Exact sum rule for transversely polarized DIS

We have shown how, working in a field-theoretic framework, one can derive expressions for the even moments of the valence parts of $g_{1,2}\mathrm{}\left(x\right)\mathrm{}$. These expressions cannot be written as matrix elements of 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 $g_{1,2}\mathrm{}\left(x\right)\mathrm{}$. But, most importantly, for the case $n=2\mathrm{}$ we have proved rigorously that the hadronic matrix element vanishes, yielding an exact sum rule. We have argued that the convergence properties of this sum rule are good and are a further test of QCD.