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===== Impact of aN3LO PDFs on ggF PDF-TH uncertainty =====

==== Comments ====

Pavel: will it be possible to compare the MSHT and NNPDF NNLO and aN3LO luminosities with the |y_X | < 2.5 cut? This is most relevant for the measurements.

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=== Settings ===

**Energies:** 
$\sqrt{s}= 7, 8, 13, \mathbf{13.6}, 14, 33, 100\ \mathrm{TeV}$

**PDF Sets:**
^ NNLO PDFs      ^ aN3LO PDFs        ^
| MSHT           | MSHT              |
| NNPDF4.0       | NNPDF4.0 (prelim) |
| PDF4LHC21      |                   |

**Coefficient Function:**
Always N3LO (even when using NNLO PDFs)

**iHixs Runcard (defines defaults):** 
[[https://phystev.cnrs.fr/wiki/_media/2023:groups:smhiggs:ggf-an3lo:run3.txt|run3.card]] 
[[https://phystev.cnrs.fr/wiki/_media/2023:groups:smhiggs:ggf-an3lo:13.6_125.09_log.txt|log]]
[[https://phystev.cnrs.fr/wiki/_media/2023:groups:smhiggs:ggf-an3lo:13.6_125.09_out.txt|out]]

=== 1. ggF Stacked Error Plots (iHixs) ===

Recreate plots similar to Fig. 2 in [[https://arxiv.org/pdf/1802.00827.pdf|iHixs]], showing only the uncertainties: 
  * $\delta(PDF)$, 
  * $\delta(PDF_{TH})$, 
  * $\delta(\mathrm{scale})$
or (for aN3LO PDFs):
  * $\delta(PDF+PDF_{TH})$, 
  * $\delta(\mathrm{scale})$.
We will omit uncertainties not relevant to the study (namely $\delta(EW)$, $\delta(\alpha_s), \delta(1/m_t), \delta(t,b,c)$.
Note that we can get away with varying both muR & muF instead of only muR (as recommended by MSHT20 aN3LO) since the muF variation is anyway tiny. see e.g. Fig. 2 (top) in [https://arxiv.org/pdf/2209.06138.pdf].

=== 2. Cross Section Tables ===

Main production code is iHixs (validated on HTL with ggH and n3loxs).

Produce tables with:
  * $\sigma$ (total cross section)
  * $\delta(\sigma)$ breakdown of the errors on the cross section (see Study 1 for details)

Comments made by observing preliminary results presented in the initial session: The aN3LO/NNLO k-factor for MSHT ~0.98 and for NNPDF looked ~1 (warning: preliminary). The NNPDF NNLO result is above the MSHT result and so at aN3LO we may see an even larger disagreement.

=== 3. Gluon Lumi Comparison ($\sqrt{s}=13.6\ \mathrm{TeV}$) ===

Ratio plots for PDFs and Luminosities.

^                          ^ Order            ^ Sets          ^
|  $xg(x)$                 | aN3LO/NNLO       | MSHT/MSHT     |
|  :::                     | :::              | NNPDF/NNPDF   |
|  $\mathcal{L}_{gg}$      | aN3LO/NNLO       | MSHT/MSHT     |
|  :::                     | :::              | NNPDF/NNPDF   |
|  $\mathcal{L}_{gg}$      | NNLO/NNLO        | MSHT/PDF4LHC  |
|  :::                     | :::              | NNPDF/PDF4LHC |

The luminosity plots should also be created with $|y_X| < 2.5$ to see the impact of the Higgs rapidity cut.
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