Study on the systematic effects on $b \to c$ inclusive semileptonic decays

This paper investigates systematic uncertainties in the lattice QCD calculation of inclusive $B_s \to X_c \, l \nu_l$ decays, proposing a hybrid method that treats ground-state contributions as exclusive to isolate and suppress errors in excited-state reconstructions, thereby advancing efforts to resolve the tension in $|V_{cb}|$ determinations.

The Gist

The Big Picture: Solving a Physics Mystery

Imagine two groups of detectives trying to solve the same crime: measuring a specific number in the universe called $|V_{cb}|$. This number tells us how likely a heavy particle (a bottom quark) is to turn into a slightly lighter one (a charm quark) while emitting a neutrino and a charged lepton.

• Group A (Exclusive) uses a very precise, "microscope" approach. They look at specific, individual outcomes of the decay. Their result is one number.
• Group B (Inclusive) uses a "wide-angle lens." They look at all possible outcomes at once, adding them up. Their result is a slightly different number.

Right now, these two numbers don't match. They are about 3 "sigma" (a statistical measure of confidence) apart. This is a big deal in physics. It could mean:

1. We are missing a piece of the puzzle (New Physics!).
2. Or, one of the methods is slightly broken due to hidden errors (Systematic Uncertainty).

This paper is about Group B (Inclusive). The authors are trying to build a new, ultra-precise "wide-angle lens" using a supercomputer simulation called Lattice QCD. Their goal is to see if the mismatch is real or just a glitch in their calculation method.

The Challenge: The "Blurry" Photo

To calculate the "wide-angle" view, the scientists have to reconstruct a complex image from a series of blurry snapshots.

1. The Snapshots (Correlation Functions): In their computer simulation, they take "photos" of particles at different moments in time.
2. The Blur (Smearing): To make the photos clearer, they apply a technique called "smearing" (like using a soft-focus filter). They have to guess how much blur is just right. Too much, and you lose detail; too little, and the image is noisy.
3. The Reconstruction (Chebyshev Method): They use a mathematical trick (Chebyshev polynomials) to piece these blurry snapshots back together into a clear picture of the total decay rate.

What They Investigated (The "Systematic Effects")

The authors asked: "What if our settings for the camera are slightly off? Does that change the final picture?" They tested three main "knobs" on their camera:

1. The Smearing Width: How much "soft focus" do we apply to the start and end of the particle's life?

   • The Test: They tried different amounts of blur.
   • The Result: On their specific computer grid, the amount of blur mattered a little bit. But when they checked on a larger grid, the blur didn't matter at all. Conclusion: The blur setting is under control.

2. The Time Gap: How long do we wait between taking the first photo and the last photo?

   • The Test: They waited for 18, 20, or 22 "time steps."
   • The Result: The final picture looked the same regardless of the wait time. Conclusion: The timing is stable.

3. The Insertion Point: Where exactly in the middle of the timeline do we take the "action" photo?

   • The Test: They moved the action photo to five different spots.
   • The Result: Again, the final picture didn't change. Conclusion: The position is stable.

The Good News: They found that the "noise" from excited, unstable states (like a particle vibrating wildly before settling down) is under control. The camera is stable.

The Tricky Part: The "Sharp Peak" Problem

There is one remaining issue. The mathematical tool they use to reconstruct the image requires a parameter called $\sigma$ (sigma). Think of $\sigma$ as the "sharpness" of the edge they are trying to draw.

• The Problem: As they try to make the edge sharper (making $\sigma$ smaller), the calculation gets noisier and the error bars get huge. It's like trying to trace a very sharp, jagged mountain peak with a thick marker; the more you try to be precise, the more you wobble.
• Why it happens: Some parts of the calculation have "sharp peaks" (mathematically), while others are "smooth hills." The sharp peaks are the ones causing the wobble.

The Solution: Separating the "Main Act" from the "Background Noise"

The authors came up with a clever trick to fix the wobble. They realized the total picture is made of two parts:

1. The Ground State (The Main Act): The most common, stable way the particle decays. This is like the main actor on stage.
2. The Excited States (The Background Noise): The rare, unstable, vibrating ways the particle decays. This is like the background dancers.

The Strategy:
Instead of trying to reconstruct the entire blurry picture at once, they split the work:

• They use old, proven techniques to calculate the "Main Act" (Ground State) perfectly. Since this part is smooth and stable, it doesn't need the tricky "sharpness" parameter.
• They use the new, tricky technique only for the "Background Noise" (Excited States).

The Result:
Because the "Main Act" makes up most of the picture, and they calculated that part perfectly, the final result is much more stable. The "wobble" caused by the sharpness parameter ($\sigma$) is significantly reduced.

Summary

This paper is a "quality control" report for a new way of measuring a fundamental physics number.

• They checked if their computer settings (blur, timing, position) were messing up the results. They weren't.
• They found a problem with how they handle "sharp" mathematical edges.
• They invented a fix: Separate the stable, easy part of the calculation from the unstable, hard part.
• By doing this, they reduced the errors and showed that their new method is robust enough to potentially solve the mystery of why the two groups of detectives (Exclusive vs. Inclusive) got different numbers.

They haven't solved the mystery yet, but they have built a much better, more reliable camera to take the next photo.

Technical Summary

1. Problem Statement

The paper addresses a long-standing tension in particle physics regarding the determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{cb}|$.

• The Discrepancy: There is a $\approx 3\sigma$ tension between "exclusive" determinations (based on specific decay channels like $B \to D^* \ell \nu$, calculated via Lattice QCD) and "inclusive" determinations (based on the total decay rate $B \to X_c \ell \nu$, calculated via perturbative QCD).
  • Exclusive: $|V_{cb}| \approx (39.46 \pm 0.53) \times 10^{-3}$
  • Inclusive: $|V_{cb}| \approx (42.16 \pm 0.51) \times 10^{-3}$
• The Goal: To resolve whether this tension stems from new physics or systematic errors in the theoretical frameworks. The authors aim to perform the first phenomenologically relevant calculation of the inclusive decay rate ($B_s \to X_c \ell \nu$) directly using Lattice QCD, thereby providing a non-perturbative check independent of the exclusive/inclusive divide.

2. Methodology

The authors utilize Lattice QCD combined with Chebyshev reconstruction to calculate the inclusive decay rate.

A. Theoretical Framework

• Decay Rate Definition: The decay rate $\Gamma$ is expressed as an integral over the hadronic tensor $W_{\mu\nu}$, which is related to the Euclidean four-point correlation function $C_{\mu\nu}$ via a Laplace transform.
• Kernel Approximation: To relate the physical decay rate to the Euclidean correlator, a Heaviside step function $\theta(\omega_{max} - \omega)$ is introduced to impose kinematic constraints. This is smoothed using a sigmoid function $\theta_\sigma$ with a smearing parameter $\sigma$.
• Chebyshev Expansion: The kernel function is approximated as a power series in $e^{-\omega}$. To handle the signal deterioration at large time separations ($t$) in lattice data, the expansion is reformulated using Chebyshev polynomials. This allows the calculation of "Chebyshev matrix elements" $\langle T_k \rangle_{\mu\nu}$, which are bounded in $[-1, 1]$ and can be treated with Bayesian priors.
• Limits: The physical result is recovered by taking the limits $\sigma \to 0$ (removing smearing) and $N \to \infty$ (increasing polynomial order).

B. Simulation Details

• Ensemble: Calculations were performed on RBC/UKQCD $N_f = 2+1$ domain wall fermion (DWF) configurations.
• Parameters:
  • Lattice size: $24^3 \times 64$ (and $32^3$ for specific checks).
  • Lattice spacing: $a \approx 0.11$ fm.
  • Pion mass: $M_\pi \approx 330$ MeV.
  • Quark actions: Möbius DWF for charm, Relativistic Heavy Quark (RHQ) action for bottom.
• Software: Grid and Hadrons libraries on DiRAC HPC resources.

3. Key Contributions and Systematic Studies

The paper focuses on quantifying and mitigating systematic uncertainties inherent to the inclusive reconstruction method.

A. Excited-State Contamination (Section 4)

The authors investigated whether contaminations from excited $B_s$ states affect the results by varying three key parameters of the four-point function:

1. Jacobi Smearing Width ($w$): Varied between 5.0, 6.0, and 6.5. Results showed no significant dependence on a larger lattice ($L=32^3$), suggesting the observed trend on the smaller lattice was negligible.
2. Source-Sink Separation ($T_{sep}$): Tested at 18, 20, and 22 lattice units. Results were consistent within errors.
3. Current Insertion Time ($t_{ins}$): Varied between 4 and 8. Results remained stable.

• Conclusion: The contribution of excited states is under control for the chosen parameters ($w=6.0, T_{sep}=20, t_{ins}=6$).

B. Smearing Parameter Dependence ($\sigma$) (Section 5)

A major systematic issue is the dependence of the result on the smearing parameter $\sigma$.

• The Problem: As $\sigma \to 0$, the convergence of Chebyshev coefficients slows, requiring higher-order terms ($N$). However, higher orders correspond to larger time separations where lattice signals are dominated by noise.
• Observation:
  • Dominant Channels: For the dominant decay channels (axial-vector currents $A_i$), the kernel function is broad, leading to mild $\sigma$ dependence.
  • Sub-dominant Channels: For channels with sharply peaked kernels (e.g., vector currents $V_0$), the dependence on $\sigma$ is strong, and errors increase significantly as $\sigma \to 0$.

C. Ground-State Separation Strategy (Section 5.1)

To mitigate the $\sigma$-dependence and reduce systematic errors, the authors propose separating the four-point function into Ground State (GS) and Excited State (ES) contributions:
$$C_{\mu\nu} = C_{\mu\nu}^{GS} + C_{\mu\nu}^{ES}$$

• Method:
  • The Ground State contribution (identified as the $D_s$ state) is calculated using standard exclusive techniques (double-exponential fits).
  • The Excited State contribution is calculated using the inclusive Chebyshev method.
• Result: Since the ground state dominates the decay, calculating it via precise exclusive methods removes the bulk of the $\sigma$-sensitive noise. The remaining inclusive calculation only needs to handle the sub-dominant excited states, where the $\sigma$-dependence is less critical.
• Outcome: This "separated-and-summed" approach significantly reduces the dependence on $\sigma$ and lowers the error bars compared to the unseparated method, particularly for channels with sharp kernels.

4. Results

• Stability: The inclusive decay rate $\bar{X}(q^2)$ is stable across variations in smearing width, source-sink separation, and current insertion times.
• Error Reduction: By separating the ground state, the systematic error associated with the $\sigma \to 0$ limit is mitigated. For the sub-dominant channel ($V_0$), the separated method shows significantly reduced sensitivity to $\sigma$ compared to the raw inclusive reconstruction.
• Limitations: At the highest values of $q^2$, the ground-state reconstruction becomes unstable, suggesting a need for Bayesian priors derived from two-point functions to constrain these regions.

5. Significance

• First Step Toward Precision: This work represents a critical step toward the first phenomenologically relevant, continuum-limit calculation of inclusive $B$-meson decays using Lattice QCD.
• Resolving the Tension: By providing a lattice-based inclusive determination of $|V_{cb}|$, this methodology offers a third, independent avenue to resolve the 3$\sigma$ tension between exclusive and inclusive measurements.
• Methodological Innovation: The proposed strategy of hybridizing exclusive and inclusive techniques (treating the ground state exclusively and excited states inclusively) provides a robust framework for controlling systematic errors in future lattice calculations of inclusive observables. This approach can be generalized to other decay processes where specific states dominate the spectrum.