24 Chapter 2: Motivation parton a in a proton with momentum fraction between x$_{a}$ and x$_{a}$ + dx , where x$_{a}$ = p$_{a}$/P$_{p}$ and something of an abuse of notation occurred by which p$_{a}$ stands for the momentum of parton a and P$_{p}$ stands for the momentum of the proton. The FF gives the probability of parton c hadronizing into h with fractional momentum between z$_{c}$ and z$_{c}$ + dz , where P$_{h}$ = z$_{c}$p$_{c}$ . All allowed combinations of a , b , and c are summed over where a + b → c + X such that c may hadronize into h . We note that we have slightly extended the factorization theorem to include the fragmentation process, which introduces the new scale µ ′ $_{F}$. Addi- tionally a renormalization scale µ$_{R}$ associated with the running coupling α$_{s}$ is included in the perturbative cross section. Parameterizations of PDFs may be found from, e.g., the$_{CTEQ}$ collabora- tion [161], GRV [162], MRST [163], or Alekhin [164]. Hadronic fragmentation functions have similarly been parameterized by, e.g., DSS [165], AKK [166], HKNS [167], and KKP [168]; [169] provides a photon FF. We note that the posted code for KKP has a bug in its leading order (LO) kaon output that has still not been corrected; ported Mathematica codes for the DSS and KKP fragmentation functions, as well as a corrected$_{Fortran}$ code for KKP can be found online [170]. Although PDFs and FFs at a specific scale must be found from experiment, their form is constrained by QCD and general sum rule con- siderations and their evolution is governed by pQCD and DGLAP dynamics [171–173]. Derivations of perturbative cross sections vary depending on the specific reaction considered. For those involving gluons and light guarks NLO calcu-