1. Concerns Regarding the Standard Model

1) According to the Standard Model (SM) of particle physics, the building blocks of the universe are quantum fields defined by abstract creation and annihilation operators sums, in contrast to the atomistic point of view where the fundamental building blocks of the universe were particles [1].

2) Paul Dirac thought that better understanding of the vacuum substructure is needed [2] [3]. In contrast to the SM, Dirac thought that electrons are not point like particles and proposed a spherical shell electron model [4].

3) Dirac thought that electrons interact strongly with the vacuum electron-positron virtual pairs and hence are never bare in contrast to the SM path integral approach, where bare electrons propagate in free space in zero-order [5] [6].

4) Harari suggested beyond the SM that all matter (protons, neutrons, electrons and also the interaction bosons, w^+/− and Z) are composite particles. Harari named the proposed substructure fundamental particles Rishons, T and V, having an electric charge of a 1/3 and a 0. Accordingly, various combinations of Rishons and their anti-particle pairs T,V,˜T,˜V create the leptonic and baryonic matter [7].

5) Erwin Schrödinger introduced the idea of a wave packet particle and found that a gaussian wave packet remains coherent with harmonic potential, however in free space, the width of the gaussian wave packet grows rapidly with time [8]. If an electron wave packet is initially localized in a region of an atomic dimension of 10⁻¹⁰ meter, the width of the wave packet doubles in about 10⁻¹⁶ second and after about a milli-second the wave packet width grows to about a kilometer [9], which is an unreasonable result for a microscopic particle. Is the Schrödinger equation and the path integral approach wrong in free space or maybe the understanding of the vacuum having a fixed potential value is wrong?

2. Addressed Questions

The main questions addressed in the paper are:

1) Does the quantum vacuum have substructure Pionic fabric with a unit cell that may be described by eight-component Spinor?

2) Are the light quarks and antiquarks u,d,˜d,˜u , the fundamental building blocks that comprise all leptonic and baryonic matter?

3) Do the non-elementary, non-point like embedded electron tetraquarks move on the Pionic fabric by rapid u and d and ˜u and ˜d quark permutations?

The Pion tetrahedron and the vacuum substructure are described in Section 3. The classical and quantum quark molecular dynamics hybrid scheme and the Pionic fabric unit cell are described in Section 4. A double well potential model for the electron and pion tetraquarks and the symmetric quark exchange reaction is described in Section 5. The embedded electron and the Pionic fabric cloud are described in Section 6. The positron tetraquarks are described in Section 7. The electron-positron creation and decay embedded in the Pionic fabric are described in Section 8. A proton embedded in the Pionic fabric cell is described in Section 9. The eight-component spinors of the vacuum and the embedded electron and positron dynamics on the Pionic fabric are described in Section 10. A lattice QCD computation of the electron and pion tetraquark mass is proposed in Section 11. Section 12 is a summary.

3. The Pion Tetrahedron and the Vacuum Substructure

We assume that the quantum vacuum is filled with exotic pion tetraquark tetrahedrons that form a Pionic fabric [10]-[14]. We note that the vacuum pion tetraquark tetrahedrons are not ordinary particles since they are composed of 50% matter and 50% antimatter. We assume that the Pionic fabric quarks and antiquarks do not annihilate each other, and that their dynamics may be modeled with classical molecular dynamics with additional quark exchange operation described below. We assume that the Pionic fabric unit cell includes two exotic tetraquarks, u˜dd˜u , each composed of the two light quarks, d and u , and their antiquark pairs, ˜d and ˜u that may be described by eight-component spinor described further in Section 10 below.

A Classical and Quantum Quark Molecular Dynamics Hybrid Scheme

The pion tetraquark molecule is assumed to be composed of a d˜d and u˜u mesons having a tetrahedron structure shown in Figure 1 inside a cubic cell, where the cell size aπ is determined by the classical and quantum hybrid quark molecular dynamics scheme described below. Two pion tetraquark tetrahedron enantiomer molecules may exist obtained by exchanging the positions of two quarks at the tetrahedron vertices that breaks dynamically the chiral symmetry assumed by effective field QCD theory [15]-[21].

 

 

Figure 1. Illustrates the two pion tetraquark tetrahedron enantiomer molecules.

The pion tetraquark Hamiltonian using a quark pair interaction model [22] is:

Hpion tetraquark=12muv2u+12m˜uv2˜u+12mdv2d+12m˜dv2d−49e24πε0ru,˜u+σu,ur˜u,u   +29e24πε0ru,˜d+σu,dr(u,˜d)−29e24πε0ru,d+σu,dru,d+29e24πε0r˜u,d   +σu,dr˜u,d−29e24πε0r˜u,˜d+σu,dr˜u,˜d−29e24πε0rd,˜d+σd,drd,˜d(1)

The classical and quantum quark molecular dynamics hybrid scheme includes in addition to solving Newtonian classical dynamics equations for the four quarks and antiquarks, quark exchange operations that occur by quantum tunneling via a barrier. We assume that an Active Gluonic Center (AGC) is created in the center of hadrons by the quark and antiquark interaction where quark and antiquark pairs exchange positions and velocities of the quark pair according to Equations 2(a)-(b) below. The quark exchanges at the AGC prevent the quarks falling into the attractive coulomb singularity at short distances rq,˜q~0 . The quarks and antiquarks continue their classical periodic trajectories following exactly the path of their pair quark and antiquark after the exchange operation.

rq(t+1)=r˜q(t),  r˜q(t+1)=rq(t)(2a)

vq(t+1)=−v˜q(t),  v˜q(t+1)=−vq(t)(2b)

An example of the pion tetraquark trajectory is shown in Figure 2 for two mesons. The meson quarks and antiquarks, u˜u and ˜dd , are attracted to each other and two quark exchange operations occur at the pion tetraquark AGC, where the classical trajectories of the ˜u (in blue) and u (in orange) occurs simultaneously with the switching of the ˜d (in green) and d quarks trajectories (in red). The exchange operations of the quark and antiquark at the AGC surface occurs instantly and coherently in a single time step and the classical trajectory continues using Newtonian classical dynamics equations.

 

 

Figure 2. Illustrates the exotic pion tetraquark tetrahedron trajectory with the quark exchange operations at the pion gluonic center (AGC).

4. The Pionic Fabric Cubic Unit Cell Symmetry

A Pionic cubic unit cell that may fill space may include two rotated and flipped pion tetraquark tetrahedrons as shown below. The eight up, down, anti-up and anti-down quarks capture the 8 vertices of the Pionic unit cell where on the 6 faces of the cubic cell there are a chargeless and colorless pions, ˜ud˜du . We assign the indices below to the first pion tetraquark molecule (1, 2, 3, 4) and the second pion tetraquark molecule (5, 6, 7, 8) (Figure 3).

The Pionic fabric cubic unit cell point group is Ci(S2) , that includes only two symmetry elements, identity and inversion, and two irreducible representations Ag and Au [22]. The Pionic fabric that may be created extending the Pionic cubic unit cell into a periodic fabric is shown in Figure 4. A d quark (5) has two ˜u and two u quarks surrounding it in the X-Y plane and two ˜d quarks in the Z direction. Similarly, a u quark (2) is surrounded by two ˜d and two d quarks in the X-Y plane and two ˜u quarks in the Z direction. The ˜d and ˜u quarks (3 and 1) have similar quark pair neighbors in the Pionic fabric.

 

 

Figure 3. Illustrates Pionic fabric cubic unit cell with C_i point group symmetry.

 

 

Figure 4. Illustrates the Pionic fabric cubic unit cell and nearest neighbors in the Pionic fabric.

A sum of pair potentials Hamiltonian model for the Pionic unit cell includes pair potentials between all 8 quarks and antiquarks [23] where the sign and magnitude of the Coulomb interaction terms qi.j and the strength of the string tensions σi,j are defined below.

Hpioncell=∑8i=112miv2i+∑8i,j=1(i<j)(qi.je24πε0ri,j+σi,jri,j)(3)

qi.j={−49for ˜uu pairs49for ˜u˜u and uu pairs−29for ud and ˜u˜d pairs29for ˜ud and ˜du pairs−19for d˜d pair19for dd and ˜d˜d pairs(4)

σi,j={σu,ufor ˜uu,˜u˜u and uu pairsσu,u2for ˜ud,˜u˜d,˜ud and u˜d pairsσu,u4for ˜dd,˜d˜d and dd pairs(5)

Assuming that the Hamiltonian energy vanishes in the ground state, an equation for the cubic cell length aπ as a function of the string tension parameter σu,u is derived.

aπ=√109(1−(1√3−1)√2)ℏcα2(264+(132+52√3)1√2)σu,u(6)

The quantum vacuum may be filled with infinite number of such zero-energy Pionic unit cells.

Next, the value of the vacuum Pionic cell string tension parameter

σu,u=1.5 KeV/fm is determined by the classical and quantum quark molecular dynamics hybrid scheme that generates periodic trajectories shown below in Figure 5 and Figure 6, such that the vibration frequency of the Pionic unit cell quarks

and antiquarks is equal to Dirac’s electron zitterbewegung frequency 2mec2ℏ . The calculated Pionic cell size is aπ=7.757×10−15 meters. Accordingly, the Pionic fabric density in free space is ρpion fabric=1 a3pion cell =2.1417×1042pion cellsm3 .

The distance between the four ˜uu and ˜dd quark pair of the pion cell are shown below and illustrate the active gluonic center (AGC) quark exchange operations where the quark exchanges occur vertically between ˜u(1) and u(6) , ˜u(5) and u(2) , ˜d(3) and d(8) , ˜d(7) and d(4) at the cusps (Figure 5).

 

 

Figure 5. Illustrates the quark and anti-quark pairs distances of the pion cell. The exchange operations occur at the cusps.

The potential and kinetic energies of the Pionic fabric cell quarks and antiquarks are shown in Figure 6. The Pionic cell vibrates in and out towards the AGC. The quarks kinetic energy grows when the quarks fall in approaching the AGC and then is reduced gradually after the quark exchange occurs at the cusps when the quarks move away from the AGC. We show that the total pion cell energy is zero, since there are infinite number of pion cells in the Pionic fabric, their total energy remains 0. However, the Pionic cells are not static and the quark dynamics are shown by the kinetic and potential energies of the Pionic fabric unit cell in Figure 6 similar harmonic oscillator kinetic and potential energy.

 

 

Figure 6. Illustrates the zero-energy Pionic fabric unit cell quarks and antiquarks oscillating potential and kinetic energies.

The X coordinate of the four quarks and anti-quarks of the first pion tetraquark, ˜u(1) , u(2) , ˜d(3) , d(4) are shown below in Figure 7 and the Z coordinate of ˜u(1) and u(6) are shown in Figure 8.

 

 

Figure 7. Illustrates the X coordinate of the four quarks and anti-quarks of the first pion tetraquark.

 

 

Figure 8. Illustrates the Z coordinate of the ˜u(1) and u(6) quarks.

The Z coordinate of ˜u(1) and u(6) quarks performing the quark exchange at the AGC are shown below. The quark exchanges occur vertically between ˜u(5) and u(2) , ˜d(3) and d(8) , and ˜d(7) and d(4) (not shown in the figure). We note that the quark molecular dynamics quark exchanges and velocity inversions according to Equations 2(a)-(b) above are equivalent symmetry operations of the Pionic fabric cubic unit cell Ci point group.

5. The Electron and Pion Tetraquarks Double Well

Assuming that the β decay is a second order scattering reaction triggered by the vacuum pion tetraquarks [10], the following β decay reaction generates a proton and a negatively charged exotic tetraquark, ˜ud˜dd (e− ), that may play the role of an embedded electron mixed with the surrounding pion tetraquarks of the Pionic fabric.