<p>We reinvestigate the interactions between static color sources in a finite temperature gluonic medium using both high resolution isotropic and anisotropic quenched lattice QCD ensembles. The underlying ill-posed inverse problem, related to the extraction of spectral functions, is attacked with a range of different methods, including Bayesian inference, Padé interpolation and model fits. Among the latter we include a tail amended Gaussian ansatz and a HTL-inspired fit ansatz. We reconfirm the presence of a dominant low lying spectral feature that supports the existence of a potential picture for the in-medium evolution of the static charges at late real times. Using the raw unmodified lattice data, all applicable methods show clear signs of screening of the real part of the potential. After applying a subtraction procedure featured in a previous study, we find however that screening disappears from the extracted potential. Paths toward the resolution of this puzzle are discussed.</p>

Norway

Rasmus Larsen

Gaurang Parkar

Alexander Rothkopf

Germany

Johannes Heinrich Weber

In-medium static inter-quark potential on high resolution quenched lattices

We study the correlator of temporal Wilson lines at nonzero temperature in <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavor lattice QCD with the aim to define the heavy quark-antiquark potential at nonzero temperature. For temperatures <math><mn>153</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><mo>≤</mo><mi>T</mi><mo>≤</mo><mn>352</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> the spectral representation of this correlator is consistent with a broadened peak in the spectral function, position, or width of which then defines the real or imaginary parts of the heavy quark-antiquark potential at nonzero temperature, respectively. We find that the potential’s real part is not screened contrary to the widely held expectations. We comment on how this fact may modify the picture of quarkonium melting in the quark-gluon plasma.

United States

Alexei Bazavov

Switzerland

Daniel Hoying

Rasmus N. Larsen

Swagato Mukherjee

Peter Petreczky

Unscreened forces in the quark-gluon plasma?

We present a simulation strategy for the real-time dynamics of quantum fields, inspired by reinforcement learning. It builds on the complex Langevin approach, which it amends with system-specific prior information, a necessary prerequisite to overcome this exceptionally severe sign problem. The optimization process underlying our machine-learning approach is made possible by deploying inherently stable solvers of the complex Langevin stochastic process and a novel optimality criterion derived from insight into so-called boundary terms. This conceptual and technical progress allows us to both significantly extend the range of real-time simulations in <math><mrow><mn>1</mn><mo>+</mo><mn>1</mn><mi>d</mi></mrow></math> scalar field theory beyond the state of the art and to avoid discretization artifacts that plagued previous real-time field theory simulations. Limitations of and promising future directions are discussed.

Daniel Alvestad

Austria

Dénes Sexty

Lattice real-time simulations with learned optimal kernels

We present a novel strategy aimed at restoring correct convergence in complex Langevin simulations. The central idea is to incorporate system-specific prior knowledge into the simulations, in order to circumvent the NP-hard sign problem. In order to do so, we modify complex Langevin using kernels and propose the use of modern auto-differentiation methods to learn optimal kernel values. The optimization process is guided by functionals encoding relevant prior information, such as symmetries or Euclidean correlator data. Our approach recovers correct convergence in the non-interacting theory on the Schwinger-Keldysh contour for any real-time extent. For the strongly coupled quantum anharmonic oscillator we achieve correct convergence up to three-times the real-time extent of the previous benchmark study. An appendix sheds light on the fact that for correct convergence not only the absence of boundary terms, but in addition the correct Fokker-Plank spectrum is crucial.

Towards learning optimized kernels for complex Langevin

Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be alternatively described as the degree in  $$q^{\pm 2}$$  <math> <msup> <mi>q</mi> <mrow> <mo>±</mo> <mn>2</mn> </mrow> </msup> </math>   of the fundamental Alexander polynomial, which formally corresponds to the case of no colors. We also pose a question if these Alexander polynomials can be arbitrary integer polynomials of a given degree. A first attempt to answer the latter question is a preliminary analysis of antiparallel descendants of the 2-strand torus knots, which provide a nice set of examples for all values of the defect. The answer turns out to be positive in the case of defect zero knots, what can be observed already in the case of twist knots. This proved conjecture also allows us to provide a complete set of C-polynomials for the symmetrically colored Alexander polynomials for defect zero. In this case, we achieve a complete separation of representation and knot variables.

Russia

E. Lanina

A. Morozov

Defect and degree of the Alexander polynomial

We generalize a complex heavy-quark potential model from an isotropic QCD plasma to an anisotropic one by replacing the Debye mass m  D  with an anisotropic screening mass depending on the quark pair alignment with respect to the direction of anisotropy. Such an angle-dependent mass is determined by matching the perturbative contributions in the potential model to the exact result obtained in the Hard-Thermal-Loop resummed perturbation theory. An advantage of the resulting potential model is that its angular dependence can be effectively described by using a set of angle-averaged screening masses as proposed in our previous work. Consequently, one could solve a one-dimensional Schrödinger equation with a potential model built by changing the anisotropic screening masses into the corresponding angle-averaged ones, and reproduce the full three-dimensional results for the binding energies and decay widths of low-lying quarkonium bound states to very high accuracy. Finally, turning to dynamics, we demonstrate that the one-dimensional effective potential can accurately describe the time evolution of the vacuum overlaps obtained using the full three-dimensional anisotropic potential. This includes the splitting of different p-wave polarizations.

China

Lihua Dong

Yun Guo

Ajaharul Islam

Michael Strickland

The complex heavy-quark potential in an anisotropic quark-gluon plasma — Statics and dynamics

We study the interactions of a static quark-antiquark pair at nonzero temperature using realistic <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavor lattice QCD calculations. The study consists of two parts. The first investigates the properties of Wilson line correlators in Coulomb gauge and compares to predictions of hard-thermal loop perturbation theory. As a second step we extract the spectral functions underlying the correlators using four conceptually different methods: spectral function fits, a hard-thermal-loop-inspired fit for the correlation function, Padé rational approximation and the Bayesian reconstruction. We find that our high-statistics Euclidean lattice data are amenable to different hypotheses for the shapes of the spectral function and we compare the implications of each analysis method for the existence and properties of a well-defined ground state spectral peak.

Dibyendu Bala

Olaf Kaczmarek

Static quark-antiquark interactions at nonzero temperature from lattice QCD

In this paper we study the group theoretic structures of colored HOMFLY polynomials in a specific limit. The group structures arise in the perturbative expansion of <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> Chern-Simons Wilson loops, while the limit is <math><mi>N</mi><mo>→</mo><mn>0</mn></math>. The result of the paper is twofold. First, we explain the emergence of Kadomsev-Petviashvily (KP) τ-functions. This result is an extension of what we did in [1], where a symbolic correspondence between KP equations and group factors was established. In this paper we prove that integrability of the colored Alexander polynomial is due to it's relation to soliton τ-functions. Mainly, the colored Alexander polynomial is embedded in the action of the KP generating function on the soliton τ-function. Secondly, we use this correspondence to provide a rather simple combinatoric description of the group factors in term of Young diagrams, which is otherwise described in terms of chord diagrams, where no simple description is known. This is a first step providing an explicit description of the group theoretic data of Wilson loops, which would effectively reduce them to a purely topological quantity, mainly to a collection of Vassiliev invariants.

V. Mishnyakov

A. Sleptsov

Perturbative analysis of the colored Alexander polynomial and KP soliton -functions

This study explores the potential of modern implicit solvers for stochastic partial differential equations in the simulation of real-time complex Langevin dynamics. Not only do these methods offer asymptotic stability, rendering the issue of runaway solution moot, but they also allow us to simulate at comparatively large Langevin time steps, leading to lower computational cost. We compare different ways of regularizing the underlying path integral and estimate the errors introduced due to the finite Langevin time steps. Based on that insight, we implement benchmark (non-)thermal simulations of the quantum anharmonic oscillator on the canonical Schwinger-Keldysh contour of short real-time extent.

Stable solvers for real-time Complex Langevin

We compute the proper real-time interaction potential between a static quark and antiquark in classical lattice gauge theory at finite temperature. Our central result is the determination of the screened real-part of this potential, and we reconfirm the presence of an imaginary part. The real part is intimately related to the back-reaction of the static sources onto the gauge fields, incorporated via Gauss’s law. Differences in the treatment of static sources in quantum and classical lattice gauge theory are discussed.