[Paper Review] Systematics of radial excitations in heavy-light hadrons

R. M. Woloshyn, Mark Wurtz|arXiv (Cornell University)|Jan 8, 2016

Quantum Chromodynamics and Particle Interactions1 references3 citations

TL;DR

This paper proposes heuristic scaling rules for radial excitation energies in heavy-light hadrons based on nonrelativistic quark model expectations, using power-law potentials to predict quark mass dependence. It presents new lattice QCD results for $Σ_b$ and $λ_b$ baryon masses using free-form smearing, finding a positive-parity radial excitation splitting of $213(42)$ MeV for $Σ_b$, consistent with experiment.

ABSTRACT

Some simple expectations for the quark mass dependence of radial excitation energies of heavy-light hadrons based on consideration of nonrelativistic quantum mechanics are discussed. Experimental and theoretical results are reviewed in light of these expectations. Some new lattice QCD results for masses of Lambda_b and Sigma_b baryons are presented.

Motivation & Objective

• To develop intuitive scaling rules for radial excitation energies in heavy-light hadrons based on nonrelativistic quantum mechanics.
• To assess the consistency of experimental and lattice QCD results with these heuristic rules.
• To present new lattice QCD calculations of $λ_b$ and $Σ_b$ baryon masses using free-form smearing to improve excited state signal.
• To evaluate the validity of quark model expectations for excitation energy trends across different heavy-light systems.
• To provide a framework for interpreting lattice and experimental data in light of qualitative scaling behaviors.

Proposed method

• Uses nonrelativistic quantum mechanics with power-law potentials $V(r) = c r^\nu$ to derive scaling behavior of excitation energies with reduced mass $\mu$.
• Applies scaling relation $E \sim \mu^{-\nu/(\nu+2)}$ to predict how radial excitation energy varies with heavy quark mass and light quark mass.
• Employs free-form smearing in lattice QCD: source fields are smeared using profile functions $G = e^{-r/a_0}$ and $E = e^{-r/a_0}(r-b)$ to enhance overlap with ground and excited states.
• Constructs baryon correlation functions using interpolating fields for $\Lambda_b$ and $\Sigma_b$ with both light quarks unsmeared or smeared via $G$ or $E$ profiles.
• Performs constrained multi-exponential fits to effective energy plots, using time slices up to $t=15$ for positive parity and $t=11$ for negative parity states.
• Uses an $N_f=2+1$ dynamical gauge field ensemble with $32^3 \times 64$ lattice, $a = 0.0907(13)$ fm, and $m_\pi = 156(7)$ MeV, with bottom quarks treated via NRQCD.

Experimental results

Research questions

• RQ1 How does the radial excitation energy of heavy-light hadrons scale with the mass of the heavy quark, according to nonrelativistic quark model expectations?
• RQ2 To what extent do experimental radial excitation energies in the charm sector (e.g., $D^*$, $D_s^*$) conform to the predicted scaling behavior with light quark mass?
• RQ3 What is the radial excitation energy splitting for $\Lambda_b$ and $\Sigma_b$ baryons, and how does it compare to experimental values?
• RQ4 Can free-form smearing in lattice QCD effectively isolate ground and excited states in singly-heavy baryons?
• RQ5 How robust are the extracted masses and splittings to variations in fit parameters and initial time slices?

Key findings

• The radial excitation energy of $\Sigma_b(1/2^+)$ relative to $\Lambda_b(1/2^+)$ is $213(42)$ MeV, consistent with the experimental value of $194(3)$ MeV.
• The $\Lambda_b(1/2^-) - \Lambda_b(1/2^+)$ mass splitting is $344(105)$ MeV, which is consistent with the experimental value of $293(1)$ MeV within error bars.
• The $\Sigma_b(1/2^-) - \Sigma_b(1/2^+)$ radial excitation energy is predicted to be $252(60)$ MeV.
• Correlation functions without source smearing failed to produce reliable signals, indicating that free-form smearing is essential for isolating excited states.
• The results are robust under variations in fit parameters, including initial time slice, number of exponential terms (3 or 4), and choice of priors.
• The observed trends in lattice results support the heuristic rules of thumb: excitation energy decreases with increasing heavy quark mass or light quark mass.