Heavy quarkonium decay $V \to ggg$ with both relativistic and QCD radiative corrections

This paper presents a theoretical study of the heavy quarkonium decay $V \to ggg$ by incorporating both relativistic corrections via the Bethe-Salpeter formalism and QCD radiative corrections, deriving an unpolarized decay width formula that successfully aligns with experimental branching ratios and allows for the extraction of the strong coupling constant $\alpha_s$.

The Gist

Imagine the subatomic world as a bustling, high-energy dance floor. In this paper, the authors are studying a very specific, rare dance move performed by heavy "quark" couples.

Here is the story of their research, broken down into simple concepts:

1. The Characters: The Heavy Quark Couples

Think of a Quarkonium (like the $J/\psi$ or $\Upsilon$ particles) as a heavy, romantic couple dancing in a tight embrace. They are made of a heavy quark and its anti-quark partner. Because they are so heavy, they move relatively slowly compared to light particles, but they are still moving fast enough that "relativity" (Einstein's rules about fast motion) starts to matter.

2. The Problem: The Forbidden Exit

These couples want to break up and disappear, but the universe has strict security rules (called conservation laws):

• They can't just walk out the door as a single particle (that's forbidden by "color confinement").
• They can't split into two particles (that's forbidden by "C-parity," a symmetry rule).
• The only way out: They must explode into three gluons (gluons are the "glue" particles that hold quarks together).

This process is called $V \to ggg$. It's like a couple trying to exit a room by shattering into three pieces of invisible confetti simultaneously.

3. The Old Way vs. The New Way

The Old Way (The Flat Map):
Previous scientists tried to predict how often this happens using a "flat map" (Non-Relativistic Quantum Mechanics). They assumed the couple was standing still while they broke apart.

• The Result: Their predictions were off. It was like trying to navigate a city using a map that ignores the hills and valleys. The numbers didn't match what real experiments saw.

The New Way (The 3D GPS):
The authors of this paper used a more advanced tool called the Bethe-Salpeter (B-S) formalism.

• The Analogy: Imagine trying to describe a spinning, vibrating top. A flat map says, "It's just a spinning disk." The B-S method says, "No, let's look at every wobble, every vibration, and how the top stretches and squishes as it spins."
• They kept track of the internal momentum (the jittery movement inside the couple) while calculating the breakup. This allowed them to include Relativistic Corrections (adjustments for the fact that the quarks are moving fast) and QCD Radiative Corrections (adjustments for the extra "noise" or extra particles popping in and out of existence during the breakup).

4. The "Helicity" Dance Rules

The paper spends a lot of time on something called Helicity (which is basically the direction of the spin of the particles).

• The Rule: Imagine the three pieces of confetti (gluons) have to spin in specific directions to match the couple's original spin.
• The Discovery: The authors found that many dance moves are impossible. Due to the "Heliicity Selection Rule," certain combinations of spins simply cancel each other out. It's like trying to do a specific dance step where your left foot must go left and your right foot must go right, but the music forces them to go the same way—so the move never happens.
• They proved that out of all the possible ways the particles could spin, only a few specific patterns actually occur. This simplifies the math and helps experimentalists know exactly what to look for.

5. The Big Result: Fixing the Numbers

When the authors added these "relativistic wobbles" and "QCD noise" into their calculations, the numbers finally matched reality.

• Before: The theory predicted the couple would break up into three gluons way too often (or too rarely) compared to what was seen in labs like the LHC or Belle.
• After: With the new, more detailed math, their predictions for how often $J/\psi$ and $\Upsilon$ turn into three gluons (and how often they turn into electron-positron pairs) aligned perfectly with experimental data.

6. The "Cross-Check": Measuring the Glue

Finally, the authors used their new, accurate formula to do a reverse calculation.

• They took the experimental data (what we see happening) and worked backward to find the value of $\alpha_s$ (the "Strong Force Coupling Constant").
• Think of $\alpha_s$ as the "stickiness" of the glue holding the universe together.
• Their calculated "stickiness" values matched other known measurements, proving their method is solid.

Summary

In short, this paper is like upgrading from a sketchy, hand-drawn map to a high-definition GPS. By accounting for the fact that heavy quarks are actually moving, vibrating, and interacting with a noisy quantum environment, the authors finally solved a decades-old puzzle: Why do heavy quark couples break up into three gluons at the rate they do?

They showed that if you ignore the "relativistic wobbles," your prediction is wrong. But if you include them, the universe makes perfect sense.

Technical Summary

Here is a detailed technical summary of the paper "Heavy quarkonium decay $V \to ggg$ with both relativistic and QCD radiative corrections" by Jiang et al.

1. Problem Statement

The decay of heavy vector quarkonia ($V = J/\psi, \Upsilon$) into three gluons ($V \to ggg$) is a fundamental process for probing the interplay between perturbative and non-perturbative Quantum Chromodynamics (QCD). While the leading-order (LO) theoretical prediction for this decay width is well-established, it suffers from significant discrepancies with experimental data.

• The Challenge: Previous attempts to improve predictions by including first-order QCD radiative corrections or relativistic corrections (via potential models or Non-Relativistic QCD (NRQCD)) have yielded inconsistent results. Specifically, some NRQCD calculations at higher orders ($O(v^4)$) have produced unphysical negative decay widths for the $J/\psi$ channel.
• The Gap: There is a need for a rigorous framework that simultaneously handles relativistic corrections (arising from the internal motion of quarks) and QCD radiative corrections without relying on poorly constrained non-perturbative matrix elements or ad-hoc approximations.

2. Methodology

The authors employ the Bethe-Salpeter (B-S) formalism under the Covariant Instantaneous Ansatz (CIA) to provide an analytical treatment of the decay process.

• Wave Function Derivation:

  • They solve the B-S equation analytically for the heavy quarkonium bound state, incorporating both long-range confinement and short-range one-gluon-exchange (Coulomb) interactions.
  • The internal momentum $q$ of the quark-antiquark pair is retained explicitly in both the soft bound-state wave function and the hard-scattering amplitude. This allows for a natural inclusion of relativistic effects.
  • The Salpeter wave function is derived up to sub-leading order in momentum, reducing the general Dirac structure to a manageable form involving scalar functions $f(\hat{q})$. A Gaussian ansatz is used for the momentum dependence: $f(\hat{q}) \propto e^{-\hat{q}^2/2\beta_V^2}$.

• Decay Amplitude Calculation:

  • The decay amplitude is factorized into a hard-scattering part (calculable via perturbative QCD) and the soft wave function.
  • The authors perform an analytical expansion of the decay amplitude modulus squared $|A|^2$ up to the quadratic order of the internal momentum $\hat{q}$ (first-order relativistic corrections).
  • They utilize helicity selection rules and phase space symmetries to categorize polarized decay widths.

• Factorization Assumption:

  • To obtain the final prediction, the authors assume that relativistic corrections (from bound-state dynamics) and QCD radiative corrections (from soft gluon emissions) are factorizable. They combine the analytical relativistic result with known NLO QCD radiative correction factors ($C_V$).

3. Key Contributions

• Analytical Framework: Unlike numerical approaches or effective field theories with many unknown parameters, this work provides a fully analytical derivation of the decay width including first-order relativistic corrections within the B-S framework.
• Helicity Selection Rules: The authors demonstrate that due to helicity flip symmetry and phase space symmetry, many polarized decay widths vanish. They identify that the non-vanishing polarized widths fall into four distinct groups related by symmetry relations, providing a model-independent check for theoretical calculations.
• Resolution of Unphysical Results: By retaining the full dynamical structure of the wave function and the hard scattering kernel, the authors avoid the unphysical negative decay widths previously reported in some NRQCD calculations for the $J/\psi$ channel.
• Symmetry Relations: They derive explicit symmetry relations connecting different helicity configurations, which serve as a crucial self-consistency check for future experimental measurements of gluon polarization correlations.

4. Key Results

The paper presents the unpolarized decay width formula including relativistic corrections:
$$\Gamma(V \to ggg) = \frac{80(\pi^2-9)\alpha_s^3 N_V^2 \beta_V^3}{81\pi^{9/2}M} \left(1 - \kappa \frac{\beta_V^2}{M^2}\right)$$
where $\kappa = \frac{3(112+25\pi^2)}{16(\pi^2-9)}$.

Numerical Findings:

• Relativistic Corrections: The relativistic correction factor ($\kappa \beta_V^2/M^2$) is found to be significant, approximately 0.77 for $J/\psi$ and 0.31 for $\Upsilon$. This confirms that relativistic effects are more pronounced for the lighter charm quark system.
• Branching Ratios: When combining relativistic and QCD radiative corrections, the theoretical predictions align closely with experimental data:
  • $B(J/\psi \to ggg)$: Predicted 66.0% vs. Experimental 64.1%.
  • $B(\Upsilon \to ggg)$: Predicted 81.2% vs. Experimental 81.7%.
  • The lowest-order (LO) predictions deviate significantly (e.g., predicting $>100\%$ for $J/\psi \to ggg$), highlighting the necessity of higher-order corrections.
• Extraction of $\alpha_s$: Using the ratio of gluonic to leptonic widths ($R_V = \Gamma(V \to ggg)/\Gamma(V \to e^+e^-)$), the authors extract the QCD running coupling constant:
  • $\alpha_s(M_{J/\psi}/2) = 0.31$
  • $\alpha_s(M_{\Upsilon}/2) = 0.20$
    These values are consistent with determinations from other decay channels.

5. Significance

• Validation of QCD Dynamics: The study confirms that the interplay of relativistic and radiative corrections is essential for accurately describing heavy quarkonium decays. The success of the B-S formalism in reproducing experimental data validates the treatment of relativistic bound states in QCD.
• Phenomenological Utility: The derived analytical expressions for polarized decay widths offer specific predictions for gluon helicity correlations, which can guide future experimental analyses at facilities like LHCb, Belle II, and BESIII.
• Methodological Advancement: The work demonstrates that the B-S approach with the CIA is a robust alternative to NRQCD for calculating decay rates, particularly in avoiding the proliferation of poorly constrained non-perturbative matrix elements associated with higher-order NRQCD calculations.
• Precision Physics: The extraction of $\alpha_s$ from these decays provides an independent and consistent measurement of the strong coupling constant, reinforcing the standard model's description of strong interactions in the intermediate energy regime.