The Multiplicity Scaling of the Fragmentation Function

The single-particle inclusive fragmentation function and the particle multiplicity are observables of fundamental importance in studying properties of quantum chromodynamics at colliders. It is well-known that at high energies, the multiplicity distribution satisfies KNO scaling in which all moments are proportional to powers of the mean multiplicity. We prove that, under weak assumptions, the leading dependence of the fragmentation function on multiplicity is itself a kind of KNO scaling in which all moments are inversely proportional to powers of the mean multiplicity. This scaling with multiplicity additionally accounts for the dominant dependence on collision energy in the fragmentation function. The proof relies crucially on properties of the fragmentation function conditioned on the total multiplicity and application of the Stieltjes moment problem. In the process, we construct a novel basis of the fragmentation function expressed as an overall exponential suppression times a series of Laguerre polynomials. We study this scaling of the fragmentation function in experimental electron-position collision data and observe that residual scale violations are significantly reduced.