Rapidity divergences and deep inelastic scattering in the endpoint region

The deep inelastic scattering cross section in the endpoint region $x\sim 1$ has been subjected to extensive analysis. We revisit this process using soft collinear effective theory, and show that in the endpoint individual factors in the factorized hadronic tensor have rapidity divergences. We regulate these divergences using a recently introduced rapidity regulator, and find that each operator matrix element requires a different scale to minimize large rapidity logarithms. However, the running in rapidity is nonperturbative and must be absorbed into the definition of the parton distribution function.

Figure 1

The $\mathcal{O}({\alpha }_s^0)$ Feynman diagram for the $n$ collinear function. The dashed lines are collinear quarks, the grey circles are vertices where momentum is injected, and the gap indicates a cut.

Figure 2

The one-loop Feynman diagrams for the $n$ collinear function: (a) is the virtual contribution, while (b) and (c) are real contributions. Two diagrams which are the mirror image of (a) and (b) are not shown. The double line represents a Wilson line which is the source of a single gluon.

Figure 3

Feynman diagrams for the one-loop evaluation of the soft function: (a) is the virtual contribution and (b) is the real contribution. There are two additional diagrams which are obtained by reflecting about a vertical axis through the middle of the diagram. The double lines indicate Wilson lines which produce the gluons, and $n$ and $\overline{n}$ label the direction of the Wilson lines. The gap between vertices indicates a cut.