A Determination of the Top Mass from a Global PDF Analysis

This paper presents an indirect determination of the top-quark pole mass as $172.80 \pm 0.26$ GeV by performing a global analysis of parton distribution functions within the NNPDF framework, which incorporates high-precision QCD and electroweak corrections, toponium contributions, and lattice constraints on $\alpha_s$ while refitting PDFs to properly correlate uncertainties across various ATLAS and CMS measurements.

The Gist

Imagine the universe is a giant, complex machine, and the Top Quark is its heaviest, most important gear. Physicists have been trying to measure the exact weight of this gear for decades. Why? Because if the gear is even slightly too heavy or too light, it changes how the entire machine (the Standard Model of physics) behaves, and it might even tell us if the universe is stable or if it could one day collapse.

For a long time, scientists measured this gear by "weighing" it directly after a collision, like trying to guess the weight of a car by looking at the debris after a crash. This paper presents a different, more clever approach: measuring the gear by how it affects the traffic around it.

Here is the story of how the authors did it, explained simply:

1. The Problem: The "Traffic Jam" of Data

The Large Hadron Collider (LHC) smashes protons together billions of times, creating a chaotic traffic jam of particles. Among the debris, top quarks are created.

• The Old Way: Scientists would look at specific crashes and try to reconstruct the top quark's mass. This is like trying to guess the weight of a specific car by looking at a single pile of scrap metal. It's hard, and the "scrap" (computer simulations) can be misleading.
• The New Way (This Paper): Instead of looking at one crash, the authors looked at the entire traffic pattern. They asked: "If the top quark were slightly heavier, how would the entire flow of particles change?"

2. The Method: The "Global Fit" (The Master Puzzle)

The authors used a massive digital puzzle called PDFs (Parton Distribution Functions). Think of the proton not as a solid ball, but as a bag of smaller particles (quarks and gluons) zipping around. The PDFs are a map showing how likely you are to find a specific particle in that bag at a specific speed.

• The Challenge: The top quark's mass, the strength of the "glue" holding particles together (called $\alpha_s$), and the map of the bag (PDFs) are all tangled together. If you change one, the others change too.
• The Solution: They used a technique called the Theory Covariance Method (TCM). Imagine trying to tune a radio. If you turn the volume knob (mass), the static (uncertainty) changes. The TCM allows them to turn all the knobs (mass, glue strength, and the map) simultaneously to find the one setting where the radio plays the clearest song (matches the experimental data perfectly).

3. The Secret Ingredients: What Made This Special?

The authors didn't just use old data; they added three "secret sauces" to their recipe:

• The "Ghost" at the Threshold (Toponium):
  Imagine two heavy magnets snapping together just before they collide. For a split second, they form a fuzzy, ghostly pair called Toponium. This happens right at the edge of the energy threshold. Previous studies mostly ignored this ghost. The authors realized this ghost actually pushes the traffic pattern slightly. By accounting for this "ghostly bump," they got a more accurate weight for the gear.
• The "Lattice" Anchor:
  They used a super-precise measurement of the "glue strength" ($\alpha_s$) from a different type of experiment (Lattice QCD) to anchor their calculations. It's like using a known, heavy rock to weigh your scale before you start.
• The "High-Res" Map:
  They upgraded their map of the proton bag to a higher level of detail (N3LO), which is like switching from a paper map to a 3D satellite view. This reduced the guesswork.

4. The Results: A Clearer Picture

When they ran the simulation with all these new ingredients, they found a very precise weight for the top quark:
172.80 GeV (with a tiny uncertainty of just 0.26 GeV).

• Why is this cool?
  • It matches the "Direct" measurements: Their "indirect" method (looking at traffic patterns) agrees perfectly with the "direct" method (weighing the scrap metal), but with higher precision.
  • It solves a conflict: Different experiments were getting slightly different answers. By looking at the whole picture and accounting for how the data points talk to each other, they smoothed out the disagreements.
  • It's "Monte Carlo" free: Direct measurements rely heavily on computer simulations (Monte Carlo) which can have hidden biases. This method relies on pure math and data, making it a very robust check.

The Analogy: The Orchestra

Think of the LHC data as a massive orchestra playing a symphony.

• Old methods tried to figure out the weight of the conductor (the Top Quark) by listening to just the violin section or just the drums.
• This paper listened to the entire orchestra at once. They realized that if the conductor is slightly heavier, the tempo of the strings changes, the bass gets deeper, and the brass gets louder in a specific way.
• By analyzing how the entire symphony shifts when the conductor's weight changes, and by accounting for the fact that the violinists and drummers are listening to each other (correlations), they could determine the conductor's weight with incredible accuracy.

The Bottom Line

This paper proves that you don't need to "catch" a particle to weigh it. By understanding how that particle influences the entire universe of data around it, and by using the most advanced mathematical tools available, we can measure the heaviest particle in the universe with a precision that rivals, and sometimes beats, direct measurement. It's a triumph of "listening to the whole room" rather than just "looking at one object."

Technical Summary

1. Problem Statement

The top quark pole mass ($m_t$) is a fundamental parameter of the Standard Model (SM) with profound implications for the stability of the electroweak vacuum and searches for Beyond Standard Model (BSM) physics. While direct measurements of $m_t$ via kinematic reconstruction at the LHC have reached sub-GeV precision, they rely heavily on Monte Carlo event generators, introducing ambiguities regarding the precise definition of the mass (e.g., pole mass vs. MC mass).

Indirect determinations using $t\bar{t}$ production cross-sections offer a theoretically cleaner alternative but face a critical challenge: the cross-section depends simultaneously on $m_t$, the strong coupling constant ($\alpha_s$), and Parton Distribution Functions (PDFs). Since PDFs themselves depend strongly on $\alpha_s$, a naive extraction of $m_t$ without simultaneously fitting these parameters leads to biases due to unaccounted correlations. Furthermore, previous indirect analyses often neglected higher-order corrections (beyond NNLO), electroweak (EW) effects, and specific threshold phenomena like toponium formation.

2. Methodology

The authors employ the NNPDF framework to perform a global fit of PDFs, $\alpha_s(m_Z)$, and $m_t$ simultaneously. The core methodological innovation is the application of the Theory Covariance Method (TCM).

• Theory Covariance Method (TCM): Instead of performing multiple fits for different parameter values (which is computationally expensive and statistically complex), TCM treats theoretical uncertainties and external parameters ($m_t, \alpha_s$) as nuisance parameters within a Bayesian framework.
  • It introduces nuisance parameters $\lambda_a$ to smear theoretical predictions.
  • It constructs a theory covariance matrix $S$ derived from variations of $m_t$ and $\alpha_s$ around prior central values.
  • This allows for the analytical extraction of the posterior distribution of $m_t$ and $\alpha_s$ from a single global PDF fit, correctly accounting for all correlations between the data, PDFs, and the parameters.
• Data Set: The analysis utilizes a comprehensive set of $t\bar{t}$ measurements from ATLAS and CMS at $\sqrt{s} = 8$ and $13$ TeV. This includes:
  • Single-differential distributions: invariant mass ($m_{t\bar{t}}$), transverse momentum ($p_T$), and rapidities ($y_t, y_{t\bar{t}}$).
  • Double-differential distributions: $(m_{t\bar{t}}, p_T)$ and $(m_{t\bar{t}}, y_{t\bar{t}})$.
  • The baseline includes the NNPDF4.0 dataset, supplemented with new $t\bar{t}$ data.
• Theoretical Predictions:
  • Order: Calculations are performed at NNLO QCD accuracy using the MATRIX code, with PDF evolution up to approximate N$^3$LO (aN$^3$LO) accuracy.
  • Corrections: Includes Electroweak (EW) corrections (LO and NLO QCD$\otimes$EW) and QED evolution (photon PDFs).
  • Missing Higher Order Uncertainties (MHOU): Estimated using the 7-point scale variation scheme.
  • Toponium: For the first time in a global PDF fit, the authors model the formation of toponium (a $t\bar{t}$ quasi-bound state) near the production threshold using Non-Relativistic QCD (NRQCD) and apply k-factors to the NNLO predictions.
• Validation: A rigorous closure test was performed using synthetic data generated from a known "truth" ($m_t^*, \alpha_s^*$) to verify that the TCM methodology is free from bias.

3. Key Contributions

1. Joint Global Determination: The first global PDF analysis to simultaneously determine $m_t$, $\alpha_s$, and PDFs using the TCM, ensuring all correlations are preserved without double-counting data.
2. Toponium Inclusion: The first analysis to explicitly include toponium corrections in the extraction of the top mass from cross-section data, modeling the threshold enhancement via NRQCD.
3. State-of-the-Art Theory: The analysis incorporates aN$^3$LO QCD evolution, mixed QCD$\otimes$QED evolution, and full EW corrections, moving beyond standard NNLO-only analyses.
4. Statistical Rigor: The paper explicitly demonstrates that combining individual measurements via simple statistical averaging (ignoring PDF correlations) underestimates uncertainties, while "PDF4LHC-like" envelope methods overestimate them. The TCM provides the correct statistical combination.
5. Lattice Constraint: The study incorporates the recent FLAG lattice determination of $\alpha_s$ as a prior to test its impact on $m_t$.

4. Key Results

Sensitivity to Observables

• Most Sensitive: Double-differential distributions in $(m_{t\bar{t}}, y_{t\bar{t}})$ and single-differential $m_{t\bar{t}}$ provide the tightest constraints on $m_t$.
• Least Sensitive: Rapidities ($y_t, y_{t\bar{t}}$) are the least sensitive to $m_t$, primarily constraining the gluon PDF.
• Correlations: A negative correlation is found between $m_t$ and $\alpha_s$ for $m_{t\bar{t}}$ and $p_T$ observables, while a positive correlation exists for rapidity-based observables.

Numerical Determinations

The authors present a hierarchy of results as theoretical corrections are added:

1. NNLO QCD (with MHOU): $m_t = 172.29 \pm 0.25$ GeV (from $m_{t\bar{t}}$ data).
2. aN$^3$LO QCD: Shifts $m_t$ down by $\sim 0.3$ GeV (due to decreased gluon luminosity near threshold).
3. QED + EW Corrections: Shifts $m_t$ up by $\sim 0.3$ GeV.
4. Toponium Corrections: Adds a significant upward shift of $\sim 0.6$ GeV for $m_{t\bar{t}}$-based extractions, as the toponium signal enhances the cross-section near the threshold.
5. FLAG Constraint: Imposing the lattice $\alpha_s$ prior lowers $\alpha_s$ but has a negligible effect on $m_t$ due to weak correlation.

Final Result:
Combining all corrections (aN$^3$LO, QED, EW, Toponium) and using the combined ATLAS+CMS $m_{t\bar{t}}$ data:
$$m_t = 172.80 \pm 0.26 \text{ GeV}$$

Comparison with Other Methods

• Consistency: The indirect determination ($172.80 \pm 0.26$ GeV) is fully consistent with the current PDG average ($172.4 \pm 0.7$ GeV) and the LHC combination of direct measurements.
• Precision: The indirect method achieves a precision ($\sim 0.26$ GeV) competitive with, and in some cases better than, direct kinematic reconstruction measurements, without suffering from MC mass definition ambiguities.
• Discrepancies: The result differs significantly from the ABMPtt determination (which found a lower mass), likely due to differences in heavy quark treatment, lack of jet data in ABMP, and absence of MHOU in that analysis.

5. Significance

This work establishes a new benchmark for precision top mass determination. By rigorously accounting for the interplay between $m_t$, $\alpha_s$, and PDFs via the TCM, the authors demonstrate that indirect cross-section measurements can yield a top mass precision of $\sim 150$ MeV (relative to the central value) when higher-order corrections and threshold effects are properly modeled.

The inclusion of toponium corrections is highlighted as crucial, as neglecting them leads to a systematic underestimation of the top mass by nearly 0.6 GeV in threshold-sensitive observables. The study validates the TCM as the superior method for global parameter extraction, avoiding the pitfalls of naive statistical averaging. These techniques are directly applicable to future determinations of other SM parameters (e.g., $W$-boson mass) and SMEFT Wilson coefficients, where handling complex correlations is essential.