  The Yang-Mills theory is part of the Standard Model of particle physics. The
lack of the mathematical understanding of the theory stands out in theoretical
physics. In order to address this problem we observe that a recently proposed
general model beyond the Standard Model resolves the energy scales of a lattice
regularized Yang-Mills theory by means of an extra dimension. The extra
dimension ensures that all intermediate length scales of the physical system
are available by definition. In this paper we study the role of the extra
dimension also in the case of the free boson field. We find that if the extra
dimension size is large, the model describes the classical motion of the
system. In the opposite limit we recover its standard quantum mechanical motion
without loss of information. Therefore, the Hilbert space of states in the
presence of the extra dimension describes physical phenomena at all energy
scales. This observation allows us to raise the description of length scales by
an extra dimension at the level of a principle for the theories beyond the
Standard Model, the only modeling constraint being the correspondence
principle. As it was shown recently, the fermion ground state energy of a gauge
invariant Hamilton operator of Dirac fermions gives a particular lattice
regularization of the Yang-Mills theory. Integration of gauge fields gives a
pure fermion theory of color singlet fermion-antifermion pairs at each lattice
site evolving along the extra dimension. Color confinement follows directly
from this property. It allows for a saddle point solution in the limit of a
large number of colors. In this paper we find that the glueball spectrum of the
Yang-Mills regularized theory is of the Hagedron type and bounded below by a
positive value. We show also that the color charge is screened and the
quark-antiquark potential is constant.
