Closing the knowledge gap in semileptonic $B\rightarrow X_c\ell\nu$ decays

This paper summarizes the current status of exclusive semileptonic $B\rightarrow X_c\ell\nu$ branching fractions, identifies unmeasured components dominated by baryonic and $D_s$ final states, and proposes specific candidates and simplified models to close the gap with inclusive measurements.

The Gist

Imagine the universe of particle physics as a giant, bustling city. In this city, heavy "B" particles are like delivery trucks that constantly break down into smaller packages. Physicists have two main ways of counting these packages:

1. The Inclusive Count: They look at the whole pile of debris and say, "Okay, we know that 100% of the time, a B particle breaks down into a charm particle and a lepton. That's the total."
2. The Exclusive Count: They try to sort the debris into specific boxes. "Here is a box with a D-meson and a pion. Here is a box with a D-meson and two pions." They add up the contents of every box they can identify.

The Problem: The Missing Packages
For a long time, the "Inclusive Count" (the total) has been significantly higher than the sum of all the "Exclusive Boxes" (the specific types they found). It's like knowing you ordered a pizza with 8 slices, but when you count the slices on the table, you only find 6.5. The missing 1.5 slices are the "semileptonic gap."

Physicists have been guessing what those missing slices are. Some thought they were just "D-mesons with extra pions" that were hard to spot. Others assumed they were exotic, heavy versions of charm particles. But the authors of this paper, Florian Herren and Raynette van Tonder, decided to do a forensic audit to find out exactly what is missing.

The Investigation: What's Actually Missing?
The authors took all the known "boxes" (measured decay rates) and compared them to the "total order" (the inclusive rate). They found that the missing pieces aren't just random noise; they have a specific identity.

• The "D-Meson" Shortfall: They found that even when you add up all the known decays involving standard D-mesons, there is still a small hole.
• The "Exotic" Surprise: The biggest surprise is that a huge chunk of the missing pizza slices (about half of the gap) isn't made of standard D-mesons at all. Instead, it's likely made of $D_s$ mesons (a heavier, "strange" cousin of the D-meson) or baryons (particles made of three quarks, like protons, but with a charm quark).

Think of it this way: You thought the missing slices were just crusts you dropped on the floor. But the audit reveals that half the missing slices are actually a completely different type of topping you didn't even know was on the pizza.

The "S-Wave" Clue
The paper also looks at specific, tricky ways these particles decay, involving something called "S-wave" interactions. Imagine two dancers (particles) trying to hold hands. Sometimes they do a simple, smooth spin (S-wave). The authors created a better mathematical model for how these dancers move.

They found that while these smooth spins do happen, they are too small to explain the missing pizza slices. They account for only about 1% of the mystery. This rules out the idea that the missing pieces are just "hard-to-see" versions of the dances we already know.

The Suspects: Who is Hiding?
Since the standard dances don't explain the gap, the authors propose a list of "suspects" that could be hiding in the shadows:

1. The "Three-Pion" Party: Decays where the charm particle is accompanied by three pions (like a party with three extra guests). These are hard to spot because the background noise is loud.
2. The "Strange" Connection: Decays involving $D_s$ mesons and Kaons. The authors suggest we need to look harder at these specific combinations.
3. The Baryon Brothers: Decays that produce charm baryons (like a Lambda-c and a proton). These are the "heavy lifters" of the missing gap.
4. The "Threshold" Effect: Some particles might only appear when they have just enough energy to be created, creating a sudden spike in numbers right at the edge of possibility.

The Solution: A Better Recipe
Currently, when scientists run computer simulations to predict what happens in particle collisions, they often just guess the missing pieces or assume they are all one type of particle. The authors argue this is like baking a cake and guessing the flavor of the missing ingredient.

They propose a new "cocktail" recipe for computer simulations. Instead of guessing, they suggest mixing in a variety of plausible candidates (the $D_s$ mesons, the baryons, the three-pion states, etc.) in reasonable proportions. This way, when experiments like Belle II or LHCb run their tests, they can see which specific "flavor" of the missing piece actually shows up.

The Bottom Line
This paper doesn't just say "there is a gap." It says, "We know exactly how big the gap is, and we know that the missing pieces are likely exotic particles like $D_s$ mesons and baryons, not just the standard particles we've been looking for."

They are handing the experimentalists a "Wanted" poster with specific descriptions of the missing particles, urging them to stop guessing and start hunting for these specific, exotic suspects to finally close the case on the missing semileptonic decays.

Technical Summary

Technical Summary: Closing the knowledge gap in semileptonic $B \to X_c \ell \nu$ decays

Problem Statement
Precise determinations of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements $|V_{ub}|$ and $|V_{cb}|$ are currently hindered by a persistent discrepancy of approximately three standard deviations between exclusive and inclusive measurement techniques. A primary source of systematic uncertainty in inclusive $B \to X_c \ell \nu$ analyses is the "semileptonic gap": the non-zero difference between the sum of all measured exclusive branching fractions and the total inclusive rate.

Current experimental treatments of this gap rely on assumptions that the unmeasured components are dominated by specific decay modes, such as $B \to D^{(*)}\eta \ell \nu$ (assumed by Belle and Belle II) or heavy excited charm states decaying via $D^{**}_{\text{heavy}} \to D^{(*)}\pi\pi$ (assumed by LHCb). However, theoretical constraints and existing data suggest these assumptions are inadequate. Specifically, the branching fraction for $B \to D\eta \ell \nu$ is predicted to be too small to account for the gap, and the required branching fraction for heavy excited states decaying to $D^{(*)}\pi\pi$ would be theoretically difficult to justify. Furthermore, existing measurements of $B \to D^{(*)}\pi\pi \ell \nu$ account for only half the observed difference. This lack of knowledge regarding non-resonant and unmeasured resonant processes limits the precision of $|V_{cb}|$ extractions and background suppression in $|V_{ub}|$ and lepton flavor universality tests (e.g., $R(D^{(*)})$).

Methodology
The authors perform a global analysis to quantify the size and composition of the semileptonic gap using a Bayesian framework. The methodology involves:

1. Data Aggregation: The analysis incorporates HFLAV averages for dominant exclusive modes ($B \to D^{(*)}\ell \nu$), branching fraction ratios for $B \to D^{(*)}\pi \ell \nu$ and $B \to D^{(*)}\pi\pi \ell \nu$ from Belle and BaBar, and semi-inclusive fractions ($f_{D/D^*/D^{**}}$) relative to the total $B \to DX \ell \nu$ rate from BaBar.
2. Extrapolation and Modeling:
   • Isospin symmetry is used to extrapolate charged pion modes to neutral pion modes.
   • Three distinct scenarios for $D^{**} \to D^{(*)}\pi\pi$ decay topologies (sequential decays, $I=0$ di-pion, and $I=1$ di-pion) are considered to estimate the extrapolation factor for di-pion modes.
   • The analysis distinguishes between final states containing $D$ mesons and those containing $D_s$ mesons or $\Lambda_c$ baryons.
3. Bayesian Fit: A nested sampling algorithm is employed to determine the posterior probability distributions for the gap components: $B(B \to DX \ell \nu)|_{\text{gap}}$ and $B(B \to D_s/\Lambda_c X \ell \nu)|_{\text{gap}}$.
4. S-Wave Form Factor Improvement: To address specific contributions to the gap, the authors refine the description of $S$-wave $B \to D\eta \ell \nu$ and $B \to D_s K \ell \nu$ decays. They utilize a coupled-channel $T$-matrix approach ($D\pi - D\eta - D_s K$) constrained by $B \to D\pi \ell \nu$ invariant mass spectra, improving upon previous models by incorporating updated lineshapes and form factors.

Key Contributions and Results

• Quantification of the Gap: The analysis determines that the total semileptonic gap amounts to approximately 1.8% of all $B$ decays per light lepton flavor (roughly 17% of all semileptonic decays).
• Composition of the Gap: The results indicate a non-vanishing branching ratio into $D_s$ or baryonic final states at the 99% confidence level.
  • For $B^+$ decays, the gap is dominated by $B \to D_s/\Lambda_c X \ell \nu$ components ($1.29 \pm 0.37\%$) compared to $B \to DX \ell \nu$ ($0.44 \pm 0.23\%$).
  • For $B^0$ decays, the $B \to DX \ell \nu$ gap is slightly larger ($1.01 \pm 0.38\%$) than the $D_s/\Lambda_c$ component ($0.83 \pm 0.42\%$), though both are compatible with isospin symmetry.
  • The combined $B \to D_s/\Lambda_c X \ell \nu$ gap is determined to be $(1.10 \pm 0.29)\%$, suggesting that more than half of the unmeasured final states contain $D_s$ mesons or $\Lambda_c$ baryons.
• Refined S-Wave Predictions: Using the improved coupled-channel description, the authors provide updated branching fractions for $S$-wave contributions:
  • $B(B^+ \to (D_s^- K^+)_S \ell^+ \nu_\ell) = (1.58 \pm 0.78) \times 10^{-4}$.
  • $B(B^+ \to (D^0 \eta)_S \ell^+ \nu_\ell) = (0.56 \pm 0.26) \times 10^{-4}$.
  • The $S$-wave $D\eta$ component is found to be small, accounting for only $\approx 1\%$ of the unobserved $B \to DX \ell \nu$ decays.
• Candidate Identification: The paper identifies several promising candidates to close the remaining gap, categorized by invariant mass:
  • Low Mass ($< 2.5$ GeV): Unaccounted contributions from the $D^*$ tail and $D^* \to D^0 \gamma$ decays.
  • 2.5 – 2.8 GeV: Three-pion modes ($B \to D^{(*)}\pi\pi\pi \ell \nu$) via intermediate $\omega$ or $\eta$, and contributions from $D^{(*)}K\bar{K}$ final states.
  • 2.8 – 3.1 GeV: Decays involving $D_s K^*$, $D_{s0}^*(2317)K$, and $D_{s1}(2460)K$, as well as $D^{(*)}f_0(980)/a_0(980)$.
  • High Mass ($> 3.1$ GeV): Baryonic final states, specifically $B^+ \to \Lambda_c^- p^+ \ell^+ \nu_\ell$.

Significance and Recommendations
The paper argues that the current practice of modeling the semileptonic gap with a single dominant mode (e.g., $D^{(*)}\eta$) is insufficient and potentially biased. The authors propose a "cocktail" approach for Monte Carlo simulations, where the gap is modeled as a mixture of plausible unobserved modes (including $D\gamma$, $D^{(*)}K\bar{K}$, $\Lambda_c p$, and various resonant states) with loosely constrained normalizations. This approach allows experimental analyses to robustly assess systematic uncertainties arising from the modeling of unmeasured modes.

The authors emphasize that direct searches for the identified candidate modes (particularly $B \to D_s^{(*)}K^{(*)}\ell \nu$, $B \to \Lambda_c p \ell \nu$, and $B \to D^{(*)}K^+K^- \ell \nu$) are essential. Such searches are not only required to close the knowledge gap in semileptonic branching fractions but also to provide critical insights into the spectroscopy of excited charm mesons and to improve the precision of $|V_{cb}|$ and $|V_{ub}|$ determinations. The paper concludes that while no single mode is expected to close the gap entirely, the sum of these identified candidates constitutes a sizeable portion of the missing rate.