86 of [1], you can see that they are as
eq 26 and 27 of [2], apart of a factor 4 \beta_0 ?!?
Unfortunately the transformation of A_qqNS is not written explicitly.

So there are good chances that A_gg transforms under an identity
but for according to Kay Schönwald A_{gq} is not correct.

[1] https://arxiv.org/pdf/2211.15337.pdf
[2] https://arxiv.org/pdf/2111.12401.pdf

[giacomomagni]

Actually inferring from this https://arxiv.org/pdf/2405.17252, the authors must know how to translate Larin scheme to M-scheme also for gluon distribution. Should we contact them?

[felixhekhorn]

yes, maybe this is the simplest way to resolve this issue ...

[giacomomagni]

Okay despite having written to M.Saragnese, I got a reply from B. 😬
He suggested us to perform the evolution in the Larin scheme and pointed out
the paper we already know.

But in the meantime, he has also linked this thesis (which I was not aware of):

https://inspirehep.net/files/f3ef75681dafc5d9982f2d4ace768c7d

From eq 4.31 and 4.32 I now understand that, up to NNLO, the only non vanishing entries
of the scheme transformation are the quark-to-quark NS and PS entries, which should already be
implemented in our code.
I asked confirmation of this understanding, let's see if we get another reply...

[Radonirinaunimi]

Oooh, those could be the relations we missed this morning indeed...

[giacomomagni]

Plot twist:

Dear Mr Magni,

Your conclusion is not true. The z_ij's apply to massless Wilson
coefficients and anomalous dimensions only. The case of massive OMEs
is more involved.

Best regards,J Bluemlein