Stabilizing complex Langevin for real-time gauge theories with an anisotropic kernel

Abstract

The complex Langevin (CL) method is a promising approach to overcome the sign problem that occurs in real-time formulations of quantum field theories. Using the Schwinger-Keldysh formalism, we study SU(Nc) gauge theories with CL. We observe that current stabilization techniques are insufficient to obtain correct results. Therefore, we revise the discretization of the CL equations on complex time contours, find a time reflection symmetric formulation and introduce a novel anisotropic kernel that enables CL simulations on discretized complex time paths. Applying it to SU(2) Yang-Mills theory in 3+1 dimensions, we obtain unprecedentedly stable results that we validate using additional observables and that can be systematically improved. For the first time, we are able to simulate non-Abelian gauge theory on time contours whose real-time extent exceeds its inverse temperature. Thus, our approach may pave the way towards an ab-initio real-time framework of QCD in and out of equilibrium with a potentially large impact on the phenomenology of heavy-ion collisions.

Early time dynamics and constraints on medium evolution

“…Previous studies of SU(2) theory in thermal equilibrium in 3+1 dimensions showed that usual CL simulations often converge to wrong values [36]. Exploiting the kernel freedom of CL, a new anisotropic kernel was introduced in [37] that improves the stability of the stochastic Langevin process and leads to correctly converged results on complex time paths. By going to real times, the CL method with the new kernel has the potential to allow us to extract transport coefficients in QCD directly and eventually to study the non-equilibrium dynamics of QCD from first principles.…”

Advancing real-time Yang-Mills: towards real-time observables from first principles

“…Details on the derivation of the CL formalism and the drift term on general complex time contours can be found in our recent work [3].…”

“…This allows the transformation of the drift and the noise term without altering the stationary solution of the stochastic differential equation, but it can drastically improve the stability of the dynamics [14,15]. In particular, we apply the anisotropic kernel that is motivated by a contour parametrizationdependent CL formulation and was derived in our earlier paper [3]. This kernel effectively rescales…”

“…In this conference contribution, we use the complex Langevin (CL) method that is based on stochastic quantization [2]. We present results from our recent studies [3,4] for non-Abelian SU(𝑁 𝑐 ) gauge theories on complex time contours. As detailed there, we introduce an anisotropic kernel that originates from a careful rederivation of the CL equation and stabilizes the CL dynamics.…”

Highly anisotropic lattices for Yang-Mills theory

“…To address this, we adopt the methodology introduced in [5], which relies on the equipartition of the chromo-electric and -magnetic fields within the confined phase of QCD. This work is mainly motivated by complex Langevin simulations of Yang-Mills theory for real-time observables [6,7] where increasing the lattice anisotropy has been shown to systematically improve stability [8,9]. This may open the door for calculating real-time observables and the QCD equation of state at finite density, which suffer from severe sign problems and currently impede direct calculations.…”