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Different aspects of the phase diagram of strongly interacting matter described by quantum chromodynamics (QCD), which have emerged from the recent studies using lattice gauge theory techniques, are discussed. A special emphasis is given on understanding the role of the anomalous axial U(1) symmetry in determining the order of the finite temperature chiral phase transition in QCD with two massless quark flavors and tracing its origin to the topological properties of the QCD vacuum.

Mass spectra of bottomonium states are computed using the Instanton Induced potential obtained from Instanton Liquid Model for QCD vacuum and incorporating a stronger confinement term. Spin dependent interactions through confined one gluon exchange potential are incorporated to remove the mass degeneracy. The mass spectra of the $$b\bar{b}$$ b b ¯ states up to 4S states are found to be in good agreement with the values reported by PDG(2020). Mixing of nearby isoparity states are also studied. We found the state $$\varUpsilon (10{,}860)$$ Υ ( 10 , 860 ) as an admixture of $$5^3S_1$$ 5 3 S 1 and $$6^3D_1$$ 6 3 D 1 Upsilon states with mixing angle $$\theta = 39.98^{\circ }$$ θ = 39 . 98 ∘ and the mixed state di-leptonic decay width is found to be 0.25 keV as against the width of $$0.31 \pm 0.07$$ 0.31 ± 0.07 keV reported by PDG. Further the state $$\varUpsilon (11{,}020)$$ Υ ( 11 , 020 ) is also found to be the admixture of $$6^3S_1$$ 6 3 S 1 and $$5^3D_1$$ 5 3 D 1 Upsilon states with the mixing angle $$\theta = 51.69^{\circ }$$ θ = 51 . 69 ∘ and the di-leptonic decay width of the mixed state is obtained as 0.14 keV which is very close to the width of $$0.13 \pm 0.03$$ 0.13 ± 0.03 keV reported by PDG. Present results indicates that addition of confinement to the instanton potential is crucial for the determination of the mass spectroscopy of heavy hadrons.

Within the domain model of QCD vacuum the properties of a statistical ensemble of almost everywhere homogeneous Abelian (anti-)self-dual gluon fields representing the physical vacuum of quantum chromodynamics are investigated. The two-point correlation function of the topological charge density is calculated and the topological susceptibility is found. It is shown that such vacuum fields ensure the implementation of the area law for the Wilson loop, i.e. the confinement of static quarks.

Isobaric $^{96}_{44}$Ru+$^{96}_{44}$Ru and $^{96}_{40}$Zr+$^{96}_{40}$Zr collisions at $\sqrt{s_{_{NN}}}=200$ GeV have been conducted at the Relativistic Heavy Ion Collider to circumvent the large flow-induced background in searching for the chiral magnetic effect (CME), predicted by the topological feature of quantum chromodynamics (QCD). Considering that the background in isobar collisions is approximately twice that in Au+Au collisions (due to the smaller multiplicity) and the CME signal is approximately half (due to the weaker magnetic field), we caution that the CME may not be detectable with the collected isobar data statistics, within $\sim$2$\sigma$ significance, if the axial charge per entropy density ($n_5/s$) and the QCD vacuum transition probability are system independent. This expectation is generally verified by the Anomalous-Viscous Fluid Dynamics (AVFD) model. While our estimate provides an approximate "experimental" baseline, theoretical uncertainties on the CME remain large., Comment: 6 pages, 1 multi-panel figure. v2: version after addressing referee comments. Corrected a factor 2 error in the quoted multiplicity; corrected a factor 2 error in the A-dependence exponent of the magnetic field; conclusions unchanged. v3: match published version

While it is known that the QCD vacuum in a magnetic background exhibits both diamagnetic and paramagnetic characteristics in the low-energy domain, a systematic investigation of the corresponding phases emerging in the pion-dominated regime is still lacking. Here, within two-flavor chiral perturbation theory, taking into account the pion-pion interaction, we analyze the subtle interplay between zero- and finite-temperature portions in the magnetization and magnetic susceptibility. The dependence of the magnetic susceptibility on temperature and magnetic field strength in the paramagnetic and diamagnetic phase is non-monotonic. Our low-energy analysis complements lattice QCD that is currently operating at higher temperatures and stronger magnetic fields., Comment: 17 pages, 5 figures

Gauge fields control the dynamics of fermions, also a back reaction of fermions on the gauge field is expected. This back reaction is investigated within the vortex picture of the QCD vacuum. We show that the center vortex model reproduces the string tension of the full theory also with the presence of fermionic fields., New data and new evaluations with different coaouthors. This version is submitted to MDPI / Universe

We calculate the production cross section and the transverse single-spin asymmetry for pion in ${p}^{\ensuremath{\uparrow}}+p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}+X$. Our computation is based on existence of the instanton induced effective quark-gluon and quark-gluon-pion interactions with a strong spin dependency. In this framework we calculate the cross section without using fragmentation functions. We compare predictions of the model with data from RHIC. Our numerical results, based on the instanton liquid model for QCD vacuum, are in agreement with unpolarized cross section data. The asymmetry grows with the transverse momentum of pion ${k}_{t}$ in accordance with experimental observations. It reaches a value $\ensuremath{\sim}10%$ but at higher ${k}_{t}$ than experiment shows.

The current state of knowledge of the order parameters of the spontaneously broken chiral symmetry, the quark condensate and the pseudoscalar decay constant in the chiral limit, is briefly reviewed, based on available phenomenological fits and lattice QCD calculations. We argue that while the theory is pretty well understood in the two-flavour case, there is still a gap in the knowledge of the characteristics of the QCD vacuum in the three-flavour one. Our constraints for the three-flavour parameters obtained by a Bayesian statistical analysis of the decays of eta to three pions are presented. Possible implications of a new analysis of subthreshold parameters of pion-pion scattering is outlined.

What happens to the QCD vacuum when a time-periodic circularly polarized laser field with a sufficiently large intensity and frequency is applied? Based on the Floquet formalism for periodically driven systems and the systematic low-energy effective theory of QCD, we show that for a sufficiently large frequency and above a critical intensity, the QCD vacuum is unstable against the chiral soliton lattice of pions; a crystalline structure of topological solitons that spontaneously breaks parity and continuous translational symmetries. Our work would pave the way for novel "Floquet vacuum engineering.", Comment: 7 pages; v2: added discussion of Weyl/Dirac semimetals, typos fixed, published version

Journal of high energy physics 05(5), 231 (2021). doi:10.1007/JHEP05(2021)231, We consider the short-distance expansion of the product of two gluon field strength tensors connected by a straight-line-ordered Wilson line. The vacuum expectation value of this nonlocal operator is a common object in studies of the QCD vacuum structure, whereas its nucleon expectation value is known as the gluon quasi-parton distribution and is receiving a lot of attention as a tool to extract gluon distribution functions from lattice calculations. Extending our previous study [1], we calculate the three-loop coefficient functions of the scalar operators in the operator product expansion up to dimension four. As a by-product, the three-loop anomalous dimension of the nonlocal two-gluon operator is obtained as well., Published by SISSA, [Trieste]

The center vortex model of the QCD vacuum is very successful in explaining the non-perturbative properties of QCD, especially confinement, chiral symmetry breaking and the topological charge of vacuum configurations. On the other hand, the center vortex model still suffers from a Gribov problem: Direct maximal center gauge and center projection can lead to an underestimation of the string tension in smooth configurations or after persistent simulated annealing. We discuss methods to identify center regions, whose boundaries evaluate to center elements, and want to improve the vortex detection: these regions might help to recognize vortices in configurations where maximal center gauge lost the vortex finding property.

Keeping in view of our earlier study, the multiflux-tube structure of topologically excited magnetically condensed dual QCD vacuum has been extended in the thermal domain to analyze the possible mechanism behind the QCD phase transition. The stability and annihilation of color flux-tubes in the intermediate energy regime has been investigated for QGP formation by discussing the flux-tube interaction process under varying temperature conditions at different hadronic boundaries by computing critical parameters of phase transition and condensed monopole density. Their implications in quark pair creation relevant to elaborate the phase structure of QCD and gain insight into the mechanism of the parton-hadron phase transition has also been discussed.

In this review we will concentrate on the nonperturbative effects in the properties of hadrons made from the light-heavy and heavy-heavy quarks in the framework of Instanton Liquid Model (ILM) of QCD vacuum. We briefly discuss the main features of ILM and its applicability in the heavy quark sector. The properties of gluonic systems, light and heavy quark correlators in the instanton background are also analyzed. Consideration of the both, perturbative and nonperturbative, gluon effects in the instanton background for the single heavy quark will lead to the mass shift due to the direct-instanton nonperturbative and ILM modified perturbative contributions, respectively. For the interacting heavy quark-antiquarks, the potential consists the direct instanton induced part and the one-gluon exchange (OGE) perturbative part. OGE interactions are screened at large distances due to the nonperturbative dynamics. We discuss the estimations of instanton contributions in the phenomenological Cornell type potential model. As related to the experimental data we discuss the charmonium properties and the role of instanton effects in their spectra and transitions. We discuss also the main features of heavy-light quarks systems in the ILM. As an example, it is considering the process of pions emission by exited heavy quarkonium states., Comment: Talk presented at ICNFP 2020. arXiv admin note: text overlap with arXiv:2006.01545

In this review, we provide a short outlook of some of the currently most popular pictures and promising approaches to non-perturbative physics and confinement in gauge theories. A qualitative and by no means exhaustive discussion presented here covers such key topics as the phases of QCD matter, the order parameters for confinement, the central vortex and monopole pictures of the QCD vacuum structure, fundamental properties of the string tension, confinement realisations in gauge-Higgs and Yang-Mills theories, magnetic order/disorder phase transition among others., Comment: 57 pages, 7 figures; an invited review for the Special Issue "Modern Approaches to Non-Perturbative QCD and other Confining Gauge Theories", in Universe 7 (2021) no. 9, 330

The glueballs lead to gluon and QCD monopole condensations as by-products of color confinement. A color dielectric function $G(|\phi|)$ coupled with Abelian gauge field is properly defined to mediate the glueball interactions at confining regime after {\it spontaneous symmetry breaking} (SSB) of the gauge symmetry. The particles are expected to form through quark gluon plasma (QGP) hadronization phase where the free quarks and gluons start clamping together to form hadrons. The QCD-like vacuum $\langle\eta^2m_\eta^2F^{\mu\nu}F_{\mu\nu}\rangle$, confining potential $V_c(r)$, string tension $\sigma$, penetration depth $\lambda$, superconducting and normal monopole densities ($n_s\,n_n$) and the effective masses ($m_\eta^{2}$ and $m_A^{2}$) will be investigated at finite temperature $T$. We also calculate the strong `running' coupling $\alpha_s$ and subsequently the QCD $\beta$-function. {\it Dual superconducting} nature of the QCD vacuum will be investigated based on monopole condensation., Comment: 21 pages, 11 figures

We investigate the quark number fluctuations up to the fourth order in the matter composed of two light flavor quarks with isospin symmetry and at finite temperature and finite chemical potential using the Dyson-Schwinger equation approach of QCD. In order to solve the quark gap equation, we approximate the dressed quark-gluon vertex with the bare one and adopt both the Marris-Tandy (MT) model and the infrared constant (Qin-Chang) model for the dressed gluon propagator. Our results indicate that the second, third, and forth order fluctuations of net quark number all diverge at the critical end point (CEP). Around the CEP, the second order fluctuation possesses obvious pump while the third and fourth order ones exhibit distinct wiggles between positive and negative. For the MT model and the Qin-Chang model, we give the pseudo-critical temperature at zero quark chemical potential as $T_{c}=146$ MeV and $150$ MeV, and locate the CEP at $({\mu_{E}^{q}}, {T_{E}^{}}) = (120, 124)$ MeV and $(124,129)$ MeV, respectively. In addition, our results manifest that the fluctuations are insensitive to the details of the model, but the location of the CEP shifts to low chemical potential and high temperature as the confinement length scale increases., Comment: 10 pages, 6 figures

Using the soft pion theorem, crossing, and the dispersion relations for the two-pion distribution amplitude (DA), we argue that the second Gegenbauer moment, the ρ-meson DA (a2(ρ)), is most probably negative. This result is at variance with the majority of the model calculations for a2(ρ). Using the instanton theory of the QCD vacuum, we computed a2(ρ) at a low normalization point and obtain for the ratio a2(ρ)/M3(π) a definitely negative value in the range of a2(ρ)/M3(π)∈[−2,−1]. The range of values corresponds to a generous variation of the parameters of the instanton vacuum. The value of the second Gegenbauer moment of pion DA is positive in the whole range and is compatible with its the most advanced lattice measurement. It seems that the topologically nontrivial field configurations in the QCD vacuum (instantons) lead to qualitatively different shapes of the pion and the ρ-meson DAs.

The rapidity anomalous dimension (RAD), or Collins-Soper kernel, defines the scaling properties of transverse momentum dependent distributions and can be extracted from the experimental data. I derive a self-contained nonperturbative definition that represents RAD without reference to a particular process. This definition makes possible exploration of the properties of RAD by theoretical methods on one side, and the properties of QCD vacuum with collider measurements on another side. To demonstrate these possibilities, I compute the power correction to RAD, its large-b asymptotic, and compare these estimations with recent phenomenological extractions., Comment: 5 pages, 3 figures; v2. minor changes, update to the journal version

Experiments with paramagnetic ground or metastable excited states of molecules (ThO, HfF$^+$, YbF, YbOH, BaF, PbO, etc.) provide strong constraints on electron electric dipole moment (EDM) and coupling constant $C_{SP}$ of contact semileptonic interaction. We compute new contributions to $C_{SP}$ arising from the nucleon EDMs due to combined electric and magnetic electron-nucleon interaction. This allows us to improve limits from the experiments with paramagnetic molecules on the $CP$-violating parameters, such as the proton EDM, $|d_p|< 1.1\times 10^{-23} e\cdot $cm, the QCD vacuum angle, $|\bar \theta, Comment: 27 pages. V2: major corrections, V3: comments on uncertainties and other minor corrections, journal version

In this paper we improve the existing order parameter for monopole condensation in gauge theory vacuum, making it gauge-invariant from scratch and free of the spurious infrared problems which plagued the old one. Computing the new parameter on the lattice will unambiguously detect weather dual superconductivity is the mechanism for color confinement. As a byproduct we relate confinement to the existence of a finite correlation length in the gauge-invariant correlator of chromo-electric field strengths., 12 pages, 2 figures Revised version. Presentation improved in Sect V and Section VI. No substantial change of the content

We aim to show the effects of the magnetic monopoles and instantons in quantum chromodynamics (QCD) on observables; therefore, we introduce a monopole and anti-monopole pair in the QCD vacuum of a quenched SU(3) by applying the monopole creation operator to the vacuum. We calculate the eigenvalues and eigenvectors of the overlap Dirac operator that preserves the exact chiral symmetry in lattice gauge theory using these QCD vacua. We then investigate the effects of magnetic monopoles and instantons. First, we confirm the monopole effects as follows: (i) The monopole creation operator makes the monopoles and anti-monopoles in the QCD vacuum. (ii) A monopole and anti-monopole pair creates an instanton or anti-instanton without changing the structure of the QCD vacuum. (iii) The monopole and anti-monopole pairs change only the scale of the spectrum distribution without affecting the spectra of the Dirac operator by comparing the spectra with random matrix theory. Next, we find the instanton effects by increasing the number density of the instantons and anti-instantons as follows: (iv) The decay constants of the pseudoscalar increase. (v) The values of the chiral condensate, which are defined as negative numbers, decrease. (vi) The light quarks and the pseudoscalar mesons become heavy. The catalytic effect on the charged pion is estimated using the numerical results of the pion decay constant and the pion mass. (vii) The decay width of the charged pion becomes wider than the experimental result, and the lifetime of the charged pion becomes shorter than the experimental result. These are the effects of the monopoles and instantons in QCD., Comment: 58 pages, 26 figures, and 28 tables

It is shown that gauge field configurations with higher topological charge modify the structure of the QCD vacuum, which is reflected in its dependence on the CP-violating topological phase $\theta$. To explore this, topological susceptibilities and the production of axion dark matter are studied here. The former characterize the topological charge distribution and are therefore sensitive probes of the topological structure of QCD. The latter depends on the effective potential of axions, which is determined by the $\theta$-dependence of QCD. The production of cold dark matter through the vacuum realignement mechanism of axions can therefore be affected by higher topological charge effects. This is discussed qualitatively in the deconfined phase at high temperatures, where a description based on a dilute gas of instantons with arbitrary topological charge is valid. As a result, topological susceptibilities exhibit a characteristic temperature dependence due to anharmonic modifications of the $\theta$-dependence. Furthermore, multi-instanton effects give rise to a topological mechanism to increase the amount of axion dark matter., Comment: 19 pages, 13 figures. v3: detailed derivation of multi-instanton effects

We revisit the path integral computation of the Casimir energy between two infinite parallel plates placed in a QED vacuum. We implement perfectly magnetic conductor boundary conditions (as a prelude to the dual superconductor picture of the QCD vacuum) via constraint fields and show how an effective gauge theory can be constructed for the constraint boundary fields, from which the Casimir energy can be simply computed, in perfect agreement with the usual more involved approaches. Gauge invariance is natural in this framework, as well as the generalization of the result to $d$ dimensions. We also pay attention to the case where the outside of the plates is not the vacuum, but a perfect magnetic (super)conductor, disallowing any dynamics outside the plates. We find perfect agreement between both setups., Comment: 18 pages, 1 figure. Comments added and relevant references included

By duality approach, we study condensate mass of QCD vacuum via dilaton wall background in presence of parameter $c$ which represents the gluon condensation in holographic set up. First from Wilson line calculation we find $m^2_0$ (condensate parameter in mixed nonlocal condensation) whose behavior mimics that of QCD. The value of $m^2_0$ that we find by this approach, is in agreement with QCD data. In the second step we consider produced mass m via Schwinger effect mechanism in presence of parameter $c$. We show that generally gluon condensation contribute mass dominantly and produced mass via Schwinger effect is suppressed by $m_0$ ., 14 pages, 3 figures

For a proton in the infinite momentum frame, its wave function contains a zero-momentum part (light-front zero-modes) originated from the modification of the QCD vacuum in the presence of the valence quarks, exhibiting a light-front long-range order in the quantum state. This ``non-travelling'' component of the proton contributes to its fundamental properties, including the mass and spin, as well as the naive-time-reversal-odd correlations between transverse momentum and spin. The large momentum effective theory (LaMET) provides a theoretical approach to study the physical properties of the proton's zero-mode ``condensate'' through lattice simulations., 5 pages, no figure

The global $SU(3)$ color symmetry and its physical consequences are discussed. The N\"{o}ther current is actually governed by the conserved matter current of color charges if the color field generated by this charge is properly polarized. The color field strength of a charge can have a uniform part due to the nontrivial QCD vacuum field and the nonzero gluon condensate, which implies that the self-energy of a system with a net color charge is infinite and thereby cannot exist as a free state. This is precisely what the color confinement means. Accordingly, the Cornell type potential with the feature of the Casimir scaling is derived for a color singlet system composed of a static color charge and an anti-charge. The uniform color field also implies that a hadron has a minimal size and a minimal energy. Furthermore, the global $SU(3)$ color symmetry requires that the minimal irreducible color singlet systems can only be $q\bar{q}$, $qqq$, $gg$, $ggg$, $q\bar{q}g$, $qqqg$ and $\bar{q}\bar{q}\bar{q}g$, etc., as such a multi-quark systems can only exist as a molecular configurations if there are no other binding mechanisms., Comment: 5 pages, 3 figures. Comments are welcome. Typos corrected. Submitted to Chin. Phys. C

The magnetic polarizability of the neutral pion has been calculated in the background magnetic-field formalism of Lattice QCD. In this investigation, the chiral extrapolation of these lattice results is considered in a formalism preserving the exact leading nonanalytic terms of chiral perturbation theory. The $n_f = 2 + 1$ numerical simulations are electro-quenched, such that the virtual sea-quarks of the QCD vacuum do not interact with the background field. To understand the impact of this, we draw on partially quenched chiral perturbation theory and identify the leading contributions of quark-flow connected and disconnected diagrams. While electro-quenching does not impact the leading-loop contribution to the magnetic polarizability, the loops which generate the leading term have yet to be considered in lattice QCD simulations. Lattice QCD results are used to constrain the analytic terms in the chiral expansion and supplementing those with the two-loop result from chiral perturbation theory enables an evaluation of the polarizability at the physical quark mass. The resulting magnetic polarizability of the neutral pion is $\beta_{\pi^0}=3.44(19)^{\rm stat}(37)^{\rm syst}\times 10^{-4}$ fm$^3$, which lies just above the $1 \sigma$ error bound of the experimental measurement., Comment: 8 pages, 5 figures

We demonstrate that electron electric dipole moment experiments with molecules in paramagnetic state are sensitive to $P,T$-violating nuclear forces and other $CP$-violating parameters in the hadronic sector. These experiments, in particular, measure the coupling constant $C_{SP}$ of the $CP$-odd contact semileptonic interaction. We establish relations between $C_{SP}$ and different $CP$-violating hadronic parameters including strength constants of the $CP$-odd nuclear potentials, $CP$-odd pion-nucleon interactions, quark-chromo EDM and QCD vacuum angle. These relations allow us to find limits on various $CP$-odd hadronic parameters., Comment: 13 pages. arXiv admin note: text overlap with arXiv:2004.10359

We study the vacuum topology of 2+1 flavor QCD above the chiral crossover transition, at $T \lesssim 1.2~T_c$, on lattices of size $32^3\times 8$. Since overlap fermions have exact chiral symmetry and an index theorem even on a finite lattice, we use them to detect the topological content of gauge fields generated using domain wall fermion discretization for quarks. We further use different periodicity phases along the temporal direction for the valence overlap quarks, which allows us to probe different topological structures present in the gauge field ensembles, through its zero modes. This procedure provides strong evidences that fermion zero-modes can be quantitatively understood to arise due to different species of instanton-dyons. We estimate their relative abundances from the Dirac-eigenvalue density and resolve the so called "topological clusters" via multi-parameter fits to their density, providing therefore an understanding of the interactions between instanton-dyons. The typical separation between dyons we obtain, is $\sim 0.3$ fm. Surprisingly, it emerges out from this study that a semi-classical description of the fermionic zero modes in the QCD vacuum is quite accurate just above $T_c$., 14 pages, 11 Figures

We consider the modifications of gluon properties in the instanton liquid model (ILM) for the QCD vacuum. Rescattering of gluons on instantons generates the dynamical momentum-dependent gluon mass M g ( q ) . First, we consider the case of a scalar gluon, no zero-mode problem occurs and its dynamical mass M s ( q ) can be found. Using the typical phenomenological values of the average instanton size ρ = 1 / 3 fm and average inter-instanton distance R = 1 fm we get M s ( 0 ) = 256 MeV . We then extend this approach to the real vector gluon with zero-modes carefully considered. We obtain the following expression M g 2 ( q ) = 2 M s 2 ( q ) . This modification of the gluon in the instanton media will shed light on nonperturbative aspect on heavy quarkonium physics.

We propose and examine a new high-density composite of $\Lambda^* \equiv K^-p = (s \bar{u}) \otimes (uud)$, which may be called {\it Kaonic Proton Matter (KPM)}, or simply, {\it $\Lambda^* $-Matter}, where substantial shrinkage of baryonic bound systems originating from the strong attraction of the $(\bar{K}N)^{I=0}$ interaction takes place, providing a ground-state neutral baryonic system with a huge energy gap. The mass of an ensemble of $(K^-p)_m$, where $m$, the number of the $K^-p$ pair, is larger than $m \approx 10$, is predicted to drop down below its corresponding neutron ensemble, $(n)_m$, since the attractive interaction is further increased by the Heitler-London type molecular covalency, as well as by chiral symmetry restoration of the QCD vacuum. Since the seed clusters ($K^-p$, $K^-pp$ and $K^-K^-pp$) are short-lived, the formation of such a stabilized relic ensemble, $(K^-p)_m$, may be conceived during the Big-Bang Quark Gluon Plasma (QGP) period in the early universe before the hadronization and $quark$-$anti$-$quark$ annihilation proceed. At the final stage of baryogenesis a substantial amount of primordial ($\bar{u},\bar{d}$)'s are transferred and captured into {\it KPM}, where the anti-quarks find places to survive forever. The expected {\it KPM} state may be {\it cold, dense and neutral $\bar{q} q$-hybrid ({\it Quark Gluon Bound (QGB)}) states, $[s(\bar{u} \otimes u) ud]_m$,} to which the relic of the disappearing anti-quarks plays an essential role as hidden components. The {\it KPM} may also be produced during the formation and decay of neutron stars., Comment: 5 pages, 5 figures

A survey of experimental results obtained by studying chiral effects in nucleus–nucleus collisions at various energies is presented. The entire body of these experimental results confirms indirectly a topologically nontrivial structure of the QCD vacuum and the possibility of a local violation of discrete QCD symmetries at finite temperature. A significant decrease in the asymmetry of the distribution of electrically charged particles with respect to the reaction plane in heavy-ion collisions in the energy range of $$\sqrt {sNN} $$ GeVmay be indicative of a possible transition to the region where hadron states dominate over quark–gluon degrees of freedom in a mixed phase formed in nucleus–nucleus collisions at intermediate energies.

We present recent results from lattice QCD relevant for the study of strongly interacting matter as it is produced in heavy ion collision experiments. The equation of state at non-vanishing density from a Taylor expansion up to $6^{th}$ order will be discussed for a strangeness neutral system and using the expansion coefficients of the series limits on the critical point are estimated. Chemical freeze-out temperatures from the STAR and ALICE Collaborations will be compared to lines of constant physics calculated from the Taylor expansion of QCD bulk thermodynamic quantities. We show that qualitative features of the $\sqrt{s_{NN}}$ dependence of skewness and kurtosis ratios of net proton-number fluctuations measured by the STAR Collaboration can be understood from QCD results for cumulants of conserved baryon-number fluctuations. As an example for recent progress towards the determination of spectral and transport properties of the QGP from lattice QCD, we will present constraints on the thermal photon rate determined from a spectral reconstruction of continuum extrapolated lattice correlation functions in combination with input from most recent perturbative calculations., Comment: 8 pages, 11 figures, to be published in Nuclear Physics A, proceedings of the XXVI international conference on ultrarelativistic heavy-ion collisions, Quark Matter 2017, February 5-11, 2017