  A Theorem on the minimal specific energy for a system with \pm 1 charged
particles interacting through the Yukawa pair potential v is proved which may
stated as follows. Let v be represented by scale mixtures of d-dimensional
Euclid's hat (cutoff at short scale distances) with d\geq 2. For any even
number of particles n, the interacting energy U_{n} divided by n, attains an
n-independent minimum at a configuration with zero net charge and particle
positions collapsed altogether to a point. For any odd number of particles n,
the ratio U_{n}/(n-1) attains its minimum value, the same of the previous
cases, at the configuration with \pm 1 net charge and particle positions
collapsed to a point. This Theorem is then used to resolve an obstructive
remark of an unpublished paper (Remark 7.5 of \cite{Guidi-Marchetti}) which,
whether the standard decomposition of the Yukawa potential into scales were
adopted, would impede a direct proof of the convergence of the Mayer series of
the two-dimensional Yukawa gas for the inverse temperature in the whole
interval [4\pi ,8\pi ) of collapse. In the present paper, it is proven
convergence up to the second threshold 6\pi and its given explanations on the
mechanism that allow it to be extend up to 8\pi . The paper distinguishes the
matters concerning stability from those related to convergence of the Mayer
series. In respect to the latter the paper dedicates to the Cauchy majorante
method applied to the density function of Yukawa gas in the interval of
collapses. It also dedicates to the proof of the main Theorem and estimates of
the modified Bessel functions of second kind involved in both representations
of two-dimensional Yukawa potential: standard and scale mixture of the Euclid's
hat function.
