<p>We consider resurgence for the nonconformal Bjorken flow with Fermi-Dirac and Bose-Einstein statistics in the extended relaxation-time approximation. First, we examine the full formal transseries expanded around the equilibrium and then construct the resurgent relation by looking at the structure of Borel-transformed ordinary differential equations. We form a conjecture of the resurgent relation based on the consideration that the Stokes constants constituting the resurgent relation originate only from singularities of dissipative variables on the Borel plane and that the other variables such as temperature and chemical potential become Borel nonsummable through nonlinear terms with the dissipative variables. We numerically check the conjecture for fundamental variables by explicitly evaluating the values of the dominant Stokes constant depending on the initial conditions and the particle mass. We also comment on some issues related to transseries structure and resurgence, such as the case of broken U(1) symmetry, the massless case, generalized relaxation time, and attractor solutions.</p>

Poland

Japan

Syo Kamata

Resurgence for the nonconformal Bjorken flow with Fermi-Dirac and Bose-Einstein statistics

We study exact Wentzel–Kramers–Brillouin analysis (EWKB) for a <math><mi>P</mi><mi>T</mi></math> symmetric quantum mechanics (QM) defined by the potential <math><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>P</mi><mi>T</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>g</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn><mi>K</mi></mrow></msup><mo>(</mo><mi>i</mi><mi>x</mi><msup><mrow><mo>)</mo></mrow><mrow><mi>ϵ</mi></mrow></msup></mrow></math> with <math><mi>ω</mi><mo>∈</mo><msub><mi>R</mi><mrow><mo>≥</mo><mn>0</mn></mrow></msub></math>, <math><mi>g</mi><mo>∈</mo><msub><mi>R</mi><mrow><mo>&gt;</mo><mn>0</mn></mrow></msub></math> and <math><mi>K</mi><mo>,</mo><mi>ϵ</mi><mo>∈</mo><mi>N</mi></math> to clarify its perturbative/nonperturbative structure. In our analysis, we mainly consider the massless cases, i.e., <math><mi>ω</mi><mo>=</mo><mn>0</mn></math>, and derive the exact quantization conditions (QCs) for arbitrary <math><mo>(</mo><mi>K</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo></math> including all perturbative/nonperturbative corrections. From the exact QCs, we clarify full transseries structure of the energy spectra with respect to the inverse energy level expansion, and then formulate the Gutzwiller trace formula, the spectral summation form, and the Euclidean path integral. For the massive cases, i.e., <math><mi>ω</mi><mo>&gt;</mo><mn>0</mn></math>, we show the fact that, by requiring the existence of the solution of the exact QCs, the path of analytic continuation in EWKB is uniquely determined for a given <math><mi>N</mi><mo>=</mo><mn>2</mn><mi>K</mi><mo>+</mo><mi>ϵ</mi></math>, and in consequence the exact QCs, the energy spectra, and the three formulas are all perturbative. Similarities to Hermitian QMs and resurgence are also discussed as additional remarks.

Exact quantization conditions and full transseries structures for <math><mi>P</mi><mi>T</mi></math> symmetric anharmonic oscillators

We consider exact Wentzel-Kramers-Brillouin analysis to a <math><mi>P</mi><mi>T</mi></math> symmetric quantum mechanics defined by the potential, <math><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>ω</mi><mn>2</mn></msup><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>g</mi><msup><mi>x</mi><mn>2</mn></msup><mo>(</mo><mi>i</mi><mi>x</mi><msup><mo>)</mo><mrow><mi>ϵ</mi><mo>=</mo><mn>2</mn></mrow></msup></math> with <math><mi>ω</mi><mo>∈</mo><msub><mi>R</mi><mrow><mo>≥</mo><mn>0</mn></mrow></msub></math>, <math><mi>g</mi><mo>∈</mo><msub><mi>R</mi><mrow><mo>&gt;</mo><mn>0</mn></mrow></msub></math>. We in particular aim to verify a conjecture proposed by Ai-Bender-Sarkar (ABS), that pertains to a relation between <math><mi>D</mi></math>-dimensional <math><mi>P</mi><mi>T</mi></math>-symmetric theories and analytic continuation (AC) of Hermitian theories concerning the energy spectrum or Euclidean partition function. For the purpose, we construct energy quantization conditions by exact Wentzel-Kramers-Brillouin analysis and write down their transseries solution by solving the conditions. By performing alien calculus to the energy solutions, we verify validity of the ABS conjecture and seek a possibility of its alternative form by Borel resummation theory if it is violated. Our results claim that the validity of the ABS conjecture drastically changes depending on whether <math><mi>ω</mi><mo>&gt;</mo><mn>0</mn></math> or <math><mi>ω</mi><mo>=</mo><mn>0</mn></math>: If <math><mi>ω</mi><mo>&gt;</mo><mn>0</mn></math>, then the ABS conjecture is violated when exceeding the semiclassical level of the first nonperturbative order, but its alternative form is constructable by Borel resummation theory. The <math><mi>P</mi><mi>T</mi></math> and the AC energies are related to each other by a one-parameter Stokes automorphism, and a median resummed form, which corresponds to a formal exact solution, of the AC energy (resp. <math><mi>P</mi><mi>T</mi></math> energy) is directly obtained by acting Borel resummation to a transseries solution of the <math><mi>P</mi><mi>T</mi></math> energy (resp. AC energy). If <math><mi>ω</mi><mo>=</mo><mn>0</mn></math>, then, with respect to the inverse energy level-expansion, not only perturbative/nonperturbative structures of the <math><mi>P</mi><mi>T</mi></math> and the AC energies but also their perturbative parts do not match with each other. These energies are independent solutions, and no alternative form of the ABS conjecture can be reformulated by Borel resummation theory.

Exact WKB analysis for <math><mrow><mi>P</mi><mi>T</mi></mrow></math>-symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture

We discuss all-order transseries in one of the simplest quantum mechanical systems: a U(1) symmetric single-degree-of-freedom system with a first-order time derivative term. Following the procedure of the Lefschetz thimble method, we explicitly evaluate the path integral for the generating function of the Noether charge and derive its exact transseries expression. Using the conservation law, we find all the complex saddle points of the action, which are responsible for the nonperturbative effects and the resurgence structure of the model. The all-order power-series contributions around each saddle point are generated from the one-loop determinant with the help of the differential equations obeyed by the generating function. The transseries are constructed by summing up the contributions from all the relevant saddle points, which we identify by determining the intersection numbers between the dual thimbles and the original path integration contour. We confirm that the Borel ambiguities of the perturbation series are canceled by the nonperturbative ambiguities originating from the discontinuous jumps of the intersection numbers. The transseries computed in the path-integral formalism agrees with the exact generating function, whose explicit form can be obtained in the operator formalism thanks to the integrable nature of the model. This agreement indicates the nonperturbative completeness of the transseries obtained by the semiclassical expansion of the path integral based on the Lefschetz thimble method.

Toshiaki Fujimori

Tatsuhiro Misumi

Muneto Nitta

Norisuke Sakai

All-order resurgence from complexified path integral in a quantum mechanical system with integrability

Quantum deformed potentials arise naturally in quantum mechanical systems of one bosonic coordinate coupled to <math><msub><mi>N</mi><mi>f</mi></msub></math> Grassmann valued fermionic coordinates, or to a topological Wess-Zumino term. These systems decompose into sectors with a classical potential plus a quantum deformation. Using exact Wentzel-Kramers-Brillouin, we derive exact quantization condition and its median resummation. The solution of median resummed form gives physical Borel-Ecalle resummed results, as we show explicitly in quantum deformed double- and triple-well potentials. Despite the fact that instantons have finite actions, for generic quantum deformation, they do not contribute to the energy spectrum at leading order in semiclassics. For certain quantized quantum deformations, where the alignment of levels to all order in perturbation theory occurs, instantons contribute to the spectrum. If deformation parameter is not properly quantized, their effect disappears, but higher-order effects in semiclassics survive. In this sense, we classify saddle contributions as fading and robust. Finally, for quantum deformed triple-well potential, we demonstrate the perturbative/nonperturbative relation, by computing period integrals and Mellin transform.

Naohisa Sueishi

United States

Mithat Ünsal

Exact WKB analysis for SUSY and quantum deformed potentials: Quantum mechanics with Grassmann fields and Wess-Zumino terms

We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with N minima on S 1. We describe the Stokes graphs of a general potential problem as a network of Airy-type or degenerate Weber-type building blocks, and provide a dictionary between the two. The two formulations are equivalent, but with their own pros and cons. Exact-WKB produces the quantization condition consistent with the known conjectures and mixed anomaly. The quantization condition for the case of N-minima on the circle factorizes over the Hilbert sub-spaces labeled by discrete theta angle (or Bloch momenta), and is consistent with ’t Hooft anomaly for even N and global inconsistency for odd N. By using Delabaere-Dillinger-Pham formula, we prove that the resurgent structure is closed in these Hilbert subspaces, built on discrete theta vacua, and by a transformation, this implies that fixed topological sectors (columns of resurgence triangle) are also closed under resurgence.

Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on S 1

Abstract We study a resurgence structure of a quantum field theory with a phase transition to uncover relations between resurgence and phase transitions. In particular, we focus on three-dimensional  supersymmetric quantum electrodynamics (SQED) with multiple hypermultiplets, where a second-order quantum phase transition has recently been proposed in the large-flavor limit. We provide interpretations of the phase transition from the viewpoints of Lefschetz thimbles and resurgence. For this purpose, we study the Lefschetz thimble structure and properties of the large-flavor expansion for the partition function obtained by the supersymmetric localization. We show that the second-order phase transition is understood as a phenomenon where a Stokes and an anti-Stokes phenomenon occur simultaneously. The order of the phase transition is determined by how saddles collide at the critical point. In addition, the phase transition accompanies an infinite number of Stokes phenomena due to the supersymmetry. These features are appropriately mapped to the Borel plane structures as the resurgence theory expects. Given the lessons from SQED, we provide a more general discussion on the relationship between the resurgence and phase transitions. In particular, we show how the information on the phase transition is decoded from the Borel resummation technique.

Masazumi Honda

Takuya Yoda

Quantum phase transition and resurgence: Lessons from three-dimensional Supersymmetric Quantum Electrodynamics

There are two well-known approaches to studying nonperturbative aspects of quantum mechanical systems: saddle point analysis of the partition functions in Euclidean path integral formulation and the exact-WKB analysis based on the wave functions in the Schrödinger equation. In this work, based on the quantization conditions obtained from the exact-WKB method, we determine the relations between the two formalism and in particular show how the two Stokes phenomena are connected to each other: the Stokes phenomenon leading to the ambiguous contribution of different sectors of the path integral formulation corresponds to the change of the “topology” of the Stoke curves in the exact-WKB analysis. We also clarify the equivalence of different quantization conditions including Bohr-Sommerfeld, path integral and Gutzwiller’s ones. In particular, by reorganizing the exact quantization condition, we improve Gutzwiller’s analysis in a crucial way by bion contributions (incorporating complex periodic paths) and turn it into an exact result. Furthermore, we argue the novel meaning of quasi-moduli integral and provide a relation between the Maslov index and the intersection number of Lefschetz thimbles.

China

On exact-WKB analysis, resurgent structure, and quantization conditions

Nonequilibrium Green’s functions provide an efficient way to describe the evolution of the energy-momentum tensor during the early-time preequilibrium stage of high-energy heavy-ion collisions. Besides their practical relevance they also provide a meaningful way to address the question when and to what extent a hydrodynamic description of the system becomes applicable. Within the kinetic theory framework we derive a new method to calculate time-dependent nonequilibrium Green’s functions describing the evolution of energy and momentum perturbations on top of an evolving far-from-equilibrium background. We discuss the approach toward viscous hydrodynamics along with the emergence of various scaling phenomena for conformal systems. By comparing our results obtained in the relaxation time approximation to previous calculations in Yang-Mills kinetic theory, we further address the question which macroscopic features of the energy-momentum tensor are sensitive to the underlying microscopic dynamics.

Mauricio Martinez

Germany

Philip Plaschke

Stephan Ochsenfeld

Sören Schlichting

Hydrodynamization and nonequilibrium Green’s functions in kinetic theory

In this work we introduce the generic conditions for the existence of a non-equilibrium attractor that is an invariant manifold determined by the long-wavelength modes of the physical system. We investigate the topological properties of the global flow structure of the Gubser flow for the Israel-Stewart theory and a kinetic model for the Boltzmann equation by employing Morse-Smale theory. We present a complete classification of the invariant submanifolds of the flow and determine all the possible flow lines connecting any pair of UV/IR fixed points. The formal transseries solutions to the Gubser dynamical system around the early-time (UV) and late-time (IR) fixed points are constructed and analyzed. It is proven that these solutions are purely perturbative (or power-law asymptotic) series with a finite radius of convergence. Based on these analyses, we find that Gubser-like expanding kinetic systems do not hydrodynamize owing to the failure of the hydrodynamization process which heavily relies on the classification of (non)hydrodynamic modes in the IR regime. This is in contrast to longitudinal boost-invariant plasmas where the asymptotic dynamics is described by a few terms of the hydrodynamic gradient expansion. We finally compare our results for both Bjorken and Gubser conformal kinetic models.