⚛️ phenomenology

Next-to-leading-order QCD corrections to the radiative decays $\chi_{bJ}\to J/\psi+\gamma$

This paper calculates the next-to-leading-order QCD corrections to the radiative decays $\chi_{bJ}\to J/\psi+\gamma$ within the NRQCD framework, revealing substantial enhancements for the $J=0,1$ channels and a significant reduction for $J=2$ , while providing updated branching fraction predictions and estimating observable event yields for future experiments at Belle II and the LHC.

Original authors: Yu-Dong Zhang, Zai-Fu Jiang

Published 2026-10-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yu-Dong Zhang, Zai-Fu Jiang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep within the heart of matter, protons and neutrons are built from smaller, more fundamental particles called quarks. These quarks are bound together by the strong force, the most powerful interaction in nature, which acts like an invisible glue. When a heavy quark and its corresponding heavy antiquark meet, they can form a short-lived, exotic atom known as a quarkonium. Scientists study these fleeting atoms to understand how the strong force works when it binds heavy particles together. One specific family of these atoms, made of bottom quarks, is called bottomonium. Another family, made of charm quarks, is called charmonium. A particularly interesting question in physics is how a heavy bottomonium atom can transform into a lighter charmonium atom while emitting a flash of light, or a photon. This process is rare and difficult to predict because it involves a complex interplay between the known laws of electromagnetism and the more mysterious, powerful strong force.

For years, physicists have been able to calculate the basic probability of this transformation using the simplest possible models. However, nature is rarely that simple. To get a truly accurate prediction, scientists must account for the subtle, higher-order effects that occur when particles interact multiple times before the final event. This is where a team of researchers from Jingchu University of Technology in China has made a significant step forward. They performed a highly detailed calculation to predict exactly how often a specific type of bottomonium, known as the chi-b, decays into a charmonium particle called the J/psi and a photon. Their work goes beyond the basic models by including the next level of complexity in the mathematics of the strong force, a step known as next-to-leading-order corrections.

The researchers focused on three different versions of the chi-b particle, which differ slightly in their internal spin and energy. They calculated the probability of each version decaying into the J/psi and a photon. In their calculations, they had to account for two distinct ways this transformation can happen. The first way is purely electromagnetic, where the bottom quarks simply rearrange themselves and emit light. The second way involves the strong force, where the quarks exchange gluons, the carriers of the strong force, before turning into the final particles. The team found that these two pathways do not just happen side by side; they interfere with each other, much like two sound waves that can either amplify or cancel each other out depending on their timing. This interference is crucial for getting the right answer.

The results of this new, more precise calculation revealed surprising differences between the three types of chi-b particles. For the first two types, the new calculations showed that the decay happens much more frequently than previously thought, with the probability increasing significantly when the complex strong-force effects are included. However, for the third type, the story is different. The inclusion of these higher-order effects actually reduces the probability of decay by about thirty-six percent. This means that the third type of particle is less likely to make this transformation than earlier, simpler models suggested. The researchers also discovered that the way they treat the emission of the photon matters. They tested two different mathematical approaches to describe how the photon is created, and found that while the methods gave different results at the basic level, they agreed very closely once the more complex corrections were applied. This agreement gives them confidence that their predictions are robust.

The study also explored how the results change if the masses of the quarks are slightly different, a common source of uncertainty in such calculations. They found that the predictions are sensitive to the mass of the charm quark, particularly for the second and third types of chi-b particles. As the mass of the charm quark increases, the likelihood of the decay drops sharply. This sensitivity highlights the importance of knowing the exact mass of these fundamental particles to make precise predictions. Furthermore, the team calculated how many of these decay events might be observed in real-world experiments. They estimated that at the Large Hadron Collider, the world's most powerful particle accelerator, roughly 19,000 events from the first type and 48,000 events from the third type could be detected if the machine runs for a standard amount of time. These numbers are large enough to be interesting, but they are small enough that detecting them requires very careful analysis to separate the signal from background noise.

Looking toward the future, the researchers noted that current experiments at the Belle facility in Japan have not yet seen these decays, largely because the number of bottomonium atoms they have collected is too small to produce enough events to be noticeable. However, the next generation of this experiment, Belle II, is designed to collect a much larger sample of these particles. With this increased data, the chances of actually observing these rare decays and confirming the team's theoretical predictions become much brighter. This work provides a reliable theoretical guide for experimentalists, telling them exactly what to look for and how often they should expect to see it. By refining the predictions for these rare events, the study helps sharpen our understanding of the fundamental forces that govern the behavior of matter at the smallest scales.

Technical Summary: Next-to-Leading-Order QCD Corrections to $\chi_{bJ} \to J/\psi + \gamma$

Problem Statement
The paper addresses the theoretical calculation of radiative decays of bottomonium P-wave states ( $\chi_{bJ}$ with $J=0, 1, 2$ ) into a charmonium vector meson ( $J/\psi$ ) and a photon. While leading-order (LO) calculations within Nonrelativistic QCD (NRQCD) factorization exist, they exhibit significant dependence on the renormalization scale and do not fully account for the interplay between Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD) amplitudes. Specifically, the QED amplitude (photon fragmentation) and the pure QCD amplitude (one-loop) start at different orders in coupling constants ( $\alpha^{3/2}$ vs. $\alpha^{1/2}\alpha_s^2$ ). A complete Next-to-Leading-Order (NLO) treatment requires calculating one-loop corrections to the QED amplitude and two-loop corrections to the pure QCD amplitude to ensure a consistent expansion and to properly include interference effects, which are numerically significant.

Methodology
The authors perform the calculation within the NRQCD factorization framework, separating the decay amplitude into perturbatively calculable short-distance coefficients (SDCs) and nonperturbative long-distance matrix elements (LDMEs).

Helicity Amplitudes: The analysis determines all six independent helicity amplitudes for the process $\chi_{bJ} \to J/\psi + \gamma$ , utilizing the helicity amplitude formalism to derive polarized and unpolarized decay widths.
Perturbative Calculation: The calculation involves matching full QCD amplitudes to NRQCD amplitudes. The authors employ FeynArts for diagram generation, FeynCalc and FormLink for algebraic manipulation, and FIRE for integration-by-parts (IBP) reduction.
- The QED contribution (photon fragmentation $\chi_{bJ} \to \gamma + \gamma^* \to \gamma + J/\psi$ ) is calculated at tree level and one-loop.
- The pure QCD contribution is calculated at one-loop (LO) and two-loop (NLO).
- Ultraviolet and infrared divergences are regularized in $D=4-2\epsilon$ dimensions, with renormalization performed in the on-shell scheme for masses/fields and the $\overline{\text{MS}}$ scheme for $\alpha_s$ .
Photon Fragmentation Schemes: Two treatments for the $\gamma^* \to J/\psi$ $γ^{*} \to J / ψ$ transition are compared:
1. QEDI: The conventional NRQCD treatment using the $J/\psi$ LDME.
2. QEDII: An alternative scheme where the transition is described by the experimentally determined $J/\psi$ decay constant ( $f_{J/\psi}$ ). In this scheme, the perturbative matching correction associated with the transition is subtracted from the QED SDC to avoid double counting.
Numerical Inputs: The calculation uses pole masses $m_b = 4.7$ GeV and $m_c = 1.5$ GeV. The renormalization scale $\mu_R$ is set to $m_b$ with variations between $2m_c$ and $2m_b$ to estimate uncertainty.

Key Results

Decay Widths and Branching Fractions:
- The NLO corrections substantially enhance the decay widths for $\chi_{b0}$ and $\chi_{b1}$ (by factors of $\sim 2.8$ and $\sim 1.8$ , respectively) but reduce the $\chi_{b2}$ width by approximately 36% compared to LO.
- The central NLO branching fractions (at $\mu_R = m_b$ ) are calculated as $1.03 \times 10^{-6}$ , $1.45 \times 10^{-5}$ , and $1.99 \times 10^{-6}$ for $J=0, 1, 2$ , respectively.
- Despite the $\chi_{b0}$ and $\chi_{b1}$ having comparable decay widths at NLO ( $\sim 1.18$ eV and $\sim 1.14$ eV), the $\chi_{b1}$ branching fraction is significantly larger due to its smaller total decay width.
Interference Effects:
- For $\chi_{b0}$ , the inclusion of the QED amplitude leads to sizable destructive interference, reducing the pure QCD width by $\sim 35\%$ at NLO.
- For $\chi_{b2}$ , the QED and interference contributions result in an enhancement of $\sim 20\%$ .
- The effect on $\chi_{b1}$ is minimal (a few percent).
Scheme Dependence: Comparing the QEDI and QEDII schemes, the authors find that while LO predictions differ (up to 30% for $\chi_{b0}$ ), the NLO unpolarized decay widths agree within a few percent (better than 2% for $\chi_{b1}$ and $\chi_{b2}$ ), indicating robustness against the choice of fragmentation scheme at this order.
Scale and Mass Dependence:
- NLO corrections significantly reduce the renormalization scale dependence, particularly for $\chi_{b2}$ . However, a sizable dependence remains for $\chi_{b0}$ , suggesting higher-order corrections may still be relevant for this channel.
- Branching fractions decrease as the mass ratio $r = m_c/m_b$ increases. This dependence is mild for $\chi_{b0}$ but pronounced for $\chi_{b1}$ and $\chi_{b2}$ .

Significance and Outlook
The paper claims to provide the first systematic investigation of NLO QCD corrections to $\chi_{bJ} \to J/\psi + \gamma$ . The primary significance lies in stabilizing the theoretical predictions by reducing scale uncertainties and quantifying the crucial interference between QED and QCD amplitudes.

Regarding experimental prospects, the authors estimate that for LHC collisions at $\sqrt{s} = 14$ TeV with an integrated luminosity of $100 \text{ fb}^{-1}$ , approximately $1.9 \times 10^4$ and $4.8 \times 10^4$ $\ell^+\ell^-\gamma$ events (from $J/\psi \to \ell^+\ell^-$ ) could be observed from $\chi_{b0}$ and $\chi_{b2}$ decays, respectively. These yields motivate experimental searches at the LHC. Conversely, measurements using existing Belle $\Upsilon(2S)$ samples are deemed challenging due to small expected yields (e.g., $\sim 6$ events for $\chi_{b0}$ ). The authors suggest that a substantially larger $\Upsilon(2S)$ sample at Belle II would improve the prospects for observing these radiative decays, particularly for the $\chi_{b1}$ channel.

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