Higher Radial & Orbital Excitations

• Higher radial and orbital excitations are quantum states with extra nodes or increased angular momentum, serving as benchmarks for nonperturbative QCD and atomic structure tests.
• Advanced methodologies like variational GEVP and structured light excitation enable precise analysis of these states in atomic, hadronic, and multiquark systems.
• Experimental observations in heavy-light meson spectra and exotic tetraquark decays critically validate theoretical models and drive progress in understanding complex quantum dynamics.

Higher radial and orbital excitations refer to quantum states of composite systems—such as atoms, hadrons, or multiquark states—in which the constituent particles exhibit additional nodes (radial excitations) or higher orbital angular momentum (orbital excitations) relative to the ground state. These excitations are manifest across atomic, hadronic, and multiquark spectra, and their characterization is pivotal for understanding the structure, spectra, and interaction mechanisms governing complex quantum systems. Experimental identification and theoretical modeling of such excitations provide stringent tests of nonperturbative QCD, atomic structure theory, and potential models.

1. Definitions and Spectroscopic Framework

Quantum states are classified by their radial quantum number $n$ and orbital angular momentum $l$ (or $L$ in hadronic/atomic notation). States with $n > 1$ but unchanged $l$ are termed higher radial excitations; those with $l > 0$ (i.e., $P$-wave, $D$-wave, etc.) are orbital excitations. In multi-body systems, further complexity arises due to possible excitations in relative coordinates ("$\rho$-mode," "$\lambda$-mode") and color-spin coupling, as relevant for tetraquarks and complex hadrons (Wang et al., 2021).

In the atomic context, Rydberg states refer to highly excited configurations with large $n$; in hadronic physics, higher radial and orbital excitations are systematically mapped via quantum numbers $n\,{}^{2S+1}L_J$ for mesons and baryons. For exotic multiquark systems, the assignment involves $J^{PC}$, internal color structure, and sometimes exotic quantum numbers (e.g., $0^{--}$, $1^{-+}$) forbidden to simple $q\bar q$ configurations.

2. Theory and Model Descriptions

Atomic Systems and Rydberg States

Atomic Rydberg states are accurately modeled using single-particle Schrödinger equations with effective potentials incorporating core polarization and quantum defect corrections. The effective potential of Marinescu et al. includes a nonlocal, core-polarized term and reproduces both inner-core electronic behavior and the asymptotic hydrogenic tail (Sanayei et al., 2015).

Radial and orbital excitations obey

$E_{n,j,l} = -\frac{1}{(n - \Delta_{j,l})^2} ,$

where $n$ is the principal quantum number, $\Delta_{j,l} = \delta_l + \eta_{j,l}$ is the quantum defect, and $\eta_{j,l}$ is the fine-structure correction. Uniform WKB-type approximations (Langer form near the outer turning point; Fock ansatz near $r \to 0$) allow for analytic expressions for radial wavefunctions across the full range of $r$, including for large $n$ (Sanayei et al., 2015).

A subtle "anomaly" for $l=3$ in Rb, Cs arises due to a tiny classically allowed region inside the core, causing three turning points in the effective potential and invalidating standard two-point WKB treatments. Numerical solutions restore agreement with experiment (Sanayei et al., 2015).

Hadronic Spectroscopy

Hadronic models, notably the relativized Godfrey–Isgur (GI) quark model with modified ("screened") linear confinement, predict mass spectra and decay properties of higher radial and orbital excitations in systems such as open-charm ($D$) mesons (Song et al., 2015). The effective Hamiltonian comprises a central potential (Coulomb + screened linear), smeared spin–spin, and Breit–Fermi corrections. Radial excitations correspond to changes in $n$, and orbital excitations increase $L$.

For heavy-light mesons in the static limit, interpolating operators such as $\bar q \gamma_k h(x)$ project onto $S$-wave ($1^-$) configurations, while $\bar q \nabla_k h(x)$, with a covariant derivative, access $P$-wave ($1^+$) and multi-hadron thresholds (Blossier et al., 2016). The Generalized Eigenvalue Problem (GEVP) approach is used to extract the spectrum from a matrix of two-point correlators with various Gaussian smearings and Dirac structures.

Multiquark States and Tetraquarks

Fully-charmed tetraquarks ($cc\bar c\bar c$) exhibit a spectrum of S-wave radial excitations ($n=2,3,\dots$) and $P$-wave orbital excitations. The nonrelativistic four-quark model incorporates one-gluon-exchange (OGE) and scalar linear confinement, matched to the charmonium spectrum. In this framework, two types of $L=1$ orbital excitations are distinguished: $\rho$-mode (inside one diquark or antidiquark cluster) and $\lambda$-mode (between clusters), each with possible color configurations $3_c \otimes \bar 3_c$ or $6_c \otimes \bar 6_c$ (Wang et al., 2021).

An essential finding is that the $6_c \otimes \bar 6_c$ configuration is typically more deeply bound than $3_c \otimes \bar 3_c$ for the lowest states, due to stronger inter-cluster attraction.

3. Experimental Spectra and Assignments

Charmed Meson Family

The modified GI model reproduces the observed ordering and approximate mass values of the higher $D$ meson excitations. For example, $D(2550)/D_J(2580)$ is identified as $2^1S_0$, and $D^*(2600)/D^*_J(2650)$ as mixed $2^3S_1$–$1^3D_1$ (mostly $2^3S_1$) (Song et al., 2015). Orbital excitations such as $D_J(2740)/D(2750)$ correspond to $1D'(2^-)$ assignments, with mixing angles reflecting heavy-quark symmetry.

Predicted masses for yet-unobserved higher excitations (2P, 2D, 1F, etc.) cluster in the 3.05–3.15 GeV region, with widths and dominant decay modes calculable in the $^3P_0$ quark pair creation model.

Selected Higher Excitations in the $D$-meson Family

State	Theoretical $M$ (GeV)	Assignment/Comment
$2^1S_0$	2.534	$D(2550)/D_J(2580)$
$2^3S_1$	2.593	$D^*(2600)/D^*_J(2650)$
$1^3D_3$	2.779	$D^*(2760)/D^*_J(2760)$
$3^1S_0$	2.976	$D_J(3000)$
$3^3S_1$	3.015	$D^*_J(3000)$
$1^3F_2$	3.053	Not yet observed

Tetraquarks and Exotic Signals

The $T_{cc\bar c\bar c}$ spectrum calculated with the parameters fixed to reproduce charmonium benchmarks results in S-wave ground states at 6405–6547 MeV depending on $J^{PC}$, with first radial excitations at 6867–6917 MeV. Low-lying $P$-wave states ($\rho$-modes in $6_c \otimes \bar 6_c$) appear at 6589–6608 MeV, with the mass gap between $1S$ and $1P$ notably smaller than between $1S$ and $2S$ states (Wang et al., 2021).

The observed $X(6900)$ state in $J/\psi J/\psi$ at LHCb coincides with these predictions, suggesting a possible identification as either a $2S$ ($0^{++}$, $2^{++}$) or $P$-wave ($1^{-+}$, $2^{-+}$) tetraquark (Wang et al., 2021).

4. Methodologies for Excitation Extraction

Lattice QCD Techniques

In heavy-light mesons, radial and orbital excitations are isolated via matrices of two-point correlators—constructed from interpolators with different smearings and Dirac structures—and the variational GEVP method. The use of derivative-based interpolators, such as $\bar q \nabla_k h$, couples strongly to $P$-wave and two-hadron states, making the inclusion or exclusion of these operators a diagnostic for the nature of extracted excitations (Blossier et al., 2016). Charge-density and axial-density spatial distributions further serve as diagnostics; nodes in axial densities confirm the one-particle nature of the radial excitation, while drifting integrated vector charge signals multihadron contamination.

Numerical extractions yield, for the $B^*$ system, a first radial–ground splitting of approximately $680 \pm 40$ MeV and an orbital–ground splitting of $1.3 \pm 0.1$ GeV (E5 ensemble, $a=0.065$ fm).

Atomic Excitation with Structured Light

Laguerre–Gauss beams, which carry photon orbital angular momentum (OAM) $\ell_0$, can induce atomic transitions to higher-$l$ Rydberg states inaccessible by conventional dipole selection rules (Rodrigues et al., 2015). The selection rules are generalized: $$\Delta m = q + \ell_0, \quad |\ell-\ell^*| \leq \ell' \leq \ell + \ell^*,$$ allowing, for example, excitation from $\ell=1$ to $\ell'=3$ and beyond in a single-photon process. The radial matrix elements scale as $n^{-3 + \alpha(\ell_0)}$ with $\alpha(0) = 0$, $\alpha(1) \approx -0.02$, etc., such that for $\ell_0=1,2,3$, $p$ in $|R|^2 \propto n^p$ is $-3.3, -2.8, -2.4$, respectively. Rates for such transitions are substantial (5–20% of usual dipole rates for $P \to D$) and can be fully calculated with the tabulated coupling coefficients (Rodrigues et al., 2015).

5. Decay Properties and Phenomenological Implications

Higher radial and orbital excitations frequently decay via OZI-allowed strong processes. The $^3P_0$ (quark pair creation, QPC) model gives partial and total widths, predicting, for instance, the main decay channels and branching ratios of $D^*(2600)$ to be consistent with experiment, although total widths tend to be underpredicted—a known issue of the model (Song et al., 2015). For Rydberg states in atoms, properties such as the Fermi-contact hyperfine splitting constants scale precisely with excitation number: $$A_{n,1/2,0}^{\mathrm{(HFS)}} (n-\delta_0)^3 = \mathrm{const}$$ (Sanayei et al., 2015).

Exotic multiquarks—such as tetraquark $1^{-+}$ or $0^{--}$ states—are predicted to have distinctive decay modes ($J/\psi J/\psi$ for $1^{-+}$) and mass gaps (P-wave–S-wave) that facilitate identification through channels inaccessible to ordinary mesons (Wang et al., 2021).

6. Challenges and Future Prospects

Extraction of higher excitations is complicated by proximity to multi-hadron thresholds and the potential for operator bases to have suppressed overlap with certain states, leading to "missing" levels in variational analyses (Blossier et al., 2016). Optimal disentanglement requires a judicious basis of interpolators, thorough analysis of density distributions, and, where relevant, the inclusion of explicit two-hadron operators.

For heavy-quark and tetraquark systems, future studies will benefit from lighter pion mass ensembles in lattice QCD, explicit inclusion of scattering operators, and systematic continuum/chiral extrapolation of energy splittings. In atomic physics, further exploration of OAM-carrying photon interactions with complex atoms or ions may extend the reach of high-$l$ excitations beyond current limits.

A plausible implication is that identification of narrow, parity-exotic states in the predicted mass bands by future experiments will provide critical evidence for the nature and dynamics of higher radial and orbital excitations, both in conventional and exotic hadrons.