Analysing Toponium at the LHC using Recursive Jigsaw Reconstruction

This paper proposes using Recursive Jigsaw Reconstruction within the MadGraph5_aMC@NLO framework to resolve combinatorial ambiguities and reconstruct toponium states at the LHC's $t\bar{t}$ threshold, demonstrating that specific angular variables can enhance signal sensitivity by up to 15% compared to current strategies.

The Gist

Imagine you are a detective trying to solve a very specific crime scene at a massive, high-speed racetrack called the Large Hadron Collider (LHC).

The Crime Scene: The "Toponium" Mystery

Recently, scientists at the LHC (specifically the ATLAS and CMS teams) noticed something strange. When they smash protons together at incredible speeds, they sometimes see a pair of heavy particles called top quarks behaving like they are holding hands. Instead of just flying apart, they seem to form a temporary, ghostly "dance couple" right before they break up. Scientists call this couple Toponium.

Finding this couple is tricky because:

1. They are shy: When the top quarks break up, they turn into other particles, including two neutrinos. Neutrinos are like invisible ghosts; they pass right through detectors without leaving a trace.
2. The mess: The collision creates a chaotic pile of debris (jets of particles and leptons). Because the neutrinos are missing, it's hard to know exactly how the original top quarks were moving or how they were paired up. It's like trying to figure out how two dancers were moving just by looking at the confetti they left behind, while knowing two of the dancers vanished into thin air.

The Old Tools vs. The New Tool

Previously, detectives used standard geometric tricks (like drawing ellipses) to guess where the missing ghosts were. But these methods sometimes got confused, mixing up which particle belonged to which top quark.

In this paper, the authors (Aman, Amelia, and Paul) propose a new, smarter detective tool called Recursive Jigsaw Reconstruction.

The Analogy: The Jigsaw Puzzle
Imagine the collision event is a giant, broken jigsaw puzzle.

• The Pieces: You have visible pieces (the b-jets and leptons) and invisible pieces (the neutrinos).
• The Problem: You don't know which visible piece goes with which invisible piece, and you don't know the shape of the invisible pieces.
• The Solution: The "Jigsaw" method uses a set of logical rules (like "this piece must fit with that piece because of how heavy they are") to try every possible combination. It builds a "decay tree" (a family tree of the particles) and recursively tests different scenarios until it finds the one that makes the most physical sense.

They tested four different "rules" for solving this puzzle and found that Method A (which assumes the two top quarks have the same mass) worked the best. It was like finding the one key that unlocked the correct picture of the event.

The New Strategy: Finding the Signal in the Noise

Once they could reconstruct the events better, they needed a way to tell the difference between:

• The Background: Normal top quark pairs that just happen to be near each other (like random people bumping into each other in a crowd).
• The Signal: The special "Toponium" couple (like a couple dancing a specific, synchronized routine).

The old strategy used two specific angles to spot the signal. The authors proposed two new variables (new ways of measuring angles and relationships between the particles) that act like a better magnifying glass.

• Variable 1 ($N_{chel}$): A modified version of an old angle measurement, but calculated from a different "perspective" (a different reference frame).
• Variable 2 ($\Delta\phi$): The difference in the "compass direction" (azimuthal angle) of the two reconstructed top quarks.

The Results: A 15% Boost

When they applied these new tools to their simulated data (which mimics the LHC's Run 3 conditions):

• The old method could spot the Toponium signal with a confidence level of about 12.4 sigma (a statistical way of saying "we are almost 100% sure this is real").
• The new method, using the Jigsaw reconstruction and the new variables, boosted this confidence to 15.5 sigma.

What does this mean?
It's like upgrading from a standard flashlight to a high-powered laser. The new method makes the "Toponium" signal shine much brighter against the background noise. The authors estimate this improves the sensitivity by 15%.

Why Does This Matter?

If we can see Toponium more clearly, we can learn more about the fundamental forces of nature. It helps us understand how the heaviest known particle (the top quark) interacts with itself. It's a step toward understanding the "glue" that holds the universe together at the smallest scales.

In summary:
The authors invented a smarter way to solve a particle physics puzzle. By using a "Jigsaw" algorithm to reconstruct missing pieces and new angles to measure the dance of the particles, they can spot a rare, ghostly particle couple (Toponium) much more easily than before. This helps physicists get closer to confirming the existence of this exotic state of matter.

Technical Summary

1. Problem Statement

Recent results from the ATLAS and CMS experiments at the Large Hadron Collider (LHC) indicate an excess of events near the top-quark pair ($t\bar{t}$) production threshold in the dilepton decay channel. This excess is consistent with the existence of toponium, a spin-0 pseudoscalar quasi-bound state.

Reconstructing this signal is technically challenging due to two primary factors:

1. Missing Energy: The dilepton decay mode ($t\bar{t} \to b\bar{b}\ell^+\nu\ell^-\bar{\nu}$) involves two neutrinos that escape detection. Their presence is only inferred via missing transverse momentum ($E_T^{miss}$), and their longitudinal momentum components are unknown due to the unknown initial parton energy in proton-proton collisions.
2. Combinatorial Ambiguity: The final state contains two $b$-jets and two oppositely charged leptons. Determining which $b$-jet pairs with which lepton to reconstruct the parent top quarks introduces significant combinatorial ambiguity.

Existing methods, such as the Ellipse Method (ATLAS) and the Sonnechain Method (CMS), rely on analytical or geometric solutions but may not fully optimize the discrimination between the toponium signal and the Standard Model (SM) $t\bar{t}$ background.

2. Methodology

The authors propose a new analysis strategy utilizing Recursive Jigsaw Reconstruction (RJR), implemented via the RestFrames package.

A. Simulation and Event Generation

• Signal (Toponium): Generated using MadGraph5_aMC@NLO with Non-Relativistic QCD (NRQCD) Green's function re-weighting to simulate the bound state near the $t\bar{t}$ threshold ($M_{t\bar{t}} \approx 320\text{--}360$ GeV).
• Background (SM $t\bar{t}$): Generated at Next-to-Leading Order (NLO) in QCD, decayed via MadSpin, and showered/hadronized using Pythia8.
• Configurations: Simulations cover LHC Run 2 ($\sqrt{s}=13$ TeV, $140 \text{ fb}^{-1}$) and Run 3 ($\sqrt{s}=13.6$ TeV, $300 \text{ fb}^{-1}$).
• Pre-selection: Events require two oppositely charged leptons ($e, \mu$) and two $b$-jets with $p_T > 25$ GeV and $|\eta| < 2.5$.

B. Reconstruction Strategy

The RJR algorithm constructs a decay tree ($t\bar{t} \to bW(\ell\nu)bW(\ell\nu)$) and applies "jigsaw rules" to resolve ambiguities:

1. Pairing: The algorithm pairs leptons and $b$-jets by minimizing a function of invariant masses ($f = \min\{M^2_{\ell a, ba} + M^2_{\ell b, bb}\}$).
2. Neutrino Splitting: The missing transverse momentum (dineutrino system) is split into two neutrinos based on specific kinematic constraints. The authors tested four distinct methods:
   • Method A: $M_{top}^a = M_{top}^b$ (Equal reconstructed top masses).
   • Method B: $M_W^a = M_W^b$ (Equal reconstructed W masses).
   • Method C: Minimize $\sum M_{top}^2$.
   • Method D: Minimize $|\Delta M_{top}|$.
   • Result: Method A was identified as optimal for reconstructing consistent top quark masses.

C. New Variables for Discrimination

The authors introduce two new angular variables to enhance signal sensitivity, replacing the traditional $c_{hel} \otimes c_{han}$ binning strategy:

1. $\Delta\phi(t, \bar{t})$: The difference in azimuthal angle between the reconstructed top and anti-top quarks.
2. $N_{chel}$: A modified version of the helicity variable $c_{hel}$. Unlike the standard $c_{hel}$, which boosts leptons to the $t\bar{t}$ center-of-mass frame, $N_{chel}$ is calculated without boosting the individual top quarks to the $t\bar{t}$ frame, altering the kinematic sensitivity.

3. Key Contributions

• Application of RJR to Toponium: This is the first application of the Recursive Jigsaw Reconstruction method specifically to the search for the toponium bound state in the dilepton channel.
• Optimization of Reconstruction: Systematic comparison of four RJR methods, establishing that equalizing reconstructed top masses (Method A) yields the best kinematic resolution.
• Novel Kinematic Variables: Proposal and validation of $N_{chel}$ and $\Delta\phi(t, \bar{t})$ as superior discriminators compared to the standard $c_{hel} \otimes c_{han}$ approach used in previous ATLAS/CMS analyses.
• Phase Space Optimization: Definition of a 9-region phase space (Cartesian product of bins for the new variables) to maximize signal significance.

4. Results

The analysis was performed using pre-selection criteria and stricter event selection (vetoing low-mass dileptons, Z-boson regions, and requiring $E_T^{miss} > 40$ GeV).

• Baseline Strategy ($c_{hel} \otimes c_{han}$):
  • In the optimal region for Run 3, the significance reached 12.4$\sigma$.
• Proposed Strategy ($N_{chel} \otimes \Delta\phi(t, \bar{t})$):
  • In the optimal region ($N_{chel} \in (0.4, 1)$ and $\Delta\phi(t, \bar{t}) \in (-2, 2)$), the significance reached 15.5$\sigma$ (Run 3).
  • This represents a ~15% improvement in sensitivity over the current strategy.
• Run 2 Comparison: The proposed method also showed improvement for Run 2 conditions, achieving 10.2$\sigma$ compared to lower values with the baseline method.
• Optimization: Further optimization of the $N_{chel}$ cut (specifically $N_{chel} > 0.0$) after applying strict event selection yielded a significance of 14.2$\sigma$, still outperforming the baseline.

5. Significance

• Enhanced Discovery Potential: The proposed method significantly increases the statistical significance of the toponium signal, potentially allowing for a more definitive observation or tighter constraints on the existence of this quasi-bound state.
• Physics Insight: The ability to access kinematic variables in the rest frames of intermediate particles provides new tools to study the spin and entanglement properties of the $t\bar{t}$ system, distinguishing the spin-0 toponium from the spin-1 gluon-mediated SM background.
• Future Applications: The Recursive Jigsaw Reconstruction framework offers a robust, scalable approach for analyzing complex final states with missing energy and combinatorial ambiguities, promising improved discrimination for other new physics searches at the LHC.

In conclusion, the paper demonstrates that combining Recursive Jigsaw Reconstruction with novel angular variables ($N_{chel}$ and $\Delta\phi$) offers a powerful new strategy for isolating the toponium signal, potentially resolving the current excess observed at the LHC with higher confidence.