Gluon condensate

While I am coping with a revision of a paper of mine asked by a referee, I realized that these solutions of Yang-Mills equations implied by a Smilga’s choice give a proof of existence of a gluon condensate. This in turn means that a lot of phenomenological studies carried out since eighties of the last century are sound as are also their conclusions. E.g. you can check this paper where the authors find a close agreement with my findings about glueball spectrum. The ideas of these authors are founded on the concept of gluon and quark condensates. As they conclusions agree with mine, I have taken some time to think about this. My main conclusion is the following. If you have a gluon condensate, the theory should give \langle F\cdot F\rangle\ne 0 being F_{\mu\nu}^a the usual gluon field. So, let us work out this classically. Let us consider a scalar field mapped on the gluon field in such a way to have

A_\mu^a(t)=\eta_\mu^a \Lambda\left(\frac{2}{3g^2}\right)^\frac{1}{4}{\rm sn}\left[\Lambda\left(\frac{3g^2}{2}\right)^\frac{1}{4}t,i\right]

being sn a Jacobi snoidal function, and \eta_\mu^a a constant array of elements obtained by a Smilga’s choice. When you work out the product F\cdot F the main contribution will come from the quartic term producing a term \langle \phi(t)^4 \rangle. Classically, we substitute the average with \frac{1}{T}\int_0^T dt being the period T=4K(i)/[\Lambda\left(\frac{3g^2}{2}\right)^\frac{1}{4}]. The integration is quite straightforward and gives

\langle \phi(t)^4 \rangle=\frac{\Gamma(1/4)^2}{18K(i)\sqrt{2\pi}}\frac{\Lambda^4}{4\pi\alpha_s}

I will evaluate this average in order to see if the order of magnitude is the right one with respect to the computations carried out by Kisslinger and Johnson. But the fact that this average is indeed not equal zero is a proof of existence of the gluon condensate directly from Yang-Mills equations.

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