The proton spin in the chiral bag model: Casimir contributions and Cheshire Cat Principle

The flavor singlet axial charge has been a source of study in the last years due to its relation to the so called {\it Proton Spin Problem}. The relevant flavor singlet axial current is anomalous, i.e., its divergence contains a piece which is the celebrated UA(1) anomaly. This anomaly is intimately associated with the η′ meson, which gets its mass from it. When the gauge degrees of freedom of QCD are confined within a volume as is presently understood, the UA(1) anomaly is known to induce color anomaly leading to 'leakage' of the color out of the confined volume (or bag). For consistency of the theory, this anomaly should be canceled by a boundary term. This ``color boundary term' inherits part or most of the dynamics of the volume (i.e., QCD). In this paper, we exploit this mapping of the volume to the surafce via the color boundary condition to perform a complete analysis of the flavor singlet axial charge in the chiral bag model using the Cheshire Cat Principle. This enables us to obtain the hitherto missing piece in the axial charge associated with the gluon Casimir energies. The result is that the flavor singlet axial charge is small independent of the confinement (bag) size ranging from the skyrmion picture to the MIT bag picture, thereby confirming the (albeit approximate) Cheshire Cat phenomenon