Gluonic vacuum field

In particle physics, the interaction between hadrons is determined by the gluonic field. Here we consider the vacuum state gluonic field.

This article always uses Clifford algebra, Lorentz gauge condition, and units with h = c = 1. Only static spherical symmetrical potentials are considered.

The gluonic and electromagnetic fields are both four vector fields with gradient symmetry. For clarity, look first at the electromagnetic field.

The electrostatic interaction between particles decreases at large distances. In this area as first approach only lagrangian of the free field <math>L_0 <math> can be taken into account. Because <math>L_0 <math> have square dependence from forces electrostatic potential at big distance is

<math>\phi =\phi_+ +\frac{G}{ R} <math>

This potential Cavendish and Coulomb found experimentally 200 years ago and it is base of electrodynamics.

Other general approaches also give a Coulomb-like potential. In Lagrangian formalism with coherence condition, any field may exist in vacuum state. It mean that the equations for interacting field are the same as for the free field. If Lagrangian <math> L_0<math> is of square form then the potential of any four vector field with gradient symmetry is

<math>A\gamma_0=\phi_+ +{Q\over R}+{d\over R^2}e_R <math>

The last term may be taken off if use gradient symmetry directly.

For the gluonic field, a Coulomb-like potential is not valid because gluonic forces disappear at short distances (it is true also for the nonlinear Coulomb field) and at infinity they do not decrease.

Hence we must conclude that the Lagrangian for a free gluonic field is not square form. In this case another not Coulomn like vacuum state exist and have no matter how complicated is lagrangian. It is

<math>\dot\phi^2-(div\vec A)^2=const<math>

Lorentz gauge condition fixed its form. It is

<math>\phi =\phi_0 +kR <math>

and it is potential of Gluonic vacuum field.

Similarly to electric gauge constant <math>k <math> determines the scale strong interaction and it is impossible to find it via coherence condition.

Consider bound states for this field. For scalar particles the Klein-Gordon equation is valid. Then equation for radial part of wave function particle with mass M, energy E, orbital moment l is

<math>\ddot F+{2\over R}\dot F={l(l+1)\over R^2}-k^2R^2+2kR+M^2-E^2<math>

At infinity the effective potential energy is equal to minus infinity. This transmutes particles into resonances and a restriction <math>ImE<0 <math> is needed. Look for solutions in standard form

<math>F\sim R^l\prod_{0}^N(R-R_n) exp\left(AR+{1\over 2}BR^2\right) <math>

Then <math>D=-ik<math> <math>A=iE <math>. Such phases are taking because in the observable region t > R.

Mass spectrum boson resonances is

<math>M^2=-ik (2N+2l+3) <math>

where N is radial quantum number. Energy levels and zeros wave function are

<math>iE=\sum_{0}^N <math><math>{1\over R_n}<math>

<math>\sum_{k\ne m}{1\over R_n-R_k}<math><math> +iE+{l+1\over R_n} =ikR_n <math>

For N=0 energy E=0 but because vacuum potential <math> \phi<math> is unknown it is relative zero. For fermions use the Dirac equation. Then the mass spectrum is similar, but the algebraic equations are more complicated.

The linear dependence of square mass hadronic resonances on moment are well known from experiment and the Regge pole model. This dependence is experimantal source of the string model of strong interaction.

In the potential model, the calculation of mass spectrum is as simple as possible. Maybe similar calculations were made when quark model of hadrons appeared.

But in this work we find and use potential of gluonic vacuum field. Existence such field follow from lagrangian formalism which is supplemented by coherence condition.

True connection between mass and moment of resonances give sure confirmation about the existence of the gluonic vacuum field.