If a gauge field in quantum chromodynamics (QCD) has a singularity of the Dirac type, the non-Abelian Bianchi identity is violated. The violation of the non-Abelian Bianchi identity (VNABI) <inline-formula><tex-math id='TM0001' notation='LaTeX'>$J_{\mu }(x)$</tex-math></inline-formula> is equal to 8 degenerate Abelian magnetic monopole currents <inline-formula><tex-math id='TM0002' notation='LaTeX'>$k^a_{\mu }(x)\ \ (a=1\sim 8)$</tex-math></inline-formula>. Abelian monopoles could exist without any artificial Abelian projection. When <inline-formula><tex-math id='TM0003' notation='LaTeX'>$J_{\mu }$</tex-math></inline-formula> condenses in the vacuum as suggested in lattice QCD simulations, color confinement of QCD is realized by the Abelian dual Meissner effect. It is important to study the effect of VNABI besides the role of the color confinement mechanism. It is found that VNABI affects topological features of QCD. Firstly, the topological charge density <inline-formula><tex-math id='TM0004' notation='LaTeX'>$\rho _t(x)=\text{Tr}G_{\mu \nu }(x)G^{*}_{\mu \nu }(x)$</tex-math></inline-formula> is not expressed by a total derivative of the Chern&#8211;Simons density <inline-formula><tex-math id='TM0005' notation='LaTeX'>$K_{\mu }(x)$</tex-math></inline-formula>, but has an additional term <inline-formula><tex-math id='TM0006' notation='LaTeX'>$L(x)= 2\text{Tr}(J_{\mu }(x)A_\mu (x))$</tex-math></inline-formula>, i.e. <inline-formula><tex-math id='TM0007' notation='LaTeX'>$\partial _\mu K_\mu (x)=(g^2/16\pi ^2)(\rho _t(x)-L(x))$</tex-math></inline-formula>. Secondly, the axial <inline-formula><tex-math id='TM0008' notation='LaTeX'>$U(1)$</tex-math></inline-formula> anomaly is similarly modified as <inline-formula><tex-math id='TM0009' notation='LaTeX'>$\partial _{\mu }j_{\mu }^5(x)=2m\bar{\psi }(x)\gamma _5\psi (x)+(g^2/8\pi ^2)(\rho _t(x)-L(x))$</tex-math></inline-formula> while the Atiyah&#8211;Singer index theorem is formally unchanged. However, the integrated term <inline-formula><tex-math id='TM0010' notation='LaTeX'>$\Lambda =(g^2/16\pi ^2)\int d^4xL(x)$</tex-math></inline-formula> is seemingly neither integer nor gauge-invariant. Hence, if <inline-formula><tex-math id='TM0011' notation='LaTeX'>$\Lambda$</tex-math></inline-formula> remains nonzero, VNABI is not allowed in QCD. Abelian monopoles as VNABI are regarded as a generalization of the Dirac monopole to a <inline-formula><tex-math id='TM0012' notation='LaTeX'>$U(1)$</tex-math></inline-formula> subgroup in non-Abelian gauge theories as discussed by Wu&#8211;Yang. When we make use of similar arguments, the additional term <inline-formula><tex-math id='TM0013' notation='LaTeX'>$\Lambda$</tex-math></inline-formula> is proved to vanish theoretically. We also evaluate the term <inline-formula><tex-math id='TM0014' notation='LaTeX'>$\Lambda$</tex-math></inline-formula> in the framework of Monte-Carlo simulations on <inline-formula><tex-math id='TM0015' notation='LaTeX'>$24^4$</tex-math></inline-formula> lattices at the lattice spacing between <inline-formula><tex-math id='TM0016' notation='LaTeX'>$0.05\sim 0.17$</tex-math></inline-formula> fm adopting a tadpole-improved <inline-formula><tex-math id='TM0017' notation='LaTeX'>$SU(2)$</tex-math></inline-formula> gluonic action. To reduce ultraviolet fluctuations, we introduce a partial gauge fixing like the Maximal Center gauge. The term <inline-formula><tex-math id='TM0018' notation='LaTeX'>$\Lambda$</tex-math></inline-formula> is largely fluctuating around zero and <inline-formula><tex-math id='TM0019' notation='LaTeX'>$|\Lambda |$</tex-math></inline-formula> decreases as <inline-formula><tex-math id='TM0020' notation='LaTeX'>$\beta$</tex-math></inline-formula> becomes larger. Then the gradient flow method is applied. The term <inline-formula><tex-math id='TM0021' notation='LaTeX'>$\Lambda$</tex-math></inline-formula> tends to vanish rapidly after small gradient flow time <inline-formula><tex-math id='TM0022' notation='LaTeX'>$t_{\rm flow}$</tex-math></inline-formula>. This suggests that the term <inline-formula><tex-math id='TM0023' notation='LaTeX'>$\Lambda$</tex-math></inline-formula> becomes zero in the continuum limit. Hence, VNABI could exist in QCD. The biggest effect of VNABI on QCD topology seems to be that self-dual instantons cannot be a classical solution of QCD at space-time points where VNABI occurs. One has to find an alternative mechanism explaining integer topological charge etc. A new interesting relation is derived with respect to the topological charge. When <inline-formula><tex-math id='TM0024' notation='LaTeX'>$\Lambda =0$</tex-math></inline-formula>, an Abelian counter term <inline-formula><tex-math id='TM0025' notation='LaTeX'>$Q_a\equiv (g^2/16\pi ^2)\int d^4x \text{Tr}(f_{\mu \nu }f_{\mu \nu }^{*})$</tex-math></inline-formula> written by Abelian field strengths is proved theoretically to satisfy <inline-formula><tex-math id='TM0026' notation='LaTeX'>$Q_a = 3 Q_t$</tex-math></inline-formula> where <inline-formula><tex-math id='TM0027' notation='LaTeX'>$Q_t=(g^2/16\pi ^2)\int d^4x \rho _t(x)$</tex-math></inline-formula>. The fact suggests that the Abelian magnetic and electric fields in the presence of <inline-formula><tex-math id='TM0028' notation='LaTeX'>$J_{\mu }$</tex-math></inline-formula> condensation could have a role in explaining the topological charge <inline-formula><tex-math id='TM0029' notation='LaTeX'>$Q_t$</tex-math></inline-formula> in place of instantons.

Japan

Tsuneo Suzuki

Violation of non-Abelian Bianchi Identity and QCD Topology

We study theoretically the structure and formation of the deeply bound <inline-formula><tex-math id='TM0002' notation='LaTeX'>$\bar{p}$</tex-math></inline-formula> atoms. We find that the widths of the atomic states are narrower than the level spacing even for deeply bound states so that well isolated deeply bound <inline-formula><tex-math id='TM0003' notation='LaTeX'>$\bar{p}$</tex-math></inline-formula> atoms are expected to exist. We also find <inline-formula><tex-math id='TM0004' notation='LaTeX'>$\bar{p}$</tex-math></inline-formula> nuclear states with huge widths. For the observation of deep <inline-formula><tex-math id='TM0005' notation='LaTeX'>${\bar{p}}$</tex-math></inline-formula> atomic states, we investigate theoretically the <inline-formula><tex-math id='TM0006' notation='LaTeX'>$(\bar{p}, p)$</tex-math></inline-formula> reactions for <inline-formula><tex-math id='TM0007' notation='LaTeX'>$^{12}$</tex-math></inline-formula>C, <inline-formula><tex-math id='TM0008' notation='LaTeX'>$^{16}$</tex-math></inline-formula>O, and <inline-formula><tex-math id='TM0009' notation='LaTeX'>$^{31}$</tex-math></inline-formula>P target nuclei. We find that the momentum transfer of the <inline-formula><tex-math id='TM0010' notation='LaTeX'>$( {\bar{p}},p)$</tex-math></inline-formula> reaction is small and the formation of <inline-formula><tex-math id='TM0011' notation='LaTeX'>${\bar{p}}$</tex-math></inline-formula> atomic states can be observed as discrete peak structures in the <inline-formula><tex-math id='TM0012' notation='LaTeX'>$( {\bar{p}},p)$</tex-math></inline-formula> spectrum. We conclude that <inline-formula><tex-math id='TM0013' notation='LaTeX'>$(\bar{p}, p)$</tex-math></inline-formula> reactions are very well suited to <inline-formula><tex-math id='TM0014' notation='LaTeX'>$\bar{p}$</tex-math></inline-formula> atom formation and the spectra of the reaction are expected to provide valuable new information on <inline-formula><tex-math id='TM0015' notation='LaTeX'>${\bar{p}}$</tex-math></inline-formula> atoms and the <inline-formula><tex-math id='TM0016' notation='LaTeX'>${\bar{p}}$</tex-math></inline-formula>&#8211;nucleus interaction.

N Miyazaki

J Yamagata-Sekihara

S Hirenzaki

Structure and Formation of the Deeply Bound <italic>&#175;p</italic> Atoms

In this letter, we construct an explicit rephasing transformation that converts an arbitrary unitary matrix into the Kobayashi–Maskawa (KM) parametrization and identify all independent CP phases in the mixing matrix as the arguments of its matrix elements. Furthermore, by applying this rephasing transformation to the fermion diagonalization matrices <inline-formula><tex-math id='TM0001' notation='LaTeX'>$U^{\nu , e}$</tex-math></inline-formula>, we show that the Majorana phases are represented by fermion-specific phases <inline-formula><tex-math id='TM0002' notation='LaTeX'>$\delta ^{\nu , e}_{\rm KM}$</tex-math></inline-formula> and their relative phases. In particular, by neglecting the 3-1 elements <inline-formula><tex-math id='TM0003' notation='LaTeX'>$U_{31}^{\nu ,e}$</tex-math></inline-formula> of the diagonalization matrices for the two fermions, the KM phase <inline-formula><tex-math id='TM0004' notation='LaTeX'>$\delta _{\rm KM}$</tex-math></inline-formula> is concisely expressed as <inline-formula><tex-math id='TM0005' notation='LaTeX'>$\delta _{\rm KM} = \arg \left[1 + ({U^{e * }_{21} U^{\nu }_{21} / U^{e * }_{11} U^{\nu }_{11} }) \right] + \arg \left[ - { U_{32}^{e *} U_{32}^{\nu } / U^{e * }_{22} U^{\nu }_{22} } \right]$</tex-math></inline-formula> by fermion-specific rephasing invariants representing two relative phases.

Masaki J S Yang

Explicit Rephasing to Kobayashi–Maskawa Representation and Fundamental Phase Structure of CP Violation

In this analysis, we examine the angular distribution of low-mass dimuon pairs produced in simulated electron–positron collisions at the proposed International Linear Collider (ILC), which operates at a center-of-mass energy of 500 GeV and has an integrated luminosity of 4 ab <inline-formula><tex-math id='TM0003' notation='LaTeX'>$^{-1}$</tex-math></inline-formula>. Our focus is on the <inline-formula><tex-math id='TM0004' notation='LaTeX'>$\cos\theta _{\text{CS}}$</tex-math></inline-formula> variable, which is defined in the Collins&#8211;Soper frame. In the Standard Model, the production of low-mass dimuon pairs is primarily driven by the Drell&#8211;Yan process, which exhibits a notable forward&#8211;backward asymmetry. However, many scenarios beyond the Standard Model predict different shapes for the <inline-formula><tex-math id='TM0005' notation='LaTeX'>$\cos\theta _{\text{CS}}$</tex-math></inline-formula> distribution. This angular distribution can be valuable for distinguishing between these models, especially in the event of observing excesses beyond the Standard Model expectations. We utilized the mono-Z <inline-formula><tex-math id='TM0006' notation='LaTeX'>$^{\prime }$</tex-math></inline-formula> model to interpret the simulated data. In the absence of any discoveries of new physics, we establish upper limits at the 95% confidence level on the masses of various particles in the model, which includes the spin-1 <inline-formula><tex-math id='TM0007' notation='LaTeX'>$\mathrm{Z}^{\prime }$</tex-math></inline-formula> boson and fermionic dark matter.

Egypt

S Elgammal

Spin Identification of the Mono-Z′ Resonance in Muon-Pair Production at the ILC with Simulated Electron–Positron Collisions at √s = 500 GeV

We study a quasi-two-zero neutrino texture based on a type-II seesaw model with modular <inline-formula><tex-math id='TM0002' notation='LaTeX'>$A_4$</tex-math></inline-formula> symmetry to evade the cosmological bound on the sum of neutrino masses while keeping some predictability in the neutrino sector. Working on three fixed points for the modulus, we discuss predictions of the model and show the allowed points satisfying the cosmological bound on neutrino mass from both cosmic microwave background (CMB) and CMB + baryon acoustic oscillation data.

China

Takaaki Nomura

Hiroshi Okada

Quasi-Two-Zero Texture in Type-II Seesaw at Fixed Points from Modular <italic>A</italic> <sub>4</sub> Symmetry

For general one-matrix models in the large- <italic>N</italic> limit, we introduce the cap amplitude <inline-formula><tex-math id='TM0002' notation='LaTeX'>$\psi (b)$</tex-math></inline-formula> as the expansion coefficient of the 1-form <inline-formula><tex-math id='TM0003' notation='LaTeX'>$ydx$</tex-math></inline-formula> on the spectral curve. We find that the dilaton equation&#160;for the discrete volume <inline-formula><tex-math id='TM0004' notation='LaTeX'>$N_{g,n}$</tex-math></inline-formula> of the moduli space of genus- <italic>g</italic> Riemann surfaces with <italic>n</italic> boundaries is interpreted as gluing the cap amplitude along one of the boundaries. In this process, one of the boundaries is capped and the number of boundaries decreases by one. In a similar manner, the genus- <italic>g</italic> free energy <inline-formula><tex-math id='TM0008' notation='LaTeX'>$F_g$</tex-math></inline-formula> is obtained by gluing the cap amplitude to <inline-formula><tex-math id='TM0009' notation='LaTeX'>$N_{g,1}$</tex-math></inline-formula>.

Kazumi Okuyama

Cap Amplitudes in Random Matrix Models

This paper proposes a numerical method using neural networks to solve the path integral problem in quantum mechanics for arbitrary potentials. The method is based on a “radial basis function” expansion of the interaction term that appears in the Euclidean path integral formalism. By constructing a corresponding “multilayered perceptron”-type neural network with exponential nonlinearities in the hidden layer, the original path integral can be approximated by a linear combination of Gaussian path integrals that can be solved analytically. The method has been tested for the double-well potential that includes a quadratic and a quartic term, giving very good, within a few percent, agreement between the true and estimated bound state wave functions that are extracted from the propagator at large Euclidean times. The proposed method can also be used to describe potentials that have imaginary parts, which is tested for a simple Gaussian path integral with complex frequencies, where the model uncertainty stays below 1% for both the real and imaginary parts of the propagator.

South Korea

Gábor Balassa

On the Solution of Euclidean Path Integrals with Neural Networks

We investigate the trilinear Higgs boson coupling derived from the functional forms of various extended Higgs potentials. In light of experimental constraints on Higgs boson couplings, we focus on extended Higgs models in which the trilinear Higgs boson coupling is predominantly determined by the Standard Model Higgs field. Such models include the nearly aligned Higgs Effective Field Theory, classically scale-invariant models, pseudo-Nambu–Goldstone boson scenarios, tadpole-induced models, and others. We also consider higher-order corrections, including top quark and new particle contributions that are often neglected, and discuss their impact on the trilinear Higgs boson coupling. Finally, we show to what extent the functional forms of the Higgs potentials can be probed at future colliders.

Nagisa Hiroshima

Mitsuru Kakizaki

Shuhei Ohzawa

Classifying Extended Higgs Models through the Trilinear Higgs Boson Coupling Measurement at Future Colliders

We study collective excitations in a relativistic collisionless plasma composed of massless fermions subject to an external magnetic field. We include dynamical electromagnetism and fluctuations of the chiral charge, while maintaining a vanishing net chiral charge. Within the framework of chiral kinetic theory, we find that the chiral anomaly gives a correction to the transverse photon self-energy and that a novel type of dynamical screening can emerge. In the strong magnetic field limit, we also show that the collective mode has a gap using the lowest Landau level approximation. We further discuss implications of the anomalous dynamical screening for neutron star phenomenology.

Sota Hanai

Anomalous Dynamical Screening of Relativistic Plasma in a Magnetic Field

We investigate crosscap states in 2D rational conformal field theories (RCFTs), with an emphasis on the role of noninvertible symmetries. In particular, we argue for the existence of crosscap states labeled by each Verlinde line in the RCFT, extending previous constructions involving simple currents. Evidence for the existence of these new states is obtained by deriving a generalized Cardy condition incorporating both crosscaps and topological defects, which we check in some concrete examples. Finally, we briefly discuss how these crosscap states transform under the action of Verlinde lines, as well as the connection to mixed anomalies between parity and internal symmetries.