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arxiv: 2606.11061 · v1 · pith:BEQGZO3Mnew · submitted 2026-06-09 · ✦ hep-ph

Endpoint Logarithms in the NLO Mueller-Navelet Jet Vertex: Threshold Matching and BLM/MOM Prescription Sensitivity

Lei Wang This is my paper

Pith reviewed 2026-06-27 12:39 UTC · model grok-4.3

classification ✦ hep-ph
keywords Mueller-Navelet jetsNLO jet vertexendpoint logarithmsthreshold matchingBFKL resummationazimuthal observablesCMS jet dataBLM prescription
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The pith

A BFKL-preserving threshold match for the NLO Mueller-Navelet vertex moves one CMS azimuthal ratio toward data while leaving the others unchanged.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper isolates the endpoint region where the jet momentum fraction approaches one inside the next-to-leading-order forward jet vertex and builds a matching procedure that keeps the exact NLO result. The procedure uses a conservative exponent that resums only the ordinary threshold logarithms and leaves the BFKL kernel term untouched, thereby avoiding mixed logarithms that would spoil the high-energy resummation. When this matched vertex is inserted into fixed-order calculations and compared with CMS measurements of azimuthal moments, one ratio shifts in the direction favored by the data but the remaining observables stay essentially the same. A reader cares because the construction shows how endpoint effects can be added without destroying the BFKL structure that controls the high-energy behavior.

Core claim

Starting from the small-cone NLO vertex, the quark and gluon plus distributions are isolated and a BFKL-aware threshold matching scheme is constructed that preserves exact NLO accuracy. The conservative Scheme-II exponent resums only the ordinary endpoint logarithms and leaves the χ(n,γ)lnN̄ term in the fixed-order coefficient. In fixed-baseline CMS tests this matched vertex is a controlled deformation of an optimized-NLL pointwise BLM/MOM calculation: it moves R21 = C2/C1 in the high-ΔY direction favored by CMS, but it does not improve C1/C0 or R32 = C3/C2 in the same prescription. A coefficient-projected table-BLM diagnostic improves the absolute moments but lowers R21 and is sensitive to

What carries the argument

the conservative Scheme-II exponent that resums only ordinary endpoint logarithms while retaining the χ(n,γ)lnN̄ term in the fixed-order coefficient

If this is right

  • The matched vertex shifts R21 = C2/C1 toward the high-ΔY region preferred by CMS data.
  • The same matching leaves C1/C0 and R32 = C3/C2 essentially unchanged under the BLM/MOM prescription.
  • A coefficient-projected table-BLM diagnostic raises the absolute moments yet lowers R21 and reveals sensitivity to the large-|ν| tail.
  • The endpoint matching procedure remains internally consistent at NLO.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The single-ratio improvement suggests that a refined treatment of the large-|ν| region might bring the remaining observables into agreement.
  • The same separation technique could be applied to other processes where both threshold and high-energy logarithms appear.
  • Direct comparison with additional LHC jet data sets would test whether the partial improvement is specific to the current five observables.

Load-bearing premise

The endpoint region can be systematically separated from BFKL energy-scale terms without generating an uncontrolled tower of mixed endpoint-BFKL logarithms.

What would settle it

An explicit calculation that includes the mixed endpoint-BFKL logarithms and finds that they alter the NLO coefficient by an amount larger than the deformation produced by Scheme-II matching would show that the separation is not controlled.

Figures

Figures reproduced from arXiv: 2606.11061 by Lei Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. Single-leg threshold factors in symmetric kinematics. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. PDF-weighted fixed-NLO and threshold-matched ver [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Factorization-scale scan of the PDF-weighted full-vertex prototype. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Harmonic and [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Pointwise-BLM/MOM benchmark against the digitized optimized-NLL DSW curves. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Threshold-matched band on the pointwise-BLM/MOM benchmark. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Coefficient-projected table-BLM comparison for the absolute moments. [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Coefficient-projected table-BLM comparison for normalized ratios. [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Matched-vertex sector decomposition on the table-BLM baseline. [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Quadrature convergence of the coefficient-projected table-BLM prediction. Legend entries denote [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
read the original abstract

The endpoint region $\zeta\to1$ of the NLO forward jet vertex has not been systematically separated from BFKL energy-scale terms in Mueller-Navelet phenomenology. Starting from the small-cone NLO vertex, we isolate the quark and gluon plus distributions and construct a BFKL-aware threshold matching scheme that preserves exact NLO accuracy. The conservative Scheme-II exponent resums only the ordinary endpoint logarithms and leaves the $\chi(n,\gamma)\ln\bar N$ term in the fixed-order coefficient, avoiding an uncontrolled tower of mixed endpoint-BFKL logarithms. In fixed-baseline CMS tests, this matched vertex is a controlled deformation of an optimized-NLL pointwise BLM/MOM calculation: it moves $R_{21}=C_2/C_1$ in the high-$\Delta Y$ direction favored by CMS, but it does not improve $C_1/C_0$ or $R_{32}=C_3/C_2$ in the same prescription. A coefficient-projected table-BLM diagnostic improves the absolute moments but lowers $R_{21}$ and is sensitive to the large-$|\nu|$ tail. Thus the endpoint matching is internally consistent and phenomenologically informative, while the present setup does not provide a simultaneous description of all five CMS azimuthal observables.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript isolates the ζ→1 endpoint region of the NLO Mueller-Navelet jet vertex, constructs a BFKL-aware threshold matching scheme (conservative Scheme-II) that resums only ordinary endpoint logarithms while retaining the χ(n,γ)lnN̄ term inside the fixed-order coefficient, and claims this preserves exact NLO accuracy without generating an uncontrolled tower of mixed endpoint-BFKL logarithms. Phenomenological tests against CMS azimuthal observables show that the matched vertex deforms an optimized-NLL BLM/MOM result in a direction favored by data for R21 but does not simultaneously improve C1/C0 or R32; a coefficient-projected table-BLM diagnostic is also introduced and shown to be sensitive to the large-|ν| tail.

Significance. If the internal consistency of Scheme-II is verified, the work supplies a controlled deformation of existing NLL BFKL vertices that isolates threshold effects while maintaining NLO accuracy, offering a concrete tool for refining high-ΔY predictions in forward-jet phenomenology. The explicit retention of the χ(n,γ)lnN̄ term and the table-BLM diagnostic provide falsifiable diagnostics of prescription sensitivity that can be checked against future data or higher-order calculations.

major comments (1)
  1. [Abstract] Abstract (and the section defining Scheme-II): the central claim that retaining the χ(n,γ)lnN̄ term inside the fixed-order coefficient 'avoids an uncontrolled tower of mixed endpoint-BFKL logarithms' is load-bearing for the assertion of preserved exact NLO accuracy. An explicit expansion of the matched vertex when inserted into the full Mueller-Navelet kernel, or a direct check of the convolution with the BFKL Green function at NLO, is required to confirm that no additional mixed logarithms appear; the abstract provides no such verification.
minor comments (2)
  1. The description of the 'coefficient-projected table-BLM diagnostic' would benefit from an explicit equation or table reference showing how the projection is performed and how the large-|ν| tail is regulated.
  2. Notation for the plus distributions isolated from the small-cone NLO vertex should be cross-referenced to the original NLO expressions to aid reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting the need to strengthen the verification of NLO accuracy in Scheme-II. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the section defining Scheme-II): the central claim that retaining the χ(n,γ)lnN̄ term inside the fixed-order coefficient 'avoids an uncontrolled tower of mixed endpoint-BFKL logarithms' is load-bearing for the assertion of preserved exact NLO accuracy. An explicit expansion of the matched vertex when inserted into the full Mueller-Navelet kernel, or a direct check of the convolution with the BFKL Green function at NLO, is required to confirm that no additional mixed logarithms appear; the abstract provides no such verification.

    Authors: We agree that an explicit verification of the absence of mixed logarithms strengthens the central claim. In the construction of conservative Scheme-II the threshold matching is applied solely to the ordinary endpoint logarithms isolated from the plus distributions of the small-cone NLO vertex; the χ(n,γ)lnN̄ term, which encodes the leading BFKL kernel structure, is deliberately retained inside the fixed-order coefficient. This separation ensures that the resummation does not act on the BFKL-sensitive piece and therefore does not generate an additional tower of mixed endpoint-BFKL logarithms when the matched vertex is inserted into the Mueller-Navelet kernel. In the revised manuscript we will add an explicit NLO expansion of the matched vertex together with a direct statement confirming that its convolution with the NLO BFKL Green function reproduces the original NLO result plus only the resummed ordinary endpoint logarithms. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from NLO vertex

full rationale

The paper starts from the small-cone NLO vertex, isolates plus distributions, and defines Scheme-II by construction to resum ordinary endpoint logs while retaining the χ(n,γ)lnN̄ term in the fixed-order coefficient. This choice is presented as the mechanism that avoids mixed endpoint-BFKL towers, but the construction itself is the input and the claim of exact NLO preservation follows directly from the definition without reducing a separate prediction to a fit. CMS data serve as an external benchmark for phenomenological tests; BLM/MOM and table-BLM are applied as standard scale prescriptions rather than parameters tuned inside the derivation. No self-citation chain, ansatz smuggling, or renaming of known results is load-bearing for the central consistency claim. The derivation therefore remains independent of its outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit list of free parameters, axioms, or invented entities; the scheme is presented as preserving NLO accuracy without introducing new postulated objects.

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