REVIEW
Three-loop rapidity anomalous dimension for jet-veto cross sections
High Energy Physics - Phenomenology · Submitted 2 Oct 2026 · 71 references
Pith's verdict: First computation of the three-loop jet-veto rapidity anomalous dimension, validated by three independent checks; the D-dimensional clustering worry dissolves on reading Appendix A, and the paper deserves proper peer review.
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Referee report
The paper
The paper claims
First calculation of the three-loop rapidity anomalous dimension for jet-veto cross sections, supplying the last missing ingredient for N3LL resummation of colour-singlet production at the LHC.
The reading
What carries the argument
The argument is carried by a purpose-built object the paper calls the RSV (Reference Simplified Variable) soft function: an unphysical but infrared-safe observable whose measurement function acts only on correlated clusters of soft emissions and is independent of the jet radius $R$. Because both the RSV and the transverse-momentum-dependent (TMD) soft function obey non-abelian exponentiation and are sensitive only to cluster transverse momenta, the RSV soft function is linked to the known three-loop TMD soft function by a Bessel transform in Eqs. (2.37)–(2.38), which fixes the analytic part $\gamma_{\nu,\rm RSV}$ of the anomalous dimension. The physical soft function is then written as $S = S_{\rm RSV} + \Delta S$, where the remainder $\Delta S$ vanishes for single-emission final states, has singularities two perturbative orders softer than the full soft function, and is evaluated semi-numerically as a sum of triple-independent (III), double-correlated (CCI), and triple-correlated (CCC) phase-space integrals. This decomposition is what turns the three-loop clustering calculation into a set of finite numerical integrals, most of them performed in $D=4$ with the singular CCI and CCC pieces handled in $D=4-2\epsilon$.
Extended reading notes
Core claim
The paper's central claim is that the three-loop rapidity anomalous dimension $\gamma_\nu^{(3)}(R^2)$ for jet-veto cross sections is now known: it is decomposed into seven colour structures in Eq. (5.1) — $C_RC_FT_Rn_f$, $C_RC_AT_Rn_f$, $C_RC_A^2$, $C_RT_R^2n_f^2$, $C_R^2C_A$, $C_R^2T_Rn_f$, and $C_R^3$ — and evaluated semi-numerically for the anti-$k_t$, $k_t$, and Cambridge-Aachen algorithms. Two structurally new facts appear at this order: the anomalous dimension depends on the jet algorithm for the first time at three loops, although the numerical spread among the three algorithms is mild, and its small-$R$ behaviour is organised into triple-correlated pieces scaling as $\ln^2 R^2$, double-correlated pieces scaling as $R^2\ln R^2$, and triple-independent pieces scaling as $R^4$. The computation is validated by the numerical cancellation of infrared poles in every colour structure and by reproduction of the known leading-logarithmic small-$R$ limit. The paper further reports that at $p_T^{\rm veto}=30$ GeV with five active flavours the three-loop term changes the two-loop result by roughly 15–40% for $R\in[0.1,0.8]$, with the larger shifts at smaller radii.
Load-bearing premise
The load-bearing premise is that the $D$-dimensional continuation of the jet-clustering algorithm, with azimuthal separations defined by the arccos of a $(D-2)$-dimensional dot product, correctly describes how the physical algorithm resolves emissions when three or more particles are present, and a wrong continuation would directly shift the computed $R$-dependence of $\gamma_\nu^{(3)}(R^2)$.
Editorial extensions
If this is right
- $N^3\mathrm{LL}$ resummation for jet-vetoed colour-singlet production is complete in terms of its ingredients; the paper states that only the phenomenological implementation and matching to fixed order remain, planned in a follow-up study.
- The new three-loop term changes jet-veto predictions by a K-factor of roughly 15–40% at $p_T^{\rm veto}=30$ GeV for $R\in[0.1,0.8]$, so third-order rapidity logarithms are numerically relevant for LHC observables such as jet-vetoed Higgs production.
- For the first time the rapidity anomalous dimension depends on the jet algorithm within the generalised-$k_t$ family; the reported spread among anti-$k_t$, $k_t$, and Cambridge-Aachen is mild, but the jet algorithm becomes a genuine parameter of the resummation at this order.
- The small-$R$ leading-logarithmic limit is reproduced but is a poor approximation to the full result even at $R\simeq0.1$, so the complete three-loop value, rather than its $\ln^2 R^2$ asymptotics, is needed for quantitative phenomenology.
- The RSV-based semi-numerical strategy extends to other three-loop soft functions and anomalous dimensions, including jet vetoes in heavy-quark-pair production and other observables with clustering effects in SCET II.
Reading between the lines
- If the Bessel-transform link between RSV and TMD soft functions survives, the four-loop TMD rapidity anomalous dimension could be converted into a jet-veto prediction in the same way; the paper does not consider this, but nothing in its construction prevents it.
- The large gap between the leading-logarithmic small-$R$ limit and the full result suggests that clustering logarithms are numerically important well below $R=0.1$; one testable consequence is that jet-veto slicing procedures that extrapolate the $R\to0$ limit should incorporate this anomalous dimension rather than its LL approximation.
- The mild algorithm dependence found at three loops could grow at four loops, and a cheap check of the framework would be to compute $\gamma_\nu^{(3)}$ for an additional member of the generalised-$k_t$ family (such as the $p=0.5$ variant) and compare it with the interpolation between the three published points.
Original abstract
We present the first calculation of the third-order corrections to the rapidity anomalous dimension for jet-veto cross sections. This is a crucial ingredient for the resummation of these scattering observables in key processes such as colour-singlet and heavy-quark-pair production at hadron colliders. More precisely, our results enable the next-to-next-to-next-to-leading logarithmic (N$^3$LL) resummation of the jet-veto cross section for colour singlet production at the LHC, with immediate phenomenological relevance. The anomalous dimension is extracted from a calculation of the three-loop soft function in soft-collinear effective field theory, and is presented as a function of the jet radius $R$ for the clustering algorithms of the generalised-$k_t$ family. To handle the complexity associated with the clustering algorithm, we formulate a semi-numerical computational approach. This strategy is itself a result of the paper, and can be extended to the calculation of other anomalous dimensions and soft functions at three loops.
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
We present the first calculation of the third-order corrections to the rapidity anomalous dimension for jet-veto cross sections. This is a crucial ingredient for the resummation of these scattering observables in key processes such as colour-singlet and heavy-quark-pair production at hadron colliders. More precisely, our results enable the next-to-next-to-next-to-leading logarithmic (N$^3$LL) resummation of the jet-veto cross section for colour singlet production at the LHC, with immediate phenomenological relevance. The anomalous dimension is extracted from a calculation of the three-loop soft function in soft-collinear effective field theory, and is presented as a function of the jet radius $R$ for the clustering algorithms of the generalised-$k_t$ family. To handle the complexity associated with the clustering algorithm, we formulate a semi-numerical computational approach. This strategy is itself a result of the paper, and can be extended to the calculation of other anomalous dimensions and soft functions at three loops.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present the first calculation of the third-order corrections to the rapidity anomalous dimension for jet-veto cross sections. | contribution 0.98 | 0.97 | 0.73 | asserted 1.00 |
| s1 | This is a crucial ingredient for the resummation of these scattering observables in key processes such as colour-singlet and heavy-quark-pair production at hadron colliders. | interpretation 0.62 | 0.15 | 0.09 | asserted 1.00 |
| s2 | More precisely, our results enable the next-to-next-to-next-to-leading logarithmic (N$^3$LL) resummation of the jet-veto cross section for colour singlet production at the LHC, with immediate phenomenological relevance. | interpretation 0.92 | 0.89 | 0.48 | asserted 0.99 |
| s3 | The anomalous dimension is extracted from a calculation of the three-loop soft function in soft-collinear effective field theory, and is presented as a function of the jet radius $R$ for the clustering algorithms of the generalised-$k_t$ family. | method 0.72 | 0.94 | 0.11 | asserted 1.00 |
| s4 | To handle the complexity associated with the clustering algorithm, we formulate a semi-numerical computational approach. | method 0.99 | 0.91 | 0.03 | asserted 1.00 |
| s5 | This strategy is itself a result of the paper, and can be extended to the calculation of other anomalous dimensions and soft functions at three loops. | contribution 0.54 | 0.87 | 0.08 | asserted 0.93 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:2610.03472 by the authors.
References
Discussion
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