Extracting $B_s\to D_s^*\ell\nu_\ell$ form factors

This paper demonstrates the extraction of the four form factors describing exclusive semileptonic $B_s\to D_s^*\ell\nu_\ell$ decays using the narrow width approximation, based on RBC/UKQCD lattice QCD simulations with 2+1 flavor gauge field ensembles employing Shamir domain-wall fermions for light, strange, and charm quarks and the relativistic heavy quark action for bottom quarks.

The Gist

Imagine the universe as a giant, incredibly complex machine. For decades, physicists have been trying to understand how the gears of this machine turn, specifically focusing on the "flavor" of the particles that make up matter. One of the biggest mysteries right now is a discrepancy in how we measure the behavior of bottom quarks (heavy particles) turning into charm quarks (slightly lighter particles).

Think of this like trying to weigh a specific ingredient in a recipe. You can either weigh the whole bowl of soup and subtract everything else (the "inclusive" method), or you can try to weigh just the carrot you are interested in (the "exclusive" method). Currently, these two methods give slightly different answers, and nobody knows why. This paper is about trying to get a more precise measurement of that "carrot" to solve the mystery.

Here is a simple breakdown of what the authors are doing:

1. The Goal: Measuring the "Shape" of a Particle Transformation

When a bottom quark decays into a charm quark, it doesn't just vanish; it sprouts a new particle (a $D^*$ meson) and shoots out a pair of invisible particles (a lepton and a neutrino).

To understand this process, physicists need to know the "form factors."

• The Analogy: Imagine the particle decay is like a dancer spinning. The "form factors" describe the dancer's shape, speed, and how they move their arms at every moment of the spin. If you don't know the exact shape of the dancer, you can't predict how fast they spin or how much energy they use.
• The authors are calculating these four specific "dance moves" (form factors) for a specific type of decay involving a "strange" version of the particles ($B_s \to D^*_s$).

2. The Method: The "Digital Universe" (Lattice QCD)

Since we can't put a single quark on a scale in a real lab, the authors use a supercomputer to simulate the universe.

• The Grid: They create a digital grid (a lattice) that acts like graph paper for space and time.
• The Simulation: They drop virtual particles onto this grid and watch how they interact.
• The Ingredients: They use different "flavors" of virtual quarks (light, strange, and heavy bottom quarks) and run the simulation on different grid sizes (some coarse, some very fine) to see how the results change as the grid gets sharper.

3. The Challenge: Noise and "Excited States"

In these simulations, the signal you want (the clean dance move) is often buried under "noise" (static) or confused by "excited states" (the dancer wobbling before they start their perfect spin).

• The Fix: The authors developed a clever way to filter out the wobbling. Instead of just looking at the final result, they analyze the whole time sequence of the simulation. By accounting for the "wobbles" (excited states), they can widen the window of time they look at, making their measurement much more confident and precise.

4. The Process: From Raw Data to Real Physics

The paper describes a multi-step recipe:

1. Raw Measurement: They calculate ratios of data points from their supercomputer simulations.
2. Blinding: To prevent human bias (like unconsciously tweaking numbers to match what you hope the answer is), they apply a secret "blindfold" (a multiplication factor) to their results. They don't know the true answer yet!
3. Interpolation: They have data for a few specific "virtual masses" of the charm quark. They use math to draw a smooth curve between these points to guess what the result would be for the exact physical mass of a real charm quark.
4. Continuum Limit: They repeat this on different grid sizes and mathematically shrink the grid size to zero. This removes the "pixelation" of their digital universe, giving them the smooth, real-world answer.

5. Why This Matters

Currently, the authors have completed all the hard work up to the final step. They have the "blinded" results.

• The Stakes: Once they remove the blindfold, these new, highly precise numbers will be fed into the global physics community.
• The Impact: If these new numbers help resolve the tension between the "inclusive" and "exclusive" measurements, it confirms our current understanding of the universe. If they don't resolve it, it might be the smoking gun for New Physics—evidence of particles or forces we haven't discovered yet!

Summary

Think of this paper as a team of master chefs using a high-tech, simulated kitchen to perfect a recipe for a very difficult dish (the $B_s$ decay). They have tested their ingredients, refined their cooking technique to remove the "burnt bits" (noise), and are now ready to taste the final dish. Once they take off the blindfold, they will tell the world exactly how the dish tastes, which might finally explain why the rest of the restaurant (the Standard Model) seems to be missing a secret ingredient.

Technical Summary

1. Problem Statement and Motivation

The paper addresses two critical tensions in the flavor sector of the Standard Model (SM):

1. The $|V_{cb}|$ Tension: A discrepancy exists between inclusive determinations (summing over all semileptonic final states) and exclusive determinations (specific final states like $B \to D^* \ell \nu$) of the CKM matrix element $|V_{cb}|$.
2. Lepton Flavor Universality (LFU) Tension: Experimental measurements of the ratios $R(D^{(*)}) = \mathcal{B}(B \to D^{(*)} \tau \nu_\tau) / \mathcal{B}(B \to D^{(*)} \ell \nu_\ell)$ deviate from SM predictions, suggesting potential new physics.

To resolve these tensions, precise theoretical predictions for hadronic matrix elements are required. While $B \to D^*$ decays have been studied extensively, this work focuses on the $B_s \to D^*_s \ell \nu_\ell$ channel. This channel is phenomenologically interesting and, on the lattice, computationally advantageous because the heavier strange quark reduces numerical costs and improves precision compared to light-quark systems.

2. Methodology

The authors utilize Lattice Quantum Chromodynamics (LQCD) to calculate the four form factors ($V, A_0, A_1, A_2$) describing the semileptonic decay.

Lattice Setup and Ensembles

• Collaboration: RBC/UKQCD.
• Action: 2+1 flavor Shamir Domain-Wall Fermions (DWF) for light and strange quarks, and Iwasaki gauge action.
• Heavy Quarks:
  • Charm: Simulated using Möbius DWF (optimized for heavy quarks).
  • Bottom: Simulated using the Relativistic Heavy Quark (RHQ) action.
• Ensembles: Six gauge field ensembles across three lattice spacings ($a^{-1} \approx 1.78, 2.38, 2.78$ GeV) and pion masses ranging from 268 to 433 MeV.
• Approximation: The $D^*_s$ meson is treated as a QCD-stable particle using the narrow width approximation (width $< 2.1$ MeV).

Correlation Functions and Matrix Elements

• Extraction: Form factors are extracted from ratios of 3-point to 2-point correlation functions (Eq. 4) in the limit of large Euclidean time separation.
• Renormalization: The lattice form factors are renormalized using a mostly non-perturbative approach (Eq. 9), combining non-perturbative calculation of flavor-diagonal parts with 1-loop perturbative corrections for the residual heavy-light current.
• Blinding: An unknown multiplicative blinding factor is applied to the renormalization factors to prevent bias during analysis.

Analysis Strategy

1. Excited State Control: To mitigate contamination from excited states, the authors employ fits that explicitly include excited state contributions, allowing for a wider range of time slices to be used in the fit compared to simple ground-state fits.
2. Two-Step Extrapolation:
   • Step 1 (Charm Mass Interpolation/Extrapolation): Data from different charm-like quark masses and momenta on a single ensemble are combined using a correlated fit ansatz (Eq. 12) to interpolate/extrapolate to the physical charm quark mass.
   • Step 2 (Chiral-Continuum Extrapolation): The resulting form factors are extrapolated to the physical pion mass and the continuum limit ($a \to 0$) using a global fit ansatz (Eq. 13) that includes terms for pion mass, $D^*_s$ energy, and lattice spacing squared.

3. Key Contributions

• First Extraction for $B_s \to D^*_s$: The paper presents the first lattice determination of the four form factors for the $B_s \to D^*_s$ transition using the RBC/UKQCD ensembles.
• Robust Excited State Treatment: The authors demonstrate that including excited state contributions significantly enlarges the fit range and reduces ambiguity in extracting the ground state signal (shown in Fig. 2).
• Validation of Dispersion Relations: Preliminary tests on the coarse ensemble (C1) confirmed that the lattice dispersion relation matches the measured energies within statistical uncertainty, validating the lattice setup.
• Systematic Workflow: The paper outlines a complete workflow from raw correlation functions to chiral-continuum extrapolated form factors, establishing a framework for future high-precision results.

4. Results

• Form Factors: The authors successfully extracted the renormalized form factors $\tilde{V}(q^2)$, $\tilde{A}_0(q^2)$, and $\tilde{A}_1(q^2)$ (with $\tilde{A}_2$ being statistically less precise and currently secondary).
• Data Coverage: Preliminary results are presented for four of the six planned ensembles (C1, C2, M3, F1S).
• Visualizations:
  • Fig. 4: Shows the blinded renormalized form factors as a function of $q^2$ for various ensembles and charm masses.
  • Fig. 5: Illustrates the interpolation/extrapolation surfaces for the $V$ form factor on coarse (C1) and fine (F1S) ensembles.
  • Fig. 6: Presents the preliminary chiral-continuum fit results for the blinded form factors.
• Current Status: The results are blinded (the true scale is hidden). The analysis includes statistical uncertainties but has not yet incorporated systematic uncertainties (e.g., higher-order perturbative corrections, finite volume effects).

5. Significance

This work provides a crucial theoretical input for the interpretation of experimental data from LHCb, Belle II, and B-factories. By providing precise lattice QCD calculations for $B_s \to D^*_s$ form factors:

• It offers an independent check on the $|V_{cb}|$ extraction, potentially helping to resolve the inclusive-exclusive tension.
• It contributes to the theoretical prediction of $R(D^*_s)$, aiding in the search for Lepton Flavor Universality violation.
• It demonstrates the feasibility and precision of using $B_s$ decays on the lattice, leveraging the computational efficiency of the strange quark to achieve high-precision results that complement $B^0$ studies.

The paper serves as a comprehensive status report, confirming that the analysis pipeline is functional and ready for the inclusion of the remaining ensembles and the unblinding of the final results.