Reconstructing Toponium using Recursive Jigsaw Reconstruction

This paper proposes a method using Recursive Jigsaw Reconstruction, enhanced by two new variables, to effectively reconstruct the toponium bound state near the $\ttbar$ threshold region and improve sensitivity to its signal based on recent ATLAS and CMS findings.

The Gist

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

The Mystery: The "Ghost" Couple

In this racetrack, scientists smash protons together to create heavy particles called top quarks. Usually, these top quarks are like lonely runners; they are created, they zip around for a split second, and then they vanish into other particles.

However, the ATLAS and CMS experiments (two giant teams of detectives) noticed something strange. Sometimes, instead of running away alone, two top quarks seem to hold hands and form a temporary, ghostly couple called Toponium. It's like a "quasi-bound state"—a couple that dances together for a tiny moment before breaking up.

The Problem:
When this Toponium couple breaks up, it turns into two "W" particles, which then turn into leptons (like electrons or muons) and neutrinos.

• The leptons are easy to see; they leave a trail.
• The neutrinos are the ultimate ghosts. They pass right through the detectors without leaving a single trace.

Because the neutrinos are missing, the detectives can't see the whole picture. It's like trying to figure out the speed of a car crash by only seeing the broken headlights, while the rest of the car has vanished into thin air. This makes it very hard to prove that the Toponium couple actually existed.

The Solution: The "Recursive Jigsaw"

To solve this, the authors of this paper (Aman, Amelia, and Paul) brought in a special tool called Recursive Jigsaw Reconstruction.

Think of the collision event as a giant, shattered jigsaw puzzle.

1. The Pieces: You have the visible pieces (the leptons and jets) and you know there are invisible pieces (the neutrinos) hiding somewhere.
2. The Rules: The "Jigsaw" method is a set of strict rules about how these pieces must fit together based on the laws of physics (like conservation of energy and momentum).
3. The Recursive Part: The computer tries to fit the pieces together. If a piece doesn't fit, it tries a different angle. It does this over and over again (recursively), building a complete picture of the crash from the fragments.

They tested four different ways to solve this puzzle (Methods A, B, C, and D). They found that Method A was the best, like finding the one specific way to assemble the puzzle that makes the picture look perfectly clear and consistent.

The Secret Weapons: Two New Clues

Even with the best puzzle solver, it's still hard to tell the difference between a "Toponium couple" and just two random top quarks that happened to be near each other. So, the team invented two new "clues" (variables) to help separate the signal from the noise:

1. The "Angle of the Dance" ($\Delta\phi$):
   Imagine the two top quarks are dancing. Do they spin in perfect sync, or are they moving in opposite directions? This variable measures the angle between them. Toponium tends to dance in a very specific, tight pattern compared to random top quarks.

2. The "Handshake Strength" ($N_{chel}$):
   This is a bit more complex. Imagine you are watching the dance from a moving train (a specific frame of reference). You look at how the two dancers are leaning toward or away from each other. This variable measures the "handshake" between the particles. Toponium has a very strong, specific handshake that random pairs don't have.

The Big Reveal

The team took millions of simulated crash events and applied their Jigsaw puzzle solver and their two new clues. They divided the data into nine different "zones" based on the angle and the handshake strength.

The Result:
They found one "Golden Zone" where the Toponium signal was incredibly loud and clear. In this specific zone:

• The angle was almost straight (they were dancing in sync).
• The handshake was strong.

When they looked at this zone, the evidence for Toponium was overwhelming. They calculated a "significance" score of 15.3 sigma.

What does 15.3 sigma mean?
In the world of science, a "5 sigma" result is the gold standard for claiming a discovery (it means there's only a 1 in 3.5 million chance it's a fluke). A 15.3 sigma result is like winning the lottery every single day for a year. It is an almost mathematical certainty that the Toponium couple exists.

The Takeaway

This paper shows that by using a clever "Jigsaw" method to reconstruct missing pieces and focusing on two specific "dance moves" (angle and handshake), scientists can finally see the elusive Toponium particle clearly. It's a new, powerful way to understand the fundamental building blocks of our universe.

Technical Summary

1. Problem Statement

Recent data from the ATLAS and CMS experiments at the Large Hadron Collider (LHC) suggest an excess of events near the top-quark pair ($t\bar{t}$) production threshold in the dileptonic decay channel. This excess is consistent with the existence of a toponium bound state (a spin-0 pseudo-scalar $t\bar{t}$ system).

However, reconstructing the $t\bar{t}$ dileptonic final state is technically challenging due to:

• Missing Energy: The presence of two neutrinos in the final state ($t\bar{t} \to b\bar{b}W^+W^- \to b\bar{b}\ell^+\nu\ell^-\bar{\nu}$) which escape detection.
• Combinatorial Ambiguity: Difficulty in correctly pairing the two $b$-jets with the two leptons to reconstruct the parent top quarks.
• Current Limitations: Existing reconstruction methods (Ellipse Method by ATLAS, Sonnenschein Method by CMS) struggle to optimally distinguish the toponium signal from the dominant $t\bar{t}$ background.

2. Methodology

The authors propose a novel reconstruction strategy utilizing Recursive Jigsaw Reconstruction (RJR) implemented via the RestFrames package.

A. Simulation and Pre-selection

• Simulation: Monte Carlo samples were generated for LHC Run 3 conditions ($\sqrt{s} = 13.6$ TeV, $L = 300 \text{ fb}^{-1}$).
  • Signal: Toponium samples generated using Non-Relativistic QCD (NRQCD) Green's function reweighting in MadGraph5_aMC@NLO and Pythia8.
  • Background: $t\bar{t}$ background generated using MadGraph5_aMC@NLO, MadSpin, and Pythia8.
  • Parameters: Top mass set to 173 GeV, width $\Gamma_t = 1.49$ GeV.
• Event Selection: Events were pre-selected requiring:
  • Two oppositely charged leptons ($ee, \mu\mu, e\mu$) with $p_T > 25$ GeV and $|\eta| < 2.5$.
  • At least two $b$-jets with $p_T > 25$ GeV and $|\eta| < 2.5$.
  • Separation $\Delta R(b\text{-jet}, \ell) > 0.4$.

B. Recursive Jigsaw Reconstruction (RJR)

The RJR algorithm constructs a decay tree and applies kinematic constraints recursively to solve for the momenta of invisible particles (neutrinos). The authors tested four constraint methods:

1. Method A: $M_{top}^a = M_{top}^b$ (Equal reconstructed top masses).
2. Method B: $M_W^a = M_W^b$ (Equal reconstructed W masses).
3. Method C: Minimize $\Sigma M_{top}^2$.
4. Method D: Minimize $\Delta M_{top}$.

Selection: Method A was identified as optimal because it yielded the most consistent top quark mass distribution.

C. Novel Discriminating Variables

To enhance sensitivity to the toponium signal, the authors introduced two specific variables:

1. $\Delta\phi(t\bar{t})$: The difference in the azimuthal angle between the two reconstructed top quarks.
2. $N_{chel}$: A modified version of the "chel" variable. It is defined as the scalar product of the lepton momenta, calculated by:
   • Boosting the lepton to the $t\bar{t}$ Center-of-Mass (CoM) frame.
   • Subsequently boosting it to the parent top-quark frame.
   • Note: Unlike previous methods, the top quark is not boosted to the CoM frame in this specific procedure.

3. Key Contributions

• Application of RJR to Toponium: This is the first application of the Recursive Jigsaw Reconstruction method specifically for distinguishing toponium bound states from $t\bar{t}$ continuum background in the dileptonic channel.
• Optimization of Reconstruction: Systematic comparison of four RJR constraint methods, establishing Method A ($M_{top}^a = M_{top}^b$) as the superior approach for this physics case.
• New Phase Space Variables: The introduction and validation of $\Delta\phi(t\bar{t})$ and $N_{chel}$ as highly effective discriminators.
• Binning Strategy: The definition of a 9-bin phase space grid based on the correlation of the two new variables to isolate the signal region.

4. Results

• Reconstruction Performance: Method A provided the sharpest invariant mass distribution ($M_{t\bar{t}}$) for the signal.
• Variable Discrimination:
  • Truth-level and reconstructed distributions showed distinct separation between toponium and $t\bar{t}$ background for both $\Delta\phi(t\bar{t})$ and $N_{chel}$.
  • Correlation plots (Figure 6) revealed that toponium events cluster differently compared to background events in the 2D phase space.
• Optimal Region: By analyzing the 9 regions defined by $\Delta\phi(t\bar{t}) \in \{-6, -2, 2, 6\}$ and $N_{chel} \in \{-1, -0.4, 0.4, 1\}$, the authors identified an optimal signal region:
  • $\Delta\phi(t\bar{t}) \in [-2, 2]$
  • $N_{chel} \in [0.4, 1]$
• Significance: In this optimal region, the analysis achieved a statistical significance of 15.3$\sigma$ (calculated as $S/\sqrt{S+B}$), assuming no systematic uncertainties or detector resolution effects.

5. Significance

The paper demonstrates that Recursive Jigsaw Reconstruction, combined with the novel variables $\Delta\phi(t\bar{t})$ and $N_{chel}$, offers a powerful alternative to existing reconstruction techniques. The method significantly improves the ability to isolate the toponium bound state signal from the overwhelming $t\bar{t}$ background. This approach could provide crucial insights into the physics phenomenology of the $t\bar{t}$ threshold region, potentially confirming the existence of toponium and shedding light on the nature of the strong interaction near the top-quark mass scale.