Understanding Bell locality tests at colliders

This paper demonstrates that while local hidden variable theories were previously considered untestable in collider experiments, they can now be disproved for decaying particles like $\mu^+ \mu^-$ and $\tau^+ \tau^-$ pairs by applying Bell-like inequalities under a small set of mild assumptions.

The Gist

Imagine you are a detective trying to solve a mystery about the fundamental nature of reality. The mystery is this: Is the universe "local," meaning things only affect their immediate neighbors, or is it "non-local," meaning particles can be mysteriously connected across vast distances, defying our everyday intuition?

For decades, physicists have tried to prove that the universe is non-local using "Bell tests." These are like high-stakes poker games where two players (particles) are separated by a huge distance. If they win the game too often, it proves they are cheating by communicating instantly (quantum entanglement) rather than just guessing.

However, there's a catch. In the world of high-energy physics (like at the Large Hadron Collider, or LHC), the "players" are unstable particles like muons or taus. They live for a split second (less than a trillionth of a second) before exploding into other particles. Because they die so fast, we can't stop them, spin them around, or ask them to "choose" a measurement setting like we do in low-energy lab experiments.

The Problem:
For years, skeptics (Local Hidden Variable Theorists) said, "You can't prove anything here! Since you can't control the measurement settings, you can't rule out the possibility that the particles just had a pre-agreed plan (hidden variables) to act a certain way." They argued that the data from colliders could always be faked by a clever, local trickster.

The Paper's Solution:
This paper by Aguilar-Saavedra, Casas, and Moreno says, "Wait a minute. We can still catch the trickster, but we need to make a few very reasonable assumptions."

Here is the breakdown of their idea using simple analogies:

1. The "Broken Toy" Analogy

Imagine you have two identical, mysterious toys (Particle A and Particle B) that are entangled. You can't see inside them, and you can't touch them. But you know that when they break, they shoot out two smaller pieces (Daughter particles) in specific directions.

• The Old View: "We can't tell if the toys were connected because we don't know exactly how they break. Maybe the breaking mechanism is random and unrelated to the connection."
• The New View: The authors say, "Let's assume a few simple rules about how these toys break:"
  1. Symmetry: The laws of physics are the same everywhere (Poincaré invariance).
  2. Independence: The way Toy A breaks doesn't magically influence how Toy B breaks after they are separated.
  3. Real Spins: The toys have a real "spin" direction (like a spinning top), even if we can't see it directly.
  4. Consistent Breaking: If a toy has a specific spin, it always breaks in the same statistical pattern. A "North-spinning" toy always shoots its pieces East; a "South-spinning" one shoots them West.

If we accept these four "mild" rules, a magical connection appears: The direction the broken pieces fly tells us exactly what the spin of the original toy was.

2. The "Translator" (The $\alpha$ Factor)

The authors introduce a "translator" variable called $\alpha$ (spin-analyzing power).

• Think of $\alpha$ as a measure of how "honest" the broken pieces are.
• If $\alpha$ is high, the direction the piece flies is a very accurate map of the toy's spin.
• If $\alpha$ is low, the pieces fly randomly, and we learn nothing.

The paper shows that for certain particles (like muons and taus), we can actually measure this "honesty" ($\alpha$) using other experiments where we do know the spin (like the decay of pions). Once we know $\alpha$, we can translate the messy data of "where the pieces flew" into a clear picture of "what the spins were doing."

3. The "Continuous" Game

Standard Bell tests usually ask for "Yes/No" answers (Binary). But in a collider, the pieces fly in a continuous range of angles.

• The Analogy: Imagine a standard Bell test is like flipping a coin (Heads/Tails). A collider test is like throwing a dart at a giant circular board.
• The authors developed a new set of rules (inequalities) specifically for "dart-throwing" (continuous variables). They proved that even with darts, if the players are connected by quantum magic, they will hit the board in a pattern that is impossible for any local trickster to replicate, even with the "mild assumptions" we made earlier.

4. The Target: Muons and Taus

The paper suggests two specific targets for this detective work:

• Muons ($\mu$): These are like long-lived ghosts. They travel too far to catch easily in current detectors, making them hard to test right now.
• Taus ($\tau$): These are the "sprinters." They die almost instantly, but because they decay so fast, we can reconstruct their "rest frame" (where they were before they exploded) very accurately. The authors argue that Tau pairs are the perfect candidates to finally prove that local hidden variable theories are wrong, even without being able to "choose" measurement settings.

The Big Picture

What does this mean for us?
For decades, we thought collider experiments were "loophole-ridden" and couldn't truly test the spooky nature of quantum mechanics. This paper says, "Not so fast."

By making a few common-sense assumptions about how particles decay, we can use the debris from high-energy crashes to prove that the universe is indeed non-local. If the data from the LHC (or future colliders) violates these new "continuous" Bell inequalities, it means no local hidden variable theory can explain the universe, even if we can't control the measurement settings.

In short: We can't stop the spinning top to check its direction, but if we watch the way it shatters, and we trust the laws of physics, we can still prove that the two tops were dancing together in a way that defies local reality.

Technical Summary

1. Problem Statement

For decades, it has been established that Local Hidden Variable Theories (LHVT) cannot be definitively disproven by high-energy collider experiments involving decaying particles. This limitation arises because:

• Lack of Direct Control: Collider experiments cannot perform direct spin measurements or select measurement settings randomly for individual particles, as the particles decay too quickly ($<10^{-12}$ s) to allow for space-like separation of measurement events.
• Reconstruction Dependence: Current analyses reconstruct spin density matrices from angular distributions of decay products, assuming the Standard Model (SM) and Quantum Mechanics (QM). This circularity means that observing entanglement (via the Peres-Horodecki criterion) does not constitute a genuine test of Bell nonlocality, as it presupposes QM.
• The "Loophole": Previous work (e.g., Refs. [57, 60]) demonstrated that any collider experiment measuring only final-state momenta can be reproduced by an explicit LHVT (specifically a Kasday construction), rendering standard Bell tests impossible without additional assumptions.

The core problem addressed is: Under what minimal, non-QM assumptions can one certify the violation of Bell inequalities in a collider environment?

2. Methodology

The authors propose a framework to bridge the gap between measurable momentum correlations of decay products and the unmeasurable spin correlations of the parent particles, without assuming the full formalism of QM.

A. The Four Mild Assumptions for LHVT

To relate momentum correlations ($P_{ij}$) to spin correlations ($C_{ij}$) within an LHVT, the authors introduce four specific assumptions:

1. Poincaré Invariance: The laws of physics are invariant under Lorentz transformations.
2. Independent Decays: The decays of the two parent particles ($A$ and $B$) are independent of each other (a necessary consequence of locality if decays are space-like separated).
3. Spin as an Element of Reality: The spin of each particle is a definite vector with a specific orientation (EPR sense), rather than a quantum operator.
4. Spin-Dependent Decay Distribution: If a particle has a specific spin vector $\vec{s}$, the probability distribution of its decay products' momenta depends only on $\vec{s}$ and is independent of other hidden variables.

B. Derivation of the Momentum-Spin Relation

Using these assumptions, the authors derive a linear relationship between the momentum correlation tensor $P_{ij} = \langle p_{ai} p_{bj} \rangle$ and the spin correlation tensor $C_{ij} = \langle s_{Ai} s_{Bj} \rangle$:
$$P_{ij} = \frac{\alpha_a \alpha_b}{9} C_{ij}$$
Here, $\alpha$ represents the spin analyzing power, a coefficient determined by the decay dynamics. In QM, this is derived from the expansion of the decay distribution in Legendre polynomials; in the LHVT framework, it is a parameter of the probability distribution function.

C. Continuous-Variable Bell Inequalities

A critical technical hurdle is that standard CHSH inequalities apply to binary observables ($\pm 1$), whereas the spin vectors in the LHVT framework are continuous variables.

• The authors derive a CHSH-like inequality for continuous vector variables (Eq. 11 in the paper).
• They prove that if the continuous spin correlations violate the standard binary CHSH bound (2), they necessarily violate the continuous-variable bound, which is even stricter (e.g., $\sqrt{2}$ for maximally entangled states with orthogonal settings).
• This allows the use of reconstructed spin correlations to test locality without needing to discretize the spin measurements.

D. Experimental Determination of $\alpha$

The relation $P_{ij} \propto C_{ij}$ depends on the unknown coefficients $\alpha_a$ and $\alpha_b$. To test Bell inequalities, these must be determined experimentally without assuming QM. The authors propose two methods:

1. Known Spin Direction: If the spin direction of the parent is known (e.g., from angular momentum conservation in a specific decay), measuring the angular distribution of the decay products yields $\alpha$ directly.
2. Anti-aligned Spins: For pairs produced in a singlet-like state (anti-aligned spins), the average angle between the decay momenta $\langle \hat{p}_a \cdot \hat{p}_b \rangle$ is directly proportional to $-\alpha_a \alpha_b / 9$.

3. Key Contributions

• Bridging Momentum and Spin: The paper provides a rigorous derivation showing that momentum correlations in decay products can serve as a proxy for spin correlations in LHVTs, provided the four mild assumptions hold.
• Continuous-Variable Bell Bounds: The derivation of Eq. (11) extends Bell inequalities to continuous variables, resolving the theoretical mismatch between continuous spin vectors in LHVTs and binary outcomes in standard Bell tests.
• Model-Independent $\alpha$ Extraction: The authors identify specific decay channels (lepton decays) where the spin analyzing power $\alpha$ can be determined using only classical conservation laws and symmetry arguments (parity, helicity), bypassing the need for QM assumptions.
• Feasibility Analysis: The paper identifies $\mu^+\mu^-$ and $\tau^+\tau^-$ pairs as the primary candidates for such tests, analyzing the experimental challenges (decay lengths, neutrino reconstruction).

4. Results and Feasibility

• Muon Pairs ($\mu^+\mu^-$): While theoretically viable, testing Bell inequalities with muons is experimentally challenging at current colliders (LHC) due to their long lifetime ($2.2 \times 10^{-6}$ s), which prevents them from decaying within the detector volume. It would require far detectors or muon cooling to preserve coherence. Additionally, the presence of multiple neutrinos in $\mu \to e \nu \bar{\nu}$ complicates rest-frame reconstruction.
• Tau Pairs ($\tau^+\tau^-$): These are the most promising candidates. With a short lifetime ($2.9 \times 10^{-13}$ s), taus decay promptly within the detector.
  • Spin Analysis: For $\tau$ decays, $\alpha$ can be determined using the decay of $D_s^\pm \to \tau^\pm \nu$ (assuming $D_s$ is a scalar and neutrinos have definite helicity).
  • Measurement: The paper suggests testing CHSH correlations in directions orthogonal to the $\tau$ momentum to minimize reconstruction uncertainties.
• Conclusion on LHVT: If the experimentally reconstructed spin correlations (derived from momentum data and the derived $\alpha$) violate the continuous-variable CHSH inequality, then any LHVT satisfying assumptions (1)–(4) is falsified.

5. Significance

This work fundamentally shifts the perspective on Bell tests at colliders. It moves away from the pessimistic view that "colliders cannot test Bell inequalities" to a constructive framework where Bell nonlocality can be certified under a specific set of physically motivated, non-quantum assumptions.

• Philosophical Impact: It clarifies the distinction between entanglement (a property of the quantum state) and nonlocality (a property of correlations). It demonstrates that one can test nonlocality without assuming the validity of QM, provided one accepts that spin is a real vector and decays are independent.
• Experimental Roadmap: It provides a concrete path for future collider experiments (specifically at the LHC and future lepton colliders) to perform genuine tests of local realism using $\tau$-lepton pairs, potentially offering the first collider-based evidence against local hidden variable theories that satisfy the proposed mild assumptions.
• Theoretical Rigor: By deriving bounds for continuous variables, the paper resolves a technical inconsistency in previous attempts to apply Bell tests to collider data, ensuring that the statistical tests used are mathematically sound for the variables actually being measured.