Contact Term, its Holographic Description in QCD and Dark Energy

Abstract

In this work we study the well known contact term, which is the key element in resolving the so-called $U(1)_A$ problem in QCD. We study this term using the dual Holographic Description. We argue that in the dual picture the contact term is saturated by the D2 branes which can be interpreted as the tunnelling events in Minkowski space-time. We quote a number of direct lattice results supporting this identification. We also argue that the contact term receives a Casimir -like correction \sim (\Lqcd R)^{-1} rather than naively expected \exp(-\Lqcd R) when the Minkowski space-time ${\cal R}_{3,1}$ is replaced by a large but finite manifold with a size $R$. Such a behaviour is consistent with other QFT-based computations when power like corrections are due to nontrivial properties of topological sectors of the theory. In holographic description such a behaviour is due to massless Ramond-Ramond (RR) field living in the bulk of multidimensional space when power like corrections is a natural outcome of massless RR field. In many respects the phenomenon is similar to the Aharonov -Casher effect when the "modular electric field" can penetrate into a superconductor where the electric field is exponentially screened. The role of "modular operator" from Aharonov -Casher effect is played by large gauge transformation operator $\cal{T}$ in 4d QCD, resulting the transparency of the system to topologically nontrivial pure gauge configurations. We discuss some profound consequences of our findings. In particular, we speculate that a slow variation of the contact term in expanding universe might be the main source of the observed Dark Energy.Comment: Final version to appear in Phys. Rev. D. Comments added on interpretation of the "topological Casimir effect" from 5d viewpoint where it can be thought as conventional Casimir effec