Gaining (Mutual) Information about Quark/Gluon Discrimination

TL;DR

This work addresses the challenge of quark/gluon discrimination by reframing jet tagging with mutual information, separating true discrimination power from mere correlations. It introduces the generalized angularities ${\lambda^{\kappa}_{\beta}}$ to explore angular and energy-weighted radiation patterns across IRC-safe and IRC-unsafe regimes, and provides both parton-shower and analytic resummation insights. A key finding is that Casimir scaling at LL yields a universal truth overlap dependent on $C_A/C_F$, while NLL and nonperturbative effects (captured by weighted-energy functions) reveal meaningful differences and complementarities when combining observables. The study offers concrete predictions and methodological guidance for optimizing quark/gluon tagging and suggests targeted experimental measurements to refine nonperturbative inputs.These results advance understanding of when adding observables improves discrimination, quantify the extent of information gain from pairing angularities, and connect perturbative predictions with nonperturbative inputs, providing a framework for robust quark/gluon tagging in jet substructure.

Abstract

Discriminating quark jets from gluon jets is an important but challenging problem in jet substructure. In this paper, we use the concept of mutual information to illuminate the physics of quark/gluon tagging. Ideal quark/gluon separation requires only one bit of truth information, so even if two discriminant variables are largely uncorrelated, they can still share the same "truth overlap". Mutual information can be used to diagnose such situations, and thus determine which discriminant variables are redundant and which can be combined to improve performance. Using both parton showers and analytic resummation, we study a two-parameter family of generalized angularities, which includes familiar infrared and collinear (IRC) safe observables like thrust and broadening, as well as IRC unsafe variants like $p_T^D$ and hadron multiplicity. At leading-logarithmic (LL) order, the bulk of these variables exhibit Casimir scaling, such that their truth overlap is a universal function of the color factor ratio $C_A/C_F$. Only at next-to-leading-logarithmic (NLL) order can one see a difference in quark/gluon performance. For the IRC safe angularities, we show that the quark/gluon performance can be improved by combining angularities with complementary angular exponents. Interestingly, LL order, NLL order, Pythia 8, and Herwig++ all exhibit similar correlations between observables, but there are significant differences in the predicted quark/gluon discrimination power. For the IRC unsafe angularities, we show that the mutual information can be calculated analytically with the help of a nonperturbative "weighted-energy function", providing evidence for the complementarity of safe and unsafe observables for quark/gluon discrimination.

Figures (21)

• Figure 1: Visualization of the space of observables ${\lambda^{\kappa}_{\beta}}$, which includes several well-known jet observables used in quark/gluon discrimination: the line $\kappa=1$ corresponds to the IRC safe angularities ${e_{\beta}}$, the origin $(\beta,\kappa)=(0,0)$ to multiplicity, and (0,2) to $p_T^D$. Here, "width" at (1,1) refers also to broadening and girth, and "mass" at (2,1) refers to jet-mass-squared divided by energy (i.e. thrust).
• Figure 2: Left: The mutual information $I(A;B)$ between observables $A$ and $B$ is visualized as the area of the shaded overlap region in information space. In keeping with the set-theoretic relation in Eq. (\ref{['eq:Hinequality']}), the region labelled $A$ has area $H(A)$, the region labelled $B$ has area $H(B)$, the union $A \cup B$ has area $H(A,B)$, and the intersection $A \cap B$ has area $I(A;B)$. Right: As a special case, we can consider the mutual information $I(T;A)$ between observable $A$ and the truth $T$ (i.e. the truth overlap).
• Figure 3: The mutual information $I(T;A,B)$ between observables $A$, $B$, and the truth $T$ are shown as the area of their respective intersection. Though the mutual information between $A$ and $B$ (i.e. their correlation) is the same for both figures, their overlap with the truth differs. In the left figure, the mutual information with the truth (shaded) is complementary, such that combining $A$ and $B$ increases the truth overlap. This is not so in the right figure.
• Figure 4: Parton shower study of quark/gluon discrimination for Pythia 8 (left) and Herwig++ (right). Top: quark/gluon discrimination power of ${\lambda^{\kappa}_{\beta}}$ as characterized by the truth overlap $I(T;{\lambda^{\kappa}_{\beta}})$. Bottom: improvement in discrimination power from supplementing multiplicity with ${\lambda^{\kappa}_{\beta}}$, $\Delta I(T;{\lambda^{0}_{0}} \to {\lambda^{0}_{0}},{\lambda^{\kappa}_{\beta}}) \equiv I(T;{\lambda^{0}_{0}},{\lambda^{\kappa}_{\beta}}) - I(T;{\lambda^{0}_{0}})$. The small solid boxes correspond to the dots indicated in Fig. \ref{['fig:lambdaspace']}, the wide dashed box indicates the IRC safe angularities $e_\beta$, and "LL" in light yellow indicates the result from Casimir scaling (i.e. $I(T;{\lambda^{\kappa}_{\beta}}) \simeq 0.1$ from Eq. (\ref{['eq:LLang_base']})).
• Figure 5: The quark/gluon truth overlap for an individual IRC safe angularity ${e_{\beta}}$ as a function of angular exponent $\beta$. The transverse momentum of the jets is $p_T > 400$ GeV and the jet radius is $0.6$. Left: comparing the NLL truth overlap to the baseline LL result. Right: comparing the Pythia 8 and Herwig++ samples.