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Hard-jet correlations probe parton energy loss and the microscopic structure of the quark-gluon plasma formed in ultra-relativistic heavy-ion collisions. The correlation of high-$p_\mathrm{T}$ jets with other jets, hadrons, or electroweak bosons, offers differential sensitivity to medium-induced effects such as momentum broadening, color decoherence, and medium response in different types of nuclear reactions. Such correlations can also be used to study cold nuclear matter effects arising in $p$+A collisions. This proceeding summarizes recent advances achieved by studying hard-jet correlations in large and small systems discussed at Hard Probes 2024, complementing the experimental jet overview.

Let $G$ be a complete edge-weighted graph on $n$ vertices. To each subset of vertices of $G$ assign the cost of the minimum spanning tree of the subset as its weight. Suppose that $n$ is a multiple of some fixed positive integer $k$. The $k$-matching problem is the problem of finding a partition of the vertices of $G$ into $k$-sets, that minimizes the sum of the weights of the $k$-sets. The case $k=3$ has been shown to be NP-hard [Johnsson et al.,1998]. In the Euclidean version, the vertices of $G$ are points in the plane and the weight of an edge is the Euclidean distance between its endpoints. We call this problem the Euclidean $k$-matching problem. We show that, for every fixed $k \ge 3$, the Euclidean $k$-matching is NP-hard. This resolves an open problem in the literature and provides the first theoretical justification for the use of known heuristic methods in the case $k=3$. We also show that the problem remains NP-hard if the trees are required to be paths.

Contact forces introduce discontinuities into robot dynamics that severely limit the use of simulators for gradient-based optimization. Penalty-based simulators such as MuJoCo, soften contact resolution to enable gradient computation. However, realistically simulating hard contacts requires stiff solver settings, which leads to incorrect simulator gradients when using automatic differentiation. Contrarily, using non-stiff settings strongly increases the sim-to-real gap. We analyze penalty-based simulators to pinpoint why gradients degrade under hard contacts. Building on these insights, we propose DiffMJX, which couples adaptive time integration with penalty-based simulation to substantially improve gradient accuracy. A second challenge is that contact gradients vanish when bodies separate. To address this, we introduce contacts from distance (CFD) which combines penalty-based simulation with straight-through estimation. By applying CFD exclusively in the backward pass, we obtain informative pre-contact gradients while retaining physical realism.

According to the "hard-steps" model, the origin of humanity required "successful passage through a number of intermediate steps" (so-called "hard" or "critical" steps) that were intrinsically improbable with respect to the total time available for biological evolution on Earth. This model similarly predicts that technological life analogous to human life on Earth is "exceedingly rare" in the universe. Here, we critically reevaluate the core assumptions of the hard-steps model in light of recent advances in the Earth and life sciences. Specifically, we advance a potential alternative model where there are no hard steps, and evolutionary novelties (or singularities) required for human origins can be explained via mechanisms outside of intrinsic improbability. Furthermore, if Earth's surface environment was initially inhospitable not only to human life, but also to certain key intermediate steps in human evolution (e.g., the origin of eukaryotic cells, multicellular animals), then the "delay" in the appearance of humans can be best explained through the sequential opening of new global environmental windows of habitability over Earth history, with humanity arising relatively quickly o

Overview of recent theoretical advances presented at Hard Probes 2024 in the study of hard-soft correlation in heavy ion collisions and small systems.

The hot and dense QCD matter, known as the Quark-Gluon Plasma (QGP), is explored through heavy-ion collision experiments at the LHC and RHIC. Jets and heavy flavors, produced from the initial hard scattering, are used as hard probes to study the properties of the QGP. Recent experimental observations on jet quenching and heavy-flavor suppression have strengthened our understanding, allowing for fine-tuning of theoretical models in hard probes. The second conference, HOT QCD Matter 2024, was organized to bring the community together for discussions on key topics in the field. This article comprises 15 sections, each addressing various aspects of hard probes in relativistic heavy-ion collisions, offering a snapshot of current experimental observations and theoretical advancements. The article begins with a discussion on memory effects in the quantum evolution of quarkonia in the quark-gluon plasma, followed by an experimental review, new insights on jet quenching at RHIC and LHC, and concludes with a machine learning approach to heavy flavor production at the Large Hadron Collider.

A mini-review on the status of global analyses of nuclear parton distribution functions is given, focusing on the most relevant constraints for the hard-probes phenomenology in ultra-relativistic heavy-ion collisions.

The strength of modern generative models lies in their ability to be controlled through text-based prompts. Typical "hard" prompts are made from interpretable words and tokens, and must be hand-crafted by humans. There are also "soft" prompts, which consist of continuous feature vectors. These can be discovered using powerful optimization methods, but they cannot be easily interpreted, re-used across models, or plugged into a text-based interface. We describe an approach to robustly optimize hard text prompts through efficient gradient-based optimization. Our approach automatically generates hard text-based prompts for both text-to-image and text-to-text applications. In the text-to-image setting, the method creates hard prompts for diffusion models, allowing API users to easily generate, discover, and mix and match image concepts without prior knowledge on how to prompt the model. In the text-to-text setting, we show that hard prompts can be automatically discovered that are effective in tuning LMs for classification.

In order for quantum computations to be done as efficiently as possible it is important to optimise the number of gates used in the underlying quantum circuits. In this paper we find that many gate optimisation problems for approximately universal quantum circuits are NP-hard. In particular, we show that optimising the T-count or T-depth in Clifford+T circuits, which are important metrics for the computational cost of executing fault-tolerant quantum computations, is NP-hard by reducing the problem to Boolean satisfiability. With a similar argument we show that optimising the number of CNOT gates or Hadamard gates in a Clifford+T circuit is also NP-hard. Again varying the same argument we also establish the hardness of optimising the number of Toffoli gates in a reversible classical circuit. We find an upper bound to the problems of T-count and Toffoli-count of $\text{NP}^{\text{NQP}}$. Finally, we also show that for any non-Clifford gate $G$ it is NP-hard to optimise the $G$-count over the Clifford+$G$ gate set, where we only have to match the target unitary within some small distance in the operator norm.

We highlight the STAR experiment's hard probes results, including jets and heavy flavor production in heavy-ion collisions to study the properties of the quark-gluon plasma. Various jet-substructure observables in proton-proton collisions are presented to explore both perturbative and non-perturbative regimes of quantum chromodynamics. Finally, we discuss the STAR experiment's hard probes physics program for ongoing data taking.

The in-medium dispersion of hard partons, encoded in their so-called asymptotic mass, receives large non-perturbative contributions from classical gluons, i.e. soft gluons with large occupation numbers. Here, we discuss how the analytical properties of thermal amplitudes allow for a non-perturbative determination of the infrared classical contribution through lattice determinations in the dimensionally-reduced effective theory of hot QCD, EQCD. We show how these lattice determinations need to be complemented by perturbative two-loop matching calculations between EQCD and QCD, so that the unphysical (classical) ultraviolet behavior of EQCD is replaced by its proper quantum QCD counterpart. We show how lattice and perturbative EQCD are in good agreement in the UV and present an outlook on the two-loop quantum QCD contribution.

Preference analysis is widely applied in various domains such as social choice and e-commerce. A recently proposed framework augments the relational database with a preference relation that represents uncertain preferences in the form of statistical ranking models, and provides methods to evaluate Conjunctive Queries (CQs) that express preferences among item attributes. In this paper, we explore the evaluation of queries that are more general and harder to compute. The main focus of this paper is on a class of CQs that cannot be evaluated by previous work. These queries are provably hard since relate variables that represent items being compared. To overcome this hardness, we instantiate these variables with their domain values, rewrite hard CQs as unions of such instantiated queries, and develop several exact and approximate solvers to evaluate these unions of queries. We demonstrate that exact solvers that target specific common kinds of queries are far more efficient than general solvers. Further, we demonstrate that sophisticated approximate solvers making use of importance sampling can be orders of magnitude more efficient than exact solvers, while showing good accuracy. In ad

The CERN SuperProtoSynchrotron (SPS) represents an ideal facility for fixed-target heavy-ion experiments exploring the phase diagram of strongly interacting matter in the region $200\leμ_{\rm B}\le500$ MeV. It can deliver high-intensity beams ($>10^6$ Pb/s), allowing a study of rare probes of the Quark-Gluon Plasma, including electromagnetic and hard processes. The NA61/SHINE experiment is currently active and plans to perform a first direct measurement of open charm production in Pb--Pb collisions at top SPS energy and possibly at lower energies. The project of a new experiment, NA60+, based on a muon spectrometer coupled to a vertex spectrometer is currently being developed, for the study of dimuon and heavy quark production, and a Letter of Intent was recently submitted. In this contribution the physics motivation for the studies of rare probes, the existing and planned experimental set-ups and their expected physics performance will be discussed.

We introduce a highly structured family of hard satisfiable 3-SAT formulas corresponding to an ordered spin-glass model from statistical physics. This model has provably "glassy" behavior; that is, it has many local optima with large energy barriers between them, so that local search algorithms get stuck and have difficulty finding the true ground state, i.e., the unique satisfying assignment. We test the hardness of our formulas with two Davis-Putnam solvers, Satz and zChaff, the recently introduced Survey Propagation (SP), and two local search algorithms, Walksat and Record-to-Record Travel (RRT). We compare our formulas to random 3-XOR-SAT formulas and to two other generators of hard satisfiable instances, the minimum disagreement parity formulas of Crawford et al., and Hirsch's hgen. For the complete solvers the running time of our formulas grows exponentially in sqrt(n), and exceeds that of random 3-XOR-SAT formulas for small problem sizes. SP is unable to solve our formulas with as few as 25 variables. For Walksat, our formulas appear to be harder than any other known generator of satisfiable instances. Finally, our formulas can be solved efficiently by RRT but only if the pa

Variable selection in linear models plays a pivotal role in modern statistics. Hard-thresholding methods such as $l_0$ regularization are theoretically ideal but computationally infeasible. In this paper, we propose a new approach, called the LAGS, short for "least absulute gradient selector", to this challenging yet interesting problem by mimicking the discrete selection process of $l_0$ regularization. To estimate $β$ under the influence of noise, we consider, nevertheless, the following convex program [\hatβ = \textrm{arg min}\frac{1}{n}\|X^{T}(y - Xβ)\|_1 + λ_n\sum_{i = 1}^pw_i(y;X;n)|β_i|] $λ_n > 0$ controls the sparsity and $w_i > 0$ dependent on $y, X$ and $n$ is the weights on different $β_i$; $n$ is the sample size. Surprisingly, we shall show in the paper, both geometrically and analytically, that LAGS enjoys two attractive properties: (1) LAGS demonstrates discrete selection behavior and hard thresholding property as $l_0$ regularization by strategically chosen $w_i$, we call this property "pseudo-hard thresholding"; (2) Asymptotically, LAGS is consistent and capable of discovering the true model; nonasymptotically, LAGS is capable of identifying the sparsity in th

This manuscript is the outcome of the subgroup ``PDFs, shadowing and $pA$ collisions'' from the CERN workshop ``Hard Probes in Heavy Ion Collisions at the LHC''. In addition to the experimental parameters for $pA$ collisions at the LHC, the issues discussed are factorization in nuclear collisions, nuclear parton distributions (nPDFs), hard probes as the benchmark tests of factorization in $pA$ collisions at the LHC, and semi-hard probes as observables with potentially large nuclear effects. Also, novel QCD phenomena in $pA$ collisions at the LHC are considered. The importance of the $pA$ program at the LHC is emphasized.

The question of whether the known virial coefficients are enough to determine the packing fraction $η_\infty$ at which the fluid equation of state of a hard-sphere fluid diverges is addressed. It is found that the information derived from the direct Padé approximants to the compressibility factor constructed with the virial coefficients is inconclusive. An alternative approach is proposed which makes use of the same virial coefficients and of the equation of state in a form where the packing fraction is explicitly given as a function of the pressure. The results of this approach both for hard-disk and hard-sphere fluids, which can straightforwardly accommodate higher virial coefficients when available, lends support to the conjecture that $η_\infty$ is equal to the maximum packing fraction corresponding to an ordered crystalline structure.

Shadow removal from a single image is generally still an open problem. Most existing learning-based methods use supervised learning and require a large number of paired images (shadow and corresponding non-shadow images) for training. A recent unsupervised method, Mask-ShadowGAN~\cite{Hu19}, addresses this limitation. However, it requires a binary mask to represent shadow regions, making it inapplicable to soft shadows. To address the problem, in this paper, we propose an unsupervised domain-classifier guided shadow removal network, DC-ShadowNet. Specifically, we propose to integrate a shadow/shadow-free domain classifier into a generator and its discriminator, enabling them to focus on shadow regions. To train our network, we introduce novel losses based on physics-based shadow-free chromaticity, shadow-robust perceptual features, and boundary smoothness. Moreover, we show that our unsupervised network can be used for test-time training that further improves the results. Our experiments show that all these novel components allow our method to handle soft shadows, and also to perform better on hard shadows both quantitatively and qualitatively than the existing state-of-the-art sha

The description of the initial state of heavy ion collisions, which covers the description of the incoming nuclei, the initial hard and soft interactions, the resulting spatial geometry of the produced matter, as well as the dynamic approach to a medium well described by hydrodynamics, has important consequences for the study of hard and electromagnetic probes. I will review new developments presented at Hard Probes 2020 that have an impact on these aspects of our understanding of the initial state of heavy ion and smaller system collisions.