Is $\mu$ the renormalization or factorization scale in the DGLAP equations?

The DGLAP equations read $$\frac{\partial f_i(x,\mu^2)}{\partial\ln\mu^2}=\sum_j\int^1_x\frac{dz}{z}P_{ij}(z,\alpha_s(\mu^2))f_j\left(\frac{x}{z},\mu^2\right),$$ where the $$f_i$$ are the parton distribution functions (PDFs), $$P_{ij}$$ are the so-called splitting kernels and $$x,z$$ are longitudinal momentum fractions.

But what is $$\mu$$? In this paper on p.26 John Collins says it is the renormalisation scale, which enters the PDF via dimensional regularization. But I have already seen other authors claim that it is a factorization scale, e.g. here on p.2.

So, which one is it?

1 Answer

Following T. Plehn (p.7) the UV-renormalisation introduces a dependence on the renormalisation scale $$\mu$$ and then the IR-regularisation introduces a further dependence on the factorisation scale $$\mu_F$$. However, one is free to choose $$\mu$$, such that one can set $$\mu:=\mu_F$$ R. Brock et al.(p.104), D.E. Soper(p.38), W.K. Tung(p.19). Hence, J.C. Collins' lack of mentioning the factorisation scale in most of his papers and most notably his book Foundations of Perturbative QCD.