Light and strange quark masses with $N_f = 2 + 1$ Wilson fermions

This paper presents an updated determination of light and strange quark masses in QCD with $N_f=2+1$ dynamical flavors using $O(a)$-improved Wilson fermions on CLS ensembles, featuring non-perturbative renormalization and a controlled extrapolation to the physical point across five lattice spacings.

The Gist

Imagine the universe is built out of tiny, invisible Lego bricks. The most fundamental of these bricks are quarks. There are different types of quarks, but the ones that make up the protons and neutrons in your body (and everything else you see) are the "up" and "down" quarks. There's also a slightly heavier cousin called the "strange" quark.

The big mystery physicists have been trying to solve is: How heavy are these bricks?

This paper is a progress report from a team of scientists (led by Fernando P. Panadero) who are trying to weigh these quarks with extreme precision. Here is how they did it, explained in everyday terms.

1. The Problem: You Can't Weigh a Ghost

Quarks are never found alone in nature; they are always glued together inside particles like protons. Because of this, you can't just put a quark on a kitchen scale. To find their mass, scientists have to use a giant, complex mathematical simulation called Lattice QCD.

Think of this simulation as a giant 3D grid (like a massive chessboard, but in 4 dimensions). They fill this grid with digital "quarks" and "gluons" (the glue holding them together) and watch how they interact. By tweaking the "weight" settings in the simulation until the digital protons and pions (a type of particle) look exactly like the real ones in the lab, they can figure out what the quark masses must be.

2. The Tool: A Digital Microscope

The team used a specific type of simulation setup called Wilson fermions. Imagine trying to take a photo of a fast-moving car. If your camera shutter is too slow, the car looks blurry. In physics, the "grid size" of the simulation is like the shutter speed.

• Coarse grid: The car is blurry (inaccurate).
• Fine grid: The car is sharp (accurate).

This paper is special because the team didn't just use one grid size. They used five different grid sizes, getting progressively finer (down to a size of 0.038 femtometers—that's 0.000000000000000038 meters!). This is like taking photos of the car with five different cameras, from a blurry phone camera to a high-end professional lens, to make sure the final image is perfect.

3. The Journey: From "Raw Data" to "Real Mass"

When the simulation runs, it gives them "bare" masses. These are like raw ingredients straight out of the factory—they aren't ready to eat yet. They need to be "cooked" (renormalized) to match real-world physics.

• The Schrödinger Functional Scheme: Think of this as a universal translator. The scientists run their simulation in a controlled, small "box" (a finite volume) to get a clean measurement. Then, they use a mathematical "ladder" (Renormalization Group running) to climb from that small box up to a very high-energy scale where the math becomes simple and predictable.
• The Result: Once they reach the top of the ladder, they can translate their findings back down to the standard units physicists use (MeV at 2 GeV).

4. The New Stuff: Why This Paper Matters

The authors are updating a previous study they did. Here is what they improved:

• More Data: They added six new sets of simulations (ensembles).
• Real-World Physics: Previous simulations often used "fake" heavy pions to make the math easier. This time, they included simulations where the pion mass is exactly the same as in the real world.
• Better Extrapolation: Imagine trying to guess the temperature of a room by measuring it at the door, the window, and the middle. If you only measure at the door, your guess might be wrong. By measuring at many different "distances" (lattice spacings) and "temperatures" (pion masses), they can draw a smooth curve to the exact physical point with much higher confidence.

5. The "Model Averaging" Trick

One of the hardest parts of this science is deciding which mathematical formula to use to connect the dots. Different formulas give slightly different answers.

• The Solution: Instead of picking just one formula, the team used a technique called Model Averaging.
• The Analogy: Imagine you are trying to guess the winner of a race. Instead of trusting just one expert, you ask 20 different experts. Some are experts on speed, some on endurance. You give more weight to the experts who have been right in the past. The final prediction is a weighted average of all of them. This reduces the risk of being wrong because you relied on just one bad guess.

The Bottom Line

The team has successfully weighed the light and strange quarks with much higher precision than before.

• They reduced the error margin by about 50-60% compared to their last attempt.
• Their results agree perfectly with the global average (the "FLAG" average), which is like the "gold standard" of particle physics.

In short: They built a better, sharper digital microscope, took more pictures from different angles, and used a smarter way to combine the results. They now know the weight of the universe's building blocks with incredible accuracy, helping us understand why the universe is the way it is.

Technical Summary

1. Problem Statement

The determination of light ($m_{ud}$) and strange ($m_s$) quark masses is a fundamental task in the Standard Model of particle physics. While these parameters are essential for precision phenomenology, their calculation from first principles requires a non-perturbative formulation of Quantum Chromodynamics (QCD).

• Challenge: Lattice QCD simulations introduce systematic errors related to the finite lattice spacing (cutoff effects) and unphysical quark masses (chiral extrapolation).
• Goal: To provide a high-precision determination of these masses using $N_f=2+1$ dynamical flavors (up, down, and strange) with $O(a)$-improved Wilson fermions, improving upon previous results by extending the dataset to include physical pion masses and finer lattice spacings.

2. Methodology

A. Simulation Setup and Ensembles

• Action: The simulations utilize $O(a)$-improved Wilson fermions and a tree-level Symanzik-improved gauge action.
• Ensembles: The analysis employs a subset of ensembles from the Coordinated Lattice Simulations (CLS) effort.
  • Flavors: $N_f = 2+1$ (degenerate up/down and strange).
  • Lattice Spacings: Five distinct spacings ranging from $a \approx 0.085$ fm down to $a \approx 0.038$ fm.
  • Mass Range: Pion masses ($m_\pi$) span from 420 MeV down to the physical value.
  • Trajectory: The simulations follow a constant trace of the quark mass matrix ($\text{Tr}(M_q) = \text{const}$) trajectory to maintain fixed lattice spacing while varying quark masses.
• New Data: This update includes six new ensembles compared to the previous work [1], specifically adding a finer lattice spacing ($a \approx 0.038$ fm) and ensembles with physical kinematics.

B. Renormalization Scheme

• Bare Mass Extraction: Bare quark masses are extracted from pseudoscalar-pseudoscalar and axial-pseudoscalar correlation functions using the PCAC (Partially Conserved Axial Current) Ward identity.
• Non-Perturbative Renormalization (NPR):
  • The renormalization is performed using the Schrödinger Functional (SF) scheme developed by the ALPHA Collaboration.
  • This allows for a non-perturbative running of the coupling and masses from a hadronic scale ($\mu_{had} \approx 233$ MeV) to a high-energy perturbative scale.
• RG Invariance: The masses are converted to Renormalization Group Invariant (RGI) masses, which are scheme-independent.
• Final Conversion: The RGI masses are converted to the $\overline{\text{MS}}$ scheme at 2 GeV using 4-loop perturbation theory, running from $N_f=3$ to $N_f=4$ flavors.

C. Extrapolation Strategy

To reach the physical point, the authors perform a simultaneous chiral-continuum extrapolation using a Model Averaging procedure to handle systematic uncertainties.

• Chiral Fits: Three functional forms are tested:
  1. [Tay n]: Taylor expansion in pion mass around the $SU(3)$ symmetric point.
  2. [$\chi$ptm]: Inspired by Next-to-Leading Order (NLO) $SU(3)$ Chiral Perturbation Theory ($\chi$PT), including chiral logarithms.
  3. [$\chi$ptr]: Fits performed on ratios of observables to separate chiral and cutoff effects.
• Cutoff Effects: Parametrized as proportional to $a^2$ or massive terms ($a^2 \phi^2$).
• Model Averaging: Instead of selecting a single "best" fit, the authors use the Takeuchi Information Criterion (TIC) to assign weights to different fit models (varying fit ranges, cutoff terms, and data cuts). The final result is a weighted average of these models, providing a robust estimate of systematic errors.

3. Key Contributions

1. Dataset Expansion: The inclusion of physical pion mass ensembles and a finer lattice spacing ($a \approx 0.038$ fm) allows for a controlled extrapolation to the physical point without relying solely on heavy pion extrapolations.
2. Systematic Error Control: The application of a rigorous Model Averaging technique across various chiral ansatzes and cutoff parametrizations provides a more reliable estimation of systematic uncertainties compared to traditional "best-fit" approaches.
3. Precision Improvement: The study achieves a significant reduction in total uncertainty compared to the collaboration's previous work [1].

4. Results

The paper reports the determination of light and strange quark masses in the $\overline{\text{MS}}$ scheme at 2 GeV:

• Light Quark Mass ($m_{ud}$): $3.387(39)$ MeV (consistent with the FLAG average).
• Strange Quark Mass ($m_s$): $92.4(1.0)$ MeV (consistent with the FLAG average).
• Precision:
  • Relative Precision: $\approx 1.8\%$ for $m_{ud}$ and $\approx 1.9\%$ for $m_s$.
  • Error Reduction: The total error is reduced by approximately 50–60% compared to the previous determination [1].
• Error Budget: The dominant sources of uncertainty are statistical/extrapolation errors and the RGI running, while model averaging systematics contribute roughly 10–14%.

5. Significance

• Validation of Standard Model: The results show excellent agreement with the Flavour Lattice Averaging Group (FLAG) review, reinforcing the consistency of the Standard Model parameters derived from different lattice actions and collaborations.
• Methodological Benchmark: The successful application of model averaging to chiral-continuum extrapolations sets a new standard for handling systematic uncertainties in high-precision lattice QCD.
• Future Directions: The authors plan to extend this work using a mixed-action setup (similar to Ref. [5]) to further cross-check cutoff effects and improve the control over the extrapolation to the physical point.

In summary, this work represents a state-of-the-art determination of light and strange quark masses, leveraging high-statistics CLS ensembles, non-perturbative renormalization, and advanced statistical methods to achieve sub-2% precision.