[Paper Review] B $ o$ $\pi$ form factor with 2 flavours of $O(a)$ improved Wilson quarks

Felix Bahr, Fabio Bernardoni+4 more|arXiv (Cornell University)|Jan 1, 2012

Particle physics theoretical and experimental studies6 references7 citations

TL;DR

This paper presents a lattice QCD study of the B → π form factor f+(q²) using Nf = 2 O(a)-improved Wilson fermions and non-perturbative renormalization in the static limit of HQET. Employing stochastic all-to-all propagators with time-dilution and gauge smearing to suppress excited-state and noise contamination, the authors achieve a 5–10% precision on f+(q²) at large q², enabling a fit to the z-expansion parametrization that determines |Vub| with ~15% uncertainty, offering a path to resolve the 3σ tension in |Vub| determinations.

ABSTRACT

The determinations of $|V_{\ m ub}|$ from the exclusive branching ratios of $B\ o \ au \ u$ and $B \ o \\pi l \ u$ tend to show a tension at the level of $3\\sigma$ \\cite{Beringer:1900zz}. On the theoretical side they depend on the lattice computation of the hadronic matrix elements $f_{\ m B}$ and the $B\ o \\pi$ form factor $f_+(q^2)$. To understand the tension, improved precision and a careful analysis of the systematics involved are necessary. Working towards this goal, we present preliminary lattice results of the ALPHA collaboration for the $B\ o \\pi$ form factor $f_+(q^2)$ with $N_{\ m f}=2$ flavours of $O(a)$-improved Wilson fermions. Our computation uses HQET in the static limit, pion masses ranging down to $\\sim250$ MeV, large volumes with $m_\\pi L >4$, three lattice spacings, and non-perturbative renormalization. We describe the techniques adopted to reduce the statistical noise (stochastic all-to-all with full time dilution) and the contamination from excited states (smearing for the B and the pion). We estimate the size of the chiral and continuum extrapolations. We discuss the impact our result could have to clarify the above mentioned discrepancy in the determination of $|V_{\ m ub}|$.

Motivation & Objective

• To reduce theoretical uncertainty in the exclusive determination of |Vub| by improving the lattice computation of the B → π form factor f+(q²).
• To address systematic challenges in lattice B → π form factor calculations, including signal-to-noise ratio and finite-T effects.
• To test the feasibility of non-perturbative renormalization and matching in HQET for B-meson decays with Nf = 2 dynamical quarks.
• To provide a precise lattice determination of f+(q²) at large momentum transfer to constrain |Vub| and resolve the 3σ tension with inclusive and B→τν determinations.

Proposed method

• Uses the static limit of Heavy Quark Effective Theory (HQET) to describe B-meson decays, enabling non-perturbative renormalization and continuum extrapolation.
• Computes the three-point function ratio R(tπ, tB) using stochastic all-to-all propagators with full time-dilution to enhance signal-to-noise ratio.
• Applies gauge smearing to both B and π interpolating fields to suppress contamination from excited states and improve ground-state dominance.
• Employs a two-parameter fit ansatz (eq. 4.1) to model finite-T effects in the T-periodic time direction, accounting for contributions from π-B intermediate states.
• Uses non-perturbative renormalization and matching for the vector current, with perturbative 3-loop matching for the current analysis.
• Performs a z-expansion parametrization of f+(q²) to fit experimental data and extract |Vub| as a free parameter.

Experimental results

Research questions

• RQ1 Can a precise lattice determination of f+(q²) be achieved with Nf = 2 O(a)-improved Wilson fermions and non-perturbative renormalization in the static HQET framework?
• RQ2 To what extent do finite-T effects and excited-state contamination limit the extraction of f+(q²) on finite lattices, and how can they be corrected?
• RQ3 What is the achievable precision on f+(q²) and |Vub| using current lattice techniques, and can it resolve the 3σ tension in |Vub| determinations?
• RQ4 How do O(a) discretization and 1/mb corrections affect the form factor, and can they be systematically controlled in future work?

Key findings

• The form factor f+(q²) is extracted with a precision of 5–10% on the O7 ensemble, which is closest to the physical point.
• The two-parameter fit (eq. 4.1) successfully models finite-T effects, with the red curve in Figure 2 showing good agreement with data.
• A plateau in the form factor is observed at early tπ due to pion smearing, while B-meson excited-state contamination remains significant, requiring careful tB selection.
• The z-expansion parametrization of f+(q²) allows a fit to experimental data, yielding |Vub| with a statistical uncertainty of approximately 15%.
• The results are consistent across different tB slices and fit ranges, indicating robustness of the signal extraction method.
• The authors project that including O(1/mb) and O(a) improvements, along with non-perturbative matching, will reduce |Vub| uncertainty to the 5–10% level in future work.