Hadronic FL wrong behaviour at small $z$ and small $xi$

[niclaurenti]

The hadronic FL (computed integrating adani in dis_tp) shows a wrong behaviour at small-$z$ and small-$xi$.
The following plot shows $z=0.001$ for the bottom FL ($m_b=4.92; GeV$).
FL_massive_comporders_5_0.001.pdf
It is clear that while the NLO and NNLO go to zero as $Q$ approaches 1, the N3LO does not. Moreover, the N3LO has a huge band.
I attach also F2 for reference, where the hadronic structure function goes to zero smoothly
F2_massive_comporders_5_0.001.pdf

After some investigation I found that this is completely due to the asymptotic limit, that is combined with the threshold to form the final approximation: see

adani/src/ApproximateCoefficientFunction.cc

Line 387 in 1444f55

return asy * damp_asy + thresh * damp_thr;

Instead, the asymptotic limit is computed as a combination of high scale ($Q^2 \gg m_b^2$), high energy ($z \rightarrow 0) and the high scale limit of the high energy.
The original idea was to compute
asy = h.s. + (h.e. - h.s.h.e.) (called additive form)
This assures that the asymptotic limit approaches the exact function for $z\rightarrow 0$ and $Q^2 \gg m_b^2$.
This function is what is used in the F2 case.
For FL instead, we saw that
asy=h.s.*h.e./h.e.h.s. (called multiplicative form)
works better.
At NNLO this is fine, but an N3LO, h.e.h.s. can cross the zero. In order to still use the multiplicative form we used

adani/src/AsymptoticCoefficientFunction.cc

Line 269 in 1444f55

Value AsymptoticCoefficientFunction::CL_3_asymptotic(

in which we basically used two modifications of the multiplicative form that prevent the denominator from going to zero.
After some investigation we saw that the modification n 2 is creating the large band, but we still don't understand why the central value does not go to zero for small x and small Q.
Marco proposed that there is some mistake in the high scale limit for FL (either g or q) like a problem in the normalization. But in this case also the NNLO would be wrong (I took NNLO and N3LO from the same paper) but if you take a look at this plot (where black=exact, yellow=h.e., blue dashed=h.s. and blue dotted=h.e.h.s)
N2LO_highenergy_xi=10.pdf
it is clear that everything approaches the correct curve at large eta (small z) (i.e. exact approaches h.e. and h.s. approaches h.e.h.s)