Exclusive semileptonic and nonleptonic $J/\psi$ decays

This paper investigates exclusive semileptonic and nonleptonic $J/\psi$ decays using a relativistic quark model to calculate form factors and predict branching fractions on the order of $10^{-9}$ to $10^{-12}$, which are then compared with existing theoretical predictions and experimental upper bounds.

The Gist

Imagine the J/ψ meson as a tiny, incredibly heavy, and very stable "atom" made of two very heavy particles (a charm quark and its anti-particle) holding hands so tightly that they rarely let go. Usually, this "atom" falls apart by exploding into pure energy or light (strong and electromagnetic forces). It's like a super-stable boulder that only breaks if hit by a sledgehammer or a laser.

However, there is a very rare, almost magical way for this boulder to break: through the Weak Force. This is the force responsible for things like radioactive decay. In the world of particle physics, this is like the boulder spontaneously turning into a different, lighter stone and a ghostly particle (a neutrino) without being hit by anything.

This paper is a detailed theoretical recipe book written by physicists Galkin and Sukhanov. They are trying to predict exactly how often this rare "weak decay" happens and what the resulting pieces look like, before experimentalists actually find them in a lab.

Here is a breakdown of their work using simple analogies:

1. The Challenge: Seeing the Invisible

The J/ψ meson is so stable that its weak decays are incredibly rare—like finding a specific grain of sand on all the beaches on Earth. The authors know that with new, massive particle colliders (like the BESIII experiment in China), scientists are finally collecting enough "sand" (data) to potentially spot these rare events.

But to spot them, you need a map. You need to know exactly what the "grain of sand" looks like if it does appear. That's what this paper provides: a precise map of the mathematical shapes (called form factors) that describe how the J/ψ transforms into other particles.

2. The Tool: The "Relativistic Quark Model"

To draw this map, the authors use a sophisticated tool called the Relativistic Quark Model.

• The Analogy: Imagine trying to predict the path of a dancer spinning very fast. If you use slow-motion physics (Newtonian), you get the wrong answer. You need to use "relativistic" physics, which accounts for the fact that things moving near the speed of light behave differently (time slows down, mass changes, etc.).
• The Method: They treat the J/ψ not as a static ball, but as two dancers (quarks) spinning and vibrating. They calculate the "overlap" of their dance moves (wave functions) before and after the decay.
• The Twist: They didn't just look at the dancers standing still; they calculated how the dance looks when the whole group is zooming across the stage (moving reference frames). This "relativistic boost" is crucial because the particles are moving so fast that ignoring it would ruin the prediction.

3. The Two Types of Decay They Predicted

The paper looks at two ways the J/ψ can decay:

A. The "Semileptonic" Decay (The Soloist and the Ghost)

• What happens: One of the charm quarks in the J/ψ turns into a lighter quark (like a strange or down quark). This releases a new charm meson (like a D-meson) and a pair of particles: a charged lepton (an electron or muon) and a ghostly neutrino.
• The Analogy: Imagine a heavy couple (J/ψ) dancing. One partner suddenly changes into a lighter partner, and in the process, they toss a ball (the lepton) and a whisper (the neutrino) into the air.
• The Result: The authors calculated the probability of this happening. They found it's very rare (about 1 in 10 billion), but not impossible. They provided a detailed chart showing how the energy of the "ball" (lepton) changes as the decay happens.

B. The "Nonleptonic" Decay (The Group Dance)

• What happens: The charm quark decays, but instead of a ghostly neutrino, it creates a spray of other particles (like pions or kaons).
• The Analogy: The heavy couple breaks up, and instead of throwing a ball, they suddenly spawn a whole new group of smaller dancers (pions/kaons) who join the dance floor.
• The Challenge: This is harder to calculate because the "new dancers" interact with each other in messy, complex ways.
• The Solution: The authors used a "Factorization" trick. They assumed that the messy interaction could be split into two simpler parts: the main couple breaking up, and the new group forming independently. They also used a mathematical limit (imagining there are infinite colors in the universe) to simplify the math, which is a standard trick in this field.
• The Result: These decays are even rarer than the semileptonic ones (1 in a trillion to 1 in a quadrillion).

4. The Comparison: Theory vs. Reality

The authors didn't just guess; they compared their "recipe" with other chefs (other theoretical models) and the current "taste tests" (experimental data).

• The "Flavor" Differences: They found their predictions for the "shape" of the decay (the form factors) were quite different from some other models. For example, some other models predicted the decay would happen "upside down" (negative values), while their model predicted it would be "right side up" (positive).
• The Verdict: Their numbers are generally smaller than some other theories but fit well with the most advanced computer simulations (Lattice QCD).

5. The Big Picture: Why Does This Matter?

Currently, no one has actually seen a J/ψ decay this way yet. The experiments have only set "upper limits" (saying, "It happens less than X times").

However, the authors are optimistic. They say:

"We have built a very precise map. The next generation of particle factories (like the Super Tau-Charm Facility) will have enough data to finally find these rare events. When they do, we can check our map. If our map matches reality, it proves our understanding of how quarks dance is correct. If it doesn't match, it might mean there is 'New Physics'—something completely unknown and exciting happening!"

Summary

This paper is a high-precision forecast for a rare particle event.

• The J/ψ is a stable boulder.
• The Weak Decay is the boulder spontaneously turning into something else.
• The Authors used advanced "relativistic" math to predict exactly how often this happens and what the debris looks like.
• The Goal: To give experimentalists a target to aim for, so they can either confirm our current understanding of the universe or discover something entirely new.

It's like writing a detailed weather forecast for a storm that hasn't happened yet, so that when the storm arrives, we know exactly what to expect.

Technical Summary

1. Problem Statement

The $J/\psi$ meson, a bound state of charm and anti-charm quarks ($c\bar{c}$), has a mass below the $D\bar{D}$ production threshold, resulting in a narrow total decay width dominated by strong and electromagnetic interactions. Consequently, weak decays of the $J/\psi$ are heavily suppressed in the Standard Model (SM), with predicted inclusive branching fractions on the order of $10^{-8}$ or lower.

Despite this suppression, the accumulation of over 10 billion $J/\psi$ events by the BESIII experiment and future data from the Super Tau Charm Facility (STCF) make the search for these rare decays feasible. Detecting $J/\psi$ weak decays serves two critical purposes:

1. Testing the SM: Verifying predictions for heavy quarkonium weak decays.
2. Probing New Physics: Significant deviations from SM predictions could indicate contributions from physics beyond the Standard Model.

The primary theoretical challenge lies in calculating the hadronic matrix elements of the weak current, which are governed by non-perturbative Quantum Chromodynamics (QCD). These elements are parameterized by form factors that must be determined reliably across the entire kinematic range, not just at specific points like zero recoil.

2. Methodology

The authors employ a Relativistic Quark Model (RQM) based on the quasipotential approach to calculate the exclusive decay rates.

• Wave Functions: The initial ($J/\psi$) and final ($D$ or $D_s$) meson wave functions are obtained by solving the relativistic Schrödinger-like quasipotential equation. The potential used is a QCD-motivated mixture of short-range one-gluon exchange and long-range scalar/vector confinement (a relativistic generalization of the Cornell potential).
• Relativistic Effects: The calculation fully accounts for relativistic effects, including:
  • Boosting meson wave functions from the rest frame to the moving frame (using Wigner rotations).
  • Contributions from intermediate negative-energy states.
  • The dependence of the reduced mass and eigenvalues on the meson mass.
• Form Factor Calculation: The hadronic matrix elements are expressed as overlap integrals of the initial and final meson wave functions. The authors calculate the invariant form factors ($V, A_0, A_1, A_2$) for the $J/\psi \to D^{(*)}$ transitions across the full accessible kinematic range of momentum transfer squared ($q^2$).
• Nonleptonic Decays: For nonleptonic decays, the authors utilize the factorization approximation in the limit of an infinite number of colors ($N_c \to \infty$). This approach neglects non-factorizable color-octet contributions, which are assumed to be suppressed, and uses effective Wilson coefficients ($a_1$ and $a_2$) derived from the renormalization group improved perturbation theory.

3. Key Contributions

• Complete Relativistic Treatment: Unlike previous models that often rely on non-relativistic approximations or simple Gaussian wave functions, this work consistently incorporates relativistic kinematics and dynamics, including negative-energy state contributions.
• Full Kinematic Range: The form factors are determined for the entire $q^2$ range, allowing for precise integration over the phase space without relying on extrapolations from specific kinematic points (like $q^2=0$).
• Systematic Comparison: The paper provides a comprehensive comparison of their calculated form factors and branching fractions against a wide array of existing theoretical approaches, including Lattice QCD (LQCD), QCD Sum Rules (QCDSR), Bethe-Salpeter (BS), and various Quark Models (CCQM, CLFQM, BSW).

4. Results

A. Form Factors

The form factors for $J/\psi \to D$ and $J/\psi \to D_s$ transitions were fitted to a pole-like expression (Eq. 35) using parameters $\sigma_{1,2,3}$.

• Discrepancies: The authors' values for the vector form factor $V(0)$ are significantly lower (by a factor of >2) than many other models (e.g., LQCD, QCDF).
• Sign Differences: The axial-vector form factor $A_2(0)$ is positive in their model (and LQCD/CCQM), whereas it is negative in BS, QCDSR, and QCDF models.
• Behavior: The form factors in this model generally exhibit a more rapid growth with $q^2$ compared to models using simple Gaussian wave functions.

B. Branching Fractions

The calculated branching fractions are extremely small, ranging from $10^{-9}$ to $10^{-12}$.

• Semileptonic Decays ($J/\psi \to D^{(*)} \ell \nu$):

  • CKM-favored transitions ($c \to s$) have branching fractions of order $10^{-10}$ (e.g., $J/\psi \to D_s^- e^+ \nu_e \approx 1.99 \times 10^{-10}$).
  • CKM-suppressed transitions ($c \to d$) are an order of magnitude smaller ($\sim 10^{-11}$).
  • Results are in good agreement with LQCD and CCQM predictions but differ from CLFQM (which predicts values >2x larger).

• Nonleptonic Decays ($J/\psi \to D^{(*)} P/V$):

  • The largest predicted branching fraction is for $J/\psi \to D_s^+ \rho^- + D_s^- \rho^+$, estimated at $\sim 1.36 \times 10^{-9}$.
  • Color-suppressed modes (involving neutral mesons like $\pi^0, \rho^0$) are further suppressed.
  • All theoretical predictions are currently four orders of magnitude below the existing experimental upper bounds set by the PDG and BESIII.

5. Significance and Conclusion

• Experimental Outlook: While current experimental upper limits are far above the theoretical predictions, the massive statistics expected from the BESIII experiment and the future Super Tau Charm Facility (STCF) suggest that the first observation of $J/\psi$ weak decays is imminent.
• Theoretical Benchmark: The precise calculation of form factors and branching ratios in this paper provides a crucial benchmark for future experimental measurements. A discrepancy between future data and these SM predictions would be a strong indicator of "New Physics."
• Model Validation: The study highlights the sensitivity of weak decay predictions to the choice of hadronic model (wave functions and relativistic treatment), emphasizing the need for consistent relativistic approaches in heavy quarkonium physics.

In summary, this paper provides a rigorous, relativistic calculation of $J/\psi$ weak decays, establishing a solid theoretical baseline for the upcoming era of high-statistics charmonium physics.