Nonperturbative Parameters of Inclusive $\Lambda_b$ Decays from Small Velocity Sum Rules

We investigate the nonperturbative parameters entering the heavy quark expansion for the inclusive decay rates of the $\Lambda_b$ baryon, focusing on the kinetic term $\hat{\mu}_\pi^2$ and the Darwin term $\hat{\rho}_D^3$. Recent evaluations have yielded an unexpectedly large Wilson coefficient associated with the dimension-six Darwin operator, which motivates a more detailed examination of the associated hadronic matrix element. We constrain $\hat{\mu}_\pi^2$ and $\hat{\rho}_D^3$ using Small Velocity Sum Rules for the inclusive semileptonic decay $\Lambda_b \to X_c e^- \bar{\nu}_e$. These sum rules relate moments of hadronic structure functions, computed via the Operator Product Expansion, to hadronic matrix elements of relevant exclusive transitions. Truncating the hadronic side to include the lowest-lying charmed baryon states with spin-parity $J^P = 1/2^+, 1/2^-, 3/2^-$, we employ the corresponding lattice QCD form factors near zero recoil as inputs. By analysing zeroth and higher moments, we derive inequalities that define an allowed region in the $(\hat{\mu}_\pi^2, \hat{\rho}_D^3)$ plane. The derived constraints are consistent, within uncertainties, with estimates from spectroscopic relations and the nonrelativistic constituent quark model. This work refines the determination of key HQE parameters for $\Lambda_b$, with implications for precision predictions of heavy baryon lifetimes and inclusive decays of heavy baryons in general.