Spin Correlations in $t{\bar t}$ Production and Decay at the LHC in QCD Perturbation Theory

This paper demonstrates that QCD perturbation theory, by including only the leading terms in the $\alpha_s/v$ expansion for $t\bar{t}$ production, is sufficient to resolve recent tensions between theoretical predictions and experimental data on spin correlations without requiring full non-relativistic resummation or bound-state effects.

The Gist

The Big Picture: A Dance of Heavy Twins

Imagine the Large Hadron Collider (LHC) as a giant, high-speed dance floor. Inside, protons smash together, creating heavy "twins" called top quarks and anti-top quarks. These twins are the heaviest particles in the universe, and they are incredibly short-lived—they vanish almost instantly.

Physicists are fascinated by how these twins "spin" (a quantum property like a spinning top) as they are created and how they decay. Specifically, they want to know: Do the twins spin in sync, or do they spin in opposite directions?

Recent experiments (by ATLAS and CMS) measured this "spin dance" and found something strange: the twins seemed to be dancing in perfect sync more often than our best computer simulations predicted. This created a "tension" or a mismatch between theory and reality.

The Old Theory: "The Ghostly Couple"

To fix this mismatch, many physicists proposed a radical idea. They suggested that just before the twins vanish, they might briefly form a bound state—a temporary, ghostly couple held together by a strong force, similar to how an electron orbits a proton in a hydrogen atom.

In the world of quantum mechanics, this ghostly couple is called $\eta_t$. The theory went like this:

• "The simulations are missing this ghostly couple. If we add the effects of this couple to our math, the spin dance will match the data perfectly."
• This required using complex, "all-or-nothing" math (resummation) to account for the fact that these twins move very slowly and stick together tightly near the edge of creation (the "threshold").

The Authors' New Idea: "The Blur Effect"

The authors of this paper (Nason, Re, and Rottoli) say: "Hold on. We don't need to look for a ghost."

Here is their argument, explained with an analogy:

The Analogy: The Foggy Camera

Imagine you are trying to take a photo of a race car crossing the finish line.

• The Reality: The car slows down right at the line, maybe even stops for a split second (the "bound state").
• The Experiment: The LHC detectors are like a camera with a very blurry lens. They cannot see the split-second stop. They only see the car's speed averaged over a wide, fuzzy area.

The authors argue that because the experimental "lens" is so blurry (the mass measurement isn't precise enough to see the tiny details of the bound state), we don't need to calculate the complex "ghostly couple" physics.

Instead, we can just use standard, simpler math (perturbation theory). They show that if you look at the "blurry" picture (the integrated data), the complex effects of the bound state cancel out with other effects. The result is that you can just add a few simple "correction terms" to the standard math, and it works perfectly.

The "Magic" Math Trick

In the paper, they use a "Toy Model" (a simplified version of the problem) to prove their point.

• Imagine a ball rolling in a valley. Sometimes it gets stuck in a tiny dip (a bound state).
• If you measure the ball's energy with a super-precise ruler, you see the dip.
• If you measure it with a ruler that has thick, fuzzy markings (like the LHC data), the dip disappears into the noise.

They discovered that when you use the "fuzzy ruler," the math simplifies dramatically. The "ghostly couple" contribution is exactly half of what people thought, and it gets cancelled out by the "free-moving" particles.

The Result: You don't need to solve the difficult "bound state" equation. You just need to add three simple correction terms to the standard equations:

1. A small boost.
2. A slightly larger boost.
3. A tiny, sharp spike (which turns out to be negligible for the big picture).

The Outcome: The Tension Vanishes

When the authors applied these simple corrections to the computer simulations (Monte Carlo generators):

• Before: The theory predicted less spin correlation than the data showed.
• After: The theory matched the data almost perfectly.

The Conclusion:
The "tension" between the data and the theory wasn't because we were missing a "ghostly couple." It was because we were using the wrong scale for our math. We were trying to use a microscope (bound state physics) to look at something that required a wide-angle lens (integrated cross-sections).

By using the right "wide-angle" math, the mystery is solved. The top quarks are dancing exactly as the Standard Model predicted; we just needed to stop overcomplicating the math near the finish line.

Summary for the General Public

• The Problem: Experiments saw top quarks spinning in sync more than computers predicted.
• The Proposed Fix: "Maybe they form a temporary ghostly pair!"
• The Authors' Fix: "No, the experiments are too blurry to see the ghost. If we just adjust the standard math to account for the 'fuzziness' of the measurement, the prediction matches the data perfectly."
• The Lesson: Sometimes, the most complex solution (finding a new particle state) is unnecessary. The answer was hiding in the details of how we measure the data.

Technical Summary

1. Problem Statement

Recent experimental data from ATLAS and CMS regarding spin correlations in top-antitop ($t\bar{t}$) production at the LHC show a tension with standard theoretical predictions. Specifically, the measured correlations are stronger than those predicted by state-of-the-art Monte Carlo (MC) generators (e.g., POWHEG, MiNNLO) at NLO or NNLO accuracy.

To resolve this, previous literature suggested that the discrepancy arises because standard perturbative QCD (pQCD) fails to account for non-relativistic dynamics near the production threshold ($M_{t\bar{t}} \approx 2m_t$). It was argued that one must include the effects of the $\eta_t$ pseudoscalar bound state (a $t\bar{t}$ bound state) or perform a full all-order resummation of non-relativistic corrections (scaling as powers of $\alpha_s/v$, where $v$ is the top velocity).

The Authors' Challenge: The authors argue that this "full resummation" and explicit bound-state modeling are unnecessary and potentially incorrect for the specific observables measured at the LHC. They posit that because experimental analyses use integrated cross-sections with relatively large invariant mass cuts ($M_{cut} \approx 380\text{--}400$ GeV), the physics is dominated by velocities $v \gg \alpha_s$. Consequently, the non-relativistic enhancements can be treated as a convergent perturbative expansion rather than requiring a bound-state resummation.

2. Methodology

A. Theoretical Framework: Perturbative Expansion vs. Resummation

The authors utilize the Optical Theorem, expressing the integrated cross-section as a contour integral of the forward scattering amplitude in the complex energy plane.

• Key Insight: For an integrated cross-section up to a cut $M_{cut}$, the contour can be deformed to a circle with radius $M_{cut} - 2m_t$. This region is far from the threshold singularity ($v \sim \alpha_s$).
• Result: The integral is sensitive to velocities $v \sim \sqrt{(M_{cut}-2m_t)/m_t}$, which are large enough ($v \approx 0.29$) that the series of enhanced terms $(\alpha_s/v)^n$ converges. Therefore, a full resummation (which sums to all orders to handle $v \sim \alpha_s$) is not required; a fixed-order expansion of the threshold corrections suffices.

B. The Toy Model

To validate this approach, the authors first analyze a 1D quantum mechanical toy model: a particle in a delta-function potential.

• They demonstrate that while the spectral density $\rho(E)$ contains a bound state pole (for attractive potentials) and a continuum, the integral of the spectral density up to a high energy cut has a well-defined perturbative expansion in powers of the coupling $\lambda$.
• Crucial Finding: The contribution of the bound state to the integrated cross-section is exactly halved ($\theta(\lambda) \to 1/2$) when combined with the continuum contributions in the perturbative expansion. This "magic factor" arises because the bound state only exists for attractive potentials, while the perturbative expansion must be valid for both signs of the coupling.

C. Application to $t\bar{t}$ Production

The authors generalize the toy model to the $t\bar{t}$ system, treating it as a non-relativistic Coulomb problem with color factors:

• Color Singlet ($gg$ channel): Attractive potential ($C_F \alpha_s$), supports bound states.
• Color Octet ($gg$ and $q\bar{q}$ channels): Repulsive potential, no bound states.
• Spectral Density: They derive the perturbative expansion for the spectral density $\rho_l(E)$ up to $\mathcal{O}(\alpha_s^3)$. The expansion replaces the Born velocity factor $v$ with:
  $$v \to v + \frac{b_l}{2} + \frac{b_l^2}{12v} + \frac{\zeta(3)}{4\pi^2} b_l^3 m \delta(E)$$
  where $b_l$ depends on the color channel ($b_1 = \pi C_F \alpha_s$, $b_8 = -\pi \alpha_s / 2N_c$).
• Interpretation:
  • $\mathcal{O}(\alpha_s/v)$ and $\mathcal{O}(\alpha_s^2/v^2)$: Standard Coulomb corrections (no bound state).
  • $\mathcal{O}(\alpha_s^3)$: Contains the bound state contribution, but it is exactly reduced by a factor of 2 due to the cancellation with continuum contributions (the "1/2 factor" from the toy model).

D. Implementation in Monte Carlo

The authors propose a method to incorporate these corrections into existing MC generators (POWHEG, MiNNLO) without double-counting:

1. Scale Choice: The strong coupling $\alpha_s$ in the threshold terms must be evaluated at a scale related to the top momentum at the mass cut ($S(M) \approx \frac{1}{4}\sqrt{M^2 - 4m^2}$), not the hard production scale.
2. Reweighting: They compute the threshold corrections using a fixed reference $\alpha_0$ and then reweight the events based on the running coupling $\alpha_s(S(M))$.
3. Subtraction: When augmenting NLO/NNLO generators, they subtract the threshold terms already included in the generator (evaluated at the hard scale) to avoid double counting, adding only the difference evaluated at the threshold scale.

3. Key Contributions

1. Resolution of the "Bound State" Debate: The paper demonstrates that for integrated observables with large mass cuts, explicit modeling of the $\eta_t$ bound state is not necessary. The effects are fully captured by a fixed-order perturbative expansion of the threshold corrections.
2. The "1/2 Factor": A rigorous derivation showing that the bound state contribution to the integrated cross-section is effectively halved due to the interplay with the continuum, a feature often missed in phenomenological models that simply add a Breit-Wigner resonance.
3. Scale Sensitivity: The authors highlight that the large discrepancy in previous studies was partly due to using the hard production scale for threshold corrections. Using the correct threshold scale (momentum at the cut) significantly alters the magnitude of the corrections.
4. Practical Implementation: A concrete recipe for correcting NLO and NNLO generators to include these threshold effects without requiring a full non-relativistic QCD (NRQCD) resummation framework.

4. Results

The authors applied their method to the $D$ observable (a measure of spin correlation, $D = -3\langle C_{hel} \rangle$) and compared the results with ATLAS and CMS data.

• NLO+PS Generators (POWHEG-hvq):

  • The nominal generator underestimates the spin correlation (predicts $D \approx -0.45$ vs. data $\approx -0.54$).
  • Adding the threshold corrections (up to $\mathcal{O}(\alpha_s^3)$) with the correct scale shifts the prediction to $D \approx -0.49$ (ATLAS) and $-0.47$ (CMS).
  • Conclusion: The tension with data is significantly reduced, though a small discrepancy remains, potentially due to the approximate treatment of top decays in the generator.

• NNLO+PS Generators (MiNNLO-ttbar):

  • The nominal NNLO prediction is already closer to data ($D \approx -0.46$).
  • Adding the threshold corrections yields $D \approx -0.47$, showing good agreement with data.
  • The corrections are smaller here because the NNLO generator already includes some threshold resummation effects.

• Differential Mass Spectrum:

  • The authors tested the correction against the $t\bar{t}$ invariant mass distribution. While the corrections move the prediction in the right direction (increasing the cross-section near threshold), they are insufficient to fully resolve the deficit in the first bin of the CMS mass spectrum. This suggests other effects (e.g., higher-order corrections or different PDF sets) may be at play for the differential spectrum, even if the integrated spin correlations are resolved.

5. Significance

• Paradigm Shift: The paper challenges the prevailing view that "bound state modeling" is the only way to fix spin correlation tensions. It argues that for inclusive measurements, standard pQCD with proper scale choices is sufficient.
• Clarification of NRQCD: It clarifies the relationship between full NRQCD resummation and fixed-order perturbation theory, showing that for large cuts, the latter is a valid approximation of the former.
• Phenomenological Impact: The work provides a practical, computationally cheap method to improve the accuracy of top quark spin correlation predictions at the LHC, bringing theory and experiment into much better agreement without the need for complex, non-standard bound-state generators.
• Community Education: The authors explicitly aim to disseminate the "1/2 factor" insight, noting that it has been known in mathematical physics (e.g., in $g-2$ calculations) but overlooked in recent top-quark phenomenology.

In summary, the paper resolves the spin correlation tension by demonstrating that the "missing" physics is not a missing bound state resonance, but rather the correct perturbative treatment of threshold enhancements with appropriate scale choices in integrated cross-sections.