Advancing Tools for Simulation-Based Inference

The authors report on several advances in the application of Simulation-Based Inference (SBI) tools for particle physics analyses at the LHC. This work builds on the rapid development of these tools (particularly in the "MadMiner" approach) for realistic physics applications, and presents several new breakthroughs in algorithms for extracting the likelihood that improve the speed and reliability of training networks for these inference problems. These developments: fractional smearing, morphing-aware learning, and the novel use of equivariant networks in SBI, are clear breakthroughs in SBI applications, with practical implications for LHC analyses, and present several clear directions for follow-up works and numerous applications. The paper is clearly written, and the key improvements are clearly demonstrated with the use of toy models and a well-chosen example.

My primary questions/comments are with regards to understanding the scope of these improvements, and the possibilities for future applications. While the factorization in Eq. (19) is not limited to only SMEFT examples, this form doesn't necessarily hold when moving beyond leading order. In fact, for exactly the WZ process chosen as an application in this paper, the NLO corrections spoil this form: in contrast to many other diboson/Higgs processes where an NLO K-factor is sufficient, WZ production has a radiation zero that is particularly sensitive to anomalous couplings but which is spoiled by QCD radiation (see arXiv:1909.11576), leading to corrections that change as a function of both the kinematics and the theory parameters.

The authors apply a jet veto in their reconstruction level analysis, so I believe their results in Section 4 are unaffected by this, but perhaps the authors can comment on whether their techniques can be applied to full NLO generation (since this at least in principle now straightforward with MG5 and SMEFT@NLO)? A parton-level analysis is probably ill-defined, but it seems to me that the morphing-aware learning would still help in getting stable results at the reconstruction level, at least compared to derivative learning where the expansion in the theory parameters may be very unstable? Fractional smearing also seems as though it would be especially useful in this case, since it would help interpolate to the amplitude with the additional hard parton. If nothing else, perhaps the authors can comment on whether their methods would provide a reasonable, conservative estimate of the likelihood in the case the factorization assumed at parton level breaks down?

I have a couple of other brief questions on the content of the paper:

• Can the authors elaborate a bit on why the exact form of the loss in Eq. (17) is important? What about the problem changes if a different exponent or computing the MSE of log r as opposed to r is used instead that doesn't result in convergence?

• Can the authors briefly comment why L-GATr only yields marginal improvements at parton level, but significant ones at reconstruction level?

Finally, some trivial edits for a revised version:

• There's a small typo below Eq. (38), the second sentence presumably starts with "Geometric"

• In the discussion below Eq. (45) it's stated that the second and third operators modify the WWZ-coupling, but the third operator modifies the W and Z couplings to quarks.

• There's a sentence fragment in the first paragraph of pg. 17: "The parton shower, the detector resolution, and the missing energy loss information."

Aside from these points, I am happy to recommend the paper for publication in SciPost.

Publish (easily meets expectations and criteria for this Journal; among top 50%)