_{1}

A chemical non-equilibrium equation for binding of massless quarks to antiquarks, combined with the spatial correlations occurring in the condensation process, yields a density dependent form of the double-well potential in the electroweak theory. The Higgs boson acquires mass, valence quarks emerge and antiparticles become suppressed when the system relaxes and symmetry breaks down. The hitherto unknown dimensionless coupling parameter to the superconductor-like potential becomes a re-gulator of the quark-antiquark asymmetry. Only a small amount of quarks become “visible”—the valence quarks, which are 13% of the total sum of all quarks and antiquarks—suggesting that the quarks-antiquark pair components of the becoming quark-antiquark sea play the role of dark matter. When quark-masses are in-weighted, this number approaches the observed ratio between ordinary matter and the sum of ordinary and dark matter. The model also provides a chemical non-equilibrium explanation for the information loss in black holes, such as of baryon number.

Two ways for explaining the origin of mass, QCD and confinement of quarks and the Higgs-mechanism in the electroweak (EW) theory have been discussed by Wilczek: “Superficially those mechanisms appear quite different, but at a fundamental level they are essentially the same” [

The Sakharov constraints [

Quark-gluon interaction is strong at distances of about a nucleon diameter (10^{−15} m), but weakens at high energies (temperatures) where quarks interact at shorter distances. Already at about 150 MeV (~2 × 10^{12} degrees K), nuclear matter boils down to a quark- gluon plasma (QGP) [

This paper identifies and suggests solutions to some of these problems that underlie the standard model. Section 2 describes the chemical non-equilibrium equation for binding of quarks (fermions) to antiquarks (antifermions). The binding equation, combined with a coherent (local) formulation of the strong spatial (non-local) correlations between the condensing particles, as shown in Section 3, yields the Ginsburg-Landau (GL) like potential used in EW theory, however, with a density-dependent order parameter. Section 4 provides an explanation as to how mass, dark matter, and valence quarks emerge by suppression of antiquarks. It is shown that the coupling to the GL- like potential becomes a regulator of the

The chemical non-equilibrium conditions exclude usual quantum field theory methods, such as the Bethe-Salpeter equation and eikonal type models [

The rate-equation for chemical non-equilibrium binding of massless quarks to antiquarks when the system cools down, is given by

k and

Equation (1) also obeys the initial boundary constraints

and that the initial “free” quark and antiquark amplitudes, q_{0} and

After insertion of these constraints, Equation (1) reads

where

The solution to Equation (3) is

where the “short-hand” notations _{K} and _{K}, _{0},

To quantify the screening effect and study the emergence of valence quarks, mass and dark matter by suppression of antimatter when the system cools down, however, Equation (3) must be first combined with the spatial correlations between the

However, to make the emerging particles point-like, Equation (5) must be contracted and synchronized to a fictitious “centre of mass”,

After a certain time, _{s}, which can be factorized out also from higher order terms. If ψ_{s} is identified with

which can be combined with Equation (3). Equation (6), which has the form of a Bose- Einstein distribution, also corresponds formally to the grand partition function, with

To obtain the relaxation dynamics and time evolution of the correlated system, Equation (6) must be linked to Equation (3). The time derivative of Equation (6),

combined with Equation (3) then yields

which has the solutions

To create a baryon with a small finite number (one or two) of valence quarks (_{p} is the number of stably

bound _{p}, and_{p} is also proportional to the number of valence quarks of a certain flavour, Q_{f}, because _{f} identical infinite sequences, each of which is then rearranged as in

By the topological quantization, the stationary zero order term of unstable _{p} of massive pairs.

By the contraction of all (x, t) to a fictitious “centre of mass” (x, t)―an approximation needed to derive the dynamics and to make all particles point-like―the internal structure and dynamics of the system were neglected. However, in principle, Equation (5)

could correspond to any arbitrary structure and dynamics.

For simplicity the time dependent solution in Equation (9) is interpreted as a travelling wave on a string-like 1D lattice of

hence

This V(φ), which corresponds to a continuum approximation dynamics [^{2} as the hitherto unknown dimensionless parameter [^{2} has become a regulator of the antiparticle suppression as will be further explained below.

The derivative of Equation (11) yields the spatial part of the equation of motion

albeit for a particle with imaginary mass. This flaw too is restored by the displacement

and the corresponding equation of motion reads

The mass term

However, the corresponding field displacement caused by the relaxation and condensation of the actual many-body system, does not take place until after all quarks except for a small number of valence quarks (

This chemical non-equilibrium model yields a quark-antiquark asymmetry, _{H} = 2gφ_{0}, and with m_{H} = 125.09 GeV and φ_{0} = 174.22 GeV [

This determines in turn the dimensionless parameter which now yields the

The topological quantization, _{p} ® ¥; m_{p} ® 0). The condensation implied by Equation (6) can then proceed in two different ways, leading to absence (

Since quarks are initially free and massless, at infinite energies essentially only photons contribute to the infinite energies to the initial binding of quarks by antiquarks and to the induction of mass, and gluons contribute later at finite energies. The number of valence quarks Q_{f} ≤ 2, with flavour f and mass m_{f}, is related to the number N_{p} of

These model derivations, which should hold for each separate flavour, also hold for charged leptons, which contribute similarly to the ratio between ordinary and dark matter. For instance, a condensate of massless electrons and positrons can produce para-positronium spin-zero atoms with a lifetime τ_{p} that decreases from infinity to 2ħ/(m_{e}c^{2}α^{5})~1.25 × 10^{−}^{10} s when the electron mass m_{e} increases from zero to 511 keV, α being the fine structure constant. Since τ_{p} is about 10^{12} times longer than the lifetime of a Higgs boson, τ_{H} = 1.56 × 10^{−}^{22} s, such a lepton condensate can thus also contribute to the GL-like potential and to the Higgs boson. Leptons probably contribute much less than baryons to the total mass of the Universe [

The actual model can also explain the binding of massless neutrinos (n_{K}) to massless anti-neutrinos (^{0}-boson. The coupling g then acquires the same form as given by Sakharov without proof in 1967, ^{−}^{9} [^{2} = 0.13, explaining why all antiparticles are so rarely observed outside laboratories. As is well-known neutrino flavour oscillations yield information about the neutrino mass differences [

A softly bound condensate of up and down ^{−}^{18} s much longer than that of a Higgs boson, can hence also contribute to the scalar field and the Higgs boson. Since g equals the Higgs boson mass

To derive

The dark mass candidates available in this model are thus identified as the ^{2} = 0.159 in agreement with the value observed [

If quarks and antiquarks in the stationary state can acquire mass before symmetry breakdown, their masses should have opposite signs and their gravitational contributions, one attractive and one repulsive, should cancel if quarks and antiquarks are not too far separated. However, at larger separations, such a repulsive form of gravitation could eventually drive the expansion of the universe. The same reasoning should hold for leptons.

A density-dependent, hence lyotropic [_{0} and

It is still not possible to predict the mass

This chemical non-equilibrium type of dynamics is assumed to have controlled the relaxation dynamics after big bang when baryonic matter with a “surplus” of valence quarks was frozen out from a hot gaseous Universe with equal amounts of massless quarks and antiquarks. It is assumed to also control the relaxation after high-energy proton-proton collisions, and probably also essential parts of the dynamics in black holes.

Obviously, it would have been preferable to obtain the binding of quarks to antiquarks by exchange of gauge particles in 4D. However, the emergence of valence quarks―an increasing number of quarks relative to the number of antiquarks―implies chemical non-equilibrium conditions. In combination with strong spatial correlations, here represented by Equation (6), that emerge when quarks condense via a plasma phase [

The Nielsen-Olesen (NO) string [_{Z} of the gauge vector field A, and one with a radius equal to the coherence length ξ = 1/m_{H}, the inverse of the Higgs boson mass (

The inverse mass m_{Z} of the Z-boson plays the role of penetration length in this density-dependent superconductor-like model [

Spontaneous symmetry breaking occurs in many condensed matter systems such as superconductors, ferromagnets and crystals. In the actual chemical non-equilibrium den- sity-dependent [

as topological defects [

Given that 4.9% of all mass is visible (m_{v}), the actual model yields 25.9% dark mass (m_{d}), which leaves 69.2% dark energy (e_{d}). The model thus gives the correct order of magnitude for the ratio between visible and non-visible mass. However, since both the visible and the dark masses started from zero at the big bang, this model should comply better with inflation than an exploding Universe containing massive objects from start. The actual model suggests that the dark energy equals the kinetic energy of the dark mass. This energy, which was neglected together with the structure in Equation (6), is estimated to_{d} = 0.259 yields a dark energy e_{d} ≈ 0.606.

Similar to the “atomistic” theory of matter and electricity proposed by Einstein and Rosen [^{2}), but in a different manner. When energy increases, the chemical non-equilibrium model dynamics goes backwards. M then becomes replaced by_{0} according to Equation (15), R > 0 being a factor to be determined. This also reduces the amplitude of gravitational waves from black hole events, such as formation and coalesce of black holes, making such waves extremely hard to detect [

When mass vanishes, the dust-like particles accelerate to the velocity of light and their clocks stop ticking, at least until they become rematerialized. The question is whether the negative gravitation and negative energy are sufficient to permit rematerialized particles to return to our own world sheet from the interior of black holes [

Clearly, this chemical non-equilibrium interaction is but an attempt to model what actually takes place in the real Universe, however, it seems to be first to go beyond the grand canonical ensemble in a system containing strong spatial correlations, and it also seems to work reasonably well. Observe that transport theories have not succeeded to solve this type of chemical non-equilibrium problem, and have thus not solved the problem for lattice QCD [

It is also interesting to compare the

I thank Ludvig Faddeev for stimulating discussions on this matter many years ago at CERN, where I first got the idea for the model.

Matsson, L. (2016) Higgs-Like Mechanism by Confinement of Quarks in a Chemical Non-Equilibrium Mo- del. World Journal of Mechanics, 6, 441-455. http://dx.doi.org/10.4236/wjm.2016.611031