REVIEW
Resummation of threshold double logarithms in gluon fragmentation into $^1S_0^{[1,8]}$ quarkonia
High Energy Physics - Phenomenology · Submitted 30 Sep 2026 · 43 references
Pith's verdict: Clean, honest extension of threshold resummation to the 1/N-suppressed spin-singlet channels, with the all-orders exponent backed only by a one-loop check that the author explicitly concedes does not generalize.
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Referee report
The paper
The paper claims
The paper derives an all-orders resummation of the threshold double logarithms that appear in gluon fragmentation functions for producing spin-singlet S-wave quarkonia, the $^1S_0^{[1]}$ and $^1S_0^{[8]}$ channels, and shows the resummed…
The reading
What carries the argument
The load-bearing object is the soft-function organization of threshold gluon emissions: soft gluons attach to adjoint Wilson lines $\Phi_p$, $\Phi_k$, and $\Phi_n$, and the double poles that generate the threshold double logarithms come from planar one-gluon exchanges between these lines, which exponentiate in Mellin space. Because producing a spin-singlet state requires a non-soft final-state gluon of momentum $k$, the paper keeps that gluon hard and shows that with the bookkeeping scale choice $\mu^2=M^2(1-z)$, only the soft region contributes to the double logarithms, so the non-factorizable collinear-region terms can be discarded without affecting the leading double-logarithmic terms.
Extended reading notes
Core claim
The central claim is that the threshold double logarithms of gluon fragmentation into $^1S_0^{[1,8]}$ quarkonia exponentiate in Mellin space as $\tilde d_{\rm Resum}(N)=\exp[J_N]\tilde d_{LO}(N)$, with exponent $J_N^{[8]}=\frac{\alpha_s C_A}{\pi}\int_0^1\frac{dz}{z}z^N\left(\frac{-2\log(1-z)}{1-z}\right)_+$ for the color-octet channel and $J_N^{[1]}=\frac12 J_N^{[8]}$ for the color-singlet channel. The double-log coefficients are fixed by one-gluon exchanges among three adjoint Wilson lines: only $\Phi_k\!-\!\Phi_n$ for the singlet, and $\Phi_p\!-\!\Phi_n$, $\Phi_p\!-\!\Phi_k$, and $\Phi_k\!-\!\Phi_n$ for the octet, with the lightlike-lightlike exchange contributing twice as much because both directions produce collinear poles. The next-to-leading-order double logarithms, $-(\alpha_s C_A/2\pi)\log^2(1-z)$ for the singlet and $-\alpha_s C_A/\pi\log^2(1-z)$ for the octet, reproduce the independent fixed-order results exactly. Because the exponential vanishes faster than any power of $N$, the inverse Mellin transform yields smooth $z$-space fragmentation functions that vanish at $z=1$.
Load-bearing premise
The all-orders result rests on the premise that leading threshold double logarithms arise only from planar soft-gluon exchanges among the three Wilson lines, that these double poles do not depend on the Wilson-line directions so the momenta over the final-state cut decouple, and that the collinear-region terms, which the author states cannot be generalized to arbitrarily many gluon exchanges, produce no double logarithms once the scale is chosen as $\mu^2=M^2(1-z)$.
Editorial extensions
If this is right
- The resummed $^1S_0^{[1,8]}$ gluon fragmentation functions vanish smoothly at $z=1$, eliminating the unphysical sign flip and double-logarithmic divergence that fixed-order next-to-leading-order results develop near threshold.
- The color-octet channel is suppressed more strongly at threshold than the singlet, because its exponent $J_N^{[8]}$ is twice the singlet exponent.
- Even after DGLAP evolution to high factorization scales, threshold resummation considerably softens the $z$-shapes, which can matter for differential observables such as quarkonium momentum distributions in jets if the spin-singlet channel dominates production.
- Most of the effect of next-to-leading-order corrections to the threshold-resummed functions is a change in overall normalization, which can be absorbed into phenomenologically determined NRQCD matrix elements and so has little phenomenological impact.
- The method extends to other $1/N$-suppressed channels, including longitudinally polarized $^3S_1^{[8]}$ and $^3P_J^{[1,8]}$ contributions and the relativistic correction to the $^3S_1^{[1]}$ channel, where the extra collinear gluon adds another lightlike direction and may enlarge the exponent.
Reading between the lines
- The exponent's dependence only on $C_A$, with no quark-flavor content, suggests the resummation is insensitive to the heavy-quark mass scale at this accuracy, so the same form should hold with the quark mass replaced by any hard scale of order the pair invariant mass.
- A two-loop computation of the singlet or octet fragmentation function near threshold would be the sharpest test: if any non-planar or collinear-region configuration generates a double logarithm beyond the exponentiated form, the assumed simple factorization breaks down.
- The bookkeeping-scale trick of setting $\mu^2=M^2(1-z)$ to discard collinear-region terms may generalize to other power-suppressed fragmentation channels whose kinematic threshold leaves finite remnant distributions.
- For jet-substructure phenomenology, the strong suppression near $z=1$ implies that spin-singlet channels contribute mostly at moderate $z$; a measurement of $z^3$ times the fragmentation function in jets could separate this contribution from the harder $^3S_1^{[8]}$ and $^3P_J^{[1,8]}$ channels.
Original abstract
I investigate the threshold double logarithmic behavior of the fragmentation function of a gluon into $^1S_0^{[1,8]}$ quarkonia, which is suppressed by $1/N$ in Mellin space. While threshold double logarithms have been resummed to all orders in the strong coupling for gluon fragmentation functions for quarkonia at leading power in $1/N$, it has not been known how to resum them for channels suppressed by powers of $1/N$. Threshold resummation can be important for the $^1S_0^{[1,8]}$ channels, as the tree-level gluon fragmentation functions have finite yet nonzero values at the kinematical threshold. I derive a formalism for resumming the threshold double logarithms for these channels at the leading double logarithmic level and discuss the phenomenological impact.
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
I investigate the threshold double logarithmic behavior of the fragmentation function of a gluon into $^1S_0^{[1,8]}$ quarkonia, which is suppressed by $1/N$ in Mellin space. While threshold double logarithms have been resummed to all orders in the strong coupling for gluon fragmentation functions for quarkonia at leading power in $1/N$, it has not been known how to resum them for channels suppressed by powers of $1/N$. Threshold resummation can be important for the $^1S_0^{[1,8]}$ channels, as the tree-level gluon fragmentation functions have finite yet nonzero values at the kinematical threshold. I derive a formalism for resumming the threshold double logarithms for these channels at the leading double logarithmic level and discuss the phenomenological impact.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | I investigate the threshold double logarithmic behavior of the fragmentation function of a gluon into $^1S_0^{[1,8]}$ quarkonia, which is suppressed by $1/N$ in Mellin space. | contribution 0.97 | 0.87 | 0.13 | asserted 0.98 |
| s1 | While threshold double logarithms have been resummed to all orders in the strong coupling for gluon fragmentation functions for quarkonia at leading power in $1/N$, it has not been known how to resum them for channels suppressed by powers of $1/N$. | gap 0.99 | 0.07 | 0.08 | asserted 1.00 |
| s2 | Threshold resummation can be important for the $^1S_0^{[1,8]}$ channels, as the tree-level gluon fragmentation functions have finite yet nonzero values at the kinematical threshold. | background 0.73 | 0.17 | 0.06 | asserted 0.62 |
| s3 | I derive a formalism for resumming the threshold double logarithms for these channels at the leading double logarithmic level and discuss the phenomenological impact. | contribution 0.99 | 0.96 | 0.68 | asserted 1.00 |
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Figures
Figures reproduced from arXiv:2609.39029 by the authors.
References
Discussion
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