Systematic Improvement of Parton Showers with Effective Theory

TL;DR

The paper develops a tower of Soft-Collinear Effective Theories (SCET_i) to systematically classify and compute power corrections to parton showers. By matching QCD to SCET_1 and iteratively descending to SCET_N, the authors separate hard-scattering and jet-structure corrections and derive operator-based replacement rules and reweighting formulas to implement NLO(lambda) accuracy in exclusive cross sections. They show how LO LL shower physics emerges from SCET_i, including LL Sudakovs and angular-ordered soft emissions, and provide detailed expressions for subleading corrections, including interference structure and RG running. Although the full NLL_exp resolution and nonabelian extensions are not complete, the framework offers a transparent path to incorporating higher-order kinematic and amplitude-level corrections into shower algorithms, with concrete steps toward practical MC implementation. The work thus enables principled, systematic improvements to parton showers beyond strict strong ordering, with clear connections to known splitting functions and Sudakov resummation.

Abstract

We carry out a systematic classification and computation of next-to-leading order kinematic power corrections to the fully differential cross section in the parton shower. To do this we devise a map between ingredients in a parton shower and operators in a traditional effective field theory framework using a chain of soft-collinear effective theories. Our approach overcomes several difficulties including avoiding double counting and distinguishing approximations that are coordinate choices from true power corrections. Branching corrections can be classified as hard-scattering, that occur near the top of the shower, and jet-structure, that can occur at any point inside it. Hard-scattering corrections include matrix elements with additional hard partons, as well as power suppressed contributions to the branching for the leading jet. Jet-structure corrections require simultaneous consideration of potential 1 -> 2 and 1 -> 3 branchings. The interference structure induced by collinear terms with subleading powers remains localized in the shower.

Figures (24)

• Figure 1: Different kinematic configurations of a final state with a quark, antiquark, and gluon are described by different SCET operators. In ( I), the quark and the gluon are collinear to the direction $n_0$, represented by their sharing a common cone. In ( II), the vectors $q'_1$ and $k'_1$ are too far apart to be collinear. The Feynman diagrams show that collinear particles can come from Lagrangian insertions, whereas non-collinear ones arise exclusively from higher-multiplicity operators. The Feynman diagram in ( I) only depicts the first term on the RHS of Eq. (\ref{['eq:AI']}).
• Figure 2: The same three-parton process as seen in two different SCETs, $\mathrm{SCET}_i$ and $\mathrm{SCET}_{i+1}$. Above: Kinematic configuration of the quarks and gluon. The solid cones represent the regions considered collinear to the vectors drawn. Below: Feynman diagrams for the corresponding amplitude. Note that in $\mathrm{SCET}_{i+1}$ we have removed a degree of freedom that propagates in $\mathrm{SCET}_i$. The amplitude thus comes from a higher dimension operator $\mathcal{O}^{(1)}_{i+1}$, rather than from a time-ordered product of $\mathcal{L}_{\mathrm{SCET}_i}$ with $\mathcal{O}^{(0)}_{i}$, as it did in $\mathrm{SCET}_i$.
• Figure 3: Operators that reproduce strongly-ordered gluons are constructed through a series of matching computations with emissions in different $\mathrm{SCET}_j$. The horizontal dashed arrows refer to the radiation of a gluon from a time-ordered product of the $\mathrm{SCET}_j$ Lagrangian with the operator creating fields at the point marked by $\otimes$. The diagonal solid arrows denote the matching onto a higher multiplicity operator in $\mathrm{SCET}_{j+1}$.
• Figure 4: The opening angle of the light grey (blue) cone is $\sim\lambda^{2i}$, and the opening angle of the dark grey (red) one is $\sim\lambda^{2(i+1)}$. The particle with momentum $p$ is collinear to both $n$ and $n'$ in $\mathrm{SCET}_i$, but only to $n'$ in $\mathrm{SCET}_{i+1}$. RPI$_i$ allows us to move the field label, $n$, to any location inside the appropriate cone for $\mathrm{SCET}_i$ while keeping the theory invariant.
• Figure 5: Momentum labels for single ($A$) and double ($B$) gluon emission.