Nonperturbative Parameters of Inclusive $Λ_b$ Decays from Small Velocity Sum Rules

TL;DR

The paper addresses the precise determination of nonperturbative HQE parameters for inclusive $\\Lambda_b$ decays, motivated by a surprisingly large Darwin coefficient. It employs Small Velocity Sum Rules to relate moments of the hadronic structure tensor calculated in the OPE to exclusive hadronic matrix elements, using lattice QCD form factors for the three lowest charmed-baryon states near zero recoil and truncating the hadronic sum to these states. By deriving zeroth-moment inequalities and higher-moment constraints (via the $Y_2$ and $Z_2$ combinations), it maps an allowed region in the $(\\hat{\\mu}_\\pi^2,\\hat{\\rho}_D^3)$ plane and checks compatibility with spectroscopic relations and the nonrelativistic quark model. The results provide a refined determination of HQE inputs for the $\\Lambda_b$ and have implications for precision predictions of heavy-baryon lifetimes and inclusive decays, notably regarding how a larger Darwin term would affect the $\\Lambda_b$ width and lifetime ratios.

Abstract

We investigate the nonperturbative parameters entering the heavy quark expansion for the inclusive decay rates of the $Λ_b$ baryon, focusing on the kinetic term $\hatμ_π^2$ and the Darwin term $\hatρ_D^3$. 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 $\hatμ_π^2$ and $\hatρ_D^3$ using Small Velocity Sum Rules for the inclusive semileptonic decay $Λ_b \to X_c e^- \barν_e$. 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 $(\hatμ_π^2, \hatρ_D^3)$ 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 $Λ_b$, with implications for precision predictions of heavy baryon lifetimes and inclusive decays of heavy baryons in general.

Figures (4)

• Figure 1: Singularity structure of $h_{\alpha\beta}$ for a fixed value of $\mathbf{q}$ in the complex $\epsilon$-plane: Semileptonic decay cut (solid blue), adjacent cut (dashed red), distant cut (dotted green). Contour enveloping the cut on the positive real axis is denoted by $\mathcal{C}_1$. The semileptonic cut receives contributions from charmed states $X_c$ in $\Lambda_b\to X_c\ell\bar{\nu}$ and extends from $\epsilon_{\text{min}}=0$ to $\epsilon_{\text{max}}=m_{\Lambda_b}-E_{\Lambda_c}-\vert\vec{q}\vert$. The point $\epsilon_0$ (marked by $*$) indicates a location on the negative real axis, sufficiently far from the cuts and with $|\epsilon_0| \gg \Lambda_{\text{QCD}}$ where the OPE can be applied.
• Figure 2: Region in the $(\hat{\mu}_\pi^2, \hat{\rho}_D^3)$ plane that satisfies the sum-rule inequalities for $\frac{dZ_2}{d\vert\vec{q}\vert^2}$ (light green) and $\frac{dZ_3}{d\vert\vec{q}\vert^2}$ (gray).
• Figure 3: Allowed region in the $(\hat{\mu}_\pi^2, \hat{\rho}_D^3)$ plane where the sum-rule inequalities are satisfied. This region is defined by imposing that the cumulative probability—derived from matching lattice QCD determinations of the hadronic moments with the corresponding OPE predictions—exceeds a $50\%$ threshold (see the text in Sec. \ref{['Numerical Analysis of the Constraints']} for more details). All values are given in the kinetic scheme with the cut-off scale $\mu = 0.75~\text{GeV}$.
• Figure 4: Interplay of the three most constraining inequalities. The allowed region in the $(\hat{\mu}_\pi^2,\hat{\rho}_D^3)$ plane is defined by requiring the OPE predictions to exceed the central lattice values of the hadronic moments. $Z_2$ denotes the derivative of the relevant moment at zero recoil (see Eq. \ref{['Eq:SRZ3']}).