The Calculation of the Two-Loop Spin Splitting Functions $P_{ij}^{(1)}(x)$

Abstract

We present the calculation of the two-loop spin splitting functions $P_{ij}^{(1)}(x)(i,j=q,g)$ contributing to the next-to-leading order corrected spin structure function $g_1(x,Q^2)$. These splitting functions, which are presented in the \MSbs, are derived from the order ${\alpha }_s^2$ contribution to the anomalous dimensions ${\gamma }_{ij}^m(i,j=q,g)$. The latter correspond to the local operators which appear in the operator product expansion of two electromagnetic currents. Some of the properties of the anomalous dimensions will be discussed. In particular our findings are in agreement with the supersymmetric relation ${\gamma }_{qq}^m+{\gamma }_{gq}^m-{\gamma }_{qg}^m-{\gamma }_{gg}^m=0$ up to order ${\alpha }_s^2$.

Comments: Updated to match the journal version. LaTeX, 34 pages, 6 figures (in Postscript file). Corrected formulas: (3.66), (3.67), (3.75), (3.76), (3.88), (4.97), (4.108). Modified renormalization discussions. Corrected conclusion and discussion about validity of supersymmetric relation. Figures improved