NLO $Q^2$-evolution of the nucleon's transversity distribution $h_1(x, Q^2)$

Abstract

We present a calculation of the two-loop anomalous dimension for the transversity distribution $h_1(x,Q^2)$, $\gamma^{h(1)}_n$, in the MS scheme of the dimensional regularization. Because of the chiral-odd nature, $h_1$ does not mix with the gluon distributions, and thus our result is the same for the flavor-singlet and nonsinglet distributions. At small $n$ (moment of $h_1$), $\gamma^{h(1)}_n$ is significantly larger than $\gamma^{f,g(1)}_n$ (the anomalous dimension for the nonsinglet $f_1$ and $g_1$), but approaches $\gamma^{f,g(1)}_n$ very quickly at large $n$, keeping the relation $\gamma^{h(1)}_n > \gamma^{f,g(1)}_n$. This feature is in parallel to the relation between the one-loop anomalous dimension for $f_1$ $(g_1)$ and $h_1$. We also show that this difference in the anomalous dimension between $h_1$ and $g_1$ leads to a drastic difference in the $Q^2$-evolution of those distributions in the small $x$ region.