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Quark flavor physics with lattice QCD

This paper provides an overview of quark flavor physics presented at Lattice 2023, focusing on lattice-QCD inputs for determining CKM matrix parameters and reviewing selected processes used to search for physics beyond the Standard Model, while excluding topics covered in other plenary talks.

Original authors: Stefan Meinel

Published 2026-07-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Stefan Meinel

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic Lego set. For decades, physicists have been trying to figure out exactly how the pieces snap together. They know there are three "generations" of building blocks—like three different sets of Legos that look almost identical but come in slightly different sizes and weights. The lightest set makes up the matter we see every day, while the heavier sets are like rare, exotic bricks that only show up in high-energy crashes. But here's the mystery: why are there three sets at all? And why do they mix together in such a specific, strange way when they interact?

This mixing is controlled by a secret code called the CKM matrix. Think of it as a cosmic recipe book that tells a heavy brick how likely it is to turn into a lighter one. If the recipe is slightly off, the universe might look very different, or perhaps new, invisible forces are sneaking into the kitchen. Scientists are obsessed with checking this recipe because if they find a single ingredient that doesn't match the Standard Model (the current best cookbook), it could mean there's a whole new universe of physics waiting to be discovered. To read this recipe accurately, however, they need to understand the messy, sticky glue that holds the bricks together—a force called the Strong Interaction. This glue is so complicated that you can't just do the math on a piece of paper; you need to simulate the entire universe on a giant grid, a method known as Lattice QCD.

This paper, presented by Stefan Meinel at the Lattice 2023 conference, is essentially a progress report on how well we are reading that cosmic recipe book using these super-computer simulations. The author walks us through the latest attempts to measure the "flavor" of quarks—the different types of these building blocks—and how they transform into one another. The main finding is that while our simulations are getting incredibly precise, there are some stubborn disagreements between different teams of scientists. Specifically, when trying to measure how often a bottom quark turns into a charm quark, different groups are getting slightly different answers. This isn't a failure; it's a sign that the simulations are reaching a level of detail where tiny, previously ignored errors are finally showing up.

The paper also highlights a fascinating tension in the data. When scientists look at how particles decay in the real world (the experimental data) and compare it to what the simulations predict, they sometimes see a mismatch. For instance, in the decay of certain heavy particles, the experiments suggest the "recipe" might be slightly different than the Standard Model predicts, hinting at new physics. However, the author is careful to note that these hints are not yet a confirmed discovery. Some of the discrepancies might just be because our simulations of the "glue" aren't quite perfect yet. The paper explicitly rules out the idea that we have solved these puzzles; instead, it argues that we are in a critical phase where we must refine our calculations to see if the "new physics" is real or just a mirage caused by imperfect math.

One of the most vivid parts of the story involves the "Wolfenstein parameters," which are just four numbers that summarize the entire mixing recipe. The paper shows that while we have pinned down two of these numbers with great confidence, the other two are still a bit wobbly. This wobble matters because it affects our ability to predict rare events, like a particle decaying into two muons. If the recipe is slightly off, we might miss a signal of a new force. The author points out that recent updates to the simulations have actually made some of these numbers less certain for a moment, because the new, more detailed calculations revealed that previous groups had underestimated their own errors. It's like realizing your ruler was slightly bent, so you have to re-measure everything.

The paper also dives into the "rare" decays—those weird, one-in-a-billion moments where particles do something unexpected. These are the prime hunting grounds for new physics. The author explains that while we have made huge strides in calculating the "glue" for these events, there are still some tricky parts, like particles that are unstable and fall apart before we can measure them. The simulations are starting to tackle these difficult cases, but it's a work in progress. The overall tone is one of cautious excitement: we are getting closer to the truth, but the path is full of bumps, and we need to keep refining our tools to make sure we aren't fooling ourselves.

In the end, this paper is a call to action for the physics community. It says that the tools we have are powerful, but they need to be sharper. The disagreements between different simulation groups are not a problem to be ignored; they are a roadmap telling us where to focus our energy next. Whether the universe holds a secret new force or just a more complex version of what we already know, the answer lies in getting these numbers right. The journey to decode the flavor of the universe is far from over, but thanks to these lattice simulations, we are finally reading the recipe with a clarity that was impossible just a few years ago.

Technical Summary: Quark Flavor Physics with Lattice QCD

Problem Statement
The Standard Model (SM) of particle physics contains three generations of fermions, leading to flavor mixing described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix. A central challenge in flavor physics is determining the four Wolfenstein parameters ( $\lambda, A, \bar{\rho}, \bar{\eta}$ ) that define this matrix with high precision. Discrepancies between experimental measurements and SM predictions in flavor-changing processes could signal physics beyond the Standard Model (BSM). However, extracting these parameters requires precise knowledge of non-perturbative Quantum Chromodynamics (QCD) matrix elements, which cannot be calculated using perturbation theory. Lattice QCD is the primary tool for computing these hadronic matrix elements, such as decay constants and form factors, which are essential inputs for global CKM fits and tests of lepton-flavor universality.

Methodology
The paper reviews the status of lattice QCD calculations as presented at the Lattice 2023 conference. The methodology involves:

Lattice Simulations: Utilizing various lattice actions (e.g., HISQ, Domain Wall, Fermilab clover, RHQ) and dynamical quark flavors ( $N_f = 2+1$ or $2+1+1$ ) to simulate QCD.
Form Factor Extractions: Calculating hadronic form factors for semileptonic decays (e.g., $B \to D^{(*)}$ , $B \to \pi$ , $B_s \to K$ , $D \to \pi/K$ ) across the full kinematic range ( $q^2$ ), often employing $z$ -expansions to parameterize the $q^2$ dependence.
Global Fits: Combining lattice results with experimental data to extract CKM matrix elements ( $|V_{ud}|, |V_{us}|, |V_{cb}|, |V_{ub}|$ ) and constrain the unitarity triangle.
Renormalization and Matching: Addressing systematic uncertainties through non-perturbative renormalization schemes (e.g., RI/sMOM, gradient flow) and matching to continuum schemes ( $\overline{\text{MS}}$ ).
Handling Resonances and Finite Volume: Applying the Lellouch-Lüscher formalism for unstable resonances (e.g., $B \to \rho$ ) and managing finite-volume effects in inclusive decays and excited-state contamination.

Key Contributions and Results

Determination of $\lambda$ ( $|V_{ud}|$ and $|V_{us}|$ ):
- The parameter $\lambda$ is primarily determined via $|V_{ud}|$ from superallowed nuclear $\beta$ -decays and $|V_{us}|$ from kaon decays.
- Lattice calculations of the ratio of decay constants $f_{K^\pm}/f_{\pi^\pm}$ and the semileptonic form factor $f_+(K \to \pi, q^2=0)$ have reached $\sim 0.2\%$ precision.
- Tension: A significant tension exists between the $(|V_{us}|, |V_{ud}|)$ values derived from lattice QCD (specifically $N_f=2+1+1$ calculations) and the CKM unitarity relation. A new preliminary result by Yamazaki et al. using large lattices ( $>10$ fm) suggests a value for $f_+(K \to \pi, 0)$ that alleviates this tension.
- Inclusive hadronic $\tau$ decays were also discussed as an alternative method, with a preliminary non-perturbative lattice calculation of the $\bar{u}d$ channel yielding a $|V_{ud}|$ with 0.4% uncertainty.
Determination of $A$ ( $|V_{cb}|$ ):
- $|V_{cb}|$ is extracted from inclusive and exclusive $B$ meson decays. Exclusive determinations rely on $B \to D^{(*)} \ell \nu$ form factors.
- New Calculations: Three major collaborations (Fermilab/MILC, HPQCD, JLQCD) have published results for $B \to D^*$ form factors with full kinematic dependence (not just at zero recoil).
- Discrepancy: A long-standing discrepancy persists between inclusive and exclusive determinations of $|V_{cb}|$ . The exclusive values (using lattice form factors) remain significantly lower than inclusive values.
- Slope Issues: Comparisons of lattice predictions with experimental differential decay rates show that HPQCD and Fermilab/MILC predict a steeper slope for the form factor combination than observed, while JLQCD results show better agreement. This slope discrepancy affects the extracted $|V_{cb}|$ values.
Determination of $\bar{\rho}$ and $\bar{\eta}$ ( $|V_{ub}|$ and Unitarity Triangle):
- $|V_{ub}|$ is determined from $b \to u$ semileptonic decays ( $B \to \pi, B_s \to K, \Lambda_b \to p$ ).
- $B \to \pi$ Form Factors: The FLAG average was updated to include 2022 JLQCD results (using fully relativistic Möbius domain-wall fermions). This inclusion increased the $\chi^2/d.o.f.$ of the fit, leading to rescaled uncertainties and a shift in the extracted $|V_{ub}|$ from $3.74(17) \times 10^{-3}$ to $3.64(16) \times 10^{-3}$ .
- $B_s \to K$ Form Factors: A new 2023 RBC/UKQCD calculation revealed a significant dependence on the chiral-continuum extrapolation basis. Extrapolating $f_0$ directly yields higher values than constructing it from extrapolated $f_\perp$ and $f_\parallel$ . This has led to increased tension between collaborations and larger uncertainties in the FLAG average.
- Global Fits: The UTfit global fit yields $\lambda = 0.22519(83)$ , $A = 0.828(11)$ , $\bar{\rho} = 0.161(10)$ , and $\bar{\eta} = 0.347(10)$ .
Constraints from Mixing and CP Violation:
- Lattice inputs for $B^0_{(s)}$ - $\bar{B}^0_{(s)}$ mixing ( $B$ -bag parameters) and Kaon CP violation ( $\epsilon_K, \epsilon'_K$ ) are crucial.
- $\epsilon_K$ Tension: A significant tension exists between the SM prediction for $\epsilon_K$ (using lattice inputs) and the experimental value when using the exclusive $|V_{cb}|$ determination; this tension disappears when using the inclusive $|V_{cb}|$ .
- Renormalization: New progress was reported on non-perturbative renormalization of four-quark operators using gradient flow and position-space schemes, aiming to reduce matching uncertainties.
Tests of Lepton Flavor Universality (LFU) and Rare Decays:
- LFU Ratios ( $R(H_c)$ ): Experimental measurements of $R(D^{(*)})$ and $R(\Lambda_c)$ show hints of LFU violation, though recent LHCb results for $R(\Lambda_c)$ are below SM predictions. No single BSM explanation fits all ratios within $2\sigma$ .
- $b \to s \ell^+ \ell^-$ : Deviations in $b \to s \mu^+ \mu^-$ observables (e.g., $P'_5$ , branching fractions) persist, suggesting a shift in the Wilson coefficient $C_9^\mu$ . Recent corrections to LHCb electron identification have shifted $R_K^{(*)}$ to be consistent with unity, but the branching fraction anomalies remain, favoring scenarios where both $C_9^\mu$ and $C_9^e$ are shifted.
- Rare Kaon Decays: Theoretical predictions for $K \to \pi \nu \bar{\nu}$ are extremely precise (1.5–4% QCD uncertainty), dominated by non-local matrix elements. Lattice calculations for these non-local contributions are a priority.

Significance and Claims
The paper asserts that lattice QCD is an essential component of the flavor physics program, providing the non-perturbative inputs necessary to interpret high-precision experimental data. The author highlights that while significant progress has been made, tensions between different lattice collaborations (e.g., in $B \to \pi$ , $B_s \to K$ , and $B \to D^*$ form factors) indicate that systematic uncertainties, particularly regarding chiral-continuum extrapolations and excited-state contamination, may have been underestimated.

The paper claims that resolving these tensions is critical because the uncertainty in the Wolfenstein parameter $A$ (derived from $|V_{cb}|$ ) is the dominant source of uncertainty in SM predictions for many flavor-changing neutral current processes. Consequently, the discrepancies between exclusive and inclusive $|V_{cb}|$ and $|V_{ub}|$ determinations, as well as the tension in $\epsilon_K$ , remain open questions that could either point to new physics or reveal deficiencies in current lattice methodologies. The author emphasizes the value of having multiple groups using different methods to cross-check results and notes that many new calculations with improved methods are underway to address these challenges.

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