One of the forthcoming major challenges in particle physics is the experimental determination of the Higgs trilinear self-coupling. While efforts have largely focused on on-shell double- and single-Higgs production in proton-proton collisions, off-shell Higgs production has also been proposed as a valuable complementary probe. In this article, we design a hybrid neural simulation-based inference (NSBI) approach to construct a likelihood of the Higgs signal incorporating modifications from the Standard Model effective field theory (SMEFT), relevant background processes, and quantum interference effects. It leverages the training efficiency of matrix-element-enhanced techniques, which are vital for robust SMEFT applications, while also incorporating the practical advantages of classification-based methods for effective background estimates. We demonstrate that our NSBI approach achieves sensitivity close to the theoretical optimum and provide expected constraints for the high-luminosity upgrade of the Large Hadron Collider. While we primarily concentrate on the Higgs trilinear self-coupling, we also consider constraints on other SMEFT operators that affect off-shell Higgs production.

Determining the form of the Higgs potential is one of the most exciting challenges of modern particle physics. Higgs pair production directly probes the Higgs self-coupling and should be observed in the near future at the High-Luminosity LHC. We explore how to improve the sensitivity to physics beyond the Standard Model through per-event kinematics for di-Higgs events. In particular, we employ machine learning through simulation-based inference to estimate per-event likelihood ratios and gauge potential sensitivity gains from including this kinematic information. In terms of the Standard Model Effective Field Theory, we find that adding a limited number of observables can help to remove degeneracies in Wilson coefficient likelihoods and significantly improve the experimental sensitivity.

Studying potential BSM effects at the precision frontier requires accurate transfer of information from low-energy measurements to high-energy BSM models. We propose to use normalising flows to construct likelihood functions that achieve this transfer. Likelihood functions constructed in this way provide the means to generate additional samples and admit a ``trivial'' goodness-of-fit test in form of a $\chi^2$ test statistic. Here, we study a particular form of normalising flow, apply it to a multi-modal and non-Gaussian example, and quantify the accuracy of the likelihood function and its test statistic.

We propose the NFLikelihood, an unsupervised version, based on Normalizing Flows, of the DNNLikelihood proposed in Ref. [1]. We show, through realistic examples, how Autoregressive Flows, based on affine and rational quadratic spline bijectors, are able to learn complicated high-dimensional Likelihoods arising in High Energy Physics (HEP) analyses. We focus on a toy LHC analysis example already considered in the literature and on two Effective Field Theory fits of flavor and electroweak observables, whose samples have been obtained throught the HEPFit code. We discuss advantages and disadvantages of the unsupervised approach with respect to the supervised one and discuss possible interplays of the two.