Fully double-logarithm-resummed cross sections

TL;DR

The paper tackles the problem of large double logarithms in semi-inclusive hadron production cross sections at small momentum fraction x. It first revisits the massive-gluon scheme to derive DL contributions to timelike splitting functions and the gluon coefficient function, then performs dimensional regularization to obtain a complete DL-resummed expression in the standard \overline{MS} scheme. The main result is a closed-form, DL-resummed gluon coefficient function in MSbar, $C_g(\omega,a_s)=\frac{K_I}{C_A}\left[(1+16C_A a_s/\omega^2)^{-1/4}-1\right]$, with a corresponding x-space form and a DL+NLO matching formula that aligns with fixed-order NNLO DLs. This resummation mitigates small-x instabilities at NLO and has practical implications for precision predictions and global fits of fragmentation functions, motivating further DL+NLO analyses.

Abstract

We calculate the complete double logarithmic contribution to cross sections for semi-inclusive hadron production in the modified minimal-subtraction scheme by applying dimensional regularization to the double logarithm approximation. The full double logarithmic contribution to the coefficient functions for inclusive hadron production in electron-positron annihilation is obtained in this scheme for the first time. Our result agrees with all fixed order results in the literature, which extend to next-next-to-leading order.

Figures (2)

• Figure 1: The gluon coefficient function in $e^+ e^-$ annihilation at NLO in Eq. (\ref{['coeffNLO']}) (blue lower line), the complete DL contribution to the gluon coefficient function in Eq. (\ref{['coeffDLpre']}) (violet line) and the DL-resummed NLO result in Eq. (\ref{['coeffDLpNLO']}) (orange line). The value $a_s=0.18$ was used, because this corresponds to the result for $a_s(Q^2)$ at LO using $Q=14$ GeV, the value for $Q$ used in Ref. Albino:2005gd.
• Figure 2: Contribution to the inclusive single hadron production cross section in $e^+ e^-$ annihilation due to the gluon channel corrections in Eq. (\ref{['gluonmultiplicity']}) at NLO and with the inclusion of the DLs at all orders in perturbation theory. As in Fig. \ref{['fig:coefficient']}, the value $a_s(Q^2)=0.18$ was chosen in the calculation of $C_g(z,a_s(Q^2))$, and the gluon FF was taken to be that obtained at $Q=14$ GeV in the LO global fit of Ref. Albino:2005gd.