Three-gluon decays of radially excited quarkonia $\psi(2S)$ and $\Upsilon(2S)$ with both relativistic and QCD radiative corrections

This paper presents a comprehensive Bethe-Salpeter analysis of three-gluon decays in radially excited quarkonia $\psi(2S)$ and $\Upsilon(2S)$, demonstrating that their nodal wave function structures cause slow relativistic convergence in gluonic widths while enabling precise extraction of harmonic oscillator parameters that yield excellent agreement with experimental branching ratios.

The Gist

Here is an explanation of the paper using simple language and creative analogies.

The Big Picture: Cracking the "Bouncy" Atom

Imagine the atom's nucleus as a tiny, vibrating drum. Inside this drum, heavy particles called quarks dance around each other. Usually, they dance in a simple, smooth circle (this is the ground state, like the J/ψ particle). But sometimes, they get excited and jump to a higher energy level. When they do this, their dance pattern changes: instead of a smooth circle, their path looks like a figure-eight with a knot or a node in the middle where the probability of finding them drops to zero.

The scientists in this paper are studying two specific "excited" dancers: the $\psi(2S)$ (a charm-quark pair) and the $\Upsilon(2S)$ (a bottom-quark pair). They want to understand how these dancers break apart into three "gluons" (particles that carry the strong force, kind of like glue).

The Problem: The Knot Makes the Math Break

For decades, physicists have tried to predict how fast these excited particles decay. They use a mathematical tool called a "Taylor expansion," which is like trying to draw a complex curve by adding up simple straight lines.

• The Smooth Dance (Ground State): For the simple, smooth dancers, this method works great. You only need a few straight lines to get a perfect picture.
• The Knotted Dance (Excited State): For the excited dancers with the "knot" in their wave function, the math gets messy. When the scientists tried to use the standard method (adding up just the first few "straight lines" or corrections), the math went haywire. It predicted a negative probability, which is impossible in the real world. It's like a weather forecast predicting a "negative 50% chance of rain."

Why did this happen?
The "knot" in the wave function causes a destructive interference. Imagine two waves crashing into each other and canceling each other out. In the math, the "knot" cancels out the signal so hard that the standard approximation fails completely. The paper shows that for these specific particles, the "knot" makes the decay process incredibly sensitive to the details of the dance, which the simple math missed.

The Solution: A New "Smart" Formula

The authors didn't just throw up their hands. They used a sophisticated framework called the Bethe-Salpeter formalism (think of it as a high-definition 3D map of the particle's interior) to build a new, more accurate model.

1. Mapping the Knot: They created a specific mathematical shape (a harmonic oscillator) that explicitly includes the "knot" in the middle.
2. The Phenomenological Fix: Since the standard math breaks down, they introduced a "phenomenological treatment." In plain English, this means they built a smart shortcut. They took the known rules of physics and tweaked them to ensure the math stays positive and realistic, even when the "knot" causes chaos. It's like adding a safety net to a trampoline so that even if the jumper does a crazy flip, they don't fall through the floor.

The Big Discovery: Two Different Speeds

The most fascinating result of the paper is a comparison between two ways these particles can decay:

1. Decaying into Light ($e^+e^-$): When the particle turns into an electron and a positron, the math is easy. The "knot" doesn't bother it much. The standard approximation works perfectly, and the result converges (settles down) very quickly.
2. Decaying into Glue ($ggg$): When the particle turns into three gluons, the math is hard. The "knot" causes massive interference. The standard approximation fails completely, and you need the "smart shortcut" (higher-order corrections) to get the right answer.

The Analogy:
Imagine trying to predict the path of a ball rolling down a smooth hill vs. a ball rolling through a dense, tangled forest.

• The Light Decay is the smooth hill. You can guess the path easily.
• The Gluon Decay is the tangled forest. The "knot" in the tree roots (the wave function) makes the ball bounce unpredictably. If you only look at the first few steps, you'll think the ball goes backward (negative probability). You need to see the whole forest to understand where it actually goes.

The Results: Hitting the Bullseye

After applying their new "smart formula," the authors calculated the branching ratios (the percentage of time the particle decays one way vs. another).

• Old Math: Predicted impossible negative numbers for the excited charm particle.
• New Math: Predicted values that match experimental data almost perfectly.
  • For the $\psi(2S)$, they predicted a ~9.4% chance of decaying into three gluons. Experiments see ~10.6%.
  • For the $\Upsilon(2S)$, they predicted ~54.5%. Experiments see ~58.8%.

This is a huge success. It proves that their model correctly understands how the "knot" in the particle affects its behavior.

The Takeaway: Tuning the Instrument

Finally, the authors used their successful predictions to "tune" a specific parameter called $\beta_V$ (the harmonic oscillator parameter). Think of this as the "tightness" of the drum skin.

• Previous models guessed the tightness was all over the place.
• By using the new, accurate math, they found the drum skin is actually tighter (smaller $\beta$ value) than many people thought.

Summary in One Sentence

This paper fixes a broken mathematical prediction for excited heavy particles by acknowledging that their internal "knots" cause chaotic interference, proving that to understand these tiny quantum dancers, you can't just use simple approximations—you need a full, high-definition view of their complex, knotted dance.

Technical Summary

Here is a detailed technical summary of the paper "Three-gluon decays of radially excited quarkonia $\psi(2S)$ and $\Upsilon(2S)$ with both relativistic and QCD radiative corrections."

1. Problem Statement

The paper addresses a longstanding theoretical challenge in Quantum Chromodynamics (QCD): the reliable prediction of three-gluon decay widths ($V \to ggg$) for radially excited heavy quarkonia, specifically the vector states $\psi(2S)$ and $\Upsilon(2S)$.

• The Core Issue: Unlike ground states ($J/\psi$, $\Upsilon(1S)$), radially excited states possess a nodal structure in their wave functions. This node leads to significant destructive interference when the wave function is convoluted with the hard-scattering amplitude.
• The Failure of Standard Approximations: Standard perturbative expansions in the relative momentum $\hat{q}$ (specifically up to $\hat{q}^2$ order) fail catastrophically for these states. The destructive interference causes the calculated decay width to become negative and unphysical, indicating a breakdown of the low-order relativistic expansion.
• The Gap: While relativistic corrections are known to be crucial, a fully relativistic calculation incorporating the nodal structure analytically within the decay channel $V \to ggg$ has been missing, leaving a gap in understanding the dynamics of excited quarkonia.

2. Methodology

The authors employ a comprehensive theoretical framework combining the Bethe-Salpeter (B-S) formalism with a novel phenomenological treatment of higher-order corrections.

• Bethe-Salpeter Formalism:

  • The bound state is described using the covariant instantaneous ansatz (CIA).
  • Wave Function Construction: They construct analytic harmonic oscillator wave functions that explicitly include the $2S$nodal structure. The scalar function$f(\hat{q})$ is modeled as:
    $$f(\hat{q}) \propto \left(1 - \frac{2\hat{q}^2}{3\beta_V^2}\right) e^{-\frac{\hat{q}^2}{2\beta_V^2}}$$
    where $\beta_V$ is the harmonic oscillator parameter and the term in parentheses represents the node.
  • Dirac Structure: The Salpeter wave function is expanded to include all independent Dirac structures up to $O(\hat{q}^2)$, ensuring a consistent relativistic treatment of spin-orbit couplings.

• Calculation of Decay Amplitudes:

  • The decay amplitude for $V \to ggg$ is calculated by convolving the B-S wave function with the perturbative hard-scattering kernel ($q\bar{q} \to ggg$).
  • Polarized Widths: The authors derive model-independent relations among polarized decay widths based on helicity-flip and phase-space symmetries, classifying non-vanishing contributions into four distinct sets ($\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$).

• Phenomenological Improvement (Key Innovation):

  • Recognizing that the standard $\hat{q}^2$ expansion yields negative widths, the authors introduce a phenomenological treatment inspired by Ref. [15].
  • Instead of a simple Taylor expansion, they replace the correction factor with a form that preserves the correct low-momentum limit ($\hat{q}^2$) while effectively incorporating partial higher-order contributions at large momentum.
  • This form ensures the decay width remains positive definite and improves convergence by naturally suppressing pathological behavior from the wave function tail.

• Radiative Corrections:

  • Both relativistic corrections (internal motion) and QCD radiative corrections (additional gluon emissions, $O(\alpha_s)$) are included.
  • The ratio $R_V = \Gamma(V \to ggg) / \Gamma(V \to e^+e^-)$ is constructed to cancel non-perturbative factors (wave function at the origin), providing a cleaner observable.

3. Key Contributions

1. Analytic Solution for Excited States: First complete analytic calculation of $V \to ggg$ decays for radially excited states ($\psi(2S), \Upsilon(2S)$) using B-S wave functions with explicit nodal structures.
2. Resolution of Unphysical Results: Identification and resolution of the "negative width" problem. The paper demonstrates that the failure of the $\hat{q}^2$ approximation is due to destructive interference from the node, which is only resolved by including higher-order relativistic effects via their phenomenological ansatz.
3. Symmetry Relations: Derivation of rigorous, model-independent relations among polarized decay widths, providing a self-consistency check for the calculation.
4. Convergence Analysis: A detailed comparison showing that while leptonic decays ($V \to e^+e^-$) converge rapidly at $\hat{q}^2$ order, gluonic decays ($V \to ggg$) converge extremely slowly due to the multi-gluon convolution sensitivity to the nodal structure.

4. Results

The numerical results, presented for both $\psi(2S)$ and $\Upsilon(2S)$, highlight a stark contrast between standard approximations and the improved method:

• Failure of $\hat{q}^2$ Order:

  • For $\psi(2S)$, the $\hat{q}^2$-order calculation yields a branching ratio $B(\psi(2S) \to ggg) \approx -88.8\%$ and a ratio $R_{\psi(2S)} \approx -111.8$. These are unphysical.
  • For $\Upsilon(2S)$, the $\hat{q}^2$ result is $13.8%$, significantly underestimating the experimental value.

• Success of Improved Treatment:

  • Using the phenomenologically improved expressions, the predictions align excellently with experimental data:
    • $B(\psi(2S) \to ggg) \approx 9.4\%$ (Exp: $10.6 \pm 1.6%$).
    • $B(\Upsilon(2S) \to ggg) \approx 54.5\%$ (Exp: $58.8 \pm 1.2%$).
  • The ratio $R_{\psi(2S)}$ is predicted to be $11.8$(Exp:$13.4 \pm 2.4$), and$R_{\Upsilon(2S)}$is$28.5$(Exp:$30.8 \pm 3.2$).

• Leptonic vs. Gluonic Convergence:

  • The leptonic width $\Gamma(V \to e^+e^-)$ shows rapid convergence; the $\hat{q}^2$ result is already very close to the improved result and matches experiment.
  • The gluonic width requires the higher-order treatment to become physical, confirming that the nodal structure drastically affects multi-gluon processes but has a milder effect on single-photon processes.

• Extraction of Parameters:

  • By fitting the ratio $R_V$ to experimental data, the authors extract the harmonic oscillator parameters:
    • $\beta_{\psi(2S)} \in (360, 430)$ MeV.
    • $\beta_{\Upsilon(2S)} \in (540, 670)$ MeV.
  • These values lie at the lower end of ranges typically found in phenomenological models (e.g., light-front models or potential models), suggesting that fully relativistic treatments yield more localized wave functions in momentum space.

5. Significance

• Dynamics of Excited States: The work provides new insights into how the internal nodal structure of excited quarkonia amplifies sensitivity to relativistic effects, particularly in multi-gluon channels.
• Benchmark for Models: The extracted $\beta_V$ values serve as a concrete benchmark for non-perturbative models. The finding that fully relativistic B-S calculations favor smaller $\beta_V$ suggests that many effective models may be overestimating the width parameter to compensate for missing relativistic dynamics.
• Theoretical Robustness: The study establishes that reliable predictions for excited heavy quarkonia cannot rely solely on low-order relativistic expansions. It demonstrates the necessity of incorporating higher-order effects, either through full higher-order calculations or effective phenomenological treatments that respect the nodal structure.
• Experimental Connection: The predictions for polarized decay widths offer new observables (transverse vs. longitudinal dominance) that can be tested via angular distributions in hadronic final states (e.g., $V \to p\bar{p}$), linking theoretical wave function details to measurable hadronic correlations.