Non linear Regge trajectories of quarkonia from holography

Technical Summary: Nonlinear Regge Trajectories of Quarkonia from Holography

Problem Statement
Standard holographic models, such as the soft-wall AdS/QCD model, successfully describe light mesons by producing linear Regge trajectories where the squared mass scales linearly with the radial excitation number ($m_n^2 \sim n$). However, this linear behavior fails to accurately describe heavy quarkonia (charmonium and bottomonium). Previous phenomenological studies utilizing the quadratic form of the spinless Salpeter-type equation (QSSE) with a Cornell potential suggest that for heavy mesons, the squared masses should scale as $m_n^2 \sim n^{2/3}$ for large radial quantum numbers $n$. Furthermore, existing holographic attempts to fit heavy meson masses often struggle to simultaneously reproduce the correct mass spectrum and decay constants, or rely on ad-hoc mass shifts that lack precision.

Methodology
The authors propose a bottom-up AdS/QCD model for heavy vector mesons that incorporates three specific theoretical modifications to the standard soft-wall framework:

1. Incorporation of Heavy Quark Masses: To account for the non-negligible masses of constituent quarks, the authors solve the inverse Schrödinger problem. They derive a specific dilaton background $\phi_I(z)$ that generates a potential containing a constant mass shift term $M_q^2$. This ensures the spectrum takes the form $m_n^2 = M_q^2 + \text{excitation terms}$, where $M_q$ represents the sum of the constituent quark masses.
2. Nonlinear Regge Trajectories via WKB and Langer Correction: To reproduce the $n^{2/3}$ scaling, the authors employ the WKB approximation with the Langer correction. The standard WKB method is insufficient for potentials containing a $1/z^2$ term (analogous to the centrifugal barrier). By applying the Langer transformation, the authors derive a quantization condition that relates the power of the potential term to the Regge trajectory exponent. They demonstrate that a potential term proportional to $z$ (linear in $z$) yields the desired $m_n^2 \sim n^{2/3}$ behavior.
3. Adjustment of Subleading Terms and Decay Constants:
   • To reproduce the specific intercept parameter $c_0$ in the trajectory $m_n^2 = \beta(n + c_0)^{2/3} + c_1$, a $1/\sqrt{z}$ term is added to the potential.
   • To fit experimental decay constants without disrupting the mass spectrum, an additional term proportional to $e^{-\kappa z}/\sqrt{z}$ is introduced. This term modifies the potential's behavior at small $z$ (where decay constants are determined) while decaying exponentially at large $z$ (preserving the mass spectrum).

The resulting Schrödinger-like potential is:
$$V(z) = \frac{3}{4z^2} - \frac{1}{2}\frac{\kappa^2}{\sqrt{\kappa z}} - \frac{B\kappa^2}{\sqrt{\kappa z}}e^{-\kappa z} + M_q^2 + \kappa^3 z$$
where $B$ is a dimensionless parameter tuned to fit decay constants.

Key Results
The model was fitted to experimental data for charmonium and bottomonium states (masses and decay constants) using the Particle Data Group (PDG) values. The fitting procedure utilized a bootstrap method with 2000 sets of data to determine parameter uncertainties.

• Mass Spectrum: The model successfully reproduces the nonlinear Regge trajectory form $m_n^2 = \beta(n + c_0)^{2/3} + c_1$.
  • For charmonium, the combined normalized root mean square error (CNRMSE) for masses and decay constants is 14.8% (with mass errors alone at 3%).
  • For bottomonium, the CNRMSE is 11.9% (with mass errors alone at 2.3%).
• Physical Parameters: The fitted parameters yield physically meaningful values:
  • The parameter $M_q$ is interpreted as the sum of constituent quark masses, yielding $m_c \approx 1.55$ GeV and $m_b \approx 4.62$ GeV.
  • The parameter $\kappa$ is related to the string tension $\sigma$, yielding $\sigma_c \approx 0.125$ GeV$^2$ and $\sigma_b \approx 0.254$ GeV$^2$. These values are consistent with expectations from potential models.
• Decay Constants: The inclusion of the $e^{-\kappa z}$ term allows the model to fit decay constants for both charmonium and bottomonium with reasonable accuracy (errors of 17.7% and 11.8% respectively), a feature where previous tangent models showed discrepancies.

Significance and Claims
The authors claim that this work represents a significant improvement over previous holographic models for heavy mesons. The primary contributions are:

1. Theoretical Rigor: The use of the Langer correction within the WKB approximation allows for the derivation of the correct $n^{2/3}$ Regge trajectory behavior directly from the holographic potential, rather than assuming it phenomenologically.
2. Simultaneous Fit: The model successfully fits both the mass spectrum and the decay constants of heavy vector mesons within a single framework, resolving a tension present in earlier approaches where fitting one often compromised the other.
3. Bridge Between Formalisms: The work establishes a connection between the QSSE approach (based on the Cornell potential) and gauge/gravity duality, providing a consistent AdS/QCD framework where the holographic parameters ($\kappa$ and $M_q$) map directly to the string tension and constituent quark masses found in potential models.

The paper concludes that while the model is a "bottom-up" approach, it provides a robust framework for studying heavy mesons and suggests future applications in analyzing thermal dissociation in quark-gluon plasma.