[Paper Review] Radially Excited States of 1P Charmonia and X(3872)

Ying Chen, Chuan Liu+3 more|ArXiv.org|Jan 25, 2007

Advanced Chemical Physics Studies4 citations

TL;DR

This lattice QCD study investigates radially excited charmonium states using quenched anisotropic lattices with improved gauge and Wilson fermion actions. Employing the sequential empirical Bayes (SEB) method, it determines the first excited state masses in $0^{++}$, $1^{++}$, and $1^{+-}$ channels as 3.825(88), 3.853(57), and 3.858(70) GeV, respectively, and observes a radial node in the Bethe-Salpeter amplitude of the $1^{++}$ state, supporting the assignment of X(3872) as the first radial excitation of $\chi_{c1}$. The results are consistent with experimental mass measurements and indicate relativistic effects are significant in these states.

ABSTRACT

The excited states of charmonia are numerically investigated in quenched lattice QCD with improved gauge and Wilson fermion actions formulated on anisotropic lattices. Through a constrained curve fitting algorithm, the masses of the first excited states in $0^{++}$, $1^{++}$, and $1^{+-}$ channels are determined to be 3.825(88), 3.853(57), and 3.858(70) GeV, respectively. Furthormore, a node structure is also observed in the Bethe-Salpeter amplitude of the $1^{++}$ first excited state. These observations indicate that X(3872) could be the first radial excitation of $χ_{c1}$.

Motivation & Objective

• To investigate radially excited charmonium states in quenched lattice QCD with improved actions on anisotropic lattices.
• To determine the masses of the first excited states in $0^{++}$, $1^{++}$, and $1^{+-}$ channels using a robust fitting method.
• To examine the internal structure of the $1^{++}$ excited state via Bethe-Salpeter amplitudes to assess its radial node structure.
• To evaluate whether X(3872) can be interpreted as the first radial excitation of $\chi_{c1}$ based on mass and wave function structure.

Proposed method

• Uses quenched lattice QCD with improved gauge and Wilson fermion actions on anisotropic lattices ($\beta = 2.4, 2.6, 2.8$) to enhance temporal resolution.
• Employs the sequential empirical Bayes (SEB) method, a constrained curve fitting algorithm, to extract masses from correlation functions with multiple exponential terms.
• Constructs two-point correlation functions using wall-source quark propagators and Coulomb gauge-fixed fields to reduce noise.
• Calculates Bethe-Salpeter (BS) amplitudes $\Phi_n(r)$ from spatially separated quark-antiquark operators to probe radial wave function structure.
• Applies a constrained-curve-fitting algorithm using known ground state masses as priors to extract $\Phi_n(r)$ for $n=1$ (ground) and $n=2$ (excited) states.
• Performs statistical averaging over spatial separations $|\mathbf{r}| \leq 6\sqrt{3}a_s$ to improve signal-to-noise ratio in BS amplitude analysis.

Experimental results

Research questions

• RQ1 Is the $1^{++}$ excited state in charmonium spectra consistent with being the first radial excitation of $\chi_{c1}$?
• RQ2 What is the mass of the first excited state in the $1^{++}$ channel, and how does it compare to the X(3872) mass?
• RQ3 Does the Bethe-Salpeter amplitude of the $1^{++}$ excited state exhibit a radial node, as expected for a $2P$ state?
• RQ4 Can the SEB method reliably extract excited state masses in the $0^{++}$, $1^{++}$, and $1^{+-}$ channels from lattice correlation functions?
• RQ5 How do the results compare to non-relativistic quark model predictions and experimental observations for X(3872)?
• RQ6 What is the role of relativistic effects in the structure of radially excited charmonia, as captured by lattice QCD?

Key findings

• The mass of the first excited state in the $1^{++}$ channel is determined to be 3.853(57) GeV, consistent with the experimental mass of X(3872) within statistical errors.
• The $0^{++}$ and $1^{+-}$ first excited states have masses of 3.825(88) GeV and 3.858(70) GeV, respectively.
• The Bethe-Salpeter amplitude $\Phi_2(r)$ for the $1^{++}$ excited state exhibits two nodes at $r/a \sim 2.5$ and $r/a \sim 8$, indicating a radial node consistent with a $2P$ state.
• The $\Phi_1(r)$ amplitude for the ground state shows one node at $r/a \sim 4.8$, consistent with the $1P$ radial wave function's maximum.
• The observed radial node structure in the $1^{++}$ state supports the interpretation that X(3872) is the first radial excitation of $\chi_{c1}$.
• The results are obtained using a relativistic lattice QCD formalism that includes relativistic effects, improving upon non-relativistic quark model predictions.