[Paper Review] Lattice Calculations of B to K/K*l+l- form factors

Ran Zhou|arXiv (Cornell University)|Jan 4, 2013

Particle physics theoretical and experimental studies17 references3 citations

TL;DR

This paper presents advanced lattice QCD calculations of $B\to K/K^*l^+l^-$ form factors using dynamical $N_f=2+1$ flavor gauge configurations, achieving ~5% total uncertainty at large $q^2$ by combining chiral-continuum extrapolation, $z$-expansion, and constrained fits. It improves upon previous quenched results by eliminating sea-quark effects and enabling precise, first-principles predictions across the full $q^2$ range for new physics searches.

ABSTRACT

This paper gives a brief review on the recent lattice QCD calculations of the B to K/K*l+l- semi-leptonic decay form factors.

Motivation & Objective

• To provide precise, first-principles lattice QCD form factors for $B\to K/K^*l^+l^-$ decays to test the Standard Model and probe new physics.
• To overcome the limitations of previous quenched calculations, which suffered from 15%-20% uncertainties and lacked sea quark effects.
• To enable accurate comparison between experimental data and SM predictions by controlling systematic errors in form factor calculations.
• To develop and apply advanced fitting techniques—such as $z$-expansion and $SU(2)$ HMS $\chi$PT—for reliable extrapolation across the $q^2$ spectrum.
• To provide a cross-checkable framework using multiple collaborations (FNAL/MILC, HPQCD, Cambridge/W&M/Edinburgh) with differing actions and methods.

Proposed method

• Uses $N_f=2+1$ flavor gauge ensembles from the MILC collaboration with lattice spacings from 0.045 fm to 0.12 fm, reducing discretization errors.
• Employs different valence quark actions: asqtad-improved staggered for light quarks and either Sheikholeslami-Wohlert (SW) or (moving)-NRQCD for the $b$ quark to control heavy quark discretization effects.
• Applies $z$-expansion with unitarity and heavy quark power counting constraints to parametrize $q^2$-dependence of form factors, enabling systematic improvement.
• Uses chiral-continuum extrapolation via $SU(2)$ HMS $\chi$PT for form factors at low $q^2$, validated with $z$-expansion fits.
• Combines statistical and systematic error budgets, including finite volume, scale, and renormalization uncertainties, to achieve total uncertainties of ~5% at large $q^2$.
• Uses modified $z$-expansion with heuristic discretization and light quark mass terms to perform combined chiral, continuum, and shape fits in HPQCD and Cambridge/W&M/Edinburgh analyses.

Experimental results

Research questions

• RQ1 Can lattice QCD provide form factors for $B\to K/K^*l^+l^-$ decays with sub-10% uncertainty across the full $q^2$ range, including large $q^2$?
• RQ2 How do different valence quark actions (SW, NRQCD, asqtad) affect the precision and consistency of $B\to K/K^*l^+l^-$ form factor calculations?
• RQ3 To what extent can $z$-expansion with unitarity constraints reduce truncation errors in $q^2$ extrapolation of form factors?
• RQ4 What is the impact of including dynamical sea quarks on the accuracy and systematic error control in rare $B$ decays compared to quenched approximations?
• RQ5 Can the combination of $z$-expansion, $\chi$PT, and constrained fits yield a systematically improvable and reliable parametrization of form factors across $q^2$?

Key findings

• The FNAL/MILC collaboration achieves a total uncertainty of ~5% in the form factors at large $q^2$ ($q^2 \gtrsim 15~\text{GeV}^2$), significantly improving over the 15%-20% uncertainty in earlier quenched calculations.
• Form factors at large $q^2$ are obtained directly from lattice data using chiral-continuum extrapolation with $SU(2)$ HMS $\chi$PT, while low $q^2$ regions are described via $z$-expansion fits.
• The HPQCD collaboration reports form factor measurements with ~1% statistical accuracy at large $q^2$ on $a \approx 0.12~\text{fm}$ ensembles, with future results expected on finer $a \approx 0.09~\text{fm}$ ensembles.
• The Cambridge/W&M/Edinburgh collaboration provides preliminary results for $T_{1,2}$ form factors in $B\to K^*l^+l^-$ using the same modified $z$-expansion method, enabling consistent $q^2$ extrapolation.
• The use of $N_f=2+1$ dynamical fermions removes the major systematic error from the quenched approximation, enabling more reliable predictions for new physics searches.
• The combination of multiple collaborations using different actions and methods provides a robust cross-check, increasing confidence in the precision of the form factor predictions.