Jets in Effective Theory: Summing Phase Space Logs

TL;DR

The paper uses Soft Collinear Effective Theory to resum large phase-space logarithms in Sterman-Weinberg dijet decays of the Z boson. By constructing a two- and three-jet operator basis, performing renormalization, and applying a Phase Space Renormalization Group, it connects jet definitions in SCET to the full SW cuts and produces a resummed dijet rate that accounts for both collinear and ultrasoft radiation. A phase-space Wilson coefficient S2 and explicit running of the two- and three-jet operators enable a complete leading-log resummation, with mixing between O2 and O3 key to capturing Sudakov double logs. The framework clarifies how to define jets in SCET and sets the stage for higher-precision jet phenomenology within an effective-theory formalism.

Abstract

We demonstrate how to resum phase space logarithms in the Sterman-Weinberg (SW) dijet decay rate within the context of Soft Collinear Effective theory (SCET). An operator basis corresponding to two and three jet events is defined in SCET and renormalized. We obtain the RGE of the two and three jet operators and run the operators from the scale $μ^2 = Q^2$ to the phase space scale $ μ^2_δ= δ^2 Q^2$. This phase space scale, where $δ$ is the cone half angle of the jet, defines the angular region of the jet. At $ μ^2_δ$ we determine the mixing of the three and two jet operators. We combine these results with the running of the two jet shape function, which we run down to an energy cut scale $μ^2_β$. This defines the resumed SW dijet decay rate in the context of SCET. The approach outlined here demonstrates how to establish a jet definition in the context of SCET. This allows a program of systematically improving the theoretical precision of jet phenomenology to be carried out.

Figures (4)

• Figure 1: The imaginary part of the forward scattering amplitude in QCD gives the $q \, \bar{q}$ final state. The operator insertion is the weak neutral current $j_{nc} = \bar{q} \, \Gamma^{\sigma} \, q \, \epsilon_{\sigma}$ with $\Gamma^{\sigma} = \gamma^{\sigma} \, (g_V + g_A \, \gamma_5)$. The wavefunction renormalizaton graphs are not drawn as Landau gauge can be used so that the wavefunction contributions vanish. The sum of imaginary part of the diagrams is gauge independent at ${\cal{O}}(\alpha_s)$.
• Figure 2: The three body $q\, \bar{q} \, g$ contributions.
• Figure 3: One gluon external state diagrams for the operators $O_i$ in the effective theory. Collinear gluons are drawn as springs with lines, usoft gluons as springs.
• Figure 4: The mixing of the three and two jet operators.