Independent factorization and renormalization scales are inconsistent with simultaneously preserving RG invariance, Ward identities, and PDF sum rules in generalized pole subtraction schemes.
What exactly is a parton density?
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I give an account of the definitions of parton densities, both the conventional ones, integrated over parton transverse momentum, and unintegrated transverse-momentum-dependent densities. The aim is to get a precise and correct definition of a parton density as the target expectation value of a suitable quantum mechanical operator, so that a clear connection to non-perturbative QCD is provided. Starting from the intuitive ideas in the parton model that predate QCD, we will see how the simplest operator definitions suffer from divergences. Corrections to the definition are needed to eliminate the divergences. An improved definition of unintegrated parton densities is proposed.
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What are the consequences of independent factorization and renormalization scales?
Independent factorization and renormalization scales are inconsistent with simultaneously preserving RG invariance, Ward identities, and PDF sum rules in generalized pole subtraction schemes.