Resummed azimuthal decorrelation and transverse momentum imbalance of dijets at the LHC

This paper presents a theoretical study of azimuthal decorrelation and transverse momentum imbalance in LHC dijet production using the recoil-free winner-take-all scheme, deriving factorization formulae in soft-collinear effective theory to perform high-precision resummation of global and non-global logarithms while demonstrating the observables' robustness against non-perturbative effects.

The Gist

Imagine you are at a massive, chaotic dance party (the Large Hadron Collider, or LHC). Two people (protons) crash into each other at incredible speeds. Usually, when they crash, they spit out two energetic dancers (jets of particles) who immediately start spinning away in exactly opposite directions, like a perfect pair of ice skaters pushing off each other.

In a perfect, frictionless world, these two dancers would stay perfectly aligned, 180 degrees apart. But this is a real party. There are other people bumping into them, wind blowing, and music vibrating the floor. These little nudges cause the dancers to wobble, drift, or lose a bit of their balance.

This paper is about measuring exactly how much those dancers wobble and how unbalanced they get, and using those measurements to understand the invisible "wind" (Quantum Chromodynamics, or QCD) that pushes them around.

Here is the breakdown of their discovery, translated into everyday language:

1. The Problem: The "Messy" Dance Floor

Physicists want to measure two things:

• The Wobble (Azimuthal Decorrelation): How far off the perfect 180-degree line are the dancers?
• The Imbalance (Transverse Momentum): If one dancer leans left, does the other lean right to compensate, or is there a net push to the side?

The problem is that the "wind" (soft radiation) is tricky. In standard physics calculations, trying to predict these wobbles is like trying to calculate the path of a leaf in a hurricane while also accounting for every single person bumping into the leaf. The math gets so messy with "non-global logarithms" (NGLs)—which are like complex, tangled feedback loops—that the predictions usually break down or become very imprecise.

2. The Solution: The "Winner-Take-All" Rule

To fix this, the authors introduced a special rule for defining who the "dancers" actually are. Instead of averaging the position of all the particles in a jet (which is like averaging the positions of a whole crowd to find the center), they used a Winner-Take-All (WTA) scheme.

The Analogy: Imagine a group of people trying to decide which way to walk.

• Standard Method: Everyone votes, and you take the average direction. If a few people get bumped by the wind, the average direction shifts slightly.
• Winner-Take-All Method: You only listen to the loudest, strongest person in the group. If the wind blows the weaker people around, the loudest person doesn't care; they keep walking straight.

By using this "loudest voice" rule, the authors found that the messy "feedback loops" (NGLs) that usually ruin the math disappear completely for the "Wobble" measurement. It's like putting on noise-canceling headphones; the chaotic background noise vanishes, leaving a clear signal.

3. The Twist: The "Small Radius" Puzzle

However, when they looked at the "Imbalance" (how much the dancers drift sideways), the noise-canceling headphones didn't work perfectly if the dancers were very close together (a small "jet radius").

In this specific scenario, the "wind" creates a new kind of mess, but only in a tiny corner of the dance floor. The authors had to invent a new way to look at the problem. They realized they needed to split the "wind" into three different layers:

1. Global Wind: The general breeze affecting the whole room.
2. Collinear-Soft Wind: A gentle breeze hugging the dancers' legs.
3. Ultra-Collinear-Soft Wind: A microscopic draft right at the dancers' toes.

By separating the wind into these three layers, they could calculate the "Imbalance" with high precision, even in that tricky small-radius corner.

4. The Prediction vs. Reality

The authors used this new, cleaner math to predict exactly what the dancers should do. They then compared their predictions to a computer simulation (PYTHIA 8), which acts like a virtual reality version of the dance party.

The Result:

• Their predictions matched the simulation almost perfectly.
• They found that the "dancers" are surprisingly robust. Even when you add in the messy effects of the dancers turning into real particles (hadronization) or bumping into other groups (multi-parton interactions), the "Wobble" and "Imbalance" measurements stay stable.
• It's like saying, "Even if the floor is slippery and people are shoving, if we use the 'Winner-Take-All' rule to pick our dancers, we can still predict their path with incredible accuracy."

Why Does This Matter?

This isn't just about dancing.

• Calibration: It helps physicists calibrate their detectors better, ensuring they measure energy correctly.
• Probing the Proton: These measurements help us see the "3D map" of the inside of a proton, revealing how quarks and gluons move.
• New Physics: By understanding the "standard" wind so perfectly, if we ever see a dancer doing something weird that our math can't explain, we might have discovered a new force of nature or a new particle.

In short: The authors found a clever way to ignore the noise in a chaotic system, allowing them to predict the behavior of high-energy particle collisions with unprecedented precision. They turned a messy, tangled knot of math into a clean, straight line.

Technical Summary

1. Problem Statement

The study addresses the theoretical prediction of two key observables in inclusive dijet production at the Large Hadron Collider (LHC):

• Azimuthal decorrelation ($\delta\phi$): The deviation from the back-to-back configuration of two jets in the azimuthal plane.
• Transverse momentum imbalance ($q_T$): The magnitude of the vector sum of the transverse momenta of the two jets.

Core Challenges:

• Large Logarithms: In the back-to-back limit ($\delta\phi \to 0$ or $q_T \to 0$), fixed-order perturbation theory breaks down due to large logarithms ($\ln(q_T/Q)$ or $\ln \delta\phi$). These require all-order resummation.
• Non-Global Logarithms (NGLs): Standard jet definitions (like the $E$-scheme) introduce NGLs. These arise from soft radiation sensitive to the jet boundary, creating complex factorization structures that have historically limited theoretical predictions to Next-to-Leading Logarithmic (NLL) accuracy.
• Small Jet Radius ($R \ll 1$) Complexity: For $q_T$, NGLs persist even with improved jet definitions in the small-$R$ limit, requiring a sophisticated treatment of soft radiation modes.

2. Methodology

The authors employ Soft-Collinear Effective Theory (SCET) combined with the Winner-Take-All (WTA) recombination scheme to define jet axes.

A. The WTA Scheme

Instead of the standard $E$-scheme, the jet axis is defined by the WTA scheme, which is recoil-free.

• Impact on $\delta\phi$: The WTA scheme completely eliminates NGLs from the factorization of azimuthal decorrelation.
• Impact on $q_T$: While WTA removes standard NGLs, NGLs re-emerge specifically in the small jet radius limit ($R \ll 1$) as jet-radius logarithms ($\ln R$).

B. Factorization and Refactorization

The paper derives distinct factorization theorems for the two observables:

1. For $\delta\phi$: A standard TMD-like factorization involving Hard, Beam, Jet, and Soft functions. The absence of NGLs allows for a clean resummation.
2. For $q_T$ (Small-$R$ limit): The soft function requires refactorization into three distinct modes to handle the interplay between global soft radiation and radiation near the jet boundary:
   • Global Soft: Independent of jet definition details.
   • Collinear-Soft (cS): Radiation near the jet boundary but with virtuality $q_y^2$.
   • Ultra-Collinear-Soft (ucs): Radiation localized near the jet boundary with virtuality $q_T^2 R^2$.

This refactorization allows the separation of global logarithms ($\ln(q_T/p_T)$) from the non-global $\ln R$ terms.

C. Resummation Accuracy

• Global Logarithms: Resummed to Next-to-Next-to-Leading Logarithmic (NNLL) accuracy for both observables.
• Non-Global Logarithms ($q_T$ only): The $\ln R$ terms arising from the cS/ucs interplay are resummed to Leading Logarithmic (LL) accuracy using a multiplicity-mixing evolution matrix (approximated via the Dasgupta-Salam parametrization).

D. Matching and Power Corrections

• Fixed-Order Matching: The resummed results are matched to Leading Order (LO) fixed-order calculations (generated via NLOJET++) using a modified additive matching scheme.
• Power Corrections: The authors investigate the structure of subleading power corrections. They find that for these jet-dependent observables, power corrections behave as $O(\ln O)$ (where $O$ is $\delta\phi$ or $q_T$) rather than vanishing as $O \to 0$, which is distinct from inclusive observables like Drell-Yan.
• Non-Perturbative Effects: A non-perturbative Sudakov form factor is included, and results are compared with PYTHIA 8 simulations to assess hadronization and Multi-Parton Interaction (MPI) effects.

3. Key Contributions

1. Extension to Dijets: Successfully extends the WTA-based resummation framework from Vector Boson + Jet ($V+$jet) processes to the more complex inclusive dijet sector, which involves a larger number of partonic channels and non-trivial color structures.
2. Refactorization of $q_T$ Soft Function: Provides a rigorous derivation of the $q_T$ soft function refactorization in the small-$R$ limit, isolating the origin of NGLs to the jet radius logarithms within the soft sector.
3. NNLL Precision: Achieves NNLL accuracy for $\delta\phi$ and a joint NNLL (global) + LL (NGL) accuracy for $q_T$, significantly improving upon previous NLL limitations.
4. Linearly Polarized Gluons: Clarifies that while linearly polarized gluons contribute at NLO for $V+$jet, their impact in the dijet case only emerges at NNLO, justifying their omission at the current accuracy level.
5. Power Correction Analysis: Identifies and fits the specific logarithmic structure of non-singular power corrections ($aO + bO \ln O$) and proposes an effective evolution matrix to stabilize the matching in the deep infrared region.

4. Results

• Theoretical Uncertainty: The transition from NLL to NNLL results in a substantial reduction in theoretical scale uncertainties for both $\delta\phi$ and $q_T$. The NNLL bands lie consistently within the NLL envelopes, demonstrating excellent perturbative convergence.
• NGL Impact: For $q_T$, the resummation of NGLs (via the $U_{NG}$ factor) has a measurable but relatively mild numerical impact on the distribution shape for $R=0.5$.
• Comparison with PYTHIA 8:
  • The analytical predictions match PYTHIA 8 parton-level results well.
  • Crucially, the observables are found to be robust against non-perturbative effects. The differences between parton-level, hadron-level, and hadron-level+MPI PYTHIA results are negligible, indicating that $\delta\phi$ and $q_T$ (defined with WTA axes) are dominated by perturbative QCD dynamics.
• Matching Stability: The modified matching scheme, utilizing an effective evolution matrix and a transition function, successfully removes unphysical Sudakov effects in the tail of the distribution while stabilizing the infrared region against divergent power corrections.

5. Significance

• Precision Phenomenology: This work provides a high-precision theoretical baseline for LHC dijet measurements, essential for testing QCD dynamics and constraining Parton Distribution Functions (PDFs).
• TMD Extraction: The robustness of these observables against non-perturbative effects makes them ideal candidates for extracting Transverse-Momentum-Dependent (TMD) parton distribution functions, particularly in polarized proton-proton collisions where single-spin asymmetries are studied.
• Methodological Advancement: The successful application of the WTA scheme and the refactorization of soft functions for small-$R$ jets offers a template for analyzing a wide class of jet substructure observables where NGLs have traditionally been a bottleneck.
• Future Applications: The techniques developed here, particularly the handling of small-$R$ refactorization and the treatment of logarithmic power corrections, are directly applicable to future high-precision studies of jet physics and potential factorization-breaking effects.