1207.2621 (Vladimir Sauli)
Vladimir Sauli
When generalizing recent various quantum mechanical models of heavy quarkonia to Quantum Filed theoretical approach based on Bethe-Salpeter equation one is faced to the solutions that do not exist in nonrelativistic limit. Mainly, there is unexpected doubling of the spectrum when comparing to the experimentally known spectrum as well as the ones obtained from the solution of the Schroedinger equation. These additional states are not apriory unphysical as both of them have the same symmetry. Our study strongly suggests that these solutions appear due to the sensitivity of BSE to the details of the analytical form of the constituents quark and antiquark propagators, more specifically they are consequence of using unconfining free propagators. To show this explicitly we develop and describe the efficient method of the numerical solution for quarkonium BSE and numerically solve it for the case of pseudoscalar charmonia. For the bare propagators of constituents we are able to find BSE solution for arbitrarily high excited state without any 3-dimensional reduction. Unlike to the Schroedinger equation the excited states are not orthogonal. Using free charm quark propagators we observe that the ground state $\eta_c$ is situated slightly above naive quark-antiquark threshold $2m_c$, thus the all excited states are situated above this threshold. In the second part of the paper we consider the model of BSE with confined propagators and show the influence on the spectrum directly in the Minkowski space for the first time.
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http://arxiv.org/abs/1207.2621
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