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

+ 1 like - 0 dislike

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 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?

asked Jul 28, 2020 in Theoretical Physics by twening (70 points) [ no revision ]
recategorized Jul 31, 2020 by Dilaton

1 Answer

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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.

answered Jul 28, 2020 by twening (70 points) [ no revision ]

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