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Can someone help explain how the equations 30, 31 and 34 were obtained in this paper.

At a conceptual level I am wondering looking at equation 34 as to if they mean that $\lambda$ is somehow the anomalous dimension of the single-trace operators in this critical U(N) model - although looking at equation 30 it seemed that $\lambda$ is just the Hubbard-Stratonivich auxiliary field (though its not clear to me as to how is equation 30 derived). That factor of $(l + \frac{1}{2})$ is confusing me everywhere.

Also what they call as $1/g$ in equation 30 is what I would think of as the threshold value of the translationaly invariant condensate at which the symmetry breaks. So it is like an order parameter and I can't see that as something that was there in the UV lagrangian as a coupling constant as they seem to imagine it here.

Although I understand how equation 1 in this paper was derived but I still don;t see how equation 35 somehow follows immediately from it.

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