Characterization of the quantum state of top quark pairs produced in proton-proton collisions at $\sqrt{s}$ = 13 TeV using the beam and helicity bases

Using 138 fb$^{-1}$ of 13 TeV proton-proton collision data collected by the CMS experiment, this study characterizes the quantum state of top quark-antiquark pairs by measuring spin correlations in the beam and helicity bases to decompose the system into Bell and spin eigenstates and evaluate properties like purity, entropy, and entanglement, with all results aligning with Standard Model predictions.

The Gist

The Big Picture: Catching the Ghosts Before They Vanish

Imagine you are at a massive, high-speed car race (the Large Hadron Collider at CERN). Two cars crash head-on, and in that split second, they create two incredibly heavy, unstable "ghost cars" called top quarks.

These ghost cars are special. They are so heavy and unstable that they vanish almost instantly—faster than a blink of an eye (about $10^{-25}$ seconds). In fact, they vanish so fast that they don't even have time to "wear a coat" (a process called hadronization that other particles do). Because they vanish before they can change their "mood," whatever spin or orientation they had when they were born is perfectly preserved in the debris they leave behind.

This paper is about a team of scientists (the CMS Collaboration) who acted like forensic detectives. They looked at the debris from 138 trillion collisions to figure out the quantum relationship between these two ghost cars. Specifically, they wanted to know: Are these two particles "entangled"?

What is "Entanglement"? (The Magic Dice Analogy)

In the quantum world, "entanglement" is like having two magic dice that are separated by the entire universe. If you roll one and it lands on a 6, the other one instantly becomes a 1, no matter how far away it is. They aren't just random; they are linked in a way that defies our normal understanding of cause and effect.

For a long time, physicists thought entanglement only happened with tiny, simple particles like electrons. This paper confirms that even the heaviest, most complex particles (top quarks) can be entangled.

The Two Ways to Look at the Spin

To understand how these particles are linked, the scientists had to look at them from two different angles, like looking at a spinning top from the side versus looking at it from above.

1. The Helicity Basis (The "Direction" View):
   Imagine looking at the top quarks as they fly away from the crash site. This view is great for seeing what happens when the cars are moving very fast (high energy). The scientists had already done this measurement before.

2. The Beam Basis (The "Head-On" View):
   This is the new discovery in this paper. Imagine looking at the collision straight down the tunnel where the proton beams are traveling. This view is like looking at the cars just as they are about to crash or just as they are barely moving apart. It gives a different perspective, especially when the particles are moving slowly (near the "production threshold").

The Analogy: Think of a spinning coin. If you look at it from the side (Helicity), you see the edge. If you look at it from the top (Beam), you see the face. Both views tell you the coin is spinning, but they reveal different details about how it is spinning relative to the table.

What Did They Find?

The scientists reconstructed the "quantum state" of these top quark pairs. Think of the quantum state as a recipe card that tells you exactly how the two particles are mixed together. They broke this recipe down into three main ingredients:

1. Purity (How "Pure" is the Mix?):
   Imagine a glass of water. If it's pure water, it's 100% water. If you add salt, sugar, and pepper, it's a "mixture."

   • The scientists found that in certain conditions (when the top quarks are moving fast), the "recipe" is very pure—it's almost entirely one specific type of quantum link.
   • In other conditions (when they are slow), the recipe is a messy mixture of different links.

2. Entropy (How Confused is the System?):
   Entropy is a measure of uncertainty.

   • Low Entropy: You know exactly what's happening (like a pure glass of water).
   • High Entropy: You are very confused about what's happening (like a messy mixture).
   • The data showed that when the top quarks are moving fast, the system is very "ordered" (low entropy). When they are slow, the system is more "chaotic" (high entropy).

3. Entanglement (The "Magic Link"):
   This is the big question. Are the two particles linked?

   • The Result: Yes! The scientists found strong evidence that the top quarks are entangled.
   • In the "fast" region (high energy), the evidence was overwhelming (more than 5 standard deviations—basically, a certainty).
   • In the "slow" region (near the crash threshold), they found evidence of entanglement too, though it was slightly less obvious (3 standard deviations).

Why Does This Matter?

You might ask, "Why do we care if two heavy particles are holding hands in the quantum world?"

1. Testing the Rules of the Universe: It proves that quantum mechanics (the weird rules of the tiny world) applies even to the heaviest particles we know. It's like proving that the laws of gravity work the same way on a pebble as they do on a mountain.
2. New Physics: If the measurements had not matched the Standard Model (our current best theory of physics), it would have been a huge discovery, suggesting there are new, unknown forces at play. Since they did match, it confirms our current theories are incredibly robust.
3. Future Tech: Understanding entanglement in complex systems is a stepping stone toward future technologies, like quantum computers.

The Bottom Line

The CMS team took a massive dataset of particle collisions and performed a delicate quantum autopsy. They confirmed that even the universe's heaviest particles, which live for a fraction of a nanosecond, can be "spooky action at a distance" partners. They mapped out exactly how these particles are linked, how "pure" their connection is, and how much "confusion" exists in their relationship.

Everything they measured matched the predictions of the Standard Model perfectly, meaning our current understanding of the universe is holding up strong, even in these extreme, high-speed crashes.

Technical Summary

1. Problem and Motivation

The top quark is the heaviest known fundamental particle with an extremely short lifetime ($\sim 10^{-25}$ s), which is shorter than the QCD hadronization time scale. Consequently, top quarks decay before their spin information can be scrambled by strong interactions. This unique property allows the spin correlations of top-antitop ($t\bar{t}$) pairs to be preserved in the angular distributions of their decay products.

While previous studies (including by ATLAS and CMS) have successfully observed quantum entanglement in $t\bar{t}$ systems using the helicity basis (optimal for high invariant masses), there was a need to characterize the quantum state in the beam basis. The beam basis provides complementary sensitivity, particularly near the production threshold where $gg \to t\bar{t}$ events are predominantly produced in a spin-singlet configuration. The goal of this paper is to extend the spin correlation measurements to the beam basis and use these results to decompose the $t\bar{t}$ system into quantum eigenstates, calculating purity, entropy, and entanglement markers to test Standard Model (SM) predictions.

2. Methodology

Data and Event Selection:

• Dataset: Proton-proton collision data collected by the CMS experiment at $\sqrt{s} = 13$ TeV during 2016–2018.
• Integrated Luminosity: $138 \text{ fb}^{-1}$.
• Final State: Events containing a $t\bar{t}$ pair reconstructed in the "lepton+jets" channel (one top decays leptonically to an electron or muon, the other hadronically).
• Reconstruction: A global particle-flow algorithm reconstructs all particles. The $t\bar{t}$ system is reconstructed, and kinematic variables are calculated.

Reference Frames and Definitions:

• Helicity Basis: The quantization axis is defined by the top quark direction in the $t\bar{t}$ rest frame. A discrete rotation of $\pi$ is applied if the scattering angle $\theta > \pi/2$ to account for Bose-Einstein symmetry of the initial gluon-gluon state.
• Beam Basis: The $z$-axis is aligned with the proton beam direction. The $x$-axis points to the center of the LHC ring, and the $y$-axis is vertical. No additional rotation is applied.
• Spin Correlation Matrix ($C_{ij}$):
  • In the helicity basis, all components are measured.
  • In the beam basis, due to symmetry, the matrix is diagonal with $C_{\perp} = C_{xx} = C_{yy}$ and $C_{\parallel} = C_{zz}$. Polarization coefficients are zero.

Analysis Technique:

• Template Fitting: The spin correlation coefficients are extracted by fitting the angular distributions of the decay products to templates generated from Monte Carlo (MC) simulations.
• Density Matrix ($\rho$): The system is described by a density matrix decomposed using the Fano–Bloch formalism:
  $$\rho = \frac{1}{4} \left( I \otimes I + \sum_i P_i \sigma_i \otimes I + \sum_j P_j I \otimes \sigma_j - \sum_{ij} C_{ij} \sigma_i \otimes \sigma_j \right)$$
• State Decomposition: The eigenvectors of $\rho$ are identified with:
  • Bell States in the helicity basis: $|\Phi^\pm\rangle$ and $|\Psi^\pm\rangle$.
  • Spin States in the beam basis: Singlet $|\Psi^-\rangle$ (pseudoscalar) and Triplets $|\uparrow\uparrow\rangle, |\Psi^+\rangle, |\downarrow\downarrow\rangle$ (vector).
• Quantum Observables:
  • Purity ($\gamma$): $\text{Tr}(\rho^2)$, measuring how mixed the state is.
  • Von Neumann Entropy ($S$): $-\text{Tr}(\rho \log_2 \rho)$, measuring uncertainty.
  • Entanglement Marker ($\Delta E$): Based on the Peres–Horodecki criterion: $\Delta E = C_{11} + |C_{22} + C_{33}| > 1$ indicates entanglement.

Simulation and Systematics:

• Predictions are generated using POWHEG V2 (NLO) combined with PYTHIA 8.240 (parton shower) and the CP5 tune.
• Comparisons are made with MINNLO+PYTHIA (NNLO) and POWHEG+HERWIG 7.1.
• Systematic uncertainties (ME scale, PDF, experimental) are evaluated using pseudo-experiments sampling from the covariance matrix of the measured coefficients.

3. Key Contributions

1. First Beam Basis Measurements: This paper presents the first experimental determination of spin correlation coefficients ($C_{\perp}$ and $C_{\parallel}$) in the beam basis for $t\bar{t}$ production at the LHC.
2. Quantum State Decomposition: It provides the first decomposition of the $t\bar{t}$ density matrix into Bell states (helicity) and spin singlet/triplet states (beam) across various kinematic regions ($m(t\bar{t})$ vs. $|\cos\theta|$ and $p_T(t)$ vs. $|\cos\theta|$).
3. Quantum Information Metrics: It reports the first experimental values for the purity, von Neumann entropy, and entanglement marker of the $t\bar{t}$ system in both bases.
4. Threshold Physics: It investigates the region near the $t\bar{t}$ production threshold ($m(t\bar{t}) < 400$ GeV) to search for bound-state-like enhancements (toponium, $\eta_t$), though current uncertainties prevent a definitive distinction between models.

4. Results

• Spin Correlation Coefficients: The measured coefficients in the beam basis show excellent agreement with Standard Model predictions (POWHEG+PYTHIA, MINNLO+PYTHIA). P-values from $\chi^2$ tests range from 0.22 to 0.79, indicating no significant deviation.
• State Decomposition:
  • High Mass ($m(t\bar{t}) > 800$ GeV): In the helicity basis, the $|\Phi^-\rangle$ Bell state contributes $\sim 70\%$ for low $|\cos\theta|$.
  • Threshold Region ($m(t\bar{t}) < 400$ GeV): Both bases show a dominant contribution (60–70%) from the spin-singlet $|\Psi^-\rangle$ state, consistent with the expectation that $gg \to t\bar{t}$ near threshold produces a singlet.
  • Toponium Search: A simulation including a pseudoscalar toponium state ($\eta_t$) was compared to data. The expected enhancement of the $|\Psi^-\rangle$ state was found to be smaller than the experimental uncertainty, so the two models cannot be distinguished.
• Purity and Entropy:
  • Purity is highest (lowest entropy) in the helicity basis at high $m(t\bar{t})$ and low $|\cos\theta|$, and in the beam basis at low $m(t\bar{t})$ and high $|\cos\theta|$.
  • All measured values are consistent with SM predictions within 2 standard deviations.
• Entanglement:
  • Helicity Basis: Entanglement is observed with $>5\sigma$ significance in the high-mass, low-angle region (consistent with previous results).
  • Beam Basis: Evidence of entanglement is found in the threshold region ($m(t\bar{t}) < 400$ GeV) with a significance of $>3\sigma$ for $|\cos\theta| > 0.7$.

5. Significance

This work represents a major step forward in quantum information physics at high-energy colliders. By characterizing the $t\bar{t}$ system in the beam basis, the CMS collaboration has demonstrated that quantum entanglement is a robust feature of top quark production across different kinematic regimes and reference frames.

The ability to measure purity, entropy, and entanglement markers directly from collider data validates the use of high-energy physics experiments as testbeds for quantum mechanics. The results provide stringent constraints on the density matrix of the $t\bar{t}$ system, confirming Standard Model predictions and ruling out significant deviations that would imply new physics or non-standard quantum decoherence mechanisms. Furthermore, the observation of entanglement near the production threshold opens new avenues for studying non-relativistic QCD effects and potential bound-state phenomena (toponium) in future high-luminosity runs.