Effects of the quantityσTSon the spin structure functions of nucleons in the resonance region

In this paper, we investigate the effects of the quantity ${\sigma }_{\mathrm{TS}}$ on the spin structure functions of nucleons in the resonance region. The Schwinger sum rule for the spin structure function $g_2(x, Q^2$ at the real photon limit is derived for the nucleon treated as a composite system, and it provides a crucial constraint on the longitudinal transition operator which has not been treated consistently in the literature. The longitudinal amplitude $S_{\frac{1}{2}}$ is evaluated in the quark model with the transition operator that generates the Schwinger sum rule. The numerical results of the quantity ${\sigma }_{\mathrm{TS}}$ are presented for both spin structure functions $g_1(x, Q^2)$ and $g_2(x, Q^2)$ in the resonance region. Our results show that this quantity plays an important role in the low $Q^2$ region, which can be tested in future experiments at CEBAF.