We present numerical solutions in a one-dimensional setting of quantum master equations that have been recently derived. We focus on the dynamics of a single heavy quark-antiquark pair in a Quark-Gluon Plasma in thermal equilibrium, in the so-called quantum Brownian regime where the temperature of the plasma is large in comparison with the spacing between the energy levels of the $Q\bar{Q}$ system. The one-dimensional potential used in the calculations has been adjusted so as to produce numbers that are relevant for the phenomenology of the charmonium. The equations are solved using different initial states and medium configurations. Various temperature regimes are studied and the effects of screening and collisions thoroughly analyzed. Technical features of the equations are analyzed. The contributions of the different operators that control the evolution are discussed as a function of the temperature. Some phenomenological consequences are addressed.
We compute the master integrals relevant for the two-loop corrections to pseudo-scalar quarkonium and leptonium production and decay. We present both analytic and high-precision numerical results. The analytic expressions are given in terms of multiple polylogarithms (MPLs), elliptic multiple polylogarithms (eMPLs) and iterated integrals of Eisenstein series. As an application of our results, we obtain for the first time an analytic expression for the two-loop amplitude for para-positronium decay to two photons at two loops.
We derive the quantum master equations for heavy quark systems in a high-temperature quark- gluon plasma in the Lindblad form. The master equations are derived in the influence functional formalism for open quantum systems in perturbation theory. These master equations have a wide range of applications, such as decoherence of a heavy quarkonium and Langevin dynamics of a heavy quark in the quark-gluon plasma. We also show the equivalence between the quarkonium master equations in the recoilless limit and the Schroedinger equations with stochastic potential.
By employing the method of differential equations, we compute the various types of two-loop master integrals involved in CP-even heavy quarkonium exclusive production and decays. All the integrals presented in this paper can be casted into canonical forms and expressed in terms of Goncharov polylogarithms and Harmonic polylogarithms. These master integrals are frequently used in the calculation of NNLO corrections of the heavy quarkonium production processes, as $γ^*γ\rightarrow Q\bar{Q}$, $e^+e^-\rightarrow γ+ Q\bar{Q}$, and $H/Z^0\rightarrow γ+ Q\bar{Q}$, and decay processes. They are also a…
The problem of the evolution of a heavy quarkonium in a medium can be recast as that of a quantum dissipative system. Within the framework of the master-equation approach to open quantum systems, we consider the real-time dynamics of quarkonia. We find that in a plasma at fixed temperature, the populations of the various quarkonium states evolve together, while their momentum distribution satisfies a Fokker-Planck equation.
In this paper we study the real-time evolution of heavy quarkonium in the quark-gluon plasma (QGP) on the basis of the open quantum systems approach. In particular, we shed light on how quantum dissipation affects the dynamics of the relative motion of the quarkonium state over time. To this end we present a novel non-equilibrium master equation for the relative motion of quarkonium in a medium, starting from Lindblad operators derived systematically from quantum field theory. In order to implement the corresponding dynamics, we deploy the well established quantum state diffusion method. In tu…
Dissociation of quarkonium in quark-gluon plasma (QGP) is a long standing topic in relativistic heavy-ion collisions because it has been believed to signal one of the fundamental natures of the QGP -- Debye screening due to the liberation of color degrees of freedom. Among recent new theoretical developments is the application of open quantum system framework to quarkonium in the QGP. Open system approach enables us to describe how dynamical as well as static properties of QGP influences the time evolution of quarkonium in a coherent way. Currently, there are several master equations for quark…
In last decades, it has been realized that the next-to-leading order corrections may become very important, and sometimes requisite, for some processes involving quarkoinum production or decay, e.g., $e^+e^- \to J/ψ+ η_c$ and $J/ψ\to 3γ$. In this article, we review some basic steps to perform automated one-loop computations in quarkonium process within the Non-relativistic Quantum Chromodynamics (NRQCD) factorization framework, and we give an introduction to some related public tools or packages and their usages in each step. We start from generating Feynman diagrams and amplitudes with \texts…
In this paper, we critically examine hierarchies between energy scales that determine quarkonium dynamics in the quark gluon plasma. A particularly important role is played by the ratio of the binding energy of species ($E_b$) and the medium scales; temperature ($T$) and Debye mass ($m_D$). It is well known that if these ratios are much larger than one then the dominant process governing quarkonium evolution is dissociation by thermal gluons (gluo-dissociation). On the other hand, if this ratio is much smaller than one then quarkonium dynamics is dominated by screening and Landau damping of th…
The in medium dynamics of heavy particles are governed by transport coefficients. The heavy quark momentum diffusion coefficient, $κ$, is an object of special interest in the literature, but one which has proven notoriously difficult to estimate, despite the fact that it has been computed by weak-coupling methods at next-to-leading order accuracy, and by lattice simulations of the pure SU(3) gauge theory. Another coefficient, $γ$, has been recently identified. It can be understood as the dispersive counterpart of $κ$. Little is known about $γ$. Both $κ$ and $γ$ are, however, of foremost import…
We show a completely analytical approach to the decoherence induced by a zero temperature environment on a Brownian test particle. We consider an Omhic environment bilinearly coupled to an oscillator and compute the master equation. From diffusive coefficients, we evaluate the decoherence time for the usual quantum Brownian motion and also for an upside-down oscillator, as a toy model of a quantum phase transition.
Quantum vs. semiclassical description of in-QGP quarkonia in the quantum Brownian regime
SUBATECH, IMT Atlantique, Nantes Université, CNRS/IN2P3（南特大学大西洋高等矿业学院国家科学研究中心/法国国家核子与亚原子物理研究所）；Institute of Physics – University of Silesia（西里西亚大学物理研究所）；Institute of Nuclear Physics, Polish Academy of Sciences（波兰科学院核物理研究所）；Institut de physique théorique, Université Paris-Saclay, CNRS, CEA（巴黎萨克雷大学理论物理研究所国家科学研究中心/法国原子能和替代能源委员会）
The emergence of an effective field theory out of equilibrium is studied in the case in which a light field --the system-- interacts with very heavy fields in a finite temperature bath. We obtain the reduced density matrix for the light field, its time evolution is determined by an effective action that includes the \emph{influence action} from correlations of the heavy degrees of freedom. The non-equilibrium effective field theory yields a Langevin equation of motion for the light field in terms of dissipative and noise kernels that obey a generalized fluctuation dissipation relation. These a…
I review recent applications of the open quantum system framework in the understanding of quarkonium suppression in heavy-ion collisions, which has been used as a probe of the quark-gluon plasma for decades. The derivation of the Lindblad equations for quarkonium in both the quantum Brownian motion and the quantum optical limits and their semiclassical counterparts is explained. The hierarchy of time scales assumed in the derivation is justified from the separation of energy scales in nonrelativistic effective field theories of QCD. Physical implications of the open quantum system approach are…
Optimal observables for (non-)equilibrium quantum metrology from the master equation
Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela（加利西亚高能物理研究所，圣地亚哥-德孔波斯特拉大学）；Department of Physics, Korea University（韩国大学物理系）
When analyzing the particle-like excitations in quantum field theory it is natural to regard the field mode corresponding to the particle momentum as an open quantum system, together with the opposite momentum mode. Provided that the state of the field is stationary, homogeneous and isotropic, this scalar two-mode system can be equivalently represented in terms of a pair of quantum Brownian oscillators under a Gaussian approximation. In other words, the two-mode system behaves as if it were interacting linearly with some effective environment. In this paper we build the details of the effectiv…
Master equations of the Lindblad type have recently been derived to describe the dynamics of quarkonium states in the quark-gluon plasma. Because their full resolution in three dimensions is very challenging, the equations are often reduced to one dimension. The main ingredient of these equations is the complex potential that describes the binding of the heavy quark-antiquark pairs and their interactions with the medium. In this work, we propose a one-dimensional complex potential parameterized to reproduce at best two key properties -- the temperature-dependent masses of the eigenstates and t…
On the basis of the closed-time path formalism of non-equilibrium quantum field theory, we derive the real-time quantum dynamics of heavy quark systems. Even though our primary goal is the description of heavy quarkonia, our method allows a unified description of the propagation of single heavy quarks as well as their bound states. To make calculations tractable, we deploy leading-order perturbation theory and consider the non-relativistic limit. Various dynamical equations, such as the master equation for quantum Brownian motion and time-evolution equation for heavy quark and quarkonium forwa…
Heavy-flavor particles are believed to provide valuable probes of the medium produced in ultrarelativistic collisions of heavy nuclei. In this article we review recent progress in our understanding of the interactions of charm and bottom quarks in the Quark-Gluon Plasma (QGP). For individual heavy quarks, we focus on elastic interactions for which the large quark mass enables a Brownian motion treatment. This opens a unique access to thermalization mechanisms for heavy quarks at low momentum, and thus to their transport coefficients in the quark-gluon fluid. Different approaches to evaluate he…
We introduce a density tensor hierarchy for open system dynamics, that recovers information about fluctuations lost in passing to the reduced density matrix. For the case of fluctuations arising from a classical probability distribution, the hierarchy is formed from expectations of products of pure state density matrix elements, and can be compactly summarized by a simple generating function. For the case of quantum fluctuations arising when a quantum system interacts with a quantum environment in an overall pure state, the corresponding hierarchy is defined as the environmental trace of produ…
