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In Wikipedia, it says that "The evidence for QCD condensates comes from two eras, the pre-QCD era 1950–1973 and the post-QCD era, after 1974. The pre-QCD results established that the strong interactions vacuum contains a quark chiral condensate, while the post-QCD results established that the vacuum also contains a gluon condensate."

I believe the quark condensate is a pretty standard story we learn from, for example, any text on QCD (see Dynamics of the Standard Model by Donoghue et all). How about the glon condensate? What is the proper expression and analytic computation of gluon condensate?

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I think the term gluon condensate" usually means $<F_{\mu\nu} F^{\mu\nu}/4>$. This is not really a condensate however, because it does not break any symmetry. Furthermore, since the operator $F_{\mu\nu} F^{\mu\nu}$ mixes with the identity operator under renormalization, it is a non-universal quantity, so it is not clear what it means physically.

There are various QCD "sum rules" that make use of this quantity, but they depend on a separation of "perturbative" from "non-perturbative" contributions. I've never understood how this can be done rigorously, and I am not alone in this skepticism, however the practitioners of this art claim that they know what they are doing, and I am not expert to argue with them.

This paper: arXiv:hep-ph/9502326v1, seems to have useful things to say about the issue.

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