International Journal of Modern Physics A

Shortly after his escape, in 1968, from the DDR to the Western world, Harald Fritzsch made a fast career in physics while investigating the strong force in elementary particles. Inspired by Gell-Mann, Yang and Mills and many others, he contributed to the rise of the Standard Model, and in particular the developments of theories for the color confinement mechanism.

A written account of my talk delivered at the Harald Fritzsch Memorial Symposium in Munich is given. The emphasis is on Harald’s achievements specifically in quark physics and in quantum chromodynamics. Beyond that exemplary results for low-energy baryons are discussed, as obtained over the past years in the framework of the relativistic constituent-quark model based on Goldstone-boson-exchange dynamics.

We discuss aspects of a promising top-down origin of flavor symmetries in particle physics. Modular transformations originating from string theory dualities are shown to play a crucial role. We introduce the notion of an “eclectic” flavor scheme that unifies traditional flavor symmetries, modular symmetries and ${𝒞}{𝒫}$-transformations. It exhibits the phenomenon of “Local Flavor Unification” with enhanced flavor symmetries at fixed points or lines in moduli space. Successful fits of masses and mixing angles of quarks and leptons are found in the vicinity of these points and lines.

A survey is given on the current status of the theoretical description of unpolarized and polarized deep-inelastic scattering processes in Quantum Chromodynamics at large virtualities.

Quantum Chromodynamics predicts that the strong coupling strength ${\alpha }_{\mathrm{\text{s}}}$ decreases with increasing energy or momentum transfer, and vanishes at asymptotically high energies. The history and the status of experimental tests of asymptotic freedom are summarised in this review.

The spontaneous breakdown of an approximate symmetry implies that the spectrum of the theory contains approximately massless particles. The hidden symmetry very strongly constrains their masses. A numerical evaluation of these constraints on the lattice would allow a more precise determination of the quark mass ratios $m_u:m_d:m_s$ and thereby reduce some of the uncertainties encountered in precision flavour physics.

Dedicated to the memory of our colleague, Harald Fritzsch, who, together with Murray Gell-Mann, introduced the color quantum number as the exact symmetry responsible for the strong interaction, thus establishing quantum chromodynamics (QCD) as a fundamental non-Abelian gauge theory. A basic understanding of hadron properties, however, such as confinement and the emergence of a mass scale, from first principles QCD has remained elusive: Hadronic characteristics are not explicit properties of the QCD Lagrangian and perturbative QCD, so successful in the large transverse momentum domain, is not applicable at large distances. In this article, we shall examine how this daunting obstacle is overcome in holographic QCD with the introduction of a superconformal symmetry in anti de Sitter (AdS) space which is responsible for confinement and the introduction of a mass scale within the superconformal group. When mapped to light-front coordinates in physical spacetime, this approach incorporates supersymmetric relations between the Regge trajectories of meson, baryon and tetraquark states which can be visualized in terms of specific $SU{(3)}_C$ color representations of quarks. We will also briefly discuss here the implications of holographic models for QCD color transparency in view of the present experimental interest.

The proton spin puzzle has inspired a vast programme of experiments and theoretical work challenging our understanding of QCD and its role in the structure of hadrons. The proton’s internal spin structure is connected to chiral symmetry and, through gluon degrees of freedom in the flavor singlet channel, to the physics of the ${\eta }^{\prime }$ meson. Why do quarks contribute just about one third of the proton’s spin? Why are ${\eta }^{\prime }$ mesons and their interactions so sensitive to gluonic degrees of freedom? We review the status of these topics and some key observables for forthcoming experiments.

We have updated theoretical studies of heavy vector quarkonium production, including $J/\mathrm{\Psi }$, ${\mathrm{\Psi }}^{\prime }$ and $\mathrm{\Upsilon }$, in the exclusive W-boson decays. Particularly, in the standard model the branching fraction of $W^{-}\to J/\mathrm{\Psi }{\ell }^{-}{\bar{\nu }}_{\ell }$ ($\ell =e$ or $\mu$) has been predicted to be about $8.5\times 10^{-7}$, which is substantially larger than those of two-body hadronic radiative W decays. Thus in the future high-luminosity experimental facilities, this rare channel could be very useful to search for the exclusive W-boson decays containing the hadronic final state. Furthermore, the surprisingly large decay rate can be explained by an electromagnetic fragmentation formalism. We have analyzed the lepton fragmentation and the photon fragmentation processes, and calculated their contributions to the differential decay rate of $W^{-}\to J/\mathrm{\Psi }{\ell }^{-}{\bar{\nu }}_{\ell }$ in the fragmentation limit. It is found that the fragmentation contribution agrees well with the result from the full calculation.

The Standard Model does not constrain the form of the Yukawa matrices and thus the origin of fermion mass hierarchies and mixing pattern remains puzzling. On the other hand, there are intriguing relations between the quark masses and their weak mixing angles, such as the well-known relationship $tan{\theta }_C=\sqrt{m_d/m_s}$ for the Cabibbo angle, which may point towards specific textures of Yukawa matrices hypothesized by Harald Fritzsch at the end of the 70s. Though the original ansatz of Fritzsch is excluded by the experimental data, one can consider its minimal modification which consists in introducing an asymmetry between the 23 and 32 entries in the down-quark Yukawa matrix. We show that this structure is perfectly compatible with the present precision data on quark masses and CKM mixing matrix, and theoretically it can be obtained in the context of $\mathrm{\text{SU}}(5)$ model with inter-family $\mathrm{\text{SU}}{(3)}_H$ symmetry. We also discuss some alternative approaches which could give a natural description of the fermion mass spectrum and weak mixing pattern.

We review aspects of the modular invariance approach to the flavour problem. Harald Fritzsch was among the first to realise the existence and the fundamental nature of the quark and lepton flavour problems in particle physics, that symmetries may be the key to the solution(s) of these problems and to propose in 1978 and 1979 a solution to the quark flavour problem in the form of the Fritzsch quark mass matrices with texture zeros. After introducing the general ingredients of the modular invariance approach, we describe the formalism that allows to construct models in which fermion (charged-lepton and quark) mass hierarchies follow solely from the properties of the modular forms, avoiding the fine-tuning of the constant parameters present in the fermion mass matrices and the need to introduce extra fields. Focusing on the lepton sector, we show how the indicated formalism can be used in lepton flavour models to obtain the charged lepton mass hierarchies without parameter fine-tuning. Only a very limited number of models satisfy the non-fine-tuning requirement in explaining the observed fermion mass hierarchies. We review next the general conditions under which also the peculiar pattern of Pontecorvo, Maki, Nakagawa and Sakata (PMNS) lepton mixing consisting of two large and one small angles can be reproduced without fine-tuning. We then give an example of a phenomenologically viable “minimal” modular lepton flavour model in which both the lepton mixing and charged lepton mass hierarchies are generated without fine-tuning. We conclude by listing some of the open questions and problems in the modular invariant approach to the flavour problem.

We review the modular flavor symmetric models of quarks and leptons focusing on our works. We present some flavor models of quarks and leptons by using finite modular groups and discuss the phenomenological implications. The modular flavor symmetry gives interesting phenomena at the fixed point of modulus. As a representative, we show the successful texture structure at the fixed point $\tau =\omega$. We also study CP violation, which occurs through the modulus stabilization. Finally, we study SMEFT with modular flavor symmetry by including higher dimensional operators.

An attempt has been made to discuss some of the developments, in a personalized historical perspective, regarding a viable answer to the ‘Flavor Riddle’ through a minor addition to the Fritzsch’s original proposal about texture zero mass matrices.

The strong mass hierarchy, flavor mixing pattern and CP violation in the quark sector have been long-standing puzzles in the Standard Model. Thanks to the elegant neutrino oscillation experiments in the past few decades, now we know that neutrinos are massive and lepton flavor mixing is significant. The flavor puzzles have extended to the lepton sector. In this article, we explain how the idea of texture zeros applies to understand lepton flavor mixing, and emphasize that it is particularly useful for exploring the underlying dynamics or symmetries for lepton flavor mixing.

This article is a brief review of the work on black hole quantum pressure published in 2021 in Physical Review D with Folkert Kuipers. We highlight how to define the volume of a black hole making the link to previous works where such a definition had appeared in the context of black holes with a cosmological constant.

The possibility that the vacuum energy density (VED) ${\rho }_{\mathrm{\text{vac}}}$ could be time dependent in the expanding Universe is intuitively more reasonable than just a rigid cosmological constant for the entire cosmic history. The dynamics of ${\rho }_{\mathrm{\text{vac}}}={\rho }_{\mathrm{\text{vac}}}(H)$ as a function of the Hubble rate, $H(t)$, most likely contributes to alleviate cosmological problems and tensions, having also implications on the so-called fundamental ‘constants’ of Nature, which should be slowly drifting with the cosmic expansion owing to the fluctuations of the quantum vacuum. This includes the gravitational ‘constant’ G, but also the gauge and Yukawa couplings as well as the particle masses themselves (both of dark matter and baryonic matter). The subtle exchange of energy involved is the basis for the “micro and macro connection”. Herein, I discuss not only this connection as a possibility but also show that it is in fact a generic prediction of QFT in cosmological spacetime which is fully compatible with general covariance. This fact has not been pointed out until recently when an appropriate renormalization framework for the VED has been found which is free from the usual conundrums associated with the cosmological constant problem.

The principles of time and space translation invariance, which result in energy and momentum conservation, are among the most fundamental and widely accepted symmetry assumptions in physics. However, it is judicious to subject such assumptions to experimental and observational scrutiny. Thus, we initiate this process by specifying a simple periodic time dependence that contravenes time translation invariance in QED, and establishing phenomenological constraints on it. Besides observational and experimental constraints on time-varying couplings, we emphasise probes of energy conservation violation such as spontaneous photon and electron pair production and the $e\to e\gamma$ process.

This article is a reflection and reminiscence of Harald Fritzsch’s contributions to the High Energy Particle Physics scene in Singapore, namely in 1978, 2008 to 2022. It also captures rare moments of Harald’s fun, light-hearted times and conversations beneath his stoic demeanor.