Systematic analysis of the D-wave charmonium states with the QCD sum rules

This paper employs QCD sum rules to systematically analyze D-wave charmonium states, predicting masses that align with recent experimental observations of $\psi(3770)$, $\psi_2(3823)$, and $\psi_3(3842)$ while also estimating the mass of the yet-unobserved $\eta_{c2}$ state.

The Gist

Imagine the universe is a giant, bustling construction site. At the center of this site, there are tiny, heavy bricks called quarks. Specifically, there's a type of heavy brick called the "charm" quark. When a charm quark and its anti-brick (an antiquark) get together, they snap together to build a tiny, short-lived structure called charmonium.

Think of charmonium like a tiny, vibrating guitar string. Just like a guitar string can vibrate in different ways to produce different notes (low, high, sharp, flat), these quark pairs can vibrate in different patterns. Physicists call these patterns "waves."

• S-wave: The simplest vibration, like a string plucked gently in the middle.
• P-wave: A slightly more complex vibration, like a string vibrating in two loops.
• D-wave: A more complex, "higher-order" vibration, like a string vibrating in three or more loops.

For a long time, physicists have been very good at identifying the simple notes (S-wave and P-wave). But the D-wave notes have been a bit of a mystery. We know they should exist, but finding them in the noisy, chaotic construction site of particle physics has been tricky. Some notes have been found, but others are still hidden, or we aren't sure which note is which.

The Problem: A Noisy Construction Site

The authors of this paper, Qi Xin and Zhi-Gang Wang, wanted to solve a puzzle: What are the exact "notes" (masses) of these D-wave charmonium structures?

They looked at four specific types of D-wave structures:

1. $\psi_1$ (Spin-triplet, 1D): A specific vibrating pattern.
2. $\psi_2$ (Spin-triplet, 2D): Another pattern.
3. $\psi_3$ (Spin-triplet, 3D): A third pattern.
4. $\eta_{c2}$ (Spin-singlet, 1D): A fourth pattern that hasn't been seen yet.

The Tool: The "QCD Sum Rule" Calculator

To figure this out without building a giant new particle collider, the authors used a theoretical tool called QCD Sum Rules.

Imagine you are trying to guess the weight of a hidden box inside a sealed, opaque suitcase. You can't open it, but you can shake the suitcase, listen to the sound, and feel how the weight shifts.

• The Suitcase: The complex laws of Quantum Chromodynamics (QCD), the physics of how quarks stick together.
• The Shaking: Mathematical equations that mix the known rules of physics with the "noise" of the vacuum (empty space isn't actually empty; it's bubbling with energy).
• The Result: By crunching these numbers, the authors can predict exactly how heavy the box (the particle) must be to make the math work.

They didn't just guess; they calculated the "weight" (mass) of these four D-wave particles with high precision, accounting for the tiny vibrations of the vacuum itself.

The Big Reveal: Matching the Notes

Once they calculated the theoretical weights, they compared them to the "real-world" particles that experiments (like those at the LHC, BESIII, and Belle) have actually found.

Here is what they discovered:

1. The $\psi(3770)$ Match: They calculated a mass of 3.77 GeV. This perfectly matches a particle already found called $\psi(3770)$.

   • Conclusion: We can now be very confident that the $\psi(3770)$ is indeed a D-wave particle. It's not just a random bump; it's a specific, complex vibration of charm quarks.

2. The $\psi_2(3823)$ Match: They calculated 3.82 GeV. This matches a recently discovered particle called $\psi_2(3823)$.

   • Conclusion: This confirms that the $\psi_2(3823)$ is the second type of D-wave particle.

3. The $\psi_3(3842)$ Match: They calculated 3.84 GeV. This matches the $X(3842)$ (now likely called $\psi_3(3842)$).

   • Conclusion: This confirms the third type of D-wave particle.

4. The Missing Note ($\eta_{c2}$): They calculated a mass of 3.83 GeV for a particle called $\eta_{c2}$.

   • The Twist: Nobody has seen this one yet! It's like a ghost note. The authors are saying, "We know it should be here, weighing 3.83 GeV. Experimentalists, go look for it in this specific range!"

Why Does This Matter?

Think of the periodic table of elements. For a long time, we knew there were gaps where elements should be, but we hadn't found them yet. Once we found them, our understanding of chemistry became complete.

This paper does the same for the "Periodic Table of Quarks."

• It fills in the gaps: By confirming that the particles we found are indeed D-wave states, the authors have completed a major section of the charmonium family tree.
• It gives a roadmap: By predicting the mass of the missing $\eta_{c2}$, they are handing experimental physicists a "Wanted" poster with a specific description.
• It proves the theory: The fact that their mathematical "shaking of the suitcase" matched the real-world findings so perfectly shows that our understanding of how quarks stick together (QCD) is solid.

In a Nutshell

The authors used advanced math to predict the "weight" of four complex, vibrating quark structures. They found that three of them match particles we've already discovered, confirming our understanding of how they work. For the fourth one, which is still hiding, they provided a precise target for scientists to hunt down in future experiments. It's a victory for theoretical physics, turning a blurry picture of the subatomic world into a sharp, clear image.

Technical Summary

Here is a detailed technical summary of the paper "Systematic analysis of the D-wave charmonium states with the QCD sum rules" by Qi Xin and Zhi-Gang Wang.

1. Problem Statement

Charmonium states (bound states of a charm quark and its antiquark) serve as an ideal laboratory for studying Quantum Chromodynamics (QCD) in the non-perturbative regime. While S-wave and P-wave charmonium states have been extensively studied using QCD sum rules, the D-wave charmonium states remain less systematically explored theoretically, despite recent experimental discoveries.

The paper addresses the need to:

• Theoretically predict the mass spectrum of D-wave charmonium states ($1^3D_1, 1^3D_2, 1^1D_2, 1^3D_3$).
• Provide a rigorous identification of recently observed experimental resonances (e.g., $\psi(3770)$, $\psi(3823)$, $X(3842)$) as conventional D-wave charmonia.
• Predict the properties of the unobserved spin-singlet state $\eta_{c2}$ ($1^1D_2$).
• Calculate decay constants to facilitate future studies of strong and radiative decays.

2. Methodology

The authors employ the QCD Sum Rules (QCDSR) approach, a non-perturbative method that connects hadronic properties to QCD fundamental parameters via the Operator Product Expansion (OPE).

• Interpolating Currents: The authors construct specific local interpolating currents containing covariant derivatives ($\overleftrightarrow{D}_\mu$) to represent the D-wave nature of the states:
  • $J_\mu(x)$ for $1^3D_1$($\psi_1$,$J^{PC}=1^{--}$).
  • $J^1_{\mu\nu}(x)$ for $1^3D_2$($\psi_2$,$J^{PC}=2^{--}$).
  • $J^2_{\mu\nu}(x)$ for $1^1D_2$($\eta_{c2}$,$J^{PC}=2^{-+}$).
  • $J_{\mu\nu\rho}(x)$ for $1^3D_3$($\psi_3$,$J^{PC}=3^{--}$).
• Correlation Functions: Two-point correlation functions are defined for these currents. The analysis involves:
  • Hadronic Side: Inserting a complete set of intermediate states to isolate the ground state contributions, parameterized by masses ($M$) and decay constants ($f$).
  • QCD Side: Performing the OPE up to dimension-6 condensates. This includes the perturbative term, the gluon condensate $\langle \frac{\alpha_s}{\pi} G G \rangle$, and the three-gluon condensate $\langle g_s^3 G G G \rangle$.
• Sum Rule Derivation:
  • The dispersion relation is used to match the hadronic and QCD representations.
  • Borel transformation is applied to suppress higher resonance contributions and continuum states.
  • The masses are extracted by differentiating the sum rules with respect to the Borel parameter $1/T^2$.
• Parameters:
  • Charm quark mass: $m_c(m_c) = 1.275 \pm 0.025$ GeV.
  • Two sets of condensate values (Set I and Set II) were tested to ensure stability against variations in non-perturbative parameters.
  • Continuum thresholds ($\sqrt{s_0}$) were tuned to ensure pole dominance (40–70%) and OPE convergence.

3. Key Contributions

• Systematic Framework: This work provides the first comprehensive QCD sum rule analysis specifically targeting the full set of D-wave charmonium states (both spin-triplet and spin-singlet) within a single consistent framework.
• Robustness Check: The authors demonstrate that the predicted masses are remarkably stable despite significant variations in the three-gluon condensate values between the two parameter sets used. This confirms that the results are dominated by perturbative contributions.
• Decay Constants: The paper calculates the decay constants ($f_{\psi_1}, f_{\psi_2}, f_{\eta_{c2}}, f_{\psi_3}$), which are crucial inputs for calculating strong, leptonic, and radiative decay widths, a topic previously lacking sufficient theoretical data for D-wave states.
• Experimental Identification: The study offers strong theoretical support for assigning specific experimental resonances to the D-wave multiplet.

4. Key Results

The authors obtained the following mass predictions and decay constants (with uncertainties derived from input parameters):

State	$J^{PC}$	Predicted Mass (GeV)	Experimental Candidate	Experimental Mass (GeV)
$\psi_1$	$1^{--}$| **3.77 ± 0.09** |$\psi(3770)$	~3.77
$\psi_2$	$2^{--}$| **3.82 ± 0.09** |$\psi_2(3823)$/$X(3823)$	~3.82
$\eta_{c2}$	$2^{-+}$	3.83 ± 0.09	Unobserved	N/A
$\psi_3$	$3^{--}$| **3.84 ± 0.08** |$X(3842)$	~3.84

• $\psi(3770)$: The prediction $M_{\psi_1} = 3.77 \pm 0.09$ GeV supports identifying the $\psi(3770)$ as the conventional $1^3D_1$ charmonium state, rather than a mixture or exotic state.
• $\psi_2(3823)$: The prediction $M_{\psi_2} = 3.82 \pm 0.09$ GeV is consistent with observations by Belle, BESIII, and LHCb, confirming its nature as a $1^3D_2$ state.
• $X(3842)$: The prediction $M_{\psi_3} = 3.84 \pm 0.08$ GeV supports the identification of the LHCb-observed $X(3842)$ as the $1^3D_3$ state.
• $\eta_{c2}$: The paper predicts the mass of the unobserved spin-singlet $1^1D_2$ state to be 3.83 ± 0.09 GeV. The authors suggest this state is a prime candidate for future detection at BESIII, LHCb, or Belle II.
• Decay Constants: The calculated decay constants (e.g., $f_{\psi_1} \approx 13.8$ GeV$^4$) provide the necessary phenomenological parameters for future decay width calculations.

5. Significance

• Validation of the Quark Model: The results reinforce the validity of the traditional quark model in the D-wave sector, showing that the observed resonances fit well within the expected mass spectrum of $c\bar{c}$ states.
• Guidance for Experiments: By predicting the mass of the unobserved $\eta_{c2}$ and providing precise mass ranges for the triplet states, the paper guides experimentalists on where to look for missing states and how to interpret new data near open-charm thresholds.
• Theoretical Consistency: The study highlights that QCD sum rules, when applied with high-order condensates and careful parameter selection, yield results consistent with various potential models (Bethe-Salpeter, Godfrey-Isgur, etc.) but with a direct connection to QCD fundamentals.
• Foundation for Decay Studies: The provided decay constants are essential for the next step in understanding these states: calculating their decay widths and branching ratios to distinguish them from potential tetraquark or molecular interpretations.

In conclusion, this paper successfully bridges the gap between theoretical QCD predictions and recent experimental discoveries in the D-wave charmonium sector, offering a robust theoretical basis for the identification of $\psi(3770)$, $\psi(3823)$, and $X(3842)$, while predicting the existence of the $\eta_{c2}$ state.