Charmonium radiative transitions to dileptons from lattice QCD: The case of $h_c \to \eta_c \ell^+\ell^-$ and $\chi_{c1} \to J/\psi\,\ell^+\ell^-$

This paper presents the first fully dynamical lattice QCD calculations of the dilepton decay rates and differential distributions for the charmonium transitions $h_c \to \eta_c \ell^+\ell^-$ and $\chi_{c1} \to J/\psi\,\ell^+\ell^-$, yielding results that agree with experimental data for $\chi_{c1}$ decays but predict a $h_c \to \eta_c e^+ e^-$ rate approximately $3\sigma$ higher than current BESIII measurements.

The Gist

Imagine the subatomic world as a bustling, high-energy dance floor. On this floor, heavy particles called charmonia (made of a charm quark and its anti-particle) are constantly spinning, jumping, and changing partners. Sometimes, they shed energy by flashing a light (a photon) to move to a lower energy state. This is like a dancer taking a bow and flashing a camera.

But sometimes, instead of just flashing a single camera, the dancer flashes a "virtual" light that instantly splits into a pair of new dancers: an electron and a positron (or a muon and an antimuon). This is called a Dalitz decay.

This paper is a massive, high-precision calculation by a team of physicists using a method called Lattice QCD (Quantum Chromodynamics). Think of Lattice QCD as a super-accurate 3D grid or a "pixelated universe" where they simulate the laws of physics from scratch, without guessing or using shortcuts. They wanted to predict exactly how often these heavy particles perform this specific "split-flash" dance move.

Here is the breakdown of their work in everyday terms:

1. The Two Main Dances

The team focused on two specific transitions (dance moves):

• The $h_c$ to $\eta_c$ move: A particle with a "twist" (spin) changes into a rounder, calmer particle, releasing a pair of leptons.
• The $\chi_{c1}$ to $J/\psi$ move: Another twisted particle changes into a very famous, stable particle (the $J/\psi$), also releasing a lepton pair.

2. The "Virtual" vs. "Real" Flash

In previous studies, physicists only looked at the "Real Flash" (emitting a single, real photon).

• The Real Flash: Like a camera flash that goes pop and is gone. It only tells you about the "sideways" movement of the dancer.
• The Virtual Flash (Dalitz Decay): This is like a flash that splits into two new people before disappearing. Because this "flash" is virtual (off-shell), it has a hidden dimension. It allows the physicists to see a "longitudinal" movement—a forward-and-back motion—that is completely invisible in the real flash.

The paper is special because it's the first time anyone has calculated these specific "split-flash" moves using the fundamental laws of physics (QCD) directly, rather than relying on models or guesses.

3. How They Did It (The Simulation)

To do this, the team used a supercomputer to run simulations on a grid (a lattice).

• The Ingredients: They used a digital recipe with four types of "sea quarks" (light, strange, and charm) moving around, just like they do in the real universe.
• The Resolution: They ran the simulation at four different "zoom levels" (lattice spacings). Imagine taking a photo of a face at low resolution, medium, high, and ultra-high. They then mathematically "zoomed out" to the infinite resolution (the continuum limit) to get the perfect, blur-free answer.
• The Math: They calculated "Form Factors." Think of these as the dance choreography manuals. They describe exactly how the particles twist and turn at every possible speed.

4. The Results: What Did They Find?

They calculated the "decay rate," which is essentially the probability of this dance happening.

• The Good News ($\chi_{c1}$): For the $\chi_{c1}$ particle, their prediction matched the experimental data from the BESIII lab in China almost perfectly. It's like they predicted the score of a basketball game, and the actual game ended with that exact score. This gives them great confidence in their method.
• The Mystery ($h_c$): For the $h_c$ particle, their prediction was about 30% higher than what the BESIII lab has measured so far.
  • The Analogy: Imagine the lab measured a dancer spinning 10 times a minute, but the simulation says they should be spinning 13 times.
  • The Implication: This isn't necessarily a failure. It might mean the experimental measurement needs to be more precise, or it could hint at new physics we don't understand yet. The authors are essentially saying, "Our math says it should be faster; please check your stopwatch again!"

5. Why Does This Matter?

• A New Ruler: By calculating these rates so precisely, they provide a "ruler" for future experiments. If a new particle (like a "dark photon" or a hidden force) exists, it might mess with these dance moves. If the experiment deviates from this new, precise ruler, scientists will know they found something new.
• Seeing the Invisible: They also predicted how the particles behave at different speeds (differential distributions). This allows experimentalists to look at the "shape" of the decay, not just the total number, to test if the "longitudinal" movement (the hidden dimension) behaves as the Standard Model predicts.

Summary

Think of this paper as a team of master architects building a perfect, scale-model universe to predict how heavy particles dance.

• They built the model using the most fundamental bricks of reality (QCD).
• They predicted the dance moves for two specific particles.
• One prediction matched reality perfectly (validating their model).
• The other prediction disagreed with current measurements, suggesting either the measurement needs a tune-up or a new mystery is waiting to be solved.

This work sets a new "gold standard" for how we understand the internal structure of heavy particles and provides a benchmark for the next generation of particle physics experiments.

Technical Summary

1. Problem Statement

Electromagnetic transitions in heavy quarkonia (bound states of heavy quark-antiquark pairs) serve as sensitive probes of their internal structure. While radiative decays involving real photons (e.g., $\chi_{c1} \to J/\psi \gamma$) have been extensively studied, decays involving a virtual photon that converts into a lepton-antilepton pair (Dalitz decays, e.g., $\chi_{c1} \to J/\psi \ell^+\ell^-$) offer additional physical insights.

• The Gap: In Dalitz decays, the virtual photon is off-shell ($q^2 > 0$). This allows access to longitudinal transition form factors (e.g., $C_1(q^2)$ and $\tilde{F}_2(q^2)$) which vanish in the real-photon limit ($q^2=0$).
• The Challenge: Previous theoretical studies relied on phenomenological models or effective field theories (NRQCD, pNRQCD) that require truncation of velocity expansions, introducing unquantified systematic uncertainties. A first-principles determination of the full $q^2$-dependence of these form factors directly from Quantum Chromodynamics (QCD) was lacking.
• Experimental Context: Recent experimental data from the BESIII collaboration has measured these branching fractions with increasing precision, revealing potential tensions with Standard Model expectations (particularly for $h_c \to \eta_c e^+e^-$) and opening the door to searches for new physics (e.g., dark photons).

2. Methodology

The authors performed the first fully dynamical lattice QCD study of these processes using the following setup:

• Lattice Configurations: Simulations were conducted using $N_f = 2+1+1$ dynamical Wilson–Clover twisted-mass fermions generated by the Extended Twisted Mass Collaboration (ETMC).
  • Ensembles: Four lattice spacings were used ($a \approx 0.057$ fm to $0.091$ fm).
  • Quark Masses: Simulations were performed at physical values for $u, d, s,$ and $c$ quark masses, except for the coarsest lattice where the lightest pion mass was slightly heavier ($m_\pi \approx 175$ MeV).
• Form Factor Decomposition:
  • $\chi_{c1} \to J/\psi \gamma^*$: The matrix element is parameterized by three form factors: $E_1(q^2)$, $M_2(q^2)$, and the longitudinal form factor $C_1(q^2)$.
  • $h_c \to \eta_c \gamma^*$: The matrix element is parameterized by $F_1(q^2)$ and the longitudinal form factor $\tilde{F}_2(q^2)$.
• Computational Strategy:
  • Correlation Functions: Three-point correlation functions were computed in the rest frame of the decaying meson.
  • Momentum Injection: To cover the full kinematic range of $q^2$ (from $q^2=0$ to $q^2_{max}$), the authors employed partially-quenched twisted boundary conditions to inject arbitrary spatial momenta.
  • Mixed-Action Setup: To optimize the extraction of form factors, a mixed-action approach was used with specific choices of the Wilson parameter ($r = \pm 1$) for valence quarks to avoid unwanted mixing with lighter states (Osterwalder-Seiler regularization for $\chi_{c1}$ and Twisted-Mass regularization for $h_c$).
  • Noise Reduction: Advanced statistical techniques were employed, including:
    • Correlated Subtractions: Subtracting the $k=0$ contribution to reduce noise for form factors that vanish at zero momentum.
    • Preconditioning: Using high-statistics data at $q^2=0$ to reconstruct ratios at non-zero $q^2$, significantly reducing computational cost.
• Continuum Extrapolation: Results were extrapolated to the continuum limit ($a \to 0$) using linear fits in $a^2$, employing Bayesian Akaike Information Criterion (BAIC) to average fits including and excluding the coarsest lattice.

3. Key Contributions

1. First-Principles Calculation: This is the first unquenched lattice QCD determination of the full $q^2$-dependence of form factors for $h_c \to \eta_c \ell^+\ell^-$ and $\chi_{c1} \to J/\psi \ell^+\ell^-$.
2. Longitudinal Form Factors: The study successfully extracted the longitudinal form factors ($C_1$ and $\tilde{F}_2$) which are inaccessible in real-photon radiative decays.
3. Differential Predictions: The authors provided predictions not just for total decay rates, but for differential decay widths ($d\Gamma/dq^2$) and angular observables sensitive to the longitudinal components.
4. Benchmarking: The results serve as rigorous Standard Model benchmarks for interpreting experimental data and searching for deviations indicative of new physics (e.g., light mediators).

4. Key Results

Total Decay Widths (Continuum Limit):
The authors computed the partial decay widths for electron and muon channels:

• $\chi_{c1}$ Channel:
  • $\Gamma(\chi_{c1} \to J/\psi e^+e^-) = 2.869(90)$ keV
  • $\Gamma(\chi_{c1} \to J/\psi \mu^+\mu^-) = 0.1993(72)$ keV
  • Comparison: These results show excellent agreement with experimental data from PDG and BESIII (within $\sim 1\sigma$). The lepton flavor universality ratio also matches experiment.
• $h_c$ Channel:
  • $\Gamma(h_c \to \eta_c e^+e^-) = 5.45(19)$ keV
  • $\Gamma(h_c \to \eta_c \mu^+\mu^-) = 0.635(22)$ keV
  • Comparison: The prediction for the electron channel is approximately $2.5\sigma$ to $3\sigma$ larger than the current experimental determination (BESIII/PDG). This tension suggests a need for improved experimental precision or a re-evaluation of the $h_c$ total width.

Differential and Angular Observables:

• The $q^2$ dependence of all form factors was found to be smooth.
• The longitudinal form factor $C_1(q^2)$ for $\chi_{c1}$ exhibits a non-zero slope in $q^2$, whereas $M_2(q^2)$ is proportional to $|k|^2$.
• A bin-by-bin comparison of the $q^2$-differential distribution for $\chi_{c1} \to J/\psi e^+e^-$ with BESIII data showed good overall agreement after efficiency corrections, though tensions of $2-3\sigma$ were observed in specific low-$q^2$ bins (17, 37, and 47 MeV).

5. Significance

• Theoretical Validation: The work validates the lattice QCD framework for off-shell electromagnetic transitions, demonstrating its capability to handle complex kinematic ranges and longitudinal polarization states without model assumptions.
• New Physics Searches: By providing precise Standard Model predictions for the full kinematic range, the study enables more sensitive searches for "dark sector" physics (e.g., dark photons) in charmonium decays. Any deviation in the differential spectra or angular distributions from these lattice results could signal new mediators.
• Experimental Guidance: The identified tension in the $h_c \to \eta_c e^+e^-$ channel highlights a critical area for future experimental investigation. The paper provides the necessary theoretical tools (differential distributions and angular observables) for BESIII and future charm factories to refine these measurements.
• Methodological Advance: The techniques developed for extracting longitudinal form factors and handling twisted boundary conditions in mixed-action setups set a precedent for future lattice studies of other off-shell processes.