CP invariance is a very attractive solution to the strong CP problem in quantum chromodynamics (QCD). This solution requires the vanishing <inline-formula><tex-math id='TM0001' notation='LaTeX'>${\rm arg}\, [{\rm det}\, M_d\, {\rm det} M_u]$</tex-math></inline-formula>, where <inline-formula><tex-math id='TM0002' notation='LaTeX'>$M_d$</tex-math></inline-formula> and <inline-formula><tex-math id='TM0003' notation='LaTeX'>$M_u$</tex-math></inline-formula> are the mass matrices for the down- and up-type quarks, respectively. It happens if we have several zeros in the quark mass matrices. We proceed with a systematic construction, in this paper, of two-zeros textures for the down-type quark mass matrix while the mass matrix for the up-type quarks is always diagonal. We find only three types of the mass matrices can explain the observed Cabibbo&#8211;Kobayashi&#8211;Maskawa (CKM) matrix, the masses of the quarks and the charged leptons, and the small enough vacuum angle <inline-formula><tex-math id='TM0004' notation='LaTeX'>$\theta &lt; 10^{-10}$</tex-math></inline-formula>. We extend the mass construction to the neutrino sector and derive predictions on the CP-violating parameter <inline-formula><tex-math id='TM0005' notation='LaTeX'>$\delta _{\mathrm{ CP}}$</tex-math></inline-formula> in the neutrino oscillation and the mass parameter <inline-formula><tex-math id='TM0006' notation='LaTeX'>$m_{\beta \beta }$</tex-math></inline-formula>. It is extremely remarkable that the normal (NH) and inverted (IH) hierarchies in the neutrino masses are equally possible in the case where we introduce only two right-handed neutrinos <italic>N</italic>s. Furthermore, we have a strict prediction that the <inline-formula><tex-math id='TM0008' notation='LaTeX'>$\delta _{\mathrm{ CP}} \simeq 200^\circ$</tex-math></inline-formula> or <inline-formula><tex-math id='TM0009' notation='LaTeX'>$250^\circ$</tex-math></inline-formula> in the NH case. If it is the case we can naturally explain the positive sign of the baryon asymmetry in the present universe.

Japan

Morimitsu Tanimoto

China

Tsutomu T Yanagida

Axionless Solution to the Strong CP Problem—Two-Zeros Textures of the Quark and Lepton Mass Matrices and Neutrino CP Violation

We propose a systematic approximation scheme to calculate general string-tree level <italic>n</italic>-point hard string scattering amplitudes ( <inline-formula><tex-math id='TM0002' notation='LaTeX'>$\text{HSSA}$</tex-math></inline-formula>) of open bosonic string theory. This stringy scaling loop expansion contains a finite number of vacuum diagram terms at each loop order of scattering energy due to a vacuum diagram contraint and a topological graph constraint. In addition, we calculate the coefficient and give the vacuum diagram representation and its Feynman rules for each term in the expansion of the <inline-formula><tex-math id='TM0003' notation='LaTeX'>$\text{HSSA}$</tex-math></inline-formula>. As an application to extending our previous calculation of <italic>n</italic>-point leading-order stringy scaling behavior of <inline-formula><tex-math id='TM0005' notation='LaTeX'>$\text{HSSA}$</tex-math></inline-formula>, we explicitly calculate some examples of 4-point next-to-leading-order stringy scaling violation terms.

not found

Sheng-Hong Lai

Jen-Chi Lee

Yi Yang

The Stringy Scaling Loop Expansion and Stringy Scaling Violation

I found an extended duality (triality) between Dirac fermions in periodic spacetime metrics, nonrelativistic fermions in gauge fields (e.g. Harper–Hofstadter model), and in periodic scalar fields on a lattice (e.g. Aubry–André model). This indicates an unexpected equivalence between spacetime metrics, gauge fields, and scalar fields on the lattice, which are understood as different physical representations of the same mathematical object, the quantum group <inline-formula><tex-math id='TM0001' notation='LaTeX'>$\mathcal {U}_q(\mathfrak {sl}_2)$</tex-math></inline-formula>. This quantum group is generated by the exponentiation of two canonical conjugate operators, namely a linear combination of position and momentum (periodic spacetime metrics), the two components of the gauge-invariant momentum (gauge fields), and position and momentum (periodic scalar fields). Hence, on a lattice, Dirac fermions in a periodic spacetime metric are equivalent to nonrelativistic fermions in a periodic scalar field after a proper canonical transformation. The three lattice Hamiltonians (periodic spacetime metric, Harper&#8211;Hofstadter, and Aubry&#8211;Andr&#233;) share the same properties, namely fractal phase diagrams, self-similarity, <italic>S</italic>-duality, topological invariants, flat bands, and topologically quantized current in the incommensurate regimes. In essence, this work unveils an unexpected link between gravity and gauge fields, opens new avenues for studying analog gravity&#8212;e.g. the Unruh effect and universe expansions/contractions, suggests the existence of an <italic>S</italic>-duality of spacetime curvatures, and hints at novel pathways to quantized gravity theories.

Pasquale Marra

Gauge Fields Induced by Curved Spacetime

We study quasi-stationary states in quantum mechanics using the exact Wentzel–Kramers–Brillouin (WKB) analysis as a nonperturbative framework. Whereas previous works focused mainly on stable systems, we explore unstable states such as resonances. As a concrete example, we analyze the inverted Rosen–Morse potential, which exhibits barrier resonance. This model allows exact solutions, enabling a direct comparison with exact WKB predictions. We provide a simple analytic picture of resonance and demonstrate consistency between exact and WKB-based results, extending the applicability of exact WKB analysis to nonpolynomial potentials.

Okuto Morikawa

Shoya Ogawa

Nonperturbative Formulation of Resonances in Quantum Mechanics Based on Exact WKB Method

In this paper, we examine the properties of the solutions obtained by the Schwinger–Dyson equation (SDE). As a simple example, we consider a 2D model including four-fermion interaction. It is shown that when this model is solved by an iterative method using the SDE at finite density, multiple solutions depending on the initial input values are obtained. We investigate the reasons for this situation by examining the convergence of the solutions obtained by the iterative method. We also consider the solution of the SDE using alternative methods. Furthermore, we compare these results with analytical solutions at zero temperature and discuss the relation with the behavior of the effective potential. We extend these considerations to finite temperatures. Based on these analyses, we consider ways to avoid spurious solutions that appear in the iterative method.

Hidekazu Tanaka

Shuji Sasagawa

Properties of Solutions by the Schwinger–Dyson Equation at Finite Temperature and Density—A Four-Fermion Interaction Model

We propose a new type of lepton seesaw model introducing a modular <italic>A</italic> <sub>4</sub> flavor symmetry in which isospin doublet vector fermions play an important role in constructing seesaw mechanisms for both the charged-lepton mass matrix and the neutrino one. The charged-lepton mass matrix is induced via the Dirac seesaw with a two-by-two block mass matrix. On the other hand, the neutrino mass matrix is generated via a seesaw with four-by-four block mass matrix where the right-handed neutral fermions are also added. Remarkably, the neutral fermion mass matrix provides us with a new type of seesaw formula for the active neutrino mass. Through our chi-squared analysis, we discuss some tendencies of observables, focusing on two fixed points <inline-formula><tex-math id='TM0001' notation='LaTeX'>$\tau =i,\ \omega$</tex-math></inline-formula>.

Takaaki Nomura

Hiroshi Okada

A New Type of Lepton Seesaw Model in a Modular <italic>A</italic> <sub>4</sub> Symmetry

Randomly connected tensor networks (RCTN) are dynamical systems defined by summing over all the possible networks of tensors. Because of the absence of fixed lattice structure, RCTN is not expected to have renormalization procedures. In this paper, however, we consider RCTN with a real tensor, and it is proven that a Hamiltonian vector flow of a tensor model in the canonical formalism with a positive cosmological constant has the properties that a renormalization group (RG) flow of RCTN would have: The flow has fixed points on phase transition surfaces; every flow line is asymptotically terminated by fixed points at both ends, where an upstream fixed point has higher criticality than a downstream one; the flow goes along phase transition surfaces; there exists a function that monotonically decreases along the flow, analogously to the <inline-formula><tex-math id='TM0001' notation='LaTeX'>$a$</tex-math></inline-formula>- and <inline-formula><tex-math id='TM0002' notation='LaTeX'>$c$</tex-math></inline-formula>-functions of RG. A complete classification of fixed points is given. Although there are no cyclic flows in the strict sense, they exist, if infinitesimal jumps are allowed near fixed points.

Naoki Sasakura

Renormalization Group-Like Flows in Randomly Connected Tensor Networks

We investigate the axial anomaly in Hamiltonian lattice gauge theory. The definition of axial charge operators is ambiguous, especially between conserved and nonconserved axial charges. While these charges appear to differ only by a higher-order term in lattice spacing, they do not coincide in the continuum limit. We demonstrate, through analytical and numerical calculations in <inline-formula><tex-math id='TM0001' notation='LaTeX'>$1+1$</tex-math></inline-formula> dimensions, that the conserved axial charge correctly reproduces the axial anomaly relation in continuous spacetime. Our finding serves as a valuable lesson about the doubler artifact in the Hamiltonian time evolution of lattice gauge theory.

Yoshimasa Hidaka

Arata Yamamoto

Anomaly of conserved and nonconserved axial charges in Hamiltonian lattice gauge theory

The gradient flow exact renormalization group (GFERG) is a variant of the exact renormalization group (ERG) for gauge theory that is aimed at preserving gauge invariance as manifestly as possible. It achieves this goal by utilizing the Yang–Mills gradient flow or diffusion for the block-spin process. In this paper, we formulate GFERG by the Reuter equation in which the block spinning is done by Gaussian integration. This formulation provides a simple understanding of various points of GFERG, unresolved thus far. First, there exists a unique ordering of functional derivatives in the GFERG equation that removes the ambiguity of contact terms. Second, perturbation theory of GFERG suffers from unconventional ultraviolet (UV) divergences if no gauge fixing is introduced. This explains the origin of some UV divergences that we have encountered in perturbative solutions to GFERG. Third, the modified correlation functions calculated with the Wilson action in GFERG coincide with the correlation functions of diffused or flowed fields calculated with the bare action. This shows the existence of a Wilson action that reproduces precisely the physical quantities computed by the gradient flow formalism (up to contact terms). We obtain a definite ERG interpretation of the gradient flow. The formulation given in this paper provides a basis for further perturbative/nonperturbative computations in GFERG, preserving gauge invariance maximally.

United States

Hidenori Sonoda

Hiroshi Suzuki

A Basis of the Gradient Flow Exact Renormalization Group for Gauge Theory

Particle–antiparticle interpretation under spatially inhomogeneous external fields within the framework of quantum field theory is a nontrivial problem. In this paper, we focus on the two interpretations established by Gavrilov and Gitman (Phys. Rev. D <bold>93</bold>, 045002 (2016)) and by Nakazato and Ochiai (Prog. Theor. Exp. Phys. <bold>2022</bold>, 073B02 (2022)), both of which give consistent results for vacuum instability and pair production. To shed light on their differences, a pair production under a potential step assisted by a weak and oscillating electric field is discussed. It is shown that the potential step and the oscillating field, each insufficient for vacuum decay, can produce pairs when combined. In addition, the two pictures give rise to quantitative differences in the number of created pairs at the second-order perturbation of the oscillating field. This might provide a clue to investigate the correct particle&#8211;antiparticle interpretation by comparing the result with numerical simulations or experiments.