[Paper Review] Rare $B$ decays using lattice QCD form factors

R. R. Horgan, Z. Liu+2 more|arXiv (Cornell University)|Jan 2, 2015

Particle physics theoretical and experimental studiesPhysics and Astronomy21 references31 citations

TL;DR

This paper presents an updated lattice QCD calculation of B →K∗, Bs →φ, and Bs →K∗ form factors using unquenched 2+1 flavor ensembles with physical-bottom quark masses and high-statistics correlators. The key improvement lies in enforcing all kinematic constraints at q² = 0 and q² = t⁻, and estimating key correlations between form factor parameters, leading to a more precise determination of the longitudinal polarization fraction FL, while other observables show negligible changes compared to prior results.

ABSTRACT

In this write-up we review and update our recent lattice QCD calculation of $B o K^*$, $B_s o φ$, and $B_s o K^*$ form factors [arXiv:1310.3722]. These unquenched calculations, performed in the low-recoil kinematic regime, provide a significant improvement over the use of extrapolated light cone sum rule results. The fits presented here include further kinematic constraints and estimates of additional correlations between the different form factor shape parameters. We use these form factors along with Standard Model determinations of Wilson coefficients to give Standard Model predictions for several observables [arXiv:1310.3887]. The modest improvements to the form factor fits lead to improved determinations of $F_L$, the fraction of longitudinally polarized vector mesons, but have little effect on most other observables.

Motivation & Objective

• To improve the precision and reliability of lattice QCD form factors for rare B decays by incorporating full kinematic constraints and correlations.
• To reduce theoretical uncertainties in observables like the differential branching fraction and polarization fractions by using first-principles QCD calculations.
• To provide a more robust foundation for testing the Standard Model and probing new physics via B →K* and Bs →φ decays.
• To address open issues in lattice calculations, particularly threshold effects and nonfactorizable contributions from charmonium resonances.
• To support global fits to experimental data by providing form factors with controlled systematic uncertainties and improved parametrization.

Proposed method

• Used unquenched 2+1 flavor MILC gauge configurations with physical-bottom quarks and high-statistics correlators (30,000+ estimates per ensemble).
• Employed nonrelativistic QCD for bottom quarks in the B(s) rest frame, accurate to O(v⁴), with perturbative matching to physical currents at O(αₛ², αₛΛQCD/mb, Λ²QCD/mb²).
• Applied a novel parametrization of form factors F(t) that includes quark mass dependence via ∆x and ∆xs, and enforces kinematic constraints at q² = 0 and q² = t⁻ using fake data points with controlled uncertainties.
• Implemented simultaneous fits to vector/axial and tensor/pseudotensor form factor sets to ensure consistency with equations of motion at endpoints.
• Estimated correlations between form factor parameters by fitting subsets of 3–4 form factors, focusing on pairs with kinematic constraints (A₀–A₁₂, T₁–T₂).
• Used bootstrapped fits to extract form factors and propagate statistical and systematic uncertainties, with physical results obtained at mπ, mηs = physical values.

Experimental results

Research questions

• RQ1 How can lattice QCD form factors for B →K* and Bs →φ decays be improved to reduce theoretical uncertainties in rare B decay observables?
• RQ2 To what extent do kinematic constraints at q² = 0 and q² = t⁻ affect the precision of polarization fractions like FL?
• RQ3 What is the impact of including cross-correlations between form factor parameters on the determination of angular observables?
• RQ4 How do the updated form factors affect the comparison between Standard Model predictions and experimental data on differential branching fractions?
• RQ5 What are the dominant open theoretical challenges in lattice QCD calculations of rare B decays, particularly regarding nonlocal matrix elements and threshold effects?

Key findings

• The inclusion of all four kinematic constraints (at q² = 0 and q² = t⁻) in simultaneous fits significantly improves the consistency of form factor parametrizations with the equations of motion.
• The longitudinal polarization fraction FL is shifted and more precisely determined compared to previous results, due to the enforced A₁₂(t⁻)/A₁(t⁻) constraint.
• The differential branching fraction predictions show negligible changes compared to prior work, with experimental data still consistently below the theoretical prediction in low-recoil bins.
• The central value of S₃ in Bs →φμ⁺μ⁻ shifts by less than 1σ, and the Wilson coefficient fit yields CNP₉ = −1.1 ± 0.5 and C′₉ = 1.2 ± 0.9, consistent with earlier results.
• Correlation estimates between A₀ and A₁₂, and between T₁ and T₂, were found to be essential, while broader cross-correlations could not be reliably determined due to poorly constrained covariance matrices.
• The study identifies nonfactorizable contributions from charmonium resonances (e.g., ψ(4160)) as a major open issue, with observed contributions larger than expected, challenging theoretical control.