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The integration of deep learning techniques and physics-driven designs is reforming the way we address inverse problems, in which accurate physical properties are extracted from complex data sets. This is particularly relevant for quantum chromodynamics (QCD), the theory of strong interactions, with its inherent limitations in observational data and demanding computational approaches. This perspective highlights advances and potential of physics-driven learning methods, focusing on predictions of physical quantities towards QCD physics, and drawing connections to machine learning(ML). It is shown that the fusion of ML and physics can lead to more efficient and reliable problem-solving strategies. Key ideas of ML, methodology of embedding physics priors, and generative models as inverse modelling of physical probability distributions are introduced. Specific applications cover first-principle lattice calculations, and QCD physics of hadrons, neutron stars, and heavy-ion collisions. These examples provide a structured and concise overview of how incorporating prior knowledge such as symmetry, continuity and equations into deep learning designs can address diverse inverse problems across different physical sciences., Comment: 14 pages, 5 figures, submitted version to Nat Rev Phys

I make a brief review about the QCD phases and the equation of state inferred from the neutron star data. Along the temperature axis at low baryon density, the QCD phase transition is a smooth crossover, and it is a natural extension of our imagination to postulate a similar crossover along the density axis at low temperature. Even without phase transitions, the inferred thermodynamic properties of neutron star matter turn out to be highly nontrivial already at twice of the nuclear saturation density. I also give some discussions about the substantiation of quark matter by means of the gravitational wave signals including the multi-messenger prospect., Comment: 8 pages, 7 figures; a proceedings contribution to Aspects of Criticality II, Wroclaw, Poland

Magnetic fields are ubiquitous across different physical systems of current interest; from the early Universe, compact astrophysical objects and heavy-ion collisions to condensed matter systems. A proper treatment of the effects produced by magnetic fields during the dynamical evolution of these systems, can help to understand observables that otherwise show a puzzling behavior. Furthermore, when these fields are comparable to or stronger than \Lambda_QCD, they serve as excellent probes to help elucidate the physics of strongly interacting matter under extreme conditions of temperature and density. In this work we provide a comprehensive review of recent developments on the description of QED and QCD systems where magnetic field driven effects are important. These include the modification of meson static properties such as masses and form factors, the chiral magnetic effect, the description of anomalous transport coefficients, superconductivity in extreme magnetic fields, the properties of neutron stars, the evolution of heavy-ion collisions, as well as effects on the QCD phase diagram. We describe recent theory and phenomenological developments using effective models as well as LQCD methods. The work represents a state-of-the-art review of the field, motivated by presentations and discussions during the "Workshop on Strongly Interacting Matter in Strong Electromagnetic Fields" that took place in the European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*) in the city of Trento, Italy, September 25-29, 2023., Comment: 325 pages-long review of recent topics of interest in the field of magnetic field effects on QED and QCD matter. To be susbmitted to PNPP

This is a pedagogical review of the possible connection between the stochastic quantization in physics and the diffusion models in machine learning. For machine-learning applications, the denoising diffusion model has been established as a successful technique, which is formulated in terms of the stochastic differential equation (SDE). In this review, we focus on an SDE approach used in the score-based generative modeling. Interestingly, the evolution of the probability distribution is equivalently described by a particular class of SDEs, and in a particular limit, the stochastic noises can be eliminated. Then, we turn to a similar mathematical formulation in quantum physics, that is, the stochastic quantization. We make a brief overview on the stochastic quantization using a simple toy model of the one-dimensional integration. The analogy between the diffusion model and the stochastic quantization is clearly seen in this concrete example. Finally, we discuss how the sign problem arises in the toy model with complex parameters. The origin of the difficulty is understood based on the Lefschetz thimble analysis. We point out that the SDE is not invariant under the variable change which induces a kernel and a special choice of the kernel guided by the Lefschetz thimble analysis can reduce the sign problem., Comment: 10 pages, 13 figures: Contribution to Special Topics in J.Phys.Soc.Jpn

We present analytical and numerical calculations for the photon polarization tensor at finite temperature and density in a constant magnetic field. We first discuss the tensor decomposition in the presence of the magnetic field which breaks rotational symmetry. Then, we analytically perform all the momentum integrations and numerically take the Landau level sum. We present the real and imaginary parts of the photon polarization tensor as functions of the momenta, the chemical potential, and the finite temperature. As an application, we consider the real photon limit and estimate the photon decay rate in the hot and dense medium. We specifically quantify the difference between the X-mode and the O-mode with the polarization orthogonal and parallel to the magnetic field. As long as the magnetic field is weak, the decay rate of the X-mode photon is larger than that of the O-mode photon, while the O-mode becomes dominant due to the Landau level suppression of the X-mode at strong magnetic field., Comment: 28 pages, 13 figures

In a unified perturbative treatment from the high-density side, we compute the speed of sound and the trace anomaly as functions of the chemical potential $\mu$ for the two-color diquark superfluid, the pion-condensed high-isospin matter, and the 2SC quark matter. We find that the corrections induced by the gap energy $\Delta$ involve nontrivial interplay between the dimensionless magnitude $|\Delta/\mu|$ and the derivative $|\partial\Delta/\partial\mu|$. Even though the gap equation has a common structure for these phases of our interest, different numerical constants cause drastic changes in the speed of sound corrections. As long as $|\Delta/\mu|$ is dominant over the derivative, the gap effects increase the speed of sound, which is consistent with the expected behavior of exhibiting a peak at intermediate density. We then discuss the trace anomaly which is pushed down generally by the gap effects and turns out to be negative widely in the high density regime. For demonstration of non-perturbative enhancement, we take account of the instanton-induced interaction. We confirm a further increase in the speed of sound for the two-color diquark superfluid and the pion-condensed high-isospin matter, while the speed of sound for the 2SC quark matter deviates far below the conformal limit due to the derivative contribution., Comment: 12 pages, 12 figures

We establish thermodynamic stability and gauge invariance in the magnetovortical matter of Dirac fermions under the coexistent rotation and strong magnetic field. The corresponding partition function clarifies the importance of orbital angular momentum to bulk thermodynamics in addition to the conventional contribution from anomaly-related spin effects. In particular, we make an experimentally testable prediction that the orbital contribution should preponderate over the spin contribution, and thus flip the sign of the induced charge and current in the magnetovortical matter when the magnetic field strength is increased., Comment: 6 pages (main) + 5 pages (supplemental), 1 figure

We study observational signatures of the hadron-quark crossover in binary-neutron-star mergers by systematic numerical-relativity simulations. We employ two equations of state (EoSs) for matter consistent with inference from the observational data. In the crossover scenario the EoS is softened in a density realized in binary-neutron-star mergers and is smoothly continued to quark matter. In the phase transition scenario without crossover, the EoS remains stiff and a first-order phase transition takes place in a density out of reach of mergers. A GW170817-like system forms a remnant massive neutron star in both scenarios, and only the crossover scenario makes it collapse into a black hole due to the softening while gravitational-wave emission is strong. This difference is clearly reflected in the sudden shutdown of gravitational waves. For a given EoS, the lifetime of the merger remnant is determined primarily by the total mass of the system. These features pave the way for clarifying details of hadron-quark transition by compiling a wide variety of events observed with future gravitational-wave detectors. The mass of the accretion disk surrounding the remnant black hole is affected not only by the lifetime of the remnant but also by the mass ratio of the system. Electromagnetic emission associated with the disk outflow will also be useful for detailed investigation of the hadron-quark transition., Comment: 25 pages, 17 figures

We investigate detailed properties of imaginary rotating matter with gluons and quarks at high temperature. Previously, we showed that imaginary rotation induces perturbative confinement of gluons at the rotation center. We perturbatively calculate the Polyakov loop potential and find inhomogeneous confinement above a certain threshold of imaginary angular velocity. We also evaluate the quark contribution to the Polyakov loop potential and confirm that spontaneous chiral symmetry breaking occurs in the perturbatively confined phase., Comment: 15 pages, 12 figures

Score-based diffusion models have proven effective in image generation and have gained widespread usage; however, the underlying factors contributing to the performance disparity between stochastic and deterministic (i.e., the probability flow ODEs) sampling schemes remain unclear. We introduce a novel formulation of diffusion models using Feynman's path integral, which is a formulation originally developed for quantum physics. We find this formulation providing comprehensive descriptions of score-based generative models, and demonstrate the derivation of backward stochastic differential equations and loss functions.The formulation accommodates an interpolating parameter connecting stochastic and deterministic sampling schemes, and we identify this parameter as a counterpart of Planck's constant in quantum physics. This analogy enables us to apply the Wentzel-Kramers-Brillouin (WKB) expansion, a well-established technique in quantum physics, for evaluating the negative log-likelihood to assess the performance disparity between stochastic and deterministic sampling schemes., Comment: 27 pages, 14 figures

We discuss the machine-learning inference and uncertainty quantification for the equation of state (EoS) of the neutron star (NS) matter directly using the NS probability distribution from the observations. We previously proposed a prescription for uncertainty quantification based on ensemble learning by evaluating output variance from independently trained models. We adopt a different principle for uncertainty quantification to confirm the reliability of our previous results. To this end, we carry out the MC sampling of data to infer an EoS and take the convolution with the probability distribution of the observational data. In this newly proposed method, we can deal with arbitrary probability distribution not relying on the Gaussian approximation. We incorporate observational data from the recent multimessenger sources including precise mass measurements and radius measurements. We also quantify the importance of data augmentation and the effects of prior dependence., Comment: 34 pages, 2 tables, 13 figures, v2: minor correction, references added

We discuss a novel mechanism for the proto-neutron star acceleration assisted by the chiral separation effect which induces an axial vector current in a dense medium. We consider the process of neutrinos scattering off the background axial vector current of electrons. We show that anisotropy of either magnetic field or density in momentum space is essential for nonzero recoil and we call this mechanism the chiral anisotropy conversion. Assuming a strong magnetic field $B \simeq 10^{12}$ T and anisotropy by $\sim 10\%$, we find that the chiral anisotropy conversion can yield the velocity of order of typical pulsar kicks, i.e., $v_{\mathrm{kick}} \gtrsim 1000$ km/s., Comment: 6 pages, 4 figures

We analyze the helicity conservation following from the axial Ward identity for the massless fermion. We discuss the pair production of a fermion, $\ell$, and an anti-fermion, $\bar{\ell}$, from the annihilation process of photons, $\gamma$'s and conclude that the chirality production is prohibited in any annihilation process of on-shell $\gamma$'s. We demonstrate this selection rule in the tree-level process of $\gamma+\gamma\to \ell+\bar{\ell}$. We next consider the off-shell effect on the chirality production in the presence of the background magnetic field. We point out that the particle conversion process under the magnetic field induces the polarization of an incident $\gamma$, which can also be interpreted intuitively from the helicity conservation., Comment: 9 pages, 4 figures

We estimate the nuclear saturation density and the binding energy in a nuclear liquid from precision data on the coupling of the four-quark scattering vertex in the vector channel, computed within functional QCD. We show that this coupling is directly related to the density-density potential and the latter is used for the estimates. In a first qualitative computation we find a saturation density of 0.2 fm${}^{-3}$ and an upper bound for the binding energy of 21.5 MeV, in agreement with the empirical values of 0.16 fm${}^{-3}$ and 16 MeV, respectively. We also use the scattering vertex for constructing an emergent low-energy effective theory for the liquid gas transition from QCD correlation function, whose coupling parameters can be determined within QCD. As a first consistency check of this construction we estimate the in-medium reduction of the nucleon pole mass., Comment: 13 pages, 4 figures

We propose a novel approach to probe primordial inhomogeneity in hot and dense matter which could be realized in non-central heavy-ion collisions. Although the Hanbury Brown and Twiss (HBT) interferometry is commonly used to infer the system size, the cluster size should be detected if substructures emerge in space. We demonstrate that a signal peak in the HBT two-particle correlation stands at the relative momentum corresponding to the spatial scale of pseudo one-dimensional modulation. We assess detectability using the data prepared by an event generator (AMPT model) with clustering implemented in the particle distribution., Comment: 5 pages, 4 figures

We investigate the chiral anomaly in a Floquet system under a time-periodic electric field in (1+1) dimensions. Using the van~Vleck high-frequency expansion, we analytically calculate the chiral current and the pseudo-scalar condensate for massless/massive fermions and how they are balanced with the topological charge. In the high-frequency limit, we find that finite-mass effects are suppressed and the topological charge is dominated by the chirality production. Our calculations show that the information about the chiral anomaly is stored not in the static Floquet Hamiltonian but in the periodic kick operator. The computational steps are useful as the theoretical foundation for higher-dimensional generalization., Comment: 29 pages; v2: minor changes, to appear in Annals of Physics

We discuss the baryon properties under a strong magnetic field. We adopt the Skyrme model and calculate the magnetic field dependence of the mass and the pressure distribution in the soliton. We elucidate a magnetically induced contribution to the pressure sum rule and interpret it as an extra confining force. We also quantize the soliton to estimate the difference between the proton and the neutron and find a simple relation between the pressure and the mass differences., Comment: 15 pages, 4 figures

This paper presents a comprehensive review of both the theory and experimental successes of Quantum Chromodynamics, starting with its emergence as a well defined theory in 1972-73 and following developments and results up to the present day. Topics include a review of the earliest theoretical and experimental foundations; the fundamental constants of QCD; an introductory discussion of lattice QCD, the only known method for obtaining exact predictions from QCD; methods for approximating QCD, with special focus on effective field theories; QCD under extreme conditions; measurements and predictions of meson and baryon states; a special discussion of the structure of the nucleon; techniques for study of QCD at high energy, including treatment of jets and showers; measurements at colliders; weak decays and quark mixing; and a section on the future, which discusses new experimental facilities or upgrades currently funded. The paper is intended to provide a broad background for Ph.D. students and postdocs starting their career. Some contributions include personal accounts of how the ideas or experiments were developed., Comment: Invited volume for the EJPC; 567 pages if text/figures and 4783 references occupying about 160 additional pages. arXiv abstract abridged, for the complete abstract please see the full text

We perturbatively compute the Polyakov loop potential at high temperature with finite imaginary angular velocity. This imaginary rotation does not violate the causality and the thermodynamic limit is well defined. We analytically show that the imaginary angular velocity induces the perturbatively confined phase and serves as a new probe to confinement physics. We discuss a possible phase diagram that exhibits adiabatic continuity from the perturbative confinement to the confined phase at low temperature. We also mention subtlety in the analytical continuation from imaginary to real angular velocity by imposing a causality bound., Comment: 6 pages, 5 figures; the conjectured phase diagram modified for our adiabatic continuity scenario, typos corrected, minor elaborations

We discuss an interpretation that a peak in the sound velocity in neutron star matter, as suggested by the observational data, signifies strongly-coupled conformal matter. The normalized trace anomaly is a dimensionless measure of conformality leading to the derivative and the non-derivative contributions to the sound velocity. We find that the peak in the sound velocity is attributed to the derivative contribution from the trace anomaly that steeply approaches the conformal limit. Smooth continuity to the behavior of high-density QCD implies that the matter part of the trace anomaly may be positive definite. We discuss a possible implication of the positivity condition of the trace anomaly on the $M$-$R$ relation of the neutron stars., Comment: 6 pages, 3 figures; accepted in PRL

The cores of neutron stars (NSs) near the maximum mass realize the most highly compressed matter in the universe where quark degrees of freedom may be liberated. Such a state of dense matter is hypothesized as quark matter (QM) and its presence has awaited to be confirmed for decades in nuclear physics. Gravitational waves from binary NS mergers are expected to convey useful information called the equation of state (EOS). However, the signature for QM with realistic EOS is not yet established. Here, we show that the gravitational wave in the post-merger stage can distinguish the theory scenarios with and without a transition to QM. Instead of adopting specific EOSs as studied previously, we compile reliable EOS constraints from the ab initio approaches. We demonstrate that early collapse to a black hole after NS merger signifies softening of the EOS associated with the onset of QM in accord with ab initio constraints. Nature of hadron-quark phase transition can be further constrained by the condition that electromagnetic counterparts need to be energized by the material left outside the remnant black hole., Comment: 20 pages, 5 figures

We investigate the $\theta$-vacuum structure and the 't Hooft anomaly at $\theta=\pi$ in a simple quantum mechanical system on $S^1$ to scrutinize the applicability of the functional renormalization group (fRG) approach. Even though the fRG is an exact formulation, a naive application of the fRG equation would miss contributions from the $\theta$ term due to the differential nature of the formulation. We first review this quantum mechanical system on $S^1$ that is solvable with both the path integral and the canonical quantization. We discuss how to construct the quantum effective action including the $\theta$ dependence. Such an explicit calculation poses a subtle question of whether a Legendre transform is well defined or not for general systems with the sign problem. We then consider a deformed theory to relax the integral winding by introducing a wine-bottle potential with the finite depth $\propto g$, so that the original $S^1$ theory is recovered in the $g\to\infty$ limit. We numerically solve the energy spectrum in the deformed theory as a function of $g$ and $\theta$ in the canonical quantization. We test the efficacy of the simplest local potential approximation (LPA) in the fRG approach and find that the correct behavior of the ground state energy is well reproduced for small $\theta$. When the energy level crossing is approached, the LPA flow breaks down and fails in describing the ground state degeneracy expected from the 't Hooft anomaly. We finally turn back to the original theory and discuss an alternative formulation using the Villain lattice action. The analysis with the Villain lattice at $\theta=\pi$ indicates that the nonlocality of the effective action is crucial to capture the level crossing behavior of the ground states., Comment: 30 pages, 6 figures

We propose an interpolating equation of state that satisfies phenomenologically established boundary conditions in two extreme regimes at high temperature and low baryon density and at low temperature and high baryon density. We confirm that the hadron resonance gas model with the Carnahan-Starling excluded volume effect can reasonably fit the empirical equation of state at high density up to several times the normal nuclear density. We identify the onsets of strange particles and quantify the strangeness contents in dense matter. We finally discuss the finite temperature effects and estimate the thermal index $\Gamma_{\rm th}$ as a function of the baryon density, which should be a crucial input for the core-collapse supernova and the binary neutron star merger simulations., Comment: 9 pages, 6 figures; v2 published in PLB

We calculate the electric conductivity $\sigma$ in deconfined QCD matter using a holographic QCD model, i.e., the Sakai-Sugimoto Model with varying magnetic field $B$ and chiral anomaly strength. After confirming that our estimated $\sigma$ for $B=0$ is consistent with the lattice-QCD results, we study the case with $B\neq 0$ in which the coefficient $\alpha$ in the Chern-Simons term controls the chiral anomaly strength. Our results imply that the transverse conductivity, $\sigma_\perp$, is suppressed to be $\lesssim 70\%$ at $B\sim 1\,\mathrm{GeV}^2$ as compared to the $B=0$ case when the temperature is fixed as $T= 0.2\,\mathrm{GeV}$. Since the Sakai-Sugimoto Model has massless fermions, the longitudinal conductivity, $\sigma_\parallel$, with $B\neq 0$ should diverge due to production of the matter chirality. Yet, it is possible to extract a regulated part out from $\sigma_\parallel$ with an extra condition to neutralize the matter chirality. This regulated quantity is interpreted as an Ohmic part of $\sigma_\parallel$. We show that the longitudinal Ohmic conductivity increases with increasing $B$ for small $\alpha$, while it is suppressed with larger $B$ for physical $\alpha=3/4$ due to anomaly induced interactions., Comment: 9 pages, 2 figures; references added

We elucidate magnetic effects in the skyrmion system to probe into the dense nuclear matter. We find a deformed $\pi^0$ dipole structure of a baryon induced by a magnetic field. We then extend our scope to stacked Skyrme crystal layers to scrutinize phases of magnetized nuclear matter. We observe a quantized magnetic flux and identify a phase transition from a crystalline state to a $\pi^0$ domain wall corresponding to a topological transmutation from $\pi_3(S^3)$ to $\pi_1(S^1)$. We establish the phase diagram, which could be explored also in analogous systems with two-component Bose-Einstein condensates., Comment: 6 pages, 5 figures

We discuss the emergence of a low-energy effective theory with quarks, mesons, diquarks and baryons at vanishing and finite baryon density from first principle QCD. The present work also includes an overview on diquarks at vanishing and finite density, and elucidates the physics of transitional changes from baryonic matter to quark matter including diquarks. This set-up is discussed within the functional renormalisation group approach with dynamical hadronisation. In this framework it is detailed how mesons, diquarks, and baryons emerge dynamically from the renormalisation flow of the QCD effective action. Moreover, the fundamental degrees of freedom of QCD, quarks and gluons, decouple from the dynamics of QCD below the respective mass gaps. The resulting global picture unifies the different low energy effective theories used for low and high densities within QCD, and allows for a determination of the respective low energy constants directly from QCD., Comment: 30 pages, 17 figures

We discuss the effect of rapid rotation on the phase diagram of hadronic matter. The energy dispersion relation is shifted by an effective chemical potential induced by rotation. This suggests that rotation should lower the critical temperature of chiral restoration, but it is still controversial how the deconfinement temperature should change as a function of angular velocity. We adopt the hadron resonance gas model as an approach free from fitting parameters. We identify the deconfinement from the thermodynamic behavior and find that rotation decreases the deconfinement temperature. We also discuss the spatial inhomogeneity of the pressure and give a semi-quantitative estimate of the moment of inertia., Comment: 7 pages, 3 figures

We discuss deep learning inference for the neutron star equation of state (EoS) using the real observational data of the mass and the radius. We make a quantitative comparison between the conventional polynomial regression and the neural network approach for the EoS parametrization. For our deep learning method to incorporate uncertainties in observation, we augment the training data with noise fluctuations corresponding to observational uncertainties. Deduced EoSs can accommodate a weak first-order phase transition, and we make a histogram for likely first-order regions. We also find that our observational data augmentation has a byproduct to tame the overfitting behavior. To check the performance improved by the data augmentation, we set up a toy model as the simplest inference problem to recover a double-peaked function and monitor the validation loss. We conclude that the data augmentation could be a useful technique to evade the overfitting without tuning the neural network architecture such as inserting the dropout., Comment: 45 pages, 25 figures

We discuss the Hard Dense Loop resummation at finite quark mass and evaluate the equation of state (EoS) of cold and dense QCD matter in $\beta$ equilibrium. The resummation in the quark sector has an effect of lowering the baryon number density and the EoS turns out to have much smaller uncertainty than the perturbative QCD estimate. Our numerical results favor smooth matching between the EoS from the resummed QCD calculation at high density and the extrapolated EoS from the nuclear matter density region. We also point out that the speed of sound in our EoS slightly exceeds the conformal limit., Comment: 9 pages, 6 figures; Figs. 5 and 6 are updated, results unchanged

Superfluid vortices are quantum excitations carrying quantized amount of orbital angular momentum in a phase where global symmetry is spontaneously broken. We address a question of whether magnetic vortices in superconductors with dynamical gauge fields can carry nonzero orbital angular momentum or not. We discuss the angular momentum conservation in several distinct classes of examples from crossdisciplinary fields of physics across condensed matter, dense nuclear systems, and cosmology. The angular momentum carried by gauge field configurations around the magnetic vortex plays a crucial role in satisfying the principle of the conservation law. Based on various ways how the angular momentum conservation is realized, we provide a general scheme of classifying magnetic vortices in different phases of matter., Comment: 35 pages, 5 figures

We discuss a puzzle in relativistic spin hydrodynamics; in the previous formulation the spin source from the antisymmetric part of the canonical energy-momentum tensor (EMT) is crucial. The Belinfante improved EMT is pseudo-gauge transformed from the canonical EMT and is usually a physically sensible choice especially when gauge fields are coupled as in magnetohydrodynamics, but the Belinfante EMT has no antisymmetric part. We find that pseudo-transformed entropy currents are physically inequivalent in nonequilibrium situations. We also identify a current induced by the spin vorticity read from the Belinfante symmetric EMT., Comment: 6 pages, no figure