Comparison of lattice QCD results for inclusive semi-leptonic decays of $B$ mesons with the OPE

Abstract

A recently proposed approach to compute the inclusive semi-leptonic decay rate on the lattice allows for the calculation of various quantities (differential distributions and moments) in different subchannels. We systematically compare the lattice QCD results for an unphysically light bottom quark with OPE-based predictions, including leading order perturbative and O 1/𝑚 3 power corrections, and explore possible strategies to decrease the systematic uncertainty.

Composition of the inclusive semi-leptonic decay of B meson

“…The present results can already been used to compare with the analytic OPE calculations. Even though the 𝑏-quark mass is not tuned to the physical value, the comparison would still yield useful tests of the theoretical approaches [6].…”

“…𝑗 can be easily computed for arbitrary kernel operator 𝐾 (𝜔). Since the Chebyshev polynomials 𝑇 * 𝑗 (𝑒 −𝜔 ) are constructed from polynomials (𝑒 −𝜔 ) 𝑚 with positive integer 𝑚's, they are related to the transfer matrix 𝑒 − Ĥ 𝑡 appearing in (6). The forward Compton amplitude (6) can therefore be used to approximate the target quantity (5).…”

“…This contribution mainly describes the lattice computatin, and the comparison with the OPE calculation for the same set of parameters is separately presented in [6].…”

A lattice QCD perspective on weak decays of b and c quarks Snowmass 2022 White Paper

“…The new method also opens the door for further applications, such as moments of the lepton energy and the hadronic invariant mass. Exploratory numerical studies now exist [284,289,290], raising hopes for future work with physical parameters and controlled systematic uncertainties. These calculations may play a significant role in resolving the tension between inclusive and exclusive determinations of CKM matrix elements.…”

Composition of the inclusive semi-leptonic decay of $B$ meson

“…The coefficients * can be easily computed for arbitrary kernel operator ( ). Since the Chebyshev polynomials * ( − ) are constructed from polynomials ( − ) with positive integer 's, they are related to the transfer matrix − ˆ appearing in (6). The forward Compton amplitude (6) can therefore be used to approximate the target quantity (5).…”