RIKEN FY2025 Incentive Research Projects

1. Applicant (Project leader)

iTHEMS, Okuto Morikawa

2. Title of research project

Resurgence and non-perturbative formulation of resonances in quantum theory

3. Research project outline (approximately 100 words)

In recent years, there has been progress in understanding quasi-stationary states in quantum theory (or non-perturbative phenomena) through resurgence theory. On the other hand, for systems that cannot be normalized, such as quasi-stationary resonant states, various regularization prescriptions in nuclear phenomenology have achieved a certain level of success. The aim of this research is to attempt a non-perturbative formulation of resonances using exact WKB analysis; we hope this can provide a unified and systematic understanding of each regularization method. Namely, its broad picture from analyticity under resurgence is applicable to any quasi-stationary potential. Then, results of the exact WKB analysis are consistent with the observations from traditional S-matrix poles and other modified Hilbert spaces.

4. Contents of the research project (Please do not change the size (A4) of pages) 

[4-1] Research project objective and method (background, originality, significance, goals) (3-page limit):

• Background on particle and nuclear physics and difficulty of non-perturbative effect

 Quantum theory is the most plausible description of microscopic physics, such as atoms, nuclei, and elementary particles. There are, however, a few systems which can be solved exactly. Moreover, a relativistic framework for dynamical elementary particles, called quantum field theory, has not been defined rigorously in the mathematical sense. For small fluctuations from an exactly solvable model, perturbation theory, a universal approach to general physics, provides an effective method exploring phenomenological consequences. Almost any matter that we can observe, on the other hand, is the composition due to strong interactions. Such a fundamental and realistic structure is truly controlled by non-perturbative phenomena.

 A simple non-perturbative effect is tunneling through “hump” of potentials, such as the double-well potential. See the right figure. From the field theoretical point of view, the configuration called instanton of the classical solution moving in different local minima gives rise to quantum tunneling, whose evaluation of Feynman path integral is, for instance, given by a phase rotation of the complexified coupling constant. The total result of its evaluation should be regular as an analytic function of the coupling constant, while the perturbation series has no convergence region, that is, appears to be meaningful only for no interactions. Actually, this fact is quite natural because non-small perturbation causes instability of the original vacuum; the perturbation series is just an asymptotic series. Nevertheless, the higher-order perturbation can provide rich physical observations, which is related to the non-perturbative effect by instanton. Combining the perturbative and non-perturbative estimates in an appropriate way insures the regularity of the total system. This structure is called resurgence (or resurgent structure).

 Recently, some mathematical ideas of resurgence have been applied to quasi-stationary vacua in quantum physical systems. Topics are quite broad: bion effect (recently discovered classical solution), renormalon problem (more problematic behavior of perturbation especially for gauge field theory), supersymmetric and other exact-solvable models, symmetry preserving/breaking in the large N approximation, vacuum decay in astrophysics, and fundamental understanding of quasi-stationary states in quantum mechanics. The last one, which includes the double-well potential in quantum mechanics, may be described by a non-perturbative formulation, exact WKB analysis. This is an impressive approach to any quantum mechanical system via analyticity for solutions of Riccati differential equation. We discuss its analytical structure as Stokes graph on the complexified coordinate space, where we depict the so-called Stokes curve; moving in each region surrounded by Stokes curves occurs a drastically and discretely definite “transition” of solutions (Stokes phenomenon). For an end point of three Stokes curves, turning point, nontrivial path (or cycle) observes both of stability and instability, and gives expected solutions with correctly quantized energy. Analyticity on the complex plane determines the usual quantum systems as harmonic oscillator and also non-perturbative quantum phenomena as tunneling.

 When every physicist learns quantum mechanics textbook, a “hollow” of potential traps its physical state in quantized (discrete) energy spectrum, called bound state; unbound potential system seems to be non-physical. In contrast to this common knowledge, there exist many physical states which are unbounded but quasi-bounded, that is, infer quantized energy and finite decay rate from a partially bound and flat potential. A typical potential is depicted as the right figure. In fact, we know such observations in the real world: radioisotopes, neutron-rich nuclei, exotic hadrons, non-equilibrium systems, and non-Hermitian systems. We call this type of physical state a quasi-stationary resonant state, or resonance.

 Despite of this universality of resonance as mentioned above, every standard technique to solve quantum systems suffers from the difficulty of normalization of “would-be physical” states. Precisely speaking, this is because completion of state space (defining a required Hilbert space) is nonsense in such an open system as itself. Especially, in nuclear phenomenology, various regularization prescriptions of normalizability have been proposed in order to compute each ideal observable, where we extend and modify the notion of the Hilbert space, such as complex scaling method, tight binding approximation, Zel’dovich regularization, rigged Hilbert space and so on. While nuclear physics has achieved a certain high level of success theoretically and experimentally, it is indeed unclear how the definition of modified Hilbert space is related to each other. Also, we would have no explicit and transparent understanding of the whole resonance as physics because the completeness of the Hilbert space can be proved by using highly mathematical techniques [1].

 The aim of this research is to attempt the non-perturbative formulation of resonances using exact WKB analysis; we hope this can provide a unified and systematic understanding of each regularization method. Namely, the broad picture of the exact WKB analysis from analyticity under resurgent structure is applicable to any quasi-stationary potential even for resonance. If so, then, results of the exact WKB analysis should be consistent with the observations from traditional S-matrix poles and the above other modified Hilbert spaces.

• Originality of the field and technology between mathematics and particle/nuclear physics

 Resurgence was developed as the theory of differential equation. The research of exact WKB analysis had given for a kind of differential equation by mathematicians from 1980s [2]. Applications of resurgence technique to physics has been done mainly by particle physicists. This movement and tendency were started quite recently; the exact WKB analysis for the double-well potential was discussed in Ref. [3] a few years ago. Then, we now have a clearer understanding of quantum mechanical differential equation, Schrodinger equation, and match analyticity with quantization condition and other physical observations. On the other hand, nuclear physics has a very long history and a triumph of microscopic phenomenological description of quantum theory, and has paved the way for the nuclear resonance, e.g., in Ref. [4]. Of course, there is a large gap between phenomenological success and the mathematically defined background even in the present. The application of the exact WKB analysis is interesting for understanding it not to be hindered by any conceptual problem.

 This project is not so easy. At first, we wonder what is each step of the exact WKB analysis as the physical viewpoint. The naïve (usual) analyticity under Stokes phenomenon misses real-valued physics, and so some kind of principle should be clarified in the sense of unstable physics. If everything becomes clear, our exact WKB analysis under physical principles provides a completely non-perturbative formulation and definite computational advantages of arbitrary quantum mechanical system.

 The exact WKB analysis is not just a regularization prescription but a non-perturbative quantum framework. There are many and fine enough steps between physical observations, physical principles, and mathematical rigor. Every property, which is described by an appropriate nuclear phenomenological regularization, should be well governed simultaneously. Also, such a non-perturbative framework has provided computational approach with somehow rigorous numerical error estimates. We hope, when further developed, the numerical approach based on this research will clear our path to broader quantum phenomena and more on macroscopic physics.

• Significant and concrete goal of the project and task planning

 We aim to formulate quantum mechanics in a non-perturbative way. Any quantum mechanical system can be applicable; in particular, resonant state is an intriguing issue because of its mysterious structure. We now focus on the exact WKB analysis to do this. Main topics are as follows: the physical interpretation of Stokes graph should be explained in general; it should be clear how we choose a path for analytical continuation; integrating any reasonable value along the path is consistent with solving the Schrodinger equation. For resonant state, we need some modifications of solver of Schrodinger equation, whose Hilbert spaces are not understood from each other. To confirm the consistency in phenomenology will be achieved by using the exact WKB analysis.

 The research plan of my project is given in the following.

Reproducing the exact solution for the Rosen-Morse potential

 It is known that the quantum mechanics with the Rosen-Morse potential, 1/〖cosh〗^2 (x), is exactly solvable. Apparently there is no hollow but this system possesses resonance energies; this is called a barrier resonance. No explicit explanation for this phenomenon has been given, although the exact solution is resonant.

 Now, in the complex plane, the Rosen-Morse potential looks like the right figure (the potential Re[V(x)] is seen as a function of Re[x] and Im[x]). Then, we can see many hollows periodically at Im[x]≠0. Our proposal is that if Re[V(x)] possesses local minima which are not stable physical states, then the Hilbert space includes resonant states. This is because the structure (definition) of Stokes curve is sensitive to sign flips of √(V-E) with energy E.

 We can easily compute the resonance energy up to the leading order of the exact WKB analysis, as E=〖(1-in/√2)〗^2 with an integer n. Due to the truncation of perturbation series, this is not completely same as the exact value of energy E=c〖(a-i(2n+1))〗^2, where c and a are the definite constants, but those have almost similar structure. After computing the exact WKB analysis completely, we will have the exact resonance energy.

Usefulness of practical computation for other known resonances

 As a simple generalization, we can consider the double Rosen-Morse potential, 1/(〖cosh〗^2 (x-a) )+1/(〖cosh〗^2 (x+a) ), which have an apparent hollow and possesses the resonant states (not exactly solvable). Some preceding researches gave the resonance energy, and also we can. For other famous cases, we can check the consistency with our approach of the exact WKB analysis.

Comparison with the complex scaling method

 A well-used analysis for nuclear resonance is given by the complex scaling method. Here the radial direction is complexified as r→re^iθ. For large enough θ, the completeness of a modified Hilbert space can be proven. Now, as in the right figure, continuum states (scattering) are rotating by 2θ while all states are in an upper region on the line connecting the bound states and the continuum states. Then, the normalizability is recovered. This idea is a little similar to the exact WKB analysis because in both cases we consider the analyticity of the given potential on the complex plane. We want to show this similarity with a practical meaning. We hope the exact WKB analysis provides the physical interpretation of the complex scaling.

Hilbert space theory and completeness in the exact WKB analysis

 The above similarity to the complex scaling method leads us to defining a complete norm space from the perspective of the exact WKB analysis. Reference [1] is good preceding research for us. The exact WKB analysis answers the solution of the Schrodinger equation, that is, the wave function. We need to prove that the space of wave functions should be normalizable and complete, and then it turns out that such a modification of the Hilbert space for resonance is physically required.

Consistency check with every modified Hilbert space

Finally, this project will go to a unified understanding of each regularized Hilbert space.

Numerical approach to arbitrary quantum mechanics

 Optionally, we can develop a numerical method in the exact WKB method. The good point is that this can be applied to arbitrary quantum mechanical system. A universal computational method is important as itself. At least, because the exact WKB method is a hybrid of perturbation theory and the Borel resummation, error estimate of asymptotic series is valid. Moreover, I addressed differences and precisions between many other resummation techniques in Ref. [5]. The core part, algorithm, is left as a future problem.

References

[1] J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971) 269–279; E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971) 280–294.

[2] A. Voros, Ann. Inst. Henri Poincaré 39, no. 3, (1983) 211–338.

[3] N. Sueishi, S. Kamata, T. Misumi, and M. Unsal, JHEP 12 (2020) 114.

[4] Y. B. Zel’dovich, Sov. Phys. JETP 12 (1961) 542.

[5] O. Morikawa, https://github.com/o-morikawa/Resummation

[4-2] Difference between your proposed research project for the Incentive Research Projects grant and the work you are currently undertaking in your laboratory (0.5-page limit):

 In this decade, the notion of symmetry, a fundamental tool of physics, has been developed. This idea provides a new approach to analysis of non-perturbative effect from a quite different perspective. At first, I studied the vacuum structure of gauge field theory from this generalized symmetry, while I and collaborators made use of the so-called ‘t Hooft anomaly matching, which bridges weekly and strongly coupled field theories. In spite of its success, however, I found this approach is not transparent because of highly mathematical “consensus” by experts. One explicit and clear consideration for everybody can be deduced by the most well-established non-perturbative framework, lattice gauge theory. Then, I have implemented the generalized symmetry and proved non-perturbative phenomena from the lattice regularized point of view. In those studies, there is no complicated technology (i.e., hybrid of physical sense and mathematical techniques) and no subtlety because simple computational operations are insured by the completely non-perturbative regularization. This is my current research work.

 On the other hand, around 2020, my research interests went to another non-perturbative formulation, that is, resurgence. At that time, the industry of understanding non-perturbative effect from perturbation theory was much confusing in field theory. In resurgence theory, while perturbative and non-perturbative analyses are ambiguous, the total observation is well-defined due to ambiguity cancellation mechanism. The conjecture that bion ambiguity is cancelled against renormalon problem had been focused so much, but there were many positive and negative results to it. I resolved this issue and found that the original conjecture is not correct. Then, when I had been invited at a lot of places, I learned many various non-perturbative techniques, ideas and theories of resurgence theory. This project for the Incentive Research Projects Grant was started after I have noticed the similar problem of recent non-Hermitian physics (or traditional nuclear physics) to analyticity under resurgence, especially, the exact WKB analysis.

[4-3] Roles to be fulfilled by the project leader and project member(s) (0.5-page limit):

None.

[4-4] Breakdown of grant amount, rationale for each item and execution plan (0.5-page limit)

 The maximum amount, 3 million yen, is needed. Actually, there are a few researchers who understand the exact WKB analysis as physics and can solve the Schrodinger equation along this method. Also, the number of nuclear physicists who can solve/compute the resonant state is not so many in a sense of technicians understanding mathematical completion and analyticity of hypergeometric function. This means that I should have some long-term business trips to visit, e.g., Prof. Dunne (U. of Connecticut) and Prof. Unsal (U. of North Carolina); I need much personnel expenses too.

One business trip: 500 thousand yen (abroad)

Twice travels/year at abroad indicates 2 million yen. Also, any analytic computation requires the use of special functions, mainly hypergeometric function. To this end, Mathematica is an important powerful tool; this license cost 300 thousand yen.

 I hope that many particle/nuclear physicists and mathematicians are interested in this research project. I want to create a big research network and some actual studying groups (at first, in Japan). A diversified workshop will be held at RIKEN or Kyushu U. (theoretical nuclear physics group).

Workshop support: 700 thousand yen

Totally, 3 million yen will be used.

[4-5] List of major research papers within the past 5 years (since 2020) (2-page limit):

Applying the large N approximation to infamous renormalon problem

1. K. Ishikawa, O. Morikawa, A. Nakayama, K. Shibata, H. Suzuki and H. Takaura,  ``Infrared renormalon in the supersymmetric CP^{N-1} model on RxS^1,''  PTEP 2020, no.2, 023B10 (2020).

2. M. Ashie, O. Morikawa, H. Suzuki, H. Takaura and K. Takeuchi,  ``Infrared renormalon in SU(N) QCD(adj.) on R^3xS^1,''  PTEP 2020, no.2, 023B01 (2020).

3. K. Ishikawa, O. Morikawa, K. Shibata, H. Suzuki and H. Takaura,  ``Renormalon structure in compactified spacetime,''  PTEP 2020, no.1, 013B01 (2020).

4. K. Ishikawa, O. Morikawa, K. Shibata and H. Suzuki,  ``Vacuum energy of the supersymmetric CP^{N-1} model on RxS^1 in the 1/N expansion,''  PTEP 2020, no.6, 063B02 (2020).

5. M. Ashie, O. Morikawa, H. Suzuki and H. Takaura,  ``More on the infrared renormalon in SU(N) QCD(adj.) on R^3xS^1,''  PTEP 2020, no.9, 093B02 (2020).

Identification of resurgent structure in compactified spacetime

1. O. Morikawa and H. Takaura,  ``Identification of perturbative ambiguity canceled against bion,''  PLB 807, 135570 (2020).

Non-perturbative analysis of phase structure by generalized symmetry and ‘t Hooft anomaly

1. O. Morikawa, H. Wada and S. Yamaguchi,  ``Phase structure of linear quiver gauge theories from anomaly matching,'' PRD 107, 045020 (2023).

Lattice non-perturbative implementation of generalized symmetry

1. M. Abe, O. Morikawa and H. Suzuki,  ``Fractional topological charge in lattice Abelian gauge theory,''  PTEP 2023, no.2, 023B03 (2023).

2. M. Abe, O. Morikawa, S. Onoda, H. Suzuki and Y. Tanizaki,  ``Topology of SU(N) lattice gauge theories coupled with Z_N 2-form gauge fields,''  JHEP 08, 118 (2023).

3. Y. Honda, O. Morikawa, S. Onoda and H. Suzuki,  ``Lattice realization of the axial U(1) non-invertible symmetry,''  PTEP 2024, no.4, 043B04 (2024).

Observations of non-perturbative effect under generalized symmetry from lattice description

1. N. Kan, O. Morikawa*, Y. Nagoya and H. Wada,  ``Higher-group structure in lattice Abelian gauge theory under instanton-sum modification,''  EPJC 83, no.6, 481 (2023) [Erratum: EPJC 84, no.1, 22 (2024)].

2. M. Abe, O. Morikawa* and S. Onoda,  ``Note on lattice description of generalized symmetries in SU(N)/Z_N gauge theories,''  PRD 108, 014506 (2023).

Lattice construction of magnetic objects and observation of Witten effect

1. M. Abe, O. Morikawa, S. Onoda, H. Suzuki and Y. Tanizaki,  ``Magnetic operators in 2D compact scalar field theories on the lattice,''  PTEP 2023, no.7, 073B01 (2023).

2. O. Morikawa, S. Onoda and H. Suzuki,  ``Yet another lattice formulation of 2D U(1) chiral gauge theory via bosonization,''  PTEP 2024, no.6, 063B01 (2024).

[4-6] List of other funding since 2023 and relation with your proposed research project for the Incentive Research Projects grant (2-page limit):

As mentioned in [4-2], in recent years, I have studied the recent development of non-perturbative analysis via generalized symmetry, and tried to implement it in lattice gauge theory, a completely regularized framework. To do this task, my work has been supported by the following grants:

• Japan Society for the Promotion of Science (JSPS)

Grant-in-Aid for Scientific Research Grant (KAKENHI)

Title: Nonperturbative method for quantum field theory and its application to superstring theory

Number: JP22KJ2096

Period: 4/2021-3/2024

• Grant-in-Aid for Special Postdoctoral Researcher at RIKEN

Title: Rigorous formulation of generalized symmetry from the viewpoint of lattice gauge theory

Period: 4/2024-present

My research proposed here, a quite new project, is supposed to be supported by the Incentive Research Projects Grant from 2025.