Everything visible in our universe is composed of electrons, protons, and neutrons. Protons and neutrons are composed of up quarks and down quarks. The Standard Model of particle physics assumes that electrons, up quarks, and down quarks are point particles (with infinite matter density). In contrast, this holographic model treats charged fundamental fermions as spheres with non-zero holographic radius, to determine proton and neutron masses from electron mass and thus from fundamental constants.

The radius of the event horizon of any observer in our vacuum dominated universe, with cosmological constant $\Lambda =1.088\times {10}^{-56}{\text{cm}}^2$, is $R_H=\sqrt{3/\Lambda }=1.661\times {10}^{28}\text{cm}$. Using PDG 2024 data [1] with Hubble constant $H_0=67.4\text{km}/\left(\text{sec}\cdot \text{Mpc}\right)$, critical energy density ${\rho }_{crit}=\frac{3H_0^2}{8\pi G}=8.533\times {10}^{-30}\text{g}/{\text{cm}}^{\text{3}}$, gravitational constant $G=6.67430\times {10}^{-8}{\text{cm}}^{\text{3}}/\left(\text{g}\cdot {\text{sec}}^{\text{2}}\right)$, and vacuum energy fraction ${\Omega }_{\Lambda }=0.685$, mass of the observable universe within the event horizon is $M_H=\frac{4}{3}\pi \left(1-{\Omega }_{\Lambda }\right){\rho }_{crit}{R}_H^3=5.155\times {10}^{55}\text{g}=\left(0.187\text{g}/{\text{cm}}^{\text{2}}\right)R_H^2$.

Based on quantum mechanics, general relativity, thermodynamics, and Shannon information theory, holographic analysis [2] with Planck length $l_P=\sqrt{\frac{\hslash G}{c^3}}=1.616\times {10}^{-33}\text{cm}$ finds only $N=\frac{\pi }{\ln \left(2\right)}{\left(\frac{R_H}{l_P}\right)}^2=4.741\times {10}^{122}$ bits of information on the event horizon will ever be available to describe the observable universe within the event horizon, and the mass associated with a bit of information on the event horizon is $m_{bit}=M_H/N=1.078\times {10}^{-67}\text{g}$. This holographic analysis assumes the bits of information describing systems with definite mass m within the universe are available on spherical surfaces around the systems withradius $r=\sqrt{\frac{m}{M_H}}{R}_H$, therefore holographic radii of particles with definite mass $m$ are $r=\sqrt{\frac{m}{0.187}}\text{cm}$.

This model, with electron holographic radius $r_e$, specifies holographic radii of the three lowest mass Standard Model fermions as $r_e$, $r_u=2r_e$ and $r_d=3r_e$, to predict up quark mass $m_u=4m_e$ and down quark mass $m_d=9m_e$. These up and down quark masses are within quark mass estimate ranges, based on intricate lattice quantum chromodynamic calculations, shown in “ideograms” on pages 2 and 4 of PDG 2024 “Particle Listings, Light Quarks.”

Electron holographic radius $r_e=\sqrt{\frac{m_e}{0.187}}\text{cm}$ is the radius of a spherical holographic screen accommodating $N_e=\frac{\pi }{\text{ln}\left(2\right)}{\left(\frac{r_e}{l_P}\right)}^2$ bits of information. Electrostatic potential energy of electron charge $e$ and positron charge $-e$ separated by distance $2r_e$ is $V=-\frac{e^2}{2r_e}=-\frac{\alpha \hslash c}{2r_e}$ where $\alpha =\frac{e^2}{\hslash c}$ is the fine structure constant. Two adjacent spheres with holographic radius $r_e$, a precursor for electron-positron pair production, have total energy $E=2m_e{c}^2-\frac{\alpha \hslash c}{2r_e}=0$ when $r_e=\frac{\alpha \hslash c}{4m_e{c}^2}$. Two equations for $r_e$ result in $\frac{\alpha \hslash c}{4m_e{c}^2}=\sqrt{\frac{m_e}{M_H}}{R}_H$ and the equation for electron mass $m_e={\left[\frac{{\left(\alpha \hslash \right)}^2}{32}\left(\frac{1-{\Omega }_{\Lambda }}{G{\Omega }_{\Lambda }}\sqrt{\frac{\Lambda }{3}}\right)\right]}^{1/3}\text{g}$.

PDG 2024 values predict electron mass 0.5% higher than actual electron mass to three significant figures. Setting $\Lambda =1.08800\times {10}^{-56}{\text{cm}}^{-2}$ and increasing ${\Omega }_{\Lambda }$ by 0.5% to 0.6883855, within PDG 2024 error bars, specifies electron mass to six significant figures. Since gravitational constant G is only known to six significant figures, the calculation cannot be extended to greater precision until G is measured more precisely.

The model considered in this paper describes protons and neutrons as an up or down quark bound to a diquark (composed of an up and down quark). Forces between quarks and diquarks are specified as attractive forces $-K_p{\left(e/r\right)}^2$ within protons and $-K_n{\left(e/r\right)}^2$ within neutrons.

Schrodinger equations for two particles bound by $-K{\left(e/r\right)}^2$ forces have the same form as the hydrogen atom Schrodinger equation, so two particles bound by a $-K{\left(e/r\right)}^2$ force have Bohr radius $a=\frac{{\hslash }^2}{\mu K{e}^2}=\left(\frac{\hslash c}{\mu c^2}\right)\left(\frac{\hslash c}{K{e}^2}\right)=\frac{\hslash c}{\alpha K\mu c^2}$, where $\mu$ is reduced mass of the two particles.

If quarks within protons were unbound, their masses would be $4U_p{m}_e$ and $9U_p{m}_e$. Corresponding unbound quark masses for quarks within neutrons would be $U_{n}4m_e$ and $U_{n}9m_e$.

Then reduced mass for the proton bound system is ${\mu }_p=\frac{\left(4U_p{m}_e+9U_p{m}_e\right)4U_p{m}_e}{4U_p{m}_e+4U_p{m}_e+4U_p{m}_e}=\frac{52}{17}{U}_p{m}_e$ and reduced mass for the neutron bound system is ${\mu }_n=\frac{\left(4U_n{m}_e+9U_n{m}_e\right)9U_n{m}_e}{9U_n{m}_e+9U_n{m}_e+4U_n{m}_e}=\frac{117}{22}{U}_n{m}_e$.

Hydrogen atom van der Waals radius is radius of the sphere surrounding all mass of a hydrogen atom. Free hydrogen atom van der Waals radius [3] is 3.16 times hydrogen atom Bohr radius. So, this model treats nucleon holographic radii $r_p$ and $r_n$ as 3.16 times Bohr radii $a_p$ and $a_n$, resulting in $r_p=3.16a_p=\frac{3.16\hslash c}{\alpha K_p{U}_p{m}_e{c}^2}=\frac{3.16\hslash c}{\alpha C_p{m}_e{c}^2}$ and $r_n=3.16a_n=\frac{3.16\hslash c}{\alpha K_n{U}_n{m}_e{c}^2}=\frac{3.16\hslash c}{\alpha C_n{m}_e{c}^2}$, with composition factors $C_p=K_p{U}_p$ and $C_n=K_n{U}_n$. Thus $C_p=5591$, $C_n=5570$, proton mass $m_p=1.87{\left(\frac{\hslash }{\alpha c{C}_p}\right)}^2\left(\frac{1}{m_e^2}\right)=\frac{0.986}{m_e^2}=0.986{\left[\frac{{\left(\alpha \hslash \right)}^2}{32}\left(\frac{1-{\Omega }_{\Lambda }}{G{\Omega }_{\Lambda }}\sqrt{\frac{\Lambda }{3}}\right)\right]}^{-2/3}$, and neutron mass $m_n=1.87{\left(\frac{\hslash }{\alpha c{C}_n}\right)}^2\left(\frac{1}{m_e^2}\right)=\frac{0.979}{m_e^2}=0.979{\left[\frac{{\left(\alpha \hslash \right)}^2}{32}\left(\frac{1-{\Omega }_{\Lambda }}{G{\Omega }_{\Lambda }}\sqrt{\frac{\Lambda }{3}}\right)\right]}^{-2/3}$.

Proton and neutron masses are therefore determined by composition factors $C_p$ and $C_n$ Planck’s constant $\hslash$, gravitational constant $G$, fine structure constant $\alpha$, cosmological constant $\Lambda$, and vacuum energy fraction ${\Omega }_{\Lambda }$.

Holographic analysis only applies to systems in the universe with definite mass. The three neutrinos in the Standard Model of particle physics oscillate between three mass states when travelling within the universe, so holographic analysis does not apply to neutrinos. The only lengths characteristic of neutrinos are their Compton wavelengths $\lambda =\frac{\hslash }{mc}$. If electron neutrinos are vacuum energy excitations, with radius $r=\frac{1}{4}\frac{\hslash }{mc}$ and the lowest energy density in the universe (cosmic vacuum energy density ${\rho }_v=5.83\times {10}^{-30}\text{g}/{\text{cm}}^{\text{3}}$), electron neutrinos have mass $m_1={\left[\frac{\pi }{6}{\rho }_v{\left(\frac{\hslash }{c}\right)}^3\right]}^{\frac{1}{4}}=2.02\times {10}^{-36}\text{g}=0.0013\text{eV}$. Neutrino oscillation data [4] [5] then predict $m_2=\sqrt{m_1^2+7.42\times {10}^{-5}{\left(\text{eV}\right)}^2}=0.00871\text{eV}$ and $m_3=\sqrt{0.5{\left(m_1+m_2\right)}^2+2.517\times {10}^{-3}{\left(\text{eV}\right)}^2}=0.0507\text{eV}$, resulting in neutrino mass sum = 0.0607 eV, consistent with minimum neutrino mass sum [6] .

Electron, proton, and neutron masses have been considered independent fundamental constants of nature. Electron mass is determined by Planck’s constant $\hslash$, gravitational constant $G$, fine structure constant $\alpha$, cosmological constant $\text{Λ}$, and vacuum energy fraction ${\Omega }_{\Lambda }$. This analysis specifies proton and neutron masses by composition factors $C_p=5591$ and $C_n=5570$ and electron mass. Therefore electron, proton, and neutron masses are not fundamental constants.

The author is grateful for the hospitality of the Rutherford Institute, Russell Street, Clontarf, NSW, where this analysis was initiated.

Holographic analysis cannot address details of matter accumulation into large scale structures, but can account for masses of large-scale structures at specific times, with the following results:

Massive early galaxies

It was thought galaxies would not be seen at redshifts $z>10$, because there was not time for them to develop so early in the history of the universe. Therefore, many were surprised when JWST found JADES-GS-z14-0 with mass ${10}^9{M}_{\odot }$ at redshift $z=14$. In contrast, holographic analysis [7] predicts average galactic mass $~{10}^9{M}_{\odot }$ for galaxies expected at redshifts $z$ between 10 and 20.

Supermassive black holes in galaxy centers

Holographic analysis ^[8] predicts central supermassive black holes (SMBH) with mass $M_{SMBH}\left(z\right)=\sqrt{M_g\left(z\right)M_{sc}\left(z\right)}$ in galaxies at redshift $z$ with isothermal matter density distribution within their holographic radii, where $M_g\left(z\right)$ is galactic mass and $M_{sc}\left(z\right)$ is mass of a star cluster within the galaxy.

Formation of first stars (Population III) at redshift z = 66

Holographic analysis [8] , updated with PDG 2024 parameters, finds first stars (population III stars with mass $~300M_{\odot }$) formed at redshift = 66, consistent with observations.

Upper bound on supercluster, galaxy, and black hole masses

Holographic analysis [8] , updated with PDG 2024 parameters, shows: