Gluon fragmentation function in polarised $\Lambda$ hyperon production: The Method of Factorisation

We discuss the polarised fragmentation functions of quarks and gluons in Perturbative Quantum Chromodynamics. The Altarelli-Parisi evolution equations governing these fragmentation functions are presented. We find that the first moment of the polarised gluon fragmentation function to order $\alpha_s$ is as important as that of the quark and anti-quark fragmentation functions. The $\Lambda$ production in polarised $e^+ e^-$ annihilation where this can be realised is discussed. We propose the factorisation method to compute the gluonic contribution to $\Lambda$ production to order $\alpha_s$. With appropriate operator definitions for the quark and the gluon fragmentation functions, we find that the hard part of the factorisation formula is free of any infrared singularities confirming the factorisation.