# Is the starting distribution in the solution of DGLAP IR-bare?

On p.27 of this paper by John Collins, he says that when defining PDFs in terms of partonic number operators, one acquires an IR-divergent bare PDF (eq. 52). The residue of the IR-divergent term is proportional to the DGLAP splitting kernels: $$f_{a/b}(\xi,\epsilon)=\delta_{ab}\delta(1-\xi)-\frac{1}{\epsilon}\frac{\alpha_s}{\pi}P^{(1)}_{a/b}(\xi)+\mathcal{O}(\alpha^2_s).$$

Now, when we solve the DGLAP equations formally, we find the solution $$f_i(x,\mu^2)=f_i(x,\mu^2_0)+\sum_j\int^{\mu^2}_{\mu^2_0}d\ln\mu^{\prime2}\int^1_x\frac{dz}{z}P_{ij}(z,\alpha_s(\mu^{\prime2}))f_j(\frac{x}{z},\mu^{\prime2}).$$

Is the $$f_i(x,\mu^2_0)$$ the same IR-bare PDF from above? Can I see the splitting kernels as a kind of IR counter terms to cancel the remaining IR divergence in the bare PDF?

• Absolutely yes! The PDF absorbs the IR divergences from the splitting functions so that the total result is finite. Collins goes through all the details carefully in his book Renormalization. – WAH May 3 at 19:03
• @WAH To which pages are you referring to specifically? I have tried to find a section on PDFs or DGLAP in said book but could not find anything on the renormalisation of PDFs. – Thomas Wening May 6 at 8:14