Collinear factorization violation and effective field theory

The factorization of amplitudes into hard, soft and collinear parts is known to be violated in situations where incoming particles are collinear to outgoing ones. This result was first derived by studying limits where noncollinear particles become collinear. We show that through an effective field theory framework with Glauber operators, these factorization-violating effects can be reproduced from an amplitude that is factorized before the splitting occurs. We confirm results at one loop, through single Glauber exchange, and at two loops, through double Glauber exchange. To approach the calculation, we begin by reviewing the importance of Glauber scaling for factorization. We show that for any situation where initial-state and final-state particles are not collinear, the Glauber contribution is entirely contained in the soft contribution. The contributions coming from Glauber operators are necessarily nonanalytic functions of external momentum, with the nonanalyticity arising from the rapidity regulator. The nonanalyticity is critical so that Glauber operators can both preserve factorization when it holds and produce factorization-violating effects when they are present.

Figure 1

Left: Poles in the complex $k^{-}$ plane for $I_{\mathrm{full}}$. Right: The complex $k^{-}$ plane of $I_{\mathrm{full}}$ in the region around $k^{\mu }=0$. In this region, all components of $k^{\mu }$ are less than $\kappa Q$. The integration contour, $-\kappa Q\le k^{-}\le \kappa Q$ (the dashed red line on the real axis) can be deformed (to the solid green contour) to avoid the Glauber region (hashed blue) for any values of $|k^{+}|<\kappa Q$ and $|{\overrightarrow{k}}_{\perp }|<\kappa Q$.

Figure 2

Example Glauber diagrams that can be ignored for the two-loop splitting amplitude.

Figure 3

Example Glauber-soft and Glauber-collinear mixing diagrams that may break strict collinear factorization.

Figure 4

Double Glauber exchange diagrams that involve two collinear sectors.

Figure 5

Double Glauber diagrams involving three collinear sectors.