Regge Trajectories in Baryon Excitations

• Regge trajectories in baryon excitations are equations relating the squared mass, spin, and excitation quantum numbers, revealing insights into QCD confinement.
• String-theoretic and holographic approaches, including quark–diquark models, offer predictive mass relations across light and heavy baryon families.
• Methodological modifications such as quark mass effects and diquark clustering yield non-linear trajectories that refine baryon classification and resonance predictions.

Regge trajectories in baryon excitations constitute a cornerstone of contemporary hadron spectroscopy, providing a quantitative relation between mass, spin, and excitation quantum numbers for baryons. Historically motivated by the phenomenology of meson spectra, Regge trajectories have proven to be deeply connected to the confining dynamics of Quantum Chromodynamics (QCD) in both string-theoretic and field-theoretic frameworks. Their forms, universality, and deviations—arising from quark masses, dynamical effects, and internal degrees of freedom such as diquark correlations—yield powerful insights into the structure and classification of baryon resonances across the light, strange, charm, bottom, and even triply heavy sectors.

1. Foundations of Regge Trajectories in Baryon Spectroscopy

Regge trajectories embody the empirical observation that the squared mass $M^2$ of hadronic resonances increases approximately linearly with the total angular momentum $J$ (so-called Chew–Frautschi plots) and, for radially excited states, with the principal quantum number $n$. The generic forms are: $$J = \alpha M^2 + \alpha_0 \quad \text{(orbital/excitation trajectories)}$$

$$n = \beta M^2 + \beta_0 \quad \text{(radial trajectories)}$$

Values of trajectory slopes $\alpha$ and $\beta$ reflect underlying QCD dynamics—specifically the string tension in confining models or the strength of the linear potential in effective quark models. In baryons, these relations may be further modified by quark mass effects, diquark clustering, and internal excitation modes.

For light baryons, experimental families (such as the $N$ and $\Delta$ series) exhibit well-defined approximately linear Regge trajectories in both $J$ and $n$ (Oudichhya et al., 2022), while heavy (charm, bottom) baryons show similar behavior—with slopes extracted from fits to experimental and theoretical mass spectra (Ebert et al., 2011, Thakkar et al., 2016, Jia et al., 2019).

2. String--Theoretic and Holographic Approaches

String models posit baryons as rotating relativistic strings with massive endpoints representing constituent quarks or diquarks (1305.3985, Sonnenschein et al., 2014). This framework leads to

$$J \simeq \alpha' M^2 + \alpha_0, \quad \text{with} \quad \alpha' = \frac{1}{2\pi \gamma}$$

for the canonical quark–diquark configuration, where $\gamma$ is the string tension. Notably, the slope $\alpha'$ extracted from baryon data ($\sim 0.9$--$0.95\,\text{GeV}^{-2}$) aligns closely with that for mesons, suggesting a universal QCD string tension for hadrons (Sonnenschein et al., 2014).

More sophisticated holographic constructions treat baryons as D-brane configurations in a warped AdS${}_5\times S^5$ background. In "Holographic Approach to Regge Trajectory and Rotating D5 brane" (Ghoroku et al., 2011), the baryon vertex is built from a wrapped D5 brane, leading to two special configurations:

• Point vertex: All angular momentum carried by $N_c$ strings, with slope $\alpha'_B=(2\alpha'_M)/(N_c(1+\beta))$; for $N_c=3$, this slope is significantly less than observed $\Delta/N$ slopes.
• Split (spinning) vertex: The vertex itself rotates and contributes dominantly, yielding an effective tension

$\tau_B = \frac{2N_c}{3\pi}\sin^3\theta_c\tau_M$

and Regge slope

$\alpha'_B = \frac{1}{2\pi \tau_B}$

The split vertex configuration can match experimental slopes, resolve the baryon mass shift relative to mesons, and agrees with universal QCD scale setting via the gauge condensate $\langle F^2 \rangle$.

In addition, holographic softwall models (traditionally used for glueballs and mesons) have been adapted for baryons, with analytically solvable mass spectra yielding linear Regge trajectories for both even (pomeron-like) and odd (odderon-like) spins (Capossoli et al., 2015).

3. Diquark Picture and Excitation Modes

A key advance in baryon Regge phenomenology is the explicit use of diquark clustering, organizing baryons as bound states of a quark and a diquark ("quark–diquark models") (Ebert et al., 2011, Masjuan et al., 2017, Chen et al., 2023). This leads to two distinct types of internal excitations:

• $\lambda$-mode excitations: Excitations between the diquark and the light or heavy quark. For doubly and triply heavy baryons, the $\lambda$-mode mass spectrum follows

$$M \sim (x_\lambda + c_0)^{1/2} \quad (x_\lambda = L, N_r)$$

as in the heavy–light quark systems (Song et al., 18 Feb 2025).

• $\rho$-mode excitations: Internal excitations of the diquark subsystem (orbital $l$ or radial $n_r$ quantum numbers). These heavy–heavy systems follow a steeper dependence,

$$M \sim (x_\rho + c_0)^{2/3} \quad (x_\rho = l, n_r)$$

with trajectories distinctly shaped (concave in $(M^2, x)$ space) compared to the nearly linear trajectories for light baryons (Xie et al., 25 Jul 2024, Song et al., 18 Feb 2025).

The explicit formulas enable straightforward spectral estimates for both modes and clarify the internal structure of doubly and triply heavy baryons.

4. Modifications Due to Quark Mass and Non-Linearity

While linear Regge trajectories well characterize mesonic systems, baryons often show significant non-linearities. Accounting for finite constituent quark masses in string models and three-body geometries yields parabolic or concave Regge curves

$J \ne \alpha' M^2 + \alpha_0$

but rather $J$ as a non-linear function of $M^2$, where effective slope and intercept become functions of the mass distribution and velocity factors (Ranjan et al., 2011, Chen et al., 2023, Xie et al., 25 Jul 2024). In the low mass and angular momentum regime, degeneracies arise: different baryons with distinct internal quark compositions can exhibit identical $M$ and $J$ (Ranjan et al., 2011). This complicates quantum number assignments and indicates a breakdown of uniqueness in classical Regge mapping for baryons.

Concavity in Regge trajectories becomes pronounced as constituent masses grow; nearly linear behavior is recovered only in the chiral/light-quark limit (Xie et al., 25 Jul 2024).

5. Universal, Non-Universal, and Phenomenological Aspects

The extent to which Regge trajectory slopes are universal—for both orbital ($J$) and radial ($n$) excitations—has been rigorously tested via linear regression fits, incorporating resonance widths as uncertainties (Masjuan et al., 2012, Masjuan et al., 2017). For light mesons, radial slopes are statistically larger than angular slopes, refuting strict universality ($M^2 = a(n+J)+c$) at the $>4\sigma$ level (Masjuan et al., 2012). Similarly, baryonic slopes differ between families and excitation modes, with flavor and diquark content modulating the trajectory parameters (Oudichhya et al., 2022, Ebert et al., 2011, Masjuan et al., 2017).

Key relations for slopes/intercepts in baryon families are given by additivity rules: $$\frac{1}{\alpha'_{i i q}} + \frac{1}{\alpha'_{j j q}} = \frac{2}{\alpha'_{i j q}}$$

$a_{i i q}(0) + a_{j j q}(0) = 2a_{i j q}(0)$

These relations permit analytical predictions for missing states and enable systematic assignment of quantum numbers for newly observed baryons (Oudichhya et al., 2022).

A plausible implication is that the diquark picture provides a universal organizational principle for baryon Regge trajectories, both for low-lying states and for highly excited hadrons.

6. The Large-N$_c$ Expansion and Operator Analysis

The $1/N_c$ expansion method enables systematic classification and analysis of baryon states, leading to hierarchical mass operators involving orbital, spin, and flavor structures (Matagne et al., 2013). By treating all $N_c$ quarks equally, distinct Regge trajectories for symmetric and mixed symmetric multiplets emerge, with the leading singlet term's contribution varying linearly with band number $N$. This approach yields:

• Discrete Regge trajectories for [56]-plets (symmetric) and [70]-plets (mixed symmetry).
• Clarified roles for spin and isospin operators in baryon mass splitting, with implications for multiplet assignment and trajectory separation. This methodology informs model-independent interpretations of the baryon spectrum and refines the connection between operator expectations and observed Regge patterns.

7. Predictive Power and Implications for Experimental Baryon Spectroscopy

Modern quantitive fits—using relativistic quark models, hCQM, and string-based approaches—show that both radial and orbital excitations of baryons (Λ, Σ, Ξ, Ω, N, Δ, along with charmed and bottom families) follow nearly linear or concave Regge trajectories with slopes extracted from experimental data (Ebert et al., 2011, Thakkar et al., 2016, Menapara et al., 2022, Oudichhya et al., 2022, Jia et al., 2019). These fits facilitate quantum number assignments for poorly characterized states (e.g., Σ(2250), Ξ(1950)), cross-verification with alternative models, and predictions of yet-unobserved resonances in the heavy sector. Mass splitting analyses—especially in the heavy diquark regime—reveal smaller level spacings for diquark excitations compared to light-quark excitations, a manifestation of adiabatic separation in baryon dynamics (Song et al., 2022).

The systematic application of Regge trajectory analysis, incorporating diquark structures, mass effects, and symmetry considerations, continues to be an essential tool for interpreting the experimental baryon spectrum, classifying new states, and refining theoretical models for hadronic excitations.