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

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$$\frac{\partial f_i(x,\mu^2)}{\partial\ln\mu^2}=\sum_j\int^1_x\frac{dz}{z}P_{ij}(z,\alpha_s(\mu^2))f_j\left(\frac{x}{z},\mu^2\right),$$
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 https://arxiv.org/pdf/hep-ph/0409313.pdf on p.26 John Collins says it is the renormalisation scale, which enters the PDF via dimensional regularisation. But I have already seen other authors claim that it is a factorisation scale, e.g. https://arxiv.org/pdf/hep-ph/0703156.pdf on p.2.

So, which one is it?

Following https://arxiv.org/abs/0810.2281 (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 $\mu:=\mu_F$ https://inspirehep.net/files/a07787f3653b2a41945961ad44fbe926 (p.104), https://arxiv.org/pdf/hep-ph/9702203.pdf (p.38), https://icecube.wisc.edu/~aya/simulation/prompt/ref/pqcd-theory/IntroPqcd-long.pdf (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.
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