Toward Multi-Differential Cross Sections: Measuring Two Angularities on a Single Jet

TL;DR

This work extends jet-physics beyond single observable cross sections by rigorously formulating the double differential cross section for two recoil-free angularities on a single jet. Using soft-collinear effective theory, it proves factorization at the phase-space boundaries but shows that a universal bulk factorization is not possible with only soft and collinear modes, motivating an interpolation between boundaries. The authors construct a next-to-leading-logarithmic (NLL) interpolation that enforces boundary constraints and introduces a novel k_T logarithmic structure in the bulk, arguing for near-uniqueness under exponentiation assumptions. Comparisons with Monte Carlo show qualitative agreement, supporting the relevance of the approach for correlations among observables and for observables built from ratios, such as N-subjettiness or energy correlation functions. The framework suggests broad applicability to phenomenology and provides a path toward systematic study of multi-differential cross sections in QCD.

Abstract

The analytic study of differential cross sections in QCD has typically focused on individual observables, such as mass or thrust, to great success. Here, we present a first study of double differential jet cross sections considering two recoil-free angularities measured on a single jet. By analyzing the phase space defined by the two angularities and using methods from soft-collinear effective theory, we prove that the double differential cross section factorizes at the boundaries of the phase space. We also show that the cross section in the bulk of the phase space cannot be factorized using only soft and collinear modes, excluding the possibility of a global factorization theorem in soft-collinear effective theory. Nevertheless, we are able to define a simple interpolation procedure that smoothly connects the factorization theorem at one boundary to the other. We present an explicit example of this at next-to-leading logarithmic accuracy and show that the interpolation is unique up to $α_s^4$ order in the exponent of the cross section, under reasonable assumptions. This is evidence that the interpolation is sufficiently robust to account for all logarithms in the bulk of phase space to the accuracy of the boundary factorization theorem. We compare our analytic calculation of the double differential cross section to Monte Carlo simulation and find qualitative agreement. Because our arguments rely on general structures of the phase space, we expect that much of our analysis would be relevant for the study of phenomenologically well-motivated observables, such as $N$-subjettiness, energy correlation functions, and planar flow.

Figures (8)

• Figure 1: Summary of the results of the factorization theorem of the double differential cross section of angularities. The factorization theorems exists near the boundaries of the allowed phase space where the double differential cross section reduces to the appropriate single differential cross section plus terms that integrate to 0. The bulk of the phase space is described by an interpolating function.
• Figure 2: The allowed phase space of the double differential cross section of angularities $e_\alpha$ and $e_\beta$. The angular exponent $\alpha$ is fixed to be $2$ and $\beta$ is varied. For a given value of $\beta$, the phase space consists of the respective shaded region and all shaded regions above.
• Figure 3: Illustration of the double cumulative distribution evaluated on the boundaries of phase space. Left: Evaluated on the boundary $e_\alpha^\beta = e_\beta^\alpha$ which reduces the double cumulative distribution to $\Sigma(e_\alpha)$. Right: Evaluated on the boundary $e_\alpha = e_\beta$ which reduces the double cumulative distribution to $\Sigma(e_\beta)$.
• Figure 4: Angularities phase space divided into boundary regions $\alpha$ and $\beta$ in which different factorization theorems live, defined by the soft modes' virtuality. The virtuality of the soft modes in region $\alpha$ ($\beta$) is $\lambda^{2\alpha}$ ($\lambda^{2\beta}$). The dividing line of the regions is $e_\alpha = e_\beta^\kappa$, where $\kappa\in[1,\alpha/\beta]$.
• Figure 5: Illustration of the interpolation of the logarithmic structure between the boundaries of the phase space. Collinear logarithms ($\log e_\beta^{1/\beta}$) always interpolate between the jet functions defined on the boundaries and soft logarithms ($\log e_\alpha$) always interpolate between the soft functions. $k_T$ logarithms ($\log e_\alpha^\frac{1-\beta}{\alpha-\beta} e_\beta^\frac{\alpha-1}{\alpha-\beta}$) interpolate between the double differential jet and soft functions.