Measurement of the Z $\to$ $\mu^+\mu^-$ angular coefficients in pp collisions at $\sqrt{s}$ = 13 TeV as functions of transverse momentum and rapidity

This paper presents a measurement of the eight angular polarization coefficients ($A_0$ to $A_7$) for Drell-Yan $Z \to \mu^+\mu^-$ production in 13 TeV proton-proton collisions using 140 fb$^{-1}$ of CMS data, providing double-differential results in transverse momentum and rapidity that are compared with next-to-next-to-leading order QCD predictions.

The Gist

The Big Picture: A Cosmic Pinball Machine

Imagine the Large Hadron Collider (LHC) at CERN as the world's most powerful pinball machine. Scientists smash protons together at nearly the speed of light. Usually, these collisions create a chaotic mess of particles. However, sometimes, the collision creates a heavy, unstable particle called a Z boson, which immediately splits apart into two muons (heavy cousins of electrons).

This paper is about CMS, one of the giant detectors watching this pinball machine. The team didn't just count how many times this happened; they wanted to understand how the muons flew out. Did they shoot straight out? Did they spin? Did they favor one direction over the other?

The Goal: Measuring the "Spin" of the Collision

The scientists measured eight different numbers (labeled $A_0$ through $A_7$). Think of these numbers as a detailed report card on the "posture" or polarization of the Z boson before it exploded.

• The Analogy: Imagine a firework exploding in the sky. If it explodes perfectly symmetrically, the sparks fly out in a perfect sphere. If it's tilted or spinning, the sparks might shoot out more to the left, or more upward, or in a spiral.
• The Measurement: The eight coefficients ($A_0$ to $A_7$) tell us exactly what that "shape" of the explosion looks like. They reveal if the Z boson was spinning, wobbling, or if it was "stretched" in a certain direction.

How They Did It: The "Double-Check"

The team looked at 140 trillion collisions (140 fb⁻¹ of data) recorded between 2016 and 2018. They didn't just look at the whole pile of data; they sliced it up like a loaf of bread to see if the "spin" changed depending on how hard the protons hit each other.

1. Speed (Transverse Momentum): They looked at muons that were moving slowly sideways versus those moving very fast.
2. Angle (Rapidity): They looked at muons flying straight ahead versus those flying at a sharp angle.

By measuring the angles of the muons in these specific slices, they could calculate the eight coefficients with extreme precision.

The Rules of the Game: The "Lam-Tung" Rule

The paper discusses a famous rule in physics called the Lam-Tung relation.

• The Analogy: Think of a rule that says, "If you throw a ball straight up, it must come straight down." In the world of particle physics, at the simplest level of calculation, two of these coefficients ($A_0$ and $A_2$) should cancel each other out perfectly ($A_0 - A_2 = 0$).
• The Reality: The paper confirms that this rule holds up well at low speeds, but as the collisions get more energetic (higher momentum), the rule starts to break down. This isn't a failure; it's a feature! It tells us that the "messy" parts of the collision (like extra particles being kicked out) are starting to matter.

The Results: Data vs. Theory

The scientists compared their measurements against the best computer simulations available (the "theoretical predictions").

• The Good News: For most of the coefficients, the real-world data matched the computer models very well. This means our current understanding of how these particles interact is solid.
• The Interesting Tension: In the middle range of speeds, the data for one specific coefficient ($A_0$) was slightly higher than the computer predicted (about 3 standard deviations off). It's like if a weather forecast predicted a 50% chance of rain, but it actually rained 80% of the time. It's not a disaster, but it suggests the computer model might be missing a tiny detail.
• The "Ghost" Coefficients: Three of the coefficients ($A_5, A_6, A_7$) are supposed to be zero or very close to it. The data showed they were indeed tiny, consistent with zero, though one of them ($A_6$) showed a tiny, faint hint of being non-zero. This is like hearing a whisper in a quiet room; it's there, but you need very sensitive ears to hear it.

Why This Matters

This paper is essentially a high-precision calibration check for the laws of physics.

1. Understanding the "Glue": These measurements help us understand the "partonic dynamics"—how the tiny building blocks inside the proton (quarks and gluons) behave when they collide.
2. Testing the Theory: By comparing the "spin" of the Z boson against complex math (Quantum Chromodynamics), the scientists are stress-testing our understanding of the universe. If the math doesn't match the data, it means we need to invent new physics.
3. The Benchmark: This paper provides a new, ultra-precise "ruler" for future experiments. Any new theory must be able to explain these eight numbers.

Summary

In short, the CMS team took a massive snapshot of particle collisions, measured the exact angles of the resulting particles, and calculated eight numbers that describe the "spin" of the event. They found that while our current theories are mostly correct, there are tiny, fascinating discrepancies in the middle-speed range that keep physicists on their toes, ensuring the search for a deeper understanding of the universe continues.

Technical Summary

1. Problem and Motivation

The Drell–Yan (DY) process ($pp \to \gamma^*/Z \to \ell^+\ell^-$) is a fundamental probe for understanding the underlying partonic dynamics of proton-proton collisions and the production mechanisms of electroweak vector bosons. The angular distribution of the final-state leptons in the rest frame of the boson encodes information about the polarization states of the virtual photon or Z boson.

This distribution is parameterized by eight angular polarization coefficients, $A_0$ through $A_7$, defined in the Collins–Soper (CS) frame.

• Physics Goals:
  • $A_0$ and $A_2$: Parametrize longitudinal polarization and interference between transverse states. Their difference ($A_0 - A_2$) tests the Lam–Tung relation ($A_0 - A_2 = 0$), which holds at Leading Order (LO) in QCD but is expected to be violated by higher-order QCD radiation and intrinsic parton transverse momentum.
  • $A_4$: Related to the forward-backward asymmetry ($A_{FB}$) and used to measure the effective weak mixing angle ($\sin^2 \theta_{eff}^W$).
  • $A_1, A_3$: Encode interference between longitudinal and transverse states and depend on axial/vector couplings.
  • $A_5, A_6, A_7$: Define T-odd asymmetries. These are expected to be zero at LO and NLO but can be non-zero at Next-to-Next-to-Leading Order (NNLO) due to one-loop processes.

While previous measurements existed at $\sqrt{s}=8$ TeV and in the forward region (LHCb), a comprehensive, double-differential measurement of the full set of coefficients ($A_0$–$A_7$) at $\sqrt{s}=13$ TeV with high statistics was needed to constrain theoretical models and test NNLO QCD predictions with greater precision.

2. Methodology

Data and Event Selection

• Dataset: Proton-proton collision data collected by the CMS detector at $\sqrt{s} = 13$ TeV during 2016–2018, corresponding to an integrated luminosity of 140 fb$^{-1}$.
• Signal: Dilepton events ($Z \to \mu^+\mu^-$) with an invariant mass $81 < m_{\mu\mu} < 101$ GeV (Z pole region).
• Kinematic Cuts:
  • Dimuon transverse momentum ($p_T^{\mu\mu}$): 0 to 400 GeV.
  • Dimuon rapidity ($|y_{\mu\mu}|$): $< 2.4$.
  • Muon isolation and identification criteria applied to ensure high purity.
  • Trigger requirements: Single-muon triggers with $p_T > 24$ or $27$ GeV depending on the data-taking period.

Simulation and Corrections

• Signal Simulation: Generated using POWHEG v2.0 + MiNNLOPS interfaced with PYTHIA 8 (for parton showering/hadronization) and PHOTOS++ (for QED final-state radiation). The NNPDF3.1 PDF set was used.
• Background Estimation: Monte Carlo simulations for diboson (WW, WZ, ZZ), $t\bar{t}$, and $W$+jets processes. Background contributions are small (0.03% at low $p_T$ rising to ~2.9% at high $p_T$).
• Corrections:
  • Scale Factors: Derived via "tag-and-probe" to correct for trigger, identification, and isolation efficiencies.
  • Pileup Reweighting: To match the simulated pileup distribution to data.
  • Kinematic Reweighting: Simulated events were reweighted in $(p_T^{\mu\mu}, y_{\mu\mu})$ bins to match the data kinematic distributions, minimizing model dependence.
  • Prefiring Correction: Addressed misalignment of bunch crossings in the Level-1 trigger.

Extraction of Coefficients

The extraction utilizes a template method to account for detector acceptance, resolution, and migration between $p_T$ bins.

1. Definition: The differential cross-section is expanded as a sum of trigonometric polynomials $f_i(\theta^*, \phi^*)$ weighted by coefficients $P_i$ (where $A_i = P_i/P_8$).
2. Templates: Nine 2D templates ($T_i$) were constructed from reconstructed MC events. Each template corresponds to a specific angular term $f_i$, weighted to remove polarization information from the event generation while preserving detector effects.
3. Fitting: A two-dimensional binned maximum likelihood fit (using MINUIT) was performed on the observed angular distributions ($\cos \theta^*, \phi^*$) in each of the 16 kinematic bins (8 $p_T$ bins $\times$ 2 rapidity bins). The fit extracts the coefficients $P_i$ by minimizing the difference between data and the linear combination of templates.

3. Key Contributions

• First Full Set at 13 TeV: This is the first measurement of the complete set of angular coefficients ($A_0$–$A_7$) in the central rapidity region ($|y_{\mu\mu}| < 2.4$) at $\sqrt{s}=13$ TeV.
• Double-Differential Precision: Results are provided in 8 bins of $p_T^{\mu\mu}$ and 2 bins of $|y_{\mu\mu}|$, offering a multidimensional view of the angular structure.
• NNLO Comparison: The results are compared against state-of-the-art theoretical predictions at NNLO in QCD (using POWHEG+MiNNLOPS and MADGRAPH5_aMC@NLO).
• T-Odd Asymmetry Constraints: Provides the most stringent constraints to date on the T-odd coefficients ($A_5, A_6, A_7$) in the central region.

4. Results

General Agreement

• The measured coefficients generally agree with NNLO QCD predictions within uncertainties.
• The large dataset (140 fb$^{-1}$) significantly reduced statistical uncertainties compared to previous Run 1 measurements.

Specific Observations

• $A_0$ and $A_2$:
  • $A_0$ shows a discrepancy of about 3 standard deviations in the intermediate $p_T$ range (10–35 GeV) between data and the MADGRAPH5_aMC@NLO prediction, while POWHEG+MiNNLOPS agrees well.
  • $A_2$ is slightly overestimated by MC simulations at high $p_T$ ($>50$ GeV).
• Lam–Tung Relation ($A_0 - A_2$):
  • The relation is violated, as expected due to higher-order QCD effects.
  • POWHEG+MiNNLOPS predictions agree well with the measured $A_0 - A_2$ values in the 10–35 GeV range.
  • Both MC predictions underestimate the measurement for $p_T > 35$ GeV.
• $A_1$:
  • Significantly overestimated by POWHEG+MiNNLOPS in the $1 < |y_{\mu\mu}| < 2.4$ region for $p_T < 35$ GeV.
  • MADGRAPH5_aMC@NLO also overestimates $A_1$ at low $p_T$ in both rapidity regions.
• T-Odd Coefficients ($A_5, A_6, A_7$):
  • Values are of the order of $10^{-3}$ and consistent with zero in most bins.
  • The most significant deviation is for $A_6$ in the bin $55 < p_T < 80$ GeV and $1.2 < |y_{\mu\mu}| < 2.4$, showing a 2.8$\sigma$ deviation from zero. This is consistent with NNLO expectations but smaller in magnitude than previous ATLAS observations at 8 TeV.
• Systematic Uncertainties:
  • Dominant for $A_0$ and $A_1$ at low $p_T$ ($<55$ GeV) and for $A_2, A_3, A_4$ up to $p_T < 200$ GeV.
  • Statistical uncertainties dominate for other coefficients and higher $p_T$ regions.

5. Significance

• Theoretical Validation: The results provide a rigorous benchmark for NNLO QCD calculations, confirming that higher-order corrections are essential to describe the angular structure of Drell–Yan production.
• Parton Dynamics: The measurements offer new constraints on Parton Distribution Functions (PDFs) and the modeling of intrinsic transverse momentum of partons.
• Future Physics: The precision and multidimensional binning established here serve as a critical reference for future phenomenological studies and for testing even higher-order calculations (e.g., N$^3$LO).
• Detector Performance: Demonstrates the CMS detector's capability to measure subtle angular correlations with high precision, even in the presence of complex detector effects and pileup.

In summary, this paper represents a milestone in precision electroweak physics, confirming the validity of NNLO QCD descriptions of the Z boson production mechanism while highlighting specific kinematic regions where theoretical models may require further refinement.