We report on recent theory progress in understanding the production of heavy quarkonium in heavy-ion collisions based on the in-medium heavy-quark potential extracted from lattice QCD simulations. On the one hand, the proper in-medium potential allows us to study the spectral properties of heavy quarkonium in thermal equilibrium, from which we estimate the

The bound states of heavy quarks and antiquarks, so-called heavy quarkonia, have matured into a high precision tool in heavy-ion collisions (HIC) at accelerator facilities, such as the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC). The availability of experimental data of unprecedented accuracy for both bottomonium

The STAR collaboration at RHIC has observed overall suppression of bottomonium states in

The goal for theory thus must be to provide a first principles description of this intricate phenomenology. As the temperatures encountered in current heavy-ion collisions are relatively close to the chiral crossover transition, genuinely non-perturbative methods are called for and, in this article, I discuss one possible route how first principles lattice QCD simulations can contribute to gain insight into the equilibrium and non-equilibrium properties of heavy quarkonium in HIC.

Let us start with the question of what are the properties of heavy quarkonium in thermal equilibrium? That is, we consider the idealized setting of immersing a heavy quark and antiquark pair in an infinitely extended QCD medium at a fixed temperature and wait until full kinetic equilibration is achieved. Then, we ask for the presence or absence of in-medium bound eigenstates and their properties, such as their in-medium mass and stability.

These questions may be answered in the modern language of quantum field theory by computing so-called in-medium meson spectral functions, which encode the particle properties as well defined peak structures. The position of the peaks along the frequency axis encodes the mass of the particle, while their width is directly related to the inverse lifetime of the state. At higher frequencies, the open heavy-flavor threshold manifests itself in the spectral function as broad continuous structures, often with a steep onset. In a thermal setting, the peak width not only encodes the decay of the bound state into gluons but also carries a contribution from processes that: (1) excite the color singlet bound state into another singlet state due to thermal fluctuations; and (2) transform the singlet state into a color octet state due to the absorption of a medium gluon. On the level of the spectral function, these three contributions cannot be disentangled. Once we have access to the in-medium meson spectral function, we argue that phenomenologically relevant processes, such as the production of

There are currently two viable options to determine the in-medium quarkonium spectra in QCD and both involve lattice QCD simulations. For the first and direct one, we can compute the current–current correlators of a heavy meson in the Euclidean time domain, in which the simulation is carried out. In particular, for bottomonium, it is customary to use a discretization of the heavy quarks, which is derived from a non-relativistic effective field theory (EFT) (see, e.g., [

The second possibility is to take a detour and instead of the spectral function compute first the potential acting in between a static quark and antiquark at finite temperature. Using this in general complex valued potential one can solve a Schrödinger equation for the unequal time correlation function of meson color singlet wavefunctions, i.e., for the meson forward current–current correlator, whose imaginary part then yields the in-medium spectral function. This approach on the one hand provides us with a very precise determination of the spectral function, however it does not yet include finite velocity or spin dependent corrections, since only the static potential is used in the computation. At

Today we are in the fortunate position of not having to rely anymore on model potential for the description of heavy quarkonium. Indeed, over the past decade, it has become possible to derive the inter quark potential directly from QCD using a chain of EFTs [

It is the process of matching that allows us to connect back to QCD. We need to select a correlation function in the EFT and find the corresponding correlation function in QCD with the same physics content. Once we set them equal at the scale at which the EFT is supposed to reproduce the microscopic physics, we can express the non-local Wilson coefficients of the former in terms of correlation functions of QCD. For static quarks, it can be shown that the unequal time singlet wave function correlation function is related to the rectangular Wilson loop

If the function

This genuinely real-time definition of the potential was first evaluated at high temperature in resummed perturbation theory by Laine et al. [

We may now ask how to evaluate the real-time definition of the potential in non-perturbative lattice QCD, as these simulations are carried out in artificial Euclidean time. It is here that the technical concept of spectral function again finds application [

The quantity accessible on the lattice is the imaginary time Wilson correlator, which is governed by the same spectral function, just with a different integral transform. Let me first note that using the spectral decomposition, inserted in the r.h.s. of Equation (

The central challenge lies in extracting the spectral function from lattice simulations, which amounts to solving an ill-posed inverse problem. In the past, this required the application of Bayesian inference [

The lattice data on which the latest determination of the potential is based were obtained in a collaboration with the HotQCD and TUMQCD collaboration [

In

While it might be tempting to use the lattice values of

In

How can such spectral functions help us to learn about quarkonium production in HICs? Note that we are considering a fully thermalized scenario here, which applies, if at all, for charmonium. Note further that what is measured in experiment are not the decay dileptons from the in-medium states but the decays of vacuum states long after the QGP ceases to exist. Thus, any information of in-medium quarkonium needs to be translated into a modification of the yields of produced vacuum states at hadronization. The process of hadronization is among the least well known stages of a HIC and a first principles understanding of its dynamics has thus far not been achieved. Therefore, we continue with the phenomenological ansatz of instantaneous freezeout introduced in [

That is, we compute the weighted area under the in-medium

Up to this point, we have only considered equilibrium aspects of quarkonium. In a HIC, this will always constitute only an approximation to the genuine non-equilibrium physics occurring. Therefore, we wish to learn more about the real-time dynamics of quarkonium states exploiting the fact that we already have access to the in-medium potential extracted on the lattice. A promising route towards a microscopic understanding of quarkonium real-time dynamics is offered by the open-quantum systems approach, a technique developed originally in the context of condensed matter theory.

The overall system consisting of the heavy quark and antiquark, as well as the medium degrees of freedom is of course closed and described by a hermitean Hamiltonian. The overall density matrix evolves according to the von Neumann equation

Our goal however is to investigate the properties and dynamics of the heavy quarkonium sub system coupled to the thermal bath. To this end, we may trace out all medium degrees of freedom from the density matrix of the full system, ending up with

Over the past five years, it has become possible to derive the master equation for

The first part is related to a real valued in-medium potential term, while the second and third implement the fluctuation–dissipation relation for the heavy quarkonium. They are intimately related to the imaginary part of the interquark potential. The last term assures that the master equation for

The above expression for

The operators

Together with collaborators from Japan, we have investigated the effects of the Lindblad operators on the real-time dynamics of heavy quarkonium in a simple one-dimensional setting [

While the evolution of each realization of the ensemble proceeds via a norm preserving evolution operator, the ensemble average of the wave function washes out according to a Schrödinger equation with a complex valued potential. This mechanism provides a unitary microscopic implementation of quarkonium real-time dynamics, which reproduces the imaginary part of the interquark potential for the unequal time correlation function of wavefunctions.

Note that there is a new physical scale present in this approach, which is the correlation length of the noise induced by the medium. Depending on the size of the quarkonium bound state compared to this correlation length, the noise may be able to efficiently destabilize the bound state or not. This phenomenon is known as decoherence. That is, the noise provides an additional mechanism to dissociate a heavy quarkonium particle over time, which acts in addition to the screening of the real-valued potential.

In

While the stochastic potential provides a conceptually attractive microscopic implementation of the complex inter-quark potential, it can only be the first step towards understanding heavy quarkonium in-medium dynamics. It does not account for dissipation effects and thus does not allow the quarkonium to thermalize with its surroundings. This means that the stochastic potential description is only applicable to early times in the evolution. Incorporation of the full Linblad equation is work in progress and we have successfully tested it in the single heavy quark case [

In this article, I have showcased recent progress in our understanding of in-medium heavy quarkonium in the context of heavy-ion collisions. In thermal equilibrium, it has become possible to derive a complex valued real-time in-medium potential from QCD based on EFT methods. Its evaluation in lattice QCD simulations is challenging as it involves the reconstruction of spectral functions from Wilson correlators. The most recent determination has been performed on realistic

This work is part of and partially supported by the DFG Collaborative Research Centre “SFB 1225 (ISOQUANT)”.

The author declares no conflict of interest.

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