# Generalized sum rules for spin-dependent structure functions of the nucleon

### Abstract:

The Drell-Hearn-Gerasimov and Bjorken sum rules are special examples of dispersive sum rules for the spin-dependent structure function [iopmath latex="$G_1(\nu, Q^2)$"]*G*

_{1}(,

*Q*

^{2}) [/iopmath] at [iopmath latex="$Q^2 = 0$"]

*Q*

^{2}= 0 [/iopmath] and [iopmath latex="$\infty$"] [/iopmath] . We generalize these sum rules through studying the virtual-photon Compton amplitudes [iopmath latex="$S_1(\nu, Q^2)$"]

*S*

_{1}(,

*Q*

^{2}) [/iopmath] and [iopmath latex="$S_2(\nu, Q^2)$"]

*S*

_{2}(,

*Q*

^{2}) [/iopmath] . At small [iopmath latex="$Q^2$"]

*Q*

^{2}[/iopmath] , we calculate the Compton amplitudes at leading order in chiral perturbation theory; the resulting sum rules will be able to be tested against data soon available from the Jefferson Laboratory. For [iopmath latex="$Q^2 \gg\Lambda_{\rm QCD}^2$"]

*Q*

^{2}>>

_{QCD}

^{2}[/iopmath] , the standard twist-expansion for the Compton amplitudes leads to the well known deep-inelastic sum rules. Although the situation is still relatively unclear in a small intermediate- [iopmath latex="$Q^2$"]

*Q*

^{2}[/iopmath] window, we argue that chiral perturbation theory and the twist-expansion alone already provide strong constraints on the [iopmath latex="$Q^2$"]

*Q*

^{2}[/iopmath] -evolution of the [iopmath latex="$G_1(\nu, Q^2)$"]

*G*

_{1}(,

*Q*

^{2}) [/iopmath] sum rule from [iopmath latex="$Q^2 = 0$"]

*Q*

^{2}= 0 [/iopmath] to [iopmath latex="$\infty$"] [/iopmath] .

**Document Type:** Miscellaneous

Publication date: January 1, 2001