Higher orders inA(αs)/[1−x]+of nonsinglet partonic splitting functions

We develop a simplified method for obtaining higher orders in the perturbative expansion of the singular term $A\left({\alpha }_s\right)/[1-x{]}_{+}$ of nonsinglet partonic splitting functions. Our method is based on the calculation of eikonal diagrams. The key point is the observation that the corresponding cross sections exponentiate in the case of two eikonal lines, and that the exponent is directly related to the functions $A\left({\alpha }_s\right)$ due to the factorization properties of parton distribution functions. As examples, we rederive the one- and two-loop coefficients $A^{\mathrm{}\left(1\right)\mathrm{}}$ and $A^{\mathrm{}\left(2\right)\mathrm{}}.$ We go on to derive the known general formula for the contribution to $A^{\mathrm{}\left(n\right)\mathrm{}}$ proportional to $N_f^{n-1},$ where $N_f$ denotes the number of flavors. Finally, we determine the previously uncalculated term proportional to $N_f$ of the three-loop coefficient $A^{\mathrm{}\left(3\right)\mathrm{}}$ to illustrate the method. Our answer agrees with the existing numerical estimate. The exact knowledge of the coefficients $A^{\mathrm{}\left(n\right)\mathrm{}}$ is important for the resummations of large logarithmic corrections due to soft radiation. Although only the singular part of the splitting functions is calculable within our method, higher-order computations are much less complex than within conventional methods, and even the calculation of $A^{\mathrm{}\left(4\right)\mathrm{}}$ may be possible.