Evolve parton densities without inventing their initial shape

A daughter observed at momentum fraction x may originate from a parent with fraction x divided by z, where z is the daughter’s share of the parent longitudinal momentum. Since the parent fraction cannot exceed one, the convolution runs from z equal to x to one. Its Jacobian supplies a factor one over z. The kernel index tells which parent produces which daughter, so writing the index order explicitly avoids interchanging quark-to-gluon and gluon-to-quark kernels.

At leading order the overall coefficient is alpha s divided by two pi when differentiating with respect to logarithm of factorization scale squared. The PDF’s scale is not an elapsed physical time. Radiation migrates momentum between x bins and between species as the resolution convention changes. Unpolarized PDFs have a probability-like leading-order interpretation, but renormalized distributions and subtractions beyond leading order are scheme dependent. Individual higher-order PDF values should not be treated as directly observable positive event counts.

For a gluon producing one selected quark flavor, the leading kernel is one half times z squared plus one minus z squared. For a quark producing a gluon, the kernel has a soft singularity as the gluon share goes to zero. Quark retention has a complementary endpoint singularity as the radiated gluon becomes soft. Integrating only real emission would produce divergent probabilities and would incorrectly deplete conserved valence quantum numbers without a compensating unresolved contribution.

The full quark-retention kernel includes a plus distribution and a delta term. The plus prescription subtracts the endpoint value of a test function before integrating the singular denominator. It is a distributional statement, not an instruction to evaluate infinity at z equal to one. Writing the kernel as two over one minus z with the plus prescription, followed by regular minus one minus z terms, makes its integral especially transparent. The remaining delta coefficient restores the required zero valence moment.

Integrating a nonsinglet valence density over x gives a conserved net flavor number. The zero integral of its evolution kernel expresses that conservation. Weighting by x instead measures momentum, which can leave the quark sector and enter the gluon sector. For each parent, the momentum-weighted sum over all daughter kernels must vanish. These constraints are stringent checks on endpoint coefficients and flavor multiplicities; replacing kernels by their positive real parts violates them even if individual emissions look plausible.

Evolution requires an initial scale, boundary densities, a coupling prescription, perturbative order, and heavy-flavor matching rules. Very small x or endpoint regions may need logarithmic resummation, and low virtualities admit power corrections outside leading twist. A finite-state Markov illustration can conserve a total probability, but its two bins are not x-dependent parton species and its transition rates are not DGLAP kernels. Its useful role is to expose why apparent loss from one component must be balanced elsewhere.