An exact sum rule for transversely polarized DIS

The Operator Product Expansion provides expressions for the $n^{th}$ moments of $g_1(x)$ and $g_2(x)$ in terms of hadronic matrix elements of local operators for $n =$ odd integer. In some cases these matrix elements are expected to be small leading to approximate sum rules for the {\em odd\/} moments of $g_{1,2}(x)$. We have shown how, working in a field-theoretic framework, one can derive expressions for the {\em even\/} moments of the {\em valence\/} parts of $g_{1,2}(x)$. These expressions cannot be written as matrix elements of {\em local\/} operators and do not coincide with the analytic continuation to $n=$ even integer of the OPE results. Just as for the OPE one can in some cases argue that the hadronic matrix elements should be small, leading to approximate sum rules for the moments of the valence parts of $g_{1,2}(x)$. But, most importantly, for the case $n=2$ we have proved rigorously that the hadronic matrix element vanishes, yielding the exact ELT sum rule \int^1_0 dx\, x\left[g^V_1(x)+2g^V_2(x)\right]=0. We have argued that the convergence properties of this sum rule are good and have discussed how it can be used to get information about $g_2(x)$ and as a further test of QCD.