problems with scheme B scale variations

[felixhekhorn]

in pineko we have a problem with scheme B scale variations NNPDF/pineko#19 (we should move pineko to NNPDF to make @giacomomagni able to see it ...)

• our first guess was to remove the scale variations of the strong coupling to "no correction is applied here to the "resummation" scale," [as said by @alecandido] - which we didn't do before (see here ) and indeed we know that for scheme A = LHA paper shifting was correct (since we're matching there); this fix turns out to be not sufficient
• out of fun I tried to invert the scale ratio (xiev -> 1/xiev) and surprisingly enough this brought scheme B and C much closer: from 90% off to mostly O(10%) off
• then I started looking at the scheme B implementation of eko and there is something I don't understand: why is the expanded correction not similar to the exponentiated correction ? i.e. one has beta0 the other not?
• however, looking at Eq. 3.33 of MHOU I can't see an immediate error ...

PS: side note: https://github.com/N3PDF/eko/blob/341e6323e680d2320ae4400da61c92a445160416/doc/source/theory/pQCD.rst#L126
there is no such thing as a PDF set in eko - the "whole operator" is meant

[giacomomagni]

Hi @felixhekhorn, I don't have an answer straight to this issue.
If I recall correctly the difference between the two methods (expanded vs exponentiated) scales at NLO as:
as_0 * \gamma_0 * k when using the fully linearised (truncated) solution. See this test
Can we check this is true also in our case?

The factor beta0 should be compensated by the truncated solution integral which contains a factor 1/beta0

you are right about the docs, I'll implement a fix.

[felixhekhorn]

• most likely a missing minus sign from Eq. 3.33
• in scheme B the starting scale has to be kept fixed (so no effective shifting - in practice we might do and undo the thing); in scheme A the current implementation is correct
• pineko needs to request for scheme B the central scale (and not the shifted one as at the moment of writing is doing); effectively eko is taking fully care of the SV

[giacomomagni]

If recall there were many different signs wrt to the paper,
for instance the beta function expansion was different see eq. 3.3, so and at the end turned out the sign were the same, but it's possible that I missed it.

Edit: Another difference was in 3.28

[alecandido]

(we should move pineko to NNPDF to make @giacomomagni able to see it ...)

What if we made pineko public to begin with?

[giacomomagni]

It's already public! I'm able to see it :)

[giacomomagni]

This should be closed right?

[alecandido]

I definitely think so. @andreab1997 @felixhekhorn ?

[andreab1997]

I definitely think so. @andreab1997 @felixhekhorn ?

Yes we can close this