⚛️ phenomenology

Recent progress in decays of $b$ and $c$ hadrons

This paper reviews recent theoretical and experimental progress in the decays of $b$ and $c$ hadrons, focusing on calculations addressing neutral current anomalies, resolving the exclusive-inclusive discrepancy in CKM element determinations, and assessing the status of lepton universality ratios.

Original authors: Aoife Bharucha

Published 2026-02-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Aoife Bharucha

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, incredibly complex Lego set. For decades, physicists have been building models of how these Lego bricks (particles) interact using a specific instruction manual called the Standard Model. This manual has been incredibly accurate, predicting almost everything we've seen in our experiments.

However, in the last ten years, a few pieces of the Lego set have started behaving strangely. They aren't following the instructions perfectly. This paper, written by physicist Aoife Bharucha, is a report card on the progress made in understanding these "glitches" and the tools we are using to fix our manual.

Here is a breakdown of the paper's main points using simple analogies:

1. The "Glitches" (The B Anomalies)

Think of heavy particles like B-mesons and D-mesons as heavy delivery trucks. Usually, these trucks drop off packages (decay) in a very predictable way.

The Problem: Recently, scientists noticed that these trucks are dropping off their packages to certain types of customers (leptons like electrons and muons) at rates that don't match the instruction manual.
The Mystery: Sometimes, the trucks seem to prefer delivering to "tau" customers over "muon" customers, even though the manual says they should treat them equally. This is called a violation of Lepton Universality.
Why it matters: If the manual is wrong, it means there are new, invisible forces or particles (Physics Beyond the Standard Model) helping the trucks make these deliveries. This is the "Holy Grail" of modern physics.

2. The "Measuring Tape" Problem (CKM Elements)

To understand the trucks, we need to know exactly how heavy they are and how fast they move. In physics, we use numbers called CKM elements (specifically $V_{cb}$ and $V_{ub}$ ) as our measuring tapes.

The Conflict: There are two ways to measure these numbers:
1. The "Exclusive" Method: Counting specific, individual delivery routes (e.g., Truck A going to House B). This is like counting every single brick in a wall.
2. The "Inclusive" Method: Counting the total weight of all deliveries in a neighborhood without looking at individual houses. This is like weighing the whole truck.
The Discrepancy: For years, these two methods have given slightly different answers. It's like if you counted the bricks in a wall and got 100, but weighed the wall and calculated it should have 110 bricks. This paper reviews the latest attempts to reconcile these two numbers, which is crucial because if our "tapes" are wrong, we might be misinterpreting the glitches mentioned above.

3. The "Map" (Form Factors)

To predict how a truck behaves, we need a detailed map of the road it travels. In particle physics, this map is called a Form Factor.

The Challenge: The road isn't smooth; it's full of bumps and potholes caused by the strong nuclear force (QCD). Calculating this map is like trying to draw a perfect map of a city while it's being built and demolished at the same time.
The Progress: The paper highlights huge improvements in our ability to draw these maps using supercomputers (Lattice QCD) and mathematical tricks (Light-Cone Sum Rules). We are getting much better at predicting exactly how the trucks should behave, which helps us tell if the "glitches" are real new physics or just a bad map.

4. The "Detective Work" (Rare Decays)

Some trucks take very rare, dangerous routes that almost never happen (like a truck driving through a wall).

The Strategy: Scientists look at these rare routes (like $B \to K \mu^+ \mu^-$ ) because they are very sensitive to new physics. If a new invisible force exists, it might nudge the truck slightly off course.
The Tools: The paper discusses "Angular Observables." Imagine the truck dropping off a package and spinning in a specific direction. By measuring the angle of that spin, physicists can tell if the truck was nudged by a ghost (new physics) or if it just hit a bump (standard background noise).
The Debate: There is a heated debate in the physics community: Are the weird angles we see caused by new physics, or just because our calculation of the "bumps" (long-distance QCD effects) isn't perfect yet? The paper reviews the latest efforts to settle this.

5. The Future: New Eyes on the Scene

The paper ends with a look at the future. We are currently in a "golden age" of data collection.

Belle II and LHCb: These are like two new, ultra-high-definition cameras being set up to film the trucks.
The Goal: With more data, we will be able to see the glitches with much higher precision. If the "glitches" persist and the "maps" are proven accurate, we will finally have proof of new physics. If the glitches disappear, it means our maps just needed a little more tweaking.

Summary

In short, this paper is a status report on a massive detective story.

The Crime: Particles are acting weird.
The Suspects: New, undiscovered forces of nature.
The Investigation: Physicists are using better math, faster computers, and more powerful microscopes to figure out if the suspects are real or if the witnesses (our measurements) are just confused.

The excitement in the field is palpable because, for the first time in decades, we have strong hints that the "Instruction Manual" of the universe might need a new chapter.

1. Problem Statement

The paper addresses the current state of heavy flavor physics, focusing on the significant discrepancies between experimental measurements and Standard Model (SM) predictions observed over the last decade. The core problems identified are:

The "B Anomalies": Persistent deviations in semileptonic $b \to s$ and $b \to c$ transitions. These include branching ratios, angular observables (notably $P'_5$ in $B \to K^*\mu^+\mu^-$ ), and lepton flavor universality (LFU) ratios ( $R_K$ , $R_{K^*}$ , $R_D$ , $R_{D^*}$ ).
The $|V_{cb}|$ and $|V_{ub}|$ Tension: A long-standing discrepancy between the "inclusive" determination of CKM matrix elements (based on summing all final states using the Operator Product Expansion) and the "exclusive" determination (based on specific decay channels like $B \to D^*\ell\nu$ or $B \to \pi\ell\nu$ ). The exclusive values are consistently lower than inclusive values, creating a tension of roughly $2.4\sigma$ for $|V_{cb}|$ .
Theoretical Uncertainties: The need to better understand long-distance QCD contributions (non-local matrix elements) in rare decays and to improve the precision of hadronic form factors calculated via Lattice QCD (LQCD) and Light-Cone Sum Rules (LCSR).

2. Methodology

The review synthesizes recent theoretical and experimental progress across several domains:

Theoretical Framework:
- Form Factors: Utilization of LQCD (for high $q^2$ ) and LCSR (for low $q^2$ ) to calculate hadronic form factors describing transitions between heavy and light mesons/baryons. The paper reviews parameterizations like CLN and BGL used to extrapolate these factors across the full kinematic range.
- Effective Field Theory (EFT): Analysis of decays using the Weak Effective Hamiltonian, focusing on Wilson coefficients ( $C_i$ ) for Flavor Changing Neutral Currents (FCNCs) and Charged Currents (CC).
- QCD Factorization (QCDf) & OPE: Application of QCD factorization for exclusive decays and the Operator Product Expansion (OPE) for inclusive decays to separate perturbative short-distance physics from non-perturbative long-distance effects.
Experimental Data Integration:
- Aggregation of results from LHCb, Belle, BaBar, and CMS.
- Focus on differential branching fractions, angular distributions, and LFU ratios ( $R_M = \Gamma(B \to M \ell \nu) / \Gamma(B \to M \mu \nu)$ ).
Global Fits: Discussing how experimental data constrains Wilson coefficients to identify potential Beyond Standard Model (BSM) contributions.

3. Key Contributions and Results

A. Neutral Current Transitions ( $b \to s$ , $c \to u$ )

Rare Decays: The paper details the status of $B \to K^{(*)}\mu^+\mu^-$ and $B_s \to \phi\mu^+\mu^-$ . The $P'_5$ anomaly (a deviation in an angular observable) remains a $2.5\sigma$ effect, though recent updates suggest some tension reduction.
Non-Local Effects: A major contribution is the discussion on "long-distance" contributions (charm loops) in $b \to s\ell^+\ell^-$ . The paper reviews phenomenological models (Breit-Wigner resonances) and analyticity-based approaches to constrain these non-local matrix elements, concluding their uncertainty is likely $\mathcal{O}(10\%)$ .
Lepton Universality: The ratios $R_K$ and $R_{K^*}$ are measured to be $2.1-2.5\sigma$ below SM predictions. The paper notes that QED corrections have been ruled out as the cause, pointing toward potential BSM physics.
Radiative Decays: Progress in measuring photon polarization in $B \to K^*\gamma$ and $B \to K\pi\pi\gamma$ via up-down asymmetries and time-dependent CP asymmetries.

B. Charged Current Transitions ( $b \to c$ , $b \to u$ )

The $|V_{cb}|$ Puzzle:
- Inclusive: Determinations using the kinetic mass scheme and moments of lepton energy/hadronic mass distributions yield $|V_{cb}| \approx 42 \times 10^{-3}$ with $\sim 1.2\%$ uncertainty.
- Exclusive: The dominant channel $B \to D^*\ell\nu$ yields lower values ( $\approx 39 \times 10^{-3}$ ). The paper highlights that using the BGL parameterization (instead of CLN) improves agreement with inclusive results but creates tension with Heavy Quark Effective Theory (HQET) predictions for form factor slopes.
- New Channels: Recent LQCD calculations for $B_s \to D_s^*$ and baryonic decays ( $\Lambda_b \to \Lambda_c$ ) are providing independent checks that may help resolve the tension.
The $|V_{ub}|$ Puzzle:
- Similar to $|V_{cb}|$ , inclusive and exclusive determinations differ by $\sim 2\sigma$ .
- Exclusive determinations rely heavily on $B \to \pi\ell\nu$ form factors from LQCD (e.g., HPQCD, FNAL/MILC, JLQCD).
- The paper emphasizes the ratio $|V_{ub}|/|V_{cb}|$ measured via baryonic ( $\Lambda_b \to p\ell\nu$ ) and $B_s$ decays at LHCb, which offers a theoretically clean constraint independent of some systematic uncertainties.

C. Lepton Universality Ratios ( $R_D$ and $R_{D^*}$ )

Experimental averages for $R_D$ and $R_{D^*}$ show a combined significance of $3.3\sigma$ deviation from the SM.
Theoretical predictions rely on LQCD form factors at non-zero recoil. Recent updates from Fermilab-MILC have reduced uncertainties, but a full resolution of the tension awaits more precise lattice data and Belle II measurements.

D. Charm Sector ( $c \to u$ )

The paper reviews the precise determination of $|V_{cd}|$ and $|V_{cs}|$ from $D$ meson decays, which are now consistent with unitarity constraints.
Rare $c \to u\ell^+\ell^-$ decays are discussed as probes for BSM physics, though they suffer from larger long-distance QCD backgrounds compared to $B$ decays.

4. Significance and Future Prospects

BSM Sensitivity: The anomalies in $b \to s$ and $b \to c$ transitions represent the most compelling hints of physics beyond the Standard Model currently available. The paper argues that if these persist, they likely indicate new particles or interactions (e.g., Leptoquarks or $Z'$ bosons) affecting the Wilson coefficients $C_9$ and $C_{10}$ .
Role of Upcoming Experiments:
- Belle II: With an expected dataset of $50 \text{ ab}^{-1}$ , Belle II will provide crucial measurements of electron modes (essential for $R_K$ ) and angular observables with percent-level precision.
- LHCb Upgrade II: With $300 \text{ fb}^{-1}$ , LHCb aims to reduce statistical uncertainties on angular observables and LFU ratios to the sub-percent level.
Theoretical Imperatives: The resolution of the $|V_{cb}|$ $∣ V_{c b} ∣$ and $|V_{ub}|$ $∣ V_{u b} ∣$ tensions requires:
1. Higher-order perturbative corrections in inclusive calculations.
2. Lattice QCD calculations at non-zero recoil with reduced uncertainties.
3. Better modeling of non-local hadronic effects in rare decays.

Conclusion

The paper concludes that heavy flavor physics is at a critical juncture. While the "B anomalies" and CKM tensions remain unresolved, the convergence of high-precision lattice calculations and massive experimental datasets from Belle II and LHCb in the coming years promises to either confirm the existence of new physics or force a re-evaluation of our understanding of QCD dynamics in heavy meson decays.