The role of Glauber Exchange in Soft Collinear Effective Theory and the Balitsky-Fadin-Kuraev-Lipatov Equation

Abstract

In soft collinear effective theory (SCET) the interaction between high energy quarks moving in opposite directions involving momentum transfer much smaller than the center-of-mass energy is described by the Glauber interaction operator which has two-dimensional Coulomb-like behavior. Here, we determine this $n$-$\bar{n}$ collinear Glauber interaction operator and consider its renormalization properties at one loop. At this order a rapidity divergence appears which gives rise to an infrared divergent (IR) rapidity anomalous dimension commonly called the gluon Regge trajectory. We then go on to consider the forward quark scattering cross section in SCET. The emission of real soft gluons from the Glauber interaction gives rise to the Lipatov vertex. Squaring and adding the real and virtual amplitudes results in a cancelation of IR divergences, however the rapidity divergence remains. We introduce a rapidity counterterm to cancel the rapidity divergence, and derive a rapidity renormalization group equation which is the Balitsky-Fadin-Kuraev-Lipatov Equation. This connects Glauber interactions with the emergence of Regge behavior in SCET.

Figures (4)

• Figure 1: Leading order contribution to forward quark-quark scattering at high energy: $(a)$ QCD diagram, $(b)$ SCET diagram (dashed lines indicate collinear quarks, and dotted lines Glauber gluons).
• Figure 2: Leading order Feynman diagrams corresponding to operators that couple collinear and soft degrees of freedom via Glauber exchange: $(a)$ collinear quark coupling to a soft gluon, $(b)$ collinear quark coupling to a soft quark (solid line). Not shown is the collinear quark coupling to a soft ghost.
• Figure 3: One loop Feynman diagrams contributing to the renormalization of ${\cal O}^{n\bar{n}}_{G}$. The double line in the diagrams in $(a)$ indicate soft gluon emission from a Wilson line. These diagrams have a rapidity divergence which gives the gluon Regge trajectory. The diagrams in $(b)$ have no rapidity divergence, but have UV divergences. The first two diagrams involve soft gluons and soft quarks (the soft-ghost loop diagram is not shown), and the UV divergence in these diagrams is cancelled by a soft Lagrangian counter-term. The last diagram involves the exchange of a collinear gluon (spring with a line) and the UV divergence is cancelled by a collinear Lagrangian counter-term.
• Figure 4: Real emission of soft gluons from the $n$-$\bar{n}$ Glauber interaction: (a) emission from the soft Wilson lines, (b) emission from the Glauber gluon.