## Gauge Field Theory For Strong Interactions

This post presents a brief introduction to the gauge field theory of strong interactions, developed recently by the authors, based only on the ${SU(3)}$ gauge invariance, the principle of Lorentz invariance, the principle of representation invariance (PRI) and the principle of interaction dynamics (PID). The field theory leads to strong interaction potential formulas, provides duality between spin-1 gluons and dual spin-0 gluons, and offers a theoretical explanation of the quark confinement and asymptotic freedom.

### 1. Fundamental Symmetries and Strong Charges

Strong interaction is one of the four fundamental interactions of Nature, the others being the electromagnetic, the weak and the gravitational interactions. The strong interaction is responsible for binding protons and neutrons together to form atoms, and for binding quarks together to form hadrons (baryons and mesons).

The current theory for the strong interactions is the quantum chromodynamics (QCD), described by a non-Abelian ${SU(3)}$ gauge theory. The ${SU(3)}$ gauge theory consists of

• the eight ${SU(3)}$ gauge fields, representing the strong interaction potentials:

$\displaystyle S^k_{\mu}=(S^k_0,S^k_1,S^k_2,S^k_3) \qquad \text{ for } 1\leq k\leq 8,$

• three copies of Dirac spinors, representing different flavors of quarks:

$\displaystyle \Psi=(\psi_1, \psi_2, \psi_3)^T,$

which satisfy the Dirac equations:

$\displaystyle \left[i{\rm \gamma}^{\mu}D_{\mu} - m \right] \Psi =0, \ \ \ \ \ (1)$

where

$\displaystyle D_{\mu}=\partial_{\mu}+igS^a_{\mu}\tau_a,$

where ${\{\tau_1, \cdots ,\tau_8\}}$ is a basis of the set of traceless Hermitian matrices with ${\lambda^j_{kl}}$ being the structure constants.

Elements in ${SU(3)}$ can be written as

$\displaystyle \Omega =e^{i\theta^a\tau_a}\in SU(3).$

Then the  ${SU(3)}$ gauge transformation takes the following form:

$\displaystyle (\tilde{\Psi}, \ \ \tilde{G}^a_{\mu}\tau_a, \ \ \tilde m) =\left( \Omega\Psi, \ \ G^a_{\mu}\Omega\tau_a\Omega^{-1}+\frac{i}{g}(\partial_{\mu}\Omega)\Omega^{-1}, \ \ \Omega m\Omega^{-1} \right). \ \ \ \ \ (2)$

Principle of Gauge Invariance. The strong interactions obey the gauge invariance:

• the Dirac equations are gauge covariant, and
• the Lagrangian action of the interaction fields is gauge invariant.

In classical quantum chromodynamics (QCD), the gauge invariance of the strong interactions refers to the color charge of the quarks cannot be distinguished, and consequently, the energy contribution of different flavors of quarks is invariant under the ${SU(3) }$ phase (gauge) transformation.

As presented in the previous post, we can choose different basis for the set of traceless Hermitian matrices, leading to the principle of representation invariance (PRI), first discovered and postulated by (Ma-Wang, 2012). Namely, consider the following representation transformation of ${SU(3)}$:

$\displaystyle \tilde{\tau}_a=x^b_a\tau_b, \quad X=(x^b_a) \text{ is a nondegenerate complex matrix.} \ \ \ \ \ (3)$

Then it is easy to verify that ${\theta^a}$, ${S^a_{\mu}}$, and and ${\lambda^c_{ab}}$ are ${SU(3)}$-tensors under the representation transformation (3).

PRI (Ma-Wang, 2012): The ${SU(3)}$ gauge theory of strong interactions must be invariant under the representation transformation (3):

• the Yang-Mills action of the gauge fields is invariant, and
• the corresponding gauge field equations are covariant.

By PRI, there is an ${SU(3)}$ vector ${\alpha_a}$ associated with the representation transformation (3) such that the following contraction

$\displaystyle S_\mu = \alpha_a S^a_\mu \ \ \ \ \ (4)$

defines the (total) strong interaction potential, and ${S_0}$ is the ${SU(3)}$ strong interaction charge potential, and ${F=-g_s \nabla S_0}$ is the strong force. Consequently, PRI implies that ${g_s}$ plays the role of the strong charge, as the electric charge ${e}$ in the ${U(1)}$ abelian gauge theory for quantum electrodynamics (QED).

### 2. Action

The field strengths for the ${SU(3)}$ gauge theory for strong interactions are naturally defined by

$\displaystyle S_{\mu\nu}=S^a_{\mu\nu} \tau_a = \frac{i}{g_s}\left[D_{\mu},D_{\nu}\right],$

which implies that

$\displaystyle S^a_{\mu\nu} = \partial_{\mu}S^a_{\nu}-\partial_{\nu}S^a_{\mu}+g\lambda^a_{bc}S^b_{\mu}S^c_{\nu}. \ \ \ \ \ (5)$

Then the Lagrangian action density should be functions of the contraction

$\displaystyle {\mathcal G}^s_{kl}S^k_{\mu\nu}S^{\mu\nu l},$

where ${\mathcal G_{ab}=\frac12 \text{Tr}(\tau_a \tau_b^\dagger)}$. Consequently by the simplicity of laws of physics, the action of an ${SU(3)}$ gauge theory for strong interactions takes the following standard form of the Yang-Mills action:

$\displaystyle \mathcal{L}_S=-\frac{1}{4} {\mathcal G}^s_{kl}S^k_{\mu\nu}S^{\mu\nu l}+\bar{\Psi}\left[i\gamma^{\mu}(\partial_{\mu}+ig_sS^k_{\mu}\tau_k)-m \right] \Psi. \ \ \ \ \ (6)$

Geometrically, ${D_\mu}$ and ${S_{\mu\nu}}$ are the connection and curvature tensors on the complex spinor bundle ${M\otimes_p (\mathbb C^4)^3}$. By construction, it is clear that ${\mathcal L_S}$ is Lorentz, gauge and representation invariant, and is dictated by these three symmetries.

### 3. Field Equations and Dual Gluon Fields

With the Lagrangian action at our disposal, the classical Yang-Mills equations used in the classical QCD follow from the least action principle also called the principle of Lagrangian dynamics.

However, as we have demonstrated in the previous posts, for the four fundamental interactions, the principle of Lagrangian dynamics should be replaced the principle of interaction dynamics (PID), which takes the variation of the Lagrangian action under energy-momentum conservation constraint. As indicated in the previous post, by PID and PRI, we derive the following field equations for strong interactions (Ma-Wang, 2012), with proper scaling:

$\displaystyle \partial^{\nu}S^k_{\nu\mu}-\frac{g_s}{\hbar c}f^k_{ij}g^{\alpha\beta}S^i_{\alpha\mu}S^j_{\beta}- g_s Q^k_\mu =\left[\partial_{\mu}-\frac{1}{4}k^2_sx_{\mu}+\frac{g_s\delta}{\hbar c}S_{\mu}\right]\phi^k_s, \ \ \ \ \ (7)$

$\displaystyle \partial^{\mu}\partial_{\mu}\phi^k_s-k^2\phi^k_s+\frac{1}{4}k^2_sx_{\mu}\partial^{\mu}\phi^k_s+\frac{g_s\delta}{\hbar c}\partial^{\mu}(S_{\mu}\phi^k_s) =-g_s\partial^{\mu}Q^k_{\mu}-\frac{g_s}{\hbar c}f^k_{ij}g^{\alpha\beta}\partial^{\mu}(S^i_{\alpha\mu}S^j_{\beta}), \ \ \ \ \ (8)$

$\displaystyle \left[ i\gamma^{\mu}\left( \partial_{\mu}+i\frac{g_s}{\hbar c}S^l_{\mu}\tau_l\right) -\frac{mc}{\hbar}\right] \Psi =0, \ \ \ \ \ (9)$

for ${1 \le k \le 8}$, where ${\delta}$ is a parameter, ${S_{\mu}}$ is as in (4), and

$\displaystyle Q^k_\mu= \Psi \gamma_{\mu}\tau_k\Psi.$

Here we have taken the representation basis ${\tau_k}$ to be the Gell-Mann matrices.

The right-hand side of the field equations (7) above is due to PID. In other words, PID induces a natural duality between the interaction fields ${S^k_{\mu}}$, representing the spin-1 massless gluons, and their corresponding spin-0 dual gluon fields ${\phi^S_k}$. Namely, corresponding to the eight gluon fields ${S^k_{\mu}\ (1\leq k\leq 8)}$, there are eight dual gluon fields ${\phi^k}$, which we call the scalar gluons due to ${\phi^k_s}$:

$\displaystyle \text{ gluons}\ g_k\ \leftrightarrow\ \text{ scalar\ gluons}\ g^k_0 \qquad 1\leq k\leq 8.$

In the weakton model [8], we realize that these dual gluons possess the same weakton constituents, but different spins, as the gluons.

The above duality can be viewed as a duality of the strong forces. We start with the physical significance of the parameters ${k_s}$ and ${\delta}$. Usually, ${k_s}$ and ${\delta}$ are regarded as masses of the field particles. However, when (7)-(9) are viewed as the field equations for the interaction forces, ${k^{-1}}$ represents the range of attracting force for the strong interaction, and ${\left(\frac{g_s\phi^0_s}{\hbar c}\delta \right)^{-1}}$ is the range of the repelling force, where ${\phi^0}$ is a ground state of ${\phi=\alpha_a \phi^a}$ and ${\alpha_a}$ is as given in (4).

In fact, we have shown in [5] that for a particle with ${N}$ strong charges ${g_s}$ of the elementary particles, its strong interaction potential is given by

$\displaystyle \Phi_s=Ng_s(\rho)\left[\frac{1}{r}-\frac{A}{\rho}(1+kr)e^{-kr}\right],\qquad g_s(\rho )=\left(\frac{\rho_w}{\rho}\right)^3g_s, \ \ \ \ \ (10)$

where ${\rho_w}$ is the radius of the elementary particle (i.e. the ${w^*}$ weakton), ${\rho}$ is the particle radius, ${k>0}$ is a constant with ${k^{-1}}$ being the strong interaction attraction radius of this particle, and ${A}$ is the strong interaction constant, which depends on the type of particles.

It is clear that the first part, involving ${1/r}$, of the strong interaction potential is repulsive and is due to the gluon fields, and the second main term

$\displaystyle - Ng_s(\rho)\frac{A}{\rho}(1+kr)e^{-kr}$

gives rise to attractions, responsible for quark confinement. This term is due to the dual gluon fields. This observation  leads immediately to the following consequences.

First, in order for a gauge theory to describe the quark confinement, the inclusion of the dual scalar fields are inevitably necessary. Hence the observed quark confinement phenomena can be viewed as another physical evidence for PID.

Second, strong forces display both repulsive and attractive behaviors. For example, as the distance between two quarks increases, the strong force changes from repulsive, to asymptotically free, and then to attractive.

Third, strong forces/interactions are layered, and we shall explore more physical consequences of the layered potentials in the future posts.

Tian Ma & Shouhong Wang