Quantum Chromodynamics of the Nucleon in Terms of Complex Probabilistic Processes

In the observable universe, ordinary matter, and star formations in particular, are composed primarily of strongly interacting particles of protons and neutrons called nucleons. In this regard, one of the most important tasks of modern nuclear physics is the study of the structure of the nucleon and its excited states from the point of view of effective degrees of freedom and, at a more fundamental level, the emergence of these from QCD [1]. After postulating the quark structure of strongly interacting particles by Gell-Man, Zweig and Fritzsch [2, 3, 4], based on the ideas of symmetry and invariance in the system of particles and fields, Feynman, Ravndahl and Kislinger [5] was proposed a three-quark representation of nucleons within the framework of a four-dimensional relativistic quantum oscillator model. Note that the main idea underlying the three-quark nucleon model is that quarks, interacting through the potential of a four-dimensional harmonic oscillator, cannot move away from each other and become free. In a series of papers [6, 7, 8], the authors discussed a quantum harmonic oscillator formalism to study the important features of hadronic structures in relativistic quark models. An important achievement of these models is that they allow a covariant-probabilistic interpretation of the wave functions under consideration [9].

Despite the obvious successes of the relativistic model of a four-dimensional oscillator in describing the structure and internal motion of a nucleon with the effect of quark confinement, it still remains insufficiently realistic. The fact is that the nucleon model under consideration does not take into account the continuous processes of colored gluon exchange between quarks. Moreover, the model does not take into account the spontaneous breaking of chiral symmetry, which is responsible for the generation of nucleon mass from more elementary light quarks. The latter is obviously a strong simplification of the problem. The difficulties of the model become even more obvious when we have to consider nucleons in nuclei or dense and superdense stellar formations. Recall that in this case the main characteristics of the processes of gluon exchange between quarks, as well as the properties of the quark sea, change, which in turn directly affects the structure and other parameters of nucleons.

To solve a number of the above problems, this paper considers a relativistic three-quark dynamical system immersed in colored gluon fields, which, in turn, generate a quark-antiquark sea. At the same time, we describe the interactions between quarks using a four-dimensional harmonic oscillator, which ensures confinement of quarks, their asymptotic freedom at short distances and chiral symmetry. We formulate the mathematical problem in terms of a complex probabilistic process that satisfies a stochastic differential Equation (SDE) of the Langevin-Klein-Gordon-Fock type. Note that this formulation of the problem allows us to take into account both elastic and inelastic processes of gluon exchange between quarks as well as self-action, type of gluon-antigluon interactions. It is shown that for the case when fluctuations of the color quark-antiquark sea are characterized by complex processes of the Markov-Gauss type, it is possible to construct a mathematically closed non-perturbative theory of the nucleon with additional six-dimensional compact sub-spaces. In particular, when calculating the mathematical expectation of various parameters of the nucleon, we perform averaging over additional sub-spaces, which breaks the chirial symmetry [10] but at the same time maintains the color of the nucleon. Recall that for all parameters of the nucleon we obtain double integral representations, where the integrand contains a function that is a solution to a system of two coupled second-order partial differential equations (PDEs).

In conclusion, we note that considering the nucleon as a complex self-consistent system of “three quarks + a sea of quarks-antiquarks”, as a problem of self-organization within the framework of the developed concept, allows us to go beyond the framework of perturbation theory, which is very important for obtaining new non-trivial results in the field of quantum chromodynamics (QCD), which is essentially a non-perturbative theory.

The manuscript is organized as follows:
In Section 2, briefly outlines the well-known formulation of the problem of the internal motion of a nucleon as a three-quark dynamical system within the framework of the Klein-Gordon-Fock equation using a four-dimensional model of a relativistic oscillator for quark interactions [6]. An exact relativistically invariant solution of the wave function of the internal motion of a nucleon is presented.

In Section 3, the problem of the internal motion of a nucleon immersed in a colored quarks sea is mathematically formulated in the framework of a complex probabilistic process that satisfies the Langevin-Klein-Gordon-Fock type SDE. It is shown that a complex probabilistic process in the model of a four-dimensional relativistic oscillator, after a convenient transformation of coordinates, is written in factorized form, as a product consisting of three independent functions.

In Section 4, a system of stochastic Equations for gluon fields is derived taking into account the synchronization of four-dimensional events in the Minkowski space. The conditions of complex stochastic processes for generators of colored gluon fields, in the form of Markov-Gaussian processes, are determined.

In Section 5, taking into account the SDE system, an Equation for the distribution of gluon fields in the limit of statistical equilibrium is derived. It is shown that the solution for the field distribution is factorized into the product of three two-dimensional distributions, each of which describes the states of gluon fields of a certain color and anticolor. It is shown that the additional six-dimensional sub-space generated by the SDEs system is factorized as a direct product of three two-dimensional sub-spaces.

In Section 6, the geometric and topological features of the emerging two-dimensional sub-spaces are analyzed. It is shown that each of the subspaces is generated by an algebraic equation of the fourth degree and is described by non-commutative geometry. It is also proven that the typological features of these manifolds are characterized by the Betti number $n\leq 4$.

In Section 7, using a two-dimensional distribution Equation for gluon fields, a Fokker-Planck measure of the functional space is constructed.

In Section 8, the mathematical expectation of the total wave function of a three-quark dynamical system immersed in a colored quark-antiquark sea is defined in the form of a functional integral representation. Using the generalized Feynman-Kac theorem, functional integrals were calculated. As a result, a factorized representation in the form of three double integrals was obtained for the mathematical expectation of the wave function of a nucleon internal motion.

In Section 9, using the mathematical expectation of the total wave function of a nucleon, its radius and mass are determined depending on the constants characterizing the fluctuation powers of colored gluon fields. Two independent equations are derived that make it possible to uniquely calculate two independent constants characterizing the gluon fields of a nucleon when the nucleon is in a free state.

In Section 10, discusses in detail the possibilities of representation for a more correct description of the quantum state of the nucleon, taking into account the processes of gluon exchange between quarks. The issues of the state of a nucleon in the case of their immersion in dense and superdense stellar formations are discussed in the light of changes in the spectrum and power of fluctuations of gluon fields.

In Section 11, discusses the issues of color synchronization of valence quarks or the problem of preserving the white color of the nucleon, which, in turn, is closely related to the problem of the three-particle interaction of valence quarks.

As is known, the main assumption about the dynamics of the internal motion of a nucleon is that the motion of three quarks is described by the relativistic Klein–Gordon–Fock equation in the light front formalism, where the interaction between quarks is carried out through a four-dimensional harmonic potential ensuring confinement of quarks in the nucleon [5]. Taking this work into account, the following Lorentz-invariant Equation was proposed to describe the state of a three-quark dynamical system [6]:

where ${{x}}_{a},{{x}}_{b}$ and ${{x}}_{c}$ are the four-dimensional space-time coordinates of quarks $\bf{a},\bf{b}$ and $\bf{c}$ (see Fig. 1), in addition, $\Omega_{0}$ is the some constant, $m_{0}$ denotes the sum of the rest masses of three quarks, and $\Box_{\zeta}$ denotes the d’Alembert operator acting on the $\zeta$-th quark:

Recall that below all calculations will be carried out in units; $\hslash=c=1.$

By performing the following coordinate transformations:

the Equation (2) can be reduced to diagonal form:

where $\xi=(\xi_{0},\xi_{1},\xi_{2},\xi_{3})$ and, accordingly, the d’Alembert operator in new coordinates has the form $\Box_{\xi}=\partial_{\xi_{0}}^{2}-\sum_{k=1}^{3}\partial^{2}_{\xi_{k}}.$

Now representing the solution of the Equation (3) in factored form:

from (3) we obtain three new Equations:

where $m_{0}^{2}=\lambda_{01}+\lambda_{02}+\lambda_{03}$.

The solution to the first Equation of the system (5) can be represented as:

where $P^{\nu}(\nu=0,1,2,3)$ is a four vector.
Substituting this solution into the first Equation of the system (5), it is easy to find that $P^{2}=3\lambda_{01}.$ In addition, from (5) follows that $\lambda_{01}=\lambda_{02}=(1/3)\Omega^{2}_{0}=(1/3)m_{0}^{2}.$

The remaining two equations describe the quantum motions of two independent four-dimensional oscillators, described by the wave functions $\Psi_{2}(v)$ and $\Psi_{2}(w)$, respectively. In the rest frame, where $P^{\nu}=(m_{0},{\bf 0})=(m_{0},0,0,0)$, the ground state wave function normalized in relative coordinates $v$ and $w$, can be written as:

which may be written in covariant form:

As for the wave function of the excited state, as shown in the work [6], it can be represented in the form:

where $N_{n_{k}}$ is the normalization constant of the one-dimensional oscillator, ${\bf n}=(n_{1},n_{2},n_{3})$ the set of quantum number, and $H_{n}(x)$ denotes the Hermite polynomials. Recall that the wave function of a four-dimensional oscillator has the form:

Note that we obtain exactly the same function for another oscillator described by the function $\Psi_{3}({\bf m};w)$.

As can be seen from expressions (7) and (9), the wave function of the nucleon is localized along the coordinates $v$ and $w$, while the three-quark dynamical system performs translational motion along the coordinates $u$.

Since nucleons consist of combinations of two types of light quarks $\bf u$ and $\bf d$ (see Fig. 1), it is natural to expect that the interaction of these quarks should be carried out by gluons of different colors. In particular, when the colors of two quarks are known, the interaction between them will be carried out by gluons of these colors and corresponding antigluons. Note that by antigluon we mean the same gluon that has an anticolor. Having gone through all the combinations, it becomes obvious that a nucleon, consisting of three quarks of different colors without taking into account the spins of the quarks, is immersed in a six-color quark-antiquark sea (see Fig. 2). Taking this into account, we must construct a consistent non-perturbative theory of self-organization of a quark system and its random multi-color quark-antiquark environment. Based on the experience of studying similar problems of nonrelativistic quantum mechanics [11], we can rewrite Equation (1) in the following form:

where $m^{2}(q_{a},q_{b},q_{c})$ is some space-time complex random function, the form of which will be indicated below. Recall that the valence quarks $\bf a$, $\bf b$ and $\bf c$ (see Fig. 1) in a nucleon are in three different quantum states or colors, so the mathematical expectation of the mixed color of a nucleon should be white.

Using coordinate transformations (2) the Equation (10) can be written as:

where $\breve{m}^{2}(u,v,w)=m^{2}(x_{a},x_{b},x_{c}).$

To clarify the function $\breve{m}^{2}(u,v,w)$, we will assume that it can be represented in the form:

where $\lambda_{1}(s_{u})$, $\lambda_{2}(s_{v})$ and $\lambda_{3}(s_{w})$ some complex random processes depending on an proper space-time events:

Recall that since we are considering a relativistic problem, it is natural to use a 4-vector, which defines a chronologized sequence of space-time events in Minkowski space, as a parameter describing the evolution of a dynamical system.

Finally, based on the symmetry of Equation (11)-(12), the solution to the total wave function of the system can be represented as the following product:

Substituting (14) into (11), taking into account (12), we obtain three new equations:

It should be noted that the system of equations (15), due to the presence of random generators $\lambda_{k}(s_{\xi}),(k=1,2,3)$ in them in this form, still requires definition and reduction to canonical form, i.e. to the form of a first-order differential equation of Langevin type.

Now our main task will be to construct the mathematical expectation of the total wave function of a nucleon on the light cone, taking into account gluon exchanges between quarks and the influence of the quark-antiquark sea:

where $\mathbb{E}[\cdot\cdot\cdot]$ denotes the mathematical expectation of a random variable, in addition, recall that the sets $(x_{a},x_{b},x_{c})$ and $(u,v,w)$ denote 4 -vectors of Minkowski space-time.

In view of the foregoing, the solution of each of the Equations (15) can be represented as:

where $\Psi_{k}(\xi)$ is a regular function, a solution to one of the Equations of the system (5) and $\Lambda_{k}(\xi,s_{\xi})$ denotes a complex probabilistic process, which can be conveniently represented as a sum consisting of real and imaginary terms:

Substituting a solution of the form (17) into the corresponding Equations of the system (15), we obtain:

where

Note that the operator $\nabla_{M}=(\partial_{\xi_{0}},-\partial_{\xi_{1}},-\partial_{\xi_{2}},-\partial_{\xi_{3}})$ denotes the gradient in Minkowski space-time $\mathbb{R}^{4}$.

Now we define random functions $\lambda_{k}(s_{\xi})$ that characterize the properties of colored and anticolored gluon fields, representing them as the sum of real and imaginary terms:

For definiteness, we will assume that these functions satisfy Markov-Gaussian random processes or white noise correlation relations:

where $\upsilon=(i,r)$.

We assume that the random generators $f_{j}^{(r)}$ and $f_{j}^{(i)}$ characterize elastic and inelastic processes of exchange of gluons and antigluons of a given color between two specific quarks.

Finally, using the expressions (17)-(21), we can obtain the following system of six non-linear Langevin-type SDEs:

Recall that for further study of the system of Equations (23) we synchronized the evolutionary parameters, that is, we took the smallest parameter among them $s=\min\{s_{u},s_{v},s_{w}\}$.

Thus, the system of Equations (23) describes colored gluon fields under the influence of three valence quarks, and also takes into account the self-actions of gluons. Now it is important to use the SDEs (23) to obtain a regular equation that describes the distribution of gluon fields in the limit of statistical equilibrium. Note that this is fundamentally important for a consistent analytical construction of the problem.

When gluons are exchanged between two valence quarks of different colors, since the overall white color of the nucleon must be conserved, this must affect the dynamics of the third valence quark to preserve the color of the nucleon. It follows that any processes of gluon exchange between two quarks make the interaction in the nucleon three-particle. The latter means that the probabilities of the distribution of gluon fields of all six colors should be interconnected and combined. Taking into account the above, it is necessary to represent the distribution of gluon fields in the following form:

where ${\bm{\vartheta}}(s)=\bigl{\{}\bm{\phi}(s),\bm{\varphi}(s),\bm{\theta}(s)\bigr{\}}\in{\Xi}_{\{{\bm{\vartheta}}(s)\}}$ and $\langle\cdot\cdot\cdot\rangle$ denotes the functional integration over the functional space ${\Xi}_{\{{\bm{\vartheta}}(s)\}}$, whose measure will be defined below, in addition, $\vartheta_{0j}=\vartheta_{j}(s_{0})$ denotes the field value in $s_{0}=0$.

Using the SDE system (22), it is possible to strictly prove that the probability distribution of gluon fields satisfies the following Fokker–Planck type PDE (see [14, 15]):

where the evolution operator $\widehat{\mathcal{L}}({\bm{\vartheta}},s|u,v,w)$ has the following form:

where $\bar{\varepsilon}_{k}^{(\upsilon)}=c_{k}^{2}\varepsilon_{k}^{(\upsilon)},\,(k=1,2,3)$.

Note that in the Equation (26) the follwing notations are made ${\bm{\phi}}=(\phi_{1},\phi_{2}),$ ${\bm{\varphi}}=(\varphi_{1},\varphi_{2})$ and ${\bm{\theta}}=(\theta_{1},\theta_{2})$, in addition:

where $a^{(r)}_{k}=Re\{a\}$ and $a^{(i)}_{k}=Im\{a\},\,(k=1,2,3).$

It is important to note that the Equation (25)-(26) takes into account gluon-antigluon interactions. Recall that this is reflected in the mixed terms $\sigma_{2}({\bm{\phi}},s|u),\,\pi_{2}({\bm{\varphi}},s|v)$ and $\omega_{2}({\bm{\theta}},s|w)$. It is important to note that since colored gluon fields generate a sea of colored quarks-antiquarks, the indicated distributions also describe a quark sea.

The symmetry of the Equation (25)-(26) allows us to represent its solution in factorized form:

where each of the probability distributions $\mathcal{P}_{\bm{\phi}}({\bm{\phi}},s|u),\,\,\mathcal{P}_{\bm{\varphi}}({\bm{\varphi}},s|v)$ and $\mathcal{P}_{{\bm{\theta}}}({\bm{\theta}},s|w)$ is defined on the corresponding two-dimenssional manifold. In other words, the additional six-dimenssional sub-space $\Xi^{6}_{\{{\bm{\vartheta}}\}}$ in the limit of the statistical equilibrium can be represented as the following decomposition:

If the nucleons are free, then we assume that the powers of gluon fluctuations are relatively small, i.e. $\bar{\varepsilon}_{k}^{(r)},\bar{\varepsilon}_{k}^{(i)}\sim 1$ for all $k=1,2,3$. In this case, we can safely assume that all additional two-dimensional sub-spaces are Euclidean; $\Xi^{2}_{{\bm{\phi}}}\cong\Xi^{2}_{{\bm{\varphi}}}\cong\Xi^{2}_{{\bm{\theta}}}\cong\mathbb{R}^{2}=(-\infty,+\infty){\bf\times}(-\infty,+\infty).$

An important feature of the equation (26), which describes the distribution of fields of colored gluons, or rather the sea of quarks-antiquarks, in the limit of statistical equilibrium, is quantized. Recall that this follows from the dependence of the function $a_{k}(\xi,s_{\xi}),\,(k=1,2,3)$ on the quantum state of quarks (see the definition of the function (19)), which is characterized by six quantum numbers.

In the case when the fluctuation’s powers for some reason take on large values; $\bar{\varepsilon}_{k}^{(r)},\,\bar{\varepsilon}_{k}^{(i)}\gg 1$ for all $k=1,2,3$ it is necessary to conduct a comprehensive analysis to identify the geometric and topological features of the two-dimensional manifolds; $\Xi^{2}_{\{{\bm{\chi}}\}},\,\,({\bm{\chi}}={\bm{\phi}},{\bm{\varphi}},{\bm{\theta}})$.

Theorem. If a dynamical system is described by any of the SDEs from (19), then in the limit of statistical equilibrium the functional space ${\Xi}_{\{{\bm{\chi}}(s)\}},\,\{{\bm{\chi}}(s)={\bm{\phi}}(s),{\bm{\varphi}}(s),{\bm{\theta}}(s)\}$ is compactified into the two-dimensional manifold $\Xi^{2}_{\{{\bm{\chi}}\}},\,\,({\bm{\chi}}={\bm{\phi}},{\bm{\varphi}},{\bm{\theta}})$, which is characterized by non-commutative geometry.