Group-Variation Equations for the Coefficients in the 1/N Expansions of Physical Quantities in SU(N) Gauge Theories in D=3+1

The coefficients in the 1/N expansions of the vacuum expectation values and correlation functions of Wilson loops, in continuum SU(N) gauge theories in 3+1 dimensions, are shown to be determined by a closed and complete set of equations, called the Group-Variation Equations, that exhibit a simple and robust mechanism for the emergence of massive glueballs and the Wilson area law. The equations predict that the cylinder-topology minimal-area spanning surface term in the two-glueball correlation function, when it exists, must be multiplied by a pre-exponential factor, which for large area A of the minimal-area cylinder-topology surface, decreases with increasing A at least as fast as $1\mathrm{/}\ln (\sigma A)$. If this factor decreases faster than $1/\ln(\sigma A)$,thenthemass$m_{0^{++}}$ofthelightestglueball,andthecoefficient$\sigma$ of the area in the Wilson area law, are determined in a precisely parallel manner, and the equations give a zeroth-order estimate of $m_{0^{++}}\mathrm{/}\sqrt{\sigma }$ of 2.38, about 33% less than the best lattice value, without the need for a full calculation of any of the terms in the right-hand sides. The large distance behaviour of the vacuum expectation values and correlation functions is completely determined by terms called island diagrams, the dominant contributions to which come from islands of \emph{fixed} size of about $1\mathrm{/}\sqrt{\sigma }$. The value of $\sigma$ is determined by the point at which $\mathrm{\mid }\beta (g)\mathrm{/}g\mathrm{\mid }$ reaches a critical value, and since the large distance behaviour of all physical quantities is determined by islands of the fixed size $1\mathrm{/}\sqrt{\sigma }$, the running coupling $g^2$ never increases beyond the value at which $\mathrm{\mid }\beta (g)\mathrm{/}g\mathrm{\mid }$ reaches the critical value.