New high-precision $b$, $c$, and $s$ masses from pseudoscalar-pseudoscalar correlators in $n_f=4$ lattice QCD

This paper presents new, high-precision determinations of the $\overline{\mathrm{MS}}$ masses for the $b$, $c$, and $s$ quarks using $n_f=4$ lattice QCD simulations with HISQ discretization and QED corrections, achieving some of the most accurate values to date while providing a detailed analysis of discretization errors and isospin corrections.

The Gist

Imagine the universe is built out of tiny, invisible LEGO bricks. In the world of particle physics, these bricks are called quarks. They are the fundamental ingredients that make up protons and neutrons, which in turn make up everything we see around us.

However, quarks come in different "flavors" and have different weights (masses). Some are light as a feather (like the "up" and "down" quarks), while others are incredibly heavy, like the bottom (b), charm (c), and strange (s) quarks.

Knowing the exact weight of these heavy quarks is crucial. It's like knowing the exact weight of a specific engine part; if you get the number wrong, your calculations for how a car (or a particle collider) will behave will be off. This is especially important for understanding the Higgs boson, the particle that gives other particles their mass.

The Problem: The "Pixelated" Universe

To weigh these quarks, scientists use a method called Lattice QCD. Imagine trying to measure the smooth curve of a circle using a grid of square graph paper. The finer the grid (the smaller the squares), the more accurate your measurement of the curve.

In this paper, the scientists are using a super-fine grid (a lattice) to simulate the universe on a computer. The challenge is that the bottom quark is so heavy that it requires an incredibly fine grid to measure accurately. If the grid squares are too big, the heavy quark looks "jagged" and distorted, leading to errors.

Think of it like trying to draw a tiny, detailed ant on a piece of paper using only thick, chunky crayons. You can't get the details right. You need a very sharp pencil (a fine lattice) to capture the ant's shape.

The Solution: A New "Super-Pencil"

The researchers used a special mathematical tool called the HISQ action. You can think of this as a "super-pencil" that is designed specifically to draw heavy objects without them looking jagged, even if the grid isn't perfect.

They also used a new trick: instead of trying to weigh the heavy quark directly (which is hard because it moves so fast and is so heavy), they looked at how it behaves in a specific type of "dance" called a correlator. By watching the rhythm of this dance over different time intervals (called moments), they could extract the weight with extreme precision.

It's like trying to guess the weight of a heavy truck by listening to the sound of its engine. If you listen to the engine for a short time, the sound is messy. But if you listen to the rhythm over a long period, the pattern becomes clear, and you can calculate the weight perfectly.

The Results: The Most Accurate Weighing Ever

The team, known as HPQCD, combined their super-fine grid simulations with data from the MILC collaboration (who provided the "LEGO sets" or gluon configurations). They also accounted for QED (electromagnetism), which is like adding the effect of static electricity to their calculation to make it even more precise.

Here are their new, ultra-precise measurements:

• The Bottom Quark ($b$): Weighs about 4.192 GeV. This is the most accurate measurement of this particle ever made.
• The Charm Quark ($c$): Weighs about 0.981 GeV.
• The Strange Quark ($s$): Weighs about 0.083 GeV.

Why Does This Matter?

1. The Higgs Boson: The bottom quark is the main way the Higgs boson decays (breaks apart). If we know the bottom quark's weight perfectly, we can test if the Higgs boson behaves exactly as the Standard Model of physics predicts. If it doesn't, it could mean there is new physics waiting to be discovered.
2. Future Colliders: Scientists are planning massive new particle colliders (like the Future Circular Collider). To design these machines and interpret their data, they need to know the weights of these quarks with incredible accuracy. This paper provides the "blueprint" numbers they need.
3. A New Strategy: The authors found a clever shortcut. Because the bottom quark is so heavy, its weight is less sensitive to the "grid size" errors than the lighter charm quark. So, they measured the bottom quark first, and then used the known ratio between the bottom and charm quarks to calculate the charm quark's weight. It's like weighing a heavy anchor first, and then using that to calibrate a scale for a much lighter feather, getting a more accurate result for the feather than if you tried to weigh it directly.

In a Nutshell

This paper is a triumph of computational physics. The scientists built a virtual universe with a grid so fine and used such clever mathematical tricks that they managed to weigh the heaviest particles in nature with a precision never seen before. They didn't just find a number; they refined the ruler we use to measure the fundamental building blocks of our reality.

Technical Summary

1. Problem Statement

Accurate determination of heavy-quark masses ($b$, $c$, and $s$) is critical for Standard Model phenomenology, particularly for precision studies of the Higgs boson (where the decay $H \to b\bar{b}$ is dominant and proportional to $m_b^2$). While lattice Quantum Chromodynamics (QCD) offers a first-principles approach, simulating $b$-quarks is notoriously difficult due to their large mass ($m_b \approx 4.2$ GeV).

• The Challenge: On a lattice with spacing $a$, discretization errors scale as $(am_h)^2$. For $b$-quarks, $am_b$ can approach or exceed 1 on standard lattices, leading to massive discretization errors that typically require prohibitively fine lattice spacings ($a < 0.03$ fm) to control.
• The Gap: Previous high-precision results often lacked full inclusion of Quantum Electrodynamics (QED) effects or relied on extrapolations that introduced significant systematic uncertainties. There was a need for a method that could handle large $am_b$ values while maintaining high precision and incorporating QED corrections.

2. Methodology

The authors employ a sophisticated lattice QCD analysis using Highly Improved Staggered Quark (HISQ) action and MILC collaboration gluon configurations.

A. Lattice Setup and Ensembles

• Sea Quarks: Simulations use $n_f = 4$ dynamical flavors ($u, d, s, c$) with HISQ sea quarks.
• Ensembles: Four gluon ensembles with lattice spacings ranging from 0.032 fm to 0.059 fm (labeled EF-5, UF-5, SF-P, SF-5). The inclusion of the 0.032 fm spacing is crucial for controlling $b$-quark discretization errors.
• Tuning: Quark masses are tuned to reproduce experimental meson masses ($J/\psi$, $\eta_b$, $\pi^0$). The $b$-quark mass is tuned to the experimental $\eta_b$ mass ($9.3987$ GeV).

B. Theoretical Framework: Moments of Correlators

Instead of directly simulating the heavy quark mass, the authors analyze moments of the pseudoscalar current-current correlator ($G_n$):
$$G_n \equiv \sum_t (t/a)^n G(t)$$
They utilize reduced moments $R_n$ to minimize systematic errors:
$$R_n(m_h) \equiv \left( \frac{G_n}{G_n^{(0)}} \right)^{1/(n-4)}$$
where $G_n^{(0)}$ is the lowest-order perturbative result.

C. Handling Discretization Errors ($am_h$)

A key innovation is the exploitation of the non-relativistic limit for high-order moments ($n \ge 14$).

• Suppression Mechanism: For non-relativistic systems, the HISQ action suppresses $am_h$ errors by a factor of $(v/c)^4$, where $v$ is the heavy quark velocity.
• Result: While low-order moments ($n=6$) suffer from large $am_h$ errors, higher moments ($n=14$ to $22$) are effectively free of these errors even on coarser lattices where $am_b \approx 1.2$. This allows the use of larger lattice spacings than previously thought possible for $b$-quarks.

D. Perturbation Theory and Renormalization

• The lattice moments are matched to continuum perturbation theory (MS scheme) using a series expansion $r_n(\alpha_s)$.
• The analysis includes perturbative coefficients up to third order ($\alpha_s^3$) and estimates fourth-order truncation errors.
• Renormalization scales $\mu_n$ are optimized for each moment to minimize perturbative coefficients.

E. QED Corrections

The study includes quenched QED corrections (valence quarks interact with QED, sea quarks do not).

• Corrections are calculated to keep the $\eta_b$ mass fixed at its experimental value when QED is turned on.
• A separate analysis (Appendix C) determines the QED shift required for the fictitious $\eta_s$ meson mass to tune the strange quark mass correctly.

3. Key Contributions

1. Validation of HISQ for Heavy Quarks: The paper provides a rigorous demonstration that the HISQ action is suitable for $b$-quark simulations even when $am_b > 1$, provided high-order non-relativistic moments are used. This challenges the conventional wisdom that $am_b \ll 1$ is strictly required.
2. Error Budget Analysis: A detailed breakdown shows that the dominant errors are no longer from lattice spacing but from perturbative coefficients and sea-quark tuning priors.
3. Indirect Determination of $c$ and $s$ Masses: Instead of calculating $c$ and $s$ masses independently, the authors derive them from the highly precise $b$-mass using non-perturbative mass ratios ($m_b/m_c$ and $m_b/m_s$). This leverages the fact that $b$-mass uncertainties are less sensitive to lattice spacing errors than $c$-mass uncertainties.
4. Inclusion of QED: The results are corrected for QED effects, a necessary step for matching the precision required by future colliders (like the ILC or FCC).

4. Key Results

The authors report the following values (with QED corrections included):

• Bottom Quark Mass ($m_b$):

  • $m_b(m_b, n_f=5) = \mathbf{4.1923(63) \text{ GeV}}$
  • This is the most precise lattice determination to date and agrees well with the Particle Data Group (PDG) average.
  • Uncertainty: 0.15%.

• Charm Quark Mass ($m_c$):

  • Derived from $m_b$ and the ratio $m_b/m_c$.
  • $m_c(3 \text{ GeV}, n_f=4) = \mathbf{0.9813(34) \text{ GeV}}$
  • This is significantly more accurate than previous direct lattice determinations.
  • Uncertainty: 0.35%.

• Strange Quark Mass ($m_s$):

  • Derived from $m_b$ and the ratio $m_b/m_s$, including QED/isospin corrections to the $\eta_s$ mass.
  • $m_s(3 \text{ GeV}, n_f=4) = \mathbf{83.39(26) \text{ MeV}}$
  • This is the most accurate determination by any method.
  • Uncertainty: 0.31%.

5. Significance and Impact

• Higgs Physics: The precision achieved (0.15% for $m_b$ and 0.35% for $m_c$) meets and exceeds the requirements for future high-precision Higgs coupling measurements at proposed colliders (ILC, FCC).
• Methodological Shift: The paper establishes that using high-order moments ($n \ge 14$) with the HISQ action is a superior strategy for heavy quarks compared to using Non-Relativistic QCD (NRQCD) or extremely fine lattices alone. It effectively decouples the precision of the mass determination from the lattice spacing uncertainty.
• Standard Model Tests: These precise mass values reduce theoretical uncertainties in Standard Model predictions, allowing for more stringent tests of the theory and searches for New Physics.
• QED Integration: The successful integration of quenched QED corrections demonstrates a pathway for future lattice calculations to reach sub-percent precision where electromagnetic effects are non-negligible.

In conclusion, this work represents a major advancement in lattice QCD, pushing the precision of heavy-quark mass determinations to a level where they are limited by perturbative theory and input parameters rather than lattice discretization artifacts.