We analyze briefly in this note group structure of the heterotic E₈ × E₈ string theory in 10D and compare its symmetry structure to the composite models of quarks and leptons introduced by Pati [1] and us [2] [3] [4] . The former model proposes gauge interactions between preons, while we utilize topological concepts for preon interactions.

In spite of its immense experimental success, the standard model (SM) has its problems: it does not include gravity and supersymmetry (no sparticles found), it requires many arbitrary constants and parameters that are not derived from the theory itself, it does not explain the origin of lower scales such as M_W and the origin of the number of generations, to mention some of them. String theory can in principle solve this kind of problems, but a working phenomenology is missing.

Pati [1] and we [3] [4] believe that the main cause of the SM difficulties is the choice of the fundamental fields as quarks and leptons. The quarks, the leptons and the Higgs particle should rather be interpreted as preon composites. As a bonus, we obtain a preliminary concordance between the preon models considered here and string theory.

We have introduced a topological Chern-Simons (CS) interaction for preon binding (our preons are called from now on chernons) while the Pati model uses a non-abelian metacolor gauge force. CS model is preferred because we want to have background independent model, and secondly, the preonic phase of matter is assumed to occur at very high energy scale where no regular spacetime may not exist [5] . In addition, baryon asymmetry and the “hiddenness” of supersymmetry can be explained naturally [5] .

We think that a topological approach to preon models is more far reaching than the S U ( 4 ) M in [1] with respect to string theory. The Calabi-Yau string manifold is topologically non-trivial. Topology affects the spectrum of particles and determines whether the theory contains massless particles, like scalars and the graviton. Our use of a topological model is, at this stage, far from string theory as such, but it may be a first step, among others, to stringy particle phenomenology with applications to cosmology.

This note is organized as follows. Section 2 is divided into two subsections. We first take in Subsection 2.1 a glimpse in heterotic E₈ × E₈ string group theory. In Subsection 2.2 we recap the Pati preon model. In the next Section 3, we present a summary of the features of our preon model. The differences in the preon models considered here are pointed out. Conclusions are given in Section 3.

We begin with a concise recap of heterotic string theory E₈ × E₈ in 10D. This provides a yardstick for finding similar structures between strings and preons/chernons.

The chernon model was developed originally from bottom up so that three chermons of certain charge (namely ±1/3) can produce the quark charges 2/3 and −1/3. Later we found that the model resembles closely the global supersymmetric Wess-Zumino model [9] , which can be extended to supergravity.¹

The next phase in developing the model was to make it function in early cosmology. Introducing a binding energy scale near reheating Λ_cr ≥10¹² - 10¹⁶ GeV for the model, T-duality was introduced [10] . Thirdly, above Λ_cr the universe was in a form having topological interactions. Any metric needs not necessarily be defined. Therefore, 2 + 1 dimensional Chern-Simons interactions between the chernons, inside a 3 + 1 dimensional spacetime, were introduced. Chernons with certain spontaneous symmetry breaking turned out to give a natural explanation for baryon asymmetry [3] [4] . Furthermore, the chernon model explains why supersymmetry has not been observed in nature [5] . The preon interaction is the main difference between the Pati model(s) and ours. Nevertheless, both models have similar numerical behavior in UV and IR.

Chern-Simons-Maxwell (CSM) models have been studied in condensed matter physics (for original references see [3] [4] ). In this note we apply the CSM model in particle physics phenomenology in the early universe. The following CS-QED action has been defined

S CS-QED SSB = ∫ d 3 x { − 1 4 F μ ν F μ ν + 1 2 M A 2 A μ A μ − 1 2 ξ ( ∂ μ A μ ) 2 + ψ ¯ + ( i ∂ − m e f f ) ψ +     + ψ ¯ − ( i ∂ + m e f f ) ψ − + 1 2 θ ϵ μ ν α A μ ∂ n u A α + ∂ μ H ∂ μ H − M H 2 H 2           + ∂ μ θ ∂ μ θ − M θ 2 θ 2 − 2 y v ( ψ ¯ + ψ + − ψ ¯ − ψ − ) H − e 3 ( ψ ¯ − { + } A ψ − { + } + ψ ¯ − { + } A ψ − { + } ) } (1)

where the mass parameters

M A 2 = 2 v 2 e 3 2 ,   m e f f = m c h + y v 2 ,   M H 2 = 2 v 2 ( ζ + 2 λ v 2 ) ,   M θ 2 = ξ M A 2 (2)

depend on the SSB mechanism: v², the vacuum expectation value, and y is the parameter that measures the coupling between fermions and Higgs scalar. Being a free parameter, v² indicates the energy scale of the spontaneous breakdown of the U(1) local symmetry. The Proca mass M A 2 represents the mass acquired by the photon through the Higgs mechanism. The Higgs mass, M H 2 , is associated with the real scalar field. The Higgs mechanism also contributes to the chernon mass m_ch, resulting in an effective mass m_eff. There are two photon mass-terms in (1), the Proca and the topological one. Now the idea of (1) is that the parameters of the model can be chosen such that the photon mass is high enough to create an attractive, very short range Yukawa force which is stronger than the repulsive Coulomb force between equal charge chernons.

The chernon-chernon scattering amplitude in the non-relativistic approximation is obtained by calculating the t-channel exchange diagrams of the Higgs scalar and the massive gauge field. The propagators of the two exchanged particles and the vertex factors are calculated from the action (1). The gauge invariant effective potential for the scattering considered is obtained in [11] [12]

V M C S ( r ) = e 2 2 π [ 1 − θ m c h ] K 0 ( θ r ) + 1 m c h r 2 { l − e 2 2 π θ [ 1 − θ r K 1 ( θ r ) ] } 2 (3)

where K 0 ( x ) and K 1 ( x ) are the modified Bessel functions and l is the angular momentum ( l = 0 in this note).

One sees from (3) the first term may be positive or negative while the second term is always positive. The function K 0 ( x ) diverges as x → 0 and approaches zero for x → ∞ and K 1 ( x ) has qualitatively similar behavior. For our scenario we need negative potential between all, including equal charge chernons. We can achieve this if we require the condition

θ ≫ m c h (4)

The attractive equal charge potential plays the key role when chernons begin to form quarks and leptons [3] [4] .

For applications to condensed matter physics, one must require θ ≪ m e , and the scattering potential given by (3) then comes out positive [13] .

Neither the preon nor the chernon binding interactions are fully studied. We discuss briefly our case. The global attractive potential (3) at tree level is obtained after the introduction of the Higgs mechanism in the context of the MCS-QED [13] . This should allow approximate Yukawa bound states of chernons below the scale Λ_cr and free chernons above the scale Λ_cr. The eigenvalues of a Yukawa potential in Schrödinger equation are E = − 1 2 μ g 2 ϵ n , l with

ϵ n l = − 1 / n 2 + 2 δ − 1 / 2 ( 3 n 2 − l ( l + 1 ) ) δ 2 (5)

where μ is the reduced mass of the particle, δ = a 0 / D , a₀ is the Bohr radius of the bound state, D the range of the potential and g the strength of the force [14] . The bound state mass would be of the order of chernon mass. The energy scale of the composite formation is, however, Λ_cr. This may be the point where smooth continuum theory changes, see Figure 1. We should seek some more general spacetime framework [15] . This question is beyond the scope of this note.

In this note we have sought a connection between strings and chernons. We compare first the two models of Pati and ours, discussed in subsections 2.2 and 2.3, respectively. In Pati’s model, the existences of vacuum solutions M⁴ × K at the tree level are known. Such a compactification leaves an unbroken N = 1 local supersymmetry at the compactification scale. It breaks E₈ × E₈ into E₈ × E₆ if K is simply connected and E₆ breaks at the compactification scale to a subgroup G 0 = S U ( 4 ) M × G ˜ . Pati identifies the symmetry group S U ( 4 ) M with the metacolor gauge symmetry. The field theory with this gauge group is asymptotically free in UV and confining in IR [16] . The spectrum of states of S U ( 4 ) M may be too rich for a preon model. Numerically, our model has similar UV/IR behavior as the Pati model.²

The global symmetry analysis of our model with one fermion generation is weaker than Pati’s. Our model can be extended to include supergravity, but we have chosen the binding interaction from Chern-Simons model. The number of generations depends on the compactification of the topology of K. We have done standard quantum mechanics for approximate binding at scale m_ch. At scales above Λ_cr we believe the topology spacetime manifold changes, presumably towards string topology. On the other hand, we have found the following experimental support for the chernon model: 1) it reduces to the standard model at accelerator energies [2] , 2) it has proposal for the dark sector (not observed/understood yet) [3] [4] , 3) it explains the baryon asymmetry in the universe in a natural way [3] [4] , 4) it provides T-dual topological evolution of the universe [10] , and 5) it explains why supersymmetry has not been found in experiments [5] .

These agreements are found using phenomenological methods rather than by mathematically rigorous proofs. It is our understanding that supersymmetry and string theory have some direct support from experiments.

A crucial question for our model is: Can Chern-Simons theory be derived from the topology of K in some limit? If so, this may provide a way to study aspects of string theory using Chern-Simons theory as a tool. This has been studied in [17] [18] .³

The authors declare no conflicts of interest regarding the publication of this paper.

Raitio, R. (2024) Stringy Phenomenology with Preon Models. Journal of High Energy Physics, Gravitation and Cosmology, 10, 1-7. https://doi.org/10.4236/jhepgc.2024.101001