We study Koide equation sequentially in a chain of mass triplets. We notice at
least a not previously published triplet whose existence allows to
built a quark mass hierarchy from fixed yukawas for
top and up quarks. Also, the new triplets are used to build mass predictions either
descending from the experimental values or top and bottom, or ascending
from the original triplet of charged leptons.
N. Bilic - "Hamiltonian Method in the Braneworld" 2/3SEENET-MTP
This document discusses braneworld models and the Randall-Sundrum model. It begins by introducing the relativistic particle and string actions used to describe dynamics in higher dimensions. It then summarizes the two Randall-Sundrum models: RS I contains two branes separated in a fifth dimension to address the hierarchy problem, while RS II has the negative tension brane sent to infinity and observers on a single positive tension brane. Finally, it derives the RS II model solution, using Gaussian normal coordinates and imposing junction conditions at the brane.
N. Bilic - "Hamiltonian Method in the Braneworld" 1/3SEENET-MTP
This document provides an overview of the Hamiltonian method in braneworld cosmology. It begins with introductory lectures on preliminaries like Legendre transformations, thermodynamics, fluid mechanics, and basic cosmology. It then covers braneworld universes, strings and branes, and the Randall-Sundrum model. The document concludes with applications of the Hamiltonian method to topics like quintessence, dark energy/matter unification, and tachyon condensates in braneworlds.
N. Bilic - "Hamiltonian Method in the Braneworld" 3/3SEENET-MTP
This document discusses various models for dark energy and dark matter unification, including quintessence, k-essence, phantom quintessence, Chaplygin gas, and tachyon condensates. It provides field theory descriptions and equations of state for these models. It also discusses issues like the sound speed and structure formation problems that arise for some unified dark energy/dark matter models. Modifications to address these issues, such as generalized Chaplygin gas and variable Chaplygin gas models, are presented.
I. Cotaescu - "Canonical quantization of the covariant fields: the Dirac fiel...SEENET-MTP
The document discusses the canonical quantization of covariant fields on curved spacetimes, specifically the de Sitter spacetime. It introduces covariant fields that transform under representations of the spin group SL(2,C) and have covariant derivatives ensuring gauge invariance. Isometries of the spacetime generate Killing vectors and induce representations of the external symmetry group, which is the universal covering group of isometries and combines isometries with gauge transformations. Generators of these representations provide conserved observables that allow canonical quantization analogous to special relativity. The paper focuses on applying this framework to the Dirac field on de Sitter spacetime.
This document outlines a presentation on thermodynamics of anti-de Sitter black holes as regularized fidelity susceptibility. It introduces concepts like entanglement entropy, the holographic principle, and holographic entanglement entropy. It then discusses fidelity susceptibility and holographic complexity. The document derives formulas for these quantities for Reissner-Nordström anti-de Sitter black holes. Specifically, it derives an expression for the difference in entanglement entropy between the pure background and metric deformation. It also computes the holographic complexity and fidelity susceptibility volumes for RN black holes through integrals involving the metric function.
I. Antoniadis - "Introduction to Supersymmetry" 1/2SEENET-MTP
Supersymmetry (SUSY) is a symmetry that relates bosonic and fermionic degrees of freedom. It extends the Poincaré algebra by including spinorial generators (supercharges) that transform bosonic fields into fermionic fields and vice versa. SUSY provides motivations like natural elementary scalars, gauge coupling unification, and a dark matter candidate. SUSY is formulated using superspace, which extends spacetime by Grassmann coordinates. Chiral and vector superfields contain the bosonic and fermionic components of supermultiplets and allow constructing SUSY invariant actions.
This document provides an overview of quantum electrodynamics (QED). It begins by discussing cross sections and the scattering matrix, defining cross section as the effective size of target particles. It then derives an expression for cross section in terms of the transition rate and flux of incident particles. Next, it summarizes the derivation of the differential cross section and decay rate formulas in QED using relativistic quantum field theory and Feynman diagrams. It concludes by briefly reviewing the historical development of QED and the equivalence of the propagator approach and other formulations.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
N. Bilic - "Hamiltonian Method in the Braneworld" 2/3SEENET-MTP
This document discusses braneworld models and the Randall-Sundrum model. It begins by introducing the relativistic particle and string actions used to describe dynamics in higher dimensions. It then summarizes the two Randall-Sundrum models: RS I contains two branes separated in a fifth dimension to address the hierarchy problem, while RS II has the negative tension brane sent to infinity and observers on a single positive tension brane. Finally, it derives the RS II model solution, using Gaussian normal coordinates and imposing junction conditions at the brane.
N. Bilic - "Hamiltonian Method in the Braneworld" 1/3SEENET-MTP
This document provides an overview of the Hamiltonian method in braneworld cosmology. It begins with introductory lectures on preliminaries like Legendre transformations, thermodynamics, fluid mechanics, and basic cosmology. It then covers braneworld universes, strings and branes, and the Randall-Sundrum model. The document concludes with applications of the Hamiltonian method to topics like quintessence, dark energy/matter unification, and tachyon condensates in braneworlds.
N. Bilic - "Hamiltonian Method in the Braneworld" 3/3SEENET-MTP
This document discusses various models for dark energy and dark matter unification, including quintessence, k-essence, phantom quintessence, Chaplygin gas, and tachyon condensates. It provides field theory descriptions and equations of state for these models. It also discusses issues like the sound speed and structure formation problems that arise for some unified dark energy/dark matter models. Modifications to address these issues, such as generalized Chaplygin gas and variable Chaplygin gas models, are presented.
I. Cotaescu - "Canonical quantization of the covariant fields: the Dirac fiel...SEENET-MTP
The document discusses the canonical quantization of covariant fields on curved spacetimes, specifically the de Sitter spacetime. It introduces covariant fields that transform under representations of the spin group SL(2,C) and have covariant derivatives ensuring gauge invariance. Isometries of the spacetime generate Killing vectors and induce representations of the external symmetry group, which is the universal covering group of isometries and combines isometries with gauge transformations. Generators of these representations provide conserved observables that allow canonical quantization analogous to special relativity. The paper focuses on applying this framework to the Dirac field on de Sitter spacetime.
This document outlines a presentation on thermodynamics of anti-de Sitter black holes as regularized fidelity susceptibility. It introduces concepts like entanglement entropy, the holographic principle, and holographic entanglement entropy. It then discusses fidelity susceptibility and holographic complexity. The document derives formulas for these quantities for Reissner-Nordström anti-de Sitter black holes. Specifically, it derives an expression for the difference in entanglement entropy between the pure background and metric deformation. It also computes the holographic complexity and fidelity susceptibility volumes for RN black holes through integrals involving the metric function.
I. Antoniadis - "Introduction to Supersymmetry" 1/2SEENET-MTP
Supersymmetry (SUSY) is a symmetry that relates bosonic and fermionic degrees of freedom. It extends the Poincaré algebra by including spinorial generators (supercharges) that transform bosonic fields into fermionic fields and vice versa. SUSY provides motivations like natural elementary scalars, gauge coupling unification, and a dark matter candidate. SUSY is formulated using superspace, which extends spacetime by Grassmann coordinates. Chiral and vector superfields contain the bosonic and fermionic components of supermultiplets and allow constructing SUSY invariant actions.
This document provides an overview of quantum electrodynamics (QED). It begins by discussing cross sections and the scattering matrix, defining cross section as the effective size of target particles. It then derives an expression for cross section in terms of the transition rate and flux of incident particles. Next, it summarizes the derivation of the differential cross section and decay rate formulas in QED using relativistic quantum field theory and Feynman diagrams. It concludes by briefly reviewing the historical development of QED and the equivalence of the propagator approach and other formulations.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
I. Antoniadis - "Introduction to Supersymmetry" 2/2SEENET-MTP
1) The Supersymmetric Standard Model (SSM) extends the particle content of the Standard Model by introducing supersymmetric partners for each particle, called sparticles. This includes gluinos, winos, binos, squarks, sleptons, and higgsinos as sparticle counterparts to gluons, W/Z bosons, photons, quarks, leptons, and Higgs bosons.
2) The SSM Lagrangian contains additional terms beyond the Standard Model including gauge interactions between fermions and gauginos, quartic scalar interactions, and a superpotential with Yukawa-like couplings.
3) Spontaneous supersymmetry breaking is required to make sparticles heavy enough to have evaded detection
This document is a dissertation proposal by Rishideep Roy at the University of Chicago in November 2014. The proposal is to generalize results on extreme values and entropic repulsion for two-dimensional discrete Gaussian free fields to a more general class of Gaussian fields with logarithmic correlations. Specifically, the proposal plans to find the convergence in law of the maximum of these log-correlated Gaussian fields under minimal assumptions, as well as obtain finer estimates on entropic repulsion which relates to the behavior of these fields near hard boundaries. The proposal provides background on related works and outlines the key steps to be taken, including proving expectations and tightness of maxima, invariance of maximum distributions under perturbations, approximating the fields, and
Alexei Starobinsky - Inflation: the present statusSEENET-MTP
This document summarizes a presentation on inflation and the present status of inflationary cosmology. It discusses the key epochs in the early universe, including inflation, and how inflation solved issues with prior models. Observational evidence for inflation is presented, including measurements of the primordial power spectrum and constraints on the tensor-to-scalar ratio. Simple single-field inflation models are shown to match observations. The document also discusses the generation of primordial perturbations from quantum fluctuations during inflation and how this provides the seeds for structure formation.
This document provides an overview of string theory and superstring theory. It discusses the following key points:
1) A Calabi-Yau manifold is a smooth space that is Ricci flat and represents a deformation that smooths out an orbifold singularity from a space-time perspective.
2) In the 1960s, particle physics was dominated by S-matrix theory, which focused on scattering matrix properties rather than fundamental fields. S-matrix theory assumed analyticity, crossing, and unitarity of scattering amplitudes.
3) Early string theory models treated particles as vibrating strings to address limitations of S-matrix theory for high spin particles. This led to the development of bosonic string theory and super
A Calabi-Yau manifold is a smooth space that represents a deformation which smooths out an orbifold singularity. This document discusses superstring theory and fermions in string theories. It introduces the spinning string action and shows that the Neveu-Schwarz model contains a tachyon ground state while the Ramond model contains massless fermions. Combining the two sectors using the Gliozzi-Scherk-Olive projection results in a model with N=1 supersymmetry in ten dimensions.
I am Samantha K. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, McGill University, Canada
I have been helping students with their homework for the past 8 years. I solve assignments related to Statistics.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
Primordial gravitational waves from inflation could be detected using the cosmic microwave background. Gravitational waves passing through the last scattering surface would induce B-mode polarization patterns in the CMB. Future experiments aim to detect these B-modes as a signature of primordial gravitational waves, which would provide insights into inflation and the very early universe.
D. Mladenov - On Integrable Systems in CosmologySEENET-MTP
Lecture by Prof. Dr. Dimitar Mladenov (Theoretical Physics Department, Faculty of Physics, Sofia University, Bulgaria) on December 7, 2011 at the Faculty of Science and Mathematics, Nis, Serbia.
This document contains the chapter outline and answers to questions for a physics and measurement chapter. The chapter outline lists the main topics covered, including standards of length, mass and time, dimensional analysis, and significant figures. The answers to questions section provides worked solutions to sample problems related to these topics, such as density calculations, unit conversions, and dimensional analysis questions.
This document summarizes a presentation about reconstructing inflationary models in modified f(R) gravity. It discusses the current status of inflation based on Planck data, reviews how inflation works in f(R) gravity, and describes two approaches - the direct approach of comparing models to data and the inverse approach of smoothly reconstructing models from observational quantities like the scalar spectrum index. A key model discussed is the simple R + R^2 model that can match current measurements of the spectral index and tensor-to-scalar ratio.
1) The document discusses the problem of a particle sliding off a moving hemisphere, using conservation of momentum and energy equations to derive an expression for the particle's horizontal velocity vx as a function of the angle θ.
2) Setting the derivative of vx equal to zero yields a cubic equation that determines the angle θ at which vx is maximized, corresponding to the particle losing contact with the hemisphere.
3) For the special case where the particle and hemisphere masses are equal (ratio r = 1), the cubic equation can be solved to find θ ≈ 42.9 degrees.
This document provides an analysis of quantizing energy levels for the Schwarzschild gravitational field using the prescriptions of Old Quantum Theory. It begins with introducing the Schwarzschild metric and deriving the Lagrangian for motion in a central gravitational field. It then applies the Bohr-Sommerfeld quantization rule to obtain expressions for angular momentum and energy as quantized values. Integrating these expressions provides a very simple formula for the quantized energy levels in the Schwarzschild field within the Old Quantum Theory framework.
- The document derives the second order Friedmann equations from the quantum corrected Raychaudhuri equation (QRE), which includes quantum corrections terms.
- One correction term can be interpreted as dark energy/cosmological constant with the observed density value, providing an explanation for the coincidence problem.
- The other correction term can be interpreted as a radiation term in the early universe that prevents the formation of a big bang singularity and predicts an infinite age for the universe by avoiding a divergence in the Hubble parameter or its derivative at any finite time in the past.
This document contains a chapter outline and solutions to problems for a physics textbook. The chapter outline lists topics such as standards of length, mass and time, matter and model-building, density and atomic mass. The solutions provide worked examples and calculations for problems related to these topics, such as calculating density using mass and volume, dimensional analysis, and unit conversions.
This document contains a chapter outline and answers to questions about physics and measurement. The chapter outline lists topics like standards of length, mass and time, density and atomic mass, and dimensional analysis. The answers to questions section provides explanations and calculations in response to multiple choice and free response questions about these topics. For example, it explains why atomic clocks and pulsars can serve as highly accurate time standards, and it calculates densities, masses, numbers of atoms, and rates of change using the relevant physics equations and units.
Calculando o tensor de condutividade em materiais topológicosVtonetto
This document describes a new efficient numerical method to calculate the longitudinal and transverse conductivity tensors in solids using the Kubo-Bastin formula. The method expands Green's functions in terms of Chebyshev polynomials, allowing both diagonal and off-diagonal conductivities to be computed for large systems in a single step at any temperature or chemical potential. The method is applied to calculate the conductivity tensor for the quantum Hall effect in disordered graphene and a Chern insulator in Haldane's model on a honeycomb lattice.
This document discusses different notions of convergence for sequences of graphs as studied in graph theory, statistical physics, and probability. It addresses three main notions of convergence for both dense and sparse graphs:
1) Left convergence, which requires subgraph counts to converge.
2) Convergence of quotients, which requires properties like MaxCut to converge as graphs are colored and collapsed.
3) Right convergence, which requires free energies of graphical models on the graphs to converge.
For sparse graphs with bounded degrees, the document shows these three notions are not equivalent, and introduces a new notion of large deviation convergence, which implies the other three notions. The large deviation principle characterizes the probability distribution of random color
This document outlines an approach to studying time correlations of conserved fields in anharmonic chains using nonlinear fluctuating hydrodynamics. It introduces the BS model, which has two conserved fields - displacement and potential energy. The dynamics of these fields can be approximated by a two-component stochastic Burgers equation. Classifying the universality classes of this equation's correlation functions allows insights into the original anharmonic chain model. Numerical results for specific potentials are also discussed.
1) The document provides an overview of the contents of Part II of a slideshow on modern physics, which covers topics such as charge and current densities, electromagnetic induction, Maxwell's equations, special relativity, tensors, blackbody radiation, photons, electrons, scattering problems, and waves.
2) It aims to provide a brief yet modern review of foundational concepts in electromagnetism and set the stage for introducing special relativity, quantum mechanics, and matter waves for undergraduate students.
3) The overview highlights that succeeding chapters will develop tensor formulations of electromagnetism and special relativity from first principles before discussing applications like blackbody radiation and early quantum models.
This document presents a model for matching theoretical predictions of the relationship between supermassive black hole mass (MBH) and stellar velocity dispersion (σ) to observational data. The model uses a spherical Hernquist profile for stars within a Dehnen-McLaughlin dark matter halo. It finds that for a stellar to dark matter mass ratio (fe) of 6, the model agrees well with observed MBH-σ and MBH-bulge mass relations for large elliptical galaxies with σ ≥ 150 km/s.
This document describes a study using integral equation theory and Monte Carlo simulation to determine the structure and thermodynamics of a colloidal solution where particles interact via Yukawa or Sogami potentials. The authors use the hybridized mean spherical approximation within the integral equation theory to calculate properties like the pair correlation function, structure factor, internal energy and pressure. They find good quantitative agreement between results from integral equation theory using a Sogami potential and results from Monte Carlo simulation. The theoretical results are also compared to experimental data and show agreement when using a Sogami potential.
I. Antoniadis - "Introduction to Supersymmetry" 2/2SEENET-MTP
1) The Supersymmetric Standard Model (SSM) extends the particle content of the Standard Model by introducing supersymmetric partners for each particle, called sparticles. This includes gluinos, winos, binos, squarks, sleptons, and higgsinos as sparticle counterparts to gluons, W/Z bosons, photons, quarks, leptons, and Higgs bosons.
2) The SSM Lagrangian contains additional terms beyond the Standard Model including gauge interactions between fermions and gauginos, quartic scalar interactions, and a superpotential with Yukawa-like couplings.
3) Spontaneous supersymmetry breaking is required to make sparticles heavy enough to have evaded detection
This document is a dissertation proposal by Rishideep Roy at the University of Chicago in November 2014. The proposal is to generalize results on extreme values and entropic repulsion for two-dimensional discrete Gaussian free fields to a more general class of Gaussian fields with logarithmic correlations. Specifically, the proposal plans to find the convergence in law of the maximum of these log-correlated Gaussian fields under minimal assumptions, as well as obtain finer estimates on entropic repulsion which relates to the behavior of these fields near hard boundaries. The proposal provides background on related works and outlines the key steps to be taken, including proving expectations and tightness of maxima, invariance of maximum distributions under perturbations, approximating the fields, and
Alexei Starobinsky - Inflation: the present statusSEENET-MTP
This document summarizes a presentation on inflation and the present status of inflationary cosmology. It discusses the key epochs in the early universe, including inflation, and how inflation solved issues with prior models. Observational evidence for inflation is presented, including measurements of the primordial power spectrum and constraints on the tensor-to-scalar ratio. Simple single-field inflation models are shown to match observations. The document also discusses the generation of primordial perturbations from quantum fluctuations during inflation and how this provides the seeds for structure formation.
This document provides an overview of string theory and superstring theory. It discusses the following key points:
1) A Calabi-Yau manifold is a smooth space that is Ricci flat and represents a deformation that smooths out an orbifold singularity from a space-time perspective.
2) In the 1960s, particle physics was dominated by S-matrix theory, which focused on scattering matrix properties rather than fundamental fields. S-matrix theory assumed analyticity, crossing, and unitarity of scattering amplitudes.
3) Early string theory models treated particles as vibrating strings to address limitations of S-matrix theory for high spin particles. This led to the development of bosonic string theory and super
A Calabi-Yau manifold is a smooth space that represents a deformation which smooths out an orbifold singularity. This document discusses superstring theory and fermions in string theories. It introduces the spinning string action and shows that the Neveu-Schwarz model contains a tachyon ground state while the Ramond model contains massless fermions. Combining the two sectors using the Gliozzi-Scherk-Olive projection results in a model with N=1 supersymmetry in ten dimensions.
I am Samantha K. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, McGill University, Canada
I have been helping students with their homework for the past 8 years. I solve assignments related to Statistics.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
Primordial gravitational waves from inflation could be detected using the cosmic microwave background. Gravitational waves passing through the last scattering surface would induce B-mode polarization patterns in the CMB. Future experiments aim to detect these B-modes as a signature of primordial gravitational waves, which would provide insights into inflation and the very early universe.
D. Mladenov - On Integrable Systems in CosmologySEENET-MTP
Lecture by Prof. Dr. Dimitar Mladenov (Theoretical Physics Department, Faculty of Physics, Sofia University, Bulgaria) on December 7, 2011 at the Faculty of Science and Mathematics, Nis, Serbia.
This document contains the chapter outline and answers to questions for a physics and measurement chapter. The chapter outline lists the main topics covered, including standards of length, mass and time, dimensional analysis, and significant figures. The answers to questions section provides worked solutions to sample problems related to these topics, such as density calculations, unit conversions, and dimensional analysis questions.
This document summarizes a presentation about reconstructing inflationary models in modified f(R) gravity. It discusses the current status of inflation based on Planck data, reviews how inflation works in f(R) gravity, and describes two approaches - the direct approach of comparing models to data and the inverse approach of smoothly reconstructing models from observational quantities like the scalar spectrum index. A key model discussed is the simple R + R^2 model that can match current measurements of the spectral index and tensor-to-scalar ratio.
1) The document discusses the problem of a particle sliding off a moving hemisphere, using conservation of momentum and energy equations to derive an expression for the particle's horizontal velocity vx as a function of the angle θ.
2) Setting the derivative of vx equal to zero yields a cubic equation that determines the angle θ at which vx is maximized, corresponding to the particle losing contact with the hemisphere.
3) For the special case where the particle and hemisphere masses are equal (ratio r = 1), the cubic equation can be solved to find θ ≈ 42.9 degrees.
This document provides an analysis of quantizing energy levels for the Schwarzschild gravitational field using the prescriptions of Old Quantum Theory. It begins with introducing the Schwarzschild metric and deriving the Lagrangian for motion in a central gravitational field. It then applies the Bohr-Sommerfeld quantization rule to obtain expressions for angular momentum and energy as quantized values. Integrating these expressions provides a very simple formula for the quantized energy levels in the Schwarzschild field within the Old Quantum Theory framework.
- The document derives the second order Friedmann equations from the quantum corrected Raychaudhuri equation (QRE), which includes quantum corrections terms.
- One correction term can be interpreted as dark energy/cosmological constant with the observed density value, providing an explanation for the coincidence problem.
- The other correction term can be interpreted as a radiation term in the early universe that prevents the formation of a big bang singularity and predicts an infinite age for the universe by avoiding a divergence in the Hubble parameter or its derivative at any finite time in the past.
This document contains a chapter outline and solutions to problems for a physics textbook. The chapter outline lists topics such as standards of length, mass and time, matter and model-building, density and atomic mass. The solutions provide worked examples and calculations for problems related to these topics, such as calculating density using mass and volume, dimensional analysis, and unit conversions.
This document contains a chapter outline and answers to questions about physics and measurement. The chapter outline lists topics like standards of length, mass and time, density and atomic mass, and dimensional analysis. The answers to questions section provides explanations and calculations in response to multiple choice and free response questions about these topics. For example, it explains why atomic clocks and pulsars can serve as highly accurate time standards, and it calculates densities, masses, numbers of atoms, and rates of change using the relevant physics equations and units.
Calculando o tensor de condutividade em materiais topológicosVtonetto
This document describes a new efficient numerical method to calculate the longitudinal and transverse conductivity tensors in solids using the Kubo-Bastin formula. The method expands Green's functions in terms of Chebyshev polynomials, allowing both diagonal and off-diagonal conductivities to be computed for large systems in a single step at any temperature or chemical potential. The method is applied to calculate the conductivity tensor for the quantum Hall effect in disordered graphene and a Chern insulator in Haldane's model on a honeycomb lattice.
This document discusses different notions of convergence for sequences of graphs as studied in graph theory, statistical physics, and probability. It addresses three main notions of convergence for both dense and sparse graphs:
1) Left convergence, which requires subgraph counts to converge.
2) Convergence of quotients, which requires properties like MaxCut to converge as graphs are colored and collapsed.
3) Right convergence, which requires free energies of graphical models on the graphs to converge.
For sparse graphs with bounded degrees, the document shows these three notions are not equivalent, and introduces a new notion of large deviation convergence, which implies the other three notions. The large deviation principle characterizes the probability distribution of random color
This document outlines an approach to studying time correlations of conserved fields in anharmonic chains using nonlinear fluctuating hydrodynamics. It introduces the BS model, which has two conserved fields - displacement and potential energy. The dynamics of these fields can be approximated by a two-component stochastic Burgers equation. Classifying the universality classes of this equation's correlation functions allows insights into the original anharmonic chain model. Numerical results for specific potentials are also discussed.
1) The document provides an overview of the contents of Part II of a slideshow on modern physics, which covers topics such as charge and current densities, electromagnetic induction, Maxwell's equations, special relativity, tensors, blackbody radiation, photons, electrons, scattering problems, and waves.
2) It aims to provide a brief yet modern review of foundational concepts in electromagnetism and set the stage for introducing special relativity, quantum mechanics, and matter waves for undergraduate students.
3) The overview highlights that succeeding chapters will develop tensor formulations of electromagnetism and special relativity from first principles before discussing applications like blackbody radiation and early quantum models.
This document presents a model for matching theoretical predictions of the relationship between supermassive black hole mass (MBH) and stellar velocity dispersion (σ) to observational data. The model uses a spherical Hernquist profile for stars within a Dehnen-McLaughlin dark matter halo. It finds that for a stellar to dark matter mass ratio (fe) of 6, the model agrees well with observed MBH-σ and MBH-bulge mass relations for large elliptical galaxies with σ ≥ 150 km/s.
This document describes a study using integral equation theory and Monte Carlo simulation to determine the structure and thermodynamics of a colloidal solution where particles interact via Yukawa or Sogami potentials. The authors use the hybridized mean spherical approximation within the integral equation theory to calculate properties like the pair correlation function, structure factor, internal energy and pressure. They find good quantitative agreement between results from integral equation theory using a Sogami potential and results from Monte Carlo simulation. The theoretical results are also compared to experimental data and show agreement when using a Sogami potential.
Bag of Pursuits and Neural Gas for Improved Sparse CodinKarlos Svoboda
This document proposes a new method called Bag of Pursuits and Neural Gas for learning overcomplete dictionaries from sparse data representations. It improves upon existing methods like MOD and K-SVD by employing a "bag of pursuits" approach that considers multiple sparse coding approximations for each data point, rather than just the optimal one. This allows the use of a generalized Neural Gas algorithm to learn the dictionary in a soft-competitive manner, leading to better performance even with less sparse representations. The bag of pursuits extends orthogonal matching pursuit to retrieve not just the single best sparse code but an approximate set of the top sparse codes for each point.
- The document discusses congruence of triangles and various criteria for determining if two triangles are congruent.
- There are four criteria presented: 1) SAS (two sides and included angle are equal), 2) ASA (two angles and a non-included side are equal), 3) SSS (all three sides are equal), and 4) RHS (right triangle with hypotenuse and one side equal).
- Theorems are proved that the angles opposite equal sides of a triangle are equal, and the sides opposite equal angles of a triangle are equal.
Slides of the talk on Koide Formula. Video should be available at http://viavca.in2p3.fr/alejandro_rivero.html
or directly al flv
http://viavca.in2p3.fr/video/alejandro_rivero.flv
A Numerical Method For Friction Problems With Multiple ContactsJoshua Gorinson
This document summarizes a numerical method for solving friction problems involving multiple contact surfaces. It begins by reviewing previous work on solving differential equations with discontinuities. The author then describes extending their previous method to handle problems with multiple contacts. Indicator functions are used to represent the regions of contact. Linear complementarity problems (LCPs) are solved to determine changes in the active contact surfaces. The method assumes the indicator functions and vector fields are smooth. Convergence results are proven showing the method can achieve high-order accuracy.
We first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.
Helium gas with Lennard-Jones potential in MC&MDTzu-Ping Chen
Helium gas with Lennard-Jones potential in MC&MD
- The document discusses using Monte Carlo and Molecular Dynamics simulations to model helium gas atoms interacting via the Lennard-Jones potential.
- Initial MC simulations of many helium atoms did not converge well. Simulations were then reduced to just four atoms, allowing analysis of compressibility and heat capacity as functions of volume and temperature.
- For four atoms in a tetrahedral configuration, equations were derived relating potential energy, pressure, compressibility, and heat capacity to volume and temperature in the low-temperature regime.
This document provides solutions to problems from Chapter 1 of an introductory fluid mechanics textbook. The key information is:
1) Problem 1.10 asks if the Stokes-Oseen formula for drag on a sphere is dimensionally homogeneous. The formula contains terms with dimensions of force, viscosity, diameter, velocity, density, and the student confirms it is homogeneous.
2) Problem 1.12 asks for the dimensions of the parameter B in an equation relating pressure drop, viscosity, radius, and velocity in laminar pipe flow. The student determines B has dimensions of inverse length.
3) Problem 1.13 calculates the efficiency of a pump given values for volume flow rate, pressure rise, and input power
Within the framework of the general theory of relativity (GR) the modeling of the central symmetrical
gravitational field is considered. The mapping of the geodesic motion of the Lemetr and Tolman basis on
their motion in the Minkowski space on the world lines is determined. The expression for the field intensity
and energy where these bases move is obtained. The advantage coordinate system is found, the coordinates
and the time of the system coincide with the Galilean coordinates and the time in the Minkowski space.
This document discusses light-matter interaction using a two-level atom model. It describes how an atom with only two energy levels can be modeled as a two-dimensional quantum mechanical system. The interaction of such a two-level atom with an electromagnetic field is then derived, leading to Rabi oscillations between the atomic energy levels driven by the field. Dissipative processes require a statistical description using the density operator formalism.
1) The document discusses modeling gravitational fields within general relativity by mapping geodesic motion in curved spacetime to motion along worldlines in Minkowski spacetime.
2) It presents equations showing the equivalence between motion in a gravitational field described by a metric and motion in flat spacetime under the influence of a force field.
3) As an example, it models the Schwarzschild metric by considering dust moving radially in Minkowski spacetime, recovering the Schwarzschild solution from Einstein's field equations in vacuum.
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π−
and
Σ−
momentum
vectors
can
be
determined
by
drawing
tangents
to
the
π−
and
Σ−
tracks
at
the
point
of
the
Σ−
decay.
We
can
then
measure
the
angle
between
the
tangents
using
a
protractor.
Alternative
method
which
does
not
require
a
protractor
is
also
possible.
Let
AC
and
BC
be
the
tangents
to
the
π−
and
Σ−
tracks
respectively.
Drop
a
perpendicular
(AB)
and
measure
the
distances
AB
and
BC.
The
ratio
AB/BC
gives
the
tangent
of
the
angle180◦−θ.
It
should
be
noted
that
only
some
of
the
time
will
the
angle
θ
exceed
90◦
as
shown
here.
Determining
the
uncertainty
of
Measurements
In
part
B,
It
is
asked
to
estimate
the
uncertainty
of
your
measurements
of
𝜃
and
r.
Uncertainty
of
measurement
is
the
doubt
that
exists
about
the
result
of
any
measurement.
You
might
think
that
well-‐made
rulers,
clocks
and
thermometers
should
be
trustworthy,
and
give
the
right
answers.
But
for
every
measurement
-‐
even
the
most
careful
-‐
there
is
always
a
margin
of
doubt.
It
is
important
not
to
confuse
the
terms
‘error’
and
‘uncertainty’.
Error
is
the
difference
between
the
measured
value
and
the
‘true
value’
of
the
thing
being
measured.
Uncertainty
is
a
quantification
of
the
doubt
about
the
measurement
result
Since
there
is
always
a
margin
of
doubt
about
any
measurement,
we
need
to
ask
‘How
big
is
the
margin?’
and
‘How
bad
is
the
doubt?’
Thus,
two
numbers
are
really
needed
in
order
to
quantify
an
uncertainty.
One
is
the
width
of
the
margin,
or
interval.
The
other
is
a
confidence
level,
and
states
how
sure
we
are
that
the
‘true
value’
is
within
that
margin.
You
can
increase
the
amount
of
information
you
get
from
your
measurements
by
taking
a
number
of
readings
and
carrying
out
Casimir energy for a double spherical shell: A global mode sum approachMiltão Ribeiro
In this work we study the configuration of two perfectly conducting spherical shells. This is a problem of basic importance to make possible development of experimental apparatuses that they make possible to measure the spherical Casimir effect, an open subject. We apply the mode sum method via cutoff exponential function regularization with two independent parameters: one to regularize the infinite order sum of the Bessel functions; other, to regularize the integral that becomes related, due to the argument theorem, with the infinite zero sum of the Bessel functions. We obtain a general expression of the Casimir energy as a quadrature sum. We investigate two immediate limit cases as a consistency test of the expression obtained: that of a spherical shell and that of two parallel plates. In the approximation of a thin spherical shell we obtain an expression that allows to relate our result with that of the proximity-force approximation, supplying a correction to this result.
- The document discusses a braneworld model where a 3-brane moves in a 5-dimensional anti-de Sitter bulk. The brane behaves effectively as a tachyon field with an inverse quartic potential.
- When the backreaction of the radion field (related to fluctuations of the brane position) is included, the tachyon Lagrangian is modified by its interaction with the radion. This results in an effective equation of state at large scales that describes "warm dark matter".
- The model extends the second Randall-Sundrum braneworld model to include nonlinear effects from the radion field, which distorts the anti-de Sitter geometry.
The document proposes a new infinity differentiable weight function, Wε(r), for smoothed particle hydrodynamics (SPH) methods. Wε(r) is based on a smeared-out Heaviside function and satisfies properties required of weight functions, including compact support, continuity, and asymptotic unity. The consistency and estimation errors of SPH interpolation using Wε(r) are analyzed, showing it provides C1 consistency and O(h4) order accuracy. The new weight function is developed to address limitations of existing weight functions like cubic and quartic splines in SPH approaches.
This document is a project report submitted by Shubham Patel for the partial fulfillment of an M.Sc. in Physics. The report introduces Galilean electromagnetism and constrained Hamiltonian systems. In part one, the report discusses various Galilean limits of Maxwell's equations including the electric limit, magnetic limit, and Carrollian limit. It also discusses formulations of these limits that are invariant under different systems of units. In part two, the report discusses Maxwell's field theory from a Hamiltonian perspective and constraints that arise in the formulation. It also discusses a higher order field tensor Lagrangian and its Hamiltonian formulation.
This document summarizes a numerical study of the structure and thermodynamics of colloidal suspensions using the variational method and integral equation theory. The interactions between colloid particles are modeled using either a Yukawa or Sogami potential. Results from the integral equation theory using a Sogami potential are found to be in good agreement with Monte Carlo simulation results and experimental data. The variational method and integral equation theory are used to calculate structural properties like the pair correlation function and thermodynamic properties.
Similar to Koide equations for quark mass triplets. (20)
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Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
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it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
1. Koide equations for quark mass triplets.
Alejandro Rivero∗
Institute for Biocomputation and Physics of Complex Systems (BIFI),
University of Zaragoza,Mariano Esquillor, Edificio I + D , 50018 Zaragoza ,Spain
(Dated: February 6, 2013)
We study Koide equation sequentially in a chain of mass triplets. We notice at least a not
previously published triplet whose existence allows to built a quark mass hierarchy from fixed
yukawas for top and up quarks. Also, the new triplets are used to build mass predictions either
descending from the experimental values or top and bottom, or ascending from the original triplet
of charged leptons.
PACS numbers: 12.15Ff
I. ORIGIN OF KOIDE FORMULA
An early [3, 8] intuition of the mass of the d quark as
the result of Cabibbo mixing with s quark, md ∼ θ2
c ms,
guided the efforts of modellers in the seventies [4, 23, 24]
to derive either this relationship or a close one, namely
tan θc =
md
ms
(1)
Particularly, Harari, Haut and Weyers [10] used a dis-
crete symmetry to produce not only the above equation
but also two exact equations
mu = 0;
md
ms
=
2 −
√
3
2 +
√
3
(2)
His model was a sequential symmetry breaking, using
the trivial and bidimensional irreducible representations
of the permutation group S3 The work was promptly cri-
tiqued [5] because at the end it was unclear how do they
get the mass eigenvalues; they rotate the mass matrix to
select a particular representation and then they disregard
non diagonal terms. Besides they did not provide an ex-
plicit representation for the Higgs sector (As for mu = 0,
it was not really a problem: other QCD mechanism could
happen later, to give it a mass of order mdms/MΛ). Still,
the result contributed to sustain an interest on the use
of discrete symmetry to predict structures in the mass
matrix.
Following explicitly this topic, in 1981 Koide [12, 13]
attempted another approach, incorporating the symme-
try in the context of a model of preons. Given that such
preons were components both of quarks and leptons, the
result was a formula for Cabibbo angle using the mass
of charged leptons and, as a by-product, a formula (eq.
(17) of [14]) linking their masses:
(
√
m1 +
√
m2 +
√
m3)2
m1 + m2 + m3
=
3
2
(3)
∗ arivero@unizar.es
The formula predicts a tau mass of 1776.96894(7)
MeV, perfectly matching the experimental [1] value of
1776.82 ± 0.16 MeV. In 1981, the measurement was still
1783 ± 4 MeV, so in some sense Koide’s work was a real
prediction, even if its composite character was ruled out
by experiments.
This formula (3), considered independently of a un-
derlying model, is referred as “Koide formula” or “Koide
equation”, and a tuple of three masses fulfilling it is called
a “Koide triplet” or “Koide tuple”
The masses (2) are a Koide triplet.
II. MODERN RESEARCH ON KOIDE
TRIPLETS
The use of a composite model can be substituted by
an ad-hoc Higgs sector with discrete symmetries. Such
models of ”flavons” have been revisited by Koide in later
work [16, 17]. In some proposals [21, 22], more symmetry
is added in order to protect Koide equation from renor-
malisation running.
Only with the SM, the equation is not protected, and
its running under the renormalisation group has been
studied for leptons as well as for some quark triplets. A
detailed study is for instance [25]. Generically, it is no-
ticed that the running up to GUT scale corrects the equa-
tion (3) by less than a ten per cent in the most extreme
case, and that in the case of charged leptons the equation
works better for pole masses. The reason for the stability
of the formula is that we are considering mass quotients
and the correction becomes proportional, for the case of
leptons, to mµ/mτ , and similarly for triplets of quarks
with same charge. For some quark triplets studied pre-
viously [18, 25], the general results indicate too that the
match is slightly better in the zone of low energy, and
that the discrepancy remains stable along the running.
Koide formula has had two relevant rewritings. First,
Foot [7] suggested an exact angle between the vector of
square roots of the masses and the permutation invariant
tuple:
(
√
m1,
√
m2,
√
m3)∠(1, 1, 1) = 45◦
(4)
More recently, when generalised to neutrinos [2, 6], it
2. 2
was seen that some fits with negative signs of the square
root could be need, and so nowadays a presentation more
agnostic about such signs is preferred,
√
mk = M
1
2
0 1 +
√
2 cos
2π
3
k + δ0 (5)
For the charged lepton, we have then three masses fit-
ted with two parameters
Meµτ = 313.8MeV, δeµτ = 0.222 (6)
and as we move δ0, we can produce zero and negative
values for some square root.
D. Look and M. D. Sheppeard [19] proposed (5) in a
generalised way, allowing for any real value λ0 instead of√
2. Sheppeard noticed [20] that the angles δ0 have some
extra regularity, compared to its value for leptons, when
the match is applied to the up type (uct) and the down
type (dbs) quarks: δeµτ = 3δuct and δdsb = 2δuct.
This insight was revised by Zenczykowski [27]. He
notes that the value of δ0 in (5) can be extracted directly
from
1
√
2
(
√
m2 −
√
m1) = 3M0 sin δ0 (7)
1
√
6
(2
√
m3 −
√
m2 −
√
m1) = 3M0 cos δ0 (8)
and then he studies its experimental value independently
of the whole formula, confirming
δeµτ = 3δuct (9)
The above equations are related to the initial research
on mixing angles, as Koide used the combination of
(7),(8) to produce Cabibbo angle, while a third condi-
tion
1
√
3
(
√
m3 +
√
m2 +
√
m1) = 3M0 (10)
joined to the previous two, implies Koide formula from a
pythagorean condition on the LHS of the three equations
(7)2
+ (8)2
= (10)2
(11)
The above equations relate to the doublet and singlet
irreps of S3, where δ0 can be understood a rotation of
the basis for the bidimensional irrep.
Generalisations of Koide equation to neutrinos have
been addressed in the literature but without a conclusive
result. We do not address them in this paper. Last, ex-
tensions to six-quark equations where done by Goffinet[9,
p. 78] and more recently by Kartavsev[11], but a definite
method to build tuples with an arbitrary number of par-
ticles has not been built.
III. RECENT EMPIRICAL FINDINGS
Two years ago, Rodejohann and Zhang [18] noticed
that if one forgets the prejudice of using equal-charge
quarks, then the triplet (c, b, t) matches Koide equation
accurately.
We can inquire if this success is repeatable and we
can keep connecting triplets of quarks with alternating
isospin. Note that already the first solution of Harari et
al involved quarks of different charge.
To explore this possibility, we solve (3) to get one of
the masses as a function of the other two
m3 = (
√
m1 +
√
m2) 2 − 3 + 6
√
m1m2
(
√
m1 +
√
m2)2
2
(12)
And then iterate a waterfall from experimental values
of top and bottom [1]:
mt = 173.5 GeV
mb = 4.18 GeV
mc(173.5, 4.18) = 1.374 GeV
ms(4.18, 1.374) = 94 MeV
mq(1.374, 0.094) = 0.030 MeV
mq (0.094, 0.000030) = 5.5 MeV
(13)
Experimental mc and ms are, respectively, 1.275 ±
0.025 GeV and 95 ± 5 MeV. So a new Koide triplet, not
detected previously, is apparent: (s, c, b). The reason of
the miss, besides the need of alternate isospin, is that the
solution to in this case has a negative sign for
√
ms.
This has also the collateral effect that
−
√
ms,
√
mc,
√
mb is almost orthogonal to the lep-
tonic tuple
√
mτ ,
√
mµ,
√
me. From the view of Foot’s
cone, they are in opposite generatrices.
That we can choose such negative signs in the search
for Koide triplets can be inferred for the rewritting (5)
above, as they are produced forcefully for some values
of the angle. Furthermore, the formalism of equations
(7),(8) and (10) allows to justify, in a model independent
way, that the sign of the square root can be arbitrary
with the only condition of keeping the same choosing in
the three equations. Formally, their LHS can be seen as
a row sum of the product matrix Q,
Q =
0 1√
2
− 1√
2
2√
6
− 1√
6
− 1√
6
1√
3
1√
3
1√
3
√
m3 0 0
0
√
m2 0
0 0
√
m1
(14)
The matrix Q being constructed so that QQ+
(and
Q+
Q too) is a mass matrix having m1, m2, m3 as eigen-
values, independently of the choosing of signs (or phase)
in the square root of the diagonal matrix. Of course, the
argument to impose the pythagorean equation in the sum
of each row will be model dependent.
In the obtained chain of equations, we infer that q and
q correspond respectively to the masses of up and down
3. 3
quarks, and so the other two triplets are (usc) and (dus)
A zero mass for the up quark allows to suspect that we are
seeing some symmetry which is only slightly displaced.
Plus, a situation where the up quark is massless at some
stage of the breaking constitutes an escape from the QCD
θ problem.
We can question if there is some other lost triplet be-
tween the 20 possible combinations. From the empir-
ical masses in [1] we can check that only two triplets
meet Koide equation (for either sign) with a 1% of toler-
ance: (cbt) and (scb). Next, (usc), (dsb) and (dsc) can fit
within a 10%, and the rest of the triplets go worse. If we
use instead the GUT-level masses of [26], the five better
matches, within a 10% of tolerance, are still the same but
now neither of them are in the 1% accuracy, again sug-
gesting that the formula works better in the electroweak
scale than at GUT energies.
Comparing the “waterfall” chain with this exploration,
the disagreement comes only with the last step, (dus),
obviously because of the non zero mass of the up quark
in the measured data. But the alternatives really only
match Koide equation within a ten percent and break
the pattern of chaining, that here was proceeding reg-
ularly (generation-wise and isospin-wise). We will com-
ment briefly on these alternatives later; their main differ-
ence comes in the way they manage to derive the down
quark mass, and the two next sections do not involve it.
IV. A PREDICTION UPSTREAM
In the extra symmetry revised by Zenczykowski [27],
the angle δ0 of charged leptons was three times the an-
gle for up-type quarks. A similar proportionality is also
happening here. For the triplet bottom-charm-strange,
the fitted parameters are
Mscb = 941.2MeV, δscb = 0.671 (15)
Here not only the angle, but also the mass seems re-
lated by a factor three.
Were we to accept the relationships Mscb = 3Meµτ
and δscb = 3δeµτ , the set of Koide equations would be
the most predictive semi-empirical formula for standard
model masses: taking electron and muon as input, we
can build the parameters of the lepton triplet and then
seed them to produce the (scb) triplet. The chain pre-
dicts good values for down, strange, charm, bottom, and
top quarks, as well as for the tau lepton. For the top
quark, with usual values of me and mµ as input, the pre-
dicted mass is 173.264 GeV, right in the target of current
averages, from Tevatron and LHC, of the pole mass.
For reference, the complete comparison between exper-
iment and prediction is:
exp. pred.
t 173.5 ± 1.0 173.26
b 4.18 ± 0.03 4.197
c 1.275 ± 0.025 1.359
τ 1.77682(16) 1.776968
s 95 ± 5 92.275
d ∼ 4.8 5.32
u ∼ 2.3 .0356
(16)
Where we do not quote error widths in the prediction
side because the input data, electron and muon mass,
is known with a precision higher than the rest of the
measurements, so all shown digits are exact. The table
can be taken more as an illustration of the power of Koide
triplets than as a working prediction; the experimental
quark masses are not the the pole masses, except for
the top quark, and the relationships Mscb = 3Meµτ and
δscb = 3δeµτ are ad-hoc as explained.
V. RESULTS WITH SIMPLE INPUTS
To understand the factors of three in the last section,
it is instructive to examine a case without experimental
inputs, considering instead that the Yukawa coupling of
the top quark is exactly one and the Yuwaka coupling
of the up quark is exactly zero. The point of asking for
yu = 0 is that there is the possibility of having some
extra symmetry that becomes restored in this limit.
Recall from the standard model that when yt = 1, the
mass of the top is 174.1 GeV, via mq = yq v /
√
2; we are
going to use this proportionality instead of quoting the
adimensional value of the yukawa coupling.
We should consider the four possible solutions in each
step of equation (12). A fast exploration with a computer
shows that it is possible to try different combinations of
triplets to get the goal yu = 0, but that the best matching
with data is the case of our sequential chain, and that it
is also the case with the greatest gap between top and
bottom. Thus we are going to use this fact as a postulate
to select the solution, in two steps:
First, we ask the difference between the top and the
next quark to be the greatest possible within Koide equa-
tion. This amounts to set the angle δcbt = 0. The masses
of bottom and charm are then the same, 2.56 GeV. The
next triplet has an angle, δscb = 60◦
and predicts a mass
for the strange quark of 150 MeV. And then the next
triplet has an angle of 12.62◦
and predicts a small but
non zero mass for the next quark: 0.810 MeV. All these
steps are unambiguous.
So this case with a single input, yt = 1, already pro-
vides a good match with the standard model patterns.
Now we proceed to smoothly move the parameters in or-
der to approach this yu to zero, fixng it at the cost of
4. 4
losing exact angles in the upper mass triplets. We get:
mt 174.1 GeV
mb 3.64 GeV
mτ mc 1.698 GeV
mµ ms 121.95 MeV
me mu 0 MeV
md 8.75 MeV
(17)
In the above table, we have chosen by hand a position for
lepton masses. This has some value: contemplate simply
the lower part of the table, written now as
mb 4Mscb
mτ mc
2+
√
3
2 Mscb
mµ ms
2−
√
3
2 Mscb
me mu 0
(18)
It can be seen that in this case Meµτ = Musc = Mscb/3,
giving some ground to the ad-hoc use of this factor in the
previous section; and also that the angles are one of them
three times the other: δeµτ = 15◦
and δscb = 45◦
, then
agreeing both with our observation and Zenczykowski’s.
VI. DISCUSSION
A question about the validity of Koide triplets is if they
are equations for the pole masses or for running masses,
and if the later, for which scale should they be applied.
Looking at the running, it can be argued that they can
be used by modellers in any situation, as they are good at
electroweak scale and still approximate enough at GUT
scale, but that their better agreement seems to happen
with pole masses (or, for quarks, for mq(mq) values).
This was already known for the lepton triplet, and here
we have provided an ”upstream prediction” going from
the e, µ pole mass as input to the experimental top mass,
which is also a pole mass.
Consider (s, c, b). For MS masses in current data [1],
the quotient LHS/RHS of (3) is 0.986. The value con-
temporary to [26] was 0.974. Using running masses from
[26], the quotient at MZ is 0.949, and at GUT scale it is
0.947.
Pole masses can be more natural on models where mix-
ing is the origin of Koide equation, while running masses
are appropriate for models where a symmetry breaking
at some scale produces the mass pattern.
There is the question of the model itself. We do not aim
in this paper to propose such; there are some attempts
in the literature to include more symmetry. Now, we can
say some words about the fact that we seem to have not
one, but four equations in the quark sector.
We believe that S4 = V4 S3 is an interesting candi-
date for a bigger group. It is the group of rotations of a
cube, and in fact we can classify our triplets by drawing
such cube with quarks in the faces, such that u, c, t meet
in a vertex, with respective opposite faces b, d, s. The
vertexes, arranged by oppositeness across the diagonal,
are
bds usc scb cbt
uct btd tdb dus
(19)
These four diagonal axis, have associated S3 sub-
groups, and each axis has at least one of the triplets
we are interested on. It can be noticed that an axis is
“overloaded” with the triplet dus, and this can become
an argument to substitute it by bds. Some other substi-
tutions are possible, and while we can not concrete the
option without resorting to an explicit model, the point
of being able to select here, from the 20 possible triplets,
the ones having a best matching with data, is an argu-
ment to use S4 at least for an initial organisation.
Lets conclude by reviewing what we have found in this
paper: we have done a complete exploration of Koide
quark triplets. We have found at least a new exact tuple
for Koide equation, for quarks s, c, b, and two more, ap-
proximate, for u, s, c and d, u, s; even if the later is not
really new in the literature and we could be in doubt
about substituting it for d, s, b.
We noticed that these tuples allow to build a chain of
equations, starting from c, b, t from [18]; we have verified
its empirical validity and we have suggested yet two other
methods to reproduce the mass scales of the Standard
Model fermions: either to fix the yukawas of top and up
quark to the natural values 1 and 0 or, independently,
to propose a proportionality between the parameters of
the charged leptons tuple and those of the s, c, b tuple.
Both methods have some virtue: if the last method is
applied using as input the experimental values of two
lepton masses, all the quark masses, except the up quark,
coincide with the experimental measurements within one
or two sigmas at most, and a very good prediction of
the top quark pole mass is reached. And in the former
method, with fixed yt = 1, and yu = 0, we could tell
that the only input from experiment is Fermi constant
and the number of generations, and that Koide triplets
provide all the necessary levels for mass.
ACKNOWLEDGMENTS
The author wants to acknowledge C. Brannen, M.
Porter and J. Yablon by their support and comments.
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