Resummation of soft gluon logarithms in the DGLAP evolution of fragmentation functions

We define a general scheme for the evolution of fragmentation functions which resums both soft gluon logarithms and mass singularities in a consistent manner and to any order, and requires no additional theoretical assumptions. Using the double logarithmic approximation and the known perturbative results for the splitting functions, we present our scheme with the complete contribution from the double logarithms, being the largest soft gluon logarithms. We show that the resulting approximation is more complete than the modified leading logarithm approximation even with the fixed order contribution calculated to leading order only, and find, after using it to fit quark and gluon fragmentation functions to experimental data, that this approximation in our scheme gives a good description of the data from the largest $x_p$ values to the peak region in $\xi =\ln (1/x_p)$, in contrast to other approximations. In addition, we develop a treatment of hadron mass effects which gives additional improvements at large $\xi$.

Figure 1

$P_{gg}(x,a_s)$ calculated in the $\mathrm{DL}+\mathrm{LO}+\mathrm{LO}\delta$ scheme, $P_{gg}(x,a_s)$ calculated to $O(a_s)$ in the FO approach (labeled “LO”), and $P_{gg}^{\mathrm{DL}}(x,a_s)$ (labeled “DL”). $a_s=0.118/(2\pi )$.

Figure 2

Fit to data as described in Table . Some of the data sets used for the fit are shown, together with their theoretical predictions from the results of the fit. Data to the right of the vertical dotted lines have not been used in the fit. Each curve is shifted up by 0.8 for clarity.

Figure 3

Fit to data as described in Table .

Figure 4

Comparison with data from the fit described in Table .

Figure 5

Comparison with data from the fit described in Table .

Figure 6

Comparison of the gluon FF in the $\mathrm{DL}+\mathrm{LO}+\mathrm{LO}\delta$ scheme to gluon jet measurements from OPAL. The position of the cut of Eq. (60) is shown to indicate where the gluon is constrained. $Q=2E_{\mathrm{jet}}$.

Figure 7

As in Fig. 6, but from a fit in which the gluon jet data below the cut are included in the fit.

Figure 8

Fit to data as described in Table .

Figure 9

As in Fig. 7, but with a fitted hadron mass.

Figure 10

Fit to data as described in Table .