Next-to-leading-order QCD corrections to the radiative decays $χ_{bJ}\to J/ψ+γ$

Yu-Dong Zhang, Zai-Fu Jiang

Oct 21, 2026arXiv:2610.29181v1

hep-ph

Scorecard· 16/16

4.5/10 impact

Solid, technically demanding two-loop NRQCD calculation filling a specific gap in the NLO bottomonium-decay series, but incremental, with niche audience and no measurement to confront.

Abstract

Within nonrelativistic QCD (NRQCD) factorization, we calculate the next-to-leading-order (NLO) QCD corrections to $χ_{bJ}\to J/ψ+γ$ with $J = 0, 1, 2$ . We determine all six independent helicity amplitudes, including the one-loop corrections to the QED amplitude and the two-loop corrections to the QCD amplitude, and retain the interference between the QED and QCD amplitudes in the decay widths. The NLO corrections substantially enhance the $χ_{b0}$ and $χ_{b1}$ decay widths, while reducing the $χ_{b2}$ decay width by approximately $36 %$ . Including the QED amplitude leads to sizable destructive interference for $χ_{b0}$ and an enhancement for $χ_{b2}$ , but has only a small effect on the $χ_{b1}$ decay width. Two treatments of photon fragmentation, based on the NRQCD matrix element and the experimentally determined $J / ψ$ decay constant, yield NLO unpolarized decay widths that agree at the few percent level or better. Using the decay constant scheme at $μ_{R} = m_{b}$ , we obtain NLO central branching fractions of $1.03\times10^{-6}$ , $1.45\times10^{-5}$ , and $1.99\times10^{-6}$ for the three channels, respectively. The NLO corrections reduce the renormalization scale dependence, particularly for $χ_{b2}$ , although a sizable dependence remains for $χ_{b0}$ . At fixed $m_{b}$ , the branching fractions decrease with increasing $r = m_{c} / m_{b}$ , more strongly for $χ_{b1}$ and $χ_{b2}$ than for $χ_{b0}$ . For $p p$ collisions at $\sqrt{s}=14\,{\rm TeV}$ and an integrated luminosity of $100\,{\rm fb}^{-1}$ , we estimate about $1.9\times10^4$ and $4.8\times10^4$ $\ell^+\ell^-γ$ events from $χ_{b0}$ and $χ_{b2}$ decays, respectively. Measurements with the existing Belle $Υ (2 S)$ sample remain challenging because of the small expected yields. A larger $Υ (2 S)$ sample at Belle II would improve the prospects for observing these radiative decays.

AI Impact Assessments

(1 model)

Core Contribution

The paper computes NLO QCD corrections to the radiative decays χbJ→J/ψ+γ (J=0,1,2) in NRQCD at leading order in velocity. This requires one-loop corrections to the photon-fragmentation (QED) amplitude and two-loop corrections to the pure-QCD amplitude, which starts at one loop. The authors obtain all six independent helicity amplitudes and keep the QED–QCD interference. They also compare two treatments of the γ*→J/ψ transition: the NRQCD LDME (QED I) and the measured J/ψ decay constant with the matching correction subtracted (QED II). The main results are:

NLO corrections enhance Γ(χb0) by about 2.8× and Γ(χb1) by about 1.8×, and reduce Γ(χb2) by about 36%.

The QED interference is strongly destructive for χb0, constructive for χb2, and small for χb1.

Scheme dependence drops to about 2% or less at NLO.

Branching fractions are 1.0×10⁻⁶, 1.45×10⁻⁵, and 2.0×10⁻⁶.

The scale dependence is reduced, especially for χb2, but remains large for χb0.

The paper gives yield estimates for the LHC and Belle/Belle II.

This is the natural next entry in a series of NLO/NNLO exclusive bottomonium decay calculations (Υ→ηc/χcJ+γ, Υ→J/ψ+ηc/χcJ, ηb→J/ψ+J/ψ, ηb→J/ψ+γ). It fills a specific gap and is technically nontrivial because of the two-loop P-wave amplitude with two mass scales.

Methodological Rigor

The workflow is standard and sound:

Diagram generation with FeynArts, with FeynCalc/FormLink for the algebra.

Partial fractioning and IBP reduction via FIRE, giving 14 one-loop and 552 two-loop master integrals.

Numerical evaluation of the two-loop master integrals with auxiliary-mass flow.

Covariant projectors and the method of regions (hard region).

Poles cancel, and there is an analytic r→0 asymptotic check on the LO QCD SDCs.

The helicity-amplitude formalism is clearly laid out. Weaknesses:

There is no independent cross-check of the two-loop results beyond pole cancellation, for example against an existing NLO calculation or a known limit such as ηb→J/ψ+γ.

The large and sizable negative K-factors, and the persistent μR dependence for χb0, indicate poor perturbative convergence.

The scale variation (2mc to 2mb) is the only theoretical uncertainty propagated into the tables. The uncertainties from the LDME (potential-model wave functions varying by roughly 2×), from relativistic corrections, and from mc and mb are discussed only separately.

Heavy-quark masses are fixed at mb=4.7 GeV and mc=1.5 GeV for consistency with earlier work. The LDMEs come from a potential model, not from data.

The paper does not treat the double-logarithms ln²r, which may matter given that r≈0.32 is not very small.

The QED I/QED II comparison is a useful consistency test, but the reported agreement partly reflects cancellations. Using fixed α=1/131 is a choice rather than a derivation.

The 36% reduction for χb2 and the 2.8× enhancement for χb0 may be sensitive to the scheme and mass choices.

Potential Impact

The impact is mostly confined to the quarkonium-theory community. These are first-principles NLO predictions for channels not yet observed. The expected branching fractions are about 10⁻⁶ to 10⁻⁵, and LHC yields are claimed to be large (~10⁴ ℓ⁺ℓ⁻γ events). That claim is optimistic because it ignores backgrounds and efficiencies, and the production cross sections are taken from a 2005 estimate. Belle yields are tiny (about 6, 160, and 22 events), so a measurement is unlikely in the near term. Belle II prospects require a large Υ(2S) sample. The paper provides:

Analytic asymptotics.

Tables of SDCs that can be reused.

A demonstration of the decay-constant scheme to tame photon-fragmentation uncertainty.

It will be cited by follow-up NRQCD work, such as relativistic corrections, NNLO, and resummation of ln r, and by experimental feasibility studies. It is unlikely to change the field's practice.

Timeliness & Relevance

The paper is timely within precision-NRQCD work. Multi-loop tools (AMFlow, FIRE) have made two-loop quarkonium calculations routine, and the Belle II and LHC bottomonium programs motivate predictions. The paper is part of an ongoing series, and an ηb→J/ψ+γ NLO calculation appeared only recently, which makes the χbJ extension timely. The interplay between large NLO corrections and the lack of data on these channels is a mild motivation. Interest is limited by the absence of measurements, since there are no current anomalies to explain.

Strengths

It is a complete treatment of all helicity channels, with QED–QCD interference at NLO.

The paper includes a careful scheme comparison.

Mass-ratio dependence is mapped out in tables and figures.

Asymptotic analytic expressions are provided.

The paper is clearly organized and its formulas are fairly self-contained.

Limitations

It is an incremental extension of an established method.

Large K-factors and the sizable χb0 scale dependence undermine "precision" claims.

LDME and mass uncertainties are not folded into the final predictions.

No code or ancillary files are mentioned. The two-loop SDCs are given only numerically at one mass point, plus asymptotics, which limits reuse.

The experimental outlook is pessimistic for existing data.

Only the leading-velocity term is included, and relativistic corrections are known to be large for charmonium.

Overall

This is a competent, technically demanding but expected next step in NRQCD higher-order calculations. Its value is as a reference prediction for a future measurement and as a validation of the decay-constant scheme. Its expected influence is modest and confined to quarkonium phenomenology.

Rating:4.5/ 10

Significance 4Rigor 6Novelty 4Clarity 7

Generated Oct 22, 2026

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