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arxiv: 2604.02859 · v1 · submitted 2026-04-03 · ✦ hep-ph

Recognition: 1 theorem link

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The Fate of Ultra-Collinear Modes in On-Shell Massive Sudakov Form Factors

Marvin Schnubel , Jakob Schoenleber , Robert Szafron
Authors on Pith no claims yet

Pith reviewed 2026-05-13 19:00 UTC · model grok-4.3

classification ✦ hep-ph
keywords Sudakov form factorultra-collinear modesSCET_IIgauge invarianceQCD factorizationjet functionsoft functionresummation
0
0 comments X

The pith

Ultra-collinear modes cancel to all orders in on-shell massive Sudakov form factors due to gauge invariance.

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

In the on-shell massive Sudakov form factor in QCD, multi-loop diagrams produce an infinite tower of ultra-collinear momentum regions. These contributions cancel exactly to all orders because of gauge invariance. The leading-power SCET_II factorization formula therefore requires no modification. Computations of the soft function and massive jet functions through two loops with the eta rapidity regulator support NNLL resummation that includes hierarchies of fermion masses. A gauge-boson mass regulator truncates the tower and makes the ultra-collinear and ultra-soft modes explicit and factorizable.

Core claim

For the on-shell form factor in QCD the contributions from ultra-collinear modes cancel to all orders as a consequence of gauge invariance, so the leading-power SCET_II factorization formula is unchanged. Using the eta rapidity regulator the soft function and the massive jet function of the quark and gluon Sudakov form factors are computed through two loops and logarithms are resummed at NNLL accuracy including hierarchies of fermion masses. With a gauge-boson mass regulator the infinite tower of modes is truncated and ultra-collinear and ultra-soft modes become manifest and factorize explicitly, providing a direct EFT derivation of the regulated infrared dependence.

What carries the argument

Gauge invariance, which enforces exact cancellation of the infinite tower of ultra-collinear contributions in on-shell QCD kinematics.

If this is right

  • The leading-power SCET_II factorization formula for the Sudakov form factor remains valid without ultra-collinear corrections.
  • Two-loop results for the soft and massive jet functions allow consistent NNLL resummation of large logarithms with fermion mass hierarchies included.
  • The eta regulator hides the ultra-collinear modes through cancellation while the gauge-boson mass regulator renders them explicit and factorized.
  • The infrared dependence of the form factor can be derived directly from the effective theory once the regulator truncates the mode tower.

Where Pith is reading between the lines

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

  • Similar cancellations may simplify higher-order calculations in other on-shell QCD processes that involve Sudakov logarithms.
  • The regulator dependence shows that infrared mode structure in effective theories can be made manifest or hidden by the choice of regulator.
  • Extensions to processes with different kinematics could test whether the gauge-invariance mechanism persists beyond strict on-shell conditions.

Load-bearing premise

The cancellation holds specifically for on-shell kinematics in QCD where gauge invariance applies directly without off-shell effects or extra infrared structures.

What would settle it

An explicit three-loop or higher calculation of the on-shell Sudakov form factor that yields a non-vanishing ultra-collinear contribution would disprove the all-order cancellation.

read the original abstract

Individual multi-loop diagrams for the massive Sudakov form factor contain an infinite tower of ultra-collinear momentum regions. We show that, for the on-shell form factor in QCD, these contributions cancel to all orders as a consequence of gauge invariance, so the leading-power SCET$_{\rm II}$ factorization formula is unchanged. Using the $\eta$ rapidity regulator, we compute the soft function and the massive jet function of the quark and gluon Sudakov form factors through two loops and resum logarithms at NNLL accuracy, including hierarchies of fermion masses. We also show that with a gauge-boson mass regulator, the infinite tower of modes is truncated and ultra-collinear and ultra-soft modes become manifest and factorize explicitly, providing a direct EFT derivation of the regulated infrared dependence.

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 paper claims that for the on-shell massive Sudakov form factor in QCD, contributions from the infinite tower of ultra-collinear momentum regions in multi-loop diagrams cancel to all orders as a consequence of gauge invariance, leaving the leading-power SCET_II factorization formula unchanged. Using the η rapidity regulator, the authors compute the soft function and massive jet functions for both quark and gluon cases through two loops, perform NNLL resummation including hierarchies of fermion masses, and demonstrate that a gauge-boson mass regulator truncates the tower, making ultra-collinear and ultra-soft modes manifest and explicitly factorizing.

Significance. If the all-order cancellation holds, the result is significant for SCET applications to massive QCD processes, as it confirms that no additional leading-power corrections arise from ultra-collinear modes and validates the standard SCET_II factorization for on-shell kinematics. The two-loop computations with explicit mass hierarchies and the NNLL resummation provide concrete, reproducible results for phenomenology. The gauge-boson mass regulator demonstration offers a direct EFT derivation of regulated infrared dependence, which is a clear strength for higher-order work.

major comments (1)
  1. [Abstract] Abstract and the section presenting the all-order cancellation: the claim that ultra-collinear contributions cancel to all orders rests on gauge invariance, but the manuscript bridges from finite-order Ward identities to the infinite tower primarily via two-loop verification with the η regulator. An explicit general argument (e.g., inductive use of the Ward identity or reference to a theorem covering the full perturbative series in on-shell kinematics) is needed to secure the central claim against possible residual overlapping or mass-induced structures at higher orders.
minor comments (2)
  1. The two-loop expressions for the jet and soft functions should be collected in a dedicated appendix or ancillary file to facilitate direct comparison with existing literature and reproduction.
  2. [Resummation] Clarify in the resummation section how the NNLL logarithms are organized when multiple distinct fermion mass hierarchies are present; a concrete numerical example would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the single major comment below and will revise the manuscript to incorporate an explicit general argument for the all-order cancellation.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the section presenting the all-order cancellation: the claim that ultra-collinear contributions cancel to all orders rests on gauge invariance, but the manuscript bridges from finite-order Ward identities to the infinite tower primarily via two-loop verification with the η regulator. An explicit general argument (e.g., inductive use of the Ward identity or reference to a theorem covering the full perturbative series in on-shell kinematics) is needed to secure the central claim against possible residual overlapping or mass-induced structures at higher orders.

    Authors: We agree that an explicit general argument strengthens the central claim. The manuscript establishes the cancellation as a direct consequence of gauge invariance of the on-shell form factor, which must hold order by order. To make this fully rigorous, we will add an inductive argument in the revised section on all-order cancellation: the one-loop case follows immediately from the Ward identity on the external legs. Assuming cancellation through (n-1) loops, the n-loop ultra-collinear regions in individual diagrams are constrained by the same Ward identity (transversality of gluon attachments and current conservation for on-shell quarks/gluons), forcing their sum to vanish in the gauge-invariant amplitude. This inductive step rules out residual overlapping or mass-induced leading-power contributions. We will also cite standard results on region cancellations in gauge-invariant on-shell amplitudes. The two-loop η-regulator computation remains as a non-trivial consistency check. These changes will be implemented in the relevant section and reflected in the abstract. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The central claim—that ultra-collinear modes cancel to all orders due to gauge invariance—is presented as following from an external principle (Ward identities applied to on-shell QCD kinematics) rather than being defined by the paper's own fitted parameters or self-referential equations. Two-loop explicit computations with the η regulator serve as consistency checks, not as the source of the all-order result. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the argument are exhibited in the abstract or described structure. The gauge-boson mass regulator comparison provides an independent explicit factorization, keeping the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard QCD gauge invariance as a domain assumption and the use of the eta rapidity regulator for infrared handling; no new entities or fitted parameters are introduced.

axioms (1)
  • domain assumption Gauge invariance of QCD
    Invoked to establish cancellation of ultra-collinear modes to all orders in the on-shell form factor.

pith-pipeline@v0.9.0 · 5433 in / 1222 out tokens · 50900 ms · 2026-05-13T19:00:05.347885+00:00 · methodology

discussion (0)

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Reference graph

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