Next-to-leading order Q^2-evolution of the transversity distribution h_1(x, Q^2)

TL;DR

This work computes the two-loop anomalous dimension $\gamma^{h(1)}_n$ for the transversity distribution $h_1(x,Q^2)$ in the MS scheme, enabling the next-to-leading order $Q^2$-evolution of $h_1$ to be implemented alongside $f_1$ and $g_1$. Using twist-2 tensor operators and a two-loop renormalization framework akin to prior $f_1$ and $g_1$ analyses, the authors derive a comprehensive expression for $\gamma^{(1)}_n$ in terms of harmonic sums, and demonstrate that $\gamma^{h(1)}_n$ is larger than $\gamma^{f(1)}_n$ at small moments but rapidly converges to it at large $n$. The numerical results show sizable NLO effects for $h_1$, particularly influencing small-$x$ evolution, with explicit evolution behavior for the tensor charge and the first moments. These findings complete the NLO twist-2 anomalous dimensions for nucleon distributions and have important implications for phenomenology and experimental access to transversity.

Abstract

We present a calculation of the two-loop anomalous dimension for the transversity distribution h_1(x,Q^2), $γ^{h(1)}_n$, in the MS scheme of the dimensional regularization. Due to 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), $γ^{h(1)}_n$ is significantly larger than $γ^{f(1)}_n$ (the anomalous dimension for the nonsinglet f_1), but approaches $γ^{f(1)}_n$ very quickly at large n, keeping the relation $γ^{h(1)}_n > γ^{f(1)}_n$. This feature is in parallel to the relation between the one-loop anomalous dimension for f_1 and h_1.