Jet Shapes with the Broadening Axis

TL;DR

This work proposes the broadening axis as a recoil-free reference for jet observables, enabling a beta-independent factorization theorem that smoothly interpolates between thrust-like and broadening-like regimes. By exploiting SCET and soft/collinear factorization, the authors derive event-wide and jet-based angularity resummations to NLL and NLL' accuracy, show equality of anomalous dimensions with other recoil-free observables, and validate results against Monte Carlo and CAESAR. They also discuss practical recoil-free jet algorithms, notably the winner-take-all axis, and provide a historical perspective linking to spherocity and prior broadening analyses. The findings indicate that broadening-axis observables are robust, perturbatively tractable, and potentially advantageous for jet substructure studies at the LHC, with future work aimed at non-perturbative effects and broader phenomenological applications.

Abstract

Broadening is a classic jet observable that probes the transverse momentum structure of jets. Traditionally, broadening has been measured with respect to the thrust axis, which is aligned along the (hemisphere) jet momentum to minimize the vector sum of transverse momentum within a jet. In this paper, we advocate measuring broadening with respect to the "broadening axis", which is the direction that minimizes the scalar sum of transverse momentum within a jet. This approach eliminates many of the calculational complexities arising from recoil of the leading parton, and observables like the jet angularities become recoil-free when measured using the broadening axis. We derive a simple factorization theorem for broadening-axis observables which smoothly interpolates between the thrust-like and broadening-like regimes. We argue that the same factorization theorem holds for two-point energy correlation functions as well as for jet shapes based on a "winner-take-all axis". Using kinked broadening axes, we calculate event-wide angularities in e+e- collisions with next-to-leading logarithmic resummation. Defining jet regions using the broadening axis, we also calculate the global logarithms for angularities within a single jet. We find good agreement comparing our calculations both to showering Monte Carlo programs and to automated resummation tools. We give a brief historical perspective on the broadening axis and suggest ways that broadening-axis observables could be used in future jet substructure studies at the Large Hadron Collider.

Figures (14)

• Figure 1: Kinked broadening axes $\hat{b}_L$ and $\hat{b}_R$. While we will use the thrust axes $\hat{t}_L$ and $\hat{t}_R$ to partition the event into left ($H_L$) and right ($H_R$) hemispheres, we measure the angularities with respect to the broadening axis in each hemisphere.
• Figure 2: Hierarchy of modes in the effective theory for $\beta > 1$ (left) and $\beta < 1$ (right). Shown is the light-cone momentum plane $(n\cdot p, \bar{n}\cdot p)$ and the dots correspond to the hard ($H$), left-collinear ($J$), right-collinear ($\bar{J}$), and soft ($S$) modes. The curves indicate the natural invariant mass scale for the hard ($\mu_H$), jet ($\mu_J$), and soft ($\mu_S$) functions. For $\beta > 1$, the soft modes have smaller invariant mass than the collinear modes, but this hierarchy is inverted for $\beta < 1$. The arrows indicate the direction of renormalization group evolution if modes are factorized at the jet scale (as we will do in this paper).
• Figure 3: Relationship between the anomalous dimensions of observables $e_A$ and $e_B$ that share the same soft, small angle behavior (and the same hard function). The observable $e_{A'}$ ($e_{B'}$) is constructed to have the same soft behavior as $e_A$ ($e_B$), albeit with angles measured with respect to broadening axes. But this then implies that $e_{A'}$ and $e_{B'}$ have the same jet function (up to power corrections). By consistency of the anomalous dimensions, this means that $e_A$ and $e_B$ have the same jet and soft anomalous dimensions.
• Figure 4: Event-wide broadening ($\tau^{(1)}$, left) and thrust ($\tau^{(2)}$, right) in Pythia 8.165. We test three different axes choices: broadening axes, winner-take-all axes, and thrust axes. The effect of recoil is seen clearly in the left figure, where the recoil-sensitive thrust axes give a larger value of $\tau^{(1)}$ compared to the recoil-free axes. In the right figure, all the curves are quite similar in the Sudakov peak region, since the effect of recoil is power-suppressed for $\tau^{(2)}$.
• Figure 5: Same as Fig. \ref{['fig:mc_axes']}, but for the jet-based observable with $R= 0.8$. As expected, the qualitative effect of recoil is the same as for the event-wide observables.