Projected Energy Correlators: Two-Loop Jet Functions and NNLL Resummation

The Gist

Imagine you are at a massive, chaotic fireworks display. When a firework explodes, it sends sparks flying in all directions. Physicists call this an "event." For decades, scientists have been trying to understand the rules governing how these sparks fly, which helps them understand the fundamental forces of the universe (specifically, the strong force that holds atoms together).

This paper is about a new, highly precise way of measuring those fireworks.

The Old Way: Counting Two Sparks

Previously, scientists mostly looked at the Energy-Energy Correlator (EEC). Imagine you have two detectors, and you measure the angle between just two sparks. You ask: "How often do two sparks land at this specific angle?" This has been a classic tool for decades, like using a ruler to measure the width of a river. It's useful, but it only gives you a one-dimensional view of a very complex explosion.

The New Way: Measuring the Whole Shape

This paper introduces a more advanced tool called Projected N-point Energy Correlators. Instead of just looking at two sparks, imagine you are looking at a group of 3, 4, 5, or even 6 sparks at once.

The scientists don't just measure the angles between every single pair (which would be a messy, impossible calculation). Instead, they use a clever trick: they find the widest angle among that group of sparks and ignore the rest.

• The Analogy: Imagine a group of friends standing in a circle. Instead of measuring the distance between every pair of friends, you just measure the distance between the two friends who are standing the furthest apart.
• The Result: This simplifies the math while still capturing the complex "shape" of the explosion. The paper calculates these measurements for groups of up to 6 sparks (N=6) with extreme precision.

The "Two-Loop" Challenge: Fixing the Blurry Lens

In physics, calculations are done in layers of precision.

• Level 1 (LO): A rough sketch.
• Level 2 (NLO): A detailed drawing.
• Level 3 (NNLL): A high-definition, 3D model that accounts for tiny, invisible wiggles in the data.

To get to this "High-Definition" level (NNLL), the authors had to solve a massive mathematical puzzle called the two-loop jet function.

• The Metaphor: Imagine trying to predict exactly how a jet of water will spray out of a hose. At first, you just guess. Then you add wind speed. Finally, you have to account for the microscopic turbulence inside the hose itself.
• The Achievement: The authors calculated these "microscopic turbulence" rules for groups of 4, 5, and 6 sparks. This is the "secret sauce" that allows them to make predictions that are accurate enough to be trusted by experimentalists.

The "Fuzzy" Edge: When Math Meets Reality

There is a catch. The math works perfectly when the sparks are flying very close together (the "collinear" limit). But as they get further apart, the math starts to break down because of non-perturbative effects.

• The Analogy: Think of a smooth, mathematical curve representing a road. But as you get to the edge of the map, the road turns into a muddy, bumpy dirt path. The math can't describe the mud perfectly.
• The Solution: The authors added a "correction factor" (represented by $\Omega_1$) to account for this muddy, messy reality. They showed that as you look at groups of more sparks (higher N), this "muddy" part of the road starts to appear earlier in the measurement.

Why Does This Matter?

The paper claims two main things:

1. Precision Control: They have now brought these complex "multi-spark" measurements under strict mathematical control. They are no longer just guessing; they have a precise formula.
2. A New Tool for $\alpha_s$: One of the biggest mysteries in physics is the exact strength of the strong force (called $\alpha_s$). Different experiments give slightly different answers, causing a "tension" in the scientific community.
   • The authors show that by looking at these higher-point correlators (3, 4, 5, 6 sparks), they can extract the value of $\alpha_s$ with a different set of errors than previous methods.
   • The Metaphor: If you are trying to find the weight of a hidden object, you can weigh it on a scale (Method A), or you can measure how much it sinks in water (Method B). If both methods give the same answer, you are confident. If they disagree, you know something is wrong. This paper provides a brand new "scale" to weigh the strong force, helping scientists resolve the disagreement between different measurements.

Summary

The authors have built a new, ultra-precise mathematical microscope. They figured out how to measure the shape of particle explosions using groups of up to 6 particles, calculated the complex "noise" that usually ruins these measurements, and showed that this new method is a powerful way to test our understanding of the universe's fundamental forces. They compared their math to computer simulations (Pythia8 and Herwig7) and found that while the math works well for simple cases, the complex simulations still struggle to match the precision of these new formulas, suggesting the simulations need an upgrade.

Technical Summary

Technical Summary: Projected Energy Correlators: Two-Loop Jet Functions and NNLL Resummation

Problem Statement
Energy correlators (ENCs) are observables defined as correlation functions of energy flow operators that probe the underlying quark–gluon dynamics of Quantum Chromodynamics (QCD). While the two-point energy-energy correlation (EEC) has been computed to high orders and serves as a precision tool for determining the strong coupling constant $\alpha_s$, higher-point correlators ($N > 2$) offer a family of observables with complementary systematics. However, theoretical predictions for projected $N$-point energy correlators beyond $N=3$ have been limited to Leading Logarithmic (LL) or Next-to-Leading Logarithmic (NLL) accuracy. A critical bottleneck for achieving Next-to-Next-to-Leading Logarithmic (NNLL) precision for $N=4, 5, 6$ was the lack of the two-loop jet function, which encodes the specific $N$-point dependent collinear dynamics. Furthermore, a comprehensive understanding of these observables requires the inclusion of leading non-perturbative power corrections and their interplay with perturbative resummation.

Methodology
The authors employ the framework of QCD collinear factorization, where the cumulative distribution of the projected $N$-point energy correlator factorizes into a process-dependent hard function and a universal jet function. The hard function is governed by timelike DGLAP evolution, while the jet function satisfies a modified evolution equation dependent on the $N$-point measurement.

To achieve NNLL accuracy, the authors computed the missing two-loop jet function boundary constants ($c^{[N]}_{2,0}$) for $N=4, 5, 6$. The calculation was decomposed into two distinct sectors based on the measurement definition:

1. Contact Terms: These involve configurations where detectors are placed on fewer than $N$ particles (specifically $r=1, 2$ particles for the $N$-point observable). These terms were calculated analytically using Integration-by-Parts (IBP) reduction and differential equations. The authors utilized tools such as LiteRed, FIRE6, Canonica, and Libra to reduce master integrals and solve the resulting differential systems, expressing results in terms of harmonic and classical polylogarithms.
2. Three-Particle Terms: These correspond to configurations where detectors are placed on three distinct particles ($r=3$). In the collinear limit, these are generated by the double-real emission contribution. The authors computed these semi-analytically by factorizing the matrix element into a Born process and a $1 \to 3$ timelike splitting kernel. The phase space integrals were evaluated using a combination of analytic methods for the singular regions and Monte Carlo integration (using the Cuba library) for the finite parts, specifically handling "squeezed" singularities where two partons become collinear.

The authors matched these resummed predictions to fixed-order results:

• For $e^+e^- \to q\bar{q}$, they matched to NLO results obtained numerically from the Event2 dipole-subtraction program.
• For Higgs decay $H \to gg$, they matched to NLO results from Eerad3.
• Leading non-perturbative corrections, parameterized by universal soft matrix elements $\Omega_{1q}$ and $\Omega_{1g}$, were included. These corrections were incorporated via a jet boundary modification that relates the $N$-point jet function to the $(N-1)$-point jet function, governed by the same anomalous dimensions.

Key Contributions

• Two-Loop Jet Functions: The primary contribution is the semi-analytic computation of the two-loop jet function constants for projected $N$-point energy correlators with $N=4, 5, 6$ for both quark and gluon jets. These constants are provided as numerical values with Monte Carlo uncertainties.
• NNLL Resummation: The authors present the first NNLL collinear resummation for $N$-point projected energy correlators up to $N=6$, matched to NLO fixed-order predictions.
• Non-Perturbative Framework: The work includes a systematic treatment of leading power corrections ($\mathcal{O}(\Lambda_{\text{QCD}}/Q)$) for the entire family of $N$-point correlators, demonstrating how these corrections scale with $N$ and the process (quark vs. gluon).
• Validation: The calculated two-loop jet constants were cross-checked against known $N=2$ and $N=3$ results and verified against numerical fixed-order codes (Event2 and Eerad3) in the small-angle limit.

Results

• Perturbative Convergence: The matched distributions show good perturbative convergence from NLL to NNLL across most of the kinematic range. However, deviations appear at very small angles ($x_L$), where the perturbative expansion becomes sensitive to non-perturbative effects.
• Non-Perturbative Effects: The inclusion of leading non-perturbative corrections is essential to reproduce the behavior of parton-shower simulations (Pythia8 and Herwig7) in the small-angle region. The onset of non-perturbative dominance shifts to larger $x_L$ as $N$ increases, consistent with the scaling $N \Omega_{1\kappa} / (Q\sqrt{x_L})$.
• Process Dependence: Comparisons between $e^+e^-$ annihilation and $H \to gg$ decay reveal that the gluon channel transitions to the non-perturbative regime at larger angles than the quark channel, reflecting the larger value of the gluon soft matrix element $\Omega_{1g}$.
• Ratios: The paper analyzes ratios of $N$-point correlators to the EEC ($N=2$). These ratios cancel the classical $1/x_L$ scaling, isolating the anomalous scaling determined by twist-two operators. The logarithmic slopes of these ratios are found to be highly sensitive to $\alpha_s$, with sensitivity increasing with $N$.
• Comparison with Simulations: While the theoretical predictions with non-perturbative corrections agree well with Pythia8 and Herwig7 for $N \le 4$ in $e^+e^-$, significant discrepancies remain for $N > 4$ and for the $H \to gg$ channel. The authors attribute this partly to the lack of precision $H \to gg$ data for tuning parton showers.

Significance
The paper establishes that higher-point projected energy correlators are now under quantitative control at NNLL accuracy. This development opens the door to precision extractions of the strong coupling constant $\alpha_s$ using a family of observables with systematics complementary to traditional event shapes and lattice QCD. By providing the necessary theoretical ingredients (two-loop jet functions and NNLL resummation) for $N=4, 5, 6$, the work enables future phenomenological studies at $e^+e^-$ colliders and Higgs factories (such as FCC-ee and CEPC). The ability to study the $N$-dependence of these observables offers a unique handle on disentangling perturbative and non-perturbative physics, potentially helping to resolve longstanding tensions in $\alpha_s$ determinations. The authors also note the potential for extending these results to track-based observables and to the analytic continuation of the correlator to non-integer $N$.