REVIEW
Next-to-leading-order QCD corrections to the radiative decays $\chi_{bJ}\to J/\psi+\gamma$
High Energy Physics - Phenomenology · Submitted 21 Oct 2026 · Published version · 48 references
Pith's verdict: Solid NLO calculation for a set of rare bottomonium decays, carefully done and cross-checked, but the missing relativistic corrections (O(v^2)) for P-wave χbJ could be comparable to the NLO QCD effects—a caveat the authors themselves state but do not quantify.
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Referee report
The paper
The paper claims
NLO QCD corrections dramatically alter χbJ → J/ψ + γ decay widths, enhancing χb0 and χb1 while reducing χb2 by 36%.
The reading
What carries the argument
The central machinery is the NRQCD factorization formula (7), which expresses each helicity amplitude as a product of a dimensionless short-distance coefficient (SDC) and nonperturbative long-distnace matrix elements (LDMEs). The SDCs are obtained via perturbativ matching by replacing the physical quarkonia with free on-shell heavy-quark pairs in the 3S1 and 3P[1]J states, and extracting the hard part of the amplitude. The calculation uses FeynArts to generate diagrams, FIRE for integration-by-parts reduction, and the auxiliary-mass-flow method for two-loop master integrals. The photon-fragmentation contribution is handled in two ways—QEDI uses the NRQCD LDME for J/ψ, while QEDII uses the J/ψ decay constant from experiment, with a subtractive matching correction to avoid double counting. The six independent helicity amplitudes are determined, and the NLO corrections include both one-loop QED and two-loop QCD contributions.
Extended reading notes
Core claim
The paper’s central discovery is that the NLO QCD corrections to χbJ → J/ψ + γ are sizable and qualitatively different across the three spin states. At leading order in the heavy-quark velocity expansion ($v^{2}$ → 0), the six helicity amplitudes are determined analytically or numerically. The NLO corrections enhance the unpolarized decay widths of χb0 and χb1 by factors ≈ 2.8 and ≈ 1.8, respectively, and suppress that of χb2 by ≈ 36%. The QED amplitude (photon fragmentation) interferes destructively with the QCD amplitude for χb0, reducing the pure QCD width by 35% at NLO, and constructively for χb2, increasing it by 20%; it has little effect on χb1. Two treatments of the γ* → J/ψ transition — the standard NRQCD matrix element (QEDI) and the experimentally measured J/ψ decay constant (QEDII) — yield unpolarized decay widths that agree within about 2% at NLO, demonstrating that the scheme dependence is under control. The predicted NLO branching fractions in the QEDII scheme, using mb = 4.7 GeV, mc = 1.5 GeV, and μ_R = mb, are Br(χb0) = 1.03 × $10^{{-6}}$, Br(χb1) = 1.45 × $10^{{-5}}$, and Br(χb2) = 1.99 × $10^{{-6}}$. The NLO corrections also reduce the renormalization-scale dependence, especially for χb2, although a sizable scale variation persists for χb0.
Load-bearing premise
The calculation works at leading order in the heavy-quark velocity expansion (v² → 0) for both the initial and final quarkonia, meaning that relativistic corrections, which are expected to be of order 10% for P-wave bottomonia, are omitted.
Editorial extensions
If this is right
- The predicted branching fractions and event yields (≈ 1.9 × 10^4 ℓ⁺ℓ⁻γ events from χb0 and ≈ 4.8 × 10^4 from χb2 at LHC with 100 fb⁻¹) provide concrete targets for experimental searches, especially at LHC and a future large Υ(2S) sample at Belle II.
- The strong dependence of the branching fractions on the ratio r = m_c/m_b (over ~50% change for χb1 and χb2 when mc is varied from 1.3 to 1.7 GeV) means that precision measurements could constrain the charm- and bottom-quark masses.
- The large NLO correction and remaining renormalization-scale dependence in the χb0 channel indicate that next-to-next-to-leading-order (NNLO) corrections may be necessary for a reliable prediction of that channel.
- The agreement between the QEDI and QEDII schemes at the few-percent level at NLO validates the approach of using the J/ψ decay constant to resum large logarithms associated with the photon fragmentation.
- The small full width of χb1 implies that its branching fraction is the largest despite its decay width being comparable to χb0; hence χb1 → J/ψ + γ is the most promising channel for detection.
Reading between the lines
- If the negative NLO correction to χb2 is confirmed by experiment, it would provide a nontrivial test of the NRQCD counting rules for P-wave quarkonia, where the LO QCD amplitude starts at one loop and receives large cancellations.
- The neglected relativistic corrections (of order v^2 ≈ 10 % for P-wave bottomonia) could shift the predicted branching fractions by amounts comparable to the remaining scale uncertainty, especially for χb0; future work should include them to sharpen the predictions.
- The study implies that the photon fragmentation (γ*) contribution is not negligible even in the NLO corrections, establishing a template for treating mixed QED-QCD amplitudes in other heavy-quarkonium decays such as ηb → J/ψ + γ.
- The event-yield estimates suggest that LHC may be the first experiment to observe these decays, possibly before Belle II accumulates a sufficiently large Υ(2S) dataset.
Original abstract
Within nonrelativistic QCD (NRQCD) factorization, we calculate the next-to-leading-order (NLO) QCD corrections to $\chi_{bJ}\to J/\psi+\gamma$ 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 $\chi_{b0}$ and $\chi_{b1}$ decay widths, while reducing the $\chi_{b2}$ decay width by approximately $36\%$. Including the QED amplitude leads to sizable destructive interference for $\chi_{b0}$ and an enhancement for $\chi_{b2}$, but has only a small effect on the $\chi_{b1}$ decay width. Two treatments of photon fragmentation, based on the NRQCD matrix element and the experimentally determined $J/\psi$ decay constant, yield NLO unpolarized decay widths that agree at the few percent level or better. Using the decay constant scheme at $\mu_R=m_b$, 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 $\chi_{b2}$, although a sizable dependence remains for $\chi_{b0}$. At fixed $m_b$, the branching fractions decrease with increasing $r=m_c/m_b$, more strongly for $\chi_{b1}$ and $\chi_{b2}$ than for $\chi_{b0}$. For $pp$ collisions at $\sqrt{s}=14\,{\rm TeV}$ and an integrated luminosity of $100\,{\rm fb}^{-1}$, we estimate about $1.9\times10^4$ and $4.8\times10^4$ $\ell^+\ell^-\gamma$ events from $\chi_{b0}$ and $\chi_{b2}$ decays, respectively. Measurements with the existing Belle $\Upsilon(2S)$ sample remain challenging because of the small expected yields. A larger $\Upsilon(2S)$ sample at Belle II would improve the prospects for observing these radiative decays.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Within nonrelativistic QCD (NRQCD) factorization, we calculate the next-to-leading-order (NLO) QCD corrections to $\chi_{bJ}\to J/\psi+\gamma$ 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 $\chi_{b0}$ and $\chi_{b1}$ decay widths, while reducing the $\chi_{b2}$ decay width by approximately $36\%$. Including the QED amplitude leads to sizable destructive interference for $\chi_{b0}$ and an enhancement for $\chi_{b2}$, but has only a small effect on the $\chi_{b1}$ decay width. Two treatments of photon fragmentation, based on the NRQCD matrix element and the experimentally determined $J/\psi$ decay constant, yield NLO unpolarized decay widths that agree at the few percent level or better. Using the decay constant scheme at $\mu_R=m_b$, 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 $\chi_{b2}$, although a sizable dependence remains for $\chi_{b0}$. At fixed $m_b$, the branching fractions decrease with increasing $r=m_c/m_b$, more strongly for $\chi_{b1}$ and $\chi_{b2}$ than for $\chi_{b0}$. For $pp$ collisions at $\sqrt{s}=14\,{\rm TeV}$ and an integrated luminosity of $100\,{\rm fb}^{-1}$, we estimate about $1.9\times10^4$ and $4.8\times10^4$ $\ell^+\ell^-\gamma$ events from $\chi_{b0}$ and $\chi_{b2}$ decays, respectively. Measurements with the existing Belle $\Upsilon(2S)$ sample remain challenging because of the small expected yields. A larger $\Upsilon(2S)$ sample at Belle II would improve the prospects for observing these radiative decays.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Within nonrelativistic QCD (NRQCD) factorization, we calculate the next-to-leading-order (NLO) QCD corrections to $\chi_{bJ}\to J/\psi+\gamma$ with $J=0,1,2$. | contribution 0.98 | 0.97 | 0.07 | asserted 1.00 |
| s1 | 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. | method 0.62 | 0.97 | 0.08 | asserted 1.00 |
| s2 | The NLO corrections substantially enhance the $\chi_{b0}$ and $\chi_{b1}$ decay widths, while reducing the $\chi_{b2}$ decay width by approximately $36\%$. | result 1.00 | 0.95 | 0.42 | asserted 1.00 |
| s3 | Including the QED amplitude leads to sizable destructive interference for $\chi_{b0}$ and an enhancement for $\chi_{b2}$, but has only a small effect on the $\chi_{b1}$ decay width. | result 0.99 | 0.93 | 0.10 | asserted 1.00 |
| s4 | Two treatments of photon fragmentation, based on the NRQCD matrix element and the experimentally determined $J/\psi$ decay constant, yield NLO unpolarized decay widths that agree at the few percent level or better. | result 0.81 | 0.93 | 0.07 | asserted 1.00 |
| s5 | Using the decay constant scheme at $\mu_R=m_b$, 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. | result 1.00 | 0.96 | 0.15 | asserted 1.00 |
| s6 | The NLO corrections reduce the renormalization scale dependence, particularly for $\chi_{b2}$, although a sizable dependence remains for $\chi_{b0}$. | result 0.91 | 0.95 | 0.09 | asserted 1.00 |
| s7 | At fixed $m_b$, the branching fractions decrease with increasing $r=m_c/m_b$, more strongly for $\chi_{b1}$ and $\chi_{b2}$ than for $\chi_{b0}$. | result 1.00 | 0.89 | 0.05 | asserted 1.00 |
| s8 | For $pp$ collisions at $\sqrt{s}=14\,{\rm TeV}$ and an integrated luminosity of $100\,{\rm fb}^{-1}$, we estimate about $1.9\times10^4$ and $4.8\times10^4$ $\ell^+\ell^-\gamma$ events from $\chi_{b0}$ and $\chi_{b2}$ decays, respectively. | result 0.96 | 0.92 | 0.04 | asserted 0.64 |
| s9 | Measurements with the existing Belle $\Upsilon(2S)$ sample remain challenging because of the small expected yields. | interpretation 0.40 | 0.27 | 0.04 | asserted 0.94 |
| s10 | A larger $\Upsilon(2S)$ sample at Belle II would improve the prospects for observing these radiative decays. | future work 0.99 | 0.16 | 0.07 | hedged possibility 0.63 |
What was examined, and what was not
- arXiv listing abstract v1
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Figures
Figures reproduced from arXiv:2610.29181 by the authors.
References
Discussion
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