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In the AdS/QCD correspondence, glueballs operators are given, for example, by $\text{Tr}[F_{\mu \nu}F^{\mu \nu}]$ for $0^{++}$ or $\text{Tr}[F_{\mu \nu}\widetilde{F}^{\mu \nu}]$ for $0^{-+}$. However, in this link1, figure 1, it is stated that glueball operators are made from linear combinations of these Wilson loops. My question is: Is there a link between these two approaches? Or phrased differently: Can we deduce glueball operators in AdS/QCD from those Wilson loops?


  1. C. J. Morningstar and M. Peardon, "Glueball spectrum from an anisotropic lattice study", Phys. Rev. D 60, 034509 (1999), arXiv:hep-lat/9901004.
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  • $\begingroup$ I'm not sure where you see a difference between the two approaches. E.g. one might also say that in the standard model you have operators $e$, $\mu$, $\tau$ for the three leptons, or one might say that "lepton states are created by linear combinations of $e$, $\mu$, $\tau$". It's just a different emphasis of talking about specific particles vs. a group of particles, but where is the difference? $\endgroup$ – ACuriousMind Sep 24 '20 at 15:14
  • $\begingroup$ My confusion comes from the fact that Wilson loops are expressed as $W_p = \text{Tr}\left[\mathcal{P}\,e^{i \int_{\partial p} A}\right]$, while in AdS/QCD they are expressed as $ \text{Tr}[F_{\mu \nu}F^{\mu \nu}]$ etc.. Especially I don't know how to recover the lasts from the Wilson loops on differents contours. $\endgroup$ – Jeanbaptiste Roux Sep 24 '20 at 15:28
  • $\begingroup$ For future reference: It would have helped me to understand the question better if you had said a bit more explicitly that the problem is the connection between $\mathrm{tr}(F^2)$ and the Wilson loops, not the "linear combination" part. Also, please consider mentioning author and title of papers you refer to in the post itself to guard against link rot $\endgroup$ – ACuriousMind Sep 24 '20 at 15:54
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The "glueball operators are Wilson loops" is the non-perturbative viewpoint amenable to lattice computations, while the "glueball operators are quadratics in the field strength" is the perturbative viewpoint.

The connection between them comes from reasoning like in "Interaction of Wilson loops in confining vacuum" by Shevchenko and Simonov, where they show that in the limit of large Wilson loops - compared to the gluon correlation length - and to lowest order in perturbation theory (i.e. the coupling constant $g$), we have schematically that $$ \langle W \rangle \propto \langle \mathrm{tr}(F^2)\rangle,$$ where $W$ is a Wilson loop and $F^2 = F_{\mu\nu}F^{\mu\nu}$ (this is their eqs. (25) + (26)).

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  • $\begingroup$ So, if I want to calculate the two-point function for the $0^{-+}$, I ``just'' have to calculate the correlation function between to Wilson loops, given a certain type of contour? $\endgroup$ – Jeanbaptiste Roux Sep 24 '20 at 16:21
  • $\begingroup$ @JeanbaptisteRoux Yes, I think so. (I don't really know a lot about glueballs in particular, so if you have another specific question about them, feel free to ask it as a new question on the site!) $\endgroup$ – ACuriousMind Sep 25 '20 at 7:57
  • $\begingroup$ @JeanbaptisteRoux Yes; that's mostly correct. The situation in 2 spatial dimensions is well summarized in arxiv.org/pdf/hep-lat/9804008.pdf by Michael J. Teper. In particular see page 9 and page 10 where they lay out how to construct glueball operators. It's not just a correlation function between two wilson loops; it's the correlation function between the linear combinations of the traces of parity transformed/rotated wilson loops. It's the real part of the trace for C (Charge Parity)+ glueballs, the imaginary part for C- glueballs. $\endgroup$ – chuckstables Nov 5 '20 at 3:09

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